Abstract

By taking into account the curved travel path instead of the conventional abrupt optical line path in the bent region of three-dimensional embedded structures, we propose two kinds of microprism-type waveguide bends and systematically derive precise phase compensation formulas. The simulation results obtained with the fast Fourier-transform beam propagation method indicate that the normalized transmitted powers are greater than 95% even though the bent angle is as large as 10°.

© 1998 Optical Society of America

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References

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  1. T. Tamir, Guided-Wave Optoelectronics (Springer-Verlag, Berlin, 1989).
  2. H. Sasaki, N. Mikoshiba, “Normalized power transmission in single mode optical branching waveguides,” Electron. Lett. 17, 136–138 (1981).
    [CrossRef]
  3. T. Shiina, K. Shiraishi, S. Kawakami, “Waveguide-bend configuration with low-loss characteristics,” Opt. Lett. 11, 736–738 (1986).
    [CrossRef] [PubMed]
  4. M. L. Wu, P. L. Fan, J. M. Hsu, C. T. Lee, “Design of ideal structures for lossless bends in optical waveguides by conformal mapping,” J. Lightwave Technol. 14, 2604–2614 (1996).
    [CrossRef]
  5. S. K. Korotky, E. A. J. Marcatili, J. J. Veselka, R. H. Bosworth, “Greatly reduced losses for small-radius bends in Ti:LiNbO3 waveguides,” Appl. Phys. Lett. 8, 92–94 (1986).
    [CrossRef]
  6. K. Hirayama, M. Koshiba, “A new low-loss structure of abrupt bend in dielectric waveguides,” J. Lightwave Technol. 10, 563–569 (1992).
    [CrossRef]
  7. H. B. Lin, J. Y. Su, Y. P. Liao, W. S. Wang, “Study and design of step-index channel waveguide bends with large-angle and low-loss characteristics,” IEEE J. Quantum Electron. 31, 1131–1138 (1995).
    [CrossRef]
  8. P. L. Fan, M. L. Wu, C. T. Lee, “Analysis of abrupt bent waveguides by the beam propagation method and the conformal mapping method,” J. Lightwave Technol. 15, 1026–1031 (1997).
    [CrossRef]
  9. C. T. Lee, M. L. Wu, J. M. Hsu, “Beam propagation analysis for tapered waveguides: taking account of the curved phase-front effect in paraxial approximation,” J. Lightwave Technol. 15, 2183–2189 (1997).
    [CrossRef]
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    [CrossRef]
  14. C. T. Lee, P. L. Fan, “Beam propagation analysis of fast mode-conversion evolution bent waveguides with apexes-linked microprisms,” IEEE Microwave Guided Wave Lett. 7, 338–340 (1997).
    [CrossRef]

1997 (3)

C. T. Lee, P. L. Fan, “Beam propagation analysis of fast mode-conversion evolution bent waveguides with apexes-linked microprisms,” IEEE Microwave Guided Wave Lett. 7, 338–340 (1997).
[CrossRef]

P. L. Fan, M. L. Wu, C. T. Lee, “Analysis of abrupt bent waveguides by the beam propagation method and the conformal mapping method,” J. Lightwave Technol. 15, 1026–1031 (1997).
[CrossRef]

C. T. Lee, M. L. Wu, J. M. Hsu, “Beam propagation analysis for tapered waveguides: taking account of the curved phase-front effect in paraxial approximation,” J. Lightwave Technol. 15, 2183–2189 (1997).
[CrossRef]

1996 (1)

M. L. Wu, P. L. Fan, J. M. Hsu, C. T. Lee, “Design of ideal structures for lossless bends in optical waveguides by conformal mapping,” J. Lightwave Technol. 14, 2604–2614 (1996).
[CrossRef]

1995 (1)

H. B. Lin, J. Y. Su, Y. P. Liao, W. S. Wang, “Study and design of step-index channel waveguide bends with large-angle and low-loss characteristics,” IEEE J. Quantum Electron. 31, 1131–1138 (1995).
[CrossRef]

1992 (1)

K. Hirayama, M. Koshiba, “A new low-loss structure of abrupt bend in dielectric waveguides,” J. Lightwave Technol. 10, 563–569 (1992).
[CrossRef]

1986 (2)

S. K. Korotky, E. A. J. Marcatili, J. J. Veselka, R. H. Bosworth, “Greatly reduced losses for small-radius bends in Ti:LiNbO3 waveguides,” Appl. Phys. Lett. 8, 92–94 (1986).
[CrossRef]

T. Shiina, K. Shiraishi, S. Kawakami, “Waveguide-bend configuration with low-loss characteristics,” Opt. Lett. 11, 736–738 (1986).
[CrossRef] [PubMed]

1983 (1)

1981 (1)

H. Sasaki, N. Mikoshiba, “Normalized power transmission in single mode optical branching waveguides,” Electron. Lett. 17, 136–138 (1981).
[CrossRef]

1980 (2)

Bosworth, R. H.

S. K. Korotky, E. A. J. Marcatili, J. J. Veselka, R. H. Bosworth, “Greatly reduced losses for small-radius bends in Ti:LiNbO3 waveguides,” Appl. Phys. Lett. 8, 92–94 (1986).
[CrossRef]

Fan, P. L.

P. L. Fan, M. L. Wu, C. T. Lee, “Analysis of abrupt bent waveguides by the beam propagation method and the conformal mapping method,” J. Lightwave Technol. 15, 1026–1031 (1997).
[CrossRef]

C. T. Lee, P. L. Fan, “Beam propagation analysis of fast mode-conversion evolution bent waveguides with apexes-linked microprisms,” IEEE Microwave Guided Wave Lett. 7, 338–340 (1997).
[CrossRef]

M. L. Wu, P. L. Fan, J. M. Hsu, C. T. Lee, “Design of ideal structures for lossless bends in optical waveguides by conformal mapping,” J. Lightwave Technol. 14, 2604–2614 (1996).
[CrossRef]

Feit, M. D.

Fleck, J. A.

Hirayama, K.

K. Hirayama, M. Koshiba, “A new low-loss structure of abrupt bend in dielectric waveguides,” J. Lightwave Technol. 10, 563–569 (1992).
[CrossRef]

Hsu, J. M.

C. T. Lee, M. L. Wu, J. M. Hsu, “Beam propagation analysis for tapered waveguides: taking account of the curved phase-front effect in paraxial approximation,” J. Lightwave Technol. 15, 2183–2189 (1997).
[CrossRef]

M. L. Wu, P. L. Fan, J. M. Hsu, C. T. Lee, “Design of ideal structures for lossless bends in optical waveguides by conformal mapping,” J. Lightwave Technol. 14, 2604–2614 (1996).
[CrossRef]

Kawakami, S.

Korotky, S. K.

S. K. Korotky, E. A. J. Marcatili, J. J. Veselka, R. H. Bosworth, “Greatly reduced losses for small-radius bends in Ti:LiNbO3 waveguides,” Appl. Phys. Lett. 8, 92–94 (1986).
[CrossRef]

Koshiba, M.

K. Hirayama, M. Koshiba, “A new low-loss structure of abrupt bend in dielectric waveguides,” J. Lightwave Technol. 10, 563–569 (1992).
[CrossRef]

Lee, C. T.

C. T. Lee, P. L. Fan, “Beam propagation analysis of fast mode-conversion evolution bent waveguides with apexes-linked microprisms,” IEEE Microwave Guided Wave Lett. 7, 338–340 (1997).
[CrossRef]

P. L. Fan, M. L. Wu, C. T. Lee, “Analysis of abrupt bent waveguides by the beam propagation method and the conformal mapping method,” J. Lightwave Technol. 15, 1026–1031 (1997).
[CrossRef]

C. T. Lee, M. L. Wu, J. M. Hsu, “Beam propagation analysis for tapered waveguides: taking account of the curved phase-front effect in paraxial approximation,” J. Lightwave Technol. 15, 2183–2189 (1997).
[CrossRef]

M. L. Wu, P. L. Fan, J. M. Hsu, C. T. Lee, “Design of ideal structures for lossless bends in optical waveguides by conformal mapping,” J. Lightwave Technol. 14, 2604–2614 (1996).
[CrossRef]

Liao, Y. P.

H. B. Lin, J. Y. Su, Y. P. Liao, W. S. Wang, “Study and design of step-index channel waveguide bends with large-angle and low-loss characteristics,” IEEE J. Quantum Electron. 31, 1131–1138 (1995).
[CrossRef]

Lin, H. B.

H. B. Lin, J. Y. Su, Y. P. Liao, W. S. Wang, “Study and design of step-index channel waveguide bends with large-angle and low-loss characteristics,” IEEE J. Quantum Electron. 31, 1131–1138 (1995).
[CrossRef]

Marcatili, E. A. J.

S. K. Korotky, E. A. J. Marcatili, J. J. Veselka, R. H. Bosworth, “Greatly reduced losses for small-radius bends in Ti:LiNbO3 waveguides,” Appl. Phys. Lett. 8, 92–94 (1986).
[CrossRef]

Mikoshiba, N.

H. Sasaki, N. Mikoshiba, “Normalized power transmission in single mode optical branching waveguides,” Electron. Lett. 17, 136–138 (1981).
[CrossRef]

Pulker, H. K.

Saijonmaa, J.

Sasaki, H.

H. Sasaki, N. Mikoshiba, “Normalized power transmission in single mode optical branching waveguides,” Electron. Lett. 17, 136–138 (1981).
[CrossRef]

Shiina, T.

Shiraishi, K.

Su, J. Y.

H. B. Lin, J. Y. Su, Y. P. Liao, W. S. Wang, “Study and design of step-index channel waveguide bends with large-angle and low-loss characteristics,” IEEE J. Quantum Electron. 31, 1131–1138 (1995).
[CrossRef]

Sze, S. M.

S. M. Sze, VLSI Technology (McGraw-Hill, New York, 1983).

Tamir, T.

T. Tamir, Guided-Wave Optoelectronics (Springer-Verlag, Berlin, 1989).

Veselka, J. J.

S. K. Korotky, E. A. J. Marcatili, J. J. Veselka, R. H. Bosworth, “Greatly reduced losses for small-radius bends in Ti:LiNbO3 waveguides,” Appl. Phys. Lett. 8, 92–94 (1986).
[CrossRef]

Wang, W. S.

H. B. Lin, J. Y. Su, Y. P. Liao, W. S. Wang, “Study and design of step-index channel waveguide bends with large-angle and low-loss characteristics,” IEEE J. Quantum Electron. 31, 1131–1138 (1995).
[CrossRef]

Wu, M. L.

C. T. Lee, M. L. Wu, J. M. Hsu, “Beam propagation analysis for tapered waveguides: taking account of the curved phase-front effect in paraxial approximation,” J. Lightwave Technol. 15, 2183–2189 (1997).
[CrossRef]

P. L. Fan, M. L. Wu, C. T. Lee, “Analysis of abrupt bent waveguides by the beam propagation method and the conformal mapping method,” J. Lightwave Technol. 15, 1026–1031 (1997).
[CrossRef]

M. L. Wu, P. L. Fan, J. M. Hsu, C. T. Lee, “Design of ideal structures for lossless bends in optical waveguides by conformal mapping,” J. Lightwave Technol. 14, 2604–2614 (1996).
[CrossRef]

Yevick, D.

Appl. Opt. (2)

Appl. Phys. Lett. (1)

S. K. Korotky, E. A. J. Marcatili, J. J. Veselka, R. H. Bosworth, “Greatly reduced losses for small-radius bends in Ti:LiNbO3 waveguides,” Appl. Phys. Lett. 8, 92–94 (1986).
[CrossRef]

Electron. Lett. (1)

H. Sasaki, N. Mikoshiba, “Normalized power transmission in single mode optical branching waveguides,” Electron. Lett. 17, 136–138 (1981).
[CrossRef]

IEEE J. Quantum Electron. (1)

H. B. Lin, J. Y. Su, Y. P. Liao, W. S. Wang, “Study and design of step-index channel waveguide bends with large-angle and low-loss characteristics,” IEEE J. Quantum Electron. 31, 1131–1138 (1995).
[CrossRef]

IEEE Microwave Guided Wave Lett. (1)

C. T. Lee, P. L. Fan, “Beam propagation analysis of fast mode-conversion evolution bent waveguides with apexes-linked microprisms,” IEEE Microwave Guided Wave Lett. 7, 338–340 (1997).
[CrossRef]

J. Lightwave Technol. (4)

M. L. Wu, P. L. Fan, J. M. Hsu, C. T. Lee, “Design of ideal structures for lossless bends in optical waveguides by conformal mapping,” J. Lightwave Technol. 14, 2604–2614 (1996).
[CrossRef]

P. L. Fan, M. L. Wu, C. T. Lee, “Analysis of abrupt bent waveguides by the beam propagation method and the conformal mapping method,” J. Lightwave Technol. 15, 1026–1031 (1997).
[CrossRef]

C. T. Lee, M. L. Wu, J. M. Hsu, “Beam propagation analysis for tapered waveguides: taking account of the curved phase-front effect in paraxial approximation,” J. Lightwave Technol. 15, 2183–2189 (1997).
[CrossRef]

K. Hirayama, M. Koshiba, “A new low-loss structure of abrupt bend in dielectric waveguides,” J. Lightwave Technol. 10, 563–569 (1992).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Lett. (1)

Other (2)

S. M. Sze, VLSI Technology (McGraw-Hill, New York, 1983).

T. Tamir, Guided-Wave Optoelectronics (Springer-Verlag, Berlin, 1989).

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Figures (9)

Fig. 1
Fig. 1

Microprism-type bent waveguide: (a) top view of type I, (b) top view of type II, (c) 3-D structure.

Fig. 2
Fig. 2

Dependences of refractive indices n pn and half-lengths L pn of the microprisms on the widths of analytic space W An for (a) microprism-type I and (b) microprism-type II with a bent angle of θ = 10°.

Fig. 3
Fig. 3

Dependences of the widths of analytic space W An and half-lengths of microprisms L pn on refractive indices n pn for (a) microprism-type I and (b) microprism-type II with a bent angle of θ = 10°.

Fig. 4
Fig. 4

(a) Dependences of W A1 and L p1 on bent angle θ with a fixed n p1 of 1.6. (b) Dependences of n p1 and L p1 on bent angle θ with a fixed W A1 of 591 for microprism-type I.

Fig. 5
Fig. 5

(a) Dependences of W A2 and L p2 on bent angle θ with a fixed n p2 of 1.6. (b) Dependences of n p2 and L p2 on bent angle θ with a fixed W A2 of 558 for microprism-type II.

Fig. 6
Fig. 6

Field intensity distributions for (a) microprism-type I and (b) microprism-type II and the normalized output intensities for (c) microprism-type I and (d) microprism-type II with a bent angle of θ = 10°.

Fig. 7
Fig. 7

Output field intensity contours and cross sections for (a) microprism-type I and (b) microprism-type II and 3-D plots of output field intensities for (c) microprism-type I and (d) microprism-type II with a bent angle of θ = 10°.

Fig. 8
Fig. 8

Phase distributions for (a) microprism-type I and (b) microprism-type II with a bent angle of θ = 10°.

Fig. 9
Fig. 9

Dependences of normalized transmitted power efficiency η on bent angle θ for microprism-types I and II.

Equations (10)

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

W An = 2 R An = 2 L fn tan   θ / 2 - W g ,
Δ ϕ b = β Δ d b = β BB ͡ - AA ͡ = k 0 n eff θ W p ,
W eff = W g + 1 α 1 x + 1 α 2 x ,
Δ ϕ p 1 = n p 1 - n s k 0 Δ d p 1 = n p 1 - n s k 0 AA ͡ - CC ͡ = n p 1 - n s k 0 θ   W A 1 - W p 2 × cos θ / 2 - m   cos 2 θ / 2 1 - m   cos 2 θ / 2 ,
n p 1 - n s W A 1 - W p 2 × cos θ / 2 - m   cos 2 θ / 2 1 - m   cos 2 θ / 2 = n eff W p .
Δ ϕ p 2 = n p 2 - n s k 0 AA ͡ = n p 2 - n s k 0 θ   W A 2 - W p 2 .
n p 2 = n s + 2 n eff W p W A 2 - W p .
n p 1 - n s W A 1 - W p 2 × cos θ / 2 - m   cos 2 θ / 2 1 - m   cos 2 θ / 2 = n s W p ,
n p 2 = n s + 2 n s W p W A 2 - W p .
L pn = W An - W p 2 sin θ / 2 ,     n = 1 ,   2 .

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