Abstract

The asymptotic theory of eikonal approximation permits electromagnetic fields to be described locally in terms of plane waves and the trajectory to be described as space–time rays. Properties of these space–time rays in a dispersive medium can be studied through the effective index of refraction defined by the higher-order space–time eikonal equation. From the analysis it is shown that the diffraction effects due to the finite size of the beam and the dispersion effects due to the finite pulse width can be treated in a unified manner in the generalized space–time eikonal equation. The space–time rays of a Gaussian pulse in a dispersive medium can be readily solved analytically and shown to provide results similar to those obtained by the Fourier transformation method.

© 1998 Optical Society of America

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References

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  1. A. Sommerfeld, J. Runge, “Anwendung der Vektorrechnung auf die Grundlagen der Geometrischen Optik,” Ann. Phys. (Leipzig) 35, 277–298 (1911).
    [CrossRef]
  2. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1975), Chap. 3.
  3. J. B. Keller, “Geometrical theory of diffraction,” J. Opt. Soc. Am. 52, 116–130 (1962).
    [CrossRef] [PubMed]
  4. J. B. Keller, W. Streifer, “Complex rays with application to Gaussian beams,” J. Opt. Soc. Am. 61, 40–43 (1971).
    [CrossRef]
  5. S. Choudhary, L. B. Felsen, “Asymptotic theory for inhomogeneous waves,” IEEE Trans. Antennas Propag. 21, 827–842 (1973).
    [CrossRef]
  6. Yu. A. Kravtsov, Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, Berlin, 1990), Chap. 2.
  7. Y. A. Kravtsov, L. A. Ostrovsky, N. S. Stepanov, “Geometrical optics of inhomogeneous and nonstationary dispersive media,” Proc. IEEE 62, 1492–1510 (1974).
    [CrossRef]
  8. M. Kline, I. W. Kay, Electromagnetic Theory and Geometrical Optics (Interscience, New York, 1965), Chap. 6.
  9. L. B. Felsen, “Rays, dispersion surfaces and their uses for radiation and diffraction problems,” SIAM Rev. 12, 425–448 (1970).
    [CrossRef]
  10. H. Guo, X. Deng, “Differential geometrical methods in the study of optical transmission (scalar theory). I. Static transmission case,” J. Opt. Soc. Am. A 12, 600–606 (1995).
    [CrossRef]
  11. H. Guo, X. Deng, “Differential geometrical methods in the study of optical transmission (scalar theory). II. Time-dependent transmission theory,” J. Opt. Soc. Am. A 12, 607–610 (1995).
    [CrossRef]
  12. F. John, Partial Differential Equations (Springer-Verlag, New York, 1978), Chap. 1.
  13. S. Choudhary, L. B. Felsen, “Analysis of Gaussian beam propagation and diffraction by inhomogeneous wave tracking,” Proc. IEEE 62, 1530–1541 (1974).
    [CrossRef]
  14. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), Chap. 7.
  15. D. Marcuse, “Pulse distortion in single-mode fibers,” Appl. Opt. 19, 1653–1660 (1980).
    [CrossRef] [PubMed]

1995 (2)

1980 (1)

1974 (2)

Y. A. Kravtsov, L. A. Ostrovsky, N. S. Stepanov, “Geometrical optics of inhomogeneous and nonstationary dispersive media,” Proc. IEEE 62, 1492–1510 (1974).
[CrossRef]

S. Choudhary, L. B. Felsen, “Analysis of Gaussian beam propagation and diffraction by inhomogeneous wave tracking,” Proc. IEEE 62, 1530–1541 (1974).
[CrossRef]

1973 (1)

S. Choudhary, L. B. Felsen, “Asymptotic theory for inhomogeneous waves,” IEEE Trans. Antennas Propag. 21, 827–842 (1973).
[CrossRef]

1971 (1)

1970 (1)

L. B. Felsen, “Rays, dispersion surfaces and their uses for radiation and diffraction problems,” SIAM Rev. 12, 425–448 (1970).
[CrossRef]

1962 (1)

1911 (1)

A. Sommerfeld, J. Runge, “Anwendung der Vektorrechnung auf die Grundlagen der Geometrischen Optik,” Ann. Phys. (Leipzig) 35, 277–298 (1911).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1975), Chap. 3.

Choudhary, S.

S. Choudhary, L. B. Felsen, “Analysis of Gaussian beam propagation and diffraction by inhomogeneous wave tracking,” Proc. IEEE 62, 1530–1541 (1974).
[CrossRef]

S. Choudhary, L. B. Felsen, “Asymptotic theory for inhomogeneous waves,” IEEE Trans. Antennas Propag. 21, 827–842 (1973).
[CrossRef]

Deng, X.

Felsen, L. B.

S. Choudhary, L. B. Felsen, “Analysis of Gaussian beam propagation and diffraction by inhomogeneous wave tracking,” Proc. IEEE 62, 1530–1541 (1974).
[CrossRef]

S. Choudhary, L. B. Felsen, “Asymptotic theory for inhomogeneous waves,” IEEE Trans. Antennas Propag. 21, 827–842 (1973).
[CrossRef]

L. B. Felsen, “Rays, dispersion surfaces and their uses for radiation and diffraction problems,” SIAM Rev. 12, 425–448 (1970).
[CrossRef]

Guo, H.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), Chap. 7.

John, F.

F. John, Partial Differential Equations (Springer-Verlag, New York, 1978), Chap. 1.

Kay, I. W.

M. Kline, I. W. Kay, Electromagnetic Theory and Geometrical Optics (Interscience, New York, 1965), Chap. 6.

Keller, J. B.

Kline, M.

M. Kline, I. W. Kay, Electromagnetic Theory and Geometrical Optics (Interscience, New York, 1965), Chap. 6.

Kravtsov, Y. A.

Y. A. Kravtsov, L. A. Ostrovsky, N. S. Stepanov, “Geometrical optics of inhomogeneous and nonstationary dispersive media,” Proc. IEEE 62, 1492–1510 (1974).
[CrossRef]

Marcuse, D.

Ostrovsky, L. A.

Y. A. Kravtsov, L. A. Ostrovsky, N. S. Stepanov, “Geometrical optics of inhomogeneous and nonstationary dispersive media,” Proc. IEEE 62, 1492–1510 (1974).
[CrossRef]

Runge, J.

A. Sommerfeld, J. Runge, “Anwendung der Vektorrechnung auf die Grundlagen der Geometrischen Optik,” Ann. Phys. (Leipzig) 35, 277–298 (1911).
[CrossRef]

Sommerfeld, A.

A. Sommerfeld, J. Runge, “Anwendung der Vektorrechnung auf die Grundlagen der Geometrischen Optik,” Ann. Phys. (Leipzig) 35, 277–298 (1911).
[CrossRef]

Stepanov, N. S.

Y. A. Kravtsov, L. A. Ostrovsky, N. S. Stepanov, “Geometrical optics of inhomogeneous and nonstationary dispersive media,” Proc. IEEE 62, 1492–1510 (1974).
[CrossRef]

Streifer, W.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1975), Chap. 3.

Ann. Phys. (Leipzig) (1)

A. Sommerfeld, J. Runge, “Anwendung der Vektorrechnung auf die Grundlagen der Geometrischen Optik,” Ann. Phys. (Leipzig) 35, 277–298 (1911).
[CrossRef]

Appl. Opt. (1)

IEEE Trans. Antennas Propag. (1)

S. Choudhary, L. B. Felsen, “Asymptotic theory for inhomogeneous waves,” IEEE Trans. Antennas Propag. 21, 827–842 (1973).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Proc. IEEE (2)

Y. A. Kravtsov, L. A. Ostrovsky, N. S. Stepanov, “Geometrical optics of inhomogeneous and nonstationary dispersive media,” Proc. IEEE 62, 1492–1510 (1974).
[CrossRef]

S. Choudhary, L. B. Felsen, “Analysis of Gaussian beam propagation and diffraction by inhomogeneous wave tracking,” Proc. IEEE 62, 1530–1541 (1974).
[CrossRef]

SIAM Rev. (1)

L. B. Felsen, “Rays, dispersion surfaces and their uses for radiation and diffraction problems,” SIAM Rev. 12, 425–448 (1970).
[CrossRef]

Other (5)

F. John, Partial Differential Equations (Springer-Verlag, New York, 1978), Chap. 1.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), Chap. 7.

M. Kline, I. W. Kay, Electromagnetic Theory and Geometrical Optics (Interscience, New York, 1965), Chap. 6.

Yu. A. Kravtsov, Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, Berlin, 1990), Chap. 2.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1975), Chap. 3.

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Equations (28)

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2x2-n02c22t2E(x, t)=0,
E(x, y, z, t)=eˆϕ(x, y, z, t)exp[-ik0L(x, y, z, t)],
·(ϕ2L)+n0ictϕ2 n0Lict=0,
(L)2+n0icLt2-1k02ϕ2ϕ+n0ic2 2ϕt2=0,
=xˆ x+yˆ y+zˆ z.
·(ϕ2L)=0,
=xxˆ+yyˆ+zzˆ+n0icttˆ
(L)2=n02c2Lt2.
k2=n02c2w2,
k·k=n02+c2w2ϕ2ϕ-n02c22ϕt2 w2c2=nG2 w2c2.
k·k=n02+c2w2ϕ(2ϕ) w2c2,
Hk2-n02+c2w2ϕ2ϕx2-n02c22ϕt2 w2c2=0.
dxdu=Hk=2k,
dkdu=-Hx=x1ϕ2ϕx2-n02c22ϕt2,
dtdu=-Hw=2n02c2w,
dwdu=Ht=-t1ϕ2ϕx2-n02c22ϕt2,
dxdt=-H/kH/w=c2n02kw,
dkdt=+H/xH/w=c22n02wx1ϕ2ϕx2-n02c22ϕt2,
dwdt=-H/tH/w=-c22n02wt1ϕ2ϕx2-n02c22ϕt2.
d2xdt2=c42n04w2x+kwt1ϕ2ϕx2-n02c22ϕt2.
vg=c2n02kwx=0=acn0
d2Xdt2=vg2(1-a2)22k2X1ϕ2ϕX2.
ϕ(X, 0)=A exp(-X2/σ02).
d2Xdt2=4vg2k2(1-a2)2 X0σ04.
X(t)=X01+2vg2(1-a2)2k2σ04t2,
X(t)=X01+4vg2(1-a2)2k2σ04t21/2,
σ=σ01+vg2 t2LD21/2,
ϕ(X, t)=Aσ0/σ exp(-X2/σ2),

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