## The math of Mie scattering | ECE532 Biomedical Optics ©1998 Steven L. Jacques, Scott A. Prahl Oregon Graduate Institute |

Consider a source, a spherical scattering particle, and an observer whose three positions define a plane called the **scattering plane**. Incident light and scattered light can be reduced to their components which are parallel or perpendicular to the scattering plane. As shown in the following figure, the parallel and perpendicular components can be experimentally selected by a linear polarization filter oriented parallel or perpendicular to the scattering plane.

The Scattering matrix describes the relationship between incident and scattered electric field components perpendicular and parallel to the scattering plane as observed in the "far-field" (ref: Bohren and Huffman):

The above expression simplifies in practical experiments:

- The exponential term, -exp(-ik(r-z))/ikr, is a transport factor that depends on the distance between scatterer and observer. If one measures scattered light at a contant distance r from the scatterer, eg., as a function of angle or orientation of polarization, then the transport factor becomes a constant.

- The total field (E
_{tot}) depends on the incident field (E_{i}), the scattered field (E_{s}) , and the interaction of these fields (E_{int}). If one observes the scattering from a position which avoids E_{i}, then both E_{i}and E_{int}are zero and only E_{s}is observed.

- For "far-field" observation of E
_{s}at a distance r from a particle of diameter d such that kr >> n_{c}^{2}, k = 2/, n_{c}= d/, the scattering elements S_{3}and S_{4}equal zero (see Eq. 4.75, Bohren and Huffman).

- Practical experiments measure intensity, I = <E E*> = (1/2)a
^{2}, where E = a exp(-i), and a is amplitude and is phase of the electric field.

Hence for practical scattering measurements, the above equation simplifies to the following:

**ANGULAR SCATTERING PATTERN OF POLARIZED LIGHT** Mie theory calculates the angular dependence of the two elements, S_{1}() and S_{2}(), of the Scattering matrix, from which the scattered intensities of polarized light are computed (see example). The scattering pattern is also used to calculate the anisotropy, g, of scattering by the particle.

**EFFICIENCIES OF SCATTERING AND ABSORPTION** Mie theory calculates the efficiencies Q_{s} and Q_{a} of scattering and absorption, respectively, such that µ_{a} = Q_{a}A and µ_{s} = Q_{s}A, where A is the geometrical cross-sectional area a^{2} for a sphere of radius a.

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