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Polarization efficiency: size/shape/orientation effects

The behaviour of the polarization efficiency $P/\tau$ (see Eq. (14)) for non-absorbing and absorbing spheroids is shown in Fig. 11 for the case when a maximum polarization is expected (PF alignment, $\alpha = 90\hbox{$^\circ$}$). The chosen refractive indices are typical for water ice and soot. The

Figure 11: Polarization efficiency against $x_V$ for prolate and oblate spheroids with $m=1.3+0.0i$ and $1.7+0.7i$, picket fence orientation, $\alpha = 90\hbox{$^\circ$}$. The size/shape effects are illustrated. After Voshchinnikov et al. ([2000]).

size/shape dependence of the ratio $P/\tau$ for these two materials is different. The polarization efficiency grows with the aspect ratio $a/b$ and is larger for oblate particles in comparison with prolate ones of the same volume but the polarization reversal takes place for disk-like particles. The last effect depends on the imaginary part of the refractive index and appears for absorbing prolate particles of large sizes as well.

In Table 1, the values of the parameter $x_{V} = 2\pi r_{V}/\lambda$ at which the polarization cross section $C_{\rm pol}/\pi r^2_V$ reach a maximum are presented for icy particles at $\alpha = 90\hbox{$^\circ$}$.

Table 1: Values of $x_{V} = 2\pi r_{V}/\lambda$ at which the circular polarization changes sign $(x^c_{V})$ and the linear polarization cross-section reaches a maximum $(x^{\rm pol}_{V})$ for spheroids and infinitely long circular cylinders with $m= 1.31 + 0.01i$ and $\alpha = 90\hbox{$^\circ$}$. Adapted from Voshchinnikov ([1990]).

$a/b$	$x^{\rm pol}_{V}$	$$\frac{C_{\rm pol}}{\pi r^{2}_{V}}$$	$$\frac{Q_{\rm pol}}{Q_{\rm ext}}$$	$~r^{\rm pol^\ast}_{V}, \,{\mu}\rm {m}$	$s^{\ast\ast}, \, {\mu}\rm {m}$	$x^c_{V}$	$$\frac{x^c_{V}}{x^{\rm pol}_{V}}~$$
Prolate spheroid
1.5	3.45	0.1036	0.057	0.30	0.26	3.40	0.99
2.0	3.44	0.1536	0.092	0.30	0.23	3.41	0.99
3.0	3.15	0.2036	0.157	0.28	0.19	3.42	1.08
5.0	3.71	0.3510	0.241	0.32	0.19	3.76	1.01
$\infty^{\ast\ast\ast}$	1.94	0.3552	0.184	--	0.19	2.00	1.03
Oblate spheroid
1.5	3.05	0.1210	0.067	0.27	0.31	2.97	0.97
2.0	3.01	0.2237	0.124	0.26	0.34	3.05	1.01
3.0	3.39	0.4052	0.176	0.30	0.43	3.38	1.00
5.0	3.34	0.6342	0.298	0.29	0.50	3.41	1.02

$^{\ast} r^{\rm pol}_{V}$ is the radius of equivolume sphere corresponding to $x^{\rm pol}_{V}$ if $\lambda=\lambda_{\max}=0.55\,{\mu}\rm {m}$; $^{\ast\ast} s = b$ for prolate spheroids and $s = a$ for oblate spheroids; $^{\ast\ast\ast}$ For infinitely long cylinders the following quantities are given: the parameter $x^{\rm pol}_{\rm cyl} = 2\pi r_{\rm cyl}/\lambda$ corresponding to the maximum linear polarization cross-section $Q^{E}_{\rm ext} - Q^{H}_{\rm ext}$, polarizing efficiency, the cylinder radius $r_{\rm cyl}$ for $\lambda=0.55\,{\mu}\rm {m}$ multiplied by the factor 1.145 (see Sect. 1.1), the parameter $x^c_{\rm cyl}$ at which the circular polarization changes the sign, and the ratio $x^c_{\rm cyl}/x^{\rm pol}_{\rm cyl}$.

These values $(x^{\rm pol}_{V})$ were obtained without smoothing the curves of $C_{\rm pol}/\pi r^2_V(x_V)$. From Table 1, one can see that the growth of $a/b$ leads to an increase of the polarization cross-sections and the polarizing efficiency of the medium, but the particle volume does not change strongly. With increasing $a/b$, the optical properties of prolate spheroids and infinitely long cylinders become similar.

It is also seen from Fig. 11 that relatively large particles produce no polarization independently of their shape. For absorbing particles, it occurs at smaller $x_V$ values than for non-absorbing ones. However, the position at which the ratio $P/\tau$ reaches a maximum is rather stable in every panel of Fig. 11 independently of $a/b$.

Figure 12: Extinction and linear polarization factors, polarization efficiency and circular polarization factors against $x_V$ for prolate spheroids with $m=1.3+0.0i$ and $a/b=2$, picket fence orientation. The effect of variations of particle orientation is illustrated.

This effect is broken if one considers tilted radiation incidence (Figs. 12, 13). The angular dependence of the extinction and linear polarization factors in Eq. (14) differs: if $\alpha$ decreases, the position of the maximum for $Q_{\rm ext}$ shifts to smaller values of $x_V$ while that for $Q_{\rm pol}$ shifts to larger $x_V$ (Fig. 12, upper panels). As a result, the maximum polarization efficiency for prolate spheroids is reached for smaller $x_V$ in the case of tilted orientation (Fig. 12, lower left panel). And the picture is reversed for oblate particles (see Fig. 13).

Figure 13: Polarization efficiency as a function of the size parameter $2 \pi a/\lambda$ for the prolate and oblate homogeneous spheroids with $m = 1.5+0.0i$ and $a/b=2$. The behaviour of extinction efficiencies is shown in Fig. 4. The effect of variations of particle orientation is illustrated.

Thus, it should be emphasized that for particles larger than the radiation wavelength, the linear polarization is expected to be rather small. This does not allow one to distinguish between the particle properties like refractive index, shape, orientation. Even in the case of ideal (PF) orientation, large particles (possibly available in dark clouds, - see Fig. 32 and discussion in Sect. 3.3.1 in Voshchinnikov [2002]) do not polarize the transmitted radiation. So, the decrease of the ratio $P/A_{\rm V}$ with the rise of $R_{\rm V}$ like found by Clayton and Cardelli ([1988]) should imply only that large grains are not efficient at producing polarization and is not connected with the change of grain shape or their less efficient alignment.

However, there is a possibility of reducing of the polarization efficiency associated with growth of the spherical icy mantles on non-spherical cores in dark clouds. In Fig. 14, this effect is illustrated for spheroidal particles with astrosil core and water ice mantle. The influence of variations of the mantle shape for particles of different sizes is shown. The shape of the core was fixed and for each curve the shape of the mantle remains the same. In this case, the ratio of the core volume to that of the particle does not change. For the values of $a/b$(core) used, it is rather small (from 0.11 to 0.004) and, therefore, the core's influence appears for particles of small radii only. For particles larger than $\sim 0.05\,{\mu}\rm {m}$, the polarization seems to be mainly determined by the shape of the particle mantle.

Values of the ratio $P/\tau$ presented in Figs. 11-14 are usually much larger than the upper limit for the interstellar polarization given by Eq. (3.42) in Voshchinnikov [2002]: . In order to reduce the ratio $P/\tau$, the imperfect orientation of dust grains should be considered. Note that the PDG orientation (in comparison with PF) should reduce the polarization of prolate grains only as PF and PDG orientations for oblate grains are equivalent (see Eq. (16)). It is interesting that the polarization efficiency created by rotating ellipsoidal particles is lower than that of oblate spheroids and sometimes even prolate spheroids (Fig. 15)⁴. Taking into account the problems with grain alignment in dark clouds (e.g., Lazarian et al., [1997]), the hope to solve the problem of the origin of polarization with the aid of more complex three-dimensional particles looks like unfounded.

Figure: Polarization efficiency as a function of particle equivolume radius $r_V$ for prolate core-mantle spheroids with $m({\rm core})=1.679+0.03i$ and $m({\rm mantle})=1.3+0.0i$, PF orientation, $\alpha = 90\hbox{$^\circ$}$. The effect of variations of mantle shape is illustrated.

Figure 15: Polarization efficiency as a function of size parameter $x_V$ for prolate and oblate spheroids and ellipsoids with $m=1.70+0.03i$, PDG orientation, $\Omega= 60\hbox{$^\circ$}$. The effect of variations of particle shape is illustrated. Adapted after Il'in et al. ([2002]).