No abstract

The time-varying state of polarization of an optical channel modified by the intensity of other channels in a three-channel WDM system is measured. It compares well with theory expanded from a previously published two-channel model. A change in the polarization state of an optical signal caused by the nonlinear refractive index in a wavelengthdivision multiplexed (WDM) transmission system has been previously reported [1,2]. If the optical powers of the WDM channels are modulated, cross-polarization modulation degrades the degree of polarization of the signal. This can have deleterious effects in many optical systems: in combination with polarization-dependent gain or loss it can cause interchannel crosstalk [4], it can degrade the performance of polarization-mode dispersion (PMD) compensators [2,3], and in systems employing polarization modulation [5,6] it can cause crosstalk between the channels. Although cross-polarization modulation can seriously degrade signal quality, it can easily be overlooked in system experiments. Laboratory demonstrations often use co-polarized channels to maximize crosstalk due to other non-linearities [5], but this minimizes cross-polarization modulation. Because this phenomenon can be a serious impairment, it is important to understand and model it for multi-channel WDM systems.In previously published models of the cross-polarization modulation with two WDM channels, the two channels precess about an average Stokes vector. This model predicted measurements of a RF-tone modulated two-channel system very well for a variety of optical modulation depths and modulation frequencies [1]. A typical WDM communication system, however, has many more than two channels. Here we extend the previous model to a general WDM multi-channel system and compare it to measurements of a three-channel WDM system. One implication of this model is that the depolarization caused by cross-polarization modulation is deterministic, and therefore cannot be treated as random noise in the state of polarization when considering other nonlinear optical effects such as cross-phase modulation.

Coning motions are analyzed for an unusual spinning re-entry body whose roll moment of inertia is nearly as large as its moment of inertia in pitch or yaw. The analysis treats the rotational motion during re-entry caused by dynamic mass unbalance and by initial angle of attack and angular velocity. Even though some of the motions are complicated, analytical results are presented which are useful in predicting many facets of the motion. Initial angle of attack excites a fairly simple coning motion with the body symmetry axis spiraling in toward the velocity vector at a rate increasing with dynamic pressure. Initial angular velocity leads to a more complex sequence of cycloidal motions. During early re-entry, mass unbalance causes a beat phenomenon peculiar to a body with such moment-of-inertia characteristics. Later in the re-entry period, there is a severe build-up in angle of attack in the region where the natural pitching frequency of the body is nearly equal to its spin rate.Nomenclature total angle-of-attack amplitude complex angle of attack = a + i ft angle-of-attack projection on pitch plane angle-of-attack projection on yaw plane normal force coefficient slope of normal force coefficient = axial force coefficient aerodynamic force vector acceleration due to gravity altitude pitch/yaw moment of inertia roll moment of inertia I -IxO'xyz C) = nonspinning body axes = time derivative applied moment vector mass spin rate stability parameter = dynamic pressure [(n/2) 2 + Q'p" reference area time velocity vector static margin angle-of-attack projection in XZ plane angle-of-attack projection in XY plane pitch flight-path angle pitch flight-path angle at re-entry pitch flight-path angle change =7 -70 mass unbalance angle r -n/2 • r + n/2 yaw flight-path angle air density pitch attitude angle roll attitude angle yaw attitude angle natural frequency of nonspinning vehicle complex angular velocity = co y + earth-fixed axes translating axes (\\OXYZ)