TIPOS DE PLÁSTICOS DEGRADABLES

The co-ordinate axes are defined as a right handed set, with the x vertical (positive direction upwards), y horizontal (positive direction to the right), and with light travelling along the z direction with the source behind the observer. This is illustrated in Figure 2-1. When a polariser is set at 0°, the electric vector vibrates along the x axis.

-ve rotation +ve rotation

Figure 2-1 Co-ordinate axes defined in this work

Light propagates along the z direction

2.2

Elliptical polarisation

The two components of a light wave vibrating along the x and y axes can be represented as [Born & Wolf, 1999, p.25 et seq.]:

= a cos[cüt -A z-t-y) equation 2-1

E y = b cos{cot — Az — y j equation 2-2

Where a and b are the amplitudes of the x and y components respectively, k is the wave constant {=2 tc/X), cû is the angular frequency (=2tc/ where/ is the frequency in hertz),

and t is the time in seconds. If the absolute phase of thex component is defined as and that o f the y component is ^ then ^ is the phase difference between the orthogonal vibrations (defined so (f>=(f>x- (f>y). When sin <f> is negative, the x vibration leads the y, so the light is defined as right-handed (see Appendix 2A). This represents the tip of the electric vector tracing a clockwise rotation when the observer looks towards the source [Azzam & Bashara, 1979, p.39]. Adding these two vibrations leads to the equation of an ellipse oriented at an angle or to the co-ordinate axis [Bom & Wolf, 1999, p.25 et seq.] - equation 2-3. It can be seen that the sign of <f> does not alter this equation.

2 f E '\( E equation 2-3 y - 2cos^ ’ a y = sin (f> \ o j

This is illustrated in Figure 2-2. a is defined so -180°< a <180° An azimuth angle o f +30° is equivalent to one of -150° '.

' This is valid because in the relevant equations (namely equation 2-5 and equation 2-6) cos û: or sin a is either squared, in which case the sign doesn’t matter, or multiplied by sin or cos a. In the latter case either both the sine and cosine have the same sign, or one is negative when the other is positive; so dividing (or multiplying) one by the other will give the correct sign. As an example, taking a value of a to be +60° is equivalent to taking it to be -120°. At 60°, both sine and cosine are positive, and at -120° both are negative, so dividing one by the other will be positive.

Chapter 2 Polarisation

Figure 2-2 Ellipse oriented at an angle alpha to the co-ordinate axes

The azimuth angle of the ellipse, a, is given by equation 2-4 [Bom & Wolf, 1999, p.28].

1

a - —tan-1

2ab cos(j) _{equation 2-4}

The magnitudes of the major and minor axes of the ellipse can be calculated from the original amplitudes of the vibrations, and the phase difference between them (equation 2-5 and equation 2-6 [Collett 1993, p.28]).

.2 _ 2 „ O______ :_______A COS ÛT sin a 2 COS ûf sin a COS ^ sin ^

. 2

ab

(o')

2

sin 6% cos a 2 cos a sin or cos sin é : — + •

equation 2-5

equation 2-6

a"- b^ ab {b 'Ÿ

In this thesis, polarisation is defined using the azimuth angle of the ellipse, a, and its

ellipticity, e, defined so that ^ ~ ~ p2 - should be noted that this is the square o f the

reciprocal of the definition in some textbooks.

2,3

Representations o f polarisation

Two mathematical representations of polarised light have been developed. The first originated in the 1850s and was developed by G.G. Stokes [Hecht, 1998, pp.366-368; Azzam & Bashara, 1979, pp.55 et seq.], and the second was developed in 1941 by R. Clark Jones [Hecht, 1998, pp.368-369]. Each method has its advantages and disadvantages: Stokes vectors

are more mathematically complex than Jones vectors because the Mueller calculus that is needed to transform one [4 x 1 ] Stokes vector into another involves [4 x 4] matrices, whereas the matrices needed to transform the [ 2 x 1 ] (complex) Jones vectors are only [2 x 2]. Additionally, because Stokes vectors do not include phase information, they do not describe coherent light. However, the Stokes representation can be used when the light is not completely polarised, and the intensity based definition means that it is possible to look at a Stokes vector and immediately have some knowledge about the state of polarisation (see s.2.3.7).

Jones calculus does not describe partially polarised light, or optical systems which depolarise. This disadvantage can be circumvented if partially polarised light is thought o f as a combination of completely polarised light and completely unpolarised light [Clarke & Grainger, 1971, p.31; Chen & Wolff, 1998). It is also not normally easy to tell the state of polarisation simply by looking at a Jones vector - calculations have to be performed on the vector before the exact state of polarisation is known. Jones calculus is covered in [Jones, 1941] and [Jones, 1942]. A complete discussion of the difference between Stokes and Jones vectors can be found in [Azzam & Bashara, 1979, pp. 13-65].

Many papers have covered the modelling of liquid crystal cells using Jones matrices (such as [Gooch & Tarry, 1975], [Raynes. & Tough, 1985], [Raynes, 1987]). A Jones matrix model o f the LCTVD, which has been developed by researchers at UCL, is used in this work [Kilpatrick et al., 1998]. Consequently Jones calculus is initially used in this thesis. However, Stokes vectors will also be used in Chapter 6, so this representation is also discussed in this chapter (section 2.3.7).