Snell's Law Research Articles

Colloidal Photonic Crystals Containing Silver Nanoparticles with Tunable Structural Colors

Polystyrene (PS) colloidal photonic crystals (CPhCs) containing silver nanoparticles (AgNPs) present tunable structural colors. PS CPhC color films containing a high concentration of AgNPs were prepared using self-assembly process through gravitational sedimentation method. High-concentration AgNPs were deposited on the bottom of the substrate and acted as black materials to absorb background and scattering light. Brilliant structural colors were enhanced because of the absorption of incoherent scattering light, and color saturation was increased by the distribution AgNPs on the PS CPhC surfaces. The vivid iridescent structural colors of AgNPs/PS hybrid CPhC films were based on Bragg diffraction and backward scattering absorption using AgNPs. The photonic stop band of PS CPhCs and AgNPs/PS hybrid CPhCs were measured by UV–visible reflection spectrometry and calculated based on the Bragg–Snell law. In addition, the tunable structural colors of AgNPs/PS hybrid CPhC films were evaluated using color measurements according to the Commission International d’Eclairage standard colorimetric system. This paper presents a simple and inexpensive method to produce tunable structural colors for numerous applications, such as textile fabrics, bionic colors, catalysis, and paints.

Refraction in electrically thin inhomogeneous media.

This work presents a new formulation for refraction from flat electrically thin lenses and reflectors comprised of inhomogeneous material. Inhomogeneous electrically thin flat lenses and reflectors cannot make use of the Snell law since this classical formulation works solely at interfaces of planar homogeneous media. The refraction of a perpendicularly incident plane wave at a planar interface is physically explained through the phase advance of the rays within the medium. The Huygens principle is then used to construct the refracted wavefront. The formulation is validated using numerical full wave simulation for several examples where the refractive angle is predicted with good accuracy. Furthermore, the formulation gives a physical insight of the phenomenon of refraction from electrically thin inhomogeneous media.

Representation of complex geological velocity model based on level set method

The difficulties in the expression of geological velocity model are the accurate description of complex geological phenomena and the effective support for the forward and inverse applications. We presented a novel representation method for describing velocity models. Velocity discontinuity was embedded into model in the form of zero-value contour of an implicit function. Multi-copy strategy was to properly handle the contradictory interpolation of velocity value at the boundary of discontinuities. The representation method can be applied to describe very complicated velocity models such as the faulted interface with non-manifold nature and multiple value surfaces. In applications based on the ray theory,signed distance field and velocity discontinuity binary tree enable fast access to velocity information of arbitrary point and easy detection of ray-interface intersection. The model can make the ray propagate positively and honor Snell's law strictly when facing velocity interface and thus improve the accuracy of ray path. Two groups of ray tracing examples verify the effectiveness of the method.

대기 효과를 고려한 전파 전달의 수학적 모델링 및 응용

본 논문에서는 전파 전달 특성을 결정하는 굴절 계수(refractive index)가 고도에 따른 함수라는 사실을 이용하여, 대기권에서의 전파 전달을 수학적으로 표현하는 방법을 제안하였다. 제안한 방법은 전파가 다른 굴절 계수를 가지는 매질로 입사될 때 입사각과 굴절각의 관계를 나타내는 스넬의 법칙(Snell's law)으로부터 유도할 수 있다. 모의실험에서는 굴절 계수의 제곱을 고도에 대한 일차 다항식으로 모델링함으로써 전파 전달을 도식화 하고, 여러 개의 레이다를 이용하여 각각의 레이다에서 측정한 발사각도 정보를 통해 전파 전달 특성을 나타내는 변수들, 즉, 모델링한 다항식의 계수들을 추정할 수 있음을 보인다. In this paper, we propose a method that represents a mathematical form of wave propagation by using the fact the refractive index determining wave propagation characteristic is a function of altitude. Proposed method uses Snell's law that expresses relationship between incident angle and refraction angle when incident wave passes medium having a different refractive index. We present the simulation results about wave propagation by setting the square of refractive index in the form of the polynomial for altitude and show that it is possible to estimate the coefficients of the polynomial through the angle information from vertical axis of multiple radar systems.

Retrieval of refractive index fields in two-dimensional gradient-index elements from external deflectometry data.

In a previous work, we presented a numerical method for retrieving inhomogeneous refractive index fields in rectangular gradient-index elements from boundary positions and internal boundary slopes associated with a set of interrogating probe beams that transit the medium. The present work extends this method to external boundary beam slopes without knowledge of the refractive index along the surface of the optical element, requiring minimal additional information (outside of beam position and slope data) such as a single known index point inside the medium. The inverse problem is cast as a linear algebraic system describing the deflection of probe beams inside the optical material, and an iterative inversion algorithm is used to generate an index field that produces the boundary value data. By incorporating Snell's law into the system equation through surface values derived from tentative reconstructions of the refractive index, we show in simulation that a series of inversion cycles applied to the system equation accurately recovers the index profile used to generate the test data.

Dynamic Control of the Extraordinary Optical Scattering in Semicontinuous 2D Metamaterials

Coherent control of the reflection and refraction in semicontinuous metasurfaces is realized. Both the reflection and refraction efficiency of a single-layered metasurface can be greatly improved with this method. By controlling the relative phase between the signal beam and the control beam, the Snell's law can be dynamically manipulated. This method may open possibilities to the practical application of metasurfaces.

Three-dimensional ray-tracing model for the study of advanced refractive errors in keratoconus.

We propose a numerical three-dimensional (3D) ray-tracing model for the analysis of advanced corneal refractive errors. The 3D modeling was based on measured corneal elevation data by means of Scheimpflug photography. A mathematical description of the measured corneal surfaces from a keratoconus (KC) patient was used for the 3D ray tracing, based on Snell's law of refraction. A model of a commercial intraocular lens (IOL) was included in the analysis. By modifying the posterior IOL surface, it was shown that the imaging quality could be significantly improved. The RMS values were reduced by approximately 50% close to the retina, both for on- and off-axis geometries. The 3D ray-tracing model can constitute a basis for simulation of customized IOLs that are able to correct the advanced, irregular refractive errors in KC.

Direct design of aspherical lenses for extended non-Lambertian sources in three-dimensional rotational geometry.

Illumination design used to redistribute the spatial energy distribution of light source is a key technique in lighting applications. However, there is still no effective illumination design method for extended sources, especially for extended non-Lambertian sources. What we present here is to our knowledge the first direct method for extended non-Lambertian sources in three-dimensional (3D) rotational geometry. In this method, both meridional rays and skew rays of the extended source are taken into account to tailor the lens profile in the meridional plane. A set of edge rays and interior rays emitted from the extended source which will take a given direction after the refraction of the aspherical lens are found by the Snell's law, and the output intensity at this direction is then calculated to be the integral of the luminance function of the outgoing rays at this direction. This direct method is effective for both extended non-Lambertian sources and extended Lambertian sources in 3D rotational symmetry, and can directly find a solution to the prescribed design problem without cumbersome iterative illuminance compensation. Two examples are presented to demonstrate the effectiveness of the proposed method in terms of performance and capacity for tackling complex designs.

Geometry of Refractions and Reflections Through a Biperiodic Medium

The behavior of light rays obeying Snell's law in a medium made up of two materials with different refractive indices and which are arranged in a periodic chessboard pattern is described. The analysis is in some ways analogous to the study of rational billiards and uses a return map on one surface to prove, among other things, that the number of angles with which any individual ray intersects the lattice is bounded and that if the ratio of refractive indices is large enough, then the dynamics can be described by interval exchange maps.

High-efficiency wavefront control with based on helical metamaterials

Metamaterials or metasurfaces have been widely studied to manipulate the propagation of light by controlling the wavefront. In previous work, more and more structures were designed to study the reflected or the transmitted light. However, as far as we know, it is rarely reported how to efficiency tailor the wavefront, especially for transmitted light. Helical metamaterial, which has a relatively strong coupling effect among the helical nanowires, may provide an alternative to the wavefront control. In this study, a kind of complementary helical metamaterial with a left-handedness and a right-handedness helixes coupled to each other is proposed. The complementary helical metamaterial has a strong circular conversion dichroism, and it is expected to be a good candidate for generating phase shift and controlling wavefront with high efficiency. Using the finite-difference time-domain method, we find that this kind of helix has a high circular polarization conversion in a broadband, which often implies a high efficiency of the transmitted light. Moreover, it is also found that the structure will introduce a controllable phase shift（) between the incident and the transmitted light whose polarizations are orthogonal to each other. By calculating the surface current density of the helix, the performance of high circular polarization conversion is explained. Meanwhile, we also find that the phase shift has a linear relationship with the initial angle of the helix（), which is =2. This relationship can be explained exactly by Jones calculus. According to the generalized Snell's law, the refracted beam can have an arbitrary direction by designing a suitable constant gradient of phase discontinuity. And then, by arranging 12 helixes in an array with a constant phase gradient along the X-axis, the phenomenon of anomalous refraction with a high efficiency（64%) is observed in the near infrared range（1.0-1.4 m). The angle of the anomalous refraction is in good agreement with the theoretical value. Compared with the metasurface, the helical metamaterial has a relatively complex structure. But with the development of the nanotechnology, there are several methods that can complete the propagations of nano helical structures, such as the direct laser writing, the glancing angle deposition, and the molecular self-assembly techniques. And by carefully designing the structure parameters of the helix, this kind of complementary helical metamaterial is expected to be an ideal candidate not only for traditional optics but also for biological detection and medical science.

Seismic Refraction Method to Study Subsoil Structure

Seismic waves are used in many fields. It can be applied on determine subsoil structure, and materials. In this work use seismograph for the measurement seismic wave propagation velocity in the real geological (subsoil layers) medium. These investigations are applied in petroleum research institute Egypt. Predication results of subsoil layers thickness and seismic wave’s velocity analysis are obtained. The seismograph recorded receiving sample data from geophones and by “SeisImager” software is extract the final seismogram. The seismograph is dependent on Snell law for wave’s propagation. The obtain result of P and S waves of this work are the same as P and S waves references. P waves are shake ground in the direction they are propagating (longitudinal waves), and S waves are shake perpendicularly or transverse to the direction of propagation.

Design of reflective linear-circular polarization converter based on phase gradient metasurface

Manipulating the propagating direction and polarization state of electromagnetic wave is always fascinating and used in a wide field. One of the approaches to achieving this aim is typically based on steering the propagation phase of wave traveling inside an optical medium, such as dielectric lens. Nevertheless, this approach creates new problems, such as high loss, bulky volume and fabrication difficulty. Recently, metasurface was found to be a two-dimensional equivalence of metamaterial, which attracted a great deal of attention because of its unique properties and capability of manipulating and controlling electromagnetic waves on a sub-wavelength scale. So metasurface serves as an alternative approach to dealing with the loss and fabrication issues, and opens a door for bridging the gap between the fundamental research of the artificial structures and their device applications. A reflective phase gradient metasurface (PGM) achieving the linear-to-circular (LTC) polarization conversion and anomalous reflection simultaneously is designed in this paper. Firstly, the conventional cross-shaped structure is modified for enlarging the phase range. Then, six modified cross-shaped structures are designed cautiously to serve as quarter wave-plates, and achieve 60 phase difference between adjacent structures. The reflection phase difference between x-and y-direction components is 90, and their magnitudes are both equal to 0.5. Secondly, a one-dimensional PGM is constructed by distributing six modified cross-shaped quarter wave-plates one by one. Furthermore, an LTC polarization converter with an area of 216 mm216 mm is designed by placing 366 one-dimensional PGMs periodically. The mirror reflectivity and axial ratio are simulated and measured to verify the performances of LTC polarization conversion and anomalous reflection. The measured sample is fabricated by printing circuit board technique through using FR4 substrate, and a free space method is adopted in measurement in the anechoic chamber. In addition, the operating bandwidth can be evaluated from the reflective power density spectra. The measured results of mirror reflectivity, reflective power density spectra and axial ratio characteristic are in good agreement with the corresponding simulations, which shows that the mirror reflectivity is lower than -10 dB; the axial ration is lower than 2 dB within the frequency band of 13.8-14.7 GHz. Meanwhile, the theoretical reflection angles from the generalized Snell law are consistent with the CST microwave studio simulated results and measured results. Compared with the reported LTC polarization converters, the proposed LTC polarization converter not only achieves polarization conversion, but also can manipulate the output wave direction, thereby it has an important promising application value for microwave engineering and communication system.

Nondestructive testing and characterization of residual stress field using an ultrasonic method

To address the difficulty in testing and calibrating the stress gradient in the depth direction of mechanical components, a new technology of nondestructive testing and characterization of the residual stress gradient field by ultrasonic method is proposed based on acoustoelasticity theory. By carrying out theoretical analysis, the sensitivity coefficients of different types of ultrasonic are obtained by taking the low carbon steel(12%C) as a research object. By fixing the interval distance between sending and receiving transducers, the mathematical expressions of the change of stress and the variation of time are established. To design one sending-one receiving and oblique incidence ultrasonic detection probes, according to Snell law, the critically refracted longitudinal wave (LCR wave) is excited at a certain depth of the fixed distance of the tested components. Then, the relationship between the depth of LCR wave detection and the center frequency of the probe in Q235 steel is obtained through experimental study. To detect the stress gradient in the depth direction, a stress gradient LCR wave detection model is established, through which the stress gradient formula is derived by the relationship between center frequency and detecting depth. A C-shaped stress specimen of Q235 steel is designed to conduct stress loading tests, and the stress is measured with the five group probes at different center frequencies. The accuracy of ultrasonic testing is verified by X-ray stress analyzer. The stress value of each specific depth is calculated using the stress gradient formula. Accordingly, the ultrasonic characterization of residual stress field is realized. Characterization results show that the stress gradient distribution is consistent with the simulation in ANSYS. The new technology can be widely applied in the detection of the residual stress gradient field caused by mechanical processing, such as welding and shot peening.

Corneal powers measured with a rotating Scheimpflug camera

Background/aimKeratometry measures the anterior corneal curvature only. Corneal power is calculated by multiplication with the keratometric refractive index, which takes into account the average negative posterior corneal power. The aim...

Broadband manipulation of acoustic wavefronts by pentamode metasurface

An acoustic metasurface with a sub-wavelength thickness can manipulate acoustic wavefronts freely by the introduction of abrupt phase variation. However, the existence of a narrow bandwidth and a low transmittance limits further applications. Here, we present a broadband and highly transparent acoustic metasurface based on a frequency-independent generalized acoustic Snell's law and pentamode metamaterials. The proposal employs a gradient velocity to redirect refracted waves and pentamode metamaterials to improve impedance matching between the metasurface and the background medium. Excellent wavefront manipulation based on the metasurface is further demonstrated by anomalous refraction, generation of non-diffracting Bessel beam, and sub-wavelength flat focusing.

Angularly symmetric splitting of a light beam upon reflection and refraction at an air-dielectric plane boundary.

Conditions for achieving equal and opposite angular deflections of a light beam by reflection and refraction at an air-dielectric boundary are determined. Such angularly symmetric beam splitting (ASBS) is possible only if the angle of incidence is >60° by exactly one third of the angle of refraction. This simple law, plus Snell's law, leads to several analytical results that clarify all aspects of this phenomenon. In particular, it is shown that the intensities of the two symmetrically deflected beams can be equalized by proper choice of the prism refractive index and the azimuth of incident linearly polarized light. ASBS enables a geometrically attractive layout of optical systems that employ multiple prism beam splitters.

Two-dimensional polymer grating and prism on Bloch surface waves platform.

A one-dimensional photonic crystal sustaining Bloch Surface Waves (BSWs) is used as a platform for two-dimensional integrated optics. The dielectric platform shows low loss, long propagation distance and high surface field enhancement. In order to study the potential of the platform for future photonic chips, polymer ultra-thin prisms and gratings (~100 nm) are engineered on the top of the platform. This polymer layer modifies the BSWs effective index enabling a direct manipulation of light. The BSW deflection effects caused by surface prisms are observed in the near-field and Snells law is verified; then the BSW diffractions through surface gratings are experimentally and theoretically characterized. The results show a robust platform that can be used for integrated optics that includes different optical components. One of the main advantages is that these 2D photonic devices can have arbitrary shapes, which is difficult to obtain in 3D.

One-way acoustic mirror based on anisotropic zero-index media

We have designed a one-way acoustic mirror comprising anisotropic zero-index media. For acoustic beam incident at a particular angle, the designed structure behaves like a high-efficient mirror that redirects almost all the incident energy into another direction predicted by the Snell's law, while becoming virtually transparent to beams propagating reversely along this output path. Furthermore, the mirror can be tailored to work at arbitrary incident angle by simply adjusting its geometry. Our design, with undirectional reflection functionality and flexible working angle, may offer possibilities in space isolations and have deep implication in various scenarios like ultrasound imaging or noise control.

Ultrathin Metasurface for Controlling Electromagnetic Wave With Broad Bandwidth

In this paper, ultrathin metasurface based on phase discontinuities is proposed to manipulate electromagnetic waves in the microwave region. The gradual phase changes accumulated along the transmission path in the conventional lens are substituted with abrupt phase changes at the interface of the metasurface, which are realized by the Pancharatnam-Berry phase element. An ultrathin metasurface with a linear phase gradient is designed, fabricated, and measured. The phenomenon of anomalous refraction can be observed in a broad bandwidth, in which the cross-polarization conversion efficiency keeps approaching the theoretical limit. The measured result shows that the anomalous refraction angle can be well predicted by the generalized Snell's laws, and the efficiency keeps >20% in a broad bandwidth of 24%. Our design makes a remarkable progress in the cross-polarization conversion efficiency of the transmission-type metasurface, which is beneficial to put metasurface into practical application.

Uniform refraction in negative refractive index materials.

We study the problem of constructing an optical surface separating two homogeneous, isotropic media, one of which has a negative refractive index. In doing so, we develop a vector form of Snell's law, which is used to study surfaces possessing a certain uniform refraction property, in both the near- and far-field cases. In the near-field problem, unlike the case when both materials have positive refractive indices, we show that the resulting surfaces can be neither convex nor concave.