See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/252270693 Total internal reflection scattering Article in Applied Physics Letters · October 2004 DOI: 10.1063/1.1801681 CITATIONS READS 6 1,546 3 authors: Marco A.C. Potenza Doriano Brogioli University of Milan Università degli Studi di Milano-Bicocca 128 PUBLICATIONS 1,307 CITATIONS 98 PUBLICATIONS 3,121 CITATIONS SEE PROFILE Margherita Giglio IEO - Istituto Europeo di Oncologia 153 PUBLICATIONS 2,335 CITATIONS SEE PROFILE Some of the authors of this publication are also working on these related projects: Nonequilibrium fluctuations View project capmix View project All content following this page was uploaded by Doriano Brogioli on 26 March 2014. The user has requested enhancement of the downloaded file. SEE PROFILE APPLIED PHYSICS LETTERS VOLUME 85, NUMBER 14 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 microscopy.4 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 intensity. 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 a) 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. 0003-6951/2004/85(14)/2730/3/$22.00 2730 © 2004 American Institute of Physics Downloaded 15 Oct 2004 to 159.149.47.51. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp Appl. Phys. Lett., Vol. 85, No. 14, 4 October 2004 Potenza, Brogioli, and Giglio 2731 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 = 2n1 / 0 and k2 = 2n2 / 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 2n222 2 2 ␦ + ⵜ ␦ = − ␦ − ␦n, xy z2 c2 c2 共1兲 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 2 ␦共q,z兲 = − 共k22 − q2兲␦共q,z兲 + F共q,z兲, z2 共2兲 where q = 共qx , qy兲 is the projection of the scattering wave vector onto the interface plane and F共q,z兲 = − 2n22 exp共− 冑q2i − k22z兲␦n共q − qi,z兲. c2 共3兲 The solution is found by means of the arbitrary constants method and two integration constants are defined through the 1 冑k22 − q2 冕 ⬁ ␦n共q − qi,z⬘兲 0 ⫻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 ⬀ i 冑q2 − k22 冕 ⬁ ␦n共q − qi,z⬘兲 0 ⫻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 ⬀ 1 冑兩q2 − k22 FQ关␦n共x,y,z = 0兲兴, 共6兲 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. Downloaded 15 Oct 2004 to 159.149.47.51. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp 2732 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. 1 C. Allain, D. Ausserre, and F. Rondelez, Phys. Rev. Lett. 49, 1694 (1982). K. H. Lan, N. Ostrowsky, and D. Sornette, Phys. Rev. Lett. 57, 17 (1986). 3 B. Lin, S. Rice, and D. A. Weitz, J. Chem. Phys. 99, 8308 (1993). 4 D. C. Prieve and J. Y. Walz, Appl. Opt. 32, 1629 (1993). 5 H. Matsuoka, Macromol. Rapid Commun. 22, 51 (2001). 2 Downloaded 15 Oct 2004 to 159.149.47.51. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp View publication stats
