1. 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.

2. 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.

3. 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.

4. 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 ?

5. 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

6. 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 ?

7. 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.

8. Shack Hartmann Wavefront sensorα 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

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

10. 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

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

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

13. 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

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

15. 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.

16. 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

17. 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.

18. 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

19. 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.

20. 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.

21. + 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)

22. 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

23. 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?