190 Basic Instrumentation for Optical Measurement be random. Many light sources in nature are unpolarized, such as sunlight, whereas optical signals from lasers are mostly polarized. An optical signal can usually be divided into a fully polarized part and a completely unpolarized part. The degree of polarization (DOP) is often used to describe the polarization characteristics of a partially polarized light, which is defined as DOP ¼ P polarized P polarized þ P unpolarized ð2:6:4Þ where P polarized and P unpolarized are the powers of the polarized part and unpolarized part, respectively. DOP is therefore the ratio between the power of the polarized part and the total power. DOP is equal to zero for an unpolar- ized light and is equal to unity for a fully polarized light. 2.6.2 The Stokes Parameters and the Poincare Sphere The state of polarization of a lightwave signal is represented by the maxi- mum amplitudes E x0 , E y0 , in the x and y directions, and a relative phase f between them, as shown in Equation 2.6.1. In addition, if the signal is not completely polarized, the DOP also has to be considered. Because of the multi- ple parameters, the representation and interpretation of polarization states are often confusing. A Stokes vector is one of the most popular tools to describe the state of polarization of an optical signal. A Stokes vector is determined by four independent Stokes parameters, which can be represented by optical powers in various specific reference polarization states. Considering an optical signal with the electrical field given by Equation 2.6.1, the four Stokes parameters are defined as S 0 ¼ P S 1 ¼ jE x0 j 2 À jE y0 j 2 S 2 ¼ 2jE x0 jjE y0 jcos f S 3 ¼ 2jE x0 jjE y0 jsin f ð2:6:5Þ ð2:6:6Þ ð2:6:7Þ ð2:6:8Þ where P is the total power of the optical signal. For an ideally polarized light- wave signal, since there are only three free variables, E x0 , E y0 , and f, in Equa- tion 2.6.1, the four Stokes parameters should not be independent and the total optical power is related to S 1 , S 1 and S 1 by 2 2 2 S 1 þ S 2 þ S 3 P ð2:6:9Þ S 0 represents the total power of the optical signal, S 0 ¼ jE x0 j 2 þ jE y0 j 2 ¼ P. How- 2 2 2 ever, if the optical signal is only partially polarized, S 1 þ S 2 þ S 3 only represents