Circular-polarization vector

Plane-polarized radiation comprises two circularly polarized vectors of equal intensity, one right-handed and the other left-handed (Fig. B3.5.3A), which are separately measured in the CD spectrometer by means of a photoelastic modulator. A chromophore situated... 

There are two known standard methods for decomposition of any smooth (differentiable) vector field. One is that attributed to Helmholtz, which splits any vector field into a lamellar (curl-free) component, and a solenoidal (divergenceless) component. The second, which divides a general vector field into lamellar and complex lamellar parts, is that popularized by Monge. However, the relatively recent discovery by Moses [7] shows that any smooth vector field— general or with restraints to be determined—may also be separable into circularly polarized vectors. Furthermore, this third method simplifies the otherwise difficult analysis of three-dimensional classical flow fields. The Beltrami flow field, which has a natural chiral structure, is particularly amenable to this type analysis. 

As shown on Figure 14B, the sum of the two circularly polarized vectors describes an ellipse after interacting with the sample. The ellipticity 6 is defined as the arctangent of the ratio of the axes of this ellipse ... 

State I ) m the electronic ground state. In principle, other possibilities may also be conceived for the preparation step, as discussed in section A3.13.1, section A3.13.2 and section A3.13.3. In order to detemiine superposition coefficients within a realistic experimental set-up using irradiation, the following questions need to be answered (1) Wliat are the eigenstates (2) What are the electric dipole transition matrix elements (3) What is the orientation of the molecule with respect to the laboratory fixed (Imearly or circularly) polarized electric field vector of the radiation The first question requires knowledge of the potential energy surface, or... 

Note that there are other possible ways to impose a time-dependent Bx. The one described in eqn (5.8) corresponds to a circularly polarized field initially aligned along the x-axis and rotating about the z-axis in a counterclockwise direction. The vector product of eqn (5.7) then becomes ... 

If the plane of polarization is defined as xz, the electric vector can be represented in terms of the unit vector i in the direction of x as E = E0i cos 0. This plane polarized wave may be decomposed into two circularly polarized components propagating in the same direction. A simple description of these waves are given by... 

The meaning of circular polarization is explained in terms of Figure 1, showing the vector sum i cos 9 + j sin 8 for a half wave. 

The composite vector is seen to spiral around the z-axis and in projection moves anti-clockwise in a circle around the z-axis. The other component which is the mirror image of the first, performs a clockwise circular motion in projection along z. The decomposition into circularly polarized components can also be formulated in complex notation, Er = E0e t6. 

The state of polarization is determined by the pair of complex numbers e and e2 the quantities ei 2 and e2 2 represent probability densities of a definite (linear or circular) polarization of the photon as determined by the unit vectors Xi and x2- Since ej and e2 are related by the normalization condition... 

Since the spin operator commutes with the momentum operator, it is possible to speak of states of definite momentum p and spin component /x. The components of the polarization vector may be chosen in such a way that e = XP- The two possible polarizations correspond to only two values of the component of spin angular momentum y,. The third value is excluded by the condition of tranversality. If the z-axis is directed along p, then x0 s excluded. The two vectors Xi and X2> corresponding to circular polarization are equivalent, respectively to Xi and X-i- Thus, the value17 of the spin component y = 1 corresponds to right circular polarization, while /z = — 1 corresponds to left circular polarization. 

The application of circularly polarized rf fields in ENDOR spectroscopy (CP-ENDOR) will be described in Sect. 4.6. Circularly polarized rf fields may be obtained by the vector... 

Figure 7 shows an aberration-free intensity distribution at the focus of a typical objective lens similar to that used for DLW lithography. Calculations were carried out using a vectorial Debye theory, which accounts for the polarization effects. For the linearly polarized wave it can be seen that the spot is elongated along the polarization vector. To reduce this asymmetry, a X/4-plate can be used to convert the polarization of the incident beam to circular, which can be interpreted as a combination of two mutually perpendicular linearly polarized components. Thus, width of the photomodified line becomes independent of the beam scanning direction in the sample. 

Equation (2.78) describes an ellipse, the vibration ellipse (Fig. 2.11). If A = 0 (or B = 0), the vibration ellipse is just a straight line, and the wave is said to be linearly polarized , the vector B then specifies the direction of vibration. (The term plane polarized is also used, but it has become less fashionable in recent years.)If A = B and A B = 0, the vibration ellipse is a circle, and the wave is said to be circularly polarized. In general, a monochromatic wave of the form (2.77) is elliptically polarized. 

Experiment IV. Circular Polarizers. We need to introduce one more set of basis vectors and L ... 

These basis vectors represent right-circularly and left-circularly polarized waves and are orthonormal in the sense that... 

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. 

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.

The handedness, or chirality, inherent in foundational electrodynamics at the U(l) level manifests itself clearly in the Beltrami form (903). The chiral nature of the field is inherent in left- and right-handed circular polarization, and the distinction between axial and polar vector is lost. This result is seen in Eq. (901), where , is a tensor form that contains axial and polar components of the potential. This is precisely analogous with the fact that the field tensor F, contains polar (electric) and axial (magnetic) components intermixed. Therefore, in propagating electromagnetic radiation, there is no distinction between polar and axial. In the received view, however, it is almost always asserted that E and A are polar vectors and that is an axial vector. 

In the most general case, the above two equations mean that the electric field vector traces an ellipse in the j>z plane. There are two special cases of note in this general situation. If the phase difference between the two components of the field (5K -5.) is zero or some integral multiple of 7r, the ellipse flattens to a line. If the phase difference is t/2 or any odd integral multiple of tt/2 and the amplitudes of the two components are equal, the ellipse is rounded to a circle. In the former case we speak of the radiation as being plane polarized, and in the latter case as being circularly polarized. 

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