## Abstract

A new method to measure the light polarization state and the birefringent media parameters is proposed. We have used the setup described previously, consisting of two pairs of the linear Wollaston and circular compensators which form a set of two spatially modulated elliptical compensators. We have modified this setup introducing some carrier frequencies in all compensators and assuming that the second linear one would introduce the frequency which is a multiplicity of the basis frequency of the first linear compensator. Both of these modifications allow calculating all polarization parameters of polarized light or birefringent medium from only one measured intensity distribution of the light outcoming the described setup. They allow measuring not only the parameters of homogeneous beams/mediums but also *x*,*y*-distributions of all desired parameters, like azimuth and ellipticity angles of the light or first medium eigenvector and the phase difference introduced by this medium. The proposed calculation method comprises of Fourier analysis of obtained intensity distribution with some manipulation of coordinate system and filtration of obtained data. This method is claimed to be simple and fast enough to be treated as a real-time method.

©2009 Optical Society of America

## 1. Introduction

The main aim of all polarimetric systems is the measurement of the light polarization state. Even when these systems are constructed for measurement of the birefringent media parameters they effectively measure the changes in the polarization states of light passing through the investigated medium. The most important issue here is the proper selection of used testing light beams as well as the quickness and accuracy of the algorithms used to obtain desired quantities. So far many different polarimetric systems have been proposed. The methods which use rotating retardation plates and rotating polarizers/analyzers [1–6] belong to the classical systems. Latter systems use liquid crystal variable retarders [7–11] and more sophisticated methods of acquiring and processing obtained data like the interferometric configurations [12–15] or Fourier transforms [16–20]. The special class of polarization measurements is the imaging polarimetry [12,19,20], in which the spatial (2-D) distribution of birefringent media parameters could be obtained.

There are some tendencies in modern polarimetry which consist in increasing the measurements accuracy, the measurement process automatization and designing smaller and smaller compact devices without mechanically rotating or even active elements (for example liquid-crystal retarders) in the setup. The constructions of polarimeters based on birefringent wedge prisms [18,19,21] are interesting examples of these tendencies. Such wedges introduce a spatially modulated phase shift (instead of time modulation in the case of rotated retarders or liquid crystals) which allows total elimination of all active elements in the measurement setup.

In the present paper we describe a method of measurements of the light polarization state as well as the birefringent media parameter realized in the modified version of our previously proposed setup - spatial elliptical polariscope [21]. The main aim of the introduced modification is to obtain a carrier fringe in the light intensity distribution at the output of the setup which allows to apply the computer Fourier transform to calculate desired quantities. Using the method similar to the one proposed in [16] we can obtain 2-D distributions of all desired light or medium parameters from the analysis of only one light intensity distribution recorded at the output of the setup. This should enable us to carry out some dynamic measurements as there are no active elements in the setup which would have to be changed during the measurement process. We present the description of the proposed calculation method as well as some numerical simulations which should confirm its usefulness.

## 2. Carrier frequencies of spatial elliptical polariscope

Let us briefly remind the construction of spatial elliptical polariscope described in [21]. The setup consists of two modules, which we called all polarization state spatial generator (*PL _{1}C_{1}*) and polarization state spatial analyzer (

*C*)-Fig. 1.

_{2}L_{2}ABoth modules consist of the same set of birefringent prisms: classical, linearly birefringent Wollaston compensators (*L*
_{1,2}) and analogically constructed circular compensators (*C*
_{1,2}). P denotes the linear polarizer in the input of the setup while A stands for the linear analyzer in the output. All compensators are formed of two wedges made of linearly or circularly birefringent material, respectively. The wedge inclinations in linear and circular compensators in both *PL _{1}C_{1}* and

*C*elements are perpendicular to each other.

_{2}L_{2}AAs we have shown previously, due to the different phase shifts between its eigenvectors introduced in different points of *x*,*y*-plane, the first element (*PL _{1}C_{1}*) creates a light field with spatially modulated azimuth and ellipticity angles. Analogically, the second element (

*C*) can simply analyze all polarization states because there is always a point on

_{2}L_{2}A*x*,

*y*-plane in which the analyzer eigenvector is perpendicular to the Stokes vector of incident light. We can describe both elements in the same way: they have spatially modulated eigenvectors (azimuth and ellipticity angles) on

*x*,

*y*-plane. This modulation comes from variable thickness of all birefringent wedges and could be controlled by their inclinations. One can see these spatial modulations by simply putting every compensator between crossed (or parallel) linear polarizers and observing characteristic fringe patterns at the output. In our previous paper [21] we assumed that these spatial modulations have the same value in both linear compensators as well as the same (but maybe different from linear) in both circular compensators. Now we have decided to modify the setup by introducing different values of spatial frequency to linear compensators.

Let the shearing angles of all four compensators (*i*=*L*
_{1},*C*
_{1},*C*
_{2},*L*
_{2}) be denoted as *α _{i}* and their birefringences as Δ

*n*. We can then describe the spatial frequencies of all compensators as

_{i}where *λ* denotes the light wavelength and Λ* _{i}* are the distances between the above mentioned, characteristic fringes in all compensators. Let us also denote as

*δ*the phase shifts introduced by appropriate compensator, which linearly depends on the proper

_{i}*x*or

*y*spatial coordinate

Now we can describe both the *PL _{1}C_{1}* and

*C*elements behavior by introducing the

_{2}L_{2}A*V*⃗

*and*

_{PLC}*V*⃗

*eigenvectors of both modules which are, of course, spatially modulated. Assuming the light passing through the setup is fully polarized and all the setup elements are nondichroic we can replace all the elements transmission coefficients with one constant*

_{CLA}*T*and introduce a kind of truncated Stokes vector (without the first element responsible for light intensity) as follows

We have omitted the first element of Stokes vector due to some practical reasons. Firstly, this element is simply equal to 1 for all compensators’ and polarizers’ eigenvectors; secondly we have found this vectors’ form as more suitable for further calculations (see Eqs. (5) and (11) below).

The main extension of the proposed configuration is that now we essentially want to use their carrier frequencies as a new possibility to measure the light polarization state as well as birefringent media parameters. To do that we have assumed that: 1) all compensators can introduce the phase shifts of the multiplicity value of 2*π* and 2) the second linear compensator (*L _{2}*) introduces different phase shift than the first one (

*L*). In fact, the second condition is not important while explaining our first method: how to determine the light polarization state distribution?

_{1}## 3. Light polarization state distributions determination using Fourier analysis method

First, let us use the *C _{2}L_{2}A* module only to determine the light polarization state parameters. We believe that, as opposed to the method proposed in [21], we should determine not only the polarization parameters of homogeneous light beam but also the distribution of these parameters in the

*x*,

*y*plane. Assuming that the examined light is heterogeneous (in the sense of polarization parameters, not the intensity) yet fully polarized we can introduce the analogous truncated Stokes vector

*V*⃗, representing its

*x*,

*y*-distribution

where *α* (*x*, *y*) and *ϑ*(*x*, *y*) denote the azimuth angle and ellipticity angle *x*,*y*-distributions, respectively. Now we have omitted the first element of Stokes vector of the light due to the fact that this light is fully polarized and the polarization’s information is included only in presented three elements.

The intensity distribution *I* of the light outcoming from the analyzer *A* could be simply written as

where *I*
_{0}(*x*, *y*) denotes the intensity distribution of incident light beam and *T* denotes the common transmission coefficient of all the setup elements. Using this vectors representation (·stands for scalar product) makes the Eq. (5) simple and elegant; however, if we multiply *V*⃗_{CLA} by *V*⃗ we will obtain a long formula which consists of many sine and cosine functions with *α* (*x*, *y*), *ϑ*(*x*, *y*), ${\delta}_{{L}_{2}}\left(x\right)$ and ${\delta}_{{C}_{2}}\left(y\right)$ as arguments. Nevertheless, taking into account the Eq. (2) we can rewrite this formula in the representation known from Fourier formalism

where *c*.*c*. stands for complex conjugation and

(as a function of *α*(*x*, *y*), *ϑ*(*x*, *y*) and *I*
_{0}
*T*
^{2}) denotes the appropriate coefficients in this Fourier-like expansion. We can explicitly write two most important coefficients as

which can allow us to calculate the desired *α*(*x*, *y*) and *ϑ*(*x*, *y*) distributions. The main problem would now be: how to obtain (measure) these *a*
_{m,n} coefficients? We applied the method proposed in [16]. In the first step we made the numerical Fourier transform of measured intensity distribution *I*(*x*, *y*) which also allows us to obtain the Fourier transforms *A*
_{m,n}(*u*,*v*) of *a*
_{m,n}(*x*,*y*) coefficients. Then the computation algorithm should be as follows:

a) shifting the center of the coordinate system of the *u*,*v*-Fourier domain into the point representing given carrier frequency (red dots in Fig. 2; blue dots denote the frequencies which are uninteresting from the point of view of further computations);

b) masking the area near this frequency with low frequencies filter (green circles in Fig. 2) which means: filling the outside area with zeroes;

d) calculating the reverse Fourier transform, which gives us the desired *a*
_{m,n}(*x*,*y*) coefficient. We have repeated these steps to obtain all desired *a*
_{m,n}(*x*,*y*) coefficients. Note that only some of the systems carrier spatial frequencies are shown in Fig. 2 (only for the positive value of *u*,*v*-Fourier coordinates).

Having measured chosen *a*
_{m,n}(*x*,*y*) coefficients one can immediately obtain the desired *α*(*x*, *y*) and *ϑ*(*x*, *y*) distributions from

The spatial resolution of the *α*(*x*, *y*) and *ϑ*(*x*, *y*) measurements depends of course on the used carrier frequencies which in turn depend on the compensators shearing value. Higher frequencies can cause decreasing the measurements accuracy connected with the resolution of CCD camera used for data acquisition. The golden mean should be found which depends on the quality of used devices and expected spatial changes of measured light beam.

Let us emphasize that *I*
_{0}(*x*,*y*)*T*
^{2} term disappears during calculation process which simply means that the proposed method requires only one measurement of the intensity distribution of the output light. No additional measurements are required to calibrate the setup and nor any changes in measurement setup have to be made during the entire measurement process. This means that the process could be extremely short and the only time limitations could come from electronics used (CCD camera, computer). This should allow using the proposed setup in real-time measurements for fast changing light polarization states. We hope that the method described, which uses carrier frequencies, should be more interesting when applying to full polariscopic setup to determine the birefringent media properties distribution.

## 4. Determination of birefringent medium properties distributions

Let us now come back to the spatial elliptical polariscope (Fig. 1). As we have shown in [21] when we put homogeneous birefringent medium between spatial generator *PL _{1}C_{1}* and spatial analyzer

*C*we can obtain a specific intensity distribution at the end of the setup, in which two black points simply “show” us the azimuth and ellipticity angle of the first and second medium eigenvector. Also the phase difference

_{2}L_{2}A*γ*introduced by examined medium could be calculated from the intensity distribution analysis. Now we want to modify this setup and use the compensators with bigger shearing to obtain carrier fringes in the setup. Let us introduce the truncated version of the examined medium Stokes vector

*V*⃗

_{f}where *α _{f}* and

*ϑ*denote the azimuth and ellipticity angles of the first medium eigenvector. Now the normalized intensity distribution

_{f}*I*(

*x*,

*y*) at the output of the setup could we written as

where

and × stands for vector product. The Eq. (11) again seems to be simple but, in fact, when we calculate all vector products, it becomes a very long and complicated formula which consists of many sine and cosine functions with *a _{f}*(

*x*,

*y*),

*ϑ*(

_{f}*x*,

*y*),

*γ*(

*x*,

*y*), ${\delta}_{{L}_{2}}\left(x\right)$ and ${\delta}_{{C}_{2}}\left(y\right)$ as arguments. Nevertheless, we can again rewrite this formula in the representation well known from Fourier formalism (see Section 3, Eq. (6))

where the coefficients *a*
_{m,n}(*x*,*y*) are now the functions of *α _{f}*(

*x*,

*y*),

*ϑ*(

_{f}*x*,

*y*),

*γ*(

*x*,

*y*) and

*I*

_{0}(

*x*,

*y*)

*T*

^{2}(compare Eq. (7)). Due to the additional quantity

*γ*(

*x*,

*y*) there are much more different

*a*

_{m,n}(

*x*,

*y*) coefficients than in Eq. (6). And, unfortunately, if the carrier frequencies of all compensators are the same, all these coefficients contain only one cosine functions of

*γ*(

*x*,

*y*). This simply means that if we use the same procedure to obtain the desired birefringent parameters distributions as described in Section 3 then the term cos(

*γ*) could be treated formally as the constant factor

*I*

_{0}

*T*

^{2}and would disappear. It is easy to show that the solution is common to all measurements which involve Fourier transforms: another carrier spatial frequency in the setup should help. Then we propose the second modification of the setup: the second linear compensator (

*L*) should have the carrier frequency twice as big as the first one (

_{2}*L*). Although there are probably more possible configurations of both linear compensators frequencies as well as circular ones, we have carried out the detailed calculations for this particular arrangement. Now the explicit forms of some interesting

_{1}*a*

_{m,n}(

*x*,

*y*) coefficients are given by

We could again apply the algorithm described in Section 3. The positions of Fourier transforms *A*
_{m,n}(*u*,*v*) of *a*
_{m,n}(*x*,*y*) coefficients in this case are shown in Fig. 3.

Note that due to the multiplied carrier spatial frequency of the second linear compensator (in comparison to the first one) there are different *m*,*n* frequencies in *u*,*v*-Fourier plane.

And again, having measured the chosen value of *a*
_{m,n}(*x*,*y*) coefficients one can immediately obtain the desired *α*(*x*, *y*), *ϑ*(*x*, *y*) and *γ* (*x*, *y*) distributions from the following equations

Note that due to the only mathematical (not measured) equality of *a*
_{1,2} and *a*
_{3,2} coefficients we can use their mean value to increase measurements accuracy. It is likely that other sets of *a*
_{m,n} (*x*,*y*) coefficients could be chosen for even different ratio of linear compensators frequency. We would like to emphasize that the present paper describes only the main idea of the method; however, useful for some specific realization. We have chosen this realization to test the proposed method as well as the proposed algorithm by making some numerical simulations of possible measurements.

## 5. Numerical simulations and algorithm testing

We have decided to test the proposed algorithm through numerical simulations. We assumed that the input intensity distribution *I*
_{0}(*x*, *y*) could be represented by contaminated Gaussian beam to simulate all possible inaccuracies in the light beam forming module (Fig. 4(a)). Then we carried out a simulation of the light passing through our setup with some compensators carrier frequencies. Figure 4(b) presents the intensity distribution *I*(*x*, *y*) at the output of the system with a simulated birefringent medium inside the polariscope. Afterwards, according to the algorithm described in detail in Section 3, we made the Fourier transform of this intensity distribution and chose the proper points in frequency domain. These points became new origins of the coordinate system and only values of Fourier transform from the area inside the circles were chosen for next computations. This situation is illustrated in Fig. 4(c).

We have simulated a complex birefringent medium inside our setup to test the influence of the all parameters distributions: phase difference *γ*(*x*, *y*), ellipticity angle *ϑ*(*x*, *y*) and azimuth angle *α*(*x*, *y*). We assumed that our slightly complicated medium introduces linearly changing *γ*(*x*, *y*) in one direction, also linearly (but with different frequency) changing *ϑ*(*x*, *y*) in perpendicular direction and it has the same value of *α*(*x*, *y*) in whole input plane. This last condition could be a bit confusing but it should test the method’s stability in the sense of the possible cross-influences of all desired parameters in the proposed scheme of calculations. Next three figures (Fig. 5, Fig. 6, and Fig. 7) present obtained results: assumed (a) and calculated (b) distributions of all three important birefringent medium parameters *α*,*ϑ*,*γ* as well as the differences between calculated and assumed parameters (c).

We can see that all the parameters were calculated in accordance with the input data. As is always the case in polarization measurements the values of the ellipticity angle *ϑ* near to ± 45° cause the biggest errors – in these cases the azimuth angle *α* is indefinite and light blue areas in Fig. 5, which denotes the azimuth angles of even about 20°, have no physical meaning. Also the accuracy of phase shifts *γ* determination (Fig. 7) for these extreme values of *ϑ* decreases together with the accuracy of ellipticity angle *ϑ* determination for small (negligible) values of phase shift *γ*. And, finally, the accuracy of ellipticity angle *ϑ* determinations itself (Fig. 6) also decreases near this ± 45° special value. In fact, all these conclusions could be easily expected knowing the common problems of all polarization-based measurements.

We have only presented some theoretical considerations followed by numerical simulations which include only the possible imperfection of the input contaminated Gaussian beam. There are more sources of possible errors such as the imperfections of the incident angles of the beam passing through all the setup elements (we assumed normal incidence both in theoretical analysis and simulations); the inaccuracy in manufacturing of all Wollaston compensators (spatial constancy of the compensators wedge as well as their transmission coefficients); the influence of multiple reflections (more than 20 surface in full setup). However there are too many factors to take them into consideration in our numerical simulations which makes the simulation only a kind of verification of method’s behavior. We have left further analysis, including experimental verification, to our future research.

It is also obvious that possible changes of all polarization parameters distributions which could be measured are subjected to all Fourier transform conditions. The Nyquist-Shannon theorem is still valid and we have to apply the compensators with proper, large quantity of carrier frequencies to measure the light/media parameters with large heterogeneity. On the other hand, higher carrier frequencies could cause additional errors related to setup resolution (depends mainly on CCD camera resolution).

Due to the fact that only a single measurement has to be made in our setup the proposed method could be really fast and allow measuring the time-dependent processes of the birefringent parameters’ changes. Bearing in mind that no additional operations (like rotating or changing voltage supply) are required between two or more measurements, we claim this method (and the setup) as a real-time method. The time needed to obtain the final results from measured data (one intensity distribution) depends on two factors. The first one is connected with the CCD camera’s speed of data acquisition, expressed in fps (frame-per-second) value; the time needed to acquire every single distribution could reach even milliseconds. The second factor depends on the speed of the processor which realize FFT; our PC needs about 0.2 sec for every transformation while specialized calculating machines could reach milliseconds again.

## 6. Conclusions

The main aim of the presented work is a description of the new method of measuring light polarization parameters as well as birefringent media parameters in the modified setup of spatial elliptical polariscope. In fact, we have presented only a scheme of the method which could be modified depending of chosen carrier frequencies.

Our modifications of previously presented setup include introducing some carrier spatial frequencies to it. Firstly, we postulated the all compensators frequencies should be increased to obtain high carrier frequencies which allow us to measure not only the polarization parameters of homogeneous light beam/birefringent medium but also the *x*,*y*-distributions of these parameters. Secondly, we introduced two different frequencies in both linear compensators to allow independent measuring of the birefringent medium phase shift distribution. The second compensator should have the carrier frequency equal to multiplicity of the first one’s frequency. This enables measuring of all medium polarization parameters in one step without any calibration measurements as well as auxiliary measurements with some changes introduced to the setup.

Even with these modifications, the proposed setup still has all advantages of his ancestral: neither moveable parts (like rotating analyzers or retarders) nor active elements (like liquid crystal modulators) are needed. However, now we propose a different analysis of obtained data which should lead us to final solution: *x*,*y*-distributions of heterogeneous (in the sense of polarization parameters) light beams or birefringent media. Making the Fourier transform of obtained intensity distribution and applying the proposed algorithm allows measuring simultaneously all desired polarization parameters distributions: azimuth and ellipticity angle of polarized light or azimuth and ellipticity angles of birefringent medium eigenvectors as well as the distribution of the phase shift, introduced by this medium. We believe that due to the fact that only single measurement has to be made in our setup the proposed method would give a chance to measure all desired parameters really fast. We do hope that the described method and setup could be treated as a real-time method which allows measuring the changes of the light or mediums polarization parameters during some time-depended processes.

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