Extracto
ICFO
PhD Dissertation
Dynamics of optically levitated
nanoparticles in high vacuum
Author:
Jan Gieseler
February 3
rd
, 2014
(corrected June 5
th
, 2015)
Dedicated to Marie Emilia Gieseler
ii
Contents
Introduction
vii
0.1
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
0.2
Overview and state of the art . . . . . . . . . . . . . . . . .
viii
0.2.1
Optomechanics and optical trapping . . . . . . . . .
ix
0.2.2
Complex systems . . . . . . . . . . . . . . . . . . . .
x
0.2.3
Statistical physics
. . . . . . . . . . . . . . . . . . .
xi
1
Experimental Setup
1
1.1
Optical setup . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1.1
Overview of the optical setup . . . . . . . . . . . . .
2
1.1.2
Detector signal . . . . . . . . . . . . . . . . . . . . .
6
1.1.3
Homodyne measurement . . . . . . . . . . . . . . . .
9
1.1.4
Heterodyne measurement . . . . . . . . . . . . . . .
12
1.2
Feedback Electronics . . . . . . . . . . . . . . . . . . . . . .
14
1.2.1
Bandpass filter . . . . . . . . . . . . . . . . . . . . .
15
1.2.2
Variable gain amplifier . . . . . . . . . . . . . . . . .
16
1.2.3
Phase shifter . . . . . . . . . . . . . . . . . . . . . .
17
1.2.4
Frequency doubler . . . . . . . . . . . . . . . . . . .
17
1.2.5
Adder . . . . . . . . . . . . . . . . . . . . . . . . . .
18
1.3
Particle Loading
. . . . . . . . . . . . . . . . . . . . . . . .
18
1.3.1
Pulsed optical forces . . . . . . . . . . . . . . . . . .
19
1.3.2
Piezo approach . . . . . . . . . . . . . . . . . . . . .
20
1.3.3
Nebuliser . . . . . . . . . . . . . . . . . . . . . . . .
21
iii
Contents
1.4
Vacuum system . . . . . . . . . . . . . . . . . . . . . . . . .
22
1.4.1
Towards ultra-high vacuum . . . . . . . . . . . . . .
23
1.5
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
2
Theory of Optical Tweezers
27
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
2.2
Optical fields of a tightly focused beam . . . . . . . . . . . .
28
2.3
Forces in the Gaussian approximation
. . . . . . . . . . . .
32
2.3.1
Derivation of optical forces
. . . . . . . . . . . . . .
33
2.3.2
Discussion . . . . . . . . . . . . . . . . . . . . . . . .
36
2.4
Optical potential . . . . . . . . . . . . . . . . . . . . . . . .
41
2.5
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
3
Parametric Feedback Cooling
43
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
3.2
Description of the experiment . . . . . . . . . . . . . . . . .
45
3.2.1
Particle dynamics . . . . . . . . . . . . . . . . . . . .
45
3.2.2
Parametric feedback . . . . . . . . . . . . . . . . . .
46
3.3
Theory of parametric feedback cooling . . . . . . . . . . . .
48
3.3.1
Equations of motion . . . . . . . . . . . . . . . . . .
49
3.3.2
Stochastic differential equation for the energy . . . .
49
3.3.3
Energy distribution . . . . . . . . . . . . . . . . . . .
54
3.3.4
Effective temperature
. . . . . . . . . . . . . . . . .
55
3.4
Experimental results . . . . . . . . . . . . . . . . . . . . . .
56
3.4.1
Power dependence of trap stiffness . . . . . . . . . .
56
3.4.2
Pressure dependence of damping coefficient . . . . .
57
3.4.3
Effective temperature
. . . . . . . . . . . . . . . . .
59
3.5
Towards the ground state . . . . . . . . . . . . . . . . . . .
60
3.5.1
The standard quantum limit . . . . . . . . . . . . . .
62
3.5.2
Recoil heating . . . . . . . . . . . . . . . . . . . . . .
63
3.5.3
Detector bandwidth . . . . . . . . . . . . . . . . . .
65
3.6
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
iv
Contents
4
Dynamics of a parametrically driven levitated particle
69
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
4.2
Theoretical background
. . . . . . . . . . . . . . . . . . . .
70
4.2.1
Equation of motion . . . . . . . . . . . . . . . . . . .
71
4.2.2
Overview of modulation parameter space . . . . . . .
72
4.2.3
Secular perturbation theory . . . . . . . . . . . . . .
75
4.2.4
Steady state solution . . . . . . . . . . . . . . . . . .
78
4.3
Dynamics below threshold (linear regime) . . . . . . . . . .
79
4.3.1
Injection locking . . . . . . . . . . . . . . . . . . . .
80
4.3.2
Linear instability . . . . . . . . . . . . . . . . . . . .
80
4.3.3
Frequency pulling . . . . . . . . . . . . . . . . . . . .
81
4.3.4
Off-resonant modulation (low frequency) . . . . . . .
82
4.4
Dynamics above threshold (nonlinear regime) . . . . . . . .
83
4.4.1
Nonlinear frequency shift
. . . . . . . . . . . . . . .
84
4.4.2
Nonlinear instability . . . . . . . . . . . . . . . . . .
85
4.4.3
Modulation frequency sweeps . . . . . . . . . . . . .
85
4.4.4
Modulation depth sweeps . . . . . . . . . . . . . . .
87
4.4.5
Relative phase between particle and external modu-
lation . . . . . . . . . . . . . . . . . . . . . . . . . .
89
4.4.6
Nonlinear mode coupling . . . . . . . . . . . . . . . .
91
4.4.7
Sidebands . . . . . . . . . . . . . . . . . . . . . . . .
93
4.5
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
5
Thermal nonlinearities in a nanomechanical oscillator
99
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
100
5.2
Description of the experiment . . . . . . . . . . . . . . . . .
100
5.2.1
Origin of nonlinear frequency shift . . . . . . . . . .
102
5.2.2
Nonlinear spectra . . . . . . . . . . . . . . . . . . . .
103
5.3
Experimental results . . . . . . . . . . . . . . . . . . . . . .
106
5.3.1
Frequency and energy correlations . . . . . . . . . .
106
5.3.2
Pressure dependence of frequency fluctuations . . . .
108
5.3.3
Frequency stabilization by feedback cooling . . . . .
110
5.4
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . .
111
v
Contents
6
Dynamic relaxation from an initial non-equilibrium steady
state
113
6.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
113
6.2
Description of the experiment . . . . . . . . . . . . . . . . .
115
6.2.1
Average energy relaxation . . . . . . . . . . . . . . .
117
6.3
Fluctuation theorem . . . . . . . . . . . . . . . . . . . . . .
118
6.3.1
General case
. . . . . . . . . . . . . . . . . . . . . .
118
6.3.2
Relaxation from an initial equilibrium state . . . . .
121
6.3.3
Relaxation from a steady state generated by paramet-
ric feedback . . . . . . . . . . . . . . . . . . . . . . .
122
6.4
Experimental results . . . . . . . . . . . . . . . . . . . . . .
124
6.4.1
Relaxation from feedback cooling . . . . . . . . . . .
124
6.4.2
Relaxation from excited state . . . . . . . . . . . . .
127
6.5
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . .
130
Appendix
133
A
Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . .
134
A.1
Calibration factor . . . . . . . . . . . . . . . . . . . .
134
A.2
Effective temperature
. . . . . . . . . . . . . . . . .
135
A.3
Natural damping rate . . . . . . . . . . . . . . . . .
135
Bibliography
136
Acknowledgements
148
List of Publications
152
vi
Introduction
0.1
Motivation
Nanotechnology was named one of the key enabling technologies by the
european commission [Europe, 2012] and it's tremendous impact on tech-
nology was envisioned early by
20
th
century physicist R.Feynman in his now
oft-quoted talk "Plenty of Room at the bottom" [Feynman, 1960].
Nanotechnology and nanoscience deal with structures barely visible with
an optical microscope, yet much bigger than simple molecules. Matter at
this mesoscale is often awkward to explore. It contains too many atoms to
be easily understood by straightforward application of quantum mechanics
(although the fundamental laws still apply). Yet, these systems are not so
large as to be completely free of quantum effects; thus, they do not sim-
ply obey the classical physics governing the macroworld. It is precisely
in this intermediate regime, the mesoworld, that unforeseen properties of
collective systems emerge [Roukes, 2001]. To fully exploit the potential of
nanotechnology, a thorough understanding of these properties is paramount.
The objective of the present thesis is to investigate and to control the
dynamics of an optically levitated particle in high vacuum. This system
belongs to the broader class of nanomechanical oscillators. Nanomechani-
cal oscillators exhibit high resonance frequencies, diminished active masses,
low power consumption and high quality factors - significantly higher than
those of electrical circuits [Ekinci and Roukes, 2005]. These attributes make
vii
Introduction
them suitable for sensing [Chaste et al., 2012; Moser et al., 2013; Cleland
and Roukes, 1998; Yang et al., 2006; Arlett et al., 2011], transduction [Lin
et al., 2010; ?; Unterreithmeier et al., 2009] and signal processing [Liu et al.,
2008]. Furthermore, nanomechanical systems are expected to open up in-
vestigations of the quantum behaviour of mesoscopic systems. Testing the
predictions of quantum theory on meso- to macroscopic scales is one of to-
days outstanding challenges of modern physics and addresses fundamental
questions on our understanding of the world [Kaltenbaek et al., 2012].
The state-of-the-art in nanomechanics itself has exploded in recent years,
driven by a combination of interesting new systems and vastly improved fab-
rication capabilities [Verhagen et al., 2012; Eichenfield et al., 2009; Sankey
et al., 2010]. Despite major breakthroughs, including ground state cooling
[O'Connell et al., 2010], observation of radiation pressure shot noise [Purdy
et al., 2013], squeezing [Safavi-Naeini et al., 2013] and demonstrated ultra-
high force [Moser et al., 2013] and mass sensitivity [Chaste et al., 2012;
Hanay et al., 2012; Yang et al., 2006], difficulties in reaching ultra-high me-
chanical quality (Q) factors still pose a major limitation for many of the
envisioned applications. Micro-fabricated mechanical systems are approach-
ing fundamental limits of dissipation [Cleland and Roukes, 2002; Mohanty
et al., 2002], thereby limiting their Q-factors. In contrast to micro-fabricated
devices, optically trapped nanoparticles in vacuum do not suffer from clamp-
ing losses, hence leading to much larger Q-factors.
0.2
Overview and state of the art
At the beginning of the present PhD thesis (early 2009), quantum optome-
chanics had just emerged as a promising route toward observing quantum
behaviour at increasingly large scales. Thus far, most experimental efforts
had focused on cooling mechanical systems to their quantum ground states,
but significant improvements in mechanical quality (Q) factors are gener-
ally needed to facilitate quantum coherent manipulation. This is difficult
viii
Overview and state of the art
given that many mechanical systems are approaching fundamental limits of
dissipation [Cleland and Roukes, 2002; Mohanty et al., 2002]. To overcome
the limitations set by dissipation, I developed an experiment to trap and
cool nanoparticles in high vacuum.
Figure 1 summarises the content of the thesis. It consists of six chapters,
ranging from a detailed description of the experimental apparatus (chap-
ter 1) and proof-of-principle experiments (parametric feedback cooling -
chapter 3) to the first observation of phenomena owing to the unique pa-
rameters of this novel optomechanical system (thermal nonlinearities - chap-
ter 5). Aside from optomechanics and optical trapping, the topics covered
include the dynamics of complex (nonlinear) systems (chapter 4) and the ex-
perimental and theoretical study of fluctuation theorems (non-equilibrium
relaxation - chapter 6), the latter playing a pivotal role in statistical physics.
0.2.1
Optomechanics and optical trapping
Except for Ashkin's seminal work on optical trapping of much larger micro-
sized particles from the early seventies [Ashkin, 1970, 1971; Ashkin and
Dziedzic, 1976], there was no further experimental work and little theo-
retical work [Libbrecht and Black, 2004] on optical levitation in vacuum
published at the time. Still, trapping in air had been demonstrated by dif-
ferent groups [Summers et al., 2008; Omori et al., 1997]. Two years later,
Li et al. demonstrated linear feedback cooling of micron sized particles [Li
et al., 2011] in high vacuum. However, due to fundamental limits set by re-
coil heating, nanoscale particles are necessary to reach the quantum regime.
During my thesis, I developed a novel parametric feedback mechanism for
cooling and built an experimental setup, which is capable of trapping and
cooling nanoparticles in high vacuum (c.f. chapters 3 and 1). The com-
bination of nanoparticles and vacuum trapping results in a very light and
ultra-high-Q mechanical oscillator. In fact, the Q-factor achieved with this
setup is the highest observed so far in any nano- or micromechanical system.
ix
Introduction
Summary
Chapter
Trapping of nanoparticles in high
vacuum is achieved by a novel
parametric feedback scheme,
which allows for thee dimensional
cooling with a single laser beam.
Subwavelength particles can be
trapped by single beam because
the optical gradient force
dominates over the scattering
force. Demonstrated cooling from
room temperature to 50mK and
ultra-high Q-factors exceeding 100
million.
Main Result
First observation of nonlinear
thermal motion in a mechanical
oscillator, enabled by a
combination of high Q and low
mass.
Validation of fluctuation theorem
for relaxation dynamics from non-
equilibrium steady states.
Experimental
Setup
1.
Parametric
Feedback Cooling
3.
Thermal
Nonlinearities
5.
Relaxation from
non-Equilibrium
6.
Study of relaxation dynamics to
thermal equilibrium from non-
equilibrium steady states.
Study of the dynamics of a
levitated nanoparticle under
stochastic (thermal) driving.
Experimental demonstration of
parametric feedback cooling.
Description of the experimental
apparatus, which allows to trap
and cool single dielectric
nanoparticles in high vacuum.
Dynamics of
driven particle
4.
Study of the nonlinear dynamics of
a levitated nanoparticle under
deterministic parametric driving
Theory of optical
tweezers
2.
A simple model is provided to gain
insight into the optical forces and
the conditions under which single
beam trapping in vacuum can be
achieved.
Particle dynamics is well explained
by a Duffing model. The Duffing
nonlinearity has it's origin in the
shape of the optical potential.
Figure 1: Overview of the thesis. The thesis consists of the six chapters
summarised above.
0.2.2
Complex systems
Discovering new effects by either pushing existing techniques to their fun-
damental limit or by developing new ones is the main motivation behind
x
Overview and state of the art
fundamental research. The exceptional high Q-factor in combination with
the low mass of the vacuum trapped nanoparticle allowed me to observe a
novel dynamic regime in which thermal excitations suffice to drive a me-
chanical oscillator into the nonlinear regime (c.f. chapter 5). The interplay
of thermal random forces and the intrinsic nonlinearity of the oscillator
gives rise to very rich dynamics. Yet, compared to other systems where
this effect can become important (i.e. lattice vibrations in a solid state),
a vacuum trapped nanoparticle is simple enough that it can be modelled
efficiently starting from first principles, thereby making it amenable to rig-
orous theoretical analysis. Therefore, a vacuum trapped nanoparticle is an
ideal testbed to study complex nonlinear dynamics both theoretically and
experimentally.
0.2.3
Statistical physics
Due to low coupling to the environment, random forces act on a vacuum
trapped particle on timescales much larger than the characteristic timescale
of the system (i.e. the oscillation period). However, despite these random
forces being small, they still dominate the dynamics of the particle. This
insight initiated me to study fluctuation relations in the context of optome-
chanics. Fluctuation relations are a generalisation of thermodynamics on
small scales and have been established as tools to measure thermodynamic
quantities in non-equilibrium mesoscopic systems. However, it is paramount
to study the theoretical predictions on controlled experiments in order to
apply them to more complex systems.
During my PhD, I studied experimentally and theoretically non-equilibrium
relaxation of a vacuum trapped nanoparticle between initial and final steady
state distributions. In a newly formed collaboration with Prof. Christoph
Dellago (University of Vienna, Austria), we showed experimentally and the-
oretically (c.f. chapter 6) the validity of the fluctuation theorem for relax-
ation of a non-thermal initial distribution. The same framework allows also
to study experimentally non-equilibrium fluctuation theorems for arbitrary
steady states and can be extended to investigate quantum fluctuation the-
orems [Huber et al., 2008] and situations where detailed balance does not
xi
Introduction
hold [Dykman, 2012].
xii
CHAPTER 1
Experimental Setup
The experimental setup is at the very heart of all experimental results pre-
sented in this PhD thesis and its design and development constitute a main
part of the PhD work. The purpose of this chapter is to provide guidelines
for a researcher interested in reproducing a similar experimental setup.
The setup consists of an optical trap in a vacuum chamber, a loading mech-
anism, optical detection, feedback electronics and data acquisition. Data
acquisition is performed with LabView and will not be detailed in this chap-
ter. The other parts are described in detail in the following sections.
1.1
Optical setup
The optical setup serves two purposes, trapping of a nano-particle with an
optical tweezer and detection of the particle motion.
In section 1.1.1 we give a detailed description of the optical setup. Section
1.1.2 gives the mathematical analysis of the detected signal. Sections 1.1.3
and 1.1.4 discuss the differences between measuring the backscattered or the
forward scattered light as well as homodyne versus heterodyne detection.
1
Experimental Setup
1.1.1
Overview of the optical setup
The optical setup for trapping and cooling is depicted schematically in fig-
ure 1.1. The light source is an ultra-stable low noise Nd:YAG laser
1
with
an optical wavelength of
= 1064 nm (Fig. 1.1). The optical table
2
has
active vibration isolation to reduce mechanical noise.
A stable single beam optical trap is formed by focusing the laser (
80 mW
at focus) with a high NA objective
3
(c.f. chapter 2), which is mounted in-
side a vacuum chamber. To parametrically actuate the particle, the beam
passes through a Pockels cell (EOM)
4
before entering the vacuum chamber.
We use parametric actuation to either cool (c.f. chapter 3) or drive the
particle (c.f. chapter 4).
For feedback cooling the particle position must be monitored with high
precision and high temporal resolution. This is achieved with optical in-
terferometry.
The particle position is imprinted into the phase of light
scattered by the particle. Through interference of the scattered light with
a reference beam, the phase modulation induced by particle motion is con-
verted into an intensity modulation. The intensity modulation is measured
with fast balanced photodetectors. We have chosen to measure the forward
scattered light. In this configuration, the non-scattered part of the incident
beam serves as a reference. Since light scattered in the forward direction
and the transmitted beam follow the same optical path, the relative phase
between the two is fixed in the absence of particle motion. If the parti-
cle moves, the interference of scattered light and transmitted beam causes
an intensity modulation of the light propagating in forward direction. We
collimate the light propagating in forward direction with an aspheric lens
5
which is mounted on a three dimensional piezo stage
6
for alignment with
respect to the objective (Fig. 1.2b).
1
InnoLight Mephisto 1W
2
CVI Melles Griot
3
Nikon LU PLan Fluor 50x, NA
= 0.8
4
Conoptics 350-160 and amplifier M25A
5
Thorlabs AL1512-C, NA
= 0.546
6
Attocube
2×ANPx101, 1×ANPz101
2
Optical setup
Figure 1.1: Optical Detection(a) The beam from the laser source is split
in two polarisations. The bulk (horizontally polarised) passes a EOM while
the rest (vertically polarised) is frequency shifted by an AOM. Before the
vacuum chamber the two beams are recombined with a second PBS. (b) In the
chamber the light is focused with a high-NA objective to form a single beam
trap. The scattered and the transmitted light are collected with an aspheric
lens. (c) After the vacuum chamber the two polarisations are separated. The
horizontally polarised part of the beam is dumped. The vertically polarised
part of the beam is sent to three balanced photodetectors for detection of
particle motion in all three spatial directions. (d) The detector signal is
processed by a home-built electronic feedback unit and sent to the EOM.
3
Experimental Setup
To detect particle displacement along the transversal
x and y axes, we
split the collimated beam with a D-shaped mirror
1
vertically and horizon-
tally, respectively. The two parts of each beam (
200 W each) are sent
to the two ports of a fast (80MHz) balanced detector
2
. The signal propor-
tional to
z is obtained by balanced detection of the transmitted beam with
a constant reference (Fig. 1.1c). Note that the reference beam for the bal-
anced detection in
z is not interfered with the beam that carries the signal.
The interferometric signal is already contained in the signal beam. Here,
the reference beam only cancels the offset. This could also be achieved by
a standard photodetector and an electronic high pass filter.
At the detectors, the optical intensity is converted into an electrical sig-
nal. This signal is processed by an electronic feedback unit (c.f. section 1.2)
and used to drive the EOM which modulates the intensity of the trapping
laser to cool the motion of the particle. Since an intensity modulation at
the detector resulting from motion of the particle can not easily be distin-
guished from an intensity modulation due to an applied signal at the EOM,
we use an auxiliary beam for detection. The auxiliary beam is obtained by
splitting off a small fraction (ratio 1:20) of the original laser beam before
it enters the EOM. To avoid interference in the laser focus, the auxiliary
beam is cross polarised and frequency shifted by an acusto optic modula-
tor (AOM)
3
. Before the chamber the auxiliary beam is recombined with
the trapping beam with a polarising beam splitter (PBS) and separated
again after the chamber by another PBS. The trapping beam (horizontal
polarisation) is dumped because it contains both the feedback signal and
the particle motion. The auxiliary beam (vertical polarisation), which con-
tains only the particle motion, is sent to the photodetectors (Fig. 1.2c). A
photograph of the experimental setup is depicted in figure 1.2.
1
Thorlabs, BBD1-E03
2
Newport, 1817-FS
3
Brimrose 410-472-7070, rf-driver AA opto-electronic MODA110-B4-34
4
Optical setup
a
a
b
c
b
c
x
y
z
Figure 1.2: Experimental Setup (a) The laser source is split into two
polarisations. One (horizontally polarised) is used for trapping and the other
(vertically polarised) is used for detection. (b) The high-NA objective is
mounted inside the vacuum chamber. The scattered and the transmitted light
is collected with an aspheric lens which is mounted on a piezoelectric stage
for alignment. (c) The vertically polarised light is sent to three balanced
detectors to detect the particle motion is all three spatial directions.
5
Experimental Setup
1.1.2
Detector signal
In this section we look into the formalism of interferometric detection. This
helps us to get a better understanding of how particle motion is related to
the detected signal.
We consider an incident Gaussian beam polarised along
x. The electric
field at position
r = (x, y, z) is given by
E
00
(
r) = E
0
w
0
w(z)
e
-
2
w(z)2
e
i kz-(z)+
k2
2R(z)
n
x
(1.1)
with beam waist
w
0
, Rayleigh range
z
0
, radial position
= (x
2
+ y
2
)
1/2
,
wavevector
k electric field at focus E
0
and polarisation vector
n
x
.
We used the following abbreviations
w(z) = w
0
1 + z
2
/z
2
0
beam radius
(1.2)
R(z) = z 1 + z
2
0
/z
2
wavefront radius
(1.3)
(z) = arctan (z/z
0
)
phase correction.
(1.4)
The incident field excites a dipole moment
dp
(
r
dp
) =
E
00
(
r
dp
) in a
particle situated at
r
dp
with polarisability
. The induced dipole in turn
radiates an electric field
E
Dipole
(
r, r
dp
) =
2
0
G(r, r
dp
)
dp
,
(1.5)
where
G(r, r
dp
) is the dyadic Green's function [Novotny and Hecht, 2006]
of an electric dipole loacted at
r
dp
.
0
and
are the vacuum permeability
and optical frequency, respectively.
In the paraxial approximation (
z
x, y, hence z
f), the far field of
the dipole at distance
f from the focus is a spherical wave
E
dp
(
r, r
dp
) = E
dp
exp i(kf +
dp
)
n
x
(1.6)
6
Optical setup
with amplitude
E
dp
1/f and phase
dp
, which depend on the particle
position. A lens with focal length
f transforms the spherical wave into
a plane wave with
k = xk/f (c.f. Fig. 1.3). Here, x = (x, y, f) is the
position on the reference sphere. Hence, the scattered light that arrives at
the detector is a plane wave with amplitude
E
dp
= E
0
2
0
4f
e
-
2
dp
/w
0
2
(1.7)
and phase
dp
=
- k · r
dp
+ (k
- 1 /z
0
) z
dp
(1.8)
=
- k
x
x
dp
- k
y
y
dp
- k
z
z
dp
+ (k
- 1 /z
0
) z
dp
- k
x
x
dp
- k
y
y
dp
- k 1 /kz
0
- k
2
x
+ k
2
y
2k
2
z
dp
,
where we used the approximation
k
z
k(1-[k
2
x
+ k
2
y
] 2k
2
. For clarity, we
dropped the propagation phase
k(f + d), where d is the distance between
lens and detector. Both the amplitude and the phase depend on the particle
position
r
dp
(t). But for small displacements of the nano-particle (
|r
dp
|
w
0
, z
0
), the dependency of the amplitude is weak. Thus, for the sake of
simplicity we assume in the further discussion that the amplitude of the
scattered field is constant.
As argued in the previous paragraph, the particle motion is primarily
imprinted into the phase of the scattered light. Thus, a phase sensitive
measurement is required. Additionally, the scattering cross section
s
=
k
4
2
6
2
0
is only
258 nm
2
for a
a = 75 nm SiO
2
particle. This means that
the total scattered light intensity is very weak (typically a few
W ). To
read out the phase and to amplify the signal, the scattered light is interfered
with a reference beam field
E
ref
. As a result, the intensity distribution at
the detector is
I(
r, r
dp
)
|E
total
|
2
=
|E
dp
+
E
ref
|
2
(1.9)
=E
2
dp
+ 2E
dp
E
ref
cos (
dp
(
r, r
dp
) +
ref
) + E
2
ref
,
7
Experimental Setup
outgoing plane wave
outgoing spherical wave
reference sphere
focus
f
z
d
r
dp
y k
y
x k
x
detector
Figure 1.3: Light collection In the far field, light scattered from the particle
has the form of a spherical wave. A lens with focal length f maps the
spherical wave into a plane wave. Hence, each
k-vector is mapped onto a
point in the detector plane.
where
ref
depends on the relative phase between the scattered light and the
reference. If the intensity of the reference is much larger than the scattered
intensity, the first term of the last expression in (1.9) is small compared to
the other two and can be neglected.
For detection of the longitudinal displacement
z we focus the transmitted
beam on one port of a balanced detector. This amounts to integration of
(1.9) over the full detector area. Due to the symmetry of
I(
r, r
dp
), the
dependency on
x and y vanishes (c.f. Fig. 1.4b). Hence, the detector
output only depends on
z. To cancel the constant offset E
2
ref
, we focus a
second beam of equal intensity on the second port of the balanced detector.
Thus, the detector signal reads:
S
z
= 2
k
max
0
2
0
E
dp
E
ref
cos (
dp
(
r, r
dp
) +
ref
) dkd.
(1.10)
Here,
k
max
= k NA
det
is the maximum
k vector that is detected and which
depends on the numerical aperture
NA
det
of the collimating lens.
8
Optical setup
For the detection of the lateral displacement
x (y), we split the beam
vertically (horizontally) with a D-shaped mirror (c.f. Fig. 1.4a). Each of
the two parts is focused onto one of the two ports of a balanced detector.
Thus, the detector signal for
x (y) is given by:
S
x (y)
=
k
max
0
3/2 ()
/2 (0)
I(
r, r
dp
) dkd
-
k
max
0
/2 (2)
3/2 ()
I(
r, r
dp
) dkd.
(1.11)
Since the polarisation of the beam is aligned parallel (perpendicular) to the
edge of the D-shaped mirror, the symmetry of the beam is conserved. As a
consequence, the orthogonal displacement cancels and we measure a signal
that only depends on
x
dp
(
y
dp
).
1.1.3
Homodyne measurement
We now consider the situation in which the scattered light interferes with a
reference beam at the same optical frequency. This is the situation encoun-
tered when analysing the forward scattered light. In that case the reference
is simply the non-scattered transmitted beam. Alternatively, we can also
analyse the backscattered light. However, in that case, we have to manu-
ally set up a reference beam. Under the assumption that the scattered light
intensity is much weaker than the reference and that the term
E
2
ref
can be
eliminated by balanced detection, the intensity distribution is given by
2E
dp
E
ref
cos (
dp
(
r, r
dp
) +
ref
) .
(1.12)
To first approximation, the motion of the particle is harmonic, that is
dp
q
0
cos
0
with oscillation amplitude
q
0
and angular frequency
0
.
With the identity
e
ia cos(b)
= J
0
(a) + 2
k=1
i
k
J
k
(a) cos(kb),
(1.13)
we find that the spectrum of the detected signal (1.12) consists of harmon-
ics of the particle oscillation frequency
0
. The relative strength of the
9
Experimental Setup
a
b
Figure 1.4: Optical detection of the particle position. The total light
that propagates towards the detector consists of the transmitted beam (red)
and the scattered light (black). The red arrow indicated the direction of
beam propagation. (a) To detect lateral displacements, the transmitted light
is split and the two resulting beams are detected in differential mode. Be-
cause scattered light from the particle travels distances that are different for
the two detectors, the accumulated phase at each detector is different. At
the detector the scattered light interferes with the unscattered light, which
serves as reference beam, making the detector signal sensitive to changes in
the phase. Since the phase depends on the position of the particle, the de-
tector signal is proportional to the motion of the particle. (b) To detect the
motion along the optical axis, the entire transmitted beam is detected. The
total intensity at the detector is the sum of scattered light and transmitted
beam. The former has a phase that depends on the particle's position on the
optical axis. Due to interference of the two, the detector signal is sensitive
to the phase and therefore to the position of the particle. Note that this de-
tection is not dependent on lateral displacement as the phase due to lateral
displacements at one half of the detector cancels with the phase at the other
half of the detector.
harmonics is given by Bessel functions
J
k
and depends on the particle am-
plitude
q
0
. For small amplitudes only the first terms contribute significantly.
10
Optical setup
The homodyne signal (1.12) is the real part of (1.13). If
ref
=
-/2
we get an additional factor
i in (1.13). Hence, the main contribution to the
signal is
J
1
(q
0
) cos
0
t. In contrast, if the relative phase is
ref
= 0, the
main contribution is
J
2
(q
0
) cos 2
0
t. Therefore, to get a detector signal
that is a linear function of the particle displacement, the relative phase
ref
has to be fixed to
-/2.
Note that, if the phase is fixed to
0, the detector signal is
cos 2
0
t.
This is the signal required for feedback cooling. Hence, if we can interfere
the scattered light with a reference that is in-phase, we can circumvent an
electronic frequency doubler (c.f. section 1.2).
Forward detection
In forward detection the reference beam is the non-scattered transmitted
beam. In the paraxial approximation, the far-field of the transmitted Gaus-
sian is a spherical wave
E
ref
(
r) = E
ref
exp i(kR
- /2)n
x
,
(1.14)
with amplitude
E
ref
E
0
z
0
/R and phase
-/2, known as the Gouy phase
shift. The spherical wave is collected by a lens and propagates as a plane
wave to the detector. Since the transmitted beam and the forward scat-
tered light follow the same optical path, they acquire the same propagation
phase. Hence, the phase difference at the detector is given by the Gouy
phase shift
ref
=
-/2 and the detector signal is linear with respect to
particle position.
Note that in forward scattering the scattered light and the reference are
both proportional to the incident laser
E
dp
E
ref
E
0
. As a consequence,
if the trapping laser is used for detection, the intensity of the reference (the
last term of (1.9)) is time dependent because of the feedback signal. If the
detectors are not 100% balanced, they will pickup the intensity modulation
and feed it back into the feedback circuit. This results in undesired back-
action. To resolve this problem, we chose to split off a part of the incident
beam before it is modulated as explained in section 1.1.1. But it also means
11
Experimental Setup
that we do not use the scattered light from the trapping beam, which is
much more intense than the scattered light from the auxiliary beam. Hence,
the sensitivity is not as high as it potentially could be if we could use the
scattered light from the trapping beam.
Backward detection
In backward detection we use the same lens for collection as for focusing of
the light. The collected light is interfered with an independent reference.
This gives us the freedom to choose the strength of the reference field freely
and to optimise it to get the best signal-to-noise ratio (SNR). The light scat-
tered back into the objective is of the same form as the forward scattered
light of equation (1.6) because the dipole radiation pattern is symmetric.
Hence, the detector signal is given by (1.12). However, since the reference
beam and backscattered light do not follow the same optical path, the rel-
ative phase of reference and backscattered light
ref
is not fixed anymore.
It depends on the relative path difference and is therefore subject to any
noise source that alters the optical path length, for example air currents
and mechanical drift of the optical elements.
1.1.4
Heterodyne measurement
Measuring in back-reflection not only allows us to choose the intensity of
the reference field but also it's frequency. If the optical frequency of the
reference differs from that of the scattered light, it is called a heterodyne
measurement. The detector signal is then given by
2E
dp
E
ref
cos (
dp
(
r, r
dp
) +
ref
+ t) ,
(1.15)
where
is the difference in optical frequency. For = 0 (1.15) reduces
to (1.12). The spectrum of the detector signal is now shifted to
with
sidebands at
±
0
.
In order to benefit from a lower noise floor at
high frequencies, one would typically choose
0
. With a lock-in
amplifier that locks to the modulation frequency
one can measure both
12
Optical setup
left sideband
right sideband
-
m
+
m
0
2E
dp
E
ref
E
dp
E
ref
m
0
2E
dp
E
ref
E
dp
E
ref
2
0
+ 2
0
- 2
0
Figure 1.5: Spectrum of detector signal. In the heterodyne measurement
the spectrum is shifted to . The particle oscillation shows up as side-peaks
around at harmonics of the particle oscillation frequency
0
(black). A
modulation of the intensity at
m
reproduces the main structure at side-
frequencies
±
m
(red and blue). The feedback modulates the intensity
of the laser at
m
= 2
0
. As a consequence, the sidebands overlap with the
main peaks (see inset). However, since the phase of the intensity modulation
is fixed to the particle oscillation, this results only in a constant phase offset,
which is eliminated by the lock-in.
quadratures of (1.13) and extract, therefore, the phase
dp
+
ref
. To get
rid of the low frequency spurious phase
ref
, the lock-in amplifier output
has to be high-pass filtered.
Note that the bandwidth
B of the lock-in has to be large enough so
that the signal from the particle passes, that is
B
0
. This requirement
is quite demanding, given that typical particle frequencies are of the order
of
100kHz-1Mhz. For an ideal harmonic oscillator, the damping
0
deter-
mines the timescale on which the energy and the phase change. Thus, the
minimum required bandwidth for an harmonic oscillator is of the order
0
,
which is a factor
Q
10
8
less than
0
. However, since for efficient feed-
back cooling the phase stability is paramount, a higher bandwidth might be
13
Experimental Setup
necessary (still
0
) if there are additional sources of phase noise such as
frequency fluctuations due to nonlinear amplitude to frequency conversion
(c.f. chapter 5). A detection scheme with a bandwidth
B
0
0
could
be realized for instance with a phase locked loop (PLL), where the particle
itself acts as the frequency determining element.
Detection with the trapping laser
As mentioned before, we would get a much stronger signal if we could use
the scattered light from the trapping laser. Let's assume the trapping laser
is modulated at frequency
m
and modulation depth . The scattered field
is
E
dp
(1 + cos
m
t). Hence, the spectrum of the detector signal exhibits
sidebands at the modulation frequency
m
and amplitude
E
dp
E
ref
as
shown in figure 1.5. However, for small modulation
the amplitude of the
main signal
E
dp
E
ref
is much stronger than the sidebands.
If the trapping laser is used for detection, the intensity modulation is at
m
= 2
0
. Therefore, the sidebands from the intensity modulation overlap
with the main signal. But, since the feedback fixes the phase of the sideband
to the main signal, this results only in a constant phase, which is eliminated
by lock-in detection.
1.2
Feedback Electronics
For feedback cooling of the particle's center of mass (CoM) motion we use
an analog electronic feedback signal. The signal is used to modulate the
intensity of the trapping laser with an EOM.
Figure 1.6 shows the electronic feedback system. It consists of five units:
a bandpass filter, a variable amplifier, a phase shifter, a frequency doubler
and an adder. The first four units are replicas of the same electronic circuit,
each one optimised for one specific frequency. The frequencies correspond
to the three oscillation frequencies of the vacuum trapped particle. Table
1.1 shows the feedback parameters for the three axes. The fifth unit adds
14
Feedback Electronics
in
bandpass filter
variable gain
frequency doubler
and phase shifter
sum
out
in
out
gain
monitor
in
out
phase
monitor
gain
in
out
TTL in
attenuate
aux. in
Figure 1.6: The Feedback Electronics consist of of five modules.
A
bandpass filter (red), variable gain (white), frequency doubler (green), phase
shifter (same module, also green) and adder.
the three signals together and sends the resulting signal to the EOM.
The following sections describe the functionality of each of the individual
modules.
1.2.1
Bandpass filter
The signal of interest is at the natural frequency of the particle. However,
there are low frequency components because of mechanical drift and beam
pointing instability. Furthermore, the auxiliary beam and the part of the
trapping beam that leaks through the polarising beam splitter interfere at
the detector and create a high frequency beating at the AOM frequency. The
unwanted frequency components can have a detrimental effect on the output
signal because each electronic circuit has a maximum range of input voltages
before it saturates. Since the peak values at the unwanted frequencies exceed
15
Experimental Setup
axes
parameters
x
y
z
f
0
[kHz]
125
140
38
(f
0
) [
]
12
149
159
BW [kHz]
15-250
15-250
15-150
Table 1.1: Parameters of electronic feedback. Each stage of the elec-
tronic circuit is optimised for one specific particle frequency. f
0
is the fre-
quency of the frequency doubler. is the phase difference between a sinu-
soidal input at the bandpass filter and the frequency doubled output at the
adder. BW is the bandwidth of the bandpass filter. All subsequent modules
have a higher cut-off frequency.
the peak values at the signal frequency, it is necessary to filter them at an
early stage of the feedback circuit. Therefore, each of the three signals from
the balanced photodetectors is filtered by a second order bandpass filter
1
.
Because every electronic circuit adds noise, the detector signals are also
amplified at this first stage of the feedback by a factor
×20. This mitigates
the noise contribution from subsequent stages of the feedback and thereby
gives a better SNR at the output of the feedback circuit.
1.2.2
Variable gain amplifier
To get the optimum SNR at the end of the feedback circuit, it is important
amplify the signal enough to mitigate subsequent additive noise, but not
too much to saturate any of the circuits. Therefore, to tune to the optimum
gain, the bandpass is followed by a variable gain amplifier
2
from
×1 to ×10
that is controlled with a potentiometer.
1
multiple feedback, AD817
2
OPA1611AID
16
Feedback Electronics
1.2.3
Phase shifter
As shown in figure 1.7, the optical power
P has to be larger than the average
power
P
0
when the particle moves away from the trap center and smaller
if the particle comes closer. Thus, the feedback signal has to be exactly
in phase (
= 0) with respect to the particle oscillation to achieve cooling.
Since every stage (detector, electronics, electro-optic modulator) adds some
EOM
particle
/
2
0
0
t
x
P
P
0
x
0
-x
0
P
0
Figure 1.7: Feedback Phase To achieve cooling, the modulation of the
EOM has to be exactly in phase = 0 with respect to the particle oscillation.
The modulation depth
depends on the feedback gain.
phase to the feedback signal, the phase of the electronic signal has to be
tuned to the right value. This is achieved with a two stage phase shifter, as
a single stage would not be able to cover the full range of
180
. Further, it
allows us to have one control for coarse and one for fine tuning. The phase
shifter is essentially a bandpass filter with unitary gain where the center
frequency can be tuned by a potentiometer. Thus, the phase response of
that filter is not flat but has a slope that depends on the value of the
potentiometer. The operational amplifier used was AD8671ARZ.
1.2.4
Frequency doubler
The frequency doubler is integrated in the same module as the phase shifter.
To create a signal at twice the frequency of the input signal, we use a com-
17
Experimental Setup
mercial multiplier circuit
1
. For an input signal
X
in
= X
0
sin t, the resulting
signal is
X
out
= X
2
0
cos 2t /(10V ) . Hence, the device is optimised for a
single frequency where an input amplitude of
10V gives and output also at
10V .
1.2.5
Adder
To cool the particle in all three spatial dimensions, we sum the three pro-
cessed signals together
2
and send it to the EOM. The summed signal can
be switched off through a TTL signal by an analog switch
3
. This feature
is used in the relaxation experiment of chapter 6. Additionally, there is an
extra input that is added to the output. This is needed in chapter 4 to drive
and cool the particle simultaneously.
1.3
Particle Loading
Loading a particle into the optical trap is a critical first step. In a liquid
particles can be captured easily. By moving the chamber that contains the
liquid with a translation stage, the suspended particles are dragged along
and can thereby be brought into close vicinity of the trapping laser. In
contrast, particles in a gas will quickly fall down due to gravity.
The size of the optical trap is of the order of the focal volume
3
.
Therefore, a particle that passes the focus at a distance larger than
will
not be captured. Furthermore, if a particle enters the volume at high speed,
the low damping in vacuum will not slow the particle down sufficiently. The
maximum allowed speed is
v
max
0
, where
0
is the damping constant.
In water
0
is given by Stokes formula
0
= 6a/m, where a and m are
the radius and mass of the particle and
the viscosity of the surrounding
medium. In water
(water)
= 890 Pa s. Hence, the maximum velocity for a
1
AD734/AD
2
AD817
3
NC7SB3157P6X
18
Particle Loading
a = 75nm SiO
2
particle is
v
(water)
max
= 344 m/s. In air the viscosity is about
two orders of magnitude smaller
(air)
= 18 Pa s. Hence, the particle ve-
locity should not exceed
v
(air)
max
= 7 m/s. The challenge lies, therefore, in
finding a technique by which a slow single nano-particle is brought into the
focal volume.
One strategy to bring a nano-particle into the focal volume in a con-
trolled manner, is to place it onto a substrate, which can be manipulated
with high accuracy by piezo-electric actuators. However, a particle on a
substrate remains there due to dipole-dipole interactions known as van der
Waals (VdW)-forces. The VdW-force in the "DMT limit" of the Derjaguin-
Muller-Toporov theory is given by [Israelachvili]
F
VdW
= 4a,
(1.16)
where
a is the radius of the particle and is the effective surface energy.
Measurements on silica spheres on glass give
F
VdW
= 176 nN for a = 1 m
[Heim et al., 1999]. For comparison, the gravitational force for a particle
of this size is only
0.1 pN and therefore much too weak to remove the
particle.
Sections 1.3.1 and 1.3.2 discuss two approaches that aim at removing
nano-particles form a surface, pulsed optical forces and inertial forces. Both
are found to be insufficient to overcome the forces that keep the particle on
the surface. Section 1.3.3 gives a detailed description of the successfull neb-
uliser approach. The nebuliser creates an aerosol of nano-particles, which
are then trapped by the optical tweezer.
1.3.1
Pulsed optical forces
The maximum force from a typical continuous wave optical tweezer (
1 nN)
is two to three orders of magnitude weaker than the VdW-force. The optical
force depends linearly on the optical power. A pulsed laser can have a peak
power much higher than a continuous wave laser (CW-laser). Therefore, it
was suggested in [Ambardekar and Li, 2005] that a pulsed laser should have
19
Experimental Setup
a maximum force strong enough to overcome the VdW-force. Generally, the
shorter the pulse the higher the maximum peak power that can be achieved.
However, the force has to bring the particle at least to a critical distance
d
crit
, so that the VdW-force doesn't pull the particle back to the surface
once the pulse ends. Hence, assuming a constant acceleration throughout
the pulse, the minimum pulse duration is
pulse
= (2d
crit
m/F
pulse
)
1/2
.
We used a Nd:YAG laser with a pulse duration of
50 ns and a total
energy per pulse of
0.1 mJ at the sample. Thus, the maximum power
(
2kW) is three orders of magnitude higher than what can be achieved
with a CW-laser. Indeed, we managed to remove several silica particles
from a glass coverslip that was coated with a thin film (
20nm) of either
aluminium or gold. However, the laser pulse often left a crater behind.
This suggests that actually thermal forces due to light absorption in the
metal film removed the particle. Some of the released particles disappeared
from the field of view and some landed a few
m away from their initial
position. Still, we did not manage to trap any of the released particles with
a second superimposed CW laser and decided to pursue other approaches.
In retrospective, this is not too surprising. To trap a particle, we do not
only have to remove it, but also shoot it close to the optical trap with not
too much kinetic energy. From the above discussion about the trap volume
and the required particle speed, it is clear that the probability of catching
a particle is rather small.
1.3.2
Piezo approach
Already Ashkin in his pioneering experiments [Ashkin, 1971] used a piezo-
electric transducer to overcome the VdW-forces. This method was also used
to trap and cool microspheres to
mK temperatures [Li et al., 2011] and is de-
scribed in detail in reference [Li, 2011]. For a sinusoidally driven piezo with
oscillation amplitude
q
p
and frequency
p
, the force due to the particle's
inertia is
F
piezo
= m
2
p
q
p
,
(1.17)
20
Particle Loading
where
m is the mass of the particle. Consequently, the acceleration needed
to release the particle is
a
p
=
2
p
q
p
= 4a/m
a
-2
.
(1.18)
Hence, for a given maximum acceleration the piezo can provide, there is
a minimum particle size that can be shaken off. For typical values this
is
a
min
1m. Therefore, this approach can not be used to shake nano-
particles off a substrate.
1.3.3
Nebuliser
From the preceding sections it is clear that optical, gravitational and inertial
forces are too weak to remove a particle from a substrate. In this section we
describe a surprisingly simple but functional approach to load the trap with
nanoparticles [Summers et al., 2008]. We use a commercial nebuliser
1
and
a highly diluted solution of silica beads
2
of
147 nm diameter. The diluted
solution is obtained from mixing
10l initial solution (50 mg/ml) with
1ml of ethanol. The nebuliser consists essentially of a mesh on top of a piezo
element. A little bit of liquid is brought into the space between the piezo and
the mesh. The motion of the piezo pushes the liquid through the mesh. The
mesh breaks the liquid into little droplets of diameter smaller than
2 m.
Under standard humidity conditions, the droplets quickly evaporate and
only the solid component is left behind. The concentration of the solution
is such that on average there is one or no particles in a single droplet. As
shown in figure 1.8, we use a nozzle in an upside down configuration to
funnel the falling particles close to the laser focus. Through a viewport on
the side of the vacuum chamber we observe the falling particles with a CCD
camera
3
and wait until one is trapped. Then we remove the nozzle, close
the chamber and pump down. Note that because the light scattered by the
particle has a dipole radiation pattern the polarisation of the laser beam
1
Omron NE-U22-W
2
Microparticles SIO2-R-B1181
3
Hamamatsu C8484-05G01
21
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- Citar trabajo
- Jan Gieseler (Autor), 2014, Dynamics of optically levitated nanoparticles in high vacuum, Múnich, GRIN Verlag, https://www.grin.com/document/300442
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