Static closed loop observations

Here are a set of results illustrating simulated closed loop performance of the Palomar Adaptive Optics System (PALAO). This system contains a 16x16 element Shack Hartmann wavefront sensor and a deformable mirror, and is mounted on the Palomar 200 inch telescope. These simulations were performed by propagating sensing (.5 micron) and detection (2 micron) wavefronts through an atmosphere constructed from the 6 layer Cerro Pachon model. The wavefronts were then transformed using an annular aperture with inner and outer diameters 189 cm and 508 cm, respectively, and were reflected off a tip tilt and a deformable mirror. The wavefront used for sensing was passed through a lenslet array. The PSF of each lenslet was resolved using a 24x24 grid, and a classical centroid calculation was used to compute the centroids. These centroids were passed through the reconstructor currently used by PALAO to compute the tip tilt and deformable mirror residuals. These residuals were passed through an integral controller, and the output of this controller was used to drive the mirrors. The 2 micron wavefronts were propagated to the far field to form a simulated science PSF. The pixel scale of the atmospheric layers and the wavefronts were both chosen as 2 cm, leading to wavefronts of dimension 254x254. The simulation timestep was one millisecond, and the control loop ran at 1 kHz.

In these movies, the wind speed of all 6 atmospheric layers was set to zero, and the control loop was allowed to converge on a static wavefront. First the tip tilt loop was closed, and after a delay of 25 ms the deformable mirror loop was closed. The simulation ran for an additional 75 ms, for a total of 100 ms. Six panels appear in this movie. The first is the residual wavefront phase error. The second and third are the wavefront phases introduced by the tip tilt and deformable mirror, respectively. The fourth panel shows the Shack Hartmann spots on a log scale at 50 percent of full range. The fifth panel shows the PSF on a linear scale at full range. The sixth panel shows the PSF on a log scale at 10 percent of full range. A number of these simulations were performed with differing atmospheric aberrations to illustrate the character of the closed loop corrections. Strehl ratios for the closed loop PSF are also listed. Their variations may be interpreted as a measure of closed loop PSF stability.

Simulation

Strehl
First closed loop simulation 87.5%
Second closed loop simulation88.3%
Third closed loop simulation 87.2%
Fourth closed loop simulation89.7%
Fifth closed loop simulation 87.4%
Sixth closed loop simulation 89.1%
Seventh closed loop simulation 87.9%


Here is a plot of the Strehl histories for these seven simulations. These histories clearly show the deformable mirror loop closing at 25 milliseconds. Several of them have a rather odd evolution in the first 10 millseconds after the dm loop closes, but all asymptote to a constant value after about 60 milliseconds. This timescale is characteristic of the deformable mirror integral gain chosen for this simulation.

The asymptotic values of about 88% Strehl ratio are 7% lower than the 95% predicted on the basis of the contribution of fitting error to the overall error budget for PALAO. The poorer performance of the simulated AO system may result from partially illuminated subapertures. In PALAO light destined for the wavefront sensor passes through a field stop that helps to isolate cross-talk between the spots formed by the lenslet array. The field stop was not modelled in this simulation. The largest errors in the compensated wavefront phase appear to arise at the outer edge of the aperture, and this may be due to the fact that the spots from partially illuminated subapertures are being corrupted by adjacent fully illuminated subapertures. Higher fidelity models of the adaptive optical beam train may mitigate this discrepancy.

A notable feature of all of these simulations is the three lobe structure of the first Airy ring in the PSF. This actually appears quite consistently in science data taken at Palomar. I had believed that it was a non-common path aberration, but these simulations appear to indicate that it is a consequence of the reconstruction process.

Dynamic closed loop observations

Here are some dynamic closed loop simulations generated by using a finite wind speed of 10 meters per second for all 6 layers in the atmospheric model. These simulations were performed to compare two reconstructors. The first is the reconstructor currently used by PALAO, while the second is a reconstructor generated by Doug MacMartin. The current PALAO reconstructor projects out tip and tilt in centroid space, while Doug's reconstructor projects out tip and tilt in actuator space.

I made some improvements and modifications to these movies compared to those above. For plots of the wavefront phase I included a colorbar giving the numerical range of the plot in radians. I also made some improvements in my log scaling. The six panels in the movie now show the following:

First panel is the piston subtracted wavefront phase at the aperture.
Second panel is the phase compensation provided by the tip tilt mirror.
Third panel is the phase compensation provided by the deformable mirror.
Fourth panel is the residual wavefront phase error.
Fifth panel is the PSF on a log stretch.
Sixth panel is the unaberrated PSF on a log stretch.

Dynamic simulation with current PALAO reconstructor
Dynamic simulation with Doug MacMartin's reconstructor

Here is a movie showing the sensing wavefront intensities in the focal plane of the lenslet array. On the left these intensities are shown at full scale on a linear stretch. At right, they are shown at full scale on a log stretch.

Lenslet array focal plane intensities

Here is a plot showing the Strehl ratio history computed from the simulated PSF's for the two reconstructors above.

Strehl histories

I can make a few observations about these simulations. Residual tip tilt error is apparent in both the residual wavefront phase and the compensated PSF. This may be a consequence of choosing a tip tilt gain that was too small, or may be due to the finite responsivity of the tip tilt mirror. Additional simulations will clarify this point.

Speckle pinning is clearly present in the compensated PSF.

The Strehl ratio varies over the simulation from about 85 to 90 percent, with most of the variability occurring on timescales of order 30 milliseconds - the subaperture size divided by the wind velocity. There is an intermittent high frequency ringing effect in the Strehl histories. It is present at about .25 seconds, then goes away at about .5 seconds, and returns at about .75 seconds. It is possible that this ringing arises from the deformable mirror loop going slightly unstable.

The first simulation shows a strong edge effect on the deformable mirror that grows in time. The second simulation does not show such an effect. This edge effect is likely the cause of the approximately 2 percent relative Strehl degradation seen in the Strehl history. The edge effect arises from partially illuminated subapertures, but I'm unsure as to how the control loop introduces the error.

Here are plots of the tip tilt histories for the two simulations. These plots show the components of Zernike tilt on the uncompensated wavefront phase, the tip tilt mirror, and the deformable mirror as a function of time. Also shown are the histories of the sum of the tilt components on the tip-tilt mirror and the deformable mirror.

X tilt histories
Y tilt histories

Centroids provide a measurement of gradient tilt, and Doug MacMartin's hypothesis was that projecting tip and tilt in centroid space allows Zernike tip and tilt to leak onto the deformable mirror. Projecting out tip and tilt in actuator space will eliminate such leakage. The tip tilt histories above appear to support this hypothesis.

Computational Requirements

These simulations were run on a single 2.4 GHz Pentium IV processor. The computational time required to perform each of these simulations breaks down as follows. Generation of the six atmospheric phase screens required 6.75 seconds, and this dominated the overhead for setting up the simulation. Each simulated timestep required about 3.6 seconds, which broke down as follows:

Simulation Component

Time (sec)
Propagation through atmosphere and aperture .02
Tip tilt and deformable mirror compensation .78
Far field propagation to form PSF 1.6
Transformation by lenslet array .89
Generation of centroids .07
Reconstruction of residuals .003


In addition to this, writing out all of the intermediate results required to generate the movies took an additional 2.5 seconds per frame. The amount of time required for each of these components of the simulation scales differently with the parameters of the simulation (e.g. wavefront and layer pixel scales, aperture diameter, etc.). I should also emphasize that these simulations used geometric propagation through the atmosphere. Using diffractive near field propagation between atmospheric layers increases the time per step to about 8.3 seconds.