The Michelson interferometer is one of the most important instruments in wave optics. It splits a beam of light into two perpendicular paths, reflects each beam back from a mirror, and then recombines them. Because the two returning beams travel slightly different optical paths, they interfere. The resulting fringe pattern is extremely sensitive to small changes in mirror position, wavelength, or refractive index.
The key idea behind this calculator is very simple: if one Michelson mirror moves by a distance \(\Delta L\), the light in that arm travels an extra distance on the way to the mirror and the same extra distance again on the way back. So the total optical path change is
\[
2\Delta L.
\]
Each time the optical path changes by one full wavelength, the interference pattern shifts by one fringe. That is why the fringe shift is
\[
\Delta m = \frac{2\Delta L}{\lambda}.
\]
Here \(\lambda\) is the wavelength of the light. If \(\Delta m=1\), then one complete fringe passes the reference point. If \(\Delta m=0.5\), then the pattern shifts by half a fringe, which often changes a bright center into a dark one or vice versa.
This formula is one of the reasons Michelson interferometers are so powerful: a mechanical displacement much smaller than a millimeter can produce an easily observable fringe count. In other words, interference converts tiny physical motion into a measurable optical signal.
The corresponding phase change is
\[
\Delta \phi = \frac{4\pi \Delta L}{\lambda}.
\]
This is the same information as the fringe shift formula, because
\[
\Delta \phi = 2\pi \Delta m.
\]
In the calculator, the detector preview uses a normalized two-beam interference model with visibility \(V\):
\[
\frac{I}{I_0}=\frac12\left(1+V\cos\Delta\phi\right).
\]
When \(V=1\), the fringes have maximum contrast. Bright fringes can reach the full reference intensity and dark fringes can fall to zero. When \(V<1\), the contrast is reduced, which is what happens in real experiments when the beams are not perfectly balanced or not perfectly coherent.
For the sample input
\(\Delta L=0.3164\ \mu\text{m}\)
and
\(\lambda=632.8\ \text{nm}\),
first convert the displacement:
\[
\Delta L = 0.3164\times10^{-6}\ \text{m}.
\]
Then the round-trip path change is
\[
2\Delta L = 0.6328\times10^{-6}\ \text{m}.
\]
Since
\[
\lambda = 632.8\times10^{-9}\ \text{m}
= 0.6328\times10^{-6}\ \text{m},
\]
the fringe shift is
\[
\Delta m = \frac{2\Delta L}{\lambda} = 1.
\]
So exactly one fringe crosses the reference point. This is the example result.
The animation in the calculator shows the classic Michelson layout: a source beam reaches the beam splitter, one part goes to a fixed mirror, another part goes to a movable mirror, and both return to the detector. The right-side fringe panel then shows how the interference pattern translates as the movable mirror changes position.
Historically, the Michelson interferometer became famous for extremely precise measurements. It was used in the Michelson–Morley experiment, in wavelength standards, in refractive-index measurements, and in early precision determinations related to the speed of light. In modern physics, interferometric ideas appear everywhere from spectroscopy to gravitational-wave detection.
At a more advanced level, the same fringe-shift logic can be generalized to include a medium of refractive index \(n\), where optical path is \(nL\) instead of just \(L\). That leads to more refined formulas for gases, liquids, or pressure-dependent cells placed in one arm of the interferometer. But the fundamental geometric idea remains unchanged:
\[
\text{mirror motion changes optical path twice, once going out and once returning.}
\]
So the most important formulas to remember are
\[
\Delta m = \frac{2\Delta L}{\lambda},
\qquad
\Delta \phi = \frac{4\pi \Delta L}{\lambda},
\]
because they connect tiny mirror displacements directly to visible fringe motion.
