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| EXPLORING THE INTERFACE OF LIGHT AND BIOMEDICINE

  • Editors: Andrew Dunn and Anthony Durkin
  • Vol. 7, Iss. 11 — Oct. 31, 2012
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Magnetic interaction in all silicon waveguide spherical coupler device

Lei Shi and Francisco Meseguer  »View Author Affiliations


Optics Express, Vol. 20, Issue 20, pp. 22616-22626 (2012)
http://dx.doi.org/10.1364/OE.20.022616


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Abstract

The magnetic field component of light in dielectric materials generally plays a negligible role at optical frequency values. However, it is a key component of metal based metamaterials. Here we report on the dominant role of the magnetic interaction in a dielectric spherical silicon nanocavity coupled to a silicon waveguide. The analytical method, as well as the finite difference time domain (FDTD) simulation, show a three dimensional (3D) magnetic trap effect when the magnetic like Mie resonances of the nanocavity are excited.

© 2012 OSA

1. Introduction

Optical trapping forces discovered by Ashkin in the 1970s [1

1. A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24(4), 156–159 (1970). [CrossRef]

, 2

2. A. Ashkin and J. M. Dziedzic, “Observation of resonances in the radiation pressure on dielectric spheres,” Phys. Rev. Lett. 38(23), 1351–1354 (1977). [CrossRef]

] have raised great interest in the last 15 years [3

3. S. Chu, “Noble lecture: the manipulation of neutral particles,” Rev. Mod. Phys. 70(3), 685–706 (1998). [CrossRef]

5

5. M. L. Juan, M. Righini, and R. Quidant, “Plasmon nano-optical tweezers,” Nat. Photonics 5(6), 349–356 (2011). [CrossRef]

]. They have allowed providing revolutionary techniques as optical tweezers [6

6. A. Ashkin, “Optical trapping and manipulation of neutral particles using lasers,” Proc. Natl. Acad. Sci. U.S.A. 94(10), 4853–4860 (1997). [CrossRef] [PubMed]

, 7

7. F. M. Fazal and S. M. Block, “Optical tweezers study life under tension,” Nat. Photonics 5(6), 318–321 (2011). [CrossRef] [PubMed]

], laser cooling [3

3. S. Chu, “Noble lecture: the manipulation of neutral particles,” Rev. Mod. Phys. 70(3), 685–706 (1998). [CrossRef]

, 8

8. S. Chu, “Laser manipulation of atoms and particles,” Science 253(5022), 861–866 (1991). [CrossRef] [PubMed]

], as well as new tools for hot field areas like Bose-Einstein condensation [9

9. W. Ketterle, “Nobel lecture: when atoms behave as waves: Bose-Einstein condensation and the atom laser,” Rev. Mod. Phys. 74(4), 1131–1151 (2002). [CrossRef]

, 10

10. I. Bloch, “Ultracold quantum gases in optical lattices,” Nat. Phys. 1(1), 23–30 (2005). [CrossRef]

]. Optical tweezers allow controlling small objects such as colloids [5

5. M. L. Juan, M. Righini, and R. Quidant, “Plasmon nano-optical tweezers,” Nat. Photonics 5(6), 349–356 (2011). [CrossRef]

7

7. F. M. Fazal and S. M. Block, “Optical tweezers study life under tension,” Nat. Photonics 5(6), 318–321 (2011). [CrossRef] [PubMed]

] and even atoms [3

3. S. Chu, “Noble lecture: the manipulation of neutral particles,” Rev. Mod. Phys. 70(3), 685–706 (1998). [CrossRef]

, 8

8. S. Chu, “Laser manipulation of atoms and particles,” Science 253(5022), 861–866 (1991). [CrossRef] [PubMed]

]. It is agreed the optical response of matter is dominated mainly by the electrical component, rather than the magnetic component [11

11. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Wiley, 1984)

, 12

12. J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, Inc, 1962).

]. However, with the discovery of metamaterials [13

13. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004). [CrossRef] [PubMed]

15

15. R. Merlin, “Metamaterials and the Landau-Lifshitz permeability argument: large permittivity begets high-frequency magnetism,” Proc. Natl. Acad. Sci. U.S.A. 106(6), 1693–1698 (2009). [CrossRef] [PubMed]

], both magnetic and electric component of the electromagnetic (EM) field play a pivotal role. The development of metamaterials with strong magnetism at optical frequencies involves complex designs [16

16. M. Burresi, D. van Oosten, T. Kampfrath, H. Schoenmaker, R. Heideman, A. Leinse, and L. Kuipers, “Probing the magnetic field of light at optical frequencies,” Science 326(5952), 550–553 (2009). [CrossRef] [PubMed]

18

18. M. Burresi, T. Kampfrath, D. van Oosten, J. C. Prangsma, B. S. Song, S. Noda, and L. Kuipers, “Magnetic light-matter interactions in a photonic crystal nanocavity,” Phys. Rev. Lett. 105(12), 123901 (2010). [CrossRef] [PubMed]

] as well as sophisticated fabrication techniques [19

19. C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, Th. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95(20), 203901 (2005). [CrossRef] [PubMed]

, 20

20. A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, “Nanofabricated media with negative permeability at visible frequencies,” Nature 438(7066), 335–338 (2005). [CrossRef] [PubMed]

]. Even direct evidence of the resulting magnetic field has been reported using an artificial metamaterial building block fabricated on a probing tip [16

16. M. Burresi, D. van Oosten, T. Kampfrath, H. Schoenmaker, R. Heideman, A. Leinse, and L. Kuipers, “Probing the magnetic field of light at optical frequencies,” Science 326(5952), 550–553 (2009). [CrossRef] [PubMed]

18

18. M. Burresi, T. Kampfrath, D. van Oosten, J. C. Prangsma, B. S. Song, S. Noda, and L. Kuipers, “Magnetic light-matter interactions in a photonic crystal nanocavity,” Phys. Rev. Lett. 105(12), 123901 (2010). [CrossRef] [PubMed]

]. Therefore, the magnetic field should also play a pivotal role for optical forces in metamaterials [21

21. R. Zhao, P. Tassin, T. Koschny, and C. M. Soukoulis, “Optical forces in nanowire pairs and metamaterials,” Opt. Express 18(25), 25665–25676 (2010). [CrossRef] [PubMed]

, 22

22. M. Lapine, I. V. Shadrivov, D. A. Powell, and Y. S. Kivshar, “Magnetoelastic metamaterials,” Nat. Mater. 11(1), 30–33 (2011). [CrossRef] [PubMed]

]. There are two main obstacles for studying optical forces on metamaterials. Firstly, most metamaterials developed so far are substrate supported type structures [23

23. Y. Liu and X. Zhang, “Metamaterials: a new frontier of science and technology,” Chem. Soc. Rev. 40(5), 2494–2507 (2011). [CrossRef] [PubMed]

] and they are of difficult use for measuring photonic forces. Secondly, the optical absorption of metallic metamaterials would induce strong thermophoretic forces [24

24. V. Garcés-Chávez, R. Quidant, P. J. Reece, G. Badenes, L. Torner, and K. Dholakia, “Extended organization of colloidal microparticles by surface plasmon polariton excitation,” Phys. Rev. B 73(8), 085417 (2006). [CrossRef]

] thus hindering photonic force experiments.

In last few years, several groups have suggested that high refractive index dielectric structures supporting well defined electric and magnetic modes can be a new type of metamaterials [25

25. S. O'Brien and J. B. Pendry, “Photonic band-gap effects and magnetic activity in dielectric composites,” J. Phys. Condens. Matter 14(15), 4035–4044 (2002). [CrossRef]

27

27. J. A. Schuller, R. Zia, T. Taubner, and M. L. Brongersma, “Dielectric metamaterials based on electric and magnetic resonances of silicon carbide particles,” Phys. Rev. Lett. 99(10), 107401 (2007). [CrossRef] [PubMed]

]. From the theoretical side, several groups have shown high refractive index particles are good metamaterial candidates. Nieto-Vesperinas et al. [28

28. M. Nieto-Vesperinas, R. Gomez-Medina, and J. J. Saenz, “Angle-suppressed scattering and optical forces on submicronmeter dielectric particles,” J. Opt. Soc. Am. A 28(1), 54–60 (2011). [CrossRef]

] have modeled a large magnetic response of high refractive index spheres. From the fabrication side, Korgel et al. [29

29. J. T. Harris, J. L. Hueso, and B. A. Korgel, “Hydrogenated Amorphous Silicon (a-Si:H) Colloids,” Chem. Mater. 22(23), 6378–6383 (2010). [CrossRef]

], and Fenollosa et al. [30

30. R. Fenollosa, F. Meseguer, and M. Tymczenko, “Silicon colloids: from microcavities to photonic sponges,” Adv. Mater. (Deerfield Beach Fla.) 20(1), 95–98 (2008). [CrossRef]

], have reported on the synthesis of silicon colloids with spherical shape and very smooth surface. Very recently, Meseguer et al. [31

31. E. Xifré-Pérez, R. Fenollosa, and F. Meseguer, “Low order modes in microcavities based on silicon colloids,” Opt. Express 19(4), 3455–3463 (2011). [CrossRef] [PubMed]

], have shown silicon colloids show well defined low order optical Mie resonance modes. Importantly, because of both, the low absorption of silicon in the infrared (IR) region and the simplicity for handling silicon colloids [32

32. E. Xifré-Pérez, J. D. Domenech, R. Fenollosa, P. Muñoz, J. Capmany, and F. Meseguer, “All silicon waveguide spherical microcavity coupler device,” Opt. Express 19(4), 3185–3192 (2011). [CrossRef] [PubMed]

], they are ideal candidates for studying optical forces in metamaterials. Several papers about electric and magnetic contributions to optical forces on high refractive index particles have been reported [33

33. P. C. Chaumet and A. Rahmani, “Electromagnetic force and torque on magnetic and negative-index scatterers,” Opt. Express 17(4), 2224–2234 (2009). [CrossRef] [PubMed]

36

36. L. Shi, E. Xifré-Pérez, F. J. García de Abajo, and F. Meseguer, “Looking through the mirror: optical microcavity-mirror image photonic interaction,” Opt. Express 20(10), 11247–11255 (2012). [CrossRef] [PubMed]

].

Most papers concern that optical forces between particles and photonic waveguides [37

37. D. Erickson, X. Serey, Y. F. Chen, and S. Mandal, “Nanomanipulation using near field photonics,” Lab Chip 11(6), 995–1009 (2011). [CrossRef] [PubMed]

, 38

38. S. Lin, E. Schonbrun, and K. Crozier, “Optical manipulation with planar silicon microring resonators,” Nano Lett. 10(7), 2408–2411 (2010). [CrossRef] [PubMed]

], optical resonators [39

39. S. Mandal, X. Serey, and D. Erickson, “Nanomanipulation using silicon photonic crystal resonators,” Nano Lett. 10(1), 99–104 (2010). [CrossRef] [PubMed]

], and plasmonics structures [40

40. M. Righini, A. S. Zelenina, C. Girard, and R. Quidant, “Parallel and selective trapping in a patterned plasmonic landscape,” Nat. Photonics 3, 477–480 (2007).

] only consider the electric component of light, the magnetic contribution being negligible. However, when high refractive index particles enter into scene, the magnetic contribution should also carefully be considered. In this paper, we systematically discuss the optical forces acting on a silicon spherical nanocavity near a linear waveguide, in which the standing wave is generated by two coherent counterpart guided light. The resonating modes of the silicon nanocavity are tuned to the fundamental magnetic Mie modes. As the maxima for the electric and magnetic field of the standing wave are located at different spatial regions, the particle will be trapped in different spatial regions of waveguide when the wavelength of guided light is changed. Two independent methods, an analytical calculation and the FDTD simulation show that the silicon sphere can be trapped in the high magnetic and low electric field region due to the strong magnetic response of silicon sphere. This magnetic like trap effect is completely different from the electric one that usually appears in optical lattice experiments [41

41. M. Greiner and S. Fölling, “Condensed-matter physics: optical lattices,” Nature 453(7196), 736–738 (2008). [CrossRef] [PubMed]

]. Our findings may open a new road to manipulate the small object using the magnetic field of light.

2. Analytic approach and numerical method to solve the interaction between silicon sphere and photonic waveguide

2.1. Mie theory of a silicon spherical nanocavity

Firstly, we will consider a silicon sphere of radius r (215 nm in this paper), with a refractive index value of 3.5. The scattering of light by silicon sphere can be well described by the Mie theory [42

42. P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, 1990).

]. The scattering cross section (σ) can be expressed as a function of the electric and magnetic Mie coefficients an and bn as
σ=6πk02(|an|2+|bn|2),
(1)
where k0 is the wavevector of incident light. The electric (𝛼e) and the magnetic (𝛼m) polarizabilities can be defined as,
P=αeE,M=αmH.
(2)
As we are restricted to wavelength values resonating with the lowest order modes, we only take into account the contribution of the first electric and magnetic mode. Then, 𝛼e and 𝛼m can be rewritten as [28

28. M. Nieto-Vesperinas, R. Gomez-Medina, and J. J. Saenz, “Angle-suppressed scattering and optical forces on submicronmeter dielectric particles,” J. Opt. Soc. Am. A 28(1), 54–60 (2011). [CrossRef]

],

αe=i(6πk03)ε0a1,αm=i(6πk03)b1.
(3)

2.2. Optical force and optical potential on a single small silicon sphere in an arbitrary EM field

The force acting on a particle embedded in an arbitrary EM field can be calculated from the Maxwell stress tensor T [43

43. J. A. Stratton, Electromagnetic Theory (The Maple Press Company, 1941)

] as,
F=STndS=(12)Re(Sε0(En)E*+1μ0(Bn)B*(12){ε0E2+[1μ0]B2}n),
(4)
where n is the normal direction of a closed surface surrounding the silicon sphere. For a very small particle whose size is much smaller than the wavelength of incident light, the optical force can be simplified as [33

33. P. C. Chaumet and A. Rahmani, “Electromagnetic force and torque on magnetic and negative-index scatterers,” Opt. Express 17(4), 2224–2234 (2009). [CrossRef] [PubMed]

],
F=Fe+Fm+Fem.
(5)
The two first terms on the right side of the above equation come from the electric and magnetic dipole components respectively, and the third one is the interaction between them. The electric dipole component of the force can be expressed as [35

35. R. Gómez-Medina, M. Nieto-Vesperinas, and J. J. Saenz, “Nonconservative electric and magnetic optical forces on submicron dielectric particles,” Phys. Rev. A 83(3), 033825 (2011). [CrossRef]

]
Fe=(1/4)Re(αe)|E|2+(1/2)σexteRe((1/c)S)+cσexte×Lspe.
(6)
The electric like component of the force in Eq. (6) has three terms. The first one is the gradient force. The second one corresponds to the scattering force, where σexte=(k0/ε0)Im(αe) is the extinction cross section contribution from the electric dipole, c is the speed of light, and S=E×H* is the Poynting vector of incident light. Finally, the third term is the electric spin force component [42

42. P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, 1990).

], Lspe=ε0/(4iω)(E×E*) being the electric spin density of the incident light [44

44. S. Albaladejo, M. I. Marqués, M. Laroche, and J. J. Sáenz, “Scattering forces from the curl of the spin angular momentum of a light field,” Phys. Rev. Lett. 102(11), 113602 (2009). [CrossRef] [PubMed]

].

In a similar manner, the magnetic dipole component of the force can be expressed as [35

35. R. Gómez-Medina, M. Nieto-Vesperinas, and J. J. Saenz, “Nonconservative electric and magnetic optical forces on submicron dielectric particles,” Phys. Rev. A 83(3), 033825 (2011). [CrossRef]

],
Fm=(μ0/4)Re(αm)|H|2+(1/2)σextmRe((1/c)S)+cσextm×Lspm,
(7)
also with three components, where σextm=k0Im(αm) is the extinction cross section contribution from the magnetic dipole and Lspm=μ0/(4iω)(H×H*) is the magnetic spin density of the incident light [35

35. R. Gómez-Medina, M. Nieto-Vesperinas, and J. J. Saenz, “Nonconservative electric and magnetic optical forces on submicron dielectric particles,” Phys. Rev. A 83(3), 033825 (2011). [CrossRef]

].

U=Fdr.
(9)

2.3. The evanescent field of standing waves in a photonic waveguide

The waveguide discussed in the paper is a rectangular silicon waveguide on a silica substrate, which can easily be fabricated from a silicon on insulator (SOI) wafer. We consider the lowest order guided mode, the polarization direction of the electric field parallel and the magnetic field perpendicular to the substrate surface, respectively (see Fig. 1
Fig. 1 Schematic view of a spherical silicon nanocavity located above the top side of a silicon waveguide (dark red). The gap between silicon nanocavity and waveguide is 20 nm. Two counter-propagative guided light beams (yellow arrows) interfere themselves forming a standing wave. The waveguide cross section and the field distribution of Ex in the waveguide is shown in the inset of the Fig. 1. The diameter of the silicon sphere is 430 nm. The height and the width of waveguide are 220 nm and 500 nm respectively. The lowest order guided mode is used. The white dotted line is along the symmetry line of the top surface of waveguide in y direction.
). In order to obtain a standing wave pattern, two coherent monochromatic beams are coupled at both sides of the waveguide. It is well known, in the standing wave the peaks and nodes of the electric and magnetic fields are shifted each other, the electric field maxima corresponds to minima of the magnetic field component and vice versa. In this paper a small silicon sphere located near the top surface of waveguide can feel the evanescent fields of the standing wave.

Analytic method is very complicated to describe well the optical modes of a dielectric rectangular waveguide. Therefore, in the analytic approach, for simplified the problem, a slab waveguide is assumed. This approximation is acceptable because we only consider the lowest order guided mode. Later, by comparing to a full FDTD simulation, we will prove the analytic approximation is a reasonable approximation as it keeps the most important points of photonic force results.

The evanescent field distribution of the electric component outside the waveguide along the y direction (shown in Fig. 1) can be written as,
Ex=Aerz[eiβy+eiβy]=2Aerzcos(βy),
(10)
where A is the amplitude of electric field, β=neffk0 is the propagation constant along the y direction and r2=β2k02 is the decay rate of field in the z direction, neff being the effective refractive index of the waveguide for the guided light. From Eq. (10), the magnetic Hy and Hz evanescent field distribution can easily be derived,
Hy=2i/(ωμ0)Arerzcos(βy),
(11)
Hz=2β/(iωμ0)Aerzsin(βy).
(12)
From the evanescent field expressions Eq. (10-12), the optical force along the y direction acting on the silicon sphere can be written as,

Fe=Re(αe)βA2e2rzsin(2βy),
(13)
Fm=Re(αm)βA2e2rzsin(2βy)k02/(ω2μ0),
(14)
Fem=[k03/(6π)][Im(αm)Re(αe)Im(αe)Re(αm)][A2e2rzβsin(2βy)].
(15)

2.4. FDTD simulation of the force acting on the silicon sphere located near the top surface of the photonic waveguide

To verify the analytical method described above, we also have calculated the photonic forces through FDTD simulation (FDTD Solutions, Lumerical Corp.). The structure simulated is shown in Fig. 1. The refractive index of both, the silicon sphere and the silicon waveguide, is assumed as a constant value of 3.5. The radius of silicon sphere is 215 nm. The width and height of silicon rectangular waveguide is 500 nm and 220 nm respectively. The optical force is modeled by the integrating Maxwell stress tensor [21

21. R. Zhao, P. Tassin, T. Koschny, and C. M. Soukoulis, “Optical forces in nanowire pairs and metamaterials,” Opt. Express 18(25), 25665–25676 (2010). [CrossRef] [PubMed]

, 46

46. X. Yang, Y. Liu, R. F. Oulton, X. Yin, and X. Zhang, “Optical forces in hybrid plasmonic waveguides,” Nano Lett. 11(2), 321–328 (2011). [CrossRef] [PubMed]

]. We consider there is a 20 nm gap between the silicon sphere and the waveguide. The laser intensity in the waveguide is 9 mW. The Ex field distribution of the guided light shown in Fig. 1, corresponds to the lowest order mode of the waveguide. The scattering cross section has been calculated by integrating of all the light scattered by a single silicon sphere when a plane wave impinges on it. For comparison purposes, we also have calculated for polystyrene (PS) sphere (refractive index 1.6) with the same diameter value.

3. Results and discussion

Figure 2
Fig. 2 (a) Light scattering efficiency of a single silicon spherical nanocavity, 430 nm large, immersed in air as a function of the wavelength. The results of the Mie analytical method (solid line) and the FDTD method (dash line) are shown. The black line corresponds to the total scattering efficiency. The red and blue lines correspond to the magnetic and electric contributions to the total scattering efficiency. (b) The same as (a) for the same size PS sphere. (c) The magnetic (red) and electric (blue) polarizabilities as a function of wavelength of light. The solid and dash lines correspond to the real and the imaginary parts of polarizabilities. (d) The same as (c) but for a PS nanocavity.
shows the scattering efficiency values and their wavelength dependence for a silicon and a PS nanospheres with the same size. The analytic method, as well as the FDTD simulation gives very similar results. Due to the small refractive index values of PS, the scattering efficiency shows small values without resonance features. On the contrary, silicon nanocavity presents three clear resonances with huge values of the scattering cross section. At certain wavelength values (for instance at 1570 nm), the scattering cross section of silicon nanocavity are 10 times bigger than the projected surface value of the particle. Also, resonant wavelength value is around 4 times larger than the diameter of sphere. It indicates silicon nanocavities can be a good metamaterial candidate, as we later will show. In order to understand the origin of the three resonance peaks, we have plotted separately the electric (blue line) and magnetic (red line) contribution to the scattering efficiency. It is clear that the peak at 1570 nm corresponds to the magnetic dipole mode and the peak at 1200 nm corresponds to electric dipole mode. Also, in a broad wavelength range (from 1447 nm to 1721 nm in this paper), the scattering cross section is dominated by the magnetic dipole resonance. The strong scattering peak around 1100 nm is induced by the magnetic quadrupole mode [36

36. L. Shi, E. Xifré-Pérez, F. J. García de Abajo, and F. Meseguer, “Looking through the mirror: optical microcavity-mirror image photonic interaction,” Opt. Express 20(10), 11247–11255 (2012). [CrossRef] [PubMed]

].

In order to better understand the electric and the magnetic contribution to the scattering cross section, Fig. 2(c) shows the magnetic and electric polarizabilities 𝛼m and 𝛼e calculated from Eq. (3). Both 𝛼m and 𝛼e dramatically change as a function of wavelength. The negative value of 𝛼m means that the induced oscillating magnetic dipole has a 𝜋 phase shift with respect the magnetic field component of the incident light. Then, the silicon nanoparticle looks like a diamagnetic material in this region. The positive value of 𝛼m means that the induced magnetic dipole is in-phase with the incident light and the silicon sphere looks like a paramagnetic material. The behavior of 𝛼m as a function of wavelength shows a strong Lorentz-shaped resonance [47

47. W. Cai and V. Shalaev, Optical Metamaterials: Fundamentals and Applications (Springer, 2010).

]. These results have close similarities to those reported for metamaterials, this leading to infer silicon nanosphere may be an excellent dielectric, light absorption free, metamaterial in infrared region [48

48. E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, 1985).

].

Before calculating the optical force with the analytic model, we need determining the evanescent electric and the magnetic field components of the standing wave along the y direction. Figure 3(a)
Fig. 3 The upper panel shows the analytical results of the evanescent electric (blue line) and magnetic (red line) fields along the y direction. The middle panel shows the optical force on a silicon 430 nm large nanocavity located above the top surface of the planar waveguide. The gap between the nanocavity and the waveguide is 20 nm. The black line corresponds to the total optical force acting on the silicon sphere, and the red, blue and yellow dashed lines correspond to the magnetic, electric, and the interaction between the electric and the magnetic dipoles optical forces appearing in the expression shown in Eq. (5). The lower panel shows the optical potential along the y direction. Left and right panels correspond to light wavelength values of 1.6 𝜇m (a), and 1.5 𝜇m (b), respectively.
, 3(b) shows the evanescent fields distribution with the wavelength value of 1.6 𝜇m (a) and 1.5 𝜇m (b) on the top surface of the photonic waveguide (as the white dot line shown in Fig. 1) calculated from Eqs. (10)(12). Maxima of the magnetic field component correspond to minima of the electric field component and vice versa. The middle panels of Figs. 3(a), 3(b) show the optical force obtained from Eqs. (13)(15). The electric, magnetic and electric-magnetic interaction components of the optical force are shown with blue, red, and orange dashed lines respectively. The center of the silicon sphere is located 235 nm above the top of the waveguide (i.e: the gap between the sphere and the waveguide is 20 nm), where the local guided light intensity along y direction is assumed to be 1.38 mW/𝜇m2 (equivalent to the intensity at the same place above the waveguide in FDTD simulation for 1.6 𝜇m wavelength guided light). In the case of low refractive index nanocavities (PS case), the magnitude of all three components are much smaller and only the electric force component dominates over the other two components (see Eqs. (13)(15)) and Fig. 2(d)). In the case of high refractive index nanocavities, for wavelength values at the magnetic resonance (λ = 1.6 𝜇m), silicon nanospheres are trapped at the points of the waveguide where magnetic field show maxima, which are different from the locations with maxima electric field. On the contrary, when the wavelength value is shifted to 1.5 𝜇m, silicon nanocavities are trapped at different waveguide positions which correspond to the electric field maxima. Based on Eqs. (14), (15), the anomalous magnetic trap effect is originated from the positive magnetic dipole polarizabilities 𝛼m. Around λ = 1.6 𝜇m (see Fig. 2(c)) the real part of 𝛼m has a significant positive value. In this case the silicon sphere looks like a paramagnetic material in which the magnetic dipole orientated parallel to the magnetic field direction of the guided light. Therefore, the particle is trapped at the waveguide locations where the magnetic field show maxima values. On the contrary, at wavelength values of 1.5 𝜇m, the real part of 𝛼m changes to large negative values and the nanocavity looks like a diamagnetic material in which the magnetic dipole oriented antiparallel to the magnetic field direction of the guided light. Therefore, the particle is pushed out the magnetic field maxima and is addressed to the waveguide locations with maxima values of the electric field. Both cases can be clearly seen by calculating the optical potential (Eq. (9)) of the silicon sphere at the top surface of photonic waveguide along y direction (see bottom panel of Fig. 3(a), 3(b)). It can clearly be seen that the lowest optical potential corresponds to the highest magnetic field for λ = 1.6 𝜇m, while the lowest optical potential corresponds to the highest electric field position in the case of λ = 1.5 𝜇m.

Figure 4
Fig. 4 The upper panel of (a, b) shows the FDTD results of the evanescent electric (blue line) and magnetic (red line) fields along the y direction above the top surface of the waveguide. The bottom panel of (a, b) shows the optical force (black line) and the optical potential (purple line) acting on the silicon sphere 430 nm large separated 20 nm from the top surface of the waveguide in y direction (see white dashed line shown in Fig. 1). The wavelength of guided light is 1.6 𝜇m (a), and 1.5 𝜇m (b). The electric (c), and the magnetic (d) fields distributions on the equatorial xy plane of the nanocavity are shown. The field plot at (e) shows the magnetic field distribution in the xz plane cross section of the system. In all (c), (d) and (e) field plot cases, the nanocavity is located at the high magnetic field region along y direction, the wavelength of guided light being 1.6 𝜇m.
shows the FDTD results of a silicon nanocavity located above a rectangular shaped dielectric waveguide. In order to better understand the photonic forces, we plot the field distributions along the symmetry line in the y direction (white dashed line in the Fig. 1). The left and right hand panels correspond to wavelength values of λ = 1.6 𝜇m and λ = 1.5 𝜇m respectively. The bottom panel of Fig. 4(a), 4(b), shows the optical forces (black line) and the optical potential (purple line) of the silicon nanocavity separated 20 nm from the symmetry line at the top surface of waveguide along y direction. For λ = 1.6 𝜇m (left panel), guided light traps the sphere at the high magnetic field region. On the contrary, for a wavelength value of λ = 1.5 𝜇m (right panel), guided light traps the sphere at the high electric region. FDTD simulation on a realistic rectangular dielectric waveguide is qualitatively consistent to those obtained by using the analytic model in the planar waveguide case. With the help of the FDTD simulation we have plotted in Fig. 4(c), 4(d) the fields distribution in the equatorial plane of the silicon nanosphere at the magnetic resonance (λ = 1.6 𝜇m). The field map plot shows a typical pure magnetic dipole profile. Figure 4(e) shows the magnetic field distribution of the cross section of the sphere and the waveguide.

Figure 5
Fig. 5 The upper panel shows the FDTD results of the evanescent electric (blue line) and the magnetic (red line) field distribution at the highest magnetic field region along x direction. The lower panel shows the optical force (black line) and the optical potential (purple line) on a silicon sphere 430 nm large 20 nm above the top surface of waveguide along x direction. The sphere is located at the high magnetic field region along y direction, and the wavelength of the guided light is 1.6 𝜇m.
shows FDTD results on the optical force on the nanocavity along x direction, at a wavelength value of λ = 1.6 𝜇m. Here, the sphere is located at high magnetic field region along the y direction and it can freely move in x direction. In the upper panel of Fig. 5, the evanescent electric (blue) and magnetic (red) field distribution at the top surface of waveguide along x direction is shown. The magnetic field has a maximum at the waveguide center and it decays to 0 at the waveguide side. On the contrary, the electric field is 0 at the center and has a peak at both waveguide sides. These two peaks of electric field are caused by the boundary conditions. The optical force and optical potential along x direction is shown in the bottom panel of Fig. 5. The results show the particle is trapped at the center of waveguide. Furthermore, the strong field gradient along the z direction would also trap the nanoparticle, finally resulting in a full 3D optical trap for silicon nanocavity. Here it is worth considering other competing forces as van der Waals (vdW) forces that would much stronger than photonic forces when the particles are immersed in air. However we have not considered them, since this result may be extrapolated to the case where both the waveguide and the silicon particle are immersed in a liquid medium. In this case vdW forces could be much smaller than the photonic forces at reasonable realistic light intensity values [36

36. L. Shi, E. Xifré-Pérez, F. J. García de Abajo, and F. Meseguer, “Looking through the mirror: optical microcavity-mirror image photonic interaction,” Opt. Express 20(10), 11247–11255 (2012). [CrossRef] [PubMed]

].

Finally, in order to know the wavelength dependence of the electric and magnetic traps, Fig. 6
Fig. 6 The potential difference between the high magnetic field region and high electric field region at the waveguide as a function of wavelength for both the analytical model (blue line) and FDTD simulation (red line). For comparison purposes, we have also plotted the PS sphere as a blue dash line. The diameter of silicon and PS spheres are 430 nm, and the intensity of guided light in waveguide is 0.9 mW.
shows the potential difference between the high magnetic field region and high electric field region at the waveguide in KT units. The results from both, the analytic model (blue curve), and the FDTD simulation (red curve) are shown. When the difference is positive, the electric trap dominates over the magnetic one and the silicon sphere is pushed towards the high electric field region. On the contrary, when the difference is negative, the silicon sphere is trapped in the high magnetic field region. The two methods have qualitative agreement with each other, while the FDTD approach gives stronger photonic trapping values. The difference is manly resulted by two approximate assumptions used in the analytic method: one is assuming the silicon sphere size to be far less than the wavelength of the guided light; the other is using a thin film as the waveguide instead of a rectangular shape. From the values of Fig. 6 the photonic trapping effect show quite realistic values since at moderate laser intensity values (as low as 0.9 mW), the magnetic trap is about 30 times stronger (FDTD results) than the thermal energy. However, weak photonic electric traps effects appear in the case of PS nanocavities at all the wavelength values.

4. Conclusion

In this paper, we have reported on the interaction between a standing wave confined in a waveguide and the fundamental modes of a silicon nanocavity. The spatial separation between electric like and magnetic like standing waves allows us studying separately the electrical and magnetic interaction between the sphere and the waveguide. Because of the strong magnetic response of the silicon sphere, the magnetic component of the optical force acting on a silicon nanocavity can be dominant over the electric component, pushing the sphere towards the magnetic trap region of the optical waveguide. Therefore we show silicon nanosphere can be used as a probe for mapping magnetic field distribution in photonic structures. These results have important implications for optical tweezers and metamaterials research.

Acknowledgments

The authors acknowledge financial support from the following projects FIS2009-07812, Consolider 2007-0046 Nanolight, and the PROMETEO/2010/043. L. Shi thanks the financial support from the MICINN (Estancias de profesores e investigadores extranjeros en centros españoles) fellowship program. The authors thank to Dr. E. Xifre-Perez for useful discussions.

References and links

1.

A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24(4), 156–159 (1970). [CrossRef]

2.

A. Ashkin and J. M. Dziedzic, “Observation of resonances in the radiation pressure on dielectric spheres,” Phys. Rev. Lett. 38(23), 1351–1354 (1977). [CrossRef]

3.

S. Chu, “Noble lecture: the manipulation of neutral particles,” Rev. Mod. Phys. 70(3), 685–706 (1998). [CrossRef]

4.

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 21–27 (2006). [CrossRef] [PubMed]

5.

M. L. Juan, M. Righini, and R. Quidant, “Plasmon nano-optical tweezers,” Nat. Photonics 5(6), 349–356 (2011). [CrossRef]

6.

A. Ashkin, “Optical trapping and manipulation of neutral particles using lasers,” Proc. Natl. Acad. Sci. U.S.A. 94(10), 4853–4860 (1997). [CrossRef] [PubMed]

7.

F. M. Fazal and S. M. Block, “Optical tweezers study life under tension,” Nat. Photonics 5(6), 318–321 (2011). [CrossRef] [PubMed]

8.

S. Chu, “Laser manipulation of atoms and particles,” Science 253(5022), 861–866 (1991). [CrossRef] [PubMed]

9.

W. Ketterle, “Nobel lecture: when atoms behave as waves: Bose-Einstein condensation and the atom laser,” Rev. Mod. Phys. 74(4), 1131–1151 (2002). [CrossRef]

10.

I. Bloch, “Ultracold quantum gases in optical lattices,” Nat. Phys. 1(1), 23–30 (2005). [CrossRef]

11.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Wiley, 1984)

12.

J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, Inc, 1962).

13.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004). [CrossRef] [PubMed]

14.

C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development of three-dimensional photonic metamaterials,” Nat. Photonics 5, 523–530 (2011).

15.

R. Merlin, “Metamaterials and the Landau-Lifshitz permeability argument: large permittivity begets high-frequency magnetism,” Proc. Natl. Acad. Sci. U.S.A. 106(6), 1693–1698 (2009). [CrossRef] [PubMed]

16.

M. Burresi, D. van Oosten, T. Kampfrath, H. Schoenmaker, R. Heideman, A. Leinse, and L. Kuipers, “Probing the magnetic field of light at optical frequencies,” Science 326(5952), 550–553 (2009). [CrossRef] [PubMed]

17.

S. Vignolini, F. Intonti, F. Riboli, L. Balet, L. H. Li, M. Francardi, A. Gerardino, A. Fiore, D. S. Wiersma, and M. Gurioli, “Magnetic imaging in photonic crystal microcavities,” Phys. Rev. Lett. 105(12), 123902 (2010). [CrossRef] [PubMed]

18.

M. Burresi, T. Kampfrath, D. van Oosten, J. C. Prangsma, B. S. Song, S. Noda, and L. Kuipers, “Magnetic light-matter interactions in a photonic crystal nanocavity,” Phys. Rev. Lett. 105(12), 123901 (2010). [CrossRef] [PubMed]

19.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, Th. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95(20), 203901 (2005). [CrossRef] [PubMed]

20.

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, “Nanofabricated media with negative permeability at visible frequencies,” Nature 438(7066), 335–338 (2005). [CrossRef] [PubMed]

21.

R. Zhao, P. Tassin, T. Koschny, and C. M. Soukoulis, “Optical forces in nanowire pairs and metamaterials,” Opt. Express 18(25), 25665–25676 (2010). [CrossRef] [PubMed]

22.

M. Lapine, I. V. Shadrivov, D. A. Powell, and Y. S. Kivshar, “Magnetoelastic metamaterials,” Nat. Mater. 11(1), 30–33 (2011). [CrossRef] [PubMed]

23.

Y. Liu and X. Zhang, “Metamaterials: a new frontier of science and technology,” Chem. Soc. Rev. 40(5), 2494–2507 (2011). [CrossRef] [PubMed]

24.

V. Garcés-Chávez, R. Quidant, P. J. Reece, G. Badenes, L. Torner, and K. Dholakia, “Extended organization of colloidal microparticles by surface plasmon polariton excitation,” Phys. Rev. B 73(8), 085417 (2006). [CrossRef]

25.

S. O'Brien and J. B. Pendry, “Photonic band-gap effects and magnetic activity in dielectric composites,” J. Phys. Condens. Matter 14(15), 4035–4044 (2002). [CrossRef]

26.

B. I. Popa and S. A. Cummer, “Compact dielectric particles as a building block for low-loss magnetic metamaterials,” Phys. Rev. Lett. 100(20), 207401 (2008). [CrossRef] [PubMed]

27.

J. A. Schuller, R. Zia, T. Taubner, and M. L. Brongersma, “Dielectric metamaterials based on electric and magnetic resonances of silicon carbide particles,” Phys. Rev. Lett. 99(10), 107401 (2007). [CrossRef] [PubMed]

28.

M. Nieto-Vesperinas, R. Gomez-Medina, and J. J. Saenz, “Angle-suppressed scattering and optical forces on submicronmeter dielectric particles,” J. Opt. Soc. Am. A 28(1), 54–60 (2011). [CrossRef]

29.

J. T. Harris, J. L. Hueso, and B. A. Korgel, “Hydrogenated Amorphous Silicon (a-Si:H) Colloids,” Chem. Mater. 22(23), 6378–6383 (2010). [CrossRef]

30.

R. Fenollosa, F. Meseguer, and M. Tymczenko, “Silicon colloids: from microcavities to photonic sponges,” Adv. Mater. (Deerfield Beach Fla.) 20(1), 95–98 (2008). [CrossRef]

31.

E. Xifré-Pérez, R. Fenollosa, and F. Meseguer, “Low order modes in microcavities based on silicon colloids,” Opt. Express 19(4), 3455–3463 (2011). [CrossRef] [PubMed]

32.

E. Xifré-Pérez, J. D. Domenech, R. Fenollosa, P. Muñoz, J. Capmany, and F. Meseguer, “All silicon waveguide spherical microcavity coupler device,” Opt. Express 19(4), 3185–3192 (2011). [CrossRef] [PubMed]

33.

P. C. Chaumet and A. Rahmani, “Electromagnetic force and torque on magnetic and negative-index scatterers,” Opt. Express 17(4), 2224–2234 (2009). [CrossRef] [PubMed]

34.

M. Nieto-Vesperinas, J. J. Sáenz, R. Gómez-Medina, and L. Chantada, “Optical forces on small magnetodielectric particles,” Opt. Express 18(11), 11428–11443 (2010). [CrossRef] [PubMed]

35.

R. Gómez-Medina, M. Nieto-Vesperinas, and J. J. Saenz, “Nonconservative electric and magnetic optical forces on submicron dielectric particles,” Phys. Rev. A 83(3), 033825 (2011). [CrossRef]

36.

L. Shi, E. Xifré-Pérez, F. J. García de Abajo, and F. Meseguer, “Looking through the mirror: optical microcavity-mirror image photonic interaction,” Opt. Express 20(10), 11247–11255 (2012). [CrossRef] [PubMed]

37.

D. Erickson, X. Serey, Y. F. Chen, and S. Mandal, “Nanomanipulation using near field photonics,” Lab Chip 11(6), 995–1009 (2011). [CrossRef] [PubMed]

38.

S. Lin, E. Schonbrun, and K. Crozier, “Optical manipulation with planar silicon microring resonators,” Nano Lett. 10(7), 2408–2411 (2010). [CrossRef] [PubMed]

39.

S. Mandal, X. Serey, and D. Erickson, “Nanomanipulation using silicon photonic crystal resonators,” Nano Lett. 10(1), 99–104 (2010). [CrossRef] [PubMed]

40.

M. Righini, A. S. Zelenina, C. Girard, and R. Quidant, “Parallel and selective trapping in a patterned plasmonic landscape,” Nat. Photonics 3, 477–480 (2007).

41.

M. Greiner and S. Fölling, “Condensed-matter physics: optical lattices,” Nature 453(7196), 736–738 (2008). [CrossRef] [PubMed]

42.

P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, 1990).

43.

J. A. Stratton, Electromagnetic Theory (The Maple Press Company, 1941)

44.

S. Albaladejo, M. I. Marqués, M. Laroche, and J. J. Sáenz, “Scattering forces from the curl of the spin angular momentum of a light field,” Phys. Rev. Lett. 102(11), 113602 (2009). [CrossRef] [PubMed]

45.

T. Iida and H. Ishihara, “Theoretical study of the optical manipulation of semiconductor nanoparticles under an excitonic resonance condition,” Phys. Rev. Lett. 90(5), 057403 (2003). [CrossRef] [PubMed]

46.

X. Yang, Y. Liu, R. F. Oulton, X. Yin, and X. Zhang, “Optical forces in hybrid plasmonic waveguides,” Nano Lett. 11(2), 321–328 (2011). [CrossRef] [PubMed]

47.

W. Cai and V. Shalaev, Optical Metamaterials: Fundamentals and Applications (Springer, 2010).

48.

E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, 1985).

OCIS Codes
(290.4020) Scattering : Mie theory
(160.3918) Materials : Metamaterials
(350.4855) Other areas of optics : Optical tweezers or optical manipulation

ToC Category:
Optical Trapping and Manipulation

History
Original Manuscript: April 3, 2012
Revised Manuscript: June 18, 2012
Manuscript Accepted: June 29, 2012
Published: September 19, 2012

Virtual Issues
Vol. 7, Iss. 11 Virtual Journal for Biomedical Optics

Citation
Lei Shi and Francisco Meseguer, "Magnetic interaction in all silicon waveguide spherical coupler device," Opt. Express 20, 22616-22626 (2012)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-20-20-22616


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References

  1. A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett.24(4), 156–159 (1970). [CrossRef]
  2. A. Ashkin and J. M. Dziedzic, “Observation of resonances in the radiation pressure on dielectric spheres,” Phys. Rev. Lett.38(23), 1351–1354 (1977). [CrossRef]
  3. S. Chu, “Noble lecture: the manipulation of neutral particles,” Rev. Mod. Phys.70(3), 685–706 (1998). [CrossRef]
  4. D. G. Grier, “A revolution in optical manipulation,” Nature424(6950), 21–27 (2006). [CrossRef] [PubMed]
  5. M. L. Juan, M. Righini, and R. Quidant, “Plasmon nano-optical tweezers,” Nat. Photonics5(6), 349–356 (2011). [CrossRef]
  6. A. Ashkin, “Optical trapping and manipulation of neutral particles using lasers,” Proc. Natl. Acad. Sci. U.S.A.94(10), 4853–4860 (1997). [CrossRef] [PubMed]
  7. F. M. Fazal and S. M. Block, “Optical tweezers study life under tension,” Nat. Photonics5(6), 318–321 (2011). [CrossRef] [PubMed]
  8. S. Chu, “Laser manipulation of atoms and particles,” Science253(5022), 861–866 (1991). [CrossRef] [PubMed]
  9. W. Ketterle, “Nobel lecture: when atoms behave as waves: Bose-Einstein condensation and the atom laser,” Rev. Mod. Phys.74(4), 1131–1151 (2002). [CrossRef]
  10. I. Bloch, “Ultracold quantum gases in optical lattices,” Nat. Phys.1(1), 23–30 (2005). [CrossRef]
  11. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Wiley, 1984)
  12. J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, Inc, 1962).
  13. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science305(5685), 788–792 (2004). [CrossRef] [PubMed]
  14. C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development of three-dimensional photonic metamaterials,” Nat. Photonics5, 523–530 (2011).
  15. R. Merlin, “Metamaterials and the Landau-Lifshitz permeability argument: large permittivity begets high-frequency magnetism,” Proc. Natl. Acad. Sci. U.S.A.106(6), 1693–1698 (2009). [CrossRef] [PubMed]
  16. M. Burresi, D. van Oosten, T. Kampfrath, H. Schoenmaker, R. Heideman, A. Leinse, and L. Kuipers, “Probing the magnetic field of light at optical frequencies,” Science326(5952), 550–553 (2009). [CrossRef] [PubMed]
  17. S. Vignolini, F. Intonti, F. Riboli, L. Balet, L. H. Li, M. Francardi, A. Gerardino, A. Fiore, D. S. Wiersma, and M. Gurioli, “Magnetic imaging in photonic crystal microcavities,” Phys. Rev. Lett.105(12), 123902 (2010). [CrossRef] [PubMed]
  18. M. Burresi, T. Kampfrath, D. van Oosten, J. C. Prangsma, B. S. Song, S. Noda, and L. Kuipers, “Magnetic light-matter interactions in a photonic crystal nanocavity,” Phys. Rev. Lett.105(12), 123901 (2010). [CrossRef] [PubMed]
  19. C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, Th. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett.95(20), 203901 (2005). [CrossRef] [PubMed]
  20. A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, “Nanofabricated media with negative permeability at visible frequencies,” Nature438(7066), 335–338 (2005). [CrossRef] [PubMed]
  21. R. Zhao, P. Tassin, T. Koschny, and C. M. Soukoulis, “Optical forces in nanowire pairs and metamaterials,” Opt. Express18(25), 25665–25676 (2010). [CrossRef] [PubMed]
  22. M. Lapine, I. V. Shadrivov, D. A. Powell, and Y. S. Kivshar, “Magnetoelastic metamaterials,” Nat. Mater.11(1), 30–33 (2011). [CrossRef] [PubMed]
  23. Y. Liu and X. Zhang, “Metamaterials: a new frontier of science and technology,” Chem. Soc. Rev.40(5), 2494–2507 (2011). [CrossRef] [PubMed]
  24. V. Garcés-Chávez, R. Quidant, P. J. Reece, G. Badenes, L. Torner, and K. Dholakia, “Extended organization of colloidal microparticles by surface plasmon polariton excitation,” Phys. Rev. B73(8), 085417 (2006). [CrossRef]
  25. S. O'Brien and J. B. Pendry, “Photonic band-gap effects and magnetic activity in dielectric composites,” J. Phys. Condens. Matter14(15), 4035–4044 (2002). [CrossRef]
  26. B. I. Popa and S. A. Cummer, “Compact dielectric particles as a building block for low-loss magnetic metamaterials,” Phys. Rev. Lett.100(20), 207401 (2008). [CrossRef] [PubMed]
  27. J. A. Schuller, R. Zia, T. Taubner, and M. L. Brongersma, “Dielectric metamaterials based on electric and magnetic resonances of silicon carbide particles,” Phys. Rev. Lett.99(10), 107401 (2007). [CrossRef] [PubMed]
  28. M. Nieto-Vesperinas, R. Gomez-Medina, and J. J. Saenz, “Angle-suppressed scattering and optical forces on submicronmeter dielectric particles,” J. Opt. Soc. Am. A28(1), 54–60 (2011). [CrossRef]
  29. J. T. Harris, J. L. Hueso, and B. A. Korgel, “Hydrogenated Amorphous Silicon (a-Si:H) Colloids,” Chem. Mater.22(23), 6378–6383 (2010). [CrossRef]
  30. R. Fenollosa, F. Meseguer, and M. Tymczenko, “Silicon colloids: from microcavities to photonic sponges,” Adv. Mater. (Deerfield Beach Fla.)20(1), 95–98 (2008). [CrossRef]
  31. E. Xifré-Pérez, R. Fenollosa, and F. Meseguer, “Low order modes in microcavities based on silicon colloids,” Opt. Express19(4), 3455–3463 (2011). [CrossRef] [PubMed]
  32. E. Xifré-Pérez, J. D. Domenech, R. Fenollosa, P. Muñoz, J. Capmany, and F. Meseguer, “All silicon waveguide spherical microcavity coupler device,” Opt. Express19(4), 3185–3192 (2011). [CrossRef] [PubMed]
  33. P. C. Chaumet and A. Rahmani, “Electromagnetic force and torque on magnetic and negative-index scatterers,” Opt. Express17(4), 2224–2234 (2009). [CrossRef] [PubMed]
  34. M. Nieto-Vesperinas, J. J. Sáenz, R. Gómez-Medina, and L. Chantada, “Optical forces on small magnetodielectric particles,” Opt. Express18(11), 11428–11443 (2010). [CrossRef] [PubMed]
  35. R. Gómez-Medina, M. Nieto-Vesperinas, and J. J. Saenz, “Nonconservative electric and magnetic optical forces on submicron dielectric particles,” Phys. Rev. A83(3), 033825 (2011). [CrossRef]
  36. L. Shi, E. Xifré-Pérez, F. J. García de Abajo, and F. Meseguer, “Looking through the mirror: optical microcavity-mirror image photonic interaction,” Opt. Express20(10), 11247–11255 (2012). [CrossRef] [PubMed]
  37. D. Erickson, X. Serey, Y. F. Chen, and S. Mandal, “Nanomanipulation using near field photonics,” Lab Chip11(6), 995–1009 (2011). [CrossRef] [PubMed]
  38. S. Lin, E. Schonbrun, and K. Crozier, “Optical manipulation with planar silicon microring resonators,” Nano Lett.10(7), 2408–2411 (2010). [CrossRef] [PubMed]
  39. S. Mandal, X. Serey, and D. Erickson, “Nanomanipulation using silicon photonic crystal resonators,” Nano Lett.10(1), 99–104 (2010). [CrossRef] [PubMed]
  40. M. Righini, A. S. Zelenina, C. Girard, and R. Quidant, “Parallel and selective trapping in a patterned plasmonic landscape,” Nat. Photonics3, 477–480 (2007).
  41. M. Greiner and S. Fölling, “Condensed-matter physics: optical lattices,” Nature453(7196), 736–738 (2008). [CrossRef] [PubMed]
  42. P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, 1990).
  43. J. A. Stratton, Electromagnetic Theory (The Maple Press Company, 1941)
  44. S. Albaladejo, M. I. Marqués, M. Laroche, and J. J. Sáenz, “Scattering forces from the curl of the spin angular momentum of a light field,” Phys. Rev. Lett.102(11), 113602 (2009). [CrossRef] [PubMed]
  45. T. Iida and H. Ishihara, “Theoretical study of the optical manipulation of semiconductor nanoparticles under an excitonic resonance condition,” Phys. Rev. Lett.90(5), 057403 (2003). [CrossRef] [PubMed]
  46. X. Yang, Y. Liu, R. F. Oulton, X. Yin, and X. Zhang, “Optical forces in hybrid plasmonic waveguides,” Nano Lett.11(2), 321–328 (2011). [CrossRef] [PubMed]
  47. W. Cai and V. Shalaev, Optical Metamaterials: Fundamentals and Applications (Springer, 2010).
  48. E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, 1985).

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