Interaction between thin conductive fibers and microwave radiation

We study the effect of an anomalously strong interaction of electromagnetic radiation with very thin conducting fibers. Metal, semiconductor or graphite fibers, which diameter is several hundred times smaller than the radiation wavelength, strongly absorb and scatter the electromagnetic waves, which electric vector is parallel to the fiber axis. The efficiency factors of attenuation, absorption, scattering and pressure of radiation for fibers with a diameter of several micrometers in the centimeter and millimeter ranges reach several thousand. We determine the nature of this effect. It occurs when the transverse dimension of the fiber is comparable to the thickness of the skin layer. Then the electric field of the incident wave fills the entire volume of the fiber, and the absorbed power is the highest. We also provide a theoretical analysis of such effect, determine conditions for its existence: the ratio between the radiation wavelength and the diameter of the fiber, the value of conductivity. We also provide an experimental study of the effect, when unfocused radiation beam with a wavelength of 8 mm at the output of waveguide with a 7.2×3.4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$7.2 \times 3.4$$\end{document} mm cross section transfers more than 10% of the beam power to a 12 μm diameter graphite fiber. We make an analysis of the mathematical model of the process of heating of graphite fiber by a microwave radiation beam. Then we solve a heat conduction equation and find the temperature distribution along the fiber, which is in agreement with the measured one.


Introduction
The problem of diffraction of electromagnetic waves by cylinder is one of the most famous in electrodynamics. The results of its solution are presented in classical monographs (Van de Hulst 1981;Kerker 1969;Bohren et al. 1983) and in many papers. An analysis of the diffraction pattern makes it possible to obtain information about the size of the cylinder cross section, its shape, and the refractive index (Lazarev et al. 1988 The impact of the wave on the cylinder depends on the ratio between the radiation wavelength λ and the cylinder diameter D. The strongest effect is, as usual, when the cylinder cross-section is comparable with the wavelength. In this case, various resonances arise, and the interaction between wave and cylinder increases. Objects that are small compared to the wavelength and located on the wave propagation path, as usual, have no effect on the diffraction pattern. But in works (Kuzmichev et al. 2003;Kokodii 2006;Akhmeteli 2004Akhmeteli , 2006 there are described the effect of strong absorption of electromagnetic radiation in very thin conducting cylinders (metal wires, semiconductor and graphite fibers), when the radiation wavelength is several hundred times larger than the diameter of the cylinder.
In papers (Akhmeteli et al. 2013;Akhmeteli et al. 2015;Kokodii et al. 2017) described are cases when the wire is located along the radiation beam axis: the effect of strong absorption of radiation by the wire is also observed. The paper (He et al. 2011) describes an experiment, where the wire is located in the waveguide.
In all above works, the physical nature of the observed effect has not been explained. But its understanding is necessary both for physics and for the practical use of such effect: for transferring the energy of electromagnetic radiation to small targets without beam focusing, for creating protective screens in the microwave range, etc.
In this paper we present our results of studies of the nature of the effect mentioned above. We study the case of normal incidence of incident wave. The case of a longitudinal arrangement of the wire in the radiation beam is even less clear, since in this case the electric wave vector is perpendicular to the wire, and there are no conditions for the occurrence of current in it. This is a separate task.

Absorption and scattering of electromagnetic radiation by a thin wire
Calculations and experiments in works mentioned above show, that for some ratios between radiation wavelength and cylinder diameter, the absorption, scattering and radiation pressure can be very large. In particular, this is observed in centimeter and millimeter ranges, when the cylinder diameter is equal to several micrometers, that is, it is hundred and thousand times smaller than the wavelength. Under the normal incidence, the electric vector must be parallel to the axis of the cylinder (E-wave).
In Fig. 1 presented are dependences of the absorption efficiency factor of a platinum wire on its diameter for several wavelengths, which are calculated using the formulas from works (Van de Hulst 1981;Kerker 1969;Bohren et al. 1983). At some values of D∕ one can see a maximum of absorption. If = 8 mm, then the maximum is Q abs =962 , at = 10 cm Q abs = 2615 , at = 1 m Q abs = 7928 . The position of the maximum Q abs also depends on the value of D∕ : D max = 4.1 μm at = 1 m, D max = 1.5 µm at = 0.1 m, D max = 0.3 µm at = 8 mm.
In the case of the H-wave, the mentioned above effect is absent. Under the same conditions, Q abs < 0.01 . Therefore, in further investigations we will consider the E-wave case.
The absorption of radiation energy in a wire is characterized by the absorption efficiency factor Q abs = P abs P , where P is the radiation power that hit the wire, P abs is the radiation power absorbed in it. The absorbed power can be either less than the power that hit the cylinder, or more than it. This is explained by the interaction of the incident wave with the oscillating systems of atoms and molecules of the substance and the resulting resonances. Graphs in Fig. 1 have some similarity with resonance curves.
We have solved two problems: 1. We have found ratios between the wavelength and the cylinder diameter, when the most efficient transfer of radiation energy occurs. 2. We have found the intensity distribution of the electromagnetic field inside the wire and clarified the conditions, when the power density in wire is maximum.
The absorption efficiency factor can be found by the equation (Van de Hulst 1981;Kerker 1969;Bohren et al. 1983) where J l (z) is Bessel function, H (2) l (z) is Hankel function of the 2nd kind, m = niκ is complex refractive index, ρ = πD/λ, D is diameter of the wire, λ is radiation wavelength in free space.
In the microwave range, the complex refractive index of a conducting medium is given by the following formula [1]: Where σ is specific conductivity, ω is circular frequency, ε 0 is free space dielectric constant.
We assume that the wire is very thin and the conditions ρ < < 1, |mρ|< < 1 are fulfilled.
(1) In this case, one can restrict series (1) to the first term and coefficients b l to the first term of the expansions into a series of Bessel and Hankel functions. After some transformations, we get In this expression the unity in the sum in the denominator can be neglected, and it can be simplified to Let us determine the position and the magnitude of the absorption maximum. From (5) one can see that the trend of the curve is determined mainly by the value ρ. The value ln( ) changes slowly. Therefore, we differentiate formula (5) with the argument ρ, also we assume ln = const . After this we obtain the equation After solving (6), we can determine the position of maximum. The second term in square brackets is much smaller than the first one, it can be neglected. Then Eq. (6) reduces to one that can be solved by the method of successive approximations: The initial value of the unknown quantity ρ must be small. Below are the results of successive approximations of (7) with the initial approximation = 0.1.
The second column of the Table 1 shows the values of n = D∕ i , where i = ∕n is the wavelength inside the wire. This parameter indicates how many wavelengths fit along the circumference of the wire cross section.
One can see, the stable value of the root of Eq. (7) is obtained quickly. The condition for the absorption maximum becomes as follow, (4) Q abs = 2 2 n 2 2 1 + 2 n 2 2 2 + 2 ln 2 .
(5) Q abs = n 2 1 + (n ) 2 ln 2 . or It means that the diameter of wire should be approximately 10 times smaller than the wavelength in the medium.
Substitution condition (8) into expressions (4) and (5) shows that the maximum value of the absorption efficiency factor is In spite of the similarity in appearance of the curves in Fig. 1 and resonance curves, this is not a resonance. In resonance, an integer number of half-waves fits on the characteristic dimension of the object. Here, the diameter of the cylinder is much smaller than the wavelength.
The fact that the nature of the interaction of a wave with a very thin wire is non-resonant was also indicated in (Akhmeteli 2006) -"the presence of absorption in cylinder results in the appearance of a deep hole in the field distribution and following diffusion of the field to the axis from a large volume of the surrounding space". A thin wire is illuminated by the flux of radiation from surrounding space (see Fig. 2a). In Fig. 2b presented is the radiation flux pattern onto the wire located far from the absorption maximum. In this case the radiation flux is much weaker.

Qualitative explanation of the effect
The effect of strong absorption of electromagnetic radiation in a very thin conducting fiber can be explained as follows. It is observed when the fiber diameter is comparable to the thickness of its skin layer. Then the field fills the entire cross section of the fiber, and the average value of the electric field density in it is maximum possible. Therefore, the volume field energy density W E = E 2 and the absorbed radiation power P abs are also the maximum possible. This power is proportional to the volume of the fiber, that is, the square of its diameter. The radiation power P incident on the fiber is proportional to its frontal cross-section, that is, the diameter ( P = IDL , where I is the radiation intensity, D is the fiber diameter, L is the length of the illuminated part of the fiber). The absorption efficiency factor Q abs = P abs ∕P increases with the fiber diameter. But when the fiber diameter becomes larger than the thickness of the skin layer, the field fills only part of the fiber volume, and the average value of the electric field density decreases with further increasing of diameter. The absorbed power and the absorption efficiency factor also decrease. This explains the presence of a flat maximum on the absorption efficiency factor graphs. There is a diameter value at which the P abs ∕P ratio is maximum. The effect of conductivity is dual. On the one hand, it must be large for the large volume energy density W E . On the other hand, it must be small to provide the thicker skin layer. In case of high conductivity, the skin layer is thin, and the absorption maximum will be in thin fibers. In case of low conductivity, the skin layer is thicker, and the absorption maximum will be in thick fibers.
The radiation wavelength also plays a role. At longer wavelengths, the skin layer is thicker, and the absorption maximum shifts to larger fiber diameters.
After substitution into (10) the expression for the skin layer thickness = √ 2∕ , we obtain Absorption increases with increasing of conductivity and skin layer thickness. But these values cannot increase simultaneously. The skin layer thickness decreases with increasing of conductivity. This leads to the appearance of a maximum at a certain value of the fiber diameter.
For platinum on frequency of 30 GHz ( = 0.01 m), = 0.94 µm. The maximum of absorption efficiency factor for platinum wire at radiation wavelength = 0.01 m can be reached for a wire diameter D = 0.45 µm. This is 2 times less than the skin layer thickness. The intensity distribution of the electric field inside the wire and near it for such case is shown in Fig. 3. The field intensity is almost the same over the entire cross section of the wire and in its vicinity, and it is much lower than the field intensity of the incident wave, which is taken equal to 1. This indicates that the interaction between radiation and wire is very strong, despite on its small diameter.

Experiment
The block scheme of the experimental installation is shown in Fig. 4. We use a backward-wave tube 1 as the source of radiation. The radiation wavelength in free space = 8 mm. The radiation, wave type is H 10 , goes through waveguide 2 with a cross section of 7.2 × 3.4 mm. At a distance of 3 mm from the outlet flange 3, there is located a frame 4 with graphite fiber 5 with a diameter D = 12 μm. The fiber is placed perpendicular to the wide wall of the waveguide at the electric field maximum. The length of the fiber is L = 70 mm, therefore, whole radiation from the waveguide reaches a fiber. Part of the radiation power is absorbed in the fiber and heats it. Heat is determined by the change of the fiber resistance, which is measured with the ohmmeter 6. The radiation power is controlled by a thermistor wattmeter 7, which is connected to the waveguide with the directional coupler 8.
The resistance of the "cold" fiber is R 0 = 4149 Ω, the resistance of the "hot" fiber is R = 4100 Ω. The length-average heating temperature of the fiber is Here R = −0.0003 K −1 is the temperature coefficient of resistance of graphite. The resistance of graphite decreases under the heating.
The power absorbed in a thin fiber can be defined as Here P = 0.02 W/(m*K). This is the linear coefficient of heat exchange of the fiber with the external environment. It determines the amount of heat, that goes into the external environment from 1 m of fiber at a temperature difference between the fiber and the environment of 1 K. For thin fibers, it does not depend on their diameter (Bosworth 1952). It was determined experimentally. The absorption coefficient is Thus, 11% of the microwave radiation energy was absorbed by the fiber, in spite of the fact that the fiber diameter is 600 times smaller than the transverse size of the radiation beam.
The radiation power incident on the fiber can be defined as where I 0 is the radiation intensity at the fiber location, D is the fiber diameter, b is the length of the illuminated area, which is equal to the size of the vertical wall of the waveguide, b = 3.4 mm. The fiber is located in the middle of the wide wall of the waveguide in the place of the electric field maximum of the H 10 wave. The radiation intensity here is where a and b are cross section dimensions of the waveguide.

Therefore and absorption efficiency factor is
The absorption efficiency factor calculated using Eqs. (1) and (2) is Q abs = 22 . Taking into account the large spread of parameters of different types of graphite, the agreement between the theoretical and experimental values of Q abs is satisfactory.
In Fig. 5 presented is a picture of graphite fiber heated by microwave radiation, obtained using a thermovision camera. One can see the waveguide flange 1, the waveguide output 2 with dimensions 7.2 × 3.4 mm, graphite fiber 3 and its heated Sect. 4. The thermovision camera was located on the side of the waveguide, so it seems that the fiber is shifted from the middle of the wide wall of the waveguide.

Mathematical modeling of the process of absorption of electromagnetic radiation by a fiber
In experiment, we have measured the length-average temperature of the fiber. But the temperature distribution along the length of the fiber is non-uniform. The temperature of the fiber part, which is just opposite the waveguide output, is high. Roughly it can be estimated as follows. The length of the heated fiber part is approximately equal to the length of the narrow wall of the waveguide, b = 3.4 mm. If we assume that the temperature T max in this part is constant, then the length-average temperature of the fiber is Q abs = P abs P = 33. It follows that when T cp = ΔT + T 0 = 59 °C ( T 0 = 20 °C is environment temperature),T max = 1214 °C. This value is overestimated, since the temperature of the heated section is not a constant along its length, but it is obvious that it is high.
We have solved the heat conduction equation for a thin wire in presence of external heat sources (Carslow et al. 1959), where A is bulk density of heat sources, W/m 3 , a = k∕(c ) is thermal diffusivity, m 2 /s, k is thermal conductivity, W/(m*K), c is specific heat capacity, J/(kg*K), ρ is density, kg/m 3 , T is heating temperature, °C, z is coordinate along the fiber axis, m, t is time, s.
Let the intensity of the radiation incident on the fiber vary along the z axis as I(z). Then the power absorbed in a part of the wire of length dz is The bulk density of absorbed power is One part of this power heats the fiber, other power dissipates in the external environment. This dissipation is carried out by two mechanisms -convection and radiation. It is reasonable to take into account only convection, since the average heating of the fiber is relatively small, and the power losses are small.
The power which dissipates from fiber part dz is where T 0 is environment temperature, α p is linear heat exchange coefficient. Volume heat loss density is Equation (20) now can be written as After substitution expressions (23) and (24) in (25), we obtain Let us set the following initial and boundary conditions: dP abs = Q abs ID dz.
Such conditions demonstrate that heat does not reach the ends of the fiber, and their temperature are equal to the environment one.
For calculations we have used the following data: graphite fiber diameter D = 12 μm, length L = 70 mm, graphite density = 2500 kg/m 3 , thermal conductivity k = 156 W/ (m*K), heat capacity L = 700 J/(kg* K). These are table values. The linear heat transfer coefficient is p = 0.02 W/(m*K).
The radiation beam falls on the middle of the fiber segment. Type of the wave in the waveguide is H 10 . Therefore, the intensity of the incident radiation along the z axis does not change, and the function I(z) can be written as Here b = 3.4 mm is the waveguide narrow wall width.
The results of solving the heat conduction equation are presented in Fig. 6. In Fig. 6, the solid line shows the results of solving Eq. (26). The dashed line shows the temperature distribution measured by the thermovision camera. The two thin dashed lines show the part of fiber heated by radiation. The temperature in maximum, determined from the Eq. (26), is 590 °C. This gives an average fiber temperature of 29 °C, which is in good agreement with the results of experiment.

Conclusions
We have shown that conductive fibers, which diameter is much smaller than the wavelength of the incident electromagnetic radiation, absorb and scatter radiation very strongly. The effect has a non-resonant character. The strongest absorption of radiation occurs when the wavelength in the fiber medium is 10 times bigger than its diameter. Strong absorption occurs when the fiber diameter is comparable to the thickness of the skin layer.
In the experiment, the radiation beam at the output of the waveguide with a cross section of 7.2 × 3.4 mm transferred 11% of the energy to a graphite fiber with a diameter of 12 μm. The temperature of the illuminated part of the fiber was about 600 °C. This effect (28) Fig. 6 Temperature distribution along the fiber can be used for transferring the electromagnetic radiation power without focusing to thin wire targets.
Funding The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.

Conflict of interest
The authors have no relevant financial or non-financial interests to disclose.