Rayleigh Range Research Articles

On the possibility of averaging the equations of an electron motion in the intense laser radiation

The problem of averaging of the relativistic motion equations of electron in the intense laser radiation, caused by the decreasing of the rate of wave phase change due to the Dopplers effect, is considered. As a result the phase can go from the fast to slow variables of the motion, so averaging over the phase becomes impossible. An analysis is presented of the conditions which are necessary for averaging of the relativistic equations of motion over the fast phase of the intense laser radiation on the base of the general principles of the averaging method. Laser radiation is considered in the paraxial approximation, where the ratio of the laser beam waist to the Rayleigh length is accepted as a small parameter. It is supposed that the laser pulse duration is of the order if the laser beam waist. In this case first-order corrections to the vectors of the laser pulse field should be taken into account. The general criterion for the possibility of the averaging of the relativistic motion equations of electron in the intense laser radiation is obtained. It is shown that an averaged description of the relativistic motion of an electron is possible in the case of a fairly moderate (relativistic) intensity and relatively wide laser beams. The known in the literature analogical criterion has been obtained earlier on the base of the numerical results.

Three-dimensional spatiotemporal nondiffracting parabolic cylinder beams

In this paper, the propagation characteristics of three-dimensional spatiotemporal nondiffracting parabolic cylinder beams in free space are studied. From the (3+1)-dimensional paraxial wave equation, we obtain an exact solution by the method of separating variables. The solution is constructed using parabolic cylinder functions, described by the three optical mode numbers. The spatiotemporal intensity distribution of the parabolic cylinder beam at five times Rayleigh range with different mode parameters is presented analytically and simulated numerically. Our results indicate that the intensity distribution can be well controlled by adjusting the values of the three mode numbers. They are of great significance for improved understanding of the formation mechanism of parabolic cylinders and other nondiffracting beams.

Two-dimensional asymmetric Laguerre-Gaussian diffraction-free beams

We find asymmetric diffraction-free Laguerre-Gaussian solutions of the two-dimensional paraxial wave equation in free space. They are constructed by the Laguerre polynomials and trigonometric functions, and described by the radial and angular mode numbers, and modulation depth. Different optical field distributions of the asymmetric diffraction-free Laguerre-Gaussian beams are presented and analyzed. By selecting appropriate angular mode numbers, we identify novel types of asymmetric diffraction-free beams, obtained by the linear superposition of solutions, such as asymmetric vortex and asymmetric necklace-ring beams. Intensity distribution of the asymmetric diffraction-free beams at four times Rayleigh range with different mode parameters is presented analytically and simulated numerically. This research provides theoretical basis for possible experimental verification and utilization of asymmetric diffraction-free beams in free space or uniform transparent dielectrics.

Electric-field-profile measurement along a probing laser path based on electric-field-induced second-harmonic generation

Electric field measurement based on electric-field-induced second-harmonic generation (E-FISHG) of gas molecules is a promising method that enables noninvasive field measurement in gases. Although the probing laser beam is focused on the target area by a lens, the signal detected by the E-FISHG method is affected by the electric field along the probing laser path. The influencing path spans many tens of times the Rayleigh length of the front and rear of the focused spot. The E-FISHG signal has been frequently calibrated under uniform electric fields. However, this can yield a significantly incorrect calibrated value if the difference between the electric-field profiles in the measurement and calibration is not considered. In this paper, we propose a method to calibrate and furthermore restore the electric-field profile along a probing optical path by measuring a series of E-FISHG signals by changing the focusing spot position. The electric-field profile along the probing laser path is successfully restored from the measured E-FISHG signal sequence, which is verified by an experiment.

Investigation on failure mechanism of electrical connectors under repetitive mechanical insertion and withdrawal operations

Electrical connector is an essential accessory component for electrical and electronic interconnection circuit. In order to investigate the degradation behavior of electrical connector, a series of repetitive mechanical insertion and withdrawal operations of electrical connector have been carried out. The results indicate that there is an increasing trend in insertion/extraction force in the initial stage. Afterwards, it becomes a gradually decreasing trend attributed to the mechanical wear of the contact components. In addition, the oxidative wear of substrate copper alloy material causes the fluctuation phenomenon of contact resistance. The relevant mathematic models for insertion/extraction force and contact resistance calculation are presented to research the dynamic insertion/extraction process. Finally, the degradation behavior and associated physical mechanisms are proposed by analysing the laser confocal photographs and parameter waveforms comparison.

Finite Volume Simulation of Natural Convection for Power-Law Fluids with Temperature-Dependent Viscosity in a Square Cavity with a Localized Heat Source

In this paper, numerical study on natural convection heat transfer for confined thermo-dependent power-law fluids is conducted. The geometry of interest is a fluid-filled square enclosure where a uniform flux heating element embedded on its lower wall is cooled from the vertical walls while the remaining parts of the cavity are insulated, without slipping conditions at all the solid boundaries. The governing partial differential equations written in terms of non-dimensional velocities, pressure and temperature formulation with the corresponding boundary conditions are discretized using a finite volume method in a staggered grid system. Coupled equations of conservation are solved through iterative Semi Implicit Method for Pressure Linked Equation (SIMPLE) algorithm. The effects of pertinent parameters, which are Rayleigh number (103 ≤ Ra ≤ 106), power-law index (0.6 ≤ n ≤ 1.4), Pearson number (0 ≤ m ≤ 20) and length of the heat source (0.2 ≤ W ≤ 0.8) on the cooling performance are investigated. The results indicate that the cooling performance of the enclosure is improved with increasing Pearson and Rayleigh numbers as well as with decreasing power-law index and heat source length.

Intense XUV pulses from a compact HHG setup using a single harmonic

We report on a compact and spectrally intense extreme-ultraviolet (XUV) source, which is based on high-harmonic generation (HHG) driven by 395 nm pulses. In order to minimize the XUV virtual source size and to maximize the XUV flux, HHG is performed several Rayleigh lengths away from the driving laser focal plane in a high-density gas jet. As a result, a high focused XUV intensity of 5 × 1013 W cm−2 is achieved, using a beamline with a length of only two meters and a modest driving laser pulse energy of 3 mJ. The high XUV intensity is demonstrated by performing a nonlinear ionization experiment in argon, using an XUV spectrum that is dominated by a single harmonic at 22 eV. Ion charge states up to Ar3+ are observed, which requires the absorption of at least four XUV photons. The high XUV intensity and the narrow bandwidth are ideally suited for a variety of applications including photoelectron spectroscopy, the coherent control of resonant transitions and the imaging of nanoscale structures.

The effect of Rayleigh length of incident Gaussian beam on the in-plane angular spin splitting

In this paper, we established the relationship model between Rayleigh length of incident Gaussian beam and in-plane angular spin splitting (IPASS) under arbitrary linear polarization, and systematically investigated the effect of Rayleigh length on the IPASS. The results show that the IPASS shifts decrease with the increase of Rayleigh length, and the smaller the Rayleigh length, the greater its impact on IPASS. The maximum IPASS shift can almost reach half of the beam divergence angle by adjusting incident angle and polarization angle. In addition, several interesting characters of IPASS are discovered.

Coherent superposition of orthogonal Hermite–Gauss modes

The coherent superposition of orthogonal modes can result in phase dependent transverse offsets, shifts of the focus position, variations of the Rayleigh length and a reduction of the beam quality factor of the coherent sum of modes in comparison to the incoherent sum. Relations for first and second order moments, the beam quality and the Rayleigh length for the superposition of Hermite–Gauss modes are derived. The Courant–Snyder formalism, which was originally developed in the context of charged particle optics, is applied to propagate an arbitrary coherent sum of orthogonal modes through a lens system. Relations of generating and observable optical functions are highlighted. In the last part of the report the elegant Hermite–Gauss solution is interpreted in terms of generating and observable functions and the solution is decomposed into a sum of standard Hermite–Gauss modes.

Investigation of methods to extract confocal function parameters for the depth resolved determination of attenuation coefficients using OCT in intralipid samples, titanium oxide phantoms, and in vivo human retinas.

The attenuation coefficient provides a quantitative parameter for tissue characterization and can be calculated from optical coherence tomography (OCT) data, but accurate determination requires compensation for the confocal function. We present extensive measurement series for extraction of the focal plane and the apparent Rayleigh length from the ratios of OCT images acquired with different focus depths and compare these results with two alternative approaches. By acquiring OCT images for a range of different focus depths the optimal focus plane difference is determined for intralipid and titanium oxide (TiO2) phantoms with different scatterer concentrations, which allows for calculation of the attenuation coefficient corrected for the confocal function. The attenuation coefficient is determined for homogeneous intralipid and TiO2 samples over a wide range of concentrations. We further demonstrate very good reproducibility of the determined attenuation coefficient of layers with identical scatter concentrations in a multi-layered phantom. Finally, this method is applied to in vivo retinal data.

Effect of focal position offset on joint integrity of AA1050 battery busbar assembly during remote laser welding

This paper aims at studying the impact of focal position offset on joint integrity of AA1050 battery busbar during remote laser welding. Welding experiments were conducted at different focal position offsets in the AA1050 L-joint with the integration of laser beam oscillation. The impact of focal position offset was evaluated by multiple joint indicators including weld geometry, thermal profile, weld porosity and mechanical strength. Results showed that defocusing of the laser beam tends to be favourable for the reduction of peak temperatures and weld porosity. However, a defocusing more than 3 mm can lead to 60% drop in bonding integrity and 40% loss of joint strength. A tolerance window of [-2 1] mm was determined for the focal position offset, corresponding to 70% and 30% of the Rayleigh length. This result is a useful reference to inform precise focal position control with respect to the accumulated variations of part tolerance, part-to-part gap and part positioning errors. This study also indicated that the thermal measurement can be a potential approach for in-process monitoring of joint integrity. Indeed, the peak temperature near the weld zone showed good linear correlation with off-line detectable joint indicators including weld depth, weld width at interface, weld tensile strength and weld porosity.

Properties of special hyperbolic Bessel-Gaussian optical beams

An explicit formula for a type of beams, which in this work are called the ``special'' hyperbolic Bessel-Gaussian (SHBG) beams, has been derived, using the method of the Hankel transform formulated in our previous work [T. Rado\ifmmode \dot{z}\else \.{z}\fi{}ycki, arXiv:2103.06988]. The fundamental properties of these beams are analyzed. The parameters that define the beam shape have been identified and related to those of the fundamental Gaussian beam. The analytical expressions for the SHBG beams include an additional parameter $\ensuremath{\gamma}$, which allows the beam's shape to be modified to some extent. In the plane perpendicular to the propagation direction, these beams exhibit the annular nature. Interestingly, initially (i.e., near the beam's spot) a single ring splits into a number of rings as one is moving along the beam. This is especially apparent for $\ensuremath{\gamma}$ close to unity, as this effect then appears for values of $z$ relatively small compared to the Rayleigh length, i.e., where the energy concentration in the beam is still high. The phase of the wave, whose behavior is in certain aspects typical of modes having the vortex character, is also studied in this paper.

Volume averaging effect in nonlinear processes of focused laser fields.

We report theoretical derivations and experimental results on the volume averaging effect of nonlinear processes in focused laser fields. This effect is considered detrimental in revealing the intensity dependence of a nonlinear process, caused by the intensity variation across the sampled volume of a focused laser. Following the treatment in the literature, we prove that if the signal dependence can be expressed as a simple power function of the laser intensity and if the detection region encompasses effectively the whole volume, volume average does not affect the final conclusion on the derived exponent. However, to reveal the detailed saturation effect of a multi-photon process, intensity selective scans involving spatial filters and displacement of the laser focus (z-scan) are required. Moreover, to fully capture the dependence of the signal on the variation of the laser intensity, the degree of spatial discrimination and the corresponding range of the z-scan need to be modeled carefully. Limitations in the dynamic range of the detector or the laser power, however, can thwart the desired scan range, resulting in erroneous fitting exponents. Using our nanosecond laser with a non-ideal Gaussian beam profile based on multiphoton ionization of argon atoms from a collimated molecular beam and from ambient argon gas, we report experimental measurements of the beam waist and Rayleigh range and compare the experimental intensity dependence of Ar+ with theoretical values. Agreements between theory and experiment are remarkable.

Attaining quantum limited precision of localizing an object in passive imaging

We investigate our ability to determine the mean position, or centroid, of a linear array of equally-bright incoherent point sources of light, whose continuum limit is the problem of estimating the center of a uniformly-radiating object. We consider two receivers: an image-plane ideal direct-detection imager and a receiver that employs Hermite-Gaussian (HG) Spatial-mode Demultiplexing (SPADE) in the image plane, prior to shot-noise-limited photon detection. We compare the Fisher Information (FI) for estimating the centroid achieved by these two receivers, which quantifies the information-accrual rate per photon, and compare those with the Quantum Fisher Information (QFI): the maximum attainable FI by any choice of measurement on the collected light allowed by physics. We find that focal-plane direct imaging is strictly sub-optimal, although not by a large margin. We also find that the HG mode sorter, which is the optimal measurement for estimating the separation between point sources (or the length of a line object) is not only suboptimal, but it performs worse than direct imaging. We study the scaling behavior of the QFI and direct imaging's FI for a continuous, uniformly-bright object in terms of its length, and find that both are inversely proportional to the object's length when it is sufficiently larger than the Rayleigh length. Finally, we propose a two-stage adaptive modal receiver design that attains the QFI for centroid estimation.

Millisecond-long suppression of spectroscopic optical signals using laser filamentation.

Ultrashort laser pulse filamentation in air can extend the delivery of focused laser energy to distances greatly exceeding the Rayleigh length. In this way, remote measurements can be conducted using many standard methods of analytical spectroscopy. The performance of spectroscopic techniques can be enhanced by temporal gating, which rejects the unwanted noise and background. In the present work, we investigate the thermal relaxation of air in the wake of single-filament plasmas using shadowgraphy. We demonstrate that the transient change in refractive index associated with relaxation of the gas can be used to reject both continuous and time-varying spectroscopic signals, including emission from laser-produced plasmas. This method can augment temporal gating of simple optical detectors.

Adjustable spin Hall effect of light in surface plasmon resonance structure with consideration of angular Imbert–Fedorov shift

For a horizontally polarized Gaussian beam incident on surface plasmon resonance structure composed of prism, silver film, and absorptive liquid, the left- and right-handed circularly polarized components of the reflected beam undergo spin-dependent spatial and angular Imbert–Fedorov (IF) shift. We investigate spin Hall effect of light (SHEL) after the reflected beam travels a distance of Rayleigh range. It is shown that the maximum total transverse shift (TTS) of one spin component at Rayleigh range is always larger than the maximum spatial IF shift, and TTS is more sensitive to variations in the parameters of the structure than spatial IF shift. Furthermore, this SHEL is adjustable by changing the complex refractive index of the absorptive liquid. This research provides a simple way to enhance SHEL and may have potential application of improving the accuracy of sensors based on SHEL.

Corneal Confocal Microscopy Predicts the Development of Diabetic Neuropathy: A Longitudinal Diagnostic Multinational Consortium Study.

Corneal nerve fiber length (CNFL) has been shown in research studies to identify diabetic peripheral neuropathy (DPN). In this longitudinal diagnostic study, we assessed the ability of CNFL to predict the development of DPN. From a multinational cohort of 998 participants with type 1 and type 2 diabetes, we studied the subset of 261 participants who were free of DPN at baseline and completed at least 4 years of follow-up for incident DPN. The predictive validity of CNFL for the development of DPN was determined using time-dependent receiver operating characteristic (ROC) curves. A total of 203 participants had type 1 and 58 had type 2 diabetes. Mean follow-up time was 5.8 years (interquartile range 4.2-7.0). New-onset DPN occurred in 60 participants (23%; 4.29 events per 100 person-years). Participants who developed DPN were older and had a higher prevalence of type 2 diabetes, higher BMI, and longer duration of diabetes. The baseline electrophysiology and corneal confocal microscopy parameters were in the normal range but were all significantly lower in participants who developed DPN. The time-dependent area under the ROC curve for CNFL ranged between 0.61 and 0.69 for years 1-5 and was 0.80 at year 6. The optimal diagnostic threshold for a baseline CNFL of 14.1 mm/mm2 was associated with 67% sensitivity, 71% specificity, and a hazard ratio of 2.95 (95% CI 1.70-5.11; P < 0.001) for new-onset DPN. CNFL showed good predictive validity for identifying patients at higher risk of developing DPN ∼6 years in the future.

Compact intense extreme-ultraviolet source

High-intensity laser pulses covering the ultraviolet to terahertz spectral regions are nowadays routinely generated in a large number of laboratories. In contrast, intense extreme-ultraviolet (XUV) pulses have only been demonstrated using a small number of sources including free-electron laser facilities and long high-harmonic generation (HHG) beamlines. Here, we demonstrate a concept for a compact intense XUV source based on HHG that is focused to an intensity of 2 × 10 14 W / c m 2 , with a potential increase up to 10 17 W / c m 2 in the future. Our approach uses tight focusing of the near-infrared (NIR) driving laser and minimizes the XUV virtual source size by generating harmonics several Rayleigh lengths away from the NIR focus. Accordingly, the XUV pulses can be refocused to a small beam waist radius of 600 nm, enabling the absorption of up to four XUV photons by a single Ar atom in a setup that fits on a modest (2 m) laser table. Our concept represents a straightforward approach for the generation of intense XUV pulses in many laboratories, providing exciting opportunities for XUV strong-field and nonlinear optics experiments, for XUV-pump XUV-probe spectroscopy and for the coherent diffractive imaging of nanoscale structures.

Superluminal light propagation in a normal dispersive medium

The propagation of a Laguerre-Gaussian (LG) beam through a dispersive medium is investigated. The effect of the doughnut-like intensity profile of the probe LG beam on the group velocity is studied. We find an analytical expression for the group velocity out of the optical axis and compare with its projection onto the propagation axis. It is turned out that the group velocity vector is along the optical axis at the waist of the beam and the Rayleigh range. We numerically and analytically investigate the effect of the helical phase front of the coupling LG field on the group velocity of the probe LG field in a four-level double V-type quantum system. Our analysis predicts a strange behavior for the group velocity of the probe LG beam inside a normal dispersive medium in the gain region so that it can exceed the speed of light in free space, leads to the gain-assisted superluminal light propagation in normal dispersion. Such an unusual propagation of the LG beam results from the distortion of its helical phase front via the classical interference of the planar and LG fields. The obtained results may find some potential applications in increasing the velocity of the information transmission in optical communications.

Coherent interaction of atoms with a beam of light confined in a light cage

Controlling coherent interaction between optical fields and quantum systems in scalable, integrated platforms is essential for quantum technologies. Miniaturised, warm alkali-vapour cells integrated with on-chip photonic devices represent an attractive system, in particular for delay or storage of a single-photon quantum state. Hollow-core fibres or planar waveguides are widely used to confine light over long distances enhancing light-matter interaction in atomic-vapour cells. However, they suffer from inefficient filling times, enhanced dephasing for atoms near the surfaces, and limited light-matter overlap. We report here on the observation of modified electromagnetically induced transparency for a non-diffractive beam of light in an on-chip, laterally-accessible hollow-core light cage. Atomic layer deposition of an alumina nanofilm onto the light-cage structure was utilised to precisely tune the high-transmission spectral region of the light-cage mode to the operation wavelength of the atomic transition, while additionally protecting the polymer against the corrosive alkali vapour. The experiments show strong, coherent light-matter coupling over lengths substantially exceeding the Rayleigh range. Additionally, the stable non-degrading performance and extreme versatility of the light cage provide an excellent basis for a manifold of quantum-storage and quantum-nonlinear applications, highlighting it as a compelling candidate for all-on-chip, integrable, low-cost, vapour-based photon delay.