Explotación sostenible de los recursos naturales

10−10 10−9 10−8 10−7 10−6 10−5

Particle / Beam ratio

Normalise d For ce (N W − 1 )

Figure 4.5: Transverse trapping force (hollow, red) and axial levitating force (solid, blue) normalised to the absorbed laser power in the respective trans- verse and axial direction recovered from the measurements of trapping stiff- ness in the transverse plane and by taking the mass of levitating particles along the axis direction. It should be noted here that the power absorbed in transverse direction is significantly lower than in axial direction. The solid lines are the power fitting to the experimental data of force normalised to laser power.

fitting to the experimental data is the same for both forces, it is_∝ρ−3.1whereρis

the particle-beam ratio.

4.4

Levitating particles with spatially variant azimuthal

and radial polarisation in the beam

First, I again demonstrate how the dependence of the photophoretic force is strongly affected by the polarisation state of the driving light field. The state of polarisation of cylindrical vector beams, such as a radially or azimuthally polarised beam shown in Fig. (2.5), can be varied while keeping the intensity and other parameters unchanged. This work[58] extends and generalises the previous experiments on transport of spherical particles in air with counter-propagating optical vortex beams having

orthogonal linear polarisations.

The experiment was carried out at atmospheric pressure with the large carbon- coated, hollow glass spheres first described inFig. (2.1) and Fig. (2.2). For this experiment, particles with a diameter of 80 µm to 110 µm were selected. At atmo- spheric pressure, in the low Knudsen regime, the photophoretic force on a spherical particle exposed to one-sided electromagnetic radiation can be approximated by [11, 31]:

F_ph=9µ_a2P_abs/4aρ_aT k_p +2k_a

, (4.1)

wherek_p,a are the thermal conductivities of the particle and the surrounding gas, µa andρa the viscosity and density of the gas, andPabsandT are the total power absorbed by the particle, and the average gas temperature.

The experiment was performed in the optical levitation geometry shown in Fig. (4.6). Either a radially or azimuthally polarised continuous wave 532 nm laser beam with a doughnut intensity profile was used to levitate particles. The radially and azimuthally polarised beam was synthesised by means of a uni-axial crystal[59]. By rotating the second half-wave plate after the pinhole, seeFig. (4.6), the state of polarisation can be rotated from azimuthal to radial, and vice versa, without affecting the intensity profile of the beam, as shown in the bottom left panel of the figure.

Particles were launched into the levitation setup from the top atz =200 mm by

flicking a paintbrush with particles adhering to it near the upper end of the trapping cuvette. At this position and height in the beam, the beam itself is wide enough for the particles to be easily placed into its dark centre. The particles are confined by the steep gradient of the hollow core doughnut mode, and fall into the trap while slowly decelerated due to the increase of photophoretic force as the beam gradually tapers down along the axis of propagation towards the focus, which marks the lower end of the trapping region. The equilibrium position, where the downward force of gravity is counterbalanced by the photophoretic force and light pressure forces, exists at different powers. At constant power, any changes introduced into the beam that affect the equilibrium position of the particle indicate that the trapping efficiency has been changed, where an uplift in the particles axial position away from the beam focus is considered an increase in the trapping efficiency. Changing the state of polarisation of the beam from azimuthal to radial, the axial force is increased due to the increase in power absorption “efficiency" as shown inFig. (2.5)and experimentally observed inFig. (4.7)at constant vertical positionzaz. This increase in absorbed power leads to the particle seeking a new equilibrium position slightly higher. This motion will ceases when the equilibrium condition is reinstated in a new stable position with

4.4. Levitating particles with spatially variant azimuthal and radial polarisation in the beam

Figure 4.6: Optical levitation measuring the polarisation sensitivity of the photophoretic force. (Top) The c-cut calcite crystal (CR) converts a circularly polarised single-charge vortex beam into a superposition of the radially and azimuthally polarised beam (see inset). The polarity and handedness of the vortex must be of opposite signs. The pinhole (P) selects either the azimuthal or radial polarisation. The polarisation state of the beam after P is varied using two half-wave plates (λ/2). The lens (L) forms the trapping beam with w0 = 27 µm. The cuvette (C) protects the trapped particles from air flows.

The power meter (PM) measures the total power in the beam. The trapped particles are observed with a CCD camera under bright field white light (WL) illumination. (Bottom left) The intensity profile for the radial (blue line) and azimuthal (red crosses) polarisations atz =40 mm above the focus, also shows

the corresponding intensity distributions recorded after an analysing polariser whose transmission axis is denoted by arrow. (Bottom right) Radial profile of the trapping beam.

20 25 30 35 40 45 50 0 100 200 Axial position (mm) Laser p ow er (mW ) Azimuthal Radial

Figure 4.7: A hollow sphere 110 µm in diameter is trapped with an azimuthally polarised (strong scattering) and radially polarised beam (weak scattering) at approximatelyz =40 mm. The measured dependency of the required laser

power maintaining a desired axial position shown above for both azimuthal and radial polarisations. The solid lines are a fit toEq. (4.1). The light ab- sorbing carbon coating is 150 nm thick based the previously discussed SEM measurements, see Sec. (2.1), and can be considered fully absorbing. The mass was estimated at 27 ng by measuring its terminal settling velocity and applying Stoke’s law. This estimate is confirmed byFig. (2.2).

a new vertical positionz_rad. Conversely, if the polarisation is switched from radial

to azimuthal, the particle will find a position closer to the beam waist where the intensity is higher. If needed, the particle can be kept in the same position, provided that the laser power is adjusted accordingly in order to compensate for the introduced polarisation-related changes in the trapping efficiency. By setting the angle between the fast axes of the two half-wave plates, seeFig. (4.6)somewhere between 0 and p/4, the beam is spirally polarised, i.e., a linear superposition of a cylindrically symmetric radial polarisation and a cylindrically symmetric azimuthal polarisation. As a result of the intermediate polarisation, the power absorbed across the particle places the particle in-between the aforementionedz_az <z_sp <z_rad. The results of

the measurements[58] are quantified inFig. (4.7). A 20%-30% difference in the power required to keep the trapped particle in the same position when switching between azimuthal and radial polarisation is observed. The effect is most pronounced when the particle geometry matches the state of polarisation, which again was predicted inFig. (2.5)and experimentally observed here. This opens up a pathway of further optimisation of photophoretic forces depending on the proposed application.