The Angular Spectrum Representation of the Pulsed Radiation Field in Spatially and Temporally Dispersive Media

  • Kurt E. Oughstun
Part of the Springer Series in Optical Sciences book series (SSOS, volume 224)


Attention is now directed to the rigorous solution of the electromagnetic field that is radiated by a general current source in a homogeneous, anisotropic, locally linear, spatially and temporally dispersive medium occupying all of space that is characterized by the space and time-dependent dielectric permittivity response tensor \( \underline {\hat {\epsilon }}(\mathbf {r}-\mathbf {r}',t-t')\), magnetic permeability response tensor \( \underline {\hat {\mu }}(\mathbf {r}-\mathbf {r}',t-t')\), and electric conductivity response tensor \( \underline {\hat {\sigma }}(\mathbf {r}-\mathbf {r}',t-t')\), where (r, t′) denotes the space–time point of the inducing field stimulus and (r, t) the space–time point of the induced field response.


  1. 1.
    J. J. Stamnes, Radiation and Propagation of Light in Crystals. PhD thesis, The Institute of Optics, University of Rochester, Rochester, New York, 1974.Google Scholar
  2. 2.
    J. A. Stratton, Electromagnetic Theory. New York: McGraw-Hill, 1941.zbMATHGoogle Scholar
  3. 3.
    L. Lorenz, “On the identity of the vibrations of light with electrical currents,” Philos. Mag., vol. 34, pp. 287–301, 1867.CrossRefGoogle Scholar
  4. 4.
    E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable. London: Oxford University Press, 1935. p. 110.zbMATHGoogle Scholar
  5. 5.
    E. T. Whittaker and G. N. Watson, Modern Analysis. London: Cambridge University Press, fourth ed., 1963. Section 6.222.Google Scholar
  6. 6.
    E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable. London: Oxford University Press, 1935. Section  6.1.
  7. 7.
    H. B. Phillips, Vector Analysis. New York: John Wiley & Sons, 1933.zbMATHGoogle Scholar
  8. 8.
    H. Weyl, “Ausbreitung elektromagnetischer Wellen über einem ebenen Leiter,” Ann. Physik (Leipzig), vol. 60, pp. 481–500, 1919.ADSCrossRefGoogle Scholar
  9. 9.
    E. T. Whittaker, “x,” Math. Ann., vol. 57, pp. 333–355, 1902.CrossRefGoogle Scholar
  10. 10.
    G. C. Sherman, A. J. Devaney, and L. Mandel, “Plane-wave expansions of the optical field,” Opt. Commun., vol. 6, pp. 115–118, 1972.ADSCrossRefGoogle Scholar
  11. 11.
    G. C. Sherman, J. J. Stamnes, A. J. Devaney, and É. Lalor, “Contribution of the inhomogeneous waves in angular-spectrum representations,” Opt. Commun., vol. 8, pp. 271–274, 1973.ADSCrossRefGoogle Scholar
  12. 12.
    A. J. Devaney and G. C. Sherman, “Plane-wave representations for scalar wave fields,” SIAM Rev., vol. 15, pp. 765–786, 1973.MathSciNetCrossRefGoogle Scholar
  13. 13.
    P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields. Oxford: Pergamon, 1966.zbMATHGoogle Scholar
  14. 14.
    J. J. Stamnes, Waves in Focal Regions: Propagation, Diffraction and Focusing of Light, Sound and Water Waves. Bristol, UK: Adam Hilger, 1986.zbMATHGoogle Scholar
  15. 15.
    M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics. New York: Wiley-Interscience, 1991.zbMATHGoogle Scholar
  16. 16.
    T. B. Hansen and A. D. Yaghjian, Plane-Wave Theory of Time-Domain Fields. New York: IEEE, 1999.CrossRefGoogle Scholar
  17. 17.
    A. J. Devaney, Mathematical Foundations of Imaging, Tomography and Wavefield Inversion. Cambridge: Cambridge University Press, 2012.CrossRefGoogle Scholar
  18. 18.
    G. N. Watson, A Treatise on the Theory of Bessel Functions. Cambridge: Cambridge University Press, second ed., 1958. Sect. 3.62, Eq. (5).Google Scholar
  19. 19.
    A. Sommerfeld, “Über die Ausbreitung der Wellen in der drahtlosen Telegraphie,” Ann. Phys. (Leipzig), vol. 28, pp. 665–737, 1909.ADSCrossRefGoogle Scholar
  20. 20.
    A. Baños, Dipole Radiation in the Presence of a Conducting Half-Space. Oxford: Pergamon, 1966. Sect. 2.12.Google Scholar
  21. 21.
    H. Ott, “Reflexion und Brechung von Kugelwellen; Effekte 2. Ordnung,” Ann. Phys., vol. 41, pp. 443–467, 1942.MathSciNetCrossRefGoogle Scholar
  22. 22.
    W. Heitler, The Quantum Theory of Radiation. Oxford: Clarendon Press, 1954.zbMATHGoogle Scholar
  23. 23.
    L. Mandel and E. Wolf, Optical Coherence and Quantum Optics. Cambridge: Cambridge University Press, 1995.CrossRefGoogle Scholar
  24. 24.
    R. F. A. Clebsch, “Über die Reflexionan einer Kugelfläche,” J. Reine Angew. Math., vol. 61, pp. 195–262, 1863.MathSciNetCrossRefGoogle Scholar
  25. 25.
    G. Mie, “Beiträge zur Optik truber Medien, speziell kollaidaler Metallosungen,” Ann. Phys. (Leipzig), vol. 25, pp. 377–452, 1908.ADSCrossRefGoogle Scholar
  26. 26.
    P. Debye, “Der lichtdruck auf Kugeln von beliegigem Material,” Ann. Phys. (Leipzig), vol. 30, pp. 57–136, 1909.ADSCrossRefGoogle Scholar
  27. 27.
    T. J. I. Bromwich, “The scattering of plane electric waves by spheres,” Phil. Trans. Roy. Soc. Lond., vol. 220, p. 175, 1920.ADSCrossRefGoogle Scholar
  28. 28.
    E. T. Whittaker, “On the partial differential equations of mathematical physics,” Math. Ann., vol. 57, pp. 333–355, 1903.MathSciNetCrossRefGoogle Scholar
  29. 29.
    A. J. Devaney and E. Wolf, “Multipole expansions and plane wave representations of the electromagnetic field,” J. Math. Phys., vol. 15, no. 11, p. 234, 1974.ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    A. J. Devaney, A New Theory of the Debye Representation of Classical and Quantized Electromagnetic Fields. PhD thesis, The Institute of Optics, University of Rochester, 1971.Google Scholar
  31. 31.
    B. A. Fuks, Introduction to the Theory of Analytic Functions of Several Complex Variables. Providence: Amer. Math. Soc., 1963.CrossRefGoogle Scholar
  32. 32.
    A. Erdélyi, “Zur Theorie der Kugelwellen,” Physica (The Hague), vol. 4, pp. 107–120, 1937.ADSCrossRefGoogle Scholar
  33. 33.
    E. L. Hill, “The theory of vector spherical harmonics,” Am. J. Phys., vol. 22, pp. 211–214, 1954.ADSMathSciNetCrossRefGoogle Scholar
  34. 34.
    J. D. Jackson, Classical Electrodynamics. New York: John Wiley & Sons, third ed., 1999.zbMATHGoogle Scholar
  35. 35.
    R. S. Beezley and R. J. Krueger, “An electromagnetic inverse problem for dispersive media,” J. Math. Phys., vol. 26, no. 2, pp. 317–325, 1985.ADSMathSciNetCrossRefGoogle Scholar
  36. 36.
    G. Kristensson and R. J. Krueger, “Direct and inverse scattering in the time domain for a dissipative wave equation. I. Scattering operators,” J. Math. Phys., vol. 27, no. 6, pp. 1667–1682, 1986.ADSMathSciNetCrossRefGoogle Scholar
  37. 37.
    G. Kristensson and R. J. Krueger, “Direct and inverse scattering in the time domain for a dissipative wave equation. II. Simultaneous reconstruction of dissipation and phase velocity profiles,” J. Math. Phys., vol. 27, no. 6, pp. 1683–1693, 1986.ADSMathSciNetCrossRefGoogle Scholar
  38. 38.
    J. P. Corones, M. E. Davison, and R. J. Krueger, “Direct and inverse scattering in the time domain via invariant embedding equations,” J. Acoustic Soc. Am., vol. 74, pp. 1535–1541, 1983.ADSCrossRefGoogle Scholar
  39. 39.
    G. Kristensson and R. J. Krueger, “Direct and inverse scattering in the time domain for a dissipative wave equation. Part III. Scattering operators in the presence of a phase velocity mismatch,” J. Math. Phys., vol. 28, pp. 360–370, 1987.ADSMathSciNetCrossRefGoogle Scholar
  40. 40.
    H. E. Moses and R. T. Prosser, “Initial conditions, sources, and currents for prescribed time-dependent acoustic and electromagnetic fields in three dimensions,” IEEE Trans. Antennas Prop., vol. 24, no. 2, pp. 188–196, 1986.ADSCrossRefGoogle Scholar
  41. 41.
    H. E. Moses and R. T. Prosser, “Exact solutions of the three-dimensional scalar wave equation and Maxwell’s equations from the approximate solutions in the wave zone through the use of the Radon transform,” Proc. Roy. Soc. Lond. A, vol. 422, pp. 351–365, 1989.ADSMathSciNetCrossRefGoogle Scholar
  42. 42.
    H. E. Moses and R. T. Prosser, “Acoustic and electromagnetic bullets: Derivation of new exact solutions of the acoustic and Maxwell’s equations,” SIAM J. Appl. Math., vol. 50, no. 5, pp. 1325–1340, 1990.ADSMathSciNetCrossRefGoogle Scholar
  43. 43.
    H. E. Moses and R. T. Prosser, “The general solution of the time-dependent Maxwell’s equations in an infinite medium with constant conductivity,” Proc. Roy. Soc. Lond. A, vol. 431, pp. 493–507, 1990.ADSMathSciNetCrossRefGoogle Scholar
  44. 44.
    R. W. P. King and T. T. Wu, “The propagation of a radar pulse in sea water,” J. Appl. Phys., vol. 73, no. 4, pp. 1581–1590, 1993.ADSCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Kurt E. Oughstun
    • 1
  1. 1.College of Engineering and Mathematical Sciences, University of VermontBurlingtonUSA

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