Abstract

The formal analogy between Hamiltonian classical mechanics and quantum mechanics on the one hand and geometrical optics and physical optics on the other hand is systematically presented. When the time is replaced by the axial coordinate of a cylindrically uniform optical system and Hamilton's principle replaced by Fermat's principle, classical particle trajectories correspond to rays and quantum-mechanical wave functions to physical-optics fields. The operator formalism of quantum mechanics then provides elegant solutions for problems associated with propagation of beams in a gradient-index multimode optical slab guide.

© 1983 Optical Society of America

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  18. C. Cohen-Tannoudji, C. B. Diu, and F. Laloë, Quantum Mechanics (Wiley, New York, 1977), Vols. I and II.
  19. H. K. Shah, "An investigation in Hamiltonian optical theory of the propagation of coherent state modes in gradient index fibers," D.Sc. Dissertation (The George Washington University, Washington, D.C., May 4, 1980).
  20. W. K. Kahn and H. K. Shah, "Gaussian beam-modes as coherent states," presented at the International Union of Radio Science Symposium, Munich, Germany, August 26–29, 1980.
  21. M. Born and E. Wolf, Principles of Optics (Pergamon, New York,1959).
  22. H. Goldstein, Classical Mechanics 2nd ed. (Addison-Wesley, Reading, Mass., 1980).
  23. P. A. M. Dirac, The Principles of Quantum Mechanics 4th ed. (Oxford at the Clarendon Press, Oxford, England, 1958).
  24. For noncommuting operators X and Y, such that their commutator [X, Y] commutes with both X and Y, we have exp(X + Y) = exp(-1/2[X, Y])eXeY.
  25. N. Marcuvitz, Waveguide Handbook, Vol. 10 of Radiation Laboratory Series (McGraw-Hill, New York, 1951).

1982

1981

1977

1973

1971

1970

J. A. Arnaud, "Nonorthogonal optical waveguides and resonators," Bell Syst. Tech. J. 49, 2311–2348 (1970).

1969

1966

1965

H. Kogelnik, "Imaging of optical modes—resonators with internallenses," Bell Syst. Tech. J. 44, 455–494 (1965).

1964

E. A. J. Marcatili, "Modes in a sequence of thick astigmatic lens-like focusers," Bell Syst. Tech. J. 43, 2887–2904 (1964).

1961

J. R. Pierce, "Modes in sequences of lenses," Proc. Natl. Acad. Sci. U.S. 47, 1808–1813 (1961).

Arnaud, J. A.

J. A. Arnaud, "Nonorthogonal optical waveguides and resonators," Bell Syst. Tech. J. 49, 2311–2348 (1970).

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976).

Born, M.

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

Cohen-Tannoudji, C.

C. Cohen-Tannoudji, C. B. Diu, and F. Laloë, Quantum Mechanics (Wiley, New York, 1977), Vols. I and II.

Deschamps, G. A.

G. A. Deschamps and P. E. Mast, "Beam tracing and applications," in Proceedings of the Symposium on Quasi-Optics (Polytechnic Press, New York, 1964), Vol. XIV, pp. 379–395.

Dirac, P. A. M.

P. A. M. Dirac, The Principles of Quantum Mechanics 4th ed. (Oxford at the Clarendon Press, Oxford, England, 1958).

Diu, C. B.

C. Cohen-Tannoudji, C. B. Diu, and F. Laloë, Quantum Mechanics (Wiley, New York, 1977), Vols. I and II.

Eichmann, G.

Felsen, L. B.

Glauber, R. J.

R. J. Glauber, Quantum Optics and Electronics, Les Houches Lectures 1964, C. DeWitt, A. Blandin, and C. Cohen-Tannoudji, eds. (Gordon and Breach, New York, 1965).

Gloge, D.

Goldstein, H.

H. Goldstein, Classical Mechanics 2nd ed. (Addison-Wesley, Reading, Mass., 1980).

Gordon, J. P.

P. K. Tien, J. P. Gordon, and J. R. Whinnery, "Focusing of a light beam of Gaussian field distribution in continuous and periodic lens-like media," Proc. IEEE 53, 129–136 (1965).

Kahn, W. K.

N. Kurauchi and W. K. Kahn, "Rays and ray envelopes within stable optical resonators containing focusing media," Appl. Opt. 5, 1023–1029 (1966).

W. K. Kahn and H. K. Shah, "Gaussian beam-modes as coherent states," presented at the International Union of Radio Science Symposium, Munich, Germany, August 26–29, 1980.

Klauder, R. J.

R. J. Klauder and E. C. G. Sudarshan, Fundamentals of Quantum Optics (Benjamin, New York, 1968).

Kogelnik, H.

H. Kogelnik, "Imaging of optical modes—resonators with internallenses," Bell Syst. Tech. J. 44, 455–494 (1965).

Kurauchi, N.

Laloë, F.

C. Cohen-Tannoudji, C. B. Diu, and F. Laloë, Quantum Mechanics (Wiley, New York, 1977), Vols. I and II.

Marcatili, E. A. J.

E. A. J. Marcatili, "Modes in a sequence of thick astigmatic lens-like focusers," Bell Syst. Tech. J. 43, 2887–2904 (1964).

Marcuse, D.

D. Gloge and D. Marcuse, "Formal quantum theory of light rays," J. Opt. Soc. Am. 59, 1629–1631 (1969).

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

Marcuvitz, N.

N. Marcuvitz, Waveguide Handbook, Vol. 10 of Radiation Laboratory Series (McGraw-Hill, New York, 1951).

Mast, P. E.

G. A. Deschamps and P. E. Mast, "Beam tracing and applications," in Proceedings of the Symposium on Quasi-Optics (Polytechnic Press, New York, 1964), Vol. XIV, pp. 379–395.

Nazarathy, M.

Pierce, J. R.

J. R. Pierce, "Modes in sequences of lenses," Proc. Natl. Acad. Sci. U.S. 47, 1808–1813 (1961).

Shah, H. K.

H. K. Shah, "An investigation in Hamiltonian optical theory of the propagation of coherent state modes in gradient index fibers," D.Sc. Dissertation (The George Washington University, Washington, D.C., May 4, 1980).

W. K. Kahn and H. K. Shah, "Gaussian beam-modes as coherent states," presented at the International Union of Radio Science Symposium, Munich, Germany, August 26–29, 1980.

Shamir, J.

Shin, S. Y.

Siegman, A. E.

Stoler, D.

Sudarshan, E. C. G.

R. J. Klauder and E. C. G. Sudarshan, Fundamentals of Quantum Optics (Benjamin, New York, 1968).

Tien, P. K.

P. K. Tien, J. P. Gordon, and J. R. Whinnery, "Focusing of a light beam of Gaussian field distribution in continuous and periodic lens-like media," Proc. IEEE 53, 129–136 (1965).

Whinnery, J. R.

P. K. Tien, J. P. Gordon, and J. R. Whinnery, "Focusing of a light beam of Gaussian field distribution in continuous and periodic lens-like media," Proc. IEEE 53, 129–136 (1965).

Wolf, E.

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

Appl. Opt.

Bell Syst. Tech. J.

J. A. Arnaud, "Nonorthogonal optical waveguides and resonators," Bell Syst. Tech. J. 49, 2311–2348 (1970).

E. A. J. Marcatili, "Modes in a sequence of thick astigmatic lens-like focusers," Bell Syst. Tech. J. 43, 2887–2904 (1964).

H. Kogelnik, "Imaging of optical modes—resonators with internallenses," Bell Syst. Tech. J. 44, 455–494 (1965).

J. Opt. Soc. Am.

Proc. Natl. Acad. Sci. U.S.

J. R. Pierce, "Modes in sequences of lenses," Proc. Natl. Acad. Sci. U.S. 47, 1808–1813 (1961).

Other

G. A. Deschamps and P. E. Mast, "Beam tracing and applications," in Proceedings of the Symposium on Quasi-Optics (Polytechnic Press, New York, 1964), Vol. XIV, pp. 379–395.

P. K. Tien, J. P. Gordon, and J. R. Whinnery, "Focusing of a light beam of Gaussian field distribution in continuous and periodic lens-like media," Proc. IEEE 53, 129–136 (1965).

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976).

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

R. J. Glauber, Quantum Optics and Electronics, Les Houches Lectures 1964, C. DeWitt, A. Blandin, and C. Cohen-Tannoudji, eds. (Gordon and Breach, New York, 1965).

R. J. Klauder and E. C. G. Sudarshan, Fundamentals of Quantum Optics (Benjamin, New York, 1968).

C. Cohen-Tannoudji, C. B. Diu, and F. Laloë, Quantum Mechanics (Wiley, New York, 1977), Vols. I and II.

H. K. Shah, "An investigation in Hamiltonian optical theory of the propagation of coherent state modes in gradient index fibers," D.Sc. Dissertation (The George Washington University, Washington, D.C., May 4, 1980).

W. K. Kahn and H. K. Shah, "Gaussian beam-modes as coherent states," presented at the International Union of Radio Science Symposium, Munich, Germany, August 26–29, 1980.

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

H. Goldstein, Classical Mechanics 2nd ed. (Addison-Wesley, Reading, Mass., 1980).

P. A. M. Dirac, The Principles of Quantum Mechanics 4th ed. (Oxford at the Clarendon Press, Oxford, England, 1958).

For noncommuting operators X and Y, such that their commutator [X, Y] commutes with both X and Y, we have exp(X + Y) = exp(-1/2[X, Y])eXeY.

N. Marcuvitz, Waveguide Handbook, Vol. 10 of Radiation Laboratory Series (McGraw-Hill, New York, 1951).

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