Cn2 profile measurement from Shack-Hartmann data Clélia Robert, Nicolas Védrenne,Vincent Michau, Jean-Marc Conan This work has been performed in the framework of a PhD thesis with my colleagues …. from ONERA the French aerospace research center in the optics department. This work is in connection with an adaptive optics program.

A new method to profile Cn2 I will talk about the development of Cn2 profile measurement from Shack-Hartmann data, specially with the use of scintillation signal to retrieve CN2 along the whole optical path.

Measurement of Cn² profile Motivation and techniques Shack-Hartmann data Exploitation to measure Cn2 profile Numerical validation The outline of my talk is the following.

How to measure ? Cn2 Profile Dimensioning systems Evaluation of performances A priori for servo-loop laws Profile knowledge: Profile from Observatoire de Haute Provence (ballon sonde) High variability Need of profile monitoring In many Adaptive Optics (AO) systems (MCAO for example), the knowledge of turbulence strength distribution in altitude, Cn2(h), is of great interest. Motivation. Cn2 profile may be obtained indirectly from meteorological parameters. It is classically also measured via optical means. How to measure ?

No sensitivity to law altitude layers (no propagation) Principles of Cn2 profiling : single source h Spectral analysis of scintillation structures TF (λh)-(1/2) PSDχ(ν) ν intensité dans la pupille No sensitivity to law altitude layers (no propagation) More operations needed (mode: « generalized») These means differ from the number of souces employed and type of data involved. Using scintillation data like in MASS (Multi-Aperture Scintillation Sensor) Cn2 is retrieved from correlations of the scintillation pattern produced by a single star in a pupil plane. Low vertical resolution. More uncertainties MASS (V. Kornilov, A. Tokovinin) SSCIDAR (D. Garnier) Single source: low vertical resolution

What about simultaneous exploitation of slopes and intensities ? Principles of Cn2 profiling : multiple source θ X h θ h Intensities: Cross-correlations of scintillation indices: G-SCIDAR (J. Vernin, V.A. Klueckers) In generalized SCIDAR, Cn profile is retrieved from correlation of the scintillation pattern produced by a binary star in a pupil plane. In SLODAR, it is retrieved from correlations of wavefront slopes measured on a binary using a Shack-Hartmann wavefront sensor (SHWFS). Slopes: Cross-correlations of wavefront slopes: SLODAR (R.W. Wilson) What about simultaneous exploitation of slopes and intensities ?

Measurement of Cn² profile Motivation and techniques Shack-Hartmann data Exploitation to measure Cn2 profile Numerical validation We present here the analytical background of a Cn2 profile measurement based on the exploitation of the correlations between SH data: Slopes correlations, scintillation correlation and correlations between slopes and Intensities correlations (which are referred further to coupling). Expected performances under weak-turbulence and measurement noise-free conditions are illustrated by computer simulations.

Shack Hartmann Wavefront sensor α α y m rm x Given a star with position in the field of view (FOV), SHWFS delivers wavefront slopes and intensities for each frame acquired. Wavefront slopes are estimated by center of gravity (COG). The COG computed on the mth subaperture focal image is a vector s_m(theta) with 2 components, for a slope measurement along the x/y axis. Star intensities, i_m(theta), are recorded in every subaperture by adding pixel intensities up. SH data: sm(θ) = wavefront slopes averaged on subaperture at rm im(θ) = averaged intensity of the incident wave on subaperture at rm

Correlations of data (intensities & slopes): h Turbulent volume N layers Small perturbations layers independant 1 layer at altitude h Perturbation phi

Correlations of data (intensities & slopes): rm Propagation + subaperture averaging h Small perturbation approximation (Rytov regime, σχ2 < 0.3) Slope on a source is a linear fonction of phi Convolution H m

Correlations of data (intensities & slopes): θ θh – rm rm Propagation + subaperture averaging h With 2 stars m

Correlations of data (intensities & slopes): θ Propagation + subaperture averaging h Correlation in 2 subapertures drom 2 stars m n dmn

Correlations of data (intensities & slopes): θ Propagation + subaperture averaging h θh Altitude of maximum sensitivity Maximum of correlation when Vertical resolution Hmax m n dmn

Correlations of data (intensities & slopes): θ Propagation + moyenne sur la sous-pupille h θh Generalisation continous profile m n dmn Measurement unknown Weighting

Correlations of Shack-Hartmann data Slopes SLODAR Shack-Hartmann ++ !! SCIDAR, MASS Intensities Coupling For two stars separated from theta, correlations of SHWFS data can be empirically estimated from a finite number of recorded frames. Slopes correlations, scintillation indices correlationsand their coupling are stacked in a single dimension covariance vector that will be our pseudo measurements.

Simultaneous exploitation: better sensitivity Complementarity of measurements Slopes Intensities Shack-Hartmann: dy n m dmn dx D= 0.4 m,16 x 16, λ = 0.5 μm Law layers High layers sensitivity: 80 % 15 % 5 % Se: Every layer contribution to slope correlations, scintillation and coupling correlations is illustrated for every 32 layers of a typical sampled profile S_e. Signal amplitude depends on layer strength only. S_e is a locally averaged version of a 2500 samples balloon sounding profile measured at Haute Provence Observatory (HPO).Sounded altitudes go from ground (HPO elevation) to 20~km. Altitudes are relative to HPO elevation. The SHWFS square subapertures are 2.5~cm width. Pupil diameter is 40~cm. Detection is performed at lambda=0.5~\mu m. SLODAR data correspond to plot (a) : the strongest signal is from the two first layers. Ground layer sensitivity is given by slopes correlations. Discrimination with upper layers relies on the scintillation and coupling signals which differ from a layer to one another. Simultaneous exploitation: better sensitivity

Measurement of Cn² profile Motivation and techniques Shack-Hartmann data Exploitation to measure Cn2 profile Numerical validation The outline of my talk is the following.

Problem statement Estimated covariances: SH data: sm(θ), im(θ): xki Pseudo data ou Single source Multiple sources So correlations of SHWFS data can be empirically estimated from a finite number of recorded frames. Slopes correlations, scintillation indices correlations and their coupling are stacked in a single dimension covariance vector C_mes. Single and multiple sources. This covariance vector is related to C_n2(h) according to Eq. where M is the interaction function. The column vectors of M are formed by the concatenation of weighting functions, for slope correlations and for scintillation and coupling correlations. They can be seen as correlations induced by a turbulent layer at altitude h. In Rytov regime, M is a linear operator and depends on SHWFS geometry, statistical properties of the turbulence, stars separation and distance between subapertures. C_d is the covariance vector of the noises affecting slope and intensity measurements; n represents fluctuation of C_mes due to the limited number of frames. Actually, C_mes will be estimated from measurements affected by detection noise that bias wavefront slope and scintillation covariances estimates. Direct problem: : covariance of detection noise (bias) : weighting functions : statistical noise on

Inversion of direct problem Calibration Subtraction of detection noise bias Pseudo-data: Covariance matrix Cnoise Limited statistic (convergence noise) Noise treatment: Minimisation of J with positivity constraint : regularisation parameter (depends on h) Criterion to minimise relatively to S (Cn2 profile) -1 Data likelihood A priori However these biases may be removed if the system is calibrated. The remaining fluctuation affecting Cmes is then reduced to statistical convergence. The covariance matrix C_conv is estimated from C_mes and C_d as the empirical covariance matrix of a Gaussian random variable vector. A sampled estimate of C_n2, may be retrieved from the inversion of Eq., assuming Gaussian noise. Under positivity constraint, S minimizes the penalized maximum likelihood criterium J: Penalization is performed using weighted Laplacian regularization Delta. Beta is a regularisation parameter. The weights are given by H, a diagonal matrix considered as an average expected profile. H describes a Night-time Hufnagel-Valley profile whose ground layer strength is adjusted to be conform to slope variances.

Measurement of Cn² profile Motivation and techniques Shack-Hartmann data Exploitation to measure Cn2 profile Numerical validation The outline of my talk is the following.

+ Simulation: Object model: binary star θ = 10 arcsec. Code PILOT Simulation of turbulent screens + Diffractive propagation 32 layers/ 400 frames + Shack-Hartmann: 16 x 16, d = 2.5 cm, λ = 0.5 μm (D = 40 cm) The ability of our methods to retrieve the Cn profile is illustrated using a numerical experiment. Sources and SHWFS parameters are the same as those used in previous slides. The source model represents two stars of same magnitude for CO-SLIDAR and a single star for SCO-SLIDAR. Turbulent wavefront slopes and intensities are obtained with a propagation program based on a split-step algorithm. The turbulent volume is decomposed into 32 phase screens whose spatial statistics follow von Karman spectrum. The profile used in the simulation is S_e. Corresponding Fried parameter is 4.9 cm and Rytov log amplitude variance is 0.04 at lambda=0.5 microns. Subaperture images are computed without detection noise. COG and intensities for each star are computed separately for the same turbulence occurrence to avoid overlapping in the subaperture images. A set of 400 frames is simulated. The inversion process is performed with 32 layers in the sampled turbulence profile S. Data: sm(θ), im(θ)

Preliminary results Both CO-SLIDAR and SCO-SLIDAR recovers the complete profile. Wavefront slope and scintillation correlations enable to separate contribution of the highest layers, responsible for the scintillation signal, from the lowest ones. Considering mid-altitude layers, in the range 4~km to 13~km, CO-SLIDAR is more precise than SCO-SLIDAR. Unlike SCO-SLIDAR, which has poor discrimination ability from ground to 7 km (where scintillation signal is the weakest), CO-SLIDAR benefits from the stars separation which enables a more precise estimation. N. Védrenne, V. Michau, C. Robert, J.-M Conan, « Improvements in Cn2 profile monitoring with a Shack-Hartmann wavefront sensor », Proc. SPIE Vol. 6303, septembre 2006. N. Védrenne, V. Michau, C. Robert, J.-M Conan, « Full exploitation of Shack-Hartmann data for Cn2 profile measurement », OL, octobre 2007

Conclusion and perspectives: Proposition of two original methods to profile Cn2 New exploitation of the Shack-Hartmann Sensitivity Validated numerically Study of nois effect (photons, detector, quantification) Processing of real data (SLODAR) Calibration Adaptation to close binary, moon edge, sun edge We have proposed here two new approaches for Cn profile measurement with a SHWFS reaping the benefit from both slope and intensity data. In CO-SLIDAR, slope correlations recorded on two separated stars deliver low altitude layers sensitivity. Scintillation correlations and coupling with slope deliver high altitude layer sensitivity. SCO-SLIDAR operates according to the same principle but on a single star. With a limited pupil size and in a single instrument, SCO-SLIDAR conjugates the advantages of MASS and DIMM. Both methods have been validated numerically. We plan to test both concepts on experimental SHWFS data recorded on a SLODAR instrument, in order to quantify their effective performances. Both methods should also be applied to unresolved binaries. Further investigations are required in order to improve data reduction. Additional numerical simulations could be performed by taking into account detection noise and the debiasing process. Outer scale influence on the precision of the methods should be investigated. Determination of wind profile Influence of external scale?