Abstract

In paper 1 [Appl. Opt. 33, this issue (1994)] we examined the polarization aberrations of rotationally symmetric systems. In this paper we extend polarization-aberration theory to include two types of tilted and decentered systems composed of rotationally symmetric elements. One type is systems with collinear centers of curvatures but with decentered pupils. The symmetry in such systems permits the analysis to proceed along lines similar to those in paper 1. The other type is systems with arbitrary tilts and decenters. In these systems the field dependencies of the aberrations from each surface are not concentric. The extension is made by use of a polarization-aberration matrix with vector, instead of scalar, arguments.

© 1994 Optical Society of America

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References

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  1. J. P. McGuire, R. A. Chipman, “Polarization aberrations. 1. Rotationally symmetric optical systems,” Appl. Opt. 33, (1994).
    [PubMed]
  2. R. A. Buchroeder, “Tilted component optical systems,” Ph.D. dissertation (Optical Sciences Center, University of Arizona, Tucson, Ariz., 1976).
  3. K. P. Thompson, “Aberration fields in tilted and decentered optical systems,” Ph.D. dissertation (Optical Sciences Center, University of Arizona, Tucson, Ariz., 1980).
  4. J. R. Rogers, “Aberrations of unobscured reflective optical systems,” Ph.D. dissertation (Optical Sciences Center, University of Arizona, Tucson, Ariz., 1983).

1994 (1)

J. P. McGuire, R. A. Chipman, “Polarization aberrations. 1. Rotationally symmetric optical systems,” Appl. Opt. 33, (1994).
[PubMed]

Buchroeder, R. A.

R. A. Buchroeder, “Tilted component optical systems,” Ph.D. dissertation (Optical Sciences Center, University of Arizona, Tucson, Ariz., 1976).

Chipman, R. A.

J. P. McGuire, R. A. Chipman, “Polarization aberrations. 1. Rotationally symmetric optical systems,” Appl. Opt. 33, (1994).
[PubMed]

McGuire, J. P.

J. P. McGuire, R. A. Chipman, “Polarization aberrations. 1. Rotationally symmetric optical systems,” Appl. Opt. 33, (1994).
[PubMed]

Rogers, J. R.

J. R. Rogers, “Aberrations of unobscured reflective optical systems,” Ph.D. dissertation (Optical Sciences Center, University of Arizona, Tucson, Ariz., 1983).

Thompson, K. P.

K. P. Thompson, “Aberration fields in tilted and decentered optical systems,” Ph.D. dissertation (Optical Sciences Center, University of Arizona, Tucson, Ariz., 1980).

Appl. Opt. (1)

J. P. McGuire, R. A. Chipman, “Polarization aberrations. 1. Rotationally symmetric optical systems,” Appl. Opt. 33, (1994).
[PubMed]

Other (3)

R. A. Buchroeder, “Tilted component optical systems,” Ph.D. dissertation (Optical Sciences Center, University of Arizona, Tucson, Ariz., 1976).

K. P. Thompson, “Aberration fields in tilted and decentered optical systems,” Ph.D. dissertation (Optical Sciences Center, University of Arizona, Tucson, Ariz., 1980).

J. R. Rogers, “Aberrations of unobscured reflective optical systems,” Ph.D. dissertation (Optical Sciences Center, University of Arizona, Tucson, Ariz., 1983).

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Figures (10)

Fig. 1
Fig. 1

Geometry of a surface in a tilted or decentered system. The pupil, local axis, central ray, object vector H, and object decenter vector h are shown.

Fig. 2
Fig. 2

Vector coordinate system.

Fig. 3
Fig. 3

(a) IR LIDAR off-axis beam expander and (b) the equivalent rotationally symmetric system. In systems with decentered pupils such as (a), one can easily calculate the aberrations by analyzing the equivalent rotationally symmetric system (b) and then decentering the pupil.

Fig. 4
Fig. 4

Diattenuation and retardance pupil aberrations of the LIDAR beam expander at the edge of the field of view. The full pupil represents the equivalent rotationally symmetric system. The decentered pupil of the LIDAR beam expander is highlighted.

Fig. 5
Fig. 5

Effective field height at a tilted or decentered element. The object vector H, object decenter vector h q and effective object vector (Hh q ) are shown.

Fig. 6
Fig. 6

Vector tilt aberration map across the pupil. Examples for (a) rotationally symmetric systems and (b) tilted or decentered systems are shown. The object vectors are superimposed over the pupil to show the effect of the tilts and decenters on the orientation of the aberration pattern.

Fig. 7
Fig. 7

Contours of constant vector tilt in the object field.

Fig. 8
Fig. 8

Vector quadratic-piston aberration map across the pupil. Examples for (a) rotationally symmetric systems and (b) tilted or decentered systems are shown. The object vectors are superimposed over the pupil to show the effect of the tilts and decenters on the orientation of the aberration pattern.

Fig. 9
Fig. 9

Contours of constant vector quadratic piston in the object field. Two nodes of zero aberration are shown.

Fig. 10
Fig. 10

IR scan-mirror system pointed at (a) π/8,(b) π/4, and (c) 3π/8. The fixed pupil of the optical system is shown.

Tables (4)

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Table 1 Optical Design of the LIDAR Off-Axis Beam Expander and the Associated Chief and Marginal Angles of Incidence

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Table 2 Aberration Coefficients for the LIDAR Beam Expander

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Table 3 Specifications for IR Scan-Mirror Systema

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Table 4 Aberration Coefficients for the IR Scan-Mirror Systema

Equations (32)

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h = h x x ^ + h y y ^ ,
H = ( H , θ ) = H x x ^ + H y y ^ ,
ρ = ( ρ , ϕ ) = ρ x x ^ + ρ y y ^ ,
H ρ = ( H ρ , θ + ϕ )
H = ( H , θ 2 )             or            ( H , θ 2 + π )
H · ρ = H x ρ x + H y ρ y = H ρ cos ϕ
2 ( A · B ) ( A 2 · C 2 ) = A 2 ( AB · C 2 ) + ( A 3 · B C 2 ) ,
2 ( A · B ) ( A · C ) = ( A 2 · BC ) + A 2 ( B · C ) ,
2 ( A · B ) ( AB · C 2 ) = A 2 ( B 2 · C 2 ) + B 2 ( A 2 · C 2 ) ,
J ( H , ρ , ϕ ) = exp [ P 0000 σ 0 + P 0200 H 2 σ 0 + P 0111 H ρ cos ϕ σ 0 + P 0020 ρ 2 σ 0 + P 1200 H 2 σ 1 + P 1111 H ρ ( cos ϕ σ 1 - sin ϕ σ 2 ) + P 1022 ρ 2 ( cos 2 ϕ σ 1 - sin 2 ϕ σ 2 ) ] .
J ( H , θ , ρ , ϕ ) = R ( - θ ) J ( H , ρ , ϕ ) R ( θ ) = exp { P 0000 σ 0 + P 0200 H 2 σ 0 + P 0111 H ρ cos ϕ σ 0 + P 0020 ρ 2 σ 0 + P 1200 H 2 ( cos 2 θ σ 1 - sin 2 θ σ 2 ) + P 1111 H ρ [ cos ( ϕ + 2 θ ) σ 1 - sin ( ϕ + 2 θ ) σ 2 ] + P 1022 ρ 2 [ cos ( 2 ϕ + 2 θ ) σ 1 - sin ( 2 ϕ + 2 θ ) σ 2 ] } ,
J ( H , ρ ) = exp ( P 0000 σ 0 + constant piston P 0200 H · H σ 0 + quadratic piston P 0111 H · ρ σ 0 + tilt P 0020 ρ · ρ σ 0 + defocus P 1200 H 2 · A + vector quadratic piston P 1111 H ρ · A + vector tilt P 1022 ρ 2 · A vector defocus ) ,
A = y ^ σ 1 - x ^ σ 2
J ( H , ρ ) = exp [ P 0000 σ 0 + P 0200 H · H σ 0 + P 0111 H · ( ρ + ρ 0 ) σ 0 + P 0020 ( ρ + ρ 0 ) · ( ρ + ρ 0 ) σ 0 + P 1200 H 2 · A + P 1111 H ρ · A + P 1022 ( ρ + ρ 0 ) 2 · A ] .
J ( H , ρ ) = exp [ q P 0000 q σ 0 + q P 0200 q ( H - h q ) · ( H - h q ) σ 0 + q P 0111 q ( H - h q ) · ρ σ 0 = q P 0020 q ρ · ρ σ 0 + q P 1200 q ( H - h q ) 2 · A + q P 1111 q ( H - h q ) ρ · A + q P 1022 q ρ 2 · A ] ,
J ( H , ρ ) = exp { P 1200 [ ( H - a 1200 ) 2 - b 1200 2 ] · A + P 1111 ( H - a 1111 ) ρ · A + P 1022 ρ 2 · A } ,
P 1200 = q P 1200 q ,
a 1200 = 1 P 1200 q P 1200 q h q ,
b 1200 2 = a 1200 2 - 1 P 1200 q P 1200 q h q 2 ,
P 1111 = q P 1111 q ,
a 1111 = 1 P 1111 q P 1111 q h q ,
P 1022 = q P 1022 q .
P 1022 ρ 2 · A .
P 1111 ( H - a 1111 ) ρ · A ,
P 1200 [ ( H - a 1200 ) 2 - b 1200 2 ] · A
( H - a 1200 ) 2 - b 1200 2 = 0
H = a 1200 ± b 1200 .
h 1 = ( α i c , 0 ) ,
h 2 = ( 0 , π 4 i c ) ,
J ( H ) = exp { P 1200 [ ( H - a 1200 ) 2 - b 1200 2 ] · A } ,
a 1200 = π 8 i c ( 1 , 1 ) ,
b 1200 = π 8 i c ( 1 , 1 ) .

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