Abstract

In this paper we determine the location of image and transform planes and the equivalent focal length for a parabolic gradient-index rod, and we consider the problem of modal propagation.

© 1985 Optical Society of America

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References

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  1. P. O. McLaughlin, J. J. Miceli, D. T. Moore, D. P. Ryan, J. M. Stagaman, “Design of Gradient-Index Binocular Objective,” Proc. Soc. Photo-Opt. Instrum. Eng. 237, 369 (1980).
  2. L. G. Atkinson, S. N. Houde-Walter, D. T. Moore, D. P. Ryan, J. M. Stagaman, “Design of a Gradient-Index Photographyc Objective,” Appl. Opt. 21, 993 (1982).
    [CrossRef] [PubMed]
  3. L. G. Atkinson, D. T. Moore, N. J. Sullo, “Imaging Capabilities of a Long Gradient-Index Rod,” Appl. Opt. 21, 1005 (1982).
    [CrossRef]
  4. T. Yamagishi, K. Fujü, I. Kitano, “Gradient-Index Rod Lens with High N.A.,” Appl. Opt. 22, 400 (1983).
    [CrossRef] [PubMed]
  5. E. W. Marchand, “Distortion in a Gradient-Index Rod,” Appl. Opt. 22, 404 (1983).
    [CrossRef] [PubMed]
  6. C. Gómez-Reino, E. Larrea, “Paraxial Imaging and Transforming in a Medium with Gradient-Index: Transmittance Function,” Appl. Opt. 21, 4271 (1982).
    [CrossRef] [PubMed]
  7. C. Gómez-Reino, M. V. Pérez, E. Larrea, “Modal Propagation in GRIN Material,” Opt. Commun. 45, 372 (1983).
    [CrossRef]
  8. C. Gómez-Reino, E. Larrea, M. V. Pérez, J. M. Cuadrado, “Transmittance Function and Modal Propagation in a Conical GRIN Rod,” Appl. Opt. 23, 1107 (1984).
    [CrossRef] [PubMed]
  9. J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), Chaps. 2 and 4.
  10. E. W. Marchand, Gradient Index Optics (Academic, New York, 1978), Chaps. 1 and 5.
  11. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980), Sec. 16.
  12. C. Gómez-Reino, M. V. Pérez, E. Larrea, “Imaging and Transforming Transmission Through Inhomogeneous Media with Revolution Symmetry,” Opt. Commun. 44, 8 (1982).
    [CrossRef]

1984 (1)

1983 (3)

1982 (4)

C. Gómez-Reino, M. V. Pérez, E. Larrea, “Imaging and Transforming Transmission Through Inhomogeneous Media with Revolution Symmetry,” Opt. Commun. 44, 8 (1982).
[CrossRef]

L. G. Atkinson, S. N. Houde-Walter, D. T. Moore, D. P. Ryan, J. M. Stagaman, “Design of a Gradient-Index Photographyc Objective,” Appl. Opt. 21, 993 (1982).
[CrossRef] [PubMed]

L. G. Atkinson, D. T. Moore, N. J. Sullo, “Imaging Capabilities of a Long Gradient-Index Rod,” Appl. Opt. 21, 1005 (1982).
[CrossRef]

C. Gómez-Reino, E. Larrea, “Paraxial Imaging and Transforming in a Medium with Gradient-Index: Transmittance Function,” Appl. Opt. 21, 4271 (1982).
[CrossRef] [PubMed]

1980 (1)

P. O. McLaughlin, J. J. Miceli, D. T. Moore, D. P. Ryan, J. M. Stagaman, “Design of Gradient-Index Binocular Objective,” Proc. Soc. Photo-Opt. Instrum. Eng. 237, 369 (1980).

Arnaud, J. A.

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), Chaps. 2 and 4.

Atkinson, L. G.

L. G. Atkinson, S. N. Houde-Walter, D. T. Moore, D. P. Ryan, J. M. Stagaman, “Design of a Gradient-Index Photographyc Objective,” Appl. Opt. 21, 993 (1982).
[CrossRef] [PubMed]

L. G. Atkinson, D. T. Moore, N. J. Sullo, “Imaging Capabilities of a Long Gradient-Index Rod,” Appl. Opt. 21, 1005 (1982).
[CrossRef]

Cuadrado, J. M.

Fujü, K.

Gómez-Reino, C.

C. Gómez-Reino, E. Larrea, M. V. Pérez, J. M. Cuadrado, “Transmittance Function and Modal Propagation in a Conical GRIN Rod,” Appl. Opt. 23, 1107 (1984).
[CrossRef] [PubMed]

C. Gómez-Reino, M. V. Pérez, E. Larrea, “Modal Propagation in GRIN Material,” Opt. Commun. 45, 372 (1983).
[CrossRef]

C. Gómez-Reino, E. Larrea, “Paraxial Imaging and Transforming in a Medium with Gradient-Index: Transmittance Function,” Appl. Opt. 21, 4271 (1982).
[CrossRef] [PubMed]

C. Gómez-Reino, M. V. Pérez, E. Larrea, “Imaging and Transforming Transmission Through Inhomogeneous Media with Revolution Symmetry,” Opt. Commun. 44, 8 (1982).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980), Sec. 16.

Houde-Walter, S. N.

Kitano, I.

Larrea, E.

C. Gómez-Reino, E. Larrea, M. V. Pérez, J. M. Cuadrado, “Transmittance Function and Modal Propagation in a Conical GRIN Rod,” Appl. Opt. 23, 1107 (1984).
[CrossRef] [PubMed]

C. Gómez-Reino, M. V. Pérez, E. Larrea, “Modal Propagation in GRIN Material,” Opt. Commun. 45, 372 (1983).
[CrossRef]

C. Gómez-Reino, E. Larrea, “Paraxial Imaging and Transforming in a Medium with Gradient-Index: Transmittance Function,” Appl. Opt. 21, 4271 (1982).
[CrossRef] [PubMed]

C. Gómez-Reino, M. V. Pérez, E. Larrea, “Imaging and Transforming Transmission Through Inhomogeneous Media with Revolution Symmetry,” Opt. Commun. 44, 8 (1982).
[CrossRef]

Marchand, E. W.

E. W. Marchand, “Distortion in a Gradient-Index Rod,” Appl. Opt. 22, 404 (1983).
[CrossRef] [PubMed]

E. W. Marchand, Gradient Index Optics (Academic, New York, 1978), Chaps. 1 and 5.

McLaughlin, P. O.

P. O. McLaughlin, J. J. Miceli, D. T. Moore, D. P. Ryan, J. M. Stagaman, “Design of Gradient-Index Binocular Objective,” Proc. Soc. Photo-Opt. Instrum. Eng. 237, 369 (1980).

Miceli, J. J.

P. O. McLaughlin, J. J. Miceli, D. T. Moore, D. P. Ryan, J. M. Stagaman, “Design of Gradient-Index Binocular Objective,” Proc. Soc. Photo-Opt. Instrum. Eng. 237, 369 (1980).

Moore, D. T.

L. G. Atkinson, D. T. Moore, N. J. Sullo, “Imaging Capabilities of a Long Gradient-Index Rod,” Appl. Opt. 21, 1005 (1982).
[CrossRef]

L. G. Atkinson, S. N. Houde-Walter, D. T. Moore, D. P. Ryan, J. M. Stagaman, “Design of a Gradient-Index Photographyc Objective,” Appl. Opt. 21, 993 (1982).
[CrossRef] [PubMed]

P. O. McLaughlin, J. J. Miceli, D. T. Moore, D. P. Ryan, J. M. Stagaman, “Design of Gradient-Index Binocular Objective,” Proc. Soc. Photo-Opt. Instrum. Eng. 237, 369 (1980).

Pérez, M. V.

C. Gómez-Reino, E. Larrea, M. V. Pérez, J. M. Cuadrado, “Transmittance Function and Modal Propagation in a Conical GRIN Rod,” Appl. Opt. 23, 1107 (1984).
[CrossRef] [PubMed]

C. Gómez-Reino, M. V. Pérez, E. Larrea, “Modal Propagation in GRIN Material,” Opt. Commun. 45, 372 (1983).
[CrossRef]

C. Gómez-Reino, M. V. Pérez, E. Larrea, “Imaging and Transforming Transmission Through Inhomogeneous Media with Revolution Symmetry,” Opt. Commun. 44, 8 (1982).
[CrossRef]

Ryan, D. P.

L. G. Atkinson, S. N. Houde-Walter, D. T. Moore, D. P. Ryan, J. M. Stagaman, “Design of a Gradient-Index Photographyc Objective,” Appl. Opt. 21, 993 (1982).
[CrossRef] [PubMed]

P. O. McLaughlin, J. J. Miceli, D. T. Moore, D. P. Ryan, J. M. Stagaman, “Design of Gradient-Index Binocular Objective,” Proc. Soc. Photo-Opt. Instrum. Eng. 237, 369 (1980).

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980), Sec. 16.

Stagaman, J. M.

L. G. Atkinson, S. N. Houde-Walter, D. T. Moore, D. P. Ryan, J. M. Stagaman, “Design of a Gradient-Index Photographyc Objective,” Appl. Opt. 21, 993 (1982).
[CrossRef] [PubMed]

P. O. McLaughlin, J. J. Miceli, D. T. Moore, D. P. Ryan, J. M. Stagaman, “Design of Gradient-Index Binocular Objective,” Proc. Soc. Photo-Opt. Instrum. Eng. 237, 369 (1980).

Sullo, N. J.

L. G. Atkinson, D. T. Moore, N. J. Sullo, “Imaging Capabilities of a Long Gradient-Index Rod,” Appl. Opt. 21, 1005 (1982).
[CrossRef]

Yamagishi, T.

Appl. Opt. (6)

Opt. Commun. (2)

C. Gómez-Reino, M. V. Pérez, E. Larrea, “Imaging and Transforming Transmission Through Inhomogeneous Media with Revolution Symmetry,” Opt. Commun. 44, 8 (1982).
[CrossRef]

C. Gómez-Reino, M. V. Pérez, E. Larrea, “Modal Propagation in GRIN Material,” Opt. Commun. 45, 372 (1983).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

P. O. McLaughlin, J. J. Miceli, D. T. Moore, D. P. Ryan, J. M. Stagaman, “Design of Gradient-Index Binocular Objective,” Proc. Soc. Photo-Opt. Instrum. Eng. 237, 369 (1980).

Other (3)

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), Chaps. 2 and 4.

E. W. Marchand, Gradient Index Optics (Academic, New York, 1978), Chaps. 1 and 5.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980), Sec. 16.

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Figures (1)

Fig. 1
Fig. 1

Parabolic GRIN rod.

Equations (39)

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n 2 ( x , y , z ) = n 0 2 [ 1 g 2 ( z ) ( x 2 + y 2 ) ] ,
g ( z ) = g 0 1 ( z / L ) 2 ,
H ¨ 1,2 ( z ) + g 2 ( z ) H 1,2 ( z ) = 0
H 1 ( o ) = H ˙ 2 ( o ) = 0 ,
H ˙ 1 ( o ) = H 2 ( o ) = 1 ,
H ˙ 1 ( z ) H 2 ( z ) H ˙ 1 ( z ) H ˙ 2 ( z ) = 1 ,
sin t = z L or t = sin 1 z L .
g ( z ) = g 0 cos 2 t ,
H ˙ ( z ) = H ( t ) L cos t ,
d t d z = d d z ( sin 1 z L ) = 1 L 1 ( z L ) 2 = 1 L cos t .
H ¨ ( z ) = d d t ( H L cos t ) d t d z = H ( t ) cos t + H ( t ) sin t L 2 cos 3 t .
H ( t ) + H ( t ) tan t + ( g 0 L ) 2 sec 2 t H ( t ) = 0 .
H ( t ) = A cos t cos [ b ln ( sec t + tan t ) + ψ ] ,
b 2 = ( g 0 L ) 2 1 .
H 1 ( t ) = L b cos t sin [ b ln ( sec t + tan t ) ] ,
H 2 ( t ) = cos t cos [ b ln ( sec t + tan t ) ] .
H 1 ( z m ) = 0 image condition
sin [ b ln ( sec t m + tan t m ) ] = 0 ,
sec t m + tan t m = L + z m L z m = exp ( m π b ) ,
z m = L 1 exp ( 2 m π b ) 1 + exp ( 2 m π b ) ,
M = H 2 ( z m ) = ( 1 ) m cos t m = ( 1 ) m 1 ( z m L ) 2 ,
H 2 ( z ˜ p ) = 0 transform condition ,
cos [ b ln ( sec t ˜ p + tan t ˜ p ) ] = 0 .
sec t ˜ p + tan t ˜ p = L + z ˜ p L z ˜ p = exp [ ( 2 p + 1 ) π 2 b ] .
z ˜ p = L · 1 exp [ ( 2 p + 1 ) π b ] 1 + exp [ ( 2 p + 1 ) π b ] ,
M ˜ = H 1 ( z ˜ p ) = ( 1 ) P L b cos t ˜ p = ( 1 ) p L 2 z ˜ p 2 b .
t ( x , y , d ) = exp ( i k n 0 d ) H 2 ( d ) exp [ i k n 0 H ˙ 2 ( d ) 2 H 2 ( d ) ( x 2 + y 2 ) ] ,
f = H 2 ( d ) n 0 H ˙ 2 ( d ) ,
H ˙ 2 ( z ) = 1 L { tan t cos [ b ln ( sec t + tan t ) ] + sec t sin [ b ln ( sec t + tan t ) ] } .
f = L 2 d 2 n 0 [ d + L b tan ( b L + d L d ) ] .
β p q = k n 0 1 z [ b g 0 L 0 z g ( z ) d z ] ( p + q + 1 ) ,
b g 0 L 0 z g ( z ) d z = b 2 ln ( L + z L z ) ,
β p q = k n 0 ( p + q + 1 ) 2 z b ln ( L + z L z ) .
b g 0 L 0 z m g ( z ) d z = m π ; m = 0,1,2
ln ( L + z m L z m ) = 2 m π b
β p q = k n 0 { 1 2 ( p + q + 1 ) k n 0 z [ b g 0 L 0 z g ( z ) d z ] } 1 / 2 = k n 0 [ 1 b ( p + q + 1 ) ln ( L + z L z ) k n 0 z ] 1 / 2 .
β p q = β p q b 2 ( p + q + 1 ) 2 ln 2 ( L + z L z ) 8 k n 0 z 2 ,
| β p q β p q | z π .
z max m max 2 λ ( p max + q max + 1 ) 2 4 n 0 ,

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