a fast wavefront reconstruction algorithm for XAO with pyramid WFS

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Preprocessed CuReD

Quality and speed performance

P-CuReD – a fast wavefront reconstruction algorithm for XAO with pyramid WFS Iuliia Shatokhina1 , Andreas Obereder2 , Matthias Rosensteiner2 , Ronny Ramlau1 1 – Industrial Mathematics Institute, JKU, Linz 2 – MathConsult GmbH

AO4ELT3, Florence, Italy 26-31 May 2013 This research was partially supported by the Austrian Science Fund (FWF): W1214-N15, project DK8.

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Preprocessed CuReD

Outline

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Problem

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Preprocessed CuReD

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Problem

Preprocessed CuReD


Problem

XAO (EPICS on E-ELT) D = 42 m telescope pyramid WFS with 200 × 200 subapertures, without / with linear / with circular modulation frame rate 3 kHz, time for reconstruction: 0.3 ms Task To determine the unknown wavefront φ from pyramid WFS data Sx , Sy Sx = Qx φ,

Sy = Qy φ

with Qx , Qy – nonlinear singular integral operators.

(1)

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Preprocessed CuReD


Linearized forward models

To linearize Qx , Qy , we use assumptions: roof approximation, infinite telescope size, small wavefront perturbations (closed loop) Sxn (x, ·) = φ(x, ·) ∗

1 , πx

sinc(αλ x) , πx J0 (αλ x) Sxc (x, ·) = φ(x, ·) ∗ , πx

Sxl (x, ·) = φ(x, ·) ∗

{n, l, c} – no / linear / circular modulation case, αλ = 2πα/λ,

α = bλ/D – modulation radius,

b – positive integer, J0 – zero-order Bessel function of the first kind.

(2) (3) (4)

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Preprocessed CuReD


P-CuReD

P-CuReD = Data Preprocessing + CuReD Two-step method

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Data preprocessing: Transform the modulated P-WFS data to SH-like data according to the analytical relation in the Fourier domain.

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CuReD: To the modified data apply the CuReD – very efficient reconstructor for SH-WFS data. Details on CuReD: Matthias Rosensteiner, Poster 13208, Fast wavefront reconstruction with the CuReD algorithm

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Preprocessed CuReD


Step 1: Data preprocessing Representation of the measurements in the Fourier domain (FSpyr )(u) = (Fφ)(u) · gpyr (u) · sinc(du)

(5)

(FSsh )(u) = (Fφ)(u) · gsh (u) · sinc(du)

(6)

u – spatial frequency, d – subaperture size. Fourier domain relation between the two sensors (FSsh )(u) = (FSpyr )(u) · gsh/pyr (u),

gsh/pyr (u) :=

gsh (u) . gpyr (u)

(7)

Fourier convolution theorem → relation between the two sensors in the space domain 1 Ssh (x) = √ Spyr (x) ∗ F −1 gsh/pyr (x) . 2π | {z } psh/pyr (x)

(8)

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Preprocessed CuReD


Fourier domain filters

SH gsh (u) = 2iπdu,

(9)

Pyramid n gpyr (u) = i sgn(u), ∀u ∈ [−ucut , ucut ]. ( i sgn(u), |u| > umod , l gpyr (u) = iu/umod , |u| ≤ umod , ( i sgn(u), |u| > umod , c gpyr (u) = 2i arcsin(u/u ), |u| ≤ umod . mod π

(10) (11)

(12)

where the parameter umod > 0 is defined as umod = α/λ = b/D, and b is a positive integer.

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Preprocessed CuReD


Step 1: SH-Pyr transmission filters n gsh/pyr (u) = 2πdu sgn(u), ∀u ( 2πdu sign(u), l gsh/pyr (u) = 2πdumod , ( 2πdu sgn(u), c gsh/pyr (u) = π 2 du , arcsin(u/u )

∈ [−ucut , ucut ] |u| > umod , |u| umod , |u| ≤ umod ,

mod

SH/pyramid transmission filters in the Fourier domain 3.5

no modulation linear modulation, α = 4λ /D circular modulation, α = 4λ /D

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Fourier SNR

2.5 2 1.5 1 0.5 0 -2.5

-2

-1.5

-1

-0.5

0

0.5

frequency, m-1

1

1.5

2

2.5

(13) (14) (15)

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Preprocessed CuReD


Step 1: Space domain kernels n psh/pyr (x) =

πx 2πdb2 π + sinc2 = sinc d d D2

πxb D

−

Analytical kernel 2

1.5

1

kernel

l psh/pyr (x)

πx πx π π − . sinc sinc2 d d 2d 2d

0.5

0

-0.5

-1

-1.5

-2

-1

0 space variable x, m

1

2

πx π . sinc2 2d 2d

(16)

(17)

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Preprocessed CuReD


Step 1: Convolution Convolve Sx row-wise and Sy column-wise with 1d kernel psh/pyr 1 Ssh (x) = √ Spyr (x) ∗ psh/pyr (x). 2π

Computationally very cheap, highly parallelizable and pipelinable.

(18)

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Preprocessed CuReD


Step 2: CuReD

To the modified data apply the CuReD – method for wavefront reconstruction from SH-WFS data CuReD has a linear complexity, is parallelizable and pipelinable Details on CuReD: Matthias Rosensteiner, Poster 13208, Fast wavefront reconstruction with the CuReD algorithm

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Preprocessed CuReD


LE Strehl

LE Strehl in K band: MVM and P-CuReD vs. the detected NGS photon flux.

ESO median atmosphere

ESO bad/good atmospheres

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Preprocessed CuReD


LE PSF

LE PSF in the K-band for median atmosphere and high-flux case.

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Preprocessed CuReD


Comparison: P-CuReD vs MVM

Complexity: Data preprocessing requires 26N operations. CuReD requires 20N operations. Parallelizable and pipelinable.

Strehl ratio Complexity Number of megaFLOP for ∼ 29600 actuators to be performed within 0.3ms Percentage of MVM flop Parallelizability and pipelinability

P-CuReD 0.965 46n

MVM 0.965 4n2

1.36 0.04% yes

3500 100% yes

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Preprocessed CuReD


Speed of the C-prototype System configuration: Intel(R) Xeon(R) CPU X5650 @ 2.67GHz, 12 Cores (dual hexacore) MVM: 402 ms Reconstruction time 2500

P-CuReD (2) P-CuReD (3) P-CuReD (4) P-CuReD (5)

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time (µs)

1500

1000

500

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0

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number of processor cores used

10

12

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Preprocessed CuReD


Paper

More quality results in: Iu. Shatokhina, A. Obereder, M. Rosensteiner, R. Ramlau. Preprocessed cumulative reconstructor with domain decomposition: a fast wavefront reconstruction method for pyramid wavefront sensor. Applied Optics, 52(12), 2640–2652 (2013). Derivation of analytical kernel for different modulation scenarios SE Strehl (convergence) LE Strehl for non-modulated case Noise propagation analysis

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Preprocessed CuReD


Summary

P-CuReD – fast reconstruction method for P-WFS with / without modulation. Linear complexity O(N). Strehl ratios are the same as / better than MVM. PSF is suitable for extrasolar planet search. Method suits the XAO requirements on ELTs.

Thanks for your attention!

Problem

Preprocessed CuReD


Summary

P-CuReD – fast reconstruction method for P-WFS with / without modulation. Linear complexity O(N). Strehl ratios are the same as / better than MVM. PSF is suitable for extrasolar planet search. Method suits the XAO requirements on ELTs.

Thanks for your attention!