18A Skew Aberration

18.1Introduction

This chapter considers the polarization aberrations of non-polarizing optical systems. If all the diattenuation and retardance are removed from an optical system, systematic polarization changes remain, named skew aberration.

Aberrations can be considered as deviations from the ideal behavior of imaging optical systems, that is, deviations from the mapping of spherical waves with uniform amplitude and polarization into spherical waves with uniform amplitude and polarization. The main aberration categories are wavefront aberration,1apodization (amplitude aberration),2–4 and polarization aberration.5 Wavefront aberration is the variation in optical path length, which is calculated in all the commercial ray tracing programs. Apodization is an amplitude aberration; different rays have different transmittances due to reflection losses and absorption. Polarization aberration, a non-uniform polarization change across wavefronts, is divided into (1) diattenuation aberration, which is polarization-dependent transmission or reflection, (2) retardance aberration, which is polarization-dependent optical path difference, and (3) skew aberration, the polarization change in the absence of diattenuation and retardance aberration.

One example of skew aberration occurs in a corner cube system where there is polarization change even when the diattenuation and retardance at three reflections are set to zero. This example is explained in detail in Section 16.11. As shown in Figure 18.1, the incident polarization (y-polarized) rotates by 120° as it propagates through the corner cube solely due to geometric transformation, that is, skew aberration. Further, the rotation is different in the three pairs of subapertures. This variation of the geometrical transformation across the pupil has a profound effect on the image formation with corner cubes as shown in Figure 16.44.

Figure 18.1Polarization state rotates due to skew aberration as the ray propagates through the non‑polarizing critical angle corner cube system.

The effect of skew aberration on point spread functions (PSFs) is examined using the Mueller point spread matrix (MPSM) (see Chapter 16 [Image Formation with Polarization Aberration]) to show how non-polarizing and ideal optical systems can have undesired polarization mixing due to skew aberration. Skew aberration’s separate origin and behavior is fascinating. In radially symmetric systems, skew rays have skew aberration but meridional rays do not. Thus, the name skew aberration is applied.

In this chapter, skew aberration is defined and algorithms are provided for its calculation. The skew aberration of a high numerical aperture (NA) lens with a large field of view (FOV) is analyzed in detail. The effect of a linear variation of skew aberration, as shown in Figure 18.2, the most common form, on a diffraction-limited MPSM is treated. Finally, the skew aberration is calculated for 2383 lens systems in the CODE V6 patent library and their statistics provide perspective on the role and importance of skew aberration.

Figure 18.2The effect of linear skew aberration transforming a uniform incident linearly polarized state (left) into a skew aberrated state (right). This linear form of skew aberration occurs in most lenses and other rotationally symmetric systems in addition to higher-order variations of skew aberration-induced polarization rotation.

18.2Definition of Skew Aberration

Skew aberration is the rotation of each ray’s polarization state between the entrance pupil and the exit pupil due to the intrinsic geometric transformation of polarization states. Skew aberration occurs even for non-polarizing optical systems. It seems odd at first that a non-polarizing optical system should have polarization aberration. As shown in Chapter 17, the propagation of polarized light in a non-polarizing optical system can be simulated with the parallel transport of vectors on a sphere, and skew aberration is related to Berry phase and Pancharatnam phase calculated using the parallel transport matricesQ in Chapter 17. Thus, if a ray has non-zero geometric transformation, its skew aberration is also non-zero.

Skew aberration is independent of the incident polarization state or polarization properties of optical elements such as polarizers, retarders, and coatings. As was shown in the critical angle corner cube reflector example in Chapter 16, non-polarizing optical system with zero diattenuation or retardance can have a significant amount of spatially varying polarization aberration. Although there is no optical path difference in the wavefront exiting the non-polarizing optical system, the wavefront may have varying optical phase similar to the six subapertures of critical angle corner cube. This phase difference can be measured by interferometers and may cause large polarization aberration.

Skew aberration is a polarization rotation, rotating the plane of polarization of linearly polarized light or the axis of elliptically polarized light. The rotation of circularly polarized light causes a phase change but not a change of polarization state. Skew aberration is distinct from diattenuation aberration and retardance aberration since its origin arises from purely geometric effects.

Since skew aberration occurs even for rays propagating through ideal, aberration-free, and non‑polarizing optical systems, the example systems chosen for this chapter are non-polarizing. Such a system refracts and reflects the incident polarization ellipses into image space without changing its ellipticity.7 For example, a refraction through a surface with ts = tp = 1, where ts and tp are Fresnel transmission coefficients for s- and p-polarization, is ideal and non-polarizing. The effect of skew aberration in radially symmetric optical systems, such as a lens or Cassegrain telescope, occurs for the off-axis fields. The polarization of the exiting wavefront relative to the incident wavefront has a steady rotation of its polarization state, which increases linearly with the distance from the meridional plane; this linear rotation continues toward the origin and is present even for paraxial ray paths. Figure 18.2 shows an example of this generic skew aberration. The polarization is not rotated along the center of the pupil, rotated clockwise on one side of the pupil and counterclockwise on the other side. The magnitude of rotation shown in here is greater than commonly encountered skew aberration. The skew aberration shown is linear but may also have quadratic, cubic, or contributions of other functional forms; Figure 18.2 shows the most common manifestation.

The skew aberration is determined solely by the ray’s propagation path, that is, its sequence of normalized propagation vectors (kIn, k1, k2, …kj, …kExit). kIn, kj, and kExit correspond to the propagation vector at the entrance pupil, after the jth surface, and at the exit pupil, respectively.

18.3Skew Aberration Algorithm

An algorithm is presented to define the polarization changes between the entrance and exit pupil of a non-polarizing optical system. Consider Figure 18.3 showing a grid of polarization states in the entrance pupil and the corresponding grid in the exit pupil. These pupils have different NAs and propagation directions. To compare two different wavefronts, a scale invariant reference polarization grid, the double pole basis vector, is selected; a grid of double pole (gIn,i) are defined on the entrance pupil, traced via parallel transport of vectors, and compared with an exit pupil double pole (gExit,i). As shown in Figure 18.4, this function does not change when the function is radially enlarged or shrunk about its center. If a double pole polarization pattern is present in the entrance and exit pupil, it is justified that there is no skew aberration and the resulting PSF would not be degraded by polarization aberration.

Figure 18.3A grid of reference vectors (gIn,i) on the entrance pupil (green) and another grid of reference vectors (gExit,i) on the exit pupil (purple) are shown in two different views.

Figure 18.4An array of double pole reference vectors on a unit wavefront sphere viewed along the chief ray’s propagation vector. These are reference vectors for defining the x-component of Jones vectors over a spherical wavefront.

The following steps are a recap of calculating the double pole reference basis vectors, which were introduced in Chapter 11, since they are used as reference vectors in this chapter. First, define a vector (gC) that is perpendicular to the center ray’s propagation vectors on the entrance and exit pupil

$$g_C=k_{\text{In},C}\times k_{\text{Exit},C}.$$18.1

18.1

The grid of reference vectors gIn,i are generated by a counterclockwise rotation of gC along axisIn,i by θIn,i. Similarly, the gExit,i grid is obtained by a counterclockwise rotation of gC along axisExit,i by θExit,i,

$$\{\begin{array}{l}{g}_{\text{In},i}=R({\theta }_{\text{In},i},\mathrm{axi}{s}_{\text{In},i})g_C\\ g_{\text{Exit},i}=R({\theta }_{\text{Exit},i},\mathrm{axi}{s}_{\text{Exit},i})g_C\end{array}$$18.2

18.2

where index i indicates the ith ray,

$$\{\begin{array}{l}{\theta }_{\text{In},i}={\cos }^{-1}(k_{\text{In},i}\cdot k_{\text{In},C}),axi{s}_{\text{In},i}=k_{\text{In},C}\times k_{\text{In},i},\\ {\theta }_{\text{Exit},i}={\cos }^{-1}(k_{\text{Exit},i}\cdot k_{\text{Exit},C}),axi{s}_{\text{Exit},i}=k_{\text{Exit},C}\times k_{\text{Exit},i},\end{array}$$18.3

18.3

and R(θ, axis) is the 3D rotation matrix for a counterclockwise rotation around axis by θ.

The pair of gIn,i and gExit,i are invariant to magnification of the system and allows the polarization change from the entrance pupil to the exit pupil to be readily visible since it is defined by an angle relative to a radially symmetric set of lines. This rotation method described here is analogous to parallel transporting gC along a great circle arc on a unit k-sphere that connects points kIn,C and kIn,i. This is the “double pole grid” described in Chapter 11, which defines the “standard” to describe a linearly polarized spherical wavefront, as shown in Figure 18.4.

Once the reference vectors gIn,i and gExit,i are established for an optical system, the system’s geometric transformation can be calculated. The geometric transformation of each ray intercept due to the change in ray propagation direction from kj−1 to kj is described by the parallel transport matrix Qj (see Chapter 17 [Parallel Transport and the Calculation of Retardance]). Qj for refracting surface j is equivalent to sliding vectors from a point kj−1 to a point kj on a unit k-sphere following the great circle arc that connects two points as shown in Figure 17.4. Qj for reflecting surface j is equivalent to inverting vectors on a point kj−1 about kj−1−kj and then moving them to a point kj by parallel transport. The cumulative geometric transformation through the system along a single ray is QTotal,i,

$$Q_{\mathrm{Total},i}={\displaystyle \prod _{j=\text{Exit}}^{\text{In}}{Q}_j=Q_{\text{Exit}}\cdots Q_j\cdots Q_1{Q}_{\text{In}}}.$$18.4

18.4

This maps the reference input polarization into ${\mathbf{g}}_{\text{Exit,}i}^{\prime }$,

$${g^{\prime }}_{\text{Exit},i}=Q_{\mathrm{Total},i}{g}_{\text{In},i}.$$18.5

18.5

The ith ray’s skew aberration is defined as the angle between the ideal vector gExit,i and the non‑polarizing system’s vevtor ${\mathbf{g}}_{\text{Exit,}i}^{\prime }$. If ${\mathbf{g}}_{\text{Exit,}i}^{\prime }$ results from a counterclockwise rotation from the gExit,i looking into the beam, the ray has a positive skew aberration.

QTotal,i serves a role for polarization evolution similar to the role of paraxial optics. Paraxial optics describes ray paths in an aberration-free system; it describes ideal systems. When a spherical wave is traced via paraxial optics, the exiting spherical wave forms an image at the ideal location free of distortion, astigmatism, and other aberrations. When real rays are traced through the same system, the results are quite complex, containing large sets of ray intercepts, optical path lengths, and so on. Optical designers can compare these real ray trace results with paraxial optics to calculate wavefront aberrations, since aberration is often defined as the deviation from the paraxial optics. These wavefront aberrations are convenient for communication with other optical designers. Similarly, QTotal,i is “paraxial optics-like” since it provides ${\mathbf{g}}_{\text{Exit,}i}^{\prime }$, which is the “natural coordinate system” for a non-polarizing optical system; other polarization changes that deviate from ${\mathbf{g}}_{\text{Exit,}i}^{\prime }$ due to polarizing elements or interactions such as coatings, Fresnel coefficients, and so on, are the polarization aberration. The skew aberration calculates a change in polarization that is natural (i.e., intrinsic) to the system, and all other polarization changes are aberrations due to light–matter interactions.

Figure 18.5 shows the form of a typical skew aberrated polarization state, ${g^{\prime }}_{\text{Exit,}i}$, for a rotationally symmetric refractive optical systems of an off-axis source along y-axis with gin,i shown in Figure 18.4. Note that in the y–z plane, ${g^{\prime }}_{\text{Exit,}i}=g_{\text{Exit},i}$, while the skew rays’ ${g^{\prime }}_{\text{Exit,}i}$ are rotated from gExit,i.

Figure 18.5The form of the skew aberration, greatly exaggerated, ${g^{\prime }}_{\text{Exit},i}$ for a rotationally symmetric refractive optical systems for gExit,i shown in Figure 18.4.

18.4Lens Example—U.S. Patent 2,896,506

In a radially symmetric optical system, meridional rays lie on a plane containing the optical axis of the system. The chief ray and marginal ray are examples of meridional rays. They remain in one plane as they reflect and refract through the system from object space to image space. Skew rays, on the other hand, do not stay in one plane. In a long lens, skew rays will spiral around the optical axis in either a purely clockwise or counterclockwise sense. Skew aberration only occurs for skew rays, not for meridional rays. When the optical system is not radially symmetric, definitions of meridional rays are no longer clearly defined but the skew aberration algorithm still applies.

Skew aberration naturally increases with NA and FOV; hence, systems with high NA and wide FOV tend to have larger skew aberration. For example, U.S. Patent 2,896,5068 has a comparatively large skew aberration. The system is rotationally symmetric with F/1.494 and a maximum FOV of 32°. Figure 18.6 shows the layout of this seven-lens system with two field angles, 0° and 20°.

Figure 18.6Optical system layout of U.S. Patent 2896506 from the CODE V lens patent library with seven lenses. Two field angles at 0° (right) and 20° (left) are shown.

Figure 18.7 (left) shows the skew aberration at the exit pupil calculated for a grid of rays from the 32° field angle. The skew aberration is generally largest at the edge of the pupil of the maximum field angle. For this example, the skew ray formed from the chief ray in the y–z plane plus the marginal ray in the x-direction at the 32° field angle has the largest skew aberration of 7.01°, shown in Figure 18.7 (left) as a gray point (point B). The skew ray at the opposite side of the pupil (point A) has −7.01° of skew aberration. The pupil is elliptical due to vignetting and pupil distortion. A meridional fan of rays through the center of the exit pupil has zero skew aberration.

Figure 18.7(Left) Skew aberration of a ray grid from the 32° field is evaluated at the exit pupil of the US lens patent 2896506 with skew aberration varying from −7.01° to +7.01°. (Right) The skew aberration along a horizontal cross section (orange dashed line in left figure) perpendicular to the meridional plane. It shows zero skew aberration for the center chief ray with a linear variation at the center of the pupil and higher-order variations toward the edge. In the meridional plane along the y-axis, the skew aberration is zero.

Figure 18.7 (right) shows that the skew aberration point A to point B has a form of a linear variation in x (circular retardance tilt1) plus a coma-like cubic variation in x. The linearly varying circular retardance causes polarization-dependent tilt in the wavefront as shown in Figure 18.8.

Figure 18.8Skew aberration map showing linearly varying skew aberration.

Consider a circularly polarized plane wave incident on a non-polarizing system with a linear skew aberration, such as Figure 18.8. Along the y-axis, the light’s phase stays unchanged. On the +x side of the pupil, the phase advances; on the −x side of the pupil, the phase retards, as shown in Figure 18.9 for left and right circularly polarized light. Thus, they focus at two shifted image points, by a fraction of an Airy disk diameter, enlarging the PSF in a way similar to astigmatism.

Figure 18.9(Left) The effect of linear skew aberration on a right circularly polarized beam and (right) a left circularly beam that acquire opposite linear phase shift across the pupil due to skew aberration.

For lens systems, skew aberration tends to be largest from the edge of the object and at the side of the pupil furthest from the meridional plane. The skew aberration contribution for patent lens 2,896,506 for the ray corresponding to point A in Figure 18.7 at each lens surface is shown in Figure 18.10. Such surface contribution plots identify surfaces with the largest skew contributions.

Figure 18.10Skew ray A’s skew aberration contribution from each lens surface sums to -7.01° through the system.

18.5Skew Aberration in Paraxial Ray Trace

Next, the skew aberration is studied in the paraxial limit. The linear skew aberration that appeared through the center of the pupil in Figure 18.7 clearly shows that skew aberration is present for paraxial rays. Paraxial optics is a method of determining the first-order properties of a radially symmetric optical system. It assumes that all ray angles and angles of incidence are small.9 More information on paraxial optics and ray tracing is found in Chapter 15.

The general paraxial ray trace procedure traces two rays, the paraxial marginal ray (from the center of the object to the edge of entrance pupil) and the full-field paraxial chief ray (from the top of the object to the center of the entrance pupil).10 Then, all other paraxial rays can be calculated from linear combinations of these two rays. For the paraxial skew ray from the top of the object and the edge of the pupil, the paraxial marginal ray height at each surface is the x-coordinate of the skew ray, the paraxial chief ray height is the y-coordinate of the skew ray, and the vertex of each surface is the z-coordinate of the skew ray.

The skew aberration calculation uses the propagation vector specified after the qth ray intercept, which is along $\left(y_{q+1}-y_q,{\overline{y}}_{q+1}-{\overline{y}}_q,t_q\right)$, where yq is the marginal ray height, ${\overline{y}}_q$ is the chief ray height, and tq is the distance between the qth and the (q + 1)th surface vertices along the axis. The normalized propagation vectorskq are

$$k_q=\frac{\left(w_q,{\overline{w}}_q,1\right)}{\sqrt{w_q^2+{\overline{w}}_q^2+1}},$$18.6

18.6

where uq is the marginal ray angle, ${\overline{u}}_q$ is the chief ray angle, wq = nquq, tq and ${\overline{w}}_q=n_q{\overline{u}}_q,t_q$ are reduced angles, and nq is the refractive index following the qth surface.

In paraxial ray trace, the calculation of the spherical polygon’s area associated with the parallel transport of skew rays, as shown in Section 17.2, reduces to a polygon on the plane perpendicular to the optical axis. Thus, the skew aberration in paraxial ray trace is proportional to the area of the polygon. By dropping the z-component of the propagation vectors, the remaining 2D propagation vectors that form the polygon can be calculated as

$$k_{2D,q}=\frac{\left(w_q,{\overline{w}}_q\right)}{\sqrt{w_q^2+{\overline{w}}_q^2+1}}\approx \left(w_q,{\overline{w}}_q\right).$$18.7

18.7

The area of the triangle shown in Figure 18.11 which connects the origin, k2D,q, and k2D,q+1 is

Figure 18.11A triangle that connects the origin and two-dimensional propagation vectors.

$$\mathrm{Are}{a}_q=\frac{1}{2}\frac{w_q{\overline{w}}_{q+1}-w_{q+1}{\overline{w}}_q}{\sqrt{w_q^2+{\overline{w}}_q^2+1}\sqrt{w_{q+1}^2+{\overline{w}}_{q+1}^2+1}}.$$18.8

18.8

Further manipulations using wq+1 = wq − yqϕq, ${\overline{w}}_{q+1}={\overline{w}}_q-{\overline{y}}_q{\varphi }_q$ results in a relationship for the area

$$\begin{array}{l}Are{a}_q=\frac{{\varphi }_q}{2}\frac{{\overline{w}}_q{y}_q-w_q{\overline{y}}_q}{\sqrt{w_q^2+{\overline{w}}_q^2+1}\sqrt{w_{q+1}^2+{\overline{w}}_{q+1}^2+1}}\\ \phantom{Are{a}_q}=\frac{H}{2}\frac{{\varphi }_q}{\sqrt{w_q^2+{\overline{w}}_q^2+1}\sqrt{w_{q+1}^2+{\overline{w}}_{q+1}^2+1}},\end{array}$$18.9

18.9

where $H={\overline{w}}_q{y}_q-w_q{\overline{y}}_q$ is the Lagrange invariant of the system.

Therefore, the paraxial skew aberration of the system is proportional to the Lagrange invariant and is closely related to the sum of the individual surface powers in Equation 18.9,

$$\mathrm{Total}\mathrm{Area}=\frac{H}{2}{\displaystyle \sum _q\frac{{\varphi }_q}{\sqrt{w_q^2+{\overline{w}}_q^2+1}\sqrt{w_{q+1}^2+{\overline{w}}_{q+1}^2+1}}}.$$18.10

18.10

Frequently, paraxial rays with large y, $\overline{y}$, w, $\overline{w}$ are traced instead of small values. Since paraxial values are small, $\sqrt{w_q^2+{\overline{w}}_q^2+1}\approx 1$ and $\sqrt{w_{q+1}^2+{\overline{w}}_{q+1}^2+1}\approx 1$. Then, Equation 18.10 simplifies to

$$\mathrm{Total}\mathrm{Area}=\frac{H}{2}{\displaystyle \sum _q{\varphi }_q}.$$18.11

18.11

The example lens in Section 18.4 demonstrates the calculation of paraxial skew aberration. A paraxial skew ray (point A in Figure 18.7) is created by adding a paraxial marginal ray in the x–z plane to a paraxial chief ray in the y–z plane, yielding the paraxial skew aberration of −4.49° ≈ 0.078. The surface-by-surface skew aberration contribution for ray A is shown in Figure 18.12.

Figure 18.12Paraxial skew ray at point A’s skew aberration contribution from each lens surface sums to −4.49° through the system.

18.6Example of Paraxial Skew Aberration

In this section, an example of paraxial skew aberration is calculated using a four-element lens relay system that is chosen such that the exiting rays are parallel to the incident rays to simplify the understanding of the skew aberration concept.

Four thin lenses with the same effective focal length f are spaced 2f from each other as shown in Figure 18.13. The object plane is 2f in front of the first lens. The first lens is the entrance pupil. The first lens images the object plane with magnification −1 at the second lens. The second lens, a field lens, images the entrance pupil onto the third lens with magnification −1, such that the third lens is a pupil image. The third lens images the object onto the fourth lens, with a magnification of one. The fourth lens is another field lens, such that all paraxial rays in the image space after the fourth lens are parallel to the corresponding incident rays.

Figure 18.13Four thin lenses (blue) with effective focal length of 100 mm are spaced by 200 mm. The object is shown in green. Paraxial chief ray is shown in red and the marginal ray is shown in blue.

Table 18.1 contains the paraxial ray trace where each lens is specified by power, not by curvature and index. Figure 18.14 (left) shows the x–y components of the propagation vectors forming a four-sided spherical polygon about the z-axis on the propagation vector sphere, which would be a square if the surface were flat. The four-sided spherical polygon subtends about 0.3 × 0.3 = 0.09 steradians. Thus, the skew aberration for this extreme skew ray is 0.09 radians ≅ 5°. The $y-\overline{y}$ diagram shows surfaces 0, 2, and 4 are where the object and its images are located since y = 0, and due to the field lens at 4, each ray exits the system parallel to the incident light.

Table 18.1Paraxial Ray Tracing Form for Four-Element Relay Lens

Figure 18.14(Left) Propagation vector sequence near the z-axis for the relay lens and (right) the $y-\overline{y}$ diagram.

The variation of the skew aberration is shown for a set of fields in Figure 18.15. All meridional rays are shown over a gray background; other rays in the white background are skew rays. There is no skew aberration on-axis since all on-axis rays in radially symmetric systems are meridional rays. For fields along the y-axis where the meridional rays are down the vertical axis, there is no rotation, and along the horizontal axis for the x fields, the skew aberration is zero.

Figure 18.15Functional form for the variation of paraxial skew aberration for a set of 13 fields for vertically polarized incident light. The skew aberration is the rotation from vertical. Meridional rays, shown over a gray background, have no skew aberration. In paraxial systems, the skew aberration increases linearly from the meridional plane and increases linearly with field.

18.7Skew Aberration’s Effect on PSF

Skew aberration creates undesired polarization components in the exit pupil and the image. Skew aberration modifies the PSF and thus image quality can be degraded, even in the absence of wavefront, retardance, and diattenuation aberrations. Typically, cross-polarized satellites form around the PSF.11–13 This section applies the methods in Chapter 16 to show the effect of skew tilt on the PSF.13

Skew aberration causes a spatially varying “circular retardance–like” behavior across the pupil. Skew aberration occurs with different functional forms, such as constant, linear, and quadratic variations, just like wavefront aberrations. Section 18.6 showed that the linearly varying skew aberration is the expected form in paraxial optics and is likely to be the dominant skew aberration component in many other systems. For example, linear variation is a significant component in Figure 18.7. At the lowest order, skew aberration has a linearly varying component, like the wavefront aberration named tilt; thus, this linear skew component is named skew tilt. Skew tilt has a Jones pupil function of the form

$$J_{\mathrm{pupil}}(u,v)=p(u,v)\left(\begin{array}{ll}\cos (u∆)& \sin (u∆)\\ -\sin (u∆)& \cos (u∆)\end{array}\right),$$18.12

18.12

where u is the x-pupil coordinate in the direction perpendicular to the meridional plane, v is the y-coordinate, and Δ is the magnitude of skew aberration at the edge of the pupil in the u-direction. p(u, v) is the pupil function of the system.

To evaluate the effect of skew aberration on the PSF, an example with Δ = π is analyzed, a value far larger than expected, but good for tutorial purposes. Let’s consider a circular aperture for

$$p(u,v)=\{\begin{array}{ll}1& u^2+v^2\le 1\\ 0& \text{otherwise}\end{array}.$$18.13

18.13

The Jones pupil is plotted in Figure 18.16 as a density plot (left) and a cross-section plot (right). This Jones pupil has purely real components. Note that the sine and cosine functions are plotted for −π ≤ uΔ ≤ π.

Figure 18.16A density plot (left) and a cross-section plot (right) of the Jones pupil for skew tilt with a circular aperture.

A two-dimensional Fourier transform of Jpupil(u,v) gives a 2 × 2 Jones amplitude response matrix (ARM) of the system, which is then converted into a 4 × 4 MPSM as shown in Figure 18.17. For more details of these calculations, see Chapter 16.

Figure 18.17A density plot (left) and a cross-section plot (right) of the MPSM of the example system.

The m03 component of the MPSM shows two peaks. Referring to Figures 18.8 and 18.9, the right circularly polarized light has been tilted to one side and the left circularly polarized light has been tilted to the other. This is seen in the two peaks in the m03, one positive (right circular) and one negative (left circular). This means unpolarized incident light produces a double image with opposite circular components. Considering the m00, m03, m30, and m33 elements, right circularly polarized incident light produces a single shifted peak to the left, and left circularly polarized incident light produces a single shifted peak to the right. Thus, the skew aberration divides the incident light into left and right circularly polarized states and shifts the peaks in opposite direction depending on the incident polarization state.

This example chose a half wave (Δ = π) of skew aberration. At this level, the two peaks are entirely shifted from each other to visually show the effect of skew tilt on the MPSM. A typical skew tilt is in order of 1/100 waves; hence, the two images would overlap with just a small shear. Similar analyses are easily performed for different aperture functions such as rectangular apertures and higher-order skew aberration than linear skew aberration.

18.8PSM for U.S. Patent 2,896,506

In this section, MPSM of the example optical system shown in Section 18.4 is further analyzed. The Jones pupil has been calculated by polarization ray tracing determining the geometric transformation for each ray using QTotal,i. Since Jones matrices are defined in local coordinates, the exit pupil reference vectors (gExit,i) are used as local coordinate (u, v) when converting each QTotal,i to a Jones matrix. The choice of local coordinates is critical since the geometric transformation of each ray can be obscured by choosing local coordinates that already contain skew aberration, and in that basis, each Jones matrix would not show any skew aberration. For example, if ${g^{\prime }}_{\text{Exit},i}$ in Equation 18.5 is used to calculate ith ray’s Jones matrix, the matrix will not show any skew aberration since ${g^{\prime }}_{\text{Exit},i}$ is already a skew aberrated local coordinate.

The Jones matrix pupil of the example high-NA lens, shown in Figure 18.18, becomes elliptical as the object moves far off-axis, 32° in this case, due to pupil distortion.

Figure 18.18Jones matrix pupil at the exit pupil of the patent lens system calculated from the parallel transport matrix of the system at the exit pupil.

As shown in Figure 18.7, the system has circular retardance-like skew aberration described in Equation 18.12, which has linear and cubic variation along the pupil u-axis. Since the skew aberration is small, cos (uΔ) ≈ 1, sin (uΔ) ≈ uΔ, the j1,1 and j2,2 components of the Jones matrix pupil are constant, and the magnitudes of j1,2 and j2,1 components are varying along the pupil u-axis.

The MPSM is plotted in two different scales in Figure 18.19; the legend of the left plot shows the full range of the MPSM values while the legend of the right plot is limited to the minimum and maximum values of the m03 component to highlight the m03 and m30 components. Note that the shape of the MPSM elements is elongated in the vertical direction and shortened along the horizontal direction due to the discrete Fourier transform of an elliptical aperture becoming elongated in the opposite direction.

Figure 18.19A MPSM of the patent lens system calculated from a discrete Fourier transform of the Jones matrix pupil. On the left, green indicates zero, and a few nonzero regions, most obviously in elements m03 and m30, are shown in darker green and yellow. On the right, the scale has been enlarged from −1 to 1 with zero mapped to black to show nonzero regions of m03, m12, m21, and m30.

The non-zero m03 and m30 components show, for unpolarized light, that the right circularly polarized component is tilted one way and the left circularly polarized component is tilted the opposite way. For this example, the shift between right and left circularly polarized light is quite small. However, the presence of off-diagonal components shows the degradation to the image quality arising from the skew aberration.

18.9Statistics—CODE V Patent Library

Skew aberration is typically a small effect, but how small can only be addressed in the context of the magnitudes of the other families of aberrations. To understand the typical magnitude of skew aberration and its importance relative to other aberrations, the skew aberration of 2382 non-reflecting optical systems in CODE V’s U.S. patent library was calculated. Figure 18.20 (left) shows the skew aberration for a non-paraxial ray from the largest defined field through the edge of the pupil along an axis perpendicular to the field point. Figure 18.20 (right) zooms in to the range from 2° to 6.5°, indicating the number of optical systems with skew aberration in each range. The mean of the overall skew aberration is 0.28° with a standard deviation of 0.47°. The maximum skew aberration in this set of lenses is 6.44°. Thus, for more than 90% of these lenses, the skew aberration corresponds to less than a hundredth of a wave of circular retardance and is insignificant.

Figure 18.20(Left) Histogram of the maximum skew aberration evaluated for 2382 non-reflecting optical systems in CODE V’s library of patented lenses. (Right) Zoomed-in view of the histogram for skew aberration of 2° to 6.5°.

18.10Conclusion

Skew aberration is the component of polarization aberration that originates from pure geometric effects associated with the sequence of propagation vectors through an optical system. Skew aberration is calculated by following the parallel transport of polarization states through non-polarizing optical systems. Skew aberration depends only on the sequence of k vectors; it does not change by varying the coatings applied to an optical system. The variation of skew aberration across the pupil often has a linear component, skew tilt, orthogonal to the meridional plane. A constant skew, similar to wavefront piston, would uniformly change the polarization and would not affect the PSF. It is the variation of skew (i.e., skew aberration) that affects the MPSM and, thus, degrades image quality. Skew aberration is typically a small effect in lenses but is a large effect in corner cubes.

Skew aberration was calculated for two example systems to demonstrate how to separate skew aberration from other retardance effects and its effect on the MPSM. Greater skew aberration is expected in systems with high NA and large FOV such as microlithography optics and other polarization-sensitive systems.

As we understand it, the polarization ray tracing algorithms in most of the optical design and ray tracing software account for polarization aberrations but do not identify skew aberration as a separate component. Thus, we expect that these PSF calculations will incorporate the effects of skew aberration.

18.11Problem Sets

1. A skew ray propagates through a sequence of four thin lenses with the following propagation vectors:

   k0 = (3, 4, 12)/13, k1 = (4, −3, 12)/13, k2 = (−3, −4, 12)/13,

   k3 = (−4, 3, 12)/13, and k4 = (3, 4, 12)/13

   1. Calculate Q matrices for the four refractions: Q1, Q2, Q3, and Q4.

   2. Find the cumulative Q matrix for the ray path. Rational or floating point numbers are acceptable.

   3. What is the skew aberration in radians?

   4. Derive a set of local coordinates for the planes transverse to k0 and k4 using yLocal = (4/5, −3/5, 0).

   5. Convert Q into a Jones matrix J using these local coordinates.

   6. Find the eigenvalues and eigenvectors for J. What type of polarization element does J behave as?

2. A 1000 mm focal length lens with a 2 cm × 2 cm square aperture has ξo = 0.3 radians of skew aberration (polarization rotation varying linearly across the pupil) such that its polarization aberration function is $J_{\mathrm{skew}}=\left(\begin{array}{c}\cos {\xi }_{o}x-\sin {\xi }_{o}x\\ \sin {\xi }_{o}x-\cos {\xi }_{o}x\end{array}\right)$, where −1 < (x, y) < 1 are normalized pupil coordinates. A collimated beam of 400 nm light is incident on-axis.

   1. How much amplitude and intensity are transmitted when the lens is located between a horizontal and a vertical linear polarizer?

   2. What is the associated wavefront aberration and apodization of the cross-polarized component?

   3. Calculate the 2 × 2 amplitude response function at the image through suitable Fourier transforms of rectangle functions. Use the Fourier transform, not discrete Fourier transform, no approximations.

References

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2E. Sklar, Effects of small rotationally symmetrical aberrations on the irradiance spread function of a system with Gaussian apodization over the pupil, J. Opt. Soc. Am. 65 (1975): 1520–15215.

3J. P. Mills and B. J. Thompson, Effect of aberrations and apodization on the performance of coherent optical systems. I. The amplitude impulse response, J. Opt. Soc. Am. A 3 (1986): 694–7035

4J. P. Mills and B. J. Thompson, Effect of aberrations and apodization on the performance of coherent optical systems. II. Imaging, J. Opt. Soc. Am. A 3 (1986): 704–7165.

5R. A. Chipman, Polarization analysis of optical systems, Opt. Eng. 28 (1989): 90–995.

6CODE V Version 10.3, Synopsys, Inc. (http://www.opticalres.com/cv/cvprodds_f.html).

7G. Yun, S. McClain, and R. A. Chipman, Three-dimensional polarization ray-tracing calculus II. Retardance, Appl. Opt. 50 (2011): 2866–28745.

8H. Azuma, High aperture wide-angle objective lens, U.S. Patent 2,896,506 (July 28, 1959).

9J. E. Greivenkamp, Field Guide to Geometrical Optics, SPIE Press (2004).

10B. R. Irving et al., Code V, Introductory User’s Guide, Optical Research Associates (2001).

11J. Ruoff and M. Totzeck, Orientation Zernike polynomials: A useful way to describe the polarization effects of optical imaging systems, J. Micro/Nanolithogr. MEMS MOEMS 8(3) (2009): 031404.

12M. Mansuripur, Effects of high-numerical-aperture focusing on the state of polarization in optical and magneto-optic data storage systems, Appl. Opt. 50 (1991): 3154–31625.

13J. P. McGuire and R. A. Chipman, Diffraction image formation in optical systems with polarization aberrations I: Formulation and example, J. Opt. Soc. Am. A 7 (1990): 1614–16265.

1 The linear variation of skew aberration adds a linear phase to the wavefront, which results in a tilt-like wavefront aberration.