Characterizing the optical trapping force

A laser-equipped dark eld microscope for optical trapping (see gure 3.1) was used to optically conne nanoparticles in solution and to perform further chemical manipulation. In order to achieve a stable trapping of nanoparticles, the property of the optical trap, especially the optical trapping forces, need to be characterized. The TEM00 mode of a focused laser beam can be approximated to be a Gaus- sian beam(Figure 4.11), and the generated electromagnetic eld is expressed as follows:[127]

E(λ, r, z) =E0(λ) r 2 π ω0(λ) ω(λ, z)exp − r 2 ω2_{(λ, z)} +i h k(λ) r 2 2R(λ, z) +z +ζ(λ, z)i (4.2) In this cylindrical coordinate system, the z−axis is along the laser propagation and the r−axis is perpendicular to the laser propagation direction. The laser focus point is set to be the center r = 0, z = 0, and the laser beam width is a function of the waist width ω0(λ) = 2λ/(π·N A):

ω(λ, z) = ω0(λ) s 1 + z z0(λ) 2 (4.3) with the Rayleigh range z0(λ) = (πω02(λ))/(nλ) and the radius of curvature of the wavefront R(z) = z[1 +z0(λ)/z], and ζ(z) = arctan(z/z0(λ)) is the Gouy phase. The electric eld at the beam center isE0(λ) =

√

2ηP /(ω0(λ)), and η is the characteristic impedance of the medium in which the laser beam is propagat- ing . In water,η=pµ/ε= 142.7Ω. P is the laser power density at the center of

the laser beam at the waist (r=z = 0). And the numerical aperture is N A= 1 for the water immersion objective.

Figure 4.11: An illustration of a Gaussian beam passing through a medium along thez−axis. The width of the beam ω(λ, z), the waist width atz = 0isω0, and the Rayleigh range

z0(λ). ù0 ù(ë,z) z z0 2ù0

Introducing the calculated eletric eld from equation 4.2, and the calculated polarizability from the modied Claussius-Mossotti equation (equation 2.37) into the equation 2.35 (_F~_{grad) and 2.36 (F}~_{sct), the optical forces in the trap are calcu-}

lated for a r=20 nm Au nanosphere (Figure 4.12).[128] The equilibrium position of the trap is close to the laser focus, where the total force is approximately zero (_F~ _{→ 0 pN) and the particle is conned near this point due to the attractive}

optical forces acting on the particle. As shown in Figure 4.12, the equilibrium position is on thez−axis of the Gaussian beam, and slightly away from the laser focus, with ze = 11 nm. The maximum trapping force is 0.262 pN.

The calculations above are based on the approximation that the trapping laser beam is a Gaussian beam. This is true for objectives with small N A (N A <1),

4.3. Characterization of the optical trap - 400 - 200 0 200 400 - 400 - 200 0 200 400 xHnmL zHnmL

r (nm)

Z (nm)

E

Figure 4.12: The total optical forces on a r=20 nm Au nanosphere in water. The laser power P=0.1W. The the contour arrows point to the direction of the total optical force (_F~ _=F~

grad+F~sct), the color bar indicates the absolute

value of the total optical force within the trap. The maximum trapping force is 0.262 pN, and the equilibrium position (where _F~_{=0 fN) is at r=0, ze=11}

nm, near the laser focus point.

the equation ω0(λ) = 2λ/(π·N A), the waist width of the laser beam ω0 =677.4 nm is obtained. While a FDTD simulation for a 1064 nm laser beam passing through a thin lens with N A=1 yields ω0 = 514 nm. Therefore the calculations above give only a rough estimation of the optical forces.

A more precise analysis of the optical trap is based on FDTD calculations. From section 2.3, the total force contains contributions of the gradient force and the scattering force, which are proportional to the real and imaginary part of the particle polarizability, respectively. When the trapping laser wavelength is xed, the polarizability of the nanoparticle of a certain size is constant, and the trapping forces can be determined by the electromagnetic eld induced by the trapping laser. FDTD provides numerical calculation of the elds and the total force at each point in the system can be calculated according to equation 2.35 and 2.36. Figures 4.13a,b compare the optical forces calculated from the Dipole Approximation and FDTD for an r=20nm Au nanosphere in the optical trap (P=0.1W). The results of the two methods match quite well while the equilibrium

position shifts from ze=11 nm with Dipole Approximation to ze=40 nm with

FDTD calculation, and the maximum force along r−axis shifted by 100 nm. This may due to the approximations in the laser beam as a Gaussian beam.

a

b

Figure 4.13: Calculated optical forces on an r=20nm Au nanosphere along (a) z−axis (r=0) and (b)r−axis (z=ze) calculated with the Dipole Approxi-

mation (red) and FDTD (blue), respectively. The equilibrium point isze=11

nm using the Dipole Approximation andze=40 nm using the FDTD calcula-

tion, respectively. The maximum optical force on the r−axis is Fr = -0.262

pN with the Dipole Approximation and Fr = -0.252 pN with the FDTD

calculation. The inset indicates the r− and z− axis. Fz points toward the

equilibrium point of the trap and Fr points toward the center of the trap.

The laser power at the focus P=0.1W.

In real experiments, the nanoparticle is not a stationary object at equilibrium, but randomly diuses within the trap. The movement of a nanoparticle in an optical trap is analogous to a spring obeying the Hooke's law.[129, 130] The particle encounters a thermal force (Ft), which drives the particle out of the

equilibrium position; a restoring force (Fr), which drives the particle back to the

equilibrium position; a damping force (Fd), which is caused by the viscous drag

force of the solution and an inertial force (Fi):

Fi+Fd+Fr =Ft, (4.4)

Equation 4.4 above can be written as:

m∂ 2_x(t) ∂t2 +γ ∂x(t) ∂t +κx(t) = p 2kBT γR(t). (4.5)

in which the thermal force Ft (Langevin force) is a random Gaussian process

4.3. Characterization of the optical trap constantkB and a noise term R(t). The inertial force is neglected in comparison

to the damping force, since the Reynolds number of nanoparticles is very small. The damping force is determined by the drag coecient γ = 6πηr according to

the Stoke's law. The restoring forceFris linearly proportional to the displacement

of the particle: Fr = κx. The trap stiness κ represents the total optical force

acting on the nanoparticle. The harmonic potential of the particle is determined by: U(x) =Fr x 2 = κx 2 x 2_{, (4.6)} -100 -50 0 50 100 0.0 0.2 0.4 P (x ) x (nm) -100 -50 0 50 100 0.0 0.2 0.4 P (y ) y (nm)

b

a

c

Figure 4.14: (a) A typical diusion trajectory of a nanorod in an optical trap over 3000 frames (within one minute) at 30◦ _{C and the probability density} over 3000 frames (1min) along the (b) x−axis and (c) y−axis. Gaussian tting to the probability density histograms (red) gives the width σx=22.6

nm, σy=23.6 nm, and the trap stiness κx=8.2 fN/nm, κy=7.5 fN/nm.

The trap stiness can be experimentally determined by recording the nanopar- ticle motion with a digital video camera and extracting the distribution of the nanoparticle positions over time. Figure 4.14 shows a typical trajectory of a nanorod in an optical trap over 3000 frames (within one minute). With suf- cient statistics, the position distribution shows a Gaussian distribution. The probability density function is then determined by:

P(x) = 1 σx √ 2π exp− (x−µ)2 2σ2 x , (4.7)

From Boltzmann statistic, the potential energy U(x) of the particle in the trap follows the distribution:

P(x) = Cexp−U(x)

kBT

in which C is a constant which normalizes the probability distribution with

R

P(x)dx= 1. Therefore, the potential energy can be experimentally plotted:

U(x) = −kBT lnP(x) +kBT lnC. (4.9)

Introducing the value of Boltzmann's constant kB = 1.3806488× 10−23 J/K,

the experimentally determinedU(x)into the equation 4.9, the trap stinessκcan

be experimentally determined. The probability density P(x) and P(y) for the trajectory in Figure 4.14a is then plotted in Figure 4.14b and c, and the stiness of the trap is then calculated: κx=8.2 fN/nm, κy=7.5 fN/nm. The spring model

provides the trap stiness in real experiment. Such model is limited to x and y

dimension, while the optical forces along the laser propagation direction (z) is

dicult to measure with the current experimental condition.