The Telescope

Chapter 2: Gathering light - the telescope 2.1.Basic Principles Astronomy centers on the study of vanishingly faint signals, often from complex fields of sources. Job number one is therefore to collect as much light as possible, with the highest possible angular resolution. So life is simple. We want to: 1.) build the largest telescope we can afford (or can get someone else to buy for us), 2.) design it to be efficient and 3.) at the same time shield the signal from unwanted contamination, 4.) provide diffraction- limited images over as large an area in the image plane as we can cover with detectors, and 5.) adjust the final beam to match the signal optimally onto those detectors. Figure 2.1 shows a basic telescope that we might use to achieve these goals. Achieving condition 4.) is a very strong driver on telescope design. The etendue is constant only for a beam passing through a perfect optical system. For such a Figure 2.1. Simple Prime-Focus Telescope system, the image FoTelescope primary mirror parameters. quality is limited by the wavelength of the light and the diffraction at the limiting aperture for the telescope (normally the edge of the primary mirror). Assuming that the telescope primary is circular, the resulting image illumination is where m = (π r0 θ / λ), r0 is the radius of the telescope aperture (=D/2 in Figure 2.1), is the wavelength of operation, and J1 is a Bessel function of the first kind. The result is the well- known Airy function (Figure 2.2), named after the British astronomer George Biddell Airy. We will illustrate a simple derivation later in this chapter. The first zero of the Bessel function occurs at m= 1.916, or for small θ, at θ Figure 2.2. The Airy Function 1.22 λ / D. The classic Rayleigh Criterion states that two point sources of equal brightness can be distinguished only if their separation is > 1.22 /D; that is, the peak of one image is no closer than the first dark ring in the other. With modern image analysis techniques, this criterion can be readily surpassed. The full width at half maximum of the central image is about 1.03 /D for an unobscured aperture and becomes slightly narrower if there is a blockage in the center of the aperture (e.g., a secondary mirror). Although we will emphasize a more mathematical approach in this chapter, recall from the previous chapter that the diffraction pattern can be understood as a manifestation of Fermat’s Principle. Where the ray path lengths are so nearly equal that they interfere constructively, we get the central image. The first dark ring lies where the rays are out of phase and interfere destructively, and the remainder of the light and dark rings represent constructive and destructive interference at increasing path differences. Although the diffraction limit is the ideal, practical optics have a series of shortcomings, described as aberrations. There are three primary geometric Figure 2.3. The most famous example aberrations: of spherical aberration, the Hubble Space Telescope before various means were taken to correct the problem. 1. spherical, occurs when an off-axis input ray is directed in front of or behind the image position for an on-axis input ray, with rays at the same off- axis angle crossing the image plane symmetrically distributed around the on-axis image. Spherical aberration tends to yield a blurred halo around an image (Figure 2.3). A sphere forms a perfect Figure 2.4. The focusing properties of a spherical reflector. image of a point source located at the center of the sphere, with the image produced on top of the source. However, for a source at infinity, the distance, F, where the reflected ray crosses the axis of the mirror is where R is the radius of curvature of the mirror and is the angle of reflectance from the surface of the mirror. That is, the reflected rays cross the mirror axis at smaller values of F the farther off-axis they impinge on the mirror. The result is shown in Figure 2.4; there is no point along the optical axis with a well-formed image. 2. coma occurs when input rays arriving at an angle from the optical axis miss toward the same side of the on-axis image no Figure 2.5. Comatic matter where they enter the telescope aperture, and with a image displayed to show interference. progressive increase in image diameter with increasing distance from the center of the field. Coma is axially symmetric in the sense that similar patterns are generated by input rays at the same off axis angle at all azimuthal positions. Comatic images have characteristic fan or comet-shapes with the tail pointing away from the center of the image plane (Figure 2.5). A paraboloidal reflector provides an example. The paraboloid by construction produces a perfect image of a point source on the axis of the mirror and at infinite distance. As shown in Figure 2.6, the focus for a ray that is off-axis by the angle is displaced from the axis of the mirror by 3. astigmatism (Figure 2.7) is a cylindrical wavefront distortion resulting from an Figure 2.6. Focusing properties of a optical system that has different focal paraboloidal mirror. planes for an off-axis object in one direction from the optical axis of the system compared with the orthogonal direction. It results in images that are elliptical on either side of best-focus, with the direction of the long axis of the ellipse changing by 90 degrees going from ahead to behind focus. Two other aberrations are less fundamental but in practical systems can degrade the entendue: 4. curvature of field, which occurs when the best images are not formed at a plane but instead on a surface that is convex or concave toward the telescope entrance aperture (see Figure 2.8). 5. distortion arises when the image scale changes over the focal plane; that is, if a set of point sources placed on a uniform Figure 2.7. Illustration of astigmatism (from grid is observed, their relative image Starizona). If we assume the astigmatism is due to an asymmetry in the optics, then the positions are displaced from the tangential plane contains both the object being corresponding grid positions at the focal imaged and the axis of symmetry, and the plane. Figure 2.9 illustrates symmetric sagittal plane is orthogonal to the tangential one. forms of distortion, but this defect can occur in a variety of other forms, such as trapezoidal. Neither of these latter two aberrations actually degrade the images themselves. The effects of field curvature can be removed by suitable design of the following optics, or by curving the detector surface to match the curvature Figure 2.8. Curvature of field of field. Distortion can be removed by resampling the images and placing them at the correct positions, based on a previous characterization of the distortion effects. Such resampling may degrade the images, particularly if they are poorly sampled (e.g., the detectors are too large to return all the information about the images) but need not if care is used both in the system design and the method to remove the distortion. Optical systems using lenses are also subject to: 6. chromatic aberration, resulting when light of different colors is not brought to the same focus. A central aspect of optics design is Figure 2.9. Distortion. that aberrations introduced by an optical element can be compensated with a following element to improve the image quality in multi-element optical systems. For example, some high performance commercial camera lenses have up to 20 elements that all work together to provide high quality images over a large field. In addition to these aberrations, the telescope performance can be degraded by manufacturing errors that result in the optical elements not having the exact prescribed shapes, by distortions of the elements in their mountings, and by mis-alignments. Although these latter effects are also sometimes termed aberrations, they are of a different class from the six aberrations we have listed and we will avoid this terminology. In operation, the images can be further degraded by atmospheric seeing, the disturbance of the wavefronts of the light from the source as they pass through a turbulent atmosphere with refractive index variations due to temperature inhomogeneity. A rough approximation of the behavior is that there are atmospheric bubbles of size r0 = 5 – 15 cm with temperature variations of a few hundredths up to 1o C, moving at wind velocities of 10 to 50 m/sec (r0 is defined by the typical size effective at a wavelength of 0.5m and called the Fried parameter). The time scale for variations over a typical size of r0 at the telescope is therefore of order 10msec. For a telescope with aperture smaller than r0, the effect is to cause the images formed by the telescope to move as the wavefronts are tilted to various angles by the passage of warmer and cooler air bubbles. If the telescope aperture is much larger than r0, many different r0-sized columns are sampled at once. Images taken over significantly longer than 10msec are called seeing-limited, and have typical sizes of /r0, derived similarly to equation (1.12) since the wavefront is preserved accurately only over a patch of diameter ~ r0 . These images may be 0.5 to 1 arcsec in diameter, or larger under poor conditions. However, since the phase of the light varies quickly over each r0-diameter patch, a complex and variable interference pattern is formed at the focal plane due to the interference among these different patches.

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