- #1
etotheipi
If light passes into a birefringent material with constant fast and slow directions, ##\hat{x}## and ##\hat{y}##, that are oriented the same way at any point in the crystal, then the electric field is$$\vec{E}(z,t) = E_0\hat{x}\cos{(\theta_0)}e^{i\omega(t - \frac{n_x}{c}z)} + E_0\hat{y}\cos{(\theta_0)}e^{i\omega(t - \frac{n_y}{c}z)}$$if the electric field is initially oriented at ##\theta_0## to the ##\hat{x}## axis before it enters the crystal. For a chiral nematic crystal, the fast and slow directions are functions of position, i.e. ##\hat{x} = \cos{(\mu z)} \hat{x}_{0} + \sin{(\mu z)} \hat{y}_0## and ##\hat{y} = -\sin{(\mu z)} \hat{x}_{0} + \cos{(\mu z)} \hat{y}_0##, where ##\mu## is just a constant. I want to find an expression for ##\vec{E}(z=w)##, where ##w## is the length of the crystal.
I'm not really sure what the best way to go about this is. I thought to try and split the crystal into ##N## slices of width ##w/N##, with lower faces at heights ##z_n = nw/N, n \in [0,N-1]##, and ##\hat{x}_n = \cos{(\mu z_n)} \hat{x}_{0} + \sin{(\mu z_n)} \hat{y}_0## and ##\hat{y}_n = -\sin{(\mu z_n)} \hat{x}_{0} + \cos{(\mu z_n)} \hat{y}_0##. Hence, the electric field at height ##z_1## would be$$\begin{align*}
\vec{E}(z = z_1,t) &= E_0\hat{x}_0\cos{(\theta_0)}e^{i\omega(t - \frac{n_x w}{cN})} + E_0\hat{y}_0\cos{(\theta_0)}e^{i\omega(t - \frac{n_y w}{cN})} \\
&= E_0(\cos{\mu z_1} \hat{x}_1 - \sin{\mu z_1} \hat{y}_1)\cos{(\theta_0)}e^{i\omega(t - \frac{n_x w}{cN})} + E_0(\sin{\mu z_1} \hat{x}_1 + \cos{\mu z_1} \hat{y}_1)\cos{(\theta_0)}e^{i\omega(t - \frac{n_y w}{cN})} \\
&= \left[ E_0 \cos{\mu z_1} \cos{\theta_0} e^{i\omega(t - \frac{n_x w}{cN})} + E_0 \sin{\mu z_1}\cos{\theta_0} e^{i\omega(t - \frac{n_y w}{cN})} \right] \hat{x}_1 +
\left[ E_0 \cos{\mu z_1} \cos{\theta_0} e^{i\omega(t - \frac{n_y w}{cN})} - E_0 \sin{\mu z_1}\cos{\theta_0} e^{i\omega(t - \frac{n_x w}{cN})} \right] \hat{y}_1
\end{align*}
$$and then between ##z_1## and ##z_2##, these two components will propagate at two different speeds along the ##\hat{x}_1## and ##\hat{y}_1## directions, so we need to write another equation for this layer, and so on. I really don't know if I'm going to get anything useful out of this, so I wondered if someone could let me know if there is a better approach? Thanks
I'm not really sure what the best way to go about this is. I thought to try and split the crystal into ##N## slices of width ##w/N##, with lower faces at heights ##z_n = nw/N, n \in [0,N-1]##, and ##\hat{x}_n = \cos{(\mu z_n)} \hat{x}_{0} + \sin{(\mu z_n)} \hat{y}_0## and ##\hat{y}_n = -\sin{(\mu z_n)} \hat{x}_{0} + \cos{(\mu z_n)} \hat{y}_0##. Hence, the electric field at height ##z_1## would be$$\begin{align*}
\vec{E}(z = z_1,t) &= E_0\hat{x}_0\cos{(\theta_0)}e^{i\omega(t - \frac{n_x w}{cN})} + E_0\hat{y}_0\cos{(\theta_0)}e^{i\omega(t - \frac{n_y w}{cN})} \\
&= E_0(\cos{\mu z_1} \hat{x}_1 - \sin{\mu z_1} \hat{y}_1)\cos{(\theta_0)}e^{i\omega(t - \frac{n_x w}{cN})} + E_0(\sin{\mu z_1} \hat{x}_1 + \cos{\mu z_1} \hat{y}_1)\cos{(\theta_0)}e^{i\omega(t - \frac{n_y w}{cN})} \\
&= \left[ E_0 \cos{\mu z_1} \cos{\theta_0} e^{i\omega(t - \frac{n_x w}{cN})} + E_0 \sin{\mu z_1}\cos{\theta_0} e^{i\omega(t - \frac{n_y w}{cN})} \right] \hat{x}_1 +
\left[ E_0 \cos{\mu z_1} \cos{\theta_0} e^{i\omega(t - \frac{n_y w}{cN})} - E_0 \sin{\mu z_1}\cos{\theta_0} e^{i\omega(t - \frac{n_x w}{cN})} \right] \hat{y}_1
\end{align*}
$$and then between ##z_1## and ##z_2##, these two components will propagate at two different speeds along the ##\hat{x}_1## and ##\hat{y}_1## directions, so we need to write another equation for this layer, and so on. I really don't know if I'm going to get anything useful out of this, so I wondered if someone could let me know if there is a better approach? Thanks
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