Near-field effects
In the case of compact aggregates/media the electromagnetic interaction becomes even more complex, because the electromagnetic field in the close vicinity of the scattering particle is inhomogeneous due to the lag of the wave within the particle with respect to the incident wave. This effect is mostly expressed if the scatterer is comparable in size to the wavelength. Direct calculations using the Lorentz-Mie theory for spherical particles show that the constant phase surface of the total field is funnel shaped in the particle vicinity (Fig. 4.1a). Consequently, the field inhomogeneity near the particle causes a rotation of the total field vector relatively to the incident field vector. This results in the formation of a Z- component of the total field that lies in the scattering plane and, consequently, reduces the scattered intensity in the back and forward scattering regions and increases the negative polarization (Tishkovets, 1998; Tishkovets et al., 1999; 2004a, b; Petrova et al., 2007). To illustrate the influence of the field inhomogeneity in the vicinity of a particle, let us consider Rayleigh test particles located on a constant phase surface near a larger particle in its inhomogeneous zone (Fig. 4.1b). First, assume that the incident field is polarized in the scattering plane (as shown in Fig. 4.1a). If the test particles were far from each other and from other particles, i.e., in the homogeneous field, their dipole moments would be parallel to the x0 axis. In this case, the intensity of the light scattered by all four test particles-dipoles would concentrate in the direction a = 0° and 180° and would be zero in the direction a = 90°. If the test particles are, however, in the inhomogeneous zone near a wavelength-sized particle, the dipole moments induced in particles 1 and 3 have a nonzero component in the direction of wave propagation, i.e., along the z0 axis. This results in decreasing intensity of the scattered light in the direction a = 0° and 180°, whereas the intensity in the direction a = 90° becomes nonzero. In both cases, the scattered wave is polarized the same way as the incident one, i.e. in the scattering plane (negatively). Now assume that the incident wave is polarized perpendicular to the scattering plane. Then particles 1 and 3 produce the radiation that is polarized perpendicular to the scattering plane and does not depend on phase angle. The radiation scattered by particles 2 and 4 has a component parallel to the z0 axis (i.e., polarized in the scattering plane) that depends on a. As a result, the intensity again decreases in the directions a = 0° and 180° and increases in side directions, and polarization gets a negative component. So, at any polarization of the incident wave, the field inhomogeneity in the vicinity of the scattering particle induces a rotation of the field vector and leads to appearance of Z-component of the total field, which affects the angular distribution of the scattered intensity and causes negative polarization (for more details, see Tishkovets, 1998; Tishkovets et al., 1999; 2004a, b; Petrova et al., 2007).
One more type of interaction of particles in the near field is the mutual shielding of particles (Tishkovets, 2008; Petrova et al., 2009). The scheme with the test dipoles (Fig. 4.1b) helps to estimate qualitatively the result of the shielding. For the sake of simplicity, let us assume that at a given polarization of the incident radiation, the dipole moment of particle 1 is oriented exactly opposite to the ksc vector. In this case, particle 1 does not radiate in the ksc direction. It does not matter whether we take the shielding into account or not. When the incident radiation is polarized in the y0z0 plane, in the case of ignoring the shielding, particle 1 would radiate like particle 3 or like all the particles in the homogeneous field. However, when the large particle shields particle 1, the latter does not radiate in the a direction, i.e., its positive polarization does not contribute to the scattered light. Thus, the shielding diminishes the positively polarized scattered radiation and diminishes the intensity in the a direction. However, in the backscattering direction, dipole 1 contributes to the scattered radiation, which induces an increase in the intensity with respect to that in the a direction. Contrary to the field inhomogeneity in the near zone, which is most noticeable for the wavelength-sized particles, the mutual shielding effect is independent of the size of the particles located in the near field.
Under the above described conditions the wave coming from one particle to another is not spherical, and the single-scattering characteristics of individual monomers, such as their phase matrix, are not applicable. In other words, in densely packed systems the scatterers become highly dependent. The influence of the interaction in the near field on intensity and polarization of the scattered light can be easily demonstrated by the models, where the near-field contribution is ignored in the calculations of the light-scattering characteristics. The example presented in Fig. 4.2 clearly shows that the interaction in the near field substantially diminishes the backscattering peak in intensity induced by the coherent backscattering effect and changes the shape of the negative polarization branch.
Contrary to the coherent backscattering mechanism, the near-field effects work in a wide angular range. In the backscattering domain they distort the manifestations of the coherent backscattering. Their influence on polarization is rather complex and significantly depends on the size parameter of monomers, their packing density, and the refractive index. For example, with increasing packing density (i.e., when the near-field effects manifest themselves more clearly), the negative branch becomes deeper and wider if the aggregate is composed of larger monomers, while it may become shallower for smaller constituents. The modeling experiments with particles of different properties show that the most permanent and noticeable manifestation of the near-field effects in polarization is the shift of the polarization minimum out of opposition (Petrova et al., 2007; 2009). In other words, while the coherent backscattering mechanism forms the negative branch with the minimum near zero phase angle, the interaction in the near field causes the shift of the polarization minimum to larger phase angles and makes the negative polarization branch more symmetric.
Fig. 4.1. (a) The scheme shows the constant phase surfaces and directions of electric field vectors (sum of the incident and scattered waves) in the close vicinity of a particle with x=4.0 and m = 1.32 + ’0.05. The incident wave propagates along the wave vector ko and is polarized in the x0z0 plane. Adapted from Tishkovets et al. (2004a). (b) The scheme for the scattering of inhomogeneous waves by the Rayleigh test particles 1 – 4. Particles 1 and 3 are in the x0z0 plane, while particles 2 and 4 are in the y0z0 plane. The incident wave propagates along the z0 axis and is polarized in the x0z0 plane. The scattered wave propagates to the direction of the phase angle a. The vectors at the Rayleigh particles show the directions of the induced dipole moments.
Fig. 4.2. The influence of the interaction in the near field on the intensity (normalized to the value at zero phase angle) and polarization in the backscattering domain for the compact cluster shown in the insert. Thick and thin curves present the models calculated with and without the near-field effects respectively. Dashed curves show intensity and polarization for the individual monomer. The parameters x1, m, and N are shown in the figure.
Due to their nature, the manifestations of the near-field effects can be more easily observed in absorbing aggregates when the packing density exceeds 10-15%. One of such examples is shown in Figs. 4.3 for the whole range of phase angles and separately for the backscattering domain. For rather small number of monomers, the conditions for diffuse scattering and coherent backscattering are applied. With increasing number of monomers, the forward-scattering peak develops, the intensity profile becomes flatter, and the polarization maximum gets depressed. Then the opposition peak in intensity grows, and the negative branch of polarization appears. However, the opposition features do not develop as quickly as in nonabsorbing aggregates (compare Fig. 3.4), because the free paths become somewhat shorter when absorbing monomers are added into the volume. Partly due to this effect, partly due to the interaction in the near field -which becomes more important with increasing packing density – the polarization minimum moves out of opposition. Further increase of the packing density makes the near-field effects even more decisive. We see that the opposition peak stops to grow, while the negative branch continues to develop; it becomes wider and deeper (the curves for N=200).
Fig. 4.3. Top panel: same as Fig. 3.1-3.3, but for the parameters listed in the plot. The packing density varied from 0.1 % to 20% (N changes from 1 to 200, respectively). Bottom panel: larger scale for the backscattering domain; the intensity is normalized to the value at zero phase angle.
Modeling of light scattering by aggregates
In this section we explore how the considered above phenomena associated with electromagnetic interaction between constituents in a complex medium affect the angular and spectral dependence of intensity and linear polarization of the scattered radiation. We show how these results can be applied to the study of cosmic dust and other types of complex particles. We also briefly consider how the cooperative effects affect circular polarization of aggregates that contain optically active materials, e.g. complex organics of biological origin.
To model electromagnetic scattering by complex dispersed systems, several methods are now available. They are based on the numerically exact solutions of the Maxwell equations. One of them, the so-called superposition T-matrix method (Mishchenko et al., 2002; Mackowski & Mishchenko, 1996), was used to obtain the intensity and linear polarization of clusters of particles discussed above. Since these computations are time and resource consuming, they cannot be presently fulfilled for very large clusters/layers of particles, such as regolith. Nevertheless, they allow us to obtain the scattering characteristics of aggregates of a restricted number of monomers that are typical for cosmic dust, and to study the dependence of the light-scattering characteristics on the size of monomers, their packing density and refractive index.
Dependence of light scattering characteristics on the physical properties of aggregates
Exploring the light scattering characteristics of aggregates, we continue to focus on the dependence of intensity and linear polarization on phase angle, i.e. photometric and polarimetric phase curves. Our goal is to find out how the phase curves depend on such characteristics of aggregates as the size and composition of the monomers, their number and arrangement in the aggregate. In the previous sections we were mainly interested in the models of complex objects that allowed us to better see specific physical phenomena such as coherent backscattering or near-field effects. This section is directed to provide a basis for the interpretation of experimental data, specifically the observations of cosmic dust. This is why in this section we use more realistic models of natural aggregates, namely the aggregates grown under ballistic process (Meakin et al. 1984). There are commonly used two types of such aggregates: Ballistic Particle-Cluster Aggregate (BPCA) that grows at collision of individual monomers with the aggregate and Ballistic Cluster-Cluster Aggregate (BCCA) that grows at collision of clusters of monomers. Examples of such aggregates are shown in Fig. 5.1. Notice that BPCAs are usually more compact than BCCAs. The packing density of ballistic aggregates is defined as the ratio of the volume taken by their monomers to the total volume of the aggregate which is the volume of a sphere of the characteristic radius A calculated as
(Kozasa et al., 1992) where ri is location of the center of the ith monomer and the total number of the monomers is N. Packing density depends on the number of monomers; as this number increases, it decreases significantly for BCCAs and slightly for BPCAs (Kolokolova et al., 2007).
The results of the modeling of the light scattering characteristics of BCCA and BPCA at some refractive indexes and monomer size are shown in Figs. 5.2 -5.3; for more results see LISA database at https://www.cps-jp.org/~lisa/. There instead of intensity I we use albedo, a characteristic that is usually used in astronomical observations to describe the reflectivity of an object. In the case of aggregates, albedo is defined as (I/I0)*(n/G) where I0is the intensity of the incident light and G is the aggregate geometric cross section (Hanner et al., 1981; Kimura et al., 2003). We show spectral dependence of albedo and polarization in two filters: 450 nm (blue filter) and 600 nm (red filter). Following astronomical definitions, if albedo or polarization have larger values in the red filter we say that they have a red color and if the values are larger in the blue filter we say that they have a blue color.
Fig. 5.1. Samples of BPCA (left) and BCCA (right) aggregates. These aggregates were used in Kimura et al. (2003, 2006) computations to model light-scattering characteristics of cometary dust.
First, notice in Fig. 5.2-5.3 the features of the modeled phase curves described in the previous sections, namely: (1) strong forward scattering resulted from the interference of the light single-scattered by individual monomers; (2) rather low values of the maximum polarization that manifests depolarizing effects of the diffuse scattering and influence of the near-field effects; (3) some, although small, backscattering enhancement; and (4) rather small but symmetric branch of negative polarization at small phase angles. The last two features indicate a serious influence of the near-field effects. This is not surprising as the monomers in aggregates touch each other, i.e. they do are located in the inhomogeneous field produced by their neighbors. As it was shown in Section 4, the near-field effects affect the shape of the intensity curve and result in a more pronounced and symmetric negative polarization branch and in diminished values of the positive polarization. The figures also show a difference between the plots obtained for aggregates of different physical properties. The most influential parameter seems to be the monomer size whose variations change the shape of the polarization phase curve and the dependence of the albedo on the wavelength. The real part of the refractive index mostly affects the maximum polarization. The imaginary part of the refractive index affects the spectral dependence of photometric phase curve and the values of albedo but does not much affect polarization. Notice also that the more compact BPCAs depolarize the light more strongly than the more porous BCCAs, although their other characteristics are rather similar.
Although the curves in Figs 5.2-5.3 resemble the typical observational curves shown in Fig.1.1, they have some characteristics that are not typical for cometary dust. Observational facts summarized in Kolokolova et al. (2004a, b) indicate that comets usually have red photometric and polarimetric colors, i.e. their albedo and polarization have larger values at longer wavelengths. Unlike the observational data, the results of the modeling shown in Figs. 5.2-5.3 always demonstrate predominantly blue photometric color. In the case of the monomers of radius 120 nm and the refractive index equal to 1.4+i0.01, the results of the modeling also demonstrate blue polarimetric color for some phase angles. Also, in the majority of plots, the value of albedo at zero phase angle is higher than the one observed, which is equal to 3 – 5% (Hanner, 2003).
Fig. 5.2. Albedo (in %) and polarization as functions of phase angle for aggregates of monomer radius equal to 90 nm. Real part of the refractive index, n, and imaginary part of the refractive index, k, are shown in the top left corner of each figure. Results for the wavelength 450 nm are shown by thick line (BCCA) and crosses (BPCA) and for 600 by thin line (BCCA) and circles (BPCA). All aggregates consisted of 128 monomers.
Fig. 5.3. The same as Fig. 5.2 but for monomers of radius 120 nm.
Our computations, summarized in Kimura et al. (2003, 2006) provided characteristics of the aggregates that satisfy the observational data for cometary dust. The best fit was achieved for the monomers of radius 100 nm and the refractive index that was determined based on in situ study of comet Halley, which is equal to 1.88+i0.47 for the wavelength A=450nm and to 1.98+i0.48 for A=600nm. It appears that for such a dark material a crucial characteristic is the number of monomers in the aggregate. Fig. 5.4 shows that increasing the number of monomers in the aggregate results in a more pronounced negative polarization branch and in a stronger depolarization of the positive polarization. This allows us to suggest that in the case of aggregates of thousands of monomers it is possible to reach the observable values of negative (~0.015) and positive (~0.3) polarization.
Fig. 5.4. Albedo (in %) and polarization as functions of phase angle depending on the aggregate size (number of monomers in the aggregate). The monomer radius is equal to 100 nm. The refractive index was taken as typical for cometary dust (based on in situ data for comet Halley) and is equal to 1.88+i0.47 for the wavelength A=450nm and 1.98+i0.48 for A=600nm. The number of monomers in the aggregate is 64 (left panel), 128 (middle panel), 256 (right panel). Development of the negative polarization is shown in the inserts. Notice also a decrease of the polarization maximum more pronounced for the shorter wavelength.
Figs. 5.2-5.4 also demonstrate that the polarimetric color is often less red in the case of more compact BPCAs. We explain this by a stronger depolarization of light in the case of more compact aggregates. Such a depolarization is even more evident from Figs. 3.3 and 4.3 where aggregates with higher packing density (more particles in the volume) always demonstrate smaller polarization maximum. Depolarization of light with increasing packing density is consistent with increasing electromagnetic interaction between the monomers resulted from both diffuse multiple scattering and near-field effects as considered in Sections 3-4 Kolokolova & Kimura (2010) showed that a measure of the depolarization can be the number of monomers covered by a single wavelength; the more monomers the wavelength covers, the more depolarized is the scattered light. It is clear that a single wavelength covers more monomers in the case of more compact aggregates. It also covers more monomers if the wavelength is longer. Thus, we can expect the scattered light to be more depolarized at longer wavelengths and the color of polarization should be blue. Blue polarimetric color is frequently observed. For example, it is typical for asteroid surfaces and interplanetary dust. However, as we already mentioned, cometary dust has a red polarimetric color. In our opinion, this is good evidence that cometary aggregates are highly porous. For porous aggregates, an increase in the wavelength may not increase the number of monomers covered by a single wavelength. Then the polarimetric color is defined by properties of individual monomers. Specifically, the monomer size parameter decreases with increasing wavelength that moves it closer to the Rayleigh regime of scattering characterized by higher polarization, thus, resulting in the red color of polarization.
An interesting observational result was reported by Kiselev et al. (2008) who summarized the observational data of spectral behavior of comet polarization and showed that cometary dust is characterized by a red polarimetric color in the visible (wavelengths of 400-800nm) but it changes to a blue polarimetric color in the near infrared (wavelengths of 1000-3000nm). They also showed that some comets exhibit a blue polarimetric color even in the visible. These observations can be interpreted based on the dependence of electromagnetic interaction on the number of monomers covered by a single wavelength. Fig. 5.5 illustrates our point. One can see there that in the case of a porous aggregate a small change in the wavelength does not change the number of particles it covers. However, at longer wavelength even in porous aggregates the number of monomers covered by a single wavelength increases causing depolarization of the scattered light.
Fig. 5.5. Illustration of the effect of increasing wavelength on the light scattering by an aggregate. In a compact aggregate (top part of the aggregate) the longer the wavelength the more monomers it covers, so the interaction between the monomers becomes stronger, and the light becomes more depolarized. This results in a decrease of polarization with wavelength, i.e. blue color of polarization. For a porous aggregate (bottom part of the aggregate), the number of monomers covered by a single wavelength does not change much as the wavelength increases, i.e. the change in the interaction between the monomers cannot overpower the change in the monomer size parameter, and so the polarization color stays red. However, as the wavelength reaches some critical value, the number of covered monomers in the porous aggregate changes significantly(as shown in the right-hand aggregate) and interaction becomes the main factor that defines the polarization color which then becomes blue.
This explains the change in the observed polarimetric color as the observations move to the near infrared. In the case of more compact aggregates, even a slight change in wavelength increases the number of covered monomers resulting in blue polarimetric color even in the visible. Thus, it is likely that the dust in the comets with blue polarimetric color, as well as asteroidal and interplanetary dust, is characterized by more compact particles. The wavelength where polarimetric color changes from red to blue may be used to determine the porosity of aggregate particles.
