THE MIE SOLUTION FOR SPHERICAL PARTICLES
Following from: Radiative properties of single particles and fibers: the hypothesis of independent scattering and the Mie theory
Leading to: Limiting cases of the general Mie theory
The scheme of the problem is shown in Fig. 1.
Figure 1. Scheme of the scattering problem for a spherical particle
The amount of scattering and absorption by a particle is usually expressed in terms of the scattering cross section C_{s} and absorption cross section C_{a}. The total amount of absorption and scattering, or extinction, is expressed in term of the extinction cross section C_{t}. The dimensionless efficiency factors are often used instead of cross sections,
Absorption and scattering of radiation by a two-layer spherical particle depend on the complex index of refraction of the core and mantle substances (m' = n'-iκ' and m'' = n''- iκ'', correspondingly) and also on diffraction parameters x' = 2πa'/λ, x'' = 2πa''/λ, where a' is the core radius, and a'' is the external radius of the particle.
According to Mie theory, the efficiency factors of scattering and extinction are expressed in the form of the following series:
where complex coefficients a_{k}, b_{k} are called the Mie coefficients. The efficiency factor of absorption is determined as a difference, Q_{a} = Q_{t} - Q_{s}. The terms in the Mie series correspond to the partial waves of different orders. The number of terms, which should be taken into account, increases with increasing the diffraction parameter. As a result, the calculations for particles of radius much greater than the wavelength are more complicated.
The angular characteristics of scattering are expressed by the following complex amplitude functions corresponding to perpendicular polarizations:
Here, μ = cosθ, where θ is the angle of scattering measured from the direction of the incident radiation (see Fig. 1), and π_{k} and τ_{k} are special angular functions defined later by Eq. (22). The scattering (phase) function for linear polarized incident radiation is
where
Angle φ is measured from the polarization plane of the incident radiation. For unpolarized (randomly polarized) incident radiation, the following relations hold true:
and the polarization degree of scattered radiation is
For calculating the asymmetry factor of scattering, one can use the Debye equation,
where the asterisk denotes a complex conjugate quantity. Remember that the asymmetry factor of scattering is defined as
According to Eq. (8), the value of μ does not depend on polarization of the incident radiation.
The Mie coefficients for two-layer spherical particles are determined by
where
and α_{k}, β_{k}, γ_{k} expressions will be given later [see Eqs. (17) and (18)].
For hollow particles, Eqs. (10) and (11) become
For homogeneous particles, the simplifications of the Mie coefficients are considerable,
For perfectly reflecting homogeneous particles (|m| → ∞),
and for two-layer particles with perfectly reflecting central core,
Functions α_{k}, β_{k}, γ_{k} are expressed by the Riccati-Bessel functions ψ_{k}, ζ_{k} (Abramowitz and Stegun, 1965),
The Riccati-Bessel functions are related to the Bessel functions and the Hankel second-kind functions in a simple manner,
Two recursion relations,
and formulas for derivatives,
can be used in calculations of the Mie coefficients. As a result, we have the following expressions for logarithmic derivatives α_{k} and β_{k}:
Special angular functions π_{k}, τ_{k} are expressed by associated Legendre polynomials,
and can be calculated following the recursion relations derived by Hosemann (1971),
For calculations employing Eqs. (19) and (23), it is necessary to know the following initial functions:
We have now all the expressions for calculations of absorption and scattering of radiation by homogeneous, hollow, or two-layer spherical particles.
The calculations of the Riccati-Bessel functions based on recursion relations (19) starting from the first functions [(24a) and (24b)] up to higher-order functions (the so-called upward recursion) may lead to significant computational errors. The latter limitation is important in the case of large values of x and |m|x when the functions are of the order close to the argument are calculated with great error. This problem was overcome by Kattawar and Plass (1967), who showed that some components of the Riccati-Bessel functions are well calculated by using downward recursion. The detailed analysis of the accuracy and stability of several algorithms for Mie scattering calculations have been performed by Wiscombe (1980). A comparison of Mie scattering subroutines can be found in the paper by Felske et al. (1983). The reader can find the early FORTRAN codes for homogeneous and various two-layer particles in appendices of the books by Bohren and Huffman (1983) and Dombrovsky (1996). The algorithms for the general case of stratified spheres were presented by Toon and Ackerman (1981) and Bhandari (1985). Additional codes for multilayered spheres are listed by Flatau (2000) and Wriedt (2000). Note that the work on improving the Mie scattering algorithms continuesd. One can find some new results in recent papers by Yang (2003), Du (2004), Li et al. (2006), and Cai et al. (2008). A more detailed bibliography on this subject can be found in the recent monograph by Dombrovsky and Baillis (2010).
Fortunately, this advanced technique of the Mie calculations should only be used mainly in the limiting cases when the general solution is degenerated and reliable estimates can be obtained on the basis of known approximations. The geometrical optics approximation for very large particles is a good illustration of the latter statement, which is a particular case of the general observation, namely, one should find an alternative physical approach in the range when the ordinary procedure leads to more and more complicated mathematics.
REFERENCES
Abramowitz, M. and Stegun, I. A. (Eds.), Handbook of Mathematical Functions, Dover, New York, 1965.
Bhandari, R., Scattering coefficients for a multilayered sphere: analytic expressions and algorithms, Appl. Opt., vol. 24, no, 13, pp. 1960-1967, 1985.
Bohren, C. F. and Huffman, D. R., Absorption and Scattering of Light by Small Particles, Wiley, Hoboken, NJ, 1983.
Cai, W., Zhao, Y., and Ma, L., Direct recursion of the ratio of Bessel functions with applications to Mie scattering calculations, J. Quant. Spectrosc. Radiat. Transfer, vol. 109, no. 16, pp. 2673-2678, 2008.
Dombrovsky, L. A., Radiation Heat Transfer in Disperse Systems, Begell House, Redding, CT, and New York, 1996.
Dombrovsky, L. A. and Baillis, D., Thermal Radiation in Disperse Systems: An Engineering Approach, Begell House, Redding, CT, and New York, 2010.
Du, H., Mie-scattering calculation, Appl. Opt., vol. 43, no. 9, pp. 1951-1956, 2004.
Felske, J. D., Chu, Z. Z., and Ku, J. C., Mie scattering subroutines (DBMIE and MIEV0): A comparison of computational times, Appl. Opt., vol. 22, no. 15, pp, 2240-2241, 1983.
Flatau, P. J., SCATTERLIB: Light Scattering Codes Library, atol.uscd.edu/~pflatau/scatlib/, 2000.
Hosemann, J. P., Computation of angular functions π_{n} and τ_{n} occurring the Mie theory, Appl. Opt., vol. 10, no. 6, pp. 1452-53, 1971.
Kattawar, G. W. and Plass, G. N., Electromagnetic scattering from absorbing spheres, Appl. Opt., vol. 6, no. 8, pp. 1377-1382, 1967.
Li, R., Han, X., Jiang, H., and Ren, K. F., Debye series for light scattering by a multilayered sphere, Appl. Opt., vol. 45, no. 6, pp. 1260-1270, 2006.
Toon, O. B. and Ackerman, T. P., Algorithms for the calculation of scattering by stratified spheres, Appl. Opt., vol. 20, no. 20, pp. 3657-3660, 1981.
Wiscombe, W. J., Improved Mie scattering algorithms, Appl. Opt., vol. 19, no. 9 pp. 1505-1509, 1980.
Wriedt, T., Electromagnetic scattering programs, www.t-matrix.de, 2000.
Yang, W., Improved recursive algorithm for light scattering by a multilayered sphere, Appl. Opt., vol. 42, no. 9, pp. 1710-1720, 2003.
References
- Abramowitz, M. and Stegun, I. A. (Eds.), Handbook of Mathematical Functions, Dover, New York, 1965.
- Bhandari, R., Scattering coefficients for a multilayered sphere: analytic expressions and algorithms, Appl. Opt., vol. 24, no, 13, pp. 1960-1967, 1985.
- Bohren, C. F. and Huffman, D. R., Absorption and Scattering of Light by Small Particles, Wiley, Hoboken, NJ, 1983.
- Cai, W., Zhao, Y., and Ma, L., Direct recursion of the ratio of Bessel functions with applications to Mie scattering calculations, J. Quant. Spectrosc. Radiat. Transfer, vol. 109, no. 16, pp. 2673-2678, 2008.
- Dombrovsky, L. A., Radiation Heat Transfer in Disperse Systems, Begell House, Redding, CT, and New York, 1996.
- Dombrovsky, L. A. and Baillis, D., Thermal Radiation in Disperse Systems: An Engineering Approach, Begell House, Redding, CT, and New York, 2010.
- Du, H., Mie-scattering calculation, Appl. Opt., vol. 43, no. 9, pp. 1951-1956, 2004.
- Felske, J. D., Chu, Z. Z., and Ku, J. C., Mie scattering subroutines (DBMIE and MIEV0): A comparison of computational times, Appl. Opt., vol. 22, no. 15, pp, 2240-2241, 1983.
- Flatau, P. J., SCATTERLIB: Light Scattering Codes Library, atol.uscd.edu/~pflatau/scatlib/, 2000.
- Hosemann, J. P., Computation of angular functions Ï€_{n} and Ï„_{n} occurring the Mie theory, Appl. Opt., vol. 10, no. 6, pp. 1452-53, 1971.
- Kattawar, G. W. and Plass, G. N., Electromagnetic scattering from absorbing spheres, Appl. Opt., vol. 6, no. 8, pp. 1377-1382, 1967.
- Li, R., Han, X., Jiang, H., and Ren, K. F., Debye series for light scattering by a multilayered sphere, Appl. Opt., vol. 45, no. 6, pp. 1260-1270, 2006.
- Toon, O. B. and Ackerman, T. P., Algorithms for the calculation of scattering by stratified spheres, Appl. Opt., vol. 20, no. 20, pp. 3657-3660, 1981.
- Wiscombe, W. J., Improved Mie scattering algorithms, Appl. Opt., vol. 19, no. 9 pp. 1505-1509, 1980.
- Wriedt, T., Electromagnetic scattering programs, www.t-matrix.de, 2000.
- Yang, W., Improved recursive algorithm for light scattering by a multilayered sphere, Appl. Opt., vol. 42, no. 9, pp. 1710-1720, 2003.