Michelson Fringe Bench

Detailed abstract

Purpose

A hands-on alignment trainer for the classic Michelson interferometer. It shows how the fringes on a screen respond to the two things you control in the lab: each arm's length (coarse, by millimetres, and fine, by fractions of a wavelength) and each end mirror's tilt (yaw and pitch). It shows both output ports side by side, so you can see that they are complementary, and why the laser's wavelength and coherence matter.

What it can do

• Tilt both end mirrors (EX and EY) in yaw and pitch, by slider or by dragging the knobs in 3D (±2000 µrad).
• Change each arm length independently: coarse ±25 mm (the mount slides on a stage stand-in) and fine ±5 µm in 5 nm steps, with the sliders or by grabbing the mirror mount or its post in 3D and dragging it along the arm (hold Shift for the fine adjust).
• Turn either end-mirror mount on its post by dragging the blue ring around the post (Shift for fine). This is a coarse yaw of the whole mount, added to the knob yaw and limited to ±2000 µrad (about ±0.11°); beyond that the two beams no longer overlap on the screen.
• Add dust (0–120 specks, 5–200 µm) on the beam-splitter and pickoff exit faces or on the expanding lenses, and watch ring-shaped "bullseyes" appear in the fringes.
• Pick any wavelength from 405 to 700 nm (presets for common lasers); the beam and fringes change colour.
• Set the beam splitter's reflectance from 10 % to 90 % (buttons for the stock 1″ Thorlabs cubes: 10:90 BS025, 30:70 BS019, 50:50 BS013, 90:10 BS028; values in between are a what-if) and put a neutral-density filter in arm Y, to see which imbalance dims the fringes and which washes them out.
• Choose the laser's coherence: single-frequency, single-mode diode (≈ 0.3 m) or multimode pointer (≈ 2 mm).
• See both output ports at once: the trans screen (after the beam splitter) and the refl screen (light returning toward the laser, picked off by a 10:90 (R:T) cube, Thorlabs BS025). Each beam is expanded by an f = −15 mm lens 80 mm after the BS or the pickoff.
• See the patterns on the white cards in 3D and in zoomable side-by-side views; read the path difference, fringe order, visibility, fringe spacing and fringe count.
• Scan the fine adjust slowly and watch fringes move; a trace shows the brightness at each pattern centre vs. fine position, trans and refl out of step by half a period.

Limitations

• The beam splitter cube is ideal and lossless: the equal glass path in both arms is left out (it cancels), with no ghost reflections; its reflectance is adjustable (10–90 %) but the same for both polarizations and both passes. The pickoff cube is an ideal 10 % reflector. Real cubes add ≈ 0.5 % ghost reflections at each AR-coated face and are usually tilted 1–2° to steer those back-reflections away from the laser and screens; that tilt and the ghosts are not modelled.
• Mirrors and lens are perfect (no surface figure error); beams are never clipped by apertures.
• Paraxial optics only; the lens is a thin lens.
• Dust specks are modelled as Gaussian-edged disks, so each bullseye shows the interference rings but not the faint Airy rings of a hard edge; dust is only placed after the arms recombine (on the BS/pickoff exit faces or the lenses), not on the end mirrors.
• Coherence is a single Gaussian visibility factor, not a full laser spectrum (no multimode revivals).
• No vibration, air currents or thermal drift: the real pattern would shimmer.
• The 90 % of returning light that passes the pickoff back into the laser is not drawn, and laser feedback is ignored.
• The refl screen is shown with its own exposure; in reality it is about 25 times dimmer than the trans screen.
• Mirror tilts can be drawn exaggerated (×20 or ×100) so the mounts visibly turn; the drawn beams then follow the exaggerated tilt and swing far off after the expanding lenses. The screens always use the true tilt.

How realistic is it?

• Best case: straight-fringe geometry, the λ/2 period of the fine adjust, the complementary trans and refl patterns, the effect of wavelength and the tilt-to-fringe relation follow textbook two-beam interference exactly within the paraxial Gaussian-beam model.
• Likely case: the coherence lengths (0.3 m and 2 mm) are illustrative; real diode lasers vary widely, and multimode lasers show visibility revivals at multiples of the cavity length that are not modelled.
• Hidden burdens: on a real table, room air and vibration make fringes drift by many fringes per second unless the setup is enclosed and short; finding equal path with a short-coherence source needs patience; stray reflections from the cube faces (≈ 0.5 % each, AR-coated) can add a faint second pattern unless the cubes are tilted 1–2°.

The modeling: core physics and how it is handled

Core physics and how it is handled

1. Two-beam interference at two ports. Each screen shows I = |E_X|² + |E_Y|² + 2·V_c·Re(E_X E_Y), where V_c is the coherence factor. For a lossless beam splitter the Stokes relation r′ = −r puts an extra π between the arms at the trans port and none at the refl port, so I_trans + I_refl is constant: the patterns are complementary.
Handled:* each arm's complex field is computed analytically at every pixel (240 × 240 per screen view, 160 × 160 for each 3D card texture). Powers per arm: 5 mW × 0.90 (pickoff cube) × 0.25 at the trans screen, and × 0.10 more at the refl screen. The π is added to arm X at the trans port.

2. Chief-ray trace with real 3D tilts. Each arm's central ray is traced from the laser through the pickoff, to the 45° beam splitter, to the tilted end mirror and back to the splitter. For the trans port it continues through the lens 80 mm after the BS to the screen 340 mm from the BS (260 mm after the lens); for the refl port it travels back to the pickoff, reflects (10 %, cube hypotenuse), passes the lens 80 mm after the pickoff and lands on the second screen 340 mm from the pickoff.
Handled: mirror normals n = (−1, −tan yaw, tan pitch) for EX and (tan yaw, −1, tan pitch) for EY; reflection d′ = d − 2(d·n)n; exact ray-plane intersections. The optical path length (OPL) is the sum of segment lengths; the thin lens adds −h²/(2f) and bends the ray by −h/f, where h is the ray's offset from the lens axis.

3. Gaussian beam per arm. The laser waist w₀ = 1.0 mm (a 2 mm beam) sits at the aperture. Complex beam parameter q = z + i·z_R, z_R = π w₀²/λ; free space q → q + d; thin lens 1/q → 1/q − 1/f; amplitude factor q_before/q_after on each free-space step (this carries the Gouy phase).
Handled: the q of each arm is propagated along its own unfolded, aligned path to each port (segment lengths taken from the aligned chief-ray trace, lens where it sits), so the two arms differ slightly in curvature when their lengths differ. With a 2 mm beam (z_R ≈ 5.9 m at 532 nm) and the lens after the recombination, this curvature difference is tiny, so the patterns are straight fringes set by the mirror tilts, not rings.

4. Field on the screen. For an offset ρ from the arm's chief-ray hit point: E = A·exp(−i k ρ²/(2q))·exp(−i k (ΔOPL + d·ρ)), with 1/q = 1/R − iλ/(πw²).
Handled: closed form. Straight fringes come from the relative tilt of the wavefronts. The number of fringes across the spot is fixed before the lens (about 2w·2θ/λ for a mirror tilt θ); the lens only magnifies the pattern (about 18.4× here).

5. Fine and coarse length. The mirror position is L = 150 mm + coarse + fine. One full bright-dark-bright cycle needs a mirror move of λ/2.
Handled: the same trace and Gaussian model are re-run with the new length (no special-case formula).

6. Coherence. V_c = exp(−(OPD/L_c)²), with OPD the chief-ray path difference.
Handled: L_c = ∞, 300 mm or 2 mm (ASSUMED illustrative values). With the multimode option, fringes vanish a millimetre or two away from equal arms.

7. Dust (Babinet's principle). A speck of radius a blocks a small disk of the local field. That equals the unobstructed field plus a weak wave of opposite sign, −E_local, leaving the speck. This wave spreads (far-field half-angle ≈ λ/(πa)) and interferes with the main beams, whose curvature at the screen differs slightly from the speck wave's, giving Fresnel-zone rings of radius ≈ √(n λ R_eff).
Handled: each speck is a Gaussian of 1/e amplitude radius a (same area as the disk) carrying −E_local of each arm separately (coherence kept per arm); its complex q is propagated from the speck (80 mm before the lens, or on the lens exit face) through the lens to the screen, its centre follows the local ray, and its phase includes the paraxial path to the screen. Specks are seeded random positions within 2.5 mm of the beam axis, log-normal sizes around the chosen diameter. Only the screen pixels within 3.5 beam radii of each speck's footprint are updated, so 120 specks per optic stay interactive.

8. Display. Colour from wavelength (Bruton-style piecewise RGB); screen brightness = card grey × (1 − 0.55·t) + laser colour × t, with t = 1 − exp(−I/I_ref) and I_ref = 0.6·(I_X,peak + I_Y,peak), so a fully constructive peak is near saturation and dark fringes show the grey card.
Handled: each screen's exposure is fixed by its own beam powers, not by the current pattern, so turning the fine adjust visibly dims and brightens the spots in opposite senses.

9. Unequal beam splitter and arm imbalance. For a lossless splitter with power reflectance R and transmittance T = 1 − R, and arm round-trip power returns t_X, t_Y (mirror reflectivity times any filter, passed twice), arm X is transmitted out and reflected back to the trans port, arm Y the reverse, so both arrive with weight R·T: I_trans = R·T·[t_X + t_Y + 2√(t_X t_Y) cos Δφ]. The peak is 4RT of the 50:50 value (1.00, 0.96, 0.84, 0.64, 0.36 for 50, 60, 70, 80, 90 % R), but the visibility V = 2√(t_X t_Y)/(t_X + t_Y) does not depend on R at all. Toward the laser the weights are T² and R²: I_refl = T² t_X + R² t_Y − 2RT√(t_X t_Y) cos Δφ, with V = 2RT/(R² + T²) for equal arms (0.92 at 60/40, 0.47 at 80/20). For t_X = t_Y = 1 the two ports add to 1 at every Δφ (energy conservation). An ND filter of round-trip transmission 25 % in one arm gives V = 0.80 at both ports.
Handled: each arm's field amplitude at each port is scaled by √(weight) before the per-pixel interference sum, so tilt, coherence and dust effects combine with the imbalance automatically. The screens and the centre-brightness trace keep the 50:50 exposure, so dimming shows. To recreate: multiply each arm's field by √(R·T·t) at the trans port and by T√t_X or R√t_Y at the refl port; keep the Stokes π on arm X at the trans port.

How the simulation is built

• src/mf_model.js: geometry, mirror normals, chief-ray trace with OPL, Gaussian q propagation, screen field, intensity map and wavelength-to-colour. Pure functions; runs in Node.
• src/mf_app.js: knob dragging, mount/post dragging along the arms (raycast onto a horizontal plane, Shift for fine), dust specks drawn on the optics; three.js bench (table, kinematic mounts on sliding carriages, laser, pickoff and beam dump, two lens mounts, two white cards with the live patterns as textures), controls, readouts, knob dragging and the centre-brightness traces.
• src/clamps.js: the shared post/fork clamp solver; the mirror posts re-clamp as the carriages slide.
• The 3D beams are drawn from the same model (chief rays plus Gaussian radius), with the tilts exaggerated only in the drawing.
• src/build.py inlines everything into michelson_fringe_bench.html.

How to recreate it

1. Set up the geometry: laser waist at x = −200 mm, pickoff at x = −130 mm and BS at the origin (both with normal (−1, 1, 0)/√2), mirrors at (L_X, 0) and (0, L_Y), lenses at y = −80 mm below the BS and below the pickoff, screens in the plane y = −340 mm at x = 0 and x = −130 mm.
2. Trace each arm's chief ray with vector reflections; sum segment lengths for OPL; apply the thin-lens bend and −h²/(2f) term where the lens sits.
3. Propagate q = i·z_R through the unfolded path for each arm, keeping the product of q_before/q_after.
4. At each screen point, evaluate E_X and E_Y with item 4 (add π to arm X at the trans port), then I with item 1.
5. Check: equal arms and no tilt give uniform spots whose brightness cycles every λ/2 of mirror travel, trans dark when refl is bright; a 600 µrad yaw of EX gives about 4–5 straight fringes across each spot at 532 nm.
6. To rerun: cd src && python build.py, then open michelson_fringe_bench.html; or node -e "require('./mf_model.js')" to use the model directly.

Key parameters