Controllable Trapping of Nanowires Using a Symmetric Slot Waveguide

Aug 8, 2013 - A controllable trapping method based on using a symmetric slot waveguide is proposed. The structure is composed of a subwavelength slot ...
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Controllable Trapping of Nanowires Using a Symmetric Slot Waveguide Nafiseh Zavareian and Reza Massudi* Laser and Plasma Research Institute, Shahid Beheshti University, Evin, GC, Tehran, Iran 19839 ABSTRACT: A controllable trapping method based on using a symmetric slot waveguide is proposed. The structure is composed of a subwavelength slot formed between two adjacent thin metallic films embedded in an infinite homogeneous dielectric medium. Generated near-field components interact with a nanowire and exert net force on it. Green’s function surface integral equation method is exploited for numerical calculation of the electric and the magnetic fields and, consequently, the radiation force acting on the nanowire. Casimir force is also obtained by calculating Maxwell stress tensor and using fluctuation−dissipation theorem. Results illustrate that depending on the width and the thickness of the slot, the radiation force and, consequently, the position of the stable equilibrium point change. By controlling the phase difference of the incident SPP waves it is possible to trap or release the nanowire at a specified position. In addition, results reveal that Casimir force moves nanowires toward the center of the slot and is maximum at the entrance of the slot with magnitude depending on the width and thickness of the slot. consists of two identical semi-infinite metallic films located close to each other in a homogeneous dielectric medium to form a nanosized slot between them. SPP waves propagate along the upper and lower sides of the semi-infinite metallic films and interact with their edges. As a result, a fraction of the energy of each SPP wave is reflected, some part of it is converted to scattering wave, and the rest propagates along the walls of the slot in the form of SPP wave. The latter part may excite surface plasmon modes of the slot. Hence, in this configuration the scattered and the excited plasmonic waves inside the slot, and not the incident SPP waves, are responsible for trapping of the sample. Trapping of the sample occurs by radiation forces originated from scattered EM waves and also by excited plasmonic waves. To calculate such force, we calculate electric and magnetic fields by exploiting Green’s function surface integral equation technique. EM fields at each arbitrary point inside a closed boundary are related to the fields and their derivatives on the boundaries via second Green’s theorem.19 The radiation force is calculated by integrating of Maxwell stress tensor over the surface of the sample that is a nanowire in this study. Furthermore, we also take into account Casimir force exerted on the nanowire. This force exists among charge neutral bodies and arises due to quantum fluctuations of the electromagnetic field. Whereas it is negligibly small for macroscopic bodies separated by distances of more than a few micrometers, it can play a significant role in the interaction of bodies separated on micrometer and submicrometer length scales.20 Such force can be calculated by using classical Dyadic Green’s function via the

1. INTRODUCTION Optical trapping of single particles, as a noninvasive manipulation method, has been widely exploited to create 3D structures of microparticles, to aggregate nanoparticles for SERS spectroscopy, to sort and manipulate living cells, and to study single biomolecules, for example, for determining mechanical and dynamical properties of DNA.1−6 Ashkin demonstrated far-field optical trapping of microparticles by radiation forces of two counter-propagating laser beams in 1970 and, next, by a single focused laser beam in 1986.7,8 Despite advantages of far-field optical trapping, this technique has important limitations such as requirement to bulk optics, for example, high numerical-aperture objective lens and focusing of light to a diffraction-limited spot. Recent advances in nano-optics have led to alternative techniques in optical trapping based on using evanescent, instead of propagating, waves that extend trapping region down to submicrometer scale.9−12 Among different proposed approaches, plasmonic traps that exploit surface plasmon waves supported by metallic nanostructures are more efficient for trapping of nano-objects.13−17 Those waves are localized ppolarized electromagnetic fields produced by coupling of electromagnetic waves with free electron charge density oscillations at the surface of metallic objects.18 The metallic objects can be patterned nanostructures, nanoparticles, nanorods, or sharp tips that are in contact with a dielectric medium. Further to field enhancement, those waves provide spatial confinement down to nanometer scale and open up new opportunities for optical trapping of nano-objects. In most proposed configurations based on surface plasmon waves, a plane wave impinges on a metallic structure and excites surface plasmon waves. We propose another configuration by using surface plasmon polariton (SPP) waves with the ability to control the trapping process. The structure under consideration © XXXX American Chemical Society

Received: January 16, 2013 Revised: June 8, 2013

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electromagnetic stress tensor and the fluctuation−dissipation theorem.21 Results show that the position of the stable equilibrium point depends on the permittivity of the nanowire and also on the width and thickness of the slot. Casimir force is attractive, pulling the nanowire toward the center of the slot, and is maximum at the entrance of the slot. Investigation of the effect of phase difference between incidents SPP waves reveals that by controlling phase difference the nanowire can be trapped or released. Hence, this structure can provide an adjustable trap for nano-objects.

where k0 is free-space wavenumber, (∇∇)ij = ∂i∂j, and g(r′, r) is scalar Green’s function.19 By using eqs 1 and 2, boundary integral equation over the boundaries of region Mi (i = 1 to 4) is written as:22

∮ dl′{[∇′ × G⃡ i(r, r′)]·[n′i × Ei(r′)]

f Ei(r) =

+ iωμ0 G⃡ i(r, r′) ·[n′i × Hi(r′)]}

(4)

In eq 4, when the observation point is located inside (outside) region Mi, the coefficient f on the left-hand side of eq 4 becomes equal to one (zero). If the observation point is located on the boundaries of the region Mi, then the Green’s function and its derivative become singular at r = r′. Considering the contribution of the singularity, the term on the left-hand side of eq 4 becomes as (1/2)Ei(r), and the righthand side is written in the form of Cauchy principal values integral. To calculate electric and magnetic fields by using eq 4, it is necessary to integrate along infinite boundaries, that is, boundaries of a semi infinite film. To do so, we use a technique similar to that introduced in ref 23 to minimize truncation error. Equation 4 gives the electric and the magnetic fields over the boundary of the nanowire that is used to calculate the time-averaged optical force acting on it. Such radiation force is calculated by integrating Maxwell stress tensor over the whole surface of the nanowire:24

2. REVIEW OF THEORY Let us consider two adjacent identical semi-infinite metallic films (M2 and M3 in Figure 1) with thickness d located in a

Figure 1. Schematic of the structure under consideration. Mi denotes the ith medium (i = 1 to 4). Origin of the Cartesian coordinate is considered at the center of the slot.

∮ dl′[ε1(E·n)E − ε1/2|E|2 n + μ0(H·n)H − μ0 /2|H

F=⟨

|2 n]⟩

homogeneous dielectric medium (M1 in Figure 1) to form a symmetric slot waveguide with width w. A nanowire, M4, is placed in the vicinity of the slot with its long axis oriented parallel to the z axis (Figure 1). Dielectric constants of the metallic film, the dielectric environment, and the nanowire are ε1, ε2, ε3, and ε4, respectively (ε2 = ε3). Because of symmetry, it is possible to simplify the structure to two dimensions. In the following subsections we introduce methods of calculation of optical force and Casimir force. A. Optical Force. Let SPP waves propagate along the upper and the lower surfaces of the semi-infinite films and scatter at their edges. Under appropriate conditions, radiation forces of the scattered and the excited waves exerted on the nanowire may trap it in the proximity of the slot. To calculate the electric and the magnetic fields at the boundary of the nanowire, Green’s function surface integral equation method (GFSIEM) can be used. To do so, we start with the wave equation for the electric field, that is, as: ∇ × ∇ × Ei(r, ω) − k 02εi(ω)Ei(r, ω) = 0

where the bracket represents the time average and n is the outwardly directed unit normal vector. We numerically solve eq 4 for incident SPP waves, each with power of P = (1/2)∫ ∞ −∞| Re⟨E × H*⟩|dy per unit length, by using boundary element method and by applying appropriate boundary conditions and considering n′i = −n′i (i = 2, 3, 4). Then, the boundaries of the structure under study are discretized and the integral in eq 4 is transformed as the summation of integrals over those elements. That results in 2 N equations for N nodes located on the boundaries. Solving the resulting system of equations gives the (tangential) electric field and the magnetic field at nodal points. Next, we write eq 4 to obtain vertical component of the electric field on the boundary of the nanowire. Those fields are used to obtain the radiation force by numerically calculating eq 5 over the surface of the nanowire. B. Casimir Force. Casimir force was predicted for the first time by Casimir in 1948.25 Different theoretical and experimental works were established to study this force.26−33 It arises from variation of the zero-point energy of the electromagnetic field in the presence of boundary surfaces. Such variation originates from the difference between the zeropoint energy in the presence and in the absence of external constraints. In other words, Casimir force is the response of the vacuum against the presence of external bodies in the form of geometry dependent fluctuations of the electromagnetic fields E and H. Casimir force is usually calculated either by summation of the energy of discrete quantum modes of the EM field of the system or by using dyadic Green’s function.26,27 In the latter formalism, which is first introduced by Lifshitz and is based on evaluating the vacuum radiation pressure on the sample, the Casimir force is obtained by calculating the vacuum expectation value of the Maxwell stress tensor in the form of:21

(1)

The corresponding dyadic Green’s function is obtained by solving the following equation: ∇ × ∇ × G⃡ i(r, r′, ω) − k 02εi(ω)G⃡ i(r, r′, ω) = I ⃡ δ(r − r′)

(2)

where r = (x, y) and r′ = (x′, y′) are, respectively, position vectors of the observation and the source points and εi(ω) (i = 1, 2, 3, 4) is the dielectric constant of different media. Dyadic Green’s function in eq 2 is given by:22 ⎛ ∇∇ ⎞ ⎟g (r′, r) G⃡ i(r′, r) = ⎜ 1⃡ + εik 02 ⎠ ⎝

(5)

(3) B

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⎡ 1 ⟨T(r, ω)⟩ = μ(r, ω)⎢⟨Hi(r)Hj(r)⟩ − δij ⎢⎣ 2

function current placed at a different point along a surface enclosing the nanowire is calculated. Introducing electric and magnetic correlation functions in eq 6 gives rise to an oscillatory Maxwell stress tensor that should be used as integrand in eq 7. Because the contribution of different frequency components is significant, numerical calculation of eq 7 becomes complicated.20 The integrand of eq 7 has no poles in the upper complex plane because of physical causality, and it is appropriate to re-express eq 7 as a complex contour integral. The standard contour is a Wick rotation that corresponds to the particular choice of ω = iξ. That gives a smooth and exponentially decaying integrand with respect to ξ. Then, eq 9 transforms to the following equation:34

⎤

∑ ⟨Hk(r)Hk(r)⟩⎥ + ε(r, ω) ⎥⎦

k

⎤ ⎡ ⎢⟨Ei(r)Ej(r)⟩ − 1 δij ∑ ⟨Ek (r)Ek (r)⟩⎥ ⎥⎦ ⎢⎣ 2 k

(6)

and then integrating it over the surface of body and also over all frequency components:34 F=

∫0

∞

dω ∮∮ ⟨T(r, ω)⟩·dS

⎡ ⎤ 1 ∇ × +ξ 2ε(r, iξ)⎥GjE(r, r′, iξ) ⎢∇ × ⎣ ⎦ μ(r, iξ)

(7)

There is a simple expression for correlation functions in eq 6 in terms of fluctuation−dissipation theorem of statistical physics.21 That theorem relates the correlation of the fields in eq 6 to the imaginary part of the dyadic Green’s function of the system. The correlation function of vector potential at zero temperature gives the following relation:21 ℏ ⟨AiE (r, ω)AjE (r′, ω)⟩ = − Im GijE(r, r′, ω) π

= δ(r − r′)ej

Dielectric constant ε(r, iξ) appearing in eq 12 is real and positive, and eq 12 is a counterpart equation for dyadic Green’s function with real frequency with dielectric constant transformed to εn = −ε(r, iξ).34 Then, to calculate GEj (r, r′, iξ) is similar to that of GEj (r, r′, ω) with dielectric constant εn instead of ε(r, iξ). A similar procedure is used to obtain GHj (r, r′, iξ).

(8)

where dyadic Green’s function can be calculated by the following relation:21

3. NUMERICAL RESULTS We numerically solve eq 4 by using dyadic Green’s function (eq 3) and BEM method to calculate electric and magnetic fields at the boundaries of the structure. Consider two semi-infinite metallic films of gold with thickness d = 500 nm forming a slot with width w = 200 nm in between to be in contact with a dielectric medium with refractive index n = 1.334. Four SPP waves with vacuum wavelength λ = 632.8 nm propagate along the upper and the lower surfaces of these films. Because of interaction with the end faces of the films, each SPP wave is partially reflected. Another part is also converted to radiation wave mostly in the forward direction (Figure 2). That is

⎡ ⎤ 1 ∇ × −ω 2ε(r, ω)⎥GjE(r, r′, ω) ⎢∇ × ⎣ ⎦ μ(r, ω) = δ(r − r′)ej

(9)

By using the definition of the electric and the magnetic fields versus vector potential (E = (−∂AE)/(∂t), B = ∇ × AE), correlation function of those fields introduced in eq 6 can be obtained from the following:21 ⟨Ei(r, ω)Ej(r′, ω)⟩ =

(12)

ℏ 2 ω Im GijE(r, r′, ω) π

⟨Hi(r, ω)Hj(r′, ω)⟩ E = −ℏ(∇ × )i S (∇′ × )Sm Im Glm (r, r′, ω)

(10)

To simplify calculation of the magnetic field correlation function in eq 10, we use a different gauge as H = (∂AH)/ (∂t), E = ∇ × AH, where the magnetic field presents the response to a magnetic-dipole current. Magnetic field correlation function can be computed from the magnetic dyadic Green’s function as:34 ℏ ⟨Hi(r, ω)Hj(r′, ω)⟩ = ω 2 Im GijH (r, r′, ω) π

Figure 2. Electric-field distribution, |E|2, resulting from propagation of four incident SPP waves, in accordance with Figure 1, with wavelength λ = 632.8 nm.

(11)

Equation 11 has the advantage that the force is given directly in terms of the dyadic Green’s function instead of its derivative. Then, Maxwell stress tensor in eq 6 can be re-expressed versus dyadic Green’s function. To calculate Casimir force by using eq 7, it is necessary to find components of dyadic Green’s function GHij (r, r, ω) on the surface of integration for each frequency component. Different numerical techniques, for example, FDTD, FDFD, BEM, and so on, have been used for calculation of those components.35 We use GFSIEM method introduced in the previous subsection to calculate components of dyadic Green’s function. To do so, the generated electric and magnetic fields in response to a delta

because of breaking translational symmetry at the end faces of semi-infinite films and consequently forming sources of freeelectron charge density. The rest of the energy of incident SPP waves enters the slot’s wall, and, if the width of the slot is appropriately chosen, then its plasmonic modes are excited (Figure 2). Assume a nanowire with radius a = 10 nm in the vicinity of the slot at position (Lx, Ly). Electric and magnetic fields over the surface of the nanowire can be calculated by using eqs 4 and 5. Following that, the exerted radiation force on the nanowire is acquired by using eq 6. We assume total power of the incident C

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Figure 3. (a) Variation of the radiation force, Fy, versus the position of nanowire along the y axis. (b) Variation of Fx versus Lx for a nanowire located at the first stable equilibrium point along the y axis, that is, y = 72 nm for gold and y = 177 nm for polystyrene nanowires, respectively.

Figure 4. Variation of the radiation force Fy versus position of gold nanowire along the y axis for different (a) gap size w for d = 500 nm and (b) thickness d of the slot for w = 200 nm. Insets show variation of size of optical trap (SOT) with w (part a) and d (part b) for the first and the second stable optical traps.

SPP waves equal to 40 mW, that is,10 mW for each of them. We have found that there is a linear relationship between optical force and power of incident SPP waves. That is due to proportionality of the Maxwell stress tensor with the square of magnitude of the electric and magnetic fields (eq 5). Then, we normalize optical force to input optical power. We consider metallic, that is, gold and silver, and dielectric nanowires, that is, SiO2 and polystyrene, with dielectric constants given in refs 36−38. Figure 3a shows radiation force along the y axis, Fy, exerted on a nanowire located at different positions along the y axis. The Figure shows that Fy has oscillatory behavior versus Ly and is zero where we call equilibrium points, for example, Ly= ±72 nm for gold and Ly= ±177 nm for polystyrene. If the nanowire is displaced in the vicinity of an equilibrium point along the y axis, for example, because of Brownian force, provided that the radiation force is sufficiently large and has appropriate direction, then the nanowire returns to equilibrium point and the trap is stable. We name this point and its counterpart along x axis, respectively, as vertical and lateral equilibrium points.39 To investigate trapping along the x axis, we calculate radiation force, Fx around the first stable vertical equilibrium point. Results illustrate that there is a stable equilibrium point in both the x and y directions inside the slot, while outside the slot, such point exists only along the y direction. Dielectric nanowires, that is, SiO2 and polystyrene in this study, have different behaviors compared with metallic nanowire. For example, whereas location (0, ±72 nm) is a vertical−lateral stable equilibrium point for gold nanowire, position (0, ±177) is only a vertical stable equilibrium point for polystyrene nanowire (Figure 3b). In the following, we study the effect of different parameters on the trapping force, equilibrium point position, and its stability. Because metallic nanowire has a stable equilibrium

point inside the slot we only consider such nanowire, for example, gold nanowire, in the following. Effects of the gap size of the slot on radiation force are shown in Figure 4a. As expected, larger gap size increases the radiation force due to more intense interaction of scattered SPP waves with the slot. Furthermore, we show that for small gap size there is no stable point along the x and y directions inside and in the vicinity of slot. For example, for w = 50 nm, the first three stable points along the y axis are only vertical stable points, whereas the fourth one is a stable vertical−lateral point. For w = 200 nm, there is a vertical−lateral stable point inside the gap, whereas for outside the gap there is no stable point. In the trapping process, in addition to stability, the size of the optical trap (SOT), which we define as the distance between two adjacent unstable points where there exists a stable point, is also important. As the inset of Figure 4a illustrates, by increasing the gap size, the size of the optical trap increases. Effects of the thickness of the slot, d, are also shown in Figure 4b. Varying the thickness d changes radiation force as well as position of stable point. That is due to the variation of interaction of scattered SPP waves with the edges of the slot. Investigating the stability of the stable points along the x axis shows that for smaller d, the number of stable points along the x and y directions is larger; for example, for d = 250 nm, the first and the second points are stable, whereas for d = 1000 nm there is not any stable point inside and at the vicinity of the slot in both directions. Investigation of dependency of the size of the optical trap on the thickness d (inset of Figure 4b) shows that by increasing d such size reduces. Thus, there is a competition between stability and size of optical trap versus d. Although the calculated size of the optical trap acquired by this structure, shown in the insets of Figure 4, is not as small as those reported elsewhere, for example, ref 40, our results show D

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Figure 5. Variation of the Casimir force per maximum optical force (Fy,O‑max) versus position of polystyrene and gold nanowires for different (a,b) gap size w and (c,d) thickness d. Variation of the Casimir force per optical force (Fx,O) versus position of gold and polystyrene nanowires along the x axis for w = 200 nm shown in the insets of parts a and b, respectively.

Figure 6. Variation of the radiation force versus position of gold nanowire for different values of power of different incident SPP waves with equal total incident power along (a) y axis and (b) x axis.

the x axis, then it is pushed toward the walls of the slot, which confirms the attractive nature of Casimir force in this structure. For occurrence of optical trapping of the nanowire, optical force should be much larger than Casimir force, and it requires a threshold incident power. Such requirement is fulfilled for the structure under study and for the total of power for incident SPP waves. Hence, one can neglect the effect of Casimir force in this study. In the following, we study the effects of different parameters of the incident SPP waves on the trapping process. In Figure 6a, effects of variation of power of different SPP waves, while total power is kept fixed (Σ4i=1Pi = constant), on trapping force along the y axis are shown. It shows that such variation changes the magnitude of radiation force and also the position of the stability point. For example, for the P1 = 40 mW case, the first stable point along the y axis is at Ly = 41.4 nm, whereas for P1,2,3,4 = 10 mW, it is at Ly = 72 nm. Figure 6b shows the radiation force along the x axis for the first stable points along the y axis in Figure 6a, which are labeled A−D. As expected, the results illustrate that for symmetrical values of power of different SPP waves, for example, for P1 = P2 = 20 mW (P1,2 = 20mW) and for P1 = P2 = P3 = P4 = 10 mW (P1,2,3,4 = 10 mW), stable trapping points exist for both directions, whereas for asymmetric values, for example, P1 = 40 mW and P1 = P3 = 20 mW (P1,3 = 20 mW) in Figure 6a, it does not exist. Hence, to

that such structure has controlling features that are suitable for trapping process. So far we only considered the force related to optical radiation. Casimir force between different elements (i.e., between two truncated metallic films and nanowire) exist that may disrupt the optical trapping process. To calculate Casimir force only between nanostructure and nanowire, we deduct the part of that force between different elements and only keep that between nanostructure and nanowire.41 Variation of the ratio of Casimir force to maximum optical force along the y axis, that is, Fy/Fy,O‑max is shown in Figure 5, where Fy,O‑max is the maximum optical force along the y axis for a similar structure. As the Figure illustrates, Casimir force pushes the nanowire toward the center of the slot along the y axis and is maximum at the upper and the lower entrances of the slot. That is because of stronger interaction of vacuum fluctuation with the structure. Such behavior has also been observed in similar structures.42,43 In addition, it shows that increasing gap size w and decreasing thickness d reduce peaks values of Casimir force, which is desirable in the optical trapping process. In the insets of Figure 5a,b, variation of Casimir force per optical force, that is, Fx/Fx,O, for a structure with d = 500 nm and w = 200 nm versus Lx (except for Lx = 0, where both of the forces Fx and Fx,O are equal to zero) is shown. Results illustrate that if the nanowire is displaced along E

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Figure 7. Electric-field distribution, |E|2, resulting from interference of four incident SPP waves with phase differences (a) φ1 = φ2 = 0°, φ3 = φ4 = 180° and (b) φ1 = φ4 = 0°, φ2 = φ3 = 180°.

Figure 8. Variation of radiation force exerted on a gold nanowire for different values of the phases of the incident SPP waves (a) along the y axis and (b) along the x axis (for the first stable point in part a). (c,d) Variation of the position of the stable point versus phase of the incident SPP waves for two different phases of incident SPP waves shown in the insets. Variation of the size of the optical trap (SOT) with the phase is shown in the insets.

also illustrates sensitive dependency of the stability location to the phase of the incident SPP waves. To better understand the effect of the phase on trapping process, variation of the position of stable equilibrium point along the y axis by the phase of the incident SPP waves for two different illumination configurations is shown in Figure 8c,d. In the insets of those Figures, the size of optical trap is also shown. As observed, the position of the stable point sensitively depends on the phase of the incident SPP waves, and it is possible to move nanowire by variation of phase. Variation of the size of the optical trap offers that such size can be adjusted by the phase of the incident waves. Hence, changing the phase difference can be used for switching from trapping to releasing modes and vice versa and also variation of optical trap size.

acquire a vertical−lateral stable point along the axis of the slot waveguide, one can use symmetrical illumination. (See the inset of Figure 6a.) Up to now, we have considered trapping of nanowires illuminated by in-phase incident SPP waves. Assume the SPP waves are out-of-phase relative to each other. Such phase difference can be introduced between incident lights used to excite SPP waves. Figure 7a,b, respectively, illustrate the electric-field pattern corresponding to φ1 = φ2 = 0°, φ3 = φ4 = 180°, and φ1 = φ3 = 0°, φ2 = φ4 = 180°. As it is observed, electric-field pattern significantly changes by varying the phase difference between SPP waves. That is due to the interference of the scattered waves and to the interference of the generated SPP waves inside the slot. The resulting change in the pattern of the electric field in Figure 7a,b confirms that the phase difference between SPP waves affects the optical force and position of the trapping point. Figure 8a,b shows variation of trapping force Fy versus Ly for different values of the phase of incident SPP waves. This Figure reveals that by changing the phase of the incident SPP waves and, consequently, changing electric and magnetic fields at the position of the nanowire, trapping point moves toward other positions. In Figure 8b, stability of the first stable points of those illustrated in Figure 8a along the x axis is shown. That

4. CONCLUSIONS We studied a plasmonic nanostructure that can be used as a device to trap nanowires. The structure consisted of a slot formed by two adjacent semi-infinite parallel metallic films. SPP waves propagate along the upper and the lower surfaces of the films and are scattered at their edges. GFSIEM method is used to calculate electric and magnetic fields at the surface of the nanowire. Radiation force exerted on the nanowire was calculated by integrating of Maxwell stress tensor over the F

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The Journal of Physical Chemistry C

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surface of the nanowire. Casimir force was also calculated by integrating of Maxwell stress tensor over the surface of the nanowire and over all frequencies, where fluctuation− dissipation theorem was used to calculate correlation functions of the electric and the magnetic fields appearing on that tensor. Variation of radiation and Casimir forces, position of the stable optical equilibrium point, and size of the optical trap were investigated versus different parameters. Results illustrated that by changing the slot’s width and thickness, the magnitude of optical and Casimir forces changes. The position of the equilibrium point and size of the optical trap can be adjusted by those parameters. Furthermore, the position of the trapping point changes by phase difference of the incident SPP waves, which leads to an adjustable trap for nanowires and also the possibility of switching from trapping to releasing modes.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

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The Journal of Physical Chemistry C

Article

(42) Pernice, W. H.P.; Li, M.; Garcia-Sanchez, D.; Tang, H. X. Analysis of Short Range Forces in Optomechanical Devices with a Nanogap. Opt. Express 2010, 18, 12615−12621. (43) McCauley, A. P.; Rodriguez, A. W.; Joannopoulos, J. D.; Johnson, S. G. Casimir Forces in the Time Domain: Applications. Phys. Rev. A 2010, 81, 012119-1−012119-10.

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dx.doi.org/10.1021/jp400495s | J. Phys. Chem. C XXXX, XXX, XXX−XXX