Higher Order Bessel Functions Research Articles

Complete description of effects on reconstruction of dynamic objects from time-averaged digital holography to high-speed digital holography

Time-averaged holography (TAH) is a specialized technique for studying objects subjected to sinusoidal vibration, characterized by presenting a Bessel J0 envelope in the object’s reconstruction, a condition that occurs when the vibration period is much shorter than the hologram exposure time. In this work, we present an analytical expression that describes the reconstruction effects when both the exposure time and the period can take arbitrary values, allowing the application of the TAH technique for exposure times as fractions of the period. We observe that the presented function contains higher-order Bessel functions. Additionally, we found that the envelope not only depends on the relationship between the exposure time and the vibration period but is also directly related to the vibration amplitude. The expression we introduce applies to conditions where exposure times are very short, possible with pulsed lasers, called high-speed holography (HSH), where the object reconstructs as if it were static. This mathematical expression serves as a bridge that continuously connects the techniques of HSH and TAH, enabling a smooth transition between both techniques.

Focused Gaussian beam in the paraxial approximation.

A focused Gaussian beam represents a case of highly practical importance in many areas of optics and photonics. We derive analytical expressions for a focused Gaussian beam in the paraxial approximation, considering an arbitrary lens filling factor. We discuss the role of higher-order Bessel functions of the first kind in defining the electric field in the focal region.

Topographic synthesis of arbitrary surfaces with vortex Jinc functions.

When a circular aperture is uniformly illuminated, it is possible to observe in the far field an image of a bright circle surrounded by faint rings known as the Airy pattern or Airy disk. This pattern is described by the first-order Bessel function of the first type divided by its argument expressed in circular coordinates. We introduce the higher-order Bessel functions with a vortex azimuthal factor to propose a family of functions to generalize the function defining the Airy pattern. These functions, which we call vortex Jinc functions, happen to form an orthogonal set. We use this property to investigate their usefulness in fitting various surfaces in a circular domain, with applications in precision optical manufacturing, wavefront optics, and visual optics, among others. We compare them with other well-known sets of orthogonal functions, and our findings show that they are suitable for these tasks and can pose an advantage when dealing with surfaces that concentrate a considerable amount of their information near the center of a circular domain, making them suitable applications in visual optics or analysis of aberrations of optical systems, for instance, to analyze the point spread function.

Deterministic Synthesis of Wide Scanning Sparse Concentric Ring Antenna Arrays

This article presents the synthesis of isophoric (equally fed), wideband wide-scan sparse concentric ring antenna arrays (CRAs) with low sidelobe level (SLL). The technique utilizes the characteristics of Bessel functions (BFs) of the first kind for representing the array factor (AF) and propose deterministic design guidelines for SLL reduction. It is shown that higher order BFs are necessary to predict nonsymmetry in the azimuth pattern of CRA. A design process for deriving ring radii and number of elements in the rings is presented to achieve highly sparse CRAs. Numerical test case studies are presented demonstrating the design procedure and its validation with the prior literature. The effect of the mutual coupling on the array performance over wideband and wide-scan volume is also presented. The results show sparser CRAs with comparable values of peak SLL (PSLL) and −3 dB beamwidth. This work may complement the existing stochastic or other efficient optimization techniques in reducing their computational time, especially for large arrays (more than hundreds of elements). This work focuses on phased array antennas for radar applications.

Refractive-diffractive dispersion compensation for optical vortex beams with ultrashort pulse durations.

Wave fields, which are described mathematically by higher order Bessel functions, carry an orbital angular momentum and thus represent particular types of optical vortex beams with helical wavefronts. For the generation of such vortex beams, one may use, for instance, diffractive spiral axicons. Diffraction, however, leads invariably to strong dispersion, which is detrimental for ultrashort pulses since it leads to severe pulse broadening. This pulse broadening can be minimized or reduced completely (at least, in a specific plane of propagation) if the pulses propagate additionally through a medium with normal refractive dispersion. The refractive-diffractive generation of ultrashort vortex pulses was demonstrated earlier for a pulse duration of approximately 8fs [Opt. Lett.37, 3804 (2012)10.1364/OL.37.003804OPLEDP0146-9592]. Here, we present an analytical description of the generation and propagation of these vortex beams and of the refractive-diffractive compensation of the dispersion.

Polarization transfer in relativistic magnetized plasmas

The polarization transfer coefficients of a relativistic magnetized plasma are derived. These results apply to any momentum distribution function of the particles, isotropic or anisotropic. Particles interact with the radiation either in a non resonant mode when the frequency of the radiation exceeds their characteristic synchrotron emission frequency, or quasi resonantly otherwise. These two classes of particles contribute differently to the polarization transfer coefficients. For a given frequency, this dichotomy corresponds to a regime change in the dependence of the transfer coefficients on the parameters of the particle s population. The derivation of the transfer coefficients involves an exact expression of the conductivity tensor of the relativistic magnetized plasma that has not been used hitherto in this context. Suitable expansions valid at frequencies larger than the cyclotron frequency allow us to analytically perform the summation over all resonances at high harmonics of the relativistic gyrofrequency. The transfer coefficients are represented in the form of two variable integrals that can be conveniently computed for any set of parameters by using Olver s expansion of high order Bessel functions. We particularize our results to a number of distribution functions, isotropic, thermal or powerlaw, with different multipolar anisotropies of low order, or strongly beamed. For isotropic distributions, the Faraday coefficients are expressed in the form of a one variable quadrature over energy, for which we provide the kernels in the high-frequency limit and in the asymptotic low-frequency limit. A similar reduction to a one-variable quadrature over energy is derived at high frequency for a large class of anisotropic distribution functions that may form a basis on which any smoothly anisotropic distribution could be expanded.

Hybrid Field/Transmission-Line Model for the Study of Coaxial Corrugated Waveguides

A transmission-line model is reformulated and combined with field theory to study the propagation characteristics in coaxial waveguides with wedge-shaped corrugations, either on the outer wall or the inner conductor. Numerical results show that this equivalent-circuit approach is in agreement with conventional full-wave methods presented in the literature. Additionally, this formulation overcomes numerical issues in the calculation of higher order Bessel functions, which usually conscript sophisticated expansion techniques.

MATLAB GUI for computing Bessel functions using continued fractions algorithm

Higher order Bessel functions are prevalent in physics and engineering and there exist different methods to evaluate them quickly and efficiently. Two of these methods are Miller's algorithm and the continued fractions algorithm. Miller's algorithm uses arbitrary starting values and normalization constants to evaluate Bessel functions. The continued fractions algorithm directly computes each value, keeping the error as small as possible. Both methods respect the stability of the Bessel function recurrence relations. Here we outline both methods and explain why the continued fractions algorithm is more efficient. The goal of this paper is both (1) to introduce the continued fractions algorithm to physics and engineering students and (2) to present a MATLAB GUI (Graphic User Interface) where this method has been used for computing the Semi-integer Bessel Functions and their zeros.

Robust Bessel-function-based method for determination of the(n,m)indices of single-walled carbon nanotubes by electron diffraction

We report a calibration-free method for the determination of chiral indices $(n,m)$ of single-walled carbon nanotubes from their electron diffraction patterns based on Bessel function analysis of the diffracted layer lines. An approach has been developed for confident identification of the orders of the Bessel functions from the intensity modulations of the diffraction layer lines, to which $(n,m)$ are correlated. In particular, we critically evaluate the effect of nanotube inclination on the validity of the method and show that the layer lines governed by high-order Bessel functions tolerate higher tilt angles than those of low-order Bessel functions and thus are favored for $(n,m)$ evaluation. The method is of particular significance in that it considerably enhances the precision of chiral indexing and makes possible the analysis of high-order Bessel functions, especially when EDPs are of relatively low pixel resolution. The technique can be extended to structural analysis of double-walled carbon nanotubes.

Polar Fourier transforms of radially sampled NMR data

Radial sampling of the NMR time domain has recently been introduced to speed up data collection significantly. Here, we show that radially sampled data can be processed directly using Fourier transforms in polar coordinates. We present a comprehensive theoretical analysis of the discrete polar Fourier transform, and derive the consequences of its application to radially sampled data using linear response theory. With adequate sampling, the resulting spectrum using a polar Fourier transform is indistinguishable from conventionally processed spectra with Cartesian sampling. In the case of undersampling in azimuth—the condition that provides significant savings in measurement time—the correct spectrum is still produced, but with limited distortion of the baseline away from the peaks, taking the form of a summation of high-order Bessel functions. Finally, we describe an intrinsic connection between the polar Fourier transform and the filtered backprojection method that has recently been introduced to process projection-reconstruction NOESY data. Direct polar Fourier transformation holds great potential for producing quantitatively accurate spectra from radially sampled NMR data.

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The spatial autocorrelation (SPAC) method requires a special circular array where several observation points are equally spaced on the circumference. In order to look for the possibility of developping a new method of array observation with fewer restriction of arrangement than the SPAC method, we proposed a formula of the complex coherence function (CCF) of the Rayleigh wave measured on a couple of observation points located at any place. This formula was derived on the basis of an analytical solution of Lamb's problem, aiming to study the relation between wave source and observation point. The formula was given as simple discrete representation consisting of the Bessel function of the first kind of zero order J0 (kr) (k: wavenumber, r: radius of array) and an infinite series with higher-order Bessel functions. In the SPAC method, by regarding the SPAC coefficient from directional average of CCFs (real part) as J0 (kr), phase velocities of Rayleigh waves (wave number k) are calculated.We first studied the relationship between the values of CCF and wave sources located far from a couple of observation points, and found that the values of CCF strongly varies depending on the direction with increase in kr, and also found that such directional properties were mainly caused by the variation of the infinite series in the formula of CCF. Furthermore, we applied the formula to the SPAC method for revealing the mechanism of the directional average and the reason why the SPAC method requires the special circular array with sensors equally spaced on a circle. The results are summarized as follows: 1) The values of the infinite series gets lower enough to be negligible after the directional average of CCFs, so that the SPAC coefficient can be approximated to J0 (kr). 2) From the inverse analysis on the condition that the values of the infinite series is equal to zero, it was found that the condition was satisfied not only in the usual SPAC arrays but also in some extra circular arrays consisting of observation points not equally spaced on the circumference.This result suggests the possibility of array design with fewer restriction of arrangement of observation points, using a new algorithm for obtaining J0 (kr) without the operation of directional average used in the SPAC method.

Evaluation of Fresnel integrals based on the continued fractions method

We present a simple algorithm for evaluating Fresnel integrals based on the continued fractions method: we use the relation between these integrals and first-kind Bessel functions of fractional order, and we apply a fast code to calculate them based on the continued fractions method. This latter code is especially useful for evaluating high order Bessel functions because it does not require recalculations using normalization relations. Comments on the same procedure but using Miller’s algorithm to evaluate the required Bessel functions are presented and a comparison with a standard code for evaluating Fresnel integrals (Numerical Recipes program FRENEL) is provided.

An efficient, preconditioned, high-order solver for scattering by two-dimensional inhomogeneous media

We consider the problem of evaluating the scattering of TE polarized electromagnetic waves by two-dimensional penetrable inhomogeneities: building upon previous work [IEEE Trans. Antennas Propag. 48 (2000) 1862] we present a practical and general fast integral equation algorithm for this problem. The contributions introduced in this text include: (1) a preconditioner that significantly reduces the number of iterations required by the algorithm in the treatment of electrically large scatterers, (2) a new radial integration scheme based on Chebyshev polynomial approximation, which gives rise to increased accuracy, efficiency and stability, and (3) an efficient and stable method for the evaluation of scaled high-order Bessel functions, which extends the capabilities of the method to arbitrarily high frequencies. These enhancements give rise to an algorithm that is much more accurate and efficient than its previous counterpart, and that allows for treatment of much larger problems than permitted by the previous approach. In one test case, for example, the present algorithm results in far-field errors of 8.9×10 −13 in a 2.12 s calculation (on a 1.7 GHz PC) whereas the original algorithm gave rise to far-field errors of 1.1×10 −8 in 88.91 s on a 400 MHz PC. In another example, the present algorithm evaluates accurately the scattering by a cylinder of acoustical size κR=256, which is of the order of 20 times larger (400 times larger in square wavelengths) than the largest scatterers that could be treated by the previous approach. Yielding, at worst, third-order far field accuracy (or substantially better, for smooth scatterers) in fast computing times ( O(N logN) operations for an N point mesh) even for discontinuous and complex refractive index distributions (possibly containing severe geometric singularities such as corners and cusps), the proposed approach is the highest-order O(N logN) solver in existence for the problem under consideration.

Electro-optic holography fringe control using diode laser current modulation

Electro-optic holography (EOH) is a whole field, laser-based, vibration measurement technique. Highly dense, unresolvable fringe patterns often limit the EOH measurement range. This paper presents a technique for increasing the measurement range of an EOH system using diode laser current modulation. This technique is known as frequency translated electro-optic holography (FTEOH). Using diode laser current modulation, the fringe patterns are based on higher-order Bessel functions and the sensitivity of the EOH system can either be increased or decreased. The amount of sensitivity reduction is limited only by the frequency limit of the signal generator that modulates the current supplied to the diode laser. EOH and FTEOH experimental results are presented and shown to closely compare.

Radiation fields from whispering-gallery modes of oxide-confined vertical-cavity surface-emitting lasers

Near- and far-field radiation patterns are analyzed for whispering-gallery-mode lasing in native-oxide-confined vertical-cavity surface-emitting lasers. Calculations from classical antenna theory based on the higher-order Bessel functions are compared with experiment and shown to be a useful predictor of the measured radiation fields. The most interesting features of the lasing modes are the multiple and uniformly distributed intensity spots in the near and far fields.

Evaluation of some algorithms and programs for the computation of integer-order Bessel functions of the first and second kind with complex arguments

The efficiency and accuracy of a few available algorithms for the computation of integer-order Bessel functions are considered. First, the computation of integer-order Bessel functions of the first kind, using the fast Fourier transform (FFT) algorithm as opposed to recurrence techniques, is investigated. It is shown that recurrence techniques are superior to the FFT technique, both in accuracy and speed. An algorithm suggested in the literature and used in commercially available software, specifically MATLAB 3.5 and MATHEMATICA 1.2, for computing integer-order Bessel functions of the second kind is revealed to be erroneous by comparing these routines with an algorithm developed by the author. It is shown that catastrophic errors result from using the erroneous algorithm to compute high-order Bessel functions with nonreal arguments.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Parseval's integral and the Jacobi expansions in series of Bessel fuinctions

AbstractThe sumsare shown to approximate J0 (z); the error terms are series in higher order Bessel functions, leading with J2M (z). Similar sums approximate J1(z). These sums may be looked on as extensions of the Jacobi expansions for cos z and sin z in series of Bessel functions. They become numerically useful for M &gt; |z|.

Slow-wave structures for M-type devices

M-type devices such as magnetron oscillators, magnetron amplifiers and backward-wave oscillators always contain a sole plate to provide the static electric field which, together with a transverse static magnetic field, provides electron focusing. The sole plate unavoidably becomes part of the RF structure. It is the purpose of this paper to show the effect of the sole plate on the RF characteristics of the more common slow-wave structures. The results of the study indicate that no slow-wave structure containing a sole plate can be truly a backward-wave structure. The fundamental space harmonic must contain a region wherein the phase and group velocities are of the same sign. These results are predicted by a field analysis of the uniform-vane, strapped magnetron anode system. Such an analysis is made possible by two simplifying assumptions. The first assumption is that the interaction space between anode and sole can be replaced by an equivalent linear structure. In this way, the derived expressions for admittance become infinite series of exponential functions rather than infinite series of untabulated higher order Bessel functions. The second assumption is that the fields peculiar to the straps can be described by TEM waves. This assumption allows one to formulate a strap admittance which is in parallel with the admittance of the interaction space and the side cavity. The resultant dispersion equation can be solved graphically for the frequency-phase shift characteristic of the composite system. This analysis predicts that the straps may be either capacitive or inductive depending upon mode number, frequency and geometry. It shows that the frequency-phase shift characteristic is not simply derivable from the unstrapped anode system by the addition of a constant phase shift. It shows further that the fundamental space harmonic must contain a region wherein the phase and group velocities are of the same sign, i.e., the anode system is not, and cannot be, a simple backward-wave structure. These results are verified in detail by a careful and complete experimental study on a standard strapped magnetron anode structure. They are further verified by a second experimental study of the very common interdigital line structure.

Coherent Electromagnetic Effects in High Current Particle Accelerators: II. Electromagnetic Fields and Resistive Losses

Coherent electromagnetic fields arising from an azimuthally modulated beam are considered. The beam is completely enclosed in a toroidal vacuum tank of rectangular cross section and highly conducting walls. Expressions are given for the image currents arising from low harmonics of the beam circulation frequency. These expressions are then used to evaluate resistive losses in the walls of the chamber. Expressions are given for fields arising from harmonics of the revolution frequency high enough that the beam may be in resonance with a characteristic mode of the vacuum chamber. The results are generalized to provide a description of the electric field in the neighborhood of a resonance. Numerical examples of resistive losses are given, indicating that these effects will not be serious for circulating currents of the order of 1 amp. Some properties of high-order Bessel functions, required for a description of the resonant chamber modes and the energy lost in their excitation, are developed in an appendix.