Abstract

A method for computing the optical path length in gradient-index media is presented. A comparison with other methods shows that this is more accurate and faster. This method is based on a series expansion of the refractive index of the medium as a function of a suitable ray-trace parameter.

© 1985 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. W. Johnson, “Measurement of Strongly Refracting, Three-Dimensional Index Distributions,” Ph.D. Thesis, The Institute of Optics, U. Rochester, New York (1979).
  2. W. H. Southwell, “Ray Tracing in Gradient-Index Media,” J. Opt. Soc. Am. 72, 908 (1982).
    [Crossref]
  3. N. Arai, “New Calculation Method of Optical Path Length through Gradient-Index Media,” in Technical Digest, Topical Meeting on Gradient-Index Optical Imaging Systems (Optical Society of America, Washington, D.C., 1983), paper B-5.
  4. S. Doric, E. Munro, “Improvements of the Ray Trace Through the Generalized Luneberg Lens,” Appl. Opt. 22, 443 (1983).
    [Crossref] [PubMed]
  5. M. P. Rimmer, “Ray Tracing in Inhomogeneous Media,” in SPIE International Technical Conference, 18 Apr. 1983, Geneva (1983).
  6. M. P. Rimmer, “Ray Tracing in Inhomogeneous Media,” in Technical Digest, Topical Meeting on Gradient-Index Optical Imaging Systems (Optical Society of America, Washington, D.C., 1984), paper ThE-C1.
  7. A. Sharma, A. K. Ghatak, “A New Method for Tracing Rays through Graded Index Media,” in Technical Digest, Topical Meeting on Gradient-Index Optical Imaging Systems (Optical Society of America, Washington, D.C., 1981), paper MA-3.
  8. A. Sharma, D. V. Kumar, A. K. Ghatak, “Tracing Rays Through Graded-Index Media: a New Method,” Appl. Opt. 21, 984 (1982).
    [Crossref] [PubMed]
  9. L. Montagnino, “Ray Tracing in Inhomogeneous Media,” J. Opt. Soc. Am. 58, 1667 (1968).
    [Crossref]
  10. D. T. Moore, “Ray Tracing in Gradient-Index Media,” J. Opt. Soc. Am. 65, 451 (1975).
    [Crossref]
  11. E. Atad, “Optical Design of a Graded Index Biocular” in Technical Digest, Topical Meeting on Gradient-Index Optical Imaging Systems (Optical Society of America, Washington, D.C., 1984), paper ThE-A3.
  12. A. D. Boardman, D. Horncastle, “Monte Carlo Study of the Dependence of the Image Quality of a Wood Lens on Refractive Index Profile Fluctuations and Mounting Errors,” in Technical Digest, Topical Meeting on Gradient-Index Optical Imaging Systems (Optical Society of America, Washington, D.C., 1984), paper FE-A5.
  13. M. J. Nadeau, “Image Analysis of Curved Gradient-Index Rods,” M. S. Thesis, Institute of Optics, U. Rochester, New York (1984).

1983 (1)

1982 (2)

1975 (1)

1968 (1)

Arai, N.

N. Arai, “New Calculation Method of Optical Path Length through Gradient-Index Media,” in Technical Digest, Topical Meeting on Gradient-Index Optical Imaging Systems (Optical Society of America, Washington, D.C., 1983), paper B-5.

Atad, E.

E. Atad, “Optical Design of a Graded Index Biocular” in Technical Digest, Topical Meeting on Gradient-Index Optical Imaging Systems (Optical Society of America, Washington, D.C., 1984), paper ThE-A3.

Boardman, A. D.

A. D. Boardman, D. Horncastle, “Monte Carlo Study of the Dependence of the Image Quality of a Wood Lens on Refractive Index Profile Fluctuations and Mounting Errors,” in Technical Digest, Topical Meeting on Gradient-Index Optical Imaging Systems (Optical Society of America, Washington, D.C., 1984), paper FE-A5.

Doric, S.

Ghatak, A. K.

A. Sharma, D. V. Kumar, A. K. Ghatak, “Tracing Rays Through Graded-Index Media: a New Method,” Appl. Opt. 21, 984 (1982).
[Crossref] [PubMed]

A. Sharma, A. K. Ghatak, “A New Method for Tracing Rays through Graded Index Media,” in Technical Digest, Topical Meeting on Gradient-Index Optical Imaging Systems (Optical Society of America, Washington, D.C., 1981), paper MA-3.

Horncastle, D.

A. D. Boardman, D. Horncastle, “Monte Carlo Study of the Dependence of the Image Quality of a Wood Lens on Refractive Index Profile Fluctuations and Mounting Errors,” in Technical Digest, Topical Meeting on Gradient-Index Optical Imaging Systems (Optical Society of America, Washington, D.C., 1984), paper FE-A5.

Johnson, G. W.

G. W. Johnson, “Measurement of Strongly Refracting, Three-Dimensional Index Distributions,” Ph.D. Thesis, The Institute of Optics, U. Rochester, New York (1979).

Kumar, D. V.

Montagnino, L.

Moore, D. T.

Munro, E.

Nadeau, M. J.

M. J. Nadeau, “Image Analysis of Curved Gradient-Index Rods,” M. S. Thesis, Institute of Optics, U. Rochester, New York (1984).

Rimmer, M. P.

M. P. Rimmer, “Ray Tracing in Inhomogeneous Media,” in SPIE International Technical Conference, 18 Apr. 1983, Geneva (1983).

M. P. Rimmer, “Ray Tracing in Inhomogeneous Media,” in Technical Digest, Topical Meeting on Gradient-Index Optical Imaging Systems (Optical Society of America, Washington, D.C., 1984), paper ThE-C1.

Sharma, A.

A. Sharma, D. V. Kumar, A. K. Ghatak, “Tracing Rays Through Graded-Index Media: a New Method,” Appl. Opt. 21, 984 (1982).
[Crossref] [PubMed]

A. Sharma, A. K. Ghatak, “A New Method for Tracing Rays through Graded Index Media,” in Technical Digest, Topical Meeting on Gradient-Index Optical Imaging Systems (Optical Society of America, Washington, D.C., 1981), paper MA-3.

Southwell, W. H.

Appl. Opt. (2)

J. Opt. Soc. Am. (3)

Other (8)

N. Arai, “New Calculation Method of Optical Path Length through Gradient-Index Media,” in Technical Digest, Topical Meeting on Gradient-Index Optical Imaging Systems (Optical Society of America, Washington, D.C., 1983), paper B-5.

G. W. Johnson, “Measurement of Strongly Refracting, Three-Dimensional Index Distributions,” Ph.D. Thesis, The Institute of Optics, U. Rochester, New York (1979).

E. Atad, “Optical Design of a Graded Index Biocular” in Technical Digest, Topical Meeting on Gradient-Index Optical Imaging Systems (Optical Society of America, Washington, D.C., 1984), paper ThE-A3.

A. D. Boardman, D. Horncastle, “Monte Carlo Study of the Dependence of the Image Quality of a Wood Lens on Refractive Index Profile Fluctuations and Mounting Errors,” in Technical Digest, Topical Meeting on Gradient-Index Optical Imaging Systems (Optical Society of America, Washington, D.C., 1984), paper FE-A5.

M. J. Nadeau, “Image Analysis of Curved Gradient-Index Rods,” M. S. Thesis, Institute of Optics, U. Rochester, New York (1984).

M. P. Rimmer, “Ray Tracing in Inhomogeneous Media,” in SPIE International Technical Conference, 18 Apr. 1983, Geneva (1983).

M. P. Rimmer, “Ray Tracing in Inhomogeneous Media,” in Technical Digest, Topical Meeting on Gradient-Index Optical Imaging Systems (Optical Society of America, Washington, D.C., 1984), paper ThE-C1.

A. Sharma, A. K. Ghatak, “A New Method for Tracing Rays through Graded Index Media,” in Technical Digest, Topical Meeting on Gradient-Index Optical Imaging Systems (Optical Society of America, Washington, D.C., 1981), paper MA-3.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1

Geometry of the ray-tracing method.

Fig. 2
Fig. 2

Geometrical interpretation of the OPL computation methods. The actual n2(t) variation is shown by —— AC and the four-term OPL method by….AC, the trapezoidal approximation method by – ····AC, the rectangular approximation method by – ······AB, and Rimmer’s method5,6 by – – – – – AD.

Fig. 3
Fig. 3

Absolute error in computation of OPL as a function of ray-trace step size Δt for different methods.

Equations (20)

Equations on this page are rendered with MathJax. Learn more.

W = n d s ,
d 2 R d t 2 = D ( R ) ,
D = 1 2 ( n 2 x , n 2 y , n 2 z ) .
T d R d t = n d R d s
W = n 2 d t .
W m + 1 = W m + t m t m + 1 n 2 ( t ) d t = W m + 0 Δ t m n 2 ( t ˜ ) d t ˜ ,
C m + 1 = 0 Δ t m n 2 ( t ˜ ) d t ˜ .
n 2 ( t ˜ ) = a 0 + a 1 t ˜ + a 2 t ˜ + a 3 t ˜ 3 + .
N m n 2 ( t m ) n 2 ( R m ) ,
S m d n 2 d t | t = t m = Δ n 2 · T | t = t m = 2 D ( R m ) · T m ,
a 0 = N m , a 1 = S m , a 2 = [ 3 ( N m + 1 N m ) ( 2 S m + S m + 1 ) Δ t m ] / Δ t m 2 , a 3 = [ ( S m + 1 + S m ) Δ t m 2 ( N m + 1 N m ) ] / Δ t m 3 . }
C m + 1 = a 0 Δ t m + 1 2 a 1 Δ t m 2 + 1 3 a 2 Δ t m 3 + 1 4 a 3 Δ t m 4 + .
C m + 1 = N m Δ t m .
W M = W 0 + Δ t m = 0 M 1 N m ,
C m + 1 = N m Δ t m + 1 2 S m Δ t m 2 = N m Δ t m + [ D ( R m ) · T m ] Δ t m 2 .
W M = W 0 + Δ t m = 0 M 1 N m + 1 2 Δ t 2 m = 0 M 1 S m .
C m + 1 = N m Δ t m + 1 2 ( N m + 1 n m ) Δ t m = 1 2 ( N m + 1 + N m ) Δ t m .
W M = W 0 + Δ t m = 0 M N m 1 2 Δ t ( N 0 + N M ) .
C m + 1 = 1 2 Δ t m ( N m + N m + 1 ) Δ t 2 12 ( S m + 1 S m ) .
W M = W 0 + Δ t m = 0 M 1 N m Δ t 2 12 ( S M S 0 ) + 1 2 Δ t ( N M N 0 ) .

Metrics