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References

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  1. R. T. Holleran, Appl. Opt. 2, 1336 (1963).
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1963

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Figures (3)

Fig. 1
Fig. 1

Geometry of a spherical wavefront refracted by a spherical surface.

Fig. 2
Fig. 2

The third-order approximation to a spherical wavefront refracted by a spherical surface is a conicoid whose deformation parameter e02 is shown as a function of the ratio wavefront vertex radius at the interface to radius of refracting sphere. The aplanatic case occurs when this ratio is (n + 1)/n, and the ratio n/(n − 1) demarcates real and virtual distances to the point of origin of the wavefront. The curve is drawn for index of refraction of 1.5.

Fig. 3
Fig. 3

Spherical back-surface provides null-test condition for a concave oblate spheroid.

Equations (20)

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y 2 = 2 R x - x 2 + e 2 x 2 ,
e 0 2 = ( 1 - R 0 / R 1 ) 2 [ n 2 - 1 - ( n 2 - n ) R 0 / R 1 ] ,
R 0 = R + t = n d R 1 / [ ( n - 1 ) d + R 1 ] ,
( d - l ) 2 = y 2 + ( d - x ) 2 ,
y 2 = 2 R 1 x - x 2 .
( 2 R 1 x - n 2 t 2 ) / ( 2 x - 2 n t ) = R 1 R 0 / [ n R 1 - ( n - 1 ) R 0 ]
e 2 ( x - t ) = R 1 - R 0 - [ ( R 1 - R 0 ) 2 + e 2 t ( 2 R 1 - t ) ] 1 2 .
x = t + 1 2 t ( R 1 - R 0 ) { - 2 R 1 + t + [ e 2 t R 1 2 / ( R 1 - R 0 ) 2 ] [ 1 - t / R 1 - e 2 t R 1 / ( R 1 - R 0 ) 2 + t 2 ] } .
e 2 [ 1 - t / R 1 - ( e 2 t R 1 ) / ( R 1 - R 0 ) 2 + ] = ( 1 - R 0 / R 1 ) 2 [ n 2 - 1 - ( n 2 - n ) R 0 / R 1 ] .
e t 2 = e 0 2 R 0 / ( R 0 - t ) ,
Δ x = y 4 ( e t 2 - e 0 2 ) / 8 ( R 0 - t ) 3 .
Δ x = [ 1 / R 0 - 1 / R 1 - e 0 2 R 0 2 / ( R 1 - R 0 ) 2 ] y 4 e 0 2 t / 8 R 0 3 .
t = 1 2 y 2 / ( 1 / R 1 - 1 / R 0 ) ,
Δ x = [ e 0 2 R 0 / ( R 1 - R 0 ) - ( 1 - R 0 / R 1 ) 2 ] y 6 e 0 2 / 16 R 0 5
Δ x = OPD = y 4 e 2 / 8 R 3 .
OPD = y 4 ( n 2 - n ) / 8 R 1 3 ,
OPD = [ e 0 1 2 - e 0 2 2 ( R 0 - t ) / R 0 ] n y 4 / 8 R 0 3 .
e r 2 = ( a - b ) / ( a + b ) .
2 y 4 e 2 / 8 R 3 - 2 y 4 e r 2 / 8 R 3 = y 4 e a 2 / 8 a 3 + y 4 e b 2 / 8 b 3 .
e 2 = 1 2 ( e a 2 / a 3 + e b 2 / b 3 + 2 e r 2 ) ( for R = 1 ) .

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