Scattering parameterization for interpreting asteroid polarimetric and photometric phase effects
© The Society of Geomagnetism and Earth, Planetary and Space Sciences, The Seismological Society of Japan 2010
Received: 5 September 2008
Accepted: 23 January 2009
Published: 7 February 2015
We derive an analytical parameterization of the amplitude and Mueller scattering matrices of a system composed of a small number of electric dipoles. The appeal of this derivation is that it provides a wide range of light-scattering polarization states with a minimal number of parameters. Such a tool can be used to interpret observations in terms of physical parameters. We aim to utilize these results in multiple-scattering studies, such as the interpretation of polarimetric and photometric phase effects of asteroids and other atmosphereless solar-system objects.
Two ubiquitous phenomena are observed near opposition for asteroids and other atmosphereless solar-system objects as well as for cometary and interplanetary dust: negative degree of linear polarization and opposition effect. The phenomena are confined to Sun-object-observer angles (phase angles) of less than 30 and 10 degrees, respectively. Sometimes they appear at extremely small phase angles, less than one degree. Negative polarization implies that the intensity polarized parallel to the Sun-object-observer plane (scattering plane) is predominating over the one perpendicular to the plane. The opposition effect is a nonlinear surge of brightness towards the backward scattering geometry. The coherent-backscattering and shadowing mechanisms have been considered as the primary causes for the phenomena.
Wide backscattering peaks and negative polarization branches have been detected consistently in extensive numerical simulations of light scattering by irregular wavelength-scale particles (e.g., Lumme and Rahola, 1998; Zubko et al., 2006a; Muinonen et al., 2007a). The phenomena are present for compact irregular particles as well as for irregular aggregates of constituent spherical or nonspherical particles. Recently, we have succeeded in uncovering internal-field characteristics that give rise to such polarization and intensity signatures (Zubko et al., 2006b, 2007; Muinonen et al., 2007a; Tyynelä et al., 2007, 2008; Muinonen and Erkkilä, 2007>) and thus have introduced a single-scattering polarization and intensity mechanism (Muinonen et al., 2007b). As to the light-scattering experiments, by measuring the single-particle scattering characteristics and those of a close-packed particulate medium of similar single particles, it has been established that the particulate media continue to exhibit single-particle polarization characteristics but that these characteristics are neutralized (e.g., Shkuratov et al., 2004).
Multiple-scattering models for planetary surfaces depend on five groups of physical parameters. First, surface roughness in length scales of several wavelengths and large numbers of particles causes surfacial or interfacial shadowing effects. Second, the porosity of the particulate medium causes volume shadowing effects—again, the length scales are large compared to the wavelength and size of the particles. Third, particle size introduces its signature into the scattering characteristics. Fourth, particle shape plays an important role in determining the detailed structure of the scattering matrix. Fifth, the complex refractive index represents the refractory optical properties of the material of which the particles are composed.
In Section 2, we review the interference effects in scattering by multiple small scatterers essential for the present study. In Section 3, we develop scattering parameterizations based on systems composed of a small number of electric dipoles. We conclude the article in Section 4 describing future prospects for the scattering parameterization developed.
2. Single-scattering Interference
Characteristics In order to illustrate single-scattering interference of relevance to the present study (Muinonen et al., 2007a; Tyynelä et al., 2007), we consider an electromagnetic plane wave (vector amplitude E0, wavelength λ, wave number k = 2π/λ, and wave vector k0) propagating along the z-axis and incident on a spherical scatterer located in the origin (Figs. 1 and 2). Consider an observer in the xz-plane (scattering plane) with the scattering angle θ describing the angular deviation from the forward-scattering direction. Thus, the phase angle is A = π - θ. In order to obtain the scattering characteristics for incident unpolarized light, the scattering problem needs to be solved for two perpendicular linear polarization states of the incident field, that is, for the ypolarized incident polarization vector perpendicular to the scattering plane and for the x-polarized incident polarization vector parallel to the scattering plane. The final scattering characteristics follow as the average of the characteristics for the two polarization states of the incident wave. To further simplify the illustration, consider the internal fields induced in the spherical particle on the x and y-axes only. In reality, interference can take place at differing depths in the direction of the z-axis and the depths do not need to be equal among the pairs of locations shown in Figs. 1 and 2. Assume, for the time being, that the six contributions depicted in Figs. 1 and 2 do not interfere with each other.
The single-scattering mechanism is based on the hypothesis that, typically, for the incident y-polarized field, the y-polarized internal-field components on the x-axis are stronger than those on the y-axis and, similarly for the incident x-polarized field, the x-polarized internal-field components on the y-axis are stronger than those on the x-axis. As a net result close to backscattering, with the x-polarized internal-field components on the y-axis predominating over other contributions via constructive interference, the degree of linear polarization for incident unpolarized light assumes negative values. This is enhanced by the first destructive interference geometry close to backscattering for the ypolarized scattered field arising from the x-axis. For both incident polarizations, the transverse internal-field components give rise to constructively interfering scattered-field components in the backward-scattering direction, resulting in a backscattering peak in the scattered intensity.
The single-scattering mechanism differs from the coherent-backscattering mechanism. In coherent backscattering, reciprocal waves travel through the same scatterers in a random medium, interfering constructively in the exact backscattering direction but not necessarily in other directions. In the single-scattering mechanism, the electric fields on the mirror locations describe the net result of all possible interactions among the electric dipoles constituting the scatterer. Last but not least, coherent backscattering is typically seen to occur in supermicron length scales; whereas, the single-scattering mechanism is relevant in submicrontomicron length scales for visible light.
3. Scattering Parameterization
In what follows, we develop an analytical parameterization based on the scattering from pairs of electric dipoles. We make use of Figs. 1 and 2 in developing the amplitude scattering matrix elements and, in particular, fix the scattering plane to be the xz-plane. We follow the geometries in Figs. 1 and 2 and assume that the dipoles are located either on the y-axis (scatterer 1) or on the x-axis (scatterer 2). Treated separately, these scatterers produce pure scattering matrices and well-defined complex amplitudescattering matrices. We assess two perpendicular polarization states of the incident wave propagating in the positive direction of the z-axis. Our present modeling relies on the differing interference characteristics along the x-axis for the perpendicular and parallel polarizations.
Let us start by studying scatterer 1 constituting a system in the direction of the y-axis (Figs. 1(b), 1(c), and 2(c)). Due to destructive interference, there is no contribution from the configuration in Fig. 1(c) in the scattering plane. Thus, the configuration in Fig. 1(b) is solely responsible for the contribution from the incident field perpendicular to the scattering plane. The configuration in Fig. 2(c) is solely responsible for the contribution from the incident field parallel to the scattering plane.
Let us continue by studying scatterer 2, constituting a system in the direction of the x-axis (Figs. 1(a), 2(a), and 2(b)). The configuration in Fig. 1(a) is responsible solely for the contribution from the incident field perpendicular to the scattering plane. However, the contributions from configurations in Figs. 2(a) and 2(b) interfere with each other. For scatterer 2, we introduce the interdipole distances d1, d2, and d3 corresponding to the configurations in Figs. 1(a), 2(a), and 2(b). Also, we assign different amplitudes and phase factors a t , ϕ t and a l , ϕ l for the transverse (Figs. 1(a) and 2(a)) and longitudinal (Fig. 2(b)) electric dipoles, respectively.
4. Coherent-backscattering Simulations
We carry out example coherent-backscattering simulations for infinitely thick spherical random media of scatterers with the parameters indicated in Fig. 3 (choosing a l = 0.8) for three single-scattering albedos ῶ=0.3, 0.6, and 0.9, and 28 extinction mean free paths kl = 30, 40, …, 100, 120, 140, … , 200, 250, 300, … , 400, 500, 600, … , 1000, 2000, 3000, … , 5000, and 10000 (cf., Boehnhardt et al., 2001; Muinonen, 2004). For ῶ=0.3 and 0.6, we sampled 100000 incident rays; whereas, for ῶ=0.9, we sampled 25000 rays due to extensive computational time.
We have succeeded in developing an analytical semiempirical representation of amplitude and Mueller scattering matrices to be utilized in inverse problems concerning asteroid photometric and polarimetric phase curves. Our first numerical coherent-backscattering simulations show that the parameterization is realistic and flexible, allowing systematic application to the existing asteroid phase-curve data. Whereas we presently account for the differing interference characteristics along the x-axis for the perpendicular and parallel polarizations, it remains as our future goal to study analytically more complicated systems, such as those composed of four dipoles.
Research supported, in part, by the Academy of Finland (project No. 127461) and by the Chancellor at the University of Helsinki.
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