**Astronomy**& Astrophysics A&A 368, 1133–1136 (2001) DOI: 10.1051/0004-6361:20010060 c ? ESO 2001 Influence of scintillations on the performance of adaptive astronomical systems with Hartmann-like wavefront sensors V. V. Voitsekhovich, V. G. Orlov, and L. J. Sanchez Instituto de Astronomia, UNAM AP 70-264 Cd. Universitaria, 04510 Mexico D.F., Mexico Received 10 November 2000 / Accepted 8 January 2001 Abstract. The influence of scintillations on image centroid measurements and on the phase reconstruction from Hartmann-like wavefront sensors is investigated quantitatively by means of computer simulations. It is shown that under the conditions of astronomical observations, the magnitude of the effect is between 10% (for excellent seeing) and 18% (for poor seeing). However, because the magnitude of the effect increases with the increasing of the turbulence strength, one can expect that under the strong-turbulence conditions the influence of scintillations on the image centroid can be quite strong. So, starting from agiven turbulence strength, it can be impossible to make a successful phase reconstruction from image centroid measurements. It is also shown that the scintillations affect in a different and complicated way the reconstruction quality of different aberrations. Nevertheless one can notice some general tendencies: The scintillations affect more strongly the reconstruction quality of the tip-tilt and high-order aberrations than the reconstruction quality of intermediate aberrations. Key words. atmospheric effects – instrumentaion: adaptive optics – techniques: high angular resolution amplitude and phase samples with the desired statistics and cross-statistics. 1. Introduction Hartmann-like wavefront sensors are often-used in astro- nomical adaptive systems for measurements of turbulence- induced wavefront distortions (Voitsekhovich et al. 1988; Rigaut et al. 1991; Jiang et al. 1993; Li et al. 1993; Colucci et al. 1994; Rigaut et al. 1997). These sensors are popular in applications because they provide a direct and simple relationship between the measurements and phase gradi- ents at the telescope aperture: It is assumed that the phase gradient averaged over a subaperture of a Hartmann mask is proportional to the corresponding image centroid off- set. This simple relationship, however, is valid if the ef- fect of amplitude fluctuations (scintillations) is not taken into account. It is widely accepted that under the weak- turbulence conditions which are of main interest for as- tronomical observations, the effect of scintillations on the image centroid is negligible (Roddier 1981) but this as- sumption has never been supported by quantitative cal- culations. In this paper we calculate the magnitude of the ef- fect of interest by means of computer simulations. The simulations are based on the recently proposed method of random wave vectors (RWV) (Kouznetsov et al. 1997; Voitsekhovich et al. 1999) that allows us to simulate the 2. Effect of scintillations on a single image centroid The set of Hartmann measurements is composed from the measurements of separated image centroids. So, at first we consider the effect of scintillations on a single image centroid, and then we investigate how the scintillations affect the quality of phase reconstruction from Hartmann data. Let the wave Ψ(ρ) pass through a thin lens of diameter d and focal length f. The centroid ρcof the image formed by this wave at the lens focal plane can be written as (Tatarski 1968): Z Z where xcand ycdenote the Cartesian coordinates of image centroid, χ(ρ) and S (ρ) are the log-amplitude and the phase of the wave Ψ, respectively, k is the wavenumber, and Gddenotes the integration over the lens aperture. However, in experiments related to the phase recon- struction from centroid measurements it is always as- sumed that the effect of amplitude fluctuations on the d2ρexp{2χ(ρ)}∇S(ρ) ρc= {xc,yc} = −f Gd (1) , k d2ρexp{2χ(ρ)} Gd Send offprint requests to: V. V. Voitsekhovich, e-mail: voisteko@astroscu.unam.mx Article published by EDP Sciences and available at http://www.aanda.org or http://dx.doi.org/10.1051/0004-6361:200100601134 V. V. Voitsekhovich et al.: Influence of scintillations on the performance Table 1. Relationship between Fried parameter r0 and image FWHM λ µ λ=0.55µm D=1 m FWHM, arcsec 1.60 1.41 1.25 1.13 1.03 0.94 0.87 0.81 0.76 D=10 m FWHM, arcsec 1.58 1.38 1.23 1.11 1.01 0.92 0.85 0.79 0.74 σ, % Fried parameter, m 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 Fig.1. Relative error σ versus the ratio of lens diameter d to the Fried parameter r0 image centroid is negligible. Mathematically this assump- tion can be written as Z where Σ denotes the lens area. Since in the problems related to propagation through atmospheric turbulence, the quantities ρcand ρ0 dom, we can define the relative error σ of image centroid measurements associated with scintillations as From a physical point of view, the error σ shows how big the relative contribution of scintillations is to the image centroid offset. In what follows, we calculate this error as a function of turbulence conditions and lens size. In order to calculate the error σ we apply the method of random wave vectors (RWV) (Kouznetsov et al. 1997; Voitsekhovich et al. 1999) that allows us to simulate the phase and log-amplitude samples with desired statis- tics and cross-statistics. The detailed description of RWV method and step-by-step simulation procedure can be found in Kouznetsov et al. (1997). The present simula- tions are performed for the Hufnagel model of C2 (turbulence strength profile) that is given by (Hufnagel 1974) So, using Eqs. (1–3) and the samples simulated by the RWV method, we calculate the relative error σ defined by Eq. (3). The number of samples used in the simulation is 5000. The simulation results are shown in Fig. 1 where the relative error σ is plotted versus the ratio of lens di- ameter d to the Fried parameter r0. In Fig. 1 we show the error of interest for three cases of seeing conditions: poor seeing (r0= 0.07 m), good seeing (r0= 0.1 m), and ex- cellent seeing (r0= 0.15 m). Because in the astronomical community it is widely accepted to characterize the seeing by the image FWHM rather than by the Fried parame- ter, we present a relationship between two parameters in Table 1. Table 1 includes two very different telescope di- ameters (1 m and 10 m) in order to stress that, for the considered range of telescope sizes, the relationship of in- terest is affected slightly by the telescope diameter. As one can see from Fig. 1, the error initially grows to a maximum magnitude, and then slowly approaches to an asymptotic level. This behavior is mainly determined by the cross-correlation between the log-amplitude and phase gradient fluctuations that have the same salient features: it starts from zero, reaches its maximum magnitude un- der some separation between the observation points, and then starts to decrease. So, in the initial phase, while the lens size is smaller than the log-amplitude phase gradi- ent cross-correlation length, the error increases because the log-amplitude and phase gradient fluctuations become more and more correlated with the increasing lens size. After reaching its maximum magnitude, the error starts to decrease due to the progressive decorrelation between the log-amplitude and phase gradient fluctuations. Physically, it means that the main contribution to the error comes from the aperture zones within which the log-amplitude and phase gradient are still correlated. With increasing lens size this contribution starts to be smaller and smaller, as is reflected in the behavior of the error. c} = −f c= {x0 d2ρ∇S(ρ), (2) ρ0 c,y0 kΣ Gd care ran- rD rD · c)2E c)2E ci (xc− x phx2 (yc− y phy2 0 0 σ =1 + (3) 2 ci nprofile n(z) = C0r−5/3 k−2 ?10 C2 0 "?z ?# ? ? ? −z −z exp + exp (4) × , z0 z1 z2 where r0 is the Fried parameter, k is the wavenumber, C0 = 1.027 10−3m−1, z0 = 4.632 103m, z1 = 103m, z2= 1.5 103m.

V. V. Voitsekhovich et al.: Influence of scintillations on the performance 1135 3. Scintillation-induced error of phase reconstruction from Hartmann data In the Hartmann test a set of image centroid measure- ments H is used to reconstruct the phase at the aper- ture. In real experiments these measurements are always affected by scintillations that can be written mathemati- cally as Z Z where ρsdenotes the center of sth subpupil, Gs denotes the integration over sth subpupil, χ and S are the log- amplitude and the phase, respectively, and N is the num- ber of subpupils. However, in order to provide a successful phase recon- struction, one always uses a scintillation-free approxima- tion H0of Hartmann measurements that can be written as Z where Σsdenotes the area of sth subpupil. So, there is always some scintillation-related error of phase reconstruction that arises from the difference between Eqs. (5) and (6). In our case, when the log- amplitude and phase fluctuations are random, this er- ror is a statistical quantity, and can be considered as a variance of the difference between the phases restored from the sets of scintillation-affected and scintillation-free Hartmann measurements. For the completeness of analy- sis it is preferable to calculate not only the total error of phase reconstruction, but also the error for each aberra- tion that allows one to see the reconstruction quality of separated aberrations. Such an investigation can be done if, for example, the Zernike phase expansion is included in the analysis (Voitsekhovich 1996). Expanding the phase S (ρ) over the set of Zernike poly- nomials, one can write d2ρexp{2χ(ρ)}∇S (ρ) H(ρs) = −f Gs , s = 1,...,N,(5) k d2ρexp{2χ(ρ)} Gs f H0(ρs) = − d2ρ∇S (ρ), s = 1,...,N, (6) Fig.2. Hartmann mask configuration kΣs Gs expansion, this procedure can be written as N L X So, denoting by al and a0 stored from scintillation-affected [H(ρs), Eq. (5)] and scintillation-free [H0(ρs), Eq. (6)] Hartmann data, respec- tively, one can write the relative scintillation-induced error σlof lth Zernike mode reconstruction as rD?al− a The simulation procedure is as follows. First, we gener- ate with the RWV method the log-amplitude and phase samples and calculate from these samples the scintillation- affected H(ρs) and scintillation-free H0(ρs) Hartmann data using Eqs. (5) and (6), respectively. Then, apply- ing the least-square procedure, we restore the Zernike co- efficients al and a0 Eq. (10) and averaging over the samples, we calculate the errors σl. The simulation has been performed with a 48-subpupil Hartmann mask for a 2 m telescope diameter. The mask configuration is shown in Fig. 2, and the number of sam- ples used in the simulation is 5000. The simulation re- sults are plotted in Fig. 3. As for the case of a single centroid error, we present the results for three cases of seeing conditions: poor seeing (r0= 0.07 m), good seeing (r0 = 0.1 m), and excellent seeing (r0 = 0.15 m). One can see that the error depends in a complicated way on the Zernike mode number that is due to a different com- plexity of the geometric structure of Zernike polynomi- als. Nevertheless, one can notice some general tendencies. X [al∇Zl(ρs/R) − H(ρs)]2= min. (9) s=1 l=2 lthe Zernike coefficients re- ?2E σl= (10) 0 l /ha2 li. ∞ X S (ρ) = alZl(ρ/R), lfor each sampling. And finally, using i=2 Z 1 al = d2ρZl(ρ/R)S (ρ), (7) πR2 GR where Zl are the two-dimensional Zernike polynomials (Noll 1976), R is the telescope radius, and GR denotes the integration over the aperture. The Zernike coefficients al to be obtained from the Hartmann data H can be written as al= T [H], (8) where the operator T is a symbolic notation for the recon- struction procedure. In the present paper we use the procedure of least- square fit to the phase gradient. In terms of the Zernike

1136 V. V. Voitsekhovich et al.: Influence of scintillations on the performance In this paper we have applied the RWV method that was developed for investigations related to weak- turbulence propagation. Because weak-turbulence condi- tions are of strong interest in astronomical observations, our quantitative analysis was restricted to this case only. Nevertheless, the results obtained allow us to draw some qualitative conclusions related to the strong-turbulence propagation that is of importance for a number of an- other applications. It was shown that the magnitude of the effect increases with increasing turbulence strength. So, one can expect that under the strong-turbulence condi- tions, the influence of scintillations on the image centroid can be quite strong, and starting from a given turbulence strength, it can make impossible a successful phase recon- struction from image centroid measurements. The quanti- tative results corresponding to the strong turbulence case can be obtained applying another simulation technique, for instance that based on propagation through multiple phase screens (Martin & Flatte 1988). 20 λ=0.55 µm 18 16 r0= 0.07 m 14 12 σl,% 10 8 6 r0= 0.1 5 m 4 r0= 0.1m 2 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 Number of Zernike mode Fig.3. Relative error σlof Zernike mode reconstruction versus the mode number For the lowest-order aberrations (tip-tilt, N2 and N3), the error is between 10–18% (depending on the seeing con- ditions). Then, for the second-order aberrations (defocus and astigmatism, NN 4–6), it drops abruptly in 2–3 steps. With the increase in the aberration number, the error in- creases (with some oscillations), and for high-order aber- rations it reaches practically the same magnitude as for the case of tip-tilt. So, generally speaking, the scintilla- tions more strongly affect the reconstruction quality of the tip-tilt and high-order aberrations, while for intermediate aberrations the effect is less pronounced. References Coluccib, D., Lloyd-Hart, M., Wittman, D., et al. 1994, PASP, 106, 1104 Hufnagel, R. E. 1974, in Optical Propagation through Turbulence, OSA Technical Washington, D. C., WA1-1 Jiang, W., Li, H., Liu, C., et al. 1993, in Proc. ICO-16 Satellite Conf. on Active and Adaptive Optics, ed. F. Merkle, ICO 16 Secretariat (Garching, Germany), 127 Kouznetsov, D., Voitsekhovich, V. V., & Ortega-Martinez, R. 1997, Appl. Opt., 36, 464 Li, H., Xian, H., & Jiang, W. 1993, in Proc. ICO-16 Satellite Conf. on Active and Adaptive Optics, ed. F. Merkle, ICO 16 Secretariat (Garching, Germany), 21 Martin, J. M., & Flatte, S. M. 1988, Appl. Opt., 27, 2111 Noll, R. J. 1976, J. Opt. Soc. Am., 66, 207. Rigaut, F., Rousset, G., Kern, P., et al. 1991, A&A, 250, 280 Rigaut, F., Ellerbroek, B. L., & Northcott, M. J. 1997, App. Opt., 36, 2856 Roddier, F. 1981, Prog. Opt., 19, 281 Tatarski, V. I. 1968, The Effects of the Turbulent Atmosphere on Wave Propagation, National Science Foundation Report TT-68-50464 Voitsekhovich, V. V. Gubin, V. B., & Mikulich, A. V. 1988, Atmo. Opt., 1, 66 Voitsekhovich, V. V., Kouznetsov, D., Orlov, V. G., & Cuevas, S. 1999, Appl. Opt. 38, 3985 Voitsekhovich, V. V. 1996, J. Opt. Soc. Am. A, 8, 1749 Digest Series, OSA, 4. Conclusions We have estimated quantitatively how scintillations af- fect the measurement of the image centroid. The results obtained show that, under the conditions of astronomi- cal observations, the magnitude of the effect is between 10% (for excellent seeing) and 18% (for poor seeing). The influence of scintillations on the quality of phase recon- struction from Hartmann data was also analyzed. The results shown that the scintillations affect in a different and complicated way the reconstruction quality of differ- ent aberrations. However, one can see a general trend: the scintillations more strongly affect the reconstruction quality of the tip-tilt and high-order aberrations than the reconstruction quality of intermediate aberrations.