Abstract

The concept of isoindicial surfaces and position variables is used to develop analytical expressions for the effective focal length and back focal length of a single gradient lens with a spherical distribution of refractive index, where the square of the refractive index is a quadratic function of the distance from the center of symmetry.

© 1992 Optical Society of America

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References

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  1. E. Marchand, Gradient Index Optics (Academic, New York, 1978).
  2. S. Dorić, “Ray tracing through gradient-index media: recent improvements,” Appl. Opt. 29, 4026–4029 (1990).
    [CrossRef]
  3. E. Marchand, Gradient Index Optics (Academic, New York, 1978), Chap. 10, p. 120.
  4. I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, San Diego, Calif., 1980), Chap. 1, p. 49, formulas (4) and (5).
  5. Reference 4, Chap. 1, p. 50, formulas (1), (4), and (5).

1990

Doric, S.

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, San Diego, Calif., 1980), Chap. 1, p. 49, formulas (4) and (5).

Marchand, E.

E. Marchand, Gradient Index Optics (Academic, New York, 1978), Chap. 10, p. 120.

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

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, San Diego, Calif., 1980), Chap. 1, p. 49, formulas (4) and (5).

Appl. Opt.

Other

E. Marchand, Gradient Index Optics (Academic, New York, 1978), Chap. 10, p. 120.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, San Diego, Calif., 1980), Chap. 1, p. 49, formulas (4) and (5).

Reference 4, Chap. 1, p. 50, formulas (1), (4), and (5).

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

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Tables (2)

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Table I Paraxial Values of the Position Variables and Their Derivatives

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Table II Singlet with Spherical Distribution of Refractive Indexa

Equations (31)

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q i + 1 = q i - y i c i ( n i + 1 - n i )
y i + 1 = y i + ( d i + 1 / n i + 1 ) q i + 1 ,
q 2 = - c 1 ( n - 1 ) ,
y 2 = 1 - ( d 2 / n ) c 1 ( n - 1 ) ,
q 3 = - c 1 ( n - 1 ) - c 2 ( 1 - n ) + ( d 2 / n ) c 1 c 2 ( n - 1 ) ( 1 - n ) .
EFL = f = - 1 / q 3 ,
BFL = s F = - y 2 / q 3 .
d 2 y d t 2 = n ( p ) n ( p ) p y , d 2 z d t 2 = n ( p ) n ( p ) p z ,
d 2 y d t 2 = n ( z - z c ) n ( z - z c ) y z - z c ,
d 2 z d t 2 = n ( z - z c ) n ( z - z c ) z - z c z = ½ [ n 2 ( z - z c ) ] / z ,
t = 0 d d z n ( z - z c ) .
n ( r ) = [ N 0 2 ± N 2 2 ( r ) 2 ] 1 / 2 ,
n ( z - z c ) = [ N 0 2 ± N 2 2 ( z - z c ) 2 ] 1 / 2 .
n ( z - z c ) = [ N 0 2 - N 2 2 ( z - z c ) 2 ] 1 / 2 ,
d 2 y d t 2 = - N 2 2 y .
y 2 = cos ( N 2 t ) - c 1 [ n ( - z c ) - 1 ] N 2 sin ( N 2 t ) , q 2 = - c 1 [ n ( - z c ) - 1 ] cos ( N 2 t ) - N 2 sin ( N 2 t ) .
d x ( a + c x 2 ) 1 / 2 = sin - 1 [ x ( - c / a ) 1 / 2 ] / ( - c ) 1 / 2 , c < 0 , a > 0 ,
t = 0 d d z [ N 0 2 - N 2 2 ( z - z c ) 2 ] 1 / 2 = { sin - 1 [ ( d - z c ) N 2 / N 0 ] + sin - 1 ( z c N 2 / N 0 ) } / N 2 ,
N 2 t = sin - 1 [ ( d - z c ) N 2 / N 0 ] + sin - 1 ( z c N 2 / N 0 ) .
cos ( sin - 1 x - sin - 1 y ) = ( 1 - x 2 ) 1 / 2 ( 1 - y 2 ) 1 / 2 + x y , sin ( sin - 1 x - sin - 1 y ) = x ( 1 - y 2 ) 1 / 2 - y ( 1 - x 2 ) 1 / 2 ,
cos ( N 2 t ) = { 1 - [ ( d - z c ) N 2 / N 0 ] 2 } 1 / 2 [ 1 - ( z c N 2 / N 0 ) 2 ] 1 / 2 - z c ( d - z c ) ( N 2 / N 0 ) 2 , sin ( N 1 t ) = { z c { 1 - [ ( d - z c ) N 2 / N 0 ] 2 } 1 / 2 + ( d - z c ) [ 1 - ( z c N 2 / N 0 ) 2 ] 1 / 2 } N 2 / N 0
cos ( N 2 t ) = [ n ( d - z c ) n ( z c ) - z c ( d - z c ) N 2 2 ] / N 0 2 , sin ( N 2 t ) = { ( d - z c ) n ( z c ) + z c n ( d - z c ) } N 2 / N 0 2 .
n ( z - z c ) = [ N 0 2 + N 2 2 ( z - z c ) 2 ] 1 / 2 ,
d 2 y d t 2 = N 2 2 y .
y 2 = cosh ( N 2 t ) - c 1 [ n ( - z c ) - 1 ] N 2 sinh ( N 2 t ) , q 2 = - c 1 [ n ( - z c ) - 1 ] cosh ( N 2 t ) + N 2 sinh ( N 2 t ) .
N 2 t = sinh - 1 [ ( d - z c ) N 2 / N 0 ] - sinh - 1 [ - z c N 2 / N 0 ] .
cosh ( sinh - 1 x - sinh - 1 y ) = ( 1 + x 2 ) 1 / 2 ( 1 + y 2 ) 1 / 2 - x y , sinh ( sinh - 1 x - sinh - 1 y ) = x ( 1 + y 2 ) 1 / 2 - y ( 1 + x 2 ) 1 / 2 ,
cosh ( N 2 t ) = { 1 + [ ( d - z c ) N 2 / N 0 ] 2 } 1 / 2 [ 1 + ( z c N 2 / N 0 ) 2 ] 1 / 2 + z c ( d - z c ) ( N 2 / N 0 ) 2 , sinh ( N 2 t ) = { ( d - z c ) { 1 + [ z c N 2 / N 0 ] 2 } 1 / 2 + z c { 1 + [ ( d - z c ) N 2 / N 0 ] 2 } 1 / 2 } N 2 / N 0
cosh ( N 2 t ) = [ n ( d - z c ) n ( z c ) + z c ( d - z c ) N 2 2 ] / N 0 2 , sinh ( N 2 t ) = [ ( d - z c ) n ( z c ) + z c n ( d - z c ) ] N 2 / N 0 2 .
q 3 = q 2 - y 2 c 2 [ 1 - n ( d - z c ) ] .
n ( r ) = ( 2 - r 2 ) 1 / 2 ,

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