Plane wave properties Jan. 25, 2018

• Intensity of wave: Average Poynting vector vec {S}_{ave} = 1/2 Re \{ vec E times vec H ^{**} \}
For plane wave | \vec E | = eta | \vec H | and hat k = hat E times hat H where eta = sqrt ( mu / epsilon ) is wave impedance:
- vec S_{ave} = hat k | E_o |^2 / {2 eta}
• General form of plane wave: - vec E = hat E E_o e ^{-j vec k cdot vec r}
Derivation of plane wave solution
Example for a plane wave propagating on axis with vec e = hat y sin ( 10^9 t + 10 z )
Example for a oblique incident plane wave, i.e. propagating off axis.
Group velocity: Signal travels in a packet on the top of a carrier. Signal speed is determined by group velocity v_g = {d omega} / {d k} (pp. 731-732 2nd Ed., pp. 769-770 3rd Ed.)
• Wave natural of light is responsible for interference, diffraction and polarization effects etc. which determine operations of passive optical devices.
• To understand the conversion between light and electricity, i.e. actions in lasers and photodetectors, we need to consider particle natural of light, photon.
Photon energy E (J) = h nu where h is the Planck’s constant and nu is the optical frequency.
Or $E(eV) = \frac {1.24 } {\lambda ( \mu m )}$ where $E(eV) = \frac {E(J) } {q }$ and q = 1.6 times 10 ^{-19}C
Linewidth (Delta lambda) relates to bandwidth (Delta nu) {Delta lambda}/lambda = {Delta nu}/nu (pp. 28-31 2nd Ed., pp. 26-29 3rd Ed.).
• Reflection and refraction: Snell’s law - theta_i = theta_r and n_1 sin theta_1 = n_2 sin theta_2
Results of matching the phase at the boundary.
• Amplitude relations: From Maxwell’s equations, we can find the reflection coef Gamma = {E_r}/{E_i} and transmission coef tau = {E_t}/{E_i}
Procedure for setting the directions at a boundary
2 polarizations TE or S-polarized (_|_ to plane of incidence) and TM or P-polarized (|| to plane of incidence).
For || polarization, it is not reflected at Brewster angle theta_B = tan^{-1} ( n_2 / n_1 )
For n_1 > n_2, totally internal reflection results when theta_1 > theta_c = sin^{-1} ( n_2 / n_1 )
Fresnel equations
- Gamma_s = - {sin ( theta_1 - theta_2 ) }/{sin (theta_1 + theta_2 )}, tau_s = Gamma_s + 1
- Gamma_p = - {tan ( theta_1 - theta_2 ) }/{tan (theta_1 + theta_2 )}, tau_p = {n_1}/{n_2} (1 - Gamma_p)

Deriving reflection coefficient for s-polarized wave
Deriving reflection coefficient for p-polarized wave
Examples for application of reflection and refraction: polarizing beam splitter, sunglass for fishing at a pond or working in a snow covering ground.