Optics 045 · Interference, coherence, cavities, and metrology

Michelson Precision Lab

A connected 3D Michelson bench couples real mirror intersections to straight and equal-inclination fringes, fixed-exposure power images and a four-step measured phase scan. Isolate displacement, vacuum wavelength or a double-pass gas-index change.

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 $\text{valid model regime}$

Validated scientific apparatus

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 follows it and the camera fringes translate. 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 signed change. Add RMS mirror jitter to wash out contrast. Drag to orbit; play never fabricates a measurement.

Ideal scalar balanced splitter and calibrated dark-port bias, fixed nondispersive ambient index, broad overlap and ideal reference steps. Gaussian jitter is an ensemble average, not detector noise. Mirror displacement is diagrammatically enlarged 4000×; no instrument precision, absolute phase order or low-coherence scan is claimed.

Physics tutorial

Audit the physics of Michelson Precision Lab

BackgroundA beam traverses each arm twice, reflecting once at its end mirror. With parallel on-axis beams, the change in optical path is $\Delta\mathrm{OPL}=2n_0\Delta x+2L_g\Delta n$ . Cyan and gold paths terminate at actual mirrors and recombine at the splitter. The camera and acquired scan share one phase model; moving the measuring mirror moves its path endpoint too.

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: An unwrapped scan gives signed cycles $N=\Delta\phi/(2\pi)$ . A single intensity image is even and periodic in phase: it cannot give the motion direction or the missing integer order. This experiment adds four calibrated reference phase steps at each scan sample. The scan starts at zero change, not at a secretly known absolute phase.

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 control is vacuum wavelength. Ambient phase index is fixed at 1.0002718; the cell length is 100 mm and its index change is relative to that ambient index. Both cell windows are ideal and stationary. The model does not calculate pressure, temperature or wavelength-dependent Ciddor corrections.

Straight fringes are real wavefront tilt

\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 beta. Unfolding its specular reflection through the splitter gives twice that output angle; the phase contains a transverse carrier and the small longitudinal correction. Coordinate u is horizontal on the camera; the output travels in the positive bench direction. Base arms are 180 mm and this spatial camera is 145 mm from the splitter. Zero tilt makes the spatial image uniform. Gas/tilt mixed terms are neglected.

Circular fringes require an angular observation

\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 inserts an ideal expanded angular illumination and a 100 mm focal-plane observation lens. Its mirrors remain parallel. The virtual-mirror separation sets radial phase variation: these are equal-inclination rings, not straight fringes drawn into circles. The camera lies 100 mm behind the observation lens. At zero separation and zero displacement the angular image is uniform. Off-axis wavelength inversion includes the cosine factor.

Measured phase, not a copied slider

\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}

The ideal reference actuator adds quarter-cycle steps. The inverse receives only four sampled intensity arrays and unwraps adjacent scan phases. Every allowed scan uses 257 samples and stays below half a cycle per step. Displacement needs known wavelength; wavelength needs independently known displacement; gas index needs known wavelength and cell length. Zero displacement cannot calibrate wavelength. No absolute integer order is claimed.

Vibration erases contrast, not energy

\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 control is zero-mean Gaussian axial mirror jitter averaged over a long exposure. The motion projection is cosine squared for the tilted spatial view and cosine for the parallel angular view. This is an analytic ensemble average, not a random movie or photon/readout-noise simulation. Contrast below 0.05 suppresses reported measurement; that threshold is a UI guard, not a precision specification. The complementary map is a model prediction, not a second acquired camera.

Common difficulties

An attractive ring pattern is not a measurement

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

Better mental modelOnly a known scan and phase-step sequence supplies signed change. Each preset isolates one unknown and requires the other quantities as calibrated inputs. Ideal forward/inverse agreement does not establish instrument accuracy. Splitter dispersion, coating phase, finite apertures, beam shear, diffraction, phase-step error, shot noise, drift and low-coherence gating are not solved here.

Run the experiment

01
Scene 1: Tilted-mirror displacement
Move the gold mirror through positive and negative displacement. Its endpoint follows it, fringes translate, and recovered scan cycles change sign. Set tilt to zero to see a uniform field.
What to observe: The bench shows actual central specular paths; diagrammatic mirror displacement is enlarged 4000 times, but its true phase is not.
02
Scene 2: Equal-inclination wavelength calibration
Select wavelength calibration. Change virtual-mirror separation to tighten the rings, then probe off axis: the recovered vacuum wavelength still agrees. Set displacement to zero and calibration becomes unavailable.
What to observe: Images use fixed linear grayscale. Half-wave motion repeats intensity while accumulated scan phase continues.
03
Scene 3: Double-pass gas refractometry
Select the gas cell. The mirror is held fixed. Change index, then raise RMS vibration: the image washes toward mid-gray and eventually the measurement is withheld.
What to observe: Play does not create motion or alter a static result. Orbit and sliders work when paused; the scan is a simulated sweep to the current setting.