Abstract

The canonical operator theory introduced recently for the description of lossless first-order optics is extended here to first-order systems with loss or gain, elucidating the relation between canonical operator and complex ray methods. The spread functions in the space, frequency, and hybrid domains are derived in terms of the <i>ABCD</i> raytransfer matrix as well as in terms of four other new matrix descriptions of geometrical optics. The fourfold correspondence among these matrices, the Hamilton characteristics, and the spread functions in the space, frequency, and hybrid domains leads to the derivation of four fundamental explicit canonical representations of the transfer operator and to their parameterization in terms of characteristic matrix elements. The relation between adjoint operators and bidirectional propagation is derived within canonical operator theory of first-order optics as well as within a more-general setting for all reciprocal (but not necessarily lossless) optical systems.

© 1982 Optical Society of America

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  1. M. Nazarathy and J. Shamir, "First-order optics—a canonical operator representation: lossless systems," J. Opt. Soc. Am. 72, 356–364 (1982).
  2. A. Hardy, "Gaussian modes of resonators containing saturable gain medium," Appl. Opt. 19, 3830–3835 (1980).
  3. P. A. Belanger, A. Hardy, and A. E. Siegman, "Resonant modes of optical cavities with phase conjugate mirrors," Appl. Opt. 19, 602–609, (1980).
  4. J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976).
  5. D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972).
  6. J. L. Synge and B. A. Griffith, Principles of Mechanics (McGraw-Hill, New York, 1959).
  7. R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1964).
  8. A. E. Siegman, "A canonical formulation for analyzing multielement unstable resonators," IEEE J. Quantum Electron.QE-12, 35–40 (1976).
  9. P. Baues, "Huygens principle in inhomogeneous media and ageneral integral equation applicable to optical resonators," Opto-electronics 1, 37–44 (1969).
  10. S. A. Collins, Jr., "Lens-system diffraction integral written in terms of matrix optics," J. Opt. Soc. Am. 60, 1168–1177 (1970).
  11. A. Hardy, "Beam propagation through parabolic index waveguideswith distorted optical axis," Appl. Phys. 18, 223–226 (1979).
  12. M. Nazarathy and J. Shamir, "Multiport theory applied to optical design and analysis," presented at the First Mediterranean Electrotechnical Conference, Melecon '81, Tel Aviv, Israel, 1981.
  13. M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1975).
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  16. M. J. Bastiaans, "The Wigner distribution function and Hamilton's characteristics of a geometrical-optical system," Opt. Commun. 30, 321–325 (1979).
  17. D. Stoler "Operator-algebraic derivation of the impulse response of a lens system," Opt. Lett. 6, 484–486 (1981).
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  19. J. A. Arnaud, "Nonorthogonal optical systems," Bell Syst. Tech. J. 49, 2311–2347 (1970).
  20. Arnaud derives differential equations for the elements of thepoint-characteristic matrix [Eq. (14)], whereas our application of Jacobi's theorem directly leads to the ABCD ray-transfer matrix.
  21. W. J. Bastiaans, "Wigner distribution function and its application to first-order optics," J. Opt. Soc. Am. 69, 1710–1716 (1979).
  22. G. A. Deschamp, "Ray techniques in electromagnetics," Proc. IEEE 60, 1022–1035 (1972).
  23. L. B. Felsen, "Evanescent waves," J. Opt. Soc. Am. 66, 751–760 (1976).
  24. P. D. Einziger and S. Raz, "On the asymptotic theory of inhomogeneous wave tracking," Radio Sci. 15, 763–771 (1980); P. D. Einziger and L. B. Felsen, "Evanescent waves and complex rays," IEEE Trans. Antennas Propag. (to be published).
  25. M. Nazarathy and J. Shamir, "Holography described by operator algebra," J. Opt. Soc. Am. 71, 529–541 (1981).
  26. M. Nazarathy and J. Shamir, "Fourier optics described by operator algebra," J. Opt. Soc. Am. 70, 150–158 (1980).
  27. W. D. Montgomery, "Unitary operators in the homogeneous wave field," Opt. Lett. 6, 314–315 (1981).
  28. A. E. Siegman, "Orthogonality properties of optical resonator eigenmodes," Opt. Commun. 31, 369–373 (1979).
  29. M. Nazarathy, A. Hardy, and J. Shamir, "Phase conjugate mirror resonators—a canonical operator analysis," J. Opt. Soc. Am. 72, 410 (A) (1982).

1982 (2)

1981 (4)

1980 (3)

1979 (4)

A. Hardy, "Beam propagation through parabolic index waveguideswith distorted optical axis," Appl. Phys. 18, 223–226 (1979).

M. J. Bastiaans, "The Wigner distribution function and Hamilton's characteristics of a geometrical-optical system," Opt. Commun. 30, 321–325 (1979).

A. E. Siegman, "Orthogonality properties of optical resonator eigenmodes," Opt. Commun. 31, 369–373 (1979).

W. J. Bastiaans, "Wigner distribution function and its application to first-order optics," J. Opt. Soc. Am. 69, 1710–1716 (1979).

1977 (1)

1976 (2)

A. E. Siegman, "A canonical formulation for analyzing multielement unstable resonators," IEEE J. Quantum Electron.QE-12, 35–40 (1976).

L. B. Felsen, "Evanescent waves," J. Opt. Soc. Am. 66, 751–760 (1976).

1970 (2)

J. A. Arnaud, "Nonorthogonal optical systems," Bell Syst. Tech. J. 49, 2311–2347 (1970).

S. A. Collins, Jr., "Lens-system diffraction integral written in terms of matrix optics," J. Opt. Soc. Am. 60, 1168–1177 (1970).

1969 (1)

P. Baues, "Huygens principle in inhomogeneous media and ageneral integral equation applicable to optical resonators," Opto-electronics 1, 37–44 (1969).

Arnaud, J. A.

J. A. Arnaud, "Nonorthogonal optical systems," Bell Syst. Tech. J. 49, 2311–2347 (1970).

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

Bastiaans, M. J.

M. J. Bastiaans, "The Wigner distribution function and Hamilton's characteristics of a geometrical-optical system," Opt. Commun. 30, 321–325 (1979).

Bastiaans, W. J.

Baues, P.

P. Baues, "Huygens principle in inhomogeneous media and ageneral integral equation applicable to optical resonators," Opto-electronics 1, 37–44 (1969).

Belanger, P. A.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1975).

Butterweck, H. J.

H. J. Butterweck, "General theory of linear, coherent optical data-processing systems," J. Opt. Soc. Am. 67, 60–70 (1977).

H. J. Butterweck, "Data-processing systems," in Progress inOptics, Vol. XX, E. Wolf, ed. (North-Holland, Amsterdam, The Netherlands, 1980).

Collins, Jr., S. A.

Deschamp, G. A.

G. A. Deschamp, "Ray techniques in electromagnetics," Proc. IEEE 60, 1022–1035 (1972).

Einziger, P. D.

P. D. Einziger and S. Raz, "On the asymptotic theory of inhomogeneous wave tracking," Radio Sci. 15, 763–771 (1980); P. D. Einziger and L. B. Felsen, "Evanescent waves and complex rays," IEEE Trans. Antennas Propag. (to be published).

Felsen, L. B.

Griffith, B. A.

J. L. Synge and B. A. Griffith, Principles of Mechanics (McGraw-Hill, New York, 1959).

Hardy, A.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1964).

Marcuse, D.

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

Montgomery, W. D.

Nazarathy, M.

Raz, S.

P. D. Einziger and S. Raz, "On the asymptotic theory of inhomogeneous wave tracking," Radio Sci. 15, 763–771 (1980); P. D. Einziger and L. B. Felsen, "Evanescent waves and complex rays," IEEE Trans. Antennas Propag. (to be published).

Shamir, J.

Siegman, A. E.

P. A. Belanger, A. Hardy, and A. E. Siegman, "Resonant modes of optical cavities with phase conjugate mirrors," Appl. Opt. 19, 602–609, (1980).

A. E. Siegman, "Orthogonality properties of optical resonator eigenmodes," Opt. Commun. 31, 369–373 (1979).

A. E. Siegman, "A canonical formulation for analyzing multielement unstable resonators," IEEE J. Quantum Electron.QE-12, 35–40 (1976).

Stoler, D.

Synge, J. L.

J. L. Synge and B. A. Griffith, Principles of Mechanics (McGraw-Hill, New York, 1959).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1975).

Appl. Opt. (2)

Appl. Phys. (1)

A. Hardy, "Beam propagation through parabolic index waveguideswith distorted optical axis," Appl. Phys. 18, 223–226 (1979).

Bell Syst. Tech. J. (1)

J. A. Arnaud, "Nonorthogonal optical systems," Bell Syst. Tech. J. 49, 2311–2347 (1970).

IEEE J. Quantum Electron. (1)

A. E. Siegman, "A canonical formulation for analyzing multielement unstable resonators," IEEE J. Quantum Electron.QE-12, 35–40 (1976).

J. Opt. Soc. Am. (9)

Opt. Commun. (2)

A. E. Siegman, "Orthogonality properties of optical resonator eigenmodes," Opt. Commun. 31, 369–373 (1979).

M. J. Bastiaans, "The Wigner distribution function and Hamilton's characteristics of a geometrical-optical system," Opt. Commun. 30, 321–325 (1979).

Opt. Lett. (2)

Opto-electronics (1)

P. Baues, "Huygens principle in inhomogeneous media and ageneral integral equation applicable to optical resonators," Opto-electronics 1, 37–44 (1969).

Other (10)

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

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

J. L. Synge and B. A. Griffith, Principles of Mechanics (McGraw-Hill, New York, 1959).

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1964).

Arnaud derives differential equations for the elements of thepoint-characteristic matrix [Eq. (14)], whereas our application of Jacobi's theorem directly leads to the ABCD ray-transfer matrix.

H. J. Butterweck, "Data-processing systems," in Progress inOptics, Vol. XX, E. Wolf, ed. (North-Holland, Amsterdam, The Netherlands, 1980).

M. Nazarathy and J. Shamir, "Multiport theory applied to optical design and analysis," presented at the First Mediterranean Electrotechnical Conference, Melecon '81, Tel Aviv, Israel, 1981.

M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1975).

P. D. Einziger and S. Raz, "On the asymptotic theory of inhomogeneous wave tracking," Radio Sci. 15, 763–771 (1980); P. D. Einziger and L. B. Felsen, "Evanescent waves and complex rays," IEEE Trans. Antennas Propag. (to be published).

G. A. Deschamp, "Ray techniques in electromagnetics," Proc. IEEE 60, 1022–1035 (1972).

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