Computational Analysis of Scattering of Electromagnetic Waves by Particles

Computational Analysis of Scattering of Electromagnetic Waves by Particles


A computational model of Mie scattering was developed using COMSOL Multiphysics® and its RF Module. It solves for the scattering off a dielectric, magnetic or metal spherical particle with radius a. The model geometry is shown in Figure 1.


MIE Scattering

Figure 1: Model geometry for Mie scattering by a spherical particle


The air domain is truncated by a perfectly matched layer (PML) inserted to limit the extent of the model to a manageable region of interest. The solution inside the domain is not affected by the presence of the PML, which lets the solution behave as if the domain was of infinite extent. This layer absorbs all outgoing wave energy without any impedance mismatch that could cause spurious reflections at the boundary. The PML is useful in maintaining the solution at the desired level of accuracy and optimizing usage of computational resources.  COMSOL also supports far-field calculations, which are done on the inner boundary of the PML domain where the near field is integrated. The surface S is used to calculate total scattered energy. An incident plane wave travels in the positive x-direction (see Figure 1), with the electric field polarized 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 functionality to calculate scattering integrals. Scattering characteristics for the three types of particles considered are shown in Figures 2, 3, and 4. The results of the computational analysis show good agreement with available experimental results (see references).


 Fig2Blog2Figure 2. Cross-section parameters and radiation force for a dielectric particle with refractive index n and relative permeability MuFig3Blog2

Figure 3. Cross-section parameters and radiation force for a magnetic particle with relative permittivity Epsilon and relative permeability MuToo

Fig4Blog2Figure 4. Cross-section parameters and radiation force for a silver particle with dielectric constants.

Simulation of Mie scattering problems enables visualization of the effects of small particles on an incident electromagnetic wave (see Figure 5) to allow better understanding of the interactions.




Figure 5: Distribution of the z-component of the electric field due to scattering of the incident electromagnetic wave by a particle of 0.1µm of 950 radius. The arrows show the time-averaged power flow of the relative fields at a frequency THz.




Mätzler, C.: MATLAB Functions for Mie Scattering and Absorption, Version 2, IAP Research Report, No. 2002-11, InstitutfürangewandtePhysik, Universität Bern, 2002.

Scattering of Electromagnetic Waves by Particles

Scattering of Electromagnetic Waves by Particles


Interaction between electromagnetic waves and particles produce unique scattering patterns that are wavelength and particle size dependent.


As electromagnetic waves propagate through matter they interact with particles or inhomogeneities and locally perturb the local electron distribution. This variation produces periodic charge separation within the particle causing oscillation of the induced local dipole moment, this periodic acceleration acts as a source of electromagnetic radiation thus causing scattering. The majority of the scattered wave oscillates at the same frequency as the incident wave and is termed elastic scattering. Interaction with the incident beam may also lead to absorption in the form of thermal energy. The combination of scattering and absorption attenuate the incident beam leading to extinction.


Scattering of electromagnetic waves by particles can be treated by two theoretical frameworks: Rayleigh scattering that is applicable to small, dielectric, non-absorbing spherical particles, and Mie scattering that provides a general solution to scattering independent of particle size. Mie scattering theory provide a generalized approach, has no particle size limitations and converges to the limit of geometric optics at large particle sizes. Consequently Mie scattering theory can be used to describe most scattering by spherical particles, including Rayleigh scattering, but due to the complexity of implementation, Rayleigh scattering theory is often preferred.


Rayleigh scattering is strongly dependent upon the size of the particle and the wavelength of the illuminating radiation. The intensity of the Rayleigh scattered radiation increases rapidly as the ratio of particle size to wavelength increases and is identical in the forward and reverse directions. The Rayleigh scattering model breaks down when the particle size becomes larger than approximately 10% of the wavelength of the incident radiation at which point Mie theory must be applied. The Mie solution is obtained through an analytical solution of Maxwell’s equations for the scattering of electromagnetic radiation by spherical particles in terms of infinite series rather than a simple mathematical expression.


Mie scattering differs from Rayleigh scattering in several respects: it is roughly independent of wavelength and it is larger in the forward direction than in the reverse direction, figure 1. The greater the particle size, the more of the light is scattered in the forward direction. In addition to explaining many atmospheric effects of light scattering, applications of Mie scattering include environmental areas such as dust particles in the atmosphere and oil droplet in water as well as medical technology to measure cell nuclei in biological systems or measurement of collagen fibers in body tissue.


Electromagnetic Waves

Figure 1: Electric field due to Mie scattering of incident wave in x direction showing enhanced scattering in forward direction.



Analysis of Mie Scattering


Implementation of analytical solutions for Mie scattering by a particle or object is complex and requires solution of Maxwell’s equations to represent the incident, scattered and internal fields. These are not simple mathematical expressions and take the form of infinite series expansion of vector spherical harmonics that allows the cross sections, efficiency factors and distributions of intensity to be predicted. Further, the influence of particle geometry, incident of the incident wave and the particle’s material properties can be investigated.


In electromagnetic wave scattering problems, the total wave decomposes into the incident and scattered wave components:

Electromagnetic Waves   (1)

Electromagnetic Waves   (2)

Maxwell’s wave equation is solved with respect to scattered electric field:

Electromagnetic Waves   (3)

and the scattered magnetic field is calculated from Faraday’s law:

Electromagnetic Waves   (4)

The time-average Poynting vector for time-harmonic fields gives the energy flux:

Electromagnetic Waves   (5)

For an incident plane wave, the magnetic field is related to the electric field by:

Electromagnetic Waves   (6)

where k is direction of the incident wave propagation, η = (μ/ε)1/2 is the characteristic impedance, ε is permittivity and μ is permeability of ambient medium. Hence, incident energy flux is calculated as

Electromagnetic Waves   (7)

Important physical quantities can be obtained from the scattered fields. One of these is the cross section, which can be defined as the net rate at which electromagnetic energy (W) crosses the surface of an imaginary sphere centered at the particle divided by the incident irradiationPinc. To quantify the rate of the electromagnetic energy that is absorbed (Wabs) and scattered (Wsca) by the particle, the absorption (σabs), scattering (σsca) and extinction cross sections are defined as:

Electromagnetic Waves   (8)


The total absorbed energy is derived by integrating the energy loss over the volume of the particle:

Electromagnetic Waves   (9)



The scattered energy is derived by integrating the Poynting vector over an imaginary sphere around the particle:

Electromagnetic Waves   (10)



where n is unit vector normal to the imaginary surface S.


Due to the particulate nature of electromagnetic waves they also carry momentum P-c and exert a force on the particle, called radiation pressure which can be obtained by integrating the Maxwell stress tensor over the surface of the sphere:

Electromagnetic Waves   (11)



where  σpr is the pressure cross-section, and <cosθ>  is the asymmetry parameter.

The radiation pressure cross section can be used to calculate force which the particle experiences in the incident direction:

Electromagnetic Waves   (12)



The total time-averaged force F acting on a particle illuminated with light can also be calculated using surface integral of the time-averaged Maxwell’s stress tensor T:

Electromagnetic Waves   (13)



Where Sp is surface enclosing particle volume Vp and n is unit normal vector to surface Sp.


In a following Blog we will demonstrate the implementation of Mie scattering into COMSOL Multiphysics and correlate the results with available analytical solution to simplified cases.


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