Astron. Astrophys. 360, 777-788 (2000)

## 2. Some concepts of light scattering and the experimental setup

In this section, we summarize some concepts from light scattering theory that are used in this work. The flux and polarization of a quasi-monochromatic beam of light can be represented by a column vector I , or Stokes vector (Van de Hulst 1957; Hovenier & van der Mee 1983), where I is proportional to the total flux of the beam. The Stokes parameters Q and U represent differences between two components of the flux for which the electric field vectors oscillate in mutual orthogonal directions. The Stokes parameter V is the difference between two oppositely circularly polarized components of the flux. A plane through the direction of propagation of the beam is chosen as a plane of reference for the Stokes parameters.

If light is scattered by an ensemble of randomly oriented particles and time reciprocity applies, as is the case in our experiment, the Stokes vectors of the incident beam and the scattered beam are related by a scattering matrix, for each scattering angle , as follows (Van de Hulst 1957, Sect. 5.22),

where the subscripts i and s refer to the incident and scattered beam, respectively, is the wavelength of the incident beam and D is the distance from the ensemble to the detector. The matrix with elements is called the scattering matrix. Its elements depend on the scattering angle, but not on the azimuthal angle. Here the plane of reference is the scattering plane, i.e., the plane containing the incident and the scattered light. The elements contain information about the size distribution, shape and refractive index of the scatterers. It follows from Eq. (1) that there are 10 different matrix elements to be determined. This number is further reduced in case a scattering sample consists of randomly oriented particles with equal amounts of particles and their mirror particles. In that case, the four elements , , , and are identically zero over the entire angle range (Van de Hulst 1957).

A schematic picture of the experimental setup used to measure the scattering matrix is shown in Fig. 1. We use either a HeNe laser (633 nm, 5mW) or a HeCd laser (442 nm, 40mW) as a light source. The laser light passes through a polarizer oriented at an angle and an electro-optic modulator oriented at an angle (angles of optical elements are the angles between their optical axes and the scattering plane, measured counterclockwise when looking in the direction of propagating of the light). The modulated light is subsequently scattered by the ensemble of randomly oriented particles located in a jet stream produced by an aerosol generator. The scattered light passes through a quarter-wave plate oriented at an angle and an analyzer oriented at an angle (both optional) and is detected by a photomultiplier tube which moves along a ring. A range in scattering angles is covered from approximately 5o (nearly forward scattering) to about 173o (nearly backward scattering). We employ polarization modulation in combination with lock-in detection to obtain all elements of the four-by-four scattering matrix up to a common constant. A more detailed description of the setup is given by Hovenier (2000).

 Fig. 1. Schematic picture of the experimental setup; P, polarizer; A, polarization analyzer; Q, quarter-wave plate; PM, photomultiplier; M electro-optic modulator.

Errors in the measured matrix elements originate from fluctuations in the measured signal or signals. For each data point at a given scattering angle, 720 measurements are conducted in about 2 seconds. The values obtained for the measured matrix elements or combinations of matrix elements are the average of several data points (about 5 or more) and the corresponding experimental error is the standard deviation in these. The resulting standard deviations are indicated by error bars in Fig. 2 and later figures. When no error bar is shown the value for the standard deviation is smaller than the symbol plotted. A monitor photomultiplier at a fixed scattering angle (see Fig. 1) is used to correct for variations in the amount of particles in the jet stream during the measurement run.

 Fig. 2. Calibration measurements with water droplets. The squares correspond to the measurements at 442 nm and the circles to the results at 633 nm. The measurements are presented together with their error bars. In case no error bars are shown, they are smaller than the symbols. The solid lines correspond to the Mie calculations at both wavelengths.

We investigated the reliability of the measurements presented in this paper by applying the Cloude coherency test (Hovenier & van der Mee 1996). For the particles of the ground piece of Allende meteorite, we had not enough sample material to measure and . To be able to apply the Cloude coherency test for these samples, we assumed these elements to be zero at all scattering angles, since they proved to be zero within the experimental errors for the other samples. We found that for all matrix elements, the values measured for scattering angles from 5o to 173o are in agreement with the Cloude coherency test within the experimental errors.

Measurements with water droplets have been done in order to test the alignment of the set-up. Since the water droplets have spherical shapes, we could compare the experimental results with those obtained from Mie calculations. The water droplets were produced by a nebulizer. For convenience, we normalize all matrix elements (except itself) to , i.e., we consider , with = 1 to 4. Instead of we have plotted the degree of linear polarization for incident unpolarized light,

where and represent the flux of the scattered light polarized perpendicular and parallel to the plane of scattering respectively. The results of all measurements and calculations presented are plotted on a logarithmic scale. We chose to normalize so that it equals 1 at . We omitted the four element ratios , , and , since we verified that these ratios do not differ from zero by more than the error bars.

A comparison between measurements with water droplets at 442 and 663 nm and Mie calculations is shown in Fig. 2. We find that there is an excellent agreement over the entire angle range measured for all scattering matrix elements. For the Mie calculations we used a log-normal size distribution (Hansen & Travis 1974) with = 1.1 , = 0.3, and a refractive index . Values for and were chosen so that the differences between the results of Mie calculations and measurements were minimized for all scattering matrix elements. Remaining differences between the measured and calculated values may be due to small aligment errors or to the fact that the size distribution of the droplets deviates somewhat from a log-normal distribution.

© European Southern Observatory (ESO) 2000

Online publication: August 17, 2000