Espectro FTIR del PEBD con sus bandas características (Rajandas, et al, 2012, p 1096)

Biréfringent materials have different refractive indices to different polarisations, and may also be called refractoanisotropic [Shurcliff, 1962, p.65]. This causes incident light falling on the biréfringent material to give rise to two refracted waves [Bom and Wolf, 1998, pp.684 et seq.].

The incident ray is separated into two orthogonally polarised components, an ordinary ray, which obeys the usual law of refraction and an extraordinary ray that does not '''. These components may be linearly polarised (for a linearly biréfringent material) or circularly or elliptically polarised. The test security marking mentioned later in this work is made of linearly biréfringent material, so this will be discussed in detail.

A biréfringent material has (at least) two values of refractive index, the major and minor principal refractive indices. A material with only two principal refractive indices is called uniaxial. This is the type that will be used in this work. The optic axis of the material is defined as the direction of propagation (relative to the crystal lattice) for which light of all vibrational directions travels at the same speed [Clarke & Grainger, 1971, p.73].

The change of refractive index with direction of light propagation can be visualised by the use of a uniaxial indicatrix, where the lengths of the (major and minor) axes represent the extraordinary and ordinary refractive indices and«„) [Collett, 1993, p.444]. A positive (i.e. n^rio) uniaxial indicatrix is shown in Figure 2-4. For a particular angle of incidence, the refractive index is calculated by taking the direction of the incident light (relative to the optic axis) to the centre of the figure, and then dropping perpendiculars to the surface [Shurcliff,

1962, p.68] (shown by and rio in the figure). The length of these normals indicates the refractive indices that are encountered by the orthogonal components of the light. The largest refractive index, rig is encountered when light is incident orthogonal to the optic axis. Light travelling along the optic axis, z has no change in its polarisation {rtx=no).

'' i.e. sin //sin r = n, and the beam lies in the plane of incidence,

the ray is not always in the plane of incidence, and its velocity is a function of its direction of travel through the material.

Chapter 2 Polarisation Optic axis

A

V

y

Figure 2-4 Index ellipsoid for a uniaxial (liquid) crystal.

The effect a biréfringent material has upon light passing through it differs considerably depending on whether the principal plane (the plane containing the optic axis) is parallel to the incident face o f the material. For completeness, the behaviour o f a biréfringent material when its optic axis is not parallel to the incident face of the material is covered in Appendix 2E. The effect when the principal section is parallel to the incident face is discussed here, as this is the aiTangement used in retarders and the security marking discussed in Chapter 8. The special case of twisted nematic liquid crystals (TNLCs) will be discussed in s. 2.4.2.b.

When the biréfringent material is arranged so that the principal plane is parallel to the surface of the material (i.e. in the xy plane), light travelling along the z direction (i.e. normal to the surface) will always have its E vector vibrating in the principal plane. Therefore its orthogonal components can always be considered as being parallel or perpendicular to the optic axis (Figure 2-5).

rie

no

Optic axis Figure 2-5 Principal plane parallel to incident face of biréfringent crystal. Light incident normally on the crystal will always > z

have its orthogonal components vibrating parallel and perpendicular to the optic axis.

Light will travel through the material undeviated, and if the incident light is unpolarised no effect will be seen as the emergent light will also be unpolarised. If the incident light is polarised, the two orthogonal components will travel through the crystal at different velocities. They will recombine on emergence from the crystal, but with a different relative phase than they had when incident on the crystal, and hence a different polarisation. The resultant polarisation depends on the incident polarisation, the retardance of the material {And, where An=n,-n,), and the relative amplitudes of the components parallel and perpendicular to the optic axis (which can be varied by rotating the biréfringent material in the xy plane).

2.4.2. b

Birefringence and liquid crystals

The exact configuration of a twisted nematic liquid crystal display (TNLCD) will be discussed in Chapter 3 (s.3.4.2), but the effect of the birefringence of the TNLCD on polarisation will be discussed here. The anisotropy of nematic liquid crystals causes them to be biréfringent. Nematic liquid crystals are uniaxial, but differ from crystalline waveplates in their effect upon polarised light because the configuration of the TN structure means that as voltage is applied across the LC cell, the optic axis tilts towards or away from the source, rather than merely rotating in the xy plane.

In a TNLC device, as the director (optic axis) tilts (with increasing voltage), the index indicatrix tilts about they (or x) axis (i.e. towards or away from the source), and so the length Hx in Figure 2-4 changes. The varying refractive index for a given direction, ly, is defined by [Yeh & Gu 1999, p.63]:

^co s^ u/ sin^ ty l equation 2-21

=

\ "'o "’e J

The change in refractive index, nx causes a change in the effective birefringence. An {AnF=nx- «J. This causes a change in phase difference between the two rays travelling parallel and perpendicular to the optic axis. Therefore there is a change in polarisation with voltage.

2.4.2.0

The different use of birefringence and liquid crystals in this thesis

Liquid crystals are used in two separate ways in this thesis: Firstly, a twisted nematic liquid crystal, as described in Chapter 3, is used either in a pixellated display or as a single pixel cell. As discussed in s.2.4.2.b, the application of a voltage to the LC cell causes a change in polarisation of light passing through it. This is caused by the re-orienting of the optic axis changing the effective birefringence. However, the application of voltage does not make the

Chapter 2 Polarisation

optic axis rotate in th exy plane (it twists in thex^ plane and tilts out of it - see s.3.4.2), the amplitudes o f the components o f the incident light which are parallel and perpendicular to the optic axis o f the LC do not change. The change in polarisation is caused by the change in«v (equation 2-21).

Secondly, a LC polymer is used to make the security marking described in Chapter 8. This nematic material is also biréfringent, but is arranged in such a way so that its optic axis varies spatially across the marking in theory plane. This too changes polarisation, but it does so not by changing the birefringence, An, but by changing the orientation o f the optic axis ( 6

in equation 2-23).