Abstract

We give a brief review of the theory of quantum and optical universal invariants, i.e., certain combinations of the second- and higher-order moments (variances) of quantum-mechanical operators or the transverse phase-space coordinates of optical paraxial beams that are preserved in time (or along the axis of the beam) independently of the concrete form of the coefficients of the Hamiltonian or the parameters of the optical system, provided that the effective Hamiltonian is either a generic quadratic form of the generalized coordinate-momenta operators or a linear combination of generators of certain finite-dimensional algebras. Using the phase-space representation of quantum mechanics (paraxial optics) in terms of the Wigner function, we elucidate the relation between the quantum invariants and the classical universal integral invariants of Poincaré and Cartan. The specific features of the Gaussian beams are discussed as examples.

© 2000 Optical Society of America

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  69. V. I. Man’ko, K. B. Wolf, “The influence of aberrations on the Gaussian beam propagation,” in Classical and Quantum Effects in Electrodynamics, A. A. Komar, ed., Proc. P.N. Lebedev Phys. Inst.176, 127–168 (1988).
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    [CrossRef]

2000 (4)

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

R. Fedele, M. A. Man’ko, V. I. Man’ko, “Wave-optics applications in charged-particle-beam transport,” J. Russ. Laser Res. 21, 1–33 (2000).
[CrossRef]

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

V. V. Dodonov, O. V. Man’ko, “Universal invariants in quantum mechanics and physics of optical and particle beams,” J. Russ. Laser Res. 21, 438–464 (2000).
[CrossRef]

1999 (3)

1998 (2)

R. Fedele, V. I. Man’ko, “Quantumlike corrections and semiclassical description of charged-particle beam transport,” Phys. Rev. E 58, 992–1001 (1998).
[CrossRef]

F. Encinas-Sanz, J. Serna, C. Martı́nez, R. Martı́nez-Herrero, P. M. Mejı́as, “Time-varying beam quality factor and mode evolution in TEA CO2 laser pulses,” IEEE J. Quantum Electron. 34, 1835–1838 (1998).
[CrossRef]

1997 (2)

Q. Cao, X. M. Deng, “Spatial parametric characterization of general polychromatic light beams,” Opt. Commun. 142, 135–145 (1997).
[CrossRef]

R. Martı́nez-Herrero, P. M. Mejı́as, “On the fourth-order spatial characterization of laser beams: new invariant parameter through ABCD systems,” Opt. Commun. 140, 57–60 (1997).
[CrossRef]

1996 (3)

K. B. Wolf, “Wigner distribution function for paraxial polychromatic optics,” Opt. Commun. 132, 343–352 (1996).
[CrossRef]

N. M. Atakishiyev, S. M. Chumakov, A. L. Rivera, K. B. Wolf, “On the phase space description of quantum nonlinear dynamics,” Phys. Lett. A 215, 128–134 (1996).
[CrossRef]

H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, E. F. Yelden, “Propagation characteristics of coherent array beams from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32, 400–407 (1996).
[CrossRef]

1995 (3)

1994 (4)

L. H. Yeh, Y. S. Kim, “Correspondence between the classical and quantum canonical transformation groups from an operator formulation of the Wigner function,” Found. Phys. 24, 873–884 (1994).
[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]

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]

M. A. Porras, “Experimental investigation on aperture-diffracted laser beam characterization,” Opt. Commun. 109, 5–9 (1994).
[CrossRef]

1993 (3)

1992 (7)

G. Dattoli, C. Mari, M. Richetta, A. Torre, “On the generalized Twiss parameters and Courant–Snyder invariant in classical and quantum optics,” Nuovo Cimento 107, 269–287 (1992).
[CrossRef]

M. J. Bastiaans, “ABCD law for partially coherent Gaussian light, propagating through first-order optical systems,” Opt. Quantum Electron. 24, S1011–S1019 (1992).
[CrossRef]

R. Martı́nez-Herrero, P. M. Mejı́as, M. Sanchez, J. L. H. Neira, “Third- and fourth-order parametric characterization of partially coherent beams propagating through ABCD optical systems,” Opt. Quantum Electron. 24, S1021–S1026 (1992).
[CrossRef]

H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, S1027–S1049 (1992).
[CrossRef]

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

T. F. Johnston, M. W. Sasnett, J. L. Doumont, A. E. Siegman, “Laser-beam quality versus aperture size in a cw argon-ion laser,” Opt. Lett. 17, 198–200 (1992).
[CrossRef] [PubMed]

G. Dattoli, L. Giannessi, C. Mari, M. Richetta, A. Torre, “Formal quantum theory of electronic rays,” Opt. Commun. 87, 175–180 (1992).
[CrossRef]

1991 (6)

R. Fedele, G. Miele, “A thermal-wave model for relativistic-charged-particle beam propagation,” Nuovo Cimento D 13, 1527–1544 (1991).
[CrossRef]

A. E. Siegman, “Defining the effective radius of curvature for a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
[CrossRef]

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

M. Kauderer, “First-order sources in first-order systems: second-order correlations,” Appl. Opt. 30, 1025–1035 (1991).
[CrossRef] [PubMed]

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

P. A. Belanger, “Beam propagation and the ABCD ray matrices,” Opt. Lett. 16, 196–198 (1991).
[CrossRef] [PubMed]

1990 (2)

D. D. Holm, W. P. Lysenko, J. C. Scovel, “Moment invariants for the Vlasov equation,” J. Math. Phys. 31, 1610–1615 (1990).
[CrossRef]

F. Neri, G. Rangarajan, “Kinematic moment invariants for linear Hamiltonian systems,” Phys. Rev. Lett. 64, 1073–1075 (1990).
[CrossRef] [PubMed]

1988 (1)

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

1987 (1)

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian–Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987).
[CrossRef] [PubMed]

1986 (2)

V. V. Dodonov, V. I. Man’ko, “Phase space eigenfunctions of multidimensional quadratic Hamiltonians,” Physica A 137, 306–316 (1986).
[CrossRef]

M. Garcı́a-Bullé, W. Lassner, K. B. Wolf, “The metaplectic group within the Heisenberg–Weyl ring,” J. Math. Phys. 27, 29–36 (1986).
[CrossRef]

1985 (1)

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]

1984 (1)

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]

1981 (3)

R. E. Turner, R. F. Snider, “A phase space moment method for classical and quantum dynamics,” Can. J. Phys. 59, 457–470 (1981).
[CrossRef]

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

K. B. Wolf, “On time-dependent quadratic Hamiltonians,” SIAM J. Appl. Math. 40, 419–431 (1981).
[CrossRef]

1980 (3)

E. S. Hernández, B. Remaud, “Quantal fluctuations and invariant operators for a general time-dependent harmonic oscillator,” Phys. Lett. A 75, 269–272 (1980).
[CrossRef]

B. Remaud, E. S. Hernández, “Constants of motion and non-stationary wave functions for the damped time-dependent harmonic oscillator,” Physica A 103, 35–54 (1980).
[CrossRef]

E. Collett, E. Wolf, “Beams generated by Gaussian quasi-homogeneous sources,” Opt. Commun. 32, 27–31 (1980).
[CrossRef]

1978 (1)

V. V. Dodonov, V. I. Man’ko, “Integrals of motion of pure and mixed quantum systems,” Physica A 94, 403–412 (1978).
[CrossRef]

1974 (1)

L. C. Lee, “Wave propagation in a random medium: a complete set of the moment equations with different wave numbers,” J. Math. Phys. 15, 1431–1435 (1974).
[CrossRef]

1969 (1)

H. R. Lewis, W. B. Riesenfeld, “An exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in a time-dependent electromagnetic field,” J. Math. Phys. 10, 1458–1473 (1969).
[CrossRef]

1958 (1)

E. D. Courant, H. S. Snyder, “Theory of alternating-gradient synchrotron,” Ann. Phys. (N.Y.) 3, 1–48 (1958).
[CrossRef]

1946 (1)

M. A. Leontovich, V. A. Fock, “Solution of the problem of propagation of electromagnetic waves along the earth’s surface by the method of parabolic equation,” Zh. Eksp. Teor. Fiz. 16, 557–573 (1946).

1934 (1)

H. P. Robertson, “An indeterminacy relation for several observables and its classical interpretation,” Phys. Rev. 46, 794–801 (1934).
[CrossRef]

Alda, J.

Arnaud, J. A.

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

Arnold, V. I.

V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer, New York, 1978).

Atakishiyev, N. M.

N. M. Atakishiyev, S. M. Chumakov, A. L. Rivera, K. B. Wolf, “On the phase space description of quantum nonlinear dynamics,” Phys. Lett. A 215, 128–134 (1996).
[CrossRef]

Bacry, H.

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

Baker, H. J.

H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, E. F. Yelden, “Propagation characteristics of coherent array beams from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32, 400–407 (1996).
[CrossRef]

Bastiaans, M. J.

M. J. Bastiaans, “ABCD law for partially coherent Gaussian light, propagating through first-order optical systems,” Opt. Quantum Electron. 24, S1011–S1019 (1992).
[CrossRef]

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

Belanger, P. A.

Bernabeu, E.

Cadilhac, M.

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

Cao, Q.

Q. Cao, X. M. Deng, “Spatial parametric characterization of general polychromatic light beams,” Opt. Commun. 142, 135–145 (1997).
[CrossRef]

Chountasis, S.

S. Chountasis, A. Vourdas, “The extended phase-space: a formalism for the study of quantum noise and quantum correlations,” J. Phys. A 32, 6949–6961 (1999).
[CrossRef]

Chumakov, S. M.

N. M. Atakishiyev, S. M. Chumakov, A. L. Rivera, K. B. Wolf, “On the phase space description of quantum nonlinear dynamics,” Phys. Lett. A 215, 128–134 (1996).
[CrossRef]

Collett, E.

E. Collett, E. Wolf, “Beams generated by Gaussian quasi-homogeneous sources,” Opt. Commun. 32, 27–31 (1980).
[CrossRef]

Courant, E. D.

E. D. Courant, H. S. Snyder, “Theory of alternating-gradient synchrotron,” Ann. Phys. (N.Y.) 3, 1–48 (1958).
[CrossRef]

Dattoli, G.

G. Dattoli, L. Giannessi, C. Mari, M. Richetta, A. Torre, “Formal quantum theory of electronic rays,” Opt. Commun. 87, 175–180 (1992).
[CrossRef]

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V. V. Dodonov, O. V. Man’ko, “Universal invariants in quantum mechanics and physics of optical and particle beams,” J. Russ. Laser Res. 21, 438–464 (2000).
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V. V. Dodonov, V. I. Man’ko, “Phase space eigenfunctions of multidimensional quadratic Hamiltonians,” Physica A 137, 306–316 (1986).
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V. V. Dodonov, V. I. Man’ko, “Integrals of motion of pure and mixed quantum systems,” Physica A 94, 403–412 (1978).
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V. V. Dodonov, O. V. Man’ko, “Universal invariants of paraxial optical beams,” in Group Theoretical Methods in Physics, Proceedings of the Third Seminar, Yurmala, Latvia, May 1985, V. V. Dodonov, M. A. Markov, V. I. Man’ko, eds. (VNU Science, Utrecht, The Netherlands, 1986), Vol. 2, pp. 523–530.

V. V. Dodonov, O. V. Man’ko, “Universal invariants of the paraxial optical beams,” in Computing Optics, Fundamentals, E. P. Velikhov, A. M. Prokhorov, eds. (International Center of Scientifical and Technical Information, Moscow, 1987), pp. 84–90.

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.

V. V. Dodonov, V. I. Man’ko, “Universal invariants of quantum systems and generalized uncertainty relations,” in Group Theoretical Methods in Physics, Proceedings of the Second International Seminar, Zvenigorod, Russia, November 24–26, 1982, M. A. Markov, V. I. Man’ko, A. E. Shabad, eds. (Nauka, Moscow, 1983), Vol. 2, pp. 11–33 (in Russian) [(Harwood Academic, London, 1985), Vol. 1, pp. 591–612 (in English)].

V. V. Dodonov, V. I. Man’ko, “Density matrices and Wigner functions of quasiclassical quantum systems,” in Group Theory, Gravitation and Elementary Particle Physics, A. A. Komar, ed., Proc. P.N. Lebedev Phys. Inst.167, 7–101 (1987).

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A. J. Dragt, F. Neri, G. Rangarajan, “General moment invariants for linear Hamiltonian systems,” Phys. Rev. A 45, 2572–2585 (1992).
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F. Encinas-Sanz, J. Serna, C. Martı́nez, R. Martı́nez-Herrero, P. M. Mejı́as, “Time-varying beam quality factor and mode evolution in TEA CO2 laser pulses,” IEEE J. Quantum Electron. 34, 1835–1838 (1998).
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R. Fedele, M. A. Man’ko, V. I. Man’ko, “Wave-optics applications in charged-particle-beam transport,” J. Russ. Laser Res. 21, 1–33 (2000).
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R. Fedele, V. I. Man’ko, “Quantumlike corrections and semiclassical description of charged-particle beam transport,” Phys. Rev. E 58, 992–1001 (1998).
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G. Dattoli, L. Giannessi, C. Mari, M. Richetta, A. Torre, “Formal quantum theory of electronic rays,” Opt. Commun. 87, 175–180 (1992).
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Gluckstern, R. L.

A. J. Dragt, R. L. Gluckstern, F. Neri, G. Rangarajan, “Theory of emittance invariance,” in Frontiers of Particle Beams: Observation, Diagnosis and Correction: Proceedings of a Topical Course, Held by the Joint US–CERN School on Particle Accelerators at Anacapri, Isola di Capri, Italy, October 20–26, 1988, M. Month, S. Turner, eds. (Springer-Verlag, Berlin, 1989), pp. 94–121.

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H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, E. F. Yelden, “Propagation characteristics of coherent array beams from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32, 400–407 (1996).
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E. S. Hernández, B. Remaud, “Quantal fluctuations and invariant operators for a general time-dependent harmonic oscillator,” Phys. Lett. A 75, 269–272 (1980).
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D. D. Holm, W. P. Lysenko, J. C. Scovel, “Moment invariants for the Vlasov equation,” J. Math. Phys. 31, 1610–1615 (1990).
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H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, E. F. Yelden, “Propagation characteristics of coherent array beams from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32, 400–407 (1996).
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H. R. Lewis, W. B. Riesenfeld, “An exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in a time-dependent electromagnetic field,” J. Math. Phys. 10, 1458–1473 (1969).
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D. D. Holm, W. P. Lysenko, J. C. Scovel, “Moment invariants for the Vlasov equation,” J. Math. Phys. 31, 1610–1615 (1990).
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R. Fedele, M. A. Man’ko, V. I. Man’ko, “Wave-optics applications in charged-particle-beam transport,” J. Russ. Laser Res. 21, 1–33 (2000).
[CrossRef]

Man’ko, O. V.

V. V. Dodonov, O. V. Man’ko, “Universal invariants in quantum mechanics and physics of optical and particle beams,” J. Russ. Laser Res. 21, 438–464 (2000).
[CrossRef]

V. V. Dodonov, O. V. Man’ko, “Universal invariants of the paraxial optical beams,” in Computing Optics, Fundamentals, E. P. Velikhov, A. M. Prokhorov, eds. (International Center of Scientifical and Technical Information, Moscow, 1987), pp. 84–90.

V. V. Dodonov, O. V. Man’ko, “Universal invariants of paraxial optical beams,” in Group Theoretical Methods in Physics, Proceedings of the Third Seminar, Yurmala, Latvia, May 1985, V. V. Dodonov, M. A. Markov, V. I. Man’ko, eds. (VNU Science, Utrecht, The Netherlands, 1986), Vol. 2, pp. 523–530.

Man’ko, V. I.

R. Fedele, M. A. Man’ko, V. I. Man’ko, “Wave-optics applications in charged-particle-beam transport,” J. Russ. Laser Res. 21, 1–33 (2000).
[CrossRef]

R. Fedele, V. I. Man’ko, “Quantumlike corrections and semiclassical description of charged-particle beam transport,” Phys. Rev. E 58, 992–1001 (1998).
[CrossRef]

V. V. Dodonov, V. I. Man’ko, “Phase space eigenfunctions of multidimensional quadratic Hamiltonians,” Physica A 137, 306–316 (1986).
[CrossRef]

V. V. Dodonov, V. I. Man’ko, “Integrals of motion of pure and mixed quantum systems,” Physica A 94, 403–412 (1978).
[CrossRef]

V. I. Man’ko, K. B. Wolf, “The influence of aberrations on the Gaussian beam propagation,” in Classical and Quantum Effects in Electrodynamics, A. A. Komar, ed., Proc. P.N. Lebedev Phys. Inst.176, 127–168 (1988).

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.

V. I. Man’ko, K. B. Wolf, “Symplectic and Euclidean groups of transformations in optics,” in Theory of the Interaction of Multilevel Systems with Quantized Fields, V. I. Man’ko, M. A. Markov, eds., Proc. P.N. Lebedev Phys. Inst.209, 163–189 (1996).

V. V. Dodonov, V. I. Man’ko, “Universal invariants of quantum systems and generalized uncertainty relations,” in Group Theoretical Methods in Physics, Proceedings of the Second International Seminar, Zvenigorod, Russia, November 24–26, 1982, M. A. Markov, V. I. Man’ko, A. E. Shabad, eds. (Nauka, Moscow, 1983), Vol. 2, pp. 11–33 (in Russian) [(Harwood Academic, London, 1985), Vol. 1, pp. 591–612 (in English)].

V. V. Dodonov, V. I. Man’ko, “Density matrices and Wigner functions of quasiclassical quantum systems,” in Group Theory, Gravitation and Elementary Particle Physics, A. A. Komar, ed., Proc. P.N. Lebedev Phys. Inst.167, 7–101 (1987).

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G. Dattoli, C. Mari, M. Richetta, A. Torre, “On the generalized Twiss parameters and Courant–Snyder invariant in classical and quantum optics,” Nuovo Cimento 107, 269–287 (1992).
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G. Dattoli, L. Giannessi, C. Mari, M. Richetta, A. Torre, “Formal quantum theory of electronic rays,” Opt. Commun. 87, 175–180 (1992).
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F. Encinas-Sanz, J. Serna, C. Martı́nez, R. Martı́nez-Herrero, P. M. Mejı́as, “Time-varying beam quality factor and mode evolution in TEA CO2 laser pulses,” IEEE J. Quantum Electron. 34, 1835–1838 (1998).
[CrossRef]

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F. Encinas-Sanz, J. Serna, C. Martı́nez, R. Martı́nez-Herrero, P. M. Mejı́as, “Time-varying beam quality factor and mode evolution in TEA CO2 laser pulses,” IEEE J. Quantum Electron. 34, 1835–1838 (1998).
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R. Martı́nez-Herrero, P. M. Mejı́as, “On the fourth-order spatial characterization of laser beams: new invariant parameter through ABCD systems,” Opt. Commun. 140, 57–60 (1997).
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R. Martı́nez-Herrero, P. M. Mejı́as, M. Sanchez, J. L. H. Neira, “Third- and fourth-order parametric characterization of partially coherent beams propagating through ABCD optical systems,” Opt. Quantum Electron. 24, S1021–S1026 (1992).
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J. Serna, R. Martı́nez-Herrero, P. M. Mejı́as, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
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F. Encinas-Sanz, J. Serna, C. Martı́nez, R. Martı́nez-Herrero, P. M. Mejı́as, “Time-varying beam quality factor and mode evolution in TEA CO2 laser pulses,” IEEE J. Quantum Electron. 34, 1835–1838 (1998).
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R. Martı́nez-Herrero, P. M. Mejı́as, “On the fourth-order spatial characterization of laser beams: new invariant parameter through ABCD systems,” Opt. Commun. 140, 57–60 (1997).
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R. Martı́nez-Herrero, P. M. Mejı́as, M. Sanchez, J. L. H. Neira, “Third- and fourth-order parametric characterization of partially coherent beams propagating through ABCD optical systems,” Opt. Quantum Electron. 24, S1021–S1026 (1992).
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J. Serna, R. Martı́nez-Herrero, P. M. Mejı́as, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
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R. Fedele, G. Miele, “A thermal-wave model for relativistic-charged-particle beam propagation,” Nuovo Cimento D 13, 1527–1544 (1991).
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H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, E. F. Yelden, “Propagation characteristics of coherent array beams from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32, 400–407 (1996).
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R. Simon, N. Mukunda, “Gaussian Schell-model beams and generalized shape invariance,” J. Opt. Soc. Am. A 16, 2465–2475 (1999).
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R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
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R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian–Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987).
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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).
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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).
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R. Martı́nez-Herrero, P. M. Mejı́as, M. Sanchez, J. L. H. Neira, “Third- and fourth-order parametric characterization of partially coherent beams propagating through ABCD optical systems,” Opt. Quantum Electron. 24, S1021–S1026 (1992).
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Neri, F.

A. J. Dragt, F. Neri, G. Rangarajan, “General moment invariants for linear Hamiltonian systems,” Phys. Rev. A 45, 2572–2585 (1992).
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F. Neri, G. Rangarajan, “Kinematic moment invariants for linear Hamiltonian systems,” Phys. Rev. Lett. 64, 1073–1075 (1990).
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F. Neri, G. Rangarajan, “Kinematic moment invariants for linear Hamiltonian systems,” Phys. Rev. Lett. 64, 1073–1075 (1990).
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E. S. Hernández, B. Remaud, “Quantal fluctuations and invariant operators for a general time-dependent harmonic oscillator,” Phys. Lett. A 75, 269–272 (1980).
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G. Dattoli, C. Mari, M. Richetta, A. Torre, “On the generalized Twiss parameters and Courant–Snyder invariant in classical and quantum optics,” Nuovo Cimento 107, 269–287 (1992).
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R. Martı́nez-Herrero, P. M. Mejı́as, M. Sanchez, J. L. H. Neira, “Third- and fourth-order parametric characterization of partially coherent beams propagating through ABCD optical systems,” Opt. Quantum Electron. 24, S1021–S1026 (1992).
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D. D. Holm, W. P. Lysenko, J. C. Scovel, “Moment invariants for the Vlasov equation,” J. Math. Phys. 31, 1610–1615 (1990).
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F. Encinas-Sanz, J. Serna, C. Martı́nez, R. Martı́nez-Herrero, P. M. Mejı́as, “Time-varying beam quality factor and mode evolution in TEA CO2 laser pulses,” IEEE J. Quantum Electron. 34, 1835–1838 (1998).
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J. Serna, R. Martı́nez-Herrero, P. M. Mejı́as, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
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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).
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R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
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G. Dattoli, L. Giannessi, C. Mari, M. Richetta, A. Torre, “Formal quantum theory of electronic rays,” Opt. Commun. 87, 175–180 (1992).
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Equations (54)

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iψz=-12kn02ψx2+2ψy2+V(x, y; k)ψ,
V(x, y; z; k)=(k/2n0)[n02(z)-n2(x, y, z)]
E(r)=n0-1/2ψ(x, y; z)expik0zn0(ξ)dξ
iψz=12nkn02c22ψt2-2ψx2-2ψy2+kn2ψ.
iρz=j=1a+b-Δj2n0(z; kj)kj+V(rj; kj)ρ,
QˆαQˆβ-QˆβQˆα=-iΣαβ=iΣβα,
Σ=0En-En0.
Hˆ=1/2 j,k=12nBjk(z)QˆjQˆk+j=12nCj(z)Qˆj
1/2 QˆB(z)Qˆ+C(z)Qˆ,
n2-n02=aik(z)xixk+bi(z)xi,
Qˆα(z)=ΣβΛαβ(z)Qˆα(0)+δα(z).
Qαβ(z)=Σμ,νΛαμ(z)Qμν(0)Λβν(z),
t2¯=ψ*(x, y, t)t2ψ(x, y, t)dxdydt,
pt2¯=-ψ*(x, y, t)2t2ψ(x, y, t)dxdydt.
Λ˙=-ΣBΛ,Λ˙˜=Λ˜BΣ,
Λ˜(z)Σ-1Λ(z)Σ-1,Λ(z)ΣΛ˜(z)Σ,
Q(z)Qαβ(z)=Λ(z)Q(0)Λ˜(z),
D(γ; z)=det[Q(z)-γΣ]=m=02nDmγm=D(γ; 0),
det(Q-γΣ)=det(Q˜-γΣ˜)=det(Q+γΣ).
ΔD0(1)=pp¯·xx¯-(px¯)2.
ρ(x1, x2)=A exp(-ax12-a*x22+2bx1x2),
|ρ(R, τ)|=|A|exp[-2(u-b)R2-(1/2)(u+b)r2],
D2n-2(n)=i,j=1n(pipj¯·xixj¯-pixj¯·xipj¯).
D2(2)=px2¯ x2¯+py2¯ y2¯+2xy pxpy¯-2xpy¯ ypx¯-(ypy¯)2-(xpx¯)2;
D0(2)=[x2¯ y2¯-(xy¯)2][px2¯ py2¯-(px¯py¯)2]+(xpx¯)2 (ypy¯)2+(ypx¯)2 (xpy¯)2-y2¯ px2¯ (xpy¯)2-x2¯ py2¯ (ypx¯)2-y2¯ py2¯ (xpx¯)2-x2¯ px2¯ (ypy¯)2-2xpx¯ ypy¯ xpy¯ ypx¯-2xy¯ pxpy¯ (xpx¯ ypy¯+xpy¯ ypx¯)+2xy¯[px2¯ ypy¯ xpy¯+py2¯xpx¯ ypx¯]+2pxpy¯[y2¯ xpy¯ xpx¯+x2¯ypx¯ ypy¯].
xˆ2+yˆ2pˆx2+pˆy2-xˆpˆx+pˆyyˆ2=constant.
W(p, x)=(det Q)-1/2 ×exp[-(1/2)(q-q)Q-1(q-q)],
W=1Δexp-12Δ(σppx˜2+σxxp˜2-2σpxx˜p˜),
a±=T+Δ±T-Δ,T=(1/2)(σpp+σxx).
D2j(n)=k=0min(j, n-j)αkPn-j+k(n)Pn-j-k(n),
Wt=k=12nqk[(ΣBq+ΣC)kW],
Kˆ0(z)=(1/2)RˆΣ˜-1Q(z)Σ-1Rˆ,
Kˆ0(z)=-(1/2)Tr(QΣ-1QΣ-1).
L2m=Tr([QΣ-1]2m)
Kˆ2m(z)=-Rˆ[Σ-1Q(z)]1+2mΣ-1Rˆ.
Ws=2(-1)se-κsLs(2κs),s=0, 1, 2,,
κs(x, p, z)=σpp(z)x˜2+σxx(z)p˜2-2σpx(z)x˜p˜s+1/2
dΔ/dz=2n3nAn[xn¯ px¯-x2¯ pxn-1¯].
Δ(z)-Δ(0)3A4z32ν5(α+βν2+γν4),
α=(p2¯ p3x¯-p4¯ px¯)z=0,
β=(p2¯ px3¯-x2¯ xp3¯)z=0,
γ=(x4¯ xp¯-x2¯ x3p¯)z=0.
Λ˙αβ=ν,δ=1Ncανδfν(z)Λδβ.
Λ(z)gΛ˜(z)=g,Λ=Λαβ,g=gαβ,
G(γ; z)=det[Q(z)-γg]=m=0NγmGm=G(γ; 0).
Lm=Tr{[Q(z)g-1]m},g-1=gαβ.
G0=R11R22R33+2R12R23R31-R11R232-R22R312-R33R122,
G1=R11R22-R122+4(R12R33-R13R23),
G2=R12-R33.
Δ*=σpp (σqqp2-σqqqσppq)+σqq (σppq2-σpppσqqp)+σpq (σpppσqqq-σqqpσppq),
σab  c=W(p, q)δaδb  δcdpdq2π,
δaa-aˆ.
K2m(j)=(RˆΣ˜-1Q(z)Σ-1Q(z)2mΣ-1Rˆ)j.
K=Δ-2 (ppppσxx2+σxxxxσpp2+4σpppxσxxσxp+4σxxxpσppσxp+6σppxxσpx2)+Δ-1 (δpˆ)2(δxˆ)2+(δxˆ)2(δpˆ)2.

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