1. NSF Center for Adaptive Optics UCO Lick Observatory Laboratory for Adaptive Optics Tomographic algorithm for multiconjugate adaptive optics systems Donald Gavel Center for Adaptive Optics UC Santa Cruz

2. IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004 2 Multi-conjugate AO Tomography using Tokovinin’s Fourier domain approach 1 1 Tokovinin, A., Viard, E., “Limiting precision tomographic phase estimation,” JOSA-A, 18, 4, Apr. 2001, pp873-882. Measurements from guide stars: Problem as posed: Find a linear combination of guide star data that best predicts the wavefront in a given science direction, 

3. IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004 3 Least-squares solution A-posteriori error covariance:

4. IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004 4 Some preliminaries: The Hartmann wavefront sensor model can be “ignored” Since the 2 nd equation doesn’t depend on the data, M = ||f|| 2 is an appropriate scalar measurement model (the 1 st equation above). Without loss of generality, we’ll assume the WFS has a post-filter ||f|| -2. This allows us to set M=1 (up to some frequency cut-off associated with the WFS spatial sampling). The noise spectrum is modified accordingly ( 1/f spectrum). Perform an invertible transformation on the WFS data

5. IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004 5 Re-interpret the meaning of the c vector Filtered sensor data vector: The solution again, in the spatial domain and in terms of the filtered sensor data: Define the volumetric estimate of turbulence as which is the sum of back projections of the filtered wavefront measurements. The wavefront estimate in the science direction is then which is the forward propagation along the science direction through the estimated turbulence volume. Solution wavefront

6. IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004 6 The new interpretation allows us to extend the approach into useful domains Solution is independent of science direction (other than the final forward projection, which is accomplished by light waves in the MCAO optical system) The following is a least-squares solution for spherical waves (guidestars at finite altitude) An approximate solution for finite apertures is obtained by mimicking the back propagation implied by the infinite aperture solutions An approximate solution for finite aperture spherical waves (cone beams from laser guide stars) is obtained by mimicking the spherical wave back propagations

7. IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004 7 Spherical Wave Solution Turbulence at position x at altitude h appears at position at the pupil So back-propagate position x in pupil to position at altitude h Frequencies f at altitude h scale down to frequencies at the pupil Frequencies f at the pupil scale up to frequencies at altitude h Forward propagation Backward propagation Spatial domain Frequency domain

8. IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004 8 Another algorithm 2 projects the volume estimates onto a finite number of deformable mirrors 2 Tokovinin, A., Le Louarn, M., Sarazin, M., “Isoplanatism in a multiconjugate adaptive optics system,” JOSA-A, 17, 10, Oct. 2000, pp1819-1827.

9. IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004 9 MCAO tomography algorithm summary Wavefront slope measurements from each guidestar Filter Back-project Along guidestar directions Project onto DMs Actuator commands Convert slope to phase (Poyneer’s algorithm) Guide star angles DM conjugate heights Field of view References: Tokovinin, A., Viard, E., “Limiting precision tomographic phase estimation,” JOSA-A, 18, 4, Apr. 2001, pp873-882. Tokovinin, A., Le Louarn, M., Sarazin, M., “Isoplanatism in a multiconjugate adaptive optics system,” JOSA-A, 17, 10, Oct. 2000, pp1819-1827. Poyneer, L., Gavel, D., and Brase, J., “Fast wave-front reconstruction in large adaptive optics systems with use of the Fourier transform,” JOSA-A, 19, 10, October, 2002, pp2100-2111. Gavel, D., “Tomography for multiconjugate adaptive optics systems using laser guide stars,” work in progress.  k =angle of guidestar k x = position on pupil (spatial domain) f = spatial frequency (frequency domain) h = altitude H m = altitude of DM m

10. IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004 10 The MCAO reconstruction process a pictoral representation of what’s happening Propagate light from Science target Measure light from guidestars Back- Project* to volume Combine onto DMs 12 34 *after the all-important filtering step, which makes the back projections consistent with all the data

11. IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004 11 For implementation purposes, combine steps 2 and 3 to create a reconstruction matrix A simple approximation, or clarifying example: assume atmospheric layers (C n 2 ) occur only at the DM conjugate altitudes. Filtered measurements from guide star k Shifted during back projection Weighted by C n 2 

12. IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004 12 It’s a “fast” algorithm The real-time part of the algorithm requires –O(N log(N))  K computations to transform the guidestar measurements –O(N)  K  M computations to filter and back-propagate to M DM’s –O(N log(N))  M computations to transform commands to the DM’s –where N = number of samples on the aperture, K = number of guidestars, M = number of DMs. Two sets of filter matrices, A ( f )+ Iv ( f ) and P DM ( f ), must be pre-computed –One K x K for each of N spatial frequencies (to filter measurements)-- these matrices depend on guide star configuration –One M x K for each of N spatial frequencies (to compact volume to DMs)-- these matrices depend on DM conjugate altitudes and desired FOV Deformable mirror “commands”, d m ( x ) are actually the desired phase on the DM –One needs to fit to DM response functions accordingly –If the DM response functions can be represented as a spatial filter, simply divide by the filter in the frequency domain

13. IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004 13 Simulations Parameters –D=30 m –du = 20 cm –9 guidestars (8 in circle, one on axis) –z LGS = 90 km –Constellation of guidestars on 40 arcsecond radius –r 0 = 20 cm, CP C n 2 profile (7 layer) –  = 10 arcsec off axis (example science direction) Cases –Infinite aperture, plane wave –Finite aperture, plane wave –Infinite aperture, spherical wave –Finite aperture, spherical wave (cone beam)

14. IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004 14 Plane wave 129 nm rms 155 nm rms Infinite aperture Finite aperture

15. IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004 15 Spherical Wave 421 nm rms 388 nm rms 155 nm rms Infinite Aperture Finite Aperture

16. IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004 16 Movie

17. IPAM Workshop on Estimation and Control Problems in Adaptive Optics, Jan., 2004 17 Conclusions MCAO Fourier domain tomography analyses can be extended to spherical waves and finite apertures, and suggest practical real-time reconstructors Finite aperture algorithms “mimic” their infinite aperture equivalents Fourier domain reconstructors are fast –Useful for fast exploration of parameter space –Could be good pre-conditioners for iterative methods – if they aren’t sufficiently accurate on their own Difficulties –Sampling 30m aperture finely enough (on my PC) –Numerical singularity of filter matrices at some spatial frequencies –Spherical wave tomographic error appears to be high in simulations, but this may be due to the numerics of rescaling/resampling (we’re working on this) –Not clear how to extend the infinite aperture spherical wave solution to frequency domain covariance analysis (it mixes and thus cross-correlates different frequencies)