Optical Coherence

Hold a flashlight and a laser pointer side by side and shine both on a wall through a pair of fine slits. The laser paints crisp light-and-dark stripes — interference fringes. The flashlight makes a featureless smear. The waves are the same kind of electromagnetic field; what differs is coherence — how long each wave keeps a phase you can predict from one instant to the next.

What coherence really means

A perfectly monochromatic wave, E(t) = E₀·cos(ωt + φ), with a single frequency ω and a phase φ that never jumps, is the ideal coherent source. Knowing its phase now lets you predict it forever. No real source is perfect. Atoms emit light in finite bursts — wave trains — interrupted by collisions and spontaneous re-emission, each with a fresh random phase. Between phase jumps, the wave is predictable; across a jump, it is not.

Coherence quantifies that predictability. The key object is the complex degree of coherence, defined from the field's autocorrelation at time delay τ:

g(τ) = ⟨E*(t) · E(t + τ)⟩ / ⟨|E(t)|²⟩

The magnitude |g(τ)| runs from 1 (the wave at time t+τ is perfectly correlated with the wave at t — fully coherent) down to 0 (no correlation — incoherent). The delay τc at which |g(τ)| falls to roughly 1/e — or, by common convention, 0.5 — is the coherence time. Multiply by the speed of light and you get the coherence length:

L_c = c · τ_c

Bandwidth sets coherence: the Fourier link

Coherence time and spectral width are Fourier conjugates — they are two views of the same thing. A wave that lasts a long time without a phase jump must be built from a narrow band of frequencies; a wave chopped into short bursts requires a wide band. The relationship is reciprocal:

τ_c ≈ 1 / Δν          (Δν = frequency bandwidth, in Hz)
L_c ≈ c / Δν ≈ λ² / Δλ  (Δλ = wavelength bandwidth)

The middle and right forms come from differentiating ν = c/λ, so |Δν| = c·Δλ/λ². This single chain of equalities is the heart of the subject: narrow spectrum → long coherence length; broad spectrum → short coherence length. A precise prefactor (1, 1/π, or 0.44…) depends on the line shape (Lorentzian, Gaussian) and the chosen threshold, but the scaling is universal.

Real numbers across sources

The span of coherence lengths in nature is staggering — eight orders of magnitude between a household bulb and a stabilized laser.

Source	Center λ	Bandwidth	Coherence length L_c
Tungsten bulb (white)	~550 nm	Δλ ≈ 300 nm	≈ 1 µm
Red LED	633 nm	Δλ ≈ 20 nm	≈ 20 µm
Superluminescent diode (OCT)	840 nm	Δλ ≈ 50 nm	≈ 14 µm
Filtered sodium lamp	589 nm	Δλ ≈ 0.6 nm	≈ 0.6 mm
Multimode HeNe laser	633 nm	Δν ≈ 1.5 GHz	≈ 20 cm
Single-mode HeNe laser	633 nm	Δν ≈ 1 MHz	≈ 95 m
Stabilized lab laser	1064 nm	Δν ≈ 1 Hz	≈ 3 × 10⁵ km

Worked example: a red LED at λ = 633 nm with Δλ = 20 nm gives

L_c ≈ λ²/Δλ = (633e-9)² / (20e-9) ≈ 2.0e-5 m = 20 µm

So in a Michelson interferometer the LED's two arms must match to within about 20 µm — twenty wavelengths — or the fringes vanish. The single-mode laser tolerates a 95-meter mismatch.

Fringe visibility — the lab measurement

You don't measure g(τ) directly; you measure the contrast of interference fringes. Send light into a two-beam interferometer (Michelson, Mach–Zehnder, or a double slit) with a path-length difference ΔL, corresponding to delay τ = ΔL/c. The detected intensity is

I(τ) = I₁ + I₂ + 2·√(I₁·I₂) · |g(τ)| · cos(ω·τ + arg g)

The fringe visibility (Michelson contrast) is

V = (I_max − I_min) / (I_max + I_min)

For equal-intensity beams (I₁ = I₂), V = |g(τ)| exactly. So the visibility curve is the coherence envelope: V = 1 at zero path difference, decaying to 0 as ΔL exceeds the coherence length. Scanning a mirror and watching V collapse is the textbook way to measure L_c — and the principle behind Fourier-transform spectroscopy, where the visibility-versus-path-difference interferogram transforms back into the source's spectrum.

Temporal vs. spatial coherence

Coherence has two independent flavors. Confusing them is the single most common error in optics.

Property	Temporal coherence	Spatial coherence
Direction of correlation	Along propagation (between t and t+τ)	Across the wavefront (between two points)
Set by	Spectral bandwidth Δν	Source angular size
Characteristic measure	Coherence time τ_c, length L_c	Coherence area / transverse coherence width
Governing relation	L_c ≈ λ²/Δλ	van Cittert–Zernike theorem
Test instrument	Michelson interferometer	Young's double slit (vary slit separation)
Improved by	Narrowing the spectrum (filter, laser)	Pinhole / distance (smaller apparent source)

Spatial coherence asks: do two points on the wavefront share a phase relationship? A point source is perfectly spatially coherent; an extended source is not, because each emitting atom contributes an independent wave. The van Cittert–Zernike theorem says the transverse coherence width grows as the source shrinks or recedes: w ≈ λ·R/d, where d is the source diameter and R the distance. This is why even sunlight — temporally a mess — is spatially coherent over about 50 µm at Earth's surface (the Sun's 0.5° angular size), enough to make faint double-slit fringes. It is also how stellar interferometers measure star diameters: they find the slit separation that kills the fringes.

Python — coherence length and visibility

import numpy as np

c = 2.998e8  # m/s

def coherence_length_from_dlambda(lam, dlam):
    """L_c from center wavelength and spectral width (m)."""
    return lam**2 / dlam

def coherence_length_from_linewidth(dnu):
    """L_c from frequency linewidth (Hz)."""
    return c / dnu

# Red LED: 633 nm, 20 nm wide
print(coherence_length_from_dlambda(633e-9, 20e-9) * 1e6, "um")   # ~20 um

# Single-mode HeNe: 1 MHz linewidth
print(coherence_length_from_linewidth(1e6), "m")                  # ~300 m (1/Δν form)

# Gaussian-spectrum visibility envelope vs path difference
def visibility(path_diff, Lc):
    # |g(tau)| for a Gaussian line: exp(-(pi/2)(dL/Lc)^2)
    return np.exp(-(np.pi / 2) * (path_diff / Lc) ** 2)

Lc = coherence_length_from_dlambda(633e-9, 20e-9)  # 20 um
for dL_um in [0, 10, 20, 40]:
    V = visibility(dL_um * 1e-6, Lc)
    print(f"dL = {dL_um:>2} um -> V = {V:.3f}")
# dL =  0 um -> V = 1.000
# dL = 10 um -> V = 0.675
# dL = 20 um -> V = 0.208
# dL = 40 um -> V = 0.002  (fringes gone)

Where coherence matters

• Interferometry & gravitational waves. LIGO splits a stabilized 1064 nm laser, recombines the arms, and reads displacements of ~10⁻¹⁸ m from fringe shifts. Sub-hertz laser linewidth — kilometer-plus coherence length — is essential so the recombined beams still interfere after 4 km folded paths.
• Holography. Recording a hologram requires the reference and object beams to interfere stably across the whole plate; the object's depth must stay within the laser's coherence length, or the fringes (which encode the 3D information) blur.
• Optical coherence tomography (OCT). Here low coherence is the feature. A broadband superluminescent diode (L_c ≈ 10 µm) means only reflections from one thin depth slice interfere with the reference arm — depth gating that images the retina layer by layer at micron resolution.
• Telecom & coherent detection. Coherent optical receivers mix the signal with a narrow-linewidth local-oscillator laser to recover amplitude and phase, packing more bits per symbol; phase noise (finite coherence) limits the constellation density.
• Lidar & Doppler velocimetry. Coherent (heterodyne) lidar beats reflected light against a reference to read velocity from the Doppler-shifted beat frequency — only possible within the laser's coherence length.
• Stellar interferometry. Combining starlight from separated telescopes and measuring the loss of spatial-coherence fringes yields stellar diameters far below single-telescope resolution.

Common mistakes

• Conflating coherence with monochromaticity. They are linked but distinct. Monochromatic refers to a narrow spectrum; coherence refers to phase predictability. High temporal coherence implies a narrow spectrum, but a wave can be temporally incoherent yet still partly spatially coherent (sunlight) or vice versa.
• Forgetting spatial coherence. Many students compute L_c from bandwidth and assume fringes will appear. If the source is spatially incoherent (large and nearby), Young's fringes still wash out no matter how monochromatic the light is.
• Treating coherence as an on/off property. Real light is partially coherent: 0 < |g(τ)| < 1. Fringe visibility degrades smoothly, it doesn't switch off at a hard edge.
• Mismatching the prefactor and line shape. L_c ≈ λ²/Δλ is an order-of-magnitude rule. The exact factor (and the shape of the visibility decay — exponential for a Lorentzian, Gaussian for a Gaussian line) depends on the spectrum.
• Ignoring multimode structure. A multimode laser has several discrete frequencies; its visibility revives periodically at path differences equal to multiples of the cavity length, instead of decaying monotonically.
• Assuming a longer laser cavity is always more coherent. Coherence is set by linewidth, not cavity length directly; stabilization (locking the frequency) is what extends coherence, not raw power or size.