Abstract

We present a method to solve the TE-like modes for a 3D circularly bent dielectric rib waveguide. The radial part of the Helmholtz equation in cylindrical coordinates was reduced to contain only the radial component of the electric field. The effective index method and the conformal transformation were applied to transfer the structure into two straight slab waveguides. The problem of a 3D circularly bent waveguide is reduced to the problem of solving two 2D straight slab waveguides and each one is characterized by a 1D eigenvalue equation that is solvable numerically by the beam propagation method (BPM). The bending loss, propagation constant, and radial component of the electric field distribution of the modes can be obtained directly. In contrast to many commercial 3D BPM methods our method can be applied to the bent waveguide without restriction to a large radius. Examples were given to demonstrate our method on the bent rib waveguide and compared with the results of other methods.

© 2008 Optical Society of America

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References

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  1. G. T. Reed and A. P. Knights, Silicon Photonics: An Introduction (Wiley, 2004).
    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
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2007 (1)

2005 (2)

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435, 325-327 (2005).
[Crossref] [PubMed]

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433, 725-728 (2005).
[Crossref] [PubMed]

2004 (1)

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 615-618 (2004).
[Crossref] [PubMed]

2001 (1)

2000 (1)

1998 (1)

1997 (1)

P. D. Trinh, S. Yegnanarayanan, F. Coppinger, and B. Jalali, “Silicon-on-insulator (SOI) phased-array wavelength multi-demultiplexer with extremely low-polarization sensitivity,” IEEE Photon. Technol. Lett. 9, 940-942 (1997).
[Crossref]

1995 (2)

P. D. Trinh, S. Yegnanarayanan, and B. Jalali, “Integrated optical directional couplers in silicon-on-insulator,” Electron. Lett. 31, 2097-2098 (1995).
[Crossref]

L. H. Spiekman, Y. S. Oei, E. G. Metaal, F. H. Groen, P. Demeester, and M. K. Smit, “Ultrasmall waveguide bends: the corner mirrors of the future?,” IEE Proc.: Optoelectron. 142, 61-65 (1995).
[Crossref]

1994 (1)

S. Jungling and J. C. Chen, “A study and optimization of eigenmode calculations using imaginary-distance beam-propagation method,” IEEE J. Quantum Electron. QE-30, 2098-2105 (1994).
[Crossref]

1991 (1)

E. C. M. Pennings and R. J. Deri, “Simple method for estimating usable bend radii of deeply etched optical rib waveguides,” Electron. Lett. 27, 1532-1534 (1991).
[Crossref]

1984 (1)

1980 (1)

1975 (1)

M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. QE-11, 75-83 (1975).
[Crossref]

Berglund, W.

Campbell, J. C.

Chen, J. C.

S. Jungling and J. C. Chen, “A study and optimization of eigenmode calculations using imaginary-distance beam-propagation method,” IEEE J. Quantum Electron. QE-30, 2098-2105 (1994).
[Crossref]

Chilwell, J.

Cohen, O.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433, 725-728 (2005).
[Crossref] [PubMed]

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 615-618 (2004).
[Crossref] [PubMed]

Coppinger, F.

P. D. Trinh, S. Yegnanarayanan, F. Coppinger, and B. Jalali, “Silicon-on-insulator (SOI) phased-array wavelength multi-demultiplexer with extremely low-polarization sensitivity,” IEEE Photon. Technol. Lett. 9, 940-942 (1997).
[Crossref]

Csutak, S. M.

Decoster, D.

Demeester, P.

L. H. Spiekman, Y. S. Oei, E. G. Metaal, F. H. Groen, P. Demeester, and M. K. Smit, “Ultrasmall waveguide bends: the corner mirrors of the future?,” IEE Proc.: Optoelectron. 142, 61-65 (1995).
[Crossref]

Deng, H.

Deri, R. J.

E. C. M. Pennings and R. J. Deri, “Simple method for estimating usable bend radii of deeply etched optical rib waveguides,” Electron. Lett. 27, 1532-1534 (1991).
[Crossref]

Fang, A.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433, 725-728 (2005).
[Crossref] [PubMed]

Feit, M. D.

Fleck, J. A.

Gopinath, A.

Groen, F. H.

L. H. Spiekman, Y. S. Oei, E. G. Metaal, F. H. Groen, P. Demeester, and M. K. Smit, “Ultrasmall waveguide bends: the corner mirrors of the future?,” IEE Proc.: Optoelectron. 142, 61-65 (1995).
[Crossref]

Hak, D.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433, 725-728 (2005).
[Crossref] [PubMed]

Harari, J.

Harris, J. H.

M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. QE-11, 75-83 (1975).
[Crossref]

Heiblum, M.

M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. QE-11, 75-83 (1975).
[Crossref]

Hodgkinson, I.

Hunsperger, R. G.

R. G. Hunsperger, Integrated Optics: Theory and Technology (Springer-Verlag, 1991), Chap. 3.

Jalali, B.

P. D. Trinh, S. Yegnanarayanan, F. Coppinger, and B. Jalali, “Silicon-on-insulator (SOI) phased-array wavelength multi-demultiplexer with extremely low-polarization sensitivity,” IEEE Photon. Technol. Lett. 9, 940-942 (1997).
[Crossref]

P. D. Trinh, S. Yegnanarayanan, and B. Jalali, “Integrated optical directional couplers in silicon-on-insulator,” Electron. Lett. 31, 2097-2098 (1995).
[Crossref]

Jin, G. H.

Jones, R.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433, 725-728 (2005).
[Crossref] [PubMed]

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 615-618 (2004).
[Crossref] [PubMed]

Jungling, S.

S. Jungling and J. C. Chen, “A study and optimization of eigenmode calculations using imaginary-distance beam-propagation method,” IEEE J. Quantum Electron. QE-30, 2098-2105 (1994).
[Crossref]

Knights, A. P.

G. T. Reed and A. P. Knights, Silicon Photonics: An Introduction (Wiley, 2004).
[Crossref]

Knox, R. M.

R. M. Knox and P. P. Toulios, “Integrated circuits for millimeter through optical frequency range,” in Symposium on Submillimeter Waves (Polytechnic Institute of Brooklyn, 1970), pp. 497-516.

Li, R.

Liao, L.

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 615-618 (2004).
[Crossref] [PubMed]

Lipson, M.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435, 325-327 (2005).
[Crossref] [PubMed]

C. R. Pollock and M. Lipson, Integrated Photonics (Kluwer Academic, 2003), Chap. 8.

Liu, A.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433, 725-728 (2005).
[Crossref] [PubMed]

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 615-618 (2004).
[Crossref] [PubMed]

Metaal, E. G.

L. H. Spiekman, Y. S. Oei, E. G. Metaal, F. H. Groen, P. Demeester, and M. K. Smit, “Ultrasmall waveguide bends: the corner mirrors of the future?,” IEE Proc.: Optoelectron. 142, 61-65 (1995).
[Crossref]

Morse, M.

M. Paniccia, M. Morse, and M. Salib, “Integrated photonics,” in Silicon Photonics, L.Pavesi and D.J.Lockwood, eds. (Springer-Verlag, 2004), pp. 51-88.
[Crossref]

Nicolaescu, R.

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 615-618 (2004).
[Crossref] [PubMed]

Oei, Y. S.

L. H. Spiekman, Y. S. Oei, E. G. Metaal, F. H. Groen, P. Demeester, and M. K. Smit, “Ultrasmall waveguide bends: the corner mirrors of the future?,” IEE Proc.: Optoelectron. 142, 61-65 (1995).
[Crossref]

Paniccia, M.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433, 725-728 (2005).
[Crossref] [PubMed]

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 615-618 (2004).
[Crossref] [PubMed]

M. Paniccia, M. Morse, and M. Salib, “Integrated photonics,” in Silicon Photonics, L.Pavesi and D.J.Lockwood, eds. (Springer-Verlag, 2004), pp. 51-88.
[Crossref]

Pennings, E. C. M.

E. C. M. Pennings and R. J. Deri, “Simple method for estimating usable bend radii of deeply etched optical rib waveguides,” Electron. Lett. 27, 1532-1534 (1991).
[Crossref]

Pollock, C. R.

C. R. Pollock and M. Lipson, Integrated Photonics (Kluwer Academic, 2003), Chap. 8.

Pradhan, S.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435, 325-327 (2005).
[Crossref] [PubMed]

Reed, G. T.

G. T. Reed and A. P. Knights, Silicon Photonics: An Introduction (Wiley, 2004).
[Crossref]

Rong, H.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433, 725-728 (2005).
[Crossref] [PubMed]

Rubin, D.

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 615-618 (2004).
[Crossref] [PubMed]

Salib, M.

M. Paniccia, M. Morse, and M. Salib, “Integrated photonics,” in Silicon Photonics, L.Pavesi and D.J.Lockwood, eds. (Springer-Verlag, 2004), pp. 51-88.
[Crossref]

Samara-Rubio, D.

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 615-618 (2004).
[Crossref] [PubMed]

Schaub, J. D.

Schermer, R. T.

Schmidt, B.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435, 325-327 (2005).
[Crossref] [PubMed]

Smit, M. K.

L. H. Spiekman, Y. S. Oei, E. G. Metaal, F. H. Groen, P. Demeester, and M. K. Smit, “Ultrasmall waveguide bends: the corner mirrors of the future?,” IEE Proc.: Optoelectron. 142, 61-65 (1995).
[Crossref]

Spiekman, L. H.

L. H. Spiekman, Y. S. Oei, E. G. Metaal, F. H. Groen, P. Demeester, and M. K. Smit, “Ultrasmall waveguide bends: the corner mirrors of the future?,” IEE Proc.: Optoelectron. 142, 61-65 (1995).
[Crossref]

Tamir, T.

T. Tamir, Integrated Optics (Springer-Verlag, 1975), Chap. 2.

Toulios, P. P.

R. M. Knox and P. P. Toulios, “Integrated circuits for millimeter through optical frequency range,” in Symposium on Submillimeter Waves (Polytechnic Institute of Brooklyn, 1970), pp. 497-516.

Trinh, P. D.

P. D. Trinh, S. Yegnanarayanan, F. Coppinger, and B. Jalali, “Silicon-on-insulator (SOI) phased-array wavelength multi-demultiplexer with extremely low-polarization sensitivity,” IEEE Photon. Technol. Lett. 9, 940-942 (1997).
[Crossref]

P. D. Trinh, S. Yegnanarayanan, and B. Jalali, “Integrated optical directional couplers in silicon-on-insulator,” Electron. Lett. 31, 2097-2098 (1995).
[Crossref]

Vilcot, J. P.

Xu, Q.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435, 325-327 (2005).
[Crossref] [PubMed]

Yegnanarayanan, S.

P. D. Trinh, S. Yegnanarayanan, F. Coppinger, and B. Jalali, “Silicon-on-insulator (SOI) phased-array wavelength multi-demultiplexer with extremely low-polarization sensitivity,” IEEE Photon. Technol. Lett. 9, 940-942 (1997).
[Crossref]

P. D. Trinh, S. Yegnanarayanan, and B. Jalali, “Integrated optical directional couplers in silicon-on-insulator,” Electron. Lett. 31, 2097-2098 (1995).
[Crossref]

Appl. Opt. (1)

Electron. Lett. (2)

E. C. M. Pennings and R. J. Deri, “Simple method for estimating usable bend radii of deeply etched optical rib waveguides,” Electron. Lett. 27, 1532-1534 (1991).
[Crossref]

P. D. Trinh, S. Yegnanarayanan, and B. Jalali, “Integrated optical directional couplers in silicon-on-insulator,” Electron. Lett. 31, 2097-2098 (1995).
[Crossref]

IEE Proc.: Optoelectron. (1)

L. H. Spiekman, Y. S. Oei, E. G. Metaal, F. H. Groen, P. Demeester, and M. K. Smit, “Ultrasmall waveguide bends: the corner mirrors of the future?,” IEE Proc.: Optoelectron. 142, 61-65 (1995).
[Crossref]

IEEE J. Quantum Electron. (2)

M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. QE-11, 75-83 (1975).
[Crossref]

S. Jungling and J. C. Chen, “A study and optimization of eigenmode calculations using imaginary-distance beam-propagation method,” IEEE J. Quantum Electron. QE-30, 2098-2105 (1994).
[Crossref]

IEEE Photon. Technol. Lett. (1)

P. D. Trinh, S. Yegnanarayanan, F. Coppinger, and B. Jalali, “Silicon-on-insulator (SOI) phased-array wavelength multi-demultiplexer with extremely low-polarization sensitivity,” IEEE Photon. Technol. Lett. 9, 940-942 (1997).
[Crossref]

J. Lightwave Technol. (3)

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

Nature (3)

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433, 725-728 (2005).
[Crossref] [PubMed]

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 615-618 (2004).
[Crossref] [PubMed]

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435, 325-327 (2005).
[Crossref] [PubMed]

Opt. Express (1)

Other (7)

G. T. Reed and A. P. Knights, Silicon Photonics: An Introduction (Wiley, 2004).
[Crossref]

M. Paniccia, M. Morse, and M. Salib, “Integrated photonics,” in Silicon Photonics, L.Pavesi and D.J.Lockwood, eds. (Springer-Verlag, 2004), pp. 51-88.
[Crossref]

R. G. Hunsperger, Integrated Optics: Theory and Technology (Springer-Verlag, 1991), Chap. 3.

R. M. Knox and P. P. Toulios, “Integrated circuits for millimeter through optical frequency range,” in Symposium on Submillimeter Waves (Polytechnic Institute of Brooklyn, 1970), pp. 497-516.

T. Tamir, Integrated Optics (Springer-Verlag, 1975), Chap. 2.

C. R. Pollock and M. Lipson, Integrated Photonics (Kluwer Academic, 2003), Chap. 8.

RSoft CAD Environment 6.0 (RSoft Design Group, Inc., 2005), pp. 93.

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Figures (10)

Fig. 1
Fig. 1

Circularly bent rib waveguide in the cylindrical coordinate system. Radius R is measured from the center of the circle to the center of the rib. Radius R 1 is measured from the center of the circle to the outside edge of the rib.

Fig. 2
Fig. 2

(a) Geometry of the bent waveguide in the cylindrical coordinate system. (b) Equivalent straight waveguide in the conformally transformed ( u , v ) space.

Fig. 3
Fig. 3

Cross section of a symmetrical bent rib waveguide.

Fig. 4
Fig. 4

Effective refractive index distribution of the waveguide in Fig. 3.

Fig. 5
Fig. 5

Conformal transformation of the effective refractive index distribution in Fig. 4.

Fig. 6
Fig. 6

Cross section of the rib waveguide given in [13].

Fig. 7
Fig. 7

Bending loss of the bent waveguide in Fig. 6 from different methods.

Fig. 8
Fig. 8

Cross section of the rib waveguide given in [12].

Fig. 9
Fig. 9

Modal refractive index versus radius of curvature for the waveguide in Fig. 8.

Fig. 10
Fig. 10

Electric field distribution of the bent rib waveguide in Fig. 6. R = 85 μ m .

Equations (28)

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

2 E + k 2 E = 0 ,
( 2 E r 2 r 2 E ϕ ϕ 1 r 2 E r ) a r + ( 2 E ϕ + 2 r 2 E r ϕ 1 r 2 E ϕ ) a ϕ + ( 2 E y ) a y + k 2 ( E r a r + E ϕ a ϕ + E y a y ) = 0 ,
2 E r 2 r 2 E ϕ ϕ 1 r 2 E r + k 2 E r = 0 .
2 E ϕ + 2 r 2 E r ϕ 1 r 2 E ϕ + k 2 E ϕ = 0 .
2 E y + k 2 E y = 0 .
1 r r ( r E r r ) + 1 r 2 ϕ ( E r ϕ 2 E ϕ ) + 2 y 2 E r + ( n 2 ( r , y ) k 0 2 1 r 2 ) E r = 0 .
2 r 2 E r + 1 r 2 2 ϕ 2 E r + 2 y 2 E r + 1 r r E r E r r 2 + n 2 ( r , y ) k 0 2 E r = 0 .
E r ( r , ϕ , y ) = F ( r , ϕ ) J ( y ) .
1 F ( r , ϕ ) 2 r 2 F ( r , ϕ ) + 1 r 2 F ( r , ϕ ) 2 ϕ 2 F ( r , ϕ ) + 1 J ( y ) 2 y 2 J ( y ) + 1 r F ( r , ϕ ) r F ( r , ϕ ) 1 r 2 + n 2 ( r , y ) k 0 2 = 0 .
1 J ( y ) 2 y 2 J ( y ) + [ n 2 ( r , y ) n eff 2 ( r ) ] k 0 2 = 0 ,
1 F ( r , ϕ ) 2 r 2 F ( r , ϕ ) + 1 r 2 F ( r , ϕ ) 2 ϕ 2 F ( r , ϕ ) + 1 r F ( r , ϕ ) r F ( r , ϕ ) 1 r 2 + n eff 2 ( r ) k 0 2 = 0 ,
r = R 1 exp ( u R 1 ) ,
ϕ = v R 1 .
F ( u , v ) = G ( u ) exp ( γ s v ) ,
γ = γ s R 1 ,
γ = α 2 + j β ,
β = N k 0 R 1 ,
1 G ( u ) 2 u 2 G ( u ) + ( n eff 2 ( u ) k 0 2 + γ s 2 ) = 0 ,
n eff 2 ( u ) = C { n eff 2 ( r ) } exp ( 2 u R 1 ) 1 ( R 1 k 0 ) 2 ,
ε eff , G ( u i δ ) = ε eff , G ( u i + δ ) .
G corrected ( u ) = ε eff , ε eff , ε ε G ( u ) ,
E r ( r , ϕ , y ) = E r 0 ( r , y ) exp { [ ( α 2 + j β ) ] ϕ } ,
E ϕ ( r , ϕ , y ) = E ϕ 0 ( r , y ) exp { [ ( α 2 + j β ) ] ϕ } ,
E r ( r , ϕ , y ) = E r 0 ( r , y ) exp ( γ ϕ ) ,
E ϕ ( r , ϕ , y ) = E ϕ 0 ( r , y ) exp ( γ ϕ ) .
E r ϕ = γ E r 0 ( r , y ) exp ( γ ϕ ) .
( E r ϕ ) ( 2 E ϕ ) = ( α 2 4 + β 2 ) 1 2 E r 0 ( r , y ) E ϕ 0 ( r , y ) 2 .
( E r ϕ ) ( 2 E ϕ ) = ( N π R 1 ) E r 0 ( r , y ) E ϕ 0 ( r , y ) λ 0 .

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