Electromagnetic Waves & Polarization
Electromagnetic Waves & Polarization
Electromagnetic waves transport energy and momentum. Understanding polarization — the orientation of the electric field — is essential for optics, telecommunications, and quantum information. Linear, circular, and elliptical polarizations cover all cases.
Definition
A linearly polarized plane wave has \(\mathbf{E}\) oscillating along a fixed direction. Circular polarization combines two equal-amplitude orthogonal waves with a \(90°\) phase difference: \(\mathbf{E} = E_0(\hat{x}\cos\theta - \hat{y}\sin\theta)\).
Key Result
Malus's law: intensity after a linear polarizer is \(I = I_0\cos^2\theta\) where \(\theta\) is the angle between light polarization and the polarizer axis.
Example 1
A wave travels from vacuum into glass (\(n=1.5\)). Reflection coefficient at normal incidence: \(R = ((n-1)/(n+1))^2 = 0.04\) (4% reflected).
Example 2
At Brewster's angle \(\theta_B = \arctan(n_2/n_1)\), the reflected light is completely \(s\)-polarized — the basis of anti-glare coatings.
Loading em-wave-polarizer...
Practice
- Two polarizers at 60° to each other: what fraction of unpolarized light passes?
- Explain why the sky appears blue using Rayleigh scattering.
- What is the difference between group velocity and phase velocity in a dispersive medium?
- Describe total internal reflection and its applications in fiber optics.
Show Answer Key
1. Malus's law: intensity after first polarizer = $I_0/2$ (unpolarized → linear). After second at 60°: $I = (I_0/2)\cos^2 60° = (I_0/2)(1/4) = I_0/8$. Fraction transmitted: $1/8 = 12.5\%$.
2. Rayleigh scattering intensity $\propto 1/\lambda^4$. Short wavelengths (blue, $\lambda \approx 450$ nm) scatter ~5.5× more than red ($\lambda \approx 700$ nm). Sunlight entering the atmosphere scatters blue light in all directions; we see this scattered blue from the sky. At sunset, the long path through the atmosphere removes most blue, leaving red/orange.
3. Phase velocity: $v_p = \omega/k$ (speed of wavefronts). Group velocity: $v_g = d\omega/dk$ (speed of energy/information). In a dispersive medium, $v_p$ depends on $\omega$, so $v_g \neq v_p$. For normal dispersion ($dn/d\lambda < 0$): $v_g < v_p$. For anomalous dispersion: $v_g > v_p$ or even $v_g > c$ (but signal velocity $\leq c$).
4. When light hits an interface at angle $\theta > \theta_c = \sin^{-1}(n_2/n_1)$ (from denser to less dense medium), it is totally reflected. An evanescent wave penetrates the second medium but carries no net energy. Applications: fiber optics (light trapped by TIR in glass core), prisms, binoculars, fingerprint sensors.