# 物理代写|光学代写Optics代考|ECOC2022

## 物理代写|光学代写Optics代考|Free Energy and Torques by Electric and Magnetic Fields

In this section, we consider the interactions of nematic liquid crystals with applied fields (electric or magnetic); we will limit our discussion to only dielectric and diamagnetic interactions.

For a generally applied (dc, low frequency, or optical) electric field $\vec{E}$, the displacement $\vec{D}$ may be written in the form
$$\vec{D}=\varepsilon_{\perp} \vec{E}+\left(\varepsilon_{|}-\varepsilon_{\perp}\right)(n \cdot \vec{E}) n$$

The electric interaction energy density is therefore
$$\mu_E=-\int_0^E \vec{D} \cdot d \vec{E}=-\frac{1}{2} \varepsilon_{\perp}(\vec{E} \cdot \vec{E})-\frac{\Delta \varepsilon}{2}(n \cdot \vec{E})^2 .$$
Note that the first term on the right-hand side of Eq. (3.24) is independent of the orientation of the director axis. It can therefore be neglected in the director axis deformation energy. Accordingly, the free-energy density term associated with the application of an electric field is given by
$$F_E=-\frac{\Delta \varepsilon}{2}(n \cdot \vec{E})^2$$
in SI units (in cgs units, $F_E=-(\Delta \varepsilon / 8 \pi)(\hat{n} \cdot \vec{E})^2$ ). The molecular torque produced by the electric field is given by
$$\vec{\Gamma}E=\vec{D} \times \vec{E}=\Delta \varepsilon(n \cdot \vec{E})(n \times \vec{E}) .$$ Similar considerations for the magnetic field yield a magnetic energy density term $U_m$ given by $$U_m=-\int_0^M \vec{B} \cdot d \vec{M}=\frac{1}{2 \mu_0} \chi{\perp}^m B^2-\frac{1}{2 \mu_0} \Delta \chi^m(n \cdot \vec{B})^2,$$
a magnetic free-energy density (associated with director axis reorientation) $F_m$ given by
$$F_m=\frac{1}{2 \mu_0} \Delta \chi^m(n \cdot \vec{B})^2,$$
and a magnetic torque density
\begin{aligned} \vec{\Gamma}_m &=\vec{M} \times \vec{H} \ &=\Delta \chi^m(n \cdot \vec{H})(n \cdot \vec{H}) . \end{aligned}
These electric and magnetic torques play a central role in various field-induced effects in liquid crystals.

## 物理代写|光学代写Optics代考|Linear Susceptibility and Local Field Effect

In the optical regime, $\varepsilon_{|}>\varepsilon_{\perp}$. Typically, $\varepsilon_{|}$is on the order of $2.89 \varepsilon_0$ and $\varepsilon_{\perp}$ is $2.25 \varepsilon_0$. These correspond to refractive indices $n_{|}=1.7$ and $n_{\perp}=1.5$. An interesting property of nematic liquid crystals is that such a large birefringence $\left(\Delta \varepsilon_{\perp}=\varepsilon_{|}-\varepsilon_{\perp} \approx 0.2\right)$ is manifested throughout the whole optical spectral regime (from near-ultraviolet $[\approx 400 \mathrm{~nm}]$, to visible $[\approx 500 \mathrm{~nm}]$ and near-infrared $[1-3 \mu \mathrm{m}]$, to the infrared regime $[8-12 \mu \mathrm{m}$ ], i.e. from $400 \mathrm{~nm}$ to $12 \mu \mathrm{m}$ ). Figure $3.6$ shows the measured birefringence of three typical nematic liquid crystals from the UV to the far-infrared $(\lambda=16 \mu \mathrm{m})$

The optical dielectric constants originate from the linear polarization $\vec{P}$ generated by the incident optical field $\vec{E}_{\mathrm{op}}$ on the nematic liquid crystal:
$$\vec{P}=\varepsilon_0 \vec{\chi} \cdot \vec{E} .$$
From the defining equation
$$\vec{D}=\varepsilon_0 \vec{E}+\vec{P}=\overrightarrow{\bar{\varepsilon}}: \vec{E},$$

we have
$$\overrightarrow{\bar{\varepsilon}}=\varepsilon_0\left[1+\overrightarrow{\bar{\chi}}^{(1)}\right] .$$
Here $\overrightarrow{\bar{\chi}}^{(1)}$ is the linear (sometimes termed “first order”) susceptibility tensor of the nematics. $\overrightarrow{\bar{\chi}}^{(1)}$ is a macroscopic parameter and is related to the microscopic (molecular) parameter, the molecular polarizabilities tensor $\alpha_{i j}$, in the following way:
$$\begin{gathered} d_i=\alpha_{i j} E_j^{\text {loc }}, \ \vec{d}=\overrightarrow{\vec{\alpha}} \cdot \vec{E}^{\mathrm{oc}}, \ \vec{P}=N \vec{d}, \end{gathered}$$
where $d_i$ is the $i$ th component of the induced dipole $\vec{d}$ and $N$ is the number density. In Chapter 8, a rigorous quantum mechanical derivation of $\alpha$ in terms of the dipole matrix elements or oscillator strengths and the energy levels and level populations will be presented. The connection between the microscopic parameter $\alpha_{i j}$ and the macroscopic parameter $\chi_{i j}$ is the local field correction factor (i.e. the difference between the externally applied field and the actual field as experienced by the molecules). Several theoretical formalisms have been developed to evaluate the field correction factor, ranging from simplified to complex and sophisticated ones.

## 物理代写|光学代写Optics代考|Free Energy and Torques by Electric and Magnetic Fields

$$\vec{D}=\varepsilon_{\perp} \vec{E}+\left(\varepsilon_{\mid}-\varepsilon_{\perp}\right)(n \cdot \vec{E}) n$$

$$\mu_E=-\int_0^E \vec{D} \cdot d \vec{E}=-\frac{1}{2} \varepsilon_{\perp}(\vec{E} \cdot \vec{E})-\frac{\Delta \varepsilon}{2}(n \cdot \vec{E})^2 .$$

$$F_E=-\frac{\Delta \varepsilon}{2}(n \cdot \vec{E})^2$$

$$\vec{\Gamma} E=\vec{D} \times \vec{E}=\Delta \varepsilon(n \cdot \vec{E})(n \times \vec{E}) .$$

$$U_m=-\int_0^M \vec{B} \cdot d \vec{M}=\frac{1}{2 \mu_0} \chi \perp^m B^2-\frac{1}{2 \mu_0} \Delta \chi^m(n \cdot \vec{B})^2,$$

$$F_m=\frac{1}{2 \mu_0} \Delta \chi^m(n \cdot \vec{B})^2,$$

$$\vec{\Gamma}_m=\vec{M} \times \vec{H} \quad=\Delta \chi^m(n \cdot \vec{H})(n \cdot \vec{H}) .$$

## 物理代写|光学代写Optics代考|Linear Susceptibility and Local Field Effect

$$d_i=\alpha_{i j} E_j^{\text {loc }}, \vec{d}=\overrightarrow{\vec{\alpha}} \cdot \vec{E}^{o c}, \vec{P}=N \vec{d},$$

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