Figure 3.3: Dependence of the equilibrium sphere separationD on the refractive index difference ∆nfor 5µm(left picture) and 1.28µm(right picture) silica spheres. The experimental results are shown as blue data points where the errorbars indicate spread of the measurement values for 12 data sets and typically 300 measurements with at least 3 different sphere pairs. Red dots indicate the modelling results.
fields by the two spheres onto each other, the equilibrium sphere separations are expected to follow the same trend as the sphere focal length, namely that it would decrease with increasing ∆n. This trend is clearly seen for the 5µm
diameter spheres (left plot in figure 3.3), but not for the 1.28µm diameter spheres (right plot in figure 3.3). A reason for this behaviour will be given in the next section.
3.3
Bistability in optical binding
The second set of experiments was designed to explore the bistability in optical binding, see figure 3.4.
Figure 3.4: The picture shows two images of the observed bistability with different separations
D1≈8µmandD2≈19µmbetween the two spheres for constant parameters.
In particular, the stable equilibrium separationDof two 3µmdiameter silica spheres was measured as a function of the refractive index difference for a fixed fibre separations ofDf = 70, 90, 100µm±4%. The experimental results
are shown in blue in the left hand plots in figure 3.5 a), b) and c) respectively along with the corresponding numerical model results indicated by the red dots. In all of these experimental plots the blue crosses represents the overall mean value for a series of realisations, and the blue vertical bars delineate the spread in measured values. For example, in figure 3.5 b) the overall mean values of the sphere separation was taken over on average 12 data sets of about 300 measurements each with typically 3 different sphere pairs. We see that there is good overall agreement between the experimental results and the stable solutions from the numerical model; the regions of negative slope in the D versus ∆n plot being found to be unstable (see figure A.7). In particular, the experiment shows the bistability predicted by the theoretical modelling.
3.3. BISTABILITY IN OPTICAL BINDING
Figure 3.5: Experimental and theoretical results for the bistability in a two sphere system for a fixed fibre separations ofDf =70, 90, 100µm±4% corresponding to a), b), c) respectively for a variable
index mismatch ∆n. On the right the equivalent calculated potential is shown. The numbers indicate corresponding points in the graphs to aid the eye. a) The left hand plot shows the experimental (blue) and theoretical (red) data for the sphere separationD versus refractive index difference ∆nfor a fibre separation ofDf= 70µm. The right hand plot shows the corresponding theoretical plot of the potentialU
as a function ofDand ∆n. b) The left hand plot shows the experimental (blue) and theoretical (red) data for the sphere separationD versus refractive index difference ∆nfor a fibre separation of Df = 90µm.
The right hand plot shows the corresponding theoretical plot of the potentialUas a function ofDand ∆n. c) The left hand plot shows the experimental (blue) and theoretical (red) data for the sphere separation
For the fibre spacingDf = 70 in figure 3.5 a) we see that the sphere sep-
aration hovers around D = 11µm, and a bifurcation point appears around ∆n = 0.09 beyond which a new stable upper branch appears. This is also seen in the plot of the numerically generated effective potential U which is plotted on the right hand side of figure 3.5 a) as a function of D and ∆n. In particular, for ∆n = 0.06 there is a global potential minimum at around
D= 10µmand marked 4), but for ∆n >0.09 two potential minima, marked 1) and 3), are evident (along with an unstable potential maximum marked 2)). Furthermore the lower branch, which exists below the bifurcation point, exhibits the expected trend that the sphere separation decreases with in- creasing ∆n.
In contrast, for the fibre spacingDf = 90µm in figure 3.5 b) we see that the
bifurcation point is reduced to ∆n = 0.077, and both the upper and lower stable branches are equally evident in the sphere separation and potential plots. In contrary the sphere separation for the lower bistable branch has a tendency to decrease with ∆n, the sphere separation for the upper bistable branch has a tendency to increase with ∆n. Thus, in the vicinity of bista- bility of the optical binding the simple argument based on focusing that the sphere separation must decrease with increasing refractive index difference is negated, and this underlies the differences seen for the two cases in figure 3.3 with and without bistability.
As the fibre spacing is further increased to Df = 100µm in figure 3.5 c) we
see that the bifurcation point has increased to ∆n ≈0.087, but in this case it is the lower branch that appears only beyond the bifurcation point (in contrast to the case in figure 3.5 a) for Df = 70µm where it is the upper
branch that only appears beyond the bifurcation point). The plot of sphere separation versus refractive index difference is in this case mainly dominated by the upper branch with the trend forD to now increase with ∆n.
As expected from the competition of two stable solutions, the experimental fluctuations indicated by the blue vertical bars in figures 3.5 a) to c), are largest closest to the bifurcation points where new bistable branches appear. Furthermore, it is also seen that the deviation between theory and exper- iment in figures 3.5 a) to c) is largest for smaller values of ∆n. This is understood by realising that as the index mismatch decreases the net optical forces acting on the spheres also get smaller, so the equilibria are created by cancellation of ever smaller forces due to the CP fields. In this situation the numerical equilibria become more and more sensitive to the precise material parameters, whereas for larger index mismatches the equilibria are more ro-
3.3. BISTABILITY IN OPTICAL BINDING
bust against slight parameter variations.
This is due to two effects: first by raising the refractive index the respective forces on the sphere are getting smaller, figure 3.6 (right). More importantly the forces on the two beads follow equilibrium over a wider range of separa- tion distances between them figure 3.6 (left).
Figure 3.6: Left: Force acting on the spheres for different separations D between them (with a fibre separation of 90µm). The red graph shows three solutions for a refractive index mismatch of ∆nabout 0.09 the bistability point. The blue graph shows the simulation for ∆n= 0.07 the point of bifurcation, here solutions can appear between 10 to 15µm. As the forces are almost equal over a large range of separations of the spheres. The individual forces are shown on the right. Right (top): ∆n= 0.09 The individual forces on one sphere are shown; blue, the force evolving from the unperturbed field and red from the diffracted field. Right (bottom): same as above for ∆n= 0.07. Importantly the acting individual forces have decreased by≈40% from about 3pN to 1.8pN by lowering the refractive index difference.
At the bifurcation point (∆n = 0.07) there is a zero transition at D= 10µm
as shown in figure 3.6(left - blue graph). However it is not as distinct as for the bifurcation point (at ∆n= 0.09) and over a sphere separation range 10−15µm the forces from both CP fields are almost equal.
Consequently in the experiment slight differences in refractive index of the spheres and the host medium and their size can produce experimental results anywhere in this regime, even for long integration times of up to 1 minute. This can also be seen if the potential plots are compared. Especially for a
fibre separation of 100µm a large area of the potential (shown in figure 3.5 c) ) is quite shallow (dark blue area). Experimentally solutions can occur anywhere as the optical potential has no strong gradient over a wide area. Thus the observed measurement spread is dependent on fluctuation in sphere parameters which are captured by measuring different arrays and averaging over these results.
To elucidate the sensitivity, I will evaluate theoretically slight changes in sphere parameters for a refractive index difference of ∆n= 0.09.
By changing only the refractive index of both spheres by +0.3% a shift for the first stable solution of -8.2% is obtained, the second stable solution shifts by +2.1% while the unstable solution shifts by +2.6%.
Varying only the sphere diameter by +3.3% (Bangs laboratories gives the STD of the mean diameter 3µm to <10%) causes the first stable solution to shift by -4% and the second only by -0.1%, while the unstable solution changes by -15.5%.
Also slight changes of the host refractive index can shift the simulations1.
For +0.3% the first solution shifts by +17.8% and the second stable solution by -2.8% while the unstable solution is shifted by -5%.
The fibre separation can induce shifts in the data (as the system had to be re-setup for each measurement step of ∆n). Where an error of +1% shifts the first stable position by +3%, the second stable position is shifted by +0.2% and the unstable position is shifted by -0.4%.
The interplay of these parameters can change the agreement of theory and experiment significantly. However to achieve consistent results material pa- rameters were left constant for all simulations to obtain comparable results. Additionally the trapping forces acting on the spheres influence the spread of the measurements. For lower ∆n values all forces acting on the spheres are reduced in magnitude making them more susceptible to perturbations (e.g. slight flow within the open sample cell). This in turn produces a wider spread of the experimental data within the accessible theoretical regime in which the forces are following close to each other.
A closed sample cell [9] would overcome these limitations: however such a enclosed design was not realised as it would have not permitted us to in situ vary the fibre separation Df.
In the experiment these sensitivities add up and induce the relative large
1In an evaporation experiment the concentration of the host medium was found to