Abstract

The description of the precursor fields in a single-resonance Lorentz model dielectric is considered in the singular and weak dispersion limits. The singular dispersion limit is obtained as the damping approaches zero and the material dispersion becomes increasingly concentrated about the resonance frequency. The algebraic peak amplitude decay of the Brillouin precursor with propagation distance z>0 then changes from a z1/2 to a z1/3 behavior. The weak dispersion limit is obtained as the material density decreases to zero. The material dispersion then becomes vanishingly small everywhere and the precursors become increasingly compressed in the space-time domain immediately following the speed-of-light point (z,t)=(z,z/c). In order to circumvent the numerical difficulties introduced in this case, an approximate equivalence relation is derived that allows the propagated field evolution due to an ultrawideband signal to be calculated in an equivalent dispersive medium that is highly absorptive.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. A. Lorentz, Theory of Electrons (Teubner, 1906), Chap. IV.
  2. A. Sommerfeld, “Über die fortpflanzung des lichtes in disperdierenden medien,” Ann. Phys. (Leipzig) 44, 177–202 (1914).
  3. L. Brillouin, “Über die fortpflanzung des licht in disperdierenden medien,” Ann. Phys. (Leipzig) 44, 203–240 (1914).
  4. L. Brillouin, Wave Propagation and Group Velocity (Academic, 1960).
  5. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941), Sec. 5.18.
  6. K. E. Oughstun and G. C. Sherman, Electromagnetic Pulse Propagation in Causal Dielectrics (Springer-Verlag, 1994).
  7. K. E. Oughstun, Electromagnetic & Optical Pulse Propagation 1: Spectral Representations in Temporally Dispersive Media (Springer, 2006).
    [PubMed]
  8. K. E. Oughstun, Electromagnetic & Optical Pulse Propagation 2: Temporal Pulse Dynamics in Dispersive Attenuative Media (Springer, 2009).
  9. H. M. Nussenzveig, Causality and Dispersion Relations (Academic, 1972), Chap. 1.
  10. 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]
  11. 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]
  12. N. A. Cartwright and K. E. Oughstun, “Uniform asymptotics applied to ultrawideband pulse propagation,” SIAM Rev. 49, 628–648 (2007).
    [CrossRef]
  13. There are several definitions of what an ultrawideband signal is; see, for example, Section 11.2.2 of . We take here the simple physical definition to mean a temporal pulse whose frequency spectrum along the positive real frequency axis is nonzero as ω→0 and which goes to zero as 1/ω or less as ω→∞.
  14. P. D. Smith and K. E. Oughstun, “Electromagnetic energy dissipation and propagation of an ultrawideband plane wave pulse in a causally dispersive dielectric,” Radio Sci. 33, 1489–1504 (1998).
    [CrossRef]
  15. The Beer–Lambert–Bouger law was originally discovered by P. Bouger, Essai d’Optique sur la Gradation de la Lumiere (Claude Jombert, 1729) and subsequently cited by J. H. Lambert, Photometri (V. E. Klett, 1760); the result was then extended by A. Beer, Einleitung in die höhere Optik (Friedrich Viewig, 1853) to include the concentration of solutions in the expression of the absorption coefficient for the intensity of light.
  16. K. E. Oughstun, “Dynamical evolution of the Brillouin precursor in Rocard–Powles–Debye model dielectrics,” IEEE Trans. Antennas Propag. 53, 1582–1590 (2005).
    [CrossRef]
  17. M. Pieraccini, A. Bicci, D. Mecatti, G. Macaluso, and C. Atzeni, “Propagation of large bandwidth microwave signals in water,” IEEE Trans. Antennas Propag. 57, 3612–3618 (2009).
    [CrossRef]
  18. In the group velocity description, the “strength” of the material dispersion is typically measured through the derivative dnr(ω)/dω as that is what appears in the coefficients β1≡(nr+ωdnr/dω)/c and β2≡(2dnr/dω+ωd2nr/dω2)/c in the Taylor series expansion of the real propagation factor β(ω)≡R{k̃(ω)} in a hypothetical “lossless” dispersive medium. It is directly found from Eq. that this first derivative is directly proportional to the number density N so that these coefficients can always be made as small as desired at any real ω simply by choosing N sufficiently small.
  19. H. Jeong, A. M. C. Dawes, and D. J. Gauthier, “Direct observation of optical precursors in a region of anomalous dispersion,” Phys. Rev. Lett. 96, 143901 (2006).
    [CrossRef] [PubMed]
  20. B. Macke and B. Ségard, “Optical precursors in transparent media,” Phys. Rev. A 80, 011803 (2009).
    [CrossRef]
  21. H. Jeong and S. Du, “Two-way transparency in the light-matter interaction: Optical precursors with electromagnetically induced transparency,” Phys. Rev. A 79, 011802 (2009).
    [CrossRef]
  22. D. Wei, J. F. Chen, M. M. T. Loy, G. K. L. Wong, and S. Du, “Optical precursors with electromagnetically induced transparency in cold atoms,” Phys. Rev. Lett. 103, 093602 (2009).
    [CrossRef] [PubMed]
  23. B. Macke and B. Ségard, “Optical precursors with self-induced transparency,” Phys. Rev. A 81, 015803 (2010).
    [CrossRef]
  24. J. F. Chen, S. Wang, D. Wei, M. M. T. Loy, G. K. L. Wong, and S. Du, “Optical coherent transients in cold atoms: from free induction decay to optical precursors,” Phys. Rev. A 81, 033844 (2010).
    [CrossRef]
  25. H. Jeong and S. Du, “Slow-light-induced interference with stacked optical precursors for square input pulses,” Opt. Lett. 35, 124–126 (2010).
    [CrossRef] [PubMed]
  26. W. R. LeFew, S. Venakides, and D. J. Gauthier, “Accurate description of optical precursors and their relation to weak-field coherent optical transients,” Phys. Rev. A 79, 063842 (2009).
    [CrossRef]
  27. M. D. Crisp, “Propagation of small-area pulses of coherent light through a resonant medium,” Phys. Rev. A 1, 1604–1611 (1970).
    [CrossRef]
  28. B. Macke and B. Ségard, “Comment on “Direct observation of optical precursors in a region of anomalous dispersion”,” arXiv:physics/0605039.
  29. K. E. Oughstun, “Noninstantaneous, finite rise-time effects on the precursor field formation in linear dispersive pulse propagation,” J. Opt. Soc. Am. A 12, 1715–1729 (1995).
    [CrossRef]

2010

B. Macke and B. Ségard, “Optical precursors with self-induced transparency,” Phys. Rev. A 81, 015803 (2010).
[CrossRef]

J. F. Chen, S. Wang, D. Wei, M. M. T. Loy, G. K. L. Wong, and S. Du, “Optical coherent transients in cold atoms: from free induction decay to optical precursors,” Phys. Rev. A 81, 033844 (2010).
[CrossRef]

H. Jeong and S. Du, “Slow-light-induced interference with stacked optical precursors for square input pulses,” Opt. Lett. 35, 124–126 (2010).
[CrossRef] [PubMed]

2009

M. Pieraccini, A. Bicci, D. Mecatti, G. Macaluso, and C. Atzeni, “Propagation of large bandwidth microwave signals in water,” IEEE Trans. Antennas Propag. 57, 3612–3618 (2009).
[CrossRef]

B. Macke and B. Ségard, “Optical precursors in transparent media,” Phys. Rev. A 80, 011803 (2009).
[CrossRef]

H. Jeong and S. Du, “Two-way transparency in the light-matter interaction: Optical precursors with electromagnetically induced transparency,” Phys. Rev. A 79, 011802 (2009).
[CrossRef]

D. Wei, J. F. Chen, M. M. T. Loy, G. K. L. Wong, and S. Du, “Optical precursors with electromagnetically induced transparency in cold atoms,” Phys. Rev. Lett. 103, 093602 (2009).
[CrossRef] [PubMed]

W. R. LeFew, S. Venakides, and D. J. Gauthier, “Accurate description of optical precursors and their relation to weak-field coherent optical transients,” Phys. Rev. A 79, 063842 (2009).
[CrossRef]

2007

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

2006

H. Jeong, A. M. C. Dawes, and D. J. Gauthier, “Direct observation of optical precursors in a region of anomalous dispersion,” Phys. Rev. Lett. 96, 143901 (2006).
[CrossRef] [PubMed]

2005

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

1998

P. D. Smith and K. E. Oughstun, “Electromagnetic energy dissipation and propagation of an ultrawideband plane wave pulse in a causally dispersive dielectric,” Radio Sci. 33, 1489–1504 (1998).
[CrossRef]

1995

1989

1988

1970

M. D. Crisp, “Propagation of small-area pulses of coherent light through a resonant medium,” Phys. Rev. A 1, 1604–1611 (1970).
[CrossRef]

1914

A. Sommerfeld, “Über die fortpflanzung des lichtes in disperdierenden medien,” Ann. Phys. (Leipzig) 44, 177–202 (1914).

L. Brillouin, “Über die fortpflanzung des licht in disperdierenden medien,” Ann. Phys. (Leipzig) 44, 203–240 (1914).

Atzeni, C.

M. Pieraccini, A. Bicci, D. Mecatti, G. Macaluso, and C. Atzeni, “Propagation of large bandwidth microwave signals in water,” IEEE Trans. Antennas Propag. 57, 3612–3618 (2009).
[CrossRef]

Beer, A.

The Beer–Lambert–Bouger law was originally discovered by P. Bouger, Essai d’Optique sur la Gradation de la Lumiere (Claude Jombert, 1729) and subsequently cited by J. H. Lambert, Photometri (V. E. Klett, 1760); the result was then extended by A. Beer, Einleitung in die höhere Optik (Friedrich Viewig, 1853) to include the concentration of solutions in the expression of the absorption coefficient for the intensity of light.

Bicci, A.

M. Pieraccini, A. Bicci, D. Mecatti, G. Macaluso, and C. Atzeni, “Propagation of large bandwidth microwave signals in water,” IEEE Trans. Antennas Propag. 57, 3612–3618 (2009).
[CrossRef]

Bouger, P.

The Beer–Lambert–Bouger law was originally discovered by P. Bouger, Essai d’Optique sur la Gradation de la Lumiere (Claude Jombert, 1729) and subsequently cited by J. H. Lambert, Photometri (V. E. Klett, 1760); the result was then extended by A. Beer, Einleitung in die höhere Optik (Friedrich Viewig, 1853) to include the concentration of solutions in the expression of the absorption coefficient for the intensity of light.

Brillouin, L.

L. Brillouin, “Über die fortpflanzung des licht in disperdierenden medien,” Ann. Phys. (Leipzig) 44, 203–240 (1914).

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

Cartwright, N. A.

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

Chen, J. F.

J. F. Chen, S. Wang, D. Wei, M. M. T. Loy, G. K. L. Wong, and S. Du, “Optical coherent transients in cold atoms: from free induction decay to optical precursors,” Phys. Rev. A 81, 033844 (2010).
[CrossRef]

D. Wei, J. F. Chen, M. M. T. Loy, G. K. L. Wong, and S. Du, “Optical precursors with electromagnetically induced transparency in cold atoms,” Phys. Rev. Lett. 103, 093602 (2009).
[CrossRef] [PubMed]

Crisp, M. D.

M. D. Crisp, “Propagation of small-area pulses of coherent light through a resonant medium,” Phys. Rev. A 1, 1604–1611 (1970).
[CrossRef]

Dawes, A. M. C.

H. Jeong, A. M. C. Dawes, and D. J. Gauthier, “Direct observation of optical precursors in a region of anomalous dispersion,” Phys. Rev. Lett. 96, 143901 (2006).
[CrossRef] [PubMed]

Du, S.

J. F. Chen, S. Wang, D. Wei, M. M. T. Loy, G. K. L. Wong, and S. Du, “Optical coherent transients in cold atoms: from free induction decay to optical precursors,” Phys. Rev. A 81, 033844 (2010).
[CrossRef]

H. Jeong and S. Du, “Slow-light-induced interference with stacked optical precursors for square input pulses,” Opt. Lett. 35, 124–126 (2010).
[CrossRef] [PubMed]

H. Jeong and S. Du, “Two-way transparency in the light-matter interaction: Optical precursors with electromagnetically induced transparency,” Phys. Rev. A 79, 011802 (2009).
[CrossRef]

D. Wei, J. F. Chen, M. M. T. Loy, G. K. L. Wong, and S. Du, “Optical precursors with electromagnetically induced transparency in cold atoms,” Phys. Rev. Lett. 103, 093602 (2009).
[CrossRef] [PubMed]

Gauthier, D. J.

W. R. LeFew, S. Venakides, and D. J. Gauthier, “Accurate description of optical precursors and their relation to weak-field coherent optical transients,” Phys. Rev. A 79, 063842 (2009).
[CrossRef]

H. Jeong, A. M. C. Dawes, and D. J. Gauthier, “Direct observation of optical precursors in a region of anomalous dispersion,” Phys. Rev. Lett. 96, 143901 (2006).
[CrossRef] [PubMed]

Jeong, H.

H. Jeong and S. Du, “Slow-light-induced interference with stacked optical precursors for square input pulses,” Opt. Lett. 35, 124–126 (2010).
[CrossRef] [PubMed]

H. Jeong and S. Du, “Two-way transparency in the light-matter interaction: Optical precursors with electromagnetically induced transparency,” Phys. Rev. A 79, 011802 (2009).
[CrossRef]

H. Jeong, A. M. C. Dawes, and D. J. Gauthier, “Direct observation of optical precursors in a region of anomalous dispersion,” Phys. Rev. Lett. 96, 143901 (2006).
[CrossRef] [PubMed]

Lambert, J. H.

The Beer–Lambert–Bouger law was originally discovered by P. Bouger, Essai d’Optique sur la Gradation de la Lumiere (Claude Jombert, 1729) and subsequently cited by J. H. Lambert, Photometri (V. E. Klett, 1760); the result was then extended by A. Beer, Einleitung in die höhere Optik (Friedrich Viewig, 1853) to include the concentration of solutions in the expression of the absorption coefficient for the intensity of light.

LeFew, W. R.

W. R. LeFew, S. Venakides, and D. J. Gauthier, “Accurate description of optical precursors and their relation to weak-field coherent optical transients,” Phys. Rev. A 79, 063842 (2009).
[CrossRef]

Lorentz, H. A.

H. A. Lorentz, Theory of Electrons (Teubner, 1906), Chap. IV.

Loy, M. M. T.

J. F. Chen, S. Wang, D. Wei, M. M. T. Loy, G. K. L. Wong, and S. Du, “Optical coherent transients in cold atoms: from free induction decay to optical precursors,” Phys. Rev. A 81, 033844 (2010).
[CrossRef]

D. Wei, J. F. Chen, M. M. T. Loy, G. K. L. Wong, and S. Du, “Optical precursors with electromagnetically induced transparency in cold atoms,” Phys. Rev. Lett. 103, 093602 (2009).
[CrossRef] [PubMed]

Macaluso, G.

M. Pieraccini, A. Bicci, D. Mecatti, G. Macaluso, and C. Atzeni, “Propagation of large bandwidth microwave signals in water,” IEEE Trans. Antennas Propag. 57, 3612–3618 (2009).
[CrossRef]

Macke, B.

B. Macke and B. Ségard, “Optical precursors with self-induced transparency,” Phys. Rev. A 81, 015803 (2010).
[CrossRef]

B. Macke and B. Ségard, “Optical precursors in transparent media,” Phys. Rev. A 80, 011803 (2009).
[CrossRef]

B. Macke and B. Ségard, “Comment on “Direct observation of optical precursors in a region of anomalous dispersion”,” arXiv:physics/0605039.

Mecatti, D.

M. Pieraccini, A. Bicci, D. Mecatti, G. Macaluso, and C. Atzeni, “Propagation of large bandwidth microwave signals in water,” IEEE Trans. Antennas Propag. 57, 3612–3618 (2009).
[CrossRef]

Nussenzveig, H. M.

H. M. Nussenzveig, Causality and Dispersion Relations (Academic, 1972), Chap. 1.

Oughstun, K. E.

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

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

P. D. Smith and K. E. Oughstun, “Electromagnetic energy dissipation and propagation of an ultrawideband plane wave pulse in a causally dispersive dielectric,” Radio Sci. 33, 1489–1504 (1998).
[CrossRef]

K. E. Oughstun, “Noninstantaneous, finite rise-time effects on the precursor field formation in linear dispersive pulse propagation,” J. Opt. Soc. Am. A 12, 1715–1729 (1995).
[CrossRef]

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]

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]

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

K. E. Oughstun, Electromagnetic & Optical Pulse Propagation 1: Spectral Representations in Temporally Dispersive Media (Springer, 2006).
[PubMed]

K. E. Oughstun, Electromagnetic & Optical Pulse Propagation 2: Temporal Pulse Dynamics in Dispersive Attenuative Media (Springer, 2009).

Pieraccini, M.

M. Pieraccini, A. Bicci, D. Mecatti, G. Macaluso, and C. Atzeni, “Propagation of large bandwidth microwave signals in water,” IEEE Trans. Antennas Propag. 57, 3612–3618 (2009).
[CrossRef]

Ségard, B.

B. Macke and B. Ségard, “Optical precursors with self-induced transparency,” Phys. Rev. A 81, 015803 (2010).
[CrossRef]

B. Macke and B. Ségard, “Optical precursors in transparent media,” Phys. Rev. A 80, 011803 (2009).
[CrossRef]

B. Macke and B. Ségard, “Comment on “Direct observation of optical precursors in a region of anomalous dispersion”,” arXiv:physics/0605039.

Sherman, G. C.

Smith, P. D.

P. D. Smith and K. E. Oughstun, “Electromagnetic energy dissipation and propagation of an ultrawideband plane wave pulse in a causally dispersive dielectric,” Radio Sci. 33, 1489–1504 (1998).
[CrossRef]

Sommerfeld, A.

A. Sommerfeld, “Über die fortpflanzung des lichtes in disperdierenden medien,” Ann. Phys. (Leipzig) 44, 177–202 (1914).

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941), Sec. 5.18.

Venakides, S.

W. R. LeFew, S. Venakides, and D. J. Gauthier, “Accurate description of optical precursors and their relation to weak-field coherent optical transients,” Phys. Rev. A 79, 063842 (2009).
[CrossRef]

Wang, S.

J. F. Chen, S. Wang, D. Wei, M. M. T. Loy, G. K. L. Wong, and S. Du, “Optical coherent transients in cold atoms: from free induction decay to optical precursors,” Phys. Rev. A 81, 033844 (2010).
[CrossRef]

Wei, D.

J. F. Chen, S. Wang, D. Wei, M. M. T. Loy, G. K. L. Wong, and S. Du, “Optical coherent transients in cold atoms: from free induction decay to optical precursors,” Phys. Rev. A 81, 033844 (2010).
[CrossRef]

D. Wei, J. F. Chen, M. M. T. Loy, G. K. L. Wong, and S. Du, “Optical precursors with electromagnetically induced transparency in cold atoms,” Phys. Rev. Lett. 103, 093602 (2009).
[CrossRef] [PubMed]

Wong, G. K. L.

J. F. Chen, S. Wang, D. Wei, M. M. T. Loy, G. K. L. Wong, and S. Du, “Optical coherent transients in cold atoms: from free induction decay to optical precursors,” Phys. Rev. A 81, 033844 (2010).
[CrossRef]

D. Wei, J. F. Chen, M. M. T. Loy, G. K. L. Wong, and S. Du, “Optical precursors with electromagnetically induced transparency in cold atoms,” Phys. Rev. Lett. 103, 093602 (2009).
[CrossRef] [PubMed]

Ann. Phys. (Leipzig)

A. Sommerfeld, “Über die fortpflanzung des lichtes in disperdierenden medien,” Ann. Phys. (Leipzig) 44, 177–202 (1914).

L. Brillouin, “Über die fortpflanzung des licht in disperdierenden medien,” Ann. Phys. (Leipzig) 44, 203–240 (1914).

IEEE Trans. Antennas Propag.

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

M. Pieraccini, A. Bicci, D. Mecatti, G. Macaluso, and C. Atzeni, “Propagation of large bandwidth microwave signals in water,” IEEE Trans. Antennas Propag. 57, 3612–3618 (2009).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Lett.

Phys. Rev. A

B. Macke and B. Ségard, “Optical precursors in transparent media,” Phys. Rev. A 80, 011803 (2009).
[CrossRef]

H. Jeong and S. Du, “Two-way transparency in the light-matter interaction: Optical precursors with electromagnetically induced transparency,” Phys. Rev. A 79, 011802 (2009).
[CrossRef]

B. Macke and B. Ségard, “Optical precursors with self-induced transparency,” Phys. Rev. A 81, 015803 (2010).
[CrossRef]

J. F. Chen, S. Wang, D. Wei, M. M. T. Loy, G. K. L. Wong, and S. Du, “Optical coherent transients in cold atoms: from free induction decay to optical precursors,” Phys. Rev. A 81, 033844 (2010).
[CrossRef]

W. R. LeFew, S. Venakides, and D. J. Gauthier, “Accurate description of optical precursors and their relation to weak-field coherent optical transients,” Phys. Rev. A 79, 063842 (2009).
[CrossRef]

M. D. Crisp, “Propagation of small-area pulses of coherent light through a resonant medium,” Phys. Rev. A 1, 1604–1611 (1970).
[CrossRef]

Phys. Rev. Lett.

D. Wei, J. F. Chen, M. M. T. Loy, G. K. L. Wong, and S. Du, “Optical precursors with electromagnetically induced transparency in cold atoms,” Phys. Rev. Lett. 103, 093602 (2009).
[CrossRef] [PubMed]

H. Jeong, A. M. C. Dawes, and D. J. Gauthier, “Direct observation of optical precursors in a region of anomalous dispersion,” Phys. Rev. Lett. 96, 143901 (2006).
[CrossRef] [PubMed]

Radio Sci.

P. D. Smith and K. E. Oughstun, “Electromagnetic energy dissipation and propagation of an ultrawideband plane wave pulse in a causally dispersive dielectric,” Radio Sci. 33, 1489–1504 (1998).
[CrossRef]

SIAM Rev.

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

Other

There are several definitions of what an ultrawideband signal is; see, for example, Section 11.2.2 of . We take here the simple physical definition to mean a temporal pulse whose frequency spectrum along the positive real frequency axis is nonzero as ω→0 and which goes to zero as 1/ω or less as ω→∞.

The Beer–Lambert–Bouger law was originally discovered by P. Bouger, Essai d’Optique sur la Gradation de la Lumiere (Claude Jombert, 1729) and subsequently cited by J. H. Lambert, Photometri (V. E. Klett, 1760); the result was then extended by A. Beer, Einleitung in die höhere Optik (Friedrich Viewig, 1853) to include the concentration of solutions in the expression of the absorption coefficient for the intensity of light.

H. A. Lorentz, Theory of Electrons (Teubner, 1906), Chap. IV.

In the group velocity description, the “strength” of the material dispersion is typically measured through the derivative dnr(ω)/dω as that is what appears in the coefficients β1≡(nr+ωdnr/dω)/c and β2≡(2dnr/dω+ωd2nr/dω2)/c in the Taylor series expansion of the real propagation factor β(ω)≡R{k̃(ω)} in a hypothetical “lossless” dispersive medium. It is directly found from Eq. that this first derivative is directly proportional to the number density N so that these coefficients can always be made as small as desired at any real ω simply by choosing N sufficiently small.

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

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941), Sec. 5.18.

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

K. E. Oughstun, Electromagnetic & Optical Pulse Propagation 1: Spectral Representations in Temporally Dispersive Media (Springer, 2006).
[PubMed]

K. E. Oughstun, Electromagnetic & Optical Pulse Propagation 2: Temporal Pulse Dynamics in Dispersive Attenuative Media (Springer, 2009).

H. M. Nussenzveig, Causality and Dispersion Relations (Academic, 1972), Chap. 1.

B. Macke and B. Ségard, “Comment on “Direct observation of optical precursors in a region of anomalous dispersion”,” arXiv:physics/0605039.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

Numerically determined peak amplitude decay of a Heaviside unit step function signal with below resonance carrier frequency ω c 0.769 ω 0 in a single-resonance Lorentz model dielectric as a function of the relative propagation distance z / z d for several decreasing values of the phenomenological damping constant δ > 0 .

Fig. 2
Fig. 2

Average slope of the base ten logarithm of the numerical data presented in Fig. 1. The discontinuous behavior exhibited in the δ 0 / 1000 case is attributed to numerical error, which may be eliminated by increasing the number of sample points used to calculate the wave field.

Fig. 3
Fig. 3

Propagated plane wave field at ten absorption depths due to an input Heaviside unit step function modulated signal starting at time t = 0 at the plane z = 0 with below resonance angular carrier frequency ω c 0.769 ω 0 in a single-resonance Lorentz model dielectric. The cross-symbol in the plot marks the time t 0 = θ 0 z / c following the onset of the Brillouin precursor.

Fig. 4
Fig. 4

Propagated wave field at five absorption depths in a single-resonance Lorentz medium with plasma frequency ω p 1 = 3.05 × 10 14   rad / s . Notice that A ( z , t ) = 0 for all t < z / c .

Fig. 5
Fig. 5

Propagated wave field at five absorption depths in a single-resonance Lorentz medium with plasma frequency ω p 2 = 3.05 × 10 12   rad / s . Notice that A ( z , t ) = 0 for all t < z / c .

Fig. 6
Fig. 6

Propagated wave field at five absorption depths in a single-resonance Lorentz medium with plasma frequency ω p 3 = 3.05 × 10 10   rad / s . Notice that A ( z , t ) = 0 for all t < z / c .

Fig. 7
Fig. 7

Separate Sommerfeld precursor A S ( z , t ) , Brillouin precursor A B ( z , t ) , and signal A c ( z , t ) components of the propagated wave field and their superposition to yield the total propagated field A ( z , t ) at five absorption depths in a single-resonance Lorentz medium with plasma frequency ω p 2 = 3.05 × 10 12   rad / s . Notice that A ( z , t ) = 0 for all t < z / c .

Equations (27)

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

A ( z , t ) = 1 2 π C f ̃ ( ω ) e ( z / c ) ϕ ( ω , θ ) d ω ,
f ̃ ( ω ) = f ( t ) e i ω t d t ,
( 2 + k ̃ 2 ( ω ) ) A ̃ ( z , ω ) = 0 ,
k ̃ ( ω ) ω c n ( ω )
ϕ ( ω , θ ) i c z [ k ̃ ( ω ) z ω t ] = i ω [ n ( ω ) θ ]
θ c t z ,     z > 0
n ( ω ) = ( 1 ω p 2 ω 2 ω 0 2 + 2 i δ ω ) 1 / 2 .
A ( z , t ) = A S ( z , t ) + A B ( z , t ) + A c ( z , t ) ,
θ 0 n ( 0 ) = 1 + ω p 2 / ω 0 2 ,
( ω p / ω 0 ) 2 θ 1 ( ω 0 / ω p ) 2 ,
( 2 c ω p z ) 2 θ 1 ( ω 0 c ω p δ z ) 2 .
ω S P n ± ( θ 1 ) 2 δ 3 α c f i ,
θ 1 θ 0 + 2 δ 2 ω p 2 3 α c f θ 0 ω 0 4 ,
ϕ ( ω S P n + ( θ 0 ) , θ 0 ) = 0 ,
T eff 8 δ 2 ω p 2 z 3 θ 0 ω 0 4 c ,
A ( 0 , t ) = f ( t ) = u H ( t ) sin ( ω c t ) ,
z d α 1 ( ω c )
k ̃ 1 ( ω ) z 1 ω t 1 = k ̃ 2 ( ω ) z 2 ω t 2
β 1 ( ω ) z 1 ω t 1 = β 2 ( ω ) z 2 ω t 2 ,
α 1 ( ω ) z 1 = α 2 ( ω ) z 2 ,
z 2 = α 1 ( ω ) α 2 ( ω ) z 1 ,     ω
n ( ω ) = 1 + N g ( ω ) 1 + 1 2 N g ( ω )     as   N 0 ,
z 2 N 1 N 2 z 1 .
ω c ( 1 + 1 2 N 1 g r ( ω ) ) z 1 ω t 1 ω c ( 1 + 1 2 N 2 g r ( ω ) ) z 2 ω t 2 ω c ( 1 + 1 2 N 2 g r ( ω ) ) N 1 N 2 z 1 ω t 2 ,
t 2 t 1 + ( N 1 N 2 1 ) z 1 c ,
z 2 ω p 1 2 ω p 2 2 z 1 ,
t 2 t 1 + ( ω p 1 2 ω p 2 2 1 ) z 1 c .

Metrics