The interest in the separation of enantiomers of organic compounds has been recently increased in the food, medicine and cosmetic industries. As it is well know, the chirality is extremely important for living systems, because most of the biomolecules are chiral ones
Liquid crystal elastomers combine the rubbery elastic properties with the liquid crystals (LC) anisotropy . When these stiff molecules are joined to a large and flexible polymer chain, they leave their muddy liquid macroscopic form and acquire the appearance of a rubbery network. Particularly, imprinted cholesteric elastomers acquire its chirality from an induced chiral structure instead of chiral molecules. Upon cross-linking nematic polymers in a chiral solvent to form a gel, the solvent, which causes a natural twist to the nematic is thus removed but nonetheless leaves behind an imprinted cholesteric elastomer . A spectacular property of imprinted networks is their capacity to preferentially absorb and retain right or left molecules from a racemic solvent which renders them chiral pumps. Moreover the elastomersare photonic materialsin the sense that exhibit an optical band structure just as the one shown by electrons in metals and semiconductors. Elastomers provide a large spectrum of applications that exploit the possibility of externally handle a band structure by various types of agents like mechanical stress, electric fields chemical solvents for designing devices such as optical sensors, mirrors, laser and filters  .
The purpose of this chapter is to propose an optical method to determine the volume fraction of preferentially absorbed molecules by measuring the transmittance and reflectance spectra during a preferentially absorbing process. Specifically, we are interested in calculate the reflectance and transmittance of circularly polarized light for oblique incidence due to a cholesteric elastomer, assuming arbitrary values of chiral order parameter, . We have thoroughly discussed the solution of the boundary value problem for cholesteric elastomers in sections 4 and have determined the optical spectra of the system in Sec. 5
2. Elastic formulation
A cholesteric elastomer has locally a nematic structure. Consequently, the director vector is at plane and is the angle between director vector and axis (see Fig.1). The director vector has a continuous rotation as function of , which describes a periodic helical structure characterized by a pitch (or equivalent by wave number )  .
The chiral order parameter is related with the imprinting efficiency of the cholesteric phase, where is the nematic penetration depth in rubbery networks, is the twist elastic constant and is local anchoring of the director to the rubbery network. Variations in the chiral order parameterimplychanges in the robustness of chiral imprinting, which can be controlled by varying the density of cross-links in the network (affecting ) .
The energy for an elastomer formed under thepresence of a chiral solvent which is subsequently replaced with an achiral one is given by ,
The Frank free energy to twist the material is given by
associated with the first term of the eq. (1)- tend to unwind the cholesteric helix but it finds resistance to rotate in the anchoring of the rubbery network- associate with the second term of eq. (1). is proportional to rubbery elasticity modulus and the rubbery elasticity modulus is related with cross-linked density . The scale length or nematic penetration depth rubbery at which the two energy contributions are comparable is usually small m for a typical J/m, where is the anisotropy chain number or cross-linked number and J/m is the elasticity rubber modulus. The pitch is bigger than nematic penetration depth , then
There are two forms of modifying this relation, i) by increasing the nematic penetration depth which amounts to have a weaker gel or one with smaller anisotropy, ii) by enlarging the Frank free energy. The chiral imprinting will be lost if diminish, i.e. so that the director vector anchoring to rubbery will be almost null. Similarly, if the wave number or twist elastic constant are bigger the chiral imprinting will be also lost. The imprinting efficiency is reached for values of chiral parameter order lower than .
Eq. (1) can be expressed in terms of chiral parameter order as,
The equilibrium configuration can be obtained by minimizing with respect to the angle, then we have got the Euler-Lagrange equations,
which in this case turns out to be
Notice that if the solution of eq. (6) is which corresponds to an ideal or undistorted cholesteric helix. By introducing the variable , Eq.(6) transforms in the expression
which is the simple pendulum equation. Hence, for large values of , small harmonic oscillations of frequency around are expected. Nevertheless, for smaller values of anarmonic oscillations take place until abandons its oscillatory dependence and grows without limit following the dependence corresponding to a rotatory motion in a simple pendulum. This behavior can be inferred by using the general solution of eq.(6) which is given by
where is the Jacobian amplitude and is a constant related with the reduced elastic energy  . In the Fig. 2 is shown the director vector angle against position for different values of the parameter . In this case, the director vector changes the amplitude of its oscillations around the ideal case solution , while the spatial period is maintained constant to increase the values of parameter from zero until to reach the critical value in . Schematic representation of the director vector is shown in the the Fig. 3. In the Fig. 4 is shown the director vector angle against position for different values of chiral order parameter . It is interesting to note that by increasing from zero, presents oscillations without changing its amplitude, around the undistorted solution whose spatial period increases, until reaches the critical values: . After this value deviate considerably from and enlarges its effective -value for which is equivalent to have an untwisted helix than that of as is shown in the Fig. 5.
3. Optical description
As any chiral material whose axis of chiral nonhomogeneity is along the axis, the optical relative permittivity matrix of the chiral elastomer may be stated as
where is the dielectric anisotropy with and the dielectric constants in the optical regime parallel and perpendicular to the director , respectively.
Maxwell curl equations (in MKS units) for media whose dielectric tensor is only dependent, are given by
where and are the permittivity and the permeability of free space (i.e., vacuum). If a plane wave were to be incident obliquely on a Cholesteric Elastomer slab of thickness , we can describe the fields in terms of the column vector 
where the angleformed by the propagation vector and the-axisin the plane. Thus, the essential part of Maxwell curl equations can then be written as .
In this equation, the matrix can be factorized as where the 44 matrix
Equation (12) can be solved by a variety of techniques. We chose to implement the piecewise homogeneity approximation method. The final result is the computation of a transfer matrix such that
4. Boundary value problem
Suppose that plane waves are obliquely incident on a cholesteric of finite thickness from both half-spaces and , which are taken to be vacuous. As a result, the total electric and magnetic fields in these half-spaces may be written as
Continuity of the tangential components of the electric and the magnetic fields across the planes and leads to the prescriptions of the boundary values
where the matrix
By virtue of Eqs. (16), (17) and (19), we obtain the matrix algebraic equation
where the so called transfer matrix is given by
and the symbol denotes the conjugate transpose.
The result of solving Eq. (21) is best arranged as
where and , etc., are the transmission coefficients and and , etc., are the reflection coefficients. The co–polarized transmittances engendered by the waves coming from the left-hand side are denoted by and , and the cross–polarized ones by and ; and similarly for the reflectances , etc. similar definitions are valid for the waves generated by the waves coming from the right-hand side , and . The matrix involved in Eq.(23) is known as the scattering matrix and contains the same information of the transfer matrix but order in different way. It can be straightforwardly shown by writing explicitly the algebraic equations defined by Eq.(21) and solving them for , , and that both matrices are related by the expression
where and are the forward and backward projectors given by
In what follows we shall restrict our calculation to the case in which the incident waves are coming only from the left hand side that is we take .
In our calculations we have chosen the physical amounts of a real cholesteric elastomer material  . Fig. 6 shows that in the absence of solvent ( there is only a band reflection for right-circularly copolarized light with the bandwidth of nm and maximum amplitude. The center of the reflection band blue-shift whereas the width and amplitude band remain constant as the incidence angle increases. The corresponding reflectances and do no exhibit practically any band. In contrast, when there appear bands for both left- and right-copolarized reflectances and whereas the cross-polarized reflectance is almost zero. In this case the is not as well formed as in the absence of solvent because is amplitude present various peaks and the bandwidth varies as a function of the incidence angle, reaching its smaller value for angles around 45 degrees. Differently, is very thin since its bandwidth is around nm and contained in the interval where the band lies.
Fig. 7 presents the corresponding trasmittances to Fig.6. The case without solvent ( confirms consistently the complemetary behavior of an undistorted cholesteric. The co-polarized and exhibit troughs where the transmittance diminish, the one for is well defined whereas that of changes is thickness a deep. The blue-shift of the trough for large incident angles is also observed. In the Fig. 8 is shown the band reflection for left and right circularly polarized ligh and for different values of parameter, In both cases, the band reflection for left circularly polarized light is thinner than the band reflection for right circularly polarized light.
In the Fig. 9 is shown the reflectance of left and right circularly polarized light as a function of wavelenght and the chiral order parameter to oblique incidence. Complementary, In the Fig. 10 is shown the same of the Fig.9 as function of wavelenght and the parameter
We have obtained the circularly polarized reflectances and transmittances as function of the chiral order parameter of a cholesteric elastomer immersed in a racemic solvent. We have found considerably changes in the bandwidth of the reflectance for left- and right-copolarized light under the presence of the solvent which are susceptible to be detected experimentally. We have obtained a bandwidth larger than that for an undistorted elastomer () for and obliquely incidence for angles larger than 60 Also in this value we have observed a thin band reflection for both polarizations and for oblique incidence.
The modifications of the transmittance and reflectance spectra during a preferentially absorbing process suggest to utilize the optical spectra as an indirect method to determine the concentration of preferentially absorbed molecules during a segregation process.
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