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Photonics Research
Vol. 7, Issue 7, 771 (2019)

Time-domain compact macromodeling of linear photonic circuits via complex vector fitting

Yinghao Ye^{1、2、3、*}, Domenico Spina¹, Dirk Deschrijver¹, Wim Bogaerts^2、3, and Tom Dhaene¹

Author Affiliations

¹IDLab, Department of Information Technology, Ghent University-imec, Ghent, Belgium

²Photonics Research Group, Department of Information Technology, Ghent University-imec, Ghent, Belgium

³The Center for Nano- and Biophotonics (NB-Photonics), Ghent, Belgium

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DOI: 10.1364/PRJ.7.000771 Cite this Article Set citation alerts

Yinghao Ye, Domenico Spina, Dirk Deschrijver, Wim Bogaerts, Tom Dhaene. Time-domain compact macromodeling of linear photonic circuits via complex vector fitting[J]. Photonics Research, 2019, 7(7): 771 Copy Citation Text

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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.

a (t) = A (t) \cos [2 π f_{c} t + ϕ (t)],

(1)

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a_{l} (t) = A (t) e^{j ϕ (t)},

(2)

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S_{l} (s) = \sum_{k = 1}^{K} \frac{R_{k}}{s - p_{k}} + D,

(3)

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p_{k}^{VF} = - α + j β, p_{k + 1}^{VF} = - α - j β,

(4)

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R_{k}^{VF} = η + j γ, R_{k + 1}^{VF} = η - j γ,

(5)

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{\begin{matrix} \frac{d x_{l} (t)}{d t} = {Ax}_{l} (t) + {Ba}_{l} (t), \\ b_{l} (t) = {Cx}_{l} (t) + {Da}_{l} (t), \end{matrix}

(6)

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M = [\begin{matrix} M_{11} & M_{12} \\ M_{21} & M_{22} \end{matrix}],

(7)

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M_{11} = A - {BL}^{- 1} D^{H} C, M_{12} = - {BL}^{- 1} B^{H}, M_{21} = C^{H} Q^{- 1} C, M_{22} = - A^{H} + C^{H} {DL}^{- 1} B^{H}, L = D^{H} D - I_{n}, Q = {DD}^{H} - I_{n} .

(8)

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{\begin{cases} \frac{d x_{l} (t)}{d t} = (A_{VF} - j 2 π f_{c} I) x_{l} (t) + B_{VF} a_{l} (t) \\ b_{l} (t) = C_{VF} x_{l} (t) + D_{VF} a_{l} (t), \end{cases}

(9)

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a_{l} (t) = a_{l R} (t) + j a_{l I} (t) .

(10)

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{\begin{matrix} \frac{d x_{l R} (t)}{d t} = A_{R} x_{l R} (t) - A_{I} x_{l I} (t) + {Ba}_{l R} (t), \\ \frac{d x_{l I} (t)}{d t} = A_{R} x_{l I} (t) + A_{I} x_{l R} (t) + {Ba}_{l I} (t), \\ b_{l R} (t) = C_{R} x_{l R} (t) - C_{I} x_{l I} (t) + {Da}_{l R} (t), \\ b_{l I} (t) = C_{R} x_{l I} (t) + C_{I} x_{l R} (t) + {Da}_{l I} (t), \end{matrix}

(11)

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\hat{a} (t) = [\begin{matrix} a_{l R} (t) \\ a_{l I} (t) \end{matrix}], \hat{b} (t) = [\begin{matrix} b_{l R} (t) \\ b_{l I} (t) \end{matrix}], \hat{x} (t) = [\begin{matrix} x_{l R} (t) \\ x_{l I} (t) \end{matrix}],

(12)

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\begin{matrix} \hat{A} = [\begin{matrix} A_{R} & - A_{I} \\ A_{I} & A_{R} \end{matrix}], & \hat{B} = [\begin{matrix} B & 0 \\ 0 & B \end{matrix}], \\ \hat{C} = [\begin{matrix} C_{R} & - C_{I} \\ C_{I} & C_{R} \end{matrix}], & \hat{D} = [\begin{matrix} D & 0 \\ 0 & D \end{matrix}], \end{matrix}

(13)

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{\begin{matrix} \frac{d \hat{x} (t)}{d t} = \hat{A} \hat{x} (t) + \hat{B} \hat{a} (t) \\ \hat{b} (t) = \hat{C} \hat{x} (t) + \hat{D} \hat{a} (t), \end{matrix}

(14)

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\tilde{A} = T^{- 1} \hat{A} T = [\begin{matrix} A_{R} - j A_{I} & 0 \\ 0 & A_{R} + j A_{I} \end{matrix}] = [\begin{matrix} A^{*} & 0 \\ 0 & A \end{matrix}],

(15)

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\hat{M} = [\begin{matrix} {\hat{M}}_{11} & {\hat{M}}_{12} \\ {\hat{M}}_{21} & {\hat{M}}_{22} \end{matrix}],

(16)

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{\hat{M}}_{11} = \hat{A} - \hat{B} {\hat{L}}^{- 1} {\hat{D}}^{T} \hat{C}, {\hat{M}}_{12} = - \hat{B} {\hat{L}}^{- 1} {\hat{B}}^{T}, {\hat{M}}_{21} = {\hat{C}}^{T} {\hat{Q}}^{- 1} \hat{C}, {\hat{M}}_{22} = - {\hat{A}}^{T} + {\hat{C}}^{T} \hat{D} {\hat{L}}^{- 1} {\hat{B}}^{T}, \hat{L} = {\hat{D}}^{T} \hat{D} - I_{2 n}, \hat{Q} = \hat{D} {\hat{D}}^{T} - I_{2 n} .

(17)

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\begin{matrix} {\hat{M}}_{11} = [\begin{matrix} M_{11}^{R} & - M_{11}^{I} \\ M_{11}^{I} & M_{11}^{R} \end{matrix}], \end{matrix} \begin{matrix} {\hat{M}}_{12} = [\begin{matrix} M_{12} & 0 \\ 0 & M_{12} \end{matrix}], \end{matrix} \begin{matrix} {\hat{M}}_{21} = [\begin{matrix} M_{21}^{R} & - M_{21}^{I} \\ M_{21}^{I} & M_{21}^{R} \end{matrix}], \end{matrix} \begin{matrix} {\hat{M}}_{22} = [\begin{matrix} M_{22}^{R} & - M_{22}^{I} \\ M_{22}^{I} & M_{22}^{R} \end{matrix}], \end{matrix}

(18)

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\bar{M} = P^{- 1} \hat{M} P = [\begin{matrix} M^{*} & 0 \\ 0 & M \end{matrix}],

(19)

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P = [\begin{matrix} I_{m} & 0 & I_{m} & 0 \\ j I_{m} & 0 & - j I_{m} & 0 \\ 0 & I_{m} & 0 & I_{m} \\ 0 & j I_{m} & 0 & - j I_{m} \end{matrix}],

(20)

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{\begin{matrix} \frac{d x_{l} (t)}{d t} = (A - j 2 π Δ f I_{m}) x_{l} (t) + {Ba}_{l} (t), \\ b_{l} (t) = {Cx}_{l} (t) + {Da}_{l} (t), \end{matrix}

(21)

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Yinghao Ye, Domenico Spina, Dirk Deschrijver, Wim Bogaerts, Tom Dhaene. Time-domain compact macromodeling of linear photonic circuits via complex vector fitting[J]. Photonics Research, 2019, 7(7): 771

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