216 6.3  Optical Force Tools

photon detector device, approximated by a Poisson distribution. The relatively small photon

budget limits the speed of image sampling before shot noise in the detector swamps the

photon signal in each sampling time window.

For BFP detection, the focused laser beam used to generate an optical tweezer trap

propagates through a specimen flow cell and is typically recollimated by a condenser lens. The

BFP of the condenser lens is then imaged onto a QPD. This BFP image represents the Fourier

transform of the sample plane and is highly sensitive to phase changes of the trapping laser

propagating through an optically trapped bead. Since the trapping laser is highly collimated,

interference occurs between this refracted beam and the undeviated laser light propagating

through the sample. The shift in the intensity centroid of this interference pattern on the

QPD is a sensitive metric of the displacement between the bead center and the center of the

optical trap.

In contrast to bright-​field detection of the bead, BFP detection is not shot noise limited

and so the effective photon budget for detection of bead position in the optical trap is large

and can be carved into small submicrosecond sampling windows with sufficient intensity in

each to generate sub-​nanometer estimates on bead position, with the high sampling time

resolution limited only by the ~MHz bandwidth of QPD detectors. Improvements in local­

ization precision can be made using a separate BFP detector laser beam of smaller wave­

length than the trapping laser beam, coaligned to the trapping beam.

The stiffness k of an optical trap can be estimated by measuring the small fluctuations of a

particle in the trap and modeling this with the Langevin equation. This takes into account the

FIGURE 6.2  Controlling bead deflections in optical tweezers. (a) In an AOD, radio-​frequency

driving oscillations from a piezo transducer induce a standing wave in the crystal that acts as

diffraction grating to deflect an incident laser beam. (b) Schematic of sectors of a quadrant

photodiode. (c) Bead displacement in optical tweezers. (d) Bead displacements in an optical

trap, here shown with a trap of stiffness 0.15 pN/​nm, have a Lorentzian-​shaped power, resulting

in a characteristic corner frequency (here 1.4 kHz) that allows the trap stiffness to be determined,

which can also be determined from (e) the root mean squared displacement, shown here for

data of the same trapped bead (see Leake, 2001).