Astronomical Techniques - Images and Aberrations

Telescopes are almost universally needed in conjunction with detectors to limit the solid angle under study, map the detector size or resolution to angular scales of interest, and concentrate radiation on the detector. Among these, imaging systems share processes and limitations imposed by radiation properties and optical design.

First, some useful terms:

• Focal length (sometimes an effective focal length): distance from the primary optical element to the focal plane. This may be altered by intermediate elements changing the physical length of the system for given focal length. The effective focal length is then set by the rate of convergence of rays at the focal plane and the primary diameter.
• Focal ratio f: ratio between (effective) focal length F and primary element diameter D, as f = F/D. This controls how quickly extended objects can be detected through their surface brightness, and how strongly some aberrations act.
• Image scale: the mapping factor between angular distance on the sky and linear distance in the focal plane. For an effective focal length F, this is just F in linear distance per radian (or F/206264.8 in distance per arcsecond).
• Snell's law: At the interface between two media of different refractive indices n1,n2, the angles between the direction of ray propagation and the surface normal on each side of the boundary are given by n1 sin q1 = n2 sin q2. This can be shown from Fermat's principle.
• Reflection law: angle of incidence equals angle of reflection with respect to the local surface normal.

All imaging systems suffer diffraction effects due to the wave nature of radiation. The simplest example is to consider a slit aperture (which by symmetry can be analyzed one-dimensionally). The familiar result is a central maximum surrounded by a symmetric set of diffracted peaks; the book's derivation starts from Huygens wavelets, which give a very useful way of seeing how diffraction acts in a particular situation. For the astronomically more usual case of a circular aperture, the derivation is more complex, but we still see a central maximum (the Airy disk) surrounded by concentric diffracted rings. A common definition of the resolution attainable with a given optical system is the Rayleigh criterion, reached when two point sources have separation equal to the radius of the first diffraction minimum. This is close to λ/D (the Rayleigh criterion gives a factor 1.22) for aperture diameter D; a rule of thumb is that for visible light, a 100-mm aperture resolves 1 arcsecond. It is common at a variety of wavelengths to take λ/D as a measure of the diffraction limit. For example, a 43-m telescope has a diffraction limit in arcminutes numerically close to the observing wavelength in cm.

The diffraction pattern of an arbitrary aperture is closely related to its two-dimensional Fourier transform (being its power spectrum). The rings result from the sharp edges of the slit or aperture. Sometimes one wishes to minimize these rings at the expense of the width of the central peak, to detect faint companion stars or look for planets. In this case, one apodizes the aperture by placing a filter at some pupil plane (where the aperture is imaged) that gently rolls to zero transmission at the edges, so there will be little or no light outside the (broader) central pattern. You can't beat the diffraction limit, but there is scope for reshaping the details of its shape.

Aberrations:
• Chromatic: real materials have refractive index varying with wavelength n(λ), so the focal length varies with λ. Systems using refraction are thus limited in the bandwidth over which good focus can be attained. Early telescopes used very long f-ratios to minimize the false color so produced, reaching "aerial telescopes" of length 60 meters. The traditional fix involves two lenses having glass with rather different n(λ) behavior, such as so-called crown and flint glass. Generally, the focal length of a set of N lenses as a function of wavelength is a polynomial-like function of order N - so a doublet (achromat) has a parabolic form, and can be corrected over some limited wavelength range. Some camera lenses have as many as 16 elements, though most don't act on the chromatic aberration. Small telescopes with three-element objectives (apochromats) are available. Chromatic aberration leads to a circle of confusion for finite bandwidth; one usually focuses to the minimal size of this circle.
• Spherical: occurs if all parts of the primary don't have the same focal length, such as spherical mirrors exhibit. This means that there is no single point of sharp focus. The best-known case is of course the primary mirror on the Hubble Space Telescope, which shows a shape error of order 1 μm peak-to-peak, brought on by incorrect spacing of mirrors in a test apparatus. Most large telescope primaries are thought to show significant spherical aberration, often corrected by custom polishing of other optical elements.
• Coma: while a paraboloid gives sharp images on-axis, they deteriorate quickly off-axis into comet-like shapes (hence the name). This can be avoided if the optical system obeys the Abbe' sine condition for imaging (text p. 56). This is possible with multiple reflections to a good approximation.
• Distortion: scale variation with position. Some is unavoidable, since we map from angles to distances on a nearly plane surface. This is usually termed the tangent-plane approximation, which we will see again in astrometry.

Fabrication: grinding, polishing, spin casting.

Testing: Foucault test, null test versions.

Optical elements are often coated, either for optimal reflectivity or to reduce reflections at air:glass boundaries. At the interface between substances with n1,n2, minimal reflection is achieved by use of interference if an appropriately thin coating of a substance with n3 = (n1 n2)1/2 is applied. Mirror coatings may reflect the need for high reflectivity or minimum IR emission. Aluminum, sometimes with a protective coating, is common for visible work. Silver is occasionally used, and gold is excellent for IR work. These coatings are applied by evaporation in vacuum.

Design of telescopes may rely on analytic or ray tracing approaches. Ray tracing is particularly amenable to programming, requiring only the Snell and reflection laws plus descriptions of the optical components.

Mirrors require support, possible across the entire back surface because they don't have to be transparent. Thermal effects are important in thick mirrors, so glasses with low thermal expansion coefficients (first Pyrex, later Cer-Vit and Zerodur) are chosen. Even so, a measure of active support improves even conventional telescope performance, and is crucial for the new generation of thin-mirror and segmented-mirror designs.

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