Wavelength-dependent longitudinal polarizability of gold nanorod on optical torques Jiunn-Woei Liaw,1,2 Wei-Jiun Lo,3 and Mao-Kuen Kuo3,* 1
Department of Mechanical Engineering, Chang Gung University, 259 Wen-Hwa 1st Rd., Kwei-Shan, Taoyuan 333, Taiwan 2 Center for Biomedical Engineering, Chang Gung University, 259 Wen-Hwa 1st Rd., Kwei-Shan, Taoyuan 333, Taiwan 3 Institute of Applied Mechanics, National Taiwan University, 1 Sec. 4, Roosevelt Rd., Taipei 106, Taiwan * [email protected]
Abstract: This study theoretically investigates the wavelength-dependent longitudinal polarizability of a gold nanorod (GNR) irradiated by a polarized laser beam. The resultant optical torque in terms of the Maxwell stress tensor was analyzed quantitatively using the multiple multipole method. Our results indicate that the real part of the longitudinal polarizability of GNR can be either positive or negative, leading to the parallel or perpendicular modes, respectively. For the parallel and perpendicular modes, the long axis of GNR is rotated to align parallel and perpendicular, respectively, to the polarization direction of the illuminating light. The turning point between these two modes, depending on the aspect ratio (AR) and the size of GNR, nearly coincides with the longitudinal surface plasmon resonance (LSPR). The perpendicular mode ranges from the transverse SPR to LSPR, and the range of the parallel mode is broadband from LSPR to the near infrared regime. Owing to that a larger optical torque and less plasmonic heating are of concern, an efficiency of optical torque is defined to evaluate the performance of different wavelengths. Analysis results indicate that lasers with wavelength in the perpendicular mode are applicable to rotate and align a GNR of a higher AR. For example, the laser of 785 nm (the perpendicular mode) is superior to that of 1064 nm (the parallel mode, off-resonant from LSPR of 955 nm) for rotating a GNR of AR = 4 and radius 20 nm with an orientation of 45° with respect to the laser polarization. ©2014 Optical Society of America OCIS codes: (250.5403) Plasmonics; (310.6628) Subwavelength structures, nanostructures; (260.3910) Metal optics; (240.6680) Surface plasmons; (140.7010) Laser trapping.
References and links 1. 2. 3. 4. 5. 6. 7. 8.
K. Bonin, B. Kourmanov, and T. Walker, “Light torque nanocontrol, nanomotors and nanorockers,” Opt. Express 10(19), 984–989 (2002). K. C. Toussaint, M. Liu, M. Pelton, J. Pesic, M. J. Guffey, P. Guyot-Sionnest, and N. F. Scherer, “Plasmon resonance-based optical trapping of single and multiple Au nanoparticles,” Opt. Express 15(19), 12017–12029 (2007). 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). M. L. Juan, M. Righini, and R. Quidant, “Plasmon nano-optical tweezers,” Nat. Photonics 5(6), 349–356 (2011). Z. Li, S. Zhang, L. Tong, P. Wang, B. Dong, and H. Xu, “Ultrasensitive size-selection of plasmonic nanoparticles by Fano interference optical force,” ACS Nano 8(1), 701–708 (2014). P. M. Hansen, V. K. Bhatia, N. Harrit, and L. Oddershede, “Expanding the optical trapping range of gold nanoparticles,” Nano Lett. 5(10), 1937–1942 (2005). E. Messina, E. Cavallaro, A. Cacciola, M. A. Iatì, P. G. Gucciardi, F. Borghese, P. Denti, R. Saija, G. Compagnini, M. Meneghetti, V. Amendola, and O. M. Maragò, “Plasmon-enhanced optical trapping of gold nanoaggregates with selected optical properties,” ACS Nano 5(2), 905–913 (2011). M. Ploschner, M. Mazil, T. F. Kraussc, and K. Dholakia, “Optical forces near a nanoantenna,” J. Nanophotonics 4(1), 041570 (2010).
#205663 - $15.00 USD (C) 2014 OSA
Received 5 Feb 2014; revised 7 Apr 2014; accepted 20 Apr 2014; published 29 Apr 2014 5 May 2014 | Vol. 22, No. 9 | DOI:10.1364/OE.22.010858 | OPTICS EXPRESS 10858
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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.
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Received 5 Feb 2014; revised 7 Apr 2014; accepted 20 Apr 2014; published 29 Apr 2014 5 May 2014 | Vol. 22, No. 9 | DOI:10.1364/OE.22.010858 | OPTICS EXPRESS 10859
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), eL 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 PL, which depends on wavelength λ. Here, the wavelengths of TSPR and LSPR denoted by λT and λL, 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=
1 1 Re ε EE + μ HH − ( ε E ⋅ E + μ H ⋅ H ) I 2 2
(1)
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 = Ei + 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 = T ⋅ n ds,
(2)
S
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Received 5 Feb 2014; revised 7 Apr 2014; accepted 20 Apr 2014; published 29 Apr 2014 5 May 2014 | Vol. 22, No. 9 | DOI:10.1364/OE.22.010858 | OPTICS EXPRESS 10860
and M = r × T ⋅ n ds
(3)
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. PL: 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.
The absorption power (heating power) Pa of GNR in terms of the surface integral of Poynting vector is expressed as 1 Pa = − Re E × H ⋅ n ds . 2 S
(4)
The absorption efficiency is further defined as
Qa = Pa AS i
(5)
where A is the cross section area of GNR, A = r 2 (π + 4AR − 4) , and the fluence S i = Ei × 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
#205663 - $15.00 USD (C) 2014 OSA
Received 5 Feb 2014; revised 7 Apr 2014; accepted 20 Apr 2014; published 29 Apr 2014 5 May 2014 | Vol. 22, No. 9 | DOI:10.1364/OE.22.010858 | OPTICS EXPRESS 10861
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).
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.
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
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Received 5 Feb 2014; revised 7 Apr 2014; accepted 20 Apr 2014; published 29 Apr 2014 5 May 2014 | Vol. 22, No. 9 | DOI:10.1364/OE.22.010858 | OPTICS EXPRESS 10862
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 (λT) and LSPR (λL), 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. 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.
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 PL of GNR can be expressed as [14]
PL = α L Ei ⋅ e L e L
(6)
where αL 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=
1 2
Re( PL × E ) =
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i
1 4
Re(α L ) E
i 2
sin 2θ e y =
1 4
αL E
i 2
cos φ sin 2θ e y
(7)
Received 5 Feb 2014; revised 7 Apr 2014; accepted 20 Apr 2014; published 29 Apr 2014 5 May 2014 | Vol. 22, No. 9 | DOI:10.1364/OE.22.010858 | OPTICS EXPRESS 10863
where −π / 2 ≤ θ ≤ 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
Department of Mechanical Engineering, Chang Gung University, 259 Wen-Hwa 1st Rd., Kwei-Shan, Taoyuan 333, Taiwan 2 Center for Biomedical Engineering, Chang Gung University, 259 Wen-Hwa 1st Rd., Kwei-Shan, Taoyuan 333, Taiwan 3 Institute of Applied Mechanics, National Taiwan University, 1 Sec. 4, Roosevelt Rd., Taipei 106, Taiwan * [email protected]
Abstract: This study theoretically investigates the wavelength-dependent longitudinal polarizability of a gold nanorod (GNR) irradiated by a polarized laser beam. The resultant optical torque in terms of the Maxwell stress tensor was analyzed quantitatively using the multiple multipole method. Our results indicate that the real part of the longitudinal polarizability of GNR can be either positive or negative, leading to the parallel or perpendicular modes, respectively. For the parallel and perpendicular modes, the long axis of GNR is rotated to align parallel and perpendicular, respectively, to the polarization direction of the illuminating light. The turning point between these two modes, depending on the aspect ratio (AR) and the size of GNR, nearly coincides with the longitudinal surface plasmon resonance (LSPR). The perpendicular mode ranges from the transverse SPR to LSPR, and the range of the parallel mode is broadband from LSPR to the near infrared regime. Owing to that a larger optical torque and less plasmonic heating are of concern, an efficiency of optical torque is defined to evaluate the performance of different wavelengths. Analysis results indicate that lasers with wavelength in the perpendicular mode are applicable to rotate and align a GNR of a higher AR. For example, the laser of 785 nm (the perpendicular mode) is superior to that of 1064 nm (the parallel mode, off-resonant from LSPR of 955 nm) for rotating a GNR of AR = 4 and radius 20 nm with an orientation of 45° with respect to the laser polarization. ©2014 Optical Society of America OCIS codes: (250.5403) Plasmonics; (310.6628) Subwavelength structures, nanostructures; (260.3910) Metal optics; (240.6680) Surface plasmons; (140.7010) Laser trapping.
References and links 1. 2. 3. 4. 5. 6. 7. 8.
K. Bonin, B. Kourmanov, and T. Walker, “Light torque nanocontrol, nanomotors and nanorockers,” Opt. Express 10(19), 984–989 (2002). K. C. Toussaint, M. Liu, M. Pelton, J. Pesic, M. J. Guffey, P. Guyot-Sionnest, and N. F. Scherer, “Plasmon resonance-based optical trapping of single and multiple Au nanoparticles,” Opt. Express 15(19), 12017–12029 (2007). 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). M. L. Juan, M. Righini, and R. Quidant, “Plasmon nano-optical tweezers,” Nat. Photonics 5(6), 349–356 (2011). Z. Li, S. Zhang, L. Tong, P. Wang, B. Dong, and H. Xu, “Ultrasensitive size-selection of plasmonic nanoparticles by Fano interference optical force,” ACS Nano 8(1), 701–708 (2014). P. M. Hansen, V. K. Bhatia, N. Harrit, and L. Oddershede, “Expanding the optical trapping range of gold nanoparticles,” Nano Lett. 5(10), 1937–1942 (2005). E. Messina, E. Cavallaro, A. Cacciola, M. A. Iatì, P. G. Gucciardi, F. Borghese, P. Denti, R. Saija, G. Compagnini, M. Meneghetti, V. Amendola, and O. M. Maragò, “Plasmon-enhanced optical trapping of gold nanoaggregates with selected optical properties,” ACS Nano 5(2), 905–913 (2011). M. Ploschner, M. Mazil, T. F. Kraussc, and K. Dholakia, “Optical forces near a nanoantenna,” J. Nanophotonics 4(1), 041570 (2010).
#205663 - $15.00 USD (C) 2014 OSA
Received 5 Feb 2014; revised 7 Apr 2014; accepted 20 Apr 2014; published 29 Apr 2014 5 May 2014 | Vol. 22, No. 9 | DOI:10.1364/OE.22.010858 | OPTICS EXPRESS 10858
9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34.
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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.
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Received 5 Feb 2014; revised 7 Apr 2014; accepted 20 Apr 2014; published 29 Apr 2014 5 May 2014 | Vol. 22, No. 9 | DOI:10.1364/OE.22.010858 | OPTICS EXPRESS 10859
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), eL 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 PL, which depends on wavelength λ. Here, the wavelengths of TSPR and LSPR denoted by λT and λL, 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=
1 1 Re ε EE + μ HH − ( ε E ⋅ E + μ H ⋅ H ) I 2 2
(1)
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 = Ei + 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 = T ⋅ n ds,
(2)
S
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Received 5 Feb 2014; revised 7 Apr 2014; accepted 20 Apr 2014; published 29 Apr 2014 5 May 2014 | Vol. 22, No. 9 | DOI:10.1364/OE.22.010858 | OPTICS EXPRESS 10860
and M = r × T ⋅ n ds
(3)
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. PL: 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.
The absorption power (heating power) Pa of GNR in terms of the surface integral of Poynting vector is expressed as 1 Pa = − Re E × H ⋅ n ds . 2 S
(4)
The absorption efficiency is further defined as
Qa = Pa AS i
(5)
where A is the cross section area of GNR, A = r 2 (π + 4AR − 4) , and the fluence S i = Ei × 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
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Received 5 Feb 2014; revised 7 Apr 2014; accepted 20 Apr 2014; published 29 Apr 2014 5 May 2014 | Vol. 22, No. 9 | DOI:10.1364/OE.22.010858 | OPTICS EXPRESS 10861
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).
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.
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
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Received 5 Feb 2014; revised 7 Apr 2014; accepted 20 Apr 2014; published 29 Apr 2014 5 May 2014 | Vol. 22, No. 9 | DOI:10.1364/OE.22.010858 | OPTICS EXPRESS 10862
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 (λT) and LSPR (λL), 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. 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.
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 PL of GNR can be expressed as [14]
PL = α L Ei ⋅ e L e L
(6)
where αL 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=
1 2
Re( PL × E ) =
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i
1 4
Re(α L ) E
i 2
sin 2θ e y =
1 4
αL E
i 2
cos φ sin 2θ e y
(7)
Received 5 Feb 2014; revised 7 Apr 2014; accepted 20 Apr 2014; published 29 Apr 2014 5 May 2014 | Vol. 22, No. 9 | DOI:10.1364/OE.22.010858 | OPTICS EXPRESS 10863
where −π / 2 ≤ θ ≤ 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
