Optical forces on cylinders near subwavelength slits - OSA Publishing

Optical forces on cylinders near subwavelength slits: effects of extraordinary transmission and excitation of Mie resonances F. J. Valdivia-Valero and M. Nieto-Vesperinas∗ Instituto de Ciencia de Materiales de Madrid, C.S.I.C., Campus de Cantoblanco 28049 Madrid, Spain ∗ [email protected]

Abstract: We study the optical forces on particles, either dielectric or metallic, in or out their Mie resonances, near a subwavelength slit in extraordinary transmission regime. Calculations are two-dimensional, so that those particles are infinite cylinders. Illumination is with p-polarization. We show that the presence of the slit enhances by two orders of magnitude the transversal forces of optical tweezers from a beam alone. In addition, a drastically different effect of these particle resonances on the optical forces that they experience; namely, we demonstrate an enhancement of these forces, also of binding nature, at plasmon resonance wavelengths on metallic nanocylinders, whereas dielectric cylinders experience optical forces that decrease at wavelengths exciting their whispering gallery modes. Particles located at the entrance of the slit are easily bound to apertures due to the coincidence in the forward direction of scattering and gradient forces, but those particles at the exit of the slit suffer a competition between forward scattering force components and backward gradient forces which make more complex the bonding or antibonding nature of the resulting mechanical action. © 2012 Optical Society of America OCIS codes: (050.1940) Diffraction; (050.1220) Apertures; (350.4855) Optical tweezers or optical manipulation; (230.5750) Resonators; (240.6680) Surface plasmons.

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Received 22 Mar 2012; revised 25 Apr 2012; accepted 25 Apr 2012; published 30 May 2012 4 June 2012 / Vol. 20, No. 12 / OPTICS EXPRESS 13368

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1.

Introduction

The manipulation of nano-objects by means of the mechanical action of light constitutes a refinement of the development of optical tweezers, with a potential in physics and biology [1–10]. Extending optical trapping to the nanoscale requires coping with very small gradient forces and large thermal movement of the particles [3,4]. A possibility suggested in [3] to make #165256 - $15.00 USD (C) 2012 OSA


it feasible, uses the morphology dependent resonances (MDR) of a subwavelength aperture in presence of a nanoparticle. This allows stronger gradient forces with lower illuminating power. However, apart from that work, and in spite of the large amount of research on the phenomenon of extraordinary transmission, (or supertransmission), by subwavelength apertures [11–25], very few studies exist, as far as we know, [2,3], on the control of optical forces on particles near apertures. Using these systems as probes, it is of paramount interest in nano-optics to investigate this phenomenon at the subwavelength scale for e.g. sensing of nanostructures or estimation of adhesion between cell components in biology [8]. For instance, it was recently shown that the excitation of nano-object MDRs [26–28], e. g. of whispering gallery modes (WGM) [29–34] in dielectric particles or of localized surface plasmons (LSP) [35–38] in noble metal ones, placed either at the exit or at the entrance of a subwavelength aperture, contributes to enhance the supertransmission of the latter [39–41]. The excitation wavelengths of the particle resonances governs the range in which this supertransmission phenomenon works, e. g. near infrared wavelengths for high index (e. g. Silicon) dielectric particles and ultraviolet for those that are metallic. Therefore, the high localization of energy involved in this phenomenon suggests that these systems are interesting candidates to investigate the optical forces on the resonant particles at the nanoscale. This is the purpose of this paper. We shall present a study of the photonic forces exerted on either a Silicon (Si) particle or on a metallic particle, placed at the entrance or at the exit of a subwavelength slit in a metallic slab under extraordinary transmission illumination. We show that the optical force on dielectric nanoparticles with their Mie resonance excited is quite different to that of metallic ones in the same circumstances. In this way, resonantly illuminated dielectric nanoparticles near nanoapertures, experience optical forces that are generally weaker than using wavelengths out of their Mie resonance; this being due to the lower intensity distribution in their neighborhoods as a result of its localization inside as WGMs. On the other hand, metallic nanoparticles near subwavelength apertures suffer optical forces that are stronger in resonance than out of resonance; this stemming from the intensity enhancement and localization corresponding to their LSPs in the near field zone. In addition the forces between the aperture and the resonant metallic particle may be enhanced and attractive, contrary to what occurs on dielectric particles, specially when these latter are placed on the aperture exit. Calculations are done by using the Maxwell stress tensor (MST) [2, 8, 42–47]. In Section 2 we present a detail of the configuration and illumination conditions, as well as a brief description of the computations of intensities whose details were given in our previous works [40, 48]. The geometry is 2D, so that the particles are cylinders with cross section in the plane of calculations. It is well-known that this accounts for the main features of the phenomenon, except for depolarization effects [49–51]. In order to obtain supertransmission, the incident wave is p-polarized. Since, however, the FEMLAB procedure that we use does not straightforwardly yield the complex values of the space-dependent electric and magnetic fields, required for the evaluation of the MST, we report in the Appendix the details on the obtention of these complex quantities. Subsequently, Section 3 contains the study of the optical forces exerted on a Si cylinder, placed either at the exit or at the entrance of the slit. This particle being in or out from resonance excitation conditions. (Of course, when the slit is also present, the resonance of the particle shifts. We shall then refer to an illumination as ”resonant” when the particle MDR is excited in presence of the slit). In particular, when only the aperture MDR appears, we confirm the results of force enhancement reported on a dielectric particle in [3]. In addition, we investigate these forces when two of these particles are placed: one both above and one below the aperture. Finally, Section 4 investigates the photonic forces in the same configurations as in Section 3, but now when the particles are of Silver (Ag) so that LSPs may be excited. We shall see that due

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to the different effect of the excitation of a LSP from that of a WGM upon the transmitted light by the slit, the optical forces exerted on the particle are very different depending on whether this is at the entrance or at the exit of the aperture. 2.

Numerical calculations

Maxwell equations are solved by using a finite element method (FE), (FEMLAB 3.0a of COMSOL), for a 2D configuration of particles located near a slit practiced in a metallic slab. Apart from polarization effects, the essential features are like those obtained in 3D. Details of the meshing geometry and convergence which leads to accurate results are given elsewhere [40,48]. Hereafter, all refractive indices under the different wavelengths on use are taken from [52,53]. All particles in this study are considered of either Si or Silver (Ag), because of their rich Mie resonance spectrum in the near infrared and near ultraviolet, respectively. However, it should be stressed here that this is done for the sake of illustrating the effects, and that other materials can be chosen. Since the slab is thick, in order to have as small as possible skin depth and losses, its metal is assumed to be Aluminium (Al) [54]. It should be remarked in this connection that, ideally, a material which is as close as possible to a perfect conductor would exhibit the most pronounced supertransmission effects under study. For experiments with thinner slabs, other materials like noble metals: Gold (Au) or Ag may be employed.

Fig. 1. Schematic illustration of the geometry of the slit transmission and force calculations: An incident p-polarized Gaussian beam, (see main text), with magnetic vector Hz and Poynting vector Sy impinges from below an Al slab of width D and thickness h, containing and aperture of width d. Notice that D is the horizontal size of the window of the simulation space, in whose boundaries low reflection conditions are set. At those boundaries that coincide with the exterior limit of the metallic slab, a conductor condition is selected, (see [40]). When a cylinder of radius r is placed, the transmitted intensity is evaluated as follows: For dielectric (Si) particles, the time averaged energy flow norm | < Sy > | is calculated inside the cylinder circle cross section. For metallic (Ag) cylinders, one determines | < Sy > | in an annulus of exterior and interior radii: re and r, respectively. The circumference Σ of radius re , is also used to perform the integration of Eq. (1), no matter whether the cylinder is dielectric or metallic. An analogous procedure applies if the particle is placed below the aperture entrance.

In the 2D geometry dealt with here, we have employed an incident wave, linearly ppolarized, namely with its magnetic vector Hz perpendicular to the geometry of the XYplane (i. e. the plane of the images shown in this work), and propagating in the Y-axis di-

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rection, as shown in Fig. 1. For such a 2D geometry, it is well known that this choice of ppolarization (in contrast with s-polarization) is the one under which the subwavelength slit presents homogeneous eigenmodes, i. e. which transmit and may lead to extraordinary transmission [39, 40, 55–58]. In all cases the incident wave has a Gaussian profile at its focus: Hz (x, y) = |Hz0 |exp(−x2 /2σ 2 )exp(i((2π /λ )y − ω t)), |Hz0 | being the modulus, 21/2 σ corresponding to the half width at half maximum (HWHM) of the beam, and λ standing for its wavelength. In this way, the particles are cylinders with OZ axis. The geometrical parameters of the particles and the slab with the slit have been adjusted so that the excitation of morphologydependent resonances, of the slit and of the particles, match in presence of each other. The physical quantities studied are Hz (x, y) and the time-averaged energy flow < S(x, y) >. The light transmitted by the slit, and the energy concentrated in and on the cylinders, are obtained by integrating | < S(x, y) > | either in a circle which coincides with the cylinder cross section, or in an annulus surrounding it, depending on whether such a dielectric or metallic particle is placed near the slit, respectively. (Notice that if the particle is dielectric, the intensity transmitted by the slit, that couples to the particle WGM, is concentrated inside the cylinder, whereas when the nanoparticle is metallic, this transmitted intensity, coupled to a LSP, remains on the particle surface. This motivates our choice of domains of integration to estimate the intensity transmitted into the particle). On the other hand, when the slit is alone, the transmitted intensity is calculated by integrating | < Sy > | in the following form: (1) Inside a circle of radius r that coincides with the cross section of the dielectric cylinder that will subsequently be placed near the slit. (2) In an annulus limited by the radii r and re that coincides with the aforementioned annulus made on the metallic cylinder that will subsequently be placed near the slit. These intensity values are in all cases normalized to the maximum intensity of the incident Gaussian beam | < Smax > | = (1/π )mW /µ m2 . The time-averaged force on the particle is calculated by means of the time-averaged Maxwell stress tensor (MST). Its expression in SI units is [10, 59]:

< Fem > =

Z Z

∑

[ε /2 · Re{(E · n)E∗ } − ε /4 · (E · E∗ ) · n + µ /2 · Re{(H · n)H∗ }

− µ /4 · (H · H∗ ) · n] · dA,

(1)

where the surface of integration Σ surrounds the particle as seen in Fig. 1 and n stands for the outward unit normal. In our 2D geometry, Σ is the circumference of radius re , (see Fig. 1). In Eq. (1), E, H and E∗ , H∗ stand for the values of the fields and their complex conjugates, ε and µ being the electric permittivity and magnetic permeability of the surrounding medium (assumed here to be vacuum). Since the calculation with the complex values E(r) and H(r) of the real physical fields: ER (r,t) = R[E(r) exp(−iω t)] and HR (r,t) = R[H(r) exp(−iω t)] is not straightforward with FEMLAB variables, we show in the Appendix how to implement Eq. (1). Finally, the nomenclature followed to classify both the surface localized plasmon (LSP) and whispering gallery mode (WGM) resonances of the cylinders will use the subscripts (i, j), i and j standing for their angular i-th and radial j-th orders, respectively. In the case of the supertransmission resonances of the slit alone the subscripts (u, v) will be used, u and v standing for their longitudinal u-th and transversal v-th orders.

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3. 3.1.

Response in extraordinary transmission of a slit-cylinder system due to the excitation of WGMs Effects due to a WGM excited in a cylinder located either at the exit or at the entrance of the slit

Due to the potential as light concentrators that W GMs have, we first briefly study the energy transmission effects in a subwavelength slit in the infrared region, by adding dielectric cylinders at its entrance and exit planes. This will have important consequences for the understanding of the optical forces exerted on the particle, subsequently studied.

Fig. 2. (a) Detail of the time-averaged energy flux (< S(r) > in arrows and | < S(r) > | in the color spatial distribution) concentrated in the slit exit practiced in an Al slab, (slab width D = 19.920µ m, slab thickness h = 857.6nm, slit width d = 428.8nm) and in a Si cylinder (radius r = 200nm) tangent to the exit plane of the slit. The Gaussian beam at wavelength λ = 1170nm, with σ = 3µ m, incides from below. (b) Time-averaged energy flow norm | < S(r) > | versus wavelength λ transmitted by the slit alone (black curve with squares) and in presence of the dielectric cylinder (red curve with circles). The calculations of these intensities, with and without cylinder, are explained in Fig. 1 and in the main text above.

The distribution of time-averaged energy flux < S(r) > in Fig. 2(a) shows how a Si cylinder of radius r = 200nm concentrates the field transmitted by a slit of width d = 428.8nm practiced in an Al slab of thickness h = 857.6nm. The typical pattern of hot spots for the < S(r) > distribution at the slit exit corners due to charge concentration is seen. As shown in Fig. 2(b), the slit alone works near a supertransmission peak at λ = 1185nm (black curve with squares), which is close to the λ that excites a W GH2,1 resonance in the above cylinder when it is isolated (λ = 1195nm). The presence of the cylinder on the slit exit plane enhances and blue-shifts its transmission (red curve with circles), its maximum being now at λ = 1170nm, [its energy distribution is shown in Fig. 2(a)]. If the cylinder is instead located tangent to the entrance plane of the slit (which is the optimized distance for field enhancement in this configuration), its W GH2,1 is again excited at λ = 1200nm, but now there is no supertransmission peak by this particle-slit system. Instead, an almost linear transmittance growth appears as λ increases. This is seen in the black curve with squares of Fig. 3(a). Now the cylinder blocks the slit supertransmission due to its resonant behavior at a λ near that of the slit transmittance peak, (cf. red curve with circles of Fig. 3(a)). Hence, the cylinder WGM excitation “steals” a large portion of the incident energy that, in its absence, would have been transmitted by the slit. However, the suppression of transmission due to the excitation of a WGM at the entrance of the slit is partially thwarted by the presence of another WGM above the aperture exit plane. In

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Fig. 3. (a) The Si cylinder is now tangent to the entrance plane of the slit. All parameters are like in Figs. 2(a) and 2(b). Time-averaged energy flow norm | < S(r) > | against wavelength λ : transmitted by the slit (black curve with squares), and concentrated in the cylinder (red curve with circles). (b) Two cylinders are now present: one above the slit exit, like in Fig. 2(a), and another tangent to the entrance plane of the slit, like in Fig. 3(a). Time-averaged energy flow norm | < S(r) > | versus wavelengt λ concentrated in the lower cylinder (black curve with squares) and in the upper one (red curve with circles).

this case, the lower particle resonance peak in absence of upper particle, namely at λ = 1200nm, slightly redshifts to λ = 1235nm due to the presence of the upper cylinder, [compare the square curve of Fig. 3(b) with the red circle line of Fig. 3(a)]. However, the resonance peak of the upper particle is largely redshifted in Fig. 3(b) with respect to its value of Fig. 2(b). The wide lower particle resonance lineshape is not splitted by the presence of the slit. When light excites this lower WGM, there is light concentration in the lower particle and therefore does not reach the upper cylinder. Only when the lower particle allows the light to be transmitted through the slit, and due to the width of its resonance lineshape, [cf. curve with squares in Fig. 3(b)], the upper cylinder acquires a maximum intensity concentration. 3.2.

Optical forces on WGMs. Excited dielectric cylinder located either at the exit or at the entrance of the slit

Next, we analyze the behavior of the electromagnetic forces that the presence of the slit exerts on each of the above addressed cylinders. First, we consider the Si cylinder either above or below the slit as shown in Figs. 4(a) and 4(b), tangent to its exit or entrance plane, respectively. The electromagnetic forces are obtained from Eq. (1) by evaluating the integrand on a circle of radius re = 210nm surrounding the cylinder section, (see Fig. 1). As an example, the local force distribution on this circle is shown by the white arrows of Fig. 4(a) when this particle is horizontally shifted from x = 0 towards the right corner (or edge) of the slit. The resulting horizontal total force is stronger as it approaches the slit corners, where the electromagnetic energy acquires high values, [cf. Figs. 2(a) and 4(a)]. The variations of the X- and Y-components of these forces as the cylinder in Fig. 4(a) moves to the right from the aperture center x = 0 is shown in Figs. 5(a) and 5(b), respectively. As x increases, the particle is more and more horizontally attracted by the slit right corner. The growth of this attractive force magnitude is almost linear versus the lateral position when the particle is not resonant [see Fig. 5(a)]. The effect of the excitation of the W GH2,1 in the particle is better appreciated in the repulsive vertical forces. As seen in Fig. 5(b), this Y-component of the force pushes the particle off the slit exit, slightly more at x = 0 than in the corners. This is surely due to the superposition of radiation pressure Y-components when the particle is in the

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Fig. 4. (a) Detail of the spatial distribution of | < S(r) > | (colors) and local forces (white arrows) exerted on the surface of the upper dielectric cylinder of Fig. 2(a), at the same illumination (λ = 1170nm). The forces are evaluated on the circle of radius re = 210nm surrounding the cylinder section, (see Fig. 1). The particle is tangent to the exit plane of the slit, and horizontally shifted 135nm from the center towards the right edge. (b) The same for the cylinder below the slit, tangent to its entrance plane and 135nm horizontally moved towards the right. The illumination is the same as in (a), (λ = 1170nm).

Fig. 5. (a) X-component < FT x > of the time-averaged total electromagnetic force exerted on the Si cylinder above the slit [shown in Fig. 4(a)], as it moves from x = 0 to the right. Left vertical axis, black square and red circle curves stand for the case in which the cylinder in presence of the slit is out (λ = 1280nm) or in [λ = 1170nm, see Fig. 2(b)] resonance, respectively. These values are compared to those for the Si cylinder illuminated by the same Gaussian beam in absence of slit, (see the right vertical axis). The non-resonant and resonant cases are shown by the green up- and blue down-triangle curves, respectively. (b) The same study for the Y-component < FT y > of the time-averaged total electromagnetic force.

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center of the slit, case in which the configuration is symmetric with respect to the X-direction and hence the particle presents the larger scattering cross section to the energy flow exiting the slit. As clearly seen in Fig. 5(a), the lateral forces exerted on the Si rod by the light pattern at the slit exit (black squared and red circled curves, respectively) are about 102 times stronger than those exerted by the Gaussian beam alone. This is illustrated by the green up- and blue down-triangle lines of Fig. 5(a) corresponding to a non-resonant and to a resonant cylinder, respectively. Notice that the green up-triangle stright line of Fig. 5(a) illustrates the Hooke’s law negative gradient force, with spring constant (2/σ 2 )|Hz0 |2 , of a conventional optical tweezer, whose difference with the positive horizontal force when a WGM is excited, is remarkable. On the other hand, Fig. 5(b) shows that the vertical force exerted by the beam on the non-resonant particle, in absence of slit, (green up-triangle curve), is characteristic of a radiation pressure, and hugely becomes reduced when the resonance is present, (blue down-triangle line). Hence, the presence of the slit enhances the transversal forces with respect to those of a conventional optical tweezer. This allows lower incident energy to obtain the same force magnitude. Also, there is a difference in the magnitude of the vertical force depending on whether or not a Mie resonance is excited. The existence of a WGM couples the evanescent modes of the supertransmitting slit and takes a momentum in the direction opposite to that of the incident light beam. Under this p-polarized illumination, the W GH2,1 particle mode concentrates much of the field energy in the cylinder interior; lowering it outside. This causes the MST integrand of Eq. (1) to be smaller in the surrounding circle of radius ra than when there is no WGM; hence the electromagnetic forces become weaker. The result is therefore that the resonance counteracts the expelling force on the particle exerted by the light transmitted by the slit. This has consequences in photonic force microscopy of raster scanned topographies [42, 60].

Fig. 6. (a) X-component < FT x > of the time-averaged total electromagnetic force exerted on the Si cylinder below the slit (shown in Fig. 4(b)), as it moves to the right from x = 0. Black square and red circle curves stand for the case in which the cylinder in presence of the slit is out (λ = 1305nm) or in (λ = 1200nm) resonance, respectively. (b) The same for the Y-component < FT y > of the time-averaged total electromagnetic force.

If the Si cylinder is below the slit, Fig. 4(b) shows the distribution of forces obtained analogously as before. Now, Fig. 6(a) shows that the horizontal force grows as the particle approaches the slit corner. The effect of the resonance excitation extracting energy from the particle exterior is now clear for this X-component of the force, consequently being stronger when the cylinder is non-resonant than when it is. The reason is the same as that argumented before for the cylinder being above the slit. There is, though, no appreciable difference in the behavior of the Y-component of the forces whether or not there is resonance excitation, [see Fig. 6(b)], be#165256 - $15.00 USD (C) 2012 OSA


cause of the contribution of the the pushing radiation pressure added to the effect of the pulling component towards the slit corner. In particular, although we do not illustrate it here with figures for the sake of brevity, we have observed that this trapping force on the lower dielectric cylinder is larger at the transmission MDR wavelength of the slit, (slightly shifted in presence of this cylinder), than out from it. This confirms the so-called SIBA (self induced back action) effect observed in [3] when the illuminating wavelength excites a supertransmission resonance of the aperture but not a MDR of the particle. Also, a comparison of these aperture forces with those of an optical tweezer, as discussed in Figs. 5(a) and 5(b) when a beam alone illuminates the particle, highlights their enhancement by some order of magnitude due to the presence of the aperture.

Fig. 7. (a) X-component < FT x > of the time-averaged total electromagnetic force exerted on the lower Si cylinder in presence of the slit and of the upper cylinder. The lower particle moves horizontally towards the right corner of the slit. Black square and red circle curves stand for the case in which the lower cylinder is out (λ = 1290nm) or in (λ = 1200nm) resonance, respectively. (b) The same study for the Y-component < FT y > of the time-averaged total electromagnetic force. (c) Spatial distribution of Hz (r) in the configuration analyzed in (a) and (b) when the lower cylinder is resonant (λ = 1200nm). (d) The same as in (c) for the case in which no cylinder is resonant (λ = 1290nm).

To complete this part of our study, calculations of electromagnetic forces have been made for the configuration of a slit with two Si cylinders: one at its entrance and one at its exit. We address the forces acting on each cylinder separately. Figure 7(a) displays the variation of the X-component of the force exerted by the field on the lower particle, in presence of the upper one. The values of these curves (black squared and red circled curves standing for the case of non-resonant and resonant lower particle, respectively) against the variation in horizontal

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particle position look like those of Fig. 6(a) for the lower particle without the upper one. By contrast with the vertical force on the lower particle when there is no upper cylinder, [cf. Fig. 6(b)], we now see in Fig. 7(b) a stronger Y-component of the force on the lower particle on illumination at a resonant resonant wavelength. Thus, the pushing effect of the incident light beam, enhanced near the slit corners, on the lower particle is reinforced by the excitation of its W GH2,1 resonance. Figure 7(c) presents an instantaneous map of the spatial distribution of Hz (r) in the configuration studied in Figs. 7(a) and 7(b) with both Si cylinders at x = 0nm. The illumination wavelength is λ = 1200nm, which corresponds to the resonance peak of the lower cylinder, exhibited by the black square curve of Fig. 3(b). This pattern is useful to explain the antibonding nature of the pair: upper particle-slit, in contrast with the bonding characteristic of the set: lower cylinder-slit. We observe that in the former pair there is a sudden change, from positive to negative, in the sign of Hz (r) as one goes from inside the upper cylinder to the outside region which is in the exit of the slit. However, in the latter pair this sign is kept positive as one goes from inside the lower cylinder to the outside region in the slit entrance. Moreover, the resulting oscillation along OY of this magnetic vector inside the slit manifests the excitation of this latter cavity mode. Figure 7(d) presents a snapshot of the spatial distribution of Hz (r) in the same configuration, concentrating less light than in Fig. 7(c) because both cylinders are now illuminated out of resonance (λ = 1290nm). Whereas the force Y-component is attractive on the lower Si cylinder, Figs. 8(a) and 8(b) that show these forces on the upper cylinder, manifest that the force Y-component is repulsive. This variation of the sign of Hz (r) between pair of objects resembles that observed between pairs of particles in photonic molecules [61]: the same sign of the field in the WGM oscillations, facing each other in the zone where both particles are closer, is associated to a bonding force between both particles, whereas opposite signs of the field in those WGM oscillations is related to an antibonding force. We observe that the vertical force on the upper Si cylinder is one order of magnitude weaker than that on the lower one (1.18fN/nm versus 0.19fN/nm); this is due to the light power concentration from the resonance excited in the latter, which, as mentioned above, suppresses part of the light to be transmitted through the aperture.

Fig. 8. (a) X-component < FT x > of the time-averaged total electromagnetic force exerted on the upper Si cylinder, in presence of the slit and of the lower cylinder. The upper particle moves horizontally towards the right corner of the slit. Black square and red circle curves stand for the upper cylinder being out (λ = 1290nm) or in (λ = 1235nm) resonance, respectively. (b) The same for the Y-component < FT y > of the time-averaged total electromagnetic forces.

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Notice that the force components on the upper Si cylinder, shown in Figs. 8(a) and 8(b), look like those of Figs. 5(a) and 5(b) when there is no lower particle, but now with the effects of the presence or absence of particle resonance stronger and exchanged. The inclusion of the two cylinders simultaneously, adds complexity to the momentum transfer in this configuration; as we have just seen, there is no binding effect between the upper particle and the slit. The horizontal force, [cf. Fig. 8(a)], grows more in resonance of this upper particle as it moves to the right, than out of it; and the same happens for the vertical decay of the vertical force. 4. 4.1.

Response in extraordinary transmission of a slit-cylinder system due to the excitation of LSPs Effects due to a LSP excited on a metallic cylinder located either at the exit or at the entrance of the slit

Next, we study the effects on a slit supertransmitting in the ultraviolet when metallic cylinders and excitation of their LSPs are addressed. This will underline the optical forces on plasmonic nanoparticles. Those cylinders are, as before, located either at the exit or at the entrance of the slit. The illuminating Gaussian beam has σ = 1.3µ m. Figure 9(a) shows the variation of the time-averaged energy flow norm | < S(r) > | of the light transmitted through a slit of width d = 109.7nm practiced in a slab of thickness h = 219.4nm exciting a LSP of an Ag cylinder of radius r = 30nm, placed at the slab exit. This time, it is known that the particle resonance concentrates and enhances the transmitted light on its surface. The cylinder resonance is LSP2,1 , associated to a dipolar distribution of light, and presents a peak at λ = 335.1nm in this slit-particle system, (red curve with triangles), which occurs at λ = 335.1nm, slightly blue-shifted with respect to that from the slit alone, (which appears at λ = 339.7nm, as seen in this figure). This transmittance peak is significatively narrower than that of the slit alone, because of the sharp LSP2,1 lineshape [39]. We recall that according to Fig. 1, the intensity is now calculated on integration of | < S(r) > | in an annulus of radii r and re , either with or without this metallic cylinder. This annulus is shown in all the images shown from now on, and has radii: r = 30nm and re = 35nm. The circumference with this radius re is the curve Σ over which the force on the metallic cylinder is calculated according to Eq. (1). However, when the same Ag cylinder is placed below the slit entrance at an optimized distance for transmittance, (i. e. 40nm from the entrance plane of the slit), a huge enhancement in slit supertransmission is reached. This can be seen in Fig. 9(b), where a sharp peak of the timeaveraged energy flow norm at the exit of the slit appears (black curve with squares). This effect was discussed in Ref. [39], where we demonstrated that LSPs excited on metallic particles below the entrance of subwavelength slits couple with the aperture MDR modes, reinforcing the transmission by several orders of magnitude. To further analyze the transmittance enhancement by the presence of Ag plasmonic cylinders, both are now simultaneously placed at the exit and at the entrance of the slit. This is seen in Fig. 9(c), where the black square and red circle curves stand for the total energy in the annulus on the lower and on the upper cylinder, respectively. The shift between the resonant peaks of each cylinder are at λ = 326.3nm and λ = 330.6nm for the upper and lower particle, respectively. This shift is smaller than in the case of two dielectric cylinders shown in Fig. 3(b), because now the lower metallic cylinder does not ”extract” incident light energy as the WGM did; but, on the contrary, it enhances and localizes the field energy around its surface via the excited LSP and, consequently, reinforces the aperture transmittance on coupling with the slit mode.

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Fig. 9. (a) An Ag cylinder is placed tangent to the exit plane of a slit practiced in an Al slab, (slab width D = 5.096µ m, slab thickness h = 219.4nm, slit width d = 109.7nm). The radius of the cylinder is r = 30nm. The curves show the time-averaged energy flow norm | < S(r) > | against wavelength λ , transmitted by the slit alone (black curve with squares), and that concentrated on the cylinder cross-section (red curve with triangles) when the latter is placed as explained above. (cf. Fig. 1 and the text explaining the evaluation of the transmitted | < S(r) > |. The incident beam has σ = 1.3µ m. (b) The cylinder is now placed below, at 40nm from the entrance plane of the slit. The curves show: time-averaged energy flow norm | < S(r) > | versus wavelength λ transmitted by the slit (black curve with squares), and | < S(r) > | concentrated on the lower cylinder surface (red curve with circles). Notice that while the circle curve is obtained on integration of | < S(r) > | in the annulus of radii: r = 30nm and re = 35nm surrounding the lower cylinder, the annulus leading to the square curve is drawn in vacuum above the slit. (c) Both Ag cylinders are simultaneously placed with the slit. The curves show: Time-averaged energy flow norm | < S(r) > | versus wavelength λ concentrated on the surface of the lower cylinder (black curve with squares) and that on the surface of the upper cylinder (red curve with circles), respectively.

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4.2.

Electromagnetic forces on a LSP. Excited cylinder located either above or below the slit

From now on we study the effect of the wavefields discussed in Section 4.1 in presence of metallic cylinders and the aperture, on the forces exerted over these plasmonic particles near the entrance and / or exit of the subwavelength slit.

Fig. 10. Detail of the spatial distribution of | < S(r) > | (color) and of the local forces (arrows) exerted on the upper Ag cylinder, resonantly illuminated by a Gaussian beam of σ = 1.3µ m at λ = 335.1nm, evaluated on the circumference of radius re = 35nm surrounding its cross section. The cylinder is tangent to the exit plane of the slit and moved to the right 35nm from x = 0 . (b) The same for the lower Ag cylinder, at the same illumination wavelength, now 40nm down from the entrance plane of the slit and moved 20nm towards its right corner.

Figures 10(a) and 10(b) show the distribution of energy flow, and of forces on the circle of radius re = 35nm surrounding the cylinder section, when this particle is at the exit and at the entrance of the slit, respectively. The calculation of forces according to Eq. (1) is like in the previous case of dielectric particles. These images also show that | < S(r) > | exhibits the dipolar characteristics of the plasmonic resonance LSP2,1 . The same comments can be made about the magnitude variation of the local forces around the particles as those made concerning Figs. 4(a) and 4(b). We plot in Figs. 11(a) and 11(b) the horizontal and vertical components of the total electromagnetic force on the Ag cylinder above the slit, [cf. Fig. 10(a)], as it is laterally displaced from x = 0. The different behavior of the forces < FTx > and < FTy > on the cylinder illuminated in resonance is interesting. Due to the distribution of scattered light around the cylinder, the transversal force on the resonant particle drags it to the center of the slit as it laterally moves, with a magnitude which is maximum at a certain distance from the slit axis at which this force is counteracted by an attractive force towards the slit corner which dominates as the particle continues approaching it, [see the red curve with circles of Fig. 11(a)]. The radiation pressure of the field emerging from the slit pushes the particle when located at the slit center, this pushing force diminishing gradually in magnitude till, again, becomes a pulling force which attracts the particle to the slit corner, [cf. red curve in Fig. 11(b)]. It should be remarked that the transversal and vertical forces exerted by the Gaussian beam alone on this Ag cylinder illuminated out of resonance, are in a range similar to those shown in Figs. 11(a) and 11(b). However, when the resonance is excited (λ = 339.7nm), while the lateral force varies in a similar range, the vertical force is ”negative” i.e. it points against the sense of propagation of the beam, and varies between 1 and 2 f N/nm as the particle is displaced across the beam. This negative radiation pressure is a consequence of the interplay of the excited LSP on the cylinder with the scattered field surrounding it, similar to other pulling scattering forces #165256 - $15.00 USD (C) 2012 OSA


Fig. 11. (a) X-component < FT x > of the time-averaged total electromagnetic force exerted on the upper Ag cylinder in presence of the slit [cf. Fig. 10(a)], as the cylinder moves from x = 0 towards the right edge of the slit. Black square and red circle curves stand for the cylinder out (λ = 500.0nm) or in (λ = 335.1nm) resonance, respectively. (b) The same for the Y-component < FT y > of the time-averaged total electromagnetic force.

recently found [62–64]. The graphs are not shown for the sake of brevity.

Fig. 12. (a) X-component < FT x > of the time-averaged total electromagnetic force exerted on the lower Ag cylinder, in presence of the slit [cf. Fig. 10(b)], as it moves from x = 0 to the right. Black square and red circle curves stand for the cylinder illuminated out (λ = 500nm) or in (λ = 335.1nm) resonance, respectively. (b) The same for the Y-component < FT y > of the time-averaged total electromagnetic force.

The behavior of the X- and Y-components of the electromagnetic forces on the Ag particle when it is placed below the slit and shifted from its center, is shown in Figs. 12(a) and 12(b). When the particle is illuminated out of resonance, shown by the black curves with squares, the force keeps practically constant with the lateral distance, and is somewhat similar to that found on the upper particle [cf. black curves with squares of Figs. 11(a) and 11(b)]. But once again, important variations of the forces appear when the LSP2,1 mode is excited. Then, the variation of < FTx > and < FTy > is remarkable. The particle is strongly pushed horizontally more and more towards the slit right corner until a distance of maximum force is reached, then gradually diminishing as the particle is closer to this right corner, [this is shown by the red curve with circles of Fig. 12(a)]. The behavior of this force component is opposite to that on the upper Ag particle, [compare with Fig. 11(a)]. The particle is also vertically pushed towards the slit by #165256 - $15.00 USD (C) 2012 OSA


the incident light beam, being repelled by the slit corners as it gradually approaches them, as shown by the red curve with circles in Fig. 12(b). It is remarkable that these forces on the lower resonant particle cannot be explained symply by resourcing to conservative components attracting that cylinder to the slit edges, where there is more charge concentration. A metallic particle has a large scattering cross section and, hence, suffers strong radiation pressure. As a consequence, the total force results from a complex redistribution of the LSP which couples with the slit MDR as Figs. 11(a) and 11(b) show.

Fig. 13. (a) X-component < FT x > of the time-averaged total electromagnetic force exerted on the lower Ag cylinder, in presence of the slab and of the upper Ag cylinder, as shown in Fig. 11(a). This lower particle moves towards the right edge of the slit. Black square and red circle curves stand for the lower Ag cylinder out (λ = 500.0nm) or in (λ = 330.6nm) resonance, respectively. (b) The same for the Y-component < FT y > of the time-averaged total electromagnetic force. (c) Detail of a snapshot of Hz (r) at a certain time instant for λ = 330.6nm at which the lower cylinder is resonant. (d) The same as in (c) for the case in which no cylinder is resonant (λ = 500.0nm).

The force components in the configuration of a supertransmitting slit and two metallic cylinders simultaneously placed at each side, are shown in Figs. 13(a) and 13(b) for the lower Ag cylinder. In resonance of this lower cylinder, [λ = 330.6nm, see the larger peak of the black square curve in Fig. 11(b)], the forces < FTx > and < FTy > are attractive towards the slit axis and the slab, respectively. Just the opposite occurring out of resonance. Figure 13(c) illustrates the spatial distribution of Hz (r) which, like shown in Section 3.1 studying WGMs, explains the behavior of the forces. In this case, the coupling between the slit supertransmissive MDR and the lower LSP cylinder, [and also between the slit and the weakly excited upper LSP, see the red circle line of Fig. 9(b)], creates an intense and highly localized wavefield Hz (r) between lower

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Fig. 14. (a) X-component < FT x > of the time-averaged total electromagnetic force exerted on the upper Ag cylinder, in presence of the slab and of the lower cylinder, as shown in Fig. 11(a), as the upper particle moves from x = 0 to the right. Black square and red circle curves stand for the case in which the upper cylinder is illuminated out (λ = 500.0nm) or in (λ = 326.3nm) resonance, respectively. (b) The same for the Y-component < FT y > of the time-averaged total electromagnetic force.

cylinder and the slit, thus giving rise to an attractive force towards the aperture on both cylinders, i. e. like that of a bonding state of a photonic molecule. The vertical forces on the lower and upper cylinders are 0.59 f N/nm and −0.07 f N/nm, respectively. Figure 13(d) presents the spatial distribution of Hz (r) in the same configuration at a certain time, both cylinders being now out of resonance. As seen in Fig. 14(a), the upper cylinder, when this nanoparticle is out of resonance, suffers a weak transversal force as it moves to the right, being pushed towards the slit right corner, but it suffers a pulling force towards x = 0 in resonance, [see black square and red circle curves, respectively, in Fig. 14(a)]. Concerning the vertical force, the slit attracts the upper non-resonant cylinder, an effect that is enhanced on exciting a resonance of this upper cylinder, [as shown by the black square and red circle curves, respectively, in Fig. 14(b)]. This suggests that in resonance the slit allows a binding of both the upper and lower particles; the explanation is the same as that given above pertaining to the Hz (r) pattern of Fig. 13(c). The vertical forces exerted over the cylinders are now: 0.15 f N/nm on the lower one and −0.27 f N/nm on the resonant upper one. 5.

Discussion and conclusions

In this paper we have carried out a study of the mechanical action of light on nanoparticles near a subwavelength slit illuminated in its supertransmission regime. Special emphasis has been made in comparing the force on the particles when a Mie resonance is excited with that on non-resonant particles. The study has been carried out in two dimensions, so that these objects are cylinders. In order to excite the transmission modes of the slit, a p-polarized incident beam has been used. However, we believe that these results also hold for spheres in 3D. First, we have found that the presence of the slit enhances by two orders of magnitude the transversal optical forces on non-resonant cylinders, that would be obtained from an optical beam in a conventional optical tweezer configuration. Further, we have proven that whereas the morphological resonance of the slit, causing supertransmission, enhances the fields surrounding these cylinders and hence the optical forces exerted upon them, the excitation of a particle resonance has a quite different effect on these

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forces, depending on whether the cylinder is dielectric or metallic. The electromagnetic force on the particle is highly dependent on the field surrounding it. This means that WGMs excited in dielectric nanocylinders under p-polarization, either at the entrance or at the exit of the slit, have no great impact on the optical force; and even the high localization of these WGMs inside the cylinder may decrease its strength as a consequence of the very weak intensity distribution surrounding this particle. On the contrary, the excitation of LSPs in metallic nanocylinders, (whether at the entrance or exit of the slit), produce localized field energy enhancement around the cylinder surface; this reinforces the optical force, and even can produce a bonding pair: slit-cylinder. This bonding system may even be extended to one more metallic cylinder placed at the other side of the slit; then the aperture mediates between the two particles with a binding force. In this connection, it should be remarked that both the WGM and the LSP addressed in this study are electric dipole modes. This means that in principle, illuminating such a particle in front of a metallic thick slab should give rise to an attractive force, as a consequence of the distribution of charges induced in this particle as well as in its image [65]. This is the case with a metallic cylinder, then the strong external field intensity associated to a LSP helps to producing such a binding interaction. By contrast, the weak intensity surrounding a dielectric cylinder when a WGM is excited at p-polarization, should prevent this attractive force. Enhanced optical forces on dielectric cylinders are only obtained under s-polarization [66], which is the situation in which the generated WGM extends to the near field surrounding the cylinder. We believe that these results should stimulate further experiments to obtain systems in which manipulation by the mechanical action of light on nano-objects may be done taking into account their MDR resonances. Also this has consequences for raster scanning photonic force microscopy with nanometric tips-supertransmitting aperture devices. 6.

Appendix

A generic time-harmonic field is dealt with as a complex field V(r,t) = V0 (r) exp(−iω t), (V I may be either E or H), ω being the frequency and t the time. Also, V0 (r) = VR 0 (r) − iV0 (r) = |V0 (r)| exp(−iφ (r)) stands for the value of the complex field, when t = 0, which in general would have a phase φ 6= 0. The real part |V0 (r)|cos(φ (r) − iω t) of V(r,t) is the only observable quantity, whose value can be obtained from V0 (r) and V∗0 (r) by means of VR (r,t) = Re{V0 (r) exp(−iω t)} = 1/2[V0 (r) exp(−iω t) + V∗0 (r) exp(iω t)], which is a useful equation to express time-averaged quantities from their complex values, (see the corresponding expressions developed in [67] for the electric and magnetic fields). In our calculations, we need both the complex electric E0 (r) and magnetic H0 (r) fields, and their complex conjugates E∗0 (r) and H∗0 (r), in order to calculate the time-averaged value of the electromagnetic force exerted on a particle as shown in Eq. (1). Even though FEMLAB only plots real values, it allows to work with the complex values of these fields as internal variables, in such a way that a complex variable A0 (r) can be isolated by using [A(r,t)](t=0) as explained in the section “THE PHASOR VARIABLE” in the FEMLAB Documentation/User’s Guide/Modeling Physics and Equations/Variables and Expressions/Special variables, to be found in e.g. FEMLAB 3.0a version. All particles here studied are 2D (infinite cylinders), hence the integration of time-averaged local forces, previously decomposed into their components along the X and Y axis, < Flx > and< Fly >, is made along a circumference that surrounds the cross section of the cylinder, as seen in Fig. 15. Hence, by line integrating, a force per axial length unit is obtained. First, the integration surface embedding the particle is generated as close as field calculation convergence allows. Also keeping the symmetry to facilitate the computations. As seen in Eq. (1), it is important to correctly define the outwards normal to the integration curve. Once the

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Fig. 15. Schematic illustration of the geometry for the calculations of the force when a cylinder of radius r is placed near the slit. The components of the local electromagnetic force, < Flx > and < Fly >, are evaluated along the exterior circle of radius re (red curve). In our calculations re = 210nm and re = 35nm when r = 200nm (dielectric cylinder) and r = 30nm (metallic cylinder), respectively. The integration of these local forces over the four segments into which the exterior circle is divided, yields the total electromagnetic T force Cartesian components (< FT x > and < Fy >).

values of the fields and their complex conjugates are known, in order to correctly perform the calculations, n must be decomposed in its components nx and ny , taking into account that FEMLAB divides the circumference into four curved segments, equivalent to Bézier curves as Fig. 15 shows. The direction of parameterization for each curve, namely, the direction in which the parameterization variable s takes values from 0 to 1, is denoted by the arrows. The sense that FEMLAB assigns to n is always that pointing to the left relative to the parameterization direction of the segment. Hence, the orientation of the normal relative to the enclosed area, which FEMLAB initially defines for each segment, depends on which of them we are considering for the integration in Eq. (1). FEMLAB offers two additional variables, named nd and nu (the subindex d and u standing for down and up), which keep unchanged and invert, respectively, the sense of the normal. This is thoroughly explained in the sections “Normal Variables” and “Direction of the Normal Component on Interior Boundaries” in FEMLAB Documentation/User’s Guide/Modeling Physics and Equations/Variables and Expresions/Geometric Variables/TANGENT AND NORMAL VARIABLES, to be found in FEMLAB 3.0a version. Namely, for the upper segments, the normal vector which FEMLAB uses points outwards the close curve (n or nd must be used), while for the lower ones, it points inward. Thus, when Eq. (1) is evaluated over these lower segments, the sense of this n must be changed (i. e. −n or nu must be used), for the sake of physical significance. To illustrate this, we address the expressions to be evaluated in the integral of Eq. (1) over the segments of the imaginary line outside the 2D particle (c. f. the red curve in Fig. 15). To calculate the components of time-averaged local electromagnetic force < Flx > and < Fly > per axial unit length in p-polarization, only the field quantities Ex (x, y,t = 0), Ey (x, y,t = 0), |E(x, y)| and |H(x, y)| are needed. Then the computation of the integrand in Eq. (1) reads:

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d,u 2 d,u < Flx >(E) =0.5Re[εr ε0 (Ex · nd,u x + Ey · ny )Con j[Ex ] − 0.5εr ε0 |E| nx ],

< Flx >(H) =0.5Re[−0.5µr µ0 |H|2 nd,u x ], < Flx > =< Flx >(E) + < Flx >(H). d,u 2 d,u < Fly >(E) =0.5Re[εr ε0 (Ex · nd,u x + Ey · ny )Con j[Ey ] − 0.5εr ε0 |E| ny ],

< Fly >(H) =0.5Re[−0.5µr µ0 |H|2 nd,u y ], < Fly > =< Fly >(E) + < Fly >(H). (2) Notice that since H · n = 0, the third term of Eq. (1) does not contribute in Eq. (2). The superindex l denotes locality and the subindices x, y stand for the Cartesian components. The superindex d and u means that we choose nd or nu depending on whether we are integrating over the upper or lower segments, respectively. The equivalence between physical quantities and FEMLAB parameters are shown in Table 1, Table 1. Physical operators and variables needed to calculate the local forces and their equivalence in FEMLAB language.

Physical significance Re[] Conj[] εr ε0 µr µ0 Ex (x, y,t = 0) Ey (x, y,t = 0) |E(x, y)| |H(x, y)| ndx (x, y), nout x (x, y) ndy (x, y), nout y (x, y) u nx (x, y), nout x (x, y) nuy (x, y), nout y (x, y) < Flx >(E) < Flx >(H) < Flx > = < Flx >(E) + < Flx >(H) < Fly >(E) < Fly >(H) < Fly > = < Fly >(E) + < Fly >(H)

FEMLAB variable notation real() conj() epsilonr wh epsilon0 wh mur wh mu0 wh Ex wh Ey wh normE wh normH wh dnx (for upper segments) dny (for upper segments) unx (for lower segments) uny (for lower segments) Fxav tE polp Fxav tH polp Fxav polp Fyav tE polp Fyav tH polp Fyav polp

where real() and con j() are FEMLAB commands to get the real part and the conjugate of any complex expression. The FEMLAB variable names for < Flx >(E), < Flx >(H), < Flx >, < Fly >(E), < Fly >(H) and < Fly > are imposed by the user. The quantities of Eq. (2) are inserted as user-made functions into the box “Boundary Expressions...” of the label Options/Expressions/ in the Toolbar before FEMLAB develops its field #165256 - $15.00 USD (C) 2012 OSA


calculations. Then, once FEMLAB has run, the integration of the expressions of Eq. (2) is made to calculate Eq. (1) by activating the close line segments surrounding the particle, (as in the red curve of Fig. 15), in the box “Boundary Integration” of the label Postprocessing/ in the Toolbar and inserting into it the name of the corresponding variable, either < Flx > or < Fly >, and then selecting the solution at angle 0. The latter is done for these quantities to be first calculated with the complex fields at a temporal phasor value ω t = 0. Then they are integrated over the corresponding segments. Only after this is done, the X- and Y-components of the total time-averaged electromagnetic force < FTx > or < FTy > per axial unit length are, in the end, calculated. One can even obtain a graphical distribution of the local forces on the integration line by entering the user-given names of < Flx > and < Fly > in the box “Plot Parameters...” of the label Postprocessing/ in the Toolbar, as an “Arrow Plot in Boundaries”. Acknowledgments We acknowledge useful discussions on the subject with J. J. Sáenz, R. Quidant and F. Messeguer. Work supported by the Spanish MEC through FIS2009–13430–C02–C01 and Consolider NanoLight (CSD2007–00046) research grants, FJVV is supported by the last grant.

#165256 - $15.00 USD (C) 2012 OSA
