Abstract

Coupled-mode theory was used to analyze guiding in a transverse Bragg resonance (TBR) waveguide structure composed of a GaAs substrate with air holes. This analysis predicts that propagation loss will be minimized for discrete widths of the waveguide core. Although the coupled-mode theory is normally applied to structures with small index perturbations, two-dimensional finite-difference time-domain simulations of the TBR waveguide show good quantitative agreement with the coupled-mode predictions, and these results corroborate the previously predicted existence of discrete core widths for low-loss propagation.

© 2004 Optical Society of America

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References

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  1. P. Yeh, A. Yariv, “Bragg reflection waveguides,” Opt. Commun. 19, 427–430 (1976).
    [CrossRef]
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    [CrossRef]
  3. P. Yeh, A. Yariv, E. Marom, “Theory of Bragg fiber,” J. Opt. Soc. Am. 68, 1196–1201 (1978).
    [CrossRef]
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    [CrossRef]
  6. M. Notomi, A. Shinya, K. Yamada, J. Takahashi, C. Takahashi, I. Yokohama, “Structural tuning of guiding modes of line-defect waveguides of silicon-on-insulator photonic crystal slabs,” IEEE J. Quantum Electron. 83, 736–742 (2002).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  13. N. W. Ashcroft, N. D. Mermin, Solid State Physics (Harcourt, Fort Worth, Tex., 1976).

2003 (1)

2002 (3)

M. Lončar, D. Nedeljkovič, T. P. Pearsall, J. Vučković, A. Scherer, S. Kuchinsky, D. C. Allan, “Experimental and theoretical confirmation of Bloch-mode light propagation in planar photonic crystal waveguides,” Appl. Phys. Lett. 80, 1689–1691 (2002).
[CrossRef]

M. Notomi, A. Shinya, K. Yamada, J. Takahashi, C. Takahashi, I. Yokohama, “Structural tuning of guiding modes of line-defect waveguides of silicon-on-insulator photonic crystal slabs,” IEEE J. Quantum Electron. 83, 736–742 (2002).
[CrossRef]

A. Yariv, “Coupled-wave formalism for optical waveguiding by transverse Bragg reflection,” Opt. Lett. 27, 936–938 (2002).
[CrossRef]

2001 (1)

1996 (1)

S. D. Gedney, “An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices,” IEEE Trans. Antennas Propag. 44, 1630–1639 (1996).
[CrossRef]

1995 (2)

1994 (1)

J. N. Winn, R. D. Meade, J. D. Joannopoulos, “Two-dimensional photonic band-gap materials,” J. Mod. Opt. 41, 257–273 (1994).
[CrossRef]

1978 (1)

1977 (1)

A. Y. Cho, A. Yariv, P. Yeh, “Observation of confined propagation in Bragg waveguides,” Appl. Phys. Lett. 30, 471–472 (1977).
[CrossRef]

1976 (1)

P. Yeh, A. Yariv, “Bragg reflection waveguides,” Opt. Commun. 19, 427–430 (1976).
[CrossRef]

Allan, D. C.

M. Lončar, D. Nedeljkovič, T. P. Pearsall, J. Vučković, A. Scherer, S. Kuchinsky, D. C. Allan, “Experimental and theoretical confirmation of Bloch-mode light propagation in planar photonic crystal waveguides,” Appl. Phys. Lett. 80, 1689–1691 (2002).
[CrossRef]

Ashcroft, N. W.

N. W. Ashcroft, N. D. Mermin, Solid State Physics (Harcourt, Fort Worth, Tex., 1976).

Chen, J. C.

Cho, A. Y.

A. Y. Cho, A. Yariv, P. Yeh, “Observation of confined propagation in Bragg waveguides,” Appl. Phys. Lett. 30, 471–472 (1977).
[CrossRef]

Chow, E.

Devenyi, A.

Fan, S.

Gedney, S. D.

S. D. Gedney, “An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices,” IEEE Trans. Antennas Propag. 44, 1630–1639 (1996).
[CrossRef]

Joannopoulos, J. D.

Johnson, S. G.

Kuchinsky, S.

M. Lončar, D. Nedeljkovič, T. P. Pearsall, J. Vučković, A. Scherer, S. Kuchinsky, D. C. Allan, “Experimental and theoretical confirmation of Bloch-mode light propagation in planar photonic crystal waveguides,” Appl. Phys. Lett. 80, 1689–1691 (2002).
[CrossRef]

Lin, S. Y.

Loncar, M.

M. Lončar, D. Nedeljkovič, T. P. Pearsall, J. Vučković, A. Scherer, S. Kuchinsky, D. C. Allan, “Experimental and theoretical confirmation of Bloch-mode light propagation in planar photonic crystal waveguides,” Appl. Phys. Lett. 80, 1689–1691 (2002).
[CrossRef]

Marom, E.

Meade, R. D.

Mermin, N. D.

N. W. Ashcroft, N. D. Mermin, Solid State Physics (Harcourt, Fort Worth, Tex., 1976).

Mookherjea, S.

Nedeljkovic, D.

M. Lončar, D. Nedeljkovič, T. P. Pearsall, J. Vučković, A. Scherer, S. Kuchinsky, D. C. Allan, “Experimental and theoretical confirmation of Bloch-mode light propagation in planar photonic crystal waveguides,” Appl. Phys. Lett. 80, 1689–1691 (2002).
[CrossRef]

Notomi, M.

M. Notomi, A. Shinya, K. Yamada, J. Takahashi, C. Takahashi, I. Yokohama, “Structural tuning of guiding modes of line-defect waveguides of silicon-on-insulator photonic crystal slabs,” IEEE J. Quantum Electron. 83, 736–742 (2002).
[CrossRef]

Pearsall, T. P.

M. Lončar, D. Nedeljkovič, T. P. Pearsall, J. Vučković, A. Scherer, S. Kuchinsky, D. C. Allan, “Experimental and theoretical confirmation of Bloch-mode light propagation in planar photonic crystal waveguides,” Appl. Phys. Lett. 80, 1689–1691 (2002).
[CrossRef]

Sakoda, K.

K. Sakoda, “Optical transmittance of a two-dimensional triangular photonic lattice,” Phys. Rev. B 51, 4672–4675 (1995).
[CrossRef]

Scherer, A.

M. Lončar, D. Nedeljkovič, T. P. Pearsall, J. Vučković, A. Scherer, S. Kuchinsky, D. C. Allan, “Experimental and theoretical confirmation of Bloch-mode light propagation in planar photonic crystal waveguides,” Appl. Phys. Lett. 80, 1689–1691 (2002).
[CrossRef]

Shinya, A.

M. Notomi, A. Shinya, K. Yamada, J. Takahashi, C. Takahashi, I. Yokohama, “Structural tuning of guiding modes of line-defect waveguides of silicon-on-insulator photonic crystal slabs,” IEEE J. Quantum Electron. 83, 736–742 (2002).
[CrossRef]

Takahashi, C.

M. Notomi, A. Shinya, K. Yamada, J. Takahashi, C. Takahashi, I. Yokohama, “Structural tuning of guiding modes of line-defect waveguides of silicon-on-insulator photonic crystal slabs,” IEEE J. Quantum Electron. 83, 736–742 (2002).
[CrossRef]

Takahashi, J.

M. Notomi, A. Shinya, K. Yamada, J. Takahashi, C. Takahashi, I. Yokohama, “Structural tuning of guiding modes of line-defect waveguides of silicon-on-insulator photonic crystal slabs,” IEEE J. Quantum Electron. 83, 736–742 (2002).
[CrossRef]

Vuckovic, J.

M. Lončar, D. Nedeljkovič, T. P. Pearsall, J. Vučković, A. Scherer, S. Kuchinsky, D. C. Allan, “Experimental and theoretical confirmation of Bloch-mode light propagation in planar photonic crystal waveguides,” Appl. Phys. Lett. 80, 1689–1691 (2002).
[CrossRef]

Wendt, J. R.

Winn, J. N.

Xu, Y.

Yamada, K.

M. Notomi, A. Shinya, K. Yamada, J. Takahashi, C. Takahashi, I. Yokohama, “Structural tuning of guiding modes of line-defect waveguides of silicon-on-insulator photonic crystal slabs,” IEEE J. Quantum Electron. 83, 736–742 (2002).
[CrossRef]

Yariv, A.

Yeh, P.

P. Yeh, A. Yariv, E. Marom, “Theory of Bragg fiber,” J. Opt. Soc. Am. 68, 1196–1201 (1978).
[CrossRef]

A. Y. Cho, A. Yariv, P. Yeh, “Observation of confined propagation in Bragg waveguides,” Appl. Phys. Lett. 30, 471–472 (1977).
[CrossRef]

P. Yeh, A. Yariv, “Bragg reflection waveguides,” Opt. Commun. 19, 427–430 (1976).
[CrossRef]

Yokohama, I.

M. Notomi, A. Shinya, K. Yamada, J. Takahashi, C. Takahashi, I. Yokohama, “Structural tuning of guiding modes of line-defect waveguides of silicon-on-insulator photonic crystal slabs,” IEEE J. Quantum Electron. 83, 736–742 (2002).
[CrossRef]

Appl. Phys. Lett. (2)

A. Y. Cho, A. Yariv, P. Yeh, “Observation of confined propagation in Bragg waveguides,” Appl. Phys. Lett. 30, 471–472 (1977).
[CrossRef]

M. Lončar, D. Nedeljkovič, T. P. Pearsall, J. Vučković, A. Scherer, S. Kuchinsky, D. C. Allan, “Experimental and theoretical confirmation of Bloch-mode light propagation in planar photonic crystal waveguides,” Appl. Phys. Lett. 80, 1689–1691 (2002).
[CrossRef]

IEEE J. Quantum Electron. (1)

M. Notomi, A. Shinya, K. Yamada, J. Takahashi, C. Takahashi, I. Yokohama, “Structural tuning of guiding modes of line-defect waveguides of silicon-on-insulator photonic crystal slabs,” IEEE J. Quantum Electron. 83, 736–742 (2002).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

S. D. Gedney, “An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices,” IEEE Trans. Antennas Propag. 44, 1630–1639 (1996).
[CrossRef]

J. Mod. Opt. (1)

J. N. Winn, R. D. Meade, J. D. Joannopoulos, “Two-dimensional photonic band-gap materials,” J. Mod. Opt. 41, 257–273 (1994).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Commun. (1)

P. Yeh, A. Yariv, “Bragg reflection waveguides,” Opt. Commun. 19, 427–430 (1976).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. B (1)

K. Sakoda, “Optical transmittance of a two-dimensional triangular photonic lattice,” Phys. Rev. B 51, 4672–4675 (1995).
[CrossRef]

Other (1)

N. W. Ashcroft, N. D. Mermin, Solid State Physics (Harcourt, Fort Worth, Tex., 1976).

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Figures (7)

Fig. 1
Fig. 1

TBR waveguide geometry consisting of a GaAs substrate and air holes. x and z are the transverse and the longitudinal dimensions, respectively, the core width is W, and the hole radius is r=0.15a. A0 and A1 represent the inward propagating plane-wave components in the core and cladding, respectively. A2 represents the incoming field outside the cladding; B0, B1, and B2 represent similar quantities for the outward-propagating components.

Fig. 2
Fig. 2

2D FDTD simulation domain showing a sample field calculation. By taking advantage of the symmetries, we reduced the domain to the slice shown in the figure. The boundary conditions were even parity at x=0, Bloch periodic at z=0 and z=a, and perfectly matched layers at the outermost boundary. The cladding is composed of approximately ten layers.

Fig. 3
Fig. 3

Normalized transverse field decay calculated by the 2D FDTD simulation for a structure with approximately ten layers of Bragg periods. The core width W=b/4. Also shown are the exponential decays predicted by the coupled-mode theory, E(x)/E(0)exp(-|κ|x); we used the theoretical value calculated from Eq. (5), κ=-i0.1475/a, and an empirical fit, κ=-i0.1100/a.

Fig. 4
Fig. 4

Dispersion curves for varying widths of the core. Symbols, normalized frequencies from the 2D FDTD simulations; and solid curves, the corresponding dispersion curves from the coupled-mode (CM) theory in ascending order of width, W=0.18b to 0.45b, from top to bottom.

Fig. 5
Fig. 5

Plot of normalized loss constant, αa, from 2D FDTD simulations for varying core widths with the propagation constant βR as a parameter. For each βR, a minimum in the loss is apparent when W=b/4. Owing to the limits of the coupled-mode theory (see Fig. 7), only the first four values of βR are shown.

Fig. 6
Fig. 6

Plot of the normalized loss constant from 2D FDTD simulations as well as coupled-mode (CM) theory for selected core width values. The deviation seen when βRa/π2/3 is expected, as shown in Fig. 7. The solid curves corresponding to the CM theory are labeled from 1 to 4 in ascending order of width, W=0.18b to 0.45b.

Fig. 7
Fig. 7

Reciprocal lattice vectors for the triangular lattice, a1=-4π/bxˆ and a2=-2π/bxˆ-2π/azˆ. Owing to the symmetry of the triangular lattice, the coupling of k1 to k3 by a2 when βRa/π2/3 causes the deviation from theory seen in Fig. 6.

Equations (13)

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2E+μ(r)ω2E=0,
E(r, t)=E(x)E(z, t)=[A(x)exp(ikx)+B(x)exp(-ikx)]exp(iωt-iβz)
xA(x)B(x)=-(γ-iΔk)κκ*(γ-iΔk)A(x)B(x)
γ=-βRβIk,
κ=ik022¯k1ababΔ(r)exp(-i2kx)dxdz,
Δk=kW-k,
k=kb=2πb,claddingkW=¯ω2c2-βR21/2,core.
A1(W/2)B1(W/2)=121+kWkbexp(ikWW/2)1-kWkbexp(-ikWW/2)1-kWkbexp(ikWW/2)1+kWkbexp(-ikWW/2)A0(W/2)B0(W/2),
Ax-W2=-(γ-iΔk)SAW2+κSBW2sinh(Sx)+BW2cosh(Sx),
Bx-W2=κ*SAW2+(γ-iΔk)SBW2sinh(Sx)+AW2cosh(Sx),
QωE0-ΔP,
α=-2βI=-ΔEE01L=ωQΔtL=ωQvg,
vgddωn¯2c2ω2-kb2-1=c2β¯ω=βcn¯(β2+kb2)-1/2,

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