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 (redline) in the direction closely orthogonal to the direction of the original ray. 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 closely orthogonal to the direction of original ray (ray before it hits the interface) and in opposite the direction of momentum change. 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 downward 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 Oto 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 (efficiency). Q is therefore the main determinant of trapping force. It depends upon the NA, laser wavelength, light polarization state, laser mode structure, relative index of refraction, and geometry of the particle. Value of Q is in between 0<Q<1. 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.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. But over some range of NA (definitely more than 1.2)the scattering force is higher than the gradient force. The gradient force behave non-monotonically with transverse gradient field which decreases the trapping due to gradient force. And also for high NA the beam convergence is high, which results in small spotsize and Rayleigh range, which mean high divergence angle. As divergence angle gets closer to the convergence angle the ray model starts falling. So only a range of NA is best for the ray model.
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: Scattering 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.
Where θ and r are incidence and refraction angles related by Snell's law. R and T are Fresnel reflection and transmission coefficients. Since R and T are polarization dependent the force will be differet for different TE/s and TM/p polarizations. These formulas are sum over all the scattered rays and therefore exact.
Here the polarization orientation is based on the vertical (yz fig: sample plane facing us/the normal to this plane is also a normal into our eyes). In our case the laser is TM/p polarized in the sample plane. So the eqs. 1,2,3 and 4 can be used to computer simulate the trap for our case.
The absolute magnitude of the total force is the vector sub of the components:
The net trapping force is a contribution of the two forces and it is a function of θ and n. For optimum trapping θ is required (see eq. 1 & 2). Optimum θcan be achieved by using a high NA objective with over-field back aperture. In that case incidence angle θ will be equal (close) to convergence angleφ.
Force along Z-direction
The scattering force along the z-direction can be shown as:
Since scattering force is always along the z-direction the equilibrium point is always slightly below the sphere center ( see fig: S≠0). Which mean that the trap center is slightly below the bead center where the backward gradient force just balances the weak forward scattering force. Away from this point the gradient force dominates the scattering force and always has finite net trapping force.
Total force can also be computed for different forcus positions (S) along the z-direction, relative to the sphere center. In such cases incience angle can be computed through:
Where Rb is the bead radius. In ray model forces are independent of the radius Rb=1:
[math]\displaystyle{ \mathbf {\theta}= sin^{-1}(\pm S sin{\phi}) }[/math]
Where ± S is the distance between the focus f (trap canter) and the sphere center O. If focus is below the sphere center S is negative and vice-versa.
Force along Y-direction:
Fig ; in this case focus of the trapping beam is located along the y-direction. β in the angle between the y-axis and the incidence plane, γ is the convergence angle to the y-axis.α is the angle ray making to the horizontal plane. Knowing α and β we can calculate γ, and using γwe can calculate θ:
[math]\displaystyle{ cos {\mathbf {\gamma}}= cos{\alpha}cos{\beta} }[/math]
[math]\displaystyle{ \mathbf {\theta}= sin^{-1}(\pm S' sin{\gamma}) }[/math]
WhereS' is the distance between f and O along y-direction.
The net force here particularly depends on the polarization. In eqs. 1& 2 , not only R and T depends on the polarization but also the power P. For the case of an incident beam polarized perpendicular (TE) to the y-axis, for example, one first resolves the polarized electric field E into components E cos β and E sin β perpendicular and parallel to the vertical plane containing the ray. Each of these components can be further resolved into the so-called TM/p and TE/s components parallel and perpendicular to the plane of incidence in terms of these angle μ between the vertical plane and the plane of
incidence. By geometry, [math]\displaystyle{ cos {\mu} = \frac {tan {\alpha}} {tan{\gamma}} }[/math]. The fraction of power lies in the p and s components. This is a general case. In particular case if ray incidence such μ (ray is in the vertical yz-plane) is zero than the polarization can alone be determined on the bases of the vertical plane.
Knowing θ , PTMand PTE one can compute the gradient and scattering force components for p and s separately using eqs. 1 and 2 and add the results.
For arbitrary position:
Forces for the arbitrary position of focus is shown below for the first quadrant of the sphere (quadrent of the topright of the fig:). Similar expression are obtained for other quadrants.
[math]\displaystyle{ \mathbf {F_{Z}}= F_s sin{\alpha} + F_g cos {\mu} cos {\alpha} }[/math]
A.Ashkin's paper[5]shows the nice vector distribution of the force over the vertical plane yz for the circularly polarized light. Fig:
is taken from the paper, shows the gradient, scattering and total force with their relative vector directions. The gradient force depends on the location of the focus in the yz-plane and it is always acting out of the sphere center. The force is maximum at the edges. Which makes sense because stiffness (K) depends on the displacement of the bead center relative to trap center (F=-KS). Scattering force on other hand largely points in the direction of the light. When the focus is below the sphere center the two forces act in the opposite direction and net trapping force is the minus of two (this gives the z-stiffness. When the focus is above the sphere center the two forces add. Since gradient force is bigger there is always net force acting until the sphere center reaches to the equilibrium position. This force can be seen in the figure. The net movement of the free bead is long the vector direction of the net force. It is very important to keep track of this to figure-out the right geometry (angle and distances) of the trap when the DNA is attached to the bead. The net force will be in opposite direction of DNA pulling force. If DNA is pulling the bead in horizontal y-direction to the right than the restoring force (net force) will be opposite of it, in negative yz-direction (the gray arrow shows the opposition).
Electric Dipole Method
When the wavelength of the light is larger than the particle (sphere)(D < .1λ) diameter, this approach can be useful. In the fig: the laser is TM,p linearly polarized. The high intensity induces dipole moment and particle behaves like a dipole. The negative charge is stretched in the electric field direction by distance d and gives p=q.d polarization. The polarization is proportional to the electric field, and the force on the dipole is proportional to electric field gradient and polarization by Lorentz force. For TM00 Gaussian mode intensity profile, the highest intensity gradient is towards the beam axis so, so as the force on the dipole.
In Rayleigh regime, the trapping force is decomposed into two: Gradient force and Rayleigh scattering force (here is a link for a good reference [6] . The gradient force is given by Lorentz force.
Where ηm is the medium wave impedance [math]\displaystyle{ {\eta_m}=\sqrt{(\frac{\mu}{\epsilon})}=\frac{1}{n_m}\sqrt{(\frac{\mu_o}{\epsilon_o})} }[/math], nm is medium refractive index and α is the polarizability and given by (see Jackson: Classical ED sec. 4.4):
The scattering force is proportional the third power of the particle radius. Force is also proportional to the intensity gradient of the beam and in the same direction. So particle feels an attraction force towards the region of high intensity. The force is analogous to the eq.1 of the ray model. Stable trapping requires that the gradient force in the -z direction, against the direction of the incident light, be grater then scattering force. Just like RO model, increasing the NA decreases the focal spot size and increases the gradient strength. Hence, in the Rayleigh regime, trapping forces in all directions increase with higher NA. For a trapped particle, the effect of Fs is to move the equilibrium trapping position upward away from the focus.
Scattering force is given by Rayleigh scattering: Over Rayleigh regime the particle is much smaller than the spotsize, the electric field is uniform over the particle cross section and it can be taken as single point scattering source. Thus the force is given by:
Since the scattering is isotropic over the cross section; the net momentum is transferred in the forward direction. This force is similar to the force in eq.2 of RO model.
As for the small particle, the criteria for the axial stability of the trap is that R; the ratio of the backward axial gradient force to the forward-scattering force, be greater than the unity at the position of maximum axial intensity gradient. For a Gaussian beam of focal spot ω_0 this occurs at the axial position:
This condition strictly applies to the Rayleigh regime where the particle diameter 2r ≤ .2λ≈ 80 nm. In practice R requires to be larger than unity.
So there two stability conditions for the trap:
R≥1 for any spotsize.
Boltzmann factor [math]\displaystyle{ exp(-\frac{U}{KT})\ll 1 }[/math], where [math]\displaystyle{ U = n_m\alpha \frac{E^2}{2} }[/math] is the potential of the gradient force. This condition is equivalent to requiring that the time to pull a particle into the trap be much less than the time for the particle to diffuse out of the trap by Brownian motion.
And also remember that the gradient force is still polarization dependent, for linearly polarized light the gradient force will be much higher in the direction of the electric field polarization. So the stiffness along the electric field will be much higher than the stiffness in the transverse direction (the ratio of the two stiffness will depend on the ratio of the polarization). For circularly polarized light it will be distributed over the transverse plane of the particle, which is normal to the propagation vector z).In our experiments particles are much bigger than 80nm. Even though particles upto 15 nm have been trapped successfully in the Rayleigh regime this model is not useful for us. For our size (.5 to 1μm) this model's accuracy is 30%.
Electromagnetic Theory Model (Intermediate size regime)
EMT-model is applied when the particle diameter is .2λ ≤ D ≤ 1λ (λ is the wavelength in the medium). Unfortunately we follow this regime because we use .5 and 1 μm beads at 1064nm. For such particles diffraction effects are significant which we been ignoring in the previous models. And also for highly convergent (high NA) beams, the vector character of the EM field cannot be neglected (paraxial approximation / RO model uses scalar form). These are the factors make realistic forces-computing difficult in this size regime. The time-averaged force due to an arbitrary particle, is given by the following integral over the surface enclosing the particle[8]:
[math]\displaystyle{ \mathbf{F_i}=\langle \oint_S \mathbf{T_{ij}}\mathbf {n_j} da \rangle }[/math]
Where Tij is the Maxwell stress tensor [9], nj is the outward unit normal vector, and the brackets denote temporal average of the force. Where Tij is:
Where ε is the electric permittivity of the medium. The difficulty lies in deriving all six components (x,y,z or θ,ρ,φ) of E and B. at the surface of the particle, because the field includes contributions from the incident beam as well as the scattered and internal fields.
