Topics in Particle and Dispersion Science

Mie theory: Algorithms for functions of the refractive index and/or size of the sphere: a, b, A, w

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The Mie-theory requires efficient and effective algorithms for the recurrence-based calculations of the particle size- and refractive index-dependent coefficients a_n and b_n (see Mie theory: Overview), especially for spheres much larger than the wavelength of light. After Deirmendijan (1969), the a_n and b_n coefficients can be calculated as follows:

an =		[An(y) / m + n / x] Re wn(x) - Re wn-1(x) [An(y) / m + n / x] wn(x) - wn-1(x)	(1a)
bn =		[mAn(y) + n / x] Re wn(x) - Re wn-1(x) [mAn(y) + n / x] wn(x) - wn-1(x)	(1b)

where y = mx, m is the refractive index of the sphere material relative to that of the surrounding medium, x is the relative size of the sphere, and

An = -(n / y) + 1 / [(n / y) - An-1]	(2)
wn = [(2n - 1) / x] wn-1 - wn-2	(3)

where the initial values are

A0 =

sinp cosp (1 - tanh2q) + i tanhq  sin2p + cos2p tanh2q

(4)

where p and q are defined as follows:

y	= (m' - m"i) x
	= p - qi	(5)

and

w0 = sinx + i cosx	(6)
w-1 = cosx - i sinx	(7)

for the upward (i.e. increasing index n) recurrence.

Function A_n(y) is a logarithmic derivative of the Ricatti-Bessel function. Its upward recurrence calculations are known to be unstable due to the cumulative rounding error (Dave 1969). Kattwar and Plass (1967) proposed a downward recursion (see also Jones 1983 and Dave 1969 and his references 25 and 26) as a way to counteract that unstability: one simply starts with an arbitrary initial value, A_n=nD, at a maximum value of the summation index, n = nD > nmax, where nmax is the maximum value of n assuring the required accuracy of the Mie series (see Number of terms in Mie series), and calculates A_n for the decreasing index n by using a relation that can be easily derived from Eq. 2. The succesive values of A_n rapidly converge to the correct value. A convenient starting value of A_n=nD is 0. Alternatively, the initial value A_n=max can be found, usually at a lower computational cost (Wiscombe 1996, Wiscombe 1980) by using a continued-fraction algorithm of Lentz (1976).

It is worth to note that the relativity of the particle size in respect of the wavelength of light, as expressed by the definition of the relative particle size, x, does not imply that A_n(mx) or any Mie parameter that involves it can be interpreted as a function of the wavelength for a fixed sphere radius. This is because that function depends also on the particle refractive index, m, which is itself a function of the wavelength of light. Only in those wavelength regions where m varies little with the wavelength, can this relativity imply such an interpretation of the Mie parameters and then only as an approximation.

CITATION: Jonasz M. 2006. Mie theory: Algorithms for functions of the refractive index and/or size of the sphere: a, b, A, w (www.tpdsci.com/Tpc/MieCalcAw.php). In: Top. Part. Disp. Sci. (www.tpdsci.com).

HISTORY: Published: 21-Mar-2006 Modified: 02-Aug-2006 Peer-reviewed: PENDING

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