Michelson Interferometer Fringe Shift

Equivalent forms

\Delta d = \frac{\Delta N \,\lambda}{2}

\Delta N \,\lambda = 2\,\Delta d

It converts an invisible nanometer displacement into a countable number of light pulses — measurement by tally marks.

Symbol	Name	SI unit	Dimension	Range
$delta_d$	Mirror displacement Distance one interferometer mirror is translated	m	L	0 – 5000
$lambda$	Wavelength Vacuum wavelength of the source light	m	L	400 – 700
$delta_N$	Fringe shift countoutput Number of fringes that cross the detector	dimensionless	1	0 – 25

Unit systems

Symbol	SI	CGS	Imperial
$delta_d$	m	cm	in
$lambda$	m	cm	in
$delta_N$	dimensionless	dimensionless	dimensionless

Where it holds

Valid for a Michelson interferometer in vacuum or air (index

\sim 1)

with a monochromatic, well-collimated source. In a medium of index n the optical path becomes 2*n*Delta d; broadband sources blur the count beyond the coherence length.

Discovery

Albert A. Michelson · 1887

Michelson built his interferometer to hunt for the luminiferous aether; the famous null result with Edward Morley helped overturn aether theory. The same instrument let Michelson measure lengths in wavelengths of light, earning him the 1907 Nobel Prize.

Try this

How can moving a tiny mirror by a fraction of a hair's width be measured by simply counting flickers of light?

In a Michelson interferometer using 633 nm laser light, one mirror is translated by 633 nm. How many bright-fringe shifts cross the detector?

Research status: stable

Real-world applications

LIGO uses kilometer-scale Michelson interferometers to detect gravitational-wave strains of 10^-21.
Fourier-transform infrared (FTIR) spectrometers scan a mirror and Fourier-analyze the fringe record.
Precision metrology calibrates displacement stages and gauge blocks in wavelengths of light.
Optical coherence tomography images tissue by interferometric depth scanning.

Common misconceptions

Moving the mirror by one wavelength gives one fringe — it gives two, because the path change is doubled by the round trip.
The fringe count depends on the source distance — only on the mirror displacement and wavelength.
White light works as well as a laser — limited coherence length restricts white-light fringes to near-zero path difference.

Experimental verification

Mounting a mirror on a calibrated piezo stage and counting fringes against the known 633 nm HeNe wavelength reproduces displacements to nanometer accuracy; conversely, counting fringes for a known displacement measures the wavelength, as Michelson did for the meter.

Derivation

The beam in the moving arm travels to the mirror and back, so a mirror shift Delta d changes its optical path by 2*Delta d.

The interference condition advances by one fringe each time this path change equals one wavelength:

2*\Delta d = \Delta N * \lambda

, hence

\Delta N = 2*\Delta d / \lambda

.

Limiting cases

\

\Delta d =

\lambda/2⟶ \

\Delta N = 1

A half-wavelength mirror move changes the round-trip path by one full wavelength — exactly one fringe.

\Delta d \to 0⟶ \Delta N \to 0No displacement means no path change and a stationary fringe pattern.

What if…

What if you fill the moving arm with a gas of index n?

The optical path becomes 2*n*Delta d, so changing the gas pressure (and thus n) shifts fringes even with a fixed mirror — this is how interferometric refractometers measure index.

What if you switch from red (633 nm) to blue (450 nm) light?

For the same displacement Delta N increases (more, finer fringes), giving higher displacement resolution per fringe.

1

Counting fringes for a known move

Given ·

delta d:: 633
lambda:: 633

Find · delta_N

Steps

Apply $\Delta N = 2*\Delta d/\lambda$ .
Substitute $\Delta d = 633\,\mathrm{nm}$ , $\lambda = 633\,\mathrm{nm}$ .
$\Delta N = 2*633/633 = 2$ fringes.

Result ·

\Delta N = 2*\Delta d/\lambda = 2*633/633 = 2

. The mirror moved one wavelength, so two fringes cross the detector.

Young's Double Slit Interference→Thin-Film Interference→Wave Equation→