Abstract

The advent of ultrawideband, short-pulse sources has recently generated renewed interest in the problem of pulse propagation in dispersive media. When one is trying to numerically simulate transient plane-wave propagation through a dispersive medium, the main difficulties are encountered in trying to transform from the frequency domain to the time domain. We develop an asymptotic-extraction technique wherein asymptotic methods are used in conjunction with a fast Fourier transform (FFT) to overcome the limitations of each method. The basic idea behind the asymptotic-extraction technique is to extract and analytically invert as much of the signal as possible. The remainder of the signal is then transformed by use of a numerical FFT. We demonstrate that it is possible to construct functions that possess analytical inverse transforms and also asymptotically model the low- and high-frequency behavior of a field propagating in a single-resonance Lorentz medium. Extraction of the low- and high-frequency responses from the spectral representation for the waveform in essence preconditions the waveform for application of a FFT, thereby significantly reducing the number of sample points required by the FFT. It is also demonstrated that these two extracted functions, whose inverse Fourier transforms are evaluated analytically in terms of known special functions, provide a good approximation for the transient signal for the case of a square-wave-pulse-modulated source, provided that the carrier frequency resides far enough from the absorption band.

© 1998 Optical Society of America

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References

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  1. A. Sommerfeld, “Über die Fortpflanzung des Lichtes in dispergierenden Medien,” Ann. Phys. (Leipzig) 44, 177–202 (1914).
    [CrossRef]
  2. L. Brillouin, “Über die Fortpflanzung des Lichtes in dispergierenden Medien,” Ann. Phys. (Leipzig) 44, 203–240 (1914).
    [CrossRef]
  3. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  4. L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N.J., 1973).
  5. R. E. Collin, Field Theory of Guided Waves, 2nd ed. (IEEE Press, Piscataway, N.J., 1991).
  6. A. G. Lieberman, “Transient analysis of electromagnetic reflection from dispersive materials,” (U.S. GPO, Washington, D.C., 1985).
  7. K. E. Oughstun, G. C. Sherman, “Propagation of electromagnetic pulses in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. B 5, 817–849 (1988).
    [CrossRef]
  8. S. Shen, K. E. Oughstun, “Dispersive pulse propagation in a double-resonance Lorentz medium,” J. Opt. Soc. Am. B 6, 948–963 (1989).
    [CrossRef]
  9. K. E. Oughstun, G. C. Sherman, “Uniform asymptotic description of electromagnetic pulse propagation in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. A 6, 1395–1420 (1989).
    [CrossRef]
  10. P. Wyns, D. P. Foty, K. E. Oughstun, “Numerical analysis of the precursor fields in linear dispersive pulse propagation,” J. Opt. Soc. Am. A 6, 1421–1429 (1989).
    [CrossRef]
  11. K. E. Oughstun, “Pulse propagation in a linear, causally dispersive medium,” Proc. IEEE 79, 1379–1390 (1991).
    [CrossRef]
  12. G. C. Sherman, K. E. Oughstun, “Energy-velocity description of pulse propagation in absorbing, dispersive dielectrics,” J. Opt. Soc. Am. B 12, 229–247 (1995).
    [CrossRef]
  13. C. D. Hechtman, C. Hsue, “Transient analysis of a step wave propagating in a lossy dielectric,” J. Appl. Phys. 65, 3335–3339 (1989).
    [CrossRef]
  14. J. J. A. Klaasen, “Time-domain analysis of one-dimensional electromagnetic scattering by lossy media,” (Electromagnetic Pulse Group, Physics and Electronics Laboratory, Netherlands Organization for Applied Scientific Research, The Hague, 1990).
  15. H. L. Bertoni, L. Carin, L. B. Felsen, S. U. Pillai, eds., Ultra-Wideband Short-Pulse Electromagnetics (Plenum, New York, 1993).
  16. K. S. Kunz, R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC Press, Boca Raton, Fla., 1993).
  17. A. Taflove, Computational Electrodynamics. The Finite-Difference Time-Domain Method (Artech, Norwood, Mass., 1995).
  18. P. G. Petropoulos, “The wave hierarchy for propagation in relaxing dielectrics,” Wave Motion 21, 253–262 (1995).
    [CrossRef]
  19. J. B. Judkins, R. W. Ziolkowski, “Finite-difference time-domain modeling of nonperfectly conducting metallic thin-film gratings,” J. Opt. Soc. Am. A 12, 1974–1983 (1995).
    [CrossRef]
  20. S. L. Dvorak, “Exact, closed-form expressions for transient fields in homogeneously filled waveguides,” IEEE Trans. Microwave Theory Tech. 42, 2164–2170 (1994).
    [CrossRef]
  21. S. L. Dvorak, D. G. Dudley, “Propagation of ultra-wideband electromagnetic pulses through dispersive media,” IEEE Trans. Electromagn. Compat. 37, 192–200 (1995).
    [CrossRef]
  22. S. L. Dvorak, D. G. Dudley, “A comment on propagation of ultra-wideband electromagnetic pulses through dispersive media—author’s reply,” IEEE Trans. Electromagn. Compat. 38, 203–205 (1996).
  23. H.-Y. Pao, S. L. Dvorak, 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]
  24. H.-Y. Pao, S. L. Dvorak, 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]
  25. S. L. Dvorak, R. W. Ziolkowski, D. G. Dudley, “Propagation of ultra-wideband electromagnetic pulses through lossy plasmas,” Radio Sci. 32, 239–250 (1997).
    [CrossRef]
  26. L. P. Huelsman, Active and Passive Analog Filter Design: An Introduction (McGraw-Hill, New York, 1993).
  27. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1986).
  28. S. L. Dvorak, E. F. Kuester, “Numerical computation of the incomplete Lipschitz–Hankel integral Je0(a, z),” J. Comput. Phys. 87, 301–327 (1990).
    [CrossRef]
  29. M. M. Mechaik, S. L. Dvorak, “Series expansions for the incomplete Lipschitz–Hankel integral Je0(a, z),” Radio Sci. 30, 1393–1404 (1995).
    [CrossRef]
  30. K. E. Oughstun, G. C. Sherman, “Uniform asymptotic description of ultrashort rectangular optical pulse propagation in a linear, causally dispersive medium,” Phys. Rev. A 41, 6090–6113 (1990).
    [CrossRef] [PubMed]
  31. K. E. Oughstun, G. C. Sherman, Electromagnetic Pulse Propagation in Causal Dielectrics (Springer-Verlag, Berlin, 1994).

1997 (1)

S. L. Dvorak, R. W. Ziolkowski, D. G. Dudley, “Propagation of ultra-wideband electromagnetic pulses through lossy plasmas,” Radio Sci. 32, 239–250 (1997).
[CrossRef]

1996 (3)

S. L. Dvorak, D. G. Dudley, “A comment on propagation of ultra-wideband electromagnetic pulses through dispersive media—author’s reply,” IEEE Trans. Electromagn. Compat. 38, 203–205 (1996).

H.-Y. Pao, S. L. Dvorak, 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.-Y. Pao, S. L. Dvorak, 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 (5)

P. G. Petropoulos, “The wave hierarchy for propagation in relaxing dielectrics,” Wave Motion 21, 253–262 (1995).
[CrossRef]

J. B. Judkins, R. W. Ziolkowski, “Finite-difference time-domain modeling of nonperfectly conducting metallic thin-film gratings,” J. Opt. Soc. Am. A 12, 1974–1983 (1995).
[CrossRef]

M. M. Mechaik, S. L. Dvorak, “Series expansions for the incomplete Lipschitz–Hankel integral Je0(a, z),” Radio Sci. 30, 1393–1404 (1995).
[CrossRef]

G. C. Sherman, K. E. Oughstun, “Energy-velocity description of pulse propagation in absorbing, dispersive dielectrics,” J. Opt. Soc. Am. B 12, 229–247 (1995).
[CrossRef]

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

1994 (1)

S. L. Dvorak, “Exact, closed-form expressions for transient fields in homogeneously filled waveguides,” IEEE Trans. Microwave Theory Tech. 42, 2164–2170 (1994).
[CrossRef]

1991 (1)

K. E. Oughstun, “Pulse propagation in a linear, causally dispersive medium,” Proc. IEEE 79, 1379–1390 (1991).
[CrossRef]

1990 (2)

K. E. Oughstun, G. C. Sherman, “Uniform asymptotic description of ultrashort rectangular optical pulse propagation in a linear, causally dispersive medium,” Phys. Rev. A 41, 6090–6113 (1990).
[CrossRef] [PubMed]

S. L. Dvorak, E. F. Kuester, “Numerical computation of the incomplete Lipschitz–Hankel integral Je0(a, z),” J. Comput. Phys. 87, 301–327 (1990).
[CrossRef]

1989 (4)

C. D. Hechtman, C. Hsue, “Transient analysis of a step wave propagating in a lossy dielectric,” J. Appl. Phys. 65, 3335–3339 (1989).
[CrossRef]

S. Shen, K. E. Oughstun, “Dispersive pulse propagation in a double-resonance Lorentz medium,” J. Opt. Soc. Am. B 6, 948–963 (1989).
[CrossRef]

K. E. Oughstun, G. C. Sherman, “Uniform asymptotic description of electromagnetic pulse propagation in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. A 6, 1395–1420 (1989).
[CrossRef]

P. Wyns, D. P. Foty, K. E. Oughstun, “Numerical analysis of the precursor fields in linear dispersive pulse propagation,” J. Opt. Soc. Am. A 6, 1421–1429 (1989).
[CrossRef]

1988 (1)

1914 (2)

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

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

Brillouin, L.

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

Collin, R. E.

R. E. Collin, Field Theory of Guided Waves, 2nd ed. (IEEE Press, Piscataway, N.J., 1991).

Dudley, D. G.

S. L. Dvorak, R. W. Ziolkowski, D. G. Dudley, “Propagation of ultra-wideband electromagnetic pulses through lossy plasmas,” Radio Sci. 32, 239–250 (1997).
[CrossRef]

H.-Y. Pao, S. L. Dvorak, 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]

H.-Y. Pao, S. L. Dvorak, 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]

S. L. Dvorak, D. G. Dudley, “A comment on propagation of ultra-wideband electromagnetic pulses through dispersive media—author’s reply,” IEEE Trans. Electromagn. Compat. 38, 203–205 (1996).

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

Dvorak, S. L.

S. L. Dvorak, R. W. Ziolkowski, D. G. Dudley, “Propagation of ultra-wideband electromagnetic pulses through lossy plasmas,” Radio Sci. 32, 239–250 (1997).
[CrossRef]

H.-Y. Pao, S. L. Dvorak, 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.-Y. Pao, S. L. Dvorak, 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]

S. L. Dvorak, D. G. Dudley, “A comment on propagation of ultra-wideband electromagnetic pulses through dispersive media—author’s reply,” IEEE Trans. Electromagn. Compat. 38, 203–205 (1996).

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

M. M. Mechaik, S. L. Dvorak, “Series expansions for the incomplete Lipschitz–Hankel integral Je0(a, z),” Radio Sci. 30, 1393–1404 (1995).
[CrossRef]

S. L. Dvorak, “Exact, closed-form expressions for transient fields in homogeneously filled waveguides,” IEEE Trans. Microwave Theory Tech. 42, 2164–2170 (1994).
[CrossRef]

S. L. Dvorak, E. F. Kuester, “Numerical computation of the incomplete Lipschitz–Hankel integral Je0(a, z),” J. Comput. Phys. 87, 301–327 (1990).
[CrossRef]

Felsen, L. B.

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

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1986).

Foty, D. P.

Hechtman, C. D.

C. D. Hechtman, C. Hsue, “Transient analysis of a step wave propagating in a lossy dielectric,” J. Appl. Phys. 65, 3335–3339 (1989).
[CrossRef]

Hsue, C.

C. D. Hechtman, C. Hsue, “Transient analysis of a step wave propagating in a lossy dielectric,” J. Appl. Phys. 65, 3335–3339 (1989).
[CrossRef]

Huelsman, L. P.

L. P. Huelsman, Active and Passive Analog Filter Design: An Introduction (McGraw-Hill, New York, 1993).

Judkins, J. B.

Klaasen, J. J. A.

J. J. A. Klaasen, “Time-domain analysis of one-dimensional electromagnetic scattering by lossy media,” (Electromagnetic Pulse Group, Physics and Electronics Laboratory, Netherlands Organization for Applied Scientific Research, The Hague, 1990).

Kuester, E. F.

S. L. Dvorak, E. F. Kuester, “Numerical computation of the incomplete Lipschitz–Hankel integral Je0(a, z),” J. Comput. Phys. 87, 301–327 (1990).
[CrossRef]

Kunz, K. S.

K. S. Kunz, R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC Press, Boca Raton, Fla., 1993).

Lieberman, A. G.

A. G. Lieberman, “Transient analysis of electromagnetic reflection from dispersive materials,” (U.S. GPO, Washington, D.C., 1985).

Luebbers, R. J.

K. S. Kunz, R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC Press, Boca Raton, Fla., 1993).

Marcuvitz, N.

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

Mechaik, M. M.

M. M. Mechaik, S. L. Dvorak, “Series expansions for the incomplete Lipschitz–Hankel integral Je0(a, z),” Radio Sci. 30, 1393–1404 (1995).
[CrossRef]

Oughstun, K. E.

G. C. Sherman, K. E. Oughstun, “Energy-velocity description of pulse propagation in absorbing, dispersive dielectrics,” J. Opt. Soc. Am. B 12, 229–247 (1995).
[CrossRef]

K. E. Oughstun, “Pulse propagation in a linear, causally dispersive medium,” Proc. IEEE 79, 1379–1390 (1991).
[CrossRef]

K. E. Oughstun, G. C. Sherman, “Uniform asymptotic description of ultrashort rectangular optical pulse propagation in a linear, causally dispersive medium,” Phys. Rev. A 41, 6090–6113 (1990).
[CrossRef] [PubMed]

P. Wyns, D. P. Foty, K. E. Oughstun, “Numerical analysis of the precursor fields in linear dispersive pulse propagation,” J. Opt. Soc. Am. A 6, 1421–1429 (1989).
[CrossRef]

S. Shen, K. E. Oughstun, “Dispersive pulse propagation in a double-resonance Lorentz medium,” J. Opt. Soc. Am. B 6, 948–963 (1989).
[CrossRef]

K. E. Oughstun, G. C. Sherman, “Uniform asymptotic description of electromagnetic pulse propagation in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. A 6, 1395–1420 (1989).
[CrossRef]

K. E. Oughstun, G. C. Sherman, “Propagation of electromagnetic pulses in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. B 5, 817–849 (1988).
[CrossRef]

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

Pao, H.-Y.

H.-Y. Pao, S. L. Dvorak, 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]

H.-Y. Pao, S. L. Dvorak, 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]

Petropoulos, P. G.

P. G. Petropoulos, “The wave hierarchy for propagation in relaxing dielectrics,” Wave Motion 21, 253–262 (1995).
[CrossRef]

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1986).

Shen, S.

Sherman, G. C.

G. C. Sherman, K. E. Oughstun, “Energy-velocity description of pulse propagation in absorbing, dispersive dielectrics,” J. Opt. Soc. Am. B 12, 229–247 (1995).
[CrossRef]

K. E. Oughstun, G. C. Sherman, “Uniform asymptotic description of ultrashort rectangular optical pulse propagation in a linear, causally dispersive medium,” Phys. Rev. A 41, 6090–6113 (1990).
[CrossRef] [PubMed]

K. E. Oughstun, G. C. Sherman, “Uniform asymptotic description of electromagnetic pulse propagation in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. A 6, 1395–1420 (1989).
[CrossRef]

K. E. Oughstun, G. C. Sherman, “Propagation of electromagnetic pulses in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. B 5, 817–849 (1988).
[CrossRef]

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

Sommerfeld, A.

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

Stratton, A.

A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

Taflove, A.

A. Taflove, Computational Electrodynamics. The Finite-Difference Time-Domain Method (Artech, Norwood, Mass., 1995).

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1986).

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1986).

Wyns, P.

Ziolkowski, R. W.

S. L. Dvorak, R. W. Ziolkowski, D. G. Dudley, “Propagation of ultra-wideband electromagnetic pulses through lossy plasmas,” Radio Sci. 32, 239–250 (1997).
[CrossRef]

J. B. Judkins, R. W. Ziolkowski, “Finite-difference time-domain modeling of nonperfectly conducting metallic thin-film gratings,” J. Opt. Soc. Am. A 12, 1974–1983 (1995).
[CrossRef]

Ann. Phys. (Leipzig) (2)

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

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

IEEE Trans. Antennas Propag. (2)

H.-Y. Pao, S. L. Dvorak, 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.-Y. Pao, S. L. Dvorak, 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)

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

S. L. Dvorak, D. G. Dudley, “A comment on propagation of ultra-wideband electromagnetic pulses through dispersive media—author’s reply,” IEEE Trans. Electromagn. Compat. 38, 203–205 (1996).

IEEE Trans. Microwave Theory Tech. (1)

S. L. Dvorak, “Exact, closed-form expressions for transient fields in homogeneously filled waveguides,” IEEE Trans. Microwave Theory Tech. 42, 2164–2170 (1994).
[CrossRef]

J. Appl. Phys. (1)

C. D. Hechtman, C. Hsue, “Transient analysis of a step wave propagating in a lossy dielectric,” J. Appl. Phys. 65, 3335–3339 (1989).
[CrossRef]

J. Comput. Phys. (1)

S. L. Dvorak, E. F. Kuester, “Numerical computation of the incomplete Lipschitz–Hankel integral Je0(a, z),” J. Comput. Phys. 87, 301–327 (1990).
[CrossRef]

J. Opt. Soc. Am. A (3)

J. Opt. Soc. Am. B (3)

Phys. Rev. A (1)

K. E. Oughstun, G. C. Sherman, “Uniform asymptotic description of ultrashort rectangular optical pulse propagation in a linear, causally dispersive medium,” Phys. Rev. A 41, 6090–6113 (1990).
[CrossRef] [PubMed]

Proc. IEEE (1)

K. E. Oughstun, “Pulse propagation in a linear, causally dispersive medium,” Proc. IEEE 79, 1379–1390 (1991).
[CrossRef]

Radio Sci. (2)

M. M. Mechaik, S. L. Dvorak, “Series expansions for the incomplete Lipschitz–Hankel integral Je0(a, z),” Radio Sci. 30, 1393–1404 (1995).
[CrossRef]

S. L. Dvorak, R. W. Ziolkowski, D. G. Dudley, “Propagation of ultra-wideband electromagnetic pulses through lossy plasmas,” Radio Sci. 32, 239–250 (1997).
[CrossRef]

Wave Motion (1)

P. G. Petropoulos, “The wave hierarchy for propagation in relaxing dielectrics,” Wave Motion 21, 253–262 (1995).
[CrossRef]

Other (11)

J. J. A. Klaasen, “Time-domain analysis of one-dimensional electromagnetic scattering by lossy media,” (Electromagnetic Pulse Group, Physics and Electronics Laboratory, Netherlands Organization for Applied Scientific Research, The Hague, 1990).

H. L. Bertoni, L. Carin, L. B. Felsen, S. U. Pillai, eds., Ultra-Wideband Short-Pulse Electromagnetics (Plenum, New York, 1993).

K. S. Kunz, R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC Press, Boca Raton, Fla., 1993).

A. Taflove, Computational Electrodynamics. The Finite-Difference Time-Domain Method (Artech, Norwood, Mass., 1995).

A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

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

R. E. Collin, Field Theory of Guided Waves, 2nd ed. (IEEE Press, Piscataway, N.J., 1991).

A. G. Lieberman, “Transient analysis of electromagnetic reflection from dispersive materials,” (U.S. GPO, Washington, D.C., 1985).

L. P. Huelsman, Active and Passive Analog Filter Design: An Introduction (McGraw-Hill, New York, 1993).

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1986).

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

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Figures (19)

Fig. 1
Fig. 1

Real (solid curve) and imaginary (dashed curve) parts of the index of refraction as a function of frequency for Brillouin’s choice for the parameters, i.e., ωp2=20×1032 (rad/s)2, ω0=4 ×1016 rad/s, and ν=0.56×1016 Hz.

Fig. 2
Fig. 2

Spectra for the impulse response (dotted curve) for the two cases: (a) z=1.0 µm, (b) z=1.0 mm. Brillouin’s choice for the parameters was employed. The solid and dashed curves show the modified spectra after the AE terms in Eqs. (15) and (25), respectively, have been extracted. A comparison of the solid and dashed curves indicates that the AE term in Eq. (15) is significantly more accurate than the one in Eq. (25).

Fig. 3
Fig. 3

Early-time electric-field impulse response (solid curve) as a function of the delayed time for z=1.0 µm and Brillouin’s choice for the parameters. Also shown are results associated with the AE (dotted curve) and the FFT (dashed curve) terms, i.e., results computed from Eqs. (33) and (11), respectively.

Fig. 4
Fig. 4

Late-time electric-field impulse response as a function of the delayed time for z=1.0 µm and Brillouin’s choice for the parameters.

Fig. 5
Fig. 5

Late-time electric-field impulse response (solid curve) as a function of the delayed time for z=1.0 µm and Brillouin’s choice for the parameters. Also shown are results associated with the AE (dotted curve) and the FFT (dashed curve) terms, i.e., results computed from Eqs. (33) and (11), respectively.

Fig. 6
Fig. 6

Spectrum for the impulse response (dotted curve) for a highly transparent Lorentz medium, i.e., z=1.0 µm, ωp2=20×1032 (rad/s)2, ω0=4×1016 rad/s, and ν=2.0 Hz. The solid and dashed curves show the modified spectra after the AE terms in Eqs. (15) and (25), respectively, have been extracted. A comparison of the solid and dashed curves indicates that the AE term in Eq. (15) is significantly more accurate than the one in Eq. (25).

Fig. 7
Fig. 7

Early-time electric-field impulse response (solid curve) as a function of the delayed time for z=1.0 µm in a highly transparent Lorentz medium. Also shown are results associated with the AE (dotted curve) and the FFT (dashed curve) terms, i.e., results computed from Eqs. (33) and (11), respectively.

Fig. 8
Fig. 8

Late-time electric-field impulse response as a function of the delayed time for z=1.0 µm in a highly transparent Lorentz medium.

Fig. 9
Fig. 9

Late-time electric-field impulse response (solid curve) as a function of the delayed time for z=1.0 µm in a highly transparent Lorentz medium. Also shown are results associated with the AE (dotted curve) and the FFT (dashed curve) terms, i.e., results computed from Eqs. (33) and (11), respectively.

Fig. 10
Fig. 10

Normalized spectrum for the modulated source (dotted curve) for ωc=1.0×1016 rad/s, N=4, z=1.0 µm, and Brillouin’s choice for the parameters. The solid curve shows the modified normalized spectrum after the AE term in Eq. (44) has been extracted.

Fig. 11
Fig. 11

Electric-field transient responses as a function of time for ωc=1.0×1016 rad/s, N=4, NFFT=65,536, fmax=1.0×1017 Hz, Brillouin’s choice for the parameters, and five values of z.

Fig. 12
Fig. 12

Electric-field transient response (solid curve) as a function of time for ωc=1.0×1016 rad/s, N=4, NFFT=65,536, fmax=1.0×1017 Hz, z=1.0 µm, and Brillouin’s choice for the parameters. Also shown are results associated with the AE (dotted curve) and the FFT (dashed curve) terms, i.e., results computed from Eqs. (45) and (41), respectively.

Fig. 13
Fig. 13

Normalized spectrum for the modulated source (dotted curve) for ωc=1.0×1015 rad/s, N=4, z=1.0 µm, and Brillouin’s choice for the parameters. The solid curve shows the modified normalized spectrum after the AE term in Eq. (44) has been extracted.

Fig. 14
Fig. 14

Electric-field transient response (solid curve) as a function of the delayed time for ωc=1.0×1015 rad/s, N=4, NFFT=65,536, fmax=1.0×1016 Hz, z=1.0 µm, and Brillouin’s choice for the parameters. Also shown are results associated with the AE (dotted curve) and the FFT (dashed curve) terms, i.e., results computed from Eqs. (45) and (41), respectively.

Fig. 15
Fig. 15

Normalized spectrum for the modulated source (dotted curve) for ωc=1.0×1018 rad/s, N=4, z=1.0 µm, and Brillouin’s choice for the parameters. The solid curve shows the modified normalized spectrum after the AE term in Eq. (46) has been extracted.

Fig. 16
Fig. 16

Electric-field transient response (solid curve) as a function of the delayed time for ωc=1.0×1018 rad/s, N=4, NFFT=65,536, fmax=1.0×1019 Hz, z=1.0 µm, and Brillouin’s choice for the parameters. Also shown are results associated with the AE (dotted curve) and the FFT (dashed curve) terms, i.e., results computed from Eqs. (48) and (41), respectively.

Fig. 17
Fig. 17

Normalized spectrum for the modulated source (dotted curve) for ωc=5.0×1016 rad/s, N=4, z=1.0 µm, and Brillouin’s choice for the parameters. The solid curve shows the modified normalized spectrum after the AE term in Eq. (46) has been extracted. Note the large overshoot in the absorption band.

Fig. 18
Fig. 18

Normalized spectrum for the modulated source (dotted curve) for ωc=5.0×1016 rad/s, N=4, z=1.0 µm, and Brillouin’s choice for the parameters. The solid curve shows the modified normalized spectrum after the AE term in Eq. (49) has been extracted. The addition of the filter functions has eliminated the large overshoot in the absorption band.

Fig. 19
Fig. 19

Electric-field transient response plotted as a function of the delayed time for ωc=5.0×1016 rad/s, N=4, NFFT=65,536, fmax=5.0×1017 Hz, z=1.0 µm, and Brillouin’s choice for the parameters.

Equations (75)

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Ex(ξ, t)=12π-E˜x(0, ω)exp{jω[t-ξn(ω)]}dω,
ξ0,
E˜x(0, ω)=-Ex(0, t)exp(-jωt)dt
n(ω)=1+m=0M ωpm2ωom2-ω2+jωνm1/2,
n(ω)=1+ωp2ωo2-ω2+jων1/2.
Ex(0, t)=δ(t)V/m,
E˜x(0, ω)=1.
Ex(ξ, t)=I(ξ, t),
I(ξ, t)=12π-I˜(ξ, ω)exp(jωt)dω,
I˜(ξ, ω)=exp-jωξ1+ωp2ωo2-ω2+jων1/2.
I(ξ, t)=IA(ξ, t)+IF(ξ, t),ξ0,
IF(ξ, t)=12π-[I˜(ξ, ω)-I˜A(ξ, ω)]exp(jωt)dω
limω0 I˜(ξ, ω)[1-O(ω2)]exp[-jωξn(0)],
limω I˜(ξ, ω)[1-O(ω-2)]exp(-jξω2-ωp2),
n(0)=ω02+ωp2ω0.
I˜A(ξ, ω)=L(ω, ωL)exp[-jωξn(0)]+H(ω, ωH)exp(-jξω2-ωp2),
L(ω, ωL)
=ωL4(ω-jωLP)(ω-jωLP*)(ω-ωLP)(ω+ωLP*),
H(ω, ωH)
=ω4(ω-jωHP)(ω-jωHP*)(ω-ωHP)(ω+ωHP*),
n(ω)ω02-ω2+ωp2ω02-ω21/2×1-jωp2ων(ω02-ω2)(ω02-ω2+ωp2)1/2.
I˜(ξ, ω)exp-jωξω02-ω2+ωp2ω02-ω21/2×exp-ω2ξνωp22(ω02-ω2)3/2(ω02-ω2+ωp2)1/2.
ω2ξνωp22(ω02-ω2)3/2(ω02-ω2+ωp2)1/2=1.
ωLB=ω02ωp2n(0)ξν1/2.
ωLB=minω0, ω02ωp2n(0)ξν1/2
ωHB=ωpξν21/2.
ωHB=maxω02+ωp2, ωpξν21/2.
I˜A(ξ, ω)=L(ω, ωL)exp[-jωξn(0)]+H(ω, ωH)exp(-jωξ).
L(ω, ωL)=ωL4[Re(P)-Im(P)]×1Im(P)P*ω-jωLP-Pω-jωLP*+jRe(P)P*ω-ωLP+Pω+ωLP*,
H(ω, ωH)=1-ωH4[Re(P)-Im(P)]×1Im(P)Q*ω-jωHP-Qω-jωHP*+jRe(P)R*ω-ωHP+Rω+ωHP*,
Q=2 Re(P)-{4[Re(P)]2-1}P*,
R=2j Im(P)+{4[Im(P)]2-1}P*.
IA(ξ, t)=12π-L(ω, ωL)exp{jω[t-ξn(0)]}dω+δ(t-ξ)-ωpξJ1(ωpt2-ξ2)t2-ξ2u(t-ξ)-ωH4[Re(P)-Im(P)]×j[Q*e(ωHP)-Qe(ωHP*)]Im(P)-[R*e(-jωHP)+Re(jωHP*)]Re(P),
u(t)=1,t>01/2,t=00,t<0
e(α*)=[e(α)]*
IA(ξ, t)=δ(t-ξ)-ωpξJ1(ωpt2-ξ2)t2-ξ2u(t-ξ)+12[Re(P)-Im(P)]×ImωHQ*e(ωHP)-ωLP* exp{-ωLP[t-ξn(0)]}u[t-ξn(0)]Im(P)+ReωHR*e(-jωHP)-ωLP* exp{jωLP[t-ξn(0)]}u[t-ξn(0)]Re(P).
t=t-ξ,
limtξ IA(ξ, t)=δ(t-ξ)-ωp2t4+ωH4[Re(P)-Im(P)]
×Im(Q*)Im(P)+Re(R*)Re(P).
Ex(0, t)=A sin(ωct)[u(t)-u(t-T)],
T=(2πN)/ωc,
E˜x(0, ω)=-Aωcω2-ωc2[1-exp(-jωT)].
E˜x(ξ, ω)=E˜x(0, ω)I˜(ξ, ω),
Ex(ξ, t)=ExA(ξ, t)+ExF(ξ, t),ξ0,
ExF(ξ, t)=12π-[E˜x(ξ, ω)-E˜xA(ξ, ω)]exp(jωt)dω.
E˜xA(ξ, ω)=A[1-exp(-jωT)]F˜(ξ, ω).
ExA(ξ, t)=A[F(ξ, t)-F(ξ, t-T)].
F˜(ξ, ω)=-ωc exp[-jωξn(0)]ω2-ωc2.
F(ξ, t)=sin{ωc[t-ξn(0)]}u[t-ξn(0)].
F˜(ξ, ω)=-ωc exp(-jξω2-ωp2)ω2-ωc2.
ωcω2-ωc2=121ω-ωc-1ω+ωc,
F(ξ, t)=Im[e(-jωc)].
F˜(ξ, ω)=-ωc{L(ω, ωL)exp[-jωξn(0)]+H(ω, ωH)exp(-jξω2-ωp2)}ω2-ωc2.
L(ω, ωL)ω±ωc=SL±ω±ωc±ωL4[Re(P)-Im(P)]1Im(P)P*(ωc±jωLP)(ω-jωLP)-P(ωc±jωLP*)(ω-jωLP*)+jRe(P)P*(ωc±ωLP)(ω-ωLP)+P(ωcωLP*)(ω+ωLP*),
H(ω, ωH)ω±ωc=SH±ω±ωcωH4[Re(P)-Im(P)]1Im(P)Q*(ωc±jωHP)(ω-jωHP)-Q(ωc±jωHP*)(ω-jωHP*)+jRe(P)R*(ωc±ωHP)(ω-ωHP)+R(ωc±ωHP*)(ω+ωHP*),
SL±=ωL4(ωc±jωLP)(ωc±jωLP*)(ωc±ωLP)(ωcωLP*),
SH±=1+ωH (ωc2-ωH2){ωH2jωc[Re(P)+Im(P)]}+ωHωc2[1+Re(P)Im(P)](ωc±jωHP)(ωc±jωHP*)(ωc±ωHP)(ωcωHP*).
F(ξ, t)=Im(SH-e(-jωc)+SL- exp{jωc[t-ξn(0)]}×u[t-ξn(0)])+ωc2[Re(P)-Im(P)]×1Im(P)ImωHQ*e(ωHP)ωc2+(ωHP)2-ωLP* exp{-ωLP[t-ξn(0)]}u[t-ξn(0)]ωc2+(ωLP)2+1Re(P)ReωHR*e(-jωHP)ωc2-(ωHP)2-ωLP* exp{jωLP[t-ξn(0)]}u[t-ξn(0)]ωc2-(ωLP)2.
12π- exp[j(ωt-ξω2-ωp2)]dω
=δ(t-ξ)-ωpξJ1(ωpt2-ξ2)t2-ξ2u(t-ξ).
e(α)=12πj- exp[j(ωt-ξω2-ωp2)](ω-jα)dω.
e(α)=u(t-ξ){exp(-αt)cosh(ξα2+ωp2)-½[S+ exp(a+ζ)Je0(a+, ζ)+S- exp(a-ζ)Je0(a-, ζ)]},
Jen(a±, ζ)=0ζ exp(-a±t)tnJn(t)dt
a±=-αt±ξα2+ωp2ωpt2-ξ2,
ζ=ωpt2-ξ2,
S±=-αξ±tα2+ωp2ωpt2-ξ2.
Jen(a±, δ±, ζ)=δ±ζ exp(-a±t)tnJn(t)dt,
δ±=,Re(a±)0-,Re(a±)<0.
Je0(a±, ζ)=1a±2+1+Je0(a±, δ±, ζ),
Im(a±2+1)>0,Re(a±)=0andIm(a±)>1,
Im(a±2+1)<0,Re(a±)=0andIm(a±)<-1,
Re(a±2+1)<0,Re(a±)<0,
Re(a±2+1)>0,otherwise,
e(α)=u(t-ξ)2[G+ exp(a+ζ)+G- exp(a-ζ)-S+ exp(a+ζ)Je0(a+, δ+, ζ)-S- exp(a-ζ)Je0(a-, δ-, ζ)],
G±=2,S±a±2+10,S±=a±2+1.

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