Abstract

Optical devices with a slot configuration offer the distinct feature of strong electric field confinement in a low refractive index region and are, therefore, of considerable interest in many applications. In this work we investigate light propagation in a waveguide-resonator system where the resonators consist of slotted ring cavities. Owing to the presence of curved material interfaces and the vastly different length scales associated with the sub-wavelength sized slots and the waveguide-resonator coupling regions on the one hand, and the spatial extent of the ring on the other hand, this prototypical system provides significant challenges to both direct numerical solvers and semi-analytical approaches. We address these difficulties by modeling the slot resonators via a frequency-domain spatial Coupled-Mode Theory (CMT) approach, and compare its results with a Discontinuous Galerkin Time-Domain (DGTD) solver that is equipped with curvilinear finite elements. In particular, the CMT model is built on the underlying physical properties of the slotted resonators, and turns out to be quite efficient for analyzing the device characteristics. We also discuss the advantages and limitations of the CMT approach by comparing the results with the numerically exact solutions obtained by the DGTD solver. Besides providing considerable physical insight, the CMT model thus forms a convenient basis for the efficient analysis of more complex systems with slotted resonators such as entire arrays of waveguide-coupled resonators and systems with strongly nonlinear optical properties.

© 2011 OSA

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    [CrossRef]

2010 (1)

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181(3), 687–702 (2010).
[CrossRef]

2009 (5)

J. Niegemann, M. König, K. Stannigel, and K. Busch, “Higher-order time-domain methods for the analysis of nano-photonic systems,” Photon. Nanostr.: Fundam. Appl. 7(1), 2–11 (2009).
[CrossRef]

J. Niegemann, W. Pernice, and K. Busch, “Simulation of optical resonator using DGTD and FDTD,” IOP J. Opt. A: Pure Appl. Opt. 11, 114015:1–10 (2009).
[CrossRef]

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 2(3), 216–219 (2009).
[CrossRef]

K. Stannigel, M. König, J. Niegemann, and K. Busch, “Discontinuous Galerkin time-domain computations of metallic nanostructures,, Opt. Express 17(17), 14934–14947 (2009).
[CrossRef] [PubMed]

K. R. Hiremath, “Analytical modal analysis of bent slot waveguides,” J. Opt. Soc. Am. A 26(11), 2321–2326 (2009).
[CrossRef]

2006 (2)

A. M. Armani and K. J. Vahala, “Heavy water detection using ultra-high-Q microcavities,” Opt. Lett. 31(12), 1896–1898 (2006).
[CrossRef] [PubMed]

K. R. Hiremath, R. Stoffer, and M. Hammer, “Modeling of circular integrated optical microresonators by 2-D frequency domain coupled mode theory,” Opt. Commun. 257, 277–297 (2006).
[CrossRef]

2005 (5)

K. R. Hiremath, M. Hammer, S. Stoffer, L. Prkna, and J. Čtyroký, “Analytic approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37(1–3), 37–61 (2005).
[CrossRef]

T. Baehr-Jones, M. Hochberg, C. Walker, and A. Scherer, “High-Q optical resonators in silicon-on-insulator-based slot waveguides,” Appl. Phys. Lett. 86, 081101 (2005).
[CrossRef]

R. Stoffer, K. R. Hiremath, M. Hammer, L. Prkna, and J. Čtyroký, “Cylindrical integrated optical microresonators: Modeling by 3-D vectorial coupled mode theory,” Opt. Commun. 256, 46–67 (2005).
[CrossRef]

T. Baehr-Jones, M. Hochberg, G. Wang, R. Lawson, Y. Liao, P. Sullivan, L. Dalton, A. Jen, and A. Scherer, “Optical modulation and detection in slotted silicon waveguides,” Opt. Express 13(14), 5216–5226 (2005).
[CrossRef] [PubMed]

X. Ji, T. Lu, W. Cai, and P. Zhang, “Discontinuous Galerkin Time Domain (DGTD) methods for the study of 2-D waveguide-coupled microring resonators,” J. Lightwave Technol. 23(11), 3864–3874 (2005).
[CrossRef]

2004 (2)

S. Pereira, P. Chak, S. E. Sipe, L. Tkeshelashvili, and K. Busch, “All-optical diode in an asymmetrically apodized kerr nonlinear microresonator system,” Photon. Nanostr.: Fundam. Appl. 2, 181–190 (2004).
[CrossRef]

V. R. Almeida, Q. Xu, C. A. Barios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29(11), 1209–1211 (2004).
[CrossRef] [PubMed]

2003 (1)

2002 (1)

H. G. Rabus, M. Hamacher, U. Troppenz, and H. Heidrich, “High-Q channel-dropping filters using ring resonators with integrated SOAs,” IEEE Photon. Technol. Lett. 14(10), 1442–1444 (2002).
[CrossRef]

2000 (1)

J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P.-T. Ho, “Higher order filter response in coupled microring resonators,” IEEE Photon. Technol. Lett. 12(3), 320–322 (2000).
[CrossRef]

1999 (1)

1997 (1)

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[CrossRef]

Absil, P. P.

J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P.-T. Ho, “Higher order filter response in coupled microring resonators,” IEEE Photon. Technol. Lett. 12(3), 320–322 (2000).
[CrossRef]

Almeida, V. R.

Armani, A. M.

Baehr-Jones, T.

T. Baehr-Jones, M. Hochberg, C. Walker, and A. Scherer, “High-Q optical resonators in silicon-on-insulator-based slot waveguides,” Appl. Phys. Lett. 86, 081101 (2005).
[CrossRef]

T. Baehr-Jones, M. Hochberg, G. Wang, R. Lawson, Y. Liao, P. Sullivan, L. Dalton, A. Jen, and A. Scherer, “Optical modulation and detection in slotted silicon waveguides,” Opt. Express 13(14), 5216–5226 (2005).
[CrossRef] [PubMed]

Baets, R.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 2(3), 216–219 (2009).
[CrossRef]

Barios, C. A.

Bermel, P.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181(3), 687–702 (2010).
[CrossRef]

Biaggio, I.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 2(3), 216–219 (2009).
[CrossRef]

Bogaerts, W.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 2(3), 216–219 (2009).
[CrossRef]

Brener, I.

Bruce, A. J.

Busch, K.

K. Stannigel, M. König, J. Niegemann, and K. Busch, “Discontinuous Galerkin time-domain computations of metallic nanostructures,, Opt. Express 17(17), 14934–14947 (2009).
[CrossRef] [PubMed]

J. Niegemann, M. König, K. Stannigel, and K. Busch, “Higher-order time-domain methods for the analysis of nano-photonic systems,” Photon. Nanostr.: Fundam. Appl. 7(1), 2–11 (2009).
[CrossRef]

J. Niegemann, W. Pernice, and K. Busch, “Simulation of optical resonator using DGTD and FDTD,” IOP J. Opt. A: Pure Appl. Opt. 11, 114015:1–10 (2009).
[CrossRef]

S. Pereira, P. Chak, S. E. Sipe, L. Tkeshelashvili, and K. Busch, “All-optical diode in an asymmetrically apodized kerr nonlinear microresonator system,” Photon. Nanostr.: Fundam. Appl. 2, 181–190 (2004).
[CrossRef]

Cai, W.

Capuzzo, M. A.

Chak, P.

S. Pereira, P. Chak, S. E. Sipe, L. Tkeshelashvili, and K. Busch, “All-optical diode in an asymmetrically apodized kerr nonlinear microresonator system,” Photon. Nanostr.: Fundam. Appl. 2, 181–190 (2004).
[CrossRef]

Chu, S. T.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[CrossRef]

Costa, R.

Ctyroký, J.

K. R. Hiremath, M. Hammer, S. Stoffer, L. Prkna, and J. Čtyroký, “Analytic approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37(1–3), 37–61 (2005).
[CrossRef]

R. Stoffer, K. R. Hiremath, M. Hammer, L. Prkna, and J. Čtyroký, “Cylindrical integrated optical microresonators: Modeling by 3-D vectorial coupled mode theory,” Opt. Commun. 256, 46–67 (2005).
[CrossRef]

Dalton, L.

Diederich, F.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 2(3), 216–219 (2009).
[CrossRef]

Dumon, P.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 2(3), 216–219 (2009).
[CrossRef]

Esembeson, B.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 2(3), 216–219 (2009).
[CrossRef]

Foresi, J.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[CrossRef]

Freude, W.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 2(3), 216–219 (2009).
[CrossRef]

Ghatak, A.

A. Ghatak and K. Thyagarajan, Optical Electronics (Cambridge University Press, 1993).

Gomez, L. T.

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite Difference Time Domain Method , 3rd ed. (Artech House Inc., 2005).

Hamacher, M.

H. G. Rabus, M. Hamacher, U. Troppenz, and H. Heidrich, “High-Q channel-dropping filters using ring resonators with integrated SOAs,” IEEE Photon. Technol. Lett. 14(10), 1442–1444 (2002).
[CrossRef]

Hammer, M.

K. R. Hiremath, R. Stoffer, and M. Hammer, “Modeling of circular integrated optical microresonators by 2-D frequency domain coupled mode theory,” Opt. Commun. 257, 277–297 (2006).
[CrossRef]

R. Stoffer, K. R. Hiremath, M. Hammer, L. Prkna, and J. Čtyroký, “Cylindrical integrated optical microresonators: Modeling by 3-D vectorial coupled mode theory,” Opt. Commun. 256, 46–67 (2005).
[CrossRef]

K. R. Hiremath, M. Hammer, S. Stoffer, L. Prkna, and J. Čtyroký, “Analytic approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37(1–3), 37–61 (2005).
[CrossRef]

M. Hammer, K. R. Hiremath, and R. Stoffer, “Analytical approaches to the description of optical microresonator devices,” in F. Michelotti, A. Driessen, and M. Bertolotti, editors, Microresonators as building blocks for VLSI photonics , volume 709 of AIP conference proceedings, pages 48–71. American Institute of Physics, Melville, New York (2004).

Haus, H. A.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[CrossRef]

Heidrich, H.

H. G. Rabus, M. Hamacher, U. Troppenz, and H. Heidrich, “High-Q channel-dropping filters using ring resonators with integrated SOAs,” IEEE Photon. Technol. Lett. 14(10), 1442–1444 (2002).
[CrossRef]

Hesthaven, J. S.

J. S. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications , Number 54 in Texts in Applied Mathematics, (Springer-Verlag, 2007).

Hiremath, K. R.

K. R. Hiremath, “Analytical modal analysis of bent slot waveguides,” J. Opt. Soc. Am. A 26(11), 2321–2326 (2009).
[CrossRef]

K. R. Hiremath, R. Stoffer, and M. Hammer, “Modeling of circular integrated optical microresonators by 2-D frequency domain coupled mode theory,” Opt. Commun. 257, 277–297 (2006).
[CrossRef]

R. Stoffer, K. R. Hiremath, M. Hammer, L. Prkna, and J. Čtyroký, “Cylindrical integrated optical microresonators: Modeling by 3-D vectorial coupled mode theory,” Opt. Commun. 256, 46–67 (2005).
[CrossRef]

K. R. Hiremath, M. Hammer, S. Stoffer, L. Prkna, and J. Čtyroký, “Analytic approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37(1–3), 37–61 (2005).
[CrossRef]

M. Hammer, K. R. Hiremath, and R. Stoffer, “Analytical approaches to the description of optical microresonator devices,” in F. Michelotti, A. Driessen, and M. Bertolotti, editors, Microresonators as building blocks for VLSI photonics , volume 709 of AIP conference proceedings, pages 48–71. American Institute of Physics, Melville, New York (2004).

Ho, P.-T.

J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P.-T. Ho, “Higher order filter response in coupled microring resonators,” IEEE Photon. Technol. Lett. 12(3), 320–322 (2000).
[CrossRef]

Hochberg, M.

T. Baehr-Jones, M. Hochberg, C. Walker, and A. Scherer, “High-Q optical resonators in silicon-on-insulator-based slot waveguides,” Appl. Phys. Lett. 86, 081101 (2005).
[CrossRef]

T. Baehr-Jones, M. Hochberg, G. Wang, R. Lawson, Y. Liao, P. Sullivan, L. Dalton, A. Jen, and A. Scherer, “Optical modulation and detection in slotted silicon waveguides,” Opt. Express 13(14), 5216–5226 (2005).
[CrossRef] [PubMed]

Hryniewicz, J. V.

J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P.-T. Ho, “Higher order filter response in coupled microring resonators,” IEEE Photon. Technol. Lett. 12(3), 320–322 (2000).
[CrossRef]

Ibanescu, M.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181(3), 687–702 (2010).
[CrossRef]

Jen, A.

Ji, X.

Jin, J.

J. Jin, The Finite Element Method in Electromagnetics , 2nd ed. (Wiley-Interscience Publication, 2002).

Joannopoulos, J. D.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181(3), 687–702 (2010).
[CrossRef]

Johnson, S. G.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181(3), 687–702 (2010).
[CrossRef]

König, M.

J. Niegemann, M. König, K. Stannigel, and K. Busch, “Higher-order time-domain methods for the analysis of nano-photonic systems,” Photon. Nanostr.: Fundam. Appl. 7(1), 2–11 (2009).
[CrossRef]

K. Stannigel, M. König, J. Niegemann, and K. Busch, “Discontinuous Galerkin time-domain computations of metallic nanostructures,, Opt. Express 17(17), 14934–14947 (2009).
[CrossRef] [PubMed]

Koos, C.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 2(3), 216–219 (2009).
[CrossRef]

Laine, J. P.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[CrossRef]

Lawson, R.

Lenz, G.

Leuthold, J.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 2(3), 216–219 (2009).
[CrossRef]

Liao, Y.

Lipson, M.

Little, B. E.

J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P.-T. Ho, “Higher order filter response in coupled microring resonators,” IEEE Photon. Technol. Lett. 12(3), 320–322 (2000).
[CrossRef]

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[CrossRef]

Lu, T.

Madsen, C. K.

Martinelli, M.

Melloni, A.

Michinobu, T.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 2(3), 216–219 (2009).
[CrossRef]

Monguzzi, P.

Niegemann, J.

K. Stannigel, M. König, J. Niegemann, and K. Busch, “Discontinuous Galerkin time-domain computations of metallic nanostructures,, Opt. Express 17(17), 14934–14947 (2009).
[CrossRef] [PubMed]

J. Niegemann, W. Pernice, and K. Busch, “Simulation of optical resonator using DGTD and FDTD,” IOP J. Opt. A: Pure Appl. Opt. 11, 114015:1–10 (2009).
[CrossRef]

J. Niegemann, M. König, K. Stannigel, and K. Busch, “Higher-order time-domain methods for the analysis of nano-photonic systems,” Photon. Nanostr.: Fundam. Appl. 7(1), 2–11 (2009).
[CrossRef]

Nielsen, T. N.

Oskooi, A. F.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181(3), 687–702 (2010).
[CrossRef]

Pan, G. W.

G. W. Pan, Wavelets in Electromagnetics and Device Modeling , Microwave and Optical engineering, (Wiley Interscience, 2003).
[CrossRef]

Pereira, S.

S. Pereira, P. Chak, S. E. Sipe, L. Tkeshelashvili, and K. Busch, “All-optical diode in an asymmetrically apodized kerr nonlinear microresonator system,” Photon. Nanostr.: Fundam. Appl. 2, 181–190 (2004).
[CrossRef]

Pernice, W.

J. Niegemann, W. Pernice, and K. Busch, “Simulation of optical resonator using DGTD and FDTD,” IOP J. Opt. A: Pure Appl. Opt. 11, 114015:1–10 (2009).
[CrossRef]

Prkna, L.

K. R. Hiremath, M. Hammer, S. Stoffer, L. Prkna, and J. Čtyroký, “Analytic approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37(1–3), 37–61 (2005).
[CrossRef]

R. Stoffer, K. R. Hiremath, M. Hammer, L. Prkna, and J. Čtyroký, “Cylindrical integrated optical microresonators: Modeling by 3-D vectorial coupled mode theory,” Opt. Commun. 256, 46–67 (2005).
[CrossRef]

Rabus, H. G.

H. G. Rabus, M. Hamacher, U. Troppenz, and H. Heidrich, “High-Q channel-dropping filters using ring resonators with integrated SOAs,” IEEE Photon. Technol. Lett. 14(10), 1442–1444 (2002).
[CrossRef]

Roundy, D.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181(3), 687–702 (2010).
[CrossRef]

Scherer, A.

T. Baehr-Jones, M. Hochberg, C. Walker, and A. Scherer, “High-Q optical resonators in silicon-on-insulator-based slot waveguides,” Appl. Phys. Lett. 86, 081101 (2005).
[CrossRef]

T. Baehr-Jones, M. Hochberg, G. Wang, R. Lawson, Y. Liao, P. Sullivan, L. Dalton, A. Jen, and A. Scherer, “Optical modulation and detection in slotted silicon waveguides,” Opt. Express 13(14), 5216–5226 (2005).
[CrossRef] [PubMed]

Sipe, S. E.

S. Pereira, P. Chak, S. E. Sipe, L. Tkeshelashvili, and K. Busch, “All-optical diode in an asymmetrically apodized kerr nonlinear microresonator system,” Photon. Nanostr.: Fundam. Appl. 2, 181–190 (2004).
[CrossRef]

Stannigel, K.

K. Stannigel, M. König, J. Niegemann, and K. Busch, “Discontinuous Galerkin time-domain computations of metallic nanostructures,, Opt. Express 17(17), 14934–14947 (2009).
[CrossRef] [PubMed]

J. Niegemann, M. König, K. Stannigel, and K. Busch, “Higher-order time-domain methods for the analysis of nano-photonic systems,” Photon. Nanostr.: Fundam. Appl. 7(1), 2–11 (2009).
[CrossRef]

Stoffer, R.

K. R. Hiremath, R. Stoffer, and M. Hammer, “Modeling of circular integrated optical microresonators by 2-D frequency domain coupled mode theory,” Opt. Commun. 257, 277–297 (2006).
[CrossRef]

R. Stoffer, K. R. Hiremath, M. Hammer, L. Prkna, and J. Čtyroký, “Cylindrical integrated optical microresonators: Modeling by 3-D vectorial coupled mode theory,” Opt. Commun. 256, 46–67 (2005).
[CrossRef]

M. Hammer, K. R. Hiremath, and R. Stoffer, “Analytical approaches to the description of optical microresonator devices,” in F. Michelotti, A. Driessen, and M. Bertolotti, editors, Microresonators as building blocks for VLSI photonics , volume 709 of AIP conference proceedings, pages 48–71. American Institute of Physics, Melville, New York (2004).

Stoffer, S.

K. R. Hiremath, M. Hammer, S. Stoffer, L. Prkna, and J. Čtyroký, “Analytic approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37(1–3), 37–61 (2005).
[CrossRef]

Sullivan, P.

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite Difference Time Domain Method , 3rd ed. (Artech House Inc., 2005).

Thyagarajan, K.

A. Ghatak and K. Thyagarajan, Optical Electronics (Cambridge University Press, 1993).

Tkeshelashvili, L.

S. Pereira, P. Chak, S. E. Sipe, L. Tkeshelashvili, and K. Busch, “All-optical diode in an asymmetrically apodized kerr nonlinear microresonator system,” Photon. Nanostr.: Fundam. Appl. 2, 181–190 (2004).
[CrossRef]

Troppenz, U.

H. G. Rabus, M. Hamacher, U. Troppenz, and H. Heidrich, “High-Q channel-dropping filters using ring resonators with integrated SOAs,” IEEE Photon. Technol. Lett. 14(10), 1442–1444 (2002).
[CrossRef]

Vahala, K. J.

Vallaitis, T.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 2(3), 216–219 (2009).
[CrossRef]

Vorreau, P.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 2(3), 216–219 (2009).
[CrossRef]

Walker, C.

T. Baehr-Jones, M. Hochberg, C. Walker, and A. Scherer, “High-Q optical resonators in silicon-on-insulator-based slot waveguides,” Appl. Phys. Lett. 86, 081101 (2005).
[CrossRef]

Wang, G.

Warburton, T.

J. S. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications , Number 54 in Texts in Applied Mathematics, (Springer-Verlag, 2007).

Wilson, R. A.

J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P.-T. Ho, “Higher order filter response in coupled microring resonators,” IEEE Photon. Technol. Lett. 12(3), 320–322 (2000).
[CrossRef]

Xu, Q.

Zhang, P.

Appl. Phys. Lett. (1)

T. Baehr-Jones, M. Hochberg, C. Walker, and A. Scherer, “High-Q optical resonators in silicon-on-insulator-based slot waveguides,” Appl. Phys. Lett. 86, 081101 (2005).
[CrossRef]

Comput. Phys. Commun. (1)

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181(3), 687–702 (2010).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P.-T. Ho, “Higher order filter response in coupled microring resonators,” IEEE Photon. Technol. Lett. 12(3), 320–322 (2000).
[CrossRef]

H. G. Rabus, M. Hamacher, U. Troppenz, and H. Heidrich, “High-Q channel-dropping filters using ring resonators with integrated SOAs,” IEEE Photon. Technol. Lett. 14(10), 1442–1444 (2002).
[CrossRef]

IOP J. Opt. A: Pure Appl. Opt. (1)

J. Niegemann, W. Pernice, and K. Busch, “Simulation of optical resonator using DGTD and FDTD,” IOP J. Opt. A: Pure Appl. Opt. 11, 114015:1–10 (2009).
[CrossRef]

J. Lightwave Technol. (2)

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[CrossRef]

X. Ji, T. Lu, W. Cai, and P. Zhang, “Discontinuous Galerkin Time Domain (DGTD) methods for the study of 2-D waveguide-coupled microring resonators,” J. Lightwave Technol. 23(11), 3864–3874 (2005).
[CrossRef]

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

Nat. Photonics (1)

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 2(3), 216–219 (2009).
[CrossRef]

Opt. Commun. (2)

K. R. Hiremath, R. Stoffer, and M. Hammer, “Modeling of circular integrated optical microresonators by 2-D frequency domain coupled mode theory,” Opt. Commun. 257, 277–297 (2006).
[CrossRef]

R. Stoffer, K. R. Hiremath, M. Hammer, L. Prkna, and J. Čtyroký, “Cylindrical integrated optical microresonators: Modeling by 3-D vectorial coupled mode theory,” Opt. Commun. 256, 46–67 (2005).
[CrossRef]

Opt. Express (2)

Opt. Lett. (4)

Opt. Quantum Electron. (1)

K. R. Hiremath, M. Hammer, S. Stoffer, L. Prkna, and J. Čtyroký, “Analytic approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37(1–3), 37–61 (2005).
[CrossRef]

Photon. Nanostr.: Fundam. Appl. (2)

J. Niegemann, M. König, K. Stannigel, and K. Busch, “Higher-order time-domain methods for the analysis of nano-photonic systems,” Photon. Nanostr.: Fundam. Appl. 7(1), 2–11 (2009).
[CrossRef]

S. Pereira, P. Chak, S. E. Sipe, L. Tkeshelashvili, and K. Busch, “All-optical diode in an asymmetrically apodized kerr nonlinear microresonator system,” Photon. Nanostr.: Fundam. Appl. 2, 181–190 (2004).
[CrossRef]

Other (8)

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite Difference Time Domain Method , 3rd ed. (Artech House Inc., 2005).

J. Jin, The Finite Element Method in Electromagnetics , 2nd ed. (Wiley-Interscience Publication, 2002).

G. W. Pan, Wavelets in Electromagnetics and Device Modeling , Microwave and Optical engineering, (Wiley Interscience, 2003).
[CrossRef]

J. S. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications , Number 54 in Texts in Applied Mathematics, (Springer-Verlag, 2007).

T. Tamir, editor, Integrated Optics (Second Corrected and Updated Edition) , Topics in Applied Physics, vol. 7, (Springer-Verlag, 1982).

A. Ghatak and K. Thyagarajan, Optical Electronics (Cambridge University Press, 1993).

K. R. Hiremath, Coupled mode theory based modeling and analysis of circular optical microresonators , PhD thesis, University of Twente, The Netherlands (2005).

M. Hammer, K. R. Hiremath, and R. Stoffer, “Analytical approaches to the description of optical microresonator devices,” in F. Michelotti, A. Driessen, and M. Bertolotti, editors, Microresonators as building blocks for VLSI photonics , volume 709 of AIP conference proceedings, pages 48–71. American Institute of Physics, Melville, New York (2004).

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Figures (8)

Fig. 1
Fig. 1

Schematics for a slotted-resonator based 4-port device. The resonator consists of a ring of outer radius R, width w tot, and refractive index n c, and contains a low-index slot of width w slot and refractive index n slot < n c. The slot’s position inside the ring is described by the asymmetry parameter η such that the width of the inner high-index ring layer is ηw and that of the outer high-index ring layer is (1 – η)w, where w = w tot – w slot [19]. This slotted resonator is coupled to two identical straight waveguides of width w s and refractive index n s that realize input and output ports. The minimal separation between resonator and the straight waveguides are g and g̃, respectively. The entire device is embedded in a host material with a background index nb and its performance may be characterized via the power levels associated with the input ports, P I and P A (In- and Add-port), and the output ports, P T and P D (Through- and Drop-port), respectively. Within a coupled-mode theoretical approach this device is further decomposed into several functional elements. Two couplers, (I) and (II), delineated with dashed-line boxes, are connected via two identical segments of bent slotted waveguides of length L and (at positions a, b, and ã, b̃, respectively) and each of these couplers is further connected to two identical input and output port waveguides (at positions A, B, and Ã, B̃, respectively).

Fig. 2
Fig. 2

Detailed representation of the bent slotted - straight waveguide couplers delineated in Fig. 1. The coupler is defined via the domain [xl , xr ] × [zi , zo ], in which the interaction between the bent slotted waveguide and the straight waveguide takes place. Outside this coupling region, we assume that the modes of the constituent waveguides are uncoupled. The minimal separation between the waveguides is g and the amplitudes of the input modes at z = zi are denoted by a and A, respectively. The corresponding amplitudes of the output modes at z = z 0 are denoted by b and B, respectively. See Fig. 1 for further details.

Fig. 3
Fig. 3

Dependence of the straight output waveguide’s fundamental mode amplitude B s0 on the coupler length in z-direction when the input straight waveguide is excited with its fundamental mode. These results have been obtained with the ‘naive’ field-matching approach (dashed line) and with the projection correction technique (solid line). The coupler parameters (see Fig. 2) are: n c = n s = 2.1, n slot = n b = 1, w tot = 1 μm, w slot = 0.2 μm, w s = 0.4 μm, λ = 1.55 μm, g= 0.4 μm, η = 0.5, R = 5 μm, [xl , xr ] = [1μm, 8μm], zi = −8μm is fixed, and zo is varied from [−8μm, 8μm]. The numerical computations have been performed with discretizations hx = 0.005 μm and hz = 0.1 μm along the x- and z-directions respectively.

Fig. 4
Fig. 4

Transverse profiles of selected waveguide modes that are used for the CMT-based simulations of the couplers in Sec. 2.3. The first column depicts the real part of Hy and Ex for the straight waveguide TM0 mode. The TM0 modes of the bent slotted waveguides for various values of η are depicted in the second, third and fourth column (see inset). The fifth column depicts the TM1 mode for the bent slotted waveguide for η = 0.5. All these modes are computed at λ = 1.55 μm, and the corresponding effective indices are (columnwise) n eff=1.4335343, 1.53462 i 3.50971 × 10−8, 1.4221 – i 2.92095 × 10−6, 1.48579 – i 8.3777 × 10−9, and 1.19039 – i 1.69572 × 10−3 respectively.

Fig. 5
Fig. 5

Spectral response of the slotted resonator device depicted in Fig.1 for the TM polarization and various slot positions within the ring (see the text for details on the device parameters). The results of the coupled-mode theoretical (CMT) approach are compared with the results of exact numerical computations via a Discontinuous Galerkin Time-Domain (DGTD) method that has been equipped with curvilinear elements.

Fig. 6
Fig. 6

Meshes that have been used for the DGTD computations of the slotted resonator device sketched in Fig. 1. From left to right, the slot position corresponds to η = 0.4, η = 0.5, and η = 0.7 respectively. The computational domain is enclosed by perfectly matched layers as indicated by the finite-width outermost box. In order to determine the spectral response of the device a broad-band pulse for is injected in the upper left waveguide. The flux through the output ports is recorded and subsequently Fourier-transformed.

Fig. 7
Fig. 7

Spectral response of the slotted resonator device depicted in Fig. 1 for a symmetric slot position (η = 0.5) for various minimal separations g and g̃ of the straight waveguides from the slotted resonator (see the text for details on the device parameters). The results of the coupled-mode theoretical (CMT) approach are compared with the results of exact numerical computations via a Discontinuous Galerkin Time-Domain (DGTD) method that has been equipped with curvilinear elements.

Fig. 8
Fig. 8

Resonance field distributions (real part) of the magnetic field Hy of the slotted resonator device depicted in Fig. 1 for various values of the asymmetry parameter η (see the text for details on the device parameters). The three leftmost panels depicts the resonator’s TM0,30 resonances for η = 0.4, η = 0.5, and η = 0.7, respectively. The rightmost panel depicts the resonator’s TM1,25 resonance for η = 0.5. These field distributions have been obtained with the CMT model.

Equations (11)

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

( b B ) = 𝖲 ( a A ) , ( b ˜ B ˜ ) = 𝖲 ˜ ( a ˜ A ˜ ) .
𝖲 = ( 𝖲 bb 𝖲 bs 𝖲 sb 𝖲 ss ) , 𝖲 ˜ = ( 𝖲 ˜ bb 𝖲 ˜ bs 𝖲 ˜ sb 𝖲 ˜ ss ) ,
a = 𝖦 b ˜ and a ˜ = 𝖦 ˜ b ,
B = ( 𝖲 sb 𝖦 𝖲 ˜ bb 𝖦 ˜ Ω 1 𝖲 bs + 𝖲 ss ) A , B ˜ = ( 𝖲 ˜ s b 𝖦 ˜ Ω 1 𝖲 bs ) A , }
( E s q H s q ) ( x , z ) = ( E ˜ s q H ˜ s q ) ( x ) e i β s q z .
( E b p H b p ) ( x , z ) = ( E ˜ b p H ˜ b p ) ( r ( x , z ) ) e i γ b p R θ ( x , z ) ,
( E H ) ( x , z ) v = b , s i = 1 N v C vi ( z ) ( E vi H vi ) ( x , z )
( E , H ) = [ ( × E ) · H * ( × H ) · E * + i ω μ 0 H · H * + i ω ɛ 0 ɛ E · E * ] d x d z .
v = b , s i = 1 N v 𝖬 vi , wj d C vi d z v = b , s i = 1 N v 𝖥 vi , wj C vi = 0 ,
𝖬 ( z ) d C ( z ) d z = 𝖥 ( z ) C ( z ) .
d 𝖳 ( z ) d z = 𝖬 ( z ) 1 𝖥 ( z ) 𝖳 ( z ) ,

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