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the Gaussian focus shift by adding the constant term which shifts the origin to minimize the rms; the term R0 (No. 8) balances the third order spherical aberration against focus and a constant. Such balancing is, of course, well known from optical design using classical Hamilton terms.

Another excellent treatment of Zernike polynomials is given by Malacara [3.124]. He tabulates some comparative properties of a number of one-dimensional orthogonal polynomials: Legendre, Tschebyscheff, Jacobi, Laguerre and Hermite. In connection with active optics we shall introduce a further important type with great practical significance, namely natural vibration modes. Malacara also gives formulae and tables, based on work by Sumita [3.125], for conversion from Zernike polynomials to monomials and vice-versa.

3.10 Diffraction theory and its relation to aberrations

3.10.1 The Point Spread Function (PSF) due to diffraction at a rectangular aperture

Our treatment up to now has been entirely on the basis of geometrical optics for which rays are defined as the normals to a geometrical wavefront undisturbed by the physical boundary limiting its size.

Diffraction is a phenomenon arising from the wave theory of light. For a full account, the reader is referred to the standard work by Born-Wolf [3.120]. Many other excellent treatments with various emphases are available, e.g. Bahner [3.5], Welford [3.6], Marechal and Francon [3.26] and Schroeder [3.22]. Schroeder's treatment in his Chap. 10 and 11, with particular reference to the Hubble Space Telescope, not only gives the basic theory but also an admirable practical treatment of the diffraction theory of aberrations.

Practical diffraction theory is based on certain important approximations: that the divergence angles of all rays from the object to the pupil, and the convergence angles from the pupil to the image, are small; that the pupil is large compared with the wavelength of the incident light; that polarisation effects are neglected. Normally, it is also assumed that the light transmission is uniform over the pupil: if a departure from this condition is introduced, then a procedure known as apodisation is effected. The first approximation above concerning small divergence and convergence angles leads to the neglect of all quadratic terms in the expression of the optical path length in terms of the pupil and field coordinates. This distinguishes Fraunhofer and Fresnel diffraction: in Fraunhofer diffraction the quadratic terms are ignored; in Fres-nel diffraction they are taken into account. Fortunately, for most problems of optical instruments and virtually all cases of telescopes, the Fraunhofer approximations give a very good accuracy. However, for special cases of very high relative apertures or fields, the small angle approximation must be borne in mind.

For consistency with our previous notation in which a point in the pupil defined by p, $ was resolved into coordinates x, y (z being the axial direction) and the image height denoted by r/, we shall reverse the notation used by Born-Wolf and Schroeder and use that shown in Fig. 3.99 for the case of a rectangular aperture (or slit). We assume a perfectly aberration-free spherical wavefront is converging from the exit pupil E' to the image point Q0. Following Born-Wolf [3.120(b)], the Fraunhofer diffraction integral giving the complex amplitude U(Q ) at point Q' near to Q'0 is

JLXi

Fig. 3.99. Exit pupil (x,y,z) and image plane (n, Z) coordinate systems

JLXi

Fig. 3.99. Exit pupil (x,y,z) and image plane (n, Z) coordinate systems

In this equation, k = 2n/X is defined from the basic differential wave equations, while the quantities p and q are defined by p = l — lo , q = m — mo , where lo,mo and l,m are the first two direction cosines of the incident and diffracted waves respectively. It is part of the definition of the approximation of Fraunhofer diffraction that the four direction cosines only enter linearly in the above combinations, implying that the diffraction effect is unchanged if the pupil aperture is shifted in its own plane. Now i+ m e-ikpxdx =__L le-ikpxm — eikpxm 1 = 2sin(kpXm)

J-xm ikp kp from the Euler relation, and similarly for the other integral in y. The intensity I at point Q in the image plane is therefore

with Io = C2A2 = EA/A2 is the intensity at the central point Q'0 of the diffraction pattern, E being the total energy falling on the pupil and A = 4xmym being its area. According to geometrical optics, the image at Q0 of an infinitesimal conjugate object point Q0 would be an infinitesimal point, whereas diffraction theory gives the intensity distribution of (3.435).

From the definition of p and q above in terms of the differences of the direction cosines, which represent the angular shifts of Q from Q0 in the x and y planes respectively, we have for an object at infinity in the normal telescopic case

as the linear and angular separations of Q from Q0 in the y-plane. Then

Sv = qf = —y f , kVm if we define Suy as the normalized angle

Suy = kqym appearing in (3.435). Substituting for k gives

in which Ny is the f/no of the rectangular aperture in the y-direction. The function

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