On the substrate contribution to the back action trapping of plasmonic nanoparticles on resonant near-field traps in plasmonic films

Punnag Padhy; Mohammad Asif Zaman; Paul Hansen; Lambertus Hesselink

doi:10.1364/OE.25.026198

1. Introduction

Near-field optical traps have enabled the trapping and manipulation of sub-diffraction sized particles and biological samples in the steep evanescent field of various resonant structures [1–10]. Trapping of particle sizes that are 10 nm in diameter and below have been reported [11, 12]. Taking a step further, Hansen et.al and Zheng et. al [13, 14] used an array of C-shaped plasmonic apertures and engravings as traps to devise a Nano-Optical Conveyor Belt (NOCB) for optical transport of dielectric nanoparticles across a chip. Polystyrene beads with a minimum diameter of 200 nm have been successfully transported on the NOCB. Extending this mechanism to manipulate plasmonic nanoparticles is exciting, given their multitude of applications in biomedical diagnostics, sensing, imaging and Surface Enhanced Raman Scattering (SERS) [15–17]. Unlike their dielectric counterparts, plasmonic nanoparticles (such as nanospheres and nanorods) sustain localized plasmon modes at wavelengths much larger than their physical dimension. The resonance position of these modes depend on their size, shape, dielectric environment and substrate interaction [18–21]. Although, it is very well understood that the near-field trap force exerted on subwavelength particles is directly proportional to their physical volume, the contribution of the interaction with the plasmonic substrate to the net force has not been investigated in any prior study.

In resonant near-field traps the trapped nanoparticle actively influences the force acting on itself through the Self-Induced Back Action (SIBA) effect [22]. SIBA has been extensively studied and reported in the last decade for dielectric nanoparticles as well as plasmonic nanoparticles with significantly blue shifted resonance w.r.t the trap mode [23–27]. Another prior study went further and investigated the effect of tuning between the resonant modes of a plasmonic nanorod and a spherical cavity in a plasmonic film. It shows that the nanorod and the cavity mode interact to form a pair of coupled modes of the combined system, which asymptotically approach the bare cavity and nanorod modes in the limit of large detuning [28]. But, the contribution of the substrate mode to the back action effect has not been reported to date.

In this paper, we numerically study the contribution of the plasmonic substrate to the total force acting on the plasmonic nanoparticle as well as the back action effect. As an example, we use a bowtie engraved on the plasmonic film as the trap and a nanorod as the trapped nanoparticle. We establish the role of the substrate mode by analyzing the resonant modes of the coupled particle-trap system. Specifically, we look at the eigenmodes of the coupled system in the asymptotic limit of large detuning and small coupling to obtain the uncoupled modes contributing to the coupling [28, 29], and identify the role of the substrate interaction in those modes. Starting with a nanoparticle that has its localized plasmon resonance mode significantly blue detuned w.r.t the trap mode, we see that the substrate force contribution to the high frequency mode decreases and correspondingly increases for the low frequency mode, as the resonant mode of the nanoparticle is tuned to the trap mode. Under normal incidence of optical excitation the attractive substrate force on the nanoparticle is solely perpendicular to the substrate [30, 31], while the trap exerts a force with components both perpendicular and along the substrate. We highlight the contribution of the substrate interaction to the difference in the spectrum of forces acting on the plasmonic nanoparticle along these orthogonal directions. The role of the substrate interaction in the back action contribution to trapping also increases as the nanoparticle resonance is tuned closer to the trap. As the strength of the substrate interaction depends solely on the height of the nanoparticle above the substrate, the back action mechanism also differs for nanoparticle displacement perpendicular to the substrate and along it. We point out these differences in the back action effect.

A large variety of nanoparticles have to be trapped and manipulated on a chip using the same set of resonant near-field apertures/engravings to realize complex lab-on-a-chip functionalities [32]. The resonances of these particles will have varied detuning w.r.t the near-field trap resonance. So, the contribution of the substrate interaction to the net force acting on these nanoparticles will vary. Estimations of the net force acting on the nanoparticle and the back action contribution made solely on the basis of the near-field trap mode may deviate significantly from the actual values. This calls for a detailed understanding of the detuning dependent contribution of the substrate interaction to the net force acting on the nanoparticle as well as the back action effect and makes our study significant. As a large combination of particle-trap combinations are possible, we embarked on a numerical study involving plasmonic nanoparticles with varying plasma frequencies trapped in the near-field of an engraving carved on a plasmonic film to understand the nature of the interaction. This general analysis will hold good for any particle-trap systems.

2. Modal theory of interaction between nanoparticle and trap

The eigenmodes of two coupled resonators derived from the modes of the participating individual resonators with resonance frequencies ω₀ and ω₀ + δ and line-widths κ and γ respectively are [29, 33]:

ω_{\pm}^{c} (\vec{r}) = ω_{0} + \frac{δ}{2} - i \frac{(κ + γ)}{2} \pm \sqrt{g^{2} (\vec{r}) + \frac{{[δ - i (γ - κ)]}^{2}}{4}}

where

g (\vec{r})

is the coupling constant that is proportional to the spatial overlap of the mode fields of the two resonators. It is a function of

\vec{r}

which is the relative position of one resonator w.r.t the other. We then define,

ω_{\pm} (\vec{r}) = R e (ω_{\pm}^{c} (\vec{r}))

and

η_{\pm} (\vec{r}) = I m (ω_{\pm}^{c} (\vec{r}))

. Assuming δ to be positive, we see that

ω_{-} (\vec{r})

is red shifted w.r.t ω₀ and

ω_{+} (\vec{r})

is blue shifted w.r.t ω₀ + δ. In the case of a resonant plasmonic nanoparticle (resonance at ω₀ + δ) trapped on a resonant near-field trap (resonance at ω₀), the low frequency and high frequency resonant modes of the coupled system are attractive (bonding) and repulsive (antibonding) in nature respectively [34, 35]. So, generally the low frequency mode is used for stable trapping. Any change in position

(\vec{r})

of the trapped particle results in a change in

ω_{-} (\vec{r})

. It is accompanied by a change in the amount of input power at the laser excitation frequency (ω) that is coupled into the near-field trap. In the limit

| δ | ≫ | g (\vec{r}) |

the coupled modes asymptotically approach the individual participating modes. So,

ω_{-} (\vec{r}) \to ω_{0}

and

η_{-} (\vec{r}) \to κ

. In this regime, the near-field trap resonance undergoes a small shift

(δ ω_{0} (\vec{r}) = ω_{-} (\vec{r}) - ω_{0})

in ω₀ which is much smaller than the bare cavity resonance frequency

(| δ ω_{0} (\vec{r}) | ≪ ω_{0})

due to its coupling with the plasmonic nanoparticle. This small

δ ω_{0} (\vec{r})

is directly proportional to the negative of the nanoparticle polarizability given by α(ω) and the normalized spatial intensity profile of the bare cavity given by

f (\vec{r})

[26]. The normalized intensity profile has a maximum value of unity. So,

δ ω_{0} (\vec{r}) \propto - α (ω) f (\vec{r})

. Owing to the dispersive nature of the coupling [27]

(| δ ω_{0} (\vec{r}) | ≫ | η_{-} (\vec{r}) - κ |

[36]) the

\vec{r}

dependence of

η_{-} (\vec{r})

is neglected and it is assumed to be equal to κ at all

\vec{r}

. In this limit, the

\vec{r}

dependence of the input power coupled into the near-field trap is captured in the spectrum of the trap intensity profile as given by [27]:

\begin{matrix} I (\vec{r}, ω) = I_{0} \frac{η_{-}^{2} (\vec{r})}{{(ω - ω_{-} (\vec{r}))}^{2} + η_{-}^{2} (\vec{r})} \\ \approx I_{0} \frac{κ^{2}}{{(Δ - δ ω_{0} (\vec{r}))}^{2} + κ^{2}} \\ = I_{0} \frac{κ^{2}}{Δ^{2} {(1 - \frac{δ ω_{0} (\vec{r})}{Δ})}^{2} + κ^{2}} \\ \approx I_{0} \frac{κ^{2}}{Δ^{2} + κ^{2} - 2 δ ω_{0} (\vec{r}) Δ} \\ = I_{0} \frac{κ^{2}}{Δ^{2} + κ^{2}} {(1 - \frac{2 δ ω_{0} (\vec{r}) Δ}{Δ^{2} + κ^{2}})}^{- 1} \\ \approx I_{0} \frac{κ^{2}}{Δ^{2} + κ^{2}} + I_{0} \frac{κ^{2}}{Δ^{2} + κ^{2}} \frac{2 δ ω_{0} (\vec{r}) Δ}{Δ^{2} + κ^{2}} \end{matrix}

where I₀ is the spatial intensity profile of the bare cavity at ω = ω₀ and Δ = ω − ω₀. The approximation is obtained by a Taylor series expansion of

I (\vec{r}, ω)

upto the lowest order involving

δ ω_{0} (\vec{r}))

. It is valid only when

δ ω_{0} (\vec{r}) ≪ Δ

. The net gradient force

({\vec{F}}^{N})

acting on the plasmonic nanoparticle is then given by [24, 27]:

{\vec{F}}^{N} = \frac{R e (α (ω))}{4} \vec{\nabla} I (\vec{r}, ω)

The gradient of the second term in the approximation of

I (\vec{r}, ω)

is the SIBA contribution to the trap force acting on the plasmonic nanoparticle.

When the trap is an aperture/engraving in a plasmonic film, then, in addition to interacting with the trap, the plasmonic nanoparticle is expected to undergo image like interaction with the substrate that should be further enhanced by the surface plasmon mode at the interface of the plasmonic film and the suspension medium of the nanoparticle [21]. This should be accompanied by an attractive force drawing the nanoparticle towards the substrate [37]. The strength of the interaction will depend on the height of the nanoparticle above the substrate and lead to changes in its polarizability. So, α(ω) → α_s (ω, z) (assuming the surface of the plasmonic film is at the z = 0 plane). The s in the subscript indicates that it is the effective parameter after incorporating the substrate interaction. As absorption and scattering spectrum of the nanoparticle are functions of α_s (ω, z) [19, 38, 39], the change in α_s (ω, z) will be accompanied by changes in its resonance frequency and line-width [21]. So, δ → δ_s (z) and γ → γ_s (z) respectively. As α_s (ω, z) depends only on z, a difference in back action mechanism for nanoparticle displacements perpendicular to the substrate and along it is expected. For displacements along the substrate, back action will be due to change in $g_{s} (\vec{r})$ without any change in δ_s (z), γ_s (z), and α_s (ω, z), while for displacements perpendicular to the substrate, back action will be due to change in all the above. With the substrate interaction taken into account, $g_{s} (\vec{r})$ represents the coupling constant between the rod on substrate mode (the modified rod mode due to image like interaction with the plasmonic substrate) and the near-field trap mode. From the above discussion it can be concluded that δ, κ, γ, and $\vec{r}$ are the independent handles that determine the substrate contribution to the net force acting on the nanoparticle as well as the the back action effect.

3. Numerical computation of force and absorption spectrum

Optical forces acting on the plasmonic nanoparticle are evaluated using COMSOL Multiphysics, which is a commercial finite elements based simulation package [40]. The package offers built in Maxwell’s Stress Tensor to compute the forces. In the framework of Maxwell’s Stress Tensor, the forces acting on the plasmonic nanoparticle due to the near-field trap and the substrate interaction are individually given by [41]:

F_{j}^{S / T} = \int_{V} {(\vec{\nabla} \cdot {\overset{\leftrightarrow}{T}}^{S / T})}_{j} d v

where

{(\vec{\nabla} \cdot \overset{\leftrightarrow}{T})}_{j} = ε [(\vec{\nabla} \cdot \vec{E}) E_{j} + (\vec{E} \cdot \vec{\nabla}) E_{j} - 0.5 \nabla_{j} E^{2}] + \frac{1}{μ} [(\vec{\nabla} \cdot \vec{B}) B_{j} + (\vec{B} . \vec{\nabla}) B_{j} - 0.5 \nabla_{j} B^{2}]

, S/T in the superscript represent substrate or trap, j represents the co-ordinates x, y, and z respectively, and

\vec{E}

and

\vec{B}

represent the electric and magnetic field respectively.

F_{j}^{S}

can be evaluated by simulating a plasmonic nanoparticle on a plasmonic substrate, devoid of the near-field trap.

F_{j}^{T}

cannot be directly evaluated. It can only be inferred by simulating the trapping of a nanoparticle that is significantly detuned from the near-field trap resonance and obtaining the force on it in the vicinity of the trap mode. The net force on the nanoparticle under the combined effect of both the contributing components can be calculated by [38]:

\begin{matrix} F_{j}^{N} = F_{j}^{S} + F_{j}^{T} + F_{j}^{I} \\ F_{j}^{I} = F_{j}^{S - T} + F_{j}^{T - S} \\ F_{j}^{S - T / T - S} = \int_{V} {(\vec{\nabla} \cdot {\overset{\leftrightarrow}{T}}^{S - T / T - S})}_{j} d v \end{matrix}

where

{(\vec{\nabla} \cdot {\overset{\leftrightarrow}{T}}^{S - T})}_{j} = ε [(\vec{\nabla} \cdot {\vec{E}}^{S}) E_{j}^{T} + ({\vec{E}}^{S} \cdot \vec{\nabla}) E_{j}^{T} - 0.5 \nabla_{j} E^{S} E^{T}] + \frac{1}{μ} [(\vec{\nabla} \cdot {\vec{B}}^{S}) B_{j}^{T} + ({\vec{B}}^{S} . \vec{\nabla}) B_{j}^{T} - 0.5 \nabla_{j} B^{S} B^{T}]

and

{(\vec{\nabla} \cdot {\overset{\leftrightarrow}{T}}^{T - S})}_{j}

, that can be calculated analogously, are the interaction terms that arise due to the linear superposition of the fields. So,

F_{j}^{I}

is essentially the force in the j direction due to the interaction terms. These terms combine to determine the spectrum of the near-field optical forces on the plasmonic nanoparticle.

Absorption cross section spectrum is used as an indicator of the resonance position of the various structure involved in the study. It is also evaluated using COMSOL Multiphysics. It’s inbuilt resistive power loss density function is integrated over the volume of the lossy plasmonic nanoparticle and the plasmonic substrate to evaluate the total absorption energy which then is divided by the intensity of the incident plane wave to obtain the absorption cross section. Parametric sweep over the requisite range of frequencies then yields the spectrum.

In this paper, the nature of the spectrum of $F_{j}^{N}$ and absorption cross section evaluated numerically are explained using the modal coupling based theory discussed in section 2. In doing so, the contribution of the substrate interaction to $F_{j}^{N}$ acting on the nanoparticle as well as the back action effect is established. Specifically, using suitable numerical examples, we establish the role of the substrate in the particle-trap interaction by comparing the resonance position of the absorption spectrum of the coupled system w.r.t the resonance position of the absorption spectrum of the bare plasmonic nanoparticle and the near-field trap. Fixing the near-field trap, we carry out parametric scans across appropriate numerical values to study the role of δ and nanoparticle position in determining the contribution of the substrate interaction to the spectrum of $F_{j}^{N}$ as well as the back action effect. In our numerical study we go beyond the asymptotic limit $(| δ ω_{0} | ≪ ω_{0})$ , in which SIBA has generally been discussed, and study closely tuned particle-trap systems as well.

4. Structure under study

As an illustrative example, we use a plasmonic nanorod suspended in water (n_w = 1.33), trapped on a bowtie engraving filled with a high index material (n_eng = 1.58), as depicted in Fig. 1(a). The xy cross-section of the bowtie geometry is defined by $a = 245 / \sqrt{2} n m$ , b = 70 nm, and θ = 50° (Fig. 1(b)). It is engraved up to a depth of t = 180 nm into the metallic substrate (Fig. 1(c)). The nanorod has a diameter d = 20 nm. Its length (L), height from the substrate (h) and displacement from the center of the engraving along the x direction (s) are used as parameters in the study (Fig. 1(c)). So, $\vec{r} = (s, 0, h)$ . A plane wave incident normal to the substrate (travelling in the −z direction) with the electric field polarized along the x-direction is used as the excitation (Fig. 1(c)). An incident intensity of 1mW/µm² is used in all simulations.

Fig. 1 Schematic of the simulated structure. (a) Trapped nanorod on a bowtie engraving in a plasmonic substrate. (b) xy cross section of the bowtie engraving. (c) xz cross section of the trapped nanorod on the bowtie engraving.

Download Full Size | PDF

A nanorod is chosen as the trapped particle in the study as its localized plasmon resonance can be tuned just by changing L, without modifying its d. This allows us to tune the plasmon resonance of the particle and the height of its center from the substrate independently. By changing L, the strength of the interaction of its localized plasmon mode with the surface mode of the substrate and the associated red shift of the nanorod resonance can be controlled. This is essential to demonstrate a pronounced substrate contribution. As the diameter of a spherical particle increases, the higher order multipolar resonances become prominent. The presence of these additional modes complicates the interaction picture based analysis. This is easily averted by using a nanorod of very small diameter. In practice, the torque acting on the nanorod will change its orientation but to highlight the substrate effect on the force and to simplify the analysis we neglect the effect of the torque and focus on the essential features of the force. In order to evaluate the absorption spectrum of isolated nanorod and rod on substrate separate simulations are set up with only the nanorod suspended in a medium of water and the nanorod at a specific height (h) from the substrate in a medium of water respectively.

The relative permittivity of the plasmonic nanorod and the substrate are taken in accordance with Drude theory, and are given by [42, 43]:

ε (ω) = ε_{\infty} - \frac{ω_{p}^{2}}{ω^{2} + i Γ ω}

where ε_∞ is the infinity frequency limit of the dielectric constant, ω_p is the plasma frequency, Γ is the damping frequency that is related to the relaxation time (τ) of the free electron gas by Γ = 1/τ. Γ is a property of the plasmonic material making up the nanorod and the substrate. It is different from γ which is the line-width of the localized plasmon mode of the plasmonic nanorod. γ depends on the material of the nanorod as well as the geometry. The values of the Drude parameters for the material of the plasmonic nanorod and substrate used in the study are tabulated in Table 1. A range of values for ω_p of the nanorod are used in the study. This range covers most commonly used plasmonic materials [42]. Although, ε_∞ and Γ will also vary for the different plasmonic materials, for the purpose of our study, we choose to keep them constant and vary δ between the plasmonic nanorod and the bowtie engraving just by varying ω_p. Like Γ, ω_p is solely a property of the plasmonic material. It does not depend on the rod length L unlike the localized plasmon resonance frequency (ω₀ + δ) of the nanorod. ω₀ + δ is a function of ω_p of the plasmonic material of the nanorod as well as L. So, it can be changed by changing either of the parameters. The values of ε_∞ and Γ for the plasmonic nanorod correspond to that of silver [43]. The choice of ω_p and ε_∞ for the plasmonic substrate allows to get the desired range of values of δ to explain the substrate contribution to the interaction picture for the chosen values of drude parameters for the plasmonic nanorod.

Table 1. Infinity frequency limit, plasma frequency and damping frequency of dielectric constant

View Table

5. Evidence of substrate mediated back action

The eigenmodes of a coupled two level system approach the uncoupled modes asymptotically in the limit of large detuning (δ) and/or small coupling, as discussed section 2. Conversely, these eigenmodes in the asymptotic limit are a means to identify the contributing uncoupled modes to the coupled system. Two such instances of eigenmodes in the asymptotic limit are analyzed to establish the role of the substrate in the particle-trap interaction.

A nanorod of length 220 nm resonant at ω₀+δ = 1.94 × 10¹⁵ Hz is positioned at h = 40 nm and s = 20 nm. The engraving is resonant at ω₀ = 1.22 × 10¹⁵ Hz as shown in Fig. 2(a). So, δ is positive and comparable to ω₀. The large h makes the interaction of the nanorod with the substrate as well as the engraving extremely small. So, the rod on substrate mode, which depicts the interaction of the nanorod with the substrate, is slightly red shifted w.r.t the isolated rod mode, as depicted in Fig. 2(a). As a result, δ_s (h) is slightly less than δ and g_s (s, 0, h) ≈ g(s, 0, h) → 0. The resonant modes of the coupled system are shown in Fig. 2(b). As expected, ω₋ is red shifted w.r.t ω₀. ω₊ is blue shifted w.r.t ω₀ + δ_s (h) but is red shifted w.r.t ω₀ + δ. So, the contributing uncoupled modes to the coupled particle-trap system are the rod on substrate mode and the engraving mode.
A nanorod of length 350 nm resonant at ω₀+δ = 1.30 × 10¹⁵ Hz is positioned at h = 20 nm and s = 140 nm. So, δ is slightly positive, as evident from Fig. 2(d). The smaller h and the larger L make its interaction with the substrate extremely large. As a result, δ_s (h) is negative. This is evident from Fig. 2(d). The large s makes the interaction with the engraving extremely small. So, g_s (s, 0, h) → 0. Thus, the eigenmodes of the coupled system (Fig. 2(e)) asymptotically approach the modes of the participating interacting systems. As δ_s (h) is negative in this case, ω₋ is red shifted w.r.t ω₀ + δ_s (h) whereas ω₊ is blue shifted w.r.t ω₀. As in the previous case, ω₊ is red shifted w.r.t ω₀ + δ. So, the contributing uncoupled modes to the coupled particle-trap system are again the rod on substrate mode and the engraving mode.

Fig. 2 Evidence of substrate mediated back action. (a), (b), and (c) are for nanorod with L = 220 nm, h = 40 nm, and s = 20 nm. (d), (e) and (f) are for nanorod with L = 350 nm, h = 20 nm, and s = 140 nm. (a) and (d) Absorption cross section of the isolated rod (red solid), rod on substrate (red dashed) and engraving mode (black solid). (b) and (e) Absorption cross section of the coupled modes. (c) and (f) Net force acting on the nanorod. $F_{z}^{N}$ (solid) acts in the z direction and $F_{x}^{N}$ (dashed) acts in the x direction.

Download Full Size | PDF

Both the above instances combine to establish that the eigenmodes of the coupled system are due to the interaction between the engraving mode and the rod on substrate mode. This is schematically represented in Fig. 3. $F_{j}^{N}$ follows the resonant modes of the coupled system as evident from Fig. 2(b) and 2(c) and Fig. 2(e) and 2(f). This is because the plasmonic resonances maximize near-field interaction [28, 44, 45]. For normally incident excitation there cannot be any component of the optical force due to the substrate acting along the substrate, making $F_{x}^{S} = 0$ [30, 31]. This is also evident from the symmetry of the structure representing the substrate contribution in Fig. 3. $F_{x}^{T} + F_{x}^{I}$ can be both positive and negative. So, the x component of the force corresponding to the low frequency mode is attractive (bonding), while the force corresponding to the high frequency mode is repulsive (anti-bonding) in both Fig. 2(c) and 2(f). Interestingly, in spite of $F_{x}^{S}$ being 0, the substrate interaction evidently contributes to the spectral profile of $F_{x}^{N}$ because it still traces the modes of the coupled rod on substrate-trap system. It turns out that the reorganization of charges on the nanorod due to simultaneous interaction with the substrate and the near-field trap gives rise to substrate contribution to $F_{x}^{N}$ by means of $F_{x}^{I}$ . This can be understood from an analysis of the terms of ${(\vec{\nabla} \cdot {\overset{\leftrightarrow}{T}}^{S - T / T - S})}_{x}$ . $F_{z}^{T} + F_{z}^{I}$ can also be both positive and negative, as in the case of the x-direction. As the substrate exerts an attractive force in the z-direction, $F_{z}^{S} < 0$ . So, the z component of force corresponding to the low frequency mode is expectedly attractive in both cases. In the first case, the repulsion in the high frequency mode is negated by $F_{z}^{S}$ over a considerable frequency range, resulting in an asymmetric lineshape. In the second case, as the nanorod is shifted away from the engraving, along the interface of the substrate and the medium, $F_{z}^{S}$ becomes dominant. This leads to a net attractive force in the z direction at all frequencies (even for the high frequency mode). This further substantiates the role of the substrate in the near-field trapping of plasmonic nanoparticles on resonant plasmonic apertures/engravings.

Fig. 3 Interaction picture for the trapping of the nanorod on the bowtie engraving. The coupled particle-trap system arises due to the superposition of the engraving and the rod on substrate mode. The image like interaction with the substrate makes the nanoparticle polarizability a function of its height from the substrate.

Download Full Size | PDF

The usage of the rod on substrate mode allows to reduce the system involving three interacting components (substrate, nanorod and trap) into a much simpler two component system making the subsequent modal coupling based analysis, as discussed in section 2, easier. The rod on substrate mode can be obtained by simulating a nanorod at a height h above the substrate. So, the problem of understanding the contribution of the substrate interaction then reduces to analyzing the difference between the rod on substrate mode and the isolated rod mode as seen by the engraving mode. This method of analysis has been adopted in the subsequent sections.

6. Effect of substrate on the near-field optical forces

With the role of the substrate established, it is imperative to understand the precise manner in which it affects the near-field optical forces acting on the nanorod at any given ω. Here we analyze the role of the substrate as a function of the variations in δ, h, and s.

6.1. Detuning between particle and engraving resonance

The effect of δ is studied by considering a plasmonic nanorod with L = 220 nm, h = 40 nm, s = 0 nm and ω_p varying between 1.25 × 10¹⁶ Hz and 3 × 10¹⁶ Hz with appropriate intermediate values. As a result ω₀ + δ varies between 1.09 × 10¹⁵ Hz and 1.94 × 10¹⁵ Hz. As L remains constant the volume of the nanorod does not change. So, the sole contribution to back-action is expected to arise due to δ. As s = 0, the nanorod experiences a force only in the z direction. The effect on the high frequency and the low frequency modes are separately analyzed.

At ω₀ + δ = 1.94 × 10¹⁵ Hz, δ and δ_s (h) ≫ 0 as evident from Fig. 4(a). So, the high frequency mode of the coupled system (in Fig. 4(b)) is predominantly like the rod on substrate mode. Thus, $F_{j}^{N}$ corresponding to this mode is dominated by the attractive $F_{j}^{S}$ as shown in Fig. 4(c). The minute repulsive contributions from $F_{j}^{T} + F_{j}^{I}$ give an asymmetric lineshape with a slight positive peak to $F_{j}^{N}$ . As ω_p decreases, δ and δ_s (h) also decrease and the contribution of the rod on substrate mode to the high frequency mode decreases. So, the attractive and repulsive force contributions to this mode almost cancel out for ω₀ + δ = 1.44 × 10¹⁵ Hz. Reducing δ even further, leads to a repulsive force dominated high frequency mode as the rod on substrate contribution to this mode decreases further. This is accompanied by an increase in the rod on substrate mode contribution to the low frequency mode, as is evident from the increasing red shift of ω₋ w.r.t ω₀. So, an increasing contribution of $F_{j}^{S}$ to $F_{j}^{N}$ in the vicinity of this mode ensues.

Fig. 4 Effect of detuning between isolated rod mode and the engraving mode on the nature of the near-field force on the nanorod. The ω₀ + δ of the different nanorods are 1.94 × 10¹⁵ Hz (pink), 1.55 × 10¹⁵ Hz (blue), 1.44 × 10¹⁵ Hz (purple), 1.26 × 10¹⁵ Hz (olive), and 1.09 × 10¹⁵ Hz (red) (a) Absorption cross section of uncoupled isolated rod modes (solid curves), rod on substrate modes (dashed) and engraving mode (solid black). (b) Absorption cross section of coupled modes and the engraving mode. (c) $F_{Z}^{N}$ on the nanorod. The plots have been biased by −60 pN (pink), −45 pN (blue), −30 pN (purple), −15 pN (olive) to ensure legibility.

Download Full Size | PDF

The low frequency mode of the coupled particle-trap system is used for stable trapping. Generally, the resonance of the trapped particle is significantly blue shifted w.r.t the engraving mode. For such instances, the difference in the modal overlap of the isolated rod and rod on substrate modes with the engraving mode is negligible, making the contribution of the interaction components $(F_{j}^{I})$ to the trap force insignificant. This is represented in Fig. 5 for a dielectric nanorod of identical dimensions as that of the plasmonic nanorods in Fig. 4. Even with an exceptionally high dielectric constant of 100, its standing wave resonance modes are at much higher frequencies than ω₀ + δ for plasmonic nanorods with different ω_p. The large detuning also means $F_{j}^{S} + F_{j}^{I}$ is negligible in the vicinity of the low energy mode. So, the particle experiences a force solely due to the engraving $(F_{j}^{T})$ in the vicinity of the low frequency mode. As the nanoparticle approaches the engraving the refractive index in its vicinity changes. This induces a shift in the engraving mode. It is the source of the SIBA contribution to $F_{j}^{N} \approx F_{j}^{T}$ . If this change in nanoparticle position involves a change in h, it is accompanied by a shift in δ_s (h). This leads to a change in the difference between the overlap of the isolated rod and rod on substrate mode with the engraving mode. As this difference is already negligible, the contribution of a small change on top of that to the back action force is insignificant. Alternatively, the change in α_s (ω, h) is negligible. Coming back to Fig. 4, as ω₀ + δ is decreased from 1.94 × 10¹⁵ Hz to 1.26 × 10¹⁵ Hz, the difference in the modal overlap of the isolated rod and rod on substrate mode with the engraving mode becomes increasingly significant, as is evident from the solid curves and the corresponding dashed curves in Fig. 4(a). So, the change in α_s (ω, h) as seen by the engraving also becomes significant. This leads to an enhanced contribution of $F_{j}^{I}$ to $F_{j}^{N}$ in the vicinity of the low energy mode with decreasing δ. As a result, the substrate contribution to the low frequency mode of the particle-trap system increases. For such instances, as the nanoparticle approaches the engraving, ω₀ (the low frequency uncoupled mode) translates into ω₋. With the change in the particle position, apart from a shift in ω₋ akin to SIBA, the relative contribution of the rod on substrate mode and the engraving mode also varies, as the overlap of the isolated rod mode with the near-field of the engraving and the substrate mode changes. This leads to a difference in the back action mechanism for nanoparticle displacements in the x and z directions. For displacement in the x direction, back action arises because of shift in ω₋ solely due to change in g_s (s, 0, h) without a change in δ_s (h), as the rod on substrate mode remains unchanged with x position of the nanoparticle. On the other hand, for displacements in the z direction, the shift in ω₋ is caused due to change in g_s (s, 0, h) as well as δ_s (h). Due to these additional features in the back action force afforded by the significant contribution of the substrate interaction, it is apt to describe this as Substrate Mediated Self Induced Back Action. As is rather clear, for the same engraving, the substrate contribution to the low frequency trapping mode is solely dependent on δ. In the limiting case of δ ≫ 0, the substrate contribution to the low frequency mode dies down. The system converges to an isolated nanorod interacting with the engraving. So, as the nanoparticle position changes, the back action is solely due to the shift in the engraving mode. The contribution of the change in α_s (ω, h), as perceived by the engraving mode, to the back action effect is negligible. This is the regime of the widely reported SIBA effect.

Fig. 5 Self Induced Back Action Trapping of a dielectric nanorod on the bowtie engraving. The dielectric constant and the conductivity of the dielectric are taken as 100 and 10⁻² S/m respectively. The presence of the nanorod shifts the bare engraving mode (solid black) to the low frequency coupled mode of the particle-trap system (blue solid). The isolated rod (solid red) and rod on substrate (dashed green) mode have negligible difference in overlap with the bare engraving mode. The higher energy stored in the coupled particle-trap system at the operating frequency (solid red vertical) compared to the bare engraving contributes to the back action component of the near-field trap force (dashed blue) on the nanorod.

Download Full Size | PDF

6.2. Displacement along the substrate

As the nanorod is displaced away from the engraving along the x direction, δ_s (h), γ_s (h), and α_s (ω, h) remain unchanged. But g_s (s, 0, h) diminishes. So, ω_± approach ω₀ + δ_s (h) and ω₀ depending on the sign of δ_s (h), as depicted in Fig. 6(a) and 6(d) for ω₀ + δ = 1.94 × 10¹⁵ Hz and ω₀ + δ = 1.09 × 10¹⁵ Hz respectively. This leads to a decrease in $F_{j}^{T}$ and $F_{j}^{I}$ acting on the nanorod. In either case, as the nanorod is displaced away from the engraving $F_{z}^{N}$ converges to $F_{z}^{S}$ while $F_{x}^{N}$ converges to $F_{x}^{S} = 0$ as is evident from Fig. 6(b)–6(f). For the nanorod with ω₀ + δ = 1.94 × 10¹⁵ Hz, δ_s (h) > 0. So, $F_{z}^{N}$ corresponding to the high frequency mode converges to $F_{z}^{S}$ , while the one corresponding to the low frequency mode dies down as is evident from Fig. 6(b). Vice versa holds for ω₀ + δ = 1.09 × 10¹⁵ Hz, as depicted in Fig. 6(e). In this case, the back action effect is solely due to changing g_s (s, 0, h) with s. Alternatively, the back action is due to change in f (s, 0, h) that changes δω₀(s, 0, h). This changes the I(s, 0, h, ω) and in turn ${\vec{F}}^{N}$ as per Eq. (2) and Eq. (3) respectively.

Fig. 6 Variation of the near-field force on the nanorod with displacement along the substrate. (a), (b), and (c) are for nanorod with ω₀ + δ = 1.94 × 10¹⁵ Hz. (d), (e) and (f) are for nanorod with ω₀ + δ = 1.09 × 10¹⁶ Hz. Solid blue curve is for s = 0, dotted blue curve is for s = 100 nm, and dashed blue curve is for s = 200 nm. (a) and (d) Absorption cross section of rod on substrate mode (red solid), engraving mode (black solid) and the coupled modes for different s values. (b) and (e) $F_{z}^{S}$ (red solid) and $F_{z}^{N}$ for different s values. (c) and $F_{x}^{N}$ on the nanorod for different s values.

Download Full Size | PDF

6.3. Height above the substrate

As h decreases, g_s (s, 0, h) increases. So, δ_s (h) decreases, as is depicted in Fig. 7(a) for a nanorod with ω₀+δ = 1.44 × 10¹⁵ Hz. Fig. 7(b)–7(d) depict the resonant modes of the coupled system and the associated $F_{z}^{N}$ on the nanorod with decreasing h as the nanorod approaches the engraving. $F_{x}^{N} = 0$ in this analysis because s = 0. Fig. 7(c) depicts the modes of the coupled system at h = 40nm, which is the same as in Fig. 4. If the particle moves to a higher h, δ_s (h) increases, and the observed force profile is akin to plasmonic nanorods with higher δ at h = 40nm, as can be seen in Fig. 7(b), in agreement with the results of Fig. 4. The reverse happens as h decreases (Fig. 7(d)). Alternatively, it is α_s (ω, h) that determines the nature of the force spectrum at a given h. The decrease in δ_s (h) on moving closer to the substrate is negated by the increased g_s (s, 0, h) and the associated larger blue shift of ω₊ w.r.t ω₀ + δ_s (h). So, ω₊ is nearly the same at different h, in spite of the seemingly increased g(s, 0, h) as the nanorod approaches the engraving. This can only be explained by taking the substrate contribution (in the form of the rod on substrate mode) into account. As h changes, g_s (s, 0, h) as well as δ_s (h), and α_s (ω, h) change. From the electromagnetic perturbation theory point of view both f (s, 0, h) as well as α_s (ω, h) change and contribute to change in δω₀(s, 0, h). So, the shift in the low energy trapping mode, and hence the back action effect, is due to the combined effect of all.

Fig. 7 Variation of near-field force on the nanorod with height above the substrate. (a) Absorption cross section of the uncoupled isolated rod mode (solid black), engraving (dashed black), rod on substrate mode for h = 60 nm (solid blue), h = 40 nm (solid olive), and h = 20 nm (solid red). (b), (c), and (d) represent the absorption cross section of the coupled modes and $F_{z}^{N}$ acting on the nanorod for h = 60 nm, h = 40 nm, and h = 20 nm respectively.

Download Full Size | PDF

7. Impact of substrate on the back action effect

In section 6.1 we show the increasing contribution of the substrate mode to the low frequency stable trapping mode of the coupled particle-trap system as δ decreases. In section 6.3 we show that on changing h the back action effect is due to change in f (s, 0, h) and α_s (ω, h). In this section we study the back action effect with change in h for plasmonic nanorods of distinctly different δ w.r.t the engraving mode. For the case of the nanorod with large δ the contribution of the substrate to the back action effect is expected to be negligible. So, the back action effect will be predominantly SIBA. By comparing the back action effect in the case of the plasmonic nanorods for the two δ′s we study the impact of the contribution of the substrate to the back action effect.

We compute the absorption cross section and trap force spectrum for two plasmonic nanorods with ω₀ + δ values 1.44 × 10¹⁵Hz and 1.94 × 10¹⁵ trapped on the bowtie engraving resonant at ω₀ = 1.22 × 10¹⁵ Hz. The spectrums are computed for h = 20 nm, 40 nm and 60 nm in each case as shown in Fig. 8. The nanorods are centered about the trap. So, s = 0 and the ${\vec{F}}^{N}$ acts solely in the z direction. For the nanorod with ω₀ + δ = 1.44 × 10¹⁵ Hz there is a significant difference in the overlap of the rod on substrate modes at h = 20 nm and h = 40 nm with the engraving mode because of the small δ as evident from Fig. 8(a). A much smaller difference in the overlap is seen for h = 40 nm and h = 60 nm. Alternatively, we can say that there is a large increase in α_s (ω, h) in the vicinity of the spectral peak of the engraving mode as h decreases from 40 nm to 20 nm because absorption cross section is a function of the imaginary part of α_s (ω, h) [19, 38, 39]. This combined with the change in f (s, 0, h) lead to large red shifts of the intensity and trap force spectrums as the nanorod drops from h = 60 nm to h = 20 nm. Major portion of the shift occurs between h = 40 nm and h = 20 nm. In our simulations we show the red shift in the low frequency stable trapping mode of the absorption cross section and $F_{z}^{N}$ spectrum in Fig. 8(b) and 8(c) respectively. As as result of these red shifts we see that the absorption cross section at the spectral peak location for h = 60 nm marked by the dashed vertical line in Fig. 8(b) drops as the rod shifts to h = 40 nm and then to 20 nm. At h = 20 nm it is approximately 1/6 of its value at h = 60 nm as can be inferred from the dashed horizontal lines. Such a drop in the intensity can also be inferred from Eq. (2). There is also a change in $F_{z}^{N}$ as can be seen in Fig. 8(c). For the nanorod with ω₀ + δ = 1.94 × 10¹⁵ Hz there is a negligible increase in α_s (ω, h) in the vicinity of the spectral peak of the engraving mode with decrease in h because of the large δ as can be inferred from Fig. 8(d). So, the back action effect in this case should be mainly due to f (s, 0, h). This is in the regime of SIBA. In this case, as the nanorod drops from h = 60 nm to 40 nm the shift in the low frequency stable trapping mode is comparable to the previous case as can be inferred by comparing Fig. 8(b) and 8(e). But, as the nanorod drops from h = 40 nm to h = 20 nm the red shift is much smaller than in the previous case. This is attributed to the significant increase in α_s (s, 0, h) as h changes from 40 nm to 20 nm for the plasmonic nanorod with ω₀ + δ = 1.44 × 10¹⁵ Hz compared to almost a negligible change for the plasmonic nanorod with ω₀ + δ = 1.94 × 10¹⁵ Hz. f (s, 0, h) is a property of the bare engraving and remains the same in both cases. For the nanorod with ω₀ + δ = 1.94 × 10¹⁵ the absorption cross section at the spectral peak location for h = 60 nm drops to approximately 1/3 of its value as h decreases from 60 nm to 20 nm. Similar decrease in the intensity is expected from Eq. (2). The larger spectral shifts in the case of ω₀ + δ = 1.44 × 10¹⁵ Hz due to the increased substrate contribution lead to a more significant drop in the absorption cross section and near-field intensity at a suitable choice of operating frequency as the nanoparticle moves to the stable equilibrium position. This is significantly advantageous when biological samples such as proteins that are intolerant to high optical intensities are conjugated to plasmonic nanoparticles [46]. The absorption of light and the subsequent heating of the trapped nanoparticle at the stable equilibrium position is also reduced. These have been reported as the advantages of SIBA. By enhancing the spectral shifts, substrate mediated self induced back action significantly improves upon these advantages. In the case of NOCB the plasmonic nanoparticle will experience strong trapping forces under the combined influence of the substrate and near-field engraving as it moves over the substrate and will be exposed to significantly reduced optical intensity at the stable equilibrium position.

Fig. 8 Impact of substrate on the back action effect. (a), (b), and (c) are for nanorod with ω₀ + δ = 1.44 × 10¹⁵ Hz. (d), (e) and (f) are for nanorod with ω₀ + δ = 1.94 × 10¹⁶ Hz. Solid black is for isolated nanorod, dashed black is for bare engraving, blue curve is for h = 60 nm, olive curve is for h = 40 nm, red curve is for h = 20 nm. (a) and (d) Absorption cross section spectrum of rod on substrate mode at different h. (b) and (e) Absorption cross section spectrum of the coupled particle-trap system at different h. (c) and (f) Spectrum of $F_{z}^{N}$ acting on the nanorod at different h.

Download Full Size | PDF

8. Conclusion

In this paper, we numerically demonstrate the contribution of the image like interaction between a plasmonic nanoparticle and a plasmonic substrate, to the force acting on the nanoparticle trapped on an engraving in the same substrate. A study of the eigenmodes of the coupled particle-trap system in the asymptotic limit of large detuning and small coupling shows that the participating modes in the interaction are the nanoparticle on substrate mode and the trap mode, thus establishing the substrate contribution. It is found that as the localized plasmon mode of the nanoparticle is tuned closer to the resonant mode of the engraving, the difference in the overlap of the isolated nanoparticle mode and nanoparticle on substrate mode with the bare engraving mode becomes prominent, and the contribution of the substrate force to the low frequency stable trapping mode becomes significant. This substrate contribution leads to a difference in the back action mechanism for nanoparticle displacements along the substrate and perpendicular to it. For nanoparticle displacements along the substrate, the back action contribution is solely due to change in its interaction with the engraving, as the interaction with the substrate does not change. This is evident because $F_{z}^{N}$ converges to $F_{z}^{S}$ when displaced beyond the near-field of the engraving. For its displacement perpendicular to the substrate, the back action involves changes in its interaction with both the engraving as well as the substrate. Alternatively, it can be said that as the nanoparticle is displaced perpendicular to the substrate its effective polarizability changes. So, for displacements along the substrate the back action is solely due to change in position of the nanoparticle w.r.t the trap. On the other hand, for displacements perpendicular to the substrate the back action is due to combined effect of the change in position of the nanoparticle and the associated change in its polarizability. This additional contribution of the change in polarizability leads to larger spectral shifts of the nanoparticle position dependent trap intensity profile and in turn allows for steeper variation in the trapping intensity at the operating frequency. Thus, the substrate contribution enhances the efficiency of SIBA. It is also observed that, although the substrate by itself does not exert a force on the nanoparticle along the substrate, it still contributes to the particle position dependent spectral profile of this force through its interaction term with the trap mode. So, it is justified to term the back action that involves a significant contribution of the substrate interaction as Substrate Mediated Self Induced Back Action. Thus, it is necessary to take into account the substrate effect while designing near-field traps for plasmonic nanoparticles with any arbitrary detuning of the localized plasmon mode w.r.t the engraving mode, to precisely explain the observed force profile on the nanoparticle. It is especially important in the process of transport of nanoparticles between non-interacting neigboring traps of the conveyor belt, as they have to move over regions of bare plasmonic substrate. Also, for the realization of complex lab-on-a-chip functionalities a large variety of nanoparticles having varied detuning w.r.t the trap mode will have to be transported using a common network of conveyor belts on the chip. So, the nature of forces on each type of particle will change due to the varying substrate contribution to the net force. Thus, our study is significant in the context of on chip trapping and manipulation of nanoparticles for lab-on-a-chip applications.

References and links

1. K. Okamoto and S. Kawata, “Radiation force exerted on subwavelength particles near a nanoaperture,” Phys. Rev. Lett. 22(83), 4534–4537 (1999). [CrossRef]

2. A. Rahmani and P. C. Chaumet, “Optical trapping near a photonic crystal,” Opt. Express 14(13), 6353–6358 (2006). [CrossRef] [PubMed]

3. M. Ploschner, M. Mazilu, T. F. Krauss, and K. Dholakia, “Optical forces near a nanoantenna,” J. Nanophoton. 4041570 (2010). [CrossRef]

4. M. Righini, A. S. Zelenina, C. Girard, and R. Quidant, “Parallel and selective trapping in a patterned plasmonic landscape,” Nat. Phys. 3(7), 477–480 (2007). [CrossRef]

5. A. N. Grigorenko, N. W. Roberts, M. R. Dickinson, and Y. Zhang, “Nanometric optical tweezers based on nanostructured substrates,” Nat. Photon. 3(6), 365–370 (2008). [CrossRef]

6. B. J. Roxworthy, K. D. Ko, A. Kumar, K. H. Fung, E. K. C. Chow, G. L. Liu, N. X. Fang, and K. C. Toussaint Jr., “Application of plasmonic bowtie nanoantenna arrays for optical trapping, stacking and sorting,” Nano Lett. 12(2), 796–801 (2012). [CrossRef] [PubMed]

7. M. L. Juan, M. Righini, and R. Quidant, “Plasmon nano-optical tweezers,” Nat. Photon. 5(6), 348–356 (2011). [CrossRef]

8. R. Quidant, “Plasmonic tweezers - The strength of surface plasmons,” Mater. Res. Bull. 37(8), 739–744 (2012). [CrossRef]

9. T. Leest and J. Caro, “Cavity-enhanced optical trapping of bacteria using a silicon photonic crystal,” Lab Chip 13(22), 4358–4365 (2013). [CrossRef] [PubMed]

10. S. Mandal, X. Serey, and D. Erickson, “Nanomanipulation using silicon photonic crystal resonators,” Nano Lett. 10(1), 99–104 (2010). [CrossRef]

11. W. Zhang, L. Huang, C. Santschi, and O. J. F. Martin, “Trapping and sensing 10 nm metal nanoparticles using plasmonic dipole antennas,” Nano Lett. 10(3), 1006–1011 (2010). [CrossRef] [PubMed]

12. A. A. E. Saleh and J. A. Dionne, “Toward efficient optical trapping of sub-10-nm particles with coaxial plasmonic apertures,” Nano Lett. 12(11), 5581–5586 (2012). [CrossRef] [PubMed]

13. P. Hansen, Y. Zheng, J. Ryan, and L. Hesselink, “Nano-optical conveyor belt, part I: Theory,” Nano Lett. 14(6), 2965–2970 (2014). [CrossRef] [PubMed]

14. Y. Zheng, J. Ryan, P. Hansen, Y. T. Cheng, T. J. Lu, and L. Hesselink, “Nano-optical conveyor belt, part II: Demonstration of handoff between near field optical traps,” Nano Lett. 14(6), 2971–2976 (2014). [CrossRef] [PubMed]

15. G. Sotiriou, “Biomedical applications of multifunctional plasmonic nanoparticles,” WIREs Nanomed. Nanobiotechnol. 5(1), 19–30 (2013). [CrossRef]

16. D. J. Aberasturi, A. B. Serrano-Montes, and L. M. Liz-Marzán, “Modern applications of plasmonic nanoparticles: From energy to health,” Adv. Optical Mater. 3(5), 602–617 (2015). [CrossRef]

17. F. Svedberg, Z. Li, H. Xu, and M. Käll, “Creating hot nanoparticle pairs for surface-enhanced raman spectroscopy through optical manipulation,” Nano Lett. 6(12), 2639–2641 (2006). [CrossRef] [PubMed]

18. M. Meier and A. Wokaun, “Enhanced fields on large metal particles: dynamic depolarization,” Opt. Lett. 8(11), 581–583 (1983). [CrossRef] [PubMed]

19. K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: The influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, (3)668–677 (2003). [CrossRef]

20. V. Myroshnychenko, J. Rodríguez-Fernández, I. Pastorizia-Santos, A. M. Funston, C. Novo, P. Maulvaney, L. M. Liz-Marzán, and F. J. Garciá de Abajo, “Modelling the optical response of gold nanoparticles,” Chem. Soc. Rev. 37(9), 1792–1805 (2008). [CrossRef] [PubMed]

21. P. Nordlander and E. Prodan, “Plasmonic hybridization in nanoparticles near metallic surfaces,” Nano Lett. 4(11), 2209–2213 (2004). [CrossRef]

22. M. Barth and O. Benson, “Manipulation of dielectric particles using photonic crystal cavities,” Appl Phys Lett. 89(25), 253114 (2006). [CrossRef]

23. M. L. Juan, R. Gordon, Y. Pang, F. Eftekhari, and R. Quidant, “Self-induced back action optical trapping of dielectric nanoparticles,” Nat. Phys. 5(12), 915–919 (2009). [CrossRef]

24. J. Hu, S. Lin, L. C. Kimerling, and K. Crozier, “Optical trapping of dielectric nanoparticles in resonant cavities,” Phys. Rev. A 82(5), 053819 (2010). [CrossRef]

25. N. Descharmes, U. P. Dharanipathy, Z. Diao, M. Tonin, and R. Houndré, “Observation of back action and self-induced trapping in a planar hollow planar photonic crystal cavity,” Phys. Rev. Lett. 110(15), 123601 (2013). [CrossRef]

26. L. Neumeier, R. Quidant, and D.E. Chang, “Self-induced back-action optical trapping in nanophotonic systems,” New J. Phys. 17(12), 123008 (2015). [CrossRef]

27. P. Mestres, J. Berthelot, S. S. Aćimović, and R. Quidant, “Unravelling the optomechanical nature of plasmonic trapping,” Light Sci. Appl. 5, e16092 (2016). [CrossRef]

28. R. Sainidou and F. J. Garciá de Abajo, “Optical tunable surfaces with trapped particles in microcavities,” Phys. Rev. Lett. 101(13), 136802 (2008). [CrossRef] [PubMed]

29. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984), Chap. 7.

30. J. R. Arias-Gonźalez and M. Nieto-Vesperinas, “Optical forces on small particles: attractive and repulsive nature and plasmon-resonance conditions,” J. Opt. Soc. Am. A 20(7), 1201–1209 (2003). [CrossRef]

31. M. I. Petrov, S. V. Sukhov, A. A. Bogdanov, A. S. Shalin, and A. Dogariu, “Surface plasmon polariton assisted optical pulling force,” Laser Photonics Rev. 10(1), 116–122 (2016). [CrossRef]

32. P. M. Bendix, L. Jauffred, K. Norregaard, and L. B. Oddershede, “Optical trapping of nanoparticles and quantum dots,” IEEE J. Sel. Top. Quantum Electron 20(3), 4800112 (2014).

33. H. A. Haus and W. Huang, “Coupled-mode theory,” Proc. IEEE 79(10), 1505–1516 (1991). [CrossRef]

34. D. V. Thourhout and J. Roels, “Optomechanical device actuation through the optical gradient force,” Nat. Photon. 4(4), 211–217 (2010). [CrossRef]

35. M. L. Povinelli, M. Loncar, M. Ibanescu, E. J. Smythe, S. G. Johnson, F. Capasso, and J. D. Joannopoulos, “Evanescent-wave bonding between optical waveguides,” Opt. Lett. 30(22), 3042–3044 (2005). [CrossRef] [PubMed]

36. W. P. Bowen and G. J. Milburn, Quantum Optomechanics (CRC Press, 2016), Chap. 2.

37. Y. J. Zheng, H. Liu, S. M. Wang, T. Li, J. X. Cao, L. Li, C. Zhu, Y. Wang, S. N. Zhu, and X. Zhang, “Selective optical trapping based on strong coupling between gold nanorods and slab,” Appl. Phys. Lett. 98(8), 083117 (2011). [CrossRef]

38. C. F. Bohren and D. R. Huffman, Absorption and scattering of light by small particles (John Wiley and Sons, 1998), Chap. 5. [CrossRef]

39. A. Pinchuk, A. Hilger, G. Plessen, and U. Kreibig, “Substrate effect on the optical response of silver nanoparticles,” Nanotechnology 15(12), 1890–1896 (2004). [CrossRef]

40. Comsol Multiphysics ®

41. D. J. Griffiths, Introduction to Electrodynamics (Prentice-Hall, 1999), Chap. 8.

42. M. A. Ordal, R. J. Bell, R. W. Alexander, L. L. Long, and M. R. Querry, “Optical properties of fourteen metals in the infrared and far infrared: Al, Co, Cu, Au, Fe, Pb, Mo, Ni, Pd, Pt, Ag, Ti, V and W,” Appl. Opt. 24(24), 4493–4499 (1985). [CrossRef] [PubMed]

43. W. Cai and V. Shalaev, Optical metamaterials: Fundamentals and Applications (Springer, 2009), Chap. 2.

44. P. Nordlander, C. Oubre, E. Prodan, K. Li, and M. I. Stockman, “Plasmon hybridization in nanoparticle dimers,” Nano Lett. 4(5), 899–903 (2004). [CrossRef]

45. E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302(5644), 419–422 (2003). [CrossRef] [PubMed]

46. P. Bergese and K. Hamad-Schifferli, Nanomaterial Interfaces in Biology (Springer, 2013), Chap. 3. [CrossRef]

Parameter	Nanorod	Substrate
ε_∞	5	8
ω_p (Hz)	1.25 × 10¹⁶ − 3 × 10¹⁶	0.85 × 10¹⁶
Γ (H_z)	0.32 × 10¹⁴	0.32 × 10¹⁴

On the substrate contribution to the back action trapping of plasmonic nanoparticles on resonant near-field traps in plasmonic films

Abstract

1. Introduction

2. Modal theory of interaction between nanoparticle and trap

3. Numerical computation of force and absorption spectrum

4. Structure under study

5. Evidence of substrate mediated back action

6. Effect of substrate on the near-field optical forces

6.1. Detuning between particle and engraving resonance

6.2. Displacement along the substrate

6.3. Height above the substrate

7. Impact of substrate on the back action effect

8. Conclusion

References and links

Cited By

Figures (8)

Tables (1)

Equations (6)

Optics Express