Abstract

I apply the principle of finite elements, known in optics for calculating the beam propagation in waveguide structures, for calculation of meridional rays in inhomogeneous media. The plane is divided into finite elements that have a constant refractive index. The ray trajectory is calculated by a simple algorithm. Contrary to the existing methods, the model I propose in this research does not require an explicit formula for the index distribution. Only the numerical representation is sufficient, which can be a major advantage for calculation of the light propagation in real problems, such as the thermal lensing effect.

© 1996 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics, 6. ed. (Pergamon, New York, 1980), pp. 109–127.
  2. E. W. Marchand, “Ray tracing in gradient-index media,” J. Opt. Soc. Am. 60, 1–7 (1970).
    [CrossRef]
  3. E. W. Marchand, Gradient Index Optics (Academic, New York, 1978), Chaps. 4–5, pp. 43–66.
  4. E. W. Marchand, “Ray tracing in cylindrical gradient-index media,” Appl. Opt. 11, 1104–1106 (1972).
    [CrossRef] [PubMed]
  5. W. Streifer, K. B. Paxton, “Analytic solution of ray equations in cylindrically inhomogeneous guiding media. 1. Meridional rays,” Appl. Opt. 10, 769–775 (1971).
    [CrossRef] [PubMed]
  6. A. D. Pearson, W. G. French, E. G. Rawson, “Preparation of a light focusing glass rod by ion-exchange techniques,” Appl. Phys. Lett. 15, 76–77 (1969).
    [CrossRef]
  7. T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, “Selfoc lenses,” presented at IEEE/Optical Society of America Joint Conference on Laser Engineering Applications, Washington, D.C., May 1969.
  8. H. A. Buchdahl, “Rays in gradient-index media: separable systems,” J. Opt. Soc. Am. 63, 46–49 (1973).
    [CrossRef]
  9. D. T. Moore, “Ray tracing in gradient-index media,” J. Opt. Soc. Am. 65, 451–455 (1975).
    [CrossRef]
  10. L. Montagnino, “Ray tracing in inhomogenous media,” J. Opt. Soc. Am 58, 1667–1668 (1968).
    [CrossRef]
  11. A. Sharma, D. Vizia Kumar, A. K. Ghatak, “Tracing rays through graded-index media: a new method,” Appl. Opt. 21, 984–987 (1982).
    [CrossRef] [PubMed]
  12. Y. Chung, N. Dagli, “Modeling of guided-wave optical components with efficient finite-difference beam propagation methods,” IEEE Antennas Propag. Soc. Int. Symp. 1, 248–250 (1992).
    [CrossRef]
  13. T. Itoh, Finite Element Method in Numerical Techniques for Microwave and Millimeter-Wave Passive Structures (Wiley, New York, 1989), Chap. 1, pp. 2–62.
  14. W. H. Press, B. P. Flannery, S. A. Teukilsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, New York, 1989), Chap. 17, pp. 615–622.
  15. A. R. Mitchell, D. F. Griffiths, The Finite Difference Method in Partial Differential Equation (Wiley, New York, 1980), Chap. 1.
  16. A. R. Mitchell, R. Wait, The Finite Element Method in Partial Differential Equation (Wiley, New York, 1977), Chap. 7, pp. 166–190.
  17. J. B. Scarborough, Numerical Mathematical Analysis (Johns Hopkins U. Press, Baltimore, 1966), Chap. 97, pp. 299–303.
  18. B. Richerzhagen, G. Delacrétaz, R. P. Salathé, “Complete model to simulate the thermal defocusing of a laser beam focused in water,” Opt. Eng. 35 (7) (1996).
    [CrossRef]

1996

B. Richerzhagen, G. Delacrétaz, R. P. Salathé, “Complete model to simulate the thermal defocusing of a laser beam focused in water,” Opt. Eng. 35 (7) (1996).
[CrossRef]

1992

Y. Chung, N. Dagli, “Modeling of guided-wave optical components with efficient finite-difference beam propagation methods,” IEEE Antennas Propag. Soc. Int. Symp. 1, 248–250 (1992).
[CrossRef]

1982

1975

1973

1972

1971

1970

1969

A. D. Pearson, W. G. French, E. G. Rawson, “Preparation of a light focusing glass rod by ion-exchange techniques,” Appl. Phys. Lett. 15, 76–77 (1969).
[CrossRef]

1968

L. Montagnino, “Ray tracing in inhomogenous media,” J. Opt. Soc. Am 58, 1667–1668 (1968).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6. ed. (Pergamon, New York, 1980), pp. 109–127.

Buchdahl, H. A.

Chung, Y.

Y. Chung, N. Dagli, “Modeling of guided-wave optical components with efficient finite-difference beam propagation methods,” IEEE Antennas Propag. Soc. Int. Symp. 1, 248–250 (1992).
[CrossRef]

Dagli, N.

Y. Chung, N. Dagli, “Modeling of guided-wave optical components with efficient finite-difference beam propagation methods,” IEEE Antennas Propag. Soc. Int. Symp. 1, 248–250 (1992).
[CrossRef]

Delacrétaz, G.

B. Richerzhagen, G. Delacrétaz, R. P. Salathé, “Complete model to simulate the thermal defocusing of a laser beam focused in water,” Opt. Eng. 35 (7) (1996).
[CrossRef]

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukilsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, New York, 1989), Chap. 17, pp. 615–622.

French, W. G.

A. D. Pearson, W. G. French, E. G. Rawson, “Preparation of a light focusing glass rod by ion-exchange techniques,” Appl. Phys. Lett. 15, 76–77 (1969).
[CrossRef]

Furukawa, M.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, “Selfoc lenses,” presented at IEEE/Optical Society of America Joint Conference on Laser Engineering Applications, Washington, D.C., May 1969.

Ghatak, A. K.

Griffiths, D. F.

A. R. Mitchell, D. F. Griffiths, The Finite Difference Method in Partial Differential Equation (Wiley, New York, 1980), Chap. 1.

Itoh, T.

T. Itoh, Finite Element Method in Numerical Techniques for Microwave and Millimeter-Wave Passive Structures (Wiley, New York, 1989), Chap. 1, pp. 2–62.

Kitano, I.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, “Selfoc lenses,” presented at IEEE/Optical Society of America Joint Conference on Laser Engineering Applications, Washington, D.C., May 1969.

Koizumi, K.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, “Selfoc lenses,” presented at IEEE/Optical Society of America Joint Conference on Laser Engineering Applications, Washington, D.C., May 1969.

Marchand, E. W.

Matsumura, H.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, “Selfoc lenses,” presented at IEEE/Optical Society of America Joint Conference on Laser Engineering Applications, Washington, D.C., May 1969.

Mitchell, A. R.

A. R. Mitchell, R. Wait, The Finite Element Method in Partial Differential Equation (Wiley, New York, 1977), Chap. 7, pp. 166–190.

A. R. Mitchell, D. F. Griffiths, The Finite Difference Method in Partial Differential Equation (Wiley, New York, 1980), Chap. 1.

Montagnino, L.

L. Montagnino, “Ray tracing in inhomogenous media,” J. Opt. Soc. Am 58, 1667–1668 (1968).
[CrossRef]

Moore, D. T.

Paxton, K. B.

Pearson, A. D.

A. D. Pearson, W. G. French, E. G. Rawson, “Preparation of a light focusing glass rod by ion-exchange techniques,” Appl. Phys. Lett. 15, 76–77 (1969).
[CrossRef]

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukilsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, New York, 1989), Chap. 17, pp. 615–622.

Rawson, E. G.

A. D. Pearson, W. G. French, E. G. Rawson, “Preparation of a light focusing glass rod by ion-exchange techniques,” Appl. Phys. Lett. 15, 76–77 (1969).
[CrossRef]

Richerzhagen, B.

B. Richerzhagen, G. Delacrétaz, R. P. Salathé, “Complete model to simulate the thermal defocusing of a laser beam focused in water,” Opt. Eng. 35 (7) (1996).
[CrossRef]

Salathé, R. P.

B. Richerzhagen, G. Delacrétaz, R. P. Salathé, “Complete model to simulate the thermal defocusing of a laser beam focused in water,” Opt. Eng. 35 (7) (1996).
[CrossRef]

Scarborough, J. B.

J. B. Scarborough, Numerical Mathematical Analysis (Johns Hopkins U. Press, Baltimore, 1966), Chap. 97, pp. 299–303.

Sharma, A.

Streifer, W.

Teukilsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukilsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, New York, 1989), Chap. 17, pp. 615–622.

Uchida, T.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, “Selfoc lenses,” presented at IEEE/Optical Society of America Joint Conference on Laser Engineering Applications, Washington, D.C., May 1969.

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukilsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, New York, 1989), Chap. 17, pp. 615–622.

Vizia Kumar, D.

Wait, R.

A. R. Mitchell, R. Wait, The Finite Element Method in Partial Differential Equation (Wiley, New York, 1977), Chap. 7, pp. 166–190.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6. ed. (Pergamon, New York, 1980), pp. 109–127.

Appl. Opt.

Appl. Phys. Lett.

A. D. Pearson, W. G. French, E. G. Rawson, “Preparation of a light focusing glass rod by ion-exchange techniques,” Appl. Phys. Lett. 15, 76–77 (1969).
[CrossRef]

IEEE Antennas Propag. Soc. Int. Symp.

Y. Chung, N. Dagli, “Modeling of guided-wave optical components with efficient finite-difference beam propagation methods,” IEEE Antennas Propag. Soc. Int. Symp. 1, 248–250 (1992).
[CrossRef]

J. Opt. Soc. Am

L. Montagnino, “Ray tracing in inhomogenous media,” J. Opt. Soc. Am 58, 1667–1668 (1968).
[CrossRef]

J. Opt. Soc. Am.

Opt. Eng.

B. Richerzhagen, G. Delacrétaz, R. P. Salathé, “Complete model to simulate the thermal defocusing of a laser beam focused in water,” Opt. Eng. 35 (7) (1996).
[CrossRef]

Other

M. Born, E. Wolf, Principles of Optics, 6. ed. (Pergamon, New York, 1980), pp. 109–127.

T. Itoh, Finite Element Method in Numerical Techniques for Microwave and Millimeter-Wave Passive Structures (Wiley, New York, 1989), Chap. 1, pp. 2–62.

W. H. Press, B. P. Flannery, S. A. Teukilsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, New York, 1989), Chap. 17, pp. 615–622.

A. R. Mitchell, D. F. Griffiths, The Finite Difference Method in Partial Differential Equation (Wiley, New York, 1980), Chap. 1.

A. R. Mitchell, R. Wait, The Finite Element Method in Partial Differential Equation (Wiley, New York, 1977), Chap. 7, pp. 166–190.

J. B. Scarborough, Numerical Mathematical Analysis (Johns Hopkins U. Press, Baltimore, 1966), Chap. 97, pp. 299–303.

E. W. Marchand, Gradient Index Optics (Academic, New York, 1978), Chaps. 4–5, pp. 43–66.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, “Selfoc lenses,” presented at IEEE/Optical Society of America Joint Conference on Laser Engineering Applications, Washington, D.C., May 1969.

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Figures (3)

Fig. 1
Fig. 1

Step-by-step determination of the ray trajectory.

Fig. 2
Fig. 2

Principle of the finite element ray-tracing method.

Fig. 3
Fig. 3

Expanded element.

Tables (1)

Tables Icon

Table 1 Error in Period for the Runge–Kutta and FER Methods Compared with the Analytical Exact Solution

Equations (8)

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

d d s ( n ( r ) d r d s ) = n ( r ) ,
r = a z + b ,
r = i = 1 N a i Δ z + r 0 ,
α j = arcsin ( n i , j 1 n i , j sin β j 1 ) ,
β j = α j δ j , δ j = arctan [ Δ z 2 Δ r n i 1 , j n i + 1 , j 1 1 + 1 2 ( n i 1 , j n i + 1 , j + 1 ) ] .
n = n ( r ) = n 1 1 ( A r ) 2 ,
r = r 0 sin ( 2 π Λ z + π 2 )
Λ = 2 π A 1 ( A r 0 ) 2 .

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