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Tomasz Radozycki

Publications and source records attributed to Tomasz Radozycki.

At least 19 recordsLinked to original sources
arXiv2025

Generalized solution of the paraxial equation

A fairly general expression for a light beam is found as a solution of the paraxial Helmholtz equation. It is achieved by exploiting appropriately chosen complex variables which entail the separability of the equation. Next, the expression for the beam is obtained independently by superimposing shifted Gaussian beams, whereby the shift can be made either by a real vector (in which case the foci of the Gaussian beams are located on a circle) or by a complex one. The solutions found depend on several parameters, the specific choice of which allows to obtain beams with quite different properties. For several selected parameter values figures are drawn, demonstrating the spatial distribution of the energy density and phase. In special cases, the effect of a shift of the intensity peak from one branch to another and phase singularities are observed.

physics.optics↗
arXiv2025

Accurate paraxial Maxwell beams

In this work, the paraxial version of Maxwell equations is derived with the use of two Riemann-Silberstein vectors. Exact solutions of these equations are then obtained representing the paraxial electromagnetic fields. These fields satisfy the full Maxwell equations up to the order of $(λ/w_0)^2$. The solutions contain some additional terms, which turn out to be relevant for polarization, especially in the case of beams endowed with orbital angular momentum.

physics.optics↗
arXiv2024

Guiding polarizable particles in multi-hole Gaussian beams

The present paper discusses certain special Gaussian beams that, thanks to some polynomial prefactors, have uniquely designed holes in the irradiance. Such holes, or rather tubes, can constitute potential valleys for negatively polarizable particles, providing the possibility of guiding several objects of that kind, each along its own trajectory. The mechanism of creating these holes by interference of Gaussian beams which exhibit orbital angular momentum is discussed, and then the trajectories of particles moving in such a wave are numerically calculated. As it turns out, these particles, performing transverse oscillations, follow the designed tunnels of low irradiance. On the contrary, for particles with positive polarizability these areas are inaccessible.

physics.optics↗
arXiv2020

Knotted nodal lines in superpositions of Bessel-Gaussian light beams

A simple analytical way of creating superpositions of Bessel-Gaussian light beams with knotted nodal lines is proposed. It is based on the equivalence between the paraxial wave equation and the two-dimensional Schrödinger equation for a free particle. The $2D$ Schrödinger propagator is expressed in terms of Bessel functions, which allows to obtain directly superpositions of beams with a desired topology of nodal lines. Four types of knots are constructed in the explicit way: the unknot, the Hopf link, the Borromean rings and the trefoil. It is also shown, using the example of the figure-eight knot, that more complex structures require larger number of constituent beams as well as high precision both from the numerical and the experimental side. A tiny change of beam's intensity can lead to the knot "switching".

physics.optics↗
arXiv2020

Knotted trajectories of neutral and charged particles in Gaussian light beams

Making use of the equivalence between paraxial wave equation and two-dimensional Schrödinger equation, Gaussian beams of monochromatic light, possessing knotted nodal structures are obtained in an analytical way. These beams belong to the wide class of paraxial beams called the Hypergeometric-Gaussian beams [E. Karimi, G. Zito, B. Piccirillo, L. Marrucci and E. Santamato, Opt. Lett. {\bf 32}, 3053(2007)]. Four topologies are dealt with: the unknot, the Hopf link, the Borromean rings and the trefoil. It is shown in the numerical way that neutral polarizable particles placed in such light fields, upon precise tuning of the initial conditions, can be forced to follow the identical knotted trajectories. A similar outcome is also valid for charged particles that are subject to a ponderomotive potential. This effect can serve to precisely steer particles along chosen complicated pathways exhibiting non-trivial topological character, guide them around obstacles and seems to be helpful in engineering more complex nanoparticles.

physics.optics↗
arXiv2019

Manipulating neutral particles in Bessel beams: from rings, through fixed helices to 3D traps

The motion of neutral, polarizable atoms (also called neutral particles in this work) in the field of the Bessel beam is considered. It is shown in the numerical way, that the Bessel rings, i.e., the regions of high energy concentration can trap particles of positive polarizability (atoms in red-detuned beams). This trapping occurs only in the plane perpendicular to the wave propagation, and the motion along the beam is unrestricted. When the beam is superposed with the plane wave of the same frequency propagating in the same direction, the particles are guided along helices, fixed in space. The shape of these helices depends on the parameters characterizing the electromagnetic fields but not on the initial state of guided particles. Depending on the vorticity of the Bessel beam, these helices can be made left or right-handed. In the special case of zero vorticity, the helices get degenerated to the true, three-dimensional rings, which can serve as $3D$ traps. The emerging structure of potential valleys can be applied to parallel guidance or capture several independent atoms, each in its own trap.

physics.optics↗
arXiv2019

Guiding neutral polar molecules by electromagnetic vortex field --- a classical approach

It is shown, within classical mechanics, that the field of an electromagnetic vortex is capable of capturing and guiding neutral molecules endowed with a permanent electric dipole moment (PEDM). Similarly as in the case of the magnetic field applied to elementary particles or atoms, this effect turns out to be very delicate because of the small values of PEDM observed in real molecules. They amount to $2\times10^{5}\, e{\mathrm{fm}}$ (electron charge $\times$ fermi) or less, which requires the use of very strong electric fields. It has also been observed that there exists a threshold in field strength above which the particles are ejected from the trap. Trajectories of guided particles are usually quite chaotic, which is a consequence of non-linearity of the equations of motion. With a very special and precise adjustment of parameters, a regular (i.e., circular, in the transverse plane) trajectory can be obtained. The presence of an additional constant electric field pointing along the direction of the wave propagation might help to achieve the necessary tuning and realize such trajectories.

physics.optics↗
arXiv2018

Geometrical optics and geodesics in thin layers

The propagation of a light ray in thin layer (film) within geometrical optics is considered. It is assumed that the ray is captured inside the layer due to reflecting walls or total internal reflection (in the case of a dielectric layer). It has been found that for a very thin film (the length scale is imposed by the curvature of the surface at a given point) the equations describing the trajectory of the light beam are reduced to the equation of a geodesic on the limiting curved surface. There have also been found corrections to the trajectory equation resulting from the finite thickness of the film. Numerical calculations performed for a couple of exemplary curved layers (cone, sphere, torus and catenoid) confirm that for thin layers the light ray which is repeatedly reflected, propagates along the curve close to the geodesic but as the layer thickness increases, these trajectories move away from each other. Because the trajectory equations are complicated non-linear differential equations, their solutions show some chaotic features. Small changes in the initial conditions result in remarkably different trajectories. These chaotic properties become less significant the thinner the layer under consideration.

physics.optics↗
arXiv2018

Reduction of the classical electromagnetism to a two-dimensional curved surface

The reduction of the three-dimensional classical electromagnetism is performed in a twofold way. In the first case the ordinary two-dimensional electromagnetism is obtained with sources in the form of conserved electric currents flowing along the surface. The electric field is a two-vector tangent to the surface and magnetic field is a scalar quantity. In the second approach the reduced theory is that of the two-vector magnetic field and a scalar electric one. The only source coupled to the fields is now a scalar subject to no conservation law. In the redefined theory this scalar source is may be converted into an eddy magnetic current flowing in the surface. No magnetic monopoles appear. Our results can find some applications in the electrodynamics of thin layers and of metal-dielectric interfaces.

physics.class-ph↗
arXiv2017

Limitations in the $2D$ description of the electromagnetic waves propagation in thin dielectric and magnetic layers

The propagation of electromagnetic waves trapped within dielectric and magnetic layers is considered. The description within the three-dimensional theory is compared with the simplified analysis in two dimensions. Two distinct media configurations with different topology are dealt with: a plane slab and a hollow cylinder. Choosing the appropriate values for the geometrical parameters (layer thickness, radius of the cylinder) and for the electromagnetic properties of the media one can trap exactly one mode corresponding to that obtained within the two-dimensional electromagnetism. However, the symmetry between electric and magnetic fields suggests, that the two versions of the simplified electromagnetism ought to be taken into account. Its usual form is incomplete to describe all modes. It is also found that there is a domain of optimal values of parameters for which the $2D$ model works relatively correctly. In the case of a cylindrical surface we observe, however, several differences which are attributed to the curvature of the layer, and which exclude the propagation of evanescent modes. The two-dimensional electrodynamics, whichever form used, turns out still too poor to describe the so called `hybrid modes' excited in a real layer. The obtained results can be important for proper description of the propagating waves within thin layers for which $3D$ approach is not available due to mathematical complexity, and reducing the layer to a lower-dimensional structure seems the only possible option.

physics.class-ph↗
arXiv2016

Classical probability density distributions with uncertainty relations for ground states of simple non-relativistic quantum-mechanical systems

The probability density distributions for the ground states of certain model systems in quantum mechanics and for their classical counterparts are considered. It is shown, that classical distributions are remarkably improved by incorporating into them the Heisenberg uncertainty relation between position and momentum. Even the crude form of this incorporation makes the agreement between classical and quantum distributions unexpectedly good, except for the small area, where classical momenta are large. It is demonstrated that the slight improvement of this form, makes the classical distribution very similar to the quantum one in the whole space. The obtained results are much better than those from the WKB method. The paper is devoted to ground states, but the method applies to excited states too.

quant-ph↗
arXiv2015

Lorentz contraction of the equal-time Bethe-Salpeter amplitude in two-dimensional massless quantum electrodynamics

The Lorentz transformation properties of the equal-time bound-state Bethe-Salpeter amplitude in the two-dimensional massless quantum electrodynamics (the so called Schwinger Model) are considered. It is shown that while boosting a bound state (a `meson') this amplitude is subject to approximate Lorentz contraction. The effect is exact for large separations of constituent particles (`quarks'), while for small distances the deviation is more significant. For this phenomenon to appear, the full} function, i.e. with the inclusion of all instanton contributions has to be considered. The amplitude in each separate topological sector does not exhibit such properties.

hep-th↗
arXiv2014

Relative energy dependence of the Bethe-Salpeter amplitude in two-dimensional massless QED

The exact $q\bar{q}$ Bethe-Salpeter bound state amplitude is investigated in the space of relative energy $E$ for fixed value of relative position. By means of approximate analysis it is shown to possess singularities in $E$ whenever one of the quarks reaches the energy threshold for the creation of certain number of Schwinger bosons. These results are afterwards confirmed by numerical plots.

hep-th↗
arXiv2013

Relative variables of the exact fermion-antifermion bound state wave-function in the Schwinger Model

The exact Bethe-Salpeter amplitude for the fermion-antifermion bound state in the Schwinger Model is investigated. The dependence on the relative time and position in the center-of-mass frame in all contributing instanton sectors is analyzed. The same is accomplished for the relative energy and momentum variables. Several interesting properties of the amplitude are revealed. The explicit threshold structure is demonstrated.

hep-th↗
arXiv2013

Infrared self-consistent solutions of spinor QED3

Three-dimensional electrodynamics in the spinor (i.e. two-component) version is considered. With the use of the so called Salam's vertex, the infinite hierarchy of Dyson-Schwinger equations is turned into a set of four self-consistent equations for four parameters describing the infrared behavior of fermion and boson propagators. It is shown numerically, that this set of equations has solutions, at least for certain values of gauge parameter. For weak coupling (i.e. for heavy fermions) the values of all these quantities are found analytically. In the case of massless bare fermion, masses of both particles are generated leading thereby to parity breaking.

hep-th↗
arXiv2013

Infrared self-consistent solutions of bispinor QED3

Quantum electrodynamics in three dimensions in the bispinor formulation is considered. It is shown that the Dyson-Schwinger equations for fermion and boson propagators may be self-consistently solved in the infrared domain if on uses the Salam's vertex function. The parameters defining the behavior of the propagators are found numerically for different values of coupling constant and gauge parameter. For weak coupling the approximated analytical solutions are obtained. The renormalized gauge boson propagator (transverse part) is shown in the infrared domain to be practically gauge independent.

hep-th↗
arXiv2008

Instantons and the infrared behavior of the fermion propagator in the Schwinger Model

Fermion propagator of the Schwinger Model is revisited from the point of view of its infrared behavior. The values of anomalous dimensions are found in arbitrary covariant gauge and in all contributing instanton sectors. In the case of a gauge invariant, but path dependent propagator, the exponential dependence, instead of power law one, is established for the special case when the path is a straight line. The leading behavior is almost identical in any sector, differing only by the slowly varying, algebraic prefactors. The other kind of the gauge invariant function, which is the amplitude of the dressed Dirac fermions, may be reduced, by the appropriate choice of the dressing, to the gauge variant one, if Landau gauge is imposed.

hep-th↗