TPDSci Mie theory Angular patterns

Mie theory: Angular patterns

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Fig. 1, Fig. 2, Fig. 3

Angular patterns of light scattering by homogeneous spheres are symmetrical about the incident beam axis. These patterns possess certain general characteristics illustrated in Fig. 1 (polystyrene in water; effect of the sphere size) and Fig. 2 (water in air), Fig. 3 (air in water), and Fig. 4 (carbon in air; the last three figures illustrate the effect of the refractive index). These figures were created by using results of calculations performed with the MJC Light Scattering Calculator for Homogeneous Spheres, a Windows-based program that utilizes the downward recursion for the A_n function with the initial value calculated by using the Lentz algorithm (Lentz 1976). See Algorithms for functions of the refractive index and/or size of the sphere and the program's help file for more detail.

Characteristics of light scattering by spheres as predicted by Mie theory, which to certain extent are shared by light scattering patterns of other particle shapes, can be formulated as follows. First, the angular patterns narrow substantially with the increasing relative sphere size, x (see Fig. 1). Kattawar and Plass (1967) show that the half-width of the angular pattern of |S₁|² (as well as that of |S₂|², see Mie theory: Overview for definitions) follows a power law: ~x^-1.

Second, the angular patterns exhibit oscillations whose frequency increases with the sphere size (see Fig. 1). Steiner et al (1999), who examined power spectra of the unpolarized scattered intensity pattern (M₁₁) between the scattering angle, θ, of 73° and 107°, for microspheres with the relative particle size, 50 ≤ x ≤ 500, and refractive index 1.3 ≤ n ≤ 1.75, found a single dominant angular-frequency maximum in all examined cases. They established the following relationship between the angular frequency (number of oscillations per degree) corresponding to the maximum, and the relative particle size, x:

f_θ = 0.00483 x

(1)

This evolution of the angular scattering patterns has been used to determine the sphere size through the Fourier analysis of such patterns (see also Semyanov et al 2004) or by comparing theoretical and measured patterns. Ray et al (1991), who investigated the potential of the second approach, conclude that the accuracy in the determination of the sphere size and refractive index achievable with that approach is on the order of 2 parts in 10⁴ and 1 part in 10⁴ respectively, under certain stringent experimental conditions. A better accuracy can be obtained by comparing the theoretical and experimental angular pattern of optical resonances. Grainger et al (2004) provide analytical formulation of derivatives of the Mie a_n and b_n functions (see Mie theory: Algorithms for functions of the refractive index and/or size of the sphere) that are useful in algorithms for determining the size and refractive index of the sphere via fitting theoretical to experimental light scattering patterns.

Third, oscillations in the angular patterns are damped for light absorbing spheres (compare an example of a carbon sphere in air, Fig. 4, with that of a negligibly absorbing water droplet in air, Fig. 2). For light-absorbing spheres, functions |S₁|² and |S₂|², representing the scattered light linearly polarized in two orthogonal planes, tend monotonously to each other at the scattering angles greater than ~90°. See Mie theory: Overview for definitions of these functions. In fact, Sun and Lin (2006), who compared scattering matrices of dielectric (sand) and metallic (perfectly conducting) spheres and cylinders, point out that functions |S₁|² and |S₂|² by perfectly conducting particles assume the same value at the scattering angle > ~90°. This implies that the linear polarization degree, P_L = 0.

CITATION:
Jonasz M. 2006. Mie theory: Angular patterns (www.tpdsci.com/Tpc/MiePtnAng.php). In: Top. Part. Disp. Sci. (www.tpdsci.com).

HISTORY:
Published: 03-Mar-2006
Modified: 18-Apr-2006
Peer-reviewed: PENDING

Topics in Particle and Dispersion Science