1. Introduction

Since the pioneering studies by Ashkin and coworkers [1,2] it is well known that radiative “gradient” forces can either trap or expell particles from high optical intensity regions. This is an important issue concerning the optical manipulation and sorting of microparticles.

In this work we have observed the motion of metallic particles above optical waveguides where the particles are subjected to both a strong gradient of electromagnetic field which may trap them and a propagating wave which exerts a radiation pressure force able to propell them. The trapping and propelling of nanoparticles in an optical evanescent field has been first demonstrated by Kawata and Sugiura [3] and Kawata and Tani [4]. By contrast, experiments by Vilfan et al using an atomic force microscope suggested that a dielectric microsphere is repelled by an evanescent field [5], an effect that seems to contradict both the experiments by Kawata et al and further theoretical works [6]. The optical forces exerted on gold nanoparticles has been characterized by Sasaki et al [7] who showed the attractive nature of the gradient force for 250nm gold particles. Other works demonstrated that the velocity of the particles increases with the intensity of the optical field [8] that can be enhanced by confining it in narrow (including single-mode) waveguide and/or by using waveguides with a high index of refraction [9–12]. This kind of device could be combined with various integrated structure for the manipulation and sorting of particles or possibly biological cells [12].

In the present work we are mostly concerned about the effect of the polarization of the electromagnetic field on radiative forces. The influence of the polarization of the optical field has been observed by Ng et al [8,9]: at a given modal power the velocity of gold nanoparticles 10-40nm diameter was found somewhat larger for TM polarization, and this was attributed to the enhancement of the optical field in the evanescent region. In a series of theoretical works Ariaz-Gonzalez, Chaumet and Nieto-Vesperinas [6,13–15] suggested that not only the magnitude, but the sign of the radiative forces exerted on metallic microparticles subjected to non uniform electromagnetic fields could depend on the polarization of the optical field, and that this sign might change with the size of the particles. For nanocylinders they showed that this can be simply explained by considering the expressions of the polarizability of the particles, that depends of the polarization [13]. They confirmed these effects by numerical calculations carried out using the coupled dipole approximation in the case of spherical or cylindrical particles of larger size [14,15]. More recently, using Arbitrary Beam Theory Jaysing and Helleso predicted that morphological dependent resonance occuring for large (several μm diameter) dielectric particles could also yield to repelling gradient forces [10].

In the present work we give some experimental evidences confirming these predictions concerning the variation of the sign of the gradient force, for the first time to our knowledge. Our observations are complemented by exact finite element 3D numerical calculations, corresponding to our experimental conditions, of the electromagnetic field distribution in and above an optical waveguide in the presence of metallic microparticles, that yield the radiative forces applied to the microparticules. In the case of large metallic particles we show that some of the numerical and experimental observations concerning the sign of the gradient force can be qualitatively understood using elementary electromagnetism considerations.

 

Fig. 1. Experimental set-up

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2. Experimental procedures

Our experimental set-up described in [12] is depicted on Fig. 1. Briefly, a chamber containing a suspension of microparticles in water is placed over a channeled optical waveguide injected by radiation emitted by a Nd:YAG laser source operating at 1.064 μm. The motion of the particles is observed using a zoom system (with a 20X and NA=0.55 microscope objective) and a CCD camera mounted above the waveguide. The particles close to the waveguide may be trapped by the optical gradient force along the waveguide and propelled by the radiation pressure, as observed previously by several authors [4, 9–12].

A halfwave plate was incorporated to adjust the polarization of the light entering the waveguide. The effective guided power was monitored using an optical power meter at the end of the waveguide after a microscope objective and a diaphragm.

We will present experimental results using two types of optical waveguides:

-Waveguides made by silver ion exchange on the surface of a glass substrate. The difference of refraction index between glass and the exchanged region is approximately 0.01. Exchange occurs through a window whose width is about 3μm. These waveguides propagate a single mode whose size is about 4μm (vertical direction) × 5.5μm (horizontal direction), with propagation losses of about 0.05-0.25db/cm.

Waveguides made of a silicon substrate covered by a 2μm silica film (n=1.45 at 1064nm) and a silicon nitride strip (n=1.97 at 1064nm) as used in our previous work [12]. The strips are deposited by LPCVD (Low Pressure Chemical Vapor Deposition), and etched by RIE (Reactive Ion Etching). Silicon nitride film thickness is 200nm. Different waveguides with strip width in the range 1-10μm have been studied. This corresponds to multimode waveguides, the thickness of each mode being about 1μm. Propagation losses depends on the polarization, about 3db/cm with TE and 14db/cm with TM.

We used gold particles of diameters 250nm, 600nm, 1000nm with a standard deviation around 35% (Duke scientific). A cell defined by double-sided adhesive tape spacer and a cover slip was glued on the surface sample in order to form a chamber for the particles in deionized water. Motion of the particles was observed using a zoom system (with a 20X and NA=0.55 microscope objective) and a CCD camera mounted above the waveguide.

3. Results

3.1 Motion of gold particles above a silver ion waveguide

Figure 2 shows the effect of the polarization of the incident field on the motion of 1μm gold particles. For TM polarization it is clear that the particles seems to be trapped above the axis of the waveguide where they are propelled. When switching the polarization to TE the particles slow immediately, and seem to be expelled on the side of the waveguide. The process reverses when switching the polarization back to TM.

3.2 Motion of gold nanoparticles above a silicon nitride waveguide

As discussed by Gaugiran et al[12] silicon nitride waveguides are very efficient to trap and propell dielectric particles. This behavior is confirmed for small (250nm) gold particles. In this case the particles are always guided along the waveguide (whatever the polarization TE or TM). The mean value of the velocity distribution is 130μm±35μm for an injected power of only 20mW with TE polarization in a 3μm wide waveguide (but we have observed some particles as fast as 350μm/s). This indicates that these waveguides are even more efficient than Cs+ waveguides [11,16] since in the latter case 250nm gold particles have been observed to move at similar velocities, but with a modal power of 150mW. We think this is due to the higher intensity of the field at the surface of the waveguide due to the strong confinement of the guided wave that can be related to the refraction index contrast Δn between the guide and the substrate: at λ=1064nm Δn=0.03 for Cs+/glass but Δn=0.52 for silicon nitride/SiO₂ [12].

 

Fig. 2. Movie (1.92 MB) Effect of the polarization of the optical field on the propulsion and trapping of 1μm gold particles above a single mode Ag⁺ waveguide propagating 400mW of 1064nm radiation. [Media 1]

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3.3 Motion of gold microparticles above a silicon nitride waveguide

We consider now the motion of 600nm diameter gold particles. The comparison between the polarizations TE and TM is not as straightforward as in the case of Ag+ waveguide because our silicon nitride waveguides are not single mode and their propagation losses are large and depend very much on the polarization. Hence the switching of the polarization from TE to TM changes both the (generally complicated) field profile in the waveguide and its intensity.

Anyway Figs. 3 and 4 show that for TM polarization the moving particles are confined on the waveguide strip, while for TE polarization the particles tend to be trapped and guided on the side of the waveguide and are not propelled when they happen to cross the waveguide.

A more direct comparison of the effects of radiative forces on metallic particles for TE and TM polarizations can be obtained by observing the motion of a mixture of 2μm glass and 600nm gold beads. Experimental results are shown in the first two columns of Table 1. We see that the velocity of gold beads is smaller with TE compared to TM, while the glass beads behave the opposite: there is a factor of about 10 between the velocity observed with TE (8.7μm/s) and TM (0.8μm/s). This latter fact is unexpected since we anticipate from calculations presented in the next sections, in agreement with previous works [10], that for dielectric particles like glass of this size, the gradient forces are attractive and that the propelling forces are about the same for both TE and TM polarizations.

Moreover we have checked that for a given polarization the beads velocity increases approximately linearly with the injected power. Then this factor of 10 between the observed velocity of the glass beads for TE and TM polarizations can be related to the actual power of the guided optical wave in the observation region being divided by 10 with TM compared to TE polarization due to increased propagation losses. With the same power, the velocities of metallic beads would thus be about 50μm/s instead of 4.9μm/s. It is thus obvious that the effective radiation pressure exerted on the metallic beads is much larger for TM than for TE polarization.

 

Fig. 3. Movie (0.78MB) Propulsion and trapping of 600nm gold particles above a multimode silicon nitride waveguide propagating 1064nm radiation with TM polarization. The power of the optical radiation at the observation region, estimated from the observation of the motion of dielectric particles in similar conditiosn is a few mW. [Media 2]

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Table1. Comparison of the observed propelling velocity of a mixture of gold and glass beads above a silicon nitride waveguide for two polarizations of the optical field. The middle and left columns are experimental values. The values in the right column are extrapolated from the experimental values of the middle column and correspond to the values which would be observed if the optical power in the waveguide were the same as TE polarization (see text).

View Table

 

Fig. 4. Movie (0.87.MB) Propulsion and trapping of 600nm gold particles above a multimode Silicon Nitride waveguide propagating 60 mW of 1064nm radiation with TE polarization. [Media 3]

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4. Numerical simulations

To understand better our observations we have carried out numerical simulations of the electromagnetic field resulting from the interaction of the particles with the optical field injected in the waveguide. This enables us to calculate the optical forces thanks to the Maxwell stress tensor formalism [17]. Similar calculations, mostly using the coupled dipole approximation, have been carried out by Nieto-Vesperinas et al [13–15] in the case of particles subjected to an evanescent wave. In the present work calculations are carried out in our silicon-nitride waveguide, taking into account the 2D confinement of the field.

Details for the method of calculation are given in our previous publication [12]. Briefly, the calculations are based on the finite element method and the FEMLAB (Comsol) software.

The particles are either gold cylinders or spheres (ε=-53,6+i4,18 at 1064 nm or n=0.272- 7.07i [18]) of various diameters in the range 20-800nm located at an altitude of 2 nm from the surface of the waveguide, immersed in water (n₁=1.33). The waveguide geometrical properties used in the simulation correspond to silicon nitride waveguides with 1μm strip width in the case of spheres, infinite width in the case of cylinders. The wavelength is 1064nm. Only the results concerning spherical particles will be detailed below.

Results concerning the electromagnetic field energy distribution and the local flux of the Minkowski-Maxwell stress tensor [17] acting on the surface of gold beads placed above a silicon nitride waveguide are shown on Figs. 5–6. Results concerning the total force applied to the particle as a function of its size and polarization are shown on Fig. 7 for either dielectric and gold beads. The force is composed of two components, the “propelling force” directed along the direction of propagation of the incident wave, and the “gradient force” directed normally to the waveguide surface. This total force is calculated as the integral of the local flux of the Maxwell tensor along a closed surface containing the particle according to Eq. (1) [19]:

where the integration is over any closed surface including the particle (in practice a sphere located in the vicinity of the outer particle surface).

These simulations show clearly that:

-the magnitude of the radiative forces is generally much larger for metallic (gold) particles than for dielectric (glass) particles. This can be related to the large, imaginary part of the refractive index of metallic particles.

-“small” and “large” particles do not react the same with respect to the electromagnetic field: while the field in the waveguide seems almost unaffected by small particles (diameter smaller than 100nm, results not shown) it is very strongly affected by a 500nm particle.

-while the response of dielectric particles is only weakly sensitive to the polarization of the incident field, by contrast the response of medium size metallic particles is completely different for TE and TM polarization :

-the vertical, gradient force repells the particle for TE polarization and attracts the particle towards the waveguide for TM polarization. Moreover, for TE polarization the sign of the gradient force depends on the size of the particle: repulsive for large particles, attractive for small particles. This confirms the results of the calculations by [6,14,15].

-For TE polarization the electromagnetic energy density shows an accumulation of the electromagnetic density in the waveguide upwards the particle, while the electromagnetic field almost vanishes close to the particle. By contrast for TM polarization there is a strong accumulation of the electromagnetic energy density just below the particle surface, as if the wave were attracted by the particle. -for TE polarization the radiative forces applied on medium size gold particles repell them from the waveguide when they are on the axis, but attract them to the waveguide when they are on the side.

 

Fig. 5. Electromagnetic field energy density distribution (arbitrary units) and flux of Maxwell stress tensor distribution on the surface of a gold particle (500nm diameter) placed on the axis of the waveguide for TE and TM polarizations of the incident field. Numerical values of the Maxwell tensor correspond to an incident wave of power 1Watt. Side view with respect to the direction of propagation of the incident wave.

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5. Discussion

Our observations are fully consistent with the results of our numerical simulations:

-metallic particles of 250nm diameter are always trapped whatever the polarization of the optical field.

- for TM polarization 600nm metallic particles go faster than the glass particles because the radiative forces are larger. They are slower than the 250nm particles because the radiation pressure is about the same for 250nm and 600nm particles, but the viscous force increases like the radius (Stokes law).

- for TE polarization 600nm metallic particles are expelled and trapped of the side of the waveguide (Fig. 6). They are not propelled when they are above the waveguide because they undergo a repulsive gradient force which pull them away from the surface of the waveguide. As a result they undergo a reduced propelling force since it is expected that this force decreases strongly with the distance to the waveguide [6,14,15].

 

Fig. 6. Electromagnetic field energy density distribution for TE polarization of the incident filed and flux of Maxwell stress tensor distribution on the surface of a gold particle (diameter 500nm) placed on the axis (left), and of the side (right) of a silicon nitride waveguide Front view with respect to the direction of propagation of the incident wave.

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Fig. 7. Numerical calculation of the components of the radiative forces exerted on spherical particles placed on the axis of a silicon nitride waveguide. On the left, for glass particles, the forces increase with the particle radius. On the right, for gold particles and TE polarization the vertical “gradient force” is attractive (negative) at small radius but it becomes repulsive at large radius. The power of the incident electromagnetic wave is arbitrary fixed to 1Watt.

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Some qualitative arguments can be given to explain the repulsive/attractive nature of the gradient force on “large” metallic particles versus the polarization TE/TM of the incident guided wave. “Large” means that the size of the particle is very large compared to the skin depth δ. (In the case of gold at λ=1.05μm δ= 23nm [20]. Thus the electromagnetic field must vanish inside the particle. Because the normal component of the magnetic field B and the tangential component the electric field E must be continuous, these components must be zero just outside the surface of the particle. However the tangential component of B and normal component of E may not be continuous through the particle surface. This is caused by surface free currents j_f and surface free charges σ_f induced by the incident electromagnetic field (see Fig. 8). This results in Laplace and electric forces normal to the surface of the particle. Elementary electromagnetism (Fig. 8) yields the following expression for the component of the force per unit area along the normal to the surface:

where the normal is oriented outwards the particle, and the brackets denote the time average. This expression is identical to the flux of the Maxwell tensor [17,19]. It shows that the electric field induces attractive forces, while the magnetic field yields to repulsive forces, and that the repulsive or attractive nature of the radiative force depends on the relative magnitude of the electric and magnetic fields.

Let us assume to simplify that the particles are cylindrical with their axis perpendicular to the direction of propagation of the incident field: Since we have found that for large (diameter > 500nm) particles our numerical simulations predict a similar dependence with the polarization of the incident field for both spherical and cylindrical particles it is likely that the geometrical difference between cylinders and spheres is of little importance.

For TE polarization the normal component of the electric field is identically zero. Since the tangential component must be zero the amplitude of the electric field is zero at the surface of the particle. However, the magnetic field can, and has, a non zero tangential component (although our calculations predict it is relatively weak, as suggested by the small electromagnetic energy density close to the particle, cf Fig. 5). Then the radiative force tends to repell the metallic particle.

For TM polarization none of the magnetic and electric field are equal to zero close to the particle, in agreement with the accumulation of electromagnetic energy below the particle as pointed out in the latter section, see Fig. 5. It is not obvious which of these two forces (electric and magnetic) has the larger component along the direction perpendicular to the waveguide surface, but the results of the numerical calculations (and the experiments) seem to indicate that the attractive, electric force, prevails.

These arguments help to understand why the force is repulsive in the case of TE polarization and why it may behave differently in the case of TM polarization.

In the case of small particles where the skin depth is not negligeable in front of the size of the particles the previous discussion is not valid. In particular, in the Rayleigh regime where the size of the particles is small enough that the electromagnetic field is uniform inside the whole particle, the relevant parameter is the sign of the real part of the polarizability of the particle [15] whose expression for a spherical particle of radius a is given by :

where ε’ et ε’’ are respectively the real and imaginary part of the permittivity of the material relative to the surrounding medium. Then the intensity of the optical gradient force is given by the expression:

where E₀ is the amplitude of the electric field. For gold nano-particles at λ=1064nm we have ε=-53,6+i4,18 in vacuum [18]. In water ε must be divided by n₀ ² where n₀=1.33 is the refractive index of water. As a result Eqs. (3) and (4) show that the gradient force is attractive whatever the polarization, which is consistent with both our calculations and our observations. A non trivial issue is the determination of the critical radius of the particle where the sign of the gradient force reverses for TE polarization: a rigorous numerical modeling is needed to evaluate it.

 

Fig. 8. Electromagnetic surface forces applied to a metal. The discontinuity of the electric (a) and magnetic (b) fields across the skin depth is related to surface charges ans currents. Note that the forces applied on the electric surface charges still tend to attract the metal towards the dielectric if the direction of the electric field is reversed, while the magnetic forces still tend to repell the metal from the dielectric if the direction of the magnetic field is reversed.

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It is interesting to note that for cylindrical particles the polarizability depends on the polarization of the incident field [13]. For gold nanocylinders in water illuminated at λ=1064nm Eq. (4) predicts an attractive gradient force for TM polarization and a repulsive gradient force for TE polarization. Actually, numerical calculations of the radiative forces carried out for cylindrical particles in our experimental geometry show that the gradient force is always attractive for TM polarization and always repulsive for TM polarization, and that there is no reversal of the sign of the gradient force as the size of cylinder increases.

6. Conclusion

In this paper, we have demonstrated that the motion of metallic microparticles subjected to radiative forces can be very effectively controlled using the polarization of the incident radiation. While demonstrated in the case of channeled waveguides, this result could be applied to other experimental configurations, and opens the way to the development of new microsystems for particles manipulation (see e.g. [21]) and sorting applications.