Abstract

It has recently been suggested that a concentric corrector, i.e., a corrector all the surfaces of which are normal to axial rays in the paraxial region, might be used to correct field curvature alone. The theory given was not sufficiently extended to show what errors the corrector will introduce. In particular, it does affect distortion in the extended paraxial region. Further, when the corrector is placed somewhere within the system its effects on the contributions to the aberrations by those surfaces of the original system which follow the corrector must be taken into account, and these effects may in particular cases be substantial. This paper therefore provides a careful and detailed analysis of the problem under consideration, and the theory follows closely that of concentric systems developed in an earlier paper of the present series. Numerical illustrations are provided.

© 1962 Optical Society of America

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References

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  1. S. Rosin, J. Opt. Soc. Am. 49, 862 (1959). Hereafter referred to as SR.
    [Crossref]
  2. In any case I cannot understand how the sine relation can be used to infer properties of the distortion.
  3. H. A. Buchdahl, Optical Aberration Coefficients (Oxford University Press, New York, 1954). References to this work will be distinguished by the letter M.
  4. H. A. Buchdahl, J. Opt. Soc. Am. 52, 1361 (1962). Hereafter quoted as X.
    [Crossref]
  5. If the Hamiltonian method were used complications would arise at this stage which the “algebraic,” or “direct,” or “Lagrangian” method circumvents.
  6. Its specifications are given in M Sec. 37(a) and Sec. 113.
  7. AR is in fact the quantity which in M, Sec. 128 was written A(kj) if now j and k number the first and last surfaces of R when it formed part of K, i.e., not of S.

1962 (1)

1959 (1)

Buchdahl, H. A.

H. A. Buchdahl, J. Opt. Soc. Am. 52, 1361 (1962). Hereafter quoted as X.
[Crossref]

H. A. Buchdahl, Optical Aberration Coefficients (Oxford University Press, New York, 1954). References to this work will be distinguished by the letter M.

Rosin, S.

J. Opt. Soc. Am. (2)

Other (5)

In any case I cannot understand how the sine relation can be used to infer properties of the distortion.

H. A. Buchdahl, Optical Aberration Coefficients (Oxford University Press, New York, 1954). References to this work will be distinguished by the letter M.

If the Hamiltonian method were used complications would arise at this stage which the “algebraic,” or “direct,” or “Lagrangian” method circumvents.

Its specifications are given in M Sec. 37(a) and Sec. 113.

AR is in fact the quantity which in M, Sec. 128 was written A(kj) if now j and k number the first and last surfaces of R when it formed part of K, i.e., not of S.

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Tables (2)

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Table I Effects of the insertion of C in the image space of K.

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Table II Effects of the insertion of C within K.

Equations (53)

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i p = 0 ,
v p = v p = 1.
i q = - c / N .
Δ i q = - Δ v q = - ϖ .
v p S = v p K ,             v q S = v q K + C Δ v q = v q K + π C ,
δ = Δ v q 2 ,             d C = 1 2 C δ .
b ¯ p = ϖ ,             c ¯ p = 1 2 δ .
A p S = A p K ,             Ā p S = Ā p K ,             B ¯ p S = B ¯ p K + π C ,             C p S = C p K ,             C ¯ p S = C ¯ p K + d C .
G = G S - G K .
A p = Ā p = C p = 0 ,             B ¯ p = π C ,             C ¯ p = d C .
d C = π C ( v q K + 1 2 π C ) .
a ¯ q = - 1 2 ϖ ,
A q = B q = C q = 0 ;             Ā q = - 1 2 π C ;             C ¯ = C c ¯ q ( = C ¯ q C , say ) .
s 1 p = 0 ,
S 1 p = 0.
S 2 p = A p K π C .
S 1 q = - 1 2 A p K π C ,             S 2 q = - Ā p K π C , S 3 q = - 1 2 C p K π C + A p K C ¯ q C ,             S 4 q = 0 , S 5 q = 2 Ā p K C ¯ q C ,             S 6 q = C p K C ¯ q C .
S ¯ 4 p S = S 4 q S + B ¯ p S B q S - 2 B ¯ q S Ā p S + 2 v q S 2 Ā p S + v q S ( B ¯ p S - B q S ) - B ¯ q S - 1 2 ( v q S 2 + 2 v q S v p 1 v q 1 - 3 v p 1 2 v q 1 2 ) .
S ¯ 4 p S = S 4 q K + ( B ¯ p K + π C ) B q K - 2 B ¯ q K Ā p K + 2 ( v q K 2 + 2 d C ) Ā p K + ( v q K + π C ) ( B ¯ p K + π C - B q K ) - B ¯ q K - 1 2 [ v q K 2 + 2 d C + 2 ( v q K + π C ) v p 1 v q 1 - 3 v p 1 2 v q 1 2 ] .
S ¯ 4 p = 4 Ā p K d C + ( B ¯ p K - v p 1 v q 1 + 1 2 π C ) π C ,
S 1 = 0 ,             S ¯ 1 = A π C , S 2 = A π C ,             S ¯ 2 = ( 3 Ā - 1 2 v p 1 2 ) π C + 2 A d C , S 3 = A d C ,             S ¯ 3 = 3 Ā d C - 1 2 v p 1 v q 1 π C , S 4 = 2 Ā π C ,             S ¯ 4 = 4 Ā d C + ( B ¯ - v p 1 v q 1 + 1 2 π C ) π C , S 5 = 2 Ā d C + C π C ,             S ¯ 5 = ( 3 B ¯ + 2 C ) d C - ( C ¯ + 3 2 v q 1 2 ) π C + π C ( π C + 1 2 v q K ) ( π C + v q K ) - C ¯ q C , S 6 = C d C ,             S ¯ 6 = - C ¯ q K π C - ( v q K + π C ) C ¯ q C + 3 C ¯ d C + 3 2 d C 2 .
T 1 = A 2 π C ,             T ¯ 1 = ( S 1 - 1 2 v p 1 2 A ) π C + A 2 d C , T 2 = ( S 1 + 4 A Ā - 1 2 v p 1 2 A ) π C + 2 A 2 d C .
s ¯ 2 p = - 1 2 ϖ
f ( n ) p = 0 ,             f ¯ ( n ) p = κ ¯ n .
f ( 1 ) p = 0 ,             f ¯ ( 1 ) p = κ ¯ 1 ,
f ( 2 ) p = F ( 1 ) p κ ¯ 1 ,             f ¯ ( 2 ) p = κ ¯ 2 + F ¯ ( 1 ) p κ ¯ 1 .
F ( 1 ) p = 0 ,             F ¯ ( 1 ) p = 0 ,
F ( 2 ) p = 0 ,             F ¯ ( 2 ) p = 0.
( κ ¯ 2 i + κ ¯ 1 i κ ¯ 1 j ) = 0 ,
F ( 1 ) q = 0.
( A 1 ) = 0.
( Ā 1 ) = F ( 1 ) π C ,
( A 1 ) = 0 ,             ( Ā 1 ) = F ( 1 ) π C , ( B 1 ) = F ( 1 ) π C ,             ( B ¯ 1 ) = F ¯ ( 1 ) π C + 2 F ( 1 ) d C - C c Δ ( ν 1 / N 2 ) , ( C 1 ) = F ( 1 ) d C ,             ( C ¯ 1 ) = 3 F ¯ ( 1 ) d C - F ¯ ( 1 ) q K π C - ( v q K + π C ) F ¯ ( 1 ) q C .
( A 2 ) = ( F ( 1 ) ) 2 π C ,
( S 1 1 ) = 2 F ( 1 ) A π C ,
c 1 = 1.021066 ,             c 2 = 3.06748 ,             d 1 = 0.01 ,             d 2 = 0.50 ; N 1 = N 2 = 1.68743 ,             κ 11 = 0.015804 ,             κ 12 = - 0.04500.
π C = 0.75644 ,             d C = 1.06691.
A K = A L + A R .
G S = L g + C g + R g ,
L g = G L .
i q = - ϕ c / N ,             Δ v q = - Δ i q = ϕ ϖ ,
d C = ϕ π C ( v q L + 1 2 ϕ π C ) ,
χ = q - q
( v p i q - v q i p ) = ( v p i q - v q i p )
v p ( i q - i q ) = i p ( v q - v q ) .
χ = ϕ ( v q - v q ) = ϕ 2 π C ,
b ¯ p = 2 ( q + χ ) 2 a p = b ¯ p + 4 χ a ¯ p + 2 χ 2 a p .
A = 0 , Ā = χ A R , B ¯ = 2 χ 2 A R + 4 χ Ā R + π C , C = χ 2 A R + 2 χ Ā R , C ¯ = χ 3 A R + 3 χ 2 Ā R + χ ( B ¯ R + C R ) + d C .
χ * = ( p S - p ) / [ ( 1 - p l ) ( 1 - p S l ) ] .
* Ā = χ A R + χ * A S = χ A R + χ * A K ,
* A = 0 ,             * Ā = A R + A K , 1 2 ( * B ¯ - π C ) = * C = χ * 2 A K + 2 χ * Ā K + χ ( χ + 2 χ * ) A R + 2 χ Ā R * C ¯ = χ * 3 A K + 3 χ * 2 Ā K + χ * ( B ¯ K + C K ) + ( χ 3 + 3 χ * χ 2 + 3 χ χ * 2 ) A R + 3 ( χ 2 + 2 χ χ * ) A R + χ ( B ¯ R + C R ) + χ * π C .
c 1 = 0.101410 ,             c 2 = 0.102449 ;             N 1 = N 2 = 1.6000 ;             d 1 = 0.20 ,             d 2 = 0.10.
π C = 0.0003896 ,             d C = 0.005325.