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Abstract
This work presents a description of a polarimetric system for measuring the properties of birefringent media. In our reflection system the applied Stokes polarimeter acts both as a generator of the light’s selected polarization states as well as a light analyzer leaving the examined medium. The method is based on six intensity distribution measurements realized in six different configurations of polarizers/analyzers: four linear and two circular ones. Thus, we have achieved parallel polariscope for linear polarizers and the crossed polariscope for circular polarizers. Such a setup can be easily applied for linearly birefringent media properties measurements including dichroic ones. This measurement setup and the measurement method were successfully tested in a homogeneous medium and a medium with variable phase difference.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
A birefringent medium could be described in eigenwaves formalism as a medium which introduces the optical path difference $\gamma $ between its two eigenwaves. This difference can change under the influence of external factors like stresses or magnetic and electric fields. However, by measuring the values of $\gamma $ one can obtain information about these factors. In many media some other polarization parameters (or their changes) are also interesting: the azimuth angle $\alpha $ and the ellipticity angle $\vartheta $ of the first eigenvector as well as the amplitude transmission coefficients ${T_f}$ and ${T_s}$ of the fast and slow eigenwaves – these parameters become more interesting when the investigated birefringent medium is spatially heterogeneous. There are many methods for the birefringence media properties determination: ellipsometric [1–6], polarimetric [7–10] and interferometric [11]. Some of them are direct methods, based on compensation of the phase difference introduced by the examined medium using the elements which introduce known phase difference (for example the Wollaston prism). There are also indirect methods where the changes of the light polarization state after passing through the medium are measured and, thus, the optical path difference is calculated [12–15]. Most of them are based on rotating elements [16–20], liquid crystal or electro-optic modulators [21–23], aperture splitting [24] or division of amplitude [25]. The general idea of all these methods is the same: the light with known polarization parameters pass through the examined medium and the changes in their polarization states (as well as in their intensities) are measured, which finally allow obtaining the Mueller matrix of the birefringent medium [26–28]. This so-called Mueller polarimetry is relatively easy when the testing light’s polarization states generator and the analyzing setup (for example, Stokes polarimeter [29]) are placed on opposite sides of the examined medium. However, if one wants to measure, for example, the birefringent parameters of a cornea of a human eye in vivo, both parts of the setup (generator and analyzer) should be placed on the same side of the eye [30,31]. Oblique lightning of the corneal surface allows separation of both systems, but the analysis of the results obtained is hindered by the variable optical path of the incident beam components and variable reflection conditions. This is why we decided to develop a measuring setup that uses the same polarization module as a generator and polarization state analyzer.
2. Theoretical background and measurement setup
The most general schematic diagram of the proposed system looks as follows: the collimated light emerging from the coherent source S falls on the beam splitter B. Next the light passes through the polarization state generator PSG (consisting of several birefringent elements) and through the examined medium M. The mirror R, placed just behind the medium, reflects the light back through the medium and the analyzing system (the same polarization generator PSG) to the detector (here: camera C). There is another lens before the recording camera – the role of this lens is to obtain an image of interference fringes (especially in case of investigated Wollaston prism) due to the finite coherence of the source. This configuration is shown in Fig. 1.
Fig. 1. Setup scheme: S – source of light, CL – collimating lens, BS – beam splitter, PSG – polarization state generator (consisting of polarizer P and two Liquid Crystal Variable Retarders LCVR1 and LCVR2), M – examined birefringent medium, R – reflective mirror, IL – image lens, C – camera.
Automation of the measuring system assumes that the output light intensities ${I_n}$ are the only ones measureable by the camera:
where ${I_0}$ is an input light intensity (in fact, in our further consideration we assume that this quantity also contains information about energy losses in the measurement system resulting from reflections and passing through its elements), ${\vec{V}_{P,n}}$ and ${\vec{V}_{A,n}}$ are the Stokes vectors of the n-th polarization state generator/analyzer, respectively and $M = M({{T_f},{T_s},\gamma ,\alpha ,\vartheta } )$ stands for the Mueller matrix of examined birefringent medium. The R matrix is responsible for the mirror's operation – it inverts the coordinate system. Our calculations can be simplified even further if we notice that the product $M \cdot R \cdot M$ can be replaced with the substitute medium matrix ${M_e}$, whose parameters ${T_{fe}}$, ${T_{se}}$, ${\gamma _e}$, ${\alpha _e}$ and ${\vartheta _e}$ are associated with the appropriate parameters of the searched matrix M with equations:
Table 1. Stokes vectors of the generator ${\vec{V}_{P,n}}$ and analyzer ${\vec{V}_{A,n}}$: H – linearly, horizontally polarized light, V – linearly, vertically polarized light, D – linearly polarized light with the azimuth angle + 45°, A – linearly polarized light with the azimuth angle -45°, R – circularly right handed polarized light, L – circularly left handed polarized light, γ1 – phase difference introduced by LCVR1, γ2 – phase difference introduced by LCVR2.
Let us note that the generators and analyzers vectors for the linear configurations (n = 1..4) are the same which means that we obtained four parallel linear polarimeters. In the case of circular states (n = 5.6), however, due to the fact that the mirror R rotates the local coordinate system, we are dealing here with circular crossed polarimeters.
By measuring the intensities ${I_n}$ (n = 1..6) in six described generator-analyzer configurations one can obtain a system of equations for some of the elements ${m_{ij}}$ (i,j = 1..4) of examined medium’s Mueller matrix:
3. The solution for linearly birefringent medium
Before the appropriate formulas that allow calculation of the desired parameters of a linear birefringent medium are presented, one more practical problem needs to be solved. As it was mentioned in Section 2, we need to establish the intensity of the input light ${I_0}$ as well as reflection/transmission coefficients of all the elements of our setup (splitting plate, LCVR’s, polarizer, mirror). Calibration measurement seemed the most appropriate solution – measuring six intensities ${I_{n0}}$ for six generator-analyzer systems in the absence of a birefringent medium in the system. In this case, the solution for investigated medium (here: air) is known a priori: ${T_{fe}} = {T_{se}} = 1$, ${\gamma _e} = 0^\circ $, ${\alpha _e}$ – arbitrary and ${\vartheta _e} = 0^\circ $. However, we don’t calculate the quantity ${I_0}$ from these measurement but use all six measured intensities ${I_{n0}}$ as known values of the substitute medium matrix ${M_e}$ (in fact, they are matrices, as we measure the spatial distribution of birefringent media parameters and use them to calculate the desired values ($\gamma $, $\alpha $, $\vartheta $, ${T_f}$ and ${T_s}$). The formulas for the azimuth angle of the first eigenvector ${\alpha _e}$, as well as introduced phase shift ${\gamma _e}$, are quite simple:
4. Measurements results
Sample measurements were carried out, the results of which are presented in order to confirm the correctness of the method. The light source is a red high-power LED diode, behind which there is an $\lambda = 650\,nm$ interference filter with a half width of about 5 nm (to increase source coherence) and a Carl Zeiss Jena DDR Tessar 4.5/300 photographic lens. One Melles-Griot linear polarizer and two Meadowlark Liquid Crystal Variable Retarders were used to form the polarization state generator PSG. The CMOS camera (Basler aCA1440-220um) with an additional lens was used as a detector to record the light output of the setup. Here we used another photographic lens made in Poland (PZO AMAR/S 4,5/105). The size of the scanned area was 500 by 900 pixels. Birefringent parameters of two media were measured: homogeneous retardation plate (quarter wave plate made by Meadowlark Optics AQM-100-0630, operating range 555-730) and heterogeneous Wollaston prism.
4.1 Homogeneous medium
We have used a retardation plate made from quartz. Due to the fact that our sample was high-order quarter wave plate for the wavelength 633 nm, our measurement results may have been further distorted by mismatching of the working wavelength. It was measured in various azimuth configurations to see if the measurement result of the introduced phase difference $\gamma $ and transmission parameters ${T_f}$ and ${T_s}$ will be similar while the measured azimuth angle $\alpha $ should change proportionally to the set angle. After the initial approximate azimuth orientation of the sample, we changed its azimuth angle in the range of 0 to 45 degrees every 5 degrees. Figure 2 shows histograms of calculated values of azimuth angle $\alpha $, phase difference $\gamma $ and transmission coefficients ${T_f}$ and ${T_s}$ for one of the azimuth settings of the tested sample. As one can easily see, the typical Gaussian histograms appeared. The calculated standard deviations are equal: $\delta \alpha = 0.36^\circ $, $\delta \gamma = 2.3^\circ $, $\delta {T_f} = 0.004$ and $\delta {T_s} = 0.004$, respectively.
Fig. 2. Histograms of calculated values of azimuth angle $\alpha $ (a), phase difference $\gamma $ (b) and transmission coefficients ${T_f}$ (c) and ${T_s}$ (d) for chosen azimuth settings of the tested sample.
The next three graphs show the calculated parameters of the measured sample as a function of its given input azimuth angle ${\alpha _{in}}$. As mentioned above, the azimuth angle ${\alpha _{in}}$ of the tested sample was approximated, so the calculated angles are slightly different. In addition, there are deviations from the linearity of the graph shown in Fig. 3 for the extreme values of the azimuth angle of the tested sample. This is due to the known problem of poor specificity of birefringent media parameters when tested using light with a polarization state corresponding to (or similar to) its own eigenstate. One should remember that calculations of birefringent medium parameters usually are carried out in two stages and we use trigonometric functions, so the constant errors in the values of these functions translate into different error values of the determined angular values.
Fig. 3. Dependence of the calculated azimuth angle $\alpha $ as a function of the set angle ${\alpha _{in}}$.
The results of calculations of the phase difference $\gamma $ are more interesting – in many applications only this quantity changes under the influence of external factors and this is usually measured. Figure 4 shows the calculated values of $\gamma $ for different azimuthal orientations ${\alpha _{in}}$ of the measured plate. Relatively large errors in determining each $\gamma $ values result mainly from the heterogeneous illumination of the system and strongly varying intensity in the entire observed field.
Our method was also tested for sensitive to azimuthal sample setting when calculating the values of eigenwave transmission coefficients ${T_f}$ and ${T_s}$ – the results are presented in Fig. 5. In fact, these coefficients do not describe the actual transmission but account for both absorption and reflection of the whole setup. The reflection coefficients, however, depend on the polarization state of the incident light (also reflection from the mirror can affect the calculated values of these coefficients). It is not surprising then that the calculated values are not constant but to same extent depend on the azimuth alignment of the sample relative to the six test lights of the PSG.
Fig. 5. Calculated transmission coefficients ${T_f}$ and ${T_s}$ as a function of the set angle ${\alpha _{in}}$.
The presented results support the correctness of our method and its expected accuracy after eliminating the indicated errors. The obtained results also confirm the agreement of the experiment result with the presented theory.
4.2 Non-homogeneous sample
We also decided to check how the tested setup will handle the measurement of a heterogeneously birefringent object. The Wollaston prism made of quartz with small wedge angle equal to 30 arc minutes was set with the vertical wedge axis. During the measurement, we observed that in a system of two parallel polariscopes, for which the polarization directions coincided with the natural directions of both components of the prism wedges (H and V), the entire view field of the camera was illuminated almost evenly (no fringes). However, in the case of the linear polariscopes in the D and A setup and circular polariscopes (R and L), characteristic horizontal fringes appeared and was observed by the camera. The calculated value of the azimuth angle (part a) as well as value of the phase difference (part b) in the cameras entire field of view are presented in Fig. 6. We also present a section through the selected vertical line of the image – both values ($\alpha $ and $\gamma $) in one figure (part c).
Fig. 6. Calculated azimuth angle α (a) and phase difference γ (b) for Wollaston prism in whole camera field; (c) a section through chosen vertical line of the left image for both parameters.
The obtained result is, of course, presented for $\gamma $ reduced to the range of 0 to 90 degrees. Remembering that Wollaston prism gives $\gamma $ growing linearly one should have unwrapped it. However, since we were satisfied with the view of almost perfectly straight (sections) $\gamma $ dependence on the vertical coordinate, the unwrapped procedures were abandoned. The truncated peak on the observed charts was, in turn, most likely due to overexposure of the image or non-linear behavior of the camera for high illuminance.
A Wollaston prism is a specific example of a birefringent medium with a variable phase difference. The fact that it came from the bonding of two linear birefringent wedges with perpendicular azimuth angles makes determining the azimuth angle $\alpha $ of such a resultant medium quite arbitrary. Therefore, the results of $\alpha $ measurement in Fig. 6, look not impressive – there are areas where the azimuth angle differs significantly from the assumed 0°. Comparing both parameters changes, however, one can see that the alleged values, strongly differing from the set values, are in places where the phase difference $\gamma $ is equal to 0°. This makes determining their azimuth angle by definition impossible.
4.3 Problems
The specific configuration of the proposed measuring system causes unusual problems. Their proper solution will make the system’s many practical application possible. The first and most important obstacle are light reflections from the surface of all the system’s elements. The sum of these reflections, although constant and possible to determine in the system’s calibration process, is large enough to significantly exceed the value of light intensity carrying information about the examined medium. This requires the use of a special camera with high dynamics of the received signal. In the conducted sample measurements, we rotated the polarizer, the first and last element of the PSG system, by a small angle along its azimuth axis. This directed the beam reflected from this element (and this is the first, largest and most disturbing reflection) outside the camera's field of view without any perturbations in the PSG operation. Another problem to be solved in this particular measurement with Wollaston prism was transverse beam separation due to the high birefringence, which calls for an appropriate selection of the interference plane. For this purpose, an additional lens was used that projects the interference plane of the resulting fringes in the detector plane.
5. Conclusions
We described a very specific polarimetric system for measuring polarization parameters of birefringent media. The system can be dedicated to the measurements of birefringence of the eye cornea, which causes an unusual measurement configuration as well as a non-standard measurement process. Due to the lack of access to the both sides of the examined medium, we had to use the light passing through the medium twice after reflection from the mirror behind it. To avoid oblique illumination of the examined medium, we decided to use the Stokes polarimeter twice – as a generator of selected test polarization states of light and (simultaneously) as an output light analyzer. Regardless of the Stokes polarimeter configuration – in our case it is a system consisting of a linear polarizer and two suitably oriented phase shifters – the measurement capabilities of the system are, in practice, narrowed down to linear birefringent media. Nevertheless, the presented setup correctly measures not only the phase difference introduced by the medium, but also the azimuth angle of its first eigenvector and the fast and slow wave transmission coefficients. Although the measured parameters ${T_f}$ and ${T_s}$ are (in our method) not strictly transmission coefficients, but rather factors describing the selective absorption and reflection of both medium eigenwaves, their values are interesting especially in the case of dichroic media (the cornea of a human eye might be such a case).
Both the presented measurement system and the method of measurements of the birefringent medium parameters have been tested for two specific cases: a homogeneous medium and a medium with a variable phase difference. The obtained results look satisfactory and the measurement process points to possible sources of errors and suggested solutions that will help in further setup’s improvements. Of course, applying it to measurement of polarization parameters of the human eye cornea will encounter further problems – the test medium is not a flat-parallel plate but has a high focusing ability. This will probably require the modification of the lighting beam geometry and certainly – the projection system. However, the solution developed seems promising and worth further research.
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