Geometrical Optics Research Articles

Design of freeform phase diffractive optical elements based on the quadratic assignment problem.

The design of freeform phase diffractive optical elements is a challenging task, typically necessitating the use of complex differential equations or a large number of iterative calculations. This paper proposes what we believe to be a novel approach to address this problem. In this strategy, we introduce overall comparison optimization (OCO) to ensure the fast convergence of the cost function. The quadratic assignment problem (QAP) is used as the mathematical framework for designing freeform phase diffraction optics. Specifically, the ray mapping calculation problem in geometric optics is simplified as a QAP. To solve this problem, we apply the OCO method, which ensures that the cost function rapidly progresses in the "non-negative" direction, thereby facilitating fast convergence in each optimization iteration. In this manner, the proposed approach alleviates the computational burden associated with repeated evaluations of the cost function and accelerates convergence in the design process. We construct holographic masks using the OCO method and perform simulations to demonstrate the potential of the proposed method in swiftly realizing complex illumination patterns. The results show that the design model has good performance when dealing with complex illumination tasks. The conclusions obtained in this paper can be extended to the realization of phase-only holography and the solution of freeform surfaces illumination design.

The ‘thin lens’ in the light of idealisations

This paper argues that the simplifying idealisations assumed in making complex science tractable should be identified and discussed before model-building. It is suggested that alternative conceptions can persist, and learning difficulties arise for students when they do not adequately understand these idealisations. We argue that explicitly addressing idealisations for a particular physical phenomenon can ease students’ conceptual understanding. A theoretical framework for identifying and reflecting idealisations is introduced based on epistemological considerations from science philosophy. The thin lens approximation of geometric optics, a common topic taught in the introductory classroom, is used as an archetypal illustration.

Data reduction for spatially resolved reflectance anisotropy spectrometer.

We show that in spatially resolved reflectance anisotropy (RA) spectrometers, off-axis optical rays introduce a spurious signal component that cannot be addressed by optical alignment. Such a component is associated with the difference between the reflectivities s and p of the sample and depends, in a complex manner, on the incidence position of the incident light on the surface of the sample. We report a data-reduction procedure to easily identify and remove spurious RA signals associated with the off-axis optical rays, based on the singular value decomposition analysis of spatially resolved RA spectra. We validated this approach by developing a spatially resolved RA spectrometer based on an 8 × 8 multi-anode photomultiplier (PMT). The PMT allowed the use of phase-sensitive detection techniques to enhance the signal-to-noise ratio, which is essential for the evaluation of the proposed data reduction procedure.

Optical, thermal and gamma ray attenuation characteristics of tungsten oxide modified: B2O3–SrCO3–TeO2–ZnO glass series

The glass series modified by tungsten oxide was created using the compounds (75-x) B2O3– 10SrCO3– 8TeO2– 7ZnO - xWO3, where x = 0, 1, 5, 10, 22, 27, 34, and 40% mole percentage. A UV–visible spectrophotometer and thermogravimetric-differential thermal analysis (TG-DTA) methods were employed to characterize the specimen's optical and phase transition attributes, respectively. The mass-attenuation coefficient (AC) of all created glasses from BSTZW0 to BSTZ7 was estimated using Geant4 code from 0.05 to 3 MeV and compared to the XCOM software results, with a relative difference of less than 2% between the two results. The increase of WO3 percentage lead to an increase in the Linear-AC at each studied energy, and this is mainly due to the fact that the higher the percentage of WO3 in the glass increases its density which causes an increase in the Linear-AC, so an energy of 0.06 MeV, as an example, the values of the Linear-AC was 4.009, 4.509, 5.442, 6812, 8.564, 9.856, 10.999 and 11.628 cm−1 form BSTZW0 too BSTZW7, respectively. The Half-VL (value layer), Mean-FP (free path), Tenth-VL, and Radiation attenuation performance (RAP) were also calculated for the current BSTZW-glass samples and revealed that BSTZW7 had the best gamma ray attenuation performance at all discussed energies when compared to other studied glass samples.

Critical review and experimental validation of radiative transfer models for semitransparent media containing large gas bubbles

Many materials manufacturing, photobiochemical, and photoelectrochemical processes involve radiation transfer through foams and bubbly fluids. Controlling and optimizing these processes require accurate predictions of radiation transfer through semitransparent media. Previously developed models treated the heterogeneous media as homogeneous plane-parallel slabs with some effective radiation characteristics while radiation transfer was governed by the one-dimensional radiative transfer equation (RTE). Unfortunately, their accuracy and range of validity remains unclear for lack of comparison with one another or against the same set of experimental data. This study aims to critically review and assess the validity of these models both numerically and experimentally. First, predictions of the transmittance and reflectance of heterogeneous slabs containing large randomly distributed bubbles were compared with those obtained from Monte Carlo ray-tracing (MCRT) method based on geometric optics. A new hybrid model was proposed that predicts the effective scattering coefficient and asymmetry factor using the Lorenz–Mie theory and the effective absorption coefficient as the volume-weighted sum of the bubbles and medium absorption coefficients and solves the RTE using the Monte Carlo method. Its predictions showed excellent agreement with those by the MCRT method for a wide range of bubble volume fractions (0%–30%) as well as slab thicknesses (2–40 mm) and medium absorption coefficients (0.1–100 m−1). Second, microcomputed X-ray tomography scans were performed on a fused silica sample containing bubbles with mean radius of 480 μm to retrieve the exact locations, diameters, and total volume fraction of bubbles. Here also, predictions of the hybrid model using the retrieved data agreed well with experimental measurements of the spectral normal-hemispherical reflectance and transmittance of the sample for wavelengths between 0.4 and 3 μm when silica ranges from weakly absorbing to absorbing.

Optoplasmonic Effects in Highly Curved Surfaces for Catalysis, Photothermal Heating, and SERS.

Surface curvature can be used to focus light and alter optical processes. Here, we show that curved surfaces (spheres, cylinders, and cones) with a radius of around 5 μm lead to maximal optoplasmonic properties including surface-enhanced Raman scattering (SERS), photocatalysis, and photothermal processes. Glass microspheres, microfibers, pulled fibers, and control flat substrates were functionalized with well-dispersed and dense arrays of 45 nm Au NP using polystyrene-block-poly-4-vinylpyridine (PS-b-P4VP) and chemically modified with 4-mercaptobenzoic acid (4-MBA, SERS reporter), 4-nitrobenzenethiol (4-NBT, reactive to plasmonic catalysis), or 4-fluorophenyl isocyanide (FPIC, photothermal reporter). The various curved substrates enhanced the plasmonic properties by focusing the light in a photonic nanojet and providing a directional antenna to increase the collection efficacy of SERS photons. The optoplasmonic effects led to an increase of up to 1 order of magnitude of the SERS response, up to 5 times the photocatalytic conversion of 4-NBT to 4,4'-dimercaptoazobenzene when the diameter of the curved surfaces was about 5 μm and a small increase in photothermal effects. Taken together, the results provide evidence that curvature enhances plasmonic properties and that its effect is maximal for spherical objects around a few micrometers in diameter, in agreement with a theoretical framework based on geometrical optics. These enhanced plasmonic effects and the stationary-phase-like plasmonic substrates pave the way to the next generation of sensors, plasmonic photocatalysts, and photothermal devices.

Teaching Paradigm Innovation in Geometrical Optics for Cultivating Outstanding Engineers

Teaching Paradigm Innovation in Geometrical Optics for Cultivating Outstanding Engineers

Quasinormal modes from Penrose limits

We use Penrose limits to approximate quasinormal modes (QNMs) with large real frequencies. The Penrose limit associates a plane wave to a region of spacetime near a null geodesic. This plane wave can be argued to geometrically realize the geometrical optics approximation. Therefore, when applied to the bound null orbits around black holes, the Penrose limit can be used to study QNMs. For instance, this Penrose limit point of view makes manifest the symmetry that emerges in the geometrical optics approximation of QNMs in terms of isometries of the resulting plane wave spacetime. We apply the procedure to warped , Schwarzschild black holes and Kerr black holes. In the former we show explicitly how the symmetry algebra contracts to that of the limiting plane wave while in the latter we find the expected agreement with numerically computed QNMs for large (real) frequencies.

Extinction of high-power laser radiation under adverse weather conditions.

The propagation of laser radiation over long distances can be significantly affected by atmospheric extinction due to precipitation as well as aerosol particles and molecules. The knowledge of the contribution of precipitation is critical to the operation of a variety of laser-based systems. The study of high-power laser transmission around 1µm is of particular interest because several atmospheric transmission windows are located in this region. To investigate the effect of adverse weather conditions on laser transmission, free-space laser transmission experiments are conducted on the DLR test range in Lampoldshausen, Germany. A high-power laser with a wavelength of 1.03µm is used for the transmission measurements in combination with calibrated power monitors. Local weather conditions are continuously monitored by meteorological instruments during the experiments. Extinction coefficients are derived from transmission measurements showing that the extinction for snow is 7 times higher than for rain, and the extinction for drizzle/rain is 4 times higher than for rain at a given precipitation rate of 1mm/h. For a mixture of rain and snow, the extinction is comparable to that of rain, indicating that the water content strongly influences the optical properties and thus the extinction of laser radiation in mixed precipitation. A good relationship is found between the measured extinction coefficient and visibility for drizzle and rain and a slightly larger scatter of the data for snow. Furthermore, measured extinction coefficients are compared to the extinction coefficients based on the experimental size distributions of precipitation particles and geometric optics. A reasonable agreement is obtained for rain, with no improvement taking the forward-scattering into the detector aperture into account, and a much better agreement is obtained for snow when the forward-scattering contribution is included.

Reconstructing unknown inclusions for the biharmonic equation

Herein, we study an inverse problem for detecting unknown obstacles by the enclosure method using the Dirichlet–to–Neumann map for measurements. We justify the method for an penetrable obstacle case involving a biharmonic equation. We use complex geometrical optics solutions with a logarithmic phase to reconstruct some non–convex parts of the obstacle.

Solutions of the Ortvay Rudolf International Competition in Physics: Can One See One’s Own Back Around a Black Hole? (1991/12 Problem)

This paper is the fifth in a series of published solutions1–4 discussing problems of the Ortvay Rudolf international competition. The problem treated below is an exercise in general relativity and geometrical optics. Specifically, we check and interpret an often-heard statement, roughly of the form: “If you were to orbit a black hole around its photon sphere, you would see the back of your own head”. As the calculation involves general relativity, the full treatment of the problem as given in this article is appropriate for university students in advanced classes. However, by omitting the relativistic calculations and only referring to the results, one can easily formulate the problem as an exercise in geometric optics in a way that could be appropriate for introductionary level.

Geometrical optics of large deviations of Brownian motion in inhomogeneous media

Geometrical optics provides an instructive insight into Brownian motion, ‘pushed’ into a large-deviations regime by imposed constraints. Here we extend geometrical optics of Brownian motion by accounting for diffusion inhomogeneity in space. We consider three simple model problems of Brownian motion on the line or in the plane in situations where the diffusivity of the Brownian particle depends on one spatial coordinate. One of our results describes ‘Brownian refraction’: an analog of refraction of light passing through a boundary between two media with different refraction indices.

Contributed Session II: Ex-vivo human crystalline lenses geometrical changes during simulated disaccommodation.

The young crystalline lens changes its shape to focus targets at different distances, a capability (accommodation) that is lost with age. Quantifying crystalline lens morphology during accommodation and aging is essential for understanding basic mechanisms of function and disease (presbyopia and cataract). We studied the geometrical changes of the ex-vivo human crystalline lens while simulating different levels of physiological disaccommodation using a motorized lens-stretcher system designed for this purpose. 14 human lenses (11-54 y/o) from donor eyes were radially stretched mounted in the lens-stretcher and immersed in a cuvette filled with balanced salt solution for preservation purposes. Lens geometry and power were characterized using custom-developed high resolution 3-D optical coherence tomography (OCT) and laser ray tracing system coupled with the OCT. The lens thickness decreased linearly 81 microns/diopter (D) of simulated disaccommodation (-r=0.87, p<0.001), the diameter increased 76 microns/D (r=0.84, p<0.01), the radius of curvature of the anterior surface increased 0.63 mm/D (r=0.92, p<0.001), the lens surface area increased 1.9 mm2/D (r=0.62, p<0.01) and the lens volume remained constant. The same stretching displacement produced larger deformations in younger than in older lenses. Simultaneous high resolution optical and geometrical measurements of stretched lenses therefore match results in vivo and support the Helmholtz theory of accommodation.

Strongly Lensed Supermassive Black Hole Binaries as Nanohertz Gravitational-wave Sources

Supermassive black hole binary systems (SMBHBs) should be the most powerful sources of gravitational waves (GWs) in the universe. Once pulsar timing arrays (PTAs) detect the stochastic GW background from their cosmic merger history, searching for individually resolvable binaries will take on new importance. Since these individual SMBHBs are expected to be rare, here we explore how strong gravitational lensing can act as a tool for increasing their detection prospects by magnifying fainter sources and bringing them into view. Unlike for electromagnetic waves, when the geometric optics limit is nearly always valid, for GWs the wave-diffraction-interference effects can become important when the wavelength of the GWs is larger than the Schwarzchild radius of the lens, i.e., . For the GW frequency range explored in this work, the geometric optics limit holds. We investigate GW signals from SMBHBs that might be detectable with current and future PTAs under the assumption that quasars serve as bright beacons that signal a recent merger. Using the black hole mass function derived from quasars and a physically motivated magnification distribution, we expect to detect a few strongly lensed binary systems out to z ≈ 2. Additionally, for a range of fixed magnifications 2 ≤ μ ≤ 100, strong lensing adds up to ∼30 more detectable binaries for PTAs. Finally, we investigate the possibility of observing both time-delayed electromagnetic signals and GW signals from these strongly lensed binary systems—that will provide us with unprecedented multi-messenger insights into their orbital evolution.

Radio Occultation Data Analysis With Analytical Ray‐Tracing

AbstractRadio occultation experiments are a sensing technique dedicated to the remote sounding of planetary atmospheres. The technique exploits the frequency shift of a radio signal due to refraction in a planetary atmosphere. The aim is to infer the physical properties of the neutral atmosphere (e.g., pressure and temperature) and ionosphere (e.g., the electron number density). For one‐way occultations, the data processing usually relies on Abel transform algorithms when the atmosphere is spherically symmetric. For two‐way occultations, such techniques require the introduction of approximate relationships for the bending experienced by the signal to be obtained. In this context, we introduce a new method to process two‐way occultations data by spherically symmetric atmospheres using a ray‐tracing approach. However, the numerical integration of the geometrical optics equation through the atmosphere requires a significant computational time due to initial pointing issues. For this reason, our novel algorithm exploits a closed‐form solution to the equations of geometrical optics (Bourgoin et al., A&amp;A, 624, A41, 2019, https://doi.org/10.1051/0004-6361/201834962) applied to a spherically symmetric atmosphere. Within this approach, the bending is directly provided by the analytical solution and no numerical integration is required. In addition, we develop a procedure enabling us to disentangle the contributions from dispersive and neutral media in the frequency shift. This procedure is validated by comparing our vertical profiles to those obtained using Abel inversion or numerical ray‐tracing for Mars and Titan occultation experiments. We show that our algorithm provides similar results to purely numerical ray‐tracing algorithms while significantly decreasing the computational time.

Design of a high counting efficiency sensor for 1.0 cubic feet per minute laser particle counter.

Based on the theory of Mie scattering, an optical sensor for a 1.0 cubic feet per minute laser particle counter is developed in this paper. First, an illumination optical path relying on aspherical lens and cylindrical lens for laser shaping was designed, and a narrow optical spot with the illumination in the vertical direction of the photosensitive area approximately distributed as a flat top was obtained. Second, the scattering light path was designed by using the theory of Mie scattering and geometric optics method. Then, the structure of the sampling air path was designed with reference to the principle of Laval nozzle, and the particle flow trajectory was verified by simulation using the fluid dynamics software Ansys Fluent. Finally, the performance of the optical sensor was measured with polystyrene latex (PSL) particles, and the results showed that the counting efficiency of the 0.3μm particle size channel met the requirement of 50% ± 20% and the 0.5μm particle size channel met the requirement of 100% ± 10%, which complied with the ISO 21501-4 standard.

Geometric optics expansion for weakly well-posed hyperbolic boundary value problem: The glancing degeneracy

This article aims to finalize the classification of weakly well-posed hyperbolic boundary value problems in the half-space. Such problems with loss of derivatives are rather classical in the literature and appear for example in (Arch. Rational Mech. Anal. 101 (1988) 261–292) or (In Analyse Mathématique et Applications (1988) 319–356 Gauthier-Villars). It is known that depending on the kind of the area of the boundary of the frequency space on which the uniform Kreiss–Lopatinskii condition degenerates then the energy estimate can include different losses. The three first possible areas of degeneracy have been studied in (Annales de l’Institut Fourier 60 (2010) 2183–2233) and (Differential Integral Equations 27 (2014) 531–562) by the use of geometric optics expansions. In this article we use the same kind of tools in order to deal with the last remaining case, namely a degeneracy in the glancing area. In comparison to the first cases studied we will see that the equation giving the amplitude of the leading order term in the expansion, and thus initializing the whole construction of the expansion, is not a transport equation anymore but it is given by some Fourier multiplier. This multiplier needs to be invert in order to recover the first amplitude. As an application we discuss the existing estimates of (Discrete Contin. Dyn. Syst., Ser. B 23 (2018) 1347–1361; SIAM J. Math. Anal. 44 (2012) 1925–1949) for the wave equation with Neumann boundary condition.

An asymmetric graded indexed helical structured optical fibre for enhanced optical delays

Optical fibres are generally used to provide a guided path to optical signals. Optical fibres are classified as multi-mode or single-mode fibres. Multi-mode graded index (GRIN) optical fibres have been prevalent in short-range optical networks, while single-mode GRIN fibres are used for long-range links. Another potential application of optical fibre is the optical delays in numerous applications, such as quantum computing, network buffering, data synchronization, pattern correlation, etc. This paper conceptualizes a novel application of an asymmetric GRIN optical fibre for possible enhanced optical delays. An asymmetric GRIN parabolic profile with helical structure geometry has been proposed for this task. A detailed analysis has been presented to investigate the effects of various parameters of the proposed geometry. Further, a guided propagation has also been demonstrated with the help of ray optics and the Helmholtz wave equation to assess the feasibility of the proposal. The presented investigations revealed that the geometrical length of the proposed structure is significantly compact compared with the traditional GRIN fibre, proving it beneficial for optical delays.

Large aperture and defect-free liquid crystal planar optics enabled by high-throughput pulsed polarization patterning.

The liquid crystal (LC) geometrical phase optics, which is realized by the high-resolution control of the optical axis orientation in transparent micrometer-thin polymer films, is emerging as a next generation of planar optics. It features pronounced optical properties and stimuli-responsive behaviors, which could introduce appealing and new possibilities for photonic purposes. The development of fabrication techniques producing elements with large aperture sizes and arbitrarily varying molecular orientation is of significance in terms of practical utility. Here, we propose the pulsed polarization patterning technique to create large-aperture and defect-free LC geometrical phase elements. We investigated the capability of the azo photo-alignment material responding to nanosecond laser pulses and the corresponding anchoring behaviors to LCs. The threshold was reduced to one fourth of that under the continuous wave recording. The patterning resolution was found to be enhanced to around 0.71 µm, due to the ultra-fast interaction nature of the photo-alignment material with the polarized light field. We proposed the flying exposure mode to deliver high frequency modulated polarized laser pulses (8 kHz), with the precision stage moving in a uniform velocity for light-field stitching and the servo auto-focusing in the sample normal, enabling the stable and reliable polarization patterning for large aperture sizes. We further report on representative fabrication of LC polarization gratings with an aperture of 4 inch and 99.2% average diffraction efficiency.

Huygens’ equations and the gradient-flow equations in information geometry

In this paper, we revisit the relation between the gradient-flow equations and Hamilton’s equations in information geometry. By regarding the gradient-flow equations as Huygens’ equations in geometric optics, we have related the gradient flows to the geodesic flows induced by the geodesic Hamiltonian in an appropriate Riemannian geometry. The original evolution parameter [Formula: see text] in the gradient-flow equations is related to the arc-length parameter in the associated Riemannian manifold by Jacobi–Maupertuis transformation. As a by-product, the relation between the gradient-flow equation and replicator equations is found.