A new type of CGO solutions and its applications in corner scattering

We consider corner scattering for the operator ∇ · γ(x)∇ + k2ρ(x) in R2, with γ a positive definite symmetric matrix and ρ a positive scalar function. A corner is referred to one that is on the boundary of the (compact) support of γ(x) − I or ρ(x) − 1, where I stands for the identity matrix. We assume that γ is a scalar function in a small neighborhood of the corner. We show that any admissible incident field will be scattered by such corners, which are allowed to be concave. Moreover, we provide a brief discussion on the existence of non-scattering waves when γ − I has a jump across the corner. In order to prove the results, we construct a new type of complex geometric optics (CGO) solutions.

A Review of designing freeform optical surfaces by the Monge–Ampère equations

Background: The equations of Monge–Ampère type which arise in geometric optics is used to design illumination lenses and mirrors. The optical design problem can be formulated as an inverse problem: determine an optical system consisting of reflector and/or refractor that converts a given light distribution of the source into a desired target light distribution. For two decades, the development of fast and reliable numerical design algorithms for the calculation of freeform surfaces for irradiance control in the geometrical optics limit is of great interest in current research. Objective: The objective of this paper is to summarize the types, algorithms and applications of Monge–Ampère equations. It helps scholars to grasp the research status of Monge–Ampère equations better and to explore the theory of Monge–Ampère equations further. Methods: This paper reviews the theory and applications of Monge–Ampère equations from four aspects. We first discuss the concept and development of Monge–Ampère equations. Then we derive two different cases of Monge–Ampère equations. We also list the numerical methods of Monge–Ampère equation in actual scenes. Finally, the paper gives a brief summary and an expectation. Results: The paper gives a brief introduction to the relevant papers and patents of the numerical solution of Monge–Ampère equations. There are quite a lot of literatures on the theoretical proofs and numerical calculations of Monge–Ampère equations. Conclusion: Monge–Ampère equation has been widely applied in geometric optics field since the predetermined energy distribution and the boundary condition creation can be well satisfied. Although the freeform surfaces designing by the Monge–Ampère equations is developing rapidly, there are still plenty of rooms for development in the design of the algorithms.

Investigation of the parameters of the reflected wave near the Brewster angle

In free space, the relative permittivity is determined by the Brewster formula without taking into account dielectric and magnetic losses. In experimental studies, discrepancies in the angular position of the minimum of the reflected wave from dielectric materials are observed in comparison with calculations, which are known as deviations from Fresnel’s laws. By solving the task of inclined falling wave on an plate made of a dielectric material with complex of the dielectric and magnetic permittivity, the parameters of the reflected wave were calculated, according to which the angles corresponding to the minimum reflection were determined, depending on the dielectric losses of the material. From the condition that the reflected wave is equal to zero, a formula for determining the Brewster angle for a material with dielectric and magnetic losses was analytically obtained, the results of calculations for which coincided with the calculations for the reflected wave in the context of geometric optics. It is determined that in the general case, the conditions for determining the position of the minimum of the complex amplitude and the phase jump by 180° of electromagnetic waves do not coincide and can be found only when solving the task an falling wave on a plate with complex electrodynamic parameters of the material in the context of geometric optics.

Electric Transverse Emissivity of Sinusoidal Surfaces Determined by a Differential Method: Comparison with Approximation of Geometric Optics

We present in this paper a numerical study of the validity limit of the optics geometrical approximation in comparison with a differential method which is established according to rigorous formalisms based on the electromagnetic theory. The precedent studies show that this method is adopted to the study of diffraction by periodic rough surfaces. For periods much larger than the wavelength, the mechanism is analog to what happens in a cavity where a ray is trapped and undergoes a large number of reflections. For gratings with a period much smaller than the wavelength, the roughness essentially behaves as a transition layer with a gradient of the optical index. Such a layer reduces the reflection there by increasing the absorption. The code has been implemented for TE polarization. We determine by the two methods such as differential method and the optics geometrical approximation the emissivity of gold and tungsten cylindrical surfaces with a sinusoidal profile, for a wavelength equal to 0.55 microns. The obtained results for a fixed height of the grating allowed us to delimit the validity domain of the optic geometrical approximation for the treated cases. The emissivity calculated by the differential method and that given on the basis of the homogenization theory are satisfactory when the period is much smaller than the wavelength.

Concentrating Solar Power Advances in Geometric Optics, Materials and System Integration

In this paper, the technological advances in concentrating solar power are reviewed. A comprehensive system approach within this scope is attempted to include advances of highly specialized developments in all aspects of the technology. Advances in geometric optics for enhancement in solar concentration and temperature are reviewed along with receiver configurations for efficient heat transfer. Advances in sensible and latent heat storage materials, as well as development in thermochemical processes, are also reviewed in conjunction with efficient system integration as well as alternative energy generation technologies. This comprehensive approach aims in highlighting promising concentrating solar power components for further development and wider solar energy utilization.

WKB expansions for weakly well-posed hyperbolic boundary value problems in a strip: Time depending loss of derivatives

We are interested in geometric optics expansions for linear hyperbolic systems of equations defined in the strip [Formula: see text]. More precisely the aim of this paper is to describe the influence of the boundary conditions on the behavior of the solution. This question has already been addressed in [A. Benoit, Wkb expansions for hyperbolic boundary value problems in a strip: Selfinteraction meets strong well-posedness, J. Inst. Math. Jussieu 19(5) (2020) 1629–1675] for stable boundary conditions. Here we do not require that the boundary conditions lead to strongly well-posed problems but only to weakly well-posed problems (that is loss(es) of derivatives are possible). The question is thus to determine what can be the minimal loss of derivatives in the energy estimate of the solution. The main result of this paper is to show, thanks to geometric optics expansions, that if the strip problem admits a boundary in the so-called [Formula: see text]-class of [S. Benzoni-Gavage, F. Rousset, D. Serre and K. Zumbrun, Generic types and transitions in hyperbolic initial-boundary-value problems, Proc. Roy. Soc. Edinburgh Sect. A 132(5) (2002) 1073–1104] then the loss of derivatives shall be at least increasing with the time of resolution. More precisely this loss is bounded by below by a step function increasing with respect to time which depends on the minimal time needed to perform a full regeneration of the wave packet.