# Adaptive Optics Nicholas Devaney GTC project_ Instituto de

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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(i1 (r ))
                          
u2 (r )  a1 (r  d ) exp(i1 (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


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                                                2R
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 =1.35m
•   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

• 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
• 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

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