Abstract

We have studied the validity of the eikonal equation for an inhomogeneous medium. The propagation of an electromagnetic wave in a medium that is characterized by a parabolic dielectric constant variation in the transverse direction, has been studied in detail. The general path of a ray has been calculated and the irradiance distribution near the focal point has been analytically obtained.

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  1. M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970).
  2. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972).
  3. L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N. J., 1973).
  4. M. Kline and I. Kay, Electromagnetic Theory and Geometrical Optics (Interscience, New York, 1965).
  5. H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1968).
  6. E. Wolf (private communication).
  7. J. B. Keller and W. Streifer, J. Opt. Soc. Am. 61, 40 (1971).
  8. K. Koizumi, Y. Ikeda, I. Kitano, M. Furukawa, and T. Sumimoto, Appl. Opt. 13, 255 (1974).
  9. It may be added that this failure is not due to the particular form of ∊(γ) chosen, which is unbounded as γ→∞. Even for the case in which ∊(x)=∊e + ∊2a2/cosh2x/a, which is bounded, the term (1/k20ψ0)∇2ψ0 is finite and equal to ∊2a2 tanh2x/a.
  10. A. Erdelyi, Higher Transcendental Functions (McGraw-Hill, New York. 1953).
  11. G. P. Agrawal, A. K. Ghatak, and C. L. Mehta, Opt. Comm. 12, 333 (1974).
  12. A two-dimensional analysis is given here for simplicity.
  13. M. S. Sodha, A. K. Ghatak, and D. P. S. Malik, J. Phys. D 4, 1887 (1971).
  14. This implies that the neglect of terms O(1/k0) in βmn and ∂2A/∂z2 in Ref. 13, is equivalent to paraxial approximation.

Wolf, E.

E. Wolf (private communication).

Agrawal, G. P.

G. P. Agrawal, A. K. Ghatak, and C. L. Mehta, Opt. Comm. 12, 333 (1974).

Born, M.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970).

Erdelyi, A.

A. Erdelyi, Higher Transcendental Functions (McGraw-Hill, New York. 1953).

Felsen, L. B.

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N. J., 1973).

Furukawa, M.

K. Koizumi, Y. Ikeda, I. Kitano, M. Furukawa, and T. Sumimoto, Appl. Opt. 13, 255 (1974).

Ghatak, A. K.

G. P. Agrawal, A. K. Ghatak, and C. L. Mehta, Opt. Comm. 12, 333 (1974).

M. S. Sodha, A. K. Ghatak, and D. P. S. Malik, J. Phys. D 4, 1887 (1971).

Goldstein, H.

H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1968).

Ikeda, Y.

K. Koizumi, Y. Ikeda, I. Kitano, M. Furukawa, and T. Sumimoto, Appl. Opt. 13, 255 (1974).

Kay, I.

M. Kline and I. Kay, Electromagnetic Theory and Geometrical Optics (Interscience, New York, 1965).

Keller, J. B.

J. B. Keller and W. Streifer, J. Opt. Soc. Am. 61, 40 (1971).

Kitano, I.

K. Koizumi, Y. Ikeda, I. Kitano, M. Furukawa, and T. Sumimoto, Appl. Opt. 13, 255 (1974).

Kline, M.

M. Kline and I. Kay, Electromagnetic Theory and Geometrical Optics (Interscience, New York, 1965).

Koizumi, K.

K. Koizumi, Y. Ikeda, I. Kitano, M. Furukawa, and T. Sumimoto, Appl. Opt. 13, 255 (1974).

Malik, D. P. S.

M. S. Sodha, A. K. Ghatak, and D. P. S. Malik, J. Phys. D 4, 1887 (1971).

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972).

Marcuvitz, N.

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N. J., 1973).

Mehta, C. L.

G. P. Agrawal, A. K. Ghatak, and C. L. Mehta, Opt. Comm. 12, 333 (1974).

Sodha, M. S.

M. S. Sodha, A. K. Ghatak, and D. P. S. Malik, J. Phys. D 4, 1887 (1971).

Streifer, W.

J. B. Keller and W. Streifer, J. Opt. Soc. Am. 61, 40 (1971).

Sumimoto, T.

K. Koizumi, Y. Ikeda, I. Kitano, M. Furukawa, and T. Sumimoto, Appl. Opt. 13, 255 (1974).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970).

Other (14)

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970).

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972).

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N. J., 1973).

M. Kline and I. Kay, Electromagnetic Theory and Geometrical Optics (Interscience, New York, 1965).

H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1968).

E. Wolf (private communication).

J. B. Keller and W. Streifer, J. Opt. Soc. Am. 61, 40 (1971).

K. Koizumi, Y. Ikeda, I. Kitano, M. Furukawa, and T. Sumimoto, Appl. Opt. 13, 255 (1974).

It may be added that this failure is not due to the particular form of ∊(γ) chosen, which is unbounded as γ→∞. Even for the case in which ∊(x)=∊e + ∊2a2/cosh2x/a, which is bounded, the term (1/k20ψ0)∇2ψ0 is finite and equal to ∊2a2 tanh2x/a.

A. Erdelyi, Higher Transcendental Functions (McGraw-Hill, New York. 1953).

G. P. Agrawal, A. K. Ghatak, and C. L. Mehta, Opt. Comm. 12, 333 (1974).

A two-dimensional analysis is given here for simplicity.

M. S. Sodha, A. K. Ghatak, and D. P. S. Malik, J. Phys. D 4, 1887 (1971).

This implies that the neglect of terms O(1/k0) in βmn and ∂2A/∂z2 in Ref. 13, is equivalent to paraxial approximation.

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