Extinction efficiencies: general behaviour and deviations

The extinction efficiency factors are usually considered as a function of the size parameter for a fixed refractive index that, strictly speaking, does not characterize the dependence because of the wavelength dependence of (Sect. 2.1.2 in Voshchinnikov [2002]). The behaviour of is as follows: initially it rises rapidly with , has several maxima and minima and goes asymptotically to a constant with a decaying oscillation. This behaviour is typical for weakly absorbing homogeneous spheres (Fig. 1, upper panel; see also Fig. 11 in Voshchinnikov [2002]).

For large values of , the extinction factors approach the limiting
value 2 (see Eq. (2.73) and Fig. 11 in Voshchinnikov [2002]).
This would suggest that the particle will block off twice the light
falling upon it, an effect calling the ``extinction paradox''
(van de Hulst, [1957]; Bohren and Huffman, [1983]).
Its explanation lies in the fact that two different phenomena are occurring:
diffraction and the geometrical optics effects of
reflection, refraction
and absorption. The efficiency for each of these effects is one and hence

If the particle is very large, the diffraction pattern is very narrow. As laboratory detectors (including the human eye) collect light over a finite angular range, the diffracted light is registered and not measured as a loss. The measurement yields

as would be expected. However, the conditions in astronomy totally differ from those in the laboratory and the diffracted light is always lost, i.e.

The ripple-like structure in the form of small extremely sharp peaks and troughs is observed for non-absorbing particles if the real part of the refractive index is large (Fig. 1 as well as Fig. 11 in Voshchinnikov [2002]). These ripple-like fluctuations result from the resonances of virtual modes (Bohren and Huffman, [1983]). From the mathematical point of view, the scattering resonances are associated with zeros of the denominator in the expressions for Mie coefficients. The exclusion of the zeros lies in the basis of the S-approximation discussed in Sect. 2.2.4 in Voshchinnikov [2002]. Note that the ripples are washed away by any size distribution of particles.

Figure 1 illustrates the effect of variations of the real and imaginary parts of the refractive index and can be used for the prediction of extinction produced by real materials (e.g., given in Table 3 in Voshchinnikov [2002]). In particular, it is seen that the largest extinction for a given spherical particle can be obtained for dielectric particles with large values of (silicates, quartz, MgO, SiC). At the same time, the extinction by metals or carbonaceous particles dominates for very small size parameters (Fig. 1 as well as Fig. 12 in Voshchinnikov [2002]). The dilution of any material by the vacuum reduces both and (Table 4 and Fig. 5 in Voshchinnikov [2002]), which causes a shift and decrease of interference maxima and smoothing of the extinction curves (Fig. 1).

The behaviour of extinction for spherical particles described above is considered as typical for extinction factors. The deviations from it occur when the inhomogeneous or/and non-spherical particles are examined. Inhomogeneity causes variations of refraction and reflection inside a particle that changes the optical path of the rays. For non-spherical particles, refraction and reflection on the surfaces with variable curvature result in alteration of scattering by such particles in the comparison with that by spheres.

The optical properties of core-mantle
spheres were studied rather well
and seems to show no significant peculiarities
(Prishivalko *et al.*, [1984]) but already three-layered spheres
can produce anomalous extinction of light. This is illustrated in
Fig. 2 where the extinction efficiency factors
are plotted for spheres consisting of 3, 9 and 15 equivolume layers.
The layers are composed of different materials
(amorphous carbon (AC1),
astrosil and vacuum)
but the total volume
fraction of each constituent is 1/3. The optical constants
for AC1 and astrosil were taken as shown in Table 3 in Voshchinnikov [2002].
The peculiar behaviour of extinction is seen on the upper panel of
Fig. 2 for the case of particles
with the carbon core and astrosil as the outermost layer.
Here, a very rare situation is observed when the first maximum is damped
but there is very broad second maximum which is the highest. This peculiarity
disappears if the number of layers increases: the difference
in curves is hardly distinguished for particles with 15 layers
(lower panel of Fig. 2).
The given fact, noted by Voshchinnikov and Mathis ([1999]),
allows one to suggest multi-layered
particles as a new approximate model of composite
grains.
Such a model permits us to include an arbitrary
fraction of any
material and its numerical realization requires rather moderate computational
resources (see Sect. 3.2.5 in Voshchinnikov [2002]).

The dependence of the extinction efficiency factor on the parameter for prolate and oblate spheroids with the refractive index and the aspect ratio is shown in Figs. 3 and 4. The normalization used in Fig. 4 gives the possibility of comparing the factors calculated for different orientations of a particle. Large-scale variations of the factor have the same reasons as for spheres. But some features of the behaviour of the factors plotted in Figs. 3a and 4b take place: the third maximum is higher than the second one for prolate spheroids at and for oblate spheroids at if = 4 and 6. Asano ([1979]) found the similar peculiarity for particles with and 5. This phenomenon is determined by the particle shape and orientation and appears for dielectric particles with different ratios (see also Voshchinnikov, [2001]).

As seen from Fig. 3a, for prolate spheroids the values of can be rather large if radiation propagates along the major (rotation) axis of a particle ( ). This is a result of normalization by the geometrical cross-section which is small in this case (see Eq. (2.41) in Voshchinnikov [2002]). The factors can be even larger for (see Fig. 5).

As follows from this Figure, the behaviour of the factors for very elongated spheroids is rather regular, and the values of smoothly decrease with the size parameter. There are 25 maxima on the interval 0-300, which is totally distinct from the behaviour for spherical particles (cf. Fig. 11 in Voshchinnikov [2002]). Note that the size of ``equivolume'' particles considered in Fig. 5 is moderate: from Eq. (2.39) in Voshchinnikov [2002] it follows that if and . But the path of light inside a spheroidal particle is in times longer than inside a sphere of the same volume.

It is significant that the limiting value of the extinction factors for particles of any shape must be equal to 2 but as is seen from the inset in Fig. 5 this condition is far for being satisfied although the tendency to reduction of is observed.

Figures 3 and 4 also show that ripple-like fluctuations are better seen for the oblate spheroids than for the prolate ones. The ripples vanish from sight for elongated (prolate) spheroids if approaches zero. A similar picture is observed for infinite cylinders (Fig. 6). In this Figure, the extinction efficiencies for prolate spheroids and infinitely long circular cylinders are compared.

The simplest model of non-spherical particles -- infinite circular cylinders -- is not physically reasonable. However, it looks attractive to find cases when this model could be useful because calculations in this case are very simple and fast. Previous attempts to find the limits of applicability of the model of infinite cylinders were made by Martin ([1978]) and Voshchinnikov ([1990]). In both cases, particles of the same thickness were considered, i.e. spheroids and cylinders had the equal size parameters . Martin ([1978]) notes that the factors for spheroids resemble those for cylinders if for the normal incidence of radiation ( ) but in order to align the peaks in extinction, the -scale for cylinders was stretched by a factor 1.13.

Voshchinnikov and Farafonov ([2002]) established a similarity of the behaviour of the efficiency factors: both TM and TE-modes for spheroids converge to some limiting values which are close to those of infinite cylinders. This occurs if the size parameter is defined in a special way -- both the volume and the aspect ratio of a spheroid and a very long cylinder with the length are put to be the same, i.e. and . So, we need to compare the particles with size parameters and as it is shown in Fig. 6. Note that the scaling factor [ ] which arises is close to that empirically found by Martin ([1978]). For the normal and oblique incidence of radiation ( ), the efficiency factors converge to some limit values with an increase of the aspect ratio , provided spheroids of the same volume and thickness are considered. These values are close to, but do not coincide with the factors for infinite cylinders (see Voshchinnikov and Farafonov, [2002] for more details).