Abstract

By applying a coupled-mode theory to the angle degenerate modes of a planar waveguide and to the single mode of a neighboring monomode fiber, we derive an integrodifferential system describing the transfer of light intensity from the fiber to the planar waveguide. By numerically solving this system, the transmission loss of the fiber is computed as a function of various parameters such as the planar-guide thickness, the planar index, and the remaining fiber-cladding thickness.

© 1985 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. Bergh, G. Kotler, H. J. Shaw, “Single-mode fiber optic directional coupler,” Electron. Lett. 16, 26 (1980).
    [CrossRef]
  2. B. Lamouroux, A. Orszag, B. Prade, J. Y. Vinet, “Continuous laser amplification in a monomode fiber,” Opt. Lett. 8, 504 (1983).
    [CrossRef] [PubMed]
  3. A. W. Snyder, “Coupled-mode theory for optical fibers,”J. Opt. Soc. Am. 62, 1267 (1972).
    [CrossRef]
  4. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).

1983 (1)

1980 (1)

R. Bergh, G. Kotler, H. J. Shaw, “Single-mode fiber optic directional coupler,” Electron. Lett. 16, 26 (1980).
[CrossRef]

1972 (1)

Bergh, R.

R. Bergh, G. Kotler, H. J. Shaw, “Single-mode fiber optic directional coupler,” Electron. Lett. 16, 26 (1980).
[CrossRef]

Kotler, G.

R. Bergh, G. Kotler, H. J. Shaw, “Single-mode fiber optic directional coupler,” Electron. Lett. 16, 26 (1980).
[CrossRef]

Lamouroux, B.

Marcuse, D.

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

Orszag, A.

Prade, B.

Shaw, H. J.

R. Bergh, G. Kotler, H. J. Shaw, “Single-mode fiber optic directional coupler,” Electron. Lett. 16, 26 (1980).
[CrossRef]

Snyder, A. W.

Vinet, J. Y.

Electron. Lett. (1)

R. Bergh, G. Kotler, H. J. Shaw, “Single-mode fiber optic directional coupler,” Electron. Lett. 16, 26 (1980).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Lett. (1)

Other (1)

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

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 (9)

Fig. 1
Fig. 1

Geometry of the system.

Fig. 2
Fig. 2

a, Index profile of the stack; b, index profile of the unperturbed fiber; c, index profile of the unperturbed planar guide.

Fig. 3
Fig. 3

Transmission of the fiber as a function of the interaction length L for various liquid thicknesses T. nl = 1.462; D = 5 μm.

Fig. 4
Fig. 4

Transmission of the fiber as a function of the interaction length L for various distances D. n = 1.462; T = 20 μm.

Fig. 5
Fig. 5

Transmission of the fiber as a function of the interaction length L for various distances D. nl = 1.462; T = 100 μm.

Fig. 6
Fig. 6

Transmission of the fiber as a function of the interaction length L for various values of nl. T = 20 μm; D = 5 μm.

Fig. 7
Fig. 7

Transmission of the fiber as a function of the interaction length L for various values of nl. T = 100 μm; D = 5 μm.

Fig. 8
Fig. 8

Transmission of the fiber as a function of the index of refraction of the planar waveguide. Curve (a): T = 20 μm, D = 5 μm, L = 2 mm. Curve (b): T = 100 μm, D = 5 μm, L = 2 mm.

Fig. 9
Fig. 9

Notation for the unperturbed planar-waveguide modes.

Tables (2)

Tables Icon

Table 1 Numerical Results for D = 5 μm and nl = 1.462a

Tables Icon

Table 2 Numerical Results for D = 5 μm and nl = 1.462a

Equations (36)

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

curl H = i ω 0 r ( x , y , z ) E ,
curl E = - i ω μ 0 H ,
div ( 0 r E ) = 0 ,
div H = 0 ,
n c l 2 [ within regions I , III , and V ( cladding ) ] , n c o 2 [ within region II ( fiber core ) ] , n l 2 [ within region IV ( planar - guide core ) ] .
( z + i β ν ) = R 2 ( H × E ν * + H ν * × E ) · e z d x d y = i ω 0 R 2 ( r - r ) E · E ν * d x d y ,
E = E ν ( x , y ) exp ( - i β ν z )
H = H ν ( x , y ) exp ( - i β ν z ) .
E = a F ( z ) E F ( x , y ) ,
H = a F ( z ) H F ( x , y ) ,
E = ν = 1 N a g ν ( θ , z ) E g ν ( x , y , θ ) d θ ,
H = ν = 1 N a g ν ( θ , z ) H g ν ( x , y , θ ) d θ ,
d a g ν d z ( θ , z ) + i β g ν cos θ a g ν ( θ , z ) = - i ω 0 a F 4 P g ν ( θ ) II ( n c o 2 - n c l 2 ) E F E g ν * ( θ ) d x d y ,
P g ν ( θ ) = 2 π P g ν ( 0 ) β g ν cos θ .
d a F d z + i β F a F = - i ω 0 4 P F ν = 1 N - π / 2 + π / 2 d θ + IV ( n l 2 - n c l 2 ) a g ν ( θ , z ) E g ν ( θ ) E F * d x d y .
d a g ν ( θ , z ) d z + i β g ν cos θ a g ν ( θ , z ) = - i A ν ( θ ) a F ( z ) ,
d a F d z + i β F a F ( z ) = - i ν = 1 N - π / 2 + π / 2 B ν ( θ ) a g ν ( θ , z ) d θ ,
A ν ( θ ) = ω 0 4 P g ν ( θ ) II ( n c o 2 - n c l 2 ) E F · E g ν * ( θ ) d x d y
B ν ( θ ) = ω 0 4 P F IV ( n l 2 - n c l 2 ) E g ν ( θ ) · E F * d x d y .
d d z a p ( z ) = - i q = 0 N M C p q a q ( z ) .
J 0 ( U ) U J 1 ( U ) = K 0 ( W ) W K 1 ( W ) ,
E x = 0 , E y = α J 0 [ U ( r / a ) ] , H x = α n F [ ( 0 / μ 0 ) ] 1 / 2 J 0 [ U ( r / a ) ] , H y = 0 ,             for r a , E x = 0 , E y = α J 0 ( U ) K 0 ( W ) K 0 [ W ( r / a ) ] , H x = α n F [ ( 0 / μ 0 ) 1 / 2 J 0 ( U ) K 0 ( W ) K 0 [ W ( r / a ) , H y = 0 ,             for r a ,
E X = 0 , E Y = γ exp ( k 0 k 1 X )             within region 1 , E Y = γ [ cos ( k 0 k 2 X ) + k 1 k 2 sin ( k 0 k 2 X ) ]             within region 2 , E Y = γ [ cos ( k 0 k 2 d ) + k 1 k 2 sin ( k 0 k 2 d ) ] × exp [ - k 0 k 3 ( X - d ) ]             within region 3 , E Z = 0 ,
tan ( k 0 k 2 d ) = k 2 ( k 1 + k 3 ) / ( k 2 2 - k 1 k 3 ) .
P g = k 0 n g 2 ω μ 0 R E y 2 d X ,
P g = k 0 n g γ 2 4 ω μ 0 ( k 3 2 + k 2 2 ) k 2 2 ( d + 1 k 0 k 1 + 1 k 0 k 3 ) .
E = φ ( x ) exp [ - i β g ( y sin θ + z cos θ ) ] ( cos θ e ^ y - sin θ e ^ z ) ,
φ ( x ) = E Y ,
E = E ( x , y , θ ) exp [ - i z β g cos θ ] ,
E g ν ( θ ) = φ ( x ) exp [ - i β g ν y sin θ ] [ e ^ y cos θ - e ^ z sin θ ] .
Q μ ν ( θ ) = 1 / 2 0 2 π d θ R 2 [ E g ν * ( θ ) × H g μ ( θ ) ] e ^ z d x d y ,
E g ν ( θ ) = E g ν ( 0 ) exp ( - i β g ν y sin θ ) , H g μ ( θ ) = H g μ ( 0 ) exp ( - i β g μ y sin θ ) ,
P g ν ( θ ) = R d x [ E g ν ( 0 ) × H g μ ( 0 ) ] × e ^ z R d y exp [ i y ( β g μ sin θ - β g ν sin θ ) ] .
Q μ ν = P g ν ( θ ) δ μ ν ,
P g ν ( θ ) = 0 2 π d θ P g ν ( 0 ) 2 π δ ( θ - θ ) β g ν cos θ ,
P g ν ( θ ) = 2 π P g ν ( 0 ) β g ν cos θ .

Metrics