Plane wave properties |
Jan. 25, 2018 |
• Intensity of wave: Average Poynting
vector
`vec {S}_{ave} = 1/2 Re \{ vec E times vec H ^{**} \}`
For
`| \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}`
• : 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.
, 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:
`-` `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}`
2 polarizations
(`_|_` to plane of incidence) and
(`||` to plane of incidence).
For `||`
polarization, it is not reflected at
`theta_B = tan^{-1} ( n_2 / n_1 )`
For `n_1 > n_2`,
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)`
Examples for application of reflection and refraction:
, sunglass for fishing at a pond or
working in a snow covering ground.