Orígenes de las Respuestas

The trapping force can be separated into two components. The force in the x direction has four principal components to it, these are the transverse gradient force, gravity, an electrostatic attraction or repulsion and a destabilising Brownian motion. The gravita- tional and electrostatic forces have no component in the lateral, y direction, but there is still a lateral gradient force and Brownian motion. The electrostatic force has been omitted from the following analysis, due to the complexity of the force and due to the values being uncertain for small separations. The trapping forces are only considered for the case of gold particles as expressions for the optical forces for Mie particles in an evanescent field were not obtained.

Ideally a force would be obtained for Brownian motion that could be directly compared to other forces such as the optical forces and would therefore determine whether a particle would theoretically be trapped. However even with a very large trapping force the particle will still have random motion. The approach taken here is to look at the distance travelled under different force conditions and is a method modified from that of Hughes et al. [118], in their case for the application to dielectrophoretic trapping. In this approach the time taken, tbr, for a particle to be expected to travel a distance,

d, in one dimension is calculated as shown in section 4.4. Similarly the time taken, t_tr, for the trapping force, Ftr to move the particle the same distance, d, is calculated using

Stokes’ theory.

In the case of an optical trap:

tbr= 3πaηd 2 kT (4.47) t_tr = 6πaηd Ftr (4.48)

Parameter Dimension Value (S.I. units) Gradient x 20×1016 Gradient y 1.5×1016 Length x 29.3×10−9 Length y 960×10−9 Modal power - 1 Wavelength - 1.066×10−6 Effective index - 1.51

Table 4.6: Parameters used for predicting the ability to trap a gold particle.

For a particle to be trapped clearlyt_tr should be much less thant_br. τ is defined as:

τ = tbr

ttr =

F_trd

2kT (4.49)

The ratio, τ, required for a particle to be ’trapped’ is clearly rather arbitrary, and in [118] a factor of 10 is used. This method has the advantage of taking into account the distance over which the force acts. To explain this, consider a large gold particle in water in a cell. Gravity will act on it throughout the cell whilst the optical gradient force in the transverse direction will act only over a distance of a few hundred nanometers. Therefore to consider simply the magnitude of the forces is over-simplistic.

Here the parameter, d, is given by L, the parameter that was introduced in chapter 3. This explains the relevance of the parameter, ∇IL from section 3.5.5 as the trapping force is proportional to∇I.

From section 3.5.5 values for the maximum intensity gradient and the length over which this is roughly constant are obtained. The values used are detailed in table 4.6.

Values ofτ /dare plotted below in figure 4.10 along with the gradient and gravitational forces for a range of particle radii.

50 100 150 200 250 −18 −17 −16 −15 −14 −13 −12 particle radius, nm log 10 F (N) 0 50 100 150 200 250 −18 −17 −16 −15 −14 −13 −12 particle radius, nm log 10 F (N) a) b)

Figure 4.10: a) Transverse and b) lateral analysis of Rayleigh forces and Brownian

motion. The lines represent gradient (¥) and gravitational/buoyancy (¥) forces and

equivalent forces required to overcome Brownian motion with values ofτ /d(see equation

Power τ x intercept, nm y intercept, nm 1 10 160 105 1 5 122 80 1 1 66 45 0.3 10 - 169 0.3 5 198 128 0.3 1 104 70 0.1 10 - - 0.1 5 - 200 0.1 1 161 105

Table 4.7: Radii at which values of τ would be obtained for various powers in both the transverse (x) and lateral (y) dimensions (’-’ indicates the value is greater than

250nm).

Considering figure 4.10 a comment should be made about the resultant gravitation- al/buoyancy force. The force is shown to be much smaller than the corresponding gradient force for the parameters used, however it should be noted that gravity will act everywhere, in contrast to the gradient force which only acts over 29nm in this case. Figure 4.10 does not take this into account. A possible solution to this is to consider that the requirement for trapping is that the particle remains in the evanescent field of the waveguide. However as the evanescent field, by definition, tends to zero, this solution is not straightforward. An assumption that could be made is that the particle should remain within a distance of 1/e2 _{of the intensity at the surface. This would give}

a length value over which the gravity could be said to be relevant of the penetration depth,δ, of the waveguide which for the waveguides used in this project is approximately 330nm. This would have the effect of shifting the gravitational forces in figure 4.10 up 1.06 decades.

As can be seen from figure 4.10 due to the parameter Ly being much greater than Lx,

the corresponding equivalent force values ofτ are much less in the lateral dimension. If the gravity force is omitted, this difference in lengths has the effect that the particle is more likely to be trapped laterally than transversely, despite the transverse force being much greater than the lateral force.

It should be noted that this graph was produced with a modal power of 1W. However in practice powers this high are not readily achieved (see chapter 6). The effect of using different powers is simple, for example, changing the modal power to 100mW (a more reasonable value) simply shifts the optical forces down 1 decade.

Table 4.7 shows the intercepts of the optical forces with the various values ofτ for various powers. The gravity force has been omitted as exhibiting a much smaller force even for the case of the weakest power used.

From table 4.7 it can be seen that in all cases the ability to trap laterally is greater than that transversely, therefore the transverse trapping is the limiting case. For very

weak trapping (τ = 1) where the time taken for Brownian motion to move the particle out of the trap is equal to the time required for the optical force to move the particle the same distance, the radii required are 66nm, 104nm and 161nm for modal powers of 1W, 0.3W and 0.1W respectively. Therefore for an optimised waveguide a 100nm gold particle will only be trapped weakly at a modal power approaching 1W whereas a 250nm gold particle could theoretically be trapped weakly by a modal power of 300mW. In comparing the transverse forces shown here to the electrostatic forces calculated in section 4.5 it can be seen that for a 200nm gold particle the separation at which the optical trapping force is equal to the electrostatic repulsion force is given by 40, 55 and 100nm for a 10−3_{, 5×10}−4 _{and 10}−4_{M ion concentration. Clearly this height will}

also have an effect on the strength of the Rayleigh gradient force. It should again be noted however that the electrostatic force should not be used too predictively due to the uncertainty in the model.