Abstract

The coupled differential recurrence equations for the corrections to the paraxial approximation solutions in transversely nonuniform refractive-index media are established in terms of the perturbation method. All the corrections (including the longitudinal field corrections) to the paraxial approximation solutions are presented in the weak-guidance approximation. As a concrete application, the first-order longitudinal field correction and the second-order transverse field correction to the paraxial approximation of a Gaussian beam propagating in a transversely quadratic refractive index medium are analytically investigated.

© 1999 Optical Society of America

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References

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  1. M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
    [CrossRef]
  2. G. P. Agrawal, D. N. Pattanayak, “Gaussian beam propagation beyond the paraxial approximation,” J. Opt. Soc. Am. 69, 575–578 (1979).
    [CrossRef]
  3. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
    [CrossRef]
  4. M. Couture, P. A. Belanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355–359 (1981).
    [CrossRef]
  5. G. P. Agrawal, M. Lax, “Free-space propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
    [CrossRef]
  6. T. Takenaka, M. Yokota, O. Fukumitsu, “Propagation of light beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 2, 826–829 (1985).
    [CrossRef]
  7. S. Nemoto, “Nonparaxial Gaussian beams,” Appl. Opt. 29, 1940–1946 (1990).
    [CrossRef] [PubMed]
  8. C. J. R. Sheppard, S. Saghafi, “Beam modes beyond the paraxial approximation: a scalar treatment,” Phys. Rev. A 57, 2971–2979 (1998).
    [CrossRef]
  9. A. Wünsche, “Transition from the paraxial approximation to exact solutions of the wave equation and application to Gaussian beams,” J. Opt. Soc. Am. A 9, 765–774 (1992).
    [CrossRef]
  10. G. W. Forbes, D. J. Butler, R. L. Gordon, A. A. Asatryan, “Algebraic corrections for paraxial wave fields,” J. Opt. Soc. Am. A 14, 3300–3315 (1997).
    [CrossRef]
  11. Q. Cao, X. Deng, “Corrections to the paraxial approximation of an arbitrary free-propagation beam,” J. Opt. Soc. Am. A 15, 1144–1148 (1998).
    [CrossRef]
  12. See, for example, C. Vassallo, “Limitations of the wide-angle beam propagation method in nonuniform systems,” J. Opt. Soc. Am. A 13, 761–770 (1996).
    [CrossRef]
  13. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, UK, 1975), p. 10.
  14. L. W. Casperson, A. Yariv, “Gain and dispersion focusing in a high gain laser,” Appl. Opt. 11, 462–466 (1972).
    [CrossRef] [PubMed]
  15. L. W. Casperson, “Gaussian light beams in inhomogeneous media,” Appl. Opt. 12, 2434–2441 (1973).
    [CrossRef] [PubMed]
  16. H. Weber, Laser Resonators, 1st ed. (Huazhong Institute of Technology, Wuhan, China, 1983), p. 26.
  17. A. W. Snyder, J. D. Love, Optical Waveguide Theory, 1st ed. (Chapman & Hall, London, 1983), pp. 591 and 226.

1998 (2)

C. J. R. Sheppard, S. Saghafi, “Beam modes beyond the paraxial approximation: a scalar treatment,” Phys. Rev. A 57, 2971–2979 (1998).
[CrossRef]

Q. Cao, X. Deng, “Corrections to the paraxial approximation of an arbitrary free-propagation beam,” J. Opt. Soc. Am. A 15, 1144–1148 (1998).
[CrossRef]

1997 (1)

1996 (1)

1992 (1)

1990 (1)

1985 (1)

1983 (1)

G. P. Agrawal, M. Lax, “Free-space propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

1981 (1)

M. Couture, P. A. Belanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355–359 (1981).
[CrossRef]

1979 (2)

1975 (1)

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

1973 (1)

1972 (1)

Agrawal, G. P.

G. P. Agrawal, M. Lax, “Free-space propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

G. P. Agrawal, D. N. Pattanayak, “Gaussian beam propagation beyond the paraxial approximation,” J. Opt. Soc. Am. 69, 575–578 (1979).
[CrossRef]

Asatryan, A. A.

Belanger, P. A.

M. Couture, P. A. Belanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355–359 (1981).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, UK, 1975), p. 10.

Butler, D. J.

Cao, Q.

Casperson, L. W.

Couture, M.

M. Couture, P. A. Belanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355–359 (1981).
[CrossRef]

Davis, L. W.

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
[CrossRef]

Deng, X.

Forbes, G. W.

Fukumitsu, O.

Gordon, R. L.

Lax, M.

G. P. Agrawal, M. Lax, “Free-space propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Louisell, W. H.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Love, J. D.

A. W. Snyder, J. D. Love, Optical Waveguide Theory, 1st ed. (Chapman & Hall, London, 1983), pp. 591 and 226.

McKnight, W. B.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Nemoto, S.

Pattanayak, D. N.

Saghafi, S.

C. J. R. Sheppard, S. Saghafi, “Beam modes beyond the paraxial approximation: a scalar treatment,” Phys. Rev. A 57, 2971–2979 (1998).
[CrossRef]

Sheppard, C. J. R.

C. J. R. Sheppard, S. Saghafi, “Beam modes beyond the paraxial approximation: a scalar treatment,” Phys. Rev. A 57, 2971–2979 (1998).
[CrossRef]

Snyder, A. W.

A. W. Snyder, J. D. Love, Optical Waveguide Theory, 1st ed. (Chapman & Hall, London, 1983), pp. 591 and 226.

Takenaka, T.

Vassallo, C.

Weber, H.

H. Weber, Laser Resonators, 1st ed. (Huazhong Institute of Technology, Wuhan, China, 1983), p. 26.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, UK, 1975), p. 10.

Wünsche, A.

Yariv, A.

Yokota, M.

Appl. Opt. (3)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

Phys. Rev. A (5)

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
[CrossRef]

M. Couture, P. A. Belanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355–359 (1981).
[CrossRef]

G. P. Agrawal, M. Lax, “Free-space propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

C. J. R. Sheppard, S. Saghafi, “Beam modes beyond the paraxial approximation: a scalar treatment,” Phys. Rev. A 57, 2971–2979 (1998).
[CrossRef]

Other (3)

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, UK, 1975), p. 10.

H. Weber, Laser Resonators, 1st ed. (Huazhong Institute of Technology, Wuhan, China, 1983), p. 26.

A. W. Snyder, J. D. Love, Optical Waveguide Theory, 1st ed. (Chapman & Hall, London, 1983), pp. 591 and 226.

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Figures (1)

Fig. 1
Fig. 1

On-axis intensity distributions of (a) zeroth-order field ϕG(0)(0, 0, z) and (b) corrected transverse field ϕG(0)(0, 0, z)+ϕG(2)×(0, 0, z). The parameters are λ=0.6328 μm, n0=1.5, w0=1.2656 μm, q0=-i23.856 μm, and Γ=7901.39 m-1.

Equations (43)

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2E-μc2 2Et2+( log μ)××E+(E log )
=0,
2E+μk2E+( log μ)××E+(E log )
=0.
2u+(n02+n12)k2u=0,
i2n0 k ϕz+Fϕ+2ϕz2=0,
i2n0k ϕ(0)z+Fϕ(0)=0,
ϕ=ϕ(0)+m=1ϕ(2m),
i2n0 k ϕ(2m)z+Fϕ(2m)=-2ϕ(2m-2)z2(m1).
ϕ(2m)=i2n0km 1m! 2mz2m zm-2m 2m-1z2m-1 zm-1ϕ(0)
(m1),
1m! 2mz2m [zmϕ(0)]
=p=0m (2m)!(m-p)!(m+p)!p! zp m+p zm+p ϕ(0),
2(m-1)! 2m-1z2m-1 [zm-1ϕ(0)]
=2p=0m-1 (2m-1)!(m-p-1)!(m+p)!p! zp m+pzm+p ϕ(0),
ϕ(2m)=i2n0km 1m! p=1m (2m)!(m-1)!(m+p)!(m-p)!(p-1)!×zp m+pϕ(0)zm+p.
ϕ(2m)=1m! p=1m (2m)!(m-1)!(m+p)!(m-p)!(p-1)!
×i2n0k2m+pzpF m+pϕ(0).
ϕ(2)=iz2n0k 2ϕ(0) z2
ϕ(2)=-iz8n03k3 F 2ϕ(0).
x Ex+Exx+Ezz=0.
Exx+Ex x+Ezz=0.
2Ez+(n02+n12)k2Ez=0.
Ezz=in0k1+GEz,
1+G=1+m=1(-1)m+1 (2m-3)!!(2m)!! Gm,
Ez=exp(in0 k z)m=0ψ(2m+1).
Ezz=in0 k exp(inokz)1+m=1(-1)m+1×(2m-3)!!(2m)!! Gmm=0ψ(2m+1).
ϕ(0)x+ϕ(0) x+in0kψ(1)=0,
ϕ(2m)x+ϕ(2m) x+in0kψ(2m+1)+in0 kj=1m(-1)j+1
×(2j-3)!!(2j)!! Gjψ(2m-2j+1)=0,
1+m=1(-1)m+1 (2m-3)!!(2m)!! Gmm=0ψ(2m+1)
=ψ(1)+l=1ψ(2l+1)+j=1l(-1)j+1
×(2j-3)!!(2j)!! Gjψ(2l-2j+1).
ψ(1)=in0 k ϕ(0)x+ϕ(0) x,
ψ(2m+1)=in0 k ϕ(2m)x+ϕ(2m) x-j=1m(-1) j+1 (2j-3)!!(2j)!! G jψ(2m-2j+1).
κ=ϕ(0) x ϕ(0)x-11
2Ez+k2Ez+1 x Exz=0.
k2Ez1 x Exz.
ϕG(0)=q0S(z) expikn0r22q(z),
q=q0 cos(Γz)+Γ-1 sin(Γz)-q0Γ sin(Γz)+cos(Γz),
S=Γ-1 sin(Γz)+q0 cos(Γz),
ψG(1)=-xq+i2Γ2xn0k(1-Γ2r2)ϕG(0),
ϕG(2)=-iz8n03k ϕG(0)k2n04r4q4 (1+2Γ2q2+Γ4q4)-i8kn03r2q3 (Γ2q2+1)-4n02q2 (Γ2q2+2),

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