Measurement of polarization-mode dispersion - includes related articles on Jones calculus, the Poincare Sphere, and the HP 8509A/B Lightwave Polarization Analyzer - Technical

Hewlett-Packard Journal, Feb, 1995 by Brian L. Heffner, Paul R. Hernday

The polarization dispersion vector [omega] is probably the most intuitively meaningful representation of PMD because it is defined in the same real, three-dimensional space as the Poincare sphere.[2] This vector originates at the center of the Poincare sphere and points toward the principal state of polarization about which the output states of polarization rotate in a counterclockwise sense with increasing [omega]. [absolute value of] [omega] the length of R is the differential group delay. When the output state of polarization is expressed as a three-dimensional vector s composed of normalized Stokes parameters[1] locating the state of polarization on the sphere, the rotation about the principal states of polarization can be written as a cross product relation: ds/d[omega] = [omega] x s. In the most general case [omega] is a function of the optical frequency [omega].

In some optical components, such as isolators and waveguide modulators, PMD originates in crystals or wave-guides through which light propagates with different group delays for different polarizations. As the optical frequency transmitted through these components is varied, [omega] remains fixed in orientation, while [absolute value of][omega] might vary slightly as a result of chromatic dispersion in each polarization mode. Devices such as these, in which [omega] is essentially independent of [omega], are especially simple to describe and to measure. In particular, the differential group delay is constant over time and can be accurately characterized as a weak function of frequency. Such devices can be useful as standards because their characteristics can be expected to remain stable from one time and location to the next.

Modern single-mode fibers achieve a very low level of local birefringence in addition to low propagation loss, but the effects of small local birefringences along the fiber can accumulate to cause significant PMD through a fiber many kilometers long. Birefringence is a property of a dielectric describing the difference in the indexes of refraction for different polarizations. Local birefringence in a fiber can be caused by deviations from perfect circular symmetry of the fiber core, or by asymmetrical stress in the core region owing to the manufacturing process, bends in the fiber, or temperature gradients. Stresses can change with temperature and with time as the glass and the surrounding cable relax, leading to a birefringence along the fiber, length that evolves over time. This behavior leads to measurable PMD that is a function of both frequency and time, so that a statistical picture becomes the most appropriate view of PMD for the system designer.

A detailed statistical model of PMD has been developed from the basis of accumulated local birefringences, and has been experimentally confirmed.[3,4]

Each of the x, y, and z components of the polarization dispersion vector for a long fiber follows a normal distribution with zero mean. As a result, the orientation of [omega] is uniformly distributed, and the distribution of the differential group delay [delta][tau] = [absolute value of][omega] is proportional to [delta][tau.sup.2]exp(-[delta][tau.sup.2]/2 [alpha.sup.2]). This is often called a Maxwell distribution because it is the same as the Maxwell distribution of molecular speed for a gas in thermal equilibrium. The distribution has an expected value [less than][delta][tau][greater than] of [alpha][square root of]8/[pi].