Abstract

The nonlinear three-dimensional paraxial wave equation is known to be invariant under a lens transformation that describes the image of an electric field produced by a thin spherical lens. This symmetry, when coupled with other symmetries of the model, can be used for determining two sets of radiation modes by using the analysis of the nonlinear circular and planar parabolic graded-index optical guides. Mode profiles as well as their dispersion curves are calculated numerically. A variational approach is used for studying the near-modal propagation and for providing an approximate analytical description of the dispersion curves. The possibility of describing the dynamic evolution of the field in terms of coupled-radiation-mode equations is also briefly mentioned.

© 1991 Optical Society of America

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References

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  1. L. Gagnon, “Exact solutions for optical wave propagation including transverse effects,” J. Opt. Soc. Am. B 7, 1098–1102 (1990).
    [CrossRef]
  2. V. I. Talanov, “Focusing of light in cubic media,” JETP Lett. 11, 199–201 (1970).
  3. A. J. Glass, “Propagation modes in stationary self-focusing,” IEEE J. Quantum Electron. QE-10, 705–706 (1974).
    [CrossRef]
  4. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 16.
  5. S. A. Akhmanov, R. V. Khokhlov, A. P. Sukhorukov, “Self-focusing, self-defocusing and self-modulation of laser beams,” in Laser Handbook, F. T. Arecchi, E. O. Schulz-Dubois, eds. (North-Holland, Amsterdam, 1972), Chap. E3.
  6. J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 35–110 (1975).
    [CrossRef]
  7. G. Burdet, J. Patera, M. Perrin, P. Winternitz, “Sousalgèbres de Lie de l’algèbre de Schrödinger,” Ann. Sci. Math. Quèbec 2, 81–108 (1978).
  8. R. Y. Chiao, E. Garmire, C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
    [CrossRef]
  9. D. Pohl, “Vectorial theory of self-trapped light beams,” Opt. Commun. 2, 305–307 (1970).
    [CrossRef]
  10. H. A. Haus, “Higher order trapped light beam solutions,” Appl. Phys. Lett. 8, 128–129 (1966).
    [CrossRef]
  11. M. J. Ablowitz, H. Segur, Soliton and the Inverse Scattering Transform (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1981), Chap. 4.
    [CrossRef]
  12. D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A27, 3135–3145 (1983).
  13. D. Anderson, M. Bonnedal, “Variational approach to nonlinear self-focusing of Gaussian laser beams,” Phys. Fluids22, 105–109 (1979), and references therein.
    [CrossRef]
  14. J. T. Manassah, P. L. Baldeck, R. R. Alfano, “Self-focusing and self-phase modulation in a parabolic graded-index optical fiber,” Opt. Lett. 13, 589–591 (1988); “Self-focusing, self-phase modulation, and diffraction in bulk homogeneous material,” Opt. Lett. 13, 1090–1092 (1988).
    [CrossRef] [PubMed]
  15. G. P. Agrawal, Nonlinear Fiber Optics (Academic, Boston, 1989), Sec. 2.4, and references therein.
  16. Y. Silberberg, G. I. Stegeman, “Nonlinear coupling of waveguide modes,” Appl. Phys. Lett. 50, 801–803 (1987).
    [CrossRef]
  17. A. Hénault, R. J. Black, F. Gonthier, S. Lacroix, M. Cada, “Kerr-type nonlinearities in lightguides: coupled-mode field representations,” in Integrated Photonics Research, Vol. 5 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), pp. 90–91.

1990 (1)

1988 (1)

1987 (1)

Y. Silberberg, G. I. Stegeman, “Nonlinear coupling of waveguide modes,” Appl. Phys. Lett. 50, 801–803 (1987).
[CrossRef]

1983 (1)

D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A27, 3135–3145 (1983).

1978 (1)

G. Burdet, J. Patera, M. Perrin, P. Winternitz, “Sousalgèbres de Lie de l’algèbre de Schrödinger,” Ann. Sci. Math. Quèbec 2, 81–108 (1978).

1975 (1)

J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 35–110 (1975).
[CrossRef]

1974 (1)

A. J. Glass, “Propagation modes in stationary self-focusing,” IEEE J. Quantum Electron. QE-10, 705–706 (1974).
[CrossRef]

1970 (2)

V. I. Talanov, “Focusing of light in cubic media,” JETP Lett. 11, 199–201 (1970).

D. Pohl, “Vectorial theory of self-trapped light beams,” Opt. Commun. 2, 305–307 (1970).
[CrossRef]

1966 (1)

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

1964 (1)

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

Ablowitz, M. J.

M. J. Ablowitz, H. Segur, Soliton and the Inverse Scattering Transform (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1981), Chap. 4.
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, Boston, 1989), Sec. 2.4, and references therein.

Akhmanov, S. A.

S. A. Akhmanov, R. V. Khokhlov, A. P. Sukhorukov, “Self-focusing, self-defocusing and self-modulation of laser beams,” in Laser Handbook, F. T. Arecchi, E. O. Schulz-Dubois, eds. (North-Holland, Amsterdam, 1972), Chap. E3.

Alfano, R. R.

Anderson, D.

D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A27, 3135–3145 (1983).

D. Anderson, M. Bonnedal, “Variational approach to nonlinear self-focusing of Gaussian laser beams,” Phys. Fluids22, 105–109 (1979), and references therein.
[CrossRef]

Baldeck, P. L.

Black, R. J.

A. Hénault, R. J. Black, F. Gonthier, S. Lacroix, M. Cada, “Kerr-type nonlinearities in lightguides: coupled-mode field representations,” in Integrated Photonics Research, Vol. 5 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), pp. 90–91.

Bonnedal, M.

D. Anderson, M. Bonnedal, “Variational approach to nonlinear self-focusing of Gaussian laser beams,” Phys. Fluids22, 105–109 (1979), and references therein.
[CrossRef]

Burdet, G.

G. Burdet, J. Patera, M. Perrin, P. Winternitz, “Sousalgèbres de Lie de l’algèbre de Schrödinger,” Ann. Sci. Math. Quèbec 2, 81–108 (1978).

Cada, M.

A. Hénault, R. J. Black, F. Gonthier, S. Lacroix, M. Cada, “Kerr-type nonlinearities in lightguides: coupled-mode field representations,” in Integrated Photonics Research, Vol. 5 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), pp. 90–91.

Chiao, R. Y.

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

Gagnon, L.

Garmire, E.

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

Glass, A. J.

A. J. Glass, “Propagation modes in stationary self-focusing,” IEEE J. Quantum Electron. QE-10, 705–706 (1974).
[CrossRef]

Gonthier, F.

A. Hénault, R. J. Black, F. Gonthier, S. Lacroix, M. Cada, “Kerr-type nonlinearities in lightguides: coupled-mode field representations,” in Integrated Photonics Research, Vol. 5 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), pp. 90–91.

Haus, H. A.

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

Hénault, A.

A. Hénault, R. J. Black, F. Gonthier, S. Lacroix, M. Cada, “Kerr-type nonlinearities in lightguides: coupled-mode field representations,” in Integrated Photonics Research, Vol. 5 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), pp. 90–91.

Khokhlov, R. V.

S. A. Akhmanov, R. V. Khokhlov, A. P. Sukhorukov, “Self-focusing, self-defocusing and self-modulation of laser beams,” in Laser Handbook, F. T. Arecchi, E. O. Schulz-Dubois, eds. (North-Holland, Amsterdam, 1972), Chap. E3.

Lacroix, S.

A. Hénault, R. J. Black, F. Gonthier, S. Lacroix, M. Cada, “Kerr-type nonlinearities in lightguides: coupled-mode field representations,” in Integrated Photonics Research, Vol. 5 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), pp. 90–91.

Manassah, J. T.

Marburger, J. H.

J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 35–110 (1975).
[CrossRef]

Patera, J.

G. Burdet, J. Patera, M. Perrin, P. Winternitz, “Sousalgèbres de Lie de l’algèbre de Schrödinger,” Ann. Sci. Math. Quèbec 2, 81–108 (1978).

Perrin, M.

G. Burdet, J. Patera, M. Perrin, P. Winternitz, “Sousalgèbres de Lie de l’algèbre de Schrödinger,” Ann. Sci. Math. Quèbec 2, 81–108 (1978).

Pohl, D.

D. Pohl, “Vectorial theory of self-trapped light beams,” Opt. Commun. 2, 305–307 (1970).
[CrossRef]

Segur, H.

M. J. Ablowitz, H. Segur, Soliton and the Inverse Scattering Transform (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1981), Chap. 4.
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 16.

Silberberg, Y.

Y. Silberberg, G. I. Stegeman, “Nonlinear coupling of waveguide modes,” Appl. Phys. Lett. 50, 801–803 (1987).
[CrossRef]

Stegeman, G. I.

Y. Silberberg, G. I. Stegeman, “Nonlinear coupling of waveguide modes,” Appl. Phys. Lett. 50, 801–803 (1987).
[CrossRef]

Sukhorukov, A. P.

S. A. Akhmanov, R. V. Khokhlov, A. P. Sukhorukov, “Self-focusing, self-defocusing and self-modulation of laser beams,” in Laser Handbook, F. T. Arecchi, E. O. Schulz-Dubois, eds. (North-Holland, Amsterdam, 1972), Chap. E3.

Talanov, V. I.

V. I. Talanov, “Focusing of light in cubic media,” JETP Lett. 11, 199–201 (1970).

Townes, C. H.

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

Winternitz, P.

G. Burdet, J. Patera, M. Perrin, P. Winternitz, “Sousalgèbres de Lie de l’algèbre de Schrödinger,” Ann. Sci. Math. Quèbec 2, 81–108 (1978).

Ann. Sci. Math. Quèbec (1)

G. Burdet, J. Patera, M. Perrin, P. Winternitz, “Sousalgèbres de Lie de l’algèbre de Schrödinger,” Ann. Sci. Math. Quèbec 2, 81–108 (1978).

Appl. Phys. Lett. (2)

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

Y. Silberberg, G. I. Stegeman, “Nonlinear coupling of waveguide modes,” Appl. Phys. Lett. 50, 801–803 (1987).
[CrossRef]

IEEE J. Quantum Electron. (1)

A. J. Glass, “Propagation modes in stationary self-focusing,” IEEE J. Quantum Electron. QE-10, 705–706 (1974).
[CrossRef]

J. Opt. Soc. Am. B (1)

JETP Lett. (1)

V. I. Talanov, “Focusing of light in cubic media,” JETP Lett. 11, 199–201 (1970).

Opt. Commun. (1)

D. Pohl, “Vectorial theory of self-trapped light beams,” Opt. Commun. 2, 305–307 (1970).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. (1)

D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A27, 3135–3145 (1983).

Phys. Rev. Lett. (1)

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

Prog. Quantum Electron. (1)

J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 35–110 (1975).
[CrossRef]

Other (6)

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 16.

S. A. Akhmanov, R. V. Khokhlov, A. P. Sukhorukov, “Self-focusing, self-defocusing and self-modulation of laser beams,” in Laser Handbook, F. T. Arecchi, E. O. Schulz-Dubois, eds. (North-Holland, Amsterdam, 1972), Chap. E3.

D. Anderson, M. Bonnedal, “Variational approach to nonlinear self-focusing of Gaussian laser beams,” Phys. Fluids22, 105–109 (1979), and references therein.
[CrossRef]

M. J. Ablowitz, H. Segur, Soliton and the Inverse Scattering Transform (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1981), Chap. 4.
[CrossRef]

G. P. Agrawal, Nonlinear Fiber Optics (Academic, Boston, 1989), Sec. 2.4, and references therein.

A. Hénault, R. J. Black, F. Gonthier, S. Lacroix, M. Cada, “Kerr-type nonlinearities in lightguides: coupled-mode field representations,” in Integrated Photonics Research, Vol. 5 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), pp. 90–91.

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Figures (5)

Fig. 1
Fig. 1

First modes of (a) Eq. (2.14), (b) Eq. (2.15) for b = −1.

Fig. 2
Fig. 2

First modal dispersion curves for (a) Eq. (2.14), (b) Eq. (2.15) calculated numerically (solid curve) and analytically (dashed curve) by using a Gaussian trial function.

Fig. 3
Fig. 3

Potential V(W) given by Eq. (3.16) for γ = 1 (solid curve) and γ = −0.5 (dashed curve).

Fig. 4
Fig. 4

Dispersion curve for the first mode of (a) Eq. (3.1), (b) Eq. (3.2) calculated numerically (solid curve) and analytically by using Gaussian (dashed–dotted curve) and hyperbolic-secant (dashed curve) trial functions.

Fig. 5
Fig. 5

Numerical solutions of Eq. (3.2) with the initial conditions (a) U2(Z = 0) = 1.02U2,1(b = −1), (b) U2(Z = 0) = 1.737 exp(−0.834 ξ22) = approximate Gaussian solution, (c) U2(Z = 0) = U2,1(b = −1) + U2,2(b = 2).

Tables (2)

Tables Icon

Table 1 Numerical Values of the Constants ci and the Approximate Maximum Energy max given by Eq. (3.23) for Eq. (3.1)

Tables Icon

Table 2 Numerical Values of the Constants ci for Eq. (3.2)

Equations (45)

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i ψ z + ψ x x + ψ y y + ψ ψ 2 = 0 ,             = ± 1 ,
z = z 1 - λ z ,             x = x 1 - λ z ,             y = y 1 - λ z , ψ ( x , y , z ) = ψ ( x , y , z ) ( 1 - λ z ) exp ( i λ 4 x 2 + y 2 1 - λ z ) ,
C + Z + b M , J + a M ,
C + Z - J + b M , B x + Y ,
C = ( 1 / 2 ) z 2 z + ( 1 / 2 ) z ( x x + y y ) - z ψ ψ + ( i / 4 ) ( x 2 + y 2 ) ψ ψ + c . c . ,
B x = - ( 1 / 2 ) z x - ( i / 2 ) x ψ ψ + c . c . ,
J = y x - x y ,
M = - i ψ ψ + c . c . ,
z = z ,
Y = y ,
ψ 1 = M 1 ( ξ 1 ) ( 1 + z 2 ) - 1 / 2 exp ( i 4 z ξ 1 2 ) × exp ( i a θ ) exp ( - i b tan - 1 z ) ,
ψ 2 = M 2 ( ξ 2 ) ( 1 + z 2 ) - 1 / 2 exp { i 4 z [ ( z 2 - 1 ) ξ 2 2 + x 2 ] } × exp ( - i b tan - 1 z ) ,
ξ 1 = ( x 2 + y 2 1 + z 2 ) 1 / 2 ,
ξ 2 = z y + x 1 + z 2 .
M ¨ 1 + 1 ξ 1 M ˙ 1 + ( b - ξ 1 2 4 - a 2 ξ 1 2 ) M 1 + M 1 3 = 0 ,
M ¨ 2 + ( b - ξ 2 2 ) M 2 + M 2 3 = 0 ,
i U 1 Z + U 1 ξ 1 ξ 1 + 1 ξ 1 U 1 ξ 1 + 1 ξ 1 2 U 1 θ θ - ( 1 / 4 ) ξ 1 2 U 1 + U 1 U 1 2 = 0 ,
i U 2 z + U 2 ξ 2 ξ 2 - ξ 2 2 U 2 + U 2 U 2 2 = 0 ,
L = L ξ 1 d ξ 1 d θ d Z ,
L = ( i / 2 ) ( U 1 U 1 Z * - U 1 * U 1 Z ) + U 1 ξ 1 2 + ( 1 / ξ 1 2 ) U 1 θ 2 + ( 1 / 4 ) ξ 1 2 U 1 2 - ( 1 / 2 ) U 1 4 .
δ L δ U 1 * = 2 L Z U 1 Z * + 2 L θ U 1 θ * + 2 L ξ 1 U 1 ξ 1 * + 1 ξ 1 L U 1 ξ 1 * - L U 1 * = 0.
U 1 ( Z , ξ 1 , θ ) = A ( Z ) f ( ξ 1 W ( Z ) ) exp { i [ φ ( Z ) + P ( Z ) ξ 1 2 + a θ ] } ,
L = c 1 A 2 φ ˙ W 2 + c 2 A 2 P ˙ W 4 + ( c 3 + a 2 c 5 ) A 2 + 4 c 2 A 2 P 2 W 4 - ( 1 / 2 ) c 4 A 4 W 2 + ( 1 / 4 ) c 2 A 2 W 4
c 1 = x f 2 ,             c 2 = x 3 f 2 ,             c 3 = x ( d f d x ) 2 ,             c 4 = x f 4 ,             c 5 = ( 1 / x ) f 2 .
δ L δ y i = L y i - d d Z [ L ( d y i / d Z ) ] = 0 ,
δ L δ A = 0 c 1 φ ˙ + c 2 W 2 P ˙ + c 6 W 2 + 4 c 2 P 2 W 2 - c 4 A 2 + ( 1 / 4 ) c 2 W 2 = 0 ,
c 6 c 3 + a 2 c 5 ,
δ L δ φ = 0 A 2 W 2 = constant E
δ L δ P = 0 W ˙ = 4 P W
δ L δ W = 0 c 1 φ ˙ + 2 c 2 W 2 P ˙ + 8 c 2 P 2 W 2 - ( 1 / 2 ) c 4 A 2 + ( 1 / 2 ) c 2 W 2 = 0.
W ¨ = - d V ( W ) d W ,
V = γ W 2 + 1 2 W 2 + constant
γ = ( 1 / c 2 ) ( 2 c 6 - c 4 E ) .
W 0 4 = 2 γ .
b = c 6 c 1 W 0 2 - c 4 E c 1 W 0 2 + 1 4 c 2 c 1 W 0 2 .
W 0 2 = 2 3 c 2 [ c 1 b + ( c 1 2 b 2 + 3 c 2 c 6 ) 1 / 2 ] ,
E = U 1 2 ξ 1 d ξ 1 d θ = 2 π c 1 E = 4 π c 1 c 4 ( c 6 - 1 4 c 2 W 0 4 ) .
f ( x ) exp ( i a θ ) = [ exp ( - x 2 ) x exp ( - x 2 ) exp ( ± i θ ) ( 1 - 2 x 2 ) exp ( - x 2 ) ] .
E max = 4 π c 1 c 6 c 4
V = 2 c 3 c 2 W 2 - c 4 E c 2 W + 2 W 2 + constant ,
Period = 2 π / ( 16 + c 4 E c 1 c 2 W 0 3 ) 1 / 2 .
W 0 2 = c 1 b + ( c 1 2 b 2 + 60 c 2 c 3 ) 1 / 2 10 c 2 ,
E = c 1 W 0 c 4 ( - c 1 b + c 2 W 0 2 + c 3 W 0 2 ) .
U i ( Z , ξ i , θ ) = j = 1 N a j ( Z ) u j ( Z , ξ i , θ ) ,
i a 1 Z + a 1 [ N ( 40 ) a 1 2 + 2 N ( 22 ) a 2 2 ] + a 1 * a 2 2 N ( 22 ) exp ( - 4 i Z ) = 0 , i a 2 Z + a 2 [ N ( 04 ) a 2 2 + 2 N ( 22 ) a 1 2 ] + a 2 * a 1 2 N ( 22 ) exp ( - 4 i Z ) = 0 ,

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