424
J. Opt. Soc. Am. A/Vol. 9, No. 3/March 1992
M. Nieto-Vesperinas
and J. A. Sanchez-Gil
Light scattering from a random rough interface with total
internal reflection
M. Nieto-Vesperinas and J. A. Sanchez-Gil
Instituto de Optica, ConsejoSuperior de InvestigacionesCientificas, Serrano 121,Madrid 28006, Spain
Received April 18, 1991; revised manuscript received September 26, 1991; accepted October 12, 1991
The scattering of linearly polarized electromagnetic waves incident from a dielectric from a rough surface separating the dielectric from a vacuum is studied by using the extinction theorem. The angular distributions of
the ensemble average of intensity of the reflected and transmitted fields are calculated numerically for several
values of the angle of incidence, the surface statistical parameters, and the dielectric permittivity. To determine the effect of the corrugation on the transmitted evanescent waves,we also obtain the angular spectrum of
the transmitted field as a function of the momentum parallel to the surface in the nonradiative zone. The total
mean reflected and transmitted energies (reflectance and transmittance), as well as their incoherent parts in
the case of slight corrugations, are derived by integrating the angular intensity distribution over the angle of
observation. This permits the analysis of the influence of the corrugation and of the phenomenon of total internal reflection within two different systems of surface correlation length T namely, for T larger and smaller
than the wavelength. In particular, enhanced backscattering and forward transmission are predicted for surfaces with both T and the rms deviation greater than the wavelength of the incident light.
1.
INTRODUCTION
In this paper we present the results of numerical calculations of the scattering of s- and p-polarized light from
one-dimensional random rough surfaces separating two dielectric media, one of them assumed to be a vacuum, when
there may exist total internal reflection (TIR), namely,
when the incident wave propagates in the denser medium.
Research on the scattering of light and other electromagnetic waves from rough surfaces has increased in the
past few years because of both the possibility of doing experiments with surfaces of controlled statistics 4 and the
development of new numerical methods5-10 that permit the
solution of the scattering equations for one-dimensional
profiles without applying analytical approximations (e.g.,
the Kirchhoff approximation,"'1-3 the small-perturbation
method,13 2- 1 and the phase perturbation approach22'23 ) as
was previously done. Both the experiments and the numerical calculations have been shown to be able to yield
results on high-slope surfaces in penetrable and perfectly
conducting materials, showing new effects, among them
that of enhanced backscattering,'- 0'2 4 enhanced specular
reflection for symmetric profiles,25 '26 antispecular enhancement in thin films,27 and nonrefractive transmission in dielectrics.2 8 For instance, the remarkable effect
of enhanced backscattering, which involves multiple scattering, was proved to require a large reflectivity of the
surface, so that the wave could undergo several hits before
being reflected.9 2 9 30
In this paper we deal with the phenomenon of TIR,
which appears on interaction of a light wave, propagating
in a dielectric at an interface that separates this medium
from a vacuum. One expects that, when TIR takes place,
enhanced backscattering will occur in the angular distribution of mean reflected intensity from highly corrugated
surfaces. Our observations confirm that, indeed, this is
the case; however, since TIR is a selective phenomenon
0740-3232/92/030424-13$05.00
that appears only for angles of incidence above a critical
angle, the occurrence of enhanced backscattering on light
reflection at a dielectric-vacuum interface is also critical
and disappears rapidly with the increase of the angle of
incidence of the wave upon the mean plane of the random
surface. This decrease is much faster than in metals and
has to do not only with the shadowing from ridges but also
with the selective reflectivity of this system.
We shall also see that the corrugation of the interface
produces other interesting phenomena. It can lead to dramatic increases of the reflectance at angles of incidence
just below the critical angle of TIR associated with a flat
interface; conversely,extremely large increases in the total
transmitted energy through the interface are encountered
at angles slightly larger than the aforementioned critical
angle, at which no transmission would exist if the surface
were a plane. In addition, a large corrugation gives rise
to a nonrefractive transmission effect similar to the one
discussed in Ref. 28, now being the reciprocal of that effect, owing to the reciprocity in the natures of the system
under consideration in this work and the one discussed
in Ref. 28.
Since the existence of transmitted evanescent waves is
intimately related to the phenomenon of TIR, we shall go
through the angular spectrum of the transmitted field in
the nonradiative region. In a flat interface, the transition of the reflectivity about the critical angle is a steep
one; then the transmitted field above this angle becomes
evanescent, and its angular spectrum shows a peak in the
nonradiative zone. On introducing roughness, we shall
observe that this transition smooths and that the distribution of the evanescent components of the scattered
transmitted field gradually broadens, the peak structure
disappearing over the nonradiative region.
'T'he method of numerical solution of the scattering
equations is similar to that reported in Ref. 30 for light
propagating in vacuum and scattered at an interface that
© 1992 Optical Society of America
Vol. 9, No. 3/March 1992/J. Opt. Soc. Am. A
M. Nieto-Vesperinasand J. A. Sanchez-Gil
z
e(w)
k
o
A
V
-1
Iy
0k
where k = 2/A.
are related by
Hence all the scattering equations previously established
8 9 29 30
are valid just by replacing
'
for dielectric interfaces
with c'. Therefore we shall write the wave vectors Ko,
K, and Kt, in a Cartesian system of coordinates, as indicated in Fig. 1, in the form
e = 1
separates the vacuum from a dielectric. Hence the results presented here aim to supplement those of Ref. 30
and to provide a set of numerical data and conclusions for
comparison with results of other experiments and analytical methods and for further application in other studies,
e.g., on waveguide propagation. We use the extinction
theorem" as a nonlocal boundary condition to obtain the
induced sources, namely, the field and its normal derivative onto the surface. For the sake of comprehensiveness
and clarity for the reader, we summarize in Section 2 the
scattering equations to be used. The results and their
analysis concerning radiated waves are presented in Section 3. We introduce the formalism of the angular spectrum representation in Section 4, along with an analysis
of the transmitted evanescent field in several cases of interest. Finally, the main conclusions derived from this
work are outlined in Section 5.
2. DESCRIPTION OF THE SCATTERING
PROCESS
We consider a one-dimensional rough surface z = D(x),
varying only along the x coordinate, separating a semiinfinite dielectric of permittivity e in the region z > D(x)
from a rarer medium; we shall assume that the latter is a
vacuum (e = 1) occupying the half-space z < D(x) (Fig. 1).
A monochromatic, linearly polarized, plane electromagnetic wave impinges from the dielectric upon the interface
at an angle of incidence 00 with the normal to the average
plane of the surface, this normal being taken along the z
axis. We shall denote the components of the incident, reflected, and transmitted wave vectors as KO, K, and Kt,
respectively. Since the interaction at every point of the
interface depends on the relative permittivities of the two
media by the corresponding laws of reflection and refraction and also by the local Fresnel coefficients, the scattering of an electromagnetic wave incident from a dielectric
of permittivity E at an interface separating the dielectric
from a vacuum is equivalent to the interaction of a wave
incident from a vacuum with an interface separating the
vacuum from a dielectric with permittivity e-'. In fact,
we shall renormalize the wave vectors as follows: The
initial problem is diagrammatically stated in terms of the
moduli of the incident (and reflected) and transmitted
wave vectors in the form
KtJ =
o,
where ko = 27r/Ao,AObeing the wavelength in vacuum.
This problem is replaced by
JK.J =
KJ = k => KtJ = N/eik,
(la)
k(sin 00,0, -cos Oo),
Ko
Fig. 1. Illustration of the scattering geometry.
z>
One must keep in mind that A and A
Ao = AV-e.
A,
JK.J = IKI= VEk,
425
(lb)
0,0,cos 0),
K -k(sin
K _ N/7k (sin 0t, 0
-
cos
(ic)
X
t)
where the following relations are held:
2
K 2 = Ko1
k2=
2 =
(2a)
()2'
IKt12 =Clk2,
(2b)
A being the wavelength of the incident plane wave.
Since the surface depends on the x coordinate only,
there is no cross polarization for either s or p incident
waves. We shall then denote the incident electric vector
for s polarization
(TE waves):
Eli)(r) = E(I)exp(iKo
r)
(3)
Analogously, for p polarization (TM waves) the incident
magnetic vector is expressed as
Hl")(r) = JH(I) exp(iKo
(4)
r).
In Eqs. (3) and (4), r = (x, z),j is the unit vector along the
y axis, and E(') and H(")are complex constant amplitudes.
A time dependence factor, exp(-icot), and the y dependence are suppressed throughout.
For s polarization, the nonlocal boundary conditions
on the surface for the field and its normal derivative, according to the theory developed in Refs. 29 and 30, are
given by
D(x)]
+ |ax'd
E(')[x,
x
E(x') |AG,
8(X)
AXo _ GoF(x')
D
= E(x),
(5a)
= O.
(5b)
-- f dx'
EG
_
az,
X E(x')
D'(x')aG
axI
_ GF(x
)
kXJ
In Eqs. (5a) and (5b), Go and G are the Green functions,
represented by the outgoing cylindrical wave in each
medium, namely:
Go(r,r')= 7riHo(')(kr - r'l),
G(r,r') = 77iHO()(V kr - r),
(6a)
(6b)
where Ho(') is the Hankel function of the first kind and
zero order. Also, in Eqs. (5) the source functions E(x) and
426
J. Opt. Soc. Am. A/Vol. 9, No. 3/March 1992
M. Nieto-Vesperinas and J. A. Sanchez-Gil
F(x) are defined as the limiting values:
On the other hand, for p polarization, the nonlocal
E(x) = E(out)[xD(x)]
boundary conditions read as (see Refs. 29 and 30)
= E(in)[x,D(x)],
F(x) = [aEa
(7a)
H(')[x,D(x)] +
|
D)(r)]x)
x H(x)
[aE(n(r)]
z
D'(x')
-
] - GoL(x')
=
H(x), (13a)
(7b)
where the superindices (out) and (in) stand for the media
with permittivities 1 and C1 , respectively. Similarly, (+)
and (-) denote the limit approaching the interface from
47r
x {H(x')
ax,
above [z > D(x)] and from below [z < D(x)], respectively.
a/an represents a derivative with respect to the outward
normal, and y is defined as
= {1
l + [D'(x)]2 }"l2 ,
(8a)
-
D'(x')- ,]1
ax,
(13b)
H(x) = H(out)[x,D(x)]
= H(in)[x,D(x)],
L(x)
t
y [aH(in)(r)
E
known expressions 9
(14a)
[ aH(ou)(r)
=
2 30
-) exp[i(kr, - r/4)]
e 'GL(x') = 0.
In Eqs. (13) the source functions H(x) and L(x) are now
defined according to the limiting values
D'(x) = dD(x)
(8b)
dx
Once the source functions E(x) and F(x) are obtained
from Eqs. (5), the far-zone scattered fields corresponding to the angular distribution of the reflected (r) and
transmitted (t) waves are evaluated by using the well-
I
an
(x)
]z=D(-)(x)
(14b)
Once H(x) and L(x) are obtained from Eqs. (13), the far
fields, reflected and transmitted, are
Eor(r0)
x k/,7
x f
dx'
H(r)(r>,,) = exp[i(kr, - rrl4)]
dx'{k[cos 0 - D'(x')sin 0]
2 V2 7~kr>,
x E(x') - iF(x')}exp(-iK r'),
t
E( )(r_ 0t) - exp[i(/7kr
-
(9a)
x
/4)]
dx' {
H(t )(r.,02) exp[i(V
IVk[cos 0 + D'(x')sin St]
x E(x') + iF(x')}exp(-iKt r'),
(9b)
x
0 and Otbeing the angles of observation in z > D(x) and
z < D(x), respectively.
When Io c E(')l 2 L cos Oodenotes the total power flow of
the incident wave, L being the illuminated record of the
surface, the normalized angular distributions of mean
scattered power above and below the surface are30
(I"(r)(o))
= r (E(r)(r>, 0)12),
o0
r< (E(t)(r,
Io
t) %,
(lOb)
respectively.
As a criterion of numerical consistency of the results,
we take the unitarity condition:
R + T = 1,
(11)
where the average reflectance R and transmittance T are
given by
R
1r/2
=
io
T
respectively.
=
1 f
(I(r)(0))d0
(12a)
(I()(0))d0t,
(12b)
-7/2
kr< -
J dx'{?k[cos
(15a)
4
0 + D'(x')sin 0t]
x H(x') + i'L(x')}exp(-iKt
r').
(15b)
The normalized angular distributions of mean scattered intensity above and below the surface are calculated
from Eqs. (15) by using the expressions
(1Oa)
Io
(I(4,t)( ,)) =V~
Io
dx'{k[cos 0 - D'(x')sin 0]
x H(x') - iL(x')}exp(-iK- r'),
2[2r/7 kr<]1/2
x f
f
1(I(r)(0)) =
1 ( t)(
Io
rH(r)(
=)) 1
0)12),
(16a)
r< (IH(t )(r<,0t) 2 ).
(16b)
V~ I
Equations (16a) and (16b) should once again be subjected
to the unitarity condition, Eq. (11),with R and T given by
substituting them into Eqs. (12a) and (12b), respectively.
Equations (5) and (13) are solved by simulating surface
samples of a given length L. In practice, the x' integrals
are restricted to an interval of integration of length L.
Each sample is built by extracting a segment of 220 numbers from a sequence of random numbers (typically 105)
with the desired statistics obtained from a computer
routine; in our case, homogeneous and isotropic normal
statistics, zero mean, and the appropriate correlation
function, as described in Refs. 6 and 7, are considered.
Vol. 9, No. 3/March 1992/J. Opt. Soc. Am. A
M. Nieto-Vesperinas and J. A. Sanchez-Gil
This random sequence simulates the profile z = D(x).
Then the source functions E(x) and F(x) or H(x) and
L(x), appearing in integral Eqs. (5a) and (5b) or (13a) and
(13b),are obtained for each sample of length L through the
discretization imposed by this sampling.6'7 By introducing these source functions into Eqs. (9a) and (9b) or (15a)
and (15b), respectively, the far-field scattered amplitudes
are computed. Subsequently, after averaging the corresponding intensities over several samples (between 200
and 300 in our case), we arrive at the mean scattered intensities
through Eqs. (Oa) and (lOb) or (16a) and (16b),
respectively.
The correlation function c(r) of the surface profile is
chosen to be Gaussian:
C(fr)=
1
(D(x)D(x + T)) = exp
-
T
(17)
where o-is the rms deviation of the surface and T denotes
its correlation distance. The record length L of each
sample is usually approximately 20Ato 40A,depending on
the value of T.
3.
NUMERICAL RESULTS
In this section we discuss numerical results for the scattering of an s- or p-polarized
plane wave propagating
in a
dielectric of permittivity e on interaction with an interface separating this medium from a vacuum. We shall
consider two different
values of E, i.e.,
= 4 and e = 2.
As mentioned above, this problem is equivalent to the
scattering of a plane wave incident from a vacuum onto an
interface separating the vacuum from a rarer medium
with permittivities e 1 = 0.25 and E' = 0.5, respectively;
this is the notation that we shall use in what follows. As
was explicitly explained in Section 2, the wavelength in
vacuum in the original problem is A = A-.
Furthermore, from now on, the wavelength of the incident light A
constitutes our length unit; consequently, note that T and
C are actually smaller when expressed in terms of A in
the original problem than they are when given in units of A.
Two characteristic systems of the surface correlation
length will be addressed according to the size of the sur-
0, of TIR (0, = 300 for E = 0.25 and &.= 450 for eC =
0.5). Conversely, the reflectance for p polarization decreases with 00 until it reaches a zero minimum at the
Brewster angle OB (OB = 26.56° for e' = 0.25 and OB =
35.26° for E l = 0.5); then, like the reflectance
1.0
)
0.8
0.6
c)
6)
1, 0.4
6)
P0
s.waves: plane
p waves: aO5
swaves: o.0.5*
p waves: a1.9
Byws waves: a=1.9X
0.2
40
20
20
40
Angle of incidence
A. Correlation Length Larger than the
Wavelength (T > A)
1.0
Figures 2(a) and 2(b) show the average reflectance R
spectively. Three different profiles are considered: a
plane interface (cr = 0) and two random surfaces with T =
3.16A and
co = 0.5A or 1.9A.
In the case of a plane interface, we retrieve the expected behavior of the reflectances, which, apart from the
numerical error that is due to the finite record L, should
coincide with the Fresnel coefficients corresponding to an
infinite plane interface. There exists a small difference
between s and p polarization at low angles of incidence 00
and a notable increase of the reflectance for s polarization
as 0 increases, with a sharp increase at the critical angle
60
60
(degrees)
80,..
80
(a)
Reflection
[Eq. (12a)] versus the angle of incidence for s and p polarization for E = 4 (e` = 0.25) and e = 2 (E-' = 0.5), re-
for s waves,
it suffers a sharp increase at the critical angle 0,. Note a
slight departure from R = 1 for 0 > 0, because of inherent numerical errors.
It is interesting to observe how the perturbation, owing
to the surface roughness, alters these coefficients. First,
we see that, for low 0, an increase in conveys an increase in R, in both s and p polarizations. This effect,
which becomes more evident with increasing Eand which
has also been observed in the reciprocal situation of a
wave incident from a vacuum,28 results from the large local angle of incidence formed at low 00, which is due to the
high slope introduced by the roughness. Conversely, for
00near or larger than the critical angle 0,, the reflectance
decreases with o-for both polarizations. Of course, this is
due again to the alterations, introduced by the roughness,
on the local angle of incidence, which can be smaller than
the critical angle 0,. A remarkable instance is encountered for p polarization, in which it is seen that, for incidence near the Brewster angle OB, introducing a slight
roughness in the surface can produce a dramatic increase
in the reflectance; this effect is related to a similar one
considered in the reflection of radio waves by atmospheric
layers to increase the receiving signal.32
It has been shown92 30 that the high reflectivity of an
interface plays a crucial role in the possibility that the impinging wave, successively reflected at the surface, undergoes multiple hits. Since multiple scattering is required
for enhanced backscattering to take place,''10 one expects
face asperities, namely, T > A and T < A.
1.
427
-
0.8
a)
c 0.6
4-'
C)
, 0.4
0.2
0.0
o*0
-
80
(b)
Fig. 2. Average reflectance from 300 samples versus 00. The
reflectance from a plane is also included.
428
J. Opt. Soc. Am. A/Vol. 9, No. 3/March 1992
M. Nieto-Vesperinasand J. A. Sanchez-Gil
p waves: N=300
------- s waves: N=300
-
0.4
D1
0.4 -
-,
1
4-'
0.3
O'0.3
6)
._.'
0.2
0.2
6)
C)
a)
4'I
Q
0.11
'4
0.0
-
-90
-60
-30
0
30
60
Scattering angle (degrees)
90
Q 0.1
_
6)
/
W.o
0.0
-90
-30
0
30
60
90
-60
-30
0
30
60
90
Scattering angle (degrees)
0.4
0.7
-'
-60
80,
.S0.6 .
-1
0.3
0 0.5
d
. 0.4
0.2
'g0.3
CD
.J
W0.1
(4.4
X 0.2
Q)
p: 0.1
0.0 .
-90
-60
-30
0
30
60
90
0.0
-90
Scattering angle (degrees)
Scattering angle (degrees)
Fig. 3. Angular distribution of mean reflected intensity from a dielectric surface at 00 = 0, 100,200,and 40°. Average over 300 samples.
The specular direction is shown by the marks at the upper right. The backscattering direction is marked by vertical lines. The average
reflectance and transmittance are shown.
that the phenomenon of total reflection at a dielectricvacuum interface will give rise to a peak in the backscat-
tering direction of the angular distribution of mean
reflected intensity. Figures 3 and 4 confirm this assumption. These figures show the mean reflected intensities for scattering of s- and p-polarized waves incident
under 00 = 0, 200, and 400 from a medium with = 4
(C = 0.25) and = 2 (' = 0.5), respectively. The
statistical parameters of the corresponding interface take
on the values T = 4.69A and o = 1.86A. It is observed,
first, that there is a large enhanced backscattering peak
for both s and p waves at small O0when e& = 0.25, the
width of this peak being approximately A/T, as expected.
As 0 increases, this peak quickly drops; this decrease is
much more pronounced than in a metal surface. Also,
this can be interpreted as being produced by the local
angle of incidence that, owing to the high slope for low 00,
-
p waves: N=300
- --- s waves: N=300
R 4=0.06, T=0.93
0.10
R,=0.11, T=0.88
1/V=0.5
T=4.69X
.*0 0.08
co
g 0.06
° 0.04
.J
Q
:
-90
-60
LWJ l''.
-30
0
1LLJLJ
30
60
90~~~~~~.
%10.02
a)
0.0
-90
-60
-30
0
30
B0
90
Scattering angle (degrees)
Fig. 4. Same as Fig. 3 for eC1= 0.5 at 00 = 0°.
is larger than the critical angle 0, at many points of the
surface. Then both s and p waves are highly reflected.
As a result, it is likely that, for the values of and T chosen, a substantial amount of energy is reflected twice or
even more times under angles larger than the critical
angle,-thus giving a high resultant reflectivity and hence
producing enhanced backscattering. On the other hand,
as 0 increases, there is a decrease of the local angle of
incidence over every tilted surface segment facing the impinging wave; therefore the reflection at many surface
points is below the critical angle. As a consequence, the
reflectivity decreases; hence the multiple scattering also
decreases, and this must produce a rapid drop of the
backscattering peak. Of course, shadowing from surface ridges also plays a role in this process, as in metal
surfaces. In this respect, it is worth noticing that, for
C = 0.5 (Fig. 4), both the backscattering peak and the
reflectance at low Ooare considerably smaller for p than
for s waves; and they are also smaller than those for the
case of EC = 0.25 (Fig. 3). This can be interpreted on
the basis of the above analysis; namely, a comparison of
the reflectances at a small, almost flat, portion of the surface around the point of interaction for both materials
(see Fig. 2 for a plane surface) indicates that the angle at
which total reflection is reached for E- = 0.5 ( = 450) is
higher than that for E = 0.25 (0, = 300). Consequently
a hit in the former case is likely to have less reflectivity
than a hit in the latter case. The difference between s
and p scattering for
= 0.5 follows from the same rea-
soning; the local reflections take place principally at
angles slightly lower than 0, = 450, at which the ref lectivity for p waves is still influenced by the Brewster effect
and hence is much smaller than for s waves. Figure 3 also
Vol. 9, No. 3/March 1992/J. Opt. Soc. Am. A
M. Nieto-Vesperinas and J. A. Sanchez-Gil
shows that for 00 2 200 the reflectivity
is low for negative
2.0
angles 0 of observation, in contrast to that at positive
angles; of course, this is so because forward scattering
(0 > 0) occurs at surface faces whose local normals make
large angles with the incidence direction, whereas backward scattering (0 < 0) takes place mainly from faces
whose local normals are close to the direction of incidence.
Another fact revealed by Figs. 3 and 4 is the remarkable
structure of the mean angular distributions of reflected
light, which is more marked in the interface in which the
critical angle is lower (' = 0.25). The first subsidiary
maxima, placed in Fig. 3 at both sides of the backscattering peak for 0 < 100, have locations and sizes that are
similar to those already found in metal surfaces [compare
lobe of Fig. 3 at Oo ' 10° is larger for s waves than for
p waves, because it arises from scattering under local
angles that are near, but below, the critical angle; this is a
zone where the reflectance for p waves lies below that for
s waves (see Fig. 2). We can summarize the discussion by
saying that Fig. 3 suggests that the surface reflectivity
addressed in this case behaves like that of a metal for 00 '
100and 0 20°, whereas it resembles that of a dielectric
for any other angles.
Figure 5 shows what happens when o-is reduced to values at which the Kirchhoff approximation is expected to
work.7 The case displayed in this figure corresponds to
o = 0.5A and T
3.16A. Two angles of incidence, 00 = 00
and 00 = 400, below and above the critical angle 0, = 300
for a plane interface ( 1 l = 0.25) are considered. The results, as expected from Fig. 2(a), are drastically different
from each other for these two values of 00, exhibiting the
dramatic effect of TIR in the reflectivity of smooth surfaces. In Fig. 5 neither the average reflectance nor the
average transmittance is explicitly given: the former is
p waves: N=300
------- s waves: N=300
1.6
1.2
.6
0)
0.8
c) 0.4
0)
0.8
1v.
0.4
0
0.0
-so
-90
_
-60
-30
0
___
30
60
Scattering angle (degrees)
90
Same as Fig. 3 for o = 0.5A, T = 3.16A, and E- = 0.25.
The results at two angles of incidence are shown.
Fig. 5.
-60
-30
0
30
60
90
Scattering angle (degrees)
2.0
z 1.6
.4-.
1.2
^
.
. 0.4
Ev
0.0
_
-90
-60
-30
0
30
60
90
Scattering angle (degrees)
Fig. 6. Angular distribution of mean transmitted intensity from
a dielectric surface at 0 = 20° and 40°. Dashed curves, s polarization; solid curves, p polarization. Average over 300 samples.
The forward direction is shown by the mark at the upper right.
The specular direction of refraction, namely, that from Snell's
law for a plane, is marked by vertical lines. The average ref lectance and transmittance are included.
plotted in Fig. 2(a) versus 0, and the latter can be derived
from Fig. 2(a) by taking into account that the unitarity
condition [Eq. (11)]is satisfied within a 1% error.
2. Transmission
Figure 6 shows the angular distributions of mean transmitted intensity for waves refracted at a surface separating a dielectric with e = 2 (e- = 0.5) from a vacuum.
The statistical parameters are o- = 1.86Aand T = 4.69A.
Two angles of incidence
(Oo= 200 and
= 40°) were
chosen. It is observed that the roughness causes this angular distribution to concentrate about a direction of
transmission close to the forward direction, appreciably
shifted from the direction of refraction that Snell's law
would predict at a plane interface (dashed vertical line in
Fig. 6). This effect has a strong connection with another
one, already reported,2 30
' which occurs in refraction at a
dielectric interface in the reciprocal situation, namely,
when the incident wave comes from the vacuum side.
This shift has been demonstrated in Refs. 28 and 30 to be
essentially the result of single scattering. Note that, in
agreement with the reciprocity of the two cases, the angular distributions of Fig. 6 are almost specularly reflected
about the forward direction with respect to those discussed in Ref. 28 (compare with Fig. 1 of Ref. 28, taking
PZ:
0.o
e
1.2
Fig. 3 with, e.g., Fig. 7(a) of Ref. 9]. Thus we believe that
they have the same origin as in metals (constructive interference effects among multiple scattered waves at angles
of observation 0 = nA/(d), (d) being the average distance
between two scattering events and n being an integer29 ).
It should be pointed out that, although we do not show this
here, if the surface slope is increased (e.g., T = 3.16A,
keeping o- = 1.86A) a larger reflectance and enhanced
backscattering are achieved but the subsidiary maxima
have much less amplitude [in analogy once again with the
metal surface, as a comparison with Fig. 7(b) of Ref. 9
suggests]. On the other hand, the outstanding second
429
into account the different criterion used there for the sign
of the transmission angle 0,). Since the local angle of incidence of the impinging wave is low (because of the tilting of the surface segments facing it), most of the light
is transmitted close to the forward direction, with little
430
J. Opt. Soc. Am. A/Vol. 9, No. 3/March 1992
R=0.35 T=0.64
M. Nieto-Vesperinas and J. A. Sanchez-Gil
R,=0.44. T=0.55
fact, as in the cases in Refs. 28 and 30, and although it is
not shown here explicitly, our calculations show that the
Kirchhoff approximation closely reproduces Fig. 6.
U0.6
When is raised there is more reflectance, multiple
scattering becomes appreciable, and, as in the reciprocal
6 0.4
situation of light incident from a vacuum into the dielec~~~~~~~~~~~~~~~~~~~~~~~~~~I
tric,3 0 one expects the angular distribution of transmitted
light to broaden. This is precisely what happens, as is
0.2
shown in Fig. 7, where e = 4 (-1 = 0.25). The most re-~~
~~~~~~~~~~~~~~~~~I
markable phenomenon in this figure, however, occurs for
.
V
s
.
S
,
I,
I
I
o
.o
-g0
-60
-30
0
30
60
90
O = 40°. At this angle of incidence, a plane interface
Scattering angle (degrees)
would yield no transmitted wave at all [see Fig. 2(a)]; nevertheless, the presence of roughness gives rise to a noticeable distribution of transmitted light whose maximum
0.8
stands out near the forward direction.
-E
For low roughness, a drop in the transmitted light
should be observed with increasing angle of incidence 00.
Q
a0.6
This is shown by Figs. 8(a) and 8(b) for T = 3.16A, o- =
0.6A, and C = 0.25 and 0.5A, respectively, where the
6 0.4
mean incoherently transmitted intensities are plotted (the
coherent part has been removed because it is not null for
0.2
Oo< 0,,). As expected, the higher the value of E,the more
drastic the drop. However, note that, again for angles of
incidence above the critical angle [Oo= 40° in Fig. 8(a)], a
0.0_-9
0
-60
-30
0
30
60
90
slight roughness produces a nonnegligible distribution of
Scattering angle (degrees)
transmitted light. The average transmittance, as was reF ig. 7. Same as Fig. 6 for eC = 0.25.
marked in connection with Fig. 5, can be calculated by
subtracting the average reflectance (Fig. 2) from 1. The
0 6
- =20'
1
1/c=0.25
-4-'
-
i
a=1.86X
T=4.69k
4
.t4
_
.4-'
..J
a)40-'
$1.
transmitted
intensity
for lower e [cf. Fig. 8(b)] appears
again to be shifted from the direction of refraction that
would correspond to a plane interface.
M 1.6
O
Hl
I
1.2
0
1.0
0.8 -
()
C)
1 0.4
0
c 0.6
-
10
-60
-30
0
30
60
90
Scattering angle (degrees)
a
o6)0.4
6)
(a)
-o
M2.0 0
.51
- r
6
(a)
'I
a1.2 -
1.0
>,0.8
0.8 -
.
:-
C)
-
(U
-4-
C)
';U0.4_
0 0.6
co
0)
-4-
C)0.0
-O
C)
-60
-30
0
30
60
Scattering angle (degrees)
90
(b)
Fig. 8. Same as Fig. 6 for the diffuse component; = 0.5Aand
T = 3.16A. The results at two angles of incidence are shown.
energy being reflected back; this prevents multiple scattering. Moreover, most of the interactions take place at
local angles smaller than the critical angle of TIR. In
p waves: plane
s waves: plane
--
A, 0.4
C)
0.2
--I/e
'~
0.0 i
C
.
I
p waves:o0
waves: u=O.2A
.
T=0.2X
~
,=,
.
/
lo J
.
,
. ..
_l.
,
20
40
Angle of incidence
.
.
.
.
.
60
(degrees)
.
80
(b)
Fig. 9. Average reflectance from 200 samples versus O.
reflectance from a plane is also shown.
The
M. Nieto-Vesperinas and J. A. Sanchez-Gil
-------
Vol. 9, No. 3/March 1992/J. Opt. Soc. Am. A
p waves: N=200
waves:
w
N.=200
R,=0.13, T=0.87
e0=200
t'0.20
.."0.20 M
-e0
U)
6)
.4._
431
0.15
R.=0.22, T=0.72
,
:
a~~~~~~~~~~~~
0 0.15
0.15
4.
6)
1o 0.10_
.4-'
-'o'A
0.10
'D4
4 0.05
.0
(a
5
0
0
0
-90
-60
-30
Scattering
0
30
60
0.00-90
90
angle (degrees)
-60
-30
60
90
0
30
60
Scattering angle (degrees)
90
R=0.87.T=0.1
M
-75
a)
6)
.0.5
.4-
0.25 -
0
a
0
d 0.25
-60
-90
'7
-30
0
30
Scattering angle (degrees)
Fig. 10. Same as Fig. 3 for the diffuse component for a = T = 0.2Aand e = 0.25; 0 = 00, 200,and 400. Average over 200 samples.
p waves: N4=200
--- s waves: N,=200
0.06
R(=0.03, T 0=0.96
-=0.5
° 0.05
0.06
R.=0.05, T=0.90
I: 80=0° a=0.2X
T=0.2X
R,=0.03, T,=0.95
-eo200
R,=0.06, T,=0.89
CD
0.05
0.04
4.
6)
0.04
4-4
'. 0.03
'0.03
0 0.02
0o.02
6)
/
Q 0.01
0
A
O 0.00
_ _
10
-60
-30
0
30
60
Scattering angle (degrees)
O.5
o~o
~0.00
-90
90
R,=0.79 T=0.13
-60
-30
0
30
60
Scattering angle (degrees)
90
R,=0.73, T.=0.
0.4
>10.2
Ao .1
a0.0
-90
/
/I
-60
=
-30
Scattering
-'
0
30
60
angle (degrees)
90
Fig. 11. Same as Fig. 3 for the diffuse component for a- = T = 0.2A, and eC = 0.5; 00 = 00,200, and 500. Average over 200 samples.
B. Correlation Length Smaller than the Wavelength
1.
Reflection
Figures 9(a) and 9(b) show the average reflectances R
(coherent plus incoherent) [see Eq. (12a)] for s and p polarizations in two cases:
= 4 (C' = 0.25) and E = 2
(C' = 0.5), resepectively. A plane interface and a rough
surface with T = of = 0.2Aare studied. Note that, once
again, a relatively small roughness can produce large departures of the reflectances from those of a flat interface
at angles of incidence near the critical angle. These large
deviations introduced in the reflectances by the rough-
432
J. Opt. Soc. Am. A/Vol. 9, No. 3/March 1992
M. Nieto-Vesperinas and J. A. Sanchez-Gil
ness are more noticeable for p waves, especially in the interface that involves a greater change of permittivity, i.e.,
for E-l = 0.25 [Fig. 9(a)].
Figures 10 and 11 display the angular distributions of
the mean diffusely reflected intensity for the aforementioned interfaces under angles of incidence 0 = 0, 20,
40
(
= 0.25) and
= 0, 200, 50 ('
p waves: N=200
waves: N=200
-------
~
0.10
= 0.5), respec-
tively. Note the peaks that appear markedly for s polarization at 0 =+30° for E-' = 0.25 (Fig. 10) and that are
very intense at 0 +45° for C' = 0.5 (Fig. 11). They do
not change much as 0 increases moderately (see curves
for
= 0 and
= 20). It should be emphasized that
the angles 0 at which these peaks arise are near the critical angle; accordingly, this indicates a sudden increase of
-
0.10
M0.08
a~~~~~~~~~
M-0.08
a
6)
. 0.06
0.06
A
U)
a 0.04
aCd0.04
.0.02
C;0.02
a3
0.00
-so
-60
o.oo
-30
0
30
60
90
.
..
-90
Scattering angle (degrees)
0.10
.
.
-60
I
.
.
-30
.
I
o
.
I
I
30
.
I
.
60
90
Scattering angle (degrees)
R,=0.87.
.=40
M4'
0.08
a
6.)
m3
0.06
a 0.04 1.41
*;0.02
I-
0
o.ooI,
r
-g0
-60
-30
0
30
60
90
Scattering angle (degrees)
(a)
0.07
0.06
RP=0.03,T=0.96
0.07
-/c=0.5
0.05
0.05
0.04
0.04
@0.03
~0.01
0.01
0.00
-90
R=0.06, T.=0. 9
.3
.0.02
7
0.02
=0.03 T =0.95
0.=20'
.~0.06
-60
-30
0
30
60
Scattering angle (degrees)
0.00
90
-g0
-60
-30
0
60
30
90
Scattering angle (degrees)
0.07
"I 0.06
0.05
. 0.02
0 0.01
0.00
-90
-60
-30
0
30
60
90
Scattering angle (degrees)
(b)
Fig. 12. Same as Fig. 6 for the diffuse component from 200 samples for a
=
T = 0.2A; 00 =
0,
20, 40°.
M. Nieto-Vesperinas and J. A. Sanchez-Gil
Vol. 9, No. 3/March 1992/J. Opt. Soc. Am. A
0.7 r
ally less pronounced than that for the total reflectance
(see Fig. 9). On the other hand, the TITE does not exhibit a sharp decrease for 0 near but beyond the critical
angle O, as the total transmittance would do, but shows a
behavior qualitatively not much different from the TIRE
for s waves before falling on approaching grazing incidence [(Fig. 13(b)]. For p polarization, however, the
TITE behaves almost uniformly with 0, and it drops for
large angles of incidence.
p waves: T=a=0.2X
0.6
T a=.
- ---Bwaves:
0.5
0.4
'0.3
1/E=0.25
E-
0.2
1/E=0.5
0.1
0.0 I
.-.-.-.-.-.-.-.-.-.-.-
0
20
40
Angle of incidence
60
(degrees)
80
(a)
....
. ..
0.25
p
-
0.20
..
swaves: T=-0.2
H
E- 0.10
0.05
0.00
*IA 05
D
20.440
20
Angle of incidence
.
60
60
(degrees)
80
(b)
Fig. 13. (a)AverageTIRE from 200 samples versus 00. (b) Average TITE versus 00for the same surfaces.
reflectivity, larger for s than for p polarization, for waves
that are reflected under these angles. Also note the relatively larger strength of these peaks for El = 0.5. Apart
from these sidelobes, the behavior of the curves within the
interval defined by 0 < 0, resembles that obtained when
light is incident from the vacuum side; namely, the angular distributions for p polarization lean toward the backscattering direction, whereas the specular direction is the
predominant
4. INFLUENCE OF ROUGHNESS ON THE
DISTRIBUTION OF TRANSMITTED
EVANESCENT WAVES
We have seen so far how roughness affects the distribution of radiated waves. It is, however, of interest to investigate how the evanescent waves transmitted into the
rarer medium are affected by the presence of corrugation
in the interface. To do this, it is convenient to express
the scattered transmitted field in terms of its angular
spectrum representation. This is carried out from the
field transmitted into the rarer medium, which is given
by the following expressions3 0 for s and p polarization,
respectively:
aves: T=a-0.2X
/E=0.25
0.15
433
one for s waves.30
2. Transmission
Figures 12(a) and 12(b) show the diffuse part of the mean
E(r) =
3. Total Incoherently ScatteredEnergy
It is interesting to analyze explicitly the influence of
the critical angle transition on the amount of the total
energy diffusely scattered from the rough interface. In
Figs. 13(a) and 13(b) the mean total incoherently reflected energy (TIRE) and the mean total incoherently
transmitted energy (TITE), respectively, are plotted versus the angle of incidence. For the sake of comparison,
we include the results for the two different permittivities
studied throughout this work. The surface parameters
are T = = 0.2A in both cases. The TIRE [Fig. 13(a)]
behaves as expected: it abruptly increases on crossing
the critical angle of incidence and smoothly tends to disappear as long as the angle of incidence reaches the grazing
angle. However, the transition around 0, is proportion-
4r
aG[
aG]
E(x') a2z- D'(x')x
GF(x')jdx',
-
(18)
H(x ) a-
H(r) = -
-
D (x) aG
(19)
-xGL(x')'dx'.
The Green function G(r, r') [Eq. (6b)] is expanded into
plane waves by using the well-known Weyl representation 33 :
G(r,r') =
j|
K exp{i[K(x- x') + kz - z']}, (20)
where
k2 = /e'-k 2 -K
transmitted
intensity for
= 4 (
= 0.25) and e = 2
(C' = 0.5), respectively, and T = o- = 0.2A, under angles
of incidence 0o= 0, 20°, 400. The relatively smaller sen-
sitivity of the angular distributions for s waves to the
angle of incidence is remarkable.
.
1
-
k = i'K~2-
2,
K < -k 2 ,
(21a)
e-k2
2
(21b)
K2 > Elk
.
Equations (21a) and (21b) correspond to homogeneous
(propagating) waves and to inhomogeneous (evanescent)
waves, respectively.
On introducing Eq. (20) into Eqs. (18)
and (19), we obtain for z < Din, Dmindenoting the minimum value of z = D(x), for s polarization:
E(r) -
f
A,(s)(K)exp[i(Kx- kz)]dK,
(22)
where
A (s)(K)-
-4 1 k
f
dx' exp{-i[Kx' - kD(x')]}
x {i[KD'(x') + k]E(x') - F(x')}.
(23)
For p polarization we obtain
H(r)
=
f
A,(P)(K)exp[i(Kx - kz)]dK,
(24)
434
M. Nieto-Vesperinas and J. A. Sanchez-Gil
J. Opt. Soc. Am. A/Vol. 9, No. 3/March 1992
p waves: plane
-------- s waves: plane
k sin Ot,sin 0a= W/'esin 00,
In Eqs. (26) and (27) Kt =\/7
k, = k cos 0, and k, = k cos Ot.
For 0 larger than the critical angle, namely, for 0 >
sin-' (VI), Kt2 > Clk 2; consequently Eqs. (26) and (27),
together with Eqs. (22) and (24), represent a transmitted
field consisting of a unique evanescent plane wave component, propagating bounded to the surface and exponentially decreasing across the interface for z < 0. The
representation of this distribution shows a delta function in the nonradiative region, i.e., at K = K, > ok.
Figures 14(a) and 14(b) display IAt(K)2, obtained from
Eqs. (23) and (25) with D(x') = 0, for e = 0.25 and E'l =
0.5, respectively. These figures illustrate how the delta
distribution broadens in the evanescent wave region, becoming a sinc of width A/L because of the diffraction
introduced by the finite record L of the illuminated surface. From now on, the angular spectrum for p waves
(magnetic field), when plotted, is multiplied by a factor ( )'
so that it can be quantitatively compared
with that for s waves (electric field). The maxima of
these distributions are placed at the values of K predicted
For El = 0.25 and 0o = 400
by the Fresnel coefficients.
40.0
30.0
'-s 20.0
10.0
0-°.0
1.0
1.1
1.2
1.3
K/(cF1 / 2 k )
1.4
1.5
p waves: plane
-------- s waves: plane
50.0
2
CSE
40.0
30.0
20.0
10.0
[Fig. 14(a)]
0.0 i
1.0
1.1
1.2
1.3
1.4
1.5
K/(.-F/2, k )
Fig. 14. Square modulus of the angular spectrum of the transmitted field versus the parallel momentum in the nonradiative
region for plane surfaces.
K/(,E"/l2k)
k) =
=
and for e'
~
At(P)(K) = x
ik
dx' exp{-i[Kx' - kD(x')]}
{i[KD'(x')+ kH(x') - eC'L(x')}. (25)
-.
spectra are given by Eqs. (23) and (25).
0.6
equivalent
to shifting
the origin z = 0 to z = Dmin
(29)
p waves: plane
0.8
served that these angular spectra diverge for k, given
by Eq. (21b) because of the factor of the integrand
exp[-D(x')(K 2 - E-'k 2 )"12 ], which increases with K for
D(x') < 0 without limit in the nonradiative region, namely,
for evanescent components. Thus, to evaluate the angular spectra of the transmitted fields, we should get rid of
this exponential factor by changing D(x) to D(x) - Dmin,
so that the minimum of this new profile is zero. This is
sin 0o = 1.08.
The differences existing in the height of these maxima
[Figs. 14(a) and 14(b)] between s and p polarizations re-
Equations (22) and (24) constitute the angular decomposition of the transmitted field for s andp waves, respectively, at values z < Dmin. The corresponding angular
It should be ob-
(28)
= 0.5 and 00 = 500 [Fig. 14(b)]
K/(El2k) = az
where
i 0 = 1.29,
91
sin
-------- w
waves: plane
l
2
Cis
X1
=40°
4e=0.25
0.4
I
0.2
0.01
,. ... .. ..............
............
............... A
1. ,0
1.1
1.2
1.3
2
K/(C-1/ k )
1.4
1.5
1.1
1.2
1.3
2
K/(C-'/ k )
1.4
1.5
Of
course, the fields remain invariant after this change, since
At(s)(K)
and At(P)(K)
then
At(')(Dmi.n)(K) = At(s)(K)
= A,(P)(K)exp(-ikzDmin),
become
exp(-ik 5 Dmin) and At(p)(Dmin)(K)
respectively. We shall then address the plane wave expansion of the transmitted fields given by At(s)(Dmin)(K) and
At(p) (Dmin)(K).
To illustrate this discussion, let us first consider a flat
interface, D(x) = 0, and hence Dmin= 0. In this case the
angular spectra of the transmitted field are, for s and p
polarization, respectively,
At(s)(K) = B(K - Kt)k
zk
A1(p)(K) = 8(K - Kt) N'/ +7k*
(26)
(27)
3.0
N_
2:
2.0
1.0
0.0 LO
- 1.0
Fig. 15. Same as Fig. 14 with the origin shifted from z = 0 to
z = Dmin, Dmin being the minimum of the surface whose statistical parameters are a = T = 0.2A.
M. Nieto-Vesperinasand J. A. Sanchez-Gil
-p
--------
0.8
waves: N=200
waves: N.=200
- =40O
-T=0.2X
: a=0.2L
1 /=0.25
0.6
M
Vol. 9, No. 3/March 1992/J. Opt. Soc. Am. A
0.4
0.2
0.0
1.1.
1.1
1.2
1.3
K/(c- 1 /2 k )
1.4
1.5
reflection on the scattering of electromagnetic waves
from corrugated interfaces separating two dielectric
2l
media has been investigated. By using computer simulations for one-dimensional surfaces, it has been observed
that introducing a slight roughness in the interface can
2.0
give rise to a dramatic increase in the reflectance at
1.0
1.0
j
1.1
1.2
1.3
1.4
1.5
K/(&-'12 k)
Fig. 16. Average of the square modulus of the angular spectrum
of the transmitted field over 200 samples versus the parallel momentum in the nonradiative region.
produce qualitatively the distinct values taken on by the
transmission coefficients.
As we intend to study the alterations in the angular spectra introduced by the corrugation, we first plot
lAt(Dmin)(K)I2 in Fig. 15 for a flat surface, Dmin being the
minimum value of the surface whose statistical parameters are T =
= 0.2A. Note that, since the curves in
Fig. 15 are the result of the product between those
of Fig. 14 and the aforementioned exponential factor
exp[Dmin(K - ek 2 )"2 ] (of course, Dmin < 0), the first
subsidiary maxima appearing in the angular spectrum at
lower parallel momentum K are proportionally amplified with respect to the central lobe that corresponds to
larger K. These subsidiary maxima have been removed
in Fig. 15(a). Of course, if a Gaussian beam were employed to illuminate the surface, an apodization effect
with a Gaussian function would be present in Fig. 14, and
hence the subsidiary maxima and their subsequent amplification would not appear.
In Figs. 16(a) and 16(b) the average angular spectra
(IAt(Din)(K)I for e = 0.25 and e = 0.5, respectively,
are shown for the case of the surface with subwavelength T
(T = = 0.2A). We realize that the sinc structure observed in Fig. 15 for a flat interface is practically preserved. Even though the surface has a high slope, we
found in Section 3 that the transition about the critical
angle 0, still occurs and that almost total reflection exists
for o > 0, [cf. Figs. 9(a) and 9(b)]. Then it is not surprising that the angular spectra of the transmitted fields
3)
(Fig. 16) resemble those obtained for a flat surface
(Fig. 15). It
is due to the
the s waves;
lar spectrum
the latter.
CONCLUSIONS
In this paper the influence of roughness and total internal
3.0
0.0
In contrast, when the surface correlation length T exceeds a certain value (T 2 A), a slight roughness is high
enough to destroy the sharp transition about 0,. This is
confirmed first by the smooth behavior exhibited by the
total mean reflectance versus the angle of incidence in
Fig. 2 (T = 3.16Aand a-= 0.5A)and, second, by the broadening with absence of a peak structure in the angular
spectrum of the transmitted field for the same surface, as
the numerical calculations reveal. Because of the rather
uniform distribution, which spreads over the nonradiative
region, and hence its lack of a particular structure, we
have not considered this result worthy of being plotted.
5.
-p
waves: N=200
-------- s waves: N.=200
435
is also remarkable that the scattering that
corrugation affects the p waves more than
in the former, the departure of the angufrom the result for a plane is larger than in
angles of incidence near but below the critical angle, this
being most noticeable for p waves, for which the Brewster
angle is close to the critical angle (thus the reflectivity of
a flat interface for p polarization near this angle of incidence is indeed low).
For large roughness and correlation lengths greater than
or equal to the wavelength, a remarkable structure in the
angular distribution of mean scattered intensity arises.
Because of the total reflection near the specular and
backscattering directions, the interface reflectivity behaves like that of a metal for moderate angles of incidence
and observation, originating enhanced backscattering
with subsidiary maxima. Additional lobes can be observed at large angles of observation, once again because
of a selective increase of the reflectance because of the
combination of roughness and total internal reflection.
For steep slopes but subwavelength correlation distances, there are values of the refractive index for which
the reflectivity can exhibit large peaks at scattering angles
close to the critical angle of a plane interface. Nevertheless, the intimate mechanism by which the remarkable
structure present in this system is formed is not yet clear
to us. Furthermore, from the behavior of the total
incoherently scattered energy (Fig. 13) and the total reflectance (Fig. 9), we can conclude that the roughnessattenuated transition of these quantities about the critical
angle (which involves an exchange of energy from transmission to reflection) occurs predominantly between the
coherently scattered components. As an analysis of the
distribution of evanescent components reveals, the similarity in the angular spectra in the nonradiative region of
the transmitted field between these surfaces (Fig. 16) and
a plane (Fig. 15) corroborates the existence of a relatively
steep transition about the critical angle. Of course, as
the roughness increases, this angular distribution broadens over the nonradiative region.
As regards transmission for T > A, we have encountered an effect reciprocal to that reported in Ref. 28, according to which light is transmitted near the forward
direction. Since the wave is now incident from the denser
medium, this distribution appears reflected about the forward direction with respect to the one obtained when
436
M. Nieto-Vesperinas and J. A. Sanchez-Gil
J. Opt. Soc. Am. A/Vol. 9, No. 3/March 1992
light incides from the rarer medium. For angles of incidence larger than the critical angle, at which no transmitted light would be observed at a flat interface, roughness
can give rise to strong distributions of transmitted light.
It would be interesting to compare the results presented
in this paper with results of experiments with surfaces of
controlled statistics.
ACKNOWLEDGMENTS
J. A.
terial de Ciencia y Tecnologia under grant PB0278.
Sanchez-Gil acknowledges a grant from the Ministerio de
Educaci6n y Ciencia. Discussions with J. C. Dainty and
made possible by a NATO travel grant
(890528) are also appreciated.
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