Mie Scattering

5.16.2 Mie Scattering

If the scatterer is larger than the subwavelength scatterers responsible for Rayleigh scattering, being, say, several times the wavelength of light, a number of distinctive features are found to characterize the scattered radiation as compared with those in Rayleigh scattering. The scattering by such larger particles is commonly referred to as ‘Mie scattering’ (also known as ‘Lorenz-Mie-Debye scattering’ on account of the contributions of Lorenz and Debye), since it was Mie who put forward a complete theory of scattering of electromagnetic waves by a spherical particle of any given radius, where the particle may be either a conductor or a dielectric body. Since the radius (a) of the sphere in this theory can have any given value, one can consider special cases where the radius is small or large compared with the wavelength λ, or has an intermediate value comparable to λ. In the limit of small size of the scatterer, one actually recovers the results relating to Rayleigh scattering (see the results given in Sections 5.16.1.3 and 5.16.1.4).

While Mie’s theory gives precise results (in the form of infinite series expansions) for a spherical scatterer, the results are of considerable qualitative relevance for scatterers of other shapes as well. I will briefly relate here how a few important features of the scattered radiation undergo a gradual transformation as the size of the scatterer is made to increase gradually.

For a relatively large size of the scatterer, the scattered waves originating from the different parts belonging to it and emitted in any given direction possess a degree of mutual coherence, and their superposition is responsible for the distinctive features of Mie scattering. Stated differently, the scattered radiation is, in general, multipolar in nature and not simple dipolar radiation.

One striking difference from Rayleigh scattering is that as the scatterer becomes larger, the relative preponderance of the smaller wavelengths in the scattered radiation is gradually evened out until, for a size approximately 10–100 times the wavelength, all wavelengths are scattered equally. This explains the white color of clouds, where all the components of sunlight are scattered equally by the aggregates of water molecules in these clouds.

More precisely, the dependence of the scattering cross section on the parameter $\frac{a}{\lambda }$ in Mie scattering is of an oscillatory nature, especially for intermediate values of the parameter. If the total scattering cross section, obtained by integration of the differential cross section over all directions, is denoted by σ, then the ratio $\frac{\sigma }{\mathrm{\pi }{a}^2}$, which we term the scattering efficiency, varies as ${(\frac{a}{\lambda })}^4$ in the Rayleigh limit, while for more general values of the parameter, oscillations occur as shown in Fig. 5.45. Thus there occurs enhanced scattering for a sequence of values of $\frac{a}{\lambda }$ (referred to as Mie resonances), with relatively low values in between two successive enhancements. The oscillations are pronounced for scatterers of size comparable to the wavelength, and are damped at relatively large values of $\frac{a}{\lambda }$.

Fig. 5.45. Illustrating the occurrence of Mie resonances. The relative scattering cross section or the scattering efficiency $\frac{\sigma }{\mathrm{\pi }{a}^2}$ is plotted as a function of the relative size parameter $\frac{a}{\lambda }$. For small values of the size parameter (Rayleigh limit) the dependence is of the form ${(\frac{a}{\lambda })}^4$, while for larger values, with the scatterer size comparable to the wavelength, oscillations occur. The oscillations are damped for still larger values of the size parameter.

Another distinctive feature of Mie scattering is a lack of symmetry between the scattering in the forward and backward directions, the scattering in the forward direction being relatively more pronounced, which increases with an increase in the size of the scatterer. What is more, for a sufficiently large scatterer, the angular distribution of scattered radiation possesses a number of maxima and minima, resembling the maxima and minima in the intensity distribution in a diffraction pattern (see Fig. 5.46). Indeed, for a scatterer (say, of a spherical shape) of size approximately 100 times the wavelength or larger, the modification of the incident wave by the scatterer can be described as diffraction, where the wave bends around the sphere and, at the same time, fans out to a certain extent away from the forward direction.

Fig. 5.46. Angular distribution of scattered radiation (logarithmic polar plot, with the direction of incidence as the polar axis) in Mie scattering with unpolarized incident light (compare this with Fig. 5.42 for Rayleigh scattering, which is the limiting case of Mie scattering for a small scatterer). Notable features of Mie scattering are the dominance of forward over backward scattering and the maxima and minima in the angular distribution. The direction of incidence is OA (compare this with Fig. 5.42, where this direction is named ‘OC’).

Mie Scattering

In 1908 Gustav Mie developed a theory to explain the scattering of radiation by homogeneous, spherical particles at all particle diameter-to-wavelength ratios. Mie's complete solution to Maxwell's electromagnetic field equations has become important to a number of application areas, notably in the field of meteorology where the sizes of the scattering particles found in clouds are similar or larger than the wavelength of the incident light. The theory has also found use in the imaging of substances such as milk and biological tissue. The theory uses a number of assumptions, namely that the particles being measured are spherical, the scattered light is measured from a single scattering event, the optical properties (absorbance and RI) of both the particle and dispersant are known, and the particles are homogeneous. Despite Mie theory being developed at the turn of the twentieth century, and updated formulations being developed in the 1940s, only recently has modern computing power allowed the theory to be applied to light scattering experiments such as laser diffraction measurements over a large dynamic range of particles sizes. Earlier instruments used the Fraunhofer approximation to estimate particle sizes. However, while simpler to use than the Mie theory, the Fraunhofer approximation model can lead to significant errors in the estimation of the particle size fraction for particles less than 50 μm in diameter. These errors become particularly critical when the particles are transparent. The Fraunhofer approximation makes a number of assumptions above and beyond those used in Mie theory that only hold for the scattering of light by large particles. This is why modern laser diffraction instruments, which are required to measure particles ranging in size from 2000 to 0.02 μm, use Mie theory to inversely model the observed scattering intensities.

3.4.6.2 Mie scattering

Mie scattering is the source of color in glasses containing colloids of noble metals (Ag, Au, Pt, Cu), as described in Chapter 2. A brief derivation of Mie's theory is given in Appendix 2A.

When the size of the scattering medium is much larger than the wavelength of the incident light, then the absorption and the intensity of the scattered light are essentially wavelength independent and produce a white color. This explains the color of clouds in the sky and of gray and white hair, the latter due to scattering of light from microscopic (micron-sized) gas bubbles in the hair.

Mie scattering

The exact solution for RS from an arbitrary sphere was obtained in 1908, with a real mathematical “tour de force,” by Gustav Mie. Here only the main feature of the results are described, commenting on some general aspects. First of all, all solutions are fixed. The general solution is fixed only by two parameters, namely m = n_P/n_s and the dimensionless size x = ka = 2πn_sa/λ, and is formally given by

[28]$\begin{array}{l}{S}_1=-{\displaystyle \sum _n\frac{2n+1}{n(n+1)}(a_n{\pi }_n+b_n{\tau }_n)}\hfill \\ S_2=-{\displaystyle \sum _n\frac{2n+1}{n(n+1)}(a_n{\tau }_n+b_n{\pi }_n)}\hfill \end{array}$

where ${\pi }_n(\vartheta )=P_n^1(\text{cos}\vartheta )/\text{sin}\vartheta$ and ${\tau }_n(\vartheta )=\text{d}{P}_n^1(\text{cos}\vartheta )/\text{d}\vartheta$ give, in terms of the associate Lagrange functions $P_n^1(\text{cos}\vartheta )$, the angular distribution of the scattered light. The amplitude coefficients a_n and b_n are complicated functions of m and x. What is important to point out, however, is that their leading behavior is $a_n\sim x^{2n+1},b_n\sim x^{2n+3}$, so that, for small particles, taking into account only the first terms is sufficient. By increasing n, the functions π_n and τ_n display in a polar diagram an increasing number of lobes, changing in direction and sign with n. For large particles, therefore, the sum of many terms tends to cancel out scattering at most angles. The only exceptions are the lobes that are present for all n around ϑ = 0 and ϑ = 180, the latter however alternating in sign. As a result, large particles tend to scatter predominantly forward, with a residual backscattering cone (so one gets dazzled by back-lighted drops on a windshield). For very large particles, the forward lobe has a width Δϑ ∼ λ/a, witnessing the merging of Mie scattering with the classical diffraction theory. Figure 2 compares the full Mie and approximate RGD solutions for polystyrene latex spheres in water.

Figure 2. Normalized scattering intensity vs. scattering angle ϑ for polystyrene spheres (n_P = 1.59 in water for λ₀ = 633 nm. For each value of the ratio a/λ, indicated close to the curves, the full and dotted lines are, respectively, the numerical result from Mie theory and corresponding RGD analytical solution.

The coefficients a_n and b_n are finite for all real values of x. However, for strongly absorbing particles, they often present complex poles with very small values of the imaginary part. These “quasiresonances” lead to strong enhancement of specific modes, and are responsible for the beautiful colors often observed in suspensions of metallic colloids.

By exploiting the OT, the total extinction cross section can be obtained directly as a sum over the coefficients a_n, b_n:

[29]${\sigma }_{\text{ext}}=\frac{2\pi }{k^2}{\displaystyle \sum _n(2n+1)\text{Re}(a_n+b_n)}$

For small particles, σ_ext goes as λ⁻⁴ as in the RGD approximation, but by increasing x it tends to level off with substantial oscillations. In particular, for a specific (narrow) particles size range, it may happen that in the visible region σ_ext increases with λ (see Figure 3). This takes place for instance, when volcanic bursts eject huge quantities of large particles in the atmosphere, giving rise to the effect of a blue sun or moon (as the name suggests, a rare event indeed).

Figure 3. Extinction efficiency factor Q_ext = σ_ext/πa² for polystyrene spheres in water as a function of x (varied by fixing λ = 560 nm and increasing a). The steep rise at small x corresponds to the RGD regime, Q ∼ x⁴, while the rich substructure at higher x, shown in the inset, derives from the large number of contributing Mie modes. “Blue moon” effects may take place in the regions where dQ_ext/dx < 0.

The strong propensity to forward scattering for large particles also allows one to explain a puzzling fact about extinction. As shown in Figure 3, in the limit x → ∞, eqn [29] yields a value for σ_ext which is twice the particle geometric cross section σ_g = πa², that is, the particles subtract twice the expected power from the incident beam. This is not an artifact, but a real effect; what happens, however, is that half of the scattered power is concentrated in the forward lobe, so that it is collected by any finite-size detector as the eye. Such an intuitive geometrical optic concept as “shadow” is therefore based on subtle scattering effects.

A final consideration concerns transfer of momentum by radiation pressure, which is of primary importance both for optical levitation and for particle trapping by laser tweezers. For nonabsorbing particles, it is easy to show that the radiation pressure is given by $p_{\text{rad}}=I_0({\sigma }_{\text{s}}/{\sigma }_{\text{g}})(1-〈\text{cos}\vartheta 〉)$, where 〈cos ϑ〉 is averaged over the scattering pattern distribution. For a ≪ λ, one obtains 〈cos ϑ〉 ≃ 0, and the radiation pressure increases approximately as a⁴. However, for very large particles σ_s/σ_g → 2 and 〈cos ϑ → 1〉, so that radiation pressure decreases. Depending on the size and material composition of the particle, there is therefore an optimal λ for efficient momentum transfer.

3.5.1 Mie Scattering

For size parameters in the range 0.1–50, the wavelength of the radiation and the circumference of the particle are comparable. Radiation strongly interacts with the particle, and, therefore, the full Mie equations must be used. These equations have been applied extensively to the detection of raindrops by radar. The study of aerosols (smoke, dust, haze) using visible radiation falls in the Mie regime. Also in the Mie regime is the interaction of cloud droplets with infrared radiation.

A complete discussion of Mie scattering is outside the scope of this book because the scattering equations are very complicated. Insight into the results, however, can be gained as follows. If the volume absorption coefficient is divided by the number of scattering particles per unit volume and by the cross-sectional area of each scatterer, the result is the scattering efficiency (Q_s) for a single scatterer. Q_s is the ratio of the total scattered radiation (regardless of direction) to the incident radiation. Q_s is a function of the size parameter and of the index of refraction of the particle.

Many substances absorb radiation as well as scatter it. This can be conveniently taken into account by letting the index of refraction become a complex number

(3.50)$m\equiv n-i{n}^{\prime },$

where n, the real part of the index of refraction, is as defined above, and n′, the imaginary part, accounts for absorption inside the scatterers. Figure 3.19 showsn′ as a function of wavelength for water and ice. In the visible portion of the spectrum, n′ is negligibly small, but in the infrared it becomes significant.

FIGURE 3.19. Imaginary part of the index of refraction of water and ice.

[Plotted from data in Irvine and Pollack (1968).]Copyright © 1968

Figure 3.20 shows the scattering efficiency for water drops (n = 1.33) as a function of size parameter for several values of n′. Scattering efficiency in the Mie regime is quite clearly a complicated function. In clouds, there is usually a distribution of drop sizes. Suppose that N(r)dr is the number of drops per unit volume in the radius range r to r + dr. If the scatterers are sufficiently far apart (many wavelengths) that they act independently, the scattering coefficient is given by

FIGURE 3.20. Scattering efficiency (Q_s) of water spheres (n = 1.33) as a function of size parameter (χ) for several values of n′.

[Adapted from Liou (1980) and Hansen and Travis (1974). Reprinted by permission of Academic Press, Inc. and Kluwer Academic Publishers.]Copyright © 1980

(3.51)${\sigma }_s(\lambda )={\int }_0^{\propto }\pi r^2{Q}_{s}N(r)dr$

Integration over the size distribution smooths the scattering efficiency. In any case, it must be noted that scattering is a much smoother function of wavelength than is gaseous absorption.

Also of interest is the scattering phase function, which determines in which direction the radiation is scattered. Figure 3.21 shows the scattering phase function for water drops for several size parameters. As the size parameter increases, the phase function becomes strongly peaked in the forward direction; relatively little radiation is backscattered toward the source of the radiation. Finally, we note that in general scattering polarizes radiation; in some applications polarization must be taken into account.

FIGURE 3.21. Polar plots (note the logarithmic scales) of the scattering phase function of water drops for several size parameters.

[Plotted from data supplied by Steven A. Ackerman, Cooperative Institute for Meteorological Satellite Studies, University of Wisconsin–Madison.]

9.3.2 Static light scattering

Rayleigh–Gans–Debye scattering

A frequently used expression, which has a wider range of applicability, is the Rayleigh–Gans–Debye (RGD) approximation. The physical basis of RGD scattering is that a particle (of arbitrary shape) is assumed to consist of elements which behave as independent Rayleigh scatterers. This is appropriate when the parameter ρ (eqn 9.7) is very low, which implies small particles and a low value of refractive index. Although the RGD result can be used for particles considerably larger than in the case of the Rayleigh expression, the condition ρ < 1 means that the scattering coefficient Q must be very small (van de Hulst, 1957).

The RGD approximation can be written as a modified form of the Rayleigh result:

(9.10)$\frac{I_{\theta }}{I_o}=\frac{1}{r^2}{R}_{\theta }P(\theta )$

The term P (θ) is known as a form factor, and represents a correction to the Rayleigh expression which accounts for effects due to the size and shape of the particle. This term can be evaluated for many simple geometrical shapes and for particle aggregates in terms of the mass distribution.

For homogeneous spheres, of radius a, the form factor becomes:

(9.11)$P(\theta )={\left[\frac{3(\text{sinu}-\text{ucos}u)}{u^3}\right]}^2$

where u = qa and q is the scattering vector, given by:

(9.12)$q=\frac{4\pi }{\lambda }sin\frac{\theta }{2}$

For very low scattering angles, series expansion of eqn (9.11) leads to:

(9.13)$P(\theta )=1-\frac{u^2}{5}+\dots$

For homogeneous particles of arbitrary shape, with radius of gyration a_G, P(θ) can be expressed as a series expansion:

(9.14)$P(\theta )=1-\frac{{(q{a}_G)}^2}{3}+\dots$

which becomes equivalent to eqn (9.13) in the limit of low scattering angle, since the radius of gyration of a homogeneous sphere is $\surd (3/5)a.$

In the limit of zero scattering angle, the form factor becomes unity and the scattered intensity is just that given by the Rayleigh expression. For this reason, light scattering at very low forward angles from an aggregating suspension of constant volume fraction should be proportional to the average aggregate volume and such measurements have been used to measure aggregation rates (Young and Prieve, 1991). In particular, it can be shown (e.g. Zeichner and Schowalter, 1979) that the scattered light should increase with time according to:

(9.15)$\begin{array}{cc}{\left(\frac{1}{I(o)}\frac{dI}{dt}\right)}_{t\to o}& =2k_a{n}_o\end{array}$

where k_a is the aggregation rate constant, eqn (6.13), and n₀ is the initial number concentration of primary particles. I(0) is the measured light-scattering intensity for the initial suspension.

This result shows that absolute values of the aggregation rate constant, at least in the early stages, can be derived in a straightforward manner.

Light-scattering measurements at very low angles became much more feasible with the advent of lasers, because of the narrow parallel beam that can be generated. However, there are some practical difficulties associated with the technique, notably the very large scattering produced by stray dust particles. Measurements at larger angles are easier to carry out but more difficult to interpret, because of the need to consider the structure of small aggregates.

The RGD approach allows treatment of particle aggregates through a structure factor, S(θ), which is given by:

(9.16)$S(\theta )=\sum _i^k\sum _j^k\frac{\text{sin\hspace{0.17em}}q{r}_{ij}}{q{r}_{ij}}$

where r_ij is the centre-to-centre distance of a pair of particles in a k-fold aggregate and the summation is over all such pairs.

For aggregates of hard spheres it is, in principle, possible to calculate the structure factor for any assumed structure. However, for three-fold and higher aggregates the number of possible structures increases very rapidly (see Figure 6.8). Only in the case of a doublet is there a unique structure factor. It can be shown (van Zanten and Elimelech, 1992) that the scattered light intensity from a suspension of equal spheres undergoing perikinetic aggregation and following Smoluchowski kinetics increases according to:

(9.17)$I(\theta ,t)=I(\theta ,0)\left(1+2\frac{n_2}{n_o}\frac{\text{sin qd}}{{\text{qd}}_{\text{1}}}+2\sum _{k=3}^{\infty }\frac{nk}{n_o}{S}_k(\theta )\right)$

where d₁ is the diameter of primary particles (i.e. the centre-to-centre distance of two touching particles), and n₀, n₂ and n_k are the number concentrations of primary particles (initially), doublets and k-fold aggregates at time t. These concentrations are given by eqn (6.17). S_k(θ) is the structure factor for a k-fold aggregate for a scattering angle θ.

It follows that the initial rate of increase of scattered light from an aggregating dispersion is given by:

(9.18)${\left(\frac{1}{I(\theta ,0)}\frac{dI(\theta ,t)}{dt}\right)}_{t\to o}=2k_a{n}_o\frac{sinqd}{q{d}_1}$

which reduces to eqn (9.15) for very low scattering angles (when q → 0).

By carrying out measurements at different angles simultaneously, van Zanten and Elimelech (1992) were able to derive absolute coagulation rate constants for latex suspensions. This procedure should give more reliable values than those from measurements only at very low angles.

For a suspension containing aggregates, a log-log plot of normalized light-scattering intensity (or, effectively, the structure factor) against the scattering vector q can give information on the structure of the aggregates. Such a plot is shown schematically in Figure 9.4. The main features are constant scattering intensities at low and high scattering vectors with a linear decrease in the intermediate region. This behaviour arises because q represents a characteristic (inverse) length scale probed by the scattering experiment. The summand in eqn (9.16) is most sensitive to values of q such that qr_ij ≈ 1.

Figure 9.4. Form of log–log plot of normalized light-scattering intensity, or the structure factor S(q), against the scattering vector q, for an aggregated suspension. When 1/q is greater than the radius of gyration of the aggregates, a_G, or less than the primary particle radius, a₀, S(q) is independent of q. For intermediate q values, the slope of the line is –1/d_F

At low q values (low scattering angles), 1/q is much greater than the aggregate size and qr_ij ≪1 for all particle pairs in the aggregate, so that sin (qr_ij)/qr_ij ≈ 1 for all particle pairs and the sum becomes simply k². So, for an aggregating suspension the total light scattered increases as the weight-averaged aggregate size. This is just the behaviour found in the low angle limit, mentioned above. The light scattering is insensitive to the structure of an aggregate and depends only on the total volume of the constituent particles.

When q is very high, such that qa > 1, the length scale probed is much less than the size of the primary particles and the only significant contribution to the sum in eqn (9.16) comes from terms where i=j and hence r_ij; = 0. This gives S(θ) = k for a k-fold aggregate, so that the scattering is just the same as that from k isolated particles. Under these conditions aggregation of a suspension would give no change in scattered light.

In the intermediate region of q it can be shown (e.g. Schmidt, 1989) that the slope of the linear region in Figure 9.4 is –d_F, where d_F is the mass fractal dimension given by eqn (6.41). This result is obtained from an integral form of eqn (9.16), in terms of the radial distribution function of particles in an aggregate. No further details will be given here.

The range of q values accessible by scattering of visible light, bearing in mind that measurable scattering angles are typically 10–160°, is of the order of 2–30 μm⁻¹, so that only rather small aggregates (<1 μm) can be investigated. Nevertheless, this approach can give very useful information on fractal aggregates formed from primary particles with sizes less than, say, 0.1 μm. Chen and Russel (1991) used this method for studying the flocculation of silica dispersions and Hoekstra et al. (1992) were able to investigate fractal aggregates of nickel hydroxycarbonate in a similar manner.

By using radiation with shorter wavelengths (such as X-rays or neutrons), aggregate structure can be probed over shorter length scales. In this way information on, for instance, interparticle distances in flocs can be derived (Wong et al., 1988).

The Rayleigh–Gans–Debye approximation cannot be applied to the kind of aggregates formed in many practical cases, which may range in size up to 1 mm or more. For studies of large aggregates, other approaches have to be used.

Mie theory

When a particle is illuminated by a light beam, its constituent molecules become polarized by the oscillating electric field of the light wave and radiate light in all directions. If the radiated (scattered) light is of the same wavelength as the incident light, the phenomenon is known as elastic scattering, since no energy is lost from the beam. For particles of arbitrary size, the scattering pattern is complicated by the fact that scattered radiation from different parts of the same particle is subject to phase differences and hence interference effects. An exact theory was developed by Mie early in the twentieth century (see Kerker, 1969). Mie theory gives the complete angular distribution of scattered light intensity from a homogeneous spherical particle as a function of its refractive index, size and the light wavelength. For larger particles the scattering pattern becomes highly asymmetrical, with most of the light being scattered in the forward direction.

Although measurements of the angular distribution of scattered light, or the intensity at just one angle, can give information on particle size, the full Mie theory is of very limited use for aggregating suspensions, because of the nature of real aggregates. Usually, an approximate expression has to be adopted, in order to interpret light-scattering data in terms of aggregate properties.

5.1.5 Mie Scattering

Semiconductor nanoparticles have a much higher linear refractive index than the surrounding glass matrix. A laser beam is thus scattered inside this kind of composite material. For integrated optics devices, this supplementary source of loss has to be added to the normal absorption and/or waveguide losses [83]. The scattering losses depend on the refractive index difference between particles and matrix and the size and the form of the particles.

Scattering of electromagnetic radiation from spheres is described by the Mie theory. Exact Mie scattering calculations are very time consuming, but special algorithms have been developed to speed up this kind of calculation. In the case of semiconductor-doped glasses (i.e., for particles with sizes small compared to the wavelength), the small particle or Rayleigh limit is applicable. In this case the Mie scattering algorithm developed by Wiscombe can be used [200]. Another problem is the knowledge of the refractive index of the nanoparticles. This parameter is also size dependent, and only very few experimental data are available. Kyprianidou-Leodidou et al. have measured the refractive index of PbS nanocrystals at the wavelength of 1295 nm and for particle sizes from 4 to 80 nm [201]. They found that the size dependence in the 4–30-nm range is approximately linear. Based on these data, Fick et al. calculated the scattering losses for a slab waveguide of homogeneously PbS particle-doped glass [83]. Figure 19 shows the scattering losses versus the mean particle size for 1064- and 1295-nm wavelengths. Low wavelength dispersion of the refractive index was assumed, and the refractive index measured at 1295 nm was used for both wavelengths. The refractive index of the PbS crystals measured by Kyprianidou-Leodidou et al. and used for the calculation is also shown in Figure 19. The refractive index of the silica-titania matrix was fixed at 1.51, and a PbS concentration of 5 vol% was chosen. Absorption of the matrix and the semiconductor particles, as well as waveguide losses, was neglected.

Fig. 19. Contribution to the waveguide losses caused by Mie scattering as a function of the mean PbS crystal diameter, and linear refractive index of PbS nanoparticles at 1295 nm (from [201]).

From [83]. © 2000 Taylor &amp; Francis, with permission.Copyright © 2000

For the mostly studied particle size range (∼3 nm) the calculated scattering losses are about 0.047 dB/cm at 1064 nm and 0.021 dB/cm at 1295 nm. Hence, the contribution of the scattering losses caused by PbS nanoparticles to the overall waveguide losses can be neglected. However, for larger particles the calculated scattering losses of more than 1 dB/cm represent a serious limitation. With the use of high-index glasses, such as chalcogenide glasses, as the surrounding matrix, the scattering losses can be reduced considerably, and the presence of large particles should no longer be a great restriction.