Abstract

We develop a self-consistent Green’s function technique to calculate the full photonic band structure for guided modes in slab waveguides that have been textured in one dimension with a thin surface grating. The technique is conceptually simple and can easily be reduced to an eigenvalue problem by use of approximations similar to those used in conventional coupled-mode theories. We show that this approach yields the correct TM–TM coupling coefficient at oblique incidence and provides a transparent interpretation for its vanishing at a critical angle.

© 1998 Optical Society of America

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References

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  1. A. Yariv, IEEE J. Quantum Electron. QE-9, 919 (1973).
    [CrossRef]
  2. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton U. Press, Princeton, N.J., 1995).
  3. K. Wagatsuma, H. Sakaki, and S. Saito, IEEE J. Quantum Electron. QE-15, 632 (1979).
    [CrossRef]
  4. L. A. Weller-Brophy and D. G. Hall, IEEE J. Lightwave Technol. 6, 1069 (1988).
    [CrossRef]
  5. G. I. Stegeman, D. Sarid, J. J. Burke, and D. G. Hall, J. Opt. Soc. Am. 71, 1497 (1981).
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  7. C. M. deSterke and J. E. Sipe, J. Opt. Soc. Am. A 7, 636 (1990).
    [CrossRef]
  8. L. A. Weller-Brophy and D. G. Hall, Opt. Lett. 12, 756 (1987).
    [CrossRef] [PubMed]

1990 (1)

1988 (1)

L. A. Weller-Brophy and D. G. Hall, IEEE J. Lightwave Technol. 6, 1069 (1988).
[CrossRef]

1987 (2)

1981 (1)

1979 (1)

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

1973 (1)

A. Yariv, IEEE J. Quantum Electron. QE-9, 919 (1973).
[CrossRef]

Burke, J. J.

deSterke, C. M.

Hall, D. G.

Joannopoulos, J. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton U. Press, Princeton, N.J., 1995).

Meade, R. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton U. Press, Princeton, N.J., 1995).

Saito, S.

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

Sakaki, H.

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

Sarid, D.

Sipe, J. E.

Stegeman, G. I.

Wagatsuma, K.

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

Weller-Brophy, L. A.

L. A. Weller-Brophy and D. G. Hall, IEEE J. Lightwave Technol. 6, 1069 (1988).
[CrossRef]

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

Winn, J. N.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton U. Press, Princeton, N.J., 1995).

Yariv, A.

A. Yariv, IEEE J. Quantum Electron. QE-9, 919 (1973).
[CrossRef]

IEEE J. Lightwave Technol. (1)

L. A. Weller-Brophy and D. G. Hall, IEEE J. Lightwave Technol. 6, 1069 (1988).
[CrossRef]

IEEE J. Quantum Electron. (2)

A. Yariv, IEEE J. Quantum Electron. QE-9, 919 (1973).
[CrossRef]

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

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

Opt. Lett. (1)

Other (1)

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton U. Press, Princeton, N.J., 1995).

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Figures (2)

Fig. 1
Fig. 1

Schematic of a one-dimensionally textured slab waveguide.

Fig. 2
Fig. 2

Theoretical TM–TM coupling coefficients versus oblique angle of incidence: A, our model; B, local normal mode; C, ideal mode; D, relation (16).

Equations (17)

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2z2-β2+ω˜2szEβ;z=-4πω˜2mχmzEβ-mβg;z,
Eβ;z=dzgβ;z,zmχmzEβ-mβg;z.
Enz0=gnmtgχn-mz0Emz0,
gn=2πiω˜2wn1+rsnsˆnsˆn+pˆn+pˆn+2+pˆn-pˆn-2+rpnpˆn+pˆn--4πtgzˆzˆ.
Eβn=βˆn·gnmtgχn-m×Esmsˆm+Eβmβˆm+Ezmzˆ.
v=Mβ,ω˜v,
detMβ,ω˜-U=0,
gn2πiω˜2wnrsnsˆnsˆn+rpnpˆn+pˆn--4πtgzˆzˆ,
Ez=KEβ,
Esn=2πiω˜2wnrsnmtgχn-msˆn·sˆmEsm+sˆn·βˆmEβm,Eβn=2πiω˜2wnrpnβˆn·pˆn+mtgχn-m×wnω˜βˆn·sˆmEsm+wnω˜βˆn·βˆmEβm+βnω˜kKmkEβk.
rsnRsnω˜-ω˜sn,
ω˜-ω˜snEsn=2πRsnβnNTENTE2-11/2mtgχn-m×sˆn·sˆmEsm+sˆn·βˆmEβm,ω˜-ω˜pnEβn=2πRpnβnNTM2-11/2NTMmtgχn-m×βˆn·sˆmEsm+βˆn·βˆmEβm+NTM1-NTM21/2kKmkEβk.
Mβu=ω˜u,
κTM-TM=ng2πRp0ω˜p0tgNTM2-11/2χ1χ-1×-NTM2-1cos 2θ+NTM2D,
D=1+4πχ02-4π2χ1χ-1
κTM-TMχ-1-NTM2-1cos 2θ+NTM2,
κTM-TMχ-1-NTM2-1cos 2θ+NTM2g,

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