Strong optical force acting on a dipolar particle over a multilayer substrate

Shubo Wang; C. T. Chan

doi:10.1364/OE.24.002235

1. Introduction

Using optical force to manipulate small particles becomes feasible after the invention of lasers [1]. Such applications are typically limited to micro-sized particles that have reasonably large dipole polarizablity. Inducing strong optical forces on deep sub-wavelength nano-sized particles would be highly desirable, but there are many challenges. For structures such as parallel-plate cavity [2], nanowire pairs [3, 4 ] and metamaterial slabs [5], it is possible to achieve strong forces through cavity resonance [6] or evanescent-field manipulation. However, it is not easy to induce a strong force on a small particle like a nano-sphere [7, 8 ]. It has been shown, both theorectically and experimentally, that when a small particle is put near to a substrate, the evanescent wave can induce a gradient force on the particle that is much larger than the normal photon pressure [9–12 ]. In this paper we go one step further and show that if a dipolar particle is put near to a substrate with a metal-dielectric-metal sandwich structure, the force is enhanced by orders of magnitude compared to usual evanescent wave induced forces. The enhancement is attributed to a particle-substrate resonance that happens under the condition that the surface plasmon modes supported by the substrate has a flat dispersion.

We consider a subwavelength sphere of radius r located at a distance d above a metal-dielectric-metal sandwich-structured substrate, as shown in Fig. 1(a) . The thicknesses of the three layers are t ₁, t ₂, and t ₃, respectively, and we take $t_{3} \to \infty$ so the bottom layer is essentially semi-infinite along z direction. The substrate is infinitely large on the xy plane. The permittivities of the metal layers are given by the Drude model: $ε_{1} = ε_{3} = 1 - ω_{p}^{2} / (ω^{2} + i ω γ)$ , where $ω_{p}$ is the bulk plasma frequency and $γ$ denotes damping frequency. The dielectric layer has relative permittivity $ε_{2} = 12.96$ . We assume all the materials are non-magnetic. The system is excited by a plane wave of $E^{inc} = \hat{x} E_{0} Exp[- i k_{0} z - i ω t]$ with an intensity of $1 mW/ {(μm)}^{2}$ and we are interested in the optical force acting on the spherical particle.

Fig. 1 (a) A schematic picture showing the configuration of the model system. (b) Band diagram of the sandwich-structured substrate. The two dotted lines denote the light lines in vacuum and in the dielectric medium. Real [(c)] and imaginary [(d)] parts of the Green’s function vs. frequency and distance d. The white region denotes off-scale values beyond that represented by the color bar.

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The paper is organized as follows. In Sec. 2, we introduce the Green’s function method for solving the considered problem. In Sec. 3, we show the results of the force enhancement for particles with different polarizabilities. The conclusions are drawn in Sec. 4.

2. Analytical Green’s function method

Under the subwavelength condition, the Rayleigh approximation is invoked and the particle can be treated as an electric dipole with polarizabiltiy $α = i 6 π ε_{0} a_{1} / k_{0}^{3}$ , where a ₁ is the Mie scattering coefficient [13, 14 ]. The dipole moment is $p = α [E^{inc} (1 + R^{TE} e^{i 2 k_{0} d}) + μ_{0} ω^{2} \bar{G} \cdot p]$ with R ^TE being the reflection coefficient of the substrate under TE polarization and $\bar{G}$ being the Green’s function accounting for the effect of the substrate. One can easily obtain

p = \frac{α (1 + R^{TE} e^{i 2 k_{0} d})}{1 - μ_{0} ω^{2} α G_{x x}} E^{inc} = α_{eff}^{} E^{inc},

where G_xx is the element of

\bar{G}

and we define

α_{eff}^{}

as the effective polarizability (the polarizablity of the particle taking into account of the effect of substrate). Equation (1) indicates that the effective polarizability depends on the distance d and this is because the particle alters the field acting on it through the substrate. If the condition

| 1 - μ_{0} ω^{2} α G_{x x} | = 0

is fulfilled,

α_{eff}

and p will diverge. In this case the optical force acting on the particle will be strongly enhanced. Such a conditon cannot be fulfilled for a normal configuration of particle-over-substrate, say a small particle sitting on a semi-infinite metal substrate, due to the small value of

α

and

G_{x x}

. It can be reached if the substrate is made of materials with

ε \to - 1, μ \to - 1

[15, 16 ]. However, such a material can only be realized by using artificial metamaterials [17, 18 ]. Here we show that the sandwich structure in Fig. 1(a) provides an alternate way to obtain a large value of G_xx.

As G_xx accounts for the interaction between the particle and the substrate, it can have a large value if the scattered channels (Fourier components of the scattered field) of the particle hit the poles of the reflection coefficient so that the reflection becomes very strong. These poles correspond to the coupled surface plasmon modes sustained at the dielectric-metal interfaces. As the scattered channels cover all the range of k vectors, in order to hit as many poles as possible, the band of the plasmon mode has to be very flat at some frequency. This can happen when the layer thicknesses t ₁ and t ₂ are properly chosen. Figure 1(b) shows the TM band diagram of the substrate with $t_{1} = 0.1 λ_{p}$ , $t_{2} = 0.218 λ_{p}$ ( $λ_{p} = 2 π / k_{p} = 2 π c / ω_{p}$ ) and $γ = 0$ . A flat band is found at about $ω = 0.25 ω_{p}$ . At this frequency the Green’s function has a resonance behavior as shown in Figs. 1(c) and 1(d), where we show the real and imaginary parts of G_xx as a function of $ω$ and d. We note that the real part [Fig. 1(c)] undergoes dramatic changes from very large positive value to very large negative value, and at the same time the imaginary part is relatively small, hence there is a chance that $| 1 - μ_{0} ω^{2} α G_{x x} | \to 0$ . The maximum enhancement is bounded by the relative ratio of $Re [α G_{x x}]$ and $Im [α G_{x x}]$ .

To get the optical force, the induced dipole moment is first calculated using Eq. (1), where $G_{x x} = i / (8 π) \int_{0}^{\infty} k_{ρ} / k_{0 z} (R^{TE} - k_{0 z}^{2} / k_{0}^{2} R^{TM}) e^{i 2 k_{0 z} d} d k_{ρ}$ with $k_{0 z}^{2} + k_{ρ}^{2} = k_{0}^{2}$ [19]. The spectral integral is numerically evaluated using Wolfram Mathematica 10 [20], assuming the truncation $k_{ρ} \in [0, 1000 k_{0}]$ . Then the longitudinal field components at z > 0 due to the dipole radiation can be calculated as [21]

\begin{array}{l} E_{z}^{p} (ρ, ϕ, z) = \frac{p_{x} \cos ϕ}{4 π ε_{0}} \int_{0}^{\infty} k_{ρ}^{2} J_{1} (k_{ρ} ρ) [sign (z - d) e^{i k_{0, z} | z - d |} - R^{TM} e^{i k_{0, z} (z + d)}] d k_{ρ}, \\ H_{z}^{p} (ρ, ϕ, z) = \frac{ω p_{x} \sin ϕ}{4 π} \int_{0}^{\infty} \frac{k_{ρ}^{2}}{k_{0, z}} J_{1} (k_{ρ} ρ) [e^{i k_{0, z} | z - d |} + R^{TE} e^{i k_{0, z} (z + d)}] d k_{ρ}, \end{array}

where

(ρ, ϕ, z)

is the cylindrical coordinates; J ₁(x) is the cylindrical Bessel function;

sign (x) = 1

if

x \geq 0

, else

sign (x) = - 1

. The transverse field components are:

{\tilde{E}}_{t}^{p} (k_{ρ}, r) = \frac{1}{k_{ρ}^{2}} (\nabla_{t} \frac{\partial {\tilde{E}}_{z}^{p}}{\partial z} - i μ_{0} ω \hat{z} \times \nabla_{t} {\tilde{H}}_{z}^{p}), {\tilde{H}}_{t}^{p} (k_{ρ}, r) = \frac{1}{k_{ρ}^{2}} (\nabla_{t} \frac{\partial {\tilde{H}}_{z}^{p}}{\partial z} + i ε_{0} ω \hat{z} \times \nabla_{t} {\tilde{E}}_{z}^{p}) .

Here

\nabla_{t} = \nabla - \hat{z} \partial / \partial z

and

{\tilde{E}}_{z}^{p}, {\tilde{H}}_{z}^{p}

denote the integrand of each spectral component in Eq. (2). Integrating

{\tilde{E}}_{t}^{p}, {\tilde{H}}_{t}^{p}

over

k_{ρ}

gives the transverse field components

E_{t}^{p}, H_{t}^{p}

. The time-averaged optical force exerted on the particle is:

F = 1 / 2 {\sum_{i} Re [α E_{i} \partial_{z} E_{i}^{*}]}_{z = d}

, where

E_{i} = E_{i}^{p} + E_{i}^{inc} (1 + R^{TE} e^{i 2 k_{0} z})

is the total electric field, “*” denotes the complex conjugate and

\partial_{z}

denotes the derivative with respect to z Cartesian component. The subscript “z = d” indicates the derivative is calculated before assigning the value of z. Note that the bare polarizability

α

is used here and

E_{i}^{p} = E_{i}^{p} (d, z)

characterizes how the particle affects the source field in the presence of the substrate. For comparison we also calculated the force when the sandwich substrate is replaced by a semi-infinite metal of the same material. The results are also compared with that of the full-wave Maxwell tensor method [22] and we found excellent agreement between the two (shown in Sec.3).

3. Optical force enhancement due to particle-substrate resonance

3.1. Particle with a negative dipole polarizability

As the spherical particle is characterized by an electric polarizability $α$ , we will show that the force enhancement can be achieved for both cases of $α < 0$ and $α > 0$ . For the former we consider a metal particle of radius r = 15 nm (subwavelength in the considered frequency region) with relative permittivity $ε_{d} = - 2$ . The particle can have a negative polarizability in this case due to the resonant excitation of the dipole surface plasmon (the Fröhlich condition [8]). For the metal layers of the substrate we set $ω_{p} = 1.375 \times 10^{16} rad/s$ (corresponding to silver) and we consider first the lossless case with $γ = 0$ . Figure 2(a) shows the force as a function of distance d and frequency $ω$ , where a resonance is noted (the red region). The resonance force is positive (repulsive), meaning that it can block the particle from approaching the substrate. This is due to the fact that the strong particle-substrate coupling results in a field gradient along –z direction and $α < 0$ , hence the gradient force is along + z direction. For comparison, Fig. 2(b) shows the optical force acting on the same particle when the sandwich structure is replaced with a semi-infinite metal of the same material property. In this case, the force is mainly induced by the standing wave formed by the reflection of the incident wave, which has a field gradient along + z direction, hence the force is negative (attractive) as shown in Fig. 2(b). To show the force enhancement, in Fig. 2(c) we compare the force between the sandwich structure system (denoted as F) and the semi-infinite metal system (denoted as F _{semi-infinite}) for the case of d = 40 nm. The axis labels on the left-hand side show the force normalized to the normal photon pressure F ₀ acting on the same particle without the substrate. We see that the resonance force is about 3 orders of magnitude stronger than the normal photon pressure. The axis labels on the right-hand side give the ratio of F/|F _{semi-infinite}| and the maximum value is about 100. This demonstrates that the considered system can give strong enhancement of the coupling force between the particle and the substrate. To verify the results of the analytical method, we also calculated the force using full-wave Maxwell tensor method [22, 23 ] and the results are shown as circles in Fig. 2(c), where the agreement is very good. Figure 2(d) shows the magnitude of the induced dipole moment in the two cases, where about one order of magnitude of enhancement is observed. The inserted panel in Fig. 2 (d) shows the resonant electric field amplitude on the yoz-plane. We note the strong coupling field between the particle and the substrate (the field values here correspond to the plane wave with amplitude E ₀ = 1 V/m).

Fig. 2 Optical force on a spherical particle with $ε_{d} = - 2$ as a function of distance d and frequency $ω$ for the sandwich structure case [(a)] and for the semi-infinite metal case [(b)]. Comparison of the force [(c)] and the dipole moment [(d)] for the case of d = 40 nm. The circles in (c) denote the force computed using the full-wave Maxwell tensor method. The inserted panel in (d) shows the resonance amplitude of the total electric field on the yoz-plane.

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3.2. Particle with a positive dipole polarizability

We now consider the particle is made of dielectric material with $ε_{d} = 12.96$ so that $α > 0$ . Figure 3(a) shows the optical force acting on the particle as a function of d and $ω$ , where the resonance force now becomes attractive. This is due to the fact that $α > 0$ and the strong particle-substrate coupling field has a gradient along –z direction. Figure 3(b) shows the force in the semi-infinite metal system and it is repulsive because the field gradient produced by the standing wave is along + z direction. Figure 3(c) and (d) shows the comparisons of the force and the dipole moment. We see that the enhancement of F/|F _{semi-infinite}| can reach 50 at resonance and the dipole moment is also enhanced by one order of magnitude. We note that although the quality factor in this case is much larger than that of the $α < 0$ case, its enhancement is relatively smaller. The reason is that the value of $| α |$ for the dielectric particle is smaller than that of the metal particle, therefore the residue of $| 1 - μ_{0} ω^{2} α G_{x x} |$ is larger and the effective polarizability is smaller.

Fig. 3 Optical force on a dielectric particle with permittivity $ε_{d} = 12.96$ as a function of distance d and frequency for the sandwich structure case [(a)] and for the semi-infinite metallic substrate [(b)]. Comparisons of the force [(c)] and the dipole moment [(d)] in the case of d = 40 nm.

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3.3. Effect of loss

We have shown in Figs. 2 and 3 that a dramatic force enhancement can be achieved for both $α < 0$ and $α > 0$ cases when the metal-dielectric-metal substrate is lossless. In general the material loss of the substrate could compromise such kind of resonance. Here we take the $α < 0$ case as an example and study the effect of material loss by setting $γ = 0.002 ω_{p}$ [24]. We keep the other parameters unchanged and re-did the calculation. The enhancements of the optical force and the induced dipole moment are show in Figs. 4(a) and 4(b) , respectively. Figure 4(a) shows that when loss is taken into account, the enhancement of the optical force becomes smaller but it still can reach about one order of magnitude, compared to the non-structured semi-infinite metal case. The induced dipole moment is also several times larger [Fig. 4(b)].

Fig. 4 Optical force (a) and the dipole moment (b) for the case that the substrate is made of lossy silver. The other parameters are the same as those in Fig. 2(c).

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4. Conclusions

In short, we showed that a simple metal-dielectric-metal configuration can induce strong optical force acting on a nano-sized particle through particle-substrate resonance. The strong optical force is attributed to the divergence behavior of the effective dipole polarizability enabled by the flat-band property of the metal-dielectric-metal slab modes. Such a flat band results in the resonance coupling between the particle and the substrate, and hence induces an optical force which is orders of magnitude stronger than that of the non-structured system. We note that the flat-band property is also related to the recently discovered super-scattering of light by sub-wavelength structures [24]. The physics here is not limited to spherical particles. Our study may find applications in the optical force manipulations of small particles on a substrate. The strong field enhancement inside the sandwich structure may be useful in light-harvest related applications. In addition, the enhancement of the dipole moment can effectively make small particles much brighter and hence may also be applied in detecting small particles in similar systems.

Acknowledgments

This work is supported by Hong Kong RGC grant M-HKUST601/12. We thank Dr. M. Xiao for useful discussions.

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Strong optical force acting on a dipolar particle over a multilayer substrate

Abstract

1. Introduction

2. Analytical Green’s function method

3. Optical force enhancement due to particle-substrate resonance

3.1. Particle with a negative dipole polarizability

3.2. Particle with a positive dipole polarizability

3.3. Effect of loss

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (4)

Equations (3)

Optics Express