Chapter 11

Electromagnetic Waves in Vacuum

Plane waves, polarization, energy, momentum, and radiation pressure.

11.1 Plane Wave Solutions

The simplest solution to the wave equation is the monochromatic plane wave. For a wave traveling in the $+\hat{z}$ direction:

$$\tilde{\mathbf{E}} = \tilde{E}_0 e^{i(kz - \omega t)}\hat{x}, \qquad \tilde{\mathbf{B}} = \frac{\tilde{E}_0}{c} e^{i(kz-\omega t)}\hat{y}$$

with the dispersion relation $\omega = ck$, $k = 2\pi/\lambda$. Key properties:

Transverse: $\mathbf{E}$ and $\mathbf{B}$ are both perpendicular to the propagation direction $\hat{k}$.
Mutual perpendicularity: $\mathbf{E} \perp \mathbf{B}$.
Amplitude ratio: $|\mathbf{B}| = |\mathbf{E}|/c$.
Phase relation: $\mathbf{E}$ and $\mathbf{B}$ are in phase.

11.2 Energy & Intensity

For a plane wave the energy densities in E and B are equal:

$$u = \epsilon_0 E^2 = \frac{B^2}{\mu_0}$$

The time-averaged intensity (irradiance) is:

$$I = \langle S \rangle = \frac{1}{2}\frac{E_0^2}{\mu_0 c} = \frac{c\epsilon_0 E_0^2}{2} \qquad [\text{W/m}^2]$$

11.2.1 Radiation Pressure

When an EM wave is absorbed by a surface, it exerts a radiation pressure:

$$P_{\rm rad} = \frac{I}{c} \quad \text{(absorbed)}, \qquad P_{\rm rad} = \frac{2I}{c} \quad \text{(reflected)}$$

11.3 Polarization

The polarization state of a plane wave is described by the direction of the electric field vector:

Linear

$\mathbf{E} = E_0\cos(kz-\omega t)\hat{x}$

E oscillates in a fixed plane.

Circular

$\mathbf{E} = E_0[\cos(kz-\omega t)\hat{x} \pm \sin(kz-\omega t)\hat{y}]$

E rotates at constant amplitude; LCP/RCP.

Elliptical

$\mathbf{E} = E_x\hat{x} + E_y e^{i\delta}\hat{y}$

General case; linear and circular are special cases.

Any polarization state can be decomposed into two orthogonal linear polarizations or two circular polarizations. Sunlight is unpolarized (random superposition of all polarization states).

Simulation: EM Plane Wave Visualization

EM Waves in Vacuum

Visualizes E and B fields of a plane wave, Poynting vector, intensity, and all three polarization types.

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# ──────────────────────────────────────────────
# EM plane waves in vacuum
# Visualize E, B, and S = E×B/μ₀
# Demonstrate transversality and polarization
# ──────────────────────────────────────────────
mu0 = 4*np.pi*1e-7
eps0 = 8.854e-12
c = 1/np.sqrt(mu0*eps0)

# Plane wave: E = E0 cos(kz - ωt) x_hat
# B = (E0/c) cos(kz - ωt) y_hat
# Propagates in +z direction
lam = 500e-9      # 500 nm (green light)
k = 2*np.pi/lam
omega = c*k
E0 = 1.0          # amplitude [V/m]

z = np.linspace(0, 3*lam, 500)
t0 = 0   # snapshot

Ex = E0 * np.cos(k*z - omega*t0)
By = E0/c * np.cos(k*z - omega*t0)
Sx = np.zeros_like(z)   # Poynting in z direction
Sz = Ex * By / mu0      # S_z = E_x * B_y / μ₀

# Polarization: linear, circular, elliptical
z_pol = np.linspace(0, 4*lam, 300)
E_lin_x = E0 * np.cos(k*z_pol)
E_lin_y = np.zeros_like(z_pol)
E_circ_x = E0 * np.cos(k*z_pol)
E_circ_y = E0 * np.sin(k*z_pol)
E_ell_x  = E0 * np.cos(k*z_pol)
E_ell_y  = 0.5*E0 * np.sin(k*z_pol + np.pi/4)

fig = plt.figure(figsize=(15, 10))
fig.patch.set_facecolor('#0a0a1a')

# ── 3D wave visualization (z, Ex, By) ──
ax3d = fig.add_subplot(2, 3, 1, projection='3d')
ax3d.set_facecolor('#0a0a1a')
ax3d.plot(z*1e6, Ex, np.zeros_like(z), color='#ef4444', lw=2, label='E_x')
ax3d.plot(z*1e6, np.zeros_like(z), By*c, color='#3b82f6', lw=2, label='cB_y')
# Poynting direction arrows
for zi in z[::50]:
    ax3d.quiver(zi*1e6, 0, 0, 0.05e-6*1e6, 0, 0, color='#f59e0b', alpha=0.7)
ax3d.set_xlabel('z [μm]', color='#94a3b8')
ax3d.set_ylabel('E [V/m]', color='#94a3b8')
ax3d.set_zlabel('cB', color='#94a3b8')
ax3d.set_title('Plane Wave: E & B', color='white', fontsize=10)
ax3d.tick_params(colors='#94a3b8', labelsize=7)
ax3d.legend(fontsize=8)

# ── E and B vs z ──
ax2 = fig.add_subplot(2, 3, 2)
ax2.set_facecolor('#0a0a1a'); [sp.set_edgecolor('#334155') for sp in ax2.spines.values()]
ax2.plot(z*1e6, Ex, color='#ef4444', lw=2, label=r'$E_x$')
ax2.plot(z*1e6, By*c, color='#3b82f6', lw=2, linestyle='--', label=r'$cB_y$')
ax2.fill_between(z*1e6, 0, Sz/Sz.max(), alpha=0.2, color='#f59e0b', label=r'$S_z$ (norm.)')
ax2.set_xlabel('z [μm]', color='#94a3b8'); ax2.set_ylabel('Amplitude', color='#94a3b8')
ax2.set_title(r'E, cB, $S_z$ vs z', color='white', fontsize=10)
ax2.tick_params(colors='#94a3b8')
ax2.legend(facecolor='#1e293b', labelcolor='white', fontsize=8)
ax2.grid(alpha=0.2, color='#334155')

# ── Intensity vs time (time-averaged) ──
ax3 = fig.add_subplot(2, 3, 3)
ax3.set_facecolor('#0a0a1a'); [sp.set_edgecolor('#334155') for sp in ax3.spines.values()]
t_arr = np.linspace(0, 4/omega*2*np.pi, 200)
I_t = E0**2/(2*mu0*c) * (1 + np.cos(2*(k*lam - omega*t_arr)))
I_avg = E0**2/(2*mu0*c) * np.ones_like(t_arr)
ax3.plot(omega*t_arr/(2*np.pi), I_t*1e-3, color='#f59e0b', lw=2, label='Instantaneous I')
ax3.axhline(I_avg[0]*1e-3, color='white', lw=1.5, linestyle='--', label=r'$langle I
angle = cE_0^2/(2mu_0)$')
ax3.set_xlabel(r'$omega t / 2pi$', color='#94a3b8'); ax3.set_ylabel('I [kW/m²]', color='#94a3b8')
ax3.set_title('Intensity vs Time', color='white', fontsize=10)
ax3.tick_params(colors='#94a3b8')
ax3.legend(facecolor='#1e293b', labelcolor='white', fontsize=8)
ax3.grid(alpha=0.2, color='#334155')

# ── Polarization ellipses ──
for idx, (Ex_p, Ey_p, title, color) in enumerate([
    (E_lin_x, E_lin_y, 'Linear', '#60a5fa'),
    (E_circ_x, E_circ_y, 'Circular', '#34d399'),
    (E_ell_x, E_ell_y, 'Elliptical', '#a78bfa'),
]):
    ax = fig.add_subplot(2, 3, 4+idx)
    ax.set_facecolor('#0a0a1a'); [sp.set_edgecolor('#334155') for sp in ax.spines.values()]
    ax.plot(Ex_p, Ey_p, color=color, lw=2)
    ax.set_xlim(-1.5, 1.5); ax.set_ylim(-1.5, 1.5)
    ax.set_aspect('equal')
    ax.set_xlabel('$E_x$', color='#94a3b8'); ax.set_ylabel('$E_y$', color='#94a3b8')
    ax.set_title(f'{title} Polarization', color='white', fontsize=10)
    ax.tick_params(colors='#94a3b8')
    ax.grid(alpha=0.2, color='#334155')

plt.tight_layout()
plt.savefig('em_waves_vacuum.png', dpi=130, bbox_inches='tight', facecolor=fig.get_facecolor())
print(f"Speed of light: c = {c:.6e} m/s")
print(f"Wavelength: λ = {lam*1e9:.0f} nm (green)")
print(f"Frequency: f = {c/lam/1e12:.3f} THz")
print(f"Time-averaged intensity: <I> = E₀²/(2μ₀c) = {E0**2/(2*mu0*c):.2f} W/m² (for E₀=1V/m)")

Click Run to execute the Python code

First run will download Python environment (~15MB)

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