Abstract

We develop a path integral approach for analyzing the stationary light propagation in general dielectric media. The Hermitian form of the stationary Maxwell equations is transformed into a quantum mechanical problem of a spin 1 particle with spin-orbit coupling and position dependent mass. After appropriate ordering several path integral representations of a solution are constructed. First we keep the propagation of the polarization degrees of freedom in an operator form integrated over paths in a coordinate space. The use of spin 1 coherent states allows representing this part as a path integral over such states. Finally a path integral in a transversal momentum space with explicit transversality enforced at every time slice is also given. As an example the geometrical optics limit is discussed and the ray equation is recovered together with the Rytov rotation of the polarization vector.

© 2010 Optical Society of America

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  1. J. D. Joannopoulos, S. Jhohnson, J. Winn, and R. Meade, Photonic Crystals—Modeling the Flow of Light (Princeton U. Press, 2008).
  2. J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature  386, 143–149 (1997).
    [CrossRef]
  3. M. Eve, “The use of path integrals in guided wave theory,” Proc. R. Soc. London, Ser. A  347, 405–417 (1976).
    [CrossRef]
  4. G. Samelsohn and R. Mazarr, “Path-integral analysis of scalar wave propagation in multiple-scattering random media,” Phys. Rev. E  54, 5697–5706 (1996).
    [CrossRef]
  5. C. Gomez-Reino and J. Linares, “Optical path integrals in gradient-index media,” J. Opt. Soc. Am. A  4, 1337–1341 (1987).
    [CrossRef]
  6. L. S. Schulman, Techniques and Application of Path Integration (Wiley, 1981).
  7. C. Grosche and F. Steiner, Handbook of Feynman Path Integrals, Vol.  145 of Springer Tracts in Modern Physics (Springer-Verlag, 1998).
  8. J. I. Gersten and A. Nitzan, “Path-integral approach to electromagnetic phenomena in inhomogeneous systems,” J. Opt. Soc. Am. B  4, 293–298 (1987).
    [CrossRef]
  9. I. Bialynicki-Birula, “Weyl, Dirac, and Maxwell equations on a lattice as unitary cellular automata,” Phys. Rev. D  49, 6920–6927 (1994).
    [CrossRef]
  10. R. D. Nevels, J. A. Miller, and R. E. Miller, “A path integral time domain method for electromagnetic scattering,” IEEE Trans. Antennas Propag.  48, 565–573 (2000).
    [CrossRef]
  11. S. V. Shabanov, “Electromagnetic pulse propagation in passive media by path integral methods,” arXiv:math/0312296.
  12. J. Lee and A. Fornberg, “A split step approach for the 3-D Maxwell’s equations,” J. Comput. Appl. Math.  158, 485–505 (2003).
    [CrossRef]
  13. X. Xu, “Matrix differential-operator approach to the Maxwell equations and the Dirac equation,” Acta Appl. Math.  102, 237–247 (2008).
    [CrossRef]
  14. M. Freidlin, Functional Integration and Partial Differential Equations (Princeton U. Press, 1985).
  15. B. Simon, Functional Integration and Quantum Physics, Vol.  86 of Pure and Applied Mathematics (Academic, 1979).
  16. B. D. Budaev and D. B. Bogy, “Probabilistic solutions of the Helmholtz equation,” J. Acoust. Soc. Am.  109, 2260–2262 (2001).
    [CrossRef]
  17. B. D. Budaev and D. B. Bogy, “Analysis of one dimensional wave scattering by the random walk method,” J. Acoust. Soc. Am.  111, 2555–2560 (2002).
    [CrossRef]
  18. B. D. Budaev and D. B. Bogy, “Application of random walk methods to wave propagation,” Q. J. Mech. Appl. Math.  55, 209–226 (2002).
    [CrossRef]
  19. B. D. Budaev and D. B. Bogy, “Random walk approach to wave propagation in wedges and cones,” J. Acoust. Soc. Am.  114, 1733–1741 (2003).
    [CrossRef]
  20. L. Fishman and J. J. McCoy, “Derivation and application of the extended parabolic wave theories. 1. The factorized Helmholtz equation,” J. Math. Phys.  25, 285–296 (1984).
    [CrossRef]
  21. L. Fishman and J. J. McCoy, “Derivation and application of the extended parabolic wave theories. 2. Path integral representation,” J. Math. Phys.  25, 297–308 (1984).
    [CrossRef]
  22. L. Fishman, J. J. McCoy, and S. C. Wales, “Factorization and path integration of the Helmholtz equation: numerical algorithms,” J. Acoust. Soc. Am.  81, 1355–1376 (1987).
    [CrossRef]
  23. L. Fishman, “Helmholtz path integrals,” in Mathematical Modeling of Wave Phenomena, B.Nilsson and L.Fishman, eds., Vol.  834 of AIP Conference Proceedings (American Institute of Physics, 2006), pp. 25–55.
  24. V. B. Berestezki, E. M. Lifshitz, and L. P. Pitaevski, Quantum Electrodynamics, 2nd ed. (Pergamon, 1982).
  25. K. Gottfried, Quantum Mechanics (W. A. Benjamin, 1966).
  26. A. Messiah, Quantum Mechanics (North-Holland, 1966), Vol.  II.
  27. R. G. Levers and D. G. Gleaves, “Finding the roots of the acoustic wave equation,” Vol.  2 of Proceedings of the 11th IMACS World Congress on System Simulation and Scientific Computation, B.Wahlstrom, R.Henriksen, and N.P.Sundby, eds. (NFA, 1985), pp. 165–167.
  28. R. G. Levers, “Spinning the Helmholtz equation,” in Computational Acoustics Wave Propagation, D.Lee, R.L.Sternberg, and M.H.Schultz, eds. (North-Holland, 1988), pp. 287–293.
  29. T. Kashiwa, Y. Ohnuki, and M. Suzuki, Path Integral Methods (Oxford U. Press, 1997).
  30. M. Pletyukhov, C. Amann, M. Metha, and M. Brack, “Semiclassical theory of spin-orbit interactions using spin coherent states,” Phys. Rev. Lett.  89, 116601 (2002).
    [CrossRef]
  31. M. Pletyukhov and O. Zaitsev, J. Phys. A  36, 5181–5219 (2003).
    [CrossRef]
  32. C. Amann and M. Brack, “Semiclassical theory of spin-orbit interaction in the extended phase space,” J. Phys. A  35, 6009–6032 (2002).
    [CrossRef]
  33. I. Bialynicki-Birula, “Photon wave function,” Progress in Optics XXXVI (1996), pp. 245–294.
  34. I. Bialynicki-Birula, “On the wave function of the photon,” Acta Physiol. Pol.  86, 97–116 (1994).
  35. A. Auerbach, Interacting Electrons and Quantum Magnetism (Springer-Verlag, 1994).
  36. J. R. Klauder and B. S. Skagerstam, Coherent States: Applications in Physics and Mathematical Physics (World Scientific, 1985).
  37. J. R. Klauder, “Path integrals and stationary-phase approximations,” Phys. Rev. D  19, 2349–2356 (1979).
    [CrossRef]
  38. A. Altland and B. Simons, Condensed Matter Field Theory (Cambridge U. Press, 2006).
  39. H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 4th ed. (World Scientific, 2006).
  40. I. Mendas and P. Milutinovic, “Anticommutator analogues of certain identities involving repeated commutators,” J. Phys. A  23, 537–544 (1990).
    [CrossRef]
  41. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1960).
  42. M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1959).
  43. S. Keppeler, Spinning Particles—Semiclassics and Spectral Statistics (Springer, 2003).
  44. M. Kline and I. W. Kay, Electromagnetic Theory and Geometrical Optics (Wiley, 1965).
  45. S. P. Novikov and A. T. Fomenko, Basic Elements of Differential Geometry and Topology (Kluwer, 1990).

2008 (1)

X. Xu, “Matrix differential-operator approach to the Maxwell equations and the Dirac equation,” Acta Appl. Math.  102, 237–247 (2008).
[CrossRef]

2003 (3)

B. D. Budaev and D. B. Bogy, “Random walk approach to wave propagation in wedges and cones,” J. Acoust. Soc. Am.  114, 1733–1741 (2003).
[CrossRef]

M. Pletyukhov and O. Zaitsev, J. Phys. A  36, 5181–5219 (2003).
[CrossRef]

J. Lee and A. Fornberg, “A split step approach for the 3-D Maxwell’s equations,” J. Comput. Appl. Math.  158, 485–505 (2003).
[CrossRef]

2002 (4)

M. Pletyukhov, C. Amann, M. Metha, and M. Brack, “Semiclassical theory of spin-orbit interactions using spin coherent states,” Phys. Rev. Lett.  89, 116601 (2002).
[CrossRef]

C. Amann and M. Brack, “Semiclassical theory of spin-orbit interaction in the extended phase space,” J. Phys. A  35, 6009–6032 (2002).
[CrossRef]

B. D. Budaev and D. B. Bogy, “Analysis of one dimensional wave scattering by the random walk method,” J. Acoust. Soc. Am.  111, 2555–2560 (2002).
[CrossRef]

B. D. Budaev and D. B. Bogy, “Application of random walk methods to wave propagation,” Q. J. Mech. Appl. Math.  55, 209–226 (2002).
[CrossRef]

2001 (1)

B. D. Budaev and D. B. Bogy, “Probabilistic solutions of the Helmholtz equation,” J. Acoust. Soc. Am.  109, 2260–2262 (2001).
[CrossRef]

2000 (1)

R. D. Nevels, J. A. Miller, and R. E. Miller, “A path integral time domain method for electromagnetic scattering,” IEEE Trans. Antennas Propag.  48, 565–573 (2000).
[CrossRef]

1997 (1)

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature  386, 143–149 (1997).
[CrossRef]

1996 (1)

G. Samelsohn and R. Mazarr, “Path-integral analysis of scalar wave propagation in multiple-scattering random media,” Phys. Rev. E  54, 5697–5706 (1996).
[CrossRef]

1994 (2)

I. Bialynicki-Birula, “Weyl, Dirac, and Maxwell equations on a lattice as unitary cellular automata,” Phys. Rev. D  49, 6920–6927 (1994).
[CrossRef]

I. Bialynicki-Birula, “On the wave function of the photon,” Acta Physiol. Pol.  86, 97–116 (1994).

1990 (1)

I. Mendas and P. Milutinovic, “Anticommutator analogues of certain identities involving repeated commutators,” J. Phys. A  23, 537–544 (1990).
[CrossRef]

1987 (3)

1984 (2)

L. Fishman and J. J. McCoy, “Derivation and application of the extended parabolic wave theories. 1. The factorized Helmholtz equation,” J. Math. Phys.  25, 285–296 (1984).
[CrossRef]

L. Fishman and J. J. McCoy, “Derivation and application of the extended parabolic wave theories. 2. Path integral representation,” J. Math. Phys.  25, 297–308 (1984).
[CrossRef]

1979 (1)

J. R. Klauder, “Path integrals and stationary-phase approximations,” Phys. Rev. D  19, 2349–2356 (1979).
[CrossRef]

1976 (1)

M. Eve, “The use of path integrals in guided wave theory,” Proc. R. Soc. London, Ser. A  347, 405–417 (1976).
[CrossRef]

Altland, A.

A. Altland and B. Simons, Condensed Matter Field Theory (Cambridge U. Press, 2006).

Amann, C.

C. Amann and M. Brack, “Semiclassical theory of spin-orbit interaction in the extended phase space,” J. Phys. A  35, 6009–6032 (2002).
[CrossRef]

M. Pletyukhov, C. Amann, M. Metha, and M. Brack, “Semiclassical theory of spin-orbit interactions using spin coherent states,” Phys. Rev. Lett.  89, 116601 (2002).
[CrossRef]

Auerbach, A.

A. Auerbach, Interacting Electrons and Quantum Magnetism (Springer-Verlag, 1994).

Berestezki, V. B.

V. B. Berestezki, E. M. Lifshitz, and L. P. Pitaevski, Quantum Electrodynamics, 2nd ed. (Pergamon, 1982).

Bialynicki-Birula, I.

I. Bialynicki-Birula, “Weyl, Dirac, and Maxwell equations on a lattice as unitary cellular automata,” Phys. Rev. D  49, 6920–6927 (1994).
[CrossRef]

I. Bialynicki-Birula, “On the wave function of the photon,” Acta Physiol. Pol.  86, 97–116 (1994).

I. Bialynicki-Birula, “Photon wave function,” Progress in Optics XXXVI (1996), pp. 245–294.

Bogy, D. B.

B. D. Budaev and D. B. Bogy, “Random walk approach to wave propagation in wedges and cones,” J. Acoust. Soc. Am.  114, 1733–1741 (2003).
[CrossRef]

B. D. Budaev and D. B. Bogy, “Analysis of one dimensional wave scattering by the random walk method,” J. Acoust. Soc. Am.  111, 2555–2560 (2002).
[CrossRef]

B. D. Budaev and D. B. Bogy, “Application of random walk methods to wave propagation,” Q. J. Mech. Appl. Math.  55, 209–226 (2002).
[CrossRef]

B. D. Budaev and D. B. Bogy, “Probabilistic solutions of the Helmholtz equation,” J. Acoust. Soc. Am.  109, 2260–2262 (2001).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1959).

Brack, M.

C. Amann and M. Brack, “Semiclassical theory of spin-orbit interaction in the extended phase space,” J. Phys. A  35, 6009–6032 (2002).
[CrossRef]

M. Pletyukhov, C. Amann, M. Metha, and M. Brack, “Semiclassical theory of spin-orbit interactions using spin coherent states,” Phys. Rev. Lett.  89, 116601 (2002).
[CrossRef]

Budaev, B. D.

B. D. Budaev and D. B. Bogy, “Random walk approach to wave propagation in wedges and cones,” J. Acoust. Soc. Am.  114, 1733–1741 (2003).
[CrossRef]

B. D. Budaev and D. B. Bogy, “Analysis of one dimensional wave scattering by the random walk method,” J. Acoust. Soc. Am.  111, 2555–2560 (2002).
[CrossRef]

B. D. Budaev and D. B. Bogy, “Application of random walk methods to wave propagation,” Q. J. Mech. Appl. Math.  55, 209–226 (2002).
[CrossRef]

B. D. Budaev and D. B. Bogy, “Probabilistic solutions of the Helmholtz equation,” J. Acoust. Soc. Am.  109, 2260–2262 (2001).
[CrossRef]

Eve, M.

M. Eve, “The use of path integrals in guided wave theory,” Proc. R. Soc. London, Ser. A  347, 405–417 (1976).
[CrossRef]

Fan, S.

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature  386, 143–149 (1997).
[CrossRef]

Fishman, L.

L. Fishman, J. J. McCoy, and S. C. Wales, “Factorization and path integration of the Helmholtz equation: numerical algorithms,” J. Acoust. Soc. Am.  81, 1355–1376 (1987).
[CrossRef]

L. Fishman and J. J. McCoy, “Derivation and application of the extended parabolic wave theories. 1. The factorized Helmholtz equation,” J. Math. Phys.  25, 285–296 (1984).
[CrossRef]

L. Fishman and J. J. McCoy, “Derivation and application of the extended parabolic wave theories. 2. Path integral representation,” J. Math. Phys.  25, 297–308 (1984).
[CrossRef]

L. Fishman, “Helmholtz path integrals,” in Mathematical Modeling of Wave Phenomena, B.Nilsson and L.Fishman, eds., Vol.  834 of AIP Conference Proceedings (American Institute of Physics, 2006), pp. 25–55.

Fomenko, A. T.

S. P. Novikov and A. T. Fomenko, Basic Elements of Differential Geometry and Topology (Kluwer, 1990).

Fornberg, A.

J. Lee and A. Fornberg, “A split step approach for the 3-D Maxwell’s equations,” J. Comput. Appl. Math.  158, 485–505 (2003).
[CrossRef]

Freidlin, M.

M. Freidlin, Functional Integration and Partial Differential Equations (Princeton U. Press, 1985).

Gersten, J. I.

Gleaves, D. G.

R. G. Levers and D. G. Gleaves, “Finding the roots of the acoustic wave equation,” Vol.  2 of Proceedings of the 11th IMACS World Congress on System Simulation and Scientific Computation, B.Wahlstrom, R.Henriksen, and N.P.Sundby, eds. (NFA, 1985), pp. 165–167.

Gomez-Reino, C.

Gottfried, K.

K. Gottfried, Quantum Mechanics (W. A. Benjamin, 1966).

Grosche, C.

C. Grosche and F. Steiner, Handbook of Feynman Path Integrals, Vol.  145 of Springer Tracts in Modern Physics (Springer-Verlag, 1998).

Jhohnson, S.

J. D. Joannopoulos, S. Jhohnson, J. Winn, and R. Meade, Photonic Crystals—Modeling the Flow of Light (Princeton U. Press, 2008).

Joannopoulos, J. D.

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature  386, 143–149 (1997).
[CrossRef]

J. D. Joannopoulos, S. Jhohnson, J. Winn, and R. Meade, Photonic Crystals—Modeling the Flow of Light (Princeton U. Press, 2008).

Kashiwa, T.

T. Kashiwa, Y. Ohnuki, and M. Suzuki, Path Integral Methods (Oxford U. Press, 1997).

Kay, I. W.

M. Kline and I. W. Kay, Electromagnetic Theory and Geometrical Optics (Wiley, 1965).

Keppeler, S.

S. Keppeler, Spinning Particles—Semiclassics and Spectral Statistics (Springer, 2003).

Klauder, J. R.

J. R. Klauder, “Path integrals and stationary-phase approximations,” Phys. Rev. D  19, 2349–2356 (1979).
[CrossRef]

J. R. Klauder and B. S. Skagerstam, Coherent States: Applications in Physics and Mathematical Physics (World Scientific, 1985).

Kleinert, H.

H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 4th ed. (World Scientific, 2006).

Kline, M.

M. Kline and I. W. Kay, Electromagnetic Theory and Geometrical Optics (Wiley, 1965).

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1960).

Lee, J.

J. Lee and A. Fornberg, “A split step approach for the 3-D Maxwell’s equations,” J. Comput. Appl. Math.  158, 485–505 (2003).
[CrossRef]

Levers, R. G.

R. G. Levers and D. G. Gleaves, “Finding the roots of the acoustic wave equation,” Vol.  2 of Proceedings of the 11th IMACS World Congress on System Simulation and Scientific Computation, B.Wahlstrom, R.Henriksen, and N.P.Sundby, eds. (NFA, 1985), pp. 165–167.

R. G. Levers, “Spinning the Helmholtz equation,” in Computational Acoustics Wave Propagation, D.Lee, R.L.Sternberg, and M.H.Schultz, eds. (North-Holland, 1988), pp. 287–293.

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1960).

V. B. Berestezki, E. M. Lifshitz, and L. P. Pitaevski, Quantum Electrodynamics, 2nd ed. (Pergamon, 1982).

Linares, J.

Mazarr, R.

G. Samelsohn and R. Mazarr, “Path-integral analysis of scalar wave propagation in multiple-scattering random media,” Phys. Rev. E  54, 5697–5706 (1996).
[CrossRef]

McCoy, J. J.

L. Fishman, J. J. McCoy, and S. C. Wales, “Factorization and path integration of the Helmholtz equation: numerical algorithms,” J. Acoust. Soc. Am.  81, 1355–1376 (1987).
[CrossRef]

L. Fishman and J. J. McCoy, “Derivation and application of the extended parabolic wave theories. 2. Path integral representation,” J. Math. Phys.  25, 297–308 (1984).
[CrossRef]

L. Fishman and J. J. McCoy, “Derivation and application of the extended parabolic wave theories. 1. The factorized Helmholtz equation,” J. Math. Phys.  25, 285–296 (1984).
[CrossRef]

Meade, R.

J. D. Joannopoulos, S. Jhohnson, J. Winn, and R. Meade, Photonic Crystals—Modeling the Flow of Light (Princeton U. Press, 2008).

Mendas, I.

I. Mendas and P. Milutinovic, “Anticommutator analogues of certain identities involving repeated commutators,” J. Phys. A  23, 537–544 (1990).
[CrossRef]

Messiah, A.

A. Messiah, Quantum Mechanics (North-Holland, 1966), Vol.  II.

Metha, M.

M. Pletyukhov, C. Amann, M. Metha, and M. Brack, “Semiclassical theory of spin-orbit interactions using spin coherent states,” Phys. Rev. Lett.  89, 116601 (2002).
[CrossRef]

Miller, J. A.

R. D. Nevels, J. A. Miller, and R. E. Miller, “A path integral time domain method for electromagnetic scattering,” IEEE Trans. Antennas Propag.  48, 565–573 (2000).
[CrossRef]

Miller, R. E.

R. D. Nevels, J. A. Miller, and R. E. Miller, “A path integral time domain method for electromagnetic scattering,” IEEE Trans. Antennas Propag.  48, 565–573 (2000).
[CrossRef]

Milutinovic, P.

I. Mendas and P. Milutinovic, “Anticommutator analogues of certain identities involving repeated commutators,” J. Phys. A  23, 537–544 (1990).
[CrossRef]

Nevels, R. D.

R. D. Nevels, J. A. Miller, and R. E. Miller, “A path integral time domain method for electromagnetic scattering,” IEEE Trans. Antennas Propag.  48, 565–573 (2000).
[CrossRef]

Nitzan, A.

Novikov, S. P.

S. P. Novikov and A. T. Fomenko, Basic Elements of Differential Geometry and Topology (Kluwer, 1990).

Ohnuki, Y.

T. Kashiwa, Y. Ohnuki, and M. Suzuki, Path Integral Methods (Oxford U. Press, 1997).

Pitaevski, L. P.

V. B. Berestezki, E. M. Lifshitz, and L. P. Pitaevski, Quantum Electrodynamics, 2nd ed. (Pergamon, 1982).

Pletyukhov, M.

M. Pletyukhov and O. Zaitsev, J. Phys. A  36, 5181–5219 (2003).
[CrossRef]

M. Pletyukhov, C. Amann, M. Metha, and M. Brack, “Semiclassical theory of spin-orbit interactions using spin coherent states,” Phys. Rev. Lett.  89, 116601 (2002).
[CrossRef]

Samelsohn, G.

G. Samelsohn and R. Mazarr, “Path-integral analysis of scalar wave propagation in multiple-scattering random media,” Phys. Rev. E  54, 5697–5706 (1996).
[CrossRef]

Schulman, L. S.

L. S. Schulman, Techniques and Application of Path Integration (Wiley, 1981).

Shabanov, S. V.

S. V. Shabanov, “Electromagnetic pulse propagation in passive media by path integral methods,” arXiv:math/0312296.

Simon, B.

B. Simon, Functional Integration and Quantum Physics, Vol.  86 of Pure and Applied Mathematics (Academic, 1979).

Simons, B.

A. Altland and B. Simons, Condensed Matter Field Theory (Cambridge U. Press, 2006).

Skagerstam, B. S.

J. R. Klauder and B. S. Skagerstam, Coherent States: Applications in Physics and Mathematical Physics (World Scientific, 1985).

Steiner, F.

C. Grosche and F. Steiner, Handbook of Feynman Path Integrals, Vol.  145 of Springer Tracts in Modern Physics (Springer-Verlag, 1998).

Suzuki, M.

T. Kashiwa, Y. Ohnuki, and M. Suzuki, Path Integral Methods (Oxford U. Press, 1997).

Villeneuve, P. R.

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature  386, 143–149 (1997).
[CrossRef]

Wales, S. C.

L. Fishman, J. J. McCoy, and S. C. Wales, “Factorization and path integration of the Helmholtz equation: numerical algorithms,” J. Acoust. Soc. Am.  81, 1355–1376 (1987).
[CrossRef]

Winn, J.

J. D. Joannopoulos, S. Jhohnson, J. Winn, and R. Meade, Photonic Crystals—Modeling the Flow of Light (Princeton U. Press, 2008).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1959).

Xu, X.

X. Xu, “Matrix differential-operator approach to the Maxwell equations and the Dirac equation,” Acta Appl. Math.  102, 237–247 (2008).
[CrossRef]

Zaitsev, O.

M. Pletyukhov and O. Zaitsev, J. Phys. A  36, 5181–5219 (2003).
[CrossRef]

Acta Appl. Math. (1)

X. Xu, “Matrix differential-operator approach to the Maxwell equations and the Dirac equation,” Acta Appl. Math.  102, 237–247 (2008).
[CrossRef]

Acta Physiol. Pol. (1)

I. Bialynicki-Birula, “On the wave function of the photon,” Acta Physiol. Pol.  86, 97–116 (1994).

IEEE Trans. Antennas Propag. (1)

R. D. Nevels, J. A. Miller, and R. E. Miller, “A path integral time domain method for electromagnetic scattering,” IEEE Trans. Antennas Propag.  48, 565–573 (2000).
[CrossRef]

J. Acoust. Soc. Am. (4)

B. D. Budaev and D. B. Bogy, “Probabilistic solutions of the Helmholtz equation,” J. Acoust. Soc. Am.  109, 2260–2262 (2001).
[CrossRef]

B. D. Budaev and D. B. Bogy, “Analysis of one dimensional wave scattering by the random walk method,” J. Acoust. Soc. Am.  111, 2555–2560 (2002).
[CrossRef]

L. Fishman, J. J. McCoy, and S. C. Wales, “Factorization and path integration of the Helmholtz equation: numerical algorithms,” J. Acoust. Soc. Am.  81, 1355–1376 (1987).
[CrossRef]

B. D. Budaev and D. B. Bogy, “Random walk approach to wave propagation in wedges and cones,” J. Acoust. Soc. Am.  114, 1733–1741 (2003).
[CrossRef]

J. Comput. Appl. Math. (1)

J. Lee and A. Fornberg, “A split step approach for the 3-D Maxwell’s equations,” J. Comput. Appl. Math.  158, 485–505 (2003).
[CrossRef]

J. Math. Phys. (2)

L. Fishman and J. J. McCoy, “Derivation and application of the extended parabolic wave theories. 1. The factorized Helmholtz equation,” J. Math. Phys.  25, 285–296 (1984).
[CrossRef]

L. Fishman and J. J. McCoy, “Derivation and application of the extended parabolic wave theories. 2. Path integral representation,” J. Math. Phys.  25, 297–308 (1984).
[CrossRef]

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

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

J. Phys. A (3)

M. Pletyukhov and O. Zaitsev, J. Phys. A  36, 5181–5219 (2003).
[CrossRef]

C. Amann and M. Brack, “Semiclassical theory of spin-orbit interaction in the extended phase space,” J. Phys. A  35, 6009–6032 (2002).
[CrossRef]

I. Mendas and P. Milutinovic, “Anticommutator analogues of certain identities involving repeated commutators,” J. Phys. A  23, 537–544 (1990).
[CrossRef]

Nature (1)

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature  386, 143–149 (1997).
[CrossRef]

Phys. Rev. D (2)

J. R. Klauder, “Path integrals and stationary-phase approximations,” Phys. Rev. D  19, 2349–2356 (1979).
[CrossRef]

I. Bialynicki-Birula, “Weyl, Dirac, and Maxwell equations on a lattice as unitary cellular automata,” Phys. Rev. D  49, 6920–6927 (1994).
[CrossRef]

Phys. Rev. E (1)

G. Samelsohn and R. Mazarr, “Path-integral analysis of scalar wave propagation in multiple-scattering random media,” Phys. Rev. E  54, 5697–5706 (1996).
[CrossRef]

Phys. Rev. Lett. (1)

M. Pletyukhov, C. Amann, M. Metha, and M. Brack, “Semiclassical theory of spin-orbit interactions using spin coherent states,” Phys. Rev. Lett.  89, 116601 (2002).
[CrossRef]

Proc. R. Soc. London, Ser. A (1)

M. Eve, “The use of path integrals in guided wave theory,” Proc. R. Soc. London, Ser. A  347, 405–417 (1976).
[CrossRef]

Q. J. Mech. Appl. Math. (1)

B. D. Budaev and D. B. Bogy, “Application of random walk methods to wave propagation,” Q. J. Mech. Appl. Math.  55, 209–226 (2002).
[CrossRef]

Other (23)

M. Freidlin, Functional Integration and Partial Differential Equations (Princeton U. Press, 1985).

B. Simon, Functional Integration and Quantum Physics, Vol.  86 of Pure and Applied Mathematics (Academic, 1979).

S. V. Shabanov, “Electromagnetic pulse propagation in passive media by path integral methods,” arXiv:math/0312296.

L. Fishman, “Helmholtz path integrals,” in Mathematical Modeling of Wave Phenomena, B.Nilsson and L.Fishman, eds., Vol.  834 of AIP Conference Proceedings (American Institute of Physics, 2006), pp. 25–55.

V. B. Berestezki, E. M. Lifshitz, and L. P. Pitaevski, Quantum Electrodynamics, 2nd ed. (Pergamon, 1982).

K. Gottfried, Quantum Mechanics (W. A. Benjamin, 1966).

A. Messiah, Quantum Mechanics (North-Holland, 1966), Vol.  II.

R. G. Levers and D. G. Gleaves, “Finding the roots of the acoustic wave equation,” Vol.  2 of Proceedings of the 11th IMACS World Congress on System Simulation and Scientific Computation, B.Wahlstrom, R.Henriksen, and N.P.Sundby, eds. (NFA, 1985), pp. 165–167.

R. G. Levers, “Spinning the Helmholtz equation,” in Computational Acoustics Wave Propagation, D.Lee, R.L.Sternberg, and M.H.Schultz, eds. (North-Holland, 1988), pp. 287–293.

T. Kashiwa, Y. Ohnuki, and M. Suzuki, Path Integral Methods (Oxford U. Press, 1997).

L. S. Schulman, Techniques and Application of Path Integration (Wiley, 1981).

C. Grosche and F. Steiner, Handbook of Feynman Path Integrals, Vol.  145 of Springer Tracts in Modern Physics (Springer-Verlag, 1998).

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1960).

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1959).

S. Keppeler, Spinning Particles—Semiclassics and Spectral Statistics (Springer, 2003).

M. Kline and I. W. Kay, Electromagnetic Theory and Geometrical Optics (Wiley, 1965).

S. P. Novikov and A. T. Fomenko, Basic Elements of Differential Geometry and Topology (Kluwer, 1990).

J. D. Joannopoulos, S. Jhohnson, J. Winn, and R. Meade, Photonic Crystals—Modeling the Flow of Light (Princeton U. Press, 2008).

A. Altland and B. Simons, Condensed Matter Field Theory (Cambridge U. Press, 2006).

H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 4th ed. (World Scientific, 2006).

A. Auerbach, Interacting Electrons and Quantum Magnetism (Springer-Verlag, 1994).

J. R. Klauder and B. S. Skagerstam, Coherent States: Applications in Physics and Mathematical Physics (World Scientific, 1985).

I. Bialynicki-Birula, “Photon wave function,” Progress in Optics XXXVI (1996), pp. 245–294.

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