Part II, Chapter 2

Wave Optics

When the wavelength of light is comparable to the size of apertures and obstacles, diffraction and interference phenomena dominate, demanding a full wave treatment.

2.1 The Huygens-Fresnel Principle

Christiaan Huygens (1678) proposed that every point on a wavefront acts as a source of secondary spherical wavelets. Fresnel (1818) added the crucial ingredient of interference between these wavelets, producing the Huygens-Fresnel diffraction integral:

$$U(P) = \frac{1}{i\lambda} \iint_{\Sigma} U(Q)\, \frac{e^{ikr}}{r}\, K(\chi)\, dS$$

where U(Q) is the field at point Q on the aperture Σ, r is the distance from Q to the observation point P, k = 2π/λ, and K(χ) is the obliquity factor. Kirchhoff later placed this on rigorous mathematical footing using Green's theorem.

2.1.1 Fresnel Zones

The aperture can be divided into annular Fresnel zones, where successive zones contribute with alternating phase. The m-th zone boundary has radius:

$$r_m = \sqrt{m \lambda z}$$

where z is the distance to the observation point. A zone plate that blocks every other zone acts as a focusing element, with focal length f = r₁²/λ.

2.2 Fraunhofer (Far-Field) Diffraction

When the observation point is far from the aperture (or equivalently, with a focusing lens), the quadratic phase terms vanish and the diffraction pattern is the Fourier transform of the aperture function:

$$U(u, v) \propto \iint t(x, y)\, e^{-i(ux + vy)}\, dx\, dy$$

where u = kx′/z, v = ky′/z are spatial frequencies and t(x,y) is the aperture transmission function.

2.2.1 Single Slit

For a slit of width b, the Fraunhofer pattern is the sinc function:

$$I(\theta) = I_0 \operatorname{sinc}^2\!\left(\frac{\pi b \sin\theta}{\lambda}\right)$$

The first zeros occur at sinθ = ±λ/b. The central maximum contains 84% of the total power.

2.2.2 Circular Aperture — The Airy Pattern

For a circular aperture of radius a, the diffraction pattern involves the first-order Bessel function:

$$I(\theta) = I_0 \left[\frac{2J_1(ka\sin\theta)}{ka\sin\theta}\right]^2$$

Derivation: Airy Pattern

Step 1. The aperture function is a disk of radius a: t(ρ) = 1 for ρ ≤ a, else 0. In polar coordinates the Fraunhofer integral becomes:

$$U(q) \propto \int_0^a \int_0^{2\pi} e^{-iq\rho\cos\phi}\, \rho\, d\phi\, d\rho$$

Step 2. The angular integral is the integral representation of the Bessel function:

$$\int_0^{2\pi} e^{-iq\rho\cos\phi} d\phi = 2\pi J_0(q\rho)$$

Step 3. The radial integral uses the identity ∫₀^a ρ J₀(qρ) dρ = (a/q)J₁(qa):

Result:

$$U(q) \propto \frac{2J_1(qa)}{qa}, \quad I(q) = I_0 \left[\frac{2J_1(qa)}{qa}\right]^2$$

where q = k sinθ. The first zero of J₁ is at qa = 3.8317, giving the angular radius of the Airy disk: sinθ = 1.22λ/(2a).

2.3 Fresnel (Near-Field) Diffraction

When the observation distance z is not large enough to neglect the quadratic phase, we must retain the Fresnel approximation:

$$U(x', y') \propto \iint t(x,y)\, e^{i\frac{k}{2z}[(x'-x)^2 + (y'-y)^2]}\, dx\, dy$$

This is a convolution with a quadratic phase factor, equivalent to propagation via a paraxial Green's function. The Fresnel number N = a²/(λz) determines the regime: N >> 1 is geometrical optics, N ∼ 1 is Fresnel diffraction, and N << 1 is Fraunhofer diffraction.

2.3.1 Straight Edge Diffraction

The diffraction pattern from a straight edge involves Fresnel integrals C(w) and S(w):

$$C(w) = \int_0^w \cos\!\left(\frac{\pi t^2}{2}\right) dt, \quad S(w) = \int_0^w \sin\!\left(\frac{\pi t^2}{2}\right) dt$$

The Cornu spiral in the complex plane (C(w) + iS(w)) provides an elegant graphical construction for computing Fresnel diffraction patterns.

2.4 Resolution Limits

2.4.1 Rayleigh Criterion

Lord Rayleigh proposed that two point sources are just resolved when the central maximum of one coincides with the first minimum of the other. For a circular aperture of diameter D:

$$\theta_{\min} = 1.22 \frac{\lambda}{D}$$

This sets a fundamental limit on the angular resolution of telescopes, microscopes, and imaging systems. A 10-cm telescope at 550 nm has θ_min ≈ 1.3 arcseconds.

2.4.2 Sparrow Criterion

A more aggressive criterion: two sources are resolved when the combined intensity pattern has zero second derivative at the midpoint. This gives a slightly smaller limit than Rayleigh.

2.5 Thin Film Interference

When light reflects from a thin dielectric film of thickness d and refractive index n_f, the two reflected beams interfere. The phase difference (including the π phase shift at one interface) is:

$$\Delta\phi = \frac{4\pi n_f d \cos\theta_f}{\lambda} + \pi$$

Constructive interference (maximum reflection) occurs when Δφ = 2mπ, and destructive interference (antireflection) when Δφ = (2m+1)π. A quarter-wave coating (n_fd = λ/4) with n_f = √(n_s) eliminates reflection from a substrate of index n_s.

2.5.1 Fabry-Perot Interferometer

A pair of parallel partially reflecting mirrors forms a Fabry-Perot etalon. The transmitted intensity is given by the Airy function:

$$T = \frac{1}{1 + F \sin^2(\delta/2)}, \qquad F = \frac{4R}{(1-R)^2}$$

where δ = 4πnd cosθ/λ is the round-trip phase and F is the coefficient of finesse. The free spectral range (FSR) and finesse are:

$$\text{FSR} = \frac{c}{2nd}, \qquad \mathcal{F} = \frac{\pi\sqrt{R}}{1-R}$$

2.6 Python Simulation: Diffraction Patterns

This simulation computes and visualizes the Airy pattern, single-slit diffraction, Fabry-Perot transmission, and Fresnel diffraction from a straight edge.

Wave Optics: Diffraction & Interference Patterns

Python

script.py92 lines

import numpy as np
import matplotlib.pyplot as plt
from scipy.special import j1, fresnel

fig, axes = plt.subplots(2, 2, figsize=(14, 10))
fig.patch.set_facecolor("#0a0a0a")
for ax in axes.flat:
    ax.set_facecolor("#111111")

# --- Plot 1: Airy Pattern (circular aperture) ---
ax1 = axes[0, 0]
x = np.linspace(0.001, 15, 1000)
# I = [2*J1(x)/x]^2
airy = (2 * j1(x) / x)**2
ax1.plot(x, airy, color='#f59e0b', linewidth=2)
ax1.fill_between(x, airy, alpha=0.15, color='#f59e0b')
ax1.axvline(3.8317, color='#fbbf24', linestyle='--', alpha=0.5, label='First zero (3.83)')
ax1.axvline(7.0156, color='#fb923c', linestyle='--', alpha=0.5, label='Second zero (7.02)')
ax1.set_xlabel('ka sin(theta)', color='white')
ax1.set_ylabel('Normalized intensity', color='white')
ax1.set_title('Airy Diffraction Pattern', color='#fbbf24', fontsize=13)
ax1.legend(facecolor='#1a1a2e', edgecolor='#f59e0b', labelcolor='white', fontsize=9)
ax1.tick_params(colors='white')
for spine in ax1.spines.values():
    spine.set_color('#f59e0b')

# --- Plot 2: Single slit Fraunhofer diffraction ---
ax2 = axes[0, 1]
beta = np.linspace(-4*np.pi, 4*np.pi, 1000)
beta_safe = np.where(np.abs(beta) < 1e-10, 1e-10, beta)
sinc2 = (np.sin(beta_safe) / beta_safe)**2
ax2.plot(beta/np.pi, sinc2, color='#fb923c', linewidth=2)
ax2.fill_between(beta/np.pi, sinc2, alpha=0.15, color='#fb923c')
ax2.set_xlabel('beta/pi = b*sin(theta)/lambda', color='white')
ax2.set_ylabel('Normalized intensity', color='white')
ax2.set_title('Single Slit Fraunhofer Diffraction', color='#fbbf24', fontsize=13)
ax2.tick_params(colors='white')
for spine in ax2.spines.values():
    spine.set_color('#f59e0b')

# --- Plot 3: Fabry-Perot transmission ---
ax3 = axes[1, 0]
delta = np.linspace(0, 6*np.pi, 1000)
for R, ls in [(0.5, '-'), (0.8, '--'), (0.95, '-.')]:
    F_coeff = 4*R / (1-R)**2
    T = 1 / (1 + F_coeff * np.sin(delta/2)**2)
    finesse = np.pi * np.sqrt(R) / (1 - R)
    ax3.plot(delta/np.pi, T, ls, color='#f59e0b', linewidth=2, label=f'R={R}, F={finesse:.1f}')
ax3.set_xlabel('Round-trip phase delta/pi', color='white')
ax3.set_ylabel('Transmission', color='white')
ax3.set_title('Fabry-Perot Transmission', color='#fbbf24', fontsize=13)
ax3.legend(facecolor='#1a1a2e', edgecolor='#f59e0b', labelcolor='white', fontsize=9)
ax3.tick_params(colors='white')
for spine in ax3.spines.values():
    spine.set_color('#f59e0b')

# --- Plot 4: Fresnel diffraction from straight edge ---
ax4 = axes[1, 1]
w = np.linspace(-5, 5, 1000)
S, C = fresnel(w)
# Intensity from straight edge: I = 0.5*|(C(w) + 0.5) + i*(S(w) + 0.5)|^2
I_edge = 0.5 * ((C + 0.5)**2 + (S + 0.5)**2)
ax4.plot(w, I_edge, color='#f59e0b', linewidth=2)
ax4.axhline(1.0, color='gray', linestyle=':', alpha=0.5)
ax4.axhline(0.25, color='gray', linestyle=':', alpha=0.5)
ax4.axvline(0, color='#fbbf24', linestyle='--', alpha=0.3, label='Edge position')
ax4.set_xlabel('Fresnel parameter w', color='white')
ax4.set_ylabel('Normalized intensity', color='white')
ax4.set_title('Fresnel Diffraction: Straight Edge', color='#fbbf24', fontsize=13)
ax4.legend(facecolor='#1a1a2e', edgecolor='#f59e0b', labelcolor='white', fontsize=9)
ax4.tick_params(colors='white')
for spine in ax4.spines.values():
    spine.set_color('#f59e0b')

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0a0a0a')
plt.show()

# Numerical examples
D = 0.1  # 10 cm telescope
lam = 550e-9  # green light
theta_rayleigh = 1.22 * lam / D
print(f"Rayleigh limit for D={D*100:.0f} cm at {lam*1e9:.0f} nm:")
print(f"  theta_min = {theta_rayleigh:.2e} rad = {np.degrees(theta_rayleigh)*3600:.2f} arcsec")
print(f"\nFabry-Perot (R=0.95, d=1 cm, n=1):")
d_fp = 0.01; R_fp = 0.95
FSR = 3e8 / (2*1*d_fp)
finesse = np.pi*np.sqrt(R_fp)/(1-R_fp)
print(f"  FSR = {FSR/1e9:.2f} GHz")
print(f"  Finesse = {finesse:.1f}")
print(f"  Resolution = {FSR/finesse/1e6:.1f} MHz")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

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