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October 2011 - present

October 2009 - June 2011

January 1997 - March 2009

## Publications

Publications (53)

The present paper discusses the application of 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 it...

In this work, the paraxial approximation of the free Dirac equation is examined. The results are first obtained by constructing superpositions of exact solutions with suitable profiles, which are borrowed from paraxial optics. In this manner, the paraxial Dirac beams are obtained in four cases: as Gaussian, Bessel-Gaussian, modified Bessel-Gaussian...

Various superpositions of Bessel-Gaussian beams and modified Bessel Gaussian beams are considered. Two selected parameters characterizing these beams, with respect to which the superpositions are constructed, are the topological index $n$ associated with the orbital angular momentum carried by the beam, and $\chi$ related to the dilation of the bea...

An explicit formula for a new type of beams, which in this work are called the "special" hyperbolic Bessel-Gaussian (SHBG) beams, has been derived, using the method of the Hankel transform formulated in our previous work. The fundamental properties of these beams are analyzed. The parameters that define the beam shape have been identified and relat...

Various superpositions of Bessel-Gaussian beams and modified Bessel-Gaussian beams are considered. Two selected parameters characterizing these beams, with respect to which the superpositions are constructed, are the topological index n associated with the orbital angular momentum carried by the beam and χ related to the dilation of the beam. It is...

A few-parameter 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 e...

A concise method of deriving expressions for Gaussian-like solutions of the paraxial and wave equation (i.e. 3D d’Alembert equation) is presented. This method is based on the Hankel transform. Choosing some Gaussian base functions with slight modifications of the prefactor all basic beams of cylindrical character can be easily obtained. This refers...

An analytic formula for a certain type of a cylindrical beam, which might be called a γ beam, has been derived directly from the paraxial equation and independently using the method of the Hankel transform formulated in our previous work [T. Radożycki, Opt. Laser Technol. 147, 107670 (2022)]. The fundamental properties of this beam are analyzed and...

An explicit formula for a type of beams, which in this work are called the “special” hyperbolic Bessel-Gaussian (SHBG) beams, has been derived, using the method of the Hankel transform formulated in our previous work [T. Radożycki, arXiv:2103.06988]. The fundamental properties of these beams are analyzed. The parameters that define the beam shape h...

A concise method of deriving expressions for Gaussian-like solutions of the paraxial and d'Alembert equations is presented. This method is based on the Hankel transform. Choosing some Gaussian base functions with slight modifications of the prefactor all basic beams of cylindrical character can be easily obtained. This refers to Gaussian, Bessel-Ga...

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 di...

Making use of the equivalence between the 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,...

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 Schr\"odinger equation for a free particle. The $2D$ Schr\"odinger propagator is expressed in terms of Bessel functions, which allows to obt...

This textbook offers an extensive list of completely solved problems in mathematical analysis. This third of three volumes covers curves and surfaces, conditional extremes, curvilinear integrals, complex functions, singularities and Fourier series. The series contains the material corresponding to the first three or four semesters of a course in Ma...

This textbook offers an extensive list of completely solved problems in mathematical analysis. This second of three volumes covers definite, improper and multidimensional integrals, functions of several variables, differential equations, and more. The series contains the material corresponding to the first three or four semesters of a course in Mat...

This textbook offers an extensive list of completely solved problems in mathematical analysis. This first of three volumes covers sets, functions, limits, derivatives, integrals, sequences and series, to name a few. The series contains the material corresponding to the first three or four semesters of a course in Mathematical Analysis.
Based on th...

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 val...

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 plan...

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 val...

The reduction of the three-dimensional classical electromagnetism to a two-dimensional curved surface 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...

The propagation of a light ray in a 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...

The quantum-mechanical states of neutral particles endowed with the magnetic moment (such as neutrons, light atoms, or even neutrinos, although the effect will be extremely tiny) in the combination of electromagnetic vortex field together with the constant magnetic field are investigated. It is shown that this system of fields is in principle capab...

The propagation of electromagnetic waves trapped within dielectric and magnetic layers is considered. The description within the three-dimensional theory is compared to the simplified analysis in two dimensions. Two distinct media configurations of different topology are dealt with: a plane slab and a hollow cylinder. Choosing the appropriate value...

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 quant...

The propagation of electromagnetic waves trapped within dielectric and magnetic layers is considered. The description within the three-dimensional theory is compared to the simplified analysis in two dimensions. Two distinct media configurations of different topology are dealt with: a plane slab and a hollow cylinder. Choosing the appropriate value...

The motion of a neutral atom endowed with a magnetic moment interacting with the magnetic field is determined from the Ehrenfest-like equations of motion. These equations for the average values of the translational and spin degrees of freedom are derived from the Schr\"odinger-Pauli wave equation and they form a set of nine coupled nonlinear evolut...

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 sep...

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...

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 numeri...

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...

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 diffe...

Collection of problems with solutions in linear algebra.

The instanton-antiinstanton contributions to the q overline q bound
state pole in the four-point Green function in the Schwinger Model are
calculated. It is shown that these configurations, thanks to the
cancellation of all unwanted terms, are responsible for the restoration
of the perfect factorizability of the residue.

The fermion propagator of the Schwinger model is revisited from the point of view of its infrared behavior. The values of the 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 a power law one,...

We consider the quark-antiquark Green's function in the Schwinger Model with instanton contributions taken into account. Thanks to the fact that this function may analytically be found, we draw out singular terms, which arise due to the formation of the bound state in the theory -- the massive Schwinger boson. The principal term has a pole characte...

Electromagnetic waves with phase defects in the form of vortex lines combined with a constant magnetic field are shown to pin down cyclotron orbits (Landau orbits in the quantum mechanical setting) of charged particles at the location of the vortex. This effect manifests itself in classical theory as a trapping of trajectories and in quantum theory...

The influence of the nonlinear, quantum terms in the Maxwell equations on the evolution of vortex lines was analyzed. The quantum corrections led to the deformation and disappearance of the vortex ring in the considered configuration of the constantly expanded vortex ring. The topological change was expected as a result of nonlinearity introduced b...

We extend our previous analysis of the motion of vortex lines in wave mechanics to the case of more elaborate vortex patterns and to a rotating harmonic trap.

The fermion propagator and the 4-fermion Green function in the massless QED2 are explicitly found with topological effects taken into account. The corrections due to instanton sectors k=+1,-1, contributing to the propagator, are shown to be just the homogenous terms admitted by the Dyson-Schwinger equation for S. In the case of the 4-fermion functi...

The evaluation of the 4-point Green functions in the 1+1 Schwinger model is presented both in momentum and coordinate space representations. The crucial role in our calculations play two Ward identities: i) the standard one, and ii) the chiral one. We demonstrate how the infinite set of Dyson-Schwinger equations is simplified, and is so reduced, th...

Perturbation series for the electron propagator in the Schwinger Model is
summed up in a direct way by adding contributions coming from individual
Feynman diagrams. The calculation shows the complete agreement between
nonperturbative and perturbative approaches.

Assuming an ansatz for the vertex function and the electron propagator suggested by the Ward identity we solve the Dyson-Schwinger equations in quantum electrodynamics in the infrared domain. The nonperturbative results obtained in this way are in agreement with perturbation theory. In our approach the loop integrals are finite. Our procedure is se...

We present an analytical proof of the formula for the distribution of pairs created in the homogeneous electric field with step-type time dependence considered in the quoted paper. We deal with the case of large time and momenta.

We investigate multiphoton ejection probability of strongly bound electrons in relativistically intense laser fields. A solvable model of a Klein-Gordon electron bound in a finite-range separaable potential and interacting with a circularly polarized plane-wave field is used for the analysis. For binding energies of the order of several keV the rat...

Using an exactly solvable model of a bound Klein-Gordon particle in an intense laser field we obtain the above-threshold energy spectra of the ejected electron for intensities in the range I=7×1014 W/cm2 to 1020 W/cm2, and compare them with the predictions of the Schrödinger theory. The present calculations also allow us to test an approximation fo...

We analyze a three-dimensional model of a Klein-Gordon particle in a short-range separable potential and interacting with an intense plane-wave electromagnetic field. In the specific case of the circular polarization of the radiation, we find an exact solution of the Klein-Gordon equation of the system and derive analytic expressions for obtaining...

The properties of the virtual cloud around the hydrogen atom in the ground state are studied with the use of quantum field theory methods. The relativistic expression for the electromagnetic energy density around the atom, with the electron spin taken into account, is obtained. The distribution of the angular momentum contained in the cloud and the...

The influence of the center-of-mass correction and quark-pion interactions on the hadron mass spectrum is investigated. Various possibilities of treating gluonic and pionic corrections in the pressure-balance equation are discussed. With the correction for the finite size of the pion, excellent agreement with experimental data is obtained.