A Michelson interferometer splits light into two perpendicular arms, reflects the beams from two mirrors, and then recombines them to form an interference pattern.
The observed bright and dark fringes depend on the difference in optical path length between the two arms. When one arm becomes longer or shorter, the phase difference
between the returning beams changes, and the fringes move across the observation screen.
The most important quantity is the optical path difference, often written as \(\Delta L\). If the interferometer is in air or vacuum and one mirror is displaced by
a small amount \(d\), the beam travels that extra distance on the way to the mirror and again on the way back. Therefore, the total path change is
\[
\Delta L = 2d.
\]
This round-trip factor of \(2\) is the key reason Michelson fringe calculations differ from simpler one-pass wave problems. The fringe shift is then found by comparing the
path change with half a wavelength. Because a full fringe cycle corresponds to a phase change of \(2\pi\), the number of fringe intervals that pass is
\[
\Delta m = \frac{\Delta L}{\lambda/2} = \frac{2d}{\lambda}.
\]
Here \(\lambda\) is the wavelength of the light. If \(\Delta m\) is an integer, the final pattern has moved by an exact whole number of fringes. If \(\Delta m\) has a fractional part,
then the pattern ends between fringe positions. Bright fringes occur when the recombined beams satisfy a constructive-interference condition, and dark fringes occur when the phase difference
produces destructive interference. In the simplest idealized model, bright fringes line up with integer orders and dark fringes lie halfway between them, at half-integer orders.
Consider the common HeNe laser example with \(\lambda = 632.8\ \text{nm}\). If the optical path difference is \(\Delta L = 1\ \mu\text{m}\), then
\[
\Delta m = \frac{1\times 10^{-6}}{632.8\times 10^{-9}/2} \approx 3.16.
\]
So the pattern shifts by a little more than three fringe intervals. This agrees with the equivalent round-trip formula. If instead you start from a mirror displacement \(d\), you first compute
\(\Delta L = 2d\) and then evaluate the same ratio. For example, a mirror move of \(0.5\ \mu\text{m}\) would also give \(\Delta L = 1\ \mu\text{m}\), producing the same \(\Delta m \approx 3.16\).
This is why it is important to distinguish carefully between mirror displacement and optical path difference. They are not the same quantity. In a Michelson interferometer,
one is twice the other when only one mirror is moved in air or vacuum. Many mistakes in homework and lab work come from forgetting this factor of \(2\).
Michelson interferometers are historically important because they made extremely precise measurements possible. They were central to classic studies of wavelength standards,
refractive index, fine displacement measurement, and historically significant speed-of-light work. Modern interferometric methods still rely on the same idea:
phase is extraordinarily sensitive to very small path changes, so fringe counting can detect motions much smaller than a millimeter, and often much smaller than a micrometer.
This calculator uses the vacuum or air-style relation described above. In more advanced settings, the optical path can also include a refractive index factor \(n\), so the path becomes
\(nL\) rather than just \(L\). That refinement appears in university-level treatments and precision metrology, but the basic fringe-counting logic remains the same:
compare the total optical path change with the effective wavelength scale that controls the interference pattern.