Appl. Sci. Res.

Vol. 16

REFLECTION OF FROM

AND

TRANSMISSION

ELECTROMAGNETIC

A NONLINEAR

BETWEEN

TWO

ANISOTROPIC

LINEAR

by PREDHIMAN

WAVES

ISOTROPIC

KRISHAN

SLAB MEDIA

KAW

Department of Physics, Indian Institute of Technology, Hauz Khas, New Delhi-29, India Summary

In this paper, the author has studied the reflection from and transmission through a homogenous nonlinear anisotropic slab (anisotropy being due to an external magnetic field) bounded by two linear isotropic media. Nonlinear equations describing the growth of the two modes of propagation of an electromagnetic wave in the direction of the magnetic field, in an anisotropic nonlinear medium, have been set up and solved; the solutions have been used to obtain expressions for the reflected and transmitted components of the incident wave. The simpler problem of the reflection a~ld transmission from an isotropic nonlinear slab has also been discussed as a special case.

§ 1. Introduction. A n o n l i n e a r m e d i u m is r e n d e r e d anisotropic b y a n e x t e r n a l m a g n e t i c field. T h e effect of a m a g n e t i c field on the p r o p a g a t i o n of e l e c t r o m a g n e t i c w a v e s t h r o u g h nonlinear media, which are infinite or senti-infinite in extent, has b e e n i n v e s t i g a t e d t h e o r e t i c a l l y b y m a n y workers. I n the l a b o r a t o r y , in a n y experim e n t s t h a t m a y be carried out to v e r i t y the conclusions of the p r o p o s e d nonlinear theories, one can h a r d l y realise the s i t u a t i o n of an infinite or semi-infinite nonlinear m e d i u m . I t is therefore necess a r y to i n v e s t i g a t e t h e o r e t i c a l l y the effect of a m a g n e t i c field on the p r o p a g a t i o n of e l e c t r o m a g n e t i c w a v e s t h r o u g h nonlinear m e d i a of finite dimensions. Such an i n v e s t i g a t i o n would also be applicable to m i c r o w a v e m a g n e t o p l a s m a diagnostics, m i c r o w a v e c o m m u n i cation t h r o u g h p l a s m a s h e a t h s a r o u n d h y p e r s o n i c missiles in u p p e r a t m o s p h e r e , m i c r o w a v e reflection f r o m a n d t r a n s m i s s i o n t h r o u g h s e m i - c o n d u c t o r s in the presence of m a g n e t i c fields etc. d

215

216

P R E D H I M A N K R I S H A N KAW

Thus, in tile present paper the author has studied the reflection from and transmission through a homogeneous nonlinear anisotropic slab bounded b y linear isotropic media on either side. Starting from the general wave equation, appropriate nonlinear equations describing the growth of the two modes of propagation of an electromagnetic wave in a nonlinear medium have been obtained for the case when the magnetic field is acting along the direction of propagation of the wave. These have been solved and the solutions have been used to obtain expressions for the linear and nonlinear components of the reflected and transmitted parts of the electromagnetic wave. By putting the magnetic field equal to zero, the simpler case of the reflection and transmission of electromagnetic waves from an isotropic slab, has also been discussed. § 2. Growth o/ the two modes o/propagation in a nonlinear slab. Consider a slab of the nonlinear medium bounded b y the planes z = 0 and z = d in such a way that the region 0 < z < d corresponds to the nonlinear anisotropic medium and the regions --oo < < z < 0 and d < z < c~ correspond to the two linear isotropic media I and I I respectively. The components of the current density in the nonlinear medium due to an electric field E having components Ex ~ Eox exp(io>t), Eu -~ Eov exp(icot), Ez ~

(1)

0

and a magnetic field B' having the components B~=B~-0

and

B'z=B;

are given b y 1) J x = (AEox -- BEou) exp(icot)

(2)

and Jv = (BEox + AEoy) exp(io)t), where A = Ao + Al(EoxEox -[- EouEoy), B = Bo -F Bl(EoxEox -F EoyP-,ov)

(3)

are the components of the nonlinear complex conductivity tensor.

217

ELECTROMAGNETIC WAVES

For wave propagation along the z-direction (the direction of the external magnetic field) in a uniform neutral medium, Maxwell's equations reduce to ~eE Oz2.,

e# c2

bee 4re# ~t2 + c2

8J ~t

'

where e and # are the dielectric constant and the magnetic permeability of the medium and c is the velocity of light in vacuum. Writing down the x- and y-components of this equation, multiplying the latter by ~ i , adding to the first and substituting for components of E and J from equations (1), (2) and (3), we get for the two modes of propagation of the electromagnetic wave in the nonlinear medium ~2

o~2 (#'~ + i#y) + fi~*2 (#~ + i#y) =

(4a) and ~2

o~2 (#'~ - igu) + ~;;z(#~ - igy) - -

(4b) where /5£2= ~ l _ _ 4 x i

(A°+iB°) 1 = (-n'

+ ik')2,

~n(D fi;2 =

1--

# = Eo/Eoo,

- -

En(.O

(Ao -- iBo)

4ui . ~- - EXo, al

-~-

a2 =

= (-~"+

ik")2,

~ = (enid.) ½".°)z ,

A1 + iB1, A i -- iB1,

E0o is an arbitrary normalizing parameter and sn and #n are the dielectric constant and the permeability of the nonlinear medium. Similarly for wave propagation in linear media I and I I we

218

PREDHIMAN KRISHAN KAW

have 02g - -05z + ~

(sa)

= 0,

(5b)

052

where 4~i

--0"1

sn#n

)

= (--nl + ikOZ,

81a)

--

--

"

enl~n

0"2

=

(--n2+ik2)

2,

e2~

O'1 and 0"2 are the complex conductivities for the two linear media; the other symbols have their usual meanings, subscripts 1 and 2 referring to the media I and I I respectively. Solving the nonlinear equations (4) by a perturbation technique similar to that of Epstein Z), one obtains for the ordinary and extraordinary components of the electric vector in the nonlinear medium, t

t

i

gx + igv = cl exp(ifi~5) + c9 exp(--~flnS) + •

7

-}- ~{c{1 exp(ifl~5) + C~l exp(--ifl~5)} + ~ ~] Yn~(5) k=À

(6a)

and gx -- igu = ci exp(iflgS) + c~ exp(--i/3gS) + 7

+ e{ch exp(iflgS) -t-

C~1

exp(--ifl4~5)} + ~ £ Xn~(5),

(6b)

k=l

where c{, c~, cl, c{, constants and

C{1, C21,

Cll and c{1 are eight unknown arbitrary

Y~1(5) = gl

2

c1261exp{~'(2fln--~n)5}@c2 C2 exp{--z(2fln--~n)5} t2 t • 2

(7a)

Y.~(~) = al

Cl e g e x p { z ( 2 f l n + ~ ) 5 } + c 2

,2

,

el exp{--z(2fln+~n)5}

•

(7b)

WAVES

219

a1~1c2{~2~xp(~G~) + ~ ~xp(-¢.~)} ~,~ ~,~, ,

(7c)

ELECTROMAGNETIC

t

g,~a(8) =

t

~t

•

v

~t

b'n

• ~t

--t-'n

Yn4($) = a

' "~"i o ~ p { ," ( &'+ & - ~". ) e-" 1 ~i~, }+

c'2~"6"~ C x v { - ~ (' & +'& - 5 .") ~ }~"

, (7d)

Yn~(~) = =_ a_, clc~g~ exp{i(fi~-B(i q-fl;)~}q-cbcigl e x p { - i ( f l ~ - f l g

4-fi~i)~}

~,, 2

Bin -(&-B~+~) '2

2

'

, (7e)

Ynn(~) =

= a~ cici@ exp{i(B~ q- B,7 +fllt)~} q- c~c~61 exp { - i(B£ +Big q-f;)~} &t 2 - ( & +t & +,,& ) - - 2 2

, (7f)

Y~(~) = 0¢1 CiC~Oi exp{~ ( B £ - B ; ~ - f ; ; ) ~ } + c~.ci6~ e x p { - i ( f i ~ - B;g-f;~)~}

-=-

Xnl(~) : ~ -

B;~2- (2B~7_f;~) 2

, (7g)

, (7h)

a2 c126~exp{i(2fl;i +fIi)~}q-c~2~i exp{--i(2Bii q-f;i)~ } Xn~(~) = ~ B;/2 - (2fl£+f~) 2 , (7i) Xn3(~) =

a2clcg{@ exp(if;g:) ~- 6i e x p ( - - if;i~)} fl;/2_f£2 '

(7j)

X ~ , 4 ( ~ ) ~---

= --

~

2

,

,

B~ - ( B ~ + & - G )

,

~

'

(7k)

x+d~) = t ~t

a2 01c2c2 e x p {

~

(fln--fln-~fn)~}-~C2OlOl exp{--t(fin--Bnq-fn)~} tr

P

/i

1 ~t

&- 2 - ( & -. & + ~I )

2

n

2

P

t~t

e x

•

.

i

l

-2

'

f

P

t 2

prCt ~t

'

rt

•

J 2

.

•

(71)

•

(7~)

220

PREDHIMAN KRISHAN KAW

=

_ a2 cic'2cl exp{*(fin--fl~--~)~}+cicg.c'2 exp{--,(fln--fi;t--fi;)~} r

~

"

rt

~

tr ~

"

tt

• (Tn)

§ 3. Reflected and transmitted components o/an electromagnetic wave /rom a nonlinear anisotropic slab. Consider a plane polarized electromagnetic wave, propagating along the z-axis (the direction of magnetic field), to be incident normally on the plane ~ = 0, from the side of the linear medium I. We shall assume that the incident wave is plane polarized along the x-axis (without loss to generality, because we are free to choose the direction of the x-axis in the plane of wave front (i.e. xy plane) and can choose it along the direction of the electric vector.) Let the electric vector of the incident wave be given b y E~/E00 : 6~i exp(irot), (8a) where o~, = A, exp(ifll~). (8b) In medium 1, the electric vector is then given by

E1/Eoo = o~1 exp(io~t) --~ (o~, + o~r) exp(icot),

(9a)

6~r = (At q- ~arl) exp(--i/JlE),

(9b)

where where Ar and Arl are the amplitudes of the linear and nonlinear parts of the reflected wave at ~ ---- 0. The components of the electric vector in the nonlinear medium are given b y equations (6a) and (6b) and in the linear medium II, the electric vector is given b y Ez/Eoo = ~2 exp(i~ot) ---- o~t exp(io~t)

(10a)

~t ~ (At + o:Atl) exp(ifl2~),

(10b)

where and At and Atl a r e the amplitudes of the linear and nonlinear parts of the transmitted wave. It m a y be pointed out that in general, the reflected and transmitted components of the electromagnetic wave will not be polarized along the x-axis, even though the incident wave be so, because of the intricate manner in which the xand y-components of the electric vector are coupled in the nonIinear medium; it is necessary, therefore to consider both the xand y-components of ~r and g°t.~

ELECT ROMAGNETIC WAVES

221

Using the b o u n d a r y conditions that x- and y-components of E and f-l(dE/clz) are continuous across the planes z = 0 and z = d and hence equating the linear and nonlinear parts of (Cx ~2 i#v) and f - l ( d / d $ ) ( ~ =k iNv) in the linear medium I and the nonlinear m e d i u m at $ = 0 and in the nonlinear m e d i u m and the linear medium I t at $ = $0, where $o = (e,,f,,)}.

o d/c,

one obtains sixteen equations, whose solutions are

Arx / iAry

1 [(fl~

A¢

~'

fn

\ fn

f2

-iz7

'3" pin,

A r z - - iArv 1 [ ( fi~ Ai = ~" f~

fz

--

A tx + iA tv

4

filfi~

~'

fflffn

f n

f l

-~-

exp(2ifla~o) --

~'-~ -/--jr2-

'

(11b) (I Ic)

fllfl'~ exp{g(fig -- f12) ~o}, fftffn

(1 ld)

(2/.@')[(3;~fll)/(f.,#x) + ([~2f11)/(f2f1)~,

(lle)

A t . -- iAtv 4 = A~ ~" cl/A,=

\ fir,

-J

exp{i(fi~ -- f12) ~0},

A~

=

(lla) fz

/..1

ci/A~ = (2/-~')[(~;,¢h)/(f~ft) -- (/32¢h)/(f~fl)~ exp(2i/3/$o),

(I If)

c~/Ai, = (2fi~') [(fl~fll)/(fnfl) ~- (flZ/~l)/(/Agf])~,

(llg)

c~/A~ = (2/~")[(flafl~)/(fn~l) -- (fl2fil)/(#2fl)] exp(2ifl,~$o),

(llh)

A q ~ + iAr~v

ch + c',21

1

7

A r~x ~ iAr~y A~A~

c~1 --~ C~l ! 7 A~Ai " + AiA* ~ - k=l Z Xn~(O),

(11j)

222

PREDHIMAN KRISHAN KAW

At,x + iAt~v __ ch A~A, A~A, exp{i(fi~

-

f12) ~o} q-

-

ch exp(--ifl2~o) + q-~.~, ~xp{-~(~;, + &) ~o} + A~A,

7

(llk) k=l

Atiz iAt~y ci1 2 . -- A~A, exp(i(/3;;, -- 132)~'o}+ Aid~ --

+~

C~1

exp(--ififfo)

,xp{--@~; + &) ~o} +

A~A, 2 ~

fl~ ) exp(ifl~o)

& + ~;,

A + /~1

#n

=

Z

#1

k=l

-

"

#2

C21

-

X

k=l

#n

+ (&

"

(lIm)

•

•

JAn

fi~ ) exp(ifl~8o)l =

~1

+

+

X

/A1

fin

=

Y~(~o)

#1

fin

~2

=

-

•

#n

\/A2

#~

(lll)

k=l

exp(-~&8o)

Y;M~o)

#n

7

X x,MSo),

~1

-

#n

Y;,~(~o) -

--Y~,~(~o)

#2

+

fl~ ){Y~(O) + ~Yn~(O)} exp(ifl~o) , /~n

#2

fla-) exp(ifl~o) -/Zn

~1

--~-+

/zn

#1

+ fl~") exp(_iflg~eo)l = #n

(11n)

ELECTROMAGNETIC

=E

~(~o)-

#n

k=l

+~7~ +

223

WAVES

x.k(~o) + (1 lo)

/~n

/~i

and

#n

\ ) exp(--ifi;~o) -/

fl~; ) exp(ifi~;~o)l = ~2

= E

k=l

#n

/~i

@

#n

~n

X£k(~O)-

#i

Xnk(~O) @ (llp)

#n

where

2' = (G tin

\ #n

tt2

-

e p(2~o),

ttn Y£~($) --

x;~(~)

-

#1

1

d

i/~

d~

1

d

ittn de

#n

#2

{Yne(~)},

{x~(~)}.

Y£~($o), Y~,k(~o), X£~($o) and Xne(~o) are obtained b y replacing b y $o and similarly Y~(O), Yn~(O), X;~(O) and Xn~(O) are obtained b y replacing ~ b y zero.

224

PREDHIMAN KRISHAN KAW

§ 4. Reflection and transmission/rom an isotropic slab. The nonlinear slab behaves as an isotropic medium in the absence of a magnetic field. Since there is no coupling of x and y components of the electric vector in the isotropic slab, we can without loss of generality consider an electric vector having only the x-component (instead of the one defined b y (1)); thus Ex : Eox exp(io~t), Ev : Ez : O.

It can be shown (by putting the magnetic field equal to zero in the equations of reference 1 that (2) and (3) reduce to J x = AEox exp(i(ot), J v = 0,

where A = Ao + A1EozEoz, B----O.

Wave equations (4a) and (4b) lead to the same equation viz. 852 where # = gx, a ---- A 1 and 3~ =

1 -

4~i

--

Ao =

(-n

+ ik)2.

8no)

Equations (6a) and (6b) also reduce to the same equation viz. # : Cl exp(ifln$) + c2 exp(--ifln$) + a{Cll exp(ifin~) + 3

+ c21 exp(--ifln$)} + oc Z Zn~($),

(12)

k=l

where Cl, ca, Cll and c21 are four unknown arbitrary constants and z,,~(~)

=

I~

-

(2t~,, -

~,~)~

ac1212exp{i(2~n -3vfin)$} -~ ac22[iexp{--i(2~n + ~n) $} z,~(~) :

/~ -

(2/~,, + / L , ) ~

2a~2{~. exp(i~) + a~ e~p(--ilh~)}

'

225

ELECTROMAGNETIC WAVES

The electric vectors in linear media I, II and the nonlinear medium are respectively given by (9), (10) and (12), the constants involved therein being given by the following equations"

ArAi1~@

#r~fin #2fl2) (~n + fl~l) exp(2ifin~o)-

[(

--

,an

,Ul

#n

~2

'

At/A~ = (4/~){(131fln)/(#1~n) } exp{i(fln -- f12) ~o},

(13b)

c,/Ai = (2/~)[(finfiz)/(#n#l) + (#2#1)/(#2#1)], (13C) c2/Ai = (2/~)[(flnfll)/(#ntO)-- (f12fi1)/(#2#1)]exp(2ifin~o), (13d) Arl

Cll @- c21

A~A, At1

1

3

A~A, + A~A, k=iZ Zn,~(O),

(13e)

Cli

A~A, -- A~A, exp{i(#n - #~) ~o} +

+ ~ Cll

c21

exp{--i(fln H- f12) }o} -F

fln ) ( fll

[(fi2 #2

fin

= Z

k:~

#n

/*z

(13f)

fl'-)exp(ifln~o)--

#I

fin

(f12 + /~n)(fll #2

exp(--ifi2~o) a A~A~ e=,2 Zn~(~:o),

/'1

/*n

+fin)exp(--ifln~O)l = - , 7 f*n

z;~(~o)-

Z'

fil

#1

#2

z~k(~o) +

Zn,~(O)}exp(--ifln~o)],

C21

#2

#~

/~i

fin )exp(ifln~o)] = #n

(13g)

226

=Ek = l

P R E D H I M A N KRISHAN KAW

~n

+

#1

.

+(/3~

/.2

Zn~ O) -[-

#2

~tn

exp(ifln~o) /~1

(13h)

J

where

#n

/'1

#~

#n

(/3"

~)(/3"

#n

z,~(~) -

#1

~n

1

d

i#n

d~

{zn~(,)}.

/3~) exp(2//3.~o)

#2

§ 5. Discussion. The linear and nonlinear parts of the reflected and transmitted components of the electromagnetic wave incident on the nonlinear anisotropic slab are given b y equations (gb) and (10b) where the x- and y-components of the coefficients appearing in these equations are given b y some of the equations (1 1) ; for the isotropic slab also, the reflected and transmitted components are given b y equations (9b) and (10b) b u t the coefficients appearing are given b y equations (13). The ordinary and extraordinary components of the electromagnetic wave in the anisotropic nonlinear m e d i u m are given b y rest of the equations (1 1). These equations can be easily modified to include some cases of interest. Thus for free space on either side of the nonlinear anisotropic medium, we have to put /31 = fi2 = --1 and #1 = #2 ~- 1 ; for free space on one side and a dielectric on the other we put/31 = -- 1, #1 = 1, fi~ = rid and #2 = = #a where the subscript d refers to the dielectric and, for a nonlinear anisotropic medium between free space and a metallic conductor we put /31 = - - 1 , /*1 = 1, /32 ~ oO and #2 = #~I where/~M is the permeability of the given metal. For the equations of the isotropic slab also, the above modifications can be made. It is worth noting from equations (1 la), (1 lb), (1 li) and (1 lj) that the linear and nonlinear parts of the reflected component of the electromagnetic wave from the first interface depend upon the properties of the linear m e d i u m II. This, though not obvious at

ELECTROMAGNETIC WAVES

227

first, is to be expected. For, in the nonlinear medium the resultant electric vector is the one obtained b y the super-imposition of the wave transmitted from the first interface and the wave reflected from the second interface (the latter has a magnitude which depends on the characteristics of the linear m e d i u m / / ) ; therefore the use of the boundary condition at first interface causes the reflected wave from it to carry the characteristics of the linear medium II. Expressions for Ao, Bo, A1/Ao and B1/Bo in terms of the electron density, electron collision frequency with neutral molecules, electronic mass and charge and ~o, for a Lorentzian magnetoplasma have been derived by Sodha and Palumbol). Using the appropriate collision integrals, degeneracy etc. their treatment can be extended to partially ionized plasmas, plasma sheaths, semiconductors and a number of such kindred nonlinear media. The use of the theory given in the present communication enables reflection measurements at different power levels to yield values of A1/Ao and B1/Bo in various materials which should aid in an understanding of the transport processes in these materials. The present treatment is an improvement over the earlier ones in that the effect of the finite extent of the nonlinear medium has been taken into account. Resonances in the amplitudes of the reflected and transmitted components of the electromagnetic wave can be studied b y a method similar to that used b y Sodha and Palumbo 1) for an infinite magnetoplasma. The use of a magnetic field has been proposed as a means of rendering an overdense plasma relatively transparent to electromagnetic waves. The present analysis m a y be helpful in an investigation of this concept. Acknowledgement. The author is indebted to Professor M. S. Sodha for suggesting the problem and valuable guidance during the investigation. Received 22nd October, 1965.

REFERENCES 1) SODHA, M. S. and C. J. PALUMBO,Can. J. Phys. 41 (1963) 2155. 2) EPSTEI? G M., Phys. Fluids 5 (1962), 492.