Journal of Quantitative Spectroscopy and Radiative Transfer
ReviewElectromagnetic scattering by nonspherical particles: A tutorial review
Introduction
In his famous paper of 1908, Mie [1] developed a comprehensive theory of electromagnetic scattering by a perfectly spherical homogeneous particle and used it to explain several experimental facts pertaining to the optics of colloidal gold [2]. A small discrepancy between the measured and calculated polarization was interpreted by Mie as a residual effect of nonsphericity of the gold microparticles (see Fig. 1a). The final sentence of his paper reads: “For the sake of completeness of the theory, it is absolutely necessary to investigate also the behavior of ellipsoidal particles.” It had taken several decades to accomplish that simply stated goal [3].
Nonspherical particles are abundant in natural and artificial environments (Fig. 1). Furthermore, it has become universally recognized that nonsphericity (or more generally, complex morphology) of particles has a profound effect on their scattering and absorption properties [9]. Yet our knowledge and understanding of how nonspherical particles scatter and absorb electromagnetic energy remains incomplete and in some respects unsatisfactory.
The main goal of this tutorial review is to provide an accessible introduction to the subject of electromagnetic scattering by nonspherical particles and discuss the most general and typical ways in which the scattering and absorption properties of particles are affected by deviations of the particle morphology from that of a perfect sphere. Specifically, I will focus on how nonsphericity influences our way of describing and quantifying electromagnetic scattering by particles and how it is likely to affect, both qualitatively and quantitatively, the principal theoretical descriptors of scattering and the relevant optical observables.
Given how vast the subject of electromagnetic scattering by nonspherical particles has become, it can no longer be covered comprehensively in a review paper and even in one monograph. Therefore, this tutorial paper has several fundamental restrictions. First of all, I will not discuss specifically the commonly used theoretical and experimental techniques for the computation and measurement of electromagnetic scattering since those are well covered in the monographs [9], [10], [11], [12], [13], [14], [15], [16], [17], [18] as well as in recent reviews of exact numerical methods by Kahnert [19] and Wriedt [20]. Secondly, I consider only frequency-domain scattering by assuming that all sources and fields vary in time harmonically and do not discuss transient electromagnetic phenomena [11], [21], in particular the scattering of ultra-short laser pulses. Thirdly, I consider only electromagnetic scattering in the far-field zone of a particle. Near-field scattering effects are also very interesting and important, especially in connection with the rapidly developing discipline of nano-optics [22], and have become the subject of intense research. An instructive introduction to the physics of the near field is provided by [23]. Fourthly, I will focus on the effects of nonsphericity on single scattering of light by individual particles and small particle groups. In other words, manifestations of particle nonsphericity in electromagnetic scattering by large random particle groups such as clouds, particulate surfaces, and particle suspensions will not be discussed. Finally, there will be no in-depth discussion of the various applications of electromagnetic scattering by nonspherical particles in science and engineering. A plentiful source of that information is the collection of monographs [9], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34] supplemented by several special issues of JQSRT published over the past years [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49].
Essentially all quantitative illustrative examples of electromagnetic scattering given in this review are either the result of a direct measurement performed in a controlled laboratory setting or a numerically exact theoretical computation. By giving preference to exact theoretical results I do not mean to question the importance and utility of various approximate theoretical approaches: in many cases an approximate technique is the only practical way of describing electromagnetic scattering by particles. Instead, I simply follow Mie's philosophy of applying an exact theoretical approach to a somewhat idealized particle morphology rather than an approximate technique to a seemingly more realistic particle model. The virtues of this approach are rather obvious. First, a numerically exact result remains exact irrespective of what specific technique was used to obtain it and is not subject to future change or improvement besides, perhaps, infrequent attempts to re-compute it with one or two more decimals. As a consequence, the basic conclusions drawn in this review are also unlikely to change. Secondly, analyses of measurement results based on an exact theoretical technique are more definitive since they are not affected by uncertainties regarding the range of applicability and numerical accuracy of an approximate technique and/or the likely physical meaning of ad hoc model parameters not following directly from the Maxwell equations. Thirdly, the actual morphology of many natural and artificial particles is so complex that claims of the ability to model these morphologies “more realistically” with an approximate scattering technique are justified less often than not. This latter aspect of modeling electromagnetic scattering by ensembles of complex nonspherical particles will be discussed specifically in Section 11 in connection with the so-called statistical approach.
Section snippets
Electromagnetic scattering by a fixed particle
The gist of the fundamental concept of electromagnetic scattering by a fixed particle used in Mie's paper is explained in [2], [50] and will not be discussed in much detail here. In brief, we assume that the unbounded host medium surrounding the particle is homogeneous, linear, isotropic, and nonabsorbing. The particle is illuminated by a plane electromagnetic wave given bywith constant amplitudes and where E is
Far-field scattering
An important property of the dyadic Green's function is the asymptotic behavior where and . By placing the origin of the laboratory coordinate system O close to the geometrical center of the scattering particle, as shown in Fig. 4, and substituting Eqs. (1), (9) in Eq. (7), we derive [15], [55]Here is a unit vector in the incidence
Optical observables
The typically high frequency of time-harmonic electromagnetic oscillations makes it virtually impossible to measure the electric and magnetic fields associated with the incident and scattered waves using traditional optical instruments. Therefore, in order to make the theory applicable to analyses of actual optical observations, the scattering phenomenon must be characterized in terms of derivative quantities that can be measured directly (i.e., actual optical observables). The conventional
Derivative quantities
There are several derivative quantities that are often used to describe various manifestations of electromagnetic scattering. The product of the extinction cross section and the intensity of the incident plane wave yields the total attenuation of the electromagnetic power measured by detector 1 in Fig. 3b owing to the presence of the particle. This means that the extinction cross section depends on the polarization state of the incident wave and is given by [15], [55]
Scattering by a “random” particle
Strictly speaking, the above formalism applies only to scattering by a fixed particle. However, one often encounters situations in which the scattering particle moves, rotates, and perhaps changes its size and/or shape during the measurement. A typical example is the measurement of scattering by a single particle suspended in air or vacuum with one of the existing levitation techniques [71]. The particle position within the levitator trap volume and its orientation are never perfectly fixed,
Scattering by a random particle group
Although we have been so far discussing electromagnetic scattering by a “single particle”, the concept of electromagnetic scattering and all formulas of Section 2 remain valid irrespective of the specific morphology of the scattering object. In particular, they are valid for what a human eye could classify as a “collection of discrete particles”. Examples of such “many-particle” objects are clouds, particulate surfaces, and particle suspensions. In all such cases the incident field perceives a
Spherically symmetric particles
It follows from the Mie theory [1], [51], [63] (or its generalizations for radially inhomogeneous particles [16]) that the extinction, scattering, and absorption cross sections and the single-scattering albedo for a spherically symmetric particle are independent of the direction of propagation and polarization state of the incident light. Furthermore, the extinction matrix is diagonal and given byThe phase matrix satisfies the symmetry relations [93]
General effects of nonsphericity and orientation
The discussion in this section applies equally to a fixed nonspherical particle, a “random” nonspherical particle which is perfectly or preferentially oriented during the measurement (Section 6), and to a small random particle group in which particles are also perfectly or preferentially oriented (Section 7). Then, in general,
- •
the 4×4 extinction matrix K or 〈K〉ξ does not degenerate to a direction- and polarization-independent scalar extinction cross section;
- •
the (ensemble-averaged) extinction,
Mirror-symmetric ensembles of randomly oriented particles: general traits
Consider now a random particle or a small random group of particles such that the distribution of particle orientations during the measurement is uniform. Furthermore, we assume that the single random particle is mirror-symmetric (i.e., has a plane of symmetry), while each particle in the group has a plane of symmetry and/or is accompanied by its mirror counterpart. Then most of the results of Section 8 apply [15], [63]. Specifically, the ensemble-averaged extinction, scattering, and absorption
Mirror-symmetric ensembles of randomly oriented particles: quantitative traits
Of course, besides the qualitative distinctions discussed in the preceding section, there can be significant quantitative differences in specific scattering properties of randomly oriented nonspherical particles and “equivalent” (e.g., volume-equivalent or surface-equivalent) spheres. We begin by discussing the effects of nonsphericity and random orientation on MDRs. Fig. 6b summarizes the results of numerically exact T-matrix computations for monodisperse spheres and volume-equivalent,
Conclusions
There is no doubt that since the publication of Mie's seminal paper, our knowledge of electromagnetic scattering by nonspherical particles has improved profoundly. In particular, the general effects of nonsphericity are largely understood, both qualitatively and quantitatively, and a vast body of practical applications have been documented. Still much remains to be done since in some respects our knowledge of specific manifestations of particle shape and morphology in electromagnetic scattering
Acknowledgments
I thank Pat Arnott, Matthew Berg, Brian Cairns, Jacek Chowdhary, Oleg Dubovik, Joop Hovenier, Michael Kahnert, Li Liu, Daniel Mackowski, Pinar Mengüç, Martin Schnaiter, Larry Travis, Gorden Videen, Warren Wiscombe, Thomas Wriedt, and Ping Yang for numerous fruitful discussions. Martin Schnaiter has kindly contributed Fig. 9b. This research was sponsored by the NASA Radiation Sciences Program managed by Hal Maring and by the NASA Glory Mission project.
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