Dávid Juhász - Academia.edu (original) (raw)

Papers by Dávid Juhász

Research paper thumbnail of Convergence and completeness for square-well Stark resonant state expansions

Journal of Mathematical Physics, 2018

In this paper, we investigate the completeness of the Stark resonant states for a particle in a s... more In this paper, we investigate the completeness of the Stark resonant states for a particle in a square-well potential. We find that the resonant state expansions for target functions converge inside the potential well and that the existence of this convergence does not depend on the depth of the potential well, V0. By analyzing the asymptotic form of the terms in these expansions, we prove some results on the relation between smoothness of target functions and the asymptotic rate of convergence of the corresponding resonant state expansion and show that the asymptotic rate of convergence is also independent of V0, but the absolute size terms in the series asymptotically goes as V0−1.

Research paper thumbnail of Modelling pulse propagation in complex index materials using the method of multiple scales

Physica Scripta, 2021

In this paper we study pulse propagation in complex refractive index materials, modelled by a dis... more In this paper we study pulse propagation in complex refractive index materials, modelled by a dispersive wave equation, using the method of multiple scales (MMS), and perform several numerical tests to investigate its accuracy. We assume a complex valued refracting index in a noncentrosymmetric medium with a Kerr response. The key feature of our MMS solution is the linearity of the amplitude equation and the complex nature of the mode-frequency. The MMS is tested as an initial value problem using three different dispersion models. Depending on the parameters of the problem, the amplitude equation can be both well-or ill-posed. Despite the ill-posedness, the MMS solution remains a valid approximation of the solution to the original nonlinear model.

Research paper thumbnail of Convergence and completeness for square-well Stark resonant state expansions

JOURNAL OF MATHEMATICAL PHYSICS, 2018

In this paper, we investigate the completeness of the Stark resonant states for a particle in a s... more In this paper, we investigate the completeness of the Stark resonant states for a particle in a square-well potential. We find that the resonant state expansions for target functions converge inside the potential well and that the existence of this convergence does not depend on the depth of the potential well, V0. By analyzing the asymptotic form of the terms in these expansions, we prove some results on the relation between smoothness of target functions and the asymptotic rate of convergence of the
corresponding resonant state expansion and show that the asymptotic rate of convergence is also independent of V0, but the absolute size terms in the series asymptotically goes as V_0^-1.

Research paper thumbnail of Constructing a partially transparent computational boundary for UPPE using leaky modes

Journal of Mathematical Physics, 2019

In this paper we introduce a method for creating a transparent computational boundary for the sim... more In this paper we introduce a method for creating a transparent computational boundary for the simulation of unidirectional propagation of optical beams and pulses using leaky modes. The key element of the method is the introduction of an articial-index material outside a chosen computational domain and utilization of the quasi-normal modes associated with such articial structure. The method is tested on the free space propagation of TE electromagnetic waves. By choosing the material to have appropriate optical properties one can greatly reduce the reection at the computational boundary. In contrast to the well-known approach based on a perfectly matched layer, our method is especially well suited for spectral propagators.

Research paper thumbnail of Nonlinear optics

Nonlinear light-matter interactions have been drawing attention of physicists since the 1960'... more Nonlinear light-matter interactions have been drawing attention of physicists since the 1960's. Quantum mechanics played a significant role in their description and helped to derive important formulas showing the dependence on the intensity of the electromagnetic field. High intensity light is able to generate second and third harmonics which translates to generation of electromagnetic field with multiples of the original frequency. In comparison with the linear behaviour of light, the nonlinear interactions are smaller in scale. This makes perturbation methods well suited for obtaining solutions to equations in nonlinear optics. In particular, the method of multiple scales is deployed in paper 3, where it is used to solve nonlinear dispersive wave equations. The key difference in our multiple scale solution is the linearity of the amplitude equation and a complex valued frequency of the mode. Despite the potential ill-posedness of the amplitude equation, the multiple scale solu...

Research paper thumbnail of Modelling laser-matter interactions using resonant states

Studying how light interacts with materials has become important for many technological applicati... more Studying how light interacts with materials has become important for many technological applications from optical communication to developing of new materials. Therefore scientists have always tried to improve their understanding of these effects. The primary goal has always been to microscopically describe the pertinent processes. This paper provides a brief introduction into the interactions of atoms with laser fields. Precisely this interaction, photoelectric effect and the blackbody radiation were those findings which started off the development of quantum mechanics. This theory allowed better description of atoms and it will be used in this work to handle the problem we are confronting. We will consider two of the simplest potentials and let the atom interact with a strong laser pulse in these potentials. From this interaction the so called resonant states will arise. The goal of this thesis is to investigate to what extent and in what meaning these resonant states form a compl...

Research paper thumbnail of Convergence and completeness for square-well Stark resonant state expansions

Journal of Mathematical Physics, 2018

In this paper, we investigate the completeness of the Stark resonant states for a particle in a s... more In this paper, we investigate the completeness of the Stark resonant states for a particle in a square-well potential. We find that the resonant state expansions for target functions converge inside the potential well and that the existence of this convergence does not depend on the depth of the potential well, V0. By analyzing the asymptotic form of the terms in these expansions, we prove some results on the relation between smoothness of target functions and the asymptotic rate of convergence of the corresponding resonant state expansion and show that the asymptotic rate of convergence is also independent of V0, but the absolute size terms in the series asymptotically goes as V0−1.

Research paper thumbnail of Modelling pulse propagation in complex index materials using the method of multiple scales

Physica Scripta, 2021

In this paper we study pulse propagation in complex refractive index materials, modelled by a dis... more In this paper we study pulse propagation in complex refractive index materials, modelled by a dispersive wave equation, using the method of multiple scales (MMS), and perform several numerical tests to investigate its accuracy. We assume a complex valued refracting index in a noncentrosymmetric medium with a Kerr response. The key feature of our MMS solution is the linearity of the amplitude equation and the complex nature of the mode-frequency. The MMS is tested as an initial value problem using three different dispersion models. Depending on the parameters of the problem, the amplitude equation can be both well-or ill-posed. Despite the ill-posedness, the MMS solution remains a valid approximation of the solution to the original nonlinear model.

Research paper thumbnail of Convergence and completeness for square-well Stark resonant state expansions

JOURNAL OF MATHEMATICAL PHYSICS, 2018

In this paper, we investigate the completeness of the Stark resonant states for a particle in a s... more In this paper, we investigate the completeness of the Stark resonant states for a particle in a square-well potential. We find that the resonant state expansions for target functions converge inside the potential well and that the existence of this convergence does not depend on the depth of the potential well, V0. By analyzing the asymptotic form of the terms in these expansions, we prove some results on the relation between smoothness of target functions and the asymptotic rate of convergence of the
corresponding resonant state expansion and show that the asymptotic rate of convergence is also independent of V0, but the absolute size terms in the series asymptotically goes as V_0^-1.

Research paper thumbnail of Constructing a partially transparent computational boundary for UPPE using leaky modes

Journal of Mathematical Physics, 2019

In this paper we introduce a method for creating a transparent computational boundary for the sim... more In this paper we introduce a method for creating a transparent computational boundary for the simulation of unidirectional propagation of optical beams and pulses using leaky modes. The key element of the method is the introduction of an articial-index material outside a chosen computational domain and utilization of the quasi-normal modes associated with such articial structure. The method is tested on the free space propagation of TE electromagnetic waves. By choosing the material to have appropriate optical properties one can greatly reduce the reection at the computational boundary. In contrast to the well-known approach based on a perfectly matched layer, our method is especially well suited for spectral propagators.

Research paper thumbnail of Nonlinear optics

Nonlinear light-matter interactions have been drawing attention of physicists since the 1960'... more Nonlinear light-matter interactions have been drawing attention of physicists since the 1960's. Quantum mechanics played a significant role in their description and helped to derive important formulas showing the dependence on the intensity of the electromagnetic field. High intensity light is able to generate second and third harmonics which translates to generation of electromagnetic field with multiples of the original frequency. In comparison with the linear behaviour of light, the nonlinear interactions are smaller in scale. This makes perturbation methods well suited for obtaining solutions to equations in nonlinear optics. In particular, the method of multiple scales is deployed in paper 3, where it is used to solve nonlinear dispersive wave equations. The key difference in our multiple scale solution is the linearity of the amplitude equation and a complex valued frequency of the mode. Despite the potential ill-posedness of the amplitude equation, the multiple scale solu...

Research paper thumbnail of Modelling laser-matter interactions using resonant states

Studying how light interacts with materials has become important for many technological applicati... more Studying how light interacts with materials has become important for many technological applications from optical communication to developing of new materials. Therefore scientists have always tried to improve their understanding of these effects. The primary goal has always been to microscopically describe the pertinent processes. This paper provides a brief introduction into the interactions of atoms with laser fields. Precisely this interaction, photoelectric effect and the blackbody radiation were those findings which started off the development of quantum mechanics. This theory allowed better description of atoms and it will be used in this work to handle the problem we are confronting. We will consider two of the simplest potentials and let the atom interact with a strong laser pulse in these potentials. From this interaction the so called resonant states will arise. The goal of this thesis is to investigate to what extent and in what meaning these resonant states form a compl...