## Abstract

This paper describes a method for the mapping of spatially-distributed states-of-polarization of light with a simple and compact configuration. A tiny block of polarization-analyzing optics, consisting of four thin birefringent wedge prisms and a sheet analyzer, are incorporated into an imaging polarimeter, such that mesh-like multiple fringes are generated over a CCD image sensor of a video camera. Fourier analysis of the obtained fringes provides information for determining the two-dimensional distribution of the state-of-polarization. No mechanical or active elements for analyzing polarization are used, and all the parameters related to the spatially-dependent monochromatic Stokes parameters corresponding to azimuth and ellipticity angles can be determined from a single frame. The effectiveness of this method is demonstrated by a prototype incorporating calcite wedge prisms.

©2003 Optical Society of America

## 1. Introduction

Two-dimensional state-of-polarization (SOP) mapping of light plays an important role in many fields, such as photoelasticity [1], polarimetric testing of optoelectronic devices [2], machine vision [3], and remote sensing [4]. In conventional imaging polarimetry, polarization-analyzing optics, consisting of polarizers, rotators, and retarders, is usually used together with a CCD video camera. The SOP distribution is determined from several images, each of which is acquired at various settings of the polarization-analyzing optics. Unfortunately, the optical component that controls the polarization-analyzing optics, such as a rotating analyzer or a liquid-crystal retarder, generally includes either a mechanical element or an active element. These components often bring with them various problems such as vibration, heat generation, and spatial co-registration, and hence require considerable care be taken for precise operation. In addition, the incorporation of such polarization-controlling components into the imaging optics almost always results in a considerable increase in volume and complexity of the measurement system.

While recent progress in the design of elements for polarization control has remarkably enhanced the performance of imaging polarimeters, the development of imaging polarimeters without the need for mechanical or active elements for polarization control remains a beneficial objective. One approach to this objective is to use either micropolarizers or microretarders [5, 6]. With the rapid progress in microlithography, it is now possible to fabricate out of semiconductor materials subwavelength periodic structures that work as wire-grid polarizers or retarders. Imaging polarimetry can be realized with the arrayed micropolarizers or microretarders placed just in front of the CCD image sensor of a video camera. Most of the current research is confined to the infrared region because of the difficulties in the fabrication of the subwavelength structures, but some trials have been made for shorter wavelengths, namely the visible region.

Another approach for the development of imaging polarimetry without mechanical or active elements is the use of mesh-like multiple fringes. In this approach, multiple interference patterns are generated at the CCD image sensor such that each interference fringe carries the information of a different polarized component of the light under measurement. The spatial frequency filtering of each fringe allows us to demodulate the spatial distribution of the SOP. Based on this concept, one of the present authors has developed an interferometric imaging polarimeter using a reference beam consisting of two orthogonal linearly-polarized components [7–9]. A similar idea has been adopted by Colomb *et al.* for their holographic imaging polarimeter [10]. Developing the idea even further, another type of imaging polarimeter with multiple fringes was also developed using a couple of Savart plates [11, 12].

Recently, the present authors proposed a revolutionary alternative method for imaging polarimetry that obviates the need to control the polarization-analyzing optics. This method incorporates a tiny block of thin birefringent prisms and a sheet analyzer to generate multiple fringes carrying the information of the SOP under interest. The major advantage of this method is that all of the optical components required for the polarization analysis are stable and very compact, and can be incorporated within the housing of the CCD video camera. In addition, all of these components are made from standard passive materials for polarization optics. Part of this idea has been presented at previous conferences [13, 14]. The objective of this paper is to describe the principles of operation in detail as well as give the results of an experimental demonstration with a prototype incorporating calcite wedge prisms. It should also be noted that the present method evolved from channeled spectroscopic polarimetry which is capable of measurements of spectrally-resolved SOPs [15] from a single spectrum. Its fundamental idea is extended and applied to the measurement for the spatially-distributed SOPs.

## 2. Performance principle

A schematic of the developed imaging polarimeter is illustrated in Fig. 1. A beam of light from a He-Ne laser source is expanded and spatially filtered by means of a lens pair and a pinhole. It then passes through a polarizer. This expanded linearly-polarized beam is launched into a birefringent sample plate with spatial irreguralities. The transmitted wave turns into elliptically-polarized light, the SOP of which varies with position over the sample. Our objective is to determine the spatial distribution of the SOP as a function of two-dimensional *x* and *y* coordinates.

The light emerging from the sample passes through an imaging lens so that the output face of the sample is imaged over a CCD image sensor. A block of polarimetric devices, consisting of four birefringent wedge prisms PR_{1}, PR_{2}, PR_{3}, and PR_{4} and an analyzer A, is placed just in front of the CCD image sensor. The fast axes of PR_{1}, PR_{2}, PR_{3}, and PR_{4} are oriented at 0°, 90°, 45°, and -45° respectively relative to the *x*-axis, and the transmission axis of A is aligned with the *x*-axis. The planes of contact between PR_{1} and PR_{2}, and between PR_{3} and PR_{4}, are inclined with respect to the *y* and *x* axes with an angle *α*, whereas the other contact planes as well as the input face of PR_{1} are parallel to the *xy* plane. For now, we assume that the wedge angle *α* is small enough that the refraction occurring at the inclined contact surfaces is negligible.

We now formulate the analytic expression of the polarimeter. It is useful to think of the block of the polarimetric devices as composed of three elements, namely the pair of PR_{1} and PR_{2}, the pair of PR_{3} and PR_{4}, and the analyzer A, as illustrated in the blue, yellow, and gray parts of Fig. 2. The first and second elements act as retarders whose retardations linearly vary with *y* and *x* coordinates, respectively. The orthogonal principal axes of the first element are coincident with the *x* and *y* axes, whereas those of the second element are oriented at ±45° to the *x*-axes. Let *S*
_{0}(*x,y*), *S*
_{1}(*x,y*), *S*
_{2}(*x,y*), and *S*
_{3}(*x,y*) be the spatially-dependent Stokes parameters of the light emerging from the birefringent sample. Mueller calculus of the optical system allows us to derive the analytic expression for the intensity pattern formed over the CCD image sensor as

$$+\frac{1}{4}\mid {S}_{23}(x,y)\mid \mathrm{cos}\left\{2\pi U\left(x-y\right)+\mathrm{arg}\left[{S}_{23}(x,y)\right]\right\}$$

$$-\frac{1}{4}\mid {S}_{23}(x,y)\mid \mathrm{cos}\left\{2\pi U\left(x+y\right)-\mathrm{arg}\left[{S}_{23}(x,y)\right]\right\},$$

with

and where *B* and λ respectively denote the birefringence of the prisms and the wavelength of the incident light. This implies that the obtained interference pattern consists of one low-frequency component and three quasi-cosinusoidal components which can be seen as fringes in the spatial pattern. One fringe has the period 1/*U*, and the other two fringes have the period 1/(√2*U*). The low-frequency component is proportional to *S*
_{0}(*x,y*), whereas each fringe carries the information of either *S*
_{1}(*x,y*) or *S*
_{23}(*x,y*).

We next consider how to demodulate the Stokes parameters from the obtained interference pattern. The Fourier transform method modified for the extraction of complex amplitudes from an interference fringe is suited for this purpose [7–9]. The Fourier transformation of Eq. (1) immediately gives the Fourier spectrum of the interference pattern as

$$+\frac{1}{8}{A}_{23}({f}_{x}-U,{f}_{y}+U)+\frac{1}{8}{A}_{23}^{*}(-{f}_{x}-U,-{f}_{y}+U)$$

$$-\frac{1}{8}{A}_{23}^{*}({f}_{x}-U,{f}_{y}-U)-\frac{1}{8}{A}_{23}(-{f}_{x}-U,-{f}_{y}-U),$$

where

Here *f _{x}* and

*f*denote the spatial frequencies and

_{y}*ℱ*stands for the operator of the two-dimensional Fourier transformation. The seven components included in

*Ĩ*(

*f*), which are centered at (

_{x}, f_{y}*f*)=(0,0), (±

_{x}, f_{y}*U*,0), (±

*U*,∓

*U*), and (±

*U*,±

*U*), are satisfactorily separable from one another over the spatial frequency domain if the inclination angle

*α*of the birefringent wedge prisms are properly selected. It follows that any one of the components can be extracted by spatial-frequency filtering. The Fourier inversion of the first, second, and fourth terms of Eq. (4) respectively give

The phase modulations exp[*i*2*πUx*] and exp[*i*2*πU*(*x*–*y*)] included in *F*
_{1}(*x,y*) and *F*
_{23}(*x,y*) are independent from the SOP of the light that is being measured, and hence can easily be calibrated in advance by use of light with a known SOP. As a result, we can determine all the Stokes parameters from *F*
_{0}(*x,y*), *F*
_{1}(*x,y*), and *F*
_{23}(*x,y*). It should be noted that any other parameters related to the spatial distribution of the SOP of the light can be computed from the demodulated Stokes parameters. For example, the *x* and *y* dependent azimuth angle *θ* and ellipticity angle *ε* can be computed following Azzam and Bashara [16] as

## 3. Experimental procedure

To demonstrate the performance principle, a prototype of the imaging polarimeter was fabricated. Figure 3 is the photograph of the fabricated imaging polarimeter. The left-hand side of the photograph is the imaging lens, Micro Nikkor (*f*=50 mm) with an extension ring, whose magnification is set to be unity. The right-hand side of the photograph shows the CCD video camera incorporating the birefringent wedge prisms and the analyzer. It should be noted that the polarization-analyzing devices are wholly included within the housing of the CCD video camera, and cannot be seen in this photograph. The CCD video camera, Hamamatsu C3077, possesses a 2/3 inch-size CCD image sensor, and provides the NTSC-formatted analog video signal. This signal is sent to a computer to be digitized into the image consisting of 512 pixel × 512 pixel, each pixel corresponding to the area of 12.9 *µ*m square over the image sensor of the CCD video camera. The block of the four birefringent wedge prisms made of calcite and one polarcor analyzer is placed just in front of the image sensor. Its height and width are 12 mm × 12 mm, and the total thickness is less than 2 mm. The wedge angle *α* is set to be 1.5°, which determines the periods of the fringes formed over the image sensor as 1/*U*=72.5 *µ*m (5.62 pixel) and 1/(√2*U*)=51.3 *µ*m (4.07 pixel). It should be noted that the wedge angle *α* is selected so that the beam displacement at the image sensor due to the double refraction at the inclined interfaces do not exceed the pixel size.

A 90° twisted nematic liquid crystal cell was supplied for the experiment as a birefringent sample. Part of the cell is covered with patterned transparent electrodes so that electric fields can be applied across the cell. When light passes through the regions without the electrodes, the azimuth of the polarization ellipse rotates by 90°. On the other hand, the light passing through the regions with electrodes having sufficient electric fields experiences no change in its SOP. Since only part of the liquid crystal cell is covered with the transparent electrodes, the SOP of the light transmitted through the cell is spatially-dependent. The SOP distribution was measured with the fabricated imaging polarimeter.

## 4. Results and discussion

Figure 4 shows the intensity pattern *I*(*x,y*) obtained when the sample was illuminated with a linearly-polarized collimated wave whose polarization azimuth angle is -67.5°. In this figure, a 208 pixel × 190 pixel portion of the intensity pattern is extracted and enlarged for the convenience of seeing the details of the intensity pattern. The mesh-like fringes carry the information of all the spatially-dependent Stokes parameters. This figure implies that the SOP of the region inside the transparent electrode of the shape ‘A’ is different from that of the region outside the electrode. This intensity pattern is Fourier transformed to give the spectrum *Ĩ*(*f _{x}, f_{y}*) by means of the fast Fourier transform (FFT) algorithm. The power spectrum |

*Ĩ*(

*f*)|

_{x}, f_{y}^{2}is shown in Fig. 5. This spectrum consists of seven peaks. The peak around (

*f*)=(0,0) possesses the information of

_{x}, f_{y}*S*

_{0}(

*x,y*), as designated by Eq. (4). Two peaks centered at (

*f*)=(+(512/5.62),0)=(+91.1,0) and (

_{x}, f_{y}*f*)=(-91.1,0) carry the information of

_{x}, f_{y}*S*

_{1}(

*x,y*), and the rest of the peaks at (

*f*)=(+91.1,+91.1), (+91.1,-91.1), (-91.1,+91.1), and (-91.1,-91.1) are determined by

_{x}, f_{y}*S*

_{23}(

*x,y*). The seven peaks included in the spectrum

*Ĩ*(

*f*) are successfully separated from one another in the spatial frequency domain, and hence can easily be extracted from the spectrum.

_{x}, f_{y}The spatial-distribution of the Stokes parameters are demodulated from the respective peaks in the spectrum *Ĩ*(*f _{x}, f_{y}*), and then the azimuth and ellipticity angles are computed by use of Eqs. (11) and (12). Figure 6 represents the three dimensional plots of the obtained azimuth angle

*θ*(

*x,y*) and ellipticity angle

*ε*(

*x,y*), and Figure 7 shows their cross section, cut at the red dotted planes in Fig. 6. It should be noted that these paired angles are suited for the assessment of the light transmitted through the twisted nematic liquid crystal cell. That is because the twisted nematic liquid crystal does not affect the ellipticity angle of the transmitting light; only the azimuth angle changes while passing through the liquid crystal cell. As can be seen from Fig. 6, an apparent difference exists between the azimuth angles of the light which does and does not pass the electrode of the ‘A’ shape. On the other hand, the ellipticity angle is almost constant over the observation area except for the edge of the electrode. The cross section shows that the azimuth step is almost 90°, and the ellipticity angle is everywhere almost 0°. This implies that the light which has passed the electrode is a linearly-polarized wave with an azimuth of -67.5°, which is almost the same as that of the incident light, whereas the light which has not passed the electrode experiences 90° rotation in its polarization azimuth. It should be noted that some artifacts are found around the edge of the electrode; the width of the artifact is around 10 pixels. This determines the spatial resolution of the fabricated system. Aside from the artifacts, the result agrees well with the function of the 90° twisted nematic liquid crystal, and hence proves the performance principle of the present method.

One of the major advantages of this method is that no mechanically moving elements for polarization control or active elements for polarization modulation are used to measure the spatially distributed SOP. This implies that the developed polarimeter is free from the problems associated with these elements, such as vibration, heat generation, and spatial co-registration. In addition, the developed imaging polarimeter has quite a simple and compact configuration. All the polarimetric components required can be wholly incorporated within the housing of the CCD video camera without increasing its volume. Hence, this method is feasible for many types of imaging systems. Another advantage of this method is that any parameters related to the spatially-dependent SOP of the monochromatic light, including the Stokes parameters and the paired azimuth and ellipticity angles, are determined from only the single intensity pattern taken from the CCD image sensor. With conventional imaging polarimetry, several (at least three) intensity patterns taken with different settings of the polarization-analyzing optics are required to offer one SOP distribution. The conventional method thereby inevitably requires synchronization between the circuits controlling the polarization-analyzing optics and those acquiring the image from the CCD video camera. In contrast, the present method releases us from any difficulties in such synchronization, since the polarization-analyzing optics’ state does not need to be changed. This method also has the potential to reduce the image blurring effect if it is combined with a stroboscopic light source or a CCD camera equipped with an electronic shutter. The above mentioned features are quite beneficial for many application areas of imaging polarimetry. This method allows us to fabricate imaging polarimeters more simply and compactly than conventional methods, enabling the imaging polarimeters to be easily incorporated into various kinds of application systems with only a slight modification of their imaging optics. Although the present method requires somewhat complicated digital signal processing, it can easily be implemented with standard FFT routines. Therefore, this method has the potential to promote the expansion of the application areas of imaging polarimetry.

It should be noted, however, that the current configuration of the calcite wedge prisms is not suited for the application areas which require higher spatial resolution. In the method presented here, the four spatially-dependent Stokes parameters are modulated over the fringes in the single intensity pattern. This implies that the spatial variation of the SOP of the light that is being measured should vary gradually compared to the period of the fringes. If the SOP abruptly changes at a certain point in the observation area, this method cannot offer correct results around the point. That is why some artifacts are found around the edge of the electrode in the experimental results. To increase the spatial resolution, the fringe period should be minimized. However the present configuration of the calcite prisms is not feasible to offer such finely spaced fringes. Although the fringe periods can be decreased with an increase in the wedge angle *α*, this increase also causes the significant beam splitting effect due to the double refraction at the inclined interface of the prisms, much like the function of Wollaston prisms. For applications which require higher spatial resolution, another configuration using wedge prisms with smaller beam displacement should be selected [17]. Detailed discussion about the optimization of the configuration and the parameters for higher spatial resolution will be presented elsewhere.

Finally, some discussion of the applicabilities of this method to time-varying objects is in order. It is obvious that the present method is useful for stationary objects, and may also have some time-varying applications, because no rapidly controllable mechanical or active components for polarization-analyzing are required. The present method can be conveniently applied to time-variation measurements with the above-mentioned features of simplicity and compactness remaining intact. Each frame taken from the video camera offers an SOP distribution at a different time, and hence the temporal resolution to follow a change in SOP is determined only with the frame rate of the CCD video camera. It should be noted, however, that this fact does not directly suggest the superiority of the measurement speed of the present method. In general, many high-speed video cameras increase frame rate by sacrificing pixel numbers. The present method requires a video camera with many pixels to acquire the fine structure of the mesh-like intensity pattern, and is therefore not compatible with many high-speed cameras. When this method is used to follow a rapid change in SOP distribution, it is necessary to consider not only the frame rate but also the pixel numbers of the video camera.

## 5. Conclusion

A revolutionary method has been developed for the measurement of the two-dimensional SOP distribution. Introducing a tiny block of polarization-analyzing optics, consisting of four thin birefringent wedge prisms and a sheet analyzer, into the CCD video camera makes it possible to measure the SOP distribution without using mechanical or active optics for the polarization analysis. The performance principle was verified by the demonstrative experiment for the 90° twisted nematic liquid crystal cell.

## Acknowledgements

The authors are grateful to Noshi Baba and Tsuyoshi Ishigaki, for many helpful discussions. They are also grateful to Hiroyuki Kowa of Uniopt and Shinji Yasuda ofMiyuki Kogaku for the help in fabricating the polarimeter. They thank Euctace L. Dereniak, Nathan Hagen, and Tod C. Picone for their useful comments for the manuscript.

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