Snell's Law Research Articles

Design of a centimeter-scale achromatic hybrid metalens with polarization insensitivity in the visible.

Achromatic metalenses formed using previous design methods face a compromise between diameter, numerical aperture, and working wave band. To address this problem, the authors coat the refractive lens with a dispersive metasurface and numerically demonstrate a centimeter-scale hybrid metalens for the visible band of 440-700 nm. By revisiting the generalized Snell law, a universal design of a chromatic aberration correction metasurface is proposed for a plano-convex lens with arbitrary surface curvatures. A highly precise semi-vector method is also presented for large-scale metasurface simulation. Benefiting from this, the reported hybrid metalens is carefully evaluated and exhibits 81% chromatic aberration suppression, polarization insensitivity, and broadband imaging capacity.

Elastic metasurface for flexural wave refraction based on acoustic black hole

This letter presents a thin plate structure with an acoustic black hole (ABH) sub-unit to reorient the flexural wave. Different from the previously reported flexural wave metasurface, ABH sub-units are introduced into thin plates in this work, which can control the group velocity of flexural waves and realize their efficient transmission. According to generalized Snell's law, the mechanism of phase shift of transmitted waves across subwavelength sub-units is theoretically revealed. An analysis of the ABH sub-units is established by the finite element method. The deflection and focusing effect of flexural waves are demonstrated. Furthermore, adjusting the black hole section can quickly obtain the transmission phase response in the range of 2π, and it can accurately predict the phase shift and amplitude of the transmitted wave. The results show that the ABH beam-plate structure can effectively control the propagation direction of flexural waves, which provides a modern design idea and method for the manipulation and energy harvesting of the flexural wave.

On the generalized Snell's law for the design of elastic metasurfaces

As the most popular mechanism, the generalized Snell's law has been applied extensively to design metasurfaces for wave manipulation. By modulating phase profile using metasurfaces, various novel wave transformations have been demonstrated. However, it has been found that the performance of these metasurfaces cannot be fully determined by the generalized Snell's law and in some cases, the wave fields appear to be contradictory to what predicted by the generalized Snell's law. In this work, a systematic numerical study is conducted to investigate the generalized Snell's law for elastic waves. The inherent assumptions of the generalized Snell's law are examined first followed by the study of the effects of various implementation issues on the performance of the metasurfaces. In particular, the fundamental mechanism for producing wave components that do not obey the generalized Snell's lay is identified and theoretically justified. Design guidelines for metasurfaces for improved performance are provided.

Variable discrete ordinates method for radiation transfer in concentric spherical variable index media

Radiation heat transfer in participating concentric spherical media with variable refractive index along the radial direction is solved by the variable discrete ordinates method. The radiative properties, including refractive index, are assumed to be constant in each volume element, such that a radiative ray travels along straight line in each volume element, and redirects by passing the interface of volume elements according to Snell's law of refraction, or specularly reflects back into the volume element. The radiative transfer equation in spherical coordinates system is solved for different number and direction of ordinates in each volume element. The performance of the present method is examined for two cases with medium's specified temperature and radiative equilibrium with continuous linear, continuous non-linear, and discontinuous profiles of refractive index. The accuracy of the present method is checked by comparing with benchmark solutions. For the cases where the exact solution is not available, the energy conservation criterion is used to validate the solution method. The effects of radiative properties, such as wall emissivities, optical thickness, single scattering albedo, and anisotropy factor are examined by some example problems.

Asymmetric acoustic metagrating enabled by parity-time symmetry

Passive parity-time (PT) symmetric metamaterials as a fundamental carrier to investigate non-Hermitian systems have attracted significant interest, but previous studies have rarely focused on the diffraction properties of the PT system, especially the diffraction effect in the metasurfaces with phase gradient modulation. Here, by merging two concepts of phase gradient and PT symmetry together, we design and study a non-Hermitian acoustic gradient metasurface (i.e., metagrating). We show that the PT symmetry breaks spatially the symmetry of reflected wave diffraction in the acoustic metagrating, making it an asymmetric mirror. In particular, at the PT-ensured exceptional point, for incident waves from one side, such metagrating behaves as a phase gradient metasurface, in which high-efficiency retroreflection governed generalized Snell's law occurs. For an incident wave from the opposite direction, such a metagrating is a common grating, whose diffraction in the reflection follows the common diffraction equation. Our work builds a bridge connecting the phase gradient and passive PT symmetric system, enriching the methods for wavefront manipulation and exhibiting potential applications in the design of multifunctional acoustic devices.

On minimal focal distance of a focused ultrasound probe for neuromodulation.

Focal distance is a key parameter for a focused ultrasound probe, especially in mouse brain stimulation where targets are right below the skull. A closed-form solution for the minimal focal distance with a given transducer size was derived in this study to facilitate precise focal spot alignment with targets in the mouse brain. The spherical profile corresponding to the minimal focal distance does not produce accurate focusing. An iterative algorithm based on Snell's law was introduced for lens profile calculation. With a suitable step size, an accurate lens profile can be obtained for the minimal focal distance.

An ocean acoustical ray-tracing tool based on Fermat's least time principle in real a environment

Recently the Underwater Acoustics Modelling and Sonar Prediction - SeaRider tool was developed by the Portuguese Naval Academy. Based on Fermat's least-time principle, 2D and 3D models of propagation of acoustic rays in the ocean were developed. These models consist of an algebraic differential system of equations. In this work we considered additionally the possibility of calculating the trajectory, addressing the occurrence of acoustic ray reflections either on the water surface or on the ocean floor. The reflections in the boundaries were assumed specular and were modelled based on Snell's law. We also considered the possibility of calculating the trajectories, taking into account the actual data of the speed of sound in the ocean as well as the bathymetry of the bottom. For this purpose, the 2D and 3D ocean-acoustic ray tracing developed can use existing data bases (GEBCO, NOAA, WOD18 and TEOS-10), extracting information onthe topography and climatology to compute the acoustic ray trajectories in a real environment. This tool allows us, for example, to identify positions of ecosystems subjects to the effects of different sources of noise. The developed algorithms were submitted to several numerical simulations and compared with other ocean-acoustic modelling programs.

Parity-protected anomalous diffraction in optical phase gradient metasurfaces

Optical phase gradient metasurfaces (PGMs) have provided unprecedented opportunities for arbitrarily controlling wave propagation via the generalized Snell's law (GSL). However, the whole picture of wave diffraction therein has not been clearly presented, particularly for the incident angles beyond the critical angle. Although a parity-dependent diffraction effect was found in acoustic metagratings, little is known about whether this effect holds true in typical optical PGMs. Here we demonstrate the universality of the parity-dependent diffraction effect by employing some optical PGMs with popular designs, such as all-dielectric and plasmonic meta-atoms. It is first shown that the parity in optical PGMs plays a significant role in determining the diffraction physics, producing a robust reversal effect of outgoing waves from the reflection to the transmission side. As an alternative degree of freedom in PGMs, the parity-dependent diffraction effect, together with the GSL, provides a complete theory to manipulate wave fields, further advancing various explorations in unique wave phenomena and promising applications.

Separating seismic diffractions by an improved Cook-distance singular value decomposition method

When seismic waves encounter abrupt points of stratum or lithology in the process of propagation, such as fault edges, pinch-out points, or the protrusion of unconformity surfaces, the Snell law will break down and many diffractions will be generated. However, the diffraction information is typically masked by strong reflections; thus, separating diffractions is one key issue for diffraction imaging. The traditional singular value decomposition (SVD) method with the ability for wavefield decomposition and reconstruction is useful in removing strong reflections of large singular values. However, the continuously changing characteristics of the singular value of steep-slope reflections make it difficult to choose a suitable parameter for separating diffractions. To solve the problem of singular value selection in diffractions reconstruction by the SVD method, we have developed a Cook-distance SVD method by mathematically least-squares measuring the contribution of every singular value. The Cook-distance SVD can amplify the difference between continuously changing singular values and easily find the singular values representing diffractions. Synthetic examples and field data applications demonstrate that our Cook-distance SVD method can avoid parameter testing and has good performance in diffraction separation and can highlight reservoir-involved small-scale geologic edges and scatterers.

Encoding terahertz metasurface reflectors based on geometrical phase modulation

Multi-dimension and multi-freedom modulation of polarization state based on the geometrical-phase periodic encoding metasurface has important application prospects. Here, terahertz metasurface composed of specially shaped metal pattern coded particles is proposed. When the coded particles are normally incident, the amplitude reflectivity of the terahertz wave is above 80% in a range of 0.50–1.80 THz. Combined with the Pancharatnam-Berry (P-B) phase theory, 8 kinds of coded particles are designed by rotating the angle of the designed unit. Three kinds of 1-bit, 2-bit, and 3-bit periodic encoding metasurfaces with different encoding sequences are used to manipulate the reflected terahertz waves splitting into multiple-beam with different deflection angles. In addition, both reflection characteristics (including amplitude, phase, and phase coverage) of all coded particles and the angle deflection of the designed 2-bit periodic metasurface are measured by normal incidence THz time-domain spectrometer and variable incident angle THz time-domain spectrometer, respectively. Based on generalized Snell law and experimental results, the reason for the discrepancy between theoretical value and experimental value is further analyzed, which can provide a reference for the reverse design of the coded metasurfaces to meet various practical needs.

Solar Angle Model for Daylight Redirection in Prismatic Panel

An advanced complex fenestration system can utilize uniform daylight. Nonetheless, an inefficient design would increase solar heat gain and indoor temperatures, besides uneven light distribution that would cause the "cave effect." Prismatic panels are widely used as complex fenestration systems, providing uniform daylight. This paper proposes a computational model that integrates optical principles like Snell's law with environmental variables and visualizes the performance of prismatic panels in terms of redirection angle while encountering the prism refractive index and geometry at the specified geographic location. The proposed model entails a prismatic panel as a daylight system for redirecting daylight. In contrast to detailed modeling needed for simulation in software programs like Radiance, this computational tool provides a more straightforward and efficient solution for the initial design of light redirection panels that rely on the principle of refraction and evaluate their annual performance based on the angle of deviation. The model's applicability has been demonstrated by utilizing various triangular prism design examples with diverse materials in Frankfurt and Helsinki.

Determination of complex refractive index using maximum deviation angle through prism

Light-absorbing materials are widely used, and their optical properties are an important factor. Snell's law does not hold in materials that partially absorb light. Hence, the optical path in refraction is calculated from Maxwell's law. We used the optical path to obtain the deviation angle when a light passes through a prism made of light-absorbing material. The deviation angle has a local maximum. The deviation angle near the maximum is sensitive to the real and imaginary parts of the refractive index. The local maximum deviation angle and its incident angle are used to determine the complex index of refraction. This measurement has the same advantages as measuring the minimum deviation angle of a transparent prism. The detection is easy. Then, it is necessary to determine the complex refractive index from the measured maximum deviation angle and its incident angle. These two angles are plotted parametrically by varying the real part of the refractive index under a fixed imaginary part. The similar curves are drawn under the fixed real parts. Each curve has a fixed value for the real or imaginary part of the refractive index. Drawings of these many curves are made prior to measurement. Then we select the four curves that are closest to the measured local maximum deviation angle and its incident angle. Four fixed values attached to each of the four curves determine the complex index of refraction. The method in this paper can easily determine the complex refractive index and can be used for material identification.

Exact artificial boundary condition and 1D time-domain method for seismic response analysis of layered saturated poroelastic half-space

This study proposed a one-dimensional (1D) time-domain finite element method (FEM) for seismic response analysis of a layered saturated soil half-space under obliquely incident plane waves, based on Biot's u-w wave equations. The spatially two-dimensional problem was transformed into an equivalent 1D time-domain problem along the depth according to Snell's law. An exact artificial boundary condition for saturated poroelastic media was developed to simulate the wave absorption and input effects of the truncated half-space. Using the displacement and stress continuity conditions between the adjacent saturated poroelastic-medium layers and combining all finite elements and the exact artificial boundary condition, the total dynamic equation for the spatially 1D problem was obtained. The 1D dynamic equation was solved using the FEM with explicit time integration. The proposed method was verified by analyzing two engineering sites in the time domain and a comparison with the frequency-domain transfer matrix method with fast Fourier transform.

Magneto-thermo-elastic plane waves in a rotating micropolar fiber-reinforced solid/liquid media under G–L theory for non-insulated boundary: reflection and transmission

The aim of this article is to investigate the reflection and transmission of waves in a rotating micropolar fiber-reinforced thermo-elastic solid and inviscid liquid media under magnetic fields by adopting Green-Lindsay thermo-elasticity for non-insulated boundary. Incident P-wave at the rotating solid and inviscid liquid media yielded four reflected waves via the solid; quasi-longitudinal displacement (qLD), quasi-transverse displacement (qTD), quasi-transverse microrotational (qTM) and quasithermal wave, while two waves are transmitted through the inviscid liquid medium; quasi-Longitudinal transmitted (qLT) wave and quasi-thermal transmitted (qTT) wave. Normal modes approach is used simultaneously with Snell's laws and Maxwell's equations governing electromagnetic fields in determining the solution. Amplitude ratios which correspond to reflected/transmitted waves are presented analytically and graphically. We observed that rotation, magneto-thermo-elastic parameters have varied and significant degree of effects to the modulations and amplitude ratios of the waves. Also, rotation produced increased behaviors to the transmission coefficient of qLT wave at all points in the liquid medium. The study should prove useful in understanding wave phenomena in a rotating micropolar fiber-reinforced magneto-thermo-elastic-acoustic machination fields and future works about the behaviors of seismic waves resulting in fluid interaction especially in the fields of geophysics, physics, optics, oil exploration, etc.

Polarization beam splitting in a Glan-Taylor prism based on dual effects of both birefringence and Goos-Hanchen shift

With the structure of biprism, polarization beam splitting in a Glan-Taylor polarizer was explored based on the birefringence and Goos-Hanchen shift. Due to the birefringence of the light in calcite crystal, the extraordinary light worked as the position calibration. As for the ordinary light, the propagated direction tilted noticeably due to the refraction as well as Goos-Hanchen effect at the prism-air interface of in the air gap. The Snell's law and stationary-phase approach were utilized for the calculation of the beam splitting between the two orthogonal polarization elements. By choosing appropriate incident angle and initial polarization, remarkable beam splitting was realized. With this configuration, the resolution with a magnitude of 10−5° and 10−3° was achieved for the response of the incident angle and polarization detection respectively.

DIRAC'S LINEARIZATION APPLIED TO THE FUNCTIONAL, WITH MATRIX ASPECT, FOR THE TIME OF FLIGHT OF LIGHT

In this paper, we adopt a matrix treatment to solve the variational problem that consists of determining the physical path traveled by light between two points in a medium whose refractive index depends on a spatial coordinate. The considered treatment begins with the trivial repetition of the expression of the value of the considered functional, repetition expressed in the form of a matrix. Next, we adopt the trick (of Dirac) originally used as part of the construction of the dynamic equation of relativistic quantum mechanics, which allows us to rewrite the (now) matrix integrand in the expression of the value of the functional in terms of the sum of two (non-diagonal) matrices brought externally to the problem, which are determined based on some requirements. As a result of this development, we obtain two equivalent versions of Snell's law.

The Beginnings of Optimization: Snell's Law and Fermat's Principle

The study provides a brief description of the basic concepts of the theory and its history. We considered Fermat's principle of least time, proposed to substantiate Snell's experimentally established law of refraction of light. This example can be applied as the beginning of the formation of any scientific discipline. The main purpose is to show the benefits of applying three components: experiments, bold hypotheses and mathematical models.

Ray Optics for Gliders.

Control of self-propelled particles is central to the development of many microrobotic technologies, from dynamically reconfigurable materials to advanced lab-on-a-chip systems. However, there are few physical principles by which particle trajectories can be specified and can be used to generate a wide range of behaviors. Within the field of ray optics, a single principle for controlling the trajectory of light─Snell's law─yields an intuitive framework for engineering a broad range of devices, from microscopes to cameras and telescopes. Here we show that the motion of self-propelled particles gliding across a resistance discontinuity is governed by a variant of Snell's law, and develop a corresponding ray optics for gliders. Just as the ratio of refractive indexes sets the path of a light ray, the ratio of resistance coefficients is shown to determine the trajectories of gliders. The magnitude of refraction depends on the glider's shape, in particular its aspect ratio, which serves as an analogue to the wavelength of light. This enables the demixing of a polymorphic, many-shaped, beam of gliders into distinct monomorphic, single-shaped, beams through a friction prism. In turn, beams of monomorphic gliders can be focused by spherical and gradient friction lenses. Alternatively, the critical angle for total internal reflection can be used to create shape-selective glider traps. Overall our work suggests that furthering the analogy between light and microscopic gliders may be used for sorting, concentrating, and analyzing self-propelled particles.

Switchable multifunctional knob elastic metasurface for arbitrary modulation of flexural waves

Manipulating elastic waves propagating in the desired manner has attracted substantial attention, and the metasurfaces offer the important ideas to achieve this goal. In particular, it will be more advantageous if wave function switching can be achieved without changing the microstructure of the metasurface. In the paper, a knob elastic metasurface is designed to realize the regulation of flexural wave covering the 0–2π range of phase shift by rotating the angle of the knob. When the metasurface is in the turn-on state, the angles of the knob at different positions can be adjusted reasonably though the generalized Snell's law to achieve several phenomena of anomalous refraction, beam focus, and non-paraxial self-acceleration of flexural wave in a wide frequency. And when the metasurface is placed in the turn-off state, the total reflection of the flexural wave can be realized, and then bidirectional vibration isolation can be achieved.

Steering of flexural wave propagation in tunable magnetorheological elastomers metasurface by modulating magnetic field

Metasurfaces have remarkable power for steering flexural waves, and most of the existing studies for phase modulation of the metasurface concentrate on changing the geometrical parameters of the structure. However, when the external conditions change, these parameters cannot be adjusted in real-time, which is not convenient for engineering applications. Therefore, in this paper, a tunable magnetorheological elastomers (MREs) metasurface composed of MREs and three-dimensional (3D) printing material polylactic acid (PLA) is proposed to steer the propagation of incident flexural waves arbitrarily. The phase shift in a full 2π span can be achieved by manipulating the applied magnetic field to tune the modulus of the MREs. Then, based on the Generalized Snell's Law, MREs metasurfaces with different refraction angles are designed by adjusting the applied magnetic field, including positive refraction and negative refraction. In addition, wave focusing and self-bending beam effects are also investigated. Different focal spots and self-bending beam paths can be realized by adjusting the applied magnetic field. Finally, a sample of the tunable MREs metasurface is fabricated by 3D printing technology composed of host plate and MREs units to demonstrate the focusing effect experimentally. The results show that the designed tunable MREs metasurface only requires a small magnetic field to achieve deflecting of the flexural waves (the maximum magnetic field is 66.93 mT), which is easy to implement in practical engineering. The simulated and experimental results of the elastic wave focusing effect are in good agreement. The proposed metasurface provides a guiding design for elastic wave modulation.