Topics in Particle and Dispersion Science

Mie theory: Fourier analysis of particle size-dependent patterns

Prev topic | Next topic Fig. 1a, Fig. 1b Fig. 2a, Fig. 2b Fig. 3a, Fig. 3b

Even a casual look at Fig. 2 in Mie theory: Particle size-dependent patterns, showing the attenuation efficiency as a function of the particle size, suggests a well-defined oscillatory behavior of such efficiencies as functions of the microsphere diameter beyond the first maxima of these functions. Such a generalization, which neglects the optical resonances, is well described by the anomalous diffraction approximation (ADA; for example, van de Hulst 1981). The Mie functions that depend on the scattering angle, such as the M₁₁ element of the scattering matrix, exhibit a particle-size-dependent behavior (Fig. 1a and Fig. 1b) that is similar to that of the optical efficiencies.

Such a well-defined oscillatory behavior of a function implies that the frequency spectrum of the function is relatively narrow. Indeed, Fourier analysis of the particle size-dependency of, for example, the M₁₁ element of the scattering matrix confirms that impression (Fig. 2a and Fig. 2b). Such analysis of the M₁₁(x) element at four values of the real refractive index of a microsphere, m = 1.02, 1.05, 1.19, and 1.33, suggests the following conclusions. In the case of a low real refractive index (such as m = 1.05, Fig. 2a) and for most of the scattering angles in a range of 0 to 90° there is a single dominant component in the frequency spectrum of M₁₁(x). More components appear in the frequency spectra of M₁₁ at the scattering angles greater than 90°. However, as shown in Fig. 3a, the frequencies, f_x, of the dominant components almost exclusively lie on or below a line described by the following equation:

fx = af + bf θ

(1)

where a_f and b_f are almost independent of the refractive index of the microsphere (Table 1) and of the relative particle size, x.

Somewhere between m = 1.02 and 1.05 an interesting new feature in the distribution of the major frequencies with the scattering angle, θ, emerges. It is a clearly-distinguishable descending frequency branch that mirrors the ascending frequency branch existing at θ ≤ ~90° (Fig. 2a). This results in a triangular pattern centered at θ = ~90° (Fig. 3a and Fig. 3b).

As the relative refractive index of a microsphere increases, the frequency spectra of M₁₁ become more complicated as evidenced by Fig. 2b and Fig. 3b (m = 1.19, for example, polystyrene in water). Although the minimum dominant frequency peak of the spectra can still be characterized by Eq. 1 (Fig. 3b), many more frequency components appear with frequencies higher than those predicted by that equation. Such a change is caused by a significantly more complex oscillation patterns of the M₁₁ element at moderate to large values of the refractive index of a microsphere (compare Fig. 1a and Fig. 1b, as well as Fig. 2a and Fig. 2b).

This behavior of the frequency spectra has implications for the choice of an integration step in the numerical integration of the particle-size dependent scattering patterns.

CITATION: Jonasz M. 2006. Mie theory: Fourier analysis of particle size-dependent patterns (www.tpdsci.com/Tpc/MiePtnSzFT.php). In: Top. Part. Disp. Sci. (www.tpdsci.com).

HISTORY: Published: 03-Mar-2006 Modified: 02-May-2007 Peer-reviewed: PENDING

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