Paraxial Research Articles

Derivation of an in-plane amplitude equation and its paraxial approximation for elastic guided waves in plates

In addressing the construction of beam solutions for elastic guided waves in plates, we propose expressions of the amplitude function for the Lamb wave and the shear horizontal plate wave. The amplitude function is governed by an in-plane amplitude equation describing plate waves that has a same form as the two-dimensional Helmholtz equation. Therefore, beam solutions can be constructed within the same framework as this Helmholtz equation. The paraxial approximation is then applied to this in-plane amplitude equation. On the basis of this approximation, Gaussian beam solutions for both the Lamb and shear horizontal plate waves are derived. The effects of beam width, wavenumber, and guided-wave mode upon the wavefields are examined through several numerical examples.

Operator Formulation for Centered Optical Systems

Currently, there are many mathematical methods in use in geometric optics. This paper presents a new mathematical apparatus: an operator formalism, which describes centered optical systems in the paraxial approximation. This work is an ideological continuation of author’s previous research. The refraction and reflection operators of spherical surfaces are defined here. The mathematical properties of the operators are studied, and their physical interpretations are established. In addition, the relations between the lensing operator and the refraction operators of a spherical surface are determined. The behavior of rays is also considered, which helped to establish the injectivity and nondegeneracy for points with infinite coordinates. The operator formalism is helpful for finding a centered optical system that performs a given transformation. Moreover, the interchangeability of the optical operators is investigated, and it is found that each operator has a unique effect.

Low frequency twisted waves in a self-gravitating nonextensive complex plasma

Abstract The effects of dust–dust self-gravitational force and nonextensive characteristics of plasma species on the low frequency twisted waves owing to the helical wave structure in complex (dusty) plasmas are analyzed. The electrons and ions of the plasma are modelled by nonextensive q-distribution function while massive dust particles are Maxwellian distributed. The self-gravitational effects are incorporated in the Vlasov equation of kinetic theory where perturbed distribution function, electrostatic and gravitational potentials are expressed with Laguerre–Gauss functions. The governing equations of kinetic theory are solved together under paraxial approximations. The dispersion relations and damping rates of twisted dust-acoustic waves (TDAWs) are obtained for two situations; (a) super-extensivity (q < 1) and (b) sub-extensivity (q > 1). The effects of self-gravity, nonextensivity and twist parameter significantly modified the basic features of dust-acoustic waves. This study contributes to our understanding of the complex dynamics of TDAWs in interstellar dust clouds, considering the interplay of self-gravity, nonextensivity, and helical phase structures. The obtained theoretical and numerical results provide valuable insights into the behavior of these waves and offer a foundation for further investigations in this field. However, understanding of the topic can be enhanced through a combination of theoretical models, numerical simulations and observational data.

A volume-averaged model for acoustic streaming induced by focused ultrasound in soft porous media.

Equations describing acoustic streaming in soft, porous media driven by focused ultrasound are derived based on the assumption that acoustic waves pass through the porous material as if it were homogeneous. From these equations, a model that predicts the time-averaged flow on the macroscopic scale, as well as the advective transport of the trace components, is created. The model is used to perform simulations for different shapes of the focused ultrasound beam. For a given shape, and using the paraxial approximation for the ultrasound, the acoustic streaming is found to be linearly proportional to the applied ultrasound intensity, to the permeability of the porous material and to the attenuation coefficient, and inversely proportional to the liquid viscosity. Results from simulations are compared to a simplified expression stating that the dimensionless volumetric liquid flux is equal to the dimensionless acoustic radiation force. This approximation for the acoustic streaming is found to be reasonable near the beam axis for focused ultrasound beam shapes that are long in the axial direction, compared to their width. Finally, a comparison is made between the model and experimental results on acoustic streaming in a gel, and good agreement is found.

Virtual sources for structured partially coherent light fields.

A virtual source (VS) is a hypothetical source instead of an actual physical entity, but provides a distinctive perspective to understand physical fields in a source-free area. In this work, we generalize the VS theory to structured partially coherent light fields (PCLFs) by establishing the partially coherent inhomogeneous Helmholtz equation, then demonstrate that PCLFs can be generated from the incoherent extended VS in imaginary space. Especially, we put forward an understanding of the Gaussian Schell-model beam, which consists of a group of partially coherent paraxial complex rays. The mutual coherence between these rays depends on the included angle between them. In previous studies, the analytical solution of the partially coherent Airy beam was obtained with difficulty by the Huygens-Fresnel integral; however, by applying the VS, we put forward, to our knowledge, an unprecedented analytical solution for a partially coherent Airy beam. We believe this example will qualify the VS as an important perspective to understand structured PCLFs.

Accurate design of a TeO2 noncollinear acousto-optic tunable filter with refractive index correction.

The refractive index is a key factor in the design and analysis of noncollinear acousto-optic tunable filter (AOTF) devices. While previous studies have corrected and analyzed the effects of anisotropic birefringence and the rotatory property, they still rely on paraxial and elliptical approximations, which can introduce non-negligible errors (0.5° or more) into the geometric parameters of TeO2 noncollinear AOTF devices. In this paper, we address these approximations and their effects through refractive index correction. This fundamental theoretical research has significant implications for the design and application of noncollinear AOTF devices.

Propagation of intense electromagnetic pulse with a small conical phase shift induced by Axicon optics.

The conical phase shift induced by the axicon generates a non-diffracting Bessel beam. In this paper, we examine the propagation property of an electromagnetic wave focused by a thin lens and axicon waveplate combination, which induces a small amount of conical phase shift less than one wavelength. A general expression describing the focused field distribution has been derived under the paraxial approximation. The conical phase shift breaks the axial symmetry of intensity and shows a focal spot-shaping capability by controlling the central intensity profile within a certain range near focus. The focal spot-shaping capability can be applied to form a concave or flattened intensity profile, which can be used to control the concavity of a double-sided relativistic flying mirror or to generate the spatially uniform and energetic laser-driven proton/ion beams for hadron therapy.

Low frequency 3D transmission ultrasound tomography: technical details and clinical implications

A novel 3D ultrasound tomographic (3D UT) method (called volography) that creates a speed of sound (SOS) map and a reflection modality that is co-registered are reviewed and shown to be artifact free even in the presence of high contrast and thus shown to be applicable for breast, orthopedic and pediatric clinical use cases. The 3D UT images are almost isotropic with mm resolution and the reflection image is compounded over 360 degrees to create sub-mm resolution in plane. MethodsThe physics of ultrasound scattering requires 3D modeling and the concomitant high computational cost is ameliorated with a bespoke algorithm (paraxial approximation – discussed here) and Nvidia GPUs. The resulting reconstruction times are tabulated for clinical relevance. The resulting SOS map is used to create a refraction corrected reflection image at ∼3.6 MHz center frequency. The transmission data are highly redundant, collected over 360 degrees and at 2 mm levels by true matrix receiver arrays yielding 3D data.The high resolution SOS and attenuation maps and reflection images are used in a segmentation algorithm that optimally utilizes this information to segment out glandular, ductal, connective tissue, fat and skin. These volumes are used to estimate breast density, an important correlate to cancer. ResultsMultiple SOS images of breast, knee and segmentations of breast glandular and ductal tissue are shown. Spearman rho is calculated between our volumetric breast density estimates and Volpara™ from mammograms, as 0.9332. Multiple timing results are shown and indicate the variability of the reconstruction times with breast size and type but are ∼30 minutes for average size breast. The timing results with the 3D algorithm indicate ∼60 minute reconstruction times for pediatrics with two Nvidia GPUs. Characteristic variations of the glandular and ductal volumes over time are shown. The SOS from QT images are compared with literature values.The results of a multi-reader multi-case (MRMC) study are shown that compares the 3D UT with full field digital mammography and resulted in an average increase in ROC AUC of 10%. Orthopedic (knee) 3D UT images compared with MRI indicate regions of zero signal in the MRI are clearly displayed in the QT image.Explicit representation of the acoustic field is shown, indicating its 3D nature. An image of in vivo breast with the chest muscle is shown and speed of sound agreement with literature values are tabulated. Reference is made to a recently published paper validating pediatric imaging. ConclusionsThe high Spearman rho indicates a monotonic (not necessarily linear) relation between our method and industry gold standard Volpara™ density. The acoustic field verifies the need for 3D modeling. The MRMC study, the orthopedic images, breast density study, and references, all indicate the clinical utility of the SOS and reflection images. The QT image of the knee shows its ability to monitor tissue the MRI cannot. The included references and images herein indicate the proof of concept for 3D UT as a viable and valuable clinical adjunct in pediatric and orthopedic situations in addition to the breast imaging.

Customized optical chirality of vortex structured light through state and degree-of-polarization control

We show how both the ellipticity $\eta$ and degree of polarization $\textit{P}$ influences the extraordinary optical chirality properties of non-paraxial vortex beams. We find that, in stark contrast to paraxial optics and non-vortex modes, extremely rich and tuneable spatial distributions of optical chirality density can be produced by an optical vortex beam under tight focussing. We develop a theoretical description of how the optical chirality can be tailored for purpose by altering both the state $\eta$ and degree of polarization $\textit{P}$ of the input vortex mode, along with the magnitude and sign of optical orbital angular momentum via the pseudoscalar topological charge $\ell$. We expect that the results will have a significant role in both producing novel techniques and improving existing methods in chiral nano-optics and structured light photonics.

Exact and paraxial Airy propagation of relativistic electron plasma wavepackets

The different forms of propagation of relativistic electron plasma wavepackets in terms of Airy functions are studied. It is shown that exact solutions can be constructed showing accelerated propagations along coordinates transverse to the thermal speed cone coordinate. Similarly, Airy propagation is a solution for relativistic electron plasma waves in the paraxial approximation. This regime is considered in time domain, when paraxial approximation is considered for frequency, and in space domain, when paraxial approximation is considered for wavelength. In both different cases, the wavepackets remain structured in the transverse plane. Using these solutions we are able to define generalized and arbitrary Airy wavepackets for electron plasma waves, depending on arbitrary spectral functions. Examples of this construction are presented. These electron plasma Airy wavepackets are the most general solutions of this kind.

Beyond the ABCDs: A better matrix method for geometric optics by using homogeneous coordinates

Geometric optics is often described as tracing the paths of non-diffracting rays through an optical system. In the paraxial limit, ray traces can be calculated using ray transfer matrices (colloquially, ABCD matrices), which are 2 × 2 matrices acting on the height and slope of the rays. A known limitation of ray transfer matrices is that they only work for optical elements that are centered and normal to the optical axis. In this article, we provide an improved 3 × 3 matrix method for calculating paraxial ray traces of optical systems that is applicable to how these systems are actually arranged on the optical table: lenses and mirrors in any orientation or position (e.g., in laboratory coordinates), with the optical path zig-zagging along the table. Using projective duality, we also show how to directly image points through an optical system using a point transfer matrix calculated from the system's ray transfer matrix. We demonstrate the usefulness of these methods with several examples and discuss future directions to expand the applications of this technique.

The Optical Perspective of Hawking‐Like Radiation on a Curved Surface

AbstractElectrodynamics on curved surfaces, as a developed theory, has analogously become a new experimental verification of light transmission in general relativity. The thermal effect of an optical field on a specific 2D surface with constant Gaussian curvature is described in this paper. By considering the analogy between Schrodinger equation and Helmholtz equation under the paraxial approximation, the “quantized” momentum field is generated from the light transmitting on a curved surface by using the effective potential approach, and when decreasing the number of photons until is thought about, a temperature of Hawking‐like radiation is obtained. The creation process of radiation is also investigated, which is the scattering of light as it travels from a surface of positive curvature to a surface of negative curvature. The derived temperature of radiation field is also equivalent to the event horizon scattering explanation of Hawking radiation. The research may provide new perspectives for Hawking radiation and thermal lens.

4f‐Less Terahertz Optical Pattern Recognition Enabled by Complex Amplitude Modulating Metasurface Through Laser Direct Writing

AbstractOptical pattern recognition (OPR) has the advantage of single intensity detection ability for low‐cost terahertz (THz) systems of imaging or security checks. However, conventional 4f‐system‐based OPR is limited by the paraxial approximation and bulky device volumes for THz applications. Here, a full‐diffraction‐based 4f‐less OPR method is proposed using a single complex‐amplitude‐modulating metasurface, which is valid for systems with large Fresnel numbers. Moreover, a laser‐induced graphene technique is applied for processing the device. A 15 mm × 15 mm metasurface can be fabricated by one‐step laser writing in 34 s, indicating the potential of the proposed method in developing THz OPR systems with miniaturization, fast fabrication, and low‐cost.

Filamentation instability in the interaction of high-order Hermite cosh Gaussian laser beam and inhomogeneous plasmas

Based on high order paraxial theory and WKB approximation, filamentation instability in the interaction of high-order Hermite cosh Gaussian laser beam (HchG) and inhomogeneous plasma is studied. By changing the initial light intensity parameters and the non-uniformity of plasma density, the effect of different mode indices beams propagating in uniform and non-uniform plasmas on self-focusing filamentation structures is investigated. The results show that the HchG laser beam exhibits obvious self-focusing and filamentation phenomena along propagation distance in inhomogeneous plasma under the effect of the ponderomotive force. It is found that the mode index of HchG and plasma inhomogeneity have significant effects on filamentation instability, the higher order mode indices of HchG laser beam can strengthen filamentation instability, and plasma inhomogeneity also can enhance filamentation instability. The results provide a useful theoretical basis for effectively regulating the filamentation instability, increasing the peak laser power and designing narrower pulse width lasers.

Dimensional Analysis of Double-Track Microstructures in a Lithium Niobate Crystal Induced by Ultrashort Laser Pulses

Double-track microstructures were induced in the bulk of a z-cut lithium niobate crystal by 1030 nm 240 fs ultrashort laser pulses with a repetition rate of 100 kHz at variable pulse energies exceeding the critical Kerr self-focusing power. The microstructure topography was characterized by atomic force microscopy in piezoelectric response mode. The spatial positions of laser-induced modification regions inside lithium niobate in the case of laser beam propagation along the crystal optical axis can be directly predicted by simple analytical expressions under the paraxial approximation. A dimensional analysis of the morphology of the double-track structures revealed that both their length and width exhibit a monotonous increase with the pulse energy. The presented results have important implications for direct laser writing technology in crystalline dielectric birefringent materials, paving the way to control the high spatial resolution by means of effective energy deposition in modified regions.

Extreme Refractive-, Diffractive- and Hybrid-Hyperchromats: Minimizing the Equivalent Abbe Number of a Two-Lens System

This work provides a comprehensive analysis of the maximum chromatic axial split of two-element hyperchromats, with the distance between the two lenses being a key variable. Purely refractive and diffractive systems are considered, as well as hybrid layouts combining refractive and diffractive elements. In order to achieve extreme chromatic axial splitting and accordingly a minimum equivalent Abbe number for lens combinations, a three-step procedure was used. In the first paraxial step, purely optical quantities such as focal lengths of the lenses, inter-lens distances and dispersion properties of the lenses were investigated. In the second step, which also takes place in the paraxial domain, additional geometric boundary conditions such as the radii, diameters and thicknesses of the lenses are taken into account. The results of this step serve as an input for the final optimization using optical design software, which derives practical solutions for minimum equivalent Abbe numbers with diffraction-limited image quality. As a significant result, the comparison with directly cemented lens doublets shows that the introduction of a distance between the elements allows for a much stronger chromatic decomposition for refractive, diffractive and also hybrid combinations. Quantitatively, the minimum equivalent Abbe number for refractive systems is reduced from 2.5 (without spacing) to 1.79 (with spacing). For hybrid combinations, a corresponding reduction from 0.4 to 0.29 is achieved.

Numerical analysis of twisted ion-acoustic waves

Twisted ion-acoustic waves with finite orbital angular momentum (OAM) states are numerically investigated in a plasma containing electrons, positrons and dynamical ions. For this purpose, the well-known Vlasov–Poisson equations are solved together under the paraxial approximation. The perturbed quantities are assumed to follow the Laguerre–Gaussian type solutions and a generalized plasma response function for finite OAM states is derived. The generalized response function is numerically analyzed to investigate dispersion characteristics of the twisted ion-acoustic waves. Consequently, a computer code is developed which is based on the very famous numerical technique known as the Newton–Raphson method. The dependence of the (real) wave oscillation frequency and damping rate involving the twisted ion-acoustic waves is highlighted with different plasma parameters. It is found that the dynamics of ion-acoustic waves are strongly influenced by the twist parameter owing to OAM states. The results obtained in the present study are useful in the understanding of the particle transport and trapping phenomena.

Average paraxial power of a lens and visual acuity

To provide a solution for average paraxial lens power (ApP) of a lens. Orthogonal and oblique sections through a lens of power F were reduced to a paraxial representation of lens power followed by integration. Visual acuity was measured using lenses of different powers (cylinders of − 1.0 and − 2.0D) and axes, mean spherical equivalent (MSE) of S + C/2, ApP and a toric correction, with the order of correction randomised. A digital screen at 6 m was used on which a Landolt C with crowding bars was displayed for 0.3 s before vanishing. The general equation for a symmetrical lens of refractive index (n), radius of curvature R, in medium of refractive index n1, through orthogonal (theta) and oblique meridians (gamma) as a function of the angle of incidence (alpha) reduces for paraxial rays (alpha sim 0) to F_{n,R} left( {alpha ,theta ,gamma } right)left. right|_{alpha sim 0} = frac{{n - n_{1} }}{R}cos^{2} theta cos^{2} gamma. The average of this function is F_{n,R} left( {alpha ,theta ,gamma } right)left. right|_{alpha sim 0} = frac{{n - n_{1} }}{4R} providing a solution of frac{F}{4} for ApP.For central (p = 0.04), but not peripheral (p = 0.17) viewing, correction with ApP was associated with better visual acuity than a MSE across all tested refractive errors (p = 0.04). These findings suggest that frac{F}{4} may be a more inclusive representation of the average paraxial power of a cylindrical lens than the MSE.

Nonuniformly Filled Vortex Rings in Nonlinear Optics

A new type of long-lived solitary structures for paraxial optics with two circular polarizations of light in a homogeneous defocusing Kerr medium with an anomalous group velocity dispersion has been revealed numerically in the coupled nonlinear Schrödinger equations. A found hybrid three-dimensional soliton is a vortex ring against the background of a plane wave in one of the components, and the core of the vortex is filled with another component nonuniformly in azimuth angle. The existence of such quasistationary structures with a reduced symmetry in a certain parametric region is due to the saturation of the so-called sausage instability caused by the effective surface tension of a domain wall between two polarizations.

Twisted dust-acoustic waves in a self-gravitating dusty plasma with nonextensive q-distributed electrons and ions

We have investigated the twisted dust-acoustic waves (TDAWs) in an electrostatic self-gravitating dusty plasma whose electrons and ions are modelled by nonextensive q-distribution function while massive dust particles are Maxwellian distributed. A well-known kinetic theory is employed for this purpose where perturbed distribution function, electrostatic and gravitational potentials are expressed with Laguerre–Gauss functions. The governing equations of kinetic theory are solved together under paraxial approximations. The dispersion relations and instability growth rates are obtained for two situations; a) super-extensivity (q < 1) and b) sub-extensivity (q > 1). Significant modifications concerning the wave frequencies and growth rates are presented with respect to self-gravitation parameter, twist parameter, nonextensive parameter and streaming speed. It is observed that wave frequency and growth rate of TDAWs reduces in the presence of self-gravitating effects. Furthermore, the growth rates exhibit a significant enhancement in amplitude with the increase in twist parameter, q-parameter and streaming speed. Our present results may have applications in interstellar dust clouds and in the dusty plasma environments of Halley’s Comet.