An elementary geometric proof of Snell’s law from Fermat’s principle

Deviation problems in ray optics in the light of the refined unambiguous definitions of angles of incidence, reflection and refraction

Optical diffraction of binary-nanoparticle film prepared by convective deposition with vibration assistance

Brewster’s law in the light of the refined unambiguous angles of incidence, reflection, and refraction

Refinement of the theory of formation of rainbow with incorporation of the refined unambiguous angles of incidence, reflection, and refraction

Roughness is one of the principal factors influencing reflectance of a surface that takes place in radiation heat transfer. In the present work, we investigate the influence of the orientation of single-oriented roughness grooves on the directed–directed spectral reflectance of surfaces within the wavelength range of 0.2–20.0 μm. The surfaces have a mean square deviation of the roughness (σ) of 0.2 μm and 2.0 μm and the areas exceed the size of the incident light beam. The angles of reflection are equal to the angles of incidence and are equal to 50° and 70°. The degree of influence of the groove orientation on the reflectance is defined as the ratio of the reflectances of one and the same surface with the grooves oriented parallel and perpendicular to the plane of incidence. In the theoretical investigation, we engage the unidimensionally rough surface model. An experimental study by means of physical modeling was performed: the degree of influence of the groove orientation on the spectral reflection of the surface was estimated, within the investigated wavelength range, according to the degree of influence of the groove orientation on the monochromatic reflectance of the reference ground surfaces within the σ-range of 0.01–4.33 μm on the wavelength 0.405 μm. The studies performed show that the spectral range of the groove orientation influence on the surface reflectance increases with an increase in σ and a decrease in the angle of incidence; the degree of that influence has a maximum the value of which increases with the increase in the angle of incidence. We determine the conditions of agreement of the theory and the experiment.

In Marangoni–Bénard convection with mass transfer from the air to the liquid, an experimental exploration of reflections of solitary waves at walls has been carried out. Mach stems are observed for small enough incident angles. Upon increasing the incident angle the stem disappears thus leading to regular reflections with, however, the angle of reflection being generally larger than the incident angle.

On the basis of the method of reduced Rayleigh equations we present a simple and reciprocal theory of the coherent and incoherent scattering of x-rays from one- and two-dimensional randomly rough surfaces, that appears to be free from the limitations of earlier theories of such scattering based on the Born and distorted-wave Born approximations. In our approach, the reduced Rayleigh equation for the scattering amplitude(s) is solved perturbatively, with the small parameter of the theory η(ω) = 1 - ϵ(ω), where ϵ(ω) is the dielectric function of the scattering medium. The magnitude of η(ω) for x-rays is in the range from 10−6 to 10−3, depending on the wavelength of the x-rays. The contributions to the mean differential reflection coefficient from the coherent and incoherent components of the scattered x-rays are calculated through terms of second order in η(ω). The resulting expressions are valid to all orders in the surface profile function. The results for the incoherent scattering display a Yoneda peak when the scattering angle equals the critical angle for total internal reflection from the vacuum-scattering medium interface for a fixed angle of incidence, and when the angle of incidence equals the critical angle for total internal reflection for a fixed scattering angle. The approach used here may also be useful in theoretical studies of the scattering of electromagnetic waves from randomly rough dielectric-dielectric interfaces, when the difference between the dielectric constants on the two sides of the interface is small.

The number of electrons reflected elastically from a mercury surface has been observed as a function of the electron velocity and of the angles of incidence and reflection of the electrons. When the initial beam struck the surface normally the reflected current was symmetrical on both sides. Measurements were made with velocities from 20 to 70 volts and with angles of incidence from 10\ifmmode^\circ\else\textdegree\fi{} to 90\ifmmode^\circ\else\textdegree\fi{}. The total number of elastically reflected electrons decreased rapidly with increasing velocity. The intensity of the reflected electrons was not proportional to the cosine of the angle of reflection, but showed a preference toward reflection backward to the direction of the initial beam. Measurements made with the incident beam at other angles than normal to the surface showed the same preference toward reflection in the direction backward to the direction of the initial beam.

This paper presents measurements and characterization of D-band indoor channels. The measurements are performed in line-of-sight (LoS), obstructed-LoS (OLoS), and reflected non-LoS (RNLoS) environments. For OLoS sceanrio, cylindrical objects of different materials are used as an obstruction. For RNLoS, different surfaces are used as reflectors. From the large set of LoS and OLoS measured data, the parameters for single-slope path loss model with shadowing are devised. Furthermore, the analysis of multipath propagation is performed. The results show that strong multiple reflections from the transmitter and receiver electronics are present both in LoS and OLoS environments. Additionally, the results show that glass and ceramic objects in the propagation path produce surface-diffracted rays which clock-wise and counter clock-wise superposition leads to frequency-dependent path loss. Finally, the results show that the RNLoS measured path loss with aluminum plate as a reflector is very similar to free-space path loss when the angle of incidence and the angle of reflection are equal.

The refraction angles θ d of electron beams passing through aluminum and thin flat copper foils and reflection angles θ r are measured. A microtron with 7.4 MeV particles is used as a source of electrons. The angle between the particle trajectory and the target surface α is varied in the range 5°–30°. The dependences of the refraction and reflection angles on the α angle and foil thickness δ are measured. A dosimetric film is used to make pictures of cross sections of the electron beam scattered by a thin 50 μm copper foil. Image processing allows the spatial distributions of refracted and reflected particles to be obtained. The processes of relativistic electron scattering at a small angle of incidence on a flat target are simulated by the Monte Carlo method. The results of simulation are compared with experimental data. Particle scattering at a bimetallic target consisting of 200-μm aluminum and 70-μm lead layers are simulated. A dependence of the spatial-energy distributions on the order of metal layers placed along electron trajectories is found.

Gradient metasurface for four-direction anomalous reflection in terahertz

An expression of the degree of polarization (DOP) based on the geometrical optics polarimetric bidirectional reflectance distribution function model is presented. In this expression, the DOP is related to the surface roughness and decreases at different reflection angles because diffuse reflection is taken into consideration. A shadowing/masking function introduced into the specular reflection expression makes the DOP values decrease as the angle of incidence or observation approaches grazing. Different kinds of materials were measured to validate the accuracy of this DOP expression. The measured results suggest that the errors of the DOP are reduced significantly, and the polarized reflection characteristics can be described more reasonably and accurately.

This study aims to demonstrate that dielectric particles can be stacked to produce electromagnetic functions. By drawing up a hydrophilic substrate from an aqueous suspension in which fine particles are dispersed, monolayer closed-packed or ordered structures can be obtained by utilizing the meniscus attraction between the particles during the drying process. By repeating this process, particles of different sizes and materials can be stacked layer by layer. By changing the permittivity and/or diameters of the particles through layers, the reflectance of light at a particular wavelength can be increased or decreased. Aiming at selective reflection of the near infrared light, silica (SiO2) particles of φ200 nm and titanium oxide (TiO2) particles of φ33 nm were chosen and two layers were stacked on a silicon wafer. The reflectivity was measured with an original setup. Monochromatic light was focused on the sample at various angles of incidence, and the reflection intensity was measured at specified angles independent of the angle of incidence. By scanning the wavelength of the light, the reflection spectrum at specific incident and reflection angles was obtained. It was confirmed that reflectivity increased by 1.7 μm when the TiO2 oxide layer covered the SiO2 layer, as derived from theory.

Abstruct-Surface skimming bulk waves may be strongly reflected by the sides of a crystal device structure. The range of angle of incidence over which reflections are strong and over which mode conversion can occur is of interest for the utilization and for the suppression of reflections in devices. Theoretical results are presented for lithium niobate, lithium tantalate, and quartz, giving the amplitude, phase shift, reflection angle, and power reflection angle of reflected waves generated when a surface skimming bulk wave is incident on a free, planar surface at various angles of incidence. In addition, the effect on reflectivity of a thin metal film on the reflecting surface is considered. Some experimental results are presented for lithium niobate. Conclusions are drawn relevant to both the use and suppression of reflected waves.

Properties of isolated frog retinae in reflecting non-polarized and polarized light