27Summary and Conclusions

27.1Difficult Issues

Our final chapter examines the big picture of polarized light and optical systems, highlighting several slippery and difficult topics such as coherence, scattering, and depolarization. To tie together key concepts, an example telescope analysis is included, which shows how the polarization ray tracing and polarization aberration approaches are complementary.

Polarization engineering is a rapidly evolving area of optics. In such a changing environment, how can polarization engineers communicate better, make fewer mistakes, and complete projects faster? The following sections summarize concepts that have been particularly difficult to define or describe.

For polarization engineers to communicate better, the language must be clear and understood through the whole cycle of development and production, from those who specify optical systems, the optical designers who perform the analysis, the engineers who perform detailed design of all the parts and subsystems, the subcontractors and vendors making components, metrology and quality control testing these parts, to the system integration and test team.

Problems occur with the introduction of new technologies when technical terms are not understood across these boundaries. As a field matures, terms and methods of communication become standardized. A good example of the standardization of terms and methods is seen in the optics’ industries implementation of interferograms. In the 1950s, interferograms were not common and interferometers were not standard equipment in optics laboratories since coherent light sources (lasers) were not available, partially coherent sources, lightbulbs with pinholes, were not bright, and imaging sensors were primitive. Foucault tests, Ronchi tests, and similar geometric optical tests were standard. After the introduction of lasers and the spread of custom research interferometers in the 1960s and early 1970s, vendors still could not easily produce optics to interferometric specifications. The technology was new and expensive. Interferometers were custom instruments with interferograms still recorded onto film, developed chemically, and finally printed. Customers and vendors did not always agree on their interferometric tests. Because interferometry was a new technology, the language had not become standardized and technical communication remained an issue. By the 1980s, commercial interferometer technology with charge-coupled devices became affordable and widespread. Interferograms could be readily passed back and forth between departments, customers, and vendors. Measurements and terminology had become standardized and interferograms could serve as an efficient means of communication. Lens designers could predict interferograms for their optical designs; later metrology departments could confirm the performance of optics using standardized interferometers.

Mueller matrix images and Mueller matrix spectra have recently made a similar transition. In the 1970s, ellipsometers became standardized for measuring surfaces and films. But in the 1980s and 1990s, polarimeters for remote sensing and for optical system measurement were all custom instruments. In the 2000s, Mueller matrix polarimeter measurement technology was commercialized and, in the 2010s, became widespread and affordable. Now, a system integrator, such as an aerospace company or display company, can specify parts in terms of Mueller matrices, and vendors understand the specifications and can perform the tests in-house or outsource the metrology. A common language for polarization engineering has evolved, but is still not fully standardized.

Some of the more complex and difficult topics in polarization analysis are considered next.

27.2Polarization Ray Tracing Complications

27.2.1Optical System Description Complications

For an accurate polarization ray trace of an optical system, anything that makes a significant contribution to the polarization properties needs to be included. In conventional ray tracing, optical systems are defined only by the materials and shape of the optical elements. Polarization analysis requires much more information, much of which is not commonly used in conventional analysis.

Consider the complications associated with thin films, gratings, stress birefringence, and liquid crystal cells in the polarization ray trace. First, the thin films prescriptions, thicknesses, and indices must be specified and assigned to surfaces. Since many thin film designs are proprietary to the coating vendors, the prescriptions are not readily available to the optical designer; hence, measured thin film data or encrypted coating prescriptions need to be incorporated in ray tracing codes. Diffraction gratings have strong polarization effects. To include diffraction gratings in a polarization ray trace, the profile of the diffraction grating rulings is needed, and then rigorous coupled wave analysis can be performed at each ray intercept; unfortunately, the complex RCWA algorithm slows the calculation by typically a factor of one thousand per ray intercept. Grating polarization data can also be imported and interpolated, but grating vendors provide only a limited amount of measured data, angles and wavelengths, in-plane and out-of-plane, for commercial gratings; thus, vendor information is insufficient for accurately analyzing the polarization properties of most spectrometers and similar systems. Stress birefringence limits the imaging performance of many plastic lenses. For accurate analysis, the three-dimensional distribution of stress birefringence must be calculated with molding simulations or measured data. Liquid crystal cells are complex elements where the three-dimensional distribution of the director needs to be used for calculating the complex amplitude transmittance.

Often, such component information is not available, but this information shortage should not bring the polarization analysis and design process to a halt. For thin films with unknown designs, the designs for similar thin films can be used or a witness sample of the film can be measured in a polarimeter or ellipsometer. During the ray trace, the properties can be interpolated on a ray-by-ray basis. Similarly, grating and liquid crystal measurements can be used in lookup tables for polarization ray tracing. Often, a reasonable distribution of stress birefringence can be used to put an upper limit on the allowable stress and calculations can perform tolerance analysis without the actual stress distributions. For liquid crystal cells, since they are such critical optical components manufactured in enormous quantities in semiconductor fabs costing billions of dollars, imaging Muller matrix polarimetric metrology is widely deployed to measure the cell’s director distributions and to obtain accurate experimental data for polarization ray tracing models.

Thus, a successful polarization analysis requires the integration of many science models for the physics of thin films, reflection, refraction, gratings, crystals, stress, scattering, diffraction, and more.

27.2.2Elliptical Polarization Properties of Ray Paths

In propagating through optical systems, light rays encounter a sequence of polarization effects. The accumulated effect is described in the exit pupil where it is desirable to describe the ray path’s polarization properties in terms of an equivalent diattenuation and retardance. In systems of isotropic lenses and mirrors, each ray intercept acts as a homogeneous linear polarization element; the diattenuation and retardance are linear and aligned with the s- and p-planes. For skew ray paths, the ray’s s- and p-planes rotate from surface to surface. Sequences of “misaligned” linear diattenuators and linear retarders become elliptical diattenuators and elliptical retarders, thus complicating the polarization description of ray paths and wavefronts (see Section 14.3). Thus, for a general-purpose polarization ray trace, elliptical polarization must be calculated, but in lenses and other systems, the ellipticity of ray paths may be a low priority.

27.2.3Optical Path Length and Phase

The optical design of imaging systems involves calculating phase, amplitude, and polarization. Small variations of phase, in the order of half a wavelength or so, have a large impact on image quality. Small variations of amplitude or polarization state, in the order of 10%, have a much smaller impact on image quality. Hence, optical design is principally concerned with phase since it is the top priority.

To ensure that the wavefronts exiting an imaging optical system remain nearly spherical, optical design methods calculate the phase on a reference sphere in the exit pupil by calculating the optical path length (OPL) along ray paths through the optical system. To improve the transmission, antireflection coatings are added to lenses. Multiple reflections occur within these thin film structures and partial waves are generated in the forward and reverse directions. For the films to improve transmission, these partial waves must substantially constructively interfere in the forward direction and destructively interfere in the reverse direction. Thus, for an antireflection-coated optical system, a substantial number of partial waves are propagating in the forward direction. Each of these partial waves has a different OPL, amplitude, and polarization. Therefore, in a system with coatings, the OPL becomes multivalued. The OPL calculated by the ray tracing program is only the “direct” ray, without any multiple reflections, which is typically the first and usually the brightest of the partial waves. Figure 27.1 (left) shows an example of a single wavefront exiting from an uncoated lens. Figure 27.1 (middle) shows a set of partial waves exiting a lens with an ideal quarter wave coating, where each partial wave is separated by exactly one wavelength; thus, all the partial waves constructively interfere, suppressing the reflections from the lens and boosting the transmission. Each of these partial waves has a different OPL. When the wavelength changes (Figure 27.1, right), the partial waves are no longer separated by an integer number of wavelengths; thus, many more waves are seen in the figure. The lenses’ transmittance suffers since the exiting wavefronts do not completely constructively interfere. When an ultra-short laser pulse enters an optical system with coatings, a sequence of ultra-short pulses exit spread out in time, corresponding to the partial waves created by the coatings.

Figure 27.1 (Left) The spherical wavefront from the direct light through the system. (Middle) Ideal quarter wave antireflection coatings yield an additional series of partial waves separated by one wavelength with diminishing amplitude. (Right) If the wavelength changes, multiple partial waves are generated with non-integer wavelength separations.

Because of these multiple reflections, our bedrock concept of optical path length becomes more complicated, blurred by a legion of smaller partial waves; OPL has become a multivalued function. These partial waves cannot be physically separated from each other. An entering photon has a probability amplitude to be in any of the partial waves. The resulting electromagnetic wave is the vector sum of the electric field over all the partial waves and has a definite phase, amplitude, and polarization state. Thus, although the optical path length is no longer well defined, the phase is well defined and can be uniquely measured by an interferometer.

27.2.4Definition of Retardance

For similar reasons, the definition of retardance does not generalize well to systems with complex crystal assemblies and ray paths through such optical systems. When light transmits through a sequence of anisotropic materials, a set of two, four, eight, or more partial waves can exit, due to ray doubling, each with a different OPL. Again, the OPL becomes multivalued, but the amplitude, phase, and polarization state are still uniquely defined. Retardance is simple to define for a single crystal waveplate but more difficult to define in such general circumstances such as multiple birefringent plates. For a single birefringent plate, since only two waves exit, the retardance is simply the difference between the optical path lengths of the two modes. With a sequence of n anisotropic materials, 2n beams exit, each with different optical path lengths, amplitudes, and polarization states. No single “retardance” number can describe a complex set of optical path differences. A different approach to defining retardance is to consider a retarder as an element that rotates polarization states around the Poincaré sphere about an axis by an angle equal to the retardance. With this definition, the number of exiting beams does not matter and the retardance and fast axis can be specified.

27.2.5Retardance and Skew Aberration

Extending the concept of retardance to ray paths through optical systems is also complicated. These ray paths can be characterized by polarization ray tracing matrices, P. Since the entering and exiting rays need not be collinear, the eigenvectors of the P usually do not represent actual polarization states. The polarization-dependent phase changes can be divided into two categories, geometrical transformation and proper retardance. Coordinate system rotations, such as occur after sequences of refractions, can masquerade as circular retardance and are an example of geometrical transformation. Polarization changes on reflection, which masquerade as a half wave of linear retardance, are another example. The parallel transport of transverse vectors along ray paths through optical systems calculates the geometrical transformation, and the collection of geometrical transformations across a wavefront is the skew aberration.

Skew aberration is the polarization aberration of non-polarizing optical systems; skew aberration is the systematic polarization changes that remain when all the diattenuation and retardance are removed from an optical system. Skew aberration depends only on the sequence of propagation vectors. Skew tilt is the linear variation of skew aberration across the pupil orthogonal to the meridional plane as shown in Figure 27.2. A constant skew would uniformly change the polarization and would not affect the point spread function. It is the variation of skew (i.e., skew aberration) that affects the Mueller point spread matrix and thus degrades image quality. Skew aberration is typically a small effect in lenses but is a large effect in corner cubes.

Figure 27.2 The linear skew aberration transforms a uniform incident linearly polarized state into a skew aberrated state. This linear form of skew aberration occurs in most lenses and other rotationally symmetric systems.

The proper retardance refers to a physical property by which optical path length accumulation depends on the incident polarization state. P describes the polarization state changes due to diattenuation, retardance, and geometric transformations. The parallel transport matrix Q describes the associated non-polarizing optical system and thus keeps track of just the geometric transformation. To calculate the proper retardance, the geometric transformation needs to be removed, M = Q–1P. The difference in eigenvalue arguments of MR, the unitary part of the polar decomposed M, gives the proper retardance; the proper retardance cannot be assigned to a ray through an optical system in a black box whose internal ray path is unknown.

27.2.6Multi-Order Retardance

A multi-order retarder is a retarder with more than a half wave of retardance and a compound retarder is a combination of two or more retarders in sequence. When a compound retarder has fast axes that are misaligned with respect to one another, we call it a misaligned retarder. For misaligned compound multi-order retarders, more than two beams are involved and the common definition of retarders is inadequate. For these retarders, the optical path length can be multivalued as shown in Figure 27.3.

Figure 27.3 A misaligned two-element compound linear retarder has four output beams exiting with four different combinations of the fast and slow OPLs. For an aligned retarder, only the FF and SS beams exit.

Compound retarders have what at first appears to be mysterious behavior; the unwrapped retardance of compound retarders avoids the values around 2nπ and the retardance appears to jump, implying retardance discontinuities. These discontinuities are better understood in retardance space where the trajectory of the principal retardance can easily avoid the origin, which is just a single point representing the identity matrix (and the only point with zero retardance) as shown in Figure 27.4. For the unwrapped retardance, this point at the origin becomes the entire 2π sphere, 4π sphere, and so on. As the retardance increases, the trajectory appears to jump discontinuously across the 2nπ sphere in the retarder space.

Figure 27.4 The retarder space inside the π sphere (purple sphere) and the principal retardance component trajectories are plotted as the wavelength decreases (from the left figure to the right figure). The middle figure starts from the symmetric point of the ending point of the left figure, and so on. Note that the trajectories for the second and third figures miss the origin.

For small misalignment of fast axis orientation in a compound retarder, the trajectory passes close to the origin; thus, the discontinuities are small. These compound multi-order retarders have a single Jones and Mueller matrix and perform a single rotation on the Poincaré sphere. However, because of the multiple exiting beams, they do not exist in a single state of “retardance.”

27.2.7Birefringent Ray Tracing Complications

Anisotropic materials provide retardance, ray doubling, polarization separation, and other phenomena that cannot be generated by isotropic materials. The associated light–matter interactions are different from glass and other isotropic materials. An anisotropic material’s directional properties are described by the dielectric tensor and gyrotropic tensor rather than a single refractive index as in isotropic material. The two modes’ reflected and refracted rays propagate in different directions rather than in the same direction as with s- and p-polarizations. Wavefronts in anisotropic materials have more complex shapes, orthogonal to k but not orthogonal to S, and different residual aberrations compared to wavefronts in isotropic material. Propagation within the anisotropic material induces a phase difference between polarization states, the retardance.

At a birefringent interface, light divides into two eigenpolarizations, mutually orthogonal polarizations propagating in different directions with distinct refractive indices. This ray doubling characteristic of birefringent materials, which produces two new ray segments in both transmission and reflection, complicates the calculations of refracted and reflected amplitude coefficients in polarization ray tracing. The OPL associated with each ray becomes a function of polarization state.

The stress-induced birefringence in optical components is modeled as a spatially varying birefringence medium in Chapter 25. Such modeling predicts the spatial retardance variation throughout the stressed optical component and its effects on the PSF and MTF for different incident polarizations. Stress birefringence modeling is a powerful tool to predict the performance of components with residual stress to tolerance allowable stress, and optimize the molding process parameters, for example the tradeoff between processing time (time in the mold and molding temperatures) and product quality.

27.2.8Coherence Simulation

The standard ray tracing algorithms trace rays one wavelength at a time to determine wavefronts on a wavelength-by-wavelength basis; thus, these algorithms simulate coherent monochromatic light through the optical system. Often, the optical system properties are needed for partially coherent or incoherent light. The coherent, partially coherent, and incoherent simulation cases can be compared with a simple example configuration, light reflecting from the front and back surfaces of a glass plate or a retarder. In monochromatic light, the principal and reflected beams produce fringes. When the wavelength is swept, the fringes move. Thus, for light with a small spectral bandwidth, the fringe motion is small and the fringes are slightly blurred; the fringe visibility is reduced but the fringes are visible. This is the partially coherent case. For a larger bandwidth, so many different fringe patterns are superposed that in the large spectral bandwidth limit, fringes are completely lost. This is the incoherent case.

The monochromatic ray trace, either a conventional or polarization ray trace, calculates the coherent configuration; it will predict fringes from the glass plate. For an LED with a smaller spectral bandwidth, fringes may or may not be visible, but the monochromatic ray trace is always predicting fringes. Thus, for proper simulation of partial coherence (finite spectral bandwidth), the calculation is generally repeated for many wavelengths and the fringe patterns are combined to evaluate the effect of the spectral bandwidth on the reduction of fringe visibility. This is the essence of partial coherence calculations, repetition of the ray trace for a large number of wavelengths, fields, or configurations. Sometimes, shortcuts can reduce the amount of calculation, but usually the simplest procedure is just to ray trace a system a hundred times or more. Thus, individual wavelengths are ray traced to calculate optical path lengths, Jones matrices, and polarization ray tracing matrices, but light from different wavelengths, or light from different spatially incoherent sources, are added incoherently as Mueller matrices or the equivalent.

27.2.9Scattering

Evaluating the effects of scattering in an optical system is an important optical analysis task. For example, telescopes have baffles to prevent light from outside the field of view from reaching the focal plane. Before building expensive telescopes, for example, to be placed in orbit, it is common to calculate the out-of-field rejection of the baffle design. If the sun or moon is, for example, 3°, 10°, or 30° from the telescope’s axis, how much light scatters from the baffles and telescope structures and reach the focal plane? This involves tracing rays to the baffles and other scattering surfaces and then tracing many rays from each ray intercept to other scattering or optical surfaces and finding the flux along the ray paths that do reach the focal plane. At each ray intercept at each surface, a calculation of the bidirectional reflectance distribution function is commonly performed. Millions or billions of rays may need to be calculated with Monte Carlo routines to get accurate stray light estimates because of the vast number of potential ray paths.

Scattering effects can be very polarization dependent, for example, when scattering near Brewster’s angle or from diffraction gratings. In such cases, a scattering calculation needs the polarization bidirectional reflectance distribution function (PBRDF) for the surfaces of interest, a Mueller matrix function of the incident and scattering angles.1 This may be measured with a Mueller matrix polarimeter illuminating and collecting scatter from a large number of angles. Alternatively, a suitable analytic PBRDF model may exist, such as the ScatMech library from the National Institute of Standards and Technology.1,2 Scattering ray trace simulations are an example of an incoherent ray trace since the path lengths vary by thousands of wavelengths.

27.2.10Depolarization

Some optical systems have measurable depolarization, where incident polarized light exits with a reduced degree of polarization. Some optical elements, such as diffraction gratings, holographic optical elements, multimode waveguides, and liquid crystal cells, are more likely to contribute depolarization. Depolarization is a statistical process, a randomization of the polarization state often due to roughness, scattering, and variations of a material at a microscopic scale. With laser illumination through an optical system, in the absence of depolarization, the wavefront is smooth with a slowly varying amplitude, phase, and polarization. Depolarization causes the wavefront to become noisy with some high-frequency content. A rough surface or a coating that is cracked or peeling imposes many small phase variations to the wavefront, which cause such noise. For a large amount of depolarization, such as scattering from a rough surface or light transiting an integrating sphere, the emerging wavefront becomes a speckle pattern with a rapidly varying polarization state.

The phase changes from randomly scattering surfaces, integrating spheres, and the like cannot be exactly simulated with a ray tracing program. A surface with roughness of the scale of 0.1 μm would require more than (105)2 points or more than 1010 points per square centimeter to characterize for ray tracing, an impossible task.

Depolarization is not described by Jones matrices and polarization ray tracing matrices; it is a statistical process. Depolarizing optical components can be ray traced with Mueller matrices. In general, when light is incident on a depolarizing surface, the light scatters into many directions. Scattered light ray tracing programs are programmed to calculate many scattered rays, leaving an interface for each incident ray, and can be programmed to calculate with Mueller matrices. Polarization BRDF models are available for many depolarizing surfaces and can be used for each scattering calculation. Such calculations are usually performed by choosing rays, leaving a ray intercept using Monte Carlo methods, and not by following regular ray grids through an optical system.

A ray tracing calculation with Mueller matrices is an example of an “incoherent” ray trace, where incoherent refers to the ray trace, not the coherence of the light. A Mueller matrix does not track the absolute phase of the light. Mueller matrices can be added to simulate the combination of polarized or partially polarized beams that do not have a phase relationship with each other; they are mutually incoherent.

Such a calculation is appropriate for an integrating sphere illuminated with laser light. The output is a speckle pattern that cannot be exactly simulated because the description of the surface (with 1010 points per square centimeter for example) is unreasonable. Light reaches a surface following the integrating sphere from paths leaving many different areas of the sphere. The optical path lengths of rays incident on a particular point are randomly distributed over hundreds of waves of optical path length; thus, the rays are incoherently combined. A Mueller matrix ray trace can calculate the Mueller matrix, which would be measured by averaging over many many speckles, even though the exact distribution of speckles cannot be calculated. For this coherent light, an incoherent ray trace is appropriate due to the scattering nature of the system.

For analyzing an optical system with some depolarization, such as from a diffraction grating, a coherent ray trace can be performed for the majority of the light, the image-forming part. The scattered light part of the light can be represented with Mueller matrices, and a separate incoherent ray trace can be performed for this depolarized fraction.

27.3Polarization Ray Tracing Concepts and Methods

27.3.1Jones Matrices and Jones Pupil

The Jones matrix operates on Jones vectors, which describe the polarization ellipses with respect to an x–y local coordinate system in the transverse plane. To use Jones vectors and matrices in optical design for the ray tracing of highly curved beams, different local coordinate systems are required for each ray segment to define the direction of the Jones vector’s x- and y-components in space. These local coordinate systems lead to complications due to the intrinsic singularities of local coordinates.

The full description of the wavefront and polarization at the exit pupil is divided into a combination of four functions: the wavefront aberration function, amplitude function, aperture function, and polarization aberration function at the exit pupil. This combination is called the Jones pupil,

$$\mathrm{JonesPupil}(x,y)=\mathrm{aperture}(x,y)\cdot a(x,y)\cdot \mathbf{\text{J}}(x,y)\cdot e^{-2\pi iW(x,y)}.$$27.1

27.1

aperture(x,y) is an aperture function that is one inside the aperture and zero outside the aperture. a(x,y) is an amplitude function that describes the transmission along the ray path, also known as the apodization. W(x,y) is the wavefront aberration function that characterizes the optical path difference between each ray and the chief ray. J(x,y) is the Jones matrix along ray paths from the entrance pupil to the exit pupil. (x,y) is the pupil coordinate. The JonesPupil(x,y) is spatially varying and is the starting point for the polarization image formation calculations.

27.3.2P Matrix and Local Coordinates

The 3 × 3 polarization ray tracing matrices, P matrices, enable ray tracing in global coordinates, which provide an easy basis to interpret polarization properties. Diattenuation can be calculated by the singular value decomposition (SVD) of P since its singular values are the maximum and minimum amplitude transmittances.

Given the distribution of polarization states on a spherical wavefront, representing this information on a computer screen or page involves choices on how to flatten the information. Two most commonly used basis vector functions for this transformation are the dipole coordinates, a latitude and longitude system, and the double pole coordinates, a coordinate system that matches the polarization and direction changes when lenses focus collimated wavefronts.

Chapter 11 presented methods for converting the polarization representation from 3D to 2D, and vice versa, using the dipole and double pole coordinates. Figure 27.5 shows Jones pupil’s dependence on the choice of local coordinates when converting the P matrix pupil.

Figure 27.5 (Top) Jones pupil of the cell phone lens system using (left) double pole coordinates and (right) s–p coordinates to convert from the P matrix pupil. (Bottom) Local x coordinates across the exit pupil from the (left) double pole coordinates and (right) s–p coordinates.

27.3.3Generalization of PSF and OTF

A point spread function (PSF) is a fundamental metric for optical imaging systems and describes the image of a point object. For systems with polarization aberration, the PSF depends on the incident polarization state. Thus, the concept of a PSF is generalized so that it describes imaging of arbitrary polarization states by introducing an amplitude response matrix (ARM). Given a wavefront aberration function and Jones pupil function, the Mueller point spread matrix (MPSM) is calculated, which shows how the PSF varies with the incident polarization state. Similarly, the optical transfer function (OTF) of conventional optics can be extended to an optical transfer matrix (OTM) to describe how the spatial filtering of an object during imaging depends on the incident polarization state. Figure 27.6 shows the relationship between Jones pupil, ARM, MPSM, and OTM. Off-diagonal elements in ARM, MPSM, or OTM describe the coupling between polarization components in the image. The relative magnitude of the off-diagonal elements with respect to the diagonal elements tells the amount of polarization mixing occurring in the image plane. Thus, for the analysis of optical systems with polarization aberrations, matrix representations of the PSF and OTF are necessary.

Figure 27.6 Relationships between the 2 × 2 amplitude response matrix (ARM), the 4 × 4 Mueller point spread matrix (MPSM), the 4 × 4 optical transfer matrix (OTM), the 4 × 4 modulation transfer matrix (MTM), and the 4 × 4 phase transfer matrix (PTM) are shown.

27.3.4Ray Doubling, Ray Trees, and Data Structures

At an anisotropic element, one incident wavefront yields two exiting wavefronts in transmission, a system with N anisotropic elements can have 2N transmitted wavefronts. Each of the 2N wavefronts focuses at different locations, has different polarization state variation, and has differing amounts of wavefront aberrations. One incident ray exits with 2N optical path lengths. Retardance, which is defined as the optical path difference between two overlapping rays, needs to be redefined for the case of three or more overlapping wavefronts.

To simulate the propagation of wavefronts through optical systems, a large number of rays are traced to accurately sample the shape, amplitude, and polarization over the wavefront. We extend the polarization ray tracing calculation to optical systems with anisotropic materials. The ray tracing results of these wavefronts are stored in the P matrices. Then, the properties of these wavefronts, such as the wavefront aberrations, polarization aberrations including diattenuation and retardance, and the image quality are calculated from the P matrices. Because of the ray doubling behavior of the anisotropic materials, the algorithm for polarization ray tracing is further generalized to handle the exponentially increasing number of ray segments as the number of anisotropic elements increased.

To characterize wavefronts transmitted through anisotropic materials, families of rays need to be traced. When tracing large grids of rays to accurately sample the multiple exiting wavefronts, accessing the ray properties along these ray paths through a specific surface or a series of surfaces is an important step in optical design. K incident rays refracting through N anisotropic intercepts result in K × 2N exiting rays and $K\left({\displaystyle \sum _{n=1}^N{2}^n}\right)$ ray segments. These branches of ray segments can be organized by the cumulative mode label (introduced in Table 19.1) into a ray tree such as is shown in Figure 27.7.

Figure 27.7 A ray tree for one incident ray refracting through three uniaxial interfaces, where the ordinary mode is o and the extraordinary mode is e.

Such a ray tree keeps track of the multiplicity of rays generated by sequences of anisotropic elements. The cumulative mode label identifies the individual polarized wavefront of a specific sequence of ray splittings as the wavefront propagates through the series of optical surfaces. These mode labels accommodate the automation of ray doubling in a computer ray trace without imposing any assumptions and assist in sorting the exiting rays for each polarized wavefront. The data structure derived from the mode label systematically manages the multiplicity of rays generated by anisotropic elements. These structures are useful to handle many special cases unique to anisotropic interactions: inhibited refraction, total internal reflection, and conical refraction.

27.3.5Mode Combination

After tracing all the ray splitting through an optical system, parts of these exiting wavefronts are usually overlapping. A method to analyze the resultant wavefront represented by these ray trees is as follows. The eigen-wavefront produced from each separate mode sequence is first reconstructed by interpolation. As illustrated in Chapter 20, each of these reconstructed wavefronts has different aberrations, but a smooth distribution at the exit pupil. Therefore, they can be interpolated with the electric field of the sample rays.

When the eigen-wavefronts overlap at the image plane, they interfere. Each of these wavefronts produces an image that can be calculated using methods presented in Chapter 16. Then, the overall image is the summation of these eigen-images.

27.3.6Alternative Simulation Methods

This book’s emphasis has been on polarization, but other methods are necessary for related polarization problems in optical systems with nonlinear optics, waveguides, photonic devices, metamaterials, and nanostructured materials. Another class of simulation methods, wave propagation algorithms, propagates the fields through devices and optical systems, diffracting the light from the first to second surface, and so on. Wave propagation algorithms are necessary at longer wavelengths, that is, radio, where the ray approximations break down, and in waveguides where structures have dimensions on the order of the wavelength. For subwavelength structures, periodic structures can be simulated with rigorous coupled wave analysis (Chapter 23). Non-periodic structures are often simulated with finite difference time domain algorithms (FDTD), which solve the time-dependent Maxwell’s equations on a space–time grid taking small steps in space and time. First the electric field is propagated in step, then the magnetic field, and then the process is repeated. FDTD is a very general light field propagation method, but because of the amount of computation involved, the region of solution is generally limited to volumes less than 0.1 by 0.1 by 0.1 mm.3

27.4Polarization Aberration Mitigation

Optical designers have a variety of methods to change the polarization aberrations of any particular optical system.4 Fresnel aberrations and coating-induced polarization aberrations tend to be of small magnitude with low-order functional variation (constant, linear, quadratic, etc.).5 The following summarizes several mitigation approaches:

1. Reducing angles of incidence: Since the diattenuation and retardance increase quadratically for small angles of incidence, reducing the largest angles of incidence can significantly reduce polarization aberration. Reducing the range of angles at mirrors and lenses reduces the variation of retardance and diattenuation.

2. Reducing coating polarization: The optical coating prescriptions for antireflection coatings of lenses and reflection-enhancing coatings of mirrors provide design degrees of freedom (thicknesses and materials) to adjust the diattenuation and retardance. In our experience, these coating prescriptions can be adjusted to moderately reduce the polarization properties, but cannot zero out diattenuation or retardance for substantial angle and wavelength ranges. The surfaces of antireflection-coated lenses typically have one-third or less the diattenuation of uncoated lens surfaces, providing great benefit. The reflection-enhancing coatings for mirrors often increase the retardance and diattenuation of metal mirrors in some wavebands.

3. Compensating polarization elements: Polarization aberrations can be introduced in several ways. Simply placing a (spatially uniform) weak polarizer (diattenuator) and a weak retarder in the system could zero out the polarization aberration at one point in the pupil, leaving overall polarization aberrations smaller. A spatially varying diattenuator and retarder with polarization magnitude approximately equal to the cumulative diattenuation and retardance but orthogonally oriented would nearly eliminate the polarization aberration. Such a polarization plate could be considered as the matrix inverse of the Jones pupil. Such correction plates might be fabricated from liquid crystal polymers with spatially varying magnitude and orientation of diattenuation or retardance, similar to the vortex retarders used in coronagraphy.6–8 Wedged, spherical, and aspheric crystalline elements or element assemblies can provide a wide variety of compensating polarization aberrations.9 Since polarization aberrations of telescopes and fold mirrors tend to be small, spatially varying anisotropic thin films, which can only provide small retardances, could provide another path toward compensation.10

4. Crossed fold mirrors: Fold mirrors tilted about opposite axes, such that the p-polarized light exiting one mirror is s-polarized on the second, have a compensating effect for both diattenuation and retardance.11,12 A linear variation of polarization about zero will still remain across the pupil.

5. Compensating optical elements: The diattenuation of lenses has the opposite sign (greater p-transmission) compared to the diattenuation of mirrors. Thus, including lenses would reduce the diattenuation from the primary and secondary mirrors in the example system in Section 27.5. Similarly, sets of coatings might be selected to have opposite retardance contributions. Despite several concerted attempts, the author (Chipman) has not been able to change the sign of the diattenuation or retardance of an antireflection- or reflection-enhancing coating over a useful spectral bandwidth. In practice, this approach has never been very successful.

Considering these mitigation approaches, novelty, fabrication issues, scattering, tolerances, and risk must be balanced against the magnitude of the polarization aberration. For critical systems, with low levels of polarization aberration, the imperfections associated with these mitigation approaches, in particular irregularity, absorption, and scattering, can easily become worse than the problem.

To design optical systems, typically a merit function is defined to characterize the wavefront and image quality, and an optimization program adjusts the system’s constructional parameters to find acceptable configurations.

1. If polarization ray tracing parameters such as diattenuation and retardance are included in the merit function, the optimizer can balance the polarization aberrations against the wavefront aberrations and other constraints, pushing the solutions toward reduced polarization aberration.

2. Similarly, if the coating and polarization element constructional parameters are included in the optimization, the optimizer can explore the coating design space and polarization element configuration to find compensation schemes. For example, overcoated layers on aluminum will modify the polarization.

These two steps are complicated, but advanced users can apply these methods, often through the use of the optical design program’s macro languages, to evaluate polarization mitigation strategies listed above.

27.4.1Analyzing Polarization Ray Tracing Output

Polarization ray tracing is complicated! Conventional optical design produces a wavefront aberration function, a scalar function. Incorporating the effects of coatings and isotropic elements, polarization ray tracing produces arrays of polarization ray tracing matrices. These polarization ray tracing matrices can then be transformed into Jones matrices defined between spherical surfaces in the entrance pupil and exit pupil. The ray tracing problem has moved from scalar wavefront functions with one degree of freedom into an eight-dimensional representation using Jones matrices. Three dimensions are hard to visualize. Four-dimensional spaces are very difficult to visualize. Eight dimensions are truly daunting. To understand and operate on this eight-dimensional data, the designer needs to extract the most critical information. To effectively communicate the information, the quantity of information needs to be reduced, priorities need to be established, and the total information needs to be simplified without oversimplifying. Only certain polarization properties are likely to be important, but which are the most important properties vary from system to system. Buried in all the data, only a few aberration patterns may be important. Finding such features can often summarize the important information from a vast mass of calculated data.

The polarization ray tracing algorithm needs to be general purpose. The algorithm needs to make accurate calculations for a broad range of optical systems. To do this, it needs to calculate all eight degrees of freedom in the Jones calculus to handle a general set of problems. But many systems have no significant circular retardance or circular diattenuation (see Section 27.2.2). Hence, these two degrees of circular freedom can often be set aside, typically after the calculation, as having lower priority. Similarly, some systems have no significant retardance, like uncoated lenses. Other systems have no significant polarization aberration, so that only the optical path length and wavefront aberration are needed, but the polarization ray tracing calculations may be needed to verify, for example, that a set of coatings have minimal effect on the image formation of a particular lens.

The polarization analysis challenge remains. The optical designer is confronting an eight-dimensional Jones pupil function and needs effective tools in the polarization ray tracing program to quickly understand the data and prioritize the optical effects. In the next section, the application of polarization aberration expansions to polarization ray tracing data is shown to provide a method for reducing the polarization ray tracing output to a small number of parameters and simple functions. Further, if the coating design is changed or fold mirror angles changed, the system doesn’t necessarily need to be retraced; in this example, design rules were derived from the aberrations to show how the performance scales with such changes.

27.5Comparison of Polarization Ray Tracing and Polarization Aberrations

The polarization ray tracing method and polarization aberration method will be compared for an example system. Ray tracing methods treat system performance and aberrations numerically as grids of OPLs, Jones matrices, and other values. Aberration theory describes system aberrations in terms of simple closed-form functions. Both methods have their strengths and weaknesses and complement each other. Aberration theory has strengths in the simplicity of the aberration representation, representation of the aberration with a small number of parameters, the ability to describe how performance scales with numerical aperture and object size, and the ability to pinpoint at which surfaces aberration is arising and suggest methods for aberration balancing. Aberration theory becomes complicated as systems lose symmetry, become tilted or decentered, or incorporate free-form optical surface shapes. The strength of ray tracing is its ability to analyze arbitrary systems and to provide (nearly) exact answers. The aberration approach can lead to powerful design rules to assist in making optical design trade-offs.13

An example Cassegrain telescope (Figure 27.8) consisting of a primary, secondary, and fold mirror is analyzed, comparing the ray tracing and aberration methods with the goal of relating the constructional parameters of the telescope to the coating effects on the PSF. A closed-form expression for the Jones pupil is found in terms of the second-order polarization aberrations: constant (piston), linear (tilt), and quadratic (defocus) terms for the diattenuation, retardance, amplitude, and phase. With these aberrations, the entire Jones pupil can be reduced to 12 parameters, its second-order polarization aberration coefficients. These coefficients can be directly related to the mirror coatings and the angles of incidence through the system. The following are two of the aberrations to be considered: (1) The XX and YY PSF components, the two bright co-polarized components, are shifted in opposite directions. This shear PSF is related to the fold mirror coatings and image-space numerical aperture. (2) The XX and YY PSFs have different magnitudes and orientations of on-axis coating-induced astigmatism, which are related to the primary and secondary mirror angles of incidence and coatings. This example combines the polarization aberration analysis of the fold mirror of Section 12.3 with the Cassegrain telescope analysis of Section 12.5. This example was initially analyzed for coronagraphs for the imaging of exoplanets and debris disks around stars where the control of the PSF needs to be exquisite to image planets with expected brightnesses of <10−8 of the star within several Airy disk radii.

Figure 27.8 An example Cassegrain telescope system with a primary mirror at F/1.2, a Cassegrain focus of F/8, and a 90° fold mirror in the F/8 converging beam. The 90° fold mirror is folded about the x-axis. The primary mirror has a clear aperture of 2.4 m. All three mirrors are coated with aluminum with index n = 2.80 + 8.45i at 800 nm. Y-polarized light refers to the polarization in the entrance pupil in the plane of incidence of the axial ray at the fold mirror, or vertical in the picture. X-polarized light is s-polarized at the fold mirror.

Figure 27.8 shows the Cassegrain telescope and fold mirror illuminated with an on-axis collimated beam. The fold mirror tilts about the x-axis reflecting light propagating toward +z into the −y-axis. The primary mirror is parabolic. The conic constant of the hyperbolic secondary mirror is chosen to eliminate spherical aberration; there are no on-axis wavefront aberrations. Therefore, from a conventional ray trace, the on-axis system is perfect, diffraction limited. During the polarization ray trace, any deviations from ideal imaging are due to the mirrors coating polarization and are not mixed with the effects of wavefront aberration.

27.5.1Aluminum Coating and Polarization Aberration Expression

An aluminum coating has been chosen for this analysis. The amplitude and phase coefficients for aluminum at 800 nm are plotted in Figure 27.9. The diattenuation is shown in Figure 27.10 (left) where it is zero at normal incidence and increases approximately quadratically to about 0.05 at the 45° central angle of the fold mirror. Similarly, the retardance, shown in Figure 27.10 (right), increases quadratically from 0 to about 0.15 at 45°. Hence, the retardance is expected to generate about 0.15/0.05 ≈ 3 times more polarization aberration.

Figure 27.9 Reflection coefficients for the amplitude (left) and the phase (right) for angles of incidence θ between 0° and 90° for a bare aluminum mirror. ϕrs and ϕrp are the reflected phase for s- and p-polarized light. The green vertical line highlights the phase changes at 45°. The different slopes or linear phase shifts (red ϕrs and blue ϕrp lines) at the 45° incident angle cause different XX and YY image shifts.

Figure 27.10 (Left) Diattenuation and (right) retardance of bare aluminum mirror at 800 nm.

For the polarization aberration analysis, the Fresnel equations can be replaced with simpler linear and quadratic polynomial fits valid over the range of angles in use. For the on-axis mirrors, the coating diattenuation D(θ) and retardance δ(θ) functions are expanded about normal incidence in quadratic functions as

$$D(\theta )\approx a_2{\theta }^2+O({\theta }^4),\delta (\theta )\approx b_2{\theta }^2+O({\theta }^4),$$27.2

27.2

with second-order coefficients a2 for diattenuation and b2 for retardance. These quadratic fits are shown in Figure 27.11 (left) and Figure 27.12 (left). For the fold mirror, a first-order expansion about the axial ray angle of incidence θ0 = 45° suffices,

Figure 27.11 (Left) Aluminum diattenuation versus angle of incidence with quadratic fit about 0° and (right) linear fit about 45°. Solid brown lines are the exact diattenuation. Dashed black lines are the quadratic and linear fit of diattenuation (see Table 27.1). The solid green line in the right figure indicates the angle of incidence of the axial ray and the dashed green lines indicate the range of the NA = 0.06 beams.

Figure 27.12 (Left) Aluminum retardance in radians versus angle of incidence with quadratic fit about 0° and (right) linear fit about 45°. Solid magenta lines are the exact retardance. Dashed black lines are the quadratic and linear fit of retardance. The solid green line in the right figure indicates the angle of incidence of the axial ray and the dashed green lines indicate the range of the NA = 0.06 beams.

$$D(\theta )\approx a_0+a_1(\theta -{\theta }_0)+O({\theta }^2),\delta (\theta )\approx b_0+b_1(\theta -{\theta }_0)+O({\theta }^2)$$27.3

27.3

and the fits are shown in Figure 27.11 (right) and Figure 27.12 (right). The fit coefficients are tabulated in Table 27.1. These Taylor series coefficients can be evaluated at other wavelengths to describe the polarization aberration as a function of wavelength. Fits to coating design program output can provide the coefficients for other metals and for arbitrary multilayer coatings by the curve fitting method of Math Tip 13.1.

Table 27.1Aluminum Mirror Coating Fitting Coefficients

a0 = 0.049

a1 = 0.0026

a2 = 0.000024

b0 = 0.150

b1 = 0.0079

b2 = 0.000070

Since the fold mirror is in a converging beam, the non-zero slopes of the s- and p-phases are important and have been highlighted in Figure 27.9 (right). These slopes indicate linear phase shifts that move the locations of the X- and Y-polarized PSF components from the geometrical image location. Since the slopes are different with opposite sign, the corresponding image components move in different directions by a small fraction of the Airy disk radius.

27.5.2Polarization Ray Trace and the Jones Pupil

A polarization ray trace of the telescope and fold mirror was performed with Polaris-M (Airy Optics Inc.). The Jones pupil calculation is displayed in Figure 27.13. From the amplitude maps (Figure 27.13, left), it is seen that this Jones pupil is very close to the identity matrix times a constant (~0.806); the 0.806 accounts for average amplitude reflection losses from three aluminum reflections. Since the Jones pupil is close to the identity matrix, the coating-induced polarization aberrations are small. Deviations from the identity matrix are due to the mirror’s diattenuation and retardance.

Figure 27.13 Values of the Jones pupil elements, given by Equation 27.4, are displayed as color-coded images to show variation across the exit pupil. The four images on the left show the amplitude as a function of pupil position and the four images on the right show the phase. Amplitudes AXX and AYY are constant to within 2%. Since AXY and AXY are highly apodized, their diffraction patterns are significantly larger than images associated with the diagonal terms. A scale appears to the right of each box: units are amplitude (left) and phase in radians (right). The phase of a complex number changes by π when the amplitude passes through zero. This causes the phase discontinuities in ϕXY and ϕYX.

The diagonal elements, JXX and JYY, contain different amplitude variations. The overall JXX amplitude is about 5% larger than the overall JYY amplitude because of the s- and p-reflection difference at 45° as shown in Figure 27.11 (left). This will cause the x-polarized image to be about 9% brighter than the y-polarized image. The amplitude images of the Jones pupil in Figure 27.13 (left) are close to the identity matrix, only a small fraction of the light has its polarization changed. The off-diagonal JXY and JYX elements show the polarization coupling between orthogonal polarizations. This polarization cross-talk has relatively low amplitude compared to the diagonal elements. The amplitudes AXX and AYY are nearly constant (<2% variation), but the AXY and AYX are highly apodized, showing a Maltese cross pattern (dark along x- and y-axes) shifted downward.

The phases of the four elements in the Jones pupil, shown in Figure 27.13 (right), represent contributions to the wavefront aberration function from the aluminum mirrors. The Fresnel phase changes are different for the s- and p-components leading to different wavefronts for these two components. Along the y-axis and along a horizontal line below the x-axis, the amplitudes of the AXY and AYX terms change sign, so the phases ϕXY and ϕYX change by π along these lines. Since the on-axis geometrical wavefront aberration of the telescope is zero, the phase variation across the pupil is entirely from the mirror coatings. ϕXX, the wavefront aberration function of the telescope illuminated with x-polarized light and analyzed with an x-analyzer, has an overall linear variation of about 0.008 waves with a small additional deviation, which primarily has the form of astigmatism. The diagonal elements ϕXX and ϕYY have a different linear variation indicating a wavefront tilt between the XX and YY wavefronts. The source of this tilt difference is seen in Figure 27.12 (right), where the fold mirror at 45° angle of incidence has different slopes of the phase change for s-polarized light and p-polarized light. If the Fresnel phases were linear about 45°, only tilt would be introduced. The deviations from linear introduce higher-order aberrations including small amounts of astigmatism (from quadratic deviation), coma (from cubic deviation), and other aberrations.

Thus, polarization ray tracing can provide a detailed picture of the polarization aberrations. If the object moves off-axis, the coatings are modified, or the configuration of the system changed, the polarization ray trace can readily recalculate the polarization aberrations. A large number of rays have been calculated to generate this Jones pupil.

27.5.3Aberration Expression for the Jones Pupil

To develop a simpler polarization aberration model for an optical system, the Jones pupil (Figure 27.13) can be fit to a Jones matrix aberration function containing aberration terms appropriate for that particular optical system. For wavefront aberrations, defocus, spherical aberration, coma, astigmatism, and so on, are appropriate functions for a large fraction of optical systems. Similarly, for polarization aberrations, constant, linear, and quadratic functions for diattenuation, retardance, amplitude, and phase provide a good fit to many systems including this Cassegrain telescope with a fold mirror.

The Jones pupil of Figure 27.13 is a smooth function. Each element’s amplitude (left) and phase (right) has low-order variations that are well described by simple constant, linear, and quadratic (piston, tilt, and defocus) terms as functions of polar pupil coordinates ρ, the normalized radial distance, and ϕ, the azimuth measured from the x-axis. The Jones pupil description has four components: the wavefront aberration W(ρ, ϕ) from the coatings, amplitude transmission A(ρ, ϕ), and the diattenuation and retardance shown here combined into the Jones pupil function JP(ρ, ϕ)

$$\mathbf{\text{JP}}(\rho ,\varphi )=A(\rho ,\varphi )e^{i2\pi W\left(\rho ,\varphi \right)/\lambda }\mathbf{\text{J}}(\rho ,\varphi ).$$27.4

27.4

The wavefront aberration and amplitude transmission are scalar terms. Examining the aluminum mirror phase changes (Figure 27.9, right), the difference (ϕs − ϕp) is the retardance and the average of the variation, (ϕs + ϕp)/2, is the wavefront aberration. The coatings introduce small amounts of the wavefront aberrations piston w0, tilt w1, and defocus w2 with the functional forms

$$W(\rho ,\varphi )=w_0+w_1\rho \sin \varphi +w_2{\rho }^2.$$27.5

27.5

Higher-order wavefront aberrations, spherical aberration, coma, and so on, are not generated in significant amounts by the aluminum coatings in this example. Similarly, small polarization-independent amplitude variations (variations of the s- and p-average reflectance) are generated from the Fresnel aberrations, and the associated piston, tilt, and defocus are expressed with coefficients a0, a1, and a2 as

$$A(\rho ,\varphi )=a_0+a_1\rho \sin \varphi +a_2{\rho }^2.$$27.6

27.6

The diattenuation and retardance aberrations are each described by three terms yielding the six polarization aberration terms (Equation 27.7) J1, J2, … J6, listed in Equation 27.8, defined in terms of the Pauli matrices, σ1, σ2, and σ3, and the identity matrix σ0 (see Chapter 14).

Each aberration term of the Jones matrix function is defined over the pupil using polar coordinates ρ and ϕ. The magnitude of each term is specified by an aberration coefficient, d0, d1, and d2, for the diattenuation terms, and Δ0, Δ1, and Δ2 for the retardance terms. All six aberration coefficients are much less than one, so when these six terms are cascaded, only the first-order terms in σ1 and σ2 are significant, as described in Section 15.3. The result describes the Jones pupil as the sum of the six aberration terms,

$$\begin{array}{l}\mathbf{\text{J}}={\mathbf{\text{J}}}_6{\mathbf{\text{J}}}_5{\mathbf{\text{J}}}_4{\mathbf{\text{J}}}_3{\mathbf{\text{J}}}_2{\mathbf{\text{J}}}_1\\ \phantom{J}={\mathbf{\sigma }}_0+{\mathbf{\sigma }}_1\left[\frac{(d_0+i{∆}_0)-(d_1+i{∆}_1)\rho \sin \varphi +(d_2+i{∆}_2){\rho }^2\cos 2\varphi }{2}\right]\\ +{\mathbf{\sigma }}_2\frac{(d_1+i{∆}_1)\rho \cos \varphi +(d_2+i{∆}_2){\rho }^{2}sin2\varphi }{2},\end{array}$$27.7

27.7

where J1 is diattenuation piston, J2 is retardance piston, J3 and J4 are tilt, and J5 and J6 are defocus terms:

$$\begin{array}{l}{\mathbf{\text{J}}}_1={\mathbf{\sigma }}_0+\frac{d_{\text{0}}}{2}{\mathbf{\sigma }}_1,{\mathbf{\text{J}}}_2={\mathbf{\sigma }}_0+\frac{i{∆}_{\text{0}}}{2}{\mathbf{\sigma }}_1,\\ {\mathbf{\text{J}}}_3={\mathbf{\sigma }}_0+\frac{d_{\text{1}}\rho }{2}(-{\mathbf{\sigma }}_1\sin \varphi +{\mathbf{\sigma }}_2\cos \varphi ),{\mathbf{\text{J}}}_4={\mathbf{\sigma }}_0+\frac{i{∆}_{\text{1}}\rho }{2}(-{\mathbf{\sigma }}_1\sin \varphi +{\mathbf{\sigma }}_2\cos \varphi ),\\ {\mathbf{\text{J}}}_5={\mathbf{\sigma }}_0+\frac{d_{\text{2}}{\rho }^2}{2}({\mathbf{\sigma }}_1\cos 2\varphi +{\mathbf{\sigma }}_{2}sin2\varphi ),{\mathbf{J}}_6={\mathbf{\sigma }}_0+\frac{i{∆}_{\text{2}}{\rho }^2}{2}({\mathbf{\sigma }}_1\cos 2\varphi +{\mathbf{\sigma }}_{2}sin2\varphi ).\end{array}$$27.8

27.8

Equation 27.4 incorporating Equations 27.5, 27.6, and 27.7 provides an accurate expression for the example telescope’s Jones pupil.

Table 27.2 lists the aberration coefficients for the retardance, diattenuation, amplitude, and wavefront for the Jones pupil of the telescope end to end determined by curve fitting the coefficients of Equation 27.4 incorporating Equations 27.5, 27.6, and 27.7 to the Jones pupil polarization ray trace data (Figure 27.13). Each of the aberration coefficients, d0, d1, d2, Δ0, Δ1, and Δ2, is the value of the diattenuation or retardance at the edge of the pupil for that specific term.

Table 27.2Polarization Aberration Coefficients for Telescope’s Jones Pupil Shown in Figure 27.13

	Polarization Aberration Coefficients
Diattenuation	d0 = 0.050	d1 = –0.008	d2 = −0.007
Retardance	Δ0 = −0.151	Δ1 = −0.023	Δ2 = −0.022
Amplitude	a0 = 0.806	a1 = −0.002	a2 = 0.0000
Wavefront	w0 = 2.492	w1 = −0.004	w2 = 0.000

All of the polarization aberration coefficients are much less than one, so that the combination of terms in Equation 27.7 is an accurate representation for the cascaded polarization of the three elements, and the overall matrix depends little on the order of the six terms chosen. Figure 27.14 shows the Jones pupil of Figure 27.13 as approximated by the polarization aberration function. Figure 27.15 shows the small residual difference between the polarization ray trace result shown in Figure 27.13 and the polarization aberration fit shown in Figure 27.14; the fit matches the exact polarization ray tracing within 0.002 in amplitude, which is better than 0.2% of the average amplitude ~0.8.

Figure 27.14 The Jones pupil calculated using Equation 27.7 with the aberration coefficients of Table 27.2 provides an accurate representation of the Jones pupil (left) for Jones pupil amplitude and (right) for Jones pupil phase in radians.

Figure 27.15 The differences between the Jones pupil obtained by polarization ray tracing in Figure 27.13 and the aberration expansion fit in Figure 27.14 are small. In all of the plots, black represents a difference of zero; the fit generated by the aberration expansion has the same value as the ray tracing data. This residual contains small contributions from polarization aberration terms of higher order than Equation 27.7.

The phase and retardance values of Table 27.2 are provided in radians. They can be divided by 2π to express the aberration in waves. For example, the aluminum coatings have contributed Δ1/2π ~ 0.004 waves, or 4 milliwaves of polarization dependent tilt. It is seen from the last line in Table 27.2 that the wavefront aberration contributions w1 and w2 contribute less than 5 milliwaves of aberration in the present example.

Equation 27.7 is a general-purpose diattenuation and retardance aberration equation which with different coefficient values will closely approximate the Jones pupils of many camera lenses, microscope objectives, telescopes, and many other optical systems. The coefficient fitting helps determine if higher-order terms are needed to describe the system’s polarization aberrations.

27.5.4Diattenuation and Retardance Contributions

The individual diattenuation contributions from the three mirror elements were calculated during the polarization ray trace for a grid of rays. The surface-by-surface contributions are shown in the first three panels of Figure 27.16. The fourth panel in Figure 27.16 shows the cumulative diattenuation for the entire telescope as viewed looking into the exit pupil from the image plane. The primary and secondary mirrors produce diattenuation defocus, J5 = σ0 + d2 ρ2 (σ1 cos 2ϕ + σ2 sin 2ϕ)/2, a rotationally symmetric, tangentially oriented diattenuation with a magnitude that increases quadratically2 from the center of the pupil. The fold mirror introduces diattenuation tilt, J3 = σ0 + d1 ρ (–σ1 sin ϕ + σ2 cos ϕ)/2, a horizontally oriented diattenuation with a linear variation along the vertical axis, as well as having a constant offset at the center described by the term, diattenuation piston, J1 = σ0 + d0σ1/2. Since the fold mirror makes the largest contribution, the entire telescope’s diattenuation is similar to the fold mirror.

Figure 27.16 Diattenuation maps for each mirror element (the first three panels) and the cumulative diattenuation for the entire telescope (the last panel). The key in the lower right corner of each panel shows the scale of the largest diattenuation. For this example telescope, the dominant source of diattenuation is the 45° fold mirror.

Figure 27.17 shows the individual surface contributions to the retardance aberration in the first three panels and the cumulative retardance aberration in the last panel. The primary and secondary mirrors produce retardance defocus, J6 = σ0 + iΔ2 ρ2 (σ1 cos 2ϕ + σ2 sin 2ϕ)/2, a rotationally symmetric tangentially oriented fast axis, which increases quadratically from the center, while the fold mirror introduces retardance tilt, J4 = σ0 + iΔ1ρ (–σ1 sinϕ + σ2 cosϕ)/2, and retardance piston, J2 = σ0 + iΔ0σ1/2, with a vertically oriented fast axis. Since the fold mirror has the largest retardance, the resultant retardance for the entire telescope is similar to the fold mirror retardance. The cumulative linear retardance map (the fourth panel of Figure 27.17) is primarily a constant retardance with a linear variation from bottom to top, and a smaller variation of retardance orientation from left to right. Constant retardance is a constant difference in the wavefront aberration between XX and YY, a “piston” between polarization states; it changes polarization states, but this piston does not degrade image quality. The linear variation of retardance indicates a difference in the wavefront aberration tilt; X- and Y-polarizations get different linear phases, and so their images are shifted from the nominal image location by different amounts. This effect is very interesting and will be analyzed later.

Figure 27.17 Retardance maps for each mirror element (the first three panels) and the cumulative retardance (the last panel). The key in the lower right corner of each of the four panels shows the scale of the largest retardance in radians. This figure shows that the dominant source of retardance is the 90° fold mirror (third panel).

The retardance defocus from the primary and secondary mirrors causes astigmatism. For X-polarized light, the relative phase is advanced quadratically moving along the x-axis from the center to the edge of the field and is retarded quadratically moving to the edge of the field along the y-axis. This causes astigmatism arising from the different quadratic variations of ϕrs and ϕrp about the origin in Figure 27.9 (right). Hence, the X-polarized image, being astigmatic by 0.022 radians (0.012 + 0.010 or 3.4 milliwaves), becomes slightly elongated in opposite directions on either side of the best focus. Similarly for Y-polarized light, the relative phase is advanced moving along the y-axis from the center to the edge of the field and is retarded moving to the edge of the field along the x-axis. Therefore, the Y-polarized image is astigmatic with the opposite sign. For unpolarized light, the coating-induced astigmatic image is the average over the PSF of all polarization components, which is also the sum of the PSF for any two orthogonal components. Thus, when the astigmatism of X-polarized light is added to the astigmatism for Y-polarized light where the astigmatism is rotated by 90°, the combination forms a radially symmetric PSF, which is slightly larger than the unaberrated image. Inserting a polarizer will reveal the astigmatism at any particular polarization orientation. More information on retardance defocus and the associated astigmatism in Cassegrain telescopes is found in Reiley.14

These polarization aberrations affect the PSF. A detailed discussion of the image defects will be delayed until after the design rules are introduced.

27.5.5Design Rules Based on Polarization Aberrations

The analysis above was for aluminum coatings at one wavelength. With the aberration tools, the telescope’s polarization aberrations for many coatings can be accurately estimated by combining two of our expressions: (A) The Jones pupil is described in terms of six simple polarization aberration terms: constant (J1 and J2), linear (J3 and J4), and quadratic terms (J5 and J6) in Equation 27.7, and (B) the coating polarization has been described in terms of simple constant (a0 and b0), linear (a1 and b1), and quadratic terms (a2 and b2) in Equations 27.2 and 27.3. If the coating wavelength is changed, or if the coating design is changed, then a0, a1, a2, b0, b1, and b2 in Equations 27.2 and 27.3 also change. Hence, changes in (A) or (B) can be simply related to the Jones pupil polarization aberration coefficients: d0, d1, d2, Δ0, Δ1, and Δ2.

A list of design rules will be considered based on the behaviors of the aberrations of Figure 27.14 for the following list of the example telescope’s image defects:

1. Diattenuation at the center of the pupil.

2. Retardance at the center of the pupil.

3. The PSF shears between the XX- and YY-components, the two bright co-polarized components. These PSFs shift in opposite directions because of the fold mirror, thus stretching point images!

4. The polarization-dependent astigmatism, whose orientation rotates with the incident polarization state.

5. The fraction of light in the ghost PSF in components XY and YX. These dim cross-polarized components have PSFs about twice as large as the XX- and YY-components.

By analyzing a simple three-element system with aberration theory, each image defect can be related to the fold mirror angle, numerical aperture, and coating choices.

27.5.5.1Diattenuation at the Center of the Pupil

The diattenuation at the center of the pupil corresponds to the diattenuation piston term J1 with magnitude d0. This arises only from the fold mirror; the primary and secondary mirrors have zero diattenuation at the center of the pupil. The coefficient d1 is close to the value of the average diattenuation, averaging over the pupil. J1 is primarily responsible for the 9% difference in the flux transmitted to the focal plane between the incident X- and Y-polarized components. Unpolarized light has a ~4% DoP when incident at the focal plane. This is typically calibrated out when using focal plane Stokes polarimeters. The magnitude of d0 only depends on the diattenuation at the fold mirror evaluated at the axial ray’s angle of incidence θ3 (for surface 3, 45° in this case). The angle θ0 in the Taylor series expansion of Equation 27.3 is θ3 for the telescope’s fold mirror. This leads to two design rules for the average diattenuation.

• Design rule 1 The average diattenuation (diattenuation piston value) characterized by d0 is quadratic in the fold mirror angle θ3.

• Design rule 2 The diattenuation piston value, d0, is linear in the coating parameter a0 defined in Equation 27.3. If a0(λ) is calculated for a coating as a function of wavelength, the spectral variation of the average diattenuation will be proportional to a0(λ). If the aluminum is overcoated with silicon monoxide or some other material, a0(λ) can be recalculated to compare the resulting average diattenuation between coatings.

27.5.5.2Retardance at the Center of the Pupil

The retardance at the center of the pupil corresponds to the retardance piston term J2 with value Δ0. J2 also arises only from the fold mirror. The coefficient Δ0 is close to the value of the average retardance, averaging over the pupil. J2 does not change the polarization (Stokes parameters) of unpolarized light transiting the system; retardance only modifies polarized and partially polarized light. J2 introduces a constant phase difference between the X- and Y-incident components; thus, J2 does not degrade the PSF. The value of Δ0 only depends on the retardance at the fold mirror evaluated at the axial ray’s angle of incidence θ3. This leads to two more design rules.

• Design rule 3 The retardance piston characterized by Δ0 is quadratic in the fold mirror’s tilt angle θ3.

• Design rule 4 Δ0 is linear in the coating parameter b0 the retardance at the fold mirror’s nominal angle of incidence (Equation 27.3).

27.5.5.3Linear Variation of Diattenuation

The diattenuation tilt J3 with magnitude d1 describes a linear variation of diattenuation across the pupil, corresponding to unpolarized light exiting with a smaller DoP at the bottom of the pupil and a larger DoP at the top of the pupil. J3 arises only from the fold mirror; the primary and secondary mirrors do not contribute. J3 causes the X-polarized input to be brighter at the top of the pupil and be linearly dimmer toward the bottom of the pupil. Y-polarized input has the opposite variation. This apodization has a small effect on the shape and structure of the XX- and YY-PSFs, much smaller than the other polarization imaging defects. J3 does contribute significantly to the off-diagonal Jones pupil elements JXY and JYX and thus to the brightness of the ghost PSFs, IXY and IYX. The value of d1 depends on the slope of the diattenuation a1 at the fold mirror evaluated at θ3 and on the range of angles at the fold mirror, characterized by the F/# in image space, F/8, or alternatively the numerical aperture, NA = 0.06. This leads to the design rules for the diattenuation tilt J3.

• Design rule 5 The diattenuation tilt characterized by d1 is linear in the angle θ3. This follows from the slope of a quadratic function (Equation 27.2) being linear.

• Design rule 6 d1 is linear in the coating diattenuation slope parameter a1.

• Design rule 7 d1 is also linear in the coating diattenuation quadratic parameter angle a2, which follows from the slope of a quadratic function being linear.

27.5.5.4Linear Variation of Retardance, the PSF Shear between the XX- and YY-Components

The retardance tilt J4 with value Δ1 describes a linear phase variation for the XX-polarized component and an opposite linear phase variation for the YY-polarized component. This term arises only from the fold mirror; the primary and secondary mirrors do not contribute. J4 is a very important term because it shifts the images of the XX- and YY-components in opposite directions, causing the overall PSF to become elliptical. J4 also contributes to the off-diagonal Jones pupil elements JXY and JYX and thus to the brightness of the ghost PSFs, IXY and IYX. The value of Δ1 depends on the slope of the retardance at the fold mirror evaluated at θ3 and on the range of angles at the fold mirror, characterized by the numerical aperture. This leads to the design rules for the retardance tilt J4.

• Design rule 8 The retardance tilt characterized by Δ1 is linear in the fold mirror angle θ3, which follows from the slope of a quadratic function being linear.

• Design rule 9 The retardance tilt magnitude Δ1 is linear in the coating retardance slope parameter b1.

• Design rule 10 The retardance tilt magnitude Δ1 is also linear in the coating retardance quadratic parameter b2, which follows from the fact that the slope of a quadratic function, b1, is linear in the quadratic parameter.

27.5.5.5The Polarization-Dependent Astigmatism

The retardance defocus term J6 with magnitude d2 describes a quadratic variation of retardance from the center of the pupil, which is tangentially oriented. This arises primarily from the primary and secondary mirrors with a small contribution from the fold mirror. J6 causes the X-polarized input exiting into X-polarized output to become astigmatic. From the center, the phase is advanced quadratically along the x-axis and delayed quadratically along the y-axis. For YY, this astigmatism is rotated by 90°. The effect on unpolarized light is like spinning the astigmatic PSF; the PSF is rotationally symmetric, but enlarged by the astigmatism. J6 also contributes to the off-diagonal Jones pupil elements JXX and JYY and thus to the brightness of the ghost PSFs, IXY and IYX. The magnitude of d2 depends on the quadratic variation of the retardance about normal incidence b2 and on the angle of incidence of the marginal ray at the edge of the primary θ1 and secondary θ2 mirrors. This leads to the design rules for the retardance defocus J6.

• Design rule 11 The polarization-dependent astigmatism term, retardance defocus, is characterized by Δ2, and thus is quadratic in the sum of the angles θ1 + θ2 assuming identical coatings. Therefore, Δ2 is quadratic in the NA if the design F/# is scaled by just changing the entrance pupil diameter.

• Design rule 12 The retardance defocus magnitude Δ2 is quadratic in the coating retardance quadratic parameter b2.

• Design rule 13 The maximum fraction of flux coupled into the orthogonal state for weak linear polarization elements is quadratic in the diattenuation and quadratic in the retardance, and occurs for incident light polarized at 45° to the diattenuation axis or retardance fast axis.

27.5.5.6The Fraction of Light in the Ghost PSF in XY- and YX-Components

The fraction FYX of the X-polarized incident light coupled into the ghost PSF image IYX depends on the tilt coefficients d1 and Δ1 and the defocus coefficients d2 and Δ2, but not the piston coefficients. This is also equal to the fraction FXY of the Y-polarized incident light coupled into the ghost PSF image IXY. The fraction of incident flux incident in X- or Y-polarization coupled into the orthogonal polarization state is found by integrating the off-diagonal element’s (the σ2 term) magnitude squared |JYX|2 or |JXY|2 over the pupil and normalizing by π, the area of the pupil,

$$F_{XY}=\frac{{\displaystyle \underset{0}{\overset{2\pi }{\int }}{\displaystyle \underset{0}{\overset{1}{\int }}{\left|J_{XY}\right|}^2\rho d\rho d\varphi }}}{{\displaystyle \underset{0}{\overset{2\pi }{\int }}{\displaystyle \underset{0}{\overset{1}{\int }}\rho d\rho d\varphi }}}=\frac{d_1^2+{∆}_1^2}{16}+\frac{d_2^2+{∆}_2^2}{24}.$$27.9

27.9

FXY is quadratic in the tilt and defocus coefficients; hence, an order of magnitude reduction in these polarization aberrations reduces the ghost brightness by two orders of magnitude.

• Design rule 14 The fraction of the flux incident in either the X-polarization or the Y-polarization coupled into the orthogonal ghost images IYX (fraction FYX) or IXY (fraction FXY) depends quadratically on d1 and Δ1. Since d1 and Δ1 are linear in the angle θ3, FYX and FXY are quadratic in the fold mirror angle θ3 (see Design rules 5 and 8). FYX and FXY are also quadratic in the coating retardance slope parameters a1 and b1 (see Design rules 6 and 9). FXY and FYX are also quadratic in the coating diattenuation quadratic parameter a2 (see Design rules 7 and 10).

Analysis of the defocus aberrations leads to additional design rules for the fractions FYX and FXY. The diattenuation defocus, d2, and the retardance defocus, Δ2, are quadratic in the sum of the marginal ray angles θ1 + θ2 assuming identical coatings.

• Design rule 15 The fractions FYX and FXY are fourth order in the sum of the marginal ray angles θ1 + θ2 assuming identical coatings, and thus fourth order in the NA, assuming the design F/# is scaled by just changing the entrance pupil diameter. Thus, small decreases in F/# can yield large increases in ghost PSF brightness (see Design rule 11).

• Design rule 16 The fractions FYX and FXY are fourth order in the coating diattenuation quadratic parameter a2 (see Design rule 12).

Thus, these design rules describe the scaling of the polarization aberration as the pupil size changes, F/# changes, coating prescription changes, and fold mirror angle changes for a telescope of the configuration in Figure 27.8. As presented, these relations only apply to the on-axis image. The off-axis equations are more complicated, but coronagraphs and other astronomical systems usually have small enough fields that the polarization aberration variation over the field is not significant and therefore these design rules do not change over this system’s practical field of view. As more fold mirrors or other components are added to this system, these polarization aberration equations need to be generalized to relate the polarization aberrations to the coating prescriptions and optical prescription. Lam and Chipman discussed polarization aberration reduction with two- and four-mirror combinations.15 Discussions of higher-order polarization aberration terms can be found in McGuire and Chipman,16,17 Ruoff and Totzeck,18 and Sasián.19,20

27.5.6Amplitude Response Matrix

In conventional scalar image formation calculations, the amplitude response function is calculated as the Fourier transform of the exit pupil function. This electric field distribution is then squared to obtain the PSF.21 To evaluate the image formed by systems with polarization aberration, McGuire and Chipman introduced a Jones calculus version of the amplitude response function named the Amplitude Response Matrix,ARM,22,23

$$ARM=\left(\begin{array}{ll}\Im \left[J_{XX}(x,y)\right]& \Im \left[J_{XY}(x,y)\right]\\ \Im \left[J_{YX}(x,y)\right]& \Im \left[J_{YY}(x,y)\right]\end{array}\right),$$27.10

27.10

where $\Im$ is a spatial Fourier transform over each of the Jones pupil elements (see Section 16.4). For a plane wave incident on the telescope with Jones vector E, the amplitude and phase of the image are given by the matrix multiplication, ARM·E. The ARM for the three-mirror telescope of Figure 27.8 is shown in Figure 27.18. Table 27.3 summarizes the system and the parameters most relevant to this imaging calculation.

Figure 27.18 The absolute value of the amplitude of the 2 × 2 ARM at an on-axis field point is shown for the example telescope of Figure 27.8 normalized by the peak of the XX-component.

Table 27.3Parameters Associated with the Imaging Calculation

Wavelength	800 nm
Image space F/#	8
Entrance pupil diameter	2.4 m
Effective focal length	19.236 m
Number of rays across entrance pupil	65
Number of rays across the Jones pupil array	513
Spacing in the ARM and PSM viewing from object space	9.0 milliarcsec

The ARM’s diagonal elements are close to the well-known Airy disk pattern but are slightly larger due to the aberrations in ϕXX and ϕYY. Each is slightly astigmatic. Their centroids are slightly shifted due to the differences in their tilt. The off-diagonal elements have much lower amplitudes and contain interesting structure, mostly due to the fold mirror. We refer to these off-diagonal PSF images as the ghost PSFs.

For unpolarized illumination, the incident X- and Y-polarizations are incoherent with respect to each other. Thus, the output components ARMXX(X in X out) and ARMYX (X in Y out) are coherent with each other but incoherent with ARMXY and ARMYY. Hence, for unpolarized illumination, the two output X-components in the ARM are incoherent with respect to each other, as are the two output Y-components. Thus, the PSF for an unpolarized source has four additive components I = IX + IY = (|ARMXX|2 + |ARMXY|2) + (|ARMYX|2 + |ARMYY|2).

27.5.7Mueller Matrix Point Spread Matrices

The distribution of flux and polarization in the image of an incoherent point source, such as a star, can be described with a 4 × 4 Mueller matrix point spread matrix (PSM), the Mueller matrix generalization of the PSF (Section 16.5). This PSM is calculated by the transformation of the ARM’s Jones matrices into Mueller matrix functions using Equation 6.102 or 6.107. The Mueller matrix representation of polarization properties is familiar to most astronomers who make or work with astrophysical measurements of the four Stokes parameters.24–27 The example telescope’s PSM, calculated from the ARM (Figure 27.18), is shown in Figure 27.19. The contribution of each of the 16 elements varies across the PSM and changes depend on the incident Stokes parameters. Hence, each element of the matrix is shown with its contribution and appears as miniature PSFs with different shapes. An example of PSM measurements is found in McEldowney.7

Figure 27.19 The 4 × 4 point spread matrix (PSM) (left) operates on an example X-polarized incident beam with Stokes parameters (1, 1, 0, 0) (middle) to calculate the point spread function. The resultant polarization distribution is a 4 × 1 Stokes image (right) with components (IX, QX, UX, VX). The subscript X represents Stokes parameters resultant from X-polarized incident light. The normalized magnitude of each matrix element is shown on each vertical scale. The UX image indicates a small variation of polarization orientation, while VX indicates small variations of ellipticity. The red box on the left (m00, m10, m20, m30) is the resultant Stokes parameter image (I, Q, U, V) for a collimated beam of unpolarized incident light, such as an unpolarized star.

The PSF for unpolarized illumination is described by the Stokes parameter image in the first column (m00, m10, m20, m30) inside the red rectangle. Since m10, m20, and m30 are not zero, the PSF of an unpolarized star is not unpolarized. In this example, the Q-component’s 4.7 × 10−2 contribution mostly arises from the diattenuation of the fold mirror, which is reflecting more 0° (s-polarized) light than 90° (p-polarized) polarized light. The U-component (at 4.36 × 10−3) is mostly due to the diattenuation contributions at 45° and 135° from the primary and secondary seen in the first two panels in Figure 27.16. The ellipticity (from the V-component) arises when weakly polarized light reflected from the primary and secondary interacts with the retardance from the fold mirror. The spatial variations of Q, U, and V introduce polarization fluctuations in the region of the diffraction rings. Figure 27.20 maps the DoP in these zones. Such polarization fluctuations in the PSF of a star are clearly a concern when measuring the polarization of exoplanets and debris disks.

Figure 27.20 The DoP variation throughout the PSF of an unpolarized object. Regions with intensity below 0.0008 of the peak have been removed due to noise and are shown in gray.

Figure 27.19 contains a graphical equation describing the 4 × 4 PSM operating on an X-polarized incident beam (represented by the 4 × 1 matrix), which yields a 4 × 1 Stokes image (IX, QX, UX, VX), as represented by the rightmost term in the equation. For an unpolarized collimated incident beam, the resulting Stokes image is contained in the first column of the PSM, shown inside the red box. The m10 element describes the ~9% DoP for the image of the unpolarized source. The m20 element describes small variations of linear polarization orientation within the PSF while m30 characterizes even smaller ellipticity variations.

The light coupled into orthogonal components has a significant impact on the outer portions of the PSF because they arise from the highly apodized Jones pupil components AXY and AYX as shown in Figure 27.14 (left). To see this, compare the PSF arising from JXX and JYX. These PSF terms are calculated using the resultant Stokes image components on the right side of Figure 27.19 as

$$I_{XX}\propto \frac{I_X+Q_X}{2}\text{and}{I}_{YX}\propto \frac{I_X-Q_X}{2}.$$27.11

27.11

The two terms in Equation 27.11, IXX and IYX, are compared in Figure 27.21, where it is seen that the peak of IYX is about 10−5 of IXX. This “ghost PSF” should be very important in imaging applications that require contrast ratios of 10−8 or greater.

Figure 27.21 PSF of IXX and IYX calculated from Equation 27.11 is shown normalized to the peak of IXX.

Figure 27.21 compares the IXX PSF component with the ghost component IYX. Figure 27.22 shows the irradiance along an x-axis cross section, indicated by the vertical plane, through the two PSFs in log10 scale at the plane drawn through the two images shown in Figure 27.21. This ghost PSF has its light spread away from the center. In Figure 27.22, the Airy disk’s zeros of IXX are not at the same location as the zeros for the cross-coupled term IYX. Thus, the zeros of IXX are washed out by the light leakage from the non-zero IYX. The PSF IYX cannot be corrected by wavefront compensation for either the XX- or YY-components alone because most of the image spread is due to IYX’s apodization (Figure 27.13). A linear Polaroid placed at the image plane can pass IXX and remove IYX, but will still pass the other ghost IXY, and thus will not correct for this polarization aberration.

Figure 27.22 (Left) Cross sections through the Log10PSF image for IXX and IYX between −1 and +1 arc second along the x-direction. The solid and dashed black curves show IXX and IYX, respectively. Note that the first three minima of IXX (dark rings in the Airy disk) are close to the angles for the local maxima of the IYX curve. On average, the polarization-coupled IYX flux is about 10−4 below IXX. (Right) PSF for IYX shown in log10 contour with its scale on the right ranging from −2.4 to −4.1, normalized to the peak intensity of IXX. The superposed pink circles show the location of the first and second Airy dark ring of IXX.

The shape of IYX shows that the IX and QX Airy disks are not exactly on top of each other. The image plane irradiance distribution for the IYX term sits beneath the Airy diffraction pattern characteristic of the IXX term. Figure 27.22 (left) shows a slice normal to the axis at the RMS best focus through the PSF for IXX and for IYX in Figure 27.21. Figure 27.22 (right) shows a high-dynamic range image of the irradiance across the focal plane in the vicinity of the core of the PSF for IYX. The concentric pink circles superposed on Figure 27.22 (right) shows the first and second zeros of the Airy diffraction pattern of IXX. These dark rings overlay regions with non-zero IYX.

The right column in Figure 27.19 (IX, QX, UX, VX) is the Stokes parameter PSF for the X-polarized component of an incident beam. The flux of this component is IX = IXX + IYX. Similarly, the PSF IYfor a Y-polarized incident beam is calculated by multiplying the Stokes parameters (1, −1, 0, 0) to the PSM, and IY = IYY + IXY. Finally, the PSF for unpolarized incident light is (IX + IY)/2, which can also be calculated by multiplying the unpolarized Stokes parameters (1, 0, 0, 0) to the PSM.

This demonstrates that for unpolarized starlight passing into a “generic” optical system like that shown in Figure 27.8, the PSF is the sum of two nearly Airy diffraction patterns, IXX and IYY, plus two secondary or “ghost” PSFs, IYX and IYX, which originates from the system’s polarization cross-talk, the off-diagonal elements in the Jones pupil.

The Jones pupil, ARM, and PSM can be calculated in basis sets other than x and y. Here, x and y are aligned parallel and perpendicular to the fold mirror’s s-state. The resulting Jones pupil and ARM for other bases are found by Cartesian rotation of the matrices of Figures 27.13 and 27.18. The overall flux distributions, I = IX + IY, for an unpolarized source or point source of arbitrary polarization are unchanged by such a change of the basis. Similarly, the PSM is rotated by the same rotation operation as Mueller matrices, and again, the net flux for any source polarization is unchanged; the corresponding Stokes images are just rotated versions of the ones presented above. The advantage of the x- and y-basis chosen here is that the off-diagonal elements IXY and IYX have their smallest values in this basis. As the basis set rotates or becomes elliptical, the scale of these off-diagonal image components increases rapidly and quickly approaches Airy disks due to the coupling between bright diagonal elements and weak off-diagonal elements from the rotation operation. Thus, the fold mirror’s s- and p-basis (our x and y) is the best basis for viewing the value and functional form of the ghost PSF components for the example system of Figure 27.8.

27.5.8Location of the PSF Image Components

For the telescope of Figure 27.8, consider a Stokes imaging polarimeter measuring the PSF of an unpolarized star as a Stokes image. The PSFs for the X-polarized, IX = IXX + IYX, and Y-polarized light, IY = IXY + IYY, at the focal plane are very close in form to the classical Airy diffraction pattern because the polarization-induced wavefront aberration, ϕXX and ϕYY in Figure 27.13, is less than 8 milliwaves and the amplitude apodization is less than 0.015. But these two PSF images are not exactly superposed; the peaks of IX and IY are displaced from each other by 0.625 milliarcsec. The PSF cross sections through the maxima of IX, IY, and IX − IY (the star’s Stokes Q image) are shown in Figure 27.23. The shift between the IX and IY PSFs arises from the difference in slopes of the s- and p-phases in the Fresnel coefficients (red and blue tangent lines in Figure 27.9 (right), which is the cause of the overall linear variations in ϕXX and ϕYY. Their difference Q = IX – IY is sheared from IX and IY by 5.8 milliarcsec, as shown in Figure 27.23, and is due to the shift between IX and IY. These PSF shifts and PSF ellipticities are listed in Table 27.4 for a single 45° fold mirror before the focal plane. The ellipticity of the PSF image was calculated by fitting an ellipse to the PSF at the half power points.

Figure 27.23 The cross-section profiles of the IX (red) and IY (blue) PSF images in arc seconds from the center of the PSF. The black line shows Stokes Q image, the difference between the two PSFs.

Table 27.4The Shape of the PSF Calculated from PSM in Figure 27.19 Is Described by the Following Parameters: The PSF’s Flux, the Radius of Encircled Energy, the PSF Shears, and the PSF Ellipticity for X- and Y-Polarized Incident Light

Characterize the shape of PSF
PSF shear in object space:
Between IX and IY	0.625 milliarcsec
Between IX and (Q = IX − IY)	5.820 milliarcsec
Flux in PSF:
$\frac{\text{Flux of }{I}_{YX}}{\text{Flux of }{I}_{XX}}$	0.0048%
$\frac{\text{Flux of }{I}_{YY}}{\text{Flux of }{I}_{XX}}$	90.6%
$\frac{\text{Flux of }{I}_{YX}}{\text{Flux of }{I}_{XX}}$	0.0046%
$\frac{\text{Peak of }{I}_Y}{\text{Peak of }{I}_X}$	90.6%
$\frac{\text{Peak of (}{I}_X-I_Y)}{\text{Peak of }{I}_X}$	$\frac{\text{Peak of }Q}{\text{Peak of }{I}_X}=9.6\%$
Radius of 90% encircled energy in object space:
rXX = rYY	0.15 arcsec
rYX = rXY	0.36 arcsec
Ellipticity of PSF:
Unpolarized incident light	7.502 ×10−6
X-polarized incident light	0.00199
Y-polarized incident light	0.00208

In astronomical applications involving the precise measurement of the location of the centroid of the PSF, distortions of the shape of the PSF are important. Most systems incorporate multiple folds. These relay optics with multiple folds may increase the shear between the PSF’s polarization components. The variation of linear phase across the pupil, as seen in Figure 27.8, is approximately linear; thus, the shear between polarization components is linear in the F/#.

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1 Also known as the Mueller matrix bidirectional reflectance distribution function, MMBRDF.

2 Throughout, linear implies approximately linear and quadratic implies approximately quadratic.