Next Article in Journal
Evaluation of Machine Learning Algorithms for Classification of EEG Signals
Next Article in Special Issue
Distribution Path Optimization by an Improved Genetic Algorithm Combined with a Divide-and-Conquer Strategy
Previous Article in Journal
Editorial for the Special Issue “Reviews and Advances in Materials Processing”
Previous Article in Special Issue
Explainable AI (XAI) Applied in Machine Learning for Pain Modeling: A Review
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:

Exciting of Strong Electrostatic Fields and Electromagnetic Resonators at the Plasma Boundary by a Power Electromagnetic Beam

Kurnakov Institute of General and Inorganic Chemistry, Russian Academy of Sciences, Leninsky pr. 31, 119991 Moscow, Russia
Technologies 2022, 10(4), 78;
Received: 20 May 2022 / Revised: 22 June 2022 / Accepted: 27 June 2022 / Published: 29 June 2022
(This article belongs to the Special Issue 10th Anniversary of Technologies—Recent Advances and Perspectives)


The interaction of an electromagnetic beam with a sharp boundary of a dense cold semi-limited plasma was considered in the case of a normal wave incidence on the plasma surface. The possibility of the appearance of an electrostatic field outside the plasma was revealed, the intensity of which decreased according to the power law with a distance from the plasma and the center of the beam. It was possible to form cavities with a reduced electron density, being each electromagnetic resonators, which probed deeply into the dense plasma and couldexist in a stable state for a long period.

1. Introduction

The origination of many phenomena taking place during the interaction of electromagnetic radiation with a dense plasma occurs on the interface of media where the possibility of the appearance of certain effects is determined depending on the conditions and the ratio of parameters. Therefore, identifying such conditions and characterizing interactions in simple modeling cases appearsto be an important primary step toward detecting and predicting many interesting phenomena. It was within the framework of the simplest models of cold plasma with a sharp boundary that the effects of nonlinear transparency [1,2,3], complete absorption [4,5], and anomalous radiation [6] of electromagnetic radiation were investigated. The construction of such a model impliedan accurate representation of the physical essence of the phenomenon under study and those basic features of the interaction of radiation with plasma that ensured its existence.
In this work, the possibility of the formation of globe-shaped resonators, being cavities with a rarefied electron density created at the plasma boundary under the influence of a beam of powerful electromagnetic radiation was considered. The main features of this phenomenon couldbe best studied in a simple model of a semi-infinite plasma with a sharp boundary and stationary ions for the case when electromagnetic radiation normally reached it in the form of a beam with an exponential intensity distribution in the frontal plane. The possibility of forming a cavity with a low electron density followedfrom the physical essence of the interaction of radiation with a plasma. This wasdue to the fact that, on the one hand, a powerful electromagnetic flux wasable to remove electrons from a certain volume and to hold the boundary in the equilibrium against forces of the thermal pressure and the charge separation field [7]. However, on the other hand, such cavities in the plasma couldacquire, under certain conditions, the properties of an electromagnetic resonator [8]. This occurredwhen the size of the cavity, the amplitude, the frequency and spatial structure of the electromagnetic field, the thermal pressure of electrons, and other characteristics reached certain resonant values, for which the stable state of the cavity couldbe maintained for a long periodin the absence of dissipation. The formation of the surface of the cavity and the spatial structure of the electromagnetic field inside the resonator wereinterrelated processes, the parameters of which maintained equilibrium by mutual correction of their values. At the same time, depending on the ratio of the characteristics of the task, the shape of the cavity couldbe either spherical, ellipsoidal, or cylindrical. In the latter case, a situation is possible when such a cylinder crosses the entire thickness of the plasma layer so that the radiation can pass through this layer of dense plasma into a region where it could not penetrate at low values of its intensity. At the same time, for small amplitudes of the electromagnetic signal, when the plasma boundary remains flat, a nonlinear surface charge couldbe formed on it under certain conditions [9], which createdan electrostatic field outside the plasma with a large localization region, when its amplitude decreasedwith a distance from the boundary and the center of the beam according to the power law, in contrast to the strength of the electromagnetic wave field. The possibilities of such a field may arouse interest, both from the point of view of the practical application (for example, for particle acceleration) and from the standpoint of the probable need to prevent undesirable effects.

2. Basic Equations

Consider a semi-infinite plasma consisting of electrons with mass m, density ne, charge -e, and immobile ions with density ni (x ≥ 0), forming a sharp boundary, to which a beam of plane-polarized electromagnetic radiation with a frequency ω and a wave number k propagates along itsnormal on the axis 0X (Figure 1). In the region (x ≤ 0) surrounding the plasma, the following expression can be written for the intensity of E0 = {0, E0,0} of the electric field having auniform spatial distribution in azimuth in the front plane 0YZ.
E ( r , t ) =   φ e ( r ) + y ^ E 0 sin ( ω t k x ) + y ^ E 0 r sin ( ω t + k x )   ,
where φe(r) is the electrostatic potential of the surface charge formed at the plasma boundary [9,10,11], and E0r is the amplitude of the reflected electromagnetic signal.
The motion of electrons with the velocity v is described by the equation:
t v + ( v ) v = e m E + v × B v T 2 n 0 n e
where B is the strength of the magnetic field of external radiation, vT2 = Te/m, Te is the temperature of electrons, thermal pressure is taken into account in (2) only to estimate the parameters of the equilibrium state, and n0 is the equilibrium density of plasma particles in the stationary state (ne = nin0).
The field strengths in (1), (2) satisfy Maxwell’s equations:
t B = × E ,
× B = 1 c 2 t E + μ 0 e n e v ,
E = e ( n e n i ) / ε 0 .
Here, ε0 is the dielectric density of a vacuum, and μ0 is its magnetic permeability.
Due to the azimuthhomogeneity of the electromagnetic beam, it waspossible to use a cylindrical coordinate system with an axis of 0X and coordinates ρ,χ in the frontal plane (z = ρcosχ, y = ρsinχ). In this case, the amplitude of E0(y, z) woulddepend only on the coordinate ρ, and it waspossible to consider different intensity distributions in the plane of the wave front, for example, an exponential one.
E 0 ( ρ ) = E a exp {   ρ / ρ 0 } , ρ 0 = c o n s t ,   k ρ 0 > > 1 .
For harmonic analysis, the velocity v should be divided into a fast-variable vE component and a static δv(r) part (vE = e E0/mω). As a result, the following expression can be derived from Equation (2)
( δ v ) δ v 1 2 v E 2 e m φ v T 2 n 0 n e = 0
Therefore, the function F(x,ρ) defined by the formula.
F ( x , ρ ) = 1 2 δ v 2 1 2 v E 2 e m φ v T 2 n 0 n e
This is a continuous quantity both along the polar coordinate ρ and normally to the surface of the plasma (axis 0X) in the case where the velocity δv is determined by the potential ψ (δv = ∇ψ). With its help, it waspossible to estimate the change in individual physical quantities, as compared to their values at selected points.

3. Analytical and Numerical Results

For high-power radiation, the continuity of the function F(x,ρ) wasreduced to the balance of electromagnetic and thermal energy:
1 2 v E 2 ( x , ρ ) + v T 2 n 0 n e ( x , ρ ) c o n s t .
From the equilibrium ratio (9), it followedthat the total pressure (the sum of radiation and heat) of electrons wasa continuous quantity, and this balance wasobserved everywhere, including along the 0X axis and along the ρ axis. It also enabled us to understand how many electrons wereforced out of the cavern formed by the electromagnetic beam incident on the plasma. Along the polar radius ρ, the amplitude of E0changedsmoothly, and when the density of nereachedthe critical value of nc, that is ω = ωpp2 = e2ne/(m·ε0)), the plasma becameopaque to this wave field, as a result of which it droppedexponentially rapidly into the dense plasma, the density of which, in turn, increasedas rapidly, according to (9). At this increase in density on the surface (x = xb, ρ = ρb), thermal pressure vT2 dominated one side, and on the other, electromagnetic, characterized by vE2, which in a stationary state wouldbalance each other. Therefore, an approximate condition
v E 2 v T 2
wouldbe executed at this boundary to determinethe threshold value of the amplitude E0(xb,ρb) for the formation of the cavern.

3.1. Conditions for the Formation of Globe-Shaped Resonators of the Electromagnetic Field

The dynamics of the development of the cavity were represented as follows. First, a small space formednear the surface of the plasma and close to the center of the beam with a boundary separating the bulk of the electrons and having a surface shape similar to function (6). As it moveddeeper into the plasma, this cavity wasformed in accordance with the values of the plasma and radiation parameters acting at each time. Since the ions remained stationary, a bulk electric charge formed in the cavern, creating an electric field φe, which attemptedto return the displaced electrons back to their positions. The shape of the cavern varied depending on the ratio of the characteristics of the task. However, for example, in the case of a spherical cavity, its radius R in the equilibrium state was determined bycondition (8), in which the potential φe that dependedonly on the radial coordinate rhad tobe substituted from the solution of Equation (5) for the sphere, within which ne ≈ 0. In this case, one couldobtain from (5):
φ e = e 6 ε 0 n i r 2
By substituting (11) into condition (8) taken at the boundary r = R, it was possible to obtain an estimate of the magnitude of the cavity radius:
R = 6 ω p v E 2 v T 2 ~ 6 ω p v E
When a spherical resonator formed simultaneously with the electronic surface of the cavity, structural changes in the spatial distribution of electric (and magnetic) fields occurred, which beganto reflect from the curved boundary and, according to (3) and (4), were described by the equation:
Δ E + ω 2 c 2 ε ( ω ) E = 0 ,   ε ( ω ) = 1 ω p 2 ω 2 .
The general solution of this equation wasgiven in [8] for a spherical coordinate system (r, ϑ, χ), beginning in the center of the cavity. It has a cumbersome appearance, but for a spherically symmetric case, it was written in a simple form for the radial intensity Er of the electric field
E r ( r R ) = E a sin   k r k r , k = ω c ε 1 , ε 1 = ε ( ω , r R ) .
E r ( r R ) = E a 1 κ r e κ r , κ = ω c ε 2 , ε 2 = ε ( ω , r R ) .
The oscillations described by formulas (14) and (14a) did not have a wave structure along the surface of the sphere and hada frequency of ωm (m = 1,2,3,…), as defined from the following dispersion equation:
ε 2 k e κ R =   ε 1 κ   s i n ( k R ) .
For large values of the parameter ap = ωpR/c, the approximate value of the frequency of natural oscillations was in the form ωm = amc/R, where the constant am is determined from the solution of the following transcendental equation:
a p e a p = a m sin a m .
The expression (16) together with (12) allowed us to derive the value of the amplitude of the electric field and frequency, at which it was possible to form a spherical resonator with the parameters presented herein in the form of estimates. We determined that at the value of the velocity δv, the resonator moved deeply into the plasma. To do this, using expression (8), it was necessary to take the parameter values near the surface of the cavity close to the center of the beam where the velocity δv wasentirely directed along the 0X axis. The result wasthe following approximation:
δ v 2 ~ V E 2 V T 2
It should be noted that for other resonant combinations between the parameters of plasma and external radiation, an ellipsoidal form of the resonator couldbe realized. In addition, when the thickness of the plasma along the 0X axis wasnarrow, it couldhave the appearance of a cylinder through which radiation wasable to penetrate through dense plasma.

3.2. Generation of Electrostatic Fields of Surface Charge near Plasma Space by a Beam of Electromagnetic Radiation

In the case when the force effect of the electromagnetic beam wassmall, as compared to the pressure of electrons, the surface of the plasma remainedflat when interacting with the radiation. However, as shown in [9,10,11], it formeda nonlinear surface charge associated with the electrostatic field φe(r), which hada large localization region near the plasma boundary and couldaccelerate charged particles [12,13,14]. The description of this surface charge, performed in [9,10,11], was based on the theory of the potential [8,15], which could express the value of the potential φe(r) throughout space viaits value on the surface of the plasma:
φ e ( r ) = 1 2 π d y d z | x | Φ ( y , z ) ( y y ) 2 + ( z z ) 2 + x 2 3 / 2 , Φ ( y , z ) = φ e ( x = 0 , y , z ) .
The integral in (18) should be interpreted as the principal value (p.v.). It indicatedthat in the limit x→±0, when the peculiar point appearedin (18), the path of the integration must have had the form of a small sphere that surrounded this point.
In polar coordinates (χ,ρ), Equation (18), after the integration at the azimuthal angle χ for x ≤ 0 values not close to the plasma boundary, could be written as follows:
φ e ( x , ρ ) = 0 Φ ( ρ ) x ρ d ρ [ ρ 2 + ρ 2 + x 2 ] 3 / 2 .
The value of the function Φ(ρ) couldbe obtained from the equation of motion (2) at the boundary (for x = 0) under conditions when nonlinear corrections from the stationary velocity of movement of electrons in the surface charge zone couldbe neglected, and the representation (6) was valid:
Φ ( ρ ) e E 0 2 ( ρ ) m ω 2 = Φ 0 exp {   ρ / ρ 0 }
In this case, the expression (19) could be expressed as follows:
φ e ( x , ρ ) = π Φ 0 x ρ 0 β H 0 ( β ) β N 0 ( β ) 2 π , β = ρ 2 + x 2 ρ 0 2 .
Here, H0(x) and N0(x) are Struve and Neumann functions, respectively [16].
The asymptotic value of the potential in the region far from the boundary |x| > ρ was described, as follows from (21), by the formula:
φ e ( x , ρ ) 2 Φ 0 x ρ 0 ρ 2 + x 2 .
Based on (22), the electrostatic field component along the plasma Eρ = −∂ρφe decreased with increasing distance |x| proportionally 1/|x| (component Ex = −∂xφefell 1/|x|2). As compared tothe amplitude of the electromagnetic beam, the magnitude of the electrostatic strength of the field decreasedat a distance from its center not according to the exponential but according to the power law, that is, the area of its localization wasmuch larger.
As an example of the acceleration of charged particles in the electrostatic field of a surface charge (22), it was possible to consider the motion of a particle with a charge e0 and a mass M from a point (x,ρ) and calculate the final velocity of its movement at infinity. From the equation of motion for the velocity vp of the particle:
t v p + ( v p ) v p = e 0 M φ e
one can write
V p = E a ω e 0 e M m
It followedfrom (24) that a particle with a mass M in the electrostatic field of the surface charge acquireda constant velocity, which in (m/M)1/2 times wasless than the amplitude of electron oscillation at the center of the electromagnetic beam.

4. Summary and Conclusions

The electrostatic field of the surface charge that arosein the process of interaction of the electromagnetic radiation beam with the plasma appearedand affectedthe environment due to the specific movement of electrons [3,9,10,11] and the complex of conditions that supported its existence (e.g., sharp boundary, quasi-neutrality, absence of non-harmonic perturbations, etc.). The power law of the decrease inthis field in space for a distance from the boundary and from the axis of the electromagnetic beam determinedthe large size of the region of its localization. This circumstance couldbe useful for achieving practical application (e.g., for particle acceleration) or couldbe considered in cases where its effect is likely to have negative consequences. In its magnitude, the strength of this electrostatic field was comparable to the amplitude of electromagnetic oscillations, but it didnot have a spatial and temporal oscillatory structure.
For high intensities of the electromagnetic beam, when the rate of oscillation of electrons wascomparable to their thermal velocity in the plasma, the flat boundary of the electrons wascurved, which in the model of stationary ions ledto the appearance of a charge separation field. As a result of the self-consistent deformation of the surface of electrons and the spatial structure of the electromagnetic field, it waspossible, under certain conditions, to form a cavity, which wasan electromagnetic resonator where the shape of its surface and the structure of the field could exist together for a long period, unchanged. Such conditions werefound in the present work for a resonator in aspherical shape. However, under other conditions, ellipsoidal cavities and even cylindrical cavities can occur. The latter, in the case of a relatively narrowthickness of the plasma layer, wereable to ensure the passage of radiation through a non-transparent medium (in other words, burn through it). The movement of electromagnetic resonators of various shapes also contributedto the penetration of the electromagnetic radiation deeply into the dense plasma and couldbe used to create a number of special nonlinear interactions [3,17,18,19,20].It should be noted that the appearance of resonators waspossible not only in plasma, but hasalso been actively investigated in plasmonic materials, such as hyperbolic metamaterials with giant enhancements [21], metamaterial cavities with broadband strong coupling, and metamaterials with large index sensitivities [22]. The results obtained in these and other similar works couldbe useful for continuing research in plasma with similar configurations.


This research received no external funding.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.


The publication was conducted within the State Assignment on Fundamental Research to the Kurnakov Institute of General and Inorganic Chemistry of the Russian Academy of Sciences.

Conflicts of Interest

The author declares no conflict of interest.


  1. Rosenbluth, M.N.; Liu, C.S. Excitation of plasma waves by two laser beams. Phys. Rev. Lett. 1972, 29, 701. [Google Scholar] [CrossRef]
  2. Gradov, O.M.; Stenflo, L. On the parametric transparency of a magnetized plasma slab. Phys. Lett. 1981, 83, 257. [Google Scholar] [CrossRef]
  3. Gradov, O.M.; Ramazashvili, R.R.; Stenflo, L. Parametrlc transparency of a magnetized plasma. Plasma Phys. 1982, 24, 1101. [Google Scholar] [CrossRef]
  4. Aliev, Y.M.; Gradov, O.M.; Kyrie, A.Y.; Čadež, V.M.; Vuković, S. Total absorption of electromagnetic radiation in a dense inhomogeneous plasma. Phys. Rev.A 1977, 15, 2120. [Google Scholar] [CrossRef]
  5. Forslund, D.W.; Kindel, J.M.; Lee, K.; Lindman, E.L. Absorption of laser light on self-consistent plasma-density profiles. Phys. Rev. Lett. 1976, 36, 35. [Google Scholar] [CrossRef]
  6. Gradov, O.M.; Larsson, J.; Lindgren, T.; Stenflo, L.; Tegeback, R.; Uddholm, P. Anomalous radiation from a nonstationary plasma. Phys. Scr. 1980, 22, 151. [Google Scholar] [CrossRef]
  7. Berger, J.M.; Newcomb, W.A.; Dawson, J.M.; Frieman, E.A.; Kulsrud, R.M.; Lenard, A. Heating of a confined plasma by oscillating electromagnetic fields. Phys. Fluids 1958, 1, 301. [Google Scholar] [CrossRef]
  8. Stratton, J.A. Electromagnetic Theory; McGraw-Hill: New York, NY, USA, 1941. [Google Scholar]
  9. Gradov, O.M. Self-consistent plasma boundary distortions during the interaction of a normally incident electromagnetic beam and a nonlinear surface charge. Chin. J. Phys. 2021, 72, 360–365. [Google Scholar] [CrossRef]
  10. Gradov, O.M. Three-dimensional surface charge nonlinear waves at a plasma boundary. Phys. Scr. 2019, 94, 125601. [Google Scholar] [CrossRef]
  11. Gradov, O.M. Nonlinear behavior of a surface charge on the curved plasma boundary with a moving cavity. Phys. Lett. A 2020, 384, 126566. [Google Scholar] [CrossRef]
  12. Yamagiva, M.; Koga, J. MeV ion generation by an ultra-intense short-pulse laser: Application to positron emitting radionuclide production. J. Phys. D Appl. Phys. 1999, 32, 2526. [Google Scholar] [CrossRef]
  13. Kovalev, V.F.; Bychenkov, V.Y. Analytic theory of relativistic self-focusing for a Gaussian light beam entering a plasma: Renormalization-group approach. Phys. Rev. E 2019, 99, 043201. [Google Scholar] [CrossRef] [PubMed]
  14. Boyer, C.N.; Destler, W.W.; Kim, H. Controlled collective field propagation for ion-acceleration using a slow-wave structure. IEEE Trans. Nucl. Sci. 1977, 24, 1625–1627. [Google Scholar] [CrossRef]
  15. Jeffreys, H.; Swirles, B. Methods of Mathematical Physics; Cambridge University Press: Cambridge, UK, 1956. [Google Scholar]
  16. Bateman, H.; Erdélyi, A. Higher Transcendental Functions; McGraw-Hill: New York, NY, USA, 1953; Volume I–II. [Google Scholar]
  17. Ma, J.Z.G.; Hirose, A. Parallel propagation of ion solitons in magnetic flux tubes. Phys. Scr. 2009, 79, 045502. [Google Scholar] [CrossRef]
  18. Brodin, G.; Stenflo, L. Large amplitude electron plasma oscillations. Phys. Lett. A 2014, 378, 1632. [Google Scholar] [CrossRef]
  19. Vladimirov, S.V.; Yu, M.Y.; Tsytovich, V.N. Recent advances in the theory of nonlinear surface waves. Phys. Rep. 1994, 241, 1–63. [Google Scholar] [CrossRef]
  20. Vladimirov, S.V.; Yu, M.Y.; Stenflo, L. Surface-wave solitons in an electronic medium. Phys. Lett. A 1993, 174, 313. [Google Scholar] [CrossRef]
  21. Xu, H.; Zhu, Z.; Xue, J.; Zhan, Q.; Zhou, Z.; Wang, X. Giant enhancements of high-order upconversion luminescence enabled by multiresonant hyperbolic metamaterials. Photonics Res. 2021, 9, 395. [Google Scholar] [CrossRef]
  22. Gu, P.; Chen, J.; Chen, S.; Yang, C.; Zhang, Z.; Du, W.; Chen, Z. Ultralarge Rabi splitting and broadband strong coupling in a spherical hyperbolic metamaterial cavity. Photonics Res. 2021, 9, 829. [Google Scholar] [CrossRef]
Figure 1. Scheme of interaction of the electromagnetic beam with the surface of the plasma.
Figure 1. Scheme of interaction of the electromagnetic beam with the surface of the plasma.
Technologies 10 00078 g001
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Share and Cite

MDPI and ACS Style

Gradov, O.M. Exciting of Strong Electrostatic Fields and Electromagnetic Resonators at the Plasma Boundary by a Power Electromagnetic Beam. Technologies 2022, 10, 78.

AMA Style

Gradov OM. Exciting of Strong Electrostatic Fields and Electromagnetic Resonators at the Plasma Boundary by a Power Electromagnetic Beam. Technologies. 2022; 10(4):78.

Chicago/Turabian Style

Gradov, O. M. 2022. "Exciting of Strong Electrostatic Fields and Electromagnetic Resonators at the Plasma Boundary by a Power Electromagnetic Beam" Technologies 10, no. 4: 78.

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Metrics

Back to TopTop