Numerical investigation of passive optical sorting of plasmon nanoparticles

M. Ploschner; M. Mazilu; T. Čižmár; K. Dholakia

doi:10.1364/OE.19.013922

1. Introduction

The preparation of highly monodispersed colloidal solutions of plasmon nanoparticles is crucial for any application where a narrow size distribution is required particularly to exploit their plasmonic properties. This includes biomolecular sensors [1], localised heaters [2] and photo-thermal imaging [3]. As most of these applications require colloids prepared in sterile way, the application of optical sorting offers an ideal solution not only because of the non-contact nature of sorting but also because of the exceptional sensitivity on particle size, shape and refractive index. Two main optical sorting approaches exist – namely active and passive. Active sorting techniques use fluorescent signals [4], optical switches [5, 6] and real-time computation [7, 8] for particle recognition that then subsequently acts as a trigger for another part of the system where optical (or other) forces are used to separate particles of different properties into separate streams. However, the need to use a trigger may add complexity and be impractical in many instances. For this reason, the last decade has seen the emergence of passive sorting methods which rely entirely on the different physical response of various particles to an extended optical field, commonly referred to as an optical potential energy landscape. Such passive sorting offers exceptional size and refractive index sensitivity [9, 10] and has been demonstrated in a number of geometries both with and without the presence of microfluidic flow. Sorting of particles in closed chambers with static fluid has been realised both by means of moving interference pattern [11, 12] or Bessel modes [13]. However, in the majority of sorting applications, laminar fluid flow perpendicular to the optical forces is employed and the separation of particles is either realised solely by scattering force differences [14] in aperiodic optical patterns or by means of structured light fields [10, 15–19] creating periodical optical potential energy landscapes. In all these methods, the determination of the precise form of the applied optimal field, such that the maximum sorting sensitivity on size, shape and refractive index is achieved, remains an open question. To date it is also to be noted that passive optical sorting has been mainly considered solely with respect to micron-sized dielectric particles and cellular media. It is intriguing to consider how passive optical sorting may be extended to the domain of plasmon nanoparticles.

Here, we study passive optical sorting of plasmon nanoparticles and present a general two step approach that can be used to design the optimal illumination for sorting plasmon nanoparticles. In the first step, we find the optimal wavelength of illumination such that the separation is achieved with minimum temperature increase in the system. This is an important consideration in plasmonic systems as the excessive heat increases diffusion and convective effects, which is counter-productive in any sorting application. In the second step, the optimisation of illumination shape is realised using our force optical eigenmode (FOEi) [20] method, which can be readily applied to determine the optimal laser illumination field for sorting. The approach extends previous studies of forces on plasmon nanoparticles [21, 22] by considering the shape of the field, its wavelength and heating in the system at the same time. This paper is divided into three sections. In the first section, we present the FOEi method and derive formulae leading to optimised beam shape for sorting. The second part focuses on exploiting the plasmon resonances in the system and finding the optimal wavelength for sorting. The third part uses the optimal wavelength as input and extends the FOEi method to optimise the force difference over the whole sorting region.

2. Description of FOEi method

Our aim is to optimise the incident electromagnetic field E _inc in a way that maximises the exerted force F ^u = F · u in a specified direction u (Fig. 1). Since our incident field of angular frequency ω can be decomposed into a sum of μ monochromatic plane waves (e^iωt), we can write (using summation over repeating indices)

E_{i n c} = a^{μ} E_{i n c}^{μ},

where a^μ are the complex expansion coefficients and

E_{i n c}^{μ}

are the incident plane waves. The a^μ coefficients modulate both phase and amplitude of each and single plane wave independently. The scattered field generated upon interaction with the particle has the same expansion coefficients, so that we can write

E_{s c a} = a^{μ} E_{s c a}^{μ}

. The final field E, which is a sum of incident and scattered field, can then be written as

E = a^{μ} (E_{i n c}^{μ} + E_{s c a}^{μ}) = a^{μ} E^{μ},

where E ^μ is a solution of the scattering problem for an incident field given by plane wave

E_{i n c}^{μ}

.

Fig. 1 We look for an amplitude and phase a of a given set of plane waves such that the force F ^(1,0) is maximised.

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The optical-cycle averaged electromagnetic force in the direction u is given by

F^{u} = 〈 F_{i} 〉 u_{i} = \oint_{C} 〈 σ_{i j} 〉 n_{j} u_{i} d s,

where n_j is outward unit normal to an element ds of the curve C enclosing the particle for which we optimise the force and 〈·〉 is optical-cycle average. The Maxwell stress tensor, σ_ij, can be written for the final field E as [23]

\begin{array}{l} 〈 σ_{i j} 〉 & = & \frac{1}{4} [ɛ_{0} ɛ_{m} E_{i}^{*} E_{j} + μ_{0} μ_{m} H_{i}^{*} H_{j} + ɛ_{0} ɛ_{m} E_{i} E_{j}^{*} + μ_{0} μ_{m} H_{i} H_{j}^{*} - \\ - δ_{i j} (ɛ_{0} ɛ_{m} E_{k}^{*} E_{k} + μ_{0} μ_{m} H_{k}^{*} H_{k})] . \end{array}

Using Eq. (2), we can rewrite Eq. (4) as

\begin{array}{l} 〈 σ_{i j} 〉 & = & \frac{1}{4} [ɛ_{0} ɛ_{m} {(a^{μ} E^{μ})}_{i}^{*} {(a^{ν} E^{ν})}_{j} + μ_{0} μ_{m} {(a^{μ} H^{μ})}_{i}^{*} {(a^{ν} H^{ν})}_{j} + \\ + ɛ_{0} ɛ_{m} {(a^{ν} E^{ν})}_{i} {(a^{μ} E^{μ})}_{j}^{*} + μ_{0} μ_{m} {(a^{ν} H^{μ})}_{i} {(a^{μ} H^{μ})}_{j}^{*} - \\ - δ_{i j} (ɛ_{0} ɛ_{m} {(a^{μ} E^{μ})}_{k}^{*} {(a^{ν} E^{ν})}_{k} + μ_{0} μ_{m} {(a^{μ} H^{μ})}_{k}^{*} {(a^{ν} H^{ν})}_{k})] \end{array}

and after rearrangement of the expansion coefficients a^μ we obtain

\begin{array}{l} 〈 σ_{i j} 〉 & = & {(a^{μ})}^{*} (\frac{1}{4} [ɛ_{0} ɛ_{m} {(E^{μ})}_{i}^{*} E_{j}^{ν} + μ_{0} μ_{m} {(H^{μ})}_{i}^{*} H_{j}^{ν} + ɛ_{0} ɛ_{m} E_{i}^{ν} {(E^{μ})}_{j}^{*} + \\ + μ_{0} μ_{m} H_{i}^{ν} {(H^{μ})}_{j}^{*} - δ_{i j} (ɛ_{0} ɛ_{m} {(E^{μ})}_{k}^{*} E_{k}^{ν} + μ_{0} μ_{m} {(H^{μ})}_{k}^{*} H_{k}^{ν}]) a^{ν} . \end{array}

Substituting Eq. (6) into Eq. (3) gives

\begin{array}{l} F^{u} & = & {(a^{μ})}^{*} [\oint_{C} (\frac{1}{4} [ɛ_{0} ɛ_{m} {(E^{μ})}_{i}^{*} E_{j}^{ν} + μ_{0} μ_{m} {(H^{μ})}_{i}^{*} H_{j}^{ν} + ɛ_{0} ɛ_{m} E_{i}^{ν} {(E^{μ})}_{j}^{*} + \\ + & μ_{0} μ_{m} H_{i}^{ν} {(H^{μ})}_{j}^{*} - δ_{i j} (ɛ_{0} ɛ_{m} {(E^{μ})}_{k}^{*} E_{k}^{ν} + μ_{0} μ_{m} {(H^{μ})}_{k}^{*} H_{k}^{ν})] n_{j} u_{i} d s] a^{ν} \end{array}

or more simply in matrix form

F^{u} = a^{†} Ma,

where the matrix coefficients M^μν are given by the line integrals in Eq. (7) and a is the vector form of a^μ. We remark that the matrix is Hermitian (M = M ^†) and thus its eigenvalues are real. This means that the force F ^u is in a symmetric sesquilinear form, which is just an extension of quadratic form to complex numbers. Any symmetric sesquilinear form can be visualised as an ellipsoid with the length of principal axes equal to the eigenvalues λ_n of the matrix M. This has far reaching implications for our optimisation process since the surface of the ellipsoid generated by the symmetric sesquilinear form is extremized at the end points of principal axes. This means that finding the eigenvalues λ_n of the matrix M and selecting the largest one from the set extremizes our problem. The eigenvector (force optical eigenmode)

a_{n}^{m a x}

corresponding to maximum eigenvalue

λ_{n}^{m a x}

(given by

{Ma}_{n}^{m a x} = λ_{n}^{m a x} a_{n}^{m a x}

), then provides the necessary information about amplitude and phase of incident plane waves

E_{i n c}^{μ}

in Eq. (1) so that the force is maximised. Experimentally, the optimised

a_{n}^{m a x}

can be created using spatial light modulator (offering control of both phase and amplitude) in the system placed in conjugate plane [20] with respect to the back-focal plane of a microscope objective.

We remark that the above described method optimises the force on one type of particle at a single point only. However, the method can be easily extended to provide the optimised illumination for the force difference over the whole sorting region for two types of nanoparticles. This is discussed later in the paper.

Numerical considerations: We use COMSOL Multiphysics v4.1 RF module in scattering formulation to calculate the total field solutions E ^μ for corresponding incident plane waves $E_{i n c}^{μ}$ . We first find the solutions E ^μ for particle p ₁. We subsequently use the solutions E ^μ to find the elements $M_{1}^{μ ν}$ of matrix M ₁ and determine the eigenvalues and corresponding eigenvectors. The principal eigenvector gives the optimum force for particle p ₁ in a single position. The same procedure is repeated for a second type of particle, p ₂, delivering matrix M ₂. The matrices M ₁ and M ₂ then encode all the information about interactions of the incident fields with the particles. Finding the matrix elements M^μν of matrix M is computationally very intensive as combinations of N(N + 1)/2 solutions need to be constructed and integrated over a sphere boundary. Here N denotes number of plane waves in the angular spectrum representation and thus the number of pixels on spatial light modulator. As the azimuthal discretization of angular spectrum representation increases the number of combinations in 3D significantly, we have restricted the simulations to 2D to illustrate the method.

We can obtain educated estimates of 3D values from 2D values by extruding the 2D circle of radius r by d such that it creates a cylinder with a volume equal to the volume of the sphere with the same radius. The extrusion factor d is given by

π r^{2} d = \frac{4}{3} π r^{3} \to d = \frac{4}{3} r .

The validity of the COMSOL model was tested in 3D by comparing the optical forces and scattering and absorption efficiencies with Mie theory. The difference between the COMSOL model and the Mie model was less than 2 percent. In 2D, optical forces and scattering and absorption efficiencies were calculated using several independent methods to ensure the model validity.

3. Plasmon resonances in the system

We choose as our testing system gold nanoparticles [24] of radius r ₁ = 50nm and r ₂ = 40nm. We consider a substrate of glass with refractive index n_g = 1.5. The particles are assumed to be dispersed in water with n_w = 1.33 (Fig. 1).

Plasmonic resonances offer exceptional sensitivity on size. However, for nanoparticles of very similar sizes, the force difference generated solely by the plasmon resonance in the system is still rather small. In case of dielectrics, increasing the intensity offers simple solution in such a situation, however plasmonic resonances are associated with non-negligible heat generation and as such increasing the intensity may produce increased diffusion rates and convective currents, which will interfere with sorting efforts. It is thus beneficial to find a wavelength for which the sorting effects are maximised and heating is minimised. Further, multiple laser wavelengths in a counter-propagating geometry with carefully adjusted powers can be employed to exploit the small plasmon resonance differences of nanoparticles. Our method can solve for optimum illumination for each and single wavelength separately, however, the use of multiple laser wavelengths is expensive and increases complexity of the system. It is therefore of interest to optimize the force differences using a single laser wavelength.

We choose the p-polarisation for the incident plane waves as the kind of plasmon resonance supported by the sphere appears in 2D for p-polarisation. The s-polarisation would only induce movement of electrons along the infinite cylinder. Figure 2(a) shows the scattering Q_sca and absorption Q_abs efficiencies for r ₁ = 50nm gold nanoparticle. Note that the 3D efficiencies calculated from Mie theory and the corresponding 2D efficiencies (transformed to 3D) for p-polarisation follow very similar pattern, which differs only in amplitude and a slight blue shift of 2D resonance peaks with respect to 3D resonances. Figure 2(b) shows the 3D forces along the substrate acting on the gold nanoparticles calculated for plane-wave incident at near critical angle of θ = 64°. The slight shift in resonances due to the different sizes of nanoparticles creates a force difference improvement with a peak around 550nm (black curve in Fig. 2(b)).

Fig. 2 a) Scattering Q_sca and absorption Q_abs efficiencies calculated using Mie theory for 3D nanoparticle of r ₁ = 50nm and the corresponding 2D values converted to 3D equivalents. Nanoparticle is in water with n_w = 1.33; b) Forces and their respective difference ΔF acting on r ₁ = 50nm and r ₂ = 40nm gold nanoparticles. Forces are parallel to the substrate plane. The illumination is a plane-wave at near critical angle of θ = 64° with power density corresponding to 1mW/μm²; c) Speed difference Δv and temperature increase ΔT for the same illumination as in b); d) Speed difference normalised with respect to the temperature increase in the system.

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If we do not take into account the proximity of the spheres to the surface (Faxen’s correction [25]) and assume we are in a low Reynolds number regime, then the drag force is given by the Stokes equation. Neglecting inertial effects we can equal optical and drag force and obtain the expression for the particle speed in 3D as

v_{3 D} = \frac{2 F_{2 D}}{9 π η},

where the force F _2D was calculated in 2D using Eq. (8) and the dynamic viscosity of water at T = 20°C is η = 1.002 × 10⁻³ Pa · s. Figure 2(c) shows the speed difference generated by force differences in Fig. 2(b) and the average temperature increase as the particle enters the sorting field. The estimate of average temperature increase was calculated using [26] (neglecting particle movement and proximity of glass surface)

Δ T = \frac{1}{4 π} \frac{Q_{1} + Q_{2}}{κ_{0} \cdot (\frac{r_{1} + r_{2}}{2})},

where Q ₁ and Q ₂ are 3D powers of heat generation in nanoparticles and κ ₀ = 0.6W · m⁻¹ · K⁻¹ is the thermal conductivity of water. This temperature increase is reached very quickly as the particle enters the sorting field. Results in Fig. 2(c) and Fig. 2(d) suggest, that heating at wavelengths above 700nm is quite low and the sorting speed remains high. For this reason we set the vacuum wavelength for our method to λ ₀ = 700nm. Please note that a slight (30nm) blue shift in the resonances of 2D case does not have significant impact on the choice of this wavelength for the 3D scenario.

4. Optimising the force difference in region of interest (ROI)

Using the optimum wavelength from previous section we can proceed with optimisation of the beam shape. We introduce a 10nm separation between the lowest point of particle and interface, which closely mimics a typical experimental situation. The set of N incident plane waves defined by k _θ vectors (θ = 〈−70°,...,70°〉 with a step of 2°) is used for discretization (see Fig. 1). The limits of θ correspond to experimental limitation for NA = 1.4 oil immersion objective. The goal is to optimise the force difference along u = (1, 0).

Using matrices M ₁ and M ₂, the equation for force difference in a single point for our choice of particles is

Δ F = F_{1}^{u} - F_{2}^{u} = a^{†} (M_{1}^{u} - M_{2}^{u}) a = a^{†} D_{12}^{u} a .

Due to the symmetry of the system two optimum solutions exists - one optimising the force difference in +x direction and the second for −x direction. In subsequent discussions we always choose the solution optimising the force in +x direction. The field locally optimising the force difference for gold nanoparticles of our choice is on Fig. 3. Notice that the field creates a very strong field gradient in the +x direction around point x = 0, where we want to maximise the force difference for our testing particles. Also notice that the back focal plane pattern corresponding to this field has significant contributions from plane waves propagating in the opposite −x direction. Although this might seem surprising, we need to realise that the final goal of our method is to interfere the plane waves in such a way to create the strongest gradient in +x direction. Apparently the counter-propagating waves increase the number of degrees of freedom for efficient interference leading to strong gradient and are thus utilised automatically by the FOEi method. It also make sense that the increased intensity at the back focal plane appears for near critical angle plane waves as those plane waves contribute the most to the intensity near the glass/water interface. Note that the phase at the back focal plane is also significantly altered to maximise the force difference.

Fig. 3 Field optimising the force difference ΔF for gold nanoparticles of radius r ₁ = 50nm and r ₂ = 40nm in a single point x = 0 with corresponding amplitude |a^μ| and phase arg(a^μ) for each plane wave from angular spectrum. The |a^μ| and arg(a^μ) correspond to the pattern at the back focal plane of the objective. This illumination of the back focal plane forms a very strong field gradient in +x direction in the focal plane, which maximises the force difference for our testing particles. Note that scattering from the particles is not included.

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So far our method optimises the force difference locally. To expand this approach to a larger region we need to optimise ΔF over a certain range, in our case line segment defined by x = 〈−l,l〉. Displacing the particle in x-direction causes the particle to experience different relative phases between the fields E ^μ. Since we use combination of solutions to calculate matrix M this relative phase can be taken into account using

M^{μ ν} (x) = e^{i k_{w} sin (θ_{t}^{μ}) x} [M^{μ ν}] e^{- i k_{w} sin (θ_{t}^{ν}) x}

where k_w = (2π/λ ₀)n_w and

θ_{t}^{μ}

is the angle of the plane wave after the glass/water interface for each incident plane wave

E_{i n c}^{μ}

. M^μν is the matrix calculated for particle at position x = 0. Using the translation relations for matrix M^μν we obtain

\begin{array}{l} Δ F (l) & = & {(a^{μ})}^{*} [\frac{1}{2 l} \int_{- l}^{l} e^{i k_{w} x (sin (θ_{t}^{μ}) - sin (θ_{t}^{ν}))} d x] {(M_{1}^{u} - M_{2}^{u})}^{μ ν} a^{ν} \\ = & {(a^{μ})}^{*} [sinc (k_{w} l (sin (θ_{t}^{μ}) - sin (θ_{t}^{ν})) {(M_{1}^{u} - M_{2}^{u})}^{μ ν}] a^{ν} \\ = & a^{†} R_{12}^{u} (l) a . \end{array}

The illumination optimisation is then performed on the modified matrix

R_{12}^{u} (l)

, which has the same input matrices

M_{1}^{u}

and

M_{2}^{u}

for all values of l. Finding the optimum illumination for a ROI of any size is thus very efficient.

Optimal illumination for ROI sizes of l = 500nm (Fig. 4) and l = 5μm (Fig. 5) differs significantly from the single point optimised problem (Fig. 3). The optimised field corresponds in its bulk to the focusing of light into ROI. The solution is quite close to the Gaussian beam send to the edge of the back focal plane of the objective. However, the phase for plane waves above critical angle is significantly modulated and the intensity profile is not entirely Gaussian. The width of the beam at the back-focal plane optimising the l = 5μm situation is also noticeble smaller than for the case of l = 500nm. This is a direct consequence of Eq. (14). As we increase l, the off-diagonal terms in matrix $R_{12}^{u} (l)$ become less important due to the behaviour of the sinc function as l increases. In the limit l → ∞ only the diagonal terms remain. This means that the eigenmodes (eigenvectors of $R_{12}^{u} (l)$ ) in this case correspond to single plane waves as defined in our initial set. The eigenmode (plane wave) with maximum eigenvalue optimises our problem for an infinitely large ROI. The solution found by the FOEi method for l = 100mm (Fig. 6) is the plane wave near the critical angle. As the phase of a^μ for zero amplitude |a^μ| is not well defined, it is not displayed in the graph. The result validates that the FOEi method is working correctly, as the near critical angle plane wave provides the highest intensity and force difference at the interface for infinite system.

Fig. 4 Field optimising the force difference ΔF for l = 500nm. The phase of a_μ is plotted in the region where it is well-defined. Notice that the field is focused into the ROI. The left edge of the back focal plane contributes the most to the optimised field in the focal plane. The phase at the back focal plane is slightly modulated as well.

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Fig. 5 Field optimising the force difference ΔF for l = 5μm. The shape of the beam at the back-focal plane of the objective has a narrow distribution of amplitude in the proximity of critical angle.

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Fig. 6 Field optimising the force difference ΔF for l → 100mm. As the phase of a^μ for zero amplitude |a^μ | is not well defined, it is not displayed in the graph. The optimum angle plane wave is 64°, which is close to critical angle for given interface.

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The force difference is significantly increased in the cases of small ROI sizes compared with plane wave illuminated system that optimises the sorting for infinite sorting space (Fig. 7(a)). The bulk of this improvement is due to increased intensity of light in ROI, but the periodic pattern of forces indicates a more complex response of the system. This periodic pattern is not related to the discretization of k-space, where we expect periodicity to appear around 14μm.

Fig. 7 (a) Blue and green points show forces acting on individual particles in optimised field for l = 500nm. Red points show the force difference. The coloured lines show the same but for plane wave illumination optimising the infinite system. (b) The ratio ΔF/ΔF_pw as a function of ROI size. The dip around l = 14μm is due to k-space discretization of angular spectrum representation. The gain in ΔF is significant in the experimentally interesting region.

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Naturally, we do not wish the sorting ROI to become too small as the experimental realisation would become increasingly complicated. It is interesting to look at the dependence of ΔF on the size of ROI. To show the improvement compared to infinite system, we normalise ΔF by ΔF_pw, where ΔF_pw is the force difference for optimised infinite system (pw stands for plane wave). The result (Fig. 7(b)) indicates that the gain is significant for a wide range of experimentally interesting ROI sizes. The dip around l = 14μm is present due to discretization of k-space described above. The increase in ratio ΔF/ΔF_pw for l > 14μm is then equivalent to the formation of second beam focus in ROI due to onset of periodicity.

Discussion of results: The FOEi method is capable of finding the optimal beam shape for illumination such that the force difference is maximised over the whole sorting region. The computationally intensive calculation of matrices M ₁ and M ₂ is compensated by the fact that the same matrices can be used for finding optimal illumination for any size of sorting region. We note that even though the solution does not optimise the vertical force pointing towards the substrate, we found that this is the case for all our solutions. However, the sign of this force is wavelength and particle size dependent and as such the attractive vertical force is not a general feature of the method. Further, the vertical force is not constant in the sorting region, which might introduce some modulation of force difference due to the Faxen correction. To resolve this one may minimise the vertical force and use an auxiliary beam with constant vertical force over the whole sorting region. This would restrict the diffusion of particles in vertical direction in more controlled way. It is possible to modify the method to simultaneously optimise for several parameters of the system. In our case, the full optimised solution for sorting applications of plasmon nanoparticles would involve simultaneous maximalisation of force difference in ROI, minimisation of vertical force, and minimisation of heating. Such a problem reduces to finding matrices (operators) for all parameters of interest and choosing the eigenmodes optimising for such a set of parameters. It is very interesting, for plasmonic sorting in general, to find the beam shape of the field maximising the force difference and minimising the heating. This would also clearly identify the contribution of focusing to the overall improvement of force difference. This is a focus of our ongoing research.

5. Conclusion

We successfully optimised the illumination for sorting gold nanoparticles using our two step approach. Firstly, we found the optimal wavelength maximising the nanoparticle separation and minimising the temperature increase in the system. This is an important consideration in plasmonic systems as the excessive heat increases diffusion and convective effects. Secondly, we found the optimum beam shape of the illumination field for sorting using the method of force optical eigenmodes (FOEi). The applicability of the method was numerically demonstrated for the special case of sorting gold nanoparticles of different size. We plan to extend our approach and perform simultaneous optimisation of several parameters of interest for sorting applications, e.g., minimised heating, maximised force difference along substrate and minimised vertical force. This involves finding operators for all of parameters of interest and choosing the eigenmodes optimising them. This will be subject of further work along with the efficient extension of the FOEi method to 3D case.

Acknowledgments

We thank the UK Engineering and Physical Sciences Research Council for funding, KD is a Royal Society-Wolfson Merit Award Holder.

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Numerical investigation of passive optical sorting of plasmon nanoparticles

Abstract

1. Introduction

2. Description of FOEi method

3. Plasmon resonances in the system

4. Optimising the force difference in region of interest (ROI)

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (7)

Equations (14)

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