Appl. Sci. Res. 21
S e p t e m b e r 1969
REFLECTION
AND
OF PLANE BY
LAYERED
TRANSMISSION WAVES
PERIODIC
MEDIA
E. M. D E J A G E R Math. Institute Mun. University of Amsterdam, THE NETHERLANDS
and H. L E V I N E Department of Mathematics, Stanford University, Stanford (Calif.), U.S.A.
Abstract The reflection and transmission coefficients (at n o r m a l incidence) for plane parallel slabs consisting of N inhomogeneous and exactly identical parallel layers are expressed in t e r m s of the corresponding coefficients for a single layer. The analysis is effected b y two different methods, one of which stems from t h e so called i n v a r i a n t i m b e d d i n g v i e w p o i n t while the other relies on a m o r e conventional m a t r i x argument.
§ 1. Introduction
Consider a parallel sided slab, of infinite lateral extension, whose inhomogeneous aspect pertains to the normal direction between its faces; suppose t h a t the latter occupy the planes x ~ 0 and x = l, a n d t h a t the wave n u m b e r k(x) for time periodic excitations inside the slab, 0 < x < l, is an arbitrary bounded function with a sectionally continuous derivative. If the wave number takes a constant value k in the exterior of the slab, a wave train t h a t is normally incident on the face x = 0 generally coexists with reflected and t r a n s m i t t e d wave trains in x < 0, x > l, respectively, whose amplit u d e and phase factors are determined by k, l and the profile of k(x). The m a t h e m a t i c a l description of these factors is severely hampered b y a paucity of simple and explicit solutions to the re--
87
--
88
E. M. D E J A G E R A N D H. L E V I N E
duced wave equation ~"(x) -}- k2(x)F(x) = 0
(1.1)
for the scalar wave function 9(x). A fundamental approach to the solution of (1. I ) is due to Bremmer Eli and results in the series representation oo
~(x) = X
~n(x),
(1.2)
n=0
where the initial term ~0(x) takes no cognizance of reflection and ~n(x) refers to the aggregate of n-tuply reflected waves in the slab. In his study of the diffuse reflection of light from a foggy medium Ambarzumian [21 envisaged the addition of a thin surface layer of thickness A with the same physical properties as the interior portions. After pointing out the direct and indirect ways in which a surface layer affects the intensity of reflection and citing the invariance of the reflection pattern at any angle of incidence on a semi infinite medium he obtains in the limit A -+ 0 a useful (although nonlinear) integral equation for the determination of this pattern. When the medium is bounded b y two plane parallel faces the reflection and transmission factors remain unaltered after the joint addition and removal of equal thickness layers at the entrance and exit faces, respectively, and are specified b y simultaneous integral equations. Bellman and Kalaba E3] applied a similar concept in the analysis of plane wave reflection b y and transmission through an inhomogeneous plane parallel slab at normal incidence, and their technique of invariant imbedding has been generalized by MacCallum [4~. To perform the imbedding, one face of the slab, say x = l, is held fixed while the position of the other face, namely x - ~ ~, --oo < ~ < l, remains variable. If the regions x > l, x < ~ are of a homogeneous nature, with the respective values k, k4 = k(~ -- 0) for the wave number, the simultaneous wave functions ,(x) =
e ~k*(x-~) + r(~) e -ik*(x-*), t(~) e ~k(x-Z),
x < x > l,
(1.3)
place in evidence a pair of reflection and transmission coefficients r(~), t(~) which refer to excitation of the inhomogeneous slab (~, l) from the left (x < ~). By calculating the influence of an additional and infinitesimally thin layer at the entrance face these coefficients
PLANE WAVES AND LAYERED PERIODIC MEDIA
89
are found to satisfy the nonlinear, first order differential equations
dr k'(*) -d~ 2k(~)
2ik(~) r(~)
k'(~) - r2(~) 2k(~)
(1.4)
ik( ) lt( )
(1.5)
and
d~
_ E 2k(~)
2k(~)
The integrals of the preceding system which comply with the 'initial' conditions k(l -- 0) -- k r(1) =
2k(l -- 0)
k(l - - O) + k '
t(Z)=
k(l - - O) + k
(1.6)
would evidently furnish values of the coefficients r and t for the slab (0, l) on m a k i n g the assignment ~ = O. The basic equations (1.4) and (1.5) can also be established through the more conventional means of linking the exterior wave functions (i.5) with one appropriate inside the slab; it suffices, in fact, to represent the latter formally, viz., ~ < x < l
~9(X) = Clq~l(X) 2V C2~O2(X),
(1.7)
where ~01(x), F2(x) are independent solutions of the second order equation (1.1) having the properties ~l(l) = 1,
~(l) = 0
~2(/) -- 0,
~(1) = 1.
(1.8)
A definite proportionality and individual characterization
1
cl = t(~) = ~ -
c2
for the constants in the representation (1.7) is implied by the necessary continuity of ~ and its derivative .~' at x = l, and imposition of these same requirements at x = ~ yields the relations
ik~(1 -- r(~))] = \V{(~)
V~(~)J\ikt(~)J'
which are linear in r(~), t(~). The fact t h a t 91 and 92 are solutions of the differential equation (1.1) permits their elimination from (1.9) and the system (1.4), (1.5) is obtained when this is accomplished.
90
E. M. DE J A G E R AND H. L E V I N E
Similar methods of analysis are available for specifying the reflection and transmission coefficients r~, t2v of a plane parallel slab which consists of N identical and parallel layers; and they furnish, in particular, expressions for r~, tlv in terms of the coefficients rl~, tl, characteristic of a single layer under excitation from the left (+) or right (--) sides, respectively. One of these, henceforth designated as a method of imbedding and presented in § 2, leads to such connections from a pair of recurrence (rather than differential) relations that are consequent to the introduction of a single layer in front of or behind the multi-unit slab. The other, a matrix method detailed in § 3, makes sequential use of relations akin to (1.9) for each layer along with the expressions for 9~(0), ~i'(0) in terms of r~, tl. Whichever method be chosen, there is an ultimate accord in the presentation of the scattering coefficients, viz., r~ = Flv(~) rl~ =
Tlv-l(~) rl~ TN-I(~) + tlTN-2(~)
(1.10)
and
tN =
GN(~')tl
(-- 1)N-1 =
TN-I(~) +
tlTN-2(~)
tl,
(t+ = tN)
(I. I 1)
with sin(N -~ I) sin ~
T2v(~) -~
(1.12)
and -
-
cos ~ =
1
2tl
(1.13)
It appears that the essential parameter which determines the structure factors F2v, G2v is r+l r;
-
-
I
2tl and, moreover, that these are truly independent of the propagation direction of the primary wave. The expressions (1.10), (1.11) have been derived previously by Jacobsson [5] with recourse to the matrix technique, although the counterpart of the definition (1.13) given, namely COS ~" =
1 ( ~ 1 ( 0 ) -]- ~ ( 0 ) )
PLANE WAVES AND LAYERED PERIODIC MEDIA
91
does not underscore the fact that r~, tN m a y be simply calculated once the corresponding coefficients r~, tl for a single layer are known; nevertheless, some interesting features of the scattering from slabs with layered sections of general wave n u m b e r or refractive index variation are brought out in his paper. § 2. Calculation of reflection and transmission coefficients by the method of imbedding
We consider a parallel sided slab, containing n identical layers or subunits of equal thickness/, which is imbedded in a homogeneous medium with constant wave number k. For a plane wave e ikz normally incident on the left hand face (x = 0) of the slab, the external wave functions can be expressed in the form
Cn(x) =
e i/cx @ r n+ e - i k z ,
x ~ 0
t n e ik(x-nl)
X ~
•[
n = 1, 2,
(2.1}
"'"
'
The equations (2. l) define a pair of secondary wave coefficients + tn. Moreover, with a plane r~, w a v e e - i k ( x - n l ) incident on the right h a n d face (x = nl) the representations In e -ikx,
•n(X)
e -ik(x-n/) + r j e ik(x-nl)
x ~ 0 x > nl
n = 1, 2 . . . .
(2.2)
allow for an a s y m m e t r y in reflection b y the slab (r~ ¢ r +) while making explicit the equality of transmission. In order to obtain recurrence relations for the coefficients r~± and tn we regard an aggregate of (n + l) identical and parallel units as a composite slab consisting of n adjoining layers together with a single b o u n d a r y layer. If the latter faces the incident wave (Fig. 1), contributions to the reflected field arise from: i) a wave 9(0)(x) which is directly reflected b y the single layer
(0, z), if) an infinite number of waves ~(m)(x) (m > 1) which pass through the single layer (0, l) and undergo m consecutive reflections from the slab (l, (n-}- 1)l), along with ( m - 1) reflections from the single layer before passing out again through the latter. Applying the representations (2.1), (2.2) it follows that co
" n + 1~'~-ilcx ---- E ''+
m=0
~o
~ ( m ) ( x ) = ~1+ e - i k x ~-
~ tx(~'n+)m (T1)m-1 ~1 e - i k x , m= 1
x<0
92
t~. M. DE J A G E R AND H. L E V I N E
o
(n~)t
,~
7-
7
,(?)N
43)(x)
gll,i Fig. 1. T h e n a d j o i n i n g l a y e r s t o g e t h e r w i t h a s i n g l e b o u n d a r y l a y e r f a c i n g the incident wave.
and hence + r+
n+l
=
+ +
1 --
'n
r~r
+
'
n
:
(2.3)
1 .....
The transmitted field can be synthesized with component wave functions ~(m)(x) (m = O, 1, ...), indicative of passage through both the single layer and the slab along with m-fold internal reflection; in explicit terms oo
tn+l e ik(x-(n+l)l) ~-
oo
~_~ qS(m)(x) = m=O
E t l ( r + r l ) m tn eik(x-(n+l)! ), m=O
x>_(n+
1)/
and consequently
tn+l-
tltn 1 -- Yn+ f l- '
n = 1. . . . .
(2.4)
The relations (2.3), (2.4) afford, in principle, the means of expressing r + and tN in terms of the single layer scattering parameters rl~ and h. However, before proceeding to obtain the requisite forms of expression, we m a y take note of the analogous relations which stem from the viewpoint that the single layer lies at the back of the composite slab (Fig. 2). A superposition of reflected wave components appropriate to
PLANE
WAVES
AND LAYERED
PERIODIC
MEDIA
93
n~ n[ (n4-1)[
jK×
/" J
~~-(~I(x) ~,~-(21(x) )~(!(x) ~gIj~(x)
"; -"7
Fig. 2. The n adjoining layers together with a single boundary layer at the back of the composite slab. multiple scattering in this t w o - p a r t s t r u c t u r e yields oo
rn+l+ e-ikx----
~(m)(x) = r n + e-ikx @ Y~ tn(r+) m (r~) m-1 t n e -i~x,
X m=O
~'n,= 1
x<0 or ;v+
+
2
Yl+
n+l = r,~ + t,~ 1 _ r [ r ~
'
(2.5)
n = 1, . . . .
F u r t h e r m o r e , the t r a n s m i t t e d field is given b y oo
tn+l e ik(x-(n+l)z) =
Z
oo
~(m)(x) =
Z
m=O
tn(r+rX) m tl e ik(x-(n+l)l),
m=O
x~(n+
1)/
and hence tn+l
tltn =
14 ~
I -
r
1,
....
(2.6)
r; '
Comparing this with (2.4) we find
r; r+,
(2.7)
and if this result is t a k e n in conjunction with a forthcoming repre-
94
E. M. DE J A G E R A N D H. L E V I N E
sentation, viz. r + = F N ( r +, r l , h ) r +,
(2.8)
r ~ = F N ( r +, r ~ , tl) ri-
(2.9)
the relation
is obtained which reveals the structure factor Fly to be independent of the direction of incidence of the p r i m a r y wave. The determination of r + is facilitated b y writing the recurrence relation (2.3) in the form 1
an+l = c~ - - - - - ,
(2.1 O)
an
where an --
r{r + -- 1
tl
,
n ----- 1, 2 . . . .
(2.11)
and r+ r ; - - t~ - - 1
(2.12)
=
tl
A r e a d y consequence of (2.10) is the continued fraction representation (see [61)
aN ~- ot
Ii
1[
I°~
I°t
...
II lal
-
AN AN-1
,
N
>
1
(2.13)
for a slab consisting of N single layers, where the n × n d e t e r m i n a n t a --I
--I a
--i
--I
~
--I
(2.14)
An=
-- 1
al
has nonzero elements on the principal and bordering diagonals. If the expansion of A n according to minors of the elements in the last row is employed in (2.13) it appears t h a t aN =
alBN-1 -- BN-2 alB2v-2 - - B~v-3 '
N > 1
(2.15)
PLANE WAVES AND LAYERED PERIODIC MEDIA
95
with an n × n d e t e r m i n a n t c~
--1
--1
a
--1 n > 1 (2.16)
t h a t has the initial values B0 = I, B-1 = O. The latter defines a polynomial in a of degree n and satisfies the recurrence relation
n > 0
Bn(o~) = ~B,~-I(~) - - B,,_~(~),
(2.17)
which, upon the substitution c~ = 2 cos ~, takes the form of a Chebyshev polynomial T n ( $ ) . Hence, Bn(a) = Tn($)
de f
sin(n + 1) ¢ sin ~
,
n >
(2.18)
1
and aN =
alT2v-l(~) -- TN-2(g) alTN-2(¢)
(2.19)
TN-a(¢)
--
I n a s m u c h as al = ~ + tl, it is a simple m a t t e r to replace al with h in (2.19) to obtain T2v(~) q- tlTN-I(~) aN
=
TN-I($) q- tlTN-2(~)
Finally, after eliminating a~v in favor of r + t h r o u g h (2.11) we find r~ rl+
.
r~ r~-
.
TN-I($)
.
.
TN-I(¢) -~ tlT2v-2(¢)
F~v(~),
where 2 cos ~ =
r+l r ~ - - t~i - - 1
tl
in c o n f o r m i t y with (1.10), (1.13). The relations t)v tN-1
tl 1 --r~r~_
1 1
aN-1
T2/-2(~) -~- t l T N - 3 ( ~ )
TN-I(;)
÷ hTN-2(;)
'
(2.21)
96
E. M. DE JAGER AND H. LEVINE
which follow from (2.4), (2.11), and (2.20) yield
tN
tN
tl
tN-1
t2
(-- 1)iv-1
tN--1 tN-2 tl (-- A)N-1
al a2 ... aiy-1
=
=
GNg),
Tiv-l(~) 4- tlTN-2(~) and thus substantiate the characterization (1.11) of the transmission coefficient. § 3. Calculation of reflection a n d t r a n s m i s s i o n coefficients by the m a t r i x method
Let the wave function ~(x), appropriate to any state of excitation within a single layer (0, l), be represented in the form --
c l(x) +
where 91(x), 92(x) are particular solutions of the equation (1.1), that have the properties (1.8) at x = 1. If the exterior wave functions are indicative of an incoming or primary wave on the left hand side of the layer and find their expressions in the versions of (2.1) which correspond to n = 1, a continuous fit of ~s(x) and 9'(x) at the respective boundaries of the layer is implied by the two equations jointly contained in the matrix scheme
1 4-r + ']_(~i(O) ik(1 -- r + ) /
~2(O))(tl'~
(3.1)
~2(O)]\ikt,/"
\~i(0)
From (3.1) and the analogous relationships (
tl ) = ( ~ 1 ( 0 ) --iktl \91(0)
~2(0)~( l d-ri ~ ~z(O)J\--ik(1 -- r~)J
(3.2)
that pertain to incidence from the right (and thus involve the wave function k~l(X) given by (2.2)) we obtain explicit characterizations for ~sl(0), 92(0), qs{(0), and 9~(0) in terms of rl+, r~, and tl, namely 1 4-r+1 - - r ~ - - r + r ~ +t~ 2t~ =
2(o) =
~i(O)
1 4- r + 4- r~ -]- r + r l -- t~
= ik
2ikE]
2/1
(3.3)
97
PLANE WAVES AND LAYERED PERIODIC MEDIA
9
(o) =
1 --r+~ + r 1 - r + i r l q-t~ 2tl
For the case of a slab containing N identical layers the matrix (3.1) or (3.2) relates the incoming and outgoing wave amplitudes the extremities of every layer and, accordingly, the N-th power this matrix links the overall scattering coefficients in the exterior the slab. The consequent generalization of (3.1), (
92(0)~N( tN "1 9~(0)] \iktN/'
1 -I- r~ "~= (91(0) ik(l -- r~,)] \91(0)
in at of of
(3.4)
can be recast in the form
r~v)]-(91(0) TN-I(~) -- TN-s(*~)
ik(1 --
\
91(o)
9~(0) rN-l(rl) --
TN-S(n)) \ikt~v] (3.5)
where =
sin(n q- 1) ,~ sin
and 2 COS ~ =
9 1 ( 0 ) -}- 9 ~ ( 0 ) =
1 -
l+rl +
tl
after evaluating the N-th power of the (unit determinant) matrix with the aid of mathematical induction. On substitution in (3.5) for 91(0), 92(0), 91(0), and 9~(0) from (3.3) we recover the expressions (1.10), (1.11). Received 7 August 1968 REFERENCES [l]
[2] [3] [4] [5] [6]
BREMMER, H., The W.K.B. approximation as the first term of a geometric-optical series, The Theory of Electromagnetic Waves, a symposium, Interscience, New York 1951. AMBARZUMIAN,V., Compt. Rend. Ac. Sci. U.S.S.R. 38 (1943) 229. BELL~IAN, R. a n d R. KALABA, J. Math & Mech. 8 (1959) 683. MACCALLUM, C. J., I n v a r i a n t imbedding and wave propagation in inhomogeneous media, Res. rep. S.C.-4669 (1962) Seandia Corp., S a n t a Monica (Cal.). JAcoBsso~¢, R., Arkiv Fysik 31 (1966) 191. PERRON, O., Die Lehre der Kcttenbrfiche, Band 1, Teubner Verlag, Leipzig 1954.