Using diffuse light to measure the refractive index of turbid colloids

The refractive index of materials is an essential concept for most optical applications. Although the refractive index of a homogeneous medium (for which light scattering is negligible) is well understood, when scattering within a material is significant (e.g., in turbid colloidal suspensions), the idea of an effective refractive index becomes unclear and can be controversial. Indeed, we have previously (for inhomogeneous media) used scattering theory to establish that although the effective refractive index of a turbid colloid can be clearly defined, it cannot be used as a regular refractive index (i.e., as that of a homogeneous medium). For example, the effective refractive index cannot always be used with Fresnel coefficients (used to describe the behavior of light as it passes between media with different refractive indices) to calculate amplitude reflection coefficients. For the case of a highly scattering homogenized medium, this is because of the non-local nature of its optical response.^{1, 2} We have also shown, however, that the effective refractive index can be used as a regular refractive index to refract light into a turbid medium.³ Moreover, although the reflectivity of a light beam may not match the Fresnel reflection coefficients, it is possible to use the reflectivity curves (with a proper model) from a turbid particle suspension to retrieve information about the suspended particles.^{1, 4}

Despite the confusion surrounding the refractive index of scattering media, for many years researchers have found that the effective refractive index concept is useful, i.e., in many cases it provides physical insight and considerably reduces mathematical complexity. It has also been shown that the effective refractive index can be measured through simple means, if the diffusely scattered field is ignored and only the coherent beam is considered.^5–7 In many cases, a liquid (with a turbid appearance) can be introduced into a standard Abbe refractometer and a clear reading of its refractive index can be obtained. In the case of a turbid colloidal suspension, the reading of the refractive index will depend on the amount and type of the particles that scatter the light (and cause the turbid appearance). For colloidal suspensions where scattering is too strong, however, a conventional Abbe refractometer cannot be used to obtain the refractive index because insufficient light is transmitted through the sample.

In recent work, we therefore developed a modified Abbe refractometer to measure the refractive index of highly turbid colloids.⁸ In our experimental device (see Figure 1) we use a diode laser beam to illuminate the interface between a glass semi-cylindrical prism and a turbid colloid sample (which is in contact with the prism) from the prism side. We also use a simple lens and a CCD camera to measure the angular profile of the diffuse light that is backscattered into the prism, over a critical angle range.

Figure 1. Schematic illustration of the experimental device used to measure the angular intensity profile of diffuse light that is backscattered from a turbid colloidal sample into a semi-cylindrical glass prism. A CCD camera, placed in the focal plane of a simple lens, is used to measure the angular profile and to integrate the image in the direction perpendicular to the plane of incidence. θ: Viewing angle.

For cases where there is no strong scattering in the sample, the angular profile of the diffuse light transmitted back into the prism should be non-zero if the viewing angle is smaller than the critical angle. The measured angular profile is practically zero, however, if the viewing angle is larger than the critical angle. In addition, when light is mainly absorbed in the sample (i.e., there is very little scattering), the amount of light that is transmitted just below the critical angle decreases, and for viewing angles above the critical angle, the transmitted light is null. In contrast, for situations when scattering in the colloidal sample is strong (e.g., examples in Figure 2), the angular transmission profile becomes smooth around the critical angle. For angles larger than the critical angle, it is thus clear that light is being transmitted across the interface between the optical glass and the liquid in which the particles are suspended. In fact, it is not possible to explain this change in angular profile with existing theories that deal with the propagation of diffuse light within scattering media.

Figure 2. Experimentally obtained angular intensity profiles (as a function of viewing angle) of backscattered diffuse light (at a wavelength of 670nm) across the critical angle for different particle concentrations (indicated) at the interface between a colloidal sample and an optical glass (with a refractive index of 1.51). In (a) the sample is a colloidal suspension of poly(methyl methacrylate), i.e., latex, particles (radius about 225nm) in water and in (b) it is a suspension of titanium dioxide (TiO₂) particles (average radius of 120nm) in water. a.u.: Arbitrary units.

From our experimental observations for a latex colloidal solution—see Figure 2(a)—we find that as more of the scattering particles are added to the sample (and keeping everything else equal), the angular profile of the backscattered light is displaced laterally (i.e., the profile shape shifts to higher viewing angles). This is a clear result of a change in effective refractive index. It is important to note, however, that the angular intensity profile is obtained from the diffuse light, whereas the effective refractive index concept has been established only for the coherent component of light in turbid colloids.

In our work,⁸ we have thus used a previously developed model⁹—in which the angular intensity profile of backscattered diffuse light around the critical angle is related to an effective refractive index of the turbid medium⁹—to fit our experimental profiles from the highly turbid latex colloidal solutions. From our modeling results we determined a complex effective refractive index for diffuse light. Furthermore, we find very good agreement between our formulation and established calculations for the effective refractive index of turbid colloids, i.e., observed with the coherent component of light.⁵

In summary, we have successfully used a modified Abbe-type refractometer to measure the effective refractive index of highly scattering samples (i.e., turbid colloids).^{8, 9} Our proposed device—which can only be used with media that scatter light—offers a robust and straightforward way to sense highly turbid media. We have also provided theoretical and experimental evidence that a complex effective refractive index can be used to model the propagation of diffuse light with turbid colloids. In our future work, we will aim to develop a complete theory, based on a multiple-scattering formalism, to address the role of the effective refractive index of turbid colloids in the propagation of diffuse light.