Abstract

Frequency-selective optical couplers in which a periodic perturbation in refractive index induces contradirectional power transfer in parallel single-mode waveguides are analyzed. Expressions for the spectral response are obtained by solving the coupled-mode equations. It is shown that by varying the amplitude of the periodic perturbation along he direction of propagation, the sidelobes present for the case of a uniform perturbation can be substantially reduced. Calculations are made of the mode propagation constants and coupling parameters for two rectangular-core waveguides with a periodic surface corrugation in the region between them. By using these results, design parameters and theoretical performance characteristics for a coupler with high transfer efficiency, low cross talk, and low reflection efficiency in the primary waveguide are determined.

© 1980 Optical Society of America

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References

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  1. H. F. Taylor, Opt. Commun. 8, 421 (1973).
    [Crossref]
  2. R. C. Alferness, R. V. Schmidt, Appl. Phys. Lett. 33, 161 (1978).
    [Crossref]
  3. M. Kobayashi, A. Terui, Appl. Opt. 17, 3253 (1978).
    [Crossref] [PubMed]
  4. R. C. Alferness, P. C. Cross, IEEE J. Quantum Electron. QE-14, 843 (1978).
    [Crossref]
  5. C. Elachi, C. Yeh, Opt. Commun. 7, 201 (1973).
    [Crossref]
  6. See, for example, P. K. Tien, Appl. Opt. 10, 2395 (1971).
    [Crossref] [PubMed]
  7. P. Yeh, A. Yariv, C. S. Hong, J. Opt. Soc. Am. 67, 423 (1977).
    [Crossref]
  8. D. C. Flanders, H. Kogelnik, R. V. Schmidt, C. V. Shank, Appl. Phys. Lett. 24, 194 (1974).
    [Crossref]
  9. K. Aiki, M. Nakamura, J. Umeda, A. Yariv, A. Katzir, H. W. Yen, Appl. Phys. Lett. 27, 145 (1975).
    [Crossref]
  10. See, for example, A. Yariv, Quantum Electronics, (Wiley, New York, 1975), pp. 521, 522.
  11. P. Yeh, J. Opt. Soc. Am. 69, 742 (1979).
    [Crossref]
  12. See, for example, Ref. 10, p. 520.
  13. See, for example, J. D. Jackson, Classical Electrodynamics, (Wiley, New York, 1962), p. 259.
  14. E. A. J. Marcatilli, Bell Syst. Tech. J. 48, 2071 (1969).
  15. See, for example, L. Schiff, Quantum Mechanics, (McGraw-Hill, New York, 1968), p. 255.
  16. H. F. Taylor, J. Appl. Phys. 44, 3257 (1973).
    [Crossref]
  17. L. B. Stotts, Opt. Commun. 17, 133 (1976).
    [Crossref]
  18. H. Yajima, Proceedings, Symposium on Optical and Acoustical Micro-Electronics (New York, April 1974).
  19. For the definition of Ey modes, see Ref. 14.

1979 (1)

1978 (3)

R. C. Alferness, R. V. Schmidt, Appl. Phys. Lett. 33, 161 (1978).
[Crossref]

M. Kobayashi, A. Terui, Appl. Opt. 17, 3253 (1978).
[Crossref] [PubMed]

R. C. Alferness, P. C. Cross, IEEE J. Quantum Electron. QE-14, 843 (1978).
[Crossref]

1977 (1)

1976 (1)

L. B. Stotts, Opt. Commun. 17, 133 (1976).
[Crossref]

1975 (1)

K. Aiki, M. Nakamura, J. Umeda, A. Yariv, A. Katzir, H. W. Yen, Appl. Phys. Lett. 27, 145 (1975).
[Crossref]

1974 (1)

D. C. Flanders, H. Kogelnik, R. V. Schmidt, C. V. Shank, Appl. Phys. Lett. 24, 194 (1974).
[Crossref]

1973 (3)

H. F. Taylor, Opt. Commun. 8, 421 (1973).
[Crossref]

C. Elachi, C. Yeh, Opt. Commun. 7, 201 (1973).
[Crossref]

H. F. Taylor, J. Appl. Phys. 44, 3257 (1973).
[Crossref]

1971 (1)

1969 (1)

E. A. J. Marcatilli, Bell Syst. Tech. J. 48, 2071 (1969).

Aiki, K.

K. Aiki, M. Nakamura, J. Umeda, A. Yariv, A. Katzir, H. W. Yen, Appl. Phys. Lett. 27, 145 (1975).
[Crossref]

Alferness, R. C.

R. C. Alferness, R. V. Schmidt, Appl. Phys. Lett. 33, 161 (1978).
[Crossref]

R. C. Alferness, P. C. Cross, IEEE J. Quantum Electron. QE-14, 843 (1978).
[Crossref]

Cross, P. C.

R. C. Alferness, P. C. Cross, IEEE J. Quantum Electron. QE-14, 843 (1978).
[Crossref]

Elachi, C.

C. Elachi, C. Yeh, Opt. Commun. 7, 201 (1973).
[Crossref]

Flanders, D. C.

D. C. Flanders, H. Kogelnik, R. V. Schmidt, C. V. Shank, Appl. Phys. Lett. 24, 194 (1974).
[Crossref]

Hong, C. S.

Jackson, J. D.

See, for example, J. D. Jackson, Classical Electrodynamics, (Wiley, New York, 1962), p. 259.

Katzir, A.

K. Aiki, M. Nakamura, J. Umeda, A. Yariv, A. Katzir, H. W. Yen, Appl. Phys. Lett. 27, 145 (1975).
[Crossref]

Kobayashi, M.

Kogelnik, H.

D. C. Flanders, H. Kogelnik, R. V. Schmidt, C. V. Shank, Appl. Phys. Lett. 24, 194 (1974).
[Crossref]

Marcatilli, E. A. J.

E. A. J. Marcatilli, Bell Syst. Tech. J. 48, 2071 (1969).

Nakamura, M.

K. Aiki, M. Nakamura, J. Umeda, A. Yariv, A. Katzir, H. W. Yen, Appl. Phys. Lett. 27, 145 (1975).
[Crossref]

Schiff, L.

See, for example, L. Schiff, Quantum Mechanics, (McGraw-Hill, New York, 1968), p. 255.

Schmidt, R. V.

R. C. Alferness, R. V. Schmidt, Appl. Phys. Lett. 33, 161 (1978).
[Crossref]

D. C. Flanders, H. Kogelnik, R. V. Schmidt, C. V. Shank, Appl. Phys. Lett. 24, 194 (1974).
[Crossref]

Shank, C. V.

D. C. Flanders, H. Kogelnik, R. V. Schmidt, C. V. Shank, Appl. Phys. Lett. 24, 194 (1974).
[Crossref]

Stotts, L. B.

L. B. Stotts, Opt. Commun. 17, 133 (1976).
[Crossref]

Taylor, H. F.

H. F. Taylor, Opt. Commun. 8, 421 (1973).
[Crossref]

H. F. Taylor, J. Appl. Phys. 44, 3257 (1973).
[Crossref]

Terui, A.

Tien, P. K.

Umeda, J.

K. Aiki, M. Nakamura, J. Umeda, A. Yariv, A. Katzir, H. W. Yen, Appl. Phys. Lett. 27, 145 (1975).
[Crossref]

Yajima, H.

H. Yajima, Proceedings, Symposium on Optical and Acoustical Micro-Electronics (New York, April 1974).

Yariv, A.

P. Yeh, A. Yariv, C. S. Hong, J. Opt. Soc. Am. 67, 423 (1977).
[Crossref]

K. Aiki, M. Nakamura, J. Umeda, A. Yariv, A. Katzir, H. W. Yen, Appl. Phys. Lett. 27, 145 (1975).
[Crossref]

See, for example, A. Yariv, Quantum Electronics, (Wiley, New York, 1975), pp. 521, 522.

Yeh, C.

C. Elachi, C. Yeh, Opt. Commun. 7, 201 (1973).
[Crossref]

Yeh, P.

Yen, H. W.

K. Aiki, M. Nakamura, J. Umeda, A. Yariv, A. Katzir, H. W. Yen, Appl. Phys. Lett. 27, 145 (1975).
[Crossref]

Appl. Opt. (2)

Appl. Phys. Lett. (3)

D. C. Flanders, H. Kogelnik, R. V. Schmidt, C. V. Shank, Appl. Phys. Lett. 24, 194 (1974).
[Crossref]

K. Aiki, M. Nakamura, J. Umeda, A. Yariv, A. Katzir, H. W. Yen, Appl. Phys. Lett. 27, 145 (1975).
[Crossref]

R. C. Alferness, R. V. Schmidt, Appl. Phys. Lett. 33, 161 (1978).
[Crossref]

Bell Syst. Tech. J. (1)

E. A. J. Marcatilli, Bell Syst. Tech. J. 48, 2071 (1969).

IEEE J. Quantum Electron. (1)

R. C. Alferness, P. C. Cross, IEEE J. Quantum Electron. QE-14, 843 (1978).
[Crossref]

J. Appl. Phys. (1)

H. F. Taylor, J. Appl. Phys. 44, 3257 (1973).
[Crossref]

J. Opt. Soc. Am. (2)

Opt. Commun. (3)

C. Elachi, C. Yeh, Opt. Commun. 7, 201 (1973).
[Crossref]

H. F. Taylor, Opt. Commun. 8, 421 (1973).
[Crossref]

L. B. Stotts, Opt. Commun. 17, 133 (1976).
[Crossref]

Other (6)

H. Yajima, Proceedings, Symposium on Optical and Acoustical Micro-Electronics (New York, April 1974).

For the definition of Ey modes, see Ref. 14.

See, for example, Ref. 10, p. 520.

See, for example, J. D. Jackson, Classical Electrodynamics, (Wiley, New York, 1962), p. 259.

See, for example, L. Schiff, Quantum Mechanics, (McGraw-Hill, New York, 1968), p. 255.

See, for example, A. Yariv, Quantum Electronics, (Wiley, New York, 1975), pp. 521, 522.

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Figures (14)

Fig. 1
Fig. 1

Frequency-selective coupler for use in optical fiber communication.

Fig. 2
Fig. 2

Two waveguides coupled by a periodic refractive-index perturbation.

Fig. 3
Fig. 3

Dispersion curves for coupled waveguides illustrating direct and exchange Bragg coupling.

Fig. 4
Fig. 4

Efficiency of coupling of optical power from a forward-propagating mode into a backward-propagating mode, plotted as a function of the phase mismatch parameter ΔL, for three values of the coupling strength parameter κL. The coupling constant κ is assumed to be uniform along the length of the coupling region.

Fig. 5
Fig. 5

Tapering of the coupling strength by varying the lengths of the surface corrugations.

Fig. 6
Fig. 6

Efficiency of coupling from forward-propagating mode to a backward-propagating mode, plotted as a function of the phasemismatch parameter ΔL, for three values of the tapering constant T. It is assumed that 0 L κ d z = π / 2for each curve.

Fig. 7
Fig. 7

Rectangular-core dielectric waveguide.

Fig. 8
Fig. 8

Illustration of regions in which the product-of-plane-wave solutions in Eqs. (31) and (32) do not satisfy the scalar wave equation.

Fig. 9
Fig. 9

Dispersion curves for rectangular-core waveguides with a = 1.5d. Marcatilli's approximation (solid curve), variational method (dashed curve).

Fig. 10
Fig. 10

Coupled rectangular-core waveguides.

Fig. 11
Fig. 11

Dispersion curves for the first three guided modes of coupled rectangular-core waveguides as determined by the variational method. The frequency-selective coupler makes use of coupling between the two lowest-order modes, E 11 x and E 21 x.

Fig. 12
Fig. 12

Field distributions for the two guided modes of the coupled waveguides with a = c = d,b = 1.5d, and √2ns(ncns)d/λ = 0.47.

Fig. 13
Fig. 13

Power transfer spectrum for contradirectional coupling from primary to satellite waveguides calculated using parameters given in text.

Fig. 14
Fig. 14

Calculated power transfer spectra for coupling from primary-to-satellite (PS), primary-to-primary (PP), and satellite-to-satellite (SS) waveguides.

Tables (1)

Tables Icon

Table I Characteristics of a Grating Directional Coupler

Equations (55)

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ψ ( x , y , z ) = A l ψ l ( x , y ) exp ( i β l z ) ,
[ 2 + ω 2 c 2 n 2 ( x , y ) + ω 2 c 2 Δ n 2 ( x , y , z ) ] ψ = 0 ,
Δ n 2 = μ ( x , y ) P ( z ) ,
β m | β m | d A m d z = i 2 ( ω c ) 2 l m | μ | l A l exp [ i ( β l β m ) z ] P ( z ) ,
m | μ | l = ψ m * ( x , y ) μ ( x , y ) ψ l ( x , y ) dxdy ,
| β m | m | l = δ m l ,
P ( z ) = 2 cos ( 2 π λ z ) = exp [ i ( 2 π z Λ ) ] + exp [ i ( 2 π z λ ) ] .
d A l d z = κ l m exp ( i Δ l m z ) A m ,
d A m d z = κ m l exp ( i Δ l m z ) A l ,
Δ l m = 2 π Λ β l + β m ,
κ l m = i 2 ( ω c ) 2 β l | β l | m | μ | l , m , l = 1 , 2 .
2 β 1 ( ω 1 ) = ( 2 π ) / Λ , 2 β 2 ( ω 2 ) = ( 2 π ) / Λ
κ 11 = i 2 ( ω c ) 2 ψ 1 2 ( x , y ) μ ( x , y ) dxdy ,
κ 22 = i 2 ( ω c ) 2 ψ 2 2 ( x , y ) μ ( x , y ) dxdy .
δ ω 1 2 | κ 11 | c / n eff , δ ω 2 2 | κ 22 | c / n eff ,
β 1 + β 2 = ( 2 π ) / Λ .
κ 12 = i 2 ( ω c ) 2 ψ 1 ( x , y ) μ ( x , y ) ψ 2 ( x , y ) dxdy .
Δ ω 3 2 | κ 12 | c / n eff .
R = κ 2 sinh 2 ( S L ) κ 2 sinh 2 ( S L ) + S 2 ,
S 2 = | κ | 2 ( Δ / 2 ) 2 ,
R = tanh 2 ( | κ | L ) .
Δ [ ( 4 π n s ) / c ] ( ν ν 0 ) .
κ ( z ) = κ 0 exp [ T ( z L / 2 ) 2 / L ] , 0 z L = 0 elsewhere ,
R = | A m ( 0 ) / A l ( 0 ) | 2 .
n 2 ( x , y ) = { n 0 2 y > 0 n c 2 | x | < a / 2 and d < y < 0 otherwise , n s 2
[ 2 x 2 + 2 y 2 + ω 2 c 2 n 2 ( x , y ) β 2 ] { E H } = 0 ,
n 2 ( x , y ) u ( x ) + υ ( y ) n c 2 n m 2 ( x , y ) ,
u ( x ) = { n s 2 | x | > a / 2 n c 2 | x | < a / 2 ,
υ ( y ) = { n s 2 y < d n c 2 d < y < 0 n o 2 0 < y .
ψ ( x , y , z ) = U ( x ) V ( y ) exp ( i β z ) ,
[ d 2 d x 2 + ω 2 c 2 u ( x ) ] U ( x ) = β u 2 U ( x ) ,
[ d 2 d y 2 + ω 2 c 2 υ ( y ) ] V ( y ) = β υ 2 V ( y ) ,
β 2 = β u 2 + β υ 2 ω 2 c 2 n c 2 .
β 2 = β m 2 ( n c 2 n s 2 ) ( 2 π / λ ) 2 Γ ,
ψ ( x , y ) exp ( i β z ) = [ C a ϕ a ( x , y ) + C b ϕ b ( x , y ) ] exp ( i β z )
n 2 ( x , y ) = s 2 ( x , y ) + Δ n a 2 ( x , y ) + Δ n b 2 ( x , y ) ,
s 2 ( x , y ) = { n 0 2 air n s 2 substrate ,
Δ n α 2 ( x , y ) = { n α 2 n s 2 waveguide α α = a , b 0 otherwise ,
[ 2 x 2 + 2 y 2 + ω 2 c 2 s 2 ( x , y ) + Δ n α 2 ( x , y ) ] ϕ α = β α 2 ϕ α . α = a , b
C a [ β a 2 + ω 2 c 2 Δ n b 2 ( x , y ) β 2 ] ϕ a + C b [ β b 2 + ω 2 c 2 Δ n a 2 ( x , y ) β 2 ] ϕ b = 0 .
[ β a 2 β 2 + K b J a + I ( β b 2 β 2 ) J b + I ( β a 2 β 2 ) β b 2 β 2 + K a ] ( C a C b ) = 0 ,
I = ϕ b ϕ a dxdy J a = ( ω / c ) 2 ( n a 2 n s 2 ) a ϕ b ϕ a dxdy J b = ( ω / c ) 2 ( n b 2 n s 2 ) b ϕ b ϕ a dxdy K a = ( ω / c ) 2 ( n a 2 n s 2 ) a ϕ b 2 dxdy K b = ( ω / c ) 2 ( n b 2 n s 2 ) b ϕ a 2 dxdy ϕ a 2 dxdy = ϕ b 2 dxdy = 1 } .
| β a 2 β 2 + K b J a + I ( β b 2 β 2 ) J b + I ( β a 2 β 2 ) β b 2 β 2 + K a | = 0 .
β 2 = β 0 2 + K ± J 1 ± I ,
ψ = 1 [ 2 ( 1 ± I ) ] 1 / 2 ( ϕ a ± ϕ b ) .
[ 1 ½ ( h p p h ) tan h a ] [ 1 ½ ( h p p h ) tan h b ] + exp ( 2 p c ) [ ¼ ( h p + p h ) 2 ] tan h a tan h b = 0 ,
h = [ ( n c ω / c ) 2 β u 2 ] 1 / 2 , p = [ β u 2 ( n s ω / c ) 2 ] 1 / 2 ,
0 L κ 12 d z = 0.75 π ,
2 n s ( n c n s ) d / λ
λ ( β 1 β 2 ) 2 π ( n c n s )
w c
κ 12
κ 11 κ 12
κ 22 κ 12
K 12 = | κ 12 | n s ( n c n s ) λ 2 π ( d t ) 2

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