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Geometrical optics approximation

DOI: 10.1615/thermopedia.000158

GEOMETRICAL OPTICS APPROXIMATION

Leonid A. Dombrovsky

It is well known that the geometrical optics approximation (ray optics) is correct for large particles at large phase shift,

(1)

The last condition is very important because large particles of an optically “soft” substance are in the range of anomalous diffraction approximation, where interference of transmitted and diffracted radiation is considerable (see the next section). It is also clear that the geometrical optics approximation is inapplicable for calculations of scattering at special (caustic) angles, where wave effects are important (for rainbow and glory).

In the geometrical optics limit, only a half of scattering is a result of reflection and refraction. One should also take into account diffraction effects to obtain the complete picture of scattering. The geometrical optics approximation yields the well-known simple formula for the limiting value of the extinction efficiency factor at large diffraction parameter (Shifrin, 1968; van de Hulst, 1957),

(2)

When the value of x is not large enough, the geometrical optics approximation cannot provide sufficiently good accuracy, and the edge effects should be taken into account (Nussenzveig and Wiscombe, 1980). The asymptotic representation of the Mie series gives the following approximate expression for the extinction efficiency factor:

(3)

The error of Qt estimation with Eq. (3) does not exceed 10% for x > 45 and n ≥ 1.2, and decreases very rapidly with increasing of the value of x, oscillating near zero (Kokhanovsky and Zege, 1997).

In the case of a relatively small index of absorption (κ << n), the following expression for the efficiency factor of absorption was derived by Pinchuk and Romanov (1977) and Harpole (1980):

(4)

where τ0 = 2κx is the particle optical thickness, l(μ) = 2√1-(1-μ2/n2) and R||(μ), R(μ) are the Fresnel's reflection coefficients determined by the following equations (Born and Wolf, 1999):

(5)

(6)

Note that the value of R = 1 for μ ≤ μc corresponds to the total internal reflection.

For the optically thin limit (τ0 << 1), one can find the following expression (Pinchuk and Romanov, 1977):

(7)

which was also derived by Twomey and Bohren (1980). A more accurate equation was derived by Kokhanovsky (1995) by taking into account edge effects.

In the case of large spherical particles, the diffracted radiation is described by the following amplitude function (van de Hulst, 1957):

(8)

and the corresponding angular distribution of the radiation intensity is given by

(9)

Note that the picture of diffracted radiation intensity is the same as that for Fraunhofer diffraction on a circular disk or aperture of the same radius (Born and Wolf, 1999). The function f(u) is shown in Fig. 1.

Relative intensity of diffracted radiation for large spherical particles.

Figure 1. Relative intensity of diffracted radiation for large spherical particles.

Consider now the limiting case of large opaque spheres. The solution for a sphere of perfectly reflecting substance (|m|→∞, see article The Mie solution for spherical particles) gives the scattering functions such as that shown in Fig. 2. The greater the diffraction parameter, the narrower is the angular region of strong oscillation of function g(θ). One can consider that the part of the scattering function without a strong peak in the forward direction appeared due to diffraction

(10)

Scattering function for large perfectly reflecting spherical particle

Figure 2. Scattering function for large perfectly reflecting spherical particle

Equation (10) shows that radiation scattering from a large perfectly reflecting sphere is isotropic, excluding the narrow diffraction peak near the forward direction. It is also interesting to consider an important practical case of a large opaque particle with a diffusely reflecting surface. It is clear that the interaction of electromagnetic waves with such particle is not described by the Mie solution. But one can use the geometrical optics to find the scattering function. The diffuse component of the scattering function (without diffraction) is given by the formula

(11)

This function is plotted in Fig. 3. In contrast to the case of a specularly reflecting sphere, the predominant scattering to the back hemisphere takes place for large opaque spheres with diffusely reflecting surfaces.

Some applications of the geometrical optics approximation to engineering problems have been presented in the recent monograph by Dombrovsky and Baillis (2010).

Scattering function for large opaque spheres with diffusely reflecting surfaces

Figure 3. Scattering function for large opaque spheres with diffusely reflecting surfaces

REFERENCES

Born, M. and Wolf, E., Principles of Optics, 7th (expanded) ed., Cambridge University Press, Cambridge, England, 1999.

Dombrovsky, L. A. and Baillis, D., Thermal Radiation in Disperse Systems: An Engineering Approach, Begell House, Redding, CT, and New York, 2010.

Harpole, G. M., Radiative absorption by evaporating droplets, Int. J. Heat Mass Transfer, vol. 23, no. 1, pp. 17-26, 1980.

Kokhanovsky, A. A., Influence of end effects on the light absorption by weakly absorbing spheres, Opt. Spectrosc., vol. 78, no. 6, pp. 967-969 (in Russian), 1995.

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, pp. 1-21, 1997.

Nussenzveig, H. M. and Wiscombe, W. J., Efficiency Factors in Mie Scattering, Phys. Rev. Lett., vol. 45, no. 18, pp. 1490-1494, 1980.

Pinchuk, V. P. and Romanov, N. P., Absorption cross section for spherical particles of arbitrary size with moderate absorptance, J. Appl. Spectrosc., vol. 27, no. 1, pp. 109-116 (in Russian) , 1977.

Shifrin, K. S., Scattering of Light in a Turbid Medium, NASA, Washington, DC, 1968 (orig. Russian ed., Gostekhteorizdat, Moscow, 1951).

Twomey, S. and Bohren, C. F., Simple approximations for calculations of absorption in clouds, J. Atmos. Science, vol. 37, no. 9, pp. 2086-2094, 1980.

van de Hulst, H. C., Light Scattering by Small Particles, Wiley, Hoboken, NJ, 1957 (also Dover Publ., 1981).

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