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^ I J TY 73 Institute of Mathematical Sciences
:^ ^^^ X Division of Electromagnetic Research
RESEARCH REPORT No. EM153
An Accurate Boundary Condition
to Replace Transition Conditions
at DielectricDielectric Interfaces
JULIUS KANE and SAMUEL N. KARP
Contract No. AF 19(604)5238
MAY, 1 960
^rnVTE dJ^nI?? university
NCV/ YORK UNIVERSITY
INSTHTUTE OF MATHEMATICAL SCIENCES
LIBRARY
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Institute of Mathematical Sciences
Division of ELectromagnetic Research
Research Report No. EM1?3
An Accurate Boiindary Condition to Replace
Transition Conditions at DielectricDielectric Interfaces
Julius Kane and Samuel N. Karp
'=t2^__^
Julius Kane
i^ni^
Samuel N. Karp
Morris Kline Dr. Vfemer Gerbes ^
Director Contract Monitor
The research reported in this document has been sponsored
by the Electronics Research Directorate of the Air Force
Cambridge Research Center, Air Research and Development
Command, under Contract No. AF 19(60^)^238, and by the
American Petrolevm Institute.
Requests for additional copies by Agencies of the Department of Defense,
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Abstract
Most problems involving dielectrics cannot be solved exactly, e.g«,
diffraction by a dielectric wedge. It is the purpose of this work to
describe a technique by which the transition conditions at an interface
between tvzo dielectrics can be conveniently and accurately replaced by a
single mixed boundary condition. The method is useful when one is interested
in the field in only one of the dielectrics* The boundary condition takes
the form
H + Au + B ^ =
^^ as^
where v and s are the surface normal and tangent respectively. When a
single interface is present, so that there is no diffraction, it is proved
that the maximiun percentage error, over the entire far field of an arbitrarily
placets line source is small. For example, we can tabulate the maximum per
centage error for transverse electric excitation as
Index of
refraction = n
1.2
l.h
1.6
1.8
2.0
MaximiOT pet.
error in far field
i4.5
1.1
o.h
0.2
0.1
The percentage error becomes very large for n % 1 because for n = 1
there is no reflection at the interface; however the absolute error remains
small and bounded.
Additional confidence in the technique is acquired by using the boundary
condition to replace a dielectric interface in a nontrivial problem which
does involve diffraction. A comparison of the results of the approximate
version and the exactly formulated problem shows that there is excellent
agreement between the two.
Table of Contents
Pa^
Introduction i
1. Construction of the Poimdary Condition 1
A. Angular Matching S
B. Binomial Matching 5
C. Brewster's Angle Matching 6
D. Chebyshev Hatchings 7
E. Grazing Katcliing 9
F. Generalizations 10
2. Limitations and Remarks 10
3» Existence and Uniqueness 12
A. Existence 12
B, Uniqueness 17
Ii. Validity of t he Approximation 22
Proof of Reciprocity 25
5« Comparison of the Result of the New Method,
with an Exact Solution for a Problem Involving Diffraction 2?
A. Exactly Formulated Problem and Solution 27
B. Approximately Reformiilated Problem and Solution 3U
6. Conclusion hO
Appendix A UOa
Appendix B U2
Appendix C U3
Appendix D US
References hB
 1 
Introduction
Most electromagnetic problems involving more than one dielectric
Medium are not capable of exact solution. An example of such a problem
is sketched in Figure 1, One is interested in obtaining a solution of
the wave equations
(V^ + k^) u(x,y) =0, i = 1, 2, 3
for the indicated geometry when a plane wave is incident in medium I.
It is assumed that the area of prime interest is region I, The boundary
conditions to be satisfied are continuity conditions determined from
Maxwell's equations. As the problem stands, it is not solvable by available
mathematical techniques. However, one notes that the twopart
boundaryvalue problem of Figure 2 is amenable to the WienerHopf
procedure if BC, „ and BC_ are mixed boundary conditions with constant
coefficients.
The following procedure then suggests itself for obtaining an
approximate field in region I:
(1) Replace the dielectric interface between regions I and
II by a linear boundary condition BC  = C, and do the same for the
pair of regions I and III.
(2) Solve the problem of Figure 2 exactly to obtain an
approximate solution to the problem of Figure 1 in region I,
The success of this approach depends upon constructing a boundary
condition to replace a dielectric interface and validating the procedure.
il
In the present report (Part I of two parts) we discuss the formulation
and justification of the new boundary condition, and in a subsequent
report (Part II) we solve the problan of Figure 1 as an illustration
of its use.
k,
— .— = — = — ; ^^aK^wA^'^SA"
E^ r i^^i^^
mp^
^v^»
•^3
m
Figure 1: Idealized ThreeMedia Problem
iii
Figure 2: Reformulated Approximate Version of the Problem
Shown in Figure 1 in Region I.
In section 1 we determine the coefficients in the new boundary
condition by comparing the reflection coefficient it defines to the actual
Fresnel reflection coefficient appropriate to a dielectric surface. W©
present numerical data that shows that there is little error in approximating
the Fresnel coefficient if our methods are eii?)loyed. Certain limitations
and remarks concerning this procedure are presented in Section 2.
Sections 3 and h are devoted to an investigation of the utilization
of the new boundary condition at the interface of a dielectric halfspace.
In these sections it is shown that the introduction of the boundary
condition leads to a problem with a unique solution vrtiich satisfies
±r
reciprocity. Once reciprocity, uniqueness and existence have been
demonstrated we are in a position to assert that this boundary condition
leads to (l) welldefined problems, and (2) guarantees that the eiror in
the farfield of the approximate vei^ion of the standard problem of the
radiating line source above a dielectric interface is small.
Since the problem treated in sections 3 and k involves no diffraction,
confidence in the utilization of the new boundary condition is amplified in
Section 5, in which we compare an exact solution of a diffraction problem
involving two dielectrics and its approximate counterpart. We find
strikingly good agreement between the two results*
 1 
1, Construction of the Boxindaiy Condition
In order to replace a dielectric interface by a linear boundary
condition it is necessary to determine what characterizes the interface
so far as wave propagation is concerned* If one solves the twomedium
problem of a line source above a dielectric interface (Fig. 3) then one
obtains integrals of the form (cf. Bremmer, [8]).
(1)
j R(e) e^
ilq^cos 6
de
in addition to the field that would exist in the absence of the boundary.
In the integral of (l), R(&) is the appropriate Fresnel reflection
coefficient and the contour C is the familiar contour defining the
Hankel function.
G
^ — plane
■>Re«
Figure 3» Contour defining the Hankel function
 2 
The diffracted field represented by (l) then has the interpretation of
a summation of plane waves traveling in all directions, real and
imaginary, which have as a weight factor the Fresnel coefficient defined
in the complex 9plane.
One can also observe that a linear boundary condition at the
interface will define a reflection coefficient if we seek plane wave
solutions. When medium (2) is almost conducting, but only then, it is,
in fact, known that a boundary condition of the form'
u + \u ■
y
represents the phenomena fairly well. Therefore, for a more general
case we seek to include more parameters so as to improve the approximation.
For example, notice that the boundary condition
(=' (i^ * * * ^ ^'"°' ^•°
will generate for the wave equation
(3) (V^ + k^) u(x,y) = , y >
the reflection coefficient
o
cos »  (A B sin &)
(U) R(e) 5— 
cos 9 + (a B sin 9)
where & is the angle of incidence of a plane wave (cf. Fig. i^ )
 3 
"inc"
ik(x sin 9  y cos B)
incident upon the boundary at y ■ .
R u
ref
\ 'k ay K* dx* / ^
Figure U» Plane wave incident and reflected at y  0.
To derive the expression (U) one assumes a geometricsally reflected field
ti   Re
ref
ik(x sin & + y cos &)
and the amplitude coefficient R(e) is determined so as to make the total
field
^inc * ^ Vf
satisfy the boundary condition (2).
 u 
An approximation procedure is then evident: Find a boundary
condition whose reflection coefficient is a suitable approximation to the
appropriate Fresnel reflection coefficient. One is reminded that if u
represents E_ the Fresnel coefficient is ( [$] , p. U92 ff . )
(5)R^
cos e  .jly l(ri] sin^ §
^V^
^i"(^^«) k_/ ,^2 , 2
cos © ♦ rr=' W 1 ( p^ 1 sin 6
vrfiere © is the angle of refraction and k^ is the propagation constant
in the lower media y < 0. If u ■= H then the reflection coefficient is
■^:V^^ '
In (5) and (6) we assume the magnetic peimeabilities li^i^ and n^ to be
identical for convenience only. If we conpare (5) and (6) with (U) then
we see that we can approximate (5) and (6) at least for all real angles
of incidence by choosing A and B properly. To make this selection we note
that if
(7)
'^TE
^2
n
\m 
^1
^2'
1
n
 5 
then the approxijnation (U) will agree exactly with either (?) or (6)
for nonnal incidence (i.e., 9 = O). For glancing incidence (© = n/2),
the expression (U) will agree with (5) or (6) regardless of the choice
of B since both are equal to 1 for 6 » n/2.
A. Angular Matching
We can choose B in many ways to improve the approximation. For
example B may be chosen so as to make the approximation exact for some
angle 9, where < 6. < n/2. There is nothing to suggest that any angle
S is to be preferredj for the balance of Section 1 we shall discuss
possible choices and their motivations. For any choice of 9, the
Bmn. n»# ^^6 determined by matching
TEjiM
B„
1 . "'^;™ sin^ with /(l(l/n)2sin^.
where n = k^/k. is the index of refraction or
(8)
^TE,TM^®1^
^TE,
TM
1 /(l(l/n)^3in\)
sin^e.
B. Binomial Matching
If n > 1 we can expand the radical in (8) by the binomial theorem
to obtain from (7)
(9)
TE
^TM
 6 
Note from (9) that the leading terms of B^t and B_, are independent of
ill wi
e , the fitting angle. For analytical purposes it is very convenient to
choose the B's as just these leading terms
(10)
TE " 2n
'»■ ^
Ihs selection (10) will be referred to as binondal matching . Physically
it corresponds to improving the match of the reflection coefficients in
the neighborhood of 6 => 0, or normal incidence. This is illustrated
in Graph 1 where we plot the percentage error between the reflection
coefficients (5) and (6) and their approximations (U) with binomial match
coefficients for n » 1.6. Note that in the neighborhood of Brewster's
angle 9g , R(Sg) " and, hence, the percentage error (but not the
absolute) for TM approximation is necessarily unbounded. In Graph 2
we plot the absolute errors and we see that the absolute errors in the
vicinity of B remain small. Indeed the absolute errors for both the TE
and IM approximations have about the same values,
C, Brewster's Angle Matching
We can eliminate the unbounded percentage errors for the TM case
In the vicinity of Brewster's angle by choosing B so as to yield a perfect
match at Brewster's angle 9« ■ tan" n in which case (8) becomes
10.
a
6.
5.
4.
3.
 6a 
1.0
.8
.6
.5
.4
.3
.2
.10
.08
.06
.05
.04
.03
.02
.01
/
\
1 1
/
\
/
\
/
\
/
\
"TM
/
\
A
—
\
—
\

\
/
>^^
\
1
\
\
1
^
V \
1
\ \
1
\ 1
/ /^
\ 1
—
/
' /
^TE
\
—
/
' PCT ERROR AS
A
\
—
/
FUNCT ON OF ANGLE
OF INCDENCE FOR
1 1
B
IMUMIAL MAIUMNMb
1
1 1
1
1 /
1
7 .^  ^2 _
«  k \
I
/'
b)
1
/
/
/,
/
—
—
^B
= 58°
—
B
20'
30'
40'
50'
60'
70*
80'
90"
.01
.008
.006
.005
.004
.003
.002
 6b 
\
.00 1
Q^ .0008
O
(Y .0006
Ql .0005
''' .0004
LjJ .0003
.0002
O
CD
CD
<
.0001
.00008
.00006
.00005
.00004
.00003
.00002
.00001
y
\
/
\
/ ^^
— ....^
\
/
/
\
\
//
/
\
\ \
,
v^
\ \
—
//.
k^TE
\ 1
—
r
r
\
—
J
/
Jj
Ik
"TT ^ ■ — '^
//
vv
Tl
VI
//
^F
/
—
1 1
B NOM AL MATCH NG
—
II
—
,
11
1
11
1
I
1
—
/
—
/
1
1= .6

/
0^= 58°
\
20*
30«
40'
50«
60«
70"
80" 90'
 7 
(11)
®TE,TM^*B^ " *TE,TM
n^l
n
V n +1
^ TE.IM
\^ * —
2(n2+l)
Curves 3 and h illustrate the percentage and absolute errors for this
choice of ©^ for n » 1,6.
D. Chebyshev Matchings
Still another cl»ice for B is possible at least for the TE
approximation J this selection is a value of B so chosen as to minimise
the maximum percentage error for the TE reflection coefficient. We note
that the magnitude of the error changes sign for some value 9^
if we choose a value of 9» such that the raaxiimim positive percentage error
equals the maxLmura negative percentage error then we shall have the least
raaxiinum error for the intenral < & < ii/2 by the ChebysheV criterion.
The mathsBiatical analysis to find the optimum Chebyshev matching would be
 7a 
.00006
.00005
.00004
.00003
00002
.0000
o
X
O
OC
(T
LU
h
O
a.
Ld
o
QQ
<
1.0
 7b .

.8
_r
.6
r
,5

.4
^^
■\
.3
/
\
•
.2
^
\
^^
\
.10
—^
^
V
A
^
\
.08
:_,[=
z—
n~
\
A;^
A—
■ /
/
/
\
— ^ —
.06
i
/
f
\
— 1—
05
A
I
04
/\
TM
\ '
.03
> >
\
.02
TE^^
.01
■^
' BRt
EWST
er's
T NG
ANGL
E
1
.008
f
 r 1
006
.005
_(/
.004
n=l.6
,
003
1
.002
.001 L
i
B
— A
10*
20*
30* 40*
50*
60*
70'
80*
90«
 8 
interesting . Here we have done it by a trial and error processj the
values of d_ required to afford this match for the TE case are plotted in
Graph 5, The maxiBiuin associated errors for the reflection coefficient are
plotted in Graph 7 as a function of n. It is not possible to repeat
this procedure for the 1M case owing to the unbounded nature of the
percentage eiror in the vicinity of Brewster's angle, however a Chebyshev
matching is possible for both the TE and TM cases by minimizing the
absolute error, but we have not carried this out.
It seams reasonable to use the boundary condition (U) with
Brewster's angle coefficients for problems of transverse magnetic excitation.
For a problem of transverse electric excitation we can use the values of
©_ in curre (5) in formula (8) to yield a Chebyshev fit. The maximum
percentage errors that arise for real angles of incidence for these two
approximations are plotted for reference in cuirres 6 and ? as a function
of n, the index of refraction.
Even better Chebyshev fittings are possible by relaxing the demand
of a perfect fit at & = Oj this permits choosing both A and B to improve
the match.
60"
CQO
. n» .
09
RfiO
MA'
rCH NG
CHEBYSh
ANGLE f
lEV F T
=?EQU RED
(TE)
R70
Of
•
Ob
\
t^A^
• \
V
Of
•
^
•
^^v.,^^^ «
oo
r " — 
52*
()l®
9 1
1
1
1
1
1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
INDEX OF REFRACTION
100
90
80
70
60
50
40
30
20
10
9
8
7
6
1— k r
1
"^ r~ \ k 1 /"^ ^r* p~ f~
BF
REWSTERb ANbLt
F TT NG TM
1
— \
9
—
—
\^
\
\
—
\
—
\
^
:
"^
I.I
1.2
1.3
1.5
1.6
1.7
i.8
1.9 2.0
n
 8c 
o
cr
LjJ
I
o
X
<
MAX. PCT ERROR
FOR CHEBYSHEV"
FITTING TE
1.5
1.6
1.7
1.8
1.9
2.0
n
 9 
This procedure would yield a perfect fit for four values of 6 over
the interval < 9 < n. This particular matching was not carried out,
owing to the labor involved in doing it empirically, and the excellent fit
of the cruder approximations.
E. Grazing Matching
For certain problems an alternate procedure is available if a
physical inquiry is of such a nature that most of the field of interest
is associated with a particular angle of incidence. An example is a
traismitter and distant receiver — both located near the earth's surface.
The geometry of this problem is such that one would expect most of the
energy arriving at the receiver to be associated with the arrival of rays
in the neighborhood of grazing incidence. For this problem one wo\ild
choose a value of B to match the derivative of the reflection coefficient
at grazing incidence. From (8) we see that the requisite B(n/2) would be
^^^ ^T^,™ ^"/2)  A^^^
v^
 ID 
In subsections AE we have indicated several possible choices of
the coefficients in the B.C. For the balance of the report we shall
assvune that the coefficients have been chosen for a particular problem, and
confine our discussion to the question of what is true for any such choice
of A and B^ unless otherwise indicated.
F. Genera!! izations
Beside the abovementioned methods, other refinements are possible.
Thus any desired degree of accuracy can be obtained by adding additional
terms to the boundary condition: By adding such terms as
9 u 8 u
8x 3x
to the boundary condition (U) we would have generated
sin% , sin 9, ...
in the approximate reflection coefficient. The coefficients of these terms
would then be available, either to match as many terms in a series
expansion of
Vl  if sin^ 9
" n
as needed for superlative accuracy, or for use in a Chebeyshev approximation.
2, Limitations and Remarks
Our approximation procedure is limited by the restriction
k_ > k,, or n > 1, For if k^ < k, then, there arises a real angle of
incidence corresponding to total reflection for irtiich the second radical
 n 
in (1.5) and (1.6) becomes imaginary. Since (l.U) involves constant
coefficients the boundary condition is unable to reproduce the transition
from a real reflection coefficient to a complex one for real angles of
incidence. This Is clear by inspection of the reflection coefficient
(l.U). So long as both A and B are .isaiy the reflection coefficient
remains real for any real angle ©• If A and B are imaginary or complex,
then R(8) will always be complex for any range of 0.
One can also observe fran inspection of the curves 6 and 7 that the
approximation deteriorates for n ^ 1. Physically, this corresponds to an
interface with a reflection coefficient in the vicinity of zero and hence
difficult to approximate on a percentage basis. However, this fact allows
[91
one to use other approximate methods' '. However, the absolute errors
remain small and bounded. As a consequence the boundary condition (l.U)
specialized for n ■ 1 may be useful as a nodel of a perfect absorber since
the reflection coefficient it defines is so small for most of the range.
A curve of this case is drawn in Graph 8. It is unavoidable that the
reflection coefficient goes to 1 for © » n/2. This behavior is forced by
the form of (l.U) which is true even in the exact case if n / 1.
A special feature of the new boundary condition is that it involves a
second derivative in x. In Section h we show that introduction of svich a
term does not violate reciprocity, and the method of demonstration will
show that we can include additional derivatives with respect to x of even
order without sacrificing reciprocity.
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 12 
3. Existence and Uniqueness
A. Existence
In this section we shall obtain an explicit solution to the
following problem
(1) (V^ + k^) u(x,y)  Ui6(x) 6(yh), y >
p = y(x + y ) J © = arctan (y/x)
Once a solution to this problem has been obtained and discussed we shall
show that we have the only such solution. The purpose of this demonstration
and that of Section h in which reciprocity is proved is to' show that the
introduction of the new boundary condition (2) leads to a welldefined
approximate formulation of the standard problem of the radiating line
source above a dielectric interface.
A solution to the above problem can be obtained very simply by
expressing the excitation field u (x,y) by separation of variables, in
G
the form
n\ ^ f ivx + ±)/k^ y  h
(h) u^(x,y) = ^yh^pj •=  i 5 — = dV
e ' o h n ) ix w
irtiere p. = yx + (yh) and expressing the secondary image field u (x,y)
in a similar form, namely,
page 22.
 13 
.iv^?T:3
Here G(v) is to be determined by the condition that the total field (6)
(6) u^(x,y)  u^(x,y) ♦ Ug(x,y)
satisfies the boundary condition at y ■ 0. The contour C is taken in the
complex vplane along the real axis as shown in Fig, 8 which also indicates
the choice of branch cuts for Vk  v . In this section we assume k to
be real unless otherwise mentioned.
If we apply the boundary condition (2) to the combination (6) we
find that if we choose
(7) G(v)
^
■7
 (A
2
^F
v2
♦ (A
then the boundary condition (2) will be satisfied. It will follow frcro