Self-Similarity in Transverse Intensity Distributions in the Farfield Diffraction Pattern of Radial Walsh Filters

In a recent communication we reported the self-similarity in radial Walsh filters. The set of radial Walsh filters have been classified into distinct self-similar groups, where members of each group possess self-similar structures or phase sequences. It has been observed that, the axial intensity distributions in the farfield diffraction pattern of these self-similar radial Walsh filters are also self-similar. In this paper we report the self-similarity in the intensity distributions on a transverse plane in the farfield diffraction patterns of the self-similar radial Walsh filters. 1. Introduction A self-similar object is exactly or approximately similar to a part of itself; that is, the whole has the same shape or structure as one or more of the parts. Self-similarity is a typical property of fractals [1]. Derived from radial Walsh functions [2–4], radial Walsh filters form a set of orthogonal phase filters that take on values either 0 or phase, corresponding to or ？1 value of the radial Walsh functions over the prespecified annular regions of the circular filter. They may be considered as binary zone plates [5, 6] and have wide range of applications [7–15]. The order of the Walsh filter is equal to the number of zeros crossing within the specified domain. Various orders of the Walsh filters having self-similar replicating phase sequences constitute a group. It has been shown that the axial intensity distributions in the farfield diffraction pattern of a pupil with these self-similar groups of filters are also self-similar [16]. Earlier studies on fractal zone plates and diffracting apertures [17–22] were mainly confined to the case of axial intensity distribution. As a prelude to the study of the three-dimensional light distribution in the farfield region, it is necessary to study the characteristics of the intensity distribution on the transverse plane in the presence of these filters. In this paper, we report our investigations on the transverse intensity distribution on the farfield plane of a pupil with radial Walsh filters. It is noted that the transverse intensity distributions exhibited by the self-similar groups of radial Walsh filters are also self-similar. The next section deals with the farfield diffraction pattern in the transverse plane. Section 3 deals with self-similarity in the transverse intensity distribution of the self-similar radial Walsh filters, followed by the concluding section. 2. Farfield Diffraction Pattern in the Transverse Plane With reference to Figure 1, the complex amplitude distribution on a transverse plane located