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Fig. 3.101. Fraunhofer diffraction pattern of a rectangular aperture 8 mm x 7 mm, magnification 50 x, A = 579 nm. The centre was deliberately overexposed to reveal the secondary maxima (after Born-Wolf [3.120(b)] and Lipson, Taylor and Thompson, courtesy Brian Thompson)

3.10.2 Coherence

The above formulae are concerned with a point source. For an extended source, the result can be derived by integration, but here we must introduce the concept of coherence. If two beams originate from the same physical source, the phase fluctuations are normally fully correlated and the beams are mutually coherent. In beams from different sources, the phase fluctuations are uncorrelated and the beams are mutually incoherent. An intermediate situation can occur, that of partial coherence. The concept of the "degree of coherence" was introduced by Zernike [3.126] and applied further to image formation by Hopkins [3.127] [3.128]. For a coherent extended source, the complex amplitude must be integrated; for an incoherent extended source, the intensity. For a partially coherent extended source, the integration must take account of the degree of coherence between different elements of the source. Born-Wolf [3.120(b)] treats the important case of an incoherent luminous wire of infinite length diffracted by a narrow slit aperture parallel to the source, taken to be in the y-direction. Now q = m — mo, where mo is the position of a point source, so the intensity due to the line source is obtained by integrating (3.435) with respect to q. Born-Wolf show that the intensity I' is characterized by a similar function and is given by

where I» is the intensity at the centre of the pattern with p = 0.

3.10.3 The Point Spread Function (PSF) due to diffraction at a circular aperture

The derivation of the PSF in the case of the circular aperture is, in principle, similar to that for the rectangular aperture except that polar coordinates give a more natural formulation. Then, again following Born-Wolf [3.120(b)], p cos $ = x , p sin $ = y for the pupil and w cos ^ = p , w sin ^ = q for the image plane, where w = (p2 + q2)1/2 is the angular separation of a point Q' in the image plane with direction cosines p, q relative to the central point of symmetry Q0 of the pattern. The diffraction integral (3.434) becomes

and, finally

J0(kpw)p dp

2Ji (kpmw)

kpmw

where J0 and J1 are the Bessel functions of zero and first orders. The intensity I at Q' is therefore

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