Abstract

We present a rigorous vectorial coupled-mode theory (VCMT) for the isotropic waveguide structures with anisotropic disturbance by Maxwell equations and a modified dielectric tensor. The coupling coefficients are related to every element of a disturbed dielectric tensor. In the special case of isotropic disturbance and orthogonal mode coupling, our formulas can be reduced to the previous version of VCMT, in which the coupling coefficient does not include the polarization coupling term appearing in an earlier scalar coupled-mode theory with vector correction (SCMT-VC). The origin of the fundamental error found in the SCMT-VC is analyzed. Finally, we apply our theory to a special coupling between x- and y-polarized modes in a single-mode fiber.

© 2006 Optical Society of America

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  1. L. Tsang and S.-L. Chuang, "Improved coupled-mode theory for reciprocal anisotropic waveguides," J. Lightwave Technol. 6, 304-310 (1988).
    [CrossRef]
  2. T. Feng, W. YiZun, and Y. Peida, "Random coupling theory of single-mode optical fiber," J. Lightwave Technol. 8, 1235-1242 (1990).
    [CrossRef]
  3. A. W. Snyder, "Coupled-mode theory for optical fiber," J. Opt. Soc. Am. 62, 1267-1277 (1972).
    [CrossRef]
  4. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, 1972), pp. 98-106.
  5. W. P. Huang, S. T. Chu, and S. K. Chaudhuri, "A scalar coupled-mode theory with vector correction," IEEE J. Quantum Electron. 28, 184-193 (1992).
    [CrossRef]
  6. H. A. Haus and W. P. Huang, "Coupled-mode theory," Proc. IEEE 79, 1505-1518 (1991).
    [CrossRef]
  7. W. P. Huang, "Coupled-mode theory for optical waveguides: an overview," J. Opt. Soc. Am. A 11, 963-983 (1994).
    [CrossRef]
  8. W. P. Huang and S. K. Chaudhuri, "Variational coupled-mode theory of optical couplers," J. Lightwave Technol. 8, 1565-1570 (1990).
    [CrossRef]
  9. W. P. Huang and C. L. Xu, "Simulation of three-dimensional optical waveguides by a full-vector beam propagation method," IEEE J. Quantum Electron. 29, 2639-2649 (1993).
    [CrossRef]
  10. C. L. Xu, W. P. Huang, M. S. Stern, and S. K. Chaudhuri, "Full-vectorial mode calculations by finite difference method," IEE Proc. Optoelectron. 141, 281-286 (1994).
    [CrossRef]
  11. C. L. Xu and W. P. Huang, "A full-vectorial beam propagation method for anisotropic waveguides," J. Lightwave Technol. 12, 1926-1931 (1994).
    [CrossRef]
  12. F. Fogli, G. Bellanca, P. Bassi, I. Madden, and W. Johnstone, "Highly efficient full-vectorial 3-D BPM modeling of fiber to planar waveguide couplers," J. Lightwave Technol. 17, 136-143 (1999).
    [CrossRef]
  13. K. M. Lo and E. H. Li, "Solution of the quasi-vector wave equation for optical waveguides in a mapped infinite domains by the Galerkin's method," J. Lightwave Technol. 16, 937-944 (1998).
    [CrossRef]
  14. W. W. Lui, C.-L. Xu, and W.-P. Huang, "Full-vectorial wave propagation in semiconductor optical bending waveguide approximations," J. Lightwave Technol. 16, 910-914 (1998).
    [CrossRef]
  15. W. W. Lui, T. Hirono, and W.-P. Huang, "Polarization rotation in semiconductor bending waveguides: a coupled-mode theory formulation," J. Lightwave Technol. 16, 929-936 (1998).
    [CrossRef]

1999 (1)

1998 (3)

1994 (3)

C. L. Xu, W. P. Huang, M. S. Stern, and S. K. Chaudhuri, "Full-vectorial mode calculations by finite difference method," IEE Proc. Optoelectron. 141, 281-286 (1994).
[CrossRef]

C. L. Xu and W. P. Huang, "A full-vectorial beam propagation method for anisotropic waveguides," J. Lightwave Technol. 12, 1926-1931 (1994).
[CrossRef]

W. P. Huang, "Coupled-mode theory for optical waveguides: an overview," J. Opt. Soc. Am. A 11, 963-983 (1994).
[CrossRef]

1993 (1)

W. P. Huang and C. L. Xu, "Simulation of three-dimensional optical waveguides by a full-vector beam propagation method," IEEE J. Quantum Electron. 29, 2639-2649 (1993).
[CrossRef]

1992 (1)

W. P. Huang, S. T. Chu, and S. K. Chaudhuri, "A scalar coupled-mode theory with vector correction," IEEE J. Quantum Electron. 28, 184-193 (1992).
[CrossRef]

1991 (1)

H. A. Haus and W. P. Huang, "Coupled-mode theory," Proc. IEEE 79, 1505-1518 (1991).
[CrossRef]

1990 (2)

W. P. Huang and S. K. Chaudhuri, "Variational coupled-mode theory of optical couplers," J. Lightwave Technol. 8, 1565-1570 (1990).
[CrossRef]

T. Feng, W. YiZun, and Y. Peida, "Random coupling theory of single-mode optical fiber," J. Lightwave Technol. 8, 1235-1242 (1990).
[CrossRef]

1988 (1)

L. Tsang and S.-L. Chuang, "Improved coupled-mode theory for reciprocal anisotropic waveguides," J. Lightwave Technol. 6, 304-310 (1988).
[CrossRef]

1972 (1)

Bassi, P.

Bellanca, G.

Chaudhuri, S. K.

C. L. Xu, W. P. Huang, M. S. Stern, and S. K. Chaudhuri, "Full-vectorial mode calculations by finite difference method," IEE Proc. Optoelectron. 141, 281-286 (1994).
[CrossRef]

W. P. Huang, S. T. Chu, and S. K. Chaudhuri, "A scalar coupled-mode theory with vector correction," IEEE J. Quantum Electron. 28, 184-193 (1992).
[CrossRef]

W. P. Huang and S. K. Chaudhuri, "Variational coupled-mode theory of optical couplers," J. Lightwave Technol. 8, 1565-1570 (1990).
[CrossRef]

Chu, S. T.

W. P. Huang, S. T. Chu, and S. K. Chaudhuri, "A scalar coupled-mode theory with vector correction," IEEE J. Quantum Electron. 28, 184-193 (1992).
[CrossRef]

Chuang, S.-L.

L. Tsang and S.-L. Chuang, "Improved coupled-mode theory for reciprocal anisotropic waveguides," J. Lightwave Technol. 6, 304-310 (1988).
[CrossRef]

Feng, T.

T. Feng, W. YiZun, and Y. Peida, "Random coupling theory of single-mode optical fiber," J. Lightwave Technol. 8, 1235-1242 (1990).
[CrossRef]

Fogli, F.

Haus, H. A.

H. A. Haus and W. P. Huang, "Coupled-mode theory," Proc. IEEE 79, 1505-1518 (1991).
[CrossRef]

Hirono, T.

Huang, W. P.

C. L. Xu and W. P. Huang, "A full-vectorial beam propagation method for anisotropic waveguides," J. Lightwave Technol. 12, 1926-1931 (1994).
[CrossRef]

C. L. Xu, W. P. Huang, M. S. Stern, and S. K. Chaudhuri, "Full-vectorial mode calculations by finite difference method," IEE Proc. Optoelectron. 141, 281-286 (1994).
[CrossRef]

W. P. Huang, "Coupled-mode theory for optical waveguides: an overview," J. Opt. Soc. Am. A 11, 963-983 (1994).
[CrossRef]

W. P. Huang and C. L. Xu, "Simulation of three-dimensional optical waveguides by a full-vector beam propagation method," IEEE J. Quantum Electron. 29, 2639-2649 (1993).
[CrossRef]

W. P. Huang, S. T. Chu, and S. K. Chaudhuri, "A scalar coupled-mode theory with vector correction," IEEE J. Quantum Electron. 28, 184-193 (1992).
[CrossRef]

H. A. Haus and W. P. Huang, "Coupled-mode theory," Proc. IEEE 79, 1505-1518 (1991).
[CrossRef]

W. P. Huang and S. K. Chaudhuri, "Variational coupled-mode theory of optical couplers," J. Lightwave Technol. 8, 1565-1570 (1990).
[CrossRef]

Huang, W.-P.

Johnstone, W.

Li, E. H.

Lo, K. M.

Lui, W. W.

Madden, I.

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, 1972), pp. 98-106.

Peida, Y.

T. Feng, W. YiZun, and Y. Peida, "Random coupling theory of single-mode optical fiber," J. Lightwave Technol. 8, 1235-1242 (1990).
[CrossRef]

Snyder, A. W.

Stern, M. S.

C. L. Xu, W. P. Huang, M. S. Stern, and S. K. Chaudhuri, "Full-vectorial mode calculations by finite difference method," IEE Proc. Optoelectron. 141, 281-286 (1994).
[CrossRef]

Tsang, L.

L. Tsang and S.-L. Chuang, "Improved coupled-mode theory for reciprocal anisotropic waveguides," J. Lightwave Technol. 6, 304-310 (1988).
[CrossRef]

Xu, C. L.

C. L. Xu and W. P. Huang, "A full-vectorial beam propagation method for anisotropic waveguides," J. Lightwave Technol. 12, 1926-1931 (1994).
[CrossRef]

C. L. Xu, W. P. Huang, M. S. Stern, and S. K. Chaudhuri, "Full-vectorial mode calculations by finite difference method," IEE Proc. Optoelectron. 141, 281-286 (1994).
[CrossRef]

W. P. Huang and C. L. Xu, "Simulation of three-dimensional optical waveguides by a full-vector beam propagation method," IEEE J. Quantum Electron. 29, 2639-2649 (1993).
[CrossRef]

Xu, C.-L.

YiZun, W.

T. Feng, W. YiZun, and Y. Peida, "Random coupling theory of single-mode optical fiber," J. Lightwave Technol. 8, 1235-1242 (1990).
[CrossRef]

IEE Proc. Optoelectron. (1)

C. L. Xu, W. P. Huang, M. S. Stern, and S. K. Chaudhuri, "Full-vectorial mode calculations by finite difference method," IEE Proc. Optoelectron. 141, 281-286 (1994).
[CrossRef]

IEEE J. Quantum Electron. (2)

W. P. Huang, S. T. Chu, and S. K. Chaudhuri, "A scalar coupled-mode theory with vector correction," IEEE J. Quantum Electron. 28, 184-193 (1992).
[CrossRef]

W. P. Huang and C. L. Xu, "Simulation of three-dimensional optical waveguides by a full-vector beam propagation method," IEEE J. Quantum Electron. 29, 2639-2649 (1993).
[CrossRef]

J. Lightwave Technol. (8)

J. Opt. Soc. Am. (1)

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

Proc. IEEE (1)

H. A. Haus and W. P. Huang, "Coupled-mode theory," Proc. IEEE 79, 1505-1518 (1991).
[CrossRef]

Other (1)

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, 1972), pp. 98-106.

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Equations (55)

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ε ( x , y , z ) = ε 0 ( x , y ) I + ε ̃ ( x , y , z ) ,
t × E t = j ω μ 0 H z ,
t × H t = j ω ε 0 E z j ω [ ( ε ̃ z t z ̂ t ̂ ) E t + ( ε ̃ z z z ̂ z ̂ ) E z ] ,
t × E z + z ̂ × E t z = j ω μ 0 H t ,
t × H z + z ̂ × H t z = j ω ε 0 E t j ω [ ( ε ̃ x t x ̂ t ̂ ) E t + ( ε ̃ y t y ̂ t ̂ ) E t + ( ε ̃ x z x ̂ z ̂ ) E z + ( ε ̃ y z y ̂ z ̂ ) E z ] .
1 j ω ( ε 0 + ε ̃ z z ) t × ( t × H t ) = 1 ε 0 + ε ̃ z z t × ( ε 0 E z ) 1 ε 0 + ε ̃ z z t × ( ε ̃ z t z ̂ t ̂ E t + ε ̃ z z z ̂ z ̂ E z ) ,
t × ( t × E t ) = j ω μ 0 ( z ̂ × H t z j ω ε 0 E t ) + ω 2 μ 0 ( ε ̃ x t x ̂ t ̂ E t + ε ̃ y t y ̂ t ̂ E t + ε ̃ x z x ̂ z ̂ E z + ε ̃ y z y ̂ z ̂ E z ) .
t × ( ε 0 E z ) = t ε 0 × E z + ε 0 ( t × E z ) ,
t × ( ε ̃ z z E z ) = t ε ̃ z z × E z + ε ̃ z z ( t × E z ) ,
1 j ω ( ε 0 + ε ̃ z z ) t × ( t × H t ) = 1 ε 0 + ε ̃ z z t ( ε 0 + ε ̃ z z ) × E z ( j ω μ 0 H t z ̂ × E t z ) 1 ε 0 + ε ̃ z z t × ( ε ̃ z t z ̂ t ̂ E t ) .
[ E t 0 H t 0 ] = [ e t ( x , y ) h t ( x , y ) ] exp ( j β z ) ,
1 j ω μ 0 t × ( t × e t ) j β ( z ̂ × h t ) = j ω ε 0 e t ,
1 j ω ε 0 t × ( t × h t ) + 1 ε 0 ( t ε 0 ) × e z = j ω μ 0 h t + j β ( z ̂ × e t ) .
E t = μ a μ ( z ) e μ t ,
H t = μ b μ ( z ) h μ t .
μ ( t × h μ t ) b μ ( z ) = j ω ε 0 E z j ω ( ε ̃ z t z ̂ t ̂ E t + ε ̃ z z z ̂ z ̂ E z ) .
E z = μ ε 0 ε 0 + ε ̃ z z [ e μ z b μ ( z ) 1 ε 0 ( ε ̃ z t z ̂ t ̂ e μ t ) a μ ( z ) ] .
μ [ d a μ ( z ) d z j β b μ ( z ) ] ( z ̂ × e μ t ) = μ 1 ε 0 + ε ̃ z z t × ( ε ̃ z t z ̂ t ̂ e μ t ) a μ ( z ) μ 1 j ω ( 1 ε 0 1 ε 0 + ε ̃ z z ) t × ( t × h μ t ) b μ ( z ) μ 1 ε 0 ( t ε 0 ) × e μ z b μ ( z ) + μ 1 ε 0 + ε ̃ z z t ( ε 0 + ε ̃ z z ) ε 0 ε 0 + ε ̃ z z [ e μ z b μ ( z ) 1 ε 0 ( ε ̃ z t z ̂ t ̂ e μ t ) a μ ( z ) ] .
μ P μ ν d a μ d z = μ H μ ν ( 1 ) b μ + μ R μ ν ( 1 ) a μ ,
H μ ν ( 1 ) = j P μ ν β μ + K μ ν ( 1 ) ,
P μ ν = ( e μ t × h ν t * ) z ̂ d A ,
K μ ν ( 1 ) = j ω ε ̃ z z ε 0 ε 0 + ε ̃ z z e ν z * e μ z d A ,
R μ ν ( 1 ) = 1 ε 0 + ε ̃ z z t [ ( ε ̃ z t z ̂ t ̂ e μ t ) × h ν t * ] d A 1 ( ε 0 + ε ̃ z z ) 2 t ( ε 0 + ε ̃ z z ) [ ( ε ̃ z t z ̂ t ̂ e μ t ) × h ν t * ] d A .
μ P μ ν d b μ d z = μ H μ ν ( 2 ) a μ + μ R μ ν ( 2 ) b μ ,
H μ ν ( 2 ) = j P μ ν β ν + K μ ν ( 2 ) ,
K μ ν ( 2 ) = j ω e ν t * ( ε ̃ x t x ̂ t ̂ + ε ̃ y t y ̂ t ̂ ) e μ t d A + j ω 1 ε 0 + ε ̃ z z e ν t * ( ε ̃ x z x ̂ z ̂ + ε ̃ y z y ̂ z ̂ ) ( ε ̃ z t z ̂ t ̂ e μ t ) d A ,
R μ ν ( 2 ) = j ω ε 0 ε 0 + ε ̃ z z e ν t * ( ε ̃ x z x ̂ z ̂ + ε ̃ y z y ̂ z ̂ ) e μ z d A .
P ν ν d a ν d z = j P ν ν β ν b ν + μ K μ ν ( 1 ) b μ + μ R μ ν ( 1 ) a μ ,
P ν ν d b ν d z = j P ν ν β ν a ν + μ K μ ν ( 2 ) a μ + μ R μ ν ( 2 ) b μ ,
P ν ν = ( e ν t × h ν t * ) z ̂ d A .
ε ̃ x y = ε ̃ y x = ε ̃ z t = ε ̃ t z = 0 , ε 0 + ε ̃ z z = ε .
μ P μ ν ( d 2 a μ d z 2 + β μ 2 a μ ) = μ [ j β μ ( K μ ν ( 1 ) + K μ ν ( 2 ) ) a μ + ( R μ ν ( 1 ) + R μ ν ( 2 ) ) d a μ d z R μ ν ( 1 ) R μ ν ( 2 ) a μ ] .
a μ ( z ) = c μ ( z ) exp ( j β μ z ) ,
d a μ d z = ( d c μ d z + j β μ c μ ) exp ( j β μ z ) ,
d 2 a μ d z 2 ( 2 j β μ d c μ d z β μ 2 c μ ) exp ( j β μ z ) .
μ P μ ν d c μ d z exp ( j β μ z ) = μ K μ ν ( 1 ) + K μ ν ( 2 ) + R μ ν ( 1 ) + R μ ν ( 2 ) 2 c μ exp ( j β μ z ) .
h ν t * = β ω μ 0 ( z ̂ × e ν t * ) .
e μ z = j β ( t e μ t ) z ̂ ,
μ 2 P μ ν d c μ d z exp ( j β μ z ) = μ ( K μ ν ( 2 ) + R μ ν ( 1 ) + R μ ν ( 2 ) ) c μ exp ( j β μ z ) ,
R μ ν ( 1 ) = β ν ω μ 0 1 ε 0 + ε ̃ z z t [ ( ε ̃ z t z ̂ t ̂ e μ t ) e ν t * ] d A ,
P μ ν = β ν ω μ 0 e μ t e ν t * d A .
d c ν ( z ) d z exp ( j β ν z ) = μ K μ ν c μ ( z ) exp ( j β μ z ) ,
K μ ν = K μ ν ( 2 ) 2 P ν ν = j k 2 ε ̃ e ν t * e μ t d A 2 β ν e ν t e ν t * d A .
ε 0 ( ε 0 + ε ̃ z z ) 2 t ( ε 0 + ε ̃ z z ) ( h ν t * × e μ z ) d A = j 1 ω μ 0 ε 0 ε 2 e ν t * ( t ε ̃ ) ( t e μ t ) d A ,
d a y d z = j β y a y + K x y ( 2 ) + R x y ( 1 ) + R x y ( 2 ) 2 P y y a x ,
d a x d z = j β x a x + K x y ( 2 ) + R x y ( 1 ) + R x y ( 2 ) 2 P y y a y ,
P y y = β y ω μ 0 e y 2 d A ,
R x y ( 1 ) = [ β x ω μ 0 1 ε 0 + ε ̃ z z y [ ( ε ̃ z x z ̂ x ̂ e x ) e y * ] ] d A ,
R x y ( 2 ) = j ω ε 0 ε 0 + ε ̃ z z e y * ε ̃ y z y ̂ z ̂ e z d A ,
K x y ( 2 ) = j ω e y * ε ̃ y x y ̂ x ̂ e x d A + j ω 1 ε 0 + ε ̃ z z e y * ε ̃ y z y ̂ z ̂ ( ε ̃ z x z ̂ x ̂ e x ) d A .
μ [ d a μ ( z ) d z j β b μ ( z ) ] ( e μ t × h ν t * ) z ̂ d A = μ 1 ε 0 + ε ̃ z z h * ν t [ t × ( ε ̃ z t z ̂ t ̂ e μ t ) ] d A a μ ( z ) μ 1 j ω ( 1 ε 0 1 ε 0 + ε ̃ z z ) h ν t * [ t × ( t × h μ t ) ] d A b μ ( z ) μ 1 ε 0 h ν t * [ ( t ε 0 ) × e μ z ] d A b μ ( z ) + μ ε 0 ( ε 0 + ε ̃ z z ) 2 h ν t * t ( ε 0 + ε ̃ z z ) × ( e μ z ) d A b μ ( z ) μ 1 ( ε 0 + ε ̃ z z ) 2 h ν t * t ( ε 0 + ε ̃ z z ) × 1 ε 0 ( ε ̃ z t z ̂ t ̂ e μ t ) d A a μ ( z ) .
1 j ω ( 1 ε 0 1 ε 0 + ε ̃ z z ) h ν t * [ t ( t × h μ t ) ] d A = 1 j ω ( 1 ε 0 + ε ̃ z z 1 ε 0 ) h ν t * [ t × j ω ε 0 e μ z ] d A = j ω ε ̃ z z ε 0 ε ̃ z z + ε 0 e ν z * e μ z d A ε ̃ z z ε 0 ( ε 0 + ε ̃ z z ) ( t ε 0 ) ( h ν t * × e μ z ) d A + ( t ε ̃ z z ε ̃ z z + ε 0 ε ̃ z z t ( ε ̃ z z + ε 0 ) ( ε ̃ z z + ε ) 2 ) ( h ν t * × e μ z ) .
K μ ν ( 1 ) = j ω ε ̃ z z ε 0 ε 0 + ε ̃ z z e ν z * e μ z d A ε ̃ z z ε 0 ( ε 0 + ε ̃ z z ) ( t ε 0 ) ( h ν t * × e μ z ) d A + ( t ε ̃ z z ε ̃ z z + ε 0 ε ̃ z z t ( ε ̃ z z + ε 0 ) ( ε ̃ z z + ε ) 2 ) ( h ν t * × e μ z ) d A + 1 ε 0 [ ( t ε 0 ) ( h ν t * × e μ z ) ] d A ε 0 ( ε 0 + ε ̃ z z ) 2 t ( ε 0 + ε ̃ z z ) ( h ν t * × e μ z ) d A = j ω ε ̃ z z ε 0 ε 0 + ε ̃ z z e ν z * e μ z d A .
μ [ d b μ ( z ) d z j β a μ ( z ) ] [ z ̂ ( h μ t × e ν t * ) ] = j ω μ e ν t * ( ε ̃ x t x ̂ t ̂ + ε ̃ y t y ̂ t ̂ ) e μ t d A a μ ( z ) j ω μ e ν t * ( ε ̃ x z x ̂ z ̂ E z + ε ̃ y z y ̂ z ̂ E z ) d A .
μ [ d b μ ( z ) d z j β a μ ( z ) ] [ z ̂ ( e ν t * × h μ t ) ] = j ω μ e ν t * ( ε ̃ x t x ̂ t ̂ + ε ̃ y t y ̂ t ̂ ) e μ t d A a μ ( z ) j ω μ ε 0 ε 0 + ε ̃ z z e ν t * ( ε ̃ x z x ̂ z ̂ + ε ̃ y z y ̂ z ̂ ) e μ z b μ ( z ) d A + j ω μ 1 ε 0 + ε ̃ z z e ν t * ( ε ̃ x z x ̂ z ̂ + ε ̃ y z y ̂ z ̂ ) ( ε ̃ z t z ̂ t ̂ e μ t ) a μ ( z ) d A .

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