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.
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 [
Earlier studies on fractal zone plates and diffracting apertures [
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
With reference to Figure
Exit pupil and image plane in the image space of an axially symmetric imaging system.
For an aberration-free pupil having uniform amplitude,
The normalized complex amplitude distribution on the transverse plane is
For a phase filter consisting of
The normalized complex amplitude distributions on the transverse plane is
As discussed in detail in our previous communication [
Radial Walsh filters of Group I, that is, orders 1, 3, 7, 15, and 31, produce self-similar transverse intensity distribution. Orders 1 and 3 exhibit distinct dual maxima around the point
Transverse intensity distribution curves for radial Walsh filters of Group I, orders 1, 3, 7, 15, and 31.
Subgroup IIA members are radial Walsh filters of orders 2, 4, 8, and 16. Order 2 exhibits transverse intensity distribution with dual maxima and adjacent fainter side lobes as illustrated in Figure
(a) Transverse intensity distribution curves for radial Walsh filters of Group IIA, orders 2, 4, 8, and 16. (b) Transverse intensity distribution curves for radial Walsh filters of Group IIB, orders 6, 12, and 24. (c) Transverse intensity distribution curves for radial Walsh filters of Group IIC, orders 14 and 28.
As with subgroup IIA, the general trend observed for each subgroup is that, with increase in order of filters within a particular subgroup, as we move from lower to higher orders the side ripples increase in number and the intensity of the lobes near the origin and side ripples fall in magnitude. The decrease in intensity for the central lobes near the origin is more compared to the side lobes with the consequence that the side lobes shoot up in intensity to match the scale of the central lobe of that particular order. However, each subgroup has a unique self-similar pattern exhibited by its members.
Subgroup IIB contains Walsh orders 6, 12, and 24. These filters produce self-similar transverse intensity distributions with dual maxima pair placed on either side of the origin and side ripples in the form of triplet pairs placed symmetrically on either side of the dual maxima as illustrated in Figure
Subgroup IIC contains Walsh orders 14 and 28. These filters produce self-similar transverse intensity distributions as illustrated in Figure
Members of Group IIIAA, that is, radial Walsh filters of orders 5, 11, and 23, produce transverse intensity distribution curves as illustrated in Figure
(a) Transverse intensity distribution curves for radial Walsh filters of Group IIIAA, orders 5, 11, and 23. (b) Transverse intensity distribution curves for radial Walsh filters of Group IIIAB, orders 13 and 27. (c) Transverse intensity distribution curves for radial Walsh filters of Group IIIBA, orders 9 and 19.
Subgroup IIIAB contains Walsh orders 13 and 27. The self-similar transverse intensity distributions produced by these orders are depicted in Figure
Subgroup IIIBA contains Walsh filter of orders 9 and 19. These orders produce self-similar transverse intensity distribution with higher side lobes as compared to the two central lobes on either side of the origin, as illustrated in Figure
Walsh orders contained in subgroup IVAAA, that is, 10 and 20, produce self-similar transverse intensity distribution as illustrated in Figure
Transverse intensity distribution curves for radial Walsh filters of Group IVAAA, orders 10 and 20.
It may be noted in Figures
Some studies were carried out on other subgroups whose first member falls within the range (0, 31) of
The focusing properties of radial Walsh filters both in the axial and in the transverse directions portray self-similarity which can be correlated to the self-similar structures of the diffracting apertures, a characteristic exhibited by other fractal zone plates and diffracting apertures. This feature may be exploited in practice to produce complex 3D light distribution near the focus which finds application in 3D imaging, lithography, optical superresolution, optical micromanipulation, and optical tomography, to mention a few. The inverse problem where a phase filter needs to be synthesized in accordance with a prespecified transverse intensity profile may be solved using the self-similar property of the energy efficient radial Walsh filters.
The authors declare that there is no conflict of interests regarding the publication of this paper.