Self-similarity in Transverse Intensity Distributions on the Farfield Plane of Self-similar Walsh Filters
The set of radial and annular Walsh filters can be classified into distinct self-similar groups and subgroups, where members of each subgroup possess self-similar structures or phase sequences. It has been observed that, the transverse intensity distribut
- PDF / 945,670 Bytes
- 12 Pages / 439.37 x 666.142 pts Page_size
- 12 Downloads / 150 Views
Self-similarity in Transverse Intensity Distributions on the Farfield Plane of Self-similar Walsh Filters
Abstract The set of radial and annular Walsh filters can be classified into distinct self-similar groups and subgroups, where members of each subgroup possess self-similar structures or phase sequences. It has been observed that, the transverse intensity distributions in the farfield diffraction pattern of these self-similar radial and annular Walsh filters are also self-similar. In this chapter we report results of our investigations on the self-similarity in the intensity distributions on a transverse plane in the farfield diffraction patterns of the self-similar radial and annular Walsh filters. Keywords Self-similar farfield patterns
4.1
Self-similar Walsh filters
Radial Walsh Filters
The radial Walsh filter is placed on the exit pupil of an axially symmetric imaging system (Fig. 4.1) and the normalized intensity distribution on a transverse plane in the farfield of radial Walsh filters is determined by Eq. (3.12). A point on the transverse image plane in the farfield is represented by the Cartesian co-ordinate ðn; gÞ with O0 as the origin of the ðn; gÞ axes as in Fig. 4.1. The normalized transverse intensity IN ðpÞ on the farfield diffraction pattern is computed by Eq. (3.12) where p is the reduced diffraction variable or the reduced transverse distance as in Eq. (3.2). Normalized transverse intensity distribution IN ðpÞ are evaluated for all orders of radial Walsh filters from 0 to 15. Self-similarity in the transverse intensity distributions in the farfield diffraction patterns is observed for each of the self-similar groups of radial Walsh filters [1]. Illustrative results on the same are presented in the following Figs. 4.3, 4.4 and 4.5. Emphasis is given on highlighting the distinctive features observed in the transverse intensity distribution patterns of each self-similar group or subgroup of the radial Walsh filters. For a Walsh filter of zero order on the exit pupil, the transverse intensity distribution on the farfield plane is the Airy pattern where the normalized central
© The Author(s) 2018 L. Hazra and P. Mukherjee, Self-similarity in Walsh Functions and in the Farfield Diffraction Patterns of Radial Walsh Filters, SpringerBriefs in Applied Sciences and Technology, DOI 10.1007/978-981-10-2809-0_4
47
4 Self-similarity in Transverse Intensity Distributions …
48 Fig. 4.1 Farfield diffraction pattern on the image plane at O0 . Radial Walsh filter uk ðrÞ on the exit pupil
y Exit pupil
x
η Image Plane
A
n
r
P
E Z
ρmax
χ
α
ξ
O
ζ
1.0 Normalized Intensity
Fig. 4.2 The Airy pattern
0.8 0.6 0.4 0.2 0.0 -10
-8
-6
-4
-2
0
2
4
6
8
10
Reduced Transverse Distance
intensity at the origin has magnitude one as in Fig. 4.2. In presence of a Walsh filter of orders 1, 2, 3… on the exit pupil, the central intensity is zero. The Airy pattern is also shown in Fig. 4.3, superimposed on the intensity distribution pattern of Walsh order 1. There is no central lobe or central peak in the transverse intens
Data Loading...
