Mie Scattering of Electromagnetic Waves

Transcription

1 Mie cattering of Electromagnetic aves ergei Yushanov, Jeffrey. Crompton *, and Kyle C. Koppenhoefer *Corresponding author Altaim Technologies, 30 East ilson Bridge Rd, uite 40, Columbus, OH, Abstract:cattering of electromagnetic waves by spherical particles is described by the Mie solution to Maxwell s equations. Illumination of an obstacle by an electromagnetic wave excites electric charges in the obstacle to oscillate due to interaction with the electric field of the ident wave. The oscillating charge radiates electromagnetic energy in all directions; this secondary radiation is the radiation ttered by the obstacle. In addition, the excited elementary charges may orb radiation and transform part of the ident energy into other forms, such as thermal energy. Both ttering and orption remove energy from a beam of light and the beam is attenuated. In this work, ttering of an ident electromagnetic wave by a spherical particle has been analyzed using COMOL Multiphysics. The nature of the interaction has been considered for materialswith three different properties: metallic, magnetic and dielectric. The solutions provide details of the orption, ttering, exttion, pressure cross-sections, backttering and radiation force exerted on the particle by an ident plane wave.results of these analyses are compared against the Mie analytical solutions. Keywords: electromagnetic ttering, angleresolved low-coherence interferometry, Mie solution, Maxwell s equations. Introduction The Mie solution to Maxwell s equations provides the basis for measuring the size of particles through the ttering of electromagnetic radiation. Applications lude dust particles in the atmosphere, oil droplet in water, and cell nuclei in biological systems (e.g., cancer research). The Mie solution, also called Lorenz-Mie theory or Lorenz-Mie-Debye theory, provides an analytical solution of Maxwell s equations for the ttering of electromagnetic radiation by spherical particles in terms of infinite series [- 3]. cattering may be defined as the redirection of radiation from the original propagation direction typically due to interactions with molecules and particles. Reflection, refraction etc. represent specific forms of ttering. Electromagnetic waves tter when they meet an obstacle due to the electric charges in the obstacle being oscillated by the electric field of the ident wave. Accelerated charges radiate electromagnetic energy in all directions; this secondary radiation is referred to as radiation ttered by the obstacle. Thus, charge excitation and re-radiation of the electromagnetic energy constitutes ttering. In addition to re-radiating electromagnetic energy, the excited elementary charges may transform part of the ident energy into other forms, e.g. thermal energy, via orption. Both ttering and orption remove energy from an electromagnetic wave, thus attenuating the beam. This attenuation, which is called exttion, consists of ttering and orption: Exttion = ttering + orption cattering and orption are not mutually independent processes. This paper considers elastic ttering only: the frequency of the ttered wave is the same as that of ident wave. Thus, ineleastic ttering phenomena such as Brillouin and Raman ttering are excluded from this paper.. Theory In ttering problems, the total wave decomposes onto ident and ttered waves: E E E () H H H ()

2 Maxwell s wave equation is solved with respect to ttered electric field: (3) E 0 k r j E 0 r 0 The ttered magnetic field is calculated from Faraday s law: (4) H E j The Time-average oynting vector for timeharmonic fields gives the energy flux: Re E H, / m (5) For an ident plane wave, the magnetic field is related to the electric field by: H kˆ E (6) ) by the particle, the orption ( ttering (, ) and ) cross section can be defined: (8) where is ident irradiance, defined as energy flux of the ident wave, / m is energy rate orbed by particle, is ttered energy rate, The total ttering cross-section, or exttion cross-section, is the amount of energy removed from the ident field due to orption and ttering. By energy balance, the exttion cross-section is a superposition of the ttering and orption cross-sections: ext, m The energy loss rate in a particle is: (9) wherekˆ is direction of the ident wave propagation, / is characteristic impedance, is permittivity and is permeability of ambient medium. Hence, ident energy flux is calculated as E kˆ c E kˆ (7) Q loss Re 3 E j B H, / m J (0) tot here the total current is a superposition of conduction and displacement currents: Jtot E jd. Thus, Eq. (0) ludes conduction, polarization, and magnetic losses. The total orbed energy is derived by integration of energy loss over the volume of the particle: where c / is speed of light.. Cross-ection Calculations Important physical quantities can be obtained from ttered fields. One of these is the cross section, which can be defined as the net rate at which electromagnetic energy crosses the surface of an imaginary sphere centered at the particle divided by the ident irradiation. To quantify the rate of the electromagnetic energy that is orbed ( ) and ttered ( V Q loss dv Re dv, E jd E jb H V () The ttered energy is derived by integration the oynting vector over an imaginary sphere around the particle: Re E H nd nd, [] ()

3 wheren is unit vector normal to the imaginary sphere. r e ikx f e ikr r (3) where f is the ttering amplitude, is ttering angle, and k / c is the wave vector. The optical theorem states that the relation between the total cross section and the ttering amplitude. It is usually written in the form: Figure.Imaginary sphere around a particle volume V p enclosed by surface p. cattering amplitude is evaluated in the forward direction at point F.. Optical Theorem In physics, the optical theorem is a general law of wave ttering theory, which relates the forward ttering amplitude to the total cross section of the ttering object. The optical theorem has a long history and is common to all kinds of seemingly disparate ttering phenomena involving acoustic waves, electromagnetic waves and elementary particles. The theorem can be derived directly from a treatment of a lar wave []. cattering is a superposition of the ident and ttered waves, as schematically shown in Figure. 4 ext Im k f 0/ E (4) f 0 is the ttering amplitude in the where forward direction 0. Thus, the optical theorem relates exttion cross-section to the imaginary part of the ttering amplitude evaluated in the forward direction. Yet, exttion is due to the ttering and orption. cattering involves all directions whereas orption is often independent of direction, so why should the optical theorem depend on only the forward direction, and how is orption involved? Berg, and orensen [4] provide a physical explanation of these features of the optical theorem. COMOLludes a built-in far-field computation node that automatically calculates the far-field variables defined as: E far lim re (5) r Figure.Generic ttering geometry. An oming ikx plane wave e ttered off a particle. A great distance away ttered wave is approximated by a spherical wave e ikr / r. If a plane wave is ident on an object, then the wave amplitude a great distance away from the ttering object, neglecting higher order terms, is given by: The far-field variables, e.g., E far, represent the ttering amplitude rather than the physical electric field and are measured in units E far m V / m V. Thus, the exttion cross-section can be directly calculated, without evaluation of orption and ttering cross-sections, as: 4 ext ke Im E 0, m far (6)

4 .3Radiation ressure Cross-ection In addition to energy, the electromagnetic wave carries momentum / c ; therefore, a beam of light that interacts with a particle will exert a force on the particle, called radiation pressure. Debye, in 909, was probably the first to calculate the radiation force on a small particle. By integrating the Maxwell stress tensor over the surface of the sphere, he demonstrated that: pr ext cos, m (7) where is the pressure cross-section, and pr g cos is so called asymmetry parameter. This expression for pressure cross-section holds for any small particle illuminated by a beam of light []. The radiation cross section is directly related to the radiation force. The asymmetry parameter g can be interpreted as the mean direction of the ttered light and is defined as an average of the ttering angle weighted with respect to ttered power flow: cos n d cos nd (8) where cos n k ˆ is directional cosine between vector normal to the surface and direction of ident plane wave. symmetric about a ttering angle of 90. If the particle tters more light towards the forward direction ( 0 ), the factor g is positive. The factor g is negative if the ttering is directed more towards the back direction ( 80 )..4Radiation Force The radiation pressure cross section can be used to calculate force that a particle experiences in the idence direction [5]: (9) F pr, N c here the pressure cross-section is given by Eq.(7). Alternatively, this force can be calculated by integration of Maxwell s stress tensor over the surface of the particle. The timeaveraged Maxwell s stress tensor is: ˆ Re E D H B ˆ E D H B I, a T (0) where D E is electric flux density, B H is magnetic flux density, and Î is unit tensor of second rank. The total time-averaged force F acting on a particle illuminated with light can be calculated using the surface integral: F Tˆ nd N (), where is surface enclosing particle volume V, and n is unit normal vector to surface. Calculating the force based on Eq. (8) or Eq. () gives the same result. 3. Use of COMOL Multiphysics Figure 3: Model geometry for Mie ttering by a spherical particle. For a particle that tters light isotropically (i.e., the same in all directions), the factor g vanishes; it also vanishes if ttering is This model solves for the ttering off the spherical particle with radius of a 0.m. The model geometry is shown in Figure 3. Two symmetry planes enable modelingonly onequarter of the sphere. The air domain is truncated by a perfectly matched layer (ML), andfar-field calculationsare done on the inner boundary of

5 theml domain. The surface is used to calculate total ttered energy. ymmetry in magnetic field: n E 0 EC z0 For the ttered field formulation, Eqs.() and(3)are used in the model setup. The background ident plane wave is defined as: jk 0 E x 0,0, E0e () where E 0 is plane wave amplitude, k 0 / c is wavevector in the air, is circular frequency and c is speed of light in air. The ident energy flux amplitude follows from Eq. (7): c / (3) 0 E 0 The convenient normalization factor for radiation force is defined as: 4. Results ae N F, (4) The following particles of radius a 0.m are considered: dielectric, magnetic, and metal (silver). Material properties of the dielectric r, r, 0.Material properties of the magnetic particle are: r, r 8 j, 0. At optical frequencies, the metal particle is modeled as a dielectric having a complex valued permittivity. The real and imaginary parts of the complex permittivity of silver are extracted using complex refractive index frequency-dependent data [5]. particles are: j Figure 4. Cross-section parameters and radiation force for dielectric particle. An ident plane wave travels in the positive x- direction, with the electric field polarized along the z-axis. This imposes the following boundary conditions on the symmetry planes: ymmetry in electric field: n H 0 MC y0 cattering characteristics for three types of particles considered are shown in Figure 4, Figure 5, and Figure 6. Numerical results are compared with analytical solutions obtained with Matlab code developed by Mätzler [6]. All results are in good agreement with analytical solution based on Mie theory. 5. ummary A solution of the ttering of electromagnetic waves has been developed using COMOL Multiphysics. The results assume elastic

6 ttering only and do not lude Brillouin or Raman ttering. The results developed from the COMOL Multiphysics model are compared with the Mie analytical solution. In all cases considered, the analysis shows good agreement with the Mie solution. Figure 6. Cross-section parameters and radiation force for silver particle with dielectric constants shown in Fig. 7. Figure 5. Cross-section parameters and radiation force for magnetic particle.

7 ingle articles, J. Opt. oc. Am.A 5, pp (008). 5. Johnson,.B. and Christy, R.. Optical Constants of the Noble Metals, hys. Rev. B, vol. 6, pp , Mätzler, C.: MATLAB Functions for Mie cattering and Absorption, Version, IA Research Report, No. 00-, Institutfürangewandtehysik, Universität Bern, 00. Figure 7.Dielectric constants for silver as a function of photon energy: Data from Ref. [5]. 8. Acknowledgements The authors thank rofessor C. Mätzler, Institute of Applied hysics, University of Bern for providing the Matlab code to perform the Mie solution. 9. References. van de Hulst, Light cattering by mall articles, Dover, New York, 98..Bahrem, C.F., and Huffman, D.R., Absorbing and cattering of Light by mall articles, iley, New York, Quinten, M. Optical roperties of Nanoparticle ystems: Mie and Beyond, iley, New York, 0 4.Berg, M. J., orensen, C. M., and Chakrabarti, A., Exttion and the Optical Theorem. art I.