JHI (a) High Q, simclation, spiral in (b) Medium Q, simulation (c) Low Q, simulation, spiral out (d) High Q, experiment, spiral in (f) Low Q, experiment, spiral out Th14 Spiral evolution of the polarization state on the Foincare sphere. We show both simulation and experimental results. The gray ~cale denotes transmission distance, where black represents the beginningofthesystem andgrayrepresents theendofthesystem.NDte that in Figs. I(<), l(e),and I(f), the s2-axis is obscured by the sphere.(e) Medium Q, experimentFig. 1.polarization state evolution. We therefore neglect the effects of DGD and PDG and treat the fiber simply as a random rotation of the polarization ststc in Stokes space. The rotation from the fiber followed by the rotation of the polarization o ntroller can be treated as a single rotation which is then followed by the lumped PDL element. This process is then repeated ovec n round trips of the In Stokes space, the action of a polarization controller can be represented by an axis of rotation,s,,,andanangleofrotatian about that axis, y. The PDL element has a low~loss axis given by spDL, which is located on the equator of the Poincar6 sphere. It can be shown that there are two polarization eigenstates, seigr. which can be written in terms of ssru, spDL. and y, and which correspond to the b e d polarization states described in the previous section. In the presence of PDL, these eigenstates are not orthogonal, unlike the principal states of polarization when one only considers PMD. However, any arbitrary polarization state can be written as a superposition of these two eigenstates. We therefore write the input signal of the form Ein = <+E+ + c. E-in Jones space, where E+ and E-are the components of the electric field in each ofthe eigenstates. The o w put polarization state aher n round trips for large n and small FDL is then given in Stokes space by: loop. s,,,(n) = A" I c+ I ' seig+ -25" I c+ I I c_ I [tl sin(ny-0) -t cos(ny-$)I (1) wheret,= (s .,xs,,,)l/5,,,,xs,,,I,t,= [s, , x I + c I c. I scip-. ( s~~~x s~~~)~~~s~~, x s~~~I ,and+isthephasedif-ference between <+ and L. The coefficients A. B,and Care all less than one, so that A', B", and c" all decay as n becomes large. The relative rates of decay depend upon the quantity a = silu8-spDL-In the case a > 0, then A > B > C, and the polariTation state spirals toward sun+ according to Eq. 1 as n grows. These inward spirals are similar to those seen in Figs. I(a) and 1 (d). In this case. if I c+ I < I c .1 so that the initial polarization state is closer to sI i g than seig+. then the evolution will be a spiral outward away from seis.. as in the spirals showninFigs.l(c)and I ( f ) . I f a < O , t h e n A < B < C, and one still observes spirals, but the roles of seig+ and seis-reverse, so that the spirals are now attracted to sei#. and repelled by sCidt. Finally, if a = 0, then A, B, and C are all comparable, and one observes nearly circular evolution on the Poincart sphere, with very slow convergence, lke the evolutionin Figs. l(b) and l(e).
Experimental se...
