## Abstract

Previously, fifth-order algebraic raytracing formulas for radial gradients have been derived by a method of successive approximations. These formulas are here extended to the trace of optical path lengths in radial gradients. Also, the formulas are exploited to obtain algebraic formulas for third- and fifth-order aberrations of Wood and GRIN rod lenses.

© 1986 Optical Society of America

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### References

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#### Marchand, E.

E. Marchand, Gradient Index Optics (Academic, New York, 1978).

#### Other (1)

E. Marchand, Gradient Index Optics (Academic, New York, 1978).

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### Equations (54)

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$Y = Y 1 ( 1 - Z 2 2 / 2 ) + V 1 ( Z 2 + Z 4 ) + 2 h 4 ( Y 3 + Z 2 V 3 ) + 12 h 4 2 F 1 + 3 h 6 F 2 , V = V 1 ( 1 - Z 2 2 / 2 ) + Y 1 ( Z 2 + Z 4 ) + 2 h 4 ( V 3 + Z 2 W 3 ) + 12 h 4 2 G 1 + 3 h 6 G 2 .$
$n 2 = N 0 2 ( 1 - R 2 + h 4 R 4 + h 6 R 6 ) ,$
$X = g x , Y = g y , Z = g z , R 2 = X 2 + Y 2 .$
$V = q / N 0 , q = n sin γ ,$
$Y 1 = Y 0 c + V 0 s , V 1 = V 0 c - Y 0 s ; c = cos Z , s = sin Z , a = ( Y 0 2 + V 0 2 ) / 2 ; Z 2 = a Z , Z 4 = Z ( 3 a 2 - Y 0 4 h 4 ) / 2.$
$Y 3 = M 1 Y 0 3 + 3 M 2 Y 0 2 V 0 + 3 M 3 Y 0 V 0 2 + M 4 V 0 3 , V 3 = P 1 Y 0 3 + 3 P 2 Y 0 2 V 0 + 3 P 3 Y 0 V 0 2 + P 4 V 0 3 , W 3 = R 1 Y 0 3 + 3 R 2 Y 0 2 V 0 + 3 R 3 Y 0 V 0 2 + R 4 V 0 3 ,$
$A 1 = ( c 3 s + 3 A 0 ) / 4 , A 0 = ( Z + c s ) / 2 , A 2 = ( 1 - c 4 ) / 4 , B 0 = ( Z - c s ) / 2. A 3 = A 0 - A 1 , A 4 = s 4 / 4 , A 5 = ( 3 B 0 - c s 3 ) / 4 ,$
$M i = s A i - c A i + 1 , P i = c A i + s A i + 1 , R i = - s A i + c A i + 1 + δ i , δ 1 = c 3 , δ 2 = c 2 s , δ 3 = c s 2 , δ 4 = s 3 .$
$L = ∫ 0 s n d s = 1 l 0 ∫ 0 z n 2 d z = 1 g l 0 ∫ 0 Z n 2 d Z , l = l 0 = n 0 cos γ 0 ,$
$L = N 0 2 g l 0 ( Z - ∫ 0 Z R 2 d Z + h 4 ∫ 0 Z R 4 d Z + h 6 ∫ 0 Z R 6 d Z ) , R 2 = X 2 + y 2 .$
$Y ′ = Y + g C tan γ ′ , sin γ ′ = q ′ = q = N 0 V , tan γ ′ = q ( 1 - q 2 ) - 1 / 2 ≅ q + ( 1 / 2 ) q 3 + ( 3 / 8 ) q 5 .$
$Y ′ = ( Y 1 + k ′ V 1 ) ( 1 - Z 2 2 / 2 ) + ( V 1 - k ′ Y 1 ) ( Z 2 + Z 4 ) + 2 h 4 [ Y 3 + k ′ V 3 + Z 2 ( V 3 + k ′ W 3 ) ] + 12 h 4 2 ( F 1 + k ′ G 1 ) + 3 h 6 ( F 2 + k ′ G 2 ) + ( 1 / 2 ) k ′ N 0 2 V 1 2 ( V 1 + 6 h 4 V 3 - 3 Y 1 Z 2 ) + ( 3 / 8 ) k ′ N 0 4 V 1 5 ,$
$b = N 0 g , k ′ = b C .$
$m = c - k ′ s , 1 / m = c - k s , c = cos Z , s = sin Z , Z = B g , k = A b , k ′ = b C .$
$( c - A b s ) ( c - C b s ) = 1.$
$Y 1 + k ′ V 1 = m ( Y 0 - k V 0 ) , V 1 - k ′ Y 1 = m ( V 0 + k Y 0 ) .$
$Y ′ = m ( Y 0 - k V 0 ) + m ( V 0 + k Y 0 ) Z 2 + 2 h 4 ( Y 3 + k ′ V 3 ) + ( 1 / 2 ) k ′ N 0 2 V 1 3 + m ( k V 0 - Y 0 ) Z 2 2 / 2 + m ( V 0 + k Y 0 ) Z 4 + 2 h 4 Z 2 ( V 3 + k ′ W 3 ) + 12 h 4 2 ( F 1 + k ′ G 1 ) 3 h 6 ( F 2 + k ′ G 2 ) + ( 3 / 8 ) k ′ N 0 4 V 1 5 + ( 3 / 2 ) k ′ N 0 2 V 1 2 ( 2 h 4 V 3 - Y 1 Z 2 ) .$
$Y 0 = A g tan γ = A g q ( 1 - q 2 ) - 1 / 2 = A g q [ 1 + ( 1 / 2 ) q 2 + ( 3 / 8 ) q 4 ] ,$
$q = q 0 = N 0 V 0 ,$
$Y 0 = k V 0 [ 1 + ( 1 / 2 ) N 0 2 V 0 2 + ( 3 / 8 ) N 0 4 V 0 4 ] .$
$S ¯ 3 = m V 0 3 2 g [ N 0 2 ( k + k ′ / m 4 ) + Z ( 1 + k 2 ) 2 - 4 h 4 ( k 3 T 1 + 3 k 2 T 2 + 3 k T 3 + T 4 ) ] ,$
$Z = B g , V 0 = ( sin γ ) / N 0 , T i = k A i + A i + 1 .$
$S 3 = S ¯ 3 / tan γ ′ = m S ¯ 3 / N 0 V 0 ,$
$C m = ( 1 / 2 ) ( y 1 ′ + y 2 ′ ) - y 3 ′ ,$
$Y 0 = A E + H z e , V 0 = ( E - H b ) [ 1 - ( 1 / 2 ) N 0 2 ( E - H b ) 2 + ( 3 / 8 ) N 0 4 ( E - H b ) 4 ] ,$
$E = e g / ( A + z e ) , H = h g / ( A + z e ) .$
$( C m ) 3 = ( 3 / 2 ) h ( e g ) 2 ( k 1 - k ) 3 { N 0 2 ( k + k ′ / m 3 m 1 ) + Z ( 1 + k 2 ) ( 1 + k k 1 ) - 4 h 4 [ k 2 ( T 1 k 1 + T 2 ) + k ( T 2 k 1 + T 3 ) + ( T 3 k 1 + T 4 ) ] } ,$
$k 1 = - b z e , m 1 = 1 / ( c - k 1 s ) .$
$D = ( Y c ′ - m g h ) / m g h ,$
$D 3 = ( h g ) 2 2 ( k 1 - k ) 3 [ N 0 2 ( k + k ′ / m m 1 3 ) ] + Z ( 1 + k 1 2 ) ( 1 + k k 1 ) - 4 h 4 ( T 1 k 1 3 + 3 T 2 k 1 2 + 3 T 3 k 1 + T 4 ) ] .$
$Z m = - ( 1 / g ) lim E → 0 ( Y ′ - Y c ′ t ′ - t c ′ )$
$Z m = - m lim E → 0 [ ( Y ′ - Y c ′ ) / E ] .$
$( Z m ) 3 = - 3 2 b ( m g h k - k 1 ) 2 { N 0 2 ( k + k ′ / m 2 m 1 2 ) + Z [ ( 1 + k k 1 ) 2 + ( k - k 1 ) 2 / 3 ] - 4 h 4 ( k 1 2 E 1 + k E 2 + E 3 ) } ,$
$( Z s ) 3 = - 1 2 b ( m g h k - k 1 ) 2 [ N 0 2 ( k + k ′ / m 2 m 1 2 ) + Z [ ( 1 + k 2 ) + ( 1 + k 1 2 ) - 4 h 4 ( k 1 2 E 1 + k 1 E 2 + E 3 ) ] .$
$A = ( Z s - Z m ) / 2 , P = ( 3 Z s - Z m ) / 2 ,$
$A 3 = 1 2 b ( m g h k - k 1 ) 2 [ N 0 2 ( k + k ′ / m 2 m 1 2 ) + Z ( 1 + k k 1 ) 2 - 4 h 4 ( k 1 2 E 1 + k 1 E 2 + E 3 ) ] , P 3 = Z 2 b ( m g h ) 2 .$
$L 4 = N 0 2 g l 0 [ Z - ( A 0 Y 0 2 + s 2 Y 0 V 0 + 0 V 0 2 ) ] + D 0 ( Y 0 4 - V 0 4 ) - C 0 Y 0 V 0 ( Y 0 2 + V 0 2 ) - ( 1 / 2 ) h 4 ( H 1 Y 0 4 + H 2 Y 0 3 V 0 + H 3 Y 0 2 V 0 2 + H 4 Y 0 V 0 3 + H 5 V 0 4 ) } ,$
$C 0 = s ( c Z - s / 2 ) , D 0 = ( Z s 2 - B 0 ) / 2 , H 1 = 3 D 0 - 2 A 1 + A 3 , H 2 = 3 ( s 2 - C 0 ) - 11 A 2 + A 4 , H 3 = 6 ( B 0 - 3 A 3 ) , H 4 = - 3 ( A 2 - D 0 + 3 A 4 ) H 5 = - 3 ( D 0 - A 3 ) .$
$S ¯ 5 = m V 0 5 2 g { ( 3 / 4 ) N 0 4 ( k - 2 k k ′ s m 3 + k ′ m 6 ) + ( 3 / 2 ) N 0 2 Z ( 1 + k 2 ) [ k 2 + ( k ′ ) 2 / m 4 ] - 6 h 4 k N 0 2 ( T 1 k 2 + 2 T 2 k + T 3 ) + 6 h 4 k ′ N 0 2 m 3 ( k 3 P 1 + 3 k 2 P 2 + 3 k P 3 + P 4 ) + ( 3 / 4 ) Z ( 1 + k 2 ) [ ( 1 + k 2 ) 2 - h 4 k 4 ] + 2 h 4 Z m ( 1 + k 2 ) ( k 3 S 1 + 3 k 2 S 2 + 3 k S 3 + S 4 ) + 24 h 4 2 M ( k 5 U 1 + k 4 U 2 + … + U 6 ) + 6 h 6 m ( k 5 W 1 + k 4 W 2 + … + W 6 ) } .$
$S i = P i + k ′ R i , U i = L i + k ′ J i ,$
$W 1 = N 1 + k ′ Q 1 , W 2 = 5 ( N 2 + k ′ Q 2 ) , … ,$
$D 5 = ( h g ) 4 2 ( k 1 - k ) 5 { - ( 3 / 4 ) N 0 4 ( k + 2 c k ′ m m 1 2 - k ′ m m 1 5 ) + ( 3 / 2 ) N 0 2 Z k ′ k 1 ′ m m 1 3 ( 1 + k 1 2 ) - ( 1 / 2 ) N 0 2 Z ( k 1 2 + 2 k k 1 + 3 ) + 6 N 0 2 h 4 ( k 1 2 T 2 + 2 k 1 T 3 + T 4 ) + 6 N 0 2 k ′ h 4 m m 1 2 ( k 1 3 P 1 + 3 k 1 2 P 2 + 3 k 1 P 3 + P 4 ) + ( 1 / 4 ) Z 2 ( 1 + k 1 2 ) 2 ( k - k 1 ) + Z ( 1 + k k 1 ) [ ( 3 / 4 ) ( 1 + k 1 2 ) 2 - h 4 k 1 4 ] + 2 Z h 4 m ( 1 + k 1 2 ) ( k 1 3 S 1 + 3 k 1 2 S 2 + 3 k 1 S 3 + S 4 ) + 24 h 2 4 m ( k 1 5 U 1 + k 1 4 U 2 + … + U 6 ) + 6 h 6 m ( k 1 5 W 1 + k 1 4 W 2 + … + W 6 ) } .$
$k 1 ′ = b z e ′ , m 1 = c - k 1 ′ s ,$
$D = D 3 + D 5 = a 0 h 2 + a 1 h 4$
$a 0 = b 0 + b 1 h 4 , a 1 = b 2 + b 3 h 4 + b 4 h 4 2 + b 5 h 6 .$
$S = S ¯ / tan γ ′$
$tan γ ′ = N 0 V [ 1 + ( 1 / 2 ) N 0 2 V 2 ] ,$
$V = V 1 - Y 1 Z 2 + 2 h 4 V 3$
$V 1 = V 0 m [ 1 - ( 1 / 2 ) m s k N 0 2 V 0 2 ] , Y 1 = V 0 ( s + c k ) = - V 0 k ′ / m ,$
$V = V 0 m [ 1 - ( 1 / 2 ) β ′ V 0 2 ] , β ′ = m s k N 0 2 - Z ( k ′ / m ) ( 1 + k 2 ) - 4 h 4 ( k 3 P 1 + 3 k 2 P 2 + 3 k P 3 + P 4 ) .$
$S = S ¯ ( m / N 0 V 0 ) [ 1 + ( 1 / 2 ) V 0 2 ( β ′ - N 0 2 / m 2 ) ] .$
$S ¯ = S ¯ 3 + S ¯ 5 = m V 0 3 2 g ( α + β V 0 2 ) ,$
$S = ( m V 0 ) 2 2 b { α + V 0 2 [ β + ( 1 / 2 ) ( β ′ - N 0 2 / m 2 ) ] } ,$
$S 3 = ( m V 0 ) 2 α / 2 b , S 5 = m 2 V 0 4 2 b [ β + ( 1 / 2 ) ( β ′ - N 0 2 / m 2 ) ] .$