Closed loop AO observations on a 30 meter telescope with a circular aperture

Here are a set of results illustrating simulated closed loop performance of a single conjugate adaptive optics system on a 30 meter telescope. A circular aperture was used in the simulation. The wavefront sensor used in the adaptive correction was a 64x64 subaperture Shack Hartmann sensor. The reconstructor for this simulation was generated using A++. These simulations were performed by propagating sensing (.6 micron) and detection (2.2 micron) wavefronts through an atmosphere constructed from the 6 layer Cerro Pachon model. This is the model encapsulated in the class Ellerbroek_Cerro_Pachon_model, but for this simulation the default Fried parameter of .16 cm at .5 microns was overriden with the value .25 cm. The wavefronts were then masked by the circular aperture, 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 24x24 array of samples for each lenslet, and a classical centroid calculation was used to compute the centroids. These centroids were passed through the reconstructor to compute the tip tilt and deformable mirror residuals. The residuals were passed through an integral controller, and the output of this controller was used to drive the mirrors. The 2.2 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 2.5 cm, The simulation timestep was one millisecond, and the control loop ran at 1 kHz.

Static closed loop simulations

For the first test, 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 20 ms the deformable mirror loop was closed. The simulation ran for an additional 80 ms, for a total of 100 ms. Six panels appear in the movies below. The first is the residual wavefront phase error at 2.2 microns, on a linear scale that includes the full range of wavefront phase variance in the absence of any adaptive correction. The second and third frames show the surface of the tip tilt and deformable mirrors, respectively. The fourth frame shows the wavefront residuals after adaptive correction. The fifth and sixth frames show the AO compensated and unaberrated PSF's respectively. The last two frames are 1.5 arcseconds in size, and are displayed using a log stretch

Simulation

Strehl
First closed loop simulation 95.7%
Second closed loop simulation94.2%
Third closed loop simulation 96.1%
Fourth closed loop simulation95.1%
Fifth closed loop simulation95.4%


An interesting feature of these simulations is that they suggest that the control loop will take more time to lock as the telescope aperture grows. This effect appears to arise from the fact that the DM surface quickly equilibrates to remove phase aberrations in regions of the wavefront that are locally nearly flat, while regions where the phase is changing rapidly equilibrate on a slower timescale. This leaves disjoint regions in the residual wavefront phase that can be separated by many radians. Statistically, the separation between regions grows with aperture size because the Komolgorov power spectrum is climbing steeply towards longer spatial wavelengths. At the interfaces between these regions, steep phase slopes arise that generate large centroid displacements. In contrast, all centroid values interior to the regions are near zero because the wavefront phase is nearly flat. Therefore, the reconstructor can only adjust DM actuators at the interface, and it can take many cycles of the control loop to eat away these metastable regions. Roughly speaking, the fastest one could hope to eliminate a region that is of order N actuators in size is about N control loop cycles. A simple way to improve the rate of convergence would be to employ a reconstructor that sequentially locks on Zernike modes, starting with the lowest order mode. This would avoid the generation of these metastable regions altogether, as the largest modal phase aberrations would be removed first.

There are a few shortcomings of the simulation that have a strong effect on this transient behavior, and that deserve mention. First, the DM model used in the simulation does not place any limit on the displacement of adjacent actuators. Thus, the simulation allows the interface between regions separated by arbitrarily large amounts to shrink to one subaperture in width. In reality, DM's have a physical limit on the relative displacement of adjacent actuators. If I were to include this limit in the simulation, the interface would grow to be many actuators wide. A second limitation of the simulation is that the steep interfaces cause severe distortion of the lenslet PSF's, yielding questionable centroid measurements near the interface. This definitely slows convergence. DM manufacturers design the limit on the relative displacement of adjacent actuators to correspond to the wavefront phase variations anticipated from the turbulence spectrum. Likewise, Shack Hartmann wavefront sensors are designed to operate in a linear regime in which the characteristic spot deflection is set by the turbulent spectrum as well. Realistic assumptions about the DM interactuator displacement limits would result in improved centroid measurements at the region interfaces, yielding more rapid convergence. Subsequent simulations will attempt to address these issues with higher fidelity.

There are a few things to note about the compensated PSF. The existence of waffle is evident both from the residual wavefront phase aberrations and the 4 bright spots near the corners of the PSF image. There are also 4 points aligned horizontally and vertically at about half the waffle radius. These appear to be some sort of waffle harmonic, though their origin is somewhat mysterious to me. Finally, some features of the intrinsic PSF of the tiled hexagonal aperture are just barely discernable in the compensated image. In particular, the 6 points that arise from scattering off the hexagonal tiles are visible near the edges of the PSF image.

Here is a plot of the Strehl histories for these five simulations. These histories clearly show the deformable mirror loop closing at 20 milliseconds. All asymptote to a constant value. Note that the Strehl ratios are not a monotonically increasing function of time. This behavior arises because the metastable regions in the resdiual wavefront phase have random coherence properties, and as these regions evolve they can interfere with each other constructively or destructively. The existence of fringing in the PSF as the loop closes is another manifestation of this effect. The asymptotic values of about 95.5% Strehl ratio at 2.2 microns are somewhat higher than the value of 90.3% predicted for fitting error on a 30 meter circular aperture compensated by a 65x65 actuator DM with pyramidal actuator influence functions.

Dynamic closed loop simulations

In this simulation I allowed the six layer atmosphere to evolve by giving the layers relatively small wind velocities. These velocities were determined using the two component wind model from Hardy, Section 3.3.5. The properties of each layer were as follows:

Layer Altitude (km) Weight Vx (m/s) Vy (m/s)
1 0.00 .652 -5.5 -5.8
2 2.58 .172 -8.5 -5.9
3 5.16 .055 -17.6 -6.5
4 7.73 .025 -31.2 -7.3
5 12.89 .074 -28.0 -7.1
6 15.46 .022 -14.7 -6.3


Here is a movie showing the behavior of the AO system and the compensated PSF.

Dynamic closed loop simulation

The simulation runs for 1 second. First, the tip tilt loop was closed, and then 100 milliseconds later the DM loop was closed. During the first 100 milliseconds of the movie, you can see the residual wavefront phase errors drifting slowly down and to the left. Once the DM loop closes and transient effects have died away, the PSF takes a form much like those seen in the static closed loop simulations above.

Here is a movie that shows only the residual wavefront phase.

Residual wavefront phase

Here is a plot of the Strehl history for this simulation.

Computational Requirements

These simulations were performed 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 317 seconds, and this dominated the overhead for setting up the simulation. Each simulated timestep required about 94 seconds, which broke down as follows:

Simulation Component

Time (sec)
Propagation through atmosphere and aperture 18.5
Tip tilt and deformable mirror compensation 21.2
Far field propagation to form PSF 14.4
Transformation by lenslet array 37.7
Generation of centroids 1.1
Reconstruction of residuals .8


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.). These simulations used geometric propagation through the atmosphere. Using diffractive near field propagation between atmospheric layers increases the time per step substantially.