Abstract

We report the initial results for femtosecond pulse propagation and scattering interactions for a Lorentz medium obtained by a direct time integration of Maxwell's equations. The computational approach provides reflection coefficients accurate to better than 6 parts in 10,000 over the frequency range of dc to 3 × 1016 Hz for a single 0.2-fs Gaussian pulse incident upon a Lorentz-medium half-space. New results for Sommerfeld and Brillouin precursors are shown and compared with previous analyses. The present approach is robust and permits two-dimensional and three-dimensional electromagnetic pulse propagation directly from the full-vector Maxwell's equations.

© 1991 Optical Society of America

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References

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  1. K. S. Yee, IEEE Trans. Antennas Propag. AP-14, 302 (1966).
  2. A. Taflove, M. E. Brodwin, IEEE Trans. Microwave Theory Tech. MTT-23, 623 (1975).
    [Crossref]
  3. G. Mur, IEEE Trans. Electromagn. Compat. EC-23, 377 (1981).
    [Crossref]
  4. K. R. Umashankar, A. Taflove, IEEE Trans. Electromagn. Compat. EC-24, 397 (1982).
    [Crossref]
  5. A. Taflove, Wave Motion 10, 547 (1988).
    [Crossref]
  6. G. C. Liang, Y. W. Liu, K. K. Mei, IEEE Trans. Microwave Theory Tech. MTT-37, 1949 (1989).
    [Crossref]
  7. E. Sano, T. Shibata, IEEE J. Quantum Electron. 26, 372 (1990).
    [Crossref]
  8. S. T. Chu, S. K. Chaudhuri, IEEE J. Lightwave Technol. 7, 2033 (1989).
    [Crossref]
  9. R. Luebbers, F. P. Hunsberger, K. S. Kunz, R. B. Standler, M. Schneider, IEEE Trans. Electromagn. Compat. EC-32, 222 (1990).
    [Crossref]
  10. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).
  11. A. Sommerfeld, Ann. Phys. 44, 177 (1914).
    [Crossref]
  12. L. Brillouin, Ann. Phys. 44, 203 (1914).
    [Crossref]
  13. K. E. Oughstun, G. C. Sherman, J. Opt. Soc. Am. A 6, 1394 (1989).
    [Crossref]
  14. P. Wyns, D. P. Foty, K. E. Oughstun, J. Opt. Soc. Am. A 6, 1421 (1989).
    [Crossref]

1990 (2)

E. Sano, T. Shibata, IEEE J. Quantum Electron. 26, 372 (1990).
[Crossref]

R. Luebbers, F. P. Hunsberger, K. S. Kunz, R. B. Standler, M. Schneider, IEEE Trans. Electromagn. Compat. EC-32, 222 (1990).
[Crossref]

1989 (4)

K. E. Oughstun, G. C. Sherman, J. Opt. Soc. Am. A 6, 1394 (1989).
[Crossref]

P. Wyns, D. P. Foty, K. E. Oughstun, J. Opt. Soc. Am. A 6, 1421 (1989).
[Crossref]

S. T. Chu, S. K. Chaudhuri, IEEE J. Lightwave Technol. 7, 2033 (1989).
[Crossref]

G. C. Liang, Y. W. Liu, K. K. Mei, IEEE Trans. Microwave Theory Tech. MTT-37, 1949 (1989).
[Crossref]

1988 (1)

A. Taflove, Wave Motion 10, 547 (1988).
[Crossref]

1982 (1)

K. R. Umashankar, A. Taflove, IEEE Trans. Electromagn. Compat. EC-24, 397 (1982).
[Crossref]

1981 (1)

G. Mur, IEEE Trans. Electromagn. Compat. EC-23, 377 (1981).
[Crossref]

1975 (1)

A. Taflove, M. E. Brodwin, IEEE Trans. Microwave Theory Tech. MTT-23, 623 (1975).
[Crossref]

1966 (1)

K. S. Yee, IEEE Trans. Antennas Propag. AP-14, 302 (1966).

1914 (2)

A. Sommerfeld, Ann. Phys. 44, 177 (1914).
[Crossref]

L. Brillouin, Ann. Phys. 44, 203 (1914).
[Crossref]

Brillouin, L.

L. Brillouin, Ann. Phys. 44, 203 (1914).
[Crossref]

Brodwin, M. E.

A. Taflove, M. E. Brodwin, IEEE Trans. Microwave Theory Tech. MTT-23, 623 (1975).
[Crossref]

Chaudhuri, S. K.

S. T. Chu, S. K. Chaudhuri, IEEE J. Lightwave Technol. 7, 2033 (1989).
[Crossref]

Chu, S. T.

S. T. Chu, S. K. Chaudhuri, IEEE J. Lightwave Technol. 7, 2033 (1989).
[Crossref]

Foty, D. P.

Hunsberger, F. P.

R. Luebbers, F. P. Hunsberger, K. S. Kunz, R. B. Standler, M. Schneider, IEEE Trans. Electromagn. Compat. EC-32, 222 (1990).
[Crossref]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).

Kunz, K. S.

R. Luebbers, F. P. Hunsberger, K. S. Kunz, R. B. Standler, M. Schneider, IEEE Trans. Electromagn. Compat. EC-32, 222 (1990).
[Crossref]

Liang, G. C.

G. C. Liang, Y. W. Liu, K. K. Mei, IEEE Trans. Microwave Theory Tech. MTT-37, 1949 (1989).
[Crossref]

Liu, Y. W.

G. C. Liang, Y. W. Liu, K. K. Mei, IEEE Trans. Microwave Theory Tech. MTT-37, 1949 (1989).
[Crossref]

Luebbers, R.

R. Luebbers, F. P. Hunsberger, K. S. Kunz, R. B. Standler, M. Schneider, IEEE Trans. Electromagn. Compat. EC-32, 222 (1990).
[Crossref]

Mei, K. K.

G. C. Liang, Y. W. Liu, K. K. Mei, IEEE Trans. Microwave Theory Tech. MTT-37, 1949 (1989).
[Crossref]

Mur, G.

G. Mur, IEEE Trans. Electromagn. Compat. EC-23, 377 (1981).
[Crossref]

Oughstun, K. E.

Sano, E.

E. Sano, T. Shibata, IEEE J. Quantum Electron. 26, 372 (1990).
[Crossref]

Schneider, M.

R. Luebbers, F. P. Hunsberger, K. S. Kunz, R. B. Standler, M. Schneider, IEEE Trans. Electromagn. Compat. EC-32, 222 (1990).
[Crossref]

Sherman, G. C.

Shibata, T.

E. Sano, T. Shibata, IEEE J. Quantum Electron. 26, 372 (1990).
[Crossref]

Sommerfeld, A.

A. Sommerfeld, Ann. Phys. 44, 177 (1914).
[Crossref]

Standler, R. B.

R. Luebbers, F. P. Hunsberger, K. S. Kunz, R. B. Standler, M. Schneider, IEEE Trans. Electromagn. Compat. EC-32, 222 (1990).
[Crossref]

Taflove, A.

A. Taflove, Wave Motion 10, 547 (1988).
[Crossref]

K. R. Umashankar, A. Taflove, IEEE Trans. Electromagn. Compat. EC-24, 397 (1982).
[Crossref]

A. Taflove, M. E. Brodwin, IEEE Trans. Microwave Theory Tech. MTT-23, 623 (1975).
[Crossref]

Umashankar, K. R.

K. R. Umashankar, A. Taflove, IEEE Trans. Electromagn. Compat. EC-24, 397 (1982).
[Crossref]

Wyns, P.

Yee, K. S.

K. S. Yee, IEEE Trans. Antennas Propag. AP-14, 302 (1966).

Ann. Phys. (2)

A. Sommerfeld, Ann. Phys. 44, 177 (1914).
[Crossref]

L. Brillouin, Ann. Phys. 44, 203 (1914).
[Crossref]

IEEE J. Lightwave Technol. (1)

S. T. Chu, S. K. Chaudhuri, IEEE J. Lightwave Technol. 7, 2033 (1989).
[Crossref]

IEEE J. Quantum Electron. (1)

E. Sano, T. Shibata, IEEE J. Quantum Electron. 26, 372 (1990).
[Crossref]

IEEE Trans. Antennas Propag. (1)

K. S. Yee, IEEE Trans. Antennas Propag. AP-14, 302 (1966).

IEEE Trans. Electromagn. Compat. (3)

G. Mur, IEEE Trans. Electromagn. Compat. EC-23, 377 (1981).
[Crossref]

K. R. Umashankar, A. Taflove, IEEE Trans. Electromagn. Compat. EC-24, 397 (1982).
[Crossref]

R. Luebbers, F. P. Hunsberger, K. S. Kunz, R. B. Standler, M. Schneider, IEEE Trans. Electromagn. Compat. EC-32, 222 (1990).
[Crossref]

IEEE Trans. Microwave Theory Tech. (2)

G. C. Liang, Y. W. Liu, K. K. Mei, IEEE Trans. Microwave Theory Tech. MTT-37, 1949 (1989).
[Crossref]

A. Taflove, M. E. Brodwin, IEEE Trans. Microwave Theory Tech. MTT-23, 623 (1975).
[Crossref]

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

Wave Motion (1)

A. Taflove, Wave Motion 10, 547 (1988).
[Crossref]

Other (1)

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).

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Figures (4)

Fig. 1
Fig. 1

Complex permittivity of the Lorentz medium with parameters εs = 2.25ε0, ε = ε0, ω0 = 4.0 × 1016 rad/s, and δ = 0.28 × 1016 s−1.

Fig. 2
Fig. 2

Comparison of FD-TD and exact results from dc to 3 × 1016 Hz for the magnitude and phase of the reflection coefficient of a half-space made of the Lorentz medium of Fig. 1.

Fig. 3
Fig. 3

Comparison of FD-TD, asymptotic,13 and Laplace-transform14 results for the Sommerfeld precursor observed at x = 1μm in the Lorentz medium of Fig. 1 for a unit-step modulated sinusoidal excitation ωc = 1.0 × 1016 rad/s) at x = 0.

Fig. 4
Fig. 4

FD-TD results for the total signal (including the Brillouin precursor) at x = 10 μm in the Lorentz medium of Fig. 1 for the unit-step modulated sinusoidal excitation.

Equations (16)

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H y t = 1 μ 0 E z x ,
D z t = H y x .
H y n + 1 2 y ( i + 1 2 ) = H y n 1 2 ( i + 1 2 ) + Δ t μ 0 Δ x [ E z n ( i + 1 ) E z n ( i ) ] ,
E z n + 1 ( i ) = E z n ( i ) + Δ t ɛ Δ x [ H y n + 1 2 ( i + 1 2 ) H y n + 1 2 ( i 1 2 ) ] ,
ɛ ( ω ) = D z ( ω ) E z ( ω ) .
H y n + 1 2 ( i + 1 2 ) = H y n 1 2 ( i + 1 2 ) + Δ t μ 0 Δ x [ E z n ( i + 1 ) E z n ( i ) ] ,
D z n + 1 ( i ) = D z n ( i ) + Δ t Δ x [ H y n + 1 2 ( i + 1 2 ) H y n + 1 2 ( i 1 2 ) ] ,
E z n + 1 ( i ) = f ( D z n + 1 , , D z n M + 1 ; E z n , , E z n M + 1 ) .
ɛ ( ω ) = ɛ + ɛ s ɛ 1 j ω τ = D z ( ω ) E z ( ω ) ,
f ( t ) = + f ( ω ) exp ( j ω t ) d ω ,
D z + τ d D z d t = ɛ s E z + τ ɛ d E z d t .
E z n + 1 ( i ) = Δ t + 2 τ 2 τ ɛ + ɛ s Δ t D z n + 1 ( i ) + Δ t 2 τ 2 τ ɛ + ɛ s Δ t D z n ( i ) + 2 τ ɛ ɛ s Δ t 2 τ ɛ + ɛ s Δ t E z n ( i ) .
ɛ ( ω ) = ɛ ω 0 2 ( ɛ s ɛ ) ω 2 + 2 j ω δ ω 0 2 = D z ( ω ) E z ( ω ) ,
ɛ s = 2.25 ɛ 0 , ɛ = ɛ 0 , ω 0 = 4.0 × 10 16 rad / s , δ = 0.28 × 10 16 s 1 .
ω 0 2 D z + 2 δ d D z d t + d 2 D z d t 2 = ω 0 2 ɛ s E z + 2 δ ɛ d E z d t + ɛ d 2 E z d t 2 .
E z n + 1 = [ ( ω 0 2 Δ t 2 + 2 δ Δ t + 2 ) D z n + 1 4 D z n + ( ω 0 2 Δ t 2 2 δ Δ t + 2 ) D z n 1 + 4 ɛ E z n ( ω 0 2 Δ t 2 ɛ s 2 δ Δ t ɛ + 2 ɛ ) E z n 1 ] / ( ω 0 2 Δ t 2 ɛ s + 2 δ Δ t ɛ + 2 ɛ ) .

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