1. Introduction

The trapping of micron-sized particles by optical forces is now a standard method for controlling and manipulating many types of objects in solution [1], such as cells [2–3], proteins with polymer beads [4–7] as well as for separating particles [8]. The lasers can trap particles using two different mechanisms [9 ]. One is related to the change in momentum as light is scattered or absorbed by the particle, which is known as the scattering force. The other is due to the polarization of the particle in the strong electric fields associated with focused laser beams. The force arises from the interaction of the induced dipole moment with the electric field gradients associated with the optical field. This is known as the optical gradient force. For particles in solution, the optical forces need to be strong enough to overcome the random thermal motion or Brownian motion. For micron-sized particles in optical traps the Brownian motion is used for calibrating the trapping forces [10–11]. This motion can be modeled by an equation for the acceleration in the presence of viscous forces and the random fluctuating force that is responsible for the Brownian motion [11–14]. The trapping of nano-scale particles and molecules is much more difficult because the optical force falls dramatically with the particle size [15]. Our interest is to manipulate biological molecules, such as proteins, by optical means. Since proteins are typically 5 nm in diameter [16], the thermal motion can prevent trapping unless the light intensity and its gradients are large.

To produce strong field gradients, trapping methods have been proposed and demonstrated using evanescent light waves. These can be formed at an interface where there is total internal reflection or by surface plasmon resonances in metallic structures. The trapping and manipulation of gold nano-spheres with radii in the range 10-40 nm were demonstrated using the evanescent fields above a glass waveguide [17–18]. Keyi et al proposed trapping nanometre scale objects using the tip of a scanning near-field optical microscope [19]. Their calculation showed that the trapping potential was just sufficient to overcome the thermal motion of a 20 nm diameter particle. Another method was proposed by Novotny et al. [20] in which strong evanescent electric fields are created at a sharply pointed metal tip under illumination by laser light. For a 5 nm radius gold tip in water illuminated by 810 nm light their model predicts a 3000 fold increase in the intensity which is sufficient to trap a 10 nm diameter particle in the presence of Brownian motion. A variation on this geometry was proposed by Chaumet et al. [21] in which an evanescent field created by total internal reflection interacts with a nearby tungsten wire to trap particles in air. The wire probe trap was also considered for trapping fluorescent molecules [22] which was shown to be just strong enough to overcome Brownian motion near the molecule resonance. Another method for trapping using optical near fields employs a small aperture in a metal film. Okamoto and Kawata [23] used the Finite Difference Time Domain method to study this geometry and to compare the trapping force on glass spheres with buoyancy forces and Brownian forces. Chang [24] calculated the optical force acting on a molecule near a metal sphere that included the shift in the resonance of the metal sphere.

In most of these analyses, the particles are considered to be trapped when the trapping energy is comparable with or about a factor of two greater than the thermal energy. However, this is an estimate based on the average energy of the particle whereas the Brownian motion can have quite large fluctuations about this average. In this paper we consider how the thermally excited nano-particles are distributed in optical traps, such as those created using evanescent light fields. In particular we consider the particles distributed through the solution and look at the intensity required of the evanescent fields to maintain a concentration gradient against the diffusion forces that arise from the Brownian motion. In the section below we briefly review the optical forces responsible for trapping and the parameters that describe the evanescent light fields. The equation of motion of particles in the presence of optical and Brownian forces is introduced and it is transformed into a Fokker-Planck equation that describes their statistical distribution. A solution is given for the gradient force which dominates for evanescent fields. The results are applied to the trapping of nano-scale particles by the evanescent field associated with surface plasmon resonance in a thin gold film. An expression for the lifetime of nano-particles in the trap is derived based on Kramers’ calculation of the diffusion rate over a barrier.

2. Optical trapping

In this section we briefly review the trapping of particles by lasers in the limit where the particle dimension is very much smaller than the wavelength of the light (Rayleigh limit).

2.1 Optical forces

The two forces arising from the interaction of light with a dielectric particle are the gradient force and the scattering force [9, 15]. The gradient force is associated with the energy required to polarize a particle in an electric field whereas the scattering force is due to momentum transfer from the light to the particle. The gradient and scattering forces on a spherical particle of radius a due to a light beam of intensity I are given respectively by

where the effective polarizability a of the spherical particle in the supporting medium is

In Eq. (3), ϵ_s is the relative electric permittivity of the sphere and ϵ_m is that of the supporting medium. The refractive index of the medium is $n=\sqrt{{\epsilon }_m}$ and the wave number of the incident light of wavelength λ is k=2π/λ. The gradient force depends on the gradient of the light intensity, ∇I, and the scattering force term depends on the vector k I=S which is the direction of power flow determined by the Poynting vector. Here, c is the speed of light in vacuum. The intensity is related to the modulus-square of the electric field amplitude

which involves the electric permittivity of free space (in SI units).

2.2 Evanescent wave traps

In the discussion below we consider particles trapped in an evanescent field that decays exponentially from a surface. The surface is in the x-y plane with a normal in the z direction. An example would be a glass-air interface with light incident from the glassy side. For a light wave that is totally internally reflected, the transmitted intensity decays in proportion to

where the decay length

depends on the relative permittivities of the incident medium ϵ_i and the transmitting medium ϵ_t for light incident at angle θ_i to the surface normal. These formulae also apply to surface plasmon resonance in a thin metal film if the intensity I(0) corresponds to that at the interface between the metal film and the transmitting medium. The power flow (Poynting vector) above the surface for complex fields with harmonic variations in time is

The condition for total internal reflection is ϵ_i sin² θ_i > ϵ_t so that for negligible losses in the media, the z-directed term is imaginary. This term represents a power flow that oscillates in time so that its time average is zero [25]. Therefore the average power flow is parallel to the surface and decays exponentially away from it. This means that the optical scattering force is given by

This represents a force parallel to the surface which induces a flow but does not lead to trapping.

3. Motion of a particle under Brownian forces in a fluid

In this section we determine the average motion of a particle in the laser field in the presence of Brownian motion. The aim is to determine the power required to overcome the Brownian forces to ensure that a significant number of particles are trapped. The motion of a particle in a fluid subject to Brownian motion has long been studied using statistical mechanics. At the simplest level the particle is assumed to move freely in the fluid, subject only to external driving forces, a damping force due to the viscosity of the fluid, and a random fluctuating force arising from molecular collisions. From the equation of motion of a single particle it is possible to derive an equation for the evolution of a distribution of particles. The resulting equation is known as the Fokker-Planck equation [13–14, 26]. Using the formulation of Lasota and Mackey [26], we begin with a linear array of first order differential equations

where ξ_j (t) is a “white noise” vector that consists of infinitely many independent or random impulses. This is formally the derivative of a Wiener process that represents the Brownian motion. Its functional form is not known but it is a statistically independent variable with a Gaussian probability distribution. Each component of the white noise vector has a zero mean

and a delta-function correlation with time

Eq. (9) is a stochastic differential equation that is a generalized Langevin equation. The parameters x_i describe any variables that depend on time and have both deterministic and random variations. In this regard the particles that depend on the x_i can be described by a probability density distribution u(x_i,t) that obeys [13, 26]

where

Eq. (12) is the Fokker-Planck equation, or the Kolmogorov forward equation. It describes the evolution in time and space of the probability density u of the distribution of particles under random motion and in the presence of external forces.

If x_i represents the components of the position vector of a particle in a fluid subject to an applied force per unit mass f_i(x_i) and a thermal force per unit mass qξ_i(t), the equation of motion is

where γ determines the viscous damping. The correlation strength q ² can be obtained from the solution to Eq. (14) in the absence of the external force and by relating the average energy of the particle to the temperature according to the equipartition law of statistical mechanics [13]

where m is the mass of the particle, T is the absolute temperature and k_B is Boltzmann’s constant. For motion in a viscous medium, the damping is a consequence of the hydrodynamic flow around the particle which depends on the particle radius a and the viscosity η through Stokes’ law [14]. Furthermore, by relating the mean square displacement to the diffusion of the particle, an equation for the diffusion constant D is obtained which is also related to the damping

In the limit where the viscous forces are so strong that the inertial term in (14) can be ignored we obtain

Comparing Eq. (17) with Eq. (9) and using Eqs. (11), (15), and (16) shows that b_i=f_i /γand that a_ij=2δ_ijD assuming isotropic and position independent diffusion. The resultant equation is therefore

where F(r) is the force acting on the particle at position r and we have assumed that the temperature is constant. The term in brackets on the right is the probability current density, related to the particle flux. Under the steady state condition with no net flow, we can write Eq. (18) as

By defining a function g(r) = lnu(r) and representing the force as the gradient of the optical intensity F(r)=β∇I(r), where from (1) β=2παn/c , then Eq. (19) takes the form

Because both terms involve the gradient operator, the equation is simply solved, leading to a solution for Eq. (19)

This shows that the trapping of particles that are subject to Brownian motion in an optical field under the gradient force is determined by the intensity of the light field and the thermal energy. The interesting result is that, even though the trapping force depends on the gradient of the intensity, the particle density distribution just depends on the intensity. We choose the point r ₀ where the intensity is maximum so that the density is a maximum here as well. This is the centre of the trap and the intensity falls away from this point. We require the density to be very low far from the trap center where the intensity is small or zero. Then from Eq. (21) this occurs when exp(- βI(r ₀)/k_BT) ≈0 or βI(r ₀)≫ k_BT, which is what we would expect for a trapping field. That is, the trapping energy should be very much greater than the thermal energy.

The scattering force depends on the Poynting vector which can be written in terms of the intensity of the light. Because the particle flow is limited by viscosity, then this results in a constant flow parallel to the surface. There is no solution to Eq. (19) with this force which means that the assumption of zero probability current density in Eq. (18) is false, as we would expect. Therefore, the trapping by evanescent fields on a surface is due only to the gradient force. Combining Eqs. (8), (16), and (17) the steady-state velocity is given by

and the probability current density is J=u(z)v(z). Flows of gold nano-particles driven by the scattering force were observed in the evanescent fields above a waveguide [17–18].

4. Optical trapping vs. Brownian motion

In this section we investigate the trapping of nano-particles by evanescent waves in the presence of Brownian motion. For simplicity we assume the evanescent fields are in the region above a planar surface and are formed by total internal reflection or by surface plasmon resonance. We consider an experiment in which a fluid contains particles at some background density u ₀ at a point far from the trapping region. A long time after the trapping field has been applied, the density as a function of position can be found by rearranging Eq. (21) and including Eq. (5) and the gradient force from Eq. (1), yielding

This gives the steady state density of particles in the light field as a function of position. It rapidly approaches the background value for z > δ depending on the strength of the trap which depends on the intensity of the light. As an example, we investigate the trap densities for a range of particle sizes and light intensities based on Eq. (23).

4.1 Total internal reflection traps

For the evanescent wave we consider the interface to be between glass and water. The refractive index of glass is approximately $n_i=\sqrt{{\epsilon }_i}\approx 1.54$ and that of water is$n_t=\sqrt{{\epsilon }_t}\approx 1.33$. The critical angle for total internal reflection is then 59.73°. If the particles are polymers with a refractive index of 1.5, then the effective polarizability Eq. (3) is approximately α ≈ 0.083a³ where a is the radius of the particle. At room temperature, T = 293 K , the increase in density from Eq. (23) is

Clearly, the largest density is found at the glass-water interface and for this to be substantial we require 5.72×10¹¹ a ³ I(0)≫1. When this factor equals 1, the density is increased by a factor of 2.7 due to enormous light intensity is required to increase the particle density for very small particles, such as proteins. For example, a 10 mW laser focused into an area of 1 micron square would only trap particles of radius 56 nm and larger whereas a 100 mW laser would only trap particles in excess of 25 nm radius.

 

Fig. 1. The light intensity required to increase the density of particles (n=1.5) in water by a factor of 2.7 as a function of particle radius. The light intensity represents the power focussed into an area 1 square micron. The water temperature is 20 Celsius.

Download Full Size | PDF

4.2 Surface plasmon resonance traps

As discussed in the introduction, various schemes have been devised for increasing the electric field intensity using the interaction of light with metallic structures. In this section we analyze the trapping of nano-particles in the evanescent field associated with surface plasmons in a gold film. The gold film is taken to be deposited on a glass substrate and is illuminated from below by light. The top of the film is immersed in water containing the nano-particles. The surface plasmon resonance for a given wavelength occurs at a specific incidence angle (for example, see Raether [27]). The plasmon resonance produces strong absorption of the light reflected from the film. The reflection curves for two wavelengths are shown in Fig. 2(a). The optimum thickness for the resonance at wavelengths 630 nm is about 50 nm and for 850 nm is about 60 nm.

 

Fig. 2. (a). The calculated reflectivity of light from a thin gold film on a glass substrate covered with water. The gold film thickness t shown for each wavelength is close to optimum for strong surface-plasmon resonance. The refractive index of the glass was n=1.54 and that of water n=1.33; (b). The intensity (modulus square) of the electric field relative to the incident field above the gold film surface. At 630 nm wavelength the intensity increases by a factor of 29.2 and for 850 nm it increases by a factor 131.

Download Full Size | PDF

The relative intensity of the electric field of the evanescent waves above the gold film for both wavelengths is shown in Fig. 2(b). The incident light has a unit electric field amplitude. As expected there is a significant increase in the electric field intensity associated with the surface plasmons. The electric permittivity of the gold is larger at the longer wavelength so there is relatively less absorption, which accounts for the greater increase in the electric field. From Fig. 1, the surface plasmons would enable a 10 mW/μm² beam to increase by 2.7 the density of particles with radii 18 nm at 630 nm wavelength or 11 nm at 850 nm. If the nano-particles are made from gold then the relative dielectric constants are large so that α ≈ a ³. This gives another factor of 12 which would push the trapping limit down to particles of radii 7.9 and 4.8 nm at 630 nm and 850 nm wavelengths respectively. This is based on the ratio R = αI(0)/k_BT being unity so the trap is not very strong. This is a statistical measure of the increase in particle density but gives no information about how long a given particle will remain in the trap before thermal excitations expel it.

4.3 Lifetimes of particles in optical traps

The average time during which a given particle is trapped can be derived using a similar method as Kramers who derived the rate of diffusion over a barrier [28]. The rate at which a particle escapes is determined by the probability of finding a particle in the trap and the flux of particles. The flux is obtained from Eq. (18) in one dimension as follows [13],

where the gradient force is given by Eq. (20). Assuming J is constant, then Eq. (25) can be rearranged and integrated from the surface of the metal film z=0 to some point in the trap at z=b to give

At steady state the total flux is zero because there are as many particles entering as leaving the trap. This occurs in Eq. (26) because the two terms in the numerator cancel. However, if we consider one particle in the trap and assume that when it reaches point z=b it is prevented from flowing back into the trap, then u(b)=0 and Eq. (26) gives a non-zero flux. If p is the probability of funding a particle in the trap, then we have approximately

where the steady-state density Eq. (21) has been used. Since the lifetime of the particle in the trap is the ratio of the probability to the flux then

The integrals depend on the intensity Eq. (5) which varies exponentially. The solution can be written in terms of an infinite series in the integration limit b which determines the edge of the trap. This point is not clearly defined since the laser intensity continues to fall exponentially away from the surface. A useful definition is the distance where the energy of the particle in the trap equals the thermal energy. Using the ratio R=βI(0)/k_BT then this point is located at a distance b=δln(R) from the surface. Then the lifetime of the particle in the trap is given by

where

The function S(R) is shown in Fig. 3. The sums are stopped when the N ^th term contributes less than 1×10^-6of the total. Beyond R=10 the function varies approximately as S(R) ≈ 0.05 exp(0.925R), as determined by a linear regression of the logarithm of the data in this region.

Note that the coefficient δ²/D in Eq. (29) is about twice the time taken for the nano-particle to diffuse a distance equal to the decay length of the evanescent field. When R=2.5 the integral S(R)=1. The size of the trap for this condition is b ≈ 0.92δ , based on our definition in terms of the ratio R. So, for this ratio of trap energy to thermal energy, the lifetime of particles in the trap of size ~ δ is approximately twice the diffusion time over this same distance. The diffusion coefficient depends on the viscosity of the water, the temperature and the particle size Eq. (16). At 20 Celsius, the viscosity of water is about η ≈ 1 × 10^-3 N.s.m^-2 [29], then from Eq. (16), the diffusion coefficient for a 1nm radius particle in water is approximately D=2.14×10^-10m².s^-1. This varies inversely with particle radius. Using this information and Eqs. (29) and (30) the trapping time can be calculated as a function of particle size. This is shown in Fig. 4 for the two surface plasmon wave traps for a number of laser intensities. The particle refractive index was 1.5 as before. Also included is the diffusion time δ²/D.

 

Fig. 3. The integral S(R) (eqn. 30) as a function of the ratio R of the maximum trap energy to the thermal energy.

Download Full Size | PDF

 

Fig. 4. The lifetime of a particle in a surface plasmon trap as a function of particle radius and laser intensity for two different wavelengths. (a) 630 nm wavelength, 50 nm thick gold film, incidence angle 70.5 degrees; (b) 850 nm wavelength, 60 nm thick gold film, incidence angle 63.39 degrees. The temperature was 20 Celsius and the particle refractive index was 1.5. The diffusion curve gives the time taken for the nano-particle to diffuse a distance equal to twice the evanescent field decay length in the absence of any trapping forces.

Download Full Size | PDF

The lifetimes vary dramatically with particle size owing to the exponential dependence on the cube of the particle radius coupled with the linear dependence of the inverse of the diffusion coefficient. This demonstrates that the optical traps have a very sharp cut-off with particle size. The point where each curve crosses the diffusion-time curve corresponds to R=2.5 and an effective trap depth of b ≈ 0.92δ, as discussed above. Even though the maximum trap energy is 2.5 times the thermal energy, the lifetimes are still very small (much less than 1 second). This means that the trap energy must be significantly greater than the thermal energy for the trap to be effective. Using Fig. 4(a) 10 mW/μm² laser intensity at 630 nm wavelength will produce a trap with a lifetime greater than 1 second for nano-particles with radii greater than 40 nm. This is significantly larger than the 18 nm suggested in section 4.2. This shows that a simple comparison of the trap energy with the thermal energy is not a good measure of the effectiveness of an optical trap. In these examples, the function S(R) needs to be of the order 500 or more for the trap lifetimes to be at least 1 second. This means that the ratio R should be greater than 10.

5. Conclusion

The effect of thermal energy on the trapping of nano-particles by evanescent light fields has been analysed using steady-state solutions to the Fokker-Planck equation. It was shown that the probability of finding a nano-particle in the trap depends on the ratio of the optical energy of the particle to its thermal energy. An example of the trapping of polymeric particles in water by the evanescent field associated with total internal reflection in glass showed that very high intensities were required to trap small nano-particles. By placing a thin gold film on the glass, the evanescent fields can be increased by two orders of magnitude or more as a cles in a trap formed by the evanescent fields. An analysis of the lifetime of particles in the trap, based on the method used to derive Kramers’ escape rate, shows that the optical energy of the particle needs to be substantially greater than the thermal energy for effective trapping of the particles. The ability of laser light to trap very small particles in solution is limited by the thermal motion. While the laser power can be increased, this also adds heat to the trap which raises the thermal energy as well as introducing the possibility of convection forces which can disrupt the particles in the trap. This suggests that laser trapping methods are of limited use for trapping very small particles, such as proteins which have radii in the range of 2 nm.