## Abstract

Analytic ray-tracing formulas provide a rapid means of tracing rays, including skew rays, in a radial gradient. A fifth-order formula for the ray position and direction and a fourth-order formula for the optical path length give sufficiently accurate results for various applications.

© 1988 Optical Society of America

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### Equations (31)

$X = g x , Y = g y , Z = g z , R 2 = X 2 + Y 2$
$n 2 = N 0 2 ( 1 + e R 2 + h 4 R 4 + h 6 R 6 + … ) .$
$t = N 0 g / l 0 ,$
$l = n cos γ = l 0 = n 0 cos γ 0 ,$
$U = X ′ = d X / d t = p / N 0 , V = Y ′ = d Y / d t = q / N 0 ,$
$X ″ − e X = X ( 2 h 4 R 2 + 3 h 6 R 4 + … ) , Y ″ − e Y = Y ( 2 h 4 R 2 + 3 h 6 R 4 + … ) .$
$X = X 0 c + U 0 s + 2 h 4 ( X 0 W 1 + U 0 W 2 ) , Y = Y 0 c + V 0 s + 2 h 4 ( Y 0 W 1 + V 0 W 2 ) , R 2 = X 2 + Y 2 ,$
$c = cos t , s = sin t , W 1 = M 1 u + M 2 υ + M 3 w , W 2 = M 2 u + M 3 υ + M 4 w , u = X 0 2 + Y 0 2 , υ = 2 ( X 0 U 0 + Y 0 V 0 ) , w = U 0 2 + V 0 2 .$
$L = ∫ 0 σ n d σ = 1 l 0 ∫ 0 z n 2 d z = 1 N 0 g ∫ 0 t n 2 d t = N 0 g ( t − ∫ 0 t R 2 d t + h 4 ∫ 0 t R 4 d t + … ) .$
$L = N 0 g { t − ( A 0 + s 2 υ / 2 + B 0 w ) − 4 h 4 K + h 4 [ A 1 u 2 + A 3 υ 2 + A 5 w 2 + 2 ( A 2 u υ + A 3 u w + A 4 υ w ) ] } ,$
$X = X 0 A + U 0 B , Y = Y 0 A + V 0 B , U = X 0 A ′ + U 0 B ′ , V = Y 0 A ′ + V 0 B ′ ,$
$l 2 = n 2 − p 2 − q 2 = l 0 2 , E = x q − y p = E 0 .$
$X ′ = U , Y ′ = V , U ′ = X ( e + 2 h 4 R 2 + 3 h 6 R 4 + … ) , V ′ = Y ( e + 2 h 4 R 2 + 3 h 6 R 4 + … ) , L ′ = N 0 g ( 1 + e R 2 + h 4 R 4 + h 6 R 6 + … ) ,$
$e = − 1 , N 0 = 1 . 5 , g = 2 π / 67 , h 4 = 2 / 3 , h 6 = − 17 / 45 .$
$x 0 = y 0 = 0 . 1 , p 0 = 0 . 12 , q 0 = 0 . 13 , z = 10 ,$
$x , y 0 . 750554325 0 . 808204373 , p , q 0 . 0594095553 0 . 0653051455 , L 15 . 0364008 , l 2 − l 0 2 = 10 − 9 E − E 0 = 10 − 11 .$
$x , y 0 . 750554318 0 . 808204314 , p , q 0 . 0594095443 0 . 0653051336 , L 15 . 0364002 , l 2 − l 0 2 = 10 − 11 E − E 0 = 10 − 12 .$
$A 0 = ∫ 0 t c 2 d t , B 0 = ∫ 0 t s 2 d t , A 1 = ∫ 0 t c 4 d t , A 2 = ∫ 0 t c 3 s d t , … , A 5 = ∫ 0 t s 4 d t , C 1 = ∫ 0 t c 6 d t , C 2 = ∫ 0 t c 5 s d t , … , C 7 = ∫ 0 t s 6 d t .$
$A 1 = ( c 3 s + 3 A 0 ) / 4 , A 0 = ( t + c s ) / 2 , A 2 = ( 1 − c 4 ) / 4 , B 0 = ( t − c s ) / 2 , A 3 = A 0 − A 1 , A 4 = s 4 / 4 , A 5 = ( 3 B 0 − c s 3 ) / 4 ,$
$C 1 = ( c 5 s + 5 A 1 ) / 6 , C 2 = ( 1 − c 6 ) / 6 , C i + 2 = A i − C i , ( i = 1 − 5 ) .$
$E 0 = ∫ 0 t A 0 d t , F 0 = ∫ 0 t B 0 d t , E i = ∫ 0 t A i d t , ( i = 1 − 5 ) ,$
$E 1 = ( 3 E 0 + A 2 ) / 4 , E 0 = t 2 + s 2 ) / 4 , E 2 = ( t − A 1 ) / 4 , F 0 = ( t 2 − s 2 ) / 4 . E 3 = E 0 − E 1 , E 4 = A 5 / 4 , E 5 = ( 3 F 0 − A 4 ) / 4 .$
$M i = s A i − c A i + 1 , ( i = 1 − 5 ) D i = t A i − E i , ( i = 1 − 5 ) N i = s C i − c C i + 1 , ( i = 1 − 7 ) ,$
$M i 1 = ∫ 0 t c 3 M i , M i 2 = ∫ 0 t c 2 s M i , … ,$
$M 1 j = ( 3 D j + 1 + C j + 2 ) / 8 , M 2 j = ( A j + 1 + C j + 3 − D j ) / 8 , M 3 j = ( D j + 1 − C j + 2 ) / 8 , M 4 j = ( 2 A j + 1 + C j + 1 − 3 D j ) / 8 .$
$p i j = M i j + M i , j + 2 ( i = 1 − 4 , j = 1 − 2 ) , l i j = u p i j + υ p i + 1 , j + w p i + 2 , j ( i j = 1 − 2 ) .$
$K = u l 11 + ( l 12 + l 21 ) υ / 2 + w l 22 .$
$W i = u M i + υ M i + 1 + w M i + 2 ( i = 1 − 2 ) , H i = N i u 2 + N i + 2 υ 2 + N i + 4 w 2 ( i = 1 − 2 ) , K i = N i + 1 u υ + N i + 2 u w + N i + 3 υ w ( i = 1 − 2 ) , q i j = s M i j − c M i , j + 1 ( i = 1 − 4 , j = 1 − 3 ) , S i j = u q i j + υ q i + 1 , j + w q i + 2 , j ( i = 1 − 2 , j = 1 − 3 ) .$
$h 1 = 2 h 4 , h 2 = h 1 2 , h 3 = 2 h 2 , h 5 = 3 h 6 , υ 3 = υ / 2 .$
$A = c + h 1 W 1 + h 2 ( u S 11 + υ S 12 + w S 13 ) + h 3 ( u S 11 + υ 3 ( S 12 + S 21 ) + w S 22 ) + h 5 ( H 1 + 2 K 1 ) ,$
$B = s + h 1 W 2 + h 2 ( u S 21 + υ S 22 + w S 23 ) + h 3 [ u S 12 + υ 3 ( S 13 + S 22 ) + w S 23 ] + h 5 ( H 2 + 2 K 2 ) .$