Stokes’ vector definition

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... 

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... 

The purpose of such a device consists in changing the orientation of the polarization plane of a beam by 90°. That means the initial Stokes vector 1,1,0,0 of a horizontally polarized beam becomes 1,-1,0,0 after passing through the retarder. Retarders are most often birefringent crystals of definite thickness. If the fast and slow axes of such a crystal orthogonal to each other are crossed at 45° with respect to the polarization plane, the retarder rotates the latter by 90°. The Stokes-Mueller transformation corresponding to this experiment should be ... 

It then follows from the definition of the Stokes parameters that the Stokes vector of the wave modifies according to... 

The definition of the coherency and Stokes vectors explicitly exploits the transverse character of an electromagnetic wave and requires the use of a local spherical coordinate system. However, in some cases it is convenient to introduce an alternative quantity, which also provides a complete optical specification of a transverse electromagnetic wave, but is defined without explicit use of a coordinate system. This quantity is called the coherency dyad... 

Using the definition of Stokes vector, derive the Mueller matrix given by Equation (3.70) of an optical element whose Jones matrix given by Equation (3.69). 

From the definition of a particle used in this book, it follows that the motion of the surrounding continuous phase is inherently three-dimensional. An important class of particle flows possesses axial symmetry. For axisymmetric flows of incompressible fluids, we define a stream function, ij/, called Stokes s stream function. The value of Imj/ at any point is the volumetric flow rate of fluid crossing any continuous surface whose outer boundary is a circle centered on the axis of symmetry and passing through the point in question. Clearly ij/ = 0 on the axis of symmetry. Stream surfaces are surfaces of constant ij/ and are parallel to the velocity vector, u, at every point. The intersection of a stream surface with a plane containing the axis of symmetry may be referred to as a streamline. The velocity components, and Uq, are related to ij/ in spherical-polar coordinates by... 

Taking the vector curl of the right-hand side causes the first and last terms to drop out, since the curl of the gradient vanishes. However, for variable density, the left-hand side expands to long, complex, and not-too-useful expression (see Section A.14). Therefore let us restrict attention to incompressible flows, namely constant density. The curl of the incompressible Navier-Stokes equation, incorporating the definition of vorticity u = VxV, yields... 

The Q values in these tensors represent the Stokes parameters for the spin sensitivity of the detector in its x", y", z" frame. For example, Qx- describes the detector efficiency for measuring spin projections along +x" and —x", respectively (for the definition of the spin polarization vector see Section 9.2.1). 

---