Letter Cite This: ACS Photonics XXXX, XXX, XXX-XXX
pubs.acs.org/journal/apchd5
Super-Resolution Trapping: A Nanoparticle Manipulation Using Nonlinear Optical Response Masayuki Hoshina,† Nobuhiko Yokoshi,† Hiromi Okamoto,‡ and Hajime Ishihara*,†,§ †
Department of Physics and Electronics, Osaka Prefecture University, 1-1 Gakuen-cho, Naka-ku, Sakai, Osaka 599-8531, Japan Institute for Molecular Science and The Graduate University for Advanced Studies, Myodaiji, 38 Nishigonaka, Okazaki, Aichi 444-8585, Japan § Division of Frontier Materials Science, Osaka University, 1-3 Machikaneyama-cho, Toyonaka, Osaka 560-8531, Japan ‡
S Supporting Information *
ABSTRACT: Optical manipulation of nanoparticles (NPs) with nanoscale precision has been a goal of several research fields. One of the promising ways to realize this is the usage of localized surface plasmon (LSP). The electric field at hotspots near metallic structures is highly localized, which generates a sufficient force to trap NPs, and at the same time, the optical nonlinearity of NPs appears. In this Letter, we propose a scheme to superresolutionally trap the NP into a particular hotspot of the metallic nanostructure array. The scheme is based on the optical nonlinearity of NPs, and utilizes two kinds of structured light: Gaussian and Laguerre-Gaussian beams. The results show the significant role of the optical nonlinearity in LSP trappings, and they are expected to open up new degrees of freedom to manipulate NPs. KEYWORDS: optical manipulation, localized plasmon-polaritons, optical nonlinearity, Laguerre-Gaussian beam
I
limit in order to manipulate NPs within a nanoscale space. The ability to generate trapping force within a particular nanoscale space would enable us to realize super-resolution trapping to approach a few molecular dynamics for fundamental studies, as well as build up molecular scale units for application purposes. The LSP trapping is one of the solutions because the area of the hotspot is much smaller than the light wavelength, and recently, a method to cool down the center-of-mass motion of nanoparticles by using optical feedback techniques has been reported.27 On the other hand, if the trapping point is highly scannable, one can create molecular scale structures much finer than the light wavelength. For example, we can imagine that a macroscopic area of the metallic structure arrays with many hotspots works as a drawing canvas to create an integrated structure of molecular scale units, as illustrated in Figure 1, while the use of a conventional focused laser beam cannot control such fine structures because of the diffraction limit. The purpose of this letter is to theoretically propose a physical scheme to realize such super-resolution optical manipulation. An important ingredient for the realization of super-resolution trapping is the optical nonlinearity of the targeted materials. In the resonant case, the optical nonlinearity easily appears and plays a significant role,28−30 even in the conventional laser
n 1986, Ashkin et al. demonstrated the trapping of particles with a single laser beam using a radiation force.1 Their technique has been developed as optical tweezers, and it is now widely applied in a variety of fields to manipulate micrometerscale objects.2−9 In recent years, the target of optical trapping has been shifting to nanometer-scale objects that lead to the realization of highly sensitive sensing, novel nanofabrication technologies and efficient nanometrology. However, within Rayleigh scattering theory, the radiation force is proportional to the volume of the object. It follows that the exerted force on nanometer-scale objects becomes much weaker than that on micrometer-scale objects. Therefore, some strategy is necessary to realize nano-optical trapping. As an effective method, nearfield trapping has been theoretically proposed, where nanoobjects are strongly attracted by the force due to the strong field gradient.10−12 A typical approach is to trap nanoparticles (NPs) at nanogaps between the edges of the metallic structures, where the localized surface plasmons (LSPs) are excited and the electric field is strongly localized and enhanced by the optical-antenna effect.13−18 Another approach is to utilize the electronic resonance to enhance the radiation force.19−21 In general, the nanostructures have quantized electronic levels. Thus, when the light frequency coincide with their transition energies, one can make the radiation force resonantly enhanced.21−26 The next step which is required to realize a nanoscale optical manipulation is the challenge to overcome the light diffraction © XXXX American Chemical Society
Received: September 17, 2017
A
DOI: 10.1021/acsphotonics.7b01078 ACS Photonics XXXX, XXX, XXX−XXX
ACS Photonics
Letter
Figure 2. Schematic of the model of metallic structures and the molecular level scheme. Figure 1. Image showing super-resolution trapping in periodic metallic arrays. In the area irradiated only by the Gaussian beam (red beam illustrated at the far side), a single kind of molecule is trapped in many hotspots. In the area where the Laguerre-Gaussian (LG) beam overlaps (green area illustrated at the near side), molecules are excluded and trapped only in the singular point of the LG beam (red part). The inset shows a top-view of a beam spot. The green area is the overlapped part of Gaussian and LG beams. The red part is the singular point of the LG beam. If the molecules are fixed on the substrate after trapping, molecular scale structures much smaller than the light wavelength can be created with multiple kinds of molecules by the scanning of laser beams (as indicated by white arrows), and such units are integrated as patterns over the wide area of metallic arrays.
the Methods section, we present the material parameters of the NPs and discuss their validity. In the theoretical demonstration described below, the pump and manipulation lights were plane waves, and their intensities were set to 10 kW/cm2. Their energies were 1.600 and 1.547 eV, respectively. Figure 3a shows the position dependence of the radiation force exerted on the NP. The result of the manipulation light alone without nonlinearity shows that the force due to the strong field gradient attracts the NP toward the center of the nanogap, as expected. However, the magnitude of the force is largely overestimated, even with the present incident intensity because the saturation effect due to nonlinearity is not considered. This can be seen if we compare the result with the result where the nonlinearity is correctly considered (see the red line in the same figure). On the other hand, in the presence of the pump beam, the direction of the force becomes inverted at a certain position. This inversion occurs because of the population inversion of the first excited state that makes the phase of the induced polarization inverted29 (see Figure 3b). Thus, from the on/off operation of the pump light, we can control the trapping/exclusion of NPs at the nanogap. Figure 3c shows the force spectra for the cases with and without the pump light. We see that trapping/ exclusion switching can be performed just below the 0−1 transition energy of the manipulation laser. Subsequently, we assess the possibility of the superresolution optical trapping in Figure 1. To create a molecular structure, it is important to trap a few molecules in nanometerscale. On the other hand, to develop integrated molecular structures, we have to arrange the molecules to an semimacroscopic array (Figure 1). The proposed manipulation is promising to achieve compatibility of these two operations on different size scales. Here, we simultaneously apply to a NP the light with two different structures: Gaussian beam and Laguerre-Gaussian (LG) beam. In the area where the two beams overlap, the nonlinear process in Figure 2 well works, and the NP receives the repulsive force from the hotspots. However, the attractive force is exerted on the NP in the narrow region around the axis of the LG beam because the population inversion is avoided there. This scheme is reminiscent of stimulated emission depletion (STED) microscopy.39 Such a scheme shows that the usage of the nonlinear response greatly enhances the number of degrees-of-freedom for NP manipulations. Figure 4a shows the model of periodic metallic arrays used for the calculation, and the results are shown in Figure 4b−g. It is assumed that the pump (LG) and the manipulation (Gaussian) beams are focused in the paraxial approximation. Their intensities are 100 kW/cm2 and 10 kW/cm2, respectively, and their energies are 1.600 and 1.547 eV, respectively.
trapping. For example, if molecules are subjected to very strong laser light, a nearly inverted population results, and this scenario successfully illustrates the efficient trapping reported to date under the resonance condition.29 This is particularly the case at photon energies above the molecular resonance energies,22,31 which causes a repulsive force according to the linear response theory.32 Similar effects may be more prominent in LSP trapping under the resonance condition of the NPs. However, the strong feedback effect between the dielectric polarizations of the plasmons and the NPs makes the optical process quite complex,33 and the mechanism behind the appearance of the nonlinearity in the plasmon trapping remains unknown. Therefore, we first examine how the consideration of the nonlinear response affects the expectations of fundamental phenomena. For the quantitative evaluation of the radiation force, including the nonlinear response of NPs, we develop a calculation method which is based on the quantum master equation for NPs combined with the discrete dipole approximation (DDA) method34 (refer to the Methods section). Thus, we can treat the induced polarizations of NP and LSP self-consistently with the electric field of the beams. Therefore, the effects of the radiative decay (photoemission) and quenching of NP are automatically included in the result (see Methods section). Here, we consider the NP with the size of 43 nm3 that is entirely made of dye molecules.36 As shown in Figure 2, we assume that the NP is modeled by the typical three-level structure including vibronic levels. The metallic nanostructures are assumed to be gold blocks forming a nanogap. The size of each block is 80 × 80 × 12 nm3. We used the dielectric function based on the Drude model.35 The nonlinear effect and temperature dependence of the gold dielectric function are not considered because the possible saturation effect or broadening effect does not change the essence of the results in the considered intensity region, although a small quantitative modification is expected.37,38 In B
DOI: 10.1021/acsphotonics.7b01078 ACS Photonics XXXX, XXX, XXX−XXX
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attractive force works in all of the hotspots in the beam crosssection, and we see that it is impossible to create the molecular scale structures as illustrated in Figure 1. Recent reports of the LSP trapping discuss the interesting roles of the elements other than the radiation force affecting the motion of NPs. The optomechanical effect has attracted growing attention as a source to enhance the stability of the trapping.40 Besides, the effects arising from the temperature elevation near the metallic structures, that is, the convection flow and the thermophoresis are unignorable in the LSP trapping. For example, depending on the ambient conditions, the thermophoretic force is comparable to the radiation force, and sometimes works against the optical trap adversely.41 However, some studies have reported highly efficient trappings overcoming such effects,42,43 which encourage us to further study the proposed scheme in the presence of the thermal effects. In addition, the recent experiment verified an effective trapping by the population inversion for conventional laser tweezers.44 Thus, we understand that the hypothesis of the optical manipulation by nonlinearity beyond the perturbation regime is a reality. In view of these situations, we can expect feasible conditions for the trapping/exclusion switching if we use the resonance lines of NPs in the LSP trapping. In the actual experiment, the radiation force plays the main role once the particles approach the vicinity of the nanogap.17 Thus, the quantitative evaluation of the radiation force here is essential to determine whether the NPs can be trapped or excluded. In summary, we have developed a theoretical method to study the radiation force under the nonlinear optical response of targeted NPs that interact extensively with metallic nanostructures. This is important because most of the experiments including LSP trapping are thought to experience strong field intensity. As a remarkable effect arising from the nonlinearity, we showed an inversion of the force direction induced by the pumping of NPs. As a possible application of this mechanism, we have theoretically demonstrated superresolution optical trapping on periodic metal arrays. The results show that the use of nonlinearity will greatly enhance the number of degrees-of-freedom required to develop unconventional manipulation techniques of NPs.
Figure 3. Radiation force exerted on the NP near the metal structure. (a) The position dependence of the exerted force when the NP is on the x axis. (b) The position dependence of the population of the first excited level when NP is on the x axis. (c) The force spectra when the distance between the NP and the gap center is 11.3 nm.
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METHODS Theoretical Method. For the theoretical demonstrations of NP trapping, we calculated the radiation force exerted on the NP. The expression of the time-averaged radiation force is derived from the formula of Lorentz force as45
(Although the photobleaching might occur within from several to several tens of seconds after the start of irradiation of light with assumed intensity, the trapping usually occurs within much shorter time.) Their beam waists are 1 μm, and the orbital angular momentum of the LG beam is ℏ per photon (l = 1). With respect to the parameters of the NP, we assume them to be the same as in the case of Figure 3. The dielectric function and the geometric structures of the metallic dimmers are also the same as those in Figure 3. In Figure 4b−d, we see that the radiation force is directed toward the capture point near the axis of the beams, while the NP feels the repulsive force from the other hotspots. The mechanism is the same as in the explanation for Figure 3. Similar force maps can be obtained for the different levels on the z-axis (see the Supporting Information). In addition, we also show that the trapping/ exclusion switching works for any path to approach the hotspot (also see the Supporting Information). Figure 4e−g show the result obtained in the absence of the LG beam. In this case, the
⟨F (⃗ ω)⟩ =
1 Re[ 2
∫V dr[̃ ∇E(̃ r ̃, ω)*]·PNP̃ (r ̃, ω)
(1)
where E⃗ is the total electric field and P⃗ NP is the induced NP polarization. The integration is performed over the volume of the NP. We calculated E⃗ and P⃗ NP according to the following process. First, we set up the coupled equations of the master equation of the NP and Maxwell’s equation. In Maxwell’s equations, we consider the geometric information on the system; the spatial structures of metal blocks and the position of the NP. We used the DDA method34 to solve the Maxwell’s equation. In order to obtain the background field affected only by the metallic structures, we solved the following discretized integral equation, C
DOI: 10.1021/acsphotonics.7b01078 ACS Photonics XXXX, XXX, XXX−XXX
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Figure 4. (a) Schematic of the model used for calculation. The black dot is a capture point and dashed dots are anticapture points. b-g) Radiation force map in the metal arrays irradiated by laser beams. The blue (red) color refers to the force toward (away from) the gap center. (b) Both the Gaussian and LG beams are delivered. (c, d) Enlarged maps of areas 1 and 2 in (b). Black arrows show the direction map of the exerted force. (e) Only the Gaussian beam is delivered. (f, g) Enlarged maps of areas 3 and 4 in (e). The meaning of the black arrows is the same as in (c) and (d). metal
E ⃗b( ri ⃗ , ω) = E0⃗ ( ri ⃗ , ω) +
∑
level, and Leff is the effective mean free path of electrons. We used the following parameters in gold nanoblocks: ϵb = 12.0, ℏΩpl = 8.958 eV, ℏΓbulk = 72.3 meV, ℏVf = 0.9215 eV·nm, and Leff = 20 nm.35 The formal solution of the total electric field in the presence of the NP obeys the following integral equation:
⃗ ( rj⃗ , ω)Vj G⃗0( ri ⃗ , rj⃗ , ω)Pmetal
j
(2)
where ri⃗ (rj⃗ ) is the position of the i(j)th cell, and E⃗ 0 and G⃗ 0 are, respectively, the incident field and the free-space Green’s function for the Maxwell’s equation. P⃗ metal is the polarization of the metal and is described by a local susceptibility χmetal as P⃗ metal(rj⃗ ,ω) = χmetal(ω)E⃗ b(rj⃗ ,ω). In this expression, χmetal(ω) is represented by the following Drude-type dielectric function with the parameters of gold
NP
E ⃗( ri ⃗ , ω) = E ⃗b( ri ⃗ , ω) +
where G⃗ is the renormalized Green’s function including geometrical information on the metallic structure, and P⃗ NP is the NP polarization with nonlinearity. In order to derive the renormalized Green’s function of arbitrary-shaped metallic structures, we solved the following integral equation:
(ℏΩpl)2
(
ℏ2ω 2 + iℏω ℏΓbulk +
ℏVf Leff
)
(4)
j
χmetal (ω) = ϵmetal(ω) − 1 = ϵb − 1 −
⃗ ( rj⃗ , ω)Vj ∑ G⃗( ri ⃗ , rj⃗ , ω)PNP
metal
(3)
G⃗( ri ⃗ , rj⃗ , ω) = G0⃗ ( ri ⃗ , rj⃗ , ω) +
∑
G0⃗ ( ri ⃗ , rk⃗ , ω)χmetal (ω)G⃗( rk⃗ , rj⃗ , ω)Vk
k
where ϵb is the background dielectric constant of the metal, Ω is the bulk plasma frequency, Γbulk is the electron-relaxation constant of the metal, Vf is the electron velocity at the Fermi
pl
(5)
where G⃗ 0 is the free-space Green’s function. D
DOI: 10.1021/acsphotonics.7b01078 ACS Photonics XXXX, XXX, XXX−XXX
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The NP polarization should be determined by the total electric field. Then, we assume the following Hamiltonian of the NP Ĥ (t ) =
∑
ℏωaσaâ −
* Supporting Information
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsphotonics.7b01078. Force map figures for molecules approaching the hotspot from directions that are different from that in the main text (PDF).
(6)
where the index a represents excited levels of the NP, ℏωa represents the transition energy between the ground state and the state a of the NP, and σaa represents the population of state a. Here, we describe the induced polarization as →̂ PNP( r ⃗) =
∑ dkl⃗ σkl̂ δ( r ⃗ − rp⃗ ) + c. c. k
