Optics 045 · Interference, coherence, cavities, and metrology

Michelson Precision Lab

How can counting fringes measure displacement, wavelength, and refractive index? Every time the mirror moves half a wavelength, one fringe passes. With a known wavelength, the count gives the distance moved; with a known distance, it gives the wavelength. Filling a gas cell in one arm adds optical path, and the fringes that pass give the index change. A plain count misses the direction of motion; a phase-stepped readout supplies it.

On a 3D Michelson bench, the ray intersections with the mirrors produce straight fringes or equal-inclination rings, fixed-exposure power images and a four-step phase scan. Measure displacement, vacuum wavelength or a double-pass change in gas index, one at a time.

Shared camera probe / output power fractions

u_p,p_-,p_+

$0.50$

Recovered signed scan cycles / isolated unknown

N_{\mathrm{scan}},\widehat{\Delta x},\widehat\lambda_0,\widehat{\Delta n}

$50\%$

Measured modulation / scan sampling check

V_p,\max|\Delta\phi_s|/\pi

$0.00\pi$

Model regime $\text{valid model regime}$

Model checked against analytic results

Michelson Precision Lab

Mirror displacement

\Delta L

0.80\,\mu\mathrm m

Cell index change

n

0\times10^{-6}

Vacuum wavelength (not wavelength in air)

\lambda_0

632.80\,\mathrm{nm}

Actual measuring-mirror tilt (zero is parallel)

\beta

0.060\,\mathrm{mrad}

Parallel virtual-mirror axial separation (ring preset)

d_0

0\,\mathrm{mm}

Gaussian mirror-motion RMS during exposure

\sigma_x

0\,\mathrm{nm}

Shared detector centre-line probe

u_p

0\,\mathrm{mm}

Move the gold mirror: its return path moves with it, and the fringes on the camera shift. Set its tilt to zero for a uniform field. Try the parallel-mirror angular preset for rings and wavelength calibration, then the fixed-mirror gas-cell preset. Four reference-phase steps at each scan point recover the signed change. Add RMS mirror jitter to wash out the contrast. Drag to orbit. Play does not create measurements.

The beam splitter is ideal, scalar and balanced, the dark-port bias is calibrated, the surrounding index is fixed and nondispersive, the beams overlap broadly, and the reference steps are ideal. Gaussian jitter is an ensemble average, not detector noise. Mirror displacement is drawn 4000 times larger than it is. No instrument precision, absolute phase order or low-coherence scan is claimed.

Physics tutorial

Audit the physics of Michelson Precision Lab

BackgroundLight crosses each arm twice, out and back, reflecting once from the end mirror. For parallel beams on the axis, the optical path changes by $\Delta\mathrm{OPL}=2n_0\Delta x+2L_g\Delta n$ . The cyan and gold paths end on the mirrors and meet again at the beamsplitter. The camera image and the acquired scan come from the same phase model, and when the measuring mirror moves, the end of its path moves with it.

Why it mattersHow can counting fringes measure displacement, wavelength, and refractive index?

Start with the essentials

Focus question: How can counting fringes measure displacement, wavelength, and refractive index?
One-sentence intuition: Unwrapping the phase along a scan gives the number of cycles, with its sign: $N=\Delta\phi/(2\pi)$ . A single intensity image looks the same for positive and negative phase and repeats every cycle, so it cannot show the direction of motion or the missing whole-number order. At each scan sample, this experiment adds four calibrated reference phase steps. The scan starts from zero change, and the absolute phase is never given to it.

Core mathematical model

Round trip and air wavelength

\begin{aligned}N&=\frac{2n_0\Delta x+2L_g\Delta n}{\lambda_0}\\\lambda_{\mathrm{air}}&=\lambda_0/n_0\end{aligned}

The source is set by its vacuum wavelength. The surrounding air has a fixed phase index of 1.0002718. The gas cell is 100 mm long, and its index change is measured from the air value. Both cell windows are ideal and do not move. The model does not compute pressure, temperature or the wavelength-dependent Ciddor corrections.

Straight fringes come from a tilted wavefront

\begin{aligned}\Delta_{\mathrm{plane}}(u)&=n_0\bigl[2\Delta x\cos^2\beta\\&\quad-2(L_0+D)\sin^2\beta+u\sin(2\beta)\bigr]\\&\quad+2L_g\Delta n\end{aligned}

The measuring mirror tilts by an angle beta. Unfolding its reflection through the beamsplitter shows the output beam tilted by twice that angle. The phase then has a carrier that varies across the camera, plus a small correction along the beam. The coordinate u runs horizontally across the camera, and the output travels in the positive bench direction. The base arms are 180 mm long, and the camera for this spatial view sits 145 mm from the beamsplitter. With zero tilt the image is uniform. Cross terms between the gas index and the tilt are neglected.

Ring fringes appear in an angular view

\begin{aligned}\phi(\theta)&=\frac{4\pi n_0(d_0+\Delta x)\cos\theta}{\lambda_0}\\\cos\theta&=\frac{f}{\sqrt{f^2+u^2+v^2}}\end{aligned}

The wavelength preset adds ideal extended illumination over a range of angles and a 100 mm observation lens, with the camera in its focal plane 100 mm behind it. The mirrors stay parallel. The separation between one mirror and the virtual image of the other sets how the phase changes with radius. These are equal-inclination rings, where each ring marks one viewing angle; they are not straight fringes drawn as circles. With zero separation and zero displacement, the angular image is uniform. Recovering the wavelength from an off-axis point includes the cosine factor.

Recovering the phase from four phase-stepped images

\begin{aligned}p_{-,k}&=\frac{1-V\cos(\phi+k\pi/2)}2\\\phi_w&=\operatorname{atan2}(p_{-,1}-p_{-,3},\,p_{-,2}-p_{-,0})\\N&=\frac{\phi_u(1)-\phi_u(0)}{2\pi}\end{aligned}

An ideal reference actuator adds phase steps of a quarter cycle. The recovery receives only the four sampled intensity arrays and unwraps the phase from one scan point to the next. Every allowed scan has 257 samples and moves less than half a cycle per step. Measuring displacement requires a known wavelength; measuring wavelength requires a displacement known independently; measuring gas index requires a known wavelength and cell length. With zero displacement the wavelength cannot be calibrated. The absolute whole-number order is not determined.

Vibration washes out contrast but keeps the power

\begin{aligned}V&=\exp\!\left[-\frac12\left(\frac{4\pi n_0 g\sigma_x}{\lambda_0}\right)^2\right]\\p_-+p_+&=1\end{aligned}

The noise is zero-mean Gaussian jitter of the mirror along the axis, averaged over a long exposure. The jitter is projected with a factor of cosine squared in the tilted spatial view and cosine in the parallel angular view. The result is an analytic ensemble average; there is no random animation and no photon or readout noise. Below a contrast of 0.05 the Lab stops reporting a measurement. That cutoff only guards the display; it is not a precision specification. The complementary map is a model prediction, not an image from a second camera.

Common difficulties

An attractive ring pattern is not a measurement

Typical misconceptionOne fringe photograph uniquely reveals displacement, wavelength and gas index.

Better mental modelThe sign of a change comes only from a known scan with a known sequence of phase steps. Each preset solves for one unknown and needs the other quantities as calibrated inputs. Agreement between the ideal model and its inversion says nothing about the accuracy of a real instrument. The Lab does not model beamsplitter dispersion, coating phase, finite apertures, beam shear, diffraction, phase-step error, shot noise, drift or low-coherence gating.

Run the experiment

01
Scene 1: Tilted-mirror displacement
Move the gold mirror forward and back through zero. The path endpoint follows the mirror, the fringes slide, and the recovered cycle count changes sign. Set the tilt to zero to see a uniform field.
What to observe: The bench draws the true central reflected paths. The mirror displacement in the drawing is enlarged 4000 times, but the phase is not.
02
Scene 2: Equal-inclination wavelength calibration
Select wavelength calibration. Increase the virtual-mirror separation to pack the rings closer, then probe off axis; the recovered vacuum wavelength still agrees. Set the displacement to zero and the calibration is no longer available.
What to observe: The images use a fixed linear grayscale. Moving the mirror by half a wavelength brings the intensity back to where it started, while the accumulated scan phase keeps changing.
03
Scene 3: Double-pass gas refractometry
Select the gas cell; the mirror now stays fixed. Change the index, then raise the RMS vibration. The image fades toward mid-gray, and eventually no measurement is reported.
What to observe: Pressing play does not add motion or change a static result. You can orbit the view and use the sliders while paused. The scan is a simulated sweep up to the current setting.

Michelson Precision Lab

Audit the physics of Michelson Precision Lab

Core mathematical model

Round trip and air wavelength

Straight fringes come from a tilted wavefront

Ring fringes appear in an angular view

Recovering the phase from four phase-stepped images

Vibration washes out contrast but keeps the power

Common difficulties

An attractive ring pattern is not a measurement

Run the experiment

Scene 1: Tilted-mirror displacement

Scene 2: Equal-inclination wavelength calibration

Scene 3: Double-pass gas refractometry

Related experiments and articles

Interference for measuring length and recording images

A material choice improved measurement