Abstract

A fast and accurate full-wave technique based on the dual-primal finite element tearing and interconnecting method and the second-order transmission condition is presented for large-scale three-dimensional photonic device simulations. The technique decomposes a general three-dimensional electromagnetic problem into smaller subdomain problems so that parallel computing can be performed on distributed-memory computer clusters to reduce the simulation time significantly. With the electric fields computed everywhere, photonic device parameters such as transmission and reflection coefficients are extracted. Several photonic devices, with simulation volumes up to 1.9×104 (λ/navg)3 and modeled with over one hundred million unknowns, are simulated to demonstrate the application, efficiency, and capability of this technique. The simulations show good agreement with experimental results and in a special case with a simplified two-dimensional simulation.

© 2014 Optical Society of America

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2014 (1)

Y. M. Kang, M. F. Xue, A. Arbabi, J. M. Jin, L. L. Goddard, “Modal expansion approach for accurately computing resonant modes in a high-Q optical resonator,” Microw. Opt. Technol. Lett. 56(2), 278–284 (2014).
[CrossRef]

2013 (1)

2012 (6)

W. Yao, J. M. Jin, P. T. Krein, “A dual-primal finite-element tearing and interconnecting method combined with tree-cotree splitting for modeling electromechanical devices,” Int. J. Numer. Model., Electron. Netw., Devices Fields 26, 151–163 (2012).

W. Yao, J. M. Jin, P. T. Krein, “A highly efficient domain decomposition method applied to 3-D finite-element analysis of electromechanical and electric machine problems,” IEEE Trans. Energ. Convers. 27(4), 1078–1086 (2012).
[CrossRef]

B. Milanović, B. Radjenović, M. Radmilović-Radjenović, “Three-dimensional finite-element modeling of optical microring resonators,” Phys. Scr. T 149, 014026 (2012).
[CrossRef]

M. F. Xue, J. M. Jin, “Nonconformal FETI-DP methods for large-scale electromagnetic simulation,” IEEE Trans. Antenn. Propag. 60(9), 4291–4305 (2012).
[CrossRef]

A. Arbabi, L. Goddard, “Integrated Optical Resonators: Progress in 2011,” IEEE Photonics J. 4(2), 574–577 (2012).
[CrossRef]

S. J. McKeown, L. L. Goddard, “Hydrogen detection using polarization diversity via a subwavelength fiber aperture,” IEEE Photonics J. 4(5), 1752–1761 (2012).
[CrossRef]

2011 (3)

2010 (5)

A. Arbabi, Y. M. Kang, L. Goddard, “Cylindrical coordinates coupled mode theory,” IEEE J. Quantum Electron. 46(12), 1769–1774 (2010).
[CrossRef]

Z. Peng, V. Rawat, J.-F. Lee, “One way domain decomposition method with second order transmission conditions for solving electromagnetic wave problems,” J. Comput. Phys. 229(4), 1181–1197 (2010).
[CrossRef]

Z. Peng, J.-F. Lee, “Non-conformal domain decomposition method with second-order transmission conditions for time-harmonic electromagnetics,” J. Comput. Phys. 229(16), 5615–5629 (2010).
[CrossRef]

Y. M. Kang, A. Arbabi, L. L. Goddard, “Engineering the spectral reflectance of microring resonators with integrated reflective elements,” Opt. Express 18(16), 16813–16825 (2010).
[CrossRef] [PubMed]

M. M. El Gowini, W. A. Moussa, “A finite element model of a MEMS-based surface acoustic wave hydrogen sensor,” Sensors (Basel) 10(2), 1232–1250 (2010).
[CrossRef] [PubMed]

2009 (2)

Y. M. Kang, A. Arbabi, L. L. Goddard, “A microring resonator with an integrated Bragg grating: a compact replacement for a sampled grating distributed Bragg reflector,” Opt. Quantum Electron. 41(9), 689–697 (2009).
[CrossRef]

Y. J. Li, J. M. Jin, “Parallel implementation of the FETI-DPEM algorithm for general 3D EM simulations,” J. Comput. Phys. 228(9), 3255–3267 (2009).
[CrossRef]

2007 (2)

Y. J. Li, J. M. Jin, “A new dual-primal domain decomposition approach for finite element simulation of 3D large-scale electromagnetic problems,” IEEE Trans. Antenn. Propag. 55(10), 2803–2810 (2007).
[CrossRef]

Y. J. Li, J. M. Jin, “Fast full-wave analysis of large-scale three-dimensional photonic crystal devices,” J. Opt. Soc. Am. B 24(9), 2406–2415 (2007).
[CrossRef]

2006 (1)

A. Alonso-Rodriguez, L. Gerardo-Giorda, “New nonoverlapping domain decomposition methods for the harmonic Maxwell system,” SIAM J. Sci. Comput. 28(1), 102–122 (2006).
[CrossRef]

2005 (1)

Q. Xu, B. Schmidt, S. Pradhan, M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[CrossRef] [PubMed]

2004 (5)

V. R. Almeida, C. A. Barrios, R. R. Panepucci, M. Lipson, “All-optical control of light on a silicon chip,” Nature 431(7012), 1081–1084 (2004).
[CrossRef] [PubMed]

J. K. S. Poon, J. Scheuer, A. Yariv, “Wavelength-selective reflector based on a circular array of coupled microring resonators,” IEEE Photon. Technol. Lett. 16(5), 1331–1333 (2004).
[CrossRef]

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oei, H. Binsma, G.-D. Khoe, M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432(7014), 206–209 (2004).
[CrossRef] [PubMed]

T. Barwicz, M. A. Popović, P. T. Rakich, M. R. Watts, H. A. Haus, E. P. Ippen, H. I. Smith, “Microring-resonator-based add-drop filters in SiN: fabrication and analysis,” Opt. Express 12(7), 1437–1442 (2004).
[CrossRef] [PubMed]

X. Shi, L. Hesselink, “Design of a C aperture to achieve λ/10 resolution and resonant transmission,” J. Opt. Soc. Am. B 21(7), 1305–1317 (2004).
[CrossRef]

2003 (2)

C.-Y. Chao, L. J. Guo, “Biochemical sensors based on polymer microrings with sharp asymmetrical resonance,” Appl. Phys. Lett. 83(8), 1527–1529 (2003).
[CrossRef]

T. A. Ibrahim, R. Grover, L.-C. Kuo, S. Kanakaraju, L. C. Calhoun, P.-T. Ho, “All-optical AND/NAND logic gates using semiconductor microresonators,” IEEE Photon. Technol. Lett. 15(10), 1422–1424 (2003).
[CrossRef]

2000 (1)

C. T. Wolfe, U. Navsariwala, S. D. Gedney, “A parallel finite-element tearing and interconnecting algorithm for solution of the vector wave equation with PML absorbing medium,” IEEE Trans. Antenn. Propag. 48(2), 278–284 (2000).
[CrossRef]

1998 (1)

Y. Kawabe, C. Spiegelberg, A. Schülzgen, M. F. Nabor, B. Kippelen, E. A. Mash, P. M. Allemand, M. Kuwata-Gonokami, K. Takeda, N. Peyghambarian, “Whispering-gallery-mode microring laser using a conjugated polymer,” Appl. Phys. Lett. 72(2), 141–143 (1998).
[CrossRef]

1997 (1)

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

Allemand, P. M.

Y. Kawabe, C. Spiegelberg, A. Schülzgen, M. F. Nabor, B. Kippelen, E. A. Mash, P. M. Allemand, M. Kuwata-Gonokami, K. Takeda, N. Peyghambarian, “Whispering-gallery-mode microring laser using a conjugated polymer,” Appl. Phys. Lett. 72(2), 141–143 (1998).
[CrossRef]

Almeida, V. R.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, M. Lipson, “All-optical control of light on a silicon chip,” Nature 431(7012), 1081–1084 (2004).
[CrossRef] [PubMed]

Alonso-Rodriguez, A.

A. Alonso-Rodriguez, L. Gerardo-Giorda, “New nonoverlapping domain decomposition methods for the harmonic Maxwell system,” SIAM J. Sci. Comput. 28(1), 102–122 (2006).
[CrossRef]

Arbabi, A.

Y. M. Kang, M. F. Xue, A. Arbabi, J. M. Jin, L. L. Goddard, “Modal expansion approach for accurately computing resonant modes in a high-Q optical resonator,” Microw. Opt. Technol. Lett. 56(2), 278–284 (2014).
[CrossRef]

A. Arbabi, L. L. Goddard, “Measurements of the refractive indices and thermo-optic coefficients of Si3N4 and SiOx using microring resonances,” Opt. Lett. 38(19), 3878–3881 (2013).
[CrossRef] [PubMed]

A. Arbabi, L. Goddard, “Integrated Optical Resonators: Progress in 2011,” IEEE Photonics J. 4(2), 574–577 (2012).
[CrossRef]

A. Arbabi, Y. M. Kang, C.-Y. Lu, E. Chow, L. Goddard, “Realization of a narrowband single wavelength microring mirror,” Appl. Phys. Lett. 99(9), 091105 (2011).
[CrossRef]

Y. M. Kang, A. Arbabi, L. L. Goddard, “Engineering the spectral reflectance of microring resonators with integrated reflective elements,” Opt. Express 18(16), 16813–16825 (2010).
[CrossRef] [PubMed]

A. Arbabi, Y. M. Kang, L. Goddard, “Cylindrical coordinates coupled mode theory,” IEEE J. Quantum Electron. 46(12), 1769–1774 (2010).
[CrossRef]

Y. M. Kang, A. Arbabi, L. L. Goddard, “A microring resonator with an integrated Bragg grating: a compact replacement for a sampled grating distributed Bragg reflector,” Opt. Quantum Electron. 41(9), 689–697 (2009).
[CrossRef]

Ayubi-Moak, J. S.

J. S. Ayubi-Moak, S. M. Goodnick, D. Stanzione, G. Speyer, P. Sotirelis, “Improved parallel 3D FDTD simulator for photonic crystals,” in Proceedings of DoD HPCMP Users Group Conference, 319–326 (2008).
[CrossRef]

Barrios, C. A.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, M. Lipson, “All-optical control of light on a silicon chip,” Nature 431(7012), 1081–1084 (2004).
[CrossRef] [PubMed]

Barwicz, T.

Binsma, H.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oei, H. Binsma, G.-D. Khoe, M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432(7014), 206–209 (2004).
[CrossRef] [PubMed]

Calhoun, L. C.

T. A. Ibrahim, R. Grover, L.-C. Kuo, S. Kanakaraju, L. C. Calhoun, P.-T. Ho, “All-optical AND/NAND logic gates using semiconductor microresonators,” IEEE Photon. Technol. Lett. 15(10), 1422–1424 (2003).
[CrossRef]

Chao, C.-Y.

C.-Y. Chao, L. J. Guo, “Biochemical sensors based on polymer microrings with sharp asymmetrical resonance,” Appl. Phys. Lett. 83(8), 1527–1529 (2003).
[CrossRef]

Chow, E.

A. Arbabi, Y. M. Kang, C.-Y. Lu, E. Chow, L. Goddard, “Realization of a narrowband single wavelength microring mirror,” Appl. Phys. Lett. 99(9), 091105 (2011).
[CrossRef]

Chu, S. T.

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

De Vries, T.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oei, H. Binsma, G.-D. Khoe, M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432(7014), 206–209 (2004).
[CrossRef] [PubMed]

Den Besten, J. H.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oei, H. Binsma, G.-D. Khoe, M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432(7014), 206–209 (2004).
[CrossRef] [PubMed]

Ding, Y.

Dorren, H. J. S.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oei, H. Binsma, G.-D. Khoe, M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432(7014), 206–209 (2004).
[CrossRef] [PubMed]

Einat, A.

El Gowini, M. M.

M. M. El Gowini, W. A. Moussa, “A finite element model of a MEMS-based surface acoustic wave hydrogen sensor,” Sensors (Basel) 10(2), 1232–1250 (2010).
[CrossRef] [PubMed]

Foresi, J.

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

Gedney, S. D.

C. T. Wolfe, U. Navsariwala, S. D. Gedney, “A parallel finite-element tearing and interconnecting algorithm for solution of the vector wave equation with PML absorbing medium,” IEEE Trans. Antenn. Propag. 48(2), 278–284 (2000).
[CrossRef]

Gerardo-Giorda, L.

A. Alonso-Rodriguez, L. Gerardo-Giorda, “New nonoverlapping domain decomposition methods for the harmonic Maxwell system,” SIAM J. Sci. Comput. 28(1), 102–122 (2006).
[CrossRef]

Goddard, L.

A. Arbabi, L. Goddard, “Integrated Optical Resonators: Progress in 2011,” IEEE Photonics J. 4(2), 574–577 (2012).
[CrossRef]

A. Arbabi, Y. M. Kang, C.-Y. Lu, E. Chow, L. Goddard, “Realization of a narrowband single wavelength microring mirror,” Appl. Phys. Lett. 99(9), 091105 (2011).
[CrossRef]

A. Arbabi, Y. M. Kang, L. Goddard, “Cylindrical coordinates coupled mode theory,” IEEE J. Quantum Electron. 46(12), 1769–1774 (2010).
[CrossRef]

Goddard, L. L.

Y. M. Kang, M. F. Xue, A. Arbabi, J. M. Jin, L. L. Goddard, “Modal expansion approach for accurately computing resonant modes in a high-Q optical resonator,” Microw. Opt. Technol. Lett. 56(2), 278–284 (2014).
[CrossRef]

A. Arbabi, L. L. Goddard, “Measurements of the refractive indices and thermo-optic coefficients of Si3N4 and SiOx using microring resonances,” Opt. Lett. 38(19), 3878–3881 (2013).
[CrossRef] [PubMed]

S. J. McKeown, L. L. Goddard, “Hydrogen detection using polarization diversity via a subwavelength fiber aperture,” IEEE Photonics J. 4(5), 1752–1761 (2012).
[CrossRef]

Y. M. Kang, A. Arbabi, L. L. Goddard, “Engineering the spectral reflectance of microring resonators with integrated reflective elements,” Opt. Express 18(16), 16813–16825 (2010).
[CrossRef] [PubMed]

Y. M. Kang, A. Arbabi, L. L. Goddard, “A microring resonator with an integrated Bragg grating: a compact replacement for a sampled grating distributed Bragg reflector,” Opt. Quantum Electron. 41(9), 689–697 (2009).
[CrossRef]

Goodnick, S. M.

J. S. Ayubi-Moak, S. M. Goodnick, D. Stanzione, G. Speyer, P. Sotirelis, “Improved parallel 3D FDTD simulator for photonic crystals,” in Proceedings of DoD HPCMP Users Group Conference, 319–326 (2008).
[CrossRef]

Grover, R.

T. A. Ibrahim, R. Grover, L.-C. Kuo, S. Kanakaraju, L. C. Calhoun, P.-T. Ho, “All-optical AND/NAND logic gates using semiconductor microresonators,” IEEE Photon. Technol. Lett. 15(10), 1422–1424 (2003).
[CrossRef]

Guo, L. J.

C.-Y. Chao, L. J. Guo, “Biochemical sensors based on polymer microrings with sharp asymmetrical resonance,” Appl. Phys. Lett. 83(8), 1527–1529 (2003).
[CrossRef]

Haus, H. A.

Hesselink, L.

Hill, M. T.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oei, H. Binsma, G.-D. Khoe, M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432(7014), 206–209 (2004).
[CrossRef] [PubMed]

Ho, P.-T.

T. A. Ibrahim, R. Grover, L.-C. Kuo, S. Kanakaraju, L. C. Calhoun, P.-T. Ho, “All-optical AND/NAND logic gates using semiconductor microresonators,” IEEE Photon. Technol. Lett. 15(10), 1422–1424 (2003).
[CrossRef]

Huang, D.

Ibrahim, T. A.

T. A. Ibrahim, R. Grover, L.-C. Kuo, S. Kanakaraju, L. C. Calhoun, P.-T. Ho, “All-optical AND/NAND logic gates using semiconductor microresonators,” IEEE Photon. Technol. Lett. 15(10), 1422–1424 (2003).
[CrossRef]

Ippen, E. P.

Jin, J. M.

Y. M. Kang, M. F. Xue, A. Arbabi, J. M. Jin, L. L. Goddard, “Modal expansion approach for accurately computing resonant modes in a high-Q optical resonator,” Microw. Opt. Technol. Lett. 56(2), 278–284 (2014).
[CrossRef]

W. Yao, J. M. Jin, P. T. Krein, “A highly efficient domain decomposition method applied to 3-D finite-element analysis of electromechanical and electric machine problems,” IEEE Trans. Energ. Convers. 27(4), 1078–1086 (2012).
[CrossRef]

W. Yao, J. M. Jin, P. T. Krein, “A dual-primal finite-element tearing and interconnecting method combined with tree-cotree splitting for modeling electromechanical devices,” Int. J. Numer. Model., Electron. Netw., Devices Fields 26, 151–163 (2012).

M. F. Xue, J. M. Jin, “Nonconformal FETI-DP methods for large-scale electromagnetic simulation,” IEEE Trans. Antenn. Propag. 60(9), 4291–4305 (2012).
[CrossRef]

Y. J. Li, J. M. Jin, “Parallel implementation of the FETI-DPEM algorithm for general 3D EM simulations,” J. Comput. Phys. 228(9), 3255–3267 (2009).
[CrossRef]

Y. J. Li, J. M. Jin, “A new dual-primal domain decomposition approach for finite element simulation of 3D large-scale electromagnetic problems,” IEEE Trans. Antenn. Propag. 55(10), 2803–2810 (2007).
[CrossRef]

Y. J. Li, J. M. Jin, “Fast full-wave analysis of large-scale three-dimensional photonic crystal devices,” J. Opt. Soc. Am. B 24(9), 2406–2415 (2007).
[CrossRef]

M. F. Xue, J. M. Jin, “A hybrid nonconformal FETI/conformal FETI-DP method for arbitrary nonoverlapping domain decomposition modeling,” in Proceedings of IEEE AP-S Int. Symp. (2013).
[CrossRef]

Kanakaraju, S.

T. A. Ibrahim, R. Grover, L.-C. Kuo, S. Kanakaraju, L. C. Calhoun, P.-T. Ho, “All-optical AND/NAND logic gates using semiconductor microresonators,” IEEE Photon. Technol. Lett. 15(10), 1422–1424 (2003).
[CrossRef]

Kang, Y. M.

Y. M. Kang, M. F. Xue, A. Arbabi, J. M. Jin, L. L. Goddard, “Modal expansion approach for accurately computing resonant modes in a high-Q optical resonator,” Microw. Opt. Technol. Lett. 56(2), 278–284 (2014).
[CrossRef]

A. Arbabi, Y. M. Kang, C.-Y. Lu, E. Chow, L. Goddard, “Realization of a narrowband single wavelength microring mirror,” Appl. Phys. Lett. 99(9), 091105 (2011).
[CrossRef]

Y. M. Kang, A. Arbabi, L. L. Goddard, “Engineering the spectral reflectance of microring resonators with integrated reflective elements,” Opt. Express 18(16), 16813–16825 (2010).
[CrossRef] [PubMed]

A. Arbabi, Y. M. Kang, L. Goddard, “Cylindrical coordinates coupled mode theory,” IEEE J. Quantum Electron. 46(12), 1769–1774 (2010).
[CrossRef]

Y. M. Kang, A. Arbabi, L. L. Goddard, “A microring resonator with an integrated Bragg grating: a compact replacement for a sampled grating distributed Bragg reflector,” Opt. Quantum Electron. 41(9), 689–697 (2009).
[CrossRef]

Kawabe, Y.

Y. Kawabe, C. Spiegelberg, A. Schülzgen, M. F. Nabor, B. Kippelen, E. A. Mash, P. M. Allemand, M. Kuwata-Gonokami, K. Takeda, N. Peyghambarian, “Whispering-gallery-mode microring laser using a conjugated polymer,” Appl. Phys. Lett. 72(2), 141–143 (1998).
[CrossRef]

Khoe, G.-D.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oei, H. Binsma, G.-D. Khoe, M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432(7014), 206–209 (2004).
[CrossRef] [PubMed]

Kippelen, B.

Y. Kawabe, C. Spiegelberg, A. Schülzgen, M. F. Nabor, B. Kippelen, E. A. Mash, P. M. Allemand, M. Kuwata-Gonokami, K. Takeda, N. Peyghambarian, “Whispering-gallery-mode microring laser using a conjugated polymer,” Appl. Phys. Lett. 72(2), 141–143 (1998).
[CrossRef]

Krein, P. T.

W. Yao, J. M. Jin, P. T. Krein, “A dual-primal finite-element tearing and interconnecting method combined with tree-cotree splitting for modeling electromechanical devices,” Int. J. Numer. Model., Electron. Netw., Devices Fields 26, 151–163 (2012).

W. Yao, J. M. Jin, P. T. Krein, “A highly efficient domain decomposition method applied to 3-D finite-element analysis of electromechanical and electric machine problems,” IEEE Trans. Energ. Convers. 27(4), 1078–1086 (2012).
[CrossRef]

Kuo, L.-C.

T. A. Ibrahim, R. Grover, L.-C. Kuo, S. Kanakaraju, L. C. Calhoun, P.-T. Ho, “All-optical AND/NAND logic gates using semiconductor microresonators,” IEEE Photon. Technol. Lett. 15(10), 1422–1424 (2003).
[CrossRef]

Kuwata-Gonokami, M.

Y. Kawabe, C. Spiegelberg, A. Schülzgen, M. F. Nabor, B. Kippelen, E. A. Mash, P. M. Allemand, M. Kuwata-Gonokami, K. Takeda, N. Peyghambarian, “Whispering-gallery-mode microring laser using a conjugated polymer,” Appl. Phys. Lett. 72(2), 141–143 (1998).
[CrossRef]

Laine, J.

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

Lee, J.-F.

Z. Peng, J.-F. Lee, “Non-conformal domain decomposition method with second-order transmission conditions for time-harmonic electromagnetics,” J. Comput. Phys. 229(16), 5615–5629 (2010).
[CrossRef]

Z. Peng, V. Rawat, J.-F. Lee, “One way domain decomposition method with second order transmission conditions for solving electromagnetic wave problems,” J. Comput. Phys. 229(4), 1181–1197 (2010).
[CrossRef]

Leijtens, X. J. M.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oei, H. Binsma, G.-D. Khoe, M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432(7014), 206–209 (2004).
[CrossRef] [PubMed]

Levy, U.

Li, Y. J.

Y. J. Li, J. M. Jin, “Parallel implementation of the FETI-DPEM algorithm for general 3D EM simulations,” J. Comput. Phys. 228(9), 3255–3267 (2009).
[CrossRef]

Y. J. Li, J. M. Jin, “A new dual-primal domain decomposition approach for finite element simulation of 3D large-scale electromagnetic problems,” IEEE Trans. Antenn. Propag. 55(10), 2803–2810 (2007).
[CrossRef]

Y. J. Li, J. M. Jin, “Fast full-wave analysis of large-scale three-dimensional photonic crystal devices,” J. Opt. Soc. Am. B 24(9), 2406–2415 (2007).
[CrossRef]

Lipson, M.

Q. Xu, B. Schmidt, S. Pradhan, M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[CrossRef] [PubMed]

V. R. Almeida, C. A. Barrios, R. R. Panepucci, M. Lipson, “All-optical control of light on a silicon chip,” Nature 431(7012), 1081–1084 (2004).
[CrossRef] [PubMed]

Little, B. E.

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

Liu, L.

Lu, C.-Y.

A. Arbabi, Y. M. Kang, C.-Y. Lu, E. Chow, L. Goddard, “Realization of a narrowband single wavelength microring mirror,” Appl. Phys. Lett. 99(9), 091105 (2011).
[CrossRef]

Mash, E. A.

Y. Kawabe, C. Spiegelberg, A. Schülzgen, M. F. Nabor, B. Kippelen, E. A. Mash, P. M. Allemand, M. Kuwata-Gonokami, K. Takeda, N. Peyghambarian, “Whispering-gallery-mode microring laser using a conjugated polymer,” Appl. Phys. Lett. 72(2), 141–143 (1998).
[CrossRef]

McKeown, S. J.

S. J. McKeown, L. L. Goddard, “Hydrogen detection using polarization diversity via a subwavelength fiber aperture,” IEEE Photonics J. 4(5), 1752–1761 (2012).
[CrossRef]

Milanovic, B.

B. Milanović, B. Radjenović, M. Radmilović-Radjenović, “Three-dimensional finite-element modeling of optical microring resonators,” Phys. Scr. T 149, 014026 (2012).
[CrossRef]

Moussa, W. A.

M. M. El Gowini, W. A. Moussa, “A finite element model of a MEMS-based surface acoustic wave hydrogen sensor,” Sensors (Basel) 10(2), 1232–1250 (2010).
[CrossRef] [PubMed]

Nabor, M. F.

Y. Kawabe, C. Spiegelberg, A. Schülzgen, M. F. Nabor, B. Kippelen, E. A. Mash, P. M. Allemand, M. Kuwata-Gonokami, K. Takeda, N. Peyghambarian, “Whispering-gallery-mode microring laser using a conjugated polymer,” Appl. Phys. Lett. 72(2), 141–143 (1998).
[CrossRef]

Navsariwala, U.

C. T. Wolfe, U. Navsariwala, S. D. Gedney, “A parallel finite-element tearing and interconnecting algorithm for solution of the vector wave equation with PML absorbing medium,” IEEE Trans. Antenn. Propag. 48(2), 278–284 (2000).
[CrossRef]

Oei, Y.-S.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oei, H. Binsma, G.-D. Khoe, M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432(7014), 206–209 (2004).
[CrossRef] [PubMed]

Ou, H.

Panepucci, R. R.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, M. Lipson, “All-optical control of light on a silicon chip,” Nature 431(7012), 1081–1084 (2004).
[CrossRef] [PubMed]

Peng, Z.

Z. Peng, V. Rawat, J.-F. Lee, “One way domain decomposition method with second order transmission conditions for solving electromagnetic wave problems,” J. Comput. Phys. 229(4), 1181–1197 (2010).
[CrossRef]

Z. Peng, J.-F. Lee, “Non-conformal domain decomposition method with second-order transmission conditions for time-harmonic electromagnetics,” J. Comput. Phys. 229(16), 5615–5629 (2010).
[CrossRef]

Peucheret, C.

Peyghambarian, N.

Y. Kawabe, C. Spiegelberg, A. Schülzgen, M. F. Nabor, B. Kippelen, E. A. Mash, P. M. Allemand, M. Kuwata-Gonokami, K. Takeda, N. Peyghambarian, “Whispering-gallery-mode microring laser using a conjugated polymer,” Appl. Phys. Lett. 72(2), 141–143 (1998).
[CrossRef]

Poon, J. K. S.

J. K. S. Poon, J. Scheuer, A. Yariv, “Wavelength-selective reflector based on a circular array of coupled microring resonators,” IEEE Photon. Technol. Lett. 16(5), 1331–1333 (2004).
[CrossRef]

Popovic, M. A.

Pradhan, S.

Q. Xu, B. Schmidt, S. Pradhan, M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[CrossRef] [PubMed]

Pu, M.

Radjenovic, B.

B. Milanović, B. Radjenović, M. Radmilović-Radjenović, “Three-dimensional finite-element modeling of optical microring resonators,” Phys. Scr. T 149, 014026 (2012).
[CrossRef]

B. Radjenović, M. Radmilovic-Radjenović, “Excitation of confined modes in silicon slotted waveguides and microring resonators for sensing purposes,” IEEE Sens. J., doi:.
[CrossRef]

Radmilovic-Radjenovic, M.

B. Milanović, B. Radjenović, M. Radmilović-Radjenović, “Three-dimensional finite-element modeling of optical microring resonators,” Phys. Scr. T 149, 014026 (2012).
[CrossRef]

B. Radjenović, M. Radmilovic-Radjenović, “Excitation of confined modes in silicon slotted waveguides and microring resonators for sensing purposes,” IEEE Sens. J., doi:.
[CrossRef]

Rakich, P. T.

Rawat, V.

Z. Peng, V. Rawat, J.-F. Lee, “One way domain decomposition method with second order transmission conditions for solving electromagnetic wave problems,” J. Comput. Phys. 229(4), 1181–1197 (2010).
[CrossRef]

Scheuer, J.

J. K. S. Poon, J. Scheuer, A. Yariv, “Wavelength-selective reflector based on a circular array of coupled microring resonators,” IEEE Photon. Technol. Lett. 16(5), 1331–1333 (2004).
[CrossRef]

Schmidt, B.

Q. Xu, B. Schmidt, S. Pradhan, M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[CrossRef] [PubMed]

Schülzgen, A.

Y. Kawabe, C. Spiegelberg, A. Schülzgen, M. F. Nabor, B. Kippelen, E. A. Mash, P. M. Allemand, M. Kuwata-Gonokami, K. Takeda, N. Peyghambarian, “Whispering-gallery-mode microring laser using a conjugated polymer,” Appl. Phys. Lett. 72(2), 141–143 (1998).
[CrossRef]

Shi, X.

Smalbrugge, B.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oei, H. Binsma, G.-D. Khoe, M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432(7014), 206–209 (2004).
[CrossRef] [PubMed]

Smit, M. K.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oei, H. Binsma, G.-D. Khoe, M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432(7014), 206–209 (2004).
[CrossRef] [PubMed]

Smith, H. I.

Sotirelis, P.

J. S. Ayubi-Moak, S. M. Goodnick, D. Stanzione, G. Speyer, P. Sotirelis, “Improved parallel 3D FDTD simulator for photonic crystals,” in Proceedings of DoD HPCMP Users Group Conference, 319–326 (2008).
[CrossRef]

Speyer, G.

J. S. Ayubi-Moak, S. M. Goodnick, D. Stanzione, G. Speyer, P. Sotirelis, “Improved parallel 3D FDTD simulator for photonic crystals,” in Proceedings of DoD HPCMP Users Group Conference, 319–326 (2008).
[CrossRef]

Spiegelberg, C.

Y. Kawabe, C. Spiegelberg, A. Schülzgen, M. F. Nabor, B. Kippelen, E. A. Mash, P. M. Allemand, M. Kuwata-Gonokami, K. Takeda, N. Peyghambarian, “Whispering-gallery-mode microring laser using a conjugated polymer,” Appl. Phys. Lett. 72(2), 141–143 (1998).
[CrossRef]

Stanzione, D.

J. S. Ayubi-Moak, S. M. Goodnick, D. Stanzione, G. Speyer, P. Sotirelis, “Improved parallel 3D FDTD simulator for photonic crystals,” in Proceedings of DoD HPCMP Users Group Conference, 319–326 (2008).
[CrossRef]

Takeda, K.

Y. Kawabe, C. Spiegelberg, A. Schülzgen, M. F. Nabor, B. Kippelen, E. A. Mash, P. M. Allemand, M. Kuwata-Gonokami, K. Takeda, N. Peyghambarian, “Whispering-gallery-mode microring laser using a conjugated polymer,” Appl. Phys. Lett. 72(2), 141–143 (1998).
[CrossRef]

Watts, M. R.

Wolfe, C. T.

C. T. Wolfe, U. Navsariwala, S. D. Gedney, “A parallel finite-element tearing and interconnecting algorithm for solution of the vector wave equation with PML absorbing medium,” IEEE Trans. Antenn. Propag. 48(2), 278–284 (2000).
[CrossRef]

Xu, J.

Xu, Q.

Q. Xu, B. Schmidt, S. Pradhan, M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[CrossRef] [PubMed]

Xue, M. F.

Y. M. Kang, M. F. Xue, A. Arbabi, J. M. Jin, L. L. Goddard, “Modal expansion approach for accurately computing resonant modes in a high-Q optical resonator,” Microw. Opt. Technol. Lett. 56(2), 278–284 (2014).
[CrossRef]

M. F. Xue, J. M. Jin, “Nonconformal FETI-DP methods for large-scale electromagnetic simulation,” IEEE Trans. Antenn. Propag. 60(9), 4291–4305 (2012).
[CrossRef]

M. F. Xue, J. M. Jin, “A hybrid nonconformal FETI/conformal FETI-DP method for arbitrary nonoverlapping domain decomposition modeling,” in Proceedings of IEEE AP-S Int. Symp. (2013).
[CrossRef]

Yao, W.

W. Yao, J. M. Jin, P. T. Krein, “A dual-primal finite-element tearing and interconnecting method combined with tree-cotree splitting for modeling electromechanical devices,” Int. J. Numer. Model., Electron. Netw., Devices Fields 26, 151–163 (2012).

W. Yao, J. M. Jin, P. T. Krein, “A highly efficient domain decomposition method applied to 3-D finite-element analysis of electromechanical and electric machine problems,” IEEE Trans. Energ. Convers. 27(4), 1078–1086 (2012).
[CrossRef]

Yariv, A.

J. K. S. Poon, J. Scheuer, A. Yariv, “Wavelength-selective reflector based on a circular array of coupled microring resonators,” IEEE Photon. Technol. Lett. 16(5), 1331–1333 (2004).
[CrossRef]

Zhang, X.

Appl. Phys. Lett. (3)

Y. Kawabe, C. Spiegelberg, A. Schülzgen, M. F. Nabor, B. Kippelen, E. A. Mash, P. M. Allemand, M. Kuwata-Gonokami, K. Takeda, N. Peyghambarian, “Whispering-gallery-mode microring laser using a conjugated polymer,” Appl. Phys. Lett. 72(2), 141–143 (1998).
[CrossRef]

A. Arbabi, Y. M. Kang, C.-Y. Lu, E. Chow, L. Goddard, “Realization of a narrowband single wavelength microring mirror,” Appl. Phys. Lett. 99(9), 091105 (2011).
[CrossRef]

C.-Y. Chao, L. J. Guo, “Biochemical sensors based on polymer microrings with sharp asymmetrical resonance,” Appl. Phys. Lett. 83(8), 1527–1529 (2003).
[CrossRef]

IEEE J. Quantum Electron. (1)

A. Arbabi, Y. M. Kang, L. Goddard, “Cylindrical coordinates coupled mode theory,” IEEE J. Quantum Electron. 46(12), 1769–1774 (2010).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

J. K. S. Poon, J. Scheuer, A. Yariv, “Wavelength-selective reflector based on a circular array of coupled microring resonators,” IEEE Photon. Technol. Lett. 16(5), 1331–1333 (2004).
[CrossRef]

T. A. Ibrahim, R. Grover, L.-C. Kuo, S. Kanakaraju, L. C. Calhoun, P.-T. Ho, “All-optical AND/NAND logic gates using semiconductor microresonators,” IEEE Photon. Technol. Lett. 15(10), 1422–1424 (2003).
[CrossRef]

IEEE Photonics J. (2)

A. Arbabi, L. Goddard, “Integrated Optical Resonators: Progress in 2011,” IEEE Photonics J. 4(2), 574–577 (2012).
[CrossRef]

S. J. McKeown, L. L. Goddard, “Hydrogen detection using polarization diversity via a subwavelength fiber aperture,” IEEE Photonics J. 4(5), 1752–1761 (2012).
[CrossRef]

IEEE Trans. Antenn. Propag. (3)

Y. J. Li, J. M. Jin, “A new dual-primal domain decomposition approach for finite element simulation of 3D large-scale electromagnetic problems,” IEEE Trans. Antenn. Propag. 55(10), 2803–2810 (2007).
[CrossRef]

M. F. Xue, J. M. Jin, “Nonconformal FETI-DP methods for large-scale electromagnetic simulation,” IEEE Trans. Antenn. Propag. 60(9), 4291–4305 (2012).
[CrossRef]

C. T. Wolfe, U. Navsariwala, S. D. Gedney, “A parallel finite-element tearing and interconnecting algorithm for solution of the vector wave equation with PML absorbing medium,” IEEE Trans. Antenn. Propag. 48(2), 278–284 (2000).
[CrossRef]

IEEE Trans. Energ. Convers. (1)

W. Yao, J. M. Jin, P. T. Krein, “A highly efficient domain decomposition method applied to 3-D finite-element analysis of electromechanical and electric machine problems,” IEEE Trans. Energ. Convers. 27(4), 1078–1086 (2012).
[CrossRef]

Int. J. Numer. Model., Electron. Netw., Devices Fields (1)

W. Yao, J. M. Jin, P. T. Krein, “A dual-primal finite-element tearing and interconnecting method combined with tree-cotree splitting for modeling electromechanical devices,” Int. J. Numer. Model., Electron. Netw., Devices Fields 26, 151–163 (2012).

J. Comput. Phys. (3)

Z. Peng, V. Rawat, J.-F. Lee, “One way domain decomposition method with second order transmission conditions for solving electromagnetic wave problems,” J. Comput. Phys. 229(4), 1181–1197 (2010).
[CrossRef]

Z. Peng, J.-F. Lee, “Non-conformal domain decomposition method with second-order transmission conditions for time-harmonic electromagnetics,” J. Comput. Phys. 229(16), 5615–5629 (2010).
[CrossRef]

Y. J. Li, J. M. Jin, “Parallel implementation of the FETI-DPEM algorithm for general 3D EM simulations,” J. Comput. Phys. 228(9), 3255–3267 (2009).
[CrossRef]

J. Lightwave Technol. (1)

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

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

Microw. Opt. Technol. Lett. (1)

Y. M. Kang, M. F. Xue, A. Arbabi, J. M. Jin, L. L. Goddard, “Modal expansion approach for accurately computing resonant modes in a high-Q optical resonator,” Microw. Opt. Technol. Lett. 56(2), 278–284 (2014).
[CrossRef]

Nature (3)

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oei, H. Binsma, G.-D. Khoe, M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432(7014), 206–209 (2004).
[CrossRef] [PubMed]

V. R. Almeida, C. A. Barrios, R. R. Panepucci, M. Lipson, “All-optical control of light on a silicon chip,” Nature 431(7012), 1081–1084 (2004).
[CrossRef] [PubMed]

Q. Xu, B. Schmidt, S. Pradhan, M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[CrossRef] [PubMed]

Opt. Express (4)

Opt. Lett. (1)

Opt. Quantum Electron. (1)

Y. M. Kang, A. Arbabi, L. L. Goddard, “A microring resonator with an integrated Bragg grating: a compact replacement for a sampled grating distributed Bragg reflector,” Opt. Quantum Electron. 41(9), 689–697 (2009).
[CrossRef]

Phys. Scr. T (1)

B. Milanović, B. Radjenović, M. Radmilović-Radjenović, “Three-dimensional finite-element modeling of optical microring resonators,” Phys. Scr. T 149, 014026 (2012).
[CrossRef]

Sensors (Basel) (1)

M. M. El Gowini, W. A. Moussa, “A finite element model of a MEMS-based surface acoustic wave hydrogen sensor,” Sensors (Basel) 10(2), 1232–1250 (2010).
[CrossRef] [PubMed]

SIAM J. Sci. Comput. (1)

A. Alonso-Rodriguez, L. Gerardo-Giorda, “New nonoverlapping domain decomposition methods for the harmonic Maxwell system,” SIAM J. Sci. Comput. 28(1), 102–122 (2006).
[CrossRef]

Other (9)

J. S. Ayubi-Moak, S. M. Goodnick, D. Stanzione, G. Speyer, P. Sotirelis, “Improved parallel 3D FDTD simulator for photonic crystals,” in Proceedings of DoD HPCMP Users Group Conference, 319–326 (2008).
[CrossRef]

B. Radjenović and M. Radmilović-Radjenović, “Computer-aided design and simulation of optical microring resonators”, Int. J. Numer. Model., Electron. Netw., Devices Fields, available online: http://onlinelibrary.wiley.com/doi/10.1002/jnm.1920/abstract .
[CrossRef]

B. Radjenović, M. Radmilovic-Radjenović, “Excitation of confined modes in silicon slotted waveguides and microring resonators for sensing purposes,” IEEE Sens. J., doi:.
[CrossRef]

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

MUMPS, A parallel sparse direct solver, available online: http://graal.ens-lyon.fr/MUMPS/ .

CUBIT, available online: https://cubit.sandia.gov/ .

METIS, Serial graph partitioning and fill-reducing matrix ordering, available online: http://glaros.dtc.umn.edu/gkhome/metis/metis/overview .

COMSOL Multiphysics ver. 4.2, available online: http://www.comsol.com/ .

M. F. Xue, J. M. Jin, “A hybrid nonconformal FETI/conformal FETI-DP method for arbitrary nonoverlapping domain decomposition modeling,” in Proceedings of IEEE AP-S Int. Symp. (2013).
[CrossRef]

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Figures (14)

Fig. 1
Fig. 1

(a) Top view of a ring/bus structure modeled with CUBIT. The ring has the first-order grating along the upper half on the inner side. (b) Enlarged view of the rectangular region in Fig. 1(a) showing the grating.

Fig. 2
Fig. 2

Power reflection and transmission coefficients of the ring/bus structure shown in Fig. 1 from 1.924 × 10 14 Hz to 1.944 × 10 14 Hz .

Fig. 3
Fig. 3

Re ( E z ) in the z = 0 μm plane. (a) At 1.9246 × 10 14 Hz . (b) At 1.9355 × 10 14 Hz .

Fig. 4
Fig. 4

Convergence history of the iterative solution of the global interface problem using the FETI-DP method with the SOTC-TE and the FOTC.

Fig. 5
Fig. 5

Parallel speedup versus the number of processors with N s = 512. The computation time using four processors is taken as the reference.

Fig. 6
Fig. 6

(a) Top view of an ECDMRR/bus structure modeled with CUBIT. The inner ring has a first-order grating along the upper half on its outer side. (b) Enlarged view of the rectangular region in Fig. 6(a).

Fig. 7
Fig. 7

Power reflection and transmission coefficients of the full-scale ECDMRR. (a) Comparison between the measured and simulated results. (b) Comparison with the simulated result shifted by 0.78 nm.

Fig. 8
Fig. 8

Snapshot of | E | in the z = 0 μm plane. (a) With λ = 1550 .93 nm . (b) With λ = 1549 .14 nm . (c) Enlarged view of the field near the gap between the bus and the ECDMRR at λ = 1549 .14 nm .

Fig. 9
Fig. 9

(a) Top view. (b) Side view of a fiber tip sensor with palladium layer and C-shaped aperture modeled with CUBIT. (c) Detailed geometry information for the C-shaped aperture (top view).

Fig. 10
Fig. 10

Iterative convergence of the global interface system with respect to the number of subdomains.

Fig. 11
Fig. 11

(a) Re ( E x ) in the plane 1.5 μm above the aperture, with x-polarized excitation. (b) Re ( E y ) in the plane 1.5 μm above the aperture, with y-polarized excitation.

Fig. 12
Fig. 12

(a) | E x | in the aperture, with x-polarized excitation. (b) | E y | in the aperture, with y-polarized excitation.

Fig. 13
Fig. 13

(a) Re ( E x ) in the y = 0 μm plane, with x-polarized excitation. (b) Re ( E y ) in the y = 0 μm plane, with y-polarized excitation.

Fig. 14
Fig. 14

Parallel speedup versus the number of processors with Ns = 960.

Tables (2)

Tables Icon

Table 1 Computation times for the bus/ring resonator problem with 7,522,572 unknowns and 512 subdomains.

Tables Icon

Table 2 Computation times for the fiber tip sensor problem with 70,420,434 unknowns and 960 subdomains.

Equations (13)

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×( μ r 1 × E s ) k 0 2 ε r E s =j k 0 Z 0 J imp s in V s
n ^ s ×( μ r 1 × E s )+ α s n ^ s ×( n ^ s × E s ) β s ×[ n ^ s (× E s ) n ]= Λ s on Γ s
[D]=[ s y s z / s x 0 0 0 s z s x / s y 0 0 0 s x s y / s z ].
[ K rr s K rc s K cr s K cc s ]{ E r s E c s }={ f r s f c s }{ λ r s + L rc s E c s λ c s }
{ E I s }=f({ E c s },{ λ s },{ f s })
{ E c s }=g({ λ s },{ f s })
F({ E c },{λ},{f})=0
{ E c }=G({λ},{f}).
T m = S T E e m dS S T E 0 e m dS
R m = S R (E E 0 ) e m dS S R E 0 e m dS .
{ x ( ϕ ) = ( r 1 + δ sin 2 m ϕ ) cos ϕ , y ( ϕ ) = ( r 1 + δ sin 2 m ϕ ) sin ϕ , for 0 ϕ < π x ( ϕ ) = r 1 cos ϕ , y ( ϕ ) = r 1 sin ϕ , for π ϕ < 2 π
E ( x , y ) = { z ^ cos [ k y ( y y 0 ) ] exp ( j β x ) , ( | y y 0 | d / 2 ) z ^ cos ( k y d / 2 ) exp [ α ( | y y 0 | d / 2 ) ] exp ( j β x ) , ( | y y 0 | > d / 2 )
Speedup = T 4 T N p

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