model.
4.3
Dispersion
Next I want to consider the pertinent parameters for the first demonstration of the femtosecond fibre trap. To date most optical trapping experiments have incorporated continuous wave laser sources. Standard single beam traps have used femtosecond sources recently for simultaneous trapping and non- linear excitation [54]. When considering the femtosecond fibre trap, group velocity dispersion (GVD) [55] and nonlinear optical effects such as self phase modulation (SPM) [55] become an issue as theyu alter the pulse duration. In the experiment the fibre length was between 30cm and 40cm to keep these effects at a minimum.
To compare the change in pulse duration, I measured the pulse via intensity autocorrelation prior to and subsequent to propagation in a 375mm length of fibre, for an input pulse of 95f s increased to 800f s after the fibre thus showing the dramatic effects of dispersion. Before the fibre the spectrum is of Gaussian shape, after the fibre the spectrum broadened significantly due to nonlinear optic effects by a factor of approximately 2.6, see figure 4.2.
Figure 4.2: An input pulse of 95f sduration increased to 800f safter propagating in 375mmlength of fibre thus showing the dramatic effects of GVD. Before the fibre the spectrum is of Gaussian shape (red), and after the fibre (blue) the spectrum broadened significantly by a factor of approximately 2.6 due to SPM.
The spectrum is not broad enough to excite one photon fluorescence in the fluorescein, which can be observed in a supercontinuum [53].
Although the pulse duration and spectrum were increased, a two-photon flu- orescence signal was still readily observed from the dye within the sample medium, indicating that the average intensity of 40mW after the fibre, which corresponds to a pulse energy of 0.5nJ and pulse peak power of 0.6kW, are still sufficient to obtain two-photon fluorescence. A comparative mea- surement with a 30mm shorter fibre showed no significant change in pulse duration and spectrum. It should be noted that nonlinear processes occur approximately within the first fewmmof fibre [56, 55]. However with a fibre length exceeding 60cma two-photon fluorescence signal was not obtained as GVD is present over the whole length of the fibre.
4.4
Two-photon excitation
The host medium for the microparticles was prepared following the studies in the previous chapter; a de-ionised water and sucrose mixture is used to produce a variable host refractive index, and fluorescein (broad excitation band centred around 480nm and emission band centred around 530nm) was added to the sample [51, 52] with a relatively high concentration of approx- imately 150±30mg/l, since our peak pulse power is relatively low compared to that of Ref. [57]. Adding fluorescein to the host medium did not change its refractive index. It is known that fluorescein marker concentration fluc- tuations can lead to linear deviations [57] in the observed two-photon signal between experimental realisations. Also, excitation power variations, due to laser fluctuations between measurements and readjustments to the field dis- tribution to center the array via the λ/2 plate, have a quadratic influence [58, 59] on the observed two-photon signal. Since the power fluctuations were small (<5%), induced deviations were therefore approximated as linear. It is important to note that the model does not account for the experimental changes in the two-photon signal due to the variations in marker concentra- tion and excitation power. However, since the theory cannot fix the absolute fluorescence signal strengths, the numerical results were linearly scaled via factors A and C in formula 4.2, to allow for comparison of the spatial profiles of the fluorescence. Furthermore these fluctuations did not influence the ar- ray formation as it is independent of power [11] as they apply to both beams accordingly. When the Ti:Sa laser was operated in the continuous-wave (cw)
4.5. VISUALISATION OF THE LIGHT REDISTRIBUTION IN OPTICAL BINDING
regime no signal was detected by the CCD camera and this is interpreted as evidence for two-photon excitation in the experiment1.
4.5
Visualisation of the light redistribution in
optical binding
Now I will move on to discuss the visualisation of the optical binding and its accompanying light intensity redistribution. As a first step, light is permitted solely to enter one of the fibers and the helper tweezers is used to hold a single microsphere (N = 1) in the path of the single laser field. The 5.17µm silica sphere is scanned through the emerging field and the observed light pattern is compared with a simulation. The image sequence (upper part of figure 4.3, images 1) to 6)) shows good agreement with the experiment and similar simulations conducted in Ref.[60]. When the bead is not fully centred light is diffracted away from the beam path creating a cone of low intensity light emerging from the rim of the bead, which is shown in a transverse line profile plot from a to b along the y-axis in figure 4.3 (right).
1The two-photon process can be readily observed by eye - the photo of the sample cell
Figure 4.3: Top: 1) to 6) image sequence showing a 5.17µmsilica sphere being scanned into the beam path. Black and white images show the experiment, the false colour images show the corresponding theoretical simulations. Bottom: Diffraction pattern of a 5.17µmsilica sphere being offset along the y-axis by an optical tweezers. The sphere position (overlaid white circle) is at approximately 50±1µmfrom the beam waist. The insert on the right shows the intensity distribution along the y-axis from a to b at 8±1µmafter the sphere which has an offset from the beam axis of 3±0.5µm(experiment: blue dots; theory: red line). Light is diffracted away from the beam path creating a valley of low intensity light at the beam axis.
4.5. VISUALISATION OF THE LIGHT REDISTRIBUTION IN OPTICAL BINDING
To calibrate the model to the fluorescence intensity the center line inten- sity distribution was extracted from an image of the optical field emerging from the single mode fibre (modefield diameter 5.0± 0.5µm at 850nm) and compared to the theoretical simulations of an unperturbed field in water2,
shown in figure 4.4 A). At a propagation distance (z) of 55µm the optical field of the 800nmexcitation wavelength has decayed beyond the excitation threshold of the fluorescein. At this point theory and experiment start to disagree, but very good agreement for over 50µm from the fibre output is observed. This disagreement is not observed in all experiments as an equal power distribution in all experiments was not feasible (as this is dependent on the average laser power) as well as concentration variation of the dye su- crose solution can have an significant impact in the cut off behaviour of the observed signal.
Figure 4.4: A) Single beam emerging from the fibre without sphere 1) On-axis or centreline intensity distribution (red dots - experimental data; blue line - theoretical prediction). 2) Theoretical simulation. 3) Experimental observation. Diffraction pattern of a 5.17µmsilica sphere held by an optical tweezers at 54±1µmfrom the beamwaist in a single beam originating from the left side of the images. Comparison between different refractive index differences: B) ∆n= 0.07. 1) Centreline intensity distribution (dots: experimental data; line: theoretical prediction). 2) Theoretical simulation of diffraction pattern. 3) False colour images of two-photon fluorescence. C) ∆n= 0.05. 1) Centreline intensity. 2) Theoretical simulation. 3) False colour images of two-photon fluorescence.
To elucidate the influence of the refractive index difference between the sphere and the host medium on the results in figure 4.4 a comparison of a single 5.17µmdiameter sphere diffracting the fibre output beam path at a distance
of 54±1µm from the beam waist for B) ∆n = 0.07, and C) ∆n = 0.053 is
presented.
For each example plot 4.4 shows a comparison of the experimental and nu- merical on-axis fluorescence signals (y= 0), whereas plots 2 and 3 show the numerical and experimental fluorescence profiles over the yz-plane, respec- tively. This numerical labelling of the plots is used in all subsequent plots. These results show that the higher the refractive index difference the more the light is refocused after the sphere, as expected intuitively, and this ob- servation is at the heart of the interpretation of how optical binding works, at least in the Mie size regime considered here. More specifically, the focus- ing length of the sphere in the small-angle approximation (measured from the sphere center) is f = R/(2∆n) which yields f ≈ 18µm, in reasonable agreement with the results in 4.4 where a focus appears at around 70µm.