rus
Issue #2/2025
P. P. Maltsev
Interaction of Electromagnetic Radiation with the Metal Fractal Clusters. Part 2
Interaction of Electromagnetic Radiation with the Metal Fractal Clusters. Part 2
DOI: 10.22184/1993-7296.FRos.2025.19.2.102.114
The article continues the discussion of effects occurring in the polymer threads with the metallic fractal clusters (see Photonics Russia. 2025;19(1):14–27. DOI: 10.22184/1993‑7296.FRos.2025.19.1.14.27) and considers the features of capturing external radiation with a wavelength λ in the optical wavelength range by the fractal clusters made of spherical metal particles with the radius R. An explanation is given to the effect generation conditions titled “photon localization” when the condition R << λ is met.
The article continues the discussion of effects occurring in the polymer threads with the metallic fractal clusters (see Photonics Russia. 2025;19(1):14–27. DOI: 10.22184/1993‑7296.FRos.2025.19.1.14.27) and considers the features of capturing external radiation with a wavelength λ in the optical wavelength range by the fractal clusters made of spherical metal particles with the radius R. An explanation is given to the effect generation conditions titled “photon localization” when the condition R << λ is met.
Теги: metallic fractal clusters nanoscale aluminum droplets off-surface highly conductive state photon localization plasma polymer threads локализация фотонов металлические фрактальные кластеры надповерхностное высокопроводящее состояние наноразмерные капли алюминия плазма полимерные нити
Interaction of Electromagnetic Radiation with the Metallic Fractal Clusters. Part 2
P. P. Maltsev
Inter-Agency Analytical Research Center of the RAS, Moscow, Russia
The article continues the discussion of effects occurring in the polymer threads with the metallic fractal clusters (see Photonics Russia. 2025;19(1):14–27.
DOI: 10.22184/1993‑7296.FRos.2025.19.1.14.27) and considers the features of capturing external radiation with a wavelength λ in the optical wavelength range by the fractal clusters made of spherical metal particles with the radius R. An explanation is given to the effect generation conditions titled “photon localization” when the condition R << λ is met.
Key words: metallic fractal clusters, nanoscale aluminum droplets, polymer threads, plasma, off-surface highly conductive state, photon localization
Article received: January 16, 2025
Article accepted: February 03, 2025
Properties of metallic fractal clusters
The main feature of a fractal cluster (FC) determining its physical and chemical properties shall be its scale invariance: when scaled up, any small FC fragment reproduces the spatial structure of the entire cluster. The scale invariance shall lead to the fact that the spatial arrangement of not all cluster particles, but a very large group of particles, turns out to be correlated, although visually the structure may be perceived as disordered.
The second consequence of scale invariance shall be the availability of a large number of cavities in the FC with a power distribution of dimensions, making the FC a rather delicate structure.
The photographs of polymer threads with the chains of irregular aluminum nanoislands (nanodroplets) with a size of 10–1000 nm (Fig. 11) were taken using a CAMSCAN-S4 scanning electron microscope with the energy-dispersive and wave-dispersive attachments: Oxford INCA Еnergy 350 and INCA Wave 700 (Cambridge, England) at the High Technology Center of the Synchrotron Collective Use Center of the Federal State Unitary Enterprise “Lukin Research Institute for Physical Problems” of the National Research Center “Kurchatov Institute” [1, 10].
The chains of irregular aluminum nanoislands with a size of 10–1000 nm, including spherical ones with a size of 10–30 nm, located on the surface of a polymer thread made of aramid fibers, can be represented as the metallic fractal clusters.
The publications describe various cases of the occurrence of low-field electron emission with the decreased size of particles on the surface of materials up to tens of nanometers and less [11–15], as well as the generation of threads made of individual tin atoms [16]. The studies of the dispersed media influence, including any metal particles, on the propagation of electromagnetic radiation, including the optical range, have been conducted for quite a long period of time [17–28]. Various effects have been found that require further theoretical development.
For example, an external electromagnetic wave propagating in a system of low-capture scatterers, as a result of multiple rescattering is capable of “moving in loops” in a limited spatial region. The reason for this phenomenon shall be the specific interference effects that occur even in a completely disordered system of particles [24–26, 29]. The effect of capturing external radiation in the optical wavelength range by a fractal cluster, for example, on the Rayleigh particles, is called “photon localization” that can be conditionally considered as a channel for “dissipation” of incident radiation, complementing the classical ones, such as inelastic scattering and absorption.
The “photon localization” concept in a fractal cluster
The photon localization effect in a system of scatterers can be expected under certain conditions [29]. First of all, the average photon absorption length la shall be greater than the elastic scattering length ls. In addition, the elastic scattering cross section on an individual particle shall be large enough. Otherwise, the loop generation is unlikely, since otherwise there will be no conditions for “unfolding” of the photon movement.
To describe these conditions, the expressions obtained in [28] shall be used:
la = n0 σa–1 и ls = n0 σs–1,
where σa – light absorption cross-section of an individual scatterer;
σs – cross-section of light elastic scattering by an individual scatterer;
n0 – concentration of scatterers in the system.
At first glance, due to the small elastic scattering cross-section of an individual Rayleigh scatterer with the radius R, even at the maximum achievable packing factors of the parameter p = λ / ls that determines the probability of localization, this value remains negligibly low: σs / πR = R / λ4 << 1.
This is indeed the case. However, the situation is changed if the frequency of the incident quantum ω coincides with the frequency of some electromagnetic mode of a separate scatterer. For example, for the surface plasmon of a spherical metallic individual particle with the radius R the following expression can be obtained: ω1 = ω0—3, where ω0 is the classical plasma frequency of an unlimited electron gas [28].
In this case, for a spherical metallic individual particle, the cross section of light elastic scattering by the particle shall be equal to [28]:
σs = 83 πR2 2 πRλ4 ω4ω2 – ω122 + γ2 ω14
and shall have a sharp maximum value, since the plasma resonance width is equal to γ = 10–2 for many metals, and the absorption cross-section of a spherical metallic individual particle shall be equal to: σa = 8 πR2 γω4λ ω2 – ω122 + γ2 ω14–1 and the packing factor of particles in the system shall be: f = 43 πR3 n0.
In this case, the parameter p = R / λ3 fγ2 shall be equal to one for the particles with Rλ = 10−1 already at f ≈ 0.1. The photon absorption length la shall remain comparable to the elastic scattering length ls [29].
Thus, the densely packed systems of nanoscale metallic particles at the external radiation frequencies ω1 shall be the suitable candidates for observing the photon localization phenomenon in the frequency range from visible to ultraviolet.
It shall be worth noting the paper [29] that has introduced the concept of “photon localization” in a fractal cluster with a detailed description of this effect. In the paper [29], a new approach to the description of electrodynamic properties of a fractal cluster consisting of non-absorbing solid particles has been developed using the diagrammatic method. The phenomenon of renormalization (decrease) of the external radiation wavelength λ as it penetrates the fractal cluster has been studied, and the effective capacitivity of the fractal cluster εfc has been calculated.
It is shown that the fractal cluster is characterized by a set of values εfc, each of which corresponds to its own renormalization degree λ. It is shown that the effective capacitivity of the cluster is a critical function of the fractal dimension of the cluster d, and if the d value is less than some critical value, it can be extremely high.
An external photon entering the fractal cluster has the ability to “discern” the cluster scale invariance. The photon wavelength in the cluster λint becomes much smaller than the external value λ, but the photon frequency ω is not changed, since the photon speed is simultaneously decreased, i. e. as the wavelength decreases, the photon acquires the ability to “discern” the increasingly smaller structural details. The long-range correlations in the arrangement of particles of the fractal cluster, visually expressed in the cluster connectivity, shall be the reason for the decreased external radiation wavelength λ when “moving in loops” in it.
The renormalization of λ shall occur as follows. The photon with a wavelength λ of the order of the typical cluster size L incident on the cluster shall be “captured” by some sufficiently large cavity of the fractal cluster (the primary resonance cavity). This capture shall lead to an increase in the effective capacitivity of the cluster εфк, since the ε value is increased near any electromagnetic resonance. The increased ε value shall initiate, in turn, a decrease in the photon wavelength: λint = λ. The photon with a renormalized wavelength λint shall find another resonance cavity of a smaller size. The new capture shall again stimulate an increased ε value and a new decrease in λint, etc. As a result, all cluster cavities can be filled with the renormalized virtual photons, including those which λint → 0. Their effective speed shall be equal to zero.
The light localization shall be associated with the closed loops on the trajectories of virtual photons [29]. If the photon makes a closed loop, the phase incursion of its wave function shall be equal to zero. The probability amplitudes relevant to the two possible ways of the loop bypass (clockwise and counter clockwise) shall interfere constructively regardless of the disorder degree of the scatterers. Any loop shall be a return back. Since the loop formation probability due to this kind of interference is increased, scattering into the back hemisphere is also increased. In turn, it stimulates the generation of new loops, etc. The result of this self-sustaining process shall be the photon “locking” in a limited spatial region, namely “the light localization”.
Difference from the standard description of light localization
The standard description scheme for the light localization shall be the reduction of the Bethe-Salpeter equation for an irreducible four-point vertex function (four-tail function) in the momentum representation to the transport equation (radiative transfer equation) and introduction of an effective electromagnetic energy diffusion coefficient [29]. Similar to the Anderson electron localization, vanishing of this coefficient shall mean strong light localization.
In [29], the approach to solving the radiation transfer equation differs from the standard one. The constructions shall be based on the idea of localized photons as the typical virtual particles, similar to the virtual photons of quantum electrodynamics. These photons shall not be related to either the detector or the light source.
A typical virtual photon [29] shall be a closed loop of the photon propagator (convolution of two vector potentials) growing on a double line describing the electron propagation (Fig. 11a) and occurring in the second order of perturbation theory (PT) with the Hamiltonian H:
H = p – e Ac22 m,
where e and m are the electron charge and mass,
c – speed of light in vacuum,
p – electron momentum,
A – vector potential of the electromagnetic field.
This self-closed photon propagator shall give some idea of the localized photon (Fig. 12a).
Formally, the localization [29] is associated with the occurrence of a pole in a fan-shaped diagram of four tail function (Fig. 12b). Such kind of pole shall describe the bound states of a pair of interacting particles, for example, an exciton. In the considered problems of the single photon propagation, the vertex function shall describe the efficient interaction of a pair of virtual photons (Fig. 12b) bypassing a closed loop on a trajectory in two opposite directions (more precisely, the interference of amplitudes relevant to these two bypass directions).
In opposition to the common scheme in the paper [29], the Bethe-Salpeter equation has been solved directly in the coordinate representation. As a part of the proposed constructions, there is no need to introduce the diffusion coefficient of electromagnetic energy. The localization is demonstrated simply as the interference corrections to the scattering and absorption cross-sections. It is these corrections that are calculated, and it is this “meaning” that is put into the word “localization”.
Specific features of virtual photons
A classic example of virtual photons shall be the virtual photons of quantum electrodynamics emitted by a moving electron. These photons are described by the propagator or Green’s function of the Maxwell’s equations, rather than by a plane electromagnetic wave. For example, at a given frequency, the wavelength of a virtual photon λint is determined by the efficient speed of light in the medium v according to the relation ω = 2πvλint. Under the localization conditions v → 0 that tends to zero either at a very large value of ε, or at a very large value of the derivative dεdω.
The phenomenon found sheds new light on the cause of a well-known shortage of numerous efficient medium approximations [29]. As a part of these theories, in a certain range of frequencies and packing factors, the effective capacitivity of a medium consisting of small non-absorbing particles turns out to be comprehensive one, thereby allowing for the existence of some mysterious absorption in the system. However, everything fits together: the effective absorption is related to localization.
As an example demonstrating reliability of the developed diagram technique, the results of the classical Mie theory in the problem of elastic scattering of an electromagnetic wave by a spherical metallic particle are reproduced in the paper [29]. At the same time, its shortcomings, consisting in non-consideration of the spatial dispersion effects, are found, and it is shown how to overcome them.
In a system of particles, the difference between the amplitudes of forward and reverse photon passage along the route “particle a – particle b” shall be determined not only by elastic scattering, but also by the light cycling or localization between these particles. It is this cycling that is described by the complex components σab.
In the paper [29], the photon localization theory in a dense random packing of Rayleigh particles is prepared. The strong deformation of the light scattering indicatrix by a single system particle is determined, expressed by an anomalous increase in scattering into the back hemisphere, as well as a strong localization sensitivity to the incident light polarization type. The determination of frequency and concentration range of photon localization is proposed.
As expected, the equation proposed in the paper [29] coincides with the equation for the Green’s function of the Maxwell’s equations with the same gauge. The proposed formalism is then applied to calculate the probabilities of the main electrodynamic processes in the dispersed media: elastic scattering, absorption, inelastic scattering, photoelectric effect, and various three-photon processes.
Examples of light localization application
The results of light localization studies can be used to develop the “random” (powder) lasers [29]. In a random laser, the role of mirrors is played by a multiple scattering medium, namely the clusters of nanoparticles of a weakly absorbing material (for example, ZnO). Another version of a “random” laser shall be introduction of such a powder into a laser cell on liquid dyes. After irradiation by an external light source, such a laser shall provide both light amplification and retention tin the system due to the multiple rescattering process. In contrast to a regular laser, the radiation of a “random” laser is isotropic. It is possible that a “random” laser can do without an active medium at all, due to the forced emission of localized light.
Based on the obtained theory of light localization in the fractal cluster [29], the lifetime of light localized both in a single fractal cluster and in an agglomerate of fractal clusters has been calculated. It is proposed to use the stimulated localized light emission to develop a fractal microlaser that does not require any inverse population of levels and is capable of operating in a wide range of lengths. As a result, the proposals are formalized in the form of an inventor’s certificate for a method for radiation conversion into the coherent light using the micron-sized devices [20].
In the paper [29], a model is proposed that allows us to understand the cause of giant Raman scattering (GRS) of light by the molecules adsorbed on the surface of small metallic particles. It is based on the radiation localization during the multiple inelastic light scattering in an ensemble of particles. When moving along a closed trajectory, the virtual photon repeatedly exchanges energy with the propagation medium while exciting fluctuations in the charge density. The energy of these fluctuations can be an arbitrary value, less than the photon energy.
The strong local fields related to these fluctuations and available in the entire frequency range from IR to UV shall be the GRS cause. The behavioral features of these fluctuations allow us to explain the specific features of flicker noise and also explain the catalytic properties of small metallic particles.
In addition, a new model describing the liquid-metal Rehbinder effect is proposed in the paper [29] that is based on the idea of possible electromagnetic field localization in the folds of phase boundary and components of a liquid eutectic mixture filling the cracks in the solid metal surface (a typical example shall be the liquid eutectic of In and Ga on the Al surface). Since at each spatial point of the eutectic mixture three various substances are adjacent (a homogeneous melt of In+Ga, solid In and solid Ga), the system of folds of such an interface is modeled by the Wada-Brauer structure well-known in topology, namely a surface separating three various regions at each of its points.
The localized photons are capable of “switching off” the Coulomb attraction of charge fluctuations on opposite banks of a eutectic-filled crack that provides for the Van der Waals attraction of the banks. The phenomenon of localized light emission in the liquid-metal Rehbinder effect and similar emission from a Casimir gap with the broken symmetry are predicted.
The useful result of the paper [29] is possible application of another property occurred during the photon localization process to explain generation of a wide “base” in the lower part of the Cherenkov radiation cone (Fig. 12a). Such an external form of Cherenkov radiation obtained during the occurrence of breakdown on runaway electrons (the electron velocities close to the speed of light) [30–32], has been registered during a discharge on a 30 m long polymer thread with metallic fractal clusters (at an electric field strength of 30 kV/m) [30, 32]. The composite threads with metallic fractal clusters used in the experiment presumably have the properties of a metamaterial with negative capacitivity [1, 6, 8].
Having considered that the phase velocity and group velocity in metamaterials are aimed at various directions for Cherenkov radiation (“inverted” Cherenkov radiation) [8], the phase velocity shall determine the tip of Cherenkov radiation, and the group velocity shall be opposite and shall influence the formation of its “base”.
However, when photons are localized, the group velocity in the metallic fractal cluster is decreased and a strong deformation of the scattering light indicatrix by an individual particle of the system occurs that is expressed by an anomalous increased scattering into the back hemisphere and isotropic radiation [29] that can lead to a sharp expansion of the “base” of Cherenkov radiation (Fig. 13a).
The similar form of radiation can be observed for the high-altitude sprite breakdown (Fig. 13b) on the runaway electrons (the electron velocities are close to the speed of light) [31, 32], occurring due to the impact of high-energy cosmic particles on the fractals and beginning from its micro breakdowns. It is possible that the observed “reversed” Cherenkov radiation (high-altitude breakdown), directed towards the source (towards the high-energy cosmic particles), also occurs on the fractals with “lattice architecture” [31] having the properties of metamaterials with negative capacitivity.
Conclusions for part 2
Hypothetically, it can be assumed that under certain conditions, an external electromagnetic wave propagating in a system of weakly absorbing scatterers as a result of multiple rescattering is capable of “moving in loops” in a limited spatial region, and the reason for this phenomenon is specific interference effects that occur even in an absolutely disordered system of particles. This effect of capturing external radiation in the optical wavelength range by a fractal cluster has been called “photon localization” that can be considered conditionally as a channel for “dissipation” of incident radiation, complementing the classical ones, namely the inelastic scattering and absorption.
The essence of “light localization” is well illustrated by the following simple analogy, proposed in the paper [29]. Let us collect some water in a wide vessel and after some time t open a narrow drain hole. Provided that the water inflow Q into the vessel is exactly equal to its outflow, the water level in the vessel shall soon stabilize at the H mark (Torricelli formula). If the initial water level in the vessel is low, then the vessel is filled to the H mark, and if the water delay prior to the hole opening is large enough, then the excess water shall be drained.
The role of water is played by radiation, the vessel is a system of particles, and the water in the vessel is localized light. If the lifetime t of a localized photon is short, the system reacts by decreasing scattering (the vessel is filled with water), and if t is excessively large, then scattering is increased (excess water is drained from the vessel).
Financing of the paper
The study was supported by the grant No.24–29-00129 of the Russian Science Foundation,
https://rscf.ru/project/24-29-00129/.
AUTOR
Petr Maltsev, Dr. of Sciences (Tech), Professor, Leading Researcher, Inter-agency Center of Analytical Studies of the RAS; e-mail: p.p.maltsev@mail.ru; Moscow, Russia.
ORCID: 0000-0001-9160-5272
P. P. Maltsev
Inter-Agency Analytical Research Center of the RAS, Moscow, Russia
The article continues the discussion of effects occurring in the polymer threads with the metallic fractal clusters (see Photonics Russia. 2025;19(1):14–27.
DOI: 10.22184/1993‑7296.FRos.2025.19.1.14.27) and considers the features of capturing external radiation with a wavelength λ in the optical wavelength range by the fractal clusters made of spherical metal particles with the radius R. An explanation is given to the effect generation conditions titled “photon localization” when the condition R << λ is met.
Key words: metallic fractal clusters, nanoscale aluminum droplets, polymer threads, plasma, off-surface highly conductive state, photon localization
Article received: January 16, 2025
Article accepted: February 03, 2025
Properties of metallic fractal clusters
The main feature of a fractal cluster (FC) determining its physical and chemical properties shall be its scale invariance: when scaled up, any small FC fragment reproduces the spatial structure of the entire cluster. The scale invariance shall lead to the fact that the spatial arrangement of not all cluster particles, but a very large group of particles, turns out to be correlated, although visually the structure may be perceived as disordered.
The second consequence of scale invariance shall be the availability of a large number of cavities in the FC with a power distribution of dimensions, making the FC a rather delicate structure.
The photographs of polymer threads with the chains of irregular aluminum nanoislands (nanodroplets) with a size of 10–1000 nm (Fig. 11) were taken using a CAMSCAN-S4 scanning electron microscope with the energy-dispersive and wave-dispersive attachments: Oxford INCA Еnergy 350 and INCA Wave 700 (Cambridge, England) at the High Technology Center of the Synchrotron Collective Use Center of the Federal State Unitary Enterprise “Lukin Research Institute for Physical Problems” of the National Research Center “Kurchatov Institute” [1, 10].
The chains of irregular aluminum nanoislands with a size of 10–1000 nm, including spherical ones with a size of 10–30 nm, located on the surface of a polymer thread made of aramid fibers, can be represented as the metallic fractal clusters.
The publications describe various cases of the occurrence of low-field electron emission with the decreased size of particles on the surface of materials up to tens of nanometers and less [11–15], as well as the generation of threads made of individual tin atoms [16]. The studies of the dispersed media influence, including any metal particles, on the propagation of electromagnetic radiation, including the optical range, have been conducted for quite a long period of time [17–28]. Various effects have been found that require further theoretical development.
For example, an external electromagnetic wave propagating in a system of low-capture scatterers, as a result of multiple rescattering is capable of “moving in loops” in a limited spatial region. The reason for this phenomenon shall be the specific interference effects that occur even in a completely disordered system of particles [24–26, 29]. The effect of capturing external radiation in the optical wavelength range by a fractal cluster, for example, on the Rayleigh particles, is called “photon localization” that can be conditionally considered as a channel for “dissipation” of incident radiation, complementing the classical ones, such as inelastic scattering and absorption.
The “photon localization” concept in a fractal cluster
The photon localization effect in a system of scatterers can be expected under certain conditions [29]. First of all, the average photon absorption length la shall be greater than the elastic scattering length ls. In addition, the elastic scattering cross section on an individual particle shall be large enough. Otherwise, the loop generation is unlikely, since otherwise there will be no conditions for “unfolding” of the photon movement.
To describe these conditions, the expressions obtained in [28] shall be used:
la = n0 σa–1 и ls = n0 σs–1,
where σa – light absorption cross-section of an individual scatterer;
σs – cross-section of light elastic scattering by an individual scatterer;
n0 – concentration of scatterers in the system.
At first glance, due to the small elastic scattering cross-section of an individual Rayleigh scatterer with the radius R, even at the maximum achievable packing factors of the parameter p = λ / ls that determines the probability of localization, this value remains negligibly low: σs / πR = R / λ4 << 1.
This is indeed the case. However, the situation is changed if the frequency of the incident quantum ω coincides with the frequency of some electromagnetic mode of a separate scatterer. For example, for the surface plasmon of a spherical metallic individual particle with the radius R the following expression can be obtained: ω1 = ω0—3, where ω0 is the classical plasma frequency of an unlimited electron gas [28].
In this case, for a spherical metallic individual particle, the cross section of light elastic scattering by the particle shall be equal to [28]:
σs = 83 πR2 2 πRλ4 ω4ω2 – ω122 + γ2 ω14
and shall have a sharp maximum value, since the plasma resonance width is equal to γ = 10–2 for many metals, and the absorption cross-section of a spherical metallic individual particle shall be equal to: σa = 8 πR2 γω4λ ω2 – ω122 + γ2 ω14–1 and the packing factor of particles in the system shall be: f = 43 πR3 n0.
In this case, the parameter p = R / λ3 fγ2 shall be equal to one for the particles with Rλ = 10−1 already at f ≈ 0.1. The photon absorption length la shall remain comparable to the elastic scattering length ls [29].
Thus, the densely packed systems of nanoscale metallic particles at the external radiation frequencies ω1 shall be the suitable candidates for observing the photon localization phenomenon in the frequency range from visible to ultraviolet.
It shall be worth noting the paper [29] that has introduced the concept of “photon localization” in a fractal cluster with a detailed description of this effect. In the paper [29], a new approach to the description of electrodynamic properties of a fractal cluster consisting of non-absorbing solid particles has been developed using the diagrammatic method. The phenomenon of renormalization (decrease) of the external radiation wavelength λ as it penetrates the fractal cluster has been studied, and the effective capacitivity of the fractal cluster εfc has been calculated.
It is shown that the fractal cluster is characterized by a set of values εfc, each of which corresponds to its own renormalization degree λ. It is shown that the effective capacitivity of the cluster is a critical function of the fractal dimension of the cluster d, and if the d value is less than some critical value, it can be extremely high.
An external photon entering the fractal cluster has the ability to “discern” the cluster scale invariance. The photon wavelength in the cluster λint becomes much smaller than the external value λ, but the photon frequency ω is not changed, since the photon speed is simultaneously decreased, i. e. as the wavelength decreases, the photon acquires the ability to “discern” the increasingly smaller structural details. The long-range correlations in the arrangement of particles of the fractal cluster, visually expressed in the cluster connectivity, shall be the reason for the decreased external radiation wavelength λ when “moving in loops” in it.
The renormalization of λ shall occur as follows. The photon with a wavelength λ of the order of the typical cluster size L incident on the cluster shall be “captured” by some sufficiently large cavity of the fractal cluster (the primary resonance cavity). This capture shall lead to an increase in the effective capacitivity of the cluster εфк, since the ε value is increased near any electromagnetic resonance. The increased ε value shall initiate, in turn, a decrease in the photon wavelength: λint = λ. The photon with a renormalized wavelength λint shall find another resonance cavity of a smaller size. The new capture shall again stimulate an increased ε value and a new decrease in λint, etc. As a result, all cluster cavities can be filled with the renormalized virtual photons, including those which λint → 0. Their effective speed shall be equal to zero.
The light localization shall be associated with the closed loops on the trajectories of virtual photons [29]. If the photon makes a closed loop, the phase incursion of its wave function shall be equal to zero. The probability amplitudes relevant to the two possible ways of the loop bypass (clockwise and counter clockwise) shall interfere constructively regardless of the disorder degree of the scatterers. Any loop shall be a return back. Since the loop formation probability due to this kind of interference is increased, scattering into the back hemisphere is also increased. In turn, it stimulates the generation of new loops, etc. The result of this self-sustaining process shall be the photon “locking” in a limited spatial region, namely “the light localization”.
Difference from the standard description of light localization
The standard description scheme for the light localization shall be the reduction of the Bethe-Salpeter equation for an irreducible four-point vertex function (four-tail function) in the momentum representation to the transport equation (radiative transfer equation) and introduction of an effective electromagnetic energy diffusion coefficient [29]. Similar to the Anderson electron localization, vanishing of this coefficient shall mean strong light localization.
In [29], the approach to solving the radiation transfer equation differs from the standard one. The constructions shall be based on the idea of localized photons as the typical virtual particles, similar to the virtual photons of quantum electrodynamics. These photons shall not be related to either the detector or the light source.
A typical virtual photon [29] shall be a closed loop of the photon propagator (convolution of two vector potentials) growing on a double line describing the electron propagation (Fig. 11a) and occurring in the second order of perturbation theory (PT) with the Hamiltonian H:
H = p – e Ac22 m,
where e and m are the electron charge and mass,
c – speed of light in vacuum,
p – electron momentum,
A – vector potential of the electromagnetic field.
This self-closed photon propagator shall give some idea of the localized photon (Fig. 12a).
Formally, the localization [29] is associated with the occurrence of a pole in a fan-shaped diagram of four tail function (Fig. 12b). Such kind of pole shall describe the bound states of a pair of interacting particles, for example, an exciton. In the considered problems of the single photon propagation, the vertex function shall describe the efficient interaction of a pair of virtual photons (Fig. 12b) bypassing a closed loop on a trajectory in two opposite directions (more precisely, the interference of amplitudes relevant to these two bypass directions).
In opposition to the common scheme in the paper [29], the Bethe-Salpeter equation has been solved directly in the coordinate representation. As a part of the proposed constructions, there is no need to introduce the diffusion coefficient of electromagnetic energy. The localization is demonstrated simply as the interference corrections to the scattering and absorption cross-sections. It is these corrections that are calculated, and it is this “meaning” that is put into the word “localization”.
Specific features of virtual photons
A classic example of virtual photons shall be the virtual photons of quantum electrodynamics emitted by a moving electron. These photons are described by the propagator or Green’s function of the Maxwell’s equations, rather than by a plane electromagnetic wave. For example, at a given frequency, the wavelength of a virtual photon λint is determined by the efficient speed of light in the medium v according to the relation ω = 2πvλint. Under the localization conditions v → 0 that tends to zero either at a very large value of ε, or at a very large value of the derivative dεdω.
The phenomenon found sheds new light on the cause of a well-known shortage of numerous efficient medium approximations [29]. As a part of these theories, in a certain range of frequencies and packing factors, the effective capacitivity of a medium consisting of small non-absorbing particles turns out to be comprehensive one, thereby allowing for the existence of some mysterious absorption in the system. However, everything fits together: the effective absorption is related to localization.
As an example demonstrating reliability of the developed diagram technique, the results of the classical Mie theory in the problem of elastic scattering of an electromagnetic wave by a spherical metallic particle are reproduced in the paper [29]. At the same time, its shortcomings, consisting in non-consideration of the spatial dispersion effects, are found, and it is shown how to overcome them.
In a system of particles, the difference between the amplitudes of forward and reverse photon passage along the route “particle a – particle b” shall be determined not only by elastic scattering, but also by the light cycling or localization between these particles. It is this cycling that is described by the complex components σab.
In the paper [29], the photon localization theory in a dense random packing of Rayleigh particles is prepared. The strong deformation of the light scattering indicatrix by a single system particle is determined, expressed by an anomalous increase in scattering into the back hemisphere, as well as a strong localization sensitivity to the incident light polarization type. The determination of frequency and concentration range of photon localization is proposed.
As expected, the equation proposed in the paper [29] coincides with the equation for the Green’s function of the Maxwell’s equations with the same gauge. The proposed formalism is then applied to calculate the probabilities of the main electrodynamic processes in the dispersed media: elastic scattering, absorption, inelastic scattering, photoelectric effect, and various three-photon processes.
Examples of light localization application
The results of light localization studies can be used to develop the “random” (powder) lasers [29]. In a random laser, the role of mirrors is played by a multiple scattering medium, namely the clusters of nanoparticles of a weakly absorbing material (for example, ZnO). Another version of a “random” laser shall be introduction of such a powder into a laser cell on liquid dyes. After irradiation by an external light source, such a laser shall provide both light amplification and retention tin the system due to the multiple rescattering process. In contrast to a regular laser, the radiation of a “random” laser is isotropic. It is possible that a “random” laser can do without an active medium at all, due to the forced emission of localized light.
Based on the obtained theory of light localization in the fractal cluster [29], the lifetime of light localized both in a single fractal cluster and in an agglomerate of fractal clusters has been calculated. It is proposed to use the stimulated localized light emission to develop a fractal microlaser that does not require any inverse population of levels and is capable of operating in a wide range of lengths. As a result, the proposals are formalized in the form of an inventor’s certificate for a method for radiation conversion into the coherent light using the micron-sized devices [20].
In the paper [29], a model is proposed that allows us to understand the cause of giant Raman scattering (GRS) of light by the molecules adsorbed on the surface of small metallic particles. It is based on the radiation localization during the multiple inelastic light scattering in an ensemble of particles. When moving along a closed trajectory, the virtual photon repeatedly exchanges energy with the propagation medium while exciting fluctuations in the charge density. The energy of these fluctuations can be an arbitrary value, less than the photon energy.
The strong local fields related to these fluctuations and available in the entire frequency range from IR to UV shall be the GRS cause. The behavioral features of these fluctuations allow us to explain the specific features of flicker noise and also explain the catalytic properties of small metallic particles.
In addition, a new model describing the liquid-metal Rehbinder effect is proposed in the paper [29] that is based on the idea of possible electromagnetic field localization in the folds of phase boundary and components of a liquid eutectic mixture filling the cracks in the solid metal surface (a typical example shall be the liquid eutectic of In and Ga on the Al surface). Since at each spatial point of the eutectic mixture three various substances are adjacent (a homogeneous melt of In+Ga, solid In and solid Ga), the system of folds of such an interface is modeled by the Wada-Brauer structure well-known in topology, namely a surface separating three various regions at each of its points.
The localized photons are capable of “switching off” the Coulomb attraction of charge fluctuations on opposite banks of a eutectic-filled crack that provides for the Van der Waals attraction of the banks. The phenomenon of localized light emission in the liquid-metal Rehbinder effect and similar emission from a Casimir gap with the broken symmetry are predicted.
The useful result of the paper [29] is possible application of another property occurred during the photon localization process to explain generation of a wide “base” in the lower part of the Cherenkov radiation cone (Fig. 12a). Such an external form of Cherenkov radiation obtained during the occurrence of breakdown on runaway electrons (the electron velocities close to the speed of light) [30–32], has been registered during a discharge on a 30 m long polymer thread with metallic fractal clusters (at an electric field strength of 30 kV/m) [30, 32]. The composite threads with metallic fractal clusters used in the experiment presumably have the properties of a metamaterial with negative capacitivity [1, 6, 8].
Having considered that the phase velocity and group velocity in metamaterials are aimed at various directions for Cherenkov radiation (“inverted” Cherenkov radiation) [8], the phase velocity shall determine the tip of Cherenkov radiation, and the group velocity shall be opposite and shall influence the formation of its “base”.
However, when photons are localized, the group velocity in the metallic fractal cluster is decreased and a strong deformation of the scattering light indicatrix by an individual particle of the system occurs that is expressed by an anomalous increased scattering into the back hemisphere and isotropic radiation [29] that can lead to a sharp expansion of the “base” of Cherenkov radiation (Fig. 13a).
The similar form of radiation can be observed for the high-altitude sprite breakdown (Fig. 13b) on the runaway electrons (the electron velocities are close to the speed of light) [31, 32], occurring due to the impact of high-energy cosmic particles on the fractals and beginning from its micro breakdowns. It is possible that the observed “reversed” Cherenkov radiation (high-altitude breakdown), directed towards the source (towards the high-energy cosmic particles), also occurs on the fractals with “lattice architecture” [31] having the properties of metamaterials with negative capacitivity.
Conclusions for part 2
Hypothetically, it can be assumed that under certain conditions, an external electromagnetic wave propagating in a system of weakly absorbing scatterers as a result of multiple rescattering is capable of “moving in loops” in a limited spatial region, and the reason for this phenomenon is specific interference effects that occur even in an absolutely disordered system of particles. This effect of capturing external radiation in the optical wavelength range by a fractal cluster has been called “photon localization” that can be considered conditionally as a channel for “dissipation” of incident radiation, complementing the classical ones, namely the inelastic scattering and absorption.
The essence of “light localization” is well illustrated by the following simple analogy, proposed in the paper [29]. Let us collect some water in a wide vessel and after some time t open a narrow drain hole. Provided that the water inflow Q into the vessel is exactly equal to its outflow, the water level in the vessel shall soon stabilize at the H mark (Torricelli formula). If the initial water level in the vessel is low, then the vessel is filled to the H mark, and if the water delay prior to the hole opening is large enough, then the excess water shall be drained.
The role of water is played by radiation, the vessel is a system of particles, and the water in the vessel is localized light. If the lifetime t of a localized photon is short, the system reacts by decreasing scattering (the vessel is filled with water), and if t is excessively large, then scattering is increased (excess water is drained from the vessel).
Financing of the paper
The study was supported by the grant No.24–29-00129 of the Russian Science Foundation,
https://rscf.ru/project/24-29-00129/.
AUTOR
Petr Maltsev, Dr. of Sciences (Tech), Professor, Leading Researcher, Inter-agency Center of Analytical Studies of the RAS; e-mail: p.p.maltsev@mail.ru; Moscow, Russia.
ORCID: 0000-0001-9160-5272
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