Adaptive Optics

Nicholas DevaneyGTC project, Instituto de Astrofisica de Canarias

1. Principles2. Multi-conjugate3. 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

5/32120 )()(cos423.0

dhhCkr N

56

0 r 53

0 )(

airmassr

AO Concept

N.B. Measure and correct phase errors only

Modal Correction• Can write phase error as an expansion of Zernike

polynomials (for example)

• 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.

n

i rZar )()(

Modal Correction

• When j terms are perfectly corrected, the residual variance is given by

• 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...

35

0

rD

jj

Modal Compensation

• Recall that for uncompensated turbulence the structure function is given by

• Cagigal & Fernandez define a Generalised Fried parameter,0, such that for r < lc

• The Generalised Fried parameter is related to the residual variance by

35

0

88.6)(

rrrD

35

0

88.6)(

DrD

53

044.3

j

Structure function with modal compensation

Modal Compensation

• The partially corrected PSF has two components, a coherent core and a halo, with

• The width of the halo depends on the generalised Fried parameter as follows:

and the central peak Intensity is given by

)exp(1

)exp(

jH

jc

E

E

0

27.1 h

)exp(12

02

0jDD

I

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 secondwith a pixel of side a radians is given by

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

Let b=the fraction of this signal within the pixel of side a, so the Signal=bPS. From Poisson statistics:

taDNP BB22

tDHP Ss2

)( Bs PbPnoise

AO advantage in point source detection

So the snr is given by

For faint sources, with no AO assume pixel size matched to seeing; a=2/r0

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

tDHP Ss2

)( Bs

s

PbPbPsnr

Bsuncomp N

trDHsnr 20

Bscomp N

tSDHsnr 2

2

AO advantage in point source detection

The Gain in SNR from AO is given by

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

The integration time to reach a given magnitude with the same snr

These results are optimistic since AO usually reduces throughput and increases the background

0rSDGAO

)(log5.2 10 AOGM

21

AOGt

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

R

Perfectlens

Aberrated wavefront

triraru )(exp)()(

Phase Estimation using Interferometry

• Interference of two waves u1(r) and u2(r)

• Point Diffraction Interferometer (PDI)

))()(cos()()(2)()()( 21212

22

1 rrrararararI

pinhole Semi-transparent

Mach-Zehnder Interferometer

PinholeDetector 1

Detector 2

Ref: J.R.P. Angel, Nature vol. 368 p203 (1994)

Lateral Shear Interferometer

))(exp()()())(exp()()(

112

111

dridraruriraru

For small shear d

))(cos()()(2)()()(

)()()(

1112

22

1

111

drdraradrararI

drrdr

•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

then the change in Irradiance along the propagation path is given by

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

)),,(exp(),,(),,( zyxikWzyxIzyxA

).( 2WIWIzI

Shack-Hartmann wavefront sensor

fzd

rdrdrWrAzIxdxIxIxm

nC

nn

11

)()()()( 22121

r x

CFM

fz

Shack-Hartmann design

coll

tel

array

tel

ff

DD

'

’

Collimator

Telescope

microlens array

f

b’

•Also need sytem to select guide star in field:

-pair of steering mirrors-single mirror at reimaged pupil -pick-off system

•May need to include an Atmospheric Dispersion Corrector•More optics if want to use with both natural and laser guide stars (z ~ f2/H)

Shack-Hartmann sensor gain

Input tilt

Output

Curvature Sensing Recall Transport of Intensity equation

If in addition

then we have

).( 2WIWIzI

nrrI c)(

WIrr

nW

zI

c2)(

Curvature Sensing

FP1 P2f

l

12

12

IIIII

lCfI w

22

Curvature Sensing

f

l

P1

20

20

0

)(rf

flflr

rlf

Vibrating Membrane Mirror

Lenslet array

Optical fibers

Avalanchephotodiodes

(APDs)Computer

Bimorph DM

Real curvature sensor....

Pyramid Wavefront Sensor

Ff

P

Pyramid Wavefront Sensor

Ff

P

Pyramid wavefront sensor modulation

I2 I1

I3 I4

R

4

1

3241

),(

),(),(),(),(),(

ii

x

ycxcI

ycxcIycxcIycxcIycxcIycxcS

RbbSx 2

12

b1 b2

Canonical wavefront sensor

P M

F

D

M Periodic pattern of bars Ronchi testCrossed cylinder lenses Shack-Hartmann

F Knife edge Schlieren1/4 wave retarding spot Zernike phase contrastGrating Shearing InterferometerVariable curvature mirror Curvature sensorPyramid Pyramid sensor

Detectors employed in WFS

• 80-90% QE over 450-750nm

• stable geometry (up to 128x128 pixels available for AO)

• SNR for faint sources limited by readout noise – for AO 5e rms at 1 MHz– Multiple ports

• Need cooling

• 85% QE at 0.5 m• No read-out noise• Can be electronically

gated• One device = one

pixel (but faster than charge transfer)

• Need active quenching• Need cooling

CCDs APDs

Deformable mirror requirements

• Number of actuators • Actuator spacing (pupil size)• Actuator stroke (usually tip-tilt removed)

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

65

0

365.0

rD

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

baseplate

faceplate

electrode

Bimorph

Discrete actuator

Bimorph electrode size > 4x thicknessDifficult 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

• where is the correction wavelength, D is the telescope diameter. The final Strehl ratio is given by the product of these:

22

/21

1

D

SR

c

tilt

tilt

)exp( 2

hohoSR

hotilt SRSRSR

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

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.

22 1

d

nphphot

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

the wavefront error due to a finite servo bandwidth fs is

0

427.0rvfG

35

2

S

Gbw f

f

Optimal bandwidth

Note on calculating photonsSometimes see veryoptimistic 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:

where depends on the type of deformable mirror and the actuator geometry.

35

0

2

rd

fit

Influencefunction

Actuators persubaperture

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.

• 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

35

0

2 08.0

rd

fit

Optimal Subaperture Size

note: can simultaneousely optimise subaperture d and exposure time

Putting it all together

• Bright star Error Budget

or equivalently

dominated by fitting and bandwidth error

ncpvibuncoraliasingfittingbandwidthTilt SSSSSSSS 2

2222aliasfittingbandw

Error budget for GTCAOTip – 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 pathoptics

nm 30 30

Non-common paththermal/gravitational

nm 49.5 0

Calibration nm 35 35

Alignment nm 8 8

Segment vibration nm 60 60

WFS aliasing + DMfitting + Uncorrectedtelescope

nm 134.0 165.0

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