1. Introduction

Self-assembly methods [1] have been used to construct complex materials including three dimensional photonic crystals [2]. However, a fundamental constraint is that only energetically favorable structures are formed. To overcome this problem, templated self-assembly methods have been introduced [3,4]. The template guides particle arrangement, allowing the formation of diverse — albeit static — crystal structures. Here we consider how optical forces can be used to direct assembly and reconfiguration of particles on a photonic crystal slab, which serves as an optically reconfigurable template. Our calculations predict the formation of stably trapped crystal patterns that depend on the wavelength and polarization of the light source. We envision that the process of light-assisted templated self-assembly may be used for fabrication of complex photonic materials and all-optically reconfigurable photonic devices.

Optical forces have been studied extensively since the discovery of optical tweezers [5,6]. While standard optical tweezers use a focused laser beam to trap and manipulate particles, structured light fields [7,8] allow even greater control over particle deflection, transport, and sorting, with applications from the physical sciences to biology. Recently, researchers have used light fields near microphotonic devices to trap and manipulate particles. This work leverages the strong electromagnetic gradients near devices such as nanoapertures [9], gratings [10], photonic-crystal microcavities [11,12], slot waveguides [13] and plasmonic structures [14–17] to generate large optical forces. However, much of this work has considered single-particle traps.

Here we suggest the possibility of assembling multi-particle patterns. We propose to use the guided resonances of photonic crystal slabs [18] to enhance the optical forces on particles in solution. In contrast to previous work on Fabry-Perot enhancement of trapping forces above a flat substrate [19], the photonic crystal is used to form spatially distinct trapping patterns at different polarizations and wavelengths. While previous work has examined light forces between different layers of particles in colloidal photonic crystals [20,21], forces on colloidal layers above a microfabricated template were not considered.

2. Approach and results

The system we consider is shown schematically in Fig. 1 . In an experiment, light would illuminate the photonic crystal slab from below. The structured light fields above the slab give rise to optical forces on particles in solution, which can potentially result in trapping. Due to the variation of the electromagnetic field over spatial length scales comparable to the particle diameter, the dipole approximation cannot be assumed. However, the optical force can be found by using full-vectorial calculations of Maxwell’s equations to calculate the Maxwell Stress Tensor [21].

 

Fig. 1 Light-assisted templated self-assembly using a photonic crystal slab.

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For concreteness, we consider a silicon photonic crystal (ε = 11.9) with a square lattice of holes of radius 0.2a and thickness 0.6a resting on a silica substrate (ε = 2.1), where a is the lattice constant of the photonic crystal. The holes in the photonic crystal and the region above the crystal are filled with water (ε = 1.77). The incident light is an x-polarized plane wave propagating in the z direction.

We calculate the transmission through the slab numerically using the finite-difference time-domain (FDTD) method [22] in the freely available MIT MEEP package [23]. The computational resolution is 32 grid points per a. Figure 2(a) shows the normalized transmission through the photonic-crystal slab. The transmission exhibits three distinctive resonance peaks at frequencies ν = 0.4721, 0.4958, and 0.5008 (c/a). The characteristic shape of the resonance peaks is well known from previous work in the literature [18]. We will refer to the three resonances shown as R1, R2, and R3 respectively.

 

Fig. 2 (a) Normalized transmission of a silicon slab on a silica substrate for a height of 0.6a and hole radius of 0.2a, (b) Force in the z-direction for a sphere of radius 0.1a placed 0.25a above the slab surface. The sphere is positioned equidistant from the four closest holes.

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We calculate the force on polystyrene spheres (ε = 2.28) of radius 0.1a. The center of the sphere is placed at a height of 0.25a above the slab surface. We calculate the Maxwell stress tensor on the surface of an integration box of size 0.3a centered on the particle [21,24]. The calculation is done in a computational cell that includes one period of the photonic crystal slab. Periodic boundary conditions are used in the lateral directions (x and y), and perfectly matched layer boundaries are used in the z direction. The forces calculated in this manner represent the optical forces on a periodic array of polystyrene spheres above the photonic crystal slab.

We plot the dimensionless quantity F_zc/ϕ, where F_z is the z-component of the optical force, ϕ is the incident power per unit cell and c is the speed of light [see Fig. 2(b)] for a sphere positioned equidistant from the four closest holes. For each of the three resonance frequencies shown in Fig. 2(a), we observe a peak in the optical force. The sign of the force is negative, indicating that the particles are attracted toward the slab. For comparison, we calculate the radiation pressure of a plane wave on a spherical particle in the absence of the photonic crystal. We obtain a repulsive force of F_zc/ϕ ~1x10⁻⁵, three orders of magnitude smaller than the attractive optical force above the photonic crystal. We expect that even larger optical forces can be achieved by using a photonic-crystal slab with higher-Q resonances. Higher Q has been correlated with decreasing hole size [18].

The intensity for these resonances is shown in Figs. 3(a) –3(c) at a height of 0.25a above the surface of the slab. Figures 3(d)–3(f) shows the intensity at a height of 0.1a (equal to the particle radius). The intensity |E|² is normalized to the source intensity |E₀|². We observe that each resonance has a different field profile. There is a slight change in field profile with height, and the intensity is higher closer to the slab.

 

Fig. 3 Normalized intensity for normally incident x-polarized light for R1, R2, and R3, respectively at a height of (a)–(c) 0.25a above the surface of the slab and (d)–(f) 0.1a above the surface of the slab. White circles indicate hole positions.

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We now look at the spatial dependence of the force on resonance. The force at each point is calculated by placing the sphere at that point, calculating the full-vectorial electromagnetic fields, and computing the Maxwell Stress Tensor. For numerical reasons, we have calculated the forces at a height of 0.25a above the slab to insure that the Maxwell Stress Tensor integration surface does not overlap either the particle or the slab boundaries. Figures 4(a) –4(c) show the force on a vertical (x-z) slice at y = 0. The length of the arrows next to the graphs represents a value Fc/ϕ = 0.2. The vertical forces are attractive over the whole unit cell: a particle near the slab will be attracted to it.

 

Fig. 4 (a)–(c) Force in the x-z plane at y = 0, (d)–(f) Force in the x-y plane at z = 0.25a for R1, R2, and R3, respectively. The red dots represent stable trapping points and the black circles indicate hole positions.

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Figures 4(d)–4(f) show the force on a horizontal (x-y) slice 0.25a above the slab surface. The circles indicate hole positions. The arrows next to the graphs represent a value of Fc/ϕ = 0.2, and the red circles represent the stable points to which the particles are attracted. R1, R2 and R3 correspond to different stable patterns. Figure 4(d) has an additional stable point at the center of the unit cell, but it is not labeled with a red dot because it has a weak lateral force compared to other trapping points. Similarly, Fig. 4(e) has two additional weakly stable points at the bottom (or top) edge of the unit cell. We note that the spatial profile of the forces exerted on particles touching the slab surface (height 0.1a) may differ slightly from those shown here, due to the weak variation in intensity with height (Fig. 3).

Different light polarizations correspond to different self-assembled patterns, as shown in Fig. 5 . Due to the symmetry of the photonic crystal, the patterns for y-polarized light are the same as those for x-polarized light, but rotated 90° [Figs. 5(a) and 5(b)].

 

Fig. 5 (a) Pattern for x-polarization, (b) Pattern for y-polarization.

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3. Discussion

The photonic-crystal slabs studied here can be fabricated in a silicon-on-insulator wafer using standard techniques [25]. The lattice constant of the photonic crystal can be scaled to place a given guided resonance at a particular wavelength of interest [26]. For a = 760nm, for example, R1 occurs at λ = c/n = 1609nm, while R2 occurs at 1532nm. For a = 375nm, R1 is at 794nm and R2 is at 748nm. An upper bound on the power required for trapping was obtained by setting the height of the potential depth |ΔU| > 10 K_BT [5]. The power density for a = 760nm was 1 mW/unit cell for R1, which has a modest Q of ~260. The required power decreases with increasing Q. Moreover, the power calculated in this manner is likely to be an overestimate. Our results indicate that the particles will be attracted to the surface of the slab. As seen in Fig. 3, the intensity increases closer to the slab surface, producing a higher force than the values shown in Fig. 4, which were calculated for a height of 0.25a (for a particle touching the slab surface, the height is 0.1a).

In an experiment, particles may be trapped one by one, or a few at a time, in order to build up ordered patterns. We believe that the red dots shown in Figs. 4 and 5 are good predictors of the patterns that will be formed in this manner. In our computations, due to the boundary conditions on the unit cell, we are in fact calculating the optical force on periodic arrays of particles above the photonic crystal slab. However, since interparticle interactions [27] are negligible in this system, the force on an individual particle above the slab will be nearly identical.

To show that interparticle interactions are weak compared to the force on a particle from the photonic crystal, we considered the geometry shown in Fig. 6(a) . A first particle (labeled “1”, blue circle) was placed over the slab at a position of [0.3a, 0.3a, 0.25a]. A second particle was placed in one of three alternate positions (labeled “2”, green circles). Figure 6(b) shows that the presence of the particles does not visibly affect the transmission spectrum. In Figs. 6(c) and 6(d), we plot the lateral forces on particle 1 for different positions of particle 2. It can be seen that particle 2 has minimal effect on the force.

 

Fig. 6 (a) Schematic of the system (b) Transmission spectrum of photonic-crystal slab with and without particles above (c) F_x c/ϕ on particle 1 for 3 different positions of particle 2 (d) F_y c/ϕ on particle 1 for 3 different positions of particle 2. The legend indicates the position of particle 2 in (b), (c), and (d).

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As an additional check, we recalculated the force maps of Figs. 4(d)–4(f) using a larger computational cell of size 2a x 2a, with one particle per computational cell. No changes were observed in the force maps.

Lastly, we checked that when particles were placed at each of the red dots in Fig. 4(d)–4(f), the configuration was stable to perturbation. One particle per unit cell was displaced, and it was verified that the restoring force was in the appropriate direction to restore the initial pattern.

4. Conclusions

We predict that optical forces above photonic crystal slabs will lead to a variety of complex, stably trapped crystal patterns. Changing the wavelength or polarization of the incident light may be used to reconfigure the patterns.

In the system studied here, the particles do not significantly affect the transmission spectrum [Fig. 6(b)]. We have checked that even if the particles are placed inside the holes in the slab or on the slab surface, the shift in the resonance is a small fraction of the resonance width (~15% or less). For larger particles, particles of higher index, and/or higher Q slab resonances, particle trapping may begin to shift the resonance position. This opens up the possibility for intriguing phenomena such as self-induced or bistable trapping, effects which have previously been studied for particle trapping near photonic cavities [28]. This is an interesting direction for further research.

We expect the approach of light-assisted self-assembly, described here, to be of broad utility for fabrication of complex photonic materials, sensors, and filters. It is intriguing to consider whether metamaterials based on metal nanoclusters [29], for example, could be assembled in the manner we describe, using the optical response of the nanoclusters to tune or adjust assembly.