Gemini AO Program March 31, 2000Ellerbroek/Rigaut [ ]1 Scaling Multi-Conjugate Adaptive Optics Performance Estimates to Extremely Large Telescopes Brent Ellerbroek and Francois Rigaut Gemini Observatory

Gemini AO Program March 31, 2000Ellerbroek/Rigaut [ ]2 Presentation Outline MCAO modeling for 8-meter class telescopes Extension to ELT’s –Computational limitations –Restricting attention to anisoplanatism Mathematical formulation Cases considered Sample results –Normalized –Numerical Summary and plans

Gemini AO Program March 31, 2000Ellerbroek/Rigaut [ ]3 MCAO modeling for 8-meter class telescopes Comprehensive analysis/simulation models available Integrated first-order treatment of –Anisoplanatic effects (FOV; DM conjugates; LGS/NGS constellation) –DM/WFS fitting error –WFS noise –Time delay and servo control law –Reconstruction algorithm –Windshake and non-common path aberrations Results in hours to several days with a workstation

Gemini AO Program March 31, 2000Ellerbroek/Rigaut [ ]4 Sample Gemini MCAO Results Strehl vs LGS signal level, wavelength, and field offset 5 LGS, 16 2 subapertures 4 NGS, 2 2 subapertures 3 DM’s –0, 4.5, 9.0 km conjugates –17 actuators across pupil Median CP seeing

Gemini AO Program March 31, 2000Ellerbroek/Rigaut [ ]5 Modeling Limitations for ELT’s Assuming… –Fixed DM conjugates and guide star constellation –Fixed subaperture dimensions and actuator pitch Memory requirements scale as D 4 –Factors of 256/4096/20736 for D=32/64/96 m Computation requirements scale as D 6 –Factors of 4096/262144/ Simpler, less comprehensive approaches necessary for initial trade studies

Gemini AO Program March 31, 2000Ellerbroek/Rigaut [ ]6 Simplified Modeling Approach Evaluate anisoplanatic effects only –Fundamental error source determining performace vs field-of-view, DM conjugates, and NGS/LGS guide star constellation –Area of greatest uncertainty Other error terms can be approximated with simplified scaling laws Computation requirements greatly reduced

Gemini AO Program March 31, 2000Ellerbroek/Rigaut [ ]7 Problem Formulation Aperture- and FOV-averaged mean-square phase error  2 = N -1 ||T(x-HEy)|| 2 Where x : phase profile(s) to be corrected N=dim(x) T : piston removal operator y : WFS measurement vector H : DM-to-phase influence matrix E : DM command estimation matrix

Gemini AO Program March 31, 2000Ellerbroek/Rigaut [ ]8 Analysis Summary Goal: Determine  * 2 = min E E * = arg min E denotes averaging over turbulence statistics Solution  * 2 = N -1 trace[TA-C -1 (H T TB) T (H T TH) -1 (H T TB)] E * = (H T TH) -1 H T TBC -1 Where A = B = C =

Gemini AO Program March 31, 2000Ellerbroek/Rigaut [ ]9 FOV/Aperture Scaling for Kolmogorov Turbulence Scaling Cone Effect (Focus Anisoplanatism)  * 2 =k(D/r 0 ) 5/3 with k=k(C n 2 (h),h b ) MCAO (General Anisoplanatism)  * 2 =k(D/r 0 ) 5/3 with k=k(C n 2 (h),h b,h dm,  f /D,  b /D)

Gemini AO Program March 31, 2000Ellerbroek/Rigaut [ ]10 Cases Considered Turbulence profiles –Median Cerro Pachon (r 0 = 0.166m,  0 = 2.74”) –Median Mauna Kea (r 0 = 0.236m,  0 = 2.29”) Deformable mirrors –3 conjugate to 0, 4, 8 km –4 conjugate to 0, 2.67, 5.33, 8 km Guide stars and WFS –5 or 9 NGS –5 or 9 LGS 1 or 4 auxilliary low-order NGS

Gemini AO Program March 31, 2000Ellerbroek/Rigaut [ ]11 Guide Star and FOV Geometries ff bb bb bb Evaluation points in field-of- view 5 higher- order guide stars (NGS or LGS) 9 higher- order guide stars (NGS or LGS) Auxilliary tip/tilt or low- order NGS with LGS

Gemini AO Program March 31, 2000Ellerbroek/Rigaut [ ]12 Aperture Sampling Minimum points across pupil set by h 1  /D = 1/n To avoid interpolation and under sampling of turbulence n must scale with D to study performance vs aperture diameter Computations reasonable for n = 20 h 0 =0 h1h1 h2h2  D,n

Gemini AO Program March 31, 2000Ellerbroek/Rigaut [ ]13 Sample Normalized Results CP turbulence 3 DM’s 5 higher-order guide stars Solid: LGS, with different auxilliary NGS options Dashed: NGS, with different r =  b /  f values

Gemini AO Program March 31, 2000Ellerbroek/Rigaut [ ]14 Observations on Normalized Results Normalized phase variance (  * 2 /(D/r 0 ) 5/3 ) decreases with decreasing normalized beam shear ( h 2  f /D ) –For decreasing  f, the phase variance decreases proportionately –For increasing D, the reduction is countered by the increase in (D/r 0 ) 5/3 NGS MCAO performance degrades rapidly with increasing r =  b /  f LGS MCAO requires multiple tip/tilt or low order NGS Best NGS and LGS results proportional over a wide range of normalized beam shears ( h 2  /D )

Gemini AO Program March 31, 2000Ellerbroek/Rigaut [ ]15 Sample Numerical Results (CP Turbulence, 3 DM’s) Sample Numerical Results (CP Turbulence, 3 DM’s)

Gemini AO Program March 31, 2000Ellerbroek/Rigaut [ ]16 Observations on Numerical Results Would prefer better sampling of science field LGS MCAO with 4 auxilliary NGS –Performance varies slowly with D for fixed  f –  about 0.12  m for  f =1’, 5 m < D < 12.5 m –  about 0.17  m for  f =1.5’, 7.5 m < D < m –Tempting to scale curves to larger apertures NGS MCAO –Modestly superior to LGS MCAO when r =  b /  f =1 –Performance degrades rapidly with increasing r –What values of r are consistent with guide star models?

Gemini AO Program March 31, 2000Ellerbroek/Rigaut [ ]17 Summary and Plans Summary –Anisoplanatic errors evaluated analytically for MCAO on ELT’s –LGS results favorable with 3-4 auxilliary NGS –NGS results favorable for guide stars within science field Plans –Limited optimization of DM/guide star geometries –Accelerate computations for larger apertures by exploiting matrix structures DM-to-phase influence matrix sparse Turbulence statistics shift-invariant