Depth-resolved whole-field displacement measurement using wavelength scanning interferometry

Depth-resolved whole-field displacementmeasurement by wavelength-scanning electronicspeckle pattern interferometry

Pablo D. Ruiz, Jonathan M. Huntley, and Ricky D. Wildman

We show, for the first time to our knowledge, how wavelength-scanning interferometry can be used tomeasure depth-resolved displacement fields through semitransparent scattering surfaces. Temporalsequences of speckle interferograms are recorded while the wavelength of the laser is tuned at a constantrate. Fourier transformation of the resultant three-dimensional (3-D) intensity distribution along thetime axis reconstructs the scattering potential within the medium, and changes in the 3-D phasedistribution measured between two separate scans provide the out-of-plane component of the 3-D dis-placement field. The principle of the technique is explained in detail and illustrated with a proof-of-principle experiment involving two independently tilted semitransparent scattering surfaces. Results arevalidated by standard two-beam electronic speckle pattern interferometry. © 2005 Optical Society ofAmerica

OCIS codes: 120.3180, 120.4290, 120.5050, 140.3600, 170.4500.

1. Introduction

The measurement of internal displacement or strainfields within a material has many potential applica-tions, ranging from the development of an improvedunderstanding of failure mechanisms, to medical di-agnostics such as the detection of retinal disease.Although methods such as photoelastic tomography,neutron diffraction, phase contrast magnetic reso-nance imaging, and x-ray computed tomography com-bined with digital image correlation have beendeveloped in recent years, such methods have impor-tant limitations in sensitivity, in spatial resolution,or in the type of material to which they can be ap-plied.

Multiple-wavelength techniques have been used asa depth-measuring tool in optical profilometry for anumber of years. Low coherence interferometry (LCI)involves illumination of the surface with a broadbandsource and scanning the reference mirror or sample

through the required depth range.1,2 Wavelengthscanning interferometry (WSI) also involves multiplewavelengths, but these are provided sequentially by atunable light source that can remove the need for anymechanically scanning device.3–5 Optical coherencetomography (OCT) is the name given to a family oftechniques (including both LCI and WSI) that pro-vides depth-resolved microstructure images fromwithin scattering media (see, for example, Ref. 6 for arecent review of the field). Most OCT systems arepointwise devices, with mechanical scanning in oneor more lateral directions to build up cross-sectionalimages. The WSI version of OCT was proposed byFercher et al. in 1995 (Ref. 7) and demonstrated by anumber of authors (see, e.g., Ref. 8).

The first demonstrations of depth-resolved displace-ment field measurement were made recently.9–11 Inthe case of LCI,9,10 the interferometer is sensitive onlyto the movement of scattering points lying within theslice selected by the reference mirror’s position. LikeLCI, WSI provides decoupling of depth resolution anddisplacement sensitivity, but it appears to possesssome additional practical advantages over LCI, themost important being an improved signal-to-noise ra-tio.11 Reference 11 reported a brief proof-of-principleexperimental study that demonstrated the evaluationof depth-resolved displacements of a pair of specularlyreflecting transparent surfaces.

In this paper we focus on a more-general descrip-

The authors are with the Wolfson School of Mechanical andManufacturing Engineering, Loughborough University, AshbyRoad, Loughborough, Leicestershire LE11 3TU, UK. P. D. Ruiz’se-mail address is [email protected].

Received 23 November 2004; revised manuscript received 7 Feb-ruary 2005; accepted 8 February 2005.

0003-6935/05/193945-09$15.00/0© 2005 Optical Society of America

1 July 2005 � Vol. 44, No. 19 � APPLIED OPTICS 3945

tion of the WSI approach to measuring subsurfacedisplacement fields of semitransparent scatteringtest objects. In Section 2 we provide a detailed math-ematical description of the technique, explain how toobtain depth-resolved displacement fields, and intro-duce the key length scales of the technique, i.e., gaugevolume, depth range, and displacement sensitivity.In Section 3 we present a proof-of-principle experi-ment based on a WSI approach with speckled wavefronts (as opposed to the smooth wave fronts used forRef. 11) and validate the results with standard elec-tronic speckle pattern interferometry (ESPI). In Sec-tion 4 we discuss some error sources and summarizethe advantages and disadvantages of WSI comparedwith LCI for depth-resolved displacement field map-ping.

2. Wavelength-Scanning ESPI

A. Depth-Dependent Phase Shift

Consider the layout shown schematically in Fig. 1. Abeam of wavelength � is partially reflected by glass–air interface R that acts as a reference surface andby several depth slices Sj within a semitransparentscattering object. In the derivation that follows weconsider two slices only, both for reasons of mathe-matical simplicity and to facilitate a direct compari-son with the experimental results presented inSubsection 2.B. However, we also present the mainresult for the general case at the end of the section[Eq. (9)].

The phase difference between the smooth wavefront reflected at R and the speckled wave front scat-tered back from a given slice Sj is

�j(x, y) � �0j(x, y) �4�

�zj(x, y), (1)

where �0j�x, y� is the random phase term responsiblefor the speckle from slice Sj (including a phase changeinduced on reflection) and zj�x, y� is the optical pathbetween the jth slice inside the object and the refer-ence surface. �0j�x, y� can be regarded as the phase ofa complex field that results from the convolution ofthe complex amplitude at diffusing slice Sj and the

point-spread function (PSF) of the imaging system.Both are wavelength dependent, but, as we discuss inSection 3, this dependence can be safely dropped forthe subsequent analysis.

Before deformation, S1 and S2 are defined to beplane surfaces parallel to R, in which case zj�x, y� is aconstant zj. If � is tuned in an approximately linearmanner in the range �� about a central wavelength�c, � will change in a way that, to a first-order ap-proximation, can be written as

�j(x, y, t) � �0j(x, y) �4�

�czj �

4�zj��

T�c2 t. (2)

T is the time that it takes to scan range ��, and t istime ��T�2 � t � T�2�. The second term on theright-hand side of Eq. (2) is the phase difference thatis due to the optical path between the wave frontscoming from R and Sj. The third term is a time-varying linear phase shift introduced by the changingwavelength and is equivalent to 2�fjt, where fj is acarrier frequency proportional to zj:

fj �2��

T�c2 zj. (3)

Assuming that multiple reflections can be neglectedamong R, S1, and S2, the intensity that is due to theinterference of the three wave fronts is

I(x, y, t) � IR(x, y) � I1(x, y) � I2(x, y)

� 2[IR(x, y)I1(x, y)]1�2cos[�1(x, y, t)]

� 2[IR(x, y)I2(x, y)]1�2cos[�2(x, y, t)]

� 2[I1(x, y)I2(x, y)]1�2cos[�12(x, y, t)],(4)

where IR, I1, and I2 are the intensities of the wavefronts coming from surfaces R, S1, and S2, respec-tively. �1, �2, and �12 are, respectively, the phasedifferences between wave fronts reflected at R and S1,R and S2, and S1 and S2 given by Eq. (2) for opticaldistances z1, z2, and z12 � z2 � z1 between the corre-sponding surfaces.

B. Extraction of the Scattering Potential

The Fourier transform of a sampled version of signalI�x, y, t� can be written as

I(x, y, f ) ��

�I(x, y, t) �n��

(t � n�t)�W(t)

� exp(�i2�ft)dt, (5)

where t is a continuous variable, �t is the time inter-val between samples, � is the Dirac delta function,W�t� is a window function, f is the frequency variablein Fourier space, n is an integer, and � denotes aFourier-transformed variable. Substituting Eq. (4)

Fig. 1. Layout for measuring depth-resolved out-of-plane dis-placements by WSI. Displacements w1�x, y� and w2�x, y� of scatter-ing centers at surfaces S1 and S2 are measured between referencestate a (solid lines) and loaded state b (dotted curves).

3946 APPLIED OPTICS � Vol. 44, No. 19 � 1 July 2005

into Eq. (5), we obtain

I(x, y, f ) � W(f ) ��IR � I1 � I2

�t �n��

f �n�t

��IRI1

�t exp��i�1(x, y, 0) �n��

f � f1

�n�t�

�IRI2

�t exp��i�2(x, y, 0)

� �n��

f � f2 �n�t�

�IRI2

�t� exp��i�12(x, y, 0)

� �n��

f � f12 �n�t�, (6)

where the dependence of I1, I2, and IR on �x, y� hasbeen dropped for simplicity:

�1(x, y, 0) � �01(x, y) �4�z1

�c,

�2(x, y, 0) � �02(x, y) �4�z2

�c,

�21(x, y, 0) � �2(x, y, 0) � �1(x, y, 0), (7)

and � denotes convolution.Solving the convolution in Eq. (6) and retaining

only the terms within the Nyquist frequency rangef � ��1�2�t, 1�2�t , we finally obtain

I(x, y, f ) �IR � I1 � I2

�t W(f ) ��IRI1

�t

� exp��i�1(x, y, 0) W(f � f1) ��IRI2

�t

� exp��i�2(x, y, 0) W(f � f2) ��I1I2

�t

� exp��i�12(x, y, 0) W(f � f12). (8)

In the positive frequency axis the spectrum revealsfour amplitude peaks, as can be seen from Fig. 2: thedc component and three peaks that correspond to thethree cosine terms in Eq. (4), centered at frequenciesf1, f2, and f12. The amplitude of a peak will depend ofcourse on the power reflected from the surface, but ithas been assumed the same for all peaks in thisschematic figure. Provided that the distance betweenreference and sample is greater than the sampledepth, the peaks of interest (f1 and f2) are separatedfrom the unwanted peaks (f12 and the dc), resulting ina reconstruction of the true scattering potential of thesample rather than of its autocorrelation.7 One canthen obtain scattering potential I�x, y, z� from mag-nitude spectrum I�x, y, f � in Eq. (8) by simply ex-

changing the frequency axis for a position axis(relative to the reference surface) in the spatial do-main, using the relationship between z and f given byEq. (3). A more general version of Eq. (8), valid ifmultiple scattering is neglected, can be derived for Ns

scattering layers that reads as

I(x, y, f ) �1�tIR � �

j�1

Ns

IjW(f ) �1�t �

j�1

Ns

�IRIj

� exp[�i�j(x, y, 0)]W(f � fj)

�1�t �

j�1

Ns�1

�k�j�1

Ns

�IjIk exp[�i�jk(x, y, 0)]

� W(f � fjk). (9)

C. Depth-Resolved Displacements

If we evaluate Eq. (8) at f � f1, i.e., at the peakassociated with the interference between the refer-ence beam and the wave front scattered back from S1,we get

I(x, y, f1) �IR � I1 � I2

�t W(f1) ��IRI1

�t

� exp��i�1(x, y, 0) W(0) ��IRI2

�t

� exp��i�2(x, y, 0) W(f1 � f2) ��I1I2

�t

� exp��i�12(x, y, 0) W(f1 � f12). (10)

If the Fourier transform of window function W�f � isnarrow in the frequency domain relative to thedistance between peaks S1, S2, and S12 and has low-amplitude secondary lobes (as is the case for a Han-ning window W�t� � 1⁄2 � 1⁄2 cos�2��t � �T�2� �T�, forinstance), then cross-talk effects can be neglected be-tween peaks. The first, third, and fourth terms on theright-hand side of Eq. (10) can then be ignored, giving

I(x, y, f1) ��IRI1

�t exp��i�1(x, y, 0) W(0). (11)

Fig. 2. Schematic spectrum of a three-beam interference signalthat is due to reference surface R and two test surfaces, S1 and S2.


If the scattering points within slice S1 move a dis-tance w1�x, y� from their reference position z1�x, y�(state a; solid lines representing S1 and S2 in Fig. 1)to a new position (state b, dashed curves representingS1 and S2 in Fig. 1) in such a way that the specklecomponent of the phase in �0j does not completelydecorrelate between states a and b, then Eq. (2)shows that �1�x, y, 0� changes by

��1(x, y) � 4�w1(x, y)��c. (12)

This optical phase difference can be evaluated as

��1(a, b) � tan�1�N(b)D(a) � D(b)N(a)D(b)D(a) � N(b)N(a)�, (13)

where

N(a) � Im�I(x, y, f1) ,

D(a) � Re�I(x, y, f1) . (14)

The notation in Eqs. (14) means that data from ref-erence state a may be used, and similar expressionsare valid for deformed state b. ��1 is wrapped intothe ��, � interval, and therefore the appropriateinteger number � of 2� cycles has to be added to it toproduce a continuous phase distribution:

��1u(x, y) � ��1(x, y) � 2� (x, y). (15)

The displacement of the scattering points within sliceS1, w1�x, y, z1�, can be obtained from Eq. (12) as

w1(x, y, z1) � �c

��1u(x, y, z1, 0)4�

. (16)

D. Gauge Volume, Depth Range, and DisplacementSensitivity

The total phase change �� introduced between wavefronts coming from R and S1 by wavelength change�� is

��(x, y, z1) � 4�z1

��

�1�u, (17)

where �1 and �u are, respectively, the lower ��c

� ��2� and the upper ��c � ��2� limits of thewavelength scan. This phase change introduces C� ��2� modulation cycles into the interference in-tensity signal. To comply with the Shannon samplingcondition, the number of samples (i.e., cameraframes) required is

Nf � 4z1

��

�1�2. (18)

From relation (18), and assuming that the coherence

length of the laser is long enough, it follows that, fora given central wavelength and tuning range, thenumber of frames Nf determines the maximum dis-tance that a slice can be from the reference surface forits displacements to be determined. This distance isknown as the depth range and is given by

�z �Nf�c

2

4��. (19)

A most important parameter in the system is thedepth resolution, i.e., the minimum distance betweensurfaces inside the measuring volume whose corre-sponding interference signals can be fully resolved orseparated in the frequency domain. A usual resolu-tion criterion is a frequency difference f between twoneighboring peaks of at least twice the distance fromtheir centers to their first zero (see Fig. 3). The finiteduration of the window function W�t� has the effect ofbroadening a given spectral line: A rectangular win-dow of duration T, for example, results in a sinc func-tion of width f � 2�T and a Hanning window in aspectral width of f � 4�T. From Eq. (3) the depthresolution is

z � ��c

2

��, (20)

where � � 2, 4 for rectangular or Hanning windows,respectively. Although the rectangular window hassuperior depth resolution, it is accompanied by theundesirable presence of large secondary lobes in W�f�that may strongly interfere with other peaks, leadingto phase errors. This effect is strongly reduced if aHanning window is used instead.

If the layout in Fig. 1 is imaged by an optical sys-tem, lateral resolution x and y will depend on thesystem’s magnification and the spatial resolution ofthe sensor used. The lateral and depth resolutionsdefine a gauge volume of size x � y � z within themeasurement volume (Fig. 3). All scattering pointswithin the gauge volume centered at �x, y, z1� con-tribute to the interference signal at position �x, y� inthe field of view, modulated with frequencies within

Fig. 3. Slice through the gauge volume in the WSI system, show-ing depth resolution z, lateral resolution x, and average displace-ment d of scatterers.


the range f1 � f�2 � f � f1 � f�2. After deformation,the scattering points initially inside the gauge vol-ume centered at �x, y, z1� move to a new position �x� u, y � v, z1 � w�, indicated with a dashed box rulein Fig. 3. w1�x, y, z1� then represents the out-of-planecomponent of displacement vector d, which is an av-erage displacement of all scattering points inside theintersection of the solid vertical line and the dashedbox rule.

Displacement resolution �z (sometimes called thesensitivity of the technique) is decoupled from depthresolution z and depends only on the central wave-length of the tunable laser and the degree of speckledecorrelation. For out-of-plane sensitivity, as is thecase in Fig. 1, it is typically better than �c�30. An-other parameter of practical interest is the depthdynamic range, which is the number of resolved sliceswithin the depth range and is given by

DR � Nf�2�. (21)

3. Wavelength-Scanning Speckle PatternInterferometer

A proof-of-principle experiment was designed thatincludes the main features of the model in Section 2,i.e. a smooth reference wave front interfering withtwo speckle patterns coming from two semitranspar-ent scattering surfaces located one behind the other.The configuration is shown in Fig. 4 and consists oftwo independently tiltable semitransparent scatter-ing surfaces, S1 and S2, and a third fixed surface R toprovide a smooth reference wave. Surfaces R and S1were the glass–air interfaces of glass flats, antireflec-tion coated on one side to suppress the reflection fromthe second glass–air interface. S1 was coated with athin layer of enamel paint such that part of the inci-dent light underwent specular reflection at the glass–air interface, part was transmitted, and part wasbackscattered. Surface S2 consisted of a glass flatcoated with a thick layer of water-based matte paintthat produces only a backscattered field, with nospecular component. Surface S1 was mounted upon amicrometer-driven gimbal mount; S2, on a highly re-peatable piezoelectric-transducer-driven tilting stage.

The light source used was a solid-state tunablelaser (TL; New Focus Vortex 6005), the beam fromwhich was expanded by lens L1 and steered by mirrorM toward collimating lens L2 f � 500 mm. The light

reflected from surface R and the backscattered lightfrom surfaces S1 and S2 were imaged by lens L3 (Ni-kon; f � 135 mm; f�2.8, 1 D close-up lens; distancefrom L3 front surface to L2, 800 mm) onto the sensorof a high-speed camera (C; VDS HCC-1000, 1024� 1024 pixel sensor, 10 �m � 10 �m pixel size),which recorded the resultant three-beam interfer-ence patterns.

A small polarizing disk (5 mm diameter; P1) wasplaced in the center of the Fourier plane of lens L2. Asecond polarizer, P2, placed in front of lens L3, en-sured that the speckle fields and the dc referenceterm interfered in the image sensor with the samepolarization. Polarizer P1 enabled the intensity ratiobetween the dc term of the reference beam and thespeckle fields to be adjusted to maximize the inten-sity modulation.

When surfaces R and S1 are parallel, the specularreflection coming from them is focused at the samespot (dc term) in the Fourier plane of lens L2. Thissituation causes unwanted interference fringes to ap-pear in the image plane of the system that were re-moved by tilting S1, i.e., moving the dc term off axis,and blocking it with a small screen, B. The tilt angleof S1 also controls the amplitude of the speckle fieldcoming from it, which is more intense in the speculardirection and less intense away from it. With thisconfiguration, three beams interfere in the imageplane: smooth reference beam R and the backscat-tered speckle patterns from S1 and S2. The size andsuperficial density of the pigment particles used tocoat S1 and S2 were chosen to make sure that everypixel in the image sensor gathered light coming frommultiple scattering points on S1 and S2 (��100 scat-terers per pixel). If this condition were not fulfilled,then clear areas bigger than the lateral resolutionx � y would not contribute to modulation informa-tion, and therefore no displacement informationcould be retrieved there.

We tuned laser wavelength � in an approximatelylinear manner about a center wavelength, �c, by ad-justing a voltage supplied by signal generator SG tolaser controller LC.

Standard two-beam ESPI was used to validate theresults obtained with the wavelength-scanning tech-nique. The displacements of surfaces S1 and S2 weremeasured with both techniques in the same experi-ment. The measurement procedure was as follows(see Fig. 5):

(1) With only R and S2 in the setup, record a two-beam interference speckle pattern corresponding tothe reference state of S2: I1�x, y�.

(2) Tilt S2 and record a two-beam interferencespeckle pattern (beams from R and S2), correspondingto the tilted state of S2: I2�x, y�.

(3) Move S2 back to the reference state as in step(1). This can be done with high repeatability with thepiezoelectric-transducer-driven mount. Place surfaceS1 between R and S2. Record a sequence of three-beam interference speckle patterns (beams from R,S1, and S2) corresponding to the reference state of S1

Fig. 4. Wavelength-scanning speckle pattern interferometer. PC,personal computer; other abbreviations defined in text.


and S2, using the wavelength-scanning method:I3�x, y, t�.

(4) Place a black screen between surfaces S1 andS2 and record a two-beam interference speckle pat-tern (beams from R and S1) corresponding to the ref-erence state of S1: I4�x, y�.

(5) Tilt S1 and record a two-beam interferencespeckle pattern (beams from R and S1), correspondingto the tilted state of S1: I5�x, y�.

(6) Tilt S2 and remove the black screen. Record asequence of three-beam interference speckle patterns(beams from R, S1, and S2) corresponding to the tiltedstates of S1 and S2, using the wavelength-scanningmethod: I6�x, y, t�.

With this procedure the time between the acquisitionof the two-beam interference patterns was minimizedand resulted in well-modulated subtraction fringepatterns. The drawback was that the time betweenthe acquisition of I3�x, y, t� and of I6�x, y, t� wasenough to introduce some speckle decorrelation andreduce the quality of the phase maps obtained fromthem.

In steps (3) and (6) the laser wavelength was tunednear �c � 635.05 nm at a rate ��T � 0.029 nm s�1

(�� 0.116 nm, T � 4 s). A sequence of 405 inter-ferograms was recorded with the camera running at101.4 frames�s. The optical path difference betweensurfaces R and S1 and between S1 and S2 was approx-imately 53 and 17 mm, respectively. Figure 6(a)shows normalized intensity I3�x, y, t� measured forone pixel in a region of interest of 200 � 200 pixels inthe field of view; Fig. 6(b) shows the positive side ofthe frequency spectrum of I3�x, y, t�, where the peaksof interest are present and are averaged over columnx � 100, y � 1, 2, . . .200. Peaks S1, S2, and S1S2correspond to the interference signal for surfaces Rand S1, R and S2, and S1 and S2, respectively. Beforethe Fourier transform, the mean value of the inten-sity signal was subtracted and the signal multipliedby a Hanning window. The peak frequencies for S1

and S2 are within 5% and 12%, respectively, of thoseexpected when Eq. (3) is used.

There are two low-frequency peaks of high ampli-tude close to the dc term in Fig. 6(b). These peaks aremost likely due to multiple reflections between thefaces of the glass used as reference surface R. Theshape of peaks S1 and S2 (some power loss to lowerfrequencies) is probably due to a nonlinear behaviorof the wavelength with the tuning voltage, which wasassumed to be linear in the model described in Sec-tion 2.

We obtained the phase difference that is due to tiltof surfaces S1 and S2 by evaluating phase difference��1�x, y, 0� with Eq. (12), between the states beforeand after tilt, using I3�x, y, t� and I6�x, y, t�. Figures7(a) and 7(b) show the measured wrapped phase dif-

Fig. 5. Experimental procedure for measuring displacement ofsurfaces S1 and S2 by using two-beam speckle pattern interferom-etry (2B) for validation purposes, and WSI. Subscripts a and bmean reference state and deformed state, respectively. See furtherdescription in the text.

Fig. 6. (a) Intensity signal I3�x0, y0, t� recorded at a pixel �x0, y0� inthe region of interest. (b) Average frequency spectrum of intensitysignal I3�x, y, t� recorded at all the pixels from one column in theregion of interest for x � 100 and y � 1, 2, . . . , 200. Peaks S1, S2,and S1S2 correspond to the interference signals for surfaces R andS1, R and S2, and S1 and S2, respectively.


ference maps for the movement of S1 and S2, respec-tively. Each fringe represents an out-of-planedisplacement w � �c�2 � 317 nm. Using a Hanningwindow with the experimental system described inSection 3 yields depth resolution z � 7 mm. In thisexample, 200 � 200 pixels sample a field of view of10 mm � 10 mm. The gauge volume is therefore ap-proximately 50 �m � 50 �m � 7 mm. The system’sdepth range is �z � 350 mm. The rms phase error ofthe difference between the unwrapped phase maps ofFigs. 7(a) and 7(b) and their corresponding best-fitplanes is 0.14 rad for S1 and 0.53 rad for S2. Forout-of-plane sensitivity, these values are equivalentto ��c�88 and ��c�23, respectively.

The validation measurement from two-beamESPI was performed at a fixed wavelength �c �635.05 nm. Figure 8(a) shows subtraction fringesthat correspond to the square of the difference be-tween interferograms I1�x, y� and I2�x, y�. Thesefringes represent the tilt of surface S2. Figure 8(b)shows the fringes corresponding to the tilt of surfaceS1 obtained from subtraction of interferogramsI4�x, y� and I5�x, y�. It can be seen that both fringepatterns are consistent with the phase distributionsmeasured by WSI shown in Figs. 7(a) and 7(b). Animportant point to stress is that for both the reference[Figs. 7(a) and 8(a)] and the deformed [Figs. 7(b) and8(b)] states, S1 and S2 in Fig. 7 were measured simul-taneously whereas in Fig. 8 they were measured se-

Fig. 7. Wrapped phase difference maps of (a) S1 and (b) S2 ob-tained by WSI, showing the tilt introduced for each surface. Blackrepresents �� rad; white; �� rad.

Fig. 8. Correlation fringes obtained for surfaces (a) S1 and (b) S2

by use of two-beam speckle pattern interferometry.


quentially and required one of the surfaces to beremoved for the measurement of the other.

4. Discussion

A. Error Sources

As is also the case in LCI and OCT, WSI as presentedin this paper relies on single-scattered photons com-ing from each depth slice to produce the interferencesignal. Multiply scattered photons travel shorter (orlonger) optical paths and therefore result in lower (orhigher) modulation frequencies. Figure 9 illustratesthis effect when there are only two slices and a scat-tered photon traveling a longer optical path that thedirect photons: Photons first scattered at S2 and thenat S1 travel toward the imaging system parallel to theoptical axis and are detected as coming from point�x, y� in the object. In the spectrum, these photonsproduce a peak whose frequency fm is higher thanthat which corresponds to direct photons singly scat-tered at S2, f2. The effect is visible in the experimentalresults as a low-amplitude tail to the right of thepeaks in Fig. 6(b).

A necessary condition for measuring depth-resolved displacements of semitransparent scatter-ing surfaces is that the size and density of thescattering particles within the material be such thatevery pixel in the image sensor gathers light comingfrom many scattering points on each depth slice. Ifthis condition were not fulfilled, clear areas in a par-ticular depth slice bigger than the lateral resolutionx � y would not contribute to modulation informa-tion from that slice and therefore no displacementinformation could be retrieved from that region.

Speckle decorrelation is another factor that mayprevent us from measuring depth-resolved displace-ments. In Section 2 we showed that phase difference�� can be obtained, provided that the speckle phasein �0 does not change significantly between referencestate a and deformed state b. In the schematic shownin Fig. 3 this requires that the in-plane displacementof the gauge volume be smaller than the lateral res-olution of the system.

We mentioned briefly in Subsection 2.A that

speckle phase term �0j�x, y� in Eq. (1) could be re-garded as the phase of a complex field that resultsfrom the convolution of the complex amplitude atdiffusing slice Sj and the PSF of the imaging system.The phase difference between the randomly distrib-uted scatterers at Sj is wavelength dependent. Theeffect of tuning the wavelength by �� is a change inthe relative phases of the scattering centers by afraction ��, which is completely negligible in oursystem (�2 � 10�4; Section 3). If a Ti:sapphire or dyelaser is used instead, for example with �� 30 nmand � � 600 nm, the effect will become more signif-icant �� 1�20�, and it should be considered inthe spectral analysis.

The diameter of the PSF of the imaging system islinear with the wavelength, and therefore a change�� will change the size of the PSF by the same factor��, the consequence being a slight change in thespeckle size. In our current setup the limited depth offield of imaging lens L3 means that one of the surfacesS1 or S2—but not both—is mapped in sharp focusonto the image sensor. Depending on the depth offield of lens L3, the phase may appear slightly blurredfor surfaces that are out of focus. Perspective distor-tions may also be a problem for deep samples butcould be overcome by use of a telecentric objective, aswas done in the research reported in Ref. 3.

Finally, Fig. 7(b) shows a higher noise content thanFig. 7(a), as might be expected for a diffusing surfaceilluminated and viewed through a second diffusingsurface. This suggests that a more precise descriptionof the method should consider the sources of errorsand the performance of the system according to theposition occupied by each slice within the diffusingmedium.

B. Advantages and Disadvantages of WSI

In WSI, the depth range of the displacement field islimited not by the mechanical scan range of the ref-erence arm of the interferometer as in LCI but bythe number of frames that can be recorded in thetime that it takes to scan the whole tuning range ofthe source. Ultimately, this time is determined by thelaser power, the framing rate of the camera, and thenumber of frames that can be stored. Depth rangesthat would be regarded as unfeasible for LCI (of theorder of 1 m or more) could be measured by WSI.

In systems with broadband light sources, disper-sion may be a significant cause of fringe contrastreduction. In WSI the fringes are produced at highvisibility at all times by a single wavelength andtherefore a reduction in data quality as a result ofdispersion does not arise. The limited dynamic range(typically 256 gray levels) of whole-field image sen-sors based on CCD or complementary metal-oxidesemiconductor technology limits the performance ofWSI to a much lesser extent than that of LCI. If z isthe slice thickness and Z is the overall thickness of asample containing uniformly distributed scatterers,then only the fraction z�Z of scattered photons con-tributes to signal modulation in the case of LCI; the

Fig. 9. When there are two scattering slices, multiply scatteredphotons (dashed line) travel longer paths and therefore modulatethe intensity signal with higher frequencies than direct photons(solid and dotted lines that indicate reflection at S2, S1, and R).


rest of the photons merely produce a dc offset to theintensity image. For example, when z�Z � 1�100,with an 8-bit camera the signal would be no morethan two to three gray levels deep and thereforerather susceptible to noise. WSI, however, ensuresthat the full dynamic range of the camera is utilized.

The main disadvantage is that images need to beacquired for all wavelengths before even a singleslice can be selected. The use of a high-speed cam-era on this demonstration system nevertheless al-lowed all data acquisition to be performed in �4 s(at �100 frames�s).

The depth resolution in our system was z �7 mm, limited by the laser tuning range �� 0.116 nm. However, with the ring dye lasers that areavailable today, which can output �c � 625 nm witha tuning bandwidth of �� 30 nm, the depth reso-lution could be as small as z � 10 �m, which is moreinteresting for the small-scale applications that arecurrently tackled with broadband source OCT.

5. Conclusions

We have demonstrated that wavelength scanning in-terferometry can be used to measure depth-resolveddisplacement fields through semitransparent scatter-ing media. The presence of speckled wave fronts in-troduces some limitations compared with the smoothwave-front case reported in Ref. 11. In particular, thein-plane displacements of the depth slices should notexceed the lateral resolution of the system if one is toprevent speckle decorrelation. The size and density ofthe scattering particles within the material have tobe such that every pixel in the image sensor gatherslight coming from many scattering points on eachdepth slice. Multiple scattering has the effect ofbroadening the spectral lines. The depth resolutiondoes not depend on the type of wave front involved(smooth or speckled) and is determined mainly by thetuning range and the central wavelength of thesource. Given a coherence length of the laser that is

long enough, the depth range is ultimately deter-mined by the laser power, the framing rate of thecamera, and the number of frames that can be stored.

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Depth-resolved whole-field displacement measurement using wavelength scanning interferometry

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