Chapter 6 – Forces 215
molecular force (i.e., generating a force clamp), which allows, for example, details of the kin
etics of molecular unfolding and refolding to be explored in different precise force regimes.
An alternative method to generating multiple optical traps involves physically splitting the
incident laser beam into separate paths using splitter cubes that are designed to transmit a
certain proportion (normally 50%) of the beam and reflect the remainder from a dielectric
interface angled at 45° to the incident beam so as to generate a reflected beam path at 90°
to the original beam. Other similar optical components split the beam on the basis of its
linear polarization, transmitting the parallel (p) component and reflecting the perpendicular
(s) component, which has an advantage over using nonpolarizing splitter cubes in permitting
more control over the independent laser powers in each path by rotating the incident E-field
polarization vector using a half-wave plate (see Chapter 3). These methods can be used to
generate independently steerable optical traps.
The same principle can be employed to generate more than two optical traps; however, in
this case, it is often more efficient to use either a digital micromirror array or a spatial light
modulator (SLM) component. Both optical components can be used to generate a phase
modulation pattern in an image plane conjugate to the Fourier plane of the sample’s focal
plane, which results in controllable beam deflection into, potentially, several optical traps,
which can be manipulated not only in x and y but also in z. Such approaches have been used
to generate tens of relatively weak traps whose position can be programmed to create an
optical vortex effect, which can be used to monitor fluid flow around biological structures.
The primary disadvantage of digital micromirror array or SLM devices is that they have rela
tively low refresh bandwidths of a few tens of Hz, which limit their utility to monitoring only
relatively slow biological processes, if mobile traps are required. But they have an advantage
in being able to generate truly 3D optical tweezers, also known as holographic optical traps
(Dufresne and Grier, 1998).
Splitting light into a number of N traps comes with an obvious caveat that the stiffness of
each trap is reduced by the same factor N. However, there are many biological questions that
can be addressed with low stiffness traps, but the spatial fluctuations on trapped beads can
be >10 nm, which often swamps the molecular level signals under investigation. The theor
etical upper limit to N is set by the lowest level of trap stiffness, which will just be sufficient
to prevent random thermal fluctuations pushing a bead out of the physical extent of the trap.
The most useful multiple trap arrangement for single biomolecule investigations involves
two standard Gaussian-based force gradient traps, between which a single biomolecule is
tethered.
6.3.3 CHARACTERIZING DISPLACEMENTS AND FORCES IN OPTICAL TWEEZERS
The position of an optically trapped bead can be determined using either the bright-field
image of the bead onto a charge-coupled device (CCD) camera or quadrant photodiode
(QPD) or, more commonly, to use a laser interferometry method called back focal plane
(BFP) detection. The position of the center of a bead can be determined using similar centroid
determination algorithms to those discussed previously for super-resolution localization
microscopy (see Chapter 4). QPDs are cheaper than a CCD camera and have a significantly
higher bandwidth, allowing determination of x and y from the difference in voltage signals
between relevant halves of the quadrant (Figure 6.2b) such that
(6.6)
x
V
V
V
V
y
V
V
V
V
=
+
(
) −
−
(
)
(
)
=
+
(
) −
−
(
)
(
)
α
α
2
3
1
4
1
2
3
4
where α is a predetermined calibration factor. However, bright-field methods are shot noise
limited—shot noise, also known as Poisson noise, results from the random fluctuations of
the number of photons detected in a given sampling time window and of the electrons in the