Abstract

We give a quantitative meaning to the often-used expression: in spatial solitons, diffraction is balanced by nonlinear refractive-index effects. After pointing out how the electric field shape and its derivatives relate to the magnetic field, we show that the balance of magnetic and electric stored energies describes what happens in a variety of linear and nonlinear waves, including linear guided waves, nonlinear self-guided waves in quadratic and cubic media, and in cases where several frequencies are present and energy exchange may occur. We give exact, electromagnetic theory results and also explain how the energy balance works in the slowly varying approximation.

© 2001 Optical Society of America

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References

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  1. M. Segev, “Optical spatial solitons,” Opt. Quantum Electron. 30, 503–533 (1998); Yu. S. Kivshar, “Bright and dark spatial solitons in non-Kerr media,” Opt. Quantum Electron. 30, 571–614 (1998).
    [CrossRef]
  2. M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1970).
  3. R. W. P. King and S. Prasad, Fundamental Electromagnetic Theory and Applications (Prentice-Hall, Englewood Cliffs, N.J., 1986).
  4. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).
  5. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984).
  6. R. W. Boyd, Nonlinear Optics (Academic, Boston, 1992).
  7. D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972).
  8. H. Kogelnik, “Theory of dielectric waveguides,” in Topics in Applied Physics, T. Tamir, ed. (Springer-Verlag, Berlin, 1979), Chap. 2.
  9. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).
  10. N. N. Akhmediev and A. Ankiewicz, Solitons. Nonlinear Pulses and Beams (Chapman & Hall, London, 1997).
  11. A. D. Boardman, P. Egan, F. Lederer, U. Langbein, and D. Mihalache, “Third-order nonlinear electromagnetic TE andTM guided waves,” in Nonlinear Surface Electromagnetic Phenomena, H. E. Ponath and G. I. Stegeman, eds. (North-Holland, Amsterdam, 1991).
  12. W. J. Tomlinson, “Surface wave at a nonlinear interface,” Opt. Lett. 5, 323–325 (1980).
    [CrossRef] [PubMed]
  13. R. De La Fuente and A. Barthelemy, “Spatial solitons pairing by cross-phase modulation,” Opt. Commun. 88, 419–423 (1992).
    [CrossRef]
  14. Y. N. Karamzin and A. P. Sukhorukov, “Mutual focusing of high-power light beams in media with quadratic nonlinearity,” Sov. Phys. JETP 41, 414–420 (1976).
  15. Y. U. Kivshar, “Bright and dark spatial solitons,” Opt. Quantum Electron. 30, 571–614 (1998), and references therein.
    [CrossRef]
  16. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
    [CrossRef]
  17. N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1964).
  18. P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge University, Cambridge, England, 1990); see especially Section 5.1.1.
  19. N. Akmediev, A. Ankiewicz, and J. M. Soto-Crespo, “Does the nonlinear Schrödinger equation correctly describe beam propagation?” Opt. Lett. 18, 411–413 (1993).
    [CrossRef]
  20. A. P. Sheppard and M. Haelterman, “Nonparaxiality stabilizes three-dimensional soliton beams in Kerr media,” Opt. Lett. 23, 1820–1822 (1998).
    [CrossRef]

1998 (2)

Y. U. Kivshar, “Bright and dark spatial solitons,” Opt. Quantum Electron. 30, 571–614 (1998), and references therein.
[CrossRef]

A. P. Sheppard and M. Haelterman, “Nonparaxiality stabilizes three-dimensional soliton beams in Kerr media,” Opt. Lett. 23, 1820–1822 (1998).
[CrossRef]

1993 (1)

1992 (1)

R. De La Fuente and A. Barthelemy, “Spatial solitons pairing by cross-phase modulation,” Opt. Commun. 88, 419–423 (1992).
[CrossRef]

1980 (1)

1976 (1)

Y. N. Karamzin and A. P. Sukhorukov, “Mutual focusing of high-power light beams in media with quadratic nonlinearity,” Sov. Phys. JETP 41, 414–420 (1976).

1962 (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Akmediev, N.

Ankiewicz, A.

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Barthelemy, A.

R. De La Fuente and A. Barthelemy, “Spatial solitons pairing by cross-phase modulation,” Opt. Commun. 88, 419–423 (1992).
[CrossRef]

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

De La Fuente, R.

R. De La Fuente and A. Barthelemy, “Spatial solitons pairing by cross-phase modulation,” Opt. Commun. 88, 419–423 (1992).
[CrossRef]

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Haelterman, M.

Karamzin, Y. N.

Y. N. Karamzin and A. P. Sukhorukov, “Mutual focusing of high-power light beams in media with quadratic nonlinearity,” Sov. Phys. JETP 41, 414–420 (1976).

Kivshar, Y. U.

Y. U. Kivshar, “Bright and dark spatial solitons,” Opt. Quantum Electron. 30, 571–614 (1998), and references therein.
[CrossRef]

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Sheppard, A. P.

Soto-Crespo, J. M.

Sukhorukov, A. P.

Y. N. Karamzin and A. P. Sukhorukov, “Mutual focusing of high-power light beams in media with quadratic nonlinearity,” Sov. Phys. JETP 41, 414–420 (1976).

Tomlinson, W. J.

Opt. Commun. (1)

R. De La Fuente and A. Barthelemy, “Spatial solitons pairing by cross-phase modulation,” Opt. Commun. 88, 419–423 (1992).
[CrossRef]

Opt. Lett. (3)

Opt. Quantum Electron. (1)

Y. U. Kivshar, “Bright and dark spatial solitons,” Opt. Quantum Electron. 30, 571–614 (1998), and references therein.
[CrossRef]

Phys. Rev. (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Sov. Phys. JETP (1)

Y. N. Karamzin and A. P. Sukhorukov, “Mutual focusing of high-power light beams in media with quadratic nonlinearity,” Sov. Phys. JETP 41, 414–420 (1976).

Other (13)

N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1964).

P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge University, Cambridge, England, 1990); see especially Section 5.1.1.

M. Segev, “Optical spatial solitons,” Opt. Quantum Electron. 30, 503–533 (1998); Yu. S. Kivshar, “Bright and dark spatial solitons in non-Kerr media,” Opt. Quantum Electron. 30, 571–614 (1998).
[CrossRef]

M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1970).

R. W. P. King and S. Prasad, Fundamental Electromagnetic Theory and Applications (Prentice-Hall, Englewood Cliffs, N.J., 1986).

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

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984).

R. W. Boyd, Nonlinear Optics (Academic, Boston, 1992).

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

H. Kogelnik, “Theory of dielectric waveguides,” in Topics in Applied Physics, T. Tamir, ed. (Springer-Verlag, Berlin, 1979), Chap. 2.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

N. N. Akhmediev and A. Ankiewicz, Solitons. Nonlinear Pulses and Beams (Chapman & Hall, London, 1997).

A. D. Boardman, P. Egan, F. Lederer, U. Langbein, and D. Mihalache, “Third-order nonlinear electromagnetic TE andTM guided waves,” in Nonlinear Surface Electromagnetic Phenomena, H. E. Ponath and G. I. Stegeman, eds. (North-Holland, Amsterdam, 1991).

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Figures (1)

Fig. 1
Fig. 1

Fundamental soliton: plots of intensity shape (solid curve), f2 from Eq. (30) with q=A=1, and energy difference uE-uH scaled to unity for x=0 (dashed curve).

Equations (102)

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S=Sr+iSi=12(E×H*),
12[E exp(-iwt)+E* exp(iwt)],
×E=iwμ0H,
×H*=iwD*.
·(E×H*)=iw(μ0H·H*-E·D*).
·S=iw 12(μ0H·H*-E·D*).
Tr=ASr·nˆdA=w2 V(E·D*)imdV,
Ti=ASi·nˆdA=2wV(uH-uE)dV
=2w(UH-UE),
uE=14(E·D*)real.
uH=14(μ0H·H*),
uH=14μ0w2Ex2+Ez2,
S=Sˆ|S|(cos θS+i sin θs)
d2Edz2+μ0w2D=0,
D=εE+3ε0χ|E|2E=ε0(n2+3χ|E|2)E.
E=E0 exp(iqz)yˆ,
H=-(q/μ0w)E0 exp(iqz)xˆ,
q2=k2(n2+3χ|E0|2),
S=(q/μ0w)|E0|2zˆ,
uE=uH=14ε0(n2+3χ|E0|2)|E0|2.
E=E1 exp(-iw1t)yˆ+E2 exp(-iw2t)yˆ.
D1=ε0(n2+3χ|E1|2+6χ|E2|2)E1
E1=E10 exp(iq1z),
q12=k12(n2+3χ|E10|2+6χ|E20|2),
uE=uH=14ε0(n2+3χ|E10|2+6χ|E20|2)|E10|2,
E=[0, E(x, z), 0],H=[Hx(x, z), 0, Hz(x, z)].
S=-12(EHx*)zˆ+12(EHz*)xˆ.
E=f(x)exp(iβz),
H=-βμ0wExˆ-iμ0w d fdx exp(iβz)zˆ,
=Hxxˆ+Hzzˆ.
Si=-12μ0wf(x) d fdx xˆ.
uE-uH=-14μ0w2 ddx f dfdx,
uE-uH=-18μ0w2 d2dx2f2(x)=B(x).
E=mfm(x)exp(iβmz),
12μ0w ddx[fm(x)fm(x)]cos[(βm-βm)z]xˆ.
(UH-UE)=m(UHm-UEm)=0.
D=ε0(n2+3χ|E|2)Eyˆ,
d2fdx2+k2(n2+3χ f2)f=β2f,
f(x)=A sech(qx),
β2=k2n2+q2,
A2=2q2/3χk2.
UE=ε0n2 A22q+ε0χ A4q,
UHx=ε0n2 A22q+3ε0χ4 A4q,
UHz=ε0χ4 A4q,
UE-UH=B(x)>0for|x|<xB,
UE-UH=B(x)<0for|x|>xB,
B(x)=32ε0χA4 sech2(qx)[3 sech2(qx)-2].
sech2(qxB)=2/3.
D1=ε0(n2+3χ|E1|2+6χ|E2|2)E1 yˆ,
Ej=Aj sech(qx)exp(iβjz),
βj=kj2n2+q2,
2q2=3k12χ(A12+2A22)=3k22χ(A22+2A12).
DNL=ε0χ2[2E2E1* exp(-iwt)+E12 exp(-i2wt)+c.c.]yˆ.
2E1z2+2E1x2+k2E1=-αE2E1*,
2E2z2+2E2x2+4k2(n2/n1)2E2=-2αE12,
E1=A1 exp(iβz),E2=A2 exp(i2βz),
β2=k2+α A2,A1=±2A2.
uE1=uH1=14ε0n12A12+14ε0χ2A2A12,
uE2=uH2=14ε0n22A22+14ε0χ2A2A12.
E1=A1 sech2(qx)exp(iβz),
E2=A2 sech2(qx)exp(i2βz),
β2=4γ-1γ-1,q2=(4γ-1)3k2,
A2=6αq2,A1=±12A2.
D=ε0(n2+iεI)Eyˆ,
E=A exp[(iβ-α)z]yˆ,
β2=k2n2+α2,α=k2εI/2β.
H=-1μ0w(β+iα)A exp[(iβ-α)z]xˆ.
uH-uE=ε0α22k2A2 exp(-2αz),
E=A(z)exp[iϕ(z)]yˆ.
H=exp(iϕ)(μw) -A dϕdz+i dAdzxˆ.
Sr=12µwA2 dϕdz zˆ,
Si=14µw dA2dz zˆ.
uH-uE=18µw2 d2(A2)dz2.
E=A(x, z)exp[iϕ(z)]yˆ,
Hz=-i1μw Ax exp[iϕ(z)].
Si=14µw A2z zˆ+14µw A2x xˆ.
(UH-UE)=18µw2Izz2-Izz1,
I=-A2dx.
2Ez2+2Ex2+k02R(E)E=0,
-Ex2+Ez2dx-k02-R(E)|E|2dx
=12 - 2|E|2z2 dx.
E(x, z)=E(x, z)exp(ikz),
-k2E+2ik Ez2+2Ex2+k02R(E)E=0.
--k2|E|2-ikE* Ez-E E*zdx
--Ex2dx+-k02R(E)|E|2dx=0.
-Ezapprox2+Ex2dx=k02-R(E)|E|2dx,
Ezapprox2=k2|E|2+ikE* Ez-E E*z
=Ez2-Ez2.
Ezapprox2=k02R(E)|E|2.
E=A sech(qx)exp(iβz),
β=kn+q22k.
2ik Ez+2Ex2+3k02χ|E|2E=0.
i ψξ+12 2ψτ2+|ψ|2ψ=0,
ψ(τ, ξ)=4 cosh(3τ)+3 cosh(τ)exp(4iξ)cosh(4τ)+4 cosh(2τ)+3 cos(4ξ) exp(iξ/2).
2Ejz2+2Ejx2+k02nj2Ej+k02Dj=0,
-kj2Ej+2ikj Ejz+2Ejx2+k02njEj+k02Dj=0,
Ejx2+Ejz2dx-k02nj2|Ej|2dx
-k02 12(Ej*Dj+EjDj*)dx=12  2z2|Ej|2dx,
Ejzapprox2+Ejx2dx
=k02nj2|E|2+k02 12(Ej*Dj+EjDj*)dx,
E1=A sech(z/z0),
E2=iA tanh(z/z0),

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