The Stokes Vector

In addition to the intensity of light, defined in equation (1.53), it is desirable to formulate observable quantities that characterize both the amplitude and phase of the electric vector projected onto orthogonal directions. This is accomplished using the Stokes vector, S, with components [1,5]  

Although the Stokes vector, with its greater number of components, appears to be a more cumbersome representation of the electric vector, it is often more convenient to use than the Jones vector. This is because its components are observable quantities. For monochromatic, perfectly polarized light, the four components of the Stokes vector are not linearly independent, but related according to 

Linearly polarized light is recognized as having a Stokes vector of the form 

S3 0 for left-circularly polarized light and S3 0 for right-circularly polarized light. 

Observable quantities are the result of averaging over a duration, T, long enough to yield time-independent measurements. The average of a quantity u(f) is then  

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We may represent a beam of arbitrary polarization, including partially polarized light, by a column vector, the Stokes vector, the four elements of which are the Stokes parameters. In general, the state of polarization of a beam is changed on interaction with an optical element (e.g., polarizer, retarder, reflector, scatterer). Thus, it is possible to represent such optical elements by a 4 X 4 matrix (Mueller, 1948). The Mueller matrix describes the relation between incident and transmitted Stokes vectors by incident is meant before interaction with the optical element, and by transmitted is meant after interaction. As an example, consider the Mueller matrix for an ideal linear polarizer. Such a polarizer transmits, without change of amplitude, only electric field components parallel to a particular axis called the transmission axis. Electric field components in other directions are completely removed from the transmitted beam by some means which we need not explicitly consider. The relation between incident field components (E, E i) and field components ( l, E () transmitted by the polarizer is... 

Our objective is to design an optical device that will change the polarization from horizontal to vertical linear polarization - a rotation of the Stokes vector 5i,52,53 from // = (1,0,0 to V = (0,1,0 - and to do so independently of the wavelength. For this purpose, we require a propagation equation for the Stokes vector, obtained from Eq. (5.14) and the definitions (5.16) in much the same way that Feynman et al. [9] convert the two-state TDSE into a torque equation for combinations of products of probability amplitudes see Appendix 5.B. The equations... 

Figure 5.1 The Poincard sphere for depicting the Stokes vector. Linear polarization occurs around the equator, in the 1,2 plane, while circular polarization occurs at the poles. From Figure 3.2 of [12]. Copyright Cambridge University Press.

Traditional polarization-altering devices act with constant cp and hence with a constant Y, lying in the 1,2 plane. This produces oscillations of the Stokes vector - the analog of Rabi oscillations - in which S moves regularly away from the equatorial plane and the polarization periodically becomes elliptical. 

The position of the Stokes vector on the Poincare sphere depends on the length z = L and the wavelength X. Consequently, traditional retarders are not broadband. 

Apart from the numbering of the independent variables, this is the proposed adiabatic motion of the Stokes vector. The traditional discussion of STIRAP emphasizes that the 5-before-P pulse sequence seems counterintuitive when taken with intuition based on population transfer described by rate equations. By presenting the dynamics of STIRAP as an example of a torque equation, the motion, like that of the Stokes vector, seems very obvious. 

The definition of the Stokes vector in terms of bilinear products of field amplitudes is analogous to the definition of the two-state Bloch vector r = [u, v, iv] from products of probability amplitudes. The conventional numbering of the two vectors differs the connection is... 

As with STIRAP, we introduce an adiabatic basis [2, 3] for the Stokes vector. 

The four components of the Stokes vector are determined by four measurements which refer to three different basis systems (ex, ey and, e2 and er, e/5 respectively for more details see Section 9.2) ... 

For this reason, the Stokes vector discussed in section 1.4.3 is often more convenient since each of its elements is an observable quantity. 

The consequence of dispersion in wavelength is that the polarization properties of the electric vector will fluctuate randomly in time. The parametric mapping of the electric vector shown in Figure 1.2 will produce blurred contours and the light will be partially polarized. If the light shows no preference towards a particular polarization state, it is referred to as unpolarized, or natural light. The Stokes vector for this case is... 

The Stokes vector describing quasi-chromatic light has elements that are time averaged quantities and the equality (1.60) must be replaced by... 

Since the Maxwell equations involve the components of the Jones vector, it is normally easier to derive the Jones matrix, J, for complex, anisotropic media. Once J is obtained, it is generally convenient to transform it to the Mueller matrix representation for the purpose of analyzing the quantities measured in specific optical trains. This is because the components of the Stokes vector are observable, whereas the Jones vector components are not. Since it is the intensity of light that is normally required, only the first element of Sn,... 

These designs reveal how modulation serves to isolate specific elements of the Stokes vectors that are generated. In the (P/RP)pSG design, for example, analysis of the sig-... 

The appearance of Stokes publication, in which the concept of a vector representation of the beam description parameters was introduced, renders back to 1852. It, however, has remained for many years unnoticed (Shurcliff, 1962). The need for a general systematic and effective procedure for solving the exponentially growing quantity of optical problems has led to the rediscovery of Stokes approach to polarized radiation. Mueller s contribution consists of developing a matrix calculus for evaluation of the four elements of the Stokes vector which are a set of quantities describing the intensity and polarization of a light beam. 

The four parameters that are elements of the Stokes vector have the dimensions of intensity (W sr ). Each of them corresponds actually to a time-averaged intensity, and not to an instantaneous intensity value. Each of them shows how the intensity of a beam changes under the effect of a polarization device with standardised properties. 

Various notations have been introduced by different authors for the elements of the Stokes vector. We shall use in this chapter the symbols most frequently applied nowadays since they have gradually turned to a general convention. Thus... 

If the X- and y-amplitude components of the electric vector of the radiation are denoted by flx atid fty. and the phase shift between them by 6, then the quantitative expressions of the Stokes vector elements are defined as follows ... 

The angular brackets mean time averaging of the amplitude expressions. It is often convenient to work with the so called normalised Stokes vector obtained after dividing all Stokes elements So, S, S2, and S3 by the first one, Sq. The normalised Stokes vector displays the relative state of the beam having pas.sed one or more optical elements. Let us consider a few typical cases and their description by means of the Stokes vector. 

Since the amplitudes are independent of the phase shifts the latter remain in the brackets and give exactly the zero average value of cos 8 and sin 8 in the interval 0 < 6 < 360. In other words the unpolarized light could be considered as consisting of beams containing all possible phase shifts. Therefore, the Stokes vector of an unpolarized beam is written down as ... 

Let the transformation of the Stokes vector S for an unpolarized light beam be written down as... 

The matrices of the 45° polarizer Mp, of the retarder M and of the optically active sample Ms multiply consequently from the left the Stokes vector So of the unpolarized beam. 

The vector whose components are the parameters P is called the Stokes vector. A radiation is characterized by these P p = 1,. .., 4). In particular, the polarization vector2 id is defined by its three components... 

The coherency matrix and the Stokes vector are not the only representations of polarization and not always the most convenient ones. Two other frequently used representations are the real so-called modified Stokes column vector given by... 

We eonelude this seetion with a eaution. It is important to remember that whereas the Poynting veetor ean be defined for an arbitrary eleetromagnetie field, the Stokes parameters can only be defined for transverse fields such as plane waves discussed in the previous section or spherical waves discussed in Section 12. Quite often the electromagnetic field at an observation point is not a well-defined transverse electromagnetic wave, in which case the Stokes vector formalism cannot be applied directly. 

When two or more quasi-monochromatic beams propagating in the same direction are mixed incoherently, which means that there is no permanent phase relation between the separate beams, then the Stokes vector of the mixture is equal to the sum of the Stokes vectors of the individual beams ... 

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