12Fresnel Aberrations

12.1Introduction

This chapter explores the polarization aberrations of uncoated lenses and metal mirrors. The Fresnel equations describe the change of polarization at uncoated interfaces. In optical systems, because of the Fresnel equations, polarization variations known as polarization aberrations occur across the exit pupil, which can be called Fresnel aberrations. Such polarization aberrations are usually visualized in terms of the diattenuation and retardance variations associated with ray paths.

The aberration of an optical system is its deviation from ideal performance. In an imaging system with ideal spherical or plane wave illumination, the desired output is spherical wavefronts centered on the correct image point with constant amplitude and constant polarization state. Deviations from spherical wavefronts arise from variations of optical path length of rays through the optic due to the geometry of the optical surfaces and the laws of reflection and refraction. The deviations from spherical wavefronts are known as the wavefront aberration function. Deviations from constant amplitude arise from differences in reflection or refraction efficiency between rays. Such amplitude variations are amplitude aberration or apodization. Polarization change also occurs at each reflecting and refracting surface due to differences between the s- and p-components of the light’s reflectance and transmission coefficients. Across a set of rays, the angles of incidence and, thus, the polarization change vary, so that a uniformly polarized input beam has polarization variations when exiting.1,2 For many optical systems, the desired polarization output would be a constant polarization state with no polarization change transiting the system; such ray paths through an optical system can be described by identity Jones matrices. A useful definition of polarization aberrations is deviations from this identity matrix caused by the interaction of the light with the optical system.

In this hierarchy, wavefront aberrations are by far the most important aberration, as variations of optical path length of small fractions of a wavelength can greatly reduce the image quality. The relative priority of wavefront aberrations is so great that for the first 40 years of computer-aided optical design, amplitude and polarization aberration were not calculated by the leading commercial optical design software packages. The variations of amplitude and polarization found in high-performance astronomical systems cause much less change to the image quality than the wavefront aberrations, but as the community prepares to image and measure the spectrum and polarization of exoplanets and similar demanding tasks, these amplitude and polarization effects can no longer be ignored. For example, Stenflo3 has discussed limitations on the accuracy of solar magnetic field measurements due to polarization aberration.

In a system of reflecting and refracting elements, amplitude and polarization aberration contributions arise from the Fresnel coefficients for uncoated or reflecting metal surfaces and by related amplitude coefficients for thin film-coated interfaces. Polarization aberration, also called instrumental polarization, refers to all polarization changes of the optical system and the variations with pupil coordinate, object location, and wavelength. The term Fresnel aberrations refers to those polarization aberrations that arise strictly from the Fresnel equations, that is, systems of metal-coated mirrors and uncoated lenses.1,4–6 Multilayer coated surfaces produce polarization aberrations with similar functional forms and may have larger or smaller magnitudes.

This chapter applies the Fresnel equations to describe polarization changes that occur in several example optical systems, an uncoated lens, a fold mirror, a combination of fold mirrors, a Cassegrain telescope, and a Fresnel rhomb retarder. Further analysis of a Cassegrain telescope with a fold mirror is included in Chapter 27, where a list of design rules are derived.

12.2Uncoated Single-Element Lens

This section will show the many different ways of representing polarization aberration of a fast singlet lens and demostrate its impact on image formation. Figure 12.1 shows a set of meridional rays refracting through a fast single-element lens coming to focus on the optical axis. The aperture stop that is also the entrance pupil (gray annulus) is located in object space a small distance in front of the lens. On-axis images formed with fast (high numerical aperture) spherical lenses have large amounts of spherical aberration and highly aberrated point spread functions. To correct this, the surfaces of this lens have been aspherized, they are not spherical surfaces, in such a way that the wavefront aberration of the lens for the on-axis image is negligible (a few milliwaves, λ/1000). The prescription of the lens element shown in Figure 12.1 is listed in Table 12.1.

Figure 12.1A fast single-element lens is our first Fresnel aberration example. The entrance pupil (EP, gray) limits the ray bundle from the on-axis object point. On the right, the image of the entrance pupil is the exit pupil (XP, brown), which is associated with rays in image space, the space of the rays after they refract from the last surface. In this case, the exit pupil is virtual, located to the left of the final surface, and larger than the entrance pupil, since here the pupil magnification is greater than one.

Table 12.1Lens Prescription for a Fast Singlet in Figure 12.1

Lens refractive index	1.51674
Object location	−50 mm from stop
Stop location	−1 mm from first lens surface
Image location	10.7725 mm from seconds lens surface
Effective focal length	10 mm
Entrance pupil diameter	14 mm

From a ray trace of an on-axis object point with a grid of rays, the angle of incidence (AoI) and orientation of the plane of incidence (PoI) are calculated for each surface and plotted as a set of angle of incidence maps (AoI map) in Figure 12.2. In the AoI map, the length of each line segments represents the magnitude of the AoI, and the line’s orientation indicates the orientation of the plane of incidence (PoI), which is also the p-polarization orientation for the ray. On the left side of Figure 12.2, for the first lens surface, the AoI is zero for the ray down the center of the axis through the center of the entrance pupil, represented by a line with zero length. Moving out from the center of the pupil, the angles of incidence increase with distance from the center of the pupil (the radial pupil coordinate). Four marginal rays, rays from the on-axis field through the edge of the pupil, are represented, top, bottom, left, and right, and from the legend shown on the right, their length corresponds to an angle of incidence of ~0.45 radians ≈ 34°. These lines are all radial, indicating that the PoI for all of these rays is radially oriented and contains the optical axis. The middle AoI map is plotted for the second lens surface where the angles of incidence are a little larger; the marginal ray angle of incidence is about 0.6 radians. Again, the plane of incidence is radially oriented. For both surfaces, the angle of incidence increases approximately linearly with the radial pupil coordinate.

Figure 12.2Angle of incidence map for front (left) and back (middle) surface is radially oriented and increases approximately linearly. (Right) The scale shows the maximum angle of incidence in radians and degrees.

Math Tip 12.1Approximately Linear and Quadratic Functions

It is standard in aberration theory that when a function is described as linear, it means the function is approximately linear, in the sense that the linear term in a Taylor series dominates. This avoids repeatedly using the term approximately linear throughout the remainder of the chapter. Similar interpretation applies to the terms quadratic and cubic.

From the Fresnel equations for an air (n = 1)-to-glass (such as, n = 1.51674) interface, the corresponding diattenuation for the ray’s intercepts at the first surface is calculated, and the corresponding diattenuation magnitude and orientation are plotted as diattenuation maps in Figure 12.3 (left). In the diattenuation map, the length of each line represents the diattenuation magnitude and the line’s orientation is the orientation of the maximum transmission state. Since the p-Fresnel coefficient is larger than the s-Fresnel coefficient for refraction at air-to-glass interfaces, the diattenuation is all oriented radially in the plane of incidence. The magnitude of the diattenuation increases quadratically from zero in the middle of the pupil. At the second surface in the middle of Figure 12.3, the light is undergoing internal refraction and refraction from glass into air, and the diattenuation effects are much larger, increasing quadratically from the center of the pupil to a diattenuation of about 0.1 for the marginal rays. The right side graphs the diattenuation for the combination of the two surfaces; the magnitude is dominated by the second surface. This radially oriented linear diattenuaion increasing quadratically from the center of the pupil is the form of an uncoated lens’ polarization aberration on-axis.

Figure 12.3Diattenuation maps for the front surface, back surface, and the combination of front and back surfaces of the singlet. The scale on the right shows the maximum diattenuation of the three maps.

Figure 12.4 shows the exiting state from the singlet system shown in Figure 12.1 for 0° linearly polarized incident light, horizontally oriented electric field. Polarization aberration is visible as the amplitude and polarization state of the light vary across the pupil. Moving from the center to the right or left of the pupil, the amplitude of the light increases quadratically, since the p-transmission Fresnel coefficient increases quadratically with angle of incidence from the transmission down the axis. Moving from the center to the top or bottom, the amplitude decreases quadratically, since the light is s-polarized. Along the x-axis, the 0° polarized light is along the diattenuation axis and being preferentially transmitted. Along the y-axis, the light is polarized perpendicular to the diattenuation axis and is preferentially reflected. In the corners of the pupil, the incident 0° polarized light is at an angle to the 45° diattenuation axis, and so the exiting polarization rotates toward the diattenuation axis. The light incident parallel to or perpendicular to the diattenuation axis is in an eigenpolarization state, and the polarization state of the light exiting the lens is unchanged by the polarization aberration. Other incident states will exit the lens with the state rotated toward the transmission axis.

Figure 12.4Polarization ellipse map of the singlet’s exit pupil for a 0° polarized incident state.

When the lens is illuminated with 45° polarized light, the light incident along the ±45° directions in the pupil is in an eigenpolarization. Along the 45° directions, the amplitude increases quadratically from the center. Along the 135° directions, the amplitude decreases quadratically from the center as shown in Figure 12.5 (left). When left circularly polarized light is incident, the polarization distribution exiting the lens is shown in Figure 12.5 (right). The light transitions from circular in the middle of the pupil to increasingly elliptical at the edge of the pupil. The major axis of each ellipse is aligned with the diattenuation axes shown in Figure 12.3.

Figure 12.5Exiting states from the singlet for the 45° (left) and left circularly polarized (right) incident states.

If the lens is placed in between crossed linear polarizers (oriented at 0° and 90°), as shown in Figure 12.6 (left), the 0° component of the light is removed and only the 90° component is transmitted, as shown in Figure 12.7 (left). The pupil is now dark along the x- and y-axes and the transmitted light is the brightest at the edge of the pupil along the ±45° directions. The corresponding with one “n” flux distribution (amplitude squared) is referred to as the Maltese cross, seen in Figure 12.7 (middle). The Maltese cross is an important pattern, commonly seen with most radially symmetric optical systems placed between crossed polarizers. Along the diagonal, the amplitude of the leaked light varies as ρ squared (ρ is pupil coordinate) while the transmitted flux varies as ρ to the fourth power. If the lens is stopped down to a smaller aperture, the leaked flux rapidly diminishes. For a small numerical aperture, the Maltese cross may be difficult to observe. If the two polarizers rotate together, the Maltese cross will rotate with the polarizers. When the polarizers are parallel, as in Figure 12.6 (right), the polarization pupil map has a uniform polarization state and nearly uniform amplitude, as shown in Figure 12.7 (right).

Figure 12.6Singlet lens illuminated through a 0° polarizer and analyzed through a 90° analyzer (left) and analyzed through a 0° analyzer (right); the brown annulus is the entrance pupil.

Figure 12.7Polarization exiting the single-element lens between crossed polarizers (left) is dark along the x- and y-axes (left). This is easily seen placing most lenses in a polariscope with crossed linear polarizers. The pattern observed, called the Maltese cross, leaks light through the four diagonals of the pupil (middle). Between parallel linear polarizers (right), only a small amplitude variation is observed.

When the fast singlet lens is analyzed by polarization ray tracing a grid of rays, as described in Chapter 9, the polarization aberration function is calculated, shown in Figure 12.8 (left). This displays the elements of the polarization ray tracing matrices for the on-axis beam through the singlet. It maps the curved input spherical wavefront onto the curved exiting spherical wavefront and transforms the polarization states in three dimensions. The xy, xz, yx, and zx elements are principally concerned with these curvatures of the wavefronts. The diattenuation is most clearly visible in the xy and yx elements. The polarization aberration function can be flattened into a Jones pupil choosing the local coordinates for the incident and exiting spherical waves, that is, specifying the local x- and y-coordinates to be used for describing Jones vectors and Jones matrices over the two wavefronts as explained in Chapter 10. Using double pole coordinates with the poles at −z yields the Jones pupil of Figure 12.8 (right). In the off-diagonal elements, xy and yx, the first and third quadrants are 180° out of phase with the second and fourth quadrants. The pupil average of the xy and yx pupils is zero. This has an implication in image formation.

Figure 12.8(Left) The polarization aberration function and (right) the Jones Pupil for the fast single-element lens.

The images of an on-axis point source when the lens is placed between different pairs of linear polarizers are shown in the four elements of Figure 12.9. Each of these is an amplitude response function, the electric field amplitude distribution as described in Section 16.4. Taken together, the four functions form the amplitude response matrix, ARM, which can operate on a Jones vector for a point source to determine the Jones vector distribution in its image. The elements of the ARM are calculated by Fourier transformation of the four elements of the Jones pupil as described in Chapter 16. The J11 function of Figure 12.9 corresponds to the amplitude distribution for the lens between a 0° polarizer and 0° analyzer. This function is very close to the Airy function for the diffraction-limited image of a circular aperture. It differs because the xx element of the Jones pupil is not quite uniformly illuminated. The J21 function corresponds to Figure 12.6 (left) with the lens illuminated through a 0° polarizer and the output analyzed through a 90° analyzer (polarizer). This image consists of four main lobes surrounding a dark center, and looks nothing like an Airy disk. The center is dark because the central value of the Fourier transform is the average value of the input function; in this case, the averages of both the xy and yx Jones pupil elements are zero.

Figure 12.9Amplitude response function of the fast singlet lens contains the amplitude of the E field distributions for combinations of 0° and 90° illumination analyzed through combinations of 0° and 90° analyzers and takes the form of a Jones matrix function.

This ARM describes how the images of point sources have spatial variations of polarization state within their point spread functions. The image of a point source polarized at 0° is shown in Figure 12.10 as a Jones vector distribution. The image in the center and along the x- and y-axes is polarized at 0°; elsewhere, it is a mixture of 0° and 90° components. Figure 12.11 (right) zooms in on the center of the image’s polarization state map where the polarization orientation varies between ±1.46°.

Figure 12.10(Left) Amplitude response Jones vector (J11 and J21 of Figure 12.9) and (right) its corresponding states plotted for horizontally polarized incident light.

Figure 12.11(Left) Zoom in on the polarization states in the center of the Figure 12.10 (right). (Right) The orientation of the polarizations state varies by ±1.5° within the first zero of the Airy disk.

The image of a point source is the point spread function, PSF. In the presence of polarization aberration, the PSF is generalized to the point spread matrix, PSM, a Mueller matrix function that operates on the Stokes parameters of the object point yielding a Stokes function for the image point as described in Chapter 16. The PSM for the uncoated singlet lens is shown in two views in Figures 12.12 and 12.13 (left). Figure 12.12 is a 3D plot with the maximum and minimum values indicated; for example, the M00 element peaks at 76 while the M01, M02, M10, and M20 elements peak at 0.64, about 100 times lower. Figure 12.13 (left) is a colored raster plot showing blue for positive and red for negative values. Down the diagonal are four nearly equal functions that are close to the Airy-squared functions, approximating an identity matrix, indicating that the overall polarization of a point source is not very different from the incident polarization state. When this PSM operates on the Stokes parameters for 0° light, the Stokes parameters of the image formed are shown in Figure 12.13 (right). The S2 element, third row, shows that the red areas would be brighter when analyzed by a 45° than by a 135° polarizer, while the blue areas would have the opposite occurring.

Figure 12.12The Mueller matrix point spread function for the uncoated lens as a surface plot. Elements are scaled differently to show the details. The maximum and minimum values are shown individually in each element.

Figure 12.13The Mueller matrix PSF is shown as a raster plot (left) operating on the Stokes parameters for horizontally polarized light (vector in middle) yielding the polarization distribution in the image as Stokes parameters. The S1 distribution is nearly equal to the S0 distribution, with S2 showing a small component coupled into a 45° component (blue) and a 135° component (red), describing the spatial variation of the polarization in the image. The raster plot is saturated in order to show the details in the dimmer component.

12.3Fold Mirror

Fold mirrors coated with metals such as aluminum or silver are a significant source of polarization aberration in many systems.7 Since the angles of incidence at fold mirrors are generally large, the polarization aberrations tend to be larger than polarization aberrations in on-axis systems. The refractive indices of metals are complex; hence, the reflection causes both diattenuation and retardance. The Fresnel coefficients for gold are plotted in Figure 12.14. The s- and p-phases in Figure 12.14 are plotted in the global coordinate phase convention, phases are equal at normal incidence, not separated by π. Subtracting the phases yields the retardance graph in Figure 12.14 (bottom right). The retardance has a larger effect on the polarization state than the diattenuation in Figure 12.14 (bottom left).

Figure 12.14The magnitudes (top left) and phases (top right) for the Fresnel coefficients for gold. The difference between the phases is the linear retardance (bottom right), which increases quadratically from normal incidence.

Figure 12.15 (left) shows an example system with a grid of rays at 500 nm reflecting from a gold fold mirror (n = 0.855 + 1.895 i) over a range of angles from 5° < θ < 80°. The corresponding pupil map of the magnitude and orientation of the angle and plane of incidence are shown in Figure 12.15 (right). The polarization aberration function in Figure 12.16 maps the polarization vector of the incident into the exiting polarization vector, containing both changes of direction, phase, amplitude, and polarization. After a choice of local coordinates for incident and exiting spherical waves, the Jones pupil (Figure 12.17) is calculated, which represents the phase, amplitude, and polarization changes. Figure 12.17 (bottom row) contains a 3D surface representation of the Jones pupil using double pole coordinates1 with the pole opposite to the chief ray direction.

Figure 12.15A converging grid of rays reflecting from a gold mirror (left) has the angle of incidence pupil map on the right.

Figure 12.16The amplitude (left) and phase (right) of the P matrix pupil.

Figure 12.17(Top row) The Jones pupil for the fold mirror is brighter for the x-component (s-polarized along the y-axis) as seen in the xx magnitude than the y-component in the yy magnitude. (Bottom row) 3D plots of the Jones pupil.

From the Jones pupil, the diattenuation pupil map (Figure 12.18, left) and retardance pupil map (Figure 12.18, right) are calculated. In the retardance map, the length of the line segment represents the retardance magnitude and the line’s orientation is the fast axis orientation. Along the middle of the pupil, the eigenpolarizations are 0° and 90° rotating counterclockwise to the left side and clockwise to the right side. This is why the xy and yx elements of the Jones pupil change sign crossing the y-axis. In this example, the diattenuation and retardance are both smallest at the bottom of the pupil where the beam is approaching normal incidence and increasing toward the top. The diattenuation reaches a maximum around θin = 65° and then decreases. The retardance increases all the way to the top of the pupil where it reaches ~2.41 radians, or about 0.43 waves of retardance.

Figure 12.18The diattenuation (left) is small at the bottom of the pupil near normal incidence reaching a maximum near 65° angle of incidence. The transmission axis rotates from the left to the right side of the pupil. The retardance (middle) increases to nearly a half wave at the top of the pupil. The fast axis is orthogonal with the diattenuation’s transmission axis. The average phase between s and p polarizations is shown on the right.

The polarization state exiting for 0° and 90° incident polarizations is seen in Figures 12.19 and 12.20. The changes are the least for 0° and 90° incident polarizations since these are eigenpolarizations along the y-axis, the meridional plane, and these incident states are closest to eigenpolarizations overall across the remainder of the pupil. The reflected light becomes increasingly elliptical, moving away from the meridional plane. The helicity changes sign crossing the meridional axis.

Figure 12.19An ellipse map for incident 0° polarized light (left) and the reflected state (right). Arrowheads indicate the phases.

Figure 12.20An ellipse map for incident 90° polarized light (left) and the reflected state (right). The reflected phase has a large change from the bottom to the top of the pupil of more than 3π/4.

Consider incident states that decomposed into equal amounts of 0° and 90°, such as 45° and circular polarized light. For these states, the change of polarization upon reflection is the largest. Figure 12.21 shows pupil maps for 45° light incident and the corresponding reflected states. The polarization changes are very large in the upper half of the pupil. Moving up the meridional axis, the light becomes increasingly elliptically polarized and the major axis of the ellipse rotates from 45° to 135°. Swindell discusses the handedness of this reflected state.8

Figure 12.21An ellipse map for incident 45° polarized light (left) and the reflected state (right). The ellipses are drawn looking into the beams; thus, near the bottom, 45° polarized light reflects as 135°.

The modulus of the ARM2 is plotted in Figure 12.22. When viewed between two 0° polarizers, the 11 element, a close comparison would show that the image is not that close to an Airy disk since it is slightly elliptical and the diffraction rings are not complete along the x-axis. Between two 90° polarizers, the amplitude in the 22 element is observed, which resembles the 11 image rotated by 90°. The peaks of the 11 and 22 images are shifted slightly in opposite directions owing to the opposite linear components of the s- and p-phase shifts seen in Figure 12.14 (top right). In Figure 12.22, the peak of J22 is slightly behind the black x-axis cross section while the peak of J11 is slightly in front of this cross section. The image between crossed polarizers is very interesting. The pupil averaged value of the xy and yx Jones pupil elements of Figure 12.17 is zero; thus, the center of the 12 and 21 ARM elements is zero. Both image components have two main lobes straddling the center along the meridional plane. The energy missing from the diffraction rings of the 11 and 22 elements is found in the rings of the 21 and the 12 elements. The 22 element is more aberrated than the 11 element due to the larger phase variation of the p-component seen in Figure 12.14.

Figure 12.22The modulus of the ARM for the fold mirror. The centers of the 11 and 22 responses are both shifted in opposite directions with respect to the geometric center by small amounts since their phases, as seen in Figures 12.14 and 12.17, change in opposite directions.

The polarization structure of the image formed with 0° incident light is shown (Figure 12.23) as a 3D plot of the x- and y-components (left) and pupil polarization ellipse map (right). The central lobe is wider than it is tall, similar to elliptically shaped images formed with astigmatism. The polarization state is seen to be mostly slightly elliptical with a major axis rotating steadily counterclockwise from the left to right sides of the pupil.

Figure 12.23The amplitude response Jones vector images formed from 0° polarized incident light are shown on the left, with the x-amplitude on top and the y-amplitude at the bottom. The peak of the y-amplitude is 1/8 the peak of the x-amplitude. The corresponding ellipse maps are shown in three different scalings. The polarization ellipse map on the left shows a polarization variation across the Airy disk. By excluding the higher amplitude inside the first ring, the second ellipse map rescales to show the ellipticity at the edge of the first ring. The third ellipse map zooms into the center region, showing the steady polarization rotation from left to right.

The polarization changes to the image structure are much larger with 45° light incident as shown in Figure 12.24. The light at the center of the central lobe is elliptically polarized with the major axis at 135° due to reflection.

Figure 12.24The amplitude response Jones vector images formed from 45° polarized incident light are shown on the left, with the modulus of the x-amplitude on top and the y-amplitude at the bottom. The peak of the y-amplitude is 1/1.2 the peak of the x-amplitude. The corresponding ellipse maps are shown in three different scalings. The polarization ellipse map on the left shows a polarization variation across the Airy disk. The second ellipse map is rescaled to show the ellipticity at the edge of the first ring. The third ellipse map zooms into the center region showing the steady polarization rotation from left to right.

Transforming the Jones pupil of Figure 12.17 into a Mueller matrix yields the point spread matrix in Figure 12.25 shown in 3D (left) and raster plot (right) views. The first column shows the Stokes parameters when the object point is unpolarized. The center of the image is nearly unpolarized, because the centers of the (1,0), (2,0), and (3,0) functions are nearly zero. The linearly polarized component rotates around the image center from the 10 and 20 functions, while the top has left ellipticity and the bottom has right ellipticity from the (3,0) element.

Figure 12.25Mueller point spread matrix plotted in two different forms.

The response for 0° illumination is found by adding the zeroth and first columns. Its flux is calculated by adding the (0,0) and (0,1) elements, and it is seen that the peak of the 0° image has moved toward +x. Subtracting the (0,0) and (0,1) elements, it is seen that the peak of the 90° image has moved toward −y but with 90° illumination, and the brightest part of the 90° image has moved toward −x. The (2,2) element is negative in the center because 45° reflects as 135° as seen in Figure 12.24. From the 00 and (0,2) elements, it is seen that for 45° and 135° illumination, the image peaks shift in opposite directions in y; similarly, image peak shifts occur for right and left circularly polarized light.

The integral of the (0,0) element provides the net flux (unnormalized) in the image for unpolarized illumination; about 8% of the light has been lost to absorption. Other integrals provide the net flux through various polarizer combinations. For example, when the mirror is preceded by a 45° polarizer (transmission axis at (1, 1, 0)/$\sqrt{2}$ in global coordinates) and followed by a 135° polarizer ((0, 1, 1)/$\sqrt{2}$) after reflection, the net flux is the integral of M00 + M02 − M20 − M22, which is twice the integral of M02.

This section has shown that the polarization aberrations due to a gold mirror are substantial. Similar results are obtained for aluminum and other mirrors. A large numerical aperture was used to make the effects large. For smaller numerical apertures, the magnitude of the effects approximately linearly. This example demonstrates why fold mirrors cause many of our polarization problems in optical systems.

12.4Combination of Fold Mirror Systems

Section 12.3 discussed the polarization changes of fold mirrors caused by diattenuation and retardance associated with the Fresnel coefficients. The resultant polarization aberration can be compensated by using individual components with aberrations of the opposite sign, minimizing the resultant aberration. Here, sequences of one, two, and four gold-coated fold mirrors are considered, to understand configurations that reduce the polarization aberrations.

Figure 12.26 shows a flat mirror M1 reflecting a 0.2 NA converging beam. This beam is scanned by rotating the mirror about the z-axis; this corresponds to the incident ray at the center of the beam keeping an angle of incidence of 45°. A grid of rays evenly sample the converging wavefront, and this incident beam propagates from the xy-plane toward the z-direction. In Figure 12.26, the mirror is rotated about the z-axis by 180° to five orientations, and the mirror normals are tabulated in Table 12.2.

Figure 12.26Ray trace figures showing a converging beam reflecting from a fold mirror (blue) toward a detector (gray). The image location scans as the orientation of the fold mirror rotates around the incident beam, centered on the z-axis, by 0°, 45°, 90°, 135°, and 180° from (0, 1, 1).

Table 12.2Surface Normals for the Mirror of Figure 12.6

Scan Angle	0°	45°	90°	135°	180°
Surface normal	$\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\sqrt{2}$}\right.\left(\begin{array}{c}0\\ 1\\ 1\end{array}\right)$	$\frac{1}{2}\left(\begin{array}{c}1\\ 1\\ \sqrt{2}\end{array}\right)$	$\frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\ 0\\ 1\end{array}\right)$	$\frac{1}{2}\left(\begin{array}{c}1\\ -1\\ \sqrt{2}\end{array}\right)$	$\frac{1}{\sqrt{2}}\left(\begin{array}{c}0\\ -1\\ 1\end{array}\right)$

The plane of incidence of each ray rotates as M1 scans. An AoI map of the mirror viewed from the entrance space is shown in Figure 12.27. Shorter lines in Figure 12.27 correspond to a ray moving closer to normal incidence. The ray at the center has a 45° AoI. The AoI varies linearly from across the pupil in one direction, while in the perpendicular direction, the PoI rotates from one side to the other linearly. Therefore, as M1 scans around, this AoI pattern rotates.

Figure 12.27The one-fold mirror system’s angle of incidence map shown in Figure 12.26. The legend on the right shows the largest incident angle in degrees for all frames.

Figure 12.28 shows the view from the entrance space of the diattenuation for reflection from gold for each ray. The diattenuation map is oriented orthogonal to the AoI map since the state with maximum reflectance is along the s-polarization. As shown in Figure 12.14 (bottom left), the magnitude of diattenuation increases quadratically with AoI as the rays move from normal incidence.

Figure 12.28Diattenuation map for the scanning fold mirror. The maximum diattenuation is shown in the scale on the right.

Figure 12.29 shows the variations of retardance across the pupil, the retardance maps, resulting from scanning M1. Since ϕrp < ϕrs for the gold mirror, the fast axis for reflection is p-polarized. Thus, the retardance map has the same orientation as the AoI map, which is orthogonal to the diattenuation axes shown in Figure 12.29. As shown in Figure 12.14, the retardance of gold varies quadratically with incident angle.

Figure 12.29Retardance map for the scanning fold mirror. The maximum retardance in radians is shown in the scale on the right.

To reduce the polarization aberrations of the fold mirror, another fold mirror can be added in the reflection path. The second mirror is oriented to have the s-polarized component of the axial ray on the first mirror become p-polarized on the second mirror, and vice versa. This is the crossed mirror configuration. The polarization aberration for one incident ray in a converging beam can be completely compensated. About this ray, such a mirror pair has small and linearly varying polarization aberration about the center.

In this two-fold mirror system, the first mirror M1 is fixed with surface normal at (0, 1, 1), while the second mirror M2 rotates about the incident beam along the y-axis. M2’s surface normal rotates about the M1–M2 axis by 180° from (1, 1, 0), as shown across Figure 12.30. The M2’s surface normals are listed in Table 12.3.

Figure 12.30Ray trace figures showing a converging beam reflecting from fold mirror M1 (blue), then a scanning fold mirror M2 (magenta), and then incident on a detector (gray).

Table 12.3Surface Normals of M2 in Figure 12.30

Scan Angle	0°	45°	90°	135°	180°
Surface normal	$\frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\ 1\\ 0\end{array}\right)$	$\frac{1}{2}\left(\begin{array}{c}1\\ \sqrt{2}\\ -1\end{array}\right)$	$\frac{1}{\sqrt{2}}\left(\begin{array}{c}0\\ 1\\ -1\end{array}\right)$	$\frac{1}{2}\left(\begin{array}{c}-1\\ \sqrt{2}\\ -1\end{array}\right)$	$\frac{1}{\sqrt{2}}\left(\begin{array}{c}-1\\ 1\\ 0\end{array}\right)$

As shown in Figure 12.31, when M2 scans, its plane of incidence changes with respect to M1. The PoI axes shown on M2 are propagated back to the entrance space of M1; therefore, all maps are compared directly in object space. The center ray’s incident angles at both mirrors are kept at 45°. As M2 scans about M1, the p-polarization at the center ray of the two mirrors changes from orthogonal, to parallel, and back to orthogonal. In the crossed mirror configuration (scan angle 0° and 180°), the PoIs between M1 and M2 are close to orthogonal across the beams and then become less parallel as M2 scans.

Figure 12.31The two fold mirrors’ angle of incidence maps shown in Figure 12.30. M2 scans about the y-axis while M1 is fixed. The crossed mirror configuration occurs at 0° and 180°. The legend on the right shows the largest incident angle in degrees for all frames. The M2 maps have been propagated back to the entrance space of M1.

The overall P matrix is calculated for each ray path within the field. The diattenuation and retardance are then obtained through data reduction of these P matrices, as explained in Chapters 9 and 17. The diattenuation of M1 and M2 and the cumulative diattenuation from both mirrors as M2 scans are shown in Figure 12.32. All maps directly compare because all the diattenuation axes are viewed from the entrance space. At a 90° scan angle, the diattenuation orientations of M1 and M2 are close to parallel and the overall diattenuation magnitude is maximized. At 0° and 180° scan angles, the diattenuation orientations are closest to orthogonal over the pupil; hence, the cumulative diattenuation reaches a minimum. The corresponding retardance maps shown in Figure 12.33 have similar variations, except that they are approximately orthogonal to the diattenuation maps.

Figure 12.32Diattenuation map for M1 and M2, and the total system. The maximum diattenuation is shown in the scale on the right.

Figure 12.33Retardance map for M1 and M2, and the total system. The maximum retardance is shown in the scale on the right in radians.

At 0° and 180° scan angles, the polarizations of the two mirrors are nearly orthogonal with similar magnitudes; thus, the maximum diattenuation and retardance cancel each other, producing complete cancellation at the center with a linear variation of diattenuation and retardance through the center. Their axes change by 90° crossing the center. Such polarization aberration forms are termed linear diattenuation tilt and linear retardance tilt9 where the axis orientation varies through 180° moving around the node; these will be discussed in Chapter 15.

The null of the cumulative polarization moves away from the center when M2 scans away from 0°. This null moves outside the field to the upper left when M2 scans to 45°. It returns to the center of the field when M2 scans through 180°. When M2 scans to 90°, the overall polarization magnitude of the two mirrors adds, and doubles the overall aberration when compared to the single mirror aberration.

The diattenuation of the individual mirrors is linear, not elliptical, everywhere. However, a small amount of elliptical diattenuation and retardance are present in the cumulative aberration and are observed in Figures 12.32 and 12.33 toward the edge of the pupil as slight ellipticities. Such elliptical polarization occurs when two linear diattenuators or retarders have axes neither parallel nor perpendicular and thus generate some circular polarization.10 When the polarization axes are 45° or 135° apart, the maximum circular component is generated. This coupling of linear into circular polarization is discussed in Section 14.3.

Polarization aberrations of a single fold mirror are reduced by the crossed mirror configuration. However, a linear variation remains even though their plane of incidence is orthogonal at the center. Further polarization reduction can be achieved by four mirrors arranged as a double-crossed mirror system. In the system of Figure 12.34, M1 and M2 are fixed, and their normals are at (0, 1, 1) and (−1, −1, 0), respectively. The third mirror M3 and fourth mirror M4 are fixed with respect to each other, but the pair scan around the x-axis, which is the axis of the light exiting the M1–M2 assembly. The normals of M3 and M4 start at (1, 1, 0) and (0, 1, 1). As shown across Figure 12.34, the M3–M4 detector assembly rotates about the x-axis through 180°. The normals of M3 are listed in Table 12.4. Figure 12.35 shows the angle of incidence maps for all four mirrors. M1 and M2 are fixed in the crossed mirror configuration while M3 and M4 rotate as a pair in the crossed mirror configuration.

Figure 12.34Ray trace figures showing a converging beam reflecting from M1 (blue), M2 (magenta), M3 (yellow), then M4 (green), and finally reaching a detector (gray). M1–M2 assembly is fixed, while the M3–M4 assembly scans about the x-axis.

Table 12.4Surface Normals of M3 in Figure 12.34

Scan Angle	0°	45°	90°	135°	180°
Surface normal	$\frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\ 1\\ 0\end{array}\right)$	$\frac{1}{2}\left(\begin{array}{c}\sqrt{2}\\ 1\\ 1\end{array}\right)$	$\frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\ 0\\ -1\end{array}\right)$	$\frac{1}{2}\left(\begin{array}{c}\sqrt{2}\\ -1\\ -1\end{array}\right)$	$\frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\ -1\\ 0\end{array}\right)$

Figure 12.35The angle of incidence maps of the four mirrors in Figure 12.34. M1 and M2 are fixed while M3 and M4 scan as a pair. The legend on the right shows the largest incident angle for all frames in degrees.

The retardance and diattenuation maps for the double-crossed mirror system are shown in Figures 12.36 and 12.37. Since M1 is orthogonal to M2, and M3 is orthogonal to M4 in all scan configurations, there is always a polarization null at the center of the field while the polarizations increase linearly toward the edge of the field. At 0° scan angle, M3–M4 and M1–M2 have the same magnitude and orientation in both the diattenuation and retardance; hence, the cumulative polarizations are twice that of the individual two-mirror assembly and are dominated by linear tilt. As M3–M4 scans from 0° to 180°, the overall aberration steadily decreases as cos2(ϕ/2), where ϕ is the scan angle. At a 180° scan angle, the aberration maps of M1–M2 and M3–M4 are nearly orthogonal to each other across the field and the overall polarization aberration is minimized; the linear variations from the two-mirror assemblies across the field are opposite and compensate. The magnitude of the residual aberration is quadratic across the field for this four-mirror combination at 180° scan angle and well corrected for a large field of view.

Figure 12.36Diattenuation maps for the M1–M2 assembly (top row) and M3–M4 assembly (middle row), and the total diattenuation from all four mirrors (bottom row). The maximum diattenuation is shown in the scale on the right.

Figure 12.37Retardance maps for the M1–M2 assembly (top row) and M3–M4 assembly (middle row), and the total system (bottom row). The maximum retardance is shown in the scale on the right in radians.

A summary of the cumulative polarizations of the single, the two crossed, and the double-crossed gold mirror systems is tabulated in Tables 12.5 and 12.6.

Table 12.5For a 0.2 NA Illumination, the Maximum, Average, and Chief Ray Diattenuation of the Single, the Two-Crossed, and the Double-Crossed Gold Mirror Systems

Maximum Diattenuation
	0°	45°	90°	135°	180°
4 Mirrors	0.3629	0.3198	0.2812	0.2376	0.2055
2 Mirrors	0.1921	0.3591	0.4242	0.3591	0.1921
1 Mirror	0.3164	0.3375	0.3164	0.3375	0.3164
Average Diattenuation
	0°	45°	90°	135°	180°
4 Mirrors	0.2147	0.2015	0.1684	0.1256	0.0927
2 Mirrors	0.1221	0.3086	0.4169	0.3086	0.1221
1 Mirror	0.2255	0.2255	0.2255	0.2255	0.2255
Chief Ray Diattenuation
	0°	45°	90°	135°	180°
4 Mirrors	0	0	0	0	0
2 Mirrors	0	0.3090	0.4218	0.3090	0
1 Mirror	0.2212	0.2212	0.2212	0.2212	0.2212

Table 12.6For a 0.2 NA Illumination, the Maximum, Average, and Chief Ray Retardance in Radians of the Single, the Two-Crossed, and the Double-Crossed Gold Mirror Systems

Maximum Retardance
	0°	45°	90°	135°	180°
Double-crossed mirrors	1.4119	1.2654	0.9919	0.8800	0.3587
Crossed mirrors	0.7068	1.1315	1.3258	1.1315	0.7068
Single mirror	1.0416	1.2205	1.0416	1.2205	1.0416
Average Retardance
	0°	45°	90°	135°	180°
Double-crossed mirrors	0.7699	0.7104	0.5613	0.3569	0.1836
Crossed mirrors	0.3926	0.9097	1.2470	0.9097	0.3926
Single mirror	0.6465	0.6465	0.6465	0.6465	0.6465
Chief Ray Retardance
	0°	45°	90°	135°	180°
Double-crossed mirrors	0	0	0	0	0
Crossed mirrors	0	0.8435	1.1939	0.8435	0
Single mirror	0.5969	0.5969	0.5969	0.5969	0.5969

These results for gold mirrors generalize to most reflective coatings. For most of the common reflective coatings, the diattenuation and retardance maps are similar in form to Figures 12.28, 12.29, 12.32, 12.33, 12.36, and 12.37 of the gold mirrors shown here, except for a change of scale; the polarization maps are primarily linear and quadratic with the incident angle. Figure 12.38 shows the diattenuation and retardance of uncoated metal mirrors with real and imaginary refractive indices spanning most dielectric and metal surfaces. Over this range of refractive indices, the retardance has similar functional forms with a predominantly increasing linear component at 45°. On the other hand, the diattenuation also has similar functional forms but varying magnitude. At 45°, the behavior is predominantly linear and increasing except for the upper left case in Figure 12.38.

Figure 12.38Retardance maps in radians (left) and diattenuation maps (right) as a function of incident angle for metal reflections for a 3 × 3 grid of refractive indices n + iκ.

To generate the polarization patterns with the form shown in Figures 12.28, 12.29, 12.32, 12.33, 12.36, and 12.37 but with arbitrary magnitudes from a spherical wavefront that reflects from a flat mirror, the following conditions apply. (1) The variation of the diattenuation or retardance is predominantly linear over the range of incident angles, and (2) the diattenuation or retardance magnitude is much less than one. If the variations are not linear, the two mirror aberrations will not be linear, and then the compensated four-mirror aberrations (Figures 12.36 and 12.37, bottom right) will not be quadratic. The aberrations in the present example already start to challenge these assumptions.

An example coating that does not satisfy condition (1) is shown in Figure 12.39, where the diattenuation and retardance variation are not linear about 45°. Regarding (1), the nonlinear higher-order Taylor series expansion terms of the diattenuation and retardance create nonlinearities in diattenuation and retardance maps. Regarding (2), as the magnitudes increase, the interactions of non-aligned diattenuation and retardance between surfaces generate larger amounts of elliptical retardance and diattenuation, which are observed as the small ellipticities in the bottom rows of Figures 12.32 and 12.33. Operating fold mirrors at higher incident angles, toward grazing incidence, generates larger retardance.

Figure 12.39Diattenuation and retardance as a function of angle reflecting from 0.430 μm Al2O3 coating (n = 1.6033) on a silver substrate (n + iκ = 0.0314 + i 5.308) at 500 nm.

In conclusion, the magnitude of diattenuation or retardance of two mirrors is reduced when cascading components with orthogonal diattenuation axes (maximum transmission axis) or orthogonal retardance axes (fast axis). The polarization aberration of one mirror is well compensated by another mirror crossed such that the p-polarized light exiting the first mirror is s-polarized on the second mirror. This crossed mirror configuration corrects the polarization aberration at one point, leaving a residual linear variation across the field.

Additional mirrors, a four-mirror system constructed from two pairs of crossed mirrors, can further reduce the polarization aberration of a crossed mirror assembly. The minimum polarization is obtained when the linear variation of M1–M2 is opposite from the linear variation of M3–M4. Such a configuration produces zero polarization at the center of the field with a small residual quadratic variation. This double-crossed mirror system has the lowest polarization aberrations across a large field of view among the one-, two-, and four-mirror systems shown in this section.

The polarization aberrations of the metal mirrors are a function of Fresnel coefficients calculated from the metal refractive index. Increasing refractive index decreases the magnitude of retardance. Decreasing the imaginary part of the refractive index and increasing the real part of the refractive index increases the magnitude of diattenuation. The polarization magnitudes are scaled by the metal coating, while the patterns of the polarization orientations remain the same with the polarization nulls at the center of the four-mirror configurations.

12.5Cassegrain Telescope

The polarization aberrations of radially symmetric mirror systems, like Cassegrain telescopes, have radially symmetric patterns of diattenuation and retardance. Figure 12.40 shows the Cassegrain telescope, which will be analyzed as an example. Table 12.7 lists the telescope’s specifications. Collimated light enters from the right along the optical z-axis and reaches the primary mirror, shown as a cyan paraboloid in Figure 12.40 (right), which serves as the aperture stop and entrance pupil. The reflected light converges onto the secondary mirror (shown as a purple hyperboloid) and reflects and passes through a hole in the primary mirror, coming to focus at an image plane behind the primary mirror. The radii and conic constants are chosen so that an on-axis collimated beam has zero geometrical wavefront aberration; all of the on-axis rays have the same optical path length from the entrance pupil to the exit pupil and image point. The angle of incidence over the pupil is plotted for the primary and secondary mirrors in Figure 12.41 (top); the angles increase linearly from the center of the pupil and are radially oriented. Applying the Fresnel coefficients for aluminum (Figure 12.41, bottom) to the angles of Figure 12.41 yields the diattenuation pupil maps of Figure 12.42 for the primary and secondary. The combined diattenuation effect from the entrance to exit pupil (12.42 right) is tangentially oriented and increases quadratically. Similarly, applying the Fresnel phase coefficients yields the retardance pupil maps of Figure 12.43. The combination of Figures 12.42 (right) and 12.43 (right) yields the Jones pupil of Figure 12.44.

Figure 12.40Cassegrain telescope with collimated light entering from the left, reflecting from a paraboloid-shaped primary mirror (cyan), then secondary mirror (purple), and focusing behind the primary mirror. This system has zero wavefront aberration, but the metal coatings induce polarization aberration.

Table 12.7Cassegrain Telescope Specifications

Primary Mirror
Radius R1	5688 mm
Conic constant κ1	−1
F/#	2.3862
Secondary mirror
Radius R2	1228.28 mm
Conic constant κ2	−1.780
Mirror separation	2317.80
Image plane	3674.77 from secondary mirror
Entrance pupil diameter	8323.3 mm
Exit pupil diameter	1743.44, −485.50 from secondary mirror
Aluminum index	0.958 + 6.690 i
Wavelength λ	0.55 μm

Figure 12.41(Top) The angle of incidence functions at the primary mirror (top left) and secondary mirror (top right). (Bottom) The magnitudes and phases for the Fresnel coefficients for aluminum.

Figure 12.42Pupil maps of the diattenuation on the primary mirror (left) and secondary mirror (middle) are tangentially oriented because the s-component has the greater reflectance. The map on the right is the combined diattenuation from the entrance to the exit pupil. The diattenuation reaches more than 0.04 for the marginal rays at the edge of the pupil.

Figure 12.43Retardance pupil maps for the primary (left), secondary (middle), and combination (right) are radially oriented; the p-polarized light exits before the s-polarized component. The retardance increases quadratically from the center of the pupil to about λ/20 for the marginal rays.

Figure 12.44The magnitude of the Jones pupil (left) and the phase (right) for the Cassegrain telescope on-axis shows how the x- and y-components of the incident light will be apodized and coupled by the aluminum reflections. The reflectance of x-polarized light into x-polarized light increases toward the top and bottom of the pupil where it is s-polarized and decreases toward the left and right sides. The pattern for y-polarized light into y-polarized is rotated by 90°. The reflectance of x into y and y into x is zero along the x- and y-axes and increases along the ±45° directions. These results differ from the single-element lens due to the metal reflections.

The phase changes due to the coatings (Figure 12.44, right) on the diagonal (x into x and y into y) are toroidal, curving upward quadratically along one axis and downward along the other. This toroidal form indicates that astigmatism is introduced by the aluminum reflection’s retardance!! For x into x, the peak-to-valley astigmatism is about 0.3 radians. For y into y, the magnitude is the same, but the astigmatism orientation is rotated by 90°. Hence, when the Cassegrain telescope is placed between parallel polarizers, astigmatism is measured in the exiting wavefront, and this astigmatism rotates with the polarizer pair. Such telescope-associated astigmatism is discussed by Reiley11 in his dissertation.12

This mirror-induced astigmatism is surprising to many familiar with wavefront aberration theory. For radially symmetric systems, there is no astigmatism on-axis. Instead, for lenses, Cassegrain telescopes, and other systems, the astigmatism increases quadratically from a zero (node) in the center of the field of view. Thus, conventionally, when this astigmatism was observed in interferograms, one would suspect that the astigmatism might have been polished into one of the mirrors, that the mirror was not radially symmetric. In such a case, the astigmatism’s orientation should rotate with the offending mirror. But because this is polarization aberration, if the mirrors are rotated, the astigmatism doesn’t rotate; it rotates when the polarization is rotated. Further, if the coating is changed from aluminum to gold or some other metal, the magnitude of the astigmatism changes, which would not happen with wavefront aberrations.

In addition to introducing astigmatism, the mirror’s retardance and diattennuation couple light into the orthogonal polarization state, as is indicated in the off-diagonal elements of the Jones pupil (Figure 12.44). Like the uncoated lens, this coupled polarization is seen to be apodized into the Maltese cross form, with an amplitude at the edge of the pupil of ~0.13 of the incident light for this F/#, wavelength, and metal. With crossed polarizers, when this Maltese cross light is brought to focus in the image plane, it forms the interesting point spread function components shown in the off-diagonal elements of Figure 12.45 containing four islands of light with dark bands oriented horizontally and vertically through the center of the PSF. The magnitude is plotted here; the pairs of islands on opposite diagonals have opposite amplitudes, positive and negative, and the absolute value is easier to visualize. Thus, when 0° linearly polarized light is incident, a 0° linear polarizer analyzes the J11 amplitude, which is astigmatic (at a scale not obvious in this figure). When the analyzing polarizer is rotated to 90°, a much weaker pattern of four islands is observed, the J21 amplitude. For intermediate orientations, the image passes through a continuum of intermediate patterns.

Figure 12.45The amplitude response functions for the Cassegrain telescope show the spatial variations of polarization state in the image of point objects for several polarization combinations. For x into x and y into y, the amplitude distributions are close to the Airy disk, but slightly elliptical due to the astigmatism induced by the differences between the metals’ s- and p-phases. The x into y and y into x responses show four islands around the first Airy disk.

The PSFs of the Cassegrain telescope can be measured by placing the telescope in a Mueller matrix imaging polarimeter, measuring its response to incoherent plane waves. This Mueller matrix point spread matrix is simulated in Figure 12.46. The first column contains the Stokes image of the unpolarized PSF. The total flux in the M00 element is rotationally symmetric because the astigmatism of all the polarization orientations averages to a radially symmetric distribution, which is larger than the diffraction-limited pattern. The M10 and M20 elements show that away from the center, which is unpolarized, the PSF becomes partially polarized with a spatially varying polarization.

Figure 12.46The point spread matrix for the Cassegrain telescope’s on-axis image shows the polarization distributions within the PSF for arbitrary incoherent incident polarization states. The surface plots (left) are individually scaled to show fine details in the polarization distributions. The polarization coupling due to retardance, M13, M23, M31, and M32, is much larger than the coupling due to diattenuation, M01, M02, M10, and M20. The raster plots (right) are overexposed in the center to reveal the faint details in the polarization structure.

12.6Fresnel Rhomb

One clever application of the Fresnel equations is the Fresnel rhomb, a total internal reflection (TIR)-based quarter wave retarder.13Figure 12.47 (left) shows a Fresnel rhomb, a prism with a parallelogram cross section and two TIRs. For a refractive index around n = 1.5, the retardance associated with TIR reaches a maximum near an eighth of a wave at an angle of incidence close to 52°. Thus, two internal reflections can provide approximately a quarter of a wave of retardance. The Fresnel rhomb is easy and inexpensive to fabricate since it doesn’t use birefringent materials, just a block of glass chosen with minimal stress birefringence. The beam enters and exits in parallel directions. Because of the lateral displacement between entering and exiting rays, it is inconvenient to rotate a Fresnel rhomb.

Figure 12.47(Left) A Fresnel rhomb (green) is a prism with a parallelogram cross section and two internal reflections. A ray path (red) shows a ray at normal incidence (for minimum diattenuation) at the entering and exiting faces. (Middle) A refractive index n = 1.4965 gives a π/4 retardance at wavelength 589 nm. (Right) For index n = 1.4965, a maximum retardance is attained at incident angle θin = 51.79°.

Figure 12.47 examines some design choices for the Fresnel rhomb. Figure 12.47 (middle) plots the TIR retardance versus angle of incidence for refractive indices in the range of 1.495 to 1.5. A horizontal black line indicates one eighth wave of retardance (~0.785), the target retardance. Figure 12.47 (right) shows that for a refractive index of 1.4965 and an angle of 51.79°, the retardance curve is just tangent to the eighth wave retardance line; thus, retardance is compensated in angle at this condition. One glass with a refractive index close to 1.4965 is Schott N-PK52A. Slightly larger refractive indices yield two eighth wave crossings a fraction of a degree apart.

Since the retardance depends on the refractive index, the wavelength dependence of the Fresnel rhomb depends only on the variation of refractive index with wavelength. This is plotted for Schott N-PK52A in Figure 12.48 showing that the retardance varies by less than 2.5° (0.04 radians) over the visible. This is much better than birefringent waveplates where the retardance varies as the variation of the birefringence times the thickness divided by the wavelength δ = 2π Δn t/λ.14 For waveplates, the 1/λ term yields a rapid variation of retardance with wavelength. Therefore, the retardance of Fresnel rhombs is far more achromatic than waveplates. Using the polarization ray trace, the variation of retardance with wavelength (left) and angle of incidence (right) for the Schott N-PK52A rhomb is plotted in Figure 12.48.

Figure 12.48(Left) A polarization ray trace through an N-PK52A Fresnel rhomb shows the evolution of a 45° polarized incident state exiting as right circularly polarized, depicted as blue polarization ellipses. (Middle) The ray trace yields the retardance spectrum (in blue). The red line represents the target quarter wave retardance. (Right) The retardance versus incident angle is shown from 0° to 5° in air at the entrance surface.

Many clever variations of Fresnel rhombs have been developed, often to modify the achromatization or control the field variations of retardance. A summary of these Fresnel rhomb-related designs is found in the work of Bennett.14

12.7Conclusion

Systems of uncoated lenses and/or metal mirrors naturally have polarization aberrations that follow from the Fresnel equations. These polarization aberrations have a form similar to the angle of incidence maps for the surfaces. These examples, shown in this chapter, were evaluated by the polarization ray tracing method presented in Chapter 9 using the Polaris-M program. A grid of rays are traced and their polarization ray tracing matrices are calculated. The diattenuation and retardance magnitude and orientation are calculated for the rays to represent the polarization aberrations. These parameters are then flattened using the algorithms of Chapter 11 and represented in the double pole coordinate system so that the diattenuation and retardance maps can be shown on a flat page or computer screen. For the singlet lens and Cassegrain telescope examples shown, the polarization aberrations are not particularly large because their effect on the point spread function is small, and the wavefront aberrations are a much higher priority.

To understand the magnitude of the polarization aberrations and determine when it is a problem, it is necessary to perform a polarization ray trace. This will show if the polarization aberrations are small enough to not be a concern, or to determine if the polarization aberrations may cause problems, such as loss of resolution or leakage at polarizers.

This chapter presents the polarization aberrations of uncoated systems as a first step toward understanding more complex interfaces presented in later chapters. In high-quality optical systems, such as lenses for movie production, LC projectors, microlithography, and other applications, uncoated surfaces such as those analyzed in this chapter are the exception, not the rule. Most lens surfaces are antireflection coated. Many mirrors have multilayer reflection-enhancing coatings. Other coatings are incorporated as spectral filters, beam splitters, and other functions. In that case, the examples here, the single-element lens, multiple fold mirrors, and Cassegrain telescope, would have different magnitudes of the diattenuation and retardance, although the form would be similar, because the diattenuation and retardance of thin films are still aligned with the s- and p-planes. Thus, they need to be polarization ray traced using their specified coatings to determine the new magnitudes of the diattenuation and retardance maps. To gain some understanding of such cases, Chapter 13 analyzes some typical optical thin films to provide examples and guidance for including the coatings in the polarization ray trace and polarization aberration specification.

12.8Problem Sets

1. The flux distribution (amplitude squared), seen in Figure 12.7 (left) is referred to the Maltese cross. The Maltese cross is an important pattern, commonly seen with most radially symmetric optical systems placed between crossed polarizers. Along the diagonal, the amplitude of the leaked light varies as ρ2 while the transmitted flux varies as ρ4, where ρ is pupil coordinate. If the lens is stopped down to a smaller aperture, the leaked flux rapidly diminishes. For a small numerical aperture, the Maltese cross may be difficult to observe. If the two polarizers rotate together, the Maltese cross will rotate with the polarizers.

   1. Consider the Maltese cross, seen in Figure 12.7 (center). Why does the pattern rotate when the two polarizer are rotated together? What happens if just one polarizer is rotated by a small amount? If the lens alone is rotated between crossed polarizers, what happens to the pattern?

   2. Describe the pattern of the transmitted flux when a cylindrical lens is rotated between crossed polarizers.

   3. Consider the total transmitted flux in the Maltese cross. If the diameter of the pupil is reduced to one half, by what fraction is the transmitted flux reduced?

2. Find the refractive indices for which the intensity reflection coefficients at normal incidence equal the intensity transmission coefficients.

3. Show that if a beam is incident on a plane parallel plate from air at Brewster’s angle, that it is incident on the rear surface at Brewster’s angle.

4. Find the Jones matrix for a real half wave retarder, fast axis at 45°, of sapphire at λ = 589 nm where nO = 1.76817 and nE = 1.76009. Chapter 5 (Jones Matrices and Polarization Properties) presented the Jones matrices for ideal retarders; now, consider a real retarder. Find the thickness t and calculate the absolute phases for the ordinary and extraordinary rays. Because of these optical path lengths, the retarder’s Jones matrix will not be in symmetric, fast axis unchanged, or slow axis unchanged form (Table 5.4). Evaluate the Fresnel equations at normal incidence for the ordinary and extraordinary modes and include the small resulting diattenuation in the Jones matrix.

5. Show that, for reflection in air from a dielectric, both Rs(θ) and Rp(θ) approach the same value as θ approaches 90°. Find the slopes of Rs(θ) and Rp(θ) as θ approaches 90°.

6. An excellent fit to the Fresnel amplitude coefficients rs, rp, ts, and tp, for n0 = 1 and n1 = 1.5 over the entire angle of incidence range 0 < θ < 90° can be obtained with an even fourth-order fit as shown in Figure 12.49. Perform the least squares polynomial fit with f(θ) = a0 + a2 θ2 + a4 θ4 using the method of Section 8.5. For each amplitude coefficient, make a table of its values at 0°, 2°, 4°, … 90°. Then, fit each tabulated function f(θ) to find the coefficients a0, a2, and a4. For example, the first fit is rs = −8 × 10−9 θ4 − 0.0000306 θ2 − 0.203699.

7. Verify $r_p=\frac{r_s(r_s-\cos 2\theta )}{1-r_s\cos 2\theta }$ and $r_s=\frac{1}{2}\cos 2\theta (1-r_p)+\sqrt{r_p+\frac{1}{4}{\cos }^{2}2\theta {(1-r_p)}^2}$. Hint: Start with the rs as a function of θ, and rp as a function of θ.15

Figure 12.49Comparison of Fresnel amplitude coefficients to a set of fourth-order fits yields very close fits, almost on top of each other.

References

1H. K. and S. Inoué, Diffraction images in the polarizing microscope, J. Opt. Soc. Am. 49 (1959): 191–198.

2R. A. Chipman, Polarization analysis of optical systems, Opt. Eng. 28(2) (1989): 90–99.

3J. O. Stenflo, The measurement of solar magnetic fields, Rep. Prog. Phys. 41.6 (1978): 865.

4R. A. Chipman, Polarization aberrations, PhD dissertation, College of Optical Sciences, University of Arizona (1987).

5R. A. Chipman, Polarization analysis of optical systems II, Proc. SPIE 1166 (1989): 79–99.

6J. P. McGuire and R. A. Chipman, Polarization aberrations. 1. Rotationally symmetric optical systems, Appl. Opt. 33 (1994): 5080–5100 andPolarization aberrations. 2. Tilted and decentered optical systems. Appl. Opt. 33 (1994): 5101–5107.

7D. J. Reiley and R. A. Chipman, Coating-induced wave-front aberrations: On-axis astigmatism and chromatic aberration in all-reflecting systems, Appl. Opt. 33 (1994): 2002–2012.

8W. Swindell, Handedness of polarization after metallic reflection of linearly polarized light, J. Opt. Soc. Am. 61 (1971): 212–215.

9R. A. Chipman, Polarization aberrations, PhD dissertation, The University of Arizona (1987).

10D. B. Chenault and R. A. Chipman, Measurements of linear diattenuation and linear retardance spectra with a rotating sample spectropolarimeter, Appl. Opt. 32 (1993): 3513–3519.

11D. J. Reiley and R. A. Chipman, Coating-induced wave-front aberrations: On-axis astigmatism and chromatic aberration in all-reflecting systems, Appl. Opt. 33(10) (1994): 2002–2012.

12D. J. Reiley, Polarization in Optical Design, dissertation, Physics, University of Alabama in Huntsville (1993).

13R. J. King, Quarter-wave retardation systems based on the Fresnel rhomb principle, J. Sci. Instrum. 43.9 (1966): 617.

14J. M. Bennett, A critical evaluation of rhomb-type quarter wave retarders, Appl. Opt. 9.9 (1970): 2123–2129.

15R. M. A. Azzam, Direct relation between Fresnel’s interface reflection coefficients for the parallel and perpendicular polarizations, J. Opt. Soc. Am. 69(7) (1979).

1 Section 11.4, our basic coordinate system for spherical waves.

2 Described in Section 16.4.