Trap calibration

The displacement, x, of a bead in a trap displays a Hookean proportionality to force,

F, for small displacements F = −αx [67]. The trap stiffness, α, is typically on the order of tens of pN µm−1_{. Calibrating an optical trap by determining this constant of}

proportionality allows precision force measurements to be performed, it also allows a trap to be compared to other optical traps. These are important points when testing a

novel trap type and important for later chapters, as in Section 7.2.2. There are many techniques for finding the trap stiffness that possess benefits and limitations.

Applying hydrodynamic drag by movement of the stage or optical trap can calibrate an optical trap (Figure 2.13). If small movements are made thenα can be determined from the bead displacement [67].

α = Fdrag

x =

βvf luid

x (2.6)

β is the Stokes’ drag and vf luid is the velocity of the fluid. Usually a piezo-driven stage

is moved sinusoidally and the appropriate force variation applied to determine α. The dependence on β necessitates calculation of drag in the system, which is dependent on medium viscosity and the distance between the bead and the coverslip. When the bead is close to the coverslip, Faxen’s correction must be applied, which corrects for a spherical body of diameter, r, moving at a distance, z, from a boundary in a viscous fluid (viscosityν) [67]: β = 6πνr 1− 9 16 r z +1₈ r_z3− 45 256 r z 4 − 1 16 r z 5 (2.7)

If the bead displacement is large then the linear proportionality between force and displacement is broken. Whileα is not measurable in this case, breaking this linearity allows the Q value to be determined. Increasing the stage or trap velocity to a point whereFtrap < Fdrag will cause the bead to be lost from the trap. The velocity at which

the trap was translated is then proportional to the Q value [68].

Q= βvf luidc

nmP

(2.8)

The escape method is useful when only a basic detector and stage are available to quan- tify the efficiency of a trap but, without knowledge of α, accurate force-displacement experiments are not feasible.

A

B

C

v

x

F

F

F

v

> v

Figure 2.13– Exerting hydrodynamic drag on an optically trapped bead can be used to calibrate an optical trap (A). Moving the optical trap or stage at a known velocity,v1exerts a drag force on the bead, causing it to be displaced a distance x, directly proportional to the force applied, from the centre of the trap (B), allowing determination of the trap stiffness. Applying a larger drag force (greater velocityv2) breaks this proportionality and the bead will fall from the trap when the drag force exceeds the force exerted by the trap. The force at which the bead escapes can be used to determine the Q value.

If we track the position of a trapped bead, the nature of these fluctuations can determine the strength of the optical trap [10].

The power spectrum (Sxx(f)) of the bead’s displacement follows a Lorentzian dis-

tribution [67]. Sxx(f) = kBT π2_β(f2 0 +f2) (2.9)

kB is the Boltzmann constant, T is the temperature. The rolloff frequency, f0, can

be determined from fitting this distribution to experimental data and is proportional to the trap stiffness following α = 2πβf0 (Figure 2.14D). Again, Faxen’s correction

(Equation 2.7) must be applied. Measurement of α by this method is independent of noise and misalignment in the optical system is evident in changes to the shape of the curve. A large bandwidth detector is, however, required to allow for Nyquist sampling of the roll-off frequency (at least 10f0 is suggested, usually kHz). A quadrant

photodiode (QPD) provides large bandwidths but also requires another calibration step to link signal to bead position (Figures 2.14A and B), a high-speed camera typically achieves lower bandwidths but position calibration is trivial.

A second method of determining α from the bead displacement considers the vari- ance, which provides a drag-independent measure ofα following equipartition [10].

1

2kBT = 1 2αhx

2_{i (2.10)}

hx2_{i is equivalent to the variance (Figure 2.14C). Equipartition is a simple method to}

implement but requires an assumption of the temperature of the trap and the variance increases in the presence of drift or noise in the system, entangling these properties with α. There is no hard limit for the detector bandwidth, unlike the power spectrum method (although a low bandwidth detector does filter the movement and lead to a lower variance [10]).

The fibre-based optical trap developed in Section 7.2 possesses a relatively low NA that does not allow axial trapping. The equipartition method is independent of drag and is therefore highly suitable for calibrating this system. To avoid cumbersome posi- tion calibration with a QPD, a high speed camera is used to image bead displacement, simplifying displacement measurements but at the cost of potential noise and misalign- ment and reduced bandwidth filtering the movement.