Diffractive Optics II Zone Plates

Abstract

Zone plates may be considered to be circular diffraction gratings with radially increasing line densities. Their focusing properties have long been well known,⁽¹⁾ and they offer perhaps the best chance in the foreseeable future of diffraction-limited imaging using soft X rays, where they are used in transmission. Figure 8.1 shows the requirement for focusing, namely that radiation emitted from an object point P is focused to an image point P’. The complex disturbance at P’ due to a spherical wavefront emitted by P is

$$Y\left( {P'} \right) = A\exp \left[ {ik(R + R')} \right]/(R + R')$$

((8.1))

where A is the amplitude at unit radius from P, R is the perpendicular distance from P to the focusing device, R’ is the perpendicular distance from the focusing device to P’, and k is the wavenumber $( = 2\pi /\lambda )$. In equation (8.1) the periodic time factor has been omitted since soft X-ray detectors only respond to time-averaged signals, and an inclination factor expressing the amplitude variation with distance has been ignored since (due to the small sizes of the optical components) only small angle diffraction need be considered. Using the Huygens-Fresnel wave propagation principle, each point on the wavefront emitted by P is considered to be a point source of spherical wavelets; all such points are in phase. The total effect at P’ is found by summing the effects from all points on the wavefront with relative phases at P’ determined solely by the optical distances from the wavefront to P’. If two signals on the wavefront have an optical path difference to P’ of an integral number of wavelengths, they will contribute in phase (i.e., interfere constructively), while if the optical path difference is an odd-integral number of half-wavelengths, they will contribute completely out of phase (i.e., interfere destructively). The wavefront may be imagined as being divided into a set of Fresnel (or half-period) zones by spheres differing in radius by $\lambda /2$ and centered on P’ Adjacent zones will then transmit radiation with opposite signs of phase—the resultant disturbance from these zones gives the propagating spherical wave in free space. If alternate zones are blocked, then the disturbances at P’ will have the same sign of phase and the summation shows an increase proportional to the number of zones. Note that it is not necessary for the distance R + R’ between P and P’ to be an integral number of half-wavelengths; in Figure 8.1, δ is introduced to take account of this, i.e., δ corresponds to an overall arbitrary phase.