Abstract

We examine an application of scalar field theory to the prediction of modal properties of highly localized optical fields that propagate stably along a nonlinear channel waveguide. By comparison with results previously computed with the exact vectorial approach, serious limitations on the validity of the scalar approximation are found in the high-power regime, where nonlinearities are expected to play an essential role in forming the stationary mode. As in the vectorial calculation, a catastrophic transition from a symmetric to an asymmetric mode is confirmed for the channel waveguide with a linear core embedded in a self-focusing nonlinear cladding.

© 1992 Optical Society of America

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References

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  1. G. I. Stegeman and E. M. Wright, “All-optical waveguide switching,” Opt. Quantum Electron. 22, 95–122 (1990).
    [Crossref]
  2. W. J. Tomlinson, “Surface wave at a nonlinear interface,” Opt. Lett. 5, 323 (1980).
    [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. K. Hayata and M. Koshiba, “Full vectorial analysis of nonlinear-optical waveguides,” J. Opt. Soc. Am. B 5, 2494–2501 (1988);in this paper the authors found a typographical error in Eq. (1) and errors in total powers on Figs. 2–5 and 7–9. The errata are as follows:(a)The first term on the left-hand side of Eq. (1) should be replaced by ∇ × ([K′]∇ × H).(b)In the caption of Fig. 2, P= 30 μ W should be replaced by P= 15 μW.(c)The quantity P, μW on the abscissas of Figs. 3–5, 7, and 8 should be replaced by 2P, μW.(d)In the caption of Fig. 9, P= 160 μW should be replaced by P= 80 μW.
    [Crossref]
  5. D. I. Abakarov, A. A. Akopyan, and S. I. Pekar, “Contribution to the theory of self focusing of light in nonlinearly polarizing media,” Sov. Phys. JETP 25, 303–305 (1967).
  6. D. Pohl, “Vectorial theory of self-trapped light beams,” Opt. Commun. 2, 305–308 (1970).
    [Crossref]
  7. Y. Chen and A. W. Snyder, “TM type self-guided beams with circular cross-section,” Electron. Lett. 27, 565–566 (1991).
    [Crossref]
  8. Y. Chen, “TE and TM families of self-trapped beams,” IEEE J. Quantum Electron. 27, 1236–1241 (1991).
    [Crossref]
  9. J.-L. Archambault, S. Lacroix, and A. W. Snyder, “HEImself-guided modes: vector solutions and stability,” in Nonlinear Guided-Wave Phenomena, Vol. 15 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), paper TuD4–1.
  10. A. W. Snyder, J. D. Mitchell, L. Poladian, and F. Ladouceur, “Self-induced optical fibers: spatial solitary waves,” Opt. Lett. 16, 21–23 (1991).
    [Crossref] [PubMed]
  11. M. Koshiba, K. Hayata, and M. Suzuki, “Approximate scalar finite-element analysis of anisotropic optical waveguides,” Electron. Lett. 18, 411–413 (1982).
    [Crossref]
  12. M. Koshiba, K. Hayata, and M. Suzuki, “On accuracy of approximate scalar finite-element analysis of dielectric optical waveguides,” Trans. Inst. Electron. Commun. Eng. Jpn. E66,157–158 (1983).
  13. K. Hayata, M. Koshiba, and M. Suzuki, “Lateral mode analysis of buried heterostructure diode lasers by the finite-element method,” IEEE J. Quantum Electron. QE-22, 781–788 (1986).
    [Crossref]
  14. J. H. Marburger, “Self-focusing: theory,” Prog. Quant. Electron. 4, 35–110 (1975), and references therein.
    [Crossref]
  15. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), pp. 303–333, and references therein.
  16. V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, “Quality-factor and nonlinear properties of optical whispering-gallery modes,” Phys. Lett. A 137, 393–397 (1989).
    [Crossref]
  17. Y. Sakai, R. J. Hawkins, and S. R. Friberg, “Soliton-collision interferometer for the quantum nondemolition measurement of photon number: numerical results,” Opt. Lett. 15, 239–241 (1990).
    [Crossref] [PubMed]
  18. K. Hayata and M. Koshiba, “Multidimensional complex whispering galleries: basic concept and their applicability to quantum nondemolition measurements,” Tech. Res. Rep. (Institute of Electrical Engineers of Japan/Institute of Electronics, Information, and Communication Engineers, Tokyo, 1991), paper EMT-91-67.
  19. K. Hayata and M. Koshiba, “Mutual guiding assistance between eigenmodes of nonlinearly coupled TE and TM waves,” Trans. Inst. Electron. Inform. Commun. Eng. E 74, 2890–2897 (1991).

1991 (4)

Y. Chen and A. W. Snyder, “TM type self-guided beams with circular cross-section,” Electron. Lett. 27, 565–566 (1991).
[Crossref]

Y. Chen, “TE and TM families of self-trapped beams,” IEEE J. Quantum Electron. 27, 1236–1241 (1991).
[Crossref]

A. W. Snyder, J. D. Mitchell, L. Poladian, and F. Ladouceur, “Self-induced optical fibers: spatial solitary waves,” Opt. Lett. 16, 21–23 (1991).
[Crossref] [PubMed]

K. Hayata and M. Koshiba, “Mutual guiding assistance between eigenmodes of nonlinearly coupled TE and TM waves,” Trans. Inst. Electron. Inform. Commun. Eng. E 74, 2890–2897 (1991).

1990 (2)

1989 (1)

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, “Quality-factor and nonlinear properties of optical whispering-gallery modes,” Phys. Lett. A 137, 393–397 (1989).
[Crossref]

1988 (1)

1986 (1)

K. Hayata, M. Koshiba, and M. Suzuki, “Lateral mode analysis of buried heterostructure diode lasers by the finite-element method,” IEEE J. Quantum Electron. QE-22, 781–788 (1986).
[Crossref]

1983 (1)

M. Koshiba, K. Hayata, and M. Suzuki, “On accuracy of approximate scalar finite-element analysis of dielectric optical waveguides,” Trans. Inst. Electron. Commun. Eng. Jpn. E66,157–158 (1983).

1982 (1)

M. Koshiba, K. Hayata, and M. Suzuki, “Approximate scalar finite-element analysis of anisotropic optical waveguides,” Electron. Lett. 18, 411–413 (1982).
[Crossref]

1980 (1)

1975 (1)

J. H. Marburger, “Self-focusing: theory,” Prog. Quant. Electron. 4, 35–110 (1975), and references therein.
[Crossref]

1970 (1)

D. Pohl, “Vectorial theory of self-trapped light beams,” Opt. Commun. 2, 305–308 (1970).
[Crossref]

1967 (1)

D. I. Abakarov, A. A. Akopyan, and S. I. Pekar, “Contribution to the theory of self focusing of light in nonlinearly polarizing media,” Sov. Phys. JETP 25, 303–305 (1967).

Abakarov, D. I.

D. I. Abakarov, A. A. Akopyan, and S. I. Pekar, “Contribution to the theory of self focusing of light in nonlinearly polarizing media,” Sov. Phys. JETP 25, 303–305 (1967).

Akopyan, A. A.

D. I. Abakarov, A. A. Akopyan, and S. I. Pekar, “Contribution to the theory of self focusing of light in nonlinearly polarizing media,” Sov. Phys. JETP 25, 303–305 (1967).

Archambault, J.-L.

J.-L. Archambault, S. Lacroix, and A. W. Snyder, “HEImself-guided modes: vector solutions and stability,” in Nonlinear Guided-Wave Phenomena, Vol. 15 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), paper TuD4–1.

Braginsky, V. B.

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, “Quality-factor and nonlinear properties of optical whispering-gallery modes,” Phys. Lett. A 137, 393–397 (1989).
[Crossref]

Chen, Y.

Y. Chen and A. W. Snyder, “TM type self-guided beams with circular cross-section,” Electron. Lett. 27, 565–566 (1991).
[Crossref]

Y. Chen, “TE and TM families of self-trapped beams,” IEEE J. Quantum Electron. 27, 1236–1241 (1991).
[Crossref]

Friberg, S. R.

Gorodetsky, M. L.

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, “Quality-factor and nonlinear properties of optical whispering-gallery modes,” Phys. Lett. A 137, 393–397 (1989).
[Crossref]

Hawkins, R. J.

Hayata, K.

K. Hayata and M. Koshiba, “Mutual guiding assistance between eigenmodes of nonlinearly coupled TE and TM waves,” Trans. Inst. Electron. Inform. Commun. Eng. E 74, 2890–2897 (1991).

K. Hayata and M. Koshiba, “Full vectorial analysis of nonlinear-optical waveguides,” J. Opt. Soc. Am. B 5, 2494–2501 (1988);in this paper the authors found a typographical error in Eq. (1) and errors in total powers on Figs. 2–5 and 7–9. The errata are as follows:(a)The first term on the left-hand side of Eq. (1) should be replaced by ∇ × ([K′]∇ × H).(b)In the caption of Fig. 2, P= 30 μ W should be replaced by P= 15 μW.(c)The quantity P, μW on the abscissas of Figs. 3–5, 7, and 8 should be replaced by 2P, μW.(d)In the caption of Fig. 9, P= 160 μW should be replaced by P= 80 μW.
[Crossref]

K. Hayata, M. Koshiba, and M. Suzuki, “Lateral mode analysis of buried heterostructure diode lasers by the finite-element method,” IEEE J. Quantum Electron. QE-22, 781–788 (1986).
[Crossref]

M. Koshiba, K. Hayata, and M. Suzuki, “On accuracy of approximate scalar finite-element analysis of dielectric optical waveguides,” Trans. Inst. Electron. Commun. Eng. Jpn. E66,157–158 (1983).

M. Koshiba, K. Hayata, and M. Suzuki, “Approximate scalar finite-element analysis of anisotropic optical waveguides,” Electron. Lett. 18, 411–413 (1982).
[Crossref]

K. Hayata and M. Koshiba, “Multidimensional complex whispering galleries: basic concept and their applicability to quantum nondemolition measurements,” Tech. Res. Rep. (Institute of Electrical Engineers of Japan/Institute of Electronics, Information, and Communication Engineers, Tokyo, 1991), paper EMT-91-67.

Ilchenko, V. S.

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, “Quality-factor and nonlinear properties of optical whispering-gallery modes,” Phys. Lett. A 137, 393–397 (1989).
[Crossref]

Koshiba, M.

K. Hayata and M. Koshiba, “Mutual guiding assistance between eigenmodes of nonlinearly coupled TE and TM waves,” Trans. Inst. Electron. Inform. Commun. Eng. E 74, 2890–2897 (1991).

K. Hayata and M. Koshiba, “Full vectorial analysis of nonlinear-optical waveguides,” J. Opt. Soc. Am. B 5, 2494–2501 (1988);in this paper the authors found a typographical error in Eq. (1) and errors in total powers on Figs. 2–5 and 7–9. The errata are as follows:(a)The first term on the left-hand side of Eq. (1) should be replaced by ∇ × ([K′]∇ × H).(b)In the caption of Fig. 2, P= 30 μ W should be replaced by P= 15 μW.(c)The quantity P, μW on the abscissas of Figs. 3–5, 7, and 8 should be replaced by 2P, μW.(d)In the caption of Fig. 9, P= 160 μW should be replaced by P= 80 μW.
[Crossref]

K. Hayata, M. Koshiba, and M. Suzuki, “Lateral mode analysis of buried heterostructure diode lasers by the finite-element method,” IEEE J. Quantum Electron. QE-22, 781–788 (1986).
[Crossref]

M. Koshiba, K. Hayata, and M. Suzuki, “On accuracy of approximate scalar finite-element analysis of dielectric optical waveguides,” Trans. Inst. Electron. Commun. Eng. Jpn. E66,157–158 (1983).

M. Koshiba, K. Hayata, and M. Suzuki, “Approximate scalar finite-element analysis of anisotropic optical waveguides,” Electron. Lett. 18, 411–413 (1982).
[Crossref]

K. Hayata and M. Koshiba, “Multidimensional complex whispering galleries: basic concept and their applicability to quantum nondemolition measurements,” Tech. Res. Rep. (Institute of Electrical Engineers of Japan/Institute of Electronics, Information, and Communication Engineers, Tokyo, 1991), paper EMT-91-67.

Lacroix, S.

J.-L. Archambault, S. Lacroix, and A. W. Snyder, “HEImself-guided modes: vector solutions and stability,” in Nonlinear Guided-Wave Phenomena, Vol. 15 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), paper TuD4–1.

Ladouceur, F.

Maradudin, A. A.

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.

Marburger, J. H.

J. H. Marburger, “Self-focusing: theory,” Prog. Quant. Electron. 4, 35–110 (1975), and references therein.
[Crossref]

Mitchell, J. D.

Pekar, S. I.

D. I. Abakarov, A. A. Akopyan, and S. I. Pekar, “Contribution to the theory of self focusing of light in nonlinearly polarizing media,” Sov. Phys. JETP 25, 303–305 (1967).

Pohl, D.

D. Pohl, “Vectorial theory of self-trapped light beams,” Opt. Commun. 2, 305–308 (1970).
[Crossref]

Poladian, L.

Sakai, Y.

Shen, Y. R.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), pp. 303–333, and references therein.

Snyder, A. W.

Y. Chen and A. W. Snyder, “TM type self-guided beams with circular cross-section,” Electron. Lett. 27, 565–566 (1991).
[Crossref]

A. W. Snyder, J. D. Mitchell, L. Poladian, and F. Ladouceur, “Self-induced optical fibers: spatial solitary waves,” Opt. Lett. 16, 21–23 (1991).
[Crossref] [PubMed]

J.-L. Archambault, S. Lacroix, and A. W. Snyder, “HEImself-guided modes: vector solutions and stability,” in Nonlinear Guided-Wave Phenomena, Vol. 15 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), paper TuD4–1.

Stegeman, G. I.

G. I. Stegeman and E. M. Wright, “All-optical waveguide switching,” Opt. Quantum Electron. 22, 95–122 (1990).
[Crossref]

Suzuki, M.

K. Hayata, M. Koshiba, and M. Suzuki, “Lateral mode analysis of buried heterostructure diode lasers by the finite-element method,” IEEE J. Quantum Electron. QE-22, 781–788 (1986).
[Crossref]

M. Koshiba, K. Hayata, and M. Suzuki, “On accuracy of approximate scalar finite-element analysis of dielectric optical waveguides,” Trans. Inst. Electron. Commun. Eng. Jpn. E66,157–158 (1983).

M. Koshiba, K. Hayata, and M. Suzuki, “Approximate scalar finite-element analysis of anisotropic optical waveguides,” Electron. Lett. 18, 411–413 (1982).
[Crossref]

Tomlinson, W. J.

Wright, E. M.

G. I. Stegeman and E. M. Wright, “All-optical waveguide switching,” Opt. Quantum Electron. 22, 95–122 (1990).
[Crossref]

Electron. Lett. (2)

Y. Chen and A. W. Snyder, “TM type self-guided beams with circular cross-section,” Electron. Lett. 27, 565–566 (1991).
[Crossref]

M. Koshiba, K. Hayata, and M. Suzuki, “Approximate scalar finite-element analysis of anisotropic optical waveguides,” Electron. Lett. 18, 411–413 (1982).
[Crossref]

IEEE J. Quantum Electron. (2)

K. Hayata, M. Koshiba, and M. Suzuki, “Lateral mode analysis of buried heterostructure diode lasers by the finite-element method,” IEEE J. Quantum Electron. QE-22, 781–788 (1986).
[Crossref]

Y. Chen, “TE and TM families of self-trapped beams,” IEEE J. Quantum Electron. 27, 1236–1241 (1991).
[Crossref]

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

Opt. Commun. (1)

D. Pohl, “Vectorial theory of self-trapped light beams,” Opt. Commun. 2, 305–308 (1970).
[Crossref]

Opt. Lett. (3)

Opt. Quantum Electron. (1)

G. I. Stegeman and E. M. Wright, “All-optical waveguide switching,” Opt. Quantum Electron. 22, 95–122 (1990).
[Crossref]

Phys. Lett. A (1)

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, “Quality-factor and nonlinear properties of optical whispering-gallery modes,” Phys. Lett. A 137, 393–397 (1989).
[Crossref]

Prog. Quant. Electron. (1)

J. H. Marburger, “Self-focusing: theory,” Prog. Quant. Electron. 4, 35–110 (1975), and references therein.
[Crossref]

Sov. Phys. JETP (1)

D. I. Abakarov, A. A. Akopyan, and S. I. Pekar, “Contribution to the theory of self focusing of light in nonlinearly polarizing media,” Sov. Phys. JETP 25, 303–305 (1967).

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

M. Koshiba, K. Hayata, and M. Suzuki, “On accuracy of approximate scalar finite-element analysis of dielectric optical waveguides,” Trans. Inst. Electron. Commun. Eng. Jpn. E66,157–158 (1983).

Trans. Inst. Electron. Inform. Commun. Eng. E (1)

K. Hayata and M. Koshiba, “Mutual guiding assistance between eigenmodes of nonlinearly coupled TE and TM waves,” Trans. Inst. Electron. Inform. Commun. Eng. E 74, 2890–2897 (1991).

Other (4)

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), pp. 303–333, and references therein.

K. Hayata and M. Koshiba, “Multidimensional complex whispering galleries: basic concept and their applicability to quantum nondemolition measurements,” Tech. Res. Rep. (Institute of Electrical Engineers of Japan/Institute of Electronics, Information, and Communication Engineers, Tokyo, 1991), paper EMT-91-67.

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.

J.-L. Archambault, S. Lacroix, and A. W. Snyder, “HEImself-guided modes: vector solutions and stability,” in Nonlinear Guided-Wave Phenomena, Vol. 15 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), paper TuD4–1.

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Figures (6)

Fig. 1
Fig. 1

Cross-sectional view of a nonlinear channel waveguide. Light propagates along the z axis. The linear refractive indices of the channel (rectangular core) and the surrounding medium (cladding) are nco and ncl, respectively. At least one of the two media is intensity dependent. Nonlinearities are represented by two parameters, a and b.

Fig. 2
Fig. 2

Mesh profiles for the cross section of Fig. 1. The hatched regions indicate the core. Quadratic triangular elements of isoparametric type are used for the mesh division, with the numbers of the elements and the nodal points being NE and NP, respectively. (a) Mesh A (quarter division; NE = 48, NP = 117), (b) mesh B (quarter division; NE = 192, NP = 425), (c) mesh C (full division; NE = 120, NP = 273), (d) mesh D (full division; NE = 480, NP = 1025).

Fig. 3
Fig. 3

Convergence behavior of the effective refractive index for a rectangular channel waveguide with a nonlinear core and a linear cladding: P = 5 μW, βD = 20. (a) Mesh A, (b) mesh B. The solid, dashed, and dashed-dotted curves indicate, respectively, b = a, b = a/3, and b = −a/2. For visual aid the converged values obtained by the vectorial wave analysis with mesh A are indicated by the left-pointing arrows.4 In Fig. 3(a) the three curves and arrows are indistinguishable in the scale of the ordinate.

Fig. 4
Fig. 4

Same as Fig. 3 but for doubled input power, P = 10 μW. The vectorial values for b = a and b = −a/2 are indicated by the upper and lower left-pointing arrows, respectively4; the one for b = a/3 is exactly intermediate between the two.

Fig. 5
Fig. 5

Convergence behavior of the effective refractive index for a rectangular channel waveguide with a linear core and a nonlinear cladding: P = 80 μW, βD = 10. (a) Mesh C, (b) mesh D. The meanings of the three curves and the left-pointing arrow (for b = a with mesh C) are as in Fig. 3. After N = 5 a catastrophic transition from the lower to the upper branch occurs. The former corresponds to the symmetric mode’s being unstable, whereas the latter corresponds to an asymmetric mode (QSP) with robust stability. The field distributions are shown in Fig. 6.

Fig. 6
Fig. 6

Converged field profiles for Fig. 5(a) with P = 80 μW: (a) b = a, (b) b = a/3, (c) b = −a/2. TV indicates teravolts (1012 V). The rectangular core is not to scale. Despite the symmetry mode input, the final field shape exhibits pronounced asymmetry, demonstrating the symmetry-breaking instabilities and the subsequent bifurcations of the nonlinear system far from equilibrium. Evidence for a QSP is observed.

Equations (42)

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Δ T e x + ( ɛ x B 2 1 ) e x = 0 ,
Δ T ɛ x ( ɛ z ) 1 2 / X 2 + 2 / Y 2 ,
ɛ x = ɛ x + a | e x | 2 + b | e z | 2 ,
ɛ z = ɛ z + b | e x | 2 + a | e z | 2 ,
e z = i ɛ x ( ɛ z ) 1 e x / X .
[ K ( e x , e z ) ] { e x } B 2 [ M ( e x , e z ) ] { e x } = { 0 } ,
[ K ( e x , e z ) ] = e e [ ɛ x ɛ z 1 { S X } { S X } T + { S Y } { S Y } T + { S } { S } T ] d X d Y ,
[ M ( e x , e z ) ] = e e ɛ x { S } { S } T d X d Y ,
[ e x ] co = [ e x ] cl ,
[ e x co / Y ] = [ e x cl / Y ]
[ e x ] co = [ e x ] cl ,
[ ɛ x ɛ z 1 e x co / X ] = [ ɛ x ɛ z 1 e x cl / X ] ,
e x = { S } T { e x } e ,
e z / Y = B 1 g x ,
i e x e z / X = B 1 g y ,
e x / Y = B 1 g z ,
g z / Y + i g y = ( ɛ 0 B ) 1 d x ,
i g x g z / X = 0 ,
g y / X g x / Y = ( ɛ 0 B ) 1 d z ,
d x / X i d z = 0 ,
g x / X + g y / Y i g z = 0 ,
d j = ɛ 0 ɛ j e j for j = x , z ,
e z = i ( ɛ z ) 1 [ ɛ x e x / X + ( ɛ x / X ) e x ] ,
g y = i ( B 1 ɛ x e x + B 2 e x / Y 2 ) .
Δ T e x + δ 1 e x / X + ( δ 2 + ɛ x B 2 1 ) e x = 0 ,
δ 1 = 2 ɛ z 1 ɛ x / X ,
δ 2 = ɛ z 1 2 ɛ x / X 2 .
Δ T ( TE ) e x + ( ɛ x B TE 2 ) B s 2 e x = 0 ,
Δ T ( TM ) g x + ( 1 ɛ y 1 B TM 2 ) B s 2 g x = 0 ,
Δ T ( TE ) ɛ x ( ɛ z ) 1 2 / X 2 + 2 / Y 2 ,
Δ T ( TM ) [ ( ɛ y ) 1 / X ] / X + [ ( ɛ z ) 1 / Y ] / Y ,
ɛ x = ɛ x + a x x | e x | 2 + a x y | e y | 2 + a x z | e z | 2 ,
ɛ y = ɛ y + a y x | e x | 2 + a y y | e y | 2 + a y z | e z | 2 ,
ɛ z = ɛ z + a z x | e x | 2 + a z y | e y | 2 + a z z | e z | 2 ,
e y = i B TM ɛ y 1 g x ,
e z = B TM ( ɛ z ) 1 ( g x / Y i B TE e x / X ) .
[ K TE ( e x , e y , e z ) ] { e x } B TE 2 [ M TE ] { e x } = { 0 } ,
[ K TM ( e x , e y , e z ) ] { g x } B TM 2 [ M TM ( e x , e y , e z ) ] { g x } = { 0 } ,
[ K TE ( e x , e y , e z ) ] = e e [ ɛ x ( ɛ z ) 1 { S X } { S X } T { S Y } { S Y } T + B s 2 ɛ x { S } { S } T ] d X d Y ,
[ M TE ] = e e B s 2 { S } { S } T d X d Y ,
[ K TM ( e x , e y , e z ) ] = e e [ ɛ y 1 { S X } { S X } T ɛ z 1 { S Y } { S Y } T + B s 2 { S } { S } T ] d X d Y ,
[ M TM ( e x , e y , e z ) ] = e e B s 2 ( ɛ y ) 1 { S } { S } T d X d Y .

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