1. Introduction

The capabilities of accurate generation and detection of polarized light over wide spectral ranges are required in numerous scientific fields [1–4]. For this purpose, compensators are utilized to impose accurately known differences in phase shifts between orthogonal electric field components upon light transmission through such devices. In-line compensator designs that are rotatable without beam deviation are desired to enable polarization modulation. The ultimate solution to the strongly chromatic retardance, limited zero-order spectral range, and oscillatory chromatic artifacts of rotatable monoplate and biplate compensators is the replacement of these devices by nearly-achromatic compensators based on in-line rhomb designs [2]. The disadvantage of beam re-direction imposed by one and two reflection rhombs is overcome by in-line rotatable three-reflection devices. Such devices simultaneously enable reduction in stress-induced birefringence owing to their shorter path lengths compared to in-line four-reflection dual rhombs [2,5–8] which can be designed to be self-compensating for deviation angle and thus more accurately rotatable [9]. The in-line three-reflection rhomb compensator design reported by King [10,11] uses a symmetric arrangement of the total internal reflections (TIRs). In this design, one intermediate internal reflection from a (fused silica)/MgF₂/ambient two-interface structure occurs between two internal reflections from (fused silica)/ambient single interfaces to achieve nearly quarter wave operation over a wide spectral range.

In an ideal nearly-achromatic rhomb compensator, all internal reflections are total without dissipation and no polarization modification occurs in transmission upon beam entry into and exit from the device as well as along the light beam paths within the rhomb medium. In this case, the device performance is characterized by a Jones or Mueller matrix with a single wavelength dependent phase retardance that is an accumulation of the differences in phase shifts upon reflection for optical electric field components along orthogonal fast and slow axes of the device. This retardance spectrum is a well-understood function of the angles of incidence at the internal interfaces and the refractive indices of the materials with which the beam interacts [12]. For the King rhomb [10,11], for example, these materials include fused silica and thin film polycrystalline MgF₂, assumed to be optically isotropic under ideal circumstances.

An analysis of the ideal performance of a rhomb compensator shows that it is only weakly chromatic in comparison with monoplate and biplate devices. Non-ideality due to dissipative internal reflections results in dichroic behavior that can be well characterized if generated by isotropic laminar structures at interfaces and contributes additional weak chromatic behavior [13]. In practice for these devices, however, stress-induced birefringence due to strain within the rhomb material contributes to non-ideal behavior that is comparably more difficult to characterize as a result of its spatial variations along the beam path. Such birefringence is a third source of chromatic behavior that can be quite strong for high stresses and long beam paths, the latter occurring in dual Fresnel rhombs, for example. Although extensive literature exists on the design of rhombs [2,5,9] with the early work reviewed in Ref. [2], the effects of these non-idealities are rarely considered or analyzed in detail. The goal of this study is to address such a deficiency in the characterization of rhomb compensators.

The rhomb design by King will be the focus of the present study as an illustration of the how the non-idealities observed in rhomb design compensators can be characterized. In this study, we have developed a comprehensive approach for determining the non-ideality characteristics of rhombs that have limited their wider adoption as compensators in optical instrumentation. We have demonstrated methods for complete characterization of the dichroic and stress-induced birefringence effects that lead to deviations of the Mueller matrix of the in-line three-reflection rhomb from its ideal form. This approach uses a novel combination of state-of-the-art multichannel Mueller matrix spectroscopy in transmission mode in the rhomb’s operational configuration as well as multichannel spectroscopic ellipsometry performed externally in reflection mode, the latter to characterize the internally reflecting interfaces. As a result, we identify a set of wavelength-independent parameters that fully characterize the rhomb’s spectroscopic Mueller matrix. This article is structured as described in the following two paragraphs.

Section 2 focuses on the experimental methodologies in three subsections. The first subsection describes the design of the King rhomb and the Mueller matrix measurements performed on the device using a dual rotating-compensator ellipsometer. In the second subsection, the Mueller matrix elements $m_{12}$ and $m_{44}$ from this measurement are presented and compared to the results of simulations assuming ideal performance of the rhomb as a compensator. The $m_{12}$ element reveals the presence of a dichroic effect whereas the $m_{44}$ element reveals the presence of stress induced birefringence along the four beam paths. The third subsection describes the origins of the spectrum in the stress-optic coefficient of fused silica used in the data analysis of this study.

Sections 3–5 develop the background material required for a complete analysis of the Mueller matrix of the rhomb including the effects of the identified non-idealities. Through mathematical development and simulations, Section 3 shows that when the stress birefringence variation with distance along each beam path is weak, the Mueller matrix of the rhomb is sensitive only to the accumulated retardance and average azimuthal angle of the fast principal axis along the path. Section 4 presents a first order analysis of the Mueller matrix of the complete rhomb that provides an understanding of the effects of the non-idealities on each of the Mueller matrix elements. Section 5 describes the application of multiple angle of incidence spectroscopic ellipsometry measurements performed externally on the ambient/MgF₂-film/(fused silica) structure using a single rotating-compensator multichannel ellipsometer. These measurements supplement the Mueller matrix measurements enabling a structural and optical property analysis that accounts for the observed dichroic effect and enables accurate prediction of the retardance contribution associated with this reflection. Section 6 applies the results of Sections 3 and 5 for a complete least squares regression analysis of the measured Mueller matrix of the rhomb in terms of wavelength independent structural and optical parameters. Section 7 presents a summary of the major conclusions, emphasizing the application of the methodology for accurate descriptions of the polarization generation and detection performance of rhombs acting as compensators.

2. Experimental procedure

2.1 King rhomb design and measurement

The King rhomb device studied here is fabricated from fused silica and is bounded by two rectangular parallel faces 8 mm $\times$ 12 mm and a rectangular base 8 mm $\times$ 48 mm as shown in Fig. 1(a) [14]. Symmetric internally reflecting (fused silica)/ambient interfaces are designed for angles of incidence and reflection of 72.25$^{\circ }$ for the input and output beams of the device, respectively. The base of the rhomb is coated with polycrystalline MgF₂ to a nominal thickness of ~ 50 nm with a corresponding internal angle of incidence of 54.5$^{\circ }$ for the beam at the (fused silica)/MgF₂ interface. Using an adhesive, the rhomb is affixed to offsets, ~ 1 mm in thickness, which are in turn attached to an aluminum base plate, allowing the rhomb to rotate about its optical axis. These offsets are placed near the two ends of the rhomb resulting in an air gap at the center of the rhomb, ensuring a (fused silica)/MgF₂/ambient structure for the second internal reflection. The light beam passing through the rhomb in the straight-through operational configuration of the device is ~ 2.5 mm in diameter and its center is positioned ~ 4.0 mm relative to each side and ~ 7.8 mm relative to the base of the device. The path length of the beam within the rhomb is ~ 53.0 mm, with symmetric contributions of ~ 13.0 mm for one axial beam path and ~ 13.5 mm for one meridional beam path.

 

Fig. 1. (a) King rhomb physical structure optimized for a retardance of $\delta = \delta _F - \delta _S \approx 90^{\circ }$ over a wide spectral range; (b) and (c) measured (green) and ideal predicted (red) spectra for two representative Mueller matrix elements relevant for evaluation of dichroism and stress-induced birefringence. The measured spectrum for element $m_{12}$ exhibits non-ideal dichroic behavior whereas the measured spectrum for element $m_{44}$ displays perturbations from the ideal retardance due to birefringence in the fused silica resulting from residual and mounting stress. The ideal retardance is calculated using three non-dissipative reflections from atomically smooth and abrupt interfaces where each total internal reflection occurs. The dichroic angle $\psi _c$ and retardance $\delta$ associated with the Mueller matrix elements $m_{12}$ and $m_{44}$ in (b) and (c) are shown in (d) and (e), respectively.

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This rhomb is structured to generate three total internal reflections (TIRs), cumulatively producing a nearly achromatic phase difference of $\delta _p - \delta _s \sim 90^{\circ }$ between the $p$ and $s$ orthogonal electric field components of the incoming beam over a wide spectral range. This is a consequence of the different phase shifts experienced by the $p$ and $s$ components of the optical electric field upon each oblique reflection. Here the $p$ and $s$ components oscillate parallel and perpendicular, respectively, relative to the plane of incidence which is common to the three internal reflections within the rhomb [see Fig. 1(a)]. For the two meridional beam paths within the rhomb, the $p$-$s$ coordinate system of the incoming beam is rotated about the fixed $s$ direction so that the $p$ component of the field remains perpendicular to the beam direction. For each of the three reflections, the magnitude of phase shift experienced by the $p$ component, $\delta _p$, is greater than that experienced by the complementary $s$ component, $\delta _s$. Considering the complete device as a compensator, rather than as individual reflections, the principal azimuthal axes by convention are parallel to the plane formed by the beam path, described as the fast axis, and perpendicular to the plane, described as the slow axis [15]. These fast and slow axes together with the +$z$-axis defined by the direction of the coaxial input and output beams form a right-handed coordinate system. Throughout this work, the convention based on the time dependence of exp($i \omega t$) for the optical electric field is applied, where $\omega$ is the angular frequency of field.

The material components of this rhomb, fused silica and thin film MgF₂, exhibit negligible absorption over the spectral range of interest and thus, under ideal circumstances, the rhomb is not dichroic. The magnitude of the $p$ and $s$ component phase shifts for each reflection at a given wavelength is a function of the angle of incidence and the refractive indices of the material components that establish the interfaces [10]. Thus, refractive index dispersion introduces weak wavelength dependence to the phase shift difference between the $p$ and $s$ components for each internal reflection.

For characterization of the surface and film structures generating the TIRs within the rhomb, a variable angle single rotating-compensator multichannel ellipsometer was used in reflection mode [16]. Extensive measurements were performed from the film side at the exposed ambient/MgF₂/(fused silica) interfaces of the device after removing the aluminum mounting plate. The ellipsometry angles ($\psi,\Delta$) of this structure were measured over the spectral range spanning from 210 nm to 1650 nm. Here, tan$\psi$ is the relative (reflected relative to incident) ${p}/{s}$ ratio of electric field amplitudes, and $\Delta$ is the phase shift (reflected relative to incident) $p$-$s$ difference of electric field phases. In this case, $p$ and $s$ are the directions parallel and perpendicular to the plane of incidence established by the incident and reflected beams in the film side ellipsometry measurement. Measurements were performed at multiple angles of incidence of 60$^{\circ }$, 65$^{\circ }$, 70$^{\circ }$, and 75$^{\circ }$ at a location in close proximity to that of the second TIR within the rhomb.

A dual rotating-compensator multichannel ellipsometer [17] in the straight-through transmission mode was applied to obtain the complete normalized Mueller matrix of the King rhomb in its operational configuration as a compensator. In this measurement, the rhomb serves as the sample, and the plane that contains the transmission axis of the fixed first polarizer of the instrument as well as the light beam input to the rhomb serves as the reference plane. Thus, the reference coordinate system from which the rhomb zero azimuth (or fast axis azimuth) is measured is defined by the transmission-extinction ($t$-$e$) axes of the fixed polarizer. These axes along with the +$z$-axis of the beam direction form a right handed coordinate system. Measurements were performed on the rhomb when oriented at various azimuthal angles $\alpha$ with respect to the reference coordinate system. Such measurements were collected for wavelengths over a spectral range from 210 nm to 1690 nm, resulting in 15 normalized Mueller matrix element spectra [17]. For the analysis reported here, a truncated set of spectra was necessary due to a small but nontrivial discontinuity in some of the experimental matrix element spectra at a wavelength ~ 1000 nm. This is attributed to the wavelength crossover point of the dual multichannel detectors used in this instrument. As a result, each Mueller element spectrum in this analysis consists of 771 points spanning a spectral range from 230 nm to 1000 nm. The presence of such errors due to the high sensitivity of the measurement to optical element alignment demonstrates the need for rhomb compensators that enable high accuracy rotation and data acquisition over wide spectral ranges.

In this work, the analysis is limited to $\alpha = 0 ^{\circ }$ in which case the plane of incidence within the rhomb is set to coincide with the reference plane. The two planes were superimposed as closely as possible by monitoring the Mueller matrix element $m_{22}$, which is given simply by

2.2 Initial assessment of rhomb optical non-idealities

Representative measured Mueller elements $m_{12}$ and $m_{44}$ are obtained with an intended azimuthal angle of $\alpha =0^{\circ }$ and are shown as the green lines in Figs. 1(b) and 1(c), respectively. These matrix elements provide direct insights into non-idealities of the King rhomb compensator studied here. For an ideal compensator, $m_{12}=0$ and $m_{44}=\cos \delta$ where $\delta = \delta _F -\delta _S$ is the cumulative phase retardance of the device. In this initial assessment, $m_{44}$ is used since it does not depend on $\alpha$, thus avoiding potential effects of misalignment of the fast axis of the rhomb. When dichroism is present, however, $m_{12}=-\cos 2\psi _c$, where $\psi_c$ is the compensator dichroic angle. The predicted results for $\psi _c$ and $\delta$ for an ideal compensator are shown as the red lines in Figs. 1(d) and 1(e) where $\delta$ is obtained from the rhomb geometry and tabulated component material indices of refraction [19,20]. These predictions lead to the results for the Mueller matrix elements given by the red lines in Figs. 1(b) and 1(c), respectively. The difference between the measured and predicted results for $m_{12}$ arises from dissipative absorption associated with internal reflection that generates a non-zero dichroic effect in $m_{12}$, and the corresponding difference for $m_{44}$ arises from stress-induced birefringence of the fused silica along the beam path that contributes an additional phase retardance term.

We have developed a simulation based on Mueller matrix analysis to model the measurements of the rhomb accounting for non-idealities based on both dissipative absorption in the internal reflections and stress-induced birefringence along the beam path. In addition, we have performed a theoretical analysis of the accuracy of the model used to evaluate the stress-induced birefringence along a given path length within the rhomb by considering the medium as a large collection of thin wave-plates with phase retardances and fast principal stress axis azimuthal angles for each plate that vary in a defined step-wise fashion along the beam path [21].

2.3 Stress birefringence of fused silica

Optically isotropic solids subjected to mechanical stress experience anisotropy characterized by a refractive index ellipsoid whose principal axes align with the principal stress axes. For a light beam traversing a distance $d$ within a medium subjected to homogeneous orthogonal principal stresses $\sigma _1$ and $\sigma _2$, the magnitude of the accumulated optical retardance is given by

Ordinarily, in describing the wavelength dispersion of the real part of the complex dielectric function $\varepsilon _1 = n^2$ of a non-absorbing medium, a Sellmeier expression is used as will be described in Section 5. Starting from the Sellmeier expression, $n$ and thus $n^3$ can be approximated as Cauchy dispersion expressions under the assumption that the minimum in the wavelength range is much larger than the wavelengths at which the electronic resonances occur. For the wavelength range of interest here, 230 - 1000 nm, we similarly assume $q_{11}$ and $q_{12}$ for fused silica follow Cauchy expressions. Thus, we model $C(\lambda )$ using such an expression on the basis that both $n^3$ and $q_{11} - q_{12}$ [24] are each expressible using such dispersion equations. Experimental determination of $C(\lambda )$ for fused silica has been undertaken by Sinha [25] who compiled data over a wavelength range from 230 nm to 1000 nm from a variety of sources. A least squares fit of the data compiled by Sinha using three terms of the Cauchy equation yielded

To investigate the applicability of the expression for $C(\lambda )$ in our work, a rectangular cuboid of fused silica (12 mm $\times$ 25 mm $\times$ 3 mm) [26] was subjected to uniaxial compressive stress in the long direction using a vice-like apparatus incorporating a load cell for measuring the applied force. A dual rotating-compensator spectroscopic ellipsometer was set in the transmission mode and the sample was probed normal to the largest area rectangular face, resulting in a 3 mm beam path. The location of the measurement on the cuboid was ~ 12.5 mm from both contact areas where the uniaxial stress was applied. These straight-through measurements of the fused silica cuboid were obtained under an applied force of ~ 79 N over an area of (3 $\times$ 12) mm². Orthogonal fast ($F$) and slow ($S$) axes are associated with stress-induced birefringence such that one of these two axes is aligned along the uniaxial stress direction.

Here and in the following two paragraphs, we describe the measurements performed on the fused silica cuboid under stress in order to verify the fast axis azimuthal orientation for the resulting phase retardance as measured with respect to the reference coordinate system of the dual rotating-compensator ellipsometer [17]. The goal is to identify an accurate zero-angle azimuthal orientation of the cuboid fast axis relative to the defined reference coordinate system of this ellipsometer so that an accurate phase retardance spectrum $\delta _c$ can be determined for the cuboid under stress. The reference axes were established as usual to be the $t$-$e$ coordinate system of the fixed first polarizer (such that the $t$-axis is vertical in our case; see inset Fig. 2). Three straight-through Mueller matrix measurements were taken through the center of the largest area rectangular face of the cuboid with the sample rotated so that the applied uniaxial stress is oriented at different azimuthal angles $\alpha$ with respect to the $t$-axis.

 

Fig. 2. Straight-through measurements of the Mueller matrix element $m_{42}$ as a function of wavelength performed to identify the fast axis orientation for a fused silica cuboid under uniaxial stress. Measurements start with the applied stress aligned along the $t$-axis of the reference coordinate system, which is the $t$-$e$ frame of the fixed first polarizer of the ellipsometer, corresponding to zero azimuthal angle (red). The cuboid is then rotated to an arbitrary positive angle (green) and an arbitrary negative angle (blue). By convention, an azimuthal angle is measured in the counterclockwise positive sense looking towards the light source.

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From such measurements the Mueller matrix element $m_{42}$ can be expressed as

Thus, according to the expression in the previous paragraph, $m_{42}$ can be used to resolve the direction of the fast axis since $\psi _c$ is very close to 45$^{\circ }$ and $\delta _c$ is positive by definition. One measurement was performed with $\alpha =0^{\circ }$ by ensuring $m_{42}$ values of zero across the entire spectrum. Thus, for this measurement of the $m_{42}$ spectrum the uniaxial stress axis is aligned along the $t$-axis. The results of this measurement are shown as the red data points in Fig. 2. Two succeeding measurements were made with this stressed cuboid rotated through small angles $\alpha$ such that the stress axis was at positive and negative $\alpha$ values measured with respect to the experimentally recognized $t$-axis. Viewing along the beam path toward the light source, the positive and negative values correspond to counter-clockwise and clockwise cuboid rotations, respectively. Figure 2 shows Mueller element $m_{42}$ spectra for positive and negative $\alpha$ as the green and blue data points, respectively. Because $\delta _c = \delta _F-\delta _S > 0$, the sign of $m_{42}$ in Eq. (4) must be the same as that of $\alpha$ for a weakly dichroic retarder with $\psi _c \approx 45 ^{\circ }$, assuming the correct orientation of the cuboid is selected. It can be observed from the results in Fig. 2 that this requirement is satisfied, and the fast axis of the induced retardance correctly aligns with the direction of the applied uniaxial stress.

In the next step to be described here, the ellipsometric angles $(\psi _{\exp }, \delta _{\exp })$ derived from the experimental Mueller matrix elements of the fused silica cuboid are depicted in Fig. 3(a) for $\alpha =0^{\circ }$. These angles are obtained from

 

Fig. 3. (a) Measured retardance $\delta _{\exp }$ (green circles) and best fit retardance $\delta _{\mathrm {fit}}$ (red line), the former from a straight-through measurement of a fused silica cuboid under an applied force of ~ 79 N over an area of ~ (3 $\times$ 12) mm². The location of the measured data is ~ 12.5 mm from either contact area with an estimated stress of ~ 0.6 MPa. In addition to the retardance and its best fit in (a), the associated dichroic angle $\psi _{\exp }$ is also shown (blue circles). The plot in (b) shows the spectrum in the difference between the experimental and best fit retardance data.

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The resulting spectral distribution of the fitting error is shown in Fig. 3(b). The magnitude of the discrepancy is most pronounced in the UV, reaching a maximum of ~ 1$^{\circ }$ at 230 nm and stabilizing at ~ 0.2$^{\circ }$ for wavelengths greater than 500 nm, with a variation between the experimental data and fit of ~ 2% at 230 nm and ~ 3% at 1000 nm. In fact, the nearly constant fitting error of ~ 0.2$^{\circ }$ in Fig. 3(b) above 500 nm suggests much better agreement is possible over a wavelength range restricted to the visible. Given the high quality of these measurements, based on the accurate results for $\psi _{\exp }$, the discrepancies in Fig. 3(b) are likely to arise from several possible sources. Variations may occur in the properties of the fused silica measured in the different studies, in the stress magnitude and direction over the area of the light beam used to probe the fused silica, or in the direction of the beam relative to the principal stress axes in each experiment. It also seems likely that an improved description of the wavelength dependence of $C(\lambda )$ over that of Sinha may be motivated by these results in future work. The overall discrepancy in Fig. 3(b) is deemed sufficiently small, however, to validate application of the stress-optic coefficient reported by Sinha in the King rhomb analysis performed here.

3. Stress simulation in fused silica

It is a challenge to interpret the stress-induced polarization modification that a polarized light beam incurs upon propagation through an otherwise optically-isotropic medium. Such an interpretation is sought in terms of the inherent residual thermally and mechanically generated stresses within the medium. The challenge is a result of the relatively small but spatially variable phase retardance contributions along with the accompanying variable index ellipsoid orientation that arise from the forces present. These contributions and associated orientational variations can then accumulate over long path lengths to generate significant polarization state modifications of the propagating beam. One approach for interpretation is to subdivide the path traversed by the polarized light beam within the optical medium into $N$ equipartitioned components, each generating a distinct local retardance with an associated fast axis azimuthal angle [21]. Using this construct, a total retardance and average azimuthal angle of the thick optical plate can be obtained using Jones calculus as will be described next. This requires manipulating the product of the $N$ retardance matrices along the path, each with its pair of matrices for positive and negative rotation by the local fast axis azimuthal angle. A pictorial representation of the concept is given in Fig. 4, whereby a light beam travelling in the $z$ direction encounters stress magnitudes and azimuthal angles that vary as a function of the distance along $z$.

 

Fig. 4. Construct for determining the cumulative retardance and average azimuthal angle induced by stress in an otherwise isotropic medium having a total thickness of $d_p$. The medium is characterized by arbitrarily assigned slowly-varying spatial functions of path distance $z$ including the retardance $\delta (z)$ and the principal azimuthal angle $\theta (z)$, the latter describing the fast axis direction relative to an ellipsometer reference $t$-$e$ coordinate system. To simulate these spatially varying functions for analysis, the plate is decomposed into $N$ equipartitioned components each of length $\Delta z$ characterized by $\delta _n (z_n)$ as in Eq. (7) and $\theta _n (z_n)$ values for the $n$^th component as shown. A probing polarized light beam traveling along the $z$ direction encounters variable stress related functions starting with ($\delta _1$, $\theta _1$) upon entering and ending with ($\delta _N$, $\theta _N$) upon exiting the medium.

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In order to evaluate the required approach for modeling the optical effects of stress in media, the following case study presents an analysis of the Jones and corresponding Mueller matrices resulting from light beam propagation within a medium of assumed non-uniform retardance and undulating azimuthal angle bounded by an initial fast axis azimuth $\theta _1$ and a final fast axis azimuth $\theta _N$. Each equipartitioned component has a width along the beam path of $\Delta z = d_p/N$ where $d_p$ is the total plate thickness, a retardance of

By ensuring that $\Delta z$ is sufficiently small, the matrix product in Eq. (8) can be evaluated to first order in the retardance values, i.e. by neglecting the products of two or more factors of $\delta _n$. With this approach, we obtain matrix element expressions in terms of the component retardances and azimuthal angles as

Assuming no depolarization, a Mueller-Jones matrix exists and the corresponding unnormalized Mueller matrix for $\mathbf {J}_p$ in Eq. (12) takes the form:

The preceding analysis demonstrates that the retardance and azimuthal angle profiles shown in Fig. 4 can be replaced by the accumulated retardance and average angle, the latter two uniformly spanning the beam path length. The equivalent optical element exhibits properties similar to a retarder with a retardance $\delta$ and fast axis direction $\bar \theta$ with respect to the $t$-$e$ reference frame. Accordingly, the Mueller matrix for the plate can then be treated as a single stress-induced retarder and can be written as

Simulations were undertaken in order to verify the analytical reasoning of the previous paragraphs. In these simulations, a polarized light beam traverses a plate of fused silica 50 mm in thickness. The plate was equipartitioned into 1000 components such that the $n$^th component was assigned a retardance spectrum $\delta _n$ and an azimuthal angle $\theta _n$, the former derived from a stress value $\lvert \Delta \sigma _n \rvert$, where $n=\{1,2,\ldots,1000\}$. Thus, the retardance spectrum for each component was calculated according to Eq. (7).

Smoothly varying test functions for the stress and azimuthal angles as functions of beam distance within the plate were generated using B-spline curves. The resulting Mueller matrix for the plate was simulated using the following equation similar to Eq. (8) with $N=1000$ and $\Delta z = 0.05$ mm but with arbitrarily generated variations for the component dependent $\delta _n$ and $\theta _n$:

 

Fig. 5. The plot on the top depicts stress (red) and azimuthal angle (green) profiles as functions of component number used in the simulation of a beam traversing a 50 mm thick fused silica plate equipartitioned into 1000 components according to Eq. (17). The points on the bottom plots show the corresponding lower right 3 $\times$ 3 Mueller matrix spectra generated using Eq. (17). The solid blue line depicts the analytically derived results using the matrix elements of Eq. (14) with $\delta =N \bar \delta$ from Eq. (7), $\lvert \overline {\Delta \sigma }\rvert$ = 15 kPa, and $\bar \theta =35^{\circ }$.

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Next, Eq. (15) implies a family consisting of multiple data sets, each set including different profiles in $\lvert \Delta \sigma _n \rvert$ and $\theta _n$ that result in the same spectrum in $\delta = N \bar \delta$ and the same value of $\bar \theta$, and thus, the same Mueller matrix element spectra as other members of the family if the assumptions leading to Eq. (15) are valid. A graphic illustration of this implication is shown in Fig. 6. Four carefully chosen but different combinations of stress and azimuthal angle profiles in Fig. 6(a-d), each with closely similar $\delta = N \bar \delta$ and $\bar \theta$ were used to simulate data in the Mueller matrix elements using Eq. (17). As a practical consequence of the results from Fig. 6, experimentally determined Mueller matrices cannot be analyzed to identify the true nature of the underlying retardance and stress profiles if the profiles vary only weakly from their average values.

 

Fig. 6. (Right) An overlay of Mueller elements simulated from (top) four plots (a)-(d) with spatially different stress (red) and azimuthal angle (green) profiles generated to share common values of $\lvert \overline {\Delta \sigma }\rvert$ = 15 kPa, and $\bar \theta =35^{\circ }$.

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Finally, we apply least squares regression to extract the accumulated retardance and mean azimuthal angle from Mueller matrix element spectra simulated using Eq. (17). Figure 7 depicts four different pairs of test stress and azimuthal angle variations $\{ \lvert \Delta \sigma _n \rvert, \theta _n\}$ as functions of component number $n$ within the fused silica plate that have been used to generate different conditions of stress in the plate. Each curve can be characterized analytically by its accumulated retardance $\delta = N \bar \delta$ as well as its average azimuthal angle $\bar \theta$. The achromatic prefactor is given by $\delta _{pf} = N \bar \delta _{{pf}\!,n}$, where $\bar \delta _{{pf}\!,n}$ is obtained from the relationship $\delta _{{pf}\!,n}=2 \pi \Delta z \lvert \Delta \sigma _n \rvert$. The resulting Mueller matrix ${\mathbf {m}_p}^{'}$ simulated for each pair $\{ \lvert \Delta \sigma _n \rvert, \theta _n\}$ in Fig. 7 was used as input in a least squares fitting routine modeled using the right hand side of Eq. (15). The fit parameters of interest include the pre-factor of the accumulated retardance $\delta _{{pf}(\mathrm {fit})}$ and the azimuthal angle $\bar \theta _{\mathrm {fit}}$ fit which is averaged over the plate thickness. The accumulated retardance over the full thickness could then be calculated using the formula $\delta _{\mathrm {fit}}= \delta _{{pf}(\mathrm {fit})}C(\lambda )/\lambda$.

 

Fig. 7. Stress (red) and azimuthal angle (green) profiles as functions of component number used in simulations of a beam traversing a 50 mm thick fused silica plate equipartitioned into 1000 components. Plots (a) and (b) depict nonlinear stress and azimuthal angle profiles with positive (ccw) angles over the entire beam path. These plots also share identical sets of $\delta$ and $\bar \theta$. Plot (c) depicts nonlinear stress and azimuthal angle profiles with negative (cw) angles over the entire beam path. Plot (d) depicts nonlinear stress and azimuthal angle profiles with an angle variation spanning both negative and positive values. Applying Eq. (17) to these profiles and fitting the resulting Mueller matrix spectra using Eq. (15) gives the results in Table 1.

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Table 1. Simulation results based on Eq. (17) showing the actual accumulated retardance pre-factors and average azimuthal angles given by $\delta _{pf}$ and $\bar \theta$ from Fig. 7 along with the corresponding best fit results based on the analysis of Eq. (15).

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Results obtained in an analysis of the input to the simulation of Eq. (17), $\delta _{pf}$ and $\bar \theta$, as well as the corresponding results from the fitting routine of Eq. (15), $\delta _{{pf}\!(\mathrm {fit})}$ and $\bar \theta _{\mathrm {fit}}$, are compared in Table 1. Two important observations can be made from the best fitting results. First, the accumulated retardance of the plate is recoverable, for all practical purposes, even when subject to varying azimuthal angles. Second, the average azimuthal angle is also recoverable even when its total variation is significant compared to the total accumulated retardance $\delta$. The validity of the first order expansions in $\delta _n$ that lead to Eq. (11a) can be assured by using very thin components along the beam path since $\delta _n$ is proportional to $\Delta z$. The fitting equations applied in this analysis derived from Eq. (12) are based on the assumption that the deviations of $\Delta \delta _n$ and $\Delta \theta _n$ from their average values can be likewise treated in first order. The deviations of the best fit results from the values assumed in the simulation are expected to arise from this latter assumption. In Fig. 7, the deviations in $\lvert \Delta \sigma _n \rvert$ from $\lvert \overline {\Delta \sigma } \rvert$ are no more than 20 kPa and the deviations in $\theta _n$ from $\bar \theta$ are no more than $10^{\circ }$. Evidently from Table 1, one can conclude that these deviations are sufficiently small that a first order treatment of $\Delta \delta _n$ and $\Delta \theta _n$ from their averages is valid.

4. First order stress analysis of the King rhomb

Based on the schematic of Fig. 8, the Mueller matrix describing the King rhomb can be formulated using the following matrix product

 

Fig. 8. Seven sources of optical retardance and a single source of dichroism incurred by a light beam traversing the King rhomb including stress-induced birefringence along the four beam paths {$\delta _n, n = 1,3,5,7$}, two ideal TIRs from (fused silica)/ambient interfaces with {$\delta _n, n = 2,6$}, and a single TIR at the MgF₂/ambient interface of a dissipative (fused silica)/MgF₂/ambient structure, the latter with dichroic and retardance angles of {$\psi _4$, $\delta _4$}. The resulting eight parameters at a given wavelength and four average fast axis angles {$\alpha _n, n=1,3,5,7$} define the Mueller matrix product of Eq. (18).

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Treated to the first order, the stress Mueller matrix with retardance $\delta _n \ll 1$ is given by:

The matrices for the retardance incurred upon TIR include

Table 2. Expected normalized Mueller matrix elements for the three reflection King rhomb corresponding to Eq. (18) and Fig. 8 with the rhomb fast-slow ($F$-$S$) frame aligned with the $t$-$e$ frame of the first fixed polarizer of the ellipsometer used for the measurement. The dichroic contribution from the second TIR appears in the six matrix elements $m_{12}$, $m_{13}$, $m_{14}$, $m_{21}$, $m_{31}$, and $m_{41}$. The four elements $m_{11}$, $m_{12}$, $m_{21}$, and $m_{22}$ are exempt from the effects of stress-induced retardance along the optical path.

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An inspection of the elements of the normalized Mueller matrix in Table 2 provides insights into the dichroic contribution and the distribution of stress-induced retardance from the individual paths of the beam through the rhomb. The 2 $\times$ 2 upper left block of the matrix including elements $m_{11}$, $m_{12}$, $m_{21}$, and $m_{22}$ are unaffected by stress. The elements $m_{12}$ and $m_{21}$ provide two results for the dichroic angle that can be averaged according to $\psi _4=\frac {1}{2}\cos ^{-1}[-(m_{12}+m_{21})/2]$. An example of the dichroic effect free of the influence of stress-induced retardance is clearly demonstrated in Fig. 1(b) which shows the $m_{12}$ element of the actual King compensator in comparison with the ideal (non-dichroic) result of $m_{12}=0$. Inspection of the lower right block elements $m_{33}$, $m_{34}$, $m_{43}$, and $m_{44}$ show the first order terms in the stress-induced retardance. These are added to the sine and cosine functions of the combined retardance due to the three TIRs. An example of the effect of this addition is clearly demonstrated in Fig. 1(c) which shows that the $m_{44}$ element of the actual King compensator deviates from the ideal result obtained assuming no stress-induced retardance terms. The remaining eight Mueller matrix elements forming the upper right and lower left blocks ($m_{13}$, $m_{14}$, $m_{23}$, $m_{24}$, $m_{31}$, $m_{32}$, $m_{41}$, $m_{42}$) show functional dependences on stress in individual paths of the rhomb.

It is worth noting that each Mueller matrix element in Table 2 lends itself to stress parameterization in a nonlinear least squares minimization scheme. For example in one of the two simplest analyses, $m_{13}$ can be parameterized in the form

5. Rhomb ellipsometry analysis

The normalized Mueller matrix spectra obtained by directing the light beam straight through the King rhomb device revealed unexplained dichroic behavior in the form of nonzero matrix elements $m_{12}$ and $m_{21}$. Given the spectral range of the Mueller matrix measurement performed here, the optical absorption is negligible for the fused silica and MgF₂ materials used in the construction of this device, and so these materials cannot contribute to the observed dichroism. Laboratory contamination, possibly leading to the formation of absorbing overlayers at the (fused silica)/ambient interfaces, was considered as a potential source for at least some component of the observed dissipative effect. External reflection ellipsometric measurements and associated modeling performed on these ambient/(fused silica) interfaces, however, revealed no detectable absorbing overlayers. In contrast, a second set of ellipsometric measurements with the beam reflected externally from the intermediate ambient/MgF₂-film/(fused silica) structure did in fact exhibit weak absorption as described in the following paragraphs, indicating that this structure is the source of the dichroic behavior observed in transmission through the rhomb.

Using fixed parametric expressions for the indices of refraction versus wavelength for the MgF₂ layer and the bulk fused silica from existing an database [19,20], an initial least squares regression analysis was performed on the ellipsometric spectra $(\psi, \Delta )$ acquired in external reflection from the ambient/MgF₂-film/(fused silica) structure at multiple angles of incidence. The structure yielding the best fitting simulation required a thin absorbing layer at the interface between the fused silica medium and the MgF₂ layer. Thus, the resulting model explored here was an ambient/MgF₂-film/absorbing-layer/(fused silica substrate) structure with the thicknesses of the MgF₂ and absorbing layers as fitting parameters. In this first analysis, the unknown complex dielectric function $\varepsilon=\varepsilon_{1}-i \varepsilon_{2}$ of the absorbing layer was modeled as the sum of a two parameter Sellmeier term, two Lorentz oscillators, and a constant additive real term. Each Lorentz oscillator was described with three parameters, a characteristic resonance energy, amplitude, and broadening. As a result, the four pairs of $(\psi, \Delta )$ spectra measured at the different angles of incidence were modeled with 11 free parameters, including the two layer thicknesses and the nine optical property parameters that describe the absorbing layer.

In this first analysis, it was possible to obtain a satisfactory 11 parameter fit of the ellipsometric spectra measured externally at the four angles of incidence. Next, the spectra in the normalized Mueller matrix elements $m_{12}$ and $m_{21}$ obtained from the three-reflection straight-through measurement of the rhomb were analyzed in a second least squares regression procedure. These matrix elements were averaged to extract the internal spectrum in the characteristic dichroic angle $\psi _4$. The structure yielding this spectrum in $\psi _4$, as probed by the internal light beam traversing the rhomb, consists of (fused silica)/absorbing-layer/MgF₂-film/ambient, having established the intermediate TIR as being solely responsible for the measured dichroism. Because the measured $\psi _4$ spectrum was derived from the Mueller matrix representing the entire path of the light beam within the rhomb, the associated phase shift difference $\delta$ that would normally be derived from the lower right block of the matrix has multiple undetermined contributions and so could not be used as input to this second analysis procedure. Consequently, only the $\psi _4$ spectrum was incorporated into this second fitting procedure. The thickness and optical properties of the absorbing layer were obtained in the best fit of $\psi _4$, a ten parameter fit in which case the optical parameters of the adjacent MgF₂ layer and fused silica as well as the MgF₂ thickness were fixed, the latter from the first analysis.

In a third analysis procedure, the new absorbing layer best fitting parameters from the second procedure were introduced as fixed values in the analysis of the external reflection data comprised of the multiple angle of incidence ellipsometric ($\psi$, $\Delta$) spectra whereas the MgF₂ film thickness was varied. The second and third analysis procedures were iterated until convergence of the fitting parameters occurred for both internal (fused silica side) and external (film side) reflection analysis. This approach was applied because the internal reflection data were more sensitive to the absorbing layer parameters through $\psi _4$ whereas they were less sensitive to the MgF₂ layer parameters due to the inability to extract $\delta _4$. In contrast, the multiple angle external reflection data were more sensitive to non-absorbing layer parameters through $\Delta$, whereas they were less sensitive to the absorbing layer parameters. The outcome of this approach was a single consistent structural and optical model for the intermediate reflection within the rhomb that accounts for the four pairs of experimental ellipsometric spectra at different angles of incidence and the single spectrum that characterizes the dichroism of the compensator.

The resulting fit parameters for the absorbing layer yield a thickness on the order of a few nanometers and dielectric function spectra $\varepsilon _1$ and $\varepsilon _2$ that are distinctly different than the adjacent materials in the stack. In fact, the resulting shapes of $\varepsilon _1$ and $\varepsilon _2$ are closely similar to those of amorphous silicon (a-Si), thus allowing a simplified treatment of the absorbing layer by replacing the Sellmeier term and two Lorentz oscillators having a total of eight parameters with a single Tauc-Lorentz oscillator having four parameters. The origin of the a-Si containing interface layer is unclear at present, but possibly related to the nature of the fused silica surface preparation prior to the MgF₂ deposition process or to inadvertent interactions that occur between fused silica or its adsorbates and the MgF₂ thin film during or after deposition.

Once the interface layer was identified as containing unoxidized a-Si, it was further modeled using the Bruggeman effective medium approximation (EMA) as a mixture of fused silica and a-Si using the a-Si volume percentage as a free parameter. Thus, the reconstructed stack for further data analysis was an ambient/MgF₂-film/(a-Si + fused silica interface)/(fused silica substrate). The absorbing interface optical model involved seven parameters in all, adding the volume percentage of a-Si in the layer as a compositional parameter in addition to the four Tauc-Lorentz oscillator parameters, the constant additive real value, and the layer thickness. With this simpler model for the interface layer, the optical parameters of the fused silica and MgF₂ could be varied from the fixed literature values used in the iterative analysis described in the previous paragraph. The results of this new iterative analysis based on an a-Si interface component is presented in the next two paragraphs, first through a discussion of the non-absorbing components followed by a discussion of the a-Si containing component.

The spectrum in the real part of the dielectric function for the MgF₂ film was modeled as a Sellmeier term with two free parameters, yielding an amplitude and resonance energy of 134.81 $\pm$ 38.16 eV² and 12.85 $\pm$ 0.87 eV, respectively. A constant contribution to the real part of the dielectric function $\varepsilon _0$ was used as well, yielding a best fitting value of 1.10 $\pm$ 0.12. Thus, a three parameter model was used to describe the optical properties of this film. Similarly, the optical response of the fused silica was modeled as two Sellmeier terms with one variable and three fixed parameters. The first term led to a fixed amplitude of 81.50 eV² and a resonance at 10.04 $\pm$ 0.17 eV, whereas the second term was found to be very weak and led to a fixed amplitude of 0.016 eV² and a fixed resonance at zero energy. A constant contribution $\varepsilon _0$ of 1.30 $\pm$ 0.03 was also used, resulting in a two parameter model for the optical response of the fused silica. The refractive indices derived from the fits for MgF₂ and fused silica agree reasonably well with results reported in literature. In particular, the index of refraction for the MgF₂ layer agrees closely with the published extraordinary index of refraction [27] and that of the fused silica matches the published result within 0.6% over the spectral range of interest, 210 - 1690 nm [28]. The final free parameter associated with the non-absorbing components is the MgF₂ thickness, with the analysis yielding a best fit value of 48.77 $\pm$ 0.30 nm as indicated in the structural model in Fig. 9 (lower panel).

 

Fig. 9. (Top) The complex dielectric function ($\varepsilon _1$, $\varepsilon _2$) spectra of the (a-Si + fused silica) effective medium interfacial layer obtained with a six parameter optical model including a single Tauc-Lorentz oscillator, a constant contribution to $\varepsilon _1$, and a volume percentage of a-Si in the (a-Si + fused silica) two-component mixture. (Bottom) A schematic of the second reflection structure for the King rhomb is depicted. The two best fitting structural parameters are presented as deduced in the analysis of ellipsometric and Mueller matrix spectra. These include the thicknesses of the absorbing interface and the MgF₂ layers along with their confidence limits on the right side of the schematic.

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For the absorbing layer, a Tauc-Lorentz (TL) oscillator with four parameters, along with a constant real contribution $\varepsilon _0$ as a fifth parameter, was used to describe the complex dielectric function of the a-Si absorbing component at the (fused silica)/MgF₂ interface that accounts for the dichroism of the King rhomb. This parametric form was adopted to describe the a-Si component of the interface layer with the second component of the layer being fused silica. The number of free parameters used in the fitting procedure of the (fused silica)/absorbing-layer/MgF₂/ambient structure could then be greatly reduced by employing a standard TL model for the a-Si component [29]. The best fit optical response of the interfacial layer is shown in Fig. 9 (upper panel) and was obtained with fixed standard TL oscillator parameters including an amplitude of 154.80 eV, a resonance energy of 3.72 eV, a bandgap energy of 1.28 eV, and a constant real contribution of 0.27 [29,30], as well as a variable TL broadening parameter with a best fit value of 5.75 $\pm$ 0.26 eV. In addition, two variable parameters were determined that describe the absorbing layer structure and composition as shown in Fig. 9 (lower panel) including the layer thickness and the volume percentage of a-Si in the Bruggeman EMA of the (fused silica + a-Si) composite, the latter strongly affecting the complex dielectric function of the layer. Best-fit values include a layer thickness of 1.53 $\pm$ 0.27 nm and an a-Si volume percentage of 24.50 $\pm$ 1.76 % in the EMA of the absorbing layer. The effective thickness or volume per planar area of the absorbing a-Si component is (0.245)(1.53 nm) = 0.37 nm. Based on this value, one possible interpretation of the absorbing layer is a modulation region at the (fused silica)/MgF₂ interface with a peak-to-peak amplitude given by the absorbing layer thickness and a spatial period much larger than the atomic scale. A single layer of Si-Si bonding with an associated thickness of ~ 0.4 nm would account for the observed absorption. Furthermore the presence of a correspondingly modulated fused silica surface ~ 1.5 nm thick as a (fused silica + void) composite would have a negligible effect on the TIR behavior of the (fused silica)/ambient interface.

Figure 10 summarizes the results of the final fits to the multiple angle of incidence ellipsometric spectra obtained by external reflection. These fits apply the absorber layer properties of Fig. 9 as described in the previous paragraph. Also shown in Fig. 10 is the fit to the three-reflection straight-through experimental data in $\psi _4$ using the same structure as in the multiangle ellipsometric analysis, but with an inverted sequence of media. The lower right graph in Fig. 10 shows the derived angle $\delta _4$ as a function of energy which correctly describes the phase shift that occurs upon the second TIR. The high quality of the fits in Fig. 10 supports the assignment of the physical structure and interface optical properties illustrated in Fig. 9. The results also demonstrate the potential sensitivity of the performance of polarization-based optical elements to the fabrication process. In fact, the (fused Si)/MgF₂/ambient structure is used in many such applications [2], and unless taken into account, a chemically reduced SiO₂ interface to the MgF₂ as in Fig. 9, even at the sub-monolayer level, will lead to deviations in the polarization generation and detection properties from predictions. As will be demonstrated in the next section, the structure of Fig. 9 can be incorporated as the intermediate internal reflection component in a complete analysis of the Mueller matrix of the King rhomb. This matrix describes the effect of transmission of a light beam of any incident polarization state through the rhomb in its operational configuration.

 

Fig. 10. Ellipsometric angles ($\psi$, $\Delta$) measured by external reflection at multiple angles of incidence for the ambient/MgF₂/(a-Si+fused silica)/(fused silica) structure of the King rhomb (points). The corresponding fits are shown as the solid red lines. The plot at the upper right shows the dichroic angle $\psi _4$ derived from the normalized Mueller elements $m_{12}$ and $m_{21}$ measured from the three-reflection straight-through configuration for the rhomb. The solid red line in this case is obtained as a best fit using the same structure (but inverted in sequence) for the second internal reflection as was used in the analysis of the multiangle ellipsometric measurements. The plot at the lower right shows the predicted phase shift $\delta _4$ that accompanies the dichroic angle $\psi _4$ for the second TIR from the (fused silica)/(a-Si+fused silica)/(MgF₂-film)/ambient structure. This prediction of $\delta _4$ was obtained from a simulation using the results of the least squares fit of the ten spectra including those in the ellipsometric angles along with $m_{12}$ and $m_{21}$.

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6. Rhomb complete Mueller matrix analysis

With a complete analysis of the second TIR structure characterized by the two layer thicknesses and the component material dielectric functions, including that of the bulk fused silica, then a full Mueller matrix analysis of the King rhomb $\mathbf {M}_{\mathrm {rhomb}}$ can be performed applying Eq. (18). A cascade of 15 Mueller matrices fully describe the rhomb, including 12 that describe the four beam paths each with a cumulative retardance along with positive and negative average fast axis rotation angles in accordance with Eq. (15). Excluding the Mueller matrix elements of $\mathbf {M}_{\mathrm {rhomb}}$ that reflect only the dichroic behavior of the rhomb, as modeled in the previous section, the remaining 13 experimental matrix element spectra of the normalized Mueller matrix of the rhomb are subjected to least squares regression to determine the parameters that characterize the stress-induced birefringence along the four beam paths.

Thus, the model applied to simulate and then fit the 13 measured spectra was generated using the exact result of the series of matrix multiplications in the form of Eq. (18). In this model, the optical retardance at each reflection can be calculated; thus the spectra in the Mueller matrix elements of $\mathbf {M}_2$, $\mathbf {M}_4$, and $\mathbf {M}_6$ were determined previously and fixed. $\mathbf {M}_4$ was determined as described in Section 5, and $\mathbf {M}_2$ and $\mathbf {M}_6$ were determined from the angle of incidence and index of refraction spectrum of the fused silica as obtained in the external reflection ellipsometry experiment also described in Section 5. The cumulative stress-induced retardance {$\delta _n(\lambda ); 1,3,5,7$} along the path of each beam segment $n$ was modeled according to Eq. (7). The azimuthal angles of the principal (fast) axes of the stress birefringence for the four individual paths of the beam were constrained to a single average value which serves as a variable in the analysis. This value is given by $\alpha _n=\bar {\theta }_n=\langle \alpha \rangle$ with $n = 1,3,5,7$, where $\bar {\theta }_n$ represents the average azimuthal angle along the beam path labeled $n$ defined as in Eq. (15), also assuming small angular variations along each path. This approach is physically reasonable as a starting point, given that the rhomb is fabricated from a single block of fused silica.

In reality, however, the orientations of the fast axes along the different paths of the rhomb can differ not only due to different beam path directions, but also due to stress non-uniformity within the rhomb, for example generated by the mounting needed to stabilize the rhomb for rotation. Allowing such differences by using four unconstrained azimuthal angles in the least squares fitting procedure generated unphysical results attributable to the correlation between the two retardance variables $\delta _{{pf}\!,n}$ and $\sin 2\alpha _n$ for a given beam path. The approach used here to mitigate this problem was to constrain the magnitude of the angle $\langle \alpha \rangle$ to be the same for each path but then permit different possible $\pm$ orientations for each of the four paths. In this alternative least squares regression analysis approach, $\lvert \langle \alpha \rangle \rvert$ remains a variable.

The MSE from this procedure and the confidence limits in the best fitting parameters, the latter reflecting the correlation coefficients, were used together as figures of merit to identify the best fast axis orientation for each path. Improved fitting could be obtained with positive values of $\langle \alpha \rangle$ for paths 1 and 7, and negative values for paths 3 and 5. This may reflect the behavior of mounting-induced stress. Additionally, 13 offset-fit parameters were introduced to account for small but non-negligible systematic experimental errors in the Mueller matrix spectra. All offsets were below 0.01 in magnitude, ranging from values of 0.0003 for $m_{13}$ and $m_{43}$ to 0.008 for $m_{23}$ and $m_{42}$. The presence of such errors can be identified from spectra in the elements of the normalized Mueller matrix measured in straight-through without the rhomb in place, which should be equal to the identity matrix. Finally, although the $F$-$S$ frame of the rhomb is aligned with the fixed polarizer $t$-$e$ reference frame by ensuring that $m_{22}=1$ over the entire spectral range, a small difference between the two frames can be identified upon incorporation of the Mueller matrix offsets in the complete analysis of the rhomb. The difference is defined as a rotation angle $\theta _{\mathrm {rot}}$ that brings the $F$-$S$ system of the rhomb into the $t$-$e$ system of the ellipsometer so that the complete Mueller matrix has the form of $\mathbf {M}_{\mathrm {rhomb}}^{'}=\mathbf {R}(-\theta _{\mathrm {rot}})\,\mathbf {M}_{\mathrm {rhomb}}\,\mathbf {R}(\theta _{\mathrm {rot}})$ where $\mathbf {M}_{\mathrm {rhomb}}$ describes the measured Mueller matrix and $\mathbf {M}_{\mathrm {rhomb}}^{'}$ is converted to the $t$-$e$ frame used for the best fitting simulation.

The derived results are shown in Table 3 for the signs and magnitude of the average stress birefringence azimuthal angle $\langle \alpha \rangle$ (the latter common to all four paths), the phase retardance prefactor for each individual path length, the stress in each path, and the phase retardance at selected wavelengths. The 13 measured Mueller matrix elements and their best fits using the model are shown in Fig. 11.

 

Fig. 11. Experimental and corresponding best fits for the spectra in the 13 normalized Mueller matrix elements of the King rhomb that provide the parameters characterizing the stress-induced birefringence along the four beam paths within the rhomb. The two Mueller matrix elements $m_{12}$ and $m_{21}$ were not used in the fitting routine as these provide information on only the dichroic behavior of the rhomb.

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Table 3. Results for the relevant best fitting parameters that characterize the King rhomb including the sign and average magnitude of the azimuthal angle $\langle \alpha \rangle$ (the latter common to all four paths), the retardance prefactors given by $\delta _{{pf}\!,n}=2 \pi d \lvert \sigma _{\tiny {1},n} - \sigma _{\tiny {2},n} \rvert = 2 \pi d \lvert \Delta \sigma _n \rvert$, and the deduced stress $\lvert \Delta \sigma _n \rvert =\lvert \sigma _{\tiny {1},n} - \sigma _{\tiny {2},n} \rvert$ for each of the beam paths. Also included in the fit is the offset angle $\theta _{\mathrm {rot}}$ required to rotate the $F$-$S$ coordinate system of the rhomb into alignment with the $t$-$e$ coordinate system of the fixed polarizer of the ellipsometer used for straight-through measurement of the rhomb. These results were obtained from the least squares regression analysis of 13 Mueller matrix elements of the rhomb (excluding the elements $m_{12}$ and $m_{21}$). The average angle attributed to the fast axis of the stress is ~ 135$^{\circ }$. The deduced average stress magnitude over the cumulative path length of ~ 53 mm within the rhomb is 15398 $\pm$ 2046 Pa. The derived retardances for three representative wavelengths (250 nm, 500 nm, and 1000 nm) are also shown for each path.

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The first order expressions in Table 2 provide insights into the fitting procedure and the depth of information that can be extracted. Theoretically, the matrix elements $m_{13}$, $m_{14}$, $m_{31}$, and $m_{41}$ can provide the four spectra in $\delta _n \sin 2\alpha _n; n=1,3,5,7,$ or equivalently, the four values of $\delta _{{pf}\!,n} \sin 2\alpha _n; n=1,3,5,7,$ given that $C(\lambda )/\lambda$ is known. In practice, however, as can be observed from Fig. 11, these matrix elements are small and very weakly dependent on wavelength, being proportional to the product of the dichroism and the retardance generated by stress-induced birefringence. The four elements $m_{23}$, $m_{24}$, $m_{32}$, and $m_{42}$ also provide information on $\delta _{{pf}\!,n} \sin 2\alpha _n; n=1,3,5,7$; however, without suppression by the dichroic factor. The stronger dependence of these matrix elements on wavelength in Fig. 11 highlights this observation.

Even so, limitations are expected in the extraction of the four stress-induced retardance projections from $m_{23}$, $m_{24}$, $m_{32}$, and $m_{42}$. In first order, these four matrix elements are not independent, however, and so cannot be used by themselves to extract the four projections. In fact, to first order, it can be shown that

There are two sources of strain within the in-line rotatable rhombs that give rise to stress-induced birefringence along the beam path. These include residual built-in strain within the fused silica material and mounting-induced strain due to the forces required to hold the rhomb in place which enables rotation. One might expect that in fabrication, the optical axis of the rhomb is oriented for alignment with the principal stress direction. As a result, there would be negligible stress-induced birefringence effects along the axial paths and offsetting effects along the two meridional paths. The fact that larger retardance prefactors in Table 3 occur along the axial paths near the mounting adhesive suggests that the strain due to mounting dominates. The indication that the azimuthal angles of the fast axes are in opposite directions for the two axial rays compared to the meridional rays is also a characteristic of mounting-induced stress.

Irrespective of the mounting and nature of the stress-induced birefringence over the beam path, the exact simulation for the Mueller matrix of the rhomb given in terms of the best fitting parameters in Table 3, including the appropriate signs of $\langle \alpha \rangle$ along the beam paths 1, 3, 5, and 7, is sufficient for predicting the performance of the compensator in both fixed and rotating applications. These parameters are derived including corrections for offsets arising from the errors in the Mueller matrix measurements themselves and errors arising from an inexact orientation of the reference frame of the rhomb relative to the reference frame of the ellipsometer. Further analysis of the nature and origins of the stress in the rhomb may be possible by simultaneously analyzing the Mueller matrices obtained at different orientations of the $F$-$S$ frame of the rhomb relative to the $t$-$e$ reference frame. It may also be possible to extract further information, for example, on other possible physical mechanisms such as residual and stress-induced circular birefringence. As a result of the close fits to the Mueller matrix elements for $\alpha =0^{\circ }$ in Fig. 11, however, introduction of additional mechanisms beyond dissipation upon internal reflection and linear birefringence induced by beam path stress is not warranted for this data set.

7. Summary

The $p$-$s$ difference between the phase shifts for the orthogonal optical electric field components that occurs upon total internal reflection (TIR) can be leveraged for use in compensating rhomb devices exhibiting a nearly constant phase retardance over the ultraviolet to the near-infrared spectral range. The three-TIR configuration of the in-line rotatable rhomb first described by King and studied in this research was designed specifically for this purpose. The first and third TIRs of this rhomb occur at (fused silica)/ambient interfaces whereas the intermediate reflection occurs from a (fused silica)/MgF₂-film/ambient structure wherein the thickness of the MgF₂ layer is ~ 50 nm. Two sources of deviations from the ideal description of the rhomb performance were studied using straight-through Mueller matrix ellipsometry in the operational configuration supplemented by ambient side ellipsometry, both measurements performed over the 210 nm to 1650 nm range.

The first source identified here was weak dichroic behavior in the TIR from the (fused silica)/MgF₂-film/ambient structure due to dissipative absorption within the layer stack. An interface layer having an effective thickness (or volume per planar area) on the monolayer scale (~ 0.4 nm) was identified between the MgF₂ layer and the bulk fused silica. Analysis of this TIR structure using Mueller matrix ellipsometry was facilitated by external multiangle spectroscopic ellipsometry data acquired by illumination from the ambient side of the rhomb. Modeling of both sets of data suggests that the absorption is caused by Si-Si bonding arising from reduction of the SiO₂ of the fused silica surface/interface, and thus, can be modeled with the optical properties of amorphous silicon. The generated dichroic activity is unfavorable to the performance of the rhomb when used in its intended application in straight-through as an inline compensator unless its effect is taken into account in the Jones or Mueller matrix description of the device.

Whereas the source of the first deviation from ideality may be related to the design and processing of the rhomb coating and so may be avoidable, the second deviation is innate to the fused silica that is ground and polished to form the rhomb. Fused silica residual stress and rhomb stress due to mounting, which is required when the rhomb is rotated, generate birefringence and combine to introduce intrinsic wavelength dependent components to the net retardance of the device in addition to those generated by the three TIRs. A theoretical basis supported by best fits of stress birefringence simulations in bulk fused silica has resulted in an applicable procedure for determination of average stress, accumulated stress-induced retardance, and average fast axis orientation along the four path segments within the rhomb. These characteristics derive from an analysis of Mueller matrix measurements in the straight-through operational configuration of the rhomb.

The Mueller matrix formalism was applied to describe the operation of the King rhomb, and a first order analysis provided insights into the effects of both dichroism and individual path retardances on each of the 15 elements of the normalized Mueller matrix. This analysis identified the origins of the dominant deviations from the ideal description for each of the matrix elements. Finally, least-squares regression analysis was applied to model the normalized Mueller matrix elements measured over the wavelength range from 230 to 1000 nm and to extract wavelength independent parameters that characterize the retardance generated by stress-induced birefringence present in individual paths of the device. In this analysis, fixed spectra in the optical properties of the component materials of fused silica, MgF₂, and a-Si were used along with fixed wavelength independent parameters that describe the thin film structure and beam path geometry. The best-fit resulting Mueller matrix model for the device is critically important for accurate simulations of the device performance as a fixed or rotating compensator in instruments developed for spectroscopic polarimetry and ellipsometry.