Interference fringes are easiest to observe when the light is nearly monochromatic. Real sources, however, usually contain a small spread of wavelengths rather than a single perfectly defined value. Because different spectral components accumulate phase at slightly different rates, the interference contrast gradually decreases as the path difference becomes larger. This is the physical idea behind coherence length.
A common estimate for temporal coherence length is
\[
l_c \approx \frac{\lambda^2}{\Delta\lambda},
\]
where \(\lambda\) is the central wavelength and \(\Delta\lambda\) is the spectral bandwidth. This formula is widely used as a first approximation for quasi-monochromatic light. It shows two important trends immediately:
\[
l_c \propto \lambda^2
\qquad\text{and}\qquad
l_c \propto \frac{1}{\Delta\lambda}.
\]
So narrower bandwidth means longer coherence length, while broader bandwidth means shorter coherence length. That is why laser light usually shows much stronger interference over long path differences than broadband white light.
For the sample input
\(\lambda=550\ \text{nm}\)
and
\(\Delta\lambda=1\ \text{nm}\),
convert both to SI units:
\[
\lambda=550\times10^{-9}\ \text{m},
\qquad
\Delta\lambda=1\times10^{-9}\ \text{m}.
\]
Then
\[
l_c \approx \frac{(550\times10^{-9})^2}{1\times10^{-9}}
= 3.025\times10^{-4}\ \text{m}.
\]
Therefore
\[
l_c \approx 0.3025\ \text{mm},
\]
which is about \(0.3\ \text{mm}\), matching the sample output.
Coherence length is related to coherence time through the speed of light:
\[
\tau_c \approx \frac{l_c}{c}.
\]
This gives the approximate time duration over which the light field remains phase-correlated. In practical interferometry, coherence length is especially useful because it tells us roughly how closely the two arm lengths of an interferometer must match if we want to see clear fringes.
The calculator also includes a simple educational visibility model. In real sources, the exact visibility envelope depends on the spectral shape. For example, a Gaussian spectrum produces a Gaussian coherence envelope, while other line shapes give different formulas. To keep the tool easy to use, the preview assumes
\[
V(\Delta x)=e^{-(\Delta x/l_c)^2},
\]
where \(\Delta x\) is the path difference. This captures the main idea: when the path difference is much smaller than the coherence length, visibility stays high; when the path difference becomes comparable to or larger than the coherence length, the fringe contrast drops strongly.
The phase difference associated with the path difference is
\[
\Delta\phi = \frac{2\pi\Delta x}{\lambda}.
\]
If two equal-amplitude beams interfere, an educational normalized intensity model is
\[
\frac{I}{I_0}=\frac12\left(1+V\cos\Delta\phi\right).
\]
Here the cosine term describes the rapid bright–dark oscillation, while the visibility factor \(V\) controls how strong those oscillations remain. In other words, the phase sets where the fringe is, and the coherence envelope sets how clearly it can still be seen.
The animation in this calculator shows two delayed wave packets approaching the detector. When the packets overlap strongly, the detector panel shows high contrast. When the delay is large enough that the packets overlap poorly, the interference contrast fades. This gives an intuitive picture of why limited bandwidth reduces visible fringe contrast.
At a more advanced university level, coherence is treated using correlation functions, spectral power density, and temporal coherence theory. One then distinguishes more carefully between coherence length, coherence time, phase coherence, and spatial coherence. But the simple estimate
\[
l_c \approx \frac{\lambda^2}{\Delta\lambda}
\]
remains a very useful first tool for laboratory optics. It helps answer a practical question quickly:
for a source with a given bandwidth, how much path mismatch can I tolerate before fringes fade away?
So the main formulas to remember are
\[
l_c \approx \frac{\lambda^2}{\Delta\lambda},
\qquad
\tau_c \approx \frac{l_c}{c},
\qquad
V(\Delta x)\approx e^{-(\Delta x/l_c)^2},
\]
which together connect source bandwidth, path difference, and interference visibility.
