Rotating Au nanorod and nanowire driven by circularly polarized light

Jiunn-Woei Liaw; Ying-Syuan Chen; Mao-Kuen Kuo

doi:10.1364/OE.22.026005

1. Introduction

As early as 1936, an optical torque induced by a circularly polarized (CP) light irradiating a quartz plate has been discovered by Beth [1]. This experiment can be explained by the transfer from the spin angular momentums of CP light’s photons to this birefringent plate [2,3]. In principle, the light-matter interaction is too weak to offer an enough optical torque for driving a dielectric nanoparticle to spin or rotate. However, the implementation of a light-mill, particularly in nanoscale, has attracted a lot of attentions [4–14]. In 2010, two works independently showed the possibility of light-driven rotating plasmonic nanostructures: an Ag nanowire (NW) driven by CP light [6], and a planar gammadion Au plate by linearly polarized (LP) light [7]. For the former, the spin angular momentum of photons in CP light is transferred to the irradiated Ag NW. Each photon of CP light has a spin angular momentum of h/2π, where h is Planck’s constant. For the latter, the surface plasmon resonance (SPR) mode shape of the metallic nanostructure converts the linear momentum of LP-light photons into the angular momentum of the nanostructure. The linear momentum of a photon is given by h/λ, where λ is the wavelength. For both structures, the resultant optical torque is enhanced due to the plasmonic polarizability of metallic nanostructures resulting from the collective electrons oscillation. Additionally, optical tweezers has become an important tool for noninvasive manipulation to induce optical force and torque for trapping, moving, aligning and rotating nanostructures in the past decade [15–34]. Recently, a spinning spherical Au nanoparticle (NP) irradiated by CP laser beam (830 nm) was found to have a high rotation speed, up to several kHz [12]. In contrast, a previous experiment using the same laser showed that the rotation speed of Ag NW is only several Hz [6]. Moreover, another light source, a Laguerre−Gauss (or called optical vortex) beam, can also be utilized as an illumination source to generate a rotating Ag NW [13] or Au NP [14] due to the orbital angular momentum of photons. Because of the plasmonic effect, the induced optical torque can be larger enough to overcome Brownian motion for NR or NP in liquid. Another advantage of light-driven rotation of Au or Ag NP/NW is that the rotation speed can be remotely controlled by adjusting the light power. On the other hand, the trapping and alignment of Au and Ag nanorod (NR) or NW induced by a LP laser beam have been studied [6,19–22]. In particular, the parallel and perpendicular alignments of plasmonic NR/NW were pointed out experimentally [6] and theoretically [35]. The turning point with a null torque between the two alignment modes is exactly at the longitudinal surface plasmon resonance (LSPR) of NR/NW, depending on the aspect ratio (AR) [35]. Moreover, the wavelength-dependent optical torque exerted on Au NR also depends on the long-axis orientation w.r.t. the light polarization. Using the NW’s polarizability for alignment, we can rotate a linear polarizer, controlling the polarization direction of a LP light, to rotate NW. However, if the induced optical torque and polarizer’s rotation speed do not match, NW could not catch up the polarizer’s rotation [36].

In this paper, the wavelength-dependent optical torque provided by CP light for continuously driving Au NR/NW to rotate is investigated theoretically. The multiple multipole (MMP) method is used to solve Maxwell equations for the exterior and interior electromagnetic (EM) fields [35,37]. Using these EM fields, the Maxwell’s stress tensor on the surface of NR/NW is obtained. The optical force and torque, in turn, are calculated using the integrals of surface traction, in terms of Maxwell’s stress tensor. In addition, the dissipation behavior of Au NR/NW is also analyzed. Some typically sized Au NR and NW are studied quantitatively to illustrate the optical-torque performances. In particular, the differences between the mechanical responses of Au NR/NW to CP and LP lights are discussed. Although a CP plane wave, rather than a focused Gaussian beam for experiment, is used for analysis in this paper, the results are sufficient to estimate the CP light-driven rotation of Au NR/NW in real experiment. For example, we will explain why the rotation speed of Ag NW is several Hz while that of Au NP is as high as several kHz reported in previous experiments [6,12].

2. Theory

The shape of NR/NW is assumed to be a circular cylinder with two hemispherical end-caps. The radius of the cross section is denoted by r, and the total length by l; the AR is l/2r. Throughout this paper, the NR/NW is assumed to lie on the xz plane irradiated by a normally incident CP plane wave propagating along y-direction, as shown in Fig. 1, where the unit wavenumber vector is $e_{k} = e_{y}$ . A CP plane wave is a superposition of two linearly polarized plane waves at the same frequency with perpendicular electric fields and 90° phase difference. The time harmonic factor of exp(−jωt) for Maxwell’s equations is used. Here ω is the angular frequency, and $j = \sqrt{- 1}$ . The incident electric field of CP light is given by

E^{i} = Ε_{0} (e_{x} \pm j e_{z}) / \sqrt{2}

The corresponding magnetic field is

H^{i} = e_{k} \times E^{i} / η

, where the intrinsic impedance of EM wave in the surrounding medium is

η = \sqrt{μ / ε}

. The merit of CP light is that each photon possesses not only linear momentum of h/λ but also spin angular momentum h/2π. The total EM fields in the exterior area are the linear sum of the incident and scattered fields:

Ε = Ε^{i} + Ε^{s}

and

H = H^{i} + H^{s}

. The scattered fields are solved by using MMP method [35]. The induced optical force F_o and torque M_o about the center of Au NR caused by the EM field are expressed by the surface integrals,

F_{o} = \int_{S} T \cdot n d s

M_{o} = \int_{S} r \times (T \cdot n) d s

where T is Maxwell’s stress tensor given by

T = \frac{1}{2} Re {ε E \bar{E} + μ H \bar{H} - \frac{1}{2} (ε E \cdot \bar{E} + μ H \cdot \bar{H}) I}

, S is the surface of NR/NW, n is the outward normal vector, and r is the position vector of any point on S w.r.t. the center of NR or NW. Here I is the unit tensor, ε and μ are the permittivity and permeability of the surrounding medium respectively, Re is the real part of a complex number, and the symbol of over-bar denotes the complex conjugate. The absorption (dissipation) power of NR/NW irradiated by a CP plane wave can be expressed by a surface integral of Poynting vector,

P_{a} = - \frac{1}{2} Re {\int_{S} E \times \bar{H} \cdot n d s}

where S represents the whole surface of the lossy NR/NW. The absorption behavior, leading to the plasmonic heating, is due to the Ohmic loss in metal [30,35]. The absorption efficiency of NR/NW is further defined as

Q_{a} = P_{a} / A S^{i}

, where

S^{i} = | E^{i} \times {\bar{H}}^{i} | / 2

is the fluence of a CP plane wave, and A is the projection area of NR/NW. The values of the optical force, torque and absorption power are linearly proportion to the fluence, because this is a linear problem. This is to say that a larger laser power can raise not only the optical torque but also the absorption (heating) power. Therefore, we further define an efficiency of optical torque as

M_{e} = M_{o} / (Q_{a} S^{i})

to evaluate the performances of optical torque at different wavelength under the same plasmonic heating power.

Fig. 1 Configuration of Au NR/NW lying on a plane, irradiated by a CP plane wave.

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In this paper, we use Drude-Lorentz model to fit the frequency-dependent relative permittivity of Au in the visible-light to NIR regime (600 nm to 1900 nm) given in [38],

ε_{A u} (ω) = ε_{\infty} - \frac{ω_{p}^{2}}{ω (ω + j δ)} - \frac{χ ω_{0}^{2}}{(ω^{2} - ω_{0}^{2}) + 2 j γ ω}

The coefficients are listed in Table 1. Using this new model, we can extrapolate the permittivity of Au in the regime of NIR to MIR (i.e. 1900 nm to 2700 nm). This model helps us further analyze the mechanical responses of Au NW in the NIR to MIR regime.

Table 1. Permittivity Coefficients of Drude-Lorentz Model of Au

View Table

3. Numerical results and discussion

In the following analysis, a right-handed/clockwise CP plane wave is used for illumination. The surrounding medium is water. The fluence is assumed 25 MW/cm² for various wavelengths. Typical sized NR and NW are simulated to illustrate quantitatively the wavelength-dependent optical torque induced by CP light for Au NR/NW rotation. Additionally, the results of LP light are also provided to compare with those of CP light.

3.1. Au NR

The optical torque and absorption efficiency of a typical Au NR of r = 10 nm and AR = 3 in water are shown in Fig. 2(a). The LSPR of this Au NR is at 710 nm from the absorption efficiency. It is obvious that the optical torque is in accordance with the absorption efficiency. For the CP light (red-solid line), the induced maximum torque occurs at LSPR, due to the collective motion of electrons along Au NR. In contrast, a null torque is induced by the LP light (red-dash line) at LSPR. In addition, the absorption efficiency of NR to CP light coincides with that to LP light at an orientation angle θ = 45°; their plasmonic heating performances are the same. As photons of CP light impinge Au NR, the transferred angular momentum enables this NR rotate. Since CP light carries both linear and angular momentum, not only the optical torque but also optical force are induced. Figure 2(b) shows the optical force in y-direction and extinction efficiency; they are in accordance with each other. This is because that both the absorption and scattering of photons cause the linear momentum change, leading to an optical force pushing this particle downstream. The surface charge $| E \cdot n |$ and traction $(e_{r} \times (T \cdot n)) \cdot e_{y}$ distributions on surface at LSPR (710 nm) are plotted in Fig. 3. From Fig. 3(b), we can observe that the positive torque zone (red color) is dominant over the negative torque zone (blue color) at each end of NR. Therefore, NR is driven to rotate clockwise. We have verified numerically that CP light can provide a constant optical torque for arbitrarily oriented NR. This behavior is very different from that of LP light; the optical torque induced by LP light depends very much on the NR orientation w.r.t. the polarization.

Fig. 2 (a) Optical torque (red) and absorption efficiency (blue) for Au NR (r = 10 nm, AR = 3) versus wavelength. (b) Optical force (red) and extinction efficiency (blue). Solid line: CP light. Dash line: LP light at θ = 45°. Fluence: 25 MW/cm².

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Fig. 3 (a) Surface charge $| E \cdot n |$ and (b) traction distributions $(e_{r} \times (T \cdot n)) \cdot e_{y}$ on Au NR (r = 10 nm, AR = 3) at 710 nm (LSPR).

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We also calculated the optical torque of a left-handed/anti-clockwise CP wave, not shown here; the results are in the opposite sign. This is to say that the rotation direction of a light-driven Au NR depends on the handedness of CP light. Moreover, we calculated the optical torque upon a dielectric NR of the same size; the optical torque on Au NR is two orders of magnitude larger than that on a high-k dielectric NR (say n = 2.5). Furthermore, we studied the size effect of NR on optical torque. The optical torques and absorption efficiencies of NR with different radii (r = 10 nm, 15 nm, 20 nm) are shown in Fig. 4(a) for the same AR = 3. The corresponding efficiencies of optical torque are shown in Fig. 4(b); the performances look like lowpass filters with a cutoff wavelength at 600 nm. These curves (the ratio of optical torque to absorption power) are not straight lines, indicating that the cause of optical torque is not only the absorbed photons but also the scattered photons. Moreover, these curves show that a large Au NR has a higher efficiency of optical torque. Additionally, the LSPR of Au NR is red-shifted as the radius increases for fixed AR, as shown in Fig. 4(a).

Fig. 4 (a) Optical torques (solid lines) and absorption efficiencies (dash lines) versus wavelength of CP light for Au NR (r = 10, 15, 20 nm, AR = 3). (b) Efficiency of optical torque versus wavelength of CP light.

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3.2. Au NW

A typical Au NW of r = 20 nm and AR = 12 (total length: 480 nm) is analyzed. The optical torque and absorption efficiency induced by CP light are shown in Fig. 5(a). The first LSPR is at 2170 nm, and the second LSPR is at 840 nm. The maximum optical torque exerted on Au NW occurs at the first LSPR, as the fluence of CP light is the same. In contrast, null torques are induced by the LP light (dash line) at these LSPRs, where θ = 45°. The advantage of Au NW is that they could be used to acquire torque from a broadband light source instead of a single-frequency laser. Again, we found that the plasmonic heating power provided by CP light is the same with that by LP light. The efficiency of optical torque is shown in Fig. 5(b). From the efficiency of optical torque, we found that Au NW behaves as a lowpass filter with a cutoff wavelength of 600 nm, again. In addition, there is a dip at the second LSPR, 840 nm. The efficiency of optical torque of Au NW is almost a constant, around 05 to 0.6 nN-nm/(MW/cm²), which is much smaller than those of Au NRs as shown in Fig. 4(b). Since the lengths of most Au NWs are larger than 1 μm, the corresponding LSPRs are within the MIR regime, say > 2500 nm. Although the maximum optical torque exerted on a specific Au NW occurs at LSPR, there are rare MIR lasers available. Fortunately, most NIR lasers (e.g. 1064 nm) can be used for rotating Au NW with the same performance in the efficiency of optical torque, as shown in Fig. 5(b). Generally, the optical torque on Au NW is two orders of magnitude larger than that on a high-k dielectric NW. In addition, the optical force downstream and extinction efficiency are plotted in Fig. 5(c). Our results again demonstrate that the interactions of CP light with Au NW will inevitably involve the exchange of angular momentum and linear momentum. The surface charge $| E \cdot n |$ and traction $(e_{r} \times (T \cdot n)) \cdot e_{y}$ distributions on the surface of Au NW (r = 20 nm, AR = 12) at 1064 nm are shown in Fig. 6. Again, we can observe from Fig. 6(b) that the positive torque zone (red color) is dominant over the negative torque zone (blue color) on NW. Therefore, NW is driven to rotate clockwise.

Fig. 5 (a) Optical torque (red) and absorption efficiency (blue) for Au NW (r = 20 nm, AR = 12) versus wavelength. (b) Efficiency of optical torque. Solid line: CP light. Dash line: LP light at θ = 45°. (c) Optical force (red) and extinction efficiency (blue). Fluence: 25 MW/cm².

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Fig. 6 (a) Surface charge $| E \cdot n |$ and (b) traction $(e_{r} \times (T \cdot n)) \cdot e_{y}$ distributions on Au NW (r = 20 nm, AR = 12) at 1064 nm.

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Furthermore, we evaluated the optimal AR of NR/NW for obtaining the maximum optical torque or the highest rotation speed. In fact, a NW can be regarded a highly elongated NR. Figure 7(a) shows that the LSPR of Au NR/NW with r = 20 nm is red-shifted as AR increases. The optical torques upon NR/NW of different ARs, irradiated by a CP light at their LSPR, are poltted (solid line) in Fig. 7(a). Numerical results illustrate that the optical torque at LSPR increases as NR’s AR increases. Moreover, the optical torques versus AR provided by different CP light sources of 830 nm, 1064 nm and 1500 nm are shown in Fig. 7(b), where r = 20 nm. According to these curves, there is an optimal AR of NR/NW to obtain the maximum torque for a specific wavelength. When the LSPR of a NR/NW coincides with the wavelength of CP light source, the maximum torque is induced. For example, the Au NRs of AR = 3.4, 5 and 8 have the maximum optical torques at 830 nm, 1064 nm and 1500 nm, respectively, because their LSPRs coincide with these wavelengths. Of interest is that for 830 nm there is another peak at AR = 12, besides at AR = 3.4. This is because that the second LSPR of Au NW of AR = 12 is at 840 nm. However, the highest rotation speed relies on not only the optical torque but also the viscous torque. During the rotation of Au NR in water, the resistant viscous torque from the drag force of water upon the NR is induced. Therefore we need to take into account the viscous torque for determining the rotation speed. As the steady-state (terminal) rotation speed reaches, the viscous torque will balance with the optical torque. The fluidic viscous torque upon a rotating NR/NW is proportional to the third power of the length; the viscous torque can be simply expressed as [13],

M_{v} = - C η_{v} l^{3} Ω

where Ω is the angular velocity vector, η_v is the dynamic viscosity of liquid, and C is a coefficient positively related to the radius of the cross section. When the steady-state rotation reaches,

M_{o} + M_{v} = 0

. As a result, the steady-state angular velocity is linearly proportional to

| M_{o} | / {AR}^{3}

. Hence, we also plotted the values of optical torque divided by AR³ in Fig. 7(c) for different CP light sources of 830 nm, 1064 nm and 1500 nm. Figure 7(c) indicates that there is a corresponding AR of NR/NW to obtain the highest rotation speed under the irradiance of a specific laser. For example, the optimal AR are 3.4, 5 and 8 for 830 nm, 1064 nm and 1500 nm, respectively, to obtain the highest rotation speed. The results still show that the highest rotation speed occurs at LSPR, as the maximum torque does. Moreover, the profiles of

| M_{o} | / {AR}^{3}

imply that under the same irradiance of a specific NIR (e.g. 830 nm) CP laser beam the rotation speed of Au NR could be much higher than that of Au NW (l > 1 μm, AR > 20). The phenomenun is attributed to the viscous torque. We also used Mie theory to calculate and compare the wavelength-dependent optical torques on spherical Au and Ag NPs illuminated by CP wave. According to the numerical results, the torque on Au NP is always larger than that on Ag NP with the same size. For example, when the fluence is 25 MW/cm² and λ = 830 nm, the torque on Au NP of r = 200 nm is 1.196 nN-nm, five times larger than that on Ag NP, 0.233 nN-nm; the absorption efficiencies of Au and Ag NPs are 0.0864 and 0.0168, respectively. This result illustrates that the stronger absorption behavior of Au NP leads to the large optical torque compared to Ag NP. Our analysis more or less explains why the difference is as large as three orders of magnitude in the reported experiments; the rotation speed of Ag NW (r = 50 nm and l = 2 to 5 μm; i.e. AR > 20) irradiated by a 830-nm laser is about several Hz [6], while that of a spherical Au NP (r = 200 nm, AR = 1) by the same laser is as high as several kHz [12]. Of course, raising the light power we always can reinforce the optical torque to increase rotation speed. In summary, our analysis indicates that Au NR/NW irradiated by CP light behaves as a light-driven nanomotor, whereas as irradiated by LP light it performs as an optical wrench with a lever arm l.

Fig. 7 (a) Optical torques $| M_{o} |$ (solid line) on Au NR/NW (r = 20 nm) with various ARs irradiated by a CP light at LSPR (dash line). (b) $| M_{o} |$ and (c) $| M_{o} | / {AR}^{3}$ versus AR at 830 nm, 1064 nm and 1500 nm. Fluence: 25 MW/cm².

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4. Conclusion

The wavelength-dependent optical torque provided by a right-handed or left-handed CP laser beam for driving Au NR/NW to continuously rotate was studied theoretically. Using MMP method, we quantitatively calculated the optical torques and plasmonic heating powers for typical Au NR and NW irradiated by a CP plane wave, and compared these results with those by a LP one. From these results, we concluded that the maximum optical torque occurs at LSPR of a specific NR/NW under the irradiance of CP light. In other words, for a specific wavelength CP light the maximum optical torque is induced for a NR/NW, whose LSPR is just at this wavelength. In contrast, a null torque is induced by a LP light at LSPR. The rotation direction of Au NR/NW depends on the handedness of CP light. The profile of the optical-torque spectrum is consistent with that of the absorption-efficiency one. This implies that the angular momentums of the absorbed photons are transferred to Au NR/NW. In fact, not only the absorbed photons but also scattered ones contribute the optical torque. An efficiency of optical torque was proposed to evaluate the performances of CP light of different wavelengths for rotating NR/NW under the condition of same heating power. The terminal rotation speed will reach to induce a viscous torque, depending on the rotation speed and the third power of length, for balancing with the optical torque provided by CP light. Therefore, the larger the laser power, the faster the rotation speed of the NR/NW is. Our finding demonstrates the feasibility of using a CP laser beam to induce a continuously rotating Au NR/NW as a light-driven nanomotor for a variety of applications in optofluidics or lab-on-a-chip systems. For example, a light-driven plasmonic NR/NW can serve as a stirring bar (a local vortex) to blend the fluid for mixing. Recently, rotating magnetic nanorods (e.g. Pt-Ni) driven by magnetic torque have been used to investigate the mechanical properties of living cells [39]. The light-driven Au NR/NW could be another alternative for the biophysics studies.

Acknowledgments

The research was supported by Ministry of Science and Technology, Taiwan (NSC 103-2221-E-182-033-MY2, 102-2221-E-182-049, 102-2221-E-002-056-MY3) and Chang Gung Memorial Hospital (CMRPD290043).

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Rotating Au nanorod and nanowire driven by circularly polarized light

Abstract

1. Introduction

2. Theory

3. Numerical results and discussion

3.1. Au NR

3.2. Au NW

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (7)

Tables (1)

Equations (6)

Optics Express