Part III: Optics | Chapter 8

Interference

When waves meet: constructive and destructive superposition

8.1 Superposition and Coherence

When two waves overlap, the resulting field is the sum of the individual fields. The intensity of the superposition depends on the phase differencebetween the waves:

$$I = I_1 + I_2 + 2\sqrt{I_1 I_2}\cos\Delta\phi$$

The cross term (interference term) vanishes when averaged over time if the phase difference fluctuates randomly — this is the case for incoherentsources (e.g., two light bulbs). Stable interference requires coherent sources.

Temporal Coherence

Related to the monochromaticity (spectral width) of the source. The coherence time is $\tau_c \sim 1/\Delta\nu$ and the coherence length is$l_c = c\tau_c$. A laser has $l_c$ of meters; a thermal source, micrometers.

Spatial Coherence

Related to the angular size of the source. Points on a wavefront maintain a fixed phase relationship over a transverse coherence width$d_c \sim \lambda/\theta_s$ (van Cittert-Zernike theorem).

8.2 Young's Double-Slit Experiment

Derivation 1: Double-Slit Interference Pattern

Two narrow slits separated by distance $d$ are illuminated by a coherent plane wave. The path difference to a point at angle $\theta$ on a distant screen is $\Delta = d\sin\theta$. The phase difference is:

$$\Delta\phi = \frac{2\pi}{\lambda}d\sin\theta$$

Double-Slit Intensity

$$I(\theta) = 4I_0\cos^2\!\left(\frac{\pi d \sin\theta}{\lambda}\right)$$

Bright fringes (maxima) at $d\sin\theta = m\lambda$ ($m = 0, \pm 1, \pm 2, \ldots$). Dark fringes (minima) at $d\sin\theta = (m + \frac{1}{2})\lambda$. The fringe spacing on a screen at distance $L$ is $\Delta y = \lambda L/d$.

Historical significance: Young's 1801 experiment was the definitive demonstration that light behaves as a wave. It was one of the most important experiments in physics history, later re-examined in the quantum context where even single photons or electrons produce an interference pattern.

8.3 Thin-Film Interference

Derivation 2: Reflected Intensity from a Thin Film

A thin film of thickness $t$ and refractive index $n$ creates interference between light reflected from the top and bottom surfaces. The optical path difference is $2nt\cos\theta_t$, plus a possible $\lambda/2$phase shift from reflection at a denser medium:

Thin-Film Conditions (Normal Incidence)

$$\text{Constructive: } 2nt = \left(m + \frac{1}{2}\right)\lambda \quad (\text{one phase shift})$$$$\text{Destructive: } 2nt = m\lambda \quad (\text{one phase shift})$$

This assumes one reflection has a phase shift (denser medium) and one does not. If both or neither reflections produce a phase shift, the conditions swap.

Thin-film interference explains the colors of soap bubbles, oil slicks on water, and the iridescent colors of butterfly wings and beetle shells. The reflected color depends on the film thickness, which varies across the surface, creating colorful patterns.

8.4 The Michelson Interferometer

Derivation 3: Fringe Visibility and Coherence

A beam splitter divides light into two arms. Mirrors reflect the beams back, and they recombine at the beam splitter. The path difference between the two arms is $\Delta = 2(d_1 - d_2)$:

$$I = 2I_0\left[1 + \cos\!\left(\frac{2\pi\Delta}{\lambda}\right)\right]$$

For a source with finite spectral width, the fringe visibility decreases with increasing path difference. The visibility is the Fourier transform of the source spectrum (Wiener-Khintchine theorem):

Fringe Visibility

$$\mathcal{V} = \frac{I_{\max} - I_{\min}}{I_{\max} + I_{\min}} = |\gamma(\Delta)|$$

where $\gamma(\Delta)$ is the complex degree of temporal coherence. The coherence length $l_c = \lambda^2/\Delta\lambda$ is the path difference at which fringes disappear. This is how the Michelson interferometer measures spectral widths (Fourier transform spectroscopy).

8.5 The Fabry-Perot Interferometer

Derivation 4: Multi-Beam Interference

Two parallel, highly reflective mirrors separated by distance $d$ form an optical cavity. Light bounces many times between the mirrors, creating multi-beam interference. The transmitted intensity is:

Airy Function (Fabry-Perot Transmission)

$$T = \frac{1}{1 + F\sin^2(\delta/2)} \quad \text{where } \delta = \frac{4\pi n d \cos\theta}{\lambda}$$

The coefficient of finesse is $F = 4R/(1-R)^2$ where $R$is the mirror reflectivity. The finesse $\mathcal{F} = \pi\sqrt{F}/2$measures the sharpness of the transmission peaks.

Derivation 5: Resolving Power

The Fabry-Perot can resolve two closely spaced wavelengths. The free spectral range (FSR) is the spacing between adjacent transmission peaks:

$$\Delta\nu_{\text{FSR}} = \frac{c}{2nd}$$$$\delta\nu = \frac{\Delta\nu_{\text{FSR}}}{\mathcal{F}}$$$$\text{Resolving power: } R = \frac{\lambda}{\delta\lambda} = m\mathcal{F}$$

where $m$ is the order number. With high-reflectivity mirrors ($R > 0.99$), the finesse can exceed 100, giving resolving powers of $10^6$ or higher — far better than a prism or grating.

8.5b Newton's Rings

When a plano-convex lens of radius $R$ rests on a flat glass surface, the air gap varies with radial distance $r$. The gap thickness is approximately $d(r) \approx r^2/(2R)$ for $r \ll R$.

Thin-film interference in the air gap produces concentric circular fringes called Newton's rings. The radii of the dark rings (in reflected light) are:

Newton's Rings

$$r_m = \sqrt{m\lambda R} \quad m = 0, 1, 2, \ldots$$

The center is dark because the air gap is zero and reflection introduces a phase shift. By measuring the ring radii and knowing $R$, one can determine$\lambda$ (or vice versa). This was actually a key experiment used by Newton himself, who ironically interpreted the results using his corpuscular theory.

8.5c Two-Beam vs Multiple-Beam Interference

The fundamental distinction between interferometers:

Two-Beam (Michelson, Young)

Intensity: $I = 2I_0(1 + \cos\delta)$. Fringes are sinusoidal ($\cos^2$). The fringe contrast is always 100% for equal-amplitude beams. Simple to align but moderate resolving power.

Multiple-Beam (Fabry-Perot)

Intensity follows the Airy function with sharp peaks. As mirror reflectivity increases, peaks become narrower while maintaining the same spacing (FSR). The finesse$\mathcal{F}$ can exceed 10,000 with modern dielectric mirrors.

The Fabry-Perot transmission can be understood as a geometric series. The total transmitted field is the sum of all partially transmitted beams:

$$E_t = E_0 t^2 \sum_{n=0}^{\infty} r^{2n} e^{in\delta} = \frac{E_0 t^2}{1 - r^2 e^{i\delta}}$$

Taking the modulus squared gives the Airy function derived earlier. The key insight is that multi-beam interference produces constructive interference only at very specific phase values, hence the extreme wavelength selectivity.

8.5d Worked Example: Resolving the Sodium D Lines

Problem: The sodium D lines are at 589.0 nm and 589.6 nm. Design a Fabry-Perot interferometer to resolve them. Determine the required finesse and mirror spacing.

Solution

Required resolving power:$R = \lambda/\delta\lambda = 589.3/0.6 \approx 982$

Choose mirror spacing $d = 1$ mm. The order number at 589 nm is:$m = 2d/\lambda = 2 \times 10^{-3}/(589 \times 10^{-9}) \approx 3396$

Required finesse: $\mathcal{F} = R/m = 982/3396 \approx 0.29$

This is easily achievable even with low-reflectivity mirrors! However, we also need the FSR to exceed the line separation. The FSR in wavelength is:$\Delta\lambda_{\text{FSR}} = \lambda^2/(2d) = 0.17$ nm.

Since $\Delta\lambda_{\text{FSR}} = 0.17 < 0.6$ nm, the lines would overlap from different orders. Solution: reduce $d$ to about 0.3 mm, giving $\text{FSR} = 0.58$ nm and $m \approx 1019$, requiring $\mathcal{F} \approx 1$. Even easier!

8.5e Coherence Theory

The mutual coherence function provides a rigorous description of partial coherence:

$$\Gamma_{12}(\tau) = \langle E_1^*(t)E_2(t + \tau) \rangle$$

The normalized version, $\gamma_{12}(\tau) = \Gamma_{12}(\tau)/\sqrt{\Gamma_{11}(0)\Gamma_{22}(0)}$, is the complex degree of coherence. The fringe visibility in any interference experiment is $\mathcal{V} = |\gamma_{12}|$.

The van Cittert-Zernike theorem connects spatial coherence to the angular size of the source: the mutual coherence function is the Fourier transform of the source's intensity distribution. This is exploited in stellar interferometry to measure angular diameters of stars too small to resolve directly.

Michelson stellar interferometer: Michelson measured the angular diameter of Betelgeuse (0.047 arcseconds) in 1920 using two mirrors on a 20-foot beam mounted on the 100-inch Hooker telescope. Modern optical interferometers like VLTI achieve milliarcsecond resolution using baselines of hundreds of meters.

8.6 Applications

LIGO Gravitational Wave Detection

LIGO uses a Michelson interferometer with 4 km arms to detect gravitational waves. The interferometer measures displacements of $\sim 10^{-19}$ m — one ten-thousandth the diameter of a proton. The 2015 detection confirmed Einstein's prediction and earned the 2017 Nobel Prize.

Laser Cavities

A laser is essentially a Fabry-Perot cavity filled with a gain medium. The cavity selects resonant frequencies (longitudinal modes), and the finesse determines the laser linewidth. The output coupler is a partially transmitting mirror.

Optical Coherence Tomography

OCT uses a Michelson interferometer with a broadband source to image subsurface structures in biological tissue (e.g., the retina). The coherence length determines the depth resolution, typically a few micrometers.

Wavelength Division Multiplexing

Fabry-Perot filters separate closely spaced wavelength channels in fiber-optic communication. Each channel carries independent data, and the high finesse allows dense packing of channels (DWDM: 100+ channels in the C-band).

8.6b White-Light Interference and Channeled Spectra

When a broadband (white light) source illuminates a thin film or interferometer, different wavelengths produce constructive interference at different angles or film thicknesses. This creates the vivid colors seen in:

Soap Bubbles

The film thickness varies across the surface due to gravity, creating bands of color. At a given point, the reflected color is the wavelength satisfying$2nt = (m + \frac{1}{2})\lambda$. As the film thins before bursting, it appears black (too thin for any visible wavelength to interfere constructively).

Oil Slicks

Thin oil films on water ($n_{\text{oil}} \approx 1.4$) show similar colors. The absence of a phase shift at the oil-water interface (since$n_{\text{oil}} > n_{\text{water}}$ is not always true for all oils) changes which colors are enhanced vs suppressed.

When the output of a Michelson interferometer is dispersed by a spectrometer, one observes a channeled spectrum: a sinusoidal modulation of the spectral intensity. The modulation period is $\Delta\nu = c/(2\Delta)$ where $\Delta$is the path difference. Measuring the channel spacing is a precise method for determining thin-film thicknesses.

8.6c The Sagnac Interferometer and Ring Laser Gyroscopes

In a Sagnac interferometer, a beam is split and sent around a closed loop in both directions. When the loop rotates at angular velocity $\Omega$, the co-rotating and counter-rotating beams acquire a phase difference:

Sagnac Phase Shift

$$\Delta\phi = \frac{8\pi A\Omega}{\lambda c}$$

where $A$ is the enclosed area. This is used in ring laser gyroscopes and fiber-optic gyroscopes for inertial navigation in aircraft and spacecraft. Modern fiber-optic gyroscopes use many loops of fiber (large effective $A$) and can detect Earth's rotation rate.

8.7 Historical Context

Thomas Young (1801): Performed the double-slit experiment, providing the first conclusive evidence for the wave nature of light. His presentation to the Royal Society was initially met with hostility from supporters of Newton's particle theory.

Albert Michelson (1887): With Edward Morley, used his interferometer to search for the "luminiferous ether." Their null result was a key puzzle that led to Einstein's special relativity. Michelson received the 1907 Nobel Prize for his optical precision instruments.

Charles Fabry and Alfred Perot (1899): Developed the multi-beam interferometer that bears their names. Its extraordinarily high resolving power made it essential for spectroscopy and later for laser physics.

Dennis Gabor (1948): Invented holography, which records and reconstructs the complete wavefront (amplitude and phase) using interference. He received the 1971 Nobel Prize after lasers made practical holography possible.

8.7b From Classical to Quantum Interference

The double-slit experiment takes on profound significance in quantum mechanics. When performed with single electrons (Tonomura et al., 1989) or single photons, individual particles arrive at apparently random positions on the detector. Yet after accumulating many events, the interference pattern emerges:

Which-Path Information

If a detector at the slits determines which slit each particle passes through, the interference pattern disappears. Complementarity: wave behavior and particle behavior are mutually exclusive.

Quantum Eraser

If the which-path information is "erased" before detection, the interference pattern reappears. This delayed-choice quantum eraser experiment shows that the interference is not destroyed by the measurement apparatus itself, but by the availability of distinguishing information.

Feynman famously said the double-slit experiment contains "the only mystery" of quantum mechanics. The probability amplitude for each path is a complex number, and the total probability is $|\psi_1 + \psi_2|^2$, not$|\psi_1|^2 + |\psi_2|^2$. The cross term gives interference, and its existence has been verified for electrons, neutrons, atoms, and even molecules as large as C$_{60}$ (buckminsterfullerene).

Matter-wave interferometry: Atom interferometers exploit the wave nature of matter for precision measurements. They can measure gravitational acceleration to 1 part in $10^{10}$, test the equivalence principle, and detect gravitational waves at frequencies inaccessible to LIGO. The AION and MAGIS experiments aim to create atom interferometers with kilometer-scale baselines.

8.8 Python Simulation

This simulation visualizes the double-slit pattern, thin-film colors, Michelson fringes, and the Fabry-Perot Airy function for different finesses.

Interference: Double-Slit, Thin Film, Michelson, and Fabry-Perot

Python

script.py135 lines

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

# ============================================================
# Interference: Double-Slit, Thin Film, Michelson, Fabry-Perot
# ============================================================

fig, axes = plt.subplots(2, 2, figsize=(12, 10))
fig.suptitle('Interference Phenomena', 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: Double-slit interference pattern ---
ax1 = axes[0, 0]
lam = 550e-9  # green light
d_slit = 0.2e-3  # slit separation
L_screen = 1.0  # screen distance
y = np.linspace(-10e-3, 10e-3, 1000)  # position on screen
theta = np.arctan(y / L_screen)

# Pure interference (point slits)
I_double = 4 * np.cos(np.pi * d_slit * np.sin(theta) / lam)**2

# With single-slit envelope (slit width a = 50 um)
a_slit = 50e-6
beta = np.pi * a_slit * np.sin(theta) / lam
sinc_env = np.ones_like(beta)
mask = np.abs(beta) > 1e-10
sinc_env[mask] = (np.sin(beta[mask]) / beta[mask])**2
I_with_env = I_double * sinc_env

ax1.plot(y * 1e3, I_double / 4, color='#22d3ee', linewidth=1, alpha=0.4, label='Point slits')
ax1.plot(y * 1e3, I_with_env / 4, color='#f97316', linewidth=2, label='With slit width')
ax1.plot(y * 1e3, sinc_env, color='white', linewidth=1, linestyle='--', alpha=0.3, label='Envelope')
ax1.set_xlabel('Position on screen (mm)')
ax1.set_ylabel('Intensity / I_0')
ax1.set_title("Young's Double Slit (d=0.2mm, a=50um)")
ax1.legend(fontsize=7, facecolor='#0a0a0a', edgecolor='#333', labelcolor='white')

# --- Panel 2: Thin-film reflectance for different thicknesses ---
ax2 = axes[0, 1]
wavelengths = np.linspace(380, 780, 500)
n_film = 1.33  # soap film

for thickness, color, alpha_val in [(200, '#22d3ee', 1.0), (300, '#f97316', 0.9),
                                     (400, '#a855f7', 0.8), (500, '#4ade80', 0.7)]:
    delta = 4 * np.pi * n_film * thickness / wavelengths  # phase difference
    # Assuming air-film-air (phase shift at both surfaces for n_film > 1 > 1... only top)
    R = 4 * 0.02 * np.sin(delta / 2)**2 / (1 - 0.02)**2  # simplified for low R
    R_approx = 2 * 0.02 * (1 - np.cos(delta))  # thin film approximation
    ax2.plot(wavelengths, R_approx, color=color, linewidth=2, alpha=alpha_val,
             label='t = ' + str(thickness) + ' nm')

ax2.set_xlabel('Wavelength (nm)')
ax2.set_ylabel('Reflectance (approx)')
ax2.set_title('Thin-Film Interference (Soap Film)')
ax2.legend(fontsize=8, facecolor='#0a0a0a', edgecolor='#333', labelcolor='white')

# --- Panel 3: Michelson interferometer fringes ---
ax3 = axes[1, 0]
delta_d = np.linspace(-5e-6, 5e-6, 1000)  # path difference in m

# Monochromatic
I_mono = 1 + np.cos(2 * np.pi * 2 * delta_d / lam)

# Finite bandwidth (Gaussian spectral profile)
dlam = 5e-9  # spectral width
sigma_k = 2 * np.pi * dlam / lam**2
coherence_env = np.exp(-2 * (sigma_k * delta_d)**2)
I_broad = 1 + coherence_env * np.cos(2 * np.pi * 2 * delta_d / lam)

ax3.plot(delta_d * 1e6, I_mono / 2, color='#22d3ee', linewidth=1.5, alpha=0.5, label='Monochromatic')
ax3.plot(delta_d * 1e6, I_broad / 2, color='#f97316', linewidth=2, label='Delta_lam = 5 nm')
ax3.plot(delta_d * 1e6, (1 + coherence_env) / 2, color='white', linewidth=1, linestyle='--', alpha=0.3)
ax3.plot(delta_d * 1e6, (1 - coherence_env) / 2, color='white', linewidth=1, linestyle='--', alpha=0.3)
ax3.set_xlabel('Path difference (um)')
ax3.set_ylabel('Intensity / I_0')
ax3.set_title('Michelson Interferometer')
ax3.legend(fontsize=8, facecolor='#0a0a0a', edgecolor='#333', labelcolor='white')

# --- Panel 4: Fabry-Perot Airy function ---
ax4 = axes[1, 1]
delta_fp = np.linspace(0, 4 * np.pi, 1000)

for R_val, color, label in [(0.3, '#22d3ee', 'R=0.3 (F=2.4)'),
                              (0.7, '#f97316', 'R=0.7 (F=31)'),
                              (0.9, '#a855f7', 'R=0.9 (F=360)'),
                              (0.97, '#4ade80', 'R=0.97 (F=4300)')]:
    F_coeff = 4 * R_val / (1 - R_val)**2
    T_fp = 1 / (1 + F_coeff * np.sin(delta_fp / 2)**2)
    finesse = np.pi * np.sqrt(F_coeff) / 2
    ax4.plot(delta_fp / np.pi, T_fp, color=color, linewidth=2, label=label)

ax4.set_xlabel('delta / pi')
ax4.set_ylabel('Transmission')
ax4.set_title('Fabry-Perot Airy Function')
ax4.legend(fontsize=7, facecolor='#0a0a0a', edgecolor='#333', labelcolor='white')

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("INTERFERENCE: RESULTS")
print("=" * 60)
print("\n--- Young's Double Slit ---")
lam_nm = lam * 1e9
fringe_spacing = lam * L_screen / d_slit
print("  Wavelength = " + str(round(lam_nm, 0)) + " nm")
print("  Slit separation = " + str(d_slit * 1e3) + " mm")
print("  Fringe spacing = " + str(round(fringe_spacing * 1e3, 3)) + " mm")
print("\n--- Thin Film ---")
print("  Film index n = " + str(n_film))
print("  First bright (reflected): 2nt = lambda/2 => t = " + str(round(lam_nm / (4 * n_film), 1)) + " nm")
print("\n--- Michelson ---")
l_coherence = lam**2 / (5e-9) * 1e6
print("  Coherence length (Delta_lam=5nm) = " + str(round(l_coherence, 1)) + " um")
print("\n--- Fabry-Perot ---")
for R_val in [0.3, 0.7, 0.9, 0.97]:
    F_val = 4 * R_val / (1 - R_val)**2
    finesse_val = np.pi * np.sqrt(F_val) / 2
    print("  R = " + str(R_val) + ": Finesse = " + str(round(finesse_val, 1)))
print("=" * 60)

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Summary

Core Equations

Double-slit: $I = 4I_0\cos^2(\pi d\sin\theta/\lambda)$
Thin film: $2nt = (m+\frac{1}{2})\lambda$ (constructive)
Coherence length: $l_c = \lambda^2/\Delta\lambda$
Fabry-Perot: $T = 1/(1 + F\sin^2\delta/2)$
Finesse: $\mathcal{F} = \pi\sqrt{R}/(1-R)$

Key Insights

Interference requires coherent sources
Two-beam vs multi-beam: sharper peaks with more beams
Coherence limits the path difference for fringes
Fabry-Perot: highest resolving power in optics
LIGO: interference at $10^{-19}$ m precision

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