NB: from now on we concentrate on seeing, as scintillation for large telescopes is unimportant

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1 b) intensity changes: scintillation!i/i on the ground is proportional to h!", i.e. # h e -h/h this function has maximum at h = H = 8.5 km! scintillation comes mostly from high layers! seeing and scintillation are usually uncorrelated seeing:!" # e -h/h scintillation :!I # h e -h/h also these results are confirmed by observations: a typical cell with %10 cm at h=8.5 km has angular diam. 2! planets don t scintillate (% Mars: ~5-15,Jupiter: ~40-50, Venus: ) the seeing and scintillation fluctuations are usually uncorrelated scintillation amplitude + frequency for small telescope apertures: scintillation amplitude!i/i - drops above 100 Hz - decreases strongly with D tel this fits with typical cell sizes (5-25 cm) and wind speeds at 8-10 km (~100 km/h)!i/i is largest for D $ % turbulence cell (e.g. human eye!) D tel > 1m! scintillation becomes negligible (averages out over many cells) more realistic description of wavefront distortions by atmospheric turbulence a proper description should take into account: atmospheric turbulence in reality is a 3-D field of cells with a distribution of cell sizes and temperature variations the atmosphere is in hydrostatic equilibrium, but not isothermal i.e. T = T(h) NB: from now on we concentrate on seeing, as scintillation for large telescopes is unimportant Kolmogorov (1941): in a turbulent gas flow the kinetic energy of the turbulence eddies with spatial frequencies f is # f -5/3 this power law holds between scale L u ( outer scale of turbulence = scale at which turbulence is generated) and L l (lower scale, where turbulence dissipates by viscosity! very small) on the basis of Kolmogorov turbulence, Tatarski (1961) developed the theory that is commonly used in astronomical seeing models most important parameters: Lit.: Beckers: Ann. Rev. A&A Vol.31, 13, 1993 NB: ongoing dispute about L u! for Paranal: median L u * 22 m D T (!r) & < T(r +!r) T(r) 2 > (in K 2 ) = variance in T for two points!r apart for Kolmogorov turbulence: D T (!r ) = C 2 T!r 2/3 (C 2 T = structure constant of T-variations )!T '!( '!n! structure function for n: D n (!r ) = C 2 n!r 2/3 where C n = 7.8x10-5 (P/T 2 ) C T (at ) = 0.5 µ, P in mbar)

2 fluctuations in n cause fluctuations in phase and amplitude (amplitude + scintillation: can be neglected for large telescopes)!integrated effect of all phase fluctuations along light path through atmosphere is equivalent with phase screen in front of observer that makes originally flat wavefronts corrugated for Kolmogorov turbulence the phase structure function at the entrance of the telescope is: D " (!x) = < "(x +!x) "(x) 2 > = 6.88 r 0-5/3!x 5/3 (in rad 2 ) with r 0 = the coherence length (= Fried parameter ) : r 0 (), z) = [0.423 (2,/)) 2 sec z - C n2 (h)dh ] -3/5 = ) 6/5 cos 3/5 z [ - C n2 (h)dh ] -3/5 convention: unless stated otherwise r 0 & r 0 () = 0.5 µ, z = 0 ) other ): scale #) 6/5 r 0 is related to the isoplanatic angle. 0 = radius of sky area where wavefronts can be considered as coherent (flat). 0 = r 0 /H (H = average distance to seeing layer) seeing from thick layers: H = secz { - h 5/3 C n2 dh / - C n2 dh } 3/ wind velocity! characteristic timescale: / 0 = r 0 /V wind typical values at ) = 0.5 µ : r 0 *10 cm (r 0 * size of average seeing cell)! typical seeing-dominated PSF (for r 0 < D tel.): d (FWHM) * )/r 0 * 1 if seeing layer at h=10 km!. 0 * 0.7 with V wind = 10 m/s: / 0 = s, or f 0 = 300 Hz incoming flat wavefront turbulence often occurs in 3 domains: corrugated wavefront r 0 (h 1 ). 0 (h 1 ) h 1 =10-12 km boundary troposphere-stratosphere wind-shear turbulence (jet streams!) this limits best seeing at best sites (when turb. at h 2 and h 3 negligible): r 0 * 20 cm 0 )/r 0 * 0.5,. 0 (h 1 ) * 1.4 high V wind (~100 km/h)! / 0 *2 ms h r 0 (h 3 ),. 0 (h 3 ) r 0 (h 2 ). 0 (h 2 ) NB: if there is turbulence in the lowest layers this usually dominates seeing because (n-1) h # ( h # e -h/h h 2 * 1km h 3 < 30 m planetary boundary layer turbulence from convection driven by daily solar heating if r 0 (h 2 ) = r 0 (h 1 ). 0 (h 2 ) = 10x. 0 (h 1 ) lower wind speeds! / 0 * 5 ms surface layer turbulence from wind-surface interaction (+ man-made!) small h!. 0 (h 3 ) up to few arcmin / 0 * 10 ms

3 III IMAGE QUALITY IMAGE SHARPENING TECHNIQUES III.1 DIFFRACTION-PATTERN - RAYLEIGH LIMIT light is a wave phenomenon! even for a perfect optical system the image of a pointsource is not a mathematical point simple application of the Huygens principle for all points in the aperture plane of a lens: in A: all Huygens wavelets from the lens in phase!maximum in B: )/2 optical path difference over D/2!minimum we thus get the Airy diffraction pattern here in the far field case! Fraunhofer approximation. small! sin. *. = )/D, AB = f)/d with correction for round aperture:. = 1.22 )/D, AB = 1.22 f)/d Rayleigh limit: 2 point sources can no longer be separated when they are closer than angle. (radians) in arcseconds:. = 2.52x105 )/D B A Ex.1: human eye in daylight D = 2.5 mm ) = 0.5 µm 1. = 502 f = 17 mm! AB = mm actual resolution is ~1! close to diffraction-limited! Ex.2: ELT : D= 42 m, f = 420 m, ) = 0.5 µm!. = , AB = mm but: seeing! more about the Airy pattern when measuring an optical diffraction pattern, we usually detect intensity (radiance) or flux, and not the EM fieldstrength intensity = (fieldstrength) 2 i.e. always positive f/d & focal ratio F! linear % Airy disk = 2.44 F) so: ) in µm! % Airy disk = 2.44 F) in microns in order to calculate a diffraction pattern at the focal point we have to add (amplitude, phase) of all Huygens wavelets from the entrance pupil mathematically this means: we calculate the Fourier transform (diffr. pattern) image plane = FT [(irradiance) pupil plane ] FT 2! intensity very important rule in optics: if IP = irradiance image plane, PP = irradiance pupil plane PP "FT! IP

4 diffraction patterns for non-circular apertures can become very complex! JWST full-scale model (D tel = 6.5 m) comparison of diffraction patterns for uniform circular mirror, hexagonal, JWST (greyscale log steps, range 10 4 ) (Figures: JWST web site) wavefront errors degrade diffraction-limited performance in astronomical optics there are 3 sources of wavefront errors: 1) intrinsic optical aberrations (inherent to the design, even if all optical surfaces are mathematically perfect) 2) aberrations due to optical manufacturing errors 3) seeing image quality is frequently described by Strehl ratio S: S& ratio (peak observed PSF) / (peak ideal diffraction pattern) relation of S with r.m.s. wavefront error 3 in units of ): S! e -(2"#)2 III.2 OPTICAL DEFORMATIONS AND WAVEFRONT ERRORS deformations of optical surfaces and wavefront errors are usually described by Zernike polynomials these are 2-dimensional orthogonal functions that have the advantage that they can be translated easily into the classical optical aberrations (spherical aberration, coma, astigmatism,.) Zernike polynomials describe the surface shape in polar coordinates and are ordered according to radial degree n and azimuthal degree m:

5 the first 15 Zernike terms: The terminology 3rd order, 5th order etc. has the following origin: Snell s law of refraction: sin 4 = (n /n)sin 5 series development for sin: sin 4 = 4 43/(3!) + 45/(5!) 47/(7!) + paraxial approximation: 4 very small! only first term of series: 4 = (n /n) 5 this gives aberration-free solutions, but valid only for thin beams close to the optical axis next step: include 43 term! simple description of aberrations possible: 3rd order aberrations D shapes of the first 15 Zernike s: 21 m n top views of Zernikes

6 III.3 CORRECTION OF WAVEFRONT ERRORS: ACTIVE AND ADAPTIVE OPTICS modern large optical telescopes use two related techniques for wavefront error (WFE) correction: ACTIVE OPTICS: correction of WFE due to telescope deformations ADAPTIVE OPTICS: correction of WFE due to atmospheric turbulence a) ACTIVE OPTICS classical telescopes with D up to ~ 5 m : optical shape stability was achieved passively by thickness/diameter ratio d/d~ 1:8 a 10-m mirror would weigh 200 tonnes! telescope flexures become impossible to avoid new strategies for 8-10 m telescopes developed in the 1970 s: a) segmented mirrors (e.g. Keck 10-m telescopes) b) thin monolythic mirrors (d/d ~1:50) (e.g. 8-m mirrors of VLT, Gemini, Subaru) c) both a) and b) require computer-controlled mirror supports: active optics key role in development of active optics: 3.5-m New Technology Telescope by team R.Wilson at ESO! basis for the VLT design the many computer-controlled supports get their correction signals from: 1) the finite element flexure model 2) the WFE information from the wavefront sensor via a reference star telescope WFE varies slowly! updates ~1x /min are sufficient if basic mirror shape is good, excellent WFE correction (< )/10) is possible with ~ 5 active supports per m 2

7 b) ADAPTIVE OPTICS ( AO ) Lit.: J.M.Beckers: Adaptive Optics for Astronomy Ann.Rev.A&A 31,13,1993 F.Roddier: Adaptive Optics in Astronomy, Cambridge Univ. Press 1999 AO has been developed specifically for correction of atmospheric seeing seeing varies rapidly (up to ~ 200 Hz) with small scale (~10 cm at )=500 nm)! closed loop AO correction requires: fast computer-controlled servo-systems fast + sensitive wavefront sensors with high resolution high density of active mirror supports ()=500 nm : ~ 100/m 2 in entrance pupil) the idea of AO was proposed in 1953 (!) by Babcock, but at that time the necessary technology did not exist main obstacle: fast correction can not be done with telescope mirrors! small, rapidly deformable mirror needed principle of AO system: first working AO systems: secret Star Wars program of US Air Force, end 1970 s after 1992 declassified! development for astronomical applications Note: amplitudes of wavefront slopes ( tip-tilt ) are much larger than for higher order WFE! tip-tilt and deformable mirrors separated with AO, resolution 0.1 (80 km) examples of gain in image sharpness by adaptive optics no AO with AO AO-image of Solar sunspot region (G-band , Swedish Solar Telescope, La Palma) no AO, resolution 1 (800 km)

8 WAVEFRONT SENSORS both for Active Optics and Adaptive Optics a wavefront sensor (WFS) is essential most important types: 1. Shack-Hartmann 2. Curvature sensor 3. Shearing Interferometer 4. Pyramid sensor Shearing Interferometer WFS Shack-Hartmann WFS Curvature WFS (Roddier) Pyramid WFS (Ragazzoni) DEFORMABLE MIRRORS adaptive optics became possible only with rapidly deformable mirrors (DM) technical challenge: high speed (up to ~ 200 Hz) many elements (large D tel and/or short )! N ~ ) common DM types: segmented mirror and continuous facesheet mirror actuators are usually piezo s or electromagnetic coils segmented DM +: easier manufacture and repair! cheap -: discontinuities and diffraction continuous DM -: difficult manufacture and repair! expensive +: no discontinuities and diffraction

9 adaptive optics at work ADAPTIVE OPTICS WITH NATURAL GUIDE STARS AO wavefront correction requires a reference point source most stars are point sources, but to get (WFS signal) >(WFS detector noise) we need sufficient N photons /(area A, time constant /) 1 st order estimate: A = area of aperture with diameter r 0 ())! # () 6/5 ) 2 # ) / = seeing time constant / 0 ())! # ) 3.6 6/5! limiting magnitude of natural guide stars for AO depends strongly on )! sky coverage with natural guide stars drops steeply at shorter wavelengths example: (Beckers, 1993) NB: this is independent of D tel!

10 LASER GUIDE STARS incomplete sky coverage for AO with natural guide stars (NGS) is serious limitation! artificial point sources needed this can be solved with laser guide stars (LGS) 2 techniques are used: a) Rayleigh scattering in layers up to ~ 20 km b) scattering on mesospheric sodium layer at h * 90 km pro s and con s: 6 H L 1 H L d 2 H T D Sodium LGS d = D[1- (H T /H L )]! focal anisoplanatism ( FA-error ) Rayleigh LGS turbulent layers 6 typically: 20 W ' V=6.5 mag important LGS-NGS differences : LGS is injected from below! subject to image motions! tip-tilt correction not possible with LGS, but needs NGS finite LGS distance! parallax problems and FA-errors NB: 1) tip-tilt correction can be done with WFS signal from large isoplanatic angle! faint NGS is sufficient 2) FA-errors are different for different turbulent layers! need for: high-order AO for large D tel and/or short ) large AO-corrected field solution of FA-errors development of advanced AO systems with - multiple NGS/LGS - multiple WFS for individual turbulent layers: multi-conjugate AO ultimate goal: full tomography of turbulent air column