Abstract

We give a complete description of the calculation of the eikonal coefficients of symmetric optical systems. First, we show how to express the eikonal coefficients of each component as functions of the component data and the position of the object and the stop. We decompose each eikonal into meridional and sagittal subseries, and we give a proof of the uniqueness of this decomposition. Second, we give simple formulas to calculate system eikonal coefficients of the sixth order in the direction cosines and a fast algorithm to obtain a good estimate of coefficients of orders higher than the sixth. We conclude with a discussion on pseudoaberrations and their uses and also a generalization of the Seidel formulas for the addition of aberrations.

© 1988 Optical Society of America

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References

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  1. C. H. F. Velzel, J. L. F. de Meijere, “Characteristic functions and the aberrations of symmetric optical systems. I. Transverse aberrations when the eikonal is given,” J. Opt. Soc. Am. A 5, 246–250 (1988).
    [CrossRef]
  2. C. H. F. Velzel, J. L. F. de Meijere, “Characteristic functions and the aberrations of symmetric optical systems. II. Addition of aberrations,” J. Opt. Soc. Am. A 5, 251–256 (1988).
    [CrossRef]
  3. T. Smith, “The changes in aberrations when the object and stop are moved,” Trans. Opt. Soc. London 23, 311–322 (1921–1922).
  4. C. H. F. Velzel, “Image formation by a general optical system, using Hamilton’s method,” J. Opt. Soc. Am. A 4, 1342–1348 (1987).
    [CrossRef]
  5. G. C. Steward, The Symmetric Optical System (Cambridge U. Press, London, 1927).
  6. J. A. Grace, A. Young, The Algebra of Invariants (Cambridge U. Press, Cambridge, 1903).
  7. R. J. Pegis, “The modern development of Hamiltonian optics,” in Progress in Optics I, E. Wolf, ed. (Elsevier/North-Holland, Amsterdam, 1964).
  8. O. Stavroudis, Modular Optical Design, Vol. 28 of Springer Series in Optics (Springer-Verlag, New York, 1982).
    [CrossRef]
  9. D. Shafer, “Optical design methods: your head as a personal computer,” in Geometrical Optics: Critical Review of Technology, R. E. Fischer, W. H. Price, W. J. Smith, eds., Proc. Soc. Photo-Opt. Instrum. Eng.531, 49–58 (1985).
    [CrossRef]
  10. D. Shafer, “Optical design and relaxation response,” in Recent Trends in Optical Systems Design: Computer Lens Design Workshop, C. Londono, R. E. Fischer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.766, 2–8 (1987).
    [CrossRef]

1988 (2)

1987 (1)

de Meijere, J. L. F.

Grace, J. A.

J. A. Grace, A. Young, The Algebra of Invariants (Cambridge U. Press, Cambridge, 1903).

Pegis, R. J.

R. J. Pegis, “The modern development of Hamiltonian optics,” in Progress in Optics I, E. Wolf, ed. (Elsevier/North-Holland, Amsterdam, 1964).

Shafer, D.

D. Shafer, “Optical design methods: your head as a personal computer,” in Geometrical Optics: Critical Review of Technology, R. E. Fischer, W. H. Price, W. J. Smith, eds., Proc. Soc. Photo-Opt. Instrum. Eng.531, 49–58 (1985).
[CrossRef]

D. Shafer, “Optical design and relaxation response,” in Recent Trends in Optical Systems Design: Computer Lens Design Workshop, C. Londono, R. E. Fischer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.766, 2–8 (1987).
[CrossRef]

Smith, T.

T. Smith, “The changes in aberrations when the object and stop are moved,” Trans. Opt. Soc. London 23, 311–322 (1921–1922).

Stavroudis, O.

O. Stavroudis, Modular Optical Design, Vol. 28 of Springer Series in Optics (Springer-Verlag, New York, 1982).
[CrossRef]

Steward, G. C.

G. C. Steward, The Symmetric Optical System (Cambridge U. Press, London, 1927).

Velzel, C. H. F.

Young, A.

J. A. Grace, A. Young, The Algebra of Invariants (Cambridge U. Press, Cambridge, 1903).

J. Opt. Soc. Am. A (3)

Trans. Opt. Soc. London (1)

T. Smith, “The changes in aberrations when the object and stop are moved,” Trans. Opt. Soc. London 23, 311–322 (1921–1922).

Other (6)

G. C. Steward, The Symmetric Optical System (Cambridge U. Press, London, 1927).

J. A. Grace, A. Young, The Algebra of Invariants (Cambridge U. Press, Cambridge, 1903).

R. J. Pegis, “The modern development of Hamiltonian optics,” in Progress in Optics I, E. Wolf, ed. (Elsevier/North-Holland, Amsterdam, 1964).

O. Stavroudis, Modular Optical Design, Vol. 28 of Springer Series in Optics (Springer-Verlag, New York, 1982).
[CrossRef]

D. Shafer, “Optical design methods: your head as a personal computer,” in Geometrical Optics: Critical Review of Technology, R. E. Fischer, W. H. Price, W. J. Smith, eds., Proc. Soc. Photo-Opt. Instrum. Eng.531, 49–58 (1985).
[CrossRef]

D. Shafer, “Optical design and relaxation response,” in Recent Trends in Optical Systems Design: Computer Lens Design Workshop, C. Londono, R. E. Fischer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.766, 2–8 (1987).
[CrossRef]

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Tables (3)

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Table 1 Coefficients ejklEjkl · J for a Spherical Surface as a Function of X for l = 1, …, 5

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Table 2 Coefficientsa of the Transformation Dj = ΣkcjkEk

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Table 3 Coefficientsa of the Transformation D j ( p ) = j c jj E j ( p )

Equations (55)

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E c = r ( n N n N ) [ 1 + ( n L n L ) 2 + ( n M n M ) 2 ( n N n N ) 2 ] 1 / 2 ,
u L 2 + M 2 , υ L L + M M , w L 2 + M 2 .
E c = E j k l u j υ k w l j k ,
E j k l = 1 J n = 0 l 1 c n ( j k l ) X n ,
J ( n n ) / n n r ,
X n n / ( n n ) 2 .
E = E j k l u j υ k w l j k ,
U ( u 2 G υ + G 2 w ) / ( S G ) , V ( u ( S + G ) υ + S G w ) / ( S G ) , W ( u 2 S υ + S 2 w ) / ( S G ) ,
E = D j k l U j V k W l j k .
E ( l ) = ( x u + y υ + w ) l .
c j k l l ! / j ! k ! ( l j k ) ! ,
u = S 2 U 2 S G V + G 2 W , υ = S U ( S + G ) V + G W , w = U 2 V + W .
D j k l = c j k l ( S 2 x + S y + 1 ) j [ 2 S G x ( S + G ) y 2 ] 2 × ( G 2 x + G y + 1 ) l j k .
D 101 = S 2 E 101 + S E 011 + E 001 , D 011 = 2 S G E 101 ( S + G ) E 011 2 E 001 , D 001 = G 2 E 101 + G E 011 + E 001 .
E ( l ) = j , k [ E j k l ( 0 ) u j υ k w l j k + ( u w υ 2 ) E j , k , l 2 ( l ) u j υ k w l j k 2 + + ( u w υ 2 ) p E j , k , l 2 p ( p ) u j υ k w l j k 2 p + ] .
U W V 2 = ( S G ) 2 ( u w υ 2 ) .
j , k E j , k , l 2 p ( p ) u j υ k w l j k 2 p = ( t 2 u + 2 t υ + w ) l 2 p
t 2 j + k E j k l ( p ) / 2 k c j k l .
E j 1 , k 1 , l ( p ) / 2 k 1 c j 1 , k 1 l = E j 2 , k 2 , l ( p ) / 2 k 2 c j 2 , k 2 l ,
( x u + y υ + w ) l = ( t 0 2 u + 2 t 0 υ + w ) l + ( u w υ 2 ) ( t 1 2 u + 2 t 1 υ + w ) l 2 + + ( u w υ 2 ) p × ( t p 2 u + 2 t p υ + w ) l 2 p +
( x u + y υ + w ) l = ( t 2 u + 2 t υ + w ) l + ( u w υ 2 ) ( x u + y υ + w ) l 2
c j k l x j y k = 2 k c j k l t 2 j + k + c j 1 , k l 2 x j 1 y k c j , k 2 l 2 x j y k 2 .
E j k l = E j k l ( 0 ) + e j 1 , k , l 2 e j , k 2 , l 2 ,
( t 2 u + 2 t υ + w ) 2 = t 4 u 2 + 4 t 3 u υ + 2 t 2 u w + 4 t 2 υ 2 + 4 t υ w + w 2 ,
E 102 = E 102 ( 0 ) + e 000 , E 022 = E 022 ( 0 ) e 000 .
e 000 = 1 3 ( 2 E 102 E 022 )
E 102 ( 0 ) = 1 2 E 022 ( 0 ) = 1 3 ( E 102 + E 022 ) .
E ( l ) = j , k [ D j k l ( 0 ) U j V k W l j k + ( U W V 2 ) ( S G ) 2 D j , k , l 2 ( 1 ) U j V k W l j k 2 + + ( U W V 2 ) p ( S G ) 2 p D j , k , l 2 p ( p ) × U j V k W l j k 2 p + ] .
D j , k , l 2 p ( p ) = c j , k l 2 p ( 2 ) k ( S t p + 1 ) 2 j + k ( G t p + 1 ) 2 l 2 j k 2 p .
D ¯ j k l = S 2 2 j + k G 2 l 2 j k D 1 , j k l + D 2 , j k l .
s 1 S 1 S 1 G 1 , g 1 G 1 S 1 G 1 , s 2 = g 2 1 S 2 G 2
p i , j k l 2 g i ( j + 1 ) D i , j + 1 , k , l + 1 + s i ( k + 1 ) D i , j , k + 1 , l + 1 , q i , j k l 2 s i ( l j k + 1 ) D i , j , k , l + 1 + g i ( k + 1 ) D i , j , k + 1 , l + 1 .
p j k l S 2 2 j + k + 1 G 2 2 l 2 j k p 1 , j k l + p 2 , j k l , q j k l S 2 2 j + k G 2 2 l 2 j k + 1 q 1 , j k l + q 2 , j k l .
( p q ) j k l = j = 0 j k = 0 k l = 1 l 1 p j , k , l · q j j , k k , l l .
d j k l = [ ( p 2 ) j 1 , k , l 1 + 2 ( p q ) j , k , l 1 + ( q 2 ) j , k , l 1 ] / 2 Δ l
Σ j k l = S 2 2 j + k G 2 2 l 2 j k ( g 1 p 1 , j k l + s 1 q 1 , j k l ) + g 2 ( p 2 , j k l + q 2 , j k l ) ,
d j k l = d j k l + ( Σ d ) j k l / Δ l , l 3 ,
D ¯ j k l = i = 1 n σ i 2 j + k γ i 2 l 2 j k D i , j k l ,
σ i = σ i 1 / S i , σ 1 = S , γ i = γ i 1 / G i , γ 1 = G ,
J 1 , n σ i = J 1 , i + S J i + 1 , n ,
σ i = ( J 1 , i + S J i + 1 , n ) / J 1 , n , S i = ( J 1 , i 1 + S J i , n ) / ( J 1 , i + S J i + 1 , n ) ;
J 1 , n 2 l E ¯ j k l = i = 1 n D i , j k l * J 1 , i 2 j + k J i + 1 , n 2 l 2 j k ( 1 ) k ,
S i * J i , n / J i + 1 , n , G i * J 1 , i 1 / J 1 , i ,
D i , j k l * = c j k l ( S i * 2 x + S i * y + 1 ) j ( 2 S i * G i * x ( S i * + G i * ) y 2 ) k × ( G i * 2 x + G i * y + 1 ) l j k
λ ¯ = Σ ( ξ p j k l + ξ p q j k l ) ( ξ 2 + η 2 ) j ( ξ ξ p + η η p ) k ( ξ p 2 + η p 2 ) l j k , μ ¯ = Σ ( η p j k l + η p q j k l ) ( ξ 2 + η 2 ) j ( ξ ξ p + η η p ) k ( ξ p 2 + η p 2 ) l j k ,
J 1 , i = A i / n 1 n i .
l i = n 1 B i / A i .
l i = n i C i / A i .
A i D i = B i C i 1.
C i + 1 = C i t i A i / n i , A i + 1 = A i + K i + 1 C i + 1 , D i + 1 = D i t i B i / n i , B i + 1 = B i + K i + 1 D i + 1 .
A 1 = K 1 , B 1 = C 1 = 1 , D 1 = 0.
Δ l i = n i ( t i n i C i A i n i · 1 K i + 1 ) .
G 1 , i = ( s J 1 , i + B i / n i ) 1 .
G 1 , i + 1 = ( s J 1 , i + 1 + B i + 1 / n i + 1 ) 1 .
G i + 1 = G 1 , i + 1 / G 1 , i .

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