Polarization ellipse

Essentially, this measurement determines the tilt angle of the polarization ellipse (see below).)... 

In order to illustrate the mixed state, an example with five sample wavelets will be discussed in detail. Each wavelet is represented by its components ax and ay in the Cartesian basis (optical definition, see Section 9.2.2). If the polarization vector is described by a polarization ellipse with major and minor axes a = cos y and b = sin y, by a tilt angle X of this ellipse against a fixed coordinate frame (see Fig. 1.15), and by the direction of rotation of the electric field vector indicated by the sign of y, the components ax and cty follow from... 

Figure 1.15 Tilted collision frame at the sample. The photon beam direction defines the z-axis the x- and y-axes are aligned with the major (a) and minor (b) axes of the polarization ellipse which lies in the plane perpendicular to the direction of the photon beam. X is the tilt angle between the x-axis and the plane of the storage ring. The direction of the emitted electron is described by the polar and azimuthal angles 0 and measured in the tilted...

It is convenient to make these substitutions, i.e, to replace the general Stokes parameters and S2 by St and a tilt angle A (which implies S2 = 0). S, is then defined as the greatest possible value of the Stokes parameter for linear polarization or, alternatively, by the excess of linear polarization intensity along the major (a) and minor (b) axes of the polarization ellipse, i.e.,... 

The tilt angle X describes how the polarization ellipse is tilted against the former reference system (the plane of the storage ring). With these quantities equs. (1.50) can be replaced by the simpler form... 

Half wave plates are used to rotate the principal axis of the polarization ellipse (see section 2.4.2). Half wave plates for applications involving single wavelengths can be fabricated in precisely the same manner as quarter wave plates, but with thicknesses that are twice those specified by equation (9.1). 

Maker et observed self-induced rotation of the polarization ellipse of elliptically polarized laser light ... 

Fig. 3. The polarization ellipse referred to space-fixed axes x and y. The propagation direction z is out of the plane of the paper, n is the ellipticity and 0 is the azimuth...

The parameters of the polarization ellipse are thus expressed in terms of the Stokes parameters as follows. The major and minor axes are given by... 

The polarization ellipse along with a designation of the rotation direction (right- or left-handed) fully describes the temporal evolution of the real electric vector at a fixed point in space. This evolution can also be visualized by plotting the curve in (9,tp,t) coordinates described by the tip of the electric vector as a function of time. For example, in the case of an elliptically polarized plane wave with right-handed polarization, the curve is a right-handed helix with an elliptical projection onto the Ocp - plane centered around... 

The second and third of these measurements can be made with a precisely oriented linear polarizer, while the fourth assumes the additional use of a wave plate. Stokes vector components can be expressed by means of the parameters of the polarization ellipse ... 

A natural generalization of the previous geometry is an elhptically polarized (EP) plane wave incident perpendicularly on a layer of nematic that initially has homeotropic aligmnent (see Fig. 6 in the previous chapter). The ellipticity —7t/4 < X < 7t/4 is related to the ratio between the minor and the major axis of the polarization ellipse. The case X = 0 iX = 7t/4] corresponds to a linearly [circularly] polarized light. The sign of x determines the handedness of the polarization, thus it is sufficient to choose X > 0 only. The main difference with the CP light discussed in the previous section is the broken rotational invariance around the z-axis. 

The lengths of the principal semi-major axes can be calculated from the above two equations. The sense of the revolution of an elliptical polarization is determined by the sign of sin 5. If sin > 0, the endpoint of the electric vector revolves clockwise (the light is coming toward the observer). If sin 5 <0, the endpoint of the electric vector revolves counterclockwise. The ellipticity of the polarization ellipse is define by... 

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

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