1. Introduction

For a plane electromagnetic wave propagating in a homogenous medium, the energy densities of the electric and magnetic components are uniform and locally the same; however, in the presence of a scatterer, the energy density within the scatterer is not uniform. In this study we investigate the relationship between electric and magnetic energy densities within and in the vicinity outside small scatterers.

A great deal of research has been done on the internal electric field within infinite cylinders and spheroids [1–3], as well as in irregularly shaped particles [4]. However, these previous studies concentrated on the electric field or intensity within and in the vicinity outside the particles. To our knowledge no one has shown the corresponding magnetic field distribution in the same region, which may be important in many practical applications.

In this study, we computed the energy densities associated with both the electric and magnetic fields for two particle geometries; namely a sphere and an ellipsoid, with two different refractive indices. The sizes of these particles are comparable with the incident wavelength, which rules out geometric optics. We therefore use the Lorenz-Mie theory for the sphere and the finite-difference time-domain (FDTD) [5–8] technique for the ellipsoid to perform the numerical simulation involved in this study. However various other methods have been developed for computing the scattering properties of nonspherical particles and these were recently reviewed by Mishchenko et al. [9]. The three-dimensional FDTD computational program that we used was developed by Yang et al. [7] and has been enhanced by using the Uniaxial Perfectly Matched Layer (UPML) boundary condition [10]. The validation of the improved FDTD computational program has been reported by Li et al. [11] by comparing with the exact solution for the scattering of light by spheres. This paper proceeds as follows: presented in Section 2 are the particle morphologies and the definitions of the electric and magnetic energy densities; the results of the simulations are shown in Section 3; and finally, the potential applications of this study are discussed in Sec.4.

 

Fig. 1. Particle geometries used in this study: a homogenous sphere with a diameter of 1.0 µm and a homogenous ellipsoid with a major axis of 1.56 µm and a minor axis of 0.8 µm. The two particles have the same volume.

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2. Models and definitions

We consider two particle shapes; namely a sphere and an ellipsoid as shown in Fig. 1. The diameter of the sphere is 1 µm. The aspect ratio of the ellipsoid is 1:0.51, and has the same volume as the sphere.

The illuminating light source in the present simulation is an unpolarized plane wave, and therefore polarization effects are not considered. Since the electric and magnetic fields are time-dependent, we consider the temporally averaged values of the fields. The electric and magnetic energy densities of an electromagnetic wave are defined as follows [12]:

where ε and µ are the permittivity and permeability of the medium respectively. <E²> and <H²> indicate the temporally averaged field values. The densities defined in Eqs. (1) and (2) are proportional to the incident irradiance that is set to unity in this study.

3. Results of simulation

Figure 2 shows the distributions of both the electric and magnetic energy densities inside and in the vicinity outside the sphere described in Fig. 1, which were computed on a vertical cross section through the center of the particle and parallel to the incident radiation. The incident wavelength and the refractive index of the sphere are λ=0.3µm and m=1.34, respectively. Due to the large range in values (as in Fig. 3), a logarithmic scale is used. Evidently, the intensities inside the particle are not uniformly distributed. Both of the electric and magnetic fields are focused in the forward direction along the incident light. The overall patterns of the energy density distributions for the two field components are similar, and the differences between the electric and magnetic energy densities are essentially quite small except in a focal region shown in the panel (c) in Fig. 2.

 

Fig. 2. Internal and near-field electric and magnetic energy densities and their differences. The incident wavelength and refractive index for the simulation are λ=0.3µm and m=1.34, respectively. (a) The electric energy density; (b) the magnetic energy density; and (c) the differences between the two densities (the electric energy density minus the magnetic energy density). One should note the jet-like behavior outside the particle in the forward direction.

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Figure 3 shows a case for a refractive index of m=2.0. As in the previous case, both the energy densities are focused in the forward direction near the edge of the scatterer, where significant maxima are noticed for both the electric and magnetic energy densities. Additionally, the high energy density region moves toward the back of the sphere. The differences between the two energy densities are quite large in the focal region.

 

Fig. 3. Same as Fig. 2 except for refractive index of m=2.0. Also we note similar jet-like pattern as in Fig. 2.

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Figure 4 shows the distributions of the two energy densities and their differences for an ellipsoid. The ellipsoid has a major axis of 1.56 µm and a minor axis of 0.8µm, and has the same volume as the sphere defined for Figs. 2 and 3. The incident wavelength and the refractive index of the scattering particle are chosen as λ=0.3 µm and m=1.34, respectively, which is same as the case in Fig. 2. The incident light is parallel to the major axis of the ellipsoid (as shown in Fig. 4). Similar to the cases in Figs. 2, the fields are also focused in the forward direction, but the energy density maxima located inside the particle are stronger in the case for the ellipsoid. The differences of the two energy densities are not substantial except in the focal region, as is evident from the panel (c) in Fig. 4.

 

Fig. 4. Same as Fig.3, except that the shape is an ellipsoid.

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Shown in Fig. 5 are the results similar to those in Fig. 4, except that the ellipsoid is illuminated with broadside incidence. The energy density distributions are similar to a case where the incident light passes through a convex lens; however, there is not an explicit focal point in the present results. Inside the ellipsoid, the energy density differences are noticed primarily near the front boundary and in the nearby region. Outside the particle, the energy density differences are insignificant.

 

Fig. 5. Same as Fig.4, except that the incidence is perpendicular to the symmetric axis.

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To investigate the sensitivity of size, we doubled the sphere diameter to 2µm and keep m=1.34. The results were similar to Fig. 2; the only difference being a stronger jet-like tail. We also examined the effect of absorption on the patterns. To do this we took m=1.34+i0.05 for the 1µm diameter sphere. The results were again similar to Fig. 2; however the field strength was reduced over the entire area.

4. Discussion and conclusions

The distributions of both the electric and magnetic energy densities are essential to the study of light-induced reactions, such as laser induced Raman or Fluorescence. In most cases, only the electric field is considered because the electric dipole transitions are more important in studying the interactions of radiation with matter. Note that the electric dipole transitions are normally 10⁴~10⁵ stronger than the magnetic dipole transitions. However, in some cases the electric dipole transition is forbidden, such as for the case involving the 1s to 2s transition in an atom, in which the magnetic dipole transition plays an important role and then the magnetic field distribution must be considered. As shown by the present results, the distributions of electric and magnetic energy densities are not the same inside a scattering particle, and the local differences of these two energy densities can be quite large in a certain region within the scattering particle, particularly for cases involving large refractive indices.

Laser related remote detecting technologies (such as FAST CAR [13]) are very important methods to detect biological spores or aerosols. Since biological spores may have large refractive index component parts (core, shell, etc), the non-uniform distribution of internal electric and magnetic fields and their local differences should be considered when they are being detected.

The highly concentrated radiation (shown in Fig. 3) inside the scatterer may alter the physical structure locally because the field intensity is magnified hundreds of times.

In cases where the size of the spherical scatterers is so small that geometric optics is invalid, plane wave incident light is still focused in the forward direction similar to a convex lens (although the points of maximum radiance may lie inside the particle). This phenomenon is described in previous works [1–3, 14] for electric fields, and in this paper, we demonstrated that similar results can be obtained with magnetic fields.

The jet-like behavior of the near-field intensity shown in Figs. 2 and 3 has also been described in [15], and may be applied to near-field scanning techniques [16]. It may also be useful for studying fluorescence and Raman effects.