Scattering of Plane Electromagnetic Waves by Lattices of Spherical Dielectric Resonators with Degenerate Lower Types of Natural Oscillations Текст научной статьи по специальности «Электротехника, электронная техника, информационные технологии»

Visiiyk NTIJU KP1 Servia Radiolekhnika Radioaparat.obuduuannia, "2023, Iss. 91, pp. 12—17

UDC 021.372

Scattering of Plane Electromagnetic Waves by Lattices of Spherical Dielectric Resonators with Degenerate Lower Types of Natural Oscillations

Trubin A. A.

Educational and Research Institute of Telecommunication Systems of the National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute", Kyiv, Ukraine

E-mail: alrubín'J&gmaíl. com

The problem of scattering of plane electromagnetic waves on lattices of spherical dielectric resonators (DRs) with low magnetic oscillations is considered. The results of theoretical calculations of complex coefficients of mutual coupling of spherical dielectric resonators in open space for cases of excitation of degenerate types of oscillations are presented. The expressions found coincide with those obtained earlier for the special case of oscillations of resonators excited along or perpendicular to the line connecting their centers. The main regularities of the change in the coupling coefficients wit.li variations in the coordinates of the resonators in the transverse plane are considered. Analytical expressions for c-funct.ions are found for the field of fundamental magnetic oscillations of a resonator and a plane wave in open space. On the basis of the obtained formulas, with the help of the perturbation theory, the characteristics of the scattering of plane waves on a square lattice of spherical DRs with basic degenerate magnetic oscillation types are calculated and studied. The distribution of the scattering field in the wave zone of the grating is studied for different angles of incidence. The regions of variation of the angles of incidence are determined, in which the scattering amplitude of a lattice constructed on the basis of spherical DRs differs most noticeably from DR lattices of other shapes with nondegenerate types of oscillations. The polarization characteristics of scattered waves in the far zone of the lattice are calculated. It is noted that, in contrast to the lattices of pseudorotating cylindrical DRs with the main magnetic types of oscillations, lattices based on spherical resonators are characterized by a more complex distribution of the polarization of scattered waves. In the wave zone of the lattice, scattered waves of all three types of polarization, linear, circular, elliptical, can be observed. The obtained results significantly expand the possibilities of developers, since allow us to create electrodynamic models of lattices, as well as other devices in the millimeter and infrared ranges, built on the basis of the use of spherical resonators wit.li oscillations of the main types. Such lattices can be used in antennas, passive reflectors, and other devices of modern optical communication systems.

Keywords: dielectric resonator: lattice: coupling coefficient: c-function: scattering amplitude DOI: 10.20535/RADAP. 2023.91.12-17

Introduction

Today, spherical dielectric resonators (DRs) are being actively studied as one of the main resonant elements of various devices in the optical and infrared wavelength ranges. The reason for this is the existence of relatively simple methods for manufacturing controlled dielectric samples of nanometer dimensions. The optical properties of metamaterials based on spherical microparticles fl 5. 14]. various lattices [G. 12. 16. 18]. and small spatial clusters known as photonic molecules [7. 15. 19] are studied. Spherical dielectric resonators with whispering gallery modes are already being used as sensors [9]. in optical delay lines, in multiplexers, lasers [10.11.13.17.20]. etc. In connection with the foregoing, a wide class of problems on the

scattering of electromagnetic waves by spherical DR lattices remains relevant [8.21].

Despite the fact that the fields of spherical DRs are described by relatively simple analytical expressions, the calculation of more complex structures, based on them, faces significant computational difficulties. At the same time, obtaining exact analytical solutions usually leads to cumber some computational structures [21]. Analytical modeling using perturbation theory is also associated with a number of difficulties arising from the high sensitivity of the output parameters to the relative frequencies of partial resonators in the structure, as well as the uncertainty in the choice of basis functions. However, the construction of electrodynamic models provides valuable information about the behavior of systems of coupled resonators under different scattering conditions.

The purpose of tho article is to derive the necessary analytical relationships for constructing an electrodynamic model of the scattering of plane electromagnetic waves on the lattices of spherical DRs with the main magnetic types of oscillations ffimi (m = 0, ±1). Analysis of wave scattering on a planar lattice of spherical DRs.

To carry out calculations using perturbation theory [23]. the coefficients of mutual coupling of resonators with different types of degenerate oscillations were generalized. Calculations of c-fnnctions describing the degree of interaction of natural oscillations with the field of a plane wave are carried out. An electrodynamic model of the process of scattering by a planar lattice of spherical DRs is constructed.

1 Coupling coefficients calculation for main oscillations of the spherical dielectric resonators

For expanding the scattered field of the lattice in terms of natural oscillations of the resonator system, it's necessary to know the coupling coefficients for arbitrary coordinates and types of oscillations [22]. Mutual coupling coefficients of the spherical dielectric microresonators in the open space are not studied in full detail. On the basis of the analytical relations [25]. we have obtained formulas for the most general cases of excitation for magnetic oscillations H1m1, which we will give a more symmetrical form:

for Az > 2ro : kxx = ia? (p, q) x

2h^(koAr) +

2Ax2 -Ay2 -Az2

vyy

= (p,q)x

x{ 2h0](koAr) + ia1 (p, q)x x{ 2h0\koAr) + 1

Ar2

2Ay2 - Ax2 -Az2'

Ar2

2Az2 -Ax2 - Ay2

Ar2

h2\k0 Ar h2p(k0 Ar h 2\k0 Ar]

Kxy = 3iaf (p,q) x AAAh2\koAr);

= 3ia-1 (p, q) x

AxAz, Ar2

h^(ko Ar);

KyZ = 3ia?(p,q) x AjA^h2)(koAr).

Ar2

(1)

Here the function a^ (p,q) determining coupling dependence on the dielectric parameters for a given type of resonator oscillations [ ], also p = klr0 and

q = k0r0 are the characteristic parameters; r0 - radius; k0 = w/c; kl = k0 are the wave numbers; w - circuit frequency; c - speed of light; elr - dielectric permittivity of the resonator; Ax = x\ — x2; Ay = yi —yr, Az = zx —zi; Ar = Ax2 + Ay2 + Az2; (xs,ys,zs) coordinates of the resonator centers (see Fig. l,a); hn\z) = (-K/2z)1/2H^+1/2(z) is the spherical Hankel functions of the second kind [23]. Indices in kuv denote the direction of the magnetic field at the center of each of the partial resonators in a given coordinate system, which is characteristic of one its degenerate oscillations of the magnetic type Hlml (u, v = x,y, z).

In a particular case Ax = Ay = 0, the obtained relations coincide with those found earlier [22]. As follows from (1). the formulas actually coincide with each other with an appropriate permutation of the coordinates. For example:

Kyy(Ax, Ay, Az) = Kxx(Ay, Ax, Az);

kzz(Ax, Ay, Az) = Kxx(Az, Ay, Ax);

Kxz(Ax, Ay, Az) = Kxy(Ax, Az, Ay);

Kyz(Ax, Ay, Az) = Kxy(Ay, Az, Ax).

This becomes obvious if we take into account the maximum spatial symmetry of the shape of both resonators.

The calculated dependences of the coupling on the coordinates of the resonators are shown in Figs. 1 (b-k) for the relative permittivity of the resonators elr = 36 and k0Az = 2q. Note here that these coupling functions are even on a plane (k0Ax, k0Ay) for oscillations kuu (Fig. ,b-g) and odd for osdilations kuv (see ( ) and Fig. l.h-k).

2 Calculation and analysis of c-functions

C-fnnctions determine the degree of influence of an external field on a partial dielectric resonator. Snch an interaction is determined by the distribution of the external field

E+ = (x0 cosal +y0 cosa2 + z0 cosa3)Ae~lk°r,

A - (2)

H + = (xo cosfti +yo cos^2 + Z0 cosfo) — e lk°r

wo

onto the field of natural oscillations of the resonator. Here x0, y0, z0 are unit vectors directed along the axis x, y, z, respectively; w0 = 120-k - wave resistance of open space; A - amplitude; k0 = (x0 cos + y0 cos + z0 cos^3)k0 - wave vector; r = (x0x + y0y + z0z).

K

x Z

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Tpyfiiii O. O.

Fig. 1. Two spherical DR in a given rectangular coordinate system (x,y,z) (a). Dependences of the coupling coefficients of the magnetic oscillations H1m1 of a spherical DR on the mutual distance between there centers for k0Az = 2q (b-k) (k0 = oj/c; Ax = x1 — x2; Ay = y1 — y2). The indices in kuv denote the direction of the

magnetic field at the center of each resonator (u,v = x,y, z)

We have found an analytical expression for the c-function for the simple case of excitation of oscillations of magnetic types H1m1 in a spherical resonator in a coordinate system(x',y', z') whose z'-axis is directed along the wave vector k0:

c+ = 2niroSmiA* x

x{ji(p) | kn^—jM I H1(p)]}{:;)ns

(3)

Here (0,0,2/) is the resonator center coordinate; m = 0, ±1 - azimuthal number of fundamental natural oscillations of the DR; Smn - Kronecker symbol; jn(z) = (tt /2z)1/2Jn+i/2( )

* - complex conjugate symbol. If the magnetic field of natural oscillations in the center of the resonator is directed along the y axis, this case corresponds to the upper value in the curly bracket, and if it is directed along the x axis, to the lower one.

As can be seen from ( ), the function c+ is nonzero only for oscillations Himi with m = 1 The c+-function is proportional to the projection of the magnetic field of the incident plane wave on the direction of the magnetic field of natural oscillations in the center of the resonator.

3 Construction of a model for the scattering of plane waves by an lattice of spherical DRs

As an example, let's build a scattering model on a 10x10 square lattice shown in Fig. , a,b. To simplify the calculation of the amplitudes of natural oscillations of the DR system, we pass to the coordinate system, the axis 2 of which is directed along the vector k0 and use (3). The amplitudes of the resonators are found using the perturbation theory [22], after which the

field scattered by the grating is calculated by going back to the coordinate system of the lattice. In this case, we take into account all three types of degenerate oscillations of each of the resonators that arise in the lattice, both duo to interaction with the incident wave and duo to the coupling between the resonators.

Figure 2 shows the angular dependences of the squared modulus of scattering amplitude If (0k ,ifik |0,y>)|2 for different cases. Here (0k ,ifk ) is

the direction of the vector k0 in the lattice coordinate system.

The obtained data showed that lattices built on the basis of spherical DRs differ from lattices of other resonator shapes in terms of scattering characteristics, primarily at angles of incidence close to -n/4 < 0k < 3-k/A mid <^k = 0 (Fig. , f-1). In cases where the incident wave is directed at 0k > 3-k/A mid ifk = 0 to the lattice plane, these differences are minimal (Fig. 2. c-e).

x

amplitude |/ (Ok,fk |0,<^)|2 for d = Xo/4 (Xo = 2n/ko); Ok = n; <fk = 0 (c); Ok = 0, 9k; ipk = 0 (d); Ok = 0, 75n; fk = 0 (e,f) ; Ok = 0, 5-n; ifk = 0 (g,h); and for Ok = 0.75n; ifk = 0,1n (i,j); Ok = 0.75n; ifk = 0, 2n (k,l); for s-scattering - (c-e, g,i,k) for p-scattering (f, h,j,l); straight lines show the direction of the incident wave

16

Tpyñiii O. O.

4 Polarization of scattered waves by DR lattices

In this work, wo also studied the general regularities of the change in the polarization of the scattered waves. For this purpose, the electric component of the field in the wave zone was represented as the Jones vector in the spherical coordinate system (r, 6, p), with a minimum number of independent parameters [24]:

ex = (cos 0 • n0v + eiS sin 0 • noe) • ei(ut-k°r)/r,

here n0n0g - the unit vectors are oriented in the p

90 P(6)

(a)

Fig. 3. Angular dependencies 0(6) (a); 6(6)

Dependencies calculated by us 0(6), 6(6), for a particular case of scattering (Fig. 2.o) are shown in Figs. 3. As it can be seen from the obtained data, spherical DR lattices differ from DR gratings with no degenerate oscillations in the general case by the presence of all three types of polarization: linear, circular. and elliptical. Only in a few special cases of incidence the scattered wave remain linearly polarized. In the general case, we can formulate the statement that for arrays of pseudo-rotating cylindrical DRs [26] with fundamental magnetic oscillations, the scattered waves will be linearly polarized.

90 5(6)

(b)

(b) for p-scattering in the plane pk =0,-k

Conclusions

An analytical expressions for the coupling coefficients of the spherical microresonator in the open space has been obtained. Expressions for c-functions are found for spherical DRs with the main magnetic types of degenerate natural oscillations and a plane wave. Based on the perturbation theory, the scattor-

x

aro calculated. Thus, the data obtained show that the characteristics of scattering on lattices of spherical DRs have a more complex structure associated with the excitation of several types of degenerate oscillations of this type of resonators.

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Розсиовання плоских електромагш-тних хвиль репптками сферичних дь електричних резонатор!в з виродже-ними нижними типами власних коли-вань

Трубгн О. О.

Розглядаеться задача розсиовання плоских електромагштиих хвиль па ретштках д!електричпих резопатор!в (ДР) сферичпо! форми !з пижчими коливаппями магшт-пого типу. Наведено результати теоретичпих розрахуп-к!в комплекспих коеф!ц!епт!в взаемпого зв'язку сфери-чпих д!електричпих резопатор!в у в!дкритому простор! для випадк!в порушеппя вироджепих тип!в коливапь. Зпайдеш вирази зб!гаються з отримапими paninie для окремого випадку коливапь резопатор!в. як! збуджую-ться вздовж або перпендикулярно прямш. яка з'едпуе i'x цептри. Розгляиуто ocuoBiii закопом1рпост1 змши кое-фщ!епт1в зв'язку шд час вар1ацп коордешат резопатор!в у поперечп1й площиш. Зпайдеш пов! апал1тич1П вирази с-фупкцш (3) для поля осповпих магштпих колршапь резонатора i плоско! хвшн у в!дкритому простор!. На шдстав! отримашьх формул, за допомогою теорп збу-репь розраховапо та досл1джепо характеристики розс!-юваппя плоских хвиль па квадратпих реш!тках сфери-чпих ДР з ОСПОВ1ШМИ вироджмшми магштпими тршами колршапь. Досл1джепо розпод!л поля розсиовання у хви-льов!й зош реш!тки для pi3imx кут!в падшпя. Визиачепо облает! змши кут!в падшпя. в яких ампл!туда розс!ю-ваппя реш!тки. побудоваио! па основ! сферичних ДР. пайб!льш пом!тпо в!др!зпяеться в!д реш!ток ДР шших форм !з певироджепими тршами коливапь. Розраховапо характеристики поляризацп розс!япих хвиль у дальшй зон! реш!тки. Зазпачепо. що па в!дмшу в!д грат псев-дооберталышх ДР цил!пдричпо1 форми з осповпими магштшми тршами колршапь. реш!тки. побудоваш па основ! сфернч1шх резопатор!в. характеризуються б!льш складп!шнм розиод!лом поляризац!! розояпих хвиль. У хвильов!й зон! ретштки можуть спостер!гатися розс!яп! хвил! вс!х трьох трш!в поляризац!! л!п!йпо!. кругово!. ел!птичпо!. Отримап! результати зпачпо розширюють можлршост! розробпик!в. оск!льки дозволяють створю-вати електродршам!чп! модел! реш!ток. а також шших пристро!в м!л!метрового та шфрачервопого д!апазоп!в. побудова1шх па основ! застосуваппя сферичних резо-патор!в з коливаппями осповпих тишв. Так! ретштки можуть бути використаш в аптепах. паершпих в!дби-вачах. а також в шших пристроях сучаспих оптичпих систем зв'язку.

Клюноог слова: д!електричпий резонатор: реш!тка: коеф!ц!еит зв'язку: с-фупкц!я: ампл!туда розс!юваппя