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medium is very small a subtle refocusing effect was readily observed and is shown in figure 4.5. Importantly here the refocusing effect is less pronounced as with a higher refractive index difference shown in the previous section with spheres. This indicates that diffractive refocusing is the underlying principle of optical binding between cells as well.

4.7

Optically bound arrays

Next I want to consider the case of optically bound arrays with CP laser fields. The number of spheres in an array and the associated separation is dependent on various parameters such as wavelength, waist size and separation, refrac- tive index difference and sphere diameter which has been investigated in the previous chapters; here the focus is the field redistribution that accompanies optical binding and its measurement using two-photon fluorescence. In the previous chapter it was observed that the optically bound array spacing can vary due to variations in sphere size and/or refractive index, even for spheres from the same batch that should nominally have the same properties. To avoid this only a single realisation is considered of each array, so that the array spacing is fixed, and this allows for comparison between theory and experiment.

Optical binding with CP fields arises from the fact that the net force acting on each sphere has two components deriving from the force exerted from each laser field. Considering the case of two spheres for illustration, a given sphere will experience a direct force from the field emanating from the clos- est laser fibre end, and a second oppositely directed force from the refocused laser field emanating from the other fibre. Balancing of these two forces is the usual explanation of how optical binding can arise, and the extension to more particles follows. Figure 4.6 A) is for example of an optically bound array of two spheres (N = 2), and shows the intensity distribution profile for two 3µm spheres with a separation of 8µm, and very good agreement is obtained between the numerical and experimental profiles.

Figure 4.6: A) Two 3µm sphere array with a separation of 8µm at ∆n=0.06. A1) Centreline intensity distribution showing the full waist separation of 72µm(blue - experimental data; red - theoretical prediction). A2) Theoretical simulation of diffraction pattern in a 2 sphere array. A3) False colour image of two-photon fluorescence. B) Same array as in A) with right field blocked at this image the spheres got a separation of 9µmthe array centre point has got approximately 32µmdistance to the beam waist. B2) Theoretical simulation of diffraction pattern in a 2 sphere guiding configuration. B3) False colour image of two-photon fluorescence beam coming from left side of picture. C) Three 3µmsphere array with a separation of 5µmat a ∆n=0.05 with a waist separation of 100µm. C1) Centreline intensity comparison between theory and experiment which is being cut off at 60µm. C2) and C3) theoretical and experimental images of diffraction pattern. D) Three 3µmsphere array with a separation of 12µm∆n=0.02 with a waist separation of 90µm. D1) Centreline intensity comparison between theory and experiment. C2) and C3) theoretical and experimental images of diffraction pattern. E) Four 3µmsphere array with a separation of 12µmat a ∆n=0.01 with a waist separation of 85µm. D1) Centreline intensity plot. D2) Theoretical image matching D). F) Two 5µmsphere array with a separation of 11µm∆n=0.04 with a waist separation of 90µm. D1) Centreline intensity comparison between theory and experiment. C2) and C3) theoretical and experimental images of diffraction pattern.

4.7. OPTICALLY BOUND ARRAYS

In particular, the profiles clearly show that the intensity is refocused after the spheres, in keeping with the physical picture of optical binding. Fig- ure 4.6 B) is the same as figure 4.6 A) except the right laser field has been blocked, causing the particles to be propelled to the right due to imbalance of the optical forces now acting on the spheres, and for an elapsed time such that the particle spacing had increased to 9µm. Good agreement is obtained between theory and experiment, and this example further shows that two- photon fluorescence can be used as a tool to obtain real-time monitoring of the dynamics of optically bound arrays. To obtain binding for larger arrays the refractive index difference needed to be lowered in order to inhibit the col- lapse of the array into a closed chain [17, 11]. Qualitatively this phenomena may be explained as follows: a lower refractive index difference subsequently causes less light being refocused by a sphere (as was shown in figure 4.4 onto its nearest neighbour in the array. For the case of 3µm sized spheres this means that by decreasing the refocusing effect of each individual sphere, the balance of the forces from both CP fields is still maintained for a higher num- ber of spheres N. The results in figures 4.6 C) and D) are for optically bound arrays with N = 3. Figure 4.6 C) shows three spheres bound in an array, when ∆n is changed to 0.05. The separation of the spheres decreased to 5µm

and did not permit the optical field to emerge further between the spheres. If the refractive index difference is further lowered the separation of the three sphere array increases to 12µm thus permitting the fields in between the spheres to evolve further. It can be clearly seen that the refocussing effect has decreased significantly as both exit peaks of the intensity at either end of the array have decayed off in comparison to C). With a refractive index difference of 0.01, N = 4 spheres can be bound in an array whilst having a very large spacing of 12µm , as shown in figure 4.6 E). Here again the reduced refocusing effect can be clearly seen at each end of the array, where the on-axis intensity peak has significantly decreased in comparison to figures 4.6 A) or C).

Additionally a two 5µmsphere array with a separation of 11µmat ∆n = 0.05 is shown. Here the field between the two spheres has hardly evolved even though the refractive index difference is relative high.

These examples amply demonstrate that imaging of two-photon fluorescence is a reliable tool for visualising the redistribution of intensity in optical bind- ing of arrays, and that our model for the field propagation is valid in the Mie size regime considered here.