Scattering of Electromagnetic Waves by Particles

MIE SCATTERING | COMPUTATIONAL ELECTROMAGNETICS AltaSim Technologies, OH, USA

Scattering of Electromagnetic Waves by Particles Particles can be characterized by the unique scattering patterns produced by their interaction with electromagnetic waves. Optical scattering measurements cover a broad range of applications such as meteorology, particle sizing, biomedical, and metamaterials.

BY SERGEI YUSHANOV, JEFFREY S. CROMPTON, AND KYLE C. KOPPENHOEFER, ALTASIM TECHNOLOGIES

As electromagnetic waves propagate influence of particle geometry, the through matter they interact with angle of incidence of the wave, and particles or inhomogeneities that the particle material properties can perturb the local electron distribution. be investigated. This variation produces periodic In electromagnetic wave scattering charge separation within the particle, problems, the total wave decomposes causing oscillation of the induced into the incident and scattered wave local dipole moment. This periodic components. Important physical acceleration acts as a source of quantities can be obtained from the electromagnetic radiation, thus FIGURE 1. Electric field due to Mie scattered fields. One of these is the causing scattering. scattering of the incident wave in the cross section, which can be defined as x-direction showing enhanced scattering the net rate at which electromagnetic PARTICLE SIZE MATTERS in the forward direction. energy crosses the surface of an Scattering of electromagnetic waves imaginary sphere centered at the by particles can be illustrated by two particle, divided by the incident theoretical frameworks: Rayleigh Mie scattering differs from Rayleigh irradiation (Pinc). To quantify the scattering that is applicable to small, scattering in several respects: it is rate of the electromagnetic energy dielectric, non-absorbing spherical mostly independent of wavelength absorbed (Wabs) and scattered (Wsca) particles, and Mie scattering that and is larger in the forward direction by the particle, the absorption (sabs), provides a general solution to than in the reverse direction (see scattering (ssca), and extinction (sext) scattering that is independent Figure 1). The greater the particle cross sections are defined as: of particle size. Mie scattering size, the more light is scattered Wabs Wsca theory converges to the limit of forward. In addition to the many σabs = , σσsca ==, extaσσbs + sca P Pinc geometric optics at large particle atmospheric effects of light scattering, inc sizes. Consequently, Mie scattering applications of Mie scattering include The total absorbed energy is derived theory can be used to describe most environmental models such as dust by integrating the energy loss over the scattering by spherical particles, particles in the atmosphere and oil volume of the particle. The scattered including Rayleigh scattering, droplets in water, as well as medical energy is derived by integrating the but due to the complexity of technology used to measure cell nuclei Poynting vector over an imaginary implementation, Rayleigh scattering in biological systems or the collagen sphere around the particle. theory is often preferred. fibers in body tissue. The Rayleigh scattering model breaks down when the particle size MIE SCATTERING becomes larger than approximately Implementation of analytical solutions 10 percent of the wavelength of the for Mie scattering by a particle or incident radiation, at which point object is complex and requires solving Mie theory must be applied. The Mie Maxwell’s equations to represent solution is obtained by analytically the incident, scattered, and internal solving Maxwell’s equations for the fields. These take the form of infinite scattering of electromagnetic radiation series expansion of vector spherical by spherical particles; it is modeled in harmonics, allowing the cross sections, terms of infinite series rather than a efficiency factors, and distributions of FIGURE 2. Model geometry for Mie simple mathematical expression. intensity to be predicted. Further, the scattering by a spherical particle.

42 | COMSOL NEWS | 2014 MIE SCATTERING | COMPUTATIONAL ELECTROMAGNETICS

COMPUTATIONAL to limit the extent of the model to a level of accuracy and optimizing usage ELECTROMAGNETICS manageable region of interest. The of computational resources. COMSOL A computational model of Mie solution inside the domain is not also supports far-field calculations, scattering was developed using affected by the presence of the PML, which are done on the inner boundary COMSOL Multiphysics® and its RF which lets the solution behave as if of the PML domain where the near Module. It solves for the scattering the domain was of infinite extent. This field is integrated. The surface S is used off of a dielectric, magnetic, or metal layer absorbs all outgoing wave energy to calculate total scattered energy. spherical particle with radius a. The without any impedance mismatch An incident plane wave travels in model geometry is shown in Figure 2. that could cause spurious reflections the positive x-direction (see Figure The air domain is truncated by a at the boundary. The PML is useful in 2), with the electric field polarized perfectly matched layer (PML) inserted maintaining the solution at the desired along the z-axis. Perfect magnetic conductor (PMC) and perfect electric conductor (PEC) boundary conditions are used on the x-z and x-y symmetry planes, respectively. The plane wave incident on the sphere is defined by its amplitude, wave vector in the air, and circular frequency. COMSOL conveniently provides all the necessary

FIGURE 3. Cross-section parameters and radiation force for a dielectric particle with refractive index n = 5 – 0.4j and relative permeability µ = 1.

FIGURE 6. Distribution of the z-component of the electric field due to scattering of the incident electromagnetic wave by a particle of 0.1µm radius. The arrows show the time-averaged power flow of the relative fields at a frequency of 950 THz.

FIGURE 4. Cross-section parameters and radiation force for a magnetic particle with functionality to calculate scattering relative permittivity e = 1 and relative permeability µ = 8 – 2j. integrals. Scattering characteristics for the three types of particles considered are shown in Figures 3, 4, and 5. The results of the computational analysis show good agreement with available experimental results1. Simulation of Mie scattering problems enables visualization of the effects of small particles on an incident electromagnetic wave (see Figure 6) to allow better understanding of the interactions. n

References 1Mätzler, C., MATLAB Functions for Mie Scattering and Absorption, Version 2, IAP Research Report, FIGURE 5. Cross-section parameters and radiation force for a silver particle with (Bern: InstitutfürangewandtePhysik, Universität, dielectric constants. 2001), No. 2002-11.

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