Characterizing the Atmospheric Turbulence & Systems engineering François Wildi Observatoire de Genève Credit for most slides : Claire Max (UC Santa Cruz) Adaptive Optics in the VLT and ELT era

A simplifying hypothesis about time behavior Almost all work in this field uses “Taylor’s Frozen Flow Hypothesis” –Entire spatial pattern of a random turbulent field is transported along with the wind velocity –Turbulent eddies do not change significantly as they are carried across the telescope by the wind –True if typical velocities within the turbulence are small compared with the overall fluid (wind) velocity Allows you to infer time behavior from measured spatial behavior and wind speed:

Cartoon of Taylor Frozen Flow From Tokovinin tutorial at CTIO: /~atokovin/tutorial/ /~atokovin/tutorial/

Order of magnitude estimate Time for wind to carry frozen turbulence over a subaperture of size r 0 (Taylor’s frozen flow hypothesis):  0 ~ r 0 / V Typical values: – = 0.5  m, r 0 = 10 cm, V = 20 m/sec   0 = 5 msec – = 2.0  m, r 0 = 53 cm, V = 20 m/sec   0 = 265 msec – = 10  m, r 0 = 36 m, V = 20 m/sec   0 = 1.8 sec Determines how fast an AO system has to run

But what wind speed should we use? If there are layers of turbulence, each layer can move with a different wind speed in a different direction! And each layer has different C N 2 ground V1V1 V4V4 V2V2 V3V3 Concept Question: What would be a plausible way to weight the velocities in the different layers? Concept Question: What would be a plausible way to weight the velocities in the different layers?

Rigorous expressions for  0 take into account different layers f G  Greenwood frequency  1 /  0 What counts most are high velocities V where C N 2 is big Hardy § 9.4.3

Short exposures: speckle imaging A speckle structure appears when the exposure is shorter than the atmospheric coherence time  0 Time for wind to carry frozen turbulence over a subaperture of size r 0

Structure of an AO image Take atmospheric wavefront Subtract the least square wavefront that the mirror can take Add tracking error Add static errors Add viewing angle Add noise

atmospheric turbulence + AO AO will remove low frequencies in the wavefront error up to f=D 2/n, where n is the number of actuators accross the pupil By Fraunhoffer diffraction this will produce a center diffraction limited core and halo starting beyond 2D/n 2D/nf PSD(  )

The state-of-the art in performance: The state-of-the art in performance: Diffraction limit resolution LBT FLAO PSF in H band. Composition of two 10s integration images. It is possible to count 10diffraction rings. The measured H band SR was at least 80%. The guide star has a mag of R =6.5, H=2.5 with a seeing of 0.9 arcsec V band correcting 400 KL modes

Composite J, H, K band image, 30 second exposure in each band Field of view is 40”x40” (at 0.04 arc sec/pixel) On-axis K-band Strehl ~ 40%, falling to 25% at field corner Anisoplanatism: how does AO image degrade as you move farther from guide star? credit: R. Dekany, Caltech

More about anisoplanatism: AO image of sun in visible light 11 second exposure Fair Seeing Poor high altitude conditions From T. Rimmele

AO image of sun in visible light: 11 second exposure Good seeing Good high altitude conditions From T. Rimmele

What determines how close the reference star has to be? Turbulence has to be similar on path to reference star and to science object Common path has to be large Anisoplanatism sets a limit to distance of reference star from the science object Reference Star Science Object Telescope Turbulence z Common Atmospheric Path

Expression for isoplanatic angle  0 Strehl = 0.38 at  =  0  0 is isoplanatic angle  0 is weighted by high-altitude turbulence (z 5/3 ) If turbulence is only at low altitude, overlap is very high. If there is strong turbulence at high altitude, not much is in common path Telescope Common Path

Isoplanatic angle, continued Simpler way to remember  0 Hardy § 3.7.2

Review r 0 (“Fried parameter”) –Sets number of degrees of freedom of AO system  0 (or Greenwood Frequency ~ 1 /  0 )  0 ~ r 0 / V  0 ~ r 0 / V where –Sets timescale needed for AO correction  0 (or isoplanatic angle) –Angle for which AO correction applies

Systems engineering

Issues for designer of AO systems Performance goals: –Sky coverage fraction, observing wavelength, degree of compensation needed for science program Parameters of the observatory: –Turbulence characteristics (mean and variability), telescope and instrument optical errors, availability of laser guide stars In general, residual wavefront error is the quality criterion in AO  tot 2 =    3 2   tot 2 =    3 2 

Elements of an adaptive optics system Phase lag, noise propagation DM fitting error Measurement error Not shown: tip- tilt error, anisoplanatism error Non-common path errors

Hardy Figure 2.32

Adaptive Optics wavefront errors parameters The wavefront error depends on design: –The number of degrees do freedom (i.e. +/- nb of actuators) of the deformable mirror. –The lag (delay) in the control system –The noise in the wavefront sensor which depends on WFS type (and the guide star magnitude) –The size of the field of view –Side effects like WFS non-ideality, NCPA, disturbances like vibrations And operations –integration time on wavefront sensor, wavelength, guide star mag. & offset

Wavefront phase variance due to  0 = f G -1 –If an AO system corrects turbulence “perfectly” but with a phase lag characterized by a time  then Wavefront phase variance due to  0 –If an AO system corrects turbulence “perfectly” but using a guide star an angle  away from the science target, then Wavefront errors due to  0,  0 Hardy Eqn 9.57 Hardy Eqn 3.104

Deformable mirror fitting error Accuracy with which a deformable mirror with subaperture diameter d can remove aberrations  fitting 2 =  ( d / r 0 ) 5/3 Constant  depends on specific design of deformable mirror For deformable mirror with continuous face-sheet,  = 0.28 For segmented mirror that corrects tip, tilt, and piston (3 degrees of freedom per segment)  = Those mirrors are being phased out

Dependence of Strehl on and number of DM degrees of freedom Deformable mirror fitting error only

Decreasing fitting error Assume bright natural guide star No meas’t error or iso-planatism or bandwidth error Deformable mirror fitting error only Reminder #1: Dependence of Strehl on and number of DM degrees of freedom (fitting)

Classical PIEZO actuators

MICROGATE: INSTALLING THE SHELL SHELL TRANSPORT BOX USED FOR SHELL INSTALLATION

MICROGATE: SYSTEM LEVEL ELECTROMECHANICAL AND ENVIRONMENTAL (-15°C) TEST

30 Existing MEMS mirror (sufficient for Hybrid-MOAO) Boston Micromachines 32x32 actuator, 0.6  m MEMS device, pitch 300  m

Basics of wavefront sensing Measure phase by measuring intensity variations Difference between various wavefront sensor schemes is the way in which phase differences are turned into intensity differences General box diagram: Guide star Turbulence Telescope Optics Detector Recon- structor Wavefront sensor Transforms aberrations into intensity variations Computer

Types of wavefront sensors “Direct” in pupil plane: split pupil up into subapertures in some way, then use intensity in each subaperture to deduce phase of wavefront. REAL TIME –Slope sensing: Shack-Hartmann, pyramid sensing –Curvature sensing “Indirect” in focal plane: wavefront properties are deduced from whole-aperture intensity measurements made at or near the focal plane. Iterative methods - take a lot of time. –Image sharpening, multi-dither –Phase diversity

Shack-Hartmann wavefront sensor concept - measure subaperture tilts CCD f Pupil plane Image plane

WFS implementation Compact Time-invariant

How to reconstruct wavefront from measurements of local “tilt”

Effect of guide star magnitude (measurement error) Because of the photons statistics, some noise is associated with the read-out of the Shack-Hartmann spots intensities

Effect of guide star magnitude (measurement error) Decreaing measurement error Assumes no fitting error or other error terms bright star dim star INTERLUDE: Ooh ooh ooh..... [Chord progression 2x with electric Fill 1]

Pyramid wavefront sensors

Image motion or “tip-tilt” also contributes to total wavefront error Turbulence both blurs an image and makes it move around on the sky (image motion). –Due to overall “wavefront tilt” component of the turbulence across the telescope aperture Can “correct” this image motion either by taking a very short time-exposure, or by using a tip-tilt mirror (driven by signals from an image motion sensor) to compensate for image motion (Hardy Eqn one axis) image motion in radians is indep of

Scaling of tip-tilt with and D: the good news and the bad news In absolute terms, rms image motion in radians is independent of  and  decreases slowly as D increases: But you might want to compare image motion to diffraction limit at your wavelength: Now image motion relative to diffraction limit is almost ~ D, and becomes larger fraction of diffraction limit for small

Effects of turbulence depend on size of telescope Coherence length of turbulence: r 0 (Fried’s parameter) For telescope diameter D  (2 - 3) x r 0 : Dominant effect is "image wander" As D becomes >> r 0 : Many small "speckles" develop Computer simulations by Nick Kaiser: image of a star, r 0 = 40 cm D = 1 m D = 2 m D = 8 m

Error budget so far   tot 2 =  fitting 2 +  anisop 2 +  temporal 2  meas 2  calib 2 Still need to work on these two √ √ √

Error Budgets: Summary Individual contributors to “error budget” (total mean square phase error):  anisop 2 = (  /  0 ) 5/3 –Anisoplanatism  anisop 2 = (  /  0 ) 5/3  temporal 2 = 28.4 (  /  0 ) 5/3 –Temporal error  temporal 2 = 28.4 (  /  0 ) 5/3  fitting 2 =  ( d / r 0 ) 5/3 –Fitting error  fitting 2 =  ( d / r 0 ) 5/3 –Measurement error –Calibration error,..... In a different category: 1/2 = 2.56 (D/r 0 ) 5/6 ( /D) radians 2 –Image motion 1/2 = 2.56 (D/r 0 ) 5/6 ( /D) radians 2 Try to “balance” error terms: if one is big, no point struggling to make the others tiny

Frontiers in AO technology New kinds of deformable mirrors with > 5000 degrees of freedom Wavefront sensors that can deal with this many degrees of freedom (ultra) Fast computers Innovative control algorithms “Tomographic wavefront reconstuction” using multiple laser guide stars New approaches to doing visible-light AO

We want to relate phase variance to the “Strehl ratio” Two definitions of Strehl ratio (equivalent): –Ratio of the maximum intensity of a point spread function to what the maximum would be without aberrations –The “normalized volume” under the optical transfer function of an aberrated optical system

Relation between phase variance and Strehl Maréchal Approximation –Strehl ~ exp(-   2 ) where   2 is the total wavefront variance –Valid when Strehl > 10% or so –Under-estimate of Strehl for larger values of   2