Abstract

The asymptotic description of the coupled spatial and temporal evolution of a pulsed ultrawideband electromagnetic beam field as it propagates through a dispersive, attenuative material that occupies the half-space zz0 is obtained from the angular spectrum of plane waves representation. This angular-spectrum representation expresses the wave field as a superposition of both homogeneous and inhomogeneous plane waves. The paraxial approximation of the spatial part of this representation for nontruncated beam fields results in a description that explicitly displays the temporal evolution of the pulsed-beam field through a single-contour integral that is of the same form as that obtained for a pulsed plane-wave field propagating in the positive z direction in a lossy, dispersive medium. The accuracy of this paraxial approximation is shown to improve as the material’s attenuation increases.

© 2001 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. K. E. Oughstun, G. C. Sherman, Electromagnetic Pulse Propagation in Causal Dielectrics (Springer-Verlag, Berlin, 1994).
  2. W. H. Carter, “Electromagnetic beam fields,” Opt. Acta 21, 871–892 (1974).
    [CrossRef]
  3. G. C. Sherman, “Diffracted wave fields expressible by plane-wave expansions containing only homogeneous waves,” Phys. Rev. Lett. 21, 761–764 (1968).
    [CrossRef]
  4. G. C. Sherman, “Diffracted wave fields expressible by plane-wave expansions containing only homogeneous waves,” J. Opt. Soc. Am. 59, 697–711 (1969).
    [CrossRef]
  5. K. E. Oughstun, “The angular spectrum representation and the Sherman expansion of pulsed electromagnetic beam fields in dispersive, attenuative media,” Pure Appl. Opt. 7, 1059–1078 (1998).
    [CrossRef]
  6. J. J. Stamnes, Waves in Focal Regions (Hilger, London, 1987), Sec. 15.2.
  7. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Sec. 5.18.
  8. Both cgs and mks units are employed in this paper through the use of a conversion factor that appears in the double brackets ‖⋯‖ in each affected equation. If this factor is included in the equation it is then in cgs units, provided that∊0=μ0=1,whereas if this factor is omitted the equation is in mks units. If no such factor appears the equation is correct in both systems of units.
  9. H. M. Nussenzveig, Causality and Dispersion Relations (Academic, New York, 1972), Chap. 1.
  10. T. Melamed, L. B. Felsen, “Pulsed-beam propagation in lossless dispersive media. I. Theory,” J. Opt. Soc. Am. A 15, 1268–1276 (1998).
    [CrossRef]
  11. T. Melamed, L. B. Felsen, “Pulsed-beam propagation in lossless dispersive media. II. A numerical example,” J. Opt. Soc. Am. A 15, 1277–1284 (1998).
    [CrossRef]
  12. J. E. K. Laurens, K. E. Oughstun, “Electromagnetic impulse response of triply-distilled water,” in Ultra-Wideband, Short-Pulse Electromagnetics 4. E. Heyman, B. Mandelbaum, J. Shiloh, eds. (Plenum, New York, 1999), pp. 243–264.
  13. P. D. Smith, K. E. Oughstun, “Ultrawideband electromagnetic pulse propagation in triply-distilled water,” in Ultra-Wideband, Short-Pulse Electromagnetics 4, E. Heyman, B. Mandelbaum, J. Shiloh, eds. (Plenum, New York, 1999), pp. 265–276.
  14. J. McConnell, Rotational Brownian Motion and Dielectric Theory (Academic, London, 1980).
  15. P. Debye, Polar Molecules (Dover, New York, 1929).
  16. E. T. Copson, Asymptotic Expansions (Cambridge U. Press, Cambridge, 1971), Chap. 2.
  17. M. Born, E. Wolf, Principles of Optics, 7th ed., expanded (Cambridge U. Press, Cambridge, 1999), Chap. 8.
  18. K. E. Oughstun, “Dynamical structure of the precursor fields in linear dispersive pulse propagation in lossy dielectrics,” in Ultra-Wideband, Short-Pulse Electromagnetics 2, L. Carin, L. B. Felsen, eds. (Plenum, New York, 1995), pp. 257–272.
  19. K. E. Oughstun, G. C. Sherman, “Propagation of electromagnetic pulses in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. B 5, 817–849 (1988).
    [CrossRef]
  20. K. E. Oughstun, G. C. Sherman, “Uniform asymptotic description of electromagnetic pulse propagation in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. A 6, 1394–1420 (1989).
    [CrossRef]
  21. S. Shen, K. E. Oughstun, “Dispersive pulse propagation in a double-resonance Lorentz medium,” J. Opt. Soc. Am. B 6, 948–963 (1989).
    [CrossRef]
  22. J. A. Solhaug, K. E. Oughstun, J. J. Stamnes, P. D. Smith, “Uniform asymptotic description of the Brillouin precursor in a Lorentz model dielectric,” Pure Appl. Opt. 7, 575–601 (1998).
    [CrossRef]
  23. K. E. Oughstun, G. C. Sherman, “Uniform asymptotic description of ultrashort rectangular optical pulse propagation in a linear, causally dispersive medium,” Phys. Rev. A 41, 6090–6113 (1990).
    [CrossRef] [PubMed]
  24. G. C. Sherman, K. E. Oughstun, “Description of pulse dynamics in Lorentz media in terms of the energy velocity and attenuation of time-harmonic waves,” Phys. Rev. Lett. 47, 1451–1454 (1981).
    [CrossRef]
  25. C. M. Balictsis, K. E. Oughstun, “Uniform asymptotic description of ultrashort Gaussian pulse propagation in a causal, dispersive dielectric,” Phys. Rev. E 47, 3645–3669 (1993).
    [CrossRef]
  26. K. E. Oughstun, C. M. Balictsis, “Gaussian pulse propagation in a dispersive, absorbing dielectric,” Phys. Rev. Lett. 77, 2210–2213 (1996).
    [CrossRef] [PubMed]
  27. C. M. Balictsis, K. E. Oughstun, “Generalized asymptotic description of the propagated field dynamics in Gaussian pulse propagation in a linear, causally dispersive medium,” Phys. Rev. E 51, 1910–1921 (1997).
    [CrossRef]
  28. K. E. Oughstun, H. Xiao, “Failure of the quasimonochromatic approximation for ultrashort pulse propagation in a dispersive, attenuative medium,” Phys. Rev. Lett. 78, 642–645 (1997).
    [CrossRef]
  29. H. Xiao, K. E. Oughstun, “Failure of the group velocity description for ultrawideband pulse propagation in a double-resonance Lorentz model dielectric,” J. Opt. Soc. Am. B 16, 1773–1785 (1999).
    [CrossRef]
  30. J. A. Solhaug, J. J. Stamnes, K. E. Oughstun, “Diffraction of electromagnetic pulses in a single-resonance Lorentz medium,” Pure Appl. Opt. 7, 1079–1101 (1998).
    [CrossRef]

1999

1998

J. A. Solhaug, J. J. Stamnes, K. E. Oughstun, “Diffraction of electromagnetic pulses in a single-resonance Lorentz medium,” Pure Appl. Opt. 7, 1079–1101 (1998).
[CrossRef]

T. Melamed, L. B. Felsen, “Pulsed-beam propagation in lossless dispersive media. I. Theory,” J. Opt. Soc. Am. A 15, 1268–1276 (1998).
[CrossRef]

T. Melamed, L. B. Felsen, “Pulsed-beam propagation in lossless dispersive media. II. A numerical example,” J. Opt. Soc. Am. A 15, 1277–1284 (1998).
[CrossRef]

K. E. Oughstun, “The angular spectrum representation and the Sherman expansion of pulsed electromagnetic beam fields in dispersive, attenuative media,” Pure Appl. Opt. 7, 1059–1078 (1998).
[CrossRef]

J. A. Solhaug, K. E. Oughstun, J. J. Stamnes, P. D. Smith, “Uniform asymptotic description of the Brillouin precursor in a Lorentz model dielectric,” Pure Appl. Opt. 7, 575–601 (1998).
[CrossRef]

1997

C. M. Balictsis, K. E. Oughstun, “Generalized asymptotic description of the propagated field dynamics in Gaussian pulse propagation in a linear, causally dispersive medium,” Phys. Rev. E 51, 1910–1921 (1997).
[CrossRef]

K. E. Oughstun, H. Xiao, “Failure of the quasimonochromatic approximation for ultrashort pulse propagation in a dispersive, attenuative medium,” Phys. Rev. Lett. 78, 642–645 (1997).
[CrossRef]

1996

K. E. Oughstun, C. M. Balictsis, “Gaussian pulse propagation in a dispersive, absorbing dielectric,” Phys. Rev. Lett. 77, 2210–2213 (1996).
[CrossRef] [PubMed]

1993

C. M. Balictsis, K. E. Oughstun, “Uniform asymptotic description of ultrashort Gaussian pulse propagation in a causal, dispersive dielectric,” Phys. Rev. E 47, 3645–3669 (1993).
[CrossRef]

1990

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

1989

1988

1981

G. C. Sherman, K. E. Oughstun, “Description of pulse dynamics in Lorentz media in terms of the energy velocity and attenuation of time-harmonic waves,” Phys. Rev. Lett. 47, 1451–1454 (1981).
[CrossRef]

1974

W. H. Carter, “Electromagnetic beam fields,” Opt. Acta 21, 871–892 (1974).
[CrossRef]

1969

1968

G. C. Sherman, “Diffracted wave fields expressible by plane-wave expansions containing only homogeneous waves,” Phys. Rev. Lett. 21, 761–764 (1968).
[CrossRef]

Balictsis, C. M.

C. M. Balictsis, K. E. Oughstun, “Generalized asymptotic description of the propagated field dynamics in Gaussian pulse propagation in a linear, causally dispersive medium,” Phys. Rev. E 51, 1910–1921 (1997).
[CrossRef]

K. E. Oughstun, C. M. Balictsis, “Gaussian pulse propagation in a dispersive, absorbing dielectric,” Phys. Rev. Lett. 77, 2210–2213 (1996).
[CrossRef] [PubMed]

C. M. Balictsis, K. E. Oughstun, “Uniform asymptotic description of ultrashort Gaussian pulse propagation in a causal, dispersive dielectric,” Phys. Rev. E 47, 3645–3669 (1993).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th ed., expanded (Cambridge U. Press, Cambridge, 1999), Chap. 8.

Carter, W. H.

W. H. Carter, “Electromagnetic beam fields,” Opt. Acta 21, 871–892 (1974).
[CrossRef]

Copson, E. T.

E. T. Copson, Asymptotic Expansions (Cambridge U. Press, Cambridge, 1971), Chap. 2.

Debye, P.

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

Felsen, L. B.

Laurens, J. E. K.

J. E. K. Laurens, K. E. Oughstun, “Electromagnetic impulse response of triply-distilled water,” in Ultra-Wideband, Short-Pulse Electromagnetics 4. E. Heyman, B. Mandelbaum, J. Shiloh, eds. (Plenum, New York, 1999), pp. 243–264.

McConnell, J.

J. McConnell, Rotational Brownian Motion and Dielectric Theory (Academic, London, 1980).

Melamed, T.

Nussenzveig, H. M.

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

Oughstun, K. E.

H. Xiao, K. E. Oughstun, “Failure of the group velocity description for ultrawideband pulse propagation in a double-resonance Lorentz model dielectric,” J. Opt. Soc. Am. B 16, 1773–1785 (1999).
[CrossRef]

J. A. Solhaug, K. E. Oughstun, J. J. Stamnes, P. D. Smith, “Uniform asymptotic description of the Brillouin precursor in a Lorentz model dielectric,” Pure Appl. Opt. 7, 575–601 (1998).
[CrossRef]

J. A. Solhaug, J. J. Stamnes, K. E. Oughstun, “Diffraction of electromagnetic pulses in a single-resonance Lorentz medium,” Pure Appl. Opt. 7, 1079–1101 (1998).
[CrossRef]

K. E. Oughstun, “The angular spectrum representation and the Sherman expansion of pulsed electromagnetic beam fields in dispersive, attenuative media,” Pure Appl. Opt. 7, 1059–1078 (1998).
[CrossRef]

C. M. Balictsis, K. E. Oughstun, “Generalized asymptotic description of the propagated field dynamics in Gaussian pulse propagation in a linear, causally dispersive medium,” Phys. Rev. E 51, 1910–1921 (1997).
[CrossRef]

K. E. Oughstun, H. Xiao, “Failure of the quasimonochromatic approximation for ultrashort pulse propagation in a dispersive, attenuative medium,” Phys. Rev. Lett. 78, 642–645 (1997).
[CrossRef]

K. E. Oughstun, C. M. Balictsis, “Gaussian pulse propagation in a dispersive, absorbing dielectric,” Phys. Rev. Lett. 77, 2210–2213 (1996).
[CrossRef] [PubMed]

C. M. Balictsis, K. E. Oughstun, “Uniform asymptotic description of ultrashort Gaussian pulse propagation in a causal, dispersive dielectric,” Phys. Rev. E 47, 3645–3669 (1993).
[CrossRef]

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

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

K. E. Oughstun, G. C. Sherman, “Uniform asymptotic description of electromagnetic pulse propagation in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. A 6, 1394–1420 (1989).
[CrossRef]

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

G. C. Sherman, K. E. Oughstun, “Description of pulse dynamics in Lorentz media in terms of the energy velocity and attenuation of time-harmonic waves,” Phys. Rev. Lett. 47, 1451–1454 (1981).
[CrossRef]

K. E. Oughstun, “Dynamical structure of the precursor fields in linear dispersive pulse propagation in lossy dielectrics,” in Ultra-Wideband, Short-Pulse Electromagnetics 2, L. Carin, L. B. Felsen, eds. (Plenum, New York, 1995), pp. 257–272.

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

P. D. Smith, K. E. Oughstun, “Ultrawideband electromagnetic pulse propagation in triply-distilled water,” in Ultra-Wideband, Short-Pulse Electromagnetics 4, E. Heyman, B. Mandelbaum, J. Shiloh, eds. (Plenum, New York, 1999), pp. 265–276.

J. E. K. Laurens, K. E. Oughstun, “Electromagnetic impulse response of triply-distilled water,” in Ultra-Wideband, Short-Pulse Electromagnetics 4. E. Heyman, B. Mandelbaum, J. Shiloh, eds. (Plenum, New York, 1999), pp. 243–264.

Shen, S.

Sherman, G. C.

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

K. E. Oughstun, G. C. Sherman, “Uniform asymptotic description of electromagnetic pulse propagation in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. A 6, 1394–1420 (1989).
[CrossRef]

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

G. C. Sherman, K. E. Oughstun, “Description of pulse dynamics in Lorentz media in terms of the energy velocity and attenuation of time-harmonic waves,” Phys. Rev. Lett. 47, 1451–1454 (1981).
[CrossRef]

G. C. Sherman, “Diffracted wave fields expressible by plane-wave expansions containing only homogeneous waves,” J. Opt. Soc. Am. 59, 697–711 (1969).
[CrossRef]

G. C. Sherman, “Diffracted wave fields expressible by plane-wave expansions containing only homogeneous waves,” Phys. Rev. Lett. 21, 761–764 (1968).
[CrossRef]

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

Smith, P. D.

J. A. Solhaug, K. E. Oughstun, J. J. Stamnes, P. D. Smith, “Uniform asymptotic description of the Brillouin precursor in a Lorentz model dielectric,” Pure Appl. Opt. 7, 575–601 (1998).
[CrossRef]

P. D. Smith, K. E. Oughstun, “Ultrawideband electromagnetic pulse propagation in triply-distilled water,” in Ultra-Wideband, Short-Pulse Electromagnetics 4, E. Heyman, B. Mandelbaum, J. Shiloh, eds. (Plenum, New York, 1999), pp. 265–276.

Solhaug, J. A.

J. A. Solhaug, J. J. Stamnes, K. E. Oughstun, “Diffraction of electromagnetic pulses in a single-resonance Lorentz medium,” Pure Appl. Opt. 7, 1079–1101 (1998).
[CrossRef]

J. A. Solhaug, K. E. Oughstun, J. J. Stamnes, P. D. Smith, “Uniform asymptotic description of the Brillouin precursor in a Lorentz model dielectric,” Pure Appl. Opt. 7, 575–601 (1998).
[CrossRef]

Stamnes, J. J.

J. A. Solhaug, K. E. Oughstun, J. J. Stamnes, P. D. Smith, “Uniform asymptotic description of the Brillouin precursor in a Lorentz model dielectric,” Pure Appl. Opt. 7, 575–601 (1998).
[CrossRef]

J. A. Solhaug, J. J. Stamnes, K. E. Oughstun, “Diffraction of electromagnetic pulses in a single-resonance Lorentz medium,” Pure Appl. Opt. 7, 1079–1101 (1998).
[CrossRef]

J. J. Stamnes, Waves in Focal Regions (Hilger, London, 1987), Sec. 15.2.

Stratton, J. A.

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

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 7th ed., expanded (Cambridge U. Press, Cambridge, 1999), Chap. 8.

Xiao, H.

H. Xiao, K. E. Oughstun, “Failure of the group velocity description for ultrawideband pulse propagation in a double-resonance Lorentz model dielectric,” J. Opt. Soc. Am. B 16, 1773–1785 (1999).
[CrossRef]

K. E. Oughstun, H. Xiao, “Failure of the quasimonochromatic approximation for ultrashort pulse propagation in a dispersive, attenuative medium,” Phys. Rev. Lett. 78, 642–645 (1997).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Acta

W. H. Carter, “Electromagnetic beam fields,” Opt. Acta 21, 871–892 (1974).
[CrossRef]

Phys. Rev. A

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

Phys. Rev. E

C. M. Balictsis, K. E. Oughstun, “Uniform asymptotic description of ultrashort Gaussian pulse propagation in a causal, dispersive dielectric,” Phys. Rev. E 47, 3645–3669 (1993).
[CrossRef]

C. M. Balictsis, K. E. Oughstun, “Generalized asymptotic description of the propagated field dynamics in Gaussian pulse propagation in a linear, causally dispersive medium,” Phys. Rev. E 51, 1910–1921 (1997).
[CrossRef]

Phys. Rev. Lett.

K. E. Oughstun, H. Xiao, “Failure of the quasimonochromatic approximation for ultrashort pulse propagation in a dispersive, attenuative medium,” Phys. Rev. Lett. 78, 642–645 (1997).
[CrossRef]

K. E. Oughstun, C. M. Balictsis, “Gaussian pulse propagation in a dispersive, absorbing dielectric,” Phys. Rev. Lett. 77, 2210–2213 (1996).
[CrossRef] [PubMed]

G. C. Sherman, K. E. Oughstun, “Description of pulse dynamics in Lorentz media in terms of the energy velocity and attenuation of time-harmonic waves,” Phys. Rev. Lett. 47, 1451–1454 (1981).
[CrossRef]

G. C. Sherman, “Diffracted wave fields expressible by plane-wave expansions containing only homogeneous waves,” Phys. Rev. Lett. 21, 761–764 (1968).
[CrossRef]

Pure Appl. Opt.

K. E. Oughstun, “The angular spectrum representation and the Sherman expansion of pulsed electromagnetic beam fields in dispersive, attenuative media,” Pure Appl. Opt. 7, 1059–1078 (1998).
[CrossRef]

J. A. Solhaug, K. E. Oughstun, J. J. Stamnes, P. D. Smith, “Uniform asymptotic description of the Brillouin precursor in a Lorentz model dielectric,” Pure Appl. Opt. 7, 575–601 (1998).
[CrossRef]

J. A. Solhaug, J. J. Stamnes, K. E. Oughstun, “Diffraction of electromagnetic pulses in a single-resonance Lorentz medium,” Pure Appl. Opt. 7, 1079–1101 (1998).
[CrossRef]

Other

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

J. J. Stamnes, Waves in Focal Regions (Hilger, London, 1987), Sec. 15.2.

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

Both cgs and mks units are employed in this paper through the use of a conversion factor that appears in the double brackets ‖⋯‖ in each affected equation. If this factor is included in the equation it is then in cgs units, provided that∊0=μ0=1,whereas if this factor is omitted the equation is in mks units. If no such factor appears the equation is correct in both systems of units.

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

J. E. K. Laurens, K. E. Oughstun, “Electromagnetic impulse response of triply-distilled water,” in Ultra-Wideband, Short-Pulse Electromagnetics 4. E. Heyman, B. Mandelbaum, J. Shiloh, eds. (Plenum, New York, 1999), pp. 243–264.

P. D. Smith, K. E. Oughstun, “Ultrawideband electromagnetic pulse propagation in triply-distilled water,” in Ultra-Wideband, Short-Pulse Electromagnetics 4, E. Heyman, B. Mandelbaum, J. Shiloh, eds. (Plenum, New York, 1999), pp. 265–276.

J. McConnell, Rotational Brownian Motion and Dielectric Theory (Academic, London, 1980).

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

E. T. Copson, Asymptotic Expansions (Cambridge U. Press, Cambridge, 1971), Chap. 2.

M. Born, E. Wolf, Principles of Optics, 7th ed., expanded (Cambridge U. Press, Cambridge, 1999), Chap. 8.

K. E. Oughstun, “Dynamical structure of the precursor fields in linear dispersive pulse propagation in lossy dielectrics,” in Ultra-Wideband, Short-Pulse Electromagnetics 2, L. Carin, L. B. Felsen, eds. (Plenum, New York, 1995), pp. 257–272.

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 (14)

Fig. 1
Fig. 1

Angular-frequency dependence of (a) the real and (b) the imaginary parts of the complex wave number k˜(ω)=β(ω)+iα(ω) for the Rocard–Powles model of triply distilled water at 25 °C. The values marked with a cross (×) in each diagram indicate the real and the imaginary values at the angular frequencies of ωc=2π×107 rad/s in the HF, ωc=2π×109 rad/s in the UHF, and ωc=2π×1011 rad/s in the EHF regions of the electromagnetic spectrum that are used in several examples in this paper.

Fig. 2
Fig. 2

(a) Real and (b) imaginary parts of the complex direction cosine m(ωc)={exp[i2ψ(ωc)]-(p2+q2)}1/2 at ωc=2π×107 rad/s plotted as functions of p with q=0. The solid curves depict the exact behavior, whereas the dashed curve depicts the behavior obtained with the paraxial approximation.

Fig. 3
Fig. 3

(a) Magnitude and (b) phase of the complex direction cosine m(ωc)={exp[i2ψ(ωc)]-(p2+q2)}1/2 at ωc=2π×107 rad/s plotted as functions of p with q=0. The solid curves represent the exact behavior, whereas the dashed curve depicts the behavior obtained with the paraxial approximation.

Fig. 4
Fig. 4

(a) Real and (b) imaginary parts of the propagation kernel G(p, q, ωc)=exp[ik(ωc)m(ωc)Δz] at ωc=2π×107 rad/s with Δz=0.1 m plotted as functions of p with q=0. The solid curves depict the exact behavior, whereas the dashed curves depict the behavior obtained with the paraxial approximation.

Fig. 5
Fig. 5

Same as is shown in Fig. 2 but with ωc=2π×109 rad/s.

Fig. 6
Fig. 6

Same as is shown in Fig. 3 but with ωc=2π×109 rad/s.

Fig. 7
Fig. 7

Same as is shown in Fig. 4 but with ωc=2π×109 rad/s.

Fig. 8
Fig. 8

Same as is shown in Fig. 2 but with ωc=2π×1011 rad/s.

Fig. 9
Fig. 9

Same as is shown in Fig. 3 but with ωc=2π×1011 rad/s.

Fig. 10
Fig. 10

Same as is shown in Fig. 4 but with ωc=2π×1011 rad/s.

Fig. 11
Fig. 11

(a) Real and (b) imaginary parts of the propagation kernel G(p, q, ωc)=exp[ik(ωc)m(ωc)Δz] at ωc=2π×109 rad/s with Δz=1.0 m plotted as functions of p with q=0. The solid curves depict the exact behavior, whereas the dashed curves depict the behavior obtained with the paraxial approximation.

Fig. 12
Fig. 12

Same as is shown in Fig. 11 but with Δz=10.0 m.

Fig. 13
Fig. 13

(a) Fresnel number evolution for (b) the Sommerfeld precursor field evolution plotted as functions of θ=ct/z at a fixed propagation distance of z=10zd=0.1153 m with ωc=5.00×1014 rad/s in a single-resonance Lorentz model dielectric with ω0=9.14×1014 rad/s, b0=5.00×1013 rad/s, and δ0=1.43×1013 rad/s.

Fig. 14
Fig. 14

(a) Fresnel number evolution for (b) the Brillouin precursor field evolution plotted as functions of θ=ct/z at a fixed propagation distance of z=10zd=0.1153 m with ωc=5.00×1014 rad/s in a single-resonance Lorentz model dielectric with ω0=9.14×1014 rad/s, b0=5.00×1013 rad/s, and δ0=1.43×1013 rad/s.

Equations (40)

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

U˜0(kT, ω)=-dt--dxdyU0(rT, t)×exp[-i(kT  rT-ωt)],
U0(rT, t)=14π3RC+dω--dkxdkyU˜0(kT, ω)×exp[i(kT  rT-ωt)],
U(rT, z, t)=(1/4π3)RC+dω--dkxdkyU˜0(kT, ω)×exp{i[kT  rT+γ(ω)Δz-ωt]}
E^0(kT, ω)=-[c/ωμc(ω)]k˜+(ω)×B˜0(kT, ω),
B˜0(kT, ω)=(c/ω)k˜+(ω)×E˜0(kT, ω)
k˜+(ω)=1^xkx+1^yky+1^zγ(ω)
k˜(ω)=(k˜+  k˜+)1/2=1/cω[μc(ω)]1/2,
γ(ω)=[k˜2(ω)-kT2]1/2,
c(ω)=(ω)+i4πσ(ω)/ω
U˜0(kT, ω)exp{i[kT  rT+γ(ω)Δz-ωt]}
=U˜0(kT, ω)exp{i[k˜+(ω)  Δr-ωt]}
ks=k(ω)p,ky=k(ω)q,γ(ω)=k(ω)m,
k˜(ω)=β(ω)+iα(ω)=k(ω)exp[iψ(ω)],
m(ω)={exp[i2ψ(ω)]-(p2+q2)}1/2,
m(ω)=R[m(ω)]0,m(ω)=I[m(ω)]0
m(ω)=12 {cos[2ψ(ω)]-(p2+q2)}+12{cos[2ψ(ω)]-(p2+q2)}2+1sin2[2ψ(ω)]1/21/2,
m(ω)=sin[2ψ(ω)]2m(ω),
m(ω)=[(p2+q2)-1]1/2,
m(ω)=[(p2+q2)+1]1/2,
exp[ik˜+(ω)  Δr]=exp[-k(ω)m(ω)Δz]×exp{ik(ω)[px+qy+m(ω)Δz]}.
m(ω)|(p, q)R><m(ω)|(p, q)R<
m(ω)|(p, q)R>>m(ω)|(p, q)R<,
U(r, t)=(1/4π3)RC+dω--k2(ω)U˜0(p, q, ω)×exp{ik(ω)[px+qy+m(ω)Δz]}dpdq,
m(ω)exp[iψ(ω)]-(1/2)(p2+q2)exp[-iψ(ω)],
G(p, q, ω)exp[ik(ω)m(ω)Δz]exp[ik˜(ω)Δz]exp[-k˜*(ω)Δz(p2+q2)/2],
(ω)=+j=12aj(1-iτjω)(1-iτfjω)
U(r, t)=(1/π)RC+F˜(r, ω)exp{i[k˜(ω)Δz-ωt]}dω,
F˜(r, ω)-i[k˜(ω)/2πΔz]--U^0(x, y, ω)×exp{i[k˜(ω)/2Δz]×[(x-x)2+(y-y)2]}dxdy
U^0(x, y, ω)=-U0(x, y, t)exp(iωt)dt
F˜(r, ω)i[k˜(ω)/2πΔz]--U^0(x, y, ω)×exp{-[α(ω)/2Δz][(x-x)2+(y-y)2]}exp{i[β(ω)/2Δz][(x-x)2+(y-y)2]}dxdy,
U(r, t)=(1/π)RC+F˜(r, ω)exp[(Δz/c)ϕ(ω, θ)]dω,
ϕ(ω, θ)=i(c/Δz)[k˜(ω)Δz-ωt]=iω[n(ω)-θ]
U(rT, z, t)=US(rT, z, t)+UB(rT, z, t)+Um(rT, z, t)+Uc(rT, z, t),
Us(rT, z, t)(c/2πΔz)1/2Rj=±[-ϕ(ωdj, θ)]-1/2×F˜(rT, ωdj-ωc)exp[(Δz/c)ϕ(ωdj, θ)],
UB(rT, z, t)(c/2πΔz)1/2R{[-ϕ(ωn+, θ)]-1/2×F˜(rT, ωn+-ωc)exp[(Δz/c)ϕ(ωn+, θ)]}
UB(rT, z, t1)Γ(1/3)2π31/62cΔz|ϕ(ωn, θ1)|1/3×R{iF˜(rT, ωn-ωc)×exp[(Δz/c)ϕ(ωn, θ1)]}
UB(rT, z, t)(c/2πΔz)1/2Rj=±[-ϕ(ωnj, θ)]-1/2×F˜(rT, ωnj-ωc)exp[(Δz/c)ϕ(ωnj, θ)]
U(rT, z, t)=UB(rT, z, t)+Um(rT, z, t)+Uc(rT, z, t)
Nj(θ)=(a2/2πΔz)β[ωsp(θ)],
n(ω)=1-b02(ω2-ω02)+2iδ0ω1/2.

Metrics