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 Optical Society of America
Optical trapping forces discovered by Ashkin in the 1970s [1, 2] have raised great interest in the last 15 years [3–5]. They have allowed providing revolutionary techniques as optical tweezers [6, 7], laser cooling [3, 8], as well as new tools for hot field areas like Bose-Einstein condensation [9, 10]. Optical tweezers allow controlling small objects such as colloids [5–7] and even atoms [3, 8]. It is agreed the optical response of matter is dominated mainly by the electrical component, rather than the magnetic component [11, 12]. However, with the discovery of metamaterials [13–15], 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 [16–18] as well as sophisticated fabrication techniques [19, 20]. Even direct evidence of the resulting magnetic field has been reported using an artificial metamaterial building block fabricated on a probing tip [16–18]. Therefore, the magnetic field should also play a pivotal role for optical forces in metamaterials [21, 22]. There are two main obstacles for studying optical forces on metamaterials. Firstly, most metamaterials developed so far are substrate supported type structures  and they are of difficult use for measuring photonic forces. Secondly, the optical absorption of metallic metamaterials would induce strong thermophoretic forces  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 [25–27]. From the theoretical side, several groups have shown high refractive index particles are good metamaterial candidates. Nieto-Vesperinas et al.  have modeled a large magnetic response of high refractive index spheres. From the fabrication side, Korgel et al. , and Fenollosa et al. , have reported on the synthesis of silicon colloids with spherical shape and very smooth surface. Very recently, Meseguer et al. , 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 , 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 [33–36].
Most papers concern that optical forces between particles and photonic waveguides [37, 38], optical resonators , and plasmonics structures  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 . 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 . The scattering cross section (σ) can be expressed as a function of the electric and magnetic Mie coefficients an and bn as28],
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  as,33],35]Eq. (6) has three terms. The first one is the gradient force. The second one corresponds to the scattering force, where is the extinction cross section contribution from the electric dipole, c is the speed of light, and is the Poynting vector of incident light. Finally, the third term is the electric spin force component , being the electric spin density of the incident light .
In a similar manner, the magnetic dipole component of the force can be expressed as ,35].
The magnetic contribution to the optical force is related to 𝛼m. For nanocavities with low refractive index values, like silica or polystyrene, 𝛼m present very small values and, therefore, the magnetic component of the photonic force is far less than the electric one. However, for a silicon sphere with large magnetic response, the magnetic component should be considered carefully. Furthermore, 𝛼m can take positive and negative values, and silicon nanocavity may behave as a paramagnetic or diamagnetic metamaterial depending on the wavelength of the light. Similar effect can also be observed in electric gradient force case . Finally, the force induced from the interaction between electric dipole and magnetic dipole can be written as ,Eq. (6) over the space as,
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 ). 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,Eq. (10), the magnetic Hy and Hz evanescent field distribution can easily be derived,Eq. (10-12), the optical force along the y direction acting on the silicon sphere can be written as,
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, 46]. 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 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 .
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 . 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 .
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) , 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 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 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 .
Finally, in order to know the wavelength dependence of the electric and magnetic traps, Fig. 6 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.
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.
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.
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