Accurate wavefront reconstruction for real-time AO with pyramid

Austrian Adaptive Optics

Accurate wavefront reconstruction for real-time AO with pyramid wavefront sensors Victoria Hutterer joint work with Andreas Obereder, Ronny Ramlau & Iuliia Shatokhina Industrial Mathematics Institute, Johannes Kepler University (JKU), Linz, Austria. Johann Radon Institute for Computational and Applied Mathematics, Linz, Austria.

Bologna, June 05, 2018

Principles of Adaptive Optics V. Hutterer

Pyramid sensor model

Wavefront reconstruction using pyramid sensors

Dealing with telescope spiders

Accurate wavefront reconstruction for real-time AO with pyramid wavefront sensors

Conclusion


Outline 1 Principles of Adaptive Optics 2 Pyramid sensor model 3 Wavefront reconstruction using pyramid sensors

Linear wavefront reconstruction Non-linear wavefront reconstruction 4 Dealing with telescope spiders 5 Conclusion






Conclusion


Extremely Large Telescope (ELT) • The world’s biggest eye on the sky: largest optical/near-infrared telescope in the world (primary mirror ∼ 39 m)

Credit: ESO






Conclusion


Extremely Large Telescope (ELT) • The world’s biggest eye on the sky: largest optical/near-infrared telescope in the world (primary mirror ∼ 39 m) • on top of Cerro Armazones (3060m) in the Atacama Desert of northern Chile

Credit: ESO Principles of Adaptive Optics V. Hutterer





Conclusion


Extremely Large Telescope European Southern Observatory (ESO)

• • • •

2005: first plans for ELT 2008: Austria joins ESO 2014: decision to build the ELT ∼ 2026: first light

Credit: ESO Principles of Adaptive Optics V. Hutterer





Conclusion


The world’s biggest eye on the sky






Conclusion


Adaptive Optics (AO) Adaptive Optics is a technique for correcting optical distortions arising during the imaging process.

components of AO system: • deformable mirror • wavefront sensor • control system

Credit: Claire Max, UCSC Principles of Adaptive Optics V. Hutterer





Conclusion


Pyramid Wavefront Sensor • chosen sensor for ELT instruments METIS, MICADO, and EPICS • aim: direct imaging of exoplanets • DM update 3000 times per second • 0.3 ms time for reconstruction • 29600 DM actuators • 28800 active subapertures Credit: ESO






Conclusion


Inverse Problem

Wavefront sensor measuring process: s = PΦ + η

s ... wavefront sensor measurements P ... wavefront sensor operator Φ ... incoming wavefront η ... unpredictable noise

Reconstruction of the unknown wavefront from given sensor measurements and further calculation of optimal mirror deformation is an Inverse Problem. Principles of Adaptive Optics V. Hutterer





Conclusion


Pyramid and roof WFS






Conclusion



sx (x, y ) =

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

[I1 (x, y ) + I4 (x, y )] − [I2 (x, y ) + I3 (x, y )] I0 I0 – average intensity per subaperture.

sy (x, y ) =






Conclusion



Credit: C. V´ erinaud

sx (x, y ) =

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

[I1 (x, y ) + I4 (x, y )] − [I2 (x, y ) + I3 (x, y )] I0 I0 – average intensity per subaperture.

sy (x, y ) =






Conclusion


Pyramid sensor model in distributional sense

Φ ∈ H11/6 R2



(atmospheric Kolmogorov model)




Conclusion



Φ ∈ H11/6 R2


ψaper (x, y ) = XΩy ×Ωx (x, y ) · exp (−iΦ(x, y ))






Conclusion



Φ ∈ H11/6 R2


ψaper (x, y ) = XΩy ×Ωx (x, y ) · exp (−iΦ(x, y )) XΩy ×Ωx (x, y ) → XΩ y ×Ωx



(Sobolev smoothing)




Conclusion



Φ ∈ H11/6 R2


ψaper (x, y ) = XΩy ×Ωx (x, y ) · exp (−iΦ(x, y )) XΩy ×Ωx (x, y ) → XΩ y ×Ωx ψdet (x, y ) =


(Sobolev smoothing)

1 hPSFpyr ((x, y ) − (·, ·)) , ψaper i 2π





Conclusion



Φ ∈ H11/6 R2


ψaper (x, y ) = XΩy ×Ωx (x, y ) · exp (−iΦ(x, y )) XΩy ×Ωx (x, y ) → XΩ y ×Ωx

(Sobolev smoothing)

1 ψdet (x, y ) = hPSFpyr ((x, y ) − (·, ·)) , ψaper i 2π I (x, y ) = ψdet (x, y ) · ψdet (x, y )






Conclusion


PSF of the glass pyramid PSFpyr ∈ H

−1−

R

2

1 1 1 π m n m+n+1 · δx δy + · i · (−1) · vx δy + · i · (−1) · δx vy + · (−1) · vxy 2 2 2 2π 1 1 1 vx := p.v . and vy := p.v . as well as vxy := p.v . x y xy

mn

PSFpyr =

Credit: C. V´ erinaud Principles of Adaptive Optics V. Hutterer





Conclusion


PWFS models • non-modulated PWFS Z sin[Φ(x 0 , y ) − Φ(x, y )] 0 1 sx (x, y ) = dx 2π x − x0 Ω(y )

1 2π 3

+

Z

Z

Z

sin[Φ(x 0 , y 0 ) − Φ(x, y 00 )] 00 0 0 dy dy dx (x − x 0 )(y − y 0 )(y − y 00 )

Ω(y ) Ω(x) Ω(x)

• PWFS with circular modulation Z sin[φ(x 0 , y ) − φ(x, y )]J0 [αλ (x 0 − x)] 0 1 dx sxc (x, y ) = 2π x − x0 Ω(y )

+

Z

1 2π 3

Z

sin[φ(x 0 , y 0 ) − φ(x, y 00 )]f (x 0 − x, y 0 − y 00 ) 00 0 0 dy dy dx , (x − x 0 )(y − y 0 )(y − y 00 )

Z


f (˜ x , y˜ ) := Principles of Adaptive Optics V. Hutterer

1 T

TR/2

cos[αλ x˜ sin(2πt/T )] cos[αλ y˜ cos(2πt/T )] dt

−T /2 Pyramid sensor model




Conclusion



1 2π 3

+

Z

Z

Z




+

Z

1 2π 3

Z


Z



1 T

TR/2






Conclusion



1 2π 3

+

Z

Z

Z




+

Z

1 2π 3

Z


Z



1 T

TR/2






Conclusion


Roof WFS approximation: Model with no modulation sx (x, y ) =

Z

1 2π

sin[Φ(x 0 , y ) − Φ(x, y )] 0 dx x − x0

Ω(y )

Assumptions: • roof wavefront sensor • substitute four-sided prism by two orthogonally placed two-sided prisms • two signal sets sx and sy are independent and contain information about Φ only in x- and only in y -direction correspondingly

• small wavefront distortions (as expected in closed loop), Φ 1 → sin Φ ' Φ • ignore second term =⇒Task: invert finite Hilbert Transform Principles of Adaptive Optics V. Hutterer





Conclusion



Z

1 2π

sin[Φ(x 0 , y ) − Φ(x, y )] 0 dx x − x0

Ω(y )







Conclusion



Z

1 2π

sin[Φ(x 0 , y ) − Φ(x, y )] 0 dx x − x0

Ω(y )







Conclusion


Finite Hilbert Transform Reconstructor (FHTR) The simplified roof sensor measurements are computed by Z 1 Φ(x 0 , y ) 0 sx (x, y ) ≈ dx . 2π x − x0 Ωy

Using the finite Hilbert transform operator in x-direction one gets 1 sx (x, y ) ≈ − (Tx Φ)(x, y ). 2






Conclusion


Finite Hilbert Transform Reconstructor (FHTR) The simplified roof sensor measurements are computed by Z 1 Φ(x 0 , y ) 0 sx (x, y ) ≈ dx . 2π x − x0 Ωy

Using the finite Hilbert transform operator in x-direction one gets 1 sx (x, y ) ≈ − (Tx Φ)(x, y ). 2 Inversion via direct inversion formula of finite Hilbert transform






Conclusion


Singular value type reconstructor (SVTR) The simplified roof sensor measurements are computed by Z 1 Φ(x 0 , y ) 0 sx (x, y ) ≈ dx . 2π x − x0 Ωy

Using the finite Hilbert transform operator in x-direction one gets 1 sx (x, y ) ≈ − (Tx Φ)(x, y ). 2 Inversion via SVD of finite Hilbert transform operator






Conclusion


Non-linear wavefront reconstruction Non-linear pyramid sensor model

The non-linear pyramid sensor operator Px : H11/6 R2 → L2 R2 is given by Px Φ(x, y ) := XΩy ×Ωx (x, y )

1 π

sin[Φ(x 0 , y ) − Φ(x, y )]k {n,c} (x 0 − x) dx 0 x0 − x

Z Ωy

+

1 XΩy (x) π3

Z Z Z

sin[Φ(x 0 , y 0 ) − Φ(x, y 00 )]l {n,c} (x 0 − x, y 00 − y 0 ) dy 00 dy 0 dx 0 . (x 0 − x)(y 0 − y )(y 00 − y )

Ωy Ωx Ωx






Conclusion


Non-linear Landweber Iteration for Pyramid Sensors (LIPS)

∗

Φk+1 = Φk + ωR0 (Φk ) (s − R (Φk )) ,

k = 0, 1, 2, . . .

The Fr´echet derivative R0x of the roof sensor operator without modulation is given by Z

1 R0x (Φ) ψ (x, y ) = π

cos [Φ(x 0 , y ) − Φ(x, y )] [ψ(x 0 , y ) − ψ(x, y )] dx 0 x0 − x

Ωy

The adjoint R0x (·)∗ of the Fr´echet derivative of the roof sensor operator without modulation is given by 1 R0x (Φ)∗ ψ (x, y ) = − π

Z

cos [Φ(x 0 , y ) − Φ(x, y )] [ψ(x 0 , y )+ψ(x, y )] dx 0 . x0 − x

Ωy






Conclusion


Warm restart of the system

Figure: Effect of the warm restart technique






Conclusion


Comparison linear vs non-linear

Figure: LE Strehl ratios in the K-band for the non-modulated sensor






Conclusion


Comparison linear vs non-linear

Figure: LE Strehl ratios in the K-band for the modulated sensor






Conclusion


Reconstruction qualities Algorithm SCAO mod 0 10000 1 M4 7

Modulation (λ/D) Photon flux (ph/pix/it) Frame rate (kHz) Mirror geometry Telescope spiders

≈ 0.62 [1]

Interaction matrix inversion: modal Interaction matrix inversion: zonal

Quality in end-to-end simulations (OCTOPUS) (LE Strehl ratios in the K-band) SCAO SCAO SCAO XAO mod 4 mod 0 mod 4 mod 0 10000 10000 600 50 0.5 1 0.5 3 M4 M4 M4 Fried 7 3 3 7 0.888

0.859

0.890

0.890

0.894

0.865

0.878

Preprocessed CuReD (P-CuReD) Conv. with Linearized Inverse Filter (CLIF) Pyramid FTR (PFTR) Finite Hilbert Transform Rec. (FHTR) Singular Value Type Reconstructor (SVTR)

0.871

0.887

0.779 0.74

– –

Conjugate Gradient for Normal Eq (CGNE). Steepest Descent (SD) Steepest Descent-Kaczmarz (SD-K) Linear Landweber iteration (LIPS) Linear Kaczmarz-Landweber iteration (KLIPS)

0.842 0.841 0.841 0.84 0.842

0.86 0.858 0.858 0.86 0.858

Non-linear Landweber iteration (LIPS)

0.853

0.834

Non-linear Kaczmarz-Landweber iteration (KLIPS)

0.853

0.826

– –

XAO mod 4 50 3 Fried 7 0.960

0.916 0.88 0.88 0.853 0.884

0.961 0.94 0.94 – –

0.901

0.897

0.903

[1] M. Le Louarn et. al., Latest AO simulation results for the E-ELT, poster AO4ELT5.






Conclusion


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? Principles of Adaptive Optics V. Hutterer





Conclusion


Telescope spiders






Conclusion


Split Approach

Split Approach −−−−−−−−−−−→

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






Conclusion


Conclusions & Outlook GO BACKWARDS!

Algorithm MVM P-CuReD FTR CLIF PFTR HTR TCR FHTR SVTR CGNE SD SD-K linear LIPS linear KLIPS phase retrieval JR method quasi-Newton method non-linear LIPS non-linear KLIPS


Pyramidal mask phase transmission 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3


Linearity non-linear linear 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

pyramid 3

Sensor roof

3 3 3 3 3 3 3 3 3 (3 (3 ( ( ( ( (

) ) ) ) )

( ( ( ( (

3 3 3 3 3

) ) ) ) )

one-term 3 3 3 3 3 3 3 3 3 3 3 3 3

3 ) )


3 3

Modulation yes no 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 (3) 3 3 3 3 3



Conclusion


Hands-on experiences






Conclusion


Hands-on experiences






Conclusion


Interested in details? Clare, R. M. and Lane, R. G., Phase retrieval from subdivision of the focal plane with a lenslet array, Appl. Opt. 43, 4080-4087 (2004). Korkiakoski, V., V´ erinaud, C., Louarn, M. L., and Conan, R., Comparison between a model-based and a conventional pyramid sensor reconstructor, Applied Optics 46(24), 6176-6184 (2007). Quir´ os-Pacheco, F., Correia, C., and Esposito, S., Fourier transform-wavefront reconstruction for the pyramid wavefront sensor, Proc. AO4ELT1, 07005 (2010). Frazin, R. A., Efficient, nonlinear phase estimation with the nonmodulated pyramid wavefront sensor, J.Opt. Soc. Am. A 35, 594-607 (2018). Iu. Shatokhina, A. Obereder, R. 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). Iu. Shatokhina, R. Ramlau. Convolution and Fourier transform based reconstructors for pyramid wavefront sensor, Applied Optics 56(22), 6381-6390 (2017). Iu. Shatokhina, V. Hutterer, R. Ramlau. Two novel algorithms for wavefront reconstruction from pyramid sensor data: Convolution with Linearized Inverse Filter and Pyramid Fourier Transform Reconstructor, Proc. AO4ELT5 (2017). V. Hutterer, R. Ramlau. Wavefront Reconstruction from Non-modulated Pyramid Wavefront Sensor Data using a Singular Value Type Expansion, Inverse Problems, 34, 035002 (2018). A. Obereder, S. Raffetseder, Iu. Shatokhina, V. Hutterer. Dealing with Spiders on ELTs using a Pyramid WFS to overcome residual piston effects, Proc. SPIE 10703 (2018). V. Hutterer, Iu. Shatokhina, A. Obereder, R. Ramlau. Wavefront reconstruction for ELT-sized telescopes with pyramid wavefront sensors, Proc. SPIE 10703 (2018). Principles of Adaptive Optics V. Hutterer





Conclusion