Optics 052 · Interference, coherence, cavities, and metrology

Stellar Interferometer

A connected 3D stellar interferometer moves two sky-pointing collectors and a two-mirror delay carriage, records complementary phase-stepped photodiode powers and fits uniform-disk diameter or projected binary separation from a baseline scan.

Recovered visibility / squared visibility

|\widehat V|,|\widehat V|^2

$0.50$

Sky-projected baseline / tracked geometric delay

B_\perp,g

$50\%$

Fitted angular scale / record residual

\widehat\theta_{\mathrm{UD}},|\widehat s_\parallel|

$0.00\pi$

Model regime $\text{valid model regime}$

Validated scientific apparatus

Stellar Interferometer

Physical telescope separation

B

60\,\mathrm m

Uniform-disk angular diameter (not radius)

\theta_{\mathrm{UD}}

4\,\mathrm{mas}

Monochromatic vacuum wavelength

\lambda

1650\,\mathrm{nm}

Target angle from vertical

\zeta

20^{\circ}

Ground baseline azimuth

\psi

0^{\circ}

Mirror trim after automatic delay tracking

\Delta d

0\,\mu\mathrm m

True binary angular separation

s

6\,\mathrm{mas}

Binary position angle in the sky tangent plane

\beta

30^{\circ}

Secondary / primary flux (known for projection fit)

r

1

Start with the single star: widen the telescope separation and the recorded fringe contrast falls. Switch to First null: contrast returns with reversed phase beyond the zero. Shorten wavelength or enlarge angular diameter to move that zero inward. Rotate the ground baseline or tilt the target to change the sky projection and tracking delay. Mirror trim shifts phase, not monochromatic visibility. In Binary, turn the separation perpendicular to the projected baseline: the pair becomes indistinguishable from one point. The grey curve is fitted to the mint recorded points, never read from the sky drawing. Drag telescope A in the Optical paths view; sliders are keyboard accessible. Play only moves route markers.

Far-field monochromatic scalar teaching model, spatially incoherent disk or two unresolved components; equal collected powers and ideal calibrated phase steps. The 0.25 m collecting apertures have flat angular coupling here, not a solved telescope PSF. 81 records from zero to 200 m physical baseline are pre-acquired at fixed orientation; the zero-baseline endpoint is a mathematical calibration, not overlapping hardware. No limb darkening, bandwidth, atmosphere, noise, polarization, throughput calibration, reconstructed sky image or observing-precision claim. Separation uses three world units per metre; hardware is enlarged. Delay carriage travel maps tracked OPD at 0.7 world units per metre plus 15 per micrometre of trim; these are separate schematic scales, not lengths used for phase. Mirror trim adds twice its physical travel to optical path. Raw power plots use a fixed linear scale. Detector-face brightness is a schematic indicator with a fixed 15% floor, not a calibrated power display.

Physics tutorial

Audit the physics of Stellar Interferometer

BackgroundA stellar surface is not a laser. Different surface points emit independently, but a copy of each point reaches both telescopes. Adding their intensity fringes gives $V=\widetilde I(\mathbf b/\lambda)/\widetilde I(\mathbf0)$ . This normalized Fourier integral is the van Cittert–Zernike relation. It links brightness in the sky to mutual coherence between collectors, not to a directly photographed stellar disk.

Why it mattersHow can visibility versus baseline reveal a star too small for either telescope to resolve?

Start with the essentials

Focus question: How can visibility versus baseline reveal a star too small for either telescope to resolve?
One-sentence intuition: For a uniformly bright circular disk, angular diameter appears in $V_{\mathrm{UD}}=\frac{2J_1(x)}{x},\quad x=\pi B_\perp\theta_{\mathrm{UD}}/\lambda$ . Contrast vanishes at the first Bessel zero, then returns with a phase reversal. Recording multiple baselines and fitting the recovered squared visibility reveals a diameter much smaller than either collector can image alone.

Core mathematical model

Physical separation is not sky-projected baseline

\begin{aligned}\mathbf B&=B(\cos\psi,0,\sin\psi)\\\mathbf s&=(\sin\zeta,\cos\zeta,0)\\g&=\mathbf B\cdot\mathbf s\\\mathbf b&=B(\cos\psi\cos\zeta,\sin\psi)\\B_\perp&=|\mathbf b|\end{aligned}

Up is vertical; the target lies in the first horizontal–vertical plane. Zenith angle is measured from vertical. Ground azimuth starts along that horizontal axis. The sky tangent basis is the increasing-zenith direction and the other horizontal direction. Optical delay tracks the longitudinal component; resolving power uses the transverse component. They are not interchangeable.

A disk is an incoherent sum, not one coherent ball

\begin{aligned}V_{\mathrm{UD}}&=2J_1(x)/x\\x&=\pi B_\perp\theta_{\mathrm{UD}}/\lambda\\B_{\perp,0}&=1.21967\,\lambda/\theta_{\mathrm{UD}}\end{aligned}

The angular control is a diameter, not a radius. Visibility is the modulus, while the signed real coherence is negative in the second lobe. That sign means a half-cycle fringe shift; it does not mean negative power. The first zero is approximately 3.831706 in the Bessel argument. A uniformly bright disk is not a limb-darkened stellar atmosphere.

Two complementary detector powers form the record

\begin{aligned}C&=V e^{4\pi i\Delta d/\lambda}\\P_\pm/P_{\mathrm{col}}&=\tfrac12[1\pm\operatorname{Re}(Ce^{-i\phi})]\\P_++P_-&=P_{\mathrm{col}}\end{aligned}

Both input arms carry half of the total collected power. An ideal lossless combiner has two output photodiodes. The displayed power is linear and normalized to that total, not separately rescaled for each baseline. The tracked delay matches the target centre; positive mirror trim adds twice its travel to channel A. In a monochromatic model trim shifts fringe phase without reducing contrast. Finite bandwidth would change this.

Recover coherence before fitting the angular size

\begin{aligned}\phi_j&=2\pi j/64\\D_j&=(P_{+,j}-P_{-,j})/P_{\mathrm{col}}\\\widehat C&=\frac2{64}\sum_{j=0}^{63}D_j e^{i\phi_j}\\|\widehat V|^2&=|\widehat C|^2\end{aligned}

Every baseline stores 64 phase-calibrated pairs. The Fourier coefficient comes from their differences. The fitting function accepts those recovered records, known wavelength and a candidate parameter range; it never accepts the true diameter. Eighty-one baselines span zero to 200 metres at a fixed direction. The zero-baseline endpoint is an ideal calibration, not overlapping physical telescopes. Weak contrast changes make the inference sensitive to omitted noise; the residual is numerical fit error, not observational uncertainty.

A binary modulates visibility along its projection

\begin{aligned}q&=2\pi\mathbf b\cdot\boldsymbol s/\lambda\\V_{\mathrm{bin}}&=\frac{e^{iq/2}+r e^{-iq/2}}{1+r}\\|V_{\mathrm{bin}}|^2&=\frac{1+r^2+2r\cos q}{(1+r)^2}\end{aligned}

These are two mutually incoherent unresolved components, not finite disks or an orbital simulation. Their positions are symmetric around the tracked midpoint; the known flux ratio can be unequal. Rotating the separation perpendicular to the sky-projected baseline removes its modulation. With no secondary flux, squared visibility cannot identify separation; ideal referenced phase can still encode the primary's offset. This fit uses squared visibility along one direction and returns only an absolute projected separation, not a signed sky vector.

One fitted scale is not a reconstructed stellar photograph

\widehat a=\operatorname*{arg\,min}_a\sum_m\left[|\widehat V_m|^2-|V(B_{\perp,m};a)|^2\right]^2

The known sky disk and position diagram are reference only. Mint dots are reconstructed from stored photodiode powers; the grey model is fitted to those dots. Gold shows the signed real coherence after removing the known mirror trim. No two-dimensional image is reconstructed from this line of baselines. Mirror surfaces and detector faces are geometric; collector size, relays, carriage travel and route speeds are deliberately enlarged or schematic. No atmospheric phase loss, limb darkening, detector noise or real precision is claimed.

Common difficulties

A vanished fringe does not mean a vanished star

Typical misconceptionLonger baseline simply removes light; negative visibility means the detector receives negative power.

Better mental modelTotal collected power is fixed. Different stellar surface points cancel the interference term, not their background powers. Past the first null the interference term reverses sign and contrast returns. Complementary detector powers remain nonnegative and sum to the collected total.

Run the experiment

01
Scene 1: Single-star diameter
Single star: change diameter and wavelength; compare raw phase powers, recovered contrast and fitted diameter. Then tilt the target and see the projected baseline differ from ground separation.
What to observe: Gold and cyan are false-colour direction guides. The incoming guides represent only the central target direction, not a single coherent extended-star emitter. Moving dots are not photons, physical wavefront spacing or acquisition timing.
02
Scene 2: First visibility null
First null: sweep baseline across the zero and into the second lobe. Trim the mirror and confirm phase shifts while visibility stays fixed.
What to observe: Physical baseline maps to three world units per metre. Carriage centre maps geometric delay at 0.7 per metre and trim at 15 per micrometre; both are schematic display mappings. The optical model uses physical units, not drawn lengths.
03
Scene 3: Binary-star separation
Binary: rotate its position angle and the baseline independently. Make them perpendicular, then set secondary flux to zero. Compare lost size information with the visible reference pair.
What to observe: The narrow-band scalar apparatus assumes 0.25 m collectors with flat angular response and ideal mode matching. It omits finite-bandwidth delay envelopes, noise, atmosphere, polarization, throughput drift, limb darkening and image reconstruction. Pause, orbit and reduced motion never alter the acquired records.