Abstract

We demonstrate the use of stochastic collocation to assess the performance of photonic devices under the effect of uncertainty. This approach combines high accuracy and efficiency in analyzing device variability with the ease of implementation of sampling-based methods. Its flexibility makes it suitable to be applied to a large range of photonic devices. We compare the stochastic collocation method with a Monte Carlo technique on a numerical analysis of the variability in silicon directional couplers.

© 2016 Chinese Laser Press

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References

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  1. M. Streshinsky, R. Ding, Y. Liu, A. Novack, C. Galland, A.-J. Lim, G. Q. Lo, T. Baehr-Jones, and M. Hochberg, “The road to affordable, large-scale silicon photonics,” Opt. Photon. News 24(9), 32–39 (2013).
    [Crossref]
  2. W. Bogaerts, M. Fiers, and P. Dumon, “Design challenges in silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 20, 1–8 (2014).
    [Crossref]
  3. L. Chrostowski, X. Wang, J. Flueckiger, Y. Wu, Y. Wang, and S. T. Fard, “Impact of fabrication non-uniformity on chip-scale silicon photonic integrated circuits,” in Optical Fiber Communication Conference (Optical Society of America, 2014), paper Th2A.37.
  4. S. K. Selvaraja, W. Bogaerts, P. Dumon, D. Van Thourhout, and R. Baets, “Subnanometer linewidth uniformity in silicon nanophotonic waveguide devices using CMOS fabrication technology,” IEEE J. Sel. Top. Quantum Electron. 16, 316–324 (2010).
    [Crossref]
  5. G. S. Fishman, Monte Carlo: Concepts, Algorithms, and Applications (Springer-Verlag, 1996).
  6. D. Spina, F. Ferranti, G. Antonini, T. Dhaene, and L. Knockaert, “Efficient variability analysis of electromagnetic systems via polynomial chaos and model order reduction,” IEEE Trans. Compon. Packag. Manuf. Technol. 4, 1038–1051 (2014).
    [Crossref]
  7. P. R. Johnston, “Defibrillation thresholds: a generalised polynomial chaos study,” in Conference in Computing in Cardiology, Cambridge, MA (Sept.7–10, 2014).
  8. K.-K. K. Kim and R. D. Braatz, “Generalized polynomial chaos expansion approaches to approximate stochastic receding horizon control with applications to probabilistic collision checking and avoidance,” in IEEE International Conference on Control Applications, Dubrovnik, Croatia (Oct.3–5, 2012).
  9. D. Cassano, F. Morichetti, and A. Melloni, “Statistical analysis of photonic integrated circuits via polynomial-chaos expansion,” in Advanced Photonics (Optical Society of America, 2013), paper JT3A.8.
  10. T. Weng, Z. Zhang, Z. Su, Y. Marzouk, A. Melloni, and L. Daniel, “Uncertainty quantification of silicon photonic devices with correlated and non-Gaussian random parameters,” Opt. Express 23, 4242–4254 (2015).
    [Crossref]
  11. M. S. Eldred, “Recent advance in non-intrusive polynomial-chaos and stochastic collocation methods for uncertainty analysis and design,” in Proceedings of 50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Palm Springs, California (May, 2009).
  12. N. Agarwal and N. R. Aluru, “Weighted Smolyak algorithm for solution of stochastic differential equations on non-uniform probability measures,” Int. J. Numer. Meth. Eng. 85, 1365–1389 (2011).
    [Crossref]
  13. N. Agarwal and N. R. Aluru, “Stochastic analysis of electrostatic mems subjected to parameter variations,” J. Microelectromech. Syst. 18, 1454–1468 (2009).
    [Crossref]
  14. D. Xiu, “Fast numerical methods for stochastic computations: a review,” Commun. Comput. Phys. 5, 242–272 (2009).
  15. W. A. Weiser and S. E. Zarantonello, “A note on piecewise linear and multilinear table interpolation in many dimensions,” Math. Comput. 50, 189–196 (1988).
    [Crossref]
  16. B. Ganapathysubramanian and N. Zabaras, “Sparse grid collocation schemes for stochastic natural convection problems,” J. Comput. Phys. 225, 652–685 (2007).
    [Crossref]
  17. V. Barthelmann, E. Novak, and K. Ritter, “High dimensional polynomial interpolation on sparse grid,” Adv. Comput. Math. 12, 273–288 (2000).
    [Crossref]
  18. E. Novak and K. Ritter, “High dimensional integration of smooth functions over cubes,” Numer. Math. 75, 79–97 (1996).
    [Crossref]
  19. E. Novak and K. Ritter, “Simple cubature formulas with high polynomial exactness,” Constr. Approx. 15, 499–522 (1999).
    [Crossref]
  20. A. Der Kiureghian and P. L. Liu, “Structural reliability under incomplete probability information,” J. Eng. Mech. 112, 85–104 (1986).
    [Crossref]
  21. M. Loeve, Probability Theory, 4th ed. (Springer-Verlag, 1977).
  22. W. Bogaerts, D. Taillaert, B. Luyssaert, P. Dumon, J. Van Campenhout, P. Bienstman, D. Van Thourhout, R. Baets, V. Wiaux, and S. Beckx, “Basic structures for photonic integrated circuits in silicon-on-insulator,” Opt. Express 12, 1583–1591 (2004).
    [Crossref]
  23. L. Chrostowski and M. Hochberg, Silicon Photonics Design: From Devices to Systems (Cambridge University, 2015).

2015 (1)

2014 (2)

W. Bogaerts, M. Fiers, and P. Dumon, “Design challenges in silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 20, 1–8 (2014).
[Crossref]

D. Spina, F. Ferranti, G. Antonini, T. Dhaene, and L. Knockaert, “Efficient variability analysis of electromagnetic systems via polynomial chaos and model order reduction,” IEEE Trans. Compon. Packag. Manuf. Technol. 4, 1038–1051 (2014).
[Crossref]

2013 (1)

M. Streshinsky, R. Ding, Y. Liu, A. Novack, C. Galland, A.-J. Lim, G. Q. Lo, T. Baehr-Jones, and M. Hochberg, “The road to affordable, large-scale silicon photonics,” Opt. Photon. News 24(9), 32–39 (2013).
[Crossref]

2011 (1)

N. Agarwal and N. R. Aluru, “Weighted Smolyak algorithm for solution of stochastic differential equations on non-uniform probability measures,” Int. J. Numer. Meth. Eng. 85, 1365–1389 (2011).
[Crossref]

2010 (1)

S. K. Selvaraja, W. Bogaerts, P. Dumon, D. Van Thourhout, and R. Baets, “Subnanometer linewidth uniformity in silicon nanophotonic waveguide devices using CMOS fabrication technology,” IEEE J. Sel. Top. Quantum Electron. 16, 316–324 (2010).
[Crossref]

2009 (2)

N. Agarwal and N. R. Aluru, “Stochastic analysis of electrostatic mems subjected to parameter variations,” J. Microelectromech. Syst. 18, 1454–1468 (2009).
[Crossref]

D. Xiu, “Fast numerical methods for stochastic computations: a review,” Commun. Comput. Phys. 5, 242–272 (2009).

2007 (1)

B. Ganapathysubramanian and N. Zabaras, “Sparse grid collocation schemes for stochastic natural convection problems,” J. Comput. Phys. 225, 652–685 (2007).
[Crossref]

2004 (1)

2000 (1)

V. Barthelmann, E. Novak, and K. Ritter, “High dimensional polynomial interpolation on sparse grid,” Adv. Comput. Math. 12, 273–288 (2000).
[Crossref]

1999 (1)

E. Novak and K. Ritter, “Simple cubature formulas with high polynomial exactness,” Constr. Approx. 15, 499–522 (1999).
[Crossref]

1996 (1)

E. Novak and K. Ritter, “High dimensional integration of smooth functions over cubes,” Numer. Math. 75, 79–97 (1996).
[Crossref]

1988 (1)

W. A. Weiser and S. E. Zarantonello, “A note on piecewise linear and multilinear table interpolation in many dimensions,” Math. Comput. 50, 189–196 (1988).
[Crossref]

1986 (1)

A. Der Kiureghian and P. L. Liu, “Structural reliability under incomplete probability information,” J. Eng. Mech. 112, 85–104 (1986).
[Crossref]

Agarwal, N.

N. Agarwal and N. R. Aluru, “Weighted Smolyak algorithm for solution of stochastic differential equations on non-uniform probability measures,” Int. J. Numer. Meth. Eng. 85, 1365–1389 (2011).
[Crossref]

N. Agarwal and N. R. Aluru, “Stochastic analysis of electrostatic mems subjected to parameter variations,” J. Microelectromech. Syst. 18, 1454–1468 (2009).
[Crossref]

Aluru, N. R.

N. Agarwal and N. R. Aluru, “Weighted Smolyak algorithm for solution of stochastic differential equations on non-uniform probability measures,” Int. J. Numer. Meth. Eng. 85, 1365–1389 (2011).
[Crossref]

N. Agarwal and N. R. Aluru, “Stochastic analysis of electrostatic mems subjected to parameter variations,” J. Microelectromech. Syst. 18, 1454–1468 (2009).
[Crossref]

Antonini, G.

D. Spina, F. Ferranti, G. Antonini, T. Dhaene, and L. Knockaert, “Efficient variability analysis of electromagnetic systems via polynomial chaos and model order reduction,” IEEE Trans. Compon. Packag. Manuf. Technol. 4, 1038–1051 (2014).
[Crossref]

Baehr-Jones, T.

M. Streshinsky, R. Ding, Y. Liu, A. Novack, C. Galland, A.-J. Lim, G. Q. Lo, T. Baehr-Jones, and M. Hochberg, “The road to affordable, large-scale silicon photonics,” Opt. Photon. News 24(9), 32–39 (2013).
[Crossref]

Baets, R.

S. K. Selvaraja, W. Bogaerts, P. Dumon, D. Van Thourhout, and R. Baets, “Subnanometer linewidth uniformity in silicon nanophotonic waveguide devices using CMOS fabrication technology,” IEEE J. Sel. Top. Quantum Electron. 16, 316–324 (2010).
[Crossref]

W. Bogaerts, D. Taillaert, B. Luyssaert, P. Dumon, J. Van Campenhout, P. Bienstman, D. Van Thourhout, R. Baets, V. Wiaux, and S. Beckx, “Basic structures for photonic integrated circuits in silicon-on-insulator,” Opt. Express 12, 1583–1591 (2004).
[Crossref]

Barthelmann, V.

V. Barthelmann, E. Novak, and K. Ritter, “High dimensional polynomial interpolation on sparse grid,” Adv. Comput. Math. 12, 273–288 (2000).
[Crossref]

Beckx, S.

Bienstman, P.

Bogaerts, W.

W. Bogaerts, M. Fiers, and P. Dumon, “Design challenges in silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 20, 1–8 (2014).
[Crossref]

S. K. Selvaraja, W. Bogaerts, P. Dumon, D. Van Thourhout, and R. Baets, “Subnanometer linewidth uniformity in silicon nanophotonic waveguide devices using CMOS fabrication technology,” IEEE J. Sel. Top. Quantum Electron. 16, 316–324 (2010).
[Crossref]

W. Bogaerts, D. Taillaert, B. Luyssaert, P. Dumon, J. Van Campenhout, P. Bienstman, D. Van Thourhout, R. Baets, V. Wiaux, and S. Beckx, “Basic structures for photonic integrated circuits in silicon-on-insulator,” Opt. Express 12, 1583–1591 (2004).
[Crossref]

Braatz, R. D.

K.-K. K. Kim and R. D. Braatz, “Generalized polynomial chaos expansion approaches to approximate stochastic receding horizon control with applications to probabilistic collision checking and avoidance,” in IEEE International Conference on Control Applications, Dubrovnik, Croatia (Oct.3–5, 2012).

Cassano, D.

D. Cassano, F. Morichetti, and A. Melloni, “Statistical analysis of photonic integrated circuits via polynomial-chaos expansion,” in Advanced Photonics (Optical Society of America, 2013), paper JT3A.8.

Chrostowski, L.

L. Chrostowski, X. Wang, J. Flueckiger, Y. Wu, Y. Wang, and S. T. Fard, “Impact of fabrication non-uniformity on chip-scale silicon photonic integrated circuits,” in Optical Fiber Communication Conference (Optical Society of America, 2014), paper Th2A.37.

L. Chrostowski and M. Hochberg, Silicon Photonics Design: From Devices to Systems (Cambridge University, 2015).

Daniel, L.

Der Kiureghian, A.

A. Der Kiureghian and P. L. Liu, “Structural reliability under incomplete probability information,” J. Eng. Mech. 112, 85–104 (1986).
[Crossref]

Dhaene, T.

D. Spina, F. Ferranti, G. Antonini, T. Dhaene, and L. Knockaert, “Efficient variability analysis of electromagnetic systems via polynomial chaos and model order reduction,” IEEE Trans. Compon. Packag. Manuf. Technol. 4, 1038–1051 (2014).
[Crossref]

Ding, R.

M. Streshinsky, R. Ding, Y. Liu, A. Novack, C. Galland, A.-J. Lim, G. Q. Lo, T. Baehr-Jones, and M. Hochberg, “The road to affordable, large-scale silicon photonics,” Opt. Photon. News 24(9), 32–39 (2013).
[Crossref]

Dumon, P.

W. Bogaerts, M. Fiers, and P. Dumon, “Design challenges in silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 20, 1–8 (2014).
[Crossref]

S. K. Selvaraja, W. Bogaerts, P. Dumon, D. Van Thourhout, and R. Baets, “Subnanometer linewidth uniformity in silicon nanophotonic waveguide devices using CMOS fabrication technology,” IEEE J. Sel. Top. Quantum Electron. 16, 316–324 (2010).
[Crossref]

W. Bogaerts, D. Taillaert, B. Luyssaert, P. Dumon, J. Van Campenhout, P. Bienstman, D. Van Thourhout, R. Baets, V. Wiaux, and S. Beckx, “Basic structures for photonic integrated circuits in silicon-on-insulator,” Opt. Express 12, 1583–1591 (2004).
[Crossref]

Eldred, M. S.

M. S. Eldred, “Recent advance in non-intrusive polynomial-chaos and stochastic collocation methods for uncertainty analysis and design,” in Proceedings of 50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Palm Springs, California (May, 2009).

Fard, S. T.

L. Chrostowski, X. Wang, J. Flueckiger, Y. Wu, Y. Wang, and S. T. Fard, “Impact of fabrication non-uniformity on chip-scale silicon photonic integrated circuits,” in Optical Fiber Communication Conference (Optical Society of America, 2014), paper Th2A.37.

Ferranti, F.

D. Spina, F. Ferranti, G. Antonini, T. Dhaene, and L. Knockaert, “Efficient variability analysis of electromagnetic systems via polynomial chaos and model order reduction,” IEEE Trans. Compon. Packag. Manuf. Technol. 4, 1038–1051 (2014).
[Crossref]

Fiers, M.

W. Bogaerts, M. Fiers, and P. Dumon, “Design challenges in silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 20, 1–8 (2014).
[Crossref]

Fishman, G. S.

G. S. Fishman, Monte Carlo: Concepts, Algorithms, and Applications (Springer-Verlag, 1996).

Flueckiger, J.

L. Chrostowski, X. Wang, J. Flueckiger, Y. Wu, Y. Wang, and S. T. Fard, “Impact of fabrication non-uniformity on chip-scale silicon photonic integrated circuits,” in Optical Fiber Communication Conference (Optical Society of America, 2014), paper Th2A.37.

Galland, C.

M. Streshinsky, R. Ding, Y. Liu, A. Novack, C. Galland, A.-J. Lim, G. Q. Lo, T. Baehr-Jones, and M. Hochberg, “The road to affordable, large-scale silicon photonics,” Opt. Photon. News 24(9), 32–39 (2013).
[Crossref]

Ganapathysubramanian, B.

B. Ganapathysubramanian and N. Zabaras, “Sparse grid collocation schemes for stochastic natural convection problems,” J. Comput. Phys. 225, 652–685 (2007).
[Crossref]

Hochberg, M.

M. Streshinsky, R. Ding, Y. Liu, A. Novack, C. Galland, A.-J. Lim, G. Q. Lo, T. Baehr-Jones, and M. Hochberg, “The road to affordable, large-scale silicon photonics,” Opt. Photon. News 24(9), 32–39 (2013).
[Crossref]

L. Chrostowski and M. Hochberg, Silicon Photonics Design: From Devices to Systems (Cambridge University, 2015).

Johnston, P. R.

P. R. Johnston, “Defibrillation thresholds: a generalised polynomial chaos study,” in Conference in Computing in Cardiology, Cambridge, MA (Sept.7–10, 2014).

Kim, K.-K. K.

K.-K. K. Kim and R. D. Braatz, “Generalized polynomial chaos expansion approaches to approximate stochastic receding horizon control with applications to probabilistic collision checking and avoidance,” in IEEE International Conference on Control Applications, Dubrovnik, Croatia (Oct.3–5, 2012).

Knockaert, L.

D. Spina, F. Ferranti, G. Antonini, T. Dhaene, and L. Knockaert, “Efficient variability analysis of electromagnetic systems via polynomial chaos and model order reduction,” IEEE Trans. Compon. Packag. Manuf. Technol. 4, 1038–1051 (2014).
[Crossref]

Lim, A.-J.

M. Streshinsky, R. Ding, Y. Liu, A. Novack, C. Galland, A.-J. Lim, G. Q. Lo, T. Baehr-Jones, and M. Hochberg, “The road to affordable, large-scale silicon photonics,” Opt. Photon. News 24(9), 32–39 (2013).
[Crossref]

Liu, P. L.

A. Der Kiureghian and P. L. Liu, “Structural reliability under incomplete probability information,” J. Eng. Mech. 112, 85–104 (1986).
[Crossref]

Liu, Y.

M. Streshinsky, R. Ding, Y. Liu, A. Novack, C. Galland, A.-J. Lim, G. Q. Lo, T. Baehr-Jones, and M. Hochberg, “The road to affordable, large-scale silicon photonics,” Opt. Photon. News 24(9), 32–39 (2013).
[Crossref]

Lo, G. Q.

M. Streshinsky, R. Ding, Y. Liu, A. Novack, C. Galland, A.-J. Lim, G. Q. Lo, T. Baehr-Jones, and M. Hochberg, “The road to affordable, large-scale silicon photonics,” Opt. Photon. News 24(9), 32–39 (2013).
[Crossref]

Loeve, M.

M. Loeve, Probability Theory, 4th ed. (Springer-Verlag, 1977).

Luyssaert, B.

Marzouk, Y.

Melloni, A.

T. Weng, Z. Zhang, Z. Su, Y. Marzouk, A. Melloni, and L. Daniel, “Uncertainty quantification of silicon photonic devices with correlated and non-Gaussian random parameters,” Opt. Express 23, 4242–4254 (2015).
[Crossref]

D. Cassano, F. Morichetti, and A. Melloni, “Statistical analysis of photonic integrated circuits via polynomial-chaos expansion,” in Advanced Photonics (Optical Society of America, 2013), paper JT3A.8.

Morichetti, F.

D. Cassano, F. Morichetti, and A. Melloni, “Statistical analysis of photonic integrated circuits via polynomial-chaos expansion,” in Advanced Photonics (Optical Society of America, 2013), paper JT3A.8.

Novack, A.

M. Streshinsky, R. Ding, Y. Liu, A. Novack, C. Galland, A.-J. Lim, G. Q. Lo, T. Baehr-Jones, and M. Hochberg, “The road to affordable, large-scale silicon photonics,” Opt. Photon. News 24(9), 32–39 (2013).
[Crossref]

Novak, E.

V. Barthelmann, E. Novak, and K. Ritter, “High dimensional polynomial interpolation on sparse grid,” Adv. Comput. Math. 12, 273–288 (2000).
[Crossref]

E. Novak and K. Ritter, “Simple cubature formulas with high polynomial exactness,” Constr. Approx. 15, 499–522 (1999).
[Crossref]

E. Novak and K. Ritter, “High dimensional integration of smooth functions over cubes,” Numer. Math. 75, 79–97 (1996).
[Crossref]

Ritter, K.

V. Barthelmann, E. Novak, and K. Ritter, “High dimensional polynomial interpolation on sparse grid,” Adv. Comput. Math. 12, 273–288 (2000).
[Crossref]

E. Novak and K. Ritter, “Simple cubature formulas with high polynomial exactness,” Constr. Approx. 15, 499–522 (1999).
[Crossref]

E. Novak and K. Ritter, “High dimensional integration of smooth functions over cubes,” Numer. Math. 75, 79–97 (1996).
[Crossref]

Selvaraja, S. K.

S. K. Selvaraja, W. Bogaerts, P. Dumon, D. Van Thourhout, and R. Baets, “Subnanometer linewidth uniformity in silicon nanophotonic waveguide devices using CMOS fabrication technology,” IEEE J. Sel. Top. Quantum Electron. 16, 316–324 (2010).
[Crossref]

Spina, D.

D. Spina, F. Ferranti, G. Antonini, T. Dhaene, and L. Knockaert, “Efficient variability analysis of electromagnetic systems via polynomial chaos and model order reduction,” IEEE Trans. Compon. Packag. Manuf. Technol. 4, 1038–1051 (2014).
[Crossref]

Streshinsky, M.

M. Streshinsky, R. Ding, Y. Liu, A. Novack, C. Galland, A.-J. Lim, G. Q. Lo, T. Baehr-Jones, and M. Hochberg, “The road to affordable, large-scale silicon photonics,” Opt. Photon. News 24(9), 32–39 (2013).
[Crossref]

Su, Z.

Taillaert, D.

Van Campenhout, J.

Van Thourhout, D.

S. K. Selvaraja, W. Bogaerts, P. Dumon, D. Van Thourhout, and R. Baets, “Subnanometer linewidth uniformity in silicon nanophotonic waveguide devices using CMOS fabrication technology,” IEEE J. Sel. Top. Quantum Electron. 16, 316–324 (2010).
[Crossref]

W. Bogaerts, D. Taillaert, B. Luyssaert, P. Dumon, J. Van Campenhout, P. Bienstman, D. Van Thourhout, R. Baets, V. Wiaux, and S. Beckx, “Basic structures for photonic integrated circuits in silicon-on-insulator,” Opt. Express 12, 1583–1591 (2004).
[Crossref]

Wang, X.

L. Chrostowski, X. Wang, J. Flueckiger, Y. Wu, Y. Wang, and S. T. Fard, “Impact of fabrication non-uniformity on chip-scale silicon photonic integrated circuits,” in Optical Fiber Communication Conference (Optical Society of America, 2014), paper Th2A.37.

Wang, Y.

L. Chrostowski, X. Wang, J. Flueckiger, Y. Wu, Y. Wang, and S. T. Fard, “Impact of fabrication non-uniformity on chip-scale silicon photonic integrated circuits,” in Optical Fiber Communication Conference (Optical Society of America, 2014), paper Th2A.37.

Weiser, W. A.

W. A. Weiser and S. E. Zarantonello, “A note on piecewise linear and multilinear table interpolation in many dimensions,” Math. Comput. 50, 189–196 (1988).
[Crossref]

Weng, T.

Wiaux, V.

Wu, Y.

L. Chrostowski, X. Wang, J. Flueckiger, Y. Wu, Y. Wang, and S. T. Fard, “Impact of fabrication non-uniformity on chip-scale silicon photonic integrated circuits,” in Optical Fiber Communication Conference (Optical Society of America, 2014), paper Th2A.37.

Xiu, D.

D. Xiu, “Fast numerical methods for stochastic computations: a review,” Commun. Comput. Phys. 5, 242–272 (2009).

Zabaras, N.

B. Ganapathysubramanian and N. Zabaras, “Sparse grid collocation schemes for stochastic natural convection problems,” J. Comput. Phys. 225, 652–685 (2007).
[Crossref]

Zarantonello, S. E.

W. A. Weiser and S. E. Zarantonello, “A note on piecewise linear and multilinear table interpolation in many dimensions,” Math. Comput. 50, 189–196 (1988).
[Crossref]

Zhang, Z.

Adv. Comput. Math. (1)

V. Barthelmann, E. Novak, and K. Ritter, “High dimensional polynomial interpolation on sparse grid,” Adv. Comput. Math. 12, 273–288 (2000).
[Crossref]

Commun. Comput. Phys. (1)

D. Xiu, “Fast numerical methods for stochastic computations: a review,” Commun. Comput. Phys. 5, 242–272 (2009).

Constr. Approx. (1)

E. Novak and K. Ritter, “Simple cubature formulas with high polynomial exactness,” Constr. Approx. 15, 499–522 (1999).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (2)

W. Bogaerts, M. Fiers, and P. Dumon, “Design challenges in silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 20, 1–8 (2014).
[Crossref]

S. K. Selvaraja, W. Bogaerts, P. Dumon, D. Van Thourhout, and R. Baets, “Subnanometer linewidth uniformity in silicon nanophotonic waveguide devices using CMOS fabrication technology,” IEEE J. Sel. Top. Quantum Electron. 16, 316–324 (2010).
[Crossref]

IEEE Trans. Compon. Packag. Manuf. Technol. (1)

D. Spina, F. Ferranti, G. Antonini, T. Dhaene, and L. Knockaert, “Efficient variability analysis of electromagnetic systems via polynomial chaos and model order reduction,” IEEE Trans. Compon. Packag. Manuf. Technol. 4, 1038–1051 (2014).
[Crossref]

Int. J. Numer. Meth. Eng. (1)

N. Agarwal and N. R. Aluru, “Weighted Smolyak algorithm for solution of stochastic differential equations on non-uniform probability measures,” Int. J. Numer. Meth. Eng. 85, 1365–1389 (2011).
[Crossref]

J. Comput. Phys. (1)

B. Ganapathysubramanian and N. Zabaras, “Sparse grid collocation schemes for stochastic natural convection problems,” J. Comput. Phys. 225, 652–685 (2007).
[Crossref]

J. Eng. Mech. (1)

A. Der Kiureghian and P. L. Liu, “Structural reliability under incomplete probability information,” J. Eng. Mech. 112, 85–104 (1986).
[Crossref]

J. Microelectromech. Syst. (1)

N. Agarwal and N. R. Aluru, “Stochastic analysis of electrostatic mems subjected to parameter variations,” J. Microelectromech. Syst. 18, 1454–1468 (2009).
[Crossref]

Math. Comput. (1)

W. A. Weiser and S. E. Zarantonello, “A note on piecewise linear and multilinear table interpolation in many dimensions,” Math. Comput. 50, 189–196 (1988).
[Crossref]

Numer. Math. (1)

E. Novak and K. Ritter, “High dimensional integration of smooth functions over cubes,” Numer. Math. 75, 79–97 (1996).
[Crossref]

Opt. Express (2)

Opt. Photon. News (1)

M. Streshinsky, R. Ding, Y. Liu, A. Novack, C. Galland, A.-J. Lim, G. Q. Lo, T. Baehr-Jones, and M. Hochberg, “The road to affordable, large-scale silicon photonics,” Opt. Photon. News 24(9), 32–39 (2013).
[Crossref]

Other (8)

P. R. Johnston, “Defibrillation thresholds: a generalised polynomial chaos study,” in Conference in Computing in Cardiology, Cambridge, MA (Sept.7–10, 2014).

K.-K. K. Kim and R. D. Braatz, “Generalized polynomial chaos expansion approaches to approximate stochastic receding horizon control with applications to probabilistic collision checking and avoidance,” in IEEE International Conference on Control Applications, Dubrovnik, Croatia (Oct.3–5, 2012).

D. Cassano, F. Morichetti, and A. Melloni, “Statistical analysis of photonic integrated circuits via polynomial-chaos expansion,” in Advanced Photonics (Optical Society of America, 2013), paper JT3A.8.

M. S. Eldred, “Recent advance in non-intrusive polynomial-chaos and stochastic collocation methods for uncertainty analysis and design,” in Proceedings of 50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Palm Springs, California (May, 2009).

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Figures (8)

Fig. 1.
Fig. 1. Upper plot shows the perspective view of a symmetric DC. Red arrows present the flow of light. Part of the light is coupled from bottom waveguide to the above one. Cross section is amplified in the lower plot. The mean width and thickness of the DC are w0 and t0, respectively. The width w and thickness t of the fabricated DC are indicated as dashed boxes. The refractive indexes are nsi=3.44, nSiO2=1.45.
Fig. 2.
Fig. 2. 2D contour plot of field coupling coefficient versus waveguide width and thickness.
Fig. 3.
Fig. 3. Flow chart of the proposed technique.
Fig. 4.
Fig. 4. Top: the red exes (×) represent the interpolation nodes for the normalized independent random variables ξ1 and ξ2 used to build the SC model. Bottom: the blue circles (°) are the corresponding values for the correlated random variables w and t used to compute the coupling coefficients in Fimmwave.
Fig. 5.
Fig. 5. Sampling points used to perform the MC analysis through direct Fimmwave simulations for the correlated random variables (w,t). The corresponding values for the independent random variables (ξ1,ξ2) are used to evaluate the SC model computed.
Fig. 6.
Fig. 6. Blue circles (°): coupling coefficient computed via the MC analysis for the 10000 (w,t) samples shown in Fig. 5. Red (×)-markers: corresponding values obtained by evaluating the SC model.
Fig. 7.
Fig. 7. PDF and CDF of the coupling coefficient for λ=1.55  μm. The blue solid and red dashed line are PDF and CDF obtained by means of the SC model, respectively, while the blue circles and red squares represent the same quantities computed by means of the MC analysis.
Fig. 8.
Fig. 8. PDF and CDF of the 3 dB-coupling length for λ=1.55  μm. The blue solid and red dashed line are PDF and CDF obtained by means of the SC model, respectively, while the blue circles and red squares represent the same quantities computed by means of the MC analysis.

Tables (2)

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Table 1. Performance Summary of SC and MC Simulation

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Table 2. Computation Time of SC and MC Simulation

Equations (21)

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Y(ξ)=i=1QY(ξi)Li(ξ),
Li(ξ)=i=1,ijQξξiξjξi,
Y(ξ)=i1=1Qk1iN=1QkNY(ξi1k1,,ξiNkN)(Li1k1LiNkN),
Q=n=1NQkn.
μ(Y(ξ))=ΩY(ξ)W(ξ)dξ,
μ(Y(ξ))=Ωi=1QY(ξi)Li(ξ)W(ξ)dξ,
K(z)=sin2(κz+κ0).
κ=πλ(neff_oneff_e),
Wη=12πdet(C)1/2exp(12(ημ)TC1(ημ)),
C=[(w0σw)2ρw0σwt0σtρw0σwt0σt(t0σt)2],
η=μ+VE1/2ξ,
l3dB=arcsin(sqrt(0.5))/κ.
C=VEVT
Wη=12πdet(E)1/2exp(12(ημ)TVE1VT(ημ)).
Wx=12πdet(E)1/2exp(12xTE1x),
x=VT(ημ).
x=E1/2ξ,
U(ξ)=i=1QY(ξi)Li(ξ),
Y(ξ;)=Uk1UkN=i1=1Qk1iN=1QkNY(ξi1k1,,ξiNkN)(Li1k1LiNkN),
Aq,N(ξ)=qN+1|k|q(1)q|k|(N1qk)(Uk1UkN),
Hq,N=qN+1|k|qΘ1k1××ΘNkN.

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