Contents - Max-Planck-Institut für Physik komplexer Systeme

2.23 Measuring the Complete Force Field of an Optical Trap
MARCUS JAHNEL, MARTIN BEHRNDT, ANITA JANNASCH, ERIK SCHÄFFER,
STEPHAN W. GRILL
Optical traps are widely used to investigate forces and
displacements encountered on the molecular level, for
example in the movement of molecular motors. To
measure or apply forces in such studies requires the
precise knowledge of the trapping force field, which
is typically approximated as linear in the displacement
of the trapped microsphere. Close to the trap center
this linear assumption is valid and successfully used to
set the scale of the measured forces and displacements
when performing a thermal calibration. Here, thermal
fluctuations of the microsphere around the center
of the optical trap are measured and compared
with theories of Brownian fluctuations in a confining
harmonic potential [1, 2]. However, as this assumed
linear force-displacement relation breaks down
at larger microsphere displacements, applications demanding
high forces at low laser intensities can probe
the light-microsphere interaction beyond the linear
regime. Thus, an exact mapping of the complete optical
force field is not only necessary to determine the validity
of the linear approximation, but enables the use of
the full force range of an optical trap.
Here, we measured the full non-linear force and displacement
response of an optical trap in two dimensions
[3]. We used a dual-beam optical trap setup
with back-focal-plane photodetection [4] that allowed
for independent adjustment of the positions and intensities
of two optical traps [5]. A strong, thermallycalibrated
trap [1], TC, in its linear operating regime
acted as a precise sensor of force (and position) for
analyzing a weaker, uncalibrated trap of interest, TA.
While keeping TC stationary, we scanned TA in 10nm
steps over the whole microsphere-interaction regime.
At each step—with stationary traps and measurement
times long enough to average over thermal
fluctuations—the balance of the optical forces yields
〈FC(r)〉tav = −〈FA(r)〉tav . To ensure the validity of the
linear force-displacement approximation for the calibration
trap to within 5 % (see below), we adjusted
the relative intensities in both traps such that TC had
at least a five times higher trap stiffness than TA. This
then allows for measuring the full force-displacement
relation of TA. Under the above conditions, we have
ˆκC〈r − d〉tav = 〈FA(r)〉tav , (1)
where the diagonal matrix ˆκC contains the trap stiffness
of TC in the x- and y-direction. The distance
vector between the two traps is denoted by d and is
changed by scanning TA. Allowing the microsphere
to relax to its new equilibrium position, we sampled
the complete optical force profile for arbitrary displacement
r relative to the center of TA. We determined a
two-dimensional map of the optical forces exerted by
TA on a polystyrene microsphere of diameter 1.26µm
[Fig. 1(a)]. The net force, obtained by combining the
parallel (Fx) and perpendicular (Fy) force components
relative to the trap polarization, demonstrates that for
this microsphere size the optical forces are nearly radially
symmetric [Fig. 1(b)]. We therefore restrict our
remaining discussion to cross-sections of the force map
in x [dashed line in Fig. 1(c)]. Fitting the experimental
data using numerical calculations based on the Tmatrix
method [6] showed excellent agreement of our
measurements with Mie scattering theory [Fig. 1(c)].
Close to the origin, a constant trap stiffness—assuming
Hooke’s law—is expected. However, numerical differentiation
of the measured force curve shows that the
trap stiffness continuously deviated from its value at
the origin, κ0 = 72 ± 3 pN/µm [Fig. 1(d)]. Displacing
the microsphere from the center, the trap stiffness
increased moderately within the first 300 nm towards
a maximum, before it fell off, and eventually became
negative. In this region, the analogy between optical
traps and mechanical springs fails; the trap stiffness is
negative for a decreasing, yet, still restoring force.
Displacement y (nm)
−1000 0 1000
−1000 0 1000
(a) (c) Data
Theory
Force Fx (pN)
(b) Polarization
Radial force (pN)
−1000 0 1000
Displacement x (nm)
5 10 15 20 25 30 35 −30 −15 0 15 30
Force Fx (pN)
Stiffness (pN/ µ m)
20
0
−20
100
−100 0
∅ = 1.26µ m
(d)
−1000 0 1000
Displacement x (nm)
Figure 1: 2D maps of (a) optical forces on a 1.26 µm microsphere
in the direction of
q
polarization and of (b) the magnitude of the radial
force |F | = F2 x + F2 y . Force magnitudes are color-coded by
corresponding heat maps. (c) Complete force response along the
polarization axis [dashed line in (a)]. (d) Numerical differentiation,
κ(x) = − ∂x F(r), yielded the trap stiffness with respect to the microsphere
displacement.
86 Selection of Research Results