Abstract

The use of automatically computed aberration functions for an exact determination of the geometric illuminance and caustic surfaces is demonstrated. Numerical results for two optical systems studied using aberration coefficients to the thirteenth order are presented.

© 1981 Optical Society of America

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References

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  1. M. Herzberger, J. Opt. Soc. Am. 37, 485 (1947).
    [CrossRef] [PubMed]
  2. M. Herzberger, Modern Geometrical Optics (Interscience, New York, 1958).
  3. K. Miyamoto, in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1961), Vol. 1, p. 31.
    [CrossRef]
  4. D. L. Shealy, D. G. Burkhard, Opt. Acta 20, 287 (1973).
    [CrossRef]
  5. D. G. Burkhard, D. L. Shealy, J. Opt. Soc. Am. 63, 299 (1973).
    [CrossRef]
  6. D. L. Shealy, D. G. Burkhard, Appl. Opt. 12, 2955 (1973).
    [CrossRef] [PubMed]
  7. D. L. Shealy, D. G. Burkhard, Opt. Acta 22, 485 (1975).
    [CrossRef]
  8. D. G. Burkhard, D. L. Shealy, Appl. Opt. 20, 897 (1981).
    [CrossRef]
  9. T. B. Andersen, Appl. Opt. 19, 3800 (1980).
    [CrossRef] [PubMed]
  10. T. B. Andersen, Appl. Opt. 20, 2754 (1981).
    [CrossRef] [PubMed]
  11. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1975), Chap. 1.5.
  12. Lu Sheng-Dong, Zhang Zhen-Bo, Pub. Beijing Astron. Obs. No. 4, 48 (1979) (in Chinese).

1981 (2)

1980 (1)

1979 (1)

Lu Sheng-Dong, Zhang Zhen-Bo, Pub. Beijing Astron. Obs. No. 4, 48 (1979) (in Chinese).

1975 (1)

D. L. Shealy, D. G. Burkhard, Opt. Acta 22, 485 (1975).
[CrossRef]

1973 (3)

1947 (1)

Andersen, T. B.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1975), Chap. 1.5.

Burkhard, D. G.

Herzberger, M.

M. Herzberger, J. Opt. Soc. Am. 37, 485 (1947).
[CrossRef] [PubMed]

M. Herzberger, Modern Geometrical Optics (Interscience, New York, 1958).

Miyamoto, K.

K. Miyamoto, in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1961), Vol. 1, p. 31.
[CrossRef]

Shealy, D. L.

Sheng-Dong, Lu

Lu Sheng-Dong, Zhang Zhen-Bo, Pub. Beijing Astron. Obs. No. 4, 48 (1979) (in Chinese).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1975), Chap. 1.5.

Zhen-Bo, Zhang

Lu Sheng-Dong, Zhang Zhen-Bo, Pub. Beijing Astron. Obs. No. 4, 48 (1979) (in Chinese).

Appl. Opt. (4)

J. Opt. Soc. Am. (2)

Opt. Acta (2)

D. L. Shealy, D. G. Burkhard, Opt. Acta 20, 287 (1973).
[CrossRef]

D. L. Shealy, D. G. Burkhard, Opt. Acta 22, 485 (1975).
[CrossRef]

Pub. Beijing Astron. Obs. No. (1)

Lu Sheng-Dong, Zhang Zhen-Bo, Pub. Beijing Astron. Obs. No. 4, 48 (1979) (in Chinese).

Other (3)

M. Herzberger, Modern Geometrical Optics (Interscience, New York, 1958).

K. Miyamoto, in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1961), Vol. 1, p. 31.
[CrossRef]

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1975), Chap. 1.5.

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Figures (7)

Fig. 1
Fig. 1

Meridional section of the caustic surface of the f/1.5 thick bi-convex lens for a point object at infinity in the direction (ξ0,η0) = (0,0.4). Full-line curve is the positive sheet, dashed curve is the negative sheet. Focal length is T0000 = 1.5, and paraxial focus is at z = 0.

Fig. 2
Fig. 2

Natural logarithm of the illuminance E0 as a function of yΓ+1 along the line xΓ+1 = 0 on the image plane z = Δ = −0.20 for unit incident illuminance on the entrance pupil.

Fig. 3
Fig. 3

Same as Fig. 2 on the image plane z = Δ = −0.15.

Fig. 4
Fig. 4

Same as Fig. 2 on the image plane z = Δ = −0.10.

Fig. 5
Fig. 5

Same as Fig. 2 on the Gaussian image plane z = Δ = 0.

Fig. 6
Fig. 6

Meridional section of the caustic surface of the 1.5-m telescope for the on-axis image. Explanation to curves is the same as for Fig. 1.

Fig. 7
Fig. 7

Same as Fig. 6 for the (ξ0,η0) = (0,0.003) off-axis image. Point indicated at the center of the figure is the point Pg on the best focal surface minimizing the spot radius kg for that object direction.

Tables (1)

Tables Icon

Table I Optical Specifications of the Simple Bi-Convex Lens

Equations (41)

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ρ = x 0 2 + y 0 2 , ψ = ξ 0 2 + η 0 2 , κ = x 0 ξ 0 + y 0 η 0 .
( x Γ + 1 y Γ + 1 ) = S ( ρ , ψ , κ ) ( x 0 y 0 ) + T ( ρ , ψ , κ ) ( ξ 0 η 0 ) ,
( ξ Γ + 1 ξ Γ + 1 ) = V ( ρ , ψ , κ ) ( x 0 y 0 ) + W ( ρ , ψ , κ ) ( ξ 0 η 0 ) ,
V = V , W = W , S = S + V Δ , T = T + W Δ .
E 0 = σ 1 + ψ | d S Ω d S Π | = σ 1 + ψ | d S Π d S Ω | - 1 .
d S Π d S Ω = x Γ + 1 x 0 y Γ + 1 y 0 - x Γ + 1 y 0 y Γ + 1 x 0 .
d S Π d S Ω = S ( S + 2 ρ S ρ + κ S κ + 2 κ T ρ + ψ T κ ) + 2 ( ρ ψ - κ 2 ) ( S ρ T κ - S κ T ρ ) .
d i + f i + 1 - f i N i + 1 ,
E 1 = E 0 · t 1 t Γ · exp { - i = 1 Γ 1 N i + 1 [ k i d i + ( N i + 1 N i k i - 1 - k i ) f i ] } ,
t i = 2 μ i 2 cos 2 θ i { 1 ( cos θ i + μ i cos θ i ) 2 + 1 ( μ i cos θ i + cos θ i ) 2 } ,
d S Π d S Ω = Λ 0 ( ρ , ψ , κ ) + Λ 1 ( ρ , ψ , κ ) Δ + Λ 2 ( ρ , ψ , κ ) Δ 2 ,
Λ 0 ( ρ , ψ , κ ) = S ( S + 2 ρ S ρ + κ S κ + 2 κ T ρ + ψ T κ ) + 2 ( ρ ψ - κ 2 ) ( S ρ T κ - S κ T ρ ) ,
Λ 1 ( ρ , ψ κ ) = V ( S + 2 ρ S ρ + κ S κ + 2 κ T ρ + ψ T κ ) + S ( V + 2 ρ V ρ + κ V κ + 2 κ W ρ + ψ W κ ) + 2 ( ρ ψ - κ 2 ) ( S ρ W κ - S κ W ρ + V ρ T κ - V κ T ρ ) ,
Λ 2 ( ρ , ψ , κ ) = V ( V + 2 ρ V ρ + κ V κ + 2 κ W ρ + ψ W κ ) + 2 ( ρ ψ - κ 2 ) ( V ρ W κ - V κ W ρ ) ,
Δ caust ( ρ , ψ , κ ) = - Λ 1 ± Λ 1 2 - 4 Λ 0 Λ 2 2 Λ 2 .
Δ caust - ( ρ , ψ , κ ) = - S ( ρ , ψ , κ ) V ( ρ , ψ , κ ) , negative sheet , ρ ψ - κ 2 = 0 Δ caust + ( ρ , ψ , κ ) = - S + 2 ρ S ρ + κ S κ + 2 κ T ρ + ψ T κ V + 2 ρ V ρ + κ V κ + 2 κ W ρ + ψ W κ , positive sheet , ρ ψ - κ 2 = 0
Δ caust ( y 0 ) = { - S ( y 0 ) V ( y 0 ) , negative sheet - S ( y 0 ) + y 0 d S d y 0 V ( y 0 ) + y 0 d V d y 0 , positive sheet ,
y Γ + 1 ( y 0 ) = [ S ( y 0 ) + V ( y 0 ) · Δ caust ( y 0 ) ] y 0
Δ caust + ( y 0 ) = - d d y 0 [ y 0 S ( y 0 ) ] d d y 0 [ y 0 V ( y 0 ) ] ,
y Γ + 1 ( y 0 ) = y 0 S ( y 0 ) + y 0 V ( y 0 ) Δ caust + ( y 0 ) .
y 0 S ( y 0 ) = n = 1 S n n 00 y 0 2 n + 1 , y 0 V ( y 0 ) = n = 0 V n n 00 y 0 2 n + 1 ,
Δ caust + ( 0 ) = 0 ;             d y Γ + 1 d y 0 = 0 ,
d 2 n + 1 Δ caust + d y 0 2 n + 1 ( 0 ) = 0 , n 1 ;             d 2 n y Γ + 1 d y 0 2 n ( 0 ) = 0 , n 0 ,
d 2 n Δ caust + d y 0 2 n ( 0 ) = - ( 2 n + 1 ) ! S n n 00 V 0000 - i = 1 n - 1 ( 2 n 2 i ) ( 2 n - 2 i + 1 ) ! × V n - i , n - i , 0 , 0 V 0000 d 2 i Δ caust + d y 0 2 i ( 0 ) , n 1
d 2 n + 1 y Γ + 1 d y 0 2 n + 1 ( 0 ) = i = 1 n ( 2 n ) ! ( 2 i - 1 ) ! V n - i , n - i , 0 , 0 d 2 i Δ caust + d y 0 2 i ( 0 ) , n 1.
d 2 j Δ caust + d y 0 2 j ( 0 ) = 0 ; d 2 j + 1 y Γ + 1 d y 0 2 j + 1 ( 0 ) = 0 for all j < n ,
d 2 n Δ caust + d y 0 2 n ( 0 ) = - ( 2 n + 1 ) ! S n n 00 V 0000 , d 2 n + 1 y Γ + 1 d y 0 2 n + 1 ( 0 ) = - 2 n ( 2 n + 1 ) ! S n n 00 .
G ( ρ , ψ , κ ) = n = 0 j = 0 n k = 0 j G n , n - j , j - k ρ n - j ψ j - k κ k ,
G ρ = n = 0 j = 0 n k = 0 j ( n - j ) G n , n - j , j - k , k ρ n - j - 1 ψ j - k κ k = n = 0 j = 0 n k = 0 j ( n + 1 - j ) G n + 1 , n + 1 - j , j - k , k ρ n - j ψ j - k κ k ,
G ρ = n = 0 j = 0 n k = 0 j k G n , n - j , j - k , k ρ n - j ψ j - k κ k - 1 = n = 0 j = 0 n k = 0 j ( k + 1 ) G n + 1 , n - j , j - k , k + 1 ρ n - j ψ j - k κ k .
a i = 1 N i + 1 [ k i d i + ( N i + 1 N i k i - 1 - k i ) f i ]
a = i = 1 Γ a i .
A = A 0000 + X ,
exp ( A ) = e A 0000 ( 1 + n = 1 m 1 n ! X * n ) ,
Z ( ρ , ψ , κ ) = σ · t 1 t Γ 1 + ψ exp [ - a ( ρ , ψ , κ ) ] ,
E 1 = Z ( ρ , ψ , κ ) / Λ 0 ( ρ , ψ , κ ) .
S ( y 0 ) = 1 - ( y 0 / R ) 2 - 1 1 - 2 ( y 0 / R ) 2 ,
R · V ( y 0 ) = 2 1 - ( y 0 / R ) 2 1 - 2 ( y 0 / R ) 2 .
Δ caust + ( y 0 ) R = ½ - ½ [ 1 + 2 ( y 0 / R ) 2 ] 1 - ( y 0 / R ) 2 ,
y Γ + 1 ( y 0 ) R = y 0 R [ S ( y 0 ) + V ( y 0 ) Δ caust + ( y 0 ) ] = y 0 R [ ( y 0 / R ) 2 - 2 ( y 0 / R ) 4 1 - 2 ( y 0 / R ) 2 ] = ( y 0 R ) 3 ,
Δ caust + R = ½ { 1 - [ 1 + 2 ( y Γ + 1 R ) 2 / 3 ] 1 - ( y Γ + 1 R ) 2 / 3 } .

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