Formation of discrete space-variant subwavelength dielectric gratings for polarimetric measurements

Gabriel Biener, Avi Niv, Yuri Gorodetski, Vladimir Kleiner, and Erez Hasman

Optical Engineering Laboratory, Faculty of Mechanical Engineering, Technion-Israel Institute of Technology, Haifa, Israel 32000

ABSTRACT We present a unique method for real-time polarization measurement by use of a discrete space-variant subwavelength grating. The formation of the grating is done by discrete orientation of the local subwavelength grooves. The complete polarization analysis of the incident beam is determined by spatial Fourier transform of the near-field intensity distribution transmitted through the discrete subwavelength dielectric grating followed by a subwavelength metal polarizer. We discuss a theoretical analysis based on Stokes-Muller formalism and experimentally demonstrate our approach with polarization measurements of infrared radiation at a wavelength of 10.6µm. Moreover, a new far-field polarimetry approach is presented along with preliminary experimental results. Keywords: polarization, gratings, polarization-sensitive devices, polarimetry.

1. INTRODUCTION Optical polarimetry measurement has been widely used for a large range of applications such as ellipsometry1, bio-imaging2, imaging polarimetry3 and optical communications4. A commonly used method is the measuring of the time-depended signal once the beam is transmitted through a photoelastic modulator5 or a rotating quarter wave plate (QWP) followed by an analyzer6. The polarization state of the beam can be derived by Fourier analysis of the detected signal. This method, however, requires a sequence of consecutive measurements, thus making it impractical for real-time polarization measurement in an application such as adaptive polarization-mode dispersion compensation in optical communication. Moreover, it involves either mechanically or electronically active elements resulting in a complicated and cumbersome device. An increasing demand for faster and simpler methods has led to the development of the simultaneously four-channel ellipsometer7. Using this method, the beam is split into four channels; each is analyzed using different polarization optics while the real-time polarization state is calculated from the measured intensities. The main drawback of this method is its high sensitivity to statistical errors due to the low number of measurements. We have demonstrated a real-time near-field polarimetry by spatial discrete rotating of the groove orientation of a subwavelength dielectric grating.8 The grating of this type of element is divided into equal sized zones. The subwavelength grooves are of uniform orientation and period at each zone and are rotated at discrete angles, respective to each zone. The resulting elements are unlimitted in their dimensions and have uniform optical parameters. We named such elements as discrete space-variant subwavelength dielectric gratings (DSGs). Unlike other methods based on Fourier analysis, no active elements are required to determine the polarization state of an incident beam. Our method is less sensitive to statistical errors due to the increased number of measurements, and is suitable for real-time applications and can be used in compact configurations. It is possible to integrate our polarimeter on a two-dimensional detector array for lab-on-chip applications to achieve high throughput and low cost commercial polarimeter for biosensing. *gabbyb@tx.technion.ac.il phone: 972-4-8292151 fax: 972-4-8295711

Micromachining Technology for Micro-Optics and Nano-Optics III, edited byEric G. Johnson, Gregory P. Nordin, Thomas J. Suleski, Proceedings of SPIE Vol. 5720

(SPIE, Bellingham, WA, 2005) 0277-786X/05/$15 · doi: 10.1117/12.590143

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2. THEORY The concept of the near-field polarimetry based on subwavelength gratings is presented in Fig. 1. A uniformly polarized light is incident upon a polarization sensitive media (e.g. biological tissue, optical fiber, wave plate etc.) and then transmitted through a DSG, which acts as a space-variant wave plate, followed by a polarizer. The resulting intensity distribution is imaged onto a camera and captured for further analysis. It will show further that the emerging intensity distribution is uniquely related to the polarization state of the incoming beam. This dependence is given by a spatial Fourier series analysis, whereby the resulting Fourier coefficients completely determine the polarization state of the incoming beam. The DSGs are considered as wave plates with constant retardation and space varying fast axes, the orientation of which is denoted by ),( yxθ . It is convenient to form such space varying wave plates by use of a subwavelength grating. When the period of a subwavelength periodic structure is smaller than the incident wavelength, only the zero order is a propagating order, and all other orders are evanescent. The subwavelength periodic structure behaves as a uniaxial crystal with the optical axes parallel and perpendicular to the subwavelength grooves9-21. Therefore, by fabricating

Fig 1: Schematic presentation of near-field spatial Fourier-transform polarimetry based on a discrete space-variant subwavelength dielectric grating followed by a subwavelength metal polarizer. Also shown, the measured intensity distribution captured in a single camera frame when the fast axis of the rotating QWP was at angle 20 deg. (We used a CO2 laser that emitted linear polarized light and replaced the polarization-sensitive medium with a rotating QWP).

d

θ(x)

x

π

d/N

π/N

(a) (b)x

gK

θ

d Λ

Fig. 2: (a) The magnified geometry of the discrete space-variant dielectric grating with a number of discrete levels 4=N . (b) The discrete rotation angle of the subwavelength grating as a function of the x coordinate; the inset illustrates the local groove orientation.

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quasi-periodic subwavelength structures, for which the orientation of the subwavelength grooves is changed along the length of the element, space-variant wave plates can be formed. The creation of a DSG is done by discrete orientation of the local subwavelength grating as illustrated in Fig. 2(a). The DSG is obtained by dividing the element into N equal sized zones, along the x-axis, where each zone consists of a uniform orientation as well as a uniform subwavelength grating period. The orientation of the grooves is defined by the angle, θ, between the grating vector gK of the subwavelength grating (perpendicular to the grooves) and the x-axis, therefore θ is a function of the x coordinate [ )(xθ ] . The grating period, d, is defined as the distance between the nearest zones having identical orientations. We consider the period of grating d as larger than the incident wavelength λ , whereby the local subwavelength period of the grooves Λ , is smaller than the incident wavelength. Figure 2(a) presents a DSG with a period that consists of N=4 zones of uniform orientation of the subwavelength grating. We named N as the number of discrete levels. The polarization state within Stokes representation is described by a Stokes vector S=(S0,S1, S2, S3)T, where S0 is the intensity of the beam, whereas S1, S2, S3 represent the polarization state. In general, 2

322

21

20 SSSS ++≥ , where the

equality holds for fully polarized beams 6. The polarization state emerging from an optical system (i.e. wave plates, polarizers etc.) is linearly related to the incoming polarization state through, S’=MS, where M is a 4-by-4 real Mueller matrix of the system and S, S’ are the Stokes vectors of the incoming and outgoing polarization states respectively. The optical system under consideration consists of a DSG followed by a polarizer. This composite element can be described, in Cartesian coordinates, by Mueller matrix M=PR(-θ )WR(θ ), (1) where, )(θR is the Mueller matrix that represents rotation of the axis frame by angle θ, W is the Mueller matrix of a transversally uniform retarder 8, with retardation φ and real transmission coefficients for two eigen-polarizations tx, ty. Finally, P is the Mueller matrix of an ideal horizontal polarizer. The outgoing intensity can be related to the incoming polarization state of the beam by calculating the Mueller matrix given by Eq. (1) and using the linear relation between the incoming and the outgoing stokes vectors, yielding,

,))}(4sin)(4cos)(()(2sin)(

)(2cos)()({)('

2121

32

01121

041

0

xSxSCAxDSBS

xSSBSCAASxS

θθθ

θ

+−++−

+++++= (2)

where φφ sin2,cos2,, 2222

yxyxyxyx ttDttCttBttA ==−=+= . Equation (2) describes the intensity of the outgoing beam as a truncated Fourier series with coefficients that depend on the Stokes parameters of the incident beam. In our case θ represents the discrete rotation angle of the retarder, (subwavelength grating), as a function of the location along the x-axis. Figure 2(b) illustrates θ (x) for a number of discrete levels N=4. Note that θ (x) is periodic in π. Inserting the discrete angle θ (x) and using the Fourier analysis, the first coefficients of Eq. (2) yields,

,)('4)(0

0121

0 dxxSd

SCAASd

∫=++ (3a)

( ) ,2cos)('8)(0

0132101 dxd

xxSd

bDSBSaSSBd

∫

=−++

π (3b)

( ) ( ) ,2sin)('8

00101132 dx

dxxS

dbSSBaDSBS

d

∫

=++−

π (3c)

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( ) ,4cos)('16)(0

01211 dxd

xxSd

fSeSCAd

∫

=+−

π (3d)

( ) ,4sin)('16)(0

01211 dxd

xxSd

eSfSCAd

∫

=+−

π (3e)

where ( ) ππ 22sin1 NNa = , ( ) ππ NNb 21 sin= , ( ) ππ 44sin1 NNe = and ( ) ππ 22sin2

1 NNf = . These equations are a linear combination of the Stokes parameters of the incident beam. In order to extract S0 - S3, Eqs. (3a)-(3e) should represent 4 independent equations, which are obtained for 5≥N . However, a larger number of discrete levels, N, is desirable, in which case a larger portion of the intensity is represented by the first harmonics of every Fourier series expanded from every trigonometric term written in the right side of Eq. (2) and, therefore, a larger signal to noise ratio can be obtained. We note that the grating coefficients A, B, C, and D should be determined by direct measurement of the subwavelength grating parameters, tx, ty, and φ, or by performing a suitable calibration process.

3. REALIZATION PROCESDURE AND EXPERIMENTAL RESULTS

The DSG element for CO2 laser radiation of 10.6 µm wavelength was fabricated upon 500 µm-thick GaAs wafer with Λ=2 µm, d=2.5 mm, and N=16. The dimensions of the element were 30 mm×3 mm and consisted of 12 periods of d. First, a binary chrome mask of the grating was fabricated using high-resolution laser lithography. The pattern was then transferred by contact lithography to 500µm thick GaAs wafers (refractive index 3.27) onto which a 2000Å thick layer of SiNx had previously been deposited. The SiNx deposition was performed using plasma enhanced chemical vapor deposition (PECVD) at 900mTorr and 300ºC. At this stage, adhesion of a 700Å Ni layer was used for the lift-off process. Next, the SiNx layer was etched through the Ni stripes, which served as a mask. The etching was performed by reactive ion etching (RIE) for 30sec at room temperature with CF4 and O2 at gas flow rates of 35sccm and 15sccm, respectively, and at a pressure of 80mTorr. The GaAs was etched by electron cyclotron resonance (ECR) with the etched SiNx layer serving as a mask. The ECR conditions were: 20sccm of Cl2, 5sccm of Ar, 75W RF power, 600W microwave power and 100ºC. The remaining SiNx was removed using HF acid, resulting in an approximate 2.5 µm groove depth. Finally, an antireflection coating was applied to the backside of the wafer. Figure 3 shows a scanning electron microscopy images of three cross sections of the subwavelength grooves, which we had fabricated.

Following the measurements, we used the setup depicted in Fig. 1 to test our concept for polarization measurements, whereby, a subwavelength metal wire grating was used as a polarizer10. First, a calibration process was performed by illuminating the DSG followed by a polarizer with horizontal, vertical and right-handed circularly polarized light. The calibration process yielded a DSG having tx=0.9, ty=0.8 and retardation φ=0.3π rad. These values are close to both the theoretical predictions of tx=0.87, ty=0.81 and φ=0.352π rad that were achieved using a rigorous coupled wave analysis 22 for a groove profile depicted in Fig. 3, and to the direct measurement of the optical parameters using ellipsometric techniques 6, which resulted in tx=0.940, ty=0.833 and φ=0.281π rad with a standard deviation of about 0.02 for tx and ty

and of 0.022 rad for φ. We found that in our etching process, the errors in the etching depth were about 5% of the nominal depth (3.3 µm). Therefore, we can relate the deviations of tx, ty and φ to the spatial errors in the etching process.

1µm1µm

Fig 3: Scanning electron microscopy images of three cross sections of the subwavelength grooves.

1µm

2.52µm

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In order to test the ability of our device in conducting polarization measurements of fully polarized light, we used a CO2 laser that emitted linearly polarized light at a wavelength of 10.6µm and replaced the polarization-sensitive medium with a rotated QWP. The images were captured by a Spiricon Pyrocam III, 124×124 two-dimensional pyroelectric detection array, at a rate of 24 Hz. Figure 4 shows the experimental and theoretical azimuthal angle, ψ, and ellipticity χ as a function of the rotation angle of the QWP. The measurements yield a standard deviation error with respect to the theoretical prediction of 2.6° and 0.6° for the azimuthal angle and the ellipticity angle, respectively. The errors of the polarization measurements result mainly from systematic errors such as the nature of the algorithm, unprecision in the rotation of the QWP, spatial inhomogeneity in the QWP retardation, which is about ±2°, spatial inhomogeneity in the retardation of the subwavelength grating, which is about ±1.3° (8% of the nominal phase), low resolution of the IR camera and dynamic range, as well as statistical noise due to spatial and temporal fluctuation of the light emitted from the laser, shot noise and amplifier noise of the IR camera, pixel response nonuniformity, and quantization noise. The standard deviations in the azimuthal angle and ellipticity for a series of successive measurements were 0.2° and 0.07°, respectively.

In order to demonstrate the use of a DSG for polarimetry of partially polarized beams, two independent CO2 lasers of orthogonal linear polarization states were combined by use of the setup depicted in Fig. 5(a). The degree of polarization (DOP) is defined by

02

32

22

1 /DOP SSSS ++= . For incoherent beam summation, the Stokes vector of the resulting beam is the sum of the Stokes of the combined beams 6. In the case of two orthogonal linear polarized beams, the DOP is given by )/()(DOP 2121 IIII +−= , were 1I and 2I are the intensities of the horizontally and vertically polarized beams, respectively. Figure 6 shows the measured and predicted DOP as a function of the intensity ratio 21 / II . Figure 5(b) shows the experimental intensity distributions for two extreme cases. The first is for equal intensities ( 21 II = ), in which the measured DOP is of 0.0059, indicating unpolarized light. The second is for illuminating a single laser only (i.e. 02 =I ), in which the measured DOP results in 0.975, indicating fully polarized light. This experiment shows the

Fig 4: Measured (circles) and predicted results (solid curves) values for (a) azimuthal and (b) elipticity angles as a function of orientation of the QWP.

(a) (b)

Spatial Fourier transform Polarimeter

Polarization beamsplitter

I2 Laser 2

Laser 1 I1

TM

TE

DOP=0.059

DOP=0.975

rela

tive

inte

nsity

x(mm) Fig 5: (a) A schematic presentation of the setup used for measuring the polarization state of a partial polarized light. (b) Calculated (solid curves) and measured (circles) intensity cross-sections for two extremes, I1 = I2 (DOP=0.059) and I2 = 0 (DOP=0.975).

(a) (b)

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ability to obtain all four Stokes parameters simultaneously, thereby, emphasizing the good agreement between prediction and measurement for partially polarized light.

4. FAR-FIELD POLARIMETRY

Gori 23 proposed measuring the Stokes parameters by means of a polarization grating that was comprised of a linear polarizer, whose orientation varied periodically along a line. His analysis was done using Jones calculus. A more general case of polarization grating of a periodically rotating wave plate analyzed by using Jones calculus is presented in refs 8 and 10-18. The beam emerging from a polarization grating comprises three polarization orders. The first maintains the original polarization state and phase of the incident beam, the second is right handed circularly polarized and has a phase modification of )(2 xθ , and the third has a polarization direction and phase modification opposite to those of the former polarization order. Gori assumed )(xθ to be a linear function of x, in which case the second and the third polarization orders become 1 and –1 diffraction orders, respectively, and the first polarization order becomes the zeroth diffraction order. Gori showed that the decomposition performed by the polarization grating can be used for evaluating the Stokes parameters of a light beam. Suppose that a beam that is to be analyzed is sent through the polarization grating followed by a polarizer. At a suitable distance from the grating the three diffraction orders are spatially separated. By setting the polarizer orientation angle at the two values of 0° and 45° and measuring the corresponding intensities of the undiffracted-order beam together with the intensities of the 1 and –1 diffracted beams (at any angle of the polarizer) the Stokes parameters of the beam can be obtained. Figure 7 depicts the far-field intensities measured of a beam being passed through the polarization grating, a lens and a polarizer oriented at 0° and 45°. The polarization grating, which was realized by dielectric space-variant subwavelength gratings, was illuminated by a CO2 laser at the wavelength of mµλ 6.10= . Figure 7(c) shows a scanning electron microscopy image of a region on the subwavelength structure upon GaAs wafer. The measurements were taken in the focal plane of the lens as illustrated in the inset of Fig. 7(b). The polarization state of the measured incident beam was generated by transmitting a linearly polarized beam oriented at 0° through a QWP with its fast axis oriented at 20° with respect to the x axis. The measured azimuthal angle, ψ , and the ellipticity, χ of the incident beam were 16° and 20.7°, respectively, which is in good agreement with the predicted results.

5. CONCLUSIONS We have theoretically analyzed and experimentally demonstrated the use of a computer-generated space-variant discrete subwavelength dielectric grating for real- time polarization measurement. The discrete subwavelength gratings are unlimited in their dimensions, have uniform optical properties (tx, ty and φ), and in general, are lightweight, compact and highly efficient. Both the space-variant subwavelength dielectric grating and the polarizer were realized using

Fig 6: Calculated (solid curve) and measured (circles) DOP as a function of the intensity ratio of the two independent lasers havingorthogonal linear polarization states.

0 2 4 6

0.0

0.2

0.4

0.6

0.8

1.0D

OP

I1 / I2

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photolithographic techniques commonly used in the production of micro-electric devices. Therefore, the camera and the gratings could be combined into a single chip, resulting in a very small device for polarization measurements.

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10µm

Fig 7: Cross section of the measured intensity in the Fourier plane of a beam emerging from a spatially rotating QWP polarimeter followed by a polarizer oriented in (a) 0° and (b) 45°. The polarization of the incident beam was generated by illuminating a QWP oriented at 20° with a horizontal linear polarized CO2 laser beam at the wavelength of mµλ 6.10= . The inset depicted in (b) describes the concept of the far-filed polarimetry. The far-field polarimeter contains a discrete space-variant subwavelength dielectric polarization grating (D), a lens (L) and a polarizer (P). (c) is a scanning electron microscope image of a region on the subwavelength structure.

(b)

x (mm)x (mm)

Nor

mal

ized

Inte

nsity

(c)

-1

01

-1

0

1

PLDPolarizer in 45°

-1

0

1

(a) Polarizer in 0°

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