Numerical modeling of a linear photonic system for accurate and efficient time-domain simulations

Yinghao Ye; Domenico Spina; Yufei Xing; Wim Bogaerts; Tom Dhaene

doi:doi:10.1364/PRJ.6.000560

Photonics Research, 2018, 6 (6): 06000560, Published Online: Jul. 2, 2018

Numerical modeling of a linear photonic system for accurate and efficient time-domain simulations Download： 560次

论文大纲

Yinghao Ye ^1,*Domenico Spina ¹Yufei Xing ²Wim Bogaerts ²Tom Dhaene ¹

Author Affiliations

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

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

Wavelength filtering devices Systems design Photonic integrated circuits

Abstract

In this paper, a novel modeling and simulation method for general linear, time-invariant, passive photonic devices and circuits is proposed. This technique, starting from the scattering parameters of the photonic system under study, builds a baseband equivalent state-space model that splits the optical carrier frequency and operates at baseband, thereby significantly reducing the modeling and simulation complexity without losing accuracy. Indeed, it is possible to analytically reconstruct the port signals of the photonic system under study starting from the time-domain simulation of the corresponding baseband equivalent model. However, such equivalent models are complex-valued systems and, in this scenario, the conventional passivity constraints are not applicable anymore. Hence, the passivity constraints for scattering parameters and state-space models of baseband equivalent systems are presented, which are essential for time-domain simulations. Three suitable examples demonstrate the feasibility, accuracy, and efficiency of the proposed method.

1. INTRODUCTION

In recent years, silicon photonic devices and circuits have had rapid development both in complexity and functionality thanks to an increasingly mature manufacturing process. At the same time, several computer-aided design (CAD) tools have emerged both in academic and industrial areas to analyze the behavior of silicon photonic designs.

Time-domain simulation of integrated photonic circuits is an essential part in the design flow, since it gives the most intuitive assessment of system performance. For some basic photonic components, such as waveguides, time-domain simulations can be analytically addressed because of the simple underlying physical principles and equations. However, time-domain simulations cannot be performed analytically when considering more complex devices or effects caused by parasitic elements. In such scenarios, different simulation techniques, such as finite-difference time-domain (FDTD) ^[1], time-domain traveling wave (TDTW) ^[2,3], the split-step method (SSM) ^[4], coupled mode theory (CMT) ^[5], or convolution-based methods ^[6], operate on the component or circuit level. However, a trade-off between efficiency and accuracy has to be made when using these techniques ^[5].

In the electronic field, a similar problem holds for distributed devices, such as nonuniform transmission lines or microstrip filters, since no compact circuit models are readily available for time-domain simulations ^[7]. A popular solution is based on a frequency-domain system identification technique named the vector fitting (VF) algorithm ^[8], which is able to build stable and passive rational models of the scattering parameters of the devices under study. Then, these frequency-domain models can be directly converted to an equivalent state-space representation in the time domain. This technique is widely applied for electronic systems, for example in Refs. ^{[8–13" target="_self" style="display: inline;">–13]}.

Since the VF method is developed for linear, time-invariant, passive systems and is based on their transfer function representation (e.g., scattering parameters), it is immediately applicable to passive photonic devices and circuits ^[14]. However, compared to electronic systems, the frequency range of interest for photonic systems is typically around [187,200] THz, corresponding to a wavelength range of [1.5,1.6] μm. Such a wide range at high frequencies has a direct impact on the modeling and simulation processes, which can become very time and/or memory consuming.

To address this problem, a novel modeling method is proposed in this paper, which is based on baseband equivalent signal and system representation. In particular, the proposed modeling approach computes an accurate baseband equivalent state-space representation, starting from the scattering parameters of the photonic system under study evaluated at optical frequencies. However, such an equivalent state-space model is complex-valued and not physically realizable. Furthermore, the stability and passivity constraints on scattering parameters and state-space models of complex-valued systems, which are fundamental properties for time-domain simulations, appear yet to be missing in the literature. In this paper, we rigorously discuss these conditions for the proposed baseband equivalent systems based on the classic definitions of stability and passivity to validate the feasibility of the proposed time-domain simulation method. The proposed technique offers two main advantages: (1) the modeling process is based on the scattering parameters, which makes it a widely applicable method for generic linear passive photonic components and circuits; (2) the state-space representation is a continuous time-domain model, which can be efficiently simulated in the time domain without involving convolution, fast Fourier transform (FFT), or inverse fast Fourier transform (IFFT), thereby making this method robust and accurate.

The paper is organized as follows. Section 2 presents an overview of the “standard” modeling approach based on the VF algorithm, while Section 3 introduces the baseband equivalent signals and systems and describes the novel proposed modeling framework. The stability and passivity constraints of such systems are discussed in Section 4. A practical guideline for the application of the proposed modeling approach is given in Section 5, while Section 6 validates the proposed method by means of three pertinent numerical examples. Conclusions are drawn in Section 7.

2. CONVENTIONAL STATE-SPACE MODELING OF PHOTONIC SYSTEMS

In both electronics and photonics, the scattering matrix is widely used to describe the behaviors of passive devices and circuits where $s$ is the Laplace variable, $a (s)$ and $b (s)$ are the forward wave and backward wave, respectively, and $S (s)$ is the scattering matrix of the system under study, which can be obtained through simulations or measurements. The aim of the rational modeling is to find a Laplace-domain model of Eq. (1) in a pole-residue form as where $D \in R^{n \times n}, R_{k} \in C^{n \times n}, k = 1, \dots, K$ , $n$ and $K$ being the number of ports of the system under study and the number of poles used to approximate the scattering parameters, respectively. Typically, all the elements $S_{i j} (s)$ of the scattering matrix representation in Eq. (2) use a common denominator polynomial and pole set $[p_{1}, p_{2}, \dots, p_{K}]$ , where such poles are either real quantities or complex conjugate pairs ^[8]. The identification of poles $p_{k}$ and residue matrices $R_{k}$ can be performed via the VF algorithm ^{[8,15–18" target="_self" style="display: inline;">–18]}, starting from a set of the scattering parameters under study obtained for $s_{r} = j 2 π f_{r}$ with $r = 1, \dots, R$ .

However, it is important to note that the sign convention $e^{j w t}$ is commonly used in the electronics field to represent the incident and reflected waves in Eq. (1), while $e^{- j w t}$ is sometimes adopted in the optics field ^[19,20]. Hence, the scattering matrix defined with one sign convention is the complex conjugate of the other. The VF algorithm is based on the assumption that the sign convention $e^{j w t}$ is adopted, since it has been originally developed for electromagnetic problems. In the case when $e^{- j w t}$ is used to define the scattering parameters under study, a simple solution is to apply the VF algorithm to the complex conjugate of the scattering matrix.

Then, the rational model in Eq. (2) can be transformed to state-space form by a simple rearrangement ^[17,21] where $A \in C^{m \times m}$ , $B \in R^{m \times n}$ , $C \in C^{n \times m}$ , $D \in R^{n \times n}$ , $m = n K$ , and $I$ is the identity matrix in this paper. In particular, $A$ is a diagonal matrix with all the poles as diagonal elements, while $C$ contains all the residues, and they can be always converted to real matrices as long as the poles and residues are real or complex conjugate pairs ^[17].

Now, it is straightforward to convert Eq. (3) to an equivalent state-space representation in the time domain ^[21] as where $x (t) \in R^{m \times 1}$ is the state vector.

Note that fundamental properties for time-domain simulations, such as the stability and passivity of models in Eq. (4), must be assured ^[13]. While the stability of VF models can be guaranteed by construction by means of suitable pole-flipping schemes ^[8], their passivity can be checked and, eventually, enforced only after the rational model is computed by adopting passivity enforcement techniques. Indeed, due to the unavoidable numerical approximations, the rational model computed might be non-passive. Several robust passivity enforcement methods have been proposed in the literature; see for example Refs. ^{[1618" target="_self" style="display: inline;">–18]}. Now, time-domain simulations can be carried out by solving the first-order system of ordinary differential equations (ODE) in Eq. (4) via suitable numerical techniques ^[22,23]. These approaches iteratively solve Eq. (4) for a discrete set of values of the time, which are chosen via suitable algorithms (i.e., fixed or adaptive time-step). In particular, the computational cost of solving Eq. (4) depends on three main elements: •

the bandwidth of the signals considered, which define the maximum time-step $Δ t_{max}$ that can be adopted: $Δ t_{max}$ must be smaller than the highest frequency component of the signals considered;

•

the numerical technique adopted to solve Eq. (4);

•

the number of poles $K$ and of ports $n$ of the system under study, which directly determine the number of states $m = n K$ .

The modeling technique described so far allows one to simulate any generic linear and passive system in the time or frequency domain, and it has found extensive applications in electronic engineering problems ^{[8–13" target="_self" style="display: inline;">–13]}. However, when it comes to photonic circuits, one substantial difference arises with respect to the electronic domain: the range of frequency of interest is typically around [187, 200] THz, corresponding to a wavelength of $U_{l} (f)$ , or even higher frequencies for shorter wavelengths. This has a major impact on the modeling and simulation complexity of the approach described so far. Indeed, a large number of poles $K$ can be required to accurately model the scattering parameters in the chosen frequency range, and the passivity enforcement phase can become computationally prohibitive. Furthermore, the corresponding ODE in Eq. (4) will have a large number of equations, and a small time-step (of the order of femtoseconds) must be adopted to solve it.

In order to tackle these issues, a novel approach based on baseband equivalent state-space models is proposed in this contribution.

3. BASEBAND EQUIVALENT STATE-SPACE MODELS FOR THE TIME-DOMAIN SIMULATION OF PHOTONIC SYSTEMS

The basic concepts of baseband equivalent signals and systems are first introduced in Section 3.A, given their importance in the definition of the proposed modeling approach, which is described in Section 3.B.

3.2 A. Baseband Equivalent Signals and Systems

The excitation signal of photonic systems is often an amplitude and/or phase-modulated signal with optical carrier and electronic modulating signals, which can be written in the following form: $u (t) = A (t) \cos [2 π f_{c} t + ϕ (t)],$ (5)where $A (t)$ is the time-varying amplitude or envelope of the modulated signal, and $ϕ (t)$ is the time-varying phase. In electronics or radio-frequency (RF) applications, both $A (t)$ and $ϕ (t)$ relate to electronic signals, such as voltage, current, or electric field. In photonics, the optical carrier frequency $f_{c}$ is much higher than the bandwidth of $A (t)$ and $ϕ (t)$ , given that the wavelength of light is much smaller than that of RF signals, so the representation in Eq. (5) is often called a bandpass signal.

An analytic complex-valued representation of the real-valued signal in Eq. (5), called the analytic signal, is introduced here as ^[24] $u_{a} (t) = u (t) + j H [u (t)] = A (t) e^{j [2 π f_{c} t + ϕ (t)]},$ (6)where $H [u (t)]$ is the Hilbert transform of $u (t)$ . In the frequency domain, Eq. (6) becomes $U_{a} (f) = 2 U (f) Step (f),$ (7)where $U_{a} (f)$ and $U (f)$ are the Fourier transform of $u_{a} (t)$ and $u (t)$ , respectively, and $Step (f)$ is a unit step function defined by $Step (f) = {\begin{cases} 1, & f > 0, \\ \frac{1}{2}, & f = 0, \\ 0, & f < 0 . \end{cases}$ (8)

Now, the corresponding baseband equivalent signal of the bandpass signal is defined as $u_{l} (t) = u_{a} (t) e^{- j 2 π f_{c}} = A (t) e^{j ϕ (t)},$ (9) $U_{l} (f) = 2 U (f + f_{c}) Step (f + f_{c}),$ (10)which can be considered as the complex envelope optical signal representation and is widely used in photonics and optical fiber communication. The relations between $u (t)$ , $H [u (t)]$ , and $u_{l} (t)$ in the time and frequency domains are ^[24] $u (t) = Re [u_{l} (t) e^{j 2 π f_{c} t}],$ (11) $H [u (t)] = Im [u_{l} (t) e^{j 2 π f_{c} t}],$ (12) $U (f) = \frac{1}{2} U_{l}^{*} (- f - f_{c}) + \frac{1}{2} U_{l} (f - f_{c}),$ (13)where the superscript * denotes the complex conjugate operator.

In the frequency domain, the concepts of analytic signal and baseband equivalent signal are intuitive: $U (f)$ has a symmetric spectrum with respect to the positive and negative frequencies, while $U_{a} (f)$ has only a non-zero spectrum in the positive frequencies around the carrier frequency; shifting the spectrum of $U_{a} (f)$ in the direction of the negative frequencies of $f_{c}$ [or equivalently in the time domain by multiplying $u_{a} (t)$ with $e^{- j 2 π f_{c} t}$ ] leads to $U_{l} (f)$ . Such relations are illustrated in Fig. 1.

Numerical modeling of a linear photonic system for accurate and efficient time-domain simulations Download： 560次

1. INTRODUCTION

2. CONVENTIONAL STATE-SPACE MODELING OF PHOTONIC SYSTEMS

3. BASEBAND EQUIVALENT STATE-SPACE MODELS FOR THE TIME-DOMAIN SIMULATION OF PHOTONIC SYSTEMS

3.2 A. Baseband Equivalent Signals and Systems

Fig. 1. Spectra of bandpass signal U(f), analytic signal Ua(f), and baseband equivalent signal Ul(f).

Fig. 2. Time-domain simulation of the bandpass system and baseband equivalent system.

3.3 B. Realization of Baseband Equivalent Signals and Systems

4. PASSIVITY OF THE BASEBAND EQUIVALENT SYSTEM

4.4 A. Passivity Constraints on Scattering Parameters of Baseband Equivalent Systems

4.5 B. Fast Passivity Assessment of Baseband Equivalent Systems

5. PROPOSED MODELING FRAMEWORK OF THE PHOTONIC SYSTEM FOR TIME-DOMAIN SIMULATIONS

Fig. 3. Flow chart of the proposed modeling framework for the time-domain simulation of photonic systems.

Fig. 4. Frequency ranges of Models A, LA, B, and LB.

Table 1. Comparison of Different Modeling Strategies

6. NUMERICAL EXAMPLE

6.2 A. Mach–Zehnder Interferometer

Fig. 5. Example MZI. The geometric structure of the MZI under study.

Fig. 6. Example MZI. The electronic signal and amplitude modulated optical signal for the MZI.

Fig. 7. Example MZI. Comparison of the magnitude (top) and phase (bottom) of the MZI scattering parameters extracted via Caphe (full blue line) and Model A (red dashed line), where the green dots represent the corresponding absolute error.

Fig. 8. Example MZI. Comparison of the magnitude (left) and phase (right) of the MZI scattering parameters extracted via Caphe (full blue line) and Model LB (red dashed line), where the green dots represent the corresponding absolute error.

Fig. 9. Example MZI. The output at port P3 of the MZI, where the red line is the absolute value of the complex signal obtained by the time-domain simulation of Model LB, the blue line is the corresponding signal from Model A, and the marker × denotes the same signal from the analytic model.

Table 2. Example MZIa

6.3 B. Ring Resonator

Fig. 11. Example RR. The geometric structure of the double ring resonator.

Fig. 12. Example RR. Comparison of the magnitude (top) and phase (bottom) of the ring resonator scattering parameters extracted via Caphe (full blue line) and Model A (red dashed line), where the green dots represent the corresponding absolute error.

Fig. 13. Example RR. Comparison of the magnitude (left) and phase (right) of the ring resonator scattering parameters extracted via Caphe (full blue line) and Model LB (red dashed line), where the green dots represent the corresponding absolute error.

Fig. 14. Example RR. The modulating signals: in-phase part I and quadrature part Q.

Fig. 15. Example RR. The output at port P3 of the double ring resonator, where the red line is the absolute value of the complex signal obtained by the time-domain simulation of Model LB, and the blue line is the corresponding signal from Model A.

Fig. 16. Example RR. The output at port P4 of the double ring resonator. Left: the output of Model A. Right: the recovered bandpass output from Model LB.

Fig. 17. Example RR. A zoom of the output at port P4 of the double ring resonator around t=45.6 ps (the green rectangular area in Fig. 16). The blue line is used for Model A, while the red dash line is the recovered bandpass output from Model LB.

6.4 C. Lattice Filter

Fig. 18. Example LF. The geometric structure of the MZI lattice filter.

Fig. 19. Example LF. Pseudo-random sequence of 1000 bits for t∈[0,1] ns.

Fig. 20. Example LF. Shift of the center frequency of the passband of the lattice filter due to the tolerances of the manufacturing process.

Fig. 21. Example LF. The eye diagrams at port P4 of the baseband equivalent systems of the lattice filter with passband center frequency 195.11, 195.05, and 194.98 THz (from left to right).

7. CONCLUSION

Article Outline

相关论文

相关资讯

关于本站 Cookie 的使用提示

全站搜索

Fig. 1. Spectra of bandpass signal $U (f)$ , analytic signal $U_{a} (f)$ , and baseband equivalent signal $U_{l} (f)$ .

Fig. 9. Example MZI. The output at port P3 of the MZI, where the red line is the absolute value of the complex signal obtained by the time-domain simulation of Model LB, the blue line is the corresponding signal from Model A, and the marker $\times$ denotes the same signal from the analytic model.

Table 2. Example MZI^a

Fig. 17. Example RR. A zoom of the output at port P4 of the double ring resonator around $t = 45.6 ps$ (the green rectangular area in Fig. 16). The blue line is used for Model A, while the red dash line is the recovered bandpass output from Model LB.

Fig. 19. Example LF. Pseudo-random sequence of 1000 bits for $t \in [0, 1]$ ns.