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

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2011

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]

2010

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

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]

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]

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]

2009

2008

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]

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]

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]

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]

2007

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]

2006

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]

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

2005

2004

2001

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]

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]

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]

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

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]

1998

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. 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]

1996

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

1995

1982

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]

1975

1973

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

Arnold, N.

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]

Asano, S.

Atwater, H. A.

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]

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

Backman, V.

Barton, J. P.

Bertsch, M.

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]

Boneberg, J.

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]

Bonod, N.

Callahan, D. M.

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]

Challener, W. A.

Chen, Z.

Dereux, A.

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

Devilez, A.

Ferrand, P.

French, R. H.

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]

Gérard, D.

Girard, C.

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

Grandidier, J.

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]

Hägglund, C.

Hanuska, A. R.

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]

Itagi, A. V.

Kasemo, B.

Kattawar, G. W.

Kirby, P.

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

Kirby, R.

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

Koenderink, A. F.

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]

Kofler, J.

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]

Kooi, P. S.

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]

Lagarias, J. C.

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]

Lecler, S.

Lee, J. Y.

Leiderer, P.

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]

Leong, M. S.

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]

Li, C.

Li, L. W.

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]

Luque, A.

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]

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]

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]

Marti, A.

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]

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]

Martí, A.

Mazeron, P.

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

McNeil, L. E.

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]

Mendes, M. J.

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]

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]

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]

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]

Mertens, H.

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]

Meyrueis, P.

Milster, T. D.

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

Mosbacher, M.

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]

Muller, S.

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

Munday, J. N.

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]

Münzer, H. J.

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]

Pasquali, M.

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]

Pennypacker, C. R.

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

Peumans, P.

Pianta, M.

Polman, A.

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

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]

Popov, E.

Purcell, E. M.

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

Ravey, J. C.

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]

Reeds, J. A.

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]

Richard, N.

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]

Rigneault, H.

Schmidt, H. K.

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]

Stout, B.

Taflove, A.

Takakura, Y.

Tan, K. Y.

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]

Tobias, I.

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]

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Figures (7)

Fig. 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.

Fig. 2
Fig. 2

Total electric field intensity |Et |2 distribution, in units of the incident field intensity |E0 |2, produced by DMPs (outlined with a gray dashed line) with the same size (Req = 1.5λ) but distinct aspect ratio (b/a) and real part of the relative index (nr ). The length unit is λ. The field distributions were computed on a vertical cross-section through the center of the particle, coplanar with the yz plane defined by the incident wave (K0 ,E0 ). The solid curve on the right of each image corresponds to |Et |2 along the vertical z axis. The fields in the central plots for spheres (b/a = 1) were computed with Mie theory. For comparison, the cross dots in these plots correspond to the values obtained with the spheroidal separation-of-variables method considering spheroids with b/a = 1.0001.

Fig. 3
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).

Fig. 4
Fig. 4

Same as Fig. 3(b) but for the optimal spheroid geometry with a fixed nr = 1.33 and variable Req and b/a. The dimensions of the spheroid semi-axes are: a = 4.093λ and b = 3.162λ.

Fig. 5
Fig. 5

Same as Fig. 3(b) but for the optimal spheroid parameters obtained with tunable Req , b/a and nr . The dimensions of the spheroid semi-axes are: a = 1.099λ and b = 1.605λ. The bottom inset shows the peak |Et |2 along y at the bottom of the particle (z = -a). LY is the peak full width at |Et |2=|Et |2 MAX/e 2.

Fig. 6
Fig. 6

(a) 3D plot of the optimization function P(R,nr ) for spheres. The white circle is at the spot of maximum P obtained with the optimization algorithm. (b) Same as (a) but for the optimization of spheroids with fixed nr = 1.33 and tunable Req and b/a. (c) |Et |2 distribution, in linear scale, corresponding to the optimum spheroid parameters indicated by the white circle in (b). The spheroid semi-axes are: a = 0.729λ and b = 2.464λ.

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).

Tables (2)

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Table 1 Characteristics of the Optimal Spheroids That Maximize the Scattered Field Intensity a

Tables Icon

Table 2 Characteristics of the Optimal Spheroidal Parameters That Maximize Function P a

Equations (3)

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C = K d 2
f = n r R c 2 2 ( n r 1 ) 1 a + n r ( R c a )
P ( R e q , b / a , n r ) = 1 π a b | E S | 2 | E 0 | 2 f ( η , ξ ) d η d ξ

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