1. Introduction

1.1 Background

High resolution space-based optical imaging systems require large apertures to achieve high angular resolution. To ameliorate the problem of increased mass and volume associated with traditional monolithic telescopes, a design involving a nine-telescope array (U.S. patent #5,905,591) has been built and demonstrated in a testbed known as “Star-9” [1]. Previously, another nine-telescope array, in a Y-configuration, was built at the Lockheed Martin Advanced Technology Center to demonstrate basic operations and closed-loop phasing with phase diversity [2]. Star-9 represents a second-generation testbed where the hardware design and development are flight-traceable; the array itself fits in a smaller package to show how large-aperture systems can fit into small volumes; a real-time control system is implemented to allow higher bandwidth and field-of-regard steering; and automated computer programs can bring the system from a state of gross misalignment to within capture limits for handoff to the control system.

1.2 System overview

This paper describes two software programs, “Autoalign” and “Autophase,” that help to automate the operation of a distributed-aperture imaging system. The imaging system consists of nine Maksutov-Cassegrain afocal telescopes arranged in a compact bundle (see Fig. 1 ). The telescopes each include a relay section consisting of actively controlled mirrors. Each relay section has four actively controlled mirrors which together can adjust piston, tip, and tilt of each telescope. Light entering the telescopes comes from a collimated scene projector. The scenes can be white-light point sources or white-light extended objects. Light coming through the telescopes travels through the relay sections and converge on a central faceted mirror assembly. Then, the beams from the telescopes all pass through a combiner telescope to form a focused image of the object. A defocused (by a known amount) image is also formed by use of beamsplitters.

 

Fig. 1 Left, a photo of the Star-9 lab; right, the ray trace of Star-9.

Download Full Size | PDF

A phase diversity algorithm uses the focused and defocused image to compute the maximum-likelihood joint estimate of the object scene and the tip, tilt, and piston aberrations in the system [3]. These estimates are fed back to the control system, which handles the actively controlled relay sections to correct for the aberrations. For phase diversity to provide accurate estimation, the tip/tilt errors need to be with +/− 1 pixel (0.32 μR) p-v, and piston aberrations must be within about ½ wave (λ_c = 635 nm) p-v. It is the purpose of auto-align and auto-phase to bring the aberrations down to these limits.

1.3 System concept of operation

In closed-loop operation mode, phase diversity estimates wavefront error in all telescopes simultaneously, and sends tip, tilt and piston commands to the mirror control system. Prior to this mode, Autoalign and Autophase are run in place of phase diversity. Like phase diversity, the automation tools acquire camera data (only the focused image), and send tip/tilt/piston corrections to the mirror control system. The main difference is that these corrections are corrected for one telescope at a time. See Fig. 2 .

 

Fig. 2 (a) Progression of operational modes. (b) Data flow of Autoalign and Autophase.

Download Full Size | PDF

2. Tip/tilt Alignment (Autoalign)

2.1 Concept of operation

Autoalign must be run before Autophase. Tip and tilt errors manifest themselves on the focused image plane as lateral shifts in the image. Autoalign measures and eliminates these shifts. This is done by first obtaining an image from one of the telescopes, designated as the “reference” telescope, with all of the other telescopes closed off by actuated shutters. After the image is obtained, the reference telescope is closed by its shutter, and another telescope is opened. An image with this telescope is taken, and if there are tip/tilt errors between the two, then there will be a lateral shift measured between the two images. Tip/tilt commands are sent to the control system, and when the control system settles, another image is taken to verify that the tip/tilt error was eliminated. Autoalign proceeds until all telescopes are tip/tilt aligned. Autoalign works on both point sources and extended scenes.

2.2 Image processing

We use normalized cross-correlation in obtaining shift errors between images. Here, we let $f_{jk}$ be the reference image, and $g_{jk}$ be the non-reference image. The $\left(j,k\right)$ subscripts index a pixel location in an image. Assume that $g_{jk}$ is a shifted, noisy version of$f_{jk}$, i.e., $g_{jk}=f_{j-u,k-v}+n_{jk}$, with $n_{jk}$ the noise value at pixel location $\left(j,k\right)$ and $\left(u,v\right)$ the shift amounts. The normalized cross-correlation between the two images is calculated in Eq. (1) as

This represents a surface whose peak at some value $\left(u_{\max },v_{\max }\right)$ is taken as the indication of the displacement between the two images. In practice, u and v are taken to be integer pixel displacements so that $C_{uv}$ is calculated at sampled values. To obtain subpixel resolution, the 3 by 3 samples surrounding $\left(u_{\max },v_{\max }\right)$ are interpolated using a bi-cubic spline and the location of the peak value of the spline surface is taken to be the displacement between the images.

2.3 Autoalign experimental results

Using simulated images of point-sources with known shifts, we found that the accuracy depended greatly on the image peak-to-noise ratio (PNR), defined in Eq. (2) as

As implemented, the range of Autoalign is limited to the size of the image (128 x 128 pixels). The potential range can be extended by implementing a search algorithm that scans all possible tip/tilt positions of the non-reference telescope until some portion of its image correlates well with a representative portion of the reference telescope image.

3. Piston Phasing (Autophase)

3.1 Concept of operation

After Autoalign is finished, Autophase runs. This tool opens the shutters for two telescopes, and closes the rest. One of the telescopes is the reference telescope, which does not move in piston, and the other telescope’s piston, or path length, is adjusted according to Autophase’s commands until it matches that of the reference telescope. This process is repeated for each of the other non-reference telescopes, with the reference telescope always being one of the two telescopes that are open at any given time. Both point sources and extended scenes can be used, but the fine-phasing portion only works on point sources.

3.2 Coarse phasing with MTF metric

If the initial piston error is much greater than the coherence length of the scene spectrum (~10-15 waves at λ_c = 635 nm and Δλ = 90 nm), then white-light interference fringe contrast in the image is essentially zero [4]. As piston error approaches zero, the fringe contrast gradually increases to some maximal value; the algorithm’s goal is to find this maximum. Interference fringe contrast is likened to high spatial frequency content in an image [5], and a scalar metric for evaluating the amount of high spatial frequency is desired.

For point sources, the method is to compute the modulation transfer function (MTF), and take the height of the MTF side lobes (See Fig. 3 ). For extended scenes, the metric is the magnitude of the image Fourier transform, taken at the same (u, v) spatial frequency location where MTF side lobes would appear. Since the software computation in both cases is the same, we always refer to the result as the “MTF metric,” regardless of the scene type.

 

Fig. 3 The Modulation Transfer Function (MTF) with two apertures. The arrow indicates the height of the MTF side lobe, which can be used as a scalar metric of point-source phasing error.

Download Full Size | PDF

What Autophase does is to command piston of the non-reference telescope in 1-wave steps until the record of MTF metric data indicates that a global maximum has been traversed. The region around the maximum resembles a 1-D Gaussian function (Fig. 4 ), and the algorithm computes the least-squares Gaussian fit to the logarithm of the MTF metric data and adjusts the piston to where the Gaussian is maximized. The least-squares formulation begins with the definition of an ideal Gaussian function:

 

Fig. 4 MTF metric as a function of piston between two telescopes. This data was oversampled for illustrative purposes. The piston locations are tighter, and each have 10 MTF metric measurements (dots); solid line is the average.

Download Full Size | PDF

Here, m(x) represents the measured MTF metric sampled at discrete values of piston, or x. The units of x are in waves of piston. We manipulate Eq. (3) by taking the logarithm of both sides:

This in effect creates a linear equation on which we can perform a least-squares estimate of the quantities$[-a_3],[2a_2{a}_3],[-a_2{}^2{a}_3+\ln (a_1)].$ Once these quantities are obtained, then it is possible to solve for the optimal piston location where$\widehat{x}=a_2.$

3.3 Fine phasing with PSF metric (point-sources only)

After moving to the top of the MTF metric curve, it is possible to refine the algorithm to achieve better accuracy by using another metric (still with only two telescopes open). The general concept is to measure the symmetry of the PSF [6]. The implementation here is to measure pixel intensities within the PSF side lobes. We denote one of the side lobe (row, col) locations as “inner,” or I, and the other as “outer”, or O [see Fig. 5(a) ]. The locations are computed by finding the globally maximum pixel location and projecting a line segment from there parallel to the two-telescope pupil baseline. I is at reference telescope end, and O is at the adjusted telescope end of the line segment.

 

Fig. 5 (a) the 2-telescope PSF. The dots on the PSF are markers for locations “O” and “I”, showing which pixels were used in the PSF metric computation. (b) the PSF metric plotted as a function of pathlength difference between two telescopes. Dots are individual measurements; line is the mean.

Download Full Size | PDF

The PSF metric provides an estimate of piston in waves by Eq. (5). When the piston error is zero, the side lobes are equal, and the metric reads zero. As the piston error approaches a half wave, one of the side lobes diminishes to zero, and the metric changes to + or - ½ wave, depending on the direction of the piston error. For piston errors beyond ½ wave, the metric wraps around [see Fig. 5(b)] due to the maximum pixel location “jumping” from one lobe to the next. The relative narrowness of the broadband spectral bandwidth (90 nm) delays attenuation of the metric until piston errors exceed about 10-15 waves.

The PSF metric is used iteratively as a feedback signal to correct for residual piston error.

3.4 Autophase experimental results

Repeatability experiments with point-sources show that Autophase has a repeatability of about 100 nm, RMS in pathlength. For starting conditions of more than 3 waves, there were occasional instances of whole-wave errors. What happens here is that the best-fit Gaussian is slightly off center, so the peak of the fit falls more than a ½ wave from the true point of zero-pathlength error. Use of the PSF metric then leads to errors that are one wave from correct phasing. One way to solve this problem is to run again. If the results of both runs agree, then the probability of a whole-wave error is now equal to the probability of Autophase finding the wrong wave in two independent trials, which we infer to be about 1%. Whether or not whole-wave errors are present, phase diversity always operates stably after running Autoalign and Autophase.

Repeatability experiments on extended scenes show that Autophase is good to about 150 nm RMS or better. A sequence of PSF’s and MTF’s with all telescopes open after running both tools is seen in Fig. 6 . Also shown is an extended scene after running the tools separately on it.

 

Fig. 6 PSF, MTF, and extended scene from start to finish of automatic alignment/phasing program

Download Full Size | PDF

4. Conclusion

The methods and results of automatically aligning and phasing a distributed-aperture imaging system are presented. Autoalign is run first, and it runs by taking an image through the reference telescope and then aligning the remaining telescopes by comparing their image offsets to the reference telescope image. Autophase is run next, and it works by maximizing the image spatial frequency content using the MTF metric, and if the scene is a point source, it fine-tunes the phasing error by using the PSF metric. After both programs are run, then all of the telescopes are opened, and phase diversity is run to regulate the tip/tilt and piston aberrations. This demonstration of automated alignment and phasing greatly improves the feasibility and technology readiness of distributed aperture imaging systems.