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ApplPhysLett 2004

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Total internal reflection scattering
Article in Applied Physics Letters · October 2004
DOI: 10.1063/1.1801681
3 authors:
Marco A.C. Potenza
Doriano Brogioli
University of Milan
Università degli Studi di Milano-Bicocca
Margherita Giglio
IEO - Istituto Europeo di Oncologia
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4 OCTOBER 2004
Total internal reflection scattering
Marco A. C. Potenza,a) D. Brogioli, and M. Giglio
Dipartimento di Fisica “Aldo Pontremoli,” Università di Milano and INFM, via Celoria 16, I-20133,
Milano, Italy
(Received 11 May 2004; accepted 9 August 2004)
We show that the wave front of a total internal reflected beam is perturbed by fluctuations in the
region probed by the evanescent wave, and light is scattered both above and below the critical angle.
While singly scattered light is related to the two-dimensional Fourier transform at the boundary,
multiply scattered light originating from very turbid samples can appear only below the critical
angle. We show that the very weak scattered light above the critical angle is due to a double tunnel
effect, and it is solely due to single scattering at the surface. Applications are discussed. © 2004
American Institute of Physics. [DOI: 10.1063/1.1801681]
A plane wave propagating in a medium of index of refraction n1 is totally reflected at a flat boundary with a medium of index n2 if n1 ⬎ n2 and the incidence angle ␪i is
larger than ␪c = arcsin共n2 / n1兲. An evanescent wave (EW) is
generated close to the boundary in medium 2, the depth of
the perturbation being a small fraction of the light wavelength. Total internal reflection (TIR) can be frustrated by
placing a slab of material of higher index of refraction close
enough to the boundary, so to allow the incoming beam to
tunnel through. Light however can also be scattered from the
EW by microscopic particles that come close enough to the
boundary. The random interference of the light scattered by
many particles gives rise to a speckle field, that can be studied by means of intensity fluctuation spectroscopy.1–3 Alternatively, light scattered from a single object can be utilized
for the so-called frustrated total internal reflection
Geometrical optics arguments predict that the light scattered in medium 2 is refracted in medium 1 with directions
inside a cone of angular width ␪c around the normal to the
plane. So the possibility of observing light scattered close to
a TIR beam is ruled out by geometrical optics, the semispace
outside ␪c being a forbidden region for the light. The present
letter addresses the question whether any light is actually
scattered around a TIR beam, and what type of information
can be gathered by the measurement of the scattered light
We will present some preliminary observations of light
scattered both in the permitted and in the forbidden region
from a very turbid sample placed at the glass interface. A
theory will then be presented to describe all the observed
features in the two regions. Perhaps the most striking result
of the present work is that the light scattered in the forbidden
region must be singly scattered light, since multiple scattered
light must appear in the permitted region only. We believe
the method will prove to be of interest for the study of very
turbid systems, like colloids, emulsions, and foams. We call
this method total internal reflection scattering (TIRS).
The optical setup is shown in Fig. 1. TIR occurs at the
interface between medium 1, glass, with refractive index n1
共z ⬍ 0兲, and medium 2, the sample, with refractive index
n2 ⬍ n1 共z ⬎ 0兲. The incidence angle is ␪i ⬎ ␪c. The scattered
Electronic mail: marco.potenza@unimi.it
light falls onto a ground glass screen (S), and the scattered
intensity distribution is recovered by imaging the screen onto
a CCD sensor with an ordinary photocamera objective. The
bright TIR beam is strongly attenuated in front of the screen
plane. The light source is a 10 mW He-Ne laser (the wavelength is ␭0 = 6328 Å) and the beam is about 0.65 mm in
diameter. The refraction at the exit face of the prism distorts
the scattered intensity distribution. When this effect is accounted for, the scattering appears symmetric around the TIR
beam. All the data shown in the following have been corrected for this effect.
Measurements have been performed on highly concentrated suspensions of polystyrene spheres and foams (shaving cream).
Highly concentrated (about 10% by volume) monodisperse polystyrene spheres suspensions have been left sedimentating over the top, horizontal surface of the right-angle
glass prism. The images in Fig. 2 represent the scattered
intensity distribution for 5 and 10 ␮m samples. The data
have been taken at steady state, when presumably sedimentation process had already come to completion. The typographically added white line indicates the border line ␪ = ␪c.
So above it we have the ␪ ⬎ ␪c condition (the forbidden region) and below it we have the ␪ ⬍ ␪c (permitted region).
The following points can be made. (1) There is indeed
scattered light around the main TIR beam in the forbidden
region. The scattered intensity is in the form of concentric
rings around the beam axis, and appreciable changes in the
FIG. 1. Experimental setup and coordinate system. The sample at the top of
a right-angle prism scatters light around the total internal reflected beam.
The scattered light (dotted lines) is collected onto a screen (S) and the
pattern recorded by the CCD camera. The dashed line ␪ = ␪c separates the
permitted region A 共␪ ⬍ ␪c兲 and the forbidden region B 共␪ ⬎ ␪c兲.
© 2004 American Institute of Physics
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Appl. Phys. Lett., Vol. 85, No. 14, 4 October 2004
Potenza, Brogioli, and Giglio
boundary conditions, thus yielding two different regimes.
A. 兩q 兩 ⬍ 兩k2兩 (permitted region): The scattered field, here
referred to as ␦␸A, propagates through the medium 2 and
refracts inside medium 1 with an angle ␪ ⬍ ␪c. The field
evaluated at z = 0 is proportional to
␦␸A共z = 0,q兲 ⬀
FIG. 2. The far field intensity distribution of TIRS light from sedimented
polystyrene spheres of (a) 5 ␮m and (b) 10 ␮m in diameter. The central spot
is the attenuated TIR beam and the curved line is ␪ = ␪c.
scattering intensity occur along each ring, the intensity peaking close to the boundary line. (2) The shape of the pattern
below the ␪ = ␪c line follows the pattern above. (3) The first
ring is wider in Fig. 2(b) than in 2(a). By reducing the beam
diameter, the intensity becomes speckled, and when the
beam is smaller than 150 ␮m, the scattering pattern breaks
into-well defined spots shown in Fig. 3.
We now present a theory which attempts to describe in a
quantitative way the previous results. Let the refractive index
of the scattering sample be n共x , y , z兲 = n2 + ␦n共x , y , z兲, where
兩␦n兩 Ⰶ n2. The incident field is assumed to be a plane wave
impinging on the surface at an angle ␪i. Let the EW in medium 2 be ␸2共x , y , z兲, qi = k1 sin ␪i the spatial frequency of
the incident wave on the xy plane, k1 = 2␲n1 / ␭0 and k2
= 2␲n2 / ␭0. The refractive index fluctuations cause a perturbation of the optical field in medium 2, ␸ = ␸2 + ␦␸, where
兩␦␸兩 / ␸2 Ⰶ 1. By neglecting the second-order terms in the
general wave equation one obtains:
n22 2
⳵ z2
where the ⵜxy operator denotes derivatives along x and y
axes. We now Fourier transform the fields in the xy plane.
The wave equation for ␦␸共q , z兲 then becomes
␦␸共q,z兲 = − 共k22 − q2兲␦␸共q,z兲 + F共q,z兲,
⳵ z2
where q = 共qx , qy兲 is the projection of the scattering wave
vector onto the interface plane and
F共q,z兲 = −
exp共− 冑q2i − k22z兲␦n共q − qi,z兲.
The solution is found by means of the arbitrary constants
method and two integration constants are defined through the
冑k22 − q2
␦n共q − qi,z⬘兲
⫻exp共− 冑q2i − k22z⬘兲exp共− 冑k22 − q2z⬘兲dz⬘ , 共4兲
where the numerical factors have been dropped.
B. 兩q 兩 ⬎ 兩k2兩 (forbidden region): In this case the field,
␦␸B, does not propagate through medium 2. The fluctuations
of refractive index are then sources of EWs, which tunnel
back through the interface and give rise to the propagating
waves in medium 1 共␪ ⬎ ␪c兲.
We obtain for the scattered field at the surface:
␦␸B共z = 0,q兲 ⬀
冑q2 − k22
␦n共q − qi,z⬘兲
⫻exp共− 冑q2i − k22z⬘兲exp共− 冑q2 − k22z⬘兲dz⬘ . 共5兲
In medium 1, for each q ⬍ k1 both the solution ␦␸A and ␦␸B
give source to propagating waves, which direction is given
by the wave vector 共q , 冑k21 − q2兲.
It is quite convenient to introduce the scattering vector
Q = q − qi, that is the projection onto the interface, xy plane of
the difference between the scattered and the TIR wave vector. In the limit Q Ⰶ k2, Eqs. (4) and (5) can be cast in a very
suggestive form. Since the last exponential term equals 1 and
the argument of the integral does not vanish over a small
depth determined by the EW, the integration region is effectively reduced to a distance of a few hundred nanometers. As
a consequence, the two solutions contained in Eqs. (4) and
(5) yield the remarkable result:
␦␸共q,k1z兲 ⬀
冑兩q2 − k22 FQ关␦n共x,y,z = 0兲兴,
that is the two-dimensional Fourier transform 共F兲 of the index fluctuations at the interface, evaluated at the scattering
wave vector Q. This is very similar to the classical, twodimensional scattering approximation.
Equation (6) accounts for the intensity distributions presented in Figs. 2 and 3 and shows that the sphere packing at
the interface occurs as a clustering of hexagonal lattice
patches of approximately 150 ␮m size. It also accounts for
the intensity peaking close to the border, due to the vanishing
of the denominator on the right-hand side term. Finally, the
large number of peaks in Fig. 3 is due to the fact that only
the very bottom portion of the spheres close to the boundary
contributes to the scattering, and consequently the form factor is much wider than the structure factor (an ordinary transmission scattering would show only few peaks).
Finally we discuss the capability of TIRS to study very
turbid samples avoiding troubles associated with the multiple
scattering. For a highly diffusive sample the component ␦␸A
of the scattered field may propagate through, be multiply
scattered, and partially re-enter in the permitted region. In
case of rampant multiple scattering, the intensity distribution
is expected to be much broader than for single scattering. On
the contrary, scattered light appearing in the forbidden region
must be scattered only once by a very thin layer close to the
FIG. 3. TIRS light from polystyrene spheres 30 ␮m in diameter with a
beam spot of 150 ␮m. Scattering appears to be due to a single hexagonal
crystal lattice.
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Appl. Phys. Lett., Vol. 85, No. 14, 4 October 2004
FIG. 4. TIRS distributions for a foam (shaving cream) for ␪i ⬃ ␪c (a) ␪i
− ␪c = 10° (b). (a) The scattering occurs both above and below the border
line, where the lobe is wider because of multiple scattering. (b) The scattering occurs primarily in the forbidden region, and the scattering lobe is narrow (single scattering). The 8 bit dynamical range of the images is represented on a logarithmic scale, and scattered intensity levels are different for
the two figures.
boundary (light can only originate from a double tunnel effect). We report in the following evidence of the different
behavior of the scattered light in the two regions. Figure 4
shows two scattered intensity distributions obtained with a
dense shaving cream foam. In both figures a curved, dark
line has been added to indicate the border line, and the scat-
Potenza, Brogioli, and Giglio
tered intensity is mapped with gray tones.5 The strong refracted beam is blocked (dark patch at center). The permitted
region is below, while the forbidden is above the dark line.
Figure 4(a) refers to ␪i ⬃ ␪c, while ␪i is appreciably larger
than ␪c in Fig. 4(b). The following observations can be made:
(1) the scattering pattern in Fig. 4(a) is wider in the permitted
region (below) than in the forbidden one (above). (2) The
width of the scattering pattern in the forbidden region in Fig.
4(a) is similar to that in Fig. 4(b). (3) The integrated scattered power in the forbidden region is a very small fraction
of that scattered in the permitted region.
All the above appears to support the idea that the light
scattered in the forbidden region is indeed due to single scattering alone. We believe that this could be of use in analyzing and monitoring turbid samples in a simple way.
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