Abstract

We study theoretically the propagation of a transient electromagnetic pulse in an unbounded homogeneous, isotropic, reciprocal, chiral medium, using the standard Fourier transform technique. We assume linear, causal, dispersive chiral media with single-resonance dispersion. For the permittivity and permeability functions a single-resonance Lorentz model is assumed, and for the chirality admittance function a single-resonance model consistent with Condon’s model of optical activity is used. Propagation of a TEM plane wave of a step sinusoidal signal is considered. We analyze the transient behavior of the electric-field components at a distant location and at various times of observation. In particular the role of the chirality of the medium and its effects on the dynamical evolution of the transient pulse in such a medium are highlighted, and the similarities and the differences between the transient signal propagation in chiral and nonchiral dispersive media are investigated. We show that several novel phenomena result from the presence of chirality in the medium. Among them are the development of a cross-polarized component, the splitting of instantaneous frequency (which suggests that there are effectively two first precursors and two second precursors for each field component of the transient pulse), and the buildup of the main signal in two stages. Other notable effects are also discussed. Physical insights into these results are provided.

© 1993 Optical Society of America

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  1. D. F. Arago, “Sur une modification remarquable qu’éprouvent les rayons lumineux dans leur passage à travers certains corps diaphanes, et sur quelques autres nouveaux phénomènes d’optique,” Mém. Inst. 1, 93–134 (1811).
  2. J. B. Biot, “Mémoire sur un nouveau genre d’oscillations que les molécules de la lumière éprouvent, en traversant certains cristaux,” Mém. Inst. 1, 1–372 (1812).
  3. L. Pasteur, “Sur les relations qui peuvent exister entre la forme cristalline, la composition chimique et le sens de la polarisation rotatoire,” Ann. Chim. Phys. 24, 442–459 (1848).
  4. A. Fresnel, “Mémoire sur la double refraction que les rayons lumineux éprouvent en traversant les aiguilles de cristal de roche suivant des directions paralleles à l’axe,” Oeuvres 1, 731–751 (1822).
  5. K. F. Lindman, “Über eine durch ein isotropes System yon Spiralformigen resonatoren erseugte Rotationspolarisation der Elektromagnetischen wellen,” Ann. Phys. 63, 621–644 (1920).
    [CrossRef]
  6. D. L. Jaggard, A. R. Mickelson, C. H. Papas, “On electromagnetic waves in chiral media,” Appl. Phys. 28, 211–216 (1979).
    [CrossRef]
  7. D. L. Jaggard, N. Engheta, “Chirality in electrodynamics: modeling and applications,” in Directions in Electromagnetic Wave Modelling,H. L. Bertoni, L. B. Felsen, eds. (Plenum, New York, 1991), pp. 485–493.
  8. C. F. Bohren, “Scattering of electromagnetic waves by an optically active cylinder,” J. Colloid Interface Sci. 66, 105–109 (1978).
    [CrossRef]
  9. S. Bassiri, N. Engheta, C. H. Papas, “Dyadic Green’s function and dipole radiation in chiral media,” Alta Freq. 2, 83–88 (1986).
  10. A. Lakhtakia, ed., Selected Papers on Natural Optical Activity (SPIE, Bellingham, Wash., 1990).
  11. A. Lakhtakia, V. V. Varadan, V. K. Varadan, Time-Harmonic Electromagnetic Fields in Chiral Media (Springer-Verlag, Berlin, 1989).
  12. S. Bassiri, C. H. Papas, N. Engheta, “Electromagnetic wave propagation through a dielectric-chiral interface and through a chiral slab,” J. Opt. Soc. Am. A 5, 1450–1459 (1988).
    [CrossRef]
  13. D. L. Jaggard, N. Engheta, J. C. Liu, “Chiroshields: a Salisbury/Dallenbach shield alternative,” Electron. Lett. 26, 1332–1334 (1990).
    [CrossRef]
  14. P. Pelet, N. Engheta, “The theory of chirowaveguides,” IEEE Trans. Antennas Propag. 38, 90–98 (1990).
    [CrossRef]
  15. M. S. Kluskens, E. H. Newman, “Scattering by a multilayer chiral cylinder,” IEEE Trans. Antennas Propag. 39, 91–96 (1991).
    [CrossRef]
  16. P. Pelet, N. Engheta, “Chirostrip antenna: line source problem,” J. Electromag. Waves Appl. 6, 771–793 (1992).
  17. P. L. E. Uslenghi, “Scattering by an impedance sphere coated with a chiral layer,” Electromagnetics 10, 201–211 (1990).
    [CrossRef]
  18. R. D. Graglia, P. L. E. Uslenghi, C. L. Yu, “Electromagnetic oblique scattering by a cylinder coated with chiral layers and anisotropic jump-immittance sheets,” J. Electromag. Waves Appl. 6, 695–719 (1992).
  19. W. S. Weiglhofer, “Isotropic chiral media and scalar Hertz potentials,” J. Phys. A 21, 2249–2251 (1988).
    [CrossRef]
  20. R. G. Rojas, “Integral equations for the scattering by three dimensional inhomogeneous chiral bodies,” J. Electromag. Waves Appl. 6, 733–750 (1992).
  21. A. H. Sihvola, I. V. Lindell, “Chiral Maxwell–Garnett mixing formula,” Electron. Lett. 26, 118–119, 1990.
    [CrossRef]
  22. M. I. Oksanen, S. A. Tretyakov, I. V. Lindell, “Vector circuit theory for isotropic and chiral slabs,” J. Electromag. Waves Appl. 4, 613–643 (1990).
  23. A. Toscano, L. Vegni, “Spectral dyadic Green’s function formulation for planar integrated structures with a grounded chiral slab,” J. Electromag. Waves Appl. 6, 751–769 (1992).
  24. X. Sun, D. L. Jaggard, “Accelerated particle radiation in chiral media,” J. Appl. Phys. 69, 34–38 (1991).
    [CrossRef]
  25. M. W. Kowarz, N. Engheta, “Spherical chirolenses,” Opt. Lett. 15, 299–301 (1990).
    [CrossRef] [PubMed]
  26. T. Guire, V. V. Varadan, V. K. Varadan, “Effect of chirality on reflection of EM waves by planar dielectric slab,” IEEE Trans. Electromag. Compat. 32, 300–303 (1990).
    [CrossRef]
  27. N. Engheta, guest ed., special issue on wave interactions with chiral and complex media, J. Electromag. Waves Appl. 6, 537–798 (1992).
  28. P. G. Zablocky, N. Engheta, “Transient response of chiral materials,” presented at the 1991 North American Radio Science/International IEEE AP-S/U RSI Symposium, London, Ontario, Canada, June 24–28, 1991.
  29. N. Engheta, P. G. Zablocky, “A step towards determining transient response of chiral materials: Kramers–Kronig relations for chiral parameters,” Electron. Lett. 26, 2132–2134 (1990).
    [CrossRef]
  30. N. Engheta, P. G. Zablocky, “Effect of chirality on the transient signal wave front,” Opt. Lett. 16, 1924–1926 (1991).
    [CrossRef] [PubMed]
  31. G. Kristensson, S. Rikte, “Transient scattering in bi-isotropic media,” presented at the 1991 Symposium on Progress in Electromagnetics Research, Cambridge, Mass., July 1–5, 1991.
  32. G. Kristensson, S. Rikte, “Transient wave propagation in reciprocal bi-isotropic media at oblique incidence,” Tech. Rep. LUTEDX/(TEAT-7019)/1–25 (Lund Institute of Technology, Department of Electromagnetic Theory, Lund, Sweden, 1992).
  33. S. Rikte, “Propagation of transient electromagnetic waves in dispersive reciprocal bi-isotropic media,” Tech. Rep. LUTEDX/(TEAT-1005)/1–12 (Lund Institute of Technology, Department of Electromagnetic Theory, Lund, Sweden, 1992).
  34. A. Sommerfeld, “Uber die Fortpflanzung des Lichtes in disperdierenden Medien,” Ann. Phys. 44, 177–202 (1914).
    [CrossRef]
  35. L. Brillouin, Wave Propagation and Group Velocity (Academic, New York, 1960).
  36. L. B. Felsen, Transients Electromagnetic Fields (Springer-Verlag, New York, 1976).
    [CrossRef]
  37. K. E. Oughstun, “Pulse propagation in a linear, causally dispersive medium,” Proc. IEEE 79, 1379–1390 (1991).
    [CrossRef]
  38. R. S. Elliott, “Pulse waveform degradation due to dispersion in waveguide,” IEEE Trans. Microwave Theory Tech. MTT-5, 254–257 (1956).
  39. J. R. Wait, “Propagation of electromagnetic pulses in terrestrial waveguides,” IEEE Trans. Antennas Propag. AP-13, 904–918 (1965).
    [CrossRef]
  40. R. E. Haskell, C. T. Case, “Transient signal progation in lossless, isotropic plasmas,” IEEE Trans. Antennas Propag. AP-15, 458–464 (1967).
    [CrossRef]
  41. L. Felsen, “Transients in dispersive media, part I: theory,” IEEE Trans. Antennas Propag. AP-17, 191–200 (1969).
    [CrossRef]
  42. E. Heyman, L. B. Felsen, “Complex-source pulsed-beam fields,” J. Opt. Soc. Am. A 6, 806–817 (1989).
    [CrossRef]
  43. G. Kristensson, “Direct and inverse scattering problems in dispersive media-Green’s functions and invariant imbedding techniques,” in Direct and Inverse Boundary Value Problems,R. Kleinman, R. Kress, E. Martensen, eds., Meth. Verf. Math. Phys.37, 105–119 (1991).
  44. G. Kristensson, R. J. Krueger, “Direct and inverse scattering in the time domain for a dissipative wave equation: Part I: scattering operators; Part II: simultaneous reconstruction of dissipation and phase velocity profiles,” J. Math. Phys. 27, 1667–1693 (1986).
    [CrossRef]
  45. R. W. Ziolkowski, D. K. Lewis, B. D. Cook, “Evidence of localized wave transmission,” Phys. Rev. Lett. 62, 147–150 (1989).
    [CrossRef] [PubMed]
  46. C. A. Emeis, L. J. Oosterhoff, G. De Vries, “Numerical evaluation of Kramers-Kronig relations,” Proc. R. Soc. London Ser. A 297, 54–65 (1967).
    [CrossRef]
  47. E. U. Condon, “Theories of optical rotatory power,” Rev. Mod. Phys. 9, 432–457 (1937).
    [CrossRef]
  48. N. Engheta, D. L. Jaggard, “Chiral microstructures: their frequency response,” presented at the 1990 IEEE APS/URSI Symposium, Dallas, Texas, May 7–10, 1990.
  49. Here it must be noted that notations ωp2 and ωm2 used for the two constant numerators in Eqs. (2) are chosen solely to show that the dimension of the numerators is inverse seconds to a power of 2. These two numerators can indeed be written in terms of parameters of polarized (or magnetized) microstructures in the medium. However, all those details are lumped into these two parameters ωp2 and ωm2. In particular, we note that ωm2 may also be negative for diamagnetic effects.
  50. In the present study of transient response of a single-resonance chiral material, we treat the general case where ωp, ωm, and αc are nonzero parameters. However, in several cases, in order to highlight the physics of the problem while reducing the mathematical complexity, we consider ωm = 0, i.e., the nonmagnetic single-resonance chiral material, which is also consistent with Condon’s model of optical activity.
  51. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).
  52. L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N.J., 1973).
  53. C. Bender, S. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978).
  54. As is well known in the method of steepest descent, the asymptotic expression of I± given in relation (8) is valid when ωs is not near any singularities and when the functions multiplying the exponential terms are slowly varying functions of ω near ωs. However, if these conditions are not satisfied, different asymptotic forms must be used, as is done in other parts of this paper.
  55. J. R. Wait, “Propagation of pulses in dispersive media,” Radio Sci. J. Res. 69D, 1387–1401 (1965).
  56. H. Baerwald, “Über die Fortpflanzung von Signalen in disperdierenden Medien,” Ann. Phys. 7, 731–760 (1930).
    [CrossRef]
  57. D. B. Trizna, T. A. Weber, “Brillouin revisited: signal velocity definition for pulse propagation in a medium with resonant anomalous dispersion,” Radio Sci. 17, 1169–1180 (1982).
    [CrossRef]
  58. K. E. Oughstun, P. Wyns, D. Foty, “Numerical determination of the signal velocity in dispersive pulse propagation,” J. Opt. Soc. Am. A 6, 1430–1440 (1989).
    [CrossRef]
  59. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).
  60. The approximations used in obtaining the expression of k± for |ωs| → ωz1 and |ωs| > ωz1can be explained as follows: The second term on the right-hand side of Eq. (24) is approximated by standard techniques,41 which result in the second term on the right-hand side of k±. It is the first term of k±that deserves some attention. The zeroth-order approximation to this term, ±αcω2/(ωc2− ω2), is obtained by replacing ω with ωz1. While this is a reasonable approximation, it would result in the saddle points υs± = Vωz1(2ct/z), which lack the time shift ∓2Uωz1 in ct/z∓ 2Uωz1. If we had chosen not to replace ω with ωz1, but had required only that terms of order O(υ3) and higher be neglected and that ωc2− ωz12≫ 2ωz1υ2, we would have had υs± = Vωz1/{2(ct/z ∓ 2U(ωz1+ ωz12/(ωc2 − ωz12)]}. While this is certainly a more rigorous expression, we choose the approximation used here because it shows the time shift without introducing a complicated expression for the saddle points.
  61. P. G. Zablocky, “Transient electromagnetic wave propagation, refraction, and radiation in chiral media,” Ph.D. dissertation (University of Pennsylvania, Philadelphia, Pa., 1992).

1992 (5)

P. Pelet, N. Engheta, “Chirostrip antenna: line source problem,” J. Electromag. Waves Appl. 6, 771–793 (1992).

R. D. Graglia, P. L. E. Uslenghi, C. L. Yu, “Electromagnetic oblique scattering by a cylinder coated with chiral layers and anisotropic jump-immittance sheets,” J. Electromag. Waves Appl. 6, 695–719 (1992).

A. Toscano, L. Vegni, “Spectral dyadic Green’s function formulation for planar integrated structures with a grounded chiral slab,” J. Electromag. Waves Appl. 6, 751–769 (1992).

R. G. Rojas, “Integral equations for the scattering by three dimensional inhomogeneous chiral bodies,” J. Electromag. Waves Appl. 6, 733–750 (1992).

N. Engheta, guest ed., special issue on wave interactions with chiral and complex media, J. Electromag. Waves Appl. 6, 537–798 (1992).

1991 (4)

X. Sun, D. L. Jaggard, “Accelerated particle radiation in chiral media,” J. Appl. Phys. 69, 34–38 (1991).
[CrossRef]

N. Engheta, P. G. Zablocky, “Effect of chirality on the transient signal wave front,” Opt. Lett. 16, 1924–1926 (1991).
[CrossRef] [PubMed]

K. E. Oughstun, “Pulse propagation in a linear, causally dispersive medium,” Proc. IEEE 79, 1379–1390 (1991).
[CrossRef]

M. S. Kluskens, E. H. Newman, “Scattering by a multilayer chiral cylinder,” IEEE Trans. Antennas Propag. 39, 91–96 (1991).
[CrossRef]

1990 (8)

D. L. Jaggard, N. Engheta, J. C. Liu, “Chiroshields: a Salisbury/Dallenbach shield alternative,” Electron. Lett. 26, 1332–1334 (1990).
[CrossRef]

P. Pelet, N. Engheta, “The theory of chirowaveguides,” IEEE Trans. Antennas Propag. 38, 90–98 (1990).
[CrossRef]

P. L. E. Uslenghi, “Scattering by an impedance sphere coated with a chiral layer,” Electromagnetics 10, 201–211 (1990).
[CrossRef]

M. W. Kowarz, N. Engheta, “Spherical chirolenses,” Opt. Lett. 15, 299–301 (1990).
[CrossRef] [PubMed]

T. Guire, V. V. Varadan, V. K. Varadan, “Effect of chirality on reflection of EM waves by planar dielectric slab,” IEEE Trans. Electromag. Compat. 32, 300–303 (1990).
[CrossRef]

N. Engheta, P. G. Zablocky, “A step towards determining transient response of chiral materials: Kramers–Kronig relations for chiral parameters,” Electron. Lett. 26, 2132–2134 (1990).
[CrossRef]

A. H. Sihvola, I. V. Lindell, “Chiral Maxwell–Garnett mixing formula,” Electron. Lett. 26, 118–119, 1990.
[CrossRef]

M. I. Oksanen, S. A. Tretyakov, I. V. Lindell, “Vector circuit theory for isotropic and chiral slabs,” J. Electromag. Waves Appl. 4, 613–643 (1990).

1989 (3)

1988 (2)

1986 (2)

S. Bassiri, N. Engheta, C. H. Papas, “Dyadic Green’s function and dipole radiation in chiral media,” Alta Freq. 2, 83–88 (1986).

G. Kristensson, R. J. Krueger, “Direct and inverse scattering in the time domain for a dissipative wave equation: Part I: scattering operators; Part II: simultaneous reconstruction of dissipation and phase velocity profiles,” J. Math. Phys. 27, 1667–1693 (1986).
[CrossRef]

1982 (1)

D. B. Trizna, T. A. Weber, “Brillouin revisited: signal velocity definition for pulse propagation in a medium with resonant anomalous dispersion,” Radio Sci. 17, 1169–1180 (1982).
[CrossRef]

1979 (1)

D. L. Jaggard, A. R. Mickelson, C. H. Papas, “On electromagnetic waves in chiral media,” Appl. Phys. 28, 211–216 (1979).
[CrossRef]

1978 (1)

C. F. Bohren, “Scattering of electromagnetic waves by an optically active cylinder,” J. Colloid Interface Sci. 66, 105–109 (1978).
[CrossRef]

1969 (1)

L. Felsen, “Transients in dispersive media, part I: theory,” IEEE Trans. Antennas Propag. AP-17, 191–200 (1969).
[CrossRef]

1967 (2)

C. A. Emeis, L. J. Oosterhoff, G. De Vries, “Numerical evaluation of Kramers-Kronig relations,” Proc. R. Soc. London Ser. A 297, 54–65 (1967).
[CrossRef]

R. E. Haskell, C. T. Case, “Transient signal progation in lossless, isotropic plasmas,” IEEE Trans. Antennas Propag. AP-15, 458–464 (1967).
[CrossRef]

1965 (2)

J. R. Wait, “Propagation of electromagnetic pulses in terrestrial waveguides,” IEEE Trans. Antennas Propag. AP-13, 904–918 (1965).
[CrossRef]

J. R. Wait, “Propagation of pulses in dispersive media,” Radio Sci. J. Res. 69D, 1387–1401 (1965).

1956 (1)

R. S. Elliott, “Pulse waveform degradation due to dispersion in waveguide,” IEEE Trans. Microwave Theory Tech. MTT-5, 254–257 (1956).

1937 (1)

E. U. Condon, “Theories of optical rotatory power,” Rev. Mod. Phys. 9, 432–457 (1937).
[CrossRef]

1930 (1)

H. Baerwald, “Über die Fortpflanzung von Signalen in disperdierenden Medien,” Ann. Phys. 7, 731–760 (1930).
[CrossRef]

1920 (1)

K. F. Lindman, “Über eine durch ein isotropes System yon Spiralformigen resonatoren erseugte Rotationspolarisation der Elektromagnetischen wellen,” Ann. Phys. 63, 621–644 (1920).
[CrossRef]

1914 (1)

A. Sommerfeld, “Uber die Fortpflanzung des Lichtes in disperdierenden Medien,” Ann. Phys. 44, 177–202 (1914).
[CrossRef]

1848 (1)

L. Pasteur, “Sur les relations qui peuvent exister entre la forme cristalline, la composition chimique et le sens de la polarisation rotatoire,” Ann. Chim. Phys. 24, 442–459 (1848).

1822 (1)

A. Fresnel, “Mémoire sur la double refraction que les rayons lumineux éprouvent en traversant les aiguilles de cristal de roche suivant des directions paralleles à l’axe,” Oeuvres 1, 731–751 (1822).

1812 (1)

J. B. Biot, “Mémoire sur un nouveau genre d’oscillations que les molécules de la lumière éprouvent, en traversant certains cristaux,” Mém. Inst. 1, 1–372 (1812).

1811 (1)

D. F. Arago, “Sur une modification remarquable qu’éprouvent les rayons lumineux dans leur passage à travers certains corps diaphanes, et sur quelques autres nouveaux phénomènes d’optique,” Mém. Inst. 1, 93–134 (1811).

Arago, D. F.

D. F. Arago, “Sur une modification remarquable qu’éprouvent les rayons lumineux dans leur passage à travers certains corps diaphanes, et sur quelques autres nouveaux phénomènes d’optique,” Mém. Inst. 1, 93–134 (1811).

Baerwald, H.

H. Baerwald, “Über die Fortpflanzung von Signalen in disperdierenden Medien,” Ann. Phys. 7, 731–760 (1930).
[CrossRef]

Bassiri, S.

S. Bassiri, C. H. Papas, N. Engheta, “Electromagnetic wave propagation through a dielectric-chiral interface and through a chiral slab,” J. Opt. Soc. Am. A 5, 1450–1459 (1988).
[CrossRef]

S. Bassiri, N. Engheta, C. H. Papas, “Dyadic Green’s function and dipole radiation in chiral media,” Alta Freq. 2, 83–88 (1986).

Bender, C.

C. Bender, S. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978).

Biot, J. B.

J. B. Biot, “Mémoire sur un nouveau genre d’oscillations que les molécules de la lumière éprouvent, en traversant certains cristaux,” Mém. Inst. 1, 1–372 (1812).

Bohren, C. F.

C. F. Bohren, “Scattering of electromagnetic waves by an optically active cylinder,” J. Colloid Interface Sci. 66, 105–109 (1978).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).

Brillouin, L.

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

Case, C. T.

R. E. Haskell, C. T. Case, “Transient signal progation in lossless, isotropic plasmas,” IEEE Trans. Antennas Propag. AP-15, 458–464 (1967).
[CrossRef]

Condon, E. U.

E. U. Condon, “Theories of optical rotatory power,” Rev. Mod. Phys. 9, 432–457 (1937).
[CrossRef]

Cook, B. D.

R. W. Ziolkowski, D. K. Lewis, B. D. Cook, “Evidence of localized wave transmission,” Phys. Rev. Lett. 62, 147–150 (1989).
[CrossRef] [PubMed]

De Vries, G.

C. A. Emeis, L. J. Oosterhoff, G. De Vries, “Numerical evaluation of Kramers-Kronig relations,” Proc. R. Soc. London Ser. A 297, 54–65 (1967).
[CrossRef]

Elliott, R. S.

R. S. Elliott, “Pulse waveform degradation due to dispersion in waveguide,” IEEE Trans. Microwave Theory Tech. MTT-5, 254–257 (1956).

Emeis, C. A.

C. A. Emeis, L. J. Oosterhoff, G. De Vries, “Numerical evaluation of Kramers-Kronig relations,” Proc. R. Soc. London Ser. A 297, 54–65 (1967).
[CrossRef]

Engheta, N.

P. Pelet, N. Engheta, “Chirostrip antenna: line source problem,” J. Electromag. Waves Appl. 6, 771–793 (1992).

N. Engheta, P. G. Zablocky, “Effect of chirality on the transient signal wave front,” Opt. Lett. 16, 1924–1926 (1991).
[CrossRef] [PubMed]

N. Engheta, P. G. Zablocky, “A step towards determining transient response of chiral materials: Kramers–Kronig relations for chiral parameters,” Electron. Lett. 26, 2132–2134 (1990).
[CrossRef]

M. W. Kowarz, N. Engheta, “Spherical chirolenses,” Opt. Lett. 15, 299–301 (1990).
[CrossRef] [PubMed]

P. Pelet, N. Engheta, “The theory of chirowaveguides,” IEEE Trans. Antennas Propag. 38, 90–98 (1990).
[CrossRef]

D. L. Jaggard, N. Engheta, J. C. Liu, “Chiroshields: a Salisbury/Dallenbach shield alternative,” Electron. Lett. 26, 1332–1334 (1990).
[CrossRef]

S. Bassiri, C. H. Papas, N. Engheta, “Electromagnetic wave propagation through a dielectric-chiral interface and through a chiral slab,” J. Opt. Soc. Am. A 5, 1450–1459 (1988).
[CrossRef]

S. Bassiri, N. Engheta, C. H. Papas, “Dyadic Green’s function and dipole radiation in chiral media,” Alta Freq. 2, 83–88 (1986).

D. L. Jaggard, N. Engheta, “Chirality in electrodynamics: modeling and applications,” in Directions in Electromagnetic Wave Modelling,H. L. Bertoni, L. B. Felsen, eds. (Plenum, New York, 1991), pp. 485–493.

N. Engheta, D. L. Jaggard, “Chiral microstructures: their frequency response,” presented at the 1990 IEEE APS/URSI Symposium, Dallas, Texas, May 7–10, 1990.

P. G. Zablocky, N. Engheta, “Transient response of chiral materials,” presented at the 1991 North American Radio Science/International IEEE AP-S/U RSI Symposium, London, Ontario, Canada, June 24–28, 1991.

Felsen, L.

L. Felsen, “Transients in dispersive media, part I: theory,” IEEE Trans. Antennas Propag. AP-17, 191–200 (1969).
[CrossRef]

Felsen, L. B.

E. Heyman, L. B. Felsen, “Complex-source pulsed-beam fields,” J. Opt. Soc. Am. A 6, 806–817 (1989).
[CrossRef]

L. B. Felsen, Transients Electromagnetic Fields (Springer-Verlag, New York, 1976).
[CrossRef]

L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N.J., 1973).

Foty, D.

Fresnel, A.

A. Fresnel, “Mémoire sur la double refraction que les rayons lumineux éprouvent en traversant les aiguilles de cristal de roche suivant des directions paralleles à l’axe,” Oeuvres 1, 731–751 (1822).

Graglia, R. D.

R. D. Graglia, P. L. E. Uslenghi, C. L. Yu, “Electromagnetic oblique scattering by a cylinder coated with chiral layers and anisotropic jump-immittance sheets,” J. Electromag. Waves Appl. 6, 695–719 (1992).

Guire, T.

T. Guire, V. V. Varadan, V. K. Varadan, “Effect of chirality on reflection of EM waves by planar dielectric slab,” IEEE Trans. Electromag. Compat. 32, 300–303 (1990).
[CrossRef]

Haskell, R. E.

R. E. Haskell, C. T. Case, “Transient signal progation in lossless, isotropic plasmas,” IEEE Trans. Antennas Propag. AP-15, 458–464 (1967).
[CrossRef]

Heyman, E.

Jackson, J. D.

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

Jaggard, D. L.

X. Sun, D. L. Jaggard, “Accelerated particle radiation in chiral media,” J. Appl. Phys. 69, 34–38 (1991).
[CrossRef]

D. L. Jaggard, N. Engheta, J. C. Liu, “Chiroshields: a Salisbury/Dallenbach shield alternative,” Electron. Lett. 26, 1332–1334 (1990).
[CrossRef]

D. L. Jaggard, A. R. Mickelson, C. H. Papas, “On electromagnetic waves in chiral media,” Appl. Phys. 28, 211–216 (1979).
[CrossRef]

D. L. Jaggard, N. Engheta, “Chirality in electrodynamics: modeling and applications,” in Directions in Electromagnetic Wave Modelling,H. L. Bertoni, L. B. Felsen, eds. (Plenum, New York, 1991), pp. 485–493.

N. Engheta, D. L. Jaggard, “Chiral microstructures: their frequency response,” presented at the 1990 IEEE APS/URSI Symposium, Dallas, Texas, May 7–10, 1990.

Kluskens, M. S.

M. S. Kluskens, E. H. Newman, “Scattering by a multilayer chiral cylinder,” IEEE Trans. Antennas Propag. 39, 91–96 (1991).
[CrossRef]

Kowarz, M. W.

Kristensson, G.

G. Kristensson, R. J. Krueger, “Direct and inverse scattering in the time domain for a dissipative wave equation: Part I: scattering operators; Part II: simultaneous reconstruction of dissipation and phase velocity profiles,” J. Math. Phys. 27, 1667–1693 (1986).
[CrossRef]

G. Kristensson, “Direct and inverse scattering problems in dispersive media-Green’s functions and invariant imbedding techniques,” in Direct and Inverse Boundary Value Problems,R. Kleinman, R. Kress, E. Martensen, eds., Meth. Verf. Math. Phys.37, 105–119 (1991).

G. Kristensson, S. Rikte, “Transient scattering in bi-isotropic media,” presented at the 1991 Symposium on Progress in Electromagnetics Research, Cambridge, Mass., July 1–5, 1991.

G. Kristensson, S. Rikte, “Transient wave propagation in reciprocal bi-isotropic media at oblique incidence,” Tech. Rep. LUTEDX/(TEAT-7019)/1–25 (Lund Institute of Technology, Department of Electromagnetic Theory, Lund, Sweden, 1992).

Krueger, R. J.

G. Kristensson, R. J. Krueger, “Direct and inverse scattering in the time domain for a dissipative wave equation: Part I: scattering operators; Part II: simultaneous reconstruction of dissipation and phase velocity profiles,” J. Math. Phys. 27, 1667–1693 (1986).
[CrossRef]

Lakhtakia, A.

A. Lakhtakia, V. V. Varadan, V. K. Varadan, Time-Harmonic Electromagnetic Fields in Chiral Media (Springer-Verlag, Berlin, 1989).

Lewis, D. K.

R. W. Ziolkowski, D. K. Lewis, B. D. Cook, “Evidence of localized wave transmission,” Phys. Rev. Lett. 62, 147–150 (1989).
[CrossRef] [PubMed]

Lindell, I. V.

A. H. Sihvola, I. V. Lindell, “Chiral Maxwell–Garnett mixing formula,” Electron. Lett. 26, 118–119, 1990.
[CrossRef]

M. I. Oksanen, S. A. Tretyakov, I. V. Lindell, “Vector circuit theory for isotropic and chiral slabs,” J. Electromag. Waves Appl. 4, 613–643 (1990).

Lindman, K. F.

K. F. Lindman, “Über eine durch ein isotropes System yon Spiralformigen resonatoren erseugte Rotationspolarisation der Elektromagnetischen wellen,” Ann. Phys. 63, 621–644 (1920).
[CrossRef]

Liu, J. C.

D. L. Jaggard, N. Engheta, J. C. Liu, “Chiroshields: a Salisbury/Dallenbach shield alternative,” Electron. Lett. 26, 1332–1334 (1990).
[CrossRef]

Marcuvitz, N.

L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N.J., 1973).

Mickelson, A. R.

D. L. Jaggard, A. R. Mickelson, C. H. Papas, “On electromagnetic waves in chiral media,” Appl. Phys. 28, 211–216 (1979).
[CrossRef]

Newman, E. H.

M. S. Kluskens, E. H. Newman, “Scattering by a multilayer chiral cylinder,” IEEE Trans. Antennas Propag. 39, 91–96 (1991).
[CrossRef]

Oksanen, M. I.

M. I. Oksanen, S. A. Tretyakov, I. V. Lindell, “Vector circuit theory for isotropic and chiral slabs,” J. Electromag. Waves Appl. 4, 613–643 (1990).

Oosterhoff, L. J.

C. A. Emeis, L. J. Oosterhoff, G. De Vries, “Numerical evaluation of Kramers-Kronig relations,” Proc. R. Soc. London Ser. A 297, 54–65 (1967).
[CrossRef]

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C. Bender, S. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978).

Oughstun, K. E.

Papas, C. H.

S. Bassiri, C. H. Papas, N. Engheta, “Electromagnetic wave propagation through a dielectric-chiral interface and through a chiral slab,” J. Opt. Soc. Am. A 5, 1450–1459 (1988).
[CrossRef]

S. Bassiri, N. Engheta, C. H. Papas, “Dyadic Green’s function and dipole radiation in chiral media,” Alta Freq. 2, 83–88 (1986).

D. L. Jaggard, A. R. Mickelson, C. H. Papas, “On electromagnetic waves in chiral media,” Appl. Phys. 28, 211–216 (1979).
[CrossRef]

Pasteur, L.

L. Pasteur, “Sur les relations qui peuvent exister entre la forme cristalline, la composition chimique et le sens de la polarisation rotatoire,” Ann. Chim. Phys. 24, 442–459 (1848).

Pelet, P.

P. Pelet, N. Engheta, “Chirostrip antenna: line source problem,” J. Electromag. Waves Appl. 6, 771–793 (1992).

P. Pelet, N. Engheta, “The theory of chirowaveguides,” IEEE Trans. Antennas Propag. 38, 90–98 (1990).
[CrossRef]

Rikte, S.

G. Kristensson, S. Rikte, “Transient scattering in bi-isotropic media,” presented at the 1991 Symposium on Progress in Electromagnetics Research, Cambridge, Mass., July 1–5, 1991.

G. Kristensson, S. Rikte, “Transient wave propagation in reciprocal bi-isotropic media at oblique incidence,” Tech. Rep. LUTEDX/(TEAT-7019)/1–25 (Lund Institute of Technology, Department of Electromagnetic Theory, Lund, Sweden, 1992).

S. Rikte, “Propagation of transient electromagnetic waves in dispersive reciprocal bi-isotropic media,” Tech. Rep. LUTEDX/(TEAT-1005)/1–12 (Lund Institute of Technology, Department of Electromagnetic Theory, Lund, Sweden, 1992).

Rojas, R. G.

R. G. Rojas, “Integral equations for the scattering by three dimensional inhomogeneous chiral bodies,” J. Electromag. Waves Appl. 6, 733–750 (1992).

Sihvola, A. H.

A. H. Sihvola, I. V. Lindell, “Chiral Maxwell–Garnett mixing formula,” Electron. Lett. 26, 118–119, 1990.
[CrossRef]

Sommerfeld, A.

A. Sommerfeld, “Uber die Fortpflanzung des Lichtes in disperdierenden Medien,” Ann. Phys. 44, 177–202 (1914).
[CrossRef]

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X. Sun, D. L. Jaggard, “Accelerated particle radiation in chiral media,” J. Appl. Phys. 69, 34–38 (1991).
[CrossRef]

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A. Toscano, L. Vegni, “Spectral dyadic Green’s function formulation for planar integrated structures with a grounded chiral slab,” J. Electromag. Waves Appl. 6, 751–769 (1992).

Tretyakov, S. A.

M. I. Oksanen, S. A. Tretyakov, I. V. Lindell, “Vector circuit theory for isotropic and chiral slabs,” J. Electromag. Waves Appl. 4, 613–643 (1990).

Trizna, D. B.

D. B. Trizna, T. A. Weber, “Brillouin revisited: signal velocity definition for pulse propagation in a medium with resonant anomalous dispersion,” Radio Sci. 17, 1169–1180 (1982).
[CrossRef]

Uslenghi, P. L. E.

R. D. Graglia, P. L. E. Uslenghi, C. L. Yu, “Electromagnetic oblique scattering by a cylinder coated with chiral layers and anisotropic jump-immittance sheets,” J. Electromag. Waves Appl. 6, 695–719 (1992).

P. L. E. Uslenghi, “Scattering by an impedance sphere coated with a chiral layer,” Electromagnetics 10, 201–211 (1990).
[CrossRef]

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T. Guire, V. V. Varadan, V. K. Varadan, “Effect of chirality on reflection of EM waves by planar dielectric slab,” IEEE Trans. Electromag. Compat. 32, 300–303 (1990).
[CrossRef]

A. Lakhtakia, V. V. Varadan, V. K. Varadan, Time-Harmonic Electromagnetic Fields in Chiral Media (Springer-Verlag, Berlin, 1989).

Varadan, V. V.

T. Guire, V. V. Varadan, V. K. Varadan, “Effect of chirality on reflection of EM waves by planar dielectric slab,” IEEE Trans. Electromag. Compat. 32, 300–303 (1990).
[CrossRef]

A. Lakhtakia, V. V. Varadan, V. K. Varadan, Time-Harmonic Electromagnetic Fields in Chiral Media (Springer-Verlag, Berlin, 1989).

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A. Toscano, L. Vegni, “Spectral dyadic Green’s function formulation for planar integrated structures with a grounded chiral slab,” J. Electromag. Waves Appl. 6, 751–769 (1992).

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J. R. Wait, “Propagation of pulses in dispersive media,” Radio Sci. J. Res. 69D, 1387–1401 (1965).

J. R. Wait, “Propagation of electromagnetic pulses in terrestrial waveguides,” IEEE Trans. Antennas Propag. AP-13, 904–918 (1965).
[CrossRef]

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D. B. Trizna, T. A. Weber, “Brillouin revisited: signal velocity definition for pulse propagation in a medium with resonant anomalous dispersion,” Radio Sci. 17, 1169–1180 (1982).
[CrossRef]

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M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).

Wyns, P.

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R. D. Graglia, P. L. E. Uslenghi, C. L. Yu, “Electromagnetic oblique scattering by a cylinder coated with chiral layers and anisotropic jump-immittance sheets,” J. Electromag. Waves Appl. 6, 695–719 (1992).

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N. Engheta, P. G. Zablocky, “Effect of chirality on the transient signal wave front,” Opt. Lett. 16, 1924–1926 (1991).
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N. Engheta, P. G. Zablocky, “A step towards determining transient response of chiral materials: Kramers–Kronig relations for chiral parameters,” Electron. Lett. 26, 2132–2134 (1990).
[CrossRef]

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R. W. Ziolkowski, D. K. Lewis, B. D. Cook, “Evidence of localized wave transmission,” Phys. Rev. Lett. 62, 147–150 (1989).
[CrossRef] [PubMed]

Alta Freq. (1)

S. Bassiri, N. Engheta, C. H. Papas, “Dyadic Green’s function and dipole radiation in chiral media,” Alta Freq. 2, 83–88 (1986).

Ann. Chim. Phys. (1)

L. Pasteur, “Sur les relations qui peuvent exister entre la forme cristalline, la composition chimique et le sens de la polarisation rotatoire,” Ann. Chim. Phys. 24, 442–459 (1848).

Ann. Phys. (3)

K. F. Lindman, “Über eine durch ein isotropes System yon Spiralformigen resonatoren erseugte Rotationspolarisation der Elektromagnetischen wellen,” Ann. Phys. 63, 621–644 (1920).
[CrossRef]

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[CrossRef]

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[CrossRef]

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D. L. Jaggard, A. R. Mickelson, C. H. Papas, “On electromagnetic waves in chiral media,” Appl. Phys. 28, 211–216 (1979).
[CrossRef]

Electromagnetics (1)

P. L. E. Uslenghi, “Scattering by an impedance sphere coated with a chiral layer,” Electromagnetics 10, 201–211 (1990).
[CrossRef]

Electron. Lett. (3)

D. L. Jaggard, N. Engheta, J. C. Liu, “Chiroshields: a Salisbury/Dallenbach shield alternative,” Electron. Lett. 26, 1332–1334 (1990).
[CrossRef]

A. H. Sihvola, I. V. Lindell, “Chiral Maxwell–Garnett mixing formula,” Electron. Lett. 26, 118–119, 1990.
[CrossRef]

N. Engheta, P. G. Zablocky, “A step towards determining transient response of chiral materials: Kramers–Kronig relations for chiral parameters,” Electron. Lett. 26, 2132–2134 (1990).
[CrossRef]

IEEE Trans. Antennas Propag. (5)

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[CrossRef]

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[CrossRef]

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[CrossRef]

IEEE Trans. Electromag. Compat. (1)

T. Guire, V. V. Varadan, V. K. Varadan, “Effect of chirality on reflection of EM waves by planar dielectric slab,” IEEE Trans. Electromag. Compat. 32, 300–303 (1990).
[CrossRef]

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P. Pelet, N. Engheta, “Chirostrip antenna: line source problem,” J. Electromag. Waves Appl. 6, 771–793 (1992).

M. I. Oksanen, S. A. Tretyakov, I. V. Lindell, “Vector circuit theory for isotropic and chiral slabs,” J. Electromag. Waves Appl. 4, 613–643 (1990).

A. Toscano, L. Vegni, “Spectral dyadic Green’s function formulation for planar integrated structures with a grounded chiral slab,” J. Electromag. Waves Appl. 6, 751–769 (1992).

N. Engheta, guest ed., special issue on wave interactions with chiral and complex media, J. Electromag. Waves Appl. 6, 537–798 (1992).

R. G. Rojas, “Integral equations for the scattering by three dimensional inhomogeneous chiral bodies,” J. Electromag. Waves Appl. 6, 733–750 (1992).

J. Math. Phys. (1)

G. Kristensson, R. J. Krueger, “Direct and inverse scattering in the time domain for a dissipative wave equation: Part I: scattering operators; Part II: simultaneous reconstruction of dissipation and phase velocity profiles,” J. Math. Phys. 27, 1667–1693 (1986).
[CrossRef]

J. Opt. Soc. Am. A (3)

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[CrossRef]

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D. F. Arago, “Sur une modification remarquable qu’éprouvent les rayons lumineux dans leur passage à travers certains corps diaphanes, et sur quelques autres nouveaux phénomènes d’optique,” Mém. Inst. 1, 93–134 (1811).

J. B. Biot, “Mémoire sur un nouveau genre d’oscillations que les molécules de la lumière éprouvent, en traversant certains cristaux,” Mém. Inst. 1, 1–372 (1812).

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Opt. Lett. (2)

Phys. Rev. Lett. (1)

R. W. Ziolkowski, D. K. Lewis, B. D. Cook, “Evidence of localized wave transmission,” Phys. Rev. Lett. 62, 147–150 (1989).
[CrossRef] [PubMed]

Proc. IEEE (1)

K. E. Oughstun, “Pulse propagation in a linear, causally dispersive medium,” Proc. IEEE 79, 1379–1390 (1991).
[CrossRef]

Proc. R. Soc. London Ser. A (1)

C. A. Emeis, L. J. Oosterhoff, G. De Vries, “Numerical evaluation of Kramers-Kronig relations,” Proc. R. Soc. London Ser. A 297, 54–65 (1967).
[CrossRef]

Radio Sci. (1)

D. B. Trizna, T. A. Weber, “Brillouin revisited: signal velocity definition for pulse propagation in a medium with resonant anomalous dispersion,” Radio Sci. 17, 1169–1180 (1982).
[CrossRef]

Radio Sci. J. Res. (1)

J. R. Wait, “Propagation of pulses in dispersive media,” Radio Sci. J. Res. 69D, 1387–1401 (1965).

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E. U. Condon, “Theories of optical rotatory power,” Rev. Mod. Phys. 9, 432–457 (1937).
[CrossRef]

Other (20)

N. Engheta, D. L. Jaggard, “Chiral microstructures: their frequency response,” presented at the 1990 IEEE APS/URSI Symposium, Dallas, Texas, May 7–10, 1990.

Here it must be noted that notations ωp2 and ωm2 used for the two constant numerators in Eqs. (2) are chosen solely to show that the dimension of the numerators is inverse seconds to a power of 2. These two numerators can indeed be written in terms of parameters of polarized (or magnetized) microstructures in the medium. However, all those details are lumped into these two parameters ωp2 and ωm2. In particular, we note that ωm2 may also be negative for diamagnetic effects.

In the present study of transient response of a single-resonance chiral material, we treat the general case where ωp, ωm, and αc are nonzero parameters. However, in several cases, in order to highlight the physics of the problem while reducing the mathematical complexity, we consider ωm = 0, i.e., the nonmagnetic single-resonance chiral material, which is also consistent with Condon’s model of optical activity.

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

L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N.J., 1973).

C. Bender, S. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978).

As is well known in the method of steepest descent, the asymptotic expression of I± given in relation (8) is valid when ωs is not near any singularities and when the functions multiplying the exponential terms are slowly varying functions of ω near ωs. However, if these conditions are not satisfied, different asymptotic forms must be used, as is done in other parts of this paper.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).

The approximations used in obtaining the expression of k± for |ωs| → ωz1 and |ωs| > ωz1can be explained as follows: The second term on the right-hand side of Eq. (24) is approximated by standard techniques,41 which result in the second term on the right-hand side of k±. It is the first term of k±that deserves some attention. The zeroth-order approximation to this term, ±αcω2/(ωc2− ω2), is obtained by replacing ω with ωz1. While this is a reasonable approximation, it would result in the saddle points υs± = Vωz1(2ct/z), which lack the time shift ∓2Uωz1 in ct/z∓ 2Uωz1. If we had chosen not to replace ω with ωz1, but had required only that terms of order O(υ3) and higher be neglected and that ωc2− ωz12≫ 2ωz1υ2, we would have had υs± = Vωz1/{2(ct/z ∓ 2U(ωz1+ ωz12/(ωc2 − ωz12)]}. While this is certainly a more rigorous expression, we choose the approximation used here because it shows the time shift without introducing a complicated expression for the saddle points.

P. G. Zablocky, “Transient electromagnetic wave propagation, refraction, and radiation in chiral media,” Ph.D. dissertation (University of Pennsylvania, Philadelphia, Pa., 1992).

G. Kristensson, S. Rikte, “Transient scattering in bi-isotropic media,” presented at the 1991 Symposium on Progress in Electromagnetics Research, Cambridge, Mass., July 1–5, 1991.

G. Kristensson, S. Rikte, “Transient wave propagation in reciprocal bi-isotropic media at oblique incidence,” Tech. Rep. LUTEDX/(TEAT-7019)/1–25 (Lund Institute of Technology, Department of Electromagnetic Theory, Lund, Sweden, 1992).

S. Rikte, “Propagation of transient electromagnetic waves in dispersive reciprocal bi-isotropic media,” Tech. Rep. LUTEDX/(TEAT-1005)/1–12 (Lund Institute of Technology, Department of Electromagnetic Theory, Lund, Sweden, 1992).

G. Kristensson, “Direct and inverse scattering problems in dispersive media-Green’s functions and invariant imbedding techniques,” in Direct and Inverse Boundary Value Problems,R. Kleinman, R. Kress, E. Martensen, eds., Meth. Verf. Math. Phys.37, 105–119 (1991).

P. G. Zablocky, N. Engheta, “Transient response of chiral materials,” presented at the 1991 North American Radio Science/International IEEE AP-S/U RSI Symposium, London, Ontario, Canada, June 24–28, 1991.

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

L. B. Felsen, Transients Electromagnetic Fields (Springer-Verlag, New York, 1976).
[CrossRef]

D. L. Jaggard, N. Engheta, “Chirality in electrodynamics: modeling and applications,” in Directions in Electromagnetic Wave Modelling,H. L. Bertoni, L. B. Felsen, eds. (Plenum, New York, 1991), pp. 485–493.

A. Lakhtakia, ed., Selected Papers on Natural Optical Activity (SPIE, Bellingham, Wash., 1990).

A. Lakhtakia, V. V. Varadan, V. K. Varadan, Time-Harmonic Electromagnetic Fields in Chiral Media (Springer-Verlag, Berlin, 1989).

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Figures (10)

Fig. 1
Fig. 1

Plot of real and imaginary parts of (normalized) wave numbers k±(ω) and the asymptote k0 = ω/c versus positive real frequency. The parameters chosen here are ωp = ωc, ωm = 0, and αc = 0.1 ωc. These plots are obtained for real frequencies on the contour on the real axis (just above all singularities). For the negative real-frequency regions, we note that k±(−ω) = −k*(ω). Note that at ωc, k+ is infinite, whereas k is continuous and finite with a value ωp2/(2αcc). The asymptote k0 will at higher real frequencies end up between k+ and k, with k above it (for parameters chosen here). As the frequency increases, both wave numbers approach the asymptote. The small region of frequency between ωc and ωz2 (ωz2 = 1.0051ωc) is enlarged to show the details that are otherwise not visible. The wave numbers are real for real frequencies between ωc and ωz2. The real part of k+ tends to +∞ at ωc − 0 and to −∞ at ωc − 0. Im k± starts from zero at ωz2 and rises rapidly and then declines to zero at ωz1.

Fig. 2
Fig. 2

Sketch of the locations of singularities (zeros and poles), branch cuts, and the path of integration C a distance δ above the singularities and branch cuts. Note that the sketch is not to scale. For the parameters chosen in the example, ωz2 is much closer to ωc, and −ωz2 is much closer to −ωc, than shown. Also, the branch cuts can lie on the real ω axis.

Fig. 3
Fig. 3

Plots of the x and y components of electric field of the leading edge of the first precursor as a function of ct/z are shown in (a) and (b); (c) is a plot of Ex versus Ey (with ct/z as a parameter). Here ω0 = 1.7ωc, ωp = ωc, ωm = 0, αc = 0.1ωc, and z/c = 100/ωc. In these plots, the quantities with dimension of radian frequency are normalized to ωc (i.e., ωc = 1 rad/s is assumed). The plot of Ex versus Ey traces the motion of the tip of the electric field vector (as seen by the observer facing the approaching transient wave) as a function of time. The starting and stopping values of ct/z [given in (a) and (b)] are labeled in (c).

Fig. 4
Fig. 4

Plots of real and imaginary parts of (normalized) dk±/dω versus positive real frequency for ωp = ωc, ωm = 0, and αc = 0.1 ωc. The intersections of these curves with the horizontal line ct/z give the real positive saddle points. We note that dk±(ω)/dω = dk*(−ω)/dω. Note that, because of the rapid variation of these functions around ωc and ωz2, some of the features cannot easily be shown to scale in the figure. For instance, note that Re(dk±/dω) → ∞ as ωωz2 − 0 and as ωωz1 + 0 for positive frequencies, dk+/dω → ∞ as ωωc but dk/dω does not go to ∞ at ω = ωc (as seen from the right-hand inset). Im(dk±/dω) → ∞ as ω = ωz2 + 0 and decreases rapidly (to the negative value) for ω > ωz2 to meet the points shown on the graph, and then goes to −∞ as ωωz1 − 0.

Fig. 5
Fig. 5

(a), (b) Plots of the x and y components of the electric fields of the first precursors versus ct/z: the solid curves represent contributions from the saddle points ωs1+ (and ωs2−), and the dashed curves represent those from ωs1− (and ωs2+); the frequency splitting can easily be noticed here. (c), (d) The sums of the solid and dashed curves in (a) and (b), respectively. Plots (e), (g), and (f) illustrate Ex versus Ey components (with ct/z as a parameter) for the solid, the dashed, and the sum curves, respectively. The plots show the motion of the tip of the electric field as ct/z increases (as seen by an observer facing the approaching wave). The plots are made for the same parameter values used in Fig. 3.

Fig. 6
Fig. 6

Sketches of the loci of X± ≡ Re[±(ω)] = 0 in the complex ω plane for various values of ct/z. Shaded and unshaded areas represent regions where X± < 0 and X± > 0, respectively. The left- and right-hand columns illustrate plots for X+ and X, respectively. The path of integration C is deformed to go through the four saddle points in the regions where X± ≤ 0. Note that, because of chirality, X± are not symmetric functions with respect to the imaginary axis of ω, and we have X±(−ω) = −X±(ω). The detailed behavior of these sketches around the small region ωc < ω < ωz2 is not shown here since the region is extremely small.

Fig. 7
Fig. 7

(a), (b) Plots of the x and y components of the electric fields of the second precursors versus ct/z: the solid curves represent contributions from the saddle points ωs3+ (and ωs4−), and the dashed curves represent those from ωs3− (and ωs4+); the frequency splitting can easily be noticed here. The arrival time of the second precursor is ct/z = 1.4076. (a) and (b) do not reflect the transition through the arrival time [i.e., relations (13) are not plotted here]. (c), (d) The sums of the solid and dashed curves in (a) and (b), respectively, along with the fields at transition times around ct/z = 1.4076 [i.e., with relations (13) included]. Plots (e), (g), and (f) illustrate Ex versus Ey components (with ct/z as a parameter) for the solid, the dashed, and the sum curves, respectively. The plots show the motion of the tip of the electric field as ct/z increases (as seen by an observer facing the approaching wave). The plots are made for the same parameter values used in Fig. 3. The discontinuities shown in (e) and (g) are due to failure of the regular steepest-descent technique at ϕ±″ = 0.

Fig. 8
Fig. 8

Various aspects of the main signal buildup in chiral materials are shown for the same parameter values used in Fig. 3. Plots of Ex and Ey versus ct/z are given in (a) and (b). Here the solid and dashed curves represent contributions from positive ωs+ and ωs, respectively. The dashed curve reaches half of its steady state value at ct/z = 1.744, whereas the solid curve reaches its own half at ct/z = 1.936. Therefore the main signal effectively arrives in two steps. The sums of solid and dashed curves for Ex and Ey are given in (c) and (d), respectively. Ex versus Ey (with ct/z as a parameter) are given in (e), (g), and (f) for the solid, dashed and sum curves, respectively. The curves show the motion of the tip of the electric field as ct/z increases (as seen by an observer facing the approaching wave). If we wait long enough (e) and (g) will finally show the electric fields of RCP and LCP eigenmodes, and (f) will illustrate the field of a linearly polarized plane wave whose plane of polarization is rotated from the effect of chirality.

Fig. 9
Fig. 9

(a), (b) Residual electric-field components from saddle points approaching resonance frequency ωc − 0. In (c) a plot of Ex versus Ey is given, with ct/z as a parameter. The values of the parameters are the same as those in Fig. 3. The instantaneous frequency approaches ωc with time, and the electric field is RCP, as expected.

Fig. 10
Fig. 10

(a), (b) Residual electric-field components from saddle points approaching cutoff ωz1. The values of the parameters are the same as those in Fig. 3. The solid and dashed curves shown in (a) and (b) represent contributions from ωs+ and ωs, respectively. (c), (d) The sums of the solid and dashed curves in (a) and (b), respectively. Ex versus Ey (with ct/z as a parameter) is shown in (e), (g), and (f) for solid, dashed, and sum curves, respectively.

Equations (52)

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ξ c r ( ω ) = α c ω / ( ω c 2 ω 2 ) ,
ɛ c r ( ω ) = 1 + i = 0 N [ ω p i 2 / ( ω i 2 ω 2 ) ] , μ c r ( ω ) = 1 + j = 0 N [ ω m j 2 / ( ω j 2 ω 2 ) ] ,
ɛ c r ( ω ) = 1 + [ ω p 2 / ( ω c 2 ω 2 ) ] , μ c r ( ω ) = 1 + [ ω m 2 / ( ω c 2 ω 2 ) ] ,
I ± = + i δ + + i δ [ ω 0 / ( ω 0 2 ω 2 ) ] exp { i [ k ± ( ω ) z ω t ] } d ω ,
( 0 , ω ) ω 1 q = 0 ( ω 0 ω ) 2 q + 1 .
I ± ω 0 exp ( i α c z / c ) ω 2 q = 0 ( ω 0 / ω ) 2 q × exp [ i ω τ i β / ω ] d ω ,
I ± 2 π ω 0 exp ( i α c z / c ) [ 2( c t / z 1 ) / ( ω p 2 + ω m 2 α c 2 ) ] 1 / 2 × q = 0 ( 1 ) q [ 2 ( c t / z 1 ) ω 0 2 / ( ω p 2 + ω m 2 α c 2 ) ] q × J 2 q + 1 { [ 2( t z / c )( ω p 2 + ω m 2 α c 2 ) z / c ] 1 / 2 } ,
E x y ( z , t ) = ω 0 cos sin ( α z / c ) [ 2 ( c t / z 1 ) / ( ω p 2 + ω m 2 α c 2 ) ] 1 / 2 × q = 0 ( 1 ) q [ 2 ( c t / z 1 ) ω 0 2 / ( ω p 2 + ω m 2 α c 2 ) ] q × J 2 q + 1 { [ 2( t z / c )( ω p 2 + ω m 2 α c 2 ) z / c ] 1 / 2 } .
E x y ( z , t ) = ω 0 cos sin ( α z / c ) [ 2( c t / z 1 ) / ( ω p 2 + ω m 2 α c 2 ) ] 1 / 2 × J 1 { [ 2( t z / c )( ω p 2 + ω m 2 α c 2 ) z / c ] 1 / 2 } .
E x y ( z , t ) = ω 0 ( t z / c ) cos sin ( α z / c )
I ± ω 0 ( ω 0 2 ω s ± 2 ) 1 { 2 π / [ | ϕ ± ( ω s ± ) | z ] } 1 / 2 × exp [ i ( z ϕ ± ( ω s ± ) + p ) ] ,
k ± ( 1 / c ) [ α c ( α c / ω 2 ) ( ω c 2 ω m 2 ) + ω ( γ / ω ) ] ,
τ + γ z / c / ω s ± 2 ± 2 α c ( ω c 2 ω m 2 ) ( z / c ) / ω s ± 3 = 0
ω s 1 ± ( γ z / c τ ) 1 / 2 + ± ( α c / γ ) ( ω c 2 ω m 2 ) ( γ z / c τ ) 1 / 2 ± 3 ( α c / γ ) ( ω c 2 ω m 2 ) + ( γ z / c τ ) 1 / 2 ,
ω s 2 ± ( γ z / c τ ) 1 / 2 + ( α c / γ ) ( ω c 2 ω m 2 ) ( γ z / c τ ) 1 / 2 ± 3 ( α c / γ ) ( ω c 2 ω m 2 ) ( γ z / c τ ) 1 / 2 ,
E x y ( z , t ) ω 0 2 π ( ω s 1 + ) 2 { 1 / [ z | ϕ ( ω s 1 + ) | ] } 1 / 2 × cos sin [ α c z c + α c ω s 1 + 2 ( ω c 2 ω m 2 ) z / c + γ z / c ω s 1 + + τ ω s 1 ± + π 4 ] ω 0 2 π ( ω s 1 ) 2 { 1 / [ z | ϕ ( ω s 1 ) | ] } 1 / 2 × cos sin [ α c z c + α c ω s 1 2 ( ω c 2 ω m 2 ) z / c γ z / c ω s 1 τ ω s 1 π 4 ] ,
ϕ ( ω s 1 ± ) = 6 ( α c / c ) ( ω c 2 ω m 2 ) / ω s 1 ± 4 ( 2 γ / c ) / ω s 1 ± 3 ,
Δ ω ω s 1 ω s 1 + [ ( 2 α c z / c τ ) ( ω c 2 ω m 2 ) ] / [ 9 ( α c 2 / γ 2 ) × ( ω c 2 ω m 2 ) 2 ( γ z / c τ ) ] .
n [ ( 1 + ω p 2 / ω c 2 ) ( 1 + ω m 2 / ω c 2 ) ] 1 / 2 , Γ [ ω p 2 + ω m 2 + α c 2 + 2 ω m 2 ( ω p 2 + α c 2 ) / ω c 2 + ω m 4 α c 2 / ω m 4 ] ω m 4 , χ α c ( 1 + ω m 2 / ω c 2 ) / ω c 2 .
ω s 3 ± = ( 2 χ n / 3 Γ ) ( 1 + { 1 + ( 3 Γ / 2 χ 2 n ) [ ( c t / z ) n ] } 1 / 2 ) ,
ω s 4 ± = ( 2 χ n / 3 Γ ) ( 1 { 1 + ( 3 Γ / 2 χ 2 n ) [ ( c t / z ) n ] } 1 / 2 ) ,
E x y ( z , t ) ± ( ω 0 / 2 π ) { 2 π / [ | ϕ + ( ω s 3 + ) | z } 1 / 2 × exp { ( 4 χ n z / 9 Γ c ) [ ( n c t / z ) ( 2 χ 2 n / 3 Γ ) ] × [ ( 3 Γ / 2 χ 2 n ) ( n c t / z ) 1 ] 1 / 2 } × Re Im [ ( ω 0 2 ω s 3 + 2 ) 1 exp { i ( 2 χ n z / 3 Γ c ) ( ( 4 χ 2 n z / 9 Γ ) + ( c t / z n ) ) } ] .
( 2 χ n z 3 Γ c [ 4 χ 2 n 9 Γ + ( n c t / z ) ] + tan 1 { [ 3 Γ ( n c t / z ) / 2 χ 2 n 1 ] 1 / 2 1 3 Γ ( n c t / z + 3 Γ ω 0 2 / 2 n ) / 4 χ 2 n } ) ,
ϕ ± ( ω ) ϕ ( ω n s ± ) + ϕ ± ( ω n s ± ) ( ω ω n s ± ) + 1 6 ϕ ± ( 3 ) ( ω n s ± ) ( ω ω n s ± ) 3 ,
ϕ ± ( ω n s ± ) = ± ( 8 χ 3 n 2 / 27 Γ 2 c ) ± ( 2 χ n / 3 Γ ) ( t / z n / c ) , ϕ ± ( ω n s ± ) = ( 2 χ 2 n / 3 Γ c ) ( t / z n / c ) ,
E x y ( z , t ) ± ω 0 ω 0 2 ω n s + 2 [ 2 | ϕ + ( 3 ) ( ω n s + ) | z ] 1 / 3 × Ai { z ϕ + ( ω n s + ) [ 2 z ϕ + ( 3 ) ( ω n s + ) ] 1 / 3 } × cos sin [ z ϕ + ( ω n s + ) ] ,
Δ t = [ ( n 1 ) ( 2 χ 2 / 3 Γ ) ] ( z / c ) Δ t ( 2 χ 2 / 3 Γ ) ] ( z / c ) ,
E x y ( z , t ) ± 1 2 π ω 0 ω 0 2 ω s 3 + 2 [ 2 π | ϕ + ( ω s 3 + ) | z ] 1 / 2 × cos sin [ z ϕ + ( ω s 3 + ) + π / 4 ] + 1 2 π ω 0 ω 0 2 ω s 3 2 [ 2 π | ϕ ( ω s 3 ) | z ] 1 / 2 × cos sin [ z ϕ ( ω s 3 ) + π / 4 ] ,
I ± 1 2 π ω 0 ω 0 + ω s ± β 1 | β 1 | exp [ i z ϕ ± ( ω 0 ) ] × [ z | ϕ ± ( ω s ± ) | / 2 ] 1 / 2 | β 1 | exp ( β 1 | β 1 | Y ) 2 d Y ,
I ± 2 2 π ω 0 ω 0 ω s ± β 2 | β 2 | exp [ i z ϕ ± ( ω 0 ) ] × [ z | ϕ ± ( ω s ± ) | / 2 ] 1 / 2 | β 2 | exp ( β 2 | β 2 | Y ) 2 d Y ,
E x s ± 1 2 ω 0 ω 0 + ω s sin [ z ϕ ( ω 0 ) ] ± 1 π ω 0 ω 0 + ω s { cos [ z ϕ ( ω 0 ) + π / 4 ] C ( υ ) sin [ z ϕ ( ω 0 ) + π / 4 ] S ( υ ) ] } ( ω 0 ω s ) ,
E y s 1 2 ω 0 ω 0 + ω s cos [ z ϕ ( ω 0 ) ] ± 1 π ω 0 ω 0 + ω s { sin [ z ϕ ( ω 0 ) + π / 4 ] C ( υ ) cos [ z ϕ ( ω 0 ) + π / 4 ] S ( υ ) ] } ( ω 0 ω s ) ,
E x pole = ½ sin [ z ϕ ( ω 0 ) ] ,
E y pole = ½ cos [ z ϕ ( ω 0 ) ] ,
E x 1 2 ω 0 ω 0 + ω s sin [ z ϕ ( ω 0 ) ] 1 π ω 0 ω 0 + ω s { cos [ z ϕ ( ω 0 ) + π / 4 ] C ( υ ) sin [ z ϕ ( ω 0 ) + π / 4 ] S ( υ ) ] } ( ω 0 < ω s ) ,
E y 1 2 ω 0 ω 0 + ω s cos [ z ϕ ( ω 0 ) ] 1 π ω 0 ω 0 + ω s { sin [ z ϕ ( ω 0 ) + π / 4 ] C ( υ ) + cos [ z ϕ ( ω 0 ) + π / 4 ] S ( υ ) ] } ( ω 0 < ω s ) ,
E x 1 2 ω 0 ω 0 + ω s sin [ z ϕ ( ω 0 ) ] + 1 π ω 0 ω 0 + ω s { cos [ z ϕ ( ω 0 ) + π / 4 ] C ( υ ) sin [ z ϕ ( ω 0 ) + π / 4 ] S ( υ ) ] } 1 2 sin [ z ϕ ( ω 0 ) ] ( ω 0 > ω s ) ,
E x 1 2 ω 0 ω 0 + ω s cos [ z ϕ ( ω 0 ) ] + 1 π ω 0 ω 0 + ω s { sin [ z ϕ ( ω 0 ) + π / 4 ] C ( υ ) + cos [ z ϕ ( ω 0 ) + π / 4 ] S ( υ ) ] } + 1 2 sin [ z ϕ ( ω 0 ) ] ( ω 0 > ω s ) ,
E x pole + = ½ sin [ z ϕ + ( ω 0 ) ] ,
E y pole + = ½ cos [ z ϕ + ( ω 0 ) ] ,
E x + 1 2 ω 0 ω 0 + ω s + sin [ z ϕ + ( ω 0 ) ] 1 π ω 0 ω 0 + ω s + { cos [ z ϕ + ( ω 0 ) + π / 4 ] C ( υ + ) sin [ z ϕ + ( ω 0 ) + π / 4 ] S ( υ + ) ] } ( ω 0 < ω s + ) ,
E y + 1 2 ω 0 ω 0 + ω s + cos [ z ϕ ( ω 0 ) ] + 1 π ω 0 ω 0 + ω s + { sin [ z ϕ + ( ω 0 ) + π / 4 ] C ( υ + ) + cos [ z ϕ + ( ω 0 ) + π / 4 ] S ( υ + ) ] } ( ω 0 < ω s + ) ,
E x + 1 2 ω 0 ω 0 + ω s + sin [ z ϕ + ( ω 0 ) ] + 1 π ω 0 ω 0 + ω s + { cos [ z ϕ + ( ω 0 ) + π / 4 ] C ( υ + ) sin [ z ϕ + ( ω 0 ) + π / 4 ] S ( υ + ) ] } 1 2 sin [ z ϕ + ( ω 0 ) ] ( ω 0 > ω s + ) ,
E y + 1 2 ω 0 ω 0 + ω s + cos [ z ϕ + ( ω 0 ) ] 1 π ω 0 ω 0 + ω s + { sin [ z ϕ + ( ω 0 ) + π / 4 ] C ( υ + ) + cos [ z ϕ ( ω 0 ) + π / 4 ] S ( υ + ) ] } 1 2 cos [ z ϕ + ( ω 0 ) ] ( ω 0 > ω s + ) ,
E x steady = ½ sin [ z ϕ + ( ω 0 ) ½ sin [ z ϕ ( ω 0 ) ] , E y steady = ½ cos [ z ϕ + ( ω 0 ) + ½ cos [ z ϕ ( ω 0 ) ] ,
k + α c ω 2 ω c c ( ω c ω ) 0 < ω < ω c , ω ω c 0 ,
k α c ω 2 ω c c ( ω c ω ) ω c < ω < 0 , ω ω c + 0 ,
I + 1 2 exp ( i ω c t i 2 α c z / c ) × ω 0 ω 0 2 ( ω c μ / τ ) 2 × exp ( i α c z ω c c μ 1 / 2 τ 1 / 2 + i τ 1 / 2 μ 1 / 2 ) d μ μ τ ,
E x y ( z , t ) 1 2 π ω 0 ω 0 2 [ ω c ( α c z ω c / c τ ) 1 / 2 ] 2 ( α c z ω c c ) 1 / 4 τ 3 / 4 cos sin [ ω c t + 2 α c z / c 2 ( α c z ω c / c ) 1 / 2 τ 1 / 2 π / 4 ] .
k ± = ω c { ± α c ω ω c 2 ω 2 + [ ( ω ω z 1 ) ( ω ω z 2 ) ( ω + ω z 1 ) ( ω + ω z 2 ) ( ω c 2 ω 2 ) 2 ] 1 / 2 } ,
ω z 2 1 = { 2 ω c 2 + ω p 2 α c 2 ± [ ( ω p 2 α c 2 ) 2 4 ω c 2 α c 2 ] 1 / 2 2 } .
E x y ( z , t ) 1 2 π ω 0 ω 0 2 ( υ s + 2 + ω z 1 ) 2 [ V 2 ω z 1 2 ( c t / z 2 U ω z 1 ) 3 z / c ] 1 / 2 × cos sin [ ω z 1 t U ω z 1 2 z / c V 2 ω z 1 2 z / c 4 ( c t / z 2 U ω z 1 ) 3 z / c + π / 4 ] ± 1 2 π ω 0 ω 0 2 ( υ s 2 + ω z 1 ) 2 [ V 2 ω z 1 2 ( c t / z + 2 U ω z 1 ) 3 z / c ] 1 / 2 × cos sin [ ω z 1 t + U ω z 1 2 z / c V 2 ω z 1 2 z / c 4 ( c t / z 2 U ω z 1 ) 3 z / c + π / 4 ] ,

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