scopic particles
Arthur Ashkin first demonstrated controlled manipulation of microscopic particles using laser beams in 1970 [39]. This first experiment used two counter-propagating beams in order to hold particles in the centre where the radiation pressure was bal- anced. In 1986 Ashkin et al published the first single-beam optical trap which would later be coined “optical tweezers”. Since then optical manipulation has grown hugely as a field with many different applications including sorting [40, 41], measurement of biological forces [42] and even rotation of microscopic particles in vacuum [43].
1.5.1
Trapping in the Mie Regime
Optical trapping depends on the size of the particle. Particles which are much smaller than the wavelength of the trapping laser are said to be in the Rayleigh regime, whilst particles larger than the wavelength of the trapping laser are said to be in the Mie regime. In this thesis the aim is to optically trap cells which are several microns in diameter and hence fall within the Mie regime.
For particles within the Mie regime, a ray-optics approach can be used to model the optical forces. In 1992, Ashkin calculated the forces of a single-beam optical trap [44]. We consider a dielectric particle with a diameter > λ, a refractive index greater than the surrounding medium and a trapping beam with a Gaussian profile. For a ray-optics approach the incident beam can be decomposed into a series of rays of light each with intensity, direction and polarisation. An illustration of the forces exerted on a particle using a ray-optics approach is shown in Figure 1.6. Each ray
1. MANIPULATION AND PORATION OF MICROSCOPIC PARTICLES USING LIGHT
of light incident on the particle at an angle of incident, θ, undergoes a change in momentum. As momentum must be conserved, a force is exerted on the particle.
For light incident on a particle with power P and an angle of incidenceθ, it can be shown that the forces acting on the particle are [44]:
Fz =Fs = n1P c 1 +Rcos 2θ−T 2[cos(2θ−2r) +Rcos 2θ] 1 +R2+ 2Rcos 2r (1.2) FY =Fg = n1P c Rsin 2θ−T 2[sin(2θ−2r) +Rsin 2θ] 1 +R2+ 2Rcos 2r (1.3) where Fz is the force acting in the direction of the incident ray (a single ray
of the scattering force) and FY is the component acting perpendicular to the ray
(a single ray of the gradient force). R and T are the reflection and transmission coefficients of the surface at angleθ, r is the distance from the beam axis andn1 is
the refractive index of the surrounding medium. This is illustrated in Figure 1.5. The gradient force is named so due to the gradient of the incoming beam intensity (typically Gaussian in shape). The trapping beam has highest intensity in the centre and so rays in the centre of the beam contibute more toward the net force than rays at the edges of the trapping beam. The net result is a force acting on the particle towards the centre of the trapping beam. This is shown in Figure 1.6 (a). In the case of a highly focused beam (ie using a high NA objective), there is also a gradient force acting in the axial direction which gives rise to 3D trapping. The scattering force, which acts in the direction of propagation, causes slight displacement of the trapped particle in the axial direction from the centre of the focused beam. The scattering force can be used for trapping either by holding a particle against a coverslip or to counter act the drag force in a fluid flow, as is the case in optical chromatography [45].
1. MANIPULATION AND PORATION OF MICROSCOPIC PARTICLES USING LIGHT
Figure 1.5: Ray optics illustration of a single beam optical trapIllustration of an incident ray and the resultant gradient and scattering force contributions, Fg
and Fs in a single beam optical trap with a focus f. Reprinted from ‘Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime’, 61, A. Ashkin, Biophysical Journal, 569-582, 1992, with permission from Elsevier. [44]
The total force experienced by the particle is equal to the sum of all the rays incident upon it. For a Gaussian profile the intensity is not uniform and so a particle which lies off-axis will feel a net force towards the point of maximum intensity. The total force can be expressed as:
F =Qn1P
c (1.4)
where Qis a dimensionless ‘quality’ factor (Qmag = (Q2s+Q2g)
1
2. Qs and Qg are
the quality factors for the scattering force and the gradient force.).
In order to trap in three-dimensions with a single beam, a high-NA objective lens is required (NA >1.2). Without a high NA, trapping in the z-direction is not possible unless the conditions for optical levitation are met where the downward force of gravity is balanced by the upward force from incident light [46].
1. MANIPULATION AND PORATION OF MICROSCOPIC PARTICLES USING LIGHT
Gaussian beam profile Net force
(a) Lateral confinement (b) Axial confinement
Force Force
Net force
Figure 1.6: Illustration of optical tweezers(a) If the particle is off axis com- pared to the laser beam (Gaussian profile), the net force will be towards the most intense part at the centre. (b) 3D trapping is obtained by using a tightly focused laser which results in a net force toward the laser focus.
The workstation described in Chapter 3 and Chapter 4 is designed to optically trap cells in 3D. As cells are typically&20µm in diameter they fall within the Mie regime and can be trapped in 3D when using a high-NA objective.