Balancing polarization aberrations in crossed fold mirrors Wai Sze Tiffany Lam* and Russell Chipman College of Optical Sciences, University of Arizona, 1630 University Blvd., Tucson, Arizona 85721, USA *Corresponding author: [email protected] Received 15 December 2014; revised 24 February 2015; accepted 2 March 2015; posted 3 March 2015 (Doc. ID 230735); published 8 April 2015
The polarization aberrations of a fold mirror can be compensated by orienting a second fold mirror’s p-polarization with the first mirror’s s-polarization. This crossed-mirror configuration compensates the polarization for a single angle to zero and leaves a linear variation of diattenuation and retardance for a spherical wavefront. Two sets of crossed fold mirrors when properly oriented compensate the remaining linear variation and leave a much reduced quadratic variation in a large compensated field of view. © 2015 Optical Society of America OCIS codes: (080.0080) Geometric optics; (080.1010) Aberrations (global); (260.0260) Physical optics; (260.5430) Polarization; (310.3915) Metallic, opaque, and absorbing coatings; (310.5448) Polarization, other optical properties. http://dx.doi.org/10.1364/AO.54.003236
1. Introduction
Fold mirrors are common optical components used in many complex optical systems. For example, space telescopes use scanning fold mirrors to direct light from different directions to detectors. Many systems use multiple fold mirrors for compactness [1–3]. Metal mirrors and thin-film-coated mirrors induce polarization variations, in both diattenuation and retardance, as a function of incident angle [4–8]. These polarization aberrations are readily observed by illuminating fold mirrors with converging or diverging light and observing the leakage between crossed polarizers or by Mueller matrix imaging polarimetry [9]. The mirrors reflect more s- than p-polarized light. So one longstanding issue with fold mirrors in spectroradiometers is that the system measures an irradiance higher or lower than the actual irradiance, depending on the incident polarization state. These polarization aberrations have been an annoyance in many applications such as liquid crystal projectors and microlithography. They have also been a 1559-128X/15/113236-10$15.00/0 © 2015 Optical Society of America 3236
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particular problem in remote sensing spectrometers, which seek to accurately measure irradiance of a scene without bias toward any particular polarization state [10,11]. Pseudo-depolarizers are one solution and have been used to reduce calibration problems due to unequal s- and p-polarized output in the NASA MODIS-T spectrometer [12,13]. Another way to reduce polarization aberration across the field, as described in this paper, is to compensate the polarization aberration between fold mirrors [14–16]. In the following discussion, the polarization properties will be described in terms of diattenuation and retardance. Upon reflection from mirrors, there is a reflected electric field difference in amplitude and phase between s- and p-polarized light. The orientation of the s- and p-polarized components is a function of the light propagation direction kˆ and the orientation of the reflecting surface, sˆ
kˆ × ηˆ jkˆ × ηˆ j
and pˆ kˆ × s;ˆ
(1)
where ηˆ is the surface normal of the interface defined as pointing toward the transmitted medium.
The electric field oscillation of p-polarized light is in the plane of incidence (PoI), while s-polarized light is orthogonal to the PoI. Diattenuation D and retardance δ of a mirror are defined as D
jrs j2 − jrp j2 jrs j2 jrp j2
and δ ϕrp − ϕrs ;
(2)
where rs and rp are the Fresnel reflection coefficients for s- and p-polarized light reflected from a mirror or the corresponding coefficients calculated for a thin film coated interface. The phase shifts of the reflected s- and p-polarized light are ϕs Argrs
and ϕp π Argrp ;
(3)
where Arga is the argument of complex number a. The phase shift of p-polarized light has an additional factor of π due to the standard definition of the Fresnel coefficients, which incorporate a coordinate change upon reflection. The complex amplitude reflection coefficients for gold at 765 nm with refractive index 0.17183 4.7493i [17] are plotted in Fig. 1 as a function of angle. At normal incidence (θ 0°), s- and p-polarized light are degenerate, so their amplitude and phase change upon reflection are identical. The corresponding D and δ are shown in Fig. 2, where both D and δ increases approximately quadratically with incident angles. Diattenuation and retardance can be represented graphically with line segments indicating magnitude and orientation. The orientation of diattenuation corresponds to the orientation of the polarization state with the maximum transmission. The s-polarized light has a higher reflection than p-polarized light,
so the orientation of reflection diattenuation is along the s-polarization state. The orientation of retardance is the fast axis, which corresponds to the polarization state with the smaller phase shift. In the metal reflection shown in Fig. 1, the fast axis corresponds to the p-polarized light, which has a smaller phase delay. Consider a converging beam reflecting from a 45° fold mirror, as shown in Fig. 3 (left) and (middle), the orientations of the s- and p-polarized components rotate slightly from one side of the field to the other side of the field due to the changing PoI of the converging beam at the flat mirror. Due to the changing incident angle across the converging field, the diattenuation and retardance induced by the fold mirror vary across the field. The orientation of polarization has a dependence on the PoI orientation, which allows the polarization variation across the field to be adjusted. By rotating the fold mirror by 20°, as shown in Fig. 3 (right), the chief ray of the same converging beam in Fig. 3 (middle) propagates at an angle 20° to the y axis. Since the PoI rotates with the orientation of the fold mirror, the orientations of the s- and p-polarization states also rotate. Therefore, the amount of polarization variation across the field can be compensated with more than one mirror. A way to visualize and understand these rotations of polarization states and the compensation of polarization between multiple mirrors in three-dimensions is to project the states and polarization properties into the entrance space. A 3D polarization ray tracing algorithm [18–20] is employed to ray trace a converging beam through systems of one, two, and four fold mirrors. The ray tracing algorithm is demonstrated in the Appendix A for two fold mirrors. The polarization aberrations of these systems for a converging beam are considered in the following sections, and the magnitude and orientation of the resultant aberrations across the
Fig. 1. Fresnel reflection coefficients: amplitude (left) and phase (right) as a function of incident angle for an air/gold interface at 765 nm. Red for s-polarized light and blue for p-polarized light.
Fig. 2. Reflection diattenuation and retardance as a function of incident angle at 765 nm for an air/gold interface.
Fig. 3. Converging beam is simulated as a grid of rays. The chief ray of the beam reflects from a fold mirror at 45°. The s- and p-polarization components are represented by the red and blue lines at each ray. Left and middle show two different views for the fold mirror whose surface normal (green arrow) is oriented at (0, 1, 1). With the same viewing angle as middle, right shows the same grid of incident rays reflecting from a rotated fold mirror. This mirror’s surface normal has rotated about the z axis at the center of the mirror by 20° from (0, 1, 1) (dashed green arrow) to −0.242; 0.664; 0.707 (solid green arrow). 10 April 2015 / Vol. 54, No. 11 / APPLIED OPTICS
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beams will be compared. 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.” Similar interpretations apply to the terms quadratic and cubic. 2. Polarization Aberration of One Fold Mirror
Consider a flat mirror, labeled as M1, reflecting a 0.2 NA converging beam, as shown in Fig. 4. The beam is scanned by rotating the mirror about the z axis, while the chief ray maintains a 45° angle of incidence. The converging beam is simulated by a grid of rays evenly sampling the converging wavefront. The incident beam starts at the x–y plane propagating in the positive z direction. Figure 4 shows that the mirror is scanning about the z axis by 180° to five different orientations. As the mirror scans, it sweeps the converging beam around in the x–y plane about the z axis. The orientations of the surface normal of M1 are tabulated in Table 1 for the scan angles of Fig. 4. As M1 scans about the incident beam, the PoI of each ray rotates; this is shown in the angle of incidence (AoI) map in Fig. 5. This AoI map of M1 is viewed from the entrance space. The length of each line segment in the map represents the magnitude of the AoI. The orientation of a line indicates the
Surface Normal of M 1 in Fig. 4
Table 1.
Scan Angle Surface Normal
0° p1 2
0! 1 1
45° 1 2
1 ! p1 2
90° p1 2
1! 0 1
135° 1 2
1 ! −1 p 2
180° p1 2
0 ! −1 1
orientation of PoI, which is also the orientation of the p-polarized component for each incident ray. A shorter line corresponds to a ray closer to normal incidence. The chief ray’s AoI is 45°. The AoI increases linearly from one side of the pupil to the other, while in the perpendicular direction of the field, the PoI rotates linearly from one side to the other. As M1 scans around, this AoI pattern simply rotates. The diattenuation for each ray path is plotted in Fig. 6, where the length of each line represents the diattenuation magnitude, and the line’s orientation corresponds to the orientation of the maximum transmission state viewed from the entrance space. Since the maximum transmission state is the s-state, the orientation in the diattenuation map is orthogonal to the orientation in the AoI map. The magnitude of diattenuation increases quadratically with AoI from normal incidence as shown in Fig. 2 (left). The variations in retardance resulting from scanning M1 are shown in the retardance maps in Fig. 7. The length of the line segment represents the retardance magnitude, and the line’s orientation is the fast axis orientation. For the gold mirror, ϕrp < ϕrs , so the reflective fast axis is p-polarized. The retardance
Fig. 5. AoI map of the one fold mirror system shown in Fig. 4. The largest incident angle shown in all the frames is 61°.
Fig. 6. Diattenuation map of the scanning fold mirror in Fig. 4. The maximum diattenuation among all five configurations is 0.0206.
Fig. 4. Plots of a converging beam reflecting from a fold mirror (blue) to a detector (gray) in five different configurations are shown, viewed from the same reference point. The orientation of the fold mirror rotates around the incident beam, centered on the z axis, by 0°, 45°, 90°, 135°, and 180°. The detectors of all five orientations are shown with a lighter gray in all five plots to show how the scanning mirror sweeps the converging beam in the x–y plane. The surface normals of the fold mirror in each of the five scan angles are tabulated in Fig. 1. 3238
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Fig. 7. Retardance map of the scanning fold mirror shown in Fig. 4. The maximum retardance shown is 0.631 radians.
map has the same orientation as the AoI map and is orthogonal to the diattenuation map. As shown in Fig. 2, the gold’s retardance varies quadratically with AoI. 3. Polarization Aberration of Two Fold Mirrors
To reduce the gold mirror’s polarization aberrations shown in Figs. 6 and 7, another fold mirror can be added in the reflected beam path. The polarization aberration of a single fold mirror can be compensated by a second mirror oriented such that the s-polarized ray on the first mirror becomes p-polarized on the second mirror and vice versa. This is the crossedmirror configuration. The polarization aberration for one incident angle (for example, the chief ray of the converging beam) can be completely compensated. Such a mirror pair produces a small and linearly varying polarization aberration across the converging beam about the chief ray. The polarization aberrations of the two-mirror configuration will be analyzed. The second mirror M2 scans through a converging beam reflected from the first mirror M1, as shown in Fig. 8. M1 is fixed with its normal oriented at (0, 1, 1) and reflects the converging beams to the −y direction. Then the normal of M2 rotates about the chief ray of the reflected beam by 180° from (1, 1, 0). As M2 rotates about the M1–M2 axis (y axis), M2 sweeps the reflected beam around in the x–z plane. The orientation of all the normals of M2 shown in Fig. 8 are listed in Table 2. As M2 rotates, its PoI also rotates, as shown in Fig. 9. The AoI axes of M2 shown in Fig. 9 have been propagated back to the entrance space of M1, so all the AoI maps are compared directly in the same space. The incident angles of the chief ray at both mirrors are kept at 45°. As M2 scans the converging beam around, the relative orientation of the ppolarized states of the chief ray at the two mirrors
Fig. 8. Converging beam, shown as a grid of rays, emerges from the y–z plane and reflects from a fixed mirror M1 (blue) with normal (0, 1, 1). The reflected beam reflects from a scanning mirror M2 (magenta) with varying normal tabulated in Table 2, and then reaches a detector (gray). All five configurations shown are viewed from the same reference point.
Surface Normal of M 2 in Fig. 8
Table 2.
Scan Angle Surface Normal
0° p1 2
1! 1 0
45° 1 2
! p1 2 −1
90° p1 2
0 ! 1 −1
135° 1 2
−1 ! p 2 −1
180° p1 2
−1 ! 1 0
changes from orthogonal, to parallel, and back to orthogonal. The crossed-mirror configuration occurs at the 0° and 180° scan angles, in which the planes of incidence across the beam at M1 and M2 are closest to orthogonal. To evaluate the collective polarization properties of combinations of mirrors, the cumulative polarization ray tracing P matrix is calculated for each ray path (see Appendix A). Then the diattenuation and retardance of each P matrix are obtained through data reduction [19,20]. The diattenuations for the two-mirror systems in Fig. 8 are shown in Fig. 10. The diattenuations of M1 and M2 as M2 scans are shown in the top two rows of Fig. 10. The cumulative diattenuations of the combination of M1 and M2 are shown in the bottom row. All the diattenuation axes shown are viewed from the entrance space of M1, so all the maps are directly compared. At the 90° scan
Fig. 9. AoI maps of the two fold mirrors in Fig. 8. M1 is fixed as M2 scans. The crossed-mirror configuration occurs at the 0° and 180° scan angles. The maximum incident angle shown in all the maps is 61°. In order to view the AoI maps of both mirrors in the same space, the lines in the AoI maps of M2 are propagated back to the entrance space of M1.
Fig. 10. Diattenuation maps of M1, M2, and the combination of both mirrors for the systems shown in Fig. 8. The largest diattenuation shown is 0.0226, which occurs at the 90° scan angle. 10 April 2015 / Vol. 54, No. 11 / APPLIED OPTICS
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angle, the diattenuation orientations of M1 and M2 are nearly parallel to each other, which produces a maximum magnitude of cumulative diattenuation. At the 0° and 180° scan angles, the diattenuation orientations are closest to orthogonal over the pupil, and the total diattenuation reaches a minimum. Figure 11 contains the corresponding retardance maps. The cumulative diattenuation and cumulative retardance have similar variations but are approximately orthogonal to each other. At the 0° and 180° scan angles where the maximum diattenuation and retardance cancellation occurs, the polarization contributions from the two mirrors are orthogonal with equal magnitude, producing a node of complete cancellation at the chief ray. The cumulative diattenuation and retardance at these two scan angles have their magnitudes vary linearly across the field through the null, and their orientations rotate by 90°. Such polarization aberration is termed linear diattenuation tilt and linear retardance tilt [21] where the orientation of the axes varies through 180° when moving through 360° around the pupil. As M2 scans away from 0°, the null of the cumulative polarization moves away from the chief ray. As M2 scans to 45°, the null moves to the upper left and outside of this field. As M2 scans toward 180°, the null returns to the center of the field. At the 90° scan angle, the overall polarization magnitude from M1 and M2 adds instead of compensating, so the overall aberration is approximately twice the aberration of the single mirror. The states of maximum transmission of M1 and M2 are linearly polarized at all the scan angles and correspond to linear diattenuation. However, in the cumulative aberration, a small amount of circular diattenuation and circular retardance are present and can be observed in Figs. 10 and 11 toward the edge of the pupil as slight ellipticities. Such elliptical diattenuation and elliptical retardance are described in [22] where two linear diattenuators or retarders with axes neither parallel nor perpendicular induce circular polarization. The maximum circular component is generated when
Fig. 12. Converging beam reflects from M1 (blue), the second fold mirror M2 (magenta), the third fold mirror M3 (yellow), then the fourth fold mirror M4 (green) and reaches a detector (gray). M3 and M4 are fixed with respect to each other and scan as a group around the x axis. All five configurations shown are viewed from the same reference point.
the polarization axes are 45° or 135° apart; thus, it is easiest to observe in these frames. Polarization aberrations are improved by the crossed-mirror configuration, but a linear variation remains even when their planes of incidence are orthogonal at the center. 4. Polarization Aberration of Four Fold Mirrors
Further polarization reduction can be achieved by four fold mirrors arranged in two pairs of compensating mirrors, as a double-crossed mirror system. In the systems shown in Fig. 12, M1 and M2 are fixed as the first crossed-mirror configuration with their normals oriented at (0, 1, 1) and −1; −1; 0, respectively. The orientations of the third mirror M3 and fourth mirror M4 are also fixed with respect to each other as the second crossed-mirror configuration. The chief ray of the beam reflected from M1–M2 propagates to the x direction and reflects from M3. Then, the beam is scanned around the y–z plane as M3–M4 rotates as a group pivoted at the center of M3 about the M2–M3 axis (x axis). At the 0° scan angle, the normals of M3 and M4 are oriented at (1, 1, 0) and (0, 1, 1), respectively. The orientation of the rotating M3 is shown in Table 3. In order to keep M3–M4 as a fixed arm, not only does the orientation of M4 rotate, but also its location rotates about the center of M3. Therefore, when the system scans around, the entire M3–M4-detector assembly rotates about the x axis through 180°, as shown across Fig. 12. The AoI for all four fold mirrors are mapped in Fig. 13. M1 and M2 are fixed in the crossed-mirror configuration. Since M3 and M4 are attached to each Table 3.
Scan Angle Fig. 11. Retardance maps of M1, M2, and the combination of both mirrors in the systems shown in Fig. 8. The maximum retardance shown is 0.662 radians, which occurs at the 90° scan angle. 3240
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Surface Normal
Surface Normal of M 3 in Fig. 12
0° p1 2
1! 1 0
1 2
45° p ! 2 1 1
90° p1 2
1 ! 0 −1
1 2
135° p ! 2 −1 −1
180° p1 2
1 ! −1 0
Fig. 15. Retardance maps of M1–M2 (top row) and M3–M4 (middle row) assemblies, and the cumulative retardance from all four mirrors (bottom row) of Fig. 12. The maximum retardance is 0.730 radians, which occurs for the 0° scan angle. Fig. 13. AoI maps of the four fold mirrors in Fig. 12. The largest incident angle shown for all mirrors is 61°.
other, as the PoI of M3 rotates by 180°, the PoI of M4 also rotates by 180°. During the rotation, the PoI of M3 and M4 are kept orthogonal to each other as the second crossed-mirror configuration. The retardance and diattenuation for the fourmirror system are mapped in Fig. 14 and Fig. 15. Since M1 is orthogonal to M2, and M3 is orthogonal to M4 in all scan configurations, there is a polarization null at the center of the field, and the polarizations increase linearly toward the edge of the field. At the 0° scan angle, M3–M4 and M1–M2 have the same magnitude and orientation in both the diattenuation and retardance across the field. Therefore, the cumulative polarization is twice that of the individual twomirror assembly and is 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 the 180° scan angle, the aberration maps of M1–M2 and M3–M4 are nearly orthogonal to each other across the field. Since the linear variations from the two-mirror assemblies across the field
are opposite, they compensate and the overall polarization aberration is a minimum. The residual aberration of this four-mirror combination at the 180° scan angle is quadratic and well corrected for a large field of view. A summary of the cumulative polarizations of the one-, two-, and four-fold mirror systems is tabulated in Tables 4 and 5. 5. Generalization to Other Coatings
These results for fold mirrors generalize to most reflective coatings. Depending on the coating, the diattenuation maps and retardance maps are similar in form to the maps of fold mirrors shown in this paper. For the majority of typical reflective coatings, the maps of diattenuation and retardance are similar to Figs. 6, 7, 10, 11, 14, and 15, except for a change of scale; the diattenuation maps and retardance maps are predominantly linear and quadratic variations as a function of the incident angle. For uncoated Table 4. Maximum, Average and Chief Ray Diattenuation for the One-, Two-, and Four-Fold Mirror Systems with 0.2 NA Illumination
Maximum Diattenuation Scan Angle
0°
45°
90°
135°
180°
4 mirrors 2 mirrors 1 mirror
0.0238 0.0119a 0.0176
0.0214 0.0192 0.0206
0.0170 0.0225 0.0176
0.0151 0.0192 0.0206
0.0066a 0.0119a 0.0176
Scan Angle 4 mirrors 2 mirrors 1 mirror
0° 0.0130 0.0067a 0.0110
135° 0.0061 0.0155 0.0110
180° 0.0033a 0.0067a 0.0110
Scan Angle 4 mirrors 2 mirrors 1 mirror
0° 0 0a 0.0102
135° 0 0.0144 0.0102
180° 0a 0a 0.0102
Average Diattenuation 45° 0.0120 0.0155 0.0110
90° 0.0095 0.0212 0.0110
Chief Ray Diattenuation
Fig. 14. Diattenuation maps of M1–M2 (top row) and M3–M4 (middle row) assemblies, and the cumulative diattenuation from all four mirrors (bottom row) of Fig. 12. The maximum diattenuation shown is 0.0238, which appears in the cumulative diattenuation map for the 0° scan angle.
45° 0 0.0144 0.0102
90° 0 0.0204 0.0102
a The crossed-mirror configuration of two mirrors and the double-crossed mirror configuration of four mirrors.
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Table 5. Maximum, Average, and Chief Ray Retardance (in Radians) for the One-, Two-, and Four-Fold Mirror Systems with 0.2 NA Illumination
Maximum Retardance Scan Angle
0°
45°
90°
135°
180°
4 mirrors 2 mirrors 1 mirror
0.7298 0.3650a 0.5273
0.6660 0.5701 0.6309
0.5143 0.6623 0.5273
0.4638 0.5701 0.6309
0.1525a 0.3650a 0.5273
Scan Angle 4 mirrors 2 mirrors 1 mirror
0° 0.3950 0.1987a 0.3200
135° 0.1785 0.4521 0.3202
180° 0.0833a 0.1987a 0.3200
135° 0 0.4123 0.2921
180° 0a 0a 0.2921
Average Retardance 45° 0.3644 0.4521 0.3202
90° 0.2863 0.6175 0.3200
Chief Ray Retardance Scan Angle 4 mirrors 2 mirrors 1 mirror
0° 0 0a 0.2921
45° 0 0.4123 0.2921
90° 0 0.5842 0.2921
a Crossed-mirror configuration of two mirrors and the doublecrossed mirror configuration of four mirrors.
metal mirrors, the diattenuation (Fig. 16) and retardance (Fig. 17) calculated from the Fresnel equations are similar over the range of metallic complex refractive indices. In order for spherical wavefronts at flat mirrors to generate patterns of the forms shown in Figs. 6, 7, 10, 11, 14, and 15, the following conditions apply: (1) Variations of the diattenuation or retardance need to be predominantly linear over the range of angles of
Fig. 16. Diattenuation as a function of incident angle for reflection from metals with a range of refractive indices n iκ. These real and imaginary refractive indices are chosen to span most dielectric and metal surfaces. In each case, the diattenuation has similar functional form but varying magnitude for this range of indices. At 45°, the behavior is predominantly linear and increasing except for the upper left case. Between n iκ 0.4 0.5i and 0.4 1.5i, the linear component at 45° must go to zero, and the leading term in that vicinity will be quadratic. 3242
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Fig. 17. Retardance in radians for reflection from metals as a function of incident angle for a range of refractive indices n iκ. Over this range of real and imaginary refractive indices, the retardance has similar functional form with a predominantly increasing linear component at 45°.
incidence, and (2) diattenuation magnitude or retardance magnitude needs to be much less than one. When the variations are not predominantly linear, the two-mirror aberrations will not be predominately linear, and the compensated four-mirror aberrations, Figs. 14 and 15 (both bottom row right) will not be quadratic. The aberrations of the present examples already begin to challenge these assumptions. Two examples of coatings, which do not satisfy condition (1), are shown in Fig. 18, where the diattenuation variation is not linear (top) and the retardance variation is not linear (bottom).
Fig. 18. Reflection diattenuation and retardance as a function of angle for (top two) 0.430 μm and (bottom two) 0.548 μm Al2 O3 coating with n 1.6033 on a silver substrate with n iκ 0.0314 5.308i at 765 nm. These are examples of coatings where the diattenuation variation at 45° is not linear (0.430 μm coating) or where the retardance is not linear (0.548 μm) as a function of angle.
with a quick guide to the priority of this problem and how it scales with field of view. The functional form of the polarization aberration is similar over a large range of numerical apertures, angles, and coatings, as was addressed in Section 5. Appendix A
Fig. 19. Grid of rays is traced to simulate a converging beam reflecting from two fold mirrors. The ray trace details are provided for the thicker ray shown at the edge of the field.
Regarding (1), the nonlinearities (higher-order Taylor series expansion terms) of the diattenuation and retardance cause nonlinearities in diattenuation and retardance maps. Regarding (2), as the magnitudes become larger, the interactions of diattenuation and retardance between surfaces generate elliptical retardance and diattenuation, as can be seen in the small ellipticities in the bottom rows of Figs. 10 and 11. Operating fold mirrors at larger angles of incidence, toward grazing incidence, will tend to generate such large retardance values. 6. Conclusions
Metal mirrors induce polarization changes in the reflected light, caused by diattenuation and retardance, which are calculated from Fresnel coefficients. These fold-mirror-induced polarization aberrations have been problematic in many optical systems (Section 1). To bring the optical designs within specification, some systems have added pseudo-depolarizers, while others have used a crossed fold mirror configuration. As shown here, the crossed-mirror configuration provides significant reduction in residual polarization with zero diattenuation and retardance at one point in the field of view. Using the methods presented here, the remaining polarization aberrations can be calculated and compared to specifications. The form of the crossed-mirror polarization aberration is a straightforward linear variation of diattenuation and retardance across the pupil. For many systems, this crossed-mirror configuration should meet specification. If a further polarization reduction is needed, one (of many) possibility is to combine a pair of crossed-mirror systems (Section 4). The final columns of Tables 4 and 5 show that the double-crossed fold mirror system leads to another factor of two reduction in the polarization over the field of view for the example system. As with any example, the exact values of this calculation are not as important as their magnitude and their functional form. Every system is expected to have different numerical apertures, angles, coatings, and other details. The magnitudes from this example provide an optical engineer approaching this issue
All the calculations in this paper were performed using a 3D polarization ray tracing algorithm [18–20]. An example ray trace calculation is presented in this section. Using the two fold mirror system, shown in Fig. 19, with gold coatings, a ray at the top of the field reflecting from M1 then M2 is traced. The diattenuation and retardance are calculated individually for each reflection as well as for the cumulative effect of the two reflections. At the first reflection, the incident propagation vector kˆ 1 −0.195; −0.195; p0.961 reflects from ˆ M1 with normal η1 0; 1; 1∕ 2 into kˆ 2 −0.195; −0.961; 0.195. The incident and exiting angles are both 57.184°. With Eq. (1), the incident sˆ 1 and pˆ 1 are −0.973; 0.164; −0.164 and −0.126; −0.967; −0.221, and the exiting sˆ01 and pˆ 01 are −0.973; 0.164; −0.164 and 0.126; −0.221; −0.967. The amplitude coefficients rs1 and rp1 for the first reflection are 0.992e2.918i and 0.975e−0.751i . The polarization ray tracing matrix P1 representing the first reflection for this particular ray is P1 Oout;1 ·J1 ·O−1 in;1 0 0 0 1 0 1 1 0 s1x p1x k2x s1x s1y s1z rs1 0 0 B C B C C B @ s01y p01y k2y A · @ 0 rp1 0 A · @ p1x p1y p1z A k1x k1y k1z 0 0 1 s01z p01z k2z 0 1 −0.8890.219i 0.1060.046i −0.3610.054i B C @ 0.361−0.054i 0.314−0.137i −0.863−0.039i A: −0.106−0.046i 0.655−0.628i 0.314−0.137i (A1) O−1 in;1
operates on the inciIn Eq. (A1), the matrix dent electric field E in global coordinates and calculates the (ˆs1 , pˆ 1 , kˆ 1 ) basis components for the incident light, (Es , Ep , 0), which is a projection of E onto the s–p incident local coordinates. The 3 × 3 matrix, 0 1 rs1 0 0 (A2) J1 @ 0 rp1 0 A; 0 0 1 an extended Jones matrix, applies the Fresnel amplitude and phase change to the s and p components. Then matrix Oout;1 rotates the resultant field from the local (ˆs01 , pˆ 01 , kˆ 2 ) coordinates back to the global Cartesian coordinates. In this process, the electric field E in (ˆs1 , pˆ 1 , kˆ 1 ) basis are mapped to (rs1 sˆ 01 , rp1 pˆ 01 , kˆ 2 ). Since both orthogonal matrices, Oin;1 and Oout;1 , are defined by a right-handed orthonormal basis, two of the three exiting bases, kˆ 2 and pˆ 01 , have a π phase change due to the changing local coordinate in 10 April 2015 / Vol. 54, No. 11 / APPLIED OPTICS
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reflection, where kˆ 2 is understood as reflection and the π phase in pˆ 01 is added to Argrp1 , as in Eq. (3). The diattenuation and retardance associated with the P matrix are calculated by the algorithms described in [19,20]. Diattenuation is calculated by the singular values of P. The singular value decomposition of P1, X † V1 P1 U1 0
1
−0.195 −0.973ei2.918 0.126e−i0.750
1
The Fresnel reflection coefficients rs2 and rp2 for the second reflections are 0.988e2.80i and 0.982e−0.507i . Then, −0.3520.081i −0.855−0.028i 0.365−0.056i P2 0.792−0.450i −0.3520.081i 0.054−0.052i −0.0540.052i −0.3650.056i −0.8530.327i and Q2
B C @ −0.961 0.164ei2.918 −0.221e−i0.750 A 0.195 −0.164ei2.918 −0.967e−i0.750 1 0 1 1 0 0 −0.195 −0.973 −0.126 † B C B C · @ 0 0.992 0 A · @ −0.195 0.164 −0.967 A ; 0
0
0
0.975
0.961 −0.164 −0.221 (A3)
has singular values 0.992 and 0.975, which correspond to jrs j and jrp j in Eq. (2). So the diattenuation is 0.018. Its maximum transmission axis is −0.973; 0.164; −0.164 which is the second column of V1 and sˆ 1 in the entrance space, or e2.918i −0.973; 0.164; −0.164 which is the second column of U1 and e2.918i sˆ 01 in the space following the M1. To calculate the retardance induced by M1, the geometrical transformation of the ray path is first removed from P1 ; this in essence straightens out the path. The geometrical transformation matrix Q1 for M1 [20] is Q1 Oout;1 · Ireflect1 · O−1 in;1 0 0 1 0 1 1 0 0 s1x p1x k2x s1x s1y s1z 1 0 0 B C B C C B @ s01y p01y k2y A · @ 0 −1 0 A · @ p1x p1y p1z A k1x k1y k1z 0 0 1 s01z p01z k2z 0 1 1 0 0 B C @ 0 0 −1 A: (A4) 0 −1
0
Q1 describes a nonpolarizing reflection. The M1 retardance is calculated from the eigenvalues of −1 −1 U1Q · V−1 1Q , where Q1 · P1 U1Q · Σ1Q · V1Q by singular value decomposition. The phase difference between the eigenvalues is 0.527 radians, the M1 retardance magnitude. The eigenvector corresponding to the smaller eigenphase is (0.126, 0.967, 0.221), which is the fast axis and p-polarization in the entrance space. For the second reflection, the P2 matrix and the corresponding diattenuation and retardance are calculated in the same manner. The surface normal of p M2 is ηˆ 2 −1; −1; 0∕ 2. The reflected kˆ 3 is (0.961, 0.195, 0.195) and the reflection angle is 35.16°. The incident sˆ 2 and pˆ 2 are 0.239; −0.239; −0.941 and 0.951; −0.137; 0.277. The exiting sˆ 02 and pˆ 02 are 0.239; −0.239; −0.941 and −0.137; 0.951; −0.277. 3244
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0 −1 0
−1 0 0
!
! 0 0 : 1
The corresponding diattenuation is 0.006 with maximum reflection axis 0.239; −0.239; −0.941 along s-polarization, and the retardance is 0.169 radians with the fast axis 0.951; −0.137; 0.277 along p-polarization in the entrance space of M2. Combining the effects from both mirrors, the cumulative retardance and diattenuation are obtained from the cumulative P matrix, which is calculated by cascading P1 and P2 through matrix multiplication: P P2 · P1
−0.056 − 0.124i −0.109 − 0.165i 0.967 − 0.059i ! −0.737 0.625i 0.008 − 0.006i 0.055 0.125i 0.013 − 0.013i −0.468 0.820i 0.110 0.163i
and
0
0 Q Q2 · Q2 @ −1 0
0 0 −1
1 1 0 A: 0
Applying the same method as described previously, the overall diattenuation is 0.012 with maximum transmission axis 0.977e−2.447i ; 0.118e0.445i ; 0.175e−2.414i , and retardance is 0.365 radians with fast axis 0.076e0.186i ; 0.974; 0.213e0.013i in the entrance space. This algorithm is used for all polarization ray trace calculations reported in this paper. References 1. J. B. Breckinridge, “Self-induced polarization anisoplanatism,” Proc. SPIE 8860, 886012 (2013). 2. J. C. Carson, B. D. Kern, J. T. Trauger, and J. B. Breckinridge, “The effects of instrumental elliptical polarization on stellar point spread function fine structure,” Proc. SPIE 6265, 62653M (2006). 3. J. B. Breckinridge and B. R. Oppenheimer, “Polarization effects in reflecting coronagraphs for white-light applications in astronomy,” Astrophys. J. 600, 1091–1098 (2004). 4. P. Drude, “Zur Elektronentheorie der Metalle,” Ann. Phys. 306, 566–613 (1900). 5. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), Chap. 14. 6. J. Sasián, “Polarization fields and wavefronts of two sheets for understanding polarization aberrations in optical imaging systems,” Opt. Eng. 53, 035102 (2014). 7. D. J. Reiley and R. A. Chipman, “Coating-induced wave-front aberrations: on-axis astigmatism and chromatic aberration in all-reflecting systems,” Appl. Opt. 33, 2002–2012 (1994).
8. G. Anzolin, A. Gardelein, M. Jofre, G. Molina-Terriza, and M. W. Mitchell, “Polarization change induced by a galvanometric optical scanner,” J. Opt. Soc. Am. A 27, 1946–1952 (2010). 9. M. Jofre, G. Anzolin, F. Steinlechner, N. Oliverio, J. P. Torres, V. Pruneri, and M. W. Mitchell, “Fast beam steering with full polarization control using a galvanometric optical scanner and polarization controller,” Opt. Express 20, 12247–12260 (2012). 10. R. A. Chipman, D. M. Brown, and J. P. McGuire, “Polarization aberration analysis of the advanced x-ray astrophysics facility telescope assembly,” Appl. Opt. 31, 2301–2313 (1992). 11. L. R. Bissonnette, G. Roy, and F. Fabry, “Range–height scans of lidar depolarization for characterizing properties and phase of clouds and precipitation,” J. Atmos. Ocean. Technol. 18, 1429–1446 (2001). 12. S. C. McClain, P. W. Maymon, and R. A. Chipman, “Design and analysis of a depolarizer for the NASA MODerate-resolution imaging spectrometer-tilt (MODIS-T),” Proc. SPIE 1746, 375–385 (1992). 13. J. P. McGuire, Jr. and R. A. Chipman, “Analysis of spatial pseudodepolarizers in imaging systems,” Opt. Eng. 29, 1478–1484 (1990).
14. L. L. Smith and P. M. Koch, “Use of four mirrors to rotate linear polarization but preserve input–output collinearity,” J. Opt. Soc. Am. A 13, 2102–2105 (1996). 15. E. J. Galvez and P. M. Koch, “Use of four mirrors to rotate linear polarization but preserve input–output collinearity. II,” J. Opt. Soc. Am. A 14, 3410–3414 (1997). 16. E. J. Galvez, M. R. Cheyne, J. B. Stewart, C. D. Holmes, and H. I. Sztul, “Variable geometric-phase polarization rotators for the visible,” Opt. Commun. 171, 7–13 (1999). 17. D. W. Lynch and W. R. Hunter, “Gold (Au),” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, 1985). 18. R. A. Chipman, “Mechanics of polarization ray tracing,” Proc. SPIE 1746, 62 (1992). 19. G. Yun, K. Crabtree, and R. A. Chipman, “Three-dimensional polarization ray-tracing calculus I: definition and diattenuation,” Appl. Opt. 50, 2855–2865 (2011). 20. G. Yun, S. C. McClain, and R. A. Chipman, “Threedimensional polarization ray-tracing calculus II: retardance,” Appl. Opt. 50, 2866–2874 (2011). 21. R. A. Chipman, “Polarization aberrations,” Ph.D. dissertation (The University of Arizona, 1987). 22. D. B. Chenault and R. A. Chipman, “Measurements of linear diattenuation and linear retardance spectra with a rotating sample spectropolarimeter,” Appl. Opt. 32, 3513–3519 (1993).
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The polarization aberrations of a fold mirror can be compensated by orienting a second fold mirror’s p-polarization with the first mirror’s s-polarization. This crossed-mirror configuration compensates the polarization for a single angle to zero and leaves a linear variation of diattenuation and retardance for a spherical wavefront. Two sets of crossed fold mirrors when properly oriented compensate the remaining linear variation and leave a much reduced quadratic variation in a large compensated field of view. © 2015 Optical Society of America OCIS codes: (080.0080) Geometric optics; (080.1010) Aberrations (global); (260.0260) Physical optics; (260.5430) Polarization; (310.3915) Metallic, opaque, and absorbing coatings; (310.5448) Polarization, other optical properties. http://dx.doi.org/10.1364/AO.54.003236
1. Introduction
Fold mirrors are common optical components used in many complex optical systems. For example, space telescopes use scanning fold mirrors to direct light from different directions to detectors. Many systems use multiple fold mirrors for compactness [1–3]. Metal mirrors and thin-film-coated mirrors induce polarization variations, in both diattenuation and retardance, as a function of incident angle [4–8]. These polarization aberrations are readily observed by illuminating fold mirrors with converging or diverging light and observing the leakage between crossed polarizers or by Mueller matrix imaging polarimetry [9]. The mirrors reflect more s- than p-polarized light. So one longstanding issue with fold mirrors in spectroradiometers is that the system measures an irradiance higher or lower than the actual irradiance, depending on the incident polarization state. These polarization aberrations have been an annoyance in many applications such as liquid crystal projectors and microlithography. They have also been a 1559-128X/15/113236-10$15.00/0 © 2015 Optical Society of America 3236
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particular problem in remote sensing spectrometers, which seek to accurately measure irradiance of a scene without bias toward any particular polarization state [10,11]. Pseudo-depolarizers are one solution and have been used to reduce calibration problems due to unequal s- and p-polarized output in the NASA MODIS-T spectrometer [12,13]. Another way to reduce polarization aberration across the field, as described in this paper, is to compensate the polarization aberration between fold mirrors [14–16]. In the following discussion, the polarization properties will be described in terms of diattenuation and retardance. Upon reflection from mirrors, there is a reflected electric field difference in amplitude and phase between s- and p-polarized light. The orientation of the s- and p-polarized components is a function of the light propagation direction kˆ and the orientation of the reflecting surface, sˆ
kˆ × ηˆ jkˆ × ηˆ j
and pˆ kˆ × s;ˆ
(1)
where ηˆ is the surface normal of the interface defined as pointing toward the transmitted medium.
The electric field oscillation of p-polarized light is in the plane of incidence (PoI), while s-polarized light is orthogonal to the PoI. Diattenuation D and retardance δ of a mirror are defined as D
jrs j2 − jrp j2 jrs j2 jrp j2
and δ ϕrp − ϕrs ;
(2)
where rs and rp are the Fresnel reflection coefficients for s- and p-polarized light reflected from a mirror or the corresponding coefficients calculated for a thin film coated interface. The phase shifts of the reflected s- and p-polarized light are ϕs Argrs
and ϕp π Argrp ;
(3)
where Arga is the argument of complex number a. The phase shift of p-polarized light has an additional factor of π due to the standard definition of the Fresnel coefficients, which incorporate a coordinate change upon reflection. The complex amplitude reflection coefficients for gold at 765 nm with refractive index 0.17183 4.7493i [17] are plotted in Fig. 1 as a function of angle. At normal incidence (θ 0°), s- and p-polarized light are degenerate, so their amplitude and phase change upon reflection are identical. The corresponding D and δ are shown in Fig. 2, where both D and δ increases approximately quadratically with incident angles. Diattenuation and retardance can be represented graphically with line segments indicating magnitude and orientation. The orientation of diattenuation corresponds to the orientation of the polarization state with the maximum transmission. The s-polarized light has a higher reflection than p-polarized light,
so the orientation of reflection diattenuation is along the s-polarization state. The orientation of retardance is the fast axis, which corresponds to the polarization state with the smaller phase shift. In the metal reflection shown in Fig. 1, the fast axis corresponds to the p-polarized light, which has a smaller phase delay. Consider a converging beam reflecting from a 45° fold mirror, as shown in Fig. 3 (left) and (middle), the orientations of the s- and p-polarized components rotate slightly from one side of the field to the other side of the field due to the changing PoI of the converging beam at the flat mirror. Due to the changing incident angle across the converging field, the diattenuation and retardance induced by the fold mirror vary across the field. The orientation of polarization has a dependence on the PoI orientation, which allows the polarization variation across the field to be adjusted. By rotating the fold mirror by 20°, as shown in Fig. 3 (right), the chief ray of the same converging beam in Fig. 3 (middle) propagates at an angle 20° to the y axis. Since the PoI rotates with the orientation of the fold mirror, the orientations of the s- and p-polarization states also rotate. Therefore, the amount of polarization variation across the field can be compensated with more than one mirror. A way to visualize and understand these rotations of polarization states and the compensation of polarization between multiple mirrors in three-dimensions is to project the states and polarization properties into the entrance space. A 3D polarization ray tracing algorithm [18–20] is employed to ray trace a converging beam through systems of one, two, and four fold mirrors. The ray tracing algorithm is demonstrated in the Appendix A for two fold mirrors. The polarization aberrations of these systems for a converging beam are considered in the following sections, and the magnitude and orientation of the resultant aberrations across the
Fig. 1. Fresnel reflection coefficients: amplitude (left) and phase (right) as a function of incident angle for an air/gold interface at 765 nm. Red for s-polarized light and blue for p-polarized light.
Fig. 2. Reflection diattenuation and retardance as a function of incident angle at 765 nm for an air/gold interface.
Fig. 3. Converging beam is simulated as a grid of rays. The chief ray of the beam reflects from a fold mirror at 45°. The s- and p-polarization components are represented by the red and blue lines at each ray. Left and middle show two different views for the fold mirror whose surface normal (green arrow) is oriented at (0, 1, 1). With the same viewing angle as middle, right shows the same grid of incident rays reflecting from a rotated fold mirror. This mirror’s surface normal has rotated about the z axis at the center of the mirror by 20° from (0, 1, 1) (dashed green arrow) to −0.242; 0.664; 0.707 (solid green arrow). 10 April 2015 / Vol. 54, No. 11 / APPLIED OPTICS
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beams will be compared. 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.” Similar interpretations apply to the terms quadratic and cubic. 2. Polarization Aberration of One Fold Mirror
Consider a flat mirror, labeled as M1, reflecting a 0.2 NA converging beam, as shown in Fig. 4. The beam is scanned by rotating the mirror about the z axis, while the chief ray maintains a 45° angle of incidence. The converging beam is simulated by a grid of rays evenly sampling the converging wavefront. The incident beam starts at the x–y plane propagating in the positive z direction. Figure 4 shows that the mirror is scanning about the z axis by 180° to five different orientations. As the mirror scans, it sweeps the converging beam around in the x–y plane about the z axis. The orientations of the surface normal of M1 are tabulated in Table 1 for the scan angles of Fig. 4. As M1 scans about the incident beam, the PoI of each ray rotates; this is shown in the angle of incidence (AoI) map in Fig. 5. This AoI map of M1 is viewed from the entrance space. The length of each line segment in the map represents the magnitude of the AoI. The orientation of a line indicates the
Surface Normal of M 1 in Fig. 4
Table 1.
Scan Angle Surface Normal
0° p1 2
0! 1 1
45° 1 2
1 ! p1 2
90° p1 2
1! 0 1
135° 1 2
1 ! −1 p 2
180° p1 2
0 ! −1 1
orientation of PoI, which is also the orientation of the p-polarized component for each incident ray. A shorter line corresponds to a ray closer to normal incidence. The chief ray’s AoI is 45°. The AoI increases linearly from one side of the pupil to the other, while in the perpendicular direction of the field, the PoI rotates linearly from one side to the other. As M1 scans around, this AoI pattern simply rotates. The diattenuation for each ray path is plotted in Fig. 6, where the length of each line represents the diattenuation magnitude, and the line’s orientation corresponds to the orientation of the maximum transmission state viewed from the entrance space. Since the maximum transmission state is the s-state, the orientation in the diattenuation map is orthogonal to the orientation in the AoI map. The magnitude of diattenuation increases quadratically with AoI from normal incidence as shown in Fig. 2 (left). The variations in retardance resulting from scanning M1 are shown in the retardance maps in Fig. 7. The length of the line segment represents the retardance magnitude, and the line’s orientation is the fast axis orientation. For the gold mirror, ϕrp < ϕrs , so the reflective fast axis is p-polarized. The retardance
Fig. 5. AoI map of the one fold mirror system shown in Fig. 4. The largest incident angle shown in all the frames is 61°.
Fig. 6. Diattenuation map of the scanning fold mirror in Fig. 4. The maximum diattenuation among all five configurations is 0.0206.
Fig. 4. Plots of a converging beam reflecting from a fold mirror (blue) to a detector (gray) in five different configurations are shown, viewed from the same reference point. The orientation of the fold mirror rotates around the incident beam, centered on the z axis, by 0°, 45°, 90°, 135°, and 180°. The detectors of all five orientations are shown with a lighter gray in all five plots to show how the scanning mirror sweeps the converging beam in the x–y plane. The surface normals of the fold mirror in each of the five scan angles are tabulated in Fig. 1. 3238
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Fig. 7. Retardance map of the scanning fold mirror shown in Fig. 4. The maximum retardance shown is 0.631 radians.
map has the same orientation as the AoI map and is orthogonal to the diattenuation map. As shown in Fig. 2, the gold’s retardance varies quadratically with AoI. 3. Polarization Aberration of Two Fold Mirrors
To reduce the gold mirror’s polarization aberrations shown in Figs. 6 and 7, another fold mirror can be added in the reflected beam path. The polarization aberration of a single fold mirror can be compensated by a second mirror oriented such that the s-polarized ray on the first mirror becomes p-polarized on the second mirror and vice versa. This is the crossedmirror configuration. The polarization aberration for one incident angle (for example, the chief ray of the converging beam) can be completely compensated. Such a mirror pair produces a small and linearly varying polarization aberration across the converging beam about the chief ray. The polarization aberrations of the two-mirror configuration will be analyzed. The second mirror M2 scans through a converging beam reflected from the first mirror M1, as shown in Fig. 8. M1 is fixed with its normal oriented at (0, 1, 1) and reflects the converging beams to the −y direction. Then the normal of M2 rotates about the chief ray of the reflected beam by 180° from (1, 1, 0). As M2 rotates about the M1–M2 axis (y axis), M2 sweeps the reflected beam around in the x–z plane. The orientation of all the normals of M2 shown in Fig. 8 are listed in Table 2. As M2 rotates, its PoI also rotates, as shown in Fig. 9. The AoI axes of M2 shown in Fig. 9 have been propagated back to the entrance space of M1, so all the AoI maps are compared directly in the same space. The incident angles of the chief ray at both mirrors are kept at 45°. As M2 scans the converging beam around, the relative orientation of the ppolarized states of the chief ray at the two mirrors
Fig. 8. Converging beam, shown as a grid of rays, emerges from the y–z plane and reflects from a fixed mirror M1 (blue) with normal (0, 1, 1). The reflected beam reflects from a scanning mirror M2 (magenta) with varying normal tabulated in Table 2, and then reaches a detector (gray). All five configurations shown are viewed from the same reference point.
Surface Normal of M 2 in Fig. 8
Table 2.
Scan Angle Surface Normal
0° p1 2
1! 1 0
45° 1 2
! p1 2 −1
90° p1 2
0 ! 1 −1
135° 1 2
−1 ! p 2 −1
180° p1 2
−1 ! 1 0
changes from orthogonal, to parallel, and back to orthogonal. The crossed-mirror configuration occurs at the 0° and 180° scan angles, in which the planes of incidence across the beam at M1 and M2 are closest to orthogonal. To evaluate the collective polarization properties of combinations of mirrors, the cumulative polarization ray tracing P matrix is calculated for each ray path (see Appendix A). Then the diattenuation and retardance of each P matrix are obtained through data reduction [19,20]. The diattenuations for the two-mirror systems in Fig. 8 are shown in Fig. 10. The diattenuations of M1 and M2 as M2 scans are shown in the top two rows of Fig. 10. The cumulative diattenuations of the combination of M1 and M2 are shown in the bottom row. All the diattenuation axes shown are viewed from the entrance space of M1, so all the maps are directly compared. At the 90° scan
Fig. 9. AoI maps of the two fold mirrors in Fig. 8. M1 is fixed as M2 scans. The crossed-mirror configuration occurs at the 0° and 180° scan angles. The maximum incident angle shown in all the maps is 61°. In order to view the AoI maps of both mirrors in the same space, the lines in the AoI maps of M2 are propagated back to the entrance space of M1.
Fig. 10. Diattenuation maps of M1, M2, and the combination of both mirrors for the systems shown in Fig. 8. The largest diattenuation shown is 0.0226, which occurs at the 90° scan angle. 10 April 2015 / Vol. 54, No. 11 / APPLIED OPTICS
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angle, the diattenuation orientations of M1 and M2 are nearly parallel to each other, which produces a maximum magnitude of cumulative diattenuation. At the 0° and 180° scan angles, the diattenuation orientations are closest to orthogonal over the pupil, and the total diattenuation reaches a minimum. Figure 11 contains the corresponding retardance maps. The cumulative diattenuation and cumulative retardance have similar variations but are approximately orthogonal to each other. At the 0° and 180° scan angles where the maximum diattenuation and retardance cancellation occurs, the polarization contributions from the two mirrors are orthogonal with equal magnitude, producing a node of complete cancellation at the chief ray. The cumulative diattenuation and retardance at these two scan angles have their magnitudes vary linearly across the field through the null, and their orientations rotate by 90°. Such polarization aberration is termed linear diattenuation tilt and linear retardance tilt [21] where the orientation of the axes varies through 180° when moving through 360° around the pupil. As M2 scans away from 0°, the null of the cumulative polarization moves away from the chief ray. As M2 scans to 45°, the null moves to the upper left and outside of this field. As M2 scans toward 180°, the null returns to the center of the field. At the 90° scan angle, the overall polarization magnitude from M1 and M2 adds instead of compensating, so the overall aberration is approximately twice the aberration of the single mirror. The states of maximum transmission of M1 and M2 are linearly polarized at all the scan angles and correspond to linear diattenuation. However, in the cumulative aberration, a small amount of circular diattenuation and circular retardance are present and can be observed in Figs. 10 and 11 toward the edge of the pupil as slight ellipticities. Such elliptical diattenuation and elliptical retardance are described in [22] where two linear diattenuators or retarders with axes neither parallel nor perpendicular induce circular polarization. The maximum circular component is generated when
Fig. 12. Converging beam reflects from M1 (blue), the second fold mirror M2 (magenta), the third fold mirror M3 (yellow), then the fourth fold mirror M4 (green) and reaches a detector (gray). M3 and M4 are fixed with respect to each other and scan as a group around the x axis. All five configurations shown are viewed from the same reference point.
the polarization axes are 45° or 135° apart; thus, it is easiest to observe in these frames. Polarization aberrations are improved by the crossed-mirror configuration, but a linear variation remains even when their planes of incidence are orthogonal at the center. 4. Polarization Aberration of Four Fold Mirrors
Further polarization reduction can be achieved by four fold mirrors arranged in two pairs of compensating mirrors, as a double-crossed mirror system. In the systems shown in Fig. 12, M1 and M2 are fixed as the first crossed-mirror configuration with their normals oriented at (0, 1, 1) and −1; −1; 0, respectively. The orientations of the third mirror M3 and fourth mirror M4 are also fixed with respect to each other as the second crossed-mirror configuration. The chief ray of the beam reflected from M1–M2 propagates to the x direction and reflects from M3. Then, the beam is scanned around the y–z plane as M3–M4 rotates as a group pivoted at the center of M3 about the M2–M3 axis (x axis). At the 0° scan angle, the normals of M3 and M4 are oriented at (1, 1, 0) and (0, 1, 1), respectively. The orientation of the rotating M3 is shown in Table 3. In order to keep M3–M4 as a fixed arm, not only does the orientation of M4 rotate, but also its location rotates about the center of M3. Therefore, when the system scans around, the entire M3–M4-detector assembly rotates about the x axis through 180°, as shown across Fig. 12. The AoI for all four fold mirrors are mapped in Fig. 13. M1 and M2 are fixed in the crossed-mirror configuration. Since M3 and M4 are attached to each Table 3.
Scan Angle Fig. 11. Retardance maps of M1, M2, and the combination of both mirrors in the systems shown in Fig. 8. The maximum retardance shown is 0.662 radians, which occurs at the 90° scan angle. 3240
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Surface Normal
Surface Normal of M 3 in Fig. 12
0° p1 2
1! 1 0
1 2
45° p ! 2 1 1
90° p1 2
1 ! 0 −1
1 2
135° p ! 2 −1 −1
180° p1 2
1 ! −1 0
Fig. 15. Retardance maps of M1–M2 (top row) and M3–M4 (middle row) assemblies, and the cumulative retardance from all four mirrors (bottom row) of Fig. 12. The maximum retardance is 0.730 radians, which occurs for the 0° scan angle. Fig. 13. AoI maps of the four fold mirrors in Fig. 12. The largest incident angle shown for all mirrors is 61°.
other, as the PoI of M3 rotates by 180°, the PoI of M4 also rotates by 180°. During the rotation, the PoI of M3 and M4 are kept orthogonal to each other as the second crossed-mirror configuration. The retardance and diattenuation for the fourmirror system are mapped in Fig. 14 and Fig. 15. Since M1 is orthogonal to M2, and M3 is orthogonal to M4 in all scan configurations, there is a polarization null at the center of the field, and the polarizations increase linearly toward the edge of the field. At the 0° scan angle, M3–M4 and M1–M2 have the same magnitude and orientation in both the diattenuation and retardance across the field. Therefore, the cumulative polarization is twice that of the individual twomirror assembly and is 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 the 180° scan angle, the aberration maps of M1–M2 and M3–M4 are nearly orthogonal to each other across the field. Since the linear variations from the two-mirror assemblies across the field
are opposite, they compensate and the overall polarization aberration is a minimum. The residual aberration of this four-mirror combination at the 180° scan angle is quadratic and well corrected for a large field of view. A summary of the cumulative polarizations of the one-, two-, and four-fold mirror systems is tabulated in Tables 4 and 5. 5. Generalization to Other Coatings
These results for fold mirrors generalize to most reflective coatings. Depending on the coating, the diattenuation maps and retardance maps are similar in form to the maps of fold mirrors shown in this paper. For the majority of typical reflective coatings, the maps of diattenuation and retardance are similar to Figs. 6, 7, 10, 11, 14, and 15, except for a change of scale; the diattenuation maps and retardance maps are predominantly linear and quadratic variations as a function of the incident angle. For uncoated Table 4. Maximum, Average and Chief Ray Diattenuation for the One-, Two-, and Four-Fold Mirror Systems with 0.2 NA Illumination
Maximum Diattenuation Scan Angle
0°
45°
90°
135°
180°
4 mirrors 2 mirrors 1 mirror
0.0238 0.0119a 0.0176
0.0214 0.0192 0.0206
0.0170 0.0225 0.0176
0.0151 0.0192 0.0206
0.0066a 0.0119a 0.0176
Scan Angle 4 mirrors 2 mirrors 1 mirror
0° 0.0130 0.0067a 0.0110
135° 0.0061 0.0155 0.0110
180° 0.0033a 0.0067a 0.0110
Scan Angle 4 mirrors 2 mirrors 1 mirror
0° 0 0a 0.0102
135° 0 0.0144 0.0102
180° 0a 0a 0.0102
Average Diattenuation 45° 0.0120 0.0155 0.0110
90° 0.0095 0.0212 0.0110
Chief Ray Diattenuation
Fig. 14. Diattenuation maps of M1–M2 (top row) and M3–M4 (middle row) assemblies, and the cumulative diattenuation from all four mirrors (bottom row) of Fig. 12. The maximum diattenuation shown is 0.0238, which appears in the cumulative diattenuation map for the 0° scan angle.
45° 0 0.0144 0.0102
90° 0 0.0204 0.0102
a The crossed-mirror configuration of two mirrors and the double-crossed mirror configuration of four mirrors.
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Table 5. Maximum, Average, and Chief Ray Retardance (in Radians) for the One-, Two-, and Four-Fold Mirror Systems with 0.2 NA Illumination
Maximum Retardance Scan Angle
0°
45°
90°
135°
180°
4 mirrors 2 mirrors 1 mirror
0.7298 0.3650a 0.5273
0.6660 0.5701 0.6309
0.5143 0.6623 0.5273
0.4638 0.5701 0.6309
0.1525a 0.3650a 0.5273
Scan Angle 4 mirrors 2 mirrors 1 mirror
0° 0.3950 0.1987a 0.3200
135° 0.1785 0.4521 0.3202
180° 0.0833a 0.1987a 0.3200
135° 0 0.4123 0.2921
180° 0a 0a 0.2921
Average Retardance 45° 0.3644 0.4521 0.3202
90° 0.2863 0.6175 0.3200
Chief Ray Retardance Scan Angle 4 mirrors 2 mirrors 1 mirror
0° 0 0a 0.2921
45° 0 0.4123 0.2921
90° 0 0.5842 0.2921
a Crossed-mirror configuration of two mirrors and the doublecrossed mirror configuration of four mirrors.
metal mirrors, the diattenuation (Fig. 16) and retardance (Fig. 17) calculated from the Fresnel equations are similar over the range of metallic complex refractive indices. In order for spherical wavefronts at flat mirrors to generate patterns of the forms shown in Figs. 6, 7, 10, 11, 14, and 15, the following conditions apply: (1) Variations of the diattenuation or retardance need to be predominantly linear over the range of angles of
Fig. 16. Diattenuation as a function of incident angle for reflection from metals with a range of refractive indices n iκ. These real and imaginary refractive indices are chosen to span most dielectric and metal surfaces. In each case, the diattenuation has similar functional form but varying magnitude for this range of indices. At 45°, the behavior is predominantly linear and increasing except for the upper left case. Between n iκ 0.4 0.5i and 0.4 1.5i, the linear component at 45° must go to zero, and the leading term in that vicinity will be quadratic. 3242
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Fig. 17. Retardance in radians for reflection from metals as a function of incident angle for a range of refractive indices n iκ. Over this range of real and imaginary refractive indices, the retardance has similar functional form with a predominantly increasing linear component at 45°.
incidence, and (2) diattenuation magnitude or retardance magnitude needs to be much less than one. When the variations are not predominantly linear, the two-mirror aberrations will not be predominately linear, and the compensated four-mirror aberrations, Figs. 14 and 15 (both bottom row right) will not be quadratic. The aberrations of the present examples already begin to challenge these assumptions. Two examples of coatings, which do not satisfy condition (1), are shown in Fig. 18, where the diattenuation variation is not linear (top) and the retardance variation is not linear (bottom).
Fig. 18. Reflection diattenuation and retardance as a function of angle for (top two) 0.430 μm and (bottom two) 0.548 μm Al2 O3 coating with n 1.6033 on a silver substrate with n iκ 0.0314 5.308i at 765 nm. These are examples of coatings where the diattenuation variation at 45° is not linear (0.430 μm coating) or where the retardance is not linear (0.548 μm) as a function of angle.
with a quick guide to the priority of this problem and how it scales with field of view. The functional form of the polarization aberration is similar over a large range of numerical apertures, angles, and coatings, as was addressed in Section 5. Appendix A
Fig. 19. Grid of rays is traced to simulate a converging beam reflecting from two fold mirrors. The ray trace details are provided for the thicker ray shown at the edge of the field.
Regarding (1), the nonlinearities (higher-order Taylor series expansion terms) of the diattenuation and retardance cause nonlinearities in diattenuation and retardance maps. Regarding (2), as the magnitudes become larger, the interactions of diattenuation and retardance between surfaces generate elliptical retardance and diattenuation, as can be seen in the small ellipticities in the bottom rows of Figs. 10 and 11. Operating fold mirrors at larger angles of incidence, toward grazing incidence, will tend to generate such large retardance values. 6. Conclusions
Metal mirrors induce polarization changes in the reflected light, caused by diattenuation and retardance, which are calculated from Fresnel coefficients. These fold-mirror-induced polarization aberrations have been problematic in many optical systems (Section 1). To bring the optical designs within specification, some systems have added pseudo-depolarizers, while others have used a crossed fold mirror configuration. As shown here, the crossed-mirror configuration provides significant reduction in residual polarization with zero diattenuation and retardance at one point in the field of view. Using the methods presented here, the remaining polarization aberrations can be calculated and compared to specifications. The form of the crossed-mirror polarization aberration is a straightforward linear variation of diattenuation and retardance across the pupil. For many systems, this crossed-mirror configuration should meet specification. If a further polarization reduction is needed, one (of many) possibility is to combine a pair of crossed-mirror systems (Section 4). The final columns of Tables 4 and 5 show that the double-crossed fold mirror system leads to another factor of two reduction in the polarization over the field of view for the example system. As with any example, the exact values of this calculation are not as important as their magnitude and their functional form. Every system is expected to have different numerical apertures, angles, coatings, and other details. The magnitudes from this example provide an optical engineer approaching this issue
All the calculations in this paper were performed using a 3D polarization ray tracing algorithm [18–20]. An example ray trace calculation is presented in this section. Using the two fold mirror system, shown in Fig. 19, with gold coatings, a ray at the top of the field reflecting from M1 then M2 is traced. The diattenuation and retardance are calculated individually for each reflection as well as for the cumulative effect of the two reflections. At the first reflection, the incident propagation vector kˆ 1 −0.195; −0.195; p0.961 reflects from ˆ M1 with normal η1 0; 1; 1∕ 2 into kˆ 2 −0.195; −0.961; 0.195. The incident and exiting angles are both 57.184°. With Eq. (1), the incident sˆ 1 and pˆ 1 are −0.973; 0.164; −0.164 and −0.126; −0.967; −0.221, and the exiting sˆ01 and pˆ 01 are −0.973; 0.164; −0.164 and 0.126; −0.221; −0.967. The amplitude coefficients rs1 and rp1 for the first reflection are 0.992e2.918i and 0.975e−0.751i . The polarization ray tracing matrix P1 representing the first reflection for this particular ray is P1 Oout;1 ·J1 ·O−1 in;1 0 0 0 1 0 1 1 0 s1x p1x k2x s1x s1y s1z rs1 0 0 B C B C C B @ s01y p01y k2y A · @ 0 rp1 0 A · @ p1x p1y p1z A k1x k1y k1z 0 0 1 s01z p01z k2z 0 1 −0.8890.219i 0.1060.046i −0.3610.054i B C @ 0.361−0.054i 0.314−0.137i −0.863−0.039i A: −0.106−0.046i 0.655−0.628i 0.314−0.137i (A1) O−1 in;1
operates on the inciIn Eq. (A1), the matrix dent electric field E in global coordinates and calculates the (ˆs1 , pˆ 1 , kˆ 1 ) basis components for the incident light, (Es , Ep , 0), which is a projection of E onto the s–p incident local coordinates. The 3 × 3 matrix, 0 1 rs1 0 0 (A2) J1 @ 0 rp1 0 A; 0 0 1 an extended Jones matrix, applies the Fresnel amplitude and phase change to the s and p components. Then matrix Oout;1 rotates the resultant field from the local (ˆs01 , pˆ 01 , kˆ 2 ) coordinates back to the global Cartesian coordinates. In this process, the electric field E in (ˆs1 , pˆ 1 , kˆ 1 ) basis are mapped to (rs1 sˆ 01 , rp1 pˆ 01 , kˆ 2 ). Since both orthogonal matrices, Oin;1 and Oout;1 , are defined by a right-handed orthonormal basis, two of the three exiting bases, kˆ 2 and pˆ 01 , have a π phase change due to the changing local coordinate in 10 April 2015 / Vol. 54, No. 11 / APPLIED OPTICS
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reflection, where kˆ 2 is understood as reflection and the π phase in pˆ 01 is added to Argrp1 , as in Eq. (3). The diattenuation and retardance associated with the P matrix are calculated by the algorithms described in [19,20]. Diattenuation is calculated by the singular values of P. The singular value decomposition of P1, X † V1 P1 U1 0
1
−0.195 −0.973ei2.918 0.126e−i0.750
1
The Fresnel reflection coefficients rs2 and rp2 for the second reflections are 0.988e2.80i and 0.982e−0.507i . Then, −0.3520.081i −0.855−0.028i 0.365−0.056i P2 0.792−0.450i −0.3520.081i 0.054−0.052i −0.0540.052i −0.3650.056i −0.8530.327i and Q2
B C @ −0.961 0.164ei2.918 −0.221e−i0.750 A 0.195 −0.164ei2.918 −0.967e−i0.750 1 0 1 1 0 0 −0.195 −0.973 −0.126 † B C B C · @ 0 0.992 0 A · @ −0.195 0.164 −0.967 A ; 0
0
0
0.975
0.961 −0.164 −0.221 (A3)
has singular values 0.992 and 0.975, which correspond to jrs j and jrp j in Eq. (2). So the diattenuation is 0.018. Its maximum transmission axis is −0.973; 0.164; −0.164 which is the second column of V1 and sˆ 1 in the entrance space, or e2.918i −0.973; 0.164; −0.164 which is the second column of U1 and e2.918i sˆ 01 in the space following the M1. To calculate the retardance induced by M1, the geometrical transformation of the ray path is first removed from P1 ; this in essence straightens out the path. The geometrical transformation matrix Q1 for M1 [20] is Q1 Oout;1 · Ireflect1 · O−1 in;1 0 0 1 0 1 1 0 0 s1x p1x k2x s1x s1y s1z 1 0 0 B C B C C B @ s01y p01y k2y A · @ 0 −1 0 A · @ p1x p1y p1z A k1x k1y k1z 0 0 1 s01z p01z k2z 0 1 1 0 0 B C @ 0 0 −1 A: (A4) 0 −1
0
Q1 describes a nonpolarizing reflection. The M1 retardance is calculated from the eigenvalues of −1 −1 U1Q · V−1 1Q , where Q1 · P1 U1Q · Σ1Q · V1Q by singular value decomposition. The phase difference between the eigenvalues is 0.527 radians, the M1 retardance magnitude. The eigenvector corresponding to the smaller eigenphase is (0.126, 0.967, 0.221), which is the fast axis and p-polarization in the entrance space. For the second reflection, the P2 matrix and the corresponding diattenuation and retardance are calculated in the same manner. The surface normal of p M2 is ηˆ 2 −1; −1; 0∕ 2. The reflected kˆ 3 is (0.961, 0.195, 0.195) and the reflection angle is 35.16°. The incident sˆ 2 and pˆ 2 are 0.239; −0.239; −0.941 and 0.951; −0.137; 0.277. The exiting sˆ 02 and pˆ 02 are 0.239; −0.239; −0.941 and −0.137; 0.951; −0.277. 3244
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0 −1 0
−1 0 0
!
! 0 0 : 1
The corresponding diattenuation is 0.006 with maximum reflection axis 0.239; −0.239; −0.941 along s-polarization, and the retardance is 0.169 radians with the fast axis 0.951; −0.137; 0.277 along p-polarization in the entrance space of M2. Combining the effects from both mirrors, the cumulative retardance and diattenuation are obtained from the cumulative P matrix, which is calculated by cascading P1 and P2 through matrix multiplication: P P2 · P1
−0.056 − 0.124i −0.109 − 0.165i 0.967 − 0.059i ! −0.737 0.625i 0.008 − 0.006i 0.055 0.125i 0.013 − 0.013i −0.468 0.820i 0.110 0.163i
and
0
0 Q Q2 · Q2 @ −1 0
0 0 −1
1 1 0 A: 0
Applying the same method as described previously, the overall diattenuation is 0.012 with maximum transmission axis 0.977e−2.447i ; 0.118e0.445i ; 0.175e−2.414i , and retardance is 0.365 radians with fast axis 0.076e0.186i ; 0.974; 0.213e0.013i in the entrance space. This algorithm is used for all polarization ray trace calculations reported in this paper. References 1. J. B. Breckinridge, “Self-induced polarization anisoplanatism,” Proc. SPIE 8860, 886012 (2013). 2. J. C. Carson, B. D. Kern, J. T. Trauger, and J. B. Breckinridge, “The effects of instrumental elliptical polarization on stellar point spread function fine structure,” Proc. SPIE 6265, 62653M (2006). 3. J. B. Breckinridge and B. R. Oppenheimer, “Polarization effects in reflecting coronagraphs for white-light applications in astronomy,” Astrophys. J. 600, 1091–1098 (2004). 4. P. Drude, “Zur Elektronentheorie der Metalle,” Ann. Phys. 306, 566–613 (1900). 5. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), Chap. 14. 6. J. Sasián, “Polarization fields and wavefronts of two sheets for understanding polarization aberrations in optical imaging systems,” Opt. Eng. 53, 035102 (2014). 7. D. J. Reiley and R. A. Chipman, “Coating-induced wave-front aberrations: on-axis astigmatism and chromatic aberration in all-reflecting systems,” Appl. Opt. 33, 2002–2012 (1994).
8. G. Anzolin, A. Gardelein, M. Jofre, G. Molina-Terriza, and M. W. Mitchell, “Polarization change induced by a galvanometric optical scanner,” J. Opt. Soc. Am. A 27, 1946–1952 (2010). 9. M. Jofre, G. Anzolin, F. Steinlechner, N. Oliverio, J. P. Torres, V. Pruneri, and M. W. Mitchell, “Fast beam steering with full polarization control using a galvanometric optical scanner and polarization controller,” Opt. Express 20, 12247–12260 (2012). 10. R. A. Chipman, D. M. Brown, and J. P. McGuire, “Polarization aberration analysis of the advanced x-ray astrophysics facility telescope assembly,” Appl. Opt. 31, 2301–2313 (1992). 11. L. R. Bissonnette, G. Roy, and F. Fabry, “Range–height scans of lidar depolarization for characterizing properties and phase of clouds and precipitation,” J. Atmos. Ocean. Technol. 18, 1429–1446 (2001). 12. S. C. McClain, P. W. Maymon, and R. A. Chipman, “Design and analysis of a depolarizer for the NASA MODerate-resolution imaging spectrometer-tilt (MODIS-T),” Proc. SPIE 1746, 375–385 (1992). 13. J. P. McGuire, Jr. and R. A. Chipman, “Analysis of spatial pseudodepolarizers in imaging systems,” Opt. Eng. 29, 1478–1484 (1990).
14. L. L. Smith and P. M. Koch, “Use of four mirrors to rotate linear polarization but preserve input–output collinearity,” J. Opt. Soc. Am. A 13, 2102–2105 (1996). 15. E. J. Galvez and P. M. Koch, “Use of four mirrors to rotate linear polarization but preserve input–output collinearity. II,” J. Opt. Soc. Am. A 14, 3410–3414 (1997). 16. E. J. Galvez, M. R. Cheyne, J. B. Stewart, C. D. Holmes, and H. I. Sztul, “Variable geometric-phase polarization rotators for the visible,” Opt. Commun. 171, 7–13 (1999). 17. D. W. Lynch and W. R. Hunter, “Gold (Au),” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, 1985). 18. R. A. Chipman, “Mechanics of polarization ray tracing,” Proc. SPIE 1746, 62 (1992). 19. G. Yun, K. Crabtree, and R. A. Chipman, “Three-dimensional polarization ray-tracing calculus I: definition and diattenuation,” Appl. Opt. 50, 2855–2865 (2011). 20. G. Yun, S. C. McClain, and R. A. Chipman, “Threedimensional polarization ray-tracing calculus II: retardance,” Appl. Opt. 50, 2866–2874 (2011). 21. R. A. Chipman, “Polarization aberrations,” Ph.D. dissertation (The University of Arizona, 1987). 22. D. B. Chenault and R. A. Chipman, “Measurements of linear diattenuation and linear retardance spectra with a rotating sample spectropolarimeter,” Appl. Opt. 32, 3513–3519 (1993).
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