1. Introduction

The measurements of the light polarization state and birefringent media parameters play an important role in many fields of research like, for example, polarimetric testing of optoelectronic devices or photoelasticity. Several polarimetric systems have been proposed so far, which could be classified into the following groups: rotating retardation plates methods [1–3], rotating polarizers/analyzers methods [4–7] (with a kind of a hybrid method [8]), liquid crystal variable retarders methods [9–17], methods based on the interferometric configuration [18–21] or based on Fourier transform [22–26]. In fact, all these methods are based on the measurements of the change in the polarization state of light passing through the investigated medium. This means that the determination of the light polarization state becomes the main issue of all polarimetric measurements. Generally, all of them tend to increase the accuracy of measurement and to automatize the measurement process. The automatization should also be accompanied by designing smaller and smaller, compact devices without mechanically rotating elements. The good example of this tendency would be an application of arrays of photonic crystals, polarizers and/or wave-plates instead of rotating/moving elements [16,17]. Another recent tendency is to decrease the number of active elements (for example liquidcrystal retarders) in the setup. The constructions of polarimeters based on birefringent wedge prisms [24,25] are interesting examples of this tendency. Such wedges introduce a spatially modulated phase shift (instead of time modulation in case of rotated retarders or liquid crystals) which allows total elimination of all active elements in the measurement setup. Some standard techniques could be applied to the image obtained in these polarimeters to increase the accuracy or to simplify the data analysis (like Fourier analysis in [24]).

In the present paper we describe a compact, simple setup for determining the polarization state of the spatially uniform light wave and the birefringent homogeneous media parameters with the use of birefringent wedge prisms. Instead of the compensators used in the setup described above, for example, in [25] (Savart plates), [24] (a special kind of wedges), [16] (array of photonic crystals) and [17] (array of polarizers and wave-plates) we have used well known Wollaston compensators [27] as well as new Wollaston-like prisms made of optically active medium (for example quartz). No complicated image analysis is needed in case of the measurement of light polarization state and only some simple intensity measurements are necessary to measure the optical path difference introduced by birefringent medium. The theoretical analysis of the setup behavior and some numerical simulations are presented in this paper.

2. All polarization state spatial generator

Let us briefly present the behavior of the commonly used Wollaston compensator in the polariscopic setup. This compensator consists of two linearly birefringent wedges with orthogonal eigenvectors of both parts. If we place this compensator behind the linear polarizer with the azimuth angle equal to 45° (with respect to the azimuth angles of both Wollaston’s eigenvectors, assumed as 0° and 90°, respectively), we could obtain a light field with different polarization states. The azimuth angles of all states are equal to the initial value of 45°. The ellipticity angle of this light takes values from the range ±45° and it changes along the line parallel to the plane of wedges’ inclination. In most setups an analyzer with an azimuth angle equal to -45° (135°) is placed behind the Wollaston compensator which allows us to observe the set of parallel interference fringes. They shift when a linearly birefringent medium is additionally placed between the Wollaston compensator and the analyzer. This feature is used in the common application of this compensator - the measurement of the optical path difference in linearly birefringent media. However, we should always remember that it becomes a kind of a spatial polarization modulator which only modulates the ellipticity angle.

Taking into account consideration the above presented facts, we proposed a different type of a compensator: a circular one. By analogy, it should spatially modulate the azimuth angle of the light polarization state without changing the ellipticity angle. So one can construct this compensator of two wedges, made of circularly birefringent material. Quartz is a good example; when we cut the parallel plates with surfaces orthogonal to its optical axis, it appears to be a pure circularly birefringent medium (this kind of the birefringence is commonly called the optical activity). There are two types of quartz, existing in nature, with opposite signs of birefringence, which means that there exist two types of optical activity: right- and left-handed (and finally, the clockwise and counterclockwise rotation of the incident light azimuth plane). This means that this kind of Wollaston-like prism compensator made of both types of quartz could also be treated as a spatial polarization modulator - but now the azimuth angle of light would be modulated, while the ellipticity angle still would have the same, initial value. By the analogy to the classical setup with a Wollaston compensator one can use this new, circular compensator in the setup in which the optical path difference in circularly birefringent media could be measured.

The application of this circular compensator becomes more interesting, if we use it in combination with the Wollaston compensator. The scheme of the proposed device, which we called an all polarization state spatial generator, is presented in Fig. 1. The setup consists of three elements: the linear polarizer P with the azimuth angle equal to 45°; the Wollaston compensator L with the azimuth angles of both component prisms equal to 0° and 90°, respectively, and the shearing angle in x-direction; and the new circular compensator C with the shearing angle in y-direction and opposite optical activity of two wedges forming this compensator. Now the first compensator modifies the ellipticity angle of emerging light in x-direction while the second one modifies the azimuth angle of the light, but in perpendicular, y-direction. Let us write the space dependence of the phase differences for both compensators in the following form:

 

Fig. 1. Scheme of all polarization state spatial generator. P - linear polarizer, L - Wollaston compensator, C - circular wedge compensator.

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where Λ_L and Λ_L denote the characteristic space distances between repeatable polarization states for both compensators (the fringe distance in the classical Wollaston compensator placed between crossed linear polarizers; those quantities depend on the shearing angle of both compensators). Then, the relation between the polarization parameters of light emerging from the presented setup and their xy-coordinates could be written as:

The π4 term appears due to the chosen orientation of the input polarizer compared with the coordinate system - for x=y=0 (the center of both compensators, namely the point in which all four birefringent wedges have the same thickness) the polarization state of emerging light is the same as the polarization state of incident light, which is assumed as linearly polarized with azimuth angle equal to 45°.

Equations (2a)–(2b) show that we can obtain all possible polarization states in xy output plane of our setup which we called PLC (polarizer - linear compensator - circular compensator). To receive all the azimuth angles of the range [-90°, 90°] and the ellipticity angles of the range [-45°, 45°] it is enough to use zero-order L and C prisms, i.e. they should generate only the one-period phase shift variations.

This means that we constructed a simple and easy to arrange setup which could be called an all polarization states spatial generator. To show the usefulness of such construction we first use it in different, backward configuration as a kind of a new polarization state analyzer.

3. Polarization state spatial analyzer

Let us use the described above configuration in reverse direction - instead of PLC we denote it as CLA which means the following order: circular compensator - linear compensator - analyzer (Fig. 2). We called this setup the analyzer because the sequence of CLA elements is reversed with regard to PLC ones. Due to the similar construction of PLC and CLA modules, the analyzer eigenvectors are spatially modulated, i.e. their ellipticity and azimuth angles depend on x- and y- coordinates, respectively. If the polarized light with the azimuth angle α and the ellipticity angle ϑ is passing through this setup, there is only one point in xy-plane in which these parameters are matched to the parameters of the first eigenvector of CLA module. The intensity of light emerging from the analyzer reaches the maximum value in this special point. And there will be exactly one point in which those parameters are matched to the second eigenvector of CLA module (with orthogonal parameters α+90° and -ϑ) - now the intensity of light reaches the minimum value.

 

Fig. 2. Scheme of polarization state spatial analyzer. C - circular wedge compensator, L - Wollaston compensator, A - linear analyzer.

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Let us calculate (using for example Stokes and Muller formalism) the intensity distribution I(x,y) on the output plane of our analyzer as a function of α and ϑ parameters:

where I ₀ denotes the input light intensity and T_CLA is a combined transmission coefficient for all elements, and δ_L(x,y), δ_C(x,y) are described by Eqs. (1). Following these remarks it is obvious that there will be a single point on xy-plane in which this intensity is equal to zero (with xy-coordinates fulfilling the conditions):

Some exemplary results of the numerical simulations of this analyzer behavior are illustrated in Fig. 3. We have rescaled both axis descriptions in α,ϑ coordinates according to Eqs. (2) to make the analysis of those figures more comfortable. The intensity distribution I(x,y) is presented for the same azimuth angle α of incident light (here equal to -20°) and three different values of ellipticity angle: 15°, 35° and 45°. The points representing these initial values of the light polarization state are marked as white circles. As it was expected, the accuracy of the polarization parameters determination depends on the ellipticity angle of the incident light; for ϑ=0° [Fig. 3(a)] and ϑ=15° [Fig. 3(b)] there is a strongly separated minimum in I(x,y), while for ϑ=35° [Fig. 3(c)], this minimum spreads out on a wide area. For ϑ=45° [Fig. 3(d)] which simply means circularly polarized light, there is a line of minimums, parallel to α axis - as it was expected, so there is no possibility (and no necessity) to determine the azimuth angle at all.

 

Fig. 3. The output light intensity I distribution after the elliptical analyzer for constant azimuth angle α=-20° and variable ellipticity angles of the examined light: a) ϑ=0°,b) ϑ=15°, c) ϑ=35°, d) ϑ=45°. The points with the minimum light intensity are marked as white circles.

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There is also another interesting fact referring to this characteristic minimum intensity point. If we use Jones formalism and calculate the phase distribution in xy output plane of our space analyzer, we can easily show that this point becomes an optical vortex - the phase singularity [28] in the optical field. Figure 4 shows the phase distribution in xy output plane.

 

Fig. 4. The phase distribution of the output light for the intensity distribution presented in Fig. 3(b).

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This could be an interesting property of the proposed analyzer, but not studied in the present work.

4. Spatial elliptical polariscope

Let us put together the polarization state spatial generator and the analyzer as in a polariscopic setup. We can treat both PLC and CLA elements as a kind of a spatial elliptical polarizer and analyzer and put them in both classical configurations: parallel or crossed (which depends on the azimuthal orientations of the birefringent axis in all wedges and the physical orientations of the wedge inclinations). We will use the crossed polariscope in our further considerations. As it was mentioned in Section 2, if we illuminate the PLC module with a monochromatic, plane wave, the light emerging from the PLC element becomes fully polarized with the azimuth angle and the ellipticity angle changes linearly from point to point in xy-plane. Placing the CLA module with orthogonal azimuths of all the elements (while the shearing angles of respective elements are parallel) allows us to make a kind of an elliptical crossed polariscope. In all points in xy-plane the first PCA eigenvector corresponds to the second CLA eigenvector, which means that the light intensity will be equal to zero in all points of the detection plane (CCD camera).

If we put an elliptically birefringent medium between PLC and CLA (Fig. 5) this medium changes, in general, the polarization state of the light entering CLA. If the polarization state of the light entering the birefringent medium is of the same α and ϑ parameters as the first or the second medium eigenvector, these parameters remain the same in the light emerging from the examined medium. This causes no change in light intensity in two points in detection plane - and this intensity still remains minimal in these points. It allows us to see directly the α and ϑ parameters in xy-coordinates which could be simply scaled in α and ϑ units.

 

Fig. 5. Scheme of the spatial elliptical polariscope. PLC - all polarization state spatial generator, M - elliptically birefringent medium, CLA - polarization state spatial analyzer, CCD - CCD camera.

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There is still one important parameter to determine - the phase shift γ introduced by the birefringent medium. A simple analysis of the intensity distribution on I(x,y) allows us to conclude that:

1) the phase shift γ dependence of the light intensity maximum value in detection plane is given (as in all commonly used polariscopes) by the formula:

where C is a constant dependent on the transmission coefficients of all setup elements and it should be determined in a calibrating process;

2) the intensity distribution I(x,y) reaches the maximum value in many points which lie on the curve with a shape dependent on α and ϑ parameters; however, one of these points lie exactly in the middle of the section connecting the characteristic points with the minimum intensity of the output light.

Figure 6 shows the numerical simulations of the proposed spatial elliptical polariscope for the assumed values of the birefringent medium first eigenvector parameters α=75° and ϑ=-30° and phase shift equal to 100°, respectively. The characteristic points of the minimum intensity representing two eigenvectors are marked as white circles, while the point with the maximum intensity is marked as a black circle. As in case of a polarization state spatial analyzer, described in Section 3, both axis descriptions have been rescaled in α, ϑ coordinates, which allows simply to observe the azimuth and the ellipticity angles of two medium eigenvectors. The measurement of the light intensity in this chosen point allows us also to obtain the last parameter - the phase shift γ.

 

Fig. 6. The output light intensity I (a) and phase (b) distributions for elliptically birefringent medium with the azimuth angle α=75° and the ellipticity angle ϑ=-30°. Points representing parameters of two medium eigenwaves (and where the output light intensity reaches the minimum) were marked as white circles, while the point where the output light intensity reaches the maximum value - as black circle.

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The accuracy of the measurements of all α, ϑ and γ parameters depends on their value, as in all commonly used setups. As it was mentioned in Section 3, the accuracy of the azimuth angle α determination depends on the value of the ellipticity angle ϑ and makes the determination almost impossible for ϑ near to ±45°; however, the azimuth angle becomes an unimportant parameter in case of (near) circular media. And also the accuracy of the phase shift γ measurement decreases for the values of γ for which the sinus function reaches its extreme values.

5. Conclusions

The main aim of the presented work is a description of the new setup established to measure the birefringent media parameters. However, both the synthesis and analysis modules used in the final construction could be useful devices, worth to be analyzed and applied in future.

We should emphasize that the term spatial used in the name of all the presented constructions stands for their spatially changed optical path differences. In fact, the light polarization state and the birefringent medium measured in the proposed setups should be homogeneous (spatially-uniform). To measure the local birefringent medium parameters or light polarization state parameters distributions the image polarimetry methods should be applied.

The construction of all polarization state spatial generator PLC could be useful in all the methods in which some initial values of the incident light polarization parameters are necessary. Also in case in which no continuous modulation of the azimuth angle and ellipticity angle is required. Some simple optical system could be used to allow projection of the chosen polarization states onto the incident plane of examined medium and a kind of the polarization state scanning is still available.

The authors are convinced that the proposed construction of a polarization state spatial analyzer presented in this work could be really useful not only for its potential accuracy but mainly for its simplicity and clarity. Neither moveable parts (like rotating analyzers or retarders) nor active elements (like liquid crystal modulators) are needed. No complicated analysis of output light should be made, like the fringe analysis or Fourier transform. However, there is still a possibility to apply some image analysis technique to increase the setup accuracy. Such techniques are useful in case of another possible application of the proposed device, not yet developed by the authors. If, for example, two incoherent light beams with different polarization states pass through our analyzer, the intensity distribution I(x,y) will take a more complicated form; however, there still will be some characteristic points in this distribution (maybe differentiated only in phase image as optical vortices), which allows us to measure the polarization state of both beams.

Finally, both synthesizer and analyzer advantages were taken into account in the process of the construction of a new spatial elliptical polariscope. Maintaining the simplicity of the observation and measurements of the azimuth and ellipticity angle of examined birefringent medium, this setup allows us to measure the most important parameter - the phase shift γ introduced by the birefringent medium between its first and second eigenvectors. In the most precise version this measurement requires some kind of an image analysis, but it is always nice to simply see the birefringence α, ϑ and γ parameters on the CCD camera image.