zmax gg z’max
f
2f
1Figure 4.12: A system of lenses can be used after the axicon to re-size a Bessel beam. This is necessary to get micron-sized transverse spatial features. Misalignment of these lenses along the direction of propagation combined with the introduction of further spherical aberrations can introduce or enhancetilt in the Bessel beam.
to the fact that the radial intensity profiles for the Bessel beam’s individual rings are very slightly asymmetric. To prove this, I decided to create a computer model of the system.
4.6
Computer modelling
At the outset, it was not clear why particles migrated to the centre in an untilted Bessel beam. To get a better understanding of why particles hopped preferentially in one direction, I felt it necessary to model the potential energy landscapes particles would experience while moving under the influence of the Bessel beam. Initially, I began work on my own ray optics model to calculate the forces exerted on the particles by the Bessel beam, however a friend and collaborator of our group, Karen Volke-Sepulveda, had already developed a computational model and in the end I opted to base my approach on her tried-and-tested force calculating software. In the next section I will review her work and describe in full how the model works.
4.6.1
Ray-optics model
In the Mie regime, the interaction of particles with an optical field can be modelled effectively using ray optics. Although a geometrical analysis is generally regarded as being appropriate for particles with diametersd >10λ, I found that our ray-optics model correctly predicted the general behaviour of slightly smaller particles (2λ < d <7λ) in our Bessel beam. In this
CAMERA IR LASER MICROSCOPE OBJECTIVE SAMPLE STAGE AXICON IR FILTER DIELECTRIC MIRROR LENS
LENS COLLIMATIONOPTICS
COMPUTER
Dx
Figure 4.13: Set-up for studying particle motion in a tilted Bessel beam. A Bessel beam illuminates the sample cell from below. Tilt is generated by moving one of the collimation lenses in the optical train along the direction of propagation.
section I review the ray-optics model that was used to calculate both the optical forces and potential energy landscape experienced by a spherical particle moving in a Bessel beam as a function of the particle’s size. As stated, our model is an evolution of previous work by Karen Volke-Sepulveda [33, 15, 18] which itself was influenced by earlier ray-optics models by Gussgardet al.[34] and Ashkin [35].
The position of the centre of mass of the particle within the Bessel beam is given in cylindrical coordinates by (r, z) (since the beam is circularly symmetric, the azimuthal coordinate, φB, is unnecessary). For the ray optics analysis, our goal is to establish the
nature of interaction of the light field with individual area elements of the particle’s surface. The positions of these elements are given in spherical coordinates (R, φ, θ), where R is the radius of the particle, and the origin of the coordinate system is located at the centre of the sphere. These two coordinate systems can then be related through the expressions
r0= q
4.6. Computer modelling 108
Figure 4.14: Sequence showing 2.3µm silica sphereshopping down the washboard potential of atilted Bessel beam, into the beam core. When they reach the core the radiation pressure of the beam, which illuminates the sample from below, is sufficient to guide the particles to the top of the sample chamber where they collect in a cloud.
and
z0=z+Rcosθ (4.14)
The total force F~ acting on a dielectric spherical particle due to the optical field is then given by ~ F =nlR 2 2c Z π/2 0 Z 2π 0 I(r0, z0) Rsin 2θ−T sin 2(θ−θt) +Rsin 2θ 1 +R2+ 2Rcos 2θ t dArˆ (4.15)
where nl is the refractive index of the surrounding liquid and I(r0, z0) is the intensity of
the optical field at the surface area element under consideration. Since for our model of the Bessel beam we assume that the optical field is paraxial, the angle of incidence of the incoming ray with the surface area element is trivially the polar angle of the surface area element in the coordinate system of the sphere: θ. The angle of transmission,θt, is related
toθby Snell’s law: θt= sin−1(nl/nm) sinθ wherens is the refractive index of the material
making up the homogeneous dielectric sphere.
To evaluate equation 4.15, numerical integration with Gauss-Legendre quadrature is used to sum contributions over the particle’s illuminated hemisphere. To get an idea of how the particle behaves as a function of its radial positionrin the Bessel beam, we run the program iteratively, slowly varying the valuer. The resulting plot of forceF(r) versusrcontains all
the information about the particle we need, with equilibrium points located at the roots of the plot where the slope is negative. It can however be informative to study distribution of the potential energy as a function ofr. To generate plots of potential energyU(r) as a function ofr, we use a Runge-Kutta approach to integrateF(r) with respect to r:
U(r) =− Z rmax
0
F(r′)dr′ (4.16)
wherermaxis the maximum transverse radial position in the Bessel beam that we consider.
For the work presented in this thesis, we calculated F(r) and U(r) from r = 0 to r = rmax = 20µm. As we will see, our model was able to explain fully the reason why 2.3µm
particles migrate towards the core of the Bessel beam. It also explains the existence of intermediate inter-ring equilibrium positions for particular particle sizes (section 4.10.1), as well as predicting offset equilibrium positions (section 4.10.3). I was able to verify all of this behaviour experimentally.
4.6.2
A 2.3 micron silica particle in a Bessel beam
As discussed in section 4.5.3, I observed that 2.3µm silica particles migrated towards the core in an untilted Bessel beam with a ring spacing slightly larger than the sphere diameter. Initially I was surprised by this, having expected that atilt would have to be present in the optical intensity distribution to facilitate a bias in the particle movement.
4.6. Computer modelling 110
Figure 4.15: (a) Plots showing the predicted radial force and potential energy plots for a 2.3µm silica sphere moving in a Bessel beam, whose profile is also shown. (b) Close inspec- tion reveals a slight difference in heights for adjacent potential energy barriers, enabling transitions towards the core to dominate those away from it, even for an untilted Bessel beam.
Figure 4.15(a) shows the predicted curves for radial force and potential energy for a 2.3µm silica particle in a given Bessel beam. The potential energy curve appears to vary in a manner commensurate with the oscillations of the Bessel function. Closer inspection (figure 4.15(b)) shows that adjacent potential energy barriers can in fact vary in height, with the barrier closer to the core being slightly lower. This occurs for all rings, even at distance from the core, however the magnitude of the difference between consecutive barriers increases as the particle approaches the beam core. The difference in height can be attributed to the fact that the individual oscillations of the Bessel function are not strictly symmetrical. For the 2.3µm particles, which are only slightly smaller than the rings of the Bessel beam, the bias is reasonably pronounced. For smaller particles, the asymmetry is still present however their smaller sampling area means they will experience smaller differences between consecutive barrier heights. Before we look at the experimental confirmation of the behaviour of a 2.3µm silica particle, I briefly review some of the physics that determines the particle’s response to the applied optical potential.