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

Abstract

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.

Self-similarity in azimuthal Walsh filters and corresponding far-field diffraction characteristics : a unique study to control tightly focused fields and coupling of light into metamaterials, plasmonic structure and waveguides

“…It has been observed that like radial and annular varieties, the whole set of azimuthal Walsh functions does not exhibit self-similar structures. But only distinct groups of azimuthal Walsh functions demonstrate self-similarity among their individual constituents like radial and annular Walsh functions [48][49]. Azimuthal Walsh filters are observed to possess a unique rotational self-similarity exhibited among adjacent orders which is also reported in this research work.…”

“…To obtain the higher orders of azimuthal Walsh functions, the technique as proposed by Hazra and Mukherjee [48] is used whereas rotational self-similarity between specific orders has also been observed. From 0 ( ) , 1 ( ) can be derived by dividing the interval (0, 2π) in two equal sectors: the first sector is within (0, π) over which the phase is 0 and the second sector is within (π, 2π) over which the phase is π.…”

Multiple-plane image formation by Walsh zone plates

“…For instance, the irradiances shown in the red box for 8 ≤ k ≤ 15 are replicated on a smaller scale in the two green boxes for 16 ≤ k ≤ 31 and even on smaller scale in the four blue boxes for 32 ≤ k ≤ 63. The property of self-similarity in axial intensity distributions of radial Walsh filters was also reported in the far field diffraction patterns [20]. We have extended this property to the axial irradiances provided by WZPs working as aperiodic diffractive lenses.…”

“…Let us start revising the concept of radial Walsh filters [19,20]. It was based on the Walsh functions represented in Fig.…”

“…For example, these filters have been proved to be useful in adaptive optics and apodization problems [16,17], and for tailoring the resolution of microscopic systems [18]. Earlier studies on radial Walsh filters have shown that the corresponding axial and transverse intensity distributions in the far-field diffraction patterns present self-similar properties [19,20].…”