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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  1. E. W. Marchand, “Fifth-Order Analysis of GRIN Lenses,” Appl. Opt. 24, 4371 (1985). Note that Eqs. (A6) should read E1 = (3E0 + A2)/4.
    [Crossref] [PubMed]
  2. W. Streifer, K. B. Paxton, “Analytic Solution of Ray Equations in Cylindrically Inhomogeneous Guiding Media. 1: Meridional Rays,” Appl. Opt. 10, 769 (1971).
    [Crossref] [PubMed]
  3. K. Paxton, W. Streifer, “Analytic Solution of Ray Equations in Cylindrically Inhomogeneous Guiding Media. 2: Skew Rays,” Appl. Opt. 10, 1164 (1971).
    [Crossref] [PubMed]
  4. E. Marchand, Gradient Index Optics (Academic, New York, 1978).
  5. J. D. Rees, “Non-Gaussian Imaging Properties of GRIN Fiber Lens Arrays,” Appl. Opt. 21, 1009 (1982).
    [Crossref] [PubMed]
  6. E. Marchand, “Third-Order Aberrations of the Photographic Wood Lens,” J. Opt. Soc. Am. 66, 1326 (1976).
    [Crossref]
  7. N. Yamamoto, K. Iga, “Evaluation of Gradient-Index Rod Lenses by Imaging,” Appl. Opt. 19, 1101 (1980).
    [Crossref] [PubMed]
  8. E. W. Marchand, “Distortion in a Gradient-Index Rod,” Appl. Opt. 22, 404 (1983).
    [Crossref] [PubMed]
  9. T. Sakamoto, “GRIN Lens Profile Measurement by Ray Trace Analysis,” Appl. Opt. 22, 3064 (1983).
    [Crossref] [PubMed]

1985 (1)

1983 (2)

1982 (1)

1980 (1)

1976 (1)

1971 (2)

Iga, K.

Marchand, E.

Marchand, E. W.

Paxton, K.

Paxton, K. B.

Rees, J. D.

Sakamoto, T.

Streifer, W.

Yamamoto, N.

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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 ) ] .

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