Chapter 13

Reflection & Transmission (Fresnel Equations)

13.1 Boundary Conditions

When a plane wave hits a planar interface between media $(n_1, \mu_1)$ and $(n_2, \mu_2)$, we match tangential E and H across the boundary (from Maxwell's equations). By Snell's law: $n_1\sin\theta_i = n_2\sin\theta_t$.

13.2 Fresnel Equations

Separating into s-polarization (TE: E ⊥ plane of incidence) and p-polarization (TM: E ∥ plane):

s-polarization (TE)

$$r_s = \frac{n_1\cos\theta_i - n_2\cos\theta_t}{n_1\cos\theta_i + n_2\cos\theta_t}$$$$t_s = \frac{2n_1\cos\theta_i}{n_1\cos\theta_i + n_2\cos\theta_t}$$

p-polarization (TM)

$$r_p = \frac{n_2\cos\theta_i - n_1\cos\theta_t}{n_2\cos\theta_i + n_1\cos\theta_t}$$$$t_p = \frac{2n_1\cos\theta_i}{n_2\cos\theta_i + n_1\cos\theta_t}$$

Reflectance $R = r^2$, Transmittance $T = (n_2\cos\theta_t)/(n_1\cos\theta_i)\,t^2$, and $R + T = 1$ (energy conservation).

13.3 Special Angles

Brewster's Angle

$$\tan\theta_B = \frac{n_2}{n_1} \implies r_p = 0$$

p-polarized light is completely transmitted at Brewster's angle. Used in laser Brewster windows.

Critical Angle (TIR)

$$\sin\theta_c = \frac{n_2}{n_1} \quad (n_1 > n_2)$$

For $\theta_i > \theta_c$, total internal reflection: $R = 1$. Basis of optical fibers.

Simulation: Fresnel Equations

Fresnel Equations & TIR

Plots Rs, Rp, Ts, Tp vs angle for air-glass interface, shows Brewster's angle and total internal reflection.

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

# ──────────────────────────────────────────────
# Fresnel Equations: reflectance and transmittance
# for s and p polarizations at a dielectric interface
# n1 = 1 (air), n2 = 1.5 (glass)
# ──────────────────────────────────────────────
n1 = 1.0    # air
n2 = 1.5    # glass

theta_i = np.linspace(0, np.pi/2 - 0.001, 500)
cos_i = np.cos(theta_i)
sin_i = np.sin(theta_i)

# Snell's law: n1 sin θ_i = n2 sin θ_t
sin_t = n1/n2 * sin_i
cos_t = np.sqrt(np.maximum(0, 1 - sin_t**2))

# Fresnel equations
r_s = (n1*cos_i - n2*cos_t) / (n1*cos_i + n2*cos_t)
r_p = (n2*cos_i - n1*cos_t) / (n2*cos_i + n1*cos_t)
t_s = 2*n1*cos_i / (n1*cos_i + n2*cos_t)
t_p = 2*n1*cos_i / (n2*cos_i + n1*cos_t)

# Reflectance and Transmittance
Rs = r_s**2
Rp = r_p**2
Ts = 1 - Rs
Tp = 1 - Rp

# Brewster's angle
theta_B = np.arctan(n2/n1)

# Critical angle (TIR for n1→n2, reverse)
n1r, n2r = 1.5, 1.0
theta_c = np.arcsin(n2r/n1r)

# TIR: n1=1.5 (glass) → n2=1.0 (air)
sin_t_tir = n1r/n2r * sin_i
cos_t_tir = np.sqrt(np.maximum(0, 1 - sin_t_tir**2 + 0j)).real
r_s_tir = (n1r*cos_i - n2r*cos_t_tir) / (n1r*cos_i + n2r*cos_t_tir + 1e-20)
r_p_tir = (n2r*cos_i - n1r*cos_t_tir) / (n2r*cos_i + n1r*cos_t_tir + 1e-20)
Rs_tir = np.abs(r_s_tir)**2
Rp_tir = np.abs(r_p_tir)**2

fig, axes = plt.subplots(2, 2, figsize=(12, 9))
fig.patch.set_facecolor('#0a0a1a')
for ax in axes.flat:
    ax.set_facecolor('#0a0a1a')
    for sp in ax.spines.values(): sp.set_edgecolor('#334155')

deg = np.degrees(theta_i)

# Reflectance
axes[0,0].plot(deg, Rs, color='#60a5fa', lw=2, label='$R_s$ (TE)')
axes[0,0].plot(deg, Rp, color='#f59e0b', lw=2, label='$R_p$ (TM)')
axes[0,0].axvline(np.degrees(theta_B), color='#34d399', lw=1.5, linestyle='--', label=f'Brewster {np.degrees(theta_B):.1f}°')
axes[0,0].set_xlabel('Angle of incidence θ_i [°]', color='#94a3b8')
axes[0,0].set_ylabel('Reflectance R', color='#94a3b8')
axes[0,0].set_title('Fresnel: n₁=1 → n₂=1.5', color='white', fontsize=11)
axes[0,0].tick_params(colors='#94a3b8')
axes[0,0].legend(facecolor='#1e293b', labelcolor='white', fontsize=9)
axes[0,0].grid(alpha=0.2, color='#334155')

# Transmittance
axes[0,1].plot(deg, Ts, color='#60a5fa', lw=2, label='$T_s$')
axes[0,1].plot(deg, Tp, color='#f59e0b', lw=2, label='$T_p$')
axes[0,1].set_xlabel('θ_i [°]', color='#94a3b8'); axes[0,1].set_ylabel('Transmittance T', color='#94a3b8')
axes[0,1].set_title('Transmittance: n₁=1 → n₂=1.5', color='white', fontsize=11)
axes[0,1].tick_params(colors='#94a3b8')
axes[0,1].legend(facecolor='#1e293b', labelcolor='white', fontsize=9)
axes[0,1].grid(alpha=0.2, color='#334155')

# TIR
axes[1,0].plot(deg, Rs_tir, color='#60a5fa', lw=2, label='$R_s$ (glass→air)')
axes[1,0].plot(deg, Rp_tir, color='#f59e0b', lw=2, label='$R_p$ (glass→air)')
axes[1,0].axvline(np.degrees(theta_c), color='#ef4444', lw=1.5, linestyle='--', label=f'TIR θ_c = {np.degrees(theta_c):.1f}°')
axes[1,0].set_xlabel('θ_i [°]', color='#94a3b8'); axes[1,0].set_ylabel('Reflectance', color='#94a3b8')
axes[1,0].set_title('Total Internal Reflection: n₁=1.5→n₂=1', color='white', fontsize=11)
axes[1,0].tick_params(colors='#94a3b8')
axes[1,0].legend(facecolor='#1e293b', labelcolor='white', fontsize=9)
axes[1,0].grid(alpha=0.2, color='#334155')

# n2 dependence at θ=45°
n2_arr = np.linspace(1.0, 3.0, 200)
theta45 = np.pi/4
R_45 = []
for n2_v in n2_arr:
    s_t = np.sin(theta45)/n2_v
    if s_t > 1: R_45.append(1.0); continue
    c_t = np.sqrt(1 - s_t**2)
    c_i = np.cos(theta45)
    rs = (c_i - n2_v*c_t)/(c_i + n2_v*c_t)
    rp = (n2_v*c_i - c_t)/(n2_v*c_i + c_t)
    R_45.append(0.5*(rs**2 + rp**2))
axes[1,1].plot(n2_arr, R_45, color='#a78bfa', lw=2)
axes[1,1].set_xlabel('n₂', color='#94a3b8'); axes[1,1].set_ylabel('Avg Reflectance (θ=45°)', color='#94a3b8')
axes[1,1].set_title('Reflectance vs n₂ at θ=45°', color='white', fontsize=11)
axes[1,1].tick_params(colors='#94a3b8')
axes[1,1].grid(alpha=0.2, color='#334155')

plt.tight_layout()
plt.savefig('fresnel.png', dpi=130, bbox_inches='tight', facecolor=fig.get_facecolor())
print(f"Brewster's angle: θ_B = arctan(n2/n1) = {np.degrees(theta_B):.2f}°")
print(f"At Brewster: R_p = {Rp[np.argmin(np.abs(theta_i-theta_B))]:.6f} (should be ~0)")
print(f"Critical angle (glass→air): θ_c = {np.degrees(theta_c):.2f}°")

Click Run to execute the Python code

First run will download Python environment (~15MB)

← Waves in Matter Waveguides →