# 物理代写|光学代写Optics代考|ELEC-E5730

## 物理代写|光学代写Optics代考|SCATTERING FROM DIRECTOR

From Eqs. (5.6) to (5.8), one can see that crucial parameters involved in light scattering are the wave vectors $\mathbf{k}i$ and $\mathbf{k}_f$, the scattering wave vector $\mathbf{q}$, the director axis orientation $\hat{n}$, and its fluctuations $\delta \mathbf{n}$ from its equilibrium $\hat{n}_0$. The problem of analyzing light scattering in nematic liquid crystals (NLCs) can be greatly simplified if the coordinate system is properly defined in terms of the initial orientation of the director axis $\hat{n}_0$ with respect to the scattering wave vector $\mathbf{q}$ [1]. We shall consider here the exemplary and most illustrative case of light scattering in NLC, where the director axis $\hat{n}$ points generally in one direction; fluctuations $\delta \mathbf{n}$ can thus be analyzed by simpler geometrical consideration, unlike other ordered phases such as Blue-phase liquid crystals with complicated director axis configurations and defect network. As shown in Figure 5.2, the director axis fluctuation is decomposed into two orthogonal components $\delta \mathbf{n}_1$ and $\delta \mathbf{n}{\mathbf{2}}$, along the unit vectors $\hat{e}_1$ and $\hat{e}_2$, respectively. Note that one of them, $\delta \mathbf{n}_1$ is in the plane defined by $\mathbf{q}$ and $\hat{n}_0$ (taken as $\hat{z}$ ), and the other, $\delta \mathbf{n}_2$ is perpendicular to the $q-z$ plane.

In this case, one can express $\delta \mathbf{n}1(\mathbf{r})$ and $\delta \mathbf{n}_2(\mathbf{r})$ in terms of their Fourier components as $$\delta n{1,2}(\mathbf{r})=\sum_q n_{1,2}(\mathbf{q}) \exp (i \mathbf{q} \cdot \mathbf{r}) .$$
where
$$n_{1,2}(\mathbf{q})=\frac{1}{V} \int n_{1.2}(\mathbf{r}) \mathrm{e}^{-i \mathbf{q} \cdot \mathbf{r}} d V .$$
The total free energy associated with this director axis deformation is, from Eq. (3.6), given by
\begin{aligned} F_{\text {total }} &=\frac{1}{2} \int K_1\left(\frac{\partial n_1}{\partial x_1}+\frac{\partial n_2}{\partial x_2}\right)^2+K_2\left(\frac{\partial n_1}{\partial x_2}-\frac{\partial n_2}{\partial x_1}\right)^2 \ &+K_3\left[\left(\frac{\partial n_1}{\partial x_3}\right)^2+\left(\frac{\partial n_2}{\partial x_3}\right)^2\right] d V \end{aligned}

## 物理代写|光学代写Optics代考|LIGHT SCATTERING IN THE ISOTROPIC PHASE

In the isotropic phase, director axis orientations are random. The optical dielectric constant, a thermal average, therefore, becomes a scalar parameter. The fluctuations in $\varepsilon$, in this case, are due mainly to fluctuations in the density of the liquid caused by temperature fluctuations. As detailed in later chapters dealing with nonlinear optical responses of liquid crystals, fluctuations in the optical dielectric constant can also be due to laser-induced individual (birefringent) molecular reorientation, electrostriction, or electronic transitions among the molecular energy levels.

Denoting the average dielectric constant in the isotropic phase by $\varepsilon$ and denoting the local change in the volume by $u(r)$, the dielectric constant may be expressed as
\begin{aligned} \varepsilon &=\bar{\varepsilon}+\frac{d \varepsilon}{d V} u(\mathbf{r}) \ &=\bar{\varepsilon}+\varepsilon^{\prime} u(\mathbf{r}) \end{aligned}
The compressional energy associated with the volume change is
$$\begin{gathered} F_u=\frac{1}{2} \int W|u(\mathbf{r})|^2 d V \ =\frac{V W}{2} \sum_q|u(\mathbf{q})|^2, \end{gathered}$$
where $u(\mathbf{q})$ is the Fourier transform of $u(\mathbf{r})$, in analogy to our previous analysis of $n(\mathbf{r})$ and $n(\mathbf{q})$, and $W$ is the isothermal compressibility. Applying the equipartition theorem, we get
$$\left\langle|u(\mathbf{q})|^2\right\rangle=\frac{k_B T}{W V} .$$
From Eq. (5.8) and noting that $\delta \varepsilon_{f i}(\mathbf{q}) \equiv \hat{f} \cdot \hat{i} \varepsilon^{\prime} u(\mathbf{q})$, the scattering amplitude $a_{f i}$ is given by
$$\alpha_{f i}^2=\left(\frac{\omega^2}{c^2} V \hat{f} \cdot \hat{i} \varepsilon^{\prime}\right)^2|u(\mathbf{q})|^2$$

## 物理代写|光学代写Optics代考|SCATTERING FROM DIRECTOR

$$n_{1,2}(\mathbf{q})=\frac{1}{V} \int n_{1.2}(\mathbf{r}) \mathrm{e}^{-i \mathbf{q} \cdot \mathbf{r}} d V$$

$$F_{\text {total }}=\frac{1}{2} \int K_1\left(\frac{\partial n_1}{\partial x_1}+\frac{\partial n_2}{\partial x_2}\right)^2+K_2\left(\frac{\partial n_1}{\partial x_2}-\frac{\partial n_2}{\partial x_1}\right)^2+K_3\left[\left(\frac{\partial n_1}{\partial x_3}\right)^2+\left(\frac{\partial n_2}{\partial x_3}\right)^2\right] d V$$

## 物理代写|光学代写Optics代考|LIGHT SCATTERING IN THE ISOTROPIC PHASE

$$\varepsilon=\bar{\varepsilon}+\frac{d \varepsilon}{d V} u(\mathbf{r}) \quad=\bar{\varepsilon}+\varepsilon^{\prime} u(\mathbf{r})$$

$$F_u=\frac{1}{2} \int W|u(\mathbf{r})|^2 d V=\frac{V W}{2} \sum_q|u(\mathbf{q})|^2,$$

$$\left\langle|u(\mathbf{q})|^2\right\rangle=\frac{k_B T}{W V} .$$

$$\alpha_{f i}^2=\left(\frac{\omega^2}{c^2} V \hat{f} \cdot \hat{i} \varepsilon^{\prime}\right)^2|u(\mathbf{q})|^2$$

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