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Astronomical telescopes have long been tools of research, allowing astrophysicists and planetary scientists to gather the data they need to test hypotheses about the nature and behaviour of objects and phenomena in space. Increasingly such telescopes are finding applications in industry and defence, monitoring the trajectories of artificial satellites and receiving their optical communications signals. It should come as no surprise that the optical design of telescopes can vary significantly with application. In this article we look at some of the main optical properties of telescopes, how varying them affects telescope performance, and how this variation is used to meet the needs of different remote sensing applications.
Fundamentally a telescope is a collection of optics that are used to produce images of objects observed at a distance. Many telescopes produce magnified images to improve our ability to resolve details. While a telescope’s function is to collect and concentrate photons emitted or reflected by remote objects, it’s worth remembering that light is merely the carrier of the information or signal that we are ultimately trying to observe or record.
Figure 1 - Diagrams showing how the light paths from an object (in this case a cat) to a telescope objective change as the object is moved further from the telescope. Ultimately the object is optically at infinity and forms a point source image.
As shown in Figure 1, an object placed in front of a telescope objective (e.g. a cat in front of a lens) will have its image formed at the focal plane of the lens. As the object is moved further away from the objective two things happen. First, the wavefront of the light emitted or reflected by the object becomes increasingly planar by the time it hits the objective (i.e. light rays from the object are increasingly parallel at the objective). Second, the object covers a smaller portion of the telescope’s field of view even if its physical size doesn’t change. In the extreme case, where an object is effectively infinitely far from the objective, its light rays are effectively parallel when they reach the telescope and its angular size is effectively zero (i.e. it is a point source). Most astronomical objects, unless very large or comparatively close to the Earth, fall into this category. A telescope focus their light to a small point at the focal plane.
The fundamental properties of each telescope include the diameter of its objective and its focal length. From these two properties we can derive several other useful parameters that describe how a telescope focuses light to form an image.
Figure 2 - Diagram of the optical properties of two telescopes with identical field stops but different focal lengths. The telescope with a shorter focal length can access a larger angular Field of View.
Here we'll demonstrate how adjusting a telescope’s properties can change the way that is focuses light from a point source. As mentioned above, a telescope is not able to focus light from a point source to a point image with an angular size smaller the telescope’s diffraction limit, given by θ=1.22 λ/D. We can see from this formula, that for a telescope to produce a perfect point image of a point source with infinitely small size it would need to have an infinitely large objective. The distribution formed by the image of a point source at the focal plane of a telescope is known as the telescope’s Point Spread Function (PSF).
Assuming there are no external wavefront aberrations (e.g. from Earth’s turbulent atmosphere), most telescopes, having circular objectives, will have theoretical PSFs that are Airy disks with a bright central peak and concentric dark and bright rings of decreasing intensity [3]. For a reflecting telescope with primary and secondary mirrors with diameters of M_1 and M_2 respectively, the intensity of a telescope’s Airy disk PSF can be approximated as a function of distance R from the optical axis with the following formula [4].
In figures 3-5 this formula has been used to generate model Airy disks for different configurations of reflecting telescope. In each case, the top and bottom rows of subplots show identical Airy disks but with different scaling (log scale on top; linear scale on bottom). The central Airy disk in each figure is that produced by a telescope with M_1=1.0 m, f#=10, and a secondary mirror that obscures the central 10% of the primary by area (equivalent to M_2=0.32 m). In all cases the light has wavelength λ=0.55 μm. White lines indicate the scale of a 0.1 arcsecond angular distance in each image, and scales around the border of each image indicate the physical size of each Airy disk in microns. For each Airy disk the diffraction limit θ is the distance from the brightest point of the central peak to the darkest point of the first dark ring around the central peak. Scaling limits are the same for all linearly scaled plots in each figure.
Figure 3 - Demonstration of how a telescope's Airy disk changes with primary mirror diameter.
Figure 3 shows how changing the size of a telescope’s primary mirror can change its PSF. Halving the size of the primary mirror to 0.5 m, as seen in the left pair of frames, reduces its collecting power by a factor of 4. This results in a much fainter image. Following the Rayleigh criterion, reducing the size of the primary increases the telescope’s diffraction limit and reduces its peak resolution (Note that the Airy disk on the left is larger relative to the 0.1” reference bar than it is in the central image.). Although the PSF’s angular size has been doubled, its physical spot size is unchanged. Similarly, if we double the diameter of the primary mirror the collecting power is increased by a factor of 4 and the diffraction limit is halved, improving both the telescope’s sensitivity and its peak resolution.
Figure 4 - Demonstration of how a telescope's Airy disk changes with focal ratio.
Figure 4 shows how changing the focal ratio of a telescope can change its PSF. Reducing the telescope’s f# from 10 to 6 brings the image to focus over a shorter distance. This makes the focal spot physically smaller in the focal plane but does not change the diffraction limit of the telescope; angular resolution has stayed the same. Shrinking the physical size of the image concentrates light over a smaller area, increasing the intensity of the Airy disk relative to those imaged with larger f#. Notably, the field of view within the focal plane is now also larger. Increasing f# from 10 to 14 increases the magnification of the image and spreads light over a wider area of the focal plane, making the image fainter. Although the image is physically magnified, the angular resolution is unchanged.
Figure 5 - Demonstration of how a telescope's Airy disk changes with the size of a central obstruction.
Figure 5 shows how changing the size of a telescope’s secondary mirror changes its PSF. The changes shown here apply to changing the size of any circular central obstruction of the primary. Removing the central obstruction in the images on the left shows what the Airy disk looks like for a refracting telescope which has no secondary mirror. In this case both image contrast and the central peak of the airy disk are at their highest. The image on the right shows what happens to the PSF if the secondary mirror is increased in size until the central 50% of the primary mirror’s area is obscured. In this scenario the contrast of the PSF is greatly reduced as a greater proportion of light is focused into the outer rings of the Airy disk instead of the central peak.
In practice, telescope PSFs are almost never pure airy disks. This is partly because telescopes are typically constructed with more optical components than just the objective and a secondary mirror. Downstream optics in the telescope or instruments will impart their own effects onto the imaged PSF, even if these effects are small. Secondary mirrors are typically suspended above the primary, requiring supporting struts or wires to cross the view of the primary mirror. For a telescope focused at infinity, such structures are way out of focus and cannot be seen directly when images are taken. Support structures for the secondary mirror do change the diffraction properties of the telescope, however, and cause the formation of diffraction spikes in the telescope’s PSF that reduce contrast and can clearly be seen in images of bright point sources. In cases where maximising contrast is important (e.g. solar astronomy, or observation of night sky targets with low surface brightness), use of an off-axis telescope design may be preferable to minimise diffraction effects. A telescope’s PSF will also be significantly different if its objective is segmented. Very large telescopes with apertures much larger than 8 metres can only be constructed with an objective that is created with an array of small tessellated hexagonal mirrors. The hexagonal shape of these mirrors produces a diffraction pattern and PSF with hexagonal rotational symmetry instead of a round one like an Airy disk.
It is worth noting that a telescope’s PSF will also vary with the distance between a point source image and the telescope’s optical axis. Optical aberrations such as coma an astigmatism typically increase with distance from the optical axis. Telescopes designed for wide field imaging will include additional corrective optics to minimise these aberrations and maximise the useful FoV where stellar PSFs remain unaberrated within certain tolerances. At some point, however, the edge of the telescope’s useful FoV will be reached, beyond which images will be too optically distorted to be useful. The point at which this occurs for any given telescope depends significantly on its optical design and, by extension, its primary function or application.
In ground-based astronomy, it is only possible to image a telescope’s theoretical PSF at very high frame rates or with the assistance of an adaptive optics system. In all other cases the shape of the PSF will be defined by rapidly varying atmospheric turbulence that smears starlight over a wider area of the focal plane. Seeing limited PSFs in long-exposure images take the form of Moffat distributions [5]:
Here α and β are terms governed by the properties of the atmospheric turbulence at the time of the observation [6]. The Moffat distribution has a shape like that of the well-known Gaussian distribution, except that it has more pronounced wings (i.e. it’s fatter at its base).
Figure 6 compares the Airy disk of a telescope to a Moffat distribution containing the same amount of light for seeing conditions of 0.5’’. Note that a seeing FWHM of 0.5’’ is excellent for ground based observing and can only be regularly achieved at the very best observing sites in the world. Seeing values of >1’’ are much more common under natural conditions in most locations. Atmospheric seeing has the effect of smearing the PSF over the focal plane as photons are integrated. This reduces a telescope’s peak angular resolution from the diffraction limit to the Full Width at Half Maximum (FWHM) of the Moffat distribution. The amplitude of the seeing-limited PSF is also greatly reduced, and sensitivity suffers as a result.
Figure 6 - Comparison of a diffraction-limited Airy disk PSF and a seeing-limited PSF described by a Moffat distribution.
By rearranging the formula of the Rayleigh criterion, we calculate the telescope aperture required to achieve a given theoretical angular resolution: D=1.22 λ/θ.
Because 0.5’’ angular resolution is close to the best angular resolution achievable at optical wavelengths in seeing-limited conditions anywhere on Earth, there is a corresponding telescope aperture above which no practical improvement in angular resolution is possible for seeing-limited long exposure imaging and spectroscopy. By converting 0.5’’ to radians and plugging that into D=1.22 λ/θ we get an aperture of only 0.28 m for a wavelength of 550 nm. This means that under the best natural seeing conditions the largest optical telescopes in the world perform no better in terms of angular resolution than the amateur astronomer’s 11 inch telescope. While very large telescopes provide obvious benefits in terms of raw collecting power, specialist techniques and instrumentation such as interferometry and adaptive optics are needed to unlock the theoretically superior resolving power afforded by a large aperture.
In research, telescopes tend to fall into two broad categories that span the full gamut of telescope sizes. The first of these is the traditional high magnification telescope designed with a large f# to achieve high angular resolution over a relatively narrow field of view. Double figure focal ratios are common for such telescopes, and upper limits on their FoVs are typically <10’. For many applications, including high resolution imaging and spectroscopy, a small FoV is completely acceptable, making these telescopes versatile and widely used.
The second main telescope type is the astrographic telescope, which leverages a small f# to image very wide areas of the sky in a single shot. In extreme cases the FoV of astrographic telescopes may be as high as 60°, making them well suited to wide field imaging, all sky surveys, and time domain astronomy.
Astrographic telescopes are not well suited to applications other than imaging, and as a result are less versatile in application. Wide field multi-object spectroscopy instrumentation is undergoing active development but is extremely complex and inefficient without the inclusion of advanced robotic systems that correctly configure the spectrograph on the fly for each field observed [e.g. 7-9]. Only a small number of cutting-edge facilities have achieved this capability so far. Adaptive optics is even more difficult to implement over very wide FoVs, as starlight collected near the edges of a wide field image will have experienced different atmospheric distortions to that collected near its centre [10].
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