Part II: Electromagnetic Waves | Chapter 5

Wave Propagation

How electromagnetic waves interact with boundaries and media

5.1 Boundary Conditions

Derivation 1: EM Boundary Conditions from Maxwell's Equations

At an interface between two media, Maxwell's equations impose constraints on the fields. Applying Gauss's law and Faraday's law to infinitesimal pillboxes and loops across the boundary gives:

Electromagnetic Boundary Conditions

$$\epsilon_1 E_{1\perp} = \epsilon_2 E_{2\perp} \quad \text{(normal D continuous)}$$

$$B_{1\perp} = B_{2\perp} \quad \text{(normal B continuous)}$$

$$E_{1\parallel} = E_{2\parallel} \quad \text{(tangential E continuous)}$$

$$\frac{1}{\mu_1}B_{1\parallel} = \frac{1}{\mu_2}B_{2\parallel} \quad \text{(tangential H continuous)}$$

These boundary conditions, applied to incident, reflected, and transmitted plane waves, determine the reflection and transmission coefficients at any interface.

5.2 Reflection and Refraction: Snell's Law

Derivation 2: Snell's Law from Phase Matching

When a plane wave hits an interface, the boundary conditions must hold at every point on the surface and at all times. This requires phase matching: the spatial variation of all three waves (incident, reflected, transmitted) along the interface must be identical.

$$k_i \sin\theta_i = k_r \sin\theta_r = k_t \sin\theta_t$$

Since $k_i = k_r = \omega n_1/c$ (same medium) and $k_t = \omega n_2/c$:

Snell's Law

$$\theta_r = \theta_i \quad \text{(law of reflection)}$$$$n_1 \sin\theta_i = n_2 \sin\theta_t \quad \text{(Snell's law of refraction)}$$

Total Internal Reflection

When light goes from a denser to a rarer medium ($n_1 > n_2$), there exists a critical angle beyond which no transmitted wave exists:

$$\theta_c = \arcsin\!\left(\frac{n_2}{n_1}\right)$$

For $\theta_i > \theta_c$, the wave is totally reflected. However, an evanescent wave penetrates into the second medium, decaying exponentially with distance. This evanescent field is exploited in fiber optics, prism coupling, and total internal reflection fluorescence microscopy.

5.3 Fresnel Equations

Derivation 3: Reflection and Transmission Coefficients

Applying the boundary conditions for the tangential components of $\vec{E}$and $\vec{H}$ to the incident, reflected, and transmitted waves gives the Fresnel equations. For the two polarizations:

s-polarization (TE: E perpendicular to plane of incidence)

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

p-polarization (TM: E parallel to plane of incidence)

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

Brewster's Angle

The p-polarization reflection vanishes when $r_p = 0$, which occurs at Brewster's angle:

$$\theta_B = \arctan\!\left(\frac{n_2}{n_1}\right)$$

At Brewster's angle, reflected light is purely s-polarized. This is used in laser cavities (Brewster windows) and polarizing filters.

5.4 Wave Impedance and Impedance Matching

Derivation 4: Characteristic Impedance

The wave impedance of a medium relates the electric and magnetic field amplitudes:

Wave Impedance

$$Z = \frac{E}{H} = \sqrt{\frac{\mu}{\epsilon}} = \frac{Z_0}{n}$$

where $Z_0 = \sqrt{\mu_0/\epsilon_0} \approx 377\,\Omega$ is the impedance of free space. At normal incidence, the reflection coefficient becomes:$r = (Z_2 - Z_1)/(Z_2 + Z_1)$.

To eliminate reflection (impedance matching), one can insert a quarter-wave layer of intermediate impedance $Z_m = \sqrt{Z_1 Z_2}$ and thickness$d = \lambda/(4n_m)$. This is the principle behind anti-reflection coatings on lenses and optical fibers.

$$n_{\text{coating}} = \sqrt{n_1 n_2}, \quad d = \frac{\lambda_0}{4n_{\text{coating}}}$$

5.5 Waveguides

Derivation 5: Modes in a Rectangular Waveguide

A rectangular waveguide of dimensions $a \times b$ confines EM waves by total reflection from conducting walls. The fields must satisfy the wave equation with boundary conditions $E_{\text{tangential}} = 0$ at the walls.

For TE (transverse electric) modes, the solutions give discrete allowed wavevectors:

$$k_x = \frac{m\pi}{a}, \quad k_y = \frac{n\pi}{b}, \quad m,n = 0, 1, 2, \ldots$$

Each mode $\text{TE}_{mn}$ has a cutoff frequency below which it cannot propagate:

Waveguide Cutoff and Dispersion

$$\omega_{mn}^{\text{cutoff}} = c\sqrt{\left(\frac{m\pi}{a}\right)^2 + \left(\frac{n\pi}{b}\right)^2}$$$$k_z = \sqrt{\frac{\omega^2}{c^2} - \left(\frac{m\pi}{a}\right)^2 - \left(\frac{n\pi}{b}\right)^2}$$

Below the cutoff, $k_z$ becomes imaginary and the mode is evanescent. The lowest-order mode ($\text{TE}_{10}$) has the lowest cutoff and is the dominant mode used in most waveguide applications.

The phase and group velocities in a waveguide are:

$$v_p = \frac{c}{\sqrt{1 - (\omega_c/\omega)^2}} > c, \quad v_g = c\sqrt{1 - (\omega_c/\omega)^2} < c$$$$v_p \cdot v_g = c^2$$

5.5b Evanescent Waves and Tunneling

Beyond the critical angle, the "transmitted" wave has an imaginary component of $k_z$, meaning it decays exponentially into the second medium:

$$\vec{E}_t \propto e^{ik_x x} e^{-\kappa z} \quad \text{where } \kappa = k_1\sqrt{\sin^2\theta_i - (n_2/n_1)^2}$$

The penetration depth is $d_p = 1/\kappa$, typically on the order of one wavelength. Although no net energy flows into the second medium (time-averaged Poynting vector perpendicular to the surface is zero), the evanescent field is real and measurable.

Frustrated Total Internal Reflection

If a second high-index medium is brought within the penetration depth, energy can "tunnel" across the low-index gap — this is frustrated TIR, the optical analog of quantum mechanical tunneling:

$$T \propto e^{-2\kappa d}$$

where $d$ is the gap width. This is used in:

Beam Splitters

FTIR beam splitters with controlled gap thickness provide variable reflection/transmission ratios without absorption losses.

TIRF Microscopy

Total Internal Reflection Fluorescence microscopy excites only fluorophores within ~100 nm of the surface, enabling single-molecule imaging at cell membranes.

5.5c Worked Example: Fiber Optic Design

Problem: Design a step-index optical fiber with core index$n_1 = 1.48$ and cladding index $n_2 = 1.46$. Find: (a) the critical angle for TIR, (b) the numerical aperture, and (c) the maximum acceptance half-angle for light entering the fiber from air.

Solution

(a) Critical angle: $\theta_c = \arcsin(n_2/n_1) = \arcsin(1.46/1.48) = 80.6°$

(b) Numerical aperture: $\text{NA} = \sqrt{n_1^2 - n_2^2} = \sqrt{1.48^2 - 1.46^2} = \sqrt{0.0588} = 0.242$

Light entering within this cone will be guided by TIR. Single-mode fibers use a much smaller core (8–10 $\mu$m) to support only the fundamental mode, eliminating intermodal dispersion.

5.5d Transmission Lines

A transmission line (coaxial cable, parallel wires, microstrip) guides EM waves in the TEM mode. The telegrapher's equations relate voltage and current:

$$\frac{\partial V}{\partial x} = -L'\frac{\partial I}{\partial t}, \quad \frac{\partial I}{\partial x} = -C'\frac{\partial V}{\partial t}$$

These combine to give the wave equation for voltage and current, propagating at$v = 1/\sqrt{L'C'}$ with characteristic impedance$Z_0 = \sqrt{L'/C'}$. When a line of impedance $Z_0$is terminated with load $Z_L$:

Reflection on a Transmission Line

$$\Gamma = \frac{Z_L - Z_0}{Z_L + Z_0}$$

For a matched load ($Z_L = Z_0$), $\Gamma = 0$ and there is no reflection. A short circuit ($Z_L = 0$) gives $\Gamma = -1$; an open circuit ($Z_L = \infty$) gives $\Gamma = +1$. This is directly analogous to reflection of EM waves at an interface.

5.6 Applications

Optical Fibers

Total internal reflection confines light within a glass fiber. Modern single-mode fibers carry terabits per second over thousands of kilometers, forming the backbone of the internet. The critical angle determines the acceptance cone (numerical aperture).

Anti-Reflection Coatings

Quarter-wave coatings on camera lenses, eyeglasses, and solar cells reduce reflective losses. Multi-layer coatings can achieve broadband anti-reflection, improving transmission from about 96% to over 99.9%.

Microwave Waveguides

Rectangular metal waveguides carry microwave signals in radar systems, particle accelerators, and satellite communications. The cutoff frequency determines the operating bandwidth for single-mode propagation.

Dielectric Mirrors

Alternating quarter-wave layers of high and low refractive index create highly reflective mirrors (R > 99.999%). These are essential for laser cavities and gravitational wave detectors like LIGO.

5.6b Multi-layer Dielectric Coatings

Modern optical coatings use multiple alternating layers of high-index ($n_H$) and low-index ($n_L$) dielectrics, each a quarter-wavelength thick. For a stack of $N$ pairs, the peak reflectivity is:

Multi-layer Mirror Reflectivity

$$R = \left[\frac{1 - (n_H/n_L)^{2N}(n_H^2/n_s)}{1 + (n_H/n_L)^{2N}(n_H^2/n_s)}\right]^2$$

where $n_s$ is the substrate index. With $n_H = 2.3$ (TiO$_2$) and $n_L = 1.38$ (MgF$_2$), just 20 pairs give$R > 99.99\%$. Such mirrors are essential for laser cavities and LIGO.

The transfer matrix method provides a systematic way to compute the reflection and transmission of any multilayer structure. Each layer is represented by a 2x2 matrix, and the total system matrix is the product of all layer matrices:

$$\mathbf{M}_j = \begin{pmatrix} \cos\delta_j & -i\sin\delta_j/\eta_j \\ -i\eta_j\sin\delta_j & \cos\delta_j \end{pmatrix}$$

where $\delta_j = 2\pi n_j d_j/\lambda$ is the phase thickness and$\eta_j$ is the admittance. This method handles any number of layers, including graded-index and rugate filters.

5.7 Historical Context

Willebrord Snellius (1621): Discovered the law of refraction empirically, though it was not published until after his death. Ibn Sahl had actually discovered the law in 984 CE in Baghdad, but his work was lost to the West.

Augustin-Jean Fresnel (1823): Derived the reflection and transmission coefficients for polarized light at an interface, supporting the wave theory of light over Newton's corpuscular theory. His equations remain fundamental to optics today.

Sir David Brewster (1815): Discovered the polarizing angle experimentally. At Brewster's angle, reflected light is completely polarized, a result later explained by Fresnel's theory.

Lord Rayleigh (1897): Analyzed electromagnetic wave propagation in hollow metallic tubes, laying the theoretical groundwork for microwave waveguides developed during World War II for radar.

5.7b Polarization Effects at Interfaces

The Fresnel equations show that s- and p-polarizations behave differently at interfaces, with important consequences:

Glare Reduction

Light reflected from flat surfaces (road, water) at near-Brewster's angle is predominantly s-polarized. Polarizing sunglasses block this polarization, dramatically reducing glare. Fishermen use polarized lenses to see below the water surface.

Ellipsometry

By measuring the change in polarization state upon reflection, ellipsometry determines thin-film thickness and optical constants with sub-nanometer precision. This is the standard metrology tool in semiconductor fabrication.

Goos-Hanchen Shift

At total internal reflection, the reflected beam is shifted laterally along the interface by a small amount ($\sim \lambda$) called the Goos-Hanchen shift. This occurs because the evanescent wave carries energy along the surface before re-emerging.

Surface Plasmon Resonance

At a metal-dielectric interface, p-polarized light can excite surface plasmon polaritons when the phase-matching condition is met. SPR sensors detect molecular binding events and are widely used in biosensing and drug discovery.

5.8 Python Simulation

This simulation visualizes the Fresnel coefficients, total internal reflection, Brewster's angle, and waveguide dispersion.

Wave Propagation: Fresnel Equations, TIR, Coatings, and Waveguides

Python

script.py156 lines

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

# ============================================================
# Wave Propagation: Fresnel, Snell, and Waveguides
# ============================================================

fig, axes = plt.subplots(2, 2, figsize=(12, 10))
fig.suptitle('Wave Propagation at Interfaces', fontsize=16, fontweight='bold', color='white')
fig.patch.set_facecolor('#0a0a0a')

for ax in axes.flat:
    ax.set_facecolor('#0a0a0a')
    ax.tick_params(colors='white')
    ax.xaxis.label.set_color('white')
    ax.yaxis.label.set_color('white')
    ax.title.set_color('white')
    for spine in ax.spines.values():
        spine.set_color('#333')

# --- Panel 1: Fresnel coefficients (air to glass, n1=1, n2=1.5) ---
ax1 = axes[0, 0]
n1, n2 = 1.0, 1.5
theta_i = np.linspace(0, np.pi/2 - 0.001, 500)
theta_t = np.arcsin(n1 / n2 * np.sin(theta_i))

rs = (n1 * np.cos(theta_i) - n2 * np.cos(theta_t)) / (n1 * np.cos(theta_i) + n2 * np.cos(theta_t))
rp = (n2 * np.cos(theta_i) - n1 * np.cos(theta_t)) / (n2 * np.cos(theta_i) + n1 * np.cos(theta_t))
Rs = rs**2
Rp = rp**2

theta_B = np.arctan(n2 / n1)

ax1.plot(np.degrees(theta_i), Rs, color='#22d3ee', linewidth=2.5, label='R_s (TE)')
ax1.plot(np.degrees(theta_i), Rp, color='#f97316', linewidth=2.5, label='R_p (TM)')
ax1.axvline(x=np.degrees(theta_B), color='#4ade80', linewidth=1.5, linestyle='--',
            label='Brewster angle = ' + str(round(np.degrees(theta_B), 1)) + ' deg')
ax1.set_xlabel('Angle of incidence (degrees)')
ax1.set_ylabel('Reflectance')
ax1.set_title('Fresnel: Air to Glass (n=1.5)')
ax1.legend(fontsize=7, facecolor='#0a0a0a', edgecolor='#333', labelcolor='white')
ax1.set_xlim(0, 90)
ax1.set_ylim(0, 1)

# --- Panel 2: Total internal reflection (glass to air) ---
ax2 = axes[0, 1]
n1_tir, n2_tir = 1.5, 1.0
theta_c = np.arcsin(n2_tir / n1_tir)

theta_below = np.linspace(0, theta_c - 0.001, 300)
theta_t_below = np.arcsin(n1_tir / n2_tir * np.sin(theta_below))
rs_below = ((n1_tir * np.cos(theta_below) - n2_tir * np.cos(theta_t_below)) /
            (n1_tir * np.cos(theta_below) + n2_tir * np.cos(theta_t_below)))**2
rp_below = ((n2_tir * np.cos(theta_below) - n1_tir * np.cos(theta_t_below)) /
            (n2_tir * np.cos(theta_below) + n1_tir * np.cos(theta_t_below)))**2

theta_above = np.linspace(theta_c + 0.001, np.pi/2 - 0.001, 300)

ax2.plot(np.degrees(theta_below), rs_below, color='#22d3ee', linewidth=2.5, label='R_s')
ax2.plot(np.degrees(theta_below), rp_below, color='#f97316', linewidth=2.5, label='R_p')
ax2.plot(np.degrees(theta_above), np.ones_like(theta_above), color='#22d3ee', linewidth=2.5)
ax2.plot(np.degrees(theta_above), np.ones_like(theta_above), color='#f97316', linewidth=2.5)
ax2.axvline(x=np.degrees(theta_c), color='#ef4444', linewidth=2, linestyle='--',
            label='Critical angle = ' + str(round(np.degrees(theta_c), 1)) + ' deg')
ax2.set_xlabel('Angle of incidence (degrees)')
ax2.set_ylabel('Reflectance')
ax2.set_title('Total Internal Reflection: Glass to Air')
ax2.legend(fontsize=7, facecolor='#0a0a0a', edgecolor='#333', labelcolor='white')
ax2.set_xlim(0, 90)
ax2.set_ylim(0, 1.1)

# --- Panel 3: Anti-reflection coating effectiveness ---
ax3 = axes[1, 0]
n_glass = 1.5
n_air = 1.0
wavelengths = np.linspace(400, 800, 500)  # nm
lambda0 = 550  # design wavelength
n_coat = np.sqrt(n_air * n_glass)  # ideal coating index

# Reflectance of bare surface
R_bare = ((n_glass - n_air) / (n_glass + n_air))**2

# Quarter-wave coating reflectance
d_coat = lambda0 / (4 * n_coat)  # physical thickness
delta = 2 * np.pi * n_coat * d_coat / wavelengths  # phase thickness

r12 = (n_air - n_coat) / (n_air + n_coat)
r23 = (n_coat - n_glass) / (n_coat + n_glass)
R_coated = (r12**2 + r23**2 + 2 * r12 * r23 * np.cos(2 * delta)) / (1 + r12**2 * r23**2 + 2 * r12 * r23 * np.cos(2 * delta))

ax3.plot(wavelengths, np.ones_like(wavelengths) * R_bare * 100, color='#f97316', linewidth=2,
         linestyle='--', label='Bare surface: ' + str(round(R_bare * 100, 1)) + '%')
ax3.plot(wavelengths, R_coated * 100, color='#22d3ee', linewidth=2.5,
         label='MgF2 coating (n=1.22)')
ax3.axvline(x=lambda0, color='#4ade80', linewidth=1, linestyle=':', alpha=0.5)
ax3.set_xlabel('Wavelength (nm)')
ax3.set_ylabel('Reflectance (%)')
ax3.set_title('Anti-Reflection Coating on Glass')
ax3.legend(fontsize=8, facecolor='#0a0a0a', edgecolor='#333', labelcolor='white')
ax3.set_ylim(0, 6)

# --- Panel 4: Waveguide dispersion ---
ax4 = axes[1, 1]
c_val = 1.0
a, b = 2.286, 1.016  # standard WR-90 waveguide (cm), normalized

# TE10 mode
fc_10 = c_val / (2 * a)
# TE20 mode
fc_20 = c_val / a
# TE01 mode
fc_01 = c_val / (2 * b)

f = np.linspace(0.01, 3 * fc_10, 1000)

for fc, label, color in [(fc_10, 'TE10', '#22d3ee'), (fc_20, 'TE20', '#f97316'), (fc_01, 'TE01', '#a855f7')]:
    mask = f > fc
    kz = np.zeros_like(f)
    kz[mask] = (2 * np.pi * f[mask] / c_val) * np.sqrt(1 - (fc / f[mask])**2)
    ax4.plot(f[mask] / fc_10, kz[mask] / (2 * np.pi / a), color=color, linewidth=2.5, label=label)

# Light line
ax4.plot(f / fc_10, 2 * np.pi * f / c_val / (2 * np.pi / a), color='white', linewidth=1,
         linestyle=':', alpha=0.5, label='Light line')

ax4.set_xlabel('f / f_c(TE10)')
ax4.set_ylabel('k_z (normalized)')
ax4.set_title('Waveguide Dispersion Curves')
ax4.legend(fontsize=8, facecolor='#0a0a0a', edgecolor='#333', labelcolor='white')
ax4.set_xlim(0, 3)
ax4.set_ylim(0, 3)

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0a0a0a')
plt.close()
print("Plot saved to output.png")

print("\n" + "=" * 60)
print("WAVE PROPAGATION: RESULTS")
print("=" * 60)
print("\n--- Fresnel (Air to Glass, n=1.5) ---")
print("  Normal incidence R = " + str(round(((n2 - n1)/(n2 + n1))**2, 4)))
print("  Brewster angle = " + str(round(np.degrees(theta_B), 2)) + " degrees")
print("\n--- Total Internal Reflection ---")
print("  Critical angle (glass to air) = " + str(round(np.degrees(theta_c), 2)) + " degrees")
print("\n--- Anti-Reflection Coating ---")
print("  Ideal coating index = sqrt(1 * 1.5) = " + str(round(n_coat, 4)))
print("  Bare glass reflectance = " + str(round(R_bare * 100, 2)) + "%")
print("  Min coated reflectance = " + str(round(np.min(R_coated) * 100, 4)) + "% at design wavelength")
print("\n--- Waveguide (WR-90) ---")
print("  TE10 cutoff: f_c = c/(2a)")
print("  Single-mode band: f_c(TE10) < f < f_c(TE20)")
print("=" * 60)

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Summary

Core Equations

Snell's law: $n_1\sin\theta_i = n_2\sin\theta_t$
Brewster: $\theta_B = \arctan(n_2/n_1)$
Critical angle: $\theta_c = \arcsin(n_2/n_1)$
Impedance: $Z = \sqrt{\mu/\epsilon}$
Waveguide cutoff: $\omega_c = c\pi\sqrt{(m/a)^2+(n/b)^2}$

Key Insights

Boundary conditions determine reflection/transmission
Brewster angle: zero p-polarization reflection
TIR enables fiber optics
Quarter-wave coatings cancel reflections
Waveguide phase velocity exceeds c (no contradiction)

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