Coherence length

Coherence length quantifies the propagation distance over which an optical wave remains phase-correlated with itself. For a source with spectral width $\Delta\nu$ (FWHM) and corresponding wavelength bandwidth $\Delta\lambda$:

where $n$ is the refractive index of the medium of propagation. The associated coherence time is $\tau_c = \ell_c / (c/n) = 1/\Delta\nu$.

For Gaussian-distributed spectra, the relation has a $\sqrt{\pi \ln 2}$ factor that is often absorbed into convention; for Lorentzian spectra (typical for stable lasers), $\ell_c = c / (\pi n \Delta\nu)$.

Typical coherence lengths at 1550 nm in air:

Operational meaning. Two beams from the same source can produce interference fringes only if their path-length difference is less than $\ell_c$. For OCT imaging (deliberately low-coherence), this sets axial resolution. For interferometric sensing or coherent communication, $\ell_c$ must exceed the round-trip path of the interferometer or unrepeated link.

Distinction from $1/e$ vs $1/e^2$ vs FWHM definitions. Authors variously define $\ell_c$ from full-width-half-maximum, $1/e$, or $1/e^2$ of the intensity autocorrelation. The functional form depends on the spectral profile (Gaussian, Lorentzian, sinc). The numerical values reported in different sources for the "same" source can disagree by factors of 2–3 from definition differences. For careful work, the autocorrelation function or visibility-vs-delay curve is preferable to a single coherence length number.

For broadband sources used in low-coherence interferometry (LED, SLD, swept laser), the deliberately short coherence length provides the depth gating that enables tomographic imaging. For coherent communication, the long coherence length (DFB and beyond) is what enables phase-sensitive demodulation.