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Optical Trapping

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Since their invention just over 20 years ago, optical traps have emerged as a powerful tool with broad-reaching applications in biology and physics. Capabilities have evolved from simple manipulation to the application of calibrated forces on-and the measurement of nanometer-level displacements of-optically trapped objects. We review progress in the development of optical trapping apparatus, including instrument design considerations, position detection schemes and calibration techniques, with an emphasis on recent advances. We conclude with a brief summary of innovative optical trapping configurations and applications.
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Optical trapping
Keir C. Neuman and Steven M. Block
a)
Department of Biological Sciences, and Department of Applied Physics, Stanford University,
Stanford, California 94305
Abstract
Since their invention just over 20 years ago, optical traps have emerged as a powerful tool with broad-
reaching applications in biology and physics. Capabilities have evolved from simple manipulation
to the application of calibrated forces on—and the measurement of nanometer-level displacements
of—optically trapped objects. We review progress in the development of optical trapping apparatus,
including instrument design considerations, position detection schemes and calibration techniques,
with an emphasis on recent advances. We conclude with a brief summary of innovative optical
trapping configurations and applications.
I. INTRODUCTION
Arthur Ashkin pioneered the field of laser-based optical trapping in the early 1970s. In a series
of seminal papers, he demonstrated that optical forces could displace and levitate micron-sized
dielectric particles in both water and air,
1
and he developed a stable, three-dimensional trap
based on counterpropagating laser beams.
2
This seminal work eventually led to the
development of the single-beam gradient force optical trap,
3
or “optical tweezers,” as it has
come to be known.
4
Ashkin and co-workers employed optical trapping in a wide-ranging series
of experiments from the cooling and trapping of neutral atoms
5
to manipulating live bacteria
and viruses.
6,7
Today, optical traps continue to find applications in both physics and biology.
For a recent survey of the literature on optical tweezers see Ref. 8. The ability to apply
picoNewton-level forces to micron-sized particles while simultaneously measuring
displacement with nanometer-level precision (or better) is now routinely applied to the study
of molecular motors at the single-molecule level,
9–19
the physics of colloids and mesoscopic
systems,
20–29
and the mechanical properties of polymers and biopolymers.
18,20,30–43
In
parallel with the widespread use of optical trapping, theoretical and experimental work on
fundamental aspects of optical trapping is being actively pursued.
4,20,44–48
In addition to the
many excellent reviews of optical trapping
9,49–53
and specialized applications of optical traps,
several comprehensive guides for building optical traps are now available.
54–60
For the
purpose of this review, we will concentrate on the fundamental aspects of optical trapping with
particular emphasis on recent advances.
Just as the early work on optical trapping was made possible by advances in laser technology,
4
much of the recent progress in optical trapping can be attributed to further technological
development. The advent of commercially available, three-dimensional (3D) piezoelectric
stages with capacitive sensors has afforded unprecedented control of the position of a trapped
object. Incorporation of such stages into optical trapping instruments has resulted in higher
spatial precision and improved calibration of both forces and displacements. In addition, stage-
based force clamping techniques have been developed that can confer certain advantages over
other approaches of maintaining the force, such as dynamically adjusting the position or
stiffness of the optical trap. The use of high-bandwidth position detectors
61
improves force
a)
Electronic mail: sblock@stanford.edu.
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calibration, particularly for very stiff traps, and extends the detection bandwidth of optical
trapping measurements. In parallel with these technological improvements, recent theoretical
work has led to a better understanding of 3D position detection
62–64
and progress has been
made in calculating the optical forces on spherical objects with a range of sizes.
65,66
II. PRINCIPLES OF OPTICAL TRAPPING
An optical trap is formed by tightly focusing a laser beam with an objective lens of high
numerical aperture (NA). A dielectric particle near the focus will experience a force due to the
transfer of momentum from the scattering of incident photons. The resulting optical force has
traditionally been decomposed into two components: (1) a scattering force, in the direction of
light propagation and (2) a gradient force, in the direction of the spatial light gradient. This
decomposition is merely a convenient and intuitive means of discussing the overall optical
force. Following tradition, we present the optical force in terms of these two components, but
we stress that both components arise from the very same underlying physics (see theoretical
progress, below for a unified expression). The scattering component of the force is the more
familiar of the two, which can be thought of as a photon “fire hose” pushing the bead in the
direction of light propagation. Incident light impinges on the particle from one direction, but
is scattered in a variety of directions, while some of the incident light may be absorbed. As a
result, there is a net momentum transfer to the particle from the incident photons. For an
isotropic scatter, the resulting forces cancel in all but the forward direction, and an effective
scattering cross section can be calculated for the object. For most conventional situations, the
scattering force dominates. However, if there is a steep intensity gradient (i.e., near the focus
of a laser), the second component of the optical force, the gradient force, must be considered.
The gradient force, as the name suggests, arises from the fact that a dipole in an inhomogeneous
electric field experiences a force in the direction of the field gradient.
67
In an optical trap, the
laser induces fluctuating dipoles in the dielectric particle, and it is the interaction of these
dipoles with the inhomogeneous electric field at the focus that gives rise to the gradient trapping
force. The gradient force is proportional to both the polarizability of the dielectric and the
optical intensity gradient at the focus.
For stable trapping in all three dimensions, the axial gradient component of the force pulling
the particle towards the focal region must exceed the scattering component of the force pushing
it away from that region. This condition necessitates a very steep gradient in the light, produced
by sharply focusing the trapping laser beam to a diffraction-limited spot using an objective of
high NA. As a result of this balance between the gradient force and the scattering force, the
axial equilibrium position of a trapped particle is located slightly beyond (i.e., down-beam
from) the focal point. For small displacements (~150 nm), the gradient restoring force is simply
proportional to the offset from the equilibrium position, i.e., the optical trap acts as Hookean
spring whose characteristic stiffness is proportional to the light intensity.
In developing a theoretical treatment of optical trapping, there are two limiting cases for which
the force on a sphere can be readily calculated. When the trapped sphere is much larger than
the wavelength of the trapping laser, i.e., the radius (a) λ, the conditions for Mie scattering
are satisfied, and optical forces can be computed from simple ray optics (Fig. 1). Refraction
of the incident light by the sphere corresponds to a change in the momentum carried by the
light. By Newton’s third law, an equal and opposite momentum change is imparted to the
sphere. The force on the sphere, given by the rate of momentum change, is proportional to the
light intensity. When the index of refraction of the particle is greater than that of the surrounding
medium, the optical force arising from refraction is in the direction of the intensity gradient.
Conversely, for an index lower than that of the medium, the force is in the opposite direction
of the intensity gradient. The scattering component of the force arises from both the absorption
and specular reflection by the trapped object. In the case of a uniform sphere, optical forces
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can be directly calculated in the ray-optics regime.
68,69
The extremal rays contribute
disproportionally to the axial gradient force, whereas the central rays are primarily responsible
for the scattering force. Thus, expanding a Gaussian laser beam to slightly overfill the objective
entrance pupil can increase the ratio of trapping to scattering force, resulting in improved
trapping efficiency.
69,70
In practice, the beam is typically expanded such that the 1/e
2
intensity
points match the objective aperture, resulting in ~87% of the incident power entering the
objective. Care should be exercised when overfilling the objective. Absorption of the excess
light by the blocking aperture can cause heating and thermal expansion of the objective,
resulting in comparatively large (~μm) axial motions when the intensity is changed. Axial
trapping efficiency can also be improved through the use of “donut” mode trapping beams,
such as the TEM
01
**
mode or Laguerre-Gaussian beams, which have intensity minima on the
optical propagation axis.
69,71–73
When the trapped sphere is much smaller than the wavelength of the trapping laser, i.e., a
λ, the conditions for Raleigh scattering are satisfied and optical forces can be calculated by
treating the particle as a point dipole. In this approximation, the scattering and gradient force
components are readily separated. The scattering force is due to absorption and reradiation of
light by the dipole. For a sphere of radius a, this force is
F
scatt
=
I
0
σn
m
c
, (1)
σ
=
128
π
5
a
6
3
λ
4
(
m
2
1
m
2
+ 2
)
2
,
(2)
where I
0
is the intensity of the incident light, σ is the scattering cross section of the sphere,
n
m
is the index of refraction of the medium, c is the speed of light in vacuum, m is the ratio of
the index of refraction of the particle to the index of the medium (n
p
/n
m)
, and λ is the wavelength
of the trapping laser. The scattering force is in the direction of propagation of the incident light
and is proportional the intensity. The time-averaged gradient force arises from the interaction
of the induced dipole with the inhomogeneous field
F
grad
=
2
πα
cn
m
2
I
0
,
(3)
where
α
=
n
m
2
a
3
(
m
2
1
m
2
+ 2
)
(4)
is the polarizability of the sphere. The gradient force is proportional to the intensity gradient,
and points up the gradient when m>1.
When the dimensions of the trapped particle are comparable to the wavelength of the trapping
laser (a ~ λ), neither the ray optic nor the point-dipole approach is valid. Instead, more complete
electromagnetic theories are required to supply an accurate description.
74–80
Unfortunately,
the majority of objects that are useful or interesting to trap, in practice, tend to fall into this
intermediate size range (0.1–10λ). As a practical matter, it can be difficult to work with objects
smaller than can be readily observed by video microscopy(~0.1 μm), although particles as
small as ~35 nm in diameter have been successfully trapped. Dielectric microspheres used
alone or as handles to manipulate other objects are typically in the range of ~0.2–5 μm, which
is the same size range as biological specimens that can be trapped directly, e.g., bacteria, yeast,
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and organelles of larger cells. Whereas some theoretical progress in calculating the force on a
sphere in this intermediate size range has been made recently,
65,66
the more general
description does not provide further insight into the physics of optical trapping. For this reason
we postpone discussion of recent theoretical work until the end of the review.
III. DESIGN CONSIDERATIONS
Implementing a basic optical trap is a relatively straightforward exercise (Fig. 2).
55,58
The
essential elements are a trapping laser, beam expansion and steering optics, a high NA
objective, a trapping chamber holder, and some means of observing the trapped specimen.
Optical traps are most often built by modifying an inverted microscope so that a laser beam
can be introduced into the optical path before the objective: the microscope then provides the
imaging, trapping chamber manipulation, and objective focus functions. For anything beyond
simply trapping and manually manipulating objects, however, additional elements become
necessary. Dynamic control of trap position and stiffness can be achieved through beam
steering and amplitude modulation elements incorporated in the optical path before the laser
beam enters the objective. Dynamic control over position and stiffness of the optical trap has
been exploited to implement position-and force-clamp systems. Position clamps, in which the
position of a trapped object is held constant by varying the force, are well suited for stall force
measurements of molecular motors.
39,49,81–83
Force clamps, in which the force on a trapped
object is fixed by varying the position of the trap, are well suited for displacement
measurements.
49,56,81,84,85
Incorporation of a piezoelectric stage affords dynamic
positioning of the sample chamber relative to the trap, and greatly facilitates calibration.
Furthermore, for the commonly employed geometry in which the molecule of interest is
attached between the surface of the trapping cell and a trapped bead “handle,” piezoelectric
stages can be used to generate a force clamp.
86–88
The measurement of force and displacement
within the optical trap requires a position detector, and, in some configurations, a second, low
power laser for detection. We consider each of these elements in detail.
A. Commercial systems
Commercial optical trapping systems with some limited capabilities are available. Cell
Robotics
89
manufactures a laser-trapping module that can be added to a number of inverted
microscopes. The module consists of a 1.5 W diode pumped Nd:YVO
4
laser (λ = 1064 nm)
with electronic intensity control, and all of the optics needed to both couple the laser into the
microscope and manually control the position of the trap in the specimen plane. The same
module is incorporated into the optical tweezers workstation, which includes a microscope, a
motorized stage and objective focus, video imaging, and a computer interface. Arryx
Incorporated
90
manufactures a complete optical trapping workstation that includes a 2 W diode
pumped solid-state laser (λ=532 nm), holographic beam shaping and steering, an inverted
microscope, a motorized stage, and computer control. Holographic beam shaping provides
control over the phase of the trapping laser,
91,92
which allows multiple, individually
addressable, optical traps in addition to high order, complex trapping beams. An integrated
optical trap is also available from PALM Microlaser Technologies,
93
either alone or
incorporated with their microdissection system. The PALM system employs an infrared
trapping laser and computer control of the stage, similar to the other optical trapping systems.
The commercial systems tend to be expensive, but they offer turnkey convenience at the price
of flexibility and control. None of the systems currently comes equipped with position detection
capabilities beyond video imaging, and only one (Arryx) provides dynamic control over the
trap position, but with an unknown update rate (~5 Hz or less). Overall, these systems are
adequate for positioning and manipulating objects but are incapable, without further
modifications, of ultrasensitive position or force measurements. As commercial systems
become increasingly sophisticated and versatile, they may eventually offer an “off-the-shelf”
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option for some optical trapping applications. In deciding between a commercial or custom-
built optical trap, or among commercial systems, several factors should be considered. Basic
considerations include cost, maximum trap force and stiffness, choice of laser wavelength
(important for biological samples), specimen or trap positioning capability, optical imaging
modes, position-detection capabilities, and sample geometry. In addition, flexibility and the
possibility to upgrade or improve aspects of the system should also be considered. How easily
can the optical system be modified or adapted? Can the functionality be upgraded? Perhaps
the most fundamental question concerns the decision to buy or to build. Whereas building a
basic optical trap is now standard practice in many labs, it requires a certain familiarity with
optics and optical components (in relation to the complexity of the optical trap), as well as a
significant time investment for the design, construction, and debugging phases. These factors
should be weighed against the potential benefits of reduced cost, increased flexibility and
greater control of home-built optical traps.
B. Trapping laser
The basic requirement of a trapping laser is that it delivers a single mode output (typically,
Gaussian TEM
00
mode) with excellent pointing stability and low power fluctuations. A
Gaussian mode focuses to the smallest diameter beam waist and will therefore produce the
most efficient, harmonic trap. Pointing instabilities lead to unwanted displacements of the
optical trap position in the specimen plane, whereas power fluctuations lead to temporal
variations in the optical trap stiffness. Pointing instability can be remedied by coupling the
trapping laser to the optical trap via an optical fiber, or by imaging the effective pivot point of
the laser pointing instability into the front focal plane of the objective. Both of these solutions
however, trade reduced pointing stability against additional amplitude fluctuations, as the fiber
coupling and the clipping by the back aperture of the microscope objective depend on beam
pointing. Thus, both power and pointing fluctuations introduce unwanted noise into any
trapping system. The choice of a suitable trapping laser therefore depends on several
interdependent figures of merit (power, power stability, pointing stability, thermal drift,
wavelength, mode quality, etc.).
Output power of the trapping laser and the throughput of the optical system will determine the
maximum attainable stiffness and force. As discussed above, trapping forces depend on
multiple parameters and are difficult to calculate for most conditions of practical interest.
Generally speaking, maximum trapping forces on the order of 1 pN per 10 mW of power
delivered to the specimen plane can be achieved with micron-scale beads.
9
As a specific
example, trapping a 0.5 μm polystyrene (n=1.57) sphere in water with a TEM
00
1064 nm laser
that overfills a 1.2 NA objective by ~10% (1/e
2
intensity points matched to the aperture radius),
gives a stiffness of 0.16 pN/nm per W of power in the specimen plane. In practice, laser power
levels can range from a few mW to a Watt or more in the specimen plane, depending on details
of the laser and setup, objective transmittance, and the desired stiffness.
Wavelength is an important consideration when biological material is trapped, particularly for
in vivo trapping of cells or small organisms.
94
There is a window of relative transparency in
the near infrared portion of the spectrum(~750–1200 nm), located in the region between the
absorption of proteins in the visible and the increasing absorption of water towards the infrared.
9
Substantial variation with wavelength of optical damage to biological specimens is observed
even within the near infrared region (Fig. 3), with damage minima occurring at 970 and 830
nm
95–97
for bacterial cells of Escherichia coli. If damage or “opticution”
98
of biological
specimens is not a concern, then the choice of wavelength becomes less critical, but the
potential effects of heating resulting from light absorption by the medium or the trapped particle
should certainly be considered.
99–101
The optimal choice of trapping wavelength will also
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depend on the transmission of the objective used for optical trapping (discussed below), as well
as the output power available at a given wavelength.
In practice, a variety of lasers has been employed for optical trapping. The factors discussed
above, along with the cost, will determine the final selection of a trapping laser. The laser of
choice for working with biological samples is currently the neodymium:yttrium–aluminum–
garnet (Nd:YAG) laser and its close cousins, neodymium: yttrium–lithium–fluoride (Nd:YLF),
and neodymium: yttrium–orthovanadate (Nd:YVO
4)
. These lasers operate in the near infrared
region of the spectrum at 1.047, 1.053, or 1.064 μm, which helps to limit optical damage. Diode
pumped versions of these lasers offer high power (up to 10 W or even more) and superior
amplitude and pointing stability. An additional advantage of diode-pumped solid-state (DPSS)
lasers is that the noise and heat of the laser power supply can be physically isolated from the
laser itself and the immediate region of the optical trap. The output of the pump diodes can be
delivered to the laser head via an optical fiber bundle, in some cases up to 10 m in length. The
main drawback of such DPSS lasers is their cost, currently on the order of $5–10 K per W of
output power. Diode lasers afford a lower-cost alternative in a compact package and are
available at several wavelengths in the near infrared, but these devices are typically limited to
less than ~250 mW in a single-transverse mode, the mode required for efficient trapping. Diode
lasers also suffer significantly from mode instabilities and noncircular beams, which
necessitates precise temperature control instrumentation and additional corrective optics. By
far the most expensive laser option is a tunable cw titanium:sapphire (Ti:sapphire) laser
pumped by a DPSS laser, a system that delivers high power(~1 W) over a large portion of the
near infrared spectrum(~750–950 nm), but at a current cost in excess of $100 K. The large
tuning range is useful for parametric studies of optical trapping, to optimize the trapping
wavelength, or to investigate the wavelength-dependence of optical damage.
95
A Ti:sapphire
laser is also employed for optical trapping in vivo
94
since it is the only laser currently available
that can deliver over ~250 mW at the most benign wavelengths (830 and 970 nm).
95
In optical trapping applications where no biological materials will be trapped, any laser source
that meets the basic criteria of adequate power in the specimen plane, sufficient pointing and
amplitude stability, and a Gaussian intensity profile, may be suitable. Optical traps have been
built based on argon ion,
3
helium-neon,
102
and diode laser sources,
103,104
to name a few.
The DPSS lasers employed in our lab for biological work supply ~4 W of power at 1064 nm
with power fluctuations below 1% –2% and a long-term pointing stability of ±50 μrad.
C. Microscope
Most optical traps are built around a conventional light microscope, requiring only minor
modifications. This approach reduces the construction of an optical trap to that of coupling the
light from a suitable trapping laser into the optical path before the objective without
compromising the original imaging capabilities of the microscope. In practice, this is most
often achieved by inserting a dichroic mirror, which reflects the trapping laser light into the
optical path of the microscope but transmits the light used for microscope illumination.
Inverted, rather than upright, microscopes are often preferred for optical trapping because their
stage is fixed and the objective moves, making it easier to couple the trapping light stably. The
use of a conventional microscope also makes it easier to use a variety of available imaging
modalities, such as differential interference contrast and epi-fluorescence.
With more extensive modifications, a position detector can be incorporated into the trapping
system. This involves adding a second dichroic mirror on the condenser side of the microscope,
which reflects the laser light while transmitting the illuminating light. In order to achieve the
mechanical stability and rigidity required for nanometer scale position measurements, more
extensive modifications of the microscope are generally required.
50,59
In the current
generation of optical traps, the rotating, multiobjective turret is conventionally replaced with
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a custom-built single objective holder, along with a mount for the dichroic mirror. The original
stage is removed and the microscope is modified to accommodate a more substantial stage
platform, holding a crossed-roller bearing stage (for coarse movement) mounted to a
piezoelectric stage with feedback (for fine movement). Finally, the condenser assembly is
attached to a fine focus transport (similar to that used for the objective) that is then mounted
to the illumination column by a rigid aluminum beam.
59
An alternative to the redesign and retrofitting of a commercial microscope is to build the entire
optical trap from individual optical components.
57,103,104
This approach is slightly more
involved, as the entirety of the imaging and trapping optical paths have to be designed and
built. The increase in complexity, however, can be offset by increased flexibility in the design
and a wider choice of components, greater access to the optical paths, and reduced cost.
D. Objective
The single most important element of an optical trap is the objective used to focus the trapping
laser. The choice of objective determines the overall efficiency of the optical trapping system
(stiffness versus input power), which is a function of both the NA and the transmittance of the
objective. Additionally, the working distance and the immersion medium of the objective (oil,
water, or glycerol) will set practical limits on the depth to which objects can be trapped.
Spherical aberrations, which degrade trap performance, are proportional to the refractive index
mismatch between the immersion medium and the aqueous trapping medium. The deleterious
effect of these aberrations increases with focal depth. The working distance of most high NA
oil immersion objectives is quite short (~0.1 mm), and the large refractive index mismatch
between the immersion oil (n=1.512) and the aqueous trapping medium (n ~ 1.32) leads to
significant spherical aberrations. In practice, this limits the maximum axial range of the optical
trap to somewhere between 5 and 20 μm from the coverglass surface of the trapping chamber.
104
Trapping deeper into solution can be achieved with water immersion objectives that
minimize spherical aberration
105
and which are available with longer working distances. A
high NA objective (typically, 1.2–1.4 NA) is required to produce an intensity gradient sufficient
to overcome the scattering force and produce a stable optical trap for microscopic objects, such
as polystyrene beads. The vast majority of high NA objectives are complex, multielement
optical assemblies specifically designed for imaging visible light, not for focusing an infrared
laser beam. For this reason, the optical properties of different objectives can vary widely over
the near infrared region (Fig. 4).
9,95
Generally speaking, objectives designed for general
fluorescence microscopy display superior transmission over the near infrared compared to most
general-purpose objectives, as do infrared-rated objectives specifically produced for use with
visible and near infrared light (Table I). Given the wide variation in transmission characteristics
for different objectives, an objective being considered for optical trapping should be
characterized at the wavelength of the trapping light. Manufacturers rarely supply the
transmission characteristics of objectives outside the visible portion of the spectrum. When
transmission characteristics in the near infrared are provided, the figures may represent an
overestimate, since the throughput of the objective is often measured using an integrating
sphere, which also registers scattered light that is not well focused, and hence does not
contribute to trapping. To measure the effective transmission of a high NA objective accurately,
the dual objective method is preferred,
9,95,106
in which two identical, matched objectives are
used to focus and then recollimate the laser beam (the transmission of a single objective is the
square root of the transmission for the objective pair). Furthermore, because the transmission
may depend on the degree to which light is bent, the laser beam should be expanded to fill the
objective rear aperture. It should be noted that the extremely steep focusing produced by high
NA objectives can lead to specular reflection from surfaces at the specimen plane, so simply
measuring the throughput of an objective by placing the probe of a power meter directly in
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front of the objective lens results in an underestimation of its transmission. This approach is
not recommended.
E. Position detection
Sensitive position detection lies at the heart of quantitative optical trapping, since nanoscale
measurements of both force and displacement rely on a well-calibrated system for determining
position. Position tracking of irregularly shaped objects is feasible, but precise position and
force calibration are currently only practical with spherical objects. For this purpose,
microscopic beads are either used alone, or attached to objects of interest as “handles,” to apply
calibrated forces. The position detection schemes presented here were primarily developed to
track microscopic silica or polystyrene beads. However, the same techniques may be applied
to track other objects, such as bacterial cells.
107–109
1. Video based position detection—For simple imaging of a trapped particle, a video
camera mounted to the camera port of the microscope (or elsewhere) often suffices. By digitally
processing the signal acquired from the camera, and knowing the size subtended by a single
pixel (e.g., by calibrating the video picture against a distance standard, such as a ruled objective
micrometer), the position of a trapped object can be determined with subpixel accuracy
(typically, to within ~5 nm or better), using any of several centroid-finding algorithms.
110–
112
Video tracking of trapped objects using such algorithms has been implemented in real
time,
113,114
but this approach is restricted to video acquisition rates (typically ~25–120 Hz),
and the precision is ultimately limited by video timing jitter (associated with frame acquisition)
or variations in illumination. In principle, temporal resolution could be improved through the
use of high speed video cameras. Burst frame rates in excess of 40 kHz can be achieved with
specialized complementary metal oxide semiconductor (CMOS) cameras, for example.
However, the usefulness of high speed cameras can be limited by computer speed or memory
capacity. Current CPU speed limits real-time position tracking to ~500 Hz,
115
while practical
storage considerations limit the number of high-resolution frames that can be stored to ~10
5
,
which corresponds to less than 2 min of high-speed video at 1 kHz. Even if these technological
hurdles are overcome, high-speed video tracking is ultimately limited by the number of
recorded photons (since shorter exposures require more illumination), so spatial resolution
decreases as the frame rate increases. Generally speaking, the signal-to-noise ratio is expected
to decrease as the square root of the frame rate. The discrepancy between the low video
bandwidth(~100 Hz) and the much higher intrinsic bandwidth of even a relatively weak optical
trap(~kHz) results in aliasing artifacts, and these preclude the implementation of many of the
most effective calibration methods. Furthermore, video-based methods are not well suited to
the measurement of the relative position of an object with respect to the trap center, further
complicating force determination.
2. Imaging position detector—Several alternative (nonvideo) methods have been
developed that offer precise, high-bandwidth position detection of trapped objects. The
simplest of these is to image directly the trapped object onto a quadrant photodiode (QPD).
56,116,117
The diode quadrants are then summed pairwise, and differential signals are derived
from the pairs for both x and y dimensions. If desired, the differential signals can be normalized
by the sum signal from the four quadrants to reduce the dependence of the output on the total
light intensity. Direct imaging of a trapped particle is typically restricted to a small zone within
the specimen plane, and requires careful coalignment of the trap with the region viewed by the
detector. Moreover, the high magnification required to achieve good spatial resolution results
in comparatively low light levels at the QPD, ultimately limiting bandwidth and noise
performance.
49,50
The latter limitation has been addressed by the use of a diode laser operating
just below its lasing threshold, acting as a superbright, incoherent illumination source.
56
Imaging using laser illumination is considered impractical because of the speckle and
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interference that arise from coherent illumination over an extended region. Various laser phase-
randomization approaches may relieve this restriction, but these typically carry additional
disadvantages, most often reduced temporal bandwidth.
3. Laser-based position detection—Laser-based position detection is appealing, because
it is possible to use a single laser for both trapping and position detection. Unlike the imaging
detector scheme described above, laser-based detection requires the incorporation of a dichroic
mirror on the condenser side of the microscope to couple out the laser light scattered by the
specimen. Furthermore, the detector and its associated optics (lens, filters) must be stably
mounted on (or next to) the condenser to collect the output light. Two different laser-based
position detection schemes have been developed. The first relies on polarization interferometry.
9,49,50,118,119
This method is quite analogous to differential interference contrast (DIC)
microscopy, and it relies on a subset of the DIC imaging components within the microscope.
Incoming plane polarized laser light is split by a Wollaston prism into two orthogonal
polarizations that are physically displaced from one another. After passing through the
specimen plane, the beams are recombined in a second Wollaston prism and the polarization
state of the recombined light is measured. A simple polarimeter consists of a quarter wave plate
(adjusted so that plane-polarized light is transformed into circularly polarized light) followed
by a polarizing beam splitter. The intensity in each branch of the beam splitter is recorded by
a photodiode, and the normalized differential diode signal supplies the polarization state of the
light. A bead centered in the trap introduces an equal phase delay in both beams, and the
recombined light is therefore plane polarized. When the bead is displaced from its equilibrium
position, it introduces a relative phase delay between the two beams, leading to a slight elliptical
polarization after the beams are recombined. The ellipticity of the recombined light can be
calibrated against physical displacement by moving a bead a known distance through the
optical trap. This technique is extraordinarily sensitive
118
and is, in theory, independent of the
position of the trapped object within the specimen plane, because the trapping and detection
laser beams are one and the same, and therefore intrinsically aligned. In practice, however,
there is a limited range over which the position signal is truly independent of the trap position.
A further limitation of this technique is that it is one dimensional: it is sensitive to displacement
along the Wollaston shear axis, providing position detection in a single lateral direction.
A second type of laser-based position detection scheme—back focal plane detection—relies
on the interference between forward-scattered light from the bead and unscattered light.
59,
64,120–122
The interference signal is monitored with a QPD positioned along the optical axis
at a plane conjugate to the back focal plane of the condenser (rather than at an imaging plane
conjugate to the specimen). The light pattern impinging on the QPD is then converted to a
normalized differential output in both lateral dimensions as described above. By imaging the
back focal plane of the condenser, the position signal becomes insensitive to absolute bead
position in the specimen plane, and sensitive instead to the relative displacement of the bead
from the laser beam axis.
120
As with the polarization interferometer, the detection beam and
the optical trap are intrinsically aligned, however the QPD detection scheme can supply
position information in both lateral dimensions.
Laser-based position detection schemes have also been implemented with a second, low-power
detection laser.
49,50,59,81
The experimental complication of having to combine, spatially
overlap, and then separate the trapping and detection beams is frequently outweighed by the
advantages conferred by having an independent detection laser. Uncoupling trapping and
detection may become necessary, for example, when there are multiple traps produced in the
specimen plane, or if the absolute position of a bead is the relevant measure, rather than the
relative position of a bead from the optical trap. When dynamic position control of the optical
trap is implemented (see below), a separate detection laser permits rapid position calibration
of each trapped particle, and greatly simplifies position measurements in situations in which
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the trap is being moved.
50
The choice of a laser for position sensing is less constrained than
that of a trapping laser, and only a few mW of output power suffice for most detection schemes.
The total power should be kept as low as feasible to prevent the detection light from generating
significant optical forces itself, thereby perturbing the trap. A detection laser wavelength
chosen to match the peak sensitivity of the photodetector will minimize the amount of power
required in the specimen plane. Separating the detection and trapping wavelengths facilitates
combining and separating the two beams, but increases the constraints on the dichroic mirror
that couples the laser beams into the microscope. We have found that combining two beams
of similar wavelength is most easily accomplished with a polarizing beamsplitter, i.e., the
beams are orthogonally plane polarized and combined in the polarizer before entering the
microscope. Since the trapping and detection wavelengths are closely spaced, a single
reflection band on the coupling dichroic mirror suffices to couple both beams into and out of
the microscope. A holographic notch filter in front of the position detector provides ~6 orders
of magnitude of rejection at the trapping wavelength, permitting isolation and measurement of
the much less intense detection beam.
4. Axial position detection—The detection schemes described above were developed to
measure lateral displacement of objects within the specimen plane, a major focus of most
optical trapping work. Detecting axial motion within the optical trap has rarely been
implemented and has not been as well characterized until recently. Axial motion has been
determined by: measuring the intensity of scattered laser light on an overfilled photodiode;
123–126
through two-photon fluorescence generated by the trapping laser;
127–130
and by
evanescent-wave fluorescence at the surface of a coverglass.
131,132
Although these various
approaches all supply a signal related to axial position, they require the integration of additional
detectors and, in some cases, fluorescence capability, into the optical trapping instrument. This
can be somewhat cumbersome, consequently the techniques have not been widely adopted.
The axial position of a trapped particle can also be determined from the total laser intensity in
the back focal plane of the condenser.
62,64
The axial position signal derives from the
interference between light scattered by the trapped particle and the unscattered beam. On
passing through a focus, the laser light accumulates a phase shift of π, known as the Gouy
phase.
133
The axial phase shift is given by ψ(z) =tan
1
(z/z
0
), where z
0
is the Rayleigh range
(z
0
= πw
0
2
/λ, where w
0
is the beam waist and λis the wavelength of light), and z is the axial
displacement from the focus.
133
Light scattered by a particle located near the focus will
preserve the phase that it acquired prior to being scattered, whereas unscattered light will
accumulate the full Gouy phase shift of π. The far-field interference between the scattered and
unscattered light gives rise to an axial position-dependent intensity, which can be measured,
for example, at the back focal plane of the condenser (see below and Fig. 8). This is the axial
counterpart, in fact, of the lateral interference signal described above. Axial position detection
can be achieved through a simple variant of quadrant photodiode-based lateral position
detection. Recording the total incident intensity on the position detector supplies the axial
position of trapped particle relative to the laser focus.
63,64
In contrast to lateral position
detection, axial position detection is inversely proportional to the NA of the detector.
62,63
When a single detector is used to measure both lateral and axial position simultaneously, an
intermediate detector NA should be used to obtain reasonable sensitivity in all three
dimensions.
5. Detector bandwidth limitations—Position detection based on lasers facilitates high
bandwidth recording because of the high intensity of light incident on the photodetector.
However, the optical absorption of silicon decreases significantly beyond ~850 nm, therefore
position sensing by silicon-based photodetectors is intrinsically bandwidth limited in the near
infrared.
61,134
Berg-Sørenson and co-workers
134
demonstrated that the electrical response
of a typical silicon photodiode to infrared light consists of both a fast and a slow component.
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The fast component results from optical absorption in the depletion region of the diode, where
the optically generated electron hole pairs are rapidly swept to the electrodes. This represents
the intended behavior of the diode, and is valid at wavelengths that are readily absorbed by the
active material, i.e., λ < ~ 1 μm. At longer wavelengths, however, a slow component also
appears in the diode response, due to absorption of light beyond the depletion region. Electron–
hole pairs generated in this zone must diffuse into the depletion region before flowing on to
the electrodes, a much slower process. Infrared light is poorly absorbed by silicon, resulting
in a greater proportion of the incident light being absorbed beyond the depletion region,
increasing the relative contribution of the slow component. Thus, the output of the diode
effectively becomes lowpass filtered (f
3dB
~ 8–9 kHz at 1064 nm) in an intensity-, wavelength-,
and reverse bias-dependent manner.
134
In principle, the effect of this lowpass filtering could
be calculated and compensated, but in practice, this approach is complicated by the intensity
dependence of the parasitic filtering. One workaround would be to employ a detection laser at
a wavelength closer to the absorption maximum of silicon, i.e., shorter than ~850 nm. Two
other solutions include using nonsilicon-based detectors employing different photoactive
materials, or using silicon-based photo-detectors with architectures that minimize the parasitic
filtering. Peterman and co-workers measured the wavelength dependence of parasitic filtering
in a standard silicon-based detector. They also reported an increased bandwidth at wavelengths
up to 1064 nm for an InGaAs diode as well as for a specialized, fully depleted silicon detector.
61
We have found that one commercial position sensitive detector (PSD) (Pacific Silicon
Detectors, which supplies output signals similar to those from a QPD, although operating on
a different principle), does not suffer from parasitic filtering below ~ 150 kHz with 1064 nm
illumination (Fig. 5).
F. Dynamic position control
Precise, calibrated lateral motion of the optical trap in the specimen plane allows objects to be
manipulated and moved relative to the surface of the trapping chamber. More significantly,
dynamic computer control over the position and stiffness of the optical trap allows the force
on a trapped object to be varied in real time, which has been exploited to generate both force
and position clamp measurement conditions.
50,81
Additionally, if the position of the optical
trap is scanned at a rate faster than the Brownian relaxation time of a trapped object, multiple
traps can be created by time sharing a single laser beam.
49
We consider below the different
beam-steering strategies.
1. Scanning mirrors—Traditional galvanometer scanning mirrors benefited from the
incorporation of feedback to improve stability and precision. Current commercial systems
operate at 1–2 kHz with step response times as short as 100 μs, and with 8 μrad repeatability.
The comparatively slow temporal response limits their usefulness for fast-scanning
applications, but their low insertion loss and large deflection angles make them a low-cost
option for slow-scanning and feedback applications. Recent advances in feedback-stabilized
piezoelectric (PZ) systems have resulted in the introduction of PZ scanning mirrors. For the
time being, PZ mirrors represent only a slight improvement over galvanometers, with effective
operation up to 1 kHz, but just 50 mrad deflection range, and only slightly better resolution
and linearity than galvanometers.
2. Acousto-optic deflectors—An acousto-optic deflector (AOD) consists of a transparent
crystal inside which an optical diffraction grating is generated by the density changes associated
with an acoustic traveling wave of ultrasound. The grating period is given by the wavelength
of the acoustic wave in the crystal, and the first-order diffracted light is deflected through an
angle that depends on the acoustic frequency through Δθ=λf/ν, where λ is the optical
wavelength, and ν and f are the velocity and frequency of the acoustic wave, respectively (ν/f
is the ultrasound wavelength). The diffraction efficiency is proportional to depth of the grating,
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and therefore to the amplitude of the acoustic wave that produced it. AODs are thereby able
to control both the trap position (through deflection) and stiffness (through light level). The
maximum deflection of an AOD is linearly related to its operating frequency range, and
maximum deflections of somewhat over 1° are possible at 1064 nm. AODs are fast: their
response times are limited, in principle, by the ratio of the laser spot diameter to the speed of
sound within the crystal (~1.5 μs/mm laser diameter for TeO
2
crystals, slightly less for
Li
6
NbO
3
crystals). In practice, however, the response time of an optical trapping instrument
is often limited by other components in the system. A pair of AODs can be combined in an
orthogonal configuration to provide both x and y deflections of the optical trap. Due to optical
losses in the AODs (an ~80% diffraction efficiency is typical), however, this scheme results
in an almost 40% power loss. In addition to mediocre transmission, the diffraction efficiency
of an AOD will often vary slightly as a function of its deflection. The resulting position-
dependent stiffness variation of the optical trap can either be tolerated (if within acceptable
margins for error), calibrated out,
53
or minimized by the selection of a particular range of
operating deflections over which the diffraction efficiency is more nearly constant. In practice,
however, every AOD needs to be characterized carefully before use for deflection-dependent
changes in throughput.
3. Electro-optic deflectors—An electro-optic deflector (EOD) consists of a crystal in
which the refractive index can be changed through the application of an external electric field.
A gradient in refractive index is established in one plane along the crystal, which deflects the
input light through an angle θlV/w
2
, where V is the applied voltage, l is the crystal length,
and w is the aperture diameter. Deflections on the order of 20 mrad can be achieved with a
switching time as short as 100 ns, sufficient for some optical trapping applications. Despite
low insertion loss (~1%) and straightforward alignment, EODs have not been widely employed
in optical trapping systems. High cost and a limited deflection range may contribute to this.
G. Piezoelectric stage
Piezoelectric stage technology has been improved dramatically through the introduction of
high-precision controllers and sensitive capacitive position sensing. Stable, linear,
reproducible, ultrafine positioning in three dimensions is now readily achievable with the latest
generation of PZ stages. The traditional problems of hysteresis and drift in PZ devices have
been largely eliminated through the use of capacitive position sensors in a feedback loop. With
the feedback enabled, an absolute positional uncertainty of 1 nm has been achieved
commercially. PZ stages have had an impact on practically every aspect of optical trapping.
They can provide an absolute, NIST-traceable displacement measurement, from which all other
position calibrations can be derived. Furthermore, these stages permit three-dimensional
control of the position of the trap relative to the trapping chamber, which has previously proved
difficult or inaccurate.
39
The ability to move precisely in the axial dimension, in particular,
permits characterization of the longitudinal properties of the optical trap and can be used to
eliminate the creep and backlash typically associated with the mechanical (gear based) focusing
mechanism of the microscope. Position and force calibration routines employing the PZ stage
are faster, more reproducible, and more precise than previously attainable. Finally, a
piezoelectric stage can be incorporated into a force feedback loop
86,135–137
permitting
constant-force records of essentially arbitrary displacement, ultimately limited by the stage
travel (~100 μm) rather than the working range of the position detector (~0.3 μm), the latter
being the limiting factor in feedback based on moving the optical trap.
50,59
Stage-based force-
feedback permits clamping not only the transverse force, but also the axial force, and hence
the polar angle through which the force is applied. Despite these advantages, PZ stages are not
without their attendant drawbacks. They are comparatively expensive: a 3D stage with
capacitive feedback position sensing plus a digital controller costs roughly $25,000.
Furthermore, communication with the stage controller can be slower than for other methods
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of dynamically controlling trap position (e.g., AODs or EODs), with a maximum rate of ~50
Hz.
59
H. Environmental isolation
To achieve the greatest possible sensitivity, stability, and signal-to-noise ratio in optical
trapping experiments, the environment in which the optical trapping is performed must be
carefully controlled. Four environmental factors affect optical trapping measurements:
temperature changes, acoustic noise, mechanical vibrations, and air convection. Thermal
fluctuations can lead to slow, large-scale drifts in the optical trapping instrument. For typical
optical trapping configurations, a 1 K temperature gradient easily leads to micrometers of drift
over a time span of minutes. In addition, acoustic noise can shake the optics that couple the
laser into the objective, the objective itself, or the detection optics that lie downstream of the
objective. Mechanical vibrations typically arise from heavy building equipment, e.g.,
compressors or pumps operating nearby, or from passing trucks on a roadway. Air currents
can induce low-frequency mechanical vibrations and also various optical perturbations (e.g.,
beam deflections from gradients in refractive index produced by density fluctuations in the
convected air, or light scattering by airborne dust particles), particularly near optical planes
where the laser is focused.
The amount of effort and resources dedicated to reducing ambient sources of noise should be
commensurate with the desired precision in the length and time scale of the measurements.
Slow thermal drift may not affect a rapid or transient measurement, but could render
meaningless the measurement of a slower process. Several methods of reducing noise and drift
have been employed in the current generation of optical traps.
The vast majority of optical trapping instruments have been built on top of passive air tables
that offer mechanical isolation (typically, 20 dB) at frequencies above ~2–10 Hz. For rejection
of lower frequencies, actively servoed air tables are now commercially available, although we
are not yet aware of their use in this field. Acoustic noise isolation can be achieved by ensuring
that all optical mounts are mechanically rigid, and placing these as close to the optical table as
feasible, thereby reducing resonance and vibration. Enclosing all the free-space optics will
further improve both mechanical and optical stability by reducing ambient air currents.
Thermal effects and both acoustical and mechanical vibration can be reduced by isolating the
optical trapping instrument from noisy power supplies and heat sources. Diode pumped solid
state lasers are well suited to this approach: since the laser head is fiber coupled to the pump
diodes, the power supply can be situated outside of the experimental room. A similar isolation
approach can be pursued with noisy computers or power supplies, and even illumination
sources, whose outputs can be brought to the instrument via an optical fiber. Further
improvements in noise performance and stability may require more substantial modifications,
such as acoustically isolated and temperature controlled experimental rooms situated in low-
vibration areas. The current generation of optical trapping instruments in our lab
59,138
are
housed in acoustically quiet cleanrooms with background noise less than the NC30 (OSHA)
rating, a noise level roughly equivalent to a quiet bedroom. In addition, these rooms are
temperature stabilized to better than ±0.5 K. The stability and noise suppression afforded by
this arrangement has paved the way for high-resolution recording of molecular motor
movement, down to the subnanometer level.
85–87
IV. CALIBRATION
A. Position calibration
Accurate position calibration lies at the heart of quantitative optical trapping. Precise
determination of the displacement of a trapped object from its equilibrium position is required
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to compute the applied force (F=−αx, where F is the force, α is the optical trap stiffness, and
x is the displacement from the equilibrium trapping position), and permits direct measurement
of nanometer-scale motion. Several methods of calibrating the response of a position detector
have been developed. The choice of method will depend on the position detection scheme, the
ability to move the trap and/or the stage, the desired accuracy, and the expected direction and
magnitude of motion in the optical trap during an experiment. The most straightforward
position calibration method relies on moving a bead through a known displacement across the
detector region while simultaneously recording the output signal. This operation can be
performed either with a stuck bead moved by a calibrated displacement of the stage, or with a
trapped bead moved with a calibrated displacement of a steerable trap.
Position determination using a movable trap relies on initial calibration of the motion of the
trap itself in the specimen plane against beam deflection, using AODs or deflecting mirrors.
This is readily achieved by video tracking a trapped bead as the beam is moved.
49
Video
tracking records can be converted to absolute distance by calibrating the charge coupled device
(CCD) camera pixels with a ruled stage micrometer (10 μm divisions or finer),
49,50
or by
video tracking the motion of a stuck bead with a fully calibrated piezoelectric stage.
59
Once
the relationship between beam deflection and trap position is established, the detector can then
be calibrated in one or both lateral dimensions by simply moving a trapped object through the
detector active area and recording the position signal.
50,59,81
Adequate two-dimensional
calibration may often be obtained by moving the bead along two orthogonal axes in an “X”
pattern. However, a more complete calibration requires raster scanning the trapped bead to
cover the entire active region of the sensor.
59
Figure 6 displays the two-dimensional detector
calibration for a 0.6 μm bead, raster scanned over the detector region using an AOD-driven
optical trap. A movable optical trap is typically used with either an imaging position detector,
or a second low-power laser for laser-based detection (described above). Calibrating by moving
the trap, however, offers several advantages. Position calibration can be performed individually
for each object trapped, which eliminates errors arising from differences among nominally
identical particles, such as uniform polystyrene beads, which may exhibit up to a 5% coefficient
of variation in diameter. Furthermore, nonspherical or nonidentical objects, such as bacteria
or irregularly shaped particles, can be calibrated on an individual basis prior to (or after) an
experimental measurement. Because the object is trapped when it is calibrated, the calibration
and detection necessarily take place in the same axial plane, which precludes calibration errors
arising from the slight axial dependence of the lateral position signals.
Laser-based detection used in conjunction with a movable trap affords additional advantages.
Because the trapping and detection lasers are separate, the focal position of the two can be
moved relative to one another in the axial dimension. The maximum lateral sensitivity and
minimum variation of lateral sensitivity with axial position occur at the focus of the detection
laser. The axial equilibrium position of a trapped object, however, lies above the focus due to
the scattering force. Since the detection and trapping lasers are uncoupled, the focus of the
detection laser can be made coincident with the axial position of the trapped object, thereby
maximizing the detector sensitivity while minimizing the axial dependence of the lateral
sensitivity.
59
An additional benefit to using an independent detection laser is that it can be
more weakly focused to a larger spot size, since it does not need to trap, thereby increasing the
usable detection range. Beyond the added complication and cost of building a movable trap,
calibrating with a movable trap has some important limitations. The calibration is limited to
the two lateral dimensions, which may be inadequate for experiments where the trapped bead
is displaced significantly in the axial dimension.
39,82
Due to the ~4–6-fold lower trap stiffness
in the axial dimension, a primarily lateral force pulling an object out of the trapping zone may
result in a significant axial displacement. In practice, this situation arises when the trapped
object is tethered to the surface of the trapping chamber, e.g., when a bead is attached by a
strand of DNA bound at its distal end to the coverglass.
39,82,88,135–137
Accurate
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determinations of displacement and trapping force in such experiments require axial, as well
as lateral, position calibration.
Position calibration is most commonly accomplished by moving a bead fixed to the surface
through the detection region and recording the detector output as a function of position.
Traditionally, such calibrations were performed in one or two lateral dimensions. The advent
of servo-stabilized, 3D piezoelectric (pz) stages has made such calibrations more accurate,
easier to perform and—in conjunction with an improved theoretical understanding of the axial
position signal—has permitted a full 3D position calibration of an optical trap.
62–64
Whereas
full 3D calibration is useful for tracking the complete motion of an object, it is cumbersome
and unnecessary when applying forces within a plane defined by one lateral direction and the
optical axis. When the trapped object is tethered to the surface of the trapping chamber, for
example, it is sufficient to calibrate the axial and the single lateral dimension in which the force
is applied. Figure 7 displays the results of such a two-dimensional (“xz”) position calibration
for a 0.5 μm bead stuck to the surface of the trapping chamber. The bead was stepped through
a raster scan pattern in x (lateral dimension) and z (axial dimension) while the position signals
were recorded. Using a stuck bead to calibrate the position detector has some limitations and
potential pitfalls. Because it is difficult, in general, to completely immobilize an initially
trapped particle on the surface, it is not feasible to calibrate every particle. Instead, an average
calibration derived from an ensemble of stuck beads must be measured. Furthermore, the stuck-
bead calibration technique precludes calibrating non-spherical or heterogeneous objects, unless
these can be attached to the surface (and stereospecifically so) prior to, or after, the experimental
measurements. Due to the axial dependence of the lateral position signals (“xz crosstalk”),
using a stuck bead to calibrate only the lateral dimension is prone to systematic error. Without
axial position information, it is difficult to precisely match the axial position of a stuck bead
with the axial position of a trapped bead. Optically focusing on a bead cannot be accomplished
with an accuracy better than ~100 nm, which introduces uncertainty and error in lateral position
calibrations for which the axial position is set by focusing. Therefore, even when only the
lateral dimensions are being calibrated, it is useful to measure the axial position signal to ensure
that the calibration is carried out in the appropriate axial plane.
1. Absolute axial position and measurement of the focal shift—The absolute axial
position of a trapped object above the surface of the trapping chamber is an important
experimental parameter, because the hydrodynamic drag on an object varies nonlinearly with
its height above the surface, due to proximal wall effects (see below and Ref. 9). Absolute axial
position measurements may be especially important in situations where the system under
investigation is attached to the surface and to a trapped object, as is often the case in biological
applications. Force–extension relationships, for example, depend on the end-to-end extension
of the molecule, which can only be determined accurately when the axial position of the trapped
object with respect to the surface is known. Axial positioning of a trapped object depends on
finding the location of the surface of the chamber and moving the object relative to this surface
by a known amount. The problem is complicated by the focal shift that arises when focusing
through a planar interface between two mismatched indices of refraction e.g., between the
coverglass (n
glass
~ 1.5) and the aqueous medium (n
water
~ 1.3).
139–144
This shift introduces
a fixed scaling factor between a vertical motion of the chamber surface and the axial position
of the optical trap within the trapping chamber. The focal shift is easily computed from Snell’s
law for the case of paraxial rays, but it is neither straightforward to compute nor to measure
experimentally when high NA objectives are involved.
144
Absolute axial position
determination has previously been assessed using fluorescence induced by an evanescent wave,
131
by the analysis of interference or diffraction patterns captured with video,
113,145
or
through the change in hydrodynamic drag on a trapped particle as it approaches the surface.
39
These techniques suffer from the limited range of detectable motion for fluorescence-based
methods, and by the slow temporal response of video and drag-force-based measurements.
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The position detector sum signal (QPD or PSD output), which is proportional to the total
incident intensity at the back focal plane of the condenser, provides a convenient means of both
accurately locating the surface of the trapping chamber and measuring the focal shift. In
conjunction, these measurements permit absolute positioning of a trapped object with respect
to the trapping chamber surface. The detector sum signal as a function of axial stage position
for both a stuck bead and a trapped bead are shown in Fig. 8. The stuck bead trace represents
the axial position signal of a bead moving relative to the trap. As the bead moves through the
focus of the laser (marked on the figure), the phase of light scattered from the bead changes
by 180° relative to the unscattered light, modulating the intensity distribution at the back focal
plane of the condenser. The region between the extrema of the stuck-bead curve is well
described by the expression for axial sensitivity derived by Pralle and co-workers:
62
I
z
I
(
z
)
(
1 +
(
z
z
0
)
2
)
1
/
2
sin tan
−1
(
z
/
z
0
) , (5)
where an overall scaling factor has been ignored, z is the axial displacement from the beam
waist, and z
0
= πw
0
2
/λ is the Raleigh length of the focus, with beam waist w
0
at wavelength
λ. The phase difference in the scattered light is described by the arctangent term, while the
prefactor describes the axial position dependent intensity of the scattered light. The fit returns
a value for the beam waist, w
0
=0.436 μm. The equilibrium axial position of a trapped bead
corresponds to a displacement of 0.379 μm from the laser focus. A stuck bead scan can also
be useful for determining when a free bead is forced onto the surface of the cover slip.
As a trapped bead is forced into contact with the surface of the chamber by the upward stage
motion, the free and stuck bead signals merge and eventually become indistinguishable (Fig.
8). The approximate location of the surface with respect to the position of a trapped bead can
be determined by finding the point at which both curves coincide. Brownian motion of the
trapped bead, however, will shift this point slightly, in a stiffness-dependent manner that will
introduce a small uncertainty in the measured position of the surface. The scattering peak in
Fig. 8, however, serves as an easily identifiable fiducial reference from which the trapped bead
can be moved an absolute distance by subsequent stage motion. In this manner, trapped particles
can be reproducibly positioned at a fixed (but uncertain) distance relative to the surface. In
order to obtain a precise location of the trapped particle above the surface, both the position of
the scattering peak with respect to the surface and the focal shift must be determined. This may
be accomplished, for example, by a one-time measurement of the drag on a trapped bead at a
series of positions above the scattering peak. The interaction of a sphere with the boundary
layer of water near a surface leads to an increase in the hydrodynamic drag β, which can be
estimated by Faxen’s law for the approximate drag on a sphere near a surface:
9
β
=
6
πηa
1
9
16
(
a
h
)
+
1
8
(
a
h
)
3
45
256
(
a
h
)
4
1
16
(
a
h
)
5
,
(6)
which depends only on the bead radius a, the distance above the surface h, and the viscosity
of the liquid η. By measuring the rolloff frequency or the displacement of the trapped bead as
the stage is oscillated (see below), the drag force can be determined at different axial stage
positions relative to the scattering peak and normalized to the calculated asymptotic value, the
Stokes drag coefficient, 6πηa. The resulting curve (Fig. 9) is described by a two parameter fit
to Eq. (6): a scaling parameter that represents the fractional focal shift and an offset parameter
related to the distance between the scattering peak and the coverglass surface. The fit
parameters from the curve in Fig. 9 allow absolute positioning of a trapped particle with respect
to the surface. The uncertainty in the axial position amounts to roughly 3% of the bead-surface
separation, with the residual uncertainty largely due to the estimate of the focal shift (which
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leads to a relative rather than an absolute uncertainty). The position of the surface, calculated
from the fit parameters of Fig. 9, is indicated in Fig. 8. The focal shift was 0.82±0.02, i.e., the
vertical location of the laser focus changed by 82% of the vertical stage motion.
The periodic modulation of the axial position signal as a trapped bead is displaced from the
surface (Fig. 8) can be understood in terms of an étalon picture.
146
Backscattered light from
the trapped bead reflects from the surface and interferes with forward-scattered and unscattered
light in the back focal plane of the condenser. The phase difference between these two fields
includes a constant term that arises because of the Gouy phase and another term that depends
on the separation between the bead and the surface. The spatial frequency of the intensity
modulation is given by d =λ/(2n
m
), where d is the separation between the bead and the cover
slip, λ is the vacuum wavelength of the laser, and n
m
is the index of refraction of the medium.
This interference signal supplies a second and much more sensitive means to determine the
focal shift. The motion of the stage (d
s
) and motion of the focus (d
f
) are related through a scaling
parameter f
s
equal to the focal shift d
f
= f
s
d
s
. The interference signal is observed experimentally
by stage translations. The measured spatial frequency will be given by d
s
= λ/(2n
m
f
s
), which
can be rearranged to solve for the focal shift f
s
=λ/(2n
m
d
s
). The focal shift determined in this
manner was 0.799±0.002, which is within the uncertainty of the focal shift determined by
hydrodynamic drag measurements (Fig. 9). The true focal shift with a high NA lens is more
pronounced than the focal shift computed in the simple paraxial limit, given (from Snell’s law)
by the ratio of the indices of refraction: n
m
/n
imm
=0.878 for the experimental conditions, where
n
m
is the index of the aqueous medium (1.33) and n
imm
is the index of the objective immersion
oil (1.515). The discrepancy should not be surprising, because the paraxial ray approximation
does not hold for the objectives used for optical trapping.
146
The interference method
employed to measure the focal shift is both easier and more accurate than the drag-force method
presented earlier.
2. Position calibration based on thermal motion—A simple method of calibrating the
position detector relies on the thermal motion of a bead of known size in the optical trap.
122
The one-sided power spectrum for a trapped bead is
9
S
xx
(
f
) =
k
B
T
π
2
β
(
f
0
2
+
f
2
)
,
(7)
where S
xx
( f) is in units of displacement
2
/Hz, k
B
is Boltzmann’s constant, T the absolute
temperature, β is the hydrodynamic drag coefficient of the object (e.g., β=6πηa for Stokes drag
on a sphere of radius a in a medium with viscosity η), and f
0
is the rolloff frequency, related
to the trap stiffness through f
0
= α(2πβ)
1
for a stiffness α (see below). The detector, however,
measures the uncalibrated power spectrum S
νν
( f), which is related to the true power spectrum
by S
νν
(f) =ρ
2
·S
xx
(f), where ρ represents the linear sensitivity of the detector (in volts/unit
distance). The sensitivity can be found by considering the product of the power spectrum and
the frequency squared S
xx
(ff
2
, which asymptotically approaches the limit k
B
T(π
2
β)
1
for f
f
0
. Inserting the relationship between the displacement power spectrum and the uncalibrated
detector spectrum in this expression and rearranging gives
ρ
=
S
νν
(
f
)
π
2
β
/
k
B
T
1
/
2
. (8)
This calibration method has been shown to agree to within ~20% of the sensitivity measured
by more direct means, such as those discussed above.
122
An advantage to the method is that
it does not require any means of precisely moving a bead to calibrate the optical trap. However,
the calibration obtained by this method is valid only for small displacements, for which a linear
approximation to the position signal is valid. In addition, the system detection bandwidth must
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be adequate to record accurately the complete power spectrum without distortion, particularly
in the high frequency regime. System bandwidth considerations are treated more fully in
conjunction with stiffness determination, discussed below.
B. Force calibration–stiffness determination
Forces in optical traps are rarely measured directly. Instead, the stiffness of the trap is first
determined, then used in conjunction with the measured displacement from the equilibrium
trap position to supply the force on an object through Hooke’s law: F=−αx, where F is the
applied force, α is the stiffness, and x is the displacement. Force calibration is thus reduced to
calibrating the trap stiffness and separately measuring the relative displacement of a trapped
object. A number of different methods of measuring trap stiffness, each with its attendant
strengths and drawbacks, have been implemented. We discuss several of these.
1. Power spectrum—When beads of known radius are trapped, the physics of Brownian
motion in a harmonic potential can be exploited to find the stiffness of the optical trap. The
one-sided power spectrum for the thermal fluctuations of a trapped object is given by Eq. (7),
which describes a Lorentzian. This power spectrum can be fit with an overall scaling factor
and a rolloff frequency, f
0
= α(2πβ)
1
from which the trap stiffness (α) can be calculated if the
drag (β) on the particle is known (Fig. 10). For a free sphere of radius a in solution far from
any surfaces, the drag is given by the usual Stokes relation β=6πηa, where η is the viscosity
of the medium. For a bead trapped nearer the surface of the trapping chamber, additional drag
arises from wall effects and must be considered: Faxen’s law [Eq. (6)] is appropriate for
estimating the drag due to lateral motion. Axial stiffness is also measured via the power
spectrum of the axial position signal, but the corrections to the axial drag due to wall effects
are larger than those for the lateral drag. The drag on a sphere moving normal to a surface
is
147
β
=
β
0
4
3
sinh
α
n
=1
n
(
n
+ 1)
(2
n
1)(2
n
+ 3)
×
2 sinh (2
n
+ 1)
α
+ (2
n
+ 1) sinh 2
α
4sinh
2
(
n
+
1
2
)
α
(2
n
+ 1)
2
sinh
2
α
1 ,
(9)
where
α
= cosh
−1
(
h
a
)
= ln
{
h
a
+
(
h
a
)
2
1
1
/
2
}
,
h is the height of the center of the sphere above the surface, and β
0
=6πηa is the Stokes drag.
The sum converges fairly quickly and ~10 terms are required to achieve accurate results.
Whereas it is tempting to measure trap stiffness well away from surfaces to minimize
hydrodynamic effects, spherical aberrations in the focused light will tend to degrade the optical
trap deeper in solution, particularly in the axial dimension. Spherical aberrations lead to both
a reduction in peak intensity and a smearing-out of the focal light distribution in the axial
dimension.
Determining the stiffness of the optical trap by the power spectrum method requires a detector
system with sufficient bandwidth to record faithfully the power spectrum well beyond the
rolloff frequency (typically, by more than 1 order of magnitude). Lowpass filtering of the
detector output signal, even at frequencies beyond the apparent rolloff leads directly to a
numerical underestimate of the rolloff frequency and thereby to the stiffness of the optical trap.
Errors introduced by low pass filtering become more severe as the rolloff frequency of the trap
approaches the rolloff frequency of the electrical filter. Since the trap stiffness is determined
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solely from the rolloff of the Lorentzian power spectrum, this method is independent of the
position calibration, per se. In addition to determining the stiffness, the power spectrum of a
trapped bead serves as a powerful diagnostic tool for optical trapping instruments: alignment
errors of either the optical trap or the position detection system lead to non-Lorentzian power
spectra, which are easily scored, and extraneous sources of instrument noise can generate
additional peaks in the power spectrum.
The measurement and accurate fitting of power spectra to characterize trap stiffness was
recently investigated by Berg-Sørensen and Flyvbjerg,
148,149
who developed an improved
expression for the power spectrum that incorporates several previously ignored corrections,
including the frequency dependence of the drag on the sphere, based on an extension of Faxen’s
law for an oscillating sphere [Faxen’s law, Eq. (6), only holds strictly in the limit of constant
velocity]. These extra terms encapsulate the relevant physics for a sphere moving in a harmonic
potential with viscous damping. In addition to this correction, the effects of finite sampling
frequency and signal filtering during data acquisition (due to electronic filters or parasitic
filtering by the photosensor) were included in fitting the experimental power spectrum. The
resulting fits determine the trap stiffness with an uncertainty of ~1% and accurately describe
the shape of the measured spectra. This work underscores the importance of characterizing and
correcting the frequency response of the position detection system to obtain accurate stiffness
measurements. Figure 10 illustrates a comparison between the fit obtained with the improved
fitting routine and an uncorrected fit.
The power spectrum of a trapped bead can also be used to monitor the sample heating due to
partial absorption of the trapping laser light. Heating of the trapping medium explicitly changes
the thermal kinetic energy term (k
B
T) in the power spectrum [Eq. (7)] and implicitly changes
the drag term as well, β=6πη(T)a, through its dependence on viscosity, which is highly
temperature dependent. Peterman and co-workers were able to assess the temperature increase
as a function of trapping laser power by determining the dependence of the Lorentzian fit
parameters on laser power.
100
2. Equipartition—The thermal fluctuations of a trapped object can also be used to obtain the
trap stiffness through the Equipartition theorem. For an object in a harmonic potential with
stiffness α:
1
2
k
B
T
=
1
2
α x
2
,
(10)
where k
B
is Boltzmann’s constant, T is absolute temperature, and x is the displacement of the
particle from its trapped equilibrium position. Thus, by measuring the positional variance of a
trapped object, the stiffness can be determined. The variance ‹x
2
› is intimately connected to
the power spectrum, of course: it equals the integral of the position power spectrum, i.e., the
spectrum recorded by a calibrated detector. Besides its simplicity, a primary advantage of the
Equipartition method is that it does not depend explicitly on the viscous drag of the trapped
particle. Thus, the shape of the particle, its height above the surface, and the viscosity of the
medium need not be known to measure the trap stiffness (although, in fairness, both the particle
shape and the optical properties of the medium will influence the position calibration itself).
The bandwidth requirements of the position detection system are the same as for the power
spectral approach, with the additional requirement that the detector must be calibrated. Unlike
the power spectral method however, the variance method does not provide additional
information about the optical trap or the detection system. For this reason, care should be taken
when measuring the stiffness with the Equipartition method. Because variance is an
intrinsically biased estimator (it is derived from the square of a quantity, and is therefore always
positive), any added noise and drift in position measurements serve only to increase the overall
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variance, thereby decreasing the apparent stiffness estimate. In contrast, low pass filtering of
the position signal results in a lower variance and an apparent increase in stiffness.
3. Optical potential analysis—A straightforward extension of the Equipartition method
involves determining the complete distribution of particle positions visited due to thermal
motions, rather than simply the variance of that distribution. The probability for the
displacement of a trapped object in a potential well will be given by a Boltzmann distribution
P
(
x
)
exp
(
U
(
x
)
k
B
T
)
= exp
(
αx
2
2
k
B
T
)
, (11)
where U(x) is the potential energy and k
B
T is the thermal energy. When the potential is
harmonic, this distribution is a simple Gaussian parametrized by the trap stiffness α. When the
potential is anharmonic, the position histogram can be used, in principle, to characterize the
shape of the trapping potential by taking the logarithm and solving for U(x). In practice, this
approach is not especially useful without a considerable body of low-noise/low-drift position
data, since the wings of the position histogram—which carry the most revealing information
about the potential—hold the fewest counts and therefore have the highest relative uncertainty.
4. Drag force method—The most direct method of determining trap stiffness is to measure
the displacement of a trapped bead from its equilibrium position in response to viscous forces
produced by the medium, generated by moving the stage in a regular triangle wave or sinusoidal
pattern. Since forces arise from the hydrodynamics of the trapped object, the drag coefficient,
including any surface proximity corrections, must be known. For the case of a sinusoidal
driving force of amplitude A
0
and frequency f, the motion of the bead is
x
(
t
) =
A
0
f
f
0
2
+
f
2
exp
i
(2
πft
ϕ
) ,
ϕ
= tan
−1
(
f
0
/
f
),
(12)
where f
0
is the characteristic rolloff frequency (above), and ϕ is the phase delay. Both the
amplitude and the phase of the bead motion can be used to provide a measure of trap stiffness.
A triangular driving force of amplitude A
0
and frequency f results in a square wave of force
being applied to the bead. For each period of the motion the bead trajectory is
x
(
t
) =
βA
0
f
2
α
1 exp
(
α
β
t
)
,
(13)
where α is the trap stiffness and β is the drag coefficient of the bead, including Faxen’s law
corrections. Due to the finite response time of the stage, the exponential damping term is
convolved with the response time of the stage. Therefore, only the asymptotic value (βA
0
f/
2α) should be used to obtain a reliable estimate of trap stiffness. Drag-force measurements are
slow compared with the thermal motion of the particle, so the bandwidth requirements of the
detection system are significantly relaxed. Increasing the amplitude or the frequency of the
stage motion generates larger displacements of the trapped bead. By measuring the stiffness
as a function of bead displacement, the linear region of the trap over which the stiffness is
constant can be easily determined.
A variation on the drag force method of stiffness calibration, sometimes called step response
calibration, involves rapidly displacing the trap by a small, fixed offset and recording the
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subsequent trajectory of the bead. The bead will return to its equilibrium position in an
exponentially damped manner, with a time constant of α/β as in Eq. (13).
5. Direct measurement of optical force—The lateral trapping force arises from the
momentum transfer from the incident laser light to the trapped object, which leads to a change
in the direction of the scattered light (Fig. 1). Measuring the deflection of the scattered laser
beam with a QPD or other position sensitive detector therefore permits direct measurement of
the momentum transfer, and hence the force, applied to the trapped object—assuming that all
the scattered light can be collected.
38,57,104
An expression relating the applied force to the
beam deflection was presented by Smith et al.:
38
F=I/c·(NA)·X/R
ba
where F is the force, I is
the intensity of the laser beam, c is the speed of light, NA is the numerical aperture, X is the
deflection of the light, and R
ba
is the radius of the back aperture of the microscope objective.
In principle, this approach is applicable to any optical trapping configuration. However,
because it necessitates measuring the total intensity of scattered light, it has only been
implemented for relatively low NA, counter-propagating optical traps, where the microscope
objective entrance pupils are underfilled. In single-beam optical traps, it is impractical to collect
the entirety of the scattered light, owing to the higher objective NA combined with an optical
design that overfills the objective entrance pupil.
6. Axial dependence of lateral stiffness—Three-dimensional position detection
facilitates measurement of the axial stiffness and mapping of the lateral stiffness as a function
of axial position in the trap. Due to the high refractive index of polystyrene beads typically
used in optical trapping studies, there is a correspondingly large scattering force in the axial
direction. Consequently, the axial equilibrium position of a trapped polystyrene bead tends to
lie well beyond the focus, where the lateral intensity gradient—and hence the lateral stiffness
—are significantly reduced from their values at the focus. In experiments in which beads are
displaced from the axial equilibrium position, the change in lateral trapping strength can be
significant. The variation of lateral stiffness as a function of axial position was explored using
beads tethered by DNA (1.6 μm) to the surface of the flow chamber (Fig. 11, inset). Tethered
beads were trapped and the attachment point of the tether was determined and centered on the
optical axis.
39
The bead was then pulled vertically through the trap, i.e., along the axial
dimension, by lowering the stage in 20 nm increments. At each position, the lateral stiffness
of the trap was ascertained by recording its variance, using the Equipartition method. The axial
force applied to the bead tether can increase the apparent lateral stiffness, and this effect can
be computed by treating the tethered bead as a simple inverted pendulum.
150,151
In practice,
the measured increase in lateral stiffness (given by α
x
= F
a
/l, where α
x
is the lateral stiffness,
F
a
is the axial force on the bead, and l is the length of the tether) resulted in less than a 3%
correction to the stiffness and was thereafter ignored in the analysis. An average of 12
measurements is shown in Fig. 11, along with a fit to the lateral stiffness based on a simple
dipole and zero-order Gaussian beam model.
152
α
x
(
z
) =
8
n
m
p
cw
0
(
a
w
0
)
3
(
m
2
1
m
2
+ 2
)(
1 +
(
z
z
0
)
2
)
−2
,
(14)
where n
m
is the index of refraction of the medium, p is the laser power in the specimen plane,
c is the speed of light, m is the ratio of the indices of refraction of the bead and the medium,
and w
0
, z and z
0
are the beam diameter at the waist, the axial displacement of the particle relative
to the focus, and the Raleigh range, respectively (as previously defined). The data are well fit
by this model with the exception of the laser power, which was sixfold lower than the actual
power estimated in the specimen plane. A significant discrepancy was anticipated since it had
been previously shown that for particle sizes on the order of the beam waist, the dipole
approximation greatly overestimates the trap stiffness.
152
The other two parameters of interest
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are the beam waist and the equilibrium axial position of the bead in the trap. The fit returned
distances of 0.433 μm for the beam waist and 0.368 μm for the offset of the bead center from
the focal point. These values compare well with the values determined from the fit to the axial
position signal, which were 0.436 and 0.379 μm, respectively (see above). The variation in
lateral stiffness between the optical equilibrium position and the laser focus was substantial: a
factor of l.5 for the configuration studied.
V. TRANSCRIPTION STUDIED WITH A TWO-DIMENSIONAL STAGE-BASED
FORCE CLAMP
Our interest in extending position detection techniques to include the measurement of force
and displacement in the axial dimension arose from the study of processive nucleic acid
enzymes moving along DNA (Fig. 12). The experimental geometry, in which the enzyme
moving along the DNA pulls on a trapped bead, results in motion of the bead in a plane defined
by the direction of the lateral force and the axial dimension. In previous experiments, the effects
of axial motion had been calculated and estimated, but not directly measured or otherwise
calibrated.
39,82
Improvements afforded by three-dimensional piezoelectric stages permitted
the direct measurement of, and control over, the separate axial and lateral motions of the trapped
bead. We briefly describe this instrument and the implementation of a two-dimensional force
clamp to measure transcription by a single molecule of RNA polymerase.
87
The optical layout and detection scheme are illustrated in Fig. 13. An existing optical trap
39,
153
was modified by adding a normalizing photodetector to monitor the bleedthrough of the
trapping laser after a 45° dielectric mirror and a feedback-stabilized three-axis piezoelectric
stage (Physik Instrumente P-517.3CD and E710.3CD digital controller) to which the trapping
chamber was affixed. The optical trap was built around an inverted microscope (Axiovert 35,
Carl Zeiss) equipped with a polarized Nd:YLF laser (TFR, Spectra Physics, λ=1047 nm,
TEM
00
, 2.5 W) that is focused to a diffraction-limited spot through an objective (Plan Neofluar
100×, 1.3 NA oil immersion). Lateral position detection based on polarization interferometry
was implemented. The trapping laser passes through a Wollaston prism below the objective
producing two orthogonally polarized and slightly spatially separated spots in the specimen
plane; these act as a single trap. The light is recombined by a second Wollaston prism in the
condenser, after which it passes through a quarter-wave plate and a polarizing beamsplitter.
Two photodetectors measure the power in each polarization, and the difference between them,
normalized by their sum, supplies the lateral position signal. The sum of the detector signals
normalized by the incident laser power (from the normalizing detector) provides the axial
position signal.
62,64
The axial position signal is a small fraction of the total intensity and is
roughly comparable to the intensity noise of the laser. Normalizing the axial position signal
with reference to the instantaneous incident laser power, therefore, provides a significant
improvement in the signal-to-noise ratio. The two-dimensional position calibration of the
instrument, obtained by raster scanning a stuck bead, is shown in Fig. 7. Stiffness in the lateral
dimension was measured by a combination of rolloff, triangle-wave drag force, and variance
measurements. Stiffness in the axial dimension was measured using the rolloff method and
was found to be ~eightfold less than the lateral stiffness.
Single-molecule transcription experiments were carried out with an RNA polymerase
specifically attached to the beads and tethered to the surface of the trapping chamber via one
end of the template DNA (Fig. 12). Tethered beads were trapped, the surface position was
determined as described above, and the bead was centered over the attachment point of the
DNA tether, at a predetermined height. Once these initial conditions were established, the two-
dimensional force-clamp routine was begun. The stage was moved in both the axial and lateral
directions until the trapped bead was displaced by a predetermined distance from its
equilibrium position. Position signals were recorded at 2 kHz and boxcar averaged over 40
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points to generate a 50 Hz signal that was used to control the motion of the stage. In this fashion,
the displacement of the bead in the trap, and hence the optical force, was held constant as the
tether length changed by micron-scale distances during RNA polymerase movement over the
DNA template. The motion of RNA polymerase on the DNA can be calculated from the motion
of the stage (Fig. 14). Periods of constant motion interrupted by pauses of variable duration
are readily observed in the single-molecule transcription trace shown in Fig. 14. Pauses as short
as 1 s are readily detected (Fig. 14, inset). Positional noise is on the order of 2 nm, while drift
is less than 0.2 nm/s.
Two-dimensional stage based force clamping affords a unique advantage. Since the stiffness
in both dimensions is known, the force vector on the bead is defined and constant during an
experiment. Tension in the DNA tether opposes the force on the bead, therefore the angle of
the DNA with respect to the surface of the trapping chamber is similarly defined and constant.
More importantly, the change in the DNA tether length can be calculated from the motion of
the stage in one dimension and the angle calculated from the force in both dimensions. As a
result, such measurements are insensitive to drift in the axial dimension, which is otherwise a
significant source of instrumental error.
VI. PROGRESS AND OVERVIEW OF OPTICAL TRAPPING THEORY
Optical trapping of dielectric particles is sufficiently complex and influenced by subtle,
difficult-to-quantify optical properties that theoretical calculations may never replace direct
calibration. That said, recent theoretical work has made significant progress towards a more
complete description of optical trapping and three-dimensional position detection based on
scattered light. Refined theories permit a more realistic assessment of both the capabilities and
the limitations of an optical trapping instrument, and may help to guide future designs and
optimizations.
Theoretical expressions for optical forces in the extreme cases of Mie particles (a λ, a is
the sphere radius) and Raleigh particle (a λ) have been available for some time. Ashkin
calculated the forces on a dielectric sphere in the ray-optic regime for both the TEM
00
and the
TEM
01
* (“donut mode”) intensity profiles.
69
Ray-optics calculations are valid for sphere
diameters greater than ~10λ, where optical forces become independent of the size of the sphere.
At the other extreme, Chaumet and Nieto-Vesperinas obtained an expression for the total time
averaged force on a sphere in the Rayleigh regime
154
F
i
=
(
1
2
)
Re
αE
0
j
i
(
E
0
j
)
*
,
(15)
where
α
=
α
0
(1
2
3
ik
3
α
0
)
−1
is a generalized polarizability that includes a damping term, E
0
is the complex magnitude of the electric field, α
0
is the polarizability of a sphere given by Eq.
(4), and k is the wave number of the trapping laser. This expression encapsulates the separate
expressions for the scattering and gradient components of the optical force [Eqs. (1) and (3)]
and can be applied to the description of optical forces on larger particles through the use of the
coupled dipole method.
155
In earlier work, Harada and Asakura calculated the forces on a
dielectric sphere illuminated by a moderately focused Gaussian laser beam in the Rayleigh
regime by treating the sphere as a simple dipole.
152
The Raleigh theory predicts forces
comparable to those calculated with the more complete generalized Lorenz–Mie theory
(GLMT) for spheres of diameter up to ~w
0
(the laser beam waist) in the lateral dimension, but
only up to ~0.4λ in the axial dimension.
152
More general electrodynamic theories have been
applied to solve for the case of spheres of diameter ~λ trapped with tightly focused beams. One
approach has been to generalize the Lorenz–Mie theory describing the scattering of a plane
wave by a sphere to the case of Gaussian beams. Barton and co-workers applied fifth-order
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corrections to the fundamental Gaussian beam to derive the incident and scattered fields from
a sphere, which enabled the force to be calculated by means of the Maxwell stress tensor.
76,
77
An equivalent approach, implemented by Gouesbet and coworkers, expands the incident
beam in an infinite series of beam shape parameters from which radiation pressure cross
sections can be computed.
80,156
Trapping forces and efficiencies predicted by these theories
are found to be in reasonable agreement with experimental values.
157–159
More recently,
Rohrbach and co-workers extended the Raleigh theory to larger particles through the inclusion
of second-order scattering terms, valid for spheres that introduce a phase shift, k
0
(Δn)D, less
than π/3, where k
0
=2π/λ
0
is the vacuum wave number, Δn=(n
p
n
m
) is the difference in
refractive index between the particle and the medium, and D is the diameter of the sphere.
65,
66
For polystyrene beads (n
p
=1.57) in water (n
m
=1.33), this amounts to a maximum particle
size of ~0.7λ. In this approach, the incident field is expanded in plane waves, which permits
the inclusion of apodization and aberration transformations, and the forces are calculated
directly from the scattering of the field by the dipole without resorting to the stress tensor
approach. Computed forces and trapping efficiencies compare well with those predicted by
GLMT,
66
and the effects of spherical aberration have been explored.
65
Since the second-order
Raleigh theory calculates the scattered and unscattered waves, the far field interference pattern,
which is the basis of the three-dimensional position detection described above, is readily
calculated.
63,64
VII. NOVEL OPTICAL TRAPPING APPROACHES
Optical trapping (OT) has now developed into an active and diverse field of study. Space
constraints preclude a complete survey the field, so we have chosen to focus on a small number
of recent developments that seem particularly promising for future applications of the
technology.
A. Combined optical trapping and single-molecule fluorescence
Combining the complementary techniques of OT and single-molecule fluorescence (SMF)
presents significant technological challenges. Difficulties arise from the roughly 15 orders of
magnitude difference between the enormous flux of infrared light associated with a typical
trapping laser (sufficient to bleach many varieties of fluorescent dye through multiphoton
excitation) compared to the miniscule flux of visible light emitted by a single excited
fluorophore. These challenges have been met in a number of different ways. Funatsu and co-
workers built an apparatus in which the two techniques were employed sequentially, but not
simultaneously.
160
In a separate development, Ishijima and co-workers were able to trap beads
attached to the ends of a long (5–10 μm) actin filament while simultaneously monitoring the
binding of fluorescent Adenosine triphosphate (ATP) molecules to a myosin motor interacting
with the actin filament.
161
In this way, the coordination between the binding of ATP to myosin
and the mechanical motion of the actin filament (detected via the optical trap) was determined.
This experiment demonstrated the possibility of simultaneous—but not spatially coincident—
OT and SMF in the same microscope field of view. In a more recent development, both
simultaneous and spatially coincident OT and SMF have been achieved, and used to measure
the mechanical forces required to unzip short duplex regions [15 base pair (bp)] of double-
stranded DNA.
138
Dye-labeled hybrids were attached via a long (~1000 bp) DNA “handle”
to a polystyrene bead at one end (using the 3 end of one strand) and to the coverglass surface
at the other (using the 5 end of the complementary stand). In one experiment, the adjacent
terminal ends of the two strands of the DNA hybrid were each labeled with
tetramethylrhodamine (TAMRA) molecules. Due to their physical proximity, these dyes self-
quenched (the quenching range for TAMRA is ~1 nm). The DNA hybrid was then mechanically
disrupted (“unzipped”) by applying a force ramp to the bead while the fluorescence signal was
monitored. The point of mechanical rupture detected with the optical trap was coincident with
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a stepwise increase in the fluorescence signal, as the two dyes separated, leaving behind a dye
attached by one DNA strand to the coverglass surface, as the partner dye was removed with
the DNA strand attached to the bead. Control experiments with fluorescent dyes attached to
either, but not both, DNA strands verified that the abrupt mechanical transition was specific
for the rupture of the DNA hybrid and not, for example, due to breakage of the linkages holding
the DNA to the bead or the coverglass surface.
B. Optical rotation and torque
Trapping transparent microspheres with a focused Gaussian laser beam in TEM
00
mode
produces a rotationally symmetric trap that does not exert torque. However, several methods
have been developed to induce the rotation of trapped objects.
20,52,162
Just as the change of
linear momentum due to refraction of light leads to the production of force, a change in angular
momentum leads to torque. Circularly polarized light carries spin angular momentum, of
course, and propagating optical beams can also be produced that carry significant amounts of
orbital angular momentum, e.g., Laguerre–Gaussian modes.
163
Each photon in such a mode
carries (σ+l)Ħ of angular momentum, where σ represents the spin angular momentum arising
from the polarization state of the light and l is the orbital angular momentum carried by the
light pattern. The angular momentum conveyed by the circular polarization alone, estimated
at ~10 pN nm/s per mW of 1064 nm light, can be significantly augmented through the use of
modes that carry even larger amounts of orbital angular momentum.
164
Transfer of both orbital
and spin angular momentum to trapped objects has been demonstrated for absorbing particles.
102,165
Transfer of spin angular momentum has been observed for birefringent particles of
crushed calcite,
166
and for more uniform microfabricated birefringent objects.
167,168
Friese
and coworkers derived the following expression for the torque on a birefringent particle:
166
τ
=
ɛ
2
ω
E
0
2
{
1 cos (
kd
(
n
0
n
e
))
}
sin 2
ϕ
sin (
kd
(
n
0
n
e
)) cos 2
ϕ
sin 2
θ
,
(16)
where ɛ is the permittivity, E
0
is the amplitude of the electric field, ω is the angular frequency
of the light, ϕ describes the ellipticity of the light (plane polarized, ϕ=0; circularly polarized,
ϕ=π/4), θ represents the angle between the fast axis of the quarter-wave plate producing the
elliptically polarized light and the optic axis of the birefringent particle, k is the vacuum wave
number (2π/λ), and n
0
and n
e
are the ordinary and extraordinary indices of refraction of the
birefringent material, respectively. Theoretically, all the spin angular momentum carried in a
circularly polarized laser beam can be transferred to a trapped object when it acts as a perfect
half-wave plate, i.e., ϕ=π/4 and kd(n
0
n
e
)=π. For the case of plane polarized light, there is a
restoring torque on the birefringent particle that aligns the fast axis of the particle with the
plane of polarization.
166
Rotation of the plane of polarization will induce rotation in a trapped
birefringent particle.
Whereas the transfer of optical angular momentum is a conceptually attractive means of
applying torque to optically trapped objects, several other techniques have been employed
towards the same end. In one scheme, a high order asymmetric mode, created by placing an
aperture in the far field of a laser beam, was used to trap red blood cells: these could be made
to spin by rotating the aperture.
169
A more sophisticated version of this same technique
involves interfering a Laguerre–Gaussian beam with a plane wave beam to produce a spiral
beam pattern.
170
By changing the relative phase of the two beams, the pattern can be made to
rotate, leading to rotation in an asymmetric trapped object.
48
Alternatively, the interference
of two Laguerre–Gaussian beams of opposite helicity (l and l) creates 2l beams surrounding
the optical axis, which can be rotated by adjusting the polarization of one of the interfering
beams.
46
Additionally, a variety of small chiral objects, such as microfabricated “optical
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propellers,” can be trapped and made to rotate in a symmetric Gaussian beam due to the optical
forces generated on asymmetrically oriented surfaces.
171–174
Rotation of trapped particles is most commonly monitored by video tracking, which is
effectively limited by frame rates to rotation speeds below ~15 Hz, and to visibly asymmetric
particles (i.e., microscopic objects of sufficient size and contrast to appear asymmetric in the
imaging modality used). Rotation rates up to 1 kHz have been measured by back focal plane
detection of trapped 0.83 μm beads sparsely labeled with 0.22 μm beads to make these optically
anisotropic.
175
Backscattered light from trapped, asymmetric particles has also been used to
measure rotation rates in excess of 300 Hz.
102,166
C. Holographic optical traps
Holograms and other types of diffractive optics have been used extensively for generating
complex, high-order optical trapping beams,
20,52,162,165,176
such as the Laguerre–Gaussian
modes discussed above. Diffractive optical devices may also be used to synthesize multiple
optical traps with arbitrary intensity profiles.
20,91,177–179
A diffractive element placed in a
plane optically conjugate to the back aperture of the microscope objective produces an intensity
distribution in the specimen plane that is the Fourier transform of the pattern imposed by the
element,
177
and several computational methods have been developed to derive the holographic
pattern required for any given intensity distribution in the specimen plane.
91,92,180
Generally
speaking, diffractive elements modulate both the amplitude and the phase of the incident light.
Optical throughput can be maximized by employing diffractive optics that primarily modify
the phase but not the amplitude of the incident light, termed kinoforms.
91
Computer-generated
phase masks can also be etched onto a glass substrate using standard photolithographic
techniques, producing arbitrary, but fixed, optical traps.
Reicherter and co-workers extended the usefulness of holographic optical trapping techniques
by generating three independently movable donut-mode trapping beams with an addressable
liquid crystal spatial light modulator (SLM).
181
Improvements in SLM technology and real-
time hologram calculation algorithms have been implemented, allowing the creation of an array
of up to 400 optical traps, in addition to the creation and three-dimensional manipulation of
multiple, high order, trapping beams.
92,182,183
Multiple optical traps can also be generated
by time sharing, using rapid-scanning techniques based on AODs or galvo mirrors,
49,50
but
these are typically formed in just one or two axial planes,
184
and they are limited in number.
Dynamic holographic optical tweezers can produce still more varied patterns, limited only by
the optical characteristics of the SLM and the computational time required to generate the
hologram. Currently, the practical update rate of a typical SLM is around 5 Hz, which limits
how quickly objects can be translated.
92
Furthermore, the number and size of the pixels in the
SLM restrict the complexity and the range of motion of generated optical traps,
92
while the
pixelation and discrete phase steps of the SLM result in diffractive losses. Faster refresh rates
(>30 Hz) in a holographic optical trap have recently been reported with a SLM based on
ferroelectric, as opposed to nematic, materials.
185
Further improvements in SLM technology
should expand the possible applications of dynamic, holographic optical traps.
VIII. PROSPECTS
The nearly 2 decades that have passed since Ashkin and co-workers invented the single beam,
gradient force optical trap have borne witness to a proliferation of innovations and applications.
The full potential of most of the more recent optical developments has yet to be realized. On
the biological front, the marriage of optical trapping with single-molecule fluorescence
methods
138
represents an exciting frontier with enormous potential. Thanks to steady
improvements in optical trap stability and photodetector sensitivity, the practical limit for
position measurements is now comparable to the distance subtended by a single base pair along
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DNA, 3.4 Å. Improved spatiotemporal resolution is now permitting direct observations of
molecular-scale motions in individual nucleic acid enzymes, such as polymerases, heli-cases,
and nucleases.
86,87,186
The application of optical torque offers the ability to study rotary
motors, such as F
1
F
0
ATPase,
187
using rotational analogs of many of the same techniques
already applied to the study of linear motors, i.e., torque clamps and rotation clamps.
50
Moving
up in scale, the ability to generate and manipulate a myriad of optical traps dynamically using
holographic tweezers
20,92
opens up many potential applications, including cell sorting and
other types of high-throughput manipulation. More generally, as the field matures, optical
trapping instruments should no longer be confined to labs that build their own custom
apparatus, a change that should be driven by the increasing availability of sophisticated,
versatile commercial systems. The physics of optical trapping will continue to be explored in
its own right, and optical traps will be increasingly employed to study physical, as well as
biological, phenomena. In one ground-breaking example from the field of nonequilibrium
statistical mechanics, Jarzynski’s equality
188
which relates the value of the equilibrium free
energy for a transition in a system to a nonequilibrium measure of the work performed—was
put to experimental test by mechanically unfolding RNA structures using optical forces.
189
Optical trapping techniques are increasingly being used in condensed matter physics to study
the behavior (including anomalous diffusive properties and excluded volume effects) of
colloids and suspensions,
21
and dynamic optical tweezers are particularly well suited for the
creation and evolution of large arrays of colloids in well-defined potentials.
20
As optical
trapping techniques continue to improve and become better established, these should pave the
way for some great new science in the 21st century, and we will be further indebted to the
genius of Ashkin.
3
Acknowledgements
The authors thank members of the Block Lab for advice, suggestions, and helpful discussions. In particular, Elio
Abbondanzieri helped with instrument construction and all aspects of data collection, Joshua Shaevitz supplied Fig.
6, and Megan Valentine and Michael Woodside supplied valuable comments on the manuscript. We also thank Henrik
Flyvbjerg, Kirstine Berg-Sørensen, and the members of their labs for sharing results in advance of publication, for
critical reading of the manuscript, and for help in preparing Fig. 10. Finally, we thank Megan Valentine, Grace Liou,
and Richard Neuman for critical reading of the manuscript.
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FIG. 1.
Ray optics description of the gradient force. (A) A transparent bead is illuminated by a parallel
beam of light with an intensity gradient increasing from left to right. Two representative rays
of light of different intensities (represented by black lines of different thickness) from the beam
are shown. The refraction of the rays by the bead changes the momentum of the photons, equal
to the change in the direction of the input and output rays. Conservation of momentum dictates
that the momentum of the bead changes by an equal but opposite amount, which results in the
forces depicted by gray arrows. The net force on the bead is to the right, in the direction of the
intensity gradient, and slightly down. (B) To form a stable trap, the light must be focused,
producing a three-dimensional intensity gradient. In this case, the bead is illuminated by a
focused beam of light with a radial intensity gradient. Two representative rays are again
refracted by the bead but the change in momentum in this instance leads to a net force towards
the focus. Gray arrows represent the forces. The lateral forces balance each other out and the
axial force is balanced by the scattering force (not shown), which decreases away from the
focus. If the bead moves in the focused beam, the imbalance of optical forces will draw it back
to the equilibrium position.
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FIG. 2.
Layout of a generic optical trap. The laser output beam usually requires expansion to overfill
the back aperture of the objective. For a Gaussian beam, the beam waist is chosen to roughly
match the objective back aperture. A simple Keplerian telescope is sufficient to expand the
beam (lenses L1 and L2). A second telescope, typically in a 1:1 configuration, is used for
manually steering the position of the optical trap in the specimen plane. If the telescope is built
such that the second lens, L4, images the first lens, L3, onto the back aperture of the objective,
then movement of L3 moves the optical trap in the specimen plane with minimal perturbation
of the beam. Because lens L3 is optically conjugate (conjugate planes are indicated by a cross-
hatched fill) to the back aperture of the objective, motion of L3 rotates the beam at the aperture,
which results in translation in the specimen plane with minimal beam clipping. If lens L3 is
not conjugate to the back aperture, then translating it leads to a combination of rotation and
translation at the aperture, thereby clipping the beam. Additionally, changing the spacing
between L3 and L4 changes the divergence of the light that enters the objective, and the axial
location of the laser focus. Thus, L3 provides manual three-dimensional control over the trap
position. The laser light is coupled into the objective by means of a dichroic mirror (DM1),
which reflects the laser wavelength, while transmitting the illumination wavelength. The laser
beam is brought to a focus by the objective, forming the optical trap. For back focal plane
position detection, the position detector is placed in a conjugate plane of the condenser back
aperture (condenser iris plane). Forward scattered light is collected by the condenser and
coupled onto the position detector by a second dichroic mirror (DM2). Trapped objects are
imaged with the objective onto a camera. Dynamic control over the trap position is achieved
by placing beam-steering optics in a conjugate plane to the objective back aperture, analogous
to the placement of the trap steering lens. For the case of beam-steering optics, the point about
which the beam is rotated should be imaged onto the back aperture of the objective.
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FIG. 3.
The wavelength dependence of photodamage in E. coli compared to Chinese hamster ovary
(CHO) cells. (Solid circles and solid line, left axis, half lethal dose time for E. coli cells
(LD
50
); open circles and dashed line, right axis, cloning efficiency in CHO cells determined
by Liang et al. (Ref. 96) (used with permission). Lines represent cubic spline fits to the data).
The cloning efficiency in CHO cells was determined after 5 min of trapping at 88 mW in the
specimen plane (error bars unavailable), selected to closely match to our experimental
conditions (100 mW in the specimen plane, errors shown as ± standard error in the mean).
Optical damage is minimized at 830 and 970 nm for both E. coli and CHO cells, whereas it is
most severe in the region between 870 and 930 nm (reprinted from Ref. 95).
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FIG. 4.
Microscope objective transmission curves. Transmission measurements were made by means
of the dual-objective method. Part numbers are cross-referenced in Table I. The uncertainty
associated with a measurement at any wavelength is ~5% (reprinted from Ref. 95).
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FIG. 5.
Comparison of position detector frequency response at 1064 nm. Normalized frequency
dependent response for a silicon quadrant photodiode (QPD) (QP50–6SD, Pacific Silicon
Sensor) (open circles), and a position sensitive detector (PSD), (DL100–7PCBA, Pacific
Silicon Sensor) (solid circles). 1064 nm laser light was modulated with an acousto-optic
modulator and the detector output was recorded with a digital sampling scope. The response
of the QPD was fit with the function: γ
2
+(1−γ
2
)[1+(f/f
0
)
2
]
1
, which describes the effects of
diffusion of electron-hole pairs created outside the depletion layer (Ref. 134), where γ is the
fraction of light absorbed in the diode depletion layer and f
0
is the characteristic frequency
associated with light absorbed beyond the depletion layer. The fit returned an f
0
value of 11.1
kHz and a γ parameter of 0.44, which give an effective f
3dB
of 4.1 kHz, similar to values found
in Ref. 134 for silicon detectors. The QPD response was not well fit by a single pole filter
response curve. The PSD response, in contrast, was fit by a single pole filter function, returning
a rolloff frequency of 196 kHz. Extended frequency response at 1064 nm has also been reported
for InGaAs and fully depleted silicon photodiodes (Ref. 61).
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FIG. 6.
Lateral two-dimensional detector calibration (adapted from Ref. 59). Contour plot of the x
(solid lines) and y (dashed lines) detector response as a function of position for a 0.6 μm
polystyrene bead raster scanned through the detector laser focus by deflecting the trapping
laser with acousto-optic deflectors. The bead is moved in 20 nm steps with a dwell time of 50
ms per point while the position signals are recorded at 50 kHz and averaged over the dwell
time at each point. The x contour lines are spaced at 2 V intervals, from 8 V (leftmost contour)
to 8 V (rightmost contour). The y contour lines are spaced at 2 V intervals, from 8 V (bottom
contour) to 8 V (top contour). The detector response surfaces in both the x and y dimensions
are fit to fifth order two-dimensional polynomials over the shaded region, with less than 2 nm
residual root mean square (rms) error. Measurements are confined to the shaded region, where
the detector response is single valued.
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FIG. 7.
Axial two-dimensional detector calibration. Contour plot of the lateral (solid lines) and axial
(dashed lines) detector response as a function of x (lateral displacement) and z (axial
displacement) of a stuck 0.5 μm polystyrene bead moving through the laser focus. A stuck
bead was raster scanned in 20 nm steps in x and z. The detector signals were recorded at 4 kHz
and averaged over 100 ms at each point. The lateral contour lines are spaced at 1 V intervals,
from 9 V (leftmost contour) to 7 V (rightmost contour). The axial contour lines are spaced at
0.02 intervals (normalized units). Measurements are confined to the region of the calibration
shaded in gray, over which the surfaces of x and z positions as a function of lateral and axial
detector signals were fit to seventh order two-dimensional polynomial functions with less than
5 nm residual rms error.
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FIG. 8.
Axial position signals for a free (heavy dashed line) and stuck (light dashed line) bead as the
stage was scanned in the axial direction. All stage motion is relative to the scattering peak,
which is indicated on the right of the figure. The positions of the surface (measured) and the
focus [calculated from Eq. (5)] are indicated by vertical lines. The axial detection fit [Eq. (5)]
to the stuck bead trace is shown in the region around the focus as a heavy solid line.
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FIG. 9.
Normalized drag coefficient (β
0
/β, where β
0
is the Stokes drag on the sphere: 6πηa) as a
function of distance from the scattering peak. The normalized inverse drag coefficient (solid
circles) was determined through rolloff measurements and from the displacement of a trapped
bead as the stage was oscillated. The normalized inverse drag coefficient was fit to Faxen’s
law [Eq. (6)] with a height offset ɛ and scaling parameter δ, which is the fractional focal shift,
as the only free parameters: β
0/
β =1(9/16)×[aδ
1
(zɛ)
1
] + 1/8[aδ
1
(zɛ)
1
]
3
(45/256)
[aδ
1
(zɛ)
1
]
4
(1/16)[aδ
1
(zɛ)
1
]
5
, where a is the bead radius, z is the motion of the stage
relative to the scattering peak, β
0
is the Stoke’s drag on the bead, (6πηa), and β is the measured
drag coefficient. The fit returned a fractional focal shift δ of 0.82±0.02 and an offset ɛ of 161
nm. The position of the surface relative to the scattering peak is obtained by setting the position
of the bead center, δ(zɛ) equal to the bead radius a, which returns a stage position of 466 nm
above the scattering peak, as indicated in Fig. 8.
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FIG. 10.
Power spectrum of a trapped bead. Power spectrum of a 0.5 μm polystyrene bead trapped 1.2
μm above the surface of the trapping chamber recorded with a PSD (gray dots). The raw power
spectrum was averaged over 256 Hz windows on the frequency axis (black circles) and fit
(black line) to a Lorentzian [Eq. (7)] corrected for the effects of the antialiasing filter, frequency
dependent hydrodynamic effects, and finite sampling frequency, as described by Berg–
Sørensen and Flyvbjerg (Ref. 148). The rolloff frequency is 2.43 kHz, corresponding to a
stiffness of 0.08 pN/nm. For comparison the raw power spectrum was fit to an uncorrected
Lorentzian (dashed line), which returns a rolloff frequency of 2.17 kHz. Whereas the
discrepancies are on the order 10% for a relatively weak trap, they generally become more
important at higher rolloff frequencies.
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FIG. 11.
Axial dependence of lateral stiffness. The experimental geometry for these measurements is
depicted in the inset. A polystyrene bead is tethered to the surface of the cover glass through
a long DNA tether. The stage was moved in the negative z direction (axial), which pulls the
bead towards the laser focus, and the lateral stiffness was determined by measuring the lateral
variance of the bead. The data (solid circles) are fit with the expression for a simple dipole [Eq.
(14)], with the power in the specimen plane, the beam waist, and an axial offset as free
parameters.
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FIG. 12.
Cartoon of the experimental geometry (not to scale) for single-molecule transcription
experiment. Transcribing RNA polymerase with nascent RNA (gray strand) is attached to a
polystyrene bead. The upstream end of the duplex DNA (black strands) is attached to the
surface of a flowchamber mounted on a piezoelectric stage. The bead is held in the optical trap
at a predetermined position from the trap center, which results in a restoring force exerted on
the bead. During transcription, the position of the bead in the optical trap and hence the applied
force is maintained by moving the stage both horizontally and vertically to compensate for
motion of the polymerase molecule along the DNA (adapted from Ref. 87).
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FIG. 13.
The optical trapping interferometer. Light from a Nd:YLF laser passes through an acoustic
optical modulator (AOM), used to adjust the intensity, and is then coupled into a single-mode
polarization-maintaining optical fiber. Output from the fiber passes through a polarizer to
ensure a single polarization, through a 1:1 telescope and into the microscope where it passes
through the Wollaston prism and is focused in the specimen plane. The scattered and
unscattered light is collected by the condenser, is recombined in the second Wollaston prism,
then the two polarizations are split in a polarizing beamsplitter and detected by photodiodes A
and B. The bleedthrough on a turning mirror is measured by a photodiode (N) to record the
instantaneous intensity of the laser. The signals from the detector photodiodes and the
normalization diode are digitized and saved to disk. The normalized difference between the
two detectors (A and B) gives the lateral, x displacement, while the sum signal (A+B)
normalized by the total intensity (N) gives the axial, z displacement.
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FIG. 14.
Two-dimensional, stage based force clamp. Position record of a single RNA polymerase
molecule transcribing a 3.5 kbp (1183 nm) DNA template under 18 pN of load. The x and z
position signals were low pass filtered at 1 kHz, digitized at 2 kHz, and boxcar averaged over
40 points to generate the 50 Hz feedback signals that controlled the motion of the piezoelectric
stage. Motion of the stage was corrected for the elastic compliance of the DNA (Ref. 39) to
recover the time-dependent contour length, which reflects the position of the RNA polymerase
on the template. Periods of roughly constant velocity are interrupted by pauses on multiple
timescales. Distinct pauses can be seen in the trace, while shorter pauses (~1 s) can be discerned
in the expanded region of the trace (inset: arrows).
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TABLE I
Transmission of microscope objectives, cross-referenced with Fig. 2.
Transmission (±5%)
Part No. Manufacturer Magnification/Tube length
(mm)/Numerical aperture
Type designation 830 (nm) 850 (nm) 990 (nm) 1064 (nm)
461832 Zeiss 63/160/1.2 Water Plan NeoFluar 66 65 64 64
506038 Leica 100//1.4-0.7 Oil Plan Apo 58 56 54 53
85020 Nikon 60/160/1.4 Oil Plan Apo 54 51 17 40
93108 Nikon 60//1.4 Oil Plan Apo CFI 59 54 13 39
93110 Nikon 100//1.4 Oil Plan Apo CFI 50 47 35 32
93110IR Nikon 100//1.4 Oil Plan Apo IR CFI 61 60 59 59
93144 Nikon 100//1.3 Oil Plan Fluor CFI 67 68 61
Rev Sci Instrum. Author manuscript; available in PMC 2006 July 27.
... It works on the principle that light carries momentum and can exert force on particles. 18 Highlighting the invention of optical tweezers and their application to biological systems, its inventor, Arthur Ashkin, won the Nobel prize for Physics in 2018. For a dielectric particle kept in the focused beam of a laser source, the force exerted due to the change in momentum of laser light due to refraction pulls the particle toward the focusing spot, whereas the force due to reflection pushes the bead along the propagation direction of the beam, trapping the particle stably at a point near the focusing spot 19 (Figure 2A). ...
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RNA polymerase (RNAP) moves along DNA while carrying out transcription, acting as a molecular motor. Transcriptional velocities for single molecules of Escherichia coli RNAP were measured as progressively larger forces were applied by a feedback-controlled optical trap. The shapes of RNAP force-velocity curves are distinct from those of the motor enzymes myosin or kinesin, and indicate that biochemical steps limiting transcription rates at low loads do not generate movement. Modeling the data suggests that high loads may halt RNAP by promoting a structural change which moves all or part of the enzyme backwards through a comparatively large distance, corresponding to 5 to 10 base pairs. This contrasts with previous models that assumed force acts directly upon a single-base translocation step.
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Single fluorescent dye molecules were visualized at video-rate in aqueous solution by using total internal reflection fluorescence microscopy. This approach enabled us to directly image single myosin molecules and detect individual ATP turnover reactions. This approach also can be used to directly image single fluorescently-labeled kinesin molecules moving on an axoneme. The methods can be extended to simultaneous measurements of the ATPase reaction and force/movement by single motor protein molecules. This should provide a clear answer to the fundamental problem of how mechanical reaction is coupled to ATPase reaction.
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We describe a set of image processing algorithms for extracting quantitative data from digitized video microscope images of colloidal suspensions. In a typical application, these direct imaging techniques can locate submicrometer spheres to within 10 nn in the focal plane and 150 nn in depth. Combining information from a sequence of video images into single-particle trajectories makes possible measurements of quantities of fundamental and practical interest such as diffusion coefficients and pair-wise interaction potentials, The measurements we describe in detail combine the outstanding resolution of digital imaging with video-synchronized optical trapping to obtain highly accurate and reproducible results very rapidly. (C) 1996 Academic Press,Inc.
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Refraction of light in a specimen volume may cause aberrations that influence the imaging properties in confocal microscopy. In this paper the influence on three-dimensional resolution and geometry is experimentally investigated for a uniform specimen volume. It is found that the depth resolution is more severely affected than the lateral resolution. This is unfortunate, because even under ideal conditions the depth resolution is lower than the lateral resolution. Lateral image geometry is little affected by the specimen refractive index, whereas the depth scale can be considerably elongated or compressed. The influence of a finite detector integration time is also considered. This can give a noticeable, but not particularly severe effect on the image resolution in the line-scan direction. Because the integration time can be accurately controlled, a shorter integration time can be used when maximum resolution is essential, albeit at the price of a higher noise level. In scanning fluorescence microscopy a non-uniform scan speed may give large variations in bleaching over the specimen surface. Experiments illustrate how serious such non-uniform bleaching effects can be when a specimen area is repeatedly scanned, for example when recording optical serial sections.
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Optically induced mechanical angular alignment of trapped birefringent micrometer-sized objects resulting from the transfer of angular momentum produced by birefringence using linearly polarized light has been experimentally demonstrated. Fluorinated polyimide (PMDA/TFDB) micro-objects having a large birefringence of Δn=nslow-nfast=0.13 (refractive indices nslow=1.62, nfast=1.49), which were fabricated by micromachining (reactive ion etching) and suspended in water (n=1.33), were trapped and manipulated by radiation pressure from a single focused Gaussian beam (wavelength λ=1.064 μm, power>130 μW). The effect of the linearly polarized light on such micro-objects showed that they were angularly aligned about the laser beam axis and their angular position could be smoothly controlled by rotating the vibration plane of the electric field. We have shown that the retardation of the birefringent micro-object determines which axis (fast or slow) of the micro-object coincides with the vibration plane of the electric field of the incident light.