Abstract
In preparation for the asymptotic analysis of the exact Fourier–Laplace integral representation given either in (11.45) as
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
From the Greek isotimos, of equal worth.
- 2.
Notice that these approximate expressions for the branch point zero locations are different from those given in an earlier paper [11].
- 3.
- 4.
Notice that this result is somewhat different from (and more accurate than) that given in [11].
- 5.
This result is an extension of that given in [11].
- 6.
Recall that it is assumed here that each of the spectral functions \(\tilde{f}\)(ω) and \(\tilde{u}\)(ω − ω c ) is an analytic function of the complex variable ω, regular in the entire complex ω-plane except at a countable number of isolated points where that function may exhibit poles.
- 7.
Notice that Λ(θ) changes discontinuously with the space–time parameter θ as the path P(θ) crosses over each pole. However, each of these discontinuities is cancelled by a corresponding discontinuous change in I(z, θ).
- 8.
If u(t) is bounded and tends to zero rapidly enough such that the Fourier transform of u(t) converges uniformly for all real ω, then \(\tilde{u}\)(ω − ω c ) is an entire function of complex ω.
References
A. Sommerfeld, “Über die fortpflanzung des lichtes in disperdierenden medien,” Ann. Phys., vol. 44, pp. 177–202, 1914.
L. Brillouin, “Über die fortpflanzung des licht in disperdierenden medien,” Ann. Phys., vol. 44, pp. 204–240, 1914.
L. Brillouin, Wave Propagation and Group Velocity. New York: Academic, 1960.
K. E. Oughstun, Propagation of Optical Pulses in Dispersive Media. PhD thesis, The Institute of Optics, University of Rochester, 1978.
K. E. Oughstun and G. C. Sherman, “Uniform asymptotic description of dispersive pulse propagation,” J. Opt. Soc. Am. A, vol. 69, no. 10, p. 1448A, 1979.
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, vol. 5, no. 4, pp. 817–849, 1988.
K. E. Oughstun and G. C. Sherman, Pulse Propagation in Causal Dielectrics. Berlin: Springer, 1994.
H. M. Nussenzveig, Causality and Dispersion Relations. New York: Academic, 1972. Chap. 1.
L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media. Oxford: Pergamon, 1960. Ch. IX.
K. E. Oughstun, “Dynamical evolution of the precursor fields in linear dispersive pulse propagation in lossy dielectrics,” in Ultra-Wideband, Short-Pulse Electromagnetics 2 (L. Carin and L. B. Felsen, eds.), pp. 257–272, New York: Plenum, 1994.
S. Shen and K. E. Oughstun, “Dispersive pulse propagation in a double-resonance Lorentz medium,” J. Opt. Soc. Am. B, vol. 6, pp. 948–963, 1989.
J. E. K. Laurens and K. E. Oughstun, “Electromagnetic impulse response of triply-distilled water,” in Ultra-Wideband, Short-Pulse Electromagnetics 4 (E. Heyman, B. Mandelbaum, and J. Shiloh, eds.), pp. 243–264, New York: Plenum, 1999.
E. T. Whittaker and G. N. Watson, Modern Analysis. London: Cambridge University Press, fourth ed., 1963. p. 133.
K. E. Oughstun, “Dynamical evolution of the Brillouin precursor in Rocard-Powles-Debye model dielectrics,” IEEE Trans. Ant. Prop., vol. 53, no. 5, pp. 1582–1590, 2005.
M. Altarelli, D. L. Dexter, H. M. Nussenzveig, and D. Y. Smith, “Superconvergence and sum rules for the optical constants,” Phys. Rev. B, vol. 6, pp. 4502–4509, 1972.
H. Xiao and K. E. Oughstun, “Hybrid numerical-asymptotic code for dispersive pulse propagation calculations,” J. Opt. Soc. Am. A, vol. 15, no. 5, pp. 1256–1267, 1998.
K. E. Oughstun, J. E. K. Laurens, and C. M. Balictsis, “Asymptotic description of electromagnetic pulse propagation in a linear dispersive medium,” in Ultra-Wideband, Short-Pulse Electromagnetics (H. L. Bertoni, L. B. Felsen, and L. Carin, eds.), pp. 223–240, New York: Plenum, 1992.
J. McConnel, Rotational Brownian Motion and Dielectric Theory. London: Academic, 1980.
P. Debye, Polar Molecules. New York: Dover, 1929.
J. E. K. Laurens, Plane Wave Pulse Propagation in a Linear, Causally Dispersive Polar Medium. PhD thesis, University of Vermont, 1993. Reprinted in UVM Research Report CSEE/93/05-02 (May 1, 1993).
M. A. Messier, “A standard ionosphere for the study of electromagnetic pulse propagation,” Tech. Rep. Note 117, Air Force Weapons Laboratory, Albuquerque, NM, 1971.
P. Drude, Lehrbuch der Optik. Leipzig: Teubner, 1900. Chap. V.
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.
N. A. Cartwright and K. E. Oughstun, “Ultrawideband pulse penetration in a Debye medium with static conductivity,” in Fourth IASTED International Conference on Antennas, Radar, and Propagation, 2007.
E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable. London: Oxford University Press, 1935. Section 6.1.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Oughstun, K.E. (2009). Analysis of the Phase Function and Its Saddle Points. In: Electromagnetic and Optical Pulse Propagation 2. Springer Series in Optical Sciences, vol 144. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-0149-1_4
Download citation
DOI: https://doi.org/10.1007/978-1-4419-0149-1_4
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-0148-4
Online ISBN: 978-1-4419-0149-1
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)