Circular, Annular and Rectangular Apertures

Here is a movie showing a plane wave incident on several types of apertures from a variety of angles. Arroyo maintains the relative geometrical orientations of apertures and wavefronts by assigning a reference frame to each of these. In this movie, apertures of different shape are rotated 360 degrees in azimuth with respect to the wavefront. The apertures are then rotated 25 degrees off normal. The wavefronts appear foreshortened because they are striking the apertures off normal. Finally, the apertures are again rotated through 360 degrees in azimuth at this 25 degree angle. Note that edge pixels are weighted by their areal overlap with the aperture to avoid scalloping effects.

Aperture simulations

This demonstration was generated using the program aperture_verification, which is included in the Arroyo distribution.



Tiled Hexagonal Apertures

Arroyo has a specific class for representing compound apertures formed from packing hexagons into a regular pattern. These tiled hexagonal apertures are described by the edge length of each hexagon and the gap size between hexagons, as shown in the figure below.











Note that tiled hexagonal apertures provide only part of the functionality required to represent a true compound mirror. For example one can tip, tilt and piston the individual elements of a tiled hexagonal mirror, but this is not possible for an aperture.

Arroyo downweights pixels that straddle aperture edges by computing the areal overlap of the pixel with the aperture. This permits one to compute point spread functions even in cases for which the wavefront pixel scale is larger than the gap size, as is the case in the simulations below. The accuracy of this approximation has yet to be investigated, but clearly improves with decreasing pixel scale.

Here are two movies showing a plane wave incident on a tiled hexagonal aperture, and the resulting point spread function in the focal plane. The first movie shows an aperture chosen to match the tiling scheme for the Keck primary. The electromagnetic wavelength chosen for this simulation was 1 micron. The pixel scale of the wavefront was chosen to be 2.5 cm. The edge length of each tile was chosen to be 90 cm, and the gap size was chosen to vary between 0 and 2 cm. (The canonical value for Keck seems to be about 7 mm.) The PSF's in this movie are shown on a log stretch. The first shows a 3.16 arcsecond square area of the focal plane, where the PSF has been clipped at 1e-3 of its peak. This image is roughly Nyquist sampled. The second shows a 1.06 arcsecond square area of the focal plane, where the PSF is shown over its full range. Sampling is about 3 times Nyquist.

In the movie, a tiled hexagonal aperture with no gaps rotates about the optical axis by 60 degrees in 1 degree increments. The gap size is then increased to 2 cm in steps of .5 mm. The aperture rotates through another 60 degrees about the optical axis. The aperture is then tilted 30 degrees with respect to the optical axis, in 1 degree steps. Finally, the aperture is rotated through another 60 degrees about the optical axis.

Keck simulation


The second movie shows PSF's arising from an aperture like that proposed for the Thirty Meter Telescope (TMT) (See, for example, the CELT green book or Troy & Chanan, Applied Optics, v 42, n19 p3745). This tiled hexagonal aperture is composed of hexagonal apertures with edge lengths of .5 meters. The electromagnetic wavelength was 1 micron and the gap size varied from 0 to 10 mm. The pixel scale of the wavefront was chosen to be 2 cm. The movie shows only rotation about the optical axis. Again, the PSF's in this movie are shown on a log stretch. The first shows a roughly 1 arcsecond square area of the focal plane, where the PSF has been clipped at 1e-4 of its peak. This image is roughly Nyquist sampled. The second shows a third of an arcsecond square area of the focal plane, where the PSF has been clipped at 1e-2 of its peak. Sampling is about 3 times Nyquist.


TMT simulation


A third of an arcsecond is roughly the range of habitability for earthlike planets around stars within 10 pc. Exojupiters are expected to have brightnesses of about 1e-9 of their parent stars, while exoearths should have brightnesses of about 1e-10. Thus, if an adaptive optics system is able to remove the effects of atmospheric turbulence and telescope vibrations within a third of an arcsecond of the parent star, one might expect to be able to directly image exoearths with a thirty meter telescope in regions of the focal plane for which the PSF appears white.


Compute Time Estimates

On a 2.5 GHz pentium, Arroyo required about 4 seconds to aperture the 500x500 wavefront used for the Keck tiling scheme, and about 33 seconds to aperture the 1450x1450 wavefront used in the TMT simulation. The wavefront dimensionality is a free parameter in the simulation.

These demonstrations were generated using the program tiled_hexagonal_aperture_verification, which is included in the Arroyo distribution.