Exploration of Interferometric Detection Methods based on ...

Thèse de doctorat

de l’UTT

Yunlong ZHU

Exploration of Interferometric Detection Methods based on

Continuous Phase Modulation

Champ disciplinaire : Sciences pour l’Ingénieur

2018TROY0024 Année 2018

THESE

pour l’obtention du grade de

DOCTEUR

de l’UNIVERSITE DE TECHNOLOGIE DE TROYES

EN SCIENCES POUR L’INGENIEUR

Spécialité : MATERIAUX, MECANIQUE, OPTIQUE, NANOTECHNOLOGIE

présentée et soutenue par

Yunlong ZHU

le 11 juillet 2018

Exploration of Interferometric Detection Methods based on Continuous Phase Modulation

JURY

M. W. UHRING PROFESSEUR DES UNIVERSITES Président

M. L. CHASSAGNE PROFESSEUR DES UNIVERSITES Rapporteur

M. A. DUBOIS PROFESSEUR DES UNIVERSITES Rapporteur

M. H.-L. HSIEH ASSOCIATE PROFESSOR Examinateur

Mme L. LE JONCOUR MAITRE DE CONFERENCES Examinatrice

M. A. BRUYANT MAITRE DE CONFERENCES - HDR Directeur de thèse

Personnalités invitées

M. M. FRANÇOIS PROFESSEUR DES UNIVERSITES

M. Y. HADJAR INGENIEUR DE RECHERCHE UTT

I

Acknowledgements

First of all, I’d like to thank Prof. Aurélien Bruyant, my supervisor, for guiding me into the world of

scientific research, and for giving me a strong support whenever I met a problem. I am really impressed

by his passion for work and his optimism. For scientific research, he always has a lot of new ideas, and

I enjoyed the brainstorms with him.

Also, I’d like to thank Mr. Julien Vaillant for his help on optical setup and LabVIEW/Arduino coding;

his work on signal processing also inspired me a lot. I want to express my gratitude to the research group

in the laboratory LASMIS (Prof. Khemais Saanouni, Prof. Manuel François, Prof. Guillaume Montay,

Prof. Bruno Guelorget, Prof. Carl Labergere, Prof. Léa Le Joncour, etc.) as well for taking me as a

member of them, listening to my research progress and giving comments, and helping me with

experimentation. I also give my thanks to Dr. Tzu-Heng Wu, for his technical support on SPR sensors,

as well as Miss Zu-Yi WANG, for cooperating with me on the SPR sensing experiments and providing

several figures.

Besides, I am truly thankful that I met so many warmhearted people in L2N who helped me a lot: Prof.

Renaud Bachelot, Prof. Gilles Lerondel, Prof. Julien Proust, Prof. Christophe Couteau, Mr. Regis

Deturche, Dr. Serguei Kochtcheev, Dr. Anna Rumyantseva, Dr. Komla Nomenyo, Dr. Feng Tang, Dr.

Jiyong Wang, Dr. Wei Geng, Dr. Ying Peng, Dr. Feifei Zhang, Mr. Shijian Wang, Miss Yi Huang, Mr.

Junze Zhou, Mr. Hongshi Chen, Mr. Quan Liu, Miss Fang Dai, Miss Dandan Ge, Mr. Xiaolun Xu, Miss

Lan Zhou, Mr. Lin Pan, etc. Also, I would like to thank all my friends outside L2N; I do not think it is

necessary to make a long list here, but I am truly grateful to them for their company.

I want to give special thanks to the nice and excellent scholars that I met during the conferences / forums

/ summer schools that I have been. I feel extremely lucky to have had an unforgettable talk with Prof.

Thomas Kreis. I was really moved by his kind instruction and encouragement.

Last but not least, I want to thank my family wholeheartedly, for always being there for me. I know that

they are the few persons in the whole world who cares about me the most and truly love me no matter

what happens. Their love and their blind trust in me really helped me go through many troubles. I love

them forever.

II

Abstract

Interferometric techniques have been widely used to carry out precise measurements. Phase-shifting and

continuous phase modulation techniques are often applied to further improve the precision and to get

rid of the phase ambiguity. Phase-shifting technique is simple and effective, so it has been applied in

interferometric measurements. However, the discrete phase shifts may cause error or limit the measuring

speed.

To solve these problems and to further improve the precision, continuous phase modulation techniques

are applicable. By continuously modulating the phase of reference light beam, a temporal beating can

be recorded, from which amplitude and phase information can be obtained by analyzing this signal. To

perform this analysis, different algorithms may be used depending on the actual type of phase

modulation.

In this thesis, the mathematical expression of interference signal with continuous phase modulation is

first presented and discussed. Several common phase modulation functions and phase retrieval

algorithms are presented. We mainly focus on the use of sinusoidal phase modulation (SPM), which is

the most natural way of oscillation for most modulators. When SPM is applied, we need to face issues

related to synchronization as well as a possible additional intensity modulation. Mathematical solutions

are proposed for these two problems.

The second part of the thesis focuses on the application of continuous phase modulation in simple and

cost-efficient home-made setups. A lensless co-axis digital holography setup has been built, where SPM

can be applied. Two different algorithms are used to retrieve the phase information. The results are

compared for imaging and for measuring out-of-plane displacement. Besides, continuous phase

modulations have been applied in electronic speckle pattern interferometry (ESPI) to realize

simultaneous 2D in-plane deformation measurements. The fringe visibility is very good, and the

displacement along two different axes can be efficiently separated in the frequency domain from the

same temporal interference signal. Lastly, SPM has been introduced into surface plasmon resonance

(SPR) sensor for phase sensitive imaging purpose. Tests have been done on a home-made SPR sensor

to prove the feasibility of this method.

III

Contents

Acknowledgements ................................................................................................................................. I

Abstract ................................................................................................................................................. II

Nomenclature ..................................................................................................................................... VII

General introduction ............................................................................................................................. 1

Chapter 1 Basic methods of phase retrieval .................................................................................... 7

1.1 Fundamentals of optical interference ..............................................................................................................7

1.2 Interferometry and holography ........................................................................................................................9

1.3 Phase retrieval methods ................................................................................................................................... 10

1.3.1 Analysis of static interferograms ...................................................................................................................... 10

1.3.2 Phase shifting method ......................................................................................................................................... 11

1.3.3 Continuous phase modulation methods ....................................................................................................... 12

1.4 Different types of phase modulation function ........................................................................................... 12

1.4.1 Linear phase modulation .................................................................................................................................... 12

1.4.2 Sinusoidal phase modulation ............................................................................................................................ 13

1.4.3 Other types of phase modulation .................................................................................................................... 14

1.5 Phase demodulation techniques .................................................................................................................... 15

1.5.1 Lock-in amplifier technique (LIA algorithm) ................................................................................................ 15

1.5.2 SPM algorithm ........................................................................................................................................................ 16

1.5.3 G-LIA algorithm ..................................................................................................................................................... 17

1.5.4 f-G-LIA algorithm .................................................................................................................................................. 19

1.5.5 Integrating bucket algorithm ............................................................................................................................. 20

1.6 Comparison between different algorithms ................................................................................................. 20

1.6.1 SPM algorithm and (f-)G-LIA algorithm ........................................................................................................ 20

1.6.2 Amplitude of phase modulation ...................................................................................................................... 22

IV

1.7 Conclusion ........................................................................................................................................................... 28

Chapter 2 Advanced methods of phase retrieval .......................................................................... 29

2.1 Initial phase problem ........................................................................................................................................ 30

2.1.1 LIA algorithm ........................................................................................................................................................... 31

2.1.2 SPM algorithm ........................................................................................................................................................ 32

2.1.3 (f-)G-LIA algorithm ............................................................................................................................................... 34

2.2 The intensity modulation problem and frequency analysis .................................................................... 37

2.2.1 Frequency domain analysis ................................................................................................................................ 37

2.3 Demodulation techniques for intensity modulated signals .................................................................... 38

2.3.1 Modified SPM algorithm ..................................................................................................................................... 39

2.3.2 Determination of the initial phase ................................................................................................................... 42

2.3.3 Modified f-G-LIA algorithm ............................................................................................................................... 42

2.3.4 Modified integrating bucket algorithm .......................................................................................................... 44

2.4 Conclusion ........................................................................................................................................................... 49

Chapter 3 Application of SPM in Digital Holography and Holographic Interferometry ......... 51

3.1 Experimental method and data processing ................................................................................................ 51

3.2 Digital holography (DH) ................................................................................................................................... 54

3.3 Digital holographic interferometry (DHI) .................................................................................................... 59

3.4 Conclusion ........................................................................................................................................................... 67

Chapter 4 2D-ESPI with double phase modulations .................................................................... 69

4.1 Introduction ......................................................................................................................................................... 69

4.2 Experimental Method and Data Processing ................................................................................................ 72

4.2.1 Optical arrangement ............................................................................................................................................ 72

4.2.2 Principle of measurement ................................................................................................................................... 73

4.2.3 Set appropriate voltages ..................................................................................................................................... 80

V

4.3 Experimental details .......................................................................................................................................... 82

4.3.1 Practical requirement on the Laser .................................................................................................................. 82

4.3.2 Evaluation of exposure conditions ................................................................................................................... 82

4.3.3 Data acquisition / video recording .................................................................................................................. 84

4.3.4 Initial phase problem ............................................................................................................................................ 85

4.4 Potential for 3D displacement field measurement ................................................................................... 87

4.4.1 Linear/sawtooth phase modulations............................................................................................................... 88

4.4.2 Sinusoidal phase modulations .......................................................................................................................... 89

4.5 Results ................................................................................................................................................................... 90

4.6 Conclusion ........................................................................................................................................................... 94

Chapter 5 Application of SPM in SPR detector ............................................................................ 97

5.1 Introduction to SPR ........................................................................................................................................... 97

5.2 Principle of phase modulation through wavelength modulation ....................................................... 101

5.3 Phase extraction in wavelength modulated interferometers ............................................................... 104

5.4 Experiments ....................................................................................................................................................... 104

5.4.1 Preliminary setup: test of the algorithms .................................................................................................... 105

5.4.2 Phase-sensitive SPR sensor .............................................................................................................................. 107

5.5 Perspective: Combining shearing interferometry with SPRi ................................................................. 111

5.6 Conclusion ......................................................................................................................................................... 114

General conclusion and perspectives ............................................................................................... 116

General conclusion ................................................................................................................................................. 116

Perspectives .............................................................................................................................................................. 117

Smart detector using sinusoidal phase modulation ................................................................................................. 117

Automatic control of measuring systems .................................................................................................................... 118

Final comments ..................................................................................................................................................................... 119

VI

Résumé en français ............................................................................................................................ 121

1. Algorithmes de récupération de phase ................................................................................................... 121

2. Holographie / interférométrie holographique numérique ................................................................ 129

3. Interférométrie de speckle ......................................................................................................................... 137

4. Détections avec résonance plasmonique de surface (SPR) ............................................................... 145

Conclusions et perspectives ................................................................................................................................. 153

Appendix ............................................................................................................................................ 155

Complete derivation processes of formulae .................................................................................................... 155

Angular spectrum method ................................................................................................................................... 179

References .......................................................................................................................................... 181

VII

Nomenclature

DC: direct current (zero-frequency)

LIA: lock-in amplifier

LIA algorithm: algorithm for lock-in amplifier

SPM: sinusoidal phase modulation

SPM interferometer: sinusoidal phase modulating interferometer

SPM algorithm: traditional algorithm for SPM interferometer

G-LIA: generalized lock-in amplifier

G-LIA algorithm: algorithm for generalized lock-in amplifier

f-G-LIA algorithm: G-LIA algorithm with DC filter

DH: digital holography

DHI: digital holographic interferometry

ESPI: electronic speckle pattern interferometry

CCD: charge-coupled device

CMOS: complementary metal–oxide–semiconductor

VCSEL: vertical-cavity surface-emitting laser

SPR: surface plasmon resonance

SPRi: surface plasmon resonance imaging

LSPR: localized surface plasmon resonance

LSPRi: localized surface plasmon resonance imaging

General introduction

1


The wave nature of light was progressively discovered in the 17th and 18th. The famous “rings of

Newton” described by Sir Isaac Newton in his treatise on optics (1704) are a manifestation of this wave

nature. However, Newton did not accept this theory and the concepts were developed by other scientists

like Christiaan Huygens, Leonhard Euler or Robert Hooke. Amongst them, the polymath Thomas Young

is often regarded as the father of interferometry, with the introduction of its famous experiment referred

to as the double slit experiment (c.f. Figure 1), where “fringes produced by the interference of two

portions of light” can be observed, as quoted by T. Young, and from which wavelength can be deduced

[1].

Figure 1. Double-Slit experiment as reported in Thomas Young's "Lectures", (1807), as a proof of the

wave theory of light.

In fact, as explained in [2] this phenomenon observed by T. Young was not firstly observed using two

adjacent slits but using a simple piece of paper, namely a “slip of card” held edgewise into the sunbeam

coming from a tiny hole in a window shutter.

More advanced experiments on interferometry were then conducted by the French polytechnician

Augustin Fresnel who also completed most of the optical wave theory. He notably introduced the two

tilted “Fresnels Mirrors” (1816), illuminated by a point source (Experiment made in a camera obscura


2

with the light coming from a heliostat) allowing for the observation of more clear interferences, without

additional diffraction phenomenon compared to the T. Young’s experiments [3].

In 1851, the French astronomer Hyppolite Fizeau introduced a new setup, which would be considered

now as a Mach-Zehnder based system, as shown in Figure 2. Water flows in two opposite directions in

the two arms A1 and A2 (pipes) where the light propagates. The difference in speed of light in the two

directions of the water is inducing a continuous displacement of the interference fringes. Although the

development served another purpose, this system was most probably the first interferometer

incorporating a linear phase modulator.

Figure 2. Fizeau experiment (1851): the light of a point source can be introduced inside the

interferometer by a beam splitter G on the right. After collimation, the light is passing through two

openings (O1 and O2) forming two beams. On the left, a mirror m is reflecting back the two beams

toward the image S where the fringe pattern can be observed via additional optics.

Later on, in 1862, he also invented the so called Fizeau interferometer whose design is still in use

nowadays to inspect the 3D shape of optical surface, notably during their manufacture. At that time, we

can also mention the Jamin interferometer (1856) based on the use of thick metalized glass. The Jamin-

type beam splitters are also currently used in stable, modern, interferometers for displacement sensing

(e.g. AIMS Interferometer from Queensgate Instruments inc.).

About 20 years later, the well-known Michelson interferometer was proposed (1881) [4] and used few

years later to test the existence of Aether (A.A. Michelson & E.W. Morley, 1887) [5]. Following these

works, an amplitude splitting interferometer inspired by Jamin’s work was proposed by L. Zehnder

(1891) [6] and refined by L. Mach (1892). [7]

Since that time, many interferometers have been proposed, but the laser invention in 1960 tremendously

improved the performance of all these systems for calibration or displacement measurement, making

them essential tools for metrology. About one hundred years after the introduction of Mach-Zehnder

interferometer, precision in the order of 10-19m over measurement time of one second were for example

achieved in a controlled environment lab [8].

For a common interferometer, there are typically two coherent light beams: one of them is the signal

beam which contains the required phase information; the other one is the reference beam which do not

contain useful information. For example, in a typical Michelson interferometer [9], the signal beam is

the one reflected by the sample, while the reference beam is the one reflected by a mirror [10]. The


3

reference beam may stay still [11]; it may also make discrete phase shifts [10] / continuous phase

modulations [12,13,14] to eliminate the phase ambiguity and to improve the precision of interferometric

measurements [10]. Since interferometry is widely applied in mechanics [15,16,17], biosensing [18,19],

nanophotonics [20,21], etc., developments on the phase modulation and retrieval method are still of

great importance for the researchers working in these fields. In term of application, achieving

experimentally simple and compact systems based on versatile modulation/demodulation scheme is a

requirement for developing ubiquitous interferometric detection means with strong metrologic

performances.

In practice, when applying continuous phase modulation, different types of modulation function (linear

/ sawtooth function [22,23], sinusoidal function [14,24,25], triangular function [22,23], etc.) may be

used, mainly due to the cost, the working environment, the requirement of measurement precision, etc.

For each type of phase modulation, different algorithms may be used to retrieve the phase information

of signal [14,22,26]. Among these different modulation functions, we noticed that the sinusoidal phase

modulation, which was formally introduced into interferometer in 1986 [14,24], has great potential to

be applied in cheap and compact detectors. Sinusoidal function represents the most natural way of

oscillation; thus, it can be easily realized with high precision by using simple and cheap devices, e.g.

through a piezo-mounted mirror (achromatic) or via a sine wavelength modulation in an unbalanced

interferometer. However, as far as we know, there is not many studies aiming at making simple and

compact interferometry devices using sinusoidal phase modulation [27], as well as comparing the

performance of different phase-retrieval algorithms [28].

Based on the principle of optical interference, many techniques have been proposed and developed,

including: (digital) holography [29 ], (digital) holographic interferometry [ 30 ], (digital/electronic)

speckle pattern interferometry [31], phase-sensitive surface plasmon resonance (SPR) sensor [32], etc.

In all these techniques, the method of phase-shifting or phase modulation can be applied to enhance the

performance [33,34,35,36].

Holography is proposed by Gabor in 1948 [37,38]. The general idea is to reconstruct the original wave

front from the sample with the recorded interference fringe pattern (called “hologram”). In traditional

holography, the method of phase-shifting or phase modulation is not needed: when the reference light

is incident on the hologram, the original sample can be seen by naked eyes. This procedure is often

called reconstruction [39,30]. Later on, along with the development of semiconductor industry, digital

cameras (CCD/CMOS matrices) have been introduced to record the interference fringe pattern instead

of traditional holographic plates [40]. This way, the acquired data can be easily processed using

computers. Besides, the chemical development process of the holographic plates is no more necessary,

real-time measurement becomes possible, and the technique becomes also more economic to use since

CCD/CMOS detectors, unlike the one-time holographic plates, are of course reusable. The


4

reconstruction can also be done numerically in the computer [41,42,43]. In order to further improve the

precision of measurements and simplify the process of numerical reconstruction, the phase-shifting

technique and phase modulation techniques have also been applied to digital holography [33,44] (DH).

By using two holograms before and after certain displacement of the sample, the

displacement/deformation may be measured [30,31]. For example, when a co-axis configuration is

applied, then it is sensible to the out-of-plane displacement [28]. This technique is called holographic

interferometry (HI) [45, 30, 31]. The traditional way of doing HI is also by using holographic plates.

We may choose to make a double exposure (before and after displacement) to the same holographic

plate [45]. However, the chemical development makes the measurement impossible to be realized in

real-time. Another way is to record a hologram on a holographic plate, make the chemical development,

and put it back to the original position [30, 31]. This way, the fringes representing the

displacement/deformation can be shown in real-time. However, it is difficult to put the hologram exactly

to its original position. After the introduction of digital cameras into HI, digital holographic

interferometry (DHI) arose [30], and these practical issues were solved. Like in DH, phase-shifting

technique and phase modulation techniques have also been applied to DHI [46,34, 30].

Recently, a new phase-retrieval algorithm call “generalized lock-in amplifier (G-LIA)” was proposed,

which is able to deal with interference signal when applying sinusoidal phase modulation [22,47,48].

However, its performance in DH and DHI is to be determined, and the comparison between this

algorithm and the traditional one needs to be done [28,49].

As for the speckle pattern interferometry (SPI), it is a widely used technique based on the speckle

phenomenon to measure displacement field in mechanics [31]. Like holography/HI, the speckle pattern

is also recorded on a photosensitive plate. Likewise, when digital cameras were introduced into SPI, it

became electronic/digital speckle pattern interferometry (ESPI/DSPI) [31,50], while the phase-shifting

technique can also be applied to improve the performance of the system [51,35,34]. Recently, the phase-

shifting technique has been widely used in commercial ESPI systems for its simplicity and efficacy

[52,53]. However, these systems mainly aim at measuring 1D displacement field, since the 2D/3D

measuring systems require complicated configurations and longer data acquisition time [53,54,55]. In

this context, continuous phase modulation techniques have received only little attention [36], but could

be helpful to reduce the systems complexity and provide simultaneous high-throughput measurements

of different displacement components.

Another field of application of phase modulation techniques is the phase-sensitive surface plasmon

resonance (SPR) sensor. In a typical SPR sensor, a light beam is incident on a thin layer of metal, through

a substrate. The light reflected at a specific angle experiences a strong SPR and is collected to detect

tiny refractive index change on the superstrate side of the metal layer [56]. With proper functionalization

of the metal surface, bio-sensing (including medical diagnostic testing) can be realized [57]. By using


5

detector matrix like CCD/CMOS, SPR imaging can be made to realize high-throughput measurements

[58,59]. In fact, SPR sensors have been widely recognized as an effective way of measuring dynamic

interactions between biomolecules. Many different schemes of SPR sensor have been proposed,

including: intensity detection scheme [32], angular interrogation scheme [ 60 , 61 ], wavelength

interrogation scheme [62], phase detection scheme [63], etc. Among these schemes, the phase detection

scheme is not the most mature one technically, but it is considered by many researchers to be the most

sensitive one to extremely small refractive index change [32,64]. Many different phase detection

methods have been presented [63,65], and the phase-shifting / phase modulation techniques have been

applied [66,67,68], but most of them are not very cost-effective, or not compact enough, thus not suitable

for the emerging requirement for point-of-care testing (POST). As a cost-effective laser source, vertical

cavity surface emitting laser (VCSEL) is promising to be applied in the POST systems [69,70].

However, as a kind of laser diode, the stabilization of wavelength is needed to avoid phase drifts in

interferometric configurations [71]. Sinusoidal phase modulation can be realized by modulating the

input voltage to laser diode [72], yet the additional output intensity/power modulation which comes

along with this modulation can affect the interference signal [73,74,75].

In this thesis, theoretical works have been done on the interference signal analysis when using linear or

sinusoidal phase modulation(s). The feasibility of our phase-retrieval algorithms has been proved by the

experiments of DH/DHI, ESPI and phase-sensitive SPR sensor. By combining phase modulation

techniques with these practical applications, new possibilities of making simple, compact, cost-effective

yet precise measuring systems are shown.

In Chapter 1, the fundamentals of optical interference are presented. We give the basic mathematical

expression of interference. The meanings of the terms “interferometry” and “holography” are presented.

The traditional phase-shifting method is described. Several common types of continuous phase

modulation functions are also presented. Then the basic principles of phase-retrieval methods in phase

modulating interferometers are presented. We focus on the linear and sinusoidal phase modulation

functions, and give the mathematical detail of four phase-retrieval algorithms: LIA algorithm (algorithm

for lock-in amplifier), SPM algorithm (traditional algorithm for sinusoidal phase modulating

interferometer), G-LIA algorithm (algorithm for generalized lock-in amplifier), and f-G-LIA algorithm

(G-LIA algorithm with DC filter). Besides, a comparison between SPM algorithm and (f-)G-LIA

algorithm is done theoretically and by simulations. The amplitude of the sinusoidal phase modulation

(phase modulation depth), which is an important coefficient, is also taken into consideration.

In Chapter 2, the initial phase problem and the additional intensity modulation problem, which are two

practical ones while applying sinusoidal phase modulation in cost-effective interferometric measuring

systems, have been proposed. The mathematical expression of the interference signal affected by these

two problems is shown and analyzed in the frequency domain. For the first problem, the influence of a


6

wrong initial phase value on the final result is discussed, and a mathematical solution is given to

calculate the initial phase when it is unknown. For the second problem, modified SPM algorithm,

modified f-G-LIA algorithm and modified integrating bucket method are proposed to get rid of the

influence of the additional intensity modulation.

In Chapter 3, sinusoidal phase modulation is applied into digital holography (DH) and digital

holographic interferometry (DHI). A simple homemade lensless co-axis setup is used. The phase and

intensity images of a resolution test target is obtained. The profile of the surface can be measured with

a reasonable spatial resolution is obtained. The out-of-plane rotation of a scattering sample is also

measured using the principle of DHI. Clear fringes are obtained, and the precision is good. Different

filtering methods to get clear fringe images are also discussed. In the whole chapter, SPM and G-LIA

algorithms are compared to each other for every measurement, showing similar performance for the

considered phase modulation depths.

In Chapter 4, phase modulation techniques are applied to ESPI to realize simultaneous 2D in-plane

displacement field measurement. An innovative 3-beam electronic speckle pattern interferometry (ESPI)

configuration is proposed. With one reference beam and two modulating beams at different modulation

frequencies, the 2D displacement field can be recorded simultaneously. By analyzing the phase images

before and after deformation, clear fringes representing the displacement along two perpendicular

directions are obtained, then the 2D in-plane deformation is calculated. This method can be easily

expanded to do 3D displacement field. The core idea of mixing interference signals at different

frequencies and separate them in the frequency domain could be applied in other configurations to

simplify the setups and to realize simultaneous measurements.

In Chapter 5, sinusoidal phase modulation is applied in phase-sensitive SPR detection. A cost-effective

vertical-cavity surface-emitting laser (VCSEL) is used as the light source. The phase modulation is

carried out by modulating the input voltage to the VCSEL. The resulting intensity modulation problem

is taken into account, and the experiments show that the induced error is small. By using the software

LabVIEW to control the system, the initial phase problem is solved. A CMOS camera is used as detector,

showing the possibility to do SPR imaging in the future. Several methods to compensate the influence

of ambient temperature fluctuation are proposed, including the scheme of combining shearing

interferometry with SPR imaging.

At last, a general conclusion of this thesis is made; several perspectives to improve the performance or

to overcome the disadvantages of phase modulating interferometry are proposed, including the

promising idea of combining sinusoidal phase modulation with smart detector.

An appendix is also attached, providing the complete derivation processes of some formulae, which may

be helpful for the readers.

Chapter 1: Basic methods of phase retrieval

7

Chapter 1 Basic methods of phase retrieval

In this chapter, the basis of optical interference is described. Interferometry and holography, which are

two main applications of optical interference investigated in this work are presented. The phase-

shifting/modulating methods, which are often used to solve the phase ambiguity problem, are also

discussed. In our experimental configuration, we will use lasers that are spatially and longitudinally

single mode within relatively balanced interferometer, therefore the notion of coherence will not be

detailed. When applying different phase modulation functions, different algorithms may be used to

process the interference signal. The methods considered in this work will be presented and compared:

LIA algorithm (algorithm for lock-in amplifier), SPM algorithm (traditional algorithm for sinusoidal

phase modulating interferometer), G-LIA algorithm (algorithm for generalized lock-in amplifier), and

f-G-LIA algorithm (G-LIA algorithm with DC filter).

1.1 Fundamentals of optical interference

From the quantum mechanics concept of wave–particle duality, light can be seen as an electromagnetic

wave as well as made of particles [76]. However, when it comes to the macroscopic propagation of light,

it is more convenient to consider it as a wave; in other words, to make use of the Maxwell's equations

[77,29] to describe and predict the behaviour of light.

For two correlated or coherent waves, when they superpose, they may interact with each other to result

in a total intensity which is not only determined by the intensities of these two waves, but also by the

phase difference between these two waves. This phenomenon observed for all kind of waves including

the transverse optical waves is called interference [78,31].

When two coherent light beams interfere with each other, the total intensity is also influenced by the

phase difference between the two light beams. In fact, the information provided by the phase of the light

is usually more useful and precise than intensity when it comes to optical measurements, and the

phenomenon of interference makes it possible to obtain this phase information. To date, many

techniques based on optical interferometry have been used to carry out such precise measurements.


8

To describe the principle of interferometry, we take the simplest case of two scalar plane waves

comprising a signal field and a reference field. The signal light field which is transverse is usually

expressed by the complex quantity [29,31]. When the light consists of a single frequency, the complex

signal field is expressed as:

( ) ( ) ( )( )2 si ft t

s sE t A t e +

= (0.1)

where i is the imaginary unit, t is time, sA is the amplitude of signal light, s is the phase of

signal light, and f is the frequency of light. Despite the harmonic character of the considered wave, a

time dependence in the amplitude As(t) and phase φs(t), considering that these two quantities can change

during an experiment but on timescale much longer than a field oscillation period (by several orders of

magnitude).

The frequency of light is in the order of 1014Hz in the visible range, while for an ordinary light detector,

the GHz range typically corresponds to an upper limit of detection, notably due to capacitive effects. In

consequence, only the light intensity is detected, which is a time averaged quantity:

( ) ( ) ( ) ( )2 2

d s sI t I t E t A t = = (0.2)

I(t) is the normalized light intensity, and Id(t) is the detected light intensity value which is proportional

to I(t). In practice, Id(t) can be used to represent I(t) for all the data processing methods in this thesis,

since for most of the interferometric detections, it is not necessary to obtain the absolute light intensity.

For simplicity, only I(t) will be used hereinafter.

If we have a reference light Er with the same frequency at the same point, then the corresponding

complex field can be expressed as:

( ) ( ) ( )( )2 ri ft t

r rE t A t e +

= (0.3)

where rA is the amplitude of reference light, r is the phase of reference light. The total light

intensity can be expressed as:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )2 2 2 2 cosr s r s r s r sI t E t E t A t A t A t A t t t = + = + + − (0.4)

where 𝜂 is a coefficient representing the coherence between signal and reference. In practice, we often

have 0<𝜂<1 because of the limited coherence of the source (or because of polarization mismatch

between the signal and reference beams). For two non-coherent beams, 𝜂=0; for two completely

coherent beams, 𝜂=1.


9

Obviously, the phase information ( ) ( )r st t − now has an influence on ( )I t , which can be detected

by light detectors. This means than by optical interference, phase-sensitive detections can be achieved.

It should be noted that only the phase difference between ( )s t and ( )r t is accessible but not each

phase separately since the absolute phase of a harmonic signal is relative to a certain time origin and is

therefore arbitrary. So the detectable phase signal that can be measured is noted:

( ) ( ) ( )s s rt t t = − (0.5)

In a relatively stable state, Eq. (0.4) can be expressed as:

2 2

2 2s 1 co

22 co sr s

r s r s s s

r s

o

A AI A A A A

A AI +

= + + =

+ (0.6)

where Io=As2+Ar

2 is the average intensity. With this definition, the fringe contrast is the factor

2 2

2 r s

r s

A A

A A

+. Usually, As and Ar can be easily measured, while φs is the main measurand of interest.

1.2 Interferometry and holography

The word “interferometry” refers to the techniques which make use of the phenomenon of interference

to extract information [79].

One of the most used classical interferometric method is used for surface profiling, as can be done with

a Fizeau interferometer. The observed fringes correspond to the contour lines of the surface. A good

precision is obtained given the wavelength scale of visible light (380nm-780nm), and the fact that the

height difference between two fringes is in the order of one wavelength. The wavelength being used as

a ruler, the interferometric measurement can provide precise non-contact measurement which is an

inherent advantage of phase-sensitive detection.

In 1948, holography was proposed by Denis Gabor [37,38], as mentioned in the general introduction.

The idea is to reconstruct the original wave front via the recorded interference pattern. While strictly

speaking, the technique also belongs to interferometry, holography usually requires scattering samples,

and light reconstruction is always required to observe or to measure the sample. In this sense, it is a

rather specific technique that can be distinguished from a variety of classical interferometry. Therefore,

nowadays, the techniques based on classical interferometry are called interferometric techniques, and

the techniques based on holography are often referred to as holographic techniques. At present, both

interferometric and holographic techniques are highly developed. They have been widely used in


10

physics, astronomy, engineering and applied science, biology, medicine, etc. [79]. However, precise yet

cheap measuring methods are still needed.

1.3 Phase retrieval methods

As expressed by Eq. (0.6), cos s can be extracted, provided that sA and rA are measured (or if Io

and 𝜂 are known). However, the phase ambiguity may still be a problem in real measurements. For

example, the values 3

s n

= +

( n is an integer) all give the result of cos 0.5s = . Besides, when

the measurements of sA and rA are not precise, a large error may occur. To solve these issues, and

determine both amplitude and phase without ambiguity, the following three types of phase retrieval

methods can be used.

1.3.1 Analysis of static interferograms

In many applications of interferometry, a 2D image containing interference fringes (also called

interferogram) can be obtained. In this case, the signal is said to have a spatial carrier. Eq. (0.6), can

then be expressed as:

( ) ( ) ( ) ( ) ( ) ( )( )2 2, = , , 2 , , ,r s r s sI x y A x y A x y A x y A x y f x y + + (0.7)

These fringes are caused by the term ( )( , )sf x y . As previously, this function is simply ( )cos ,s x y in

case of interference between two harmonic plane waves. By analyzing the fringes, we can measure the

variation of ( ),s x y . The phase ambiguity remains a problem, but it is less annoying here because

when the whole image of fringes is taken into consideration with a sufficient spatial sampling, the

obtained phases will be continuous from point to point. No jumps between positive and negative phase

values will occur between neighboring points. Since we are usually interested in the phase changes

between different points, and sometimes we may have a reference point with a known phase value to

eliminate the ambiguity, this method turns out to be very practical. Besides, some noises caused by the

variation of ( ),rA x y and ( ),sA x y can be easily filtered out.

Classical analysis of static interferograms has been widely used. It is intuitive, and it only needs one

image to get the phase information at every point. Many algorithms have been proposed to improve the

speed and precision of fringe analysis. However, the precision is limited by the fact that the phase value

relies on the judgement of fringe center and the noise level of neighboring zone. Retrieving amplitude


11

and phase without ambiguity become also more complex when more beams are interfering, and when

amplitude has fast spatial variations.

1.3.2 Phase shifting method

In order to address the issues of phase ambiguity and measurement precision, phase shifting method has

been introduced [33,34,35]. By using this method, the measured phase value at one point has no more

relation with the light intensity measured at other points: no spatial carrier is needed, and the phase can

be more precisely measured point by point without ambiguity.

The idea is to add a controllable phase r to the phase of reference so that the reference becomes:

( ) ( ) ( )2 r ri ft

r rE t A t e + +

= (0.8)

And the measurable intensity becomes:

( )2 2 2 2 cosr s r s r s r sI E E A A A A = + = + + − (0.9)

For a traditional 4-step phase-shifting method, the light intensity I is detected when φr takes four

different values (φr=0, α, 2α, 3α; α is a constant, 0<α<π):

( )2 2

1= 2 cos 1.5r s r s sI A A A A + + − − (0.10)

( )2 2

2 = 2 cos 0.5r s r s sI A A A A + + − − (0.11)

( )2 2

3 = 2 cos 0.5r s r s sI A A A A + + − (0.12)

( )2 2

4 = 2 cos 1.5r s r s sI A A A A + + − (0.13)

If we suppose:

1 2 3 4X I I I I= − + + − (0.14)

1 2 3 4 1 2 3 43 3Y I I I I I I I I= + − − − + − + (0.15)

It should be noticed that when calculating the value of Y, if 1 2 3 4 0I I I I+ − − or

1 2 3 43 3 0I I I I− + − +

, then imaginary number expression should be used. Then we can obtain:

( )args X iY = − + (0.16)

2 2

sA X Y + (0.17)


12

where arg means the argument of complex number①. This method is very simple and effective, so it has

been widely used. People often take α=π/2; but in fact, α can take any value as long as 0<α<π, which is

very practical: the precise calibration of α is not necessary. In practice the phase shift can be obtained

using a piezo-actuated mirror in the path of the reference beam.

1.3.3 Continuous phase modulation methods

Instead of making the discrete phase shifts forr , we may also make a continuous phase modulation

( )r t [14,26,36]. So the intensity of light also becomes a temporal signal I(t):

( ) ( )( )2 2 2= 2 cosr s r s r s r sI t E E A A A A t + = + + − (0.18)

Obviously, the non-modulated term 2 2

r sA A+ can be filtered out with a DC filter, while the modulated

term ( )( )2 cosr s r sA A t − can be used to do the phase retrieval. Since η is often a constant over time,

it is not necessary to measure its value in most phase retrieval methods, as will be shown later in this

chapter.

In practice, since the sampling time cannot be infinitely small, I(t) is also composed of a sequence of

values with a limited length (e.g. 1 2, ,..., ;nI I I n ). But the number of samples needed to calculate

s

is often bigger than phase-shifting method (i.e. n>4).

Continuous phase modulations may be used to replace discrete phase shifts for a variety of reasons: to

achieve a higher phase resolution and suppress certain noise, to simplify the measuring system, to

improve the measurement speed, etc. [14,26,36]

Different types of phase modulation function have different advantages and disadvantages. Some typical

phase modulation methods are discussed in the next section.

1.4 Different types of phase modulation function

1.4.1 Linear phase modulation

If we apply a linear phase modulation:

① The function arg(Z) takes the argument of a complex number Z=X+iY. In practice it is often computed through the function

atan2(X,Y) or angle(X+iY) in MATLAB.


13

( ) 0 02r t t f t = = (0.19)

where 0 and 0f are constants, then the light intensity turns into:

( ) ( ) ( )2 2 2 2

0 02 cos 2 cos 2r s r s s r s r s sI t A A A A t A A A A f t = + + − = + + − (0.20)

Obviously, the signal ( )I t has only one fundamental harmonic described by 0 or 0f . The phase

can be extracted by the traditional algorithm for lock-in amplifier (hereinafter referred to as "LIA

algorithm"), which will be detailed in the next chapter.

This linear phase modulation can be carried out in some heterodyne schemes, but the setups are often

complex and expensive. This modulation can also be realized with a simple mirror mounted on a

piezoelectric actuator. However, since the extension of piezoelectric crystal is always limited, usually

the linear modulation must be replaced by an equivalent sawtooth modulation.

It should be noticed that when piezoelectric actuators are driven to make sawtooth displacements, the

precision cannot be guaranteed, especially at high frequency, where the fly-back time of the mirror

cannot be neglected. The nonlinearity and noise generated by the sudden return becomes unacceptable

when high speed measurement is required. The sudden return may also reduce the lifetime of

piezoelectric crystals. These issues can be addressed with sinusoidal phase modulations.

1.4.2 Sinusoidal phase modulation

In the context of accurate interferometric measurement, sinusoidal phase modulating (SPM)

interferometry was introduced to enable the use of sinusoidal phase modulators within optical

interferometers [14,24]. Since then, this method has been recognized as a space- and cost-efficient

approach providing accurate phase determination. Significant works and possible improvements were

proposed based on this approach, including the use of a feedback system [72], laser diodes [72,73,80]

or the integrating-bucket method [26,81].

In an SPM interferometer, the modulator often consists in a simple mirror mounted on a piezoelectric

actuator which is driven to follow a harmonic motion at an angular frequency . Such device is simple,

cheap (usually much cheaper than the heterodyne schemes for linear phase modulation) and achromatic;

therefore, it can advantageously replace various modulators such as Bragg cells [82], rotating gratings

[83] or waveplates [84].

The sinusoidal phase modulation function ( )r t can be expressed as the following function:


14

( ) ( ) ( )sin sin 2r t a t a ft = = (0.21)

where a is a constant representing the amplitude of phase modulation (phase modulation depth),

is the angular frequency of modulation, and f is the frequency of modulation. The light intensity

( )I t becomes:

( ) ( )( )2 2 2 cos sinr s r s sI t A A A A a t = + + − (0.22)

According to the Jacobi–Anger expansion, this interferometric signal ( )I t presents a number of

harmonics at n ( 0, 1, 2,n = ) [22]:

( ) ( )( )

( )( ) ( )( )( )

( ) ( ) ( )

( ) ( )( )

2 2

2 2

0 2

12 2

2 1

1

2 cos sin

2 cos sin cos sin sin sin

2 cos 2 cos

2

2 sin 2 1 sin

r s r s s

r s r s s s

n s

n

r s r s

n s

n

I t A A A A a t

A A A A a t a t

J a J a n t

A A A A

J a n t

+

=

+

−

=

= + + −

= + + +

+

= + + + −

(0.23)

To determine s from SPM interferometric signal ( )I t , both the traditional algorithm (hereinafter

referred to as "SPM algorithm") [14] and a novel detection method based on so-called "Generalized

Lock-In Amplifier" (hereinafter referred to as "G-LIA algorithm") [22] can be used. Besides, the

integrating-bucket method [26,81] can also be applied to obtain s . These algorithms are detailed later

in the forthcoming section.

1.4.3 Other types of phase modulation

Linear/sawtooth and sinusoidal phase modulations are widely used in interferometric techniques.

However, sometimes, other types of phase modulation function may be more appropriate to eliminate

noise at certain frequencies or to make the best use of the equipment. In these cases, it is recommended

to consider G-LIA algorithm as a method to obtain s since the adaptability of G-LIA algorithm is

good.


15

1.5 Phase demodulation techniques

1.5.1 Lock-in amplifier technique (LIA algorithm)

In optical interferometry, LIA algorithm is able to deal with linear phase modulation:

( ) 0 02r t t f t = = (0.24)

where the light intensity is:

( ) ( ) ( )2 2 2 2

0 02 cos 2 cos 2r s r s s r s r s sI t A A A A t A A A A f t = + + − = + + − (0.25)

As we can see here, 0 and 0f represent the beating frequency.

Now we define two functions ( )C t and ( )S t as follows:

( ) ( )0cosC t t= (0.26)

( ) ( )0sinS t t= (0.27)

Then we define two values X and Y which can be calculated by integrating over time:

( ) ( )0

T

X I t C t dt= (0.28)

( ) ( )0

T

Y I t S t dt= (0.29)

T is the integration time. In order to make use of the orthogonality of trigonometric functions to get

accurate results, the integration time T should be long enough to cover many periods of modulation,

or it should be an integer multiple of the period 02 / . X and Y can be calculated as follows:

( )( ) ( )2 2

0 0

0

2 cos cos cos

T

r s r s s r s sX A A A A t t dt TA A = + + − = (0.30)

( )( ) ( )2 2

0 0

0

2 cos sin sin

T

r s r s s r s sY A A A A t t dt TA A = + + − = (0.31)


16

Then we see that s can be obtained by taking the argument of the complex number (X+iY). By

analogy with the methods that will be presented later on, we can also define two coefficients 1M =

and 1N = , so that s is given by:

args

X Yi

M N

= +

(0.32)

We also have:

2 2

s

X YA

M N

+

(0.33)

1.5.2 SPM algorithm

In this case, we have a sinusoidal phase modulation:

( ) ( ) ( )sin sin 2r t a t a ft = = (0.34)

so the light intensity is:


As described in [14], the and 2 components of the recorded signal ( )I t can be used to

calculate s . First, we define two functions of time ( )C t and ( )S t :

( ) ( )cos 2C t t= (0.36)

( ) ( )sinS t t= (0.37)

Then we can calculate the values of X and Y using two harmonics component provided by the

Jacobi–Anger expansion (given inside Eq. (0.23)) based on the orthogonality of trigonometric functions

(the integration time T should be long enough to cover many periods of modulation, or it should be

an integer multiple of the period 2 / ):

( ) ( ) ( )2

0

2 cos

T

r s sX I t C t dt TA A J a = = (0.38)


17

( ) ( ) ( )1

0

2 sin

T

r s sY I t S t dt TA A J a = = (0.39)

where nJ is the n-th Bessel function of the first kind. If we define M and N as follows:

( )2M J a= (0.40)

( )1N J a= (0.41)

Then we can obtain the value of φs:

args

X Yi

M N

= +

(0.42)

and the value of As:

2 2

s

X YA

M N

+

(0.43)

1.5.3 G-LIA algorithm

The G-LIA idea is to use reference signals C(t) and S(t) containing the same harmonic contents (in

frequency component and relative weights) than the signal modulation induced by the phase modulation.

G-LIA algorithm was first applied to phase sensitive near-field nanoscopy [22], where the signal

amplitude is also modulated but at a frequency different from the phase modulation① via the periodic

scattering of tip frequency carrier). In such technique where the signal is low, G-LIA is interesting as

all the SPM sidebands on the low and high frequency sides of the tip frequency are contributing to the

near-field signal.

In G-LIA algorithm, for any type of phase modulation function ( )r t , ( )C t and ( )S t are defined

as:

( ) ( )( )cos rC t t= (0.44)

( ) ( )( )sin rS t t= (0.45)

① The origin of this modulation of the signal amplitude comes from the fact that the nano-probe providing the signal is

oscillating in and out the near-field region where a maximum of optical field is scattered toward the detector.


18

For a sinusoidal phase modulation described by:

( ) ( ) ( )sin sin 2r t a t a ft = = (0.46)

The light intensity also turns into:


Then we can calculate X according to Jacobi–Anger expansion and the orthogonality of

trigonometric functions (here also, the integration time T should be long enough to cover many

periods of modulation, or it should be an integer multiple of the period ( ) ( )2

0 01 2 2M J a J a= + − ):

( ) ( ) ( ) ( ) ( )( )2 2

0 0

0

2 co1 s

T

r s r s sX I t C t dt T A A J a TA A J a + += = + (0.48)

Since ( )r t is controllable, if we set 2.4048a rad= ① so that ( )0 0J a = , then we have:

( )( )01 2 cosr s sX TA A J a = + (0.49)

Likewise, we can calculate Y :

( ) ( ) ( )( )0

0

sin21

T

r s sY I t S t dt TA A J a −= = (0.50)

In G-LIA algorithm, M and N should be defined according to the type of phase modulation function

( )r t [22]. When dealing with sinusoidal modulation function, as we do now, M and N should be

defined as follows:

( )01 2M J a= + (0.51)

( )01 2N J a= − (0.52)

So that we can obtain:

args

X Yi

M N

= +

(0.53)

① Or more generally a equals to any zero of Jo(a).


19

2 2

s

X YA

M N

+

(0.54)

Obviously, LIA algorithm can be seen as a special case of G-LIA algorithm when ( )r t is a linear

function and 1M N= = . However, to overcome limitations of G-LIA, the f-G-LIA should be used as

explained in the next paragraphs.

1.5.4 f-G-LIA algorithm

As shown in the previous section, for G-LIA algorithm working with a sinusoidal phase modulation, the

DC component of ( )I t is also concerned: it consists of the non-modulated term 2 2

r sA A+ , as well as

the DC component in the modulated term ( )( )2 cosr s r sA A t − . The presence of such DC

component undoubtedly requires that ( )0 0J a = to eliminate the term 2 2

r sA A+ ; otherwise, error will

occur. This restricts the choice of amplitude of phase modulation a .

In order to avoid this problem, we can apply a DC filter to the signal ( )I t before calculating the values

of X and Y . We know that:

( ) ( )( )

( )( ) ( )( )( )

( ) ( ) ( )

( ) ( )( )

2 2

2 2

0 2

12 2

2 1

1

2 cos sin

2 cos sin cos sin sin sin

2 cos 2 cos

2

2 sin 2 1 sin

r s r s s

r s r s s s

n s

n

r s r s

n s

n

I t A A A A a t

A A A A a t a t

J a J a n t

A A A A

J a n t

+

=

+

−

=

= + + −

= + + +

+

= + + + −

(0.55)

So the filtered signal ( )I t can be expressed as:

( ) ( ) ( ) ( ) ( )( )

( )( ) ( )( )( ) ( )

2 2 1

1 1

0

2 2 cos 2 cos 2 sin 2 1 sin

2 cos sin cos sin sin sin 2 cos

r s n s n s

n n

r s s s r s s

I t A A J a n t J a n t

A A a t a t A A J a

+ +

−

= =

= + −

= + −

(0.56)

Then we can calculate the new values of X and Y :

( ) ( ) ( ) ( )( )2

0 0

0

1 2 2 cos

T

r s sX I t C t dt TA A J a J a = += − (0.57)


20

( ) ( ) ( )( )0

0

sin21

T

r s sY I t S t dt TA A J a −= = (0.58)

Obviously, M and N should be redefined as:

( ) ( )2

0 01 2 2M J a J a= + − (0.59)

( )01 2N J a= − (0.60)

This modified G-LIA algorithm is hereinafter referred as "f-G-LIA algorithm" (G-LIA algorithm with

DC filter). Strictly speaking, f-G-LIA algorithm still belongs to G-LIA algorithm. From the definitions

of M and N above, it can be seen that when ( )0 0J a = , f-G-LIA algorithm and G-LIA algorithm

are equivalent. Since f-G-LIA makes use of all the harmonics contents, it is also much preferable to use

this method rather than the traditional SPM method when the phase modulation depth is large, as the

signal is spread over a large number of harmonics in this case.

1.5.5 Integrating bucket algorithm

Another possibility of doing phase retrieval is the so-called “integrating bucket” technique [85,86]:

instead of recording the whole interference signal, it only records several values (typically 4 values)

during each period of modulation; each recorded value is the integration of light intensity over time. The

integral operations can naturally be done by making use of the integration time of the sampling process

of photosensitive elements. This way, the speed of data processing can be largely improved. But it

should be noticed that in order to carry out this method, the exposure time must be precisely controlled.

The traditional integrating bucket algorithm was proposed in 1975 to deal with the case of linear phase

modulation [85]. In 1987, the application of integrating bucket algorithm in sinusoidal phase modulating

interferometer was proposed [26]; we will give a more general mathematical expression of this algorithm

in Section 2.3.4.

1.6 Comparison between different algorithms

1.6.1 SPM algorithm and (f-)G-LIA algorithm

Although SPM algorithm and (f-)G-LIA algorithm can retrieve the same quantities, the different

definitions of ( )C t and ( )S t make them quite different from the perspective of frequency domain

analysis, as shown in the following figure.


21

Figure 1-1 Frequency analysis of: (a) Signal I(t) when a=2.4048rad or 8.0000rad; As=1, Ar=1, 𝜂=1,

𝜔=5Hz, 𝜑s is randomly set to be 0.5rad; (b) C(t) and S(t) in SPM algorithm when a=2.4048rad and

𝜔=5Hz; (c) C(t) and S(t) in G-LIA algorithm when a=2.4048rad and 𝜔=5Hz.


22

SPM algorithm uses a limited number of frequency components (see Figure 1-1(b)), typically and

2 , while ignoring the information contained in the higher harmonics. Through (f-)G-LIA algorithm,

all the available harmonics are extracted with an adequate weight (which is determined by the frequency

spectrum of ( )( )sin r t and ( )( )cos r t ) in a single step. Besides, a variety of phase modulation

functions can be used.

(f-)G-LIA algorithm has thus two interests. Firstly, it can extract phase and amplitude information with

the same procedure for a variety of phase modulation functions including the traditional ones (linear,

sine or triangular). Secondly, since the frequency components used in (f-)G-LIA and SPM algorithms

are different (see Figure 1-1(c)), the (f-)G-LIA can have a better anti-noise ability depending on the type

of noise and modulation. More precisely, we may consider that since all the useful frequency

components are used to recover the signal in (f-)G-LIA, it tends to have a better noise to signal ratio,

especially if the phase modulation depth is large so that signal is spread over a large number of

harmonics (see Figure 1-1(a)). This should be tempered by the fact that extracting weak frequency

components can be problematic if the noise is unexpectedly stronger for some of these harmonics. A

possible strategy offered by (f-)G-LIA, if the experimental conditions permit, is then to select a proper

modulation function r so that ( )( )sin r t and ( )( )cos r t have a lesser similarity with the noise

in the frequency domain, in order to reach the best anti-noise ability.

1.6.2 Amplitude of phase modulation

According to the discussion above, we know that when ( )0 0J a = (which means 2.4048a rad= if

the first zero is taken), f-G-LIA algorithm and G-LIA algorithm are equivalent, and all the algorithms

(f-G-LIA, G-LIA and SPM) can give accurate results; otherwise, only f-G-LIA and SPM algorithms can

give accurate results. This is exemplified in the following Figure 1, where a known phase value is

retrieved by the three methods.


23

Figure 1-2 Simulation results given by SPM, G-LIA and f-G-LIA algorithms at different values of a .

3s rad = − is the exact value to be detected. The sampling rate was 15 points per period of phase

modulation.

A more practical issue is to study the influence of an error on the amplitude of phase modulation a . In

other words, when the experimental modulation depth a is not the same as the a value used in the

algorithms. It is then important to know how much error it may bring.


24

Figure 1-3 Effect of an error made on the amplitude of phase modulation: simulation results given by

SPM, G-LIA and f-G-LIA algorithms at different values of a while a is always considered to be

2.63rad in the algorithms.

Figure 1-4. Same as Figure 1-3, for value of a=2.4048 rad in the algorithms.

The figures above show the simulation results concerning the impact on the retrieved φs of an error on

the phase modulation depth a in the intensity I(t). The phase to be retrieved is 3s rad = − , and the

sampling rate is 15 points per period of phase modulation.


25

As shown in Figure 1-3 , when the a value used in the references C(t) and S(t) is set to be 2.63rad .

This value corresponds to a nearly optimal value for the SPM algorithm in terms of anti-noise capability,

as mentioned in [14, 24]. The results derived from G-LIA algorithm is particularly affected, even for a

phase modulation depth 2.4a rad , where the precision is not as good as in Figure 1-2 because of

the mismatching value of a . On the contrary, for SPM and f-G-LIA algorithms, the results are still

good near 2.63a rad because they are not so dependent on the value of a , the mismatch of a

being the only problem. Given the difference of slopes in Figure 1-3, it can be seen that when a

measurement error on a exists, the f-G-LIA algorithm is less affected than SPM algorithm.

Likewise, in Figure 1-4, when a is set to be 2.4048rad , the results are still good near 2.4a rad

, and f-G-LIA algorithm remains less affected by the measurement error of a than SPM algorithm.

Another point to notice is that the results of G-LIA and f-G-LIA algorithm completely overlap because

as previously discussed, when 2.4048a rad= , f-G-LIA algorithm and G-LIA algorithm are

equivalent. So hereinafter when a is set to be 2.4048rad , only the results of SPM and G-LIA

algorithms will be shown and compared.

For SPM algorithm, the value of a also affect the anti-noise ability, so 2.63a rad= is often used

[14, 24]. In order to compare the anti-noise ability of SPM, G-LIA and f-G-LIA algorithms, simulations

were done and the results are shown by the following figures. In these simulations, white Gaussian noise

was added to the signal ( )I t resulting in a signal-to-noise ratio (SNR) of 20 dB. (Power of

noise: -17.46 dBW; power of signal ( )I t : 2.54 dBW.) For each sampling rate, 100 repetitive

measurements were made to obtain the standard deviation of s .


26

Figure 1-5 Simulation results of the standard deviation of measured s when white Gaussian noise is

added to the signal ( )I t . The phase modulation depth and the a value used in the algoritm are both set

to 2.63a rad= .


added to the signal ( )I t . 2.4048a rad= , and the measurement error of a is supposed to be 0.


27


added to the signal ( )I t . 5.5201a rad= , and the measurement error of a is supposed to be 0.

As shown in Figure 1-5, f-G-LIA algorithm is slightly less affected by the noise than SPM algorithm.

Besides, according to Figure 1-3, when the actual phase modulation depth is 2.63a rad= , both f-G-

LIA and SPM algorithms gave accurate results. In short, f-G-LIA algorithm had a slightly better

performance than SPM algorithm when 2.63a rad= .

In Figure 1-6 and Figure 1-7, the same analysis is performed when 2.4048a rad= and

5.5201a rad= , where ( )0 0J a = thus the f-G-LIA algorithm is equivalent to the G-LIA algorithm.

In Figure 1-6, the G-LIA approach is still slightly less affected by the noise than the SPM algorithm.

We note that for both cases of Figure 1-5 and Figure 1-6, the obtained standard deviations are similar

because the phase modulation depths are close from each other. However, as shown in Figure 1-7, when

the phase modulation depth increases, the obtained standard deviations for G-LIA algorithm remain at

the same level, while for SPM algorithm they have been increased a lot. In fact, according to several

simulations, we observed that: as the value of a increases, there is no significant change of the standard

deviations for G-LIA algorithm, while for SPM algorithm they keep increasing. It means that as the

phase modulation depth increases, the advantage of G-LIA algorithm on anti-noise ability will be more

and more obvious. This result can be explained by the fact that when a increases, the power of


28

high-frequency components in I(t) also increases, but these components cannot be used by SPM

algorithm, since it only makes use of the first two harmonics ω and 2ω, as shown by Figure 1-1.

1.7 Conclusion

In this chapter, the basic idea of optical interference is introduced, and the basic expression of

interference signal is given. To measure the phase difference between two laser beams, the common

phase-shifting and continuous phase modulation techniques are introduced. Traditional phase retrieval

algorithms are presented, as well as the newly proposed (f-)G-LIA algorithm. Finally, a comparison

between SPM algorithm and (f-)G-LIA algorithm is done. The results of simulations show that, even at

modest phase modulation depth, (f-)G-LIA algorithm has a slightly better anti-noise ability than SPM

algorithm regarding a white Gaussian noise. Besides, (f-)G-LIA algorithm can also be adapted to

different types of phase modulation functions. A table is shown below to make this comparison clearer.

Algorithm C(t) S(t) Involved

frequencies

Anti-noise①

ability

Type of

modulation

SPM cos(2ωt) sin(ωt) ω,2ω Good Only sinusoidal

(f-)G-LIA cos(φr(t)) sin(φr(t)) ω,2ω,3ω,…

(same as I(t))

Better Not limited

Table 1-1. Brief comparison between SPM and (f-)G-LIA algorithm.

From Section 1.5, we can see that when the value of η is a non-zero constant over time, it is not necessary

to know its true value to carry out precise measurements of φs, and the measured value of As is only

multiplied by η, since X and Y are both proportional to η. For the sake of simplicity, hereinafter we

suppose η=1 for the following chapters. Still, it should be noticed that the (As2+Ar

2) term in I(t) is not

multiplied by η, so this supposition may bring errors when 0<η<1 and the DC component of I(t) is used.

① Only the white Gaussian noise has been discussed. Depending on the spectrum of noise and the spectrum of signal, the

result may be different. Generally speaking, the more different these two spectrums are, the more obvious the advantage of

(f-)G-LIA algorithm is.

Chapter 2: Advanced methods of phase retrieval

29

Chapter 2 Advanced methods of phase retrieval

In the last chapter, we have considered the case of an interferometric signal containing solely an ideal

sinusoidal phase modulation: ( ) ( )( )2 2 2 cos sinr s r s sI t A A A A a t = + + − . Besides noises, two

issues discussed in this chapter can increase the complexity of the interferometric signal. The first one

is the non-zero initial phase of phase modulation, which occurs when the modulating components (like

the piezo-electric actuators) and the acquisition of the detector signals (like Photodiode and camera) are

not synchronized. The second one is the additional intensity modulation of laser. This problem often

occurs when the phase modulation is induced through a wavelength modulation driven by a current

modulation of the laser source, as can be done with laser diode such as VCSEL①. In such case the

algorithm presented in the previous chapter can fail.

In fact, the first issue can be solved by determining an initial phase. Section 2.1 focus on the error that

a wrong guess of the initial phase may bring. The results are useful for the trial and error method in

experiments. A mathematical method to calculate the initial phase is also introduced in Section 2.3.2.

In Section 2.2, we solve the second issue related to the intensity modulation in a mathematical way. The

resulting interference signal is analyzed in the frequency domain in Section 2.2.1. Then the mathematical

solutions for this problem as well as the initial phase problem are proposed.

In Section 2.3.1 and Section 2.3.3, SPM algorithm and f-G-LIA algorithm have been modified to solve

these problems. In Section 2.3.4, based on the analysis in Section 2.2.1, a generalized integrating bucket

algorithm is also proposed, which may largely extend the application area of integrating bucket method.

① VCSEL: vertical-cavity surface-emitting laser.


30

2.1 Initial phase problem

In all the algorithms discussed in the last chapter, ideal phase modulation functions like

( ) 0 02r t t f t = = or ( ) ( ) ( )sin sin 2r t a t a ft = = were used. However, in reality, the exact

time when the phase modulation begins may be difficult to decide, because of synchronization issue.

For example, a piezo actuator exhibits a frequency-dependent lag in its mechanical response when it is

excited with a sine signal. If we use 0t to represent the initial time difference between the beginning

of phase modulation and the beginning of signal recording, then we have: (we suppose η=1)

( ) ( )0'r rt t t = + (1.1)

( ) ( )( )2 2' 2 cos 'r s r s r sI t A A A A t = + + − (1.2)

where ( )'r t is the real phase modulation function, and ( )'I t is the real light intensity. We can

make a guess that the initial time difference is gt , then we can set:

( ) ( )' gC t C t t= + (1.3)

( ) ( )' gS t S t t= + (1.4)

Likewise, we define two values 'X and 'Y :

( ) ( )0

' ' '

T

X I t C t dt= (1.5)

( ) ( )0

' ' '

T

Y I t S t dt= (1.6)

If the guess is right, which means g ot t= , then we have:

'X X= (1.7)

'Y Y= (1.8)

Thus the final results will not be influenced. However, in fact, we often have g ot t resulting in

algorithm- dependent errors.


31

2.1.1 LIA algorithm

When g ot t , for a linear/sawtooth phase modulation:

( ) ( ) ( )0 0 0 0' 2r t t t f t t = + = + (1.9)

We have:

( ) ( )( )0' cos gC t t t= + (1.10)

( ) ( )( )0sin gS t t t= + (1.11)

( )( ) ( )( )

( )( ) ( )

( )( )( ) ( )

2 2

0 0 0

0

2 2

0 0 0

0

2 2

0 0 0 0

0

' 2 cos cos

2 cos - cos

2 cos cos

T

r s r s s g

T

r s r s g s

T

r s r s s g

X A A A A t t t t dt

A A A A t t t dt

A A A A t

t

t tt dt

= + + + − +

= + + + −

= + +

−− +

(1.12)

( )( ) ( )( )

( )( ) ( )

( )( )( ) ( )

2 2

0 0 0

0

2 2

0 0 0

0

2 2

0 0 0 0

0

' 2 cos sin

2 cos - sin

2 cos sin

T

r s r s s g

T

r s r s g s

T

r s r s s g

Y A A A A t t t t dt

A A A A t t t t dt

A A A A tt tt dt

= + + + − +

= + + + −

= + +

−− +

(1.13)

By comparing Eq. (1.12)-(1.13) with Eq. (0.30)-(0.31), we see that the final obtained phase using LIA

algorithm is:

0

' 'arg

1 1's s

X Yi

= =

−+ (1.14)

with

( )0 0 0 gt t = − (1.15)


32

This means that for the case of linear/sawtooth phase modulation, the non-corrected guess of initial

phase will result in a constant 0− added to the final detected phase. This phase shift is tolerable when

0 remains indeed the same all the time, and when we are only interested in the phase change.

2.1.2 SPM algorithm

When 0gt t , for a sinusoidal phase modulation:

( ) ( )( ) ( )( )0 0' sin sin 2r t a t t a f t t = + = + . (1.16)

Using SPM algorithm, we have:

( ) ( )( )' cos 2 gC t t t= + , (1.17)

( ) ( )( )' sin gS t t t= + . (1.18)

If we set:

( )0 gt t = − , (1.19)

then we can calculate:

( )( )( ) ( )( )

( ) ( )( )( )

2 2

0

0

2 0

2

' 2 cos sin cos 2

2 cos 2 cos

2 cos 2 cos

T

r s r s s g

r s g s

r s s

X A A A A a t t t t dt

TA A J a t t

TA A J a

= + + + −

=

+

= −

(1.20)

( )( )( ) ( )( )

( ) ( )( )( )

2 2

0

0

1 0

1

' 2 cos sin sin

2 cos sin

2 cos sin

T

r s r s s g

r s g s

r s s

Y A A A A a t t t t dt

TA A J a t t

TA A J a

= + + + − +

= −

=

(1.21)

Comparing the values of X and Y with 'X and 'Y we have:

'

cos 2X

X= , (1.22)


33

'

cosY

Y= . (1.23)

Since we have:

' '

arg's

X Yi

M N

= +

, (1.24)

args

X Yi

M N

= +

, (1.25)

the desired result 's s = is obtained only when '/

1'/

X X

Y Y= ,

'0

X

X and

'0

Y

Y ,

When θ changes from 0 to 2 , the corresponding values of 'X

X,

'Y

Y and

'/

'/

X X

Y Y are shown in

the figure below:

Figure 2-1 Initial phase (θ) problem in SPM algorithm. Relationship between (a) '/X X and ; (b)

'/Y Y and ; (c) ( ) ( )'/ / '/X X Y Y and , when goes from 0 to 2 .

This figure shows that:

(1) At point P0 (where 2n = , n is an integer), ' '

1X Y

X Y= = , so 's s = ;


34

(2) At point P2 and P4, ' '

0X Y

X Y= , so 's s = ;

(3) At point P3, where θ=(2n+1)π , with n an integer, ' '

1X Y

X Y= − = , so 's s = − ;

(4) At point P1 and P5, ' '

0X Y

X Y− = , so 's s = − .

When gets any of the values discussed above, we know that s can be measured with ambiguity (

's s n = ). When we only care about the absolute value of phase change s , then the

measurements can be carried out without problem.

However, when gets other values, then ' 'X Y

X Y , so a measurement error will certainly occur.

2.1.3 (f-)G-LIA algorithm

For G-LIA algorithm dealing with the same sinusoidal phase modulation, we have:

( ) ( )( )( )' cos sin gC t a t t= + , (1.26)

( ) ( )( )( )' sin sin gS t a t t= + , (1.27)

and:

( ) ( )

( ) ( )

0

2 2

0

0

' ' '

cos 2 cos sin cos 2 cos sin sin2 2

,

cos 2 sin cos cos 2 sin cos sin2 2

sin

sin

T

r s

s sT

r s

s s

X I t C t dt

T A A J a

a t a t

A A dt

a t a t

=

= +

+

+

+

+

(1.28)


35

( ) ( )0

0

' ' '

sin 2 cos sin cos 2

c

cos sin sin2 2

.

sin 2 sin cos cos 2 sin cos sin2 2

cos

os

T

s sT

r s

s s

Y I t S t dt

a t a t

A A dt

a t a t

=

=

−

−

+

(1.29)

Each integral contains the four following terms:

( )0 2

10 0

0

cos 2 cos sin 2 cos 2 2 cos cos 22 2 2

2 cos2

T T

n

n

a t dt J a J a n t dt

TJ a

+

=

= +

=

(1.30)

( )( )2 1

10 0

2 cos sin 2 2 cos sin 2 1 02

s2

in

T T

n

n

a t dt J a n t dt

+

−

=

= − =

(1.31)

( ) ( )0 2

10 0

0

cos 2 sin cos 2 sin 2 1 2 sin cos 22 2 2

2 sin2

T Tn

n

n

a t dt J a J a n t dt

TJ a

+

=

= + −

=

(1.32)

( ) ( )( )2 1

10 0

sin 2 sin cos 2 1 2 sin cos 2 1 0.2 2

T Tn

n

n

a t dt J a n t dt

+

−

=

= − − − =

(1.33)

So, for the G-LIA, we obtain:

( ) ( )2 2

0 0 0' 2 sin 2 cos cos2 2

r s r s sX T A A J a TA A J a J a

= + + +

(1.34)

0 0' 2 sin 2 cos sin2 2

r s sY TA A J a J a

= −

(1.35)

Similarly, for f-G-LIA algorithm, we can get:

( )2

0 0 0' 2 sin 2 cos 2 cos2 2

r s sX TA A J a J a J a

= + −

(1.36)


36

0 0' 2 sin 2 cos sin2 2

r s sY TA A J a J a

= −

(1.37)

When 2.4048a rad= so that ( )0 0J a = , for both G-LIA and f-G-LIA algorithms, we have:

0 0' 2 sin 2 cos cos2 2

r s sX TA A J a J a

= +

(1.38)

0 0' 2 sin 2 cos sin2 2

r s sY TA A J a J a

= −

(1.39)

For this phase modulation depth 2.4048a = , the values of 'X

X,

'Y

Y and

'/

'/

X X

Y Y are shown in the

figure below, when ( )0 gt t − changes from 0 to 2 ,:

Figure 2-2 Initial phase (θ) problem with the (f)-G-LIA algorithm, when 2.4048a = rad.

Relationship between: (a) '/X X and ; (b) '/Y Y and ; (c) ( ) ( )'/ / '/X X Y Y and ,

and changes from 0 to 2 .

By comparing Figure 2-2 with Figure 2-1, we can see that they are quite similar to each other: the

positions of P0 and P3 remain the same; the zero points of '/Y Y remain the same; the positions of P1,

P2, P4 and P5 only shift a little. The descriptions made on Figure 2-1 is also applicable to Figure 2-2.


37

2.2 The intensity modulation problem and frequency analysis

An important type of phase modulation is achieved by tuning the wavelength of the laser source. When

the lengths of the two-arm interferometer are not balanced, this indeed results in a phase modulation.

This phase modulation is typically carried out by modulating the input voltage to modulate the injection

current of laser diode current, resulting in a correlated output power modulation. In the linear range of

the laser diode, we have:

( ) ( )( ) ( )2 2 2 cos 1r s r s r s rI t A A A A t t = + + − + (1.40)

where μ is a coefficient determined by the property of laser. To avoid such additional intensity

modulation due to the current modulation, we note that a photothermal modulation technique can lead

to a more negligible intensity modulation [75]. However, the complexity of setup is clearly increased.

Considering the initial phase problem in SPM interferometer, this signal becomes:

( ) ( )( )( ) ( )( )2 2

0 0' 2 cos sin 1 sinr s r s sI t A A A A a t t a t t = + + + − + + (1.41)

According to the definitions of X’, Y’, θ, C’(t) and S’(t) for either SPM algorithm or (f)-G-LIA algorithm

(see Section 2.1), we can see that processing the signal described by Eq. (1.41) with C’(t) and S’(t) is

equivalent to processing the signal described by Eq. (2.42) with C(t) and S(t).

( ) ( )( ) ( )2 2 2 cos sin 1 sinr s r s sI t A A A A a t a t = + + + − + + (1.42)

To obtain s and sA from this signal, we can still carry out different algorithms, which are detailed

later in this chapter. To address this issue, we first need to perform a signal analysis in the frequency

domain in order to determine what will be the integration result of the phase demodulation algorithms.

2.2.1 Frequency domain analysis

By using Jacobi-Anger expansion and trigonometric identities, we can analyze the signal ( )I t in the

frequency domain (see Appendix for a detailed derivation):


38

( ) ( )( ) ( )

( ) ( )( ) ( )( )

( )

( )

2 2

1

2,4,... 3,5

0

,...

0,2,...

1,3,...

2 cos sin 1 sin

2 sin cos sin

cos sin

2

sin cos cos s

cos i

n

s n

i

r s r

m

s s

r s m

m

r s

m

m

m

m

m

I t A A A A a t a t

A A R t m t R m t

m t m t

A A

R m m t m m t

R R

R m m

+ +

= =

+

=

+

=

= + + + − + +

= + + + + +

−

=

+

+ +

(1.43)

where

( ) ( )

( ) ( ) ( )( )

( ) ( ) ( )( )( ) ( ) ( )( )

2 2

0 1

2 2

1 0 2

1 1

1 1

cos sin2

2 sin cos , 12

2 c

, 0

, 2,4,...

, 3,5,..

n

.

os si

2 sin cos

r ss s

r s

r ss s

r s

m s m m s

m s s

m

m m

m

R

A AJ a aJ a

A A

A Aa J a a J a J a m

A A

J a a J a J ma

J a a J ma J a

+ −

− +

+ =

= + =

+ = −

++

++ + − =

−

(1.44)

If 0 = then we have:

( )

( )

( )

2 2

0 cos2

2 cos

2 sin

.

, 0

, 2,4, ..

, 1,3,...

m m

r ss

r s

s

m s

m

R

A AJ a

A A

J a

mJ a

m

=

= =

++

=

(1.45)

As a matter of fact, according to the definition of Fourier transform, the integrations with ( )cos n t

and ( )sin n t− correspond to the real and imaginary parts of the Fourier transform of ( )I t at the

frequency of n respectively.

2.3 Demodulation techniques for intensity modulated signals

In this section, we propose several demodulation techniques to handle the problem of intensity

modulation, as well as a method to calculate the initial phase.


39

2.3.1 Modified SPM algorithm

We know that in SPM algorithm, the definitions of ( )C t and ( )S t are:

( ) ( )cos 2C t t= , (1.46)

( ) ( )sinS t t= . (1.47)

Then according to the orthogonality of trigonometric functions, we have:

( ) ( )

( ) ( ) ( ) ( )( )( )0

1 1cos 2 2 cos sin

T

r s m s m m s

X I t C t dt

TA A J a a J a J a + −

=

= −+

(1.48)

( ) ( )

( ) ( ) ( )( )

0

2 2

1 0 2cos 2 sin cos2

T

r sr s s s

r s

Y I t S t dt

A ATA A a J a a J a J a

A A

=

+= + + −

(1.49)

If we define the following coefficients:

( )2 2

0

1cos

2r sQ aT A A = + (1.50)

( )1 12 cosr sQ TA A J a= (1.51)

( ) ( )( )2 0 2cosr sQ aTA A J a J a = − (1.52)

( ) ( )3 2 cos 2r s mTA A J aQ = (1.53)

( ) ( ) ( )( )14 1cos 2r s m maTA A J a J aQ + −−= (1.54)

Then we obtain an equation set:

0 1 2sin coss sY Q Q Q = + + (1.55)

3 4cos sins sX Q Q = + (1.56)


40

The solutions of this equation set are:

4 1 0 4

2 4 1 3

cos s

Q Y Q X Q Q

Q Q Q Q

− −=

− (1.57)

3 2 0 3

1 3 2 4

sin s

Q Y Q X Q Q

Q Q Q Q

− −=

− (1.58)

So we can obtain:

( )arg cos sins s si = + (1.59)

From the derivation above, we can see that the and 2 components are used to obtain s . This

is the basic idea of SPM algorithm. When 0 = , it becomes the case described by [74].

However, according to the frequency domain analysis, the components at other harmonics also contain

information of s (see Eq. (1.44)). If we want to use other components to obtain s , it is feasible as

long as the equation set containing cos s and sin s is solvable. For example, if we want to make

use of the and 2 components, we may redefine ( )S t :

( ) ( )sin 3S t t= (1.60)

Then the Y becomes:

( ) ( )

( ) ( ) ( ) ( )( )( )0

3 2 4cos 3 2 sin cos

T

r s s s

Y I t S t dt

TA A J a a J a J a −+

=

=

(1.61)

Also 1Q and 2Q should be redefined as:

( ) ( )1 32 cos 3r sQ TA A J a= (1.62)

( ) ( ) ( )( )2 2 4cos 3r sQ aTA A J a J a = − (1.63)

Now the equation set becomes:

1 2sin coss sY Q Q = + (1.64)


41

3 4cos sins sX Q Q = + (1.65)


4 1

2 4 1 3

cos s

Q Y Q X

Q Q Q Q

−=

− (1.66)

3 2

1 3 2 4

sin s

Q Y Q X

Q Q Q Q

−=

− (1.67)

This way s can be obtained. The advantage of using 3 instead of is that: the term 0Q , which

used to be the only one containing the term ( )2 2

r sA A+ , becomes zero. This is advantageous because

as discussed in the last chapter, when η<1, which is often the case in reality, it is preferred to avoid using

the DC component in I(t) so that there is no need to measure η. Besides, 1Q , 2Q ,

3Q and 4Q all

contain r sA A so that it can be eliminated, which means it is unnecessary to measure rA and sA .

When choosing the components to be used, the low frequency components (like ω, 2ω and 3ω) should

be chosen in priority, since the high frequency components are usually very weak, which will affect the

precision of measurements. Yet the zero-frequency/DC component of signal is not recommended to be

used, because it can be easily affected (by the ambient light, by the gain/exposure of camera/detector,

etc.).

Another thing to be noticed is that the value of θ should also be considered while choosing the

components. For example, if 4

= , then ( )cos 2 0 = , which means

3 4 0Q Q= = , making 0X = .

In this case, we may choose other frequency components, or change the definition of ( )C t from

( ) ( )cos 2C t t= to ( ) ( )sin 2C t t= .

In practice, the values of θ and μ may be difficult to measure. However, it should be noticed that after a

signal I(t) is obtained, all the frequency components below the Nyquist frequency can be extracted. This

means we have an equation set containing θ, μ, sin φs , cos φs (and eventually a, Ar, As), with some

redundance, to be solved. So if we make full use of this signal, it is hopeful that we may manage to

directly solve the values of θ, μ, φs, etc., or to improve the measurement precision since we have the

redundant equations. But this part of work requires more mathematical skills, so it will not be discussed

in this thesis.


42

2.3.2 Determination of the initial phase

When θ is unknown, we can make use of the signal at the frequency of ω to obtain its value. We may

redefine:

( ) ( )cosC t t= (1.68)

( ) ( )sinS t t= (1.69)

Then according to Eq. (1.45) we have:

( ) ( ) 1

0

sin

T

r sX I t C t dt TA A R = = (1.70)

( ) ( ) 1

0

cos

T

r sY I t S t dt TA A R = = (1.71)

As long as ( ) ( ) ( )( )2 2

1 1 0 22 sin cos 02

r ss s

r s

A AR a J a a J a J a

A A

+= + + − , we have:

( )arg Y iX = + (1.72)

This method is also described in [14] for the special case when µ=0.

2.3.3 Modified f-G-LIA algorithm

According to Eq. (1.43)-(1.44), the DC-filtered signal ( )I t can be expressed as:

( ) ( ) ( ) ( )2 2

0 12 cos sin2

r sr s s s

r s

A AI t I t A A J a aJ a

A A

+

++

= −

(1.73)

We know that in f-G-LIA algorithm, the definitions of ( )C t and ( )S t are:

( ) ( )( )cos sinC t a t= (1.74)

( ) ( )( )sin sinS t a t= (1.75)

So X and Y can be calculated: (see Appendix for a detailed derivation)


43

( ) ( )

( )

( ) ( )

0

2

0 0 0

1 1 0 1

2 sin 2 cos 2 cos2

cos si

2

2 cos 2 sin 2 sin2 2 2

n2

T

r s s

r s s

X I t C t dt

TA A J a J a J a

TA A a J a J a J a J a

=

+ −

+ −

=

+

(1.76)

( ) ( )

( ) ( )

0

2 2

0 0 1

1 1

2 sin 2 cos sin cos2 2

2 cos 2 sin cos2 2 2 2

cos sin

T

r s s r s

r s s

Y I t S t dt

TA A J a J a T A A a J a

TA A a J a J a

=

= − +

+ −

+

(1.77)

If we define the following coefficients:①

( ) ( )2 2

0 1cosr sQ T A A a J a = + (1.78)

1 0 02 sin 2 cos2 2

r sQ TA A J a J a

= −

(1.79)

2 1 12 cos 22

cos si sin2 2

n2

r sQ TA A a J a J a

= −

(1.80)

( )2

0 0 03 2 sin 2 cos 22 2

r sQ TA A J a J a J a

+ −

=

(1.81)

( ) ( )14 1 0 12 cos 2s si in 22 2 2 2

cos nr sQ TA A a J a J a J a J a

+ − =

(1.82)

Then we obtain an equation set:

0 1 2sin coss sY Q Q Q = + + (1.83)

3 4cos sins sX Q Q = + (1.84)


① In practice, when the interference is not perfect (0<η<1), the value of η should be considered.


44

4 1 0 4

2 4 1 3

cos s

Q Y Q X Q Q

Q Q Q Q

− −=

− (1.85)

3 2 0 3

1 3 2 4

sin s

Q Y Q X Q Q

Q Q Q Q

− −=

− (1.86)

So we can obtain:


In practice, when the synchronization is guaranteed so that θ=0, and a is set to be 3.8317rad so that

J1(a)=0, the expressions can be simplified:

( ) ( ) ( )2

0 0 11 2 2 cos 2 sinr s s r s sX TA A J a J a TA A aJ a + +−= (1.88)

( ) ( )0 11 2 sin 2 cosr s s r s sY TA A J a TA A aJ a = − + (1.89)

Furthermore, when μ is relatively small, approximately we have:

( ) ( )2

0 01 2 2 cosr s sX TA A J a J a + − = (1.90)

( )01 2 sinr s sY TA A J a = − (1.91)

Obviously, they become the same as Eq. (0.57)-(0.58), and s can be obtained with the f-G-LIA

algorithm described in Section 1.5.4.

2.3.4 Modified integrating bucket algorithm

In this section, the intensity modulation problem is introduced into the integrating bucket algorithm, and

mathematical solutions are deduced.


45

According to the integral properties of trigonometric functions, it can be easily calculated that:

( )f t

1 0 ( )( )cos 2 1n t−

( )( )sin 2 1n t−

( )cos 2n t

( )sin 2n t

( )2

0f t dt

2

0 ( )

( )

1

2 1

n

n

− −

−

( )1

2 1n −

0 ( )1 1

2

n

n

− −

( )2

f t dt

2

0 ( )

( )

1

2 1

n

n

−

−

( )1

2 1n −

0 ( )1 1

2

n

n

− −

( )3

2 f t dt

2

0 ( )

( )

1

2 1

n

n

−

−

( )1

2 1n

−

−

0 ( )1 1

2

n

n

− −

( )2

3

2

f t dt

2

0 ( )

( )

1

2 1

n

n

− −

−

( )1

2 1n

−

−

0 ( )1 1

2

n

n

− −

Table 2-1. Integrals of trigonometric functions over quarter periods.

We define U1, U2, U3, U4 as follows:

( )21

0U I t dt

= (1.92)

( )2

2

U I t dt

= (1.93)

( )3

23U I t dt

= (1.94)

( )2

34

2

U I t dt

= (1.95)

U1, U2, U3, U4 are the values which can be directly measured, as the exposure process of digital camera

can be seen as the integration of light intensity over time. We can deduce that:


46

( )

( )

2

2,4,...

11

2

1,3,...

0

s

s1 1

22

o

i

1 1in s

n

c

m

m

mr s

m

m

m mm

U A A

R m mm m

R R

+

=

++

=

− −

=

− − + +

−

(1.96)

( )

( )

2

0

2,4,...

12

2

1,3,...

1 1

22

1 1sin co

sin

s

m

m

m

mr s

m

m

mm

U

R

A A

R m

R

mm m

+

=

++

=

− −

=

− + +

−

(1.97)

( )

( )

2

2,4,...

1

1

0

3

2

,3,...

1 1

22

1 1s

s

in cos

inm

m

m

mr s

m

m

mm

U

R

A A

R m

R

mm m

+

=

++

=

− −

=

− − + +

−

(1.98)

( )

( )

2

0

2,4,...

14

2

1,3,...

1 1

22

1 1sin co

sin

s

m

m

m

mr s

m

m

mm

U A A

R m mm

R R

m

+

=

++

=

− −

=

− − − + +

−

(1.99)

where the functions Rm (m=0,1,2,…), which contain information of φs, are defined by Eq. (1.44). If we

define three new functions as follows:

( ) 2

1

2,4,...

sin1 1

m

m

m

L mRm

+

=

− −= (1.100)

( )

1

2

2

1,3,...

1sin

m

m

m

L R mm

++

=

− −= (1.101)

3

1,3,...

1cosm

m

L R mm

+

=

= (1.102)

Then we can simplify the expressions for U1, U2, U3, U4:


47

01 1 2 322

r sU A A LR L L

= + +

− (1.103)

02 1 2 322

r sU A A L L LR

= + − +

(1.104)

03 1 2 322

r sU A A LR L L

= − −

− (1.105)

04 1 2 322

r sU A A L L LR

= + + −

(1.106)

In this equation set, since U1, U2, U3 and U4 can be measured in the experiments, we can obtain the

values of R0, L1, L2 and L3 by solving this equation. For example:

1 2 3 4

242 r s

U U U UL

A A

− + + −− = (1.107)

1 2 3 4

142 r s

U U U UL

A A

− + − += (1.108)

Besides, according to the definitions of L1 and L2, we also have:①

( )( )

( )

( ) ( )( )( )

12 2

2

2

1,3,...

1

2

1 1

1,3,...

2 sin 14 8 sin sin

14 sin cos

m

r s

m s

mr s

m

m m s

m

A AL J a m

A A m

J a J a mm

+

+

=

++

− +

=

− + − − = +

− −

+

(1.109)

( )( )

( ) ( )( )( )

2

1

2,4,...

2

1 1

2,4,...

1 14 ssin

si

8 co

1 14 sn in

m

m s

m

m

m m s

m

L J a mm

J a J a mm

+

=

+

+ −

=

− −

=

− −

+ −

(1.110)

In order to simplify these two equations, we may define:

① In practice, when the interference is not perfect (0<η<1), the value of η should be considered.


48

( )2

0

22 sinr s

r s

QA A

A A

− += (1.111)

( )( )

1

2

1

1,3,...

18 sin

m

m

m

Q J a mm

++

=

−= (1.112)

( ) ( )( )( )

1

2

2 1 1

1,3,...

14 sin

m

m m

m

Q J a J a mm

++

− +

=

−= − (1.113)

( )( ) 2

3

2,4,...

si11

n8

m

m

m

Q J a mm

+

=

− −= (1.114)

( ) ( )( )( ) 2

4 1 1

2,4,...

1 1in4 s

m

m m

m

Q J a J a mm

+

+ −

=

− −= − (1.115)

21 4P L= − (1.116)

12 4P L= (1.117)

So Eq. (1.109)-(1.110) can be expressed as:

1 0 1 2sin coss sP Q Q Q = + + (1.118)

2 3 4cos sins sP Q Q = + (1.119)

These two equations can be seen as an equation set of sinφs and cosφs since the values of P1, P2, Q0,

Q1, Q2, Q3, and Q4 can all be obtained. This equation set can be easily solved:

4 1 1 2 0 4

2 4 1 3

cos s

Q P Q P Q Q

Q Q Q Q

− −=

− (1.120)

3 1 2 2 0 3

1 3 2 4

sin s

Q P Q P Q Q

Q Q Q Q

− −=

− (1.121)

Then the signal phase can be obtained:



49

If μ=0 which is the case described by the reference [81], then, Q0 = Q2 = Q4 = 0 so we have:

2

3

cos s

P

Q = (1.123)

1

1

sin s

P

Q = (1.124)

For the integrating bucket method, it is important to choose the right value of θ. Obviously, if 0 =

then L1 = L2 = 0, making it impossible to retrieve the phase information s . In this case, we may

redefine P1 and P2 to retrieve the values of R0 and L3 instead. However, R0 often contains DC noise,

which may drastically reduce the measurement precision of s . So it is preferred to choose a good

value of θ in order to make the integrating bucket method works. Besides, the values of a and µ may

also have an influence on the final results, since they are widely present in the coefficients. When μ=0

, in order to minimize the effect of Gaussian additive noise (to have a zero mean phase error for any s ,

and to get the minimum mean square phase error), we should set 2.45a rad= and 0.98rad =

[26,81]. Recently, the values 2.08a rad= and 4 2

n

= + (n is an integer) have also been used

in a simplified algorithm [87]. This way, the phase of signal may be calculated correctly without even

knowing which image is the first one in the sequence of images.

2.4 Conclusion

In this chapter, the initial phase problem in phase modulating interferometer is mentioned and

mathematically described. This problem is not a real issue when using linear/sawtooth modulation (see

Section 2.1.1); when using a sinusoidal phase modulation, the value of the initial phase may be

calculated from the interference signal (see Section 2.3.2). When this problem cannot be eliminated, its

influence on the phase detection is discussed in Section 2.1.

The intensity modulation problem is described. To solve this issue, the affected interference signal is

analyzed in the frequency domain (see Section 2.2.1), and a modified SPM algorithm is proposed based

on it (see Section 2.3.1). Besides, we also make modifications to the f-G-LIA algorithm (see Section

2.3.3) and the integrating bucket algorithm (see Section 2.3.4) to get rid of the influence of this problem.

By solving these two practical problems, we have now better tools to apply continuous phase modulation

techniques in simple and cost-effective measuring systems, as will be shown in Chapter 3, Chapter 4

and Chapter 5.

Chapter 3: Application of SPM in DH/DHI

51

Chapter 3 Application of SPM in Digital

Holography and Holographic Interferometry

In this chapter, we detail how the G-LIA algorithm can be used in a simple and lens-less interferometric

setup to perform low-cost yet efficient 2D phase measurement, and focus on the performance of G-LIA

algorithm in this system. We first describe the optical system which can also be regarded as a phase-

shifting digital holographic microscope without optical magnification. Digital holography (DH) and

Digital Holographic Interferometry (DHI) capability is demonstrated.

3.1 Experimental method and data processing

The setup we built is a simple and compact one, adapted from a traditional Michelson interferometer.

As shown in the following figure, we replaced one mirror of classical Michelson interferometer by a

scattering sample, the other mirror by a piezo-actuated mirror, and the point detector by a CMOS camera

(without objective lens). It can be seen as a co-axis① digital holography setup.

① Co-axis setups differs from the off-axis ones because the “reference light” reflected by the mirror and the “signal light”

scattered by the sample both hit the CMOS camera from nearly the same direction.


52

Figure 3-1 Co-axis digital holography using sinusoidal phase modulation.

We restricted our study to sinusoidal phase modulation, the most convenient one in practice. We note

that for cheap phase-shifting devices like piezo-actuated mirrors, a sinusoidal modulation function is the

most practical choice. In our case the piezo hysteresis doesn't need to be considered given the short

modulation range, although it could also be included in the modulation function used to extract the phase

information.

By applying sinusoidal phase modulation, (f)-G-LIA algorithm can be compared with SPM algorithm

in the case of digital holography and digital holographic interferometry. Since when 2.4048a rad= ,

f-G-LIA and G-LIA algorithms are the same, for simplicity, SPM and G-LIA algorithms at

2.4048a rad= are compared experimentally.

The sinusoidal phase modulation was achieved by controlling the piezo-actuated mirror with a

waveform generator. We used a red laser emitting at 640nm = (DL640-070-SO, diode laser made

by CrystaLaser, 70mW). For this wavelength, the condition of 2.4048a rad= corresponds to a mirror

oscillation amplitude of about 122nm . This amplitude is adjusted by using a strain gauge sensor

attached to the piezoelectric actuator to measure its deformation and to obtain the mirror displacement.

In fact, with this experimental method, the actual amplitude of phase modulation is estimated to be

2.40 0.05a rad .

Usually at least 5 frames should be taken for each period of modulation to ensure a reasonable sampling

rate. In this experiment, we took 10 frames for each period of modulation. Hence, the maximum

modulation frequency is fixed by the frame rate of the camera given in FPS (frames per second), e.g.

the maximum modulation frequency equals to 1 Hz when recording a video at 10 FPS. For this reason,

high-speed cameras are needed when the measurement has to be done in a short time. However,

depending on the needs, low-speed and cheap cameras can also be used.


53

The beam of the red laser (DL640-070-SO, diode laser made by CrystaLaser, 640nm, 70mW) with an

attenuated power of about 1.4mW was expended up to a diameter of about 1cm to achieve a relatively

homogeneous illumination of the CMOS sensor. Then each image taken by this lens-less CMOS camera

(8-bit, variable frame rate) is an interference pattern between the reference plane wave reflected by the

mirror and the signal light scattered by the sample. The effective numerical aperture depends on the size

of CMOS matrix and the distance from the CMOS to the sample. Usually this distance is several

centimeters, so that in our setup, an effective numerical aperture of NAeff≈0.04 (This value may differ

when the size of CMOS or the distance between sample and camera changes) is obtained. The method

we used to determine NAeff is shown by the figure below.

Figure 3-2 Determination of an effective numerical aperture relative to the sample centre. Ideally, the

sample should be small compared to the CMOS matrix to keep a homogeneous spatial resolution across

the sample. Beside pixel size, a high effective numerical aperture is required to achieve a high-resolution

imaging.

From the video recorded during the reference mirror oscillation, the phase and amplitude of the scattered

signal is obtained on each pixel of the camera using both the G-LIA algorithm and the SPM algorithm

respectively (see Section 1.5) performed in the MATLAB environment. We note that the amplitude of

the signal which can also be obtained without interferometry, is less important than the phase, in the

sense that the sample image is still perceptible from the phase information only, i.e. if we consider a

constant signal amplitude on each camera pixel.

Once the complex light field on the CMOS plane is obtained, the complex light field on the sample

plane can be reconstructed using the angular spectrum method [88,89]: the angular spectrum of plane

waves was determined by 2D Fourier transform, then the plane waves were numerically retro-

propagated from the detector to the sample plane where the best focus is obtained (see Appendix).


54

For a proper operation, the recorded signal I(t) should be synchronized with C(t) and S(t). Otherwise a

non-zero initial phase θ can appear in the signal:

( ) ( )( )2 2 2 cos sinr s r s sI t A A A A a t = + + + − (1.125)

Synchronization can be done by triggering the camera recording together with the start of sinusoidal

phase modulation. Alternately, the initial phase θ can be calculated pixel by pixel with the recorded

video using the harmonic component of ω in the signal I(t) (see Section 2.3.2). Experimentally, since

the mirror movement corresponds to a rigid body oscillation, the average value of calculated initial

phases (or the calculated initial phase using the average intensity signal over many pixels) can be

considered to be the real initial phase for each pixel to reduce the error. Once we have the value of the

initial phase, we can adjust C(t) and S(t) accordingly — which is more practical than adjusting I(t) —

and carry out the algorithms discussed in Section 1.5.

3.2 Digital holography (DH)

A USAF 1951 resolution test target was observed to test the spatial resolution of our system. A camera

with a pixel size of 5.2µm5.2µm was used. The frame rate was set to 10 FPS, and the modulation

frequency was set to 1 Hz. By using only 10 images (1 period of modulation), high-quality results can

be obtained. As shown in the Figure 3-3 hereafter, although the phase image is a little noisy, our system

which is operated in a standard environment can be used to carry out the lens-less imaging, and the

profile of the surface can be measured.

Another experiment is done to compare the performance of G-LIA and SPM algorithms. The results are

shown in Figure 3-4


55

Figure 3-3 Test with USAF 1951 resolution test target. The pixel size of camera is 5.2µm5.2µm . (a)

The positive 1951 USAF test target “R1DS1P” is from the company ThorLabs [90]. (b) Phase image

obtained by G-LIA algorithm. (c) Profiles of the sixth element of the second group, which is highlighted

by the green rectangles in (a)(b). (d) 3D profile image of the sixth element of the second group.


56

Figure 3-4 Holographic images obtained with G-LIA and SPM algorithms. Left column: light intensity

images of the resolution test target. Middle column: zoomed-in light intensity images showing the first

three elements of the fifth group of the resolution test target. Right column: phase images (without 2D

unwrapping) of the resolution test target.

According to Figure 3-4, visually G-LIA and SPM algorithms give almost the same results. The smallest

distinguishable pattern is the first element of the fifth group for both of them (as shown in the middle

column of Figure 3-4), which means the spatial resolution of this setup can reach 32 LP/mm (LP: line

pair), or 15.6μm, for both G-LIA and SPM algorithms. Although this resolution is not comparable to

the current high-resolution [91] or super-resolution [92,93] digital holography microscopes [94], which

can usually reach a spatial resolution around 1μm, it is reasonable (as detailed later in this section) for

the proposed lens-less setup, which is sufficient to test the performance of the phase retrieval algorithms.

In order to show this resolution clearly and to better compare G-LIA and SPM algorithms, 1D

normalized profiles of the first element of the fifth group are represented in the following figure. The

pattern representing resolution of 32 LP/mm is clearly distinguished, and the results given by G-LIA

and SPM algorithms are nearly identical, except that the peak-to-peak values for the G-LIA algorithm

is slightly bigger (higher contrast).


57

Figure 3-5 Normalized profiles of the light intensity images using SPM and G-LIA algorithms. Two

profiles (a) and (b) were taken on the first element of the fifth group of the resolution test target.

If all the wavevectors of the light scattered by the sample are collected, then the resolution is mostly

limited by the spatial sampling given by the distance between camera pixels (5.2µm in both directions),

since a lens-less system was used and there was no optical magnification. It can be easily calculated that

32 LP/mm means 6 pixels/LP, which is close to the limit of 2 pixels/LP given by the Nyquist–Shannon

sampling theorem. It is hard to actually reach this theoretical limit for several reasons. First of all, the


58

wavevectors are not totally collected by the system with the effective numerical aperture NAeff≈0.04

(corresponding to about 10µm resolution according to the Rayleigh criterion). Also, the digital data

acquisition itself is affected by experimental noises, as well as slight measurement error on the value of

a. Besides, theoretically, the angular spectrum method that we used for light field reconstruction is based

on Fourier transform and is affected by the finite size of the sensor. Therefore, the obtained 6 pixels/LP

resolution can be reasonably accounted for.

Another thing that we can observe from Figure 3-5 is that the background level of the phase image is

not perfectly flat. However, the observed phase variations are within one wavelength which is

reasonable, since the sample is not perfectly flat. We note that the substrate positioning also induces

residual tilts with respect to the direction of incident beam which is not a perfect plane wave either.

We also made a test on the influence of spatial phase noise on the final spatial resolution. The results

are shown in the figure below.

Figure 3-6 Zoomed-in light intensity images showing the first three elements of the fifth group of the

resolution test target with 2D white Gaussian noises added to the obtained phase image. (a) Without

noise. (b) Power of noise: -20dBW. (c) Power of noise: -9dBW. (d) Power of noise: -5dBW.

From Figure 3-7 we can see that as the power of noise goes bigger, the pattern becomes more blurred.

When the power of noise reaches -5dBW, the first element of the fifth group becomes indistinguishable.

Yet we noticed that with a power of noise smaller than -9dBW, the first element of the fifth group can

be distinguished, which means the spatial resolution will not be affected when the phase noise is small.

We also considered the case when several pixels cannot provide phase information, e.g. when the pixels

are broken. For these pixels, we set the phase as zero. A test is done with different numbers of such

pixels. The results are shown in the following figure.


59

Figure 3-8 Zoomed-in light intensity images showing the first three elements of the fifth group of the

resolution test target with the phase of several pixels in the middle of this zone set to zero. The size of

the selected zone is 100×100 pixels. (a) 1×1 pixel. (b) 10×10 pixels. (c) 30×30 pixels. (d) 100×100

pixels.

As shown in Figure 3-9, if the defection is small (from 1 pixel to 10×10 pixels), then it will not have an

obvious influence on the spatial resolution; when it reaches 30×30 pixels, a ring ripple becomes obvious,

which blurred the pattern to some extent. Finally, when all the phase information in this zone is lost, as

shown in Figure 3-10 (d), the first element of the fifth group can hardly be distinguished; yet some parts

of the pattern can be observed, since the phase information at the neighboring zone is good. In fact, it is

an intrinsic property of holography: we do not need the whole hologram to reconstruct the image of

object. [31]

3.3 Digital holographic interferometry (DHI)

For digital holographic interferometry (DHI), the same methods were used to obtain the light field on

the CMOS plane and then the same numerical reconstruction was done to get the light field on the

sample surface. According to the theory of holographic interferometry [31], by subtracting the sample

light field after a small deformation/displacement ( ( )22 2 exp sE A i= ) from the one before (

( )11 1 exp sE A i= ), fringes can be observed in the image of 2 1E E− , or in the image of

( )2 1

cos s s − for a better contrast.①

① Through angular spectrum method (see Appendix), the complex field difference can be calculated at different positions,

and the corresponding fringe images can be obtained.


60

Considering the sensitivity vector① [31] of our system which is almost perpendicular to the sample

surface, these fringes mainly account for the out-of-plane displacement field. In the same way, when the

sample is purely rotated out of plane, the reconstructed plane with the best fringe visibility coincide with

the sample surface [31]. However, in practice, a minor in-plane displacement may slightly shift these

two planes away from each other.

In these experiments, the measuring system remains the same (see Figure 3-1), and videos are recorded

at different deformation/displacement states. A camera with a pixel size of 3.63µm3.63µm was used.

Its frame rate was set to 120 FPS, and the phase modulation frequency was set to 10 Hz. By using only

12 images (1 period) for each measurement, good quality results are obtained.

Figure 3-11 The one euro cent coin sample. (a) Photo of the coin. (b) Amplitude of the scattered light field at the

plane of sample surface calculated by G-LIA algorithm. (c) Phase of the scattered light field at the plane of sample

surface calculated by G-LIA algorithm.

A coin of one euro cent was used as scattering sample (see Figure 3-11). Unlike Figure 3-4, the phase

image does not exhibit distinctive patterns. It shows that the coin has a complex and optically rough

surface, inducing speckle-like scattering. A small zone of interest was selected by covering the outer

part of the zone with a black tape. The coin was fixed on a rotation stage allowing for a nearly pure out-

of-plane rotation around the vertical axis (minimum scale: 0.04°). The reconstructed planes giving the

best fringe visibility were nearly coinciding with the sample surface. (In Figure 3-11, the images were

obtained on a plane which is 70mm away from the plane of the CMOS, while in Figure 3-12 and Figure

3-13 this distance was 68mm.)

① For a certain point on the sample, the sensitivity vector is the difference between the unit vectors representing the

directions of illumination and viewing of that point.


61

Figure 3-12 and Figure 3-13 show the results obtained for rotations of about 0.02° and 0.04°, obtained

by G-LIA and SPM algorithms by using the two complex field images E1 and E2.

Figure 3-12 Image of 2 1E E− obtained by using G-LIA and SPM algorithms for out-of-plane

rotations of about 0.02° and 0.04°.


62

Figure 3-13 Image of ( )2 1

cos s s − obtained by using G-LIA and SPM algorithms for out-of-plane

rotations of about 0.02° and 0.04°.

We can see from Figure 3-12 and Figure 3-13 that the visibility of the fringes is very good. The covered

zone is much noisier than the zone of interest because the black tape could barely scatter light. The

fringe visibility is better in Figure 3-13 where only the phase information is used, but it looks naturally

noisier than Figure 3-12 near the edges where the signal is low (the phase always gives a value in its

definition interval of 2 while the amplitudes goes to zero when the signal is weak). Given the

relatively high frequency nature of this noise, these fast fluctuations can be suppressed by a low-pass

spatial filter without affecting the fringes describing the rotation (see Figure 3-15).


63

As shown in Figure 3-12 and Figure 3-13, the fringes orientation is almost vertical, i.e. parallel to the

rotation axis. The number of fringes k we obtain can be compared with the theoretical expectation for

0.02°:

2 3.63 / 1280 sin 0.02

5.07640

m pixel pixelk

nm

= (1.126)

This k value is in agreement with the Figure 3-13 showing about 5 fringes. Likewise, for a rotation of

about 0.04°, the theoretical fringe number is 10.14, while we can observe around 10.5 fringes in Figure

3-13. A sensitivity better than 0.004° (corresponding to one single fringe in the whole image) can hence

reasonably be obtained, for example 0.0004°, if a variation of one tenth of a fringe is detected with

proper algorithms of fringe analysis.

From Figure 3-12 and Figure 3-13, no difference between the results given by G-LIA and SPM

algorithms can be seen visually. In order to make a clearer comparison, profiles of the fringes presented

in the left column of Figure 3-13 are shown in Figure 3-14; then these images of fringes went through

a 2D low-pass filter so that the noise can be suppressed, and the profiles at the same position are shown

in Figure 3-15.


64

Figure 3-14 Profiles of the fringes in the unfiltered image of ( )2 1

cos s s − obtained by G-LIA and

SPM algorithms respectively for an out-of-plane rotation of about 0.02°.

Figure 3-15 Profiles of the fringes in the filtered image of ( )2 1

cos s s − obtained by G-LIA and SPM

algorithms respectively for an out-of-plane rotation of about 0.02°.


65

Further analysis of Figure 3-14 shows that although the exact values obtained by G-LIA and SPM

algorithms differ, their noise levels are nearly the same as expected from the simulations performed

before. As shown in Figure 3-15, the high frequency noise can be efficiently suppressed using a 2D

low-pass filter, and the same smooth profile was obtained by both G-LIA and SPM algorithms.

However, when a simple low-pass filter is directly applied to the image of ( )2 1

cos s s − , error will

occur on the results of φS2 - φS1. As shown in Figure 3-15, the value of ( )2 1

cos s s − never reaches

1, which is not reasonable and will affect the precision of fringe count. To solve this problem, we may

use the following phase filtering method:

1. Calculate the images ( )2 1

cos s s − and ( )2 1

sin s s − from the image of 2 1s s − ;

2. The images of ( )2 1

cos s s − and ( )2 1

sin s s − are filtered respectively;

3. The new image of phase can be obtained: ( ) ( )2 1 2 1 2 1

coarg s sins s s s s si =− − −

+ ;

4. If necessary, repeat step 1 to 3 several times (iterative method).

In Step 2, different ways of filtering are usable: we may keep using the simple low-pass filter, or we can

use the conventional 2D convolution method [95,96]. The results are shown in Figure 3-16 and Figure

3-17 respectively.


66

Figure 3-16 Profiles of the fringes in the image of filtered ( )2 1


algorithms respectively for an out-of-plane rotation of about 0.02°. Phase filtering method with low-

pass filtering was used, and the number of iterations is 1.

Figure 3-17 Profiles of the fringes in the image of filtered ( )2 1


algorithms respectively for an out-of-plane rotation of about 0.02°. Phase filtering method with 2D

convolution filtering was used, and the number of iterations is 200. The kernel used is Gaussian blur

11×11 with a standard deviation of 0.8.

From Figure 3-16 and Figure 3-17, we can see the problem of Figure 3-15 can be solved by using this

phase filtering method, and the results given by G-LIA and SPM algorithms are still almost the same.


67

In Figure 3-16 and Figure 3-17, the fringe qualities are nearly the same, however the fringe counts are

slightly different, which is caused by the different filtering methods in Step 2 as well as the different

numbers of iterations. It should be noticed that for the simple fringe image discussed here, low-pass

Fourier filtering (number of iterations: 1) is much more effective than the convolution filtering (number

of iterations: 200). However, when the fringe image is complicated and contains very fine fringes (see

Chapter 5), the convolution filtering is more suitable.

3.4 Conclusion

From the discussions above, we can see that by using sinusoidal phase modulation with G-LIA algorithm

or SPM algorithm, our lens-less digital holography imaging system can reach a spatial resolution of

32 LP/mm which is reasonable given the effective numerical aperture and the pixel size of our system.

This method has the potential to be applied in the well-established digital holographic microscopes

[97,98] with high NA objective lenses, in order to reach a much higher spatial resolution. Besides,

holographic interferometric fringes of out-of-plane rotation could be observed clearly, with highly

visible and predictable fringes given by rotation angles as small as 0.004°. Since DHI can be applied not

only in the displacement measurement but also in a lot of circumstances [99,100,101] where the change

of wavefront of signal needs to be measured, the discussed phase retrieval algorithms may also have a

wide range of applications.

It was proved that for each measurement, data from only one period of sinusoidal modulation was

enough to perform a correct analysis, with sampling rate of about 10 images/period. Compared to the

SPM algorithm, G-LIA algorithm showed similar capacity of retrieving phase information in digital

holography (or holographic interferometry) while having the potential of using a variety of other

modulation functions. This advantage may be important for any situation where another type of

modulation function becomes the most practical choice in the future. Besides, it can be expected that

when the frequency spectrum of noise is known, a certain type of modulation function may be chosen

accordingly in order to improve the anti-noise ability.

As for displacement field measurement, a limitation of the proposed configuration for DHI is that the

sensitivity vector is essentially along the vertical direction, making it mainly sensitive to out-of-plane

displacement. Yet in the next chapter, we will mainly focus on the in-plane displacement field

measurement using continuous phase modulation techniques.

Chapter 4: 2D-ESPI with double phase modulations

69

Chapter 4 2D-ESPI with double phase modulations

Electronic/Digital speckle pattern interferometry (ESPI/DSPI) is a well-established non-contact

detection method. It has been widely used to carry out precise displacement field measurements.

However, the simultaneous 2D or 3D displacement field measurements using ESPI with phase shifting

usually involve complicated and slow equipment. To solve these issues, we proposed a modified ESPI

system based on a single laser and the use of an original modulation technique: by applying two phase

modulations at different frequencies in two different illumination arms, the whole 2D displacement field

is extracted from short video sequences recorded at each deformation state. In-plane normal and shear

strains are then obtained with good quality. This system can also be further developed to measure 3D

deformation, and it has the potential to carry out faster measurements with a high-speed camera.

4.1 Introduction

Before the introduction of digital cameras and computers, the speckle phenomenon was already used to

measure displacement fields; two major types of speckle technique were known as “speckle pattern

photography” and “speckle pattern correlation interferometry” [31]. The first technique, speckle pattern

photography, which is based on the analysis of the speckle displacement, requires a single coherent

beam, and it can be used to measure relatively large in-plane deformation. The second approach, speckle

pattern photography, which is interferometric, requires at least two coherent laser beams. Thanks to the

phase information, it can measure minute in-plane deformations or out-of-plane deformations depending

on the impinging light direction.

As mentioned in the general introduction, with the introduction of digital instruments, the two

approaches have evolved to Digital Speckle Photography (DSP) and Electronic/Digital Speckle Pattern

Interferometry (ESPI/DSPI) respectively.

In-plane deformations can be easily measured by a simple ESPI measuring system [31]. In such system,

the temporal phase-shifting technique is often applied for phase retrieval to improve the performance

[35]. However, in a standard, two-beam, configuration, only one displacement component is measured.

This direction is fixed by the orientation of the laser beams.


70

Figure 4-1 Typical phase-shifting ESPI setup for 1D in-plane displacement measurement.

In order to measure the 2D in-plane displacement field (or the whole 3D displacement field), several

solutions have been proposed. The most direct one is to use DSP for the 2D in-plane measurement (or

to combine DSP with out-of-plane ESPI for the 3D measurement) [102]. Nevertheless, the sensitivity of

DSP is not as good as ESPI. A natural solution is then to combine ESPI measuring systems [103,53].

More recently, in-plane ESPI measurement systems were notably combined with out-of-plane ESPI to

perform 3D analysis using optical switches, with a limited time resolution though due to the iterative

process requirement [104,54,55]. To solve the time issue, spatial phase-shifting technique can be

applied: for example, using three lasers producing three spatial frequency carriers [104]. The latter

approach however inevitably makes the whole system substantially more complex and expensive.


71

Figure 4-2 Optical arrangements for in-plane sensitive ESPI. It can be seen as the superposition of two

1D ESPI systems with different polarization directions. Adapted from reference [103].

Figure 4-3 Optical setup of out-of-plane and in-plane combined approach. BS: Beam splitter; M:

Mirror; PZT+M: Piezo-actuated mirror; S: (optical) Switcher. Adapted from reference [53].

We propose a new technique of doing simultaneous 2D measurement using the widely recognized ESPI

technique and a single laser without combining two whole systems or using optical switches. Moreover,


72

this technique also has the potential to handle 3D measurement at relatively high speed with minor

modifications. Instead of using the traditional phase-shifting technique, we simultaneously apply two

continuous phase modulations and a short video is taken (e.g. 1 second or much less depending on

camera speed) at each deformation state. During data processing, by selecting the right frequencies, the

displacements along two different directions can be extracted separately.

4.2 Experimental Method and Data Processing

4.2.1 Optical arrangement

The proposed optical arrangement is shown in the figure below. There are three coherent laser beams

originating from a single laser: Beam-1, Beam-2 and Beam-3. The phases of Beam-1 and Beam-2 can

be modulated by the corresponding piezo-actuated mirrors.

Figure 4-4 Setup for ESPI measurement. (a) Top view; (b) 3D view. The camera is above the sample to

take photos of its surface. The height and focus of camera can be adjusted to get different magnifications.

The optics plane is a little above the sample plane so that the surface can be illuminated by laser beams.


73

Laser: CNI MSL-532 (diode-pumped solid-state laser, 532nm, 20mW). Camera: Flea®3 FL3-U3-

13S2M-CS 1/3" Monochrome USB 3.0 Camera. CL: Concave lens. CM: Concave mirror. BS: Beam

splitter. PZT+M: Piezo-actuated mirror.

4.2.2 Principle of measurement

If Beam-1 is blocked, then it becomes a typical phase-shifting ESPI system that measures deformation

along X-axis. Likewise, if Beam-2 is blocked, then it is sensitive to the deformation along Y-axis. When

none of them is blocked and two temporal phase modulation functions, F1(t) and F2(t), are applied to

Beam-1 and Beam-2 respectively, the scalar light field of the subjective speckles E(x,y) can be expressed

as:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1 1 2 2 32 , 2 , 2 ,

1 2 3, , , ,c c ci f t x y F t i f t x y F t i f t x yE x y A x y e A x y e A x y e

+ + + + + = + + (2.1)

Am(x,y) and θm(x,y) are the amplitude and the initial phase of Beam-m (m=1,2,3) at point (x,y)

respectively, and fc is the optical frequency of laser.

On the sample surface, the light intensity I(x,y) can be expressed as:

( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

2

2 2 2

1 2 3 1 2 1 1 2 2

1 3 1 1 3 2 3 2 2 3

, ,

, , , 2 , , cos , + , -

+2 , , cos , + , +2 , , cos , + ,

I x y E x y

A x y A x y A x y A x y A x y x y F t x y F t

A x y A x y x y F t x y A x y A x y x y F t x y

=

= + + + −

− −

(2.2)

After a small displacement u(x,y), the light intensity turns into:

( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

2

2 2 2

1 2 3 1 2 1 1 2 2

31 3 1 1 2 3 32 2

, ,

, , , 2 , , cos , + , -

+2 , , cos , + , +2 , , cos , + ,

I x y E x y

A x y A x y A x y A x y A x y x y F t x y F t

A x y A x y x y F t x y A x y A x y x y F t x y

=

= + + + −

− −

(2.3)

With

( ) ( ) ( ) ( )1 1 1

2, , ,x y x y x y

= + − sn n u (2.4)

( ) ( ) ( ) ( )2 2 2

2, , ,sx y x y x y

= + −n n u (2.5)

( ) ( ) ( ) ( )3 3 3

2, , ,sx y x y x y

= + −n n u (2.6)


74

where λ is the wavelength of laser, nm is the unit vector along illumination direction of Beam-m

(m=1,2,3), ns is the unit vector along viewing direction. nm and ns can be roughly considered to be the

same for every point (x,y) on the sample surface.

If we choose the following linear (or sawtooth) modulation functions:

( )1 12F t f t= (2.7)

( )2 22F t f t= (2.8)

then we have:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

2 2 2

1 2 3 1 2 1 1 2 2

1 3 1 1 3 2 3 2 2 3

, , , , 2 , , cos , +2 , -2

+2 , , cos , +2 , +2 , , cos , +2 ,

I x y A x y A x y A x y A x y A x y x y f t x y f t

A x y A x y x y f t x y A x y A x y x y f t x y

= + + + −

− −

.

(2.9)

Obviously, when f1, f2 and |f1-f2| are not equal to each other, with a lock-in detection at f1, θ1(x,y)-θ3(x,y)

can be extracted; with a lock-in detection at f2, θ2(x,y)-θ3(x,y) can be extracted [28]. The same procedure

can be carried out to obtain θ1’(x,y)-θ3’(x,y) and θ2’(x,y)-θ3’(x,y). If we set:

( ) ( ) ( ) ( ) ( )1 1 3 1 3, , , , ,C x y x y x y x y x y = − − − (2.10)

( ) ( ) ( ) ( ) ( )2 2 3 2 3, , , , ,C x y x y x y x y x y = − − − (2.11)

then according to Eq. (2.4)-(2.6), we have:

( ) ( ) ( )1

2, ,C x y x y

= −1 3n n u (2.12)

( ) ( ) ( )2

2, ,C x y x y

= −2 3n n u (2.13)

The z components of n1, n2 and n3 are almost equal, so they will cancel each other out in Eq. (2.12)-

(2.13). Concerning the x and y components of n1, n2 and n3, we can see from Fig. 1 that n1-n3 is parallel

to the Y-axis, and n2-n3 is parallel to the X-axis. So C1(x,y) and C2(x,y) can be expressed as:

( ) ( )1 , ,y yC x g uy x= (2.14)

( ) ( )2 , ,xC x y g u x y= (2.15)


75

where g is a measurable constant, ux(x,y) and uy(x,y) are the x and y components of u(x,y) respectively.

This means the 2D in-plane displacement field can be measured. The whole procedure is shown by the

flowchart hereafter.

Figure 4-5 Flowchart of the 2D displacement measurement.

It should be noticed that when piezoelectric actuators are driven to make sawtooth displacements, the

precision cannot be guaranteed, especially at high frequency, where the fly-back time of the mirror

cannot be neglected. The nonlinearity and noise generated by the sudden return becomes unacceptable

when high speed measurement is required. This issue can be addressed with sinusoidal phase

modulations such as:

( )1 1sin 2F t a f t= (2.16)

( )2 2sin 2F t a f t= (2.17)

where a is the amplitude of phase modulation. It should be noticed that f1 and f2 are not randomly chosen.

It is favorable to choose coprime integers①, as will be detailed later. Now we have:

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

2 2 2

1 2 3

1 2 1 2 1 2

1 3 1 3 1

2 3 2 3 2

, , , ,

2 , , cos , , sin 2 sin 2

+2 , , cos , , sin 2

+2 , , cos , , sin 2

I x y A x y A x y A x y

A x y A x y x y x y a f t a f t

A x y A x y x y x y a f t


= + +

+ − + −

− +

− +

(2.18)

① Which means, as a reminder, that the only positive integer that divides both of them is 1.


76

As we can see, the situation now seems more complex. In Eq. (2.18), according to the Jacobi–Anger

expansion, in the frequency domain, the third term (representing the interference between Beam-1 and

Beam-3) will be distributed at the integer multiples of f1, and the fourth term (representing the

interference between Beam-2 and Beam-3) will be distributed at the integer multiples of f2. If we set

a=2.4048rad so that J0(a)=0 (J0 is the 0th Bessel function of the first kind), then both of them will not

contain any signal at 0Hz. So in the frequency domain, they will not overlap with each other until the

least common multiple of f1 and f2, which is 63Hz in our case (f1=9Hz, f2=7Hz, 9 and 7 are coprime

integers). This means these two terms can be efficiently separated.

The second term in Eq. (2.18) represents the interference between Beam-1 and Beam-2. It may be

eliminated by playing with the polarization (e.g. linear polarization of 0° for Beam-1, 90° for Beam-2,

and 45° for Beam-3) or the temporal coherence (e.g. optical path length: Beam-1<Beam-2<Beam-3);

but these solutions would surely increase the difficulty of adjusting the measuring system. In our

method, we keep this term as it is and make use of the trigonometric formulas (sum/difference identities)

to further analyze it:

( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2 1

1 2

cos , , sin sin

cos , , cos sin cos sin

cos , , sin sin sin sin

sin , , sin sin cos sin

sin , , cos sin si

2 2

2 2

2 2

2 2

2

x y x y a a

x y x

x

f t

y a a

x y x y a a

f t

f t f t

f t f t

f t fx y x y ta

f t

a

x y y a

− + −

= −

− −

− −

+ − ( )22n sin f ta

(2.19)

If we analyze the first term in Eq. (2.19) with the Jacobi–Anger expansion, then we have:

( ) ( )

( ) ( ) ( )

( ) ( ) ( )

1 2

2

2 2 1 2 0

1 1

0 2 2 2 2

1 1

cos sin 2 cos sin 2

4 cos(2 2 )cos(2 2 )

2 cos(2 2 ) cos(2 2 )

p q

p q

q q

q q

a f t a f t

J a J a p f t q f t J a

J a J a q f t J a q f t

= =

= =

= +

+ +

(2.20)

when J0(a)=0, Eq. (2.20) becomes:

( ) ( )

( ) ( )

1 2

2 2 1 2

1 1

cos sin 2 cos sin 2

4 cos(2 2 )cos(2 2 )p q

p q

a f t a f t

J a J a p f t q f t

= =

= (2.21)

with


77

1 2

1 2 1 2

cos(2 2 )cos(2 2 )

1cos(2 2 2 2 ) cos(2 2 2 2 )

2

p f t q f t

p f t q f t p f t q f t

= + + − (2.22)

where p and q are positive integers. Through Eq. (2.22), we know that the term described by Eq. (2.21)

contains signals at the frequencies of |2pf1±2qf2| (p,q>0). However, if J0(a)≠0, then the term described

by Eq. (2.20) contains signals at 0Hz, 2pf1 and 2qf2 (the useful frequencies for SPM and G-LIA

algorithms) as well. In order to eliminate these disturbance terms with simplicity and efficiency, we

always set a=2.4048rad thus J0(a)=0 in this experiment, since the value of a can usually be easily

controlled.

Likewise, we can analyze the other three terms in Eq. (2.19) when J0(a)=0:

( ) ( )

( ) ( ) ( )( ) ( )( )

1 2

2 1 2 1 1 2

1 1

sin sin 2 sin sin 2

4 sin 2 1 2 sin 2 1 2p q

p q

a f t a f t

J a J a p f t q f t

− −

= =

= − − (2.23)

( ) ( )

( ) ( ) ( )( )

1 2

2 1 2 1 2

1 1

sin sin 2 cos sin 2

4 sin 2 1 2 cos(2 2 )p q

p q

a f t a f t

J a J a p f t q f t

−

= =

= − (2.24)

( ) ( )

( ) ( ) ( )( )

1 2

2 2 1 1 2

1 1

cos sin 2 sin sin 2

4 cos(2 2 )sin 2 1 2p q

p q

a f t a f t

J a J a p f t q f t

−

= =

= − (2.25)

with

( )( ) ( )( )( ) ( )( )

( ) ( )( )1 2

1 2

1 2

cos 2 1 2 2 1 21sin 2 1 2 sin 2 1 2

2 cos 2 1 2 2 1 2

p f t q f tp f t q f t

p f t q f t

− + − − − = −− − − −

(2.26)

( )( )( )( )

( )( )1 2

1 2

1 2

sin 2 1 2 2 21sin 2 1 2 cos(2 2 )

2 sin 2 1 2 2 2


p f t q f t

− + − =+ − −

(2.27)

( )( )( )( )

( )( )1 2

1 2

1 2

sin 2 2 2 1 21cos(2 2 )sin 2 1 2

2 sin 2 2 2 1 2


p f t q f t

+ − − =− − −

(2.28)


78

It is shown that these terms contain signals at the frequencies of |(2p-1)f1±(2q-1)f2|, |(2p-1)f1±2qf2| and

|2pf1±(2q-1)f2| respectively. Thus it can be concluded that the second term in Eq. (2.18) contains |pf1±qf2|

signals.

Since p and q are positive integers, the solutions for pf1±qf2=nf1 or pf1±qf2=nf2 (n is a positive integer)

can only be found when p or q is relatively big (at least one of them is bigger than 6), where we have

Jp(a) or Jq(a) approximately equals to zero (see Figure 4-6). So we can estimate that it will not have too

much influence on the interesting frequencies (pf1 and qf2). A simple simulation is done with f1=9Hz and

f2=7Hz, and the result is shown in Figure 4-7 to illustrate this point.


79

Figure 4-6 Chart of Jm(2.4048) for m=1,2,…,10.

Figure 4-7 The term cos(θ1-θ2+asin2πf1t-asin2πf2t) represented in the frequency domain with

t=0s,1/63s,2/63s,…,62/63s. a is set to be 2.4048rad, f1=9Hz, and f2=7Hz. Here, we have arbitrarily set

θ1=0.2rad and θ2=0.9rad.

We can now draw a direct link between the four terms in Eq. (2.18) and the frequency spectrum. The

first term corresponds to the signal at 0Hz, the third term corresponds to signals at pf1, the fourth term

corresponds to signals at qf2, and the second term corresponds to signals at other frequencies. So just

like the case when linear modulations are applied, information can be easily sorted out so that the 2D

displacement field can be measured.

In fact, by simply replacing the lock-in detection algorithm with traditional sinusoidal phase modulation

algorithm or the recently proposed generalized lock-in detection algorithm (see Chapter 2), the needed


80

phase information can be obtained. Likewise, while dealing with the same set of data, if we set the

demodulation frequency at f1 in the algorithm, then we can get C1(x,y); if we set it at f2, then we can get

C2(x,y). With C1(x,y) and C2(x,y), the 2D displacement field ux(x,y) and uy(x,y) can be obtained.

4.2.3 Set appropriate voltages

In our measuring system, the voltages are applied by a waveform generator (RIGOL DG1032Z). It has

two outputs, so we can easily modulate the two piezo-mounted mirrors at different frequencies.

In order to drive the piezo-actuated mirrors correctly so that the phase modulation functions can be

realized, it is important to know the properties of the piezoelectric crystal. Here we take the piezoelectric

crystal “PZS001” made by the company ThorLabs as an example. It has the following specifications:

Figure 4-8 Specifications of the piezoelectric crystal “PZS001” [105].

If we suppose that the relation between voltage and piezo displacement is totally linear, then we have

U kS= (2.29)

Where U is the voltage, S is the displacement caused by the deformation of piezoelectric crystal, and k

is a coefficient. According the chart of specs above, we have:

11.6

116 /100

mk nm V

V

= (2.30)


81

After measuring the incident angle of laser to the mirror (γ), we will know the optical path difference

(OPD) caused by the voltage U:

2 cosOPD S = (2.31)

If we want to have a modulation amplitude of a, then from

2

OPD a

= (2.32)

we can deduce that

4 cos

aS

= (2.33)

Now we know the relation between voltage and phase shift. To better protect the piezoelectric crystal

and to get a better performance, it is preferred to add an appropriate offset to keep the voltage always

between 0V and the recommended drive voltage limit (100V for this piezo). It should always be paid

attention that the maximum voltage should never exceed the maximum voltage (150V for this piezo).

However, apart from the existence of non-linearity, the response of the piezoelectric crystal, which has

a mechanical resonance frequency, is actually frequency-dependent in both amplitude and phase. This

effect is not perceived in our case though because the modulation frequency remains modest. Another

possibility to obtain the voltage is to build a simple interferometer (like Michelson interferometer or

Mach–Zehnder interferometer) with the piezo-mounted mirror. The relation between the voltage and the

phase change can be easily obtained by applying a slow linear/sawtooth phase modulation and measure

the period of intensity fluctuation. The influence of modulation frequency can also be corrected by

applying the actual phase modulation function and analyze the intensity signal. By using this method,

we can avoid the errors caused by the assumption that the voltage is totally proportional to piezo

displacement, and it is also possible to determine the real amplitude of phase modulation if high

modulation frequency is used.

However, it is more convenient to determine a directly on the ESPI setup. In that case, we apply a

linear/sawtooth modulation to one of the piezo-mounted mirrors, record a video, and determine the

voltage required to obtain one period of intensity fluctuation (phase change of 2π). It is the most direct

method, which can be very accurate. In practice, it may be difficult to find the period in the total intensity

variation because of the speckle phenomenon.

In our experiments, we used a single axis piezo positioner (Nano-OP30HS) made by the company Mad

City Lab for the modulation of f1=9Hz, and a piezo made by the company Thorlabs for the modulation

of f2=7Hz. The first one is calibrated individually before sale, so we know that for the piezo positioner


82

that we used, k=3.0915μm/V. The second one is calibrated by building a simple interferometer, as

mentioned earlier in this section.

4.3 Experimental details

Since the paths of the beams are relatively long, the intensity can be easily affected by noises, and several

precautions should be taken. In the following sub-sections, we provide some practical details which are

important, although a bit technical, to set up the ESPI experiments correctly.

4.3.1 Practical requirement on the Laser

First of all, the intensity of laser should be strong enough (usually >10mW) to generate clear speckle

images in a relatively large area. So the safety issue is important while building and adjusting the optical

setup (e.g. Intensity filters, protective goggles). Special attention should be paid when adjusting the

concave mirrors, since they can easily reflect the laser out of the optical plane.

Another important thing about the laser is the coherence between the beams. Just like other ESPI

systems, since the laser is expanded, a good spatial coherence is important. Besides, the temporal

coherence of laser should also be good enough because in our setup, the optical paths for the beams have

different lengths.

In order to make sure that all these three laser beams interfere well with each other, we may successively

block one of the three beams, press the table a little, and watch the live images captured by the camera

to see whether there is a fluctuation of intensity in the speckle image. Ideally the whole zone of interest

should have an obvious speckle fluctuation. When the optical arrangement is well adjusted, yet the

coherence is not good, it is most likely caused by the laser source. The coherent length of laser should

be longer than the longest optical path difference between these three beams, and the spatial coherence

of laser should also be very good.

4.3.2 Evaluation of exposure conditions

The exposure of camera should be adjusted just before the experiments. To evaluate whether the

exposure is good or not, we may use a pseudo-color image. The color look-up table (CLUT), which is

used to convert the 8-bit gray scale intensity value to the pseudo color, is shown in the figure below.


83

Figure 4-9 Color look-up table for the pseudo-color image conversion.

Experimentally, if the pseudo-color image contains mainly green/red/pink colors, and the speckle pattern

is fine and clear, then it means the adjustment of exposure is good, even though some pixels may usually

be saturated. An example is given below.

Figure 4-10 Example of pseudo-color images. (a) A bad one: the exposure is too weak. (b) A good one.

The adjustment of exposure can be done by directly adjusting the aperture of the objective lens, or by

the software FlyCapture (see figure below).


84

Figure 4-11 The “camera control dialog” of FlyCapture. The important elements are highlighted by

red rectangles. More practical details are provided in the footnote①.

With the laser power and setting we use, the ambient light does not seem to be a trouble as long as a

good pseudo-color image is observed. However, if the intensity of ambient light changes, the adjustment

should be redone to obtain a good pseudo-color image again. It should be noticed that if the ambient

light is strong and the laser is too weak so that the speckle pattern is not clear, then adjusting only the

exposure is not enough; we should use a stronger laser or reduce the ambient light.

4.3.3 Data acquisition / video recording

It is recommended to save the videos temporarily in a SSD disk whose speed is higher, which is good

for recording the videos fluently. When we set the frame rate at 63 frames per second (FPS), the real

time needed to capture 63 frames may not be exactly 1 second; but as long as there is no dropped frame,

the quality of video is usually good enough. On the contrary, if we see the sign showing that there is

① Setting on FlyCapture. First, we should set the desired “Frame Rate”. Then we can just put all the

“Auto” on. When the image becomes stable, we may close all the “Auto” and capture an image to be

evaluated by the pseudo-color image. If it is good, which is often the case, then we can begin the

measurements as soon as possible; if not, we can adjust the “Exposure”, “Shutter” and “Gain” manually

until we see a good pseudo-color image. If the software adjustment reaches its limit, we can also adjust

the aperture of the objective of camera.


85

dropped frame (this is usually caused by the bad contact of USB cable or the unstable performance of

computer) during the recording, it is preferable to redo the recording.

Figure 4-12 The “Capture video or image sequence” window of FlyCapture. The important elements

are highlighted by red rectangles.

When recording a video, it is necessary to choose the “uncompressed” format to avoid the loss of detail

information of speckle pattern. As shown in the following figure, the compression of images may cause

the loss of details.

Figure 4-13 Comparison of compressed and uncompressed pseudo-color speckle images. (a)

Compressed image. (b) Uncompressed image.

4.3.4 Initial phase problem

In our measuring system, when the outputs of the waveform generator are turned on manually, the phase

modulations begin; when the button “Start Recording” is clicked in the software FlyCapture, the video


86

recording begins. So usually we have a non-zero initial phase for each of the two phase modulations.

Since the values of these two phase modulations are coprime numbers, there is no fixed relation between

the two initial phases. So the two initial phases should be determined one by one independently.

The best solution is to control the phase modulations and the video recording with the same computing

unit. This can be done through a microcontroller such as Arduino or via an acquisition card (e.g. NI

card). Still, the camera triggering often has a delay; yet this delay can usually be regarded as a constant

when the modulation function is fixed. The synchronization issues can then be solved by introducing

this delay in C(t) and S(t).

Another choice is to calculate the values of initial phase (as described in Section 2.3.2). Theoretically,

the fact that we have two modulated beams and one non-modulated beam interfering with each other

does not affect the final results, since the signals at different frequencies are used to calculate the initial

phase of different phase modulation functions. However, the calculated initial phase image turned out

to be very noisy, and using the average value of them can hardly lead to the best fringe image. This may

be caused by the fact that in a typical digital speckle image, there are many saturated pixels, which will

largely reduce the measurement accuracy of initial phase.

In this experiment, in order to give the best result without means of synchronization, we chose the trial

and error method. The principle is simple: we try different pairs of initial phase values before and after

deformation, and choose the phase-difference image with the best fringe visibility. This process should

be carried out for both modulation functions to get the 2D deformation field.

It should be noticed that when using the linear/sawtooth phase modulations, according to the discussion

in Section 3.1.1, a wrong guess of phase modulation will result in a constant ( )0 0gt t − added to the

final detected phase. This constant is approximately the same for every point on the plane of X-Y. So

when we make different guesses of initial phase, the changes for every point in the phase-difference

image are the same, which only makes the fringes “move” in the image without any change of fringe

visibility.

Since the phase difference is proportional to the displacement (ux or uy), a residual constant will also be

added to ux and uy. However, since we only use the derivatives of ux and uy to calculate the normal strains

(εy, εx) and shear strain (γxy), this constant will not have an influence on the final results.

x

x

u

x

=

(2.34)

y

y

u

y

=

(2.35)


87

y x

xy

u u

x y

= +

(2.36)

It means that when linear/sawtooth phase modulations are used and we only care about the deformation,

then the unknown initial phase is not a problem.

When sinusoidal phase modulations are applied, and we still use the trial and error method, then

according to the discussions in Section 2.1, the sign (+/-) of the detected phases (ux and uy) may be

wrong, and there may be a phase shift of nπ ( 's s n = , n is an integer). Through the discussion

above, we already know that this phase shift of nπ will not affect the measurements of strains. However,

according to Eq. (2.34)-(2.36), the sign ambiguity of ux or uy may bring errors to the deformation

measurements. So it is necessary to check whether the final strain field is reasonable or not from the

perspective of mechanics. Otherwise, the modulators and the camera should be precisely controlled to

realize the synchronization.

4.4 Potential for 3D displacement field measurement

This method can also be extended to carry out 3D displacement field measurement without increasing

acquisition time, and without additional laser or camera. We can simply separate the laser into a fourth

coherent beam (Beam-4). This beam also hits the sample surface and interfere with Beam-1, Beam-2

and Beam-3. If the phase modulation functions for Beam-4 is F4(t), the amplitude and the initial phase

of Beam-4 are A4(x,y) and θ4(x,y) respectively, then E(x,y) becomes:

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

1 1 2 2

3 4 4

2 , 2 ,

1 2

2 , 2 ,

3 4

, , ,

, ,

c c

c c

i f t x y F t i f t x y F t

i f t x y i f t x y F t

E x y A x y e A x y e

A x y e A x y e

+ + + +

+ + +

= +

+ + (2.37)

After a small deformation, the phase of Beam-4 on the sample surface turns into θ4’(x,y):

( ) ( ) ( ) ( )44 4

2, , ,x y x y x y

= + − sn n u (2.38)

where n4 is the unit vector along illumination direction of Beam-4. We already know that:

( ) ( ) ( ) ( )33 3

2, , ,x y x y x y

= + − sn n u (2.39)

If we set

( ) ( ) ( ) ( ) ( )3 34 4 4, , , , ,C x y x y x y x y x y = − − − (2.40)


88

then we have:

( ) ( ) ( )4 4 3

2, ,C x y x y

= − n n u (2.41)

If we deliberately make sure that the z component of n4-n3 is non-zero, then we know that:

( ) ( ) ( ) ( )4 , ,, ,x x y y z zu x y u x y u x yC x y h h h= + + (2.42)

where hx, hy and hz are measurable constants. Since ux(x,y) and uy(x,y) can be obtained from C1(x,y) and

C2(x,y), hz≠0, we know that uz(x,y) can be solved. This way the 3D displacement field u(x,y) is obtained.

We may also combine digital holographic interferometry (DHI) with 2D ESPI to obtain the 3D

displacement field. In this case, the forth coherent beam directly hits the CCD/CMOS matrix and

interfere with the other three beams. This time we have:

( ) ( )3 3, ,x y x y = (2.43)

Thus

( ) ( ) ( )4 3

2, ,sC x y x y

= − n n u (2.44)

Likewise, when the z component of ns-n3 is non-zero (which is usually true), the 3D displacement field

u(x,y) can be obtained.

The only problem left is to get the correct values of C1(x,y), C2(x,y) and C4(x,y). Different types of phase

modulations may be used, and appropriate modulation frequencies should be chosen for each case.

4.4.1 Linear/sawtooth phase modulations

In the case where we use linear/sawtooth phase modulations, we set:

( )4 42F t f t= (2.45)

then we have


89

( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

2

3 3

1,2,3,4 1,2,4

1 2 1 1 2 2

4 2 4 4 2 2

1 4 1 1 4 4

,

, +2 , , cos , +2 ,

2 , , cos , +2 , -2

2 , , cos , +2 , -2

2 , , cos , +2 , -2

m m m m

m m

I x y

A x y A x y A x y x y f t x y

A x y A x y x y f t x y f t



= =

= −

+ −

+ −

+ −

(2.46)

Obviously, when f1, f2 and f4 are not equal to each other, and they are not equal to |f1-f2| or |f4-f2| or |f1-

f4|, with lock-in detections at f1, f2 and f4, we can obtain θ1(x,y)-θ3(x,y), θ2(x,y)-θ3(x,y) and θ4(x,y)-θ3(x,y)

[28]. Same procedure can be carried out to calculate θ1’(x,y)-θ3’(x,y), θ2’(x,y)-θ3’(x,y) and θ4’(x,y)-

θ3’(x,y). This way, we can get C1(x,y), C2(x,y) and C4(x,y), and thus the 3D displacement field.

The choice of is pretty simple. For example, f1=3Hz, f2=4Hz and f4=5Hz (|f1-f2|=1Hz, |f4-f2|=1Hz and |f1-

f4|=2Hz) is a good choice. Then we may set the camera frame rate at 60FPS and set the recording time

at 1 second to record every video.

4.4.2 Sinusoidal phase modulations

For the case of sinusoidal phase modulations, we set:

( )4 4sin 2F t a f t= (2.47)

Now we have:

( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

2

3 3

1,2,3,4 1,2,4

1 2 1 1 2 2

4 2 4 4 2 2

1 4 1 1

,

, +2 , , cos , + sin 2 ,

2 , , cos , + sin 2 , - sin 2

2 , , cos , + sin 2 , - sin 2

2 , , cos , + sin 2

m m m m

m m

I x y

A x y A x y A x y x y a f t x y

A x y A x y x y a f t x y a f t

A x y A x y x y a f t x y a f t

A x y A x y x y a f t

= =

= −

+ −

+ −

+ −

( )4 4, - sin 2x y a f t

(2.48)

Through similar analysis (see Section 4.2.2), we know that when J0(a)=0, the last three terms contain

|pf1±qf2|, |pf4±qf2| and |pf1±qf4| signals respectively, where p and q are positive integers.

For SPM algorithm, it is required that the last three terms do not contain f1, 2f1, f2, 2f2, f3 or 2f3 signals.

From Section 4.2.2 we know that this requirement means that: when |pf1±qf2|, |pf4±qf2| or |pf1±qf4| equals

to f1, 2f1, f2, 2f2, f3 or 2f3, the value of p or q should be relatively big (at least one of them is bigger than

6) so that Jp(a) or Jq(a) approximately equals to zero (see Figure 4-6). It is more difficult to find


90

appropriate values of f1, f2 and f3 than the 2D case, but still they exist. For example, this requirement can

be satisfied by setting f1=17Hz, f2=31Hz and f3=36Hz.

For G-LIA algorithm, it is required that the last three terms do not contain any signals at integer multiples

of f1, f2, or f3 signals. This means that when |pf1±qf2|, |pf4±qf2| or |pf1±qf4| equals to mf1, mf2 or mf3 (m is

a positive integer), the value of p or q should be relatively big (at least one of them is bigger than 6).

However, this requirement is more difficult to satisfy. For example, if we still set f1=17Hz, f2=31Hz and

f3=36Hz, then we can only guarantee that p or q is bigger than 2, not 6. Knowing that J3(2.4048)≈0.2,

this set of modulation frequencies will bring some errors. So if the results are not satisfactory, it is

recommended to look for more suitable modulation frequencies, to change the algorithm, or to change

the type of modulation functions.

4.5 Results

We used a bending specimen shown in the figure below. It is a test sample fabricated by the company

HOLO 3 [52].

Figure 4-14 The bending specimen (photo taken by a camera which is not used in the experiments). By

adjusting the micrometer screw, different deformation states can be obtained. The white rectangle

represents the zone of interest.

First, we make sure that there is already an initial contact between the micrometer screw and the bending

specimen (see Figure 4-14). Then we start the two phase modulations by turning on the two channels of

the waveform generator. A short video (1 second, 63 frames per second) is recorded by the software

FlyCapture (see Section 4.3.3). Likewise, we recorded another video after turning the micrometer screw

so that the deformation state changed. We may repeat this procedure several times to record one video

at each deformation state. (It is preferred to turn the screw along the same direction without any going


91

back during the measurements to avoid the hysteresis error.) By analyzing two videos at different

deformation states, we can measure the 2D displacement field.

When applying sinusoidal phase modulations described by Eq. (2.16)-(2.17) with f1=9Hz and f2=7Hz,

we successfully obtained phase images (C1 and C2), as shown in Figure 4-15. The fringe visibility is

very good; besides, very fine fringes can be observed on the left part of Figure 4-15(a,b).

Figure 4-15 Phase images (without filtering) showing the displacement field along Y-axis and X-axis

obtained with sinusoidal phase modulations. A phase difference of 2π represents a displacement

difference of about 385nm. The micrometer screw advanced 10μm and 50μm respectively along Y-axis.

The generalized lock-in detection [22,28,47] is used to process data.

From the obtained phase images (Figure 4-15), we can quantitatively measure the 2D deformation

(Figure 4-16). First, the original phase images (Figure 4-16(a,b)) were filtered (see Figure 4-16(c,d))

using the phase filtering method described in Section 3.3 (conventional 2D convolution filtering [95,96]

was used; number of iteration: 150; kernel: Gaussian blur 11×11 with a standard deviation of 0.8). Then

they are 2D-unwrapped using 2D Goldstein branch cut phase unwrapping algorithm [106,107] to get

smooth phase images, and the displacements uy and ux (Figure 4-16(e,f)) can be calculated by Eq. (2.14)


92

-(2.15). Through Eq. (2.34)-(2.36), the strains εy, εx and γxy can be quantitatively measured (Figure

4-16(g,h,i)) for any choice of origins for uy and ux.

During the 2D unwrapping process, an origin point is required for each wrapped phase image. When the

real origin (the point where the displacement is known to be zero) is unknown, we may randomly choose

a point in a smooth part of the wrapped phase image as the origin. This will introduce a constant phase

shift to every point in the image, but it will not affect the measurements of strains according to Eq.

(2.34)-(2.36).


93

Figure 4-16 From phase images to quantitative 2D strain field. (a,b) Unfiltered phase images (we took

the central parts of Figure 4-15(c,d) as an example). (c,d) Filtered phase images. (e,f) Displacements

uy and ux. (g,h) Normal strains εy and εx. (i) Shear strain γxy.

When applying linear/sawtooth phase modulations described by Eq. (2.7)-(2.8), similar fringes (see

Figure 4-17) are obtained, since the modulation frequencies are quite low (f1=9Hz and f2=7Hz).

However, the sawtooth approach will become much less efficient at higher speed. There are some small

differences in the fringe pattern, which are mainly due to phase noise, initial phase adjustment and the

fact that the loading processes were done manually and are not perfectly reproducible.


94

Figure 4-17 Phase images (without filtering) showing the displacement field along Y-axis and X-axis

obtained with linear/sawtooth phase modulations. A phase difference of 2π represents a displacement

difference of about 385nm. The micrometer screw advanced 50μm along Y-axis. The lock-in detection

[28] is used to process data.

4.6 Conclusion

In this chapter we have applied original double phase modulations in order to perform quasi-real-time

2D-ESPI. Compared to previous reports of 2D in-plane displacement field measurement, the proposed

approach is much simpler with only one laser and one camera, yet high-quality fringes have been

obtained.

Theoretically, we can choose any two different phase modulation functions (same type with different

parameters, or even different types) for the two piezo-mounted mirrors as long as the needed signals can

be clearly separated in the frequency domain. For different choices of phase modulation functions,

appropriate algorithms should be used to separate these signals and to obtain the phase images. For

example, LIA algorithm is suitable for linear/sawtooth phase modulations, SPM algorithm is suitable

for sinusoidal phase modulations, and G-LIA algorithm is suitable for a variety of phase modulation

functions.

In this chapter, we focused on the cases when linear/sawtooth or sinusoidal phase modulations are

applied. Generally speaking, at a relative low frequency (like f<50Hz), both linear/sawtooth or

sinusoidal phase modulations can be used (the linear one has an easier algorithm, while the sinusoidal

one is better for the protection of piezo). However, at a higher frequency, it is clear that a sinusoidal

phase modulation will guarantee a much better precision in the piezo positions.


95

While dealing with sinusoidal phase modulations, for the simplicity and efficiency of signal separation

in the frequency domain, it is required that J0(a)=0 (see Section 4.2.2), which coincides with the

requirement of G-LIA algorithm (see Section 1.5.3). According to the discussion in Section 1.6, we

know that when a=2.4048 rad so that J0(a)=0, G-LIA algorithm may have a better anti-noise ability.

This is an advantage for the G-LIA algorithm. However, since SPM algorithm needs less frequency

components to carry out the calculations, it is easier to find good frequencies of modulation for this

experiment, especially for the 3D measurements (see Section 4.4). So G-LIA and SPM algorithms both

have their own advantage, and the choice of algorithm should be done according to the experimental

conditions and requirements.

While a camera with moderate speed (63 frames per second) is used, the data acquisition time (1 second

for 2D information) is still more advantageous compared to some commercialized systems (e.g. 3.5

second for 3D information [55]). Obviously, this system has great potential to be operated at a high

speed while providing accurate results by using the sinusoidal phase modulation together with a high-

speed camera.

Although the relatively voluminous data may be a challenge for lower-end computers to carry out real-

time analysis at higher speed, this issue can be solved with enough computing resource or FPGA-based

(FPGA: Field-Programmable Gate Array) calculation. Last but not least, this approach has the potential

to carry out simultaneous ESPI measurement of 3D displacement field.

Chapter 5: Application of SPM in SPR detector

97

Chapter 5 Application of SPM in SPR detector

Surface plasmon resonance (SPR) is a well-known phenomenon which has been widely used for a

variety of detectors. SPR is very sensible to the change of refraction index, so there are many bio-sensors

based on SPR which can characterize molecular interactions. Apart from SPR bio-sensors, there are also

bio-sensors based on LSPR (Localized Surface Plasmon Resonance). Although SPR bio-sensor has been

developed for a longer time than LSPR bio-sensor, they both have their own advantages and

disadvantages, which will be only briefly detailed in this chapter. By combining 2D optical detectors

with SPR/LSPR, SPR/LSPR imaging (SPRi/LSPRi) has also been developed. This technique allows for

high throughput measurements (e.g. measure several different kinds of molecular interactions at the

same time). In this chapter, sinusoidal phase modulation is applied into phase-sensitive detection. A 2D

detector is used, so it also has the potential to do SPRi in the future.

5.1 Introduction to SPR

Surface plasmon resonance (SPR) is the resonant oscillation of conduction electrons at the interface

between negative and positive permittivity material stimulated by incident light. [108]

In a traditional SPR sensor (see figure below), a metal (usually gold or silver) thin film of about 40 to

50 nm when operating in the visible or NIR range, is coated on a piece of glass substrate to form the

required interface. A glass or plastic prism is then used as the coupling device: it is attached to the SPR

substrate with index-matching oil to realize the required incident angle (resonance angle θk), which is

needed to elicit the SPR. The resonance angle θk, named Kretschmann angle, is specific because

essentially, the wave vector of the p-polarized component of the incident light should be matched with

the resonance requirement. This angle, which is beyond the critical angle, is determined by the properties

of the two materials (metal and probed medium, cf. Figure 5-1). The metal thin film being fixed in

thickness and refractive index, the angle will vary with the refractive index value n(λ) of the probed

medium in the vicinity of the metal.


98

Figure 5-1 Typical set-up for an angular interrogation SPR biosensor. The darker ray impinging at a

specific angle on the detector is attenuated due to the SPR occurring at the gold/liquid interface. This

angle θk depends on the fluid refractive index in the immediate vicinity of the metal layer.

The Figure 5-1 shows a traditional angular interrogation scheme of SPR sensor. When the analytes in

the flow channel are captured on the surface of gold thin film, a small change of refractive index will

appear, thus the resonance angle will change accordingly. Since the absorption is maximum at the

resonance angle, the resonance angle θk is characterized by a reflection dip. This angle is determined

directly with such setup since the intensity of reflected light over a range of angles can be measured if a

1D detector array or a scanning detector is used. With such angle-resolved system, the change of

resonance angle can be monitored. The change in θk produced by a change of refractive index, gives

information on the analyte concentration, while the time trace θk(t) provides kinetic information on the

binding affinity between the selected probes and targets.

In addition to this scheme, there are also intensity detection scheme, wavelength interrogation scheme

and phase detection scheme. In intensity detection scheme, the incident angle is fixed at or very near θk,

and the intensity of reflected light is detected. While the dynamic range is reduced in this case, this is

sufficient if the change in θk is small or if only a threshold value needs to be determined [32]. In

wavelength interrogation scheme, a white light (or other broadband/tunable light sources) is used as

incident light. The incident angle is typically fixed, and the corresponding resonance wavelength can be

measured by analyzing the spectrum of the reflected light. In phase detection scheme, the incident angle

can be fixed or not, the important point being that the phase of the reflected light is detected.

In all these schemes, the ability to measure small refractive index change is a basic performance

indicator. From Table 5-1, we can see that the phase detection scheme has the best refractive index

resolution, and is also more convenient to do SPRi if a fixed angle angle is used, although the dynamic

range is limited in this case. Based on the literature review, our experiments, and simulation, it appears

indeed that phase detection is helpful to reach a higher resolution, if carefully selected metal thicknesses

are used.

Probed medium


99

SPR detection techniques have been commercialized for a long time. The first commercial SPR sensor

(called “Biacore”) was released by the company “Pharmacia Biosensor AB” in 1990 [109,110,111]. The

Biacore instruments (an example is shown in Figure 5-2(a)) have been evolving ever since, and the

brand of Biacore (now GE-Healthcare) is still one of the most recognized one in the field of SPR sensing

system; however, they are usually bulky.

SPR schemes Angular interrogation Wavelength interrogation Phase detection

Typical resolution 5×10-7 RIU 10-6 RIU 4×10-8 RIU

Record of resolution① 10-8 RIU [61] 10-6 RIU [112] 2.8×10-9 RIU [63]

Typical dynamic range 0.1 RIU >0.1 RIU 5×10-4 RIU

SPR imaging Difficult Difficult Convenient

Table 5-1 Brief comparison between several common schemes in SPR sensing. For the phase detection

a fixed angle is considered. Adapted from [32].

There are some other companies who focus on making portable and low-cost commercial SPR sensors.

For example, the SPR sensor called “Spreeta” was announced in 1997 (see Figure 5-2(b)). The size is

quite small and the cost is low, however additional signal processing system (including computer and

interface) is needed [113,114,115].

① Data given by our literature review. There may be better results that we did not know.


100

Figure 5-2 Commercial SPR/LSPR systems. (a) Biacore 8K [116]. (b) Spreeta [115]. (c) LiteChek [117].

If metal nanostructures are used instead of thin films, then the phenomenon of localized surface plasmon

resonance (LSPR) may happen. Just like SPR, LSPR is sensible to the change of refractive index.

Compared to SPR, LSPR is much easier to excite, so the strict angle control is no more needed [118].

This may be helpful to build compact and portable sensors.

In fact, the sensitivity of SPR is so high that the change of refractive index caused by ambient

temperature fluctuation will add noise to the signal. So for most SPR sensors, temperature control or

temperature compensation methods are needed. LSPR sensors, on the contrary, is usually less sensitive,

so LSPR sensors can hardly be affected by the ambient temperature fluctuation. However, LSPR is only

sensible to the refractive index change close to the metal surface due to stronger light concentration,

while SPR is sensitive to a larger volume (given by the penetration depth of the evanescent tail of the

impinging field). We can see that although LSPR is less sensitive to refractive index change, the

measurement efficiency of LSPR may be higher for very small targets. With further development, LSPR

sensors are hopeful to provide a resolution comparable to SPR sensors in certain cases, with the benefit

of a lesser temperature control and easier excitation.

Technically speaking, LSPR sensors are less mature than SPR sensors though. Still, there are already

commercial LSPR sensors, as shown in the Figure 5-2(c).

SPR imaging (SPRi) and LSPR imaging (LSPRi) is a research focus in recent years [58,119,120,121].

By replacing a point detector with a 2D detector matrix (e.g. CCD, CMOS), the throughput of

measurements can be largely improved. Through different functionalizations at different zones of the

metal surface and with the help of appropriate microfluidics design, several different kinds of molecular

interactions can be detected at the same time.

In this chapter, we will focus on an innovative, compact, phase sensitive SPR sensor. It should be noted

that the initial prototype and tests were notably developed in our lab by Julien Vaillant and the support

of Dr Tzu-Heng during his post-doctoral stay. This work has not yet been published. In the framework


101

of this thesis, we have considered experimentally and theoretically the possibility to add imaging

capability to the developed system, and we discuss some of the performance achievable with different

demodulation schemes. The system includes a common path interferometer and a sinusoidal phase

modulation. As will be detailed, the SPM is achieved through the combined use of a current-modulated

Vertical Cavity Surface Emitting Laser (VCSEL) and an anisotropic crystal. A CMOS camera is used

as detector, showing the potential to do SPRi in the future.

5.2 Principle of phase modulation through wavelength modulation

Phase modulation through wavelength modulation can be achieved when an unbalance exists between

signal and reference arm. This approach may enable low-cost and high-speed operation because, as will

be shown here, only a small wavelength change is required and laser diode can be modulated at very

high frequency. Some practical issues limit the use of this technique, mainly related to the wavelength

stability; but this issue is mostly overcome in the developed system presented in the forthcoming

sections.

In our setup, the phase modulation is obtained via an injection current modulation of the laser source.

This current modulation entails a small wavelength modulation which can be converted into a phase

modulation if the signal and reference beams have unbalanced paths. In order to modulate the current

(and therefore obtain a phase modulation), we need to modulate the input voltage.

We know that for a VCSEL, in the vicinity of the operating voltage U0, the output wavelength λ of

VCSEL approximately will exhibit a linear relationship with the input voltage U:

U = (3.1)

where τ is a constant. When applying the operating voltage U0, we have the expression of operating

wavelength λ0:

0 0U = (3.2)

If we define that:

0

0U U U

= − = −

(3.3)

Then we get:

U = (3.4)


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If we separate the laser beam in two beams, and the optical path difference is L, then we have the

resulting phase difference κ:

2 L

= (3.5)

If we take the differential on both sides, then we have:

2

2 Ldd

−= (3.6)

When applying the operating voltage U0, we have the phase difference κ0:

0

0

2 L

= (3.7)

We define that:

0 = − (3.8)

When the wavelength change Δλ is small, approximately we have:

2

0

2 L

− = (3.9)

So that

0 0 02 2

0 0

2 2L LU

− −= + = + = + (3.10)

By defining a coefficient:

2

0

2 L

−= (3.11)

We can express this approximation by:

U = (3.12)

In the case of SPM, this equation gives an expression for the phase modulation depth, i.e. a= χΔU for a

given sine voltage modulation U(t)=Uo+ΔU.sin(ωt). It shows that when the amplitude of voltage

modulation is small, then the phase modulation depth is approximately proportional to the voltage

change.


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In practice, we used a VCSEL emitting at around λ0=670nm. If we suppose that L=1mm, then we may

make a simulation to see the precision of this approximation. The results are shown by the following

figures:

Figure 5-3 Phase difference (wrapped) caused by the modulation of wavelength.

Figure 5-4 Error of the approximation described by Eq. (3.9).

Figure 5-3 shows that when the wavelength is tuned by 1nm (from 669.5nm to 670.5nm), the error of

approximation is small; besides, the phase difference changes for more than 4π in this zone, which is

big enough to carry out most phase modulation functions. From Figure 5-4, we can see that in this range

of modulation, the maximum error is about 5×10-3rad, which is often tolerable.


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In practice, in order to measure , we may directly apply a slow linear phase modulation to the

VCSEL and measure the period of the interference signal. For example, with our setup, we obtained

experimentally that 8.6 /mV rad .

When a sinusoidal phase modulation with the amplitude a=3.8317rad is needed, which corresponds to

Δκ=±3.8317rad at maximum, according to Eq. (3.12) and 8.6 /mV rad , we can get ΔU≈±33mV

at maximum, thus we can set 66mV as the peak-to-peak value of VCSEL voltage modulation. This value

should also be adjusted by observing Lissajous curves experimentally, which will be discussed in

Section 5.4, to guarantee the precision of the measurements.

5.3 Phase extraction in wavelength modulated interferometers

When a voltage modulation is applied to the VCSEL, the output power/intensity changes too, which can

be described by Eq. (1.40). To extract the phase information while using sinusoidal phase modulation,

three algorithms may be used.

The first one is the modified SPM algorithm (see Section 2.3.1). However, the coefficient μ representing

the intensity modulation should be known. It can be easily measured by applying a slow linear voltage

modulation and monitoring the output power/intensity (no interferometry). Some simple tests are done

to prove the feasibility of this method (see Section 5.4).

The second one is the modified f-G-LIA algorithm (see Section 2.3.3). In our experiments, μ is relatively

small and the amplitude of phase modulation a=3.8317rad can be achieved. So this algorithm can be

simplified into f-G-LIA algorithm (see Section 1.5.4). This method is used to carry out detections of

refractive index change (see Section 5.4).

The third one is the modified integrating bucket algorithm (see Section 2.3.4). This method requires

precise exposure time control, which is beyond the scope of this thesis.

5.4 Experiments

The operation principle is described in this part through two simplified, preliminary, setups made in this

thesis to test the phase extraction methods.


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5.4.1 Preliminary setup: test of the algorithms

Before using the CMOS camera as detector, we test the feasibility of modified SPM/f-G-LIA algorithms

by using a photo detector. A common path interferometer, which can hardly be affected by external

vibrations or noises, is built. The setup is shown in the figure below.

Figure 5-5 Preliminary setup for testing the algorithms.

The initial polarization of laser is at around 45° with respect to the vertical axis. Then the laser passes

through a birefringent crystal YVO4, which leads to an optical path difference between the 0° and the

90° polarization components of laser, although the geometrical paths are the same for them. This optical

path difference due to the crystal anisotropy can be seen as a constant L (see Eq. (3.5)). Thus the phase

modulation method described by Section 5.2 can be carried out by modulating the input voltage to the

VCSEL. A polarizer at around 45° is put after the YVO4 crystal to make the 0° and 90° laser components

interfere with each other. Then the interference signal is captured by the single photo detector. The

software LabVIEW is used to control the input voltage to the VCSEL, to obtain the signal from photo

detector, and to process the data, with the help of a NI card.


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In order to test the system setting, we add a “fake” phase change, via a small triangular voltage

modulation, which is added to the sinusoidal modulation to cause a triangular phase change (see Eq.

(3.11)-(3.12)). If the frequency of this triangular signal is much smaller than the sinusoidal phase

modulation, then during each sampling / each calculation for one phase value, the triangular signal can

be seen as a constant. Therefore, if our measuring method works fine, this triangular phase change can

be obtained.

Such simple tests are done with the modified SPM/f-G-LIA algorithms. The results are shown in the

figure below:

Figure 5-6 Measuring a sawtooth phase change in the common path interferometry with different

algorithms. (a) With modified SPM algorithm; (b) With f-G-LIA algorithm.

The retrieved phase is shown in red. From the figure above, we can see that these two algorithms both

give precise results. There is a small delay in both of them, which is caused by the data transmission and

processing. Since this delay is small and constant, it is not a problem for most measurements.

Calibration

The input voltage to the VCSEL can be calibrated using the Lissajous curve beforehand. We take the

case of the f-G-LIA algorithm as an example: if we plot all the possible points ,X Y

M N

(see Section

1.5.4 for definitions of X, Y, M and N) in a Cartesian coordinate system, then according to Eq. (0.53),


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we can get a perfect circle if all the measurements are precise. If the circle is not round enough, then the

input voltage to VCSEL should be adjusted.

By varying the added phase change in a range larger than 2π, we can get the Lissajous curves, as

exemplified in the figure below:

Figure 5-7 Lissajous curves while using different algorithms after calibrations①. (a) Modified SPM

algorithm; (b) f-G-LIA algorithm.

Generally speaking, the two Lissajous curves in Figure 5-7 are both tolerable, and similar good results

using these two algorithms have been shown in Figure 5-6. We can also see from Figure 5-7 that the

Lissajous curve given by modified SPM algorithm (see Section 2.3.1), which takes the non-zero value

of μ into account, is closer to a perfect circle than the one given by f-G-LIA algorithm (see Section

1.5.4), which requires that μ≈0 (or small enough). The non-perfect calibrations may also be one of the

reasons for that. It should be noticed that with the modified f-G-LIA algorithm (see Section 2.3.3), the

non-zero value of μ can also be taken into account. So theoretically speaking, the modified

SPM/f-G-LIA algorithms are preferred. However, f-G-LIA is simpler in practice, and when the

calibration is nicely done, it can also give very good results, as shown in Figure 5-6. Since our primary

goal is to prove the feasibility of this method, we choose to use the f-G-LIA algorithm for the following

experiments.

5.4.2 Phase-sensitive SPR sensor

A homemade SPR sensor is shown in the following figure:

① The calibrations may not be perfectly done, since they are performed manually.


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Figure 5-8 Homemade SPR sensor with CMOS detector. The gold layer is deposited on a glass slide.

Its gold side is enclosed in the micro-fluidic chamber, while the other side is coupled to the glass prism

with index-matching oil.

In Figure 5-8, the VCSEL and YVO4 crystal remain the same as in Figure 5-6. The laser beam is

collimated with a small lens fixed to the VCSEL. A rectangle prism is used to reach the required incident

angle (around 67°) to the gold layer where SPR can be excited.

Before reflection on the gold layer, a sinusoidal phase modulation is already present (see Section

5.2&5.4.1) between the 0° component (p-polarized light) which experiences the SPR and the 90°

component (s-polarized light) which serves as a reference. When liquids with different refractive indexes

go through the microfluidic channel, the reflected p-polarized light will have a phase change as well as

an intensity change, while the reflected s-polarized light remains almost the same [122,67,123]. Then

the laser beam goes through the polarizer at 45°, and the originally p-polarized light and the originally

s-polarized light will interfere with each other. The expression of this interference signal can be

described by Eq. (1.42), where the originally p-polarized light can be seen as the signal, and the

originally s-polarized light can be seen as the reference.


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In our preliminary setup, the gold layer is not functionalized, so if the gold layer is perfectly flat and

homogeneous, then theoretically every point on the surface should have the same response to the change

of refractive index. Therefore, we used the average intensity value of a matrix of pixels in a chosen zone

as the signal I(t), and apply the f-G-LIA algorithm (see Section 1.5.4) to obtain the phase

args

X Yi

M N

= +

and the relative amplitude

2 2

s

X YA

M N

+

. The results are shown in the

figure below.

Figure 5-9 Phase and amplitude detection of refractive index change using homemade SPR sensor with

CMOS detector (see Figure 5-8). Glucose solutions with different concentrations (5.5%, 5%, 3%, 2%,

1%) and de-ionization water (DI water) went through the microfluidic channel sequentially.

We can see from Figure 5-9 that phase is more sensitive than amplitude. However, the signal-to-noise

ratio is not very good, and the contrast is not very big. It is mainly due to the bad quality of the gold

layer used here. Besides, an obvious phase drift is observed. It is mainly caused by the wavelength drift

of VCSEL, and the refractive index drift when the ambient temperature changes as well. In fact, the

main reason for the wavelength drift of VCSEL is also the ambient temperature fluctuation [124].

To cancel out this drift, many methods are applicable and have been applied to more advanced

prototypes. The first method is to make use of the CMOS: if the zone of interested contains

functionalized zone and non-functionalized zone at the same time, then we can calculate the phases at


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these two zones respectively and make the subtraction. The second method is to compensate only the

wavelength drift of VCSEL: by inserting a beam-splitter between YVO4 and the prism in Figure 5-8,

we can measure the initial phase difference between the p-polarized and s-polarized light before being

reflected by the gold layer, then we can subtract this initial phase from the measured phase in real time.

This second method has been proved to be very effective. Fig. 6-10 shows the results of an experiment

done in our research group to prove this point: the second method is used to cancel out the phase drift,

a simple photodiode is used to replace the CMOS matrix, angular interrogation mechanism is added,

and a commercial SPR chip is coupled on a 45° prism, while the phase modulation method and phase

retrieval method remain the same.

Figure 5-10 Angular interrogation SPR sensor with phase and intensity detection. Sinusoidal phase

modulation is carried out by VCSEL wavelength modulation (see Section 5.2). f-G-LIA algorithm (see

Section 1.5.4) is used to extract the phase information. A commercial SPR chip is used without any

microfluidics. (a) Phase response to angle change. (b) Intensity response to angle change.

As shown in Figure 5-10(b), at SPR angle (around 44.3°), the intensity is almost zero, which proves that

the quality of this SPR chip is so good that the electromagnetic coupling is very effective. In Figure

5-10(b), we can see that the phase measurement is very stable and there is a quasi-vertical drop of phase

at SPR angle, which proves the efficiency of our method to compensate the wavelength drift of VCSEL,

as well as the feasibility of our phase detection method.

In Figure 5-10, the small zone around SPR angle where we have a quasi-vertical drop of phase can be

called “phase-sensitive zone”. We can make use of this zone to do highly sensitive SPR phase detections.

If we fix the angle at the center of this zone (SPR angle), then when a tiny change of refractive index


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takes place on the gold surface, a big phase difference can be detected. The only drawback of using this

zone is that: the light intensity of signal (p-polarized light) remains very small in this zone, while the

intensity of reference (s-polarized light) is as strong as usual, which may result in a loss of signal-to-

noise ratio to some extent. But its influence seems to be limited, since in Figure 5-10, we did not get

unnormal phase values in this phase-sensitive zone.

For all the experiments in this section, the software LabVIEW was used to control all the

outputs/inputs/calculations/displays. The advantage is that the phase modulation and intensity detection

(e.g. capturing pictures for the camera) can usually be synchronized easily. Thus we do not need to

worry about the initial phase problem (see Section 2.1).

5.5 Perspective: Combining shearing interferometry with SPRi

From the discussions in Sections 5.1 & 5.4.2, we know that the ambient temperature fluctuation may

affect the precision of SPR measurements, and two methods were proposed to solve this problem.

In this section, we propose another method which is inspired by the shearography technique: to make

SPR imaging with the help of shearing interferometry to compensate these errors, with the benefit of

using a single polarization component. First of all, we need a shearing element to make a shearing to the

original image and to superpose these two images. A modified cube beam splitter is proposed to achieve

this function. The principle is shown in the figure below:

Figure 5-11 A modified cube beam splitter working as the shearing element. The upper side and the left

side of the beam splitter are coated with metal so that the light can be efficiently reflected. If the optical

paths are the same for the separated beams (this kind of beam splitter is represented by the blue lines),

then there will no shearing between the two beams, as shown by the red lines. If one side of the beam


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splitter is thicker (see the orange lines) than the other one, then the laser beam reflected by the thicker

side (see the pink lines) will be parallel but shifted away from the other one.

Through simple geometric derivation, we deduced that the shifted beam (pink) and the other beam (red)

are parallel in the end with a distance:

2 sins d = (3.13)

Also, we can obtain the optical path difference:

2 cosL d = (3.14)

Obviously, s and L can be easily adjusted by changing d and δ.

When an expanded and collimated laser beam is incident, this shearing is applicable to the whole beam,

resulting in an image of shearing interferometry. The following figure shows the proposed shearing SPR

imaging system:

Figure 5-12 Combination of shearing interferometry and SPR imaging.

When the incident angle to the prism is fixed, presuming that the incident collimated beam is

homogeneous, we have the functions A(n) and φ(n) to represent the amplitude and phase of reflected

light (n is the refractive index). At different zones of functionalization, the values of n are often different.

For example, in Zone-1 and Zone-2, respectively we have:

( ) ( )( )12

1 1

i ft nE A n e

+= (3.15)

( ) ( )( )22

2 2

i ft nE A n e

+= (3.16)

At the places where Zone-1 is not overlapped with Zone-2, the light intensity can be expressed as:


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( )2 2

11 1 1 14I E E A n= + = (3.17)

( )2 2

22 2 2 24I E E A n= + = (3.18)

At the places where Zone-1 is overlapped with Zone-2, the light intensity can be expressed as:

( ) ( ) ( ) ( ) ( ) ( )( )2 2 2

12 1 2 1 2 1 2 1 22 cosI E E A n A n A n A n n n = + = + + − (3.19)

When the ambient temperature changes, if we suppose this change lead to the same refractive index

change nt in Zone-1 and Zone-2, then we have:

( ) ( )( )12

1 1' ti ft n n

tE A n n e + +

= + (3.20)

( ) ( )( )22

2 2' ti ft n n

tE A n n e + +

= + (3.21)

( ) ( ) ( ) ( ) ( ) ( )( )2 2 2

12 1 2 1 2 1 2 1 2' ' ' 2 cost t t t t tI E E A n n A n n A n n A n n n n n n = + = + + + + + + + − + (3.22)

As discussed in Section 5.4.2, for SPR phase detection, we make use of the “phase-sensitive zone”,

where the phase changes rapidly and almost linearly, and the intensity changes slowly (see Fig. 6-10).

So approximately we have:

( ) ( )1 1tA n n A n+ (3.23)

( ) ( )2 2tA n n A n+ (3.24)

( ) ( ) ( ) ( )1 2 1 2t tn n n n n n + − + − (3.25)

Thus

12 12'I I (3.26)

Which means that the influence of ambient temperature change can be compensated.

The intensity I12 is mainly determined by the phase difference ( ) ( )1 2n n − , making this system able

to do phase-sensitive detections. Besides, A(n1) and A(n2) can be measured where Zone-1 is not

overlapping with Zone-2. (See Eq. (3.17)-(3.18)) A simulation of a possible shearing image is shown

in the figure below:


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Figure 5-13 A possible shearing image made by simulation. It is supposed that the round zone in the

middle (Zone-2) is functionalized and that the analyte is attached to its surface evenly, while the other

part (Zone-1) is only non-functionalized gold layer. A beam-splitter of 50/50 is used, and a pure

horizontal shear is realized.

Obviously, Figure 5-13 is the simplest case for SPR imaging. SPR chips with different functionalization

zones can be used in this method, as long as the shearing distance s (see Figure 5-11) is well controlled.

5.6 Conclusion

In this chapter, we have analysed the possibility to combine SPM with SPR sensor as preliminary tests

for phase sensitive SPRi. These preliminary experiments were done to prove the feasibility of this idea.

A CMOS is used as detector to show the possibility of doing SPR imaging in the future. The initial phase

problem (see Section 2.1) is solved by using LabVIEW to control the whole measuring system.

Generally speaking, in our experiments, the goal of combining sinusoidal phase modulating

interferometer and SPR sensor is to make a cheap, simple, compact, robust yet high-precision SPR

detector. By using VCSEL as the light source, the cost can be lowered; besides, the size of VCSEL is

small, and the input voltage is low (less than 5V), which is practical for the fabrication of compact and

portable devices in the future. Phase detection is used to improve the precision. By simply modulating


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the input voltage to the VCSEL, phase modulation can be realized, which reduce the cost as well as the

complexity of measuring system. Sinusoidal phase modulation was applied to guarantee the precision

of modulation, and corresponding algorithms for signal processing have been developed to extract

information. Simple yet effective temperature compensation methods have been proposed to further

improve the stability and precision of the measuring system.

Based on this system, many methods may be applied to improve its performance in the foreseeable

future. For example, we may functionalize the gold surface differentially at different zones to realize

SPR imaging, use nanostructures (e.g. Graphene–MoS2 hybrid nanostructures [123]) to enhance the

sensitivity of SPR sensor, use holograph/grating to realize the coupling of the incident light with the

surface plasmon of the gold layer, combine SPR imaging with shearing interferometry (see Section 5.5),

use LSPR chips instead of SPR chips, combine angle interrogation with phase detection in SPR imaging,

etc.

General conclusion and perspectives

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General conclusion

Phase modulation techniques can be applied in almost every interferometric measuring systems.

Basically, by applying phase modulations, the number of sampling for each measurement increases, thus

the measurement precision can be greatly improved, and the phase ambiguity problem can also be

solved. In order to provide new possibilities for simple, cheap, reliable and precise interferometric

measurements, we have studied the signal of phase modulating interferometer theoretically and

experimentally. Sinusoidal phase modulation was specifically considered because such modulation is

typically affordable and it can be accurately achieved even at high frequency.

Sinusoidal phase modulation has been applied in holography and holographic interferometry. (f)-G-LIA

algorithm is used to retrieve phase information, and the obtained results are similar to the traditional

SPM algorithm. Besides, two linear/sawtooth or sinusoidal phase modulations are applied in an

innovative 2D-ESPI setup, making it possible to do simultaneous 2D displacement field measurement

with a single laser and camera. Sinusoidal phase modulation has also been applied to phase-sensitive

SPR sensor. Several methods to compensate the ambient temperature fluctuation have been proposed.

The feasibility has been proved by a preliminary setup.

In Chapter 1, we focused on the basic principles and mathematical expressions of interference signal.

Several phase-shifting and phase modulation techniques were introduced, and the corresponding

algorithms were presented. We highlighted the newly proposed (f-)G-LIA algorithm, and compared its

application in SPM interferometer with the traditional SPM algorithm.

In Chapter 2, two practical problems were considered: the initial phase problem, which is caused by the

non-synchronized detection and modulation, and the intensity modulation problem, which occurs in

most laser diodes when applying an injection current modulation for doing SPM. The modulated signal

was analysed in the frequency domain, and the mathematical solutions for these problems in different

algorithms have been deduced.

In Chapter 3, sinusoidal phase modulation was applied in a homemade co-axis lens-less DH/DHI system.

A reasonable spatial resolution (32 LP/mm) was obtained, and precise out-of-plane rotation

measurements (with a sensitivity better than 0.004°) have been realized. SPM algorithm and (f-)G-LIA


117

algorithm were applied to retrieve the phase information. The results given by these two algorithms have

been compared, and it can be concluded that the performance of (f-)G-LIA is only slightly superior for

the investigated phase modulation depth.

In Chapter 4, an innovative 3-beam ESPI system was built to carry out simultaneous 2D in-plane

deformation measurements. Two modulations with different frequencies have been added to the two

beams respectively. The displacement field along X-axis and Y-axis can be separated in the frequency

domain efficiently. The fringe visibility is very good for both directions. Sinusoidal phase modulation

and linear/sawtooth modulations have been tested and proved to be applicable. The possible extension

of this method to carry out 3D measurements has been discussed.

In Chapter 5, the use of sinusoidal phase modulation in phase-sensitive SPR sensor was discussed. More

precisely, a common-path design using YVO4 crystal combined with wavelength modulated VCSEL

was considered for its high stability potential and possible high modulation frequency. The modified

SPM algorithm and the f-G-LIA algorithm have been firstly tested without SPR chip. Then a preliminary

experiment using a SPR chip and a CMOS matrix was done to show the feasibility of this method. The

idea of applying shearing interferometry into SPRi was also proposed.

Perspectives

Smart detector using sinusoidal phase modulation

For SPM interferometers, the relatively complex data processing may limit the maximum measurement

speed and make it impossible to realize real-time measurements with lower-end computers.

Nevertheless, we noticed that a new kind of camera called “lock-in camera” has been proposed and

fabricated, which was originally developed for high-speed low coherence interferometry

[125,126,127,128]. Inspired by such lock-in camera, we propose to realize SPM camera by combining

SPM interferometer with smart detector array. The principle is shown by the figure below:


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Figure C-1 Flowchart of the working principle for one smart pixel in SPM camera.

We may use the waveform generator to generate two synchronized voltage signals, asin(ωt) and

acos(2ωt), where a is a positive constant representing the amplitude of phase modulation. Then the

signal of asin(ωt) is used to modulate the piezoelectric actuator (or wavelength modulation); the signals

asin(ωt) and acos(2ωt) as well as the light intensity signal I(t) enter the smart detector array to make

analogue calculations. The operations described by Eq. (0.28)-(0.29), which used to be the most time-

consuming steps, are carried out through this analogue signal treatment. [125,126] Then the images of

X and Y described by Eq. (0.28)-(0.29) are directly transferred to the computer at a speed comparable

to the frame rate of camera, and the computer can easily calculate and show the results through Eq.

(0.32) in real time. Obviously, the required time for each measurement can be greatly shortened, thus

the time issue of data acquisition and data processing should be efficiently solved.

Another advantage is that the synchronization of the system is also done automatically: the same signal

asin(ωt) is used to control the piezoelectric actuator and to make analogue calculations, and the two

outputs of waveform generator (asin(ωt) and acos(2ωt)) can usually be synchronized easily.

In a word, smart detector and SPM interferometer seems to be a perfect match even if the cost of such

smart detector may be prohibitive for a number of applications. Ideally, this approach should be certainly

pursued as a perspective beyond this thesis work.

Automatic control of measuring systems

Since the initial phase has been solved mathematically and experimentally, most of the experiments

presented in this thesis do not have synchronization control (in some cases modulation and detection

were synchronized though, especially when a point detector is used). However, in order to improve the


119

measurement precision and make automatic measurements in the future, automatic control will become

necessary. The goal is to accomplish the measurements only by using a single software in a computer

with the least manual operation after properly adjusting the optical setup. Generally speaking, this

automatic control may include: synchronization and control of the camera and modulation, calibration,

processing, image filtering, etc.

As discussed before, smart detectors may be used to fulfil the synchronization and to greatly increase

the measuring speed. Additional algorithms are also needed to find automatically the position of sample

/ find the position with the best fringe visibility in the reconstruction process of DH/DHI, as well as to

find the SPR angle in SPR sensor. To control the incident angle in SPR setup, the automatic control may

be carried out with stepping motor (which is done on recent prototypes in our group using

microcontrollers) and appropriate algorithms (to find the Kretschmann angle, to realize the

measurements at different angles, etc.).

Final comments

In this thesis, several advances based on phase modulation techniques have been proposed; besides,

linear/sawtooth and sinusoidal phase modulations have been analyzed intensively. However, there are

also many other kinds of phase modulation functions, which may be used to suppress the noise in the

interferometric signal at different frequencies, as discussed in Section 1.6. Generally speaking, a good

choice of phase modulation function can help to improve the anti-noise ability, precision, stability of a

measuring system, and even the service lifetime of modulators as well. Therefore, the choice of the

phase modulation functions, and the related parameters such as the phase modulation depth, etc. should

be considered carefully for each application.

In fact, there are many kinds of modulators in interferometry: piezo-mounted mirror, piezoelectric

optical fiber stretcher, liquid crystal on silicon-spatial light modulator (LCOS-SLM), photoelastic

modulator (PEM), free-space electro-optic modulators, laser diode with tunable wavelength, etc. Such

modulators that have different properties and different costs, can be matched with different phase

modulation functions to meet actual needs. It is important to wisely choose the type of modulator, the

type of modulation function and phase-retrieval algorithm according to the modulation frequency, and

other parameters such as the required modulation precision and range, the noise characteristics, the

wavelength of the laser or the desired cost. A well-balanced equilibrium between all these factors is

desirable to achieve high performance and cost efficient interferometric system for a considered

application.

Résumé en français

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Dans cette thèse, nous nous intéressons aux signaux interférométriques à modulation de phase continue,

dans le but de développer des dispositifs de détections performants et originaux pour des applications

en mécanique et en optique.

Nous présentons d'abord plusieurs des techniques de modulation et démodulation employées dans ce

contexte. Nous nous focalisons sur les modulations de phase sinusoïdales (SPM), qui sont

particulièrement avantageuses pour les dispositifs de modulations les plus largement accessibles (e.g.

modulateurs piézoélectriques, électro-optiques, ...). Nous proposons en particulier des solutions au

problème de synchronisation lors du processus de démodulation ainsi qu'au problème éventuel de

modulation d'amplitude concomitante à la modulation SPM.

Ces techniques de démodulations sont ensuite appliquées à trois dispositifs expérimentaux développés

au cours de cette thèse. Il s'agit d'abord d'un dispositif d’holographie digitale compact sans lentille,

mettant en œuvre une modulation SPM simple, pour de l'imagerie et de la mesure de déplacement sans

lentille. Nous utilisons ensuite une technique de modulation à double fréquence pour réaliser des

mesures bidimensionnelles de champ de déformation, à l'aide d'un dispositif ESPI (interférométrie de

speckle électronique) original. Cette approche permet une mesure simultanée selon les 2 directions du

plan à l'aide d'un seul système de laser et caméra. Finalement, nous présentons un instrument de type

SPR (Surface Plasmon Resonance) compact mettant en œuvre une détection interférométrique de type

SPM à modulation de longueur d'onde, dans lequel la modulation d'amplitude est prise en compte avec

succès.

1. Algorithmes de récupération de phase

Dans le cas d’une interférence entre deux ondes planes polarisées selon la même direction, l’intensité

en un point où les deux champs se superposent s’écrit :

( )2 2 2= 2 cosr s r s r s r sI E E A A A A + = + + − (1)


122

Où apparaissent As et ϕs l’amplitude et la phase du champ signal, qui en notation scalaire complexe

s’écrit :

( )2 si ft

s sE A e +

= (2)

Où t et f sont le temps et la fréquence de la lumière respectivement. Ar et ϕr sont l’amplitude et la phase

du champ considéré comme référence, qui en notation scalaire complexe s’écrit également:

( )2 ri ft

r rE A e +

= (3)

Nous pouvons définir la différence de phase qui est mesurable:

s s r = − (4)

Alors notre première équation devient:

( )2 2 2 cos -r s r s sI A A A A = + + (5)

Afin de mesurer s sans ambiguïté, des méthodes de « décalage de phase » ont été introduites,

consistant à ajouter une phase contrôlable r à la lumière de référence, l'intensité se transforme alors

en:

( )2 2 2 cosr s r s r sI A A A A = + + − (6)

En affectant à r certaines des valeurs différentes, s peut être mesurée sans ambiguïté. Pour

augmenter la précision, Nous pouvons également faire une modulation de phase continue ( )r t .

L'intensité de la lumière devient donc un signal temporel I(t):

( ) ( )( )2 2 2 cosr s r s r sI t A A A A t = + + − (7)

Pour une modulation de phase linéaire simple:

( ) 0 02r t t f t = = (8)

Dans ce cas une détection synchrone (LIA : Lock-in amplifier) peut être utilisée pour démoduler le

signa, c’est-à-dire retrouvé l’amplitude et la phase φs du champ signal. D’abord, nous définissons deux

fonctions C(t) et S(t) :

( ) ( )0cosC t t= (9)

( ) ( )0sinS t t= (10)


123

Ensuite, nous définissons deux quantités X et Y qui peuvent être calculées en intégrant dans le

temps l’intensité détéctée:

( ) ( )0

T

X I t C t dt= (11)

( ) ( )0

T

Y I t S t dt= (12)

T est le temps d'intégration. Afin de faire usage de l'orthogonalité des fonctions trigonométriques pour

obtenir des résultats précis, le temps d'intégration T doit être assez long pour couvrir de nombreuses

périodes de modulation, ou il doit être un multiple entier de la période 02 / .

Enfin, si nous définissons deux coefficients M et N :

1M N= = (13)

Alors nous savons que s peut être exprimée comme suit①:

args

X Yi

M N

= +

(14)

Si nous avons une modulation de phase sinusoïdale:

( ) ( ) ( )sin sin 2r t a t a ft = = (15)

Ensuite, nous pouvons choisir d'utiliser l’algorithme SPM (algorithme de démodulation

traditionnellement utilisé pour les interféromètres à modulation de phase sinusoïdale). Premièrement,

nous redéfinissons c, M et N comme suit:

( ) ( )cos 2C t t= (16)

( ) ( )sinS t t= (17)

① arg(z) prend l'argument d'un nombre complexe z=a+ib. Il peut souvent être déterminé sur un intervalle

de 2π à l’aide de la fonction atan2(a,b) ou angle(a+ib).


124

( )2M J a= (18)

( )1N J a= (19)

Et alors

args

X Yi

M N

= +

(20)

L’algorithme dit G-LIA (LIA généralisé), récemment introduit dans notre laboratoire, peut être utilisé

pour traiter de nombreux types de fonctions de modulation de phase, dont la fonction de modulation de

phase sinusoïdale. Pour tout type de fonction de modulation de phase ( )r t , C(t) et S(t) sont définis

comme suit:

( ) ( )( )cos rC t t= (21)

( ) ( )( )sin rS t t= (22)

Pour la modulation de phase sinusoïdale décrite par Eq. (15), M et N sont ainsi redéfinis:

( )01 2M J a= + (23)

( )01 2N J a= − (24)

Enfin, s peut être obtenu:

args

X Yi

M N

= +

(25)

Néanmoins, en raison de la présence d’une composante continue dans I(t) et dans C(t), cettte méthode

ne fonctionne sans précaution particulière que si on impose ( )0 0J a = , de manière à supprimer la

composante continue de C(t).

Pour s’affranchir de cette contrainte, l’algorithme f-G-LIA (algorithme G-LIA avec filtre) peut être

utilisé. Tout d'abord ( )I t passe par un filtre DC avant d’être traité. M doit alors être redéfini

comme:

( ) ( )2

0 01 2 2M J a J a= + − (26)


125

Les autres étapes de l'algorithme f-G-LIA sont les mêmes que l'algorithme G-LIA. Évidemment quand

( )0 0J a = , l'algorithme f-G-LIA et l'algorithme G-LIA sont équivalents.

Cependant, deux problèmes peuvent généralement augmenter la complexité du processus de

démodulation. Le premier est la phase initiale non nulle de la modulation de phase, qui se produit lorsque

les modulateurs (comme les actionneurs piézo-électriques) et l’acquisition ne sont pas synchronisés. Le

second est lié à l’existence d’une certaine modulation d'intensité due à la modulation de phase. Ce

problème se produit souvent lorsque la modulation de phase est assurée par une modulation de la

longueur d'onde de la source laser (comme un VCSEL). En tenant compte de ces deux problèmes, le

signal d'interférence d'un interféromètre à modulation de phase sinusoïdale devient:

( ) ( )( ) ( )2 2 2 cos sin 1 sinr s r s sI t A A A A a t a t = + + + − + + (27)

Où θ est la phase initiale de modulation, μ est la profondeur de modulation d’amplitude dont la valeur

dépends directement de la profondeur de modulation en phase a désirée et des caractéristiques du laser.

En utilisant l'expansion de Jacobi-Anger et les identités trigonométriques, nous pouvons analyser le

signal I(t) dans le domaine fréquentiel:

( ) ( )( ) ( )

( ) ( )( ) ( )( )

( )

( )

2 2

1

2,4,... 3,5

0

,...

0,2,...

1,3,...

2 cos sin 1 sin

2 sin cos sin

cos sin

2

sin cos cos s

cos i

n

s n

i

r s r

m

s s

r s m

m

r s

m

m

m

m

m

I t A A A A a t a t

A A R t m t R m t

m t m t

A A

R m m t m m t

R R

R m m

+ +

= =

+

=

+

=

= + + + − + +

= + + + + +

−

=

+

+ +

(28)

où

( ) ( )

( ) ( ) ( )( )

( ) ( ) ( )( )( ) ( ) ( )( )

2 2

0 1

2 2

1 0 2

1 1

1 1

cos sin2

2 sin cos , 12

2 c

, 0

, 2,4,...

, 3,5,..

n

.

os si

2 sin cos

r ss s

r s

r ss s

r s

m s m m s

m s s

m

m m

m

R

A AJ a aJ a

A A

A Aa J a a J a J a m

A A

J a a J a J ma

J a a J ma J a

+ −

− +

+ =

= + =

+ = −

++

++ + − =

−

(29)

Pour obtenir s de ce signal, nous pouvons utiliser un algorithme SPM modifié. Les définitions de

C(t) et S(t) restent les mêmes que l'algorithme SPM. En outre, nous définissons les coefficients suivants:


126

( )2 2

0

1cos

2r sQ aT A A = + (30)

( )1 12 cosr sQ TA A J a= (31)

( ) ( )( )2 0 2cosr sQ aTA A J a J a = − (32)

( ) ( )3 2 cos 2r s mTA A J aQ = (33)

( ) ( ) ( )( )14 1cos 2r s m maTA A J a J aQ + −−= (34)

Puis nous pouvons obtenir:

4 1 0 4 3 2 0 3

2 4 1 3 1 3 2 4

args

Q Y Q X Q Q Q Y Q X Q Qi

Q Q Q Q Q Q Q Q

− − − −= +

− − (35)

Quand est le seul paramètre inconnu, nous pouvons faire usage du signal à la fréquence de pour

obtenir sa valeur. Pour cela, nous pouvons utiliser:

( ) ( )cosC t t= (36)

( ) ( )sinS t t= (37)

Puis selon l’Eq. (28), nous avons:

( ) ( ) 1

0

sin

T

r sX I t C t dt TA A R = = (38)

( ) ( ) 1

0

cos

T

r sY I t S t dt TA A R = = (39)

Tant que R1≠0 nous avons:

( )arg Y iX = + (40)

Nous pouvons également utiliser l’algorithme f-G-Lia modifié pour obtenir s du signal ( )I t

décrit par Eq. (27). Les définitions de C(t) et S(t) restent les mêmes que l’algorithme f-G-Lia. En outre,

nous définissons les coefficients suivants:


127

( ) ( )2 2

0 1cosr sQ T A A a J a = + (41)

1 0 02 sin 2 cos2 2

r sQ TA A J a J a

= −

(42)

2 1 12 cos 22

cos si sin2 2

n2

r sQ TA A a J a J a

= −

(43)

( )2

0 0 03 2 sin 2 cos 22 2

r sQ TA A J a J a J a

+ −

=

(44)

( ) ( )14 1 0 12 cos 2s si in 22 2 2 2

cos nr sQ TA A a J a J a J a J a

+ − =

(45)

Puis nous pouvons obtenir:

4 1 0 4 3 2 0 3

2 4 1 3 1 3 2 4

args

Q Y Q X Q Q Q Y Q X Q Qi

Q Q Q Q Q Q Q Q

− − − −= +

− − (46)

Dans la pratique, lorsque la synchronisation est garantie de sorte que θ= 0, a est défini pour être 3,8317

rad, et alors J1(a) = 0, Dans le cas où μ est relativement faible, l'algorithme f-G-Lia peut être utilisé

sans modification pour obtenir s .

Un algorithme IBA (integrating bucket algoritm) modifié est également proposé pour obtenir s dans

ce cas. Tout comme l’algorithme traditionnel IBA, il prend en compte le temps d’intégration inhérent à

la mesure optique des éléments photosensibles pour faire les opérations d'intégration, ce qui peut

grandement améliorer la vitesse de traitement des données. Cependant, l'algorithme traditionnel IBA ne

traite pas du problème de modulation d'intensité. Ici, un IBA modifié est proposé pour prendre un compte

le problème de modulation d'intensité.

Nous définissons 1U , 2U , 3U et 4U comme suit:

( )21

0U I t dt

= (47)

( )2

2

U I t dt

= (48)


128

( )3

23U I t dt

= (49)

( )2

34

2

U I t dt

= (50)

Ce sont des valeurs qui peuvent être directement mesurées dans le cas de mesures sur caméra où un

temps d’intégration est défini.

Si on pose:

1 2 3 4

12 r s

U U U UP

A A

− + + −= (51)

1 2 3 4

22 r s

U U U UP

A A

− + − += (52)

( )2

0

22 sinr s

r s

QA A

A A

− += (53)

( )( )

1

2

1

1,3,...

18 sin

m

m

m

Q J a mm

++

=

−= (54)

( ) ( )( )( )

1

2

2 1 1

1,3,...

14 sin

m

m m

m

Q J a J a mm

++

− +

=

−= − (55)

( )( ) 2

3

2,4,...

si11

n8

m

m

m

Q J a mm

+

=

− −= (56)

( ) ( )( )( ) 2

4 1 1

2,4,...

1 1in4 s

m

m m

m

Q J a J a mm

+

+ −

=

− −= − (57)

Nous pouvons obtenir (IBA modifié):

4 1 1 2 0 4 3 1 2 2 0 3

2 4 1 3 1 3 2 4

args

Q P Q P Q Q Q P Q P Q Qi

Q Q Q Q Q Q Q Q

− − − −= +

− − (58)


129

2. Holographie / interférométrie holographique numérique

Pour tester les performances de l'algorithme G-LIA en holographie numérique et interférométrie

holographique numérique, nous avons proposé une configuration simple et compacte. Essentiellement,

il s'agit initialement d'un interféromètre de Michelson traditionnel. Comme indiqué sur la Figure R-1,

nous avons remplacé un miroir de l'interféromètre Michelson classique par un échantillon (diffusant),

l'autre miroir par un miroir piézo-contrôlé, et le détecteur ponctuel par un CMOS nu (sans objectif). Ce

dispositif peut être vu comme une configuration d'holographie numérique co-axiale.

Figure R-1 Holographie numérique co-axiale à modulation de phase sinusoïdale compact, bas coût,

sans système d’imagerie à lentille.

Nous avons limité notre étude à la modulation de phase sinusoïdale, Nous notons que pour l’essentiel

des modulateurs du marché, une fonction de modulation sinusoïdale est le choix le plus pratique,

modulateurs acousto-optiques mis à part, mais dont le coût est élevé. Dans notre cas, l'hystérésis du

crystal piézoélectrique n'a pas besoin d'être considérée étant donné la faible modulation appliquée, bien

qu'elle puisse également être incluse dans la fonction de modulation utilisée pour extraire les

informations de phase.

En appliquant la modulation de phase sinusoïdale, l'algorithme (f)-G-LIA peut être comparé à

l'algorithme SPM dans le cas d’holographie numériques et de l'interférométrie holographique

numérique. Nous montrons dans le manuscrit que pour 2.4048a rad= , les algorithmes f-G-LIA et g-

LIA coïncident. Nous avons choisi ici cette profondeur de modulation de phase, pour comparer ces deux

techniques au SPM traditionnels.


130

La modulation de phase sinusoïdale a été obtenue en contrôlant le miroir piezo-actionné avec un

générateur de forme d'onde. La valeur 2.4048a rad= correspond à une amplitude d'oscillation de

miroir d'environ 122nm (avec un laser à état solide rouge, pour lequel 640nm = ), ajustée

initialement à l'aide d'un pont de jauges de déformation fixé à l'actionneur piézoélectrique pour mesurer

la course du miroir. Les incertitudes estimées de notre mesure permettent en réalité de donner la

profondeur de modulation de phase avec deux à trois chiffres seulement ( 2.4a rad au lieu de

2.4048a rad= ).

A partir de la vidéo enregistrée lors de l'oscillation du miroir de référence, la phase du signal

spatialement dispersée est obtenue sur chaque pixel de la caméra à l'aide de l'algorithme G-LIA et de

l'algorithme SPM respectivement dans l'environnement MATLAB. Nous notons que l'amplitude du

signal qui peut être obtenue sans interférométrie, est moins importante que la phase, dans le sens où

l'image de l’échantillon est toujours perceptible à partir de l'information de phase seulement c'est-à-dire

si nous considérons une amplitude constante de signal sur chaque pixel de la caméra.

Une fois que le champ de lumière complexe sur le plan du CMOS est obtenu, le champ de lumière

complexe sur le plan de l'échantillon peut être reconstruit à l'aide de la méthode du spectre angulaire

[88,89]: le spectre angulaire d’ondes planes est déterminé par transformation de Fourier 2D (fft), puis

les ondes planes sont numériquement rétro-propagées (en utilisant MATLAB aussi) du détecteur au plan

d'échantillonnage où la meilleure mise au point est obtenue.

Une cible d'essai de résolution USAF 1951 a été observée pour tester la résolution spatiale de notre

système. Une caméra avec une taille de pixel de 5.2 5.2m m a été utilisé. L’échantillonnage

temporel est réglé à 10 FPS, et la fréquence de modulation a 1 Hz. Avec 10 images (1 période de

modulation), des résultats de haute qualité sont obtenus, et les performances de l’algorithme G-LIA et

SPM ont pu être comparés, comme indiqué dans la figure ci-dessous.


131

Figure R-2 Images holographiques obtenues avec l'algorithme G-LIA et SPM. Colonne de gauche:

images d'intensité lumineuse de la cible de test de résolution. Colonne du milieu: images d'intensité

lumineuse zoomées montrant les trois premiers éléments du cinquième groupe de la cible de test de

résolution. Colonne de droite: images de phase (sur l’intervalle -π à π) de la cible de test de résolution.

Selon la Figure R-2, visuellement les algorithmes G-LIA et SPM donnent presque les mêmes résultats,

et le plus petit motif distingué est le premier élément du cinquième groupe pour tous les deux (comme

indiqué dans la colonne du milieu de la Figure R-2), ce qui signifie que la résolution spatiale de cette

configuration peut atteindre 32 LP/mm (LP: Line pair) pour les algorithmes G-LIA et SPM. Avec notre

taille de pixel (5,2 μm × 5,2 μm), on détermine que 32 LP/mm correspondent à 6 pixels/LP, ce qui est

assez supérieur à la limite maximale de 2 pixels/LP donnée par le théorème d'échantillonnage de Nyquist

– Shannon. Compte tenu de l’ouverture numérique limitée par la taille du CMOS, et des bruits résiduels,

cette résolution est raisonnable. L’ouverture numérique effective, estimée géométriquement à 0.04

conduit en effet à une limite de résolution d’environ 4 pixels selon le critère de Rayleigh.

Pour l'interférométrie holographique numérique (DHI), la même méthode a été utilisée pour obtenir le

champ complexe au niveau du capteur CMOS, et la partie radiative du champ complexe en surface de

l’échantillon est reconstruit.


132

Compte tenu du vecteur de sensibilité de notre système qui est presque perpendiculaire à la surface de

l'échantillon, ces franges représentent principalement le champ de déplacement hors plan. Lorsque

l’échantillon est légèrement tourné hors du plan, le plan reconstruit avec la meilleure visibilité de frange

coïncide également avec la surface de l'échantillon [31].

Dans ces expériences, le système de mesure reste le même (voir Figure R-1), et les vidéos sont

enregistrées à différents états de déformation/déplacement. Un capteur CMOS avec une taille de pixel

carré de 3,63 µm est ici utilisé dont la cadence est réglée à 120 FPS. La fréquence de modulation est

cette fois de 10 Hz. En utilisant 12 images (1 période) pour chaque mesure, des résultats de bonne qualité

sont obtenus.

Figure R-3 Échantillon: une pièce d'un centime d'euro. (a) Photo de la pièce. (b-c) Amplitude et phase

du champ reconstruit en surface de l’échantillon depuis le champ complexe sur le capteur CMOS

déterminé par G-LIA. .

Une pièce d'un centime d'euro est ici utilisée comme échantillon diffusant (voir Figure R-3).

Contrairement à la Figure R-2, l'image de phase n'affiche pas de motifs distinctifs. Ceci indique que la

pièce a une surface complexe et optiquement rugueuse, induisant un speckle caractéristique. Une petite

zone d'intérêt a été sélectionnée en couvrant la partie extérieure de la zone avec une bande noire. La

pièce a été fixée sur un plateau de rotation permettant une rotation hors plan presque pure autour de l'axe

vertical (échelle minimale: 0,04 °). Les plans reconstruits donnant la meilleure visibilité des franges

coïncidaient presque exactement avec la surface de l'échantillon. (Figure R-3, les images ont été

obtenues sur un plan distant de 70mm environ du CMOS, tandis que dans la Figure R-4 cette distance

est de 68mm.)

La Figure R-4 montre les résultats obtenus pour des rotations d'environ 0,020° et 0,040°, obtenues par

algorithme G-LIA et SPM.


133

Figure R-4 Image de ( )2 1

cos s s − obtenus en utilisant des algorithmes G-LIA et SPM pour des

rotations hors plan d'environ 0,020° et 0,040°.

On peut voir de la Figure R-4 que la visibilité des franges est très bonne. La zone couverte est beaucoup

plus bruyante que la zone d'intérêt compte tenu du faible signal en cet endroit. Les fluctuations spatiales

haute fréquences peuvent être supprimées par un filtre spatial passe-bas sans affecter les franges

décrivant la rotation (voir Figure R-6).

Comme montré par la Figure R-4, l'orientation des franges est presque verticale, c'est-à-dire parallèle à

l'axe de rotation. Le nombre de franges que nous obtenons peut-être comparé à l’attente théorique de

0,020°:


134

2 3.63 / 1280 sin 0.02

5.07640

m pixel pixelnum

nm

= (59)

Ce nombre est en accord avec la Figure R-4 montrant environ 5 franges. De même, pour une rotation

d'environ 0,040°, le nombre de franges théoriques est 10,14, tandis que nous pouvons observer autour

de 10,5 franges sur la Figure R-4. Une sensibilité supérieure à 0,004 ° (correspondant à une frange

unique de l'image entière) peut donc être raisonnablement obtenue, ou encore 4.10-4 degré, considérant

qu’une variation d'un dixième de frange est détectable avec des algorithmes d'analyse des franges

appropriés.

Au vu de la Figure R-4, aucune différence entre les résultats des algorithmes G-LIA et SPM ne peut être

observée visuellement. Afin de faire une comparaison plus claire, les profils des franges présentés dans

la colonne de gauche de la Figure R-4 sont indiqués en Figure R-5; ensuite, ces images de franges sont

filtrées pour supprimer le bruit haute fréquences. Les profils en des endroits identiques sont affichés sur

la Figure R-6.

Figure R-5 Profils des franges dans l'image non filtrée de ( )2 1

cos s s − obtenus respectivement par

des algorithmes G-LIA et SPM pour une rotation hors plan d'environ 0,02°.


135

Figure R-6 Profils des franges dans l'image filtrée de ( )2 1

cos s s − obtenus respectivement par des

algorithmes G-Lia et SPM pour une rotation hors plan d'environ 0,02°.

Une analyse plus poussée de la Figure R-5 montre que bien que les valeurs exactes obtenues par les

algorithmes G-LIA et SPM diffèrent, leurs niveaux de bruit sont presque les mêmes. Comme le montre

la Figure R-6, le bruit à haute fréquence peut être supprimé efficacement à l'aide d'un filtre passe-bas

2D, et le même profil lisse a été obtenu à la fois par les algorithmes G-LIA et SPM.

Cependant, lorsqu'un filtre passe-bas simple est appliqué directement à l'image de ( )2 1

cos s s − , une

erreur se produira sur les résultats de 2 1s s − . Comme le montre la Figure R-6, la valeur de

( )2 1

cos s s − n'atteint jamais 1, ce qui n'est pas raisonnable et affectera la précision de la mesure du

nombre de franges. Pour résoudre ce problème, nous pouvons utiliser une méthode de filtrage de phase

itérative portant sur φs1 et φs2. Les résultats sont indiqués sur la Figure R-7.


136

Figure R-7 Profils des franges à l'image de filtrée ( )2 1

cos s s − obtenus respectivement par des

algorithmes G-Lia et SPM pour une rotation hors plan d'environ 0,02 °. Méthode de filtrage de phase

itérative avec filtrage passe-bas a été utilisée, et le nombre d’itérations est 1.

Avec la Figure R-7, on peut voir que le problème de la Figure R-6 est résolu en utilisant cette méthode

de filtrage de phase, et les résultats donnés par G-LIA et SPM sont très similaires.

Pour conclure sur cette partie, nous avons vu qu'en utilisant la modulation de phase sinusoïdale avec

l'algorithme de G-LIA ou l'algorithme de SPM, notre système d'imagerie holographie numérique sans

lentille peut atteindre une résolution spatiale raisonnable (32 LP/mm). En outre, des franges

d'interférométrie holographiques de rotation hors plan peuvent être observées clairement, avec des

franges très visibles et prévisibles, pour des angles de rotation aussi petits que 0,004 °. Il a été montré

que, pour chaque mesure, les données provenant d'une seule période de modulation sinusoïdale suffisent

pour effectuer une analyse correcte, avec un taux d'échantillonnage d'environ 10 images/période. Par

rapport à l'algorithme SPM, Le G-LIA montre une capacité similaire, pour la profondeur de modulation

sélectionnée, à récupérer l’informations de phase en holographie numérique (et en interférométrie

holographique) tout en ayant le potentiel d'utiliser une variété d'autres fonctions de modulation. Cet

avantage peut être très important dans le cas de forte profondeur de modulation de phase ou pour toute

situation où un autre type de fonction de modulation doit être considéré. Par exemple, lorsque le spectre

de fréquence du bruit est connu, un certain type de fonction de modulation peut être choisi en

conséquence afin d'améliorer la capacité anti-bruit.


137

3. Interférométrie de speckle

L'interférométrie électronique/numérique de speckle (ESPI/DSPI) est une méthode bien établie de

détection sans contact. Elle a été largement utilisée pour effectuer des mesures précises de champ de

déplacement. Toutefois, dans une configuration standard à deux faisceaux, une seule composante de

déplacement est mesurée. Cette direction est fixée par l'orientation des faisceaux laser.

Afin de mesurer le champ de déplacement 2D dans le plan (ou le champ de déplacement 3D entier),

plusieurs solutions ont été proposées. Néanmoins, elles ont toutes des défauts: soit la résolution spatiale

est limitée, soit le système est complexe et coûteux, soit la vitesse de mesure est limitée, soit les mesures

de déplacement le long de différents axes ne sont pas simultanées.

Nous proposons ici une nouvelle technique de mesure 2D simultanée, utilisant la technique ESPI

largement reconnue et un seul laser, c’est à dire sans combiner deux systèmes entiers ou de switch

optique. En outre, cette technique pourra être généralisée pour permettre d’effectuer des mesures 3D à

une vitesse relativement élevée moyennant des modifications mineures. Au lieu d'utiliser la technique

traditionnelle de décalage de phase, nous appliquons simultanément deux modulations de phase

continues et une courte vidéo est prise (par exemple 1 seconde ou beaucoup moins selon la vitesse de la

caméra) à chaque état de déformation. Pendant le traitement des données, en sélectionnant les bonnes

fréquences, les déplacements le long de deux directions différentes peuvent être extraits séparément.

La disposition optique proposée est illustrée à la Figure R-8. Il existe trois faisceaux laser cohérents

provenant d'un seul laser, dits Faisceau-1, Faisceau-2 et Faisceau-3. Les phases du Faisceau-1 et du

Faisceau-2 peuvent être modulées par les miroirs piézo-contrôlés correspondants.


138

Figure R-8. Configuration pour la mesure ESPI. (a) Vue de dessus; (b) Vue 3D. La caméra est au-dessus

de l'échantillon pour prendre des images de la surface. La hauteur et le focus de la caméra peuvent être

ajustés pour obtenir des grossissements différents. Le plan optique est un peu au-dessus du plan de

l'échantillon de sorte que la surface peut être éclairée par des faisceaux laser. Laser: CNI MSL-532

(laser pompé par diode à l'état solide, 532nm, 20mW). Caméra: Flea®3 FL3-U3-13S2M-CS 1/3"

monochrome USB 3,0 caméra. CL: lentille concave. CM: miroir concave. BS: séparateur de faisceau.

PZT+M: miroir piezo-contrôlé.

Quand deux fonctions de modulation de phase temporelle, F1(t) et F2(t), sont appliqués aux Faisceau-1

et Faisceau-2 respectivement, le champ scalaire des taches subjectives E(x,y) peut être exprimée comme:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1 1 2 2 32 , 2 , 2 ,

1 2 3, , , ,c c ci f t x y F t i f t x y F t i f t x yE x y A x y e A x y e A x y e

+ + + + + = + + (60)

Avec Am(x,y) et θm(x,y) l'amplitude et la phase initiale du Faisceau-m (m=1,2,3) au point (x,y)

respectivement, et fc est la fréquence optique du laser.


139

Sur la surface de l'échantillon, l’intensité lumineuse I(x,y) peut être exprimée comme:

( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

2

2 2 2

1 2 3

1 2 1 1 2 2

1 3 1 1 3

2 3 2 2 3

, ,

, , ,

2 , , cos , + , -

+2 , , cos , + ,

+2 , , cos , + ,

I x y E x y

A x y A x y A x y

A x y A x y x y F t x y F t

A x y A x y x y F t x y


=

= + +

+ −

−

−

(61)

Après un petit déplacement u(x,y), l'intensité lumineuse se transforme en:

( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

2

2 2 2

1 2 3

1 2 1 1 2 2

31 3 1 1

2 3 32 2

, ,

, , ,

2 , , cos , + , -

+2 , , cos , + ,

+2 , , cos , + ,

I x y E x y

A x y A x y A x y

A x y A x y x y F t x y F t



=

= + +

+ −

−

−

(62)

avec

( ) ( ) ( ) ( )1 1 1

2, , ,x y x y x y

= + − sn n u (63)

( ) ( ) ( ) ( )2 2 2

2, , ,sx y x y x y

= + −n n u (64)

( ) ( ) ( ) ( )3 3 3

2, , ,sx y x y x y

= + −n n u (65)

où λ est la longueur d'onde du laser, nm est le vecteur unité le long de la direction d'illumination du

Faisceau-m (m=1,2,3), ns est le vecteur unité le long de la direction de collection. nm et ns peuvent être

grossièrement considérés comme identiques pour chaque point (x,y) de la surface de l'échantillon.

Si nous choisissons les fonctions de modulation linéaires (ou dent de scie) suivantes:

( )1 12F t f t= (66)

( )2 22F t f t= (67)

Nous avons :


140

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

2 2 2

1 2 3

1 2 1 1 2 2

1 3 1 1 3

2 3 2 2 3

, , , ,

2 , , cos , +2 , -2

+2 , , cos , +2 ,

+2 , , cos , +2 ,



A x y A x y x y f t x y

A x y A x y x y f t x y

= + +

+ −

−

−

(68)

Avec l’algorithme LIA fonctionnant à la fréquence f1, il est clair que quand f1, f2 et |f1-f2| ne sont pas

égaux les uns aux autres, l’information de phase θ1(x,y)-θ3(x,y) peut être extraites. Avec l’algorithm LIA

fonctionnant à f2, θ2(x,y)-θ3(x,y) peut aussi être extrait [28]. La même procédure peut être effectuée pour

obtenir θ1'(x,y)-θ3'(x,y) et θ2'(x,y)-θ3'(x,y). Si nous posons:

( ) ( ) ( ) ( ) ( )1 1 3 1 3, , , , ,C x y x y x y x y x y = − − − (69)

( ) ( ) ( ) ( ) ( )2 2 3 2 3, , , , ,C x y x y x y x y x y = − − − (70)

Alors, selon les Eq. (63)-(65), nous avons:

( ) ( ) ( )1

2, ,C x y x y

= −1 3n n u (71)

( ) ( ) ( )2

2, ,C x y x y

= −2 3n n u (72)

Les composantes z de n1, n2 et n3 sont presque égales, de sorte qu'elles s'annulent les uns les autres dans

les Eq. (71)-(72). Concernant les composantes x et y de n1, n2 et n3, on peut voir à partir de la Figure

R-8 que n1-n3 est parallèle à l’axe Y, et que n2-n3 est parallèle à l’axe X. Donc C1(x,y) et C2(x,y) peuvent

être exprimées comme:

( ) ( )1 , ,y yC x g uy x= (73)

( ) ( )2 , ,xC x y g u x y= (74)

où g est une constante mesurable, ux(x,y) et uy(x,y) sont les composantes x et y de u(x,y), respectivement.

Cela signifie que le champ de déplacement 2D dans le plan peut être mesuré. Il devrait être remarqué

que lorsque les actionneurs piézoélectriques sont poussés à faire des déplacements en dents de scie, la

précision ne peut être garantie, surtout à haute fréquence, où le temps de retour du miroir ne peut pas

être négligé. La non-linéarité et le bruit générés par le retour soudain deviennent inacceptable lorsqu’une

mesure à grande vitesse est nécessaire. Ce problème est largement diminué avec des modulations de

phase sinusoïdales telles que:


141

( )1 1sin 2F t a f t= (75)

( )2 2sin 2F t a f t= (76)

Où a est l'amplitude de la modulation de phase. Il faut remarquer que f1 et f2 ne sont pas choisies au

hasard. Il est favorable de choisir des entiers premiers entre eux, comme il sera détaillé plus tard.

Maintenant, nous avons:

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

2 2 2

1 2 3

1 2 1 2 1 2

1 3 1 3 1

2 3 2 3 2

, , , ,

2 , , cos , , sin 2 sin 2

+2 , , cos , , sin 2

+2 , , cos , , sin 2


A x y A x y x y x y a f t a f t



= + +

+ − + −

− +

− +

(77)

Dans l’Eq. (77), selon l'expansion de Jacobi-Anger, dans le domaine de la fréquence, le contenu

fréquentiel du troisième terme (représentant l'interférence entre le Faisceau-1 et le Faisceau-3) est

distribué sur des multiples entiers de f1. Le quatrième terme (représentant l'interférence entre Faisceau-

2 et Faisceau-3) est lui distribué sur des composantes harmoniques de f2. Si a=2.4048rad pour que

J0(a)=0 (J0 est la 0ème fonction de Bessel du premier type), alors les deux termes ne contiendront aucun

signal à 0Hz. Ainsi, dans le domaine de la fréquence, ils ne se chevauchent pas les uns avec les autres

jusqu'au moindre multiple commun de f1 et f2, qui est 63Hz dans notre cas (f1= 9Hz, f2= 7Hz, 9 et 7 sont

premiers entre eux). Cela signifie que ces deux termes peuvent être efficacement distingués.

Le second terme dans l’Eq. (77) représente l'interférence entre le Faisceau-1 et le Faisceau-2. Nous

pouvons utiliser les formules trigonométriques (identités de somme/différence) et le développement de

Jacobi-Anger pour analyser ce terme, et tirer comme conclusion que lorsque J0(a)=0, ce terme contient

composantes aux fréquences |pf1±qf2|. Puisque p et q sont des entiers positifs, les solutions pour

pf1±qf2=nf1 ou pf1±qf2=nf2 (n étant un entier positif) ne peuvent être trouvées que lorsque p ou q sont

relativement grands (quand f1=9Hz, f2=7Hz, au moins un d'entre eux est plus grand que 6), où nous

avons Jp(a) ou Jq(a) proches de zéro. Nous pouvons donc estimer qu'il n'aura pas beaucoup d'influence

sur les fréquences intéressantes (pf1 et qf2).

Nous pouvons maintenant tracer un lien direct entre les quatre termes de l’Eq. (77) et le spectre des

fréquences. Le premier terme correspond au signal continu (fréquence nulle), le troisième terme

correspond aux signaux pf1, le quatrième terme correspond aux signaux qf2. Quant au second terme il

contient des signaux à d'autres fréquences. Ainsi, l'information utile peut être facilement obtenue afin

de mesurer le champ de déplacement 2D, par SPM ou G-LIA.


142

Ainsi, si nous définissons la fréquence de démodulation à f1 dans l'algorithmede démodulation, alors

nous pouvons obtenir C1(x,y). Si celle-ci est fixé à f2, alors nous pouvons obtenir C2(x,y). Avec C1(x,y)

et C2(x,y), le champ de déplacement 2D ux(x,y) et uy(x,y) peuvent être obtenus.

Après avoir eu les valeurs de ux(x,y) et uy(x,y), Nous pouvons calculer les composantes normale de

déformation εy(x,y), εx(x,y) et la déformation de cisaillement γxy(x,y):

x

x

u

x

=

(78)

y

y

u

y

=

(79)

y x

xy

u u

x y

= +

(80)

Cette méthode peut également être étendue pour effectuer la mesure de champ de déplacement 3D sans

augmenter le temps d'acquisition, et sans caméra supplémentaire. Nous pouvons simplement séparer le

laser en un quatrième faisceau cohérent (Faisceau-4) afin qu’il éclaire également la surface de

l'échantillon et interfère avec le Faisceau-1, le Faisceau-2 et le Faisceau-3. Si la direction d’incidence

du faisceau-4 et les fonctions de modulation de phase pour les Faisceau-1, Faisceau-2 et Faisceau-4 sont

bien choisies, la mesure de champ de déplacement 3D pourra être réalisée.

Dans notre expérience, un spécimen fabriqué par la société Holo 3 [52] est sollicité avec une vis

micrométrique. Pour différente compressions, les différents états de déformation peuvent être obtenus.

Tout d'abord, nous nous assurons qu'il y a déjà un contact initial entre la vis micrométrique et

l'échantillon. Ensuite, la double modulation est produite avec un même générateur de formes d'ondes

contrôlant la position des deux miroirs servant à moduler la phase. Une courte vidéo (1 seconde, 63

images par seconde) est enregistrée par le logiciel FlyCapture. De même, nous enregistrons une autre

vidéo après avoir tourné la vis du micromètre de sorte que l'état de déformation change. Nous pouvons

répéter cette procédure plusieurs fois pour enregistrer une vidéo à chaque état de déformation. (Il est

préférable de tourner la vis le long de la même direction sans revenir en arrière pendant les mesures pour

éviter l'erreur d'hystérésis.) En analysant deux vidéos à différents états de déformation, nous pouvons

mesurer le champ de déplacement 2D.

Par application des modulations de phase sinusoïdales décrites par les Eq. (75)-(76) avec f1=9Hz et

f2=7Hz, nous obtenons avec succès des images de phase (C1 et C2), comme illustré sur la Figure R-9. La

visibilité des franges est très bonne. En outre, des franges très fines peuvent être observées sur la partie

gauche de la Figure R-9(a,b).


143

Figure R-9 Images de phase (sans filtrage) montrant les champs de déformation le long l’axe Y et l’axe

X obtenus avec des modulations de phase sinusoïdales. Une différence de phase de 2π représente une

différence de déplacement d'environ 385nm. La vis micrométrique avance de 10μm et 50μm

respectivement le long de l’axe Y. L’algorithme G-LIA [28,47,22] est utilisé pour traiter les données.

Les images de phase obtenues (Figure R-9), nous pouvons mesurer quantitativement la déformation 2D

(Figure R-10). Tout d'abord, les images de phase originales (Figure R-10(a, b)) ont été filtrées avec une

méthode de convolution 2D conventionnelle (voir Figure R-10(c,d)) [95,96]. Ensuite, l’image de phase

2D est déroulée [106,107] pour obtenir des images de phase lisses, et les déplacements uy et ux (Figure

R-10(e,f)) peuvent être calculés par les Eq. (73)-(74). Par les Eq. (78)-(80), les déformations εy, εx et γxy

peuvent être mesurées quantitativement (Figure R-10(g,h,i)) pour tout choix d'origine de uy et ux.


144

Figure R-10 Images de phase et champ de déformation 2D. (a,b) Images de phase non filtrées (nous

avons pris les parties centrales de la Figure R-9(c,d) à titre d'exemple). (c,d) Images de phase filtrées.

(e,f) Déplacements uy et ux. (g,h) Déformations normales εy et εx. (i) Déformation de cisaillement γxy.

Lors de l'application de modulations de phase linéaire, ou dent de scie, décrites par les Eq. (66)-(67),

des franges similaires sont obtenues, puisque les fréquences de modulation sont assez basses (f1=9Hz et

f2=7Hz). Cependant, l'approche « dent de scie » deviendra beaucoup moins pertinente et efficace à une

vitesse plus élevée. Quelques petites différences dans le motif des franges sont visibles, principalement

dues au bruit de phase, à l'ajustement initial de phase et au fait que les processus de chargement sont

effectués manuellement et ne sont pas parfaitement reproductibles.

En conclusion pour cette partie, nous avons réalisé des mesures de champ de déplacement dans le plan

(2D), avec une méthode instrumentale plus simple que l’existant reposant sur un laser et une caméra,

permettant d’obtenir des franges de haute qualité. Nous avons utilisé des modulations de phase linéaire

/ en dents de scie et sinusoïdales à des fréquences relativement basses (typiquement f<50Hz). Il convient

de noter qu'aux fréquences élevées, il est préférable d'utiliser des modulations en phase sinusoïdale pour

garantir la précision des mouvements piézo-électriques. Une caméra de fréquence d’acquisition modérée


145

(63 images par seconde) est utilisée, mais le temps d'acquisition des données (1 seconde pour les

informations 2D) est néanmoins avantageux par rapport à certains systèmes commercialisés (par

exemple 3,5 secondes pour les informations 3D [55]). Nous pouvons également souligner le potentiel

de cette technique. En effet, en utilisant une modulation de phase sinusoïdale et une caméra rapide,

l’approche proposée pourra fonctionner à grande vitesse tout en fournissant des résultats quantitatifs

précis sur les champs de déplacement et déformation. Enfin, cette approche a le potentiel d'effectuer la

mesure simultanément d’ESPI du champ de déplacement 3D.

4. Détections avec résonance plasmonique de surface (SPR)

Les dispositifs dits SPR sont basés sur le phénomène bien connu de Résonance Plasmonique de Surface

(SPR), qui apparait lorsqu’une couche mince métallique d’épaisseur contrôlée est excitée en polarisation

p depuis son substrat, à un angle spécifique θk, (angle de Kretschmann), au-delà de l’angle critique. Dans

cette condition d’excitation, les SPR sont très sensibles au changement d'indice de réfraction dans le

voisinage immédiat de la couche métallique, et sont utilisés pour la détection et le suivi (cinétique)

d’interactions moléculaires entre des cibles à détecter et des sondes greffées sur la surface métallique.

Une détection SPR en lumière monochromatique peut être effectuée dans des conditions de mesures

distinctes: (a) le schéma d'interrogation angulaire (angle d’incidence variable) où l’on détecte les

variations de θk induite par les cibles, (b) le schéma de détection en intensité pure, où l’on mesure aussi

les changements d’intensité réfléchie induits par la cible, mais pour un angle fixe au voisinage de θk, ou

encore (c) une détection en phase de l’onde lumineuse réfléchie par la couche mince fonctionnalisée (à

angle fixe ou variable autours de θk). Dans tous ces schémas, la capacité à mesurer un faible changement

d'indice de réfraction dans le milieu sondé est un indicateur de performance de base.

Pour des métaux nobles comme l’Ag et l’Au et des épaisseurs bien précises (typiquement autour de

47nm pour de l’or dans le rouge ou proche infrarouge), la résonance plasmonique à θk se caractérise par

une chute très marquée de la réflexion (R<1%) et une réponse en phase extrêmement abrupte (e.g. une

variation de phase de plusieurs radians pour une variation d’angle incident de 1 milli degré). Dans ces

conditions, l’interrogation en phase est considérée comme la méthode d’interrogation la plus sensible

aux changements d’indice, au détriment malgré tout de la gamme dynamique qui est souvent limitée, à

moins que cette interrogation ne soit couplée à une mesure angulaire permettant de suivre les variations

de θk.

L'imagerie SPR (SPRi) et l'imagerie LSPR (LSPRi : localized SPRi) font l'objet de recherches intensives

ces dernières années [58,119,120,121]. En remplaçant le détecteur ponctuel par une matrice de détecteur

2D (par exemple CCD, CMOS), plusieurs zones de la surface métalliques peuvent être analysées,

augmentant ainsi le débit des mesures. En effet, en fonctionnalisant différentes zones avec différentes


146

sondes (comme des anticorps, ou des brins d’ADN), des interactions moléculaires distinctes peuvent

être détectées en même temps. Dans ce cas, la puce SPR composée typiquement d’une couche d’or

fonctionnalisée et d’un substrat transparent, comporte souvent un circuit microfluidique approprié pour

convoyer le ou les fluides à analyser.

Dans nos expériences, nous nous concentrons sur un capteur SPR en utilisant la détection de phase,

mode d’interrogation qui demeure largement sous exploité essentiellement en raison de la difficulté

d’effectuer une mesure interférométrique non bruitée dans un environnement standard. Une modulation

de phase sinusoïdale est appliquée et testée dans une configuration à chemin commun (common path).

Pour créer la modulation de phase, un laser à émission par la surface à cavité verticale (VCSEL) est

modulé en courant via la modulation d’une tension d'entrée. Il en résulte une variation légère de longueur

d’onde, suffisante pour induire une modulation de phase notable entre le faisceau signal et le faisceau

de référence dès lors que ceux-ci ne voient pas la même longueur de chemin optique (interféromètre

déséquilibré). Pour créer ce déséquilibre nous utilisons le même chemin géométrique pour les deux

faisceaux mais avec des indices différents selon les deux polarisations (milieu anisotrope). La lumière

polarisée s qui ne voit pas la résonance SPR mais qui suit le même chemin sert alors de référence. Une

caméra CMOS est utilisée comme détecteur placé après un polariseur, nécessaire pour que les deux

faisceaux (s et p) aux polarisations orthogonales interfèrent. Ce principe de fonctionnement dans son

ensemble, non encore publié, a été récemment développé suite à des travaux amorcés dans notre

laboratoire, notamment durant le séjour postdoctoral de Tzu-Heng Wu et par l’apport de Julien Vaillant

avec lesquels nous travaillons pour apporter une résolution spatiale au système. Dans le cadre de ce

travail, nous avons réalisé les dispositifs simplifiés de la Figure R-11 et de la Figure R-13 pour tester

différents algorithmes d’extraction.

Comme pour d’autres lasers spectralement monomodes, un VCSEL, autour de son point de

fonctionnement optimal, présente une accordabilité spectrale (nm/mA) relativement constante.

Cependant, lorsqu'une modulation de tension est appliquée au VCSEL, sa puissance change également,

ce qui peut être décrit par:

( ) ( )( ) ( )2 2 2 cos 1r s r s r s rI t A A A A t t = + + − + (81)

Où μ est la profondeur de modulation en intensité induite par la modulation de phase (dûe à la

modulation en courant). Pour extraire les informations de phase en utilisant une modulation de phase

sinusoïdale, trois algorithmes peuvent notamment être utilisés :

Le premier est l'algorithme SPM modifié. Cependant, le coefficient μ représentant la modulation

d'intensité doit être connu. Il peut être facilement mesuré en appliquant une modulation de phase linéaire

lente et mesurer la puissance / l'intensité de sortie directe. Quelques tests simples sont effectués pour

prouver la faisabilité de cette méthode.


147

Le second est l'algorithme f-G-LIA modifié. Dans nos expériences, μ est relativement petit et l'amplitude

de modulation de phase a=3.8317rad peut être atteinte. Cet algorithme peut donc être simplifié en

algorithme f-G-LIA. Cette méthode est utilisée pour effectuer des détections du changement d'indice de

réfraction.

Le troisième est l'algorithme IBA modifié (Integrating Bucket Algorithm). Cette méthode nécessite un

contrôle précis du temps d'exposition, et n’a pas été abordé dans le cadre de cette thèse.

Avant d'utiliser la caméra CMOS comme détecteur, nous testons la faisabilité des algorithmes SPM et

f-G-LIA modifiés en utilisant un photodétecteur ponctuel. L’interféromètre à chemin géométrique

commun, particulièrement stable, est représenté sur la Figure R-11.

Figure R-11 Configuration expérimentale réalisé pour tester les algorithmes du SPR résolu en phase.

La polarisation initiale du laser est d'environ 45° par rapport à la verticale (à 90°). La lumière traverse

un cristal biréfringent (YVO4), ce qui conduit à une différence de chemin optique entre les composantes

de polarisation 0° et 90° du laser. Cette différence de chemin optique peut être vue comme constante.

Ainsi, la méthode de modulation de phase peut être réalisée en modulant le VCSEL en courant. Un

polariseur dont l’axe est également tourné de 45° est placé après le cristal YVO4 pour que les

composantes de polarisation selon les directions 0° et 90° interfèrent l’une avec l’autre. Ensuite, le signal

d'interférence est capturé par le photodétecteur ponctuel. Le logiciel LabVIEW est utilisé pour contrôler

la tension d'entrée du VCSEL, afin d’obtenir le signal du photodétecteur et pour traiter les données, à

l'aide d'une carte NI.

Dans cette configuration, une rampe lente, triangulaire, de tension est ajoutée à la modulation

sinusoïdale, provoquant un changement de phase de forme identique. En effet, si la fréquence de ce

signal triangulaire est beaucoup plus petite que la modulation de phase sinusoïdale, alors pour chaque


148

mesure de phase, la variation de phase due à la rampe n’induit qu’un décalage constant entre chaque

point.

Des tests simples sont effectués avec des algorithmes SPM modifiés / f-G-LIA. Les résultats sont

montrés dans la figure ci-dessous:

Figure R-12 Mesure de changements de phase triangulaires dans l'interféromètre à chemin commun

avec différents algorithmes. (a) Avec l'algorithme SPM modifié; (b) Avec l'algorithme f-G-LIA.

A partir de la figure ci-dessus, nous pouvons voir que ces deux algorithmes donnent tous les deux des

résultats assez précis. Le léger retard observé entre la modulation exercée et la phase mesuré est lié à la

transmission et au traitement des données qui repose sur une intégration. Le temps d’intégration peut

être réduit dès lors que des fréquences de modulation élevées sont employées, ce qui ne pose pas de

problème particulier ici.

Le capteur SPR réalisé est montré dans la figure ci-après:


149

Figure R-13 Capteur SPR interférométrique avec détecteur CMOS. La couche d'or est déposée sur une

lame de verre. Le côté d'or est enfermé dans un chambre micro-fluidique, tandis que l'autre côté est

couplé à un prisme de verre avec via une huile d'indice.

Sur cette Figure R-13, le VCSEL et le crystal anisotrope sont les mêmes que sur la Figure R-11. Le

faisceau laser est collimaté avec une petite lentille fixée au VCSEL. Un prisme rectangulaire est utilisé

ici pour atteindre l'angle d'incidence requis (environ 67°) jusqu'à la couche d'or où la résonance SPR

peut être excitée. Une modulation de phase sinusoïdale est créée entre la composante à 0° (lumière p-

polarisée) et la composante à 90° (lumière s-polarisée). Nous considérons ici que ces deux composantes

de la lumière conservent leur caractère p et s à la réflexion, mais que la polarisation p subit une

atténuation et un déphasage caractéristique de la résonance. Ainsi, lorsque des liquides ayant des indices

de réfraction différents traversent le canal micro fluidique, la lumière p-polarisée réfléchie présente un

changement de phase ainsi qu'un changement d'intensité, tandis que la lumière réfléchie s-polarisée reste

presque la même [122,67,123]. Ensuite, le faisceau laser traverse le polariseur à 45°, et la lumière p-

polarisée et la lumière s-polarisée interfèrent l'une avec l'autre. L'expression de ce signal d'interférence

peut être décrite par l’Eq. (81), où la lumière p-polarisée peut être considérée comme le signal, et la

lumière s-polarisée peut être considérée comme la référence.

Pour cette configuration de test, la couche d'or n'est pas fonctionnalisée. Ainsi, si la couche d'or est

parfaitement plate et homogène en épaisseur, alors théoriquement chaque point de la surface doit avoir

la même réponse au changement d'indice de réfraction. Par conséquent, nous avons utilisé la valeur

d'intensité moyenne d'une matrice de pixels dans une zone choisie comme le signal I(t), et appliquons


150

l'algorithme f-G-LIA pour obtenir la phase args

X Yi

M N

= +

et l'amplitude relative

2 2

s

X YA

M N

+

. Les résultats sont affichés dans la figure ci-dessous.

Figure R-14 Détection de phase et d'amplitude du changement d'indice de réfraction à l'aide du SPR à

détecteur CMOS. Des solutions de glucose avec différentes concentrations (5,5%, 5%, 3%, 2%, 1%) et

de l'eau de désionisée traverse le canal microfluidique séquentiellement.

On peut voir sur la Figure R-14 que la phase est nettement plus sensible que l'amplitude. Néanmoins, le

rapport signal / bruit n'est pas très bon et les contrastes entre chaque concentration sont assez faible.

Ceci est principalement dû à la mauvaise qualité de la couche d'or utilisée ici. En outre, une dérive de

phase évidente est observée. Cette dérive est elle-même causée par la dérive de la longueur d'onde du

VCSEL non asservit en température et les variations d’indices de réfraction des fluides en raison de leur

coefficients thermo-réfractifs non nul. Ces dérives sont ainsi principalement le fait des fluctuations de

température ambiante [124].

Pour annuler cette dérive, de nombreuses méthodes sont applicables.

La première méthode consiste à utiliser le CMOS: si la zone d'intérêt contient à la fois la zone

fonctionnalisée et la zone non-fonctionnalisée, alors on peut calculer respectivement les phases de ces

deux zones et en faire la soustraction.


151

La seconde méthode consiste à compenser uniquement la dérive de longueur d'onde du VCSEL: en

insérant un séparateur de faisceau entre YVO4 et le prisme de la Figure R-13, on peut mesurer la

différence de phase initiale entre la lumière p-polarisée et s-polarisée avant qu’elle ne soit réfléchie par

la couche d'or. Nous pouvons alors soustraire cette phase initiale de la phase mesurée en temps réel.

La Figure R-15 montre les résultats d'une expérience réalisée dans notre groupe sur un prototype plus

avancé (interrogation interférométrique et angulaire) pour démontrer l’efficacité de cette méthode. Dans

ce cas, une puce SPR commerciale est couplée sur un prisme à 45°, tandis que le procédé de modulation

de phase et le procédé de récupération de phase restent les mêmes.

Figure R-15 Capteur SPR à interrogation angulaire avec détection de phase et d'intensité. La

modulation de phase sinusoïdale est effectuée par modulation de longueur d'onde VCSEL. L'algorithme

f-G-LIA est utilisé pour extraire les informations de phase. Une puce SPR commerciale est utilisée sans

aucune microfluidique (dans l'air). (a) Réponse de phase au changement d'angle. (b) Réponse d'intensité

au changement d'angle.

Comme le montre la Figure R-15(b), à l'angle SPR (environ 44,3°), l'intensité est presque nulle, ce qui

implique que la qualité de cette puce SPR est excellente pour la mesure de phase. Sur la Figure R-15(a),

nous pouvons voir que la mesure de phase est très stable et qu'il y a une chute de phase quasi verticale

à l'angle SPR, ce qui prouve l'efficacité de notre méthode pour compenser la dérive de longueur d'onde

du VCSEL, ainsi que la faisabilité de notre méthode de détection de phase.

Nous pouvons utiliser la zone autour de l'angle SPR pour effectuer des détections de phase SPR très

sensibles, comme indiqué par la Figure R-15. Si nous fixons l'angle θk au centre de cette zone (angle

SPR), alors un minuscule changement d'indice de réfraction (e.g. 10-7) induit une différence de phase

notable. L’inconvénient est naturellement que l'intensité lumineuse du signal (lumière p-polarisée) est


152

faible dans cette zone, tandis que l'intensité de référence (lumière s-polarisée) demeure forte que

d'habitude, ce qui n’est pas favorable à un rapport signal à bruit excellent. Il est clair que le rapport

signal à bruit déterminera ultimement la sensibilité de cette mesure de phase aux changements d’indices.

Nous pouvons proposer également une troisième méthode originale qui s'inspire de la technique de la

shearographie. Cette méthode à l’avantage de pouvoir fonctionné avec un signal entièrement p-polarisé,

comme représenté sur la figure ci-après:

Figure R-16 Combinaison d'interférométrie de type shearing et d'imagerie SPR.

On considère une puce fonctionnalisée par endroit (il faut au moins une zone de référence). Les

différentes parties du laser sont réfléchies par les différentes zones. Proche de θk, les différences de phase

peuvent être très grandes pour des différences d'intensité relativement plus faibles). Ainsi, à l’aide d’un

interféromètre de shearing comme celui représenté, nous pourrons observer l'intensité de la lumière

interférée aux zones de chevauchement, représentant la différence de phase entre les zones. En outre,

lorsque la température ambiante change, les changements de phase résultants sont presque les mêmes

pour différentes zones, de sorte que l'influence de la fluctuation de la température ambiante peut être

naturellement annulée.

En conclusion pour cette dernière partie plus exploratoire, nous avons étudié la possibilité d’utiliser

l’interférométrie à modulation de phase sinusoïdale dans un capteur SPR. Des expériences préliminaires

ont été conduites pour montrer l’applicabilité des méthodes de démodulation étudiées durant cette thèse.

Un capteur CMOS a été utilisé comme détecteur pour montrer la possibilité de faire de l'imagerie SPR

à l'avenir. Le problème de la phase initiale est résolu en utilisant LabVIEW pour contrôler l'ensemble

du système de mesure. En outre, plusieurs méthodes pour compenser l'influence de la fluctuation de la

température ambiante ont été décrites. Ces résultats ouvrent des perspectives pour le développement de

détecteurs SPR interférométriques précis, compacts, et robustes sans éléments couteux.


153

Conclusions et perspectives

Dans le but d’élargir les possibilités de mesures interférométriques alliant simplicité, stabilité, fiabilité

et précision dans des systèmes simples (peu onéreux), nous avons étudié le signal interférométrique à

modulation de phase théoriquement et expérimentalement. La modulation de phase sinusoïdale a été

particulièrement considérée car elle est facile à réaliser et garantit une bonne précision sur la modulation

de phase, même à haute fréquence.

En particulier, cette modulation de phase sinusoïdale a été appliquée en holographie et en interférométrie

holographique. L'algorithme (f)-G-LIA a été utilisé pour récupérer les informations de phase, et les

résultats obtenus sont similaires à l'algorithme SPM traditionnel, pour des profondeurs de modulation

de phase favorables. En outre, des modulations de phase linéaires (en dents de scie) ou sinusoïdales ont

été mises en œuvre dans une configuration 2D-ESPI originale, permettant de réaliser des mesures

simultanées de champ de déplacement 2D. La modulation de phase sinusoïdale est également appliquée

à un capteur SPR sensible à la phase. Plusieurs méthodes pour compenser la fluctuation de la

température ambiante ont été proposées. La faisabilité de l’approche a été démontré expérimentalement.

Quelques perspectives peuvent être dressées : le problème de la phase initiale a été résolue

mathématiquement et expérimentalement, la plupart des expériences présentées dans cette thèse

n'intégrant pas de synchronisation. Cependant, afin d'améliorer la précision de la mesure et de faire des

mesures automatiques dans le futur, il sera nécessaire de contrôler la synchronisation entre la caméra et

les modulateurs, ou même de contrôler toute la procédure de mesure.

Pour un interféromètre à modulation de phase sinusoïdale classique, les données sont relativement

volumineuses et le traitement des données est relativement complexe, ce qui peut ralentir la vitesse de

mesure lorsqu'une caméra à haute fréquence d’acquisition ou un ordinateur à faible performance est

utilisé. La méthode du IBA (integrating bucket algorithm) peut être utilisée pour améliorer cette

situation.

En outre, nous pouvons nous inspirer des nouvelles "caméra LIA" qui intègre une détection synchrone

sur chaque pixel, et proposer de combiner la modulation de phase sinusoïdale avec le traitement du

signal analogique dans les "détecteurs intelligents". De cette manière, le temps d'acquisition des données

et de traitement des données peut être considérablement raccourci. La synchronisation entre la

modulation de phase et l'acquisition de données peut également être effectuée automatiquement.

Enfin, un enseignement général est qu’il demeurera important de choisir judicieusement le type de

modulateur, le type de modulation et l'algorithme de récupération de phase en fonction de la fréquence

de modulation, de la précision de modulation requise, des caractéristiques du bruit, de la longueur d'onde


154

du laser et du coût souhaité. Ces différents facteurs doivent être considérés et bien équilibrés au sein des

prototypes à réaliser, en fonction de l’application ciblée.

Appendix

155

Appendix

Complete derivation processes of formulae

Eq. (0.30)

( )( ) ( )

( ) ( )

( ) ( )( ) ( )

( )

2 2

0 0

0

0 0

0

0 0 0

0

2

0

0

2 cos cos

2 cos cos

2 cos cos sin cos

2 cos cos

co

s n

s

i

T

r s r s s

T

r s s

T

r s s s

T

r s s

r s s

X A A A A t t dt

A A t t dt

A A t t t dt

A A t dt

TA A

= + + −

= −

=

=

=

+

Eq. (0.31)

( )( ) ( )

( ) ( )

( ) ( )( ) ( )

( )

2 2

0 0

0

0 0

0

0 0 0

0

2

0

0

2 cos sin

2 cos sin

2 cos c nsin

sin

s

os si sin

2 sin

in

T

r s r s s

T

r s s

T

r s s s

T

r s s

r s s

Y A A A A t t dt

A A t t dt

A A t t t dt

A A t dt

TA A

= + + −

= −

=

=

=

+

Appendix

156

Eq. (0.38)

( ) ( )

( )( )( ) ( )

( )( ) ( )

( )( ) ( )( )( ) ( )

( ) ( ) ( )

( ) ( )( )

0

2 2

0

0

0

0 2

1

2 1

1

2 cos sin cos 2

2 cos sin cos 2

2 cos sin cos sin sin sin cos 2

2 cos 2 cos

2

2 sin 2 1 sin

T

T

r s r s s

T

r s s

T

r s s s

n s

n

r s

n s

n

X I t C t dt

A A A A a t t dt

A A a t t dt

A A a t a t t dt

J a J a n t

A A

J a n t

+

=

+

−

=

=

= + + −

= −

= +

+

=

+ −

( )

( ) ( )

( )

0

2

2

0

2

cos 2

2 2cos cos 2

2 cos

T

T

r s s

r s s

t dt

A A J a t dt

TA A J a

=

=

Eq. (0.39)

( ) ( )

( )( )( ) ( )

( )( ) ( )

( )( ) ( )( )( ) ( )

( ) ( ) ( )

( ) ( )( )

0

2 2

0

0

0

0 2

1

2 1

1

2 cos sin sin

2 cos sin sin

2 cos sin cos sin sin sin sin

2 cos 2 cos

2

2 sin 2 1 sin

T

T

r s r s s

T

r s s

T

r s s s

n s

n

r s

n s

n

Y I t S t dt

A A A A a t t dt

A A a t t dt

A A a t a t t dt

J a J a n t

A A

J a n t

+

=

+

−

=

=

= + + −

= −

= +

+

=

+ −

( )

( ) ( )

( )

0

2

1

0

1

sin

2 2sin sin

2 sin

T

T

r s s

r s s

t dt

A A J a t dt

TA A J a

=

=

Appendix

157

Eq. (0.48)

( ) ( )

( )( )( ) ( )( )

( ) ( ) ( ) ( ) ( )( ) ( )( )

( ) ( ) ( )( ) ( )( )

( ) ( )

0

2 2

0

2 2

0 2

10 0

2 2

0

0

2 2

0

2 cos sin cos sin

2 cos 2 2 cos sin cos sin

2 cos sin cos sin

12 cos 2 sin

2

T

T

r s r s s

T T

r s n r s s

n

T

r s r s s

r s r s

X I t C t dt

A A A A a t a t dt

A A J a J a n t dt A A a t a t dt

T A A J a A A a t a t dt

T A A J a A A a

+

=

=

= + + −

= + + + −

= + + −

= + +

( )( ) ( )( )

( ) ( ) ( )( )

( ) ( ) ( )( ) ( )( )( )

( ) ( )

( ) ( ) ( )

0

2 2

0

0

2 2

0

0

2 2

0

0 2 2

1

cos

cos cos 2 sin

cos cos 2 sin cos sin 2 sin sin

cos

2 2 2 cos 2 cos 2

T

s s

T

r s r s s r s s

T

r s r s s r s s s

r s r s s

r s n s

n

t dt

T A A J a TA A A A a t dt

T A A J a TA A A A a t a t dt

T A A J a TA A

A A J a J a n t J

+

=

− + −

= + + −

= + + +

=

+

+

+ +

+ +

+

( ) ( )( )

( ) ( ) ( )

( ) ( ) ( )( )

1

10

2 2

0 0

0

2 2

0 0

2 sin 2 1 sin

cos 2 cos

2 cos1

T

n s

n

T

r s r s s r s s

r s r s s

a n t dt

T A A J a TA A A A J a dt

T A A J a TA A J a

+

−

=

−

= + +

= + +

+

+

Appendix

158

Eq. (0.50)

( ) ( )

( )( )( ) ( )( )

( ) ( ) ( )( ) ( )( ) ( )( )

( )( ) ( )( )

( )( ) ( )( )

0

2 2

0

2 2

2 1

10 0

0

0

2 cos sin sin sin

2 2 sin 2 1 2 cos sin sin sin

2 cos sin sin sin

12 sin 2 sin sin

2

T

T

r s r s s

T T

r s n r s s

n

T

r s s

T

r s s s

r

Y I t S t dt

A A A A a t a t dt

A A J a n t dt A A a t a t dt

A A a t a t dt

A A a t dt

TA A

+

−

=

=

= + + −

= + − + −

= −

= − − −

=

( )( )

( )( ) ( )( )( )

( ) ( )( ) ( ) ( ) ( )

0

0

2 1 0 2

1 10

sin 2 sin

cos sin 2 sin cos 2 sin sin

cos

2 2 sin 2 1 cos 2 2 2 cos 2 sin

cos

sin

cos

T

s s r s s

T

r s s r s s s

r s s

T

r s n s n s

n n

r s s r

A A a t dt

TA A A A a t a t dt

TA A

A A J a n t J a J a n t dt

TA A A A

+ +

−

= =

+

+ −

+ −

−

=

=

− +

+

=

( )( )

( )( )

0

0

0

2 si

si

n

21 n

T

s s

r s s

J a dt

TA A J a

= −

−

Appendix

159

Eq. (0.57)

( ) ( )

( )( ) ( )( ) ( )( )

( )( ) ( )( ) ( ) ( )( )

( )( ) ( )( )

( ) ( ) ( ) ( )

0

0

0

0

0 0

0

0 0 2

1

2 cos sin 2 cos cos sin

2 cos sin cos sin 2 cos cos sin

12 cos 2 sin cos

2

2 cos 2 cos 2

T

T

r s s r s s

T T

r s s r s s

T

r s s s

r s s n

n

X I t C t dt

A A a t A A J a a t dt

A A a t a t dt A A J a a t dt

A A a t dt

A A J a J a J a n t d

+

=

=

= − −

= − −

= − + −

− +

( )( ) ( )

( )( ) ( )( )( ) ( )

( )

( ) ( ) ( )

0

2

0

0

2

0

0

2

0

0 2

1

cos cos 2 sin 2 cos

cos cos 2 sin cos sin 2 sin sin 2 cos

2 cos cos

2 2 2 cos 2 cos 2

T

T

r s s r s s r s s

T

r s s r s s s r s s

r s s r s s

r s n s

n

t

TA A A A a t dt TA A J a

TA A A A a t a t dt TA A J a

TA A J a TA A

A A J a J a n t

+

=

+

+

= − −

= + −

= − +

+ +

+

( ) ( )( )

( ) ( )

( ) ( )( )

2 1

10

2

0 0

0

2

0 0

2 sin 2 1 sin

2 cos cos 2 co

21

s

2 cos

T

n s

n

T

r s s r s s r s s

r s s

J a n t dt

TA A J a TA A A A J a dt

TA A J a J a

+

−

=

−

+

+

= − +

= −

Appendix

160

Eq. (0.58)

( ) ( )

( )( ) ( )( ) ( )( )

( )( ) ( )( ) ( ) ( ) ( )( )

( )( ) ( )( )

( )( )

0

0

0

0 2 1

10 0

0

2 cos sin 2 cos sin sin

2 cos sin sin sin 2 cos 2 2 sin 2 1

2 cos sin sin sin

12 sin 2 sin si

2

T

T

r s s r s s

T T

r s s r s s n

n

T

r s s

r s s

Y I t S t dt

A A a t A A J a a t dt

A A a t a t dt A A J a J a n t dt

A A a t a t dt

A A a t

+

−

=

=

= − −

= − − −

= −

= − −

( )( )

( )( )

( )( ) ( )( )( )

( ) ( )( ) ( ) ( ) ( )

0

0

0

2 1 0 2

1 10

n

sin 2 sin

cos sin 2 sin cos 2 sin sin

cos

2 2 sin 2 1 cos

sin

c

2 2 2 cos 2 sin

os

T

s

T

r s s r s s

T

r s s r s s s

r s s

T

r s n s n s

n n

dt

TA A A A a t dt

TA A A A a t a t dt

TA A

A A J a n t J a J a n t dt

+ +

−

= =

−

= −

=

=

− +

+

+ −

+

=

−

( )( )

( )( )

0

0

0

cos 2 sin

21 sin

T

r s s r s s

r s s

TA A A A J a dt

TA A J a

−+

−=

Eq. (1.20)

( )( )( ) ( )( )

( )( ) ( )( )

( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( )( ) ( ) ( )( )

2 2

0

0

0

0

0 2 0

10

2 0 0

' 2 cos sin cos 2

2 cos sin cos 2

2 2 cos 2 cos cos 2

2 2cos cos 2 cos 2 cos 2 sin 2 sin 2

T

r s r s s g

T

r s s g

T

r s n s g

n

r s s g g

X A A A A a t t t t dt

A A a t t t t dt

A A J a J a n t t t t dt

A A J a t t t t t t t

+

=

= + + + − +

= − + −

= + + −

= − − −

( ) ( )( )( )

0

2 0

2

2 cos 2 cos

2 cos 2 cos

T

r s g s

r s s

dt

TA A J a t t

TA A J a

= −

=

Appendix

161

Eq. (1.21)

( )( )( ) ( )( )

( )( ) ( )( )

( ) ( )( ) ( )( )

( ) ( ) ( ) ( )( ) ( ) ( )( )

2 2

0

0

0

0

2 1 0

10

1 0 0

0

' 2 cos sin sin

2 cos sin sin

2 2 sin 2 1 sin sin

2 2sin sin sin cos cos sin

T

r s r s s g

T

r s s g

T

r s n s g

n

T

r s s g g

Y A A A A a t t t t dt

A A a t t t t dt

A A J a n t t t t dt

A A J a t t t t t t t dt

+

−

=

= + + + − +

= − + −

= − + −

= − + −

( ) ( )( )( )

1 0

1

2 cos sin

2 cos sin

r s g s

r s s

TA A J a t t

TA A J a

= −

=

Appendix

162

Eq. (1.28)

( ) ( )

( )( )( ) ( )( )( )

( ) ( ) ( )

0

2 2

0

0

0 02 2

0

02 2

0 2

' ' '

2 cos sin cos sin

2 cos sin cos sin2 2

2 cos 22

T

T

r s r s s g

Tg g

r s r s s

g

r s n

X I t C t dt

A A A A a t t a t t dt

t t t tA A A A a t a t dt

t tA A J a J a n t

=

= + + + − +

− − = + + + − +

−= + + +

( ) ( )

10

0 0

0

0 0

2 2

0


cos sin sin2 2

cos

T

n

Tg g

r s s

g g

s

r s r s

dt

t t t tA A a t a t dt

t t t ta t a t

T A A J a A A

+

=

− − + + − +

− − + + + −

= + +

+

( ) ( )

( ) ( )

0 0 0

0 02 2

0

0

0

2 2

0

sin sin2 2

cos 2 cos sin cos 2 sin cos2 2

cos 2 cos2

T

g g

s

Tg g

r s r s s s

g

r s r s

dtt t t t

a t a t

t t t tT A A J a A A a t a t dt

t ta

T A A J a A A

− − + − + −

− − = + + − + −

−

= + +

( ) ( )

0

0 0 0

2 2

0

sin cos 2 cos sin sin2

cos 2 sin cos cos 2 sin cos sin2 2

cos 2 c

sin

si

os sin co

n

ss 2 cos sin si nn i2 2

g

s sT

g g

s s

s

r s r s

t tt a t

dtt t t t

a t a t

a t a t

T A A J a A A

−

− − +

+

+

+

= + +

0 cos 2 sin cos cos 2 sinsin cos sin2 2

sT

s s

dt

a t a t

+

+

Appendix

163

Eq. (1.29)

( ) ( )

( )( )( ) ( )( )( )

( ) ( ) ( )

0

2 2

0

0

0 02 2

0

02 2

2 1

' ' '

2 cos sin sin sin

2 cos sin sin sin2 2

2 sin 2 12

T

T

r s r s s g

Tg g

r s r s s

g

r s n

Y I t S t dt

A A A A a t t a t t dt

t t t tA A A A a t a t dt

t tA A J a n t

−

=

= + + + − +

− − = + + + − +

−= + − +

10

0 0

0

0 0

0


sin sin sin2 2

0

sin sin

T

n

Tg g

r s s

g g

s

r s

dt


t t t ta t a t

A At

a t

+

=

− − + + − +

− − + + + −

= +

−− +

0 0

0 0

0

0 0

sin2 2

sin 2 cos sin sin 2 sin cos2 2

sin 2 cos sin cos 2 cos si2 2

cos

T

g g

s

Tg g

r s s s

g g

s

r s

dtt t t

a t


t t t ta t a

A A

− − + −

− − = − − −

− −

=

−

0 0 0

n sin

sin 2 sin cos cos 2 sin cos sin2 2

sin 2 cos sin cos 2 cos sin sin2 2

sin 2 sin cos cos

cos

cos

2 sicos n cos2 2

sT

g g

s s

s s

r s

s

t

dtt t t t

a t a t

a t a t

A A

a t a t

− − −

=

− +

+

−

0 sin

T

s

dt

Appendix

164

Eq. (1.43)

( ) ( )( ) ( )

( )( ) ( )( )( ) ( )

( )( ) ( )( ) ( )

2 2

2 2

2 2

2

2 cos sin 1 sin

2 cos sin cos sin sin sin 1 sin

2 cos sin cos sin sin sin 1 sin2

2

r s r s s

r s r s s s

r sr s s s

r s

r s

r s

I t A A A A a t a t

A A A A a t a t a t

A AA A a t a t a t

A A

A A

A A

= + + + − + +

= + + + + + + +

+= + + + + + +

+

=

( )( )

( )( )

( )( )

( ) ( )( )

( ) ( )( )

( )( )

( ) ( ) ( )( )

2

2 2

0 2

1

1 sin2

cos cos sin

sin sin sin

cos sin cos sin

sin sin sin sin

1 sin2

cos 2 cos 2

2 sin 2

r s

s

s

s

s

r s

r s

s n

n

r s s

a tA A

a t

a t

a t a t

a t a t

A Aa t

A A

J a J a n t

A A J

+

=

+ +

+ + + + + + + + + +

++ +

+ + +

= +

( ) ( )( )( )

( ) ( ) ( ) ( )( )

( ) ( ) ( )( )( )

( ) ( )

2 1

1

0 2

1

2 1

1

2 2 2 2

0 0

sin 2 1

cos sin 2 cos 2

sin sin 2 sin 2 1

cos2 2

2

n

n

s n

n

s n

n

r s r ss

r s r s

r s

a n t

a t J a J a n t

a t J a n t

A A A AJ a J a

A A A A

A A

+

−

=

+

=

+

−

=

− +

+ + + +

+ + − +

+ ++ + +

=

( )

( ) ( )( )

( ) ( )( )( )

( ) ( )( ) ( )

( ) ( )( )( ) ( )

2

1

2 1

1

2

1

2 1

1

cos sin

cos 2 cos 2

sin 2 sin 2 1

cos 2 cos 2 sin

sin 2 sin 2 1 sin

s

s n

n

s n

n

s n

n

s n

n

a t

J a n t

J a n t

a J a n t t

a J a n t t

+

=

+

−

=

+

=

+

−

=

+

+ +

+ − +

+ + +

+ − + +

Appendix

165

( ) ( ) ( )

( ) ( )( )

( ) ( )( )( )

( ) ( )( )( ) ( )( )( )( )

( )

2 2 2 2

0 0

2

1

2 1

1

2

1

2 1

cos cos sin2 2

cos 2 cos 2

2 sin 2 sin 2 1

cos sin 2 1 sin 2 1

sin c

r s r ss s

r s r s

s n

n

r s s n

n

s n

n

s n

A A A AJ a J a a t

A A A A

J a n t

A A J a n t

a J a n t n t

a J a

+

=

+

−

=

+

=

−

+ ++ + + +

+ +

= + − +

+ + + − − +

+

( )( )( ) ( )( )( )

( ) ( ) ( )

( ) ( )( )

( ) ( )( )

1

2 2 2 2

0 0

2,4,...

1,3,...

os 2 2 cos 2

cos cos sin2 2

cos 2 cos

2 sin 2 sin

n

r s r ss s

r s r s

s m

m

r s s m

m

n t n t

A A A AJ a J a a t

A A A A

J a m t

A A J a m t

+

=

+

=

+

=

− + − +

+ ++ + + +

+ +

= + +

( ) ( )( ) ( ) ( )( )

( ) ( )( ) ( ) ( )( )

( )

1 1

3,5,... 1,3,...

1 1

0,2,... 2,4,...

2 2

0

cos sin sin

sin cos cos

cos2

2

s m m

m m

s m m

m m

r ss

r s

r s

a J a m t J a m t

a J a m t J a m t

A AJ a a

A A

A A

+ +

− +

= =

+ +

+ −

= =

+ + − +

+ + − +

++

=

+

( )

( ) ( )( ) ( ) ( )

( ) ( )( )

( ) ( )( )

( ) ( )( ) ( ) ( )( )

1

2 2

0 2 1

2,4,...

3,5,...

1 1

3,5,... 3,5,...

sin

cos 2sin sin2

cos 2 cos

sin 2 sin

cos sin sin

s

r ss s

r s

s m

m

s m

m

s m m

m m

J a

A Aa a J a J a J a t

A A

J a m t

J a m t

a J a m t J a m t

+

=

+

=

+ +

− +

= =

++ + − + +

+ +

+ +

+ + − +

( ) ( )( ) ( ) ( )( )1 1

2,4,... 2,4,...

sin cos coss m m

m m

a J a m t J a m t + +

+ −

= =

+ + − +

Appendix

166

( ) ( )

( ) ( ) ( )( ) ( )

( ) ( ) ( )( )( ) ( )( )

( ) ( ) ( )( )( ) ( )( )

2 2

0 1

2 2

1 0 2

1 1

2,4,...

1 1

3,5,...

cos sin2

2 sin cos sin2

2

2 cos sin cos

2 sin cos sin

r ss s

r s

r ss s

r sr s

m s m m s

m

m s m m s

m

A AJ a aJ a

A A

A Aa J a J a J a t

A AA A

J a a J a J a m t

J a a J a J a m t

+

+ −

=

+

− +

=

++

++ + + − + =

+ − +

+ − +

+

+

+

( ) ( )( ) ( )( )

( )

( )

1

2,4,... 3,5,...

0,2,...

1 3, ..

0

, .

2 sin cos sin

cos sin

2

cos si

sin cos c n

n

os si

r s m

m m

m

r s

m

m

m

m

A A R t m t R m t

m t

R R

R m t

A A

R m m t

m

m t

m

m

+ +

= =

+

=

+

=

+

= + + + + +

−

= + +

Eq. (1.48)

( ) ( )

( ) ( ) ( ) ( )( ) ( )

( ) ( )

( )

( ) ( ) ( ) ( )( )( )

0

2

0

2

0

1 1

2

2

2 cos 2 sin 2 cos 2

2 cos 2 cos 2

cos 2

cos 2 2 cos si

cos 2 s

n

in 2

T

T

r s

T

r s

r s

r s m s m m s

X I t C t dt

A A t t t dt

A

R

RA t dt

TA A

TA A J a a J a J a

R

+ −+

=

= −

=

=

= −

Eq. (1.49)

( ) ( )

( ) ( )( ) ( )

( )

( ) ( ) ( )( )

0

1

0

2

1

0

1

2 2

1 0 2

2 sin cos cos sin sin

2 cos sin

cos

cos 2 sin cos2

T

T

r s

T

r s

r s

r sr s s s

r s

Y I t S t dt

A A R t t t dt

A A R t dt

TA A R

A ATA A a J a a J a J a

A A

=

= +

=

=

+ =

+ −

+

Appendix

167

Eq. (1.61)

( ) ( )

( ) ( ) ( ) ( )( ) ( )

( ) ( )

( )

( ) ( ) ( ) ( )( )( )

0

3

0

2

1

0

1

3 2 4

2 sin 3 cos 3 cos 3 sin 3 sin 3

2 cos 3 sin 3

cos 3

cos 3 2 sin cos

T

T

r s

T

r s

r s

r s s s

Y I t S t dt

A A R t t t dt

A A R t dt

TA A R

TA A J a a J a J a

=

= +

=

−+

=

=

Eq. (1.76)

( ) ( )

( )( ) ( )

( ) ( )( )( )

( )( )

( ) ( )

( )( ) ( )

0

2 2

2 2

0 0 1

2 2

2 cos sin 1 sin

cos sin2 cos sin

2

2 cos sin

sin

2 cos sin sin

2

T

r s r s sT

r sr s s s

r s

r s s

r s

r s s

X I t C t dt

A A A A a t a t

a t dtA AA A J a aJ a

A A

A A a t

A A a t

A A a t a t

A

=

+ + + − + +

+

= +− +

+ −

+ + +=

+ + − +

−

( ) ( )( )

( )( )0

0 1

cos sin

cos sin

T

r s s s

a t dt

A J a aJ a

+

According to Eq. (1.34), we can get the first term:

( )( ) ( )( )1

0

0 0

2 cos sin cos sin

2 sin 2 cos cos2 2

T

r s s

r s s

T A A a t a t dt

TA A J a J a

= + −

= +

According to the trigonometric identities and the symmetry properties of periodic functions, we know

that:

( )( ) ( )( ) ( ) ( )( ) ( )0 0 0

sin sin sin sin cos cos sin sin 0

T T T

a t t dt a t t dt a t t dt = =

According to the definition of Bessel function:

Appendix

168

( )( ) ( )( ) ( ) ( ) ( )2

1 1

0 0

2 1 2cos sin cos sin

2

T

a t t dt t a t d t J a TJ a

= = =

So with the help of the trigonometric identities, we can get the second term:

( ) ( ) ( )( )

( ) ( ) ( )( ) ( ) ( )( )

( )( )( ) ( )( )( )

( )( ) ( )( )( )

( )

2 2

2

0

2 2

0 0

02 2

0

2 2

sin cos sin

sin cos cos sin cos sin cos sin

cossin sin sin sin

2

sincos sin cos sin

2

T

r s

T T

r s

T

r s T

r s

T A A a t a t dt

A A a t a t dt t a t dt

t a t t a t dt

A A a

t a t t a t dt

A A

= + +

= + +

+ + −

= + + + + −

= +

( ) ( )( )1 1

sin0

2

0

a T J a J a

+ − +

=

The third term can be deduced as follow:

Appendix

169

( )( ) ( ) ( )( )

( )( ) ( )( ) ( ) ( )( )

( )

3

0

0

2 cos sin sin cos sin

sin2 cos sin sin cos sin sin

2

sin cos 2 sin cos cos 2 cos sin2 2 2 2

T

r s s

T

r s s s

r s s s

T A A a t a t a t dt

tA A a a t a t a t a t dt

A A a t a t a t

= + − +

+ = + − + + −

= + + − + + −

+

−

( )

( )

0

0

sin cos 2 cos sin2 2

sin cos 2 sin cos2 2

sin 2 cos sin sin 2 cos sin2 2 2 21

2s

T

sT

r s

s

s s

r s

dt

t a t

A A a dt

t a t

t a t t a t

A A a

+ + −

= + + + −

+ + + − + + − + +

=

+

0in 2 sin cos sin 2 sin cos

2 2 2 2

sin 2 cos sin2 2 2 2


2

T

s s

s

s

r s

dt

t a t t a t

t a t

t a t

A A a

+ + + − + + − + +

+ + + −

+ + − + +

=

−

−

0sin 2 sin cos

2 2 2 2

sin 2 sin cos2 2 2 2

T

s

s

dt

t a t

t a t

+ + + + −

+ + − + + −

−

where

0

0

0 0

1


sin 2 cos sin2 2

sin 2 cos sin cos cos 2 cos sin sin2 2 2 2

0 sin2

T

s

T

s

T T

s s

s

t a t dt

t a t dt

t a t dt t a t dt

TJ

+ + + −

= + −

= + − + + −

−

−

− −

+

= − −

1

2 cos2

sin 2 cos2 2

s

a

T J a

=

+

Appendix

170

0

0

0 0

1


sin 2 cos sin2 2

sin 2 cos sin cos cos 2 cos sin sin2 2 2 2

0 sin 2 c2

T

s

T

s

T T

s s

s

t a t dt

t a t dt

t a t dt t a t dt

TJ a

−

−

− −

+ − + +

= − +

= − + −

= + −

1

os2

sin 2 cos2 2

sT J a

−

=

0

0

0

0


sin 2 sin cos2 2

sin 2 sin sin2 2 2

sin 2 sin sin2 2 2 2 2

sin 2

T

s

T

s

T

s

T

s

t a t dt

t a t dt

t a t dt

t a t dt

t a

+ + + −

= + −

= + − −

= − − − − +

−

−

−

−

= −

0

0 0

1

1

sin sin2 2 2

sin 2 sin sin cos cos 2 sin sin sin2 2 2 2 2 2

0 cos 2 sin2 2

cos 2 sin2 2

T

s

T T

s s

s

s

t dt

t a t dt t a t dt

TJ a

T J a

− +

= − − + + − − +

= +

=

−

− −

+

+

Appendix

171

0

0

0

0


sin 2 sin cos2 2

sin 2 sin sin2 2 2

sin 2 sin sin2 2 2 2 2

sin 2

T

s

T

s

T

s

T

s

t a t dt

t a t dt

t a t dt

t a t dt

t a

+ − + +

= − +

= − − +

= − + − + +

−

−

−

−

= +

0

0 0

1

1

sin sin2 2 2

sin 2 sin sin cos cos 2 sin sin sin2 2 2 2 2 2

0 cos 2 sin2 2

cos 2 sin2 2

T

s

T T

s s

s

s

t dt

t a t dt t a t dt

TJ a

T J a

+ +

= + + + + +

= + − −

−

−

= −

−

+

−

So we can get:

1 1

3

1 1

1

sin 2 cos sin 2 cos2 2 2 21

2cos 2 sin cos 2 sin

2 2 2 2

sin 2 cos sin2 2 2

2

s s

r s

s s

s s

r s

T J a T J a

T A A a

T J a T J a

J a JT

A A a

+

= + −

+

+ −

+

+ −

−

=

1

1 1

1

1

1 1

2 cos2

cos 2 sin cos 2 sin2 2 2 2

2sin 2 cos2 2

22s

cos

sin

cos s

in 2 sin2 2

2 cos 2 si2 2

i2

n

s s

s

r s

s

r s

a

J a J a

J aT

A A a

J a

TA A a J a J a

+ −

=

+

= +

+

−

n sin2

s

According to Jacobi-Anger expansion and the orthogonality of trigonometric functions, the forth term

can be deduced:

Appendix

172

( ) ( )( ) ( )( )

( ) ( )( ) ( )( )

( ) ( )( ) ( ) ( ) ( )

( ) ( ) ( )( )

4 0 1

0

0 1

0

0 1 0 2

10

2

0 0 1

2 cos sin cos sin

2 cos sin cos sin

2 cos sin 2 cos 2

2 cos sin

T

r s s s

T

r s s s

T

r s s s n

n

r s s s

T A A J a aJ a a t dt

A A J a aJ a a t dt

A A J a aJ a J a J a n t dt

TA A J a aJ a J a

=

= −

= −

= − +

+

= −

+

+

+

So we get X:

( ) ( ) ( )( )

( )

1 2 3 4

0 0

1 1

2

0 0 1

2

0 0 0

2 sin 2 cos cos2 2

2 cos 2 sin sin2 2 2 2

2 cos s

0

in

2 sin 2 cos 22 2

cos sin

r s s

r s s

r s s s

r s

X T T T T

TA A J a J a

TA A a J a J a

TA A J a aJ a J a

TA A J a J a J a

= + + +

= +

+ +

−

+ −

+

+

=

( ) ( )1 1 0 1cos si

cos

2 cos 2 sin 2 sin2 2 2 2

n

s

r s sTA A a J a J a J a J a

+ −

+

Eq. (1.77)

( ) ( )

( )( ) ( )

( ) ( )( )( )

( )( )

( ) ( )

( )( ) ( )

0

2 2

2 2

0 0 1

2 2

2 cos sin 1 sin

sin2 cos sin

2

2 cos sin

sin

2 cos si

sin

n sin

2

T

r s r s sT

r sr s s s

r s

r s s

r s

r s s

Y I t S t dt

A A A A a t a t

a t dtA AA A J a aJ a

A A

A A a t

A A a t

A A a t a t

A

=

+ + + − + +

+

= +− +

+ −

+ + +=

+ + − +

−

( ) ( )( )

( )( )0

0 1

sin

cos sin

sin

T

r s s s

a t dt

A J a aJ a

+

According to Eq. (1.35), we can get the first term:

Appendix

173

( )( ) ( )( )1

0

0 0

2 cos sin sin sin

2 sin 2 cos sin2 2

T

r s s

r s s

T A A a t a t dt

TA A J a J a

= + −

= −

According to the trigonometric identities and the symmetry properties of periodic functions, we know

that:

( )( ) ( )( ) ( ) ( )( ) ( )0 0 0

sin sin sin sin cos cos sin sin 0

T T T

a t t dt a t t dt a t t dt = =

According to the definition of Bessel function:

( )( ) ( )( ) ( ) ( ) ( )2

1 1

0 0

2 1 2cos sin cos sin

2

T

a t t dt t a t d t J a TJ a

= = =

So with the help of the trigonometric identities, we can get the second term:

( ) ( ) ( )( )

( ) ( ) ( )( ) ( ) ( )( )

( )( )( ) ( )( )( )

( )( ) ( )( )( )

( )

2 2

2

0

2 2

0 0

02 2

0

2 2

sin sin sin

sin cos sin sin cos sin sin sin

coscos sin cos sin

2

sinsin sin sin sin

2

T

r s

T T

r s

T

r s T

r s

T A A a t a t dt

A A a t a t dt t a t dt

t a t t a t dt

A A a

t a t t a t dt

A A

= + +

= + +

− − +

= + + + − −

= +

( ) ( )( )

( ) ( )

1 1

2 2

1

cos0

2

cosr s

a T J a J a

T A A a J a

+ +

= +

The third term can be deduced as follow:

Appendix

174

( )( ) ( ) ( )( )

( )( ) ( )( ) ( ) ( )( )

( )

3

0

0

2 cos sin sin sin sin

sin2 sin sin sin sin sin sin

2

sin sin 2 sin cos sin 2 cos sin2 2 2 2

T

r s s

T

r s s s

r s s s

T A A a t a t a t dt

tA A a a t a t a t a t dt

A A a t a t a t

= + − +

+ = + − − + −

= + + − − + −

+

−

( )

( )

0

0

sin sin 2 cos sin2 2

sin sin 2 sin cos2 2

cos 2 cos sin cos 2 cos sin2 2 2 21

2c

T

sT

r s

s

s s

r s

dt

t a t

A A a dt

t a t

t a t t a t

A A a

+ + −

= − + + −

+ − + + − + + + −

=

+

0os 2 sin cos cos 2 sin cos

2 2 2 2

cos 2 cos sin2 2 2 2


2

T

s s

s

s

r s

dt

t a t t a t

t a t

t a t

A A a

+ − + + − + + + −

+ − + +

− + + + −

=

−

−

0cos 2 sin cos

2 2 2 2

cos 2 sin cos2 2 2 2

T

s

s

dt

t a t

t a t

+ + − + +

− + + + − −

−

Where

0

0

0 0

1


cos 2 cos sin2 2

cos 2 cos sin cos sin 2 cos sin sin2 2 2 2

cos 2 cos2

T

s

T

s

T T

s s

s

t a t dt

t a t dt

t a t dt t a t dt

TJ a

−

−

−

+ − + +

= − +

= − − −

=

−

−

1

02

cos 2 cos2 2

sT J a

−

−

=

Appendix

175

0

0

0 0


cos 2 cos sin2 2

cos 2 cos sin cos sin 2 cos sin sin2 2 2 2

cos2

T

s

T

s

T T

s s

s

t a t dt

t a t dt

t a t dt t a t dt

T

− + + + −

= − + −

= − + − + + −

−

−

= − −+

−

−

1

1

2 cos 02

cos 2 cos2 2

s

J a

T J a

+

=

+

0

0

0

0


cos 2 sin cos2 2

cos 2 sin sin2 2 2

cos 2 sin sin2 2 2 2 2

cos 2

T

s

T

s

T

s

T

s

t a t dt

t a t dt

t a t dt

t a t dt

t a

+ − + +

= − +

= − − +

= − + − + +

−

−

−

−

= +

0

0 0

1

1

sin sin2 2 2

cos 2 sin sin cos sin 2 sin sin sin2 2 2 2 2 2

sin 2 sin 02 2

sin 2 sin2 2

T

s

T T

s s

s

s

t dt

t a t dt t a t dt

TJ a

T J a

+ +

= + + − + +

= − − −

=

− −

−

+

−

Appendix

176

0

0

0

0


cos 2 sin cos2 2

cos 2 sin sin2 2 2

cos 2 sin sin2 2 2 2 2

cos

T

s

T

s

T

s

T

s

t a t dt

t a t dt

t a t dt

t a t dt

− + + + −

= − + −

= − + − −

= − − − − − +

−

−

−

−

−

=

0

0 0

1

1

2 sin sin2 2 2

cos 2 sin sin cos sin 2 sin sin sin2 2 2 2 2 2

sin 2 sin 02 2

sin 2 sin2 2

T

s

T T

s s

s

s

t a t dt

t a t dt t a t dt

TJ a

T J a

− − +

= − − − + + − − +

= − +

=

−

− −

+

+−

So we get:

1 1

3

1 1

1 1

cos 2 cos cos 2 cos2 2 2 21

2sin 2 sin sin 2 sin

2 2 2 2

cos 2 cos cos2 2 2

2

s s

r s

s s

s s

r s

T J a T J a

T A A a

T J a T J a

J a JT

A A a

+

=

−

+

− +

−

=

+

− +

1 1

1

1

1 1

c

2 cos2

sin 2 sin si

os

sin

co

n 2 sin2 2 2 2

2co

s sin

s 2 cos2 2

22cos 2 sin

2 2

2 cos 2 sin2 2 2

s s

s

r s

s

r s

a

J a J a

J aT

A A a

J a

TA A a J a J a

+ −

=

−

= −

− +

cos

2s

According to Jacobi-Anger expansion and the orthogonality of trigonometric functions, the forth term

can be deduced:

Appendix

177

( ) ( )( ) ( )( )

( ) ( )( ) ( )( )

( ) ( )( ) ( ) ( )( )

4 0 1

0

0 1

0

0 1 2 1

10

2 cos sin sin

2 cos sin sin sin

2

sin

cos sin 2 cos 2 1

0

T

r s s s

T

r s s s

T

r s s s n

n

T A A J a aJ a a t dt

A A J a aJ a a t dt

A A J a aJ a J a n t dt

−

=

= −

= −

= −

+

=

+

+ −

So we get Y:

( ) ( )

1 2 3 4

2 2

0 0 1

1 1

2 sin 2 cos sin cos2 2

2 cocos sins 2 sin cos2 2 2 2

r s s r s

r s s

Y T T T T

TA A J a J a T A A a J a

TA A a J a J a

= + + +

= − +

+ −

+

Eq. (1.96)

( )

( ) ( )

( ) ( )

21

0

2

00,2,... 1,3,...

2

02,4,... 1

0

,3,...

0

2 cos sin sin cos cos sin

2 sin sin cos cos s

cos si

2

n

ii ns n

r s m

m m

r s m

m

m

r

m

m

s

U I t dt

A A m t m t R m m t m m t dt

A A m t R m m t m m t dt

A

R m m

R R m

RA

+ +

= =

+ +

= =

=

= − + +

= − + +

+

=

( ) ( )1

2 2

2,4,... 1,3,...

s1 1 1 1

sin cosin2

m

m

m

m

m m

m R m mm m m

R

+

+ +

= =

− − − − + +

−

Eq. (1.97)

( )

( ) ( )

( ) ( )

2

2

0,2,... 1,3,...2

2,4,... 1,3,...2

0


2 sin sin cos

cos sin

sin cos sin

2

r s m

m m

r s m

m m

m

r

m

U I t dt


A A m t R m m t m m t dt

R m

R

A

m

A

m

R

+ +

= =

+ +

= =

=

= − + +

= − + +

=

+

( ) ( )1

2 2

2,4,... 1,3,...

0

1 1 1 1sins co

2in s

m m

s m

m m

mR m R m mm m m

R

+

+ +

= =

− − − + +

−

Appendix

178

Eq. (1.98)

( )

( ) ( )

( ) ( )

3

23

3

2

0,2,... 1,3,...

3

2

2,4,... 1

0

,3,...

2 cos sin sincos sin

si

cos cos sin

2 sin sin cos cos s

2

n in

r m

m

s m

m m

r s m

m m

U I t dt


A A m t R m m t m m t d

R

t

m m

R R m

+ +

= =

+ +

= =

=

= − + +

= −+ + +

=

( ) ( )1

2 2

2,4,... 1 3 . .

0

, , .

1 1 1 1sin co

2sin s

m m

r s m

m

m

m

A A m R m mm m m

R R

+

+ +

= =

− − − − + +

−

Eq. (1.99)

( )

( ) ( )

( ) ( )

2

34

2

2

3

0,2,... 1,3,...2

2

3

2,4,... 1,3,..

0

.2


2 sin sin cos cos sin

cos sin

sin

r s m

m m

r s mm

m m

m

U I t dt


A A m t R m m t m m t d

R m m

R R m t

+ +

= =

+ +

= =

=

= − + +

= − ++ +

( ) ( )1

2 2

2,4,... 1,3,..

0

.

1 1 1 12 sin cos n si

2

m m

r s m

m m

mRA A m R m mm m m

R

+

+ +

= =

− − − − − = +− +

Eq. (1.109)

( )

( ) ( ) ( )( )

( ) ( ) ( )( )( )( )

( )( )

( )

1

21 2 3 4

1 2

1,3,...

2 2

1 0 2

1

2

1 1

3,5,...

12 2

2

1

14 4 sin

2

4sin2 sin cos

2

14 2 sin cos sin

2 sin 18 sin

m

m

mr s

r ss s

r s

m

m s m m s

m

m

r s

m

mr s

U U U UP L R m

A A m

A AJ a J a J a

A A

J a J a J a mm

A AJ a m

A A m

+

+

=

+

+

− +

=

+

=

−− + + −= = − =

+−= + + −

−+ −

+ −= +

+

−

( ) ( )( )( )

,3,...

1

2

1 1

1,3,...

sin

14 sin cos

s

m

m m s

m

J a J a mm

+

+

+

− +

=

− −

+

Appendix

179

Eq. (1.110)

( )

( ) ( ) ( )( )( )( )

( )( )

( ) ( )( )( )

21 2 3 4

2 1

2,4,...

2

1 1

2,4,...

2 2

1 1

2,4,... 2,4,...

si1 1

4 42

1 14 2 cos sin

1 1 1 18 co

n

sin

sin sins 4

m

mr s

m

m s m m s

m

m m

m s m m

m

m

m

U U U UP L m

A A m

J a J a J a mm

J a m J a J a mm m

R

+

=

+

+ −

=

+ +

+ −

= =

+

− −− + − += = =

− −= −

− − − − =

+ −

sin s

Angular spectrum method

When the complex light field on the plane of z=z0 is known to be ( )0

,zE x y , then we can use 2D Fourier

transform to decompose it into plane waves along different directions described by different wave

vectors ( ), ,x y zk k k k= :

( ) ( )( )2 2 2

0

0 0

ˆ , ,x y x yi k x k y k k k z

z x y zE k k E x y e dxd y− + + − −

=

where 2

k k

= = , 2 2 2

z x yk k k k= − − .

Then the complex light field on the plane of z=z1 can be calculated by the inverse Fourier transform:

( ) ( ) ( ) ( )2 2 22 2 201 0

1 0

ˆ, ,x y x yx y

i k x k y k k k zi k k k z z

z z x y x yE x y E k k e e dk dk+ + − −− − − =

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181

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Exploration de méthodes de détection interférométriques à modulation de phase continue Dans cette thèse, nous nous intéressons aux si-gnaux interférométriques à modulation de phase continue, dans le but de développer des dispositifs de détection performants et originaux pour des applications en mécanique et en optique. Nous présentons d'abord plusieurs des techniques de modulation et démodulation employées dans ce contexte. Nous nous focalisons sur les modulations de phase sinusoïdales (SPM), qui sont particulière-ment avantageuses pour les dispositifs de modula-tion les plus largement accessibles. Nous proposons alors des solutions au problème de synchronisation ainsi qu'au problème éventuel de modulation d'amplitude concomitante à la modulation SPM. Ces techniques de démodulations sont ensuite ap-pliquées à 3 dispositifs expérimentaux développés au cours de cette thèse. Il s'agit d'abord d'un dispo-sitif d'holographie digitale compact sans lentille, mettant en œuvre une modulation SPM simple, pour l'imagerie et la mesure de déplacement. Nous utili-sons ensuite une technique de modulation à double fréquence pour réaliser des mesures bidimension-nelles de champ de déformation, à l'aide d'un dispo-sitif ESPI (interférométrie de speckle) original. Cette approche permet une mesure simultanée dans les 2 directions du plan à l'aide d'un seul système de laser et caméra. Finalement, nous présentons un instrument de type SPR (Surface Plasmon Re-sonance) compact mettant en œuvre une détection interférométrique SPM par modulation de longueur d'onde, dans lequel la modulation d'amplitude est prise en compte avec succès. Mots clés : interférométrie – modulation de phase – interférométrie holographique – interférométrie par granularité – résonance plasmonique de surface.

Yunlong ZHU Doctorat : Matériaux, Mécanique, Optique, Nanotechnologie

Année 2018

Exploration of Interferometric Detection Methods based on Continuous Phase Modulation In this thesis, interference signals with continuous phase modulations are theoretically and experimen-tally analyzed in order to develop cost-efficient solutions for sensing application in mechanics and optics. Several common phase modulation functions and phase retrieval algorithms are presented. We mainly focus on sinusoidal phase modulation (SPM), which is especially attractive for the most accessible mod-ulators (e.g. electro-optical or piezoelectrical modu-lators). In such case, the demodulation process must handle synchronization issue as well as a possible intensity modulation induced by the SPM. Mathematical solutions are proposed in this context. These demodulation techniques are then applied to three experimental devices developed during this thesis. First of all, a lens-less co-axis digital holog-raphy setup has been built, and SPM has been ap-plied for imaging and displacement measurement. Then we use a dual-frequency modulation technique to perform two-dimensional deformation field meas-urements using an original ESPI (Electronic Speckle Pattern Interferometry) device. This approach allows for simultaneous measurement of the displacement along two different axes using a single laser and a single camera. Finally, we present a compact SPR (Surface Plasmon Resonance) instrument imple-menting SPM interferometric detection through wavelength modulation, where the amplitude modu-lation is successfully taken into account. Keywords: interferometry – phase modulation – holographic interferometry – speckle metrology – surface plasmon resonance.

Ecole Doctorale "Sciences pour l’Ingénieur"

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