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Far-field Diffraction Properties of Annular Walsh Filters
Pubali Mukherjee,Lakshminarayan Hazra
Advances in Optical Technologies , 2013, DOI: 10.1155/2013/360450
Abstract: Annular Walsh filters are derived from the rotationally symmetric annular Walsh functions which 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 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. 1. 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 [1–5]. 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 [6–9]. 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
Spontaneous Resolution of Cataract and Medical Management of Corneal Opacity in an Indian Parrot—Psittacula krameri manillensis  [PDF]
Sarbani Hazra
Open Journal of Veterinary Medicine (OJVM) , 2013, DOI: 10.4236/ojvm.2013.32028
Abstract:

A 35-year-old Indian parrot (Psittacula krameri manillensis) was presented to the Department of Veterinary Surgery & Radiology, West Bengal University of Animal and Fishery Sciences with a history of blepharospasm and corneal lesion OS. Test with fluorescein dye was negative. The dense corneal opacity (macula) was identified as involving the posterior corneal layer. The further ophthalmic examination was done and hypermature phacolytic cataract was diagnosed. No other abnormality was detected. Medical management with topical nepafenac prednisolone and triple antibiotic was instituted. The corneal lesion subsided completely within one week followed by spontaneous resorption of the cataract. The treatment protocol was successfully eliminated the discomfort and intraocular lesions in the senile parrot.

Multigrid One-Shot Method for PDE-Constrained Optimization Problems  [PDF]
Subhendu Bikash Hazra
Applied Mathematics (AM) , 2012, DOI: 10.4236/am.2012.330216
Abstract: This paper presents a numerical method for PDE-constrained optimization problems. These problems arise in many fields of science and engineering including those dealing with real applications. The physical problem is modeled by partial differential equations (PDEs) and involve optimization of some quantity. The PDEs are in most cases nonlinear and solved using numerical methods. Since such numerical solutions are being used routinely, the recent trend has been to develop numerical methods and algorithms so that the optimization problems can be solved numerically as well using the same PDE-solver. We present here one such numerical method which is based on simultaneous pseudo-time stepping. The efficiency of the method is increased with the help of a multigrid strategy. Application example is included for an aerodynamic shape optimization problem.
Accuracy of Trace Formulas
Arul Lakshminarayan
Physics , 1996, DOI: 10.1007/BF02845660
Abstract: Using quantum maps we study the accuracy of semiclassical trace formulas. The role of chaos in improving the semiclassical accuracy, in some systems, is demonstrated quantitatively. However, our study of the standard map cautions that this may not be most general. While studying a sawtooth map we demonstrate the rather remarkable fact that at the level of the time one trace even in the presence of fixed points on singularities the trace formula may be exact, and in any case has no logarithmic divergences observed for the quantum bakers map. As a byproduct we introduce fantastic periodic curves akin to curlicues.
Relaxation Fluctuations in Quantum Chaos
Arul Lakshminarayan
Physics , 1998,
Abstract: Relaxation in the time correlation between operators is studied. Quantized chaotic systems are shown to have distinct relaxation fluctuations that are universal and can be usefully modelled by Random Matrix Theory. Various quantized maps are used to demonsrate these results.
The modular multiplication operator and the quantized bakers maps
Arul Lakshminarayan
Physics , 2007, DOI: 10.1103/PhysRevA.76.042330
Abstract: The modular multiplication operator, a central subroutine in Shor's factoring algorithm, is shown to be a coherent superposition of two quantum bakers maps when the multiplier is 2. The classical limit of the maps being completely chaotic, it is shown that there exist perturbations that push the modular multiplication operator into regimes of generic quantum chaos with spectral fluctuations that are those of random matrices. For the initial state of relevance to Shor's algorithm we study fidelity decay due to phase and bit-flip errors in a single qubit and show exponential decay with shoulders at multiples or half-multiples of the order. A simple model is used to gain some understanding of this behavior.
On the Quantum Baker's Map and its Unusual Traces
Arul Lakshminarayan
Physics , 1995, DOI: 10.1006/aphy.1995.1035
Abstract: The quantum baker's map is the quantization of a simple classically chaotic system, and has many generic features that have been studied over the last few years. While there exists a semiclassical theory of this map, a more rigorous study of the same revealed some unexpected features which indicated that correction terms of the order of log(h) had to be included in the periodic orbit sum. Such singular semiclassical behaviour was also found in the simplest traces of the quantum map. In this note we study the quantum mechanics of a baker's map which is obtained by reflecting the classical map about its edges, in an effort to understand and circumvent these anomalies. This leads to a real quantum map with traces that follow the usual Gutzwiller-Tabor like semiclassical formulae. We develop the relevant semiclassical periodic orbit sum for this map which is closely related to that of the usual baker's map, with the important difference that the propagators leading to this sum have no anomalous traces.
Relaxation fluctuations about an equilibrium in quantum chaos
Arul Lakshminarayan
Physics , 1997, DOI: 10.1103/PhysRevE.56.2540
Abstract: Classically chaotic systems relax to coarse grained states of equilibrium. Here we numerically study the quantization of such bounded relaxing systems, in particular the quasi-periodic fluctuations associated with the correlation between two density operators. We find that when the operators, or their Wigner-Weyl transforms, have obvious classical limits that can be interpreted as piecewise continuous functions on phase space, the variance of the fluctuations can distinguish classically chaotic and regular motions, thus providing a novel diagnostic devise of quantum chaos. We uncover several features of the relaxation fluctuations that are shared by disparate systems thus establishing restricted universality. If we consider the nonlinearity driving the chaos as pseudo-time, we find that the onset of classical chaos is indicated quantally as the relaxation of the relaxation fluctuations to a Gaussian distribution.
Barnett-Pegg formalism of angle operators, revivals, and flux lines
Arul Lakshminarayan
Physics , 2000, DOI: 10.1103/PhysRevA.62.042110
Abstract: We use the Barnett-Pegg formalism of angle operators to study a rotating particle with and without a flux line. Requiring a finite dimensional version of the Wigner function to be well defined we find a natural time quantization that leads to classical maps from which the arithmetical basis of quantum revivals is seen. The flux line, that fundamentally alters the quantum statistics, forces this time quantum to be increased by a factor of a winding number and determines the homotopy class of the path. The value of the flux is restricted to the rational numbers, a feature that persists in the infinite dimensional limit.
Shuffling cards, factoring numbers, and the quantum baker's map
Arul Lakshminarayan
Physics , 2005, DOI: 10.1088/0305-4470/38/37/L01
Abstract: It is pointed out that an exactly solvable permutation operator, viewed as the quantization of cyclic shifts, is useful in constructing a basis in which to study the quantum baker's map, a paradigm system of quantum chaos. In the basis of this operator the eigenfunctions of the quantum baker's map are compressed by factors of around five or more. We show explicitly its connection to an operator that is closely related to the usual quantum baker's map. This permutation operator has interesting connections to the art of shuffling cards as well as to the quantum factoring algorithm of Shor via the quantum order finding one. Hence we point out that this well-known quantum algorithm makes crucial use of a quantum chaotic operator, or at least one that is close to the quantization of the left-shift, a closeness that we also explore quantitatively.
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