Abstract

By generalizing the wave aberration function to include plane symmetric systems, we describe the aberration fields for a combination of plane symmetric systems. The combined system aberration coefficients for the fields of spherical aberration, coma, astigmatism, defocus and distortion depend on the individual aberration coefficients and the orientations of the individual plane symmetric component systems. The aberration coefficients can be used to calculate the locations of the field nodes for the different types of aberration, including coma, astigmatism, defocus and distortion. This work provides an alternate view for combining aberrations in optical systems.

© 2008 Optical Society of America

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Errata

Lori B. Moore, Anastacia M. Hvisc, and Jose Sasian, "Aberration fields of a combination of plane symmetric systems: erratum," Opt. Express 17, 15390-15391 (2009)
https://www.osapublishing.org/oe/abstract.cfm?uri=oe-17-17-15390

References

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  1. R. A. Buchroeder, "Tilted component optical systems," Ph.D. dissertation (University of Arizona, Tucson, Arizona, 1976).
  2. R. V. Shack, "Aberration theory, OPTI 514 course notes," College of Optical Sciences, University of Arizona, Tucson, Arizona.
  3. K. P. Thompson, "Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry," J. Opt. Soc. Am. A,  22, 1389-1401 (2005).
    [CrossRef]
  4. K. P. Thompson, "Aberration fields in tilted and decentered optical systems," Ph.D. dissertation (University of Arizona, Tucson, Arizona, 1980).
  5. J. M. Sasian, "Imagery of the Bilateral Symmetric Optical System," Ph.D. dissertation (University of Arizona, Tucson, Arizona, 1988).
  6. J. M. Sasian, "How to approach the design of a bilateral symmetric optical system," Opt. Eng. 33, 2045 (1994).
  7. M. Andrews, "Concatenation of characteristic functions in Hamiltonian optics," J. Opt. Soc. Am. 72, 1493-1497 (1982).
    [CrossRef]
  8. G. Forbes, "Concatenation of restricted characteristic functions," J. Opt. Soc. Am. 72, 1702-1706 (1982).
    [CrossRef]
  9. B. Stone and G. Forbes, "Foundations of first-order layout for asymmetric systems: an application of Hamilton's methods," J. Opt. Soc. Am. A 9, 96-109 (1992).
    [CrossRef]

2005

1994

J. M. Sasian, "How to approach the design of a bilateral symmetric optical system," Opt. Eng. 33, 2045 (1994).

1992

1982

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Eng.

J. M. Sasian, "How to approach the design of a bilateral symmetric optical system," Opt. Eng. 33, 2045 (1994).

Other

R. A. Buchroeder, "Tilted component optical systems," Ph.D. dissertation (University of Arizona, Tucson, Arizona, 1976).

R. V. Shack, "Aberration theory, OPTI 514 course notes," College of Optical Sciences, University of Arizona, Tucson, Arizona.

K. P. Thompson, "Aberration fields in tilted and decentered optical systems," Ph.D. dissertation (University of Arizona, Tucson, Arizona, 1980).

J. M. Sasian, "Imagery of the Bilateral Symmetric Optical System," Ph.D. dissertation (University of Arizona, Tucson, Arizona, 1988).

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Figures (7)

Fig. 1.
Fig. 1.

Conventions for the plane symmetry, field and aperture vectors.

Fig. 2.
Fig. 2.

Possible orientation of vectors i j as seen looking down the optical axis. Notice that the vectors are all unit vectors in different directions. (The optical axis is in and out of the page at (x,y)=(0,0).)

Fig. 3.
Fig. 3.

A line node in the field of defocus from a combination of field tilts. (Color online: Red is focused before the image plane.)

Fig. 4.
Fig. 4.

A circular node in the field of defocus from a combination of field curvature and constant defocus. (Color online: Red is focused before the image plane.)

Fig. 5.
Fig. 5.

Anamorphism can be described as an average magnification (center figure) plus an anamorphic term (figure on the right).

Fig. 6.
Fig. 6.

Astigmatic surfaces and their relationship according to the astigmatism terms in Eq. (4). Defocus from the image plane is in the ΔZ direction. (Color online: The medial surface is shown in blue)

Fig. 7.
Fig. 7.

Location and orientation of the astigmatic surfaces for the linear, constant, and quadratic astigmatism with respect to the medial surface and as mathematically represented in Table 2. Defocus from the medial surface is in the ΔZ direction. The dashed lines in the linear astigmatism figure highlight the locations of the sagittal and tangential foci. (Color online: The medial surface is blue. The tangential foci are red. The sagittal foci are green.)

Tables (7)

Tables Icon

Table 1. Coefficient and vector definitions

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Table 2. Aberration fields of a combination of plane symmetric systems

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Table 3. Aberration field components that contribute to distortion mapping errors, shown in both grid and vector plot forms. In the grid form, the dotted lines show the nominal mapping positions of a square grid. The solid line shows the distortion of the square grid. In the vector form, the vectors show the magnitude and direction of the mapping distortion. All i vectors used in creating these graphs are pointing to the right.

Tables Icon

Table 4. Aberration field components. The field of defocus is represented by circles that convey the size of the defocused image across the field. Astigmatism is represented by a projection of the astigmatic focal lines. Coma is represented by a collection of circles. Each circle represents a fixed magnitude of the aperture vector, ρ, thus the collection of circles show the magnitude and orientation of the aberration in the field. Spherical aberration is represented by circles that show the magnitude of the aberration. All i vectors used in creating these graphs are pointing to the right. (Color online: Red denotes locations in the field where the focus position is before the image plane)

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Table 5. Some distortion plots showing 2, 3, and 4 nodes. The surface maps represent the magnitude of the distortion. The points where the surfaces meet the plane are the nodes. In the vector plots, the vectors represent both the magnitude and direction of the mapping distortion. The shading represents the magnitude of the distortion. Darker shades have less distortion. All i vectors used in creating these graphs are pointing to the right.

Tables Icon

Table 6. Some distortion plots showing line and circular nodes. The surface maps represent the magnitude of the distortion. In the vector plots, the vectors represent both the magnitude and direction of the mapping distortion. The shading represents the magnitude of the distortion. Darker shades have less distortion. All i vectors used in creating these graphs are pointing to the right.

Tables Icon

Table 7. Transverse ray aberrations

Equations (41)

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W ( H , ρ ) = k , m , n W 2 k + n , 2 m + n , n ( H · H ) k ( ρ · ρ ) m ( H · ρ ) n
W ( H , ρ , i ) = k , m , n , p , q W 2 k + n + p , 2 m + n + q , n , p , q ( H · H ) k ( ρ · ρ ) m ( H · ρ ) n ( i · H ) p ( i · ρ ) q
W ( H , ρ , i ) = ( W 00000 + W 01001 ( i · ρ ) + W 10010 ( i · H ) + W 02000 ( ρ · ρ ) + W 11100 ( H · ρ ) + W 20000 ( H · H ) + W 02002 ( i · ρ ) 2 + W 11011 ( i · H ) ( i · ρ ) + W 20020 ( i · H ) 2 + W 03001 ( i · ρ ) ( ρ · ρ ) + W 12101 ( i · ρ ) ( H · ρ ) + W 12010 ( i · H ) ( ρ · ρ ) + W 21001 ( i · ρ ) ( H · H ) + W 21110 ( i · H ) ( H · ρ ) + W 30010 ( i · H ) ( H · H ) + W 04000 ( ρ · ρ ) 2 + W 13100 ( H · ρ ) ( ρ · ρ ) + W 22200 ( H · ρ ) 2 + W 22000 ( H · H ) ( ρ · ρ ) + W 31100 ( H · H ) ( H · ρ ) + W 40000 ( H · H ) 2 ) .
W ( H , ρ ) = 1 j ( W 00000 j + W 01001 j ( i j · ρ ) + W 10010 j ( i j · H ) + W 02000 j ( ρ · ρ ) + W 11100 j ( H · ρ ) + W 20000 j ( H · H ) + W 02002 j ( i j · ρ ) 2 + W 11011 j ( i j · H ) ( i j · ρ ) + W 20020 j ( i j · H ) 2 + W 03001 j ( i j · ρ ) ( ρ · ρ ) + W 12101 j ( i j · ρ ) ( H · ρ ) + W 12010 j ( i j · H ) ( ρ · ρ ) + W 21001 j ( i j · ρ ) ( H · H ) + W 21110 j ( i j · H ) ( H · ρ ) + W 30010 j ( i j · H ) ( H · H ) + W 04000 j ( ρ · ρ ) 2 + W 13100 j ( H · ρ ) ( ρ · ρ ) + W 22200 j ( H · ρ ) 2 + W 22000 j ( H · H ) ( ρ · ρ ) + W 31100 j ( H · H ) ( H · ρ ) + W 40000 j ( H · H ) 2 ) .
i cc = Σ j W 03001 j i j Σ j W 03001 j i j .
W cc = Σ j W 03001 j i j .
A = α exp ( i α ) = a ( sin α i ̂ + cos α j ̂ )
B = b exp ( i β ) = b ( sin β i ̂ + cos β j ̂ ) ,
A B = ab exp ( i ( α + β ) ) = ab ( sin ( α + β ) i ̂ + cos ( α + β ) j ̂ ) .
B * = b exp ( i β ) = b ( sin ( β ) i ̂ + cos ( β ) j ̂ ) = b ( sin β i ̂ + cos β j ̂ ) .
( A · B ) ( A · C ) = 1 2 [ ( A · A ) ( B · C ) + A 2 · B C ] ,
and ( A · B C ) = ( A B * · C ) .
W cc i cc + w lc H = 0 .
H = W cc W lc i cc .
W ca i ca 2 + W la i la H + W qa H 2 = 0 .
H = 1 2 W la W qa i la ± 1 4 ( W la W qa ) 2 i la 2 W ca W qa i ca 2 .
H = 0 and H = W la W qa i la .
H = ± W ca W qa i ca .
H = W ca W la i ca 2 i la .
W d + W dca + W ft ( i ft · H ) + W la ( i la · H ) + ( W qa + W fc ) ( H · H ) = 0 .
W ft ( i ft · H ) + W la ( i la · H ) = 0
or ( W ft i ft + W la i la ) · H = 0 .
( W ft i ft + W la i la ) · H = ( W d + W dca ) .
W d + W dca + ( W qa + W fc ) ( H · H ) W d + W dca + ( W qa + W fc ) H 2 = 0
H = ( W d + W dca W qa + W fc ) .
W fd i fd + ( W m + W ma ) H + W a i a 2 H * + ( W qdII i qd II · H ) H + ( H · H ) ( W qd I i qd I + W cd H ) = 0 .
{ W fd i fd + W qdI ( H · H ) i qdI } · ρ .
W fd i fd + W qdI i qdI ( H · H ) = 0
H = W fd W qdI .
( W m + W ma ) H + W cd ( H · H ) H ( ( W m + W ma ) + W cd ( H · H ) ) H = 0 ,
H = ( W m + W ma ) W cd .
j W 11011 j ( i j · H ) ( i j · ρ ) .
j W 11011 j ( i j · H ) ( i j · ρ ) = j ( 1 2 W 11011 j ( H · ρ ) + 1 2 W 11011 j i j 2 H * · ρ )
= W ma ( H · ρ ) + W a i a 2 H * · ρ .
j W 12101 j ( i j · ρ ) ( H · ρ ) = W la ( i la · H ) ( ρ · ρ ) + W la i la H · ρ 2 .
ε = 1 n u ' ρ W ( H , ρ )
( a · ρ ) = a
( ρ · ρ ) = 2 ρ
( a · ρ 2 ) = 2 a ρ *
( a · ρ ) ( ρ · ρ ) = ( ρ · ρ ) a + 2 ( a · ρ ) ρ
and ( ρ · ρ ) 2 = 4 ( ρ · ρ ) ρ

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