Abstract

With a perfectly matched layer boundary treatment, a semivectorial finite-difference method is used to calculate the eigenmodes of a single-mode (SM) or multimode (MM) bent rib waveguide. A detailed analysis is given for the dependence of the bending losses (including the pure bending loss and the transition loss) on geometrical parameters of the bent rib waveguide such as the rib width, the rib height, and the bending radius. The characteristics of the higher-order modes are analyzed. It is shown that the bending loss of the fundamental mode can be reduced effectively by increasing the width and height of the rib. For an integrated device, undesired effects due to the higher-order modes of a MM bent waveguide can be removed by appropriate choice of the geometrical parameters. An appropriately designed MM bent waveguide is used to reduce effectively the bending loss of the fundamental mode, and a low-loss SM propagation in a MM bent waveguide is realized when the bending losses of the higher-order modes are large enough.

© 2004 Optical Society of America

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References

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  1. S. Kim, A. Gopinath, “Vector analysis of optical dielectric waveguide bends using finite-difference method,” J. Lightwave Technol. 14, 2085–2092 (1996).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  6. J.-S. Gu, P.-A. Besse, H. Melchior, “Method of lines for the analysis of the propagation characteristics of curved optical rib waveguides,” IEEE J. Quantum Electron. 27, 531–537 (1991).
    [CrossRef]
  7. S. P. Pogossian, L. Vescan, A. Vonsovici, “The single-mode condition for semiconductor rib waveguides with large cross section,” J. Lightwave Technol. 16, 1851–1853 (1998).
    [CrossRef]
  8. M. K. Smit, E. C. M. Pennings, H. Blok, “A normalized approach to the design of low-loss optical waveguide bends,” J. Lightwave Technol. 11, 1737–1742 (1993).
    [CrossRef]
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    [CrossRef]

2002 (2)

1998 (2)

1996 (1)

S. Kim, A. Gopinath, “Vector analysis of optical dielectric waveguide bends using finite-difference method,” J. Lightwave Technol. 14, 2085–2092 (1996).
[CrossRef]

1993 (2)

T. Yamamoto, M. Koshiba, “Numerical analysis of curvature loss in optical waveguides by the finite-element method,” J. Lightwave Technol. 11, 1579–1583 (1993).
[CrossRef]

M. K. Smit, E. C. M. Pennings, H. Blok, “A normalized approach to the design of low-loss optical waveguide bends,” J. Lightwave Technol. 11, 1737–1742 (1993).
[CrossRef]

1991 (1)

J.-S. Gu, P.-A. Besse, H. Melchior, “Method of lines for the analysis of the propagation characteristics of curved optical rib waveguides,” IEEE J. Quantum Electron. 27, 531–537 (1991).
[CrossRef]

1990 (1)

1987 (1)

Alvarez-Estrada, R. F.

Besse, P.-A.

J.-S. Gu, P.-A. Besse, H. Melchior, “Method of lines for the analysis of the propagation characteristics of curved optical rib waveguides,” IEEE J. Quantum Electron. 27, 531–537 (1991).
[CrossRef]

Blok, H.

M. K. Smit, E. C. M. Pennings, H. Blok, “A normalized approach to the design of low-loss optical waveguide bends,” J. Lightwave Technol. 11, 1737–1742 (1993).
[CrossRef]

Calvo, M. L.

Dai, D.

Decoster, D.

Deng, H.

Feit, M. D.

Feng, N.-N.

Fleck, J. A.

Gopinath, A.

S. Kim, A. Gopinath, “Vector analysis of optical dielectric waveguide bends using finite-difference method,” J. Lightwave Technol. 14, 2085–2092 (1996).
[CrossRef]

Gu, J.-S.

J.-S. Gu, P.-A. Besse, H. Melchior, “Method of lines for the analysis of the propagation characteristics of curved optical rib waveguides,” IEEE J. Quantum Electron. 27, 531–537 (1991).
[CrossRef]

Harari, J.

He, S.

Huang, W.-P.

Jin, G. H.

Kim, S.

S. Kim, A. Gopinath, “Vector analysis of optical dielectric waveguide bends using finite-difference method,” J. Lightwave Technol. 14, 2085–2092 (1996).
[CrossRef]

Koshiba, M.

T. Yamamoto, M. Koshiba, “Numerical analysis of curvature loss in optical waveguides by the finite-element method,” J. Lightwave Technol. 11, 1579–1583 (1993).
[CrossRef]

Liu, S.

Melchior, H.

J.-S. Gu, P.-A. Besse, H. Melchior, “Method of lines for the analysis of the propagation characteristics of curved optical rib waveguides,” IEEE J. Quantum Electron. 27, 531–537 (1991).
[CrossRef]

Pennings, E. C. M.

M. K. Smit, E. C. M. Pennings, H. Blok, “A normalized approach to the design of low-loss optical waveguide bends,” J. Lightwave Technol. 11, 1737–1742 (1993).
[CrossRef]

Pogossian, S. P.

Smit, M. K.

M. K. Smit, E. C. M. Pennings, H. Blok, “A normalized approach to the design of low-loss optical waveguide bends,” J. Lightwave Technol. 11, 1737–1742 (1993).
[CrossRef]

Vescan, L.

Vilcot, J. P.

Vonsovici, A.

Yamamoto, T.

T. Yamamoto, M. Koshiba, “Numerical analysis of curvature loss in optical waveguides by the finite-element method,” J. Lightwave Technol. 11, 1579–1583 (1993).
[CrossRef]

Zhou, G.-R.

Zhou, Q.

IEEE J. Quantum Electron. (1)

J.-S. Gu, P.-A. Besse, H. Melchior, “Method of lines for the analysis of the propagation characteristics of curved optical rib waveguides,” IEEE J. Quantum Electron. 27, 531–537 (1991).
[CrossRef]

J. Lightwave Technol. (7)

J. Opt. Soc. Am. A (2)

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Figures (9)

Fig. 1
Fig. 1

Comparison of the pure bending loss calculated with the present FDM code, FEM,5 and MOL6 for an example (shown in the inset) considered in Ref. 5.

Fig. 2
Fig. 2

SM region and the MM region for a SOI rib waveguide.

Fig. 3
Fig. 3

Bending loss as the bending radius varies for SOI rib waveguides with different rib widths. (a) The pure bending loss. (b) The transition bending loss.

Fig. 4
Fig. 4

Transition loss as the rib width increases for different bending radii (here hr=2µm).

Fig. 5
Fig. 5

Transition loss as the rib width increases for different rib depths and bending radii.

Fig. 6
Fig. 6

Fundamental field profiles for bent waveguides with different rib widths when R=10,000µm. (a) wr=4µm, (b) wr=6µm, (c) wr=8µm.

Fig. 7
Fig. 7

Bending loss for the eigenmodes E11x, E21x, and E12x of the bent waveguide when hr=3.0 μm. (a) The pure bending loss for E11x, (b) the transition bending loss for E11x, (c) the pure bending loss for E21x, (d) the transition bending loss for E21x, (e) the pure bending loss for E12x, (f) the transition bending loss for E12x.

Fig. 8
Fig. 8

Field distribution for the E12x mode of a bent waveguide (w=6.0µm) with different radii. (a) R=10,000µm, (b) R=5000µm, (c) R=2000µm, (d) R=1000µm, (e) R=800µm, (f) R=500µm.

Fig. 9
Fig. 9

Connecting a bent waveguide with two straight waveguides.

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

PxxPxyPyxPyyExEy=1t˜x2 2z2 ExEy,
PxxEx=1t˜x2 x˜ t˜xn2 (t˜xn2Ex)x˜+2Exy˜2+n2k02Ex,
PxyEy=1t˜x2 x˜ t˜x2n2 (n2Ey)y˜-2Eyx˜y˜,
PyyEy=y˜ 1n2 (n2Ey)y˜+1t˜x x˜ t˜x Eyx˜,+n2k02Ey,
PyxEx=y˜ 1t˜xn2 (t˜xn2Ex)x˜-1t˜x x˜ t˜x Exy˜,
PxxE¯x=1t˜x2β2E¯x,
PyyE¯y=1t˜x2β2E¯y.
amn·E¯x(m-1, n)+bmn·E¯x(m, n)+cmn·E¯x(m+1, n)+dmnE¯x(m, n-1)+emnE¯x(m, n+1)=β2·E¯x(m, n),
amn=nm-12(1+x˜m-1/R)2Δx˜mΔx˜m-1/2×1+x˜m-1/Rnm-12+1+x˜m/Rnm2,
bmn=-(1+x˜m/R)nm22Δx˜m×1+x˜m+1/Rnm+12+1+x˜m/Rnm2 1Δx˜m+1/2+1+x˜m-1/Rnm-12+1+x˜m/Rnm2 1Δx˜m-1/2-(1+x˜m/R)2Δy˜n 1Δy˜n+1/2+1Δy˜n-1/2+nmn2k02(1+x˜m/R)2,
cmn=nm+12(1+x˜m+1/R)2Δx˜mΔx˜m+1/2×1+x˜m+1/Rnm+12+1+x˜m/Rnm2,
dmn=(1+x˜m/R)2Δy˜nΔy˜n-1/2,
emn=(1+x˜m/R)2Δy˜nΔy˜n+1/2,
Lα=20 log[exp(π/2×βiR)][decibels(dB)/90°].
LT=-10 log-+-+E0(x, y)E0B*(x, y)dxdy2-+-+E0(x, y)E0*(x, y)dxdy-+-+E0B(x, y)E0B*(x, y)dxdy(dB),
t<r/1-r2,

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