Difference between revisions of "Fall 2012"

Method:

Need $v_g = \frac{\partial \omega}{\partial k}$, so we have to find $\omega(k)$.

Have $v_p = \frac{\omega}{k} = \frac{c}{n(\omega)} \approx \frac{c}{\sqrt{\epsilon_r}}$

$\rightarrow$ Need $\epsilon_r$.

Have $\epsilon_r =1+\chi_e = 1 + \frac{P}{\epsilon_0 E}= 1 + \frac{n_e p_e}{\epsilon_0 E} =1+\frac{n_e ex}{\epsilon_0 E}$, where $E= E_0 e^{-i\omega t}$ and $n_e =$ electron number density.

$\rightarrow$ Need $x$.

Have $F= eE= m_e \ddot{x}$, so \[\ddot{x} = \frac{e}{m_e} E_0 e^{-i\omega t}\] Have $\ddot{x} = -\omega x$, so \[x = x_0 e^{-i\omega t}= -\frac{e}{m_e \omega^2} E_0e^{-i\omega t}\] \[\epsilon_r = 1 - \frac{n_e e^2}{\epsilon_0 m_e \omega^2} =1 - \frac{\omega_p^2}{\omega^2}\], where $\omega_p^2 = \frac{n_e e^2}{\epsilon_0 m_e}$.

From $v_p$ above we get: \[k = \frac{\omega \sqrt{\epsilon_r}}{c} = \frac{1}{c} \sqrt{ \omega^2- \omega_p^2}\] \[\frac{1}{v_g} = \frac{\partial k}{\partial \omega} = \frac{\omega}{c \sqrt{\omega^2- \omega_p^2}} =\frac{1}{c \sqrt{1- \frac{\omega_p^2}{\omega^2}}} \approx \frac{1}{c} \left[1+\frac{\omega_p^2}{2\omega^2} \right]\] \[\tau= s \left[\frac{1}{v_{g,2}} - \frac{1}{v_{g,1}}\right] \approx \frac{s \omega_p^2}{2c} \left[\frac{1}{\omega_2^2} - \frac{1}{\omega_1^2}\right]\] \[s = \frac{2c\tau}{\omega_p^2}\left[\frac{\omega_1^2\omega_2^2}{\omega_1^2-\omega_2^2}\right]\]