Abstract

A theory for treating coupling between curved transmission lines is derived as an extension of the theory of coupled parallel transmission lines. It is assumed that there exists a one-to-one correspondence between the coupled points on the two transmission lines. As an example, numerical results based on the theory are included for the characteristic interaction parameters associated with directional couplers consisting of lossless semicircular and linear transmission lines. The general expressions that are derived for the interaction matrix elements are valid in these particular cases.

© 1975 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2103 (1969).
    [CrossRef]
  2. S. E. Miller, Bell Syst. Tech. J. 33, 661 (1954).
    [CrossRef]
  3. J. R. Pierce, J. Appl. Phys. 25, 179 (1954).
    [CrossRef]
  4. A. W. Snyder, J. Opt. Soc. Am. 62, 1267 (1972).
    [CrossRef]
  5. A. Yariv, IEEE J. Quantum Electron. 9, 919 (1973).
    [CrossRef]
  6. P. D. McIntyre and A. W. Snyder, J. Opt. Soc. Am. 64, 285 (1974).
    [CrossRef] [PubMed]
  7. D. Marcuse, Bell Syst. Tech. J. 50, 1791 (1971).
    [CrossRef]
  8. M. Matsuhara and N. Kumagai, Trans. IECE Jpn. 55–C, 201, (1972).
  9. E. A. J. Marcatili, Bell. Syst. Tech. J. 48, 2071 (1969).
    [CrossRef]

1974 (1)

1973 (1)

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

1972 (2)

A. W. Snyder, J. Opt. Soc. Am. 62, 1267 (1972).
[CrossRef]

M. Matsuhara and N. Kumagai, Trans. IECE Jpn. 55–C, 201, (1972).

1971 (1)

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

1969 (2)

E. A. J. Marcatili, Bell. Syst. Tech. J. 48, 2071 (1969).
[CrossRef]

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2103 (1969).
[CrossRef]

1954 (2)

S. E. Miller, Bell Syst. Tech. J. 33, 661 (1954).
[CrossRef]

J. R. Pierce, J. Appl. Phys. 25, 179 (1954).
[CrossRef]

Kumagai, N.

M. Matsuhara and N. Kumagai, Trans. IECE Jpn. 55–C, 201, (1972).

Marcatili, E. A. J.

E. A. J. Marcatili, Bell. Syst. Tech. J. 48, 2071 (1969).
[CrossRef]

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2103 (1969).
[CrossRef]

Marcuse, D.

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

Matsuhara, M.

M. Matsuhara and N. Kumagai, Trans. IECE Jpn. 55–C, 201, (1972).

McIntyre, P. D.

Miller, S. E.

S. E. Miller, Bell Syst. Tech. J. 33, 661 (1954).
[CrossRef]

Pierce, J. R.

J. R. Pierce, J. Appl. Phys. 25, 179 (1954).
[CrossRef]

Snyder, A. W.

Yariv, A.

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

Bell Syst. Tech. J. (3)

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2103 (1969).
[CrossRef]

S. E. Miller, Bell Syst. Tech. J. 33, 661 (1954).
[CrossRef]

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

Bell. Syst. Tech. J. (1)

E. A. J. Marcatili, Bell. Syst. Tech. J. 48, 2071 (1969).
[CrossRef]

IEEE J. Quantum Electron. (1)

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

J. Appl. Phys. (1)

J. R. Pierce, J. Appl. Phys. 25, 179 (1954).
[CrossRef]

J. Opt. Soc. Am. (2)

Trans. IECE Jpn. (1)

M. Matsuhara and N. Kumagai, Trans. IECE Jpn. 55–C, 201, (1972).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

FIG. 1
FIG. 1

Optical directional couplers incorporating transmission lines with curvature.

FIG. 2
FIG. 2

Coupling between curved transmission lines.

FIG. 3
FIG. 3

Geometry of a directional coupler consisting of a straight line and a semicircular line.

FIG. 4
FIG. 4

Variation of |a11(x0, x)|2 as a function of the normalized distance 2y/πR along line 2 for β1 = β2, R/λ = 1000, and some selected values of D/λ.

FIG. 5
FIG. 5

Variation of |a11(x0, x)|2 along line 2 for R1 = 1000, D1 = 3.92, and a range of values of β2/β1.

FIG. 6
FIG. 6

Dependence of |a11(x0, xN)|2 on the normalized separation distance D1 in a directional coupler with phase-velocity synchronization.

FIG. 7
FIG. 7

Dependence of arga11 (x0, xN) and arg a 12 ( x 0 , x N ) on D1. The solid lines are for arga11 and the dotted lines are for arg a 12 .

Equations (32)

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

d Ψ 1 ( x ) = j β 1 ( x ) Ψ 1 ( x ) d x + j c 2 * ( y ) Ψ 2 ( y ) d y , d Ψ 2 ( y ) = j β 2 ( y ) Ψ 2 ( y ) d y + j c 1 * ( x ) Ψ 1 ( x ) d x ,
d ( Ψ 1 Ψ 1 * + Ψ 2 Ψ 2 * ) = j ( β 1 β 1 * ) Ψ 1 Ψ 1 * d x j ( β 2 β 2 * ) Ψ 2 Ψ 2 * d y + 2 Re { j ( c 2 * d y c 1 d x ) Ψ 1 * Ψ 2 } ,
c 1 d x = c 2 * d y .
( c 1 c 2 ) 1 / 2 = c 0 cos ( π 2 θ ) ,
c 1 = c 0 · cos ( π 2 θ ) · ( d y d x ) 1 / 2 , c 2 = c 0 · cos ( π 2 θ ) · ( d x d y ) 1 / 2 ,
d Ψ 1 d x = j β 1 Ψ 1 + j c 1 Ψ 2 , d Ψ 2 d y = j β 2 Ψ 2 + j c 2 Ψ 1 .
Ψ 1 ( x ) = ψ 1 ( x ) exp { j x 0 x β 1 ( x ) d x } , Ψ 2 ( y ) = ψ 2 ( y ) exp { j y 0 y β 2 ( y ) d y } ,
d ψ 1 d x = j c 1 exp { j ( y 0 y β 2 d y x 0 x β 1 d x ) } ψ 2 , d ψ 2 d y = j c 2 exp { j ( x 0 x β 1 d x y 0 y β 2 d y ) } ψ 1 ,
d 2 ψ 1 d x 2 + j 2 B 1 d ψ 1 d x + C 1 2 ψ 1 = 0 ,
d 2 ψ 2 d y 2 + j 2 B 2 d ψ 2 d y + C 2 2 ψ 2 = 0 ,
B 1 = 1 2 { ( β 2 d y d x β 1 ) + j 1 c 1 d c 1 d x } , B 2 = 1 2 { ( β 1 d x d y β 2 ) + j 1 c 2 d c 2 d y } , C 1 = ( c 1 c 2 d y d x ) 1 / 2 , C 2 = ( c 1 c 2 d x d y ) 1 / 2 .
[ ψ 1 ( x n ) ψ 2 ( y n ) ] = [ a 11 ( x 0 , x n ) j a 12 ( x 0 , x n ) j a 21 ( y 0 , y n ) a 22 ( y 0 , y n ) ] [ ψ 1 ( x 0 ) ψ 2 ( y 0 ) ] ,
[ Ψ 1 ( x ) Ψ 2 ( y ) ] = [ a 11 ( x 0 , x ) · exp ( j x 0 x β 1 d x ) j a 12 ( x 0 , x ) · exp ( j x 0 x β 1 d x ) j a 21 ( y 0 , y ) · exp ( j y 0 y β 2 d y ) a 22 ( y 0 , y ) · exp ( j y 0 y β 2 d y ) ] · [ Ψ 1 ( x 0 ) Ψ 2 ( y 0 ) ] .
Ψ 1 ( x ) = a 11 ( x 0 , x ) exp ( j x 0 x β 1 d x ) , Ψ 2 ( y ) = j a 21 ( y 0 , y ) exp ( j y 0 y β 2 d y ) .
a 11 a 22 + a 12 a 21 = 1 .
a 11 a 11 * + a 21 a 21 * = 1 , a 22 a 22 * + a 12 a 12 * = 1 , a 11 a 12 * = a 22 * a 21 .
a 11 = a 22 * , a 12 = a 21 * , | a 11 | 2 + | a 21 | 2 = 1 .
C 0 = a λ 1 exp ( b λ 1 l ) ,
x 2 R + D = tan y 2 R .
β 1 d x = β 2 d y .
β 2 β 1 = d x d y | x = 0 , y = 0 = 1 + D 2 R .
arg a 12 ( x 0 , x N ) = arg a 12 ( x 0 , x N ) 1 2 { ( x N x 0 ) β 1 ( y N y 0 ) β 2 }
d 2 ψ 1 d x 2 + j 2 B 1 d ψ 1 d x + c 1 2 ψ 1 = 0 ,
d 2 ψ 2 d y 2 + j 2 B 2 d ψ 2 d y + C 2 2 ψ 2 = 0 ,
B 1 = 1 2 { ( β 2 d y d x β 1 ) + j 1 c 1 d c 1 d x } , B 2 = 1 2 { ( β 1 d x d y β 2 ) + j 1 c 2 d c 2 d y } , C 1 = ( c 1 c 2 d y d x ) 1 / 2 , C 2 = ( c 1 c 2 d x d y ) 1 / 2 .
ψ 1 ( x ) = [ P · cos { ( B 1 k 2 + C 1 k 2 ) 1 / 2 · ( x x k 1 ) } + Q · sin { ( B 1 k 2 + C 1 k 2 ) 1 / 2 · ( x x k 1 ) } ] · exp { j B 1 k · ( x x k 1 ) } ,
ψ 1 ( x ) = ψ 1 ( x k 1 ) at x = x k 1 , ψ 2 ( y ) = ψ 2 ( y k 1 ) at y = y k 1 .
[ ψ 1 ( x k ) ψ 2 ( y k ) ] = [ b 11 ( x k 1 , x k ) j b 12 ( x k 1 , x k ) j b 21 ( y k 1 , y k ) b 22 ( y k 1 , y k ) ] [ ψ 1 ( x k 1 ) ψ 2 ( y k 1 ) ] ,
b 11 ( x k 1 , x k ) = [ cos { ( B 1 k 2 + C 1 k 2 ) 1 / 2 · ( x k k k 1 ) } + j B 1 k · ( B 1 k 2 + C 1 k 2 ) 1 / 2 · sin { ( B 1 k 2 + C 1 k 2 ) 1 / 2 · ( x k x k 1 ) } ] · exp { j B 1 k · ( x k k k 1 ) } , b 22 ( y k 1 , y k ) = [ cos { ( B 2 k 2 + C 2 k 2 ) 1 / 2 · ( y k k k 1 ) } + j B 2 k · ( B 2 k 2 + C 2 k 2 ) 1 / 2 · sin { ( B 2 k 2 + C 2 k 2 ) 1 / 2 · ( y k y k 1 ) } ] · exp { j B 2 k · ( y k k k 1 ) } , b 12 ( x k 1 , x k ) = c 1 ( x k 1 ) · exp { j ( y 0 y k 1 β 2 d y x 0 x k 1 β 1 d x ) } · ( B 1 k 2 + C 1 k 2 ) 1 / 2 · sin { ( B 1 k 2 + C 1 k 2 ) 1 / 2 · ( x k x k 1 ) } · exp { j B 1 k · ( x k x k 1 ) } , b 21 ( y k 1 , y k ) = c 2 ( y k 1 ) · exp { j ( x 0 x k 1 β 1 d x y 0 y k 1 β 2 d y ) } · ( B 2 k 2 + C 2 k 2 ) 1 / 2 · sin { ( B 2 k 2 + C 2 k 2 ) 1 / 2 · ( y k y k 1 ) } · exp { j B 2 k · ( y k y k 1 ) } ,
[ ψ 1 ( x n ) ψ 2 ( y n ) ] = [ a 11 ( x 0 , x n ) j a 12 ( x 0 , x n ) j a 21 ( y 0 , y n ) a 22 ( y 0 , y n ) ] [ ψ 1 ( x 0 ) ψ 2 ( y 0 ) ] ,
[ a 11 ( x 0 , x n ) j a 12 ( x 0 , x n ) j a 21 ( y 0 , y n ) a 22 ( y 0 , y n ) ] = k = 1 n [ b 11 ( x k 1 , x k ) j b 12 ( x k 1 , x k ) j b 21 ( y k 1 , y k ) b 22 ( y k 1 , y k ) ] .
k = 1 n M k = M n M n 1 M 1 ,