Abstract

The standard scalar paraxial parabolic (Fock–Leontovich) propagation equation is generalized to include all-order nonparaxial corrections in the significant case of a tensorial refractive-index perturbation on a homogeneous isotropic background. In the resultant equation, each higher-order nonparaxial term (associated with diffraction in homogeneous space and scaling as the ratio between beam waist and diffraction length) possesses a counterpart (associated with the refractive-index perturbation) that allows one to preserve the vectorial nature of the problem (·E0). The tensorial character of the refractive-index variation is shown to play a particularly relevant role whenever the tensor elements δnxz and δnyz (z is the propagation direction) are not negligible. For this case, an application to elasto-optically induced optical activity and to nonlinear propagation in the presence of the optical Kerr effect is presented.

© 2000 Optical Society of America

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References

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  1. M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
    [CrossRef]
  2. See, e.g., D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972).
  3. Yu. Savchencko and B. Ya. Zel’dovich, “Wave propagation in a guiding structure: one step beyond the paraxial approximation,” J. Opt. Soc. Am. B 13, 273–281 (1996).
    [CrossRef]
  4. M. D. Feit and J. A. Fleck, Jr., “Beam nonparaxiality, filament formation, and beam breakup in the self-focusing of optical beams,” J. Opt. Soc. Am. B 5, 633–640 (1988).
    [CrossRef]
  5. N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, “Does the NLSE correctly describe beam propagation?” Opt. Lett. 18, 411–413 (1993).
    [CrossRef] [PubMed]
  6. S. Chi and Q. Guo, “Vector theory of self-focusing of an optical beam in Kerr media,” Opt. Lett. 20, 1598–1600 (1995).
    [CrossRef] [PubMed]
  7. G. Fibich, “Small beam nonparaxiality arrests self-focusing of optical beams,” Phys. Rev. Lett. 76, 4356–4359 (1996).
    [CrossRef] [PubMed]
  8. B. Crosignani, P. Di Porto, and A. Yariv, “Nonparaxial equation for linear and nonlinear optical propagation,” Opt. Lett. 11, 778–780 (1997).
    [CrossRef]
  9. S. Blair and K. Wagner, “(2+1)-D propagation of spatio-temporal solitary waves including higher-order corrections,” Opt. Quantum Electron. 30, 697–737 (1998).
    [CrossRef]
  10. B. Crosignani, A. Cutolo, and P. Di Porto, “Coupled-mode theory of nonlinear propagation in multimode and single-mode fiber: envelope solitons and self-confinement,” J. Opt. Soc. Am. 72, 1136–1141 (1982).
    [CrossRef]
  11. V. S. Liberman and B. Ya. Zel’dovich, “Birefringence by a smoothly inhomogeneous locally isotropic medium,” Phys. Rev. E 49, 2389–2396 (1994).
    [CrossRef]
  12. A. Yu. Savchenko and B. Ya. Zel’dovich, “Birefringence by a smoothly inhomogeneous locally isotropic medium: three-dimensional case,” Phys. Rev. E 50, 2287–2292 (1994).
    [CrossRef]
  13. B. Crosignani, P. Di Porto, and A. Yariv, “Coupled-mode theory and slowly-varying approximation in guided-wave optics,” Opt. Commun. 78, 237–239 (1990).
    [CrossRef]
  14. R. Ulrich and A. Simon, “Polarization optics of twisted single-mode fibers,” Appl. Opt. 18, 2241–2251 (1979).
    [CrossRef] [PubMed]

1998 (1)

S. Blair and K. Wagner, “(2+1)-D propagation of spatio-temporal solitary waves including higher-order corrections,” Opt. Quantum Electron. 30, 697–737 (1998).
[CrossRef]

1997 (1)

B. Crosignani, P. Di Porto, and A. Yariv, “Nonparaxial equation for linear and nonlinear optical propagation,” Opt. Lett. 11, 778–780 (1997).
[CrossRef]

1996 (2)

1995 (1)

1994 (2)

V. S. Liberman and B. Ya. Zel’dovich, “Birefringence by a smoothly inhomogeneous locally isotropic medium,” Phys. Rev. E 49, 2389–2396 (1994).
[CrossRef]

A. Yu. Savchenko and B. Ya. Zel’dovich, “Birefringence by a smoothly inhomogeneous locally isotropic medium: three-dimensional case,” Phys. Rev. E 50, 2287–2292 (1994).
[CrossRef]

1993 (1)

1990 (1)

B. Crosignani, P. Di Porto, and A. Yariv, “Coupled-mode theory and slowly-varying approximation in guided-wave optics,” Opt. Commun. 78, 237–239 (1990).
[CrossRef]

1988 (1)

1982 (1)

1979 (1)

1975 (1)

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

Akhmediev, N.

Ankiewicz, A.

Blair, S.

S. Blair and K. Wagner, “(2+1)-D propagation of spatio-temporal solitary waves including higher-order corrections,” Opt. Quantum Electron. 30, 697–737 (1998).
[CrossRef]

Chi, S.

Crosignani, B.

B. Crosignani, P. Di Porto, and A. Yariv, “Nonparaxial equation for linear and nonlinear optical propagation,” Opt. Lett. 11, 778–780 (1997).
[CrossRef]

B. Crosignani, P. Di Porto, and A. Yariv, “Coupled-mode theory and slowly-varying approximation in guided-wave optics,” Opt. Commun. 78, 237–239 (1990).
[CrossRef]

B. Crosignani, A. Cutolo, and P. Di Porto, “Coupled-mode theory of nonlinear propagation in multimode and single-mode fiber: envelope solitons and self-confinement,” J. Opt. Soc. Am. 72, 1136–1141 (1982).
[CrossRef]

Cutolo, A.

Di Porto, P.

B. Crosignani, P. Di Porto, and A. Yariv, “Nonparaxial equation for linear and nonlinear optical propagation,” Opt. Lett. 11, 778–780 (1997).
[CrossRef]

B. Crosignani, P. Di Porto, and A. Yariv, “Coupled-mode theory and slowly-varying approximation in guided-wave optics,” Opt. Commun. 78, 237–239 (1990).
[CrossRef]

B. Crosignani, A. Cutolo, and P. Di Porto, “Coupled-mode theory of nonlinear propagation in multimode and single-mode fiber: envelope solitons and self-confinement,” J. Opt. Soc. Am. 72, 1136–1141 (1982).
[CrossRef]

Feit, M. D.

Fibich, G.

G. Fibich, “Small beam nonparaxiality arrests self-focusing of optical beams,” Phys. Rev. Lett. 76, 4356–4359 (1996).
[CrossRef] [PubMed]

Fleck Jr., J. A.

Guo, Q.

Lax, M.

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

Liberman, V. S.

V. S. Liberman and B. Ya. Zel’dovich, “Birefringence by a smoothly inhomogeneous locally isotropic medium,” Phys. Rev. E 49, 2389–2396 (1994).
[CrossRef]

Louisell, W. H.

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

McKnight, W. B.

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

Savchencko, Yu.

Savchenko, A. Yu.

A. Yu. Savchenko and B. Ya. Zel’dovich, “Birefringence by a smoothly inhomogeneous locally isotropic medium: three-dimensional case,” Phys. Rev. E 50, 2287–2292 (1994).
[CrossRef]

Simon, A.

Soto-Crespo, J. M.

Ulrich, R.

Wagner, K.

S. Blair and K. Wagner, “(2+1)-D propagation of spatio-temporal solitary waves including higher-order corrections,” Opt. Quantum Electron. 30, 697–737 (1998).
[CrossRef]

Yariv, A.

B. Crosignani, P. Di Porto, and A. Yariv, “Nonparaxial equation for linear and nonlinear optical propagation,” Opt. Lett. 11, 778–780 (1997).
[CrossRef]

B. Crosignani, P. Di Porto, and A. Yariv, “Coupled-mode theory and slowly-varying approximation in guided-wave optics,” Opt. Commun. 78, 237–239 (1990).
[CrossRef]

Zel’dovich, B. Ya.

Yu. Savchencko and B. Ya. Zel’dovich, “Wave propagation in a guiding structure: one step beyond the paraxial approximation,” J. Opt. Soc. Am. B 13, 273–281 (1996).
[CrossRef]

V. S. Liberman and B. Ya. Zel’dovich, “Birefringence by a smoothly inhomogeneous locally isotropic medium,” Phys. Rev. E 49, 2389–2396 (1994).
[CrossRef]

A. Yu. Savchenko and B. Ya. Zel’dovich, “Birefringence by a smoothly inhomogeneous locally isotropic medium: three-dimensional case,” Phys. Rev. E 50, 2287–2292 (1994).
[CrossRef]

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

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

Opt. Commun. (1)

B. Crosignani, P. Di Porto, and A. Yariv, “Coupled-mode theory and slowly-varying approximation in guided-wave optics,” Opt. Commun. 78, 237–239 (1990).
[CrossRef]

Opt. Lett. (3)

Opt. Quantum Electron. (1)

S. Blair and K. Wagner, “(2+1)-D propagation of spatio-temporal solitary waves including higher-order corrections,” Opt. Quantum Electron. 30, 697–737 (1998).
[CrossRef]

Phys. Rev. A (1)

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

Phys. Rev. E (2)

V. S. Liberman and B. Ya. Zel’dovich, “Birefringence by a smoothly inhomogeneous locally isotropic medium,” Phys. Rev. E 49, 2389–2396 (1994).
[CrossRef]

A. Yu. Savchenko and B. Ya. Zel’dovich, “Birefringence by a smoothly inhomogeneous locally isotropic medium: three-dimensional case,” Phys. Rev. E 50, 2287–2292 (1994).
[CrossRef]

Phys. Rev. Lett. (1)

G. Fibich, “Small beam nonparaxiality arrests self-focusing of optical beams,” Phys. Rev. Lett. 76, 4356–4359 (1996).
[CrossRef] [PubMed]

Other (1)

See, e.g., D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972).

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Equations (135)

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2E-·E+ω2c2n2(r)E
=2E+2(E· ln n)+ω2c2n2(r)E=0.
[i/z+(1/2k)2]A(r, z)=-(k/n0)δn(r)A(r, z).
E(r, z)= d2k exp(ik·r+ik2-k2 z)E˜(k),
E(r, z)=exp(ikz) d2k×exp[ik·r-i(z/2k)k2]E˜(k)exp(ikz)A(r, z),
[i/z+(1/2k)2]A(r, z)=0,
F(r, z)= d2k exp(ik·r)F˜(k, z).
×E=iωB,
×B=-i(ω/c2)ε:E,
Bz=(1/iω)×E,
Ez=(ic2/εzzω)×B-(q·E/εzz)zˆ,
iω(zˆ×B/z)=(ω2/c2)ε:E-××E+(iω/εzz)(zˆ·×B)q-(ω2/c2)(q·E/εzz)q,
(ω/ic2)(zˆ×E/z)
=(ω2/c2)B-×[(×B)/εzz]
+(ω/ic2)×(q·E/εzz)zˆ,
ε=n0+δn(r)2n02+2n0δn(r),
εn02+2n0δn(r);
q2n0δn=2n0(δnzxxˆ+δnzyyˆ);
εzzn02+2n0δnzz,
1/εzz1/n02-2δnzz/n03.
(k2/iω)(zˆ×E/z)
=k2B-××B+(2/n0)(δnzz×B)
-(2k2/iωn0)zˆ×(δn·E),
iω(zˆ×B/z)
=k2E-××E+2(k2/n0)δn:E
+(2iω/n0)(zˆ·×B)δn.
C˜(k, z)=(1/2π)2  d2r exp(-ik·r)C(r, z),
ik2ωE˜z=σ:Λ:B˜+2n0 d2kδn˜zz(k-k, z)(k:k):σ:B˜(k, z)+2k2ωn0  d2k[k:δn˜ (k-k, z)]:E˜(k, z),
ωiB˜z=σ:Λ:E˜+2k2n0 d2kσ:δn˜(k-k, z):E˜(k, z)-2ωn0 d2kσ:[δn˜(k-k, z):k]:σ+:B˜(k, z),
σ=0-110,σ+=01-10,
Λ=k2-ky2kxkykxkyk2-kx2.
E˜(k, z)=E˜(+)(k, z)exp(ihz)+E˜(-)(k, z)exp(-ihz),
B˜(k, z)=k2hω:Λ-1:σ:[E˜(+)(k, z)exp(ihz)-E˜(-)(k, z)exp(-ihz)],
E˜(+)/z=(i/n0) d2kΓ(+)(k, k, z):E˜(+)(k, z)exp[i(h-h)z]+(i/n0) d2kΓ(-)(k, k, z):E˜(-)(k, z)exp[-i(h+h)z],
E˜(-)/z=(i/n0) d2kΘ(+)(k, k, z):E˜(+)(k, z)exp[i(h+h)z]+(i/n0) d2kΘ(-)(k, k, z):E˜(-)(k, z)exp[i(h-h)z],
Γ(±)=±δn˜zz(k-k, z)[(k:k)/h]-[k:δn˜(k-k, z)]-σ:Λ:σ:{δn˜(k-k, z)/h±[δ n˜(k-k, z):k]/hh}
E˜(+)/z=(i/n0) d2kΓ(+)(k, k, z):E˜(+)(k, z)exp[i(h-h)z],
Γ(+)=δn˜zz(k-k, z)[(k:k)/h]-[k:δn˜(k-k, z)]+[k2-(k:k)]:{δn˜(k-k, z)/h-[δn˜(k-k, z):k]/hh}.
i z+k2+2E(+)
=-k2n01k2+2(δn:E(+))+1n0δnzz 1k2+2(·E(+))-1n01k2+2[·(δn:E(+))]-in0[δn·E(+)]-ik2n01k2+2δn 1k2+2(·E(+))-in01k2+2·δn 1k2+2(·E(+)),
k2+2f(r, z)
= d2k exp(ik·r)k2-k2 f˜(k, z),
1k2+2f(r, z)
= d2k exp(ik·r) 1k2-k2f˜(k, z),
k2+2=k+12k2-18k322+ k2-k2=k-12kk2-18k3k4+ ,
1k2+2=1k-12k32+38k322+ 1k2-k2=1k+12k3k2+38k3k4+
i z+12k2A
=-kn0δn:A-in0(δn·A)-in0[δn(·A)]+12n0k2(δn:A)
+1n0k[δnzz(·A)]-1n0k[·(δn:A)].
i z+12k2A=-kn0δnA+12n0k2(δnA)-1n0k(A·δn),
i ddzax=-k δnn0ax+iηay,
i ddzay=-k δnn0ay-iηax,
δn=23n2|E|2+½|Ex|2½EyEx*½EzEx*½ExEy*|E|2+½|Ey|2½EzEy*½ExEz*½EyEz*|E|2+½|Ez|2,
δn=n2|E|2+n2|Ez|2+n2(E*E),
δn=n2EzE*,
δnzz=n2|E|2+n2|Ez|2,
i z+12k2A
=-2k3n2n0|A|2A-23kn2n0|·A|2A-k3n2n0(A·A)A*+1kn2n0[(·A)|A|2]+13kn2n0(·A)2A*+n23n0k2(A|2A)+n26n0k2[(A·A)A*]-23kn2n0[·(|A|2A)]-13kn2n0{·[(A·A)A*]},
(×F)=×Fz+zˆ×Fz(×F)z=×F,
×Ez+zˆ×Ez=iωB,
×E=iωBz;
×Bz+zˆ×Bz=-iωc2ε:E-iωc2(zˆ·Ez)q,
×B=-iωc2(q·E)zˆ-iωc2εzzEz,
ε=εxxεxyεyxεyy.
Bz=1iω×E,
Ez=ic2εzzω×B-q·Eεzzzˆ,
iω(zˆ×B/z)=(ω2/c2)ε:E-××E+(iω/εzz)(zˆ·×B)q-(ω2/c2)(q·E/εzz)q,
(ω/ic2)(zˆ×E/z)=(ω2/c2)B-[(×B)/εzz]+(ω/ic2)×(q·E/εzz)zˆ.
k2iωσ: E˜z=Λ:B˜+2n01(2π)2 d2r exp(-ik·r)×(δnzz×B)-2k2iωn01(2π)2 d2r exp(-ik·r)zˆ×(δn·E),
 d2r(2π)2exp(-ik·r)×(δnzz×B)
=ik× d2r(2π)2exp(-ik·r)(δnzz×B)=ik× d2kδn˜zz(k-k, z)1(2π)2 d2r exp(ik·r)×B=- d2kδn˜zz(k-k, z)k×k
×B˜(k, z)=- d2kδn˜zz(k-k, z)
[σ:(k:k):σ]:B˜(k, z),
 d2r(2π)2exp(-ik·r)zˆ×(δn·E)=izˆ×k 1(2π)2 d2r exp(-ik·r)δn·E=izˆ×k  d2kδ n˜(k-k, z)·E˜(k, z)=i  d2kσ:k[δn˜(k-k, z)·E˜(k, z)]=i  d2kσ:[k:δn˜(k-k, z)]:E˜(k, z).
iωσ: B˜z=Λ:E˜+2 k2n0
 d2kδn˜(k-k, z):E˜(k, z)+2 iωn0 d2r(2π)2exp(-ik·r)
(zˆ·×B)δn.
 d2r(2π)2exp(-ik·r)(zˆ·×B)δn
= d2kδn˜(k-k, z) d2r(2π)2exp(-ik·r)(zˆ·×B)=i  d2kδn˜(k-k, z)[zˆ·k×B˜(k, z)]=i  d2k[δn˜(k-k, z):k]:σ+:B˜(k, z).
ik2ωE˜(+)zexp(ihz)+E˜(-)zexp(-ihz)
-k2hω[E˜(+) exp(ihz)-E˜(-) exp(-ihz)]
=-k2hω[E˜(+) exp(ihz)-E˜(-) exp(-ihz)]
+2k2ωn0 d2khδnzz(k-k, z)
(k:k):σ:Λ-1:σ:[E˜(+)(k, z)
exp(ihz)-E˜(-)(k, z)exp(-ihz)]+2k2ωn0 d2k[k:δn˜(k-k, z)]
:[E˜(+)(k, z)exp(ihz)+E˜(-)(k, z)exp(-ihz)],
iE˜(+)zexp(ihz)+E˜(-)zexp(-ihz)
=2n0 d2khδnzz(k-k, z)
(k:k):σ:Λ-1:σ:[E˜(+)
(k, z)exp(ihz)-E˜(-)
(k, z)exp(-ihz)]+2n0 d2k[k:δn˜(k-k, z)]:[E˜(+)(k, z)exp(ihz)+E˜(-)(k, z)exp(-ihz)],
h2=k2-k2,Λ-1=1h2k2(k2-k:k),
h(k:k):σ:Λ-1:σ=-k:kh,
E˜(+)zexp(ihz)+E˜(-)zexp(-ihz)
=2in0 d2kδn˜zz(k-k, z) k:kh-k:δn˜(k-k, z):E˜(+)(k, z)exp(ihz)+ d2k-δn˜zz(k-k, z) k:kh-k:δn˜(k-k, z):E˜(-)(k, z)exp(-ihz).
E˜(+)zexp(ihz)-E˜(-)zexp(-ihz)=2in0 d2k σ:Λ:σh:-δn˜(k-k, z)+δn˜(k-k, z):kh:E˜(+)(k, z)exp(ihz)+2in0 d2k σ:Λ:σh:-δn˜(k-k, z)-δn˜(k-k, z):kh:E˜(-)(k, z)exp(-ihz).
 d2k exp(ik·r) d2kδn˜zz(k-k, z)
k:kh:E˜(+)(k, z)exp(ihz)
=- d2k exp(ik·r)ik d2 kδn˜zz(k-k, z) 1hik·E˜(+)(k, z)exp(ihz)
=-  d2k exp(ik·r)  d2kδn˜zz(k-k, z) 1hik·E˜(+)(k, z)exp(ihz)=-δnzz(r, z) d2k exp(ik·r) 1hik·E˜(+)(k, z)exp(ihz)=-δnzz 1k2+2· d2k exp(ik·r)E˜(+)(k, z)exp(ihz)
=-δnzz 1k2+2(·E(+)),
 d2k exp(ik·r) d2k
[-k:δn˜(k-k, z)]:E˜(+)(k, z)exp(ihz)
=i  d2k exp(ik·r)ik d2kδn˜(k-k, z)·E˜(+)(k, z)exp(ihz)=i  d2 k exp(ik·r) d2kδn˜(k-k, z)·E˜(+)(k, z)exp(ihz)
=i(δn·E(+)),
 d2k exp(ik·r) d2k
k2-k:kh:δn˜(k-k, z)
:E˜(+)(k, z)exp(ihz)
=k2k2+2 d2k exp(ik·r)
 d2kδn˜(k-k, z)
:E˜(+)(k, z)exp(ihz)
+1k2+2 d2k exp(ik·r)ikik· d2kδn˜(k-k, z):E˜(+)(k, z)exp(ihz)
=k2k2+2(δn:E(+))+1k2+2· d2k exp(ik·r) d2kδn˜(k-k, z):E˜(+)(k, z)exp(ihz)
=k2k2+2(δn:E(+))+1k2+2[·(δn:E(+))],
d2k exp(ik·r) d2
k-k2-k:kh:δn˜(k-k, z):kh:E˜(+)(k, z)exp(ihz)
= d2k exp(ik·r) d2k-k2hδn˜(k-k, z):kh:E˜(+)(k, z)exp(ihz)+ d2k exp(ik·r) d2kk:kh: δn˜(k-k, z):kh:E˜(+)(k, z)exp(ihz).
 d2k exp(ik·r) d2k-k2hδn˜(k-k, z):kh:E˜(+)(k, z)exp(ihz)=-k2k2+2 d2k exp(ik·r) d2kδn˜(k-k, z)k·E˜(+)(k, z)hexp(ihz)=ik2k2+2δn 1k2+2 d2k exp(ik·r)ik·E˜(+)(k, z)exp(ihz)=ik2k2+2δn 1k2+2· d2k exp(ik·r)E˜(+)(k, z)exp(ihz)=ik2k2+2δn 1k2+2(·E(+)),
d2k exp(ik·r)
 d2kk:kh: δn˜(k-k, z):kh
:E˜(+)(k, z)exp(ihz)
=-1k2+2 d2k exp(ik·r)
ikik· d2kδn˜(k-k, z) k·E˜(+)(k, z)hexp(ihz)
=-1k2+2· d2k exp(ik·r) d2kδn˜(k-k, z) k·E˜(+)(k, z)hexp(ihz)
=ik2+2·δn  d2k exp(ik·r) ik·E˜(+)(k, z)hexp(ihz)
=ik2+2·δn 1k2+2· d2k exp(ik·r)E˜(+)(k, z)exp(ihz)=ik2+2·δn 1k2+2(·E(+)).
i z+k2+2E(±)
=±k2n01k2+2[δn:(E(+)+E(-))]+1n0δnzz 1k2+2[·(E(+)-E(-))]±1n01k2+2{·[δn:(E(+)+E(-))]}-in0[δn·(E(+)+E(-))]±ik2n01k2+2δn 1k2+2[·(E(+)-E(-))]±in01k2+2·δn 1k2+2[·(E(+)-E(-))].
ε=[n0+δn(r)]2n02+2n0δn(r),
Ez=ic2ω1n02-2n03δnzz×B-2n0(δn·E)zˆ.
B(+)(r, z)= d2k exp(ik·r)B˜(+)(k, z)= d2k exp(ik·r) k2hωΛ-1:σ:E˜(+)(k, z)exp(ihz),
B(+)(r, z)= d2k exp(ik·r) 1ωh(k2-k:k)σ:E˜(+)(k, z)exp(ihz)=k2ω1k2+2σ: d2k exp(ik·r)E˜(+)(k, z)exp(ihz)+1ω1k2+2 d2k exp(ik·r)ikik·[σ:E˜(+)(k, z)]exp(ihz)=k2ω1k2+2(σ:E(+))+1ω1k2+2·σ: d2k exp(ik·r)E˜(+)(k, z)exp(ihz)=k2ω1k2+2(σ:E(+))+1ω1k2+2{[·(σ:E(+))]}.
B(+)(r, z)=kωσ:E(+).
×(σ:E(+))=(·E(+))zˆ
Ez=ik1-2n023n2|E(+)|2+n2|Ez|2(·E(+))-23n2n0Ez|E(+)|2zˆik·E(+)zˆ.

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