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Abstract
Optical forces in optical tweezers enable non-contact, non-destructive trapping and manipulation of particles. One such force has been found to originate from the spin-orbit coupling of light, which produces a counter-intuitive lateral optical force on metal nanoparticles due to the spin of the electric-field components of light. Here we reveal that the spin-orbit coupling of the magnetic-field components of light also produces a lateral optical force on particles. To study this lateral force, we designed a gapped structure composed of a dielectric particle near photonic crystal surface, and found that the lateral force originates from the spin-dependent excitation of a Bloch surface wave. We further demonstrate that the lateral force can be modified by tuning the structural parameters of the gapped structure and by exploiting the magnetic resonance modes of the particle. This work should contribute to a deeper understanding of the magnetic spin-orbit coupling between light and matter and promote the development of particle manipulation on dielectric platforms.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Optical forces originate from the mechanical effect of momentum transfer from light to matter. Generally, optical forces can be categorized as one of two types: gradient forces [1,2] and scattering forces (also known as radiation pressure) [3]. Gradient forces attract particles to the high-intensity focal spot of the light field along the direction of the electric-field gradient, whereas scattering forces push particles away in the direction of light propagation.
Since the discovery of optical forces in 1970 [4], they have attracted increasing research attention worldwide [5,6]. Research into optical forces led to the invention of optical tweezers by Ashkin, which rapidly developed into an important technique for trapping and manipulating micro-nano particles [7–12]. Because optical tweezers offer non-contact, high-precision manipulation, they are now widely used in scientific research in fields such as physics [13,14], chemistry [15], biology, and medicine [16] and were awarded the Nobel Prize in 2018.
In the past decade, extraordinary optical forces have been discovered and investigated, such as the optical traction force, whose direction is opposite to radiation pressure and thus can pull particles back toward the light source [17,18]. Another counter-intuitive optical force called the “lateral force” is exerted perpendicular to the incident direction of the light [19,20] and has also attracted increasing attention in recent years. The lateral force does not require light to be focused and so can act on multiple particles at different locations over a wide region to arrange, move, and sort the particles. Research into the lateral force initially focused on chiral particles [21,22], but it was later discovered that this force could also be exerted on symmetric particles [23]. The spin-orbit coupling effect of light [23] produces a lateral force on spherical metal nanoparticles because of the angular-momentum transfer from the spin of the incident electric-field components to the unidirectional excited transverse magnetic (TM) modes, including the surface plasmon polariton mode on the metal surface and the TM guided modes in dielectric layers.
In addition to electric-field components, the magnetic-field components of circularly polarized light also have a spin state; however, the properties of such rotating magnetic-field components have been little investigated. Recently, the magnetic field of light was also reported to interact with matter through spin-orbit coupling [24,25]. To better understand the properties of magnetic spin-orbit coupling of light, we investigate herein whether magnetic spin-orbit coupling can produce a lateral optical force on a spherical particle, as is the case for the electric-field component of light. Toward this end, we designed a gapped structure consisting of a dielectric nanoparticle near surface of a one-dimensional (1D) dielectric photonic crystal (PC) to numerically study the lateral force. The results show that the spin-orbit coupling of the magnetic-field components excites a unidirectional transverse-electric–polarized (TE-polarized) Bloch surface wave (BSW), which then induces a recoil force in the lateral direction, resulting in a lateral force being exerted on the particle. We further reveal that the magnetic-field–induced lateral force can be modified by tuning the chirality of the incident light and the structural parameters of the gapped structure. Finally, we analyze how the lateral force is affected by the magnetic resonance modes of the particle, including the magnetic-dipole and -quadrupole modes, and show that the lateral force is enhanced by the magnetic-field resonances in the particle.
2. Results and discussion
2.1 Basic principle of lateral force and BSW
Figure 1 shows schematically the basic principle behind the lateral force induced by magnetic spin-orbit coupling. Figure 1(a) shows left-circularly-polarized light incident from the + x direction on a spherical dielectric particle that is close to the surface of a 1D PC structure. Upon scattering from the particle, a TE-polarized BSW is excited on the top surface of the PC structure and then propagates in the −y direction. As opposed to the metallic nanoparticle used in the studies of electric spin-orbit coupling [23], we choose here a dielectric particle because it offers strong magnetic resonances. Note that, although the particle and the incident light are both rotationally symmetric, the existence of the flat photonic crystal breaks this symmetry, which leads to a non-symmetric scattering on the top surface of the photonic crystal, thus resulting in the unidirectional excitation of a BSW. Because the momentum of the BSW is obtained from the spin angular momentum of the light via magnetic spin-orbit coupling, the BSW propagates in the + y direction upon changing the incident polarization state to right-circular polarization, as shown in Fig. 1(b). Accompanied by the unidirectional propagation of BSWs, the particle gains momentum in the opposite direction, which reflects the lateral recoil force Fy acing in the direction opposite that of the BSW propagation.
Fig. 1. Schematic diagram of lateral optical force induced by magnetic spin-orbit coupling of light. (a) A left-hand circularly polarized light is incident on a spherical dielectric particle close to a 1D PC structure surface and then excites a BSW propagating in the −y direction, while the particle is subjected to a lateral force in the + y direction. (b) Similar to panel (a) except that the incident light is right-hand circularly polarized.
Figure 2 gives more details on the gapped structure and the excited BSW. In Fig. 2(a), the 1D PC is composed of seven alternating layers of SiO2 (n = 1.46) and Si3N4 (n = 2.6). The SiO2 and Si3N4 layers are 110 nm and 66 nm thick, respectively, and a 450-nm-thick SiO2 defect layer covers the 1D PC as the defect layer. The top SiO2 defect layer is covered with water to match the usual environment of optical tweezers. Previous theoretical and experimental works have proved that such materials and optimized structural parameters can effectively excite Bloch surface waves [26,27]. A spherical silicon particle is placed 25 nm above the top SiO2 defect layer, and the light is incident on the particle in the + x direction. The red curve in Fig. 2(a) shows the intensity distribution of the TE-polarized BSW in each layer of the PC structure. The strongest BSW field occurs in the top layer and spreads into the water, so it is usually considered as the surface wave mode. Figure 2(b) shows the electromagnetic-field components of the excited BSW in the yz plane, as calculated by three-dimensional finite-difference time-domain (FDTD) simulations. We find that the excited BSW is TE polarized because it contains only Ex, Hy, and Hz components, whereas its TM-polarized components (Hx, Ey, and Ez) are near zero (not shown in Fig. 2). In the BSW, the two magnetic-field components Hy and Hz are shifted by λ/4 in the y direction (corresponding to a π/2 phase difference), so they form the spin of the magnetic field. Figure 2(c) plots the oscillation of the electric and magnetic fields of the BSW, clearly showing the transverse spin of the magnetic field of the BSW, which differs from the surface plasmon polariton excited by the spin of electric field [28]. Figure 2(d) compares how different polarizations of incident light affect the excitation of the BSW. Here we show only the Ex component of the BSW because this component does not exist in the incident light but is induced by the magnetic spin-orbit coupling. In Fig. 2(d), when the incident light is linearly polarized in the y direction, the 95-nm-radius particle scatters light simultaneously to both sides and thus excites BSWs of equal intensity on both sides. When the incident light is left-hand circularly polarized, the magnetic field of light spins in a clockwise direction and is non-symmetrically scattered onto the PC surface, exciting a BSW mainly in the −y direction. The spin of the magnetic field of the BSW is consistent with the incident light, satisfying the law of conservation of angular momentum. When the incident light is right-hand circularly polarized, the result is reversed to the left-spin case, just as shown in the bottom of Fig. 2(d). The unidirectionally propagating BSW is derived from the spin-orbit coupling of the magnetic field, so the propagating direction of the BSW can be tuned by modifying the chirality of the incident light, thereby changing the direction in which the lateral recoil force is exerted on the particle.
Fig. 2. Unidirectional excitation of TE-polarized BSW by magnetic spin-orbit coupling. (a) Schematic diagram of the gapped structure: a spherical silicon particle is placed 25 nm above the top surface of the SiO2 defect layer, the 1D PC is composed of seven alternating layers of SiO2 and Si3N4, the red curve gives the intensity of the supported TE-polarized BSW mode in each layer. (b) The electromagnetic-field components of the excited BSW in the y-z plane calculated by FDTD simulations, the dashed line represents the upper surface of the defect layer. (c) Oscillations of the electric and magnetic fields of the BSW. (d) From top to bottom, absolute value of the real part of Ex component excited by linearly polarized (LP) light, left-hand circularly polarized (LCP) light, and right-hand circularly polarized (RCP) light. The dashed line indicates the top surface of the defect layer. The wavelength of light is 633 nm, and the particle radius is 95 nm.
2.2 Effect of polarization state of incident light
To show that the spin-controlled unidirectional excitation of the BSW can produce a lateral force on a particle, we study how the polarization of the incident state affects the lateral force and the BSW (see Fig. 3). Calculation method of the lateral force is described in the Methods section. Figure 3(b) plots the lateral optical force (in the y direction) as a function of the polarization of the incident state. Here we consider that the incident polarization state varies along the blue circle on the Poincaré sphere [29], as shown in Fig. 3(a), where the north and south poles represent right and left circular polarization, respectively, and linear polarization is at the equator. The angle $\chi$ is given as follow,
Fig. 3. Effect of the incident polarization state on the lateral force and the BSW energy flow. (a) Definition of angle $\chi$ on Poincaré sphere, where the blue circle gives the variation of the incident polarization state. (b) Optical lateral force Fy as a function of polarization state of the incident light. Typical polarization states appear at the top, including left- and right-hand circular polarization and x and y linear polarization. (c) Normalized difference of BSW energy flows propagating towards the left (−y direction, PL) and right (+y direction, PR) sides on the surface as a function of polarization state of the incident light.
Figure 3(b) shows that the direction of the lateral force Fy is determined by the chirality of the incident light. When the chirality of the incident light is left-handed, the force is always positive (in the + y direction), and vice versa. The magnitude of the force reaches a positive (negative) maximum for left-hand (right-hand) circular polarization, which confirms that the lateral force derives from the spin of light. Figure 3(c) shows the difference of the BSW energy flow (Poynting vector) when the BSW propagates to the left (−y direction) and right (+y direction) on the surface. The variation of the difference in BSW energy flow is clearly the same as that of the lateral force [Fig. 3(b)]; that is, the unidirectional excitation of the BSW is always accompanied by a lateral force in the opposite direction, indicating that the recoil from the BSW energy flow produces the lateral force on the particle.
2.3 Effect of gap size
We also studied how the gap between a nanoparticle and a SiO2 defect layer affects the BSW excitation efficiency and the corresponding lateral force. With incident left-hand circularly polarized light, we changed the gap size between the particle and the multi-layer structure to calculate the lateral force Fy. The curve in Fig. 4(a) shows that the lateral force decreases with increasing gap size. Figure 4(b) shows the difference in BSW energy flow between the left and right sides for different gap sizes, which presents a similar decreasing trend as the lateral force, indicating that the drop in BSW energy flow reduces the lateral recoil force. To further investigate the gap size, we consider in Figs. 4(c) and 4(d) two typical cases with gap sizes of 30 nm and 100 nm and compare the corresponding electric-field intensity near the gap region and the corresponding component Ex of the BSW, respectively. The comparison shows that the near-field intensity in the gap region decreases with increasing gap size [Fig. 4(c)], which means a decrease in coupling efficiency between the light scattering from the particle and the BSW mode, thus resulting in weaker BSW field [Fig. 4(d)] and a smaller lateral force. It is noted that, compared to the case due to electric spin-orbit coupling in plasmonic metal structure [23], here the lateral force is weaker because it is not able to form a strong localized field in the gap region [Fig. 4(c)] due to the whole dielectric structure [30]. However, the whole dielectric structure generates little heat, and thus the Brownian motion of particles is weaker than the case of metal structure [30], showing specific advantage in particle trapping, sorting and other applications.
Fig. 4. Effect of the gap size on the lateral force and the BSW. (a) Lateral force Fy as a function of gap size from 25 to 100 nm. (b) Normalized difference of BSW energy flow between left and right sides as a function of gap size. (c) Distribution of electric-field amplitude |E| near the gap region for gap sizes of 30 nm (top panel) and 100 nm (bottom panel). The white dashed line represents the upper surface of the defect layer, and the black dashed curve gives the bottom surface of the particle. (d) |Re(Ex)| for excited BSW at the gap of 30 nm (top panel) and 100 nm (bottom panel).
2.4 Effect of particle size and magnetic resonance modes
The magnetic spin-orbit coupling in our structure relies on light scattering from a particle, so the particle size should play an important role in the scattering and thereby affect the resulting BSW and lateral force. To study this effect, we consider incident left-hand circularly polarized light and a gap of 25 nm and then change the particle radius to see how the lateral force changes [see Fig. 5(a)]. As the particle radius increases, the lateral force peaks twice, once each at the radius of 95 nm and 132.5 nm. Figure 5(d) shows the difference in BSW energy flow between the left and right sides as a function of gap size, which has a similar two-peak form as the lateral force, indicating that the change in BSW energy flow is intimately connected with the lateral recoil force.
Fig. 5. Effect of the particle size on the lateral force and the magnetic resonance modes. (a) Lateral force Fy as a function of particle radius when the incident light is left-hand circularly polarized and the gap size is 25 nm. (b) Distribution of electric current (black arrows) in xz plane for a particle radius of 95 nm (background shows Hy intensity). (c) Same as panel (b) except in xy plane and with background showing Hz intensity. (d) Difference of BSW energy flow between left and right sides as a function of particle radius. (e) Same as panel (b) except for a particle radius of 132.5 nm. (f) Same as panel (c) except for a particle radius of 132.5 nm.
To determine the physical mechanism responsible for the two peaks, we investigated the distributions of magnetic field and electric current in the particle at each peak. Figures 5(b) and 5(c) show the results for the particle at the first peak [95 nm in Fig. 5(a)] in the xz and xy planes, respectively. Figures 5(e) and 5(f) show the results for the particle at the second peak [132.5 nm in Fig. 5(a)] in the xz and xy planes, respectively. The electric current vectors in both the xz and xy planes form a single vortex (four vortices) for a particle radius of 95 nm (132.5 nm), accompanied by a magnetic-dipole (magnetic-quadrupole) resonance at the center of vortices [31,32], indicating that the magnetic-dipole and -quadrupole resonances of the particle enhance the magnetic spin-orbit coupling and thus lead to the two peaks in the lateral force. To further compare the contributions from electric and magnetic field, we calculate the enhancement factor of the electric/magnetic field (ratio of the maximum field amplitude to the incident one) at the particle radius of 50 nm, 95 nm, 120 nm, and 132.5 nm, corresponding to the two dips and two peaks in the curve of lateral force [Fig. 5(a)]. We find that at the two lateral force peaks, the calculated enhancement factor of magnetic field (5.96 at 95 nm and 5.77 at 132.5 nm) is always larger than that of electric field (1.72 at 95 nm and 2.99 at 132.5 nm), verifying that the contribution of magnetic field dominates in the lateral force. We also find that the enhancement factors of magnetic field at the two lateral force peaks (5.96 at 95 nm and 5.77 at 132.5 nm) are always larger than that at the dips (1.98 at 50 nm and 2.72 at 120 nm), proving that the lateral force achieves peaks at the resonances of magnetic field but is closed to zero without resonance. These results further confirm that the lateral force in this structure originates from the magnetic field of light.
3. Method
In this work, all electromagnetic-field results are obtained by three-dimensional finite-difference time-domain simulations (Lumerical FDTD Solutions). In FDTD simulations, the grid size is non-uniform ranging from 5 nm near material interfaces to 20 nm further from these boundaries. A perfectly matched layer is placed around the simulation area. The incident circularly-polarized light is composed by two orthogonal lineally-polarized plane-wave beams of equal intensity but with π/2 phase difference, and its polarization state can be modulated by changing the amplitudes of the two plane wave beams and the phase difference between them. The wavelength of incident light is 633 nm, and the corresponding refractive indices of different materials are chosen as: SiO2 (n = 1.46), Si3N4 (n = 2.6), silicon particle (n = 3.5), water (n = 1.333).
The optical force exerted on the particle is calculated by using the Maxwell stress tensor method [33],
4. Conclusion
In this work, we designed a gap structure consisting of a dielectric spherical nanoparticle near a 1D dielectric photonic crystal surface to numerically study whether magnetic spin-orbit coupling of light can produce a lateral force on a spherical particle, as is the case for electric spin-orbit coupling. The results show the spin-orbit coupling of the magnetic-field components of light excites a unidirectional BSW by breaking the symmetry, which then induces a recoil force in the lateral direction, which constitutes a lateral force on the particle. The lateral force can be modulated by tuning the chirality of the incident light and the structural parameters such as gap size and particle size. Finally, we analyzed how the magnetic resonance modes of the particle affect the lateral force, which shows that the magnetic-dipole and -multipole resonance modes of the dielectric particle strongly enhance the lateral force. Combined with the previous study of a lateral force induced by electric spin-orbit coupling, the present results show that both the spin of electric field and magnetic field of light can produce a unidirectional TM or TE propagation mode and produce a lateral optical force whenever spin-orbit coupling breaks the symmetry of a system.
Although this work is a theoretical research, the corresponding experiment can be implemented based on the previous experimental works of optical tweezers of Bloch surface waves [30] and the experimental works of other lateral force studies [20]. In addition, since the thermal effect of our all-dielectric system is negligible compared to other systems where plasmons and/or strongly confined electromagnetic fields are involved, the Brownian motion and convection motion due to local temperature enhancement could be weaker in our system as proved in previous experimental work [30], and thus resulting in more stable performance for optical trapping and manipulation of nanoparticles. This work should contribute to a deeper understanding of the magnetic spin-orbit coupling of light and promote the development of particle-manipulation techniques.
Funding
National Natural Science Foundation of China (grant numbers 91750205, 61935013, 61975128); Leading Talents of Guangdong Province Program (grant number 00201505); Natural Science Foundation of Guangdong Province (grant number 2016A030312010); Science and Technology Innovation Commission of Shenzhen (grant numbers JCYJ20180507182035270, KQTD2017033011044403, ZDSYS201703031605029, JCYJ2017818144338999).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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