plane wave

frequency

relative permitivity

electric amplitude

angler frequency 

\( \omega=2\pi f\)

period 

\( \omega T=2\pi\)

wave number 

\(\begin{eqnarray} \mathbf E(\mathbf r,t)=A e^{j(\mathbf k \cdot\mathbf r-\omega t)}\\ \nabla\times(\nabla\times\mathbf E)=-\frac{\partial}{\partial t}(\nabla \times \mathbf B)=-\frac{\partial}{\partial t}(\epsilon_0 \mu_0 \frac{\partial \mathbf E}{\partial t})\\ (j\mathbf k)\cdot(j\mathbf k)\mathbf E-(-j\omega)^2\epsilon_0\mu_0 \mathbf E=(k^2-\omega^2 \epsilon_0\mu_0)\mathbf{E}=0\\ \end{eqnarray}\)

reflect index 

\(n_0=\frac{ck}{\omega}=1\)

wave length 

\(k \lambda=2\pi\,\,\,\, \lambda_0=\frac{2\pi}{k}\)

phase velocity 

\(k r-\omega t=const\,\,\,\,\,k v_p-\omega=0\,\,\,\, v_p=\frac{\omega}{k}=\frac{\omega}{\omega\sqrt{\epsilon_0\mu_0}}=c\)

wave impendance 

\(\begin{eqnarray} \nabla\times \mathbf H=\frac{\partial \epsilon_0 \mathbf E}{\partial t}\,\,\,\, ik\times H=-i\omega\epsilon_0 E\,\,\,\, \mathbf E=\sqrt{\frac{\mu_0}{\epsilon_0}} \mathbf H \times \mathbf k \\ Z_0=\frac{E}{H}=\frac{k}{\omega\epsilon_0}=\sqrt{\frac{\mu_0}{\epsilon_0}}\\ |E|=Z_0 H=Z_0 \frac{|B|}{\mu_0}=c|B| \end{eqnarray} \)

averaged energy flux density  

\(W=\frac{1}{2}\epsilon_0 E_0^2\)

wave number 

\(k^2=\omega^2 \epsilon_r\epsilon_0\mu_0 \,\,\, k=k_0 \sqrt{\epsilon_r}=k_0 n\)

reflect index 

\(\mathbf n=\frac{c \mathbf k}{\omega}=\sqrt{\epsilon_r}\)

wave length 

\(k\lambda=2\pi\,\,\,\, \lambda=\frac{\lambda_0}{n}\)

phase velocity 

\(v_p=\frac{\omega}{k}=\frac{v_{p0}}{n}\)

wave impendance 

\(Z=\frac{E}{H}=\frac{k}{\omega\epsilon_0\epsilon_r}=\frac{Z_0}{n}\,\,\,\,|E|=Z H=Z\frac{|B|}{\mu_0}=\frac{c|B|}{n}\)

averaged energy flux density 

\(w=\frac{1}{2}\epsilon_0\epsilon_r E_0^2=w_0n^2\)

slowly varying critical 

\(\frac{dn}{dz}\frac{1}{k_0 n^2}\ll1\)

electrical field 

\(E_y=\frac{A}{\sqrt n} e^{\pm k_0 \int^z n dz}\)

magnetic field 

\(cB_x=\mp A \sqrt n e^{\pm k_0 \int^z n dz}\)