Abstract

A simple closed-form description of grating-assisted coupling between twin cores is presented. It is exact in the limit of weakly coupled cores and weak grating perturbations.

© 1991 Optical Society of America

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References

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  1. D. Marcuse, IEEE J. Lightwave Technol. LT-5, 268 (1987).
    [Crossref]
  2. A. W. Snyder, O. J. Davies, Proc. IEEE 58, 168 (1970).
    [Crossref]
  3. P. L. Chu, A. W. Snyder, Electron. Lett. 23, 1101 (1987).
    [Crossref]
  4. A. W. Snyder, J. Opt. Soc. Am. 62, 1267 (1972).
    [Crossref]
  5. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), pp. 314, 397–398, 545–552.
  6. A. W. Snyder, Y. Chen, A. Ankiewicz, IEEE J. Lightwave Technol. 7, 1400 (1989).
    [Crossref]
  7. W. P. Huang, H. A. Haus, IEEE J. Lightwave Technol. 7, 920 (1989).
    [Crossref]
  8. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), Chap. 10.
  9. A. W. Snyder, A. Ankiewicz, A. Altintas, Electron. Lett. 23, 1097 (1987).
    [Crossref]
  10. A. W. Snyder, IEEE Trans. Microwave Theory Tech. MT-17, 1130 (1969).
    [Crossref]
  11. A. W. Snyder, A. Ankiewicz, IEEE J. Lightwave Technol. 6, 463 (1988).
    [Crossref]
  12. A. W. Snyder, Y. Chen, Electron. Lett. 25, 502 (1989).
    [Crossref]

1989 (3)

A. W. Snyder, Y. Chen, A. Ankiewicz, IEEE J. Lightwave Technol. 7, 1400 (1989).
[Crossref]

W. P. Huang, H. A. Haus, IEEE J. Lightwave Technol. 7, 920 (1989).
[Crossref]

A. W. Snyder, Y. Chen, Electron. Lett. 25, 502 (1989).
[Crossref]

1988 (1)

A. W. Snyder, A. Ankiewicz, IEEE J. Lightwave Technol. 6, 463 (1988).
[Crossref]

1987 (3)

A. W. Snyder, A. Ankiewicz, A. Altintas, Electron. Lett. 23, 1097 (1987).
[Crossref]

D. Marcuse, IEEE J. Lightwave Technol. LT-5, 268 (1987).
[Crossref]

P. L. Chu, A. W. Snyder, Electron. Lett. 23, 1101 (1987).
[Crossref]

1972 (1)

1970 (1)

A. W. Snyder, O. J. Davies, Proc. IEEE 58, 168 (1970).
[Crossref]

1969 (1)

A. W. Snyder, IEEE Trans. Microwave Theory Tech. MT-17, 1130 (1969).
[Crossref]

Altintas, A.

A. W. Snyder, A. Ankiewicz, A. Altintas, Electron. Lett. 23, 1097 (1987).
[Crossref]

Ankiewicz, A.

A. W. Snyder, Y. Chen, A. Ankiewicz, IEEE J. Lightwave Technol. 7, 1400 (1989).
[Crossref]

A. W. Snyder, A. Ankiewicz, IEEE J. Lightwave Technol. 6, 463 (1988).
[Crossref]

A. W. Snyder, A. Ankiewicz, A. Altintas, Electron. Lett. 23, 1097 (1987).
[Crossref]

Chen, Y.

A. W. Snyder, Y. Chen, Electron. Lett. 25, 502 (1989).
[Crossref]

A. W. Snyder, Y. Chen, A. Ankiewicz, IEEE J. Lightwave Technol. 7, 1400 (1989).
[Crossref]

Chu, P. L.

P. L. Chu, A. W. Snyder, Electron. Lett. 23, 1101 (1987).
[Crossref]

Davies, O. J.

A. W. Snyder, O. J. Davies, Proc. IEEE 58, 168 (1970).
[Crossref]

Haus, H. A.

W. P. Huang, H. A. Haus, IEEE J. Lightwave Technol. 7, 920 (1989).
[Crossref]

Huang, W. P.

W. P. Huang, H. A. Haus, IEEE J. Lightwave Technol. 7, 920 (1989).
[Crossref]

Love, J. D.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), pp. 314, 397–398, 545–552.

Marcuse, D.

D. Marcuse, IEEE J. Lightwave Technol. LT-5, 268 (1987).
[Crossref]

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), Chap. 10.

Snyder, A. W.

A. W. Snyder, Y. Chen, A. Ankiewicz, IEEE J. Lightwave Technol. 7, 1400 (1989).
[Crossref]

A. W. Snyder, Y. Chen, Electron. Lett. 25, 502 (1989).
[Crossref]

A. W. Snyder, A. Ankiewicz, IEEE J. Lightwave Technol. 6, 463 (1988).
[Crossref]

A. W. Snyder, A. Ankiewicz, A. Altintas, Electron. Lett. 23, 1097 (1987).
[Crossref]

P. L. Chu, A. W. Snyder, Electron. Lett. 23, 1101 (1987).
[Crossref]

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

A. W. Snyder, O. J. Davies, Proc. IEEE 58, 168 (1970).
[Crossref]

A. W. Snyder, IEEE Trans. Microwave Theory Tech. MT-17, 1130 (1969).
[Crossref]

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), pp. 314, 397–398, 545–552.

Electron. Lett. (3)

P. L. Chu, A. W. Snyder, Electron. Lett. 23, 1101 (1987).
[Crossref]

A. W. Snyder, A. Ankiewicz, A. Altintas, Electron. Lett. 23, 1097 (1987).
[Crossref]

A. W. Snyder, Y. Chen, Electron. Lett. 25, 502 (1989).
[Crossref]

IEEE J. Lightwave Technol. (4)

A. W. Snyder, A. Ankiewicz, IEEE J. Lightwave Technol. 6, 463 (1988).
[Crossref]

A. W. Snyder, Y. Chen, A. Ankiewicz, IEEE J. Lightwave Technol. 7, 1400 (1989).
[Crossref]

W. P. Huang, H. A. Haus, IEEE J. Lightwave Technol. 7, 920 (1989).
[Crossref]

D. Marcuse, IEEE J. Lightwave Technol. LT-5, 268 (1987).
[Crossref]

IEEE Trans. Microwave Theory Tech. (1)

A. W. Snyder, IEEE Trans. Microwave Theory Tech. MT-17, 1130 (1969).
[Crossref]

J. Opt. Soc. Am. (1)

Proc. IEEE (1)

A. W. Snyder, O. J. Davies, Proc. IEEE 58, 168 (1970).
[Crossref]

Other (2)

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), pp. 314, 397–398, 545–552.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), Chap. 10.

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Figures (2)

Fig. 1
Fig. 1

Schematic of a grating-assisted coupler in planar geometry for grating (a) at the upper boundary of core 2 and (b) at the lower boundary of core 1.

Fig. 2
Fig. 2

(a) Percentage error of our formulation (which also equals that of Ref. 7) compared with the exact coupling length L for complete power transfer between cores 1 and 2 versus δβ/β [δβ = β1β2 and β ≅ 0.5 k(n1 + n2)] in the weakly coupled region where the cores are far separated and the errors are virtually independent of the separation d. The parameters are those of Marcuse,1 ρ1 = ρ2 = 1 μm, n2 = 3.23, ncl = 3.2, h = 0.1 μm, and λ = 1.5 μm, but we have changed n1 to achieve different values of δβ/β. The errors of the improved theory7 and our conventional theory are indistinguishable. Both are indicated by the solid curves. Both fail to give accurate coupling lengths when the difference between the two cores is large. The maximum differences here δβ/β = 0.0045, corresponds to a change in the waveguide parameter V by an amount (V1V2)/V1 = 0.3, with V i = k ρ i n i 2 - n cl 2. The error here is difference defined as |LexactLapproximation|/Lexact, where L i = π / C A B ( i ) , with CAB given by Eq. (6). Here i = exact uses the exact normal modes eA and eB, while i = approximate uses the modes of conventional coupled-mode theory8 or the improved theory.7 (b) Same as (a) but with Marcuse’s1 closed-form approximation [ C A B = C A B 0 in Eq. (8a)].

Equations (11)

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M = ( β 1 - β 2 ) / 2 C 12 .
δ n ( x , y , z ) = δ n ¯ ( x , y ) cos ( Ω z + Φ ) .
P 2 ( z ) = F 2 sin 2 ( C A B 2 F z ) .
F = [ 1 + ( β A - β B - Ω C A B ) 2 ] - 1 / 2 .
β A - β B β 1 - β 2 ( 1 + 1 / 2 M 2 ) .
C A B = k A δ n ¯ ( x , y ) e A e B d A / ( N A N B ) 1 / 2 .
e A ( x , y ) = e 1 ( x , y ) + ( 1 / 2 M ) e 2 ( x , y ) , e B ( x , y ) = e 2 ( x , y ) - ( 1 / 2 M ) e 1 ( x , y ) .
C A B = C A B 0 + ( 1 / 2 M ) k A δ n ¯ ( x , y ) × ( e 2 2 - e 1 2 ) d A / ( N A N B ) 1 / 2 ,
C A B = C 12 δ n ¯ δ n 2 N A N B ( 1 + δ n 2 β 1 - β 2 k η ) ,
P 2 ( L ) = 1 - 1 M 2 + 1 cos 2 ( π β A - β B C A B )
γ = C 12 / C 21 exp [ - ( W 1 / ρ 1 - W 2 / ρ 2 ) d ] ,

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