1. Introduction

Amplitude interferometry is a unique tool to observe astronomical sources with an angular resolution well beyond the diffraction limit of monolithic telescopes. The basic Michelson stellar interferometer measures the coherence factor between two independent pupils, which is linked to the Fourier component of the object intensity distribution (or object visibility V) at the spatial frequency corresponding to the baseline formed by the pupils.

One of the main challenges of Michelson stellar interferometry at optical and infrared wavelengths is the calibration of coherence factor measurements. The fringe visibility differs from the object visibility because the instrument and the atmosphere have their own interferometric efficiencies which result in an instrumental transfer function T, and an atmospheric transfer function T:

 

Thus an accurate knowledge of both transfer functions is required to obtain a good estimate of V. A well-designed interferometer is usually stable enough so that the calibration of T is not a major issue. But the interferometric efficiency of the atmosphere, which is affected by the loss of coherence caused by phase corrugations on each pupil, depends on the instantaneous state of the turbulent wavefronts. Thus T is a random variable, whose statistics is linked to the evolution of the seeing and is not even stationary. In a classical interferometer there is no way to directly calibrate T (short of sensing at all times the complete shape of the corrugated wavefronts) and the only option is a delicate, statistical calibration on a series of measurements. Then obtaining object visibilities with a relative accuracy of 10% or better is difficult to achieve; yet such a performance is insufficient for many astrophysical problems.

This difficulty can be overcome by using single-mode fibers to spatially filter the incoming beams. In single-mode fibers the normalized radiation profile is determined by the waveguide physical properties (Neumann 1988), not by the input wavefront, and the phase is constant across the guided beam. On the other hand, the intensity of the guided radiation depends on the electromagnetic field amplitude distribution in the focal plane of the telescope and may vary with time if the image is turbulent. Thus single-mode fibers force the transverse coherence of the radiation and transform wavefront phase corrugations into intensity fluctuations of the light coupled into the fibers. Unlike wavefront perturbations however, intensity fluctuations can easily be monitored and used during the data reduction process to correct each interferogram individually against the effects of atmospheric turbulence.

The correction capability was first demonstrated in a fiber unit set up between the two auxiliary telescopes of the McMath-Pierce solar tower on Kitt Peak Observatory, which transformed the telescope pair into a stellar interferometer (Coudé du Foresto et al. 1991). The prototype instrument (named FLUOR for Fiber Linked Unit for Optical Recombination) observed a dozen stars with statistical errors smaller than 1% on the object visibilities. The same fiber unit is now routinely used as part of the instrumentation in the IOTA (Infrared and Optical Telescope Array) interferometer at the Fred Lawrence Whipple Observatory on Mt Hopkins (Carleton et al. 1994). Some of the results obtained with FLUOR on IOTA can be found in Perrin et al. (1997).

This paper presents the specific data reduction procedure used to extract visibility measurements from the raw interferograms obtained with FLUOR. The procedure can also be applied (with minor modifications that are explained in Sect. 9 (click here)) even if the interferometer does not involve fiber optics. In that case, however, the spatial filtering advantage is lost.

The organization of the paper is as follows: in Sect. 2 (click here) is briefly described the conceptual design of a FLUOR-type interferometer, and the principle of interferogram correction is shown on a simple example.

Before we can derive the full analytical expression of a wide band interferogram (Sect. 5 (click here)), we need to specify two important preliminary assumptions (Sect. 3 (click here)) and to understand the photometric behavior of the system (Sect. 4 (click here)), i.e. the proportionality relationships that link the diverse outputs when light is incoherently recombined. Section 4 (click here) is specific to the use of a triple fiber coupler and can be skipped in a first reading. From the expression of a raw interferogram, obtained in Sect. 5 (click here), can be derived an expression for the corrected interferogram, which itself leads to an expression for the squared modulus of the wide band fringe visibility (Sect. 6 (click here)). Real data are affected by noise: estimation strategies and noise sources are discussed in Sect. 7 (click here). Finally, some practical considerations are developed in Sect. 8 (click here) and a generalization to non-fiber interferometers is proposed in Sect. 9 (click here).

Throughout the paper, the data reduction procedure will be illustrated with examples from actual data. They were obtained on Boo (Arcturus) with the original FLUOR unit set up between the two 0.8m telescopes (separated by 5.5m) of the McMath-Pierce tower (Coudé du Foresto et al. 1991). The unit included fluoride glass fibers and couplers, four InSb photometers, and was operated in the infrared K band (2m 2.4m). The telescopes had entirely passive optics, without even active guiding (tip-tilt correction). The sample data is a batch of 122 interferograms recorded on 7 April 1992 between 7h19 and 8h04 UT, in mediocre seeing conditions (more than 1.5arcsec).