Abstract

The field of the fundamental waveguide mode, when it propagates at extremely high intensities or when the core or cladding material has a large nonlinear coefficient, may be quite significantly distorted from that of the corresponding linear mode. We derive a variational formulation of the scalar wave equation for waveguides with arbitrary nonlinearity in the core and show that this formulation can be used to find simple, but accurate, analytical approximations for these nonlinear fields. In particular, we find Gaussian and equivalent-step-index approximations for a variety of planar waveguide and optical-fiber structures and show how they can be used to calculate quantities such as the effective area and group-velocity dispersion.

© 1991 Optical Society of America

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References

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  1. For a review and relevant references, see G. I. Stegeman and C. T. Seaton, “Nonlinear integrated optics,” J. Appl. Phys. 58, R57–R78 (1985).
    [CrossRef]
  2. A. W. Snyder and R. A. Sammut, “Fundamental (HE11) modes of graded optical fibers,” J. Opt. Soc. Am. 69, 1663–1671 (1979).
    [CrossRef]
  3. R. J. Black and C. Pask, “Developments in the theory of equivalent-step-index fibers,” J. Opt. Soc. Am. A 1, 1129–1131 (1984).
    [CrossRef]
  4. E. -G. Neumann, Single-Mode Fibers (Springer-Verlag, Berlin, 1988).
    [CrossRef]
  5. A. D. Boardman and P. Egan, “Optically nonlinear waves in thin films,” IEEE J. Quantum Electron. QE-22, 319–324 (1986).
    [CrossRef]
  6. See, for example, W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1986).
  7. R. Weinstock, Calculus of Variations (Dover, New York, 1974).
  8. R. Courant and D. Hilbert, Methods of Mathematical Physics (Wiley-Interscience, New York, 1989), Vol. 1.
  9. D. Zwillinger, Handbook of Differential Equations (Academic, San Diego, 1989).
  10. K. Okamoto and E. A. J. Marcatili, “Chromatic dispersion characteristics of fibers with optical Kerr-effect nonlinearity,” IEEE J. Lightwave Tech. 7, 1988–1994 (1989).
    [CrossRef]
  11. R. A. Sammut and C. Pask, “Accurate Gaussian and equivalent step index approximations for highly nonlinear waveguides,” in Integrated Photonics Research, Vol. 5 of 1990 OSA Technical Digest Series (Optical Society of America, Washington D.C., 1990), p. 82.
  12. S. A. Akhmanov, R. V. Khokholov, and A. P. Sukhorukov, “Self-focussing, self-defocussing and self-modulation of laser beams,” in Laser Handbook, F. T. Arecchi and E. D. Schulz-Dubois, eds. (North-Holland, Amsterdam, 1972), Vol. 2, pp. 1151–1228.
  13. R. P. Brent, Algorithms for Minimization without Derivatives (Prentice-Hall, Englewood Cliffs, N.J., 1973).
  14. K. Ogusu, “Computer analysis of general nonlinear planar waveguides,” Opt. Commun. 64, 425–430 (1987).
    [CrossRef]
  15. R. A. Sammut and C. Pask, “Group velocity and dispersion in nonlinear optical fibers,” Opt. Lett. (to be published).

1989 (1)

K. Okamoto and E. A. J. Marcatili, “Chromatic dispersion characteristics of fibers with optical Kerr-effect nonlinearity,” IEEE J. Lightwave Tech. 7, 1988–1994 (1989).
[CrossRef]

1987 (1)

K. Ogusu, “Computer analysis of general nonlinear planar waveguides,” Opt. Commun. 64, 425–430 (1987).
[CrossRef]

1986 (1)

A. D. Boardman and P. Egan, “Optically nonlinear waves in thin films,” IEEE J. Quantum Electron. QE-22, 319–324 (1986).
[CrossRef]

1985 (1)

For a review and relevant references, see G. I. Stegeman and C. T. Seaton, “Nonlinear integrated optics,” J. Appl. Phys. 58, R57–R78 (1985).
[CrossRef]

1984 (1)

1979 (1)

Akhmanov, S. A.

S. A. Akhmanov, R. V. Khokholov, and A. P. Sukhorukov, “Self-focussing, self-defocussing and self-modulation of laser beams,” in Laser Handbook, F. T. Arecchi and E. D. Schulz-Dubois, eds. (North-Holland, Amsterdam, 1972), Vol. 2, pp. 1151–1228.

Black, R. J.

Boardman, A. D.

A. D. Boardman and P. Egan, “Optically nonlinear waves in thin films,” IEEE J. Quantum Electron. QE-22, 319–324 (1986).
[CrossRef]

Brent, R. P.

R. P. Brent, Algorithms for Minimization without Derivatives (Prentice-Hall, Englewood Cliffs, N.J., 1973).

Courant, R.

R. Courant and D. Hilbert, Methods of Mathematical Physics (Wiley-Interscience, New York, 1989), Vol. 1.

Egan, P.

A. D. Boardman and P. Egan, “Optically nonlinear waves in thin films,” IEEE J. Quantum Electron. QE-22, 319–324 (1986).
[CrossRef]

Flannery, B. P.

See, for example, W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1986).

Hilbert, D.

R. Courant and D. Hilbert, Methods of Mathematical Physics (Wiley-Interscience, New York, 1989), Vol. 1.

Khokholov, R. V.

S. A. Akhmanov, R. V. Khokholov, and A. P. Sukhorukov, “Self-focussing, self-defocussing and self-modulation of laser beams,” in Laser Handbook, F. T. Arecchi and E. D. Schulz-Dubois, eds. (North-Holland, Amsterdam, 1972), Vol. 2, pp. 1151–1228.

Marcatili, E. A. J.

K. Okamoto and E. A. J. Marcatili, “Chromatic dispersion characteristics of fibers with optical Kerr-effect nonlinearity,” IEEE J. Lightwave Tech. 7, 1988–1994 (1989).
[CrossRef]

Neumann, E. -G.

E. -G. Neumann, Single-Mode Fibers (Springer-Verlag, Berlin, 1988).
[CrossRef]

Ogusu, K.

K. Ogusu, “Computer analysis of general nonlinear planar waveguides,” Opt. Commun. 64, 425–430 (1987).
[CrossRef]

Okamoto, K.

K. Okamoto and E. A. J. Marcatili, “Chromatic dispersion characteristics of fibers with optical Kerr-effect nonlinearity,” IEEE J. Lightwave Tech. 7, 1988–1994 (1989).
[CrossRef]

Pask, C.

R. J. Black and C. Pask, “Developments in the theory of equivalent-step-index fibers,” J. Opt. Soc. Am. A 1, 1129–1131 (1984).
[CrossRef]

R. A. Sammut and C. Pask, “Group velocity and dispersion in nonlinear optical fibers,” Opt. Lett. (to be published).

R. A. Sammut and C. Pask, “Accurate Gaussian and equivalent step index approximations for highly nonlinear waveguides,” in Integrated Photonics Research, Vol. 5 of 1990 OSA Technical Digest Series (Optical Society of America, Washington D.C., 1990), p. 82.

Press, W. H.

See, for example, W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1986).

Sammut, R. A.

A. W. Snyder and R. A. Sammut, “Fundamental (HE11) modes of graded optical fibers,” J. Opt. Soc. Am. 69, 1663–1671 (1979).
[CrossRef]

R. A. Sammut and C. Pask, “Accurate Gaussian and equivalent step index approximations for highly nonlinear waveguides,” in Integrated Photonics Research, Vol. 5 of 1990 OSA Technical Digest Series (Optical Society of America, Washington D.C., 1990), p. 82.

R. A. Sammut and C. Pask, “Group velocity and dispersion in nonlinear optical fibers,” Opt. Lett. (to be published).

Seaton, C. T.

For a review and relevant references, see G. I. Stegeman and C. T. Seaton, “Nonlinear integrated optics,” J. Appl. Phys. 58, R57–R78 (1985).
[CrossRef]

Snyder, A. W.

Stegeman, G. I.

For a review and relevant references, see G. I. Stegeman and C. T. Seaton, “Nonlinear integrated optics,” J. Appl. Phys. 58, R57–R78 (1985).
[CrossRef]

Sukhorukov, A. P.

S. A. Akhmanov, R. V. Khokholov, and A. P. Sukhorukov, “Self-focussing, self-defocussing and self-modulation of laser beams,” in Laser Handbook, F. T. Arecchi and E. D. Schulz-Dubois, eds. (North-Holland, Amsterdam, 1972), Vol. 2, pp. 1151–1228.

Teukolsky, S. A.

See, for example, W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1986).

Vetterling, W. T.

See, for example, W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1986).

Weinstock, R.

R. Weinstock, Calculus of Variations (Dover, New York, 1974).

Zwillinger, D.

D. Zwillinger, Handbook of Differential Equations (Academic, San Diego, 1989).

IEEE J. Lightwave Tech. (1)

K. Okamoto and E. A. J. Marcatili, “Chromatic dispersion characteristics of fibers with optical Kerr-effect nonlinearity,” IEEE J. Lightwave Tech. 7, 1988–1994 (1989).
[CrossRef]

IEEE J. Quantum Electron. (1)

A. D. Boardman and P. Egan, “Optically nonlinear waves in thin films,” IEEE J. Quantum Electron. QE-22, 319–324 (1986).
[CrossRef]

J. Appl. Phys. (1)

For a review and relevant references, see G. I. Stegeman and C. T. Seaton, “Nonlinear integrated optics,” J. Appl. Phys. 58, R57–R78 (1985).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Commun. (1)

K. Ogusu, “Computer analysis of general nonlinear planar waveguides,” Opt. Commun. 64, 425–430 (1987).
[CrossRef]

Other (9)

R. A. Sammut and C. Pask, “Group velocity and dispersion in nonlinear optical fibers,” Opt. Lett. (to be published).

R. A. Sammut and C. Pask, “Accurate Gaussian and equivalent step index approximations for highly nonlinear waveguides,” in Integrated Photonics Research, Vol. 5 of 1990 OSA Technical Digest Series (Optical Society of America, Washington D.C., 1990), p. 82.

S. A. Akhmanov, R. V. Khokholov, and A. P. Sukhorukov, “Self-focussing, self-defocussing and self-modulation of laser beams,” in Laser Handbook, F. T. Arecchi and E. D. Schulz-Dubois, eds. (North-Holland, Amsterdam, 1972), Vol. 2, pp. 1151–1228.

R. P. Brent, Algorithms for Minimization without Derivatives (Prentice-Hall, Englewood Cliffs, N.J., 1973).

E. -G. Neumann, Single-Mode Fibers (Springer-Verlag, Berlin, 1988).
[CrossRef]

See, for example, W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1986).

R. Weinstock, Calculus of Variations (Dover, New York, 1974).

R. Courant and D. Hilbert, Methods of Mathematical Physics (Wiley-Interscience, New York, 1989), Vol. 1.

D. Zwillinger, Handbook of Differential Equations (Academic, San Diego, 1989).

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Figures (6)

Fig. 1
Fig. 1

Effective index as a function of power for a step-index fiber with Kerr nonlinearity in core and cladding. Fiber parameters are nco = 1.466, ncl = 1.45, wavelength = 0.632 μm, a = 1.0 μm, and n2 = 3.2 × 10−20 m2/W Solid curve, numerical result; dashed curve, Gaussian result from Eq. (24).

Fig. 2
Fig. 2

Effective index as a function of power for a step-index fiber with Kerr nonlinearity in the core. Fiber parameters are nco = 1.57, ncl = 1.55, wavelength = 0.5145 μm, a = 1.0 μm, and n2 = 10−9 m2/W. Solid curve, numerical result; dashed curve, Gaussian result from Eq. (27).

Fig. 3
Fig. 3

(a) Normalized propagation constant, (b) group-delay parameter, and (c) dispersion parameter as a function of V for a step-index fiber with Kerr nonlinearity in the core. Power is fixed at 200 kW, and fiber parameters are as for Fig. 1. Solid curves, numerical results; dashed curves, Gaussian results from Eq. (27); dotted curves, ESI results from Eq. (31).

Fig. 4
Fig. 4

(a) Effective index and (b) spot size as a function of power for a step-index planar waveguide with Kerr nonlinearity in the core. Waveguide parameters are as for Fig. 2. Solid curves, numerical results; dashed curves, Gaussian results with Eq. (34); dotted curves, ESI results from Eq. (37).

Fig. 5
Fig. 5

(a) Effective index and (b) effective area as a function of power for a graded-index planar waveguide with Kerr nonlinearity and profile defined by Eq. (38). Waveguide parameters are as for Fig. 2. Solid curves, numerical results; dashed curves, Gaussian results from Eq. (40).

Fig. 6
Fig. 6

(a) Effective index and power for a step-index planar waveguide with saturable nonlinearity in the core. Waveguide parameters are as for Fig. 2, with Δnsat2 = 0.0315. Solid curves, numerical results; dashed curves, Gaussian results from Eq. (42).

Tables (1)

Tables Icon

Table 1 Spot Size and Effective Area in Planar Waveguides and Optical Fibers: Definitions and Approximate Forms

Equations (63)

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n ( r ) = n 0 ( r ) + n 2 I ,
I = ½ n 0 c 0 E 2 ,
n 2 ( r ) = n 0 2 ( r ) + 2 n 0 n 2 I = n 0 2 ( r ) + α ψ 2 ( r ) ,
n 2 ( r ) = n 0 2 ( r ) + f ( α ψ 2 ) ,
f ( ξ ) ξ             Kerr - law media ,
f ( ξ ) ξ 1 + ξ / Δ n sat 2             saturable , two - level system ,
f ( ξ ) Δ n sat 2 [ 1 - exp ( - ξ / Δ n sat 2 ) ]             exponential saturation .
d 2 ψ d x 2 + k 2 n 0 2 ψ + k 2 f ( α ψ 2 ) ψ = β 2 ψ ,
P = c ɛ 0 β 2 k - ψ 2 d x c ɛ 0 n 0 2 - ψ 2 d x .
J = - [ - ( ψ ) 2 + k 2 n 0 2 ψ 2 + k 2 g ( α ψ 2 ) ] d x - h 1 ( ψ , ψ ) d x ,
- ψ 2 d x - h 2 ( ψ , ψ ) d x = I ,
h = h 1 + λ h 2 ,
h ψ - d d x ( h ψ ) = 0 ,
d 2 ψ d x 2 + k 2 n 0 2 ψ + 1 2 k 2 g ψ = - λ ψ .
g ( α ψ 2 ) = 1 α 0 α ψ 2 f ( ξ ) d ξ ,
J = - ( ψ ) 2 + k 2 n 0 2 ψ 2 + k 2 g ( α ψ 2 )
ψ 2 = I ,
g ( ξ ) ξ 2 / 2 α             Kerr - law media ,
g ( ξ ) Δ n sat 2 α [ ξ - Δ n sat 2 ln ( 1 + ξ Δ n sat 2 ) ]             saturable , two - level system ,
g ( ξ ) Δ n sat 2 α { ξ - Δ n sat 2 [ 1 - exp ( - ξ Δ n sat 2 ) ] }             exponential saturation .
ϕ = [ I ψ 2 ] 1 / 2 ψ .
J ˜ = - ( ψ ) 2 + k 2 n 0 2 ψ 2 + k 2 ψ 2 I g ( α I ψ 2 ψ 2 ) ψ 2 .
β 2 = - ( ψ ) 2 + k 2 n 0 2 ψ 2 + k 2 ψ 2 f ( α ψ 2 ) ψ 2 .
d 2 ψ d r 2 + 1 r d ψ d r + k 2 n 0 2 ψ + k 2 f ( α ψ 2 ) ψ = β 2 ψ ,
P = π c 0 β k ψ 2 π c 0 n 0 ψ 2 .
ψ = ( I ω π ) 1 / 2 exp ( - x 2 2 ω 2 )             planar waveguide ,
ψ = ( 2 I ) 1 / 2 ω exp ( - r 2 2 ω 2 )             optical fiber ,
n 2 = n co 2 + α ψ 2 r a , n 2 = n cl 2 + α ψ 2 r > a ,
J = I { - 1 ω 2 + k 2 n cl 2 + V 2 a 2 [ 1 - exp [ - ( a ω ) 2 ] ] + k 2 α I 2 ω 2 } ,
ω 2 = a 2 / ln V NL 2 ,
V NL 2 = V 2 / ( 1 - k 2 α I / 2 ) .
β 2 = k 2 n co 2 - 1 a 2 ( 1 + ln V NL 2 ) + α k 2 I 2 a 2 ( 1 + ln V NL 4 ) .
n 2 = n co 2 + α ψ 2 r a , n 2 = n cl 2 r > a .
J = I ( - 1 ω 2 + k 2 n cl 2 + V 2 a 2 { 1 - exp [ - ( a ω ) 2 ] } + k 2 α I 2 ω 2 { 1 - exp [ - 2 ( a ω ) 2 ] } ) ,
V 2 exp [ - ( a ω ) 2 ] + k 2 α I 2 - 1 + k 2 α I [ ( a ω ) 2 - 1 ] exp [ - 2 ( a ω ) 2 ] = 0.
β 2 = k 2 n co 2 - 1 ω 2 - V 2 a 2 exp [ - ( a ω ) 2 ] + k 2 α I ω 2 { 1 - exp [ - 2 ( a ω ) 2 ] } .
ψ ˜ = J 0 ( U r / a e ) J 0 ( U ) r a e , ψ ˜ = K 0 ( W r / a e ) K 0 ( W ) r > a e .
U 2 = a e 2 ( k 2 n e 2 - β e 2 ) ,
W 2 = a e 2 ( β e 2 - k 2 n cl 2 ) ,
J = I W 2 J 0 2 ( U ) V e 2 a e 2 J 1 2 ( U ) [ - V e 2 + V 2 K 0 2 ( W ) [ K 0 2 ( a W a e ) - K 1 2 ( a W a e ) ] + V e 2 k 2 n co 2 a e 2 W 2 J 1 2 ( U ) J 0 2 ( U ) + 2 k 2 α I W 2 J 0 2 ( U ) V e 2 a e 2 J 1 2 ( U ) 0 a ψ ˜ 4 r d r ] ,
β 2 = W 2 J 0 2 ( U ) V e 2 a e 2 J 1 2 ( U ) [ - V e 2 + V 2 K 0 2 ( W ) [ K 0 2 ( a W a e ) - K 1 2 ( a W a e ) ] + V e 2 k 2 n co 2 a e 2 W 2 J 1 2 ( U ) J 0 2 ( U ) + 4 k 2 α I W 2 J 0 2 ( U ) V e 2 a e 2 J 1 2 ( U ) 0 a ψ ˜ 4 r d r ] .
n 2 = n co 2 + α ψ 2 x a , n 2 = n cl 2 x > a .
J = I [ - 1 2 ω 2 + k 2 n cl 2 + V 2 a 2 erf ( a ω ) + k 2 α I 2 2 π ω erf ( 2 a ω ) ]
2 π V 2 exp [ - ( a ω ) 2 ] - a ω + k 2 α I a 2 π { erf ( 2 a ω ) + 2 ( 2 π ) 1 / 2 a ω exp [ - 2 ( a ω ) 2 ] } = 0 ,
β 2 = k 2 n cl 2 - 1 2 ω 2 + V 2 a 2 erf ( a ω ) + k 2 α I ω 2 π erf ( 2 a ω ) .
ψ = A cos ( U x / a e ) cos ( U ) x a e , ψ = A exp [ - W ( x a e - 1 ) ] x > a e ,
J = I W a e ( 1 + W ) [ - U 2 a e + k 2 n co 2 a e ( 1 + 1 W ) - a e V 2 U 2 a 2 V e 2 W exp [ - 2 W ( a a e - 1 ) ] + k 2 α I W 2 ( 1 + W ) ( 3 4 + W V e 2 + W ( U 2 - W 2 ) 4 V 3 4 + U 4 2 W V e 4 { 1 - exp [ - 4 W ( a a e - 1 ) ] } ) ] ,
β 2 = W a e ( 1 + W ) [ - U 2 a e + k 2 n co 2 a e ( 1 + 1 W ) - a e V 2 U 2 a 2 V e 2 W exp [ - 2 W ( a a e - 1 ) ] + k 2 α I W ( 1 + W ) ( 3 4 + W V e 2 + W ( U 2 - W 2 ) 4 V e 4 + U 4 2 W V e 4 { 1 - exp [ - 4 W ( a a e - 1 ) ] } ) ] .
n 2 = n cl 2 + ( n co 2 - n cl 2 ) exp [ - ( x / a ) 2 ] + α ψ 2 ,
J = I [ - 1 2 ω 2 + k 2 n cl 2 + V 2 a ( ω 2 + a 2 ) 1 / 2 + k 2 α I 2 2 π ω ] ,
V 2 ( 1 + a 2 ω 2 ) - 3 / 2 - a ω + k 2 α I a 2 2 π = 0.
β 2 = k 2 n cl 2 - 1 2 ω 2 + V 2 a ( ω 2 + a 2 ) 1 / 2 + k 2 α I 2 2 π ω ,
J = I { - 1 2 ω 2 + k 2 n cl 2 + V 2 a 2 erf ( a ω ) + 2 k 2 I 0 a g [ α I w π exp ( - x 2 w 2 ) ] d x } ,
2 π V 2 exp [ - ( a ω ) 2 ] - a ω - 2 k 2 a ω 2 π 0 a ( 2 x 2 - w 2 ) × exp ( - x 2 w 2 ) f [ α I w π exp ( - x 2 2 w 2 ) ] d x ,
β 2 = k 2 n cl 2 - 1 2 ω 2 + V 2 a 2 erf ( a / ω ) + 2 k 2 ω π 0 a exp ( - x 2 w 2 ) f [ a I w π exp ( - x 2 w 2 ) ] d x .
n 2 = n co 2 + Δ n sat 2 [ 1 - exp ( - α ψ 2 Δ n sat 2 ) ] x a , n 2 = n cl 2 x > a ,
β k d β d k = ( n 0 d d k ( k n 0 ) + f ( α ψ 2 ) + k 2 d f d k ) ψ 2 ψ 2 .
b = ( β / k ) 2 - n cl 2 n co 2 - n cl 2 .
β k n cl + k ( n co - n cl ) b .
d β d k = N cl + ( N co - N cl ) d ( V ) ,
k d 2 β d k 2 = k d N cl d k + k ( d N co d k - d N cl d k ) d ( V ) + ( N co - N cl ) g ( V ) ,
d ( V ) = d ( V b ) d V ,
g ( V ) = V d 2 ( V b ) d V 2 .

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