Poincare sphere representation

Figure 1. (a) Poincare sphere representation of wave polarization and rotation (b) a Poincare... 

Figure 8.9 Schematic diagram of Poincare sphere representation and the effect of uniaxial medium on the polarization state change of a polarized incident hght.

The error due to the inexact quarter-wave plate is thus eliminated by taking the measurements of A and A with the polarizer oriented in two quadrants. Corrections for the errors from other nonideality of the optical components can be similarly worked out with the Poincare sphere representation. 

The representation of the sampling by a unipolar, single-rotation-axis, U(l) sampler of a SU(2) continuous wave that is polarization/rotation-modulated is shown in Fig. 2, which shows the correspondence between the output space sphere and an Argand plane [28]. The Argand plane, S, is drawn in two dimensions, x and v, with z = 0, and for a set snapshot in time. A point on the Poincare sphere is represented as P(t,x,y,z), and as in this representation t = 1 (or one step in the future), specifically as P(l,x,y,z). The Poincare sphere is also identified as a 3-sphere, S 1, which is defined in Euclidean space as follows ... 

The consequence of these relations is that every proper 2n rotation on S + — in the present instance the Poincare sphere—corresponds to precisely two unitary spin rotations. As every rotation on the Poincare sphere corresponds to a polarization/rotation modulation, then every proper 2n polarization/rotation modulation corresponds to precisely two unitary spin rotations. The vector K in Fig. lb corresponds to two vectorial components one is the negative of the other. As every unitary spin transformation corresponds to a unique proper rotation of S +, then any static (unipolarized, e.g., linearly, circularly or ellipti-cally polarized, as opposed to polarization-modulated) representation on S + (Poincare sphere) corresponds to a trisphere representation (Fig. 3a). Therefore... 

A /l = 7, where 7 is the identity matrix. Thus, a spin transformation is defined uniquely up to sign by its effect on a static instantaneous snapshot representation on the S+ (Poincare) sphere ... 

K thus defines a static polarization/rotation—whether linear, circular or elliptical—on the Poincare sphere. The 2, r representation of the vector K gives no indication of the future position of K that is, the representation does not address the indicated hatched trajectory of the vector K around the Poincare sphere. But it is precisely this trajectory which defines the particular polarization modulation for a specific wave. Stated differently a particular position of the vector K on the Poincare sphere gives no indication of its next position at a later time, because the vector can depart (be joined) in any direction from that position when only the static 2, r coordinates are given. 

Figure 6. Spin frame representation of a spin-vector by flagpole normalized pair representation a,b over the Poincare sphere in Minkowski tetrad (l,x,y,z) form (n representation) and for three timeframes or sampling intervals providing overall (t]. r ) a Cartan-Weyl form representation. The sampling intervals reset the clock after every sampling of instantaneous polarization. Thus polarization modulation is represented by the collection of samplings over time. Minkowski form after Penrose and Rindler [28]. This is an SU(2) Gd hx) m C over it representation, not an SO(3) Q(to, 8) in C representation over 2it. This can be seen by noting that an b or bt-z a over it, not 2n, while the polarization modulation in SO(3) repeats at a period of 2it.

Figure A-2. Representation of polarization ellipsometry by Poincare sphere.

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