Stokes vector

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... 

These later form a three-dimensional Stokes vector S [13-15] whose tip moves over the surface of a Poincare sphere as the radiation passes without attenuation along the optical axis. Figure 5.1 shows the connection between polarization and points on the Poincare sphere. Right circular polarization is represented by the north pole, left circular polarization by the south pole, linear polarizations by points in the equatorial plane, and elliptical polarization by the points between the poles and the equatorial plane. 

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. 

In these appendices, we provide connections between the present Stokes-vector approach to optical design and alternatives. To write the propagation equation in a form appropriate to the two-state TDSE we define the dimensionless slowly varying parameters... 

APPENDIX B BLOCH VECTOR ANALOG OF STOKES VECTOR... 

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. 

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... 

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... 

The polarization properties of light can be represented by the Jones or Stokes vectors, A or S, respectively. The latter prescription has the advantages of describing partial polarization and contains directly observable quantities. When light is transmitted through a polarizing element with an incident electric vector A0 or S0, the light will exit with... 

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,... 

The intensity of the light generated in this experiment is easily calculated using equations (2.5) and (2.6) combined with the appropriate Jones and Mueller matrices selected from Appendix I. The Jones and Stokes vectors, Aj and, exiting this cascade are... 

A rotary polarization modulator simply consists of an optical element that rotates uniformly at a frequency Q about the transmission axis of light. In practice, retardation plates and polarizers are used. In either case, the Mueller matrix of such a device is found by simply replacing the angle 6 by Q.t in the equations listed in Appendix I. Typical PSGs based on rotary modulators and the associated Stokes vectors, Sp G, that are produced are listed in table 8.2. 

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-... 

Stokes vector will contain components of the form sin ( 4 sinOr) and cos (AsinQt). Consequently, the temporal response is rather complicated. To complete the analysis of the Fourier content of the signal the following expansions are required ... 

In practice, the quarter-wave plates will possess some imperfection in retardation. Furthermore, the phase angle of the plates may not be zero relative to the mechanical rotation device. Both sources of error can be taken into account. For example, an imperfect quarter-wave plate with retardation S = tc/2 + (3 and a phase offset of <)> would produce the following Stokes vector for the (P/RQ) 5G,... 

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