Caustic Surface Research Articles

Spot diagram borders are always caustic curves and/or marginal rays III: segmented mirror in hexagons and trapezoids

For annular (circular and elliptical) and hexagonal exit pupils with any optical path differences (described by Zernike Polynomials), it was proven [Appl. Opt. 61, 1876 (2022)APOPAI0003-693510.1364/AO.449270; Appl. Opt. 61, C20 (2022)APOPAI0003-693510.1364/AO.435850] that the border of any spot diagram is composed of the caustic surface and/or marginal rays. In this paper, the previous results are extended to segmented (by hexagons or trapezoids) exit pupils. Examples with exact spherical, coma, and astigmatism aberrations are included, while a mathematical demonstration is also included.

Paraxial and ray approximations of acoustic vortex beams.

A compact analytical solution obtained in the paraxial approximation is used to investigate focused and unfocused vortex beams radiated by a source with a Gaussian amplitude distribution. Comparisons with solutions of the Helmholtz equation are conducted to determine bounds on the parameter space in which the paraxial approximation is accurate. A linear relation is obtained for the dependence of the vortex ring radius on the topological charge, characterized by its orbital number, in the far field of an unfocused beam and in the focal plane of a focused beam. For a focused beam, it is shown that as the orbital number increases, the vortex ring not only increases in radius but also moves out of the focal plane in the direction of the source. For certain parameters, it is demonstrated that with increasing orbital number, the maximum amplitude in a focused beam becomes localized along a spheroidal surface enclosing a shadow zone in the prefocal region. This field structure is described analytically by ray theory developed in the present work, showing that the spheroidal surface in the prefocal region coincides with a simple expression for the coordinates of the caustic surface formed in a focused vortex beam.

Caustics and wavefronts produced by multiple refraction-reflections through plane-parallel plates

We study the formation of caustics and wavefronts produced by multiple refraction-reflections through a plane-parallel transparent plate, assuming a point source placed at an arbitrary position along the optical axis. The caustic surfaces are obtained by using the envelope’s method. Subsequently, the wavefronts are directly related to the involutes, which are associated with the envelopes for all the rays. Hence by using the Malus–Dupin theorem, we obtain their respective wavefronts produced by multiple refraction-reflections through a plane-parallel transparent plate. On the other hand, we implement Huygens’ principle to obtain the wavefronts leaving the plate after undergoing multiple reflections inside the plate, which we have called zero-distance phase wavefronts. Finally, we establish the correspondence between the wavefronts obtained by Huygens’ principle and the involutes associated with caustic surfaces; they are brought in coincidence assuming parallel curves from each other.

Image formation from a concave mirror.

Computing locations and extent of images, except in the most trivial configurations or special cases, is a complex task. Even rays emanating from a point source and passing through an optical system generally fail to converge at a single image point, highlighting the care needed to establish image locations. We use three approaches to study image formation in a simple configuration, that of a point source following reflection from a spherical concave mirror. We calculate the caustic surfaces, compute cross sections of flux densities on image surfaces, and compare the results with experimentally generated light intensity fields. One of the two caustic surfaces is one dimensional while the other forms a surface. The latter undergoes a metamorphosis from a distorted cone to an open surface as the source is moved away from the axis. Cross sections of the caustic surfaces with an image plane are found to coincide with peaks in the flux density. Experimental studies validate these conclusions.

Equations for designing a dialyte with minimal TSA based on the caustic surfaces.

A method to design a dialyte based on the caustic surfaces is presented, with correction at the third-order spherical aberration. We predefined, a priori, the value for the back focal length of the separated doublet for two different wavelengths to calculate their radii of curvature, which depend on all the parameters involved in the refraction process. Additionally, we approached the caustic surface by applying a Taylor series, which provided the analytical spherical aberration coefficient at the third order. Two alternatives to design achromatic separated doublets are provided. Finally, we extended the method to design achromatic cemented triplets.

The Truth on Gravity and Terrestrial Global Warming. Part II: The Regenerative Mass

The parabolic gravitized space {Rs, Re} = e3/2 Ker-1(3/4)•(1010-1012)m, back-scattering of radiant energy at the onset of light, or the depth of gravity field, engenders a thermodynamic field upon caustic developable surfaces characterized by certain caustic developable polar (twisted ) curves, where the surrounding of their tangent lines (or rulings) envelop embodied matter or gravitized mass. In contrast to the gravity polarization (Ker = ¾) of photon nature (e, e2, e3), occurring reversible cyclical processes with excess energy, the thermal polarization (Ker = 2/3) can produce cross exchanges of heat, matter and work with the surrounding area in the inverse order, i.e. in a reversed cycle, only in the case of microscopic atomic-molecular structures of statistical nature (principle of the microscopic reversibility). At the macroscale, such a reciprocating mass process, together with its conservation, is possible only for self-regenerating matter processes. The relative mass concept and invariance of regenerative mass process along with its frozen metastable equilibrium are considered, including the time quantization (clock’s law) as the half-life of mass (gravitized matter).

An experimental study of the alteration of basalt on the surface of Venus

Characterizing the surface of Venus has been complicated by its thick atmosphere and caustic surface conditions (~470 °C, 90 bars). Several approaches, including the collection of spectral data, thermodynamic modelling, lander missions, and surface weathering laboratory experiments have progressed our view of what lies at the surface. However, surface-atmosphere interactions remain somewhat unconstrained and interpretations of the spectral data rely on an understanding of the surface-atmosphere alteration. We used a cold-seal pressure vessel apparatus pressurized with pure CO2 gas and both synthetic and natural glassy basalts specimens to simulate chemical weathering on the surface of Venus for a duration of two weeks. The extent of alteration was described from the surface of samples to depth using Rutherford Backscatter Spectroscopy (RBS), scanning electron microscopy (SEM), and transmission electron microscopy (TEM) techniques. We describe the alteration zones of reacted basalt specimens and report an enrichment of divalent cation species at the near surfaces of basalts; Ca2+ was enriched by ~5 wt% and Fe2+ was enriched by ~1–2 wt%. The enrichment at the near surface favors the production of iron (Fe) oxide(s) and carbonates on the surfaces, which form discontinuous coatings on all reacted samples in two weeks duration. Our results aid in the interpretation of radar emissivity data by constraining which alteration products should be present on the surface and suggesting timeframes necessary for their detection. Assuming that radar emissivity is able to discern weathered basalt, especially those dominated by carbonates and/or semiconducting minerals Fe oxide(s), our results suggest that basalts at Idunn Mons in Imdr Regio previously thought to be anywhere from 2.5 million to a few years old (Smrekar et al., 2010; D'Incecco et al., 2017; Filiberto et al., 2020; Cutler et al., 2020) could instead be as young as 65,000 years to 110,000 years depending on basalt type.

Multifocal wavefronts with prescribed caustics in axially symmetric optical systems

Multifocal and/or extended depth-of-focus designs are widely used in many optical applications. In most of them, the optical configuration has axial symmetry. A usual design strategy consists of exploring the optimal wavefronts that emerging out of the optical system would provide the desired multifocal properties. Those properties are closely related to light concentration on caustic surfaces. We present a systematic analysis of how to obtain those multifocal wavefronts given some prescriptions on the locations of caustics. In particular, we derive several multifocal wavefronts under archetypical prescriptions in the sagittal caustic alone, or combined with the tangential one at certain points, with some emphasis on visual optics applications.

Proof that spot diagram borders are always caustic curves and/or marginal rays II: annular (circular and elliptical) and single hexagonal exit pupils.

In a previous paper [Appl. Opt. 61, C20 (2022)] it was proven that for a circular exit pupil and any optical path differences, the border of any spot diagram is integrated by the caustic surface and/or marginal rays. In this paper, the previous results are extended to annular (circular an elliptical) as well as hexagonal (single and segmented) exit pupils. Several examples of wavefronts affected by linear combinations of orthonormal Zernike aberrations are shown.

Caustics of developable surfaces

While considering a mirror and light rays coming either from a point source or from infinity, the reflected light rays may have an envelope, called a caustic curve. In this paper, we study developable surfaces as mirrors. These caustic surfaces, described in a closed form, are also developable surfaces of the same type as the original mirror surface. We provide efficient, algorithmic computation to find the caustic surface of each of the three types of developable surfaces (cone, cylinder, and tangent surface of a spatial curve). We also provide a potential application of the results in contemporary free-form architecture design.

AUTOMATIC CONTROL OF THE ENERGY STATE OF THE WORKING LIQUID IN THE BIOREACTOR AT THE RESONANCE LEVEL

The absence of mechanical mixing devices and perfect heat and mass transfer and intensive aeration of the working fluid significantly increase the productivity and quality of the technological process, which determines the relevance of the studied problem. The aim is to increase the productivity of the technological process, which is solved by intensifying heat and mass transfer throughout the apparatus by remote artificial formation of the resonant state of the culture medium and the energy activity of the working fluid generated by this state. The task of the work is the artificial formation of the energy activity of the working fluid in the bioreactor using an ultrasonic beam. The research methodology is based on classical methods of theoretical mechanics, methods of mathematical physics, theory of thin shells, which ensures the reliability and reliability of the results. One of the possible technical solutions of artificial formation of energy activity of the working fluid in the bioreactor by means of an ultrasonic beam is given. The conditions for the occurrence of wave coincidence, which is accompanied by a sharp increase in energy activity along with minimal dissipation of radiation waves, are outlined. The nature of additional energy activity in the zones of caustic surfaces by two types of waves is explained. The process of culturing organisms, like most heterogeneous chemical reactions, directly depends on the amount of gas dissolved in the liquid and the formation and accumulation of the target product, in particular biomass. Thus, increasing the degree of dissolution of the gas in the liquid phase of the reactions will reduce its consumption and reduce energy consumption for aeration. The constructed calculation schemes and the received results allow to estimate adequacy of theoretical decisions by initial assumptions and, thus, to confirm their expediency for the further practical realization. The spatial resonance created inside the housing by ultrasonic emitters turbulizes the entire liquid-phase component of the housing and causes the gas bubbles to move without hindrance not only upwards, but also in other various directions, keeping them longer in the working fluid, thus enriching the mixture. creating conditions for intensive cell growth.

Transformations of structurally stable states of spiral beams subjected to sector perturbations

We investigated conditions for the violation of the structural stability of a spiral beam subject to sector perturbations. Based on the method of computer simulation and measurement of mode spectra, we have shown that a spiral vortex beam has a characteristic caustic surface, the intersection of which sharply changes the shape of the Poynting vector streamlines and the total topological charge of the beam. Sector beam perturbation does not almost change the streamline structure up to scale and rotation. We found that perturbation of the beam causes a change in the direction of circulation of streamlines in the region of perturbation, which is caused by the appearance of vortices with negative topological charges. Their contribution to the total energy flow is fractions of a percent. However, such perturbations do not cause changing the OAM in the beam, despite an increase in the number of vortex modes. Nevertheless, the perturbed beam remains only conditionally structurally stable due to the presence of a small fraction of optical currents with opposite circulations.

Distorted mirror images organised by cuspoid and umbilic caustics

The anatomy of the distorted and topologically disrupted images of an extended object after reflection by curved mirrors, and of their refraction analogues, e.g. in gravitational lensing, is determined by the caustic surfaces enveloping the ray family issuing from each point of the object. Simulations are presented of reflections of the same object when organised by each of the five caustics (elementary catastrophes) that are stable in three-dimensional space: the fold, cusp, swallowtail, elliptic and hyperbolic umbilic catastrophes. The object is a picture of a face, whose familiar features make the image distortions and disruptions easier to understand. An important distinction is between mirrors curved in one direction and those curved in two.

Approach for designing thick cemented doublet lenses based on the caustic surface.

A new, to the best of our knowledge, method for designing a thick-lens achromatic doublet based on the concept of a caustic surface to correct both third- and fifth-order spherical aberration is presented. We consider two different wavelengths brought into coincidence at the back focal length instead of the effective focal length as it is usually done, to calculate the radii of curvature assuming predefined values for axial thicknesses and their indices of refraction for both lenses. Alternatively, we apply Taylor's series around the optical axis, and to vanish the approximate caustic surface, we obtain the values for the conic constants, which reduce at third- and fifth-order spherical aberration. Two designs for cemented doublets are provided assuming that the lenses are cemented. Finally, we propose a method to qualitatively test doublet lenses by using null screens, considering to place the detection plane at arbitrary positions.

Borders of spot diagrams are always caustic surfaces and/or marginal rays.

In this paper, it is proven that for any optical system with a circular exit pupil and wavefronts affected by any aberrations, the borders of all leaving rays are caustic surfaces and/or marginal rays. Several examples are shown for wavefronts affected by linear combinations of Zernike aberrations.

Structural stability of spiral vortex beams to sector perturbations.

Conditions of breaking down the structural stability of a spiral vortex beam subject to sector perturbations were considered. Employing methods of computer simulation and processing experimental results, we have shown that the spiral vortex beam has a caustic surface, the intersection of which sharply changes a shape of the Poynting vector streamlines and critical points of the spiral beam. Nevertheless, the beam propagation (scaling and rotation) does not change the perturbed streamline's shape and phase pattern. We also revealed that strong beam perturbations can cause the conversion of the circulation direction of streamlines in the perturbation region, which entails the appearance of a network of optical vortices with negative topological charges. However, the beam's orbital angular momentum remains unchanging, despite increasing the information entropy (growing a number of vortex modes), so that the perturbed beam keeps new stable states.

СИНТЕЗ ДОПЛЕРОГРАМ КВАЗІВЕРТИКАЛЬНОГО ЗОНДУВАННЯ ІОНОСФЕРИ ДОПЛЕРІВСЬКИМ ВЧ РАДАРОМ ІЗ РОЗНЕСЕНИМ ПРИЙМАННЯМ

Purpose: The ionospheric channel is widely used for the communication, radio navigation, radar, direction finding, radio astronomy, and remote radio probing systems. The radio channel parameters are characterized by nonstationarity due to the dynamic processes in the ionosphere, and therefore their study is one of the topical problems of space radio physics and earth-space radio physics of geospace. This work aims at presenting the results of synthesis of temporal variations in the Doppler spectra obtained by the Doppler probing of the ionosphere at vertical and quasi-vertical incidence. Design/methotology/approach: One of the most effective methods of ionosphere research is the Doppler sounding technique. It has a high time resolution (about 10 s), a Doppler shift resolution (0.01–0.1 Hz), and the accuracy of Doppler shift measurements (~0.01 Hz) that permits monitoring the variations in the ionospheric electron density (10–4–10–3) or the study of the ionospheric plasma motion with the speed of 0.1-1 m/s and greater. The solution of the inverse radio physical problem, consisting in determination of the ionosphere parameters, often means solving the direct radio physical problem. In the Doppler sounding technique, it belongs with the construction of variations in Doppler spectra and comparing them with the Doppler spectra measurements. Findings: For the radio wave ordinary component, three echoes being produced by three rays are observed. Influence of the geomagnetic fi eld and large horizontal gradients in the electron density of δ≥10 % give rise to complex ray structures with caustic surfaces. The ionospheric disturbances traveling along the magnetic meridian form the skip zones. The longitudinal and transverse displacement of the ray reflection point attains a few tens of kilometers along the vil. Haidary to vil. Hrakove quasi-vertical radiowave propagation path, for which the great circle range is 50 km. For the vertical incidence, the signal azimuth at the receiver coincides with the traveling ionospheric disturbance azimuth. The synthesis of temporal variations in the HF Doppler spectra has been made and compared with the temporal variations in the Doppler spectra recorded with the V. N. Karazin Kharkiv National University radar. The estimate of δ=15 % obtained confirms the existence of large horizontal gradients in electron density. Conclusions: Temporal variations in Doppler spectra and in azimuth have been calculated for the vertical and quasi-vertical incidence with allowance for large horizontal gradients of the electron density caused by traveling ionospheric disturbances. Key words: ionosphere, Doppler sounding at oblique incidence, synthesis of temporal variations in HF Doppler spectra, traveling ionospheric disturbances, electron density

Shock-induced cavitation and wavefront analysis inside a water droplet

The objective of this study is to develop a basic understanding of the interaction of shock waves with density inhomogeneities. We consider the particular instance of a planar air shock impinging on a spherical water droplet and discuss to what extent this interaction can lead to the inception of cavitation inside the droplet. The effort centers on early phases of the interaction, which are analyzed using both ray theory and a hydrodynamic code. Within the context of ray theory, the occurrence of focusing is examined in detail, and parametric equations are derived for the transmitted wavefront and its multiple internal reflections. It is found that wave patterns predicted by ray calculations compare extremely well with the more accurate numerical solutions from simulations. In particular, it is shown that the internal wavefront assumes a complex time-dependent shape whose dominant feature is the existence of cusp singularities. These singular points are shown to trace out surfaces that are the caustics of the associated system of rays. From the singularities of the energy flux density of the refracted wave, the parametric equations of the caustic surface associated with the kth reflected wavefront are deduced. As a consequence of the focusing process, simulations show the formation of negative-pressure regions in the internal flow field. These low-pressure zones are identified as possible spots at which cavitation may occur, depending on the magnitude of pressure reached. Finally, numerical results provide quantitative information on the dependence of negative-pressure peaks upon incident-shock-wave strength.

Ronchi-Hartmann type null screens for testing a plano-freeform surface with a detection plane inside a caustic surface.

We have implemented an exact ray trace through a plano-freeform surface for an incident plane wavefront. We obtain two caustic surfaces and provide the critical points related to the ray tracing process. Additionally, we study the propagation of the refracted wavefronts through the plane-curved surface. Finally, by using the Ronchi-Hartmann type null screen and placing the detection plane within the caustic region, we have evaluated the shape of a plano-freeform optical surface under test, obtaining an RMS difference in sagitta value of 6.3 μm.

Caustics of Non-Paraxial Perfect Optical Vortices Generated by Toroidal Vortex Lenses

In this paper, we consider the comparative formation of perfect optical vortices in the non-paraxial mode using various optical elements: non-paraxial and parabolic toroidal vortex lenses, as well as a vortex axicon in combination with a parabolic lens. The theoretical analysis of the action of these optical elements, as well as the calculation of caustic surfaces, is carried out using a hybrid geometrical-optical and wave approach. Numerical analysis performed on the basis of the expansion in conical waves qualitatively confirms the results obtained and makes it possible to reveal more details associated with diffraction effects. Equations of 3D-caustic surfaces are obtained and the conditions of the ring radius dependence on the order of the vortex phase singularity are analyzed. In the non-paraxial mode, when small light rings (several tens of wavelengths) are formed, a linear dependence of the ring radius on the vortex order is shown. The revealed features should be taken into account when using the considered optical elements forming the POV in various applications.