Uniform asymptotic theory of edge diffraction
Citation for published version (APA):
Lewis, R. M., & Boersma, J. (1969). Uniform asymptotic theory of edge diffraction.
Journal of Mathematical
Physics
,
10
(12), 2291-2305. https://doi.org/10.1063/1.1664835
DOI:
10.1063/1.1664835
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Published: 01/01/1969
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JOURNAL
OF
MATHEMATICAL
PHYSICS
VOLUME
10,
NUMBER
12
DECEMBER
1969
Uniform Asymptotic Theory
of
Edge Diffraction *
ROBERT
M.
LEWISt
AND
JOHANNES
BoERSMA
Courant Institute
of
Mathematical Sciences, New York University
and
Department
of
Mathematics, Technological University, Eindhoven,The Netherlands
(Received
28
September 1966)
Geometrical optics fails to account for the phenomenon
of
diffraction, i.e., the existence
of
nonzero
fields in the geometrical shadow. Keller's geometrical theory
of
diffraction accounts for this phenomenon
by providing correction terms to the geometrical optics field, in the form
of
a high-frequency asymptotic
expansion.
In
problems involving screens with apertures, this asymptotic expansion fails
at
the edge
of
the
screen
and
on
shadow boundaries where the expansion has singularities. The uniform asymptotic theory
presented here provides a new asymptotic solution
of
the diffraction problem which is uniformly valid
near edges and shadow boundaries. Away from these regions the solution reduces to that
of
Keller's
theory. However, singularities
at
any caustics other than the edge are
not
corrected.
1. INTRODUCTION
Geometrical optics fails to accpunt for the phenom-
enon
of
diffraction, i.e., the existence
of
nonzero
fields in the geometrical shadow.
It is now known that
the geometrical-optics field corresponds to the leading
term
of
a high-frequency asymptotic expansion
of
the
solution
of
a boundary-value problem for the reduced
wave equation
or
Maxwell's equations, and that higher-
order terms account for diffraction. Keller's
"geo-
metrical theory
of
diffraction"
1.2
provides a systematic
means for computing these terms.
In
this paper
we
consider problems
of
diffraction
by screens. The screens may be portions
of
planes or
other smooth surfaces bounded by smooth curves,
and the prescribed incident wave may be arbitrary.
We consider here only scalar problems for the reduced
wave equation with boundary conditions
of
the first
or
second kind
(u
= 0 or
au/an
= 0) on the screen.
In
Sec. 2
we
present a brief but self-contained treat-
ment
of
Keller's geometrical theory for such problems.
This theory depends on a
"diffraction coefficient" the
value
of
which is obtained from a special ("canonical")
problem, the problem
of
diffraction
of
a plane wave
by a half plane. Sommerfeld's solution
of
this problem
is discussed in
Sec.
3 and there the diffraction coeffi-
cient is evaluated.
The geometrical theory has several shortcomings.
It
fails at the shadow boundaries
of
the incident and
reflected waves as well
as
at the edge
of
the screen
where the
"diffracted wave" becomes infinite. Further-
more, it
is
difficult to justify the determination
of
the
•
The
research in this
paper
was supported
by
the Air Force
Cambridge Research Laboratories, Office
of
Aerospace Research,
under
Contract No.
AF
19(628)3868. Reproduction in whole
or
in
part
is permitted for any purpose
of
the U.S. Government.
t
R.
M. Lewis died
on
7 November 1968.
1 J.
B.
Keller, J. Opt. Soc. Am.
52,116
(1962).
• R. M. Lewis and J. B. Keller, New
York
University Research
Report EM-194, 1964.
diffraction coefficient by comparison with the solution
of
the canonical problem, and this procedure cannot
be generalized to yield higher-order terms in the
diffracted field. These shortcomings are overcome by
the method presented in
Secs.
4 and 5
of
this paper.
Other shortcomings
of
the geometrical theory (the
failure at caustics
of
the problem) remain. Like
Keller's theory, ours is formal in the sense
that
we
do
not rigorously prove the asymptotic nature
of
the
solution obtained.
Our approach is motivated by a
new
representation
of
the solution
of
the half-plane problem.
By
using
simple concepts
of
the geometrical theory such
as
incident-, reflected-, and diffracted-phase functions,
we
show in Sec. 3 that Sommerfeld's solution can be
expressed in a remarkably simple and suggestive form.
This representation involves a special
function/which
is discussed briefly in Appendix
A.
It
is closely related
to the Fresnel integral functions.
The geometrical theory
of
diffraction is based on an
"ansatz" in the form
of
an asymptotic series involving
certain
"phase"
and "amplitude" functions.
By
inserting the series into the reduced wave equation,
one obtains the eikonal equation for the phase
function sex) and a sequence
of
transport equations
for the amplitude functions
zm(x).
These equations
can be solved by introducing lines in x-space called
"rays." Our approach is based on a new ansatz that
involves the function
f Away from the edge
of
the
screen and the
&hadow
boundaries, the
new
expression
reduces to one
of
the same form as Keller used.
Therefore the phase and amplitude functions which
appear in the new ansatz also satisfy the eikonal and
transport equations. In Keller's theory there is an
undetermined
"initial condition" for the transport
equation
of
order zero. This leads to the diffraction
coefficient.
In
our approach the initial condition is
2291
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2292
R.
M.
LEWIS
AND
J.
BOERSMA
uniquely determined by imposing the "edge condition,"
which is a
part
of
the rigorous formulation
of
the
boundary-value problem. Away from the edge and
shadow boundaries, the leading term
of
our result
reduces to Keller's, and
we
verify his expression for
the diffraction coefficient.
By
construction our solution is continuous and
finite
at
the edge
of
the screen because the edge
condition demands this.
It
is not immediately obvious
that
it is also continuous
at
the shadow boundaries.
However, in
Sec.
4
we
compute the leading term
of
our
expansion and prove that it is continuous
at
the
shadow boundaries as well as
at
the edge. (For this
reason
we
call our asymptotic solution "uniform.")
The generalization
of
this theorem to higher-order
terms has not yet been proved.
In
Sec. 5
we
compute
the next term
of
our expansion.
In
order to simplify
the calculations
we
restrict the problem
at
this point
to screens which are portions
of
planes. The com-
putation requires an expression for the Laplacian in
"ray
coordinates" which are not orthogonal. This
expression is derived in Appendix
C.
When our
result
is
evaluated away from the edge and shadow
boundaries, it again reduces to an expression
of
the
form used in the geometrical theory,
but
now the first
two terms
of
the diffracted wave are given. The second
term can be expressed in a form that involves Keller's
diffraction coefficient and a new coefficient. There
is
a
special problem (grazing incidence with boundary
condition
au/an
= 0) in which Keller's diffraction
coefficient vanishes and the second term becomes
important.
For
this. case Keller has obtained a special
diffraction coefficient by using a special canonical
problem. In this case our new coefficient reduces to
his.
In
several respects our theory is incomplete.
We
have already mentioned the unproved conjecture that
all terms are continuous
at
the shadow boundary.
There is a second unproved conjecture:
We
have
obtained the first two terms
of
the expansion,
at
least
for plane screens. (This
is
probably not an essential
restriction.)
It
seems likely that the procedure can be
continued to yield higher-order terms. But this too
is
not
obvious and has not yet been proved. [Note added
in
proof
Both conjectures were proved recently;
see
D.
S.
Ahluwalia, R. M. Lewis, and
J.
Boersma,
SIAM J. Appl. Math. 16,
703
(1968).] Furthermore,
as
we
have mentioned, our theory also fails
at
caustics
of
the incident and reflected waves and caustic points
of
the diffracted wave other than those on the edge.
Our
theory is also incomplete in another sense.
For
nonplanar screens, diffracted rays emanating from
the edge may strike another portion
of
the screen
giving rise
to
secondary reflected waves or creeping
waves. Such waves are not included in our theory.
(See the remarks
at
the end
of
Sec. 4.)
Uniform expansions which are valid
at
caustics
have recently been obtained by Kravtsov
3
and
Ludwig.4
In
fact, their work partially motivated our
approach to the problem
of
diffraction by screens. A
second motivation came from the work
of
Lewis
5
on
the uniform transition from the "forerunner" to the
"main
signal"
of
a transient wave propagating into a
dispersive medium.
The main motivation, however, came from the
recent work
of
Wolfe.
6
Wolfe considered some special
cases
of
the problems treated here, involving plane
and spherical waves incident on a screen which is a
portion
of
a plane.
For
these problems he obtained
uniform asymptotic solutions by means
of
an ansatz
involving Fresnel integrals. This ansatz, which was
given in terms
of
ray coordinates, was substituted
into the reduced wave equation which had to be
transformed
to
these same coordinates. This obscures
several important features
of
the method.
For
ex-
ample, one does
not
see
that the ansatz involves
functions that are identical to the phase and amplitude
functions
of
the geometrical theory. As a consequence
Wolfe's method is more complicated than ours.
In
addition, Wolfe relies on the use
of
the canonical
half-plane problem, since the Fresnel-integral part
of
his ansatz is derived from the uniform asymptotic
expansion
of
the solution
of
the half-plane problem
for the same incident wave. Since this problem has
been solved only for special incident waves (plane,
cylindrical, and spherical), this restricts the generality
of
his method. Nevertheless the essential features
of
our approach are contained in Wolfe's work
and
we
are very much indebted to him.
We
are
of
course also
greatly indebted to Keller, not only for his geometrical
theory
of
diffraction, but also for his continuing
interest and advice in the course
of
Wolfe's work and
our own.
In
closing this introduction
we
wish to mention
some problems closely related to the one treated here.
The problem
of
diffraction by a screen is a special
case
of
diffraction by objects which are locally wedge-
shaped. (Along the edge the screen
is
locally a zero-
angled wedge.) Such problems can be treated by
Keller's theory. The generalization
of
our method to
these problems is currently under consideration. There
3
Yu.
A.
Kravtsov,
Radiofiz.
7,
664 (1964).
4 D. Ludwig,
Commun.
Pure
Appl.
Math.
19,215
(1966).
•
R.
M. Lewis, Proceedings
of
the
U.R.S.I.
Symposium
on Electro-
magnetic Wave Theory (Delft,
The
Netherlands,
1965).
6 P. Wolfe,
"Diffraction
of
a
Scalar
Wave
by a
Plane
Screen,"
Ph.D.
thesis,
New
York
University, 1965.
Downloaded 01 Oct 2011 to 131.155.197.179. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/about/rights_and_permissions
UNIFORM
ASYMPTOTIC
THEORY
OF
EDGE
DIFFRACTION
2293
is also a fairly obvious generalization
of
our
approach
to problems
of
diffraction by screens in inhomogeneous
media. We have
not
included a treatment
of
such
problems because the
added
complications are
not
justified by the practical importance
of
the generaliza-
tion.
In
addition
it
is fairly clear
that
the method
presented here can be applied to Maxwell's equations
and
other
linear partial-differential equations,
but
this
has
not
yet been done.
Many
diffraction problems
(e.g., diffraction by a slit
or
by a circular aperture in a
plane screen) involve
"multiple diffraction" (waves
produced
at
one edge are incident
on
another). Such
problems have been treated by Keller and will be
treated by
our
method in a forthcoming sequel to this
paper. Finally there is a whole class
of
problems
of
diffraction by smooth objects
that
can
be treated by
another
part
of
Keller's theory. Recently uniform
asymptotic solutions
of
these problems have been
obtained.
7
•
8
These solutions improve
on
Keller's
theory in much the same way as the method presented
here improves
on
his theory
of
edge diffraction.
2.
KELLER'S GEOMETRICAL THEORY
OF
DIFFRACTION
In
this section
we
present a summary
of
that
part
of
Keller's theory which relates to diffraction by
an
edge
of
a screen.
Further
details are given in References
1
and
2.
It
is important for us to summarize Keller's
theory
not
only because
our
work was motivated by it,
but
because
we
make heavy use
of
his results.
In
Secs. 4
and
5,
we
use almost all the equations derived
here.
We consider asymptotic solutions
of
the reduced
wave equation
(2.1)
of
the form
00
u,-.;
eikS(x)
2 (ik)-mzm(x), k
--
00.
(2.2)
m=O
By
inserting (2.2) into (2.1), we find
that
the phase
function
sex) satisfies the eikonal equation
of
geo-
metrical optics
(Vs)2 =
1,
(2.3)
while the
amplitude functions zm(x) satisfy the recursive
system
of
transport equations
2Vs·
VZm
+
zm~s
=
-~Zm_1;
m =
0,1,2,'"
,
Z_l
==
O.
(2.4)
Solution
of
(2.3) may be described as follows: Given
a surface
(wavefront)
on
which s has the constant value
,
R.
M.
Lewis,
N.
Bleistein,
and
D. Ludwig,
Commun.
Pure Appl.
Math.
20,295
(1967).
8
D.
Ludwig,
Commun.
Pure Appl.
Math.
20,
103
(1967).
so'
we
introduce the two-parameter family
of
straight
lines
(rays) orthogonal to the surface.
If
a denotes
distance along the rays from the wavefront (measured
positively in the direction
of
increasing s), then
on
each
ray
s =
So
+
a.
(2.5)
It
is then clear
that
(2.5) satisfies (2.3).
Let
a
2
and
aa
be the two parameters
that
label the
rays
and
let us describe a ray parametrically
in
the
form
x
=
x(a)
=
x(a,
a
2
,
aa). (2.6)
If
we
set a =
aI,
then (2.6) defines a transformation
from
(aI'
a
2
,
aa)-space
to
(Xl>
X
2
, xa)-space
and
the
Jacobian
of
the transformation is
j = j(a) = j(a, a
2
, (
3
)
=
det
(OXi) ,
i,j
=
1,2,3
.
oa
j
(2.7)
For
given
Zm-l
it
is easy to see
that
(2.4) is
an
ordinary
differential equation for
Zm
along a ray. The solution
can be expressed in the form
zm(a) =
Ij(a
o
)
I!Zm(a
o
)
-!
f"
Ij(a')
1!~Zm_l(a')
da',
j(a)
2
)"0
j(a)
m = 0,
1,2,
..
'.
(2.8)
Here
zm(a) = zm[x(a, a
2
,
aa)]
is the value
of
Zm
at
a
point
a
on
a given ray. The solution (2.8) is given
in
terms
of
an
"initial value"
Zm
(a
o
)
at
some fixed
point
on
each ray.
For
m = 0
we
note
that
the second term
of
(2.8) is absent because
Z_l
==
O.
Two alternative
expressions for the ratio
of
Jacobians are sometimes
useful:
j(
a
o
) da( a
o
)
(P2
+
ao)(Pa
+ a
o
)
--=
--=
j(
a) da( a)
(P2
+
a)(P3
+ a)
(2.9)
Here
da(
a)
is the cross-sectional area
of
an
infinites-
imal tube
of
rays, while
P2(
a
2
, (
3
)
and
PaC
a
2
, (
3
)
are
the principal radii
of
curvature
of
the wavefront
a =
O.
At
the two points a = -
P2
and
a = -
P3
on
each
ray,
we
see
from
(2.9)
that
Zm
becomes infinite
and
the
integral
in
(2.8) will, in general, diverge. Such points
are called
caustic points. They lie
on
the caustic, which
is, in general, a two-sheeted surface forming the
envelope
of
the family
of
rays (the rays are tangent to
the caustic).
We
shall require
an
alternative
form
of
(2.8) which remains valid when a
o
= 0 is a caustic
point. First we rewrite (2.8) in the form
ll(a)l!
zm(a) = U(ao)l! zm(a
o
)
- t
r"ll(a')I~
~Zm_l(a')
da'.
(2.10)
)"0
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2294
R.
M.
LEWIS
AND
J.
BOERSMA
Then
we
express the integral in (2.10) in the
form
(2.11)
Here the dash denotes the
"finite
part"
9
of
a divergent
integral. (The ordinary integrals would diverge
at
a' = 0.)
Now
(2.10) becomes
li(a)l! zm(a) +
t£'
=
Ii(a
o
)I! zm(a
o
) +
i£'o.
(2.12)
The right side
of
(2.12)
is independent
of
a.
If
we
denote its value by
~m'
then
we
obtain
zin(a) =
~
-!
["
\j(a')
\!Azm_1(a')
da',
Ii(
a)l!
2
Jo
j(
a)
m = 0,
1,2,
' .
'.
(2.13)
This is the required modification
of
(2.S). The "initial
value"
~m(a2'
as) has first
to
be determined before
(2.13) is useful. We will see in Sec. 5
that
the finite-
part
integrals are a useful computational tool.
For
m = 0, the integral in (2.13)
is
again absent.
We
now consider the problem
of
diffraction by a
screen
S. The screen is a portion
of
a smooth surface.
It
is bounded by an
edge
E consisting
of
a smooth
curve
(2.14)
Here
'fJ
is
an
arclength parameter.
For
example, S
might be an infinite plane with a circular aperture
or
it
could be the complementary disk. Alternatively the
aperture may have any smooth shape.
In
general, S
need
not
be a portion
of
a plane. We consider
an
incident
wave,
00
u~
'"'-'
e
ikSI
I
(ik)-mz~,
(2.15)
m=O
which is
an
asymptotic solution
of
(2.1). Then
si
and
the
z~
satisfy the equations derived above. The total
field
u
is
a solution
of
(2.1)
and
satisfies a boundary
condition on the screen. We shall consider
simultane-
ously the two conditions
and
u = 0
on
S
au
- = N . Vu = °
on
S.
on
(2.16a)
(2. 16b)
Here N is a unit normal vector
on
S.
In
addition,
u -
u~
is required to be "outgoing."
To
solve the diffraction problem, we first set
9
Let!(E)
=
S~
g(x)
dx
have
an
asymptotic
expansion
in
(perhaps
fractional)
powers
of
E for E
->-
O.
The
coefficient
of
EO
= 1 in
the
expansion
is
called
the
finite part
of
the
integral
and
will
be
denoted
by
fo
g(x)
dx.
u =
~
+
u~.
We assume
that
the
reflected
wave
u~
has an asymptotic expansion
Then
(2.16) will be satisfied, provided
sr
=
Si
on
S
and,
for the
boundary
condition u =
0,
(2.17)
(2.1S)
z~
=
-z~
on
S;
m = 0,
1,2,
..
'.
(2.19)
For
the case
aulon
= 0, (2.19) is replaced by
::l i ::l r ::l i ::l r
Zi
~
+
zr
~
+
UZm_l
+
uZm_l
= 0
on
S,
man m
an
an
an
m = 0,
1,2,
..
'.
(2.20)
It
can be shown
that
(2.1S) implies
that
the inc;.dent
and reflected rays (which have the direction
VSI
and
Vsr,
respectively) satisfy the law
of
reflection
of
geometrical optics.
If
"p
is the angle
of
incidence
(=
angle
of
reflection), then
asrlan
= cos
"p
=
-asilan
and
(2.20) becomes
Z~
=
z~
__
1_(aZ~_l
+
aZ~_l)
on
S,
cOS"p
an an
m = 0,
1,2,
..
'.
(2.21)
Thus
sr
is determined
on
the reflected rays by (2.1S)
and
(2.5), while the functions
z~
are given by (2.S)
with zm(a
o
)
determined by (2.19)
or
(2.21). We note
that
both
u~
and
u~
are zero in their respective
"shadow
regions," i.e., where there are
no
incident
or
reflected
rays. Thus each has
an
"illuminated region" separated
from the corresponding shadow region by a
shadow
boundary
surface.
The leading term
u =
u~
+
lfo
'"'-'
z~
exp (iksi) +
z~
exp (iksr) is the geometrical-optics solution
of
the
problem, which
of
course fails to account for diffrac-
tion phenomena (nonzero fields in the shadows). The
full solution
(2.15) + (2.17) is correct only
to
first
order because, according to Keller's theory, there is an
additional
diffracted
wave
U.
Then
u =
u~
+
u~
+
u,
(2.22)
where
,
k-!
iks
~
('k)-m'
u'"'-'
e
4.
I
Zm'
(2.23)
m=O
Of
course s
and
the
2m
satisfy the equations derived
earlier for phase
and
amplitude functions. The
diffracted
rays
associated with s emanate from the
edge
E
of
the screen
and
s =
Si
on
E.
(2.24)
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