Fiber optic interferometry: stati...
arXiv:astro-ph/0508425v1 19 Aug 2005 Fiber optic interferometry: Statistics of visibility and closure phase E. Tatulli, A. Chelli Laboratoire d���Astrophysique, Observatoire de Grenoble, 38041 Grenoble cedex France Eric.Tatulli@obs.ujf-grenoble.fr Interferometric observations with three telescopes or more provide two observables: closure phase information together with visibilities measure- ments. When using single-mode interferometers, both observables have to be redefined in the light of the coupling phenomenon between the incoming wavefront and the fiber. We introduce in this paper the estimator of both so-called modal visibility and modal closure phase. Then, we compute the statistics of the two observables in presence of partial correction by Adaptive Optics, paying attention on the correlation between the measurements. We find that the correlation coefficients are mostly zero and in any case never overtakes 1/2 for the visibilities, and 1/3 for the closure phases. From this theoretical analysis, data reduction process using classical least square minimization is investigated. In the framework of the AMBER instrument, the three beams recombiner of the VLTI, we simulate the observation of a single Gaussian source and we study the performances of the interferometer in terms of diameter measurements. We show that the observation is optimized, i.e. that the Signal to Noise Ratio (SNR) of the diameter is maximal, when the full width half maximum (FWHM) of the source is roughly 1/2 of the mean resolution of the interferometer. We finally point out that in the case of an observation with 3 telescopes, neglecting the correlation between the measurements leads to overestimate the SNR by a factor of ��� 2. We infer that in any cases, this value is an upper limit. c circlecopyrt 2008 Optical Society of America OCIS codes: 030.6600, 030.7060, 070.6020, 070.6110, 120.3180 1. Introduction Thanks to the simultaneous recombination of the light arising from three telescopes, interfer- ometers such as IONIC-3T on IOTA1 or AMBER on the VLTI2 are providing closure phase 1
measurements together with the modulus of the visibility. Retrieval of phase information al- lows to scan the geometry of the source, hence opening the era of image reconstruction with infrared interferometric observations. However, with the current number of telescopes avail- able, direct image restoration requires many successive nights of observations.3 Thus, in most of the cases where the (u, v) coverage spans a relatively small number of spatial frequencies, the measurements have still to be analyzed in the light of model-fitting techniques. Furthermore, together with partial correction by Adaptive Optics (AO), many of the up- to-date interferometers are making use of waveguides that spatially filters the atmospheric corrugated wavefront, changing the turbulent phase fluctuations into random intensity vari- ations4 . The estimators that describes the visibility and the closure phase measurements obtained with such interferometers have to account for the coupling between the partially corrected wavefront and the fiber. From these appropriate estimators one can derive the statistical properties of the observables and properly investigate the performances of single- mode interferometers. In Section 2 we recall the spatial filtering properties of the waveguides in terms of inter- ferometric signal and we define the estimators of both the modal visibility and the closure phase. We investigate in Section 3 the covariance matrices of the observables with respect to atmospheric, photon and detector noises, paying particular attention to the correlation coefficients. Then, defining in Section 4 a general least square model fitting of the measure- ments, we analyze in Section 5 the ability of fiber optic interferometers to measure stellar diameters. 2. Principles of fiber optic interferometry A full analysis of the signal arising from fiber optic interferometers has been theoretically described by M`ege5 and summarized by Tatulli et al6 . We only recall here the important points for this paper, focusing on the coupling phenomenon between the incoming wavefront and the fiber, and on the observables that can be obtained from such interferometers. Figure 1 sketches the principle of a fiber optic interferometer, and reports the main technical terms that will be used all along this paper. 2.A. Spatial filtering Introducing waveguides to carry/recombine the light in an interferometer allows to perform a spatial filtering of the incoming wavefront. It means that the phase corrugation of the wave- front are changed into intensity fluctuations. In other words, the number of photometric and coherent (interferometric) photoevents at the output of the fibers depends on atmospheric fluctuations and results in the coupling between the turbulent wavefront and the fibers4,7 . Hence, spatial filtering can be seen as coupling coefficients, i.e. the fraction of (respectively 2
photometric and coherent) light that is captured by the fibers. Such coupling coefficients are mathematically described by the following equations8,9 : ��i(V���) = ��0(V��� ��� T i)f=0 (1) ��ij(V���) = ��0(V��� ��� T ij)f=fij (2) where V��� is the visibility of the source and T i and T ij are resulting in respectively the auto- correlation and cross-correlation of the aberration-corrupted pupil weighted by the fiber single mode9,10 : T i(f) = integraltext Pi(r)Pi���(r + ��f)��i(r)��i ���(r + ��f)dr integraltext |Pi(r)|2dr (3) T ij(f) = integraltext Pi(r)Pj���(r + ��f)��i(r)��j ���(r + ��f)dr integraltext Pi(r)Pj���(r)dr (4) where Pi(r) is the pupil function of the ith fiber optic telescope and ��i(r) is the aberration- corrupted wavefront incoming on the ith pupil. T i and T ij are respectively called the pho- tometric and interferometric peaks. The inverse Fourier transform of T i is called the photo- metric lobe (or antenna lobe as commonly named in radio) and the inverse Fourier transform of T ij is called the interferometric lobe. ��0 is the optimum coupling efficiency fixed by the fiber core design11 . In the case where the source is unresolved by a single telescope (i.e. is much tighter than the photometric lobe), Eq. 1 can be simplified: ��i(V���) = ��0 integraldisplay T i(f)df = ��0Si (5) where S is the instantaneous Strehl ratio12 . Moreover, if the visibility is constant over the range of the high frequency peak T ij, the interferometric coupling coefficient has also a simplified expression: |��ij(V���)|2 = ��0SiSj|V���(fij)|2 2 (6) Under these conditions, the effect of spatial filtering by the fibers in the interferometric equation and, as a result, in the observables, is entirely characterized by the instantaneous Strehl ratio statistics. 2.B. Estimation of the modal visibility We refer to Tatulli et al6 for a more complete description of the modal visibility. Note however that the expression of the coherent flux at the spatial frequency fij is given by: I(fij) = radicalbig KiKj��ij(V���) (7) 3
where Ki and Kj are the number of photoevents of telescopes i and j before entering the fiber. An estimator of the modal visibility in the Fourier space is given by the ratio of the coherent flux by the photometric ones, assuming the latter are estimated independently through dedicated photometric outputs (see Fig. 1): tildewider Vij 2 = |I(fij)|2 kikj parenleftbigg �� 1 ��� �� parenrightbigg2 (8) �� is the fraction of light selected for photometry at the output of the beam-splitter, and ki, kj are the photometric fluxes (after the fibers). 2.C. Modal bispectrum and closure phase By definition, the closure phase is the phase of the so called image bispectrum tildewide B klm. The latter consists in the ensemble average of the triple product hatwide(fkl)hatwide(flm)hatwide���(fkm) I I I . It can be expressed from Eq. 7 as: tildewide B klm = KkKlKm��kl(V���)��lm(V���)��km(V���) ��� (9) = KkKlKm��0 3 integraldisplayintegraldisplayintegraldisplay V���(f)V���(f ��� )V������(f ������ ). T kl(fkl ��� f)T lm(fkl ��� f ��� )T km��� (fkl ��� f ������ ) dfdf ��� df ������ (10) Using Roddier���s formalism13 that demonstrated the bispectrum analysis to be a generaliza- tion to the optical of the well known phase closure method currently used in radio interfer- ometry, it is straightforward to notice that the quantity T kl(f)T lm(f ��� )T km��� (f ������ ) is non zero if f ������ = f + f ��� and that in this case: T kl(f)T lm(f ��� )T km��� (f ������ ) = N(f, f ��� ) N 3(0) (11) KkKlKmV���(f)V���(f ��� )V������(f ������ ) = B���(f, f ��� ) (12) where N(f, f ��� ) is proportional to the overlap area of three pupil images shifted apart by the spacings f, f ��� and f ������ = f + f ��� , and B���(f, f ��� ) is the bispectrum of the source. Hence the modal bispectrum can be rewritten: tildewide B klm = ��0 3 integraldisplayintegraldisplay B���(f, f ��� ) N(fkl ��� f, flm ��� f ��� ) N 3(0) dfdf ��� (13) Thus, the modal bispectrum arising from fiber optic interferometers is the source bispectrum integrated over the overlap area N(fkl ��� f, flm ��� f ��� ). Hence, as the modal visibility does not equal in general the object visibility, the modal bispectrum does not coincide with the 4
source bispectrum. Nevertheless, if the source spectrum is constant over the overlap area N(fkl ��� f, flm ��� f ��� ), Eq. 13 takes a simplified form: tildewide B klm = ��0B���(fkl, 3 flm) integraldisplayintegraldisplay N(f, f ��� ) N 3(0) dfdf ��� (14) In this case, the modal bispectrum is proportional to that of the source. 3. The covariance matrices We propose to characterize the statistics of the square visibility and that of the closure phase (i.e. the bispectrum phase) by computing their respective covariance matrix. Our objective is twofold: derive the error associated to each observable and investigate the degree of de- pendency of each observable through their correlation coefficients. In order to do so, we use the spatially continuous model of photodetection introduced by Goodman14,15 where the signal is corrupted by three different types of noise: (i) the signal photon noise (ii) the ad- ditive Gaussian noise of global variance ��2 which arises from the detector and from thermal emission (iii) the atmospheric noise resulting from the coupling efficiency variations due to the turbulence. To simplify the calculations, we assume that the source is unresolved by a single aperture, such that the low frequency coupling coefficients verify Eq. 5, and that the source visibility is constant over the range of the interferometric peaks, such that the high frequency coupling coefficients verify Eq. 6. These assumptions drive to neglect the modal speckle noise regime ��� it has been shown that the modal speckle noise is rejected towards negative magnitudes and only affects very bright sources6 ��� and to only focus on ���photon noise��� and ���detector noise��� regimes . The full calculations of the covariance coefficients of the visibilities and of the closure phases are done in Appendix A. They lead to relatively complicated formulae which depend on the Strehl statistics. Using a simple analytical ap- proach, we derive in Appendix Bthe mean and the variance of the Strehl as a function of the turbulence strength and the level of AO correction. The relative error of the Strehl is bounded between two limit values10,14 : (Perfect correction) 0 ��� ��S S ��� 1 (No correction) (15) Table 1. gives the expressions of the limiting values of the variance of the visibility and the closure phase for a point source in both ���photon noise��� and ���detector noise��� regimes. The error of the visibility will be deeply used in the next section to derive the performances of the Very Large Telescope Interferometer (VLTI) with regards to single sources diameter measure- ments. Let us concentrate in this part on the correlation coefficients of the visibilities and the closure phases, respectively. Their limiting values are summarized in Table 2. Clearly, for visibilities the correlation coefficients are null in the photon noise regime when no telescope 5