Physics of the optical trap can be explained very well in terms of the size of the particle (micron sphere/bead) relative to the wavelength of the trapping light. Particle size (diameter in comparison to the wavelength helps us deciding which approach (method) is better: Ray optics Or Electric-dipole Approximation.
Different Regime
When the wavelength is larger than the diameter (λ>d: Rayleigh regime, WLDR), electric-dipole method is used, when it is smaller (λ<d: Mie regime, WSDM) ray optics method is used. But there is a discrepancy here: neither method is appropriate, when the particle is in intermediate regime (d≈λ), unfortunately this is our case. To deal it, we will have to extend Mie theory to the case of higher convergent beam, experimental this case is no different. Forces for this case can be fairly approximated thorough ray method.
Ray Optics Method
In the ray optics (geometrical optics) regime, the Gaussian beam can be decomposed into individual rays, each with its own intensity, direction and polarization, which propagates in a straight line of uniform refractive medium, and follow Snell's law[1] at the interfaces. Each ray has the characteristics of a plane wave of zero wavelengths and can change direction (reflection and refraction) at interfaces, and change polarization at dielectric interfaces following Fresnel equations [2].
A graphic view of simple ray model is shown in the figure. The trap consist of an incident parallel beam of Gaussian TEMoo mode and arbitrary polarization enters a high NA numerical aperture objective such the one we are using [3]. The rays are focused one by one to a dimensional less focal point f (original focal point).
Force Diagram
Three cases are shown: On the top when focal point is below the bead center, each ray is reflected and refracted at two points (ignoring multiple reflections and refractions inside the sphere). At the points of refraction there is a momentum change (dash lines) in the direction orthogonal to the direction of the original ray and a momentum transfer in the direction of the original ray at reflection. So there is an equal and opposite optical force for each ray when it leaves the sphere. The direction of this force (gradient force) is orthogonal to the direction of original ray (ray before it hits the interface) and in the direction of momentum transfer. At reflection the force is in the direction of the original ray (scattering force). The combination of these forces is called the trapping force. The forces act through the sphere center, in this case the resultant force will be towards the point f from the bead center vertically down and hence pull the bead down into the trap. When the point f is above the bead center the resultant force is towards f which is in the direction of the light and hence pushes the bead up to the trap center. When the point f is on the side of the bead center the force is horizontal towards the point f transverse to the direction of the light and hence the bead moves left. For simplification the resultant force can always be drawn from the bead center to the focal point f in the range of trapping distances. For computation the directions of the forces and rays are measured relative to beam axis and surface normal.
The math for the model is developed on these bases.Single beam force trapA must read paper for single beam gradient force is Forces of a single gradient laser trap on a dielectric sphere in the ray optics regine by A.Ashkin[4]. All the pictures I am using are from the link.
A ray is incident on the sphere at an angle θ to the surface normal. The linear momentum of light of wavelength λo can be expressed as:
Where p is momentum, E is energy and c is the speed of the light in vacuum. We know the relations
[math]\displaystyle{ {c}={\lambda_o}{f} }[/math] and [math]\displaystyle{ {E}={h}{f} }[/math] so
[math]\displaystyle{ \mathbf{p}= \frac{E}{c}=\frac{hn_m}{\lambda_o} }[/math] where nm is the refractive index of the medium. This is the momentum of a single photon.
Where P is the incident power and Q is the dimensionless factor. Since force is a function of incident angle and power. Power depends on the Fresnel law of refraction and reflection; this all is taken care by the dimensionless factor. [math]\displaystyle{ \frac{nP} {c} }[/math] is the incident momentum per second. Where P is the incident power and Q is the dimensionless factor.Since most scattering spheres are in the aqueous medium (water)of some refractive index nm, so the effective refractive index of the particle is [math]\displaystyle{ {n}=\frac{n_p}{n_m} }[/math]. Where np, is sphere refractive index in vacuum. In our case we use Polystyrene beads with refractive index [math]\displaystyle{ {n=1.2}=\frac{n_p=1.6}{n_m=1.33} }[/math].
The force on the particle can be divided into two: Gradient Force and Scattering Force. Total force is a combination of these two forces. Computation of the total force on the sphere consists of summing the contributions of each beam ray entering the aperture at the radius r with respect to the beam axis and angle β with respect to the y-axis (fig:). The effect of neglecting the finite size of the actual beam focus is negligible for spheres much larger than the wavelength. The point focus of the convergent beam gives the right direction and momentum of the each ray with polarization. The rays then reflect and refract at the surface (interface) of the sphere giving rise to the optical forces.
There is a discrepancy here due to the use of highly convergent Gaussian beam. We are ignoring the curvature of the phase-front, that its planner, which is not correct. Gaussian beam has a planner phase-front at the focus which changes to highly curved as beam moves to the far-field (distance larger than the Rayleigh range). Expansion angle (diffraction angle) for highly convergent beam can be as large as 30o; this does not fit with geometrical optics. Another important point is; Gaussian beam propagation formula is strictly correct only for a transverse polarized beam in the limit of small far-field diffraction angle. This formula therefore provides a poor description of a highly convergent beam used in trap.
The proper wave description of a highly convergent beam is much more complex than the Gaussian beam formula. It involves strong axial electric field components at the focus and requires use of the vector wave equations as opposed to the scalar wave equation used for Gaussian beams. So this model does not fit around and at the region of the focal spot, but it is fairly close in the far-field.In our case diffraction angle is [math]\displaystyle{ {\theta}=\frac{\lambda_o}{\pi n \omega_o}=48^o }[/math], which is larger than the convergence angle φ (65o) inside the water. Inside the bead φ is much larger to be 80o. So for this range this method is applicable.
In Fig: The force due to single ray of power P hitting a dielectric sphere at the in the incident angle θ with incident momentum of [math]\displaystyle{ \frac {nP}{c} }[/math]. The total(net) force on the sphere is the sum contributions due to reflected ray of power PR and many emergent refracted rays (due to multiple internal reflections) of power PT. The total force is divided into two: Scattered force (FZ,Fs) in the direction of the original incident ray and the gradient force (FY,Fg) orthogonal to the direction of refracted ray. The case shown in the figure is general; particularly the forces are measured in the form of their components in the x, y and z directions acting from the bead center.
