Following from: Limiting cases of the general Mie theory
In this section, we come back to optically soft particles but consider the more general solution. The specific properties of large optically soft particles have been analyzed in the greatest detail by van de Hulst (1957). It was assumed that
and one can separate transmission and diffraction. In this case, known as anomalous diffraction, the geometrical optics’ rays are supposed to pass through a particle without any deflection, but they can undergo a significant phase shift because of the long path length through a particle. In anomalous diffraction, the phase shift ρ = 2x(n-1) is assumed to be fixed and the transfer to the limit of m → 1 is considered. The following expression for the extinction efficiency factor was derived by van de Hulst (1957):
In the case of nonabsorbing particles (κ = 0), Eq. (2) is reduced to the following:
One can also find the analytical expression for the absorption efficiency factor (van de Hulst, 1957),
where τ0 = 2κx is the optical thickness of a particle. In the limiting case of small optical thickness, Eq. (4) gives the same result as that in the Rayleigh-Gans approximation. The dependences Qt(ρ) and Qa(τ0) calculated by Eqs. (3) and (4) are shown in Fig. 1.
Figure 1. Efficiency factors of extinction and absorption predicted by anomalous diffraction theory
One can see the oscillation of the curve Qt(ρ). As was shown by van de Hulst (1957), this effect is explained by interference of transmitted and diffracted radiation. It is important that positions of the maximums and minimums on the extinction curve appear to be the same at comparably large values of the refraction index n when condition |m-1| << 1 is not satisfied (van de Hulst, 1957; Dombrovsky, 1996). The monotonic function Qa(τ0) is also typical at various n (not only in the limit of n → 1) when the particle material is weakly absorbing (κ << 1). This makes anomalous diffraction a very useful approach for understanding the properties of real particles. Unfortunately, the anomalous diffraction approximation does not provide an analytical solution for the asymmetry factor of scattering ( for further details, see van de Hulst, 1957; Kokhanovsky and Zege, 1997; Perelman, 1991; Perelman and Voshchinnikov, 2002). A reader interested in the accuracy of the anomalous diffraction approximation and its applications to nonspherical particles in colloidal chemistry and atmospheric optics can find additional information in papers by Meeten (1980), Sharma (1993), Liu et al. (1996), Videen and Chýlek (1998), Sun and Fu (1999), Franssens (2001), Zhao and Hu (2003), Rysakov (2004, 2006), and Sun et al. (2008). A more detailed bibliography on this subject can be found in the recent monograph by Dombrovsky and Baillis (2010).
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.
Franssens, G. R., Retrieval of the aerosol size distribution in the complex anomalous diffraction approximation, Atmos. Environ., vol. 35, no. 30, pp. 5099-5104, 2001.
Kokhanovsky, A. A. and Zege, E. P., Optical properties of aerosol particles: A review of approximate analytical solutions, J. Aerosol Sci., vol. 28, no. 1): 1-21, 1997.
Liu, C. W., Clarkson, M., and Nicholls, R. W., An approximation for spectral extinction of atmospheric aerosols, J. Quant. Spectrosc. Radiat. Transfer, vol. 55, no. 4, pp. 519-531, 1996.
Meeten, G. H., The birefringence of colloidal dispersions in the rayleigh and anomalous diffraction approximations, J. Colloid Interface Sci., vol. 73, no. 1, pp. 38-44, 1980.
Perelman, A. Y., Extinction and scattering by soft spheres, Appl. Opt., vol. 30, no. 4, pp. 475-484, 1991.
Perelman, A. Y. and Voshchinnikov, N. V., Improved S-approximation for Dielectric Particles, J. Quant. Spectrosc. Radiat. Transfer, vol. 72, no. 5, pp. 607-621, 2002.
Rysakov, V. M., Light scattering by “soft” particles of arbitrary shape and size, J. Quant. Spectrosc. Radiat. Transfer, vol. 87, no. 3-4, pp. 261-287, 2004.
Rysakov, V. M., Light scattering by “soft” particles of arbitrary shape and size: II--Arbitrary orientation of particles in space, J. Quant. Spectrosc. Radiat. Transfer, vol. 98, no. 1, pp. 85-100, 2006.
Sharma, S. K., A modified anomalous diffraction approximation for intermediate size soft particles, Opt. Commun., vol. 100, no. 1-4, pp. 13-18, 1993.
Sun, W. and Fu, Q., Anomalous diffraction theory for arbitrary oriented hexagonal crystals, J. Quant. Spectrosc. Radiat. Transfer, vol. 63, no. 2, pp. 727-737, 1999.
Sun, X., Tang, H., and Yuan, G., Anomalous diffraction approximation method for retrieval of spherical and spheroidal particle size distributions in total light scattering, J. Quant. Spectrosc. Radiat. Transfer, vol. 109, no. 1, pp. 89-106, 2008.
van de Hulst, H. C., Light Scattering by Small Particles, Wiley, Hoboken, NJ, 1957 (also Dover Publ., 1981).
Videen, G. and Chýlek, P., Anomalous diffraction approximation limits, Atmos. Res., vol. 49, no. 1, pp. 77-80, 1998.
Zhao, J. Q. and Hu, Y. Q., Bridging technique for calculating the extinction efficiency of arbitrary shaped particles, Appl. Opt., vol. 42, no. 24, pp. 4937-4945, 2003.