1.1 How wavefront error effects the Strehl

In the last lecture we learned that the Strehl Ratio (SR) delivered by an AO system is determined by the RESIDUAL wavefront error (sigma_tot). Which is determined by all the individual residual wavefront errors added in quadrature.

sigma_tot² = sigma_aniso² + sigma_t² + sigma_rec² + sigma_fit^{2          (1)}

in the the case sigma_tot² < 1 rad²  we can express the Strehl as:

SR ~ exp(-sigma_tot²)                                                                                 (2)

for more exact expressions for the SR see page 21 from lecture 5.
 
 
 
 
 

1.2 The wavefront error due to Anisoplanatism

If the target is off-axis (at a different position on the sky w.r.t. the guide star) then the columns of air are different (click here to see movie -from the gemini AO group) and the quality of the correction is lowered.

sigma_aniso² = (theta/theta_o)^5/3    rad²                                   (page 15 lecture 5)       (3)

typically theta_o ~ r_o/h ~ 30(lambda/2.2)^6/5   arcsec
(where lambda is the science wavelength in microns)

example: if the science target (say a faint galaxy) is 15 arcsec away from a bright AO guide star, the wavefront error at K band (lambda=2.2)  is sigma_aniso² = (15/30)^5/3  = 0.31 rad²

It is very clear from equation (3) that as soon as an object gets far "off-axis" (theta > theta_o) from the guide star the correction becomes very poor, very fast. This is not surprising since the guide star really only samples the column of air directly "on-axis" (in the direction of the guide star in the sky), all other points on the sky will decrease in SR radially away from the guide star.

 A VLT NACO AO image of a young star cluster. The guide star is the central object (SR=56% at 2.2um (K band)). Note how the images become stretched and the SR falls as the angle, theta, increases away from the guide star (this is a 40x40 arcsec image from ESO).

click here for simulation of a 120 arcsec image of a cluster of stars from Francois Rigaut (Gemini AO Group) that shows a movie of Anisoplanatism

SCIENCE TIP: the closer your science target is to your guide star the better!
 
 
 
 

1.3 The wavefront error due to time delay

 sigma_t² = (deltaT/tau_o)^5/3            rad²                                                                                       (4)

where      tau_o = 0.34 (r_o/v);   where v is the "average" wind speed.
 

Clearly the faster we sample and correct the the turbulence the better the correction. Indeed one would like the time delay in out system to be zero, yet in practice there is always a delay due to:

1) mid-point of the WFS integration (long enough to collect a decent number of photons in each WFS subaperture; the brighter the guide star the better)
2) reading out the wavefront sensor (with a split frame transfer CCD this is equal to the fastest integration)
3) reconstruction of the wavefront
4) the DM actuators arriving at the commanded position

the time delay = deltaT is roughly equal to the sum of steps #1-4  (often with fast electronics/computers this is roughly equal to twice the fastest integration time of the WFS)

So there is always some loss of correction due to deltaT

example:  If we run an AO system at 550 Hz, (e.g. sampling the wavefront 550 times per second) then the time delay between measurement and correction is close to twice the integration time: delta ~ 2x(1/550 Hz) ~ 0.0036 sec.

Therefore if we have a typical wind of 8 m/s and and r_o(0.55) at V of 15 cm; then we would have an r_o(2.2) =15(lambda/0.55)^6/5 = 79 cm
Hence tau_o(2.2)= 0.34 (r_o/v)= 0.0336 sec.
So from equation 4 we see sigma_t² = (deltaT/tau_o)^5/3  = 0.024 rad²

SCIENCE TIP: for bright guide stars one wants to run the WFS as fast as is allowed by the electronics
 
 
 
 

1.4 The WFS measurement uncertainty

Since there is always error in any measurement there is a wavefront error that is generated by the error in the WFS measurement of the wavefront (sometimes this is called the reconstructor error). In a Shack-Hartmann sensor it is due to the fact that there is read-out noise every time you readout the CCD, there is error due to the thickness of the centroid spots, there is error due to photon noise, etc.

In the case of a Shack-Hartmann WFS there is error in each slope in each subaperture, but integrated over the whole pupil gives an error that we can express as:

sigma_rec² = (A/N)(1+4n²/N)(lambda_WFS/lambda)²    rad²;                                               (5)

where: A is constant of ~13 for a typical Shack-Hartman WFS (Sandler et. al 1994)
             N is the number of detected photons from the guide star per subap per sample (the brighter the star the bigger the N)
              n is the rms readnoise in electrons of the WFS CCD (usually around 3-10 e- rms)
              lambda_WFS is the wavelength of the WFS (usually ~0.7 microns)
              lambda is the science wavelength (for example K band = 2.2 microns)

example:  If we have a Ns=108 subaperture Shack-Hartmann FWS at a 6.5m telescope (like the MMT) then a V=11 magnitude guide star should give us (after all the optical losses leading to a WFS transmission=12%)

134 photons/sample if each integration is 0.0018 s (assuming we are running at 550 Hz)

a note about stellar magnitudes: if a star has a magnitude of 0 then it has 9.60x10¹⁰ photons/s/m²/um in the V band (0.55 um)
now a V=11 mag star is in fact fainter. It is 10^((11-0)/2.5)=25118 times fainter in fact. So if a V=11 guide star gives 134 photons/subap/sample then we know that a guide star of 16 magnitude is 10^{((16-11)/2.5)} = 100 times fainter = 1.34 photons/subap/sample at 550 Hz.

So if n=7 e rms; lambda_WFS = 0.7 um; lambda = 2.2 um; and we are integrating for 0.0018 sec, and we have a V=11 guide star then N=134 photons/subap/sample; and from equation (5) we see

for a V=11 guide star sigma_rec² = (A/N)(1+4n²/N)(lambdaWFS/lambda)²   = 0.024 rad²           which is pretty small

But if we have a fainter guide star say 10 times fainter (V=13.5 mag), then N=13.4 photons/subap/sample if we keep the integration as short as 0.0018 seconds.

for a V=13.5 guide star sigma_rec² = (A/N)(1+4n²/N)(lambdaWFS/lambda)²   = 1.53 rad²                 which is large!

in this second case only having 13.4 photons on each subap is far too few to overcome the CCD's read noise of 7 e rms, hence the WFS is "read-noise limited" and there is a large error in the measurement!

SCIENCE TIP: one always wants a bright guide star.

SCIENCE TIP: it's best if the WFS is not "readnoise limited". To increase N it is often required to increase deltaT and integrate longer on the WFS. For example if we increase our integration 10 times to 0.018 sec we have N=134 and sigma_rec² =0.024 rad²; however the servo lag (with DeltaT=0.036 sec) has now been increased to sigma_t² =1.12 rad² but it is still better than  sigma_rec² = 1.53 rad²
 
 
 
 

1.5 Fitting Error

Any deformable mirror (DM) will have a finite number of actuators, and any WFS will have a finite number of "modes" that are sensed. Hence any AO system is limited to finite number of "modes" which can be corrected (see page 19 of lecture 5 for list of Zernike modes). It is roughly true that the number of subapertures in an AO system (Ns) is equal to the number of modes corrected.

Since we are trying to fit a complex wavefront with a limited number of modes there will always be some residual error in the "fit". Some higher-order modes (like Zernike mode 100) will not be corrected by an AO system that senses only the lower 80 Zernike modes. Indeed, such a system will not remove any errors due to modes 81 and higher. Luckily most of the "power" in an atmospherically distorted wavefront is in the lower order modes (87% of the power is in the 2 lowest modes: tip and tilt).

an approximation for the amount of error left after fitting is:

sigma_fit² = 0.335(d/r_o)^5/3 ;                                                                                          (6)

where d=the distance between subapertures as projected on the telescope's primary mirror.

since the telescope primary (diameter=D) is a circle and the CCD/subapertaures are square we see 0.7853*(D/d)²~Ns (the number of subapertures) then we can rewrite equation (6) as:

sigma_fit^{2 ~} 0.3(D/ro)^5/3(Ns)^-5/6       rad²                                                                      (7)

in practice the fitting error is often higher than this due to difficulty matching a square grid WFS/lenslet/CCD to a round telescope primary mirror (this is especially true for Ns<50).

example: The MMT AO system currently can correct 52 modes (although physically 108 modes/subapertures are in use with the WFS).

if Ns=52 ; D=6.5 m, and ro(2.2 um)=0.79 m then our K band fitting error is

sigma_fit^{2 ~} 0.3(D/ro)^5/3(Ns)^-5/6   = 0.373 rad²

a more realistic value for the fitting error (based on real Shack-Hartmann systems with ~50 modes) is
sigma_fit^{2 ~} 0.54(D/ro)^5/3(Ns)^-5/6   = 0.67 rad²

Note that no matter how bright the guide star used, the MMT AO system correcting, say, 52 modes will always have at least 0.673 rad² of wavefront error (hence the SR is "fitting-limited" to values of ~50% at K band for all bright (V<10 mag) guide stars).

SCIENCE TIP: For bright guide stars where N is large it is best to have as large a Ns as you can afford.
SCIENCE TIP: Systems that have large Ns have small fitting errors but they cannot lock on the more common fainter guide stars.
 
 
 
 
 
 

2.0 Optimizing an AO system

Since:

SR ~ exp(-( sigma_aniso² + sigma_t² + sigma_rec² + sigma_fit² ))

therefore, from equations 3,4,5 and 7:

SR ~ exp(-((theta/theta_o)^5/3 + (deltaT/tau_o)^5/3 + (13/N)(1+4n²/N)(lambdaWFS/lambda)²  +  0.54(D/ro)^5/3(Ns)^-5/6   )))

There are two important cases to look at in detail: bright and faint guide stars.
 

2.1 Bright Guide Stars

We see that for BRIGHT guide stars (N>>n) we can optimize the SR by:
minimizing the fitting error which dominates the residual error
this cam be done by:
  1) building a fast AO system with many actuators (deltaT < 0.001 sec, Ns>1000)

Io is a good example of a bright guide "star" (V~7). The sharpest image is from NASA's Galileo spacecraft. The image to the left of that is a Keck AO image (Ns=320) at 2.2 microns. The image below that is at L band (3.5 microns). Note how the hot volcanoes are "glowing" in the L band image. the bottom right image is without AO. Click here to see a movie of Io in orbit.
 
 
 
 
 
 

2.2 Faint Guide Stars

But in the case of faint guide stars (when N~n) we have to  worry most about the reconstructor error, so to maximize the SR we need to:
minimizing the reconstructor error which dominates when (N~n)
this can be done by:
  1) building an AO system with fewer subapertures (so each subaperture gets as many photons as possible)
  2) integrating as long as possible on the WFS CCD (so N>n², at the risk of somewhat increasing phase lag errors)
  3) minimizing the WFS CCD readnoise (n<3 e rms per readout)

above is a picture of a very faint guide star (V~21,I~17). This star is almost a brown dwarf. Here we used the Gemini/Hokupa'a curvature AO system. With only Ns=36 on a D=8.2m telescope, and with n=0 (since Avalanche PhotoDiodes are used instead of CCDs) very faint guide stars can be used. However, the final SR is quite low (<5%) but this H band image (1.65 microns) still detects a possible companion (which is likely a background star) that is 10,000 times fainter. This detection would not have been possible from the ground without a "faint guide star optimized" AO system.

Conclusions:
      An AO system optimized for bright guide stars is very different from one optimized for faint guide stars.

Above we have a real example of how the performance of an AO system is effected by the brightness of the guide star (V=5-16 mag), the readnoise (n=8 or 35 e rms), the number of modes corrected (Ns=52 or 80), the integration time (deltaT=0.0036 or 0.01 seconds). This is initial engineering data from the last MMT AO run.  The observed SRs are close to that expected from theory, but are somewhat lower due to 0.02 arcsec rms vibrations in the MMT at ~18 and ~38 Hz.
 

Here we see how different AO systems have had different amounts of success in publishing refereed astronomical papers. Some very simple conclusions can be drawn: the systems that are most sensitive to faint guide stars (PUEO, UHAO) have the most publications.  The Adonis system was the only AO system in the south which made it very productive. Laser guide star systems have not yet solved the "faint-guide star problem".
 

Other lessons:
The site of the telescope. Most of these papers are produced at the best sites (Mauna Kea and Chile)

Another key to success is the ability to work from 1-2.5 microns and have a simple User Interface.

Another big advantage is being able to utilize faint guide stars. Currently Curvature AO Systems (CS) can reach fainter limiting magnitudes (R~16-17 at an 8m) compared to Shack Hartmann systems (SH).

Another key is to have access to the southern sky. Only Adonis could operate in the south.

We also find Laser Guide Stars (LGS) are not yet ready for "prime time" despite the enormous amount of work that gone into them. Only 1 published paper (from Alfa) has utilized a laser for pure science. The potential is great however, and so hopefully we soon see LGS science papers as commonplace.