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 τ_{0} = 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 Q_{t}(ρ) and Q_{a}(τ_{0}) 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 Q_{t}(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 Q_{a}(τ_{0}) 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).

#### REFERENCES

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.

#### References

- 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 â€œsoftâ€ 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 â€œsoftâ€ 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 ChÃ½lek, 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.

Heat & Mass Transfer, and Fluids Engineering