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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 17 — Aug. 15, 2011
  • pp: 16207–16222
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Light concentration in the near-field of dielectric spheroidal particles with mesoscopic sizes

Manuel J. Mendes, Ignacio Tobías, Antonio Martí, and Antonio Luque  »View Author Affiliations


Optics Express, Vol. 19, Issue 17, pp. 16207-16222 (2011)
http://dx.doi.org/10.1364/OE.19.016207


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Abstract

This paper presents a numerical study of the light focusing properties of dielectric spheroids with sizes comparable to the illuminating wavelength. An analytical separation-of-variables method is used to determine the electric field distribution inside and in the near-field outside the particles. An optimization algorithm was implemented in the method to determine the particles’ physical parameters that maximize the forward scattered light in the near-field region. It is found that such scatterers can exhibit pronounced electric intensity enhancement (above 100 times the incident intensity) in their close vicinity, or along wide focal regions extending to 10 times the wavelength. The results reveal the potential of wavelength-sized spheroids to manipulate light beyond the limitations of macroscopic geometrical optics. This can be of interest for several applications, such as light management in photovoltaics.

© 2011 OSA

1. Introduction

Near-field effects become particularly important when the size of the scattering object is smaller or on the order of the illuminating wavelength (λ), therefore outside the macroscopic regime of geometrical optics (GO) [1

1. C. Girard and A. Dereux, “Near-field optics theories,” Rep. Prog. Phys. 59(5), 657–699 (1996). [CrossRef]

3

3. M. J. Mendes, I. Tobías, A. Martí, and A. Luque, “Near-field scattering by dielectric spheroidal particles with sizes on the order of the illuminating wavelength,” J. Opt. Soc. Am. B 27(6), 1221–1231 (2010). [CrossRef]

]. The field distribution produced by objects with sizes much smaller than λ can be obtained with the electrostatic approximation (EA) [4

4. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH, 2004).

]. Scatterers in this size regime produce dipolar-like (first order) near-field patterns independently of their geometry or material. As the particle size approaches λ, higher order modes (quadrupolar, octopolar, etc.) are excited producing additional features in the field pattern which are highly dependent on the scatterer physical parameters (size, shape, material and surrounding medium) [4

4. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH, 2004).

,5

5. H. Mertens, A. F. Koenderink, and A. Polman, “Plasmon-enhanced luminescence near noble-metal nanospheres: comparison of exact theory and an improved Gersten and Nitzan model,” Phys. Rev. B 76(11), 115123 (2007). [CrossRef]

]. In this paper we focus on particles with sizes on the order of λ, which lie in an intermediate scattering regime between EA and GO – the mesoscopic regime. This regime is still rather unexplored, since the solution of electromagnetic (EM) scattering by mesoscopic objects requires the detailed calculation of the full set of Maxwell’s equations [1

1. C. Girard and A. Dereux, “Near-field optics theories,” Rep. Prog. Phys. 59(5), 657–699 (1996). [CrossRef]

,3

3. M. J. Mendes, I. Tobías, A. Martí, and A. Luque, “Near-field scattering by dielectric spheroidal particles with sizes on the order of the illuminating wavelength,” J. Opt. Soc. Am. B 27(6), 1221–1231 (2010). [CrossRef]

].

Numerical grid-based approaches, such as finite difference time-domain (FDTD) [6

6. Z. Chen, A. Taflove, and V. Backman, “Photonic nanojet enhancement of backscattering of light by nanoparticles: a potential novel visible-light ultramicroscopy technique,” Opt. Express 12(7), 1214–1220 (2004). [CrossRef] [PubMed]

8

8. J. Grandidier, D. M. Callahan, J. N. Munday, and H. A. Atwater, “Thin-film solar cells: light absorption enhancement in thin-film solar cells using whispering gallery modes in dielectric nanospheres,” Adv. Mater. (Deerfield Beach Fla.) 23(10), 1171 (2011). [CrossRef]

] and finite elements methods (FEM) [9

9. L. E. McNeil, A. R. Hanuska, and R. H. French, “Near-field scattering from red pigment particles: absorption and spectral dependence,” J. Appl. Phys. 89(3), 1898–1906 (2001). [CrossRef]

], are often employed to model the near-field distribution of complex mesoscopic structures. However, with mesh approaches it is usually difficult to resolve high field gradients that occur at the surface of particles that scatter highly resonantly with the incident light. Therefore, whenever possible, it is preferable to use analytical techniques since the fields can be calculated everywhere without needing to cope with finite mesh resolution errors. Most of the analytical studies published so far on light scattering by mesoscopic particles use Mie theory [4

4. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH, 2004).

,10

10. S. Lecler, Y. Takakura, and P. Meyrueis, “Properties of a three-dimensional photonic jet,” Opt. Lett. 30(19), 2641–2643 (2005). [CrossRef] [PubMed]

14

14. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University Press, 1999).

], which is valid for any particle size but restricted to perfectly spherical shapes. In the present work, a spheroidal coordinate separation-of-variables solution is used to study the near-field scattering properties of spheroidal particles with arbitrary size and shape [3

3. M. J. Mendes, I. Tobías, A. Martí, and A. Luque, “Near-field scattering by dielectric spheroidal particles with sizes on the order of the illuminating wavelength,” J. Opt. Soc. Am. B 27(6), 1221–1231 (2010). [CrossRef]

,15

15. L.-W. Li, X.-K. Kang, and M.-S. Leong, Spheroidal Wave Functions in Electromagnetic Theory (John Wiley & Sons, Inc., 2002).

,16

16. J. P. Barton, “Internal and near-surface electromagnetic fields for a spheroidal particle with arbitrary illumination,” Appl. Opt. 34(24), 5542–5551 (1995). [CrossRef] [PubMed]

].

Objects with sizes close to or above λ scatter the light preferentially along the incident wave propagation direction (forward scattering). Dielectric spheres and cylinders have been shown [6

6. Z. Chen, A. Taflove, and V. Backman, “Photonic nanojet enhancement of backscattering of light by nanoparticles: a potential novel visible-light ultramicroscopy technique,” Opt. Express 12(7), 1214–1220 (2004). [CrossRef] [PubMed]

,7

7. C. Li, G. W. Kattawar, P.-W. Zhai, and P. Yang, “Electric and magnetic energy density distributions inside and outside dielectric particles illuminated by a plane electromagnetic wave,” Opt. Express 13(12), 4554–4559 (2005). [CrossRef] [PubMed]

,9

9. L. E. McNeil, A. R. Hanuska, and R. H. French, “Near-field scattering from red pigment particles: absorption and spectral dependence,” J. Appl. Phys. 89(3), 1898–1906 (2001). [CrossRef]

,10

10. S. Lecler, Y. Takakura, and P. Meyrueis, “Properties of a three-dimensional photonic jet,” Opt. Lett. 30(19), 2641–2643 (2005). [CrossRef] [PubMed]

,12

12. H. J. Münzer, M. Mosbacher, M. Bertsch, J. Zimmermann, P. Leiderer, and J. Boneberg, “Local field enhancement effects for nanostructuring of surfaces,” J. Microsc. 202(1), 129–135 (2001). [CrossRef] [PubMed]

,17

17. A. Devilez, N. Bonod, J. Wenger, D. Gérard, B. Stout, H. Rigneault, and E. Popov, “Three-dimensional subwavelength confinement of light with dielectric microspheres,” Opt. Express 17(4), 2089–2094 (2009). [CrossRef] [PubMed]

19

19. A. Devilez, B. Stout, N. Bonod, and E. Popov, “Spectral analysis of three-dimensional photonic jets,” Opt. Express 16(18), 14200–14212 (2008). [CrossRef] [PubMed]

] to produce remarkably intense electric fields close to their shadow-side surface. They act as near-field lenses concentrating the light in a jet-like region located along the incidence axis, which has already been corroborated experimentally by direct imaging [20

20. P. Ferrand, J. Wenger, A. Devilez, M. Pianta, B. Stout, N. Bonod, E. Popov, and H. Rigneault, “Direct imaging of photonic nanojets,” Opt. Express 16(10), 6930–6940 (2008). [CrossRef] [PubMed]

]. The laws of such focusing are quite distinct from those of macroscopic GO lenses [11

11. J. Kofler and N. Arnold, “Axially symmetric focusing as a cuspoid diffraction catastrophe: scalar and vector cases and comparison with the theory of Mie,” Phys. Rev. B 73(23), 235401 (2006). [CrossRef]

,12

12. H. J. Münzer, M. Mosbacher, M. Bertsch, J. Zimmermann, P. Leiderer, and J. Boneberg, “Local field enhancement effects for nanostructuring of surfaces,” J. Microsc. 202(1), 129–135 (2001). [CrossRef] [PubMed]

,18

18. A. V. Itagi and W. A. Challener, “Optics of photonic nanojets,” J. Opt. Soc. Am. A 22(12), 2847–2858 (2005). [CrossRef] [PubMed]

,21

21. S. Lecler, “Light scattering by sub-micrometric particles,” PhD thesis (Louis Pasteur University, 2005).

]. In the mesoscopic regime the diffraction pattern is dictated by EM wave interference mechanisms which redistribute the energy from the regions of destructive interference to those of constructive interference [22

22. E. Hecht, Optics, 4th ed. (Addison Wesley, 2001).

], as discussed in Section 3.

In most cases of practical interest scatterers are non-spherical and can be better approximated by a spheroidal shape. However, accurate light-scattering computations for mesoscopic spheroids are complex and time consuming, and the literature in which such calculations are reported is rather scarce. The work presented in this paper contributes to fulfil this gap. The near-field light focusing properties of dielectric spheroids with arbitrary size, aspect ratio and complex refractive index are analysed; and important additional possibilities are found relative to the particular case of spheres. A small imaginary part is considered in the materials refractive index in order for the results to meet conditions attainable in practice.

The possibility to concentrate light in the near-field of dielectric mesoscopic particles (DMPs) is still little explored [11

11. J. Kofler and N. Arnold, “Axially symmetric focusing as a cuspoid diffraction catastrophe: scalar and vector cases and comparison with the theory of Mie,” Phys. Rev. B 73(23), 235401 (2006). [CrossRef]

,21

21. S. Lecler, “Light scattering by sub-micrometric particles,” PhD thesis (Louis Pasteur University, 2005).

]. This is partly due to the fact that usually only their far-field scattering properties are studied [15

15. L.-W. Li, X.-K. Kang, and M.-S. Leong, Spheroidal Wave Functions in Electromagnetic Theory (John Wiley & Sons, Inc., 2002).

,23

23. S. Asano and G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14(1), 29–49 (1975). [PubMed]

25

25. J. C. Ravey and P. Mazeron, “Light-scattering in the physical optics approximation—application to large spheroids,” J. Opt. 13(5), 273–282 (1982). [CrossRef]

]. Besides, the high energy concentration occurs only in specific cases when there is an optimized set of physical parameters that allow a pronounced constructive interference in the diffraction pattern. In any application, the absorptive nature of scatterers and the high confinement of their near-field substantially limit the parameter space where near-field structures can provide exceptional improvements to the properties of the surrounding receiving materials (emitters or absorbers depending on the application) [5

5. H. Mertens, A. F. Koenderink, and A. Polman, “Plasmon-enhanced luminescence near noble-metal nanospheres: comparison of exact theory and an improved Gersten and Nitzan model,” Phys. Rev. B 76(11), 115123 (2007). [CrossRef]

]. Therefore, a theoretical study involving a computational optimization is crucial prior to any practical implementation. In this paper, an optimization algorithm was developed that iteratively searches for the DMP parameters that provide the highest possible forward scattered field intensities along a certain region of the external medium.

This can be of interest for several applications such as nanoscale processing of materials (ablation [12

12. H. J. Münzer, M. Mosbacher, M. Bertsch, J. Zimmermann, P. Leiderer, and J. Boneberg, “Local field enhancement effects for nanostructuring of surfaces,” J. Microsc. 202(1), 129–135 (2001). [CrossRef] [PubMed]

] or photo-etching [11

11. J. Kofler and N. Arnold, “Axially symmetric focusing as a cuspoid diffraction catastrophe: scalar and vector cases and comparison with the theory of Mie,” Phys. Rev. B 73(23), 235401 (2006). [CrossRef]

]), high-resolution microscopy (e.g. biomedical diagnostics [6

6. Z. Chen, A. Taflove, and V. Backman, “Photonic nanojet enhancement of backscattering of light by nanoparticles: a potential novel visible-light ultramicroscopy technique,” Opt. Express 12(7), 1214–1220 (2004). [CrossRef] [PubMed]

,27

27. N. Richard, “Analysis of polarization effects on nanoscopic objects in the near-field optics,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(2), 026602 (2001). [CrossRef] [PubMed]

]), resonance spectroscopy (e.g. amplification of Raman and fluorescence [7

7. C. Li, G. W. Kattawar, P.-W. Zhai, and P. Yang, “Electric and magnetic energy density distributions inside and outside dielectric particles illuminated by a plane electromagnetic wave,” Opt. Express 13(12), 4554–4559 (2005). [CrossRef] [PubMed]

,28

28. J. P. Barton, “Internal and near-surface electromagnetic fields for an absorbing spheroidal particle with arbitrary illumination,” Appl. Opt. 34(36), 8472–8473 (1995). [CrossRef] [PubMed]

] signals), localized sensing techniques (e.g. enhancement of nanoparticles backscattering [9

9. L. E. McNeil, A. R. Hanuska, and R. H. French, “Near-field scattering from red pigment particles: absorption and spectral dependence,” J. Appl. Phys. 89(3), 1898–1906 (2001). [CrossRef]

,13

13. A. Devilez, N. Bonod, and B. Stout, “Near field dielectric microlenses,” Proc. SPIE 7717, 771708(2010). [CrossRef]

]), optical data storage [29

29. T. D. Milster, “Near-field optical data storage: avenues for improved performance,” Opt. Eng. 40(10), 2255–2260 (2001). [CrossRef]

], optical antennas [30

30. A. Devilez, B. Stout, and N. Bonod, “Compact metallo-dielectric optical antenna for ultra directional and enhanced radiative emission,” ACS Nano 4(6), 3390–3396 (2010). [CrossRef] [PubMed]

], among others. Nevertheless, the authors are particularly interested in its implementation in photovoltaic devices [31

31. M. J. Mendes, A. Luque, I. Tobias, and A. Marti, “Plasmonic light enhancement in the near-field of metallic nanospheroids for application in intermediate band solar cells,” Appl. Phys. Lett. 95(7), 071105 (2009). [CrossRef]

]. The incorporation of DMPs as “mesoscopic lenses” in a solar cell could lead to locally enhanced optical absorption and an overall increase in the power conversion efficiency [8

8. J. Grandidier, D. M. Callahan, J. N. Munday, and H. A. Atwater, “Thin-film solar cells: light absorption enhancement in thin-film solar cells using whispering gallery modes in dielectric nanospheres,” Adv. Mater. (Deerfield Beach Fla.) 23(10), 1171 (2011). [CrossRef]

]. The increased strength of the optical interactions with the PV material would also allow the use of a thinner (thus less expensive) photo-active region.

2. Separation-of-variables method and definitions

By applying the separation of variables to the scalar Helmholtz equation the spheroidal harmonics of EM waves can be obtained [32

32. C. Flammer, Spheroidal Wave Functions (Stanford University Press, 1957).

,33

33. L. W. Li, M. S. Leong, T. S. Yeo, P. S. Kooi, and K. Y. Tan, “Computations of spheroidal harmonics with complex arguments: a review with an algorithm,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(5), 6792–6806 (1998). [CrossRef]

]. The solution of the vector Helmholtz equations for the incident (0), internal (i) and scattered (s) electric E and magnetic H fields is determined by expanding the fields in spheroidal vector wave functions obtained from the corresponding scalar spheroidal harmonics. The boundary conditions (BCs) of continuity of the tangential fields across the spheroid surface generate a set of simultaneous linear equations that can be solved for the set of unknown expansion coefficients. The solution of this system of equations is obtained by choosing a suitable truncation number (N) for the fields expansions and then employing the orthogonality integrals approach [15

15. L.-W. Li, X.-K. Kang, and M.-S. Leong, Spheroidal Wave Functions in Electromagnetic Theory (John Wiley & Sons, Inc., 2002).

,16

16. J. P. Barton, “Internal and near-surface electromagnetic fields for a spheroidal particle with arbitrary illumination,” Appl. Opt. 34(24), 5542–5551 (1995). [CrossRef] [PubMed]

,23

23. S. Asano and G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14(1), 29–49 (1975). [PubMed]

]. The value of N is chosen to be sufficiently large for convergence of the solution. In each calculation an initial set of expansion coefficients is obtained with the truncation number N = Integer(K0α + 4). Being K0 = 2π/λ and α is the longest semi-axis of the spheroid [3

3. M. J. Mendes, I. Tobías, A. Martí, and A. Luque, “Near-field scattering by dielectric spheroidal particles with sizes on the order of the illuminating wavelength,” J. Opt. Soc. Am. B 27(6), 1221–1231 (2010). [CrossRef]

,15

15. L.-W. Li, X.-K. Kang, and M.-S. Leong, Spheroidal Wave Functions in Electromagnetic Theory (John Wiley & Sons, Inc., 2002).

]. The continuity of the fields’ tangential components is then checked at a set of points along the particle surface. If there is not enough rigour in maintaining the BCs in the computed solution, the value of N is progressively increased until an accurate match is obtained between the internal and external tangential fields at the surface.

The total computational time is roughly proportional to N3. Most of this time is spent in the determination of the orthogonality integrals to obtain the expansion coefficients. However, these calculations can be parallelized, scaling down the computational time almost proportionally to the number of CPUs used in parallel. The computational environment used to perform the calculations was Mathematica7.0; suitable due to its high-precision numerical capabilities and its packages for computation of spheroidal harmonics [3

3. M. J. Mendes, I. Tobías, A. Martí, and A. Luque, “Near-field scattering by dielectric spheroidal particles with sizes on the order of the illuminating wavelength,” J. Opt. Soc. Am. B 27(6), 1221–1231 (2010). [CrossRef]

,15

15. L.-W. Li, X.-K. Kang, and M.-S. Leong, Spheroidal Wave Functions in Electromagnetic Theory (John Wiley & Sons, Inc., 2002).

,33

33. L. W. Li, M. S. Leong, T. S. Yeo, P. S. Kooi, and K. Y. Tan, “Computations of spheroidal harmonics with complex arguments: a review with an algorithm,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(5), 6792–6806 (1998). [CrossRef]

].

Figure 1
Fig. 1 Coordinate system with origin at the center of the spheroidal particle. The spheroid has semi-axes a (z axis) and b (xy plane). Its refractive index (Np) is higher than that of the surrounding medium (Nm). The direction of illumination (K0) is collinear with the spheroid axis of symmetry (z). The scattered light can form a forward-directed lobe extending away from the particle shadow-side surface. The point of highest electric field intensity (|E|2 MAX) outside the particle is located on the z axis at a distance ZMAX from the origin. LZ is the length of the focal peak along the z axis, corresponding to the distance where the external field intensity remains above |E|2 MAX/e 2.
is a schematic drawing of the coordinate system used. At axial incidence the illuminating wave (K0,E0) can be resolved by a single polarization component, by virtue of symmetry. This wave propagates along the negative z direction with the electric field E0 parallel to y.

In this paper the electric field magnitude is given in units of the incident field amplitude (E0). The length unit used is λ, making the results given here independent of the particular wavelength of illumination.

The physical parameters involved are the size of the spheroid, its aspect ratio (b/a), and the relative refractive index given by the ratio between the refractive index of the particle and that of the ambient medium (Nr = Np/Nm). The spheroid size parameter (C) used in the separation-of-variables method is defined in terms of its inter-focal distance d [15

15. L.-W. Li, X.-K. Kang, and M.-S. Leong, Spheroidal Wave Functions in Electromagnetic Theory (John Wiley & Sons, Inc., 2002).

,32

32. C. Flammer, Spheroidal Wave Functions (Stanford University Press, 1957).

]:
C=Kd2
(1)
where K is the wavevector magnitude in the corresponding medium. Inside the particle the size parameter is Ci = NrC0, being C0 the size parameter in the external medium. Our method allows the computation of the spheroidal harmonics with complex arguments. So, a complex refractive index Nr = nr + ikr is considered; enabling us to account for light absorption in the media [4

4. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH, 2004).

]. The spheroid size parameter defined in Eq. (1) is not a measure of the spheroid size alone, but rather of the size times the eccentricity of the particle. Therefore, for non-spherical scatterers it is usual to adopt the volume-equivalent-sphere radius (Req) as the characteristic particle size [3

3. M. J. Mendes, I. Tobías, A. Martí, and A. Luque, “Near-field scattering by dielectric spheroidal particles with sizes on the order of the illuminating wavelength,” J. Opt. Soc. Am. B 27(6), 1221–1231 (2010). [CrossRef]

,24

24. E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186(2), 705–714 (1973). [CrossRef]

,31

31. M. J. Mendes, A. Luque, I. Tobias, and A. Marti, “Plasmonic light enhancement in the near-field of metallic nanospheroids for application in intermediate band solar cells,” Appl. Phys. Lett. 95(7), 071105 (2009). [CrossRef]

].

2.1 Numerical limits and computational verification

The main computational challenge of the separation-of-variables approach is the accurate calculation of the spheroidal harmonics for large and complex C values. The method used here [3

3. M. J. Mendes, I. Tobías, A. Martí, and A. Luque, “Near-field scattering by dielectric spheroidal particles with sizes on the order of the illuminating wavelength,” J. Opt. Soc. Am. B 27(6), 1221–1231 (2010). [CrossRef]

,15

15. L.-W. Li, X.-K. Kang, and M.-S. Leong, Spheroidal Wave Functions in Electromagnetic Theory (John Wiley & Sons, Inc., 2002).

,33

33. L. W. Li, M. S. Leong, T. S. Yeo, P. S. Kooi, and K. Y. Tan, “Computations of spheroidal harmonics with complex arguments: a review with an algorithm,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(5), 6792–6806 (1998). [CrossRef]

] is able to accurately determine the angular harmonics for any C, and for high number of orders N. However, the computation of the radial harmonics is limited by numerical cancelation and slow convergence. At present, there appears to be no method that is completely satisfactory for the calculation of these functions for large and complex C [34

34. P. Kirby, “Calculation of spheroidal wave functions,” Comput. Phys. Commun. 175(7), 465–472 (2006). [CrossRef]

]. In this work the well-known Wronskian relation was used to check the correctness of the computed radial functions [35

35. R. Kirby, “Calculation of radial prolate spheroidal wave functions of the second kind,” Comput. Phys. Commun. 181(3), 514–519 (2010). [CrossRef]

]. It was verified that the Wronskian relation is satisfied for absolute values of C below 60. For |C|>60 the results may not be accurate enough in some cases. Therefore, the present study is restricted to particles with |Ci|<60 in order to avoid possible computational errors associated with the radial functions.

3. Portrait of physical parameters

The field distribution produced by a mesoscopic scatterer is dictated by interference phenomena, and can be conceptually understood by picturing the object as an array of dispersion points, for instance the atoms in a solid [22

22. E. Hecht, Optics, 4th ed. (Addison Wesley, 2001).

]. As the incident light beam propagates through the particle material, planes of points transverse to the beam are progressively illuminated in phase and scattered spherical waves radiate from every point. For each point in a plane radiating spherical waves there is another point in the same plane, separated by a distance of λ/2, that scatters in opposite phase; and the waves radiated from these two centers cancel in the transverse direction. Thus, in a particle with dimensions comparable or larger than λ almost no light is scattered laterally. However, these waves interfere constructively along the direction of propagation and add up to a larger wave inside the particle which propagates along the direction of the illuminating beam, and overlaps with it. When the total internal wave reaches the bottom surface of the particle part of its energy is transmitted to the external medium, and the other part is reflected back to the particle material interfering with itself and undergoing further reflections and refractions at the particle walls.

The transmitted waves in the external medium propagate along the forward direction and interfere with the waves scattered from the borders of the particle. If the particle has a circular cross-section relative to the incoming light (such as the spheroids considered in this work - see Fig. 1) the path-length difference between the transmitted wave and the scattered waves coming from the circular perimeter of the particle can only lead to constructive interference at a region along the symmetry axis (z). This can originate jet-like “focal regions” of high electric field magnitude located close to the shadow-side surface of dielectric mesoscopic particles (DMPs) [6

6. Z. Chen, A. Taflove, and V. Backman, “Photonic nanojet enhancement of backscattering of light by nanoparticles: a potential novel visible-light ultramicroscopy technique,” Opt. Express 12(7), 1214–1220 (2004). [CrossRef] [PubMed]

,7

7. C. Li, G. W. Kattawar, P.-W. Zhai, and P. Yang, “Electric and magnetic energy density distributions inside and outside dielectric particles illuminated by a plane electromagnetic wave,” Opt. Express 13(12), 4554–4559 (2005). [CrossRef] [PubMed]

,10

10. S. Lecler, Y. Takakura, and P. Meyrueis, “Properties of a three-dimensional photonic jet,” Opt. Lett. 30(19), 2641–2643 (2005). [CrossRef] [PubMed]

12

12. H. J. Münzer, M. Mosbacher, M. Bertsch, J. Zimmermann, P. Leiderer, and J. Boneberg, “Local field enhancement effects for nanostructuring of surfaces,” J. Microsc. 202(1), 129–135 (2001). [CrossRef] [PubMed]

,16

16. J. P. Barton, “Internal and near-surface electromagnetic fields for a spheroidal particle with arbitrary illumination,” Appl. Opt. 34(24), 5542–5551 (1995). [CrossRef] [PubMed]

,18

18. A. V. Itagi and W. A. Challener, “Optics of photonic nanojets,” J. Opt. Soc. Am. A 22(12), 2847–2858 (2005). [CrossRef] [PubMed]

20

20. P. Ferrand, J. Wenger, A. Devilez, M. Pianta, B. Stout, N. Bonod, E. Popov, and H. Rigneault, “Direct imaging of photonic nanojets,” Opt. Express 16(10), 6930–6940 (2008). [CrossRef] [PubMed]

].

Figure 2 displays the total electric field (Et) intensity distribution produced by DMPs with the same size (Req = 1.5λ) but distinct shape (b/a) and material (Nr). The total field inside the particle is equal to its internal field (Et = Ei) and outside it is the sum of the scattered and incident fields (Et = ES + E0). A small imaginary part is considered in the relative refractive index (kr = 0.01i) to account for light attenuation losses of realistic dielectric materials.

In the central plots of spheres (b/a = 1) it can be seen that there is a good match between our spheroidal code and Mie theory. The bottom plots (Nr = 1.33 + 0.01i) with b/a = 0.5 and b/a = 2 are similar to those in Figs. 13 and 14 of [16

16. J. P. Barton, “Internal and near-surface electromagnetic fields for a spheroidal particle with arbitrary illumination,” Appl. Opt. 34(24), 5542–5551 (1995). [CrossRef] [PubMed]

], respectively, which consider lossless materials (kr = 0) and slightly different Req.

The field patterns of Fig. 2 present two main types of modes. The dominant mode outside the particle is the previously described transverse (EK) wave that propagates in the forward K0 direction towards the far-field. This mode leads to intensity maxima that can be located inside or outside the particle but always on the z axis. The spot of maximum field (|Et|MAX) outside the particle is called the focal point, located at a distance ZMAX from the particle center (see Fig. 1). An increase in ZMAX leads to a broadening of the focal region, and to a general decrease in the scattered field intensities. This is observed in the central and right plots of Fig. 2. The more the focus is separated from the particle surface the larger is its waist and length along the z axis (LZ) [7

7. C. Li, G. W. Kattawar, P.-W. Zhai, and P. Yang, “Electric and magnetic energy density distributions inside and outside dielectric particles illuminated by a plane electromagnetic wave,” Opt. Express 13(12), 4554–4559 (2005). [CrossRef] [PubMed]

,19

19. A. Devilez, B. Stout, N. Bonod, and E. Popov, “Spectral analysis of three-dimensional photonic jets,” Opt. Express 16(18), 14200–14212 (2008). [CrossRef] [PubMed]

]. This occurs because the region of peak constructive interference outside the particle becomes less localized, thus resulting in a wider but less intense focus.

3.1 Lens-like focusing

Interference effects can lead to regions with significant light concentration in front of the particle; as in the sphere (b/a = 1) and oblate (b/a = 2) cases of Fig. 2. A smaller volume of constructive interference implies a brighter focus since there is a higher density of electric energy. This resembles the focusing effect of a macroscopic biconvex lens. However, the characteristics (i.e. intensity, position, spatial extension) of such focal spots are distinct from those predicted by geometrical optics (GO) [11

11. J. Kofler and N. Arnold, “Axially symmetric focusing as a cuspoid diffraction catastrophe: scalar and vector cases and comparison with the theory of Mie,” Phys. Rev. B 73(23), 235401 (2006). [CrossRef]

,12

12. H. J. Münzer, M. Mosbacher, M. Bertsch, J. Zimmermann, P. Leiderer, and J. Boneberg, “Local field enhancement effects for nanostructuring of surfaces,” J. Microsc. 202(1), 129–135 (2001). [CrossRef] [PubMed]

,21

21. S. Lecler, “Light scattering by sub-micrometric particles,” PhD thesis (Louis Pasteur University, 2005).

]. The conventional GO focusing by a lens gives focal spots whose dimensions cannot be confined below the wavelength. With DMPs there is not an explicit focus; instead there is a jet-like tail of the scattered intensity along the forward direction which can have sub-wavelength dimensions [7

7. C. Li, G. W. Kattawar, P.-W. Zhai, and P. Yang, “Electric and magnetic energy density distributions inside and outside dielectric particles illuminated by a plane electromagnetic wave,” Opt. Express 13(12), 4554–4559 (2005). [CrossRef] [PubMed]

,10

10. S. Lecler, Y. Takakura, and P. Meyrueis, “Properties of a three-dimensional photonic jet,” Opt. Lett. 30(19), 2641–2643 (2005). [CrossRef] [PubMed]

,13

13. A. Devilez, N. Bonod, and B. Stout, “Near field dielectric microlenses,” Proc. SPIE 7717, 771708(2010). [CrossRef]

,18

18. A. V. Itagi and W. A. Challener, “Optics of photonic nanojets,” J. Opt. Soc. Am. A 22(12), 2847–2858 (2005). [CrossRef] [PubMed]

,19

19. A. Devilez, B. Stout, N. Bonod, and E. Popov, “Spectral analysis of three-dimensional photonic jets,” Opt. Express 16(18), 14200–14212 (2008). [CrossRef] [PubMed]

]. In GO, the focal distance f of a biconvex lens much larger than λ is obtained with the lensmaker equation [14

14. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University Press, 1999).

]:
f=nrRc22(nr1)1a+nr(Rca)
(2)
where Rc is the lens curvature radius, which is equal to b2/a at the points on the symmetry axis (z = ± a) of a spheroid. It was verified by [10

10. S. Lecler, Y. Takakura, and P. Meyrueis, “Properties of a three-dimensional photonic jet,” Opt. Lett. 30(19), 2641–2643 (2005). [CrossRef] [PubMed]

,21

21. S. Lecler, “Light scattering by sub-micrometric particles,” PhD thesis (Louis Pasteur University, 2005).

] that the values obtained with Eq. (2) only match Mie theory for spheres with radius R>20λ, and the lower is R the higher is the discrepancy. Hence, Eq. (2) is not applicable in the mesoscopic size regime; but it can be useful in a qualitative way to elucidate the dependence of the focal distance with the particle parameters. It is straightforward to deduce from Eq. (2) that f decreases with nr and increases with the spheroid aspect ratio (b/a). The same tendencies are observed (see Fig. 2) relative to ZMAX with mesoscopic spheroids [16

16. J. P. Barton, “Internal and near-surface electromagnetic fields for a spheroidal particle with arbitrary illumination,” Appl. Opt. 34(24), 5542–5551 (1995). [CrossRef] [PubMed]

]. However, the values of ZMAX in the mesoscopic regime are lower than those of f resulting from Eq. (2). The GO lens equation therefore constitutes an upper limit which can be useful for a first-order prediction of a DMP focal distance [11

11. J. Kofler and N. Arnold, “Axially symmetric focusing as a cuspoid diffraction catastrophe: scalar and vector cases and comparison with the theory of Mie,” Phys. Rev. B 73(23), 235401 (2006). [CrossRef]

,12

12. H. J. Münzer, M. Mosbacher, M. Bertsch, J. Zimmermann, P. Leiderer, and J. Boneberg, “Local field enhancement effects for nanostructuring of surfaces,” J. Microsc. 202(1), 129–135 (2001). [CrossRef] [PubMed]

].

3.2 Application in photovoltaics

The focusing characteristics of the near-field produced by DMPs can be of interest for light management in a growing list of applications, as referred in Section 1. The authors are particularly interested in the implementation of arrays of optimally designed DMPs in photovoltaic (PV) devices [8

8. J. Grandidier, D. M. Callahan, J. N. Munday, and H. A. Atwater, “Thin-film solar cells: light absorption enhancement in thin-film solar cells using whispering gallery modes in dielectric nanospheres,” Adv. Mater. (Deerfield Beach Fla.) 23(10), 1171 (2011). [CrossRef]

,36

36. H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9(3), 205–213 (2010). [CrossRef] [PubMed]

]. The aim is to use such scatterers as “mesoscopic lenses” to improve absorption in the photo-active layers of solar cells. This would enable a reduction in the amount of expensive PV material, an improvement in the conditions for charge carrier collection and a raise in the efficiency by virtue of the concentrated energy density in the PV medium [37

37. C. Hägglund and B. Kasemo, “Nanoparticle plasmonics for 2D-photovoltaics: mechanisms, optimization, and limits,” Opt. Express 17(14), 11944–11957 (2009). [CrossRef] [PubMed]

]. This is particularly suited for light trapping and concentration in novel solar cell concepts using quantum dots (e.g. intermediate band solar cells [31

31. M. J. Mendes, A. Luque, I. Tobias, and A. Marti, “Plasmonic light enhancement in the near-field of metallic nanospheroids for application in intermediate band solar cells,” Appl. Phys. Lett. 95(7), 071105 (2009). [CrossRef]

,38

38. A. Luque, A. Marti, M. J. Mendes, and I. Tobias, “Light absorption in the near field around surface plasmon polaritons,” J. Appl. Phys. 104(11), 113118 (2008). [CrossRef]

], hot carrier cells, all silicon tandem cells, etc.). DMPs should allow a significant enhancement in the absorption of a single layer of dots within a broad sunlight wavelength range.

Depending on the particular cell architecture, different strategies can be used to position the DMP array in order to take most profit from its forward scattered light. The easiest and cheapest design for practical implementation is to place the particles on the top surface of the cell [8

8. J. Grandidier, D. M. Callahan, J. N. Munday, and H. A. Atwater, “Thin-film solar cells: light absorption enhancement in thin-film solar cells using whispering gallery modes in dielectric nanospheres,” Adv. Mater. (Deerfield Beach Fla.) 23(10), 1171 (2011). [CrossRef]

,37

37. C. Hägglund and B. Kasemo, “Nanoparticle plasmonics for 2D-photovoltaics: mechanisms, optimization, and limits,” Opt. Express 17(14), 11944–11957 (2009). [CrossRef] [PubMed]

]. However, a higher interaction (photocurrent generation) between the PV medium and the scatterers’ near-field can be achieved by placing the particles inside the cell material [31

31. M. J. Mendes, A. Luque, I. Tobias, and A. Marti, “Plasmonic light enhancement in the near-field of metallic nanospheroids for application in intermediate band solar cells,” Appl. Phys. Lett. 95(7), 071105 (2009). [CrossRef]

,39

39. J. Y. Lee and P. Peumans, “The origin of enhanced optical absorption in solar cells with metal nanoparticles embedded in the active layer,” Opt. Express 18(10), 10078–10087 (2010). [CrossRef] [PubMed]

], for instance in the depletion region of the cell p-n junction. It is also possible to have several layers of DMP arrays positioned over distinct layers of a PV device, each array composed of particles designed to concentrate light the way that best suits each layer.

It is important to target the light focusing properties of these structures to the lower energy part of the solar spectrum, the infrared (IR); since the lower energy photons are harder to absorb by typical PV semiconductor materials. The higher energy photons of the visible (VIS) and ultraviolet (UV) range are easily absorbed in the first micrometers of the solar cell material, since their energy is sufficiently higher than the semiconductor bandgap. For these reasons, the complex refractive indices Nr considered in this paper have values that can be physically attainable by common dielectric/semiconductor materials in the IR range. In the optimization studies (Section 4) the domain of the real part nr was restricted between 1 and 4; and the imaginary part was taken to be kr = 0.01. Nevertheless, the main results are also given for a smaller kr = 0.001 to analyse the effect of lower light attenuation.

PV applications also have to account for the unpolarized nature of the illuminating sunlight. The incident field E0 can take any orientation orthogonal to the propagation direction K0. Therefore, it is important to have a scatterer shape that allows a response independent of such orientations. That is the case of the spheroids considered here, whose symmetry axis is K0.

4. Results of optimization studies

4.1 Optimization of local electric field intensity

The first function to be optimized is the maximum of the scattered field intensity (|ES|2/|E0|2) at any point in the external medium in front of the particle. Three particular cases are considered:

  • 1) Sphere (b/a = 1), using R and nr as variables
  • 2) Spheroid with nr = 1.33, using Req and b/a as variables
  • 3) General spheroid using Req, b/a and nr as variables

The results obtained in each case are presented in the following sub-sections and summarized in Table 1

Table 1. Characteristics of the Optimal Spheroids That Maximize the Scattered Field Intensitya

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. For the three cases the optimization converged to localized focal regions at the bottom surface of the particle in the z axis (ZMAX = a). That is the spot where the focus can be more confined outside the particle; thus allowing the highest possible field magnitude in the external medium.

4.1.1 Sphere case

We start by presenting the results for the maximization of the scattered intensity produced by spherical particles (b/a = 1) with variable radius R and nr. Since the geometry is spherical the fields were computed with Mie theory to save computational time.

For a given R, the lower is nr the higher is the distance of the focus to the particle center, as predicted by the lens Eq. (2). The focus can therefore be located inside or outside the particle depending on the value of nr. As observed in [10

10. S. Lecler, Y. Takakura, and P. Meyrueis, “Properties of a three-dimensional photonic jet,” Opt. Lett. 30(19), 2641–2643 (2005). [CrossRef] [PubMed]

,21

21. S. Lecler, “Light scattering by sub-micrometric particles,” PhD thesis (Louis Pasteur University, 2005).

], the maximum field intensity outside the particle always occurs for the nr index that sets the focal point right at the particle surface (ZMAX = R). According to Eq. (2), for macroscopic spheres the value of nr that sets f = R is nr = 2.0. With mesoscopic spheres such nr value is too high since it places the maximum intensity inside the particle [10

10. S. Lecler, Y. Takakura, and P. Meyrueis, “Properties of a three-dimensional photonic jet,” Opt. Lett. 30(19), 2641–2643 (2005). [CrossRef] [PubMed]

,11

11. J. Kofler and N. Arnold, “Axially symmetric focusing as a cuspoid diffraction catastrophe: scalar and vector cases and comparison with the theory of Mie,” Phys. Rev. B 73(23), 235401 (2006). [CrossRef]

], as shown in the central top plot of Fig. 2. A smaller nr = 1.33 (see central bottom plot of Fig. 2) is already too low because it sets the maximum Et outside the particle separated from its surface. The optimal values of nr that place the focus exactly at z = -R are plotted in Fig. 3(a)
Fig. 3 (a) Left axis (black curves) - Maximum values of |Et|2 (in units of |E0|2) outside the particle, as a function of the sphere radius (R), for kr = 0.01 and kr = 0.001. The red circle is at the value obtained with the optimization algorithm considering kr = 0.01. The line segment beneath the circle marks the interval where |Et|2 remains within a 10% difference from the maximum value. Right axis (blue curve) - Optimal nr values computed with kr = 0.01. Similar values are obtained with kr = 0.001. (b) Left - |Et|2 distribution, in logarithmic scale, for the optimal sphere parameters (R,nr) indicated in the plot. The distribution is computed on the same yz plane as those of Fig. 2. Right - |Et|2 along the z axis, in linear scale, for kr = 0.01 (black line) and kr = 0.001 (red).
(dashed line) for R ranging from 0.5λ to 10λ. The optimal nr is around 3 within such mesoscopic sizes, and tends to 2.0 as the size approaches the macroscopic GO regime [10

10. S. Lecler, Y. Takakura, and P. Meyrueis, “Properties of a three-dimensional photonic jet,” Opt. Lett. 30(19), 2641–2643 (2005). [CrossRef] [PubMed]

,11

11. J. Kofler and N. Arnold, “Axially symmetric focusing as a cuspoid diffraction catastrophe: scalar and vector cases and comparison with the theory of Mie,” Phys. Rev. B 73(23), 235401 (2006). [CrossRef]

].

The field intensities at the focal point (|Et|2 MAX) are shown in Fig. 3(a) for kr = 0.01 and kr = 0.001 (solid lines) using the optimal nr. The attenuation index kr has a quite significant impact in the peak field magnitudes that can be obtained in these optimal cases. If absorption were neglected (kr = 0), the bigger the sphere size the stronger could be the focal point intensity. The effect of absorption opposes this tendency, because the bigger the particle the more energy is dissipated in its volume and thus the less energy is available for the focus. As shown in the solid curves of Fig. 3(a), for low sizes |Et|2 MAX increases with R. However, for sufficiently big R, the effect of absorption dominates over scattering and |Et|2 MAX starts decreasing with the size. The higher is kr the lower is the optimal R value at which the maximum in the curves occurs. For kr = 0.01 such maximum is at Ropt = 5.785λ, which matches the result obtained with the optimization algorithm. The electric field intensity distribution of this case is plotted in Fig. 3(b). At the particle surface (z = -R) the intensity is |Et|2 MAX = 92.06, but with lower kr = 0.001 it is about 3.5 times higher (|Et|2 MAX = 328.3).

A DMP fabricated with the optimal parameters of Fig. 3(b) would produce a focal intensity above 0.9|Et| 2 MAX for any wavelength within λ ± 0.3λ. This broad frequency width is due to the fact that the kr = 0.01 curve in Fig. 3(a) is rather flat around Ropt (as marked by the line segment beneath the curve), and nr remains approximately constant in that interval.

4.1.2 Spheroid with nr = 1.33

A relative refractive index of 1.33 is often adopted in theoretical studies of particle scattering [4

4. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH, 2004).

,7

7. C. Li, G. W. Kattawar, P.-W. Zhai, and P. Yang, “Electric and magnetic energy density distributions inside and outside dielectric particles illuminated by a plane electromagnetic wave,” Opt. Express 13(12), 4554–4559 (2005). [CrossRef] [PubMed]

,15

15. L.-W. Li, X.-K. Kang, and M.-S. Leong, Spheroidal Wave Functions in Electromagnetic Theory (John Wiley & Sons, Inc., 2002).

,16

16. J. P. Barton, “Internal and near-surface electromagnetic fields for a spheroidal particle with arbitrary illumination,” Appl. Opt. 34(24), 5542–5551 (1995). [CrossRef] [PubMed]

,23

23. S. Asano and G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14(1), 29–49 (1975). [PubMed]

,24

24. E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186(2), 705–714 (1973). [CrossRef]

,28

28. J. P. Barton, “Internal and near-surface electromagnetic fields for an absorbing spheroidal particle with arbitrary illumination,” Appl. Opt. 34(36), 8472–8473 (1995). [CrossRef] [PubMed]

]. This is the refractive index of water in air, but can also match other possible particle/medium material combinations. For instance, a germanium particle in a gallium arsenide or silicon medium corresponds to a relative index close to that value in the infrared, which can be interesting for PV applications. In this case there is a small imaginary part in the refractive index on the order of kr = 0.01i, as considered in this work.

The optimization converged to a prolate geometry since only with b/a<1 could the focus be positioned at the particle surface given such low nr. If the shape were spherical or oblate the focal region would be located away from the surface (see bottom plots of Fig. 2) exhibiting higher length but lower peak intensity. For comparison, the maximum intensity produced by a sphere with the same Req and Nr as the prolate of Fig. 4 is |Et|2 = 33.9 at ZMAX = 4.29λ.

4.1.3 General spheroidal case

The optimization is now extended to the general case of a spheroidal particle with variable size (Req), aspect ratio (b/a) and real part of the relative index (nr). The domain of variation of nr was restricted to nr≤4.0, since most pure or compound dielectric materials have refractive indices lower than such limit at low optical frequencies, as referred in Section 3.2. The imaginary part of Nr is kept fixed (kr = 0.01).

Increasing nr also shifts the maximum field intensity towards the particle interior. Therefore, usually DMPs with big nr exhibit high internal fields but small scattered intensities in the external medium. This is, for instance, the case of spheres with any nr above the optimal values given in Fig. 3(a). Nevertheless, with spheroids there is an additional degree of freedom, the aspect ratio, which enables to keep the focus outside the particle while still using a high refractive index. This constitutes one of the main advantages of spheroids for near-field light concentration, and a key point of this paper. As displayed in Fig. 2, the effect of the aspect ratio on the location of the focal spot is opposite to the effect of nr. Thus, the focus can be kept at the bottom point of the particle (ZMAX = a) if b/a is increased together with nr. This possibility allows the achievement of near-field intensities of more than two orders of magnitude (relative to the incident intensity) and with spheroid sizes lower than those of Sections 4.1.1 and 4.1.2.

The pronounced interference caused by this high nr = 4.0 results in a more confined focal spot than that of the field distributions in Figs. 3(b) and 4 (see LZ values in Table 1). The focal peak is extremely localized in this case, having sub-wavelength dimensions both in the z (LZ = 0.13λ) and y (LY = 0.54λ - see inset in Fig. 5) directions. Spherical DMPs can also produce external focal regions that are sub-wavelength confined in the transverse xy directions, but not along the forward z axis when illuminated by a plane-wave [10

10. S. Lecler, Y. Takakura, and P. Meyrueis, “Properties of a three-dimensional photonic jet,” Opt. Lett. 30(19), 2641–2643 (2005). [CrossRef] [PubMed]

,11

11. J. Kofler and N. Arnold, “Axially symmetric focusing as a cuspoid diffraction catastrophe: scalar and vector cases and comparison with the theory of Mie,” Phys. Rev. B 73(23), 235401 (2006). [CrossRef]

,17

17. A. Devilez, N. Bonod, J. Wenger, D. Gérard, B. Stout, H. Rigneault, and E. Popov, “Three-dimensional subwavelength confinement of light with dielectric microspheres,” Opt. Express 17(4), 2089–2094 (2009). [CrossRef] [PubMed]

,19

19. A. Devilez, B. Stout, N. Bonod, and E. Popov, “Spectral analysis of three-dimensional photonic jets,” Opt. Express 16(18), 14200–14212 (2008). [CrossRef] [PubMed]

,20

20. P. Ferrand, J. Wenger, A. Devilez, M. Pianta, B. Stout, N. Bonod, E. Popov, and H. Rigneault, “Direct imaging of photonic nanojets,” Opt. Express 16(10), 6930–6940 (2008). [CrossRef] [PubMed]

]. With oblate DMPs having a high nr it is possible to achieve sub-wavelength confinement with plane-wave illumination both in the transverse and forward directions.

4.2 Optimization of electric intensity over near-field region

In view of the above, the area integral in Eq. (3) is normalized by the yz area of the particle (πab), in order to beneficiate smaller sizes (Req) in the optimization procedure.

In the following sub-sections we present the results obtained with the optimization of Eq. (3) considering the same 3 cases (sphere, spheroid with nr = 1.33 and general spheroid) of Section 4.1. The main results are summarized in Table 2

Table 2. Characteristics of the Optimal Spheroidal Parameters That Maximize Function Pa

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.

4.2.1 Sphere case

As in Section 4.1.1, the calculations for spherical particles were performed with Mie theory for faster computation. In this case the aspect ratio is fixed (b/a = 1), so the quantity P (Eq. (3)) becomes a function of only two parameters: R and nr. As before, the imaginary part of the refractive index is kept fixed (kr = 0.01).

The global P maximum obtained with the optimization is marked by a white circle in Fig. 6(a). This peak is rather flat along the R axis close to the optimal value Ropt = 1.2λ, as in the curve of kr = 0.01 in Fig. 3(a). Therefore, a sphere fabricated with the optimal parameters produces a P value above 0.9PMAX at any wavelength within λ ± 0.2λ.

4.2.2 Spheroid with nr = 1.33

The present case with a low relative refractive index Nr = 1.33 + 0.01i can be interesting for applications that benefit from a small material contrast between scatterer and surrounding medium. That is the case of PV applications in which the particles are embedded inside the PV medium. In that situation the lower is Nr the less charge carrier trapping and recombination occurs at the particle surface.

Figure 6(b) represents function P(Req,b/a) in a region around its maximum value obtained with the optimization. Function P decreases with Req due to the normalization of the integral in Eq. (3) by the spheroid area. The aspect ratio (b/a) shifts the position and spatial extension of the focal region. High aspect ratios not only reduce the overall scattered intensities but can also place the jet-like focus too far in z, outside the integration region limited by the dashed line in Fig. 1. If b/a is too small the focus moves close to the particle and becomes more spatially confined; which may produce higher peak intensities at the focal spot (such as in Fig. 4) but a lower value of the integral of ES over the chosen semi-circular region.

4.2.3 General spheroidal case

As in Section 4.1.3, the algorithm converges to the highest allowed refractive index (nr = 4.0) since the higher is nr the higher can be the scattered field outside the particle. However, the increase of nr has to be accompanied by an increase in b/a or else the focal region moves too close to the particle (or even to its interior) and loses its extension over the near-field region in front. Keeping the highest possible nr = 4.0 and large b/a, the algorithm converges to small particle sizes (Req<λ) in order to achieve a high value of the integral in Eq. (3) with a low value of the spheroid yz area (πab) in the denominator.

The spheroid parameters that allow a maximum in P(Req,b/a,nr) produce the field distribution shown in Fig. 7
Fig. 7 Left - |Et|2 distribution for the optimal parameters that maximize function P(Req,b/a,nr). The spheroid semi-axes are: a = 0.079λ and b = 2.271λ. Right - |Et|2 profiles along the z axis for kr = 0.01 (black line) and kr = 0.001 (red).
. An oblate with such extreme elongation presents a long forward scattering lobe; quite distinct from the point focus obtained with the moderately elongated oblate in Fig. 5. It can be seen in the right plot of Fig. 7 that there is almost no difference between the field intensities with kr = 0.01 and kr = 0.001, due to the small particle size.

Even higher P values could be achieved with higher values of the size parameter Ci. As referred in Section 2.2, the maximum allowed value of |Ci| was limited to 60 in order to avoid inaccuracies in the calculation of the radial spheroidal harmonics.

5. Conclusions and final remarks

The relation between the geometry and material of a spheroidal scatterer and its resulting field distribution may be very complex; especially in the mesoscopic regime where it can only be analytically determined by fully solving Maxwell’s equations using the continuity BCs across the particle surface. This can be accomplished by the spheroidal coordinate separation-of-variables method used here, described in Section 2 and further detailed in [3

3. M. J. Mendes, I. Tobías, A. Martí, and A. Luque, “Near-field scattering by dielectric spheroidal particles with sizes on the order of the illuminating wavelength,” J. Opt. Soc. Am. B 27(6), 1221–1231 (2010). [CrossRef]

,15

15. L.-W. Li, X.-K. Kang, and M.-S. Leong, Spheroidal Wave Functions in Electromagnetic Theory (John Wiley & Sons, Inc., 2002).

].

With the first function, the maximum intensity in the external medium is achieved by placing the focus right at the point of intersection between the particle surface and the z axis. The higher nr the more intense and confined can the focus be. However, the particle oblateness has to increase with nr or else the light would be focused in the interior of the particle. An optimal adjust of b/a and nr can lead to electric field intensities at the particle surface of more than 2 orders of magnitude higher than the incident intensity, with a particle size close to λ (see Fig. 5). Such pronounced intensities are achieved due to a sub-wavelength confinement of the focal peak in all directions, which is possible in plane-wave illumination with an oblate geometry and high nr. This is one of the main advantages of going beyond the simple spherical geometry with the use of spheroidal DMPs; and can be interesting for applications with localized absorbing centers, such as quantum-dot solar cells [31

31. M. J. Mendes, A. Luque, I. Tobias, and A. Marti, “Plasmonic light enhancement in the near-field of metallic nanospheroids for application in intermediate band solar cells,” Appl. Phys. Lett. 95(7), 071105 (2009). [CrossRef]

].

Pronounced near-field light amplification can also be achieved in the electrostatic regime using the surface plasmon resonance (SPR) of metallic nanoparticles (MNPs) much smaller than λ [5

5. H. Mertens, A. F. Koenderink, and A. Polman, “Plasmon-enhanced luminescence near noble-metal nanospheres: comparison of exact theory and an improved Gersten and Nitzan model,” Phys. Rev. B 76(11), 115123 (2007). [CrossRef]

,31

31. M. J. Mendes, A. Luque, I. Tobias, and A. Marti, “Plasmonic light enhancement in the near-field of metallic nanospheroids for application in intermediate band solar cells,” Appl. Phys. Lett. 95(7), 071105 (2009). [CrossRef]

,36

36. H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9(3), 205–213 (2010). [CrossRef] [PubMed]

38

38. A. Luque, A. Marti, M. J. Mendes, and I. Tobias, “Light absorption in the near field around surface plasmon polaritons,” J. Appl. Phys. 104(11), 113118 (2008). [CrossRef]

]. As shown here, the electric intensity enhancement that can be obtained with mesoscopic spheroids is as high as that predicted at the surface of MNPs sustaining SPRs (~100E0 2) in the infrared. Nevertheless, DMPs may be more suited for certain applications, such as PV, due to the following additional advantages [3

3. M. J. Mendes, I. Tobías, A. Martí, and A. Luque, “Near-field scattering by dielectric spheroidal particles with sizes on the order of the illuminating wavelength,” J. Opt. Soc. Am. B 27(6), 1221–1231 (2010). [CrossRef]

]:

First, since they are composed of dielectric material their inclusion in the interior of the photo-active medium of a solar cell should cause less current degradation (due to charge carrier recombination at the particle surface) than MNPs [8

8. J. Grandidier, D. M. Callahan, J. N. Munday, and H. A. Atwater, “Thin-film solar cells: light absorption enhancement in thin-film solar cells using whispering gallery modes in dielectric nanospheres,” Adv. Mater. (Deerfield Beach Fla.) 23(10), 1171 (2011). [CrossRef]

,37

37. C. Hägglund and B. Kasemo, “Nanoparticle plasmonics for 2D-photovoltaics: mechanisms, optimization, and limits,” Opt. Express 17(14), 11944–11957 (2009). [CrossRef] [PubMed]

]. Secondly, the scattering pattern produced by MNPs has a single dipolar (first-order) mode. DMPs excite additional higher-order modes thereby allowing very distinct near-field profiles which can be adapted to different structures of the device absorber. The intensity, spatial extension and position of the peak field intensity can be adjusted by tuning the particle parameters, as explained in this paper. Thirdly, besides higher spatial extension, there is also a higher frequency extension of the electric field peaks produced by DMPs. The SPR resonance produces sharp intensity peaks having a HWHM of about 0.1λ. With DMPs the peaks can be broader in wavelength; the examples analysed here have half widths of 0.2-0.3λ relative to only −10% of the maximum.

Appendix - Nelder and Mead optimization algorithm

It is important to define well the domain (lower and upper limits) of each n variable, in order to avoid searching in physically forbidden regions, or where it is known that F has undesired values. The optimizations performed in this work use up to 3 variables, the spheroid parameters: Req, b/a and nr. The domain allowed for the simplex was defined according to a set of lower limits (a,b>0 and nr>1) and upper limits (nr≤4.0 and |Ci|<60). The upper limit for nr was taken to be 4.0 for the reasons mentioned in Section 3.2. The upper bound set for |Ci| is due to computational constraints, as referred in Section 2.2.

Acknowledgments

This work was supported by the European Commission’s IBPOWER project (Grant No. 211640), by the Regional Government of Madrid within the project NUMANCIA-2 (Grant No. S2009/ENE1477) and by the program DENQUIBAND (Grant No. PLE2009-0045) funded by the Spanish “Ministerio de Ciencia e Innovación”. MJM also acknowledges “Universidad Politécnica de Madrid” for the scholarship Beca de Doctorado Homologada.

References and links

1.

C. Girard and A. Dereux, “Near-field optics theories,” Rep. Prog. Phys. 59(5), 657–699 (1996). [CrossRef]

2.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University Press, 2006).

3.

M. J. Mendes, I. Tobías, A. Martí, and A. Luque, “Near-field scattering by dielectric spheroidal particles with sizes on the order of the illuminating wavelength,” J. Opt. Soc. Am. B 27(6), 1221–1231 (2010). [CrossRef]

4.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH, 2004).

5.

H. Mertens, A. F. Koenderink, and A. Polman, “Plasmon-enhanced luminescence near noble-metal nanospheres: comparison of exact theory and an improved Gersten and Nitzan model,” Phys. Rev. B 76(11), 115123 (2007). [CrossRef]

6.

Z. Chen, A. Taflove, and V. Backman, “Photonic nanojet enhancement of backscattering of light by nanoparticles: a potential novel visible-light ultramicroscopy technique,” Opt. Express 12(7), 1214–1220 (2004). [CrossRef] [PubMed]

7.

C. Li, G. W. Kattawar, P.-W. Zhai, and P. Yang, “Electric and magnetic energy density distributions inside and outside dielectric particles illuminated by a plane electromagnetic wave,” Opt. Express 13(12), 4554–4559 (2005). [CrossRef] [PubMed]

8.

J. Grandidier, D. M. Callahan, J. N. Munday, and H. A. Atwater, “Thin-film solar cells: light absorption enhancement in thin-film solar cells using whispering gallery modes in dielectric nanospheres,” Adv. Mater. (Deerfield Beach Fla.) 23(10), 1171 (2011). [CrossRef]

9.

L. E. McNeil, A. R. Hanuska, and R. H. French, “Near-field scattering from red pigment particles: absorption and spectral dependence,” J. Appl. Phys. 89(3), 1898–1906 (2001). [CrossRef]

10.

S. Lecler, Y. Takakura, and P. Meyrueis, “Properties of a three-dimensional photonic jet,” Opt. Lett. 30(19), 2641–2643 (2005). [CrossRef] [PubMed]

11.

J. Kofler and N. Arnold, “Axially symmetric focusing as a cuspoid diffraction catastrophe: scalar and vector cases and comparison with the theory of Mie,” Phys. Rev. B 73(23), 235401 (2006). [CrossRef]

12.

H. J. Münzer, M. Mosbacher, M. Bertsch, J. Zimmermann, P. Leiderer, and J. Boneberg, “Local field enhancement effects for nanostructuring of surfaces,” J. Microsc. 202(1), 129–135 (2001). [CrossRef] [PubMed]

13.

A. Devilez, N. Bonod, and B. Stout, “Near field dielectric microlenses,” Proc. SPIE 7717, 771708(2010). [CrossRef]

14.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University Press, 1999).

15.

L.-W. Li, X.-K. Kang, and M.-S. Leong, Spheroidal Wave Functions in Electromagnetic Theory (John Wiley & Sons, Inc., 2002).

16.

J. P. Barton, “Internal and near-surface electromagnetic fields for a spheroidal particle with arbitrary illumination,” Appl. Opt. 34(24), 5542–5551 (1995). [CrossRef] [PubMed]

17.

A. Devilez, N. Bonod, J. Wenger, D. Gérard, B. Stout, H. Rigneault, and E. Popov, “Three-dimensional subwavelength confinement of light with dielectric microspheres,” Opt. Express 17(4), 2089–2094 (2009). [CrossRef] [PubMed]

18.

A. V. Itagi and W. A. Challener, “Optics of photonic nanojets,” J. Opt. Soc. Am. A 22(12), 2847–2858 (2005). [CrossRef] [PubMed]

19.

A. Devilez, B. Stout, N. Bonod, and E. Popov, “Spectral analysis of three-dimensional photonic jets,” Opt. Express 16(18), 14200–14212 (2008). [CrossRef] [PubMed]

20.

P. Ferrand, J. Wenger, A. Devilez, M. Pianta, B. Stout, N. Bonod, E. Popov, and H. Rigneault, “Direct imaging of photonic nanojets,” Opt. Express 16(10), 6930–6940 (2008). [CrossRef] [PubMed]

21.

S. Lecler, “Light scattering by sub-micrometric particles,” PhD thesis (Louis Pasteur University, 2005).

22.

E. Hecht, Optics, 4th ed. (Addison Wesley, 2001).

23.

S. Asano and G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14(1), 29–49 (1975). [PubMed]

24.

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186(2), 705–714 (1973). [CrossRef]

25.

J. C. Ravey and P. Mazeron, “Light-scattering in the physical optics approximation—application to large spheroids,” J. Opt. 13(5), 273–282 (1982). [CrossRef]

26.

J. P. Barton, “Internal, near-surface, and scattered electromagnetic fields for a layered spheroid with arbitrary illumination,” Appl. Opt. 40(21), 3598–3607 (2001). [CrossRef] [PubMed]

27.

N. Richard, “Analysis of polarization effects on nanoscopic objects in the near-field optics,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(2), 026602 (2001). [CrossRef] [PubMed]

28.

J. P. Barton, “Internal and near-surface electromagnetic fields for an absorbing spheroidal particle with arbitrary illumination,” Appl. Opt. 34(36), 8472–8473 (1995). [CrossRef] [PubMed]

29.

T. D. Milster, “Near-field optical data storage: avenues for improved performance,” Opt. Eng. 40(10), 2255–2260 (2001). [CrossRef]

30.

A. Devilez, B. Stout, and N. Bonod, “Compact metallo-dielectric optical antenna for ultra directional and enhanced radiative emission,” ACS Nano 4(6), 3390–3396 (2010). [CrossRef] [PubMed]

31.

M. J. Mendes, A. Luque, I. Tobias, and A. Marti, “Plasmonic light enhancement in the near-field of metallic nanospheroids for application in intermediate band solar cells,” Appl. Phys. Lett. 95(7), 071105 (2009). [CrossRef]

32.

C. Flammer, Spheroidal Wave Functions (Stanford University Press, 1957).

33.

L. W. Li, M. S. Leong, T. S. Yeo, P. S. Kooi, and K. Y. Tan, “Computations of spheroidal harmonics with complex arguments: a review with an algorithm,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(5), 6792–6806 (1998). [CrossRef]

34.

P. Kirby, “Calculation of spheroidal wave functions,” Comput. Phys. Commun. 175(7), 465–472 (2006). [CrossRef]

35.

R. Kirby, “Calculation of radial prolate spheroidal wave functions of the second kind,” Comput. Phys. Commun. 181(3), 514–519 (2010). [CrossRef]

36.

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9(3), 205–213 (2010). [CrossRef] [PubMed]

37.

C. Hägglund and B. Kasemo, “Nanoparticle plasmonics for 2D-photovoltaics: mechanisms, optimization, and limits,” Opt. Express 17(14), 11944–11957 (2009). [CrossRef] [PubMed]

38.

A. Luque, A. Marti, M. J. Mendes, and I. Tobias, “Light absorption in the near field around surface plasmon polaritons,” J. Appl. Phys. 104(11), 113118 (2008). [CrossRef]

39.

J. Y. Lee and P. Peumans, “The origin of enhanced optical absorption in solar cells with metal nanoparticles embedded in the active layer,” Opt. Express 18(10), 10078–10087 (2010). [CrossRef] [PubMed]

40.

C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms—Introduction to Quantum Electrodynamics (Wiley-Interscience, 1997).

41.

P. Mazeron and S. Muller, “Dielectric or absorbing particles: EM surface fields and scattering,” J. Opt. 29(2), 68–77 (1998). [CrossRef]

42.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optim. 9(1), 112–147 (1998). [CrossRef]

43.

M. J. Mendes, H. K. Schmidt, and M. Pasquali, “Brownian dynamics simulations of single-wall carbon nanotube separation by type using dielectrophoresis,” J. Phys. Chem. B 112(25), 7467–7477 (2008). [CrossRef] [PubMed]

OCIS Codes
(040.5350) Detectors : Photovoltaic
(260.2110) Physical optics : Electromagnetic optics
(290.5850) Scattering : Scattering, particles

ToC Category:
Scattering

History
Original Manuscript: June 3, 2011
Revised Manuscript: July 4, 2011
Manuscript Accepted: July 6, 2011
Published: August 9, 2011

Citation
Manuel J. Mendes, Ignacio Tobías, Antonio Martí, and Antonio Luque, "Light concentration in the near-field of dielectric spheroidal particles with mesoscopic sizes," Opt. Express 19, 16207-16222 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-17-16207


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References

  1. C. Girard and A. Dereux, “Near-field optics theories,” Rep. Prog. Phys. 59(5), 657–699 (1996). [CrossRef]
  2. L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University Press, 2006).
  3. M. J. Mendes, I. Tobías, A. Martí, and A. Luque, “Near-field scattering by dielectric spheroidal particles with sizes on the order of the illuminating wavelength,” J. Opt. Soc. Am. B 27(6), 1221–1231 (2010). [CrossRef]
  4. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH, 2004).
  5. H. Mertens, A. F. Koenderink, and A. Polman, “Plasmon-enhanced luminescence near noble-metal nanospheres: comparison of exact theory and an improved Gersten and Nitzan model,” Phys. Rev. B 76(11), 115123 (2007). [CrossRef]
  6. Z. Chen, A. Taflove, and V. Backman, “Photonic nanojet enhancement of backscattering of light by nanoparticles: a potential novel visible-light ultramicroscopy technique,” Opt. Express 12(7), 1214–1220 (2004). [CrossRef] [PubMed]
  7. C. Li, G. W. Kattawar, P.-W. Zhai, and P. Yang, “Electric and magnetic energy density distributions inside and outside dielectric particles illuminated by a plane electromagnetic wave,” Opt. Express 13(12), 4554–4559 (2005). [CrossRef] [PubMed]
  8. J. Grandidier, D. M. Callahan, J. N. Munday, and H. A. Atwater, “Thin-film solar cells: light absorption enhancement in thin-film solar cells using whispering gallery modes in dielectric nanospheres,” Adv. Mater. (Deerfield Beach Fla.) 23(10), 1171 (2011). [CrossRef]
  9. L. E. McNeil, A. R. Hanuska, and R. H. French, “Near-field scattering from red pigment particles: absorption and spectral dependence,” J. Appl. Phys. 89(3), 1898–1906 (2001). [CrossRef]
  10. S. Lecler, Y. Takakura, and P. Meyrueis, “Properties of a three-dimensional photonic jet,” Opt. Lett. 30(19), 2641–2643 (2005). [CrossRef] [PubMed]
  11. J. Kofler and N. Arnold, “Axially symmetric focusing as a cuspoid diffraction catastrophe: scalar and vector cases and comparison with the theory of Mie,” Phys. Rev. B 73(23), 235401 (2006). [CrossRef]
  12. H. J. Münzer, M. Mosbacher, M. Bertsch, J. Zimmermann, P. Leiderer, and J. Boneberg, “Local field enhancement effects for nanostructuring of surfaces,” J. Microsc. 202(1), 129–135 (2001). [CrossRef] [PubMed]
  13. A. Devilez, N. Bonod, and B. Stout, “Near field dielectric microlenses,” Proc. SPIE 7717, 771708(2010). [CrossRef]
  14. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University Press, 1999).
  15. L.-W. Li, X.-K. Kang, and M.-S. Leong, Spheroidal Wave Functions in Electromagnetic Theory (John Wiley & Sons, Inc., 2002).
  16. J. P. Barton, “Internal and near-surface electromagnetic fields for a spheroidal particle with arbitrary illumination,” Appl. Opt. 34(24), 5542–5551 (1995). [CrossRef] [PubMed]
  17. A. Devilez, N. Bonod, J. Wenger, D. Gérard, B. Stout, H. Rigneault, and E. Popov, “Three-dimensional subwavelength confinement of light with dielectric microspheres,” Opt. Express 17(4), 2089–2094 (2009). [CrossRef] [PubMed]
  18. A. V. Itagi and W. A. Challener, “Optics of photonic nanojets,” J. Opt. Soc. Am. A 22(12), 2847–2858 (2005). [CrossRef] [PubMed]
  19. A. Devilez, B. Stout, N. Bonod, and E. Popov, “Spectral analysis of three-dimensional photonic jets,” Opt. Express 16(18), 14200–14212 (2008). [CrossRef] [PubMed]
  20. P. Ferrand, J. Wenger, A. Devilez, M. Pianta, B. Stout, N. Bonod, E. Popov, and H. Rigneault, “Direct imaging of photonic nanojets,” Opt. Express 16(10), 6930–6940 (2008). [CrossRef] [PubMed]
  21. S. Lecler, “Light scattering by sub-micrometric particles,” PhD thesis (Louis Pasteur University, 2005).
  22. E. Hecht, Optics, 4th ed. (Addison Wesley, 2001).
  23. S. Asano and G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14(1), 29–49 (1975). [PubMed]
  24. E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186(2), 705–714 (1973). [CrossRef]
  25. J. C. Ravey and P. Mazeron, “Light-scattering in the physical optics approximation—application to large spheroids,” J. Opt. 13(5), 273–282 (1982). [CrossRef]
  26. J. P. Barton, “Internal, near-surface, and scattered electromagnetic fields for a layered spheroid with arbitrary illumination,” Appl. Opt. 40(21), 3598–3607 (2001). [CrossRef] [PubMed]
  27. N. Richard, “Analysis of polarization effects on nanoscopic objects in the near-field optics,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(2), 026602 (2001). [CrossRef] [PubMed]
  28. J. P. Barton, “Internal and near-surface electromagnetic fields for an absorbing spheroidal particle with arbitrary illumination,” Appl. Opt. 34(36), 8472–8473 (1995). [CrossRef] [PubMed]
  29. T. D. Milster, “Near-field optical data storage: avenues for improved performance,” Opt. Eng. 40(10), 2255–2260 (2001). [CrossRef]
  30. A. Devilez, B. Stout, and N. Bonod, “Compact metallo-dielectric optical antenna for ultra directional and enhanced radiative emission,” ACS Nano 4(6), 3390–3396 (2010). [CrossRef] [PubMed]
  31. M. J. Mendes, A. Luque, I. Tobias, and A. Marti, “Plasmonic light enhancement in the near-field of metallic nanospheroids for application in intermediate band solar cells,” Appl. Phys. Lett. 95(7), 071105 (2009). [CrossRef]
  32. C. Flammer, Spheroidal Wave Functions (Stanford University Press, 1957).
  33. L. W. Li, M. S. Leong, T. S. Yeo, P. S. Kooi, and K. Y. Tan, “Computations of spheroidal harmonics with complex arguments: a review with an algorithm,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(5), 6792–6806 (1998). [CrossRef]
  34. P. Kirby, “Calculation of spheroidal wave functions,” Comput. Phys. Commun. 175(7), 465–472 (2006). [CrossRef]
  35. R. Kirby, “Calculation of radial prolate spheroidal wave functions of the second kind,” Comput. Phys. Commun. 181(3), 514–519 (2010). [CrossRef]
  36. H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9(3), 205–213 (2010). [CrossRef] [PubMed]
  37. C. Hägglund and B. Kasemo, “Nanoparticle plasmonics for 2D-photovoltaics: mechanisms, optimization, and limits,” Opt. Express 17(14), 11944–11957 (2009). [CrossRef] [PubMed]
  38. A. Luque, A. Marti, M. J. Mendes, and I. Tobias, “Light absorption in the near field around surface plasmon polaritons,” J. Appl. Phys. 104(11), 113118 (2008). [CrossRef]
  39. J. Y. Lee and P. Peumans, “The origin of enhanced optical absorption in solar cells with metal nanoparticles embedded in the active layer,” Opt. Express 18(10), 10078–10087 (2010). [CrossRef] [PubMed]
  40. C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms—Introduction to Quantum Electrodynamics (Wiley-Interscience, 1997).
  41. P. Mazeron and S. Muller, “Dielectric or absorbing particles: EM surface fields and scattering,” J. Opt. 29(2), 68–77 (1998). [CrossRef]
  42. J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optim. 9(1), 112–147 (1998). [CrossRef]
  43. M. J. Mendes, H. K. Schmidt, and M. Pasquali, “Brownian dynamics simulations of single-wall carbon nanotube separation by type using dielectrophoresis,” J. Phys. Chem. B 112(25), 7467–7477 (2008). [CrossRef] [PubMed]

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