Scattering force

The third group of motors use the scattering of light as an actuation mecha- nism [102, 103], importantly the rotors need to be of appropriate shape [104]. Light diffraction or scattering by a micro object in the Mie regime on the other hand opens up an alternative solution for exerting a suitable torque on a micromotor as the operating light can be delivered by means of standard waveguides [102] or fibre optics. However this involves the careful design of such micromotors and layout of the driving light geometry to gain maximum performance of such systems. In this context it is of great importance to study the light diffraction from such a motor to investigate and understand its driving mechanism.

Here I investigate the diffraction from an optical micromotor and its rotation due to optical scattering. Experimentally by utilising two-photon imaging it is possible to visualise the diffracted light from the micromotor. Previously two-photon imaging has been used as a method to visualise the reformation of Bessel light beams [51, 52]. The numerical model developed to calculate the diffraction of a beam by a sphere in the previous chapter is modified to theoretically predict the observed diffraction pattern from the micromo- tor. This demonstrates a simple but powerful method to predict diffraction patterns of arbitrarily shaped micro objects within the constraints of the paraxial approximation. I shall present experimental and theoretical results for a micromotor being driven by the scattering force of light delivered by an optical fibre, as illustrated in figure 6.2.

6.2. THEORETICAL MODEL

Figure 6.2: 3D image of the micromotor setup. The 10µmmicromotor is held in place by a pole structure. The operating light is delivered via fibre optics (shown on the right hand side). The operation principle is similar to a water wheel, where an off center water-stream interacts with the spokes of the wheel.

carry out a theoretical performance analysis of the gear design, the layout of the light source-motor system and indeed even perform optimisation of the micromotor geometry. Firstly I want to describe the theoretical model used and then progress to discuss the experiment and data acquired.

6.2

Theoretical model

The theoretical model comprises of a monochromatic laser field of wavelength

λ propagating along z and originating from a single mode fibre, shown in figure 6.3.

Figure 6.3: (a) The set-up consisted either out of a femtosecond Ti:Sa laser for the visualisation of the light field distribution or an Ytterbium laser for the motor rotation rate experiments. The light of either laser was coupled into a fibre and split via a 50:50 splitter into equal parts (for the Ti:Sa visualisation experiment no splitter was used as the power was left constant). The light in fibre 1 was used to drive the motor whereas the light in fibre 2 was monitored via a power meter. The inset shows a not to scale side view of the experiment which was observed from underneath with a microscope and a CCD camera (see, (b)). The motor is levitated by a coverslip to be inline with the core of the fibre and hence with the emerging light field. The propagation distance z of the field emerging from the fibre is marked. For the theoretical calculations I assume the motor of 2µmheight (h) to be centred in the beam emerging from the fibre. (b) Topview brightfield image of the experimental setup. The micromotor with supporting pole structure is shown in the upper left side of the image. The motor is fabricated on top of a microscope coverslip slide (the edge of which is shown in the picture) to compensate for the 60µmthick fibre cladding. On the right the fibre end facet is shown from which the laser field emerges in the positive z-direction with z being the distance from the fibre surface to the motor. The fibre is offset from the micromotor centre (the axis of revolution) by x to allow the field to interact with the motor spokes. The beam direction is shown in the image. The positive offset in x would make the motor turn counter clockwise as indicted.

6.2. THEORETICAL MODEL

The beam waist is located at the end facet of the fibre and separated by a distance z from the outer rim of the micromotor. To give an uneven mo- mentum distribution the centre of the beam is offset by x from the axis of rotation of the motor, where the sign of x denotes from which side the motor is driven causing either an anticlockwise (+) or clockwise (-) rotation of the motor (see figure 6.3 (b)). The field is modelled as a Gaussian beam emerg- ing from the fibre facet with spotsize w0 and power P. It interacts with a

micromotor of refractive index nc = 1.62 (material SU8, approximated to a

wavelength of 1070nm) immersed in a host medium (de-ionised (DI) water) of refractive indexnh = 1.33. The micromotor is assumed to be centred ver-

tically in the beam (with y=0) and to have a height h. Notably, this method allows for simple and fast micromotor geometry evaluations as any geometry can be realised. Applying the paraxial wave theory [18] and equation A.10 it is possible to calculate the evolution of the optical fieldE(x, y, z) distorted by the refractive index variation represented by a binary image of the motor1_.

Where the colour white refers to nc, the pixel determine the grid resolution

of the propagation in z and x and can be scaled accordingly with the binary picture of the micromotor. An example is shown in figure 6.4.

Figure 6.4: Left: Binary image of the motor in the model, the pixel size of the image was normalised to represent the actual motor size. In the model the motor was located at a position z from the beam waist (located at z=0) and could be offset from the central beam propagation axis (in the figure the beam is entering the motor from the right hand side) by x. Where x=0 would donate the beam is aimed at the middle of the binary image. Right a) top view of the pole structure with motor and b) side-view.

Previous studies have used the ray optics model [105, 106, 107], with different driving geometries. Both ray optics and paraxial wave theory are approxi- mated approaches which can give sufficient accurate results when used in the appropriate circumstance. In this chapter I want to focus on the paraxial ap- proximation as I assume that the main driving mechanism for the micromotor

investigated is forward scattering of light. Here I utilise my experience in op- tical binding where the paraxial approximation has been successfully applied to model light matter interaction in the Mie regime (where the wavelength is magnitudes smaller than the object).

In contrast to the ray optics approach the paraxial approximation does not take polarisation effects into account, which are averaged in the calculations and considering that the fibre used is not polarisation maintaining I believe this as a valid approximation. Additionally the paraxial approximation used in this work is well tailored for a waveguide or fibre optics driving geometry where (in contrast to [107]) weak diverging fields are harnessed to operate the motor.

I calculate the intensity distribution by modelling the field as pill- boxes for the experimentally determined motor position (x, z). The camera observed fluorescence signal Stwo−photon(x, y) is then pro-

portional to the intensity If ield(x, y, z) of the optical field [108]:

Stwo−photon(x, y)∝

Z

I_{f ield}2 (x, y, z)dy (6.1)

The calculated fluorescence intensity distribution is shown as a greyscale im- age where white corresponds to maximum signal strength. In a next step the simulated evolution of the field within the micromotor is used to calculate the optical forceFz acting on each pillbox2 I(x, y, z) within the motor along

the propagation axis in z. This force arises due to the refractive index dif- ference between host medium nh and motornc. From the force the optically

induced torqueτ was calculated with respect to the centre of the micromotor for each individual pillbox as:

~

τ(i) =~ri×F~i (6.2)

and for the whole motor:

~

τmotor =

X

~

τ(i) (6.3)

Where ~ri are the individual position vectors for the ith pillbox with respect

to the micromotor centre andF~i being the acting force vectors. By summing

2_{Each pillbox is in xz normalised to the pixel length and width of the binary image.}

The height y of the pillbox is set by the normalisation of the grid in x, as both grids are equally spaced.