Attributes of light such as intensity and color have been extensively used for carrying information using time-division multiplexing (TDM) and WDM technologies. The polarization aspect of light, however, is less well used and is often perceived as a problem especially when it comes to exploiting it for telecommunications. Great attention has been paid to compensation of polarization-mode dispersion (PMD) for systems operating at 10 Gbit/s and especially at 40 Gbit/s.^{1} Increasing pressure is placed on component manufactures to control the polarization dependence of their devices, often to less than 0.005 dB. Polarization management through control and measurement is important not only to address problems such as PMD in fiberoptic networks, but also to test polarization dependence of various WDM components.

Consider a monochromatic plane wave of angular frequency, ω, and of wave vector, k, parallel to the *z*-axis. In an infinite medium, and in the case of homogeneous plane waves, two orthogonal electric field vectors can represent the light:

where φ_{x} and φ_{y} are the orthogonal components of the phase of the wave; *i* and *j* are unit base vectors in Cartesian coordinates. The complex vector E→, fully describes all the attributes of the lightwave while the wavelength of light is determined by ω and k. The intensity of light, I, is associated with the vector as:

However, to uniquely define the E→ vector, we must also know information about the phase. In defining the polarization aspects of light, the difference between the two-phase factors and the ratio

become relevant. The state of polarization (SOP) of light, as a description of the vectorial nature of light, is described by the so-called polarization ellipse, which we can observe by following the extremity of the E→ vector in the wave plane. We can deduce the equation of this ellipse from equation (1):

where φ = φ_{y} -φ_{x} is the phase difference between the two orthogonal components.

The most straightforward representation of the SOP of a lightwave is as described in equations (1) and (2). In this representation, a complex number represents the SOP, while a curve in a complex plane tracks the evolution of the SOP.

Another popular method of describing the SOP of completely polarized light is the Jones vector representation, which represents the SOP by a complex two-component column vector, and the optical medium by a 2 × 2 Jones matrix.^{2} In this representation, polarized light, as an electromagnetic wave, can be represented as an electric-field vector, called the Jones vector. Multiplication of the Jones vector by any complex constant does not modify the SOP, so normalized Jones vectors are convenient to work with.

Stokes parameters, an alternate representation of the states of polarization of light, are all real numbers and measurable.^{3} Stokes parameters consists of four real numbers, directly related to the intensity of light:

Here, S_{0} denotes total intensity, S_{1} refers to the difference in intensities between horizontal and vertical linearly polarized components, S_{2} refers to the difference in intensities between linearly polarized components oriented at +45° and -45°, and S_{3} refers to the difference in intensities between left- and right-circularly-polarized components. The Stokes parameters S_{0}, S_{1}, S_{2}, S_{3} are measurable quantities. Thus, the measure of these intensities allows the unambiguous determination of the states of polarization of light. This is the basic principle behind many ellipsometers.

The physical meaning of Stokes parameters is particularly important in the study of partially polarized light, in which the notion of the phase difference between components does not have much of a meaning. From the definition, none of the parameters can be greater than S_{0}, which is usually normalized to 1. For completely polarized light, we have

while for entirely unpolarized light:

The degree of polarizition is therefore defined as:

For completely polarized light, the relation always holds, so we can view the parameters (S_{1}, S_{2}, S_{3}) as the coordinates of a point on a unit sphere called the Poincaré sphere (see Fig. 1).^{4} On the Poincaré sphere, each point represents a specific polarization state. The points on the equator represent the linearly polarized states. The left and right circularly polarized states are represented by the south and north poles, respectively. Points on the lower and upper hemisphere represent the degree of left- and right-elliptically-polarized states, respectively. Any pair of antipodal points on the Poincaré sphere corresponds to states with orthogonal polarization.

This representation is particularly useful to describe the polarization transformation through an anisotropic medium. For example, we can visualize the evolution of the state of polarization when light passes through a linear birefringent plate as a curve on the Poincaré sphere (see Fig. 2). Suppose the wave plate is oriented at angle φ relative to the *x*-axis; the output state can be obtained by a rotation on the Poincaré sphere about axis Ω^{L}, which lies on the equatorial plane oriented at an angle 2φ with respect to the S_{1} axis. The amount of rotation is determined by the retardation of the wave plate.

There are several ways to measure the polarization of light. Although the measurement of S_{0}, S_{1}, S_{2} is relatively straightforward because it only involves use of a polarizer, measurement of S3 requires the use of a wave plate with a known amount of birefringence, which in the simplest case is a quarter-wave plate. Modern polarimeters involve either temporal or spatial measurements. In each case, at least four measurements are needed to measure the four Stokes parameters, either in series (temporally) or in parallel (spatially), with each approach having distinct advantages. The two basic varieties of polarimeter are those involving a single detector with multiple time-dependent measurements or multidetector simultaneous measurements.

The two varieties of polarimeter both take a set of at least four measurements, which are then transformed into the four Stokes parameters. In a typical spatial polarimeter design, the input beam is split into four different paths (see Fig. 3). Each beam can then be interrogated using a combination of polarizers and waveplates in different positions using a photodetector.

In a typical parallel polarimeter, each path can measure one of four parameters. In the simplest configuration, one path has no polarizer or wave plate, which provides the measurement of the intensity of the beam and hence the value of S_{0}. For the second path, a linear polarizer is used, which generates the S_{1} value before accounting for polarizer absorption loss. A polarizer in the 45° position generates the S_{2} value, once again considering polarizer loss. Finally, we can measure S_{3} using a circular polarizer, which is a combination of a quarter-wave plate and a polarizer, and again taking the polarizer losses into account.

Direct measurement of the Stokes parameter, which could be positive or negative, is generally not possible using the detector reading, since the detector cannot measure negative values. This scheme is the most straightforward measurement of polarization, but not the most accurate or the most reliable for commercial polarimeters, which often require high accuracy and power ranges that vary more than 50 dB.

The use of the wave plate in the polarimeter requires either a factory calibration or a periodic user calibration. Calibration may also be necessary for a particular wavelength, although use of zero-order wave plates or achromatic phase shifter reduces this requirement.^{1} Many polarimeters are expensive and bulky, but emerging alternatives can achieve high speed, high accuracy, and affordability in a compact size, which becomes increasingly important in optical networks.

Once the Stokes parameters have been measured, it is relatively easy to calculate the degree of linear polarization or the degree of circular polarization as well as the polarization content of the light, often termed degree of polarization. This parameter is often used to gauge PMD, and has been used as an optimization parameter in feedback loops to reduce PMD. Other emerging optical communications applications using polarization-shift-key encoding may also become practical as compact high-speed polarization detectors become available.

**Zhizhong (Jeff) Zhuang**, is lead engineer and **Barry Zhang** is vice president of engineering at Optellios, 250 Phillips Blvd., Suite 255, Ewing, NJ 08618. Barry Zhang can be reached at bzhang@optellios.com.

- M. Albert, WDM Solutions 4 (6), 35, (June 2002).
- R. C. Jones, J. Opt. Soc. A. 31, 488 (1941).
- G. G. Stokes, Trans. Cambridge. Phil. Soc. 9, 309 (1852).
- S. Huard, Polarization of Light, John Wiley and Sons (1997).