Flattened Gaussian Research Articles

Manufacturing of Porous Glass by Femtosecond Laser Welding.

Based on femtosecond laser glass welding, four different porous structures of welding spots were formed by the manufacturing processes of spatiotemporal beam shaping and alternating high repetition rate transformation. Compared with an ordinary Gaussian beam, the welding spot fabricated by the flattened Gaussian beam had smoother welding edges with little debris, and the bottom of the welding spot pore was flat. Instead of a fixed high repetition rate, periodically alternating high repetition rates were adopted, which induced multiple refractive indices in the welding spot pore. The welding spot pores manufactured by spatiotemporal beam shaping and alternating high repetition rate transformation have a special structure and excellent properties, which correspond to superior functions of porous glass.

Engineering a better light sheet in an axicon-based system using a flattened Gaussian beam of low order.

Lasers are fundamental tools in research and development. The shape of an incident laser beam directly affects the results, when it propagates through complex structured meso-aspheric optical elements. In conic-based systems utilizing elements such as axicons, the impact of secondary lobes is mostly overlooked, although the intensity distributions at the central spot and the side-lobes directly affect the beam properties. We investigate the interaction of two axicons (160° and 170°) with incident beams approximated by Gaussian, high-order Flattened-Gaussian, and low-order Flattened-Gaussian functions. We demonstrate that replacing an incident Gaussian beam with a low-order Flattened-Gaussian beam reduces the secondary lobes and significantly improves the uniformity of the intensity profile. We practically applied this effect in engineering a conic-aspheric-based static light-sheet microscope producing markedly improved results.

Propagation propertiesof a novel generalized flattened Hermite-cosh-Gaussian light beam

In this paper, we have introduced a general formula of a new laser beam with a nearly Flattened-top profile. The propagation properties of this beam, called Generalized Flattened Hermite-cosh-Gaussian (GFHChG) beam are studied in detail. Based on the Huygens–Fresnel integral, an analytical expression of the intensity distribution for GFHChG passing through a paraxial ABCD optical system is developed. Several particular cases such as the propagation of the flattened Gaussian beam, the cosh-Gaussian beam, the higher and squared cosh-Gaussian beam and the Hermite-cosh-Gaussian beam are deduced from our result. Numerical simulations are also given in free space and through a circular thin lens in a free space to study the comportment of this beam during propagation.

Optimal distribution of the optical intensity within a laser beam for optical wireless communications

This study is focused on analysing the optimal distribution of optical intensity within a radiated laser beam at a transmitter plane which is propagated through free space as well as through a turbulent atmosphere. To analyse the propagation of an optical wave through atmospheric turbulence, the simulation, based on the split-step beam propagation method, was utilised. The aim of this study is to determine the optimal parameters for a flattened Gaussian beam at the transmitter plane. As a result, the scintillation index should be reduced. There are a number of theoretical studies aimed at propagating the flattened beam despite the lack of experimental work in this area. Therefore the techniques for generating the flattened Gaussian beam will also be mentioned.

The Waveform Model of Laser Altimeter System with Flattened Gaussian Laser

The current waveform model of a laser altimeter is based on a Gaussian laser beam of fundamental mode, while the flattened Gaussian beam has many advantages such as nearly constant energy distribution on the center of the cross-section. Following the theory of the flattened Gaussian beam and the waveform theory of the laser altimeter, some of the primary parameters of the received waveform were derived, and a laser altimetry waveform simulator and waveform processing software were programmed and improved under the circumstance of a flattened Gaussian beam. The result showed that the bias between theoretical and simulated waveforms was less than 3% for every order mode, the waveform width and range error would increase as target slope or order number rose. Under higher order mode, the shapes of the received waveforms were no longer Gaussian, and could be fitted more precisely as a generalized Gaussian function with power bigger than 2. The flattened beam got much better performance for a multi-surface target, especially when the small surface is far from the center of the laser footprint. This article provides the waveform theoretical basis for the use of a flattened Gaussian beam in a laser altimeter.

Rectangular symmetrical mesa beams and their comparison with flattened Gaussian and multi-Gaussian beams

A two-component analytical model describing 1D, paraxial, rectangular symmetrical mesa beams of arbitrary fractional orders is proposed. Such beams possess a plane wavefront and flat-topped intensity profiles at their waists. It is shown that they are reasonably well matched with 1D flattened Gaussian and generalized multi-Gaussian beams described by more complicated models in the form of finite superpositions of an on-axis fundamental Gaussian mode mixed, respectively, either with on-axis, 1D, elegant Hermite–Gaussian modes of higher orders or with off-axis fundamental Gaussian modes. A procedure of exact optimization fitting, the main propagation characteristics and the modal content of all the above beams are considered in detail. The proposed model is shown to have the most compact and convenient mathematical description of the beams and their characteristics.

Spontaneous and sequential transitions of a Gaussian beam into diffraction rings, single ring and circular array of filaments in a photopolymer

A Gaussian beam propagating in a photopolymer undergoes self-phase modulation to form diffraction rings and then transforms into a single ring, which in turn ruptures into a necklace of stable self-trapped multimode filaments. The transitions of the beam between the three distinct nonlinear forms only occur at intensities where the beam-induced refractive index profile in the medium slowly evolves from a Gaussian to a flattened Gaussian.

Intra–cavity flat–top beam generation

We present the analytical and numerical analyses of two new resonator systems for generating flat-top-like beams. Both approaches lead to closed form expressions for the required cavity optics, but differ substantially in the design technique, with the first based on reverse propagation of a flattened Gaussian beam, and the second a metamorphosis of a Gaussian into a flat-top beam. We show that both have good convergence properties, and result in the desired stable mode.

Effects of atmospheric turbulence on the scintillation and fade probability of flattened Gaussian beams

In an attempt to mitigate the effects of the atmosphere on the coherence of an optical (laser) beam, interest has recently been shown in changing the beam shape to determine if a different power distribution at the transmitter will reduce the effects of random fluctuations in the refractive index. Here, a model is developed for the field of a flattened Gaussian beam as it propagates through atmospheric turbulence, and the resulting effects on the scintillation of the beam and beam wander are determined. A comparison of these results is made with the like effects on a standard TEM00 Gaussian beam. The theoretical results are verified by comparison with a computer simulation model for the flattened Gaussian beam (FGB). Further, a determination of the probability of fade and of mean fade time under weak fluctuation conditions is determined using the widely accepted lognormal model. Although this model has been shown to be somewhat optimistic when compared to results obtained in field tests, it has value here in allowing us to compare the effects of atmospheric conditions on the fade statistics of the FGB with those of the lowest order Gaussian beam.

Analytical solutions for the electromagnetic fields of flattened and annular Gaussian laser modes III Arbitrary length pulses and spot sizes

In the first two parts of this study, the electromagnetic field components were derived for infinitely long, flattened Gaussian laser beams [J. Opt. Soc. Am. B23, 2157 and J. Opt. Soc. Am. B23, 2166 (2006)]. These results are now extended without approximation to allow for finite laser pulses having an arbitrary duration beginning with the standard Gaussian beam profile and then generalizing these results to a flattened Gaussian. The resulting models thus allow for all pulse durations and spot sizes from infinite, paraxial beams to single-cycle, wavelength-size spots, with a savings of more than 2 orders of magnitude in computation time. Pulses having fewer than ten cycles exhibit significant modification from the monochromatic fields as a result of the finite bandwidth. Specifically, the energy in the focus is shown to decrease from the theoretical value of 86.5% to as low as 72.2% for a single-cycle pulse.

Analytical solutions for the electromagnetic fields of flattened and annular Gaussian laser modes I Small F-number laser focusing

Many laser interaction models assume that incident focused laser fields are Gaussian and use either the approximate TEM00 series model or the exact integral Gaussian angular-spectrum solution. Many practical laser systems, however, produce flat-top transverse intensity profiles, and indeed, such profiles are often desired. Here, an exact, integral solution is derived for all of the vector components having a general flattened Gaussian profile using the angular-spectrum method. This solution includes the pure and annular Gaussian modes as special cases. The resulting integrals are solved for tight focusing conditions exactly by making use of a Fourier-Gegenbauer expansion. This technique follows closely that of Sepke and Umstadter [Opt. Lett.31, 1447 (2006)] but, by redefining the expansion coefficients, the simplicity of the model is greatly enhanced and the computation time reduced by roughly a factor of 2 beyond the 2 orders of magnitude improvement obtained previously. This series solution is stable at all points and converges after S∼20w0 terms, where w0 is the 1/e waist normalized to the laser wavelength.

Laser beam shaping profiles and propagation

We consider four families of functions--the super-Gaussian, flattened Gaussian, Fermi-Dirac, and super-Lorentzian--that have been used to describe flattened irradiance profiles. We determine the shape and width parameters of the different distributions, when each flattened profile has the same radius and slope of the irradiance at its half-height point, and then we evaluate the implicit functional relationship between the shape and width parameters for matched profiles, which provides a quantitative way to compare profiles described by different families of functions. We conclude from an analysis of each profile with matched parameters using Kirchhoff-Fresnel diffraction theory and M2 analysis that the diffraction patterns as they propagate differ by small amounts, which may not be distinguished experimentally. Thus, beam shaping optics is designed to produce either of these four flattened output irradiance distributions with matched parameters will yield similar irradiance distributions as the beam propagates.

Propagation of a Laguerre–Gaussian beam through a slightly misaligned paraxial optical system

Based on the generalized diffraction integral formula for treating the propagation of a laser beam through a slightly misaligned optical system in a cylindrical coordinate system, an analytical formula for a Laguerre–Gaussian beam passing through such an optical system is derived. Furthermore, an approximate analytical formula is derived for a Laguerre–Gaussian beam passing through an apertured slightly misaligned optical system by expanding the hard aperture function as a finite sum of complex Gaussian functions. Some analytical formulas are also given for a flattened Gaussian beam by expanding its field as a superposition of a finite series of Laguerre–Gaussian beams.

The Beam Propagation Factor of Nonparaxial Truncated Flattened Gaussian Beams

By using the second-order moment of the power density, the beam width, far-field divergence angle and M2 factor of nonparaxial truncated flattened Gaussian (FG) beams are derived analytically. It is shown that the M2 factor of nonparaxial truncated FG beams depends not only on the truncation parameter δ and beam order N, but also on the initial waist-width to wavelength ratio w0/λ. The far-field divergence angle approaches an asymptotic value of θmax=63.435° when the truncation parameter δ → 0. For the special cases of N = 0 and δ → ∞ our results reduce to those of nonparaxial truncated Gaussian beams and nonparaxial untruncated FG beams, respectively.

Propagation of laser beams formed by unstable resonators with different magnifications

Laser beams generated from high-magnification on-axis unstable resonators using hard-edged axial scraper mirrors and output couplers consisting of axial spot reflectors typically have an annular distribution in the near field (i.e., a flat-top profile with a hole in the middle for an axially coupled beam). We employ a new model, based on the flattened Gaussian beam (FGB) concept, to describe the propagation of such annular near-field beams. The superposition of two FGBs, whose flatness and steepness of edges are controlled by defined parameters (i.e., the beam width and the order), is used to analyze the output beam intensity profile along the propagation axis. We apply this model to characterize the output beams of copper vapour lasers employing unstable resonators of different magnifications (M = 80, 200, and 400). We show that once the model is fitted to the beam intensity profile at any two positions along the axis of propagation, it can be used to predict the beam shape at any position.PACS Nos.: 42.55.–f, 42.55.Lt, 42.60.–v

Models for an aperture function and corresponding analytical propagation equations of a flattened Gaussian beam through an annular apertured optical system

Beam profiles that consist of a sum of complex-Gaussian functions and a sum of polynomial-Gaussian functions offset by some fixed amount are proposed as two types of models for a hard aperture function. By expanding an aperture function into two types of models, the approximate analytical expression of a flattened Gaussian beam through a paraxial ABCD optical system with an annular aperture is derived. As special cases, the corresponding closed forms for the unaperture or circular aperture or circular black screen cases are also obtained. The results provide a more convenient way of studying propagation and transformation than the usual method using diffraction integral formula directly. Numerical examples are given to illustrate the propagation properties of flattened Gaussian beams.

Vectorial nonparaxial flattened gaussian beams and their beam quality in terms of the power in the bucket

The closed-form propagation equations of vectorial nonparaxial flattened Gaussian beams in free space are derived and some special cases are discussed. The power in the bucket (PIB) is extended to the nonparaxial regime, where the intensity is replaced by the z component of the time-averaged Poynting vector, and used to characterize beam quality of nonparaxial beams in the far field. Numerical calculation results show the dependence of the PIB of vectorial nonparaxial FGBs on the waist-width to wavelength ratio w 0/ λ and beam order N.

Analytical study of a flattened Gaussian beam through multi-apertured optical imaging systems of B = 0

By using the methods of the matrix decomposition and expansion of the hard-edged aperture function into a finite sum of complex Gaussian functions, the recurrence propagation expressions for a flattened Gaussian beam (FGB) through multi-apertured optical imaging systems of B = 0 are derived and illustrated with numerical examples. Comparisons with the straightforward numerical integration of the Collins formula and with the previous work are made. It is shown that the main advantages of our methods and results are the more accuracy and great reduction of computer time.

Propagation of flattened Gaussian beams in apertured fractional Fourier transforming systems

Based on the fact that a hard-edge aperture function can be expanded into a finite sum of complex Gaussian functions, the approximate analytical expressions for the output field distribution of a flattened Gaussian beam passing through the apertured fractional Fourier transforming systems are derived. By using the approximate analytical formulae and diffraction integral formulae, some numerical simulation comparisons are done, and it is shown that our method can significantly improve the numerical calculation efficiency.

Properties of a flattened Gaussian beam in the fractional Fourier transform plane

The fractional Fourier transform (FRT) is applied to study the transformation properties of flattened Gaussian beams (FGBs). An analytical formula is derived for the FRT of FGBs based on the definition of FRT. By using the derived formula, we study the intensity distribution properties of FGB in the FRT plane. The influences of the fractional order on the intensity distribution properties of FGB with different order in the fractional Fourier plane are studied in detail. The evolution of the kurtosis parameter of FGB in the FRT plane is also studied.