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Adaptive Optics Nicholas Devaney GTC project_ Instituto de

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Adaptive Optics Nicholas Devaney GTC project_ Instituto de Powered By Docstoc
					         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


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                                                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
                 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

				
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