Circular Gaussian Research Articles

A novel method of generating qausi-non-diffracting Mahtieu beam based on axicon

A novel method of generating zero order non-diffracting Mathieu beam with an axicon is proposed. To create quasi non-diffracting Mahtieu beam, an axicon is used to focus a plane wave modulated by elliptical Gaussian amplitude. Based on the formula of diffraction integral of a plane wave modulated by elliptical Gaussian amplitude propagating through the axicon, the intensity of quasi non-diffracting beam is simulated numerically. The maximum propagation distance of the quasi-non-diffracting Mathieu beam is calculated according to a geometrical optical model. To verify the results of the theory, an experimental setup is designed.Using a cylindrical lens and a collimating and expanding system, a circular Gaussian beam can be converted in to a plane wave modulated by elliptical Gaussian amplitude. Focusing the plane wave using an axicon, a qausi-non-diffracting Mathieu beam can be generated. The exoerimental results are consistent with theoretical calcuations and numerical simulations.

The TeV blazar Markarian 421 at the highest spatial resolution

We report the results obtained for the AGN Markarian 421 by model-fitting the data in the visibility plane, studing the proper motion of jet components, the light curve, and the spectral index of the jet features. We compare the radio data with optical light curves obtained at the Steward Observatory, considering also the optical polarization information. Mrk 421 has a bright nucleus and a one-sided jet extending towards the north-west for a few parsecs. The model-fits show that brightness distribution is well described using 6-7 circular Gaussian components, four of which are reliably identified at all epochs; all components are effectively stationary except for component D, at ~0.4 mas from the core, whose motion is however subluminal. Analysis of the light curve shows two different states, with the source being brighter and more variable in the first half of 2011 than in the second half. The highest flux density is reached in February. A comparison with the optical data reveals an increase of the V magnitude and of the fractional polarization simultaneous with the enhancement of the radio activity.

Detection Performance for Distributed Radar Sensor Network

The need for mobility and the rapid deployment motivates the research of distributed radar sensor network withNTtransmitters,NRreceivers and a fusion center, where the fusion center and the receivers are communicated via wireless signaling. With the assumption of orthogonal waveforms for sensing and circular complex Gaussian noise channel model for communication, the formulae of detection probability and the false alarm probability are derived given the sensing power and communication power. Numerical results show that with the increase of the total power, the ratio between the best sensing power and the best communication power will increase. Moreover, the shape of the region in which the target detection probability is larger than a predetermined value is not regular, which should be taken care in case of network planning.

Comparative Study of Speckle Noise Reduction Approaches for Interferometric Synthetic Aperture Radar Images

Speckle noise appearing in the interferometric SAR (InSAR) phase image degrades the quality of interferogram seriously and makes interferogram reflect the scattering characteristics of the target inaccurately, reducing the capability of extracting DEM information of target areas. Therefore, speckle noise reduction plays a major role in InSAR processing by using interferogram filtering. First, according to a terrain model with the assumed geometrical parameters in InSAR system, the paper simulated an interferometric SAR phase image with noise, which can be characterized by the multilook phase distribution based on the circular Gaussian assumption [1]. Second, the paper explores three interferogram filtering algorithm to remove speckle noise: Goldstain filter, Rotating Kernel Transformation and Lee filter; the proper implementation of three methods is given. Last, the paper discusses about the performance comparatively based on experimental results and gives broad conclusions and presents recommendations.

Optimality Claims for the FML Covariance Estimator with respect to Two Matrix Norms

In this correspondence we prove two interesting properties of the fast maximum likelihood (FML) covariance matrix estimator proposed in [1] under the assumption of zero-mean complex circular Gaussian training data sharing the same covariance matrix. The new properties represent optimality claims regardless of the statistical characterization of the data and, in particular, of the multivariate Gaussian assumption for the observables. The optimality is proved with respect to two cost functions involving either the Frobenius or the spectral norm of an Hermitian matrix.

Digital Gabor and off-axis particle holography by shaped beams: A numerical investigation with GLMT

In particle field holography, the laser beam is more or less astigmatic, and the particles are usually inhomogeneous. Holograms of homogeneous and multilayered spherical particles illuminated by shaped beam with Gabor and off-axis configurations in both near and far field are investigated on the basis of the generalized Lorenz–Mie theory (GLMT). Reconstruction of the simulated holograms are demonstrated using fractional Fourier transform. Effects of recording parameters, e.g. beam shape (circular or elliptical Gaussian beam) and particle location (before, at or after the beam waist) and inner refractive index gradient using multilayer model on the hologram pattern were evaluated. For the off-axis holography, special attentions are devoted to the Moiré effect reduction using a Gaussian beam as the reference wave and to the two series of concentric rings phenomenon using the Debye series. The code developed in this work enable to investigate the effects of the incident beam, reference wave, particle inner refractive index gradient, and CCD orientation on the formation and reconstructed image of the hologram separately.

K speckle: space–time correlation function of doubly scattered light in an imaging system

The scattering of coherent monochromatic light at an optically rough surface, such as a diffuser, produces a speckle field, which is usually described by reference to its statistical properties. For example, the real and imaginary parts of a fully developed speckle field can be modeled as a random circular Gaussian process. When such a speckle field is used to illuminate a second diffuser, the statistics of the resulting doubly scattered field are in general no longer Gaussian, but rather follow a K distribution. In this paper we determine the space-time correlation function of such a doubly scattered speckle field that has been imaged by a single lens system. A space-time correlation function is derived that contains four separate terms; similar to the Gaussian case it contains an average DC term and a fluctuating AC term. However, in addition there are two terms that are related to contributions from each of the diffusers independently. We examine how our space-time correlation function varies as the diffusers are rotated at different speeds and as the point spread function of the imaging system is changed. A series of numerical simulations are used to confirm different aspects of the theoretical analysis. We then finish with a discussion of our results and some potential applications, including controlling spatial coherence and speckle reduction.

Nonlinear imaging properties under the coeffect of two wirelike opaque scatterers

Nonlinear imaging in the propagation of a flat-topped circular Gaussian beam under the coeffect of two parallel wirelike opaque scatterers is investigated through computer simulation. The formation of hot images of each scatterer is proved. Moreover, the formations of two other kinds of intense images are found; one is called interference hot image and the other is called pseudo-second-order hot image. The former corresponds to one intense image fringe whose in-beam position is at the middle point between those of the two scatterers. This image fringe is in the plane a quarter of an object distance away from the exit surface of the Kerr medium slab, and its intensity is found comparable to that of the hot image. The latter corresponds to two intense image fringes in a plane near the second-order hot-image plane predicted for single phase-typed scatterer cases. The respective in-beam positions of these image fringes are close to the respective in-beam positions of the scatterers. Interestingly, the intensities of both kinds of images are found primarily determined by the copresence rather than the in-beam positions of the scatterers. Besides, though the hot-image intensity can be lower than the corresponding single opaque scatterer case, the maximum intensity inside the Kerr medium slab increases more quickly and thus is much higher at the exit surface. This is another threat to the safe running of the Kerr media for high-power laser systems.

Angular spreading measurements using MeV ion microscopes

The sharpness of MeV ion microscope images is governed by small-angle scattering and associated lateral spreading of the ion beam in the sample. We have investigated measurement of the half-angle of the angular spreading distribution by characterising the image blurring in direct-Scanning Transmission Ion Microscopy (direct-STIM). In these tests Mylar™ foils of 0.5–6μm were used to induce angular spreading. Images were taken of an electron microscope grid using 2MeV protons with, and without, the foils in the beam path. The blurring was measured by fitting the width of a circular Gaussian point spread function to the images with and without the foil in position. The results show the half-angle width of the spreading has a square root dependence on foil thickness that lies intermediate between SRIM predictions and the theoretical estimates (Bird and Williams fits to the Sigmund and Winterbon data and Amsel et al.).

Modeling center-surround configurations in population receptive fields using fMRI

Antagonistic center-surround configurations are a central organizational principle of our visual system. In visual cortex, stimulation outside the classical receptive field can decrease neural activity and also decrease functional Magnetic Resonance Imaging (fMRI) signal amplitudes. Decreased fMRI amplitudes below baseline-0% contrast-are often referred to as "negative" responses. Using neural model-based fMRI data analyses, we can estimate the region of visual space to which each cortical location responds, i.e., the population receptive field (pRF). Current models of the pRF do not account for a center-surround organization or negative fMRI responses. Here, we extend the pRF model by adding surround suppression. Where the conventional model uses a circular symmetric Gaussian function to describe the pRF, the new model uses a circular symmetric difference-of-Gaussians (DoG) function. The DoG model allows the pRF analysis to capture fMRI signals below baseline and surround suppression. Comparing the fits of the models, an increased variance explained is found for the DoG model. This improvement was predominantly present in V1/2/3 and decreased in later visual areas. The improvement of the fits was particularly striking in the parts of the fMRI signal below baseline. Estimates for the surround size of the pRF show an increase with eccentricity and over visual areas V1/2/3. For the suppression index, which is based on the ratio between the volumes of both Gaussians, we show a decrease over visual areas V1 and V2. Using non-invasive fMRI techniques, this method gives the possibility to examine assumptions about center-surround receptive fields in human subjects.

The Performance Analysis Based on SAR Sample Covariance Matrix

Multi-channel systems appear in several fields of application in science. In the Synthetic Aperture Radar (SAR) context, multi-channel systems may refer to different domains, as multi-polarization, multi-interferometric or multi-temporal data, or even a combination of them. Due to the inherent speckle phenomenon present in SAR images, the statistical description of the data is almost mandatory for its utilization. The complex images acquired over natural media present in general zero-mean circular Gaussian characteristics. In this case, second order statistics as the multi-channel covariance matrix fully describe the data. For practical situations however, the covariance matrix has to be estimated using a limited number of samples, and this sample covariance matrix follow the complex Wishart distribution. In this context, the eigendecomposition of the multi-channel covariance matrix has been shown in different areas of high relevance regarding the physical properties of the imaged scene. Specifically, the maximum eigenvalue of the covariance matrix has been frequently used in different applications as target or change detection, estimation of the dominant scattering mechanism in polarimetric data, moving target indication, etc. In this paper, the statistical behavior of the maximum eigenvalue derived from the eigendecomposition of the sample multi-channel covariance matrix in terms of multi-channel SAR images is simplified for SAR community. Validation is performed against simulated data and examples of estimation and detection problems using the analytical expressions are as well given.

Modulation depth of Michelson interferometer with Gaussian beam

Mirror misalignment or the tilt angle of the Michelson interferometer can be estimated from the modulation depth measured with collimated monochromatic light. The intensity of the light beam is usually assumed to be uniform, but, for example, with gas lasers it generally has a Gaussian distribution, which makes the modulation depth less sensitive to the tilt angle. With this assumption, the tilt angle may be underestimated by about 50%. We have derived a mathematical model for modulation depth with a circular aperture and Gaussian beam. The model reduces the error of the tilt angle estimate to below 1%. The results of the model have been verified experimentally.

Average spreading of rectangular elliptical Gaussian beam array propagating in non-Kolmogorov turbulence

The spreading properties of the rectangular elliptical Gaussian beam (EGB) array propagating in non-Kolmogorov turbulence are investigated by using the derived analytical expression for the root-mean-square (rms) beam width. The results indicate that the rms beam width of the EGB array is smaller than that of the circular Gaussian beam array for long-distance propagation. The optimum separation distances are proved to exist and have a strong dependence on the ratio of the waist width along the long axis to that along the short axis of the EGB. Further, the rms beam width in non-Kolmogorov turbulence is different from that in Kolmogorov turbulence.

On Testing the Extent of Noncircularity

In this correspondence, we provide a multiple hypothesis test to detect the number of latent noncircular signals in a complex Gaussian random vector. Our method sequentially tests the results of individual generalized likelihood ratio test (GLRT) statistics with known asymptotic distributions to form the multiple hypothesis detector. Specifically, we are able to set a threshold yielding a precise probability of error. This test can be used to statistically determine if a given complex observation is circular Gaussian, and if not, how many latent signals in the observation are noncircular. Simulations are used to quantify the performance of the detector as compared to a detector based on the minimum description length (MDL) criterion. The utility of the detector is shown by applying it to a beamforming application using independent component analysis (ICA).

Quality parameter for coherent transmissions with Gaussian-distributed nonlinear noise

By assuming the nonlinear noise as a signal-independent circular gaussian noise, a typical case in non-dispersion managed links with coherent multilevel modulation formats, we provide several analytical properties of a new quality parameter--playing the role of the signal to noise ratio (SNR) at the sampling gate in the coherent receiver--which carry over to the Q-factor versus power (or "bell") curves. We show that the maximum Q is reached at an optimal power, the nonlinear threshold, at which the amplified spontaneous emission (ASE) noise power is twice the nonlinear noise power, and the SNR penalty with respect to linear propagation is 10Log(3/2) ≃ 1.76 dB,, although the Q-penalty is somewhat larger and increases at lower Q-factors, as we verify for the polarization-division multiplexing quadrature phase shift keying (PDM-QPSK) format. As we vary the ASE power, the maxima of the SNR vs. power curves are shown to slide along a straight-line with slope ≃-2 dB/dB. A similar behavior is followed by the Q-factor maxima, although for PDM-QPSK the local slope is around -2.7 dB/dB for Q-values of practical interest.

Cramer-Rao Lower Bounds of DOA Estimates from Square QAM-Modulated Signals

In this paper, we derive for the first time analytical expressions for the inphase/quadrature (I/Q) non-data-aided (NDA) Cramér-Rao lower bounds (I/Q NDA CRLBs) of the direction of arrival (DOA) estimates from square quadrature amplitude (QAM)-modulated signals corrupted by additive white circular complex Gaussian noise (AWCCGN) with any antenna configuration. Yet the main contribution embodied by this paper consists in deriving for the first time analytical expressions for the NDA Fisher information matrix (FIM) and then for the stochastic CRLB of the NDA DOA estimates in the case of square QAM-modulated signals. It will be shown that in the presence of any unknown phase offset (i.e., non-coherent estimation), the ultimate achievable performance on the NDA DOA estimates holds almost the same irrespectively of the modulation order. However, the NDA CRLBs obtained in the absence of the phase offset (i.e., coherent estimation) vary, in the high SNR region, from one modulation order to another.

Elliptical Gaussian solitons in synthetic nonlocal nonliear media

This paper studies analytically and numerically the dynamics of two-dimensional elliptical Gaussian solitons in a “double-self-focusing" synthetic nonlocal media featuring elliptical and circular Gaussian response with different degrees of nonlocality. Based on the variational approach, it obtains the approximately analytical solution of such Gaussian elliptical solitons. It also computes the stability of the solitons by numerical simulations.

Propagation of a General Gaussian Beam by Paraxial Ray Tracing in an Arbitrary Optical System

The propagation of a partially coherent Gaussian beam in a homogeneous medium can be explained with various methods. One of them is Gaussian shell model beams [1–8]. It uses the Gaussian Schell-model source whose irradiance distribution and degree of coherence are both Gaussian. We will describe Gaussian beams as the ray distribution L as functions of linear and angular coordinates. Of course, since the Gaussian Shell model beam can be produced by a superposition of independent coherent Gaussian beams, our concept is similar to the Gaussian Shell model concept. However, the goal of the present paper is to describe the propagation of general Gaussian beams through an arbitrary optical system by using the geometrical paraxial ray tracing method. In other words, many rays propagate through a reference plane, these rays distribute with the Gaussian profile in linear and angular direction cosine coordinates, and every ray propagates according to the laws of geometrical optics. At first, we introduce the ray distribution density of a circular symmetric Gaussian beam and investigate how that Gaussian beam propagates through an arbitrary axially symmetric optical system or in a homogeneous medium under the paraxial approximation, and we determine the physical meanings of the parameters. Also, we introduce the complex parameter q and the Lagrange invariant L for this general Gaussian beam and discover that this Gaussian beam also follows the Kogelnik ABCD law, just as a classical coherent Gaussian beam does.

On the analysis of an elliptical Gaussian flux image and its equivalent circular Gaussian flux images

For a continuous elliptical Gaussian flux image over the infinite X– Y plane, the parameter relationships between the elliptical Gaussian flux image and the equivalent circular Gaussian flux images are clearly discussed in both mathematical and graphical ways in this paper with respect to the radial power distribution around the image centre (peak flux location in image plane). This paper presents six typical methods (SIGMA-2Mean, SIGMA-1Mean, SIGMA-0Mean in Class One, SIGMA-RPeak, SIGMA-RMean, SIGMA-SqrRMean in Class Two) to give the equivalent circular Gaussian flux images to the elliptical Gaussian flux image, tries to link these circular Gaussian fitting methods to the relevant solar engineering applications, and makes some assessment comments on the elliptical/circular Gaussian modelling in solar mirror optics. By comparing the approximation performance among these six typical fitting methods, it reveals the reason for the 90% intercept over-estimation phenomenon of Francisco J. Collado’s one-point circular Gaussian fitting practice relative to the experimental flux image. The detailed algorithm for automatically finding out the major/minor axes and the image centre of the digital elliptical flux image is also provided in this paper. SIGMA-2Mean and SIGMA-SqrRMean are equivalent for an elliptical flux image, but they are applied in different ways to figure out either the reasonable intercept factor of the experimental flux image on the physical target plane respecting the aperture region of interest, or the power spillage over the limited experimental target plane. At last, this paper introduces the interpolation reconstruction of an elliptical Gaussian flux image over a rectangular domain just based on the boundary pixel values, so it is quite useful for solar engineering, such as fast simulation of a flux image concentrated by a mirror, and also instant approximation of the flux density over the receiver aperture by the linear array of radiometers around the receiver aperture, when the central receiver system is in the normal working state.

Laser Processing by Using Diffractive Optical Laser Beam Shaping Technique

Laser beam shaping and homogenization techniques are substantial to optimize a large number of laser-material processing applications and laser-material interaction studies. Diffractive optical elements (DOE) play important role in provision of the process-adapted laser beam shaping. In this paper, we present an approach for laser beam shaping by using innovative DOE. The circular Gaussian beam was transformed to a square flat-top intensity profile in the focal plane of the objective lens by using novel DOE from TOPAG Lasertechnik GmbH. About 95% of the laser energy was coupled in the main beam with a nearly perfect rectangular shape. Experimental results have been achieved by applying the shaped beam for laser microfabrication. The picosecond and nanosecond lasers with moderate beam quality (M 2 =1.5) were used in the experiment. The shaped laser beam was applied for direct laser ablation of metal film on the glass substrate, drilling in a silicon wafer, scribing of thin-film solar cells and other technical materials. This technique allows creating a rectangular-shaped flat-top intensity profile in the focal plane that shows distinct advantages in laser material processing.