Abstract

We present a simple and efficient approach for higher-order Berenger mode computation. We establish the physical mapping between radiation modes and complex Berenger modes, and theoretically prove that the higher-order substrate Berenger modes and cladding Berenger modes can converge to a cluster of complex modes with the same phase angle. This model can be explained by weighted optical path distance in both cladding and substrate, and can be implemented by adjusting parameters of perfectly matched layers. A germanium (Ge) photodetector is utilized to evaluate the merits of this method in terms of robustness, efficiency, and accuracy.

© 2014 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  9. P. Bienstman, H. Derudder, R. Baets, F. Olyslager, and D. De Zutter, IEEE Trans. Microwave Theory Tech. 49, 349 (2001).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  30. FDTD Solutions from Lumerical Solutions, Inc.

2013 (4)

2011 (1)

2010 (4)

2009 (1)

2008 (1)

2007 (1)

2006 (1)

2002 (3)

H. Rogier and D. De Zutter, J. Lightwave Technol. 20, 1141 (2002).
[CrossRef]

J. Petracek and K. Singh, IEEE Photon. Technol. Lett. 14, 810 (2002).
[CrossRef]

J. Ctyroky, S. Helfert, R. Pregla, P. Bienstman, R. Baets, R. De Ridder, R. Stoffer, G. Klaasse, J. Petracek, P. Lalanne, J. P. Hugonin, and R. M. De La Rue, Opt. Quantum Electron. 34, 455 (2002).

2001 (4)

H. Rogier and D. De Zutter, IEEE Trans. Microwave Theor. Tech. 49, 712 (2001).
[CrossRef]

H. Derudder, F. Olyslager, D. De Zutter, and S. Van den Berghe, IEEE Trans. Antennas Propag. 49, 185 (2001).
[CrossRef]

P. Bienstman and R. Baets, Opt. Quantum Electron. 33, 327 (2001).
[CrossRef]

P. Bienstman, H. Derudder, R. Baets, F. Olyslager, and D. De Zutter, IEEE Trans. Microwave Theory Tech. 49, 349 (2001).
[CrossRef]

1998 (1)

H. Derudder, D. De Zutter, and F. Olyslager, Electron. Lett. 34, 2138 (1998).
[CrossRef]

1997 (1)

W. C. Chew, J. M. Jin, and E. Michielssen, Microwave Opt. Technol. Lett. 15, 363 (1997).
[CrossRef]

1995 (1)

S. L. Lee, Y. C. Chung, L. A. Coldren, and N. Dagli, IEEE J. Quantum Electron. 31, 1790 (1995).
[CrossRef]

1994 (1)

W. C. Chew and W. H. Weedon, Microwave Opt. Technol. Lett. 7, 599 (1994).
[CrossRef]

1992 (1)

P. Benech, D. A. Khalil, and F. S. Andre, Opt. Commun. 88, 96 (1992).
[CrossRef]

1985 (1)

A. K. Ghatak, Opt. Quantum Electron. 17, 311 (1985).
[CrossRef]

1976 (1)

Ahn, D.

Andre, F. S.

P. Benech, D. A. Khalil, and F. S. Andre, Opt. Commun. 88, 96 (1992).
[CrossRef]

Baets, R.

Y. Li, L. Li, B. Tian, G. Roelkens, and R. Baets, IEEE Photon. Technol. Lett. 25, 1195 (2013).
[CrossRef]

J. Ctyroky, S. Helfert, R. Pregla, P. Bienstman, R. Baets, R. De Ridder, R. Stoffer, G. Klaasse, J. Petracek, P. Lalanne, J. P. Hugonin, and R. M. De La Rue, Opt. Quantum Electron. 34, 455 (2002).

P. Bienstman and R. Baets, Opt. Quantum Electron. 33, 327 (2001).
[CrossRef]

P. Bienstman, H. Derudder, R. Baets, F. Olyslager, and D. De Zutter, IEEE Trans. Microwave Theory Tech. 49, 349 (2001).
[CrossRef]

Beals, M.

Benech, P.

P. Benech, D. A. Khalil, and F. S. Andre, Opt. Commun. 88, 96 (1992).
[CrossRef]

Bienstman, P.

J. Ctyroky, S. Helfert, R. Pregla, P. Bienstman, R. Baets, R. De Ridder, R. Stoffer, G. Klaasse, J. Petracek, P. Lalanne, J. P. Hugonin, and R. M. De La Rue, Opt. Quantum Electron. 34, 455 (2002).

P. Bienstman, H. Derudder, R. Baets, F. Olyslager, and D. De Zutter, IEEE Trans. Microwave Theory Tech. 49, 349 (2001).
[CrossRef]

P. Bienstman and R. Baets, Opt. Quantum Electron. 33, 327 (2001).
[CrossRef]

Chen, J.

Chew, W. C.

W. C. Chew, J. M. Jin, and E. Michielssen, Microwave Opt. Technol. Lett. 15, 363 (1997).
[CrossRef]

W. C. Chew and W. H. Weedon, Microwave Opt. Technol. Lett. 7, 599 (1994).
[CrossRef]

Chiou, Y. P.

Chung, Y. C.

S. L. Lee, Y. C. Chung, L. A. Coldren, and N. Dagli, IEEE J. Quantum Electron. 31, 1790 (1995).
[CrossRef]

Coldren, L. A.

S. L. Lee, Y. C. Chung, L. A. Coldren, and N. Dagli, IEEE J. Quantum Electron. 31, 1790 (1995).
[CrossRef]

Ctyroky, J.

J. Ctyroky, S. Helfert, R. Pregla, P. Bienstman, R. Baets, R. De Ridder, R. Stoffer, G. Klaasse, J. Petracek, P. Lalanne, J. P. Hugonin, and R. M. De La Rue, Opt. Quantum Electron. 34, 455 (2002).

Dagli, N.

S. L. Lee, Y. C. Chung, L. A. Coldren, and N. Dagli, IEEE J. Quantum Electron. 31, 1790 (1995).
[CrossRef]

De La Rue, R. M.

J. Ctyroky, S. Helfert, R. Pregla, P. Bienstman, R. Baets, R. De Ridder, R. Stoffer, G. Klaasse, J. Petracek, P. Lalanne, J. P. Hugonin, and R. M. De La Rue, Opt. Quantum Electron. 34, 455 (2002).

De Ridder, R.

J. Ctyroky, S. Helfert, R. Pregla, P. Bienstman, R. Baets, R. De Ridder, R. Stoffer, G. Klaasse, J. Petracek, P. Lalanne, J. P. Hugonin, and R. M. De La Rue, Opt. Quantum Electron. 34, 455 (2002).

De Zutter, D.

H. Rogier and D. De Zutter, J. Lightwave Technol. 20, 1141 (2002).
[CrossRef]

H. Rogier and D. De Zutter, IEEE Trans. Microwave Theor. Tech. 49, 712 (2001).
[CrossRef]

P. Bienstman, H. Derudder, R. Baets, F. Olyslager, and D. De Zutter, IEEE Trans. Microwave Theory Tech. 49, 349 (2001).
[CrossRef]

H. Derudder, F. Olyslager, D. De Zutter, and S. Van den Berghe, IEEE Trans. Antennas Propag. 49, 185 (2001).
[CrossRef]

H. Derudder, D. De Zutter, and F. Olyslager, Electron. Lett. 34, 2138 (1998).
[CrossRef]

Derudder, H.

P. Bienstman, H. Derudder, R. Baets, F. Olyslager, and D. De Zutter, IEEE Trans. Microwave Theory Tech. 49, 349 (2001).
[CrossRef]

H. Derudder, F. Olyslager, D. De Zutter, and S. Van den Berghe, IEEE Trans. Antennas Propag. 49, 185 (2001).
[CrossRef]

H. Derudder, D. De Zutter, and F. Olyslager, Electron. Lett. 34, 2138 (1998).
[CrossRef]

Du, C. H.

Ghatak, A. K.

A. K. Ghatak, Opt. Quantum Electron. 17, 311 (1985).
[CrossRef]

Giziewicz, W.

Gong, Z. J.

Guo, G. C.

Hang, Z. F.

Helfert, S.

J. Ctyroky, S. Helfert, R. Pregla, P. Bienstman, R. Baets, R. De Ridder, R. Stoffer, G. Klaasse, J. Petracek, P. Lalanne, J. P. Hugonin, and R. M. De La Rue, Opt. Quantum Electron. 34, 455 (2002).

Hong, C. Y.

Huang, W. P.

Hugonin, J. P.

J. Ctyroky, S. Helfert, R. Pregla, P. Bienstman, R. Baets, R. De Ridder, R. Stoffer, G. Klaasse, J. Petracek, P. Lalanne, J. P. Hugonin, and R. M. De La Rue, Opt. Quantum Electron. 34, 455 (2002).

Jin, J. M.

W. C. Chew, J. M. Jin, and E. Michielssen, Microwave Opt. Technol. Lett. 15, 363 (1997).
[CrossRef]

Kärtner, F. X.

Khalatpour, A.

Khalil, D. A.

P. Benech, D. A. Khalil, and F. S. Andre, Opt. Commun. 88, 96 (1992).
[CrossRef]

Kimerling, L. C.

Klaasse, G.

J. Ctyroky, S. Helfert, R. Pregla, P. Bienstman, R. Baets, R. De Ridder, R. Stoffer, G. Klaasse, J. Petracek, P. Lalanne, J. P. Hugonin, and R. M. De La Rue, Opt. Quantum Electron. 34, 455 (2002).

Lalanne, P.

J. Ctyroky, S. Helfert, R. Pregla, P. Bienstman, R. Baets, R. De Ridder, R. Stoffer, G. Klaasse, J. Petracek, P. Lalanne, J. P. Hugonin, and R. M. De La Rue, Opt. Quantum Electron. 34, 455 (2002).

Lee, S. L.

S. L. Lee, Y. C. Chung, L. A. Coldren, and N. Dagli, IEEE J. Quantum Electron. 31, 1790 (1995).
[CrossRef]

Li, L.

Y. Li, L. Li, B. Tian, G. Roelkens, and R. Baets, IEEE Photon. Technol. Lett. 25, 1195 (2013).
[CrossRef]

Li, X.

Li, Y.

Y. Li, L. Li, B. Tian, G. Roelkens, and R. Baets, IEEE Photon. Technol. Lett. 25, 1195 (2013).
[CrossRef]

Liu, J.

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1983).

Lu, Y. C.

Lu, Y. Y.

Luan, F.

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, 1991).

Michel, J.

Michielssen, E.

W. C. Chew, J. M. Jin, and E. Michielssen, Microwave Opt. Technol. Lett. 15, 363 (1997).
[CrossRef]

Moussakhani, K.

Mu, J.

Olyslager, F.

H. Derudder, F. Olyslager, D. De Zutter, and S. Van den Berghe, IEEE Trans. Antennas Propag. 49, 185 (2001).
[CrossRef]

P. Bienstman, H. Derudder, R. Baets, F. Olyslager, and D. De Zutter, IEEE Trans. Microwave Theory Tech. 49, 349 (2001).
[CrossRef]

H. Derudder, D. De Zutter, and F. Olyslager, Electron. Lett. 34, 2138 (1998).
[CrossRef]

Petracek, J.

J. Ctyroky, S. Helfert, R. Pregla, P. Bienstman, R. Baets, R. De Ridder, R. Stoffer, G. Klaasse, J. Petracek, P. Lalanne, J. P. Hugonin, and R. M. De La Rue, Opt. Quantum Electron. 34, 455 (2002).

J. Petracek and K. Singh, IEEE Photon. Technol. Lett. 14, 810 (2002).
[CrossRef]

Pregla, R.

J. Ctyroky, S. Helfert, R. Pregla, P. Bienstman, R. Baets, R. De Ridder, R. Stoffer, G. Klaasse, J. Petracek, P. Lalanne, J. P. Hugonin, and R. M. De La Rue, Opt. Quantum Electron. 34, 455 (2002).

Roelkens, G.

Y. Li, L. Li, B. Tian, G. Roelkens, and R. Baets, IEEE Photon. Technol. Lett. 25, 1195 (2013).
[CrossRef]

Rogier, H.

H. Rogier and D. De Zutter, J. Lightwave Technol. 20, 1141 (2002).
[CrossRef]

H. Rogier and D. De Zutter, IEEE Trans. Microwave Theor. Tech. 49, 712 (2001).
[CrossRef]

Sammut, R.

Singh, K.

J. Petracek and K. Singh, IEEE Photon. Technol. Lett. 14, 810 (2002).
[CrossRef]

Snyder, A. W.

R. Sammut and A. W. Snyder, Appl. Opt. 15, 1040 (1976).
[CrossRef]

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1983).

Stoffer, R.

J. Ctyroky, S. Helfert, R. Pregla, P. Bienstman, R. Baets, R. De Ridder, R. Stoffer, G. Klaasse, J. Petracek, P. Lalanne, J. P. Hugonin, and R. M. De La Rue, Opt. Quantum Electron. 34, 455 (2002).

Sun, F. J.

Tian, B.

Y. Li, L. Li, B. Tian, G. Roelkens, and R. Baets, IEEE Photon. Technol. Lett. 25, 1195 (2013).
[CrossRef]

Van den Berghe, S.

H. Derudder, F. Olyslager, D. De Zutter, and S. Van den Berghe, IEEE Trans. Antennas Propag. 49, 185 (2001).
[CrossRef]

Weedon, W. H.

W. C. Chew and W. H. Weedon, Microwave Opt. Technol. Lett. 7, 599 (1994).
[CrossRef]

Xia, Y.

Xue, L. L.

Yang, L.

Zhang, H.

Zhou, Y.

Zhu, J.

Zou, C. L.

Appl. Opt. (1)

Electron. Lett. (1)

H. Derudder, D. De Zutter, and F. Olyslager, Electron. Lett. 34, 2138 (1998).
[CrossRef]

IEEE J. Quantum Electron. (1)

S. L. Lee, Y. C. Chung, L. A. Coldren, and N. Dagli, IEEE J. Quantum Electron. 31, 1790 (1995).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

J. Petracek and K. Singh, IEEE Photon. Technol. Lett. 14, 810 (2002).
[CrossRef]

Y. Li, L. Li, B. Tian, G. Roelkens, and R. Baets, IEEE Photon. Technol. Lett. 25, 1195 (2013).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

H. Derudder, F. Olyslager, D. De Zutter, and S. Van den Berghe, IEEE Trans. Antennas Propag. 49, 185 (2001).
[CrossRef]

IEEE Trans. Microwave Theor. Tech. (1)

H. Rogier and D. De Zutter, IEEE Trans. Microwave Theor. Tech. 49, 712 (2001).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

P. Bienstman, H. Derudder, R. Baets, F. Olyslager, and D. De Zutter, IEEE Trans. Microwave Theory Tech. 49, 349 (2001).
[CrossRef]

J. Lightwave Technol. (4)

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

Microwave Opt. Technol. Lett. (2)

W. C. Chew and W. H. Weedon, Microwave Opt. Technol. Lett. 7, 599 (1994).
[CrossRef]

W. C. Chew, J. M. Jin, and E. Michielssen, Microwave Opt. Technol. Lett. 15, 363 (1997).
[CrossRef]

Opt. Commun. (1)

P. Benech, D. A. Khalil, and F. S. Andre, Opt. Commun. 88, 96 (1992).
[CrossRef]

Opt. Express (8)

Opt. Quantum Electron. (3)

J. Ctyroky, S. Helfert, R. Pregla, P. Bienstman, R. Baets, R. De Ridder, R. Stoffer, G. Klaasse, J. Petracek, P. Lalanne, J. P. Hugonin, and R. M. De La Rue, Opt. Quantum Electron. 34, 455 (2002).

A. K. Ghatak, Opt. Quantum Electron. 17, 311 (1985).
[CrossRef]

P. Bienstman and R. Baets, Opt. Quantum Electron. 33, 327 (2001).
[CrossRef]

Other (3)

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, 1991).

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1983).

FDTD Solutions from Lumerical Solutions, Inc.

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Figures (4)

Fig. 1.
Fig. 1.

Multilayer waveguide structure with semi-infinite substrate and cladding truncated by PML and PRB. ai+ and ai are amplitude coefficients of the forward (+) and backward () waves at the incident interface of the ith layer, respectively. bi+ and bi are the amplitude coefficients of the forward (+) and backward () waves at the exit interface of the ith layer, respectively.

Fig. 2.
Fig. 2.

Integrated Germanium photodetector [5]. TE light is launched from a SiN slab waveguide and is coupled into a Germanium photodetector.

Fig. 3.
Fig. 3.

Modal spectrum of SiN slab on Ge. (a) Modal spectrum using UOPD, in which phase angles of C-Berenger modes differ significantly from those of S-Berenger modes. (b) Modal spectrum using WOPD, in which phase angles of C-Berenger and S-Berenger modes converge. (c) Electric field patterns of the 81st and 82nd modes using UOPD. (d) Electric field patterns of the 81st and 82nd modes using WOPD.

Fig. 4.
Fig. 4.

Relative field errors as functions of the total number of modes involved in CMMM. CMMM in the WOPD scheme shows smaller relative field error and fluctuations. Red diamonds, results from CMMM with WOPD; black squares, results from CMMM with UOPD.

Equations (25)

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

ϕm(x)=ai+ejκi(xxi)+aiejκi(xxi),κi=k0ni2neff,m2.
Bi=PiAi.
Pi=[ejκidi00ejκidi],
DiBi=Di+1Ai+1,
Di=[11ηiηi]andηi={κiforTEκi/ni2forTM.
AN+1=DN+11i=N1(DiPiDi1)D0B0=TB0.
DiPiDi1=1ηi[ηicos(κidi)jsin(κidi)jηi2sin(κidi)ηicos(κidi)].
limm|neff,m|=,limmθm=θ0.
κidi=k0|neff,mdi|ej(π2θm)1ni2/neff,m2k0|neff,mdi|[sin(θ0)+jcos(θ0)].
DiPiDi1=ejκidi2ηi[1ηi][ηi1].
T=eji=1Nκidii=0Nηi+1+ηi2ηi+1[1ηN+1ηNηN+1+ηN][1η1η0η1+η0].
ϕ(x)={aN+1+ejκN+1(x˜(x)x˜(xN+1))+aN+1ejκN+1(x˜(x)x˜(xN+1)),xN+1<x<xPMLcb0+ejκ0(x˜(x)x˜(x1))+b0ejκ0(x˜(x)x˜(x1)),xPMLs<x<x1,
x˜=0xs(τ)dτ.
s(x)={1j3λ4πnρ2dPML3ln(1R)inPML1otherwise,
d˜s,c=ds,cjλ4πns,cln1Rs,c=|d˜s,c|ejδs,c.
ϕ(xPMLc)=aN+1+ejκN+1d˜c+aN+1ejκN+1d˜c=0,
ϕ(xPMLs)=b0+ejκ0d˜s+b0ejκ0d˜s=0.
(η1+jη0cot(κ0d˜s))(ηN+jηN+1cot(κN+1d˜c))=0,
η1+jη0cot(κ0d˜s)=0,
ηN+jηN+1cot(κN+1d˜c)=0.
κ0d˜sk0|neff,md˜s|ej(θmπ2+δs).
tanθ0s=cotδs=2k0φsln(Rs),
tanθ0c=cotδc=2k0φcln(Rc),
φsln(Rs)=φcln(Rc).
ΔF(N)=x|Ey,z=0,MMM(N)Ey,z=0,FDTD|dxx|Eexcitation|dx.

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