Abstract

A recent study [J. Opt. Soc. Am. A 15, 2720 (1998)] showed that the eikonal equation can be generalized to include higher-order terms that can describe the dispersion of a pulse in a linear medium. In this companion paper, the formalism of this generalized eikonal approximation is investigated for stationary waves in both linear and nonlinear media. A local refractive index can be defined that includes higher-order terms in the form of the second derivatives of the amplitude of the wave. It is shown that wave phenomena such as diffraction, self-focusing, and self-trapping of a finite beam in linear and nonlinear media can be derived naturally as a result of the differences in the local refractive indices that arise from the spatial variation of the wave amplitude and the nonlinearity of the medium. This formalism can be readily extended to the study of other complex electromagnetic wave-propagation phenomena in both linear and nonlinear media.

© 1998 Optical Society of America

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References

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  1. B. C. Quek, B. R. Wong, 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]
  2. 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]
  3. 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]
  4. J. Durnin, “Exact solution for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
    [CrossRef]
  5. R. Y. Chiao, E. Garmire, C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
    [CrossRef]
  6. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1965).
  7. A. Yariv, Optical Electronics (Saunders, Philadelphia, Pa., 1991).
  8. P. L. Kelly, “Self-focusing of optical beams,” Phys. Rev. Lett. 15, 1005–1008 (1965).
    [CrossRef]
  9. H. A. Haus, “Higher order trapped light beam solutions,” Appl. Phys. Lett. 8, 128–129 (1966).
    [CrossRef]
  10. D. Pohl, “Vectorial theory of self-trapped light beams,” Opt. Commun. 2, 305–308 (1970).
    [CrossRef]
  11. V. E. Zakharov, A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).
  12. S. A. Akhmanov, A. P. Sukhorukov, R. V. Khokhlov, “Self-focusing and self-trapping of intense light beams in a nonlinear medium,” Sov. Phys. JETP 23, 1025–1033 (1966).

1998 (1)

1995 (2)

1987 (1)

1972 (1)

V. E. Zakharov, A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

1970 (1)

D. Pohl, “Vectorial theory of self-trapped light beams,” Opt. Commun. 2, 305–308 (1970).
[CrossRef]

1966 (2)

H. A. Haus, “Higher order trapped light beam solutions,” Appl. Phys. Lett. 8, 128–129 (1966).
[CrossRef]

S. A. Akhmanov, A. P. Sukhorukov, R. V. Khokhlov, “Self-focusing and self-trapping of intense light beams in a nonlinear medium,” Sov. Phys. JETP 23, 1025–1033 (1966).

1965 (1)

P. L. Kelly, “Self-focusing of optical beams,” Phys. Rev. Lett. 15, 1005–1008 (1965).
[CrossRef]

1964 (1)

R. Y. Chiao, E. Garmire, C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
[CrossRef]

Akhmanov, S. A.

S. A. Akhmanov, A. P. Sukhorukov, R. V. Khokhlov, “Self-focusing and self-trapping of intense light beams in a nonlinear medium,” Sov. Phys. JETP 23, 1025–1033 (1966).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1965).

Chiao, R. Y.

R. Y. Chiao, E. Garmire, C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
[CrossRef]

Deng, X.

Durnin, J.

Garmire, E.

R. Y. Chiao, E. Garmire, C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
[CrossRef]

Guo, H.

Haus, H. A.

H. A. Haus, “Higher order trapped light beam solutions,” Appl. Phys. Lett. 8, 128–129 (1966).
[CrossRef]

Kelly, P. L.

P. L. Kelly, “Self-focusing of optical beams,” Phys. Rev. Lett. 15, 1005–1008 (1965).
[CrossRef]

Khokhlov, R. V.

S. A. Akhmanov, A. P. Sukhorukov, R. V. Khokhlov, “Self-focusing and self-trapping of intense light beams in a nonlinear medium,” Sov. Phys. JETP 23, 1025–1033 (1966).

Low, K. S.

Pohl, D.

D. Pohl, “Vectorial theory of self-trapped light beams,” Opt. Commun. 2, 305–308 (1970).
[CrossRef]

Quek, B. C.

Shabat, A. B.

V. E. Zakharov, A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Sukhorukov, A. P.

S. A. Akhmanov, A. P. Sukhorukov, R. V. Khokhlov, “Self-focusing and self-trapping of intense light beams in a nonlinear medium,” Sov. Phys. JETP 23, 1025–1033 (1966).

Townes, C. H.

R. Y. Chiao, E. Garmire, C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1965).

Wong, B. R.

Yariv, A.

A. Yariv, Optical Electronics (Saunders, Philadelphia, Pa., 1991).

Zakharov, V. E.

V. E. Zakharov, A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Appl. Phys. Lett. (1)

H. A. Haus, “Higher order trapped light beam solutions,” Appl. Phys. Lett. 8, 128–129 (1966).
[CrossRef]

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

Opt. Commun. (1)

D. Pohl, “Vectorial theory of self-trapped light beams,” Opt. Commun. 2, 305–308 (1970).
[CrossRef]

Phys. Rev. Lett. (2)

R. Y. Chiao, E. Garmire, C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
[CrossRef]

P. L. Kelly, “Self-focusing of optical beams,” Phys. Rev. Lett. 15, 1005–1008 (1965).
[CrossRef]

Sov. Phys. JETP (2)

V. E. Zakharov, A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

S. A. Akhmanov, A. P. Sukhorukov, R. V. Khokhlov, “Self-focusing and self-trapping of intense light beams in a nonlinear medium,” Sov. Phys. JETP 23, 1025–1033 (1966).

Other (2)

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1965).

A. Yariv, Optical Electronics (Saunders, Philadelphia, Pa., 1991).

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Figures (2)

Fig. 1
Fig. 1

Propagation of a diffracted Gaussian beam with an initially plane wave front.

Fig. 2
Fig. 2

Propagation of a self-focusing beam with an initially plane wave front.

Equations (39)

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2E+k02n2E=0,
E=ϕ(r)exp[ik0L(r)]eˆ,
(L)2=n2+1k02ϕ2ϕ,
·(ϕ2L)=0.
(L)2=n2,
(L)2=nG2,
nG2=n2+1k02ϕ2ϕ.
E=zˆϕ(r)exp[ik0L(r)].
H=(ik0μ/)-1×E.
P=Re(E×H*)=ncμ(ϕ2L).
υg=cn02L.
(nG2)=2nG ξ(nGξ)=nG2ξξ+2nG2 ξξ.
(nG2)=ξˆ ξnG2+τˆ τnG2.
ξˆξ=τˆK,
K=12nG2nG2τ.
ϕ=A exp(-r2/σ02),
nG2=n2+1k024σ02r2σ02-1,
K1n24k02σ04r,
ω2(z)=ω021+4z2k02n2σ04.
z0=k0nσ022,
L=zˆ Lz=zˆnG,
z(φ2nG)=0.
ϕ=f(r, θ)nG1/2,
2fr2+1rfr+1r22fθ2-a2f=0,
n=n0+n2ϕ2+n4ϕ4+.
n0n2ϕ2,n4ϕ4,;
n2n02+2n0n2ϕ2,
nG2n02+2n0n2ϕ2+1k02ϕ2ϕ,
nG2=n02+2n0n2ϕ2+1k024σ02r2σ02-1,
K1k028σ04r-8n0n2σ02ϕ2r.
L=zˆ Lz=zˆnG.
2fr2+1rfr+1r22fθ2=a2f-b2f3,
2fr2+1rfr-a2f+b2f3=0,
2fy2-a2f+b2f3=0.
n02<nG2<n02+2n0n2ϕmax2,
f(y)=2absech a(y-y0)
ϕ(y)=nG2-n02n0n21/2 sech a(y-y0),
nG2=n02+n0n2ϕ2(y=y0).
1k022ϕϕ+2n0n2ϕ2(y)=nG2-n02=n0n2ϕ2(y0).

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