Spinning gold nanoparticles driven by circularly polarized light

Journal of Quantitative Spectroscopy & Radiative Transfer ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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Journal of Quantitative Spectroscopy & Radiative Transfer

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Spinning gold nanoparticles driven by circularly polarized light

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Jiunn-Woei Liaw a,b,c,n, Ying-Syuan Chen d, Mao-Kuen Kuo d,nn

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Department of Mechanical Engineering, Chang Gung University, 259 Wen-Hwa 1st Rd., Kwei-Shan, Taoyuan 333, Taiwan Center for Biomedical Engineering, Chang Gung University, 259 Wen-Hwa 1st Rd., Kwei-Shan, Taoyuan 333, Taiwan Medical Physics Research Center, Institute for Radiological Research, Chang Gung University/Chang Gung Memorial Hospital, Linkou, Taoyuan 333, Taiwan d Institute of Applied Mechanics, National Taiwan University, 1 Sec. 4, Roosevelt Rd., Taipei 106, Taiwan b c

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a r t i c l e i n f o

abstract

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Article history: Received 4 November 2015 Received in revised form 8 January 2016 Accepted 8 January 2016

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 spheroidal) 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. & 2016 Elsevier Ltd. All rights reserved.

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Keywords: Circularly polarized light Gold nanoparticle Optical torque Rotation Surface plasmon resonance Maxwell's stress tensor Absorption Scattering Surface traction Mie theory MMP Axis of revolution

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43 1. Introduction 45 Using optical forces and torques to trap, move, align and rotate microparticles and nanoparticles via optical tweezers techniques has become an important optical manipulation in the past decade [1–9]. As early as 1909, Poynting predicted the angular momentum of circularly polarized (CP) lights [10]. Beth discovered optical torques

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Corresponding author at: Department of Mechanical Engineering, Chang Gung University, 259 Wen-Hwa 1st Rd., Kwei-Shan, Taoyuan 333, Taiwan. nn Corresponding author. E-mail addresses: [email protected] (J.-W. Liaw), [email protected] (M.-K. Kuo).

induced by a CP light irradiating a quartz plate [11]. The experiment can be explained by the transfer of angular momentum through the absorption of CP-light photon flows inducing the optical torque on the irradiated plate [12–16]. Optical torques, induced by a CP light on a spherical particle, has been studied analytically by using Mie theory [17,18]. Due to the collective motion of conductive electrons in gold nanoparticles (GNPs), the localized surface plasmon resonance (LSPR) enhances the polarizability of GNPs, particularly in the regime from visible-light to near infrared. Therefore, the light-matter interaction of GNPs is profound, compared to those of dielectric nanoparticles. As a result, the induced optical forces and torques exerted on GNP are significantly strong and associated with severe heating [19–24]. A high-speed spinning

http://dx.doi.org/10.1016/j.jqsrt.2016.01.012 0022-4073/& 2016 Elsevier Ltd. All rights reserved.

61 Please cite this article as: Liaw J-W, et al. Spinning gold nanoparticles driven by circularly polarized light. J Quant Spectrosc Radiat Transfer (2016), http://dx.doi.org/10.1016/j.jqsrt.2016.01.012i

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spherical GNP induced by the illumination of a focused CP Gaussian beam has been demonstrated [25]. Additionally, a Laguerre-Gaussian beam (or called optical vortex) can also be used to constantly rotating spherical GNP due to the orbital angular momentum of photons [26]. Recently, the optical torque exerted on a GNP of thin-plate shapes (circular, rectangular, triangular or hexagonal) irradiated by a normal-incident CP light was analyzed theoretically by utilizing the finite difference time domain (FDTD) method; one of the important findings is that the scattering contribution resulting from the multipolar plasmon resonance of the symmetry-breaking nanostructure could be larger than the absorption one [27]. Various shapes of GNPs have been synthesized in the past decade, e.g. gold nanorods, bi-pyramids etc., in addition to spherical GNPs [28–30]. The behaviors of light-matter interaction of different-shaped GNPs are different due to their longitudinal surface plasmon resonance (LSPR). Optical trapping performances of gold decahedron, icosahedron, triangular and hexagonal prisms were studied [31]. Rotating a single silver nanowire by CP laser beam has been demonstrated experimentally [32,33]. Recently, optical torque that is exerted on a gold nanorod induced by CP light has been analyzed theoretically utilizing the multiple multipole (MMP) method [34,35]. This paper theoretically investigated and compared the wavelength-dependent optical torques, which are induced by a CP light, continuously driving GNPs (of spherical, and prolate or oblate spheroidal shapes) to rotate. Mie theory and MMP method were used to study optical torques exerted on a spherical GNP [16], and on a prolate/oblate spheroidal GNP [34,35], respectively. Optical forces and torques can be obtained by integrating Maxwell's stress tensor over the surface of GNP. The absorption and scattering powers of GNP were analyzed. In addition, the roles of absorption and scattering playing on the mechanical responses of GNP due to CP light incident along principal axes of GNP were investigated. Steady-state rotation (spinning) of a spherical GNP was also discussed, which was due to the optical torque provided by CP light balancing with the viscous torque from the dragging of the surrounding medium. The simulations for gold prolate and oblate spheroids can be applied to predict the light-matter interactions of the gold bi-pyramids and gold decahedrons, respectively. Through the optical manipulation, a rotating GNP may have promise in the applications of nanofluidics [37].

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2. Theory In this paper, the optical torques which are exerted on three different shapes of GNP (sphere, prolate and oblate spheroids) illuminated by a CP plane wave were studied. Spheroids have three principal axes, including an axis of revolution. Oblate spheroids and prolate spheroids are flattened and elongated, respectively, along the axes of revolution. The boundary surfaces of spheroids (assumed that axes of revolution are coincided with the z-axis, and the centers coincided with the origin of the coordinate

system) can be expressed as
x2
y2
z 2 þ þ ¼ 1; a a b

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where a and b denote the half lengths of the principal axes; a ¼b: spheres, a ob: prolate spheroids, and a4b: oblate spheroids. Mie theory was utilized to study the optical torque on a spherical GNP analytically. For the cases of prolate and oblate spheroids, MMP method was used to analyze the optical torque numerically. A CP plane wave can be treated as a superposition of two linearly polarized plane waves of the same frequency with mutually perpendicular electric fields and 90° phase difference. Let the CP plane wave propagate along the negative z-direction, the wave can be divided into two pffiffiffiorthogonal components, and written as Ei ¼ Ε0 ðe1 7je2 Þ= 2, where 7 corresponds to p the ffiffiffiffiffiffiffiffi right-handed and left-handed CP waves, and j ¼ 1. Throughout this paper, the time harmonic factor expð  jωt Þ has been omitted in Maxwell's equations, where ω is the angular frequency. The time-averaged Maxwell stress tensor T is expressed as    1 1 T ¼ Re ϵEE þμHH  ϵE  E þμH  H I ; ð2Þ 2 2

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where I is the unit tensor, and ε and μ are the permittivity and permeability of the surrounding medium, respectively. In Eq. (2), over-bar denotes a complex conjugate, and Re denotes the real part of a complex number. The total EM fields in the exterior region of GNP are the linear sums of the incident and scattered fields: Ε ¼ Εi þ Εs and H ¼ Hi þ Hs , where the superscript “i” and “s” denote the incident and scattered parts, respectively. In terms of the Maxwell's stress tensor, the induced optical force F and torque M about the center of GNP generated by the EM field are expressed by the surface integrals, Z F ¼ T UndS; ð3Þ S

Z

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ð4Þ

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where S is the surface of GNP, n is the unit outward normal vector, and r is the position vector of a generic point on S with respect to the center of GNP. In the following, the effective surface traction ðer  T U nÞ U ek inducing optical torque will be analyzed where ek is the propagating vector of the CP plane wave. The absorption (dissipation) power of GNP irradiated by a CP plane wave can be expressed as a surface integral of the Poynting vector, Z  1 P a ¼  Re E  H UndS ; ð5Þ 2 S

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r  T UndS;

The absorption (dissipation) power is associated with the plasmonic heating. The absorption efficiency of GNP is further defined as Q a ¼ P a =ASi ; ð6Þ i i i where S ¼ E  H =2 is the fluence of a circularly polarized laser beam, and A is the cross-sectional area of the GNP. Throughout this paper, λ is the wavelength of the CP plane wave in vacuum. Since the problem is linear, optical forces, optical torques, absorption powers and scattering

Please cite this article as: Liaw J-W, et al. Spinning gold nanoparticles driven by circularly polarized light. J Quant Spectrosc Radiat Transfer (2016), http://dx.doi.org/10.1016/j.jqsrt.2016.01.012i

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powers are directly proportional to the laser fluence. Accordingly, increasing the laser power increases not only the optical torque but also the absorption power (the plasmonic heating). To eliminate the effect of fluence, a ratio γ is defined as, γ ¼ ‖M=P a ‖:

ð7Þ

Eq. (7) is useful to evaluate the performance of optical torque at various wavelengths for the same plasmonic heating power.

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3. Results and discussion

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In the following analysis the optical forces and torques exerted on three different shapes of GNP (spherical, prolate and oblate spheroidal), which are irradiated by a righthanded/clockwise CP plane wave propagating along principal axes, are discussed in details. The surrounding medium is water. Mie theory is used to analyze the mechanical behavior of spherical GNPs, and the MMP method for the prolate and oblate spheroidal GNPs. The wavelength-dependent permittivity of gold is used in the simulation [38]. The fluence of the CP plane wave is assumed to be 25 MW/cm2 for various wavelengths. The dimension of a spheroid is denoted by (a, a, and b).

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Fig. 1(a) plots the resultant optical torques on spherical GNPs for r ¼35 nm and 60 nm in water irradiated by a CP light incident along the negative z-direction; the corresponding absorption efficiencies are also plotted in dashed lines. The maximum of induced torque and absorption efficiency occur roughly at 530 nm. This behavior originates from SPR of GNPs, a collective motion of free electrons in GNPs. Fig. 1(b) plots the ratios of the optical torque to absorption power, as defined in Eq. (7), versus wavelengths. The curves for r¼ 35 nm and 60 nm in Fig. 1(b) are identical and are fitted perfectly by

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γ ¼ λ=2πc:

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The results demonstrate that the ratio γ seems linearly proportional to the wavelength, but independent of the size of GNP. This result can be explained as the follows. Since the photons of CP lights carry both linear and angular momentums, both the optical force and torque are induced by the absorbed and scattered photon flows due to the conservation of linear and angular momentums. The linear momentum change contributes an optical radiation pressure as the GNP absorbs and scatters a part of incident phonons. Similarly, as incident photons impinge the GNP, the total angular momentum of the absorbed photons is transferred to GNP. However, the total angular momentum difference of the scattered photons between before and after scattering is zero for this particular case. Consequently, the scattered photons do not contribute to rotate GNP, and only the absorbed photons contribute to the optical torque due to the conservation of angular momentum. Each photon of CP lights in a dielectric medium possesses not only linear momentum of h/nλ, in

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ð8Þ

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Abraham formulation, but also spin angular momentum h/ 2πn (or h/2π), where h is the Planck's constant, λ is the wavelength in vacuum, and n is the refractive index [39,40]. Herein, our numerical experiment for the symmetric case shows γ ¼ λ=2πc, which is in agreement with the dilemma of a single photon's energy and angular momentum expressed by either (hf/n, h/2πn) or (hf, h/2π) [41]; for both expressions the ratio of photon's angular ̇ momentum to energy is the same, ϵ. In addition, the absorption spectrum of a spherical GNP shows a low-pass filter for obtaining optical torque from CP light; the cut-off wavelength is around 600 nm. The optical torques of a lefthanded/counter-clockwise CP wave are also calculated, though not shown here; the torques are equal in magnitude to those for right-handed CP wave but opposite in sign. As the radius of GNP increases, the SPR band is slightly red-shifted and broadened. Fig. 2(a) and (b) plot the effective surface-traction distributions of ðer  T  nÞ  ez on the surfaces of the spherical GNP with r¼60 nm at λ¼ 532 nm and 830 nm, respectively. Since the maximum torque on GNP of r ¼60 nm occurs roughly at 530 nm (SPR), most of the surface of GNP exhibits the positive surface traction, as shown in Fig. 2(a), inducing a positive toque to rotate the GNP clockwise for the case of λ¼ 532 nm. In contrast, for the case of λ¼ 830 nm (off-SPR), Fig. 2(b) shows that the area with positive surface-traction is only slightly larger than that with negative ones inducing a relatively weak torque, compared to the case of λ¼ 532 nm. The z-component of optical forces pushing spherical GNPs of r ¼35 and 60 nm downstream and the corresponding extinction efficiencies are shown in Supplementary material (Fig. S1). Subsequently, we take into account the viscous torque for a realistic spinning spherical GNP in water. The viscous torque on a constantly rotating sphere in a viscous fluid is proportional to the third power of radius [42], M v ¼  8πηr 3 Ω0 ;

ð8Þ

where η is the dynamic viscosity of surrounding viscous medium, r is the radius of spherical GNP, and Ω0 the angular speed of rotation. When the optical torque balances with the viscous torque, the steady-state rotation speed of GNP will reach. From Eq. (8) the steady-state angular speed is proportional to jMj=r 3 . Fig. 3 plots jMj=r 3 versus radius r for λ¼532 nm (SPR) and 830 nm (off-SPR), where the reference radius r0 is taken as 20 nm. An enlarged drawing for the case of λ ¼830 nm is also shown in the figure. Fig. 3 demonstrates that the values of jMj=r 3 for λ¼532 nm is significantly larger than those for λ¼ 830 nm for all wavelengths; this indicates that using a laser with a wavelength closer to SPR induces a faster spinning speed of the GNP. Moreover, the radii of spherical GNPs corresponding to the maximum jMj=r 3 (i.e., the highest spinning speed) are r ¼24 nm and r¼ 78 nm for λ¼ 532 nm and 830 nm, respectively. From Fig. 3, we conclude that the radius corresponding to the highest spinning speed depends on the wavelength, if the fluence is the same. Here, we consider only the size effect of GNP on the viscous torque from the surrounding medium (water), regardless of the temperature-dependence of

Please cite this article as: Liaw J-W, et al. Spinning gold nanoparticles driven by circularly polarized light. J Quant Spectrosc Radiat Transfer (2016), http://dx.doi.org/10.1016/j.jqsrt.2016.01.012i

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Fig. 2. Surface traction ðer  T  nÞ  ez distributions on a spherical GNP of r ¼60 nm at (a) 532 nm and (b) 830 nm. Fig. 1. (a) Optical torque (solid lines) and absorption efficiencies (dashed lines) of spherical GNP (r ¼35 nm and 60 nm) versus wavelength. Q6 (b) Ratio of optical torque to absorption power versus wavelength.

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viscosity. In fact, the viscosity of liquid water decreases as the temperature increases. For example, η of water is 1.002 mPa s at 20 °C, while is 0.3150 mPa s at 90 °C. Hence, the plasmonic heating of GNP is critical to the viscous torque [43,44]. If the temperature-dependence of viscosity is considered, the plasmonic heating will raise the temperature and reduce the viscosity of surrounding medium, hence the steady-state spinning speed of spherical GNPs irradiated by CP lights should be higher than that we predicted here.

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A typical prolate spheroid of (35, 35, and 60) nm is discussed, where the aspect ratio is 1.71. Fig. 4(a) plots the z-component of optical torque versus wavelength for the case of gold prolate spheroid irradiated by a CP light incident along the axis of revolution (the long axis, z-axis). In addition, the y-component of optical torque on the same

119 121 Fig. 3. M/(r/r0)3 versus r/r0 for spherical GNP of r ¼60 nm at 532 nm and 830 nm, where r0 ¼ 20 nm.

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79 Fig. 5. Surface traction ðer  T  nÞ  ey distribution of a gold prolate spheroid of (35, 35, and 60) nm irradiated by CP light incident along the short axis (y-axis) at 680 nm. (My ¼ 1.94 nN-nm).

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Fig. 4. (a) Optical torques (solid lines) and absorption efficiencies (dashed lines) for a gold prolate spheroid of (35, 35, and 60) nm versus wavelength for CP light incident along the axis of revolution (long axis, zaxis) and short axis (y-axis). (b) Ratios of optical torque to absorption power versus wavelength.

contrast, as λ4520 nm their ratios are very different. This is because that for the latter the difference of total angular momentum of the scattered photons between before and after scattering is not null anymore; both the absorbed and scattered photons contribute to the optical torque. The maximum ratio occurs at 680 nm within the longitudinal SPR band of this prolate spheroid of aspect ratio 1.71. The contribution of the scattered photons on the optical torque is larger than that of the absorbed ones. Fig. 5 plots the surface traction distribution of ðer  T  nÞ  ey on the prolate spheroid irradiated by the CP light incident along the short axis (y-axis) at 680 nm. It is obvious that the positive areas (red) are larger than the negative ones (blue) indicating that the prolate spheroid will rotate clockwise. The corresponding y-component torque is 1.94 nN-nm. For the long-axis and short-axis incident CP lights, the optical forces pushing the prolate GNP downstream and the corresponding extinction efficiencies are shown in Supplementary material (Fig. S2). These results are in accordance with the optical torques.

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spheroid irradiated by another CP light incident along the short axis (y-axis) is plotted for comparison. The corresponding absorption efficiencies are also plotted in dashed lines in Fig. 4(a). The longitudinal SPR of the latter is at 640 nm. Fig. 4(b) plots the ratios of the optical torques to absorption powers for the two cases. The formal result (the long axis) obeys the relation, γ ¼ λ=2πc, exactly the same as those of spherical ones in Fig. 1(b). Again, this is because that the total angular momentum difference of the scattered photons between before and after scattering is zero, thus only the absorbed photons contribute to the optical torque due to the conservation of angular momentum. Note that the scattering power is not zero even for this special case of axial symmetry. As λ o520 nm, the contributions from absorbed photons dominate over the scattered ones, hence the ratio of the latter (the short axis) almost overlaps with that of the former (the long axis). In

107 A typical oblate spheroid of (60, 60, and 35) nm is analyzed in details. Fig. 6(a) plots the z-component of optical torques induced by a CP light incident along the axis of revolution (the short axis, z-axis) of this oblate spheroid versus wavelength. Only the z-component of torque exists. In addition, the y-component optical torque for the case that the CP light incident along the long axis (y-axis) is plotted for comparison; the longitudinal SPR is at 600 nm. The corresponding absorption efficiencies are also plotted (dashed lines) in Fig. 6(a). Fig. 6(b) plots the ratios of the optical torque to absorption power for the two cases. We observe that the formal result (the axis of revolution) obeys the relation, γ ¼ λ=2πc, the same as those of spherical ones in Fig. 1(b). Again, this is because that the difference of total angular momentum of the scattered photons between before and after scattering is null, thus

Please cite this article as: Liaw J-W, et al. Spinning gold nanoparticles driven by circularly polarized light. J Quant Spectrosc Radiat Transfer (2016), http://dx.doi.org/10.1016/j.jqsrt.2016.01.012i

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79 Fig. 7. Surface traction ðer  T  nÞ  ey distribution of a gold oblate spheroid of (60, 60, and 35) nm irradiated by CP light incident along the long axis (y-axis) at 680 nm. (My ¼1.21 nN-nm).

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Fig. 6. (a) Optical torques (solid lines) and absorption efficiencies (dashed lines) for a gold oblate spheroid of (60, 60, and 35) nm versus wavelength for CP light incident along the axis of revolution (short axis, z-axis) and long axis (y-axis). (b) Ratios of optical torque to absorption power versus wavelength.

negative ones (blue) indicating that the oblate spheroid will rotate clockwise. The corresponding y-component torque is 1.21 nN-nm. For the long-axis and short-axis incident CP lights, the optical forces pushing the oblate GNP downstream and the corresponding extinction efficiencies are shown in Supplementary material (Fig. S3). In summary, if the CP plane wave is incident along the axis of revolution of spherical, prolate and oblate spheroids, the illumination cross section is circular. For these special cases, the total angular momentum contributed from the scattered photons is null, due to the axial symmetry. Therefore, the ratios of optical torques to absorption powers always obey the rule of γ ¼ λ=2πc. The ratio is independent of the size, aspect ratio and shape. Although the scattering power of these cases of axial symmetry is not zero, only the absorbed photons contribute to the optical torque. In contrast, if the incident direction is not along the axis of revolution, both the absorbed and scattered photons contribute to the optical torque.

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107 only the absorbed photons contribute to optical torque. However, the latter result (the long axis) is very different from the former (the short axis), because that the total angular momentum of the scattered photons is different from their total value before being scattered. This is to say that not only the absorbed photons but also the scattered ones contribute to the optical torque if the CP light is not incident along the axis of revolution (z-axis). Moreover, Fig. 6(b) indicates that the contribution of the scattered photons on the optical torque is larger than that of the absorbed ones. The maximum γ occurs at 680 nm, as shown in Fig. 6(b). Fig. 7 plots the surface traction distribution of ðer  T  nÞ  ey on this oblate spheroid irradiated by the CP light incident along the long axis (y-axis) at 680 nm. It is obvious that the positive areas (red) are larger than the

The optical torques induced by CP light, which drive three typical shapes of GNPs (spherical, prolate and oblate spheroidal) to spin continuously, were studied theoretically using Mie theory for the former and MMP method for the latter two. Results show that if the CP wave is incident along the axis of revolution, the ratios of optical torque to absorption power of these three shapes of GNP are the same. The ratio is γ ¼ λ=2πc, independent of the refractive index of surrounding medium, the size, aspect ratio, and shape of GNPs for these axial symmetrical cases. This indicates that only the absorbed photons transfer their angular momentums to the GNP due to the conservation of angular momentum, whereas the scattered photons do not contribute to optical torque. In contrast, if the incident CP wave is along other directions (asymmetric axis), both the absorbed and scattered ones contribute to optical torque.

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Therefore, the ratio does not follow the rule of γ ¼ λ=2πc any more. Take a gold prolate spheroid irradiated by a CP plane wave propagating along the short axis as an example. The ratio of the optical torque to absorption power is larger than the value of γ ¼ λ=2πc; both the absorbed and scattered photons contribute to the induced optical torque on this prolate spheroid. The maximum optical torque usually occurs within the longitudinal SPR band of a nonspherical GNP, because the maximum absorption and scattering of photons are induced. The spinning direction depends on the handedness (right-handed or left-handed) of CP lights. If considering the balance of the optical torque from CP lights with the viscous torque from surrounding medium, we demonstrated that a high-speed spinning of a spherical GNP is possible in principle; this finding is in agreement with the previous experiment of Lehmuskero et al. [25]. In addition to spherical GNP, our results for gold prolate and oblate spheroids could be used to predict the CP light-driven spinning of the gold bi-pyramids and decahedrons, respectively. Our finding may also pay the way to the noninvasive optical manipulation on rotating GNPs by using CP lights. The spinning GNPs can be utilized as a nanostirring for a variety of applications in optofluidics and lab-on-a-chip; e.g. a plasmonic GNP can serve as a stirring bar to blend the fluid for mixing or a local vortex for nanofluidics, where the angular rotation speed is tunable by adjusting the power of CP laser.

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Acknowledgments

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The research was supported by Ministry of Science and Technology, Taiwan (MOST 102-2221-E-002-056-MY3, 103-2221-E-182-033-MY2, and 104-2221-E-182-053) and Chang Gung Memorial Hospital (CIRPD2E0031).

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Appendix A. Supplementary material

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Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j. jqsrt.2016.01.012.

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