Abstract

We examine electromagnetic pulse propagation in anomalously dispersive media, using the Debye model as an example. Short-pulse, long-pulse, short-time, and long-time approximations and amplitude rate-of-decay estimates are derived with asymptotic methods. We also study the following problem: If we know only the peak amplitude and the energy density of an incident pulse, what can be said about the amplitude of the propagated pulse? We provide tight upper and lower bounds for the propagated amplitude, which may be useful in controlling the electromagnetic interference or the damage produced in dispersive media. We explain a factor-of-nine effect in the speed of pulses in a Debye model for water, which seems to have been previously unnoticed, and we also explain some observations from experimental studies of pulse propagation in biological materials. Finally, we propose some guidelines for sample size in transmission time-domain spectroscopy studies of dielectrics. Most of our results are easily extended to multiple space dimensions.

© 1996 Optical Society of America

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  1. J. C. Lin, C. K. Lam, “Coupling of Gaussian electromagnetic pulse into a muscle–bone model of biological structure,” J. Microwave Power 11, 67–75 (1976).
  2. P. Debye, Polar Molecules (Dover, New York, 1929). See p. 90.
  3. J. D. Jackson, Classical Electrodynamics2nd ed. (Wiley, New York, 1975). See Secs. 7.5e, 7.7, and 7.11.
  4. W. D. Hurt, “Measurement of specific absorption rate in human phantoms exposed to simulated Air Force radar emissions,” USAFSAM-TR-84-16 (Armstrong Laboratory, Brooks Air Force Base, Tex., 1984). The equation for σshould have 0.884f2instead of 0.8842.
  5. W. D. Hurt, “Multiterm Debye dispersion relations for permittivity of muscle,”IEEE Trans. Biomed. Eng. BME-32, 60–64 (1985).
    [CrossRef]
  6. R. J. Krueger, R. L. Ochs, “A Green’s function approach to the determination of internal fields,” Wave Motion 11, 525–543 (1989).
    [CrossRef]
  7. R. Krueger, R. Winther, “Internal field, inhomogeneous dispersive medium” and other unpublished notes and computer programs, Ames Laboratory, Ames, Iowa 50011 (personal communication, 1989).
  8. G. Kristensson, “Direct and inverse scattering problems in dispersive media—Green’s functions and invariant imbedding techniques,” in Direct and Inverse Boundary Value Problems, Vol. 37 of Methoden und Verfahren der mathematischen Physik, R. Kleinman, R. Kress, E. Martensen, eds. (Peter Lang, Frankfurt, Germany, 1991).
  9. T. M. Roberts, M. Hobart, “Energy velocity, damping, and elementary inversion,” J. Opt. Soc. Am. A 9, 1091–1101 (1992).
    [CrossRef]
  10. A. Karlsson, H. Otterheim, R. Stewart, “Transient wave propagation in composite media: Green’s function approach,” J. Opt. Soc. Am. A 10, 886–895 (1993).
    [CrossRef]
  11. S. He, Y. Hu, S. Ström, “Time domain Green functions technique for a point source over a dissipative stratified halfspace with a phase velocity mismatch at the surface,” Wave Motion 17, 241–254 (1993).
    [CrossRef]
  12. G. Kristensson, S. Rikte, “Scattering of transient electromagnetic waves in reciprocal bi-isotropic media,”J. Electromagn. Waves Appl. 6, 1517–1535 (1992).
    [CrossRef]
  13. A. Karlsson, H. Otterheim, R. Stewart, “Electromagnetic fields in an inhomogeneous plasma from obliquely incident transient plane waves,” Radio Sci. 28, 365–378 (1993).
    [CrossRef]
  14. J.-Y. Chen, C. M. Furse, O. P. Gandhi, “A simple convolution procedure for calculating currents induced in the human body for exposure to electromagnetic pulses,”IEEE Trans. Microwave Theor. Tech. 42, 1172–1175 (1994).
    [CrossRef]
  15. G. A. Kriegsmann, J. H. C. Luke, “Rapid pulse responses for scattering problems,”J. Comp. Phys. 111, 390–398 (1994).
    [CrossRef]
  16. F. John, Partial Differential Equations, 4th ed. (Springer-Verlag, New York, 1991). See pp. 152 and 160.
  17. G. B. Whitham, Linear and Nonlinear Waves (Wiley, New York, 1974).
  18. E. Zauderer, Partial Differential Equations of Applied Mathematics (Wiley, New York, 1983), pp. 218–223 and 615–618.
  19. L. Brillouin, Wave Propagation and Group Velocity (Academic, San Diego, Calif., 1960). See Chaps. II and III, which are English translations of papers by A. Sommerfeld, L. Brillouin in Ann. Phys. (1914).
  20. K. E. Oughstun, “Propagation of optical pulses in dispersive media,” Ph.D. disseration (University of Rochester, Rochester, N.Y., 1978).
  21. S. D. Conte, Carl de Boor, Elementary Numerical Analysis, 3rd ed. (McGraw-Hill, New York, 1980), p. 26.
  22. M. Reed, B. Simon, Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness (Academic, New York, 1975), pp. 28, 29, and 32.
  23. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Sec. 2.19.
  24. P. Linz, Analytical and Numerical Methods for Volterra Equations (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1985), theorems 3.1–3.3 and 3.5.
    [CrossRef]
  25. R. M. Joseph, S. C. Hagness, A. Taflove, “Direct time integration of Maxwell’s equations in linear dispersive media with absorption for scattering and propagation of femtosecond electromagnetic pulses,” Opt. Lett. 16, 1412–1414 (1991).
    [CrossRef] [PubMed]
  26. F. J. German, “The analysis of pulse propagation in linear dispersive media with absorption by the finite-difference time-domain method,” unpublished report and computer program, Department of Electrical Engineering, Auburn University, Auburn, Alabama 36849 (personal communication, 1993).
  27. C. E. Baum, “EMP simulators for various types of nuclear EMP environments: an interim categorization,”IEEE Trans. Antennas Propag. AP-26, 35–53 (1978). See Sec. II.
    [CrossRef]
  28. J.-Y. Chen, O. P. Gandhi, “Currents induced in an anatomically based model of a human for exposure to vertically polarized electromagnetic pulses,”IEEE Trans. Microwave Theor. Tech. 39, 31–39 (1991). See Figs. 3(a) and 4.
    [CrossRef]
  29. F. M. Tesche, “Prediction of the Eand Hfields produced by the Swiss mobile EMP simulator (MEMPS),” IEEE Trans. Electromagn. Compat. 4, 381–390 (1992). See Figs. 7, 10(a), 13(b), and 15(b).
    [CrossRef]
  30. In Subsec. 7.11g of Ref. 3, take large-depth limits while keeping the relative time |t− t1| fixed. Then use Eqs. (7.146)–(7.148), (7.141), and (3.89) of Ref. 3 to show that the large-depth, fixed-relative-time values of the Brillouin precursor decay as (depth)−1/3if the amplitude A(ω) is nonzero at ω= 0 and that the values decay as (depth)−2/3if A(ω) has a first-order zero at ω= 0. The error in these asymptotics is O(ω4).
  31. R. K. Adair, “Ultrashort microwave signals: a didactic discussion,” Aviation Space Environ. Med. 66, 792–794 (1995). See Figs. 1 and 2.
  32. B. Gestblom, E. Noreland, “Transmission methods in dielectric time domain spectroscopy,”J. Phys. Chem. 81, 782–788 (1977).
    [CrossRef]
  33. R. S. Beezley, R. J. Krueger, “An electromagnetic inverse problem for dispersive media,”J. Math. Phys. 26, 317–325 (1985), Sec. IV. Equate their function R(t) with our G(0, t). See also our Eq. (A3).
    [CrossRef]
  34. M. R. Querry, D. M. Wieliczka, D. J. Segelstein, “Water (H2O),” in Handbook of Optical Constants of Solids II, E. D. Palik, ed. (Academic, Boston, 1991), pp. 1059–1065.

1995 (1)

R. K. Adair, “Ultrashort microwave signals: a didactic discussion,” Aviation Space Environ. Med. 66, 792–794 (1995). See Figs. 1 and 2.

1994 (2)

J.-Y. Chen, C. M. Furse, O. P. Gandhi, “A simple convolution procedure for calculating currents induced in the human body for exposure to electromagnetic pulses,”IEEE Trans. Microwave Theor. Tech. 42, 1172–1175 (1994).
[CrossRef]

G. A. Kriegsmann, J. H. C. Luke, “Rapid pulse responses for scattering problems,”J. Comp. Phys. 111, 390–398 (1994).
[CrossRef]

1993 (3)

A. Karlsson, H. Otterheim, R. Stewart, “Transient wave propagation in composite media: Green’s function approach,” J. Opt. Soc. Am. A 10, 886–895 (1993).
[CrossRef]

S. He, Y. Hu, S. Ström, “Time domain Green functions technique for a point source over a dissipative stratified halfspace with a phase velocity mismatch at the surface,” Wave Motion 17, 241–254 (1993).
[CrossRef]

A. Karlsson, H. Otterheim, R. Stewart, “Electromagnetic fields in an inhomogeneous plasma from obliquely incident transient plane waves,” Radio Sci. 28, 365–378 (1993).
[CrossRef]

1992 (3)

G. Kristensson, S. Rikte, “Scattering of transient electromagnetic waves in reciprocal bi-isotropic media,”J. Electromagn. Waves Appl. 6, 1517–1535 (1992).
[CrossRef]

F. M. Tesche, “Prediction of the Eand Hfields produced by the Swiss mobile EMP simulator (MEMPS),” IEEE Trans. Electromagn. Compat. 4, 381–390 (1992). See Figs. 7, 10(a), 13(b), and 15(b).
[CrossRef]

T. M. Roberts, M. Hobart, “Energy velocity, damping, and elementary inversion,” J. Opt. Soc. Am. A 9, 1091–1101 (1992).
[CrossRef]

1991 (2)

R. M. Joseph, S. C. Hagness, A. Taflove, “Direct time integration of Maxwell’s equations in linear dispersive media with absorption for scattering and propagation of femtosecond electromagnetic pulses,” Opt. Lett. 16, 1412–1414 (1991).
[CrossRef] [PubMed]

J.-Y. Chen, O. P. Gandhi, “Currents induced in an anatomically based model of a human for exposure to vertically polarized electromagnetic pulses,”IEEE Trans. Microwave Theor. Tech. 39, 31–39 (1991). See Figs. 3(a) and 4.
[CrossRef]

1989 (1)

R. J. Krueger, R. L. Ochs, “A Green’s function approach to the determination of internal fields,” Wave Motion 11, 525–543 (1989).
[CrossRef]

1985 (2)

W. D. Hurt, “Multiterm Debye dispersion relations for permittivity of muscle,”IEEE Trans. Biomed. Eng. BME-32, 60–64 (1985).
[CrossRef]

R. S. Beezley, R. J. Krueger, “An electromagnetic inverse problem for dispersive media,”J. Math. Phys. 26, 317–325 (1985), Sec. IV. Equate their function R(t) with our G(0, t). See also our Eq. (A3).
[CrossRef]

1978 (1)

C. E. Baum, “EMP simulators for various types of nuclear EMP environments: an interim categorization,”IEEE Trans. Antennas Propag. AP-26, 35–53 (1978). See Sec. II.
[CrossRef]

1977 (1)

B. Gestblom, E. Noreland, “Transmission methods in dielectric time domain spectroscopy,”J. Phys. Chem. 81, 782–788 (1977).
[CrossRef]

1976 (1)

J. C. Lin, C. K. Lam, “Coupling of Gaussian electromagnetic pulse into a muscle–bone model of biological structure,” J. Microwave Power 11, 67–75 (1976).

Adair, R. K.

R. K. Adair, “Ultrashort microwave signals: a didactic discussion,” Aviation Space Environ. Med. 66, 792–794 (1995). See Figs. 1 and 2.

Baum, C. E.

C. E. Baum, “EMP simulators for various types of nuclear EMP environments: an interim categorization,”IEEE Trans. Antennas Propag. AP-26, 35–53 (1978). See Sec. II.
[CrossRef]

Beezley, R. S.

R. S. Beezley, R. J. Krueger, “An electromagnetic inverse problem for dispersive media,”J. Math. Phys. 26, 317–325 (1985), Sec. IV. Equate their function R(t) with our G(0, t). See also our Eq. (A3).
[CrossRef]

Boor, Carl de

S. D. Conte, Carl de Boor, Elementary Numerical Analysis, 3rd ed. (McGraw-Hill, New York, 1980), p. 26.

Brillouin, L.

L. Brillouin, Wave Propagation and Group Velocity (Academic, San Diego, Calif., 1960). See Chaps. II and III, which are English translations of papers by A. Sommerfeld, L. Brillouin in Ann. Phys. (1914).

Chen, J.-Y.

J.-Y. Chen, C. M. Furse, O. P. Gandhi, “A simple convolution procedure for calculating currents induced in the human body for exposure to electromagnetic pulses,”IEEE Trans. Microwave Theor. Tech. 42, 1172–1175 (1994).
[CrossRef]

J.-Y. Chen, O. P. Gandhi, “Currents induced in an anatomically based model of a human for exposure to vertically polarized electromagnetic pulses,”IEEE Trans. Microwave Theor. Tech. 39, 31–39 (1991). See Figs. 3(a) and 4.
[CrossRef]

Conte, S. D.

S. D. Conte, Carl de Boor, Elementary Numerical Analysis, 3rd ed. (McGraw-Hill, New York, 1980), p. 26.

Debye, P.

P. Debye, Polar Molecules (Dover, New York, 1929). See p. 90.

Furse, C. M.

J.-Y. Chen, C. M. Furse, O. P. Gandhi, “A simple convolution procedure for calculating currents induced in the human body for exposure to electromagnetic pulses,”IEEE Trans. Microwave Theor. Tech. 42, 1172–1175 (1994).
[CrossRef]

Gandhi, O. P.

J.-Y. Chen, C. M. Furse, O. P. Gandhi, “A simple convolution procedure for calculating currents induced in the human body for exposure to electromagnetic pulses,”IEEE Trans. Microwave Theor. Tech. 42, 1172–1175 (1994).
[CrossRef]

J.-Y. Chen, O. P. Gandhi, “Currents induced in an anatomically based model of a human for exposure to vertically polarized electromagnetic pulses,”IEEE Trans. Microwave Theor. Tech. 39, 31–39 (1991). See Figs. 3(a) and 4.
[CrossRef]

German, F. J.

F. J. German, “The analysis of pulse propagation in linear dispersive media with absorption by the finite-difference time-domain method,” unpublished report and computer program, Department of Electrical Engineering, Auburn University, Auburn, Alabama 36849 (personal communication, 1993).

Gestblom, B.

B. Gestblom, E. Noreland, “Transmission methods in dielectric time domain spectroscopy,”J. Phys. Chem. 81, 782–788 (1977).
[CrossRef]

Hagness, S. C.

He, S.

S. He, Y. Hu, S. Ström, “Time domain Green functions technique for a point source over a dissipative stratified halfspace with a phase velocity mismatch at the surface,” Wave Motion 17, 241–254 (1993).
[CrossRef]

Hobart, M.

Hu, Y.

S. He, Y. Hu, S. Ström, “Time domain Green functions technique for a point source over a dissipative stratified halfspace with a phase velocity mismatch at the surface,” Wave Motion 17, 241–254 (1993).
[CrossRef]

Hurt, W. D.

W. D. Hurt, “Multiterm Debye dispersion relations for permittivity of muscle,”IEEE Trans. Biomed. Eng. BME-32, 60–64 (1985).
[CrossRef]

W. D. Hurt, “Measurement of specific absorption rate in human phantoms exposed to simulated Air Force radar emissions,” USAFSAM-TR-84-16 (Armstrong Laboratory, Brooks Air Force Base, Tex., 1984). The equation for σshould have 0.884f2instead of 0.8842.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics2nd ed. (Wiley, New York, 1975). See Secs. 7.5e, 7.7, and 7.11.

John, F.

F. John, Partial Differential Equations, 4th ed. (Springer-Verlag, New York, 1991). See pp. 152 and 160.

Joseph, R. M.

Karlsson, A.

A. Karlsson, H. Otterheim, R. Stewart, “Transient wave propagation in composite media: Green’s function approach,” J. Opt. Soc. Am. A 10, 886–895 (1993).
[CrossRef]

A. Karlsson, H. Otterheim, R. Stewart, “Electromagnetic fields in an inhomogeneous plasma from obliquely incident transient plane waves,” Radio Sci. 28, 365–378 (1993).
[CrossRef]

Kriegsmann, G. A.

G. A. Kriegsmann, J. H. C. Luke, “Rapid pulse responses for scattering problems,”J. Comp. Phys. 111, 390–398 (1994).
[CrossRef]

Kristensson, G.

G. Kristensson, S. Rikte, “Scattering of transient electromagnetic waves in reciprocal bi-isotropic media,”J. Electromagn. Waves Appl. 6, 1517–1535 (1992).
[CrossRef]

G. Kristensson, “Direct and inverse scattering problems in dispersive media—Green’s functions and invariant imbedding techniques,” in Direct and Inverse Boundary Value Problems, Vol. 37 of Methoden und Verfahren der mathematischen Physik, R. Kleinman, R. Kress, E. Martensen, eds. (Peter Lang, Frankfurt, Germany, 1991).

Krueger, R.

R. Krueger, R. Winther, “Internal field, inhomogeneous dispersive medium” and other unpublished notes and computer programs, Ames Laboratory, Ames, Iowa 50011 (personal communication, 1989).

Krueger, R. J.

R. J. Krueger, R. L. Ochs, “A Green’s function approach to the determination of internal fields,” Wave Motion 11, 525–543 (1989).
[CrossRef]

R. S. Beezley, R. J. Krueger, “An electromagnetic inverse problem for dispersive media,”J. Math. Phys. 26, 317–325 (1985), Sec. IV. Equate their function R(t) with our G(0, t). See also our Eq. (A3).
[CrossRef]

Lam, C. K.

J. C. Lin, C. K. Lam, “Coupling of Gaussian electromagnetic pulse into a muscle–bone model of biological structure,” J. Microwave Power 11, 67–75 (1976).

Lin, J. C.

J. C. Lin, C. K. Lam, “Coupling of Gaussian electromagnetic pulse into a muscle–bone model of biological structure,” J. Microwave Power 11, 67–75 (1976).

Linz, P.

P. Linz, Analytical and Numerical Methods for Volterra Equations (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1985), theorems 3.1–3.3 and 3.5.
[CrossRef]

Luke, J. H. C.

G. A. Kriegsmann, J. H. C. Luke, “Rapid pulse responses for scattering problems,”J. Comp. Phys. 111, 390–398 (1994).
[CrossRef]

Noreland, E.

B. Gestblom, E. Noreland, “Transmission methods in dielectric time domain spectroscopy,”J. Phys. Chem. 81, 782–788 (1977).
[CrossRef]

Ochs, R. L.

R. J. Krueger, R. L. Ochs, “A Green’s function approach to the determination of internal fields,” Wave Motion 11, 525–543 (1989).
[CrossRef]

Otterheim, H.

A. Karlsson, H. Otterheim, R. Stewart, “Electromagnetic fields in an inhomogeneous plasma from obliquely incident transient plane waves,” Radio Sci. 28, 365–378 (1993).
[CrossRef]

A. Karlsson, H. Otterheim, R. Stewart, “Transient wave propagation in composite media: Green’s function approach,” J. Opt. Soc. Am. A 10, 886–895 (1993).
[CrossRef]

Oughstun, K. E.

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

Querry, M. R.

M. R. Querry, D. M. Wieliczka, D. J. Segelstein, “Water (H2O),” in Handbook of Optical Constants of Solids II, E. D. Palik, ed. (Academic, Boston, 1991), pp. 1059–1065.

Reed, M.

M. Reed, B. Simon, Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness (Academic, New York, 1975), pp. 28, 29, and 32.

Rikte, S.

G. Kristensson, S. Rikte, “Scattering of transient electromagnetic waves in reciprocal bi-isotropic media,”J. Electromagn. Waves Appl. 6, 1517–1535 (1992).
[CrossRef]

Roberts, T. M.

Segelstein, D. J.

M. R. Querry, D. M. Wieliczka, D. J. Segelstein, “Water (H2O),” in Handbook of Optical Constants of Solids II, E. D. Palik, ed. (Academic, Boston, 1991), pp. 1059–1065.

Simon, B.

M. Reed, B. Simon, Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness (Academic, New York, 1975), pp. 28, 29, and 32.

Stewart, R.

A. Karlsson, H. Otterheim, R. Stewart, “Electromagnetic fields in an inhomogeneous plasma from obliquely incident transient plane waves,” Radio Sci. 28, 365–378 (1993).
[CrossRef]

A. Karlsson, H. Otterheim, R. Stewart, “Transient wave propagation in composite media: Green’s function approach,” J. Opt. Soc. Am. A 10, 886–895 (1993).
[CrossRef]

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Sec. 2.19.

Ström, S.

S. He, Y. Hu, S. Ström, “Time domain Green functions technique for a point source over a dissipative stratified halfspace with a phase velocity mismatch at the surface,” Wave Motion 17, 241–254 (1993).
[CrossRef]

Taflove, A.

Tesche, F. M.

F. M. Tesche, “Prediction of the Eand Hfields produced by the Swiss mobile EMP simulator (MEMPS),” IEEE Trans. Electromagn. Compat. 4, 381–390 (1992). See Figs. 7, 10(a), 13(b), and 15(b).
[CrossRef]

Whitham, G. B.

G. B. Whitham, Linear and Nonlinear Waves (Wiley, New York, 1974).

Wieliczka, D. M.

M. R. Querry, D. M. Wieliczka, D. J. Segelstein, “Water (H2O),” in Handbook of Optical Constants of Solids II, E. D. Palik, ed. (Academic, Boston, 1991), pp. 1059–1065.

Winther, R.

R. Krueger, R. Winther, “Internal field, inhomogeneous dispersive medium” and other unpublished notes and computer programs, Ames Laboratory, Ames, Iowa 50011 (personal communication, 1989).

Zauderer, E.

E. Zauderer, Partial Differential Equations of Applied Mathematics (Wiley, New York, 1983), pp. 218–223 and 615–618.

Aviation Space Environ. Med. (1)

R. K. Adair, “Ultrashort microwave signals: a didactic discussion,” Aviation Space Environ. Med. 66, 792–794 (1995). See Figs. 1 and 2.

IEEE Trans. Antennas Propag. (1)

C. E. Baum, “EMP simulators for various types of nuclear EMP environments: an interim categorization,”IEEE Trans. Antennas Propag. AP-26, 35–53 (1978). See Sec. II.
[CrossRef]

IEEE Trans. Biomed. Eng. (1)

W. D. Hurt, “Multiterm Debye dispersion relations for permittivity of muscle,”IEEE Trans. Biomed. Eng. BME-32, 60–64 (1985).
[CrossRef]

IEEE Trans. Electromagn. Compat. (1)

F. M. Tesche, “Prediction of the Eand Hfields produced by the Swiss mobile EMP simulator (MEMPS),” IEEE Trans. Electromagn. Compat. 4, 381–390 (1992). See Figs. 7, 10(a), 13(b), and 15(b).
[CrossRef]

IEEE Trans. Microwave Theor. Tech. (2)

J.-Y. Chen, O. P. Gandhi, “Currents induced in an anatomically based model of a human for exposure to vertically polarized electromagnetic pulses,”IEEE Trans. Microwave Theor. Tech. 39, 31–39 (1991). See Figs. 3(a) and 4.
[CrossRef]

J.-Y. Chen, C. M. Furse, O. P. Gandhi, “A simple convolution procedure for calculating currents induced in the human body for exposure to electromagnetic pulses,”IEEE Trans. Microwave Theor. Tech. 42, 1172–1175 (1994).
[CrossRef]

J. Comp. Phys. (1)

G. A. Kriegsmann, J. H. C. Luke, “Rapid pulse responses for scattering problems,”J. Comp. Phys. 111, 390–398 (1994).
[CrossRef]

J. Electromagn. Waves Appl. (1)

G. Kristensson, S. Rikte, “Scattering of transient electromagnetic waves in reciprocal bi-isotropic media,”J. Electromagn. Waves Appl. 6, 1517–1535 (1992).
[CrossRef]

J. Math. Phys. (1)

R. S. Beezley, R. J. Krueger, “An electromagnetic inverse problem for dispersive media,”J. Math. Phys. 26, 317–325 (1985), Sec. IV. Equate their function R(t) with our G(0, t). See also our Eq. (A3).
[CrossRef]

J. Microwave Power (1)

J. C. Lin, C. K. Lam, “Coupling of Gaussian electromagnetic pulse into a muscle–bone model of biological structure,” J. Microwave Power 11, 67–75 (1976).

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

J. Phys. Chem. (1)

B. Gestblom, E. Noreland, “Transmission methods in dielectric time domain spectroscopy,”J. Phys. Chem. 81, 782–788 (1977).
[CrossRef]

Opt. Lett. (1)

Radio Sci. (1)

A. Karlsson, H. Otterheim, R. Stewart, “Electromagnetic fields in an inhomogeneous plasma from obliquely incident transient plane waves,” Radio Sci. 28, 365–378 (1993).
[CrossRef]

Wave Motion (2)

S. He, Y. Hu, S. Ström, “Time domain Green functions technique for a point source over a dissipative stratified halfspace with a phase velocity mismatch at the surface,” Wave Motion 17, 241–254 (1993).
[CrossRef]

R. J. Krueger, R. L. Ochs, “A Green’s function approach to the determination of internal fields,” Wave Motion 11, 525–543 (1989).
[CrossRef]

Other (17)

R. Krueger, R. Winther, “Internal field, inhomogeneous dispersive medium” and other unpublished notes and computer programs, Ames Laboratory, Ames, Iowa 50011 (personal communication, 1989).

G. Kristensson, “Direct and inverse scattering problems in dispersive media—Green’s functions and invariant imbedding techniques,” in Direct and Inverse Boundary Value Problems, Vol. 37 of Methoden und Verfahren der mathematischen Physik, R. Kleinman, R. Kress, E. Martensen, eds. (Peter Lang, Frankfurt, Germany, 1991).

P. Debye, Polar Molecules (Dover, New York, 1929). See p. 90.

J. D. Jackson, Classical Electrodynamics2nd ed. (Wiley, New York, 1975). See Secs. 7.5e, 7.7, and 7.11.

W. D. Hurt, “Measurement of specific absorption rate in human phantoms exposed to simulated Air Force radar emissions,” USAFSAM-TR-84-16 (Armstrong Laboratory, Brooks Air Force Base, Tex., 1984). The equation for σshould have 0.884f2instead of 0.8842.

F. John, Partial Differential Equations, 4th ed. (Springer-Verlag, New York, 1991). See pp. 152 and 160.

G. B. Whitham, Linear and Nonlinear Waves (Wiley, New York, 1974).

E. Zauderer, Partial Differential Equations of Applied Mathematics (Wiley, New York, 1983), pp. 218–223 and 615–618.

L. Brillouin, Wave Propagation and Group Velocity (Academic, San Diego, Calif., 1960). See Chaps. II and III, which are English translations of papers by A. Sommerfeld, L. Brillouin in Ann. Phys. (1914).

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

S. D. Conte, Carl de Boor, Elementary Numerical Analysis, 3rd ed. (McGraw-Hill, New York, 1980), p. 26.

M. Reed, B. Simon, Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness (Academic, New York, 1975), pp. 28, 29, and 32.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Sec. 2.19.

P. Linz, Analytical and Numerical Methods for Volterra Equations (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1985), theorems 3.1–3.3 and 3.5.
[CrossRef]

F. J. German, “The analysis of pulse propagation in linear dispersive media with absorption by the finite-difference time-domain method,” unpublished report and computer program, Department of Electrical Engineering, Auburn University, Auburn, Alabama 36849 (personal communication, 1993).

In Subsec. 7.11g of Ref. 3, take large-depth limits while keeping the relative time |t− t1| fixed. Then use Eqs. (7.146)–(7.148), (7.141), and (3.89) of Ref. 3 to show that the large-depth, fixed-relative-time values of the Brillouin precursor decay as (depth)−1/3if the amplitude A(ω) is nonzero at ω= 0 and that the values decay as (depth)−2/3if A(ω) has a first-order zero at ω= 0. The error in these asymptotics is O(ω4).

M. R. Querry, D. M. Wieliczka, D. J. Segelstein, “Water (H2O),” in Handbook of Optical Constants of Solids II, E. D. Palik, ed. (Academic, Boston, 1991), pp. 1059–1065.

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Figures (9)

Fig. 1
Fig. 1

Electric fields computed at three depths with finite-difference and Green-function methods. The time traces overlap, which validates the methods. At the three depths graphed, the fields are precisely 0 until 5.47, 8.20, and 9.60 ps, respectively.

Fig. 2
Fig. 2

Validation of magnetic-field computations.

Fig. 3
Fig. 3

Time of arrival of the peak of the Green function G(z, t) at various depths in water.

Fig. 4
Fig. 4

Predicted and computed decays of the peak field as a function of depth.

Fig. 5
Fig. 5

Predicted and computed responses, as functions of time, at various depths.

Fig. 6
Fig. 6

Time dependence of the Green function G(z, t) for water at several depths.

Fig. 7
Fig. 7

Five numerical validations of the upper-bound energy estimates for the depth-dependent peak amplitude of E(z, ·). The boldface curves are upper bounds from relation (5.5).

Fig. 8
Fig. 8

Time of arrival of the peak of the Green function G(z, t) at various depths in the medium of Ref. 32.

Fig. 9
Fig. 9

Electric fields for two models of water. The fields, which result from the same incident pulse, are computed at a 9.75-mm depth for each model.

Tables (1)

Tables Icon

Table 1 Five Incident Pulses That Satisfy the Conditions in Relation (5.4)a

Equations (40)

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ξ ( ξ c 0 z ) ( ξ + c 0 z ) E + β τ ( ξ c 1 z ) ( ξ + c 1 z ) E = 0 , z > 0 ,
( ξ 2 c 0 2 z 2 ) ( ε ξ E ) + ( ξ 2 c 1 2 z 2 ) E = 0 , z > 0 ,
( ξ + c 0 z ) E + β τ c 0 2 c 1 2 2 c 0 2 E = 0 , z < c 0 τ ,
E = g ( 0 + ) exp [ ( ε s / ε 1 2 cos 2 ϕ inc ) ( z c 0 τ ) ] .
( ξ + c 1 z ) E = τ β c 0 2 c 1 2 2 z 2 E , z > c 0 τ .
E ( z , ξ ) = β 2 π τ ( c 0 2 c 1 2 ) z 0 ξ d κ h ( κ ) ( ξ κ ) 3 / 2 × exp ( [ β 2 τ ( c 0 2 c 1 2 ) ] { [ z c 1 ( ξ κ ) ] 2 ξ κ } ) .
E δ ( z , ξ ) ~ A β 2 π τ ( c 0 2 c 1 2 ) z ξ 3 / 2 × exp { [ β 2 τ ( c 0 2 c 1 2 ) ] [ ( z c 1 ξ ) 2 ξ ] } .
[ ξ 2 ( c 0 c ) 2 z 2 ] ( τ β T p ξ E ) + [ ξ 2 ( c 1 c ) 2 z 2 ] E = 0 , z > 0.
E ( z , ξ ) = 1 2 π i B G ( s ) exp { s ξ [ 1 Φ ( s ) z c 0 ξ ] } d s ,
Φ ( s ) = s + β / τ s + β c 1 2 / τ c 0 2
E ( z , t ) = E y ( z , t ) = exp ( a z ) f ( t z c 0 ) + 0 t z / c 0 d s f ( s ) G ( z , t s ) .
0 t d s f b ( t s ) f l ( s ) = 0 t d s f l ( t s ) f b ( s ) = f b ( · ) 1 f l ( t t b t 0 ) + E ( t ) ,
| E ( t ) | t 0 f l f b 1 = const . ,
E ( z , t ) = exp ( a z ) f b ( t z / c 0 ) + sgn ( f b ) f b 1 G ( z , t t b t 0 ) + E ( z , t ) ,
| E ( z , t ) | t 0 f b 1 t G ( z , · ) ,
E ( z , t ) = exp ( a z ) f l ( t z / c 0 ) + sgn [ G b ( z , · ) ] G b ( z , · ) 1 f l ( t t b t 0 ) + E ( z , t ) ,
| E ( z , t ) | t 0 f l G b ( z , · ) 1 ,
r = [ Γ b ( z , · ) 1 / G b ( z , · ) 1 ] .
E ( z , t ) = exp ( a z ) f l ( t z / c 0 ) + sgn [ Γ b ( z , · ) ] Γ b ( z , · ) 1 f l ( t t b t 0 ) + E ( z , t ) ,
| E ( z , t ) | [ t 0 f l + ( 1 r ) f l ] G b ( z , · ) 1 ,
E ( z , t ) = exp ( a z ) f l ( t z / c 0 ) + i sgn [ G i ( b ) ( z , · ) ] G i ( b ) ( z , · ) 1 f l [ t t i ( b ) t i ( 0 ) ] × H [ t t i ( b ) ] + E ( z , t ) ,
| E ( z , t ) | i t i ( 0 ) f l G i ( b ) ( z , · ) 1 H [ t t i ( b ) ] ,
h ( z , · ) 1 = 0 d t | h ( z , t ) | , h ( z , · ) 2 = ( 0 d t | h ( z , t ) | 2 ) 1 / 2 , h ( z , · ) = least upper bound of | h ( z , t ) | ,
E ( z , · ) p exp ( a z ) f p + G ( z , · ) r f q
G ρ ( z , t ) = exp ( a z ) G ( z , t + z / c 0 ) 0 t d s G ρ ( z , s ) × exp ( a z ) G ( z , t s + z / c 0 ) .
f ( t ) = exp ( a z ) E ( z , t + z / c 0 ) + 0 t d s G ρ ( z , s ) × exp ( a z ) E ( z , t s + z / c 0 ) ,
l ( z ) f p = exp ( a z ) f p 1 + G ρ ( z , · ) 1 E ( z , · ) p ,
[ exp ( a z ) G ( z , · ) 1 ] f p l ( z ) f p E ( z , · ) p exp ( a z ) f p + G ( z , · ) r f q ,
E ( z , · ) exp ( a z ) f + min [ G ( z , · ) 1 f , G ( z , · ) 2 f 2 , G ( z , · ) f 1 ] ,
E ( z , · ) 2 2 { exp ( a z ) f 2 + min [ G ( z , · ) 1 f 2 , G ( z , · ) 2 f 1 , ] } 2 .
t E ( z , t ) = exp ( a z ) f ( t z / c 0 ) + f ( 0 + ) G ( z , t ) + 0 t d s f ( t s ) G ( z , s ) .
E ( z , t ) = exp ( a z ) f b ( t z / c 0 ) + sgn ( f b ) f b 1 × G ( z , t t b t 0 ) + E ( z , t ) ,
E ( z , t ) = exp ( a z ) f l ( t z / c 0 ) + ( 0.202 ) f l [ t 9 ( z 0.44 mm ) c 0 ] + E ( z , t ) ,
| E ( z , t ) | ( 0.202 ) [ t 0 f l + ( 1 r ) f l ] ,
f 1.00 V / m , and f 2 6.32 × 10 6 V s 1 / 2 / m , and f 1 4.00 × 10 11 V s / m ,
E ( z , · ) [ exp ( z 6.15 × 10 5 m ) V / m ] + min [ 0.202 V / m , ( 6.32 × 10 6 V s 1 / 2 / m ) F 1 ( z ) ( 4.00 × 10 11 V s / m ) F 2 ( z ) ] ,
( 1 ) f 0.740 V / m or ( 2 ) [ f 40,000 V / m and f 2 6.32 × 10 6 V s 1 / 2 / m ] or ( 3 ) [ f 50,000 V / m and f 1 2.80 × 10 11 V s / m ] ,
f n + 1 ( 0 , · ) f n + 2 ( 0 , · ) f 5 ( 0 , · ) ( 0.30 ) n f 1 ( 0 , · ) f 2 ( 0 , · ) f 5 ( 0 , · ) .
f i ( 0 , · ) f ex ( 0 , · ) f 5 ( 0 , · ) 1.4 f i ( 0 , · ) f i + 1 ( 0 , · ) f 5 ( 0 , · ) ,
R ( t ) = G ( 0 , t ) = 1 t exp [ ( b + a 2 ε cos 2 θ ) t ] × I 1 ( a t 2 ε cos 2 θ ) ,

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