Optical trapping theory

To consider the forces acting on an optically-trapped particle using a single beam, two regimes are considered, the Mie and Rayleigh regime, dependent upon the particle size.

Mie regime

When the particle size is much greater than the wavelength of light, then a simple ray-optics approach is sufficient to describe optical trapping. As light passes through a particle, it will scatter in all directions. If homogeneous light impinges on one side of a particle then the scattered light will emerge isotropically on the other side. Each photon will change direction as it scatters from the particle but the dominating effect is from photons reflected backwards off the particle. Light carries momentum so a change in direction corresponds to a change in momentum. According to Newton’s third law, the change in momentum of light will cause an equal and opposite force on the particle, pushing the particle in the direction of the light propagation [10]. This scattering force causes radiation pressure and is present in all optical traps. Homogeneous light has the

capacity to push a particle forwards; to trap a particle and manipulate it, there must be some variation in spatial intensity in the incident light, inducing a gradient force.

Considering a transparent particle with a higher refractive index than the medium, the light passing through the particle will be refracted. If the light passing through the particle is inhomogeneous in intensity then the change in momentum of light passing through will no longer be equal on either side of the particle (Figure 2.11A). The change in momentum of the higher intensity portion of the beam will be greater than the lower intensity portion, drawing the particle towards the former point by Newton’s 3rd law. If the incident light is circularly symmetric with a central intensity peak (such as a Gaussian beam) then the force acting on the particle will always be towards the centre of the beam. The particle will then be held in the centre of the laser beam [10].

For weakly focused beams, the scattering force will hold the particle against a sur- face, allowing transverse movement only. If, however, the beam is tightly focused (NA

> 1.0) then axial as well as transverse trapping can occur. Following the same ray optics principles, the gradient force acts towards the point of highest intensity axially too. This is termed an optical tweezer because the particle can now be picked up and moved in any required 3D direction [10].

The forces created by the passage of light rays of powerP and incident angle θ in a medium of refractive indexnm through a spherical bead with reflection and transmission

coefficientsR and T respectively were calculated by Ashkin [64]:

FS = nmP c 1 +Rcos2θ− T 2_{(cos(2θ−2r) +Rcos2θ)} 1 +R2_{+ 2Rcos2r} (2.1) Fg = nmP c Rsin2θ−T 2_{(sin(2θ−2r) +Rsin2θ)} 1 +R2_{+ 2Rcos2r} (2.2)

For trapping to occur, the gradient forceFg must be larger than the scattering force FS. Increasing the incident angle relatively increases the gradient force, reinforcing the

suggestion that higher NAs, contributing a greater number of high angle off-axis rays, create higher gradient forces and therefore greater trapping efficiency. The total force

F

A

B

F

a << λ

a >> λ

Figure 2.11 – Understanding optical trapping in two limits. The higher intensity in the centre of the applied beam causes the particle to be drawn towards the centre of the trap. When the particle size (a) is much larger than the wavelength of light (λ) then optical trapping can be understood using ray-optics (A). The change in momentum of light as it refracts through the off-centre particle is greater away from the point of highest intensity, the resulting change in momentum on the bead is therefore towards the trap centre. When a << λ, the bead is approximated to a point dipole. The force acting on it is then proportional to the intensity gradient when the refractive index of the bead is greater than that of the medium (B). Adapted from Dholakiaet al. [61].

on the particle is therefore:

Ftrap=

QnmP

c (2.3)

WhereQrepresents the Q value, a dimensionless measure of trapping efficiency. Theo- retically 0 < Q < 2 (from the transfer of photon momentum) but is experimentally <

0.3 [64].

Cells are generally 10-30 µm in diameter and therefore fall into this regime. Cell trapping and trapping of 10µm beads using a novel optical fibre-based trap is demon- strated in Section 7.2, where the ray optics approximation is valid.

Rayleigh regime

When a dielectric particle is much smaller than the wavelength of light, it can be approximated as a point dipole. The forces acting on this dipole from the applied electric field is [63]: FS = Inm c 128π5r6 3λ4 m2−1 m2_{+ 2} 2 (2.4) Fg = n3 mr3 2 m2₋₁ m2_{+ 2} ∇E2 (2.5)

Where r is the size of the particle and m = nsphere/nm and E is the amplitude of

the electric field. Given that the intensity of the light, I ∝ E2_{, the gradient force is}

dependent on the gradient of the light intensity whereas the scattering force depends linearly on intensity. As long as the beam is focused tightly, increasing the intensity gradient, the particle will be drawn towards the point of highest intensity whennsphere > nm (Figure 2.11B).

In between these two particle sizes is where most optical trapping occurs. The force analysis behind this parameter space (the Lorentz-Mie regime [61]) requires detailed numerical analysis for an accurate description of this regime where neither approxi- mation discussed above is valid, though often researchers use the ray optics approach which gives reasonable agreement. Trapping of 1 and 2 µm beads, which fall into this regime, are performed for calibration purposes in Section 7.2.2.

Dual-beam traps

The trapping described above uses a single beam to trap particles, if instead two counter-propagating beams are used the particle is trapped between them. The trap- ping geometry usually employed is two opposing weakly focused beams, in this case the axial trapping is provided by gradient forces and the lateral trapping is provided by the scattering forces (Figure 2.12).

particle

distance between beam waists

F

F

F

F

Figure 2.12 – Counter-propagating dual-beam traps employ scattering forces from two weakly-focused beams facing each other to hold a particle between them. The gradient force keeps the particle in the middle of the trapping region.

provide many advantages. The lower NAs used allow the use of longer working distance objectives or even optical fibres. The lack of a high intensity focal spot also reduces heating in the surrounding area. Using this trapping geometry, it is possible to trap larger objects than can be trapped with a single beam, including 100 µm beads [65], embryos [53] and microorganisms [66].

Dual-beam traps are also used for cell-stretching experiments [9], where cell popula- tions can be distinguished based on their deformability, and are useful for incorporation into microfluidic systems for high-throughput biophotonics experiments.