Abstract

In this paper we succeeded in deriving changes in the nodal positions of aberrations that belong to the fifth-order class in pupil dependence by applying a system level pupil decentration vector. Our treatment is specifically for rotationally symmetric multi-mirror optical designs that simply use an offset pupil as a means of creating an unobscured optical design. When the pupil is offset, only the vectors to determine the node locations are modified by the pupil decentration vector, while the nodal properties originally developed for titled/decentered optical systems are retained. In general, the modifications to the nodal vectors for any particular aberration type are contributed only by terms of higher order pupil dependence.

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References

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  1. R. A. Buchroeder, “Tilted component optical systems,” Ph.D. dissertation (Optical Science Center, University of Arizona, Tucson, Arizona 1976).
  2. K. P. Thompson, “Aberration fields in tilted and decentered optical systems,” Ph.D. dissertation (Optical Science Center, University of Arizona, Tucson, Arizona 1980).
  3. J. R. Rogers, W. H. Taylor and D. T. Moore, eds., “Vector aberration theory and the design of off-axis systems,” in International Lens Design Conference, W. H. Taylor and D. T. Moore, eds. Proc. SPIE 554, 76–81 (1985).
  4. J. W. Figoski, W. H. Taylor and D. T. Moore, eds., “Aberration characteristics of nonsymmetric systems,” in International Optical Design Conference, W.H. Taylor and D.T. Moore, eds. Proc. SPIE 554, 104–111 (1985).
  5. S. H. Chang and A. Prata., “Geometrical theory of aberrations near the axis in classical off-axis reflecting telescopes,” J. Opt. Soc. Am. A22(11), 2454–2464 (2005).
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  6. J. Wang, B. H. Guo, Q. Sun, and Z. W. Lu, “Third-order aberration fields of pupil decentered optical systems,” Opt. Express20(11), 11652–11658 (2012).
    [CrossRef] [PubMed]
  7. C. Menke and G. W. Forbes, “Optical design with orthogonal representations of rotationally symmetric and freeform aspheres,” Adv. Opt. Technol.2(1), 97–109 (2013).
  8. J. L. F. de Meijere and C. H. F. Velzel, “Dependence of third- and fifth-order aberration coefficients on the definition of pupil coordinates,” J. Opt. Soc. Am. A6(10), 1609–1617 (1989).
    [CrossRef]
  9. K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: spherical aberration,” J. Opt. Soc. Am. A26(5), 1090–1100 (2009).
    [CrossRef] [PubMed]
  10. J. Sasián, “Theory of sixth-order wave aberrations,” Appl. Opt.49(16), D69–D95 (2010).
    [CrossRef] [PubMed]
  11. H. H. Hopkins, The Wave Theory of Aberrations (Oxford on Clarendon, 1950).
  12. K. P. Thompson, “Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry,” J. Opt. Soc. Am. A22(7), 1389–1401 (2005).
    [CrossRef] [PubMed]
  13. L. B. Moore, A. M. Hvisc, and J. Sasian, “Aberration fields of a combination of plane symmetric systems,” Opt. Express16(20), 15655–15670 (2008).
    [CrossRef] [PubMed]

2013 (1)

C. Menke and G. W. Forbes, “Optical design with orthogonal representations of rotationally symmetric and freeform aspheres,” Adv. Opt. Technol.2(1), 97–109 (2013).

2012 (1)

2010 (1)

2009 (1)

2008 (1)

2005 (2)

1989 (1)

Chang, S. H.

de Meijere, J. L. F.

Forbes, G. W.

C. Menke and G. W. Forbes, “Optical design with orthogonal representations of rotationally symmetric and freeform aspheres,” Adv. Opt. Technol.2(1), 97–109 (2013).

Guo, B. H.

Hvisc, A. M.

Lu, Z. W.

Menke, C.

C. Menke and G. W. Forbes, “Optical design with orthogonal representations of rotationally symmetric and freeform aspheres,” Adv. Opt. Technol.2(1), 97–109 (2013).

Moore, L. B.

Prata, A.

Sasian, J.

Sasián, J.

Sun, Q.

Thompson, K. P.

Velzel, C. H. F.

Wang, J.

Adv. Opt. Technol. (1)

C. Menke and G. W. Forbes, “Optical design with orthogonal representations of rotationally symmetric and freeform aspheres,” Adv. Opt. Technol.2(1), 97–109 (2013).

Appl. Opt. (1)

J. Opt. Soc. Am. A (4)

Opt. Express (2)

Other (5)

H. H. Hopkins, The Wave Theory of Aberrations (Oxford on Clarendon, 1950).

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

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

J. R. Rogers, W. H. Taylor and D. T. Moore, eds., “Vector aberration theory and the design of off-axis systems,” in International Lens Design Conference, W. H. Taylor and D. T. Moore, eds. Proc. SPIE 554, 76–81 (1985).

J. W. Figoski, W. H. Taylor and D. T. Moore, eds., “Aberration characteristics of nonsymmetric systems,” in International Optical Design Conference, W.H. Taylor and D.T. Moore, eds. Proc. SPIE 554, 104–111 (1985).

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Figures (5)

Fig. 1
Fig. 1

Definitions of pupil vector ρ and field vector H .

Fig. 2
Fig. 2

Operation algorithm for the vector multiplication of A and B . i and j are the normalized vectors in XOY coordinate system.

Fig. 3
Fig. 3

Pupil vector conversion during the pupil offset.

Fig. 4
Fig. 4

Two nodal points of fifth-order aberration term W l=4 (2) . Nodal positions locate at H = a 242S +i b 242S and H = a 242S i b 242S , respectively.

Fig. 5
Fig. 5

The aberration field property of term Wl = 5 for a pupil-decentered optical system. Fifth-order aberration term Wl = 5is displaced in the image plane to the point located by H = a 151 S .

Equations (48)

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W j ( H,ρ,ρcosθ )= p n m ( W klm ) j ( H 2 ) p ( ρ 2 ) n ( Hρcosθ ) m = p n m ( W klm ) j H 2p+m ρ 2n+m cos m θ ,
ρ =| ρ |exp( iϕ )= ρ x x+ ρ y y,
H =| H |exp( iφ )= H x x+ H y y,
W= W j [ ( H H ),( ρ ρ ),( H ρ ) ]= j p n m ( W klm ) j ( H H ) p ( ρ ρ ) n ( H ρ ) m .
A =aexp( iα )= a x i + a y j , a x =asinα, a y =acosα,
B =aexp( iα )= a x i + a y j , a x =asinα, a y =acosα.
A B =abexp[ i( α+β ) ]=( a x a x + a x a x ) i +( a x a x + a x a x ) i =absin( α+β ) i +absin( α+β ) i .
A 2 = a 2 exp( i2α )= ( A 2 ) x i + ( A 2 ) y j ,
( A B ) 3 = A 3 3 A 2 B +3 A B 2 B 3 ,
2( A B )( A C )=( A A )( B C )+ A 2 B C .
ρ' = ρ + S ,
W Fifth = j W 060j ( ρ ρ ) 3 + j W 151j ( H ρ ) ( ρ ρ ) 2 + j W 240j ( H H ) ( ρ ρ ) 2 + j W 242j ( H ρ ) 2 ( ρ ρ )+ j W 331j ( H H )( H ρ )( ρ ρ )+ j W 333j ( H ρ ) 3 + j W 420j ( H H ) 2 ( ρ ρ )+ j W 422j ( H H ) ( H ρ ) 2 + j W 511j ( H H ) 2 ( H ρ ).
W Fifth S = j W 060j [ ( ρ + S )( ρ + S ) ] 3 + j W 151j [ H ( ρ + S ) ] [ ( ρ + S )( ρ + S ) ] 2 + j W 240Mj ( H H ) [ ( ρ + S )( ρ + S ) ] 2 + j W 242j [ H ( ρ + S ) ] 2 [ ( ρ + S )( ρ + S ) ] + j W 331Mj ( H H )[ H ( ρ + S ) ][ ( ρ + S )( ρ + S ) ]+ j W 420Mj ( H H ) 2 [ ( ρ + S )( ρ + S ) ] + j W 333j [ H ( ρ + S ) ] 3 + j W 422j ( H H ) [ H ( ρ + S ) ] 2 + j W 511j ( H H ) 2 [ H ( ρ + S ) ],
W 240M = W 240 + 1 2 W 242 ,
W 331M = W 331 + 3 4 W 333 ,
W 420M = W 420 + 1 2 W 422 ,
W l=4 (1) = W 240M ( H H ) ( ρ ρ ) 2 +3 W 151 ( H S ) ( ρ ρ ) 2 +9 W 060 ( S S ) ( ρ ρ ) 2 =[ W 240M ( H H )+3 W 151 ( H S )+9 W 060 ( S S ) ] ( ρ ρ ) 2 ,
A 240 M S =1.5 W 151 S ,
a 240 M S = A 240 M S / W 240 ,
B 240 M S =9 W 060 ( S S ),
b 240 M S = B 240M S / W 240M a 240 M S a 240 M S .
W l=4 (1) = W 240M [ ( H a 240 M S )( H a 240M S )+ b 2 40M S ] ( ρ ρ ) 2 .
A 240M = A 240M + A 240 M S = A 240M 1.5 W 151 S ,
B 240M = B 240M + B 240 M S = B 240M +9 W 060 ( S S ),
W l=4 (2) = 1 2 W 242 ( H 2 ρ 2 )( ρ ρ )+2 W 151 ( H S ρ 2 )( ρ ρ )+6 W 060 ( S 2 ρ 2 )( ρ ρ ) = 1 2 ( W 242 H 2 +4 W 151 H S +12 W 060 S 2 ) ρ 2 ( ρ ρ ).
A 242 S =2 W 151 S ,
B 242 S 2 =12 W 060 S 2 ,
a 242 S = A 242 S / W 242 ,
b 242 S 2 = B 242 S 2 / W 242 a 242 S 2 ,
W l=4 (2) = 1 2 ( W 242 H 2 2 H A 242 S + B 242 S 2 ) ρ 2 ( ρ ρ ) = 1 2 W 242 [ ( H a 242 S ) 2 + b 242 S 2 ] ρ 2 ( ρ ρ ).
A 242 = A 242 + A 242 S = A 242 2 W 151 S ,
B 242 2 = B 242 2 + B 242 S 2 = B 242 2 +9 W 060 ( S S ).
W l=5 = W 151 ( H ρ ) ( ρ ρ ) 2 +6 W 060 ( S ρ ) ( ρ ρ ) 2 =( W 151 H A 151 S ) ρ ( ρ ρ ) 2 = W 151 ( H a 151 S ) ρ ( ρ ρ ) 2 ,
A 151 = A 151 + A 151 S = A 151 +6 W 060 S .
W 242 = j W 242j ( H ρ ) 2 ( ρ ρ )= j W 242j [ H ( ρ + S ) ] 2 ( ρ + S )( ρ + S ) = W 242 [ ( H ρ ) 2 ( ρ ρ )+ ( H S ) 2 ( ρ ρ )+4( H ρ )( ρ S )( H S ) +2( H ρ )( H S )( ρ ρ )+ ( H ρ ) 2 ( S S )+2 ( H ρ ) 2 ( ρ S ) +2 ( H S ) 2 ( ρ S )+ ( H S ) 2 ( S S )+2( H ρ )( H S )( S S ) ].
W 331M = j W 331Mj ( H H )( H ρ )( ρ ρ )= j W 331Mj ( H H )[ H ( ρ + S ) ][ ( ρ + S )( ρ + S ) ] = W 331M [ ( H H )( H ρ )( ρ ρ )+2( H H )( H ρ )( ρ S )+( H H )( H ρ )( S S ) +( H H )( H S )( ρ ρ )+( H H )( H S )( S S )+2( H H )( H S )( ρ S ) ].
W 333 = j W 333j ( H ρ ) 3 = j W 333j [ H ( ρ + S ) ] 3 = W 333 [ ( H ρ ) 3 + ( H S ) 3 +3 ( H ρ ) 2 ( H S )+3( H ρ ) ( H S ) 2 ].
W 420M = j W 420Mj ( H H ) 2 ( ρ ρ )= j W 240Mj ( H H ) 2 [ ( ρ + S )( ρ + S ) ] = W 420M [ ( H H ) 2 ( ρ ρ )+2 ( H H ) 2 ( ρ S )+ ( H H ) 2 ( S S ) ].
W 422 = j W 422j ( H H ) ( H ρ ) 2 = j W 240j ( H H ) [ H ( ρ + S ) ] 2 = W 422 [ ( H H ) ( H ρ ) 2 +2( H H )( H ρ )( H S )+( H H ) ( H S ) 2 ].
W 060 = j W 060j ( ρ ρ ) 3 = j W 060j [ ( ρ + S )( ρ + S ) ] 3 = W 060 [ ( ρ ρ ) 3 + ( S S ) 3 +6( ρ S ) ( ρ ρ ) 2 +18( S S )( ρ S )( ρ ρ ) +2( ρ 3 S 3 )+9( S S ) ( ρ ρ ) 2 +9( ρ ρ ) ( S S ) 2 +6 ( S S ) 2 ( ρ S )+6( ρ ρ )( ρ 2 S 2 )+6( S S )( ρ 2 S 2 ) ].
W 151 = j W 151j ( H ρ ) ( ρ ρ ) 2 = j W 151j [ ( ρ + S )( ρ + S ) ] 2 [ H ( ρ + S ) ] = W 151 [ ( H ρ )( ρ ρ )( ρ ρ )+( H ρ )( S S )( S S )+4( S S )( H ρ )( ρ ρ ) +4( ρ S )( H S )( S S )+4( S S )( H ρ )( ρ S )+( H S )( ρ ρ )( ρ ρ ) +( H S )( S S )( S S )+4( ρ ρ )( ρ S )( H ρ )+2( H S )( ρ 2 S 2 ) +4( H S )( S S )( ρ ρ )+4( ρ S )( H S )( ρ ρ )+2( H ρ )( ρ 2 S 2 ) ].
4( ρ ρ )( ρ S )( H ρ )=2( ρ ρ )[ ( ρ ρ )( H S )+ ρ 2 H S ].
W 151 = j W 151j ( H ρ ) ( ρ ρ ) 2 = j W 151j [ ( ρ + S )( ρ + S ) ] 2 [ H ( ρ + S ) ] = W 151 [ ( H ρ )( ρ ρ )( ρ ρ )+( H ρ )( S S )( S S )+4( S S )( H ρ )( ρ ρ ) +4( ρ S )( H S )( S S )+4( S S )( H ρ )( ρ S )+3( H S )( ρ ρ )( ρ ρ ) +( H S )( S S )( S S )+2( ρ ρ )( ρ 2 H S )+2( H S )( S 2 ρ 2 ) +4( H S )( S S )( ρ ρ )+4( ρ S )( H S )( ρ ρ )+2( H ρ )( ρ 2 S 2 ) ].
W 511 = j W 511j ( H H ) 2 ( H ρ )= j W 511j ( H H ) 2 [ H ( ρ + S ) ] = W 511 [ ( H H ) 2 ( H ρ )+ ( H H ) 2 ( H S ) ].
W 240M = j W 240Mj ( H H ) ( ρ ρ ) 2 = j W 240Mj ( H H ) ( ( ρ + S )( ρ + S ) ) 2 = W 240M [ ( H H ) ( ρ ρ ) 2 +4( H H )( ρ ρ )( S S )+2( H H )( ρ 2 S 2 ) +( H H ) ( S S ) 2 +4( H H )( ρ ρ )( ρ S )+4( H H )( ρ S )( S S ) ].
A klm = j W klmj ,
B klm = j W klmj ( σ j σ j ) ,
B klm 2 = j W klmj σ j 2 ,

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