UDC ELECTROMAGNETIC

BEAM P R O P A G A T I O N

IN G Y R O T R O P I C

621. 371:538. 245

MEDIA

G. L. Gurevich, Yu. A. Otmakhov, and E. A. Rozenblyum Izvestiya VUZ. Radiofizika, Vol. 8, No. 4, pp. 725-737, 1965 The problem of electromagnetic beam propagation in an unbounded gyrotropic medium is solved, taking spatial dispersion into account. Concrete examples of electromagnetic beam propagation in a ferrite are considered both in the presence and in the absence of absorption. In solving problems on electromagnetic energy propagation in an unbounded gyrotropic m e d i u m , the propagation of plane electromagnetic waves [1-4] is considered as a first step. The next step should be the solution of the boundary problem. Although the method of solving such a problem is obvious in principle (for example, by expanding the given boundary field into plane waves), the results obtained are completely intractable in the majority of cases unless additional simplifications are made. In the limiting case when the wavelength X tends to zero, geometrical optics becomes valid, and the solution of the problem in this case is trivial. The case when the ratio of the wavelength to the characteristic field dimension X/a is small, but not equal to zero, is of interest. As will be shown below, it is possible here to obtain results in a form similar to the Fresnel formula in diffraction theory. This allows us to make immediate use of the results of this theory in a number of cases. We shall call the approximation which we use the quasi-optical approximation, and the solutions obtained i n t h i s approximation will be called beams, just as is done for isotropic media [5-8]. Although the analysis is given for ferrites*, the results can easily be applied to all types of gyrotropic media. 1. The Quasi-Optical Approximation for a Gyrotropic Medium We shall consider the boundary problem in a ferrite in the quasi-optical approximation. We shall confine ourselves first of all to the two-dimensional case. Let the z = 0 plane be the boundary dividing a ferrite from a dielectric, and let the field E(y, 0) be given in the plane z = +0. The field must be found at an arbitrary p o i n t E ( y , z). It is assumed that there are no sources present for z > 0. Four types of plane waves [2, 3] exist when spatial dispersion in a ferrite is taken into account, and the field E(y, z) can be represented as the superposition of these waves. In what follows we shall confine ourselves to considering two components of

E(y,

z)**:

E.L(y, z ) = xoEx(y , z ) + yoEy(y, z).

(1.1)

One can then write 4

E•

z) = ~ j' d'ry Ej('ry)[Xo + iej(';y)Yo] e -'h'y+~j"l~'>~l"

(1.2)

j=l Here the following symbols have been introduced: 7jz and 7y are components of the plane wave propagation constant on the z and y axes, "fy is a variable of integration, and m a y be taken to be the same for all j, and Ey('fy) and e/(Ty ) are the amplitude and ellipticity respectively of the jth plane wave. All these quantities, except E(Ty ), are known from the solution of the unbounded ferrite problem [3]. In the plane z = +0 the field (1.2) is written as

E.L(y, 0) = ~ ,~dTy E/('ry)(Xo + iejyo) e -%y.

(I. 3)

i Whence it follows that

EJ( Ty)(Xo + iejYo) = 1 (XoJ. + Y0Jy), J *The ferrite is considered to be far from resonance. ~ T h e third component can be found from the condition div D = 0.

516

(1.4)

where

Jx, y = ~ Ex, y (y', 0) e#TyY'dy '. Condition (1.4) gives two scalar relations only for finding E/(Ty). From this it is clear that when spatial dispersion is taken into account it is not sufficient for the field to be given at an arbitrary cross section if the field at a given point is to be determined unambiguously. We shall make use of the additional boundary conditions [9], which we shall write in the form

[mj(2i).aTlzMo + lto, - - t/o)] = O.

(1.5)

I

Here/712 is the amplitude of the variable part of the magneti c moment, /'/or is the constant magnetic field at the sample boundary, a is the lattice constant, k = Oe/PoMo where 0 e is the Curie temperature and P0 is the Bohr magneton, a n d / ~ o = / l f f l o --/}f2o, where Mr, 2o are the saturation magnetizations of the two ferrite sub-lattices. Expressing rnj by Ei(Ty ) [3], we obtain the necessary supplementary relations determining Ey(Ty). From these and from (1.4) we find

Ri~(Ty)Jx + RIy(Ty)Jy.

2=EI(Ty) =

( i . 6)

Here RYx(Ty) and R]y(Ty) are certain functions of Ty. We shall now assume that

Ex, y ( y ' ,

0) m a y be represented in the form

Ex, y (y', 0) = Ex, y0(y', 0) e-%oY',

(1.7)

where Ex, yo (3/, 0) is a function which varies little over distances of the order of a wavelength. Setting (1.6) and (1.7) in (1. 2) and introducing the symbol ~q ~ Ty - - 7yo, we obtain

e;(y. ,)=2.

o)+

o)l •

J

X (Xo + ielYo) exp {-- i 1~ (y ~ y') + We shall represent

T]z(N)of

(1.8)

7jz('Sl)Z-1- "fv0Y]}-

expression (1.8) as a series in ~q:

7j, = Tj,o + Pill + qJ ~i2 + ri~3 + s#i 4 + "."

(1.9)

and calculate the integral over "~ by the saddle point method. The saddle point is found from the condition

a --

aT

N(y

-

y') +

"r

,('OZ]ko = o,

(1. i0)

i. e. ,

y-

y ' + pjz = - - [2qy~o + 3 r ] %2 + 4 s j %3 + ...]z.

(1.10a)

Inverting the series (1.10a) we find

~o =

~] 4qi z

3rl

~2.

(2q]) 3 z ~

l__[gr i I~ ........ 4sj - - . . . ( ~ ] = y - - y +piz). (2qi)4\ q1 ] z3

(1. lOb)

For sufficiently large z satisfying the conditions*

*Inequalities (1.11) are the conditions which enable us to neglect the subsequent term in the asymptotic expansion of the required integral. If these inequalities are satisfied this is equivalent to the presence of a large parameter which allows us to integrate by the saddle point method. It is easy to see that kz is such a parameter for an isotropic medium.

517

z>

s; q~,

(1.11)

z>> q?,

the following expression is obtained for the field in a gyrotropic m e d i u m : 4

(1.12)

Ej_(y, z)= ~ Ej• (y, z) exp [-- i(Tizo z + TyoY)]Here

Eix (y, z) = ~dy'Ei. (y', O) Li(y -- y' --k PiZ); Ei_~ (y', O ) =

E (i)n dn n-o, ~=x, y n! d~q~

=o

+ iyo k=o kt d~qk ~= odY'k

" dY TM

Xo q-

'

"3 r-_2-J "A (9 r~ ) Ls(y --y' +pjz) = V-~-iexp [i 4q.iz ' nt-i8q ~ z2 + i \ 4 q ] - - s i X X (2qi)~za + ...

'

2~z 2@ 3ri zi + l_j_ 9r, ,c~ qi z 2q~ 12s]-- qi z~

1// [

(

)

1

"'" "

We note at once that one need not take into account terms in the field E i • ( y ' , 0) associated with derivatives of Rj= (~) and lj('~), in view of the fact that these functions vary rather slowly. Formula (1.12) can be simplified considerably. We shall assume for a start that the medium is lossless, i . e . , p, q, r, s, etc. are real numbers. Then L i (y - - y : -~ pi z) is a nonincreasing function of y ' . In addition to this let aj be the dimension of the region of y' where E I • ( y ' , 0) is significantly different from zero. So although the integration in (1.12) is formally carried out within infinite limits, the basic contribution will c o m e from the region where ]y'] ~ a ; thus when the conditions

I (y--aj+pyz)8 z"

rj I 8q~ ~ ~'

0 (y -F pyz-- aA << 1,

(2qi) ~z a 4qy sj (( % [ r~ s ~ { y -- ay W pjz ) 2 k 7 + q;Jt z << 1

(i. 13a)

(1.13b)

are satisfied, one can l e a v e only the first term in the exponent index of function Ly, and also confine oneself to the first term in the expansion of the radical appearing in L] into a series in w/z. The field (1.12) is written in the form

E•

z)= "~. r

e_~(xj~oZ+Xyoy)SEjj_(y,, O)exp[i (Y--Y'+P/Z)~ ]dy,

(i. 14)

I

Let the medium now be lossy. The coefficients p, q, G s etc. appearing in (1.12) b e c o m e complex so that e x p ( i ~ / 4 q/z), for example, m a y b e c o m e an increasing function of y ' . In this case it is impossible to assert that the region where IY'I ~ a] is significant in the integration, although the simplifications which have been m a d e were in fact based on this. However, if the functions appearing in (1.12) are analytic, it is possible to change the integration path, integrating now not along the real axis, but along the line of steepest descent. If 6r is the characteristic dimension of the region along this line in which the expression appearing under the integral is significantly different from zero, then on satisfying conditions (1.13), in which a" is now substituted for a., and p, q, r, s are complex, one m a y pass from a] ] (1.12) to (1.14). If singularities of E i . (y~, 0) are intersected when the path of integration is deformed, then the corresponding residues must be added to integral (1.14). We note that if the phase of the field is not of interest, then in the absence of losses one can propose !ess strict conditions for the a p p l i c a b i l i t y of (1.14), instead of conditions (1.13a). A c t u a l l y we shall consider the second term in the exponent index 518

rj ~a i rj (y+pyz) a iSq? -z~ = Sq? z~

3

(y,)3 ] (Y+PiZ)2 y'+aY+PJZ z" z' ( Y ' ) Z - - -Z2- J "

(1.15)

The first term in (1.15) does not depend on y', and so the factor exp[iL 5q~ ~ (Y+PJZ)a-z2 m a y be taken outside the integral sign in (1.12). Only the phase of the field is affected by not taking this term into account. In order to neglect the remaining terms for large z it suffices to require that

3(,+,,z) i z

(1.16)

8qj

Similarly one obtains the condition

4l(y-kpjz~3[9r~ z / ~4q]

~ l aj I(<~" ](4q,)4

(1.17)

Up till now we have been considering the case of large z(sj/q~ z ((1). Let z now be fairly small. Since a/is, as before, the dimension of the region of y',where E ( y ' , 0) differs significantly from zero, the width of the spectrum in q'y is of order of magnitude 1/aj. If a ] is sufficiently large then the magnitude of Ty differs from zero only in a narrow interval Ty0 close to E(Ty ). However. in this interval T]z can be represented with sufficient accuracy by a few terms of the expansion (1.6a). This is clearly allowable if

IrA

Is:l ~q4eff z ( < ~ ;

z

Carrying out the integration over ~ and assuming that once again at formula (1.14).

~eff ~-~ I/ay.

e](~])and Ryx, y (~]) are

(1.18)

sufficiently smooth functions, we arrive

When losses in the medium are taken into account the same conditions (1.13) must be satisfied for formula (1.14) to be valid, but N eft will characterize the dimension of the region where E ( y ' , 0) differs significantly from zero along the line of steepest descent. Actually the possibility of representing the field E (y, Z) in the form (1.14) is connected with the fact that the region where r ( ( 1 contributes to the field. Thus the field in a gyrotropic medium can be represented in the form (1.14) for all y when z satisfies the conditions (1.18), and for those values of y which obey the relationships (1.13) or (1.16)-(1.17), when z satisfies conditions

(1.11). It is easy to see that conditions (1.11), (i. 13), and (1.15) can be made consistent. For this it is necessary that "~eff should be sufficiently small:

~eff4 ((( q*/s*, ~eff ((( q3/r3.

(1.19)

In this case formula (1.14) is valid for all z. In addition to this for small ~qeff the radiation is produced at a small angle, so that in the absence of losses conditions (1.16)-(1.17) are satisfied in the region of y values where the field differs notably from zero. In this c a s e ( 1 . 1 4 ) gives an expression for the field, with an accuracy up to the phase, valid in a11 space where the field differs significantly from zero. We note that for an isotropie medium formula (1.14) is identical with the Fresnel approximation in diffraction theory. Conditions (1.18), (1.11) and (1.17) then take the following form (we confine ourselves to the simplest case

7yo = 0): z << - - T 8"r eff

'

z

)>

2k '

ate << 2~.

(1.20)

9.. Beam Parameters It is clear from formula (1.14) that when spatial dispersion is taken into accofint the field in a ferrite m a y be represented in the form of a sum of four fields propagating in different ways and having different polarization. We shall

519

c a l l each of these fields a beam. Beam propagation occurs in a similar manner to e l e c t r o m a g n e t i c wave propagation in an isotropic m e d i u m in the quasi-optical approximation. As is well known the field in an isotropic m e d i u m , in this approximation, has the form

E,,(y, z)=

~, ~,jEj_(y', o) exp[

ik(y--y') 2 7| dy'. 2z I

(2.1)

Whence it is clear that in the absence of losses one can use the solutions for an isotropic m e d i u m , carrying out the sub1 stitution k -+ - - - - ,

y-+y+pz

2q

everywhere except in the exponent

In the absence of losses the parameter

e -l~z where /~-+ %'jzo"

Pj has a simple physical meaning. To illustrate this we let qj tend to zero,

Then

Ej_(y, z) = ~

exp [-- i('rj.o z + TyoY)] E j i (y +&z, 0).

(2.2)

.]

It follows from this that p ] gives the tangent of the angle between the direction of motion of the jth beam and the z axis. This angle naturally coincides with the angle of inclination of the group v e l o c i t y vector with respect to the z axis. Actually, 0m ~ g r = Yo O;'y

0m

--+,o <

A tg(VgrZ~ The difference in

,

(2.3)

&o / & ~ _ _ _aTz = - - & . d-~y/OT;-- OTy

(2.4)

p] for different j leads to the beams propagating in different directions.

The p a r a m e t e r qj is responsible for the spreading of the beams. This is clear from the following concepts. The factor e x p [i(y -- y' -Jrpossesses "filtering" properties, so the width of the interval &y' contributing notably to the integral is connected with ]q]l: increasing leads to an increase in this interval, i . e . , to a widening of the y region where the field differs significantly from zero.

&z)2/4qjz]

lqj

We shall not give an explicit form of the quantities which determine the beam amplitudes E H_ (y, 0). Their analysis shows that effectively beams with s m a l l propagation constants are excited. Thus the a m p l i t u d e of the fourth beam is very small; moreover its imaginary part %'4 is large, so that we shall not concern ourselves with this beam in what follows. The third beam is in effect excited only at resonance, where 7 f ' T a , when losses are very small (e = A/-///-/.-~ 10 - 8 , which occurs at low temperatures in garnets). However even in the case when the amplitudes of the "spin" beams are sufficiently small, it is of interest to take them into account, since due to the difference in the /37 these beams can be separated from e l e c t r o m a g n e t i c beams in space. In all remaining cases if one is interested in beams of large amplitude it is sufficient to carry out the summation over indices 1, 2 in (1.14). The necessity for additional boundary conditions can be relaxed and equations (1.3) alone employed. We now turn to the parameters p / and j temporarily)

p

07. .

OTy

.

q]. in the coordinate system shown in Fig. 1 we have (omitting the index

(07/00) sin (~ - - O) - - 7 cos (a - - O) . . (07/00) c o s (~ - - O) + 7 Sin (a - - O)

ctg ~.

(2.6)

The angle between the group and phase velocities can be easily found

0r /

tg x = -- 00-

520

T.

(2.6)

We now investigate the quantity f_~---OT / T in some particular cases (we note that f ~ p for e - - 0 = ~ / 2 ) . 1. The dispersion relationships for the electromagnetic and spin branches are independent. This occurs if not very large losses are present, over the whole frequency range, and if they are absent, outside resonance [2]. We shall not give general expressions for f, but confine ourselves to the following two t cases.

a) Small gyrotropy (~>>~. - fl, ~, ~ are periodic with period ~r;

_

fi,

~ ~--- -T-

~., sin 0

4~Tc .Me)*.

The functions (Fig, 2)

for 0 ~ 0

<

Oex ;

(2.7)

where a-

~,*

1-- 2 (rq2 fx,2 ~ -T (o.

.,~)

O)u ](~/2 -- O)

"3

., Vi1+4,~.,UolH.)~..~Io~'+4(,~/2-0),

a, (T,T,)

--

fa ~

Fig. 1. Here ~ n =

% sin 20. 4o~

.for O e x ~ O . ~ (2. 7a)

}-'

(2.8)

TcH0.

The expression (2.7) is valid for COS 0>>%/2~<<1. Close to 0 --:- ~/2 this condition is violated, and there formula (2.7a) is valid. The expressions (2.7) and (2.7a) pass over into each other, so

/},,ex - ~ o V 6% b) Oyrotropy not small ( ( o ~ ? /

(2.9)

for 0, ~ex= T~ T - - -~%' V 1 + 4 ~ Mo no

- - "(clio), but O <( 1. Then

i%0 /2 ~ H 1 -- i%,02/2a~ n

0[

(2.to)

A = -L,

m

"

- - ~ . / 2 , o H + iO ~ ~ . / 2 ~

n

(2.1~)

] i

1 We note that

ILl has a sharp

-

-

i0%.d2 aWl~

m a x i m u m at 0 - - ]/r2-~o/.//mu 9

2. Now let the losses he very small (~ < 10-~). Then the dispersion relations for waves I and 3 cannot be considered independently at resonance. Using the dispersion relation derived in paper [2], and solving it by the perturbation method for small O, we obtain

§

•

-- 2

0

OI

u

~2

+ 4 ~,~#~

]

l.~.J (d.

7rrl. '

.... ~J

c

(2.12)

L

a~(%--~)

]-112

Oex/ - -4~

Here H E is the spin field. Expression (2.12) reduces to (2.10) if (eto) ~ >> 4 (o~/C ~) ate - - o0 (where mB = m n + w•), 0 is sufficiently srnall, and co = a}n .

Heag(ma

~0M

-7/r Fig. 2.

*Here 7t is the gyromagnetic ratio. 521

fora--

We shall now consider the parameter 0 ==/2

q/, restricting ourselves to some particular cases. It is easy to show that

1 [1 OaT 73

1 .072.2.

(2.13)

For small gyrotropy we have from (2.13), rejecting terms of the second order in to~/to, that

(2.14)

q,,~=--c]/-g/2m

for angles satisfying the condition COS O >> t%/2m. This condition is violated close to 0 = ~/2 so that for angles [3 = ~ / 2 - - 0 << 1

-

-

q"==

1 1 1 +- -

1--29"

%'1

V(to,/toP + 4S

to

.

(2.15)

Expression (2.14) shows that with the given accuracy spreading occurs identically for both beams and also in just the same way as in an isotropic medium. It is interesting that for 0 --- ~/2 the quantity q~ = m,/2kto B can be made quite small. This means that spreading of the corresponding beam will be small for 0 ~.~ =/2. We shall now consider the case to = to/4. We then have for 0 --- 0

1 V ql = - - 4-'-k

%.. e-l~/4 aton

1 ,

qa =

1 + to./4to.

4 k 1/1

(2.16)

-t- m,,/to M

It is clear that spreading for the extraordinary wave becomes strong at the resonant frequency. 3. Generalizationto the Three Dimensional Case We shall now consider the three dimensional case. We write the field E . ( x , y, 0)in the following form:

E•

y, O) = [xoExo(x, y, O) + yoE~,o(X, y, 0)] e x p [ - - i(7xo x + Ty0Y)],

(3.1)

where Ex, y, 0(x, y, O) is a function which varies little over distances of the order of a wavelength in the ferrite. We introduce the quantity ~]x, y = T x . y - - T x , y, 0 and expand 7jz('qx, "qy) in a series in "qx, "qy

Tjz = TjzO Jr" l~x'qjx ~ Pjy'qjy q- q/iq**+qjy~qiy+ 2 2 h/qj;qjy+...

(3.2)

Carrying out the calculations in the usual manner and using approximations similar to those in w 1, we obtain the following expression for the field:

E~_(x, y, z) = Y, - - i exp[-- i(T/.oz + 7xoX + "fyoY)] f i 2 ~ z V 4 q : . q:y -- h 2 ,J E I. (x', y', O ) d x ' d y ' X i

(3.3)

X exp i [qjy(x -- x' + pj.z)" + q:,,(y -- y" + p:yz)' + + h:(x - - x' + p:.z)(y - y' + p:yz)l/[(4q:.q:y -- h~)z]. Here similarly to (1.12)

E/. (x, y, 0) = [Xo + iei(O)yo] [G,o(X, y, O)R/,,(O) + Eyo(x, y, o)R~y(O)l.

(3.4)

Formula (3.3) is applicable for all y if

( ~x eff •

O'qx

522

"qy eft

(

o Tlz z -{- ~x eff ~

~

+ ~y eff ~

o),

o Tlz Z << ~"

(3.5)

The symbol <<0>>means that the derivatives are taken at the point ~x = 0,and ~ y ~ , 0 ; ~qx eft and ~qyeft are the effective spectral widths in the xz and yz planes respectively. We shall now determine when the three-dimensional case m a y be reduced to the two-dimensional one. We can easily see that this requires, a) that E I Z (x, y, 0) can be represented as a sum of products of functions each of which depends on one coordinate only, and b) that h i - - 0. The first condition can always be satisfied. It is easy to show that the second condition also can always be satisfied by rotating the coordinate system by a certain angle ~1 about the z axis. The case when ~! coincide for all j is of particular interest, i . e . , when h i -- 0 for all j simultaneously. Such a situation occurs, for example, in the particular case when all T/ and the z axis lie in the one plane. In this important case the three dimensional problem reduces completely to a two-dimensional one. 4. Some Examples

Making use of the results obtained we shall consider a concrete example of how electromagnetic waves propagate in a gyrotropic medium. We shall give the function E (y, O) the following form:

E ( y , O) = x o exp(-- vy = + iwy=),

(4. ~)

where v and w are certain constants (V > 0, W > 0). 1. For a start we shall not take losses into account, considering the beam parameters PI and qi to be real. Then the field at an arbitrary cross section, as determined from (1.14), can be written as

E(y, z) = ~] e - ~ s ~o~ e2, ~ / e2,j - - eL~

[_

4/ (Xo+ielyo) l~ I'

_ ~(Y + P/Z)=

X exp

(1 +" 4zvq/z) ~ + (4~q/z) = 9 +

--1

(4qtzw + 1)~ + (4qlz'v)' ] exp i { 1 arctg l + - 4wqlz +

X (4.2)

4~qs z

[(v ~ + v2~)4qi z + w](y +Pi z) 2 ] (1 +4wq I z) ~+ (4vq i z)~ ]"

In the particular case when the ferrite is magnetized along the z axis and T / ~ 0 = Tj0 -= k g - ~ -~ t.1,=( a l s o p / = 0, e I = ~ l ), the field at any cross section of the ferrite is composed of beams propagating along the z axis with propagation constants ~'10 and "f~0 . and with right-handed and left-handed circular polarizations. Since each of these spreads in its own way (Fig. 3), the total field turns out to be elliptically polarized with e11ipticity depending on the y coordinate (polarized along the x axis as the field (4. i) in the z = 0 plane). The difference in beam spreading is determined by terms of the third order in gyrotropy, so that for tu))o~B this effect is weak at not very' large distances. On the other hand if ~ ' ~ B the difference in beam spreading becomes very large. 2. Let the medium now be lossy. The parameters p! and ql are c o m p l e x : p / -----p / ~ i p ) ql = q! - - tq i 9 Substituting the field E{y, 0) as given by (4,1) in (1.14), we obtain (integrating along the line of steepest descent in accordance with w 1)

,I

IE'(y, 011

E(y, z ) = ~

1

9~ . ~ ' ~ - i e x p ( - - " .V 4.--~/z tT/~oz) exp{ [L](O~ - - LiH/) --2 2 VLl+~i

4

--

0 f

-

--

-

- 2tjpjo

I

Ci hl

(4.3)

+ qKj)]/(q +

Y

Fig. 3.

523

where

L,=v+

ql

Oi = Pl zqj -- ql(y + p'1z)

41qll~z

4 Iqtl 2 z

F

ql

=

-4lqilzz -[(y

+P}z)q I + q~Plzl,

~71=w-k- 41qil~z

1

- - - - [ 2 p ~ z ( y + p'tz)q) -- q}(y + p'jz) ~ + q} p~z'l , 41qll~z

H i --

K, = ~

1

{[(y + p;z)'--p~ 2 z~l qi + 2p~q)z(y + pjz)1.

Letting ql in (4.3) tend to zero, we find that for a lossy medium the direction of motion of the jth beam is now d e t e r mined by the p a r a m e t e r p ] (w/v)p)', which has the meaning of the tangent of the angle of inclination of the group v e l o c i t y with respect to the z axis. This p a r a m e t e r depends on the form of the function E(y, 0), in our case on the quantities v and w in (4.1). t

-

We shall determine how the field E l ( y , z ) c h a n g e s f o r p / - - 0 , when q)=q'~, q) < 0. from (4.8) that the a m p l i t u d e o f the jth b e a m is proportional to e x p

[--.rP(z)y~],

It is easily found

where

M i = v --iw. Fig, 4 gives a graph of qb(z) for 0 ~< w ~< v ( V - 2 - - 1). The__graphs of qb(z) have a similar form for all W < V ( ] ~ - 2 - -1- ! ) . The m a x i m u m of qb(z) disappears for W ~ V ( ] , / - 2 -t- 1). From Fig. 4 it is clear that the beam is focussed in the z = Zl plane, then it begins to spread, and the initial field r Z) distribution is repeated in the z - - z2 plane. For Z=Za the beam amplitude does not depend on y . F o r ' Z > Z a the amplitude increases with y, and for z - + co the field distribution is again independent of y. As has already been pointed out, the expression for the field (4.3), obtained in the quasioptical approximation is valid within an interval of y which is not very large. In the case under consideration a considerable portion of the field is concentrated outside this interval. It is interesting to trace the consequences of taking into account the following approximation. We use formula (1.12) in the exponent of which we keep terms of order (y - - y , ) i (terms of order (y _ _ y , ) a are absent, since by symmetry r i ~- 0). Calculating the integral over y' by the saddle point method, we obtain

Fig. 4.

E fly , z)= ]/ 4Gze-'~JzoZexp _(M~/NI--MI)/-- i(2q-~I)'--~j-~ j X (4.5)

X IN] + 6i s ~ M~ y2]--I/2

[

(2ql)'N~z' j

e2, 1

(X 0 + ietyo) '

e2,,--e,,,

where s t -~ q]/8k ~'. The qualitative behavior of IF(Z) = Re[isi~/(2qi) t N~z a] is represented in Fig. 5. It is clear that when iF(z) is taken into account the beam amplitude now falls off with y for z > z0. As regards z < z0,the amplitude here i n c r e a ses with y in two intervals z(O-.
524

by broken lines. We notice that when a right-hand polarized beam of frequency to - - toM and with ql determined by formula (2.16) propagates in a ferrite along the magnetic field it is damped sooner than spreading can account for. We note in conclusion that the results obtained in the paper can easily be applied to the case of active linear media with tensor parameters. The authors wish to acknowledge the interest V. I. Talanov has taken in the paper.

{ J

Fig. 5.

Z-Z,

i

.S

REFERENCES 1. A. G. Gurevich, Ferrites at Microwave Frequencies [in Russian], Fizmatgiz, Moscow, 1960. 2. M. A. Ginzburg, FTT, 2, 913, 1960. 3. B. A. Auld, J. Appl. Phys., 31, 1642, 1960. 4. G. M. Genkin, Yu. A. Otmakhov, and E. A. Rozenblyum, FTT, 5, 2968, 1963. 5. G. D. Malyuzhinets, UFN, 69, 321, 1959. 6. G. Goubau and F. Schwering, IRE Trans., AP-9, 248, 1963. 7. N. G. Bondarenko and V. I. Talanov, Izv. VUZ. Radiofizika, 7, 313, 1964. 8. V. Z. Katsenelenbaum, UFN, 83, 81, 1964. 9. W. S. Ament and G. T. Rado, Phys. Rev., 97, 1558, 1955. Fig. 6. 10 October 1964

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