Abstract

The generalized eikonal formalism has recently been shown able to describe wave phenomena, which cannot be obtained by classical geometrical optics. However, whether this generalized eikonal equation can be used to describe interference effects due to linear superposition of waves still remains unanswered. It is shown here that the generalized eikonal formalism self-consistently satisfies the superposition principle and allows the interference effect. First, it is shown analytically to comply with the superposition principle. It is also shown numerically that the formalism satisfies the linear superposition of waves. The computed trajectories provide quantitative information on optical wave propagation as well as physical insights in the formation of interference fringes. Thus the generalized eikonal formalism is expected to be readily developed for the investigation of optical beam transformation in much the same manner as in the geometrical optics.

© 2002 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics (Pergamon, London, 1975), Chap. 3.
  2. Y. A. Kravtsov, L. A. Ostrovsky, and N. S. Stepanov, “Geometrical optics of inhomogeneous and nonstationary dispersive media,” Proc. IEEE 62, 1492–1510 (1974).
    [CrossRef]
  3. M. Kline and I. W. Kay, Electromagnetic Theory and Geometrical Optics (Wiley, New York, 1965), Chap. 6.
  4. J. B. Keller, “Geometrical theory of diffraction,” J. Opt. Soc. Am. 52, 116–130 (1962).
    [CrossRef] [PubMed]
  5. J. B. Keller and W. Streifer, “Complex rays with application to Gaussian beams,” J. Opt. Soc. Am. 61, 40–43 (1971).
    [CrossRef]
  6. Yu. A. Kravtsov and Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, Berlin, 1990), Chap. 2.
  7. H. Guo and 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]
  8. H. Guo and 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]
  9. B. C. Quek, B. R. Wong, and K. S. Low, “Generalized eikonal approximation. 1. Propagation of an electromagnetic pulse in a linear dispersive medium,” J. Opt. Soc. Am. A 15, 2720–2724 (1998).
    [CrossRef]
  10. S. C. Yap, B. C. Quek, and K. S. Low, “Generalized eikonal approximation. 2. Propagation of stationary electromagnetic waves in linear and nonlinear media,” J. Opt. Soc. Am. A 15, 2725–2729 (1998).
    [CrossRef]
  11. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), Chap. 6.

1998 (2)

1995 (2)

1974 (1)

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

1971 (1)

1962 (1)

Deng, X.

Guo, H.

Keller, J. B.

Kravtsov, Y. A.

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

Low, K. S.

Ostrovsky, L. A.

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

Quek, B. C.

Stepanov, N. S.

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

Streifer, W.

Wong, B. R.

Yap, S. C.

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (4)

Proc. IEEE (1)

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

Other (4)

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

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

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), Chap. 6.

M. Born and E. Wolf, Principles of Optics (Pergamon, London, 1975), Chap. 3.

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Figures (6)

Fig. 1
Fig. 1

Ray trace of two Gaussian beams with σ0=1 mm and separated by 3σ0.

Fig. 2
Fig. 2

Interference patterns at various positions produced by two Gaussian beams with σ0=1.5 mm and separated by 3 mm.

Fig. 3
Fig. 3

Comparison of interference produced by Gaussian beams with various σ0.

Fig. 4
Fig. 4

Comparison of interference produced by Gaussian beams in media with different indices of refraction.

Fig. 5
Fig. 5

Ray trace of two Gaussian beams with σ0=0.8 mm in a medium with n0=1.5-0.1x2.

Fig. 6
Fig. 6

Ray trace of two Gaussian beams with σ0=0.8 mm in a medium with n0=1.5+0.1x2.

Tables (2)

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Table 1 Comparison Amplitude Obtained by Fresnel Integral and Generalized Eikonal Formalism

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Table 2 Comparison of Amplitudes at z=4 m Obtained by Different Methods

Equations (12)

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(L)2=n02+1k02φ2φ,
dds(L)=|L|,
·(φ2L)=0,
H=ε0μ0 L×(eˆφ)+1ik0×(eˆφ)exp(ik0L).
Re{P}=Re12 E×H*=12 ε0μ0 (φ2L).
E=1εr (L)2(eˆφ)-1k022(eˆφ)+ik0[2L(eˆφ)+2(L·)(eˆφ)]exp(ik0L).
(L1,2)2=n02+1k02φ1,22φ1,2
·(φ1,22L1,2)=0.
2Φ3=2(Φ1+Φ2)=-n02k02Φ3
(L3)2=n02+1k02φ32φ3,
·(φ32L3)=0.
Xmin=14 λx0z.

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