Abstract

The formalism of coupled-mode theory, specialized to the continuum of radiation modes, allows us to extend the standard parabolic wave equation to include nonparaxial terms and vectorial effects, and, in particular, to generalize the nonlinear Schrödinger equation that describes propagation in the presence of an intensity-dependent refractive index.

© 1997 Optical Society of America

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Corrections

B. Crosignani, P. Di Porto, and A. Yariv, "Nonparaxial equation for linear and nonlinear optical propagation: errata," Opt. Lett. 22, 1820-1820 (1997)
https://www.osapublishing.org/ol/abstract.cfm?uri=ol-22-23-1820

References

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  1. M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
    [CrossRef]
  2. M. D. Feit and J. A. Fleck, J. Opt. Soc. Am. B 5, 633 (1988).
    [CrossRef]
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    [CrossRef] [PubMed]
  4. J. M. Soto-Crespo and N. Akhmediev, Opt. Commun. 101, 223 (1993).
    [CrossRef]
  5. G. Fibich, Phys. Rev. Lett. 76, 4356 (1996).
    [CrossRef] [PubMed]
  6. A. Yu. Savchencko and B. Ya. Zel’dovich, J. Opt. Soc. Am. B 13, 273 (1996).
    [CrossRef]
  7. S. Chi and Q. Guo, Opt. Lett. 20, 1598 (1995).
    [CrossRef] [PubMed]
  8. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), Chap. 3.
  9. B. Crosignani and A. Yariv, J. Opt. Soc. Am. 1, 1034 (1984).
    [CrossRef]
  10. D. Marcuse, Bell Syst. Tech. J. 54, 985 (1975).
    [CrossRef]
  11. The set of equations given by Eq.  (8) can be generalized in a straightforward way to include backward-propagating modes, the resulting set of equations being completely equivalent to Maxwell’s equation and still first order in z.
  12. B. Crosignani, A. Cutolo, and P. Di Porto, J. Opt. Soc. Am. 72, 1136 (1982).
    [CrossRef]

1996 (2)

1995 (1)

1993 (2)

1988 (1)

1984 (1)

B. Crosignani and A. Yariv, J. Opt. Soc. Am. 1, 1034 (1984).
[CrossRef]

1982 (1)

1975 (2)

D. Marcuse, Bell Syst. Tech. J. 54, 985 (1975).
[CrossRef]

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

Akhmediev, N.

Ankiewicz, A.

Chi, S.

Crosignani, B.

Cutolo, A.

Di Porto, P.

Feit, M. D.

Fibich, G.

G. Fibich, Phys. Rev. Lett. 76, 4356 (1996).
[CrossRef] [PubMed]

Fleck, J. A.

Guo, Q.

Lax, M.

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

Louisell, W. H.

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

Marcuse, D.

D. Marcuse, Bell Syst. Tech. J. 54, 985 (1975).
[CrossRef]

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), Chap. 3.

McKnight, W. B.

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

Soto-Crespo, J. M.

Ya. Zel’dovich, B.

Yariv, A.

B. Crosignani and A. Yariv, J. Opt. Soc. Am. 1, 1034 (1984).
[CrossRef]

Yu. Savchencko, A.

Bell Syst. Tech. J. (1)

D. Marcuse, Bell Syst. Tech. J. 54, 985 (1975).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Opt. Commun. (1)

J. M. Soto-Crespo and N. Akhmediev, Opt. Commun. 101, 223 (1993).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. A (1)

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

Phys. Rev. Lett. (1)

G. Fibich, Phys. Rev. Lett. 76, 4356 (1996).
[CrossRef] [PubMed]

Other (2)

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), Chap. 3.

The set of equations given by Eq.  (8) can be generalized in a straightforward way to include backward-propagating modes, the resulting set of equations being completely equivalent to Maxwell’s equation and still first order in z.

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Equations (19)

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z+i2k2A=-ikn1δnA,
2E+2E·lnn+k02n2E=0,
Eξ, 1;rexp-iβξz=N1 exp-iξ·rxˆ-ξx/βξzˆ×exp-iβξz,
Eξ, 2;rexp-iβξz=N2 exp-iξ·r×ξxξy/βξxˆ-βξ+ξx2/βξyˆ+ξyzˆexp-iβξz,
-+-+drzˆ·Eξ, σ;r·H*ξ, σ;r=δ2ξ-ξδσ,σ, σ,  σ=1, 2.
·Eξ, σ;rexp-iβξz=0.
Er, z, t=σdξEξ, σ;r×expiωt-iβξzcξ, σ;z=σexpiωt-ikzdξEξ, σ;r×exp-iβξ-kzcξ, σ;zσexpiωt-ikzAσr, zexpiωt-ikzAr, z,
ddzcξ, σ;z=-iω0n1σdξdr×expiβξ-βξzcξ, σ;z·E*ξ, σ;r·δnr, z:Eξ, σ;r,
z+i2k2-i8k34+Aσr, zLAσ=-iω0n1drdξEξ, σ;r×E*ξ, σ;rE*ξ, σ;r:δnr, z:Ar, z.
Eξ, 1;rE*ξ, 1;r=N12[10-ξx/βξ000-ξx/βξ0ξx2/βξ2]×exp-ir-r·ξ,
Eξ, 1;rE*ξ, 1;r=12π2μ0/1×[1+ξy2-ξx2/2k2+0-ξx/k+000-ξx/k+0ξx2/k2+]×exp-ir-r·ξ,
dξexp-ir-r·ξξx,ym,  m=1, 2,
δ2r-r=12π2dξexp-ir-r·ξ.
LAx=-ik/n1δn:Ax+1/n1/xδn:Az-i1/kn12/xyδn:Ay-i1/2kn12/x2-2/y2δn:Ax,
LAy=-ik/n1δn:Ay+1/n1/yδn:Az-i1/kn12/xyδn:Ax-i1/2kn12/y2-2/x2δn:Ay,
LAz=1/n1xδn:Ax+1/n1yδn:Ay+i1/kn12δn:Az,
Az=-ikμ0ω2zz·A-zxzzAx-zyzzAy,
δn=23n2[E2+1/2Ex21/2EyEx*1/2EzEx*1/2ExEy*E2+1/2Ey21/2EzEy*1/2ExEz*1/2EyEz*E2+1/2Ez2],
Z+i22X2+2Y2-i84X4+4Y4u+iu2u=-i22X2-2Y2u2u-i3uX2u-i3u22uX2-u22u*X2,

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