Fringe visibility measures how clear an interference pattern looks. If the bright fringes are much brighter than the dark fringes, the visibility is high. If the whole pattern looks washed out and the maxima and minima are only slightly different, the visibility is low. The standard definition is
\[
V=\frac{I_{\max}-I_{\min}}{I_{\max}+I_{\min}}.
\]
Here \(I_{\max}\) is the maximum intensity of a fringe and \(I_{\min}\) is the minimum intensity. This formula gives a dimensionless number between 0 and 1. If \(V=1\), the contrast is perfect: the minima drop all the way to zero. If \(V=0\), there is no visible fringe contrast at all.
Visibility is closely related to coherence. When two interfering beams remain phase-related over the relevant path difference, the fringes stay sharp. When the path difference becomes too large compared with the coherence length, the phase relation becomes uncertain and the interference contrast decreases.
In this calculator, the coherence length \(l_c\) is assumed to be known already, and the path difference \(\Delta x\) is used to estimate the visibility. Many real sources can have different envelope shapes depending on their spectrum, so there is no single universal formula for all experiments. For this educational preview, the tool uses a simple exponential decay model:
\[
V = e^{-2\ln 2\, |\Delta x|/l_c}.
\]
This choice has a convenient interpretation:
when the path difference equals half the coherence length, the visibility becomes
\[
V=e^{-\ln 2}=0.5.
\]
That matches the sample input and makes the calculator easy to interpret visually.
For the sample values
\(l_c=0.3\ \text{mm}\)
and
\(\Delta x=0.15\ \text{mm}\),
the normalized path-difference ratio is
\[
\frac{|\Delta x|}{l_c} = \frac{0.15}{0.3} = 0.5.
\]
Substituting into the model gives
\[
V = e^{-2\ln 2 \cdot 0.5}
= e^{-\ln 2}
= 0.5.
\]
So the calculator returns a visibility of about \(0.5\), just as in the example.
If an average interference intensity \(I_0\) is used, then the maximum and minimum intensities can be reconstructed from the visibility:
\[
I_{\max}=I_0(1+V),
\qquad
I_{\min}=I_0(1-V).
\]
These formulas are easy to verify. Substituting them back into the visibility definition gives
\[
\frac{I_{\max}-I_{\min}}{I_{\max}+I_{\min}}
=
\frac{I_0(1+V)-I_0(1-V)}{I_0(1+V)+I_0(1-V)}
=
V.
\]
This is why the calculator can report not only the visibility, but also the corresponding bright-fringe and dark-fringe intensities.
Physically, you can think of the interfering beams as wave packets. When the path difference is very small, the packets overlap strongly and the fringes are crisp. As the delay increases, the overlap decreases. Less overlap means less stable phase correlation, and therefore smaller contrast. The animation in the tool is designed to show exactly this idea: the detector fringe pattern becomes more washed out as the two packets separate.
At a more advanced university level, visibility is connected to the degree of coherence and to correlation functions. One also distinguishes temporal coherence from spatial coherence. Temporal coherence is mainly about spectral width and path difference, while spatial coherence is about phase correlation across different points of a wavefront. This tool focuses only on the basic temporal-coherence-style interpretation.
So the most important formulas to remember here are
\[
V=\frac{I_{\max}-I_{\min}}{I_{\max}+I_{\min}},
\qquad
V=e^{-2\ln 2\, |\Delta x|/l_c},
\qquad
I_{\max}=I_0(1+V),
\qquad
I_{\min}=I_0(1-V).
\]
Together, these show how coherence length and path difference control interference contrast, and how that contrast directly determines the observed bright and dark fringe intensities.
