Interaction of nonlinearity and polarization mode dispersion

The derivation of the coupled nonlinear Schrödinger equation and the Manakov-PMD equation is reviewed. It is shown that the usual scalar nonlinear Schrödinger equation can be derived from the Manakov-PMD equation when polarization mode dispersion is negligible and the signal is initially in a single polarization state as a function of time. Applications of the Manakov-PMD equation to studies of the interaction of the Kerr nonlinearity with polarization mode dispersion are then discussed. There has been a flood of recent work on polarization effects in optical fibers, most of it focused on polarization mode dispersion (PMD). There are good, practical reasons for this recent interest. As the data rate per channel increases, PMD becomes an increas-ingly important limitation in communication systems. However, my own interest in polarization issues was, at least originally, focused on more fundamental questions. Since nearly all the contributions in this volume focus on the more immediate practical issues that are a consequence of polarization effects, it seemed useful to me to focus this contribution on more fundamental issues. The nice thing about fundamental issues is that even though they attract less attention than immediate practical issues, they typically stay relevant longer and can impact practical issues 10–20 years in the future. Throughout the 1970s, a considerable body of work established the basic polarization properties of optical fibers. Much of this work was summarized by Kaminow in 1981 [1]. It was found that the birefringence ∆n/n is in the range 10−4–10−9, with communication fibers in the range 10−6–10−7. It was found that the intrinsic 306 Curtis R. Menyuk birefringence is almost entirely linear, even when the fiber is twisted, because of the very small value of the electro-optic tensor in glass [2]. Remarkably, there has been no change in either the range of the birefringence values or the helicity in optical fibers in the last 20 years, although many other fiber parameters have changed considerably. The fiber-induced PMD can now be made much smaller by spinning fibers as they are drawn; the effective areas of fibers can be made much larger or smaller, affecting the strength of the nonlinearity; the dispersion can be tailored; and losses can be made smaller. In 1980, Mollenauer et al. [3] demonstrated the propagation of solitons in optical fibers for the first time. This work began a series of studies, continuing into the present day, that explored the interaction between chromatic dispersion and nonlinearity in optical fibers and their implications for communication systems. The theoretical basis for this work was the nonlinear Schrödinger equation, which had been derived by Hasegawa and Tappert in 1973 [4]. This work took no account of the birefringence in optical fibers; however, work by Botineau and Stolen [5] had made it clear that the interaction of birefringence and nonlinearity could be quite important. In their work, they measured the nonlinear polarization rotation and demonstrated its potential for nonlinear switching. As part of this work, they demonstrated that the ratio of the crossphase modulation and the self-phase modulation in optical fibers is 2/3. I note that this coefficient is only expected to be 2/3 if the fiber’s birefringence is intrinsically linear. More generally, it equals (2 + 2 | ê1 · ê2 |)/(2+ | ê1 · ê1 |), where ê1 and ê2 are the unit vectors for the birefringent eigenmodes [6]. Its value ranges from 2/3 for linearly birefringent fibers to 2 for circularly birefringent fibers. Thus, the nonlinear light evolution yields information about the linear properties of the fibers—a point to which I shall return. Given the experimental evidence of the importance of birefringence, it was natural to derive an equation that takes it into account. This equation, referred to as the coupled nonlinear Schrödinger equation, was first presented in 1987 [7]. In a form that is particularly useful for studying randomly varying birefringence, it may be written as