Abstract

We present a tool that aids in the modeling of optical circuits, both in the frequency and in the time domain. The tool is based on the definition of a node, which can have both an instantaneous input-output relation and different state variables (e.g., temperature and carrier density) and differential equations for these states. Furthermore, each node has access to part of its input history, allowing the creation of delay lines or digital filters. Additionally, a node can contain subnodes, allowing the creation of hierarchical networks. This tool can be used in numerous applications such as frequency-domain analysis of optical ring filters, time-domain analysis of optical amplifiers, microdisks, and microcavities. Although we mainly use this tool to model optical circuits, it can also be used to model other classes of dynamical systems, such as electrical circuits and neural networks.

© 2012 Optical Society of America

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  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, 687–702 (2010).
    [CrossRef]
  2. E. Lambert, M. Fiers, S. Nizamov, M. Tassaert, and W. Bogaerts, “Python bindings for the open source electromagnetic simulator Meep,” Comput. Sci. Eng. 13, 53–65 (2011).
    [CrossRef]
  3. L. M. Zhang, S. F. Yu, M. C. Nowell, D. D. Marcenac, J. E. Carroll, and R. G. S. Plumb, “Dynamic analysis of radiation and side-mode suppression in a second-order DFB laser using time-domain large-signal traveling wave model,” IEEE J. Quantum Electron. 30, 1389–1395 (1994).
  4. G. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2007).
  5. http://www.aspicdesign.com/
  6. http://www.vpiphotonics.com/optical_systems.php
  7. http://www.photond.com/products/picwave.htm
  8. http://www.rsoftdesign.com/products.php?sub=System+and+Network&itm=OptSim
  9. P. Gunupudi, T. Smy, J. Klein, and Z. Jakubczyk, “Self-consistent simulation of opto-electronic circuits using a modified nodal analysis formulation,” IEEE Trans. Adv. Packaging 33, 979–993(2010).
  10. T. Smy, P. Gunupudi, S. Mcgarry, and W. N. Ye, “Circuit-level transient simulation of configurable ring resonators using physical models,” J. Opt. Soc. Am. B 28, 1534–1543 (2011).
    [CrossRef]
  11. http://www.intec.ugent.be/caphe/
  12. K. Vandoorne, W. Dierckx, B. Schrauwen, D. Verstraeten, R. Baets, P. Bienstman, and J. Van Campenhout, “Toward optical signal processing using photonic reservoir computing,” Opt. Express 16, 11182–11192 (2008).
    [CrossRef]
  13. T. A. Davis and E. P. Natarajan, “Algorithm 8xx: Klu, a direct sparse solver for circuit simulation problems,” ACM Trans. Math. Softw. 5, 1–17.
  14. T. A. Davis, Direct Methods for Sparse Linear Systems (SIAM, 2006).
  15. K. Stanley, “Klu: a Clark Kent sparse LU factorization algorithm for circuit matrices,” SIAM Conference on Parallel Processing for Scientific Computing (PP04) (2004).
  16. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University, 2007).
  17. B. Maes, M. Fiers, and P. Bienstman, “Self-pulsing and chaos in short chains of coupled nonlinear microcavities,” Phys. Rev. A 80, 033805 (2009).
    [CrossRef]
  18. K. Vandoorne, J. Dambre, D. Verstraeten, B. Schrauwen, and P. Bienstman, “Parallel reservoir computing using optical amplifiers,” IEEE Trans. Neural Netw. 22, 1469–1481 (2011).
    [CrossRef]

2011 (3)

E. Lambert, M. Fiers, S. Nizamov, M. Tassaert, and W. Bogaerts, “Python bindings for the open source electromagnetic simulator Meep,” Comput. Sci. Eng. 13, 53–65 (2011).
[CrossRef]

T. Smy, P. Gunupudi, S. Mcgarry, and W. N. Ye, “Circuit-level transient simulation of configurable ring resonators using physical models,” J. Opt. Soc. Am. B 28, 1534–1543 (2011).
[CrossRef]

K. Vandoorne, J. Dambre, D. Verstraeten, B. Schrauwen, and P. Bienstman, “Parallel reservoir computing using optical amplifiers,” IEEE Trans. Neural Netw. 22, 1469–1481 (2011).
[CrossRef]

2010 (2)

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, 687–702 (2010).
[CrossRef]

P. Gunupudi, T. Smy, J. Klein, and Z. Jakubczyk, “Self-consistent simulation of opto-electronic circuits using a modified nodal analysis formulation,” IEEE Trans. Adv. Packaging 33, 979–993(2010).

2009 (1)

B. Maes, M. Fiers, and P. Bienstman, “Self-pulsing and chaos in short chains of coupled nonlinear microcavities,” Phys. Rev. A 80, 033805 (2009).
[CrossRef]

2008 (1)

1994 (1)

L. M. Zhang, S. F. Yu, M. C. Nowell, D. D. Marcenac, J. E. Carroll, and R. G. S. Plumb, “Dynamic analysis of radiation and side-mode suppression in a second-order DFB laser using time-domain large-signal traveling wave model,” IEEE J. Quantum Electron. 30, 1389–1395 (1994).

Agrawal, G.

G. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2007).

Baets, R.

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, 687–702 (2010).
[CrossRef]

Bienstman, P.

K. Vandoorne, J. Dambre, D. Verstraeten, B. Schrauwen, and P. Bienstman, “Parallel reservoir computing using optical amplifiers,” IEEE Trans. Neural Netw. 22, 1469–1481 (2011).
[CrossRef]

B. Maes, M. Fiers, and P. Bienstman, “Self-pulsing and chaos in short chains of coupled nonlinear microcavities,” Phys. Rev. A 80, 033805 (2009).
[CrossRef]

K. Vandoorne, W. Dierckx, B. Schrauwen, D. Verstraeten, R. Baets, P. Bienstman, and J. Van Campenhout, “Toward optical signal processing using photonic reservoir computing,” Opt. Express 16, 11182–11192 (2008).
[CrossRef]

Bogaerts, W.

E. Lambert, M. Fiers, S. Nizamov, M. Tassaert, and W. Bogaerts, “Python bindings for the open source electromagnetic simulator Meep,” Comput. Sci. Eng. 13, 53–65 (2011).
[CrossRef]

Carroll, J. E.

L. M. Zhang, S. F. Yu, M. C. Nowell, D. D. Marcenac, J. E. Carroll, and R. G. S. Plumb, “Dynamic analysis of radiation and side-mode suppression in a second-order DFB laser using time-domain large-signal traveling wave model,” IEEE J. Quantum Electron. 30, 1389–1395 (1994).

Dambre, J.

K. Vandoorne, J. Dambre, D. Verstraeten, B. Schrauwen, and P. Bienstman, “Parallel reservoir computing using optical amplifiers,” IEEE Trans. Neural Netw. 22, 1469–1481 (2011).
[CrossRef]

Davis, T. A.

T. A. Davis and E. P. Natarajan, “Algorithm 8xx: Klu, a direct sparse solver for circuit simulation problems,” ACM Trans. Math. Softw. 5, 1–17.

T. A. Davis, Direct Methods for Sparse Linear Systems (SIAM, 2006).

Dierckx, W.

Fiers, M.

E. Lambert, M. Fiers, S. Nizamov, M. Tassaert, and W. Bogaerts, “Python bindings for the open source electromagnetic simulator Meep,” Comput. Sci. Eng. 13, 53–65 (2011).
[CrossRef]

B. Maes, M. Fiers, and P. Bienstman, “Self-pulsing and chaos in short chains of coupled nonlinear microcavities,” Phys. Rev. A 80, 033805 (2009).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University, 2007).

Gunupudi, P.

T. Smy, P. Gunupudi, S. Mcgarry, and W. N. Ye, “Circuit-level transient simulation of configurable ring resonators using physical models,” J. Opt. Soc. Am. B 28, 1534–1543 (2011).
[CrossRef]

P. Gunupudi, T. Smy, J. Klein, and Z. Jakubczyk, “Self-consistent simulation of opto-electronic circuits using a modified nodal analysis formulation,” IEEE Trans. Adv. Packaging 33, 979–993(2010).

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, 687–702 (2010).
[CrossRef]

Jakubczyk, Z.

P. Gunupudi, T. Smy, J. Klein, and Z. Jakubczyk, “Self-consistent simulation of opto-electronic circuits using a modified nodal analysis formulation,” IEEE Trans. Adv. Packaging 33, 979–993(2010).

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, 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, 687–702 (2010).
[CrossRef]

Klein, J.

P. Gunupudi, T. Smy, J. Klein, and Z. Jakubczyk, “Self-consistent simulation of opto-electronic circuits using a modified nodal analysis formulation,” IEEE Trans. Adv. Packaging 33, 979–993(2010).

Lambert, E.

E. Lambert, M. Fiers, S. Nizamov, M. Tassaert, and W. Bogaerts, “Python bindings for the open source electromagnetic simulator Meep,” Comput. Sci. Eng. 13, 53–65 (2011).
[CrossRef]

Maes, B.

B. Maes, M. Fiers, and P. Bienstman, “Self-pulsing and chaos in short chains of coupled nonlinear microcavities,” Phys. Rev. A 80, 033805 (2009).
[CrossRef]

Marcenac, D. D.

L. M. Zhang, S. F. Yu, M. C. Nowell, D. D. Marcenac, J. E. Carroll, and R. G. S. Plumb, “Dynamic analysis of radiation and side-mode suppression in a second-order DFB laser using time-domain large-signal traveling wave model,” IEEE J. Quantum Electron. 30, 1389–1395 (1994).

Mcgarry, S.

Natarajan, E. P.

T. A. Davis and E. P. Natarajan, “Algorithm 8xx: Klu, a direct sparse solver for circuit simulation problems,” ACM Trans. Math. Softw. 5, 1–17.

Nizamov, S.

E. Lambert, M. Fiers, S. Nizamov, M. Tassaert, and W. Bogaerts, “Python bindings for the open source electromagnetic simulator Meep,” Comput. Sci. Eng. 13, 53–65 (2011).
[CrossRef]

Nowell, M. C.

L. M. Zhang, S. F. Yu, M. C. Nowell, D. D. Marcenac, J. E. Carroll, and R. G. S. Plumb, “Dynamic analysis of radiation and side-mode suppression in a second-order DFB laser using time-domain large-signal traveling wave model,” IEEE J. Quantum Electron. 30, 1389–1395 (1994).

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, 687–702 (2010).
[CrossRef]

Plumb, R. G. S.

L. M. Zhang, S. F. Yu, M. C. Nowell, D. D. Marcenac, J. E. Carroll, and R. G. S. Plumb, “Dynamic analysis of radiation and side-mode suppression in a second-order DFB laser using time-domain large-signal traveling wave model,” IEEE J. Quantum Electron. 30, 1389–1395 (1994).

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University, 2007).

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, 687–702 (2010).
[CrossRef]

Schrauwen, B.

K. Vandoorne, J. Dambre, D. Verstraeten, B. Schrauwen, and P. Bienstman, “Parallel reservoir computing using optical amplifiers,” IEEE Trans. Neural Netw. 22, 1469–1481 (2011).
[CrossRef]

K. Vandoorne, W. Dierckx, B. Schrauwen, D. Verstraeten, R. Baets, P. Bienstman, and J. Van Campenhout, “Toward optical signal processing using photonic reservoir computing,” Opt. Express 16, 11182–11192 (2008).
[CrossRef]

Smy, T.

T. Smy, P. Gunupudi, S. Mcgarry, and W. N. Ye, “Circuit-level transient simulation of configurable ring resonators using physical models,” J. Opt. Soc. Am. B 28, 1534–1543 (2011).
[CrossRef]

P. Gunupudi, T. Smy, J. Klein, and Z. Jakubczyk, “Self-consistent simulation of opto-electronic circuits using a modified nodal analysis formulation,” IEEE Trans. Adv. Packaging 33, 979–993(2010).

Stanley, K.

K. Stanley, “Klu: a Clark Kent sparse LU factorization algorithm for circuit matrices,” SIAM Conference on Parallel Processing for Scientific Computing (PP04) (2004).

Tassaert, M.

E. Lambert, M. Fiers, S. Nizamov, M. Tassaert, and W. Bogaerts, “Python bindings for the open source electromagnetic simulator Meep,” Comput. Sci. Eng. 13, 53–65 (2011).
[CrossRef]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University, 2007).

Van Campenhout, J.

Vandoorne, K.

K. Vandoorne, J. Dambre, D. Verstraeten, B. Schrauwen, and P. Bienstman, “Parallel reservoir computing using optical amplifiers,” IEEE Trans. Neural Netw. 22, 1469–1481 (2011).
[CrossRef]

K. Vandoorne, W. Dierckx, B. Schrauwen, D. Verstraeten, R. Baets, P. Bienstman, and J. Van Campenhout, “Toward optical signal processing using photonic reservoir computing,” Opt. Express 16, 11182–11192 (2008).
[CrossRef]

Verstraeten, D.

K. Vandoorne, J. Dambre, D. Verstraeten, B. Schrauwen, and P. Bienstman, “Parallel reservoir computing using optical amplifiers,” IEEE Trans. Neural Netw. 22, 1469–1481 (2011).
[CrossRef]

K. Vandoorne, W. Dierckx, B. Schrauwen, D. Verstraeten, R. Baets, P. Bienstman, and J. Van Campenhout, “Toward optical signal processing using photonic reservoir computing,” Opt. Express 16, 11182–11192 (2008).
[CrossRef]

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University, 2007).

Ye, W. N.

Yu, S. F.

L. M. Zhang, S. F. Yu, M. C. Nowell, D. D. Marcenac, J. E. Carroll, and R. G. S. Plumb, “Dynamic analysis of radiation and side-mode suppression in a second-order DFB laser using time-domain large-signal traveling wave model,” IEEE J. Quantum Electron. 30, 1389–1395 (1994).

Zhang, L. M.

L. M. Zhang, S. F. Yu, M. C. Nowell, D. D. Marcenac, J. E. Carroll, and R. G. S. Plumb, “Dynamic analysis of radiation and side-mode suppression in a second-order DFB laser using time-domain large-signal traveling wave model,” IEEE J. Quantum Electron. 30, 1389–1395 (1994).

ACM Trans. Math. Softw. (1)

T. A. Davis and E. P. Natarajan, “Algorithm 8xx: Klu, a direct sparse solver for circuit simulation problems,” ACM Trans. Math. Softw. 5, 1–17.

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, 687–702 (2010).
[CrossRef]

Comput. Sci. Eng. (1)

E. Lambert, M. Fiers, S. Nizamov, M. Tassaert, and W. Bogaerts, “Python bindings for the open source electromagnetic simulator Meep,” Comput. Sci. Eng. 13, 53–65 (2011).
[CrossRef]

IEEE J. Quantum Electron. (1)

L. M. Zhang, S. F. Yu, M. C. Nowell, D. D. Marcenac, J. E. Carroll, and R. G. S. Plumb, “Dynamic analysis of radiation and side-mode suppression in a second-order DFB laser using time-domain large-signal traveling wave model,” IEEE J. Quantum Electron. 30, 1389–1395 (1994).

IEEE Trans. Adv. Packaging (1)

P. Gunupudi, T. Smy, J. Klein, and Z. Jakubczyk, “Self-consistent simulation of opto-electronic circuits using a modified nodal analysis formulation,” IEEE Trans. Adv. Packaging 33, 979–993(2010).

IEEE Trans. Neural Netw. (1)

K. Vandoorne, J. Dambre, D. Verstraeten, B. Schrauwen, and P. Bienstman, “Parallel reservoir computing using optical amplifiers,” IEEE Trans. Neural Netw. 22, 1469–1481 (2011).
[CrossRef]

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

Opt. Express (1)

Phys. Rev. A (1)

B. Maes, M. Fiers, and P. Bienstman, “Self-pulsing and chaos in short chains of coupled nonlinear microcavities,” Phys. Rev. A 80, 033805 (2009).
[CrossRef]

Other (9)

http://www.intec.ugent.be/caphe/

T. A. Davis, Direct Methods for Sparse Linear Systems (SIAM, 2006).

K. Stanley, “Klu: a Clark Kent sparse LU factorization algorithm for circuit matrices,” SIAM Conference on Parallel Processing for Scientific Computing (PP04) (2004).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University, 2007).

G. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2007).

http://www.aspicdesign.com/

http://www.vpiphotonics.com/optical_systems.php

http://www.photond.com/products/picwave.htm

http://www.rsoftdesign.com/products.php?sub=System+and+Network&itm=OptSim

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Figures (5)

Fig. 1.
Fig. 1.

Structure of a node with N ports. A linear and instantaneous node is only described by a scatter matrix S. States (e.g., temperature and free carriers) can be added. In this case the node becomes nonlinear.

Fig. 2.
Fig. 2.

A coupled resonator optical waveguide (CROW). Each section is subdivided in a directional coupler and two waveguides. Port numbers are shown in the left.

Fig. 3.
Fig. 3.

Calculating the frequency response of a passive network. Using KLU, a sparse matrix solver suited for circuit matrices, we can easily calculate scatter matrices of very large networks.

Fig. 4.
Fig. 4.

Two integration routines. (a) Forward Euler integration. With larger timesteps, the accuracy decreases. (b) using adaptive stepsize and an advanced stepper routine, the solver automatically uses the optimal dt in order to maintain a desired accuracy (e.g., during switch on, the dt is reduced). Same parameters used as in [17].

Fig. 5.
Fig. 5.

(a) Topology used to simulate a complex system with ML and MC nodes. Each circle represents an SOA. Splitters are not shown. (b) The simulation time and memory usage increases linearly with the number of SOAs. Clearly there is an advantage by eliminating the ML nodes, both in terms of speed and memory usage.

Equations (12)

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

E(t)=s(t)ejωct+c.c.
sout,i(t)=jSijsin,j(t)+sext,i.
(sin,MCsin,ML)=Ctot(sout,MCsout,ML)=(CMC,MCCMC,MLCML,MCCML,ML)(sout,MCsout,ML).
sout,ML=SML,MLsin,ML,
sout,MC=SMC,MCsin,MC+sext,MC,
sin,MC=(CMC,MC+CMC,MLSML,ML(ICML,MLSML,ML)1CML,MC)sout,MC=Csout,MC.
sin,MC=(ICSMC,MC)1Csext,MC=Ssext,MC.
da(t)dt=f(a,sin,t).
(ICSMC,MC)X=C
dajdt=[i(ω0+δωj)1τ]aj+dsj;0,++dsj;1,+,
sj;1,=exp(iϕ)sj;0,++daj,
sj;1,=exp(iϕ)sj;0,++daj,

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