SISTEMAS DE CONTROL

Figure 4.10 Maker fringes in homeotropically aligned SCE13.

Wedge cell measurements of SCE13* using horizontally polarised fundamental radiation at the angle of incidence 24.28® (refracted angle = 16.4®) resulted in phase matched SHG from variation in device thickness across the wedge, as shown in figure 4.11. This configuration accessed only the d2i coefficient, and from the amplitude of the maximum, an estimate o f the coefficient was made.

« 0.14 — a 0.10 - ^ 0.08 - § 0.06 — d ^ 0.04 H

2

4.5.2 M olecular and Bulk N onlinear CoefTicients.

A measurement of the bulk nonlinear coefficient d^ff for SCE13 at 50®C can be made from the amplitude of the phase-matched peak in graph 4.10 using equations 4.16 and 4.22. At the angle of incidence of -6.5®, the value of dgff was calculated to be 0.0008 p m V ^ for eeo polarisations. The &2\

coefficient, which had not been directly measured in ferroelectric liquid crystal materials before, was estimated from figure 4.11 to be 0.0001 pm V'^. Similarly, a value of d2% was also calculated from the SHG data measured using ooo polarisations, and was equal to 0.0031 pmV"^. Both of these values are very small in comparison to other types of materials used for second harmonic generation. However, this is not surprising, as SCE13* was not developed for nonlinear optical applications, and only about 10% of the material contained in the mixture is chiral.

In crystalline or even polymer materials for SHG, there has been a great deal of success in relating values of the molecular hyperpolarisability to the bulk nonlinear coefficient The values for the hyperpolarisabilies have been calculated using ab-initio or semi-empirical models [Prasad & Williams, (1991)]. The molecular orientation may be calculated within the unit cell of the material, and combined with the volume of the unit cell, it is possible to produce values for the nonlinear d coefficients which correlate with the measured values. This should also be possible in liquid crystal materials.

Equation 2.8 in chapter 2 described the translation from the molecular nonlinear hyperpolarisability to the bulk coefficient for a crystalline material. This equation has also been adapted to describe poled polymers and EFISH measurements by changing the unit of material for calculation from the crystal unit cell to the number density of molecules (N) in a non-crystalline material [Zyss & Gudar, (1982)]. Equation 2.8 then becomes

X u l (-û >3;û> „û)2 ) = W /“* / " 7 j r { Put (-0 ),] W „ û )^ 4.28 and the orientational average of the hyperpolarisabilities is calculated using Euler angles from

~ \ P i j k ^

where ax x are the direction cosines and G(G) is a distribution function.

It would be impossible to calculate the orientation for each molecule within the liquid crystal phase, and so to simplify the calculation I shall make a number of approximations about the molecules and the chiral smectic C phase they form. The approach shall follow the theory of Zeks et al [1988], and its application by Buka et al [1989]. The microscopic model of Zeks et al assumed that the hindered rotation of the molecules around their long molecular axis can be described by a single particle potential U(\y)

t / ( i / r )

=

+U

= -a^d cosy/ -

^ c o s l y /

where \j/ = 0 when the C2 axis is perpendicular to the plane of the tilt (0) and parallel to the direction of Pg. The first term describes the polar nature of the potential and the second term is due to the biaxiality of the

molecules. Generally the second term is much larger than the first, and so the combined potential has two minima and two preferred orientations exist at \j/ = 0 and n. The minimum at \|f = 0 is deeper than at \|f = 71, and so it is more likely that the molecules will be found at this orientation.

Buka describes how the spontaneous polarisation can be related to the polar order parameter II for rigid molecules which have perfect nematic order (S=l)

P, = pli^{cosy/)

where p is the number density and |Ij^ is the transverse component of the molecular dipole moment along the C2 axis. This expression ignores any contributions to |I ^ from the longitudinal dipole moment due to molecules not having perfect order. Both the ferroelectric polarisation and the second order nonlinear coefficient have a similar requirement of noncentrosymmetry, and so have only non-zero components involving the C2 transverse polar axis of the molecules. From equation 4.28 and 4.29, the nonlinear coefficient can be found from

d,jK _UK 4.32

In the case of input and output polarisations being parallel to the y axis, then equation 4.32 is reduced to

dyyy =

f ? f } “

v ) -

There is experimental evidence to suggest that the second order nonlinear coefficient has a similar dependence on the temperature as the Pg of the bulk material. This temperature dependence was measured in the homeotropic geometry for the coefficient d2 2 of SCE13, and is shown in figure 4.12.

r 30

1.2

1.0

20

0.8

2

0.6

10

0.4

0.2

Sc-Sa transition

0.0

20

30

40

50

60

70

I

Temperature (C)

The potential U(\|/) can be used to calculate the polar order parameter from

I n