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Adaptive Optics Nicholas Devaney GTC project, Instituto de Astrofisica de Canarias 1. Principles 2. Multi-conjugate 3. Performance & challenges Outline • Background (reminder) • Concept of Adaptive Optics • Gain in Image quality • Components • Designing a system • Limitations Effect of turbulence on Images • The spatial resolution of ground based telescopes is limited to that of an equivalent diffraction-limited telescope of diameter r0 - the Fried parameter • The Fried parameter is determined by the integrated strength of turbulence along the line of sight. It therefore depends on zenith angle (). (airmass=1/cos().) • Since it is defined in terms of phase (1 rad of rms wavefront error), it also depends on wavelength r0 0.423 k (cos ) 2 1 C 2 N (h)dh 3 / 5 6 3 r0 5 r0 ( airmass ) 5 AO Concept N.B. Measure and correct phase errors only Modal Correction • Can write phase error as an expansion of Zernike polynomials (for example) ( r ) ai Z ( r ) n • Zernikes are used mostly because everyone uses them ! The first correspond to familiar Seidel aberrations (tip, tilt, defocus,Astigmatism+defocus, Coma+tilt etc.) • Useful to consider what happens as we correct n=1,2,3.... Zernikes; n is the order of correction. Modal Correction • When j terms are perfectly corrected, the residual variance is given by 5 D 3 j j r 0 • The coefficients have been calculated by Noll (JOSA 66, 207-211, 1975) • In order to determine how image quality improves as a function of the degree of compensation, first consider how the phase structure function changes... Modal Compensation • Recall that for uncompensated turbulence the structure function is given by 5 r 3 D (r ) 6.88 r 0 • Cagigal & Fernandez define a Generalised Fried parameter,0, such that for r < lc 5 D 3 D (r ) 6.88 0 • The Generalised Fried parameter is related to the residual variance by 3 3.44 5 0 j Structure function with modal compensation Modal Compensation • The partially corrected PSF has two components, a coherent core and a halo, with Ec exp( j ) EH 1 exp( j ) • The width of the halo depends on the generalised Fried parameter as follows: h 1.27 0 and the central peak Intensity is given by 0 0 2 2 I 1 exp( j ) D D Modal Correction of PSF Signal-to-Noise in AO corrected Images • Detecting faint stars against background depends on the signal-to-noise ratio (snr). This is defined as the ratio of the mean signal to the standard deviation. A detection usually requires SNR > 5. • The main sources of noise in Astronomical images are – Background noise : Sky and thermal (IR) – Detector noise – Photon noise AO advantage in point source detection Consider observing with a telescope of diameter D meters. The number of background photons detected in in t second with a pixel of side a radians is given by PB N B D 2 a 2t NB is the sky radiance in photons m-2 s-1 Sr-1 is the overall quantum efficiency For a point source of Irradiance HS photons m-2 s-1, the number of photons detected in time t is Ps H S D 2t Let b=the fraction of this signal within the pixel of side a, so the Signal=bPS. From Poisson statistics: noise (bPs PB ) AO advantage in point source detection So the snr is given by bPs snr (bPs PB ) For faint sources, with no AO assume pixel size matched to seeing; a=2/r0 Ps H S D 2t r0 t snruncomp H s D 2 NB With AO, change pixel size to match diffraction-limit; a= 2/D and the fraction of the point source flux in this pixel is given by the Strehl Ratio, S SD 2 t snrcomp H s 2 NB AO advantage in point source detection The Gain in SNR from AO is given by SD G AO r0 Example: D=10m, r0=1m, S=0.6 G=6 D=100m, r0=1m, S=0.4 G=40 In stellar magnitudes the gain is given by M 2.5 log10 (G AO ) The integration time to reach a given magnitude with the same snr 1 t 2 G AO These results are optimistic since AO usually reduces throughput and increases the background Wavefront Sensing • The vast majority of AO systems employ a wavefront sensor to measure wavefront phase errors (an alternative approach is ‘dithering’). • These are generally based on classical techniques of optical testing. Do not necessarily give quantitative measure of phase since usually works closed-loop i.e. Only need to detect null condition. • Most phase measurements are based either on Interferometry or on Propagation Phase Estimation Aberrated wavefront R Perfect lens u (r ) a(r ) exp i (r ) t Phase Estimation using Interferometry • Interference of two waves u1(r) and u2(r) 2 2 I (r ) a1 (r ) a2 (r ) 2 a1 (r ) a2 (r ) cos( 1 (r ) 2 (r )) • Point Diffraction Interferometer (PDI) pinhole Semi-transparent Mach-Zehnder Interferometer Pinhole Detector 1 Detector 2 Ref: J.R.P. Angel, Nature vol. 368 p203 (1994) Lateral Shear Interferometer u1 (r ) a1 (r ) exp(i1 (r )) u2 (r ) a1 (r d ) exp(i1 (r d )) For small shear d 1 (r d ) 1 (r ) 1 (r ) d 2 2 I (r ) a1 (r ) a2 (r d ) 2 a1 (r ) a1 (r d ) cos(1 (r ) d ) •Can vary sensitivity by adjusting d •Does not need coherent reference Wavefront sensing using propagation Most wavefront sensing techniques rely on converting wavefront gradients into measurable intensity variations. If we write the complex amplitude as A( x, y, z) I ( x, y, z) exp(ikW( x, y, z)) then the change in Irradiance along the propagation path is given by I (I .W I 2W ) z the first term is irradiance variation due to local tilt of the wavefront. The second term is due to wavefront curvature. The intensity changes are enhanced by placing a mask at one plane and measuring the resulting intensity distribution at another plane Shack-Hartmann wavefront sensor M F C r x f z xn I n 1 xI ( x )d 2 ( x ) zI n 1 A2 (r )W (r ) dr d 2 r C m 1 1 d z f Shack-Hartmann design Telescope ’ ' Dtel f microlens tel Collimator array Darray f coll •Also need sytem to select guide star in field: -pair of steering mirrors ’ b -single mirror at reimaged pupil -pick-off system •May need to include an Atmospheric f Dispersion Corrector •More optics if want to use with both natural and laser guide stars (z ~ f2/H) Shack-Hartmann sensor gain Output Input tilt Curvature Sensing Recall Transport of Intensity equation I (I .W I 2W ) z If in addition I (r rc )n then we have I W (r rc ) I 2W z n Curvature Sensing l P1 F P2 f I 2 I1 I I 2 I1 2 f 2C w I l Curvature Sensing l P1 f ( f l) r0l f 2 l r0 f f r02 Real curvature sensor.... Vibrating Membrane Mirror Bimorph DM Lenslet array Optical fibers Avalanche photodiodes Computer (APDs) Pyramid Wavefront Sensor P F f Pyramid Wavefront Sensor P F f Pyramid wavefront sensor modulation I2 I1 R b1 b2 I3 I4 S x ( xc, yc) I1 ( xc, yc) I 4 ( xc, yc) I 2 ( xc, yc) I 3 ( xc, yc) 4 b2 b1 I i ( xc, yc) Sx i 1 2R Canonical wavefront sensor P M D F M Periodic pattern of bars Ronchi test Crossed cylinder lenses Shack-Hartmann F Knife edge Schlieren 1/4 wave retarding spot Zernike phase contrast Grating Shearing Interferometer Variable curvature mirror Curvature sensor Pyramid Pyramid sensor Detectors employed in WFS CCDs APDs • 80-90% QE over 450- • 85% QE at 0.5 m 750nm • No read-out noise • stable geometry (up to • Can be electronically 128x128 pixels gated available for AO) • One device = one • SNR for faint sources pixel (but faster than limited by readout charge transfer) noise • Need active quenching – for AO 5e rms at 1 MHz • Need cooling – Multiple ports • Need cooling Deformable mirror requirements • Number of actuators • Actuator spacing (pupil size) • Actuator stroke (usually tip-tilt removed) 5 D 6 0.365 r0 on D=10m, r0=10cm at 0.5 m; 3 =1.35m • Actuator influence function, interactuator coupling • Actuator Hysterisis • Temporal response (>1kHz) • Input voltage range • Surface quality (figure, smootness, reflectivity) • Probability of failure Actuator types • Piezoelectric (PZT) – stack N elements to give range – operates over wide temperature range – hysterisis 10-20% • Electrostrictive (PMN) – low hysterisis at room temperature – long term stability – hysterisis is temperature dependent • Magnetostrictive (Terfenol-D) – 20% hysterisis – operates over large temperature range – long term stability DM types • Segmented – piston only or piston-tip-tilt • Thin plate deformable mirrors • Bimorph mirrors • Deformable secondary mirrors • Membrane mirrors • Liquid crystal mirrors DM types Discrete actuator faceplate Bimorph electrode Bimorph electrode size > baseplate 4x thickness Difficult to make high order Adaptive Secondary Mirrors • Making the secondary mirror of the telescope adaptive minimises emissivity and maximises throughput • Systems being developed for MMT and LBT • Mirror resonant frequency lower • Maintenance difficult • Calibration tricky http://caao.as.arizona.edu/caao/ Performance Limitations • The performance of real AO systems is limited by severaL sources of error. These can be studied by detailed numerical simulation or using approximate formulae. • Consider errors in wavefront tip-tilt (expressed in radians of tilt) seperately from remaining error, expressed in radians. • The corresponding Strehl ratios are given by 1 SRtilt tilt 2 2 SRho exp( ho ) 2 1 /D 2 c • where is the correction wavelength, D is the telescope diameter. The final Strehl ratio is given by the product of these: SR SRtilt SRho Sources of error • Noise in the wavefront sensor measurement • Finite number of actuators in the deformable mirror • Delay between measuring and correcting wavefront errors • Angular offset between guide source and object of interest • Uncorrectable optical errors (in the telescope & AO system) • Scintillation • ..... Noise in wavefront sensing • A general expression for the phase measurement error due to photon noise is 1 d 2 2 n ph phot where nph is the number of photons in the measurement, is the angular size of the guide source image, d is the subaperture and is the measurement wavelength. The constant depends on the details of the phase measurement. For faint sources the read noise dominates over the photon noise. Noise in wavefront sensing • Bandwidth error – The wavefront sensor has to integrate photons for a finite amount of time before a measurement can be made. In order to ensure stability, the closed-loop bandwidth should not exceed 1/6 - 1/10 of the sampling frequency. – Greenwwod defined an effective turbulence bandwidth. For a single turbulent layer moving at v ms-1 v f G 0.427 r0 the wavefront error due to a finite servo bandwidth fs is 5 f 3 bw 2 G f S Optimal bandwidth Note on calculating photons Sometimes see very optimistic estimates of throughput.... Usually will not use a standard filter in WFS Deformable mirror fitting error • Error due to the finite number of actuators in the deformable mirror. For an actuator pitch (i.e. Separation) of d, the error is given by: 5 d 3 2 fit r 0 where depends on the type of deformable mirror and the actuator geometry. Influence Actuators per function subaperture Piston only 1 1.26 Piston+tilt 3 0.14 Continuous 1 0.24-0.34 Finite Subaperture size • Finite subaperture size leads to aliasing of high-frequency wavefront errors into low-frequency errors. 5 d 3 2 0.08 fit r0 • Usually, the subaperture size is made equal to the deformable mirror actuator spacing. There is then a trade- off between snr in the wavefront sensing and fitting+aliasing errors Optimal Subaperture Size note: can simultaneousely optimise subaperture d and exposure time Putting it all together • Bright star Error Budget S STilt Sbandwidth S fitting S aliasing Suncor S vib S ncp 2 or equivalently 2 2 bandw 2 fitting 2 alias dominated by fitting and bandwidth error Error budget for GTCAO Tip – Tilt Temporal nrad 1.1 1.1 Rotator error nrad 14 0 Centroid drift nrad 19 0 Total nrad 24.0 1.1 SRtip-tilt 2.2 microns 0.946 1.0 High Order Bandwidth nm 22 22 Time delay nm 24.1 24.1 Scintillation nm 35 35 Non-common path nm 30 30 optics Non-common path nm 49.5 0 thermal/gravitational Calibration nm 35 35 Alignment nm 8 8 Segment vibration nm 60 60 WFS aliasing + DM nm 134.0 165.0 fitting + Uncorrected telescope TOTAL High-order nm 169.0 188.0 SR high-order2.2 microns 0.793 0.75 SR total 2.2 microns 0.75 0.75 What about faint stars ? • Most systems specify a sky coverage; this is tricky to verify as it depends on isoplanatic angle and on your favourite model of the sky distribution of stars • It is more practical to specify a magnitude limit for a given Strehl ratio e.g. S=0.1 For a perfect system