Theoretical and experimental study of the surface redox reaction involving interactions between the adsorbed particles under conditions of square-wave voltammetry

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Plasmonics ISSN 1557-1955Volume 8Number 2 Plasmonics (2013) 8:637-643DOI 10.1007/s11468-012-9449-y

Theoretical and Experimental Study ofSurface Plasmon Radiation Force onMicrometer-Sized Spheres

Xiaodong Wang, Kai Xiao, ChangjunMin, Qinqian Zou, Yi Hua & X.-C. Yuan

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Theoretical and Experimental Study of Surface PlasmonRadiation Force on Micrometer-Sized Spheres

Xiaodong Wang & Kai Xiao & Changjun Min &

Qinqian Zou & Yi Hua & X.-C. Yuan

Received: 4 February 2012 /Accepted: 6 September 2012 /Published online: 29 September 2012# Springer Science+Business Media, LLC 2012

Abstract We report theoretical and experimental study ofsurface plasmon (SP) radiation force on micrometer-sizedpolystyrene latex spheres with different radii. Two theoreticalmethods are introduced to numerically investigate the SPsphere system: the arbitrary beam theory for electromagneticfield distribution and the generalized Lorenz–Mie scatteringtheory for radiation force calculation. Transport velocity of thespheres in water is tested experimentally and compared withnumerical results. Our work also shows effective couplingfrom SP field to micrometer-sized spheres, with high consis-tency between experiment and calculation, which has signif-icant potential in future applications.

Keywords Surface plasmon radiation force . Arbitrary beamtheory . Generalized Lorenz–Mie scattering theory .

Biological cells sorting technology

Introduction

Single-beam optical trapping was discovered by Ashkin in1986 [1] with a gradient force realized by highly focused laserbeam. In 1992, Kawata and Sugiura found that evanescentwave is able to drive particles along the boundary betweentwo different dielectric materials [2]. Compared with the gra-dient force, radiation force generated by evanescent wave canbe used for both guiding and trapping over an extended area

where thousands of micro-particles can be organized on a flatsurface [3]. This kind of manipulation was referred to aslensless optical trapping.

It is well-known that surface plasmon resonance (SPR) canbe used to enhance evanescent waves at an interface betweendielectric material and metallic film with a p-polarized inci-dent light [4]. The incident angle must be adjusted for optimalcoupling between surface plasmon (SP) and the incident light.Amplitude of evanescent wave can be enhanced by 2–3 ordersof magnitude depending on the refractive indices of materials,thickness of metal film, and other parameters [5]. In ourexperiment, it is around 20 times enhancement.

Several theoretical methods have been developed for cal-culation about particles' transport in air or liquid environment.Almass and Brevik used the arbitrary beam theory (ABT) todeduce the radiation pressure on small particles exerted by apure evanescent wave, assuming that multiple reflectionsbetween the sphere and interface are negligible [6]. Lesterand Nieto-Vesperinas developed the research by extinctiontheorem boundary condition formalism [7]. Walz studied sucha geometry with the ray optics method [8].

Volpe et al. have demonstrated the existence of radiationpressure generated by SP resonance between metal and dielec-tric [9]. Kai et al. illustrated the SPR enhancement is signifi-cantly stronger than evanescent wave [10]. However, detaileddescription and analysis about dielectric particles transport inliquid due to SP radiation pressure is seldom reported before. Inthis paper, we present a detailed study on field distribution andtransport of a dielectric particle driven by SP radiation pressure,using both numerical and experimental methods.

In Section 2, numerical calculations on distribution ofincident and scattered fields with SPR are presented in thefirst part. In the second part, the generalized Lorenz–Miescattering theory (GLMT) is used for calculation of radiationforces on the spheres. We extend application area of theGLMT algorithm from localized Laguerre–Gaussian beam

Xiaodong Wang and Kai Xiao contributed equally to this study.

X. Wang :K. Xiao :C. Min (*) :Q. Zou :Y. Hua :X.-C. Yuan (*)Institute of Modern Optics, Key Laboratory of Optical InformationScience and Technology, Ministry of Education of China,Nankai University,Tianjin 300071, Chinae-mail: [email protected]: [email protected]

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described in [11] to SP waves. In Section 3, we show numer-ical results and comparison with experimental results. Finally,our conclusions are summarized in Section 4.

Theoretical and Numerical Analysis

Distribution of Incident and Scattered Field

Our calculations and experiments are implemented based onthe Kretschmann–Raether configuration [12] as shown inFig. 1 schematically. We choose the ABT model for fielddistribution because it is suitable for any format of incidentlight and less stringent in calculation requirement. We firstconsider the general formalism for the case of an arbitrarymonochromatic electromagnetic field with Er

(i)exp(−iωt) in-cident upon the sphere. The origin of coordinates x, y, and zis laid at center of the sphere to facilitate calculations. Thesphere is laid at a height h above the substrate. We considerthe effect of SP filed on the sphere, ignoring the influence ofthe sphere to field distribution to simplify this model.

In our discussion, we assume that the interaction is steadyso that exp(−iωt) can be ignored. The dielectric medium (airor water) is isotropic, homogeneous, nonmagnetic (μr01)and nonconductive, with the refractive index nmedium, andthe sphere (radius a) is also isotropic, homogeneous, andnonmagnetic but conductive, with a constant σsphere, so thecomplex permittivity for overall sphere is thus

"sphere ¼ "sphere þ i σsphere w=� �

: ð1Þ

So, the complex refractive index isnsphere ¼ "sphere "0=� �1 2=

:

Refractive index of the prism is nprism, thus its permittivityεprism0nprism

2. For the metal layer, its complex permittivity is

"metal ¼ "0metal þ i"

0 0metal.

The incident angle for SPR in our experiment isθinc063.56°, refractive index of prism nprism01.51, andrefractive index of water nmedium01.33. We use polysty-rene spheres (nsphere01.59) with four different radii of3.87, 5.09, 8.9, and 9.94 μm respectively. The incidentwavelength is 1.064 μm in air. A 45-nm-thin gold layeris coated above the prism for SP excitation. The com-plex permittivity of gold sphere are ε'metal0−45.4408and ε"metal01.4473, chosen from [13].

The first step in our solution is to know the incident fieldof SP waves. The governing equation for electromagneticfields is the vector Helmholtz equation [14]. We solve thisequation in a standard manner, namely the real Riccati–Bessel function in our later discussion:

y lðxÞ ¼ xjlðxÞ ¼ px2

� �1 2=JvðxÞ; ð2Þ

where jl stands for the spherical Bessel function and Jvthe ordinary Bessel function of the first kind. The relationbetween v and l is v0 l+1/2. We solve the Riccati–Besselfunction in both radial component and angular component asfollows:

EðiÞr ¼ E0

r^2

X1l¼1

Xl

m¼�l

l l þ 1ð ÞAlmy l a r^� �

Ylm θ; 8ð Þ ð3Þ

H ðiÞr ¼ H0

r^2

X1l¼1

Xl

m¼�l

l l þ 1ð ÞBlmy l a r^� �

Ylm θ; 8ð Þ ð4Þ

There is a relation above

a ¼ 2pan l= ; r^ ¼ r a= ;H0 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi"0 μ0H0=

p: ð5Þ

Fig. 1 Configuration of model

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The spherical harmonics Ylm(θ, Φ) has the expression:

Ylm θ; 8ð Þ ¼ 2l þ 1

4pl � mð Þ!l þ mð Þ!

� �1 2=

Pml cosθð Þexp imfð Þ ð6Þ

where θ is the polar angle and Φ is the azimuthal angle, asshown in Fig. 2.

During the mathematical process, normalization is takeninto consideration and the time factor exp(−iωt) has been

omitted. Our goal in solving the equation is to obtainexpressions of the orthogonal coefficients of Ylm(θ,Φ):

Alm ¼ b a=ð Þ2Eol l þ 1ð Þy l k2bð Þ

ZEðiÞr b; θ; 8ð ÞY*

lm θ; 8ð ÞdΩ ð7Þ

Blm ¼ b a=ð Þ2Hol l þ 1ð Þy l k2bð Þ

ZH ðiÞ

r b; θ; 8ð ÞY*lm θ; 8ð ÞdΩ ð8Þ

where dΩ0sinθdθdΦ. Here, the integrations are carried outover the surface of a sphere with arbitrary b, centered atorigin. If we intend to know the exact whole sphere fielddistribution, we need to make b0a in the calculation.

Mie scattering theory gives the relationship between incidentfield and scattered field. The scattered field has the same formatas incident field above. The expansion coefficients of two fieldsare dependent with each other, which are both needed whenanalyzing the radiation forces exerted on the sphere. In thedetailed numerical calculation of scattered field, we considerthe Hankel function instead of the first kind of Bessel function[14]. The coefficients have the same format as Alm and Blm,denoted as Clm and Dlm, respectively. We choose the GLMTmodel for radiation force calculation as it is designed for Gauss-ian light and there is a Matlab toolbox designed for calculationdesigned by the University of Queensland [15].

Figure 3 shows numerical calculations of the near-fielddistribution of a sphere in the SP field as a function of theheight h from the metal surface. As the sphere approached thesurface, more light travels into the surface and then refracted tothe right upper side, which may pushes the sphere to the rightbottom direction.

Fig. 2 Cartesian coordinate, origin located at the center of the sphere.Definition of polar angle θ and azimuthal angle ϕ

Fig. 3 Field distribution of asphere (radius of 3.87 μm) inSP field at different heightsfrom the metal surface: a0.5 μm, b 0.3 μm, c 0.1 μm, d0.05 μm

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Radiation Forces Exerted in Dimensions

As we discuss the situation of a sphere in stationary ormotion at a constant speed, the force component can bewritten in terms of Fx, Fy, and Fz in three dimensions. Thecoefficients Alm, Blm, Clm, and Dlm are directional compo-nents of the integration of finite radiation forces elements onthe sphere, respectively. They are deduced from the Max-well stress tensor in matrix, namely the electromagneticenergy-momentum tensor [16], and the time-average force<F> can be written as:

Fh i ¼Z

f"2Re E � nð ÞE�½ � � "

4E � E�ð Þnþ μ

2Re½μ H � nð ÞH �

� μ

4H � H�ð Þngds

ð9Þwhere ds stands for every finite part of the sphere's surface.

Numerical Simulations and Comparisonswith Experiment

Numerical Simulation

We carry out calculations by using the Matlab code with thehelp of optical tweezers computational toolbox provided by theUniversity of Queensland and the NAG Matlab toolbox. Theroutines DO1AUF and DO1ATF have been proved to beefficient for handling rapid integrands. The recurrence relationbetween l and l+1 is also taken into consideration, especiallywhen we discuss whether (l+m) is even or odd duringprogramming.

We found that it is important to truncate the expansion of asurface plasmon wave at a proper l to obtain accurate results,

as F0∑l fl. In our calculation, we terminate the series when

flmPlk¼1

fk

< 10�4: ð10Þ

For small spheres (α<10), it is easy to satisfy Eq. 10 atl020 with a rapid convergence. We can achieve a highaccuracy of 10−4 for all expansion coefficients. For largespheres, for example biological cells, α will be 80 or evengreater, which is a great challenge to the computer simula-tion. We need to choose l050 for large spheres to keepenough calculation accuracy.

Results

Illustrating an example of the driving force exerted on aparticle calculated with Eq. 9, we show in Fig. 4, thenear-field distribution of the force when the particle isat a height of 100 nm above the flat metal surface. Bothmagnitude and direction of the force on each segmentof the particle are demonstrated as green arrows in thefigure. It can be seen that the direction of the force isalong the positive x-axis and it is pulled down towardsthe metal surface (negative z-axis).

It is interesting to observe the behavior of the magnitudeand direction of the force when the particle is fixed to ahorizontal position and moved away vertically away fromthe flat surface. Illustrative examples of both vertical andhorizontal radiation forces are shown in Figs. 5 and 6,respectively. Related experimental parameters are the sameas above. In some previous reports, s-polarization light is ofgreat importance in the evanescent field, which contributesmore radiation pressure than p-polarization [6]. However, inSPR case, only p-polarization exists and dominates. The

Fig. 4 Calculated driving force exerted on the particleFig. 5 Vertical and horizontal radiation forces as a function of h withsphere radius of 3.87 μm

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vertical and horizontal forces are given in nondimensionalform as follows:

Qx ¼ Fx

"0E20a

2;Qz ¼ Fz

"0E20a

2: ð11Þ

Comparison Between Experiments and Simulations

To demonstrate our results, we set the experimental systemfor transporting polystyrene spheres in water. The incidentlight with a power of 260 mW is Gaussian light travelingthrough a polarizing plate distributed over a cross-sectionalarea with a diameter of 100 μm. The polarizing plate guar-antees the p-polarization illumination on the prism. SPRregion is illuminated by a highly focused Gaussian beamthrough an f−θ glass before entering the prism.

We derive the power illuminated on the interface as

"0E20 ¼ 0:2208 eN m2

by the Poynting vector as c 2=ð Þ

"0E02 ¼ 32:93 MW m2

. After SPR enhancement, amplitude

of electric filed on the metal layer is increased by 19.36 timeswith respect to incident light. When drag force in water equalsto radiation pressure, particles will move in a constant speed.The Stokes's formula D06πrηv can be introduced as the dragforce in water for low Reynolds number to estimate thehorizontal velocity of the sphere moving along the plate,where η stands for dynamic viscosity in the unit of 10−3kg/ms.

Figures 7 and 8 show the tracks of polystyrene spheresrecorded by a camera with interval of 3 s. Our experimentaldemonstration is similar to Volpe and Quidant's result [9]. Intheir system, a photonics force microscope has been used forlevitating particles pushed by surface plasmon radiation pres-sure. Their work has given enough proof that particles movein the same direction as that of incident beam at the interface.

Based on numerical results, particles of 3.87 μm in radiustransport at a velocity v012.56 μm/s, which is very close to

our experimental data v010.47 μm/s. We also tested otherparticles with three different sizes. Our numerical results showthat particles with a radius of separately 5.09, 8.9, and9.94 μm run at a speed of 4.56, 2.86, and 1.94 μm/s, respec-tively. Based on our experimental data, their actual transportspeeds are corresponding 3.88, 2.38, and 1.5 μm/s. Consider-ing the detailed simulation process, some reasons may lead tothe differences between numerical and experimental results.

The first obvious effect is due to thermophoresis, namelythermal diffusion. This phenomenon is sometimes calledLudwig–Soret effect and it is still not well understood so far.As electrons collectively oscillate with photons, it is of noavoidance that energy imparts to the surrounding media thatgenerate local heating [17]. It is very easy to observe convectionin our experiments caused by thermophoresis. Convection cre-ates a circulation of fluid around a volume in order to transferheat to the surrounding medium and typically drives colloidstowards the hotter regions within a chamber. To simulate theinfluences induced by thermophoresis is another complex pro-cedure and out of the scope of this paper. The next unavoidableinterference is the Brownian motion. Small particles are proneto move automatically without any external power in freedirections. The smaller sized particles, the heavier interferenceBrownian motion brings to them. Although this occurrence is

Fig. 6 Horizontal radiation force in x-direction as a function of h withfour different radii of spheres

Fig. 7 Movement image of particles with radius of 8.9 μm

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relatively small compared with radiation pressure, its accumu-lative effect on decrease of the overall calculation accuracycannot be ignored. In addition, multiple reflections betweenthe sphere and metallic surface also have small effect on theresults. Most of the scattered energy is due to the initial contact;however, for very large particles with diameter greater than100 μm, the multiple scattering effect plays an important role.

Based on the above points, we can come to a conclusionthat our simple and easy understanding theoretical modelhas proved a good agreement to experimental result. Otherphenomena cause some influence, which, however, do notplay a dominated role. Their influences on particles trans-port still need more investigations.

Explanation for Some New Findings

To confirm the excitation of SP field, we measure the reflec-tance of glass/gold/water stack. The incident beam is trans-verse magnetic (TM) polarized. The experimental resultsshow that the highest coupling efficiency between laser powerand SP field happens at 64°. Kai et al. have proved that thepower flow magnitudes are much stronger in SP field thanevanescent-deduced total internal reflection (TIR) field [10].

To look more into this, we first need to state several facts.Firstly, incident angle for peak amplitude in SP field and TIRfield is different. If this is a TIR glass/water configuration,incident angle is around 51°. Secondly, due to lacking en-hancement effect, power flow from TIR is not strong enoughto support large particles moving along the interface. Further-more, particles disturb less in SP field than TIR because ofpenetration depths. Evanescent wave has a penetration depthabout several microns while in the SP field, surface wavepower is mostly focused within several hundred nanometersfrom the interface. Thirdly, if we give more laser power to theSP field, thermal effects will be more influential than radiationpressure, thus particles will move randomly and uncontrolla-bly. So, we conclude that comparing particles movement in SPand TIR are unreasonable and meaningless.

Another new finding is about the thermal effects mentionedin the last paragraph. It is observed that at the beginning,particles move steady along the interface; however, in the centerof illumination area, particles are levitated up away from theinterface. This is rather different from our calculation andprevious experiments [9]. We think it is also due to the thermaleffect. The unequal distribution of electric field enhanced bySPR which causes thermophoresis also shows an increasingpotential from edge to the center. According to the thermaleffects, convection plays a dominant mechanism when consid-ering thermal gradients. Often the convection drives particlestowards the hotter region, namely center of illuminated area inour experiment. Particles will be levitated up and pushed bymicro-fluidics in water. Physical explanations for vertical com-ponent towards the interface rather away have been given bothin ray optics and physics optics method [9, 12]. Simply written,limitation in this physical process is the refractive index ofsphere and liquid, only if nsphere<nfluid or nsphere>>nfluid, radia-tion pressure will push particles up towards the positivez-direction. This results accord with that of Walz's.

To make a conclusion of particle transport in medium, inSP resonance, when the particles get closer to the surface,coupling efficiency of SPR energy to the particle gets great-er. For larger particles, the coupling efficiency gets down, soless energy will be coupled for particles transport. Z-com-ponent of the force exerted on the particle drags themtoward the surface, while the x-component guides the par-ticles along the surface in the positive x-direction. Thisdiscussion accords with that of Figs. 3 and 4.

Conclusion

Our research combines the ABT and GLMT in numericalcalculation of SPR force exerted onmicrometer-sized particlesand the results accord well with experiment. Calculation resultofQ is independent of incident Gaussian beam amplitude. Ourwork shows a great potential application of biological cell

Fig. 8 a–f Sorting image of particle with radius of 8.9 and 9.94 μm

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sorting technology. Even though cells are muchmore complexthan polystyrene spheres in both experiments and calculation,for example, refractive indices of subcellular components areunequal inside cells; this work provides a quantitative analysisand foundation for manipulation of biological cells in thefuture.

Acknowledgments This work was partially supported by the Na-tional Natural Science Foundation of China under grant nos.10974101, 61036013, and 61138003. XCY acknowledges the supportgiven by the Ministry of Science and Technology of China under grantno. 2009DFA52300 for China–Singapore collaborations and TianjinMunicipal Science and Technology Commission under grant no.11JCZDJC15200. The authors gratefully acknowledge the support of“National Undergraduates Innovation Experimentation Project” in Chi-na under project 10105511.

References

1. Ashkin A, Dziedizc JM, Bjorkholm JE, Chu S (1986) Observationof a single-beam gradient force optical trap for dielectric particles.Opt Lett 11:288–290

2. Kawata S, Sugiura T (1992) Movement of micrometer-sized par-ticles in the evanescent field of a laser-beam. Opt Lett 17:772–774

3. Garces-Chavez V, Dholakia K, Spalding GC (2005) Extended-areaoptically induced organization of microparticles on a surface. ApplPhys Lett 86:031106

4. Andrews DL (2008) Structured light and its applications. Elsevier,Amsterdam

5. Barnes WL, Dereux A, Ebbesen TW (2003) Surface plasmonsubwavelength optics. Nature 424:824–830

6. Almass E, Brevik I (1995) Radiation forces on a micrometer-sized sphere in an evanescent field. J Opt Soc Am B OptPhys 12:2429–2438

7. Lester M, Nieto-Vesperinas M (1999) Optical forces on micro-particles in an evanescent laser field. Opt Lett 24:936–938

8. Walz JY (1999) Ray optics calculation of the radiation forcesexerted on a dielectric sphere in an evanescent field. Appl Opt38:5319–5330

9. Volpe G, Quidant R, Badenes G, Petrov D (2006) Surface plasmonradiation forces. Phys Rev Lett 18:238101

10. Wang K, Schonbrun E, Crozier KB (2009) Propulsion of goldnanoparticles with surface plasmon polaritons: evidence of en-hanced optical force from near-field coupling between gold parti-cle and gold film. Nano Letters 9(7):2623–2629

11. Ashkin A (1992) Forces of a single-beam gradient laser trap on adielectric sphere in the ray optics regime. Biophys J 61:569–582

12. Garces-Chavez V, Quidant R, Reece PJ, Badenes G, Torner L,Dholakia K (2006) Extended organization of colloidal micropar-ticles by surface plasmon polarization excitation. PhysRev B73:085417

13. Palik ED (1985) Handbook of optical constants of solids.Academic, New York

14. Brevik I, Sivertsen TA, Almass E (2003) Radiation forces on anabsorbing micrometer-sized sphere in an evanescent field. J OptSoc Am B Opt Phys 20:1739–1749

15. Nieminen TA, Loke V (2007) Optical tweezers computationaltoolbox. J Opt A Pure Appl Opt 9:S196–S203

16. Novotny L, Hecht B (2006) Principles of nano-optics, 3rd edn.Cambridge University Press, Cambridge

17. Farahi RH, Passian A, Ferrell TL, Thundat T (2005) Marangoniforces created by surface plasmon decay. Opt Lett 30:616–618

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