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Scattering matrix, M, of a particle is measured by using an arrangement of a polarizer, setting the polarization state of incident light, and an analyzer (i.e. another polarizer) that passes scattered light with a desired polarization state only (Fig. 1). The scattered light irradiance is measured with a polarization-insensitive detector. Measurements are typically carried on a dispersion of "identical" particles (for example, Volten H et al 2005) because a single particle may scatter too little light. With nonspherical particles this usually implies random orientation of such particles.
Given that the scattering of electromagnetic radiation by a particle depends on the relative size of the particle, scattering matrices of nonspherical particles with complex shapes/structures, for which no analytical models exist, have been measured for centimeter-sized models of these particles scattering at microwave radiation wavelengths, on the order of 1 cm (for example, Gustafson BAS 1996). Needless to say, the refractive index of the model material must be the "same" as that of the particle whose scattering properties are being modeled. This is referred to as microwave analog measurements.
Each combination of the polarizer and analyzer in Fig. 1 defines the following equation:
| Ss(ξ) = [1 / (kr)2] MA M(ξ) MP Si | (1) |
where Ss and Si are the Stokes vectors of the scattered and incident light respectively, ξ denotes a direction in the reference frame tied to the particle "center", and MA and MP are respectively the Mueller matrices of the polarizer and analyzer. The incident light is assumed to be unpolarized. Eq. 1 can be normalized to the irradiance of the incident light beam so that one can assume that its Stokes vector is simply [1, 0, 0, 0]T, where T is the transpose operator.
With the arrangement shown in Fig. 1, one in fact measures the first element, S1, of the Stokes vector of the scattered light, i.e. the irradiance of light. With each combination of the polarizer and analyzer this irradiance, i.e. an element S1 of the Stokes vectors, is a linear combination of certain elements of the scattering matrix. It can be seen, for example, that by using no polarizer and analyzer with an unpolarized light of unity irradiance we have:
| Ss1 = M11 | (2) |
Hielscher AH et al 1997b show, following Bickel WS 1985, that all elements of the scattering matrix can be solved for by using 16 equations obtained with 49 combinations of the polarizer and analyzer:
| M11 = Ss1OO | (3a) |
| M12 = (Ss1HO - Ss1VO) / 2 | (3b) |
| M13 = (Ss1PO - Ss1MO) / 2 | (3c) |
| M14 = (Ss1LO - Ss1RO) / 2 | (3d) |
| M21 = (Ss1OH - Ss1OV) / 2 | (3e) |
| M22 = (Ss1HH + Ss1VV) / 4 - (Ss1HV + Ss1VH) / 4 | (3f) |
| M23 = (Ss1PH + Ss1MV) / 4 - (Ss1PV + Ss1MH) / 4 | (3g) |
| M24 = (Ss1LH + Ss1RV) / 4 - (Ss1LV + Ss1RH) / 4 | (3h) |
| M31 = (Ss1OP - Ss1OM) / 2 | (3i) |
| M32 = (Ss1HP + Ss1VM) / 4 - (Ss1HM + Ss1VP) / 4 | (3j) |
| M33 = (Ss1PP + Ss1MM) / 4 - (Ss1PM + Ss1MP) / 4 | (3k) |
| M34 = (Ss1LP + Ss1RM) / 4 - (Ss1LM + Ss1RP) / 4 | (3l) |
| M41 = (Ss1OL - Ss1OR) / 2 | (3m) |
| M42 = (Ss1HL + Ss1VR) / 4 - (Ss1HR + Ss1VL) / 4 | (3n) |
| M43 = (Ss1PL + Ss1MR) / 4 - (Ss1PR + Ss1ML) / 4 | (3o) |
| M44 = (Ss1LL + Ss1RR) / 4 - (Ss1LR + Ss1RL) / 4 | (3p) |
where the superscripts define the type of the polarizer and analyzer (in this order) as follows: O - no polarizer (see its Stokes vector), H - linear polarizer whose axis is inclined at an angle of 0° to the scattering plane (see its Stokes vector), V - linear polarizer at an angle of 90° (see its Stokes vector), P - linear polarizer at an angle of 45° (see its Stokes vector), M - linear polarizer at an angle of 135° (see its Stokes vector), R - right-handed circular polarizer (see its Stokes vector), L - left handed circular polarizer (see its Stokes vector).
Given the number of combinations of the polarizer and analyzer, this basic procedure is fairly time-consuming (for example, Kadyshevich YA et al 1977). Hence, more recent methods of the scattering matrix measurements are based on periodic modulation of the polarization states of the polarizer and analyzer and extracting the scattered light irradiances for the various polarizer-analyzer combinations from the time-dependent detector signal (for example, Mujat M and Dogariu 2001, Thompson RC et al 1979).
See also Problem 2 in Scattering matrix: Problems
| CITATION: Jonasz M. 2006. Scattering matrix measurement (www.tpdsci.com/Tpc/ScaMtxMs.php). In: Top. Part. Disp. Sci. (www.tpdsci.com). |
HISTORY: Published: 15-Oct-2006 Modified: 15-Oct-2006 Peer-reviewed: PENDING |
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