Elementary Diffraction Patterns

60 Elementary Diffraction Patterns In the previous chapter, we derived general diffraction formulae without offering any practical examples. We now examine some real situations that can be calculated in closed form. This is successful in practice only for diffraction in field-free space and we shall consider mainly the Fresnel and Fraunhofer diffraction treated in Section 58.3, therefore. 60.1 T h e o b j e c t f u n c t i o n We consider the configuration shown in Fig. 60.1 and adopt the corresponding notation. A point source Q, located at the position rs = (x~,ys,z~) with z~ =: - a < 0, emits a spherical wave C

¢~(~) -

Ir-

eikl~._,, i

(60.1)

,'~1

In the Fresnel approximation, this simplifies to ¢~(~) =

Iz

c zsl exp(iklz-zsl)exp [ - - { ( x - x s ) 2 + ( y - y s ) 2 } ]

-

21z-

z~l

(60.2)

In the plane of the diffracting object, z = 0, and with object coordinates xo, yo, this formula reduces to

~o(Xo, Yo) -- aC exp(ika)exp [ Va ik {(x°-x~)~+(y°-y~)~} ]

(60.3)

This is the wavefunction incident on the object plane, on the far side with respect to the observer. We now assume that the object is effectively plane and so thin that its extent in the z-direction can be ignored. It causes a modulation of any wave passing through it, which we can describe by a complex transmission function* O(xo, yo) - e -a(z°'y°)+irl(z°'y°) (60.4) * Note that two conventions are in use for the sign of the phase term, corresponding to the choice of sign in (57.1).