A l The Relationship between the Microscopic HyperpolarisabUity and the Bulk Susceptibility.
The macroscopic second order susceptibility is related to the molecular hyperpolarisability P by
^ ?
where there is a summation over all molecules q within the unit cell, V is the volume of the unit cell,/® are the local field factors and a// are the direction cosines of the molecular frame of reference xyz with respect to the bulk material frame XYZ. To simplify the form of a suitable symmetry group is applied which describes the symmetry of the bulk material.
The noncentrosymmetric point group oomm is applicable to systems which lack any positional ordering, but which contain a unique axis, about which there is infinitefold rotation. This point group describes the poled nematic and smectic A liquid crystal systems, to which we will restrict the following arguments. The effect of applying the oomm point group on the tensor is to limit the number of non zero components [Nye, (1964)] to four: X3 3 3, X1 1 3 = X223’ %131 = X232’ %311 = X322- Th® superscript (2)
is omitted for clarity.
There are a number of theoretical models for the electric field poling of hyperpolarisable molecules. These are the isotropic model, the Ising model [Meredith et al, (1985)], the Singer Kuzyk and Sohn model (SKS) [Singer et al, (1987)] and the Maier-Saupe model extended by van der Vorst and Pinken (MSVP) [Mohlmann & van der Vorst, (1989)]. The isotropic model assumes no preferential packing of molecules in a material, whereas the Ising model assumes molecules are cylindrical and pack perfectly to give an order parameter S equal to unity. The real life situation lies somewhere between the two models and can be described by either the SKS or MSVP models.
The SKS model utilises the axial order parameters <P2> and <P^> of the liquid crystalline host to describe the ordering and it is assumed they are insensitive to the applied field. The MSVP model is a self consistent calculation of the ordering of a system using three energy terms for the constituent molecules. These are the potential energy of a single molecule in a self ordering system described by <P2>, the energy of the permanent dipole inside a field, and the energy of the induced dipole inside a field. The SKS and MSVP models give almost identical results for poled nematic systems, the only difference being the starting conditions. The SKS model requiring independent measurements of <P2> and <P4>, and the MSVP model requiring the magnitude of the three energy terms. I shall concentrate on the SKS model.
A2 Calculation of the O rientational O rdering.
The bulk susceptibility of the material will be calculated for an orientated material at the elevated temperature at which the material was poled. When the material was cooled to lock in the induced alignment, the thermal disorder associated with the elevated temperature was also stored. It must be remembered that the energy associated with ordering is a combination of the short range intermolecular interaction as seen in nematic phases, and the externally applied Held, which is comparable to the thermal energy kT. It is assumed that the molecular nonlinear optical properties were not significantly perturbed by the neighbouring molecules. The bulk susceptibility can then be described as the statistical average of the molecular hyperpolarisability
XlJK ( - ® 3 ; (-Û), ; Û), , (Bj A2
where N is the number density of the nonlinear optical molecules and is the hyperpolarisability including local field effects. The molecular and bulk coordinate systems are related by the Euler angles (p, 0, and y , which can be transformed using the matrix a
cos 0 cos (p cos i/A - sin (p sin !//■
cos 0 sin çî cos y/" - cos ç) sin
-s in ^ c o s v '’
-c o s ^ c o s (p s in i/ - s in t p c o s i/
- c o s ^ sin sin i / - cos ç) cos i//
sin ^ sin ^
sin 6 cos (p
sin 6 sin y/
sin 0
The orientational average of the hyperpolarisabilities in the presence of a poling field Bp is of the form
( )//x ~ J )Pijk ^
where the volume element is given by
2ti I n + 1
j clQ.= j d ç j d y / j d { c o s 6 ) A4
0 0 - 1
and ,Ep ) is a distribution function. If the poling field couples with the molecular dipole moment m* which has been corrected for local field factors, and U describes the short range interaction potential between molecules which is significant in liquid crystal phases, then G(Q ,Ep ) is described by
It is assumed the molecular interactions described by the potential U are only dependent on 0 for materials with uniaxial symmetry, and it is noted that U does not distinguish between alignment parallel or
antiparallel with respect to the director n. In the case where the interaction between the applied field and the molecular dipole is smaller than the thermal energy, then the exponential can be expanded to
ex p [-(t/ - m*. / ) tr )] = [l + (m*. £ J / t r ] e x p [-(//tT ]
The function G(f2,Ep) can be expanded in terms of Legendre polynomials
G(e,E^)
=f^i^A,P,(cose)
A6 /=0 with +1A, = j d(cosd)G{6,Ep)Pi{cos6).
-1The Aj are the orientational average of Pj , denoted <P|> , and are defined as the microscopic order parameters. It is assumed that <P;> can be independently measured. The product of the transformation components and the distribution function shown in equation A3 can also be expanded in Legendre polynomials. Integrating this over cp and \|/, within the limits from A4, then the macroscopic second order susceptibility becomes
N F
X u . = - ^ [ < + < ( f 2> + C ( / ^ ) ] A7
where the order parameter coefficients are expressed in terms o f the molecular hyperpolarisability b*jjj^ and the molecular dipole moments Singer et al [1987] describe the full expansion of u(^) for all components allowed within the <»mm symmetry.
Physical insight can be gained by considering the simplified situation of a one dimensional molecule, where only and are nonzero. This approximation is not unreasonable for the rod shaped molecular systems likely to be encountered. Equation A6 can be written
N F
^ 3 3 3 = " ^ ^ z l ' ” z [ 5 + 7 - ( ^ 2 ) + ^ ( ^ 4 ) ] ^ ^ 8
and
NE
X m = X m =X}n
A9
The local field factor can be isolated in this approximation as
where is the molecular dipole moment. The local field factors have been discussed in section 2.3 for the particular case of liquid crystal materials using the long thin cylinder approximation for molecules. In
the isotropic phase the local fields at optical frequencies can be described by the Lorenz- Lorentz expression
fw = (n * 2 + 2 )/3
but in the case of liquid crystal materials, the equation must be modifîed to take into account the shape of the molecules and the difriculty in reorienting these molecules. The equation
- 1 ) 4
where L^+L2= l/ 2 and Lg=0 has been found to be a good approximation. The dipole in the presence of the local poling field can be described by the Onsager expression
r =
e