Optics 071 · Diffraction, Fourier optics, and computational imaging

Adaptive Optics Observatory

Measured Shack–Hartmann spots drive a visibly deforming reflective mirror. Compare flat and corrected star PSFs, modal fitting, delay, photon noise and stroke.

Exact peak Strehl · flat / corrected

S_{\rm flat},S_{\rm AO}

$0.50$

Piston-removed OPD RMS

\sigma_{\rm flat},\sigma_{\rm AO}

$50\%$

Control clock / normal mirror sag

t,|s|_{\max}

$0.00\pi$

Model regime $\text{valid model regime}$

Validated scientific apparatus

Adaptive Optics Observatory

Initial piston-removed OPD RMS

\sigma_{W,0}

0.30\lambda

Ripple mixing (not Kolmogorov)

b

0.12

Synthetic phase variation rate

f_{\rm synth}

0\mathrm{Hz}

Correctable modes (not actuator count)

M

10

Integrator gain

g

0.60

Additional whole-frame delay

d

0

Normal mirror stroke at sampled nodes

s_{\max}

0.80\mu\mathrm m

Assumed / actual slope response

a_G

1

Expected electrons per full lenslet

N_\gamma

1000\mathrm e^{-}

Zero: expectation; one: Poisson record

n

0

Zero: flat mirror; one: measured loop

j_{\rm loop}

1

Starting control frame (play advances)

k_0

12

Unobservable uniform piston OPD

p_0

0\lambda

A blurry star can become a diffraction core—but the mirror must earn it. Watch the reflective surface bend, inspect the measured lenslet spots, then compare the same incoming wave with a flat mirror and a measured closed loop. Pause to scrub the control history. Add latency, photon noise or an incorrect calibration; the correction can lag, saturate or become worse. The sensor never receives the answer-key phase.

Synthetic phase, not a Kolmogorov forecast. Gaussian lenslet-spot surrogate and ideal continuous modal mirror. Instantaneous scalar PSFs; piston cannot be measured.

Physics tutorial

Audit the physics of Adaptive Optics Observatory

BackgroundA lenslet divides the relayed pupil and turns mean optical-path slope into a focal-spot displacement: $\Delta x=f_{\ell}\langle\partial_x W\rangle$ . This Lab records pixels first, computes centroids second, then solves for correctable modal coefficients. The answer-key phase is not a measurement.

Why it mattersHow can measured wavefront slopes drive a mirror fast enough to sharpen a turbulent star image?

Start with the essentials

Focus question: How can measured wavefront slopes drive a mirror fast enough to sharpen a turbulent star image?
One-sentence intuition: A uniform piston changes optical phase without changing any slope: $\nabla(W+p)=\nabla W$ . Add piston and compare the spots and normalized star image. The sensor cannot recover this constant; only piston-removed RMS is scored.

Core mathematical model

From electrons to slope

\begin{aligned}\bar x_j&=\frac{\sum_i x_i D_{ji}}{\sum_i D_{ji}}\\\Delta x_j&=f_\ell\frac{\lambda}{R}\langle\partial_u w\rangle_j\end{aligned}

Wavefront is in waves; normalized pupil coordinates divide physical transverse distance by its radius. Pixels are two micrometres; lenslet focal length two millimetres. Gaussian spot surrogate with pixel integration and optional Poisson counts; finite-window centroid bias is not silently removed.

Reconstruct only measured data

\widehat{\mathbf a}=(G^TG+\epsilon I)^{-1}G^T\mathbf s

Calibrated cell-average modal gradients; fixed tiny ridge. Dark cells are omitted, not replaced by truth. The first modes are tip, tilt, defocus, astigmatism, coma, trefoil and spherical shape; no piston column.

Causal closed loop and reflective sag

\begin{aligned}\mathbf c_{k+1}&=\mathbf c_k+g\widehat{\mathbf a}_{k-d}\\w_{\rm res}&=w_{\rm in}-\sum_m c_m Z_m\\s&=\frac{\lambda\sum_m c_m Z_m}{2\cos45^\circ}\end{aligned}

Positive normal sag toward the incident beam reduces OPD. Each update follows one ten-millisecond sample; extra delay is in whole frames. Stroke globally scales modal commands at the sampled pupil nodes. Ideal quasi-static continuous mirror, not a model of independent actuators.

Compute the star, then its Strehl

\begin{aligned}I&=\frac{|\mathcal F\{A e^{2\pi i w_{\rm res}}\}|^2}{\sum |\mathcal F\{A e^{2\pi i w_{\rm res}}\}|^2}\\S&=\frac{\max I}{\max I_{\rm ideal}}\\S_{\rm M,axis}&\approx e^{-(2\pi\sigma_w)^2}\end{aligned}

Peak ratio on the sampled angular lattice, not an on-axis-only ratio. Piston changes neither image nor piston-removed RMS. The Marechal value approximates the nominal on-axis intensity at small errors; it is not a guaranteed peak-Strehl approximation when tilt shifts the star.

Common difficulties

More gain always sharpens the star?

Typical misconceptionRaise the gain and add modes until the star is perfect.

Better mental modelFinite modes cannot fit every ripple. Delayed feedback can oscillate; faint spots corrupt centroids; stroke saturates. Exact peak Strehl comes from the corrected pupil, not a gain-based empirical rule.

Run the experiment

01
Scene 1: Measured mirror correction
Inspect the actual mirror, then the WFS front. Pause playback and compare flat versus closed-loop at the same frame.
What to observe: The opaque DM folds the beam; no beam continues behind it. The splitter feeds separate physical science and WFS collectors.
02
Scene 2: Moving phase and latency
Scrub from the initial flat mirror through the first updates. The correction uses only earlier acquired spot records.
What to observe: Flat and AO PSFs share the same incoming phase, pupil, normalization and fixed display scale. They are instantaneous expectations, not time-averaged observations.
03
Scene 3: Fine ripples and photon noise
Try moving phase with delay, low photon counts, limited modes and small stroke. Add piston to test an unobservable mode.
What to observe: A small modal residual does not imply every phase feature was observed. Piston, high-order fitting error, spot-window clipping and delay remain explicit.