Abstract

Wigner’s quasi probability and related functional and operator methods of quantum mechanics have recently played an important role in optics. We present an account of some of these developments. The symmetry structures underlying the ray and wave approaches to paraxial optics are explored in some detail, and the manner in which the Wigner phase-space representation captures the merits of both approaches is brought out. A fairly self-contained analysis of the second or intensity moments of general astigmatic partially coherent beams and of their behavior under transmission through astigmatic first-order optical systems is presented. Geometric representations of the intensity moments that render the quality parameters or polynomial invariants manifest are discussed, and the role of the optical uncertainty principle in assigning unbeatable physical bounds for these invariants is stressed. Measurement of the ten intensity moments of an astigmatic partially coherent beam is considered.

© 2000 Optical Society of America

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2000 (3)

1999 (2)

K. B. Wolf, M. A. Alonso, G. W. Forbes, “Wigner functions for Helmholtz wave fields,” J. Opt. Soc. Am. A 16, 2476–2487 (1999).
[CrossRef]

R. Simon, S. Chaturvedi, V. Srinivasan, “Congruences and canonical forms for a positive matrix: application to Schweinler–Wigner extremum principle,” J. Math. Phys. 40, 3632–3640 (1999).
[CrossRef]

1998 (5)

1996 (1)

R. Simon, A. T. Friberg, E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5, 331–343 (1996).
[CrossRef]

1995 (3)

1994 (5)

G. Nemes, A. E. Siegman, “Measurement of all ten second-order moments of an astigmatic beam by the use of rotating simple astigmatic (anamorphic) optics,” J. Opt. Soc. Am. A 11, 2257–2264 (1994).
[CrossRef]

G. S. Agarwal, R. Simon, “A simple realization of fractional Fourier transformation and relation to harmonic oscillator Green’s function,” Opt. Commun. 110, 23–26 (1994).
[CrossRef]

A. T. Friberg, E. Tervonen, J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818–1826 (1994).
[CrossRef]

D. Ambrosini, V. Bagini, F. Gori, M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[CrossRef]

R. Simon, N. Mukunda, B. Dutta, “Quantum-noise matrix for multimode systems: U(n) invariance, squeezing, and normal forms,” Phys. Rev. A 49, 1567–1683 (1994).
[CrossRef] [PubMed]

1993 (9)

R. Martinez-Herrero, P. M. Mejias, H. Weber, “On the different definitions of laser beam moments,” Opt. Quantum Electron. 25, 423–428 (1993); A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed. Proc. SPIE1224, 2–14 (1990).
[CrossRef]

R. Simon, N. Mukunda, “Bargmann invariant and the geometry of the Gouy effect,” Phys. Rev. Lett. 70, 880–883 (1993); N. Mukunda, R. Simon, “Quantum kinematic approach to the geometric phase. I. General formalism,” Ann. Phys. (N.Y.) 228, 205–268 (1993) (an extensive set of references on geometric phase can be found here); “Quantum kinematic approach to the geometric phase. II. The case of unitary group representations,” Ann. Phys. (N.Y.) 228, 269–340 (1993).
[CrossRef] [PubMed]

J. Serna, G. Nemes, “Decoupling of coherent Gaussian beam with general astigmatism,” Opt. Lett. 18, 1774–1776 (1993).
[CrossRef] [PubMed]

R. Simon, N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993).
[CrossRef]

D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation: I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993); H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transforms and their optical implementation. II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
[CrossRef]

A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993); S. Abe, J. T. Sheridan, “Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation: an operator approach,” J. Phys. A 27, 4179–4187 (1994).
[CrossRef]

R. Simon, N. Mukunda, “Iwasawa decomposition for SU(1,1) and the Gouy effect for squeezed states,” Opt. Commun. 95, 39–45 (1993).
[CrossRef]

G. S. Agarwal, R. Simon, “An experiment for the study of the Gouy effect for the squeezed vacuum,” Opt. Commun. 100, 411–414 (1993).
[CrossRef]

R. Simon, K. Sundar, N. Mukunda, “Twisted Gaussian Schell-model beams. I. Symmetry structure and normal-mode spectrum,” J. Opt. Soc. Am. A 10, 2008–2016 (1993).
[CrossRef]

1992 (5)

A. J. Dragt, F. Neri, G. Rangarajan, “General moment invariants for linear Hamiltonian systems,” Phys. Rev. A 45, 2572–2585 (1992).
[CrossRef] [PubMed]

H. Weber, “Wave optical analysis of the phase space analyzer,” J. Mod. Opt. 39, 543–559 (1992); D. Onciul, “Invariance properties of general astigmatic beams through first-order optical systems,” J. Opt. Soc. Am. A 10, 295–298 (1993); S. Lavi, R. Prochaska, E. Keren, “Generalized beam parameters and transformation laws for partially coherent light,” Appl. Opt. 27, 3696–3703 (1988).
[CrossRef] [PubMed]

J. Serna, P. M. Mejias, R. Martinez-Herrero, “Beam quality dependence on the coherence length of Gaussian Schell-model fields propagating through ABCD optical systems,” J. Mod. Opt. 39, 625–635 (1992); J. Yang, D. Fan, “Intensity-moments characterization of general pulsed paraxial beams with the Wigner distribution function,” J. Opt. Soc. Am. A 16, 2488–2493 (1999).
[CrossRef]

A. T. Friberg, G. S. Agarwal, J. T. Foley, E. Wolf, “Statistical wave-theoretical derivation of the free-space transport equation of radiometry,” J. Opt. Soc. Am. B 9, 1386–1393 (1992).
[CrossRef]

H. Weber, “Propagation of the higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, S1027–S1049 (1992); D. Dragoman, “Higher-order moments of the Wigner distribution function in first-order optical systems,” J. Opt. Soc. Am. A 11, 2643–2646 (1994).
[CrossRef]

1991 (3)

1990 (1)

F. J. Narcowich, “Geometry and uncertainty,” J. Math. Phys. 31, 354–364 (1990); Q. Lin, S. Wang, J. Alda, E. Bernabeu, “Transformation of pulsed nonideal beams in a four-dimensional domain,” Opt. Lett. 18, 669–671 (1993).
[CrossRef] [PubMed]

1988 (4)

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized abcd-law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian pure states in quantum mechanics and the symplectic group,” Phys. Rev. A 37, 2028–2038 (1988).
[CrossRef]

A. T. Friberg, J. Turunen, “Imaging of Gaussian Schell-model sources,” J. Opt. Soc. Am. A 5, 713–720 (1988).
[CrossRef]

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Hamilton’s theory of turns generalized to Sp(2, R),” Phys. Rev. Lett. 62, 1331–1334 (1988); R. Simon, N. Mukunda, E. C. G. Sudarshan, “The theory of screws—a new geometric representation for the group SU(1,1),” J. Math. Phys. 30, 1000–1006 (1989).
[CrossRef]

1987 (3)

T. Pradhan, “Maxwell’s equations from geometrical optics,” Phys. Lett. A 122, 397–398 (1987).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987); R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian Wigner distributions: a complete characterization,” Phys. Lett. A 124, 223–228 (1987).
[CrossRef] [PubMed]

G. S. Agarwal, J. T. Foley, E. Wolf, “The radiance and phase-space representations of the cross-spectral density operator,” Opt. Commun. 62, 67–72 (1987).
[CrossRef]

1986 (2)

R. J. Littlejohn, “The semiclassical evolution of wave packets,” Phys. Rep. 138, 193–291 (1986).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian Maxwell beams,” J. Opt. Soc. Am. A 3, 356–360 (1986); “Cross polarization in laser beams,” Appl. Opt. 26, 1589–1593 (1987).
[CrossRef] [PubMed]

1985 (2)

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
[CrossRef] [PubMed]

E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
[CrossRef]

1984 (2)

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

N. L. Balazs, B. K. Jennings, “Wigner’s function and other distribution functions in mock phase spaces,” Phys. Rep. 104, 347–391 (1984); M. Hillery, R. F. O’Connell, M. O. Scully, E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984); V. I. Tatarskii, “The Wigner representation of quantum optics,” Usp. Fiz. Nauk. 139, 587–619 (1983) [Sov. Phys. Usp. 26, 311–327 (1983)].
[CrossRef]

1983 (1)

N. Mukunda, R. Simon, E. C. G. Sudarshan, “Paraxial wave optics and relativistic front description. II. The vector theory,” Phys. Rev. A 28, 2933–2942 (1983); N. Mukunda, R. Simon, E. C. G. Sudarshan, “Fourier optics for the Maxwell field: formalism and application,” J. Opt. Soc. Am. A 2, 416–426 (1985).
[CrossRef]

1982 (3)

1981 (2)

H. Bacry, M. Cadilhac, “The metaplectic group and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981).
[CrossRef]

E. G. C. Sudarshan, “Quantum theory of radiative transfer,” Phys. Rev. A 23, 2802–2809 (1981).
[CrossRef]

1980 (1)

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

1979 (1)

1978 (1)

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978); “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
[CrossRef]

1976 (1)

E. Wolf, “New theory of radiative energy transfer in free electromagnetic fields,” Phys. Rev. D 13, 869–886 (1976).
[CrossRef]

1974 (3)

A. Papoulis, “Ambiguity function in Fourier optics,” J. Opt. Soc. Am. 64, 779–788 (1974).
[CrossRef]

R. L. Hudson, “When is the Wigner quasi-probability nonnegative,” Rep. Math. Phys. 6, 249–252 (1974).
[CrossRef]

N. Burgoyne, R. Cushman, “Normal forms for real linear Hamiltonian systems with purely imaginary eigenvalues,” Celest. Mech. 8, 435–443 (1974); A. J. Laub, K. Meyer, “Canonical forms for symplectic and Hamiltonian matrices,” Celest. Mech. 9, 231–238 (1974).
[CrossRef]

1973 (1)

1971 (1)

M. Moshinsky, C. Quesne, “Linear canonical transformations and their unitary representation,” J. Math. Phys. 12, 1772–1780 (1971); “Canonical transformations and matrix elements,” 12, 1780–1783 (1971).
[CrossRef]

1970 (2)

S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60, 1168–1177 (1970).
[CrossRef]

G. S. Agarwal, E. Wolf, “Calculus for functions of noncommuting operators and general phase space methods in quantum mechanics: I. Mapping theorems and ordering of functions of noncommuting operators,” Phys. Rev. D 2, 2161–2186 (1970); “Calculus for functions of noncommuting operators and general phase space methods in quantum mechanics: II. Quantum mechanics in phase space,” Phys. Rev. D 2, 2187–2205 (1970).
[CrossRef]

1969 (2)

K. E. Cahill, R. J. Glauber, “Ordered expansions in boson amplitude operators,” Phys. Rev. 177, 1857–1881 (1969); “Density operators and quasiprobability distributions,” Phys. Rev. 177, 1882–1902 (1969).
[CrossRef]

D. Gloge, 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), Chap. 3.
[CrossRef]

1968 (1)

1965 (1)

H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
[CrossRef]

1936 (1)

J. Williamson, “On the algebraic problem concerning the normal forms of linear dynamical systems,” Am. J. Math. 58, 141–163 (1936).
[CrossRef]

1932 (1)

E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Agarwal, G. S.

G. S. Agarwal, R. Simon, “A simple realization of fractional Fourier transformation and relation to harmonic oscillator Green’s function,” Opt. Commun. 110, 23–26 (1994).
[CrossRef]

G. S. Agarwal, R. Simon, “An experiment for the study of the Gouy effect for the squeezed vacuum,” Opt. Commun. 100, 411–414 (1993).
[CrossRef]

A. T. Friberg, G. S. Agarwal, J. T. Foley, E. Wolf, “Statistical wave-theoretical derivation of the free-space transport equation of radiometry,” J. Opt. Soc. Am. B 9, 1386–1393 (1992).
[CrossRef]

G. S. Agarwal, J. T. Foley, E. Wolf, “The radiance and phase-space representations of the cross-spectral density operator,” Opt. Commun. 62, 67–72 (1987).
[CrossRef]

G. S. Agarwal, E. Wolf, “Calculus for functions of noncommuting operators and general phase space methods in quantum mechanics: I. Mapping theorems and ordering of functions of noncommuting operators,” Phys. Rev. D 2, 2161–2186 (1970); “Calculus for functions of noncommuting operators and general phase space methods in quantum mechanics: II. Quantum mechanics in phase space,” Phys. Rev. D 2, 2187–2205 (1970).
[CrossRef]

Alonso, M. A.

Ambrosini, D.

D. Ambrosini, V. Bagini, F. Gori, M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[CrossRef]

Bacry, H.

H. Bacry, M. Cadilhac, “The metaplectic group and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981).
[CrossRef]

Bagini, V.

D. Ambrosini, V. Bagini, F. Gori, M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[CrossRef]

Balazs, N. L.

N. L. Balazs, B. K. Jennings, “Wigner’s function and other distribution functions in mock phase spaces,” Phys. Rep. 104, 347–391 (1984); M. Hillery, R. F. O’Connell, M. O. Scully, E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984); V. I. Tatarskii, “The Wigner representation of quantum optics,” Usp. Fiz. Nauk. 139, 587–619 (1983) [Sov. Phys. Usp. 26, 311–327 (1983)].
[CrossRef]

Bastiaans, M. J.

M. J. Bastiaans, “Second-order moments of the Wigner distribution function in first-order optical systems,” Optik (Stuttgart) 88, 163–168 (1991).

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978); “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
[CrossRef]

M. J. Bastiaans, “Wigner distribution function applied to partially coherent light,” in Laser Beam Characterization, P. M. Mejias, H. Weber, R. Martinez-Herrero, A. Gonzalez-Urena, eds. (Sociedad Espanola de Optica, Madrid, 1993), pp. 65–87.

Belanger, P. A.

Borghi, V.

F. Gori, M. Santarsiero, V. Borghi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45, 539–554 (1998).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), Chap. 3.

Boyd, R. W.

R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, New York, 1983), Chap. 2.

Burgoyne, N.

N. Burgoyne, R. Cushman, “Normal forms for real linear Hamiltonian systems with purely imaginary eigenvalues,” Celest. Mech. 8, 435–443 (1974); A. J. Laub, K. Meyer, “Canonical forms for symplectic and Hamiltonian matrices,” Celest. Mech. 9, 231–238 (1974).
[CrossRef]

Cadilhac, M.

H. Bacry, M. Cadilhac, “The metaplectic group and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981).
[CrossRef]

Cahill, K. E.

K. E. Cahill, R. J. Glauber, “Ordered expansions in boson amplitude operators,” Phys. Rev. 177, 1857–1881 (1969); “Density operators and quasiprobability distributions,” Phys. Rev. 177, 1882–1902 (1969).
[CrossRef]

Castanos, O.

O. Castanos, E. Lopez-Moreno, K. B. Wolf, “Canonical transforms for paraxial wave optics,” in Lie Methods in Optics, Vol. 250 of Lecture Notes in Physics, J. Sanchez Mondragon, K. B. Wolf, eds. (Springer-Verlag, Berlin, 1986), pp. 159–182.

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960), Chap. 1.

Chaturvedi, S.

R. Simon, S. Chaturvedi, V. Srinivasan, “Congruences and canonical forms for a positive matrix: application to Schweinler–Wigner extremum principle,” J. Math. Phys. 40, 3632–3640 (1999).
[CrossRef]

Cohen, N.

Collins, S. A.

Cushman, R.

N. Burgoyne, R. Cushman, “Normal forms for real linear Hamiltonian systems with purely imaginary eigenvalues,” Celest. Mech. 8, 435–443 (1974); A. J. Laub, K. Meyer, “Canonical forms for symplectic and Hamiltonian matrices,” Celest. Mech. 9, 231–238 (1974).
[CrossRef]

Dodonov, V. V.

V. V. Dodonov, V. I. Man’ko, Invariants and the Evolution of Nonstationary Quantum Systems, in Proceedings of the Lebedev Physics Institute, M. A. Markov, ed. (Nova Science, Commack, N.Y., 1989), Vol. 183; D. D. Holm, W. P. Lysenko, J. C. Scovel, “Moment invariants for the Vlasov equation,” J. Math. Phys. 31, 1610–1615 (1990).
[CrossRef]

Dragt, A. J.

A. J. Dragt, F. Neri, G. Rangarajan, “General moment invariants for linear Hamiltonian systems,” Phys. Rev. A 45, 2572–2585 (1992).
[CrossRef] [PubMed]

Dutta, B.

R. Simon, N. Mukunda, B. Dutta, “Quantum-noise matrix for multimode systems: U(n) invariance, squeezing, and normal forms,” Phys. Rev. A 49, 1567–1683 (1994).
[CrossRef] [PubMed]

Encinas-Sanz, F.

J. Serna, F. Encinas-Sanz, G. Nemes, “Characterization of a doughnut beam using a cylindrical lens,” presented at the 5th Workshop on Laser Beam and Optics Characterization, Erice, Sicily, Italy, March 20–25, 2000.

Eppich, B.

B. Eppich, C. Gao, H. Weber, “Determination of the ten second order moments,” Opt. Laser Technol. 30, 337–340 (1998).
[CrossRef]

A. T. Friberg, C. Gao, B. Eppich, H. Weber, “Generation of partially coherent fields with twist,” in 10th Meeting on Optical Engineering in Israel, I. Shladov, S. R. Rotman, eds., Proc. SPIE3110, 317–328 (1997).
[CrossRef]

B. Eppich, A. T. Friberg, C. Gao, H. Weber, “Twist of coherent fields and beam quality,” in Third International Workshop on Laser Beam and Optics Characterization, M. Morin, A. Giesen, eds., Proc. SPIE2870, 260–267 (1996).
[CrossRef]

Foley, J. T.

A. T. Friberg, G. S. Agarwal, J. T. Foley, E. Wolf, “Statistical wave-theoretical derivation of the free-space transport equation of radiometry,” J. Opt. Soc. Am. B 9, 1386–1393 (1992).
[CrossRef]

G. S. Agarwal, J. T. Foley, E. Wolf, “The radiance and phase-space representations of the cross-spectral density operator,” Opt. Commun. 62, 67–72 (1987).
[CrossRef]

Forbes, G. W.

Friberg, A. T.

R. Simon, A. T. Friberg, E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5, 331–343 (1996).
[CrossRef]

A. T. Friberg, E. Tervonen, J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818–1826 (1994).
[CrossRef]

A. T. Friberg, G. S. Agarwal, J. T. Foley, E. Wolf, “Statistical wave-theoretical derivation of the free-space transport equation of radiometry,” J. Opt. Soc. Am. B 9, 1386–1393 (1992).
[CrossRef]

A. T. Friberg, J. Turunen, “Imaging of Gaussian Schell-model sources,” J. Opt. Soc. Am. A 5, 713–720 (1988).
[CrossRef]

A. T. Friberg, “On the existence of a radiance function for finite planar sources of arbitrary state of coherence,” J. Opt. Soc. Am. 69, 192–198 (1979).
[CrossRef]

B. Eppich, A. T. Friberg, C. Gao, H. Weber, “Twist of coherent fields and beam quality,” in Third International Workshop on Laser Beam and Optics Characterization, M. Morin, A. Giesen, eds., Proc. SPIE2870, 260–267 (1996).
[CrossRef]

A. T. Friberg, C. Gao, B. Eppich, H. Weber, “Generation of partially coherent fields with twist,” in 10th Meeting on Optical Engineering in Israel, I. Shladov, S. R. Rotman, eds., Proc. SPIE3110, 317–328 (1997).
[CrossRef]

Gao, C.

B. Eppich, C. Gao, H. Weber, “Determination of the ten second order moments,” Opt. Laser Technol. 30, 337–340 (1998).
[CrossRef]

A. T. Friberg, C. Gao, B. Eppich, H. Weber, “Generation of partially coherent fields with twist,” in 10th Meeting on Optical Engineering in Israel, I. Shladov, S. R. Rotman, eds., Proc. SPIE3110, 317–328 (1997).
[CrossRef]

B. Eppich, A. T. Friberg, C. Gao, H. Weber, “Twist of coherent fields and beam quality,” in Third International Workshop on Laser Beam and Optics Characterization, M. Morin, A. Giesen, eds., Proc. SPIE2870, 260–267 (1996).
[CrossRef]

Gase, R.

R. Gase, “Representation of Laguerre–Gaussian modes by the Wigner distribution function,” IEEE J. Quantum Electron. 31, 1811–1818 (1995).
[CrossRef]

Glauber, R. J.

K. E. Cahill, R. J. Glauber, “Ordered expansions in boson amplitude operators,” Phys. Rev. 177, 1857–1881 (1969); “Density operators and quasiprobability distributions,” Phys. Rev. 177, 1882–1902 (1969).
[CrossRef]

Gloge, D.

Gori, F.

F. Gori, M. Santarsiero, V. Borghi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45, 539–554 (1998).
[CrossRef]

D. Ambrosini, V. Bagini, F. Gori, M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[CrossRef]

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

Guillemin, V.

V. Guillemin, S. Sternberg, Symplectic Techniques in Physics (Cambridge U. Press, Cambridge, UK, 1984), Chap. 1.

Helgason, S.

S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces (Academic, New York, 1978), pp. 257–264, 401–407.

Hudson, R. L.

R. L. Hudson, “When is the Wigner quasi-probability nonnegative,” Rep. Math. Phys. 6, 249–252 (1974).
[CrossRef]

Jennings, B. K.

N. L. Balazs, B. K. Jennings, “Wigner’s function and other distribution functions in mock phase spaces,” Phys. Rep. 104, 347–391 (1984); M. Hillery, R. F. O’Connell, M. O. Scully, E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984); V. I. Tatarskii, “The Wigner representation of quantum optics,” Usp. Fiz. Nauk. 139, 587–619 (1983) [Sov. Phys. Usp. 26, 311–327 (1983)].
[CrossRef]

Kauderer, M.

M. Kauderer, Symplectic Matrices (World Scientific, Singapore, 1994).

Kogelnik, H.

H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
[CrossRef]

Littlejohn, R. J.

R. J. Littlejohn, “The semiclassical evolution of wave packets,” Phys. Rep. 138, 193–291 (1986).
[CrossRef]

Lohmann, A. W.

Lopez-Moreno, E.

O. Castanos, E. Lopez-Moreno, K. B. Wolf, “Canonical transforms for paraxial wave optics,” in Lie Methods in Optics, Vol. 250 of Lecture Notes in Physics, J. Sanchez Mondragon, K. B. Wolf, eds. (Springer-Verlag, Berlin, 1986), pp. 159–182.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (Cambridge U. Press, Cambridge, UK, 1964).

Man’ko, V. I.

V. V. Dodonov, V. I. Man’ko, Invariants and the Evolution of Nonstationary Quantum Systems, in Proceedings of the Lebedev Physics Institute, M. A. Markov, ed. (Nova Science, Commack, N.Y., 1989), Vol. 183; D. D. Holm, W. P. Lysenko, J. C. Scovel, “Moment invariants for the Vlasov equation,” J. Math. Phys. 31, 1610–1615 (1990).
[CrossRef]

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995), Chap. 5.

Marcuse, D.

Martinez-Herrero, R.

R. Martinez-Herrero, P. M. Mejias, H. Weber, “On the different definitions of laser beam moments,” Opt. Quantum Electron. 25, 423–428 (1993); A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed. Proc. SPIE1224, 2–14 (1990).
[CrossRef]

J. Serna, P. M. Mejias, R. Martinez-Herrero, “Beam quality dependence on the coherence length of Gaussian Schell-model fields propagating through ABCD optical systems,” J. Mod. Opt. 39, 625–635 (1992); J. Yang, D. Fan, “Intensity-moments characterization of general pulsed paraxial beams with the Wigner distribution function,” J. Opt. Soc. Am. A 16, 2488–2493 (1999).
[CrossRef]

J. Serna, R. Martinez-Herrero, P. M. Mejias, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
[CrossRef]

Mejias, P. M.

R. Martinez-Herrero, P. M. Mejias, H. Weber, “On the different definitions of laser beam moments,” Opt. Quantum Electron. 25, 423–428 (1993); A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed. Proc. SPIE1224, 2–14 (1990).
[CrossRef]

J. Serna, P. M. Mejias, R. Martinez-Herrero, “Beam quality dependence on the coherence length of Gaussian Schell-model fields propagating through ABCD optical systems,” J. Mod. Opt. 39, 625–635 (1992); J. Yang, D. Fan, “Intensity-moments characterization of general pulsed paraxial beams with the Wigner distribution function,” J. Opt. Soc. Am. A 16, 2488–2493 (1999).
[CrossRef]

J. Serna, R. Martinez-Herrero, P. M. Mejias, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
[CrossRef]

Mendlovic, D.

Moshinsky, M.

M. Moshinsky, C. Quesne, “Linear canonical transformations and their unitary representation,” J. Math. Phys. 12, 1772–1780 (1971); “Canonical transformations and matrix elements,” 12, 1780–1783 (1971).
[CrossRef]

Mukunda, N.

R. Simon, N. Mukunda, “Iwasawa decomposition in first-order optics: universal treatment of shape-invariant propagation for coherent and partially coherent beams,” J. Opt. Soc. Am. A 15, 2146–2155 (1998).
[CrossRef]

R. Simon, N. Mukunda, “Shape-invariant anisotropic Gaussian Schell-model beams: a complete characterization,” J. Opt. Soc. Am. A 15, 1361–1370 (1998); “Gaussian Schell-model beams and general shape invariance,” J. Opt. Soc. Am. A 16, 2465–2475 (1999).
[CrossRef]

R. Simon, N. Mukunda, “Twist phase in Gaussian-beam optics,” J. Opt. Soc. Am. A 15, 2373–2382 (1998).
[CrossRef]

K. Sundar, N. Mukunda, R. Simon, “Coherent-mode decomposition of general anisotropic Gaussian Schell-model beams,” J. Opt. Soc. Am. A 12, 560–569 (1995).
[CrossRef]

R. Simon, N. Mukunda, B. Dutta, “Quantum-noise matrix for multimode systems: U(n) invariance, squeezing, and normal forms,” Phys. Rev. A 49, 1567–1683 (1994).
[CrossRef] [PubMed]

R. Simon, K. Sundar, N. Mukunda, “Twisted Gaussian Schell-model beams. I. Symmetry structure and normal-mode spectrum,” J. Opt. Soc. Am. A 10, 2008–2016 (1993).
[CrossRef]

R. Simon, N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993).
[CrossRef]

R. Simon, N. Mukunda, “Bargmann invariant and the geometry of the Gouy effect,” Phys. Rev. Lett. 70, 880–883 (1993); N. Mukunda, R. Simon, “Quantum kinematic approach to the geometric phase. I. General formalism,” Ann. Phys. (N.Y.) 228, 205–268 (1993) (an extensive set of references on geometric phase can be found here); “Quantum kinematic approach to the geometric phase. II. The case of unitary group representations,” Ann. Phys. (N.Y.) 228, 269–340 (1993).
[CrossRef] [PubMed]

R. Simon, N. Mukunda, “Iwasawa decomposition for SU(1,1) and the Gouy effect for squeezed states,” Opt. Commun. 95, 39–45 (1993).
[CrossRef]

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized abcd-law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian pure states in quantum mechanics and the symplectic group,” Phys. Rev. A 37, 2028–2038 (1988).
[CrossRef]

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Hamilton’s theory of turns generalized to Sp(2, R),” Phys. Rev. Lett. 62, 1331–1334 (1988); R. Simon, N. Mukunda, E. C. G. Sudarshan, “The theory of screws—a new geometric representation for the group SU(1,1),” J. Math. Phys. 30, 1000–1006 (1989).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987); R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian Wigner distributions: a complete characterization,” Phys. Lett. A 124, 223–228 (1987).
[CrossRef] [PubMed]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian Maxwell beams,” J. Opt. Soc. Am. A 3, 356–360 (1986); “Cross polarization in laser beams,” Appl. Opt. 26, 1589–1593 (1987).
[CrossRef] [PubMed]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
[CrossRef] [PubMed]

E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

N. Mukunda, R. Simon, E. C. G. Sudarshan, “Paraxial wave optics and relativistic front description. II. The vector theory,” Phys. Rev. A 28, 2933–2942 (1983); N. Mukunda, R. Simon, E. C. G. Sudarshan, “Fourier optics for the Maxwell field: formalism and application,” J. Opt. Soc. Am. A 2, 416–426 (1985).
[CrossRef]

R. Simon, N. Mukunda, “The two-dimensional symplectic and metaplectic groups and their universal cover,” in Symmetries in Science VI: from the Rotation Group to Quantum Algebras, B. Gruber, ed. (Plenum, New York, 1993), pp. 659–689.

Narcowich, F. J.

F. J. Narcowich, “Geometry and uncertainty,” J. Math. Phys. 31, 354–364 (1990); Q. Lin, S. Wang, J. Alda, E. Bernabeu, “Transformation of pulsed nonideal beams in a four-dimensional domain,” Opt. Lett. 18, 669–671 (1993).
[CrossRef] [PubMed]

Nazarathy, M.

Nemes, G.

G. Nemes, A. E. Siegman, “Measurement of all ten second-order moments of an astigmatic beam by the use of rotating simple astigmatic (anamorphic) optics,” J. Opt. Soc. Am. A 11, 2257–2264 (1994).
[CrossRef]

J. Serna, G. Nemes, “Decoupling of coherent Gaussian beam with general astigmatism,” Opt. Lett. 18, 1774–1776 (1993).
[CrossRef] [PubMed]

G. Nemes, “Synthesis of general astigmatic optical systems, the detwisting procedure, and the beam quality factors for general astigmatic laser beams,” in Laser Beam Characterization, H. Weber, N. Reng, J. Ludtke, P. M. Mejias, eds. (Festkorper-Laser-Institut Berlin GmbH, Berlin, 1994), pp. 93–104.

G. Nemes, “Measuring and handling general astigmatic beams,” in Laser Beam Characterization, P. M. Mejias, H. Weber, R. Martinez-Herrero, A. Gonzalez-Urena, eds. (Sociedad Espanola de Optica, Madrid, Spain, 1993), pp. 325–358.

J. Serna, F. Encinas-Sanz, G. Nemes, “Characterization of a doughnut beam using a cylindrical lens,” presented at the 5th Workshop on Laser Beam and Optics Characterization, Erice, Sicily, Italy, March 20–25, 2000.

Neri, F.

A. J. Dragt, F. Neri, G. Rangarajan, “General moment invariants for linear Hamiltonian systems,” Phys. Rev. A 45, 2572–2585 (1992).
[CrossRef] [PubMed]

Ozaktas, H. M.

Papoulis, A.

Pradhan, T.

T. Pradhan, “Maxwell’s equations from geometrical optics,” Phys. Lett. A 122, 397–398 (1987).
[CrossRef]

Quesne, C.

M. Moshinsky, C. Quesne, “Linear canonical transformations and their unitary representation,” J. Math. Phys. 12, 1772–1780 (1971); “Canonical transformations and matrix elements,” 12, 1780–1783 (1971).
[CrossRef]

Ramee, S.

Rangarajan, G.

A. J. Dragt, F. Neri, G. Rangarajan, “General moment invariants for linear Hamiltonian systems,” Phys. Rev. A 45, 2572–2585 (1992).
[CrossRef] [PubMed]

Santarsiero, M.

F. Gori, M. Santarsiero, V. Borghi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45, 539–554 (1998).
[CrossRef]

D. Ambrosini, V. Bagini, F. Gori, M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[CrossRef]

Schempp, W.

W. Schempp, “Analog radar signal design and digital signal processing—a Heisenberg nilpotent Lie group approach,” in Lie Methods in Optics, Vol. 250 of Lecture Notes in Physics, J. Sanchez Mondragon, K. B. Wolf, eds. (Springer-Verlag, Berlin, 1986), pp. 1–27.

Serna, J.

J. Serna, G. Nemes, “Decoupling of coherent Gaussian beam with general astigmatism,” Opt. Lett. 18, 1774–1776 (1993).
[CrossRef] [PubMed]

J. Serna, P. M. Mejias, R. Martinez-Herrero, “Beam quality dependence on the coherence length of Gaussian Schell-model fields propagating through ABCD optical systems,” J. Mod. Opt. 39, 625–635 (1992); J. Yang, D. Fan, “Intensity-moments characterization of general pulsed paraxial beams with the Wigner distribution function,” J. Opt. Soc. Am. A 16, 2488–2493 (1999).
[CrossRef]

J. Serna, R. Martinez-Herrero, P. M. Mejias, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
[CrossRef]

J. Serna, F. Encinas-Sanz, G. Nemes, “Characterization of a doughnut beam using a cylindrical lens,” presented at the 5th Workshop on Laser Beam and Optics Characterization, Erice, Sicily, Italy, March 20–25, 2000.

Shamir, J.

Siegman, A. E.

Simon, R.

R. Simon, K. B. Wolf, “Structure of the set of paraxial optical systems,” J. Opt. Soc. Am. A 17, 342–355 (2000).
[CrossRef]

S. Ramee, R. Simon, “Effect of holes and vortices on beam quality,” J. Opt. Soc. Am. A 17, 84–94 (2000).
[CrossRef]

R. Simon, “Peres–Horodecki separability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2726–2729 (2000).
[CrossRef] [PubMed]

R. Simon, S. Chaturvedi, V. Srinivasan, “Congruences and canonical forms for a positive matrix: application to Schweinler–Wigner extremum principle,” J. Math. Phys. 40, 3632–3640 (1999).
[CrossRef]

R. Simon, N. Mukunda, “Twist phase in Gaussian-beam optics,” J. Opt. Soc. Am. A 15, 2373–2382 (1998).
[CrossRef]

R. Simon, N. Mukunda, “Iwasawa decomposition in first-order optics: universal treatment of shape-invariant propagation for coherent and partially coherent beams,” J. Opt. Soc. Am. A 15, 2146–2155 (1998).
[CrossRef]

R. Simon, N. Mukunda, “Shape-invariant anisotropic Gaussian Schell-model beams: a complete characterization,” J. Opt. Soc. Am. A 15, 1361–1370 (1998); “Gaussian Schell-model beams and general shape invariance,” J. Opt. Soc. Am. A 16, 2465–2475 (1999).
[CrossRef]

R. Simon, A. T. Friberg, E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5, 331–343 (1996).
[CrossRef]

K. Sundar, N. Mukunda, R. Simon, “Coherent-mode decomposition of general anisotropic Gaussian Schell-model beams,” J. Opt. Soc. Am. A 12, 560–569 (1995).
[CrossRef]

G. S. Agarwal, R. Simon, “A simple realization of fractional Fourier transformation and relation to harmonic oscillator Green’s function,” Opt. Commun. 110, 23–26 (1994).
[CrossRef]

R. Simon, N. Mukunda, B. Dutta, “Quantum-noise matrix for multimode systems: U(n) invariance, squeezing, and normal forms,” Phys. Rev. A 49, 1567–1683 (1994).
[CrossRef] [PubMed]

R. Simon, K. Sundar, N. Mukunda, “Twisted Gaussian Schell-model beams. I. Symmetry structure and normal-mode spectrum,” J. Opt. Soc. Am. A 10, 2008–2016 (1993).
[CrossRef]

R. Simon, N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993).
[CrossRef]

R. Simon, N. Mukunda, “Bargmann invariant and the geometry of the Gouy effect,” Phys. Rev. Lett. 70, 880–883 (1993); N. Mukunda, R. Simon, “Quantum kinematic approach to the geometric phase. I. General formalism,” Ann. Phys. (N.Y.) 228, 205–268 (1993) (an extensive set of references on geometric phase can be found here); “Quantum kinematic approach to the geometric phase. II. The case of unitary group representations,” Ann. Phys. (N.Y.) 228, 269–340 (1993).
[CrossRef] [PubMed]

R. Simon, N. Mukunda, “Iwasawa decomposition for SU(1,1) and the Gouy effect for squeezed states,” Opt. Commun. 95, 39–45 (1993).
[CrossRef]

G. S. Agarwal, R. Simon, “An experiment for the study of the Gouy effect for the squeezed vacuum,” Opt. Commun. 100, 411–414 (1993).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian pure states in quantum mechanics and the symplectic group,” Phys. Rev. A 37, 2028–2038 (1988).
[CrossRef]

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized abcd-law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Hamilton’s theory of turns generalized to Sp(2, R),” Phys. Rev. Lett. 62, 1331–1334 (1988); R. Simon, N. Mukunda, E. C. G. Sudarshan, “The theory of screws—a new geometric representation for the group SU(1,1),” J. Math. Phys. 30, 1000–1006 (1989).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987); R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian Wigner distributions: a complete characterization,” Phys. Lett. A 124, 223–228 (1987).
[CrossRef] [PubMed]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian Maxwell beams,” J. Opt. Soc. Am. A 3, 356–360 (1986); “Cross polarization in laser beams,” Appl. Opt. 26, 1589–1593 (1987).
[CrossRef] [PubMed]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
[CrossRef] [PubMed]

E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

N. Mukunda, R. Simon, E. C. G. Sudarshan, “Paraxial wave optics and relativistic front description. II. The vector theory,” Phys. Rev. A 28, 2933–2942 (1983); N. Mukunda, R. Simon, E. C. G. Sudarshan, “Fourier optics for the Maxwell field: formalism and application,” J. Opt. Soc. Am. A 2, 416–426 (1985).
[CrossRef]

R. Simon, N. Mukunda, “The two-dimensional symplectic and metaplectic groups and their universal cover,” in Symmetries in Science VI: from the Rotation Group to Quantum Algebras, B. Gruber, ed. (Plenum, New York, 1993), pp. 659–689.

Srinivasan, V.

R. Simon, S. Chaturvedi, V. Srinivasan, “Congruences and canonical forms for a positive matrix: application to Schweinler–Wigner extremum principle,” J. Math. Phys. 40, 3632–3640 (1999).
[CrossRef]

Starikov, A.

Stavroudis, O. N.

See, for instance, O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, New York, 1972), pp. 288–294.

Sternberg, S.

V. Guillemin, S. Sternberg, Symplectic Techniques in Physics (Cambridge U. Press, Cambridge, UK, 1984), Chap. 1.

Sudarshan, E. C. G.

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized abcd-law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian pure states in quantum mechanics and the symplectic group,” Phys. Rev. A 37, 2028–2038 (1988).
[CrossRef]

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Hamilton’s theory of turns generalized to Sp(2, R),” Phys. Rev. Lett. 62, 1331–1334 (1988); R. Simon, N. Mukunda, E. C. G. Sudarshan, “The theory of screws—a new geometric representation for the group SU(1,1),” J. Math. Phys. 30, 1000–1006 (1989).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987); R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian Wigner distributions: a complete characterization,” Phys. Lett. A 124, 223–228 (1987).
[CrossRef] [PubMed]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian Maxwell beams,” J. Opt. Soc. Am. A 3, 356–360 (1986); “Cross polarization in laser beams,” Appl. Opt. 26, 1589–1593 (1987).
[CrossRef] [PubMed]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
[CrossRef] [PubMed]

E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

N. Mukunda, R. Simon, E. C. G. Sudarshan, “Paraxial wave optics and relativistic front description. II. The vector theory,” Phys. Rev. A 28, 2933–2942 (1983); N. Mukunda, R. Simon, E. C. G. Sudarshan, “Fourier optics for the Maxwell field: formalism and application,” J. Opt. Soc. Am. A 2, 416–426 (1985).
[CrossRef]

Sudarshan, E. G. C.

E. G. C. Sudarshan, “Quantum theory of radiative transfer,” Phys. Rev. A 23, 2802–2809 (1981).
[CrossRef]

Sundar, K.

Tervonen, E.

Turunen, J.

Walther, A.

Weber, H.

B. Eppich, C. Gao, H. Weber, “Determination of the ten second order moments,” Opt. Laser Technol. 30, 337–340 (1998).
[CrossRef]

R. Martinez-Herrero, P. M. Mejias, H. Weber, “On the different definitions of laser beam moments,” Opt. Quantum Electron. 25, 423–428 (1993); A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed. Proc. SPIE1224, 2–14 (1990).
[CrossRef]

H. Weber, “Wave optical analysis of the phase space analyzer,” J. Mod. Opt. 39, 543–559 (1992); D. Onciul, “Invariance properties of general astigmatic beams through first-order optical systems,” J. Opt. Soc. Am. A 10, 295–298 (1993); S. Lavi, R. Prochaska, E. Keren, “Generalized beam parameters and transformation laws for partially coherent light,” Appl. Opt. 27, 3696–3703 (1988).
[CrossRef] [PubMed]

H. Weber, “Propagation of the higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, S1027–S1049 (1992); D. Dragoman, “Higher-order moments of the Wigner distribution function in first-order optical systems,” J. Opt. Soc. Am. A 11, 2643–2646 (1994).
[CrossRef]

A. T. Friberg, C. Gao, B. Eppich, H. Weber, “Generation of partially coherent fields with twist,” in 10th Meeting on Optical Engineering in Israel, I. Shladov, S. R. Rotman, eds., Proc. SPIE3110, 317–328 (1997).
[CrossRef]

B. Eppich, A. T. Friberg, C. Gao, H. Weber, “Twist of coherent fields and beam quality,” in Third International Workshop on Laser Beam and Optics Characterization, M. Morin, A. Giesen, eds., Proc. SPIE2870, 260–267 (1996).
[CrossRef]

Wigner, E. P.

E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

E. P. Wigner, “Quantum mechanical distribution functions revisited,” in Perspectives in Quantum Theory, W. Yourgrau, A. van der Merwe, eds. (MIT Press, Cambridge, Mass., 1971), pp. 25–36.

Williamson, J.

J. Williamson, “On the algebraic problem concerning the normal forms of linear dynamical systems,” Am. J. Math. 58, 141–163 (1936).
[CrossRef]

Wolf, E.

R. Simon, A. T. Friberg, E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5, 331–343 (1996).
[CrossRef]

A. T. Friberg, G. S. Agarwal, J. T. Foley, E. Wolf, “Statistical wave-theoretical derivation of the free-space transport equation of radiometry,” J. Opt. Soc. Am. B 9, 1386–1393 (1992).
[CrossRef]

G. S. Agarwal, J. T. Foley, E. Wolf, “The radiance and phase-space representations of the cross-spectral density operator,” Opt. Commun. 62, 67–72 (1987).
[CrossRef]

A. Starikov, E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and their radiation fields,” J. Opt. Soc. Am. 72, 923–928 (1982).
[CrossRef]

E. Wolf, “New theory of partial coherence in the space-frequency domain. Part I: Spectra and cross spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982).
[CrossRef]

E. Wolf, “New theory of radiative energy transfer in free electromagnetic fields,” Phys. Rev. D 13, 869–886 (1976).
[CrossRef]

G. S. Agarwal, E. Wolf, “Calculus for functions of noncommuting operators and general phase space methods in quantum mechanics: I. Mapping theorems and ordering of functions of noncommuting operators,” Phys. Rev. D 2, 2161–2186 (1970); “Calculus for functions of noncommuting operators and general phase space methods in quantum mechanics: II. Quantum mechanics in phase space,” Phys. Rev. D 2, 2187–2205 (1970).
[CrossRef]

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995), Chap. 5.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), Chap. 3.

Wolf, K. B.

R. Simon, K. B. Wolf, “Structure of the set of paraxial optical systems,” J. Opt. Soc. Am. A 17, 342–355 (2000).
[CrossRef]

K. B. Wolf, M. A. Alonso, G. W. Forbes, “Wigner functions for Helmholtz wave fields,” J. Opt. Soc. Am. A 16, 2476–2487 (1999).
[CrossRef]

O. Castanos, E. Lopez-Moreno, K. B. Wolf, “Canonical transforms for paraxial wave optics,” in Lie Methods in Optics, Vol. 250 of Lecture Notes in Physics, J. Sanchez Mondragon, K. B. Wolf, eds. (Springer-Verlag, Berlin, 1986), pp. 159–182.

K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979), Chaps. 7 and 9.

Am. J. Math. (1)

J. Williamson, “On the algebraic problem concerning the normal forms of linear dynamical systems,” Am. J. Math. 58, 141–163 (1936).
[CrossRef]

Bell Syst. Tech. J. (1)

H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
[CrossRef]

Celest. Mech. (1)

N. Burgoyne, R. Cushman, “Normal forms for real linear Hamiltonian systems with purely imaginary eigenvalues,” Celest. Mech. 8, 435–443 (1974); A. J. Laub, K. Meyer, “Canonical forms for symplectic and Hamiltonian matrices,” Celest. Mech. 9, 231–238 (1974).
[CrossRef]

IEEE J. Quantum Electron. (1)

R. Gase, “Representation of Laguerre–Gaussian modes by the Wigner distribution function,” IEEE J. Quantum Electron. 31, 1811–1818 (1995).
[CrossRef]

J. Math. Phys. (3)

M. Moshinsky, C. Quesne, “Linear canonical transformations and their unitary representation,” J. Math. Phys. 12, 1772–1780 (1971); “Canonical transformations and matrix elements,” 12, 1780–1783 (1971).
[CrossRef]

F. J. Narcowich, “Geometry and uncertainty,” J. Math. Phys. 31, 354–364 (1990); Q. Lin, S. Wang, J. Alda, E. Bernabeu, “Transformation of pulsed nonideal beams in a four-dimensional domain,” Opt. Lett. 18, 669–671 (1993).
[CrossRef] [PubMed]

R. Simon, S. Chaturvedi, V. Srinivasan, “Congruences and canonical forms for a positive matrix: application to Schweinler–Wigner extremum principle,” J. Math. Phys. 40, 3632–3640 (1999).
[CrossRef]

J. Mod. Opt. (4)

F. Gori, M. Santarsiero, V. Borghi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45, 539–554 (1998).
[CrossRef]

H. Weber, “Wave optical analysis of the phase space analyzer,” J. Mod. Opt. 39, 543–559 (1992); D. Onciul, “Invariance properties of general astigmatic beams through first-order optical systems,” J. Opt. Soc. Am. A 10, 295–298 (1993); S. Lavi, R. Prochaska, E. Keren, “Generalized beam parameters and transformation laws for partially coherent light,” Appl. Opt. 27, 3696–3703 (1988).
[CrossRef] [PubMed]

J. Serna, P. M. Mejias, R. Martinez-Herrero, “Beam quality dependence on the coherence length of Gaussian Schell-model fields propagating through ABCD optical systems,” J. Mod. Opt. 39, 625–635 (1992); J. Yang, D. Fan, “Intensity-moments characterization of general pulsed paraxial beams with the Wigner distribution function,” J. Opt. Soc. Am. A 16, 2488–2493 (1999).
[CrossRef]

D. Ambrosini, V. Bagini, F. Gori, M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[CrossRef]

J. Opt. Soc. Am. (9)

J. Opt. Soc. Am. A (17)

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian Maxwell beams,” J. Opt. Soc. Am. A 3, 356–360 (1986); “Cross polarization in laser beams,” Appl. Opt. 26, 1589–1593 (1987).
[CrossRef] [PubMed]

J. Shamir, N. Cohen, “Root and power transformations in optics,” J. Opt. Soc. Am. A 12, 2415–2423 (1995).
[CrossRef]

G. Nemes, A. E. Siegman, “Measurement of all ten second-order moments of an astigmatic beam by the use of rotating simple astigmatic (anamorphic) optics,” J. Opt. Soc. Am. A 11, 2257–2264 (1994).
[CrossRef]

S. Ramee, R. Simon, “Effect of holes and vortices on beam quality,” J. Opt. Soc. Am. A 17, 84–94 (2000).
[CrossRef]

R. Simon, N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993).
[CrossRef]

A. T. Friberg, E. Tervonen, J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818–1826 (1994).
[CrossRef]

J. Serna, R. Martinez-Herrero, P. M. Mejias, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
[CrossRef]

R. Simon, K. Sundar, N. Mukunda, “Twisted Gaussian Schell-model beams. I. Symmetry structure and normal-mode spectrum,” J. Opt. Soc. Am. A 10, 2008–2016 (1993).
[CrossRef]

K. Sundar, N. Mukunda, R. Simon, “Coherent-mode decomposition of general anisotropic Gaussian Schell-model beams,” J. Opt. Soc. Am. A 12, 560–569 (1995).
[CrossRef]

K. B. Wolf, M. A. Alonso, G. W. Forbes, “Wigner functions for Helmholtz wave fields,” J. Opt. Soc. Am. A 16, 2476–2487 (1999).
[CrossRef]

A. T. Friberg, J. Turunen, “Imaging of Gaussian Schell-model sources,” J. Opt. Soc. Am. A 5, 713–720 (1988).
[CrossRef]

R. Simon, K. B. Wolf, “Structure of the set of paraxial optical systems,” J. Opt. Soc. Am. A 17, 342–355 (2000).
[CrossRef]

D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation: I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993); H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transforms and their optical implementation. II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
[CrossRef]

A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993); S. Abe, J. T. Sheridan, “Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation: an operator approach,” J. Phys. A 27, 4179–4187 (1994).
[CrossRef]

R. Simon, N. Mukunda, “Twist phase in Gaussian-beam optics,” J. Opt. Soc. Am. A 15, 2373–2382 (1998).
[CrossRef]

R. Simon, N. Mukunda, “Iwasawa decomposition in first-order optics: universal treatment of shape-invariant propagation for coherent and partially coherent beams,” J. Opt. Soc. Am. A 15, 2146–2155 (1998).
[CrossRef]

R. Simon, N. Mukunda, “Shape-invariant anisotropic Gaussian Schell-model beams: a complete characterization,” J. Opt. Soc. Am. A 15, 1361–1370 (1998); “Gaussian Schell-model beams and general shape invariance,” J. Opt. Soc. Am. A 16, 2465–2475 (1999).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Acta (1)

E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
[CrossRef]

Opt. Commun. (7)

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

R. Simon, N. Mukunda, “Iwasawa decomposition for SU(1,1) and the Gouy effect for squeezed states,” Opt. Commun. 95, 39–45 (1993).
[CrossRef]

G. S. Agarwal, R. Simon, “An experiment for the study of the Gouy effect for the squeezed vacuum,” Opt. Commun. 100, 411–414 (1993).
[CrossRef]

G. S. Agarwal, R. Simon, “A simple realization of fractional Fourier transformation and relation to harmonic oscillator Green’s function,” Opt. Commun. 110, 23–26 (1994).
[CrossRef]

G. S. Agarwal, J. T. Foley, E. Wolf, “The radiance and phase-space representations of the cross-spectral density operator,” Opt. Commun. 62, 67–72 (1987).
[CrossRef]

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978); “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
[CrossRef]

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized abcd-law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

Opt. Laser Technol. (1)

B. Eppich, C. Gao, H. Weber, “Determination of the ten second order moments,” Opt. Laser Technol. 30, 337–340 (1998).
[CrossRef]

Opt. Lett. (2)

Opt. Quantum Electron. (2)

R. Martinez-Herrero, P. M. Mejias, H. Weber, “On the different definitions of laser beam moments,” Opt. Quantum Electron. 25, 423–428 (1993); A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed. Proc. SPIE1224, 2–14 (1990).
[CrossRef]

H. Weber, “Propagation of the higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, S1027–S1049 (1992); D. Dragoman, “Higher-order moments of the Wigner distribution function in first-order optical systems,” J. Opt. Soc. Am. A 11, 2643–2646 (1994).
[CrossRef]

Optik (Stuttgart) (1)

M. J. Bastiaans, “Second-order moments of the Wigner distribution function in first-order optical systems,” Optik (Stuttgart) 88, 163–168 (1991).

Phys. Lett. A (1)

T. Pradhan, “Maxwell’s equations from geometrical optics,” Phys. Lett. A 122, 397–398 (1987).
[CrossRef]

Phys. Rep. (2)

N. L. Balazs, B. K. Jennings, “Wigner’s function and other distribution functions in mock phase spaces,” Phys. Rep. 104, 347–391 (1984); M. Hillery, R. F. O’Connell, M. O. Scully, E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984); V. I. Tatarskii, “The Wigner representation of quantum optics,” Usp. Fiz. Nauk. 139, 587–619 (1983) [Sov. Phys. Usp. 26, 311–327 (1983)].
[CrossRef]

R. J. Littlejohn, “The semiclassical evolution of wave packets,” Phys. Rep. 138, 193–291 (1986).
[CrossRef]

Phys. Rev. (2)

E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

K. E. Cahill, R. J. Glauber, “Ordered expansions in boson amplitude operators,” Phys. Rev. 177, 1857–1881 (1969); “Density operators and quasiprobability distributions,” Phys. Rev. 177, 1882–1902 (1969).
[CrossRef]

Phys. Rev. A (9)

E. G. C. Sudarshan, “Quantum theory of radiative transfer,” Phys. Rev. A 23, 2802–2809 (1981).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian pure states in quantum mechanics and the symplectic group,” Phys. Rev. A 37, 2028–2038 (1988).
[CrossRef]

R. Simon, N. Mukunda, B. Dutta, “Quantum-noise matrix for multimode systems: U(n) invariance, squeezing, and normal forms,” Phys. Rev. A 49, 1567–1683 (1994).
[CrossRef] [PubMed]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
[CrossRef] [PubMed]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987); R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian Wigner distributions: a complete characterization,” Phys. Lett. A 124, 223–228 (1987).
[CrossRef] [PubMed]

A. J. Dragt, F. Neri, G. Rangarajan, “General moment invariants for linear Hamiltonian systems,” Phys. Rev. A 45, 2572–2585 (1992).
[CrossRef] [PubMed]

N. Mukunda, R. Simon, E. C. G. Sudarshan, “Paraxial wave optics and relativistic front description. II. The vector theory,” Phys. Rev. A 28, 2933–2942 (1983); N. Mukunda, R. Simon, E. C. G. Sudarshan, “Fourier optics for the Maxwell field: formalism and application,” J. Opt. Soc. Am. A 2, 416–426 (1985).
[CrossRef]

H. Bacry, M. Cadilhac, “The metaplectic group and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981).
[CrossRef]

Phys. Rev. D (2)

G. S. Agarwal, E. Wolf, “Calculus for functions of noncommuting operators and general phase space methods in quantum mechanics: I. Mapping theorems and ordering of functions of noncommuting operators,” Phys. Rev. D 2, 2161–2186 (1970); “Calculus for functions of noncommuting operators and general phase space methods in quantum mechanics: II. Quantum mechanics in phase space,” Phys. Rev. D 2, 2187–2205 (1970).
[CrossRef]

E. Wolf, “New theory of radiative energy transfer in free electromagnetic fields,” Phys. Rev. D 13, 869–886 (1976).
[CrossRef]

Phys. Rev. Lett. (3)

R. Simon, N. Mukunda, “Bargmann invariant and the geometry of the Gouy effect,” Phys. Rev. Lett. 70, 880–883 (1993); N. Mukunda, R. Simon, “Quantum kinematic approach to the geometric phase. I. General formalism,” Ann. Phys. (N.Y.) 228, 205–268 (1993) (an extensive set of references on geometric phase can be found here); “Quantum kinematic approach to the geometric phase. II. The case of unitary group representations,” Ann. Phys. (N.Y.) 228, 269–340 (1993).
[CrossRef] [PubMed]

R. Simon, “Peres–Horodecki separability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2726–2729 (2000).
[CrossRef] [PubMed]

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Hamilton’s theory of turns generalized to Sp(2, R),” Phys. Rev. Lett. 62, 1331–1334 (1988); R. Simon, N. Mukunda, E. C. G. Sudarshan, “The theory of screws—a new geometric representation for the group SU(1,1),” J. Math. Phys. 30, 1000–1006 (1989).
[CrossRef]

Pure Appl. Opt. (1)

R. Simon, A. T. Friberg, E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5, 331–343 (1996).
[CrossRef]

Rep. Math. Phys. (1)

R. L. Hudson, “When is the Wigner quasi-probability nonnegative,” Rep. Math. Phys. 6, 249–252 (1974).
[CrossRef]

Other (22)

A. E. Siegman, Lasers (Oxford U. Press, Oxford, 1986), Chap. 19.

V. V. Dodonov, V. I. Man’ko, Invariants and the Evolution of Nonstationary Quantum Systems, in Proceedings of the Lebedev Physics Institute, M. A. Markov, ed. (Nova Science, Commack, N.Y., 1989), Vol. 183; D. D. Holm, W. P. Lysenko, J. C. Scovel, “Moment invariants for the Vlasov equation,” J. Math. Phys. 31, 1610–1615 (1990).
[CrossRef]

K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979), Chaps. 7 and 9.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), Chap. 3.

O. Castanos, E. Lopez-Moreno, K. B. Wolf, “Canonical transforms for paraxial wave optics,” in Lie Methods in Optics, Vol. 250 of Lecture Notes in Physics, J. Sanchez Mondragon, K. B. Wolf, eds. (Springer-Verlag, Berlin, 1986), pp. 159–182.

S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces (Academic, New York, 1978), pp. 257–264, 401–407.

See, for instance, O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, New York, 1972), pp. 288–294.

M. Kauderer, Symplectic Matrices (World Scientific, Singapore, 1994).

R. K. Luneburg, Mathematical Theory of Optics (Cambridge U. Press, Cambridge, UK, 1964).

V. Guillemin, S. Sternberg, Symplectic Techniques in Physics (Cambridge U. Press, Cambridge, UK, 1984), Chap. 1.

E. P. Wigner, “Quantum mechanical distribution functions revisited,” in Perspectives in Quantum Theory, W. Yourgrau, A. van der Merwe, eds. (MIT Press, Cambridge, Mass., 1971), pp. 25–36.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995), Chap. 5.

R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, New York, 1983), Chap. 2.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960), Chap. 1.

M. J. Bastiaans, “Wigner distribution function applied to partially coherent light,” in Laser Beam Characterization, P. M. Mejias, H. Weber, R. Martinez-Herrero, A. Gonzalez-Urena, eds. (Sociedad Espanola de Optica, Madrid, 1993), pp. 65–87.

W. Schempp, “Analog radar signal design and digital signal processing—a Heisenberg nilpotent Lie group approach,” in Lie Methods in Optics, Vol. 250 of Lecture Notes in Physics, J. Sanchez Mondragon, K. B. Wolf, eds. (Springer-Verlag, Berlin, 1986), pp. 1–27.

G. Nemes, “Synthesis of general astigmatic optical systems, the detwisting procedure, and the beam quality factors for general astigmatic laser beams,” in Laser Beam Characterization, H. Weber, N. Reng, J. Ludtke, P. M. Mejias, eds. (Festkorper-Laser-Institut Berlin GmbH, Berlin, 1994), pp. 93–104.

B. Eppich, A. T. Friberg, C. Gao, H. Weber, “Twist of coherent fields and beam quality,” in Third International Workshop on Laser Beam and Optics Characterization, M. Morin, A. Giesen, eds., Proc. SPIE2870, 260–267 (1996).
[CrossRef]

G. Nemes, “Measuring and handling general astigmatic beams,” in Laser Beam Characterization, P. M. Mejias, H. Weber, R. Martinez-Herrero, A. Gonzalez-Urena, eds. (Sociedad Espanola de Optica, Madrid, Spain, 1993), pp. 325–358.

J. Serna, F. Encinas-Sanz, G. Nemes, “Characterization of a doughnut beam using a cylindrical lens,” presented at the 5th Workshop on Laser Beam and Optics Characterization, Erice, Sicily, Italy, March 20–25, 2000.

R. Simon, N. Mukunda, “The two-dimensional symplectic and metaplectic groups and their universal cover,” in Symmetries in Science VI: from the Rotation Group to Quantum Algebras, B. Gruber, ed. (Plenum, New York, 1993), pp. 659–689.

A. T. Friberg, C. Gao, B. Eppich, H. Weber, “Generation of partially coherent fields with twist,” in 10th Meeting on Optical Engineering in Israel, I. Shladov, S. R. Rotman, eds., Proc. SPIE3110, 317–328 (1997).
[CrossRef]

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Equations (157)

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S:ξξ=Sξ.
SΩST=Ω,Ω=00100001-10000-100.
D(d)=12×2d12×202×212×2,L(g)=12×202×2-g12×2.
S2Ω2S2T=Ω2,Ω2=01-10.
SU(1, 1)={U|Uσ3U=σ3},
Sp(2, R)SL(2, R)SU(1, 1).
Σ0=120l0-l0-10,Σ1=120l0l0-10,
Σ2=12-10 01
sp(2, R)={XaΣa|XR3}.
[Σ0, Σ1]=-Σ2,[Σ0, Σ2]=Σ1,[Σ1, Σ2]=Σ0,
S2:XaΣaS2XaΣaS2-1=XaΣa,
(X0)2-(X1)2-(X2)2=(X0)2-(X1)2-(X2)2,
sgn(X0)=sgn(X0).
S2:XX=Λ(S2)X,Λ(S2)SO(2, 1).
S2=abcd  Λ(S2)=12(a2+l0-2b2+l02c2+d2)12(a2-l0-2b2+l02c2-d2)l0-1ab+l0cd12(a2+l0-2b2-l02c2-d2)12(a2-l0-2b2-l02c2+d2)l0-1ab-l0cdl0ac+l0-1bdl0ac-l0-1bdad+bc.
Sp(2, R)/Z2SO(2, 1),Z2={1, -1}.
S=a12×2b12×2c12×2d12×2=S212×2,S2Sp(2, R).
D(g-1)L(g)D(g-1)=0g-1-g0I2×2.
F(θ)=cos θl0 sin θ-l0-1 sin θcos θ.
S˜=S200S2=12×2S2,
S˜=S200S2,
JΩ+ΩJT=0.
J=uvw-uT;u, v, wreal,v, wsymmetric.
sp(4, R)={ΩH|H=real,HT=H}
={HΩ|H=real,HT=H}.
γ0=β=iσ212×2Ω,γ1=σ1σ3,
γ2=σ1σ1,γ3=-σ312×2.
Σμ5-12K(l0)γμK(l0)-1,
Σμν-14K(l0)[γμ, γν]K(l0)-1,μ, ν=0, 1, 2, 3;
K(l0)=diag(l0, l0, 1/l0, 1/l0).
[ΣAB, ΣCD]=ηACΣBD-ηBCΣAD+ηADΣCB-ηBDΣCA,
J=XABΣAB,XBA=-XAB.
S:XABΣABSXABΣABS-1=XABΣAB.
S:XABXAB=Λ(S)CAΛ(S)DBXCD.
YA=ABCDEXBCXDE.
Sp(4, R)/Z2SO(3, 2).
H=ψ(ρ)d2ρ|ψ(ρ)|2<.
I(z)=d2ρ|ψ(ρ; z)|2.
ξ=xypxpyξˆ=xˆyˆpˆxpˆy=ρˆpˆ;
(ρˆψ)(ρ)=ρψ(ρ),
(pˆψ)(ρ)=-iƛψ(ρ).
[ξˆα, ξˆβ]=iƛΩαβ,α, β=1, 2, 3, 4.
exp(iu·ξ)exp(iu·ξˆ),uR4.
xpx=pxx12(xˆpˆx+pˆxxˆ),
ypy=pyy12(yˆpˆy+pˆyyˆ).
Uˆ(O):ψ(ρ)ψ(ρ)=[Uˆ(O)ψ](ρ).
Uˆ(O):|ψ|ψ=Uˆ(O)|ψ,
ρ|ψρ|ψ=d2ρρ|Uˆ(O)|ρρ|ψ,
ψ(ρ)ψ(ρ)=d2ρρ|Uˆ(O)|ρψ(ρ).
ξ¯a=ξˆa=I-1ψ|ξˆa|ψ.
ξ¯aI-1ψ|ξˆa|ψ=I-1ψ|Uˆ(S)ξˆaUˆ(S)|ψ.
ξˆUˆ(S)ξˆUˆ(S)=Sξˆ.
JHˆ(J)-12ξˆTH(J)ξˆ=-12ξˆTΩJξˆ
-iƛHˆ(J), -iƛHˆ(J)=-iƛHˆ([J, J]).
Uˆ(S)ξˆUˆ(S)=Sξˆ.
Uˆ(S)S,Uˆ(S)S  Uˆ(S)Uˆ(S)SS.
Mp(4)/Z2=Sp(4, R).
Tˆ0Hˆ(Σ0)=14(l0pˆx2+l0-1xˆ2),
Tˆ1Hˆ(Σ1)=14(l0pˆx2-l0-1xˆ2),
Tˆ2Hˆ(Σ2)=-14(xˆpˆx+pˆxxˆ).
12(l0pˆx2+l0-1xˆ2)|n=ƛ(n+12)|n,n=0, 1,.
|nexp-is4ƛ(l0pˆx2+l0-1xˆ2)|n=exp[-i(s/2)(n+12)]|n,
xpxxpx=exp(sΣ0)xpx=coss2l0 sins2-l0-1 sins2coss2xpx.
W(ρ, ρ)=ψ(ρ)ψ*(ρ).
W(ρ, ρ)=W*(ρ, ρ),
d2ρd2ρW(ρ, ρ)f(ρ)f*(ρ)0,f(ρ)L2(R2).
d2ρd2ρT(ρ, ρ)W(ρ, ρ)0,
Wˆ|ψψ|,W(ρ, ρ)ρ|Wˆ|ρ,
Wˆ=Wˆ,Wˆ0.
I=d2ρW(ρ, ρ)=tr Wˆ.
Wˆ=kμk|kk|,Wˆ|k=μk|k,
W(ρ, ρ)=kμkρ|kk|ρ,
S:WˆWˆ=Uˆ(S)WˆUˆ(S).
W(ρ, ρ)W(ρ, ρ)=d2ρd2ρρ|Uˆ(S)|ρW(ρ, ρ)×ρ|Uˆ(S)|ρ,
W(ρ, p)=1(2πƛ)2d2Δρ exp(-ip·Δρ)×Wρ+12Δρ, ρ-12Δρ.
W(ρ, ρ)=d2p exp[ip·(ρ-ρ)]Wρ+ρ2, p.
tr(WˆWˆ)=d2ρd2ρW(ρ, ρ)W(ρ, ρ)=(2πƛ)2d2ρd2pW(ρ, p)W(ρ, p).
d2ρd2pW(ρ, p)W(ρ, p)0,W(ρ, p)ΣW.
W(ρ, ρ)=|ψ(ρ)|2=d2pW(ρ, p),
W˜(p, p)=|ψ˜(p)|2=d2ρW(ρ, p).
I=d2ρW(ρ, ρ)=d2ρd2pW(ρ, p).
tr({xˆayˆbpˆxcpˆyd}WWˆ)trWˆ=d2ρd2pˆxaybpxcpydW(ρ, p)d2ρd2pW(ρ, p).
ψ(x)=c1 exp-(x-x0)2w2+c2 exp-(x+x0)2w2.
W(x, px)=w|c1|2ƛ2πexp-2w2(x-x0)2-w22ƛ2px2+w|c2|2ƛ2πexp-2w2(x+x0)2-w22ƛ2px2+2w|c1c2|ƛ2πexp-2w2x2-w22ƛ2px2×cos(2x0px/ƛ-α0),
WˆinWˆout=Uˆ(S)WˆinUˆ(S) Win(ξ)Wout(ξ)=Win(S-1ξ).
ψn(x)Hn(2x/w)exp(-x2/w2).
Wn(x, px)Ln4w2x2+w2ƛ2px2×exp-2w2x2+w22ƛ2px2.
W(ρ, ρ)=Idet L2πexp-14ρTLρ-14ρTLρ-12(ρ-ρ)TM(ρ-ρ)-i2ƛ(ρ-ρ)TK(ρ+ρ),
W(ξ)=(det G)1/24π2exp(-12ξTGξ)=(det V)-1/24π2exp(-12ξTV-1ξ),
V=G-1=VρρVρpVρpTVpp,
Vρρ=L-1,Vρp=-L-1KT,
Vpp=ƛ2(14L+M)+KL-1KT.
L=Vρρ-1,K=-VρpTVρρ-1,
M=ƛ-2Vpp-14Vρρ-1-ƛ-2VρpTVρρ-1Vρp,
tr({ξˆaξˆb}wey1Wˆ)=I-1d4ξξaξbW(ξ)=Vab,a, b=1, 2, 3, 4.
V=ξξT=x2xyxpxxpyyxy2ypxypypxxpxypx2pxpypyxpyypypxpy2.
x2px2ƛ2/4,y2py2ƛ2/4,
VξξT=I-1d4ξξξTW(ξ).
Γab=Γˆab=tr(ΓˆabWˆ).
Γˆabξˆaξˆb={ξˆaξˆb}weyl+i(ƛ/2)Ωab.
Γ=V+i(ƛ/2)Ω.
Vout=I-1d4ξξξTWout(ξ)=I-1d4ξξξTWin(S-1ξ).
Vout=SVinST.
VoutΩ=SVinΩS-1.
V=x2xpxpxxpx2,S2=abcd.
det V=x2px2-(xpx)2=invariant,
det Vƛ2/4.
V=l0(X0+X1)X2X2l0-1(X0-X1)=-XaΣaΩ2.
X0=12(x2/l0+l0px2)>0,
X1=12(x2/l0-l0px2),X2=xpx;
det V=(X0)2-(X1)2-(X2)2.
Vout=S2VinS2T  Xout=Λ(S2)Xin.
X=det V(cosh r, sinh r cos ϕ, sinh r sin ϕ).
XqX2+iXaXal0-1(X0-X1)=l0(X0+X1)X2-iXaXa.
q=xpx+idet Vpx2=px2xpx-idet V.
XΛ(S2)X  qaq+bcq+d.
S2Sp(2, R)  S˜2=αββ*α*SU(1, 1),
ad-bc=1  |α|2-|β|2=1.
q˜αq˜+ββ*q˜+α*.
L4/w2,M1μc2,KR-1.
V=w24w24Rw24Rƛ21w2+1μc2+w24R2.
det V=ƛ241+w2μc2.
1q=1R-i2ƛw21+w2μc21/2.
ψ(x)expix22ƛq,1q=1R-iπƛw2.
V=12XABΣABΩ,ΣBA=-ΣAB.
VinVoutSVinST=SXinABΣABΩST=XoutABΣABΩ,
XoutAB=Λ(S)CAΛ(S)DBXinCD.
YA=ABCDEXBCXDE.
C1XABXAB,
C2YAYA.
T1=-tr(VΩ)2=tr(VΩVΩT),
T2=tr(VΩ)4=tr(VΩVΩT)2.
OV={V=SVST|SSp(4, R)}
V0SVVSVT=l0κ10000l0κ20000l0-1κ10000l0-1κ2;
V0+i(ƛ/2)Ω=SVVSVT+i(ƛ/2)Ω0.
(i)κ1ƛ/2,κ2ƛ/2;
(ii)V+i(ƛ/2)Ω0;
(iii)V>0,V1/2ΩVΩTV1/2ƛ2/4;
(iv)V>0,V-(ƛ2/4)ΩV-1ΩT0.
C1=2(κ12+κ22),C2=(κ12-κ22)2;
T1=2(κ12+κ22),T2=2(κ14+κ24).
T1=C1,T2=C2+C12/4,C2=T2-T12/4.
det V=κ12κ22=(C12-4C2)/16=(T12-2T2)/8.
C1-2C2 ƛ2.
t=122ƛκ12+2ƛκ22,
Meff4=2ƛκ12ƛκ2,α=122ƛκ1-2ƛκ22.
2α(Meff4-1)2.
VFamilyI  VΩVΩT=det V14×4.
Vρρ(z1)Vρρ(z2)Vρρ(z3)=12z1z1212z2z2212z3z32Vρρ(0)Vρp(s)(0)Vpp(0).
D-1=(det D)-1z2z3(z3-z2)(z1z3)(z1-z3)z1z2(z2-z1)(z2+z3)(z2-z3)(z3+z1)(z3-z1)(z1+z2)(z1-z2)z3-z2z1-z3z2-z1.
V=L(g)VL(g)T=VρρVρp-VρρgVρpT-gVρρVpp-gVρp-VρpTg+gVρρg.
V˜=V1V12V12TV2;
V1=x2xpxxpxpx2,V12=xyxpypxypxpy,
Sp(2, R)=Mp(2)/Z2,SO(2, 1)=Sp(2, R)/Z2.
Sp(4, R)=Mp(4)/Z2,SO(3, 2)=Sp(4, R)/Z2.
tr S2>-2orS2=-12×2.
-1z 0-1,-m00-m-1,-cosh rsinh rsinh r-cosh r,

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