Snell's Law Research Articles

Three-dimensional ray tracing in P-wave azimuthal anisotropic media

Summary A new 3-D ray tracing method is developed for P-wave azimuthal anisotropic (AAN) media. We assume anisotropic media with hexagonal symmetry and take advantage of the property that the AAN symmetry axis, the phase velocity vector, and the group velocity vector are located in the same plane. The 3-D ray tracing method that combines the pseudo-bending technique and Snell's law is improved for the AAN media. We compute isotropic (ISO) and AAN rays in synthetic models and an actual 3-D P-wave AAN model of the East Japan subduction zone. The accuracy of our ray tracing code is evaluated by comparing the ray-path and travel-time differences between the ISO and AAN rays. Our results show that the AAN rays in each model bend in the right direction and satisfy Fermat's principle, so the theory and approximations adopted in the calculations are reasonable. For long rays (>350 km), the ray-path difference between the ISO and AAN rays is > 20 km, and the travel-time difference is > 0.1 s, suggesting that it is necessary and important to take azimuthal anisotropy into account in the 3-D ray tracing.

3D Multiview Holographic Display with Wide Field of View Based on Metasurface

Abstract3D display technology offers a more realistic visual experience by allowing users to perceive depth and spatial awareness, enhancing interactivity and immersion. Among various approaches, multiview display technology constructs 3D images by offering multiple smooth views along the horizontal plane. Metasurfaces, with ultra‐thin physical structures and flexible functional designs, are positioned to make significant contributions to the evolution of 3D display technologies. Here, a 3D multiview holographic display system based on metasurfaces is proposed. Using generalized Snell's law, a spatially multiplexed metasurface is designed with nine views and achieves 3D display by cascading the metasurface with a spatial light modulator (SLM). By projecting holograms onto different areas of the metasurface via the SLM, corresponding reconstructed images can be observed from specific designed directions. Thanks to the large numerical aperture (NA) of the metasurface, the system successfully achieves nine views within a 120° field of view (FOV, with a theoretical maximum of 180°), validated experimentally. This system promises potential applications in VR/AR display, surgical navigation, geographic information system (GIS), and other fields.

Acoustic metagrating focusing and Bessel vortexes

Acoustic focusing and Bessel vortexes have great potential in medical ultrasound, particle trapping, and information processing. Based on the generalized Snell's law (GSL), metasurface focusing and Bessel vortexes were achieved by using in-plane phase profiles to shape wavefronts. Recent developments in acoustic metagratings (AMs) have demonstrated an extension of the GSL capable of switching transmitted and reflected vortexes that are determined by the parity of the number of wave propagation trips. However, these metagratings were designed with a certain one-dimensional phase gradient along the azimuthal direction, and the propagation of vortexes were generally fixed into cylindrical waveguides owing to energy divergence. The propagation and manipulation of acoustic vortexes in three-dimensional (3D) free space, caused by AMs with two-dimensional (2D) aperiodic phase gradients, still pose a great challenge. Here, we experimentally demonstrate two types of switchable acoustic lenses with focusing and Bessel vortexes. Based on the GSL extension, by separating and attaching 2D dual-layer aperiodic AMs in both lenses, the switch between the reflected focusing vortex and transmitted focusing/Bessel vortex with the same focus length and topological charge in 3D free space can be observed. The designed dual-layer AMs can realize the short-range and long-range focusing vortexes in 3D free space and also have the advantage of convenient function switching, which may pave the way for designing switchable focusing vortex lenses with practical applications.

Ray and energy-flux velocities at a contact of two viscoelastic anisotropic materials

Summary A g*-Hamiltonian method for tracing real rays was developed that can handle cusps and triplication of quasi-shear wave in a general viscoelastic anisotropic medium. We demonstrate that the g*-Hamiltonian method can produce homogeneous ray-velocity vectors (with parallel real and imaginary parts) and the slowness vectors of reflected and transmitted waves at the interface based on the real Snell's law (RSL), which constrains only the continuity of the real parts of the slowness vectors, or the real slowness direction (RSD) method, which ignores the inhomogeneous component of the slowness vector. These methods are based on the characteristic lines with different Hamiltonians. Our research indicates that these methods are limited to pre-critical incidence ranges. Moreover, we derived a complex energy velocity vector (energy flux velocity) expression, which is always homogenous. We found that directions of corresponding energy velocity calculated with complex Snell's law (CSL) at a contact of two viscoelastic anisotropic materials well match the solutions of the RSL and RSD methods for all R/T waves except post-critical incidence, whereas the RSL and RSD methods fail to obtain homogenous ray velocities. The RSL and RSD methods result in discrepancies in the ray quality factor, R/T coefficients, and energy ratios, especially for post-critical incidence. However, the exact critical angle determined by the RSD method approximates the ‘critical’ angle for anelastic/inhomogeneous waves, which was a previous challenge. Our calculations suggest that the energy velocity and energy quality factor obtained with the CSL method can be used for real ray tracing at the interface of viscoelastic anisotropic media, and the complex energy flux velocity vector is always exactly homogeneous. For the post-critical incidence, the RSL and RSD methods fail because the ray quality factor drastically changes from the infinite down to near 2, which contradicts homogeneous ray velocity in elastic anisotropic materials. However, the energy quality factor for the elastic-anisotropic material is all infinite, even for post-critical incidence, which is correct. We also show that the CSL method has the same efficiency as the RSD method.

Application of Single-Frequency Arbitrarily Directed Split Beam Metasurface Reflector in Refractive Index Measurements.

We present a sensor that utilizes a modified single-frequency split beam metasurface reflector to measure the refractive index of materials ranging from one to three. Samples are placed into a cavity between a PCB-etched dielectric and a reflecting ground plane. It is illuminated using a 10.525 GHz free-space transmit horn with reflecting angles measured by sweeping a receiving horn around the setup. Predetermined changes in measured angles determined through simulations will coincide with the material's index. The sensor is designed using the Fourier transform method of array synthesis and verified with FEM simulations. The device is fabricated using PCB milling and 3D printing. The quality of the sensor is verified by characterizing 3D printed dielectric samples of various infill percentages and thicknesses. Without changing the metasurface design, the sensing performance is extended to accommodate larger sample thicknesses by including a modified 3D printed fish-eye lens mounted in front of the beam splitter; this helps to exaggerate changes in reflected angles for those samples. All the methods presented are in agreement and verified with single-frequency index measurements using Snell's law. This device may offer a viable alternative to traditional index characterization methods, which often require large sample sizes for single-frequency measurements or expensive equipment for multi-frequency parameter extraction.

Construction and experimental verification of wave velocity model for source location in goaf overlying rock strata

The geological structure of the goaf overlying rock is complex, a consequence of coal mining that has modified the original stratified structure of the sedimentary strata. To enhance the accuracy of microseismic source location in such intricate geological formations, a wave velocity model for the “three zones” goaf was constructed based on natural divisions within the strata using Snell's law and assuming a homogeneous medium. The model took into account the effects of rock deformation and fracture development, enabling the derivation of formulas for microseismic wave propagation path and travel time calculation. Additionally, the concept of equivalent wave velocity was defined. An indoor simulation test using similar materials was conducted to establish a geological model of the goaf. By comparing the errors between the theoretical and measured values of equivalent wave velocity, assessing the locating effects before and after implementing the wave velocity model of the goaf, and verifying the feasibility of the model, it was demonstrated that establishing a wave velocity model based on the characteristics of the strata structure was crucial for improving the accuracy of the microseismic source location. Notably, as the propagation path of microseismic waves in the goaf increased, the equivalent wave velocity decreased. The wave velocity structure in the goaf exhibited nonuniformity, with the relative error between the theoretical and measured values of equivalent wave velocity being limited to 10 %. The incorporation of this established wave velocity model into the location method resulted in a substantial 58.57 % increase in locating accuracy.

Guiding near-source elastic waves in a semi-infinite medium.

In this work, we propose elastic metamaterials with phase discontinuities to steer the propagation of near-source bulk waves in a semi-infinite elastic medium. Our design exploits an array of embedded subwavelength resonators with tailored masses to attain a complete phase shift spanning [Formula: see text]. This phase control allows for diverse wave functionalities, such as directional refraction and energy focusing. Through the use of dispersion diagrams and the generalized Snell's law, along with a multiple scattering formulation, we analytically demonstrate the effectiveness of our design in achieving the desired wavefront manipulation. The proposed design has the potential to advance the field of guiding elastic waves using metamaterials and find practical applications in areas such as isolating ground-borne vibrations in densely urbanized regions and energy harvesting. This article is part of the theme issue 'Current developments in elastic and acoustic metamaterials science (Part 1)'.

Refraction/reflection reversal in two-dimensional acoustic metagratings

Unlike acoustic metasurfaces that rely solely on phase gradients, acoustic metagratings (AMs) operate based on both phase gradients and grating diffraction, thus further extending the generalized Snell's law (GSL). In particular, AMs can achieve reversal of refraction and reflection based on the parity of the number of wave propagations inside the AMs. So far, discussions of this GSL extension have largely been applied to one-dimensional periodic AMs, while the designs of two-dimensional (2D) periodic AMs and their performance in three-dimensional (3D) space have been quite limited. Here, we study the GSL extension in 3D space and experimentally demonstrate a series of functional 2D periodic AMs. The designed AMs can achieve sound refraction/reflection under any incidence angle in 3D space, without restrictions to certain critical ranges; adjusting incident angles only enables the reversal of refraction and reflection. Additionally, we demonstrate two types of dual-layer sound lenses based on two AMs, whose reversal of refraction and reflection can be realized by simply attaching or separating the two AMs. Our work paves the way to complex 3D wavefront manipulation of AMs, which may find potential use in practical acoustic devices.

To a dispersion law of an electromagnetic wave in a medium having a double anisotropy

Dispersion relations for a transverse electromagnetic wave in a hypothetical medium having a low crystalline symmetry and double anisotropy are formulated phenomenological based on the general concepts of Maxwell's equations. Formalism for electrical and magnetic ordering of a medium is represented by tensor coefficients of the second rank in a general coordinate system corresponding to an external problem of incidence, reflection and refraction of a transverse wave at the interface. Based on the obtained dispersion law, the Snell law of refraction for the wave vector is displayed, as well as, some special cases for a section of a wave surface and a ray surface in the plane of incidence. Following Fresnel approach as usual, two characteristic types of linear polarization of an electric induction vector of a propagating wave are considered for a wave entering a double anisotropic medium at an arbitrary angle to the interface. Based on a character of a cross-section of a wave surface and of a ray surface for the plane of incidence, the degree of influence of anisotropy on a propagating field is discussed in terms of external/internal conic refraction of ordinary/extraordinary waves in a transparent medium. Following to initially general form of the permittivity ε^ and the magnetic permeability μ^ tensors some particular cases of a more favorable choice of coordinate system are analyzed to decrease a number of non-zero components in mentioned tensors, as well as to decrease in the number of elements in vector relations.

Angular Deviations, Lateral Displacements, and Transversal Symmetry Breaking: An Analytical Tutorial

The study of a Gaussian laser beam interacting with an optical prism, both through reflection and transmission, provides a technical tool to examine deviations from the optical path as dictated by geometric optics principles. These deviations encompass alterations in the reflection and refraction angles, as predicted by the reflection and Snell laws, along with lateral displacements in the case of total internal reflection. The analysis of the angular distributions of both the reflected and transmitted beams allows us to understand the underlying causes of these deviations and displacements, and it aids in formulating analytic expressions that are capable of characterizing these optical phenomena. The study also extends to the examination of transverse symmetry breaking, which is a phenomenon observed in the laser beam as it traverses the oblique interface of the prism. It is essential to underscore that this analytical overview does not strive to function as an exhaustive literature review of these optical phenomena. Instead, its primary objective is to provide a comprehensive and self-referential treatment, as well as give universal analytical formulas intended to facilitate experimental validations or applications in various technological contexts.

Experimental Realization of a One‐Directional Broadband Transmissive Cloak in Microwaves

AbstractHiding an isolated object in free space using a transmissive invisibility cloak has become a significant research area, propelled by advancements in metamaterials and transformation optics over the past decade. Despite the availability of various simplified methods for implementing transmissive cloaks, issues such as impedance mismatches and narrow working bandwidths often arise, posing challenges. Achieving a broadband transmissive cloak in free space has proven to be particularly arduous. This study presents a near‐perfect one‐directional broadband transmissive cloak constructed from multilayer metasurfaces of arbitrary shapes, showcasing superior performance across a broadband frequency range. The phase distribution of the metasurfaces and the efficacy of the transmissive cloak are assessed using the generalized Snell's law. An experimental near‐perfect broadband transmissive cloak is successfully demonstrated to operate within the frequency range of 8.5 to 11.2 GHz. This study contributes to reducing the density and mass of cloaks, thereby facilitating the expansion of cloaking capabilities in various directions and across different frequency bands.

Spatial Resolution Requirements for FDTD Modeling of Geoelectric Fields

AbstractTo ensure the robustness of both civilian and military infrastructure, it is important to protect electric power grids, smart grids, and other electrotechnologies from known and possibly as‐of‐yet unknown space weather hazards. Space weather can generate intense geoelectric fields at the surface of the Earth, as well as large voltage gradients across long distances of the Earth. These voltage gradients can lead to geomagnetically induced currents (GICs), which are known to produce hazards to electric power grids. The finite‐difference time‐domain (FDTD) method is a powerful and versatile method that has already been applied to the study of geoelectric fields. The advantages of FDTD over other methods are that it can account for more geometrical complexities and realistic time waveforms and that it directly solves for geoelectric fields. Snell's Law predicts that any electromagnetic waves incident on the ground should essentially propagate straight downwards into the low resistivity ground. For this reason, vertical FDTD grid resolutions of 1/3 of a skin depth were usually chosen, while the horizontal grid resolution was relaxed. We find, however, that there is another important consideration for choosing an FDTD grid resolution applied to real‐world scenarios: localized field variations due to currents generated by ground features. It turns out the grid resolution requirements are much stricter when taking this physics into account.

Acoustic Metagrating Holograms.

Metasurface holograms represent a common category of metasurface devices that utilize in-plane phase gradients to shape wavefronts, forming holographic images through the application of the generalized Snell's law (GSL). While conventional metasurfaces focus solely on phase gradients, metagratings, which incorporate higher-order wave diffraction, further expand the GSL's generality. Recent advances in certain acoustic metagratings demonstrate an updated GSL extension capable of reversing anomalous transmission and reflection, whose reversal is characterized by the parity of the number of wave propagation trips through the metagrating. However, the current extension of GSL remains limited to 1D metagratings, unable to access 2D holographic images in 3D spaces. Here, the GSL extension to 2D metagratings for manipulating waves within 3D spaces is investigated. Through this analysis, a series of acoustic metagrating holograms is experimentally demonstrated. These holographic images exhibit the unique ability to switch between transmission and reflection types independently. This study introduces an additional dimension to modern holography design and metasurface wavefront manipulation.

Selective Enhancement of Viewing Angle Characteristics and Light Extraction Efficiency of Blue Thermally Activated Delayed Fluorescence Organic Light-Emitting Diodes through an Easily Tailorable Si3N4 Nanofiber Structure.

We selectively improved the viewing angle characteristics and light extraction efficiency of blue thermally activated delayed fluorescence (TADF) organic light-emitting diodes (OLEDs) by tailoring a nanofiber-shaped Si3N4 layer, which was used as an internal scattering layer. The diameter of the polymer nanofibers changed according to the mass ratio of polyacrylonitrile (PAN) and poly(methyl methacrylate) (PMMA) in the polymer solution for electrospinning. The Si3N4 nanofiber (SNF) structure was fabricated by etching an Si3N4 film using the PAN/PMMA nanofiber as a mask, making it easier to adjust parameters, such as the diameter, open ratio, and height, even though the SNF structure was randomly shaped. The SNF structures exhibited lower transmittance and higher haze with increasing diameter, showing little correlation with their height. However, all the structures demonstrated a total transmittance of over 80%. Finally, by applying the SNF structures to the blue TADF OLEDs, the external quantum efficiency was increased by 15.6%. In addition, the current and power efficiencies were enhanced by 23.0% and 25.6%, respectively. The internal light-extracting SNF structure also exhibited a synergistic effect with the external light-extracting structure. Furthermore, when the viewing angle changed from 0° to 60°, the peak wavelength and CIE coordinate shift decreased from 20 to 6 nm and from 0.0561 to 0.0243, respectively. These trends were explained by the application of Snell's law to the light path and were ultimately validated through finite-difference time-domain simulations.

Asymptotic Antipodal Solutions as the Limit of Elliptic Relative Equilibria for the Two- and n-Body Problems in the Two-Dimensional Conformal Sphere

We consider the two- and n-body problems on the two-dimensional conformal sphere MR2, with a radius R>0. We employ an alternative potential free of singularities at antipodal points. We study the limit of relative equilibria under the SO(2) symmetry; we examine the specific conditions under which a pair of positive-mass particles, situated at antipodal points, can maintain a state of relative equilibrium as they traverse along a geodesic. It is identified that, under an appropriate radius–mass relationship, these particles experience an unrestricted and free movement in alignment with the geodesic of the canonical Killing vector field in MR2. An even number of bodies with pairwise conjugated positions, arranged in a regular n-gon, all with the same mass m, move freely on a geodesic with suitable velocities, where this geodesic motion behaves like a relative equilibrium. Also, a center of mass formula is included. A relation is found for the relative equilibrium in the two-body problem in the sphere similar to the Snell law.

Scattering of plane waves from stress-free boundary surface of a microstretch elastic solid half-space containing voids

Abstract Scattering phenomena of plane coupled waves from stress-free boundary surface of a microstretch elastic solid half-space containing uniform distribution of voids have been investigated. Using appropriate boundary conditions and Snell's law, the reflection coefficients corresponding to various reflected \textcolor{red}{sets of coupled} waves and their corresponding energy ratios have been presented. One of the boundary conditions leads to the vanishing of a cross-coupling parameter between curvature tensor and voids. Numerical computations have been carried out for a specific model and the variation of all the reflection coefficients and \textcolor{red}{their} corresponding energy ratios is depicted graphically against the angle of incidence. Some special cases have been reduced from the present model and discussed.

Mirrored transformation optics.

A mirrored transformation optics (MTO) approach is presented to overcome the material mismatch in transformation optics. It makes good use of the reflection behavior and introduces a mirrored medium to offset the phase discontinuities. Using this approach, a high-performance planar focusing lens of transmission type is designed, which has a larger concentration ratio than the other focusing lens obtained by the generalized Snell's law. The MTO will not change any functionality of the original lens and has promising potential applications in imaging and light energy harvesting.

Complex Refraction Metasurfaces for Locally Enhanced Propagation Through Opaque Media

AbstractMetasurfaces with linear phase gradients can redirect light beams. Controlling both phase and amplitude of a metasurface is proposed to extend Snell's law to the realm of complex angles, enabling a non‐decaying transmission through opaque media with complex refractive indices. This leads to the discovery of non‐diffracting and non‐decaying solutions to the wave equation in opaque media, in the form of generalized cosine and Bessel‐beams with a complex argument. While these solutions present nonphysical exponentially growing side tails, this is addressed via a windowing process, removing the side tails of the field profile while preserving significant transmission enhancement through an opaque slab on a small localized region. Such refined beam profiles may be synthesized by passive metasurfaces with phase and amplitude control at the opaque material's interface. The findings, derived from rigorous solutions of the wave equation, promise new insights and enhanced control of light propagation in opaque media.

Numerical Optimization of Metasurface Cells for Acoustic Reflection

Metamaterials and metasurfaces disclosed new degrees of freedom in controlling the acoustic field. Exploiting the generalized Snell law and the generalized law of reflection, the assembly of subwavelength unit cells is able to achieve extraordinary refraction and reflection by means of a controlled phase delay introduced in the field by the treated boundaries. The space-coiling design is one of the most powerful for cells in this metadevice class, providing effective low-thickness metasurfaces. However, space-coiling suffers from a narrow frequency operating range due to the intrinsic connection between the design operating wavelength and the characteristic dimensions of the metasurface. This work defines a procedure based on numerical optimization for designing space-coiling cells for modular acoustic metasurfaces, extending the frequency range in which the metasurface is effective. The set comprises eight different unit cells, each introducing a tailored phase shift in the reflected field that can be arranged to produce the desired acoustic effect. The broadband design is obtained by minimizing the dependency on the operating frequency of phase delay introduced by the cells, keeping the overall thickness below a quarter of the design wavelength. Results are shown for the benchmark problem of a metasurface modifying the reflection angle from a boundary.

Generalization of Snell's Law for the propagation of acoustic waves in elliptically anisotropic media

<abstract> <p>In seismic data processing, both in inversion (Inverse Processing) and modeling (Direct Processing), it is essential to consider anisotropy to unravel the geological structure of the subsoil. Besides, in most cases, the macroscopic model of anisotropy in 2D seismic surveys is elliptical and weak, with ratios of anisotropy close to one. Therefore, it is crucial to have at disposal the analytical formulas for acoustic wave propagation in elliptical anisotropic media. We presented the generalization of the Snell's Law for the case of acoustic wave propagation in elliptically anisotropic media. The generalization of the Snell's Law for acoustic anisotropic media had different applications in digital processing, raytracing, and acoustic inversion to properly consider elliptical anisotropy.</p> </abstract>