Polarization Changes in Optical Fibers induced by Self-Phase Modulation and Cross-Phase Modulation in Conjunction with Birefrigence

Polarization dependence of various fiber effects affecting signal quality in wavelength division multiplexing (WDM) systems has become a major field of research activities. Polarization– mode dispersion (PMD) and polarization–dependent loss (PDL) are quite well understood today, but there are still major open questions with respect to the interaction with nonlinear fiber effects, although the picture has already become clearer. The aim of this paper is to provide a better understanding of the impact of self–phase modulation (SPM), cross–phase modulation (XPM), and birefrigence on the state of polarization during propagation in a transmission fiber. For several configurations, closed–form expressions describing the dependence of different quantities on propagation distance are presented. Both phase modulating effects do not induce a power transfer among the channels of WDM system. However, power is exchanged among the components of the Jones vector of the individual channels. Without attenuation, this power exchange is a periodic function of the propagation distance. Furthermore, the analysis reveals that the sum of the ellipticities weighted by the corresponding fiber input power is preserved during propagation as long as the wavelength dependence of fiber attenuation can be neglected. In the presence of SPM only, the ellipticity of the channel under consideration is maintained during propagation and the powers of the different components of the Jones vector change sinusoidally as a function of a normalized propagation distance. Closed–form expressions describing the dependence of the magnitude of the power variation and the period length on initial conditions at the fiber input are derived. Conservation of the ellipticity is canceled out by the interaction with birefrigence. Neglecting fiber attenuation, a different quantity is identified that is conserved during propagation. Ellipticity as well as the power of the two components of the Jones vector change periodically versus propagation distance, but no longer sinusoidally. Furthermore, it is shown that there is a separatrix that cannot be crossed by the traces embracing the states of polarization adopted during propagation. Thus, the Poincaré sphere is split into up to three segments. The ellipticity of a channel influenced by XPM has a sinusoidal dependence on the normalized propagation distance with a period length depending on the ellipticity of the first channel only. Power variation of both components of the Jones vector of the second channel can be mathematically described by a superposition of three sine waves. The magnitude of the power exchange on initial conditions at the fiber input is pointed out. In addition, this analysis is extended for the case that a channel is affected by This work has been done at the Communication Technology Laboratory, Swiss Federal Institute of Technology (ETH), Zurich, Switzerland, in 1996. It has now been updated and translated into English. c © by Dr. Lutz Rapp, Jägerstr. 16, D–82041 Deisenhofen, Germany, Email: [email protected] SPM, XPM, and birefrigence simultaneously.