Mapping the PSF across Adaptive Optics images Laura Schreiber Osservatorio Astronomico di Bologna Email: [email protected] Abstract • Adaptive Optics (AO) has become a key technology for all the main existing telescopes (VLT, Keck, Gemini, Subaru, LBT..) and is considered a kind of enabling technology for future giant telescopes (E-ELT, TMT, GMT). • AO increases the energy concentration of the Point Spread Function (PSF) almost reaching the resolution imposed by the diffraction limit, but the PSF itself is characterized by complex shape, no longer easily representable with an analytical model, and by sometimes significant spatial variation across the image, depending on the AO flavour and configuration. • The aim of this lesson is to describe the AO PSF characteristics and variation in order to provide (together with some AO tips) basic elements that could be useful for AO images data reduction. Erice School 2015: Science and Technology with E-ELT What’s PSF • ‘The Point Spread Function (PSF) describes the response of an imaging system to a point source’ • Circular aperture of diameter D at a wavelenght λ (no aberrations) Airy diffraction disk 2 퐼휃 = 퐼0 퐽1(푥)/푥 Where 퐽1(푥) represents the Bessel function of order 1 푥 = 휋 퐷 휆 푠푛휗 휗 is the angular radius from the aperture center First goes to 0 when 휗 ~ 1.22 휆 퐷 Erice School 2015: Science and Technology with E-ELT Imaging of a point source through a general aperture Consider a plane wave propagating in the z direction and illuminating an aperture. The element ds = dudv becomes the a sourse of a secondary spherical wave. The complex field in P generated by Σ dS is: dE 0 eitkr dS r R >> a 2 2 2 푅 = 푥 + 푦 + 푧 ̴ 0 Taylor 푢2 + 푣2 2 푢푥 + 푣푦 푢푥 + 푣푦 푟 = 푧2 + (푥 − 푢) + (푦 − 푣)2 = 푅[1 + − ]1 2 ~ 푅[1 − ] 푅2 푅2 푅2 휀0 The PSF is the field intesity 퐸 푥, 푦 ≅ 푒푖(휔푡−푘푅) 푒푖푘(푢푥+푣푦)/푅푑푢푑푣 푅 distribution in the image plane aperture Erice School 2015: Science and Technology with E-ELT Imaging of a point source through a circular aperture • Spherical coordinates • …or we can define a function def .1 if u, vaperture Au,v 0 otherwise k ρ ϑ i uxvy E(x, y) Au,ve R dudv 훼 q 푘 푘 휑 • 푓 = 푥 푓 = 푦 푥 푅 푦 푅 +∞ 푖(푓푥푢+푓푦푣) 퐸 푓푥, 푓푦 = 퐴 푢, 푣 푒 푑푢푑푣 푢 = 휌 cos 휗 푣 = 휌 sin 휗 −∞ 2D Fourier = 퐹푇[퐴(푢, 푣)] transform of the 푥 = 푞 cos 휑 푦 = 푞 sin 휑 aperture function 푑푠 = 휌푑휌푑휑 푎 2휋 푘휌푞 • The field distribution in the image 푖 cos 휗−휑 퐸 푥, 푦 ∝ 휌 푒 푅 푑휌푑휗 = plane is the spatial frequency spectrum 0 0 of the aperture function 2 푅 푅 = 2휋푎 퐽 푘휌푞 1 푘휌푞 2 2 2퐽 (푘푎 sin 훼) I f x , f y E f x , f y FTAu,v 퐼 훼 = 퐼 0 [ 1 ]2 where 훼 = 푞/푅 푘푎 sin 훼 PSF computation using FFT Aperture PSF Erice School 2015: Science and Technology with E-ELT Circular apertures of different sizes D 2D 4D 8D aperture PSF Fwhm = F F/2 F/4 F/8 Erice School 2015: Science and Technology with E-ELT Circular aperture with central obstruction Obstruction = 0.0 Obstruction = 0.5 Obstruction = 0.8 Erice School 2015: Science and Technology with E-ELT Different apertures LBT JWST Erice School 2015: Science and Technology with E-ELT Diffraction in presence of aberrations . In the ideal case (red), the incident wavefront is flat; the converging optical element focuses on the focal plane the Fraunhofer diffraction pattern that would be observed at a very long distance . In the aberrated case (blue), the wavefront is distorted and the diffraction pattern on the focal plane will be different. NOTE that the aberrations can be introduced by the optical system itself In presence of aberrations, the complex field of the incoming E u,v ei u,v wave on the aperture can be expressed as: 0 The function u , v describes the wavefront distortions The intensity distribution, in Fraunhofer approximation, becomes: 2 i u,v 2 i f x , f y E f x , f y FTAu,ve Erice School 2015: Science and Technology with E-ELT Modal Representation of aberrations • Wavefront distortions at the entrance of the telescope can be represented as Piston the linear combination of a proper defined basis of functions. • Zernike polinomial are orthogonal. Tip • They are defined in polar coordinates Tilt on a unit circle as functions of azimuthal frequency m and radial Astigmatisms degree n, where m ≤ n, and n – m is Defocus even. Coma Trefoil Erice School 2015: Science and Technology with E-ELT Circular Aperture PSF in presence of aberrations (static) Z1 Z3 Z4 Z5 Z7 Z9 Z11 Aberration PSF Ideal diffraction pattern The first term (piston) does not have any effect on the image. The second term (tilt) produces a shift in the image. The higher orders terms introduce deformation in the PSF. Erice School 2015: Science and Technology with E-ELT The Strehl Ratio The Strehl Ratio (SR) is the ratio of the peak aberrated image intensity from a point source compared to the maximum attainable intensity using an ideal optical system limited only by diffraction over the system's aperture. 퐼0 푆푅 = 퐼0, 퐷퐼퐹퐹푅 Strehl Ratio Ideal PSF 2 SR≈ exp (−휎휑) Marechal approximation: SR > 0.1 where 휎휑 is the standard deviation of the phase (Wavefront error) Erice SchoolPSF with 2015: spherical Science aberration and Technology with E-ELT Circular Aperture PSF in presence of atmospheric turbulence • Before entering the atmosphere, light from stars forms plane waves (flat wavefronts). Refraction index variations in space and time due to turbulent air cells along the wavefront path, produce local variations in the wavefront phase. 3D Refractive index power spectrum Incoming flat wavefront 푘−11 3 Inomogeneous and turbulent medium Distorted wavefront • From the Power Spectrum of the Refractive Index it is possible to build a ‘phase map’, that describe, point by point, the phase delay coused by the atmospheric refracion index variation. • The Refractive index power spectrum follows a -11/3 power law (Kolmogorov) Erice School 2015: Science and Technology with E-ELT How to build a phase map (screen) • A turbolent layer can be simulated with the computer multiplyng the sqrt of power spectrum by a random phase. Frequencies (u) Power Spectrum (ps) Random phase (phase) Layer The sqrt of the power spectrum is multiplied by sin and cos of the random phase and The fourier transform (from frequency to distance space) represents tha phase map Erice School 2015: Science and Technology with E-ELT The Fried Parameter • The Fried Parameter 푟0gives a measure of the strenght of the turbulence. It is defined as the following: 2 2 −3/5 푟0 = [0.423푘 (푠푒푐휉) 푑ℎ 퐶푁 (ℎ)] Refractive index Structure function Wave number Zenith angle 2 −3/5 푟0 ∝ 휆6/5 (푠푒푐휉)[ 푑ℎ 퐶푁 (ℎ)] 2 • 푟0 gets small when turbulence is strong (퐶푁 large) • 푟0 gets bigger at longer wavelenghts (10-20 cm @ 550 nm, 50 cm @ 2200 nm) • 푟0 gets smaller as telescope looks toward the horizon • 푟0defines an aperture size over which the mean-square wavefront error is 1 rad2 Erice School 2015: Science and Technology with E-ELT Circular Aperture PSF in presence of atmospheric turbulence • We compute the istantaneus PSF remembering that: 2 i u,v 2 i f x , f y E f x , f y FTAu,ve 1 px ̴ 30 cm Erice School 2015: Science and Technology with E-ELT Circular Aperture PSF in presence of atmospheric turbulence • We compute the istantaneus PSF remembering that: 2 i u,v 2 i f x , f y E f x , f y FTAu,ve r0,λ = 0.5 m ̴ 128 pixels = 0.88’’= λ/r0 1 px ̴ 30 cm 1 px ̴ 7 mas The short exposure image consists of a large number of speckles having size λ/D Erice School 2015: Science and Technology with E-ELT Circular Aperture PSF in presence of atmospheric turbulence • To produce long exposure PSFs, we need a dynamic atmospheric model. Each of the layer is characterized by Its altitude (h), it’s altitude and its velocity. hn hn-1 The total phase delay at the telescope aperture is given by the h 푛 3 1 푙푎푦푒푟푠 in the star direction h2 h1 Erice SchoolTelescope 2015: Science aperture and Technology with E-ELT Circular Aperture PSF in presence of atmospheric turbulence • The size of the turbulence degradated image is due to: – Diffraction limit of the aperture (λ/D) – Short Esposure image spread (λ/r0) – Image motion due to overall tilt (determined by both D and r0) Short exposure PSF Long exposure PSF Diffraction limited PSF Erice School 2015: Science and Technology with E-ELT Seeing limited PSF (D/r0 ̴ 10) • The long exposure PSF (K band) looks like a 2D moffat function having FWHM = seeing D = 8 m λ/D ≈ 0.06’’ @ 2.15 흁풎 0.5 ̴ 0.6’’ 1 Moffat distribution: 퐼 푟 = 푟 [King 1971] 1+( )2훽 푟푚 whereErice School 푟푚 is 2015the :moffat Science radius and Technology and 훽 describes with E-ELT the asymptotic power low of the wings Seeing limited PSF (D = r0) • For very small apertures, atmospheric turbulence has little effect on the image size, which is determined by D. • Wavefront distortion is mainly overall tilt. Short exposure PSF Long exposure PSF Diffraction limited PSF r0 defines the diameter of a diffraction limited telescope Erice School 2015: Science and Technology with E-ELT Adaptive optics concept 1 2 3 r Deformable Distorted … 0 Mirror Wavefront Beam Splitter Control Corrected Wavefront … n-1 n System Wavefront Sensor Erice School 2015: Science and Technology with E-ELT Wavefront Sensing y Partial compensation 푛 • Zernike modes to represent the aberrated wavefront 휙 = 1 푎푗푍푗 • Subtract the measured wavefront • Compute the PSF • Integration in time TipZZ upup- Tilt toto 3005095 Incoming wavefront Fit with zernike Residual Erice School 2015: Science and Technology with E-ELT Partial compensation The fraction of light in the central core is related to the degree of correction and can be Roughly approximated with the Strehl Ratio z2 z300 z5 z50 Erice School 2015: Science and Technology with E-ELT z9 Partial compensated PSF 휆 Seeing halo ~ 푟0 Fitting error 휆 DL core ~ 퐷 Partial compensated PSF DL PSF and Partial compensated PSF Erice School 2015: Science and Technology with E-ELT What happens across the FoV? • The turbulence is measured only in the direction of the guide star, but the information is used to correct the wavefronts coming from all the directions within the FoV This causes a degradation of the correction across the FoV.