A laser is never perfectly monochromatic. Even very narrow-band lasers emit over a small but finite linewidth,
which can be described either as a frequency width \(\Delta\nu\) or as a wavelength width \(\Delta\lambda\).
That finite linewidth limits how long the optical phase remains predictable, and therefore limits the distance over which interference fringes remain clearly visible.
This is why linewidth, coherence time, and coherence length are closely connected ideas.
The simplest coherence estimate starts from the frequency linewidth.
If a source has linewidth \(\Delta\nu\), then a commonly used coherence-time estimate is
\[
\tau_c \approx \frac{1}{\Delta\nu}.
\]
This says that a narrower linewidth corresponds to a longer time over which the optical phase stays correlated.
Once the coherence time is known, the corresponding coherence length is
\[
l_c \approx c\,\tau_c=\frac{c}{\Delta\nu},
\]
where \(c\) is the speed of light.
In words: a source with a narrower linewidth has a longer coherence length, so interference can survive for larger path differences in an interferometer.
Sometimes the linewidth is specified in wavelength units instead of frequency units.
For narrow linewidths around a central wavelength \(\lambda\), one uses the differential conversion
\[
\Delta\nu \approx \frac{c\,\Delta\lambda}{\lambda^2}.
\]
Combining this with
\(l_c=c/\Delta\nu\)
gives
\[
l_c \approx \frac{\lambda^2}{\Delta\lambda}.
\]
This form is often convenient when a spectroscopic linewidth is given directly in nanometers.
It also shows that for a fixed wavelength, reducing \(\Delta\lambda\) by a factor of ten increases the coherence length by a factor of ten.
The sample case here is a HeNe laser with
\(\lambda=632.8\,\text{nm}\)
and
\(\Delta\lambda=0.001\,\text{nm}\).
Using the narrow-linewidth conversion,
\[
\Delta\nu \approx \frac{(3.00\times10^8)(1.0\times10^{-12})}{(632.8\times10^{-9})^2}
\approx 7.49\times10^8\,\text{Hz}.
\]
So the linewidth is about
\(749\,\text{MHz}\).
Then the coherence time is
\[
\tau_c \approx \frac{1}{7.49\times10^8}\approx 1.34\times10^{-9}\,\text{s},
\]
and the coherence length is
\[
l_c \approx \frac{3.00\times10^8}{7.49\times10^8}\approx 0.40\,\text{m}.
\]
So for these sample values the coherence length is about
\(0.40\) m,
not hundreds of meters.
That distinction matters because coherence length depends very strongly on the exact linewidth.
Another useful quantity is the optical quality factor,
\[
Q=\frac{\nu_0}{\Delta\nu},
\]
where
\(\nu_0=c/\lambda\)
is the central optical frequency.
A larger
\(Q\)
corresponds to a more sharply defined optical resonance or spectral line.
In interferometry, the fringe contrast usually decreases as the path difference grows.
A simple envelope model often used for intuition is
\[
V(\Delta L)\approx e^{-|\Delta L|/l_c},
\]
where \(V\) is fringe visibility and \(\Delta L\) is the path difference.
This is not the only possible model, because the exact coherence envelope depends on the line shape.
For example, Gaussian and Lorentzian spectra give different detailed visibility functions.
Still, the exponential form is a very convenient way to connect linewidth and interferometer behavior qualitatively.
At a more advanced university level, linewidth can be related to spontaneous-emission noise, cavity losses, and the Schawlow-Townes limit.
One can also distinguish between instantaneous linewidth, technical broadening, mode hopping, and long-term drift.
But for a basic estimator, the central chain of ideas is straightforward:
linewidth determines coherence time, and coherence time determines coherence length.
\[
\tau_c \approx \frac{1}{\Delta\nu},
\qquad
l_c \approx \frac{c}{\Delta\nu},
\qquad
\Delta\nu \approx \frac{c\,\Delta\lambda}{\lambda^2}.
\]
These formulas let you move back and forth between linewidth and coherence, and they explain why narrow-line lasers are so valuable in interferometry, spectroscopy, and precision metrology.
