Abstract

The theory of canonical transforms is applied to establish the mathematical foundations of the operator algebra method, leading to useful relations between geometrical ray optics and the operator representation of wave optics. The transfer operator of a general first-order system is expressed in terms of the elements of the ray-transfer <i>ABCD</i> matrix. The theory is exemplified through the discussion of basic operator subgroups and special types of systems: Fourier, Fresnel, imaging, and afocal.

© 1982 Optical Society of America

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  2. H. J. Butterweck, "General theory of linear, coherent optical data-processing systems," J. Opt. Soc. Am. 67, 60–70 (1977).
  3. G. Bonnet, "Introduction a l'optique metaxiale," Ann. Telecommun. 33, 5–8 (1978).
  4. M. Nazarathy and J. Shamir, "Fourier optics described by operator algebra," J. Opt. Soc. Am. 70, 150–158 (1980).
  5. J. Shamir, "Cylindrical lens systems described by operator algebra," Appl. Opt. 18, 4195–4202 (1979).
  6. H. J. Butterweck, "Principles of optical data-processing," in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981).
  7. M. Nazarathy and J. Shamir, "Holography described by operator algebra," J. Opt. Soc. Am. 71, 529–541 (1981).
  8. M. Nazarathy and J. Shamir, "Wavelength variation in Fourier optics and holography described by operator algebra," Isr. J. Technol. 18, 224–231 (1980).
  9. Y. Fainman, E. Lenz, and J. Shamir, "Contouring by phase conjugation," Appl. Opt. 20, 158–163 (1981).
  10. Y. Fainman, J. Shamir, and E. Lenz, "Static and dynamic behavior of speckle patterns described by operator algebra," Appl. Opt. 20, 3526–3538 (1981).
  11. D. Stoler, "Operator methods in physical optics," J. Opt. Soc. Am. 71, 334–341, (1981).
  12. M. Bacry and M. Cadilhac, "The metaplectic group and Fourier optics," Phys. Rev. A 23, 2533–2536 (1981).
  13. M. Moshinski and C. Quesne, "Linear canonical transformations and their unitary representations," J. Math. Phys. 12, 1780–1783 (1971).
  14. C. Quesne and M. Moshinski, "Canonical transformations and matrix elements," J. Math Phys. 12, 1780–1783 (1971).
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  16. K. B. Wolf, "On self-reciprocal functions under a class of integral transforms," J. Math. Phys. 18, 1046–1051 (1977).
  17. K. B. Wolf, "Canonical transforms, separation of variables, and similarity solutions for a class of parabolic differential equations," J. Math. Phys. 17, 601–613 (1976).
  18. K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979).
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  22. H. H. Arsenault, "Generalization of the principle plane concept in matrix optics" Am J. Phys. 48, 397–399 (1980).
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  32. A. Walther, "Lenses, wave optics and eikonal functions," J. Opt. Soc. Am. 59, 1325–1333 (1969).
  33. W. H. Carter, "Wave theory for a simple lens," Opt. Acta 20, 805–826 (1973).
  34. A. Yariv, Quantum Electronics (Wiley, New York, 1975).
  35. A. Yariv, "On transmission and recovery of three-dimensional image information in optical waveguides," J. Opt. Soc. Am. 66, 301–308 (1976).
  36. A. Hardy, "Gaussian modes of resonators containing saturable gain medium," Appl. Opt. 19, 3830–3835 (1980).
  37. A. Hardy, "Beam propagation through parabolic index waveguides with distorted optical axis," Appl. Phys. 18, 223–226 (1979).
  38. P. A. Belanger, A. Hardy, and A. E. Siegman, "Resonant modes of optical cavities with phase conjugate mirrors: higher order modes," Appl. Opt. 19, 479–481 (1980).
  39. W. D. Montgomery, "Unitary operators in the homogeneous wave field," Opt. Lett. 6, 314–315 (1981).
  40. W. D. Montgomery, "Basic duality in the algebraic formulation of electromagnetic diffraction," J. Opt. Soc. Am. 59, 804–811 (1969).

1981

1980

A. Hardy, "Gaussian modes of resonators containing saturable gain medium," Appl. Opt. 19, 3830–3835 (1980).

P. A. Belanger, A. Hardy, and A. E. Siegman, "Resonant modes of optical cavities with phase conjugate mirrors: higher order modes," Appl. Opt. 19, 479–481 (1980).

M. Nazarathy and J. Shamir, "Wavelength variation in Fourier optics and holography described by operator algebra," Isr. J. Technol. 18, 224–231 (1980).

H. H. Arsenault, "Generalization of the principle plane concept in matrix optics" Am J. Phys. 48, 397–399 (1980).

M. Nazarathy and J. Shamir, "Fourier optics described by operator algebra," J. Opt. Soc. Am. 70, 150–158 (1980).

1979

J. Shamir, "Cylindrical lens systems described by operator algebra," Appl. Opt. 18, 4195–4202 (1979).

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

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

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

1978

G. Bonnet, "Introduction a l'optique metaxiale," Ann. Telecommun. 33, 5–8 (1978).

1977

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

K. B. Wolf, "On self-reciprocal functions under a class of integral transforms," J. Math. Phys. 18, 1046–1051 (1977).

1976

K. B. Wolf, "Canonical transforms, separation of variables, and similarity solutions for a class of parabolic differential equations," J. Math. Phys. 17, 601–613 (1976).

A. Yariv, "On transmission and recovery of three-dimensional image information in optical waveguides," J. Opt. Soc. Am. 66, 301–308 (1976).

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

1974

K. B. Wolf, "Canonical transforms. I. Complex linear transforms," J. Math. Phys. 15, 1295–1301 (1974); "Canonical transforms. II. Complex radial transforms," J. Math. Phys., 15, 2102–2111 (1974).

1973

W. H. Carter, "Wave theory for a simple lens," Opt. Acta 20, 805–826 (1973).

1971

G. Eichmann, "Quasi-geometric optics of media with inhomogeneous index of refraction," J. Opt. Soc. Am. 61, 161–168 (1971).

M. Moshinski and C. Quesne, "Linear canonical transformations and their unitary representations," J. Math. Phys. 12, 1780–1783 (1971).

C. Quesne and M. Moshinski, "Canonical transformations and matrix elements," J. Math Phys. 12, 1780–1783 (1971).

1970

1969

1963

Arnaud, J. A.

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

Arsenault, H. H.

H. H. Arsenault, "Generalization of the principle plane concept in matrix optics" Am J. Phys. 48, 397–399 (1980).

Bacry, M.

M. Bacry and M. Cadilhac, "The metaplectic group and Fourier optics," Phys. Rev. A 23, 2533–2536 (1981).

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).

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

Belanger, P. A.

Bonnet, G.

G. Bonnet, "Introduction a l'optique metaxiale," Ann. Telecommun. 33, 5–8 (1978).

Brouwer, W.

Burch, J. M.

A. Gerrard and J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, London, 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, "Principles of optical data-processing," in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981).

Cadilhac, M.

M. Bacry and M. Cadilhac, "The metaplectic group and Fourier optics," Phys. Rev. A 23, 2533–2536 (1981).

Carter, W. H.

W. H. Carter, "Wave theory for a simple lens," Opt. Acta 20, 805–826 (1973).

Collins, Jr., S. A.

Deschamp, G. A.

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

Eichmann, G.

Fainman, Y.

Gerrard, A.

A. Gerrard and J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, London, 1975).

Hardy, A.

Lenz, E.

Marcuse, D.

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

Montgomery, W. D.

Moshinski, M.

M. Moshinski and C. Quesne, "Linear canonical transformations and their unitary representations," J. Math. Phys. 12, 1780–1783 (1971).

C. Quesne and M. Moshinski, "Canonical transformations and matrix elements," J. Math Phys. 12, 1780–1783 (1971).

Nazarathy, M.

M. Nazarathy and J. Shamir, "Holography described by operator algebra," J. Opt. Soc. Am. 71, 529–541 (1981).

M. Nazarathy and J. Shamir, "Fourier optics described by operator algebra," J. Opt. Soc. Am. 70, 150–158 (1980).

M. Nazarathy and J. Shamir, "Wavelength variation in Fourier optics and holography described by operator algebra," Isr. J. Technol. 18, 224–231 (1980).

O’Neill, E. L.

Quesne, C.

M. Moshinski and C. Quesne, "Linear canonical transformations and their unitary representations," J. Math. Phys. 12, 1780–1783 (1971).

C. Quesne and M. Moshinski, "Canonical transformations and matrix elements," J. Math Phys. 12, 1780–1783 (1971).

Shamir, J.

Siegman, A. E.

P. A. Belanger, A. Hardy, and A. E. Siegman, "Resonant modes of optical cavities with phase conjugate mirrors: higher order modes," Appl. Opt. 19, 479–481 (1980).

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

Stavroudis, O. N.

O. N. Stavroudis, The Optics of Rays, Wavefronts and Caustics (Academic, New York, 1972).

Sternberg, S.

S. Sternberg, "Some recent results on the metaplectic representation," in Group Theoretical Methods in Physics, P. Kramer and A. Rieckers, eds. (Springer-Verlag, Berlin, 1978).

Stoler, D.

Vander Lugt, A.

A. Vander Lugt, "Operator notation for the analysis and synthesis of optical data-processing systems," Proc. IEEE 54, 1055–1063 (1966).

Walther, A.

Wolf, K. B.

K. B. Wolf, "On self-reciprocal functions under a class of integral transforms," J. Math. Phys. 18, 1046–1051 (1977).

K. B. Wolf, "Canonical transforms, separation of variables, and similarity solutions for a class of parabolic differential equations," J. Math. Phys. 17, 601–613 (1976).

K. B. Wolf, "Canonical transforms. I. Complex linear transforms," J. Math. Phys. 15, 1295–1301 (1974); "Canonical transforms. II. Complex radial transforms," J. Math. Phys., 15, 2102–2111 (1974).

K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979).

Yariv, A.

Am J. Phys.

H. H. Arsenault, "Generalization of the principle plane concept in matrix optics" Am J. Phys. 48, 397–399 (1980).

Ann. Telecommun.

G. Bonnet, "Introduction a l'optique metaxiale," Ann. Telecommun. 33, 5–8 (1978).

Appl. Opt.

Appl. Phys.

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

IEEE J. Quantum Electron.

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

Isr. J. Technol.

M. Nazarathy and J. Shamir, "Wavelength variation in Fourier optics and holography described by operator algebra," Isr. J. Technol. 18, 224–231 (1980).

J. Math Phys.

C. Quesne and M. Moshinski, "Canonical transformations and matrix elements," J. Math Phys. 12, 1780–1783 (1971).

J. Math. Phys.

K. B. Wolf, "Canonical transforms. I. Complex linear transforms," J. Math. Phys. 15, 1295–1301 (1974); "Canonical transforms. II. Complex radial transforms," J. Math. Phys., 15, 2102–2111 (1974).

K. B. Wolf, "On self-reciprocal functions under a class of integral transforms," J. Math. Phys. 18, 1046–1051 (1977).

K. B. Wolf, "Canonical transforms, separation of variables, and similarity solutions for a class of parabolic differential equations," J. Math. Phys. 17, 601–613 (1976).

M. Moshinski and C. Quesne, "Linear canonical transformations and their unitary representations," J. Math. Phys. 12, 1780–1783 (1971).

J. Opt. Soc. Am.

Opt. Acta

W. H. Carter, "Wave theory for a simple lens," Opt. Acta 20, 805–826 (1973).

Opt. Commun.

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

Opt. Lett.

Phys. Rev. A

M. Bacry and M. Cadilhac, "The metaplectic group and Fourier optics," Phys. Rev. A 23, 2533–2536 (1981).

Other

K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979).

A. Vander Lugt, "Operator notation for the analysis and synthesis of optical data-processing systems," Proc. IEEE 54, 1055–1063 (1966).

H. J. Butterweck, "Principles of optical data-processing," in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981).

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

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

A. Yariv, Quantum Electronics (Wiley, New York, 1975).

O. N. Stavroudis, The Optics of Rays, Wavefronts and Caustics (Academic, New York, 1972).

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

S. Sternberg, "Some recent results on the metaplectic representation," in Group Theoretical Methods in Physics, P. Kramer and A. Rieckers, eds. (Springer-Verlag, Berlin, 1978).

A. Gerrard and J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, London, 1975).

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