Coherence and path length

Why coherence matters

Interference requires waves to maintain a predictable phase relationship. If phase varies randomly too quickly, bright and dark fringes average out and disappear. Coherence describes how well a wave maintains phase relationships over time and space.

Lasers are highly coherent compared with ordinary light sources, which is why they produce strong, stable interference patterns.

Temporal coherence

Temporal coherence describes phase correlation at different times along the same beam. A source with a very narrow frequency spread has long temporal coherence. A source with many frequencies has short temporal coherence because different frequencies drift out of phase.

The coherence time $au_c$ is roughly related to frequency bandwidth $Delta f$ by

au_csim rac{1}{Delta f}

The coherence length is

$L_c=c au_c$

in vacuum, or approximately

L_csim rac{c}{Delta f}

This is the path difference over which interference remains visible.

Wavelength bandwidth form

Since frequency and wavelength are related by $c=flambda$, a source with central wavelength $lambda$ and wavelength spread $Deltalambda$ has approximate coherence length

L_csim rac{lambda^2}{Deltalambda}

This shows why highly monochromatic light has long coherence length.

Spatial coherence

Spatial coherence describes phase correlation between different points across a wavefront. A small distant source can have good spatial coherence. A large extended source generally has poor spatial coherence because different parts emit with unrelated phases.

Young's double-slit experiment works best when the slits are illuminated coherently. If the source is too broad, fringe visibility decreases.

Path length difference

In an interferometer, two beams travel different optical paths and recombine. The relevant quantity is optical path length:

$OPL=int n,ds$

For a uniform medium,

$OPL=ns$

Interference depends on optical path difference, not just geometric length difference.

The phase difference is

Deltaphi=rac{2pi}{lambda_0}Delta(OPL)

where $lambda_0$ is vacuum wavelength.

Fringe visibility

Fringe visibility describes contrast between bright and dark fringes:

mathcal{V}=rac{I_{max}-I_{min}}{I_{max}+I_{min}}

Perfect visibility has $mathcal{V}=1$. If fringes wash out, $mathcal{V}=0$.

Visibility decreases when beams have unequal intensities, poor coherence, mixed polarizations, or path differences larger than coherence length.

Coherence and lasers

A laser emits light with narrow frequency bandwidth, strong directionality, and high spatial coherence. This makes lasers excellent for interferometry, holography, precision measurement, and communication.

However, no real laser is perfectly coherent. Linewidth, noise, and mode structure limit coherence length.

The big idea

Coherence measures whether waves maintain phase relationships well enough to interfere visibly. Temporal coherence is tied to frequency bandwidth and coherence length; spatial coherence is tied to source size and wavefront correlation. Optical path length determines phase, and fringe visibility quantifies interference quality.