Far-field Diffraction Properties of Annular Walsh Filters

AnnularWalsh filters are derived from the rotationally symmetric annularWalsh functionswhich form a complete set of orthogonal functions that take on values either +1 or −1 over the domain specified by the inner and outer radii of the annulus. The value of any annular Walsh function is taken as zero from the centre of the circular aperture to the inner radius of the annulus.The three values 0, +1, and −1 in an annular Walsh function can be realized in a corresponding annular Walsh filter by using transmission values of zero amplitude (i.e., an obscuration), unity amplitude and zero phase, and unity amplitude and π phase, respectively. Not only the order of the Walsh filter but also the size of the inner radius of the annulus provides an additional degree of freedom in tailoring of point spread function by using these filters for pupil plane filtering in imaging systems. In this report, we present the far-field amplitude characteristics of some of these filters to underscore their potential for effective use in several demanding applications like high-resolution microscopy, optical data storage, microlithography, optical encryption, and optical micromanipulation.


Introduction
Annular apertures and different types of ring-shaped apertures continue to be investigated for catering to the growing exigencies in diverse applications, for example, high resolution microscopy, optical data storage, microlithography, optical encryption, and optical micromanipulation [1][2][3][4][5]. Not only for obvious energy considerations but also for their higher inherent potential in delivering complex far-field amplitude distributions, annular phase filters are being investigated in different contexts [6][7][8][9].
A systematic study on the use of phase filters on annular pupils can be conveniently carried out with the help of annular Walsh filters derived from the annular Walsh functions. Walsh functions form a closed set of normal orthogonal functions over a given finite interval and take on values +1 or −1, except at a finite number of points of discontinuity, where they take the value zero [10,11]. The order of a Walsh function is directly related to the number of its zero crossings or phase transitions within the specified domain, and they constitute a closed set of normal orthogonal functions over the specified interval. They have the interesting property that an approximation of a continuous function over a finite interval by a finite set of Walsh functions leads to a piecewise constant approximation to the function. Walsh filters of various orders may be obtained from corresponding Walsh functions, by realizing transmission values of +1 and −1 by 0 and phase, respectively. Incidentally it may be noted that binary phase filters are being explored for many interesting light distributions [12].
Walsh functions have been used in the field of signal coding and transmission and in allied problems of information processing [13]. Two-dimensional Walsh functions in the usual rectangular coordinates have been used in digital image processing applications [14]. For treatment of problems of optical imaging, Walsh functions in polar co-ordinates have been utilised [15]. For systems with rotational symmetry about the axis, radial Walsh functions [16] have been developed as a special case of Walsh functions in polar co-ordinates and they were proved useful in the treatment of apodization problems [17,18]. It has also been observed that not only the transverse amplitude distribution on the far-field plane but also the axial distribution of amplitude/intensity in the farfield is significantly modified in presence of radial Walsh filters on unobscured apertures [19].
2 Advances in Optical Technologies Annular Walsh functions are a generalization of radial Walsh functions. For a specific central obscuration ratio, they are a complete set of orthogonal functions over the annulus. Annular Walsh filters may be considered to have ternary transmission values. The three values of transmission are zero amplitude over the central obscuration, unity amplitude and zero phase for value +1, and unity amplitude and phase for value −1 in corresponding annular Walsh functions.
In this paper, we present some results of our investigations on the far-field amplitude characteristics of annular Walsh filters of orders 0, 1, 2, and 3 with different central obscuration ratios. After a brief description of annular Walsh functions in the next section, Section 3 presents the mathematical expression for the amplitude distribution in the far-field of an exit pupil with annular Walsh filters on it. Some interesting numerical results and our observations on the same are put forward in the last two sections.

Annular Walsh Functions
To define annular Walsh function Ψ ( ) of index ≥ 0 and argument over an annular region with and 1 as inner and outer radii, respectively, it is necessary to express the integer in the form where are the bits, 0 or 1 of the binary numeral for , and (2̃) is the power of 2 that just exceeds . For all in ( , 1), Ψ ( ) is defined as where The orthogonality condition implies that where is the Kronecker delta defined as = .
(5) Figure 1 shows the first four annular Walsh functions for central obscuration = 0.3 in two dimensions.
In general, for the first , where = 2 and is a positive integer, and annular Walsh functions Ψ ( ), = 0, 1, . . . , ( − 1), the zero crossings are located at The inner and outer radii of the annulus is 0 = and = 1. Note that the set of ( − 1) zero crossing locations, , = 1, 2, . . . , ( − 1), consists of all zero crossing locations required for specifying members of this particular set of Walsh functions. An individual member of this set of Walsh functions will have the same number of zero crossings as its order.
For computational purposes it is often convenient to express an annular Walsh function Ψ ( ) as For a particular value of , the integer̃is taken such that just exceeds . ( ) are zero-one functions, also known as Walsh block functions, defined as For all annular Walsh functions of order < , the same values of , = 0, are to be used. They are unique for the particular value of , and their values are given in (7). ℎ are the elements of a ( × ) Hadamard matrix whose elements are +1 or −1 [13]. Table 1 gives the Hadamard matrix for = 2 2 . The Hadamard matrices for other values of can be obtained from the defining relation (2) of annular Walsh functions. Figure 3 shows the image space of an axially symmetric imaging system. It is well known that the complex amplitude distribution on the image plane corresponding to an axial object point is given by the far-field diffraction pattern of the pupil function over the exit pupil. The complex amplitude ( ) at a point Q in the far-field due to a point object on the axis is, apart from the multiplicative constant, given by the Hankel transform of order zero of the pupil function [20,21]:

Far-Field Amplitude Distribution
where ( ) is the circularly symmetric pupil function. The variable is the fractional coordinate for a point on the pupil sphere of center and radius , where is the axial position of the exit pupil. is equal to the ratio ( /ℎ ), where is the perpendicular distance of the point from the axis and ℎ is the maximum value of , so that 0 ≤ ≤ 1. is the reduced diffraction variable given by where is the semiangular aperture of the imaging system, is the refractive index of the image space, 2 / is the propagation constant, and (= Q ) is the actual geometrical distance of point Q on the image plane from the center of the diffraction pattern.
For a specific central obscuration ratio , annular Walsh filters of various orders can be obtained from the corresponding annular Walsh functions by realizing transmission values +1 and −1 by zero and phase filters, respectively. In presence of an annular Walsh filter on a uniform pupil, the pupil function ( ) is given by It may be noted that ( ) is binary (value either 0 or +1) only in the case of zero order annular Walsh function Ψ 0 ( ); for all other orders, ( ) is ternary with value either 0, +1, or −1.
The far-field amplitude pattern ( ) of an annular Walsh filter Ψ ( ) is given by where For the sake of underscoring amplitude variations, the scales along the ordinate and abscissa of different curves could not be maintained the same; they were tailored to bring forth the specific features. It is seen in Figures 4(a), 5(a), 6(a), and 7(a) that for the zero order annular Walsh filters, there is a central maximum of amplitude in the diffraction pattern, and the magnitude of the maximum is obviously dependent on the central obscuration . For the sake of comparison, amplitude distribution for an unobscured uniform aperture is shown alongside the same for zero order annular Walsh filters. It is well known that with higher values of obscuration, the central lobe narrows down significantly [20].  Figures 4-7 show that for Walsh filters of all orders other than zero, the central amplitude is zero. This is a consequence of orthogonality property of the Walsh functions. The dark center is surrounded by rings of oscillating amplitudes. For lower orders, this oscillation gradually decays after the first few rings, but for higher orders, the oscillations continue for many more rings. As expected from energy considerations, the amplitude of oscillation is significantly less in case of higher order filters compared to the lower order ones in each case of obscuration. From lower to higher obscuration, the ring with peak oscillatory amplitude gradually shifts away from the centre in case of higher order annular Walsh filters. Use of ternary values of transmission 0, +1, and −1 in annular Walsh filters has opened up new possibilities for generation of interesting far-field patterns. Amplitude patterns shown in Figures 6(b)-6(d) and 7(b)-7(d) corresponding to annular Walsh filters of orders 1, 2, and 3 on a pupil with large obscuration have distinct characteristics that remain to be explored further for many practical applications, for example, optical encryption. Incidentally, it is also observed that for orders higher than 3, the typical nature of amplitude distribution is more pronounced, with the frequency of oscillations in amplitude becoming increasingly higher with concomitant decrease in peak amplitude.

Concluding Remarks
Optimal combinations of annular Walsh filters can help in realizing desired amplitude distributions on the far-field plane for applications using coherent illumination. An important application is to obtain "superresolution" effect by narrowing down of the central lobe and suppression of the side ripples. Solution of the related optimization problem can make effective use of orthogonality of annular Walsh functions. Incidentally, inverse problems on pupil synthesis for prespecified intensity characteristics in the far-field plane were tackled earlier for the case of incoherent illumination by the use of radial Walsh filters for unobscured apertures [18]. It is obvious that the availability of an additional degree of freedom, namely, the obscuration ratio , is likely to expand the scope of this inverse problem.
Ready availability of high efficiency spatial light modulators has facilitated the practical realization of phase filters. [22][23][24]. Nevertheless, it is obvious that, in general, practical implementation of filters with binary or ternary phase values is relatively easier than filters with continuously varying phase. Use of Walsh function-based analysis and synthesis of pupil plane filters has become useful, for it can circumvent the tricky problem of synthesizing continuously varying phase by providing alternative analytical treatments that deal with a finite number of discrete phase levels.
Finally, it remains to investigate the axial resolution characteristics of annular Walsh filters and also their usefulness in the synthesis of three-dimensional light fields [25].