# 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
In addition, the dependence of \(\theta_{0}\) on low-frequency voltage \(V_{rf}(\approx E_{rf}L)\) is different for the NLC with positive and the negative dielectric anisotropy, when \(V_{rf}\) is higher than the Freedericks threshold \(V_{fr}=\pi\sqrt{K/\varepsilon_{0}|\epsilon_{a}^{rf}|}\). For the NLC with positive dielectric anisotropy [51], \(\theta_{0}\approx(\pi/2)\bigg{[}1-\big{(}V_{fr}/V_{rf}\big{)}^{3}\bigg{]}\), and for the NLC with negative dielectric anisotropy, we have the following  
\[\theta_{0}\approx(\pi/2)\Big{(}V_{fr}/V_{rf}\Big{)}^{3}. \tag{33}\]  
Although the molecule anchoring at the boundaries is different, the physical processes for both the positive and the negative are the same. Following the procedure to deal with the propagation of optical beams in an NLC with positive dielectric anisotropy [3; 10; 50], we obtained the equations for that in the NLC with negative dielectric anisotropy and found that the two cases can be uniformly expressed. In the presence of an externally applied (low-frequency) electric field \(E_{rf}\), the propagation of the slowly varying envelope \(\Phi\) of the optical field linearly polarized along the \(X\)-direction (the extraordinary light) and propagating along the \(Z\)-direction in the NLC-cell can be described by the following system:  
\[2 ik\frac{\partial\Phi}{\partial Z}+\
abla_{XY}^{2}\Phi+k_{0}^{2}\epsilon_{a}^ {op}\Big{[}\sin^{2}\theta-\sin^{2}\theta_{0}\Big{]}\Phi=0, \tag{34}\]  
\[2K\bigg{(}\frac{\partial^{2}\theta}{\partial Z^{2}}+\
abla_{XY}^{2}\theta \bigg{)}+\varepsilon_{0}\bigg{(}\epsilon_{a}^{rf}\vec{E}_{rf}^{2}+\epsilon_{ a}^{op}\frac{|\Phi|^{2}}{2}\bigg{)}\sin(2\theta)=0, \tag{35}\]  
where \(k=k_{0}n_{0}\) with \(k_{0}\) being the vacuum wavenumber and \(n_{0}\) being the (linear) refractive index of the extraordinary light. Although it looks the same in form for both cases, Equation (35) is in fact different for either the positive or the negative cases because the sign of \(\epsilon_{a}^{rf}\) is different. The term \(\partial_{Z}^{2}\theta\) in Equation (35) was proven to be negligible compared to \(\
abla_{XY}^{2}\theta\)[50]. Furthermore, we can write \(\theta=\hat{\theta}+(\hat{\theta}/\theta_{0})\Psi\), where \(\Psi\) represents optically induced angle perturbation. By simple substitution into Equation (35), we have the following:  
\[2K\frac{\hat{\theta}}{\theta_{0}}\
abla_{\perp}^{2}\Psi+\frac{4K}{\theta_{0}} \frac{\partial\hat{\theta}}{\partial x}\frac{\partial\Psi}{\partial x}- \varepsilon_{0}\epsilon_{a}^{rf}\vec{E}_{rf}^{2}\sin(2\hat{\theta})\bigg{[}1- \cos(2\frac{\hat{\theta}}{\theta_{0}}\Psi)+\frac{\Psi}{\theta_{0}}-\cot(2\hat {\theta})\sin(2\frac{\hat{\theta}}{\theta_{0}}\Psi)\bigg{]}\]  
\[\vec{\Gamma}_{el}=K\
abla^{2}\theta\vec{e}_{Y},\]0  
which is the exact result after direct substitution without any approximation. Supposing that the beam width is far smaller than cell thickness \(L\) and noting that \(\hat{\theta}\approx\theta_{0}\) and \(\partial_{x}\hat{\theta}\approx 0\) in the middle of the cell, we can simplify Equation (36) into the following form:\[\vec{\Gamma}_{el}=K\
abla^{2}\theta\vec{e}_{Y},\]1  
where we also made such an assumption of \(e_{a}^{op}|\Phi|^{2}/2\ll e_{a}^{rf}E_{rf}^{2}\)[3]. Multiplied by \(\theta_{0}e_{a}^{op}/E_{rf}^{2}\varepsilon_{0}e_{a}^{rf}n_{0}[1-2\theta_{0} \cot(2\theta_{0})]\), the equation can be re-expressed as follows:  
\[\vec{\Gamma}_{el}=K\
abla^{2}\theta\vec{e}_{Y},\]2  
where  
\[\vec{\Gamma}_{el}=K\
abla^{2}\theta\vec{e}_{Y},\]3 \[\vec{\Gamma}_{el}=K\
abla^{2}\theta\vec{e}_{Y},\]4 (40) \[\vec{\Gamma}_{el}=K\
abla^{2}\theta\vec{e}_{Y},\]5  
are the NRI, the Kerr coefficient and the nonlinear characteristic length for the NLC. When \(W_{mL}\to 0\), Equation (38) above will be reduced to \(n_{nl}=n_{2}|\Phi|^{2}\) for the local limit [39]. Therefore, our definition of \(n_{2}\) in Equation (40) can guarantee that its expression form is consistent when the degree of nonlocality transitions from the nonlocal case to the local case.