Abstract

Cylindrical transparent media whose refractive index decreases with increasing cylinder radius have been investigated in connection with coherent light propagation in gas waveguides. Recently, graded index glass rods (trade named SELFOC rods) have been used as imaging devices. We report here on a geometrical optical study of graded index systems used for relaying images at unit magnification. We have found that two index distributions previously studied result in large image aberrations when the presence of skew rays is taken into account. We have derived an index distribution which is “ideal” for helical skew rays. Using ray tracing methods we have examined image aberrations for various index distributions and for various rod geometries. We find that (1) no one refractive index distribution can be “ideal” for both meridional and skew rays, (2) image resolution is generally low, reaching about 1000 spots per field, and (3) the optimal index distribution varies with the ratio of rod length to radius and the relative aperture and is intermediate between the helically ideal and the meridionally ideal distributions.

© 1970 Optical Society of America

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References

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  1. D. Marcuse, S. E. Miller, Bell System Tech. J. 43, 1759 (1964).
  2. D. Marcuse, IEEE Trans. Microwave Theory Tech. 13, 111 (1965).
  3. S. E. Miller, Bell System Tech. J. 44, 2013 (1965).
  4. D. Marcuse, Bell System Tech. J. 44, 2065 (1965).
  5. D. W. Berreman, Bell System Tech. J. 44, 2117 (1965).
  6. E. A. J. Marcatili, Bell System. Tech. J. 46, 149 (1967).
  7. A. H. Carter, unpublished work.
  8. E. T. Kornhauser, A. D. Yaghjian, Radio Sci. 2, 299 (1967).
  9. W. Streifer, C. N. Kurtz, J. Opt. Soc. Amer. 57, 779 (1967).
    [CrossRef]
  10. S. Kawakami, J. Nishizawa, IEEE Trans. Microwave Theory Tech. 16, 814 (1968).
    [CrossRef]
  11. T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, paper presented at the IEEE/Opt. Soc. Amer. Joint Conf. Laser Eng. Appl., May 1969, Washington, D.C.
  12. M. Abramowitz, A. Stegun, Handbook of Mathematical Functions (U.S. Department of Commerce, Washington, 1964), p. 85.
  13. M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1959), p. 121.
  14. Original ALGOL procedure by R. Burlirsch, J. Stoer, Numerishe Mathematic 8, 1 (1966). FORTRAN IV adaptation by N. W. Clark, P. C. Crane, P. A. Fox, Numerical Mathematics Computer Programs 2, 1 (1969), Bell Telephone Laboratories, Inc., Murray Hill, New Jersey.
    [CrossRef]
  15. A. David Pearson, W. G. French, E. G. Rawson, Appl. Phys. Lett. 15, 76 (1969).
    [CrossRef]

1969 (1)

A. David Pearson, W. G. French, E. G. Rawson, Appl. Phys. Lett. 15, 76 (1969).
[CrossRef]

1968 (1)

S. Kawakami, J. Nishizawa, IEEE Trans. Microwave Theory Tech. 16, 814 (1968).
[CrossRef]

1967 (3)

E. A. J. Marcatili, Bell System. Tech. J. 46, 149 (1967).

E. T. Kornhauser, A. D. Yaghjian, Radio Sci. 2, 299 (1967).

W. Streifer, C. N. Kurtz, J. Opt. Soc. Amer. 57, 779 (1967).
[CrossRef]

1966 (1)

Original ALGOL procedure by R. Burlirsch, J. Stoer, Numerishe Mathematic 8, 1 (1966). FORTRAN IV adaptation by N. W. Clark, P. C. Crane, P. A. Fox, Numerical Mathematics Computer Programs 2, 1 (1969), Bell Telephone Laboratories, Inc., Murray Hill, New Jersey.
[CrossRef]

1965 (4)

D. Marcuse, IEEE Trans. Microwave Theory Tech. 13, 111 (1965).

S. E. Miller, Bell System Tech. J. 44, 2013 (1965).

D. Marcuse, Bell System Tech. J. 44, 2065 (1965).

D. W. Berreman, Bell System Tech. J. 44, 2117 (1965).

1964 (1)

D. Marcuse, S. E. Miller, Bell System Tech. J. 43, 1759 (1964).

Abramowitz, M.

M. Abramowitz, A. Stegun, Handbook of Mathematical Functions (U.S. Department of Commerce, Washington, 1964), p. 85.

Berreman, D. W.

D. W. Berreman, Bell System Tech. J. 44, 2117 (1965).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1959), p. 121.

Burlirsch, R.

Original ALGOL procedure by R. Burlirsch, J. Stoer, Numerishe Mathematic 8, 1 (1966). FORTRAN IV adaptation by N. W. Clark, P. C. Crane, P. A. Fox, Numerical Mathematics Computer Programs 2, 1 (1969), Bell Telephone Laboratories, Inc., Murray Hill, New Jersey.
[CrossRef]

Carter, A. H.

A. H. Carter, unpublished work.

David Pearson, A.

A. David Pearson, W. G. French, E. G. Rawson, Appl. Phys. Lett. 15, 76 (1969).
[CrossRef]

French, W. G.

A. David Pearson, W. G. French, E. G. Rawson, Appl. Phys. Lett. 15, 76 (1969).
[CrossRef]

Furukawa, M.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, paper presented at the IEEE/Opt. Soc. Amer. Joint Conf. Laser Eng. Appl., May 1969, Washington, D.C.

Kawakami, S.

S. Kawakami, J. Nishizawa, IEEE Trans. Microwave Theory Tech. 16, 814 (1968).
[CrossRef]

Kitano, I.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, paper presented at the IEEE/Opt. Soc. Amer. Joint Conf. Laser Eng. Appl., May 1969, Washington, D.C.

Koizumi, K.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, paper presented at the IEEE/Opt. Soc. Amer. Joint Conf. Laser Eng. Appl., May 1969, Washington, D.C.

Kornhauser, E. T.

E. T. Kornhauser, A. D. Yaghjian, Radio Sci. 2, 299 (1967).

Kurtz, C. N.

W. Streifer, C. N. Kurtz, J. Opt. Soc. Amer. 57, 779 (1967).
[CrossRef]

Marcatili, E. A. J.

E. A. J. Marcatili, Bell System. Tech. J. 46, 149 (1967).

Marcuse, D.

D. Marcuse, IEEE Trans. Microwave Theory Tech. 13, 111 (1965).

D. Marcuse, Bell System Tech. J. 44, 2065 (1965).

D. Marcuse, S. E. Miller, Bell System Tech. J. 43, 1759 (1964).

Matsumura, H.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, paper presented at the IEEE/Opt. Soc. Amer. Joint Conf. Laser Eng. Appl., May 1969, Washington, D.C.

Miller, S. E.

S. E. Miller, Bell System Tech. J. 44, 2013 (1965).

D. Marcuse, S. E. Miller, Bell System Tech. J. 43, 1759 (1964).

Nishizawa, J.

S. Kawakami, J. Nishizawa, IEEE Trans. Microwave Theory Tech. 16, 814 (1968).
[CrossRef]

Rawson, E. G.

A. David Pearson, W. G. French, E. G. Rawson, Appl. Phys. Lett. 15, 76 (1969).
[CrossRef]

Stegun, A.

M. Abramowitz, A. Stegun, Handbook of Mathematical Functions (U.S. Department of Commerce, Washington, 1964), p. 85.

Stoer, J.

Original ALGOL procedure by R. Burlirsch, J. Stoer, Numerishe Mathematic 8, 1 (1966). FORTRAN IV adaptation by N. W. Clark, P. C. Crane, P. A. Fox, Numerical Mathematics Computer Programs 2, 1 (1969), Bell Telephone Laboratories, Inc., Murray Hill, New Jersey.
[CrossRef]

Streifer, W.

W. Streifer, C. N. Kurtz, J. Opt. Soc. Amer. 57, 779 (1967).
[CrossRef]

Uchida, T.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, paper presented at the IEEE/Opt. Soc. Amer. Joint Conf. Laser Eng. Appl., May 1969, Washington, D.C.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1959), p. 121.

Yaghjian, A. D.

E. T. Kornhauser, A. D. Yaghjian, Radio Sci. 2, 299 (1967).

Appl. Phys. Lett. (1)

A. David Pearson, W. G. French, E. G. Rawson, Appl. Phys. Lett. 15, 76 (1969).
[CrossRef]

Bell System Tech. J. (4)

S. E. Miller, Bell System Tech. J. 44, 2013 (1965).

D. Marcuse, Bell System Tech. J. 44, 2065 (1965).

D. W. Berreman, Bell System Tech. J. 44, 2117 (1965).

D. Marcuse, S. E. Miller, Bell System Tech. J. 43, 1759 (1964).

Bell System. Tech. J. (1)

E. A. J. Marcatili, Bell System. Tech. J. 46, 149 (1967).

IEEE Trans. Microwave Theory Tech. (2)

D. Marcuse, IEEE Trans. Microwave Theory Tech. 13, 111 (1965).

S. Kawakami, J. Nishizawa, IEEE Trans. Microwave Theory Tech. 16, 814 (1968).
[CrossRef]

J. Opt. Soc. Amer. (1)

W. Streifer, C. N. Kurtz, J. Opt. Soc. Amer. 57, 779 (1967).
[CrossRef]

Numerishe Mathematic (1)

Original ALGOL procedure by R. Burlirsch, J. Stoer, Numerishe Mathematic 8, 1 (1966). FORTRAN IV adaptation by N. W. Clark, P. C. Crane, P. A. Fox, Numerical Mathematics Computer Programs 2, 1 (1969), Bell Telephone Laboratories, Inc., Murray Hill, New Jersey.
[CrossRef]

Radio Sci. (1)

E. T. Kornhauser, A. D. Yaghjian, Radio Sci. 2, 299 (1967).

Other (4)

A. H. Carter, unpublished work.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, paper presented at the IEEE/Opt. Soc. Amer. Joint Conf. Laser Eng. Appl., May 1969, Washington, D.C.

M. Abramowitz, A. Stegun, Handbook of Mathematical Functions (U.S. Department of Commerce, Washington, 1964), p. 85.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1959), p. 121.

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Figures (6)

Fig. 1
Fig. 1

The path of a light ray in a cylindrical medium is defined by two scalar equations which define r(z) and θ(z) as functions of the independent variable z. The refractive index n(r) is a function of r only. The vectors ur, uθ, and uz are a right-handed set of orthonormal unit vectors, with uz being a constant vector along the z axis.

Fig. 2
Fig. 2

If we unwrap the surface r = ro, the path of a helical ray of radius ro forms the hypotenuse of a right triangle whose sides are 2πro and L, where L is the length of the rod.

Fig. 3
Fig. 3

Transverse aberrations of a GRIN rod of radius rrod = 0.1 L and object radius robj = 0.95rrod. Aberrations are plotted as functions of γo, the initial angle which the ray makes with the z axis.

Fig. 4
Fig. 4

Transverse aberrations of a GRIN rod of radius rrod = 0.01 L and object radius robj = 0.95rrod. Aberrations are plotted as functions of γo, the initial angle which the ray makes with the z axis.

Fig. 5
Fig. 5

Transverse aberrations of a GRIN rod of radius rrod = 0.1 L and object radius robj = 0.095rrod. Aberrations are plotted as functions of γo, the initial angle which the ray makes with the z axis.

Fig. 6
Fig. 6

Transverse aberrations of a GRIN rod of radius rrod = 0.1 L, object radius robj = 0.95rrod, and with an aperture stop of radius rstop = 0.095rrod in the plane z = L/4. Aberrations are plotted as functions of γo, the initial angle which the ray makes with the z axis.

Equations (28)

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n ( r ) = n A ( 1 1 2 α 2 r 2 ) ,
n ( r ) = n A sech ( α r ) = n A ( 1 1 2 α 2 r 2 + 5 24 α 4 r 4 61 720 α 6 r 6 + + E n ( 2 n ) ! ( α r ) 2 n + ) ,
n ( r ) = n A ( 1 + α 2 r 2 ) 1 2 = n A [ 1 α 2 r 2 2 + 3 8 α 4 r 4 15 48 α 6 r 6 + + ( 1 ) n 1 · 3 · 5 ( 2 n 1 ) 2 n n ! ( α r ) 2 n + ] ,
( d / d s ) [ n ( d r / d s ) ] = n ,
d 2 r d z 2 r ( d θ d z ) 2 = 1 2 sec 2 γ 0 n 2 ( r 0 ) d d r [ n 2 ( r ) ]
r 2 ( d θ / d z ) = sec γ 0 ( x 0 cos β 0 y 0 cos α 0 ) ,
d θ d z = sec γ 0 r 0 cos β 0 r 0 2 = 2 π L ,
1 n 2 ( r 0 ) d d r 0 [ n 2 ( r 0 ) ] = 2 ( 2 π L ) 2 r 0 cos 2 γ 0 .
1 n 2 ( r 0 ) d d r 0 [ n 2 ( r 0 ) ] = 2 ( d / d r 0 ) [ n ( r 0 ) ] n ( r 0 ) = 2 d d r 0 ln [ n ( r 0 ) ] .
cos γ 0 = L [ L 2 + ( 2 π r 0 ) 2 ] 1 2 = 1 [ 1 + ( 2 π r 0 / L ) 2 ] 1 2 .
d d r 0 ln [ n ( r 0 ) ] = r 0 ( 2 π / L ) 2 1 + ( 2 π r 0 / L ) 2 .
ln [ n ( r 0 ) n A ] = 1 2 r = 0 r 0 2 ( 2 π / L ) 2 r d r 1 + ( 2 π r / L ) 2 = 1 2 ln [ 1 + ( 2 π r 0 L ) 2 ] = ln [ 1 + ( 2 π r 0 L ) 2 ] 1 2
n ( r 0 ) n A = 1 [ 1 + ( 2 π r 0 / L ) 2 ] 1 2 .
C * ( n ) = ( 1 σ ) C sech ( n ) + σ C helix ( n ) ,
( d / d s ) [ n ( d r / d s ) ] = n ,
d d s ( n d r d s ) n r ( d θ d s ) 2 = d n d r ,
n d r d s d θ d s + d d s ( n r d θ d s ) = 0 ,
d / d s [ n ( d z / d s ) ] = 0.
( d / d s ) [ n r 2 ( d θ / d s ) ] = 0.
n d z d s = n ( r 0 ) d z d s | 0
n r 2 d θ d s = n ( r 0 ) r 0 2 d θ d s | 0 ,
n ( d z / d s ) = n ( r 0 ) cos γ 0
n ( d / d s ) = n ( r 0 ) cos γ 0 ( d / d z ) .
n 2 ( r 0 ) cos 2 γ 0 [ d 2 r d z 2 r ( d θ d z ) 2 ] = n d n d r = 1 2 d d r ( n 2 ) .
r 2 cos γ 0 ( d θ / d z ) = x 0 cos β 0 y 0 cos α 0 .
d 2 r d z 2 r ( d θ d z ) 2 = 1 2 sec 2 γ 0 n 2 ( r 0 ) d d r [ n 2 ( r ) ]
r 2 d θ d z = sec γ 0 ( x 0 cos β 0 y 0 cos α 0 ) .
( d r / d z ) 0 = sec γ 0 ( cos θ 0 cos α 0 + sin θ 0 cos β 0 ) , x 0 = r 0 cos θ 0 , y 0 = r 0 sin θ 0 .

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