Abstract

Forerunners (precursors) in linear, temporally dispersive, bigyrotropic materials are investigated with time-domain techniques. Bigyrotropic materials are characterized by 12 constitutive parameters (integral kernels). Specifically, the four susceptibility dyadics are all gyrotropic with a common gyrotropic axis. Pulse propagation along this axis is analyzed with dispersive (noncoupling) wave splitting and complex, time-dependent field vectors. Two numerical examples illustrating the method are presented.

© 1998 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Sommerfeld, “Über die Fortpflanzung des Lichtes in dispergierenden Medien,” Ann. Phys. (Leipzig) 44, 177–202 (1914).
    [CrossRef]
  2. L. Brillouin, “Über die Fortpflanzung des Lichtes in dispergierenden Medien,” Ann. Phys. (Leipzig) 44, 203–240 (1914).
    [CrossRef]
  3. 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]
  4. K. E. Oughstun, G. C. Sherman, Electromagnetic Pulse Propagation in Causal Dielectrics (Springer-Verlag, Berlin, 1994).
  5. S. Shen, K. E. Oughstun, “Dispersive pulse propagation in a double-resonance Lorentz medium,” J. Opt. Soc. Am. B 6, 948–963 (1989).
    [CrossRef]
  6. L. Brillouin, Wave Propagation and Group Velocity (Academic, New York, 1960).
  7. P. G. Zablocky, N. Engheta, “Transients in chiral media with single-resonance dispersion,” J. Opt. Soc. Am. A 10, 740–758 (1993).
    [CrossRef]
  8. I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-isotropic Media (Artech House, Boston, London, 1994).
  9. I. V. Lindell, Methods for Electromagnetic Field Analysis (Clarendon, Oxford, 1992).
  10. S. Rikte, “Sommerfeld’s forerunner in stratified isotropic and bi-isotropic media,” (Lund Institute of Technology, Department of Electromagnetic Theory, P.O. Box 118, S-211 00 Lund, Sweden, 1994).
  11. J. Fridén, G. Kristensson, “Transient external 3D excitation of a dispersive and anisotropic slab,” Inverse Probl. 13, 691–709 (1997).
    [CrossRef]
  12. A. Karlsson, S. Rikte, “The time-domain theory of forerunners,” J. Opt. Soc. Am. A 15, 487–502 (1998).
    [CrossRef]
  13. A. Karlsson, G. Kristensson, “Constitutive relations, dissipation and reciprocity for the Maxwell equations in the time domain,” J. Electromagn. Waves Appl. 6, 537–551 (1992).
    [CrossRef]
  14. R. Kress, Linear Integral Equations (Springer-Verlag, Berlin, 1989).
  15. A. H. Sihvola, I. V. Lindell, “Material effects in bi-anisotropic electromagnetics,” IEICE Trans. Electron. (Japan), E78-C, 1383–1390 (1995).
  16. M. M. I. Saadoun, N. Engheta, “A reciprocal phase shifter using noval pseudochiral or Ω medium,” Microwave Opt. Technol. Lett. 5, 184–187 (1992).
    [CrossRef]
  17. I. V. Lindell, S. A. Tretyakov, A. J. Viitanen, “Plane-wave propagation in a uniaxial chiro-omega medium,” Microwave Opt. Technol. Lett. 6, 517–520 (1993).
    [CrossRef]
  18. M. Norgren, S. He, “Electromagnetic reflection and transmission for a dielectric-Ω interface and a Ω slab,” Int. J. Infrared Millim. Waves 15, 1537–1554 (1994).
    [CrossRef]
  19. M. Norgren, S. He, “Reconstruction of the constitutive parameters for an Ω material in a rectangular waveguide,” IEEE Trans. Microwave Theory Tech. 43, 1315–1321 (1995).
    [CrossRef]
  20. S. Rikte, “The theory of the propagation of TEM-pulses in dispersive bi-isotropic slabs,” Wave Motion (to be published).
  21. L. Hörmander, The Analysis of Linear Partial Differential Operators. I. Grundlehren der mathematischen Wissenschaften 256 (Springer-Verlag, Berlin, 1983).
  22. E. Beltrami, “Considerazioni idrodinamiche,” Rend. Ist. Lomb. Accad. Sci. Lett. 22, 122–131 (1889).
  23. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  24. S. A. Maksimenko, G. Y. Slepyan, A. Lakhtakia, “Gaussian pulse propagation in a linear, lossy chiral medium,” J. Opt. Soc. Am. A 14, 894–900 (1997).
    [CrossRef]
  25. H.-O. Kreiss, J. Lorenz, Initial-Boundary Value Problems and the Navier–Stokes Equations (Academic, San Diego, Calif., 1989).
  26. E. U. Condon, “Theories of optical rotatory power,” Rev. Mod. Phys. 9, 432–457 (1937).
    [CrossRef]
  27. C. Eftimiu, L. W. Pearson, “Guided electromagnetic waves in chiral media,” Radio Sci. 24, 351–359 (1989).
    [CrossRef]
  28. E. J. Post, Formal Structure of Electromagnetics (North-Holland, Amsterdam, 1962).
  29. G. Kristensson, S. Rikte, “Transient wave propagation in reciprocal bi-isotropic media at oblique incidence,” J. Math. Phys. (New York) 34, 1339–1359 (1993).
    [CrossRef]
  30. P. Drude, Lehrbuch der Optik (S. Hirzel, Leipzig, 1900).
  31. C. M. Bender, S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978).

1998 (1)

1997 (2)

J. Fridén, G. Kristensson, “Transient external 3D excitation of a dispersive and anisotropic slab,” Inverse Probl. 13, 691–709 (1997).
[CrossRef]

S. A. Maksimenko, G. Y. Slepyan, A. Lakhtakia, “Gaussian pulse propagation in a linear, lossy chiral medium,” J. Opt. Soc. Am. A 14, 894–900 (1997).
[CrossRef]

1995 (2)

M. Norgren, S. He, “Reconstruction of the constitutive parameters for an Ω material in a rectangular waveguide,” IEEE Trans. Microwave Theory Tech. 43, 1315–1321 (1995).
[CrossRef]

A. H. Sihvola, I. V. Lindell, “Material effects in bi-anisotropic electromagnetics,” IEICE Trans. Electron. (Japan), E78-C, 1383–1390 (1995).

1994 (1)

M. Norgren, S. He, “Electromagnetic reflection and transmission for a dielectric-Ω interface and a Ω slab,” Int. J. Infrared Millim. Waves 15, 1537–1554 (1994).
[CrossRef]

1993 (3)

I. V. Lindell, S. A. Tretyakov, A. J. Viitanen, “Plane-wave propagation in a uniaxial chiro-omega medium,” Microwave Opt. Technol. Lett. 6, 517–520 (1993).
[CrossRef]

P. G. Zablocky, N. Engheta, “Transients in chiral media with single-resonance dispersion,” J. Opt. Soc. Am. A 10, 740–758 (1993).
[CrossRef]

G. Kristensson, S. Rikte, “Transient wave propagation in reciprocal bi-isotropic media at oblique incidence,” J. Math. Phys. (New York) 34, 1339–1359 (1993).
[CrossRef]

1992 (2)

A. Karlsson, G. Kristensson, “Constitutive relations, dissipation and reciprocity for the Maxwell equations in the time domain,” J. Electromagn. Waves Appl. 6, 537–551 (1992).
[CrossRef]

M. M. I. Saadoun, N. Engheta, “A reciprocal phase shifter using noval pseudochiral or Ω medium,” Microwave Opt. Technol. Lett. 5, 184–187 (1992).
[CrossRef]

1989 (2)

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

C. Eftimiu, L. W. Pearson, “Guided electromagnetic waves in chiral media,” Radio Sci. 24, 351–359 (1989).
[CrossRef]

1988 (1)

1937 (1)

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

1914 (2)

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

L. Brillouin, “Über die Fortpflanzung des Lichtes in dispergierenden Medien,” Ann. Phys. (Leipzig) 44, 203–240 (1914).
[CrossRef]

1889 (1)

E. Beltrami, “Considerazioni idrodinamiche,” Rend. Ist. Lomb. Accad. Sci. Lett. 22, 122–131 (1889).

Beltrami, E.

E. Beltrami, “Considerazioni idrodinamiche,” Rend. Ist. Lomb. Accad. Sci. Lett. 22, 122–131 (1889).

Bender, C. M.

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

Brillouin, L.

L. Brillouin, “Über die Fortpflanzung des Lichtes in dispergierenden Medien,” Ann. Phys. (Leipzig) 44, 203–240 (1914).
[CrossRef]

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

Condon, E. U.

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

Drude, P.

P. Drude, Lehrbuch der Optik (S. Hirzel, Leipzig, 1900).

Eftimiu, C.

C. Eftimiu, L. W. Pearson, “Guided electromagnetic waves in chiral media,” Radio Sci. 24, 351–359 (1989).
[CrossRef]

Engheta, N.

P. G. Zablocky, N. Engheta, “Transients in chiral media with single-resonance dispersion,” J. Opt. Soc. Am. A 10, 740–758 (1993).
[CrossRef]

M. M. I. Saadoun, N. Engheta, “A reciprocal phase shifter using noval pseudochiral or Ω medium,” Microwave Opt. Technol. Lett. 5, 184–187 (1992).
[CrossRef]

Fridén, J.

J. Fridén, G. Kristensson, “Transient external 3D excitation of a dispersive and anisotropic slab,” Inverse Probl. 13, 691–709 (1997).
[CrossRef]

He, S.

M. Norgren, S. He, “Reconstruction of the constitutive parameters for an Ω material in a rectangular waveguide,” IEEE Trans. Microwave Theory Tech. 43, 1315–1321 (1995).
[CrossRef]

M. Norgren, S. He, “Electromagnetic reflection and transmission for a dielectric-Ω interface and a Ω slab,” Int. J. Infrared Millim. Waves 15, 1537–1554 (1994).
[CrossRef]

Hörmander, L.

L. Hörmander, The Analysis of Linear Partial Differential Operators. I. Grundlehren der mathematischen Wissenschaften 256 (Springer-Verlag, Berlin, 1983).

Karlsson, A.

A. Karlsson, S. Rikte, “The time-domain theory of forerunners,” J. Opt. Soc. Am. A 15, 487–502 (1998).
[CrossRef]

A. Karlsson, G. Kristensson, “Constitutive relations, dissipation and reciprocity for the Maxwell equations in the time domain,” J. Electromagn. Waves Appl. 6, 537–551 (1992).
[CrossRef]

Kreiss, H.-O.

H.-O. Kreiss, J. Lorenz, Initial-Boundary Value Problems and the Navier–Stokes Equations (Academic, San Diego, Calif., 1989).

Kress, R.

R. Kress, Linear Integral Equations (Springer-Verlag, Berlin, 1989).

Kristensson, G.

J. Fridén, G. Kristensson, “Transient external 3D excitation of a dispersive and anisotropic slab,” Inverse Probl. 13, 691–709 (1997).
[CrossRef]

G. Kristensson, S. Rikte, “Transient wave propagation in reciprocal bi-isotropic media at oblique incidence,” J. Math. Phys. (New York) 34, 1339–1359 (1993).
[CrossRef]

A. Karlsson, G. Kristensson, “Constitutive relations, dissipation and reciprocity for the Maxwell equations in the time domain,” J. Electromagn. Waves Appl. 6, 537–551 (1992).
[CrossRef]

Lakhtakia, A.

Lindell, I. V.

A. H. Sihvola, I. V. Lindell, “Material effects in bi-anisotropic electromagnetics,” IEICE Trans. Electron. (Japan), E78-C, 1383–1390 (1995).

I. V. Lindell, S. A. Tretyakov, A. J. Viitanen, “Plane-wave propagation in a uniaxial chiro-omega medium,” Microwave Opt. Technol. Lett. 6, 517–520 (1993).
[CrossRef]

I. V. Lindell, Methods for Electromagnetic Field Analysis (Clarendon, Oxford, 1992).

I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-isotropic Media (Artech House, Boston, London, 1994).

Lorenz, J.

H.-O. Kreiss, J. Lorenz, Initial-Boundary Value Problems and the Navier–Stokes Equations (Academic, San Diego, Calif., 1989).

Maksimenko, S. A.

Norgren, M.

M. Norgren, S. He, “Reconstruction of the constitutive parameters for an Ω material in a rectangular waveguide,” IEEE Trans. Microwave Theory Tech. 43, 1315–1321 (1995).
[CrossRef]

M. Norgren, S. He, “Electromagnetic reflection and transmission for a dielectric-Ω interface and a Ω slab,” Int. J. Infrared Millim. Waves 15, 1537–1554 (1994).
[CrossRef]

Orszag, S. A.

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

Oughstun, K. E.

Pearson, L. W.

C. Eftimiu, L. W. Pearson, “Guided electromagnetic waves in chiral media,” Radio Sci. 24, 351–359 (1989).
[CrossRef]

Post, E. J.

E. J. Post, Formal Structure of Electromagnetics (North-Holland, Amsterdam, 1962).

Rikte, S.

A. Karlsson, S. Rikte, “The time-domain theory of forerunners,” J. Opt. Soc. Am. A 15, 487–502 (1998).
[CrossRef]

G. Kristensson, S. Rikte, “Transient wave propagation in reciprocal bi-isotropic media at oblique incidence,” J. Math. Phys. (New York) 34, 1339–1359 (1993).
[CrossRef]

S. Rikte, “The theory of the propagation of TEM-pulses in dispersive bi-isotropic slabs,” Wave Motion (to be published).

S. Rikte, “Sommerfeld’s forerunner in stratified isotropic and bi-isotropic media,” (Lund Institute of Technology, Department of Electromagnetic Theory, P.O. Box 118, S-211 00 Lund, Sweden, 1994).

Saadoun, M. M. I.

M. M. I. Saadoun, N. Engheta, “A reciprocal phase shifter using noval pseudochiral or Ω medium,” Microwave Opt. Technol. Lett. 5, 184–187 (1992).
[CrossRef]

Shen, S.

Sherman, G. C.

Sihvola, A. H.

A. H. Sihvola, I. V. Lindell, “Material effects in bi-anisotropic electromagnetics,” IEICE Trans. Electron. (Japan), E78-C, 1383–1390 (1995).

I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-isotropic Media (Artech House, Boston, London, 1994).

Slepyan, G. Y.

Sommerfeld, A.

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

Stratton, J. A.

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

Tretyakov, S. A.

I. V. Lindell, S. A. Tretyakov, A. J. Viitanen, “Plane-wave propagation in a uniaxial chiro-omega medium,” Microwave Opt. Technol. Lett. 6, 517–520 (1993).
[CrossRef]

I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-isotropic Media (Artech House, Boston, London, 1994).

Viitanen, A. J.

I. V. Lindell, S. A. Tretyakov, A. J. Viitanen, “Plane-wave propagation in a uniaxial chiro-omega medium,” Microwave Opt. Technol. Lett. 6, 517–520 (1993).
[CrossRef]

I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-isotropic Media (Artech House, Boston, London, 1994).

Zablocky, P. G.

Ann. Phys. (Leipzig) (2)

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

L. Brillouin, “Über die Fortpflanzung des Lichtes in dispergierenden Medien,” Ann. Phys. (Leipzig) 44, 203–240 (1914).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

M. Norgren, S. He, “Reconstruction of the constitutive parameters for an Ω material in a rectangular waveguide,” IEEE Trans. Microwave Theory Tech. 43, 1315–1321 (1995).
[CrossRef]

IEICE Trans. Electron. (Japan) (1)

A. H. Sihvola, I. V. Lindell, “Material effects in bi-anisotropic electromagnetics,” IEICE Trans. Electron. (Japan), E78-C, 1383–1390 (1995).

Int. J. Infrared Millim. Waves (1)

M. Norgren, S. He, “Electromagnetic reflection and transmission for a dielectric-Ω interface and a Ω slab,” Int. J. Infrared Millim. Waves 15, 1537–1554 (1994).
[CrossRef]

Inverse Probl. (1)

J. Fridén, G. Kristensson, “Transient external 3D excitation of a dispersive and anisotropic slab,” Inverse Probl. 13, 691–709 (1997).
[CrossRef]

J. Electromagn. Waves Appl. (1)

A. Karlsson, G. Kristensson, “Constitutive relations, dissipation and reciprocity for the Maxwell equations in the time domain,” J. Electromagn. Waves Appl. 6, 537–551 (1992).
[CrossRef]

J. Math. Phys. (New York) (1)

G. Kristensson, S. Rikte, “Transient wave propagation in reciprocal bi-isotropic media at oblique incidence,” J. Math. Phys. (New York) 34, 1339–1359 (1993).
[CrossRef]

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

J. Opt. Soc. Am. B (2)

Microwave Opt. Technol. Lett. (2)

M. M. I. Saadoun, N. Engheta, “A reciprocal phase shifter using noval pseudochiral or Ω medium,” Microwave Opt. Technol. Lett. 5, 184–187 (1992).
[CrossRef]

I. V. Lindell, S. A. Tretyakov, A. J. Viitanen, “Plane-wave propagation in a uniaxial chiro-omega medium,” Microwave Opt. Technol. Lett. 6, 517–520 (1993).
[CrossRef]

Radio Sci. (1)

C. Eftimiu, L. W. Pearson, “Guided electromagnetic waves in chiral media,” Radio Sci. 24, 351–359 (1989).
[CrossRef]

Rend. Ist. Lomb. Accad. Sci. Lett. (1)

E. Beltrami, “Considerazioni idrodinamiche,” Rend. Ist. Lomb. Accad. Sci. Lett. 22, 122–131 (1889).

Rev. Mod. Phys. (1)

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

Other (13)

H.-O. Kreiss, J. Lorenz, Initial-Boundary Value Problems and the Navier–Stokes Equations (Academic, San Diego, Calif., 1989).

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

S. Rikte, “The theory of the propagation of TEM-pulses in dispersive bi-isotropic slabs,” Wave Motion (to be published).

L. Hörmander, The Analysis of Linear Partial Differential Operators. I. Grundlehren der mathematischen Wissenschaften 256 (Springer-Verlag, Berlin, 1983).

E. J. Post, Formal Structure of Electromagnetics (North-Holland, Amsterdam, 1962).

P. Drude, Lehrbuch der Optik (S. Hirzel, Leipzig, 1900).

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

R. Kress, Linear Integral Equations (Springer-Verlag, Berlin, 1989).

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

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

I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-isotropic Media (Artech House, Boston, London, 1994).

I. V. Lindell, Methods for Electromagnetic Field Analysis (Clarendon, Oxford, 1992).

S. Rikte, “Sommerfeld’s forerunner in stratified isotropic and bi-isotropic media,” (Lund Institute of Technology, Department of Electromagnetic Theory, P.O. Box 118, S-211 00 Lund, Sweden, 1994).

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 (4)

Fig. 1
Fig. 1

(a) Co-propagator kernel, (QP)co(z, t)=Re{Qc(z)Pc(z; t)}, and (b) cross-propagator kernel (QP)cross(z, t)=Im{Qc(z)Pc(z; t)} at a fixed propagation depth z in an isotropic chiral medium (ωp=100×c/z, ω0=100×c/z, ν=20×c/z, and α=-0.001×z/c). 32,768 data points were used at the equidistant discretization of the time interval 0<t<2×z/c. Several second precursor approximations obtained by the time-domain method are also shown.

Fig. 2
Fig. 2

Polar plot of (QP)cross(z, t) versus (QP)co(z, t) with time as the parameter at a fixed propagation depth z in an isotropic chiral medium. The field is propagating toward the reader. Early times correspond to large amplitudes.

Fig. 3
Fig. 3

(a) Co-propagator kernel (QP)co(z, t)=Re{Qc(z)Pc(z; t)} and (b) cross-propagator kernel (QP)cross(z, t)=Im{Qc(z)Pc(z; t)} at a fixed propagation depth z in a gyrotropic medium (ωp=100×c/z, ω0=100×c/z, ωg=100×c/z, and ν=20×c/z). 32,768 data points were used at the equidistant discretization of the time interval 0<t<1.5×z/c. Second precursor approximations obtained by the time-domain method are also shown.

Fig. 4
Fig. 4

Polar plot of (QP)cross(z, t) versus (QP)co(z, t) with time as the parameter at a fixed propagation depth z in a gyrotropic medium. (a) General view, (b) detailed view that reveals the second forerunner. The wave travels toward the reader.

Equations (128)

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

cηD(r, t)=E(r, t)+(χee*E)(r, t)+η(χem*H)(r, t),
cB(r, t)=(χme*E)(r, t)+ηH(r, t)+η(χmm*H)(r, t),
F(r, t)=F(z, t)F{E, H, D, B}.
χij(t)·JT=JT·χij(t),i, j{e, m}.
χij(t)=ITχcoij(t)+JTχcrossij(t)+uzuzχzij(t),i, j{e, m}.
F(r, t)=FT(z, t)uxFx(z, t)+uyFy(z, t).
0=cηDz(z, t)=Ez(z, t)+(χzee*Ez)(z, t)+η(χzem*Hz)(z, t),
0=cBz(z, t)=(χzme*Ez)(z, t)+ηHz(z, t)+η(χzmm*Hz)(z, t).
zET=c-1tJT·(χTme*ET+ηHT+χTmm*ηHT),
zJT·(ηHT)=c-1t(ET+χTee*ET+χTem*ηHT)+ηJTe,
ET=E++E-,
JT·(ηHT)=-Y+*E++Y-*E-,
F(t)=ITδ(t)+ITFco(t)+JTFcross(t).
2E±=Z*Y*ETZ*JT·(ηHT),
(δIT+χTmm)*Y±*Y±JT·(χTem+χTme)*Y±
=δIT+χTee.
zE±=c-1t(N±*E±)(Z*ηJTe)/2,
N±=(δIT+χTmm)*Y±JT·χTem
(N±JT·χTem)*(N±±JT·χTme)
=(δIT+χTee)*(δIT+χTmm).
2Nco±(t)+(Nco±*Nco±)(t)=χee(t)+χmm(t)+(χee*χmm)(t)-(L*L)(t),
Ncross±(t)=±κ(t).
E±(z; t)=ITEco±(z; t)+JTEcross±(z; t),
±zIT·+c-1tN±*
(±z+c-1t)E±(z, t)+c-1[ITNco±(+0)+JTNcross±(+0)]·E±(z, t)+[K±(·)*E±(z, ·)](t)=ITδ(z)δ(t),
K±(t)=ITKco±(t)+JTKcross±(t),
E±(z, t)=-12E±(z-z; t-t)·(Z*ηJTe)(z, t)dtdz.
E±(z; t)=H(±z)Q±(±z)·[ITδ(tz/c)+P±(±z; tz/c)],
czQ±(z)=-[ITNco±(+0)+JTNcross±(+0)]·Q±(z),
Q±(0)=IT,
zP±(z; t)=-K±(t)-[K±(·)*P±(z; ·)](t),
P±(0; t)=0,
Q±(z)=exp{-z[ITNco±(+0)+JTNcross±(+0)]/c}=exp[-zNco+(+0)/c]{IT cos[-zNcross±(+0)/c]+JT sin[-zNcross±(+0)/c]}.
P±(z; t)=n=1(-z)nn![(K±*)n-1K±](t).
tP±(z; t)=-F±(t)-[F±(·)*P±(z; ·)](t),
F±(t)=ztK±(t).
E±(z, t)=E±(z; t-t)·E(0, t)dt
E±(±z, t+z/c)=H(z)Q±(z)·E(0, t)+-tP±(z; t-t)·E(0, t)dt.
E±(±z; t+z/c)=H(z)[P±(z)δ](t),
P±(z)=exp-zct(ITNco±+JTNcross±)*.
P±(z1+z2)=P±(z1)·P±(z2),
P±(0)=IT,
(P±)-1(z)=P±(-z),
Fc(z, t)=p·FT(z, t)=12(ux-juy)Fc(z, t),
p·F=p(δ+Fco+jFcross)=p(δ+Fc),
zEc±=c-1t(Ec±+Nc±*Ec±)ηJce/2(Zc*ηJce)/2
2Nc±+Nc±*Nc±jNc±*(χcem-χcme)j(χcem-χcme)+χcem*χcme=χcee+χcmm+χcee*χcmm,
Nc±(t)=χcmm(t)+Yc(t)+(χcmm*Yc)(t)±jχcem(t).
Ec=Ec++Ec-,
ηHc=j(1+Yc+*)Ec+-j(1+Yc-*)Ec-.
Pc±(z)=exp-zctNc±*=Qc±(z)[1+Pc±(z; ·)*],
Qc±(z)=exp[-Nc±(+0)z/c].
zPc±(z; t)=-Kc±(t)-[Kc±(·)*Pc±(z; ·)](t),
Pc±(0; t)=0,
tPc±(z; t)=-Fc±(t)-[Fc±(·)*Pc±(z; ·)](t),
Fc±(t)=ztKc±(t),
Ec±(z, t)=Ec±(z; t-t)Ec(0, t)dt,
Ec±(±z; t+z/c)=H(z)[Pc±(z)δ](t)=H(z)Qc±(z)[δ(t)+Pc±(z; t)]
Ex(±z; t+z/c)=ux[Pco±(z)δ](t)+uy[Pcross±(z)δ](t),z>0.
[Pco±(z)δ](t)+j[Pcross±(z)δ](t)[Pc±(z)δ](t)
=Qc±(z)[δ(t)+Pc±(z; t)].
[PS,c±(z)δ](t)=Qc±(z)[δ(t)+PS,c±(z; t)],
PS,c±(z; t)=-zKc±(+0){J0[2zKc±(+0)t]+J2[2zKc±(+0)t]}H(t).
χc*=i=1sχidi-1dti-1+Rχ,s*dsdts,
Nc±*=i=1sni± di-1dti-1+Rn,s±*dsdts,
χi=(-1)i-1(i-1)!0ti-1χc(t)dt,
Rχ,s(t)=(-1)s(s-1)!t(τ-t)s-1χc(τ)dτH(t).
2nm+1±+i=0mnm-i+1±ni+1±±ji=0mnm-i+1±(χi+1me-χi+1em)
±j(χm+1me-χm+1em)+i=0mχm-i+1emχi+1me
=χm+1ee+χm+1mm+i=0mχm-i+1eeχi+1mm.
Pc(z)=exp-zcddti=1snid(i-1)dt(i-1)Ps+1(z),
Ps+1(z)=exp-zcddtRn,s*dsdts.
Pc(z)exp-zcddti=1mnid(i-1)dt(i-1)=:Tm(z; ·)*.
Tm(z; t)=12π- expjwt-zc[n1jw+n2(jw)2++nm(jw)m]dw=12π- expjwt-zc[n1(1)jw+n2(1)(jw)2++nm-2(1)(jw)m-2]dw*expzcnmnm-1mnmm 12π×- expjwt-zcnmjw+nm-1mnmmdw,
ni(1)=ni-nmminm-1mnmm-i.
Tm(z; t)=12π- expjwt-zc[n1(1)jw+n2(1)(jw)2++nm-2(1)(jw)m-2]dw*expzcnmnm-1mnmm-tnm-1mnm 1tmAmttm,
(-1)i/2 Re nim-i2>0,i=2, 4,, m
PcP2*P4**Pm*,
Pi(z; t)=expzcnim-i2ni-1m-i2inim-i2i-tni-1m-i2inim-i2 1tiAitti,
ti=(-1)i/2 zinim-i2c1/i
ni(0)=ni,
nim-i2=ni-l=i2+1m/2n2l2lin2l-12ln2l2l-i,
ni-1m-i2=ni-1-l=i2+1m/2n2l2li-1n2l-12ln2l2l-i+1.
Pc(z)P2(z; ·)*,
P2(z; t)=1t2A2t-t1t2=12πt2exp-(t-t1)22t22,
t1=n1zc,t2=-2zn2c1/2,
Pc(z)PB(z; ·)*=exp-zct(n1+n2t+n3t2)=exp-zcn3t+n23n33+n1-n223n3t-n2327n32.
-czPB(z; t)=(n1t+n2t2+n3t3)PB(z; t),
PB(0; t)=δ(t).
PB(z; t)=expn2327n32zc-n23n3[t-t1(z)]×Ai{sgn[Re(n3)][t-t1(z)]/t3(z)}t3(z),
t1(z)=n1-n223n3 zc,
t3(z)=3n3 sgn[Re(n3)]zc1/3.
cηD(r, t)=E(r, t)+(χPee*E)(r, t)+c(χPem*B)(r, t),
ηH(r, t)=(χPme*E)(r, t)+cB(r, t)+c(χPmm*B)(r, t),
Nqr(t)=0(G*E)(t)+0(κ*cB)(t),
mt2r(t)=-mνtr(t)-mω02r(t)+qE(t)-qcα×E,
(t2+νt+ω02)[(G*E)(t)+(κ*cB)(t)]
=ωp2[E(t)+αtcB(t)].
G(t)=H(t)ωp2ν0sin(ν0t)exp-νt2,κ(t)=αtG(t),
χcoem(t)=-χcome(t)=κ(t)=αtG(t),
χcoee(t)=χ(t)=G(t)-(κ*κ)(t),
χcrossem(t)=χcrossme=χcrossee=χcomm=χcrossmm=0,
gm=(-1)m-1(m-1)!0tm-1G(t)dt=(-1)m+1 ωp2ω0mν0sinm arcsinν0ω0,
κ1=0,κm+1=αgm,
χm=gm-i=1m-2κi+1κm-i.
n1=(1+χ1)1/2-1,
Re nm+1=χm+1-i=1m-1 Re nm-i+1 Re ni+12(1+n1),
Im nm+1=κm+1.
χ1=ωp2ω02,χ2=-νωp2ω04,
n1=(1+ωp2/ω02)1/2-1,
n2=-1(1+ωp2/ω02)1/2νωp22ω04+jαwp2w02,
mt2r(t)=-mνtr(t)-mω02r(t)+qE(t)+qtr(t)×B0.
(t2+νt+ω02)(χee*E)(t)
+ωguz×t(χee*E)(t)=ωp2E(t).
χee(t)=ITχcoee(t)+JTχcrossee(t)+uzuzχzee(t),
χcoee(t)=Re χcee(t),χcrossee(t)=Im χcee(t),
χcee(t)=H(t)ωp2ν0csin(ν0ct)exp-νct2,
χzee(t)=H(t)ωp2ν0sin(ν0t)exp-νt2,
χmee=(-1)m-1(m-1)!0tm-1χcee(t)dt=(-1)m+1 ωp2ω0mν0csinm arcsinν0cω0.
n1=(1+χ1ee)1/2-1
nm+1=χm+1ee-i=1m-1nm-i+1ni+12(1+n1),m1.
Pc(z1+z2; t)=Pc(z1; t)+Pc(z2; t)+[Pc(z1; ·)*Pc(z2; ·)](t)
A2k(z)=12π- exp(-ξ2k/(2k)+izξ)dξ,z.
A2k(2k-1)(z)=(-1)kzA2k(z),z,k>0.
A2k(2m)(0)=(-1)mπ(2k)2m+12k-1 Γ2m+12k,
A2k(2m-1)(0)=0,
A2k(z)=(-1)k(2k-2)!C(z-ξ)2k-2ξA2k(ξ)dξ+m=02k-2xmm!A2k(m)(0),
A2k(z)(jz)-2k-22(2k-1)[2π(2k-1)]1/2exp2k-12k(jz)2k2k-1+(-1)-2k-22(2k-1) exp2k-12k(-jz)2k2k-1

Metrics