## Light concentration in the near-field of dielectric spheroidal particles with mesoscopic sizes |

Optics Express, Vol. 19, Issue 17, pp. 16207-16222 (2011)

http://dx.doi.org/10.1364/OE.19.016207

Acrobat PDF (1371 KB)

### 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

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

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]

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]

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

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]

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. 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]

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]

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]

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]

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]

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. 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. 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]

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]

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

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

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]

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]

**27**(6), 1221–1231 (2010). [CrossRef]

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]

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]

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**(6), 1221–1231 (2010). [CrossRef]

## 2. Separation-of-variables method and definitions

**27**(6), 1221–1231 (2010). [CrossRef]

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]

**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 [15,16

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

*K*α + 4). Being

_{0}*K*= 2π/λ and α is the longest semi-axis of the spheroid [3

_{0}**27**(6), 1221–1231 (2010). [CrossRef]

^{3}. 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

**27**(6), 1221–1231 (2010). [CrossRef]

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]

**K**) can be resolved by a single polarization component, by virtue of symmetry. This wave propagates along the negative

_{0},E_{0}*z*direction with the electric field

**E**parallel to

_{0}*y*.

_{0}). The length unit used is λ, making the results given here independent of the particular wavelength of illumination.

*b/a*), and the relative refractive index given by the ratio between the refractive index of the particle and that of the ambient medium (

*N*). The spheroid size parameter (

_{r}= N_{p}/N_{m}*C*) used in the separation-of-variables method is defined in terms of its inter-focal distance

*d*[15,32]:where

*K*is the wavevector magnitude in the corresponding medium. Inside the particle the size parameter is

*C*, being

_{i}= N_{r}C_{0}*C*the size parameter in the external medium. Our method allows the computation of the spheroidal harmonics with complex arguments. So, a complex refractive index

_{0}*N*is considered; enabling us to account for light absorption in the media [4]. 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 (

_{r}= n_{r}+ ik_{r}*R*) as the characteristic particle size [3

_{eq}**27**(6), 1221–1231 (2010). [CrossRef]

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. 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

*C*values. The method used here [3

**27**(6), 1221–1231 (2010). [CrossRef]

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]

*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]

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

*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 |

*C*<60 in order to avoid possible computational errors associated with the radial functions.

_{i}|**27**(6), 1221–1231 (2010). [CrossRef]

- • The electrostatic approximation (EA) for a particle size much smaller than λ
- • Mie theory for a wavelength-sized sphere

## 3. Portrait of physical parameters

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

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]

**34**(24), 5542–5551 (1995). [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]

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]

**E**) intensity distribution produced by DMPs with the same size (

_{t}*R*= 1.5λ) but distinct shape (

_{eq}*b/a*) and material (

*N*). The total field inside the particle is equal to its internal field (

_{r}**E**) and outside it is the sum of the scattered and incident fields (

_{t}= E_{i}**E**). A small imaginary part is considered in the relative refractive index (

_{t}= E_{S}+ E_{0}*k*= 0.01i) to account for light attenuation losses of realistic dielectric materials.

_{r}*b/a*= 1) it can be seen that there is a good match between our spheroidal code and Mie theory. The bottom plots (

*N*1.33 + 0.01i) with

_{r}=*b/a*= 0.5 and

*b/a*= 2 are similar to those in Figs. 13 and 14 of [16

**34**(24), 5542–5551 (1995). [CrossRef] [PubMed]

*k*= 0) and slightly different

_{r}*R*.

_{eq}**E**⊥

**K**) wave that propagates in the forward

**K**direction towards the far-field. This mode leads to intensity maxima that can be located inside or outside the particle but always on the

_{0}*z*axis. The spot of maximum field (|

**E**|

_{t}_{MAX}) outside the particle is called the focal point, located at a distance Z

_{MAX}from the particle center (see Fig. 1). An increase in Z

_{MAX}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 (L

_{Z}) [7

**13**(12), 4554–4559 (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]

**E**//

**K**) modes are also observed close to the particle surface – the whispering gallery modes [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]

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]

*r*

^{3}) with distance (

*r*) from the surface. The solutions obtained from Helmholtz equation do not allow the existence of these longitudinal fields on the vertical

*z*axis, but that is not the case in other directions.

### 3.1 Lens-like focusing

*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]

**202**(1), 129–135 (2001). [CrossRef] [PubMed]

**13**(12), 4554–4559 (2005). [CrossRef] [PubMed]

**30**(19), 2641–2643 (2005). [CrossRef] [PubMed]

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

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

**16**(18), 14200–14212 (2008). [CrossRef] [PubMed]

*f*of a biconvex lens much larger than λ is obtained with the lensmaker equation [14]:where

*R*is the lens curvature radius, which is equal to

_{c}*b*at the points on the symmetry axis (

^{2}/a*z = ± a*) of a spheroid. It was verified by [10

**30**(19), 2641–2643 (2005). [CrossRef] [PubMed]

*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

*n*and increases with the spheroid aspect ratio (

_{r}*b/a*). The same tendencies are observed (see Fig. 2) relative to Z

_{MAX}with mesoscopic spheroids [16

**34**(24), 5542–5551 (1995). [CrossRef] [PubMed]

_{MAX}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

**73**(23), 235401 (2006). [CrossRef]

**202**(1), 129–135 (2001). [CrossRef] [PubMed]

### 3.2 Application in photovoltaics

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

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]

*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.

*N*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

_{r}*n*was restricted between 1 and 4; and the imaginary part was taken to be

_{r}*k*= 0.01. Nevertheless, the main results are also given for a smaller

_{r}*k*= 0.001 to analyse the effect of lower light attenuation.

_{r}**E**is longitudinal, energy absorption from this field cannot be determined classically such as with far-fields which are transverse [38

_{S}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]

_{S}is longitudinally polarized [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]

_{S}becomes progressively more dominant relative to the longitudinal [3

**27**(6), 1221–1231 (2010). [CrossRef]

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

**E**|

_{t}^{2}/|

**E**|

_{0}^{2}). This is the quantity represented in the field distributions displayed in this paper.

**E**can take any orientation orthogonal to the propagation direction

_{0}**K**. 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

_{0}**K**.

_{0}## 4. Results of optimization studies

*R*), aspect ratio (

_{eq}*b/a*) and real part of the relative refractive index (

*n*). The imaginary part of the relative index is kept fixed (

_{r}*k*= 0.01). Two particular functions were considered for optimization that maximize at quite distinct near-field patterns; thus providing a good illustration of the different ways in which DMPs can be used for light concentration. The results obtained for each function are given in the following Sections 4.1 and 4.2.

_{r}### 4.1 Optimization of local electric field intensity

**E**|

_{S}^{2}/|

**E**|

_{0}^{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*n*as variables_{r} - 2) Spheroid with
*n*= 1.33, using_{r}*R*and_{eq}*b/a*as variables - 3) General spheroid using
*R*,_{eq}*b/a*and*n*as variables_{r}

*z*axis (Z

_{MAX}=

*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

*b/a*= 1) with variable radius

*R*and

*n*. Since the geometry is spherical the fields were computed with Mie theory to save computational time.

_{r}*R*, the lower is

*n*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

_{r}*n*. As observed in [10

_{r}**30**(19), 2641–2643 (2005). [CrossRef] [PubMed]

*n*index that sets the focal point right at the particle surface (Z

_{r}_{MAX}=

*R*). According to Eq. (2), for macroscopic spheres the value of

*n*that sets

_{r}*f*=

*R*is

*n*= 2.0. With mesoscopic spheres such

_{r}*n*value is too high since it places the maximum intensity inside the particle [10

_{r}**30**(19), 2641–2643 (2005). [CrossRef] [PubMed]

**73**(23), 235401 (2006). [CrossRef]

*n*= 1.33 (see central bottom plot of Fig. 2) is already too low because it sets the maximum E

_{r}_{t}outside the particle separated from its surface. The optimal values of

*n*that place the focus exactly at

_{r}*z*= -

*R*are plotted in Fig. 3(a) (dashed line) for

*R*ranging from 0.5λ to 10λ. The optimal

*n*is around 3 within such mesoscopic sizes, and tends to 2.0 as the size approaches the macroscopic GO regime [10

_{r}**30**(19), 2641–2643 (2005). [CrossRef] [PubMed]

**73**(23), 235401 (2006). [CrossRef]

**E**|

_{t}^{2}

_{MAX}) are shown in Fig. 3(a) for

*k*= 0.01 and

_{r}*k*= 0.001 (solid lines) using the optimal

_{r}*n*. The attenuation index

_{r}*k*has a quite significant impact in the peak field magnitudes that can be obtained in these optimal cases. If absorption were neglected (

_{r}*k*= 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 |

_{r}**E**|

_{t}^{2}

_{MAX}increases with

*R*. However, for sufficiently big

*R*, the effect of absorption dominates over scattering and |

**E**|

_{t}^{2}

_{MAX}starts decreasing with the size. The higher is

*k*the lower is the optimal

_{r}*R*value at which the maximum in the curves occurs. For

*k*= 0.01 such maximum is at

_{r}*R*= 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 (

_{opt}*z*= -

*R*) the intensity is |

**E**|

_{t}^{2}

_{MAX}= 92.06, but with lower

*k*= 0.001 it is about 3.5 times higher (|

_{r}**E**|

_{t}^{2}

_{MAX}= 328.3).

**E**

_{t}|^{2}

_{MAX}for any wavelength within λ ± 0.3λ. This broad frequency width is due to the fact that the

*k*= 0.01 curve in Fig. 3(a) is rather flat around

_{r}*R*(as marked by the line segment beneath the curve), and

_{opt}*n*remains approximately constant in that interval.

_{r}#### 4.1.2 Spheroid with *n*_{r} = 1.33

_{r}

**13**(12), 4554–4559 (2005). [CrossRef] [PubMed]

**34**(24), 5542–5551 (1995). [CrossRef] [PubMed]

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]

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]

*k*= 0.01i, as considered in this work.

_{r}*N*= 1.33 + 0.01i) but tunable size (

_{r}*R*) and aspect ratio (

_{eq}*b/a*). The optimal geometry is plotted in Fig. 4 . The maximum field intensity (|

**E**|

_{t}^{2}

_{MAX}= 46.8) occurs at the bottom point of the particle (

*z*= -

*a*), such as in the previous sphere case. With lower attenuation (

*k*= 0.001) the maximum field at that spot increases to about twice this value (|

_{r}**E**|

_{t}^{2}= 99.2), as shown in the right plot of Fig. 4.

*b/a*<1 could the focus be positioned at the particle surface given such low

*n*. 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

_{r}*R*and

_{eq}*N*as the prolate of Fig. 4 is |

_{r}**E**|

_{t}^{2}= 33.9 at Z

_{MAX}= 4.29λ.

#### 4.1.3 General spheroidal case

*R*), aspect ratio (

_{eq}*b/a*) and real part of the relative index (

*n*). The domain of variation of

_{r}*n*was restricted to

_{r}*n*≤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

_{r}*N*is kept fixed (

_{r}*k*= 0.01).

_{r}*a*). In addition, the search moves towards the highest allowed

*n*value (

_{r}*n*= 4.0). A higher

_{r}*n*increases the optical interference inside the spheroid, since the internal light waves have smaller wavelength and suffer more refraction and reflection with the particle walls. The combination of these effects raises the intensities associated with the concentrated field at the spots of constructive interference [27

_{r}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]

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]

*n*also shifts the maximum field intensity towards the particle interior. Therefore, usually DMPs with big

_{r}*n*exhibit high internal fields but small scattered intensities in the external medium. This is, for instance, the case of spheres with any

_{r}*n*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

_{r}*n*. Thus, the focus can be kept at the bottom point of the particle (Z

_{r}_{MAX}=

*a*) if

*b/a*is increased together with

*n*. 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.

_{r}*b/a*>1) shape, since otherwise the maximum |

**E**|

_{t}^{2}would be inside the particle for such high

*n*= 4.0. The corresponding field pattern is given in Fig. 5 . As shown in the right plot, the curve corresponding to

_{r}*k*= 0.001 has a maximum |

_{r}**E**|

_{t}^{2}= 176.4, which is 1.6 times higher than the peak value of

*k*= 0.01. The ratio between the values of |

_{r}**E**|

_{t}^{2}

_{MAX}for

*k*= 0.001 and

_{r}*k*= 0.01 is lower in this case than in the previous cases of Figs. 3(b) and 4 (see Table 1). This is due to the fact that a smaller spheroid size reduces the effect of light absorption by the particle material. For the same reason, the ratio between the |

_{r}**E**|

_{t}^{2}

_{MAX}in Fig. 4 is smaller than that in Fig. 3(b).

*n*= 4.0 results in a more confined focal spot than that of the field distributions in Figs. 3(b) and 4 (see L

_{r}_{Z}values in Table 1). The focal peak is extremely localized in this case, having sub-wavelength dimensions both in the

*z*(L

_{Z}= 0.13λ) and

*y*(L

_{Y}= 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

**30**(19), 2641–2643 (2005). [CrossRef] [PubMed]

**73**(23), 235401 (2006). [CrossRef]

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]

**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]

*n*it is possible to achieve sub-wavelength confinement with plane-wave illumination both in the transverse and forward directions.

_{r}### 4.2 Optimization of electric intensity over near-field region

**E**|

_{0}^{2}, along a semi-circular region in front of the particle, in the

*yz*plane, of radius R

_{M}= 10λ:Here η and ξ are the spheroidal angular and radial coordinates, respectively; and the factor

_{M}) is on the order of the thickness of the main photocurrent generation zone in most solar cell designs, considering λ≈1μm (near-IR). As such, function P is an example of an optimization function suited to find the best spheroid parameters for implementation in PV devices. DMPs can be incorporated either on top of solar cells or embedded in their PV material, depending on the cell architecture and the preferred relative refractive index. In any case, the particles size should be as small as possible since their material always constitutes a disturbance in the PV medium which only serves for optical concentration purposes but cannot contribute (and most of the times deteriorates) to the current generation. If DMPs designed to focus IR light are placed on top of a cell the bigger their size the more significant becomes their attenuation of the lower wavelengths (VIS and UV), reducing the cell response to the higher energy part of the solar spectrum. This issue can be avoided if DMPs are placed inside the PV material. Nevertheless, in such case the bigger the DMP size the more volume of PV medium is occupied and the higher is the charge carrier trapping that may occur at the particle-medium interface.

*yz*area of the particle (π

*ab*), in order to beneficiate smaller sizes (

*R*) in the optimization procedure.

_{eq}*n*= 1.33 and general spheroid) of Section 4.1. The main results are summarized in Table 2 .

_{r}#### 4.2.1 Sphere case

*b/a*= 1), so the quantity P (Eq. (3)) becomes a function of only two parameters:

*R*and

*n*. As before, the imaginary part of the refractive index is kept fixed (

_{r}*k*= 0.01).

_{r}*R*,

*n*) is to be maximized. Figure 6(a) shows the values of P for every (

_{r}*R,n*) point within the search domain of the optimization algorithm. There are several local maxima in the function whose intensity becomes progressively lower with increasing

_{r}*R*and

*n*. The decrease of P with

_{r}*R*is mainly due to the normalization factor (π

*ab*)

^{−1}multiplying the integral in Eq. (3). As previously referred, our aim here is to find the scatterer parameters that achieve the highest possible electric field integral with a particle that occupies the least possible volume; as an exercise for possible PV applications. Nevertheless, the values of the integral alone (π

*ab*P

_{MAX}) are also given in Table 2 for the optimal parameters. The P peaks decrease with

*n*because the focal region moves towards the particle interior, thus reducing the scattered intensity in the external medium in front.

_{r}*R*axis close to the optimal value

*R*= 1.2λ, as in the curve of

_{opt}*k*= 0.01 in Fig. 3(a). Therefore, a sphere fabricated with the optimal parameters produces a P value above 0.9P

_{r}_{MAX}at any wavelength within λ ± 0.2λ.

#### 4.2.2 Spheroid with *n*_{r} = 1.33

_{r}

*N*= 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

_{r}*N*the less charge carrier trapping and recombination occurs at the particle surface.

_{r}*R*) in a region around its maximum value obtained with the optimization. Function P decreases with

_{eq},b/a*R*due to the normalization of the integral in Eq. (3) by the spheroid area. The aspect ratio (

_{eq}*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 E

_{S}over the chosen semi-circular region.

*R*width of the peak P in Fig. 6(b), within −10% of the maximum, allows a wavelength variation of λ ± 0.21λ; which is close to that reported in the previous sub-section. It can be seen in the right plot of Fig. 6(c) that there is a quite small difference between the

_{eq}**|E**

_{t}|^{2}profile of

*k*= 0.01 and

_{r}*k*= 0.001, as compared with the previous cases of Section 4.1.

_{r}#### 4.2.3 General spheroidal case

*R*) achieves a value (P

_{eq},b/a,n_{r}_{MAX}= 278.8) considerably higher than the previous two cases, as indicated in Table 2.

*n*= 4.0) since the higher is

_{r}*n*the higher can be the scattered field outside the particle. However, the increase of

_{r}*n*has to be accompanied by an increase in

_{r}*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

*n*= 4.0 and large

_{r}*b/a*, the algorithm converges to small particle sizes (

*R*<λ) in order to achieve a high value of the integral in Eq. (3) with a low value of the spheroid

_{eq}*yz*area (π

*ab*) in the denominator.

*R*) produce the field distribution shown in Fig. 7 . 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

_{eq},b/a,n_{r}*k*= 0.01 and

_{r}*k*= 0.001, due to the small particle size.

_{r}*C*. As referred in Section 2.2, the maximum allowed value of |

_{i}*C*| was limited to 60 in order to avoid inaccuracies in the calculation of the radial spheroidal harmonics.

_{i}## 5. Conclusions and final remarks

**27**(6), 1221–1231 (2010). [CrossRef]

*z*axis. The higher

*n*the more intense and confined can the focus be. However, the particle

_{r}*oblateness*has to increase with

*n*or else the light would be focused in the interior of the particle. An optimal adjust of

_{r}*b/a*and

*n*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

_{r}*n*. 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

_{r}**95**(7), 071105 (2009). [CrossRef]

*z*axis. The electric intensities are not as high as in the previous case, since the energy is distributed in a bigger volume of the external medium. This situation can be advantageous for certain applications, such as most types of thin-film solar cells [36

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

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]

**95**(7), 071105 (2009). [CrossRef]

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

_{0}

^{2}) in the infrared. Nevertheless, DMPs may be more suited for certain applications, such as PV, due to the following additional advantages [3

**27**(6), 1221–1231 (2010). [CrossRef]

**23**(10), 1171 (2011). [CrossRef]

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

## Appendix - Nelder and Mead optimization algorithm

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]

*n*variables that maximize (or minimize) a certain function

*F*. The

*n*variables are treated as the coordinates of a

*n*-dimensional space in which

*F*is defined. The algorithm starts by picking

*n*+1 initial points inside the

*n*-dimensional domain. These points are regarded as the vertices of an

*n*-simplex (

*n*-dimensional analogue of a triangle). Function

*F*is evaluated at each of these vertices and then the algorithm moves and/or redimensions the

*n*-simplex in the

*n*-dimensional space towards the vertex with better

*F*value. Function

*F*is again evaluated at the new

*n*-simplex points and the procedure is iteratively repeated as better

*F*values are found, until some desired bound is obtained.

*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:

*R*,

_{eq}*b/a*and

*n*. The domain allowed for the simplex was defined according to a set of lower limits (

_{r}*a,b*>0 and

*n*>1) and upper limits (

_{r}*n*≤4.0 and |

_{r}*C*|<60). The upper limit for

_{i}*n*was taken to be 4.0 for the reasons mentioned in Section 3.2. The upper bound set for |

_{r}*C*| is due to computational constraints, as referred in Section 2.2.

_{i}*F*until it shrinks at the maximum point found. With this method a maximum is reached within about 30-50 iterations. However, this can be a local maximum in the search domain. To ensure that a global maximum is reached, the algorithm is sequentially run with different initial point sets until the results converge to a single overall maximum. Tables 1 and 2 present the values of the overall maximum values obtained for the optimizations of |

**E**|

_{S}^{2}and P (Eq. (3)), respectively.

## Acknowledgments

## References and links

1. | C. Girard and A. Dereux, “Near-field optics theories,” Rep. Prog. Phys. |

2. | L. Novotny and B. Hecht, |

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 |

4. | C. F. Bohren and D. R. Huffman, |

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 |

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 |

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 |

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.) |

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. |

10. | S. Lecler, Y. Takakura, and P. Meyrueis, “Properties of a three-dimensional photonic jet,” Opt. Lett. |

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 |

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. |

13. | A. Devilez, N. Bonod, and B. Stout, “Near field dielectric microlenses,” Proc. SPIE |

14. | M. Born and E. Wolf, |

15. | L.-W. Li, X.-K. Kang, and M.-S. Leong, |

16. | J. P. Barton, “Internal and near-surface electromagnetic fields for a spheroidal particle with arbitrary illumination,” Appl. Opt. |

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 |

18. | A. V. Itagi and W. A. Challener, “Optics of photonic nanojets,” J. Opt. Soc. Am. A |

19. | A. Devilez, B. Stout, N. Bonod, and E. Popov, “Spectral analysis of three-dimensional photonic jets,” Opt. Express |

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 |

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

22. | E. Hecht, |

23. | S. Asano and G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. |

24. | E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. |

25. | J. C. Ravey and P. Mazeron, “Light-scattering in the physical optics approximation—application to large spheroids,” J. Opt. |

26. | J. P. Barton, “Internal, near-surface, and scattered electromagnetic fields for a layered spheroid with arbitrary illumination,” Appl. Opt. |

27. | N. Richard, “Analysis of polarization effects on nanoscopic objects in the near-field optics,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

28. | J. P. Barton, “Internal and near-surface electromagnetic fields for an absorbing spheroidal particle with arbitrary illumination,” Appl. Opt. |

29. | T. D. Milster, “Near-field optical data storage: avenues for improved performance,” Opt. Eng. |

30. | A. Devilez, B. Stout, and N. Bonod, “Compact metallo-dielectric optical antenna for ultra directional and enhanced radiative emission,” ACS Nano |

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. |

32. | C. Flammer, |

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 |

34. | P. Kirby, “Calculation of spheroidal wave functions,” Comput. Phys. Commun. |

35. | R. Kirby, “Calculation of radial prolate spheroidal wave functions of the second kind,” Comput. Phys. Commun. |

36. | H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. |

37. | C. Hägglund and B. Kasemo, “Nanoparticle plasmonics for 2D-photovoltaics: mechanisms, optimization, and limits,” Opt. Express |

38. | A. Luque, A. Marti, M. J. Mendes, and I. Tobias, “Light absorption in the near field around surface plasmon polaritons,” J. Appl. Phys. |

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 |

40. | C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, |

41. | P. Mazeron and S. Muller, “Dielectric or absorbing particles: EM surface fields and scattering,” J. Opt. |

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. |

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 |

**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

Sort: Year | Journal | Reset

### References

- C. Girard and A. Dereux, “Near-field optics theories,” Rep. Prog. Phys. 59(5), 657–699 (1996). [CrossRef]
- L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University Press, 2006).
- 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]
- C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH, 2004).
- 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]
- 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]
- 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]
- 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]
- 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]
- S. Lecler, Y. Takakura, and P. Meyrueis, “Properties of a three-dimensional photonic jet,” Opt. Lett. 30(19), 2641–2643 (2005). [CrossRef] [PubMed]
- 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]
- 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]
- A. Devilez, N. Bonod, and B. Stout, “Near field dielectric microlenses,” Proc. SPIE 7717, 771708(2010). [CrossRef]
- M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University Press, 1999).
- L.-W. Li, X.-K. Kang, and M.-S. Leong, Spheroidal Wave Functions in Electromagnetic Theory (John Wiley & Sons, Inc., 2002).
- 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]
- 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]
- A. V. Itagi and W. A. Challener, “Optics of photonic nanojets,” J. Opt. Soc. Am. A 22(12), 2847–2858 (2005). [CrossRef] [PubMed]
- 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]
- 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]
- S. Lecler, “Light scattering by sub-micrometric particles,” PhD thesis (Louis Pasteur University, 2005).
- E. Hecht, Optics, 4th ed. (Addison Wesley, 2001).
- S. Asano and G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14(1), 29–49 (1975). [PubMed]
- E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186(2), 705–714 (1973). [CrossRef]
- 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]
- 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]
- 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]
- 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]
- T. D. Milster, “Near-field optical data storage: avenues for improved performance,” Opt. Eng. 40(10), 2255–2260 (2001). [CrossRef]
- 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]
- 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]
- C. Flammer, Spheroidal Wave Functions (Stanford University Press, 1957).
- 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]
- P. Kirby, “Calculation of spheroidal wave functions,” Comput. Phys. Commun. 175(7), 465–472 (2006). [CrossRef]
- R. Kirby, “Calculation of radial prolate spheroidal wave functions of the second kind,” Comput. Phys. Commun. 181(3), 514–519 (2010). [CrossRef]
- H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9(3), 205–213 (2010). [CrossRef] [PubMed]
- C. Hägglund and B. Kasemo, “Nanoparticle plasmonics for 2D-photovoltaics: mechanisms, optimization, and limits,” Opt. Express 17(14), 11944–11957 (2009). [CrossRef] [PubMed]
- 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]
- 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]
- C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms—Introduction to Quantum Electrodynamics (Wiley-Interscience, 1997).
- P. Mazeron and S. Muller, “Dielectric or absorbing particles: EM surface fields and scattering,” J. Opt. 29(2), 68–77 (1998). [CrossRef]
- 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]
- 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]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.