Wavelength-dependent longitudinal polarizability of gold nanorod on optical torques

Jiunn-Woei Liaw; Wei-Jiun Lo; Mao-Kuen Kuo

doi:10.1364/OE.22.010858

1. Introduction

Using a focused and polarized CW laser beam to provide optical force and torque is an effective approach for optical manipulation of microparticles and nanoparticles [1–11]. By using this technique, recent works have demonstrated the feasibility for rotating and aligning gold or silver nanorods and nanowires [12–25]. Meanwhile, for a gold nanorod (GNR), there are two different optical behaviors: transverse surface plasmon resonance (TSPR) and longitudinal surface plasmon resonance (LSPR), where the LSPR wavelength is tunable depending on the aspect ratio (AR) and the size of GNR, and the TSPR wavelength is around 520 nm [26–28]. Hence, the LSPR wavelength of an elongated GNR can be tailored to locate in the near infrared (NIR) regime. Theoretically, lasers with a wavelength close to LSPR wavelength are preferred to achieve the alignment of GNR with the laser polarization. However, the maximum optical torque at the LSPR is usually accompanied with a severe photothermal effect on GNR, owing to plasmonic heating. A continuous heating may then cause thermal deformation and even melting of GNR. Consequently, lasers with NIR wavelength, e.g. Nd: YAG laser of 1064 nm, are generally used for rotating and aligning GNRs, whose LSPR is within 680 nm to 850 nm, along the laser polarization to increase the thermal stability of GNR. This is because that 1064 nm is off-resonant and red-shifted from the peak of GNR’s LSPR. Therefore the plasmonic heating on GNR is reduced, thus preventing the thermal deformation of GNR [15]. In contrast, the laser of 532 nm can provide an optical torque to align GNR with the perpendicular direction to the laser polarization [15]. However, 532 nm is close to the interband transition of gold (SPR overlaps with the absorption band), and is infeasible for aligning GNR, owing to that the plasmonic heating is severe [29]. Hence 532-nm laser can be used for the heating and printing GNR, rather than the rotating GNR. On the other hand, for gold or silver nanowires, some experimental studies have demonstrated that nanowires exhibit perpendicular behavior when a polarized laser beam is used to rotate gold or silver nanowires, where the wavelength of the adopted laser ranges from 800 nm to 1064 nm [14, 22].

We are motivated to study the mechanism and the wavelength-dependent performance of optical torque for an elongated GNR in order to clarify the above contradictory results of metal nanorods and nanowires; the nanorod is forced parallel to the polarization of the illuminating light, while the nanowire is perpendicular to the polarization of light [14]. We will propose the parallel and perpendicular modes of GNR to explain the distinctly different phenomena in this paper. In particular, both the maximum torque and the less plasmonic heating are considered. The multiple multipole (MMP) method was used to simulate the electromagnetic (EM) fields induced by a polarized plane wave illuminating a GNR [30–32]. The Maxwell stress tensor was calculated in terms of the EM fields and, then, the optical torque was analyzed quantitatively [33]. In addition, the absorption spectrum of GNR was analyzed to identify the plasmonic heating effect. It is worth noticing that this study uses a linearly polarized plane wave, rather than a non-uniform focused Gaussian beam, as the light source. Although the gradient force of the non-uniform Gaussian beam is important to induce an asymmetric local field for trapping nanoparticles or microparticles, the simplified model of a GNR irradiated by a linearly polarized plane wave is sufficient to provide some insights into the mechanism of optical torques.

2. Theory

Figure 1 schematically depicts a GNR irradiated by a z-polarized plane wave with a wavevector k incident along the y axis. The long axis of GNR lies on the xz plane. Angle θ is the orientation angle of the long axis of GNR with respect to the z axis (laser polarization direction), e_L is the unit vector along the long axis of GNR, and M is the net optical torque. GNR can be approximately regarded as an equivalent electric dipole with an induced dipole moment P_L, which depends on wavelength λ. Here, the wavelengths of TSPR and LSPR denoted by λ_T and λ_L, and Re represents the real part of a complex vector. For a harmonic field, the time-averaged Maxwell stress tensor of the external EM field is defined as [33],

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}

where

ε

and

μ

denote the permittivity and permeability of the surrounding medium, and I is the identity matrix. The E and H are the total electric and magnetic fields outside the GNR, i.e. the combination of the incident and scattered fields:

E = E^{i} + E^{s}

and

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

. In Eq. (1), the bar indicates complex conjugate. The optical force F and the induced net optical torque M exerted on GNR are

F = \int_{S} T \cdot n d s,

and

M = \int_{S} r \times T \cdot n d s

where S is the surface of GNR and n is the outward normal to the surface. In this study, the external EM fields are calculated using the MMP method. The Maxwell stress tensor is subsequently obtained using Eq. (1) and, then, the optical force and torque are obtained using Eqs. (2) and (3), respectively. Since the EM fields depend on the angle θ between the orientation of GNR and polarization of laser beam, the optical torque exerted on GNR is also θ-dependent.

Fig. 1 Configuration of GNR irradiated by a z-polarized plane wave with a wavevector k along y axis. θ: angle between long axis of GNR and z axis. λ: wavelength. P_L: dipole monent. M: net optical torque. A circle with a cross (⊗): a vector pointing into the plane, and a circle with a dot at center ⊙: a vector out of the plane. ∣∣: parallel mode, ⊥: perpendicular mode.

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The absorption power (heating power) P_a of GNR in terms of the surface integral of Poynting vector is expressed as

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

The absorption efficiency is further defined as

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

where A is the cross section area of GNR,

A = r^{2} (π + 4 AR - 4)

, and the fluence

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

. The absorption efficiency can also be calculated in terms of the EM fields obtained by the MMP method. For a linear problem, the output optical torque is linearly proportional to the power of laser. Meanwhile, the illuminating optical field also generates the plasmonic heating of GNR. The photothermal effect can usually be minimized by using a laser of off-resonant wavelength red-shifted from LSPR, e.g. 1064 nm. However, the provided optical torque is subsequently reduced. Therefore, this study evaluates the overall performance of a polarized laser beam on GNR’s rotation by defining an efficiency of optical torque as the ratio of the optical torque to the product of the absorption efficiency and laser fluence. The absolute value of efficiency of optical torque can be used to assess the wavelength-dependent optical torque per MW/cm² under the condition of the same heating power, which is useful in determining the optimal laser’s wavelength for rotating a GNR with a specific AR and radius r. Meanwhile, the positive and negative signs of efficiency of optical torque indicate the parallel and perpendicular mode, respectively. In the following analysis, the unit of the efficiency of optical torque is nN-nm/(MW/cm²).

3. Numerical results and discussion

In the following simulation, the surrounding medium is water, and the wavelength-dependent permittivity of gold is referred to Ref [34]. The shape of GNR is assumed to be a circular cylinder with two hemispherical end-caps. An elongated GNR of AR = 4 and r = 20 nm is considered first. The optical torques exerted on the GNR at various θ (15°, 30°, 45°, 60°, 75°) irradiated by a laser fluence of 25 MW/cm² versus wavelengths are shown in Fig. 2(a), the absorption efficiency in Fig. 2(b), and the efficiency of optical torque in Fig. 2(c). The order of magnitude of optical torque exerted on GNR is about 10² to 10³ pN-nm. Figure 2(a) clearly reveals two entirely different modes of torques of opposite signs: the negative and positive torques induce perpendicular and parallel modes, respectively. For the perpendicular mode, the long axis tends to align perpendicular to the laser polarization, owing to the negative polarizability. In contrast, for the parallel mode, the long axis of GNR is forced to align parallel to the laser polarization, owing to the positive polarizability. Figure 2(b) indicates that the peak of the absorption efficiency is at LSPR, 955 nm. The turning point between the perpendicular and parallel modes is very close to the LSPR (955 nm), which has a null torque. Both optical torque and the absorption efficiency depend on θ. Moreover, according to our results, the perpendicular mode ranges from TSPR (520 nm) to LSPR (955 nm), and the parallel mode ranges from LSPR to the NIR regime. In addition, bandwidths of the perpendicular and parallel polarization modes are broader than those of TSPR and LSPR modes, respectively. Also, the efficiency of the parallel mode increases with an increasing wavelength. Our results further demonstrate that the efficiency of perpendicular mode is compatible with that of the parallel mode for rotating an elongated GNR. For example, the absolute value of the efficiency of optical torque is 0.232 nN-nm/(MW/cm²) at 785 nm for θ = 45°, which is larger than the value of 0.203 nN-nm/(MW/cm²) at 1064 nm. The resultant optical torque caused by the optical force at 785 nm (perpendicular mode) rotates GNR counter-clockwise, whereas that at 1064 nm (parallel mode) rotates clockwise. This finding suggests that the perpendicular mode could be more useful than the parallel mode when rotating a GNR with high AR. The optimal wavelength of the perpendicular mode is roughly the middle value of the wavelengths of TSPR (λ_T) and LSPR (λ_L), as shown in Fig. 2(c). Figure 3 shows the optical torques and the efficiencies of optical torque versus angle θ for this GNR at 785 nm (perpendicular mode) and 1064 nm (parallel mode). The maximum torque occurs at 45° for 785 nm and 1064 nm. Additionly, the torques of the complementary angles are almost the same (Fig. 1(a) and Fig. 3(a)). In contrast, the maximum efficiencies of both wavelengths occur at 82°-89° (Fig. 3(b)), owing to that the plasmonic heating power is extremely low at a larger angle.

Fig. 2 (a) Optical torque versus wavelength for GNR of r = 20 nm and AR = 4 in water, where θ = 15°, 30°, 45°, 60°, 75° and laser fluence = 25 MW/cm². (b) Absorption efficiency and (c) efficiency of optical torque (nN-nm/(MW/cm²)) versus wavelength.

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Fig. 3 (a) Optical torques (fluence: 25 MW/cm²) and (b) efficiencies of optical torque (nN-nm/(MW/cm²)) versus angle θ for GNR of r = 20 nm and AR = 4 at 785 nm and 1064 nm.

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The above two distinctly different phenomena (i.e. perpendicular and parallel modes) can be simply explained by the longitudinal polarizability of GNR. Since the longitudinal polarizability of a GNR is dominant over the transverse one, only the former effect is considered here. The induced dipole moment P_L of GNR can be expressed as [14]

P_{L} = α_{L} E^{i} \cdot e_{L} e_{L}

where α_L is the equivalent longitudinal polarizability. Assume that the long axis of GNR lies on the xz plane (Fig. 1). Therefore, the time-averaged optical torque upon a dipole can be simply expressed as

M = \frac{1}{2} Re (P_{L} \times {\bar{E}}^{i}) = \frac{1}{4} Re (α_{L}) {| E^{i} |}^{2} \sin 2 θ e_{y} = \frac{1}{4} | α_{L} | {| E^{i} |}^{2} \cos ϕ \sin 2 θ e_{y}

where

- π / 2 \leq θ \leq π / 2

. The equivalent longitudinal polarizability

α_{L}

is a complex function of wavelength and orientation angle θ, with a phase lag ϕ with respect to the incident field Eⁱ. According to Eq. (7), the rotating direction of GNR is determined by the sign of the real part of the longitudinal polarizability,

Re (α_{L})

, i.e. cosϕ. If the sign is positive (i.e. 0<ϕ <π/2), the optical torque belongs to the parallel mode (∣∣). According to our simulation, the parallel mode of GNR is induced as λ_L<λ; GNR is positively polarized. In contrast, if the sign is negative (π/2<ϕ <π), the perpendicular mode (⊥) is induced. Our results show that the perpendicular mode of GNR is induced as λ_T<λ<λ_L; GNR is negatively polarized. The schematic illustration of the parallel mode (∣∣) and perpendicular mode (⊥) is given in Fig. 1. Moreover, the turning point between the two modes occurs at the condition with the phase lag of ϕ = π/2; i.e. the null polarizability,

Re (α_{L}) = 0

. This implies that the turning point corresponds to the peak of the imaginary part of the longitudinal polarizability,

Im (α_{L})

; i.e. the peak of absorption power. Since the peak of absorption power is physically defined as the LSPR (λ_L), the turning point is naturally at LSPR.

This study further investigates the performances of optical torque exerted on GNR with different ARs. Figure 4(a) shows the optical torques versus wavelength for GNR of r = 20 nm with various ARs (2 to 6) irradiated by a fluence of 25 MW/cm², where θ = 45°. Figures 4(b) and 4(c) show the absorption efficiencies and efficiencies of optical torque, respectively. Again, the curves in Fig. 4(c) reveal two different modes for rotating GNR: the perpendicular mode and the parallel mode. In addition, the former ranges from TSPR to LSPR, while the latter ranges from LSPR to NIR. The turning point between these two modes is very close to LSPR, depending on AR. Figure 4(d) plots the wavelengths of the turning point and LSPR of GNR of r = 20 nm versus AR (2 to 6), where θ = 45°. The two curves are correlate well with each other; the larger the AR, the more red-shifted the turning point. For example, the turning point of GNR of AR = 2 is 685 nm, while that of an elongated one of AR = 6 is 1300 nm. Interestingly, the second LSPR appears for an elongated GNRs, e.g. the second LSPR is at 600 nm for AR = 6 (Fig. 4(b)) [31,32]. Figure 4(a) indicates that this second LSPR is still located in the perpendicular mode, yet almost with a null torque. Since LSPR is red-shifted as AR increases, the turning point of a GNR of high AR (> 10) is within the middle infrared regime. Hence the perpendicular mode of a highly elongated GNR or nanowire broadens significantly. Our results more or less explain why most previous experiments of a polarized laser beam rotating gold or silver nanowire exhibited the perpendicular mode if the adopted laser’s wavelength ranged from 800 nm to NIR regime [14]. In addition, despite more higher-order LSPRs for a nanowire, the resultant toque should be negative, i.e. the perpendicular mode dominates the rotation.

Fig. 4 (a) Optical torques (fluence: 25 MW/cm²), (b) absorption efficiencies, (c) efficiencies of optical torque (nN-nm/(MW/cm²)), and (d) turning point and LSPR for GNR of r = 20 nm with different ARs (2 to 6) at θ = 45° versus wavelength.

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Figure 5 displays the efficiency of optical torque for GNR of r = 20 nm versus AR (2 to 6), as irradiated by different lasers of 685 nm, 785 nm and 1064 nm at θ = 45°. We can infer that the performance of a specific laser depends on AR if the radius of GNR remains fixed. Since the turning points of GNR (r = 20 nm) of AR = 2, 3, 4, and 5 are at 645 nm, 785 nm, 955 nm, and 1100 nm (Fig. 5), the optical-torque behavior of a specific laser changes from the parallel mode to perpendicular mode as the AR of GNR increases. When the laser’s wavelength overlaps with the turning point of a specific GNR, the resultant optical torque exerted on this GNR is nearly zero. For a highly elongated GNR (AR> 5), even the NIR laser of 1064 nm performs the perpendicular-mode rotation. This is to say that a linearly polarized laser can cause different alignments of polydisperse GNRs; three groups can be observed. Namely, the shorted GNRs are forced to be parallel to the laser polarization; the elongated ones are forced to be perpendicular to the polarization; and few GNRs of specific AR (when turning point coinciding with the laser’s wavelength) have no response. In addition, the perpendicular and parallel modes of a GNR can be observed as individually irradiated by two different-wavelength lasers (e.g., 785 nm, 1064 nm), if LSPR is located between the two wavelengths.

Fig. 5 Efficiency of optical torque (nN-nm/(MW/cm²)) for GNR of r = 20 nm at θ = 45° with various AR (2 to 6), irradiated by different lasers of 685 nm, 785 nm and 1064 nm.

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This study also examines how the size of GNR affects the efficiency of optical torque. Figures 6(a), 6(b) and 6(c) show the optical torque, absorption efficiencies and efficiencies of optical torque for GNR of AR = 4 with different radius (10 nm, 15 nm and 20 nm) irradiated by different-wavelength lasers at θ = 45°, respectively. These curves reveal that LSPR and the turning point are red-shifted as the size of GNR increases if AR remains fixed. Therefore, the output performances of a specific laser may significantly differ for differently sized GNRs of the same AR. For example, the laser of 880 nm generates parallel-mode rotation for a smaller GNR of r = 10 nm and AR = 4, whose turning point is at 825 nm. In contrast, the 880-nm laser generates perpendicular-mode motion for a larger GNR of r = 20 nm and AR = 4, whose turning point is at 955 nm. However, if the 1064-nm laser is used, all three GNRs are aligned parallel to the laser polarization. Again, this phenomenon demonstrates that the output torque of a specific laser also depends on the size of GNR, in addition to AR.

Fig. 6 (a) Optical torques (fluence: 25 MW/cm²), (b) absorption efficiencies and (c) efficiencies of optical torque (nN-nm/(MW/cm²)) of GNR of AR = 4 with different radius (r = 10 nm, 15 nm, 20 nm) at θ = 45° versus wavelength.

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4. Conclusion

This study theoretically elucidated the wavelength-dependent longitudinal polarizability of an elongated GNR irradiated by a polarized laser beam. Numerical results indicate that the induced optical torques at the perpendicular and parallel modes of GNR clearly differ from each other. If the laser’s wavelength is within the parallel-mode region, the positive polarizability is induced, explaining why the long axis of GNR tends to align with the laser polarization. In contrast, if the wavelength is within the perpendicular-mode region, the negative polarizability is induced. Therefore, the short axis of GNR is forced to align with the polarization direction, i.e. the long axis is rotated perpendicular to the laser polarization. Additionally, the wavelength of turning point between the perpendicular and parallel modes is nearly at LSPR, depending on AR and the size of GNR. The perpendicular mode ranges from TSPR to LSPR, and the parallel mode ranges from LSPR to the NIR (or even the middle infrared) regime. For both modes, the longitudinal polarizability of GNR dominates over the transverse one. Both the perpendicular and parallel modes can be utilized to rotate GNR, except for that their rotations are in opposite directions. However, if less plasmonic heating and a larger optical torque is considered, the efficiency of optical torque is defined as the ratio of the optical torque to the product of the absorption efficiency and fluence. Analysis results indicate that these lasers of wavelength within the perpendicular mode may have more available options, rather than those of the off-resonant wavelength in the parallel mode. For example, for GNR of AR = 4 and r = 20 nm with an orientation of 45° with respect to the laser polarization, the absolute value of the efficiency of optical torque of the laser of 785 nm (the perpendicular mode) is larger than that of 1064 nm (the parallel mode). However, the resultant torques of the two lasers have opposite signs. Based on the wavelength-dependent efficiency of optical torque, we can determine an optimal wavelength to rotate a specific GNR. Moreover, the optimal wavelength of the perpendicular mode is roughly the middle wavelengths of TSPR and LSPR. Using the parallel-mode or perpendicular-mode polarizability of GNR, we can induce the opposite-direction optical torques for a variety of biomedical applications, e.g. measuring the supercoiling of DNA [17]. Furthermore, it is worth future studies to examine trapping of an optical tweezers and rotation of a gold or silver nanowire located in a non-uniform site of a focused Gaussian beam [14, 22–24].

Acknowledgments

This work has been supported by National Science Council, Taiwan (NSC 102-2221-E-182-049, 102-2221-E-002-056-MY3) and Chang Gung Memorial Hospital (CMRPD290043).

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Wavelength-dependent longitudinal polarizability of gold nanorod on optical torques

Abstract

1. Introduction

2. Theory

3. Numerical results and discussion

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (6)

Equations (7)

Optics Express