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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 13, Iss. 12 — Jun. 13, 2005
  • pp: 4554–4559
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Electric and magnetic energy density distributions inside and outside dielectric particles illuminated by a plane electromagnetic wave

Changhui Li, George W. Kattawar, Peng-Wang Zhai, and Ping Yang  »View Author Affiliations


Optics Express, Vol. 13, Issue 12, pp. 4554-4559 (2005)
http://dx.doi.org/10.1364/OPEX.13.004554


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Abstract

We have studied the distribution of the electric and magnetic energy densities within and in the vicinity outside a dielectric particle illuminated by a plane electromagnetic wave. Numerical simulations were performed by using the Lorenz-Mie theory and the finite-difference time-domain method for spheres and spheroids, respectively. We found that the electric and magnetic energy densities are locally different within the scatterers. The knowledge of the two components of the electromagnetic energy density is essential to the study of the dipole (electric or magnetic) transitions that have potential applications to Raman and fluorescence spectroscopy.

© 2005 Optical Society of America

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.

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

5. K. S. Yee, “Numerical solution of initial boundary problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propagat. AP-14, 302–307 (1966).

8

8. A. Taflove and S. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method (Artech, Boston, MA, 2000).

] 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

9. M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, Eds. Light Scattering by Nonspherical Particles (Academic Press, San Diego, CA, 2000).

]. The three-dimensional FDTD computational program that we used was developed by Yang et al. [7

7. P. Yang, K. N. Liou, M. I. Mishchenko, and B.-C. Gao, 2000: An efficient finite-difference time domain scheme for light scattering by dielectric particles: application to aerosols, Appl. Opt. 39, 3727–3737. [CrossRef]

] and has been enhanced by using the Uniaxial Perfectly Matched Layer (UPML) boundary condition [10

10. Z. S. Sacks, D. M. Kingsland, R. Lee, and J. F. Lee, “A perfect matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propagat 43, 1460–1463 (1995). [CrossRef]

]. The validation of the improved FDTD computational program has been reported by Li et al. [11

11. C. Li, G. W. Kattawar, and P. Yang, “Effects of surface roughness on light scattering by small particles,” J. Quant. Spectrosc. Radiat. Transfer 89, 123–131 (2004). [CrossRef]

] 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.

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

12. J. D. Jackson, Classical Electrodynamics (John Wiley and Sons Inc.1998).

]:

ue(r)=12ε(r)<E2(r)>,
(1)
uh(r)=12μ(r)<H2(r)>,
(2)

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.

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.

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.

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.

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.

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.

The jet-like behavior of the near-field intensity shown in Figs. 2 and 3 has also been described in [15

15. Z. Chen and A. Taflove, “Photonic nanojet enhancement of backscattering of light by nanoparticles: a potential novel visible-light ultramicroscopy technique,” Opt. Express 12, 1214–1220 (2004). [CrossRef] [PubMed]

], and may be applied to near-field scanning techniques [16

16. E. Betzig and J. K. Trautman, “Near-field optics: Microscopy, spectroscopy, and surface modification beyond the diffraction limit,” Science 257, 189–195 (1992). [CrossRef] [PubMed]

]. It may also be useful for studying fluorescence and Raman effects.

Acknowledgments

References

1.

D. S. Benincasa, P. W. Barber, J.-Z. Zhang, W. -F. Hsieh, and R. K. Chang, “Spatial distribution of the internal and near-field intensities of large cylindrical and spherical scatterers,” Appl. Opt. 26, 1348–1356 (1987). [CrossRef] [PubMed]

2.

J. F. Owen, R. K. Chang, and P. W. Barber, “Internal electric field distributions of a dielectric cylinder at resonance wavelengths,” Opt. Lett. 6, 540–542 (1981). [CrossRef] [PubMed]

3.

L. G. Astafyeva and V. A. Babenko, “Interaction of electromagnetic radiation with silicate spheroidal aerosol particles,” J. Quant. Spectrosc. Radiat. Transfer 88, 9–15 (2004). [CrossRef]

4.

J. P. Barton, “Electromagnetic field calculations for an irregularly shaped, near-spheroidal particle with arbitrary illumination,” JOSA 19, 2429–2435 (2002). [CrossRef]

5.

K. S. Yee, “Numerical solution of initial boundary problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propagat. AP-14, 302–307 (1966).

6.

W. Sun, Q. Fu, and Z. Chen, 1999: “Finite-difference time-domain solution of light scattering by dielectric particles with perfectly matched layer absorbing boundary conditions,” Appl. Opt. 38, 3141–3151. [CrossRef]

7.

P. Yang, K. N. Liou, M. I. Mishchenko, and B.-C. Gao, 2000: An efficient finite-difference time domain scheme for light scattering by dielectric particles: application to aerosols, Appl. Opt. 39, 3727–3737. [CrossRef]

8.

A. Taflove and S. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method (Artech, Boston, MA, 2000).

9.

M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, Eds. Light Scattering by Nonspherical Particles (Academic Press, San Diego, CA, 2000).

10.

Z. S. Sacks, D. M. Kingsland, R. Lee, and J. F. Lee, “A perfect matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propagat 43, 1460–1463 (1995). [CrossRef]

11.

C. Li, G. W. Kattawar, and P. Yang, “Effects of surface roughness on light scattering by small particles,” J. Quant. Spectrosc. Radiat. Transfer 89, 123–131 (2004). [CrossRef]

12.

J. D. Jackson, Classical Electrodynamics (John Wiley and Sons Inc.1998).

13.

M. O. Scully, G. W. Kattawar, R. P. Lucht, T. Opatrny, H. Pilloff, A. Rebane, A. V. Sokolov, and M. S. Zubairy, “FAST CARS: Engineering a laser spectroscopic technique for rapid identification of bacterial spores,” Proc. Natl. Acad. Sci. USA 99, 10994–11001 (2002). [CrossRef] [PubMed]

14.

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988). [CrossRef]

15.

Z. Chen and A. Taflove, “Photonic nanojet enhancement of backscattering of light by nanoparticles: a potential novel visible-light ultramicroscopy technique,” Opt. Express 12, 1214–1220 (2004). [CrossRef] [PubMed]

16.

E. Betzig and J. K. Trautman, “Near-field optics: Microscopy, spectroscopy, and surface modification beyond the diffraction limit,” Science 257, 189–195 (1992). [CrossRef] [PubMed]

OCIS Codes
(140.3550) Lasers and laser optics : Lasers, Raman
(170.0170) Medical optics and biotechnology : Medical optics and biotechnology
(290.0290) Scattering : Scattering
(290.5850) Scattering : Scattering, particles
(300.2530) Spectroscopy : Fluorescence, laser-induced
(350.3450) Other areas of optics : Laser-induced chemistry

ToC Category:
Research Papers

History
Original Manuscript: March 29, 2005
Revised Manuscript: May 26, 2005
Published: June 13, 2005

Citation
George Kattawar, Changhui Li, Peng-Wang Zhai, and Ping Yang, "Electric and magnetic energy density distributions inside and outside dielectric particles illuminated by a plane electromagnetic wave," Opt. Express 13, 4554-4559 (2005)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-12-4554


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References

  1. D. S. Benincasa, P. W. Barber, J.-Z. Zhang, W. �??F. Hsieh, and R. K. Chang, �??Spatial distribution of the internal and near-field intensities of large cylindrical and spherical scatterers,�?? Appl. Opt. 26, 1348-1356 (1987) [CrossRef] [PubMed]
  2. J. F. Owen, R. K. Chang and P. W. Barber, �??Internal electric field distributions of a dielectric cylinder at resonance wavelengths,�?? Opt. Lett. 6, 540-542 (1981). [CrossRef] [PubMed]
  3. L. G. Astafyeva, V. A. Babenko, �??Interaction of electromagnetic radiation with silicate spheroidal aerosol particles,�?? J. Quant. Spectrosc. Radiat. Transfer 88, 9-15 (2004). [CrossRef]
  4. J. P. Barton, �??Electromagnetic field calculations for an irregularly shaped, near-spheroidal particle with arbitrary illumination,�?? JOSA 19, 2429-2435 (2002). [CrossRef]
  5. K. S. Yee, �??Numerical solution of initial boundary problems involving Maxwell�??s equations in isotropic media,�?? IEEE Trans. Antennas Propagat. AP-14, 302-307 (1966).
  6. Sun, W., Q. Fu, and Z. Chen, 1999: �??Finite-difference time-domain solution of light scattering by dielectric particles with perfectly matched layer absorbing boundary conditions,�?? Appl. Opt. 38, 3141- 3151. [CrossRef]
  7. Yang, P., K. N. Liou, M. I. Mishchenko, and B.-C. Gao, 2000: An efficient finite-difference time domain scheme for light scattering by dielectric particles: application to aerosols, Appl. Opt. 39, 3727-3737. [CrossRef]
  8. A. Taflove and S. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method (Artech, Boston, MA, 2000).
  9. M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, Eds. Light Scattering by Nonspherical Particles (Academic Press, San Diego, CA, 2000).
  10. Z. S. Sacks, D. M. Kingsland, R. Lee, and J. F. Lee, �??A perfect matched anisotropic absorber for use as an absorbing boundary condition,�?? IEEE Trans. Antennas Propagat 43, 1460-1463 (1995). [CrossRef]
  11. C. Li, G. W. Kattawar and P. Yang, �??Effects of surface roughness on light scattering by small particles,�?? J. Quant. Spectrosc. Radiat. Transfer 89, 123-131 (2004). [CrossRef]
  12. J. D. Jackson, Classical Electrodynamics (John Wiley and Sons Inc. 1998).
  13. M. O. Scully, G. W. Kattawar, R. P. Lucht, T. Opatrny, H. Pilloff, A. Rebane, A. V. Sokolov, and M. S. Zubairy, �??FAST CARS: Engineering a laser spectroscopic technique for rapid identification of bacterial spores,�?? Proc. Natl. Acad. Sci. USA 99, 10994-11001 (2002). [CrossRef] [PubMed]
  14. J. P. Barton, D. R. Alexander, and S. A. Schaub, �??Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,�?? J. Appl. Phys. 64, 1632-1639 (1988). [CrossRef]
  15. Z. Chen, A. Taflove, �??Photonic nanojet enhancement of backscattering of light by nanoparticles: a potential novel visible-light ultramicroscopy technique,�?? Opt. Express 12, 1214-1220 (2004). [CrossRef] [PubMed]
  16. E. Betzig and J. K. Trautman, �??Near-field optics: Microscopy, spectroscopy, and surface modification beyond the diffraction limit,�?? Science 257, 189-195 (1992). [CrossRef] [PubMed]

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