Besides systematic surface/wavefront deformations explained above (mainly due to misalignment of the optics), there are often permanent random deformations on the optic surfaces like ripples, scratches, dents, twists, misaligned panels, etc., with spatial dimensions ranging from several wavelengths to significant areas of the aperture. These deformations introduce identical deformations of the wavefront, which cannot be expressed in mathematical form (as the Zernike polynomials used above). Nevertheless, the effect on the beam pattern of this type of deformations can be analyzed in a statistical way and from a simple expression, the RUZE equation. This equation is often used to estimate the quality of a telescope, in particular as function of wavelength. The values obtained from this equation are directly related to the aperture efficiency, and beam efficiency, of the telescope, and hence are important for radiometric measurements (see Sect.1.5).
As illustrated in Figure 1.10, there are two parameters which allow a physical-optics description of the influence of random errors, i.e. the rms-value (root mean square value) of the deformations, and their correlation length L.
Random errors occur primarily on the main reflector; the other optical components of
the telescope (subreflector, Nasmyth mirror, lenses, polarizers) are relatively
small and can be manufactured with good precision. In order to explain the
rms-value , we assume that the reflector aperture is divided into many elements (i = 1,2,...N), and that for each element [i] the deformation
(i) of the reflector is known with respect to a smooth mean surface. The
rms-value of these random surface deformations is
A description of the wavefront deformation by the rms-value is incomplete since the value does not contain information on the structure of the deformations, for instance whether they consist of many dents at one part of the aperture, or many scratches at another part. A useful physical-optics description requires also a knowledge of the correlation length L of the deformations. L is a number (L D) which quantifies the extent over which the randomness of the deformations does not change. For example, the deformations of a main reflector constructed from many individual panels, which may be misaligned, often has a random error correlation length typical of the panel size, but also a correlation length of 1/3 to 1/5 of the panel size due to inaccuracies in the fabrication of the individual panels. A typical example is the 30-m telescope [Greve et al 1998].
When knowing, by one or the other method, the rms-value
correlation length L, it is possible to express the resulting beam shape in an
analytic form which describes well the real situation. The beam pattern
) of a wavefront with random deformations (
telescope may actually have several random error distributions] consists of the
degraded diffraction beam
) and the error beam
) such that