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Abstract
Mueller matrix ellipsometry has been used to precisely characterize quartz waveplates for demanding applications in the semiconductor industry and high precision polarimetry. We have found this experimental technique to be beneficial to use because it enables us to obtain absolute and precise measurement of retardation in a wide spectral range, waveplate orientation, and compound waveplate adjustment. In this paper, the necessity of including the optical activity in the Mueller matrix model and data treatment is demonstrated. Particularly, the optical activity of the quartz influences the adjustment of misalignment between the perpendicularly oriented waveplates of the compound biplate. We demonstrate that omitting the optical activity from the model leads to inaccurate values of the misalignment. In addition, the depolarization effects caused by a finite monochromator bandwidth is included in the model. Incorporation of the optical activity to the Mueller matrix model has required a development of rigorous theory based on appropriate constitutive equations. The generalized Yeh’s matrix algebra to bianisotropic media has been used for the calculation of the eigenmodes propagation in chiral materials with reduced symmetry. Based on the applied method, the authors have proposed approximated analytical form of the Mueller matrix representing optically active waveplate and biplate and provided discussion on the analytical and numerical limits of the method.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Waveplates are polarization optical components with a broad field of applicability across the semiconductor industry [1,2], quality inspection in manufacturing processes [3,4], THz spectroscopy [5,6], analytical chemistry [7–9] and polarimetry [10–13]. From a polarimetric point of view, the waveplates are commonly used in optical spectrometers as compensators [14]. It is no exception to use the waveplates during high-precision-demanding procedures like calibration of various custom experimental setups [15–18]. Then, the quality of the used waveplates directly affects the quality of acquired data [19–24] and above mentioned industrial technological properties.
Typically, industrial inspection of manufactured waveplates is based on a configuration with two crossed polarizers and the waveplate being placed between them [25]. The method determines the orientation of the waveplate fast-axis and its single-wavelength relative retardation. The waveplate retardation is given as the difference between phases of two orthogonal linearly polarized eigenmodes propagating through the plate [26,27]. Waveplates of higher order exhibit high chromatic dispersion, however, broadly used zero-order waveplate would be undesirably thin. To overcome this inconvenience, the waveplates are composed by two thicker higher order plates with crossed optical axes [28], so called biplates. However, this method may be insufficient nowadays, as the users demands on the waveplates precision fabrication raises significantly. Therefore, precise characterization of the composed waveplates in order to obtain the parameters of each of the constituting waveplate can be challenging, and proper treatment has to be established as various non-trivial effects may come to raise.
In our previous paper [29] we have shown the advantage of complete Mueller matrix inspection and we have demonstrated that spectral characterization and industrial inspection of waveplates requires the inclusion of the depolarization effects [30–32]. The Mueller calculus and Mueller matrix spectroscopic ellipsometry is a particularly advantageous [10] method for proper waveplate analysis as it is spectroscopic, non-destructive, and capable of incorporating all the mentioned phenomena due to high experimental precision [33,34]. In Ref. [29] we have also shown the presence of the nonzero circular birefringence related to optical activity (OA) using the Lu-Chipman decomposition [35] of the quartz waveplates experimental data.
In this article, the Mueller matrix spectroscopic ellipsometry in transmission configuration is used for precise quartz single waveplate and quartz waveplate biplates characterization. We exclusively measure the waveplates aligned with the optical axis perpendicular to the propagation direction and we demonstrate, that the effect of the OA can not be neglected to process the experimental data correctly. In studies [36,37], Mueller matrix ellipsometry was applied to study waveplate multiplets. Its model simplicity has an advantage in the calibration procedures involving the waveplates including their misalignment error. As long as the method doesn’t account for the OA dispersion in the waveplates, the net elliptical properties of the waveplate effectively project into other waveplate characteristics causing their quasi-oscillatory behaviour. Therefore, we derived a rigorous model of the Mueller matrix including the gyration effects from the theory based on Fedorov’s [38,39] and Yeh’s [40] work and the Condon-Fedorov’s constitution relations [38,41]. The rigorous model including the OA is compared to the experimental data for single waveplates and biplates. We further show, that the OA effect is separable and independent of other waveplate (biplates) characteristics like plates misalignment. All those parameters are obtained with very high numerical precision and great fit stability.
The paper is structured as follows: In Sec. 2, the rigorous Mueller matrix models for optically active single waveplates and waveplate biplates are derived. Section 3 describes the biplate samples and experimental configuration. Measurement analysis is divided into two steps. First of all, we model the parameters of single waveplates to use them during the biplate analysis afterwards in order to obtain precise biplate parameters including their misalignment. To demonstrate the necessity of using a Mueller matrix model with OA, we compare the experimental data with the models with and without OA. To understand the OA contribution in the Mueller matrix, in Appendix A., we generalize the standard Yeh’s matrix algebra to bianisotropic and optically active media using various constitutive relation formalism, and in particular, eigenmodes for uniaxial chiral media are calculated in Appendix B.. Detailed derivation of the proposed Mueller matrix form is given in Appendix C.. In Appendix D., we offer an algebraic analysis on the manifestation difference between the misalignment and the OA in the proposed Mueller matrices.
2. Mueller matrices of waveplates including optical activity
Mueller matrix represents the complete polarimetric information, which can be acquired from the interaction between the sample and the polarized light, including depolarizations.
2.1 Mueller matrix of a standard linear retarder
The reduced $4 \times 4$ Mueller matrix (all elements are divided by $M_{11}$) describing a single waveplate consisting of a uniaxial anisotropic material with the optical axis parallel to the waveplate surface at the normal incidence (linear retarder) is given [14] as
2.2 Mueller matrix of a single waveplate with optical activity
Natural or synthetic quartz monocrystal is very common material for the waveplate manufacturing. It belongs to uniaxial trigonal crystal symmetry [42,43] (Hermann-Maugin 32 point group) and is therefore optically active [44–46]. Across the literature, the effect of the OA in quartz (or quartz waveplates) has been studied in the configuration with the quartz optical axis being parallel (z-cut) to the direction of propagation [47] so far. Usually, the effect of the OA with optical axis perpendicular to the propagation direction is said to be experimentally negligible [48], which is inadequate for waveplate precise quality and properties control.
In our preceding work [29], the presence of non-zero circular birefringence was discovered using Lu-Chipman decomposition of experimental Mueller matrix, which we interpreted as the presence of the OA. Rigorous incorporation of the OA in waveplates requires (i) a solution of Maxwell equations based on a proper set of material equations introducing the gyration tensors $\mathbf {\hat {\alpha }}$ and $\hat {g}$ with respect to the 32 point group which leads to the wave equation of the problem. Definition of the constitutive equations and a detailed discussion of suitability of various approaches is given in Appendix A., (ii) calculation of generally elliptical eigenmodes propagating in the waveplate based on the wave equation evaluation (see Appendix B.), (iii) derivation of Mueller matrix including the OA as a function of the chirality parameter
In this Section, we show the model of the Mueller matrix including the OA effect from the Jones matrix of an elliptical retarder, see Appendix C., Eq. (33). Eigenmodes propagating through a medium described by Eq. (33) are ellipticaly polarized and are propagating with corresponding refractive indices. Since the effect of the OA is very small, the elliptical eigenmodes can be approximated as linear-like eigenmodes. Therefore, it is justifiable to linearly approximate $\kappa$, as long as it is a function of eigenmodes ellipticity, see Appendix C., Eq. (34). Moreover, the indices of refraction of the linear-like eigenmodes become equal with high precision to ordinary and extraordinary indices of refraction, respectively, see Appendix C., Eq. (31). The Mueller matrix of an elliptic retarder $\mathbf {M}_{\textrm{R,EL}}\left (\varphi ,\,\Gamma ,\,\kappa \right )$ is in the linear approximation in the form
In our approximation, we treat the quantity $\kappa$ as a real parameter. This is in a conformity with the usual quartz waveplates application range over its transparent region. The Jones matrix (33) is valid generally for any of the parameter being complex, in contrast with the Mueller matrix (5). This is due to the form (Kronecker product) of the transformation Eq. (36). Therefore, in the vicinity of absorption bands, the Eq. (5) must be calculated assuming complex $\kappa$. The measurements over this spectral region is out of the scope of the paper and will not be further elaborated.
The experimental validity condition of the linear approximation is obtained by analyzing the quadratic (erroneous) terms in the Taylor series of $\boldsymbol{\Delta}\textbf{M}_{\textrm{OA}}$ focused on the maximal possible error. The criterion is then set to $8\kappa ^2 < \sigma$, where $\sigma$ stands for an error of the experimental Mueller matrix elements. Note, that in our experimental configuration, $\sigma = 0.001$ implies the condition for $\kappa < 0.0112$. This criterion is fulfilled for the quartz waveplates [47] in the spectral range of further shown experiments.
The non-approximated matrix $\mathbf {M}_{\textrm{R,EL}}$ stands for an equivalent solution with Mueller matrices of an elliptical retarder presented in [49] [Eq. (22)], in Ref. [50] [Eq. (14)] or on page 289 in Ref. [51]. Our proposed solution applied to the subject discussed in this paper provides the opportunity to have deeper physical insight into the problematic and to get more intuitive way of the result interpretation and discussion, as we show later. Also, in contrast with Eq. (22) in Ref. [49], we reduce the number of needed fitting parameters.
2.3 Mueller matrix of a biplate
The Mueller matrix for a biplate given by the product of Mueller matrices $\mathbf {M}_{\textrm{R,EL1}}$, $\mathbf {M}_{\textrm{R,EL2}}$ each representing a standalone waveplate with its own retardation $\Gamma _{1}$, $\Gamma _{2}$, respectively, and azimuthal angle $\varphi _{1}$, $\varphi _{2}$, respectively, is in the form
Each waveplate is manufactured from the same material, therefore we assume the same optical activity dispersion, $\kappa _{1} = \kappa _{2} = \kappa$.
Note, that from now on, we will focus on the $3\times 3$ Mueller submatrix $M_{22}\dots M_{44}$ only, due to the structure of (1) and above discussed Mueller matrices.
3. Measurements and data processing
We applied spectral ellipsometric measurements using the dual rotating compensator Mueller matrix ellipsometer RC2-DI from Woollam company, which measures a full Mueller matrix in the spectral range from 0.74 to 6.42 eV (wavelength ranging from 193 to 1700 nm). The waveplates are measured in the normal-incidence transmission geometry.
We have investigated biplates consisting of a stack of two c-cut right-handed synthetic $\alpha$-quartz high-order waveplates belonging to the $32$ point group. The biplate constituting waveplates are attached to each other, having its fast axes mutually perpendicular, by perfectly polished surfaces, and therefore hold together only by the presence of Van der Waals forces. This fact allows us to split the biplate without causing damage, and measure each waveplate of the biplate separately to obtain its accurate parameters (thicknesses, retardation angles).
3.1 Single waveplate measurements
The measurements of both single waveplates constituting the biplate were performed in the normal incident transmission configuration for all azimuthal angles from $0^{\circ }$ to $360^{\circ }$ by the angle step of $5^{\circ }$. Figure 1 shows typical spectra (black curves) of the Mueller $3 \times 3$ submatrix for the high-order single waveplate. The oscillations correspond to the spectrally dependent wavelength retardation. The waveplate order $m$ (e.g. $m=9$ for $\lambda = 500$ nm) determines the oscillations density, which gets denser with increasing photon energy, while the oscillations amplitude decreases. The decreasing amplitude modulation for high photon energy is due to the dense oscillations spectral averaging caused by a finite monochromator bandwidth [52,53]. In our case, this corresponds to the spectral resolution of dispersion grating and finite pixel size of the CCD detector. The attenuation compensation is based on the incoherent summation of Mueller matrix model, which is commented in our preceding work (Eqs. (3) and (4) in [29]) in the detail. The spectral step we use in the averaging function is $\delta \lambda = 1.075\,\textrm{nm}$.
Fig. 1. Mueller $3\times 3$ submatrix of the single waveplate of thickness $d_{1} = 536.63\,\mu$m at the azimuthal angle $\varphi _{1} = 180.078^{\circ }$. The data are compared to the model of standard linear retarder (without the effect of OA) and to the model including the OA effect. Model with the OA effect disregarded is not capable to properly describe both the dispersion and the attenuation exhibiting in the first row and the first column of the Mueller submatrix off-diagonal elements. Note, that the axes of those Mueller matrix elements are scaled by the factor of $1/50$.
The data are compared to the model (1) including spectral averaging (blue curve) over high energy spectral range and shows a relatively good fit of the Mueller matrix elements $M_{33},\,M_{34},\,M_{43},\,M_{44}$. For azimuthal rotations, for which the waveplate has its axes nearly perfectly aligned with the laboratory coordinate system, the oscillation amplitude within the measured elements $M_{23},\,M_{24},\,M_{32},\,M_{42}$ decreases with decreasing photon energy over low energy spectral ranges. Together with the modulation of the oscillations dispersion within these elements, it corresponds to the effect of the optical activity dispersion in the quartz waveplate. To properly fit all the elements of the Mueller submatrix, we model the influence of OA (red curve) using (5) with the spectral averaging included. The parameters $A_{1} = 0.0277\,\textrm{nm},\,B_{1} = 105.6\,\textrm{nm}$ of (4) are taken from Ref. [47]. The fitted thicknesses for each waveplate are $d_{1} = 536.63\,\mu$m, $d_{2} = 549.03\,\mu$m, and absolute retardations were obtained. The dispersion of $n_{\textrm{o}},\,n_{\textrm{e}}$ are taken from Ref. [54,55]. Note, the OA effects are very small in comparison with the linear retardation, therefore $M_{23},\,M_{24},\,M_{32},\,M_{42}$ Mueller matrix elements are expanded 50 times. Despite the fine effect of the OA, the response in the measured Mueller matrix is well detectable and shows high sensitivity of the proposed method.
3.2 Fine control of waveplate biplate misalignment
Biplate consists of two high-order waveplates with crossed optical axes to produce an effective low-order waveplate. Possible imperfections during the biplates manufacturing can introduce a misalignment between the waveplates, as they may not be aligned to each other with their optical axes being perfectly perpendicular, see Fig. 2. The misalignment between the waveplates is then defined as
We have shown [29], that Mueller matrix approach is significantly sensitive to the waveplate misalignment. In this Section, we propose a robust and sensitive inspection method for the biplate misalignment determination based on a rigorous Mueller matrix model including OA.
Fig. 2. Schematic configuration of the waveplates constituting the biplate. Their optical axes are ideally perpendicular, $\varphi _{1} - \varphi _{2} = 90^{\circ }$. The manufacture inaccuracy causes the deviation from this perpendicularity, and introduces the missalignment $\Phi$, as shows the inset figure.
To simulate the misalignment between the waveplates of the biplate, one waveplate was placed fixed to the ellipsometer sample holder, while the second waveplate was rotated around the relative angle of $90^{\circ }$ between the waveplates.
Figure 3 shows the Mueller submatrix spectra of the biplate with controlled $\varphi _{1}$ (and therefore controlled $\Phi$) ranging from $185.7^{\circ }$ to $186.1^{\circ }$ with step $0.1^{\circ }$ ($\Phi$ is ranging from $-0.2^{\circ }$ to $0.2^{\circ }$ with step $0.1^{\circ }$). The Mueller $3\times 3$ submatrix (right part of the Fig. 3) shows the dominant effect of the linear retardation dispersion without the dense spectral oscillations, which are typical for high-order waveplates. The presence of minor oscillations corresponds to combination of the misalignment between waveplates of the biplate and its optical activity. Left part of Fig. 3 shows the detail of the Mueller submatrix first column. Each curve of the same color corresponds to different $\varphi _{1}$, and therefore to different misalignment. First, the data are compared to the Mueller matrix model of the cascade of two standard linear retarders (7) without including the OA (blue lines), $\kappa = 0$, which is insufficient for the proper analysis of the misalignment between two waveplates. In contrast, the model (7) including the OA (red lines), $\kappa \neq 0$, shows very good fit of the measured data. The slight mismatch from the data is just above experimental error of the Mueller matrix elements (0.001). At this scale, the errors are inevitable. They could origin in the calibration quality, systematic errors, non-ideal plates parallelity, waveplate miscut or precision of used material dispersion parameters. The parameters $A_{1} = 0.0277\,\textrm{nm},\,B_{1} = 105.6\,\textrm{nm}$ of (4) are taken from Ref. [47], and the waveplate thicknesses ($d_{1} = 536.63\,\mu$m, $d_{2} = 549.03\,\mu$m), and retardation dispersions were determined from the measurements of each of the separated waveplates. The fitted parameters $\varphi _{1}$ and $\varphi _{2}$ were obtained from the fit of each biplate measurement independently. The Fig. 4 shows the waveplate misalignment obtained for the data fitted to Eq. (8) point by point. The fitted values of $\varphi _{1}$, $\varphi _2$ and $\Phi$ corresponding to each curve in Fig. 3 are shown in Table 1.
Fig. 3. Experimental Mueller submatrix for biplate. Each curve of the same color represents different value of $\varphi _{1}$ ranging from $185.7^{\circ }$ to $186.1^{\circ }$ with step $0.1^{\circ }$. Comparison between data and models with and without the effect of the OA shows the significance of the the OA phenomenon in proper determination of the misalignment, which is exhibited by the presence of the minor oscillation shown in the detail of the submatrix first column. Note, that the energy range from 4.1 eV to 4.5 eV corresponds to the wavelength range from 302.4 nm to 275.5 nm.
Fig. 4. a) Fitted values of the misalignment for each measurement with various azimuthal rotations of the waveplates. b) Fit stability of the misalignment determination is demonstrated comparing the calculated angle steps to the nominal value of $0.05^{\circ }$. Red points correspond to selected measurements presented in Fig. 3.
Table 1. Fitted values of $\varphi _{1}$, $\varphi _2$ and $\Phi$ for representative Mueller submatrix spectra shown in Fig. 3.
Figure 4(b) demonstrates the misalignment fit stability. Each point was calculated as a subtraction of the misalignment from consequent values during one waveplate azimuthal rotation. The mean value of the angle step corresponds well with the actual experimental angle step of $0.05^{\circ }$, which shows high stability of the misalignment determination without any correlation to the optical activity. Red points demonstrate selected measurements presented in Fig. 3.
Ultimately, it is important to mention, that excluding the presence of the OA from the model is leading to the misinterpretation of the OA with incorrect misalignment values. We show the linear expansion of the Mueller matrix of biplate (7) for both the misalignment $(\Phi )$ and the OA $(\kappa )$ in Appendix D. to understand the significant difference between oscillations induced by $\kappa$ and $\Phi$. Both misalignment (45) and OA (46) dispersions have similar behavior in the form of minor oscillations shown in Fig. 3. These oscillations origin in the high-density trigonometric terms including only the retardation $\Gamma _1$ or $\Gamma _2$. If one tries to fit the data using the model without the OA (demonstrated for our case), the misalignment of around $0.5^{\circ }$ will tend to compensate the OA with its amplitude. Nonetheless, such model does not match the data over the whole spectral range, because $\kappa$ has spectral dependence given by Eq. (4), and thus, the amplitude of the oscillations changes. Such model without the OA also results in erroneous waveplate thicknesses, because the terms of $\textbf {m}_\kappa$ and $\textbf {m}_\Phi$ matrices are defined by terms in cosines or sines of the waveplates retardation. It is curious, that sine terms of one matrix correspond to appropriate cosine terms of second matrix and vice versa. Therefore, the fit without the OA will result in thicknesses imperfections for both waveplates by 6.4 $\mu$m to compensate the phase shift. However, their thickness difference remains the same due to the major influence of the linear retarder (1). It results in the fact, that low-density trigonometric terms of the retardation difference $\Gamma _1-\Gamma _2$ are not able to be compensated and thus make significant difference in the spectrum. Nevertheless, the sine and cosine terms are not only shifted one from another, but they also represent different functions which do not match the data, especially if both misalignment and the OA influence are present.
4. Conclusion
In this article, biplates represented by a couple of high-order waveplates with crossed optical axes have been measured by means of Mueller matrix ellipsometry in the spectral range from 193 to 1700 nm. In order to obtain accurate values of biplate parameters including the biplate misalignment, we theoretically revealed and experimentally confirmed, that it is necessary to include the effects of the optical activity into the derived model of Mueller matrix. To obtain the exact solution of the Mueller matrix, it is needed to find the eigenmodes and propagation constants of the optically active medium represented by the quartz waveplate in this case. We emphasize, that omitting the optical activity from the model would lead to the incorrect interpretation of the optical activity phenomenon with the inaccurate values of the waveplate misalignment.
We believe, that the proposed method, which has an advantage in its great experimental sensitivity, fit stability and repeatability, could stand as a fast non-destructive industrial analysis for the accurate in-situ control of the biplates misalignment during the manufacturing process. Moreover, the concept based on the rigorous model can be extended for the analysis of the waveplate multiplets. In addition, waveplates parameters obtained from precise Mueller matrix characterization can be further used for the accuracy improvement of calibration procedures of various experimental setups involving waveplates and Mueller calculus based calibration procedures.
A. Wave equation in chiral media
A.1. Covariant form of Maxwell equations
In the following, the plane monochromatic wave is described by the electric field in the form
where $\omega$ is the angular frequency and $\mathbf {k}$ is the wave vector related to the refraction vector $\mathbf {N}$ using the wave number in a vacuum $k_{0} = \omega / c$, where $c$ is the light speed in a vacuum, asThe curl Maxwell equations for the monochromatic plane waves (9) relating the electric and magnetic fields $\mathbf {E}_{0},\,\mathbf {H}_{0}$ to its displacements $\mathbf {D}_{0},\,\mathbf {B}_{0}$, respectively, can be written in the covariant form
where $\varepsilon _{0}$ and $\mu _{0}$ are the permittivity and permeability in a vacuum, respectively, andThe direction of wave propagation is given by the wave vector $\mathbf {k} = k_{0} \left [N_{x}\,\,N_{y}\,\,N_{z} \right ]^{\textrm{T}}$, and a set of constitutive relations must be chosen.
For a general anisotropic gyrotropic (bianisotropic) and magnetic media, various forms of constitutive equations have been developed [56] and compared to each other [57,58]. Nowdays, Tellegen [59] and Condon-Fedorov [38,41] material equations seem to be the most convenient and acceptable form of constitutive relations.
A.2. Tellegen constitutive relations
Tellegen relations take the form
The material tensors are in the form
where $\mathbf {\hat {\varepsilon }}_{\textrm{r}}, \mathbf {\hat {\mu }}_{\textrm{r}}$ are the relative permittivity and the relative permeability tensors, respectively, and $\mathbf {\hat {\xi }},\,\mathbf {\hat {\zeta }}$ are general second-rank coupling tensors, where we use the convention [60]Parameter $\hat {\nu }$ is related with the reciprocity of the medium, while $\hat {\eta }$ describes the optical activity. Substituting (13) into (11) and eliminating $\mathbf {H}_{0}$, the wave equation for $\mathbf {E}_{0}$ in the most general form is obtained:
A.3. Condon-Fedorov constitutive relations
For reciprocal chiral media ($\hat {\nu } = 0,\,\hat {\eta } \neq 0$) a set of Condon-Fedorov constitutive relations is commonly used:
The wave equation is then in the form
Special case for bi-gyrotropic media was described in Ref. [67].
A.4. Born-Landau constitutive equations
Equation (20) stands for a definition of the gyration tensor $\hat {g} = g_{ij}$, which describes the optical activity effect in Born-Landau [68] constitutive relations
A.5. Relation between the constitutive equations for optically active media
By the comparison of (15) with (18) we can see, that Condon-Fedorov equations are a special case of Tellegen material relations for reciprocal chiral media:
Here, we justly state, that Condon-Fedorov and Born-Landau approaches are not fully equivalent. We strongly recommend the kind reader to see the detailed discussion in Ref. [69]. However, for our particular case of transparent waveplates, both constitutive equations can be applied. Moreover, within our (and generally achievable) experimental precision, the difference is negligible and these two approaches can be used in the equivalent way providing the same results for any, even absorbing material [69].
B. Eigenmodes propagating in chiral uniaxial media of point group 32
The manifestation of the OA for light propagation along the optical axis of nonmagnetic uniaxial gyrotropic media of 32 point group (i.e. quartz) is well-known fact. We emphasize, that the OA effect in quartz is given intrinsically because of its non-centrosymmetricity. Also, the OA has similar size along the principal axes of $\mathbf {\hat {\alpha }}$ (or $\hat {g}$) [47], and is therefore non-negligible within the precision of contemporary Mueller matrix ellipsometry. Concluding, the OA is present in all possible experimental configurations (or cuts) of the quartz plate including light propagation perpendicular to the optical axis. In the following, we will discuss two important special orientations of the crystal optical axis.
For the propagation along the optical axis, the material tensors in (19) reduce to $\mathbf {\hat {\varepsilon }}_{\textrm{r}} = \textrm{diag}\left [ \varepsilon _{11}\,\,\varepsilon _{11}\,\,\varepsilon _{33} \right ]$, $\mathbf {\hat {\mu }}_{\textrm{r}} = \textrm{diag}\left [ 1\,\,1\,\,1 \right ]$, and $\mathbf {\hat {\alpha }} = \textrm{diag}\left [ \alpha _{11}\,\,\alpha _{11}\,\,\alpha _{33} \right ]$. Assuming normal incidence only, $\mathbf {N} = \left [ 0\,\,0\,\,N_{z} \right ]^{\textrm{T}}$. The refractive indices of corresponding circularly polarized eigenmodes $N_{z,\,\alpha ,\,k=\{1,2,3,4\},\,\parallel }$ have been published several times [48,57,66], and take the form
Propagation perpendicular to the optical axis is carried out introducing the rotation tensor of the Cartesian coordinate system $\mathbf {R}$, which is defined by a triad of Euler angles defined same as in [70]. In Eq. (19), any material tensor $\mathbf {a}$ is replaced by tensor $\left (\mathbf {R}\,\mathbf {a}\,\mathbf {R}^{\textrm{T}}\right )$ then. In particular, for normal incidence and propagation perpendicular to optical axis, material tensors are transformed to $\mathbf {\hat {\varepsilon }}_{\textrm{r}} = \textrm{diag}\left [ \varepsilon _{11}\,\,\varepsilon _{33}\,\,\varepsilon _{11} \right ]$, $\mathbf {\hat {\alpha }} = \textrm{diag}\left [ \alpha _{11}\,\,\alpha _{33}\,\,\alpha _{11} \right ]$, and $\mathbf {N} = \left [ 0\,\,0\,\,N_{z} \right ]^{\textrm{T}}$.
Substituting the particular forms of material tensors into (19), a following set of equations is obtained:
Because the effect of $\mathbf {\hat {\alpha }}$ is small [48,66], it is justifiable to set the term $(\mathbf {\hat {\alpha }}\mathbf {\hat {\alpha }}^{\textrm{T}})$ in (19) to zero. Using (20), following set of equations is derived:
By solving the equations that the determinants of (27) and (28) are equal to zero, normal eigenmodes $N_{z,\,k=\{1,2,3,4\}}$ propagating in bianisotropic nonmagnetic media of 32 point group perpendicular to optical axis are calculated, respectively, as follows:
Both of the solutions stand for an elliptically polarized eigenmodes. For the case of small effect of the optical activity with respect to birefrigence, the eigenmodes with propagation constants (29) and (30) become linearly-like polarized and represent ordinary and extraordinary refractive indices of linear retarder $n_{\textrm{o,\,e}}$,
and the optical activity influence comes from the ellipticity of the eigenmodes as the result of the intrinsicality of the OA. Moreover, by comparison of the wave Eq. (27) with (28) we find the symmetry between the formalism for this case [47,48],C. Mueller matrix of the chiral waveplate
Jones matrix describing a general elliptical retarder in the transmission configuration with retardation angle $\Gamma$ is given as [26,71]
With this setting, Eq. (33) becomes [48,72]
Corresponding transmission Mueller matrix of the elliptic retarder in a general azimuthal rotation $\varphi$ is derived using
The resulting expression of the approximated matrix $\mathbf {M}_{\textrm{R,EL}}\left (\varphi ,\,\Gamma ,\,\kappa \right )$ can be separated into two parts as
The gyration parameter $\kappa$ is usually derived [45,47,48,57] from the eigenvalue problem of the wave equation based on the Born-Landau constitution relations. The resultant expression with respect to the linear approximation of $\kappa$ is given as
For point group 32, scalar gyration parameter $G$ [Eq. (23)] is given by the non-zero components of $\hat {g}$ and the angle $\theta$ given by (24),
The dispersion characteristics of the components of gyration tensor $\hat {g}$ were derived in [47] and are given by the dispersion equation
where $\lambda$ is the wavelength and $A_{i},\,B_{i}$ are phenomenological constants. This dispersion model is based on the sum rule consistent model of coupled anisotropic oscillators, and was originally developed for the description of specific rotatory power dispersion [74].D. Algebraic analysis of the optical activity and misalignment effects in the Mueller matrix of biplates
Let us consider the Mueller matrix of biplate in the form of (7), which differs from an ideal biplate by the presence of the waveplate misalignment $\Phi = \varphi _1 - \varphi _2 - \pi / 2$ and the OA $\mathbf {M}_{\textrm{R,EL}}\left (\varphi ,\,\Gamma _1-\Gamma _2,\kappa \right )$ given by (5), introduced as follows:
Considering only small effects in both misalignment and OA $(\kappa ,\,\Phi \ll 1)$, we express Mueller matrix (43) as its first order Taylor expansions. The resulting form of the matrix $\textbf {M}_{\textrm{R,WB}}$ is then represented as a sum of the Mueller matrix of an ideal biplate without the OA and perturbation matrices, each representing separate contribution of the OA and the misalignment in the form of fine spectral oscillations:
Funding
Grantová Agentura České Republiky (18-22102S); European Regional Development Fund (CZ.02.1.01/0.0/0.0/16_013/0001791); EEA and Norway grants (EMP-CZ-MOP-2-013); Technology Agency of the Czech Republic (FV20020); DGS - Doctoral Grant System (CZ.02.2.69/0.0./0.0/19_073/0016945, Project DGS/TEAM/2020-030); SGS - Student Grant System (SP2020/71).
Acknowledgment
Partial support from the Czech Science Foundation (Project 18-22102S), the IT4Innovations Path to Exascale project CZ02.1.01/0.0/0.0/16013/0001791, EEA grant (EMP-CZ-MOP-2-013), the Project FV20020, doctoral grant system (CZ.02.2.69/0.0./0.0/19_073/0016945, Project DGS/TEAM/2020-030), and student grant system (SP2020/71) are acknowledged.
Disclosures
The authors declare no conflicts of interest.
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