Coherence

When two light waves overlap, they can reinforce each other to produce bright regions or cancel each other to produce dark regions — a phenomenon called interference. However, interference patterns are not always visible. For a stable, sustained interference pattern to appear, the two sources of light must satisfy specific conditions. These conditions are collectively described by the concept of coherence. Coherence is essentially a measure of how well two waves maintain a consistent relationship with each other over time and space. Think of two musicians playing the same note on their instruments. If they stay perfectly in time, the combined sound is louder and consistent — that is the analogy of constructive interference from coherent sources. If one musician gradually drifts in tempo, the loudness fluctuates unpredictably and no stable pattern emerges. Light from two independent bulbs behaves like two out-of-sync musicians: their phase relationship changes randomly and far too quickly for any sustained bright or dark fringe to form on a screen. There are two distinct aspects of coherence: temporal coherence, which relates to how monochromatic (single-wavelength) a source is, and spatial coherence, which relates to how correlated the phases are across the width of a wavefront. Both aspects must be sufficiently satisfied for clear interference fringes to be observed in practice. Understanding coherence explains why lasers produce vivid fringes while ordinary light sources require careful preparation to do the same.

Key takeaways

Phase Relationship
Temporal Coherence and Coherence Length
Spatial Coherence
Why Independent Sources Cannot Interfere
Visibility of Fringes

Phase Relationship

Two waves are said to be coherent if they maintain a constant phase difference over time. The phase difference Δφ between them may be zero (in phase) or any fixed value, but it must not change randomly. If Δφ fluctuates, the interference pattern shifts continuously and averages out, making fringes invisible. This is why a fixed phase relationship is the fundamental requirement for sustained interference.

Temporal Coherence and Coherence Length

Temporal coherence describes how consistent a wave's phase is with itself over time. A perfectly monochromatic wave (single frequency) has infinite temporal coherence. Real sources emit light over a small range of frequencies Δf, called the linewidth. The coherence time τ_c ≈ 1/Δf is the duration over which the wave remains predictably sinusoidal. Multiplied by the speed of light, this gives the coherence length l_c = c/Δf — the maximum path difference between two interfering beams that still yields visible fringes. A sodium lamp has a coherence length of roughly 0.6 mm; a laser can have a coherence length of many metres or more.

Spatial Coherence

Spatial coherence describes how correlated the phases are at different points across a wavefront at the same instant. A perfect point source produces a spherical wavefront where every point on the wavefront has a definite phase relationship with every other point — high spatial coherence. An extended source (like a large filament) is like many independent point sources; different parts of the wavefront have random phase differences relative to each other. This reduces the visibility (contrast) of interference fringes unless the source is made effectively small, as Thomas Young did by first passing light through a single narrow slit.

Why Independent Sources Cannot Interfere

Two separate light sources — even two identical lasers — cannot produce a stable interference pattern directly. Atoms emit light in random, short bursts lasting about 10⁻⁸ s. The phase of each burst is unrelated to the previous one. Because the phase difference between two independent sources changes millions of times per second, any fringe pattern shifts far faster than any detector (including the human eye) can follow, and only a uniform average brightness is seen. Coherent sources must therefore be derived from a single source, as in Young's double-slit experiment.

Visibility of Fringes

The quality of an interference pattern is quantified by the fringe visibility V = (I_max − I_min)/(I_max + I_min), where I_max and I_min are the maximum and minimum intensities of the fringes. For perfectly coherent sources of equal amplitude, V = 1 (complete contrast: bright fringes are fully bright, dark fringes are truly dark). As coherence decreases, V falls towards zero, meaning the fringes wash out into uniform illumination. Coherence therefore directly controls how observable the interference pattern is.

The Key Equation

l_c = \frac{c}{\Delta f} = c\,\tau_c

l_c — Coherence length (m): The maximum path difference between two beams for which interference fringes remain visible.
c — Speed of light (m s⁻¹): The speed of light in a vacuum, approximately 3 × 10⁸ m s⁻¹.
\Delta f — Frequency linewidth (Hz): The range of frequencies emitted by the source; a narrower linewidth means greater temporal coherence.
\tau_c — Coherence time (s): The time interval over which the wave maintains a predictable phase; τ_c = 1/Δf.

Worked Example

Problem. A helium–neon laser emits light at a wavelength of 632.8 nm with a frequency linewidth of Δf = 1.5 × 10⁹ Hz. (a) Calculate the coherence time of the laser. (b) Calculate its coherence length. (c) State whether this laser could produce visible fringes with a path difference of 15 cm between the two beams.

\tau_c \approx 6.67 \times 10^{-10}\,\text{s},\quad l_c \approx 0.20\,\text{m}

Answer. Coherence time ≈ 0.67 ns; coherence length ≈ 20 cm; fringes are visible at a 15 cm path difference because 15 cm < 20 cm.

Figure

Diagram

Origins · 1 minAugustin-Jean Fresnel and how this was discovered

Augustin-Jean Fresnel

1788–1827 · French

Fresnel made foundational contributions to the wave theory of light, developing the mathematical framework for diffraction and interference that underpins our modern understanding of coherence. His analysis of how path differences between wavefronts determine fringe visibility directly informs the concept of coherence length. His 'Fresnel biprism' experiment (1816) was a key demonstration that two coherent beams derived from a single source produce sustained interference, reinforcing the conditions required for stable fringes. Fresnel was originally trained as a civil engineer and only turned to optics in his late twenties. Despite dying at just 39 from tuberculosis, his work so thoroughly established the wave theory of light that the rival corpuscular (particle) theory was effectively abandoned within years of his publications.

Young's Double-Slit Experiment — Young's experiment is the classic demonstration of interference using two coherent sources derived from a single slit, making coherence directly observable through fringe patterns.
Path Difference and Fringe Patterns — Path difference determines whether interference is constructive or destructive; coherence length sets the maximum path difference over which fringes remain visible.
Superposition of Waves — Interference arises from the principle of superposition; coherence governs whether the superposition produces a stable, observable pattern.
Diffraction — Diffraction spreads waves through apertures and around obstacles, and spatial coherence of the incoming wavefront determines the sharpness of the resulting diffraction pattern.