6 - First-order temporal coherence in classical optics

Summary

A truly monochromatic beam of light, if it ever existed, would be perfectly coherent. Suppose that such a beam is split into two parts and each part propagated over an arbitrary distance. When the parts are finally brought together and mixed, no matter how different the two path lengths may have been, the resulting waveform will exhibit constructive and destructive interference in the form of bright and dark fringes. The coherence length of a monochromatic beam is therefore infinite, in the sense that the path-length difference can be as large as desired without hampering one's ability to create interference patterns.

Real sources of light, of course, are never monochromatic. white light restricted to the visible range of wavelengths from 400 nm to 700 nm, for example, has a coherence length of only a couple of micrometers. A green filter passing sunlight at λ₀ = 550 nm with a 10 nm bandwidth produces a beam with a coherence length of about 50 μm. The red line of cadmium (λ₀ = 643.8 nm) has a nearly Gaussian spectrum with a 0.0013 nm width at half peak intensity, leading to a coherence length of nearly 30 cm. This is similar to the coherence length of a short, inexpensive HeNe laser (λ₀ = 632.8 nm) with a few longitudinal modes and a typical bandwidth of Δf ≈ 1 GHz. A stabilized HeNe laser operating in a single longitudinal mode (Δf ≈ 100 kHz) has a coherence length of several kilometers.