Jet Propulsion Laboratory,California Institute of Technology,4800 Oak Grove Drive, Pasadena, CA 91109

The Space Interferometry Mission (SIM) will provide unprecedented
micro-arcsecond (as) precision to search for extra-solar planets
and possible life in the universe. SIM will also revolutionize our
understanding of the dynamics and evolutions of the local universe
through hundred-fold improvements of inertial astrometry
measurements. SIM has two so-called guide interferometers to
provide stable inertial orientation knowledge of the baseline, and a science
interferometer to measure target fringes. The guide and science measurements
are based on the fringe phase measurements using a CCD detector. One of
SIM's key issues is to develop a new algorithm for calculation of fringe
parameters. Not only astrometric results need that new algorithm, but
also real-time fringe tracking requires a new method to calculate
phase and visibility fast and accurately. The formulas for the phasor
algorithms for fringe estimation are presented. The signal-noise ratio
performances of the fringe quadratures are demonstrated. The
advantages of phasor algorithms for application of fast fringe
tracking and on-board data compression are discussed.

This paper presents formulas for fringe estimation based on the phasor algorithms, and demonstrate the signal-noise ratio performances of the fringe quadratures. The advantages of phasor algorithms for application of fast fringe tracking and on-board data compression are discussed.

SIM is a fringe-scanning interferometer that needs to acquire and track the white light fringe in order to equalize the paths of the two arms of the interferometer at the nanometer level. The pathlength modulation implemented by the voice coil uses a sawtooth waveform, and the detector is read out coherently using eight time bins per scan. The basic frame rate is 500 Hz, and each time bin is 0.125 ms. The path length in one arm varies linearly with the stroke, which has a length equal to the longest wavelength. Two guide-star interferometers have 4 spectral channels, and a science interferometer has 8 spectral channels. For all of spectral channels, one, or a little more than one fringe, is scanned across CCD detectors.

The fringe irradiance is written as :

where N is the mean number of photons per scan, s is the stroke, V is fringe visibility, k is the wave number (), and is the fringe phase. The modulation position where t is time for a bin, and T is the time interval of a stroke.

From the CCD measurements the accumulated photon counts for
each time bin can be written as :

The total photons per stroke are the summation of counts in all eight time bins:

For each channel we calculate the X and Y quadratures for the dithered signal. In the simple case that the modulation amplitude matches a wavelength, i.e. , we can combine the 8 bin data as follows:

, .

For most of channels the stroke is longer than their effective wavelengths. We must define:

, where s is stroke,
is wavelength;

,

.

The true quadratures,
, i.e. X and Y phasor components, and
the total fluxes are calculated as :

So the fringe visibility and phase are calculated as follows:

;

,

where angle brackets represent an average over certain
time periods of fringe tracking.

The photon counts of time bins obey Poisson distribution. The signal-noise ratio
of phasors can be expressed as

where is the dark current, and is the read noises.

Simulations are conducted for a typical case of 30 seconds of fringe tracking. For a 7th magnitude guide stars the signal-noise ratios of X, Y phasors are shown in Figure 1. For comparison, the signal-noise ratio of phasors in X and Y directions are computed for the case of four time bins(Colavita 1999). It is shown that the performances of signal-noise ratio drops significantly when the wavelength of a spectral channel is shorter than the length of the stroke. It is necessary to use new eight time bin algorithm for uniform and improved performance.

Traditional non-linear fitting techniques, or the pseudo-inverse method are much too slow, and are difficult to use for nanometer and millisecond control. The phasor algorithm also can be used for the on-board data processing in SIM. The data volumes that must be down-loaded from the spacecraft to the ground station are extremely high. Phasors computed by this algorithm can be used to compress fringe data. It is important to maximize measurement information while reducing data volume in SIM.

This paper presents the preliminary study of the phasor algorithm. Cyclic errors, the vibration effects, and nonlinear strokes are a few example of noises that will reduce the accuracy of fringe parameter determination. Those issues need to be addressed in the near future.

This work was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.

Bahcall, J. 1991, The Decade of Discovery in Astronomy and Astrophysics, National Academy Press

Colavita, M. M. 1999, PASP, 111, 111

Shao, M. et al 1988, A&A,193,357

Shaklan, S. Colavita, M. M. & Shao, M. 1992, in ESO conf. and Workshop Proc. 39, 1271

Hines, B. 2002, Fringe Tracker Software Requirements, JPL reports

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