Abstract

Intensity-dependent properties of nonlinear-optical waves guided by dielectric waveguiding structures with two-dimensional power confinement are analyzed by the vectorial finite-element method. In this approach self-consistent solutions are obtainable through an iterative procedure, and no spurious solutions are involved in the region under consideration.

© 1988 Optical Society of America

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References

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  1. G. I. Stegeman and C. T. Seaton, “Nonlinear integrated optics,” J. Appl. Phys. 58, R57–R78 (1985).
    [Crossref]
  2. J. Chrostowski and S. Chelkowski, “Analysis of an optical rib waveguide with a nonlinear substrate,” Opt. Lett. 12, 528–530 (1987).
    [Crossref] [PubMed]
  3. A. A. Maradudin, “Nonlinear surface electromagnetic waves,” in Optical and Acoustic Waves in Solids—Modern Topics, M. Borissov, ed. (World Scientific, Singapore, 1983), pp. 72–142.
  4. A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, and E. M. Wright, “Exact theory of nonlinear p-polarized optical waves,” Phys. Rev. A 35, 1159–1164 (1987).
    [Crossref] [PubMed]
  5. O. C. Zienkiewicz, The Finite Element Method, 3rd ed. (McGraw-Hill, London, 1977).
  6. M. Koshiba, K. Hayata, and M. Suzuki, “Vectorial finite-element formulation without spurious modes for dielectric waveguides,” Trans. Inst. Electron. Commun. Eng. Jpn. E67, 191–196 (1984).
  7. B. M. A. Rahman and J. B. Davies, “Penalty function improvement of waveguide solution by finite elements,” IEEE Trans. Microwave Theory Tech. MTT-32, 922–928 (1984).
    [Crossref]
  8. M. Koshiba, K. Hayata, and M. Suzuki, “Study of spurious solutions of finite-element methods in the three-component magnetic field formulation for dielectric waveguide problems,” Electron. Commun. Jpn. Part 168, 114–119 (1985).
  9. K. Hayata, M. Nagai, and M. Koshiba, “Finite-element formalism for nonlinear slab-guided waves,” IEEE Trans. Microwave Theory Tech. MTT-36, 1207–1215 (1988).
    [Crossref]
  10. K. Hayata and M. Koshiba, “Symmetry breaking instabilities and bifurcation phenomena in dielectric slab waveguides containing saturable nonlinear media,” presented at the Second Optoelectronics Conference, Tokyo, 1988.
  11. K. Hayata and M. Koshiba, “Self-focusing instability and chaotic behavior of nonlinear optical waves guided by slab structures,” Opt. Lett. (to be published).
  12. H. Haken, “The adiabatic elimination principle in dynamical theories,” in Optical Instabilities, R. W. Boyd, M. G. Raymer, and L. M. Narducci, eds. (Cambridge U. Press, Cambridge, 1986), pp. 1–19.

1988 (1)

K. Hayata, M. Nagai, and M. Koshiba, “Finite-element formalism for nonlinear slab-guided waves,” IEEE Trans. Microwave Theory Tech. MTT-36, 1207–1215 (1988).
[Crossref]

1987 (2)

J. Chrostowski and S. Chelkowski, “Analysis of an optical rib waveguide with a nonlinear substrate,” Opt. Lett. 12, 528–530 (1987).
[Crossref] [PubMed]

A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, and E. M. Wright, “Exact theory of nonlinear p-polarized optical waves,” Phys. Rev. A 35, 1159–1164 (1987).
[Crossref] [PubMed]

1985 (2)

G. I. Stegeman and C. T. Seaton, “Nonlinear integrated optics,” J. Appl. Phys. 58, R57–R78 (1985).
[Crossref]

M. Koshiba, K. Hayata, and M. Suzuki, “Study of spurious solutions of finite-element methods in the three-component magnetic field formulation for dielectric waveguide problems,” Electron. Commun. Jpn. Part 168, 114–119 (1985).

1984 (2)

M. Koshiba, K. Hayata, and M. Suzuki, “Vectorial finite-element formulation without spurious modes for dielectric waveguides,” Trans. Inst. Electron. Commun. Eng. Jpn. E67, 191–196 (1984).

B. M. A. Rahman and J. B. Davies, “Penalty function improvement of waveguide solution by finite elements,” IEEE Trans. Microwave Theory Tech. MTT-32, 922–928 (1984).
[Crossref]

Boardman, A. D.

A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, and E. M. Wright, “Exact theory of nonlinear p-polarized optical waves,” Phys. Rev. A 35, 1159–1164 (1987).
[Crossref] [PubMed]

Chelkowski, S.

Chrostowski, J.

Davies, J. B.

B. M. A. Rahman and J. B. Davies, “Penalty function improvement of waveguide solution by finite elements,” IEEE Trans. Microwave Theory Tech. MTT-32, 922–928 (1984).
[Crossref]

Haken, H.

H. Haken, “The adiabatic elimination principle in dynamical theories,” in Optical Instabilities, R. W. Boyd, M. G. Raymer, and L. M. Narducci, eds. (Cambridge U. Press, Cambridge, 1986), pp. 1–19.

Hayata, K.

K. Hayata, M. Nagai, and M. Koshiba, “Finite-element formalism for nonlinear slab-guided waves,” IEEE Trans. Microwave Theory Tech. MTT-36, 1207–1215 (1988).
[Crossref]

M. Koshiba, K. Hayata, and M. Suzuki, “Study of spurious solutions of finite-element methods in the three-component magnetic field formulation for dielectric waveguide problems,” Electron. Commun. Jpn. Part 168, 114–119 (1985).

M. Koshiba, K. Hayata, and M. Suzuki, “Vectorial finite-element formulation without spurious modes for dielectric waveguides,” Trans. Inst. Electron. Commun. Eng. Jpn. E67, 191–196 (1984).

K. Hayata and M. Koshiba, “Symmetry breaking instabilities and bifurcation phenomena in dielectric slab waveguides containing saturable nonlinear media,” presented at the Second Optoelectronics Conference, Tokyo, 1988.

K. Hayata and M. Koshiba, “Self-focusing instability and chaotic behavior of nonlinear optical waves guided by slab structures,” Opt. Lett. (to be published).

Koshiba, M.

K. Hayata, M. Nagai, and M. Koshiba, “Finite-element formalism for nonlinear slab-guided waves,” IEEE Trans. Microwave Theory Tech. MTT-36, 1207–1215 (1988).
[Crossref]

M. Koshiba, K. Hayata, and M. Suzuki, “Study of spurious solutions of finite-element methods in the three-component magnetic field formulation for dielectric waveguide problems,” Electron. Commun. Jpn. Part 168, 114–119 (1985).

M. Koshiba, K. Hayata, and M. Suzuki, “Vectorial finite-element formulation without spurious modes for dielectric waveguides,” Trans. Inst. Electron. Commun. Eng. Jpn. E67, 191–196 (1984).

K. Hayata and M. Koshiba, “Self-focusing instability and chaotic behavior of nonlinear optical waves guided by slab structures,” Opt. Lett. (to be published).

K. Hayata and M. Koshiba, “Symmetry breaking instabilities and bifurcation phenomena in dielectric slab waveguides containing saturable nonlinear media,” presented at the Second Optoelectronics Conference, Tokyo, 1988.

Maradudin, A. A.

A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, and E. M. Wright, “Exact theory of nonlinear p-polarized optical waves,” Phys. Rev. A 35, 1159–1164 (1987).
[Crossref] [PubMed]

A. A. Maradudin, “Nonlinear surface electromagnetic waves,” in Optical and Acoustic Waves in Solids—Modern Topics, M. Borissov, ed. (World Scientific, Singapore, 1983), pp. 72–142.

Nagai, M.

K. Hayata, M. Nagai, and M. Koshiba, “Finite-element formalism for nonlinear slab-guided waves,” IEEE Trans. Microwave Theory Tech. MTT-36, 1207–1215 (1988).
[Crossref]

Rahman, B. M. A.

B. M. A. Rahman and J. B. Davies, “Penalty function improvement of waveguide solution by finite elements,” IEEE Trans. Microwave Theory Tech. MTT-32, 922–928 (1984).
[Crossref]

Seaton, C. T.

G. I. Stegeman and C. T. Seaton, “Nonlinear integrated optics,” J. Appl. Phys. 58, R57–R78 (1985).
[Crossref]

Stegeman, G. I.

A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, and E. M. Wright, “Exact theory of nonlinear p-polarized optical waves,” Phys. Rev. A 35, 1159–1164 (1987).
[Crossref] [PubMed]

G. I. Stegeman and C. T. Seaton, “Nonlinear integrated optics,” J. Appl. Phys. 58, R57–R78 (1985).
[Crossref]

Suzuki, M.

M. Koshiba, K. Hayata, and M. Suzuki, “Study of spurious solutions of finite-element methods in the three-component magnetic field formulation for dielectric waveguide problems,” Electron. Commun. Jpn. Part 168, 114–119 (1985).

M. Koshiba, K. Hayata, and M. Suzuki, “Vectorial finite-element formulation without spurious modes for dielectric waveguides,” Trans. Inst. Electron. Commun. Eng. Jpn. E67, 191–196 (1984).

Twardowski, T.

A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, and E. M. Wright, “Exact theory of nonlinear p-polarized optical waves,” Phys. Rev. A 35, 1159–1164 (1987).
[Crossref] [PubMed]

Wright, E. M.

A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, and E. M. Wright, “Exact theory of nonlinear p-polarized optical waves,” Phys. Rev. A 35, 1159–1164 (1987).
[Crossref] [PubMed]

Zienkiewicz, O. C.

O. C. Zienkiewicz, The Finite Element Method, 3rd ed. (McGraw-Hill, London, 1977).

Electron. Commun. Jpn. Part (1)

M. Koshiba, K. Hayata, and M. Suzuki, “Study of spurious solutions of finite-element methods in the three-component magnetic field formulation for dielectric waveguide problems,” Electron. Commun. Jpn. Part 168, 114–119 (1985).

IEEE Trans. Microwave Theory Tech. (2)

K. Hayata, M. Nagai, and M. Koshiba, “Finite-element formalism for nonlinear slab-guided waves,” IEEE Trans. Microwave Theory Tech. MTT-36, 1207–1215 (1988).
[Crossref]

B. M. A. Rahman and J. B. Davies, “Penalty function improvement of waveguide solution by finite elements,” IEEE Trans. Microwave Theory Tech. MTT-32, 922–928 (1984).
[Crossref]

J. Appl. Phys. (1)

G. I. Stegeman and C. T. Seaton, “Nonlinear integrated optics,” J. Appl. Phys. 58, R57–R78 (1985).
[Crossref]

Opt. Lett. (1)

Phys. Rev. A (1)

A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, and E. M. Wright, “Exact theory of nonlinear p-polarized optical waves,” Phys. Rev. A 35, 1159–1164 (1987).
[Crossref] [PubMed]

Trans. Inst. Electron. Commun. Eng. Jpn. (1)

M. Koshiba, K. Hayata, and M. Suzuki, “Vectorial finite-element formulation without spurious modes for dielectric waveguides,” Trans. Inst. Electron. Commun. Eng. Jpn. E67, 191–196 (1984).

Other (5)

O. C. Zienkiewicz, The Finite Element Method, 3rd ed. (McGraw-Hill, London, 1977).

A. A. Maradudin, “Nonlinear surface electromagnetic waves,” in Optical and Acoustic Waves in Solids—Modern Topics, M. Borissov, ed. (World Scientific, Singapore, 1983), pp. 72–142.

K. Hayata and M. Koshiba, “Symmetry breaking instabilities and bifurcation phenomena in dielectric slab waveguides containing saturable nonlinear media,” presented at the Second Optoelectronics Conference, Tokyo, 1988.

K. Hayata and M. Koshiba, “Self-focusing instability and chaotic behavior of nonlinear optical waves guided by slab structures,” Opt. Lett. (to be published).

H. Haken, “The adiabatic elimination principle in dynamical theories,” in Optical Instabilities, R. W. Boyd, M. G. Raymer, and L. M. Narducci, eds. (Cambridge U. Press, Cambridge, 1986), pp. 1–19.

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Figures (12)

Fig. 1
Fig. 1

Element division for rectangular channel waveguide with nonlinear core and linear cladding (48 elements, 117 nodal points); electric wall on AB ¯ and magnetic wall on BC ¯ are imposed for the E11x mode; electric walls are assumed on AD ¯ and CD ¯.

Fig. 2
Fig. 2

Convergence behavior of the effective refractive index: b = a, P = 30 μW.

Fig. 3
Fig. 3

Dependence of the effective refractive index on total optical power.

Fig. 4
Fig. 4

Comparison of dispersion relations for different nonlinear mechanisms: A, b = a; B, b = a/3; C, b = −a/2.

Fig. 5
Fig. 5

Variation of dominant magnetic-field strength as a function of total optical power: solid, dashed, and dotted curves indicate b = a, b = a/3, and b = −a/2, respectively.

Fig. 6
Fig. 6

Element division for rectangular channel waveguide with linear core and nonlinear cladding (120 elements, 273 nodal points). Electric walls are assumed on AB ¯ , BC ¯ , CD ¯, and AD ¯.

Fig. 7
Fig. 7

Dispersion relations for the symmetric mode: A, b = a; B, b = a/3; C, b = −a/2. The solid and dashed curves indicate stability and instability, respectively.

Fig. 8
Fig. 8

Dispersion relations for the stable mode: A, b = a; B, b = a/3; C, b = −a/2.

Fig. 9
Fig. 9

Convergence behavior for multiple solutions at P = 160 μW. SM and AM indicate, respectively, symmetric mode and asymmetric mode.

Fig. 10
Fig. 10

Magnetic-field distributions for N = 1 in Fig. 9. These correspond to the linear E11y mode, i.e., the initial condition in iterations. The unit of magnetic-field components is amperes per meter. (Note that the scale of the ordinate is different among the three components.) The rectangular core is located within 4 ≤ x ≤ 8 and 3 ≤ y ≤ 5: (a) Hx, (b) Hy, (c) Hz.

Fig. 11
Fig. 11

Magnetic-field distributions for N = 12 in Fig. 9. These correspond to the nonlinear symmetric mode’s being unstable. The meaning of the axes is as in Fig. 10: (a) Hx, (b) Hy, (c) Hz.

Fig. 12
Fig. 12

Magnetic-field distributions for N = 27 in Fig. 9. These correspond to the nonlinear asymmetric mode’s being stable. The meaning of the axes is as in Fig. 10: (a)Hx, (b) Hy, (c) Hz.

Equations (27)

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× ( [ K ] H ) - k 0 2 H = 0 ,
· H = 0 ,
[ K ] = [ x 0 0 0 y 0 0 0 z ] ,
x = + a f ( E x ) + b f ( E y ) + b f ( E z ) ,
y = + b f ( E x ) + a f ( E y ) + b f ( E z ) ,
z = + b f ( E x ) + b f ( E y ) + a f ( E z ) ,
a = c 0 0 n ¯ ,
H = [ N ] T { H } e exp ( - j β z ) ,
[ N ] = [ { N } { 0 } { 0 } { 0 } { N } { 0 } { 0 } { 0 } j { N } ]
{ H } e = [ { H x } e { H y } e { H z } e ] .
{ N } = ( N 1 , N 2 , N 3 , N 4 , N 5 , N 6 ) T ,
N i = L i ( 2 L i - 1 )             for i = 1 , 2 , 3 ,
N 4 = 4 L 1 L 2 ,             N 5 = 4 L 2 L 3 ,             N 6 = 4 L 3 L 1 .
[ L 1 L 2 L 3 ] = [ x 1 x 2 x 3 y 1 y 2 y 3 1 1 1 ] - 1 [ x y 1 ] ,
( [ S ( H ) ] - ν - 2 [ T ] ) { H } = { 0 } ,
[ S ( H ) ] = e e [ B ( β ¯ ) ] * [ K ] e - 1 [ B ( β ¯ ) ] T d x ¯ d y ¯ + p 2 e e { C ( β ¯ ) } { C ( β ¯ ) } T d x ¯ d y ¯ ,
[ T ] = e e [ N ] * [ N ] T d x ¯ d y ¯ ,
[ B ( β ¯ ) ] = [ { 0 } - j { N } - { N } y j { N } { 0 } { N } x j { N } y - j { N } x { 0 } ] ,
{ C ( β ¯ ) } = ( { N } x T { N } y T { N } T ) T .
E x = ( ν Z 0 / x ) [ H y - ( j H z ) / y ¯ ] ,
E y = ( - ν Z 0 / y ) [ H x - ( j H z ) / x ¯ ] ,
E z = ( j ν Z 0 / z ) ( H x / y ¯ - H y / x ¯ ) .
P = ( 1 / 2 ) - + - + ( E x H y * - E y H x * ) d x d y = Z 0 2 k 0 β - + - + ( 1 y H x 2 + 1 x H y 2 - 1 y ( j H z ) x ¯ H x * - 1 x ( j H z ) y ¯ H y * ) d x ¯ d y ¯ ,
P = Z 0 2 k 0 β [ { H x } T [ R y ] { H x } + { H y } T [ R x ] { H y } - { H z } T [ P y ] { H x } - { H z } T [ Q x ] { H y } ] ,
[ P i ] = e e 1 i { N } x { N } T d x ¯ d y ¯ ,
[ Q i ] = e e 1 i { N } y { N } T d x ¯ d y ¯ ,
[ R i ] = e e 1 i { N } { N } T d x ¯ d y ¯ .

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