Abstract

The propagation of an electromagnetic plane-wave signal obliquely incident upon a Lorentz half-space is studied analytically. Time-domain asymptotic expressions that increase in accuracy with propagation distance are derived by application of uniform saddle point methods on the Fourier–Laplace integral representation of the transmitted field. The results are shown to be continuous in time and comparable with numerical calculations of the field. Arrival times and angles of refraction are given for prominent transient pulse features and the steady-state signal.

© 2011 Optical Society of America

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  1. A. Sommerfeld, “Über die Fortpflanzung des Lichtes in dispergierenden Medien,” Ann. Phys. 44, 177–202 (1914).
    [CrossRef]
  2. L. Brillouin, “Über die Fortpflanzung des Lichtes in dispergierenden Medien,” Ann. Phys. 44, 203–240 (1914).
    [CrossRef]
  3. L. Brillouin, Wave Propagation and Group Velocity (Academic, 1960).
  4. N. A. Cartwright and K. E. Oughstun, “Uniform asymptotics applied to ultrawideband pulse propagation,” SIAM Rev. 49, 628–648 (2007).
    [CrossRef]
  5. K. E. Oughstun, “Dynamical evolution of the Brillouin precursor in Rocard-Powles-Debye model dielectrics,” IEEE Trans. Antennas Propag. 53, 1582–1590 (2005).
    [CrossRef]
  6. N. A. Cartwright and K. E. Oughstun, “Ultrawideband pulse propagation through a lossy plasma,” Radio Sci. 44, RS4013 (2009).
    [CrossRef]
  7. M. Tanaka, M. Fujiwara, and H. Ikegami, “Propagation of a Gaussian wave packet in an absorbing medium,” Phys. Rev. A 34, 4851–4858 (1986).
    [CrossRef] [PubMed]
  8. C. M. Balictsis and K. E. Oughstun, “Generalized asymptotic description of the propagated field dynamics in Gaussian pulse propagation in a linear, causally dispersive medium,” Phys. Rev. E 55, 1910–1921 (1997).
    [CrossRef]
  9. J. Laurens and K. Oughstun, “Electromagnetic impulse response of triply-distilled water,” in Ultra-Wideband, Short-Pulse Electromagnetics 4, J.Shiloh, B.Mandelbaum, and E.Heyman, eds. (Springer, 1999), pp. 243–264.
  10. S. Dvorak and D. Dudley, “Propagation of ultra-wideband electromagnetic pulses through dispersive media,” IEEE Trans. Electromagn. Compat. 37, 192–200 (1995).
    [CrossRef]
  11. W. Colby, “Signal propagation in dispersive media,” Phys. Rev. 5, 253–265 (1915).
    [CrossRef]
  12. E. G. Skrotskaya, A. N. Makhlin, V. A. Kashin, and G. V. Skrotskii, “Formation of a forerunner in the passage of the front of a light pulse through a vacuum-medium interface,” Sov. Phys. JETP 29, 123–125 (1969).
  13. D. G. Dudley, T. M. Papazoglou, and R. C. White, “On the interaction of a transient electromagnetic plane wave and a lossy half-space,” J. Appl. Phys. 45, 1171–1175 (1974).
    [CrossRef]
  14. T. M. Papazoglou, “Transmission of a transient electromagnetic plane wave into a lossy half-space,” J. Appl. Phys. 46, 3333–3341(1975).
    [CrossRef]
  15. H. Pao, S. L. Dvorak, and D. G. Dudley, “An accurate and efficient analysis for transient plane waves obliquely incident on a conductive half space (TE case),” IEEE Trans. Antennas Propag. 44, 918–924 (1996).
    [CrossRef]
  16. H. Pao, S. L. Dvorak, and D. G. Dudley, “An accurate and efficient analysis for transient plane waves obliquely incident on a conductive half space (TM case),” IEEE Trans. Antennas Propag. 44, 925–932 (1996).
    [CrossRef]
  17. J. B. Ehrman and J. LoVetri, “Time-domain electromagnetic plane waves in static and dynamic conducting media II,” IEEE Trans. Electromagn. Compat. 37, 17–25 (1995).
    [CrossRef]
  18. A. M. Attiya, E. A. El-Diwany, and A. M. Shaarawi, “Transmission and reflection of TE electromagnetic X-wave normally incident on a lossy dispersive half-space,” in Proceedings of the Seventeenth National Radio Science Conference (2000), pp. B11.1–B11.12.
  19. E. Gitterman and M. Gitterman, “Transient processes for incidence of a light signal on a vacuum-medium interface,” Phys. Rev. A 13, 763–776 (1976).
    [CrossRef]
  20. E. L. Mokole and S. N. Samaddar, “Transmission and reflection of normally incident, pulsed electromagnetic plane waves upon a Lorentz half-space,” J. Opt. Soc. Am. B 16, 812–831(1999).
    [CrossRef]
  21. K. E. Oughstun and G. C. Sherman, Pulse Propagation in Causal Dielectrics (Springer-Verlag, 1994).
  22. J. A. Marozas, “Angular spectrum representation of ultrawideband electromagnetic pulse propagation in lossy, dispersive dielectric slab waveguides,” Ph.D. dissertation (University of Vermont, 1998).
  23. K. E. Oughstun, Electromagnetic and Optical Pulse Propagation 2 (Springer, 2006).
  24. R. A. Handelsman and N. Bleistein, “Uniform asymptotic expansions of integrals that arise in the analysis of precursors,” Arch. Ration. Mech. Anal. 35, 267–283 (1969).
    [CrossRef]
  25. C. Chester, B. Friedman, and F. Ursell, “An extension of the method of steepest descents,” Proc. Cambridge Philos. Soc. 53, 599–611 (1957).
    [CrossRef]
  26. L. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, 1973).
  27. Empirical evidence suggests that the relative error between asymptotic and numerical results is less than 10% when the pulse has propagated at least one absorption depth into the material, and that this error decreases with propagation distance.
  28. Here, we adjusted the power of γ1 from 4 to 2 in the expression for v1 given in , as well as dividing Eqs. (21) and (22) of by 2π in order to obtain better accuracy.
  29. J. G. Blaschak and J. Franzen, “Precursor propagation in dispersive media from short-rise-time pulses at oblique incidence,” J. Opt. Soc. Am. B 12, 1501–1512 (1995).
    [CrossRef]
  30. K. E. Oughstun and G. C. Sherman, “Comparison of the signal velocity of a pulse with the energy velocity of a time-harmonic field in Lorentz media,” in Proceedings of the URSI Symposium on Electromagnetic Wave Theory (1980), pp. C1–C5.

2009 (1)

N. A. Cartwright and K. E. Oughstun, “Ultrawideband pulse propagation through a lossy plasma,” Radio Sci. 44, RS4013 (2009).
[CrossRef]

2007 (1)

N. A. Cartwright and K. E. Oughstun, “Uniform asymptotics applied to ultrawideband pulse propagation,” SIAM Rev. 49, 628–648 (2007).
[CrossRef]

2006 (1)

K. E. Oughstun, Electromagnetic and Optical Pulse Propagation 2 (Springer, 2006).

2005 (1)

K. E. Oughstun, “Dynamical evolution of the Brillouin precursor in Rocard-Powles-Debye model dielectrics,” IEEE Trans. Antennas Propag. 53, 1582–1590 (2005).
[CrossRef]

2000 (1)

A. M. Attiya, E. A. El-Diwany, and A. M. Shaarawi, “Transmission and reflection of TE electromagnetic X-wave normally incident on a lossy dispersive half-space,” in Proceedings of the Seventeenth National Radio Science Conference (2000), pp. B11.1–B11.12.

1999 (2)

J. Laurens and K. Oughstun, “Electromagnetic impulse response of triply-distilled water,” in Ultra-Wideband, Short-Pulse Electromagnetics 4, J.Shiloh, B.Mandelbaum, and E.Heyman, eds. (Springer, 1999), pp. 243–264.

E. L. Mokole and S. N. Samaddar, “Transmission and reflection of normally incident, pulsed electromagnetic plane waves upon a Lorentz half-space,” J. Opt. Soc. Am. B 16, 812–831(1999).
[CrossRef]

1998 (1)

J. A. Marozas, “Angular spectrum representation of ultrawideband electromagnetic pulse propagation in lossy, dispersive dielectric slab waveguides,” Ph.D. dissertation (University of Vermont, 1998).

1997 (1)

C. M. Balictsis and K. E. Oughstun, “Generalized asymptotic description of the propagated field dynamics in Gaussian pulse propagation in a linear, causally dispersive medium,” Phys. Rev. E 55, 1910–1921 (1997).
[CrossRef]

1996 (2)

H. Pao, S. L. Dvorak, and D. G. Dudley, “An accurate and efficient analysis for transient plane waves obliquely incident on a conductive half space (TE case),” IEEE Trans. Antennas Propag. 44, 918–924 (1996).
[CrossRef]

H. Pao, S. L. Dvorak, and D. G. Dudley, “An accurate and efficient analysis for transient plane waves obliquely incident on a conductive half space (TM case),” IEEE Trans. Antennas Propag. 44, 925–932 (1996).
[CrossRef]

1995 (3)

J. B. Ehrman and J. LoVetri, “Time-domain electromagnetic plane waves in static and dynamic conducting media II,” IEEE Trans. Electromagn. Compat. 37, 17–25 (1995).
[CrossRef]

S. Dvorak and D. Dudley, “Propagation of ultra-wideband electromagnetic pulses through dispersive media,” IEEE Trans. Electromagn. Compat. 37, 192–200 (1995).
[CrossRef]

J. G. Blaschak and J. Franzen, “Precursor propagation in dispersive media from short-rise-time pulses at oblique incidence,” J. Opt. Soc. Am. B 12, 1501–1512 (1995).
[CrossRef]

1994 (1)

K. E. Oughstun and G. C. Sherman, Pulse Propagation in Causal Dielectrics (Springer-Verlag, 1994).

1986 (1)

M. Tanaka, M. Fujiwara, and H. Ikegami, “Propagation of a Gaussian wave packet in an absorbing medium,” Phys. Rev. A 34, 4851–4858 (1986).
[CrossRef] [PubMed]

1980 (1)

K. E. Oughstun and G. C. Sherman, “Comparison of the signal velocity of a pulse with the energy velocity of a time-harmonic field in Lorentz media,” in Proceedings of the URSI Symposium on Electromagnetic Wave Theory (1980), pp. C1–C5.

1976 (1)

E. Gitterman and M. Gitterman, “Transient processes for incidence of a light signal on a vacuum-medium interface,” Phys. Rev. A 13, 763–776 (1976).
[CrossRef]

1975 (1)

T. M. Papazoglou, “Transmission of a transient electromagnetic plane wave into a lossy half-space,” J. Appl. Phys. 46, 3333–3341(1975).
[CrossRef]

1974 (1)

D. G. Dudley, T. M. Papazoglou, and R. C. White, “On the interaction of a transient electromagnetic plane wave and a lossy half-space,” J. Appl. Phys. 45, 1171–1175 (1974).
[CrossRef]

1973 (1)

L. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, 1973).

1969 (2)

R. A. Handelsman and N. Bleistein, “Uniform asymptotic expansions of integrals that arise in the analysis of precursors,” Arch. Ration. Mech. Anal. 35, 267–283 (1969).
[CrossRef]

E. G. Skrotskaya, A. N. Makhlin, V. A. Kashin, and G. V. Skrotskii, “Formation of a forerunner in the passage of the front of a light pulse through a vacuum-medium interface,” Sov. Phys. JETP 29, 123–125 (1969).

1960 (1)

L. Brillouin, Wave Propagation and Group Velocity (Academic, 1960).

1957 (1)

C. Chester, B. Friedman, and F. Ursell, “An extension of the method of steepest descents,” Proc. Cambridge Philos. Soc. 53, 599–611 (1957).
[CrossRef]

1915 (1)

W. Colby, “Signal propagation in dispersive media,” Phys. Rev. 5, 253–265 (1915).
[CrossRef]

1914 (2)

A. Sommerfeld, “Über die Fortpflanzung des Lichtes in dispergierenden Medien,” Ann. Phys. 44, 177–202 (1914).
[CrossRef]

L. Brillouin, “Über die Fortpflanzung des Lichtes in dispergierenden Medien,” Ann. Phys. 44, 203–240 (1914).
[CrossRef]

Attiya, A. M.

A. M. Attiya, E. A. El-Diwany, and A. M. Shaarawi, “Transmission and reflection of TE electromagnetic X-wave normally incident on a lossy dispersive half-space,” in Proceedings of the Seventeenth National Radio Science Conference (2000), pp. B11.1–B11.12.

Balictsis, C. M.

C. M. Balictsis and K. E. Oughstun, “Generalized asymptotic description of the propagated field dynamics in Gaussian pulse propagation in a linear, causally dispersive medium,” Phys. Rev. E 55, 1910–1921 (1997).
[CrossRef]

Blaschak, J. G.

J. G. Blaschak and J. Franzen, “Precursor propagation in dispersive media from short-rise-time pulses at oblique incidence,” J. Opt. Soc. Am. B 12, 1501–1512 (1995).
[CrossRef]

Bleistein, N.

R. A. Handelsman and N. Bleistein, “Uniform asymptotic expansions of integrals that arise in the analysis of precursors,” Arch. Ration. Mech. Anal. 35, 267–283 (1969).
[CrossRef]

Brillouin, L.

L. Brillouin, Wave Propagation and Group Velocity (Academic, 1960).

L. Brillouin, “Über die Fortpflanzung des Lichtes in dispergierenden Medien,” Ann. Phys. 44, 203–240 (1914).
[CrossRef]

Cartwright, N. A.

N. A. Cartwright and K. E. Oughstun, “Ultrawideband pulse propagation through a lossy plasma,” Radio Sci. 44, RS4013 (2009).
[CrossRef]

N. A. Cartwright and K. E. Oughstun, “Uniform asymptotics applied to ultrawideband pulse propagation,” SIAM Rev. 49, 628–648 (2007).
[CrossRef]

Chester, C.

C. Chester, B. Friedman, and F. Ursell, “An extension of the method of steepest descents,” Proc. Cambridge Philos. Soc. 53, 599–611 (1957).
[CrossRef]

Colby, W.

W. Colby, “Signal propagation in dispersive media,” Phys. Rev. 5, 253–265 (1915).
[CrossRef]

Dudley, D.

S. Dvorak and D. Dudley, “Propagation of ultra-wideband electromagnetic pulses through dispersive media,” IEEE Trans. Electromagn. Compat. 37, 192–200 (1995).
[CrossRef]

Dudley, D. G.

H. Pao, S. L. Dvorak, and D. G. Dudley, “An accurate and efficient analysis for transient plane waves obliquely incident on a conductive half space (TE case),” IEEE Trans. Antennas Propag. 44, 918–924 (1996).
[CrossRef]

H. Pao, S. L. Dvorak, and D. G. Dudley, “An accurate and efficient analysis for transient plane waves obliquely incident on a conductive half space (TM case),” IEEE Trans. Antennas Propag. 44, 925–932 (1996).
[CrossRef]

D. G. Dudley, T. M. Papazoglou, and R. C. White, “On the interaction of a transient electromagnetic plane wave and a lossy half-space,” J. Appl. Phys. 45, 1171–1175 (1974).
[CrossRef]

Dvorak, S.

S. Dvorak and D. Dudley, “Propagation of ultra-wideband electromagnetic pulses through dispersive media,” IEEE Trans. Electromagn. Compat. 37, 192–200 (1995).
[CrossRef]

Dvorak, S. L.

H. Pao, S. L. Dvorak, and D. G. Dudley, “An accurate and efficient analysis for transient plane waves obliquely incident on a conductive half space (TE case),” IEEE Trans. Antennas Propag. 44, 918–924 (1996).
[CrossRef]

H. Pao, S. L. Dvorak, and D. G. Dudley, “An accurate and efficient analysis for transient plane waves obliquely incident on a conductive half space (TM case),” IEEE Trans. Antennas Propag. 44, 925–932 (1996).
[CrossRef]

Ehrman, J. B.

J. B. Ehrman and J. LoVetri, “Time-domain electromagnetic plane waves in static and dynamic conducting media II,” IEEE Trans. Electromagn. Compat. 37, 17–25 (1995).
[CrossRef]

El-Diwany, E. A.

A. M. Attiya, E. A. El-Diwany, and A. M. Shaarawi, “Transmission and reflection of TE electromagnetic X-wave normally incident on a lossy dispersive half-space,” in Proceedings of the Seventeenth National Radio Science Conference (2000), pp. B11.1–B11.12.

Felsen, L.

L. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, 1973).

Franzen, J.

J. G. Blaschak and J. Franzen, “Precursor propagation in dispersive media from short-rise-time pulses at oblique incidence,” J. Opt. Soc. Am. B 12, 1501–1512 (1995).
[CrossRef]

Friedman, B.

C. Chester, B. Friedman, and F. Ursell, “An extension of the method of steepest descents,” Proc. Cambridge Philos. Soc. 53, 599–611 (1957).
[CrossRef]

Fujiwara, M.

M. Tanaka, M. Fujiwara, and H. Ikegami, “Propagation of a Gaussian wave packet in an absorbing medium,” Phys. Rev. A 34, 4851–4858 (1986).
[CrossRef] [PubMed]

Gitterman, E.

E. Gitterman and M. Gitterman, “Transient processes for incidence of a light signal on a vacuum-medium interface,” Phys. Rev. A 13, 763–776 (1976).
[CrossRef]

Gitterman, M.

E. Gitterman and M. Gitterman, “Transient processes for incidence of a light signal on a vacuum-medium interface,” Phys. Rev. A 13, 763–776 (1976).
[CrossRef]

Handelsman, R. A.

R. A. Handelsman and N. Bleistein, “Uniform asymptotic expansions of integrals that arise in the analysis of precursors,” Arch. Ration. Mech. Anal. 35, 267–283 (1969).
[CrossRef]

Ikegami, H.

M. Tanaka, M. Fujiwara, and H. Ikegami, “Propagation of a Gaussian wave packet in an absorbing medium,” Phys. Rev. A 34, 4851–4858 (1986).
[CrossRef] [PubMed]

Kashin, V. A.

E. G. Skrotskaya, A. N. Makhlin, V. A. Kashin, and G. V. Skrotskii, “Formation of a forerunner in the passage of the front of a light pulse through a vacuum-medium interface,” Sov. Phys. JETP 29, 123–125 (1969).

Laurens, J.

J. Laurens and K. Oughstun, “Electromagnetic impulse response of triply-distilled water,” in Ultra-Wideband, Short-Pulse Electromagnetics 4, J.Shiloh, B.Mandelbaum, and E.Heyman, eds. (Springer, 1999), pp. 243–264.

LoVetri, J.

J. B. Ehrman and J. LoVetri, “Time-domain electromagnetic plane waves in static and dynamic conducting media II,” IEEE Trans. Electromagn. Compat. 37, 17–25 (1995).
[CrossRef]

Makhlin, A. N.

E. G. Skrotskaya, A. N. Makhlin, V. A. Kashin, and G. V. Skrotskii, “Formation of a forerunner in the passage of the front of a light pulse through a vacuum-medium interface,” Sov. Phys. JETP 29, 123–125 (1969).

Marcuvitz, N.

L. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, 1973).

Marozas, J. A.

J. A. Marozas, “Angular spectrum representation of ultrawideband electromagnetic pulse propagation in lossy, dispersive dielectric slab waveguides,” Ph.D. dissertation (University of Vermont, 1998).

Mokole, E. L.

Oughstun, K.

J. Laurens and K. Oughstun, “Electromagnetic impulse response of triply-distilled water,” in Ultra-Wideband, Short-Pulse Electromagnetics 4, J.Shiloh, B.Mandelbaum, and E.Heyman, eds. (Springer, 1999), pp. 243–264.

Oughstun, K. E.

N. A. Cartwright and K. E. Oughstun, “Ultrawideband pulse propagation through a lossy plasma,” Radio Sci. 44, RS4013 (2009).
[CrossRef]

N. A. Cartwright and K. E. Oughstun, “Uniform asymptotics applied to ultrawideband pulse propagation,” SIAM Rev. 49, 628–648 (2007).
[CrossRef]

K. E. Oughstun, Electromagnetic and Optical Pulse Propagation 2 (Springer, 2006).

K. E. Oughstun, “Dynamical evolution of the Brillouin precursor in Rocard-Powles-Debye model dielectrics,” IEEE Trans. Antennas Propag. 53, 1582–1590 (2005).
[CrossRef]

C. M. Balictsis and K. E. Oughstun, “Generalized asymptotic description of the propagated field dynamics in Gaussian pulse propagation in a linear, causally dispersive medium,” Phys. Rev. E 55, 1910–1921 (1997).
[CrossRef]

K. E. Oughstun and G. C. Sherman, Pulse Propagation in Causal Dielectrics (Springer-Verlag, 1994).

K. E. Oughstun and G. C. Sherman, “Comparison of the signal velocity of a pulse with the energy velocity of a time-harmonic field in Lorentz media,” in Proceedings of the URSI Symposium on Electromagnetic Wave Theory (1980), pp. C1–C5.

Pao, H.

H. Pao, S. L. Dvorak, and D. G. Dudley, “An accurate and efficient analysis for transient plane waves obliquely incident on a conductive half space (TE case),” IEEE Trans. Antennas Propag. 44, 918–924 (1996).
[CrossRef]

H. Pao, S. L. Dvorak, and D. G. Dudley, “An accurate and efficient analysis for transient plane waves obliquely incident on a conductive half space (TM case),” IEEE Trans. Antennas Propag. 44, 925–932 (1996).
[CrossRef]

Papazoglou, T. M.

T. M. Papazoglou, “Transmission of a transient electromagnetic plane wave into a lossy half-space,” J. Appl. Phys. 46, 3333–3341(1975).
[CrossRef]

D. G. Dudley, T. M. Papazoglou, and R. C. White, “On the interaction of a transient electromagnetic plane wave and a lossy half-space,” J. Appl. Phys. 45, 1171–1175 (1974).
[CrossRef]

Samaddar, S. N.

Shaarawi, A. M.

A. M. Attiya, E. A. El-Diwany, and A. M. Shaarawi, “Transmission and reflection of TE electromagnetic X-wave normally incident on a lossy dispersive half-space,” in Proceedings of the Seventeenth National Radio Science Conference (2000), pp. B11.1–B11.12.

Sherman, G. C.

K. E. Oughstun and G. C. Sherman, Pulse Propagation in Causal Dielectrics (Springer-Verlag, 1994).

K. E. Oughstun and G. C. Sherman, “Comparison of the signal velocity of a pulse with the energy velocity of a time-harmonic field in Lorentz media,” in Proceedings of the URSI Symposium on Electromagnetic Wave Theory (1980), pp. C1–C5.

Skrotskaya, E. G.

E. G. Skrotskaya, A. N. Makhlin, V. A. Kashin, and G. V. Skrotskii, “Formation of a forerunner in the passage of the front of a light pulse through a vacuum-medium interface,” Sov. Phys. JETP 29, 123–125 (1969).

Skrotskii, G. V.

E. G. Skrotskaya, A. N. Makhlin, V. A. Kashin, and G. V. Skrotskii, “Formation of a forerunner in the passage of the front of a light pulse through a vacuum-medium interface,” Sov. Phys. JETP 29, 123–125 (1969).

Sommerfeld, A.

A. Sommerfeld, “Über die Fortpflanzung des Lichtes in dispergierenden Medien,” Ann. Phys. 44, 177–202 (1914).
[CrossRef]

Tanaka, M.

M. Tanaka, M. Fujiwara, and H. Ikegami, “Propagation of a Gaussian wave packet in an absorbing medium,” Phys. Rev. A 34, 4851–4858 (1986).
[CrossRef] [PubMed]

Ursell, F.

C. Chester, B. Friedman, and F. Ursell, “An extension of the method of steepest descents,” Proc. Cambridge Philos. Soc. 53, 599–611 (1957).
[CrossRef]

White, R. C.

D. G. Dudley, T. M. Papazoglou, and R. C. White, “On the interaction of a transient electromagnetic plane wave and a lossy half-space,” J. Appl. Phys. 45, 1171–1175 (1974).
[CrossRef]

Ann. Phys. (2)

A. Sommerfeld, “Über die Fortpflanzung des Lichtes in dispergierenden Medien,” Ann. Phys. 44, 177–202 (1914).
[CrossRef]

L. Brillouin, “Über die Fortpflanzung des Lichtes in dispergierenden Medien,” Ann. Phys. 44, 203–240 (1914).
[CrossRef]

Arch. Ration. Mech. Anal. (1)

R. A. Handelsman and N. Bleistein, “Uniform asymptotic expansions of integrals that arise in the analysis of precursors,” Arch. Ration. Mech. Anal. 35, 267–283 (1969).
[CrossRef]

IEEE Trans. Antennas Propag. (3)

K. E. Oughstun, “Dynamical evolution of the Brillouin precursor in Rocard-Powles-Debye model dielectrics,” IEEE Trans. Antennas Propag. 53, 1582–1590 (2005).
[CrossRef]

H. Pao, S. L. Dvorak, and D. G. Dudley, “An accurate and efficient analysis for transient plane waves obliquely incident on a conductive half space (TE case),” IEEE Trans. Antennas Propag. 44, 918–924 (1996).
[CrossRef]

H. Pao, S. L. Dvorak, and D. G. Dudley, “An accurate and efficient analysis for transient plane waves obliquely incident on a conductive half space (TM case),” IEEE Trans. Antennas Propag. 44, 925–932 (1996).
[CrossRef]

IEEE Trans. Electromagn. Compat. (2)

J. B. Ehrman and J. LoVetri, “Time-domain electromagnetic plane waves in static and dynamic conducting media II,” IEEE Trans. Electromagn. Compat. 37, 17–25 (1995).
[CrossRef]

S. Dvorak and D. Dudley, “Propagation of ultra-wideband electromagnetic pulses through dispersive media,” IEEE Trans. Electromagn. Compat. 37, 192–200 (1995).
[CrossRef]

J. Appl. Phys. (2)

D. G. Dudley, T. M. Papazoglou, and R. C. White, “On the interaction of a transient electromagnetic plane wave and a lossy half-space,” J. Appl. Phys. 45, 1171–1175 (1974).
[CrossRef]

T. M. Papazoglou, “Transmission of a transient electromagnetic plane wave into a lossy half-space,” J. Appl. Phys. 46, 3333–3341(1975).
[CrossRef]

J. Opt. Soc. Am. B (2)

J. G. Blaschak and J. Franzen, “Precursor propagation in dispersive media from short-rise-time pulses at oblique incidence,” J. Opt. Soc. Am. B 12, 1501–1512 (1995).
[CrossRef]

E. L. Mokole and S. N. Samaddar, “Transmission and reflection of normally incident, pulsed electromagnetic plane waves upon a Lorentz half-space,” J. Opt. Soc. Am. B 16, 812–831(1999).
[CrossRef]

Phys. Rev. (1)

W. Colby, “Signal propagation in dispersive media,” Phys. Rev. 5, 253–265 (1915).
[CrossRef]

Phys. Rev. A (2)

M. Tanaka, M. Fujiwara, and H. Ikegami, “Propagation of a Gaussian wave packet in an absorbing medium,” Phys. Rev. A 34, 4851–4858 (1986).
[CrossRef] [PubMed]

E. Gitterman and M. Gitterman, “Transient processes for incidence of a light signal on a vacuum-medium interface,” Phys. Rev. A 13, 763–776 (1976).
[CrossRef]

Phys. Rev. E (1)

C. M. Balictsis and K. E. Oughstun, “Generalized asymptotic description of the propagated field dynamics in Gaussian pulse propagation in a linear, causally dispersive medium,” Phys. Rev. E 55, 1910–1921 (1997).
[CrossRef]

Proc. Cambridge Philos. Soc. (1)

C. Chester, B. Friedman, and F. Ursell, “An extension of the method of steepest descents,” Proc. Cambridge Philos. Soc. 53, 599–611 (1957).
[CrossRef]

Radio Sci. (1)

N. A. Cartwright and K. E. Oughstun, “Ultrawideband pulse propagation through a lossy plasma,” Radio Sci. 44, RS4013 (2009).
[CrossRef]

SIAM Rev. (1)

N. A. Cartwright and K. E. Oughstun, “Uniform asymptotics applied to ultrawideband pulse propagation,” SIAM Rev. 49, 628–648 (2007).
[CrossRef]

Sov. Phys. JETP (1)

E. G. Skrotskaya, A. N. Makhlin, V. A. Kashin, and G. V. Skrotskii, “Formation of a forerunner in the passage of the front of a light pulse through a vacuum-medium interface,” Sov. Phys. JETP 29, 123–125 (1969).

Other (10)

J. Laurens and K. Oughstun, “Electromagnetic impulse response of triply-distilled water,” in Ultra-Wideband, Short-Pulse Electromagnetics 4, J.Shiloh, B.Mandelbaum, and E.Heyman, eds. (Springer, 1999), pp. 243–264.

L. Brillouin, Wave Propagation and Group Velocity (Academic, 1960).

K. E. Oughstun and G. C. Sherman, Pulse Propagation in Causal Dielectrics (Springer-Verlag, 1994).

J. A. Marozas, “Angular spectrum representation of ultrawideband electromagnetic pulse propagation in lossy, dispersive dielectric slab waveguides,” Ph.D. dissertation (University of Vermont, 1998).

K. E. Oughstun, Electromagnetic and Optical Pulse Propagation 2 (Springer, 2006).

A. M. Attiya, E. A. El-Diwany, and A. M. Shaarawi, “Transmission and reflection of TE electromagnetic X-wave normally incident on a lossy dispersive half-space,” in Proceedings of the Seventeenth National Radio Science Conference (2000), pp. B11.1–B11.12.

L. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, 1973).

Empirical evidence suggests that the relative error between asymptotic and numerical results is less than 10% when the pulse has propagated at least one absorption depth into the material, and that this error decreases with propagation distance.

Here, we adjusted the power of γ1 from 4 to 2 in the expression for v1 given in , as well as dividing Eqs. (21) and (22) of by 2π in order to obtain better accuracy.

K. E. Oughstun and G. C. Sherman, “Comparison of the signal velocity of a pulse with the energy velocity of a time-harmonic field in Lorentz media,” in Proceedings of the URSI Symposium on Electromagnetic Wave Theory (1980), pp. C1–C5.

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Figures (7)

Fig. 1
Fig. 1

Geometric illustration of one component of a pulsed, TE- polarized plane wave obliquely incident on a planar interface at z = 0 .

Fig. 2
Fig. 2

Contributions ϒ S , ϒ B , and ϒ c and the total transmitted field E y = ϒ S + ϒ B + ϒ c of the electric field component of the TE- polarized plane-wave pulse with α = 10 ° at the observation point ( x , z ) = ( 0.1 μm , 0.2 μm ) .

Fig. 3
Fig. 3

Magnetic field components (a)  H x and (b)  H z for the TE-polarized plane-wave pulse. The same material and pulse parameters and observation point used for Fig. 2 are used here.

Fig. 4
Fig. 4

Early-time behavior of E y for the TE case when α = 10 ° and ( x , z ) = ( 0.1 μm , 0.2 μm ) . Our results, the first three terms of Colby’s expansion, and the numerical solution are given by the dashed, dotted, and solid curves, respectively.

Fig. 5
Fig. 5

Evolution of E y as a function of θ. Our results, the results of [19], and the numerical solution are given by the dashed, dotted, and solid curves, respectively. The same material and pulse parameters and observation point used for Fig. 2 are used here.

Fig. 6
Fig. 6

Angle of refraction of the Sommerfeld precursor α T S , the Brillouin precursor α T B , and the signal contribution α T c as a function of θ. The open circle represents the angle of refraction α T 0 of the peak amplitude point of Υ B and the asterisk represents the space–time point θ c at which the main signal velocity is defined. The same material and pulse parameters, angle of incidence, and observation point used for Fig. 2 are used here.

Fig. 7
Fig. 7

Relative velocity of the main signal v S / c = 1 / θ c for angles of incidence α = 10 ° , 30 ° , 50 ° , and 70 ° , depicted by the solid, dotted, dashed, and dashed–dotted curves, respectively. The same material and pulse parameters and observation point used for Fig. 2 are used here.

Equations (35)

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ε 2 ( ω ) / ε 0 = 1 ω p 2 ω 2 ω 0 2 + 2 i δ ω ,
E y H x H z = 1 2 π i a i a + E ˜ i ( ω ) T ( ω ; α ) T ( ω ; α ) cos α N 2 ( ω ; α ) / η 0 T ( ω ; α ) sin α / η 0 exp [ i k T ( ω ; α ) · r i ω t ] d ω .
T ( ω ; α ) = 2 1 + N 2 ( ω ; α ) .
k T ( ω ; α ) = ω c sin α , 0 , ω c N 2 ( ω ; α ) cos α ,
N 2 ( ω ; α ) = 1 ω p 2 / cos α ω 2 ω 0 2 + 2 i δ ω ,
E x E z H y = 1 2 π i a i a + E ˜ i ( ω ) T | | ( ω ; α ) cos α N 2 ( ω ; α ) / n 2 ( ω ) T | | ( ω ; α ) sin α / n 2 ( ω ) T | | ( ω ; α ) n 2 ( ω ) / η 0 exp [ i k T ( ω ; α ) · r i ω t ] d ω .
T | | ( ω ; α ) = 2 n 2 ( ω ) n 2 2 ( ω ) + N 2 ( ω ; α ) ,
E i ( t ) = u ( t ) sin ( ω c t ) ,
ϒ = 1 2 π Re { i a i a + 1 ω ^ ω ^ c g ( ω ^ ; α ) exp [ d ω 0 c ϕ ( ω ^ , θ ; α ) ] d ω ^ } ,
ϕ ( ω ^ , θ ; α ) = i ω ^ [ z d cos α N 2 ( ω ^ ; α ) ( θ x d sin α ) ] ,
d = x sin α + z cos α ,
ω ^ 1 , 2 = i δ ^ ± 1 δ ^ 2 ,
ω ^ 3 , 4 ( α ) = i δ ^ ± 1 + ω ^ p 2 / cos 2 α δ ^ 2 .
ϒ = ϒ S + ϒ B + ϒ c .
ω 0 = 4 × 10 16 s 1 , ω ^ p = 5 / 2 , δ ^ = 0.07 .
lim α π / 2 Re { ω ^ 3 ( α ) } = .
tan α T = tan α Re { N 2 ( ω ^ ; α ) } .
α T S = arctan ( tan α Re { N 2 ( ω D + ( θ ; α ) ) } ) .
θ 0 = z cos α d N 2 ( 0 ; α ) + x sin α d
α T 0 = arctan ( tan α Re { N 2 ( 0 ; α ) } ) = arctan ( sin α cos 2 α + ω ^ p 2 )
α T c = arctan ( tan α Re { N 2 ( ω ^ c ; α ) } )
Re { ϕ ( ω N + ( θ c ) , θ c ; α ) } = Re { ϕ ( ω ^ c ; α ) } .
ϒ S ( r , c t / d ) = Re { exp [ i d ω 0 c B ( θ ; α ) ] [ γ 0 ( θ ; α ) J 0 ( d ω 0 c A ( θ ; α ) ) 2 i A ( θ ; α ) γ 1 ( θ ; α ) J 1 ( d ω 0 c A ( θ ; α ) ) ] } ,
A ( θ ; α ) = Im { ϕ ( ω D + ) } , B ( θ ; α ) = i Re { ϕ ( ω D + ) } ,
γ 0 ( θ ; α ) = 1 2 [ i g ( ω D + ) ω D + ω c ( 1 2 A ) ( 4 A 3 i ϕ ( 2 ) ( ω D + ) ) 1 / 2 + i g ( ω D ) ω D ω c ( 1 2 A ) ( 4 A 3 i ϕ ( 2 ) ( ω D ) ) 1 / 2 ] ,
γ 1 ( θ ; α ) = 1 4 A [ i g ( ω D + ) ω D + ω c ( 1 2 A ) ( 4 A 3 i ϕ ( 2 ) ( ω D + ) ) 1 / 2 i g ( ω D ) ω D ω c ( 1 2 A ) ( 4 A 3 i ϕ ( 2 ) ( ω D ) ) 1 / 2 ] ,
ϒ B ( r , c t / d ) = Re { exp [ d ω 0 c α 0 ( θ ; α ) ] { 1 2 ( c d ω 0 ) 1 / 3 e i 2 π / 3 Ai ( ξ ( θ ; α ) ) · ( i ω N + ( θ ) ω c g ( ω N + ( θ ) ; α ) h + ( θ ; α ) + i ω N ( θ ) ω c g ( ω N ( θ ) ; α ) h ( θ ; α ) ) + 1 2 α 1 1 / 2 ( θ ; α ) ( c d ω 0 ) 2 / 3 e i 4 π / 3 Ai ( 1 ) ( ξ ( θ ; α ) ) · ( i ω N + ( θ ) ω c g ( ω N + ( θ ) ; α ) h + ( θ ; α ) i ω N ( θ ) ω c g ( ω N ( θ ) ; α ) h ( θ ; α ) ) } } ,
α 0 ( θ ; α ) = 1 2 [ ϕ ( ω N + ) + ϕ ( ω N ) ] , α 1 1 / 2 ( θ ; α ) = { 3 4 [ ϕ ( ω N + ) ϕ ( ω N ) ] } 1 / 3 ,
ξ ( θ ; α ) = [ α 1 e i 2 π / 3 ( c d ω 0 ) 2 / 3 ] , h ± ( θ ; α ) = ( 2 α 1 1 / 2 ϕ ( 2 ) ( ω N + ) ) 1 / 2 ,
lim θ θ 1 h ± ( θ ; α ) = ( 2 ϕ ( 3 ) ( ω N ) ) 1 / 3 h 1 ( θ 1 ; α ) ,
lim θ θ 1 1 2 [ i ω N + ω c g ( ω N + ) h + + i ω N ω ^ c g ( ω N ) h ] = i ω N ω c g ( ω N ) h 1 ,
lim θ θ 1 1 2 α 1 1 / 2 ( θ ) [ i ω N + ω c g ( ω N + ) h + i ω N ω c g ( ω N ) h ] = h 1 2 d d ω [ i g ( ω ^ ) ω ^ ω ^ c ] | ω ^ = ω N .
2 π ϒ c ( r , c t / d ) = Re { i γ u g ( ω D + ( θ ) ; α ) ( i π sgn D erfc [ i sgn D Δ D ( θ ; α ) d ω 0 c ] exp [ d ω 0 c ϕ ( ω c , θ ; α ) ] + 1 Δ D ( θ ; α ) π c d ω 0 exp [ d ω 0 c ϕ ( ω D + ( θ ) , θ ; α ) ] ) + i γ u g ( ω N + ( θ ) ; α ) ( i sgn N π erfc ( i sgn N Δ N ( θ ; α ) d ω 0 c ) exp [ d ω 0 c ϕ ( ω c , θ ; α ) ] + 1 Δ N ( θ ; α ) π c d ω 0 exp [ d ω 0 c ϕ ( ω N + ( θ ) , θ ; α ) ] ) } + u ( θ θ s ) Re { i g ( ω c ; α ) exp [ d ω 0 c ϕ ( ω c , θ ; α ) ] } ,
erfc ( ζ ) = 2 π ζ e x 2 d x ,
Δ D , N ( θ ; α ) = [ ϕ ( ω D , N + ( θ ) , θ ; α ) ϕ ( ω c , θ ; α ) ] 1 / 2 .

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