Radiophltzics and Quantum Electronice, VoL $8, No. 9, 1995

THE INFLUENCE

OF THE INTERFACE BETWEEN

ON THE STRUCTURE SCATTERED

TWO MEDIA

OF THE ELECTROMAGNETIC

BY A NONLINEAR

FIELD

HALF-WAVE LOOP

A. A. G o r b a c h e v , T. M. Z a b o r o n k o v a , a n d S. P. T a r a k a n k o v

UDC 537.874.4

In an approzimation of weak nonlinearity, we study ezperimentaIly and theoretically the scattering of a plane electromagnetic wave by a ring loop containing a nonlinear local load in the presence of a plane interface between two media. A number of patterns are detected in the structure of the field scattered at higher harmonics, which can be used in the development of nonlinear markers intended for rescue systems.

A n,,mber of applied problems have stimulated interest in the phenomena of scattering of electromagnetic waves by antennas with a nonlinear load. N. Yu. Babanov et al. [1] studied the space properties of scattering of the second harmonic of an incident field by a system of two and four nonlinear scatterers (NS) located in flee space. The main features of scattering of nonlinear products by a statistical NS system are analyzed in [2]. The present paper is devoted to the theoretical and experimental study of the influence of the interface between two media on the backscattered field of a solitary NS located in a dielectric without losses near the interface. As the NS we consider a ring loop antenna with radius b made of a thin wire with radius a (a/b << 1) loaded by a nonlinear element (NE) in the form of a semiconductor diode. The NS is located at distance h from the medi,,rn-vacu,,m interface (ideally smooth). The plane electromagnetic wave generated by a source located in vacuum at distance H from the interface is incident on the interface at angle 0 (Fig. 1). We restrict our discussion to the case of weak nonlinearity. The volt-ampere characteristic of the semiconductor diode is written in the form [3] = u - )32 u 2 + )33 u s

Ro

'

(1)

where J is the current flowing through the diode, U is the voltage acrossthe diode, R0 is the initial (at (f = 0) resistance of the diode, and )32, and )33 are the nonlinearity coefficients of the volt-ampere characteristic

(vAt). To determine the field scattered by a loop of arbitrary dimensions, we must know the real current distribution in the loop, which is found from the appropriate integro-ditferential equation [4]. The condition a << mln{b, A} (A is the wavelength in the medium) allows us to assume that the current in the loop has one azimuthal angular component and is a function of only this coordinate. We shall solve the problem in a cylindrical coordinate system (r, ~, z) in which the axis OZ is assumed to cross the loop center (see Fig. 1). All of the processes are considered to be stationary (the time dependence is assumed to be in the form ofli~t~. The electric field scattered by the loop at Z > 0 can be expressed by a superposition of the primary field E (p) of the loop (the field in the absence of an interface in a medium with permittivity e) and the secondary field J~(~) associated with reflection from the boundary. On the surface of an ideally conducting

Radiophysical Research Institute, Nizhny Novgorod. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 38, No. 9, pp. 961-968, September, 1995. Original article submitted November 22, 1994.

0033-8443/95/3809-0625515.00 (~1996 Plen,,m Publishing Corporation

625

Tt~

T

'~

Y

zl HH(b,q~ o,h) Fig. 1.

loop the following condition must hold

E(p)

o

I z=h, ~'=b

where I(~) is the current in the loop, Z is the impedance of the local load, E (i) is the angtdar component of the incident wave on the loop, s is the effective emf, which is caused by the presence of the nonlinear element and generates current at higher harmonics, and E (p) and E (~d) axe the azimuthal tangential components of (P) and ]~ (~-f). We solve Eq. (2) by the perttubatio.~ method with respect to the small parameter/~u. Then at the fundamental frequency w we have

~Cp) + ~c~)+ ~co a,, L.,,(~,) '~(~-~,o) ~,~ ~.~ = ~

C3)

At the frequency of the second 2~ and third 3w harmonics, from Eq. (2) we obtain

E(~) ~(=0 ~N 6(~- ~0) = IN~(~) ~(~ - ~o) Nw,~ § "~'Nw,r § b(1 § iRN Nu)c) "~ ZNw,

(4)

where

E2 -/~2 V~

2'

E3=/~3 V~ R____t3 P~o'

R, = Ro,

3 Vim is the voltage amplitude at the fundamental frequency, and C is capacitance of the diode. It is ass-reed that the N E is located at a point with coordinates (b,~o0,h). The subscript N = 2 or 3 depends on the harmonic number.

626

The field E(~),, is written in the sameform as in [3]. The field E(g!v, has the form

d~g O0

(B) 2

0

rx(~)=~-~ - ~o r,(~)- e~N ' (\r' 70 : ~ 2

-- N2,

71 '

;~' - N ' e 7, - eTo~

")'1 = X/A:~ -- e N 2,

l = 1, 2;

kl --- k0 V/~",

where

e is the permittivity and J0(~) is a Bessel function. Equations (4) were obtained by equating in Eq. (2) the terms of orders fl2Vtrn and flsV12m,respectively. It is ass,,med that f i s ~ ~ ~ f12 ~,,, ~ 1. Substituting in Eqs. (3) and (4) the expressions for the fields and making the appropriate transformations, we obtain equations describing the current distribution in the loop antenna at the fundamental and harmonic frequencies of the sounding signal. The solution of these equations (due to the circular symmetry of the loop) is easily found in the

form of a Fourier series in terms of the functions

cos rnT~/" The kernds of the integral equations and the sin rn~ .J

incident field E~ ) are presented in the form of corresponding Fourier series. Substituting these series into Eqs. (3)-(5) and integrating with respect to ~0', we obtain expressions for the coefficients of the current distribution in the loop. At the fundamental frequency for TE polarization of the incident wave (see Fig. 1) we have i___~_~ Z.x: Z.,

~-,

a(p)

,.0)

'

and for T M polarization we obtain

I~!~, = -

2isEob

(7)

Zo (a(mP!l"~

m,,,)

where

( ( ckoh ,

Ie,,,= T~iS~(kobsinO) exp - i

- s i n ' ,O + m 7.)}

,

Ta-mJ,,,(kobsinO) exp { - i (ko h 'je - sin2 0 + m 2 ) } kobe sinO . (e

ie

Zax?,N

~

h( m ) o !N +

-

sin 2 0)-1/z

I-~

:

Here ~ is the amplitude of the incident field, T[[ and T• are the coefficients of Fresnel transmission for the incident field, 3rn(~) and 3~m(~) are Bessel functions and their derivatives with respect to the argument, respectively, and h(m) - 2 for any m ~ 0. For m - 0 we have h(m) - 1. At the fundamental frequency we assrane that N -- 1. At the frequencies of the second and third harmonics the coefficients of the current distribution in the loop are written in the form

X,~,No, =

~EN~ ZM~,Z.~:,N ,,.(1) ~' Z o ~ ( z ~ + . . ~ + z N = ) ( ~ ! ~ + =~,N,

(S) 627

~(Ntob)'

/

. qU/~.*-I,N) '

(P)

2

(9)

e(Nkob) ~ .(1) /Z(I[1,N.) P.,+I,N q2

the

Expressions for # ~ N are easy to obtain by analogy with the appropriate expressions from [3] (N is 9 (1,2) harmonic number), which hold for N = 1, and the coefficients ~,,.N determining the influence of the

media interface are written as

AdA

..,,~c,,,) = r,,2(~,#)exp {-2koh ~/A' - N ~-,} J~(kobA) ~/A, _ N',

(10)

0

where the functions r,,z(A) are determined according to Eqs. (5). If we know the coefficientsof the current distribution in the loop, we can find the current distributionitselfby the formulas: at the fundamental frequency r

oo

/~,(~o)-/,(e! + 2 E/(.)m,~' cos m~o- 2i E m-----1

I(m0! ~ sin m~o;

(11)

m----1

at the harmonic frequencies oo

rN.(p) = ~ I~.,N.h(m)cosm(~- r

(12)

ram0

Recall that in Eq. (12) the angular coordinate ~o0 gives the location of the nonlinear element. To determine the field scattered by the loop antenna, we must solve the problem of a loop radiating with a given current distribution in the presence of a medium- vacuum interface. Omittingthe intermediate transformations, we write the final expressions for the azimuthal component of the electric field for Z < 0 (see Fig. 1) at the harmonic frequencies iZo

=

N

_

~l(A)J'(k~176

~,'k~b,.~, Tl(~,)J'.,(kob~)J'.,(ko,.~) N 2

JAz

2~/A2 - N 2

(13)

} x exp{,A'- tr 2~/A~ -

~/~- ~'~k0h} T '

N 2

Using the formulas obtained, we computed the azimuthal components of intensity of the scattered electromagnetic feld Er162 and Er at the second and third harmonics of the sounding signal. It was ass-reed in the computations that Zi~p >> ZHN~. The dependences of the azimuthal component of the field intensity on the position of the nonlinear element at a fixed depth h of loop penetration into the dielectric and on the penetration depth at different frequencies of the sounding signal were studied. The theoretical depen.dences were compared with the experimental results. In calculations as well as in the experiment, the following parameters were used: the loop diameter 2b = 1 the wire radius a/Ao = 1.5.10 -s, the wavelength of the Ao 21r' incident radiation in vacuum Ao, the angle 0 -- 73~ and the permittivity of the medium ~ = 5.

628

!

.E q)NuJ

-

!2

, clB

E t~ ~uJ m a x

/

_ oh f 'J

-30

l

,

0

I

90

I

"J

,

!

180

,

t

270

360

~O

Fig. 2.

2

~sNuj

, AB

E q~NuJ m a x 0

-8

"

~

2cu

\

-16

I !

\ -24

-

-32

\

!

0

I

x

I

\ \

I

0,125

0,25

0,375

0,50

h/j~ o

Fig. 3. Figure2 shows calculated and experimental dependences of the azimuthal component of the electric fieldat the second harmonic frequency of the sounding signalon the coordinate ~0 characterizingthe angular position of the N E when the NS is at depth h/Ao = 0.1; the calculated curve at the third harmonic is also given. Because the dynamic range of the measuring receiveris limited the experimental curve corresponding to the thirdharmonic is not shown in Fig.2. All of the curves are obtained for linear(horizontal)polarization of receiving and transmitting antennas. 629

As is evident from the plots, the calculated and experimental curves are in qualitative and quantitative agreement. The values of the scattered field can vary by 25 dB with variation of the NE position. Figure3 shows similar dependences E~,.No, of the azimuthal component of the electric field on the NS penetration depth normalized to the wavelength in free space (h/A0) for horizontal polarization of the receiving and transmitting antennas. From Fig. 3 it is seen that the curves E,,.N,~(h) have a periodic character due to the interface influence. It is noteworthy that at koh V~ :~ 1 the integral responsible for the interface contribution and hence, for the field oscillations, can be analytically calculated. As a result, to evaluate the oscillation period Th we obtain the following simple formula: Th = Ao

b

I + Aov~'

(14)

which, as is evident from n-merical calculations,works sufficientlywell up to dimensions k0b V ~ ~ I. The results of the experiments and calculations are in fair agreement for small values of h. The discrepancy between the theoretical and experimental curves with an increase of h can be attributed, in particular, to the fact that losses in the dielectricWere not allowed for in the calculations, and a certain disagreement between the values of e corresponding to experimental and predicted data was observed. A n increase in the level of the scattered signal combined with an increase in the distance between the NS and the interface as well as the oscillatorycharacter of the curve E~(h) (the oscillationperiod is known) can be important in some applications [5] and in the development of nonlinear markers designed for rescue systems based on the reception of signal harmonics. In conclusion it should be noted that the above general expressions can also be used for analysis of an NS with small electricsize,which has its own features and needs separate consideration.

REFERENCES

I. N. Yu. Babanov, A. A. Gorbachev, T.M. Zaboronkova, and S. V. Lartsov, in: Proc. 12th Int. Wroclaw Syrup. E M S , Wroclaw (1994), p. 214. 2. L.V. Vasenkova and A. A. Gorbachev, "Scatteringof higher harmonics by a statisticalsystem of nonlinear

scatterers," Izv. Vyssh. Uchebn. raved., Radiofiz., (in press). 3. A.M. Zaezdnyi, Fundamentals for Calculations of Nonlinear and Parametric Radio Engineering Circuits [in Russian], Svyaz', Moscow (1993). 4. A.Z. Fradin, Antenna-Feeder Devices [in I~ussian], Svyaz', Moscow (1977). 5. N.D. Watson, in: The measurement detection, location and suppression of external nonlinearities which affect radio systems, Conf. Electromagnet. Compatibility, England, Santhampton (1980), p. 1.

630