Abstract

A radial gradient rod is shown to be paraxially equivalent to three air-spaced thin lenses. Formulas for calculating the powers and spacings are given, and some ray traces are compared. The effective aperture stop and entrance pupil are derived.

© 1984 Optical Society of America

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References

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  1. A. Ghatak, K. Thyagarajan, “Graded Index Optical Waveguides: A Review,” Prog. Optics 18, 1 (1980).
    [CrossRef]
  2. P. J. Sands, “Inhomogeneous Lenses. III: Paraxial Optics,” J. Opt. Soc. Am. 61, 879 (1971).
    [CrossRef]
  3. F. P. Kapron, “Parabolic Index-Gradient Cylindrical Lens,” J. Opt. Soc. Am. 60, 1433 (1970).
    [CrossRef]
  4. J. D. Rees, Xerox Corp.; private communication.
  5. A. Gerrard, J. N. Burch, Introduction to Matrix Methods in Optics (Wiley, New York, 1975).
  6. E. Delano, R. J. Pegis, “Methods of Synthesis for Dielectric Multilayer Filters,” Prog. Optics 7, 67 (1969).
    [CrossRef]

1980

A. Ghatak, K. Thyagarajan, “Graded Index Optical Waveguides: A Review,” Prog. Optics 18, 1 (1980).
[CrossRef]

1971

1970

1969

E. Delano, R. J. Pegis, “Methods of Synthesis for Dielectric Multilayer Filters,” Prog. Optics 7, 67 (1969).
[CrossRef]

Burch, J. N.

A. Gerrard, J. N. Burch, Introduction to Matrix Methods in Optics (Wiley, New York, 1975).

Delano, E.

E. Delano, R. J. Pegis, “Methods of Synthesis for Dielectric Multilayer Filters,” Prog. Optics 7, 67 (1969).
[CrossRef]

Gerrard, A.

A. Gerrard, J. N. Burch, Introduction to Matrix Methods in Optics (Wiley, New York, 1975).

Ghatak, A.

A. Ghatak, K. Thyagarajan, “Graded Index Optical Waveguides: A Review,” Prog. Optics 18, 1 (1980).
[CrossRef]

Kapron, F. P.

Pegis, R. J.

E. Delano, R. J. Pegis, “Methods of Synthesis for Dielectric Multilayer Filters,” Prog. Optics 7, 67 (1969).
[CrossRef]

Rees, J. D.

J. D. Rees, Xerox Corp.; private communication.

Sands, P. J.

Thyagarajan, K.

A. Ghatak, K. Thyagarajan, “Graded Index Optical Waveguides: A Review,” Prog. Optics 18, 1 (1980).
[CrossRef]

J. Opt. Soc. Am.

Prog. Optics

A. Ghatak, K. Thyagarajan, “Graded Index Optical Waveguides: A Review,” Prog. Optics 18, 1 (1980).
[CrossRef]

E. Delano, R. J. Pegis, “Methods of Synthesis for Dielectric Multilayer Filters,” Prog. Optics 7, 67 (1969).
[CrossRef]

Other

J. D. Rees, Xerox Corp.; private communication.

A. Gerrard, J. N. Burch, Introduction to Matrix Methods in Optics (Wiley, New York, 1975).

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Figures (7)

Fig. 1
Fig. 1

Paraxial marginal ray path in a GRIN rod showing a sinusoidal ray path.

Fig. 2
Fig. 2

Paraxial ray entering and exiting heights and angles.

Fig. 3
Fig. 3

Ray paths of limiting rays for an axial object point.

Fig. 4
Fig. 4

Ray paths of limiting rays for an off-axis object point.

Fig. 5
Fig. 5

Layout of an air-spaced thin-lens equivalent.

Fig. 6
Fig. 6

Ray path comparison of two rays in a GRIN rod and three-lens equivalent. The straight lines without arrows are the ray paths in the air-spaced lens. Both entering and exiting rays are the same.

Fig. 7
Fig. 7

Shorter GRIN rod and equivalent ray paths. This system, does not have an intermediate image as does the case in Fig. 6.

Equations (44)

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n = n 0 ( 1 - A ρ 2 2 ) .
u ( z ) = u 0 n 0 cos A z - y 0 A sin A z ,
y ( z ) = u 0 n 0 A sin A z + y 0 cos A z .
u = u 0 cos A L - y 0 n 0 A sin A L ,
y = u 0 sin A L n 0 A + y 0 cos A L .
K = - u ( u 0 = 0 ) / y 0 .
K = n 0 A sin A L .
BFL = 1 / n 0 A tan A L .
u ( z ) = A R 0 cos ( A z + δ ) ,
y ( z ) = R 0 sin ( A z + δ ) ,
R 0 = y 0 2 + u 0 2 n 0 2 A             δ = tan - 1 ( y 0 n 0 A u 0 )
y 0 2 + u 0 2 n 0 2 A = R .
y 0 = y ¯ + t 0 u 0 .
u m = - y ¯ t 0 n 0 2 A ± n 0 A R 2 ( 1 + n 0 2 t 0 2 A ) - y ¯ 2 1 + n 0 2 t 0 2 A .
u m ( y ¯ = 0 ) = ± n 0 R A 1 + n 0 2 t 0 2 A ,
R 2 ( 1 + n 0 2 t 0 2 A ) - y ¯ 2 = 0 y ¯ m = R 1 + n 0 2 t 0 2 A .
L = n 0 R 2 A .
u ¯ = - y ¯ t 0 n 0 2 A 1 + n 0 2 t 0 2 A .
t p = 1 / t 0 n 0 2 A .
tan ( A t s ) = n 0 A t p = 1 / t 0 n 0 A .
( u y ) = ( cos A L - n 0 A sin A L sin A L n 0 A cos A L ) ( u 0 y 0 ) .
( u y ) = ( 1 - K t 1 - t K ) ( u 0 y 0 ) .
( 1 - K 0 1 ) ,
( 1 0 t 1 ) .
( u 3 y 3 ) = ( 1 K 3 0 1 ) ( 1 0 t 2 1 ) ( 1 - K 2 0 1 ) × ( 1 0 t 1 0 ) ( 1 - K 1 0 1 ) ( u 0 y 0 ) .
( u 3 y 3 ) = ( m 11 m 12 m 21 m 22 ) ( u 0 y 0 ) , m 11 = 1 - t 1 K 2 - K 3 [ t 2 + t 1 ( 1 - t 2 K 2 ) ] ,
m 12 = - K 1 - K 2 ( 1 - t 1 K 1 ) - K 3 [ - t 2 K 1 + ( 1 - t 1 K 1 ) ( 1 - t 2 K 2 ) ] ,
m 21 = t 2 + t 1 ( 1 - t 2 K 2 ) ,
m 22 = - t 2 K 2 + ( 1 + t 1 K 1 ) ( 1 - t 2 K 2 ) .
1 - t 1 K 2 - K 3 [ t 2 + t 1 ( 1 - t 2 K 2 ) ] = cos A L ,
K 1 + K 2 ( 1 - t 1 K 1 ) + K 3 [ - t 2 K 1 + ( 1 - t 1 K 1 ) ( 1 - t 2 K 2 ) ] = n 0 A sin A L ,
t 2 + t 1 ( 1 - t 2 K 2 ) = sin A L / n 0 A
- t 2 K 1 + ( 1 - t 1 K 1 ) ( 1 - t 2 K 2 ) = cos A L .
t 1 + t 2 = L .
1 - t 1 K 2 - K 3 sin A L n 0 A = cos A L ,
K 1 + K 2 ( 1 - t 1 K 1 ) + K 3 cos A L = n 0 A sin A L ,
t 2 + t 1 - t 1 t 2 K 2 = sin A L n 0 A ,
1 - t 2 K 2 - K 1 sin A L n 0 A = cos A L ,
t 1 + t 2 = L .
t 1 + t 2 - t 1 t 2 K 2 = sin A L n 0 A ,
1 - t 2 K 2 - K 1 sin A L n 0 A = cos A L ,
1 - t 1 K 2 - K 3 sin A L n 0 A = cos A L ,
t 1 + t 2 = L .
K 2 = n 0 A L - sin A L t 1 t 2 n 0 A , K 1 = n 0 A sin A L ( 1 - n 0 A L - sin A L t 1 n 0 A - cos A L ) , K 3 = n 0 A sin A L ( 1 - n 0 A L - sin A L t 2 n 0 A - cos A L ) .

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