JOURNAL OF MODERN OPTICS, 1995, VOL . 42, NO . 2 , 257-269
Enhanced backscattering due to total internal reflection
at a dielectric-air interface
R . E . LUNAt, E . R . MENDEZ
Centro de Investigacion Cientifica y de Educacion Superior de
Ensenada, Apdo . Postal 2732, Ensenada, B . C . 22800, Mexico
JUN Q . LU and ZU-HAN GU
Surface Optics Corporation, P .O . Box 261602,
San Diego, CA 92126, USA
(Received 20 November 1993 ; revision received 4 April 1994)
Abstract . We report the observation of enhanced backscattering in the
scattering of light from a photoresist film with a one-dimensional randomly rough
interface deposited on a flat parallel glass plate . The random interface is
illuminated from the photoresist side, entering the sample through the glass plate .
Angular scattering measurements for this sample are presented and compared
with results obtained numerically using two approximate models to describe the
interaction between the light and the sample . The observed backscattering
enhancement is believed to be due to multiple scattering processes at the
photoresist-air interface, where total internal reflection can take place .
1.
Introduction
Recently, several observations of enhanced backscattering and other related
effects in the angular distribution of light scattered from characterized randomly
rough surfaces have been reported in the literature [1-5] . In parallel, numerical
techniques to deal with surfaces with one-dimensional random profiles have been
developed and employed to study these problems [6-9] . Most of the attention has
been devoted to scattering from metallic surfaces although, more recently, rough
dielectric surfaces have also been considered [9-13] .
It has been suggested [1, 2] that for optically rough metallic surfaces, the
enhancement is due to multiple scattering of waves at the random interface . By now,
this idea seems to be well accepted and supported by the available evidence (see e .g .
the numerical work carried out in [9] and [13]) . Also in agreement with this model,
numerical calculations for p-polarized illumination show that an enhanced
backscattering peak can be observed for light scattered from a random rough
air-dielectric interface when the surface is reflective enough [9] . Similarly,
Nieto-Vesperinas and Sanchez-Gil [12] have shown through numerical work that
enhanced backscattering can be observed in the light scattered from a random
dielectric-air interface, when the light is incident from the dielectric side . They have
linked this observation with the phenomenon of total internal reflection .
t On leave from : Escuela de Ciencias Fisico-Matematicas, Universidad Autonoma de
Sinaloa, Mexico .
0950-0340/95 $1000 © 1995 Taylor & Francis Ltd .
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x3
4
Figure 1 .
Schematic diagram showing the sample geometry . The light is incident from the
side of the flat glass substrate at an angle 0 0 .
With more complex structures, on the other hand, such as rough dielectric films
deposited on flat reflecting substrates, sharp backscattering enhancement peaks have
been observed experimentally and in numerical simulations [14,15] . The mechanism responsible for the enhancement is, in this case, the constructive interference
between waves that follow reciprocal scattering paths when traversing (twice) the
randomly rough air-dielectric interface . The situation is similar to that of the double
passage configuration in which a mirror is illuminated through a thin phase screen
[16] .
Preliminary experimental results with purely dielectric samples, consisting of a
photoresist film with an interface that has two-dimensional roughness, deposited on
a glass plate, have shown the presence of a small backscattering enhancement peak
when the sample is illuminated from the side of the flat glass plate [17] . Here, we
report angular light scattering measurements from a sample that consists of a
photoresist film with a one-dimensional randomly rough interface deposited on a flat
parallel glass plate (see figure 1) . The light is incident on the sample from the side
of the parallel glass plate, and we measure the angular distribution of the reflected
light . Our results show well-defined and strong backscattering enhancement peaks .
These experimental results are compared with numerical simulations of the problem
using two approximate models to describe the sample geometry . It is argued that
the most important reflection is that arising from the photoresist-air interface, where
total internal reflection can take place, and we use this assumption to devise our
models .
In section 2 we describe, briefly, the technique employed for the fabrication of
the samples, and the instrument employed in the measurements . The approximate
models and the numerical simulation are described in section 3, and the experimental
and numerical results are presented and discussed in section 4 . Finally, a summary
of the results, together with our conclusions, are presented in section 5 .
Backscattering at dielectric-air interface
259
aperture
xl
SIDE VIEW
lens
cylindrical
lens
7
source
Figure 2 . Schematic diagram of the optical system employed for the fabrication of the
samples .
2.
Experimental details
The sample was fabricated with a variation of the technique described by Gray
[18] for the fabrication of two-dimensional randomly rough surfaces, and employed
subsequently by several authors for the fabrication of two-dimensional and quasi
one-dimensional surfaces [1-5] . Basically, the technique consists of exposing
photoresist-coated (Shipley AZ 1650) plates to speckle patterns with the appropriate
statistical properties . The procedure followed to prepare and develop the plates was
similar to the one described in [2] . The optical system employed for the fabrication
of the surface is shown schematically in figure 2 . A Gaussian beam arising from a
HeCd laser (wavelength 0 . 442 µm) is focused and cleaned by the spatial filter . The
expanding beam is converted into a converging Gaussian beam by an optical system
consisting of one or more lenses . Subsequently, a cylindrical lens and a diffuser are
introduced into the system . The cylindrical lens focuses the Gaussian beam onto a
line on the plane where the diffuser is placed . This arrangement produces elongated
`speckles' (uncorrelated areas in the radom intensity distribution) on the plane of the
photoresist-coated plate ; the correlation length of the speckle field produced is
several times larger in the x2 direction than in the x 1 one . In order to ensure far-field
conditions for the diffraction pattern of the diffuser, the photoresist-coated plate is
placed on what would be the plane of focus of the system in the absence of the
cylindrical lens and the diffuser . A rectangular mask, approximately ten speckle
correlation lengths wide (in the x2 direction), is placed just before the photoresistcoated plate, which is scanned in the x2 direction . Assuming a linear response of the
photoresist, the resulting samples should have a surface profile with an approximately Gaussian probability density function of heights and a Gaussian correlation
function . This profile should provide a good approximate to a realization of a
Gaussian random process with a Gaussian correlation function . The fabrication of
adequate samples was one of the most difficult aspects of the work reported here .
The resulting surface profile of the sample was measured with a Dektak 3030
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R . E . Luna et al .
lens
Figure 3 .
Schematic diagram of the bidirectional scatterometer employed in the measurements .
stylus machine . The stylus radius is approximately 0 . 15 gm, the sampling interval
was 0 . 1 gm, and the stylus loading was 1 mg . Statistical properties of the profile
defining the air-dielectric interface were calculated from 10 measured profiles . A
total of 2000 points were taken for each trace . The standard deviation of heights 6
was estimated to be about 1 . 0 gm, and the value of the correlation length a, as the
1 /e value of the correlation function, was 3 . 2 gm . Both the histogram of heights and
the measured correlation length were approximately Gaussian, but deviations from
the assumed model (Gaussian random process with a Gaussian correlation function)
were found in the second derivative of the measured profile .
We present angular light scattering measurements in the plane of incidence for
the characterized sample when the air-photoresist interface is illuminated through
the glass plate and the photoresist film . Scattering data were taken with an automated
bidirectional scatterometer, shown schematically in figure 3 . An expanded laser
beam (wavelength X = 0 . 6328 gm) was sent through a periscope with a series of
mirrors from which it exited horizontally toward the sample . For reflective
measurements the last mirror of the periscope obstructs the detector in the vicinity
of the backscattering direction and, in order to reduce this problem, the beam was
focused on this last mirror ; the obstruction can then be made narrower, permitting
measurements close to the backscattering direction . Thus, with this arrangement,
the surface was illuminated by a divergent beam .
To reduce the speckle noise produced in the far field of the surface it was
necessary to average spatially over many `speckles' . This was achieved by
illuminating an approximately 20 mm diameter area of the surface and by using a
field lens at the detector that integrated over an angle of about 2 . 3° . In normal
circumstances the angle subtended by the detector is smaller than the spatial
variations in the structure of the mean intensity, and it is expected that this
integration had only a small effect on the measurements . It was also verified that the
detector viewed the entire illuminated area of the diffuser, irrespective of the angle
of detection . The detector was mounted on a 73 cm long arm that rotated about the
sample in the horizontal plane . A PC-compatible computer was used to control the
position of the detector and store the data . Wide angle scans (from - 90° to 90°) were
taken with this instrument .
3.
Description of the simulation
The configuration of the sample is shown schematically in figure 1 . It essentially
consisted of a photoresist film with a one-dimensional randomly rough interface
deposited on a flat parallel glass plate . In our experiments, the light illuminated the
Backscattering at dielectric-air interface
261
rough interface after traversing the glass plate and the photoresist film . The thickness
of the glass plate was about 5 mm and the average thickness of the photoresist film,
d, was about 10 gm . By measuring various samples it was noticed that, at least over
a small range of variations in thicknesses about these values, none of these two
parameters seemed to be critical for the results .
Rigorous numerical simulations of light scattering by this complex structure
would be difficult to implement, mainly due to the comparatively large thickness of
the glass plate . However, since it is argued here that the mechanism responsible for
the enhancement is multiple scattering at the photoresist-air interface, approximate
models for describing the interaction between the incident field and the sample
should provide a reasonable description of its scattering properties . We have used
two models for describing this interaction, which are explained in the following .
In both models, the field incident on the sample was assumed to be a Gaussian
beam making an angle of incidence 0o with respect to the normal to the sample . In
the first model, the field incident on the rough photoresist-air interface, after
traversing the two flat interfaces, was calculated according to the well known laws
of refraction . The interaction between the beam and the rough photoresist-air
interface was treated quite rigorously, employing the techniques described in [9] and
[12] . This provided the angular spectrum of the scattered field inside the photoresist .
To propagate the light back, through the lower layers of the sample and into air, the
amplitude and propagating angles of the scattered field inside the photoresist were
modified to take into account the refraction that occurs at these lower interfaces . The
accuracy of this approximation depends on the relative importance of multiple
scattering effects due to these flat interfaces . In the following, we shall refer to this
model as the two-media model .
In the second model, the field incident on the photoresist film, after traversing
the glass plate, was calculated by considering the refraction due to the air-glass
interface . The interaction between the field and the film was treated rigorously [17],
and the resulting angular spectrum was then propagated through the glass plate and
into air, taking into account the refraction at the lower interface . In the following
we shall refer to this model as the film model . This model assumes that the lower,
flat interface introduces multiple scattering effects that can be neglected . It is
expected that both models give reasonable results for relatively small angles of
incidence and scattering . For large scattering angles the reflectivities of the lower
interfaces approach a value of one and this can produce significant amounts of
multiple scattering .
So, for the numerical simulation with the two-media model we assumed that the
system consisted of a one-dimensional rough interface separating air (refractive
index n„ = 1) from a semi-infinite photoresist medium . The interface is illuminated
by a Gaussian beam incident from the photoresist side with an angle of incidence
calculated using the values of the angle of incidence in air, and of the refractive
indices of glass and photoresist . For the second model we assumed that the system
consisted of a photoresist film with a one-dimensional rough interface, deposited on
a semi-infinite glass substrate . The film is illuminated by a Gaussian beam incident
from the glass side with an angle of incidence calculated using the values of the angle
of incidence in air, and of the refractive index of glass . The average film thickness,
d, was 10 gm . In all the calculation, the wavelength of the incident light (in air) was
taken to be 2 = 0 . 63 28 µm, and the indices of refraction of the photoresist film and
the glass substrate at this wavelength were taken to be np = 1 . 64 and ng = 1 . 51,
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R . E. Luna et al .
respectively . The width of the Gaussian beam was w = L cos 0 ;/4, where 0; is the angle
of incidence inside the photoresist (two-media calculations) or inside the glass (film
calculations) . The statistical parameters of the surface employed in the simulations
were those determined experimentally . That is, the standard deviation of heights
6 = 1 µm, and the correlation length a = 3 . 2 gm . The length of the rough surface was
taken to be L = 40 gm, and the number of points along the surface used for the
numerical implementation of the technique was N=400 . The total number of
surface profiles used in the averaging procedure was Np = 500 for all the results .
4.
Results and discussion
Far field experimental results for our sample are shown in figures 4-7 . Also in
these figures we show the results of the Monte Carlo type numerical simulations
obtained with our approximate models . In figure 4 (a) we show angular light
scattering measurements of the sample taken with the bidirectional scatterometer
depicted in figure 3, when the sample is illuminated by a p-polarized beam incident
at an angle of 10° with respect to the normal . The relatively strong specular
component due to the air-glass interface was blocked and it is not shown in the figure ;
a `hole' encompassing a few degrees about the specular direction is visible in the
figure . The exact retro-reflected signal was also blocked by the illuminating mirror
of the scatterometer, but this region is only a fraction of a degree wide . It is clearly
seen in this figure that there is a fairly strong backscattering peak at - 10° . It is also
worth noting that the scattering pattern is wide, and that significant amounts of
scattered light are found at large angles of scattering . Numerical results for the
incoherent component of the differential reflection coefficient ((aR P /a05 ) for p
polarization, or (aR s /a0 5) for s polarization), which represents the fraction of the
incident energy scattered per unit angle [9] are shown in figure 4 (b) . The results were
obtained with the approximate models described above, assuming p-polarized light
and an angle of incidence of Oo = 10° in air . The thicker curve represents the results
obtained with the two-media model, and the thinner one the results obtained with
the film model . It can be seen that despite the simplicity of the assumed models, the
numerical results are qualitatively similar to those shown in figure 4 (a) . As with
the experimental results, there is a clear backscattering peak and the overall shape
of the two curves is quite similar . Nevertheless, the peak obtained with the
simulation is not as pronounced as the one in the experimental curve . Comparison
of the two numerical curves in figure 4 (b) can also be illustrative, as their differences
are due to the fact that in the film calculations we are including the photoresist-glass
interface in a rigorous manner . One can see that the two curves agree quite well in
the neighbourhood of the backscattering direction, and that the overall shape of the
curves is quite similar . However, the film calculations predict a slightly lower
differential reflection coefficient than the two media calculations .
In figure 5 we show results obtained when the sample is illuminated by a
p-polarized beam incident at an angle of 20° . As before, for the experimental curve
(fig . 5 (a)) the strong specular reflection due to the air-glass interface is not shown .
Also, as in the previous case, there is a strong backscattering peak, this time at - 20° .
When compared with the 10° incidence case, we notice that the enhanced
backscattering peak is smaller but slightly wider . This is the general behaviour found
when the angle of incidence is increased . Figure 5 (b) shows the incoherent
component of the mean differential reflection coefficient of the scattered light for a
p-polarized beam incident at an angle of 20° obtained with our numerical
Backscattering at dielectric-air interface
263
Experimental Results
(a)
(b)
Figure 4 . Angular intensity distribution of the mean intensity in the far field for the case
of p polarization and an angle of incidence of 10° . (a) Measurements of the scattered
intensity and, (b) numerical results for the incoherent component of the differential
reflection coefficient calculated with our approximate models ; the bold line represents
the two-media calculations, and the thin line the film calculations .
264
R . E . Luna et al .
Experimental Results
Scattering Angle (deg .)
(a)
(b)
Figure 5 . Angular intensity distribution of the mean intensity in the far field for the case
of p polarization and an angle of incidence of 20 ° . (a) Measurements of the scattered
intensity and, (b) numerical results for the incoherent component of the differential
reflection coefficient calculated with our approximate models ; the bold line represents
the two-media calculations, and the thin line the film calculations .
Backscattering at dielectric-air interface
265
Experimental Results
0 .10
0 .08
0 .02
0 .00'.
-90
-60
-30
'
0
30
60
90
60
90
Scattering Angle (deg .)
(a)
0 .08
0 .06
r
c
m
0 .04
a
m
0 .02
0 .00
-90
-60
-30
0
30
Scattering Angle (deg .)
(b)
Figure 6 . Angular intensity distribution of the mean intensity in the far field for the case
of s polarization and an angle of incidence of 10° . (a) Measurements of the scattered
intensity and, (b) numerical results for the incoherent component of the differential
reflection coefficient calculated with our approximate models ; the bold line represents
the two-media calculations, and the thin line the film calculations .
266
R . E . Luna et al .
Experimental Results
Scattering Angle (deg .)
(a)
0 .10
0 .08
0 .02
0 .00
-90
-60
-30
0
30
60
90
Scattering Angle (deg .)
(b)
Figure 7 . Angular intensity distribution of the mean intensity in the far field for the case
of s polarization and an angle of incidence of 20 0 . ( a) Measurements of the scattered
intensity and, (b) numerical results for the incoherent component of the differential
reflection coefficient calculated with our approximate models ; the bold line represents
the two-media calculations, and the thin line the film calculations .
Backscattering at dielectric-air interface
267
simulations . A small, but well-defined peak is observed in the retro-reflection
direction, namely - 20° . As for the 10° incidence case, the numerical results are
similar to those found experimentally, with some discrepancies pertaining mainly
to the height and shape of the backscattering enhancement peak . As expected, the
main differences between the experimental results and those obtained with our
approximate models, occur at large scattering angles . Also, as in the previous case,
there are some differences between the film and two media calculations . We find that,
for most angles of scattering the two-media calculations predict larger scattering
cross-sections than the film calculations, but that the two models coincide in the
vicinity of the backscattering direction .
The corresponding experimental and numerical results for s polarization are
shown in figures 6 and 7 . There are only slight differences with the case of p
polarization and, given the similarity of the results, the same comments can be made
for this case of s-polarized illumination .
For the scattering from air-dielectric interfaces (light incident from air) with
statistical parameters and optical properties similar to those of our sample, it has been
shown numerically and experimentally that an enhanced backscattering peak is
not present, or is weak [9,17] when p-polarized light is incident on the sample . For
s-polarized light, on the other hand, a well defined peak can be observed in the
backscattering direction . This difference may be attributed to the different
reflectivities of the surface for these two kinds of polarization . With p-polarized light,
it is only when the refractive index of the dielectric is increased, so that the surface
becomes more reflective, that a backscattering peak appears . All this supports an
explanation for the enhancement based on the multiple scattering of waves at the
air-dielectric interface . On the other hand, for the analogous situation with a
dielectric-air interface (light incident from the dielectric side), the reflectivity is
much higher due to the possibility of having total internal reflection at the interface .
Numerical studies of this problem have shown that backscattering enhancement
occurs in this situation [12] . Given these circumstances, and the fact that the results
obtained with our model are in relatively close agreement with the experimental
ones, we believe that the main scattering processes occurring in our samples
take place at the dielectric-air interface . In this case, the phenomenon of total
internal reflection becomes the dominant mechanism responsible for the observed
effects .
It is to be noted that, in general, the experimental backscattering peaks obtained
experimentally are wider than the peaks obtained with the numerical simulations,
and that the numerical results show sharp oscillations around the backscattering
peak . This discrepancy could be due, at least partly, to the smoothing of the
experimental curves by the detecting aperture . The fact that the experimental
backscattering peak is stronger than those predicted by our models could be due to
the additional interaction and multiple scattering effects introduced by the lower
air-glass interface . Note that although the rigorous inclusion of the glass-photoresist
interface did not have a significant effect on the backscattering peak (compare e .g .
the two numerical curves on figures 4 (b) to 7 (b)), the reflectivity of the glass-air
interface is higher, and thus multiple scattering effects could have a non-negligible
effect in this case, even for small angles of incidence and scattering . Another
possibility for explaining these discrepancies could be the deviation, in the
higher-order statistics, of the random profile of the sample from the assumed model
of a Gaussian random process with a Gaussian correlation function .
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R . E . Luna et al .
5.
Summary and conclusions
We have presented measurements of the angular distribution of the light
scattered by a rough one-dimensional dielectric film deposited on a flat glass plate
when the incident beam impinges on the sample from the side of the glass plate . We
have observed strong backscattering peaks both with p- and s-polarized illumination . Numerical simulations of the problem employing two approximate models for
the sample and the measured statistical parameters of the rough interface, were
presented and compared with the experimental results .
Some differences have been found between the numerical calculations and the
experimental results, although the overall shape of the curves agree quite well . We
have found that the measured backscattering enhancement peaks are stronger than
those predicted by the numerical simulations . This could be due to the fact that our
models neglect the multiple scattering introduced by the flat air-glass interface .
There are also some differences concerning the width and shape of the peak . The
simulations show the presence of secondary maxima on the sides of the
backscattering enhancement peak, a feature that was not observed experimentally .
These oscillations are quite sharp and are particularly evident for angles of incidence
of 10° or smaller . The facts that the secondary maxima were not found
experimentally and that the experimental peak is wider could be partly attributed
to the smoothing of the experimental scattering curve by the detector aperture .
We believe that the reflection at the rough photoresist-air interface is responsible
for most of the features observed ; the multiple scattering occurring at this rough
interface, aided by total internal reflection effects, provides the basic mechanism for
producing the backscattering enhancement . Our belief is supported by the similarity
between the experimental and numerical results .
Acknowledgments
R . E . Luna wishes to express his gratitude for the support of the Consejo
Nacional de Ciencia y Tecnologia and the Universidad Autonoma de Sinaloa . This
work has been supported by the Consejo Nacional de Ciencia y Tecnologia through
grant F196-19205, and by the US Army Research Office through grant DAAH0493-C-0014 .
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