Mie Scattering of Electromagnetic Waves Sergei Yushanov, Jeffrey S. Crompton*, and Kyle C. Koppenhoefer *Corresponding author AltaSim Technologies, 130 East Wilson Bridge Rd, Suite 140, Columbus, OH, [email protected] Abstract:Scattering of electromagnetic waves spherical particles in terms of infinite series [1- by spherical particles is described by the Mie 3]. solution to Maxwell’s equations. Illumination of an obstacle by an electromagnetic wave excites Scattering may be defined as the redirection of electric charges in the obstacle to oscillate due to radiation from the original propagation direction interaction with the electric field of the incident typically due to interactions with molecules and wave. The oscillating charge radiates particles. Reflection, refraction etc. represent electromagnetic energy in all directions; this specific forms of scattering. Electromagnetic secondary radiation is the radiation scattered by waves scatter when they meet an obstacle due to the obstacle. In addition, the excited elementary the electric charges in the obstacle being charges may absorb radiation and transform part oscillated by the electric field of the incident of the incident energy into other forms, such as wave. Accelerated charges radiate thermal energy. Both scattering and absorption electromagnetic energy in all directions; this remove energy from a beam of light and the secondary radiation is referred to as radiation beam is attenuated. scattered by the obstacle. Thus, charge excitation and re-radiation of the electromagnetic energy In this work, scattering of an incident constitutes scattering. electromagnetic wave by a spherical particle has been analyzed using COMSOL Multiphysics. In addition to re-radiating electromagnetic The nature of the interaction has been considered energy, the excited elementary charges may for materialswith three different properties: transform part of the incident energy into other metallic, magnetic and dielectric. The solutions forms, e.g. thermal energy, via absorption. Both provide details of the absorption, scattering, scattering and absorption remove energy from an extinction, pressure cross-sections, back- electromagnetic wave, thus attenuating the beam. scattering and radiation force exerted on the This attenuation, which is called extinction, particle by an incident plane wave.Results of consists of scattering and absorption: these analyses are compared against the Mie analytical solutions. Extinction = scattering + absorption Keywords: electromagnetic scattering, angle- Scattering and absorption are not mutually resolved low-coherence interferometry, Mie independent processes. This paper considers solution, Maxwell’s equations elastic scattering only: the frequency of the scattered wave is the same as that of incident 1. Introduction wave. Thus, ineleastic scattering phenomena such as Brillouin and Raman scattering are The Mie solution to Maxwell’s equations excluded from this paper. provides the basis for measuring the size of particles through the scattering of 2. Theory electromagnetic radiation. Applications include dust particles in the atmosphere, oil droplet in In scattering problems, the total wave water, and cell nuclei in biological systems (e.g., decomposes onto incident and scattered waves: cancer research). The Mie solution, also called Lorenz-Mie E Einc Esca (1) theory or Lorenz-Mie-Debye theory, provides an analytical solution of Maxwell’s equations for the scattering of electromagnetic radiation by H Hinc H sca (2) Wsca ) by the particle, the absorption ( abs ) and Maxwell’s wave equation is solved with respect scattering ( sca ) cross section can be defined: to scattered electric field: W W (8) 1 2 (3) abs sca E k j E 0 abs , sca sca 0 r sca r 0 Pinc Pinc The scattered magnetic field is calculated from where Faraday’s law: P is incident irradiance, defined as energy flux 1 (4) inc H E 2 sca j sca of the incident wave, W / m Wabs is energy rate absorbed by particle, W The Time-average Poynting vector for time- is scattered energy rate, harmonic fields gives the energy flux: Wsca W The total scattering cross-section, or extinction 1 (5) cross-section, is the amount of energy removed P ReE H , W / m 2 from the incident field due to absorption and 2 scattering. By energy balance, the extinction cross-section is a superposition of the scattering For an incident plane wave, the magnetic field is and absorption cross-sections: related to the electric field by: 2 (9) , m 1 ˆ (6) ext abs sca H inc k Einc The energy loss rate in a particle is: ˆ 1 3 where k is direction of the incident wave Q ReJ E jB H , W / m (10) loss 2 tot propagation, is characteristic / impedance, is permittivity and is Where the total current is a superposition of permeability of ambient medium. Hence, conduction and displacement currents: incident energy flux is calculated as Jtot E jD . Thus, Eq. (10) includes conduction, polarization, and magnetic losses. 1 2 1 2 (7) The total absorbed energy is derived by P E kˆ c E kˆ inc 2 inc 2 inc integration of energy loss over the volume of the particle: where c 1/ is speed of light. W Q dV (11) abs loss V P 1 ሾ ሿ 2.1 Cross-Section Calculations ReE jD E jB H dV , ܹ 2 VP Important physical quantities can be obtained from scattered fields. One of these is the cross The scattered energy is derived by integration the section, which can be defined as the net rate at Poynting vector over an imaginary sphere around which electromagnetic energy W crosses the the particle: surface of an imaginary sphere centered at the W P ndS (12) particle divided by the incident irradiation sca sca Pinc S . To quantify the rate of the electromagnetic 1 Re E H ndS , [W] energy that is absorbed (Wabs ) and scattered ( sca sca 2 S eikr (13) wheren is unit vector normal to the imaginary r eikx f sphere. r where f is the scattering amplitude, is scattering angle, and k / c is the wave vector. The optical theorem states that the relation between the total cross section and the scattering amplitude. It is usually written in the form: Figure 1.Imaginary sphere S around a particle 4 (14) ext Imf 0 / Einc volume V p enclosed by surface S p . Scattering k amplitude is evaluated in the forward direction at point F . where f 0 is the scattering amplitude in the 2.2 Optical Theorem forward direction 0 . Thus, the optical In physics, the optical theorem is a general law theorem relates extinction cross-section to the of wave scattering theory, which relates the imaginary part of the scattering amplitude forward scattering amplitude to the total cross evaluated in the forward direction. Yet, section of the scattering object. The optical extinction is due to the scattering and absorption. theorem has a long history and is common to all Scattering involves all directions whereas kinds of seemingly disparate scattering absorption is often independent of direction, so phenomena involving acoustic waves, why should the optical theorem depend on only electromagnetic waves and elementary particles. the forward direction, and how is absorption The theorem can be derived directly from a involved? Berg, and Sorensen [4] provide a treatment of a scalar wave [1]. Scattering is a physical explanation of these features of the superposition of the incident and scattered optical theorem. waves, as schematically shown in Figure 2. COMSOLincludes a built-in far-field computation node that automatically calculates the far-field variables defined as: (15) E far lim rE sca r The far-field variables, e.g., E far , represent the scattering amplitude rather than the physical Figure 2.Generic scattering geometry. An incoming electric field and are measured in units ikx plane wave e scattered off a particle. A great E far m V / m V . distance away scattered wave is approximated by a Thus, the extinction cross-section can be directly ikr spherical wave e / r . calculated, without evaluation of absorption and scattering cross-sections, as: If a plane wave is incident on an object, then the wave amplitude a great distance away from the 4 2 (16) ext ImE far 0 , m scattering object, neglecting higher order terms, kEinc is given by: 2.3Radiation Pressure Cross-Section symmetric about a scattering angle of 90 . If the particle scatters more light towards the In addition to energy, the electromagnetic wave forward direction ( 0 ), the factor g is carries momentum P / c ; therefore, a beam of light that interacts with a particle will exert a positive. The factor g is negative if the force on the particle, called radiation pressure. scattering is directed more towards the back Debye, in 1909, was probably the first to direction ( 180 ). calculate the radiation force on a small particle. By integrating the Maxwell stress tensor over the 2.4Radiation Force surface of the sphere, he demonstrated that: The radiation pressure cross section can be used 2 (17) to calculate force that a particle experiences in pr ext cos sca , m the incidence direction [5]: 1 (19) where is the pressure cross-section, and F P , N pr c pr inc g cos is so called asymmetry parameter. This expression for pressure cross-section holds Where the pressure cross-section is given by for any small particle illuminated by a beam of Eq.(17). Alternatively, this force can be light [2]. The radiation cross section is directly calculated by integration of Maxwell’s stress related to the radiation force. tensor over the surface of the particle. The time- The asymmetry parameter g can be interpreted averaged Maxwell’s stress tensor is: as the mean direction of the scattered light and is ˆ 1 1 ˆ (20) T ReE D H B E D H BI, Pa defined as an average of the scattering angle 2 2 weighted with respect to scattered power flow: where D E is electric flux density, B H is cos Psca n dS (18) S magnetic flux density, and ˆ is unit tensor of cos I second rank.