The goal of the computational tool presented in this dissertation is to facilitate the de- sign and optimization of different optical waveguides, waveguide-based devices and sophisticated optical systems. In the process of constructing this numerical engine, the mathematical algorithms and numerical parameters are needed to be carefully se- lected, in order to ensure the computational precision and reliability. In this section, the architecture of this theoretical and numerical tool will be described.
According to the program flow shown in Fig. 4.1, every simulation towards the performance analysis encompasses the following main steps, namely, simulation setup, system evaluation and output. To be more specific, the first step is of great impor- tance but relatively simple, because it is supposed to guarantee the accuracy of the simulated results by initializing all the essential parameters. As for the third step, the results are programmed to generated automatically. Furthermore, the key part of the whole computational process is the second step, which comprises the modules of the preparation engine, the propagation simulator and the BER calculator. Notably, the
Figure 4.1: Program flow of the system evaluation model.
method of error counting is not used to verify the BER results calculated via the KLSE approaches, since this method is extremely time-consuming and subsequently makes itself infeasible especially at larger values of BER. However, the verification tests re- garding BERs have been carried out, with more details described in Sec. 4.3.6. In the following paragraphs, the thorough computational procedures of this implementation will be presented.
In the first step, the initial conditions of the whole simulation process are prepared, including the specific components of a Si photonic system and the simulation parame-
ters. To start with, a photonic system can be constructed by defining several groups of physical parameters illustrated in the first stage of Fig. 4.1: (1) Optical signal parame- ters. Generally, the signal modulation formats, the central carrier frequency, the signal power and the bit sequence pattern, as well as the white noise, all have to be properly selected in order to fulfill the research purpose; (2) Waveguides optical parameters and dimensions. In particular, the dispersion and nonlinear coefficients of Si waveguides are required to be calculated. In this dissertation, we obtain the waveguide parameters of Si-PhWs and Si-PhCWs by using Femsim by Rsoft [13] and MIT Photonic Bands (MPB) [14], respectively; (3) Receiver schematics. The mathematical expressions will be given for the direct-detection optical receiver, which contains an band-pass opti- cal filter, an ideal photodetector and a low-pass electrical filter. As for a ¯M-channel system, the ¯M sets of optical filters, photodetectors and electrical filters, as well as a demultiplexer are used in the direct-detection.
Another significant aspect of the simulation setup is the numerical parameters. Ba- sically, they can be divided into the following groups according to different numerical algorithms: (1) the temporal sampling points (or FFT points), the distance step and the time window for the SSFM method and the ODE model; (2) the Cash-Karp parameters [15] and Dormand-Prince parameters [16] for the fifth-order Runge-Kutta method (see Appendix. B); (3) the temporal sampling points, bit-rate (or bit interval), weights and abscissas of Gaussian-Hermite rule for the time-domain KLSE method; (4) the number of frequency points for both the frequency-domain and Fourier-series KLSE methods. Moreover, the output parameters determines the types of results. Apart from the es- sential simulation parameters, we also define the scanner variables. One or more input parameters can be automatically scanned in the simulation process, such as the signal power, the pulsewidth, the bit-rate and the waveguide length. Notably, the numerical parameters are not suggested to be scanned. Since the computational accuracy can only be ensured by choosing suitable values of these parameters, there is no point to con- tinue the simulation if this condition can not be satisfied. Importantly, the values of all the relevant parameters are given in Sec. 4.3.6.
Moving on now to consider the core of the numerical scheme – the system eval- uation. The system evaluation process is composed of three stages. In particular, the first stage is the preparation engine. In this stage, two decision operations will be per-
formed: one is to choose the types of simulators (or runs), i.e., the serial run and the parallel run; the other is to identify the shape of the optical signals, namely, CW and pulse. To be more specific, the parallel run is often employed in the variable scanning, in order to speed up the whole computational process. The parallel run refers to the Matlab built-in function par f or [17], where these simulations will be treated as inde- pendent tasks and then a specific number of tasks will be performed simultaneously. Additionally, the serial run is adopted in the single simulation operation or the case where the computational environment (e.g, a single processor) can not support paral- lel operations. Furthermore, the shape of optical signals decides a specific numerical routine used in the propagation section, which will be demonstrated in the next stage.
The second stage of the system evaluation engine is the propagation simulator. Take the case of single-wavelength propagation for instance. Firstly, the optical signals of Eqs. (2.6) will be placed at the input of Si waveguides. Then, the semi-analytical solutions regarding the output signals can be derived by using either the full propagation algorithm (Eqs. (2.15) and (2.20)) or the linearized version ( Eqs. (2.26) and (2.27)). Precisely, both types of simulators can be selected for the CW signals (uOOK, uPSK),
whereas only the first simulator is capable of simulating the pulsed signal (uG). In case
of the multi-wavelength co-propagation, the analysis is carried out by first extracting the mathematical expression for signals in each channel (e.g., Eq. (2.28)), and then modifying the single-channel numerical routines to the multi-channel case by using Eqs. (2.21) and Eqs. (2.30), (2.31), respectively. Therefore, the explicit numerical implementation details for full and linearized propagation models will be presented in Sec. 4.3.1 and Sec. 4.3.2, respectively.
Another important stage is the BER calculator. From Fig. 4.1, we can see that the time-domain, frequency-domain and Fourier-series KLSE calculators are available in the BER estimation. The system BERs are determined by Eqs. (3.35), (3.38) and (3.41). The numerical procedures to implement these calculators will be explained in Sec. 4.3.3, Sec. 4.3.4 and Sec. 4.3.5. Particularly, the time- and frequency-domain KLSE calculators are only suitable for CW signals, while the third KLSE calculator can be applied to all shape of optical signals. At this stage, the numerical discretization must be carefully performed in order to guarantee the convergence and accuracy. Once the simulation operation is completed, the whole evaluation will enter the output stage.
Last but not least, the output of the whole system evaluation is illustrated in the final part of Fig. 4.1. Specifically, four important output characteristics will be gen- erated, namely BER, eye diagrams, signal dynamics and noise dynamics. The BER results are exported in form of DAT files, and can be viewed via softwares like Matlab and Python. Moreover, the eye diagrams can be inspected either at the output of optical filters or at the back-end of the receiver. In addition, the signal dynamics and the noise dynamics can be exported by utilizing either the propagation simulator or the entire simulation engine, with the difference existing between the optical dynamics and the electrical dynamics correspondingly.