# Light Propagation in Confined Nematic Liquid Crystals and Device Applications
## 3 Modelling of Light Propagation in Nematic LC Waveguides  
Light propagation in LC waveguides is affected by the LC molecular orientation represented by a unity vector referred to as director **n**, as shown in Figure 8. Typical dimensions are a few nanometres along the molecular axis and about 1 nm in the perpendicular direction. Locally, a refractive index \(n_{\parallel}\) along the molecular axis and a refractive index \(n_{\perp}\) perpendicular to the molecular axis are defined. In the bulk LC, an extraordinary refractive index \(n_{e}\) and an ordinary refractive index \(n_{o}\) are defined along and perpendicularly to the director **n**, respectively, showing a birefringence ranging between 0.15 and 0.5, depending on the chemical compounds and wavelengths. The optical anisotropy and the molecular orientation of the LC molecules induce the refractive index spatial distribution \(n_{\text{LC}}\) (_x,y_) in the LCW determined by the electric field \(\textbf{E}_{\text{ac}}\) related to the ac voltage applied to the electrodes and the optical electric field \(E_{opt}\) of the propagating light beam when its intensity is relatively high. The electric field dependence enables both an electro-optic and an all-optical control. The LC molecular orientation is the result of the balance between the electric torques and the elastic restoring torque when an electric field is applied. The deformation pattern affected by the elastic torque is the result of a combination of three basic deformations, which are splay, twist and bend. The orientation of the LC is imposed by the minimum of the free energy \(F\) given by:  
\[F=F_{el}-F_{did}-F_{opt} \tag{1}\]  
in which \(F_{el}\) is the elastic term, \(F_{did}\) is the dielectric term and \(F_{opt}\) is the optical term related to the electric field of the optical beam propagating in the LC. Figure 8: Molecular director **n** and refractive index at the molecular level. Figure 7: Propagation properties of an LC:PDMS waveguide with a width of 15 \(\upmu\)m: (**a**) green laser light (wavelength = 532 nm) beam butt—coupled by using a single mode optical fiber of 125 \(\upmu\)m; (**b**) output power versus polarization at a wavelength of 1550 nm. The Oseen-Frank theory indicates that the elastic energy is given by [54]:  
\[F_{el}=\frac{1}{2}\iiint\left[k_{11}(\
abla\cdot\mathbf{n})^{2}+k_{22}(\mathbf{n} \cdot\
abla\times\mathbf{n})^{2}+k_{33}(\mathbf{n}\times\
abla\times\mathbf{n} )^{2}\right]dv \tag{2}\]  
where \(k_{11}\), \(k_{22}\) and \(k_{33}\) are the elastic constants corresponding to the splay, twist and bend, respectively, in which \(v\) is the volume. The dielectric energy is given by:  
\[F_{diel}=\frac{1}{2}\iiint\left[\varepsilon_{0}\varepsilon_{\perp,ac}|E_{ac}|^ {2}+\varepsilon_{0}\Delta\varepsilon_{ac}(\mathbf{E}_{ac}\cdot\mathbf{n})^{2} \right]dv \tag{3}\]  
where \(\varepsilon_{0}\) is the dielectric permittivity in vacuum, \(\varepsilon_{\perp,ac}\) is the dielectric permittivity perpendicular to the optical axis, \(\Delta\varepsilon_{ac}\) is the dielectric anisotropy, and \(\mathbf{E}_{ac}\) is the applied electric field at low frequency. The optical term of the energy is given by:  
\[F_{opt}=\frac{1}{2}\iiint\left[\varepsilon_{0}\varepsilon_{\perp,opt}\big{|} \mathbf{E}_{opt}\big{|}^{2}+\varepsilon_{0}\Delta\varepsilon_{opt}\big{(}\mathbf{E}_ {opt}\cdot\mathbf{n})^{2}\right]dv \tag{4}\]  
where \(\varepsilon_{0}\) is the dielectric permittivity in vacuum, \(\varepsilon_{\perp,opt}\) is the dielectric permittivity perpendicular to the optical axis, \(\Delta\varepsilon_{opt}\) is the dielectric anisotropy, and \(\mathbf{E}_{opt}\) is the electric field at the optical frequencies.