Abstract

A small-wavelength, quasi-geometric optics is developed. The approach is based on an amplitude wave function that represents both the probability amplitude of the geometrical rays and the physical-optics disturbance. All ray paths contribute phases to the amplitude function proportional to their optical paths, measured in units of normalized wavelength. However, the dominant contributions are due to those rays that satisfy Fermat’s principle. This formulation leads to an integral equation for the wave-amplitude function with a kernel function that is a path integral. The integral equation reduces, in weakly focusing paraxial media, to a solution of a Schrödinger type of partial differential equation. The classical ray variables are operators with expectation values satisfying the geometric-optics equations. The second moments of the ray operators, with minimum-uncertainty initial wave packet, define second-order optical parameters. A special case of the quasi-geometric-optics integral equation is the quasi-optical laser-mode equation, allowing the interpretation of the stationary modes as probability amplitudes. For media with arbitrary inhomogeneous index of refraction, a matrix wave equation with a two-component wave function is derived. The matrix ray operators also satisfy a correspondence principle. A physical interpretation of the new wave function is presented.

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  1. L. B. Felsen, in Proceedings of the Symposium on Quasi-Optics, edited by J. Fox (Brooklyn Polytechnical Institute Press, New York, 1964), p. 1.
  2. J. B. Keller, J. Opt. Soc. Arm. 52, 116 (1962).
  3. P. Ya. Ufimtsev, Radiotekhn. i Elektron. 7, 260 (1962).
  4. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1959), Chs. VIII, IX, and XI.
  5. V. A. Foch, Diffraction and Propagation Problems (Pergamon, New York, 1965), p. 213.
  6. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1964), p. 88.
  7. H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1950), Ch. 7.
  8. D. Gloge and D. Marcuse, J. Opt. Soc. Am. 59, 1629 (1969).
  9. I. M. Gelfand and A. M. Yaglom, J. Math. Phys. 1, 48 (1960).
  10. R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965), pp. 33, 77–78, 200–203.
  11. H. Kogelnik and T. Li, Proc. IEEE 54, 1312 (1966).
  12. L. I. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1955), pp. 25–27.
  13. A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965), p. 297.
  14. M. E. Rose, Relativistic Electron Theory (Wiley, New York, 1961), p. 32.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1959), Chs. VIII, IX, and XI.

Felsen, L. B.

L. B. Felsen, in Proceedings of the Symposium on Quasi-Optics, edited by J. Fox (Brooklyn Polytechnical Institute Press, New York, 1964), p. 1.

Feynman, R. P.

R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965), pp. 33, 77–78, 200–203.

Foch, V. A.

V. A. Foch, Diffraction and Propagation Problems (Pergamon, New York, 1965), p. 213.

Gelfand, I. M.

I. M. Gelfand and A. M. Yaglom, J. Math. Phys. 1, 48 (1960).

Gloge, D.

D. Gloge and D. Marcuse, J. Opt. Soc. Am. 59, 1629 (1969).

Goldstein, H.

H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1950), Ch. 7.

Hibbs, A. R.

R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965), pp. 33, 77–78, 200–203.

Keller, J. B.

J. B. Keller, J. Opt. Soc. Arm. 52, 116 (1962).

Kogelnik, H.

H. Kogelnik and T. Li, Proc. IEEE 54, 1312 (1966).

Li, T.

H. Kogelnik and T. Li, Proc. IEEE 54, 1312 (1966).

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1964), p. 88.

Marcuse, D.

D. Gloge and D. Marcuse, J. Opt. Soc. Am. 59, 1629 (1969).

Papoulis, A.

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965), p. 297.

Rose, M. E.

M. E. Rose, Relativistic Electron Theory (Wiley, New York, 1961), p. 32.

Schiff, L. I.

L. I. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1955), pp. 25–27.

Ufimtsev, P. Ya.

P. Ya. Ufimtsev, Radiotekhn. i Elektron. 7, 260 (1962).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1959), Chs. VIII, IX, and XI.

Yaglom, A. M.

I. M. Gelfand and A. M. Yaglom, J. Math. Phys. 1, 48 (1960).

Other (14)

L. B. Felsen, in Proceedings of the Symposium on Quasi-Optics, edited by J. Fox (Brooklyn Polytechnical Institute Press, New York, 1964), p. 1.

J. B. Keller, J. Opt. Soc. Arm. 52, 116 (1962).

P. Ya. Ufimtsev, Radiotekhn. i Elektron. 7, 260 (1962).

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1959), Chs. VIII, IX, and XI.

V. A. Foch, Diffraction and Propagation Problems (Pergamon, New York, 1965), p. 213.

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1964), p. 88.

H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1950), Ch. 7.

D. Gloge and D. Marcuse, J. Opt. Soc. Am. 59, 1629 (1969).

I. M. Gelfand and A. M. Yaglom, J. Math. Phys. 1, 48 (1960).

R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965), pp. 33, 77–78, 200–203.

H. Kogelnik and T. Li, Proc. IEEE 54, 1312 (1966).

L. I. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1955), pp. 25–27.

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965), p. 297.

M. E. Rose, Relativistic Electron Theory (Wiley, New York, 1961), p. 32.

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