Scattering of Electromagnetic Waveson Lattices of Rectangular Dielectric Resonators Текст научной статьи по специальности «Электротехника, электронная техника, информационные технологии»

Visnyk N'l'UU KP1 Seriia tiadiotekhnika tiadioaparat.obuduuannia, 2020, Iss. 83, pp. 5—10

УДК 621.372

Scattering of Electromagnetic Waveson Lattices of Rectangular Dielectric Resonators

Trubin A. A.

National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute'1, Kyiv, Ukraine

E-mail: alrubinO&gmail. com

The с-functions that determine the degree of influence of an external exciting electromagnetic field on a rectangular dielectric resonator (DR) in open space are calculated and investigated. The presence of directions with "zero” projection of the exciting electromagnetic field onto the DR field is shown. Using the perturbation theory, the spatial distribution of the scattered electromagnetic fields, which arises when a plane electromagnetic wave of p-, or s-type, is incident on a square lattice of rectangular dielectric resonators is studied. An electromagnetic model of scattering on a rectangular DR lattice is constructed. The appearance of a reflected and shadow lobe during scattering by a lattice of rectangular DRs with basic magnetic modes is demonstrated. The features of scattering by a cubic lattice with degenerate magnetic oscillations of the main type are investigated. R is shown that the degeneracy of the eigenoscillations of the resonators leads to a more complex scattering pattern: the appearance of additional lobes, as well as a change in their shape. It is noted that the shape of the spatial distribution of the scattered electromagnetic field of the grating can change noticeably with frequency variation within the frequency band of coupled oscillations of the resonators of the lattice. The obtained practical simulation results make it possible to significantly reduce the computation time and optimize complex multi-cavity structures of microwave and optical communication systems that simultaneously perform the functions of separation or combining of channels.

Key words: scattering: lattice: rectangular dielectric resonator: modeling: c-function; scattering amplitude

DOI: 10.20535/RADAP. 2021.84.5-10

Introduction

Today, lattices of rectangular dielectric resonators are actively studied as a basis for creating new nietamaterials fl 3,5,10,12], as guiding structures for manipulating photons in the terahertz range [9], as well as nanoantennas [6, 8], splitters [4, 7, 12], and in the microwave range as filters [11] and antennas [13 16,18,19]. The calculation and theoretical analysis of the properties of such lattices is carried out based on numerical solutions of Maxwell's equations, which complicate the understanding of the physical principles defining behavior of such complex structures of coupled Dielectric Resonators (DRs). The development of the physical theory of scattering by complex structures of DRs [8,17] made it possible not only to clarify their behavior in various structures, but also showed the effectiveness of this approach to a unified method for describing various devices in the microwave and optical ranges. The proposed theory was mainly used to describe DR of disk and spherical shapes: rectangular dielectric resonators were not considered. The analysis of scattering theory for rectangular DRs based on

perturbation theory was first carried out in [14]. In order to extend this theory to complex structures of rectangular DRs in open space, it is necessary to first calculate the coupling coefficients of this type of resonators [2]. The purpose of this work is to calculate and analyze the expansion coefficients of the electromagnetic field of plane wave on a rectangular DR in the open space, as well as create an electrodynamic model of a lattice of rectangular DRs. Study of the characteristics of scattering of electromagnetic waves on lattices of rectangular dielectric resonators.

1 Statement of the problem

The purpose of this article is to calculate and analysis the electromagnetic field scattered by planar lattices of rectangular DRs with lowest eigenoscillations of magnetic type. Representing the solution to the problem of scattering in the form [8], it is necessary to calculate the c-function for a rectangular DR for the case of a plane wave incidence in open space.

6

Trubin Л. Л.

2 Calculation c-functions for the plain electromagnetic waves and rectangular DRs

E +, H+

on a rectangular DR at the frequency of its magnetic oscillation Hnmi (Fig. ,a).

J- + = e-ik0(x cos 71+y cos 72 +z cos 73).

--+ = ]Joe-ik°(x cos 11+y cos 72 + z cos 73).

Eo = Eo(xo cos ai + yo cos 0.4 + zo cos 0.3).

- Eo

H0 _ — (xo cos Pi + yo cos P2 + -0 cos p3), wo

(1)

where (xo, yo,zo)

(x, y, z), Eo - amplitude vector, and wo = \Jpo/eo - is the wave impedance of the open space. Angles (ai,a2,a3), (Pi,P2,P3) define the spatial orientation of the electric and the magnetic field vectors, respectively, and the (7ь 72,73) _ defines a direction of the wave expansion. Let us calculate the projection of the electromagnetic field (E+,H+) [8] onto the field of eigenoscillations of a rectangular resonator (e, h) with magnetic oscillations Hnmt in local DR coordinate system (x',y',z') [ ]:

= h i^Po я (sinpxxl ex' = hik2 - P2 Py cospxx

i^Po cospxx'

e» = hi Ц-Щ M - »*£

cos Py y' - sin Py y'

sin^y y'

sinpz z' cospz z'

sin pz z' cos Py y' cosPz z'

0;

hx = hi

к2 - p2

PxPz

h„ = h

lk\ - №

h sin pxx

1 cos px x'

cosPxx' - sinPxx

sinpxx'

^!3z- cospxx

sin Py y cos Py y

sin^y y' cosPy y'

cosPy y - sinPy y

sin pz z' cos pzz'

cos Pz z'

- sinpz z'

cos pz z'

- sinpz z'

(2)

ko = u>ypo£o, У = шу~рф (Px,Py,Pz)-

hi ш - is the

circular frequency; po - is the magnetic permeability; eo, £1 - is the dielectric permittivity of the external space and resonator, respectively.

We need to calculate in general the integrals for t = 1, 2,... ,N

c+

ct

-1/2 £ i[^t,n](H+) + [n,ht](E+) }ds, (3)

e

Z

1

h

z

where St is the surface of the t-th DR. For a rectangular DR, it is more convenient to bring integral (3) to the form with an accuracy 1/Qy:

c+ « i/2ix(ei - £o) (et, (E+) )dv. (4)

vt

Here Qy - is the resonator Q-factor.

The results of calculating the integral ( ) in the t-DR center coordinate system (x', y', z') are presented in the form:

a) for X-position (Fig.l,b) and for the p-polarization of the incident wave:

c+ = Co • [Pyux(cos ф)-шу (cos 73) • ~~ cos 72 cos 73+ + pxwx(cos 72)<Xy (cos 7з)М]uz(cos 71), (5)

s-polarization of the incident wave:

c+ = Co • Pyux(cos 72)Шу (cos 73)<XZ (cos 71) • cos 71;

b) for Y-position (Fig.l,c) and for the p-polarization of the incident wave:

c+ = Co • [Pyivx(cos 7!)wy (cos 73) • -1cos ф cos 73+ + px^x(cos ф)wy (cos ф)М]wz(cos 72), (7)

s-polarization of the incident wave:

c+ = Co • Py ux(cos 7! )wy (cos ф)х>г (cos 72) • — cos 72;

c) for Z-position (Fig.l,d) and for the p-polarization of the incident wave:

c+ = Co • [Pyшх (cos 71 )wy (cos 72 )cos 71 -- pxwx(cos 71 )шу (cos 72) cos ф]шг (cos 73 ) •ф cos 73,

M

(9)

s-polarization of the incident wave:

c+ = -Co • [Pywx(cos71)шу(cos ^2) cos72 +

+ Px^x(cos 71 )шу (cos 72) cos 7!\<xz(cos 73) • , (10)

where co = 2(r

M = \/cos2^+cos2y2; k2

-1)hiE* zoboL.

Sir

£o ;

<xv (n) * (

1

= 2 x

Pi - ((hn)

-i[pv cospv sin(g^n)

[pv sinpv cos(qvn) -

'Шу

(n) =

x (

1

= 2 x

Pi - ((h n)

[pv sinpv cos(qvn) i[pv cospv sin(g^n)

- qvn sinpv cos(qvn)\ qvn cospv sin(g^n)\

qvncospv sin(^^n)\ qvn sinpv cos(qvn)\

) ■

(v = xpy,z).

Розслюваеея електромагштних хвиль на ренптках прямокутних д!електричних резонатор!в

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Fig. 1. Ео, Но, ко = ЕохНо unit vectors of the incident plain wave in the coordinate system of the lattice (x, y, z) (a). Three orthogonal orientations of a rectangular DR: X(b); Y (c); Z (d) relative coordinate system (x,y,z). Module |c+($, ф) | in a spherical coordinate system for three orientations of the DR with Hccc oscillation relative to the lattice coordinate system X(e,f); Z (g) for p-scattering (e); for s-scattering (f, g); of the DR with Hscc oscillation relative to the coordinate system X(h.i): Z(j. k) for p-scattering (h.j): for s-scattering (i, k)

C-functions (5 10) determine the degree of excitation of a DR by an incident plane wave in open space. The phase function is determined by the coordinates of the t-th DR center (xt, yt, zt) in a lattice:

c+ = c+eik0(xt cos zi+yt cos 72 + zt cos 73)

Fig. l.e-k shows the angular dependence of the modulus c-functions for scattering of p-type (e.h.j). s-type (f. g. i. k). Dependencies (e. h. i) in Fig. 1. correspond to the X-orientation shown in Fig. 1(b), and (g, j, k), correspond to Z-orientation (d). The lowest frequency oscillation type Hccc corresponds to ( ) with hz = hi cos Pxx' cos @yy' cos @zz-, next frequency oscillation type Hscc corresponds to ( ) with hz = hi sin ftxx' cos y' cos ftzz'. The relative dielectric constant of the resonator eir = 36; relative sizes Ьо/ао = 1 L/ао =0, 4.

The given dependences make it possible to determine the degree of interaction of the isolated DR with a plane wave, depending on the direction of its propagation. For example, in the case of p-polarization (Fig. l,e), it is seen that there are no oscillations in the DR in the plane <p = 0,k, and in the case of s-polarization (Fig. , f) at <p = l/2-к, 3/2-k.

3 Calculating scattering of electromagnetic waves by Plain Lattice of Rectangular DRs

The eigenoscillation electromagnetic field of the lattice of N coupled DRs we represented as a superposition of isolated fields of the resonators on the frequency

8

Трубш О. О.

w0 of the lowest Hccc mode:

N

e * = E bngn,

n=1

N

hs = E K^n.

(12)

The amplitudes 6* and frequencies ws of coupled DRs obtained from [8]. Representation (12) we used for solving the scattering problem of the wave (E + ,H +) on a lattice:

N

E « E + + E aS es;

S = 1 N

H « Й + + E 0s hs,

S= 1

(13)

where the unknown amplitudes as (s = 1, 2,..., N) are found from the relations [8] with considering (5 11). In this case, the coupling coefficients of rectangular resonators were calculated using the formulas [2].

The scattering electromagnetic field of the lattice (13) in the wave zone in the direction towards the observation point (0, p) represented as:

^ ^ e-ifcor

E(0,р)-Е+(0к,<fin) = eof (Ok,рк\0, <f)Eo ------. (14)

k0r

Here eo = е0(вк,Рк\0, p) - is the unit vector, defining the polarization of the scattered electric field in

the wave-zone; ($w, ) - is the fall wave direction;

f (Ok,Pk\0, Й) is the scattering amplitude.

Fig. 2,b,d illustrates the angular dependences of the squared modulus of the scattering amplitude for a plane p-type wave (b); s-type (d) on a square lattice (a) of 10 x 10 DR, (c), respectively. The dots conventionally show the centers of the resonators. The straight line shows the direction of propagation of the incident wave k0. The relative distance between the centers of adjacent resonators is A0/4 (A0 is the wavelength in free space at the frequency of Hccc resonmt oscillations w0). As can be seen from the above data, petal 1 (Fig. 2,b) is directed at an angle of ’reflection” to the surface of the grating. Petal 2 is directed along the vector k0, whence, taking into account (14), it defines the lattice ’’shadow” resulting from reflection [17]. Both petals are located in the plane of incidence.

4 Calculating scattering of electromagnetic waves by Plain Lattice of Cubic shape DRs

The sketch of coupled Cubic DRs (a0 = b0 = L) lattice is shown in Fig. 3, a. In this case, in each resonator at the lowest frequency, three eigenoscillations of the Hccc magnetic type can be simultaneously excited, differing from each other by rotation by angles of n/2.

Fig. 2. Square lattice of rectangular DRs in position Y (a); in position Z (c). Characteristics of the scattering \f (Ok ,<Pk\0, of a plane wave от a lattice (a) - (b); (c) - (d). Fall wave direction = 3/4щ рк =0

Розслюваеея електромагштних хвиль на ренптках прямокутних д!електричних резонатор!в

9

All those degenerate oscillations of the DRs are coupled to each other. This means that all 3N oscillations must be taken into account in the calculation (12 13). Fig. 3.1). c shows the result of calculating the angular scattering characteristics for a 10x10 cubic DR lattice.

An increase in the number of coupled degenerate oscillations of different ’’polarizations” leads to expansion and the appearance of additional lobes (Fig. 3.b). Additionally in some cases, frequency variation can lead to a noticeable rearrangement of the scattered electromagnetic field.

Discussion and Conclusion

In this paper, we propose model of scattering of plane electromagnetic waves by gratings of rectangular DRs in open space. The proposed model makes it much easier and faster to analyze the electromagnetic properties of complex structures of coupled DRs in open space. Unlike other analytical methods, the proposed model makes it possible to calculate all physi-

cally significant parameters of the structure with less computer time.

The performed calculations demonstrates the retention of the basic properties established for lattices of rectangulare resonators with the main magnetic types of oscillations [17].

Calculation of the spatial distribution of the values of the c-functions also makes it possible to determine the directions of the most significant interaction of the incident waves with the resonators of the array. This is especially useful in cases of higher types of oscillations, when the distribution of the DR electromagnetic field has a complex spatial structure.

As follows from our calculations, lattices made using a cubic DR are of considerable theoretical and practical interest. The indicated lattices, along with the indicated drawbacks, have a more rarefied frequency spectrum and. moreover, a lower polarization dependence.

The proposed theory can be used to calculate and analyze complex antenna structures, various dividers, multiplexers, and other communication devices in the microwave, infrared and optical wavelength ranges.

Fig. 3. A square lattice of a cubic shape DRs (a). Characteristic of the scattering of a plane wave of the p-type (b), s-type on the lattice (c) with degenerate oscillations of the DRs ($w = 3/4^; = 0)

References

[11 Ramaccia D„ Sounas D. L„ ЛШ A., Toscano A., Bi-lotti F. (2020) Phase-Induced Frequency Conversion and Doppler Direct with Time-Modulated Metasurfaces. IEEE 'transactions on Antennas and Propagation, Vol. 68. No.3, pp. 1607-1617. D01:10.1109/TAP.2019.2952469.

[2] Trubin A., Kupriianov Л. S.. Fesenko V. 1.. Tuz V. R. (2020) Coupling coefficients for dielectric cuboids located in free space. Applied Optics, Vol. 59. Issue 23. pp. 6918-6924. 1)01:10.1364/ЛО .399930.

[31 Shamkhi H. K„ Sayanskiy A., Valero Л. C„ Kupriianov A.S.. Kapitanova R. Kivshar Y. S. . Shalin A. S.. Tuz V'. R. (2019) Transparency and perfect absorption of alldielectric resonant metasurfaces governed by the transverse Kerker effect. Physical Review Materials, Vol. 3. Issue 8. pp. 1-10. D01:10.1103/PhysRevMaterials.3.085201.

[41 Li .1.. Liu C„ Wu T„ Liu Y„ Wang Y„ Yu Z„ YeH.. Yu L. (2019) Efficient Polarization Beam Splitter Based on All-Dielectric Metasurface in Visible Regi-

on. Nanoscale Research Letters, 14. Article number: 34. DOl: 10.1186/sl 1671-019-2867-4.

[51 Shi T„ Wang Y„ Deng Z.-L.. Ye X., Dai Z„ CaoY.. Guan B.-O.. Xiao S„ Li X. (2019) All-Dielectric Kissing-Dimer Metagratings for Asymmetric High Diffraction. Advanced Optical Materials, Vol. 7, Issue 24. DOl: 10.1002/adom.201901389.

[61 Bibbo L„ Liu Q„ Khan K„ Yadav A., Elshahat S„ Abood 1., Ouyang Z. (2019) Radiation-direction steerable nanoantennae. SN Applied Sciences, Vol. 1, Article number: 844. D01:10.1007/s42452-019-0882-9.

[71 Guo Z„ Zhu L„ Guo K„ Shen F„ Yin Z. (2017) High-Order Dielectric Metasurfaces for High-Efficiency Polarization Beam Splitters and Optical Vortex Generators. Nanoscale Research Letters, Vol. 12. Article number: 512. DOl: 10.1186/sl 1671-017-2279-2.

[81 Trubin A. (2016) Lattices of Dielectric Resonators. Springer Series in Advanced Microelectronics, Vol. 53, 171 p. DOF10.1007/978-3-319-25148-6.

10

Трубш О. О.

[9] Minin 1. V., Minin О. V., Pachoco-Pona V., Bnruntn М. (2015) All-dielectric periodic terajet waveguide using an array of coupled cuboids. Applied Physics Letters, Vol. 106. 001:10.1063/1.4923186.

[101 Li L., Wang .1., Wang .1., Du H„ Huang H„ Zhang .1., Qu S„ Xu Z. (2015) ЛИ-dieleetrie metamaterial frequency selective surfaces based on high-permittivity ceramic resonators. Applied Physics Letters, Vol. 106. Do"l:10.1063/1.4921712.

[Ill Hidfiurska T. V., Trubin A. A. (2014) Novel dual-band rectangular dielectric resonator filter. X International Symposium on Telecommunications (BIHTEL), pp. 1-5. DOl: 10.1109/B1HTEL.2014.6987634.

[121 L>u **., Wang .1., Xu Z„ Xia S„ Wang .1., Qu S. (2014) Band split in multiband all-dielectric left-handed metamaterials. Journal of Applied Physics, Vol. 115, Issue 23. 001:10.1063/1.4883962.

[131 Kumari R„ Behera S. K. (2013) Nine-element frequency independent dielectric resonator array for X-band applications. Microwave and Optical technology letters, Vol. 55, No. 2, pp. 400-403. DOl: 10.1002/mop.27337.

[141 Abd-Elhadv A. M„ Zainud-Deen S. H„ Mitkees A. A., Kishk A. A. (2012) Dual Sized Varying Slot Lengths Loading Dielectric Resonator ttcflcctarray. International Journal of Electromagnetics and Applications, No 2(3), pp. 46-50. DOl:10.5923/j.ijea.20120203.05.

[151 Hctosa A., Thirakoune S. (2011) Rectangular Dielectric Resonator Antennas With Enhanced Gain. IEEE Transactions on Antennas and Propagation, Vol. 59, No. 4, pp. 1385-1389. D01:10.1109/TAP.2011.2109690.

[161 Al-Zoubi A., Kishk A., Glisson A. W. (2008) Linear dielectric resonator antenna array fed by dielectric image line. IEEE Antennas and Propagation Society International Symposium, pp. 1-4. DOl:10.1109/APS.2008.4618987.

[171 'H'ubin A. A. (2009) Electromagnetic waves scattering on a lattice of Dielectric Resonators. 19 International Crimean conference 'Microwave equipment and telecommunication technologies”, pp. 405-407. DOl: 10.1109/TELSKS.2009.5339480.

[181 Luk K- Mm Leung K. W. (2003) Dielectric Resonator Antennas. Research Studies Press LTD., 388 p.

[191 Mongia R. K„ Ittipiboon A. (1997) Theoretical and Experimental Investigations on Rectangular Dielectric Resonator Antennas. IEEE ’Transactions on Antennas and Propagation, Vol. 45, No. 9, pp. 1348-1356. DOE10.1109/8.623123.

Розсчювання електромагштних хвиль на регштках прямокутних д1електрич-них резонатор!в

Трубгн О. О.

Розраховап! i досл!джеп! с-фупкцн, що вгопачають стушив впливу зовшшпього збуджуючого електромагш-тпого поля па прямокутпий д!е.лектричпий резонатор (ДР) у вцщритому простор!. Показано паявшеть па-прямкш з ’’пульовою” проекц!ею поля збуджеппя па поле ДР. За допомогою теорп збурепь, впвчепо просторовий розподгл по.л!в розсповаппя, що випикае при падшп! плоско! електромагштпо! хви.л! р- або s-типу. па ква-дратпу репптку прямокутпих д!е.лектричпих резопато-р!в. Побудовапа е.лектродипам!чпа модель розсповаппя

па репитц! ДР прямокутпо! форми. Продемопстрова-по появу в!дбито! i тшьово! пелюсткп при розс!япп! па репитц! прямокутпих ДР з осповпими магштпими видами коливапь. Дос.л!джуються особ.ливост! розсповаппя па репитц! куб!чпо! форми з вироджепими магштпими поливаниями основного типу. Показано, що вироджеп-пя власпих коливапь резопатор!в призводить до бглын складно! картин! розсповаппя: появи додаткових пелю-сток. а також змши !х форми. В!дзпачепо, що форма просторового розподглу розояпого поля ренитки може пом!тпо змшюватнея з вар!ац!ею частот в межах смугп частот зв’язапих колгшапь резопатор!в ренитки. Отри-маш практичш результат моделюваппя дозволяют зпачпо скоротптп час обчислепь i оптим!зувати складл! багаторезопаторш структурп мжрохвильових та опти-чпих систем зв’язку. як! одпочаспо викопують фупкц!! подглу або об’едпаппя капал!в.

Ключовг слова: розсповаппя; ренитка: прямокутпий д!е.лектричпий резонатор: моделюваппя; с-фупкпдя; ам-шнтуда розс!юваш1я

Рассеяние электромагнитных волн на решетках прямоугольных диэлектрических резонаторов

Трубин А. А.

Рассчитаны и исследованы с-фупкции. определяющие степень воздействия внешнего возбуждающего электромагнитного поля па прямоугольный диэлектрический резонатор (ДР) в открытом пространстве. Показано наличие направлений с "пулевой” проекцией возбуждающего поля па поле ДР. С помощью теории возмущений, изучено пространственное распределение полей рассеяния, возникающее при падении плоской электромагнитной волны р- или s-типа, па квадратную решетку прямоугольных диэлектрических резонаторов. Построена электродинамическая модель рассеяния па решетке ДР прямоугольной формы. Продемонстрировано появление отраженного и теневого лепестка при рассеянии па решетке прямоугольных ДР с основными магнитными типами колебаний. Исследуются особенности рассеяния па решетке кубической формы с вырожденными магнитными колебаниями основного типа. Показано, что вырождение собственных колебаний резонаторов приводит к более сложной картине рассеяния: появлению дополнительных лепестков, а также изменению их формы. Отмечено, что форма пространственного распределения рассеянного поля решетки может заметно меняться с вариацией частоты в пределах полосы частот связанных колебаний резонаторов решетки. Полученные практические результаты моделирования позволяют значительно сократить время вычислений и оптимизировать сложные мпогорезопаторпые структуры микроволновых и оптических систем связи, одновременно выполняющие функции разделения или объединения каналов.

Ключевые слова: рассеяние: решетка: прямоугольный диэлектрический резонатор: моделирование: с-фупкцпя; амплитуда рассеяния