Abstract

In this paper, a novel baseband macromodeling framework for linear passive photonic circuits is proposed, which is able to build accurate and compact models while taking into account the nonidealities, such as higher order dispersion and wavelength-dependent losses of the circuits. Compared to a previous modeling method based on the vector fitting algorithm, the proposed modeling approach introduces a novel complex vector fitting technique. It can generate a half-size state-space model for the same applications, thereby achieving a major improvement in efficiency of the time-domain simulations. The proposed modeling framework requires only measured or simulated scattering parameters as input, which are widely used to represent linear and passive systems. Three photonic circuits are studied to demonstrate the accuracy and efficiency of the proposed technique.

© 2019 Chinese Laser Press

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References

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2019 (1)

2018 (3)

2017 (2)

Z. Zhang, R. Wu, Y. Wang, C. Zhang, E. J. Stanton, C. L. Schow, K. T. Cheng, and J. E. Bowers, “Compact modeling for silicon photonic heterogeneously integrated circuits,” J. Lightwave Technol. 35, 2973–2980 (2017).
[Crossref]

J. Becerra, F. Vega, and F. Rachidi, “Extrapolation of a truncated spectrum with Hilbert transform for obtaining causal impulse responses,” IEEE Trans. Electromagn. Compat. 59, 454–460 (2017).
[Crossref]

2016 (2)

2015 (2)

S. Mingaleev, A. Richter, E. Sokolov, C. Arellano, and I. Koltchanov, “Towards an automated design framework for large-scale photonic integrated circuits,” Proc. SPIE 9516, 951602 (2015).
[Crossref]

C. Sorace-Agaskar, J. Leu, M. R. Watts, and V. Stojanovic, “Electro-optical co-simulation for integrated CMOS photonic circuits with VerilogA,” Opt. Express 23, 27180–27203 (2015).
[Crossref]

2014 (3)

D. Dai, M. Piels, and J. E. Bowers, “Monolithic germanium/silicon photodetectors with decoupled structures: resonant APDs and UTC photodiodes,” IEEE J. Sel. Top. Quantum Electron. 20, 43–56 (2014).
[Crossref]

J. Pond, C. Cone, L. Chrostowski, J. Klein, J. Flueckiger, A. Liu, D. McGuire, and X. Wang, “A complete design flow for silicon photonics,” Proc. SPIE 9133, 913310 (2014).
[Crossref]

P. Qiao, G.-L. Su, Y. Rao, M. C. Wu, C. J. Chang-Hasnain, and S. L. Chuang, “Comprehensive model of 1550 nm MEMS-tunable high-contrast-grating VCSELs,” Opt. Express 22, 8541–8555 (2014).
[Crossref]

2013 (1)

2012 (1)

2010 (3)

B. Gustavsen, “Fast passivity enforcement for S-parameter models by perturbation of residue matrix eigenvalues,” IEEE Trans. Adv. Packag. 33, 257–265 (2010).
[Crossref]

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

M. Jalali, M. K. Moravvej-Farshi, S. Masudy-Panah, and A. Nabavi, “An equivalent lumped circuit model for thin avalanche photodiodes with nonuniform electric field profile,” J. Lightwave Technol. 28, 3395–3402 (2010).
[Crossref]

2009 (2)

D. Deschrijver and T. Dhaene, “Fast passivity enforcement of S-parameter macromodels by pole perturbation,” IEEE Trans. Microwave Theory Tech. 57, 620–626 (2009).
[Crossref]

T. Dhaene, D. Deschrijver, and N. Stevens, “Efficient algorithm for passivity enforcement of S-parameter-based macromodels,” IEEE Trans. Microwave Theory Tech. 57, 415–420 (2009).
[Crossref]

2008 (2)

D. Deschrijver, M. Mrozowski, T. Dhaene, and D. De Zutter, “Macromodeling of multiport systems using a fast implementation of the vector fitting method,” IEEE Microwave Compon. Lett. 18, 383–385 (2008).
[Crossref]

B. Gustavsen and A. Semlyen, “Fast passivity assessment for S-parameter rational models via a half-size test matrix,” IEEE Trans. Microwave Theory Tech. 56, 2701–2708 (2008).
[Crossref]

2007 (2)

F. Xia, M. Rooks, L. Sekaric, and Y. Vlasov, “Ultra-compact high order ring resonator filters using submicron silicon photonic wires for on-chip optical interconnects,” Opt. Express 15, 11934–11941 (2007).
[Crossref]

D. Deschrijver, B. Haegeman, and T. Dhaene, “Orthonormal vector fitting: a robust macromodeling tool for rational approximation of frequency domain responses,” IEEE Trans. Adv. Packag. 30, 216–225 (2007).
[Crossref]

2006 (1)

B. Gustavsen, “Improving the pole relocating properties of vector fitting,” IEEE Trans. Power Delivery 21, 1587–1592 (2006).
[Crossref]

2005 (1)

K. Sou and O. L. De Weck, “Fast time-domain simulation for large-order linear time-invariant state space systems,” Internat. J. Numer. Methods Eng. 63, 681–708 (2005).
[Crossref]

2004 (2)

S. Grivet-Talocia, “Passivity enforcement via perturbation of Hamiltonian matrices,” IEEE Trans. Circuits Syst. I 51, 1755–1769 (2004).
[Crossref]

D. Saraswat, R. Achar, and M. S. Nakhla, “A fast algorithm and practical considerations for passive macromodeling of measured/simulated data,” IEEE Trans. Adv. Packag. 27, 57–70 (2004).
[Crossref]

1999 (2)

P. V. Mena, J. J. Morikuni, S. Kang, A. V. Harton, and K. W. Wyatt, “A comprehensive circuit-level model of vertical-cavity surface-emitting lasers,” J. Lightwave Technol. 17, 2612–2632 (1999).
[Crossref]

B. Gustavsen and A. Semlyen, “Rational approximation of frequency domain responses by vector fitting,” IEEE Trans. Power Delivery 14, 1052–1061 (1999).
[Crossref]

Achar, R.

D. Saraswat, R. Achar, and M. S. Nakhla, “A fast algorithm and practical considerations for passive macromodeling of measured/simulated data,” IEEE Trans. Adv. Packag. 27, 57–70 (2004).
[Crossref]

Akiyama, S.

Arellano, C.

S. Mingaleev, A. Richter, E. Sokolov, C. Arellano, and I. Koltchanov, “Towards an automated design framework for large-scale photonic integrated circuits,” Proc. SPIE 9516, 951602 (2015).
[Crossref]

Bahrami, H.

Balaban, P.

M. C. Jeruchim, P. Balaban, and K. S. Shanmugan, Simulation of Communication Systems: Modeling, Methodology and Techniques (Springer, 2006).

Bandinu, M.

S. Grivet-Talocia, M. Bandinu, and F. G. Canavero, “An automatic algorithm for equivalent circuit extraction from noisy frequency responses,” in IEEE International Symposium on Electromagnetic Compatibility (IEEE, 2005), pp. 163–168.

Beausoleil, R. G.

Becerra, J.

J. Becerra, F. Vega, and F. Rachidi, “Extrapolation of a truncated spectrum with Hilbert transform for obtaining causal impulse responses,” IEEE Trans. Electromagn. Compat. 59, 454–460 (2017).
[Crossref]

Bogaerts, W.

Bowers, J. E.

Canavero, F. G.

S. Grivet-Talocia, M. Bandinu, and F. G. Canavero, “An automatic algorithm for equivalent circuit extraction from noisy frequency responses,” in IEEE International Symposium on Electromagnetic Compatibility (IEEE, 2005), pp. 163–168.

Chang-Hasnain, C. J.

Chen, C.

Cheng, K. T.

Cheung, C.-M.

C.-U. Lei, C.-M. Cheung, H.-K. Kwan, and N. Wong, “Efficient complex continuous-time IIR filter design via generalized vector fitting,” in Proceedings of the International MultiConference of Engineers and Computer Scientists (IAENG, 2008).

C.-U. Lei, C.-M. Cheung, H.-K. Kwan, and N. Wong, “Efficient design of arbitrary complex response continuous-time IIR filter,” in Trends in Communication Technologies and Engineering Science, S.-I. Ao, X. Huang, and P.-K. A. Wai, eds. (Springer, 2009).

Chrostowski, L.

W. Bogaerts and L. Chrostowski, “Silicon photonics circuit design: methods, tools and challenges,” Laser Photon. Rev. 12, 1700237 (2018).
[Crossref]

J. Pond, C. Cone, L. Chrostowski, J. Klein, J. Flueckiger, A. Liu, D. McGuire, and X. Wang, “A complete design flow for silicon photonics,” Proc. SPIE 9133, 913310 (2014).
[Crossref]

Chuang, S. L.

Cone, C.

J. Pond, C. Cone, L. Chrostowski, J. Klein, J. Flueckiger, A. Liu, D. McGuire, and X. Wang, “A complete design flow for silicon photonics,” Proc. SPIE 9133, 913310 (2014).
[Crossref]

Dai, D.

D. Dai, M. Piels, and J. E. Bowers, “Monolithic germanium/silicon photodetectors with decoupled structures: resonant APDs and UTC photodiodes,” IEEE J. Sel. Top. Quantum Electron. 20, 43–56 (2014).
[Crossref]

De Weck, O. L.

K. Sou and O. L. De Weck, “Fast time-domain simulation for large-order linear time-invariant state space systems,” Internat. J. Numer. Methods Eng. 63, 681–708 (2005).
[Crossref]

De Zutter, D.

D. Deschrijver, M. Mrozowski, T. Dhaene, and D. De Zutter, “Macromodeling of multiport systems using a fast implementation of the vector fitting method,” IEEE Microwave Compon. Lett. 18, 383–385 (2008).
[Crossref]

Deschrijver, D.

D. Deschrijver and T. Dhaene, “Fast passivity enforcement of S-parameter macromodels by pole perturbation,” IEEE Trans. Microwave Theory Tech. 57, 620–626 (2009).
[Crossref]

T. Dhaene, D. Deschrijver, and N. Stevens, “Efficient algorithm for passivity enforcement of S-parameter-based macromodels,” IEEE Trans. Microwave Theory Tech. 57, 415–420 (2009).
[Crossref]

D. Deschrijver, M. Mrozowski, T. Dhaene, and D. De Zutter, “Macromodeling of multiport systems using a fast implementation of the vector fitting method,” IEEE Microwave Compon. Lett. 18, 383–385 (2008).
[Crossref]

D. Deschrijver, B. Haegeman, and T. Dhaene, “Orthonormal vector fitting: a robust macromodeling tool for rational approximation of frequency domain responses,” IEEE Trans. Adv. Packag. 30, 216–225 (2007).
[Crossref]

N. Stevens, D. Deschrijver, and T. Dhaene, “Fast automatic order estimation of rational macromodels for signal integrity analysis,” in IEEE Workshop on Signal Propagation on Interconnects (IEEE, 2007), pp. 89–92.

Dhaene, T.

Y. Ye, D. Spina, W. Bogaerts, and T. Dhaene, “Baseband macromodeling of linear photonic circuits for time-domain simulations,” J. Lightwave Technol. 37, 1364–1373 (2019).
[Crossref]

Y. Ye, D. Spina, Y. Xing, W. Bogaerts, and T. Dhaene, “Numerical modeling of a linear photonic system for accurate and efficient time-domain simulations,” Photon. Res. 6, 560–573 (2018).
[Crossref]

D. Deschrijver and T. Dhaene, “Fast passivity enforcement of S-parameter macromodels by pole perturbation,” IEEE Trans. Microwave Theory Tech. 57, 620–626 (2009).
[Crossref]

T. Dhaene, D. Deschrijver, and N. Stevens, “Efficient algorithm for passivity enforcement of S-parameter-based macromodels,” IEEE Trans. Microwave Theory Tech. 57, 415–420 (2009).
[Crossref]

D. Deschrijver, M. Mrozowski, T. Dhaene, and D. De Zutter, “Macromodeling of multiport systems using a fast implementation of the vector fitting method,” IEEE Microwave Compon. Lett. 18, 383–385 (2008).
[Crossref]

D. Deschrijver, B. Haegeman, and T. Dhaene, “Orthonormal vector fitting: a robust macromodeling tool for rational approximation of frequency domain responses,” IEEE Trans. Adv. Packag. 30, 216–225 (2007).
[Crossref]

N. Stevens, D. Deschrijver, and T. Dhaene, “Fast automatic order estimation of rational macromodels for signal integrity analysis,” in IEEE Workshop on Signal Propagation on Interconnects (IEEE, 2007), pp. 89–92.

Fiorentino, M.

Flueckiger, J.

J. Pond, C. Cone, L. Chrostowski, J. Klein, J. Flueckiger, A. Liu, D. McGuire, and X. Wang, “A complete design flow for silicon photonics,” Proc. SPIE 9133, 913310 (2014).
[Crossref]

Grivet-Talocia, S.

S. Grivet-Talocia, “Passivity enforcement via perturbation of Hamiltonian matrices,” IEEE Trans. Circuits Syst. I 51, 1755–1769 (2004).
[Crossref]

S. Grivet-Talocia, M. Bandinu, and F. G. Canavero, “An automatic algorithm for equivalent circuit extraction from noisy frequency responses,” in IEEE International Symposium on Electromagnetic Compatibility (IEEE, 2005), pp. 163–168.

S. Grivet-Talocia and B. Gustavsen, Passive Macromodeling: Theory and Applications (Wiley, 2016).

Gunupudi, P.

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

Gustavsen, B.

B. Gustavsen, “Fast passivity enforcement for S-parameter models by perturbation of residue matrix eigenvalues,” IEEE Trans. Adv. Packag. 33, 257–265 (2010).
[Crossref]

B. Gustavsen and A. Semlyen, “Fast passivity assessment for S-parameter rational models via a half-size test matrix,” IEEE Trans. Microwave Theory Tech. 56, 2701–2708 (2008).
[Crossref]

B. Gustavsen, “Improving the pole relocating properties of vector fitting,” IEEE Trans. Power Delivery 21, 1587–1592 (2006).
[Crossref]

B. Gustavsen and A. Semlyen, “Rational approximation of frequency domain responses by vector fitting,” IEEE Trans. Power Delivery 14, 1052–1061 (1999).
[Crossref]

S. Grivet-Talocia and B. Gustavsen, Passive Macromodeling: Theory and Applications (Wiley, 2016).

Haegeman, B.

D. Deschrijver, B. Haegeman, and T. Dhaene, “Orthonormal vector fitting: a robust macromodeling tool for rational approximation of frequency domain responses,” IEEE Trans. Adv. Packag. 30, 216–225 (2007).
[Crossref]

Harton, A. V.

Jakubczyk, Z. J.

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

Jalali, M.

Jeruchim, M. C.

M. C. Jeruchim, P. Balaban, and K. S. Shanmugan, Simulation of Communication Systems: Modeling, Methodology and Techniques (Springer, 2006).

Kang, S.

Klein, J.

J. Pond, C. Cone, L. Chrostowski, J. Klein, J. Flueckiger, A. Liu, D. McGuire, and X. Wang, “A complete design flow for silicon photonics,” Proc. SPIE 9133, 913310 (2014).
[Crossref]

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

Koltchanov, I.

S. Mingaleev, A. Richter, E. Sokolov, C. Arellano, and I. Koltchanov, “Towards an automated design framework for large-scale photonic integrated circuits,” Proc. SPIE 9516, 951602 (2015).
[Crossref]

Kurahashi, T.

Kwan, H.-K.

C.-U. Lei, C.-M. Cheung, H.-K. Kwan, and N. Wong, “Efficient complex continuous-time IIR filter design via generalized vector fitting,” in Proceedings of the International MultiConference of Engineers and Computer Scientists (IAENG, 2008).

C.-U. Lei, C.-M. Cheung, H.-K. Kwan, and N. Wong, “Efficient design of arbitrary complex response continuous-time IIR filter,” in Trends in Communication Technologies and Engineering Science, S.-I. Ao, X. Huang, and P.-K. A. Wai, eds. (Springer, 2009).

Lei, C.-U.

C.-U. Lei, C.-M. Cheung, H.-K. Kwan, and N. Wong, “Efficient design of arbitrary complex response continuous-time IIR filter,” in Trends in Communication Technologies and Engineering Science, S.-I. Ao, X. Huang, and P.-K. A. Wai, eds. (Springer, 2009).

C.-U. Lei, C.-M. Cheung, H.-K. Kwan, and N. Wong, “Efficient complex continuous-time IIR filter design via generalized vector fitting,” in Proceedings of the International MultiConference of Engineers and Computer Scientists (IAENG, 2008).

Leu, J.

Li, C.

Liu, A.

J. Pond, C. Cone, L. Chrostowski, J. Klein, J. Flueckiger, A. Liu, D. McGuire, and X. Wang, “A complete design flow for silicon photonics,” Proc. SPIE 9133, 913310 (2014).
[Crossref]

Masudy-Panah, S.

McGuire, D.

J. Pond, C. Cone, L. Chrostowski, J. Klein, J. Flueckiger, A. Liu, D. McGuire, and X. Wang, “A complete design flow for silicon photonics,” Proc. SPIE 9133, 913310 (2014).
[Crossref]

Mena, P. V.

Mingaleev, S.

S. Mingaleev, A. Richter, E. Sokolov, C. Arellano, and I. Koltchanov, “Towards an automated design framework for large-scale photonic integrated circuits,” Proc. SPIE 9516, 951602 (2015).
[Crossref]

Moravvej-Farshi, M. K.

Morikuni, J. J.

Morito, K.

Mrozowski, M.

D. Deschrijver, M. Mrozowski, T. Dhaene, and D. De Zutter, “Macromodeling of multiport systems using a fast implementation of the vector fitting method,” IEEE Microwave Compon. Lett. 18, 383–385 (2008).
[Crossref]

Nabavi, A.

Nakhla, M. S.

D. Saraswat, R. Achar, and M. S. Nakhla, “A fast algorithm and practical considerations for passive macromodeling of measured/simulated data,” IEEE Trans. Adv. Packag. 27, 57–70 (2004).
[Crossref]

Nomura, S.

Palermo, S.

Park, C. S.

Piels, M.

D. Dai, M. Piels, and J. E. Bowers, “Monolithic germanium/silicon photodetectors with decoupled structures: resonant APDs and UTC photodiodes,” IEEE J. Sel. Top. Quantum Electron. 20, 43–56 (2014).
[Crossref]

M. Piels, A. Ramaswamy, and J. E. Bowers, “Nonlinear modeling of waveguide photodetectors,” Opt. Express 21, 15634–15644 (2013).
[Crossref]

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IEEE Trans. Adv. Packag. (4)

B. Gustavsen, “Fast passivity enforcement for S-parameter models by perturbation of residue matrix eigenvalues,” IEEE Trans. Adv. Packag. 33, 257–265 (2010).
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D. Deschrijver and T. Dhaene, “Fast passivity enforcement of S-parameter macromodels by pole perturbation,” IEEE Trans. Microwave Theory Tech. 57, 620–626 (2009).
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Opt. Express (5)

Photon. Res. (1)

Proc. SPIE (2)

J. Pond, C. Cone, L. Chrostowski, J. Klein, J. Flueckiger, A. Liu, D. McGuire, and X. Wang, “A complete design flow for silicon photonics,” Proc. SPIE 9133, 913310 (2014).
[Crossref]

S. Mingaleev, A. Richter, E. Sokolov, C. Arellano, and I. Koltchanov, “Towards an automated design framework for large-scale photonic integrated circuits,” Proc. SPIE 9516, 951602 (2015).
[Crossref]

Other (7)

N. Stevens, D. Deschrijver, and T. Dhaene, “Fast automatic order estimation of rational macromodels for signal integrity analysis,” in IEEE Workshop on Signal Propagation on Interconnects (IEEE, 2007), pp. 89–92.

S. Grivet-Talocia, M. Bandinu, and F. G. Canavero, “An automatic algorithm for equivalent circuit extraction from noisy frequency responses,” in IEEE International Symposium on Electromagnetic Compatibility (IEEE, 2005), pp. 163–168.

S. Grivet-Talocia and B. Gustavsen, Passive Macromodeling: Theory and Applications (Wiley, 2016).

C.-U. Lei, C.-M. Cheung, H.-K. Kwan, and N. Wong, “Efficient complex continuous-time IIR filter design via generalized vector fitting,” in Proceedings of the International MultiConference of Engineers and Computer Scientists (IAENG, 2008).

C.-U. Lei, C.-M. Cheung, H.-K. Kwan, and N. Wong, “Efficient design of arbitrary complex response continuous-time IIR filter,” in Trends in Communication Technologies and Engineering Science, S.-I. Ao, X. Huang, and P.-K. A. Wai, eds. (Springer, 2009).

M. C. Jeruchim, P. Balaban, and K. S. Shanmugan, Simulation of Communication Systems: Modeling, Methodology and Techniques (Springer, 2006).

https://www.sintef.no/projectweb/vectfit/ .

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Figures (21)

Fig. 1.
Fig. 1. Spectrum of the modulated optical signal (top) and its baseband equivalent signal (bottom).
Fig. 2.
Fig. 2. Spectrum of bandpass systems (top) and the corresponding baseband equivalent systems (bottom).
Fig. 3.
Fig. 3. The simulated or measured scattering parameters at a set of discrete frequency samples (top) and the corresponding baseband scattering parameters (bottom).
Fig. 4.
Fig. 4. Flow chart of the CVF modeling approach (left branch) and the one presented in Ref. [12] (right branch).
Fig. 5.
Fig. 5. Spectrum of the model SVF(f) (top) and the model SlVF(f) represented by Model (9) (bottom).
Fig. 6.
Fig. 6. Poles of the model SVF(f) (left) and the model SlVF(f) represented by Model (9) (right).
Fig. 7.
Fig. 7. Example 7.A. The structure of the five-ring resonator filter.
Fig. 8.
Fig. 8. Example 7.A. The accuracy of the VF-based Model (9) (top) built via the technique in Ref. [12] with 108 poles and the CVF Model (6) (bottom) built via the newly proposed technique with 54 poles; the red solid lines represent the simulated scattering parameters, the blue dashed lines represent the models, while the green lines are the magnitude of the error between the two.
Fig. 9.
Fig. 9. Example 7.A. The poles of the CVF Model (6) from the proposed technique (represented by circles) and the VF-based Model (9) from the technique in Ref. [12] (represented by crosses).
Fig. 10.
Fig. 10. Example 7.A. The in-phase part I(t) and quadrature part Q(t) of the 16 QAM input signal.
Fig. 11.
Fig. 11. Example 7.A. The output signals at P1, P2, P3, and P4 obtained from baseband time-domain simulations of Models (6), (9), and (14).
Fig. 12.
Fig. 12. Example 7.A. Constellation diagrams of the transmission signal at P3 calculated from different models.
Fig. 13.
Fig. 13. Example 7.B. The schematic circuit of the ring-loaded MZI filter.
Fig. 14.
Fig. 14. Example 7.B. The accuracy of the VF-based Model (9) (top) built via the technique in Ref. [12] with 42 poles and the CVF Model (6) (bottom) built via the newly proposed technique with 21 poles; the red solid lines represent the simulated scattering parameters, and the blue dashed lines represent the models, while the green lines are the magnitude of the error between the two.
Fig. 15.
Fig. 15. Example 7.B. The poles of the CVF Model (6) (represented by circles) and the Model (9) from the VF-based technique in Ref. [12] (represented by crosses).
Fig. 16.
Fig. 16. Example 7.B. The output signals at P3 and P4 obtained from baseband time-domain simulation of Models (6), (9), and (14).
Fig. 17.
Fig. 17. Example 7.B. Constellation diagram of the transmission signal at P3 calculated from different models.
Fig. 18.
Fig. 18. Example 7.B. Constellation diagram of the transmission signal at P3 calculated from the rebuilt CVF Model (6) and the shifted Model (21), when the passband of the filter redshifts and blueshifts by 0.3 nm.
Fig. 19.
Fig. 19. Example 7.C. The structure of the Mach–Zehnder interferometer lattice filter.
Fig. 20.
Fig. 20. Example 7.C. The singular values of the scattering matrices calculated from Model (6) before and after passivity enforcement.
Fig. 21.
Fig. 21. Example 7.C. The accuracy of the VF-based Model (9) (top) built via the technique in Ref. [12] with 68 poles and the new CVF Model (6) with 34 poles (bottom); the red solid lines represent the simulated scattering parameters, and the blue dash lines represent the models, while the green lines show the error between them.

Equations (27)

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

a(t)=A(t)cos[2πfct+ϕ(t)],
al(t)=A(t)ejϕ(t),
Sl(s)=k=1KRkspk+D,
pkVF=α+jβ,pk+1VF=αjβ,
RkVF=η+jγ,Rk+1VF=ηjγ,
{dxl(t)dt=Axl(t)+Bal(t),bl(t)=Cxl(t)+Dal(t),
M=[M11M12M21M22],
M11=ABL1DHC,M12=BL1BH,M21=CHQ1C,M22=AH+CHDL1BH,L=DHDIn,Q=DDHIn.
{dxl(t)dt=(AVFj2πfcI)xl(t)+BVFal(t)bl(t)=CVFxl(t)+DVFal(t),
al(t)=alR(t)+jalI(t).
{dxlR(t)dt=ARxlR(t)AIxlI(t)+BalR(t),dxlI(t)dt=ARxlI(t)+AIxlR(t)+BalI(t),blR(t)=CRxlR(t)CIxlI(t)+DalR(t),blI(t)=CRxlI(t)+CIxlR(t)+DalI(t),
a^(t)=[alR(t)alI(t)],b^(t)=[blR(t)blI(t)],x^(t)=[xlR(t)xlI(t)],
A^=[ARAIAIAR],B^=[B00B],C^=[CRCICICR],D^=[D00D],
{dx^(t)dt=A^x^(t)+B^a^(t)b^(t)=C^x^(t)+D^a^(t),
A˜=T1A^T=[ARjAI00AR+jAI]=[A*00A],
M^=[M^11M^12M^21M^22],
M^11=A^B^L^1D^TC^,M^12=B^L^1B^T,M^21=C^TQ^1C^,M^22=A^T+C^TD^L^1B^T,L^=D^TD^I2n,Q^=D^D^TI2n.
M^11=[M11RM11IM11IM11R],M^12=[M1200M12],M^21=[M21RM21IM21IM21R],M^22=[M22RM22IM22IM22R],
M¯=P1M^P=[M*00M],
P=[Im0Im0jIm0jIm00Im0Im0jIm0jIm],
{dxl(t)dt=(Aj2πΔfIm)xl(t)+Bal(t),bl(t)=Cxl(t)+Dal(t),
L^1=([DT00DT][D00D][In00In])1=[DTDIn00DTDIn]1=[L100L1].
Q^1=[Q100Q1].
M^11=[ARAIAIAR][B00B][L100L1][DT00DT][CRCICICR]=[ARBL1DTCRAI+BL1DTCIAIBL1DTCIARBL1DTCR]=[M11RM11IM11IM11R],
M^12=[B00B][L100L1][BT00BT]=[BL1BT00BL1BT]=[M1200M12],
M^21=[CRTCITCITCRT][Q100Q1][CRCICICR]=[CRTQ1CR+CITQ1CICITQ1CRCRTQ1CICRTQ1CICITQ1CRCITQ1CI+CRTQ1CR]=[M21RM21IM21IM21R],
M^22=[ARTAITAITART]+[CRTCITCITCRT][D00D][L100L1][BT00BT]=[ART+CRTDL1BAIT+CITDL1BAITCITDL1BART+CRTDL1B]=[M22RM22IM22IM22R],

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