Abstract

The aberrations of graded-refractive-index-rod lenses (GRIN-rod lenses) are analyzed, with particular emphasis on the characteristics and parameter ranges of interest for various multimode optical fiber devices. Ray-tracing calculations are used to present a visual display of the image quality of particular lenses, and aberration theories are used to interpret and to extrapolate those results. Very simple general expressions are developed for the insertion losses of fiber devices as a function of the relevant lens parameters.

© 1980 Optical Society of America

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References

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  1. W. J. Tomlinson, Appl. Opt. 19, 1127 (1980).
    [CrossRef] [PubMed]
  2. S. E. Miller, Bell Syst. Tech. J. 44, 2017 (1965).
  3. Y. Suematsu, K. Iga, Electron. Commun. Jpn. 49, No. 9, 69 (Sept.1966).
  4. K. Iga, Appl. Opt. 19, 1039 (1980).
    [CrossRef] [PubMed]
  5. E. T. Kornhauser, A. D. Yaghjian, Radio Sci. 2, 299 (1967).
  6. E. G. Rawson, D. R. Herriott, J. McKenna, Appl. Opt. 9, 753 (1970).
    [CrossRef] [PubMed]
  7. E. G. Rawson, R. G. Murray, IEEE J. Quantum Electron. QE-9, 1114 (1973).
    [CrossRef]
  8. K. Maeda, J. Hamasaki, J. Opt. Soc. Am. 67, 1672 (1977).
    [CrossRef]
  9. W. Streifer, K. B. Paxton, Appl. Opt. 10, 769 (1971).
    [CrossRef] [PubMed]
  10. K. B. Paxton, W. Streifer, Appl. Opt. 10, 1164 (1971).
    [CrossRef] [PubMed]
  11. K. B. Paxton, W. Streifer, Appl. Opt. 10, 2090 (1971).
    [CrossRef] [PubMed]
  12. E. W. Marchand, Appl. Opt. 11, 1104 (1972).
    [CrossRef] [PubMed]
  13. M. H. Kuhn, Arch. Elektron. U. 31, 163 (1977).
  14. P. J. Sands, J. Opt. Soc. Am. 60, 1436 (1970).
    [CrossRef]
  15. D. T. Moore, J. Opt. Soc. Am. 61, 886 (1971).
    [CrossRef]
  16. D. T. Moore, P. J. Sands, J. Opt. Soc. Am. 61, 1195 (1971).
    [CrossRef]
  17. A. Gupta, K. Thyagarajan, I. C. Goyal, A. K. Ghatak, J. Opt. Soc. Am. 66, 1320 (1976).
    [CrossRef]
  18. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975). Sec. 5.3.
  19. F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1975), p. 137.
  20. W. J. Tomlinson, Appl. Opt. 16, 2180 (1977).
    [CrossRef] [PubMed]
  21. C. M. Miller, Bell Syst. Tech. J. 55, 917 (1976).
  22. R. Ishikawa, K. Kaede, M. Shikata, K. Kobayashi, K. Nishizawa, in 1979 National Convention Record (Institute of Electronics and Communication Engineers, Japan, 1979), paper S3-2 (in Japanese).

1980 (2)

1977 (3)

1976 (2)

1973 (1)

E. G. Rawson, R. G. Murray, IEEE J. Quantum Electron. QE-9, 1114 (1973).
[CrossRef]

1972 (1)

1971 (5)

1970 (2)

1967 (1)

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

1966 (1)

Y. Suematsu, K. Iga, Electron. Commun. Jpn. 49, No. 9, 69 (Sept.1966).

1965 (1)

S. E. Miller, Bell Syst. Tech. J. 44, 2017 (1965).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975). Sec. 5.3.

Ghatak, A. K.

Goyal, I. C.

Gupta, A.

Hamasaki, J.

Herriott, D. R.

Iga, K.

K. Iga, Appl. Opt. 19, 1039 (1980).
[CrossRef] [PubMed]

Y. Suematsu, K. Iga, Electron. Commun. Jpn. 49, No. 9, 69 (Sept.1966).

Ishikawa, R.

R. Ishikawa, K. Kaede, M. Shikata, K. Kobayashi, K. Nishizawa, in 1979 National Convention Record (Institute of Electronics and Communication Engineers, Japan, 1979), paper S3-2 (in Japanese).

Jenkins, F. A.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1975), p. 137.

Kaede, K.

R. Ishikawa, K. Kaede, M. Shikata, K. Kobayashi, K. Nishizawa, in 1979 National Convention Record (Institute of Electronics and Communication Engineers, Japan, 1979), paper S3-2 (in Japanese).

Kobayashi, K.

R. Ishikawa, K. Kaede, M. Shikata, K. Kobayashi, K. Nishizawa, in 1979 National Convention Record (Institute of Electronics and Communication Engineers, Japan, 1979), paper S3-2 (in Japanese).

Kornhauser, E. T.

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

Kuhn, M. H.

M. H. Kuhn, Arch. Elektron. U. 31, 163 (1977).

Maeda, K.

Marchand, E. W.

McKenna, J.

Miller, C. M.

C. M. Miller, Bell Syst. Tech. J. 55, 917 (1976).

Miller, S. E.

S. E. Miller, Bell Syst. Tech. J. 44, 2017 (1965).

Moore, D. T.

Murray, R. G.

E. G. Rawson, R. G. Murray, IEEE J. Quantum Electron. QE-9, 1114 (1973).
[CrossRef]

Nishizawa, K.

R. Ishikawa, K. Kaede, M. Shikata, K. Kobayashi, K. Nishizawa, in 1979 National Convention Record (Institute of Electronics and Communication Engineers, Japan, 1979), paper S3-2 (in Japanese).

Paxton, K. B.

Rawson, E. G.

E. G. Rawson, R. G. Murray, IEEE J. Quantum Electron. QE-9, 1114 (1973).
[CrossRef]

E. G. Rawson, D. R. Herriott, J. McKenna, Appl. Opt. 9, 753 (1970).
[CrossRef] [PubMed]

Sands, P. J.

Shikata, M.

R. Ishikawa, K. Kaede, M. Shikata, K. Kobayashi, K. Nishizawa, in 1979 National Convention Record (Institute of Electronics and Communication Engineers, Japan, 1979), paper S3-2 (in Japanese).

Streifer, W.

Suematsu, Y.

Y. Suematsu, K. Iga, Electron. Commun. Jpn. 49, No. 9, 69 (Sept.1966).

Thyagarajan, K.

Tomlinson, W. J.

White, H. E.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1975), p. 137.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975). Sec. 5.3.

Yaghjian, A. D.

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

Appl. Opt. (8)

Arch. Elektron. U. (1)

M. H. Kuhn, Arch. Elektron. U. 31, 163 (1977).

Bell Syst. Tech. J. (2)

S. E. Miller, Bell Syst. Tech. J. 44, 2017 (1965).

C. M. Miller, Bell Syst. Tech. J. 55, 917 (1976).

Electron. Commun. Jpn. (1)

Y. Suematsu, K. Iga, Electron. Commun. Jpn. 49, No. 9, 69 (Sept.1966).

IEEE J. Quantum Electron. (1)

E. G. Rawson, R. G. Murray, IEEE J. Quantum Electron. QE-9, 1114 (1973).
[CrossRef]

J. Opt. Soc. Am. (5)

Radio Sci. (1)

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

Other (3)

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975). Sec. 5.3.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1975), p. 137.

R. Ishikawa, K. Kaede, M. Shikata, K. Kobayashi, K. Nishizawa, in 1979 National Convention Record (Institute of Electronics and Communication Engineers, Japan, 1979), paper S3-2 (in Japanese).

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Figures (6)

Fig. 1
Fig. 1

Cross sections of GRIN rods, showing ray paths for an object point (a) on-axis and (b) off-axis.

Fig. 2
Fig. 2

Calculated spot diagrams for a GRIN-rod lens with a parabolic refractive-index distribution [Eq. (6)]. The assumed lens parameters are L = 32, na = 1.5. The input rays are approximately uniformly distributed over a cone with N.A. = 0.2.

Fig. 3
Fig. 3

Calculated spot diagrams for a GRIN-rod lens with a sech refractive-index distribution [Eq. (7)]. The other parameters are the same as for Fig. 2.

Fig. 4
Fig. 4

Curves of spot size as a function of lens length for a lens with a sech refractive-index distribution [Eq. (7)] for various off-axis positions. The quantity plotted Rmax is the maximum displacement of a ray from the ideal image point.

Fig. 5
Fig. 5

Calculated spot diagrams for a GRIN-rod lens with a helic refractive-index distribution [Eq. (8)]. The other parameters are the same as for Fig. 2.

Fig. 6
Fig. 6

Calculated spot diagrams for a GRIN-rod lens with the measured refractive-index distribution like that given in Eq. (9). The other parameters are the same as for Fig. 2.

Tables (2)

Tables Icon

Table I Relative Third-Order Aberration Coefficients for Various Index Distributions a

Tables Icon

Table II Numerical Values of the Loss and Critical Radius for Various Lens Lengths and Index Distributions a

Equations (17)

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n 2 ( r ) = n a 2 [ 1 ( α r ) 2 + α 2 ( α r ) 4 + α 3 ( α r ) 6 + . . . ] .
r ( z ) = r 0 cos ( α z ) + r 0 α sin ( α z ) ,
f = [ n a α sin ( α Z ) ] 1 ,
f = ( n a α ) 1 = L / ( 2 π n a ) .
N.A. ( r 0 ) = ( r rod 2 r 0 2 ) 1 / 2 2 π n a / L .
n 2 ( r ) = n a 2 [ 1 ( α r ) 2 ] .
n 2 ( r ) = n a 2 sech 2 ( α r ) = n α 2 [ 1 ( α r ) 2 + 2 3 ( α r ) 4 17 45 ( α r ) 6 + . . . ] .
n 2 ( r ) = n a 2 [ 1 + ( α r ) 2 ] 1 = n α 2 [ 1 ( α r ) 2 + ( α r ) 4 ( α r ) 6 + . . . ] .
n 2 ( r ) = n α 2 [ 1 ( α r ) 2 + 1.36 ( α r ) 4 3.0 ( α r ) 6 ] .
loss ( dB ) = 10 log 10 ( R + δ R R ) 20 2.3 δ R R ,
δ R SA 0.0035 | 1 3 / 2 α 2 | ( L 32 ) ( N.A. 0.2 ) 3 ,
loss SA ( dB ) = 0.12 | 1 3 / 2 α 2 | L R ( N.A. ) 3 .
δ R A 0.0023 | α 2 2 / 3 | ( L 32 ) ( r 0 32 0.5 L ) 2 ( N.A. 0.2 ) ,
loss A ( dB ) = 19.2 | α 2 | r 0 2 LR ( N.A. ) .
r c = 0.079 | 1 3 / 2 α 2 α 2 | 1 / 2 L ( N.A. ) .
1 3 2 α 2
1 2

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