In one line
Coherence time is the time for which a light wave keeps a fixed, predictable phase, and coherence length is the distance the wave travels in that time.
Explained simply
Think of a band playing in perfect rhythm. A good band stays in rhythm for the whole song; a beginner band loses rhythm after a few seconds. Coherence time is how long the "rhythm" of a light wave lasts before its phase jumps randomly; coherence length is how far the light has moved during that time.
Ordinary sources emit short, unrelated wave trains, so their phase changes randomly very often. A laser emits very long, unbroken wave trains, so it has long coherence time and length. Interference is possible only when path difference is smaller than the coherence length.
How it works
- Light from an atom comes as a wave train of finite duration \(\tau\) (coherence time).
- In that time it travels \(L = c\tau\) (coherence length).
- A finite wave train contains a spread of frequencies \(\Delta\nu \approx 1/\tau\). So a narrow linewidth means a long coherence time.
- Using \(\nu = c/\lambda\), \(\Delta\nu = c\,\Delta\lambda/\lambda^2\), so \(L \approx \lambda^2/\Delta\lambda\).
- Temporal coherence relates to \(\tau\) and \(L\); spatial coherence relates to the phase relationship across the beam's cross-section.
Key formula
\(L = c\,\tau\) and \(\tau \approx \dfrac{1}{\Delta\nu}\), so \(L \approx \dfrac{c}{\Delta\nu} = \dfrac{\lambda^2}{\Delta\lambda}\). Here \(L\) is coherence length (m), \(\tau\) coherence time (s), \(c = 3\times10^8\) m/s, \(\Delta\nu\) the linewidth in Hz, \(\lambda\) the wavelength and \(\Delta\lambda\) the wavelength spread (both in m).
Quick example
A laser has linewidth \(\Delta\nu = 1\) MHz. Then \(\tau = 1/10^6 = 10^{-6}\) s and \(L = 3\times10^8 \times 10^{-6} = 300\) m. A source with \(\lambda = 600\) nm and \(\Delta\lambda = 0.06\) nm gives \(L = (600\times10^{-9})^2/(0.06\times10^{-9}) = 6\times10^{-3}\) m = 6 mm.
Common mistakes
- Using \(L = \tau/c\) instead of \(L = c\tau\).
- Mixing up spatial and temporal coherence.
- Forgetting to convert nm to m when using \(\lambda^2/\Delta\lambda\).
How MU asks it
Asked as "Define coherence length and coherence time and give the relation" (3 marks) or as a numerical: "Find coherence length for a laser of linewidth ... " (3 to 5 marks). Write both definitions, \(L = c\tau\), \(\tau = 1/\Delta\nu\), then substitute with units.