Section t3, Vol. 10

Appl. sci. Res.

N O N L I N E A R REFLECTION AND REFRACTION OF ELECTROMAGNETIC WAVES AT T H E INTERFACE OF TWO MEDIA by MAHENDRA S. SODHA and CARL J. PALUMBO Republic AviationCorporation, Farmingdale,N.Y., U.S.A.

Summary In this communication the authors have considered the reflection and refraction of a plane polarized electromagnetic wave at the interface of two media, whose complex conductivities are quadratic tunctions of the amplitude of the electric vector. Explicit expressions for the refracted and reflected components (which are solutions of the nonlinear wave equation with appropriate boundary conditions) have been obtained, when the electric vector 1) lies in and 2) is normal to the plane of incidence.

§ 1. Introduction. In the linear theory of propagation, reflection and refraction of electromagnetic waves the complex conductivities of the media are assumed to be independent of the electric vector. In m a n y situations of interest this assumption is not valid and one has to consider the dependence of complex conductivity on the electric vector. The dependence of complex conductivity on the electric vector can be taken into account in two ways: 1) G e o m e t r i c a l o p t i c s a p p r o x i m a t i o n . In this method which is inconsistent with the wave equation, the dependence of complex conductivity on the electric vector is substituted in the usual expressions for refractive index and absorption coefficient and a first order differential equation is solved for the complex amplitude of the electric vector. 2) S o l u t i o n of t h e n o n l i n e a r w a v e e q u a t i o n . In this method the dependence of complex conductivity on the electric vector is substituted in the wave equation a n d a solution of the resulting nonlinear equation for the electric vector is obtained. In this communication the authors have adopted the second method to investigate the nonlinear reflection and refraction of a plane polarized electromagnetic wave at the interface of two media --

451

--

452

MAHENDRA S. SODHA AND CARL J. PALUMBO

whose complex conductivity a is given by =

--

i l,

(1)

ar : ar,O + arEo'Eo,

(2a)

a~ -- a,,o + a~Eo'Eo.

(2b)

and where E0 exp(i~ot) denotes the electric vector. The dependence of complex conductivity on the electric vector as expressed by (2) is correct only to the second order of approximation. In general there will be terms involving higher powers of Eo" Eo but in the present investigation these are not considered. S o d h a and P a l u m b o 1) recently have derived expressions for a# and a~ for a slightly ionized plasma. The method can be extended to partially ionized plasmas, semiconductors and metals by adopting the appropriate collision integrals and degeneracy, a~ and a~ can also be determined by making propagation, reflection and refraction measurements on matecials at appropriate power levels and using the appropriate theory in interpretation of the measurements. The time dependent part of conductivity has been shown (Ginzburg and Gurevich2)) to be many orders of magnitude smaller than a~ and a~ for slightly ionized gases and this may also be true of other materials; hence this has not been considered in the present investigation. The propagation of an electromagnetic wave in a medium should be investigated by solving Maxwell's equation and the material equations, which in the Gaussian system of units can be expressed as § 2. Nonlinear wave equation.

VxH

1

~

C St

D=

4~

C

J,

V×E+----B=O, C ~t

V. D

=

4 p,

V.B=O,

J = ~E, D = EE, B = #H,

(3)

NONLINEAR REFLECTION AND REFRACTION

453

where E is the electric vector, H the magnetic vector, B the magnetic induction, D the displacement vector, J the current density, p the net charge per unit volume, E the dielectric constant of the medium and # the magnetic permeability. For a plane polarized wave, propagating in a direction r in a neutral uniform medium, the equations describing the propagation of the wave, derived from (1), (2) and (3) are d2e +fl2e d~ 2

(a'~Stot'etot) e,

(4)

and d2~ d~ 2 where E = Eo exp(hot), H = Ho exp(i~ot). ~. ---- Eo/Eo,1, o~o = Ho/Ho,1,

t~2-

~

Ii_

47t/~ a'~

-

-

-

¢Ol~O0~

4~

@+/~r)]= ( - n

+ik)2,

(1)

(~t + {~) E ~0,1,

(0

g ---- -C- ('°'u°)~ r, and the subscript tot refers to the total electric field. § 3. Nonlinear reflection and re/raction o/electromagnetic waves at the interlace. Let the interface of the two media be the plane z ---- 0 and the plane x ---- 0 be the plane of incidence. Let the angles of incidence, reflection and refraction be 0,, Or and 0t, measured from the positive z axis. To proceed further we distinguish between the following two cases, referred to as I and II in the further discussion: I. The electric vector is normal to the plane of incidence i.e. along the x axis. II. The electric vector lies in the plane of incidence i.e., the magnetic vector is along the x axis.

454

M A H E N D R A S. SODHA AND CARL J. PALUMBO

C a s e I : Using (4) the propagation of the incident, reflected and t r a n s m i t t e d waves is described b y d2el

d~--~-~ + fl~e, = ala(ei + er)'(gi + gr)"gi, d2er

~r

@ fl2er ~ a~(8i @ er)" (g~ @ 8r)'er,

(6) (7)

and

d2et + fl~t = a~st'gt'st, d~

(8)

where = Y s i n 0 + Z c o s 0, ¢o

Y = -C-- (E°/~°)½Y' GO

z = -C- ('°#°)~z" The second order solution of the above equations can be written as

e = e' + ad'.

(9)

Substituting for e from (9) in equations (6) to (8) and equating the coefficients of like powers of a on both sides we obtain t

d2el

d~

+ Z~d = o,

(6~)

d~--~-r+ ~1~; = O,

(7~)

d~e; d 2et

d~-~ + ~ ; = o, d~.8i,, - -d ~ + 01~ ~

d2 ~r"

H

f)~

l!

-

t

t

t

t

(8~) t

•

al(~ + ~;).(~ + ~;).~, I

t

d~

t

(6b) (7b)

and d2

"

d~:~ + P~et ---- a2~t'et'~t.

(8b)

NONLINEAR

REFLECTION

AND REFRACTION

455

The relevant boundary conditions are ei+er=et

at

Z=0,

Yt'iy + #dry = Ygty

]

or

(1//,1) L OZ + 0Z _1 = (1/#2) - ~ - j ei = A~

at

(10a)

/

at Z = 0,

~t = 0,

(10b)

(10c)

and the appropriate radiation conditions viz. 0

at

~r = co,

(10d)

et = 0

at

~t = co,

(I 0e)

er

=

there is no backward component of ei. (10/) The first order solution, satisfying the above boundary conditions is e~ = At exp(--ifll~d (6c) e~. = Ar exp(+i/51~r)

(7c)

e~ = At exp(ifl2~t),

(8c)

where Ar

filt~2 cos 0t + fl2~1 cos 0t

At

fl1#2 cos Or -- fl2#1 cos 0t '

At At

fll/~2(cos 0t + cos Or) flt~2 cos Or -- f121~l cos 0t

The boundary conditions are satisfied for all values of y, only when fit sin 05 = --ill sin Or = --fie sin 01. (1 I) Equation (1 1) expresses the common laws of reflection and refraction and equations (6c) to (8c) are Fresnel's equations in a form, consistent with the definition of O's. From (6b), (6c) and (7c) and expressing e~ in terms of ~t we obtain 1, ~ d2s~

2

,,]

A~Ji exp{i(fl-

+ A~Xr exp(i[--fl

cos(0r

2fit)~i} - - 0l) - -

2fll] ~:t}

+ A i f t i A r exp(i[flt -- fit + fit cos(0r -- 0d] ~i} + A i A r A r exp{i[(flt -- fit) cos(0r -- 0k) -- fl~l ~t},

(6d)

M A H E N D R A S. SODHA AND CARL J. PALUMBO

456

1, ,[ d2er ,,~ ,,/ al [ - - ~ - r _t_ =

plea[

= Av4iAr exp{i[(/J1 - - 81) cos(01 - - Or) + 81~ ~r} + A,Areir exp{i[--/31 cos(0, -- Or) + 81 - - f i l l ~r} + A,A~ exp{i[lJ1 cos(0, -- Or) + 2/31! ~r} + A ~ i r exp{i[281 -- ~t] ~r},

: I d<

(7d)

,,/ = A~At exp{iC2flz -- fie cosh(--i(0t --

0t))] ~t}.

(8d)

In the above derivations the angles of incidence a n d reflection have been a s s u m e d to be real. T h e solution of t h e above equations satisfying the b o u n d a r y conditions are given b y el.' = C' exp(--iSl$/) -- C: exp{i(fll -- 281 ) $,} --- C2 exp{i[--fll cos(0r -- 0 d -- 2/31] $~} -- Ca exp{i[(fl: --/31) + 81 cos(0r -- 0,)] $,} -- C4 exp{i[(/3, -- 1~1) cos(0r -- 0 d -- 81] &},

(6e)

u

e, = C" exp(i/31~r) -- C5 exp{i[(fll --/31) cos(0i -- Or) + fill ~r} - - C6 e x p { i ~ - - / 3 1 c o s (0, - -

Or) Jr- 81 -- ~t] ~r}

-- C7 exp{i[~: cos(0, -- Or) + 281] ~r} -- Cs exp{i(2fll -- ~1) &},

(7e)

e; = C" exp(ifl2~t) -- C9 exp{i[2/3e --/~2 cosh(--i(0t -- Ot))l ~t}, (8e) where

~Ai/{(~I - 3/31)(~1 -/31)}, C2 = alA, Ar/{[~: cos(Or - - 0,) q- 3/51]. [/~1 cos(0r -C1 = a i A ,

2

01) q-/31?},

Ca = aiA,d,Ar/{[~: -- 2/31 + 81 cos(0r -- 0,)]" [~1 +/31 cos(0r -- 0,)1}, C4 al ,Arar/{[(fll /~1) COS(0r - - Oi) 2/31] COS(0 r - - 0,)" (81 - - ~1)}, C5 = a~A ,A ,Ar/{[ (~I -- 81) cos(0, -- Or)]" • [(~1-- 8it)cos(0~ -- Or) Jr- 281]}, C6 = aiA,A r~ir/{[fl: cos(0, -- Or) +/111" [81 cos(0, - - Or) @ ~1 - - 281]}, t 2 C7 = ale{iA r / { [ ~ l c o s ( 0 , - - Or) -{- 81]" [ ~ t c o s ( 0 i - - Or) @ 3/31]}, IA

~

c8 = C9 = a 2 A , , ' t t / { f l 2 - - ~ 2

__

8:]' r-B1 + a/31]}, cosh[--i(0, 0,)]}.{382 --/~ cosh[--i(0, -- 6,)]},

+

C' = C: + C2 + C~ + C4,

NONLINEAR REFLECTION AND REFRACTION

C" =

f

Ci[#i -

457

#i] cos Oi

w,

+ C2[-#1 cos(Or -- Od --#1] cos O~

01) -Jr-#1] COS Oi -~- C4[(--#1 -~- ill) cos(Or -- Oi)] cos Oi -~- C5[(fl1--#1) COS(Oi--Or)- (#2ffl COS Ot/tt2 COS Or) -~- #1] COS Or -~- C6[--/~1 COS(O/--Or) --(#2ffl COS Ot/ff2 COS Or)--]~1 -~- [}1] cos Or + C7[#1 cos(O~-@- (#2m cos 0@2 cos Or) + 2#1] COSOr + Cs[2#I - - / ~ -- (#2m cos Ot/ff2 cos Or)] cos Or -~- C3[#1 Cos(O r - -

+ C9 [ # l cos Or -- ff~ (2#2 cos Ot -- #2 qJ) l_

+

'

#2

1}

(#2#lcosOt-#lmCOSOt) {m/E#lmcosOr- thm cosOd},

C m = IC1[#1 -- #1] c o s O / - ~ C2[ - - #1 cos(Or -- Ot) -- #1] cosO/ L

+ Ca[#l cos(Or -- Od + #*] Cos O/ + C4[(#, -- #l) cos(Or -- Od] cos Oi

C5[(#1 -- #1)Cos(Oi -- Or)] cos Or + C6[-fli cos(O/- Or) --#i] cos Or -~ C7[-~-#1 c o s ( O / - - Or) -]- #I] COS Or -~ C8(#1 -- #1) COS Or @

+ C9 [ #icos 0r-- " 1 . (2#2 c o s 0 , - # ~ ) ] 1

•

• {ff2/(/~1#2 COS Or --/~2ffl COS Ot)}.

where ~ois the complex conjugate of cos Or. The second order solutions are also satisfied for all values of y, only when equation (11) is valid. Case II: From Maxwell's equations, remembering our notation we obtain ~v =

s~ =

-

if rio,1

~Jf

(~offo) ~ # 2 E o , 1

~Z

if rio, I

~'~

(~offo)~ #2Eo,i

~Y

(12a) '

(125)

These equations are valid for use in the second order analysis,

458

MAHENDRA

S. S O D H A A N D C A R L J . P A L U M B O

reported in the present communication. Using equations (5) the propagation of the incident reflected and transmitted waves is described by deo~ffI -

-

d~ de3(t°r

-

-

deW, d~

- -

+ fl~Wl = aloe(el + er)'(ei + er) Wl,

(13)

+ fl~lWr = aia(e, + er).(e, + er)'Wr,

(14)

+ ~Wt

(15)

= a~,et"~t'~t.

The second order solution of the above equations can be written as 5/0 _-- og°' + a~f".

(16)

Substituting for o~f from (16) in (13) to (15) and equating the coefficients of like powers of a we obtain

d2W~ - -

deW; -

-

-

2

-

O,

(13a)

2 i + fl13/Zr = 0.

(14a)

d234°; - d~

(15a)

2

R~ ~/.o"

'

'

~'

+ el ~ i = al(e, + e~-). (e, + e;-) W;,

d234Z:

~2~/z.

de~' d~---~t

:

+ fl2Yd, = 0,

d~

de~:"

'

-@ f l l W ,

d~

.

R2~" +

~,2~o

, :

.

.

.

,

, aeet'etWt.

(13b) (14b) (156)

The relevant boundary conditions are Wi+Wr=~,

at

(17a)

Z=0,

ety AV Cry ~ Sty or

(m/~) t-g-g- +

oz I = ~

o~i=B~

at

oz ~=0.

Z = O,

(17b) (17e)

459

NONLINEAR REFLECTION AND REFRACTION

and the appropriate radiation conditions Yfr=0

at

~r----oo,

(17d)

ogf,=O

at

~:t = o o ,

(17e)

and there is no backward component of Jr,.

(17/)

The first order solution satisfying the above boundary conditions is ogt'~ = B, exp(--iflit~d,

(13c)

~ ; = Br exp(iflit~r),

(14c)

~ ; = Bt exp(ifl2$t),

(15c)

where

Br B,

/~2/~tit COS O, "@ /~it~2 COS Ot

fl2/~it cos Or

-

-

flI#2 cos Ot '

fl2#l(COS OI -~ COS Or)

Bt B,

fl2#it cos Or -- flit#2 cos Ot

The boundary conditions are satisfied for all values of y only wtlen fll sin 0i = --ill sin 0r = --f12 sin 0t.

(11)

We can obtain e~ and s: by substituting for ~ ' from (13e) to (15e) in (12a) and (12b). Substituting these expressions for e~, and s: in (13b) to (15b) and simplifying we obtain 1 {d2dq'~ bl

2

"}

d~---T. 2 -4- ~lJ[t° i :

B/2-B, exp{/(~l -- 2fll) ~i}

- B~/~r cos(0, -- Or)exp{/[--2$z --Pit cos(0, -- 0r)~ ~,} -- B,3,Br cos(O, -- Or) exp(i~/~it + fit cos(0, -- Or) -- Bill ~*} + BiBrBr exp(i[(flit -- ~iit) cos(0, -- Or) -- Biit]~e,), b[ [ d ~

+fl2~f;

(22b)

=B~.~,.Brexp{iE(~it__[11)cos(Oi__Or)+fllj~r}

-- B,BrBr cos(0~ -- Or) exp{ii--~ + fl~-- fiit cos(0, -- Or)} ~r} --/~B2~ cos(0, -- Or)exp{i[flit cos(0, -- Or) + 2flI] ~r} + B~/~ exp{i(2fl~ -- ill) #r}, (23e)

1 {d~t b'2 d~

2 ,,} + tS~Jf t = B~Bt cosh{--i(Ot -- 0t)}.

• exp{/[2fl2 -- f12 cosh(--i(Ot -- 0t))] ~:t},

(24e)

460

MAHENDRA S. SODHA AND CARL J. PALUMBO

where

b' = a'#2 H~,I /Eo#oE'~,lfl ft. In the a b o v e d e r i v a t i o n we h a v e a s s u m e d t h a t t h e angles of incidence a n d reflection are real. T h e solution of the a b o v e e q u a t i o n s satisfying t h e b o u n d a r y c o n d i t i o n s given b y e q u a t i o n (26) is ,Y¢~' = K ' exp(--i/51~,) -- K1 exp{i(/~l -- 2fll) ~,} -- K2 exp{i[--fll cos(0r -- 0,) -- 2fll] ~,} -- K s exp{i[(/ll --/51) + fll cos(0r -- 0,)I **} - - K 4 exp{i[(fll - - ill) cos(0r - - 0i) --/511 ~i},

(22/)

.YY: = K" exp(ifllSr) - - K5 exp{i[(~l --/51) cos(0f - - Or) Jv ~1] ~r} -- K6 exp{i[--/~l + 151 --/51 cost0, -- Or)l ~r} -- K7 exp{i[fil cos(Oi--Or) + 2fl1] ~r}--Ks exp{i(2fll--~l) ~:r}, (23/) JY~[ = K " exp(i/52~t) -- K9 exp{iE(fl2 --/J2) " • cosh(--i(Ot -- 6t)) + t52] ~t},

(24/)

where K1 = biB~B,[{(~l -- 3fll)(/~t -- fit)}, K2=

--b{B~t~r/{[~i cos(0r - - 0i) -]- 3/51]" [~icos(0r - - Oi) + fllJ},

K3 ---- --biBJl,er/{Efll--2fl~ + fll CoS(Or--O,)]" [~1 JV/51 COS (Or -- 0,)]}, K 4 = bie,erlJr/{[(fl~ -- ill) cos(0r -- 0,) -- 2/51]" [(ill--ill) cos(0r-- 0,1}, K5 ---- b{BJ~,Br/{[(t~l --/51) cos(0, -- Or) + 2fill (~1 -- /51) COS(0{ -- Or)}, K6 = --biBiBrf3r/{[fll + fll cos(0, -- Or) -- 2/511 " • [~l + / 5 ~ cos(0i - Or)]},

Kv = --biB,B~/{[~l cos(0, - - Or) -~- 3fill" [~1 cos(0/ - - Or) -~- fill}, t 2 Ks : blB, Br/{(3fll --/~1)(/51 --/~1)}, K9 = b~B~l~t cosh[--i(0t -- 6t)~/{[fis -- ~2 cosh(--i(0t -- 6,))1 • • i3fl2 -- ~2 cosh(--i(Ot 0t))]}, -

-

K' - - - - K I + K 2 + K a + K 4 , K" ----- {KI(/J1 -- ill) cos 0i, + K2E--fll cos(Or - - 0~) - - ~ll cos 0~ +

+ K3E~I cos{0r -- 00 + fll~ cos 0~ +

NONLINEAR REFLECTION AND REFRACTION

461

@ K4(fll -- ]~1)" COS(0r -- 0,)" COS 0,

~ 2 COS Ot @ K5 L(~I -- ill) cos(0i -- Or) -- fl2m cos Or + K6 F--/st cos(0, -I_ + K 7 -[/~1 c o s ( 0 , -

Or)

8~m cos Ot /52#1 COS Or

+ fill cos Or

~1 -~- /51] COS Or

Or) -- 8~#2 cos Ot

]

fl2tq cos Or + 2fll cos

+ Ks [2fll -- fJ1-- fl~#2 c°s Ot 1 "/52/tl COS Or

Or

COS Or

+ K9 /51 cos Or - /5~.---7 (2/52 cos 01 - t~ ~) /51 . + h~--T

(/51f12 cos Ot __ f12fll cOS Or)l} ,

8,(/52t'1 cos Or - thin cos

01 '

and

K" = { g,(/~, -/5~) co~0, + g=[--/~ cos(0r -- 0d --th] cos0, t,.

+ K a n t cos(Or -- 0,) +/J1] cos O~ + K4(fll -- ]J1) cos(0r -- 0,) "cos Ol

-~- K5 + K6 + K7 + Ks

(/~1 --/51) cos(0, -- 0r)'COS Or [--th cos(0, -- Or) --#1] cos Or [tJ1 cos(0, -- Or) +/51] cos Or (81 --/~1) cos Or

+ K9 [ 81 cos Or -- 8~--~-2 ¢?~m " (2~ c°s °t - t~2 ~°)]} "

fl1(/~2~1 COS Or --/51J'2 cos Or) "

§ 4. Discussion. It is interesting to note that the laws of reflection and refraction as expressed by Equation (1 l) are valid, also in the present nonlinear treatment. The similarity of the results in Case I and Case II is also as noticeable in the nonlinear terms as it is in the linear terms.

462

NONLINEAR REFLECTION AND REFRACTION

The use of this theory enables reflection measurements at different power levels to yield values of a; and a~ in various materials, which should aid in an understanding of transport processes in various materials. In the linear theory a wave with arbitrary polarization can be resolved into two independent waves with their electric vectors (1) in and (2) perpendicular to, the plane of incidence; the reflected and refracted components can be obtained by superposition of the reflected and refracted components of the two independent waves. This is not valid in the nonlinear theory since the two waves are coupled through the complex conductivity of the medium, which depends on the amplitude of the electric vectors of both the waves. Received 19th December, 1962.

REFERENCES 1/ S o d h a , M. S. and C. J. P a l u m b o , Proc. Phys. Soc. 80 (1962) 1155. 2) G i n z b u r g , V. L. and A. V. G u r e v i c h , Uspekhi Fiz. Nauk. 70 (1960) 201.