Abstract

The operation of tapered velocity couplers is analyzed from the point of view of mode conversion between local-normal modes. An approximate coupled mode representation of the local-normal modes is used with a step transition model to estimate analytically power transfer. This yields an upper limit for the maximum permissible variation in mode synchronism across the coupler to achieve satisfactory coupler operation. The analysis has also been used to demonstrate the importance of obtaining a large total change in mode synchronism to minimize coupler length.

© 1975 Optical Society of America

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References

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  1. A. Ihaya, H. Furuta, H. Noda, Proc. IEEE 60, 470 (1972).
    [CrossRef]
  2. J. S. Cook, Bell Syst. Tech. J. 34, 807 (1955).
  3. M. G. F. Wilson, G. A. Teh, Electron. Lett. 9, 453 (1973).
    [CrossRef]
  4. D. Marcuse, Bell Syst. Tech. J. 49, 273 (1970).
  5. A. G. Fox, Bell Syst. Tech. J. 34, 823 (1955).
  6. W. H. Louisell, Bell Syst. Tech. J. 34, 853 (1955).
  7. W. K. Burns, A. F. Milton, IEEE, J. Quantum Electron. QE-11, 32 (1975).
    [CrossRef]
  8. W. H. Louisell, Coupled Mode and Parametric Electronics (Wiley, New York, 1960).
  9. A Yariv, IEEE J. Quantum Electron. QE-9, 919 (1973).
    [CrossRef]
  10. M. G. F. Wilson, G. A. Teh, “Tapered Optical Directional Coupler,” to be published IEEE Trans.MTT (Jan.1975).
  11. H. Yajima, “Theory and Applications of Dielectric Branching Waveguides,” Proceedings, Symposium on Optical and Acoustical Micro-Electronics (New York, 1974), to be published.
  12. N. S. Kapany, J. J. Burke, Optical Waveguides (Academic, New York, 1972).

1975 (1)

W. K. Burns, A. F. Milton, IEEE, J. Quantum Electron. QE-11, 32 (1975).
[CrossRef]

1973 (2)

A Yariv, IEEE J. Quantum Electron. QE-9, 919 (1973).
[CrossRef]

M. G. F. Wilson, G. A. Teh, Electron. Lett. 9, 453 (1973).
[CrossRef]

1972 (1)

A. Ihaya, H. Furuta, H. Noda, Proc. IEEE 60, 470 (1972).
[CrossRef]

1970 (1)

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

1955 (3)

A. G. Fox, Bell Syst. Tech. J. 34, 823 (1955).

W. H. Louisell, Bell Syst. Tech. J. 34, 853 (1955).

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

Burke, J. J.

N. S. Kapany, J. J. Burke, Optical Waveguides (Academic, New York, 1972).

Burns, W. K.

W. K. Burns, A. F. Milton, IEEE, J. Quantum Electron. QE-11, 32 (1975).
[CrossRef]

Cook, J. S.

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

Fox, A. G.

A. G. Fox, Bell Syst. Tech. J. 34, 823 (1955).

Furuta, H.

A. Ihaya, H. Furuta, H. Noda, Proc. IEEE 60, 470 (1972).
[CrossRef]

Ihaya, A.

A. Ihaya, H. Furuta, H. Noda, Proc. IEEE 60, 470 (1972).
[CrossRef]

Kapany, N. S.

N. S. Kapany, J. J. Burke, Optical Waveguides (Academic, New York, 1972).

Louisell, W. H.

W. H. Louisell, Bell Syst. Tech. J. 34, 853 (1955).

W. H. Louisell, Coupled Mode and Parametric Electronics (Wiley, New York, 1960).

Marcuse, D.

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

Milton, A. F.

W. K. Burns, A. F. Milton, IEEE, J. Quantum Electron. QE-11, 32 (1975).
[CrossRef]

Noda, H.

A. Ihaya, H. Furuta, H. Noda, Proc. IEEE 60, 470 (1972).
[CrossRef]

Teh, G. A.

M. G. F. Wilson, G. A. Teh, Electron. Lett. 9, 453 (1973).
[CrossRef]

M. G. F. Wilson, G. A. Teh, “Tapered Optical Directional Coupler,” to be published IEEE Trans.MTT (Jan.1975).

Wilson, M. G. F.

M. G. F. Wilson, G. A. Teh, Electron. Lett. 9, 453 (1973).
[CrossRef]

M. G. F. Wilson, G. A. Teh, “Tapered Optical Directional Coupler,” to be published IEEE Trans.MTT (Jan.1975).

Yajima, H.

H. Yajima, “Theory and Applications of Dielectric Branching Waveguides,” Proceedings, Symposium on Optical and Acoustical Micro-Electronics (New York, 1974), to be published.

Yariv, A

A Yariv, IEEE J. Quantum Electron. QE-9, 919 (1973).
[CrossRef]

Bell Syst. Tech. J. (4)

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

A. G. Fox, Bell Syst. Tech. J. 34, 823 (1955).

W. H. Louisell, Bell Syst. Tech. J. 34, 853 (1955).

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

Electron. Lett. (1)

M. G. F. Wilson, G. A. Teh, Electron. Lett. 9, 453 (1973).
[CrossRef]

IEEE J. Quantum Electron. (1)

A Yariv, IEEE J. Quantum Electron. QE-9, 919 (1973).
[CrossRef]

IEEE, J. Quantum Electron. (1)

W. K. Burns, A. F. Milton, IEEE, J. Quantum Electron. QE-11, 32 (1975).
[CrossRef]

Proc. IEEE (1)

A. Ihaya, H. Furuta, H. Noda, Proc. IEEE 60, 470 (1972).
[CrossRef]

Other (4)

W. H. Louisell, Coupled Mode and Parametric Electronics (Wiley, New York, 1960).

M. G. F. Wilson, G. A. Teh, “Tapered Optical Directional Coupler,” to be published IEEE Trans.MTT (Jan.1975).

H. Yajima, “Theory and Applications of Dielectric Branching Waveguides,” Proceedings, Symposium on Optical and Acoustical Micro-Electronics (New York, 1974), to be published.

N. S. Kapany, J. J. Burke, Optical Waveguides (Academic, New York, 1972).

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Figures (4)

Fig. 1
Fig. 1

(A) A tapered velocity coupler composed of two guiding layers with a linear decrease in the thickness of the top guiding layer. The first (i) and second (j) order local normal mode amplitudes are plotted at various positions along the coupler. This example is taken from the first directional coupler of Ref. 10. (B) Effective index (β/k) for the various modes as a function of position along the tapered velocity coupler of (A). Curves labeled i and j represent the local normal modes; a and b represent the conventional modes of the two three-layer waveguide taken separately. The positions corresponding to X ≡ −iΔβ/2K = ±1 for this example are also noted.

Fig. 2
Fig. 2

Model of one of the steps used to approximate a continuous decrease in the thickness of the top guiding layer of the tapered velocity coupler described in Fig. 1.

Fig. 3
Fig. 3

The amplitude ratio f = e/d of local normal mode i, gcij, and Δβij/(KK*)1/2 as a function of X = −iΔβ/2K.

Fig. 4
Fig. 4

The computer calculations and experimental data of Wilson and Teh of the power transfer ratio (P/P0) between the two waveguide layers replotted as a function of α/KK*, where α = dβ)/dz. The data points when replotted in this manner fall on a single curve. They are taken from seven separate curves and three experiments reported in Refs. 3 and 10.

Equations (44)

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d a / d z = K b exp ( i Δ β z ) ,
d b / d z = K * a exp ( i Δ β z ) ,
e y = E ( z ) ( x , z ) exp [ i α ( z ) ] ,
α ( z ) = β z + ϕ .
E γ s = ( ( β γ / 2 k ) I γ , γ ) 1 / 2 γ = i , j
I γ , δ = γ δ d x γ , δ = i , j .
A j 1 = c i j A i 0 cos ( α i 0 α j 1 ) + c j j A j 0 cos ( α j 0 α j 1 ) ,
tan α j 1 = c i j A i 0 sin α i 0 + c j j A j 0 sin α j 0 c i j A i 0 cos α i 0 + c j j A j 0 cos α j 0 ,
c i j = 2 ( β i 0 β j 1 ) 1 / 2 ( β j 0 + β j 1 ) ( β j 0 + β i 1 β i 0 + β i 1 ) I i 0 , j 1 ( I i 0 , i 0 I j 1 , j 1 ) 1 / 2 ,
e y a = a Φ a ( x ) exp ( i β a z i ϕ a )
e y b = b Φ b ( x ) exp ( i β b z i ϕ b ) .
( β i / 2 k ) 1 / 2 E i s i d Φ a + e Φ b ψ i ,
( β j / 2 k ) 1 / 2 E j s j e Φ a + d Φ b ψ j ,
d 2 + e 2 = 1 ; Φ a Φ a d x = 1 ; Φ b Φ b d x = 1 ;
f = e d = i { Δ β + [ ( Δ β ) 2 + 4 K K * ] 1 / 2 } 2 K for Δ β > 0
= X + X ( X X * ) 1 / 2 ( X X * + 1 ) 1 / 2 where X = i Δ β / 2 K ,
f = e d = 2 K i { Δ β + [ ( Δ β ) 2 + 4 K K * ] 1 / 2 } for Δ β < 0
= [ X X ( X X * ) 1 / 2 ( X X * + 1 ) 1 / 2 ] 1 .
c i j = ( β j + β i ) 2 ( β i β j ) 1 / 2 I i 0 , j 1 ( I i 0 , i 0 I j 1 , j 1 ) 1 / 2
= β i + β j 2 ( β i β j ) 1 / 2 ψ i 0 ψ j 1 d x .
Φ a 0 Φ a 1 d x = 1 ,
Φ b 0 Φ b 1 d x = 1 ,
Φ b 0 Φ a 1 d x = 0 ,
Φ a 0 Φ b 1 d x = 0 .
c i j = β i + β j 2 ( β i β j ) 1 / 2 ( d 0 e 1 + e 0 d 1 )
= ( β i + β j ) 2 ( β i β j ) 1 / 2 ( d 0 d 1 ) ( f 1 f 0 )
c i j ( β i + β j ) 2 ( β i β j ) 1 / 2 1 ( 1 + f 2 ) f X δ X ,
c i j = ( β i + β j ) 2 ( β i β j ) 1 / 2 1 ( 1 + f 2 ) f X ( i 2 K ) δ ( Δ β ) .
K = i F exp ( γ 3 d 3 ) ,
γ 3 = [ ( β a + β b 2 ) 2 n 3 2 k 2 ] 1 / 2 .
g = 1 ( 1 + f 2 ) f X = 1 2 ( X 2 + 1 ) X 0 .
Δ β i j = β i β j [ ( Δ β ) 2 + 4 K K * ] 1 / 2
2 ( K K * ) 1 / 2 ( X 2 + 1 ) 1 / 2 .
x = 1 x = 1 Δ β i j d z > π 2 ,
[ X = 1 X = 1 2 ( K K * ) 1 / 2 ( X 2 + 1 ) 1 / 2 d X ] d z d X > π 2 .
4.6 ( K K * ) 1 / 2 2 K i d z d ( Δ β ) > π 2 .
| d ( Δ β ) d z | < 5.9 K K * .
P b i P a i = e 2 d 2 β b β a 1 4 X X * = | K | 2 ( Δ β ) 2 ; Δ β > 0 .
4 X X * = ( Δ β ) 2 | K | 2 ; Δ β < 0 .
P a i P a i + P b i = 0.99 when Δ β = Δ β T / 2
= 0.012 when Δ β = Δ β T / 2 .
( Δ β T / 2 ) 2 K K * 80 ,
Δ β T / Δ Z T K K * 1.5 ,
Δ Z T 213 / Δ β T

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