## Abstract

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### Figures (3)

Fig. 1

Geometry of a spherical wavefront refracted by a spherical surface.

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

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

### Equations (20)

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