We consider the elastic scattering of light by an ensemble of scatterers with radius of gyration greater than about one-tenth of a wave; e.g., visible light scattered by an aqueous suspension of viruses or bacteria. We model the scattering molecule as an asymmetric rigid array of interacting point polarizabilities, and we include, as the source of anisotropy in the scattering ensemble, a permanent dipole moment on the molecule and a uniform electric field E" in the scattering cell. We calculate the entire Muller matrix M(e, E:, E;, E;), for scattering angles e from 0 0 to 360 0 and for E:, E;, E; nonzero one at a time, where z is the incidence axis, y is perpendicular to the scattering plane, and x is perpendicular to y and z. Our first major finding is that the nondipole elements ofM are enormously more sensitive to partial orientation than are the dipole elements. The second major finding is a set of new symmetries governing the scattering matrix for the case of axially anisotropic scattering clouds. The Perrin reciprocity symmetries for isotropic clouds may be symbolized by M(O) = PM(O), where P stands for matrix transpose plus negation of third row and column (with double negation of element 3,3). Using this operator our new axial symmetries may be expressed as PM( + 0, 0, E;. 0) = M( + e, O. -E;,O) = M( -0, 0, E;, 0). The second symmetry may be generalized to fields in any direction as M( + 0, -E:, -E;. E;) = M( -0, E:, E;, E;). We also show in Appendix A how the nonlocal polarizabilities used in the theory may be calculated by the inversion only of symmetric matrices, with a significant saving in calculation time when the number of subunits is large.
The Müller matrix representing elastic scattering of light by an isotropic suspension of particles has long been known to contain eight vanishing elements among its total of sixteen elements, provided that the particles have no handedness. We generalize the proof of this property to the case of anisotropic suspensions of particles, so that it may apply to symmetric biostructures such as viruses, undergoing electrophoresis. The matrix for the partially oriented case generally contains four crucial pieces of structural information missing from the randomly oriented case. In particular, the circular intensity difference scattering (CIDS) element M14 may be nonzero for an oriented ensemble of particles that have no helicity at all.
We have calculated the Perrin matrix for elastic light scattering by two distinct numerical methods, and we present here a critical comparison. Both methods assume the dipole array model with retarded dipole–dipole interaction between subunits. However, the average over all orientations of the model is carried out differently in the two methods. The conceptually simplest method (and the easiest to program) is rotation by Cartesian matrix (CM) algebra with three-dimensional numerical integration over Euler angles. The second method is based on rotations by Wigner matrix (WM) algebra with analytic integration, and requires an infinite sum in one index which must be truncated. The two methods agree when they are pushed far enough. Both methods require substantial calculation, but the analytic method is shown to be far superior in its convergence properties. Comparing calculations of the whole Perrin matrix of scattering curves (16 curves, 21 scattering angles) in which all curves are computed to 1% accuracy, the CPU times for the numerical integration over Euler angles is about 500 times slower.
A two-frequency beam from a Zeeman laser scatters elastically from an isotropic medium, such as randomly oriented viruses or other particles suspended in water. The Zeeman effect splits the laser line by 250 kHz, and beats can be seen electronically in the signal from a phototube that views the scattered light. There are independently rotatable half-wave and quarter-wave retardation plates in the incident beam and a similar pair in the observed scattered beam, plus a fixed linear polarizer directly in front of the detector. Each of the four retarders has two angular positions, providing a total of 16 possible polarization cases. For each of the 16 cases, there are three data to be collected: (1) the average total intensity of the scattered light, (2) the amplitude of the beats in the scattered light, and (3) the phase shift between the beats of the scattered light and those of a reference signal from the laser. When a singular value decomposition technique is used, these threefold redundant data are rapidly ransformed into a best-fit 4 × 4 Mueller scattering matrix. We discuss several different measurement strategies and their systematic and statistical errors. We present experimental results for two kinds of particle of wavelength size: polystyrene spheres and tobacco mosaic virus. In both cases the achiral retardation element M(34) of the Mueller matrix is easily measurable.
Sound scattering by zooplankton with arbitrary shape and orientation J. Acoust. Soc. Am. 92, 2392 (1992; 10.1121/1.404745Longwave properties of the orientation averaged Mueller scattering matrix for particles of arbitrary shape. I. Dependence on wavelength and scattering angleIn Part I of this paper, we started from a dipole array model of elastic light scattering, and found the longwave asymptotic formulas for all 16 elements of the orientation averaged Mueller scattering matrix. However, the Perrin symmetry of the scattering matrix was not obvious from the formulas obtained in Part I. In this paper, Part II, we carry the analysis further, finding the molecular parameter identities which result in the Perrin symmetries. The formulas we present provide a very practical method for model calculations in the longwave limit. After evaluation often sums over pairs of the dipolar subunits of the model, the orientation averaged Mueller scattering matrix, as a function of scattering angle, is given by simple trigonometric polynomials. For models containing several hundred subunits the computation is easily carried through by a desktop computer. We verify the asymptotic formulas by numerical comparison with our analytic orientation averaging program PMA T2. We use a helical model with spherical subunits, in which the first Born approximation gives excellent results for the dipole elements (symmetry DSE). In this same model the first, second, and third Born approximations are utterly worthless for calculating the helicity-retardation elements MJ3 and M 23 and their transposes, but fourth Born gives nearly the exact result.These observables therefore use four bounces to feel out the helicity of the array, and may be more sensitive to structural variations than the traditional circular difference observable M 14 , which responds after only two bounces. 4726
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