Abstract

One-dimensional propagation of a normally incident, pulsed electromagnetic plane wave upon an isotropic, spatially homogeneous, Lorentz half-space is investigated analytically. Fourier integral representations of the time-dependent reflected and transmitted fields are obtained for an arbitrary incident pulse, and detailed examinations of these fields are made when the incident field is a finite-cycle-sine function. The inversion integral for the time-dependent reflected field is expressed in terms of the pole contribution and branch-cut integrals, whereas the uniform asymptotic methodology of Oughstun and Sherman [J. Opt. Soc. Am. A 6, 1394 (1989)] is applied to the transmitted field. Only the contribution from the distant saddle points to the transmitted field is studied in detail. An example is provided that shows that the effects of including the reflection and transmission coefficients may not be ignored when microwave or optical pulses are launched across the interface. Specifically, for Brillouin’s choice of the medium’s physical parameters, the reflected field has a peak value that is 21% of the incident field’s amplitude and that corresponds to a 21% decrease in the main signal (pole contribution) of the transmitted field when the transmission coefficient is unity. This study extends past analytical formulations of the one-dimensional problem by conducting an in-depth analysis of the reflected field for a normally incident finite-cycle-sine wave and by addressing how inclusion of frequency-dependent transmission and reflection coefficients affects the fields. In particular, situations for which the effect of the frequency-dependent transmission and reflection coefficients is significant are discussed. Finally, the analysis of the reflected field should be useful for diagnostic purposes.

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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. L. Brillouin, Wave Propagation and Group Velocity (Academic, New York, 1960).
  4. K. E. Oughstun, “Propagation of optical pulses in dispersive media,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1978).
  5. K. E. Oughstun and 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]
  6. K. E. Oughstun and 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, 1394–1420 (1989).
    [CrossRef]
  7. K. E. Oughstun and 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]
  8. K. E. Oughstun and G. C. Sherman, Electromagnetic Pulse Propagation in Causal Dielectrics (Springer-Verlag, Berlin, 1994).
  9. K. Moten, C. H. Durney, and T. G. Stockham, Jr., “Electromagnetic pulse propagation in dispersive planar dielectrics,” Bioelectromagnetics (N.Y.) 10, 35–49 (1989).
    [CrossRef]
  10. R. Albanese, J. Penn, and R. Medina, “Short-rise-time microwave pulse propagation through dispersive biological media,” J. Opt. Soc. Am. A 6, 1441–1446 (1989).
    [CrossRef]
  11. D. H. Lam, “Finite difference methods for transient signal propagation in stratified dispersive media,” Tech. Rep. 3892–1 (Electro-Science Laboratory, Ohio State University, Columbus, Ohio, 1975).
  12. R. J. Luebbers and F. Hunsberger, “FDTD for Nth-order dispersive media,” IEEE Trans. Antennas Propag. 40, 1297–1301 (1992).
    [CrossRef]
  13. J. G. Blaschak and J. Franzen, “Precursor propagation in dispersive media from short-rise-time pulses at oblique incidence,” J. Opt. Soc. Am. A 12, 1501–1512 (1995).
    [CrossRef]
  14. W. Colby, “Signal propagation in dispersive media,” Phys. Rev. 5, 253–265 (1915).
    [CrossRef]
  15. J. Marozas and K. Oughstun, “Electromagnetic pulse propagation across a planar interface separating two lossy, dispersive dielectrics,” in Ultra-Wideband, Short-Pulse Electromagnetics 3, C. E. Baum, L. Carin, and A. P. Stone, eds. (Plenum, New York, 1997), pp. 217–230.
  16. A radar is defined to be UWB when its fractional bandwidth exceeds 25%, according to the executive summary of the Office of the Secretary of Defense/Defense Advanced Research Projects Agency study, “Assessment of ultra-wideband (UWB) technology,” IEEE Aerosp. Electron. Mag. 5, 45–49 (1990).
  17. B. Noel, Ultra-Wideband Radar: Proceedings of the First Los Alamos Symposium (CRC, Boca Raton, Fla., 1991).
  18. H. L. Bertoni, L. Carin, and L. B. Felsen, eds., Ultra-Wideband, Short-Pulse Electromagnetics (Plenum, New York, 1993).
  19. L. Yu. Astanin, A. A. Kostylev, Yu. S. Zinoviev, and A. Ya. Pasmurov, Radar Target Characteristics: Measurements and Applications (CRC, Boca Raton, Fla., 1994).
  20. J. D. Taylor, ed., Introduction to Ultra-Wideband Radar Systems (CRC, Boca Raton, Fla., 1995).
  21. L. Carin and L. B. Felsen, eds., Ultra-Wideband, Short-Pulse Electromagnetics 2 (Plenum, New York, 1995).
  22. H.-Y. 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]
  23. S. L. Dvorak, R. W. Ziolkowski, and D. G. Dudley, “Ultra-wideband electromagnetic pulse propagation in a homogeneous, cold plasma,” Radio Sci. 32, 239–250 (1997).
    [CrossRef]
  24. R. A. Albanese, J. W. Penn, and R. L. Medina, “An electromagnetic inverse problem in medical science,” in Invariant Imbedding and Inverse Problems, J. P. Corones, G. Kristensson, P. Nelson, and D. L. Seth, eds. (Society for Industrial and Applied Mathematics, Philadelphia, Pa. 1992), Chap. 2, pp. 30–41.
  25. R. A. Albanese, J. G. Blaschak, R. L. Medina, and J. W. Penn, “Ultrashort electromagnetic signals: biophysical questions, safety issues, and medical opportunities,” Aviat., Space Environ. Med. 65 Suppl. 5, A116–A120 (1994).
  26. R. A. Albanese, R. L. Medina, and J. W. Penn, “Mathematics, medicine and microwaves,” Inverse Probl. 10, 995–1007 (1994).
    [CrossRef]
  27. Ref. 1, p. 190.
  28. E. L. Mokole and S. N. Samaddar, “Pulsed reflection and transmission for a dispersive half space,” Publ. NRL/FR/5340–98–9874 (Naval Research Laboratory, Washington, D.C., 1998).
  29. See Ref. 3, pp. 56–57, and Ref. 8, p. 234.
  30. The fractional bandwidth BF of the single-cycle sine function is given by the formula BF=[2(ωU−ωL)/(ωU+ ωL)]×100%, where ωU and ωL, respectively, are the upper and lower frequencies of the passband of the power spectral density of the single-cycle sine, 4ωc2 sin2(πω/ωc)/(ωc2−ω2)2. If one uses the half-power criterion to determine the passband, then ωU≐1.309ωc and ωL≐ 0.411ωc. Hence BF≐104%, which is ultrawideband according to Ref. 16.
  31. L. L. Pennisi, Elements of Complex Variables (Holt, Rinehart & Winston, New York, 1963), Chap. 5, p. 185.
  32. Ref. 14, p. 264.
  33. See Ref. 4, Chap. 2, and Ref. 8, p. 141.
  34. 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]
  35. N. Bleistein and R. Handelsman, Asymptotic Expansions of Integrals (Holt, Rinehart & Winston, New York, 1975), p. 390.
  36. N. Bleistein and R. Handelsman, Asymptotic Expansions of Integrals (Dover, New York, 1986), Chap. 9.
  37. Ref. 8, p. 214.
  38. The support of a real-valued function on the real line is the closure of the set of all points at which the function is nonzero.

1997 (1)

S. L. Dvorak, R. W. Ziolkowski, and D. G. Dudley, “Ultra-wideband electromagnetic pulse propagation in a homogeneous, cold plasma,” Radio Sci. 32, 239–250 (1997).
[CrossRef]

1996 (1)

H.-Y. 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 (1)

1994 (2)

R. A. Albanese, J. G. Blaschak, R. L. Medina, and J. W. Penn, “Ultrashort electromagnetic signals: biophysical questions, safety issues, and medical opportunities,” Aviat., Space Environ. Med. 65 Suppl. 5, A116–A120 (1994).

R. A. Albanese, R. L. Medina, and J. W. Penn, “Mathematics, medicine and microwaves,” Inverse Probl. 10, 995–1007 (1994).
[CrossRef]

1992 (1)

R. J. Luebbers and F. Hunsberger, “FDTD for Nth-order dispersive media,” IEEE Trans. Antennas Propag. 40, 1297–1301 (1992).
[CrossRef]

1990 (1)

K. E. Oughstun and 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]

1989 (3)

1988 (1)

1969 (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]

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. (Leipzig) 44, 177–202 (1914).
[CrossRef]

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

Albanese, R.

Albanese, R. A.

R. A. Albanese, J. G. Blaschak, R. L. Medina, and J. W. Penn, “Ultrashort electromagnetic signals: biophysical questions, safety issues, and medical opportunities,” Aviat., Space Environ. Med. 65 Suppl. 5, A116–A120 (1994).

R. A. Albanese, R. L. Medina, and J. W. Penn, “Mathematics, medicine and microwaves,” Inverse Probl. 10, 995–1007 (1994).
[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. A 12, 1501–1512 (1995).
[CrossRef]

R. A. Albanese, J. G. Blaschak, R. L. Medina, and J. W. Penn, “Ultrashort electromagnetic signals: biophysical questions, safety issues, and medical opportunities,” Aviat., Space Environ. Med. 65 Suppl. 5, A116–A120 (1994).

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

Colby, W.

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

Dudley, D. G.

S. L. Dvorak, R. W. Ziolkowski, and D. G. Dudley, “Ultra-wideband electromagnetic pulse propagation in a homogeneous, cold plasma,” Radio Sci. 32, 239–250 (1997).
[CrossRef]

H.-Y. 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]

Durney, C. H.

K. Moten, C. H. Durney, and T. G. Stockham, Jr., “Electromagnetic pulse propagation in dispersive planar dielectrics,” Bioelectromagnetics (N.Y.) 10, 35–49 (1989).
[CrossRef]

Dvorak, S. L.

S. L. Dvorak, R. W. Ziolkowski, and D. G. Dudley, “Ultra-wideband electromagnetic pulse propagation in a homogeneous, cold plasma,” Radio Sci. 32, 239–250 (1997).
[CrossRef]

H.-Y. 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]

Franzen, J.

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]

Hunsberger, F.

R. J. Luebbers and F. Hunsberger, “FDTD for Nth-order dispersive media,” IEEE Trans. Antennas Propag. 40, 1297–1301 (1992).
[CrossRef]

Luebbers, R. J.

R. J. Luebbers and F. Hunsberger, “FDTD for Nth-order dispersive media,” IEEE Trans. Antennas Propag. 40, 1297–1301 (1992).
[CrossRef]

Medina, R.

Medina, R. L.

R. A. Albanese, J. G. Blaschak, R. L. Medina, and J. W. Penn, “Ultrashort electromagnetic signals: biophysical questions, safety issues, and medical opportunities,” Aviat., Space Environ. Med. 65 Suppl. 5, A116–A120 (1994).

R. A. Albanese, R. L. Medina, and J. W. Penn, “Mathematics, medicine and microwaves,” Inverse Probl. 10, 995–1007 (1994).
[CrossRef]

Moten, K.

K. Moten, C. H. Durney, and T. G. Stockham, Jr., “Electromagnetic pulse propagation in dispersive planar dielectrics,” Bioelectromagnetics (N.Y.) 10, 35–49 (1989).
[CrossRef]

Oughstun, K. E.

Pao, H.-Y.

H.-Y. 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]

Penn, J.

Penn, J. W.

R. A. Albanese, J. G. Blaschak, R. L. Medina, and J. W. Penn, “Ultrashort electromagnetic signals: biophysical questions, safety issues, and medical opportunities,” Aviat., Space Environ. Med. 65 Suppl. 5, A116–A120 (1994).

R. A. Albanese, R. L. Medina, and J. W. Penn, “Mathematics, medicine and microwaves,” Inverse Probl. 10, 995–1007 (1994).
[CrossRef]

Sherman, G. C.

Sommerfeld, A.

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

Stockham Jr., T. G.

K. Moten, C. H. Durney, and T. G. Stockham, Jr., “Electromagnetic pulse propagation in dispersive planar dielectrics,” Bioelectromagnetics (N.Y.) 10, 35–49 (1989).
[CrossRef]

Ziolkowski, R. W.

S. L. Dvorak, R. W. Ziolkowski, and D. G. Dudley, “Ultra-wideband electromagnetic pulse propagation in a homogeneous, cold plasma,” Radio Sci. 32, 239–250 (1997).
[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]

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]

Aviat., Space Environ. Med. (1)

R. A. Albanese, J. G. Blaschak, R. L. Medina, and J. W. Penn, “Ultrashort electromagnetic signals: biophysical questions, safety issues, and medical opportunities,” Aviat., Space Environ. Med. 65 Suppl. 5, A116–A120 (1994).

Bioelectromagnetics (N.Y.) (1)

K. Moten, C. H. Durney, and T. G. Stockham, Jr., “Electromagnetic pulse propagation in dispersive planar dielectrics,” Bioelectromagnetics (N.Y.) 10, 35–49 (1989).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

R. J. Luebbers and F. Hunsberger, “FDTD for Nth-order dispersive media,” IEEE Trans. Antennas Propag. 40, 1297–1301 (1992).
[CrossRef]

H.-Y. 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]

Inverse Probl. (1)

R. A. Albanese, R. L. Medina, and J. W. Penn, “Mathematics, medicine and microwaves,” Inverse Probl. 10, 995–1007 (1994).
[CrossRef]

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

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

Phys. Rev. (1)

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

Phys. Rev. A (1)

K. E. Oughstun and 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]

Radio Sci. (1)

S. L. Dvorak, R. W. Ziolkowski, and D. G. Dudley, “Ultra-wideband electromagnetic pulse propagation in a homogeneous, cold plasma,” Radio Sci. 32, 239–250 (1997).
[CrossRef]

Other (23)

R. A. Albanese, J. W. Penn, and R. L. Medina, “An electromagnetic inverse problem in medical science,” in Invariant Imbedding and Inverse Problems, J. P. Corones, G. Kristensson, P. Nelson, and D. L. Seth, eds. (Society for Industrial and Applied Mathematics, Philadelphia, Pa. 1992), Chap. 2, pp. 30–41.

N. Bleistein and R. Handelsman, Asymptotic Expansions of Integrals (Holt, Rinehart & Winston, New York, 1975), p. 390.

N. Bleistein and R. Handelsman, Asymptotic Expansions of Integrals (Dover, New York, 1986), Chap. 9.

Ref. 8, p. 214.

The support of a real-valued function on the real line is the closure of the set of all points at which the function is nonzero.

Ref. 1, p. 190.

E. L. Mokole and S. N. Samaddar, “Pulsed reflection and transmission for a dispersive half space,” Publ. NRL/FR/5340–98–9874 (Naval Research Laboratory, Washington, D.C., 1998).

See Ref. 3, pp. 56–57, and Ref. 8, p. 234.

The fractional bandwidth BF of the single-cycle sine function is given by the formula BF=[2(ωU−ωL)/(ωU+ ωL)]×100%, where ωU and ωL, respectively, are the upper and lower frequencies of the passband of the power spectral density of the single-cycle sine, 4ωc2 sin2(πω/ωc)/(ωc2−ω2)2. If one uses the half-power criterion to determine the passband, then ωU≐1.309ωc and ωL≐ 0.411ωc. Hence BF≐104%, which is ultrawideband according to Ref. 16.

L. L. Pennisi, Elements of Complex Variables (Holt, Rinehart & Winston, New York, 1963), Chap. 5, p. 185.

Ref. 14, p. 264.

See Ref. 4, Chap. 2, and Ref. 8, p. 141.

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

D. H. Lam, “Finite difference methods for transient signal propagation in stratified dispersive media,” Tech. Rep. 3892–1 (Electro-Science Laboratory, Ohio State University, Columbus, Ohio, 1975).

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

K. E. Oughstun, “Propagation of optical pulses in dispersive media,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1978).

J. Marozas and K. Oughstun, “Electromagnetic pulse propagation across a planar interface separating two lossy, dispersive dielectrics,” in Ultra-Wideband, Short-Pulse Electromagnetics 3, C. E. Baum, L. Carin, and A. P. Stone, eds. (Plenum, New York, 1997), pp. 217–230.

A radar is defined to be UWB when its fractional bandwidth exceeds 25%, according to the executive summary of the Office of the Secretary of Defense/Defense Advanced Research Projects Agency study, “Assessment of ultra-wideband (UWB) technology,” IEEE Aerosp. Electron. Mag. 5, 45–49 (1990).

B. Noel, Ultra-Wideband Radar: Proceedings of the First Los Alamos Symposium (CRC, Boca Raton, Fla., 1991).

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

L. Yu. Astanin, A. A. Kostylev, Yu. S. Zinoviev, and A. Ya. Pasmurov, Radar Target Characteristics: Measurements and Applications (CRC, Boca Raton, Fla., 1994).

J. D. Taylor, ed., Introduction to Ultra-Wideband Radar Systems (CRC, Boca Raton, Fla., 1995).

L. Carin and L. B. Felsen, eds., Ultra-Wideband, Short-Pulse Electromagnetics 2 (Plenum, New York, 1995).

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Figures (9)

Fig. 1
Fig. 1

Geometry and specified coordinate frame for a pulsed plane wave Eyi(z, t) in free space (z<0) that depends only on its propagation direction z and is normally incident upon the dispersive medium at z=0. The positive x direction points into the paper, and the dispersive medium occupies the half-space z>0. Free space is characterized by permittivity 0 and permeability μ0, and the dispersive medium is defined by free-space permeability μ0 and frequency-dependent relative permittivity r(ω). n is the dispersive medium’s refractive index.

Fig. 2
Fig. 2

Cut complex plane associated with the Lorentz model of refractive index n for δ<ω0, where δ and ω0 are the attenuation and resonance frequencies, respectively, of the model. The four vectors {ω-ω-, ω-ω-, ω-ω+, ω-ω+} represent the vectors from the branch points {ω-, ω-, ω+, ω+} of n to an arbitrary point ω(=ω+iω) in the complex plane. The magnitudes and angles that correspond to these vectors are {ρ, r, r, ρ} and {α, ψ, ψ, α}, respectively. The branch cuts are the horizontal line segments along ω =-iδ that connect ω+ to ω+ and ω- to ω-.

Fig. 3
Fig. 3

Closed Jordan contour C1+CFU, oriented in the counterclockwise direction, for e(z, τ) when τ<0 consists of the line segment C1 and semicircle CFU. The line segment goes from -F+ia to F+ia, and the semicircle has radius F and center at ω=ia.

Fig. 4
Fig. 4

Closed Jordan contour CFL+Σn=113Cn, oriented in the clockwise direction, for e(z, τ) when τ>0 consists of five horizontal line segments {C1, C3, C5, C9, C11}, four vertical line segments {C2, C7, C8, C13}, four circles {C4, C6, C10, C12} of radius , and one semicircle CFL of radius F. The four circles are centered about the branch points {ω-, ω-, ω+, ω+}, the horizontal line segments are the tops and bottoms of the two branch cuts, and the semicircle is centered at ω=ia.

Fig. 5
Fig. 5

Behavior of reflection coefficient R associated with a single-resonance Lorentz medium versus carrier frequency ωc: (a) real (lighter curve) and imaginary (darker curve) parts of R, (b) magnitude of R, (c) argument of R. The Lorentz medium is defined by ω0=4.0×1016 s-1, b2=20.0×1032 s-2, δ=0.28×1016 s-1.

Fig. 6
Fig. 6

Comparison of the reflected field when the medium’s boundary is perfectly conducting (lighter curves) and the reflected field from the Lorentz medium at z=-10-5 m with ω0=4.0×1016 s-1, b2=20.0×1032 s-2, δ=0.28×1016 s-1 (darker curves) for eight values of the carrier frequency ωc: (a) 0.0001ω0, (b) 0.01ω0, (c) 0.1ω0, (d) 0.25ω0, (e) ω0, (f) 1.26366ω0, (g) 13/8ω0, (h) 10ω0. The horizontal axes are normalized time θ=ct/|z|.

Fig. 7
Fig. 7

Closed Jordan contour C1+CFL, oriented in the clockwise direction, consists of line segment C1 and semicircle CFL. The line segment goes from -F+ia to F+ia, and the semicircle has radius F and center at ω=ia.

Fig. 8
Fig. 8

Early-time (lower curve) and high-frequency (upper curve) fields versus normalized time θ=ct/|z| at z=-10-5 m for ω0=4.0×1016 s-1, b2=20.0×1032 s-2, δ=0.28×1016 s-1.

Fig. 9
Fig. 9

Sommerfeld precursor and associated functions versus normalized time θ at z=10-5 m in the Lorentz medium with ω0=4.0×1016 s-1, b2=20.0×1032 s-2, δ=0.28×1016 s-1: (a) AS(z, θ, 0), (b) AS(z, θ, T0), (c) AS(z, θ)=AS(z, θ, 0)-AS(z, θ, T0), (d) AS(z, θ)-ASO(z, θ). The incident pulsed field is a single-cycle sine at ωc=1016 s-1 with T0=2π/ωc, and ASO is AS evaluated at TR(ω)1 and TI(ω)0.

Tables (2)

Tables Icon

Table 1 Values Along the Line ω = - in the Complex Plane of the Argument of the Refractive Index n and of the Angles α, ψ, α, ψ Measured from This Line to Line Segments Connecting Point ω with Branch Points ω+, ω+, ω-, and ω+ Respectivelya

Tables Icon

Table 2 Choices of Angular Carrier Frequency ωc of the Single-Cycle-Sine Incident Field for Which the Reflected Field Eyr Is Plotted in Fig. 6, As Well as Related Evaluationsa

Equations (135)

Equations on this page are rendered with MathJax. Learn more.

Eyz=μ0Hxt,
Hxz=Dyt.
Gˆ(ω)=12π-+G(t)exp(iωt)dt,
G(t)=CGˆ(ω)exp(-iωt)dω.
2Eˆyz2+k2Eˆy=0,
Hˆx=iμ0ωEˆyz.
(ω)=0z<00r(ω)z>0,
k(ω)=k0(ω)z<0k0(ω)n(ω)z>0,
n(ω)=1-b2ω2-ω02+2iδω1/2=(ω-ω+)(ω-ω-)(ω-ω+)(ω-ω-)1/2.
ω+=ω02-δ2-iδ,
ω-=-ω02-δ2-iδ,
ω+=ω12-δ2-iδ,
ω-=-ω12-δ2-iδ,
Eˆy(z, ω)=Eˆyi(z, ω)+Eˆyr(z, ω),
Hˆx(z, ω)=Hˆxi(z, ω)+Hˆxr(z, ω),
Eˆyi(z, ω)=A(ω)exp[ik0(ω)z],
Eˆyr(z, ω)=A(ω)R(ω)exp[-ik0(ω)z],
Hˆxi(z, ω)=-k0(ω)μ0ωA(ω)exp[ik0(ω)z],
Hˆxr(z, ω)=k0(ω)μ0ωA(ω)R(ω)exp[-ik0(ω)z],
Eˆy(z, ω)=Eˆyt(z, ω)=A(ω)T(ω)exp[ik0(ω)n(ω)z],
Hˆx(z, ω)=Hˆxt(z, ω)=-k0(ω)μ0ωn(ω)A(ω)T(ω)exp[ik0(ω)n(ω)z].
A(ω)=12π-+Eyi(0, t)exp(itω)dt,
R(ω)=1-n(ω)1+n(ω),
T(ω)=21+n(ω).
Eyr(z, t)=CA(ω)R(ω)exp-iωt+zcdω,
Eyt(z, t)=CA(ω)T(ω)expiωzcn(ω)-ctzdω,
Hxr(z, t)=1μ0cCA(ω)R(ω)exp-iωt+zcdω,
Hxt(z, t)=-1μ0cCn(ω)A(ω)T(ω)×expiωzcn(ω)-ctzdω.
Eyi(z, t)=sinωct-zcUt-zc-Ut-zc-T0.
U(τ)=0τ01τ>0.
A(ω)=14πexp(iωT0)S(ω)-14πS(ω)ω±ωc±in2ωc,ω=±ωc,
S(ω)=1ω-ωc-1ω+ωc.
Eyr(z, t)=e(z, tˆ-T0)-e(z, tˆ),
e(z, τ)=14π-+ia+ia exp(-iωτ)R(ω)S(ω)dω.
e(z, τ)=limF+14πC1 exp(-iωτ)R(ω)S(ω)dω=0.
ω-ω+=ρ exp(iα)-π<απ,
ω-ω+=r exp(iψ)0ψ<2π,
ω-ω-=ρ exp(iα)-π<απ,
ω-ω-=r exp(iψ)0ψ<2π.
n(ω)=ρρrr exp[i(α+α-ψ-ψ)/2]=|n(ω)|exp{i arg[n(ω)]},
R(ω)=1-ρρrr1+ρρrr+2ρρrr cosα+α-ψ-ψ2+i-2ρρrr sinα+α-ψ-ψ21+ρρrr+2ρρrr cosα+α-ψ-ψ2,
e(z, τ)=-2πi[Res(-ωc)+Res(ωc)]-14π[IB++IB-+ID++ID-],
IB+=lim0+C5R(ω)exp(-iωτ)S(ω)dω,
IB-=lim0+C3R(ω)exp(-iωτ)S(ω)dω,
ID+=lim0+C9R(ω)exp(-iωτ)S(ω)dω,
ID-=lim0+C11R(ω)exp(-iωτ)S(ω)dω.
ω2=ω02-δ2,ω3=ω12-δ2.
ρ=ω3-ω2-r,
r=r+2ω2,
ρ=ω3+ω2+r.
n(ω)=i(ω3+ω2+r)(ω3-ω2-r)r(r+2ω2)1/2,
R(ω)=[r(r+2ω2)]1/2-i[(ω3+ω2+r)(ω3-ω2-r)]1/2[r(r+2ω2)]1/2+i[(ω3+ω2+r)(ω3-ω2-r)]1/2.
IB+=-exp[-(δ+iω2)τ]×0ω3-ω2 exp(-irτ)1r+ω2-ωc-iδ-1r+ω2+ωc-iδR(ω)dr,
IB-=exp[-(δ+iω2)τ]×0ω3-ω2 exp(-irτ)1r+ω2-ωc-iδ-1r+ω2+ωc-iδR*(ω)dr,
ID+=exp[-(δ-iω2)τ]×0ω3-ω2 exp(irτ)1r+ω2+ωc+iδ-1r+ω2-ωc+iδR*(ω)dr,
ID-=-exp[-(δ-iω2)τ]×0ω3-ω2 exp(irτ)1r+ω2+ωc+iδ-1r+ω2-ωc+iδR(ω)dr.
IB++IB-+ID++ID-=Re8ib2exp[-(δ+iω2)τ]0ω3-ω2 exp(-irτ)×1r+ω2-ωc-iδ-1r+ω2+ωc-iδ×{r(r+2ω2)[b2-r(r+2ω2)]}1/2dr.
Res(ωc)=14πR(ωc)exp(-iωcτ).
-2πi[Res(-ωc)+Res(ωc)]=cos(ωcτ)Im[R(ωc)]-sin(ωcτ)Re[R(ωc)]
e(z, τ)=cos(ωcτ)Im[R(ωc)]-sin(ωcτ)Re[R(ωc)]-2πb2Rei exp[-(δ+iω2)τ]0ω3-ω2 exp(-irτ)×1r+ω2-ωc-iδ-1r+ω2+ωc-iδ×{r(r+2ω2)[b2-r(r+2ω2)]}1/2drU(τ).
e(z, τ)=cos(ωcτ)Im[R(ωc)]-sin(ωcτ)Re[R(ωc)]-2πb2exp(-δτ)[sin(ω2τ)I1(τ)-cos(ω2τ)I2(τ)]U(τ),
I1(τ)=0ω3-ω2N1(r, τ){r(r+2ω2)×[b2-r(r+2ω2)]}1/2dr,
I2(τ)=0ω3-ω2N2(r, τ){r(r+2ω2)×[b2-r(r+2ω2)]}1/2dr,
N1(r, τ)=2ωc[(r+ω2)2-ωc2-δ2]cos(rτ)+4ωcδ(r+ω2)sin(rτ)[(r+ω2-ωc)2+δ2][(r+ω2+ωc)2+δ2],
N2(r, τ)=4ωcδ(r+ω2)cos(rτ)-2ωc[(r+ω2)2-ωc2-δ2]sin(rτ)[(r+ω2-ωc)2+δ2][(r+ω2+ωc)2+δ2].
ρρrrω=ωc=(ωc2-ω12)2+4ωc2δ2(ωc2-ω02)2+4ωc2δ21/4,
α|ω=ωc=π-arctanδω3-ωc0<ωc<ω3π2ωc=ω3arctanδωc-ω3ω3<ωc,
α|ω=ωc=arctanδω3+ωc,
ψ|ω=ωc=π-arctanδω2-ωc0<ωc<ω2π2ωc=ω2arctanδωc-ω2ω2<ωc,
ψ|ω=ωc=arctanδω2+ωc.
Eyr(z, t)=|R(ωc)|sin{ωctˆ-arg[R(ωc)]}×[U(tˆ)-U(tˆ-T0)]+2πb2×{exp(-δtˆ)I(tˆ)sin[ω2tˆ-ν(tˆ)]U(tˆ)-exp[-δ(tˆ-T0)]I(tˆ-T0)sin[ω2(tˆ-T0)-ν(tˆ-T0)]U(tˆ-T0)},
I(τ)={[I1(τ)]2+[I2(τ)]2}1/2,ν(τ)=arctanI2(τ)I1(τ).
-[U(tˆ)-U(tˆ-T0)]sin(ωctˆ),
Ey,etr(z, t)={sin(ωctˆ)Re[R(ωc)]-cos(ωctˆ)Im[R(ωc)]}+2πb2exp(-δtˆ)[sin(ω2tˆ)I1(tˆ)-cos(ω2tˆ)I2(tˆ)].
Eyr(z, t)=-+ia+iaQ(ω)R(ω)A˜(ω-ωc)exp(-iωtˆ)dω--+ia+iaQ(ω)R(ω)A˜(ω+ωc)exp(-iωtˆ)dω,
Q(ω)=14π[exp(iωT0)-1],A˜(ω)=1ω.
Eyr(z, t)=Re-+ia+iaR(ω)2π(ω-ωc)exp[-iω(tˆ-T)]dω--+ia+iaR(ω)2π(ω-ωc)exp(-iωtˆ)dω.
n(ω)=1-b22φ1ω2+φ2ω4+φ3ω6+Rn1(ω-8)+ib222ψ1ω3+2ψ2ω5+2ψ3ω7+Rn2(ω-9),
R(ω)ω-ωc=b24a2ω3+a2ωcω4+a4+a2ωc2ω5+ωc(a4+a2ωc2)ω6+RF1(ω-7)-ib24a3ω4+a3ωcω5+a5+a3ωc2ω6+ωc(a5+a3ωc2)ω7+RF2(ω-8)
φ1=1,
φ2=ω02-4δ2+14b2,
φ3=ω04-12δ2ω02+12b2ω02-3b2δ2+18b4+16δ4,
ψ1=δ,
ψ2=2δ(ω02-2δ2+14b2),
ψ3=δ26ω04-32δ2ω02+3b2ω02+52b4,
a2=φ1,
a3=2ψ1,
a4=φ2+b24φ12,
a5=2ψ2+b2φ1ψ1,
a6=φ3+b242φ1φ2+b24φ13-4ψ12,
a7=2ψ3+b2φ1ψ2+ψ1φ2+3b28ψ1φ12.
-+ia+iaR(ω)2π(ω-ωc)exp(-iωτ)dω.
limF+C1+CFLexp(-iωτ)ωn+1dω=2πn!(-i)n+1τn.
-+ia+iaR(ω)2π(ω-ωc)exp(-iωτ)dω=U(τ)b24a2ωc3!τ3-a3ωc4!τ4-ωc(a4+a2ωc2)5!τ5+ωc(a5+a3ωc2)6!τ6+R1(τ7)+ia22!τ2-a33!τ3-a4+a2ωc24!τ4+a5+a3ωc25!τ5+R2(τ6),
Ey,hfr(z, t)=U(tˆ-T0)b24ωc3!(tˆ-T0)3-2δωc4!(tˆ-T0)4-ωc5!ω02-4δ2+b22+ωc2(tˆ-T0)5+2δωc6!(2ω02-4δ2+b2+ωc2)(tˆ-T0)6+R1[(tˆ-T0)7]-U(tˆ)b24ωc3!tˆ3-2δωc4!tˆ4-ωc5!ω02-4δ2+b22+ωc2tˆ5+2δωc6!(2ω02-4δ2+b2+ωc2)tˆ6+R1(tˆ7),
Ey,hfr(z, t)=-b24ωc3!tˆ3-2δωc4!tˆ4-ωc5!ω02-4δ2+b22+ωc2tˆ5+2δωc6!(2ω02-4δ2+b2+ωc2)tˆ6+R1(tˆ7).
Ey,hfr(z, t)=b24-ωc3!tˆ3+ωc5!ω02+b22+ωc2tˆ5+R1(tˆ7),
Eyt(z, t)=Eyt(z, θ)=-+ia+iaA(ω)T(ω)expzcϕ(ω, θ)dω,
ϕ(ω, θ)=iω[n(ω)-θ].
n(ω)+ωn(ω)-θ=0.
A(ω)=-(i/2)[Mˆ(ω+ωc)-Mˆ(ω-ωc)].
Eyt(z, θ)=Rei2π-+ia+iaT(ω)u˜(ω-ωc)×expzcϕ(ω, θ)dω.
u˜(ω)=exp(iωT0)-1iω,
E yt(z, θ)=A(z, θ, 0)-A(z, θ, T0),
θT0=θ-cT0z,
A(z, θ, T0)=-12πReexp(-iωcT0)-+ia+iaT(ω)ω-ωc×expzcϕ(ω, θT0)dω.
A(z, θ, T0)=-U(θT0)2πReexp(-iωcT0) PD(θ)T(ω)ω-ωc×expzcϕ(ω, θT0)dω-U(θT0)2π×Reexp(-iωcT0)PN(θ)T(ω)ω-ωc×expzcϕ(ω, θT0)dω-U(θT0)2π×Re-2πi exp(-iωcT0)× ResiduesenclosedbyCJ(θ)=AS(z, θ, T0)+AB(z, θ, T0)+AC(z, θ, T0),
Eyt(z, θ)=AS(z, θ)+AB(z, θ)+AC(z, θ).
AS(z, θ, T0)=-U(θT0)2πReexp(-iωcT0)×PD(θ)T(ω)ω-ωcexp[(z/c)ϕ(ω, θT0)]dω
U(θT0)Re(exp[izβ(θT0)]×{2α(θT0)exp[-i(π/2)]}ν{γ0(θT0)Jν[zα(θT0)]+2α(θT0)exp[-i(π/2)]γ1(θT0)Jν+1[zα(θT0)]}),
ν˜(ω)=i/ω,
α(θT0)=(i/2c)[ϕ(ωD1, θT0)-ϕ(ωD2, θT0)],
β(θT0)=(i/2c)[ϕ(ωD1, θT0)+ϕ(ωD2, θT0)],
γ0(θT0)=12v˜(ωD1(θT0)-ωc)12α(θT0)1+ν×T[ωD1(θT0)]4cα3(θT0)iϕωω[ωD1(θT0),θT0]1/2+v˜[ωD2(θT0)-ωc]-12α(θT0)1+ν×T[ωD2(θT0)]-4cα3(θT0)iϕωω[ωD2(θT0),θT0]1/2,
γ1(θT0)=14α(θT0)v˜[ωD1(θT0)-ωc]12α(θT0)1+ν×T[ωD1(θT0)]4cα3(θT0)iϕωω[ωD1(θT0),θT0]1/2-v˜[ωD2(θT0)-ωc]-12α(θT0)1+ν×T[ωD2(θT0)]-4cα3(θT0)iϕωω[ωD2(θT0),θT0]1/2.
α(θ)=2[-a1(θ)(θ-1)]1/2+O[(θ-1)3/2],
β(θ)=a0(θ)+O[(θ-1)],
4α3(θ)ψ(2)(ω±, θ)1/2=4|a1(θ)|{1+O[(θ-1)1/2]}.
ωD1(θT0)ξ(θT0)-iδ[1+η(θT0)],
ωD2(θT0)-ξ(θT0)-iδ[1+η(θT0)],
ϕ(ω, θ)iω(1-θ)-ib22(ω+iδ),
ξ(θT0)=ω02-δ2+b2θT02θT02-11/2,
η(θT0)=ξ-2(θT0)δ227+b2θT02-1.
TR(ω)=2[1+nR(ω)][1+nR(ω)]2+nI2(ω)=21+ρρrr cosα+α-ψ-ψ21+ρρrr+2ρρrr cosα+α-ψ-ψ2,
TI(ω)=-2nI(ω)[1+nR(ω)]2+nI2(ω)=-2ρρrr sinα+α-ψ-ψ21+ρρrr+2ρρrr cosα+α-ψ-ψ2,
nR[ωD1,2(θT0)]1-b2ξ-2(θT0){1+δ2ξ-2(θT0)[1-η2(θT0)]}2{1+δ2ξ-2(θT0)[1+η(θT0)]2}{1+δ2ξ-2(θT0)[1-η(θT0)]2},
nI[ωD1,2(θT0)]2δb2ξ-3(θT0)η(θT0)2{1+δ2ξ-2(θT0)[1+η(θT0)]2}{1+δ2ξ-2(θT0)[1-η(θT0)]2}.
AS(z, θ, T0)U(θT0)ξ(θT0)2bθT0-1+b221ξ2(θT0)+δ2[1-η(θT0)]21/2
×exp-zδc[θT0-1][1+η(θT0)]+b221-η(θT0)ξ2(θT0)+δ2[1-η(θT0)]2×-TR[ωD1(θT0)]{3/2δ[ξ(θT0)-ωc][1-η(θT0)]+δξ(θT0)[1+η(θT0)]}[ξ(θT0)-ωc]2+δ2[1+η(θT0)]2-TI[ωD1(θT0)]{ξ(θT0)[ξ(θT0)-ωc]-3/2δ2[1-η2(θT0)]}[ξ(θT0)-ωc]2+δ2[1+η(θT0)]2+TR[ωD1(θT0)]{3/2δ[ξ(θT0)+ωc][1-η(θT0)]+δξ(θT0)[1+η(θT0)]}[ξ(θT0)+ωc]2+δ2[1+η(θT0)]2
+TI[ωD1(θT0)]{ξ(θT0)[ξ(θT0)+ωc]-3/2δ2[1-η2(θT0)]}[ξ(θT0)+ωc]2+δ2[1+η(θT0)]2×J0zξ(θT0)cθT0-1+b221ξ2(θT0)+δ2[1-η(θT0)]2+TR[ωD1(θT0)]{ξ(θT0)[ξ(θT0)-ωc]-3/2δ2[1-η2(θT0)]}[ξ(θT0)-ωc]2+δ2[1+η(θT0)]2-TI[ωD1(θT0)]{3/2δ[ξ(θT0)-ωc][1-η(θT0)]+δξ(θT0)[1+η(θT0)]}[ξ(θT0)-ωc]2+δ2[1+η(θT0)]2-TR[ωD1(θT0)]{ξ(θT0)[ξ(θT0)+ωc]-3/2δ2[1-η2(θT0)]}[ξ(θT0)+ωc]2+δ2[1+η(θT0)]2+TI[ωD1(θT0)]{3/2δ[ξ(θT0)+ωc][1-η(θT0)]+δξ(θT0)[1+η(θT0)]}[ξ(θT0)+ωc]2+δ2[1+η(θT0)]2×J1zξ(θT0)cθT0-1+b221ξ2(θT0)+δ2[1-η(θT0)]2.
±(θ-1)exp{-(zδ/c)[θ-2(θ2+1)(θ-1)+(2θ2)-1(θ+1)2(θ-1)2]}.
±(θ-1)exp[-(2zδ/c)(θ-1)].
±(θ-1)exp[-(zδ/(2c))(θ-1)2].
1TR[ωD1(θ)]1.02,0TI[ωD1(θ)]0.00062
T(ωc)exp[(z/c)ϕ(ωc, θT0)]

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