Polarization-dependent resonant phenomena in all-dielectric scatterers: inversion of magnetic inductance and electric displacement

The theoretical description and experimental verification of resonant phenomena in electromagnetic fields generated in the near zone of all-dielectric rectangular thin sub-wavelength frames, subjected to an incident microwave, are considered. The geometry of considered problems is presented by means of arrangements of these frames in three orthogonal planes, normal, respectively, to electric component, magnetic component and wave vector of the incident wave. Such trio paves the way to design of 3D all-dielectric multiresonant microwave unit cell. Displacement currents, generated by linearly polarized electromagnetic waves in the system, lead to the formation of magnetic and electric dipoles, each of which possesses own resonant frequency. Resonant inversion of magnetic inductance and electric displacement is observed. The sliding incidence of plane wave on the frame is shown to provide the sharp and deep resonance in the components of generated field. The phase shift equal to π between the magnetic components of incident and generated waves indicates the formation of negative magnetic response. There observes the angular anisotropy of arising dipoles, manifested in values of resonance frequencies and in the dependence of depths of resonant spectral dips upon the orientation of dipoles, with respect to the direction of propagation of incident wave.


Introduction
The last 10 -15 years had witnessed a rapid growth of interest in the phenomena of resonant electromagnetic inductance for all-dielectric structures irradiated by electromagnetic waves.
Variations of magnetic components of these waves were shown to induce the displacement currents in the dielectric material; these currents, in their turn, generated the magnetic modes.
The efficiency of this generation proved to be strengthened drastically, when the frequencies of driving waves were approaching to the resonant frequencies of dielectric structures determined by their shapes, sizes and dielectric properties.Analysis of these phenomena constitutes nowadays one of the newly shaping branches in the electrodynamics of continuous media [1].
Although the dielectric ring resonators supporting the concentric GHz modes were considered several decades ago [2], the systematic research of magnetic modes became a "hot" topic later due to discovery of "artificial magnetism" [3,4].This unusual phenomenon was demonstrated by resonant excitation of magnetic modes and formation of negative magnetic inductance in alldielectric sub wavelength rings in the field of monochromatic electromagnetic wave [5,6].The resonant frequencies were calculated for isotropic spheres [7] and rotationally symmetric particles [8].The elliptically shaped plane circuit, providing the angular anisotropy of magnetic mode was examined [9] These facilities can be viewed as the generalizations of well-known Thomson oscillating circuit [10] combining the inductance and capacitive properties in one resonant LC structure.An another group of optical magnetization effects connected with Mie resonant scattering of electromagnetic waves [11] was considered for the dielectric discs [12], cylinders [13,14] and spheres [15,16].Consideration of all-dielectric magnetic and electric dipoles in the microwave range is linked usually with the design of frame aerials and their radiation patterns in the far zone.
To the contrary this paper is aimed at the theoretical analysis and experimental verification of the lowest LC resonances, both electrical and magnetic, induced in the rectangular thin all- dielectric sub wavelength circuits by the driving linearly polarized GHz microwaves.3Dstructures with predefined properties can be constructed from such circuits (see Fig. 1).ii. feasibility of creation of anisotropic radiation patterns in different spectral ranges; iii. elaboration of miniaturized sub wavelength circuits sensitive to the directivity and polarization of incident radiation [17][18][19].
Several effects contribute to the response of the system to an external irradiation [20]; each of themat its own resonant frequency.To compare and contrast the different LC resonances we consider the resonant phenomena for the rectangular frame with internal sizes and the square cross-section size h .Figure 3 shows the location of the frame in the     .The paper is organized as follows: the resonant inversions of magnetic inductance and electric displacement in the rectangular frame subject to its orientation with respect to the incident microwave are considered theoretically in Sects. 2 and 3 respectively; the experiments verifying this analysis are briefly described in Sect.4; the main results are listed in Sect. 5. Some general mathematical formulas are shown in the Appendix.

Magnetic resonances and negative magnetic inductance in the rectangular all-dielectric frames
Analysis of magnetic resonances is carried out below for two cases characterizing 3D orientation of rectangular frame: the plane of frame   y x, is either normal (Fig. 3) or parallel (see Fig. 4 and Fig. 5) to the magnetic component H  of incident wave (  z direction).2a.To examine the magnetic response of all-dielectric rectangular frame induced by the sliding incidence of microwave one has to consider the alternating magnetic flow  , stipulated by the auxiliary frame self-inductance (see Fig. 3).This flow is known to generate in the frame the electromotive force U and the curl electric field i E , directed along the frame's perimeter This field is the area of square cross-section of the frame's sides.Substitution of Eq. (1) into Eq.( 2) brings the value of displacement current in the frame The magnetic flow Considering, e.g., the orientation of frame, corresponding to the case A (see Figs. 2 and 3), we'll find Manipulations with expressions for inductance flow i  (4) and initial magnetic flow brings the equation governing the unknown flow  : Dimensionless factor K in Eq. ( 6) determining the self-inductance of rectangular frame 1 4 L a K  subject to the ratio of distances between the axes of its sides 11 / p b a  is given in Appendix (A4).Solution of Eq. ( 6) reads as Equation ( 7) describes the forced oscillations of magnetic flow  passing through the circuit, 0  is the resonant frequency The structure of the fields that arise as a result of this effect is shown in the dimensionless form for the components ,    Note, that Eqs. ( 5) and ( 6) are valid for the orientation B (see Fig. 2) due to replacement H arise in the parallel rod too.This structure resembles the pair of long inductance coils with concentric currents and axial magnetic fields.However, in the framework of thin circuit model the magnetic interaction between these coils is small: Due to smallness of sizes of square cross-section of dielectric rod h with respect to the wavelength of incident wave one can view the incident field H inside the rod as spatially homogeneous: Considering the long rod and ignoring the end effects we obtain from Eqs. ( 8) and ( 9) the equation governing z H 1 component of induced magnetic field inside the rod Note, that the directions of induced field components z H 1 inside and outside of the rod are opposite.Thus, supposing the value To find the coefficients mn  it is expedient to present the incident field (11): the coefficient 11  related to the lowest mode in the presentation (11), is . The coefficients mn  can be calculated due to orthogonality of terms in the Fourier series.Thus, the solution of Eq. ( 10), describing the lowest mode in the presentation (11) in the spectrum of oscillations of induced magnetic field, reads as: Here     is the resonant factor, 00 2 f   is the lowest resonant frequency  , one can see that these fields obey to the another pair of Maxwell equations too:

Substitution of expression for
The calculated frequency GHz 19 .4 0  f is in good agreement with the experimental data.
Note, that the values of 0 f (13) obtained for the model of a long dielectric rod, ignoring the end effects, prove to be independent upon the rod's length for both positions A and B.

Resonant pair of electric dipoles and negative electric inductance in the rectangular alldielectric frames
Let us examine the electric LC resonances excited in the pair of parallel sides of rectangular frame directed along the electric field of incident microwave E  , as is shown in Figs. 3 and 4.
The alternating field E  induces the periodically oscillating positive and negative charges Q  on the opposite ends of these rods as well as the electric fields and displacement currents propagating along the rods ( x -direction).These currents can be presented as the waves with frequency  and wave number x q .To analyze these waves, it is convenient to  and 4) and cross-section area 2 h by the equations of transmission line [10]: 0 The displacement current 4 exp Polarization of the rod by field d E induces the oscillating charges Q  on the ends of the rod: ; the resonant frequencies 0 () f  relate respectively to the parallel and anti-parallel currents induced in parallel rods In a simple case, when the plane of frame is normal to the wave vector of incident wave k  (see Fig. 4) the phases of currents excited in the parallel rods are equal; thus the currents are parallel and their mutual inductance is given by expression is determined by formulae: 21 The values of mutual inductances The resonant frequency of the pair of electric dipoles in position A computed from Eq. ( 22) is  

Brief description of experiments
Currently, when creating metamaterials with a negative magnetic response, metal elements of nanoscale values are used, the expected characteristics of which are checked in advance on a mathematical model to determine the integral field of interaction with the incident wave.Threedimensional periodic structures with a period of hundreds of nanometers are built from such elements.However, the quality factor of such elements is limited by ohmic in the metal parts.With a large dielectric constant and low dielectric losses, the dimensions of the dielectric resonators are significantly smaller than the dimensions of the metal resonators.Therefore, in recent years there has been a tendency to replace metal elements with sub-wave weakly absorbing dielectric structures from metamaterials, and the problem has arisen of controlling the magnetic components of the light field of the optical and infrared ranges using dielectric magnetic structures.Thus, the task is to create dielectric structures with the required spectral and spatial characteristics of the magnetic response in the gigahertz, terahertz and infrared ranges.
Direct experimental studies of the electromagnetic properties for individual nodes of nanoscale structures are difficult.Therefore, nanostructures with alternating magnetic susceptibility can be pre-modeled using theoretically calculated and experimentally verified parameters of the dielectric magnetic dipoles of the gigahertz range.
The purpose of the experiments was to directly measure the fields near the dielectric frame for correct comparison with the results of theoretical calculations.The main tasks that were set in the experiment: the excitation of a resonant electromagnetic response with the phase change in the reradiated wave and the inversion of the resulting field; detection of resonant frequencies when a linearly polarized TEM-wave is incident on the dielectric frame; investigation of the influence of material and object geometry on resonant frequencies.
The size of the frame was preliminarily chosen so that, on the one hand, it was large enough to measure the fields around the frame, and, on the other hand, so that it was much smaller than the wavelength of the incident wave for the region of the assumed resonant frequencies.In doing so, the studied frequency band should be accessible and convenient enough for measurements.
As a result, the following frame characteristics were selected (see Fig. 3): external dimensions is 3 cm × 2 cm (therefore Gigahertz Resonant Measurement.Agilent E5071C ENA Network Analyzer was used for the generation and registration of emission spectra of GHz-range.In one case ETS-Lindgren's model 3160-09 pyramidal horn antennas were employed as the transmitter and receiver.In the other case ETS-Lindgren's model 3115 double ridged waveguide horn antenna was applied as a transmitter.To increase the signal-to-noise ratio and decrease the influence of radio noise the additional amplifier was used.The transmitter power was 10 mw.The linear probes of high frequency electric field with the length of sensitive element 1 cm were used for measurements of electric field near by the frame.The high frequency magnetic field in the near zone was registered by the screened circular probe with 0.5 cm diameter of sensitive element.
Phase Shift Measurement.The study of the phase shift between the initial and the reradiated waves was performed using a high-speed four-channel oscilloscope Tektronix DPO73304DX with two probes of the electric field.One probe located near antenna beyond of the area of influence of dielectric frame was used as a referent one.Its signal was recorded by one channel of the oscillations detector.Another probe was located close to the frame.Its signal was fixed by the second channel.Both probes were oriented in the direction of maximum sensitivity to the electric field of the incident wave and the distance between the probes kept constant in order to avoid any supplementary phase shift between them in the measurement process.
Extensive measurements were developed and carried out by a group of experimenters from the JIHT RAS (Moscow) composed of V.Ya.Pecherkin, L.M. Vasilyak, S.P. Vetchinin, and detailed measurement results will be published by them in a separate article.Our goal was to theoretically calculate resonances and compare them with experimental data.As a result, we can state that the measured experimental values of resonant frequencies are in good agreement with the theoretical values found in previous Sects.6), ( 13), ( 18), ( 20) and ( 22).Resonant inversion of magnetic inductance and electric displacement stipulated by these dipoles occurs in the high frequency spectral ranges 0 f f  .

Conclusion
3. The turn of rectangular frame from position A to position B (see Figs.

Appendix: Formulas for capacitances and inductances
Precise formula for inductance of a wire [21]: where g r is the geometric mean distance of the cross-sectional area from itself, a r is the arithmetic mean distance of the cross-sectional area from itself, and q r is the mean square distance of the cross-sectional area from itself, D is the distance between the ends of the wire; for Self-induction of a rectilinear wire with length l and square section hh  is: When a unidirectional current is excited, the self-induction coefficient L of a rectangular frame can be found according to the general rules [21], taking into account the self-induction of two sides of the rectangular frame with the length  (A4) The capacitance of flat capacitor is , 4 where ε is the dielectric permittivity, S is the area, d is the distance between the plates.
In our case, the capacitance a c per unit length (as two pair of plates with   (A8)

Fig. 1 .Fig. 2 .
Fig. 1. 3D all-dielectric multiresonant microwave unit cell.Rectangular all-dielectric circuits are arranged in three orthogonal planes, normal respectively to electric component, magnetic component and wave vector of the driving wave.
. normal to the magnetic component of incident wave.The sizes of an auxiliary rectangular contour formed by the axes of rods constituting the rectangular frame

Fig. 3 . 1 db  and 1 da
Fig. 3. Frame parameters and the geometry of scattering for sliding incidence: the plane of frame   y x, is normal to the magnetic component of incident wave H 0 .

Fig. 4 .
Fig. 4. The geometry of scattering for normal incidence: the plane of frame   , xz is parallel to

Fig. 5 .
Fig. 5.The geometry of scattering for sliding incidence: the plane of frame   , yz is parallel to the magnetic component of incident wave.

iE
generated in the circuit with dielectric permittivity  , forms the alternating electricdisplacement i E D  , providing in its turn the curl displacement current d I in the frame[2]

i
, induced by the displacement current d I from Eq. (3), can be expressed by means of self-inductance of this frame L : unknown magnetic flow  , one has to add the inductance flow i  (4) generated by displacement current to the initial magnetic flow 0 formed by the magnetic component of incident wave

Fig. 6 .
Fig. 6.Distribution of the component x E in the plane of the frame, which is conventionally drawn by a solid line at the zero level of the field component.

Fig. 8 .
Fig. 8. Distribution of the component z H in the plane of the frame, which is conventionally drawn by a solid line at the zero level of the field component.
f  is in good agreement with the experimental data.2b.Consider now the electromagnetic inductance in a rectangular frame, when the alternating magnetic component H  of incident wave is parallel to one of the rods, forming the frame.In such geometry, shown in Figs. 4 and 5, the field H  , directed along z-axes, generates the curl electric currents and fields turn, induces the secondary magnetic field   11 0, 0, z HH in the rod itself.The similar inductive effects ensuring the generation of secondary magnetic field 2 model each rod as a segment of transmission line characterized by the capacitance a c , self- inductance a l and mutual inductance a m per unit length.Unlike the insignificant mutual inductance of curl currents in the parallel coil-like structures, discussed in the Sect.2b, the magnetic interaction of linearly polarized currents in the parallel rods is shown below to be an effect, affecting the resonant regimes in the rectangular dielectric frame.Thus, we have to examine the pair of interacting electric dipoles.Considering, e.g., the position A and ignoring the losses, one can describe the voltage U and displacement current d I

I
determined by set (15) -(16) to Eq. (17) brings the expression for d E .Solution describing the standing half-waves of field d E in this rod near by the lowest resonance, corresponding to the value 1 2

Here 0 
in(18) is the resonant frequency, ) in position B carried out by the same formulas (A7) -(A8) with the values 21 , l b d a  gives the another resonant frequency   0 4.72 GHz B f   .Thus, the LC resonant frequencies for the rectangular electric dipole, arranged in a plane, normal to the direction of wave propagation (see Fig. 4), are different in positions A and B. To the contrary, analysis of resonances in the parallel rods in the geometry of Fig. 3 is more complicated, since the phase delays φ between currents excited by the microwave with wavelength  in the parallel rods depend in this case upon the distances d between them.Thus, the mutual inductance between rods becomes equal to in mind the sub wavelength frame let us restrict ourselves by the case 0 d     02   ; herein the currents can change their directions during one period of incident wave T, providing the non-stationary piecewise variations of mutual inductances.These rapid variations impede the computation of resonant frequencies for the intermediate values of phase delay 02   .To evaluate the influence of both inductances  m and  m on the resonant frequencies    0 f in all the range 02   one can introduce the quantity M presenting some value of mutual inductance averaged over the wave period T.This quantity M , dependent upon the frequency

1 M and 2 MFig. 9 .f 1 (
Fig. 9. Approximation of frequency dispersion of mutual inductance for the pair of interacting resonant electric dipoles.

1 1
in the similar manner and using the values of inductances  m and  m in position B, we find the resonant frequency for position B:   0 5.61 GHz B f  .Comparison with the experimental spectra shows the discrepancies of computed values of resonances with the experimental ones about 10-12% and 3-4% in positions A and B respectively.The greater discrepancy of computed frequencies 0 f in position A as compared with position B is linked with the fulfillment of condition hd   , mandatory for the thin circuit model: since this condition is fulfilled better in position B   this model for position B are more correct than for position A.

2 .
Experimental and theoretical studies of electromagnetic low frequency LC resonances in all- dielectric rectangular frames excited by incident linearly polarized microwaves subject to the various orientations of these frames with respect to the polarization structure of the driving wave are carried out.The displacement currents induced by this wave in both whole frame and its parallel sides are shown to provide the formation of several magnetic and electric dipoles, characterized by different resonant frequencies.Consideration of these frames located in three orthogonal planes, normal respectively to electric component E  , magnetic component H  and wave vector k  of the driving wave (see Figs. 5, 3 and 4) had revealed the following salient features of resonant magnetic and electric dipoles: 1.The LC magnetic resonances of all-dielectric rectangular frame are interchanged by electric ones, forming the complicated spectrum of electromagnetic resonances.These resonances are habitual to the quasi stationary fields in the near zone of the scattering frame.Subject to the orientation of rectangle frame with respect to vector trio E wave, the induced magnetic and electric dipoles possess the different resonant frequencies: ( 4 and 5), accompanied by the variation of resonant frequency, indicates the angular anisotropy and frequency dispersion of induced electric dipoles.4.When the resonant oscillations are excited in the parallel sides of rectangular frame, spaced by the half length of driving wave, this circuit can be viewed as a peculiar structure -alldielectric magnetic or electric resonant quadrupole.The theory of these phenomena is elaborated from the first principles.Calculations of resonant frequencies are in good agreement with the experimental data.All-dielectric LC frames are designed and tested in the GHz range.Such frames visualizing the spectral properties of lonely oscillating element ensure the possibility to model the near fields of nanoscale LC circuits inaccessible for the direct measurements now.The practical usefulness of the obtained results is connected with extremely small losses in dielectric structures as compared with the metallic ones.Due to smallness of losses the Q-factor of all-dielectric structures is increased and the width of resonance is narrowed.
a straight wire D = l, parallel wires of length l spaced d is:

.
with parallel and anti-parallel currents respectively, the mutual inductance a m  per unit length is: can search the solution of Eq. (10) by means of and two sides of this frame with the length 1 b , as well as mutual induction in each pair (taking into account the opposite direction of currents on opposite sides framework).Denoting 11 / 1 a