Quadratic Profile Research Articles

Harmonic achromatic metalens with a wide field of view for AR/VR applications

Abstract Metalens is a cutting-edge optical marvel, revolutionizes a plethora of applications like cameras, telescope, augmented reality (AR) and virtual reality (VR). Despite having ultrathin structure, lightweight nature, and multifunctionality, achievement of both achromatic imaging and wide field of view (WFOV) remains a persistent challenge. To address these concerns, here we present a novel approach using a harmonic metalens featuring a quadratic phase profile to achieve achromatic behavior and WFOV simultaneously. This innovation enables WFOV imaging using the Fourier transform of quadratic phase profile and achromatic behavior at RGB wavelengths (blue λ = 488 nm, green λ = 532 nm and red λ = 633 nm) with a WFOV of up to ± 45 ∘ . The designed harmonic metalens is polarization-insensitive, with a numerical aperture of 0.79, and it achieves a maximum focusing efficiency of ∼50% at a wavelength of 633 nm under normal incidence. Being polarization insensitive and achromatic, proposed scheme will certainly broaden the utilization of metalenses in AR/VR, displays and applications in imaging. © 2024 Optical Society of America.

Advances in laser-based bremsstrahlung x-ray sources. II. Laser pulse propagation and guiding in nonuniform plasma media in the presence of self-focusing

An analytic Wentzel–Kramers–Brillouin model is presented of Gaussian laser pulse propagation through plasma with a quadratic transverse density profile and an arbitrarily varying, longitudinal density gradient under conditions of nonlinear self-focusing. From these solutions, it is shown that in the absence of nonlinear self-focusing and transverse nonuniformity, for exponential pre-plasma density profiles, the use of a low density coating of the laser target with electron density n0∼11 ncr (e.g., a CH foam of density 35 mg/cm3 for 1-micron laser light) maximizes laser intensity at best focus. Also, under laser and plasma conditions relevant to recent experiments on high-power laser systems, conditions are obtained for a Gaussian laser pulse to propagate stably through the pre-plasma medium. Such conditions would be expected to enhance the production of relativistic electrons from laser-target coupling, providing a possible explanation for the observed increase in MeV photon dose and enabling applications such as laser-based MeV X-ray radiography.

Analytical solutions for the advective–diffusive ice column in the presence of strain heating

Abstract. A thorough understanding of ice thermodynamics is essential for an accurate description of glaciers, ice sheets and ice shelves. Yet there exists a significant gap in our theoretical knowledge of the time-dependent behaviour of ice temperatures due to the inevitable compromise between mathematical tractability and the accurate description of physical phenomena. In order to bridge this shortfall, we have analytically solved the 1D time-dependent advective–diffusive heat problem including additional terms due to strain heating and depth-integrated horizontal advection. Newton's law of cooling is applied as a Robin-type top boundary condition to consider potential non-equilibrium temperature states across the ice–air interface. The solution is expressed in terms of confluent hypergeometric functions following a separation of variables approach. Non-dimensionalization reduces the parameter space to four numbers that fully determine the shape of the solution at equilibrium: surface insulation, effective geothermal heat flow, the Péclet number and the Brinkman number. The initial temperature distribution exponentially converges to the stationary solution. Transient decay timescales are only dependent on the Péclet number and the surface insulation, so higher advection rates and lower insulating values imply shorter equilibration timescales, respectively. In contrast, equilibrium temperature profiles are mostly independent of the surface insulation parameter. We have extended our study to a broader range of vertical velocities by using a general power-law dependence on depth, unlike prior studies limited to linear and quadratic velocity profiles. Lastly, we present a suite of benchmark experiments to test numerical solvers. Four experiments of gradually increasing complexity capture the main physical processes for heat propagation. Analytical solutions are then compared to their numerical counterparts upon discretization over unevenly spaced coordinate systems. We find that a symmetric scheme for the advective term and a three-point asymmetric scheme for the basal boundary condition best match our analytical solutions. A further convergence study shows that n≥15 vertical points are sufficient to accurately reproduce the temperature profile. The solutions presented herein are general and fully applicable to any problem with an equivalent set of boundary conditions and any given initial temperature distribution.

Microstructural and kinetics analysis of FeB–Fe2B layer grown by pulsed-DC powder-pack boriding on AISI 316 L steel

In this study, novel findings were obtained regarding the influence of a 10 A current intensity on the growth of an FeB–Fe2B layer during pulsed-DC powder-pack boriding. Boride layer formation was carried out on AISI 316 L steel at 1123–1223 K for different exposure times at each temperature, considering 10 s polarity inversion cycles. The boride layer was characterized by x-ray diffraction and high-speed Berkovich nanoindentation, the latter being used to determine the hardness and reduced Young’s modulus mappings along the depth of the layer-substrate system. Moreover, the growth kinetics of the FeB–Fe2B layer on the steel’s surface was modeled using the heat balance integral method (HBIM). This involved transforming Fick’s second law into ordinary differential equations over time, assuming a quadratic boron concentration profile in space to determine the B diffusion coefficients in FeB and Fe2B, respectively. From the Arrhenius relationship, the B activation energies in the boride layer were estimated considering the contribution of the electromigration effect; the results showed an approximately 30% reduction compared to the values obtained in the conventional powder-pack boriding for AISI 316 L steel. Finally, the theoretical layer thickness obtained by the HBIM demonstrated an error of no more than 5% against the experimental FeB and FeB + Fe2B layer thickness values.

Water-vapor absorption database using Dual Comb Spectroscopy from 300 to 1300 K Part II: Air-Broadened H2O, 6600 to 7650 cm−1

We present broadband dual frequency comb laser absorption measurements of 2 % H2O (natural isotopic abundance of 99.7 % H216O) in air from 6600 to 7650 cm−1 (1307–1515 nm) with a spectral point spacing of 0.0068 cm−1. Twenty-nine datasets were collected at temperatures between 300 and 1300 K (±0.82 % average uncertainty) and pressures ranging from 20 to 600 Torr (±0.25 %) with an average residual absorbance noise of 8.0E-4 across the spectrum for all measurements. We fit measurements using a quadratic speed-dependent Voigt profile to determine 7088 absorption parameters for 3366 individual transitions found in HITRAN2020. These measurements build on the line strength, line center, self-broadening, and self-shift parameters determined in the Part I companion of this work. Here we measure air-broadened width (with temperature- and speed-dependence) and air pressure shift (with temperature-dependence) parameters. Various trends are explored for extrapolation to weak transitions that were not covered in this work. Improvements made in this work are predominantly due to the inclusion of air pressure shift temperature dependence values. In aggregate, these updates improved RMS absorbance error by a factor of 4.2 on average, and the remaining residual is predominantly spectral noise. This updated database improves high-temperature spectroscopic knowledge across the 6600-7650 cm−1 region of H2O absorption.

Analysis of Tapered Fibre Optic Surface Plasmon Resonance Bio-Sensor Chip With Highly Perturbed Taper Profiles.

A numerical model based on the Transfer matrix method (TMM) is proposed for the first time to study the gold coated tapered fibre optic surface plasmon resonance (SPR) with eight different types of taper profiles namely linear, exponential-linear, Gaussian, quadratic, sinusoidal, error function type and highly perturbed taper profile so-called chirp type of profile. The performance in terms of sensitivity, full width at half maximum (FWHM), detection accuracy (D.A.), amplitude dip, and half power points are estimated with respect to tapering ratio and choices of taper profile. It is found that sensitivity increased almost linearly with the taper ratio of each taper choice for the account of the reduction of detection accuracy. It has been found that sensitivity is highest for the case of chirp taper profile and lowest for the case of quadratic taper profile at low taper ratio. In this study, the aqueous solution is considered for sensor development which is adulterated by biomolecules species like DNA, blood samples, etc.

An Analytical Reentry Solution Based Online Time-Coordinated A* Path Planning Method for Hypersonic Gliding Vehicles Considering No-Fly-Zone Constraint

To meet the time-coordinated requirement of hypersonic gliding vehicles to reach a single target simultaneously in the presence of no-fly-zone constraints, this paper proposes a time-coordinated A* path planning method considering multiple constraints. The path planning method is designed based on an analytical steady gliding path model and the framework of the A* algorithm. Firstly, an analytical steady gliding path model is designed based on a quadratic function-type altitude-velocity profile. It can derive the control commands explicitly according to the desired terminal altitude and velocity, thus establishing a mapping between the terminal states and the control commands. Secondly, the node extension method of the A* algorithm is improved based on the mapping. Taking the terminal states as new design variables, a feasible path-node set is produced by a one-step integration using the control commands derived according to different terminal states. This node extension method ensures the feasibility of the path nodes while satisfying terminal constraints. Next, the path evaluation function of the A* algorithm is modified by introducing a heuristic switching term to select the most proper node as a waypoint, aiming to minimize the arrival time deviation. Meanwhile, introducing the penalty items into the path evaluation function satisfies the no-fly-zone constraints, process constraints, and control variable constraints. Finally, an online time-coordinated method is proposed to determine a commonly desired arrival time for several hypersonic gliding vehicles. It eliminates the need to specify the arrival time in advance and improves the capability to deal with sudden threats, increasing the path planning method’s online application capability. The proposed method can achieve online time-coordinated multi-constraint path planning for several hypersonic gliding vehicles, whose effectiveness and superiority are verified by simulations.

Modelling surface waves on shear current with quadratic depth-dependence

The currents in the ocean have a serious impact on ocean dynamics, since they affect the transport of mass and thus the distribution of salinity, nutrients and pollutants. In many physically important situations the current depends quadratic-ally on the depth. We consider a single layer of fluid and study the propagation of the surface waves in the presence of depth-dependent current with quadratic profile. We select the scale of parameters and quantities, which are typical for the Boussinesq propagation regime (long wave and small amplitude limit) and we also derive the well known KdV model for the surface waves interacting with current.

Significance of quadratic density variation on the heat transport phenomena in Careau dusty fluid subject to Lorentz force via stretching surface

The two-dimensional boundary layer of steady mixed convective magnetohydrodynamic Carreau dusty fluid flow across a stretched sheet was investigated in this study. The primary focus of this study is to examine the impact of Carreau nanofluid on dusty flow dynamics in the presence of the Lorentz force. The partial differential equations governing the fluid flow, temperature, and concentration fields for both the fluid and dust phases are transformed into ordinary differential equations using an appropriate set of similarity transformations. Numerical outcomes are acquired employing the Runge–Kutta technique combined with the shooting method, implemented on the MATLAB platform. The numerical outcomes are compared to previous research and confirmed to be in very good accord. Finally, the impacts of relevant factors of physical and technical importance on flow and thermal transport characteristics are visually and tabulated. The results for the linear density and the quadratic density profile, it is noted that the linear density decreases compared to the quadratic density for velocity, but the opposite behavior is observed for temperature profile. The advancement of dusty nanofluids has resulted in significant enhancements in the heat transfer mechanism, which finds application in manufacturing, industry, and nanotechnology research. Research on dusty flows benefits on is beneficial in air pollution studies, dust entrainment in a cloud formed during a nuclear, vehicle emissions of smoke and other pollutants, crude oil purification, petroleum production, combustion, industrial effluent emissions, capillary blood flow, paint sparing, and, more.

A varifocal augmented reality head‐up display using Alvarez freeform lenses

AbstractA crucial requirement of augmented reality head‐up displays (AR‐HUDs) is continuously adjustable virtual image distance (VID), which allows adaptation to various depths in road environments and thereby avoids visual fatigue. However, usual varifocal components for near‐eye displays are unavailable because AR‐HUDs require the varifocal component's aperture to be larger than 10 cm. This study considers the Alvarez lenses, which change the optical power by in‐plane sliding two freeform lenses. Under the paraxial assumption, classic Alvarez lenses can create a quadratic wavefront profile, but the large aperture and extensive diopter variation range required by AR‐HUDs lead to significant aberrations. Thus, the classic paraxial Alvarez lens design is extended by co‐optimizing Alvarez lenses with high‐order surface profiles and a primary freeform mirror. Therefore, a novel varifocal AR‐HUD containing Alvarez lenses with apertures larger than 15 cm is proposed. The AR‐HUD generates a varifocal plane whose VID can be continuously adjusted between 2.5 and 7.5 m, and another focal plane with a fixed VID at 7.5 m. In addition, merely one display panel is used for compactness. Finally, an AR‐HUD prototype with a reduced volume of 9.8 L was built. The expected varifocal performance and qualified imaging quality were experimentally verified through the field of view, VID, and image sharpness.

Wide-angle metalens array with quadratic phase for terahertz polarization detection

Abstract With the advances of micro/nano fabrication technology, metasurface has become an alternative to design functional devices for manipulating electromagnetic wave. Metalens is one of the basic electromagnetic functional devices that can be applied in various fields. Currently, polarization measurement based on metalens arrays has been widely investigated, but most of them can only work for normal incident wave due to the limited field-of-view of metalens. Herein, a dielectric wide-angle metalens array (WMA) for the terahertz polarization detection is presented. The WMA is composed of three wide-angle metalenses, each wide-angle metalens is constructed by utilizing quadratic phase profile. The angle tolerance of meta-atoms which constitute the wide-angle metalens is elucidated in detail. The WMA can decompose incident terahertz wave into four channels, and the full Stokes parameters of incident wave is determined by intensities in these four channels. Simulated results show that the WMA proposed here has excellent performance for the polarization detection within incident angle of ±40°. In addition, this WMA can also be used to detect the phase gradient of incident terahertz wave, the detection error is less than 1.1%. This WMA is promising in the fields of terahertz polarization generation, detection and imaging.

Study on the film thickness and surface wave velocity of the thin liquid film formed by a round jet obliquely impinging on a horizontal plate

For a thin liquid film (in a supercritical flow) prior to the formation of a non-circular hydraulic jump formed by a round jet obliquely impinging on a horizontal plate, the time-averaged film thickness and the surface wave velocity are extracted based on the measured transient film thickness. On the one hand, the effect of many factors, including the jet velocity, impingement angle, azimuthal angle, liquid viscosity, and surface tension, on the time-averaged film thickness and surface wave velocity are discussed. When the jet Reynolds number increases to about 1.4×104, the film thickness profile suddenly increases, and the transition of liquid flow from laminar to turbulent occurs. Meanwhile, a rapid increase is observed downstream of the turbulent film thickness profile. The influence of surface tension on the time-averaged film thickness and surface wave velocity is negligible for thin liquid films before non-circular hydraulic jumps. Nonetheless, the surface tension has a significant influence on the interface profile of non-circular hydraulic jumps. Furthermore, a “crescent” kink region upstream of the jump can be identified when the surface tension is lower than 40.6 mN/m. On the other hand, experimental results are used to verify the prediction accuracy of existing approximate solutions. The laminar approximate solution with a quadratic boundary layer velocity profile can accurately predict the film thickness distribution of most laminar thin liquid films, except downstream of the thin liquid films with a dynamic viscosity higher than 9.71 mPa s. The surface wave velocities are found to be close to the predicted surface velocities of the approximate solutions.

Giant Flexoelectric Effect in Snapping Surfaces Enhanced by Graded Stiffness

Flexoelectricity is present in nonuniformly deformed dielectric materials and has size-dependent properties, making it useful for microelectromechanical systems. Flexoelectricity is small compared to piezoelectricity; therefore, producing a large-scale flexoelectric effect is of great interest. In this paper, we explore a way to enhance the flexoelectric effect by utilizing the snap-through instability and a stiffness gradient present along the length of a curved dielectric plate. To analyze the effect of stiffness profiles on the plate, we employ numerical parameter continuation. Our analysis reveals a nonlinear relationship between the effective electromechanical coupling coefficient and the gradient of Young’s modulus. Moreover, we demonstrate that the quadratic profile is more advantageous than the linear profile. For a dielectric plate with a quadratic profile and a modulus gradient of − 0.9, the effective coefficient can reach as high as 15.74 pC/N, which is over three times the conventional coupling coefficient of piezoelectric material. This paper contributes to our understanding of the amplification of flexoelectric effects by harnessing snapping surfaces and stiffness gradient design.

The droplet race: Optimization of a wettability gradient surface

Droplet behavior influenced by wettability distribution is a pertinent field of research with applications in lab-on-a-chip and heat transfer devices among others. Some have proposed patterned surfaces with controlled variation of wettability to orient the direction of the droplet motion or to increase its velocity. These patterns are arrived upon with experience and knowledge of this phenomenon. In this research paper, the authors used a mathematical approach to the physical problem by using a gradient based optimizer for maximizing droplet velocity. Given some initial conditions, the optimizer marches toward the optimum wettability distribution profile. The droplet motion is modeled in two dimensions (i.e., on the xy-plane), on a plate having a wettability distribution in one dimension (i.e., along the x axis). The single component pseudopotential model allows for the quantification of the wettability distribution as a distribution of a pseudodensity of the solid nodes of the flat plate. Starting with several monotonous analytical profiles, a quadratic convex profile allows us to reach the maximum mean velocity for the threshold droplet displacement. Different sets of initial profiles, length of the plate (L), and diameter of the droplet (D) are tested. For smaller L/D ratio, the optimal wettability distributions exhibit non-trivial features: profiles can be non-monotonous, and wettability gradient could be locally null. With the increase in the L/D ratio, these specificities tend to be less prominent and optimal profiles converge to the quadratic convex one. The main innovation and significance of the paper is that mathematical optimization algorithms have been used conjointly with a multiphase lattice Boltzmann model solver to address for the first time the droplet race defined as: “what is the best wettability profile in order for a droplet to reach a desired location as quickly as possible?”

Water-vapor absorption database using dual comb spectroscopy from 300 to 1300 K part I: Pure H2O, 6600 to 7650 cm-1

We present broadband dual frequency comb laser measurements of pure H2O absorption (natural isotopic abundance of 99.7 % H216O) from 6600 to 7650 cm−1 (1307–1515 nm) with a spectral point spacing of 0.0068 cm−1. Twenty-nine absorption spectra were collected at temperatures between 300 and 1300 K (±0.81 % average uncertainty) and pressures ranging from 0.5 to 16 Torr (±0.27 %) with an average residual absorbance noise of 7.4E-4 across the spectrum for all measurements. We fit measurements using a quadratic speed-dependent Voigt profile to determine 17,030 absorption parameters for 5986 transitions found in HITRAN2020. The measured parameters include line strength, line center, self-broadening, and self-shift, along with their power-law temperature-dependence exponents, and speed-dependent self-width parameters. A second version of the retrieved parameters with fits that do not include speed-dependence is also presented. We explore various trends to enable parameter extrapolation for weak transitions that were not covered in this work. We measured line parameters for an additional 614 features not presently catalogued in the HITRAN2020 database. In aggregate, these updates improved RMS absorbance error by a factor of 2.6 on average, and the remaining residual is predominantly spectral noise. This updated database improves high-temperature spectroscopic knowledge across the 6600–7650 cm−1 region of H2O absorption. An accompanying paper (Part II) describes similar measurements and a new database for air-broadened H2O absorption.

A two-phase Stefan problem with power-type temperature-dependent thermal conductivity. Existence of a solution by two fixed points and numerical results

A one-dimensional two-phase Stefan problem for the melting of a semi-infinite material with a power-type temperature-dependent thermal conductivity was considered. The assumption of taking thermal parameters as functions of temperature found its basis in physical and industries applications, allowing for a more precise and realistic description of phase change processes. By imposing a Dirichlet condition at the fixed face, a theoretical and approximate study was developed. Through a similarity transformation, an equivalent ordinary differential problem was obtained from which an integral problem was deduced. The existence of at least one analytical solution was guaranteed by using the Banach fixed point theorem. Due the unavailability of an analytical solution, a heat balance integral method was applied, assuming a quadratic temperature profile in space, to simulate temperature variations and the location of the interface during the melting process. For constant thermal conductivity, results can be compared with the exact solution available in the literature to check the accuracy of the approximate method.

Semi-analytical finite ray-tracing through the quadratic symmetric GRIN lens.

The propagation of light within a gradient index (GRIN) media can be analyzed with the use of differential equations for a given non-homogenous refractive index profile. Numerical methods are often necessary to perform ray-tracing in GRIN media; however, analytical solutions exist for several types of GRIN lenses. In this paper, paraxial and non-paraxial differential equations are derived to calculate the ray path in a GRIN lens. It is shown that the paraxial equation has an analytical solution for a GRIN media with a quadratic profile within the paraxial region. The analytical solution can be obtained by using Legendre polynomials or by the Frobenius method involving a power series. Using the Legendre or Frobenius solution, one can calculate the refractive indices along the ray path. A new recursive relationship is proposed to map the trajectory of light at finite heights. To illustrate the finite ray-tracing method utilizing a non-paraxial differential equation, two lenses (with spherical and elliptical iso-indicial contours) are considered. The lenses' back focal distances, for rays entering the lenses at varying finite heights, are calculated. For each lens, its spherical aberration is estimated. The effective focal length and the shape of the principal surface are also obtained. The accuracy of the results is then compared to the numerical ray-tracing using an optical design software, Zemax OpticStudio. The predicted spherical aberration for the spherical lens differs from numerical ray-tracing by less than λ14 at the marginal zone, while the error for the effective focal length is less than λ100.

The Effect of an Increasing Subglottal Stenosis Constriction That Extends From the Vocal Folds to the Inferior Border of the Cricoid Cartilage.

Acquired subglottal stenosis is an unpredicted complication that can occur in some patients who have undergone prolonged endotracheal intubation. It is a narrowing of the airway at the level of the cricoid cartilage that can restrict airflow and cause breathing difficulty. Stenosis is typically treated with endoscopic airway dilation, with some patients experiencing multiple recurrences. The study highlights the potential of computational fluid dynamics as a noninvasive method for monitoring subglottic stenosis, which can aid in early diagnosis and surgical planning. An anatomically accurate human laryngeal airway model was constructed from computerized tomography (CT) scans. The subglottis cross-sectional area was narrowed systematically using ≈10% decrements. A quadratic profile was used to interpolate the transformation of the airway geometry from its modified shape to the baseline geometry. The numerical results were validated by static pressure measurements conducted in a physical model. The results show that airway resistance follows a squared ratio that is inversely proportional to the size of the subglottal opening (R∝A-2). The study found that critical constriction occurs in the subglottal region at 70% stenosis (upper end of grade 2). Moreover, removing airway tissue below 40% stenosis during surgical intervention does not significantly decrease airway resistance.

Analysis of natural planar jump in power-law liquids—A generalized “shallow flow” approach

The study presents a generalized “shallow flow” analysis of natural planar hydraulic jump in power-law liquids. It is based on self-similar velocity profile defined as function of flow behavior index, n, and shows significant improvement over the previous analysis which assumed a quadratic velocity profile and failed for n < 0.5. Thus, the study enables a deeper understanding of the influence of n for shear thinning vis-a-vis shear thickening liquids and emphasizes that the power-law description is adequate for highly shear thinning liquids if the flow parameters are valid over the range of interest.

A Novel Optimization Strategy of Bearing Geometry with a Length to Diameter Ratio of 1.25 under Severe Operating Conditions Using Taguchi Method

Robust and well-designed rotor-bearing systems ensure safe operation and a high level of reliability under severe operating conditions. A deviation in the shaft axis with respect to the bearing longitudinal axis represents one of the most unavoidable problems in bearing systems. This deviation results from installation errors, manufacturing errors, shaft deformation under heavy loads, bearing wear, and many other causes. Each of these deviation sources has its negative consequences on the designed characteristics of the system. This work deals with the geometrical design of a journal bearing using three forms of profiles (linear (n=1), quadratic n=2 and cubic (n=3) profiles) in order to enhance bearing performance despite the presence of the inevitable shaft deviation. In addition, a wide range of bearing profile parameters are considered in the analysis to optimize the bearing profile based on the use of the Taguchi method. A general form of shaft deviation is considered to account for both horizontal and vertical deviations. A numerical solution is obtained using the finite difference method. The results show that all three suggested forms of bearing profiles elevate the film thickness significantly and also reduce the friction coefficient, but with different effects on the maximum pressure values. The Taguchi method illustrates that the optimal geometrical design parameters are the quadratic profile and the modification of one-fifth of the bearing width from both sides at a height of just less than half the radial clearance (0.4 C) at the bearing edges. These values give the best combination of the considered main bearing characteristics: the minimum film thickness, coefficient of friction, and maximum pressure. The results show that the minimum film thickness is increased by 184%, the maximum pressure is reduced by 15.1% and the friction coefficient is decreased by 6.4% due to the use of the suggested design. The outcome of this work represents an important enhancement step for the rotor bearing performance to work safely with high reliability under severe shaft deviation levels. This can be implied at the design stage of the bearing, which requires prior knowledge about the operating conditions in order to have better estimation for the levels of shaft deviation.