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					Low order modes sensing for LGS
   MCAO with a single NGS
         S. Esposito, P. M. Gori, G. Brusa
         Osservatorio Astrofisico di Arcetri

                                        Presented by: S. Esposito

                                   Conf. AO4ELT 23-26 June 2009, Paris.
Presentation overview

 Low order modes indetermination in LGS MCAO
 The technique concept
 First numerical simulations
 The achieved results

                               Conf. AO4ELT 23-26 June 2009, Paris.
         Low Order Modes indetermination
     The general configuration for an MCAO system assumes to discard the global tip-
     tilt signal measured by the LGSs. However the global tilt of LGS is generated by
     the projection of all Zernike modes given on the meta-pupils on Z2 and Z3 of the
     conical LGS footprints. So, discarding LGS global tilt some information is lost

                                                        Corrected FoV 2 arcminutes
                                                        LGS = 4
                     A single realization example
                                                        DM = 2
Residual rms [rad]

                                                        Layer = 2 (1.0km, 8km)
                        Z4,Z5,Z6                        #modes = 50

                                   Blue: layer 1.0 km        The atmospheric
                                   Red: layer 8.0 km         perturbation is perfectly
                                                             reconstructed when the
                                                             tip-tilt of the LGS is
                                                             correctly measured and
                                                             included in the MCAO
                                 Zernike mode #
                                                        Conf. AO4ELT 23-26 June 2009, Paris.
                  The technique principle
1) Recover the LGSs global tilt information. Using this information the MCAO
   reconstruction WF residuals affects only low order modes (those having
   singular eigenvalues).

2) Use measurements from a single NGS placed inside the MCAO FOV to recover
   low order modes over the various MCAO system metapupils.

  The talk will be focused on the case of 2 DMs. Initial results for 3 DMs will be
  described later.

   In all the work the wavefront sensing is simulated by decomposing the
   received wavefront into zernike polinomials.

                             AO4ELT, 22-26 June, 2009, Paris.
                                                          MCAO example configuration
Including upward tilt in                                  4 LGS located on xy axis
                                                          DM = 2
MCAO interaction matrix                                   Z4 to Z11 reconstructed
 LGS tilt signal can be achieved by                       Z2 to Z11 measured (LGS)
 taking into account the upward
 propagation of the LGS in the
 MCAO reconstructor.

   T_lgs = T_dw – T_up
                                                             -   Upward IM
                                                Downw                             Dw-Up
                                                ard IM                            IM

                  LGS displacement
                  due to layer tilt
                                                  The MCAO interaction matrix
upward                                            including upward and downward
path                       Layer perturbation     propagation is obtained by
                           over metapupil         subtracting two lines of the upward
                                                  IM to the MCAO reconstructor
                                                  (downward IM).
         Atm. layers

                                                         Conf. AO4ELT 23-26 June 2009, Paris.
                                                  Applied modes on
Total MCAO IM                                     system metapupils
                                                                           DM #1       DM #2
 MCAO example configuration                       (Z4-Z11)
 Corrected FoV 2 arcmin                                      LGS #1
 4 LGS                                               Measured
 2 DM                                                modes on
 Z4 to Z11 reconstructed modes                       WFS #1 (Z2-
 Z2 to Z11 measured (LGS)                            Z11)
                  Residual rms log scale [rad]

The IM quantified the effect on the
LGS WFS of the Zernike modes 4                               LGS #2
to 11 applied on the two
metapupils. Zernike 2 & 3 are not
applied giving null measurements
on the WFS.                                                  LGS #3

                                                 Z modes #
    3 Eigenvalues ≈ 10^-8                                    LGS #4

Error for modes higher then Z6 is negligible
                                                                   Conf. AO4ELT 23-26 June 2009, Paris.
2DM case: recovering Z4,Z5 & Z6 from NGS
We use the information from 1 NGS in the FoV to reconstruct Z4,Z5,Z6 on the 2 DM

                                         unknowns               DM 1       DM 2

    Case for LGS
    reconstructor only
    (2DM, 2L).

     3 Eigenvalues ≈ 10^-8

 a,b     are the coefficients projecting focus on metapupil in focus over LGS and
         NGS footprint.
 C_cil(Z4), C_con(Z4) are the contributions to measured Z4 due to all Zs
 over the metapupils
 Focus combination on the two DMs is badly seen because the ratio between focus
 and tilt introduced on the LGS footprint by a focus on the metapupil is constant and
 do not depend on height h of the layer.
 Using Z4,Z5 and Z6 measurement on NGS having a different footprint geometry
 permits focus measurement with a different ratio . So, ambiguity is solved.
Iso and aniso-planatic
A global tilt on the metapupil generates
the same LGS & LGS tip tilt all over the
MCAO FoV. It is called isoplanatic tilt.

       wavefront over
       the metapupil
                                           A generic mode on the
                                           metapupil generates different
                                           global tilts over the MCAO.
                                           This is the anisoplanatic tilt
                                           components in a tip tilt

                                           Conf. AO4ELT 23-26 June 2009, Paris.
Recovering isoplanatic tip tilt over the
- Z2 and Z3 applied on the system metapupils, are not seen by the LGS.
-Z2 & Z3 are recovered using tip tilt measurements of the same NGS used for
Z4,Z5 & Z6.
- NGS and LGS tilt contains isoplanatic and anisoplanatic tilt components.

                                          The anisoplanatic ones are recovered
                                          by simulating a cylindrical propagation
                                          in the MCAO estimated atmosphere.
                                          This estimate is good because
 NGS measured
                                          corrected already for Z4,Z5,Z6
 tip tilt

The recovered isoplanatic Tip / Tilt are then applied on the DM’s together with
the high order commands found directly from MCAO reconstructor.

Only the tip tilt sum on both layers is recovered but this is not relevant because
of tilt isoplanaticity.
                                                     Conf. AO4ELT 23-26 June 2009, Paris.
  Numerical simulations
      Computations done using Mathematica to obtain a formal solution for a
      function projecting Zernike modes on metapupil to stars footprints at a given

Main simulation parameters:

- r0 total = 0.12m @ 500 nm
- 2 Atm.Layers
           h = 1.0 km, r0 = 0.14m                             LGS geometry
           h = 8.0 km, r0 = 0.30m                       Green disks: LGS launching
- 2 DM’s            @ 1.0 km & 8 km;                    Red disk: EELT obstruction
- 50 Zernike modes for generating Atmosphere            Red circle: EELT pupil (42m)
- 50 Zernike modes reconstructed
           45 through MCAO Rec
           Z2,Z3,Z4,Z5,Z6 with 1 NGS and MCAO Rec
- Corrected FOV = 120 arcsec
- ELT = 42 m
- No central obscuration
- 4 LGS
- LGS proj. = 3m diam (launched from
                          telescope optics)
                                                    Conf. AO4ELT 23-26 June 2009, Paris.
                                                                      Blue: Input atm.
2DM, 2 Layers results                                                 Red: residual
                                                                      Yellow: “closed loop”
Up+Dw propagation for MCAO REC.                                        Layer 1, 1.0km
LGS global tilts measured.
1 NGS to measure: Z2,Z3,Z4,Z5,Z6 as outlined

MCAO performance: the NGS wf is
estimated using a cylindrical propagation
in the reconstructed atmosphere. This

                                                 Residual rms [rad]
estimated wf is compared with the actual
wf. This is done for all the directions in the
                                                                      Layer 2, 8.0km
The mean phase residual in the considered
2 arcmin diameter FoV is 0.19 rad coming
from Z4-Z6, input rms is 137 rad (red dots)
All other modes have a negligible residual.

  All rms is given at LGS wavelength.
                                                                       Zernike mode #
2DMs: results vs # of modes
                             Reconstruction with 50 modes and NGS measuring Z2,Z3,Z4,Z5,Z6
                             Phase rms between atm & cilindrical estimate in the MCAO FoV
    Total residual rms over FoV

                                                Single iteration (open loop)

                                                  2 iterations (closed loop)

                                               # of Zernike modes

 All rms is given at LGS wavelength.
                                                                    Conf. AO4ELT 23-26 June 2009, Paris.
                                                          No optimization of MCAO
    more layers and DMs…….
                                    3 layers at 0.1km, 6.0km ,8.5km
 Briefly…….                         r0 0.156m, 0.296m, 0.390m
 1) 2DM 3 Layer                     r0 total 0.119m
 2) 3DM 3Layer                      2DMS at 0.1km, 6.0km
                                    Residual rms over FoV 1.75 rad

                             3 layers at 0.1km, 6.0km ,8.5km
                             r0 0.16m, 0.30m, 0.39m
                             r0 total 0.12m
                             3DMS at 0.1km, 6.0km, 8.5km
                             Residual rms over FoV 1.3 rad

-7 eigenvalues 10^-8 removed from reconstructor
-3DM require three different object types to solve the ambiguity (LGS & NGS for 2DMs).
-Work in progress on this case to generalize the methods to 3 DMs Z2,Z3,Z4,Z5,Z6)

       All rms is given at LGS wavelength.           Conf. AO4ELT 23-26 June 2009, Paris.
A technique to overcome the problem of the low order modes indetermination in
LGS MCAO has been presented.

The technique consider LGS upward propagation in the IM to be able to use the
LGSs global tilt informations. In such a way all modes except Z2,Z3,Z4,Z5,Z6 are
correctly reconstructed for 2DMs case.

Then a solution is presented to use a single NGS in the FoV is to determine
Z2,Z3,Z4,Z4,Z5 over the metapupils.

The phase residual rms over the whole FoV is found 0.19 rad in the case of 2 DMs
– 2 Layers.

This results shows that using the considered technique the sky coverage of a
2DM MCAO system is highly increased, as it depends on the presence of only one
NGS in the whole Patrol Field.

Work is ongoing to evaluate the technique performance in the 3DMs case.

                                                  Conf. AO4ELT 23-26 June 2009, Paris.

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