Abstract

The paper discusses the problem of designing the aspheric surface of a Schmidt camera so as to obtain optimum performance, in an agreed sense, over the field taken as a whole. A solution is obtained in a form which allows the optimum plate-profile to be quickly determined for a Schmidt camera of given aperture-ratio and field-size, working over a given spectral range. It is shown that at apertures near f/3 the optimum plate can be obtained by slightly decreasing the strength of an ordinary “color-minimised” Schmidt plate, while in wide-field systems working at apertures near f/1, it is better to use a plate with neutral zone at the edge of the aperture.

© 1949 Optical Society of America

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Figures (2)

F. 1
F. 1

The Schmidt camera.

F. 2
F. 2

Level curves for ( e * ) 1 2 as a function of a and p.

Equations (37)

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μ = H / R .
x = H u , y = H v ;
x = μ u , y = μ v .
( N 1 ) s = 1 4 μ 4 ( r 2 r 0 2 ) 2 ( 1 + ( 5 / 2 ) r 0 2 μ 2 ) + 3 8 μ 6 ( r 2 r 0 2 ) 3 + constant + O ( μ 8 ) ,
f = 1 2 ( 1 μ 2 r 0 2 ) 1 2
2 λ = 2 μ ( 1 μ 2 r 0 2 ) 1 2 .
( N 1 ) s = 1 4 μ 4 ( r 2 a r 2 ) + O ( μ 6 ) ,
( N 1 ) s = 1 4 λ 4 ( r 2 a r 2 ) + O ( λ 6 ) ,
δ X + i δ Y = 1 4 K μ 3 φ 2 [ N + 1 2 N u + 1 2 u 2 u 2 + i 2 N v + 1 2 i u 2 u v ] ( r 4 a r 2 ) + O ( K μ 7 ) ,
n N = O ( μ 2 ) ,
δ X + i δ Y = 1 4 K μ 3 φ 2 n 1 N 1 [ n + 1 2 n u + 1 2 u 2 u 2 + i 2 n v + 1 2 i u 2 u v ] ( r 4 a r 2 ) + 1 4 K μ 3 n N N 1 ( u + i v ) × ( r 4 a r 2 ) + O ( K μ 7 ) .
δ X * + i δ Y * = 1 4 K μ 3 φ 2 [ n 0 + 1 2 n 0 u + 1 2 u 2 u 2 + i 2 n 0 v + 1 2 i u 2 u v ] ( r 4 a r 2 ) + 1 4 K μ 3 n N n 0 1 ( u + i v ) ( r 4 a r 2 ) ,
1 π H 2 x 2 + y 2 H 2 [ ( δ X ) 2 + ( δ Y ) 2 ] d x d y
E = 2 π H 2 φ 0 2 0 φ 0 φ d φ x 2 + y 2 H 2 [ ( δ X ) 2 + ( δ Y ) 2 ] d x d y = 2 π φ 0 2 0 φ 0 φ d φ u 2 + v 2 1 [ ( δ X ) 2 + ( δ Y ) 2 ] d u d v .
δ X = K μ 3 [ P u ( r 2 1 2 a ) + φ 2 u 3 ] + O ( K μ 7 ) δ Y = K μ 3 [ Q v ( r 2 1 2 a ) + φ 2 u 2 v ] + O ( K μ 7 ) } ,
P = 2 n 0 + 1 2 n 0 φ 2 + n N n 0 1 , Q = P φ 2
E = E * [ 1 + O ( μ 2 ) ] ,
E * = 2 π φ 0 2 0 φ 0 δ d φ u 2 + v 2 1 [ ( δ X * ) 2 + ( δ Y * ) 2 ] d u d v
δ X * = K μ 3 [ P u ( r 2 1 2 a ) + φ 2 u 3 ] δ Y * = K μ 3 [ Q v ( r 2 1 2 a ) + φ 2 u 2 v ] } .
( δ X * ) 2 + ( δ Y * ) 2 = K 2 μ 6 [ ( P 2 u 2 + Q 2 v 2 ) ( r 2 1 2 a ) 2 + 2 φ 2 u 2 ( P u 2 + Q v 2 ) ( r 2 1 2 a ) + φ 4 u 4 r 4 ] .
u 2 + v 2 1 ( P 2 u 2 + Q 2 v 2 ) ( r 2 1 2 a ) 2 d u d v = 1 2 ( P 2 + Q 2 ) u 2 + v 2 1 r 2 ( r 2 1 2 a ) 2 d u d v = π 16 ( P 2 + Q 2 ) ( a 2 8 3 a + 2 ) ;
u 2 + v 2 1 ( P u 2 + Q v 2 ) u 2 ( r 2 1 2 a ) d u d v = π 16 ( 3 P + Q ) ( 1 2 1 3 a ) ;
u 2 + v 2 1 u 4 r 2 d u d v = 0 2 π cos 4 θ d θ 0 l r 7 d r = 3 π 32 .
u 2 + v 2 1 [ ( δ X * ) 2 + ( δ Y * ) 2 ] d u d v = π 16 K 2 μ 6 [ ( P 2 + Q 2 ) ( a 2 8 3 a + 2 ) + 2 ( 3 P + Q ) ( 1 2 1 3 a ) φ 2 + 3 2 φ 4 ] .
α = 1 2 n 0 , p = n N n 0 1 1 φ 0 2 ;
P = ( 1 + α ) φ 2 + p φ 0 2 , Q = α φ 2 + p φ 0 2 ; 0 φ 0 φ d φ ( P 2 + Q 2 ) = φ 0 6 [ 1 6 ( 1 + 2 α + 2 α 2 ) + 1 2 p ( 1 + 2 α ) + p 2 ] , 0 φ 0 φ d φ · φ 2 ( 3 P + Q ) = φ 0 6 [ 1 6 ( 3 + 4 α ) + p ] , 0 φ 0 φ d φ · φ 4 = 1 6 φ 0 6 .
E * = 1 8 K 2 μ 6 φ 0 4 { ( a 2 8 a / 3 + 2 ) [ 1 6 ( 1 + 2 α + 2 α 2 ) ] + 1 2 p ( 1 + 2 α ) + p 2 ] + 2 ( 1 2 1 3 a ) [ 1 6 ( 3 + 4 α ) + p ] + 1 4 } ,
E 1 2 = ( E * ) 1 2 { 1 + O ( μ 2 ) } .
e * = ( a 2 8 a / 3 + 2 ) [ 1 6 ( 1 + 2 α + 2 α 2 ) ] + 1 2 p ( 1 + 2 α ) + p 2 ] + 2 ( 1 2 1 3 a ) [ 1 6 ( 3 + 4 α ) + p ] + 1 4 ,
α = 1 3.10 = 0.32 , p = n N 0.55 · ( 14 ) 2 = 356 ( n N ) .
356 ( 1.54011 1.53123 ) = 3.16 .
a = 2 r 0 2 / R 2 ,
1 2 ( e * / 2 ) 1 2 K μ 3 φ 0 2 = 0.136 ( e * ) 1 2 = 0.12 second of arc .
φ 0 = 1 6 , p = ( n N / 0.55 ) · 6 2 = 65.44 ( n N ) ,
a = 2.0 , N = 1.5383
a = 2.25 , p = n N n 0 1 1 φ 0 2 = 0.18 ,
a = 2.25 , N = n + 0.10 φ 0 2 .