Abstract

A numerical technique to obtain the wave behavior in tightly coupled multimode fibers with inhomogeneous indices is introduced in this paper. The specific problem of the coupling characteristics of two parallel multimode fibers whose index profile is parabolic is treated in detail. It was found that in spite of the fact that rather complicated coupling behavior is observed when multimodes exist, total guide power still exchanges among the fibers in a periodic manner, and the coupling length still increases monotonically as a function of the separation distance between the fibers. It has also been demonstrated that by simply specifying the index profiles of the coupling structure (provided that the profiles are slowly varying), the coupling characteristics can be generated with our technique.

© 1979 Optical Society of America

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References

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  1. C. Yeh, S. B. Dong, W. Oliver, J. Appl. Phys. 46, 2125 (1975).
    [CrossRef]
  2. D. Marcuse, Bell Syst. Tech. J. 50, 1791 (1971).
  3. A. H. Cherin, E. J. Murphy, Bell Syst. Tech. J. 54, 17 (1975).
  4. J. A. Arnaud, Bell Syst. Tech. J. 54, 1431 (1975).
  5. J. S. Cook, Bell Syst. Tech. J. 34, 807 (1955).
  6. C. Yeh, L. Casperson, B. Szejn, J. Opt. Soc. Am., to appear (1978).
  7. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  8. K. Ogawa, Bell Syst. Tech. J. 56, 729 (1977).

1977 (1)

K. Ogawa, Bell Syst. Tech. J. 56, 729 (1977).

1975 (3)

C. Yeh, S. B. Dong, W. Oliver, J. Appl. Phys. 46, 2125 (1975).
[CrossRef]

A. H. Cherin, E. J. Murphy, Bell Syst. Tech. J. 54, 17 (1975).

J. A. Arnaud, Bell Syst. Tech. J. 54, 1431 (1975).

1971 (1)

D. Marcuse, Bell Syst. Tech. J. 50, 1791 (1971).

1955 (1)

J. S. Cook, Bell Syst. Tech. J. 34, 807 (1955).

Arnaud, J. A.

J. A. Arnaud, Bell Syst. Tech. J. 54, 1431 (1975).

Casperson, L.

C. Yeh, L. Casperson, B. Szejn, J. Opt. Soc. Am., to appear (1978).

Cherin, A. H.

A. H. Cherin, E. J. Murphy, Bell Syst. Tech. J. 54, 17 (1975).

Cook, J. S.

J. S. Cook, Bell Syst. Tech. J. 34, 807 (1955).

Dong, S. B.

C. Yeh, S. B. Dong, W. Oliver, J. Appl. Phys. 46, 2125 (1975).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Marcuse, D.

D. Marcuse, Bell Syst. Tech. J. 50, 1791 (1971).

Murphy, E. J.

A. H. Cherin, E. J. Murphy, Bell Syst. Tech. J. 54, 17 (1975).

Ogawa, K.

K. Ogawa, Bell Syst. Tech. J. 56, 729 (1977).

Oliver, W.

C. Yeh, S. B. Dong, W. Oliver, J. Appl. Phys. 46, 2125 (1975).
[CrossRef]

Szejn, B.

C. Yeh, L. Casperson, B. Szejn, J. Opt. Soc. Am., to appear (1978).

Yeh, C.

C. Yeh, S. B. Dong, W. Oliver, J. Appl. Phys. 46, 2125 (1975).
[CrossRef]

C. Yeh, L. Casperson, B. Szejn, J. Opt. Soc. Am., to appear (1978).

Bell Syst. Tech. J. (5)

D. Marcuse, Bell Syst. Tech. J. 50, 1791 (1971).

A. H. Cherin, E. J. Murphy, Bell Syst. Tech. J. 54, 17 (1975).

J. A. Arnaud, Bell Syst. Tech. J. 54, 1431 (1975).

J. S. Cook, Bell Syst. Tech. J. 34, 807 (1955).

K. Ogawa, Bell Syst. Tech. J. 56, 729 (1977).

J. Appl. Phys. (1)

C. Yeh, S. B. Dong, W. Oliver, J. Appl. Phys. 46, 2125 (1975).
[CrossRef]

Other (2)

C. Yeh, L. Casperson, B. Szejn, J. Opt. Soc. Am., to appear (1978).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

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Figures (9)

Fig. 1
Fig. 1

The fiber coupler.

Fig. 2
Fig. 2

A cross-sectional view of the power density distribution for the guided wave. Note the power exchange phenomenon as the wave propagates along the coupled structure. The percent value indicates the percent of total power contained in the right-hand half of the structure (i.e., fiber 2). The distance between each frame is 24,682 μm.

Fig. 3
Fig. 3

Percent of total power in the left-hand half of the structure (i.e., fiber 1) as a function of the axial distance along the coupling structure. This is the single-mode coupling case.

Fig. 4
Fig. 4

A cross-sectional view of the power density distribution for the guided wave. Note the power exchange phenomenon as the wave propagates along the coupled structure. The percent value indicates the percent of total power contained in the right-hand half of the structure (i.e., fiber 2). The distance between each frame is 9934 μm.

Fig. 5
Fig. 5

Percent of total power in the left-hand half of the structure (i.e., fiber 1) as a function of the axial distance along the coupling structure. This is the multimode coupling case.

Fig. 6
Fig. 6

A cross-sectional view of the power density distribution for the guided wave. Note the power exchange phenomenon as the wave propagates along the coupled structure. The percent value indicates the percent of total power contained in the right-hand half of the structure (i.e., fiber 2). The distance between each frame is 7024 μm.

Fig. 7
Fig. 7

Percent of total power in the left-hand half of the structure (i.e., fiber 1) as a function of the axial distance along the coupling structure. This is the multimode coupling case.

Fig. 8
Fig. 8

Coupling length as a function of separation distance for various single-mode and multimode cases for the fiber coupler.

Fig. 9
Fig. 9

Coupling length as a function of separation distance for the single-mode coupling case for two parallel slab waveguides.

Equations (23)

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n ( r 1 , 2 ) = n a ( 1 - δ r 1 , 2 2 a 2 ) ,
u ( x , y ) = u o exp { [ - ( x + d 2 ) 2 - y 2 ] / w 2 } ,
[ 2 + k 2 n 2 ( x , z ) ] u ( x , z ) = 0 ,
{ i 2 k n o z + T 2 + k 2 [ n 2 ( x , z ) - n 0 2 ] } A ( x , z ) = - 2 A ( x , z ) z 2 ,
{ i 2 k n o z + 2 x 2 + 2 y 2 + k 2 [ n 2 ( x , z ) - n 0 2 ] } A ( x , z ) = 0.
A ( x , 0 ) = u ( x , 0 ) ,
A ( ± , z ) = 0.
u ( x , y , o ) = u o exp ( - r 2 / w 2 ) for 0 r b = 0 for r > b ,
A ( x , z ) = exp [ Γ ( x , z ) ] v ( x , z ) ,
Γ ( x , z ) = i k 2 n o z o z [ n 2 ( x , y , z ) - n 0 2 ] d z .
i 2 k n o z v ( x , z ) + exp ( - Γ ) T 2 [ exp ( Γ ) v ( x , z ) ] = 0.
( i 2 k n o z + T 2 ) v ( x , z ) = 0
v ( x , z , 0 ) = u ( x , y , 0 ) .
v ( m , n , z ) = m , n N = 0 - 1 V ( m , n , z ) exp [ i 2 π N ( m m + n n ) ] ,
[ i 2 k n o z + f ( m , n ) ( Δ x ) 2 ] V ( m , n , z ) = 0
V ( m , n , z i ) = 1 N 2 m , n N = 0 - 1 v ( m , n , z i ) exp [ Γ ( m , n , z i ) ] × exp [ - i 2 π N ( m m + n n ) ] .
T 2 v = 1 ( Δ x ) 2 [ v ( m + 1 , n , z ) - 2 v ( m , n , z ) + v ( m - 1 , n , z ) + v ( m , n + 1 , z ) - 2 v ( m , n , z ) + v ( m , n - 1 , z ) ] ,
f ( m , n ) = - 4 [ sin 2 ( π m N ) + sin 2 ( π n N ) ] .
V ( m , n , z ) = V ( m , n z i ) exp [ - i f ( m , n ) 2 k ( Δ x ) 2 n o ( z - z i ) ] ,
α = ( 2 λ a ) / [ π n a w 2 ( 2 δ ) 1 / 2 ]
n a = 2.0 , λ = 0.8 μ m , δ = 0.81 × 10 - 6 , w = 100 μ m , a = 50 μ m ,
n a = 2.0 , λ = 0.8 μ m , δ = 5 × 10 - 6 , w = 100 μ m , a = 50 μ m ,
n a = 2.0 , λ = 0.8 μ m , δ = 10 - 5 , w = 100 μ m , a = 50 μ m ,

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