Statistics – Computation
Scientific paper
Nov 1999
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1999phdt.......117d&link_type=abstract
PhD Thesis, TU Delft University, 1999
Statistics
Computation
2
Interferometry, Turbulence, Anisoplanatism, Vlti
Scientific paper
In Michelson stellar interferometry, the high-resolution information about the source structure is detected by performing observations with widely separated telescopes, interconnected to form an interferometer. At optical wavelengths, this method provides a technically viable approach for achieving angular resolutions in the milliarcsecond range, comparable to those of a 100 m diameter telescope, whose realization is beyond the immediate engineering capabilities. Considerable efforts are currently devoted to the definition of dedicated interferometric instruments, which will allow to address ambitious astronomical tasks such as high-resolution imaging, astrometry at microarcsecond level, and the direct detection of exoplanets.
Astrometry and related techniques employ the so-called wide field-of-view interferometric mode, where phase measurements are performed simultaneously at two (or more) sources; often, the actual observable is the instantaneous phase difference of the two object signals. The future success of wide-field interferometry critically depends on the development of techniques for the accurate control of field-dependent (anisoplanatic) phase errors. In this thesis, we address two aspects of this problem in detail.
The first one is theoretical in nature. For ground-based measurements, atmospheric turbulence is the largest source of random phase fluctuations between the on- and the off-axis fringes. We developed a model of the temporal power spectrum of this disturbance, whose validity is not limited to low frequencies only, as it is the case with earlier models. This extension opens the possibility of the analysis of dynamic issues, such as the determination of the allowable coherent integration time T for the off-axis fringes.
The spectrum turns out to be well approximated by a sequences of four power-law branches. In first instance, its overall form is determined by the values of the baseline length, telescope diameter, and average beam separation in the atmosphere. Due to the rapid decorrelation of the on- and off-axis phases for increasing star separation theta, the useful field for wide-field interferometry is limited to about |theta|<1', the so-called very narrow angle regime. For high-accuracy applications, this range decreases to a few arcseconds. We estimated that for the VLTI along baselines operating at lambda=2.2 mu, a turbulence-related error of less than lambda/10 rms is only available for field angles smaller than 7.3'' and 5.8'', for UT-UT and AT-AT pairs respectively.
The bulk of the spectral power is confined at relatively low frequencies, typically below 1 Hz. Both smaller star separations and larger telescope sizes contribute in lowering the spectral content at hight frequencies. We found that in general, as compared to blind observations, wide-field measurements can make use of significantly longer off-axis integration times T, even at rather big star separations. For the long UT-UT baseline operating at lambda=2.2 mu, we have calculated a 5 % fringe visibility loss is reached for T=740 ms, 2.1 s and 12.7 s for star separations of 30'', 10'', and 5'', respectively. These figures are about 2, 5 and 32 times higher than for a blind observation. Finally, we point out that for large telescopes a significant fraction of the total phase error due to anisoplanatic turbulence is contributed by wavefront modes higher than piston. Therefore, we generalized the formalism used in out study to the analysis of (Zernike) wavefront modes of arbitrary order.
This thesis also addresses an instrumental aspect of the problem of the control of anisoplanatic phase errors. A Michelson interferometric imager is suitable for wide-field operation only if the configuration of the pupil images forms a scaled replica of the total array aperture. This implies the factual coincidence of the magnification factors M and pupil rotations phi of all interferometric arms: for the VLTI, the matching accuracy requirements are as severe as dM< 1.9e-3, dphi < 3.8''.
We addressed the problem of measuring dM, dphi, to the accuracies expressed here above. In the selected approach, this is done by measuring the difference of the star separation vectors for the two interferometer arms, as measured at the corresponding pupil images. Variations of M and phi affect this quantity in orthogonal directions, which allows the simultaneous determination of both unknowns. The measurement makes use of two two-axis tilt sensors, that determine the angular separation vectors of the on- and off-axis beams, respectively, from the two interferometric arms. A 0.0075'' single-axis accuracy is required, together with a sufficiently high sensitivity for astronomical applications. This led to the choice of implementing the sensors as pupil plane devices, using the same interferometric tilt-detection principle as applied in Fine Guidance Sensors of the Hubble Space Telescope.
The main challenge was to ensure equal responses for the two sensors, to within 0.0075''. Test measurements have shown that we succeeded in controlling mismatches between the sensors (including their mutual orientations, electronic gain and phase, linearity and signal normalization) a the 0.004'' level, and in performing beam recombination without introducing errors exceeding 0.006''. Pupil rotation alignment runs confirmed a 2'' overall measurement uncertainty for dphi, about half the 3.8'' calibration requirement.
Finally, in this thesis we also developed a near-filed propagation method, intended for the diffraction-based analysis of optical systems with extremely high accuracy requirements (typically 1 deg in phase and 1.e-3 in field amplitude). Examples thereof are the nulling optics for planet detection and, outside the field of stellar interferometry, systems for the determination of the shape of mirrors for extreme-UV lithographic projection systems. The method is based on the local Fresnel approximation of the propagation integral, that we have solved analytically for rectangular domains and for triangular ones with an arched hypotenuse. This allows for an accurate computation of the field diffracted at the edges of complicated aperture shapes, without having to recur to time-consuming numerical quadrature techniques. The method has shown the ability to provide complex amplitude estimates that are consistently accurate to the specifications given above, and this in reasonable times. In a series of comparative tests, our method outperformed the Hopkins algorithm by typically a factor of fifty with respect to the computational speed.
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