# What is the meaning of this statement in Optics, Born-&-Wolf regarding coherence length?

In their treatise "Principles of Optics", Max Born and Emil Wolf state in the section on coherence length (page 317, para 1) that:

...... as the difference of optical path is increased, the visibility of fringes decreases (in general not monotonically), and they eventually, disappear. We can account for this disappearance of the fringes by supposing that the light of the spectral line is not strictly monochromatic, but is made of wavetrains of finite length ...........

I really can't understand what is the meaning of the emphasized line, does it refer of a light pulse of finite length, similar to a pulse going down a string (like the ones we see in video games!) or the width of the spectral line? Please enlighten me!

... by supposing that the light of the spectral line is not strictly monochromatic, but is made of wavetrains of finite length ...

What the authors mean is this.
The wave actually is not an ideal sinusoidal wave of infinite length, like this:

Instead, it is a superposition of wave packets, each

• having the same wavelength $$\lambda$$,
• a finite packet length $$L$$,
• and (most importantly) a random position with respect to the other packets.

(The actual shape of the wave packets is not important. I choose Gaussian packets here for convenience. But it could be any other shape of width $$L$$.)

The result of this feature is that the phase difference between two points of the wave becomes random when they are separated by a distance larger than $$L$$ (the so-called coherence length).

... does it refer of a light pulse of finite length, similar to a pulse going down a string (like the ones we see in video games!) or the width of the spectral line?

As explained above it refers to light pulses of finite length. But the coherence length $$L$$ is also related to the width $$\Delta\lambda$$ of the spectral line by $$L\approx \frac{\lambda^2}{\Delta\lambda}$$

• CORRECTION: Wave packets never have a wavelength, but always consist of infinitely many wavelengths. Which brings us to an interesting new angle of OP's question: "Wavetrains", as dealt with in Born/Wolf, are never monochromatic and therefore interference fringes will always disappear for sufficiently large optical path length differences! And since all atomic transitions have finite time durations, all emission spectra are polychromatic and, therefore, of finite coherence. Commented Mar 7, 2023 at 23:30
• This is also clear from the subsequent paragraphs in Born/Wolf, e.g. below expression (102): "[The observed intensity] is proportional to the integral of the intensities $i(\nu) = |f(\nu)|^2$ (incoherent superposition) of the monochromatic components of which a single wave train is made up." Commented Mar 7, 2023 at 23:47
• Consequently, the limiting factor here isn't the coherence time/length (which is related to the spectrum, see Wiener-Khinchin theorem), but rather the extremely short duration of said wavetrains, resulting in the superposition of wave packets of random "phases" (better: arrival times) originating from uncorrelated emission events. The term "coherence length" here is only used as an analogue to the case of continuous waves. Commented Mar 7, 2023 at 23:51
• This is the actual explanation of the mechanism by which you get a reduction in fringe visibility given by Born/Wolf (p. 318, 2nd paragraph): "With the interferometer, each monochromatic component produces an interference pattern [...], and as the path difference is increased from zero, these component patterns show increasing mutual displacement because of the difference of wavelength. The visibility of the fringes therefore decreases, and they disappear altogether when the optical path difference is sufficiently large." Commented Mar 8, 2023 at 0:00
• Important: The "difference of wavelength" here refers to the individual Fourier components of each individual wave packet! Each Fourier component individually interferes with its own copy upon traveling along both interferometer arms and produces a corresponding fringe pattern (Born/Wolf: "each monochromatic component produces an interference pattern..." and "the longer the wave trains, the narrower the frequency range over which the Fourier components have appreciable intensity"). ... Commented Mar 8, 2023 at 0:09