Mathematical Modeling and Statistical Analysis of Second Harmonic Generation Effects With Thin and Thick Crystals in Ultrahigh-Speed Optically Amplified Digital Lightwave Communication Systems

In this paper, we present an in depth analysis and discussions on the behavior and performance of using either thin or thick second harmonic generation (SHG) crystals prior to photodetector in ultrahigh-speed optically amplified digital lightwave communication systems. Our study begins by considering a conventional low-speed optically amplified lightwave receiver and study its performance in ultrahigh-speed regime and show its substantially degraded behavior, indicating the need to use advance optical nonlinear elements such as SHG prior to photodetection. In studying SHG effects, we begin by discussing the mathematical models of thin and thick SHG crystals in the context of statistical digital lightwave communication systems. In the case of thin SHG crystals and in the regime of high-speed digital systems, where Gaussian assumption is not necessarily valid, we use advanced numerical simulation based on Monte Carlo simulation to obtain its corresponding performance. However, for thick SHG crystals by invoking an advanced theorem due to Papoulis on Gaussianity approximation and a lemma due to Turin, we obtain the moment generating function of the sampled output with which, we obtain its corresponding performance using saddle point approximation method. Finally in our performance analysis, we consider electronically induced thermal noise and sketch the performance for various system parameters. The results in this paper indicate that the use of SHG crystals can enhance the system performance, when compared with the conventional receiver, by a substantial amount, i.e., five orders of magnitude. Furthermore, we highlight the existence of an optimum length for thick SHG crystals in the presence of thermal and optical amplifier noise.