Advertisement

Evolution of the Signal

  • Kurt E. Oughstun
Chapter
  • 838 Downloads
Part of the Springer Series in Optical Sciences book series (SSOS, volume 144)

Abstract

The contribution A c (z, t) to the asymptotic behavior of the propagated plane wavefield A(z, t) that is due to the presence of any simple pole singularities of the spectral function \tilde{u}(ω − ω c ), where \(A(0,t) = u(t)\sin ({\omega }_{c}t + \psi )\) with fixed angular carrier frequency ω c ≥ 0, is now condidered in some detail, with primary attention given to the oscillatory case when ω c > 0. As discussed in Sect. 12.4, the field component A c (z, t) is associated with any long-term signal that is being propagated through the dispersive material. The velocity of propagation of the signal and the transition from the total precursor field to the signal field are determined by the relative asymptotic dominance of the component fields A s (z, t), A b (z, t), A m (z, t), and A c (z, t). Consequently, discussion of these topics is deferred to Chap. 15 where the asymptotic description of the dynamical evolution of the total propagated wavefield A(z, t) is considered by combining the results of this chapter with those of Chap. 13.

Keywords

Saddle Point Pole Contribution Pole Singularity Complementary Error Function Steep Descent Path 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    F. W. J. Olver, “Why steepest descents?,” SIAM Rev., vol. 12, no. 2, pp. 228–247, 1970.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable. London: Oxford University Press, 1935. p. 110.Google Scholar
  3. 3.
    E. C. Titchmarsh, The Theory of Functions. London: Oxford University Press, 1937. Sect. 10.5.Google Scholar
  4. 4.
    L. Brillouin, “Über die fortpflanzung des licht in disperdierenden medien,” Ann. Phys., vol. 44, pp. 204–240, 1914.Google Scholar
  5. 5.
    L. Brillouin, Wave Propagation and Group Velocity. New York: Academic, 1960.zbMATHGoogle Scholar
  6. 6.
    N. Bleistein, “Uniform asymptotic expansions of integrals with stationary point near algebraic singularity,” Com. Pure Appl. Math., vol. XIX, no. 4, pp. 353–370, 1966.MathSciNetCrossRefGoogle Scholar
  7. 7.
    N. Bleistein, “Uniform asymptotic expansions of integrals with many nearby stationary points and algebraic singularities,” J. Math. Mech., vol. 17, no. 6, pp. 533–559, 1967.MathSciNetzbMATHGoogle Scholar
  8. 8.
    L. Felsen and N. Marcuvitz, Radiation and Scattering of Waves. Englewood Cliffs, NJ: Prentice-Hall, Inc., 1973.Google Scholar
  9. 9.
    N. A. Cartwright, Uniform Asymptotic Description of the Unit Step Function Modulated Sinusoidal Signal. PhD thesis, College of Engineering & Mathematical Sciences, University of Vermont, 2004.Google Scholar
  10. 10.
    N. A. Cartwright and K. E. Oughstun, “Uniform asymptotics applied to ultrawideband pulse propagation,” SIAM Review., vol. 49, no. 4, pp. 628–648, 2007.MathSciNetADSzbMATHCrossRefGoogle Scholar
  11. 11.
    L. B. Felsen and N. Marcuvitz, “Modal analysis and synthesis of electromagnetic fields,” Polytechnic Inst. Brooklyn, Microwave Res. Inst. Rep., 1959.Google Scholar
  12. 12.
    N. Bleistein and R. Handelsman, Asymptotic Expansions of Integrals. New York: Holt, Rinehart and Winston, 1975.Google Scholar
  13. 13.
    K. E. Oughstun, Propagation of Optical Pulses in Dispersive Media. PhD thesis, The Institute of Optics, University of Rochester, 1978.Google Scholar
  14. 14.
    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, vol. 6, no. 9, pp. 1394–1420, 1989.MathSciNetADSCrossRefGoogle Scholar
  15. 15.
    K. E. Oughstun and G. C. Sherman, Pulse Propagation in Causal Dielectrics. Berlin: Springer-Verlag, 1994.Google Scholar
  16. 16.
    P. D. Smith, Energy Dissipation of Pulsed Electromagnetic Fields in Causally Dispersive Dielectrics. PhD thesis, University of Vermont, 1995. Reprinted in UVM Research Report CSEE/95/07-02 (July 18, 1995).Google Scholar
  17. 17.
    K. E. Oughstun and P. D. Smith, “On the accuracy of asymptotic approximations in ultrawideband signal, short pulse, time-domain electromagnetics,” in Proceedings of the 2000 IEEE International Symposium on Antennas and Propagation, (Salt Lake City), pp. 685–688, 2000.Google Scholar
  18. 18.
    S. Dvorak and D. Dudley, “Propagation of ultra-wide-band electromagnetic pulses through dispersive media,” IEEE Trans. Elec. Comp., vol. 37, no. 2, pp. 192–200, 1995.CrossRefGoogle Scholar
  19. 19.
    N. A. Cartwright and K. E. Oughstun, “Ultrawideband pulse penetration in an isotropic collisionless plasma,” in 2007 CNC/USNC North American Radio Science Meeting, 2007.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Kurt E. Oughstun
    • 1
  1. 1.College of Engineering & Mathematical Sciences School of EngineeringUniversity of VermontBurlingtonUSA

Personalised recommendations