The problem of active controlling of the structure of fields of combined radiating systems within their near zone is studied. The characteristic size of this zone is on the order of the wavelength and is characterized by the presence of evanescent (nonpropagating) fields, which are formed, among other things, due to the interference interaction of radiators of the system. Using multipole expansions for fields and special summation formulas for such expansions allows one to obtain concise expressions convenient in carrying out numerical calculations. The results of calculations confirm that the evanescent fields’ structure plays a significant part in the process of the formation of the radiation field

Belichenko, Viktor P.

Shestakov, P. V.

Zapasnoy, Andrey S.

Optics and Spectroscopy

English

Tomsk State University Repository

602ISSN 0030-400X, Optics and Spectroscopy, 2015, Vol. 118, No. 4, pp. 602–604. © Pleiades Publishing, Ltd., 2015.Original Russian Text © V.P. Belichenko, A.S. Zapasnoy, P.V. Shestakov, 2015, published in Optika i Spektroskopiya, 2015, Vol. 118, No. 4, pp. 630–632.The Role Played by Evanescent Fields in the Process of Formationof Radiation of Combined Radiating SystemsV. P. Belichenko, A. S. Zapasnoy, and P. V. Shestakov Tomsk State University, Tomsk, 634050 Russiae-mail: bvp@mail.tsu.ruReceived August 25, 2014Abstract—The problem of active controlling of the structure of fields of combined radiating systems withintheir near zone is studied. The characteristic size of this zone is on the order of the wavelength and is charac-terized by the presence of evanescent (nonpropagating) fields, which are formed, among other things, due tothe interference interaction of radiators of the system. Using multipole expansions for fields and special sum-mation formulas for such expansions allows one to obtain concise expressions convenient in carrying outnumerical calculations. The results of calculations confirm that the evanescent fields’ structure plays a signif-icant part in the process of the formation of the radiation field.DOI: 10.1134/S0030400X15040025Continued interest in studying various interferenceeffects in counterpropagating fields of plane electro-magnetic waves, as well as in fields generated by sys-tems of radiators [1, 2], has been on display for a ratherlong time. Together with this, there are attempts beingmade to more carefully study the radiation physics ofisolated radiators. These attempts are based on distin-guishing radiating and evanescent fields with theinvolvement of multipole expansions of these fields inthe near zone of the radiators [3, 4]. There are manyreasons to believe that the structure of evanescentfields is much more complex in the so-called com-bined antennas [5]. Such antennas contain at least tworadiating centers separated by a distance of about 0.3–0.5 of the wavelength. The proposed new constructivesolutions of such antennas [6] are still based on verygeneral physical ideas presented in [5].In connection with this, the study of the specificityof the formation of evanescent fields, as well as thepossibility of controlling energy f lows of these fields inthe near zone of combined antennas, is of consider-able interest. The electric and magnetic dipoles areused in this work for the simplest model of radiatingcenters.In the system of spherical coordinates , fieldsof an electric dipole with moment  oriented alongthe  axis and of a magnetic dipole with moment oriented along the  axis (Fig. 1) can be represented[7] by the following multipole expansions and expres-sions, respectively:(1)(2)where  is an imaginary unit,  is the wavenumber, is the wave resistance of the ambient medium, are spherical Bessel and Hankel func-, ,r θ ϕepz mpy( ) ( )( ) ( ) ( )( )( )[ ] ( )( ) ( )(1) 01(2)(1) 01(2)(2)2 1 14cos ,2 14, ,(cos )[ ], ,ernn n nenn nnn np ZE n n nbrP j kr h krp ZE nbrd rj kr h kb r bdP drdd j kb rh kr r bdr∞=< >∞θ== − + +π× θ= − +π⎧ <⎪θ× ⎨θ ⎪ >⎩∑∑( ) ( ) ( )(1) (2)1(2) (2)10(2) (2)10(2) (2)102(2) (2)12(2) (2)1(cos )2 1 ,4( )sin sin ,2[ ( )]cos sin ,4[ ( )]cos ,4( )cos ,44enn nnmrmmmmdPikpH n j kr h krb dkpH h krrZkp dH rh krrZ drkp dH rh krrZ drik pE h krik pE h∞ϕ < >=θϕθϕθ= +π θ= − θ ϕπ= − θ ϕπ= − ϕπ= ϕπ= −π∑( )cos sin ,kr θ ϕi k 0Z( )2( ), ( )n nj x h xPHYSICAL OPTICSOPTICS AND SPECTROSCOPY  Vol. 118  No. 4  2015THE ROLE PLAYED BY EVANESCENT FIELDS 603tions of the second kind,  are Legendre poly-nomials, and  and  are the largest and smallestcoordinates among r and b.In the case under consideration, components of thecomplex Poynting vector are represented as a sum ofcorresponding components of isolated dipoles and aninterference component:(3)Here, in spite of the rather complicated form ofmultipole expansions (1), it turns out to be possible torepresent all quantities entering in (3) in a conciseform using the three special summation formulasdetermined in the work. In particular, especially inter-esting components of the interference component ofthe Poynting vector are determined by the followingexpressions:( )cosnP θr> r<(1) (2) int (1) (2) int(2) int, ,.r r r rS S S S S S S SS S Sθ θ θ θϕ ϕ ϕ= + + = + += +(4)(5)(6)where  and  are initial phases of currents indipoles.Figures 2a and 2b illustrate the behavior of, , and  in the field of orthog-onally oriented electric and magnetic dipoles as theobservation point moves away in the direction. The electric distance between thedipoles is , and the absolute values of dipolemoments are numerically equal. Normalization to thereal part of the radial component of the Poynting vec-tor of the isolated magnetic dipole was also carriedout.When the phase difference is zero, the behaviors of and  are similar. Both the compo-nents rapidly increase for . In the intervalint22(1)122 (2)1cos321exp ( ) [ ( )]exp ( ) ( ) ,e mrikRe mikRm ep pS ibd ei rh kr rdr r Rrei k h krR−= ϕπ⎧ ⎡ ⎤∂× ϕ − ϕ⎨ ⎢ ⎥∂ ∂θ⎩ ⎣ ⎦⎡ ⎤⎫∂− ϕ − ϕ ⎬⎢ ⎥∂θ ⎣ ⎦⎭int2 2(1) 2 2 21exp ( )cos32[ ( )] ,e me mikR ikRp pS i ibrd d d e erh kr r k rdr dr dr R Rθ− −= − ϕ − ϕ ϕπ⎧ ⎡ ⎡ ⎤⎤ ⎫× +⎨ ⎬⎢ ⎢ ⎥⎥⎩ ⎣ ⎣ ⎦⎦ ⎭{int2 2(1) 2 2 212(1)12 2exp ( )sin32[ ( )]cos2 ( )sin ,2 cos ,e me mikR ikRikRp pS i ibrd d d e erh kr r k rdr dr dr R Reh kr rr RR r b rbϕ− −−= ϕ − ϕ ϕπ⎧ ⎡ ⎡ ⎤⎤ ⎫× θ +⎨ ⎬⎢ ⎢ ⎥⎥⎩ ⎣ ⎣ ⎦⎦ ⎭⎡ ⎤⎫∂− θ ⎬⎢ ⎥∂ ∂θ ⎣ ⎦⎭= + − θeϕ mϕintRe[ ]rSintRe[ ]SθintRe[ ]Sϕ/4θ = ϕ = π2kb =intRe[ ]rSintRe[ ]Sϕ0.5kr <Fig. 1. Model of a combined radiating system.ZXbYpepmFig. 2. Dependence of  on  at ,  (1) r, (2) θ, and (3) φ; (a) 0 and (b) .2.0 (a) (b)1.51.01.00.50.58 86420kr kr1 3321264200 0−0.5−0.5−1.0Re[Sr]βRe[Sint]Re[Sr]βRe[Sint]intRe[ ]Sβ kr /4θ = ϕ = π β = Δψ = /2π604OPTICS AND SPECTROSCOPY  Vol. 118  No. 4  2015BELICHENKO et al., their behavior is of oscillating character;for , it is followed by a monotone decrease. Asis seen from Fig. 2a,  is characterized by a dif-ferent behavior.When the phase difference equals , the behaviorof interference energy f lows in the evanescent field ofthe system of electric and magnetic dipoles changesradically (Fig. 2b). In this case, the near-field interac-tion of the dipoles turns out to be minimum and thecontribution of the interference component to theradiation power is maximum.This work was supported by project no. 13-02-98025 of the regional competition of the RussianFoundation for Basic Research (r_sibir’_a) “Funda-mental Constraints in the Near-Field Electrodynam-ics of Complex Radiating Systems” (2013–2015).REFERENCES1. S. A. Afanas’ev and D. I. Sementsov, Usp. Fiz. Nauk178 (4), 377 (2008).2. E. A. Gevorkyan, Zh. Tekh. Fiz. 83 (4), 113 (2013).3. S. R. Seshadri, in Radio Science Meeting (USNC-URSINRSM) (2013), p. 1.4. S. R. Seshadri, J. Opt. Soc. America A 25 (3), 805(2008).5. V. P. Belichenko, Yu. I. Buyanov, V. I. Koshelev, andV. V. Plisko, Radiotekh. Elektron. (Moscow) 44 (2),178 (1999).6. A. Mehrdadian and K. Forooraghi, PIER C 39, 37(2013).7. G. T. Markov and A. F. Chaplin, Excitation of Electro-magnetic Waves (Radio i Svyaz’, Moscow, 1983) [inRussian].Translated by A. Nikol’skii0.5 2kr< <2kr >intRe[ ]Sθ/2π

The role played by evanescent fields in the process of formation of radiation of combined radiating systems

V. P. Belichenko

A. S. Zapasnoy

P. V. Shestakov

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The role played by evanescent fields in the process of formation of radiation of combined radiating systems

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