Optics 071 · Diffraction, Fourier optics, and computational imaging

Adaptive Optics Observatory

How can measured wavefront slopes drive a mirror fast enough to sharpen a turbulent star image? A reconstructor turns the slopes into commands for a deformable mirror, and a fast closed loop keeps flattening the wavefront as turbulence changes. The slopes come from a Shack-Hartmann sensor: each spot behind its small lens shifts in proportion to the local slope. Delay, photon noise and limited mirror travel cap the correction, and the sensor is blind to some shapes, such as uniform piston.

Spots measured by a Shack–Hartmann sensor drive a deformable mirror whose surface you can watch bend. Compare star images with a flat and a corrected mirror, and explore modal fitting, delay, photon noise and mirror 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 $\text{valid model regime}$

Model checked against analytic results

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

Adaptive optics can sharpen a blurred star into a diffraction core, but only through what the sensor measures. Watch the mirror surface bend, inspect the measured lenslet spots, then compare the same incoming wave after a flat mirror and after the measured closed loop. Pause to scrub through the control history. Add latency, photon noise or a wrong calibration, and the correction can lag, saturate or make things worse. The correction is computed only from the measured spots.

The aberration phase is synthetic, not a Kolmogorov turbulence model. Lenslet spots are approximated as Gaussians, and the mirror is an ideal continuous modal mirror. The PSFs are instantaneous and scalar, and piston cannot be measured.

Physics tutorial

Audit the physics of Adaptive Optics Observatory

BackgroundEach lenslet sees one patch of the relayed pupil and turns the average slope of the optical path across it into a sideways shift of its focal spot: $\Delta x=f_{\ell}\langle\partial_x W\rangle$ . The Lab records the camera pixels first, then finds each spot's centroid, then solves for the mode coefficients the mirror can correct. The true phase serves only as the answer key and 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: Adding the same phase everywhere, called piston, changes no slope at all: $\nabla(W+p)=\nabla W$ . Add piston and compare the spots and the normalized star image. The sensor cannot recover this constant, so only the RMS error with piston removed is scored.

Core mathematical model

From detected 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}

The wavefront is measured in waves. Normalized pupil coordinates are the transverse distance divided by the pupil radius. Pixels are two micrometres wide and the lenslet focal length is two millimetres. Each spot is modelled as a Gaussian, integrated over pixels, with optional Poisson counts. The centroid bias from the finite window stays in the result.

Reconstructing modes from measured slopes

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

The matrix holds each mode's calibrated gradient averaged over each lenslet cell, plus a tiny fixed ridge term. Dark cells are left out; the true phase never fills in for them. The modes start with tip, tilt, defocus, astigmatism, coma, trefoil and spherical shape. There is no piston column.

Closed-loop update from past data, and mirror 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 sag points along the normal toward the incoming beam and shortens the optical path. Each update follows one ten-millisecond sample, and any extra delay comes in whole frames. The stroke setting scales all modal commands at the sampled pupil points by one common factor. The mirror is an ideal, quasi-static continuous surface; individual actuators are not modelled.

Star image and Strehl ratio from the residual phase

\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}

The Strehl ratio here compares peaks over the whole sampled angular grid, not just the on-axis values. Piston changes neither the image nor the RMS with piston removed. For small errors, the Marechal value approximates the nominal on-axis intensity. When tilt moves the star, it is no longer guaranteed to approximate the peak Strehl ratio.

Common difficulties

Does more gain always sharpen the star?

Typical misconceptionJust raise the gain and add modes until the star is perfect.

Better mental modelA finite set of modes cannot fit every ripple. Feedback that arrives late can oscillate, faint spots give bad centroids, and the mirror runs out of stroke. The Lab computes the exact peak Strehl ratio from the corrected pupil, not from an empirical rule based on gain.

Run the experiment

01
Scene 1: Measured mirror correction
Look at the deformable mirror itself, then at the front of the wavefront sensor. Pause playback and compare the flat mirror with the closed loop at the same frame.
What to observe: The deformable mirror is opaque and folds the beam, so no light continues behind it. A beam splitter feeds two separate devices: the science camera and the wavefront sensor.
02
Scene 2: Moving phase and latency
Scrub from the initial flat mirror through the first few updates. Each correction uses only spot images recorded earlier.
What to observe: The flat-mirror and AO star images use the same incoming phase, pupil, normalization and fixed display scale. Each is the expected image at one instant, not an average over time.
03
Scene 3: Fine ripples and photon noise
Try moving phase with delay, low photon counts, few modes and a small stroke. Add piston to test a mode the sensor cannot see.
What to observe: A small residual in the fitted modes does not mean every phase feature was measured. Piston, high-order fitting error, clipping by the spot windows and delay all stay in the result.