We theoretically analyze the optical forces between two nearby silver nanoparticles for the case when the wavelength of the incoming light is close to the localized surface plasmon resonance (LSPR). It is shown that the optical force between the nanoparticles is enhanced by the LSPR and that it changes from attractive to repulsive for wavelengths slightly shorter than the resonance when the polarization of the incident light is parallel to the axis of the dimer. This behavior can be utilized to generate a stable separation distance between the nanoparticles. In the Rayleigh limit, the equilibrium distance is uniquely determined by the real part of the particle polarizability and the wavelength of the incident light. The results suggest that near-field optical forces can be used to manipulate and organize plasmonic nanoparticles with a tunable spatial resolution in the nanometer regime.
©2007 Optical Society of America
Optical tweezers  have become a valuable tool for manipulation and control of the position of small objects [2–4], with attractive applications in biophysics [5, 6]. However, particles with size much smaller than the wavelength of light, so-called Rayleigh particles, can move freely within the trapping volume created by a diffraction limited focused laser beam due to their rapid Brownian motion. This generally prevents precise optical manipulation of nanoparticles. One way to reduce the trapping volume, and enhance the spatial control, is to utilize optical forces created by evanescent fields, like the ones produced by light beams in total internal reflection, surface plasmon polaritons or guided light. In order to achieve 3D trapping in nanometric volumes, the localized evanescent field created by metal nanotips [7, 8], nanoapertures  or surfaces patterned with metallic nanostructures  have been proposed. Interestingly, micrometer sized gold patches have been employed to trap dielectric beads with non-focused light beams .
In the case of metallic nanoparticles, the optical forces become strongly wavelength dependent due to the electromagnetic response of the free electrons. Far to the red of the localized surface plasmon resonance (LSPR) wavelength, metal Rayleigh particles essentially behave as dielectric particles with an enhanced trapping potential, whereas strong absorption and scattering generally prevents trapping close to resonance . However, the large field enhancement and field gradients caused by the LSPR have also been proposed as a route towards generating powerful optical forces between plasmonic nanoparticles or between particles and molecules [13–16]. It was recently experimentally demonstrated that silver nanoparticles (λLSPR ≈ 380-500 nm) can be manipulated using a standard near-infrared optical tweezers set-up and that pairing of particles within the trap lead to pronounced LSPR shifts and a large increase in the surface-enhanced Raman scattering (SERS) efficiency of the particles [17–19]. In this work, we theoretically investigate such paring effects for the case when the incident wavelength is close to the LSPR. It is shown that the near-field interaction between the particles can be utilized to create a stable equilibrium distance between the particles and that the gap size is predetermined by the particle size and the incident wavelength. This effect might be utilized to either create tunable optical traps of molecular dimensions or to position particles with nanometric precision.
We first analyze the optical forces between two identical spherical silver nanoparticles immersed in water and illuminated by a plane wave polarized parallel to the axis of the dimer, i.e. y direction [Fig. 1(a)]. As a first approximation, we assume that the optical response of the nanoparticle pair can be described through a coupled point dipole analysis in which the particle polarizability is given by the Clausius-Mossotti relation α = α3 (εm - εd)/(εm + 2εd). Here, εm and εd represent the dielectric constants of silver and the external medium, respectively, and a is the particle radius. As expected, the polarizability exhibits a plasmon resonance at λLSPR ≈ 385 nm [Fig. 1(b)]. The total field acting on particle 1 and 2 can be found by solving the coupled dipole equation (CDA) , which yields:
where E inc y = E 0 e ikz represents the incident electric field, propagating in the z direction with wavevector k, and A yy = 2eiky(-1 + iky)/y 3is the dipolar coupling coefficient. The self-consistently determined total electric field induces an interparticle optical force along the y-axis, due to the interaction from the nearby particle, and a radiation force in the direction of the incident wavevector. The time-averaged total force can be computed through :
where “*” denote the complex conjugate and and are unit vectors in the y and z directions, respectively. The second term in Eq. (2) is negligible for non-absorbing particles or for situations when the amplitude of the incoming field is real, as in the case of the Gaussian beams commonly employed in optical tweezers. However, in the case of two coupled resonant nanoparticles, this term cannot be neglected, since the imaginary part of the polarizability is large close to resonance and the total field, given by Eq. (1), is intrinsically a complex quantity. If we now insert Eq. (1) into Eq. (2), the optical force in the y direction can be described as:
where α̃ = α/(1 + αA yy) represents an effective particle polarizability renormalized by the near-field interparticle coupling. For short separation distances between the particles, only the near-field term in A yy will be relevant, i.e., A yy ≈-2/y 3. Therefore, Eq. (3) can be approximated as:
This equation predicts three interesting effects: i) the optical force will exhibit a resonant behavior due to the LSPR, included in the Clausius-Mossotti polarizability α [see Fig. 1(b)]; ii) the resonance will red-shift when the nanoparticles approach each other, due to the ∣α∼∣4 factor, enclosing the near-field interaction between the particles; and iii) the optical force can be transformed from attractive to repulsive when either the distance between the particles or the incoming wavelength varies, as a consequence of the (1 - 2 Re[α]/y 3) term.
All these effects can be observed in Fig. 2(a). This figure shows the spectrum of the optical force in the y direction acting on particle 1 [see Fig. 1(a)] for various separation distances between particles, assuming a = 5 nm and an intensity of the plane-wave of 1W/μm2. In these calculations we employ the full expression of the coupling factor A yy without making any approximation. As anticipated in Eq. (4), the optical force shows a large increase in the amplitude due to the LSPR and a change of sign, passing form attractive to repulsive, for wavelengths shorter than the resonance. In addition, the spectrum of the optical force red-shifts and the amplitude of the maximum force between the particles increases as the distance between the particles is reduced.
In order to elucidate the effect of the finite size of the particles, we compare these results to those obtained using the Discrete Dipole Approximation (DDA), in which each particle is discretized in small dipoles with cubic shape, whose side is 0.25 nm. This means that each particle is composed by approximately 3.3.104 dipoles. The inset of Fig. 2(a) shows that, although the amplitude is slightly larger and the spectra are red-shifted with respect to the point dipole approximation, the optical forces display the same trend: the forces pass from attractive to repulsive at short wavelengths and the spectra red-shift when the separation distance decreases.
In contrast, F 1y exhibits a different behavior when the polarization of the incoming light is perpendicular to the dimer (x-direction). In this configuration the time averaged optical force is calculated in the same way as in Eq. (3), but interchanging the coupling coefficient A yy by A xx =e iky(1-iky-k 2 y 2)/y 3 . As Fig. 2(b) shows, for this polarization the optical force between the particles is repulsive in the whole spectral range, and the spectra blue-shift for decreasing distances, similarly to the extinction cross section of the dimer.
This peculiar behavior of the optical forces offers a way to control the separation distance between the particles. This effect can be appreciated in Fig. 3(a), where the optical force in the y direction is plotted as a function of the separation distance between nanoparticles, using the CDA. For the longest wavelengths of the incident light (415-410 nm) the optical force between nanoparticles is attractive in the whole range of separation distances shown in the figure. In contrast, for shorter wavelengths, the optical force is transformed from attractive to repulsive at a specific separation distance, and this distance increases as the wavelength becomes shorter. Accordingly, in terms of optical forces, an equilibrium separation distance between the nanoparticles can be generated. For a given wavelength of the incoming light, this equilibrium distance can be extracted directly from Eq. (4), when If 〈F 1y〉 = 0, i.e.:
If we assume a Drude dielectric function ε(ω) = 1- ω 2 p /(ω 2-iωΓ) for silver, where ω p is the plasma frequency and Γ the relaxation time of the electrons, the polarizability of the metallic particle adopts the following Lorentzian shape: α(ω) = a 3 ω 2 0/(ω 2 0 - ω 2 + iωΓ), where ω 0 = ω p /√3 is the LSPR frequency. Consequently, Eq. (5) can be rewritten as:
where we impose that d eq≥2a. As can be deduced in Eq. (6), this condition is fulfilled in a range of wavelengths slightly longer than the resonance of the single metal particle. In the inset of Fig. 3(a) we represent the wavelength dependence of d eq using the polarizability of Fig. 1(b) in Eq. (5). As can be observed, the condition d eq≥2a is satisfied in the range of wavelengths comprised between 383 and 415 nm. On the other hand, the range of tunability that can be achieved in d eq in this configuration is around 5 nm. These calculations are also compared with the equilibrium distance obtained without making approximations in A yy Both results are very similar, stressing the importance of the near-field interaction between particles in the behavior of the optical forces.
In this range of separation distances the Van der Waals force (F VdW) start to play as well a significant role in the interaction between the particles. However, as Fig. 3(a) demonstrates, when the intensity of the incident light is around 1W/μm2, the amplitude of the optical force is much larger than F VdW (calculated with the Hamaker formula and Hamaker constant for silver C = 40.10-20 J). To verify the possibility of creating a stable separation distance between the metal nanoparticles, we compute the potential energy associated to the optical and Van der Waals forces in the y direction (W y). Figure 3(b) show that the interaction between the particles yields to potential wells centered at approximately the equilibrium distance calculated with Eq. (5). In addition, for an intensity of the incoming light of 1W/μm2, which is commonly employed in optical tweezing, such potential wells are much deeper than the thermal energy KBT, where KB is the Boltzmann constant and T the absolute temperature. Consequently, it could be possible to keep the nanoparticles at specific separation distances, and this distance could be tuned by controlling the wavelength of the incoming light.
As mentioned above, when a = 5 nm the range of tunability of the equilibrium separation distance is restricted to 5 nm. However, since d eq ∝ a [see Eq. (6)] this range can be substantially expanded for bigger particles, which are, indeed, more interesting for practical applications. In addition, increasing the size significantly enhances the amplitude of the optical forces (F y ∝ a 6) and the red-shift of the optical spectrum when the particles approach each other. The inset of Fig. 4(a) displays the spectra of the optical force in the y direction for nanoparticles with a = 15 nm when their separation distance is 36 nm, computed with the CDA and DDA, assuming an intensity of the incoming light of 0.2W/μm2 For this longer separation distance both methods give similar results, showing the same trend as in smaller particles, although, as expected, the amplitude of the force is much higher. The enhancement of the optical force, together with the faster decay of F VdW at longer separation distances, allow tuning d eq in a range larger than 10 nm, employing much lower intensity of the incoming light, as the potential wells of Fig. 4(a) confirm. Even larger d eq are expected for bigger particles, however, in these cases, the point dipole approximation will not be valid and DDA or rigorous Mie Calculations will be required for the analysis of the optical forces.
The equilibrium separation distance in the y direction constitutes actually an efficient 3D optical trap [see Figs. 4(b) and 4(c)]. To show this effect, we calculate the potential energy along the x and z directions (W x and W z, respectively) assuming, for each wavelength, the equilibrium separation distance obtained in Fig. 3(a). The potential wells created in these directions are also much deeper than the thermal energy or the energy associated to the scattering force in the z direction, which is reflected in the asymmetry of the shape of the potential wells in the z direction.
The polarization of the incoming light introduces another degree of freedom to control the orientation of the dimer of metal nanoparticles. As Fig. 2(b) showed, when the polarization of the incoming light is perpendicular to the axis of the dimer, the optical force between the nanoparticles is repulsive. Thus, the rotation of the polarization of the incoming light from the y direction will tend to align the axis of the dimer parallel to the polarization of the incoming light, keeping the equilibrium separation distance between particles.
In conclusion, we have shown that the near-field interaction between resonant plasmonic nanoparticles can be exploited to create tunable 3D nano-optical traps. In such nano-traps the separation distance and orientation between particles can be controlled with the wavelength and polarization of the incoming light. These effects can find important applications in surface enhanced spectroscopies, such as surface enhanced Raman scattering, surface enhanced fluorescence, or in the improvement of the biosensing response of metal nanoparticles.
Financial support of the Swedish research council is gratefully acknowledged. The authors thank Romain Quidant, Manuel Nieto-Vesperinas, Zhipeng Li and Hongxing Xu for fruitful discussions and suggestions.
References and Links
3. M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348–350 (1998). [CrossRef]
6. S. B. Smith, Y. J. Cui, and C. Bustamante, “Overstretching B-DNA: The elastic response of individual double-stranded and single-stranded DNA molecules,” Science 271, 795–799 (1996). [CrossRef] [PubMed]
7. L. Novotny, R. X. Bian, and X. S. Xie, “Theory of nanometric optical tweezers,” Phys. Rev. Lett. 79, 645–648 (1997). [CrossRef]
09. K. Okamoto and S. Kawata, “Radiation force exerted on subwavelength particles near a nanoaperture,” Phys. Rev. Lett. 83, 4534–4537 (1999). [CrossRef]
11. M. Righini, A. S. Zelenina, C. Girard, and R. Quidant, “Parallel and selective trapping in a patterned plasmonic landscape ” Nature Physics 3, 477–480 (2007). [CrossRef]
12. J. R. Arias-Gonzalez and M. Nieto-Vesperinas, “Optical forces on small particles: attractive and repulsive nature and plasmon-resonance conditions,” J. Opt. Soc. Am. A 20, 1201–1209 (2003). [CrossRef]
17. J. Prikulis, F. Svedberg, M. Kall, J. Enger, K. Ramser, M. Goksor, and D. Hanstorp, “Optical spectroscopy of single trapped metal nanoparticles in solution,” Nano. Lett. 4, 115–118 (2004). [CrossRef]
19. F. Svedberg, Z. P. Li, H. X. Xu, and M. Kall, “Creating hot nanoparticle pairs for surface-enhanced Raman spectroscopy through optical manipulation,” Nano. Lett. 6, 2639–2641 (2006). [CrossRef] [PubMed]
20. B. T. Draine and P. J. Flatau, “Discrete-Dipole Approximation for Scattering Calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994). [CrossRef]
21. P. C. Chaumet and M. Nieto-Vesperinas, “Time-averaged total force on a dipolar sphere in an electromagnetic field,” Opt. Lett. 25, 1065–1067 (2000). [CrossRef]