## Info

Fig. 1.27 Left: Pincushion distortion. Right: Barrel distortion

In general, telescopes do not require to be corrected for distortion because this effect does not degrade the image quality and can always be removed by image processing.

However, liquid mirror telescopes (Lmts) - which are zenithal and transit instruments - require a distortion correction and, in addition, a sky projection correction when not working at the equator of the Earth. This was first solved by E.H. Richardson [76] who showed that a three-lens correction was inappropriate and thus designed dedicated four-lens correctors for large Lmts (cf. also Sect. 7.7.2).

### 1.11 Diffraction

The colored lines of a thin oil film lying on water or the colored lines of equal thickness of a very thin lens - which were recognized as interference fringes by T. Young - as well as the image showing oscillations in intensity when produced by a point source through an optical system, are evidence effects of the wave nature of light. Although easily observable, some of these phenomena were first referenced during the Renaissance by Leonardo da Vinci. It was F.M. Grimaldi, in his book (1665), who drew attention to them with accurate descriptions. Geometrical optics -or ray trace - were inadequate for deducing these phenomena, and neither could the later corpuscular theory explain the diffraction effects. In an attempt at explanation, in 1678 Huygens [80] constructed the following concept:

^ Any point of a wavefront gives rise to secondary disturbances which are spherical wavelets and, at any later instant, the light distribution may be regarded as the sum of these wavelets.

### 1.11.1 The Diffraction Theory

Inspired by Huygens' construction, Fresnel in 1816 laid down a celebrated memoir [62] in which he explained the principle of interference known as the Huygens-Fresnelprinciple. Reviewing Fresnel's prize memoir to the French Academy, Poisson deduced (1818) that, following Fresnel's theory, a bright spot should appear at the center of the shadow of a small disk, and concluded that this should be wrong since never observed. However, Arago, who was also a reviewer, carried out the experiment and observed the predicted white spot. In 1882, Kirchhoff [87] gave a more complete and sound mathematical basis to Fresnel's theory; this is known as Fresnel-Kirchhoff diffraction theory.

Let us assume a wave propagation in a vacuum. According to the wave equation (1.4), each Cartesian component W(r, t) of the field vectors E and H must satisfy the scalar wave equation

c2 dt2

where V2W is the Laplacian.

Use of an exponential function instead of trigonometric functions simplifies the calculations with scalar waves. Assuming it is strictly monochromatic of wavelength X = c/v, where the frequency v is the number of vibrations per second, we may separate the space- and time-components. Denoting i = a/—Y and ^ for a real part, we have the form

where U is the complex amplitude, 0O a phase constant, o the angular frequency, and k the wave number. In a vacuum these later quantities are o = 2nv = 2nc/X, k = o/c = 2n/X.

By substitution of W into (1.81), we find that the space-dependent part U must satisfy the equation

Considering a wave passing through a plane aperture stop (Fig. 1.28) whose area is A(x,y), a solution of the space-dependent term is expressed by the Fresnel-Kirchhoff diffraction integral

in which Uo is a constant, p, p' the respective distances from the source point P and from the point P' - where the disturbance is determined - to the elementary surface dS in the area A, and (n,p), (n,p') the angles with the normal n to the plane A. Setting the origin O of the x,y plane at any point inside the aperture A, we will assume that points P and P' are at large distances from the origin and that the angles generated by the lines PO and OP' with PP' are moderate. Thus, the term pp' may be replaced by qq and the term into the square bracket replaced by 2 cos 5 where 5 is the angle between PP' and the z axis. These two terms will not vary appreciably over the aperture, whilst the exponent term p+p' will change by many wavelengths. The diffraction integral of the disturbance at P' then becomes

Expanding p and p' as functions of q, q and power series of x/q, y/q, x/q and y/q', the wave disturbance is

\ f(x,y) = rxx + ryy + gix2 + g2xy + g3y2 + •••, (1.85b)

where, in a displacement of the element dS into the aperture plane, rx(P') and ry (P') are the Cartesian components of the relative variation of the optical path p + p' with respect to q + q', whilst gi(P'), g2, g3 are functions of the next order relative variations.

The Fresnel-Kirchhoff diffraction theory may lead to integrations by considering only the first two terms of f (x,y) if assuming that the aperture area A is small; this simplest case is called the Fraunhofer diffraction. When the quadratic terms cannot be neglected, such as for the diffraction of an infinite long edge, this is the

Fresnel diffraction case. A detailed account on these two cases is given by Born and Wolf [17].

When observing a diffracted wave near an aperture edge, one can see a broad intensity oscillation of the first fringes at the edge. A useful interpretation of this phenomena, which was investigated by Sommerfeld (1896), is to consider that the observed pattern is the superposition result of the incident wave with a locally cylin-dric edge wave such as was generated from the edge.

### 1.11.2 Diffraction from a Circular Aperture

Denoting p, 9 the polar coordinates of a current point into the plane of a circular aperture of radius p = a, the coordinates of this point are p cos 9 = x, p sin 9 = y.

From the definition of rx and ry in (1.85b), it follows that the polar coordinates r, y of a diffracted ray at the image plane are r cos y = rx, r sin y = ry.

After substitution of f (x,y) = rxx + ryy into (1.85a), the diffraction integral becomes i'a /*2n

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