Abstract

In a paper probably to be published in Optika i Spektroskopiya the wave aberration for sagittal focus for the arbitrary surface of rotational symmetry has been carried out on the base of the astigmatic beam invariant Ds = nusds. The resulting expression for the wave aberration has been reformulated into three terms which, in the Seidel region, go over into astigmatism (the first) and into the Petzval curvature (the second) while the third disappears.

© 1966 Optical Society of America

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References

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  1. H. H. Hopkins, Wave Theory of Aberrations (Oxford University Press, New York, 1950).
  2. H. H. Hopkins (private communication).
  3. M. Gaj, Pomiary Automatyka Kontrola 9, 9 (1965).
  4. M. Gaj, submitted to Optika i Spektroskopiya.

1965 (1)

M. Gaj, Pomiary Automatyka Kontrola 9, 9 (1965).

Gaj, M.

M. Gaj, Pomiary Automatyka Kontrola 9, 9 (1965).

M. Gaj, submitted to Optika i Spektroskopiya.

Hopkins, H. H.

H. H. Hopkins (private communication).

H. H. Hopkins, Wave Theory of Aberrations (Oxford University Press, New York, 1950).

Pomiary Automatyka Kontrola (1)

M. Gaj, Pomiary Automatyka Kontrola 9, 9 (1965).

Other (3)

M. Gaj, submitted to Optika i Spektroskopiya.

H. H. Hopkins, Wave Theory of Aberrations (Oxford University Press, New York, 1950).

H. H. Hopkins (private communication).

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Figures (1)

Fig. 1
Fig. 1

Illustration of Eq. (28).

Equations (44)

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Δ ( n / s ) = Δ ( n cos I ¯ ) / r s , Δ ( n / x ) = Δ ( n ) / r ,
H s = n u s η s ,
( h s ) D = h D .
D s = n u s d s ,
Δ [ N ¯ ( H ¯ / H s ) W s ] = 1 2 A [ z ¯ Δ ( u s ) + h s Δ ( N ¯ ) ] + 1 2 h Δ ( n u s ) ( 1 - z ¯ c - r s c cos G ¯ ) ,
N ¯ k ( n k u k η ¯ k / H s ) ( W s ) k = N ¯ k + 1 ( n k + 1 u k + 1 η ¯ k + 1 / H s ) ( W s ) k + 1 .
N ¯ p H ¯ p ( W s ) p - N ¯ 1 H ¯ 1 ( W s ) 1 = 1 2 H s l = 1 ρ A 1 [ z ¯ 1 Δ 1 ( u s ) + ( h s ) 1 Δ 1 ( N ¯ ) ] + 1 2 l = 1 ρ h 1 Δ 1 ( n u s ) [ 1 - z ¯ 1 c 1 - ( r s ) 1 c 1 cos G ¯ 1 ] .
4 F = 1 2 h Δ ( n u s ) ( 1 - z ¯ c - r s c cos G ¯ ) W ¯ s = N ¯ H ¯ W s / H s .
H s = n u s η s = n u s ( y ¯ + s M ¯ ) = y ¯ n u s + h s n M ¯ .
u s = ( H s - h s n M ¯ ) / y ¯ n .
Δ ( u s ) = ( 1 / y ) H s Δ ( 1 / n ) - ( h s / y ¯ ) Δ ( M ¯ ) .
Δ ( W ¯ s ) = 1 2 A [ ( z ¯ / y ¯ ) H s Δ ( 1 / n ) + ( h s / y ¯ ) ( y ¯ Δ N ¯ - z ¯ Δ M ¯ ) ] + 1 4 F .
H = B h - Δ h ¯ ,
Δ ( W ¯ s ) = 1 4 [ ( 2 z ¯ / y ¯ h ¯ ) ( B h - H ) H s Δ ( 1 / n ) + 2 ( A h s / y ) ( y ¯ Δ N ¯ - z ¯ Δ M ¯ ) ] + 1 4 F .
H s = n u s η s = n u s ( y ¯ + M ¯ s ) = n ( g s - i s ) y ¯ + n h s M ¯ ,
u s = g s - i s , g s = h s c s cos I ¯ , i s = h s c s cos I ¯ - ( h s / s )
A s = n i s , B s = n sin I ¯ .
H s = n y h s c s cos I ¯ - A s y ¯ + n h s [ sin I ¯ cos G ¯ - cos I ¯ sin G ¯ ] .
H s = - A s y ¯ + B s h s cos G ¯ .
Δ ( W s ) = 1 4 { 2 z ¯ y ¯ h ¯ H H s Δ [ - ( 1 / n ) ] + 2 z ¯ B y ¯ h ¯ ( B s h s cos G ¯ - A s y ¯ ) h Δ ( 1 / n ) + ( 2 A h s / y ¯ ) [ y ¯ ( cos G ¯ cos I ¯ + sin G ¯ sin I ¯ ) + z ¯ ( sin G ¯ cos I ¯ - cos G ¯ sin I ¯ ) ] } + 1 4 F .
Δ ( W s ) = 1 4 { ( 2 z ¯ / y ¯ h ¯ ) H H s Δ [ 1 - ( 1 / n ) ] + 2 z ¯ y ¯ h ¯ B B s h h s cos G ¯ Δ ( 1 / n ) + ( 2 A h s / y ¯ ) ( y ¯ cos G ¯ + z ¯ sin G ¯ ) Δ cos I ¯ - ( 2 z ¯ / h ¯ ) A s B h Δ ( 1 / n ) + ( 2 A h s / y ¯ ) B s Δ ( 1 / n ) ( y ¯ sin G ¯ - z ¯ cos G ¯ ) } + 1 4 F .
K = ( 2 z ¯ / y ¯ h ¯ ) H H s Δ [ 1 - ( 1 / n ) ] , F ¯ = - ( 2 z ¯ / h ¯ ) A s B h Δ ( 1 / n ) + 2 ( A h s / y ¯ ) B s Δ ( 1 / n ) ( y ¯ sin G ¯ - z ¯ cos G ¯ ) ,
S = ( 2 z ¯ / y ¯ h ¯ ) B B s h h s cos G Δ ( 1 / n ) + 2 ( A h s / y ¯ ) ( y ¯ cos G ¯ + z ¯ sin G ¯ ) , Δ ( W ¯ s ) = 1 4 ( K + S + F + F ¯ ) .
S = ( 2 z ¯ / y ¯ h ¯ ) B B s h h s cos G ¯ Δ ( 1 / n ) + 2 A h s Δ cos I ¯ .
K = k = 1 ρ K k = k = 1 ρ ( 2 z ¯ k / y ¯ k h ¯ k ) H H s Δ k [ - ( 1 / n ) ] = 2 H H s k = 1 ρ ( z ¯ k / y ¯ k h ¯ k ) Δ k [ - ( 1 / n ) ] .
K = H H s k = 1 ρ { 1 / [ h k ( cos 1 2 G ¯ k ) 2 ] } Δ k ( 1 / n ) .
S = k = 1 ρ S k = 2 k = 1 ρ ( z ¯ k / y ¯ k h ¯ k ) B k ( B s ) k cos G ¯ k Δ k ( 1 / n ) + 2 k = 1 ρ A k ( h s ) k Δ k ( cos I ) .
S = k = 1 ρ S k = 2 k = 1 ρ { z ¯ k y ¯ k h ¯ k B k ( B s ) k cos G ¯ k Δ k ( 1 n ) + 2 A k ( h s ) k y ¯ k ( y ¯ k cos G ¯ k + z ¯ k sin G ¯ k ) } .
H s = n D M ¯ D h D ,
( u s ) 1 = ( H s / H ¯ 1 ) u 1 .
( h s ) 1 = ( l 1 - z ¯ 1 / N ¯ 1 ) ( u s ) 1 = ( l 1 - z ¯ 1 / N ¯ 1 ) ( H s / H ¯ 1 ) u 1 = ( h 1 - z ¯ 1 u 1 / N ¯ 1 ) ( H s / H ¯ 1 ) .
Δ k ( n u s ) = ( h s ) k ( c s ) k Δ k ( n cos I ¯ ) , ( u s ) k = ( u s ) k + 1 ( h s ) k + 1 = ( h s ) k + [ ( u s ) k / M ¯ k ] ( y ¯ k - y ¯ k + 1 ) .
( h s ) k + 1 = ( h s ) k + [ ( u s ) k / N ¯ k ] ( d k - z ¯ k + z ¯ k + 1 ) .
y ^ k + 1 = y ^ k + M ¯ k d ¯ k .
[ y ¯ k + 1 / ( h s ) k + 1 ( h s ) k ] - [ y ¯ k / ( h s ) k + 1 ( h s ) k ] = [ M ¯ k d ¯ k / ( h s ) k + 1 ( h s ) k ] = [ n k M ¯ k ( h s ) E ¯ k d ¯ k n k ( h s ) k + 1 ( h s ) k ( h s ) E ¯ k ] ,
( h s ) E k = ( h s ) E ¯ k + 1
( h s ) E ¯ k ( h s ) k = H s n k M ¯ k ( h s ) k = n k ( u s ) k y ¯ k + n k ( h s ) k M ¯ k n k M ¯ k ( h s ) k = 1 + y ¯ k + 1 ( u s ) k M ¯ k ( h s ) k ,             ( h s ) E ¯ k + 1 ( h s ) k + 1 = n k + 1 ( u s ) k + 1 y ¯ k + 1 + n k + 1 ( h s ) k + 1 M ¯ k + 1 n k + 1 M ¯ k + 1 ( h s ) k + 1 = 1 + y ¯ k + 1 ( u s ) k + 1 M ¯ k + 1 ( h s ) k + 1 .
y ¯ k + 1 ( h s ) k + 1 ( h s ) E ¯ k ( h s ) k - y ¯ k ( h s ) k ( h s ) E ¯ k + 1 ( h s ) k + 1 = y ¯ k + 1 ( h s ) k + 1 [ 1 + y ¯ k ( u s ) k M ¯ k ( h s ) k ] - y ¯ k ( h s ) k [ 1 + y ¯ k + 1 ( u s ) k + 1 M ¯ k + 1 ( h s ) k + 1 ] = y ¯ k + 1 ( h s ) k + 1 - y ¯ k ( h s ) k = H s d k n k ( h s ) k ( h s ) k + 1 .
( h s ) k + 1 - ( h s ) k = [ ( u s ) k / M ¯ k ] ( y ¯ k - y ¯ k + 1 ) .
[ ( u s ) k / M ¯ k ] = [ H s / n k y ¯ k M ¯ k ] - [ ( h s ) k / y ¯ k ] .
( h s ) k + 1 = ( h s ) k + ( H s n k M ¯ k - ( h s ) k ) ( 1 - y ¯ k + 1 y ¯ k ) = y ¯ k + 1 y ¯ k ( h s ) k + H s n k M ¯ k ( 1 - y ¯ k + 1 y ¯ k ) = y ¯ k + 1 y ¯ k [ y ¯ k y ¯ k - 1 ( h s ) k - 1 + H s n k - 1 M ¯ k - 1 ( 1 - y ¯ k y ¯ k - 1 ) ] + H s n k M ¯ k ( 1 - y ¯ k + 1 y ¯ k ) = y ¯ k + 1 y ¯ k - 1 ( h s ) k - 1 + H s y ¯ k + 1 n k - 1 M ¯ k - 1 ( 1 y ¯ k - 1 y ¯ k - 1 ) + H s y ¯ k + 1 n k M ¯ k ( 1 y ¯ k + 1 - 1 y ¯ k ) = y ¯ k + 1 y ¯ k - 2 ( h s ) k - 2 + H s y ¯ k + 1 l = k - 2 k 1 n 1 M ¯ 1 ( 1 y ¯ 1 + 1 - 1 y ¯ 1 ) × ( h s ) k + 1 = y ¯ k + 1 y ¯ k - p ( h s ) k - p + H s y ¯ k + 1 l = k - p k 1 n 1 M ¯ 1 ( 1 y ¯ + 1 - 1 y ¯ l ) .
( h s ) k + 1 = ( y ¯ k + 1 / y ¯ k - b ) h D + H s y ¯ k + 1 l = k - b k ( 1 / n l M ¯ l ) [ ( 1 / y ¯ l + 1 ) - ( 1 / y ¯ l ) ] .
( h s ) k + 1 = y ¯ k + 1 y ¯ k - b h D + H s y ¯ k + 1 l = k - b k 1 n l N ¯ l y ¯ l y ¯ l + 1 ( d l - z ¯ l + z ¯ l + 1 ) .
( u s ) k = H s - n k ( h s ) k M ¯ k n k y ¯ k .

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