We report near-optimal tracking of freely diffusing fluorescent particles in a quasi-two-dimensional geometry via photon counting and real-time feedback. We present a quantitative statistical model of our feedback network and find excellent agreement with the experiment. We monitor the motion of a single fluorescent particle with a sensitivity of 15 nm/sqrt Hz while collecting fewer than 5000 fluorescence photons/s. Fluorescent microspheres (diffusion coefficient 1.3 μm^2/s) are tracked with a root-mean-square tracking error of 170 nm, within a factor of 2 of the theoretical limit set by photon counting shot noise

Berglund, Andrew J.

Mabuchi, Hideo

McHale, Kevin

English

Caltech Authors - Main

January 15, 2007 / Vol. 32, No. 2 / OPTICS LETTERS 145Feedback localization of freely diffusing
fluorescent particles near the optical
shot-noise limit
Andrew J. Berglund, Kevin McHale, and Hideo Mabuchi
Physical Measurement and Control 266-33, California Institute of Technology, Pasadena, California 91125, USA
Received August 7, 2006; revised October 10, 2006; accepted October 12, 2006;
posted October 18, 2006 (Doc. ID 73852); published December 23, 2006
We report near-optimal tracking of freely diffusing fluorescent particles in a quasi-two-dimensional geom-
etry via photon counting and real-time feedback. We present a quantitative statistical model of our feedback
network and find excellent agreement with the experiment. We monitor the motion of a single fluorescent
particle with a sensitivity of 15 nm/Hz while collecting fewer than 5000 fluorescence photons/s. Fluores-
cent microspheres (diffusion coefficient 1.3 m2/s) are tracked with a root-mean-square tracking error of
170 nm, within a factor of 2 of the theoretical limit set by photon counting shot noise. © 2006 Optical Society
of America
OCIS codes: 180.2520, 180.5810, 300.2530.A series of recent experimental and theoretical inves-
tigations has begun a new era in closed-loop control
of the Brownian motion of individual fluorescent
particles.1–8 All of these methods use a particle’s fluo-
rescence to sense its deviation from a target position
and use a feedback network to compensate for the
particle’s random motion, either by active tracking or
trapping. However, a diffusing particle cannot be
tracked arbitrarily well at a finite fluorescence count
rate, since an overly aggressive feedback controller
will feed photon counting noise back into the system.8
In this Letter we demonstrate a feedback control sys-
tem capable of tracking the planar Brownian motion
of a freely diffusing particle near this optical shot-
noise limit, and we introduce a quantitative model
for interpreting experimental statistics.
We use the home-built optical microscope and
tracking apparatus described in a previous
publication,5 with slight modifications. In short,
(nominally) 60 and 210 nm diameter fluorescent mi-
crospheres diffuse freely in a thin 1 m aqueous
layer between glass microscope coverslips mounted
on a piezoelectric translation stage. We detect the x
and y position of a particle by translating our excita-
tion laser in a circular pattern about the optic z axis
of the microscope. The laser rotates at 0=2
8 kHz, a frequency that is large compared with the
characteristic diffusion times across our laser focus
(45 and 170 ms, respectively, for the 60 and 210 nm
diameter microspheres). Our beam waist is w
=1 m, and the rotating laser focus is offset from the
optic axis by a distance r=0.6 m. When a particle
diffuses away from the optic axis, its fluorescence is
modulated as the laser sweeps past it at frequency
0, and we detect the amplitude and phase of this
modulation by using a lock-in amplifier. The result-
ing cosine and sine quadrature signals provide an er-
ror signal proportional to the x and y components of
the particle’s position. These two error signals are fil-
tered by two analog controllers and sent to a high-
voltage amplifier to drive the piezoelectric stage con-
taining the entire bulk sample. In this way, we0146-9592/07/020145-3/$15.00 ©translate the bulk sample to compensate for the par-
ticle’s Brownian motion; the particle is then locked to
the optic axis, and the time-dependent position of the
sample stage becomes a measure of the particle’s dif-
fusion trajectory.
We simultaneously record the position of the
sample stage and the arrival times of fluorescence
photons during a typical tracking trajectory. A single
shot of the experiment is shown in Fig. 1. For the
measurements presented here, we used a sample
containing roughly equal concentrations of 60 and
210 nm fluorescently labeled microspheres. Because
the 210 nm microspheres contain more dye mol-
ecules, they appear much brighter than the 60 nm
microspheres when illuminated by the same laser in-
tensity. To track both types of particle in a compa-
rable way, we use an electronic servo to adjust the ex-
citation laser power (by modulating the rf power to
an acousto-optic modulator) to lock the total photon
count rate to 5600 s−1. The photon count rate due to
background light was approximately 1000 s−1, inde-
Fig. 1. (Color online) Data from a typical run of the experi-
ment. The upper plot shows the rate of photon detections.
In the lower plot, the upper curves are the x and y positions
of the sample stage during the same trajectory. The lower
curve on the lower plot shows the excitation laser power (in
arbitrary units) necessary to lock the number of detected
photons to 5600 s−1. The plots show three regions, in which
no particle is present (until 17 s), a 60 nm diameter par-
ticle is tracked (until 74 s), and a 210 nm diameter particle
is tracked (until 90 s).
2006 Optical Society of America
146 OPTICS LETTERS / Vol. 32, No. 2 / January 15, 2007pendent of excitation intensity, leaving the rate of
fluorescence photon arrivals at 4600 s−1 during track-
ing.
In this Letter we are interested in recovering sta-
tistical quantities such as the tracking error and the
particle’s diffusion coefficient from the position of the
sample stage during a tracking trajectory. Let the
time-dependent x coordinate of the sample stage po-
sition be Xt, and let its mean-square deviation over
a time interval t be denoted
MSDt = Xt + t − Xt2, 1
where   represents an ensemble average. When t
is much larger than the inverse feedback bandwidth,
we expect MSDt	2Dt, the mean-square devia-
tion of a freely diffusing particle with diffusion coef-
ficient D; however, because of the inevitable roll-off of
the feedback response at short times, the sample
stage will not move appreciably for small t. Thus we
expect the mean-square deviation to probe the feed-
back bandwidth parameters at small t and the par-
ticle’s diffusion coefficient at large t. In Fig. 2 we
show the measured value of MSDt /2t, for two
different trajectories, together with fits to Eq. (5)
based on the model described below.
It was shown in Ref. 8 that for sufficiently good
tracking, our experiment may be modeled as a linear
control system. We take as our model the linear feed-
back network shown in Fig. 3, where the tracking
controller is represented by the (Laplace space)
transfer function Cs and the piezoelectric stage is
represented by Ps. We consider the control system
Fig. 2. (Color online) MSDt /2t as defined by Eq. (1)
for two typical [210 nm diameter (upper) and 60 nm diam-
eter (lower) microspheres] tracking trajectories together
with fits to Eq. (5). At long times, the curves approach the
particle diffusion coefficient, while the short-time behavior
depends on the tracking feedback performance.
Fig. 3. Block diagram used in the text to represent the
particle tracking control system.in one dimension only, since the x and y positions of aBrownian particle are statistically independent. The
particle is driven by Brownian motion, and we repre-
sent its velocity, Ut, by
Ut = 2D
dW
dt
, 2
where D is the diffusion coefficient and dW is a sto-
chastic Wiener increment.9,10 The time integral of
Ut is the position of the particle at time t. The po-
sition of the sample stage is denoted Xt, and the er-
ror signal Et is the difference between the stage po-
sition Xt and the particle’s position. Finally,
Gaussian white noise Nt=ndW /dt with a noise den-
sity n is added to the error signal. The noise Nt
must be included in our model to account for photon
counting fluctuations, which give rise to a nonzero
noise density n for all finite fluorescence intensities.8
Of course, any excess technical noise beyond these
fundamental limits may also be captured by this pa-
rameter. The resulting system is a linear, two-input
Ut ,Nt, two-output Xt ,Et system, which we
investigate below.
For our experiment, the controller is a simple inte-
grator with time constant c, and the piezoelectric
stage may be sufficiently represented by a first-order
low-pass filter with a roll-off frequency p /2: Cs
=c /s, Ps=1/ 1+s /p. Because the system is lin-
ear, the total output of the system is just the sum of
the output resulting from each input considered in
isolation. As an example of the response to a single
input, consider the sample stage position Xt when it
is driven by the particle’s diffusion Ut, with Nt
=0. For our second-order model, we require the veloc-
ity of the sample stage X˙t=dXt /dt. Using stan-
dard results, we find the following state-space real-
ization of the system11:
dt =Atdt +Utdt, X˙t =Ct, 3
where t is a two-component internal state vector
and
A = 
− p − p
c 0
, C = 0 p. 4
The components of t are simply scaled multiples of
Xt and X˙t. Upon insertion of Eq. (2), Eq. (3) be-
comes a linear stochastic differential equation, the
multivariable Ornstein–Uhlenbeck process.9 The re-
sulting solution for X˙t is a Gaussian process whose
statistics are given by a Fokker–Planck equation,
which is exactly solvable.9,10 In this way we may cal-
culate the response of any output signal due to any
input signal, and by superposition we may construct
the full output signal due to all inputs. Of particular
interest is the mean-square deviation of the sample
stage position. We find
January 15, 2007 / Vol. 32, No. 2 / OPTICS LETTERS 147MSDt = 2Dt −
2Dc
p
CA−2
I − n2A22D I − eAt

1/c 00 1/pCT. 5
For a stable feedback system A0, we find
MSDt	2Dt for large values of t (compared
with the feedback time scale). The measurement
noise contributes a dimensionless correction matrix
n2 /2DA2, which may be used to assess the influence
of measurement noise on the system. For the 22
system considered here, the matrix exponentials in
Eq. (5) may be calculated explicitly; however, the re-
sulting expression is rather complicated and provides
little additional insight.
We may extract the particle’s localization from the
measured motion of the sample stage. We define this
localization (along one axis only) to be L=Et2,
which is the standard deviation of the particle’s dis-
tance from the sample stage position, i.e., the track-
ing error. For our system,
L =D
 1
c
+
1
p
 + n2c2 . 6
We establish the parameters n, c, p, and D by fit-
ting the curves in Fig. 2 to Eq. (5). We then calculate
each particle’s localization by using Eq. (6). The re-
sulting values of L are plotted versus the measured
diffusion coefficient D in Fig. 4. To confirm the valid-
ity of this fitting technique for estimating L, we gen-
erated Monte Carlo simulations of the second-order
control system driven by diffusion and measurement
noise, using typical values of the parameters n, c, p,
and D. We then estimated L and D for the resulting
data. These results are shown in gray in Fig. 4.
The dashed curves in Fig. 4 show the shot-noise lo-
calization limit derived in Ref. 8 for our experimen-
Fig. 4. (Color online) Measured localization L versus D for
62 individual tracking trajectories (dark dots). The lighter
dots are the results of simulations described in the text.
The dashed curves are the localization limit based on opti-
cal shot noise. Typical fit parameters for the 210 nm beads
(red online, dark dots clustered in the lower left corner of
the plot) are D=1.3 m2/s, n=15 nm/Hz, c=111 Hz, p
=343 Hz. The fitted value of c is commensurate with the
bandwidth of the analog integrator used for tracking con-
trol, while p serves primarily to represent the phase delay
of our piezoelectric stage at high frequencies 100 Hz.tally determined values of the beam waist w, rotation
radius r, and fluorescence count rate for a particle lo-
cated at the maximum laser intensity (approximately
15.1 kHz and 9.8 kHz for the 60 and 210 nm par-
ticles, respectively). For the 210 nm diameter beads,
the mean inferred localization is 170 nm, and the
shot-noise limited value is 117 nm. The noise density
n for these beads is found to be 15 nm/Hz, while the
shot-noise limited value is 9.4 nm/Hz. The 60 nm
beads are more poorly localized, owing to their faster
diffusion, but the large spread in the inferred values
of L for these particles arises from noise in the fitting
procedure, not from a true spread in the underlying
tracking error. This claim is supported by the large
spread in the localization values inferred from the
Monte Carlo simulations, whose underlying localiza-
tions were tightly clustered near 300 nm. (Amore de-
tailed data analysis including fluorescence fluctua-
tions shows a mean localization of 340 nm for the
60 nm particles, but these results are beyond the
scope of this analysis.) Furthermore, the observed
values of L for the 60 nm particles stray even beyond
the simulations; however, the linear control-system
analysis assumes the particles to be well localized
relative to r and w,8 and this assumption is not par-
ticularly valid for the 60 nm beads. While it is diffi-
cult to analyze the statistics of this nonlinear track-
ing, it is a feature of our experiment that the
feedback network is robust enough to track in this re-
gime.
K. McHale acknowledges support from an NDSEG
fellowship. This work was supported by NSF grant
CCF-0323542 and by the Institute for Collaborative
Biotechnologies through grant DAAD19-03-D-0004
from the Army Research Office. Any opinions, find-
ings, and conclusions or recommendations expressed
in this material are those of the authors, and do not
necessarily reflect the views of either funding agency.
A. Berglund’s e-mail address is berglund@
caltech.edu.
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Feedback localization of freely diffusing fluorescent particles near the optical shot-noise limit

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Feedback localization of freely diffusing fluorescent particles near the optical shot-noise limit

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