Multiple Multipole Method Research Articles

Super skew scattering in two-dimensional Dirac material systems with a flat band

The phenomenon of super scattering was previously found to arise in massless pseudospin-1 two-dimensional Dirac material systems with a flat band. Here we report the phenomenon of super skew scattering of massive pseudospin-1 quasiparticles, which does not arise in the corresponding massless system. In particular, the scatterer is electrically generated with a certain geometric shape, and the mass is induced by gap opening between the Dirac cones. Even for a circular scatterer, the occurrence of resonant states inside it can induce a sizable anomalous Hall current, which is associated with the gap opening. The striking finding is that a significant reduction in the scatterer size and/or the potential height does nothing to weaken the skew scattering and, for certain resonant states, even tends to strengthen the scattering. This phenomenon of super skew scattering in Dirac materials with a flat band is in stark contrast to the scattering of massive pseudospin-$\frac{1}{2}$ quasiparticles from the same configuration, where skew scattering is significantly weaker and a reduction in the scatterer strength can quickly diminish it. The phenomenon is established analytically for the case of a circular scatterer in the framework of continuum Hamiltonian, and is found to be robust for an elliptical scatterer, which is solved numerically by adopting the multiple-multipole method to massive pseudospin-1 scattering. Calculations of the electronic transport properties in the Lieb lattice system reveal the occurrence of a large anomalous Hall current as well, paving the way for experimental observation and test of super skew scattering. Because of the ``skew'' nature that is absent in massless pseudospin-1 systems, the phenomenon of super skew scattering in massive systems can be exploited for applications in novel electronic or photonic Hall devices.

Scattering of Dirac electrons from a skyrmion: Emergence of robust skew scattering

We study electron scattering from a closed magnetic structure embedded in the top surface of a topological insulator (TI). Outside the structure there is a uniform layer of ferromagnetic insulator (FMI), leading to a positive effective mass for the Dirac electrons. The mass inside can be engineered to be negative, leading to a skyrmion structure. The geometric shape of the structure can be circular or deformed, leading to integrable or chaotic dynamics, respectively, in the classical limit. For a circular structure, the relativistic quantum scattering characteristics can be calculated analytically. For a deformed structure, we develop an efficient numerical method, the multiple multipole method, to solve the scattering wavefunctions. We find that anomalous Hall effect as characterized by strong skew scattering can arise, which is robust against structural deformation due to the resonant modes. In the short (long) wavelength regime, the resonant modes manifest themselves as confined vortices (excited edge states). The origin of the resonant states is the spin phase factor of massive Dirac electrons at the skyrmion boundary. Further, in the short wavelength regime, for a circular skyrmion, a large number of angular momentum channels contribute to the resonant modes. In this regime, in principle, classical dynamics are relevant, but we find that geometric deformations, even those as severe as leading to fully developed chaos, have little effect on the resonant modes. The vortex structure of the resonant states makes it possible to electrically ``charge'' the skyrmion, rendering feasible to manipulate its motion electrically. In the long wavelength regime, only the lowest angular momentum channels contribute to the resonant modes, making the skew scattering sharply directional. These phenomena may find applications for information storage and in Hall devices based on dynamic skyrmion.

Plasmon-enhanced optical bending and heating on V-shaped deformation of gold nanorod

The plasmon-enhanced optical bending and heating on the V-shaped deformation of a straight gold nanorod (GNR), irradiated by a linear polarized light at the longitudinal surface plasmon resonance, are studied theoretically to explain the finding in previous experiment. Multiple multipole method is employed to calculate the optical load and heating numerically, and an elastic beam model is used to analyze the bending moment and stress in the GNR theoretically. According to our analysis, we think, first, the plasmonic heating softens the GNR to reduce the yield strength of gold, and the non-uniform optical load induces a maximum bending moment at the middle cross section of a freestanding GNR. Then an irreversible breakpoint of the plastic hinge at the middle of GNR is developed to form a V-shaped GNR. The photothermal deformation of V-shaped GNR involving multidisciplinary interplay is worth for further investigation.

Light-driven self-assembly of hetero-shaped gold nanorods

Light-driven self-assembly and coalescence of two nearby hetero-shaped gold nanorods (GNRs) with different lengths are studied theoretically. The optical forces and torques, in terms of Maxwell’s stress tensor, upon these GNRs provided by a linearly polarized (LP) plane wave are analyzed using the multiple multipole (MMP) method. Numerical results show that the optical torque dominates their alignments and the optical force their attraction. The most likely outcome of the plasmon-mediated light–matter interaction is wavelength dependent. Three different coalescences of the two GNRs could be induced by a LP light in three different wavelength regimes, respectively. For example, the side-by-side coalescence of two GNRs with radius of 15 nm and different lengths (120 and 240 nm) is induced in water as irradiated by a LP light at 633 nm, the T-shaped one at 1064 nm, and the end-to-end one at 1700 nm. The plasmonic attractive force and heating power densities inside GNRs with different gaps are also studied; the smaller the gap, the larger the attractive force and heating power. The results imply that the plasmonic coalescence and heating of two discrete GNRs may cause the local fusion at the junction of the assembly and the subsequent annealing (even recrystallization). Because the heating makes the two discrete GNRs fused to become a new nanostructure, the plasmonic coalescence of optical manipulation is irreversible.

Wavelength-Dependent Plasmon-Mediated Coalescence of Two Gold Nanorods

Plasmon-mediated coalescence of two nearby gold nanorods (NRs) suspended in water induced by the illumination of a linearly polarized (LP) light was studied theoretically. We analyzed the coupled optical forces and torques in terms of Maxwell’s stress tensor upon two identical NRs irradiated by a LP plane wave using the multiple multipole method to estimate the optomechanical outcome. Numerical results show that the light-matter interaction can perform attraction or repulsion, depending on their initial configurations. For the attraction, the end-to-end or side-by-side coalescence of the two gold NRs could be caused by the LP light, depending on the wavelength. For example, the side-by-side coalescence of two adjacent NRs of r = 15 nm and L = 120 nm is most likely induced by 800-nm LP laser beam, whereas the end-to-end coalescence by 1064-nm or 1700-nm LP laser. These distinct phenomena are attributed to the perpendicular or parallel alignment of NR to the polarization of LP light in different wavelength ranges. The magnitude of optical force, proportional to the light’s fluence, could be stronger than van der Waals force. The estimation based on quasi-static model without considering the fluid dynamics may provide an insight to optical manipulation on the self-assembly of gold colloid.

Maxwell stress induced optical torque upon gold prolate nanospheroid

This study theoretically analyzes the surface traction on an elongated Au prolate nanospheroid to examine the resultant optical torque exerted by an optical tweezers. The multiple multipole method is applied to evaluate quantitatively the electromagnetic field induced by a linearly polarized plane wave illuminating a nanospheroid, then obtaining the surface traction in terms of Maxwell stress tensor. The optical torque is calculated by the surface integral of the cross product of position vector and traction over the nanospheroid’s surface. Our results show that two pairs of positive and negative traction zones at the two apexes of the nanospheroid play a critical role. Furthermore, the resultant optical torque is wavelength-dependent. If the wavelength is shorter than the longitudinal surface plasmon resonance (LSPR) of the nanospheroid, the optical torque rotates the long axis of nanospheroid perpendicular to the polarization direction of the incident wave. In contrast, if the wavelength is longer than the LSPR the long axis is pushed parallel to the polarization direction. The turning point with a null torque, between the perpendicular and parallel modes, is at the LSPR. The optical performance of Au nanospheroid is equivalent to that of Au NR with the same volume and aspect ratio, but the LSPR of Au NR is little red-shifted from that of an equivalent prolate spheroid.

Spinning gold nanoparticles driven by circularly polarized light

This study theoretically examines a spinning gold nanoparticle (GNP) driven by circularly polarized (CP) plane waves. The wavelength-dependent optical torques which were exerted on three different shapes of GNPs (spherical, prolate and oblate spheroids) were analyzed by utilizing Mie theory for the former and the multiple multipole method for the latter two, respectively. Numerical results show that both the absorbed and scattered photons contribute to optical torques in most cases. For the case that the CP wave is incident along the long axis of an oblate spheroid or the short axis of a prolate one, the scattering effect in optical torque is more pronounced than the absorption one. This phenomenon is significant especially when the wavelength of the CP wave is close to the longitudinal surface plasmon resonance band of the GNP. In contrast, when the CP wave is incident along the axes of revolution of these shapes of GNPs, the ratio of optical torque to absorption power is directly proportional to the wavelength. Moreover, this ratio is independent of the size and even the aspect ratio of GNPs. This result suggests that only the absorbed photons contribute to optical torques, but not the scattered ones, due to the conservation of angular momentum for cases of rotational symmetry.

Metal-Enhanced Fluorescence of Silver Island Associated with Silver Nanoparticle.

The coupling plasmon of a hybrid nanostructure, silver island (SI) associated with silver nanoparticle (SNP), on metal-enhanced fluorescence (MEF) was studied theoretically. We used the multiple multipole method to analyze the plasmon-mediated enhancement factor on the fluorescence of a molecule immobilized on SNP and located in the gap zone between SI and SNP; herein, the SI was modeled as an oblate spheroid. Numerical results show that the enhancement factor of the hybrid nanostructure is higher than that of a SNP or a SI alone due to the coupled gap mode. This finding is in agreement with the previous experimental results. In addition, the plasmon band of the structure is broadband and tunable, which can be red-shifted and broadened by flattening or enlarging SI. Based on this property, the hybrid nanostructure can be tailored to obtain the optimal enhancement factor on a specific molecule according to its excitation spectrum. Moreover, we found that there is an induced optical force allowing SNP be attracted by SI. Consequently, the gap is reduced gradually to perform a stronger MEF effect.

Plasmon-mediated binding forces on gold or silver homodimer and heterodimer

This study theoretically investigates plasmon-mediated optical binding forces, which are exerted on metal homo or heterodimers, induced by the normal illumination of a linearly polarized plane wave or Gaussian beam. Using the multiple multipole method, we analyzed the optical force in terms of Maxwell׳s stress tensor for various interparticle distance at some specific wavelengths. Numerical results show that for a given wavelength there are several stable equilibrium distances between two nanoparticles (NPs) of a homodimer, which are slightly shorter than some integer multiples of the wavelength in medium, such that metal dimer acts as bonded together. At these specific interparticle distances, the optical force between dimer is null and serves a restoring force, which is repulsive and attractive, respectively, as the two NPs are moving closer to and away from each other. The spring constant of the restoring force at the first stable equilibrium is always the largest, indicating that the first stable equilibrium distance is the most stable one. Moreover, the central line (orientation) of a dimer tends to be perpendicular to the polarization of light. For the cases of heterodimers, the phenomenon of stable equilibrium interparticle distance still exists, except there is an extra net photophoretic force drifting the heterodimer as one. Moreover, gradient force provided by a Gaussian beam may reduce the stability of these equilibriums, so larger NPs are preferred to stabilize a dimer under illumination of Gaussian beam. The finding may pave the way for using optical manipulation on the gold or silver colloidal self-assembly.

Theoretical study of optical torques for aligning Ag nanorods and nanowires

The optical torque that is exerted on an Ag nanorod (NR) by the irradiance of a linearly polarized laser was studied theoretically. The multiple multipole method was used to calculate the induced electromagnetic field, and then the surface traction in terms of Maxwell׳s stress tensor was integrated over the NR׳s surface. The maps of surface traction on the Ag NR in two (parallel and perpendicular) modes of alignment in various wavelength regimes were discussed. The turning point between the two modes was demonstrated to coincide with the longitudinal surface plasmon resonance (LSPR) of the Ag NR. For example, the optical torques that are induced by different lasers (1064nm, 633nm and 532nm) on Ag NR (with a radius 10nm and an aspect ratio of 3) in water were analyzed to demonstrate the wavelength-dependent performance. The NR was aligned parallel and perpendicular to the polarization of the 1064nm and 532nm lasers, respectively. A laser with a wavelength of 633nm, which was close to the LSPR, induced a null torque, and caused severe plasmonic heating. In contrast, over the entire spectrum, only the parallel mode of dielectric NRs was observed. The optical torque on Ag NR is two orders of magnitude greater than that of a high-k dielectric NR of equal size. For a highly elongated Ag NR (including nanowire), the perpendicular mode, rather than the parallel one, is induced even irradiated by a 1064nm NIR laser, because its LSPR wavelength exceeds 1064nm.

Approach for a fast computation of induced magnetic field created by naval vessels

Integral equation method is widely used in magnetic field modeling of ships, but its nonsymmetric and dense co- efficient matrix results in low computation efficiency and large memory, which limits its application in large scale problems. An approach for a fast computation of induced magnetic field created by ships was presented based on multiple layers fast multipole method and vector integral equation method. The numerical implementation of the proposed approach was described in detail. It is shown by an analytical sphere example and a simplified mock-up experiment that the proposed approach can reduce computation time and memory cost in induced magnetic field modeling of ferromagnetic objects.

An implementation of multiple multipole method in the analyse of elliptical objects to enhance backscattering light

In this paper, we present dielectric elliptical shapes modelling with respect to a highly confined power distribution in the resulting nanojet, which has been parameterized according to the beam waist and its beam divergence. The method is based on spherical bessel function as a basis function, which is adapted to standard multiple multipole method. This method can handle elliptically shaped particles due to the change of size and refractive indices, which have been studied under plane wave illumination in two and three dimensional multiple multipole method. Because of its fast and good convergence, the results obtained from simulation are highly accurate and reliable. The simulation time is less than minute for two and three dimension. Therefore, the proposed method is found to be computationally efficient, fast and accurate.

Rotating Au nanorod and nanowire driven by circularly polarized light.

The wavelength-dependent optical torques provided by a circularly polarized (CP) plane wave driving Au nanorod (NR) and nanowire (NW) to rotate constantly were studied theoretically. Using the multiple multipole method, the resultant torque in terms of Maxwell's stress tensor was analyzed. Numerical results show that the optical torque spectrum is in accordance with the absorption spectrum of Au NR/NW. Under the same fluence, the maximum optical torque occurs at the longitudinal surface plasmon resonance (LSPR) of Au NR/NW, accompanied by a severe plasmonic heating. The rotation direction of the light-driven NR/NW depends on the handedness of CP light. In contrast, the optical torque exerted on Au NR/NW illuminated by a linearly polarized light is null at LSPR. Due to the plasmonic effect, the optical torque on Au NR/NW by CP light is two orders of magnitude larger than that on a dielectric NR/NW of the same size. The steady-state rotation of NR/NW in water, resulting from the balance of optical torque and viscous torque, was also discussed. Our finding shed some light on manipulating a CP light-driven Au NR/NW as a rotating nanomotor for a variety of applications in optofluidics and biophysics.

Wavelength-dependent longitudinal polarizability of gold nanorod on optical torques.

This study theoretically investigates the wavelength-dependent longitudinal polarizability of a gold nanorod (GNR) irradiated by a polarized laser beam. The resultant optical torque in terms of the Maxwell stress tensor was analyzed quantitatively using the multiple multipole method. Our results indicate that the real part of the longitudinal polarizability of GNR can be either positive or negative, leading to the parallel or perpendicular modes, respectively. For the parallel and perpendicular modes, the long axis of GNR is rotated to align parallel and perpendicular, respectively, to the polarization direction of the illuminating light. The turning point between these two modes, depending on the aspect ratio (AR) and the size of GNR, nearly coincides with the longitudinal surface plasmon resonance (LSPR). The perpendicular mode ranges from the transverse SPR to LSPR, and the range of the parallel mode is broadband from LSPR to the near infrared regime. Owing to that a larger optical torque and less plasmonic heating are of concern, an efficiency of optical torque is defined to evaluate the performance of different wavelengths. Analysis results indicate that lasers with wavelength in the perpendicular mode are applicable to rotate and align a GNR of a higher AR. For example, the laser of 785 nm (the perpendicular mode) is superior to that of 1064 nm (the parallel mode, off-resonant from LSPR of 955 nm) for rotating a GNR of AR = 4 and radius 20 nm with an orientation of 45° with respect to the laser polarization.

Metal enhanced fluorescence of Ag-nanoshell dimer

The plasmon modes of Ag-nanoshell dimer on metal enhanced fluorescence (MEF) are studied theoretically. The amplified excitation rate of a dimer (two identical Ag nanoshells) illuminated by a plane wave for exciting a molecule located at the gap center is calculated. Subsequently, the apparent quantum yield of the emission of the excited molecule affected by the dimer is investigated. The multiple multipole method is used for the both simulations. Finally, the enhancement factor of the dimer on the overall photoluminescence of the molecule in terms of the two parameters is evaluated. Our results show that Ag-nanoshell dimer is a dual-band photoluminescence enhancer for MEF at the bonding dipole and quadrupole modes. The former is broadband, and the latter narrowband. Both bands depend on the gap size. Moreover, the average enhancement factor of Ag-nanoshell dimer for MEF with a Stokes shift is discussed.

Bi-dipole excited plasmonic modes of an elongated gold nanorod

The first and higher-order longitudinal surface plasmon resonance (SPR) modes of an elongated gold nanorod (GNR) induced by the photoluminescence of two quantum dots (QDs) respectively located at the two ends were studied theoretically. Two configurations of a GNR combined with a symmetric or anti-symmetric bi-dipole were simulated and analyzed using the multiple multipole method. The results show that the local maxima of the radiative and nonradiative powers of the bi-dipole are at these modes. When the aspect ratio (AR) of GNR exceeds a specific value, not only the first mode but also the second, third and even fourth modes are generated. For example, for an elongated GNR (radius: 30nm, AR=7) in water, the first, second, third and fourth modes are at 1800nm, 930nm, 680nm and 600nm, respectively. These SPR modes depend on the AR as well as the radius of GNR. The larger the AR is, the more the red-shift of these modes will be. In addition, the red-shift increases as the radius increases. Moreover, the odd modes are induced by the anti-symmetric bi-dipole, but suppressed by the symmetric one. On the contrary, the even modes are induced by the symmetric bi-dipole, but suppressed by the anti-symmetric one. In comparison with the scattering and absorption cross sections of GNR irradiated by a plane wave, the high-order modes, particularly the even modes, can be easily induced by the bi-dipole. Moreover, the mutual excitation rate of the two QDs is also enhanced through these modes of GNR.

Size-Dependent Surface Enhanced Fluorescence of Gold Nanorod: Enhancement or Quenching

The size-dependent surface enhanced fluorescence (SEF) of a gold nanorod (GNR) on a vicinal molecule is studied systematically by using the multiple multipole method. The enhancement factors (EFs) of GNRs of different sizes (radius: a = 7 or 10 nm) with the same aspect ratio (AR), e.g., 3 or 3.43, on the fluorescence of a molecule at different locations with various orientations excited at the longitudinal surface plasmon resonance (SPR) of a GNR are discussed in detail. The numerical results show that the EF of a GNR is sensitive not only to the molecular location and orientation but also to the size. The effective EF (EEF), the average of the EF over all possible orientations at a specific location, is further calculated. According to the EEF, the proximity of the end-cap of a GNR is a strong enhancing zone, while the waist area of the GNR is a relatively weak zone. Moreover, for the same AR, the EEF of a bigger GNR (a ≥ 10 nm) is larger than that of a smaller one. Hence, the SEF performance of a bigger GNR on the fluorescence can be a very strong enhancement if the molecule is close to the end-cup excited at the longitudinal SPR. On the contrary, the performance of a smaller GNR could be a quenching if the molecule is near the waist. In addition, the Stokes-shift effect of fluorescence on the EF is also discussed.

Theoretical investigation of plasmonic enhancement of silica-coated gold nanorod on molecular fluorescence

The plasmonic enhancement or quenching effects of a silica-coated gold nanorod (GNR@SiO2) on the fluorescence of a molecule doped in the silica layer are studied using the multiple multipole method. The enhancement factors (EF) of a GNR with a typical aspect ratio of 3 coated by a 13nm silica layer upon the fluorescence of a molecule embedded at different locations with various orientations irradiated by a plane wave are analyzed, particularly at the longitudinal surface plasmon resonance (SPR) of GNR. The numerical results show that the EF of a GNR@SiO2 on the fluorescence is sensitive to the molecular location and orientation. Furthermore, an effective EF (EEF), which is an average of EF over all possible orientations at a specific location, is calculated. According to EEF, the proximities of the end-caps of a GNR are strong enhancing zones. In contrast, the waist area is the weak zone. Moreover, a bigger GNR (a=10nm) possesses a higher EEF than a smaller one (a=7nm) for the same aspect ratio and the molecular relative location. Hence, a strong enhancement on the fluorescence is obtained using bigger GNR, if the molecule is near the end-cup and the dipole orientation is along the long axis. On the contrary, the consequence could be quenching, if the molecule is near the waist of a small GNR. The Stokes shift of fluorescence can also affect the EF, except the excitation wavelength.

On the Validity of the Effective Index Method for Long Period Grating Photonic Crystal Fibers

This paper focuses on the investigation of modal characteristics and sensing properties of long period grating photonic crystal fibers (LPG-PCFs). An improved effective index method is employed with an objective to study its limitations for various designs of LPG-PCFs. Results so obtained with the above method are compared with the corresponding values of multiple multipole (MMP) method results which points the range of validity and applicability of the improved effective index method to LPG-PCFs. It is shown that this method is excellent when the surrounding media is assumed to be air. However, it becomes less accurate when the fiber is immersed into a liquid with a refractive index close to that of the cladding.

PLASMONIC EFFECT OF GOLD NANOSPHEROID ON SPONTANEOUS EMISSION

The plasmonic efiects of a gold prolate nanospheroid on the spontaneous emission of an adjacent emitter, regarded as an oscillating electric dipole, at the excitation and emission stages are studied respectively by using the multiple multipole method. The numerical results show that when an irradiating light is at the longitudinal surface plasmon resonance frequency of the nanospheroid and with a polarization parallel to the long axis, the strongest excitation rate occurs at the proximity of the long-axis vertex. In addition, if the emitter is at this region, and its orientation is also parallel to the long axis, the apparent quantum yield of the emission is the maximum, compared to the other locations and orientations. Therefore, for this case the overall enhancement factor of a nanospheroid on an emitter's spontaneous emission is the maximum. In contrast, the emitter's emission could be quenched, if it is near the short-axis vertex.