Wavefront reconstruction with pyramid sensors and

Introduction - Pyramid Wavefront Sensor

Pupil fragmentation

Split Approach

Summary

Wavefront reconstruction with pyramid sensors and pupil fragmentation Andreas Obereder, Victoria Hutterer, Iuliia Shatokhina Johann Radon Institute for Computational and Applied Mathematics, Linz, Austria. Industrial Mathematics Institute, JKU, Linz, Austria.

Durham, March 22, 2018

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Pupil fragmentation

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Outline

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Pupil fragmentation Performance of modal MAP Performance of model-based algorithms Performance of zonal MMSE

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Split Approach

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Pupil fragmentation

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Pyramid and Roof WFS

´ Credit: C. Verinaud [I1 (x, y ) + I2 (x, y )] − [I3 (x, y ) + I4 (x, y)] sx (x, y) = I0 sy (x, y) =

[I1 (x, y ) + I4 (x, y)] − [I2 (x, y) + I3 (x, y)] I0

I0 – average intensity per subaperture. ´ Credit: C. Verinaud.

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Pupil fragmentation

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Pupil fragmentation

Split Approach

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Telescope spiders • pupil fragmentation & disconnectedness of data (wavefront information) • differential piston effects between the segments • if not properly handled extremely poor wavefront reconstruction

Figure: residual phase in radians (K-band)

How much quality do we loose in the presence of spiders? How can we make existing reconstruction methods feasible? Can we reach the same quality as obtained without spider legs? 6 / 24


Pupil fragmentation

Split Approach

Summary

Performance of modal MAP

Performance of modal MAP (as in Octopus)

Setting: frame rate 1 kHz, median atmosphere, high photon flux, 1 second simulation, sensing in K-band (!!!) Quality without spiders: 0.888 Quality with spiders: 0.848-0.859 Optimized parameters: number of photons in calibration (0.5, 5, 10, 20, 50, 100, 200) illumination threshold (0.4, 0.5, 0.75, 0.95) number of modes (2000, 2102, 2400, 3200, 4000, all=4204)

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Pupil fragmentation

Split Approach

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Performance of model-based algorithms

Adapting model-based reconstructors for segmented mirrors First approaches: • measurement/phase interpolation • sophisticated measurement extensions, e.g., generate data using the sensor forward model • data smoothing • assume zero measurements under spider legs • actuator coupling/slaving • segment coupling using boundary integrals • use less illuminated subapertures (under spider legs) adapt existing model-based reconstructors

Should we use “dark“ subapertures under the spider legs?

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Pupil fragmentation

Split Approach

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Performance of zonal MMSE


create ”PokeMatrix” using a bilinear basis on the Fried-Geometry (zonal ansatz) use all subapertures illuminated at least 45% of their area skip all actuators creating a ”small feedback only” invert using −1 −1 T −1 M †MAP = (M T Cη−1 M + µ0 CΦ ) M Cη .

(1)

play around with the parameters, smooth the reconstruction, use DM leakage, smooth the DM shape, ...do whatever helps you, derive the according theory later... inter-/extrapolate on the real actuator positions (no, we do not project on the DM)

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Pupil fragmentation

Split Approach

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Active subapertures and DM actuator positions

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...

Figure: residual wavefront reconstruction

Figure: DM actuator values and extended DM ”shape” 11 / 24


Pupil fragmentation

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Pupil fragmentation

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Split Approach

Split Approach = piston-free WF reconstruction + direct segment piston reconstruction

Requests: • compoundable with all existing reconstruction methods • providing high reconstruction quality • low computational complexity

Split Approach −−−−−−−−−−−→

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Pupil fragmentation

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Split Approach

→ divide aperture Ω into segments Ωi

Preconditions: • method for piston-free WF reconstruction on segments Ωi exists (e.g. P-CuReD) • piston pi on every segment is calculated independently by direct segment piston reconstructors General idea: Φ=

nseg X

Φi + pi

i=1

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Pupil fragmentation

Split Approach

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Direct Segment Piston Reconstructor (DSPR) I

Single-Poke Approach: • use full interaction matrix M of the system • invert M in a clever way (regularization) → M † • extract piston information on every segment by matrix Q p = QΦ = QM † s =: C 1 s

We cut down the computational expensive interaction-matrix-based MVM approach to a fast direct segment piston reconstruction method. matrix dimensions: na × 2nsub −→ nseg × 2nsub

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Pupil fragmentation

Split Approach

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Direct Segment Piston Reconstructor (DSPR) II

Segment-Poke Approach: • describe piston p as p(x, y) =

nseg X

ci XΩi (x, y)

i=1

• generate segment-poke matrix C 2 such that ∀i = 1, 2, . . . nseg

PXΩi = si and

C 2 = s1 , s2 , . . . , snseg

• solve C 2 p = s in a least-squares sense → C †2 • calculate piston on every segment by p = C †2 s

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Simulation settings

METIS-like telescope diameter central obstruction AO system

37m 30% SCAO

sensing band

K

evaluation band

K √

pyramid WFS modulation subapertures Von Karman atmospheric model simulated layers atmosphere

4 74 × 74 √ 35 median

outer scale L0 (m)

25

Fried radius r0 (m)

0.157

photon flux

600 ph/subap/frame

frame rate

500 Hz

mirror geometry

ELT M4

simulation time (s)

2

simulation environment: Octopus reconstruction method: P-CuReD algorithm (implemented on segments)

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Pupil fragmentation

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Which reconstruction quality is attainable?

Benchmark - Reconstruction without spiders: • we simulate and reconstruct without taking spider legs into account

Benchmark - Houdini Method for piston: • we calculate the piston information on segments directly from the incoming phase (which is not known in practice)

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Pupil fragmentation

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Direct Piston Reconstructors - Summary

LE Strehl without spiders with spiders

P-CuReD P-CuReD + Houdini P-CuReD + DSPR I P-CuReD + DSPR II

0.885 ≤ x ≤ 0.895 x − 0.004 x − 0.007 x − 0.02

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Pupil fragmentation

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No island effects!

Figure: Residual pistons on segments in radians (K-band)

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Pupil fragmentation

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• in the DSPR methods the non-linear pyramid model including interference effects can be taken into account • the used matrices are dense but only of dimension nseg × 2ns • linear computational complexity • computational expensive steps are computed off-line test stability with respect to varying parameters (atmosphere, photon flux, ...)

• usage of less illuminated subapertures under spider legs: illumination factor 75 % for P-CuReD, 45 % for DSPR • frequency dependent gain control

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Summary Spiders don’t seem to be a problem for the METIS instrument.

Here comes a little R-Band sensing teaser :-)

Figure: residual piston on segments

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Future Work - Enhancement

• different/broader sensing bands • different spider thickness • improve stability of reconstruction in combination with M4 • optimize number/size of subapertures for pyramid sensor • extension of existing algorithms taking the full pyramid sensor model into account • stability for non-modulated sensor • run algorithms on test benches and telescopes

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