Abstract

A new derivation of the coupled-mode equations and coupling coefficient for the troublesome but important case of the Bragg reflection of TM guided waves in a corrugated, planar optical waveguide is presented. The technique solves the (approximate) wave equation directly and clarifies those features of the analysis that are essential to treat the problem correctly.

© 1990 Optical Society of America

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References

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  1. H. A. Haus, W. P. Huang, A. W. Snyder, Opt. Lett. 14, 1222 (1989), and references therein.
    [CrossRef] [PubMed]
  2. W. Streifer, D. R. Scifres, R. D. Burnham, IEEE J. Quantum Electron. QE-12, 74 (1976).
    [CrossRef]
  3. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974); A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).
  4. L. A. Weller-Brophy, D. G. Hall, Opt. Lett. 12, 756 (1987).
    [CrossRef] [PubMed]
  5. L. A. Weller-Brophy, D. G. Hall, IEEE J. Lightwave Technol. 6, 1069 (1987).
    [CrossRef]
  6. G. I. Stegeman, D. Sarid, J. J. Burke, D. G. Hall, J. Opt. Soc. Am. 71, 1497 (1981).
    [CrossRef]
  7. E. Popov, L. Mashev, Opt. Acta 32, 265 (1985).
    [CrossRef]
  8. H. G. Kogelnik, in Integrated Optics, Vol. 7 of Topics in Applied Physics, T. Tamir, ed. (Springer-Verlag, New York, 1975), pp. 66–79.
    [CrossRef]
  9. K. Wagatsuma, H. Sakaki, S. Saito, IEEE J. Quantum Electron. QE-15, 632 (1979).
    [CrossRef]

1989 (1)

1987 (2)

L. A. Weller-Brophy, D. G. Hall, Opt. Lett. 12, 756 (1987).
[CrossRef] [PubMed]

L. A. Weller-Brophy, D. G. Hall, IEEE J. Lightwave Technol. 6, 1069 (1987).
[CrossRef]

1985 (1)

E. Popov, L. Mashev, Opt. Acta 32, 265 (1985).
[CrossRef]

1981 (1)

1979 (1)

K. Wagatsuma, H. Sakaki, S. Saito, IEEE J. Quantum Electron. QE-15, 632 (1979).
[CrossRef]

1976 (1)

W. Streifer, D. R. Scifres, R. D. Burnham, IEEE J. Quantum Electron. QE-12, 74 (1976).
[CrossRef]

Burke, J. J.

Burnham, R. D.

W. Streifer, D. R. Scifres, R. D. Burnham, IEEE J. Quantum Electron. QE-12, 74 (1976).
[CrossRef]

Hall, D. G.

Haus, H. A.

Huang, W. P.

Kogelnik, H. G.

H. G. Kogelnik, in Integrated Optics, Vol. 7 of Topics in Applied Physics, T. Tamir, ed. (Springer-Verlag, New York, 1975), pp. 66–79.
[CrossRef]

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974); A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

Mashev, L.

E. Popov, L. Mashev, Opt. Acta 32, 265 (1985).
[CrossRef]

Popov, E.

E. Popov, L. Mashev, Opt. Acta 32, 265 (1985).
[CrossRef]

Saito, S.

K. Wagatsuma, H. Sakaki, S. Saito, IEEE J. Quantum Electron. QE-15, 632 (1979).
[CrossRef]

Sakaki, H.

K. Wagatsuma, H. Sakaki, S. Saito, IEEE J. Quantum Electron. QE-15, 632 (1979).
[CrossRef]

Sarid, D.

Scifres, D. R.

W. Streifer, D. R. Scifres, R. D. Burnham, IEEE J. Quantum Electron. QE-12, 74 (1976).
[CrossRef]

Snyder, A. W.

Stegeman, G. I.

Streifer, W.

W. Streifer, D. R. Scifres, R. D. Burnham, IEEE J. Quantum Electron. QE-12, 74 (1976).
[CrossRef]

Wagatsuma, K.

K. Wagatsuma, H. Sakaki, S. Saito, IEEE J. Quantum Electron. QE-15, 632 (1979).
[CrossRef]

Weller-Brophy, L. A.

L. A. Weller-Brophy, D. G. Hall, IEEE J. Lightwave Technol. 6, 1069 (1987).
[CrossRef]

L. A. Weller-Brophy, D. G. Hall, Opt. Lett. 12, 756 (1987).
[CrossRef] [PubMed]

IEEE J. Lightwave Technol. (1)

L. A. Weller-Brophy, D. G. Hall, IEEE J. Lightwave Technol. 6, 1069 (1987).
[CrossRef]

IEEE J. Quantum Electron. (2)

W. Streifer, D. R. Scifres, R. D. Burnham, IEEE J. Quantum Electron. QE-12, 74 (1976).
[CrossRef]

K. Wagatsuma, H. Sakaki, S. Saito, IEEE J. Quantum Electron. QE-15, 632 (1979).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Acta (1)

E. Popov, L. Mashev, Opt. Acta 32, 265 (1985).
[CrossRef]

Opt. Lett. (2)

Other (2)

H. G. Kogelnik, in Integrated Optics, Vol. 7 of Topics in Applied Physics, T. Tamir, ed. (Springer-Verlag, New York, 1975), pp. 66–79.
[CrossRef]

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974); A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

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Figures (1)

Fig. 1
Fig. 1

Geometry for the analysis of the interaction of a guided wave with a corrugation of length L and period Λ. The waveguide layer has an index of refraction nf, the upper medium has an index of refraction nc, and the lower medium has an index of refraction ns. The upper surface at x = d is corrugated for 0 ≤ zL according to d = h + (Δh)cos(K0z), with K0 = 2π/Λ. I, R, and T designate the incident, reflected, and transmitted guided waves, respectively.

Equations (18)

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E ( x , y , z , t ) = f ( z ) exp ( i β y y ) E L ( x , z ) exp ( iωt ) ,
{ 2 + [ ( x ) 0 ] k 0 2 } E ( x , y , z , t ) = Δ ( x , z ) 0 k 0 2 E ( x , y , z , t ) .
[ d 2 d x 2 + n 2 ( x ) k 0 2 ] E i ( x ) = β 2 E i ( x ) ,
E i ( x ) ( d 2 d z 2 + β z 2 ) f ( z ) k 0 2 [ Δ ( x , z ) 0 ] f ( z ) E L ( x , z ) .
E ( x ) = x ˆ E ix ( x ) + [ y ˆ ( β y β ) z ˆ ( β z ˆ / β ) ] E iz ( x ) .
( d 2 d z 2 + β z 2 ) f ( z ) = ( 4 μ 0 ω I N ) K ( z ) f ( z ) ,
K ( z ) = ω 4 Δ ( x , z ) E L ( x , z ) E * ( x ) d x
I N = E i ( x ) E * ( x ) d x .
( d 2 d z 2 + β z 2 ) f ( z ) = Q ( z )
f ( z ) = g ( z , z ) Q ( z ) d z .
g ( z , z ) = ( 1 2 i β z ) exp ( i β z | z z | ) .
f ( z ) = A + ( z ) exp ( i β z z ) + A ( z ) exp ( i β z z ) ,
A + ( z ) = 1 2 i β z z exp ( i β z z ) Q ( z ) d z .
A ( z ) = 1 2 i β z z exp ( i β z z ) Q ( z ) d z .
d A ± ( z ) d z = [ 2 μ 0 ω I N K ( z ) i β z ] f ( z ) exp ( i β z z ) .
d A ± ( z ) d z = ± ( θ ) A ( z ) exp ( i 2 δz ) ,
κ ( θ ) = [ π λ Δ h h eff n f 2 N 2 N q c cos ( θ ) ] [ N 2 n f 2 ( N 2 n c 2 1 ) cos ( 2 θ ) ] for TM 0 TM 0 .
κ ( θ ) = ( π λ Δ h h eff n f 2 N 2 N ) cos ( 2 θ ) cos ( θ ) for TE 0 TE 0 .

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