Scattering by an Isotropic Homogeneous Sphere

An isotropic, homogeneous sphere is the simplest finite scatterer, the theory of scattering by which is attached to the name of Gustav Mie. So firm is this attachment that in defiance of logic and history every particle under the sun has been dubbed a "Mie scatterer", and Mie scattering has been promoted from a particular theory of limited applicability to the unearned rank of general scattering process.

Mie was not the first to solve the problem of scattering by an arbitrary sphere. It would be more correct to say that he was the last. He gave his solution in recognizably modern notation and also addressed a real problem: the colors of colloidal gold. For these reasons his name is attached to the sphere scattering problem even though he had illustrious predecessors, most notably Lorenz (not to be confused with Lorentz). This is an example in which eponymous recognition has gone to the last discoverer rather than to the first.

Mie scattering is not a physical process; Mie theory is one among many. Strictly speaking, it isn't even exact because it is based on continuum electromagnetic theory, itself approximate, and on illumination by a plane wave infinite in lateral extent.

Scattering by a sphere can be determined using various approximations and methods bearing little resemblance to Mie theory: Fraunhofer theory, geometrical optics, anomalous diffraction theory, coupled-dipole, T-matrix method, etc. Thus is a sphere a Mie scatterer or an anomalous diffraction scatterer or a coupled-dipole scatterer? The possibilities are endless.

When a physical process can be described by several different theories, it is inadvisable to attach the name of one of them to it.

There is no distinct boundary between so-called Mie and Rayleigh scatterers. Mie theory includes Rayleigh theory (for spheres), which is a limiting theory strictly applicable only as the size of the particle shrinks to zero. Even for spheres uncritically labeled "Rayleigh spheres", there are always deviations between the Rayleigh and Mie theories. By hobbling one's thinking with a supposedly sharp boundary between Rayleigh and Mie scattering, one risks throwing some interesting physics out the window. Whether a particle is a Mie or Rayleigh scatterer is not absolute. A particle may be graduated from Rayleigh to Mie status merely by a change of wavelength of the illumination. One often encounters statements about Mie scattering by cylinders, spheroids, coated spheres and other nonspherical or inhomogeneous particles. Judged historically, these statements are nonsense. Mie never considered any particles other than homogeneous spheres.

Logic would seem to demand that if a particle is a Mie scatterer, then Mie theory can be applied to scattering by it. This fallacious notion has caused and will continue to cause mischief, and is probably the best reason to cease referring to "Mie particles" or "Mie scatterers". Using Mie theory for scattering by particles other than spheres, especially near the backward direction, is risky.

More often than not, a better term than Mie or Rayleigh scattering is available. If the scatterers are molecules, molecular scattering is better than Rayleigh scattering (itself an imprecise term): the former term refers to an agent, the latter to a theory. Mie scatterer is just a needlessly aristocratic name for a humble sphere. Whenever Mie scatterer is replaced with sphere, the result is clearer. If qualifiers are needed, one can add small or large compared with the wavelength or comparable with the wavelength.

Briefly, the solution to the problem of scattering by an arbitrary homogeneous sphere illuminated by a plane wave can be obtained by expanding the incident, scattered, and internal electric and magnetic fields in series of vector spherical harmonics (general solutions to the equations of the electromagnetic field in spherical coordinates). The coefficients of these expansion functions are chosen so that the tangential components of the fields are continuous across the surface of the sphere. Thus this scattering problem is formally identical to reflection and refraction because of interfaces, although the sphere problem is considerably more complicated because the scattered and internal fields are not plane waves.

Observable quantities are expressed in terms of the complex scattering coefficients an and bn in the expansions of the scattered electric and magnetic fields. For example, the cross sections are infinite series:

The scattering coefficients can be written

where ^n and are Riccati-Bessel functions and the logarithmic derivative is

The size parameter x is ka, where a is the radius of the sphere and k is the wavenumber of the incident radiation in the surrounding medium (assumed nonabsorbing); m is the complex refractive index (discussed in the previous subsection) of the sphere relative to the (real) refractive index of the surrounding medium. Equations (3.134)-(3.136) are one among many ways of writing the scattering coefficients, some of which are more suited to computations than others. Scattering in any direction for any state of polarization of the incident illumination is also determined by the scattering coefficients.

A good rule of thumb is that the number of terms required for convergence of the series in Eqs. (3.132) and (3.133) is approximately the size parameter (2na/X). Raindrops are of order 1 mm, and hence their size parameter at visible wavelengths is of order 10,000. The details of rainbows do emerge from Mie theory but at the cost of summing 10,000 terms, which is why we often resort to approximations such as geometrical optics, which sheds light on some but not all features of rainbows (see Sec. 8.4.2). To describe all their features we have to resort to Mie theory, and even it isn't good enough because raindrops are not spheres (cloud droplets are), and their departures from sphericity have observable consequences. And this leads us to the dubious notion of an "equivalent sphere", the search for which rivals the quest of alchemists for recipes for transforming base metals into gold. Alas, just as there is no such recipe, there is no such thing as an equivalent sphere, one with all the same scattering and absorbing properties as a non-spherical particle. In the first place, such a particle is defined by what it is not, the only characteristic shared by all non-spherical particles. A non-cat is any animal that is not a cat, which leaves us with a menagerie housing everything from elephants and giraffes to porcupines and shrews. Because a collection of randomly oriented non-spherical particles has the same symmetry as a sphere, it is sometimes argued that the (ensemble averaged) scattering properties of the collection are the same as those of suitably chosen "equivalent spheres". Not true. The error here resides in confusing the symmetry of an ensemble of particles with that of a single particle. No matter what incantation is used for conjuring the properties of an equivalent sphere, differences between scattering and absorption by it and by a non-spherical particle always exist. Sometimes these differences are huge, sometimes not. Beware of equivalent spheres.

During the Great Depression mathematicians were put to work computing tables of trigonometric and other functions. The results of their labors now gather dust in libraries. Today, these tables could be generated more accurately in minutes on a pocket calculator. A similar fate has befallen Mie calculations. Before fast computers were inexpensive, tables of scattering functions for limited ranges of size parameter and refractive index were published. Today, these tables could be generated in minutes on a personal computer. Algorithms are more valuable and enduring than tables of computations, which are mostly useless except as checks for someone developing and testing algorithms. The primary tasks in Mie calculations are computing the functions in Eqs. (3.134) and (3.135) and summing series such as

Eqs. (3.132) and (3.133). Nowadays Mie codes abound. You can find them in books, on the Internet, and probably at upscale supermarket checkout stands.

Cross sections versus radius or wavelength convey physical information; efficiencies versus size parameter convey mathematical information. The size parameter is a variable with less physical content than its components, the whole being less than the sum of its parts. Moreover, cross sections versus size parameter (or its inverse) are not equivalent to cross sections versus wavelength. Except in dreamland, refractive indices vary with wavelength, and the Mie coefficients depend on x and m, wavelength being explicit in the first, implicit in the second.

Particles Much Smaller than the Wavelength

For sufficiently small x and \m\ x, the volumetric extinction and scattering cross sections for spheres are approximately



where v is the particle volume and A is the wavelength in the (negligibly absorbing) material surrounding the sphere. Similar equations hold, in particular the volume dependence, for small, homogeneous particles of other shapes. Note the similarity of Eq. (3.138) to Eq. (3.114) for reflection by a thin slab. These equations are the source of a nameless paradox, which is disinterred from time to time, a corpse never allowed to rest in peace. If the sphere is nonab-sorbing (m real), Eq. (3.137) yields a vanishing extinction cross section whereas Eq. (3.138) yields a non-vanishing scattering cross section, and yet extinction can never be less than scattering. Both equations were obtained from power series in the size parameter x. Equation (3.137) is the first term in the series. To be consistent, both cross sections must be expanded to the same order in x. When this is done, the paradox vanishes. In fact, it never existed.

Yet another pointless paradox arises from the curious definition of the radar backscatter-ing cross section as 4n times the differential scattering cross section in the backward direction. For a small sphere this leads to a backscattering cross section 50% greater than its total scattering cross section, which at first glance certainly is cause for head scratching. The radar reflectivity is the sum of the radar backscattering cross sections of all the scatterers in a unit volume.

Figure 3.9 shows the scattering and absorption cross sections of a water droplet 20 pm in diameter over six wavelength decades. For wavelengths much greater than the diameter, scattering is a linear function (on a logarimthic plot) of wavelength with slope approximately -4, in accordance with Eq. (3.138). At these wavelengths, extinction is dominated by absorption, and so according to Eq. (3.137), absorption should decrease (again, on a logarithmic plot) linearly with slope -1. This is not quite what occurs (see Fig. 3.9) because the complex refractive index also varies with wavelength (Fig. 3.8) in this region.

Although Eq. (3.138) is the scattering cross section of a small particle, it still must contain enough molecules that it can be assigned a refractive index (a molecule cannot). Nevertheless,

let's throw caution to the wind and extrapolate this equation to molecular sizes. If we use the refractive index of water and a particle diameter of 0.3 nm, we obtain a scattering cross section in the middle of the visible spectrum of about 0.7 x 10-19 pm2. That for air (an average over all molecules but predominantly nitrogen) is about 4.6 x 10-19 pm2. This isn't bad agreement considering that the scattering cross section of water vapor is less than that of air (nitrogen and oxygen), molecular diameters are not well defined, and that the scattering cross section depends on the sixth power of diameter.

Particles Much Larger than the Wavelength

Another paradox has a name: the extinction paradox. For a compact particle, such as a sphere, the extinction cross section has the limiting value lim CWt = 2G, (3.139)

where a is a linear dimension of the particle (for a sphere its radius) and G its projected geometrical cross sectional area. The factor 2 in Eq. (3.139) has caused people to sweat: according to geometrical optics it should be 1. Equation (3.139) seems to imply that a large (compared with the wavelength) particle is too big for its britches by a factor of two. Geometrical optics, according to which a beam of light is imagined to be a bundle of rays, is reckoned to be a good approximation for objects much larger than the wavelength. Thus every ray that intersects a particle with geometrical area G should be either absorbed or deviated, whereas rays that lie outside this shadow region should pass unscathed. The catch here is that rays don't exist and no matter how large a finite particle is, it always exhibits some departures, possibly small, from geometrical optics, in this instance very close to the forward direction.

We noted previously (and show in Fig. 3.13) that scattering is more peaked in the forward direction the larger the particle. Theory counts scattered light as removed from a (monodirec-tional) beam no matter how small the scattering angle. To measure the full extinction cross section of an indefinitely large particle would require a detector with vanishingly small acceptance angle. But any real detector necessarily collects some of the near-forward scattered light, which reduces the extinction cross section from its theoretical maximum. When near-forward scattered light is included in the measurement, the limiting extinction cross section [Eq. (3.139)] drops to G, as expected on the basis of intuition molded by geometrical optics. The distinction here is between the real and the ideal. A real detector measures the extinction cross section where Cext is the ideal (theoretical) extinction cross section and integration is over the acceptance solid angle of the detector.

In Fig. 3.9 the two asymptotes, G and 2G, are shown by dotted lines. At sufficiently short wavelengths, where extinction is dominated by scattering, the scattering cross section does indeed approach the asymptote 2G. But note also that over a range of intermediate wavelengths the scattering and absorption cross sections are each approximately equal to G.

Figure 3.9: Scattering (top panel) and absorption (bottom panel) cross sections for a water droplet of diameter 20 |m. The two horizontal dotted lines in the top panel indicate the geometrical cross sectional area and twice this area; the horizontal dotted line in the bottom panel indicates the geometrical cross sectional area.


Figure 3.9: Scattering (top panel) and absorption (bottom panel) cross sections for a water droplet of diameter 20 |m. The two horizontal dotted lines in the top panel indicate the geometrical cross sectional area and twice this area; the horizontal dotted line in the bottom panel indicates the geometrical cross sectional area.

Those Elusive Small Droplets

We state in Section 1.4.2 that the question of what is the average size of a cloud droplet should be greeted with a horselaugh but only sketched the reason why. We can now expand further on this armed with the results of this section.

Cloud droplets are large compared with the wavelengths of visible radiation but small compared with those of microwaves and radar. At visible wavelengths droplet scattering cross sections are proportional to the square of droplet diameter. Extinction is dominated by scattering (Fig. 3.9) and extinction is nearly the asymptotic value [Eq. (3.139)]. At microwave wavelengths, however, extinction is dominated by absorption, and hence from Eq. (3.137), the absorption cross section is proportional to droplet diameter cubed. But at these wavelengths, the scattering cross section, and hence also the radar backscattering cross section, is propor-

Atmospheric Isotropic Interruption

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Diameter (pm)

Figure 3.10: The top panel shows an in situ measurement of the droplet number size distribution in a stratus cloud. From this distribution the distribution functions for cross sectional area, volume, and volume squared follow.

0 20 40 60 80 100 120 140 160 180 200

Diameter (pm)

Figure 3.10: The top panel shows an in situ measurement of the droplet number size distribution in a stratus cloud. From this distribution the distribution functions for cross sectional area, volume, and volume squared follow.

tional to droplet diameter to the sixth power. These different dependences on size have profound consequences for inferring droplet properties remotely from radiation measurements.

Direct measurements made using an airplane flying through (water) stratus clouds near State College, Pennsylvania are displayed in Fig. 3.10, which shows the number distribution of droplets, from which area (diameter squared), volume (diameter cubed), and volume squared (diameter to the sixth power) distributions are calculated. The number density distribution (droplets per unit volume) peaks at about 6 pm. For diameters greater than about 40 pm, the number density is about six decades smaller than at the peak. Scattering of visible radiation peaks at about 16 pm, at which diameter the number density is more than a factor of two less than its peak value. The peak in the volume distribution is shifted even further, to about 20 pm. And although the peak in the volume squared distribution is shifted to only about 22 pm, droplets larger than about 40 pm contribute more to radar backscattering signals than the smaller droplets even though they are a million times more abundant.

Consider how depressing the message Fig. 3.10 conveys to anyone with hopes of inferring droplet size distributions remotely from radiation measurements. The number distribution is greatest for the smaller sizes, whereas properties on which any scheme for remotely inferring sizes might be based are distributed quite differently. For example, if the smallest droplets (< 6 pm) were to be removed entirely from the cloud, the volume, and even more so the volume-squared distributions would hardly miss them. What is needed in order to detect (remotely) the smaller droplets is some electromagnetic property that depends only on number density or, at worst, on diameter. But such a property does not seem to exist.

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