Abstract

More-accurate expressions are obtained for the coupling coefficient of the dielectric rectangular-waveguide directional coupler. The previous expressions may overestimate the coupling coefficient by as much as a factor of 2 in some cases of interest.

© 1983 Optical Society of America

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References

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  1. R. C. Alferness, R. V. Schmidt, E. H. Turner, “Characteristics of Ti-diffused lithium niobate optical directional couplers,” Appl. Opt. 18, 4012–4016 (1979).
    [Crossref] [PubMed]
  2. E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071–2102 (1969).
  3. A. Yariv, Introduction to Optical Electronics, 2nd ed. (Holt, Rinehart & Winston, New York, 1976), pp. 391–393.
  4. G. B. Hocker, W. K. Burns, “Mode dispersion in diffused channel waveguides by the effective index method,” Appl. Opt. 16, 113–118 (1977).
    [Crossref] [PubMed]
  5. D. Marcuse, Light Transmission Optics, 2nd ed. (Van Nostrand Reinhold, New York, 1982), pp. 425–429.
  6. N. S. Kapany, J. J. Burke, Optical Waveguides (Academic, New York, 1972), pp. 64–68.
  7. W. Streifer, E. Kapon, “Application of the equivalent-index method to DH diode lasers,” Appl. Opt. 18, 3724–3725 (1979).
    [PubMed]

1979 (2)

1977 (1)

1969 (1)

E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071–2102 (1969).

Alferness, R. C.

Burke, J. J.

N. S. Kapany, J. J. Burke, Optical Waveguides (Academic, New York, 1972), pp. 64–68.

Burns, W. K.

Hocker, G. B.

Kapany, N. S.

N. S. Kapany, J. J. Burke, Optical Waveguides (Academic, New York, 1972), pp. 64–68.

Kapon, E.

Marcatili, E. A. J.

E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071–2102 (1969).

Marcuse, D.

D. Marcuse, Light Transmission Optics, 2nd ed. (Van Nostrand Reinhold, New York, 1982), pp. 425–429.

Schmidt, R. V.

Streifer, W.

Turner, E. H.

Yariv, A.

A. Yariv, Introduction to Optical Electronics, 2nd ed. (Holt, Rinehart & Winston, New York, 1976), pp. 391–393.

Appl. Opt. (3)

Bell Syst. Tech. J. (1)

E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071–2102 (1969).

Other (3)

A. Yariv, Introduction to Optical Electronics, 2nd ed. (Holt, Rinehart & Winston, New York, 1976), pp. 391–393.

D. Marcuse, Light Transmission Optics, 2nd ed. (Van Nostrand Reinhold, New York, 1982), pp. 425–429.

N. S. Kapany, J. J. Burke, Optical Waveguides (Academic, New York, 1972), pp. 64–68.

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Figures (3)

Fig. 1
Fig. 1

Geometry of the slab-coupling problem. The field variation is cos hx inside the guides and exp(±px) outside the guides.

Fig. 2
Fig. 2

Dielectric rectangular-waveguide directional coupler. The field variation in the x direction is cos hx in the region with index n1, exp(±p3x) in the region with index n3, and exp(±p5x) in the region with index n5.

Fig. 3
Fig. 3

Coupling lengths L2 = π/2κy (new) and L1 = π/2κM (Marcatili) and the ratio R = L2/L1 versus wavelength. (n1/n3) = 1.005, 2a = 2b = 2d = 2 μm, and E 11 y-mode excitation.

Equations (14)

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κ TE = h 2 p exp ( 2 p d ) β a ( h 2 + p 2 ) ( 1 + 1 p a ) ,
κ TM = ( n 1 / n 2 ) 2 h 2 p exp ( 2 p d ) β a [ h 2 + ( n 1 / n 2 ) 4 p 2 ] [ 1 + 1 p a ( n 1 / n 2 ) 2 h 2 + p 2 h 2 + ( n 1 / n 2 ) 4 p 2 ] ,
κ M = h 2 p 5 exp ( 2 p 5 d ) β a ( h 2 + p 5 2 ) .
2 h a = m π tan 1 ( h / p 3 ) tan 1 [ h p 5 tanh ( p 5 d + γ ) ] ,
2 h a = [ m π tan 1 ( h / p 3 ) tan 1 ( h / p 5 ) ] 2 ( h / p 5 ) 1 1 + ( h / p 5 ) 2 exp ( 2 p 5 d )
κ = 1 2 ( β sym β antisym ) h 0 β 0 | h sym , antisym h 0 |
u = exp ( 2 p 5 d ) ,
h = h ( u = 0 ) + d h d u | u = 0 u = h 0 + h u .
2 h a = 1 1 + ( h 0 / p 30 ) 2 ( h / p 3 ) 1 1 + ( h 0 / p 50 ) 2 × [ ( h / p 3 ) ± 2 ( h 0 / p 50 ) ] ( at u = 0 ) .
h 2 + p i 2 = ( n 1 2 n i 2 ) k 0 2 , i = 3 , 5 ( for all u ) ,
p i = ( h / p i ) h , i = 3 , 5 ( for all u ) .
h = h 0 p 50 a ( h 0 2 + p 50 2 ) ( 1 + 1 2 p 30 a + 1 2 p 50 a )
κ y = h 2 p 5 exp ( 2 p 5 d ) β a ( h 2 + p 5 2 ) ( 1 + 1 2 p 3 a + 1 2 p 5 a ) ( E m l y modes ) .
κ x = ( n 1 / n 5 ) 2 h 2 p 5 exp ( 2 p 5 d ) β a [ h 2 + ( n 1 / n 5 ) 4 p 5 2 ] × [ 1 + 1 2 p 3 a ( n 1 / n 3 ) 2 h 2 + p 3 2 h 2 + ( n 1 / n 3 ) 4 p 3 2 + 1 2 p 5 a ( n 1 / n 5 ) 2 h 2 + p 5 2 h 2 + ( n 1 / n 5 ) 4 p 5 2 ] 1 ( E m l x modes ) ,

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