Poincare sphere

The longitudinal angle a of the point representing the polarization on the Poincare sphere is given by 

Therefore /i = 2t/. If we know the azimuthal angle cp and the ellipticity angle v of the polarization ellipse, the Stokes vector is 

If we know the angle x = otm AJAy) and the phase difference 8, then the Stokes vector is 

The points corresponding to some special polarization states are as follows  

Two diametrically opposed points on the sphere correspond to states with orthogonal polarization. 

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Podolsky method, Renner-Teller effect, triatomic molecules, Hamiltonian equations, 612—615 Poincare sphere, phase properties, 206 Point group symmetry ... 

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. 

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. 

Figure 10. Time-resolved measurements of a very large polarization fluctuation, where the size of the fluctuation is about half (in fact, 45%) of that of a complete polarization switch [84],, V, , v2 and Si are the normalized Stokes parameters representing the polarization state on the Poincare sphere [84],...

It is weliknown that all static polarizations of a beam of radiation, as well as all static rotations of the axis of that beam, can be represented on a Poincare sphere [25] (Fig. la). A vector can be centered in the middle of the sphere and pointed to the underside of the surface of the sphere at a location on the surface that represents the instantaneous polarization and rotation angle of a beam. Causing that vector to trace a trajectory over time on the surface of the sphere represents a polarization modulated (and rotation modulated) beam (Fig. lb). If, then, the beam is sampled by a device at a rate that is less than the rate of modulation, the sampled output from the device will be a condensation of two components of the wave, which are continuously changing with respect to each other, into one snapshot of the wave, at one location on the surface of the sphere and one instantaneous polarization and axis rotation. Thus, from the viewpoint of a device sampling at a rate less than the modulation rate, a two-to-one mapping (over time) has occurred, which is the signature of an SU(2) field. 

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

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 5. The left side [SO(3)] describes the symmetry of the trajectory K on the Poincare sphere the right side describes the symmetry of the associated 6i( K%) and Oity, %) which are functions of the v /,% angles on the Poincare sphere. (Adapted from Penrose and Rindler [28].)...

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.

It is well known that states in a two-dimensional system can be described by means of the Stokes parameters and visualized by means of the Poincare sphere. The density matrix of any two-state system can be written in the form... 

We note that any pure state in is coherent. The interpretation of the parameter a is very simple its module is proportional to the polar coordinate, while its argument cp is the azimuthal coordinate of the representative Poincare sphere point. 

For the case of two-dimensional CS, there have been computed quantities such as the mean values and variances of the various operators, including N and 4> quadratures and their commutators [16,17]. Most of these quantities can easily be displayed on the Poincare sphere and expressed by means of the Stokes parameters. We find that the following mean values and variances are given respectively by... 

FIGURE 2.4 Configuration space of light polarisation ( Poincare sphere ). Thick polarisation arrow in foreground, thin arrow in background. 

For completely polarized light, the normalized Stokes parameters satisfy the condition + 52 + 51 = 5o = l. Therefore the point with the coordinates (Si,S2,S3) is on the surface of a unit sphere in 3-D space. This sphere is known as the Poincare sphere and is shown in Figure 3.6. 

We consider how the three-component Stokes vector S evolves on the Poincare sphere under the action of retardation films. The Mueller matrix of a uniform uniaxial retarder with the retardation angle F and the slow axis making the angle (p with the x axis is given by (from Equations (3.77) and (3.78))... 

One of the reasons to use the Poincare sphere is that the effect of retardation films and the evolution of the polarization state can be easily visualized. We consider polarization conversion in the following few special cases. 

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.

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