Abstract

Analysis of a directional coupler by use of rigorous coupled-mode theory is presented. The coupling process is explained by the mutual coupling of modes of a single waveguide that is due to the presence of the other waveguide. It is explicitly shown by example that the radiation modes that take part in the coupling are just those needed to form the guided modes of the composite double-guide structure.

© 1994 Optical Society of America

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References

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  1. S. E. Miller, Bell Syst. Tech. J. 48, 2059 (1969).
  2. See, for example,D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972), Chap. 10.
  3. A. Hardy, W. Streifer, J. Lightwave Technol. LT-3, 1135 (1985).
    [Crossref]
  4. W. Streifer, M. Osinski, A. Hardy, Proc. Soc. Photo-Opt. Instrum. Eng. 385, 178 (1978).
  5. J.-P. Weber, L. Thylén, S. Wang, IEEE J. Quantum Electron. 24, 537 (1988).
    [Crossref]
  6. J. Čtyroký, M. Hofman, J. Janta, J. Schröfel, IEEE J. Quantum Electron. QE-20, 400 (1984).
    [Crossref]
  7. H. Kogelnik, in Guided-Wave Optoelectronics, T. Tamir, ed., Vol. 26 of Springer Series in Electronics and Photonics (Springer-Verlag, Berlin, 1988), pp. 74–79.

1988 (1)

J.-P. Weber, L. Thylén, S. Wang, IEEE J. Quantum Electron. 24, 537 (1988).
[Crossref]

1985 (1)

A. Hardy, W. Streifer, J. Lightwave Technol. LT-3, 1135 (1985).
[Crossref]

1984 (1)

J. Čtyroký, M. Hofman, J. Janta, J. Schröfel, IEEE J. Quantum Electron. QE-20, 400 (1984).
[Crossref]

1978 (1)

W. Streifer, M. Osinski, A. Hardy, Proc. Soc. Photo-Opt. Instrum. Eng. 385, 178 (1978).

1969 (1)

S. E. Miller, Bell Syst. Tech. J. 48, 2059 (1969).

Ctyroký, J.

J. Čtyroký, M. Hofman, J. Janta, J. Schröfel, IEEE J. Quantum Electron. QE-20, 400 (1984).
[Crossref]

Hardy, A.

A. Hardy, W. Streifer, J. Lightwave Technol. LT-3, 1135 (1985).
[Crossref]

W. Streifer, M. Osinski, A. Hardy, Proc. Soc. Photo-Opt. Instrum. Eng. 385, 178 (1978).

Hofman, M.

J. Čtyroký, M. Hofman, J. Janta, J. Schröfel, IEEE J. Quantum Electron. QE-20, 400 (1984).
[Crossref]

Janta, J.

J. Čtyroký, M. Hofman, J. Janta, J. Schröfel, IEEE J. Quantum Electron. QE-20, 400 (1984).
[Crossref]

Kogelnik, H.

H. Kogelnik, in Guided-Wave Optoelectronics, T. Tamir, ed., Vol. 26 of Springer Series in Electronics and Photonics (Springer-Verlag, Berlin, 1988), pp. 74–79.

Marcuse, D.

See, for example,D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972), Chap. 10.

Miller, S. E.

S. E. Miller, Bell Syst. Tech. J. 48, 2059 (1969).

Osinski, M.

W. Streifer, M. Osinski, A. Hardy, Proc. Soc. Photo-Opt. Instrum. Eng. 385, 178 (1978).

Schröfel, J.

J. Čtyroký, M. Hofman, J. Janta, J. Schröfel, IEEE J. Quantum Electron. QE-20, 400 (1984).
[Crossref]

Streifer, W.

A. Hardy, W. Streifer, J. Lightwave Technol. LT-3, 1135 (1985).
[Crossref]

W. Streifer, M. Osinski, A. Hardy, Proc. Soc. Photo-Opt. Instrum. Eng. 385, 178 (1978).

Thylén, L.

J.-P. Weber, L. Thylén, S. Wang, IEEE J. Quantum Electron. 24, 537 (1988).
[Crossref]

Wang, S.

J.-P. Weber, L. Thylén, S. Wang, IEEE J. Quantum Electron. 24, 537 (1988).
[Crossref]

Weber, J.-P.

J.-P. Weber, L. Thylén, S. Wang, IEEE J. Quantum Electron. 24, 537 (1988).
[Crossref]

Bell Syst. Tech. J. (1)

S. E. Miller, Bell Syst. Tech. J. 48, 2059 (1969).

IEEE J. Quantum Electron. (2)

J.-P. Weber, L. Thylén, S. Wang, IEEE J. Quantum Electron. 24, 537 (1988).
[Crossref]

J. Čtyroký, M. Hofman, J. Janta, J. Schröfel, IEEE J. Quantum Electron. QE-20, 400 (1984).
[Crossref]

J. Lightwave Technol. (1)

A. Hardy, W. Streifer, J. Lightwave Technol. LT-3, 1135 (1985).
[Crossref]

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

W. Streifer, M. Osinski, A. Hardy, Proc. Soc. Photo-Opt. Instrum. Eng. 385, 178 (1978).

Other (2)

See, for example,D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972), Chap. 10.

H. Kogelnik, in Guided-Wave Optoelectronics, T. Tamir, ed., Vol. 26 of Springer Series in Electronics and Photonics (Springer-Verlag, Berlin, 1988), pp. 74–79.

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Figures (5)

Fig. 1
Fig. 1

Schematic of the directional coupler between metallic walls.

Fig. 2
Fig. 2

Field distributions of the first several modes of the single waveguide.

Fig. 3
Fig. 3

Field distribution of several lowest-order super-modes of a directional coupler. The window width is 40 μm

Fig. 4
Fig. 4

Relative power of normal modes in the coupler.

Fig. 5
Fig. 5

Coupling length of a directional coupler for three window widths versus the effective index of the highest-order mode of the generic waveguide. The exact value is calculated independently by use of the transfer matrix method.

Equations (13)

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d 2 e m ( x ) d x 2 + ( k 2 n 2 - β m 2 ) e m ( x ) = 0 , e m ( - d / 2 ) = e m ( d / 2 ) = 0 ,
- d / 2 d / 2 e m 2 ( x ) d x = 1 .
E m ( x ) = ( 2 P Z 0 N m ) 1 / 2 e m ( x ) , H m ( x ) = - ( 2 N m P Z 0 ) 1 / 2 e m ( x ) ,
k n g 2 - N 2 w 2 = arctan [ ( n s 2 - N 2 n g 2 - N 2 ) 1 / 2 × cot k n s 2 - N 2 d - w 2 ] + m 2 ,
E ( x , z ) = 2 Z 0 P m a m ( z ) 1 N m e m ( x ) ,
d a m ( z ) d z = i β m a m ( z ) + i n κ m n a n ( z ) ,             n = 1 , 2 .
κ m n = k 2 N m N n - d / 2 d / 2 e m ( x ) Δ ɛ ( x ) e n ( x ) d x ,
Δ ɛ = { n g 2 - n s 2 for x w / 2 + s , 3 w / 2 + s 0 elsewhere ,
d a ( z ) d z = i k M · a ( z ) ,
a ( z ) = V · exp ( i k N z ) · V - 1 · a ( 0 ) ,
E ( x , z ) = 2 Z 0 P m , n V m n V 0 n 1 N m exp ( i k N n z ) e m ( x ) = 2 Z 0 P n V 0 n 1 N n exp ( i k N n z ) f n ( x ) ,
a m ( z ) = n V m n V 0 n exp ( i k N n z )
f n ( x ) = N n m V m n 1 N m e m ( x ) .

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