Geometrical Optics Research Articles

High-power pulsed gyrotron oscillators operating in the short millimeter-wave range typically utilize high-order cavity modes, requiring efficient and compact quasi-optical mode converters to maximize their performance and efficiency. This work details the design and optimization of a compact quasi-optical mode converter aimed at transforming TE22,6 at 210 GHz has been developed. The converter integrates a high-efficiency Denisov-type launcher, designed using geometric optics and coupled-mode theory, along with a mirror system comprising a quasi-elliptical, an elliptical, and a parabolic mirror that is optimized through vector diffraction theory. The design utilizes MATLAB for parametric analysis and Feldberechnung für Körper mit beliebiger Oberfläche simulations to evaluate the electromagnetic field distribution within the launcher and at the output window. Through precise optimization of the launcher radius, perturbation amplitude, and perturbation length, we achieved efficient mode conversion and beam shaping within a compact 127 mm design. Notably, the cut length is 20 mm, constituting 15.75% of the total launcher length, ensuring a well-defined Gaussian beam profile with minimal diffraction losses. Simulation results demonstrate that the scalar Gaussian mode content and vector Gaussian mode content at the output window are 99.2% and 98.6%, respectively, indicating high mode purity and efficient power transfer. The shortened launcher length and optimized design parameters provide a compact and efficient solution, outperforming earlier models while maintaining high performance. This highlights the effectiveness of the proposed design in achieving high-power, high-frequency millimeter-wave applications with minimal size and high efficiency.

Abstract The initial-to-final-state inverse problem consists in determining a quantum Hamiltonian assuming the knowledge of the state of the system at some fixed time, for every initial state. This problem was formulated by Caro and Ruiz and motivated by the data-driven 
prediction problem in quantum mechanics. Caro and Ruiz analysed the question of uniqueness for Hamiltonians of the form -Δ + V with an electric potential V = V(t, x) that depends on the time and space variables. In this context, they proved that uniqueness holds in dimension n > 1 whenever the potentials are bounded and have super-exponential decay at infinity. Although their result does not seem to be optimal, one would expect at least some degree of exponential decay to be necessary for the potentials. However, in this paper, we show that by restricting the analysis to Hamiltonians with time-independent electric potentials, namely V = V(x), uniqueness can be established for bounded integrable potentials exhibiting only super-linear decay at infinity, in any dimension n > 1. This surprising improvement is possible because, unlike Caro and Ruiz's approach, our argument avoids the use of complex geometrical optics (CGO). Instead, we rely on the construction of stationary states at different energies---this is possible because the 
potential does not depend on time. These states will have an explicit leading term, given by a Herglotz wave, plus a correction term that will vanish as the energy grows. Besides the significant relaxation of decay assumptions on the potential, the avoidance of CGO solutions is important in its own right, since such solutions are not readily available in more complicated geometric settings.

Shadow imaging is commonly used to record projectile movement for diagnostics, guidance, and control. Moving shadows from targets traversing guided paths contain rich information that can be extracted to provide quantitative measures of guide-system deviations and assess the integrity of the transport mechanism. Laser velocimetry, a popular method for measuring projectile velocity, provides additional insight into target motion. The velocity field is mapped to the transverse and angular deviations from the calibrated trajectory. The shadow-image geometry is related to the same deviations based on geometric optics. These two measurement techniques are combined to quantify target deviation in real time, allowing deviations in different degrees of freedom to be determined independently and facilitating guidance-error analysis throughout the entire trajectory. Understanding the shadow-edge position is essential for measuring deviations from guide systems using shadow imaging. The standard approach employs a laser at a fixed position; however, the additional use of laser velocimetry makes it preferable to let the laser move with the target. When shadow imaging is used to measure guide-system deviations, simple geometric optics governs the shadow and the time shape of velocity pulses. A mathematical framework identifies the target deviations that affect the width and duration of the pulses. These relationships serve as the theoretical foundation for the approach (Axisa, Mule’Stagno, & Demicoli, 2025; Orihara, & Momose, 2025; Rada Giacaman, 2022; Abosinnee, Bencsik, & Abedi, 2025).

A recent covariant formulation, that includes nonperturbative effects from loop quantum gravity (LQG) as self-consistent effective models, has revealed the possibility of nonsingular black hole solutions. The new framework makes it possible to couple scalar matter to such LQG black holes and derive Hawking radiation in the presence of quantum spacetime effects while respecting general covariance. Standard methods to derive particle production both within the geometric optics approximation and the Parikh-Wilczek tunnelling approach are therefore available and confirm the thermal nature of Hawking radiation. The covariant description of scale-dependent decreasing holonomy corrections maintains Hawking temperature as well as universality of the low-energy transmission coefficients, stating that the absorption rates are proportional to the horizon area at leading order. Quantum-geometry effects enter the thermal distribution only through subleading corrections in the graybody factors. Nevertheless, they do impact energy emission of the black hole and its final state in a crucial way regarding one of the main questions of black-hole evaporation: whether a black-to-white-hole transition, or a stable remnant, is preferred. For the first time, a first-principles derivation, based on a discussion of backreaction, finds evidence that points to the former outcome.

Wavefront sensing in the EUV to soft X-ray regime is critical for high-resolution imaging, precision metrology, and beam diagnostics in advanced light sources, including HHG sources, SR, and XFEL facilities. Hartmann and Talbot wavefront sensors, while similar in structural configuration, operate based on distinct physical principles—geometric optics and near-field diffraction, respectively. In this study, we present a unified theoretical framework based on Fresnel diffraction to model and compare the performance of both sensing approaches. Numerical simulations, supported by visible-light experiments with a He-Ne laser, demonstrate the validity of the model under realistic conditions. The proposed framework facilitates the design and optimization of X-ray wavefront sensors, enabling improved beam quality characterization and more efficient beamline alignment. This work provides a practical methodology for wavefront-based diagnostics in advanced photon science facilities, contributing to enhanced experimental precision and operational efficiency.

Numerical simulation of the X-ray Laue diffraction intensity distribution in reciprocal space from perfect and wedge multilayers was performed. It is shown that for spatially bounded X-ray beams, diffraction at the edges of collimators or slits in the diffraction scheme should be taken into account. A comparison of numerical reciprocal space mapping simulations using the geometrical optics approximation and the Fresnel approximation was conducted.

Physics‐Driven Deep Neural Networks for Solving the Optimal Transport Problem Associated With the Monge–Ampère Equation

In this paper, we study the stability in determining a time-dependent nonlinear coefficient appearing in the third-order in time Jordan-Moore-Gibson-Thompson equation of Westervelt type. By using the finite difference technique and constructing some geometric optics solutions to the linearized equation, together with the light-ray transform, a stability result of determining the time-dependent nonlinear term is obtained from the knowledge of the Neumann boundary data and only one terminal data.

The original Calderón problem consists in recovering the potential (or the conductivity) from the knowledge of the related Neumann to Dirichlet map (or Dirichlet to Neumann map). Here, we first perturb the medium by injecting small-scaled and highly heterogeneous particles. Such particles can be bubbles or droplets in acoustics or nanoparticles in electromagnetism. They are distributed, periodically for instance, in the whole domain where we want to do reconstruction. Under critical scales between the size and contrast, these particles resonate at specific frequencies that can be well computed. Using incident frequencies that are close to such resonances, we show that (1) the corresponding Neumann to Dirichlet map of the composite converges to the one of the homogenised medium. In addition, the equivalent coefficient, which consists in the sum of the original potential and the effective coefficient, is negative valued with a controllable amplitude; (2) as the equivalent coefficient is negative valued, then we can linearise the corresponding Neumann to Dirichlet map using the effective coefficient’s amplitude; (3) from the linearised Neumann to Dirichlet map, we reconstruct the original potential using explicit complex geometrical optics solutions (CGOs).

In this paper, a method for measuring the liquid refractive index based on liquid crystal on silicon (LCoS) is presented. The liquid measuring chamber is composed of two optical lenses, an inlet, and an outlet. By filling the liquid with different refractive indices, the liquid measuring chamber has different focal lengths according to geometric optics. As a reflective micro-display device, the LCoS is another key component, in which the liquid crystal (LC) molecules are arranged in a nematic phase. The programming function of the LCoS, based on the interaction of incident light and the LC molecules in the LCoS, enables the LCoS to form as a digital lens. By using the measuring chamber in the optical system and adjusting the focal lengths of the digital lens, which is loaded on the LCoS, the refractive index of the liquid can be calculated accurately when the sensor receives the focused spot. The experimental results verify the feasibility of our proposed method.

The aim of this study is to prove that the first-order term in a Fourier series of the corneal refractive power, a decentration component, is identical to a prismatic refractive component. A model using a thin round wedge prism, similar to one of the Risley prisms, is constructed. The prismatic refractive power PDφ at an angle φ in an arbitrary oblique section is formulated using geometrical optics. The discrete Fourier transform is applied to data calculated from complex equations for prisms made of glass or corneal tissue to approximate this relationship based on the amplitude, frequency, and initial phase. The approximated equation is represented by PDφ ≈ −|PD0|⁎cosφ, where |PD0| is the nominal power of the prism, which is identical to the first-order term in the Fourier series of the corneal refractive power. I term the prism with this power profile the virtual prism. These results prove that corneal refractive power has a prismatic refractive component, which is the first-order term in the Fourier series of the corneal refractive power. These findings suggest that strabismus of corneal origin exists and that the decentration component, a type of aberration that was previously thought to be uncorrectable, can be treated with some methods.Supplementary InformationThe online version contains supplementary material available at 10.1038/s41598-025-18727-y.

Abstract In this study, we consider a beam summation method adapted from the semiclassical regime of quantum mechanics to study the classical properties of thin light bundles in gravity. In Newtonian paraxial optics, this method has been shown to encapsulate the wave properties of the light beams. In our case, the wave function assigned to the light bundle can be viewed as a coarse-grained description that captures information about the dynamics of superposed bundles within the geometric optics regime. We investigate two solutions of the null bundle wave function that differ by their origin: (i) a point source and (ii) a finite source. It is shown that while the wave function in the point source case contains the same information as the standard thin null bundle framework, the finite source case corresponds to a Gaussian beam. The novel aspect of this work arises from our geometric construction of covariant Gaussian beams, which can be applied in any spacetime. Additionally, the effects of a finite source on cosmological distances are discussed. With this framework, one can model light propagation from coherent sources while avoiding the mathematical singularities of the standard thin null bundle formalism. We explicitly demonstrate the caustic-avoidance property of Gaussian beams in the analytically tractable example of a Barriola-Vilenkin monopole spacetime.

Diffractive sail components may be used in part or whole for in-space propulsion and attitude control. A sun-facing hybrid diffractive solar sail having reflective front facets and transmissive side facets is described. This hybrid design seeks to minimize the undesirable scattering from side facets. Predictions of radiation pressure are compared for analytical geometrical optics and numerical finite difference time domain approaches. Our calculations across a spectral irradiance band from 0.5 to 3 μm suggest the transverse force in a sun facing configuration reaches 48% when the refractive index of the sail material is 1.5. Diffraction measurements at a representative optical wavelength of 633 nm support our predictions.

Abstract In this study, we present a comprehensive analysis of a modified Frolov black hole (BH) model that incorporates two types of topological defects, a global monopole (GM) and a cloud of strings (CS). This composite BH solution is examined from multiple theoretical perspectives to explore the impact of these modifications on the BH’s geometric, thermodynamic and dynamical properties. We begin by studying the geometrical optics of the spacetime, focusing on the motion of null geodesics. Key features, such as the effective potential, photon sphere, the force acting on photons and the stability of circular photon orbits, are analyzed in detail. Our results show that the presence of GM and CS significantly affects the spacetime geometry and photon dynamics. In addition, the thermodynamic behavior of the modified BH is also investigated. We derive essential quantities such as the Hawking temperature and entropy, demonstrating how the inclusion of GM and CS leads to deviations from the standard thermodynamic relations observed in classical BH solutions. These deviations may offer valuable insights into quantum gravity and the role of topological defects in BH physics. Furthermore, we examine the BH shadow as an observational signature of the underlying geometry. Our analysis shows that the Frolov parameter tends to reduce the apparent size of the shadow, while the presence of topological defects, particularly GM and CS, enlarges it. In addition, we investigate the perturbative dynamics of the BH by studying both scalar (spin-0), fermionic (spin-1/2) and electromagnetic (spin-1) fields through the massless Klein–Gordon and Maxwell equations, respectively. Using the Wentzel–Kramers–Brillouin approximation, we compute the quasinormal modes (QNMs) for scalar and electromagnetic field perturbations. The results confirm the stability of the BH under small perturbations and show that the QNM frequencies and damping rates are strongly influenced by the Frolov parameter, electric charge, GM and CS.

Achieving uniform intensity distribution is essential for various laser applications such as material processing. This paper presents the design, simulation, and experimental validation of a segmented beam-shaping integrator mirror aimed at transforming an incident laser beam into a uniform line-shaped spot. The mirror surface is composed of multiple connected parabolic segments. A geometric optics computational method, implemented using Python code, was developed to determine the unique parameters and boundaries for each segment, based on input specifications including the working distance (f), the input aperture size (D), the target spot size (d), and the number of segments (s). For a design case with D=49.5mm, f=350mm, d=20mm, and s=7, the segment parameters were calculated. The calculated design was modeled in SolidWorks, and its performance was simulated using Zemax ray tracing, predicting a shaped spot closely matching the 20mm target size in the segmented direction and an expected size (approx. 1.4mm) in the orthogonal direction. Experimental validation was conducted using a 4kW fiber laser equipped with a fiber core diameter of 400µm and a numerical aperture of 0.15, along with a collimating lens with a 100mm focal length. The measured spot size at the target plane was 20.39mm×1.41mm (1/e2 width), showing excellent agreement with both the design specification and the simulation results. This work successfully demonstrates the effectiveness of the integrator mirror design method and fabrication process for creating high-performance beam-shaping integrator optics for high-power laser systems.

The aim of this study is to develop a computational model for the propagation of shortwave radio signals in an inhomogeneous ionosphere, taking into account the Earth’s sphericity. We present a software suite implementing a ray-tracing method based on geometrical optics. A distinctive feature of the developed model is its use of realistic electron-density profiles within a multilayer ionospheric structure, enabling the simulation of refraction effects as radio waves traverse layers of varying density. The algorithm incorporates mechanisms for handling total internal reflection and for transforming coordinates between geodetic and Cartesian systems. The model supports the integration of empirical data from the International Reference Ionosphere (IRI2016), both directly and via radial basis functions to accelerate computations. Implementation is done in Python using scientific libraries and 3D visualization tools. Simulation results allow for the visualization of radio-wave trajectories at various frequencies and for analyzing the influence of ionospheric parameters on their propagation. A comparative analysis with existing modeling approaches—such as the parabolic equation method and the normal-mode method—demonstrates that the proposed approach achieves high predictive accuracy with significantly lower computational cost. The results can be used for the design and optimization of long-distance radio-communication systems, as well as for investigating physical processes in the ionosphere.

Geometric and wave optics in a topologically charged Perry-Mann type wormhole with disclinations

Ray and wave optics in an optical wormhole

The line-of-sight blockage is one of the main challenges in sub-terahertz wireless networks. Interestingly, the extended near-field range of sub-terahertz nodes gives rise to near-field wavefront shaping as a feasible remedy to tackle this issue. Recently, Airy beams emerged as one promising solution that opens significant opportunities to circumvent blockers with unique self-accelerating properties and curved trajectories. Yet, to unleash the full potential of curved beams in practice, one fundamental challenge remains: How to find the best beam trajectory? In principle, an infinite number of trajectories can be engineered. To find the optimal trajectory, we develop a physics-informed machine-learning framework for Airy beam shaping based on a detailed understanding of near-field electromagnetics, ray optics, and wave optics. The experimental results indicate that Airy beams, when correctly configured, can substantially increase the link budget under high-blockage scenarios even compared to near-field beam focusing, providing insight into coverage expansion and blind-spot reduction.

Abstract Active adjustment technology is used to solve the problem of reduced electrical performance of large reflector antennas caused by environmental factors. This technology is crucial for the operation of antennas under high-frequency working conditions.This paper proposes a full-path active adjustment strategy for dual-reflector antennas. This strategy takes into account the working mode of the adjustment mechanism under comprehensive influencing factors and achieves the optimal receiving performance at the full elevation by changing different adjustment algorithms.Firstly, the relationship between the displacement of reflector and the wavefront phase was established based on geometric optics. Secondly, three adjustment algorithms of the double reflector antenna were compared and analyzed: based on the standard, the fit and the optimal parabolic surface, and the calculation process of the adjustment amount was derived. An adjustment strategy model for multiple working conditions was proposed by introducing the elevation and the complexity coefficient and combining three adjustment algorithms. Finally, a finite element analysis was conducted on the dual-reflector antenna with a diameter of 110m, and the advantages and disadvantages of different adjustment algorithms were compared. The results show that strategy model not only achieves the optimal state of the antenna at the full elevation, but also shortens the adjustment amount of the adjustment mechanism and improves the working efficiency of the antenna under various working conditions.