Coherence length

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

where $n$ is the refractive index of the medium of propagation. The associated coherence time is ${\tau }_c={\mathrm{\ell }}_c\mathrm{/}(c\mathrm{/}n)=1\mathrm{/}\mathrm{\Delta }\nu$.

The simple form $c\mathrm{/}\mathrm{\Delta }\nu$ is a convention. Mandel's definition, the integral of $\mathrm{\mid }{g}^{(1)}(\tau ){\mathrm{\mid }}^2$ over all delays (see temporal coherence), gives ${\tau }_c=1\mathrm{/}(\pi \mathrm{\Delta }\nu )\approx 0.318\mathrm{/}\mathrm{\Delta }\nu$ for a Lorentzian line, typical of stable single-frequency lasers, and ${\tau }_c=\sqrt{2\ln 2\mathrm{/}\pi }\mathrm{/}\mathrm{\Delta }\nu \approx 0.664\mathrm{/}\mathrm{\Delta }\nu$ for a Gaussian line, with ${\mathrm{\ell }}_c=c{\tau }_c\mathrm{/}n$ in each case. For broadband Gaussian sources, optical coherence tomography quotes $(2\ln 2\mathrm{/}\pi ){\lambda }_0^2\mathrm{/}\mathrm{\Delta }\lambda \approx 0.44{\lambda }_0^2\mathrm{/}\mathrm{\Delta }\lambda$, the axial resolution in air. The coherence length calculator evaluates both conventions for Lorentzian, Gaussian and flat-top spectra and plots fringe visibility against path difference.

Typical coherence lengths at 1550 nm in air, from ${\mathrm{\ell }}_c=c\mathrm{/}\mathrm{\Delta }\nu$:

Operational meaning. Two beams from the same source can produce interference fringes only if their path-length difference is less than ${\mathrm{\ell }}_c$. For OCT imaging (deliberately low-coherence), this sets axial resolution. For interferometric sensing, ${\mathrm{\ell }}_c$ must exceed the path-length difference between the interferometer arms. Coherent communication has a different requirement: the receiver's local oscillator and carrier-recovery DSP track phase over a block of symbols, so what matters is the combined transmitter and local-oscillator linewidth relative to the symbol rate, not a coherence length compared with the link length.

Distinction from $1\mathrm{/}e$ vs $1\mathrm{/}{e}^2$ vs FWHM definitions. Authors variously define ${\mathrm{\ell }}_c$ from full-width-half-maximum, $1\mathrm{/}e$, or $1\mathrm{/}{e}^2$ 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), the deliberately short coherence length provides the depth gating that enables tomographic imaging. For coherent communication, a narrow linewidth (DFB and beyond) is what enables phase-sensitive demodulation.