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Mie theory: Instability of function A

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There are two major areas of instability of the upward recursion for function A_n of the Mie theory (see Eq. 2 in Algorithms for functions of the refractive index and/or size of the sphere):

1. real refractive index m < 1, as for an air bubble in water, m = 0.752, as shown in Fig. 1. Here, as opposed to m > 1, the asymptotic range of the A_n function (Eq. 1 in Tests for function A) is reached for the summation index, n < n_max, with the latter defined by Eq. 1 in Number of terms in Mie series. This instability persists even for the small particles.
2. complex refractive index, m, as for a carbon sphere in air, m = 1.95 - 0.79i, as shown in Fig. 2. In contrast to the real m < 1 case, the instability is supressed for the small particles.

Fig. 1. Example of the instability of function A_n(mx), where n is the Mie series summation index, for real refractive index m < 1, here for an air bubble in water with a relative refractive index m = 0.752 (refractive index of water is taken to be 1.33 - 0i, that of air is taken to be 1.0 - 0i) and relative size x = 250: symbols represent values obtained with the downward recurrence, and the blue curve represents the results of an upward recurrence, departing from the asymptote (magenta, Eq. 1 in Tests for function A) at about n = 230. Function A_n, defined by Eq. 2 of the Algorithms for functions of the sphere size and refractive index, is real for real m. For air bubbles, A_n remains unstable at x as low as 25. The function was calculated in the downward recurrence case with the MJC Light Scattering Calculator for Homogeneous Spheres, the upward recurrence results were obtained with a separate program.

Fig. 2. Example of the instability of the real part of function A_n(mx), where n is the Mie series summation index, for complex refractive index m, as for a carbon sphere in air. The refractive index of "voidless" carbon is taken to be 1.95 - 0.79i, that of air is taken to be 1.0 - 0i. The relative size x = 250. Symbols represent values obtained with the downward recurrence. The blue curve represents the results of an upward recurrence, the latter becoming unstable beyond about n = 225. In contrast to the real m < 1 case, the instability is supressed for small particles. Function A_n is defined by Eq. 2 of the Algorithms for functions of the sphere size and refractive index. The function was calculated in the downward recurrence case with the MJC Light Scattering Calculator for Homogeneous Spheres, the upward recurrence results were obtained with a separate program.

CITATION: Jonasz M. 2006. Mie theory: Instability of function A (www.tpdsci.com/Tpc/MieErrA.php). In: Top. Part. Disp. Sci. (www.tpdsci.com).

HISTORY: Published: 12-Mar-2006 Modified: 19-Apr-2006 Peer-reviewed: PENDING

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