Light, polarisation

An interesting point concerns polarisation effects in the Raman spectra, which are commonly observed in low-dimensional materials. Since CNTs are onedimensional (ID) materials, the use of light polarised parallel or perpendicular to the tube axis will give information about the low dimensionality of the CNTs. The availability of purified samples of aligned CNTs would allow us to obtain the symmetry of a mode directly from the measured Raman intensity by changing the experimental geometry, such as the polarisation of the light and the sample orientation, as discussed in this chapter. 

Figure 2.41 The direction of the electric vector for light polarised in the S- and P-directions before being incident at a planar surface at an angle 6, and the resolution of the electric vector of the P-polarised light into its components parallel and perpendicular to the reflective surface.

Dust particles appear to become oriented in interstellar magnetic fields, aligning themselves with the field lines to produce preferential absorption of light polarised... 

Fig. 9.20 Lower panel, real and imaginary refractive indices of a PDA-TS crystal at room temperature for light polarised parallel to the polymer chains. Upper panel, reflection spectrum recorded under the same conditions. Reprinted with permission of Wiley-VCH from Bloor and Preston (1976).

Fig. 9.21 Upper curve electro-reflectance spectrum of a PDA-DCH crystal at 8 K for light polarised parallel to the polymer chains. (Note the scale change at 2.3 eV.) Lower curve first derivative of the reflection spectrum. Reprinted with permission from Weiser (1992). Copyright 1992 by the American Physical Society.

Optical activity is observable in any direction for crystals belonging to the two cubic enantiomorphic classes 23 and 432, but, in general, optical activity can only be observed in certain symmetry limited directions. For example, optical activity in the other enantiomorphic classes is only readily observed in a direction fairly close to an optic axis. This is because the different refractive indices that apply to light polarised vertically and horizontally masks the effect in directions further from an optic axis. In the non-enantiomorphic groups, no optical activity is found along an inversion axis or perpendicular to a mirror plane. Thus no optical activity occurs along the optic axis... 

FIGURE 1 Absorption edge of several polytypes at 4.2 K light polarised E Lc [5]. 

The total intensity is then I(t) = Ip(t) + 2 I/t). The factor of 2 results from the geometrical fact that light polarised longitudinally to the optical axis of observation is not detected [308, 549]. The situation for a microscope lens of high NA is different. Both focusing and detection under high NA results in the conversion of transversal E vectors into longitudinal ones, and vice versa [18, 426, 465, 466, 551], see Fig. 5.9. The total intensity is therefore I (t) = Ip(t) + kig (t), with k<2. 

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

It seems that a better suited approach should make use of a series of actual measurements on an evolving material quantum system. In fact, two-level atoms, equivalent to spin systems, show a formal analogy with light polarisation. Whereas the configuration space of polarisation transforms according to symmetry group SO(3), the symmetry of spin transformation is SU(2), which is a double covering of SO(3), and locally isomorphic with the latter one [17]. Thus, similar visualisations of the dynamics of both systems apply. 

Polymers are not, however, always randomly oriented and the phenomenon of orientation is discussed in the next two chapters, chapters 10 and 11. One of the ways used to obtain information about the degree of orientation of the polymer is to measure either the refractive indices of the polymer for light polarised in different directions or its birefringences, i.e. the differences in refractive index for light polarised in different directions. In chapter 11 the theory of the method is described in terms of the polarisability of structural units of the polymer. This polarisability is a second-rank tensor like the molecular polarisability referred to in the earlier sections of this chapter and, insofar as the assumption of additivity in section 9.2.3 holds, it is in fact the sum of the polarisability tensors of all the bonds in the unit. Since, however, the whole basis of the method is that the structural units are anisotropic, the tensors must be added correctly, taking account of the relative orientations of the bonds, unlike the treatment used to calculate the refractive index of PVC in example 9.1, where scalar bond refractions are used. 

Equations (9.40) can be used to evaluate the refractive indices for the crystals of any polymer with orthorhombic symmetry, provided that the co-ordinates of all the atoms within the crystal structure are known, so that all the bond orientations can be calculated. The value of an for the crystal is substituted into the Lorentz Lorenz equation (9.9) in place of a and the value of n obtained is assumed to be equal to the refractive index for light polarised with the electric vector parallel to OXi. Similar calculations are performed to calculate the indices for light polarised parallel to OX2 and... 

Starting from the Clausius-Mosotti equation (9.7) and assuming that it can be applied to each principal direction Ox, in an orthorhombic crystal, show that the Lorentz-Lorenz equation can be written in a form that leads to the equation , = (1 + 2x,)/(l — x,), where , is the refractive index for light polarised parallel to Ox,- and X,- = q ,-/(3 oF), with a,- equal to the polarisability per unit cell for light polarised parallel to Ox,- and V the volume of the unit cell. 

Assuming that the backbone of the polyethylene molecule in the crystal has the CCC angle 112° and that the HCH angle is tetrahedral, calculate the polarisability of each molecule per translational repeat length in the three symmetry directions. Use the result in problem 9.6 to deduce the refractive indices for light polarised parallel and perpendicular to the chain axis in the crystal. (Data for the polyethylene unit cell are given in section 4.4.1.)... 

A particular sample of a uniaxially oriented polymer is composed of structural units each of which is transversely isotropic with respect to an axis Oz within the unit. The value of (cos 0) is 0.65, where 6 is the angle between the Oz axis of a typical unit and the draw direction. There are 4.2 x 10 structural units per m in the polymer and the polarisabilities of an individual unit for light polarised parallel and perpendicular to Oz are 2.6 x 10 and 1.9 x 10 F m, respectively. Calculate the polarisabilities of the sample for light polarised parallel and perpendicular to the draw direction and hence the birefringence of the sample. 

A highly uniaxially oriented polymer sample in the form of a thin film has mean optical polarisabilities of 2.3 x 10 and 1.7 x 10 F m per struetural unit for light polarised parallel and perpendicular to the draw direction, respectively it transmits 50% of infrared radiation at a particular wavelength 2 when the radiation is polarised parallel to the draw direction and transmits almost all... 

---