Phasing a hierarchy of triplet apertures

 An early algorithm for adaptive optics, not relying on wavefront continuity, was proposed by Muller and Buffington ([Muller & Buffington 1974]). It is based on the maximisation of a sharpness function, defined as:  

$$S=\int{B^2_{(x,y)}dxdy},$$ (2)

where B is the intensity of the image at a point (x,y).

The mirror actuators are adjusted each in turn, with repeated iterations performed faster than the seeing lifetime, to maximise the sharpness function. The convergence however tends to be trapped in secondary maxima. To overcome this problem, Ribak ([Ribak 1990]) used simulated annealing: noise is added to the actuator signals and gradually reduced until the system falls in a state of minimum energy, reached when the sharpness function is maximised.

Hamaker, O'Sullivan & Noordam [(1977)] have later remarked that, according to Parseval's theorem, maximising the sharpness function is equivalent to maximising the integrated square modulus in the image's Fourier transform, which is the pupil's auto-correlation function multiplied by the object's Fourier transform. The maximum occurs for uniform pupil phases. However, if the aperture is diluted, the intensity of the autocorrelation peaks remains invariant if they do not overlap. This indicates that the sharpness approach cannot work if the exit pupil aperture is highly diluted, as would be the case for long-baseline Fizeau interferometers. The densified exit pupils considered here can be made to meet the peak overlap requirement.

We have tried to improve the convergence and to avoid the simulated annealing, by using a hierarchy of aperture triplets. The 27-aperture interferometer is initially divided in 9 separate interferometers, each combining images from 3 adjacent apertures, then re-arranged as 3 separate interferometers having 9 apertures each and finally re-arranged as a single interferometer having 27 apertures. If observations are performed from Earth the steps of re-arrangements are cycled at high speed, within the life-time of seeing, by tilting elements of the active mirror support to ``explode'' the image into separated triplet components. More delicate tip-tilt and piston adjustments of the same elements provide the phase adjustments, as described in Fig. 2.

 

Figure 2:  Hierarchical arrangement of actuators carrying 27 mirrors for phasing according to the triplet algorithm. For clarity, the system is sketched as a line array, and without the tip-tilt correctors carrying each elementary mirror. Each triplet of mirrors is carried by a plate, itself carried by3 actuators, arranged as a triangle to provide piston and tip-tilt corrections. 3 such plates, carrying a total of 9 mirrors, are themselves carried by a larger plate, also carried by 3 actuators. A group of 9 mirrors can be phased by adjusting both lateral triangle plates, in terms of tip-tilt and of piston. The criterion used for this 6-parameter adjustment is the sharpness value. If the star is unresolved, centering the honeycomb fringe patterns from each mirror triplet, and the finer honeycomb from the triplet of triplets is also suitable. Once the 3 groups of 9 mirrors are independently phased, the global phasing of 27 mirrors can be obtained similarly by applying tip-tilt and piston corrections to both lateral plates carrying 9 mirrors. Finally, the diffraction functions from each elementary mirror may be centered with their tip-tilt actuators (not shown). Phasing is achieved modulo $2 \pi$, but the use of several wavelengths can remove the ambiguity and provide zero path difference.

The monochromatic interference pattern from 3 sub-apertures is generally a honeycomb-like interference function, multiplied by the diffraction function. Adding 3 such patterns from adjacent groups creates a finer honeycomb pattern within the coarse one. The triplet grouping can be pursued at higher levels of the hierarchy to include 27, 81, 243, etc. apertures. The closest apertures are grouped first to benefit from the fact that the object is less resolved.

For a non-resolved star, phasing is directly achievable by centering a cell of the honeycomb pattern appearing at each level of triplet formation. This requires adjusting 2 phases, those of 2 among the 3 groups of apertures at every step of the routine (Fig. 3). In monochromatic light and for 3 groups of apertures, the honeycomb pattern can be centered by extracting the phase values in the 6 peaks of its Fourier transform. In polychromatic light, 3-dimensional Fourier transforms of the $x,y, \lambda$ data cubes provided by the spectro-imaging camera can provide similar information with full removal of the modulo $2 \pi$ phase ambiguity. This algorithm has not been tested yet, but a 2-aperture analog is routinely used at the ``Grand Interféromètre à 2 Télescopes'' (GI2T) ([Morel & Koechlin 1998]) for fringe acquisition.

 

Figure 3: Steps of phasing sequence for 27 apertures, using 3 wavelengths on a point source. Top row: 3 elements of the entrance aperture (top-left) and their interference pattern obtained before (middle) and after (right) correcting optical path errors. Middle row: combining 3 such groups of apertures provides finer fringes, which can be centered by adjusting the global phases of 2 groups. Bottom row: one more similar step corrects the optical path errors among the 27 apertures, thus producing a narrow interference peak. With a continuous spectrum rather than just 3 wavelengths, the speckles of the central bottom image would be appreciably smoothed.  

More simply, the central white cell appearing in a honeycomb pattern amidst the coloured lateral cells can also be identified, even with a broad-band monochrome camera, from its higher contrast. Centering all the white cells from the various triplet groups onto a common fiducial mark corrects directly the optical path errors. On resolved objects, such honeycomb centering should not generally succeed since de-centered honeycomb components can be present. Indeed, imaging an object through a diluted triplet aperture causes a convolution with the honeycomb-like spread function. If the object is itself a honeycomb-like grid pattern matching this function, then the convolution preserves the image contrast, a situation which may be qualified as ``honeycomb resonance''. With 3 different triplets now forming a single aperture, 3 different grid components of the object add their images, but they do not generally have a common centering. Hence the need for another criterion, sharpness for example, to adjust the tip-tilt of 2 pupil triplets before adjusting their global piston. Six parameters should therefore be adjusted at each level in the hierarchical procedure.

When calculating the sharpness integral, the diffraction envelope of the sub-images causes unwanted biases, which tend to center the brighter zones. The bias can be avoided by ``flattening'' the diffraction peak: each recorded sub-image must be first divided by the known diffraction function, within the peak area.

Somewhat surprisingly, our simulations (Fig. 4) on resolved objects worked well without these precautions, using a simplified version of the routine: only piston adjustments were made at each level, and no flattening of the diffraction envelope was carried out. It is however likely that some object shapes require the complete routine. Also, no trapping in secondary maxima of the sharpness function was observed, but the reasons are unclear. Further investigations are desirable in these respects.

 

Figure 4: Effect of photon noise on the phasing algorithm; a- simulated star cluster having 27 stars of different colour; b- interference function; c- theoretical image with zero path difference. The images were obtained with a single iteration of the phasing algorithm; d- image obtained with 2200 total photons obtained from 3 wavelengths, Strehl ratio = 0.3; e- case of 29,000 photons, Strehl ratio = 0.7; f- case of $18 \cdot 10^6$ photons, Strehl ratio = 0.8.  

A single camera can receive all partial images and the fully combined image, using either temporal or spatial separations as shown in Fig. 5. Rather than using the actuators for exploding the image into the various groups at a fast cycling rate, beam-splitters can provide a parallel display of these triplet, ninetuplet, etc. groups ([Labeyrie 1998]). Mapping the $x,y, \lambda$ information is efficiently achievable with a spectro-imaging attachment such as the combination of a grism and a micro-lens array developed by Bacon, and co-workers ([1995]) at the Observatoire de Marseille.

 

Figure 5: Picture of partial and fully combined images with 9 groups of 3 apertures (external ring), 3 groups of 9 (intermediate ring), 1 group of 27 (centre), and the corresponding diffraction patterns, as they would appear when formatted on a single camera. The object used in the simulation is the galaxy M51. The simulation was performed with photon noise. In the final image a Strehl ratio of 0.84 was obtained with 250,000 photons.