Tunable optical forces exerted on a black phosphorus coated dielectric particle by a Gaussian beam

Yang Yang; Xing Jiang; Banxian Ruan; Xiaoyu Dai; Yuanjiang Xiang

doi:10.1364/OME.8.000211

1. Introduction

Optical tweezers have already been an indispensable tool in manipulating microscopic particles since the pioneering work of A. Ashkin [1]. This non-contact technique is widely applied in multiple disciplines such as optics [2–8], atomic physics [9] and biological science [10]. Now, optical manipulations has been extended to more sophisticated systems including nanostructures [11], chiral particles [12,13], near-field optics [14], and quantum cavity mechanics [15]. Optical force plays a crucial role in these applications [16, 17].

Currently there is a considerable interest in two dimensional (2D) materials due to their extraordinary physical properties [18]. A rediscovered 2D material, single-layer black phosphorus or phosphorene has been successfully fabricated and immediately became the new focus of scientific research [19–22]. Unlike the zero-gap semimetal graphene, black phosphorene (BP) is a semiconductor with a direct bandgap and the band gap is also tunable [23–25]. Moreover, phosphorene exhibits high carrier mobility [26], high on/off current ratio [27] and remarkable anisotropic optical properties [28, 29], it is a promising material for applications in nano-optics and optoelectronics [30].

In view of the distinctive optical properties of BP, in this work, we propose a method that wrapping dielectric particles with BP layers to improve the interaction between the microscopic particle and electromagnetic waves and enhance the optical manipulation. Utilizing the Lorenz–Mie theory, we systematically study on the optical forces exerted on a BP coated particle by a focused Gaussian beam. The detailed results and discussions are shown as follows.

2. Theory

In the Lorenz–Mie theory framework, the general electromagnetic wave is expanded into vector wave functions in the spherical coordinate system with the origin coincides with particle center. The expansion coefficients for the incident electromagnetic field in arbitrary beam theory (ABT) are [31]:

\begin{array}{l} A_{l m} = \frac{1}{l (l + 1) ψ_{l} (α_{2})} \int_{0}^{2 π} \int_{0}^{π} E_{r}^{(i)} (R, θ, ϕ) Y_{l m}^{*} (θ, ϕ) \sin θ d θ d ϕ, \\ B_{l m} = \frac{1}{l (l + 1) ψ_{l} (α_{2})} \int_{0}^{2 π} \int_{0}^{π} H_{r}^{(i)} (R, θ, ϕ) Y_{l m}^{*} (θ, ϕ) \sin θ d θ d ϕ . \end{array}

where R is the particle radius, θ is the polar angle and ϕ is the azimuthal angle in the spherical coordinate,

Y_{l m} (θ, ϕ)

is the spherical harmonics. The following procedure is to obtain the electromagnetic wave solution for the scattering and internal fields of the BP coated dielectric particle by applying boundary conditions at the surface of the sphere.

Here, the BP monolayer coating is considered as a surface current around the spherical particle. The boundary conditions at the interface r = R are: the tangential component of the total electric field E is continuous and the discontinuity of tangential component of the total magnetic field H is proportional to the surface current density K, then boundary conditions can be expressed as:

\begin{array}{l} e_{r} \times (E_{i} + E_{s} - E_{w}) = 0. \\ e_{r} \times (H_{i} + H_{s} - H_{w}) = K = σ (ω) E_{∥} . \end{array}

where

e_{r}

is the normal vector,

E_{∥}

is the tangential component of the total electric field. Enforcing the continuous boundary conditions, the expansion coefficients of the scattered electromagnetic field by the BP coated particle are derived as [32]:

a_{l} = - \frac{\tilde{n} ψ_{l} (α_{1}) {ψ^{'}}_{l} (α_{2}) - ψ_{l} (α_{2}) {ψ^{'}}_{l} (α_{1}) + (1 / n_{2}) S (ω) \cdot {ψ^{'}}_{l} (α_{1}) {ψ^{'}}_{l} (α_{2})}{\tilde{n} ψ_{l} (α_{1}) {ξ^{'}}_{l} (α_{2}) - ξ_{l}^{(1)} (α_{2}) {ψ^{'}}_{l} (α_{1}) + (1 / n_{2}) S (ω) \cdot {ψ^{'}}_{l} (α_{1}) {ξ^{'}}_{l} (α_{2})},

b_{l} = - \frac{\tilde{n} {ψ^{'}}_{l} (α_{1}) ψ_{l} (α_{2}) - ψ_{l} (α_{1}) {ψ^{'}}_{l} (α_{2}) + (1 / n_{2}) S (ω) \cdot ψ_{l} (α_{1}) ψ_{l} (α_{2})}{\tilde{n} {ψ^{'}}_{l} (α_{1}) ξ_{l} (α_{2}) - ψ_{l} (α_{1}) ξ_{l}^{(1)}^{'} (α_{2}) + (1 / n_{2}) S (ω) \cdot ψ_{l} (α_{1}) ξ_{l} (α_{2})},

where the variables

α_{1} = k_{1} R, α_{2} = k_{2} R, \tilde{n} = n_{1} / n_{2} .

k = ω/c is the wave number in vacuum. k₁ and k₂ are the wave vector inside and outside the particle.

ξ_{l}^{(1)} = ψ_{l} - i χ_{l},

ψ_{l}

and

χ_{l}

are the Riccati-Bessel functions.

Also the expansion coefficients for the interior electromagnetic field inside the BP coated particle are obtained as [32]:

c_{l} = \tilde{n} \frac{ψ_{l} (α_{2}) {ξ^{'}}_{l} (α_{2}) - {ψ^{'}}_{l} (α_{2}) ξ_{l}^{(1)} (α_{2})}{\tilde{n} ψ_{l} (α_{1}) {ξ^{'}}_{l} (α_{2}) - ξ_{l}^{(1)} (α_{2}) {ψ^{'}}_{l} (α_{1}) + (1 / n_{2}) S (ω) \cdot {ψ^{'}}_{l} (α_{1}) {ξ^{'}}_{l} (α_{2})},

d_{l} = \frac{{ψ^{'}}_{l} (α_{2}) ξ_{l} (α_{2}) - ψ_{l} (α_{2}) ξ_{l}^{(1)}^{'} (α_{2})}{\tilde{n} {ψ^{'}}_{l} (α_{1}) ξ_{l} (α_{2}) - ψ_{l} (α_{1}) ξ_{l}^{(1)}^{'} (α_{2}) + (1 / n_{2}) S (ω) \cdot ψ_{l} (α_{1}) ξ_{l} (α_{2})} .

The optical force F exerted on a spherical particle is determined by integrating the Maxwell’s stress tensor of the illuminated electromagnetic wave over the surface enclosing the particle, it can be expressed as a series over the coefficients A_lm, B_lm , a_lm and b_lm [33], where

a_{l m} = a_{l} A_{l m}, b_{l m} = b_{l} B_{l m} .

The optical properties of a BP monolayer is described by employing the semi-classical Drude model [34, 35]. The direction-dependent conductivities can be expressed as:

σ_{j j} = \frac{i D_{j}}{π (ω + i η / ℏ)}, D_{j} = \frac{π e^{2} n}{m_{j}},

where D_j is the Drude weight, j represents the x and y axis directions, ω is the angular frequency of the incident wave, and η is the electron relaxation rate of BP. n is the carrier density which is determined by electron doping. The electron masses in the x-direction (armchair) and y-direction (zigzag) directions of BP layers can be described by

m_{x} = \frac{ℏ^{2}}{\frac{2 γ^{2}}{Δ} + η_{c}}, m_{y} = \frac{ℏ^{2}}{2 v_{c}},

where

η_{c}

and

v_{c}

are related to the effective masses, γ describes the effective coupling of band edges, and Δ is the energy band gap. Through fitting the known anisotropic mass, the parameters for monolayer BP are: γ = 4a/π eVm,

η = 10 eV,

Δ = 2 eV,

η_{c} = ℏ^{2} / 0.4 m_{0}

and

η_{c} = ℏ^{2} / 1.4 m_{0},

respectively. a = 0.223 nm stands for the scale length of BP and π/a is the width of Brillouin Zone. m₀ is the electron mass, and ħ is the reduced Planck's constant. Since the anisotropic conductivity of black phosphorus, unless the incident wave is polarized along one of the crystal axes, the reflected polarization will be different from the incident one. However, when the surface conductivity is much smaller than the free-space admittance as the considered case here,

σ_{j j} ≪ \sqrt{ε_{0} * μ_{0}},

this polarization rotation can be neglected.

The schematic of scattering problem is shown in Fig. 1. A focused Gaussian beam with waist radius w₀ = 2 μm is incident upon a BP coated polystyrene particle with radius R. Incident power P = 10 mW. The ambient medium is air. The refractive index of polystyrene particle is n₁ = 1.59. The particle is placed near the center of beam waist of (x = w₀/2, y = 0, z = 0).

Fig. 1 A focused Gaussian beam with waist radius w₀ is incident upon a BP-coated polystyrene spherical particle with radius R. The angle between the polarization of incident Gaussian wave and the main armchair direction in the scattering cross section plane is θ.

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Due to the anisotropic surface conductivity, the optical feature of BP layer is a function of the angle between the polarization of incident wave and the BP crystal axes. When the BP layer is wrapped a spherical particle, the anisotropy of surface conductivity is treated as follows. We consider the scattering cross section of the BP coated particle that is perpendicular to the polarization of incident wave. In this perpendicular cross section plane, the main armchair and zigzag axes are selected as the armchair and zigzag axes that are through the center of spherical particle. Suppose the angle between the polarization of incident wave and the main armchair direction in the scattering cross section plane is θ, then the surface conductivity of BP is $σ = σ_{x x} \cos^{2} (θ) + σ_{y y} \sin^{2} (θ) .$ $σ_{x x}$ and $σ_{y y}$ are the conductivity along the armchair and zigzag directions, respectively. It is legitimate to make such an approximation, because most armchair and zigzag axes in the cross section plane of the BP coating are along the direction of the main armchair axis and zigzag axis on the whole, they account for the major contribution of the BP conductivity. The dimensionless surface conductivity S(ω) of the particle can be modified by tuning the 2D conductivity of BP thin film, which has the relationship $S (ω) = i σ / ε_{0} ω R,$ $ε_{0}$ is the dielectric constant in vacuum.

3. Results and discussion

First, we investigate the variation of vertical optical force spectra F_z of the BP-coated polystyrene particle exerted by a focused Gaussian beam with different particle size and carrier density. The results are shown in Fig. 2(a) and (b). The mathematical description of the electromagnetic field of Gaussian beam for evaluation of the integral in Eq. (1) is given in [36]. As shown in Fig. 2(a), with the increasing of particle size, the magnitude of resonant peaks is enhanced, the corresponding wavelength of the resonant peaks red shifts. In Fig. 2(b), with the carrier density increasing, the magnitude of resonant peaks increases gradually and the corresponding resonant wavelength blue shifts. Figure 2(c) presents the variation of horizontal optical force F_x with the lateral displacements of particle with respect to the beam center at wavelength λ = 6 nm. It can be seen that the horizontal force can be tuned by characterizing the carrier density of BP layer, and this bidirectional oriented gradient force F_x would pull the particle to the center of beam waist laterally. It demonstrates that the optical force and resonant wavelength can be modified effectively by tuning the carrier density of the BP layer effectively, which can be quantified by chemical doping or applying gate voltage. This behavior can be well applied in the dynamic optical manipulations.

Fig. 2 Vertical optical force spectra F_z of the BP-coated polystyrene particle exerted by Gaussian beam with different: (a) particle radius, while carrier density n = 3 × 10¹³ cm⁻²; (b) carrier density, while particle radius R = 100 nm. (c) Horizontal optical force F_x as a function of lateral displacement of particle with respect to the beam center at wavelength λ = 6 nm.

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In the following we investigate the impacts of some unique characteristics of the BP layers on the optical force. Due to the anisotropic properties of BP, the surface conductivity of BP is a function of angle between the polarization of incident wave and the armchair or zigzag direction. The angle between the polarization of and the armchair direction is θ, then the optical conductivity of BP is $σ = σ_{x x} \cos^{2} (θ) + σ_{y y} \sin^{2} (θ) .$ Fig. 3(a) shows the angular dependence of optical force spectra on the anisotropic conductivity of BP coating. As we can see, when the angle θ between the polarization of incident wave and the armchair direction increases, the magnitude of the resonant peaks decreases gradually and the shape of peaks is broadened, meanwhile the resonant wavelength red shifts. This is because the surface conductivity along the armchair direction is much larger than the zigzag direction, the larger optical conductivity would excite stronger plasmonic resonant peaks with narrower width. Also, the resonant wavelength red shifts to larger wavelength with smaller exciting energy.

Fig. 3 Variation of the vertical optical force spectra of the BP-coated polystyrene particle with different: (a) angle between the polarization of incident wave and the armchair direction; (b) number of BP layers; (c) electron relaxation rate η. (d) Optical force spectra of a BP-coated Drude particle.

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BP bulk possesses a direct band gap of 0.3 eV located at the Z point [19, 20]. This direct gap moves to the Γ point and increases to 1.5–2 eV when the thickness decreases from bulk to few layers and eventually monolayer. In the GW approximation, the band gap of the N-layer BP film can be described by $E_{g}^{N} = (1.7 / N^{0.73} + 0.3) eV$ [23]. Figure 3(b) shows the variation of optical force spectra as a function of BP layer numbers (N up to 3). As we can see, with the layer number increasing, the band gap of BP decreases, the magnitude of optical force resonant peaks increases slightly, the corresponding resonant wavelength blue shifts. The reason is that when the band gap of BP reduced, the conductivity along both the armchair and zigzag directions is enhanced. The enhanced conductivity generates the stronger plasmonic resonant peaks, meanwhile the resonant peaks blue shifts.

We also examine the impact of electron relaxation rate η of the BP monolayer on the optical force spectra shown in Fig. 3(c). It shows that when the electron relaxation rate is reduced, the magnitude of optical forces is greatly enhanced. Meanwhile, the width of resonant peak is reduced, the peak becomes quite sharper. This variation trend is easy to understand. The electron relaxation rate characterizes the transport properties of the BP monolayer, which is inversely proportional to electron relaxation time. The carriers with low scattering rate and long relaxation time possess the high mobility, which would generate larger magnitude plasmonic resonant peaks with sharper shape.

The BP coating can also be applied for Drude particle. The Drude model well describes the metal's dispersion and permittivity: $ε (ω) = ε_{\infty} - ω_{p}^{2} / (ω^{2} + i ω γ),$ where ε_∞ is the dielectric constant at the infinite frequency. ω_p and γ are the bulk plasma frequency and electron collision frequency, respectively. Here we choose the silver particle, of which the physical parameters are $ε_{\infty} = 3.7,$ ω_p = 9.1 eV and γ = 0.018 eV [37]. We show the vertical optical force spectra of BP monolayer coated silver particle in Fig. 3(d), and compare it with a bare silver particle (in inset). It is seen for a 100 nm radius silver particle, the surface plasmon resonance is excited around wavelength of 0.5-1.5 μm. When coated by a monolayer BP, the resonant wavelength is tuned into the infrared wavelength band of about 6.46 um. It demonstrates that the BP coating could also be used to adjust the surface plasmon resonance of a Drude particle efficiently.

In order to get a deeper physical insight into the resonant mechanism of optical force of the BP coated dielectric particle in the infrared band, we plot the electric field |E| distribution of the BP coated polystyrene particle on resonance (R = 100 nm, n = 3 × 10¹³ cm⁻², incident wavelength λ = 5.981 μm), and compare it with a bare polystyrene particle with the same exciting condition in Fig. 4(b) and 4(a), respectively. The unit of electric field magnitude is statvolt/cm. As we can see, on resonance the field intensity of the BP coated polystyrene particle is one order of magnitude larger than that of bare polystyrene particle, this greatly enhanced field intensity generates the resonant peak on the optical force spectra. Furthermore, indicated by the Poynting vectors in Fig. 4(b), the incident wave flows into the center of BP coated polystyrene particle with coupling, induces the enhanced and more localized electric field around the particle. While for the bare polystyrene particle in Fig. 4(a), the incident wave nearly passes through particle.

Fig. 4 Comparison of the electric field distributions of (a) bare polystyrene particle and (b) BP coated polystyrene on resonance. (c) (d): whispering gallery mode field distributions of the BP coated polystyrene particle on the electric multipole resonance of l = 6 and l = 10, respectively.

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Figure 4(c)-4(d) shows the whispering gallery mode (WGM) field distributions of the BP coated polystyrene particle on the electric multipole resonance of l = 6 and l = 10, respectively. The emerge of WGM field pattern inside the BP coated particle originates from the resonance of multipole coefficients $a_{l}$ and $b_{l}$ of the partial electric and magnetic scattering waves, which is provided for $Re (a_{l}) = 1$ or $Re (b_{l}) = 1$ , respectively [38–40]. It is noticeable that on the multipolar resonance, most incident wave is coupled into the BP coated polystyrene particle. The wave undergoes total internal reflection inside the particle and a traveling wave along the surface occurs. The mutual interference of the internal travelling wave generates a lap of spots along surface that exhibits the WGM pattern, the number of spots of WGM is equal to the multipole resonance order.

In addition to Gaussian beam, we proceed to investigate the interaction of BP coated particle with complex profile beams and exploring the high order localized surface modes exciting inside the particle. Here we explore the Laguerre Gaussian (LG) beam, of which the wave front is helical and carries the angular orbital momentum (OAM). It has cylindrical symmetry and a special helical phase $\exp (i s ϕ),$ where s is the azimuthal mode index [41]. Suppose the incident LG beam is polarized along x direction and propagating along z direction, the vector potential A can be expressed by:

\begin{array}{l} A = {\hat{e}}_{x} u (r, θ, z) \exp (i k z), \\ u (r, θ, z) = A_{p s} (\frac{w_{0}}{w_{z}}) f_{p s} (ρ) \exp (- i \frac{k r^{2}}{2 R (z)}) \exp (i s ϕ) \exp [i (2 p + s + 1) ζ (z)] . \end{array}

where the parameters are related with the Rayleigh range

z_{R} = k w_{0}^{2} / 2

,

\begin{array}{l} w_{z} = w_{0} {[1 + {(\frac{z}{z_{R}})}^{2}]}^{1 / 2}, ζ (z) = \tan^{- 1} (\frac{z}{z_{R}}), R (z) = \frac{z^{2} + z_{R}^{2}}{z}, \\ f_{p s} (ρ) = ρ^{s} L_{p}^{s} (ρ^{2}) \exp (- ρ^{2} / 2), ρ = \frac{\sqrt{2} r}{w_{z}}, A_{p s} = \frac{2}{w_{0} \sqrt{1 + δ_{0 s}}} \sqrt{\frac{p!}{π (p + s)!}} . \end{array}

_{$L_{p}^{s} (ρ^{2})$}is the generalized Laguerre polynomial. Then the electromagnetic fields of LG beam can be derived from the Maxwell’s equations [42].

\begin{array}{l} H = (1 / μ) \nabla \times A, \\ E = - i ω [A + (1 / k^{2}) \nabla (\nabla \cdot A)] . \end{array}

The extinction power of electromagnetic wave illuminating on the nano-particle are [33]:

W_{e x t} = Re (E_{i} \times H_{s}^{*} + E_{s} \times H_{i}^{*}) = - \frac{c}{8 π} k^{2} Re \sum_{l m} l (l + 1) [Re (A_{l m} a_{l m}^{*} + B_{l m} b_{l m}^{*})] .

The normalized extinction coefficient Q_ext is

Q_{e x t} = W_{e x t} / (I_{i} \cdot σ) .

where I_i is the intensity of light,

σ = π R^{2}

is the geometric cross-section of a spherical particle.

Figure 5 shows the calculated optical extinction spectra Q_ext of a BP coated dielectric particle excited by LG beam of order p = 0 with vortex phase: s = 0~5. The lines in the figure represent the extinction spectra contributed by all modes and discrete plasmon modes of dipole, quadrupole, octupole, l = 4, l = 5, respectively. As we can see, with the vortex phase s of LG beam increasing, the normalized value Q_ext of the whole extinction spectra decreases. Furthermore, the low order localized surface plasmon modes disappear gradually, and the high order surface plasmon modes are excited in the infrared wavelength band. Especially, the dipole mode dominates for helical phase s = 0, 1, 2 (LG₀₀, LG₀₁, LG₀₂), while the octupole mode emerges and dominates gradually with s continuous growing, and the l = 5 mode is excited and dominates for s = 5 (LG₀₅). Meanwhile, the corresponding resonant wavelength blue shifts when the order of surface plasmon mode increasing. It illustrates that the multi-polar surface plasmons of the BP coated dielectric particle could be excited and modified by tuning the helical phase factor of LG beam in the infrared wavelength band.

Fig. 5 Optical extinction spectra Q_ext of a BP coated dielectric particle excited by LG beam of order p = 0 with vortex phase factor: s = 0~5. The lines in the figure represent the extinction spectra contributed by all modes and discrete plasmon modes of dipole, quadrupole, octupole, l = 4, l = 5, respectively.

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The reason is that the OAM carried by the LG beam is associated with the helical wavefront resulting in an azimuthal component to the Poynting vector of the field. An Laguerre-Gaussian mode possesses an OAM of $s ℏ$ per photon [41], the high order LG beam carry more of orbital angular momenta with annular regions of high optical intensity. The OAM of LG beam can be transferred to the illuminated particle due to light scattering [43, 44], the mutual interaction of the OAM of different order with the BP coated particle give rise to the corresponding localized surface plasmons mode exciting. This multi-polar surface plasmons resonant behavior of the BP coated particle opens the path to new studies in the area of optomechanics in vacuum and optical rotations [45, 46].

4. Conclusions

In summary, we give a full description of the optical forces exerted on a BP coated dielectric particle by a focused Gaussian beam. The optical force of the BP coated particle can be tuned effectively by modifying the characteristics of BP layer. The resonant behavior and WGM field distribution of the BP coated particle are investigated. Moreover, the multi-polar surface plasmons of the BP coated dielectric particle excited by the LG beam through the OAM transfer are also investigated. The proposed work could find rich potential applications in the dynamic optical tweezers and nano-optics.

Funding

National Natural Science Foundation of China (Grant Nos. 61505111 and 11604216); China Postdoctoral Science Foundation (Grant Nos. 2017M622746 and 2016M600667); Science and Technology Planning Project of Guangdong Province (Grant No. 2016B050501005); Educational Commission of Guangdong Province (Grant No. 2016KCXTD006); Guangdong Natural Science Foundation (Grant No. 2015A030313549).

References and links

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

2. L. Jauffred, S. M. Taheri, R. Schmitt, H. Linke, and L. B. Oddershede, “Optical Trapping of Gold Nanoparticles in Air,” Nano Lett. 15(7), 4713–4719 (2015). [CrossRef] [PubMed]

3. V. Kajorndejnukul, W. Q. Ding, S. Sukhov, C. W. Qiu, and A. Dogariu, “Linear momentum increase and negative optical forces at dielectric interface,” Nat. Photonics 7(10), 787–790 (2013). [CrossRef]

4. M. Nieto-Vesperinas, “The optical torque: Electromagnetic spin and orbital angular momenta conservation laws and their significance,” Phys. Rev. A 92(4), 043843 (2015). [CrossRef]

5. S. Kawata and T. Sugiura, “Movement of micrometer-sized particles in the evanescent field of a laser beam,” Opt. Lett. 17(11), 772–774 (1992). [CrossRef] [PubMed]

6. C. W. Qiu, W. Ding, M. R. C. Mahdy, D. Gao, T. Zhang, F. C. Cheong, A. Dogariu, Z. Wang, and C. T. Lim, “Photon momentum transfer in inhomogeneous dielectric mixtures and induced tractor beams,” Light Sci. Appl. 4(4), e278 (2015). [CrossRef]

7. D. L. Gao, W. Q. Ding, M. N. Vesperinas, X. M. Ding, M. Rahman, T. H. Zhang, C. T. Lim, and C. W. Qiu, “Optical manipulation from the microscale to the nanoscale: fundamentals, advances and prospects,” Light Sci. Appl. 6(9), e17039 (2017). [CrossRef]

8. O. M. Maragò, P. H. Jones, P. G. Gucciardi, G. Volpe, and A. C. Ferrari, “Optical trapping and manipulation of nanostructures,” Nat. Nanotechnol. 8(11), 807–819 (2013). [CrossRef] [PubMed]

9. S. Stellmer, B. Pasquiou, R. Grimm, and F. Schreck, “Laser cooling to quantum degeneracy,” Phys. Rev. Lett. 110(26), 263003 (2013). [CrossRef] [PubMed]

10. M. Soltani, J. Lin, R. A. Forties, J. T. Inman, S. N. Saraf, R. M. Fulbright, M. Lipson, and M. D. Wang, “Nanophotonic trapping for precise manipulation of biomolecular arrays,” Nat. Nanotechnol. 9(6), 448–452 (2014). [CrossRef] [PubMed]

11. T. Cao, L. Mao, D. Gao, W. Ding, and C. W. Qiu, “Fano resonant Ge₂Sb₂Te₅ nanoparticles realize switchable lateral optical force,” Nanoscale 8(10), 5657–5666 (2016). [CrossRef] [PubMed]

12. S. B. Wang and C. T. Chan, “Lateral optical force on chiral particles near a surface,” Nat. Commun. 5, 3307–3314 (2014). [PubMed]

13. A. C. Durand, J. A. Hutchison, C. Genet, and T. W. Ebbesen, “Mechanical separation of chiral dipoles by chiral light,” New J. Phys. 15(12), 123037 (2013). [CrossRef]

14. J. Berthelot, S. S. Aćimović, M. L. Juan, M. P. Kreuzer, J. Renger, and R. Quidant, “Three-dimensional manipulation with scanning near-field optical nanotweezers,” Nat. Nanotechnol. 9(4), 295–299 (2014). [CrossRef] [PubMed]

15. R. Sainidou and F. J. García de Abajo, “Optically tunable surfaces with trapped particles in microcavities,” Phys. Rev. Lett. 101(13), 136802 (2008). [CrossRef] [PubMed]

16. M. Li, W. H. P. Pernice, and H. X. Tang, “Reactive cavity optical force on microdisk-coupled nanomechanical beam waveguides,” Phys. Rev. Lett. 103(22), 223901 (2009). [CrossRef] [PubMed]

17. K. C. Neuman and A. Nagy, “Single-molecule force spectroscopy: optical tweezers, magnetic tweezers and atomic force microscopy,” Nat. Methods 5(6), 491–505 (2008). [CrossRef] [PubMed]

18. A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6(3), 183–191 (2007). [CrossRef] [PubMed]

19. L. Li, Y. Yu, G. J. Ye, Q. Ge, X. Ou, H. Wu, D. Feng, X. H. Chen, and Y. Zhang, “Black phosphorus field-effect transistors,” Nat. Nanotechnol. 9(5), 372–377 (2014). [CrossRef] [PubMed]

20. A. S. Rodin, A. Carvalho, and A. H. Castro Neto, “Strain-Induced Gap Modification in Black Phosphorus,” Phys. Rev. Lett. 112(17), 176801 (2014). [CrossRef] [PubMed]

21. H. Liu, Y. Du, Y. Deng, and P. D. Ye, “Semiconducting black phosphorus: synthesis, transport properties and electronic applications,” Chem. Soc. Rev. 44(9), 2732–2743 (2015). [CrossRef] [PubMed]

22. X. Wang, A. M. Jones, K. L. Seyler, V. Tran, Y. Jia, H. Zhao, H. Wang, L. Yang, X. Xu, and F. Xia, “Highly anisotropic and robust excitons in monolayer black phosphorus,” Nat. Nanotechnol. 10(6), 517–521 (2015). [CrossRef] [PubMed]

23. V. Tran, R. Soklaski, Y. F. Liang, and L. Yang, “Tunable Band Gap and Anisotropic Optical Response in Few-layer Black Phosphorus,” Phys. Rev. B 89, 235319 (2014). [CrossRef]

24. A. N. Rudenko and M. I. Katsnelson, “Quasiparticle band structure and tight-binding model for single- and bilayer black phosphorus,” Phys. Rev. B 89(20), 201408 (2014). [CrossRef]

25. J. Quereda, P. San-Jose, V. Parente, L. Vaquero-Garzon, A. J. Molina-Mendoza, N. Agraït, G. Rubio-Bollinger, F. Guinea, R. Roldán, and A. Castellanos-Gomez, “Strong Modulation of Optical Properties in Black Phosphorus through Strain-Engineered Rippling,” Nano Lett. 16(5), 2931–2937 (2016). [CrossRef] [PubMed]

26. J. Qiao, X. Kong, Z. X. Hu, F. Yang, and W. Ji, “High-mobility transport anisotropy and linear dichroism in few-layer black phosphorus,” Nat. Commun. 5, 4475–4481 (2014). [CrossRef] [PubMed]

27. H. Liu, A. T. Neal, Z. Zhu, Z. Luo, X. Xu, D. Tománek, and P. D. Ye, “Phosphorene: an unexplored 2D semiconductor with a high hole mobility,” ACS Nano 8(4), 4033–4041 (2014). [CrossRef] [PubMed]

28. F. Xia, H. Wang, and Y. Jia, “Rediscovering black phosphorus as an anisotropic layered material for optoelectronics and electronics,” Nat. Commun. 5, 4458–4463 (2014). [CrossRef] [PubMed]

29. T. Low, A. S. Rodin, A. Carvalho, Y. Jiang, H. Wang, F. Xia, and A. H. Castro Neto, “Tunable optical properties of multilayer black phosphorus thin films,” Phys. Rev. B 90(7), 075434 (2014). [CrossRef]

30. N. Youngblood, C. Chen, S. J. Koester, and M. Li, “Waveguide-integrated black phosphorus photodetector with high responsivity and low dark current,” Nat. Photonics 9(4), 247–252 (2015). [CrossRef]

31. J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64(4), 1632–1639 (1988). [CrossRef]

32. Y. Yang, Z. Shi, J. F. Li, and Z. Y. Li, “Optical forces exerted on a graphene-coated dielectric particle by a focused Gaussian beam,” Photon. Res. 4(2), 65–69 (2016). [CrossRef]

33. J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66(10), 4594–4602 (1989). [CrossRef]

34. Z. Liu and K. Aydin, “Localized surface plasmons in nanostructured monolayer black phosphorus,” Nano Lett. 16(6), 3457–3462 (2016). [CrossRef] [PubMed]

35. T. Low, R. Roldán, H. Wang, F. Xia, P. Avouris, L. M. Moreno, and F. Guinea, “Plasmons and screening in monolayer and multilayer black phosphorus,” Phys. Rev. Lett. 113(10), 106802 (2014). [CrossRef] [PubMed]

36. J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66(7), 2800–2802 (1989). [CrossRef]

37. P. Johnson and R. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]

38. P. Chylek, J. T. Kiehl, and M. K. W. Ko, “Optical levitation and partial-wave resonances,” Phys. Rev. A 18(5), 2229–2233 (1978). [CrossRef]

39. B. R. Johnson, “Theory of morphology-dependent resonances: shape resonances and width formulas,” J. Opt. Soc. Am. A 10(2), 343–352 (1993). [CrossRef]

40. Y. Yang, W. P. Zang, Z. Y. Zhao, and J. G. Tian, “Morphology-dependent resonance of the optical forces on Mie particles in an Airy beam,” Opt. Express 21(5), 6186–6195 (2013). [CrossRef] [PubMed]

41. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992). [CrossRef] [PubMed]

42. R. K. Arora and Z. Lu, “Graphical study of Laguerre-Gaussian beam modes,” IEE Proc., Microw. Antennas Propag. 141(3), 145–150 (1994). [CrossRef]

43. M. Mazilu, Y. Arita, T. Vettenburg, J. M. Auñón, E. M. Wright, and K. Dholakia, “Orbital-angular-momentum transfer to optically levitated microparticles in vacuum,” Phys. Rev. A 94(5), 053821 (2016). [CrossRef]

44. S. Thanvanthri, K. T. Kapale, and J. P. Dowling, “Arbitrary coherent superpositions of quantized vortices in Bose-Einstein condensates via orbital angular momentum of light,” Phys. Rev. A 77(5), 053825 (2008). [CrossRef]

45. Y. Arita, M. Mazilu, and K. Dholakia, “Laser-induced rotation and cooling of a trapped microgyroscope in vacuum,” Nat. Commun. 4, 2374–2380 (2013). [CrossRef] [PubMed]

46. N. Lo Gullo, S. McEndoo, T. Busch, and M. Paternostro, “Vortex entanglement in Bose-Einstein condensates coupled to Laguerre-Gauss beams,” Phys. Rev. A 81(5), 053625 (2010). [CrossRef]

Tunable optical forces exerted on a black phosphorus coated dielectric particle by a Gaussian beam

Abstract

1. Introduction

2. Theory

3. Results and discussion

4. Conclusions

Funding

References and links

Cited By

Figures (5)

Equations (13)

Optical Materials Express