# Unique Features of Nonlocally Nonlinear Systems with Oscillatory Responses
## 6 Optical Beams in NLC with Negative Dielectric Anisotropy  
### Evolution Equation for Optical Beams in NLC  
The physical mechanism of nonlinearity in NLC is the optically induced molecular reorientation and nonlocality comes from the interactions between NLC molecules [3]. We consider the propagation of optical beams in the sample cell of NLC shown in Figure 15. NLC molecules exhibit anisotropy in both the low-frequency domain and the optical frequency domain, which is expressed by \(\epsilon_{a}^{rf}(=\epsilon_{\parallel}-\epsilon_{\perp})\) and \(\epsilon_{a}^{op}(=n_{\parallel}^{2}-n_{\perp}^{2})\), respectively. Anisotropy \(\epsilon_{a}^{rf}\) can be either positive [10; 48] or negative [48; 49], while \(\epsilon_{a}^{op}\) is always positive. When an external low-frequency electrical field \(\vec{E}_{rf}=E_{rf}\vec{e}_{X}\) and an optical field \(\vec{E}_{op}=E_{op}\vec{e}_{X}\) are present, the molecular director \(\vec{n}\) expressed by \(\vec{n}(\theta)=(\sin\theta,0,\cos\theta)\) (\(\theta\) is the tilt angle between \(\vec{n}\) and \(z\)) will be reorientated. NLC molecules approach the equilibrium state when the three torques are balanced, i.e., \(\vec{\Gamma}_{op}+\vec{\Gamma}_{lf}+\vec{\Gamma}_{el}=0\), where \(\vec{\Gamma}_{op}\), \(\vec{\Gamma}_{lf}\) and \(\vec{\Gamma}_{el}\) are the optical field-induced torque, low-frequency electrical field induced torque and the elastic torque, respectively. Using the one-constant approximation [48], we can have the following:  
\[\vec{\Gamma}_{el}=K\
abla^{2}\theta\vec{e}_{Y},\]  
where \(K\) is the NLC average elastic constant. We can also obtain the following [48]:  
\[\vec{\Gamma}_{lf} = \vec{D}_{rf}\times\vec{E}_{rf}=\epsilon_{0}\epsilon_{a}^{rf}\sin \theta\cos\theta E_{rf}^{2}\vec{e}_{Y},\] \[\vec{\Gamma}_{op} = <\vec{D}_{op}\times\vec{E}_{op}>=\frac{1}{2}\epsilon_{0}\epsilon_ {a}^{op}\sin\theta\cos\theta|E_{op}|^{2}\vec{e}_{Y},\]  
where \(\vec{D}_{rf}=\epsilon_{0}\epsilon_{\perp}\vec{E}_{rf}+\epsilon_{0}\epsilon_{a }^{rf}(\vec{n}\cdot\vec{E}_{rf})\vec{n}\) and \(\vec{D}_{op}=\epsilon_{0}n_{\perp}^{2}\vec{E}_{op}+\epsilon_{0}\epsilon_{a}^ {op}(\vec{n}\cdot\vec{E}_{op})\vec{n}\) are the electric displacements, and \(<>\) denotes the time average. Figure 15: \(X\)-\(Z\) cross-section of the planar cell of the NLC with negative dielectric anisotropy, and the cell can be considered invariant along \(Y\). Two pieces of glasses sandwiches NLC. Polymer coatings provide molecules anchoring \(\theta|_{X=L/2}=\theta|_{X=-L/2}=\pi/2\) at the boundaries, and ITO films allow for the application of low frequency bias. An optical beam propagating inside the cell along \(Z\) induces an index perturbation. In the absence of light, from the balance of torques of \(\vec{\Gamma}_{el}\) and \(\vec{\Gamma}_{lf}\), the pretilt angle \(\hat{\theta}\) produced by the low-frequency electric field \(\vec{E}_{rf}\) can be determined, which is the only function of \(X\)[48]  
\[2K\frac{d^{2}\hat{\theta}}{dX^{2}}+\varepsilon_{0}\epsilon_{a}^{rf}\vec{E}_{rf} ^{2}\sin(2\hat{\theta})=0. \tag{32}\]  
If \(\epsilon_{a}^{rf}>0\) (the NLC with positive dielectric anisotropy), \(d^{2}\hat{\theta}/dX^{2}<0\), which indicates that the angle at the center (\(X=0\)) of the cell \(\theta_{0}(=\hat{\theta}|_{X=0})\) is at the maximum and decreases as it proceeds closer to the boundaries. The opposite happens for the case that \(\epsilon_{a}^{rf}<0\) (the NLC with negative dielectric anisotropy), the angle at the center of the cell \(\theta_{0}\) is at the minimum and increases as it approaches the boundaries. As a result, NLC molecules should be anchored at the boundaries in a manner where \(\theta|_{X=L/2}=\theta|_{X=-L/2}=0\) for the positive dielectric anisotropy, as was observed in Refs. [3; 10; 50], and where \(\theta|_{X=L/2}=\theta|_{X=-L/2}=\pi/2\), as shown in Figure 15, for negative dielectric anisotropy.