Abstract

A coupled-mode formulation based on complex local modes is developed for tapered and longitudinally varying optical waveguides. Different from the conventional coupled-mode theory that requires integration over the entire spectrum of radiation modes, the new formulation treats the radiation fields via discrete complex modes similarly to the guided modes. Accuracy, convergence, and scope of validity for the solutions of the complex coupled-mode equations are investigated in detail for a typical single-mode waveguide taper. It is demonstrated that the complex coupled-mode theory has overcome the difficulties of the conventional theory in simulation of radiation field effects while preserving the simplicity and intuitiveness of this popular method.

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  1. H. Yanagawa, T. Shimizu, S. Nakamura, and I. Ohyama, J. Lightwave Technol. 10, 587 (1992).
    [CrossRef]
  2. O. Mitomi, K. Kasaya, and H. Miyazawa, IEEE J. Quantum Electron. 30, 1787 (1994).
    [CrossRef]
  3. J. Haes, J. Willems, and R. Bates, Proc. SPIE 2212, 685 (1994).
    [CrossRef]
  4. J. D. Love, W. M. Henry, W. J. Stewart, R. J. Black, S. Lacroix, and F. Gonthier, IEE Proc. J. Optoelectron. 138, 343 (1991).
    [CrossRef]
  5. D. Marcuse, Bell Syst. Tech. J. 49, 273 (1970).
  6. A. R. Nelson, Appl. Opt. 14, 3012 (1975).
    [PubMed]
  7. J. A. Fleck, J. R. Morris, and M. D. Feit, Appl. Phys. 10, 129 (1976).
    [CrossRef]
  8. T. Nakamura and N. Suzuki, IEEE Photonics Technol. Lett. 12, 143 (2000).
    [CrossRef]
  9. C. T. Lee, M. L. Wu, L. G. Sheu, P. L. Fan, and J. M. Hsu, J. Lightwave Technol. 15, 403 (1997).
    [CrossRef]
  10. P. G. Suchoski, Jr. and R. Ramaswamy, J. Opt. Soc. Am. A 3, 194 (1986).
    [CrossRef]
  11. P. G. Suchoski, Jr. and R. Ramaswamy, IEEE J. Quantum Electron. QE-23, 205 (1987).
    [CrossRef]
  12. D. Marcuse, Theory of Dielectric Optical Waveguides(Academic, 1974).
  13. A. W. Snyder, IEEE Trans. Microwave Theory Tech. 18, 383 (1970).
    [CrossRef]
  14. A. F. Milton and W. K. Burns, IEEE J. Quantum Electron. 13, 828 (1977).
    [CrossRef]
  15. W. Huangand J. Mu, Opt. Express 17, 19134 (2009).
    [CrossRef]
  16. Y. C. Lu and W. P. Huang, Opt. Express 18, 713 (2010).
    [CrossRef] [PubMed]
  17. N. Song, J. Mu, and W. Huang, J. Lightwave Technol. 28, 761 (2010).
    [CrossRef]

2010 (2)

2009 (1)

2000 (1)

T. Nakamura and N. Suzuki, IEEE Photonics Technol. Lett. 12, 143 (2000).
[CrossRef]

1997 (1)

C. T. Lee, M. L. Wu, L. G. Sheu, P. L. Fan, and J. M. Hsu, J. Lightwave Technol. 15, 403 (1997).
[CrossRef]

1994 (2)

O. Mitomi, K. Kasaya, and H. Miyazawa, IEEE J. Quantum Electron. 30, 1787 (1994).
[CrossRef]

J. Haes, J. Willems, and R. Bates, Proc. SPIE 2212, 685 (1994).
[CrossRef]

1992 (1)

H. Yanagawa, T. Shimizu, S. Nakamura, and I. Ohyama, J. Lightwave Technol. 10, 587 (1992).
[CrossRef]

1991 (1)

J. D. Love, W. M. Henry, W. J. Stewart, R. J. Black, S. Lacroix, and F. Gonthier, IEE Proc. J. Optoelectron. 138, 343 (1991).
[CrossRef]

1987 (1)

P. G. Suchoski, Jr. and R. Ramaswamy, IEEE J. Quantum Electron. QE-23, 205 (1987).
[CrossRef]

1986 (1)

1977 (1)

A. F. Milton and W. K. Burns, IEEE J. Quantum Electron. 13, 828 (1977).
[CrossRef]

1976 (1)

J. A. Fleck, J. R. Morris, and M. D. Feit, Appl. Phys. 10, 129 (1976).
[CrossRef]

1975 (1)

1970 (2)

A. W. Snyder, IEEE Trans. Microwave Theory Tech. 18, 383 (1970).
[CrossRef]

D. Marcuse, Bell Syst. Tech. J. 49, 273 (1970).

Bates, R.

J. Haes, J. Willems, and R. Bates, Proc. SPIE 2212, 685 (1994).
[CrossRef]

Black, R. J.

J. D. Love, W. M. Henry, W. J. Stewart, R. J. Black, S. Lacroix, and F. Gonthier, IEE Proc. J. Optoelectron. 138, 343 (1991).
[CrossRef]

Burns, W. K.

A. F. Milton and W. K. Burns, IEEE J. Quantum Electron. 13, 828 (1977).
[CrossRef]

Fan, P. L.

C. T. Lee, M. L. Wu, L. G. Sheu, P. L. Fan, and J. M. Hsu, J. Lightwave Technol. 15, 403 (1997).
[CrossRef]

Feit, M. D.

J. A. Fleck, J. R. Morris, and M. D. Feit, Appl. Phys. 10, 129 (1976).
[CrossRef]

Fleck, J. A.

J. A. Fleck, J. R. Morris, and M. D. Feit, Appl. Phys. 10, 129 (1976).
[CrossRef]

Gonthier, F.

J. D. Love, W. M. Henry, W. J. Stewart, R. J. Black, S. Lacroix, and F. Gonthier, IEE Proc. J. Optoelectron. 138, 343 (1991).
[CrossRef]

Haes, J.

J. Haes, J. Willems, and R. Bates, Proc. SPIE 2212, 685 (1994).
[CrossRef]

Henry, W. M.

J. D. Love, W. M. Henry, W. J. Stewart, R. J. Black, S. Lacroix, and F. Gonthier, IEE Proc. J. Optoelectron. 138, 343 (1991).
[CrossRef]

Hsu, J. M.

C. T. Lee, M. L. Wu, L. G. Sheu, P. L. Fan, and J. M. Hsu, J. Lightwave Technol. 15, 403 (1997).
[CrossRef]

Huang, W.

Huang, W. P.

Kasaya, K.

O. Mitomi, K. Kasaya, and H. Miyazawa, IEEE J. Quantum Electron. 30, 1787 (1994).
[CrossRef]

Lacroix, S.

J. D. Love, W. M. Henry, W. J. Stewart, R. J. Black, S. Lacroix, and F. Gonthier, IEE Proc. J. Optoelectron. 138, 343 (1991).
[CrossRef]

Lee, C. T.

C. T. Lee, M. L. Wu, L. G. Sheu, P. L. Fan, and J. M. Hsu, J. Lightwave Technol. 15, 403 (1997).
[CrossRef]

Love, J. D.

J. D. Love, W. M. Henry, W. J. Stewart, R. J. Black, S. Lacroix, and F. Gonthier, IEE Proc. J. Optoelectron. 138, 343 (1991).
[CrossRef]

Lu, Y. C.

Marcuse, D.

D. Marcuse, Bell Syst. Tech. J. 49, 273 (1970).

D. Marcuse, Theory of Dielectric Optical Waveguides(Academic, 1974).

Milton, A. F.

A. F. Milton and W. K. Burns, IEEE J. Quantum Electron. 13, 828 (1977).
[CrossRef]

Mitomi, O.

O. Mitomi, K. Kasaya, and H. Miyazawa, IEEE J. Quantum Electron. 30, 1787 (1994).
[CrossRef]

Miyazawa, H.

O. Mitomi, K. Kasaya, and H. Miyazawa, IEEE J. Quantum Electron. 30, 1787 (1994).
[CrossRef]

Morris, J. R.

J. A. Fleck, J. R. Morris, and M. D. Feit, Appl. Phys. 10, 129 (1976).
[CrossRef]

Mu, J.

Nakamura, S.

H. Yanagawa, T. Shimizu, S. Nakamura, and I. Ohyama, J. Lightwave Technol. 10, 587 (1992).
[CrossRef]

Nakamura, T.

T. Nakamura and N. Suzuki, IEEE Photonics Technol. Lett. 12, 143 (2000).
[CrossRef]

Nelson, A. R.

Ohyama, I.

H. Yanagawa, T. Shimizu, S. Nakamura, and I. Ohyama, J. Lightwave Technol. 10, 587 (1992).
[CrossRef]

Ramaswamy, R.

P. G. Suchoski, Jr. and R. Ramaswamy, IEEE J. Quantum Electron. QE-23, 205 (1987).
[CrossRef]

P. G. Suchoski, Jr. and R. Ramaswamy, J. Opt. Soc. Am. A 3, 194 (1986).
[CrossRef]

Sheu, L. G.

C. T. Lee, M. L. Wu, L. G. Sheu, P. L. Fan, and J. M. Hsu, J. Lightwave Technol. 15, 403 (1997).
[CrossRef]

Shimizu, T.

H. Yanagawa, T. Shimizu, S. Nakamura, and I. Ohyama, J. Lightwave Technol. 10, 587 (1992).
[CrossRef]

Snyder, A. W.

A. W. Snyder, IEEE Trans. Microwave Theory Tech. 18, 383 (1970).
[CrossRef]

Song, N.

Stewart, W. J.

J. D. Love, W. M. Henry, W. J. Stewart, R. J. Black, S. Lacroix, and F. Gonthier, IEE Proc. J. Optoelectron. 138, 343 (1991).
[CrossRef]

Suchoski, P. G.

P. G. Suchoski, Jr. and R. Ramaswamy, IEEE J. Quantum Electron. QE-23, 205 (1987).
[CrossRef]

P. G. Suchoski, Jr. and R. Ramaswamy, J. Opt. Soc. Am. A 3, 194 (1986).
[CrossRef]

Suzuki, N.

T. Nakamura and N. Suzuki, IEEE Photonics Technol. Lett. 12, 143 (2000).
[CrossRef]

Willems, J.

J. Haes, J. Willems, and R. Bates, Proc. SPIE 2212, 685 (1994).
[CrossRef]

Wu, M. L.

C. T. Lee, M. L. Wu, L. G. Sheu, P. L. Fan, and J. M. Hsu, J. Lightwave Technol. 15, 403 (1997).
[CrossRef]

Yanagawa, H.

H. Yanagawa, T. Shimizu, S. Nakamura, and I. Ohyama, J. Lightwave Technol. 10, 587 (1992).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. (1)

J. A. Fleck, J. R. Morris, and M. D. Feit, Appl. Phys. 10, 129 (1976).
[CrossRef]

Bell Syst. Tech. J. (1)

D. Marcuse, Bell Syst. Tech. J. 49, 273 (1970).

IEE Proc. J. Optoelectron. (1)

J. D. Love, W. M. Henry, W. J. Stewart, R. J. Black, S. Lacroix, and F. Gonthier, IEE Proc. J. Optoelectron. 138, 343 (1991).
[CrossRef]

IEEE J. Quantum Electron. (3)

O. Mitomi, K. Kasaya, and H. Miyazawa, IEEE J. Quantum Electron. 30, 1787 (1994).
[CrossRef]

P. G. Suchoski, Jr. and R. Ramaswamy, IEEE J. Quantum Electron. QE-23, 205 (1987).
[CrossRef]

A. F. Milton and W. K. Burns, IEEE J. Quantum Electron. 13, 828 (1977).
[CrossRef]

IEEE Photonics Technol. Lett. (1)

T. Nakamura and N. Suzuki, IEEE Photonics Technol. Lett. 12, 143 (2000).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

A. W. Snyder, IEEE Trans. Microwave Theory Tech. 18, 383 (1970).
[CrossRef]

J. Lightwave Technol. (3)

N. Song, J. Mu, and W. Huang, J. Lightwave Technol. 28, 761 (2010).
[CrossRef]

C. T. Lee, M. L. Wu, L. G. Sheu, P. L. Fan, and J. M. Hsu, J. Lightwave Technol. 15, 403 (1997).
[CrossRef]

H. Yanagawa, T. Shimizu, S. Nakamura, and I. Ohyama, J. Lightwave Technol. 10, 587 (1992).
[CrossRef]

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

Opt. Express (2)

Proc. SPIE (1)

J. Haes, J. Willems, and R. Bates, Proc. SPIE 2212, 685 (1994).
[CrossRef]

Other (1)

D. Marcuse, Theory of Dielectric Optical Waveguides(Academic, 1974).

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Figures (4)

Fig. 1
Fig. 1

Geometry of a linear taper waveguide structure and its staircase approximation.

Fig. 2
Fig. 2

Power transmission of a linear taper.

Fig. 3
Fig. 3

Convergence behavior of grid size ( Δ d ).

Fig. 4
Fig. 4

Convergence behavior of number of modes.

Equations (24)

Equations on this page are rendered with MathJax. Learn more.

( e t m × h t n ) · z ^ d a = 0.
e n ( r t , z ) = [ e t n ( r t , z ) + e z n ( r t , z ) ] exp ( j β n z ) ,
h n ( r t , z ) = [ h t n ( r t , z ) + h z n ( r t , z ) ] exp ( j β n z ) .
1 2 ( e t m × h t m * ) · z ^ d a = 1.
E ( r t , z ) = E t + E z ,
H ( r t , z ) = H t + H z .
E t ( r t , z ) = ( a n ( z ) + b n ( z ) ) e t n ( r t , z ) ,
H t ( r t , z ) = ( a n ( z ) b n ( z ) ) h t n ( r t , z ) ,
E z ( r t , z ) = ( a n ( z ) b n ( z ) ) e z n ( r t , z ) ,
H z ( r t , z ) = ( a n ( z ) + b n ( z ) ) h z n ( r t , z ) .
d d z a m = j β m a m κ m n a n χ m n b n ,
d d z b m = j β m a m κ m n b n χ m n a n ,
κ m n = 1 4 N m ( e t n z × h t m + e t m × h t n z ) · z ^ d a ,
χ m n = 1 4 N m ( e t n z × h t m e t m × h t n z ) · z ^ d a ,
N m = 1 2 ( e t m × h t m ) · z ^ d a .
κ m n = 1 4 N m ω ε 0 β n β m ( n 2 z e t m · e t n ) · z ^ d a ,
χ m n = 1 4 N m ω ε 0 β n + β m ( n 2 z e t m · e t n ) · z ^ d a .
d ( z ) = d in + z tan θ .
κ m n ( z ) = ω ε 0 4 N m n c o 2 n c l 2 β n β m tan θ · ( e t m · e t n ) x = d .
d d z a 1 = j β 1 a 1 κ 1 n a n ,
d d z a n = j β n a n κ n 1 a 1 .
T = 1 n m a n a m * P m n ,
P m n = 1 4 ( e t m × h t n * + e t n * × h t m ) · z ^ d a .
T = 1 n | a n | 2 .

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