laser-propulsion of microparticles in liquid- filled
TRANSCRIPT
LASER-Propulsion of Microparticles in Liquid-
Filled Hollow-Core Photonic-Crystal Fibers
-Going against the Flow-
LASER-Antrieb von Mikropartikeln entlang des Hohlkernsflussigkeitsgefullter Photonischer Kristallfasern
-Gegen den Strom-
Der Naturwissenschaftlichen Fakultatder Friedrich-Alexander-Universitat Erlangen-Nurnberg
zurErlangung des Doktorgrades Dr. rer. nat.
vorgelegt von
Martin Konrad Garbosaus Eschweiler
Als Dissertation genehmigt von der Naturwissen-schaftlichen Fakultat der Friedrich-Alexander-Universitat
Erlangen-Nurnberg
Tag der mundlichen Prufung: 20.07.2011
Vorsitzenderder Promotionskommission: Prof. Dr. Rainer Fink
Erstberichterstatter: Prof. Dr. Philip St.J. Russell
Zweitberichterstatter: Prof. Dr. Miles J. Padgett
Abstract
This thesis demonstrates that microparticles can be controllably loaded into and
propelled along the hollow core of a liquid-filled, single-mode photonic crystal fiber
by means of optical forces. A setup is designed, combining conventional single-beam
laser tweezers and a coupling stage, in order to analyze an ensemble of microparticles
and selectively launch a single particle with desired properties. The utilized fibers
guide light in a fundamental mode due to a photonic band gap when the entire
structure is filled with liquid. The resulting low loss and axially constant mode
profile are used to propel particles over distances of several meters at a constant
speed. Particles can be guided along reconfigurable fiber paths due to the low
optical bend loss of the fiber. The flow in the fiber core can be precisely controlled,
allowing to balance the optical forces with fluidic forces, thus keeping the particle
position fixed.
The particle speed, as well as the flow necessary to hold a particle stationary
against the optical force exerted by 1W of laser power are examined over a large
particle size range. The experiments show that the drag force is strongly increased
for particle sizes in the order of the fiber core diameter, due to wall-proximity effects.
A ray-optics theory combined with numerical simulations for the drag force is used
in order to compare the experimental results, showing good agreement.
Furthermore, a Doppler-based interferometric technique is used to accurately
measure the speed of propagating particles without the need of imaging the side-
scattered light off the particle. Due to the improved accuracy and the extended
range, small periodic speed fluctuations due to intermodal beating between the
fundamental and the first higher order mode can be detected. It is found that the
beat period only depends on the utilized wavelength and the fiber core diameter.
The observed speed fluctuations are in excellent agreement with the ones predicted
by electrodynamic theory of modes in cylindrical waveguides. This demonstrates
that propagating particles can be used to investigate the mode profile continuously
along the fiber in a destruction-free manner.
The Doppler-based interferometric technique is also used to investigate the dy-
namics of particles launched off the fiber core in a horizontal fiber. The measured
time constant of the launching process is about 10 times larger than expected from
theory, indicating that additional effects take place. Particle spinning, which is not
taken account of in theory, is proposed to delay a particle launch due to the Magnus
effect.
Zusammenfassung
In der vorliegenden Arbeit wird gezeigt, dass Mikropartikel mit Hilfe von optischen
Kraften kontrolliert in flussigkeitsgefullte Photonische-Kristallfasern geladen und
entlang des Hohlkerns geleitet werden konnen. Ein experimenteller Aufbau wird ent-
wickelt, welcher konventionelle Einzelstrahl-LASER-Pinzetten mit einer Faserkop-
pelplattform in einem flussigen Medium kombiniert. Dieser erlaubt die Analyse eines
Ensembles von Mikropartikeln und einen selektiven Start von ausgewahlten Teilchen
mit gewunschten Eigenschaften. Die eingesetzten Fasern leiten Licht in einer Fun-
damentalmode aufgrund einer photonischen Bandlucke, wenn die gesamte Faser-
struktur mit Flussigkeit gefullt ist. Die daraus resultierenden, exzellenten Trans-
missionseigenschaften, minimale Krummungsverluste und das axial unveranderliche
Modenprofil ermoglichen den optischen Teilchentransport uber mehrere Meter ent-
lang beliebiger Bahnen mit konstanter Geschwindigkeit. Ein prazise kontrollierter
Flussigkeitsfluss im Hohlkern ermoglicht es die optischen Krafte gegen viskose Krafte
auszubalancieren und das Teilchen stationar zu halten.
Die Teilchengeschwindigkeit, als auch der Fluss, welcher benotigt wird um ein
Teilchen stationar zu halten fur die optische Kraft bei 1W LASER-Leistung, werden
fur einen weiten Großenbereich ermittelt. Bei Teilchengroßen nahe des Hohlkern-
durchmessers zeigen die Experimente einen starken Anstieg des Stromungswiderstan-
des aufgrund von Wandeffekten. Ein Strahlenmodell wird zusammen mit numeri-
schen Simulationen der Stromungskrafte benutzt, um die experimentell erhaltenen
Daten zu vergleichen. Hierbei stimmen beide sehr gut uberein.
Daruberhinaus wird ein Doppler-basierter, interferometrischer Ansatz genutzt,
um die Teilchengeschwindigkeit prazise zu messen, ohne auf die Beobachtung des
vom Teilchen gestreuten Lichts angewiesen zu sein. Aufgrund der verbesserten
Genauigkeit und der hohen Reichweite konnen kleine, periodische Geschwindigkeits-
fluktuationen nachgewiesen werden, die durch eine intermodale Schwebung zwischen
der Fundamentalmode der Faser und der nachsthoheren Mode entstehen. Hierbei
hangt die Periode der Geschwindigkeitsfluktuationen lediglich von der verwende-
ten LASER-Wellenlange und vom Faserkerndurchmesser ab. Zudem stimmen die
gemessenen Ergebnisse exzellent mit elektrodynamischen Berechnungen fur zylin-
drische Wellenleiter uberein. Dies beweist, dass optisch angetriebene Mikropartikel
dazu benutzt werden konnen, das Modenprofil entlang einer Hohlkernfaser kon-
tinuierlich zu messen, ohne dabei die Faser zu zerstoren.
Des Weiteren wird das Doppler-basierte Verfahren benutzt, um das dynamische
Verhalten von Teilchen zu untersuchen, welche in horizontalen Faserstucken von der
Hohlkernwand gestartet werden. Die gemessene Zeitkonstante fur den Startprozess
ist dabei etwa zehnfach großer als theoretisch vorhergesagt, was darauf hin deutet,
dass weitere Effekte stattfinden. Es ist naheliegend, dass Teilchenrotation, welche
nicht von der Theorie berucksichtigt wird, aufgrund des Magnuseffekts den Start
des Teilchens verlangsamt.
Contents
1 Introduction 3
2 Instrumentation 7
2.1 Microfluidic setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Optical setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.1 Laser tweezers trap . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.2 Optical guidance path . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Doppler velocimetry setup . . . . . . . . . . . . . . . . . . . . . . . . 13
3 Properties of liquid-filled hollow-core photonic crystal fibers 17
3.1 Guiding mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2 Scaling law for liquid-filled HCBGF . . . . . . . . . . . . . . . . . . . 21
3.3 Bend loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3.1 Theoretical description of bend loss mechanisms . . . . . . . . 26
3.3.2 Experimental bend loss characterization . . . . . . . . . . . . 28
4 Theory 33
4.1 Optical force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.1.1 Ray optics model . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.1.2 Modeled results . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.2 Fluidic force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2.1 Microfluidic theory . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2.2 Microfluidic flow profile in the hollow core . . . . . . . . . . . 51
4.2.3 Particle effects on flow rate . . . . . . . . . . . . . . . . . . . 53
2 CONTENTS
4.3 Optofluidic balancing . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5 Particle guidance in liquid-filled photonic crystal fibers 59
5.1 Particle characterization and launching . . . . . . . . . . . . . . . . . 59
5.2 Horizontal particle guidance . . . . . . . . . . . . . . . . . . . . . . . 63
5.3 Vertical particle guidance . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.3.1 Balancing against gravity . . . . . . . . . . . . . . . . . . . . 69
5.3.2 Optical flow mobility in vertical hollow-core PCFs . . . . . . . 71
5.4 Particle guidance around bends . . . . . . . . . . . . . . . . . . . . . 74
6 Doppler velocimetry 77
6.1 Measurement procedure . . . . . . . . . . . . . . . . . . . . . . . . . 78
6.2 Doppler based particle tracking . . . . . . . . . . . . . . . . . . . . . 79
6.3 Intermodal beating . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.4 Delayed particle lifting . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.5 Multi-particle tracking . . . . . . . . . . . . . . . . . . . . . . . . . . 92
7 Conclusions and recommendations 95
7.1 Experimental and theoretical conclusions . . . . . . . . . . . . . . . . 95
7.2 Recommendations and outlook . . . . . . . . . . . . . . . . . . . . . . 96
8 Acknowledgements 101
Chapter 1
Introduction
Arthur Ashkin, the founder of the field of optical micromanipulation, discovered in
1970 [1] that micron sized objects could be accelerated by the radiation pressure of
a laser beam or even stably trapped by two counterpropagating or perpendicularly
arranged [2] beams. Also optical levitation of droplets against gravity could be
demonstrated [3]. While many interesting experiments were carried out [4, 5, 6],
the field of optical micromanipulation was revolutionized by the introduction of
single beam optical tweezers in 1986 [7, 8], where microparticles are trapped a single
tightly focused laser beam. Biological applications like the artificial fertilization
of mammalian cells and microsurgery became possible [9]. Simmons et al. have
shown that the force on a microparticle can be determined from its displacement
from the trap position [10]. This allows to investigate mechanical properties on
the cell level like the measurement of the forces applied by bacteria flaggela [11] or
even molecular studies where the kinetic behavior of the motor protein kinesin is
studied [12]. Cell properties like the cytoskeletal associations and surface dynamics
of integrins [13] and the neuronal growth cone membrane mechanical properties [14]
could also be investigated. The laser tweezers toolbox was restocked by Padgett et
al. who used circularly polarized light to build an optical spanner. This tool uses
the angular momentum of light and is capable of holding a microparticle in place
while spinning it [15, 16]. Another breakthrough was done by Grier et al. in 2001
by introducing electronically controlled refractive elements into the tweezers setup
[17], capable to create several independent, arbitrarily positioned dynamic trapping
4 Introduction
sites [18] in three dimensions [19, 20]. A user friendly computer interface for the
trap steering [21] creates a unique link between the laboratory and the micron sized
world. Recently, even the use of commercially available touch pad computers for an
intuitive simultaneous steering of several laser tweezers was demonstrated [22].
The synthesis, measurement and manipulation of micrometer-sized objects are of
great importance in many fields, with examples being catalysis, cell biology, quan-
tum dots, colloidal chemistry and paint design. Optical trapping combined with
microfluidics [23, 24, 25] has been used, e.g., to size sort dielectric particles [26].
When microfluidics are combined with waveguides, novel applications such as op-
tical transportation and optical chromatography become possible. In this context,
the evanescent edging field of a single guided optical mode has been used to propel
particles over short distances ∼0.1mm on planar waveguides [27, 28] and in Si slot
waveguides [29]. This approach has the disadvantage that the transverse optical field
decays exponentially from the surface, making stable optical trapping difficult. Fur-
thermore, the particles are guided very close to the waveguide surface, resulting in
asymmetric drag forces. A hollow, symmetric waveguide, where a particle is trapped
in its center is ideal, as demonstrated by Renn et al. for atoms [30], microparticles
and water droplets [31], optically guided in glass capillaries. However, the high loss
of hollow capillaries [32] (143 dB/m for a hollow glass capillary with 40 �m core at
780 nm) strongly limits the propagation length. The use of hollow-core photonic
crystal fibers (HC-PCFs) [33, 34] which can guide light due to a photonic band gap
(PBG) [35] reduces the loss dramatically. Benabid et al. have demonstrated the
guidance of polystyrene particles along the hollow core of a 15 cm long PCF [33].
These examples where light is guided in a hollow waveguide were all performed
in air. However, the extension to liquid-filled hollow waveguides is highly desireable
since aqueous media host most of the biological processes. Mandal et al. have
shown that it is possible to use hollow core PCFs as thin-walled capillaries if only
the core is filled with liquid, while leaving the photonic crystal cladding filled with
air [36]. Particles can be propelled along the core by light which is guided due to
total internal reflection (TIR). Unfortunately the number of modes which can be
5
excited for the given parameters is over 14.6 k which can be easily calculated from
the waveguide parameter [37]. This results in random transverse intensity patterns
that are difficult to control and in general axially varying, making quantitative
predictions on an investigated particle impossible.
This thesis describes experiments in which microparticles are guided along the
hollow core of a liquid-filled, single mode photonic crystal fiber. A theory is de-
veloped and compared to the experimentally observed propagation, allowing for
quantitative studies on the particle properties. In addition, it is shown that the
waveguiding properties of HC-PCFs can be used to interferometrically measure the
position and speed of particles as they propagate along the fiber.
6 Introduction
Chapter 2
Instrumentation
Controlled optical guidance of microparticles inside liquid-filled hollow-core PCFs
requires the combination of several techniques. Firstly, the PCF needs to be com-
pletely filled with a suitable liquid and the microfluidic flow inside its core has to be
controlled precisely. Secondly, high resolution optical imaging and optical tweezers
are crucial to select and launch a microparticle with desired geometrical properties,
from an ensemble. Thirdly the position of the particle along the fiber has to be
pinpointed over time in order to analyze the optofluidic dynamics of the system.
This chapter will discuss the techniques and setups that allow for highly controlled
particle guidance.
2.1 Microfluidic setup
The hollow core of a PCF inherently provides a microfluidic channel when filled
with a viscous medium. This medium requires to have low optical absorption at
the wavelength of the laser used (here 1064 nm for a Nd:YAG laser). Additionally
a comparable refractive index, viscosity and density to water are desireable, since
water hosts most of the biological phenomena and therefore is of high interest. D2O
is chosen since it has an absorption of only 0.04 dB/m [38] which is one order of
magnitude less than H2O. Since abrupt changes in refractive index along the fiber
cause strong scattering out of the core, the D2O has to fill the core and all cladding
holes uniformly over the entire fiber length. A single micron sized air-bubble in the
core already suffices to completely suppress light guidance.
8 Instrumentation
Therefore the liquid has to be prepared in two steps. Firstly it is evacuated and
agitated, using a commercial vortexer. This ensures that the concentrations of all
solved gases in the liquid are reduced to a minimum since, according to Henry’s law,
the concentration of a solved gas is directly proportional to its pressure. Omitting
this step results in gas bubbles in the capillaries of the fiber.
In a second step the degassed liquid is pushed through a 0.2�m filter disc. This
ensures that any remaining contaminations are smaller than the smallest fiber capil-
laries and can be rinsed through the fiber. Experimental experience shows that this
filtering is sufficient to maintain excellent mode propagation, although occasionally
small scattering points can be observed propagating with the microfluidic flow.
Once the liquid is prepared, it can be filled into a custom built pressure cell
(see figure 2.1). This cell is made from poly(chloroethanediyl) (PVC). All other
components are made from polyether ether ketone (PEEK), glass (PCF and the
window) or rubber rings for sealing. Several metal prototypes, including stainless
steel, have failed due to corrosion. Even perfectly filtered D2O developed impurities
in the cell which clogged up the PCF. Various glues have also been tested in order to
build PVC cells. These failed for the same reason wherefore a PVC cell was designed
that connects all parts mechanically, using rubber o-rings and threads, only. All
connectors and tubings are made from PEEK, a material with excellent mechanical
and chemical resistance properties. The geometrical design is chosen such that the
dead volume inside the cell is minimized to ∼ 50μl. This can further be improved
using a microfluidic approach [39], where the cell is replaced by a microfluidic cavity
and the tubing by microfluidic channels.
Once the cell is filled and rinsed with filtered liquid, a PCF is inserted and re-
cleaved by removing the window and sliding the fiber out of the cell. It is then
pulled back into the cell and locked by the PEEK connector. Once the window is
mounted again, the fiber can be filled by applying pressure to the cell. Typically
pressures of ∼10 bar are applied for the filling process. Lengths of up to 2m can
be filled overnight. Hereby it is crucial that no residual air is present in the cell,
since it would immediately dissolve in the liquid due to the high pressure and create
2.2 Optical setup 9
Figure 2.1: Pressure cell fabricated from PVC. A PCF can be inserted using commerciallyavailable PEEK connectors. Optical access is provided by a window, indicated by the redlaser light. The cell can be connected to PEEK tubing from the top in order to fill it witha desired liquid (here D2O) or to adjust the pressure inside the cell.
bubbles inside the fiber in regions of lower pressure. The second end of the fiber
is located on a stage that needs to be accessible with a pipette in order to apply
microparticles to it and has to meet optical requirements, which will be discussed
next.
2.2 Optical setup
Combining fiber coupling with the principle of laser tweezers inherently arises the
conflict between the optimum choice of optics for both. The numerical aperture (NA)
for a single beam optical tweezers trap is usually in the order of unity whereas the
optimum fiber coupling is achieved for values around 0.1. Therefore the incoupling
is built independently from the tweezers setup in two perpendicular arms, providing
optimum conditions for both and highly flexible imaging. Consequently the stage
holding the other end of the fiber must have two orthogonal transparent surfaces,
needs to provide liquid environment and has to be accessibility with a pipette. A
simple but very robust solution is depicted in figure 2.2. A thickness 0 (100�m x
20mm x 20mm) microscope slide is cleaved into a 4mm strip and glued to the
end of a thickness 1 (170�m x 25mm x 50mm) microscope slide, providing two
perpendicular optically flat surfaces. A liquid-filled and properly cleaved PCF is
10 Instrumentation
Figure 2.2: A: Particle launching stage built from a thickness 1 (170 �m x 25mm x 50mm)microscope slide as base. A custom built (100 �m x 4mm x 20mm) slide is glued to itsedge, forming a vertex which holds a liquid droplet containing microparticles. Light canbe coupled into a horizontal PCF from the side. A magnification of the liquid sample isshown in B. Optical tweezers can manipulate and investigate microparticles from below.
placed horizontally, making sure that the distance to the vertex is less than ∼400 �m.
This is necessary in order to not clip the 0.1NA coupling beam since the fiber core
is only ∼50 �m above the horizontal slide. In a last step a D2O droplet and a dilute
microparticle sol of desired properties are placed in the vertex, creating a fluidic
environment with microparticles around the fiber entrance.
2.2.1 Laser tweezers trap
Laser tweezers are a link between our macroscopic and the microscopic world and
allow precise optical analysis and micromanipulation of microparticles. Applying
this technique to an ensemble of microparticles requires a good mobility of the
optical tweezers in order to scan a large area, occasionally covered with particles.
Since the field of view given by a high NA objective is not more than several 100�m,
the entire tweezers setup needs to be on a movable stage in order to extend its range
2.2 Optical setup 11
as can be seen in figure 2.3. The most important advantage of a fiber coupled
Figure 2.3: Simplified schematic of the optical tweezers setup. It is driven by laser light ofdesired wavelength which is coupled in by a single mode fiber (SMF). Laser light passes acold mirror (CM) and is focused into the sample space using a Nikon CFI Plan 100 x waterdipping microscope lens. A long-range stage moves the entire setup vertically, providing20mm x 20mm scanning range. Vertical displacement of the focus is achieved with a x-y-zstage. The lens is also utilized to image the sample space. Visible light from the sampleis reflected off the CM and imaged by a CCD camera, connected to a computer.
setup is the possibility to translate the tweezers trap, in the x-y-direction over a
long range, without changing the alignment. Moving the microscope lens in the z-
direction also does not change the lateral position of the trap focus, once the beam
is perfectly collimated and coaligned with the microscope lens axis. Furthermore,
the use of a single mode fiber (SMF) allows for high flexibility in the choice of laser
wavelength. A Nikon CFI Plan 100 x water dipping microscope objective provides a
high numerical aperture of 1.1, is optimized for visible and IR transmission and has
an extremely long working distance of 2.5mm. Additionally the water immersion
technique minimizes spherical aberrations as the focus is translated further into the
liquid. The range of the stage and the working distance of the microscope lens allow
to sample a macroscopic volume of ∼2.3mm x 20mm x 20mm. Visible light from
the sample passes the microscope lens and is reflected off a cold mirror (CM) which
12 Instrumentation
transmits IR radiation but reflects light with shorter wavelengths. Residual IR light
is filtered from the visible light, which passes a variable zoom lens and is imaged
onto a CCD chip which is connected to a computer. This allows online monitoring
of the processes taking place in the sample volume. The image in figure 2.3 shows
Duke Scientific 9005 borosilicate microparticles in H2O which have a mean diameter
of 5 �m. The optical tweezers were used to pick up single borosilicate spheres and
arrange them spelling MPI, by simply dropping each particle in the right position.
2.2.2 Optical guidance path
The incoupling setup necessary to efficiently launch light into liquid-filled PCFs
undergoes continuous evolution, depending on the current research. Figure 2.4 shows
Figure 2.4: Simplified schematic of the optical guidance path. The user can choosebetween a Nd:YAG and a tunable Ti:Sapph source using a flip mirror (FM). A λ/2 plateand a polarizing beam splitter (PBS) are used to couple a variable fraction of the lightto the tweezers setup. The transmitted light is split by another PBS into 2 arms whichare coupled into both fiber ends. A computer-controlled electro-optic modulator (EOM)can change the power ratio between both arms within 10 ns. The mode profile in bothdirections of the fiber can be monitored using BSs and CCD cameras. A reservoir isconnected to the pressure cell (PC) in order to precisely control the flow inside the PCFby changing the pressure head (PH)
2.3 Doppler velocimetry setup 13
it early 2011 where bidirectional mode-launching is possible. The setup can be driven
either by a Nd:YAG laser or a tunable Ti:Sapph laser. Both are coaligned and can
be selected using a flip mirror (FM). A λ/2 plate rotates the polarization direction of
the beam which is then split in a polarizing beam splitter (PBS). Depending on the
orientation of the λ/2 plate a variable fraction can be coupled to the tweezers setup.
The transmitted light passes an electro-optical modulator (Linos LM 0202 Pockels
cell) and another PBS. The computer controlled EOM can change the polarization
of the beam and thus the splitting ratio within 10 ns. Both arms are coupled into a
liquid-filled PCF using 4 x 0.1NA lenses, whose NAs match that of the fiber. Beam
splitters (BSs) and CCD cameras are used to image the mode profiles on both sides
of the fiber. The BSs can be replaced by wedge prisms in order to have better
transmission for the incoupled light and less reflected light from the fiber in order
to not saturate the CCD cameras.
2.3 Doppler velocimetry setup
All quantitative particle guidance experiments require information of particle po-
sition over time. The quality of this information is determined by the spatial and
temporal resolution of the measurement technique. The temporal resolution δt is
given by the time between two subsequent position measurements. It can be trans-
lated into a spatial resolution δxt, given by the displacement of the particle within
that time, propagating at speed Vp
δxt = δt · Vp. (2.1)
For experiments in liquid environment the particle speed usually does not exceed
several mm/s. For a camera with a maximum sampling frequency of 100Hz and
1mm/s particle speed this yields δxt = 10 �m. The spatial resolution of a camera
system δxcam with given field of view (FOV) and number of pixels in one dimension
np is given by:
δxcam =FOV
np. (2.2)
14 Instrumentation
A typical value for 2.5mm FOV and 1024 pixels is δxcam = 2.5 �m. However, the
ultimate limit for optical imaging with a light source of wavelength λ in a medium
with refractive index n is the Rayleigh limit δxRayleigh
δxRayleigh =λ
2 · n =400 nm
2 · 1.33 = 150 nm. (2.3)
For this limit the FOV reduces to ∼100 �m and only short range measurements are
possible. In order to avoid this trade off, a Doppler velocimeter was set up that can
determine the particle position along the fiber with 200 nm accuracy over distances
of meters. Figure 2.5 explains the working principle. As the particle propagates
along the fiber axis, it scatters back light which is Doppler shifted to ωD. This light
couples to a low-loss mode of the PCF and is guided back to the core entrance where
it is mixed with light of original frequency ω0, reflected at the core entrance. The
resulting intensity beating is picked up by a photo diode (PD) and processed. The
Figure 2.5: Schematic of the Doppler velocimetry setup; BS beam splitter; PD photo-diode. A borosilicate microsphere is propelled along the D2O-filled fiber at speed Vp byoptical forces. Backscattered light has a Doppler-shifted frequency νD and is mixed withunshifted light of frequency ν0 at the core entrance.
Doppler-shifted frequency νD is given by:
νD = ν0
(1− 2 · Vp · n
c
). (2.4)
2.3 Doppler velocimetry setup 15
Here c is the speed of light in vacuum. Using the prosthaphaeresis formulas we sum
up the electric field components of the shifted and unshifted wave:
E0 · [cos(2πν0t) + cos(2πνDt)] = 2E0 · cos(π (ν0 − νD) t) · cos(π (ν0 + νD) t) (2.5)
≈ 2E0 · cos(π2 · Vp · n · ν0c
t) · cos(2πν0t) (2.6)
= 2E0 · cos(2πVp · nλ0
t) · cos(2πν0t). (2.7)
In equation 2.7 the beat frequency of the electric field can be identified with Vp·nλ0
.
The beat frequency of the light intensity νB, or the squared electric field, is then
given by 2·Vp·nλ0
and is proportional to the particle speed. It can therefore be used to
directly measure the particle speed, given the known scaling factor 2nλ0.
Another perspective on this is a Fabry-Perot approach. The optical path differ-
ence δxopt for light reflected at the core entrance compared to light reflected at the
particle position xp determines the reflectivity of the system, where
δxopt = 2n · xp. (2.8)
Maximum reflectivity is achieved if both beams interfere in phase or if δxopt = m · λ0,
where m is an integer number. The distance the particle has to travel between two
beats (m=1), LB is therefore given by:
LB =λ0
2n. (2.9)
The beat frequency νB is given by the inverse time T it takes for the particle to
travel the distance δxfringe is identical to equation 2.7:
νB = T−1 =Vp
LB
=2 ·Vp · n
λ0
. (2.10)
The beat frequency can be used to determine the particle speed and position with
a high accuracy, without the need of imaging it (see more details in chapter 6).
16 Instrumentation
Chapter 3
Properties of liquid-filledhollow-core photonic crystal fibers
The guidance of light along conventional optical fibers is most commonly accom-
plished by making use of a high refractive index core embedded in a low refractive
index cladding. Light is kept in the high index region as it propagates along the fiber
due to total internal reflection (TIR). However, guidance in a hollow core would be
highly desirable since it has numerous advantages, amongst many high thresholds
for nonlinear effects [40, 41] allowing the guidance of light at high powers without
the occurence of nonlinear effects, polarization maintenance and the possibility to
load the hollow core with gases, liquids and even particles. For example a hollow
capillary can be used to guide atoms [30] or micron sized water droplets and solid
particles [31] by means of optical forces. Unfortunately the mode in a capillary is
strongly attenuated. The loss calculated for the fundamental mode [32] at 1064 nm
in a hollow glass capillary with a refractive index of 1.45 and changing core radius is
displayed in figure 3.1. It lies in the order of several thousand dB/m for typical radii
of about 10�m. 2D photonic band gaps would offer an elegant solution to prevent
the light from escaping a hollow core. Unfortunately, these can only occur for a
minimum refractive index contrast of 2.66 between the different media in hexagonal
lattices [42]. For square lattices this value is even larger. Therefore human made
2D photonic crystals are usually fabricated from crystalline materials such as Si or
GaAs [43, 44, 45, 46, 47, 48] and require the use of lithographic methods [49, 50]
which are not suitable for the extreme aspect ratios in optical fibers. However, in
18 Properties of liquid-filled hollow-core photonic crystal fibers
Figure 3.1: Optical loss for the fundamental mode at 1064 nm in a glass capillary with arefractive index of 1.45 over core radius, calculated from [32].
1995 Birks et al. [51] have shown that this value can be dramatically reduced if
the light is propagating transversely to the photonic crystal pattern. The field of
photonic crystal fibers was discovered.
3.1 Guiding mechanisms
Brechet et al. [52] demonstrated that even a photonic crystal fiber made from Ge-
doped silica (n=1.457) and pure silica (n=1.450) is sufficient for band gap guidance.
Out-of-plane band gaps can occur for these extremely small index contrasts because
of a large transverse wavevector contrast rather than a large absolute index contrast.
This can be explained in the wavevector diagram of a PCF (see figure 3.2 where n1
and n2 are the refractive indices of the high- and low-index materials, respectively).
The contrast in transverse wavevector kt1/kt2 can be arbitrarily large by choosing the
longitudinal wavevector β slightly smaller than the total wavevector in the optically
thin medium kn2, even for small index contrasts n1/n2 [53]:
kt1kt2
=
√k2n2
1 − β2
k2n22 − β2
. (3.1)
Full 2D photonic band gaps in the photonic crystal of a honeycomb structure
made of air holes in fused silica with 45% air-filling fraction were predicted by
Birks et al. [51], allowing for guidance of light in a hollow-core. Figure 3.3 shows
3.1 Guiding mechanisms 19
Figure 3.2: Wavevector diagram (in longitudinal kl and transverse kt direction) for aPCF made from a high-index material (n1) and a low-index material (n2). The transversewavevector contrast kt1/kt2 can be arbitrarily large as the longitudinal wave componentβ approaches kn2.
the propagation diagram for such a structure, normalized to the hole spacing Λ.
Light can propagate along a defined longitudinal direction in a medium with given
refractive index nm, as long as the longitudinal wavevector β is smaller than the
total wavevector in this medium, given by the product of the free space wavevector
and the refractive index of the medium k · nm. In figure 3.3 the white region above
the light line, given by β = k · 1, indicates where light is free to propagate in all
media. Below the light line, however, light cannot propagate in air since β > k · 1.Therefore light with β- and k-values in the yellow region can only propagate in silica
with refractive index nSi and the photonic crystal (PC) with refractive index nPC.
The green region indicates where light is no longer able to propagate in the PC and
is purely restricted to silica. In the cut-off region, light is evanescent in all directions.
20 Properties of liquid-filled hollow-core photonic crystal fibers
Figure 3.3: Propagation diagram for a hexagonal PCF made from fused silica with 45%air-filling fraction, normalized to the hole spacing Λ. In the white region light is free topropagate in all media. In the yellow region, guidance is only possible in the PC and inthe silica. Green indicates where the light can only propagate in silica. The wave is purelyevanescent in the grey cut-off area. Full 2D band gaps are indicated by the red fingers,light cannot propagate in the PC. The label TIR shows a point where guidance in a silicacore due to total internal reflection is possible. BG indicates a region where light can beguided due to a full photonic band gap [54].
The different regimes are shown schematically in figure 3.4 for a solid core and a
hollow-core PCF. In yellow regions, the entire fiber acts like a waveguide due to its
higher refractive index compared to air. The guidance mechanism is total internal
reflection. In green regions light cannot be guided in the PC and in air. In a solid
core PCF, light can be guided in the silica core due to total internal reflection (see
figure 3.3 labelled TIR) since the average refractive index of the PC is too small for
light to enter it. For a hollow-core PCF however, it is not possible to guide light
since the refractive index in the core is even smaller. The only way to guide light
there, is to take advantage of the band gap regions in figure 3.3 (see label BG) where
light can propagate in air and silica, but cannot enter the PC due to the photonic
band gap.
3.2 Scaling law for liquid-filled HCBGF 21
Figure 3.4: Schematic of light guidance corresponding to figure 3.3 for a solid core (top)and a hollow-core (bottom) PCF. Yellow: light can propagate in the entire fiber, but isevanescent in air; green: light can only propagate in solid glass, solid core PCFs can guidelight due to total internal reflection; red: light is evanescent in the PC region due to theband gap, both fibers can guide light in the core since propagation in glass and air ispossible.
3.2 Scaling law for liquid-filled HCBGF
For the trapping and guidance of particles in liquid, it is necessary to use a fiber
that is capable to guide light in a liquid-filled hollow core. An intuitive approach
where only the core of a hollow-core photonic crystal fiber is filled with the liquid
of higher refractive index was demonstrated by Mandal et al. [36]. The photonic
crystal cladding was sealed by laser fusion, preventing the liquid from entering the
PC structure. However, the resulting waveguide operates only due to total internal
reflection, like a thin-walled capillary in air, filled with liquid, resulting in a highly
multimodal waveguide. The number of supported modes can be calculated to be
larger than 14.6 k. Figure 3.5A shows the principle, where only the core of a fiber
fabricated at the Max-Planck Institute for the Science of Light, is filled with water,
resulting in 2500 possible modes. Therefore this approach can only be used to give
qualitative results, since the mode profile is strongly varying along the propagation
length. A better approach is to use a hollow-core photonic band gap fiber HC-BGF,
tailored especially to guide light of a given wavelength when the entire fiber structure
is immersed in liquid (see figure 3.5B). Such a waveguide guides light in a single
fundamental mode along the entire length of the fiber, when properly designed. In
22 Properties of liquid-filled hollow-core photonic crystal fibers
Figure 3.5: A: Guidance due to total internal reflection when only the core is filled withliquid. The number of modes can be calculated to be about 2500 at 1064 nm for the givenfiber and water in the core. B: The entire structure is filled, allowing for single modeguidance due to a band gap.
order to fabricate a fiber with the proper design, it is crucial to precisely understand
how the guidance properties of a photonic crystal fiber change, once all holes are
filled with a liquid of given refractive index. Therefore the principles of band gap
scaling will be discussed in the following.
In a z-invariant structure of given transverse refractive index distribution n(r),
the modal magnetic field distribution h(r) has to satisfy the vector wave equation(∇2t + k2n(r)2 − β2
)h(r) = (∇t × h(r))× (∇t ln n(r)2
), (3.2)
where ∇t is the transverse Laplacian operator [55]. The global length scale in
this equation is defined by the periodicity of the PC structure Λ. All solutions
to this wave vector equation can be scaled as long as the global length scale Λ is
scaled analogously. This is due to the fact that equation 3.2 lacks any absolute
length scale prior to including the defined periodicity Λ in the refractive index n(r).
Equation 3.2 can be further simplified if one assumes only a small refractive index
contrast, making the transverse refractive index gradient negligible and thus null
out the RHS. This weakly guiding scalar approximation yields:(∇2t + k2n(r)2 − β2
)h(r) = 0. (3.3)
Re-writing in normalized Cartesian coordinates X = x/Λ and Y = y/Λ and assuming
a step-index profile f(X,Y), (f(X,Y) = 1 ∀ n ∈ n1, 0 else) the equation further
3.2 Scaling law for liquid-filled HCBGF 23
generalizes to
(∇2t + k2(n2
1 − n22)f(X,Y)Λ
2 − (β2 − k2n22)Λ
2)Ψ(X,Y) = 0, (3.4)
where ∇2t = ∂2/∂X2 + ∂2/∂Y2 and Ψ(X,Y) is an eigenfunction of the problem in
normalized coordinates. Introducing the frequency parameter v2 and eigenvalue w2
v2 = Λ2k2(n21 − n2
2
)(3.5)
w2 = Λ2(β2 − k2n2
2
), (3.6)
brings the vector wave equation into a compact and normalized form:
∇2tΨ+
(v2f − w2
)Ψ = 0. (3.7)
If the parameters k, Λ, n1 and n2 are varied, all photonic states scale such, that
the frequency parameter v2 and the eigenvalue w2 remain constant. A scaling law
for the filling medium of the holes with refractive index nm can be derived from
equation 3.5. For a fixed geometry Λ and high refractive index n1, but changing low
refractive index n2, the original wavelength λ0 is shifted to a new value λ in order
to conserve v2, and thus the photonic state:
v2 = Λ2(2π/λ0)2(n21 − n2
2
)= Λ2(2π/λ)2
(n21 − n2
m
)(3.8)
Equation 3.8 simplifies, using the initial refractive index contrast N0 = n1/n2 and
introducing the new refractive index contrast N = n1/nm:
λ = λ0
√1− N−2
1− N−20
(3.9)
This scaling law is derived assuming a small refractive index contrast. However,
comparison to experimental results shows that it can still provide good qualitative
predictions how a bandgap changes in a fiber, once its air holes are filled with a
liquid of given refractive index which will be shown next.
For the particle guidance experiments, a fiber is especially designed to guide
1064 nm light in a single, fundamental mode when filled with D2O, obeying the
24 Properties of liquid-filled hollow-core photonic crystal fibers
scaling law from equation 3.9. The transmission loss of the filled fiber is measured
using a cut-back technique where the transmission of a fiber is measured before and
after reducing its length. Figure 3.6A shows the transmission spectra of a 110 cm
piece and the same piece cut back to a length of 34 cm. The band gap region where a
single mode can be guided with low loss and is robust against fiber bends is indicated
in grey, agreeing with the theoretically expected wavelength range. The loss of the
Figure 3.6: A: Transmission measured for 110 cm and for 34 cm fiber length. The band gapregion of the fiber is marked in grey. B: Fiber loss determined from A. The comparison tothe absorption spectrum of bulk D2O indicates that the total loss is governed by absorptionin D2O.
fiber is calculated from the transmission spectra and is found to be αdB = 0.05 dB/cm
at 1064 nm (see figure 3.6B). This value is extremely low compared to the 19.6 dB/cm
for a D2O-filled glass capillary, calculated [32] for the same parameters. It is mainly
due to the absorption of D2O which lies at 0.04 dB/cm for the wavelength used and
is still one magnitude lower than for H2O.
Using a 4 x 0.1NA objective lens whose numerical aperture is matched to the
3.3 Bend loss 25
fiber, coupling efficiencies of ∼89% are achieved. The optical power along the fiber
core can be determined by measuring the power Pout exiting the fiber and calculating
back to the desired position zp, using the measured loss:
Popt = Pout · 10(Lfiber−zp)·αdB/10, (3.10)
where Lfiber is the length of the fiber. The mode profile is examined, proving
fundamental-mode band gap guidance. Figure 3.7 shows a measured near field image
of the mode exiting the fiber.
Figure 3.7: Measured mode profile overlaid with a SEM image of the fiber core. Bessel J20fits (red curves) are performed along the two yellow axes, indicating excellent agreementwith theory.
The cross-sectional intensity distributions along two axes are in excellent agree-
ment with a Bessel J20 profile which is zero at the core boundary, as expected from
waveguide theory for the fundamental mode [32].
3.3 Bend loss
One property of hollow-core photonic band gap fibers is their small bend loss
[56, 57, 58], making them extremely interesting for applications such as optical fiber
26 Properties of liquid-filled hollow-core photonic crystal fibers
gyroscopes [59] where the Sagnac effect is being used in order to measure rotations
precisely. This property is also highly desirable for the guidance of microparticles
along flexible paths, allowing their delivery into places difficult to access or to in-
vestigate inertia-effects for fast propagating particles in evacuated fibers.
3.3.1 Theoretical description of bend loss mechanisms
The mechanism of bend loss was explained by Birks et al. [60] and can be understood
by looking at the effective index (neff = βk0, where k0 is the vacuum wave vector)
profile of a guided mode with mode index nmode. The effective index profile for a
simple step index fiber is depicted in figure 3.8A. nmode is above the cladding index
nclad, but below the core index ncore, allowing it to propagate in the core. Figure 3.8B
Figure 3.8: Schematic plots of effective index neff against displacement r from the fiberaxis along the radius of curvature for a step-index fiber. A: Light is guided in a straightpiece due to TIR since the mode index nmode is between the core index ncore and thecladding index nclad. B: The fiber is bent, causing the index profile to skew. Light cantunnel into the cladding as nmode ≤ neff in the cladding region (yellow arrow) and escapecentrifugally.
3.3 Bend loss 27
shows the profile for the same but bent fiber. A bend in the fiber skews the index
profile with a slope that is inversely proportional to the bend radius. The cladding
index is increased on the outside of the bend, causing the mode to leak centrifugally
to the cladding material via a radiation caustic, as indicated by the yellow arrow.
A smaller bend radius as well as a mode index nmode closer to nclad bring the caustic
closer to the core, causing the mode to leak more efficiently.
For a hollow-core BGF the schematic is inverted, as the low-index material is
located in the core (see figure 3.9) and guidance takes place due to a band gap [60].
The effective refractive index of a guided mode in a straight piece (see figure 3.9A)
has to lie within a bandgap. Light is guided in the core since no photonic states
Figure 3.9: Schematic plots of effective index neff against displacement r from the fiberaxis along the radius of curvature for a PCF. A: Light cannot escape from the core modesince no photonic states are available in the photonic crystal cladding. Two modes areconsidered, one with mode index nA close to the upper band and one with mode indexnB close to the lower band. B: The scheme is skewed for a bent PCF. Light can leakas the mode indices exit the band gap. A mode close to the upper band edge will leakstronger towards the bend, whereas a mode close to the lower band edge will rather leakcentrifugally which was confirmed experimentally by Birks et al. [60].
28 Properties of liquid-filled hollow-core photonic crystal fibers
in the photonic crystal structure can be occupied. Two cases will be considered,
one where the mode index is close to the upper band edge (nA) and one where it is
close to the lower band edge (nB). As the structure is curved, the diagram becomes
skewed (see figure 3.9B) analogous to the step-index fiber. Light can leak into the
cladding where the mode index exits the band gap. The closer this point is to
the core, the higher the probability is for the light to tunnel out. Therefore, more
light will be scattered centrifugally for the mode with index nB. The behavior of a
mode close to the upper band gap (nA) is rather counterintuitive, but was indeed
confirmed experimentally by Birks et al. [60]. Only little light exits centrifugally,
most of it exits the fiber towards the center of the bend.
3.3.2 Experimental bend loss characterization
The manifestation of the bend loss can be seen in figure 3.10 where the same previ-
ously used D2O-filled fiber is bent and fixed to a stage using transparent adhesive
tape. Broadband supercontinuum light, incoming from the bottom right is guided
Figure 3.10: Light from a broadband supercontinuum source is launched into a D2O-filled hollow-core PCF. Transparent adhesive tape is used to fix a bent piece of fiber. Theregion of smallest bend radius rb ≈ 3.75mm is indicated. The same piece is imaged usingan infrared CCD camera without filter, using a 900 nm bandpass filter and a 1000 nmlongpass filter, indicating a higher bend loss for 900 nm light, compared to light above1000 nm wavelength.
along the curved fiber. Bright scattering indicates losses due to the bend. The
images are recorded detecting the entire spectrum, only light around 900 nm and
3.3 Bend loss 29
using a long pass filter transmitting wavelengths larger than 1000 nm. The images
show that light around 900 nm is more sensitive to bends compared to light above
1000 nm. It escapes the fiber even before reaching the region of smallest bend ra-
dius rb ≈ 3.75mm. Wavelengths longer than 1000 nm are coupled to the leaky modes
later. This indicates that the bend loss for the given D2O-filled fiber is larger for
light around 900 nm, compared to light of wavelengths longer than 1000 nm.
In order to quantify the given bend loss, a series of experiments is performed
where a u-turn in the fiber is created by winding it around cylinders of different
radius. For each radius and changing wavelength the mode profile is measured and
compared to a straight piece. Hereby the incoupling is kept unchanged in order to
maintain identical conditions. Figure 3.11 shows the resulting mode intensity pro-
files. The intensity profiles for a straight piece exhibit a Bessel J20 shape, indicating
Figure 3.11: Mode intensity profiles for different wavelengths, given a straight piece andu-turn bends of 4mm and 1.5mm radius. The loss in the core depends on wavelength andbend radius. For long wavelengths light is coupled to the 6 core surrounds as the fiber isbent.
single fundamental mode guidance over a broad wavelength range. As the fiber is
bent to 4mm radius, a shorter range of wavelengths is guided, indicating that the
band gap narrows. The loss dramatically increases for wavelengths around 900 nm.
30 Properties of liquid-filled hollow-core photonic crystal fibers
At 1064 nm and 1100 nm, light is coupled to modes in the 6 core surrounds. This
can be seen better in figure 3.12 where the radial intensity distribution functions are
plotted. For 1064 nm and 1100 nm the light intensity in the cladding region for radii
Figure 3.12: Mode intensity radial distribution function found in the profiles in figure 3.11.For 1064 nm Bessel J20 fits were performed, indicating excellent fundamental single modeguidance. For a bend radius of 1.5mm, close to the critical damaging radius, opticalguidance is inhibited for all wavelengths. The inlays show zoom-ins of the indicatedcladding region.
larger than the core radius increases upon bending (shown in the zoom inlays, where
the red curve has the highest intensity). This is a clear indication that the bend
induces coupling between the fundamental core mode and lossy cladding modes. It
also shows that cladding states are available close to the long wavelength edge of
the band gap, whereas no states can be found in the vicinity of the short wavelength
edge of the band gap. Bending the fiber down to a radius of 1.5mm, which is very
close to the critical curvature where the fiber snaps, leads to a breakdown in core
3.3 Bend loss 31
transmission. This is due to the fact that the band diagram (see figure 3.9) is skewed
so strongly that the band gap disappears, even for a mode in the center of it.
Therefore the bend loss for 4mm bend radius is investigated in figure 3.13. The
Figure 3.13: Bend loss in the core region per loop for 4mm bend radius.
large loss for 900 nm and 1100 nm is due to the band gap narrowing, discussed in
chapter 3.3 and can be understood by regarding the band diagram in figure 3.9. For
a straight piece, the diagram is horizontal and modes in the band gap cannot enter
the photonic crystal cladding. As the diagram is skewed due to bending, modes
closer to the band edges can tunnel more efficiently to the cladding. Modes in the
center of the gap need to tunnel over a long distance in order to reach cladding
states. For this reason their tunneling probability is low and they will be guided in
the hollow core. As modes closer to the band edges are lost, the band gap narrows.
This band gap narrowing confirms the results of Birks et al. [60] for solid-core PCFs.
The bend loss for the used laser wavelength of 1064 nm is only 1.4 dB/loop for
a bend radius of 4mm. Furthermore, the guided mode is not distorted by the bend
and exhibits a Bessel J20 shape, as expected for the fundamental mode. This allows
to guide light and particles robustly around sharp bends, as will be shown later.
32 Properties of liquid-filled hollow-core photonic crystal fibers
Chapter 4
Theory
The liquid-filled core of a PCF creates a unique environment for single micron-sized
particles, combining optical forces, microfluidic forces, surface effects and gravity
(see figure 4.1). A precise analysis of all these effects is necessary in order to un-
derstand the complex dynamics taking place. The optical force can be divided into
scattering force (mainly responsible for axial force) and gradient force (holding the
particle on the fiber axis). The microfluidic force strongly deviates from Stokes’
drag formula due to the strong interaction between the core walls and the liquid. It
strongly depends on the particle size and radial position inside the core.
Figure 4.1: Scheme of forces acting on a particle inside the hollow core of a liquid-filledphotonic crystal fiber. A parabolic microfluidic flow speed profile with maximum speedVmax and a particle speed Vp are assumed. Light from a fundamental Bessel J20 modeimpinges on the particle and exerts optical forces, holding it on the fiber axis and pushingit along the core. The direction of the gravitational force depends on the orientation ofthe fiber.
34 Theory
4.1 Optical force
In order to determine the optical forces on a spherical particle in a waveguide, a ray
optics model similar to Ashkin’s model [8] is evaluated, where a bunch of parallel
rays is incident on a sphere with radius r. Although parallel rays are assumed
and the size of the used microparticles is only ∼ 2 - 20 times larger than the laser-
wavelength used, the ray optics model still very well explains the measured results
quantitatively. It determines the momentum transfer of each ray onto the particle
and then integrates over all rays, accounting for the intensity profile of the optical
mode.
4.1.1 Ray optics model
The theory assumes light rays parallel to the fiber axis which are refracted and
reflected at the interface between sphere and surrounding medium (see figure 4.2).
Due to the symmetry of the sphere, the path for each ray remains in a plane that
Figure 4.2: Model for a ray of light incident on a spherical particle. The angle of incidenceα only depends on the distance of the incoming ray to the particle center d and the particleradius r. The angle of refraction β is given by Snell’s laws.
includes the incoming ray trajectory and the sphere center. The angles α and β
only depend on the distance of the incident ray trajectory to the sphere center d,
the particle radius r and on Snell’s law:
4.1 Optical force 35
α = arcsin
[d
r
]and β = arcsin
[d · nm
r · ns
]. (4.1)
Here nm and ns are the refractive indices of the surrounding medium and the
sphere. The reflection coefficients Rs and Rp are given by Fresnel’s laws. For s-
polarized light (electric field oscillates perpendicular to the image plane) and p-
polarized light (electric field in plane) one obtains:
Rs =
[nmcosα− nscosβ
nmcosα + nscosβ
]2and Rp =
[nmcosβ − nscosα
nmcosβ + nscosα
]2. (4.2)
Figure 4.3: Reflection coefficients for s- and p-polarized light for a light ray hitting asphere at given displacement from its center. The p-polarized light is perfectly refractedinto the sphere medium when it impinges at Brewster’s angle (49.6◦; d/r=0.76 for nm=1.33and ns=1.56).
The first ray (see figure 4.2 I) that escapes from the sphere is simply reflected at
the surface and induces a relative momentum transfer along the fiber axis:
ΔpzI/p0 = Rs/p · nm · [1 + cos (2α)]. (4.3)
The total momentum of the incident ray in vacuum is p0 = h/λ, where h is Planck’s
constant and λ is the wavelength of the light in vacuum. For every following N-th
escaping ray the momentum transfer along the fiber axis is given by:
ΔpzN/p0 = RN−2s/p · (1− Rs/p
)2 · nm ·[1 + (−1)N−1 · cos (2α− 2β · (N− 1))
]. (4.4)
The radial relative momentum transfer for the first escaping ray is given by:
ΔprI/p0 = −Rs/p · nm · sin (2α). (4.5)
36 Theory
For the N-th escaping ray one finds:
ΔprN/p0 = RN−2s/p · (1− Rs/p
)2 · nm · (−1)N sin (2α− 2β · (N− 1)). (4.6)
Summing up all reflections one yields the following expressions for the relative mo-
mentum transfer in axial and radial direction, analogous to Ashkin et al. [8]:
Δpz/p0 = Rs/p · nm · [1 + cos (2α)]+
+∞∑
N=2
RN−2s/p
(1− Rs/p
)2nm
[1 + (−1)N−1 cos (2α− 2β (N− 1))
],
(4.7)
Δpr/p0 = −Rs/p · nm · sin (2α)+
+∞∑
N=2
RN−2s/p · (1− Rs/p
)2 · nm · (−1)N sin (2α− 2β · (N− 1)).(4.8)
Equations 4.7 and 4.8 are plotted in figure 4.4 for a borosilicate sphere (ns = 1.56)
in water (nm = 1.33) for s-, p-polarization and their average. One can see that
the radial component of the optical force is point symmetric with respect to the
sphere center and nulls out upon integration over d for intensity distributions that
are axially symmetric around the particle center. For asymmetric distributions,
however, the sphere will be pulled into regions of high light intensities, since the
radial force always (for |d/r| < 0.999) points away from the particle center. An
equilibrium position is found when the high intensity region hits the sphere center
and the net radial force vanishes. The axial force component always points in the
propagation direction of the incoming rays, causing the particle to move away from
the light source. Rays close to the rim of the sphere contribute most to the force,
where rays close to the center only transfer around 5% of their momentum. In the
latter case, the light is efficiently transmitted through the sphere due to the near
normal incidence on the sphere surface. Due to symmetry, the force always acts
on the geometrical center of the sphere which is usually also the center of mass.
Consequently, no torque can be applied, even for asymmetric intensity distribution.
This is due to the fact that each escaping ray and the incoming ray are symmetric
to an axis that contains the sphere center. Thus the optical force acts along this
4.1 Optical force 37
Figure 4.4: Relative axial (blue) and radial (purple) momentum transfer from a ray to asphere with ns = 1.56 (borosilicate) in water (nm = 1.33) with respect to the total incidentray momentum for s-, p-polarization and an average of both. Note, that the axial forceis always positive, thus pushing the sphere away from the light source. The radial force,however, always (for |d/r| < 0.999) points away from the sphere center. This explainswhy, for these parameters, the radial force will pull the sphere into regions of high lightintensity.
symmetry axis and no tangential net force can occur on the sphere surface. Since
the model is based on symmetry grounds, this is only valid for a perfectly spherical
object.
From equation 4.7 and 4.8 one can calculate the direction of the force on the
center of the sphere for each ray. The angle with respect to the optical axis ϕ and
the total relative momentum transfer are plotted in figure 4.5. Interestingly, the
radial optical force pushes the particle away from the ray axis when light impinges
very closely to the rim of the particle. This is due to the fact that rays hit the
38 Theory
Figure 4.5: A: Total relative momentum transfer, which is directly proportional to theoptical force. B: Angle of the force ϕ, acting on the sphere center, relative to the opticalaxis.
sphere surface under a very shallow angle, thus increasing the reflection coefficients
dramatically. The first ray which is simply reflected off the sphere surface at a small
angle, dominates over all other beams exiting the sphere and causes it to move away
from the ray. However, this effect is very small, as the total force vanishes for rays
close to the particle rim (|d/r| > 0.999).
Taking into account an arbitrary intensity distribution I(φ,d), where φ is the
azimuthal coordinate in the particle’s coordinate system, the total axial and radial
forces can be obtained by integration:
Fz =1
c
∫ r
0
∫ 2π
0
dI(φ, d) ·Δpz/p0 dφ dd (4.9)
and
Fr =1
c
∫ r
0
∫ 2π
0
dI(φ, d) ·Δpr/p0 dφ dd, (4.10)
where c is the speed of light in vacuum.
4.1.2 Modeled results
The formulae for axial and radial force can now be evaluated for all arbitrary in-
tensity distributions. Euser et al. have shown that many modes can be excited
4.1 Optical force 39
in PCFs [61]. However, the loss increases with mode order, thus the fundamental
mode will dominate as a mixture of modes propagates along a PCF. Therefore, the
following analysis will focus on three lowest order modes. These are the fundamental
LP01 mode which is radially symmetric and has a Bessel J20 intensity profile with
a maximum in the center. The modes with the next higher loss are the doughnut
shaped TE01, TM01 and EH21 mode. A superposition of TE01 or TM01 with the
EH21 mode yields again a linearly polarized mode. This axially symmetric LP11
mode has a two lobe shape and exhibits a loss, which is by definition identical to
the doughnut shaped modes. Depending on which transverse mode superposes with
the EH21 mode, the polarization is either parallel (TM01 + EH21) or perpendicular
(TE01 + EH21) to the line including the two maxima.
The electric and magnetic field distributions for cylindrical dielectric waveguides
by Marcatili and Schmeltzer [32] are used for the analysis. Figure 4.6 shows the
intensity distributions for a cylindrical waveguide with a refractive index of 1.33
and 17 �m diameter (indicated by the white dashed line).
The intensity distributions are calculated for 1W of optical power and 1064 nm
light, matching the parameters of the hollow-core PCF used for most experiments.
They are cross-correlated to the averaged momentum transfer matrix for a sphere
of 6.5 �m diameter and a refractive index of 1.56, as described in equations 4.9 and
4.10. These parameters are chosen, because mostly borosilicate spheres with similar
sizes were launched. The axial propulsion force and the radial trapping potential
which is calculated by integration over the radial force, are depicted in the middle
and bottom row of figure 4.6. The black dotted circle indicates where the particle
center is located when it touches the core wall. Positions beyond this circle cannot be
reached. The fundamental LP01 mode exhibits a maximum axial force of 192 pN/W
when it is located on the optical axis. While in this position, the trapping potential
is minimized to -0.67 fJ/W. The steep and deep potential indicates stiff and stable
trapping of the sphere on the fiber axis, while the propulsion force is maximized.
These properties in concert with the lowest possible loss make this mode the ideal
choice for particle guidance.
40 Theory
Figure 4.6: Top: Calculated mode intensity profiles for fundamental LP01 mode, doughnutshaped TE01 mode and two lobed LP11 mode normalized to 1W optical power. Middleand bottom row: Trapping potential and axial force on a 6.5 �m sphere (see indicatedleft in scale) of refractive index 1.56 in a 17 �m core (indicated by white dashed lines)of refractive index 1.33. The physically relevant positions where the particle does notintersect the core wall are indicated by the black dotted circles.
Since the doughnut shaped TE01 mode has an intensity minimum on the fiber
axis, and its maximum intensity is roughly two times smaller compared to the LP01
mode, its propulsive force only reaches a value of 105 pN/W on the optical axis.
The axial optical force is constant within 10% in the region where the particle can
be located without intersecting the core wall (black dotted circle). The trapping
potential is very flat in this region and exhibits a local maximum on the fiber axis,
indicating that a particle of given parameters will be weakly trapped in a ring around
the fiber axis. Collisions with the core wall are very likely. Since the axial force is
high and constant within 10%, and the trapping potential is flat, this mode would
be ideal to study small perturbations is radial force, possibly of fluidic origin.
4.1 Optical force 41
The LP11 mode exhibits the highest field intensity of the investigated modes
since the field distribution is very strongly condensed to its two lobes. Since it is a
superposition of two doughnut shaped modes with different polarization properties,
its maximum field intensity is exactly twice as large as the maximum value of the
TE01 mode. The axial force for this mode reaches a saddle point on the fiber
axis. Translating the particle towards an intensity maximum increases the axial
force as the particle interacts stronger with the mode. Moving it radially in the
perpendicular direction decreases the axial force, as the interaction decreases. The
potential exhibits two minima, indicating that the particle can be trapped in two
stable positions. However, the potential barrier before the particle crashes into the
core wall is small. Therefore, the particle is again likely to collide with the core wall,
as it propagates in the field of this mode.
Since these investigations are only valid for one particle size, another set of
calculations is performed. Here, all parameters are identical except that the particle
radius is varied (see figure 4.7). The axial force, trapping potential and the trap
stiffness are investigated for particles displaced along a line including all intensity
maxima, as indicated by the white dotted line in the small mode inlays.
The physically accessible regions before the particle crashes into the core wall
lie within the white dotted lines in each plot. Stable trapping positions where the
potential exhibits a local minimum are indicated by solid white lines.
For a LP01 mode the sphere is trapped on the fiber axis for all diameters. The
axial force increases with sphere diameter, since a larger part of the mode interacts
with the particle, and reaches a maximum of 283 pN/W at 10.7�m diameter. For
even larger sizes, the axial force decreases as the periphery of the particle, which most
efficiently transfers the photon momentum to the particle, interacts with regions of
low field intensity. The potential depth increases with particle size as a result of
the inreased radial optical force, since a larger part of the mode interacts with the
particle (also see later, figure 4.9). The trap stiffness shows an increasing trend for
the same reason. It is defined by the curvature of the potential (or the second radial
position derivative), has units of spring constant and indicates how strongly the
42 Theory
Figure 4.7: Axial force, trapping potential and trap stiffness (along the axis includingall intensity maxima; see small inlays) for a particle with a refractive index of 1.56 andchanging diameter in a medium with a refractive index of 1.33. The dotted lines indicatethe boundary at which the particle crashes into the core wall. Stable trapping positionswhere the potential exhibits local minima are indicated by the solid white lines.
radial force changes as the sphere is displaced radially by a given distance. A large
positive value together with a potential minimum (zero radial force, indicated by
white solid lines) is equivalent to a strongly trapped particle.
In the field of the TE01 mode, two stable trapping positions, where the potential
has a local minimum, are found for particles below 10.4 �m diameter. Due to the
rotational symmetry of the mode, particles will be trapped on a ring around the fiber
4.1 Optical force 43
axis but are free to move along the ring. Particles larger than 10.4 �m cannot resolve
the mode pattern and will be trapped on the fiber axis. At 12.3 �m diameter the
axial force reaches a maximum of (350 pN/W) which, somewhat counterintuitively,
is larger compared to the LP01 mode (283 pN/W). This is due to the fact that
for the doughnut shaped mode, regions of high intensity intersect with the rim of
the particle, which most efficiently transfers the photon momentum to the particle.
The light intensity is distributed more efficiently to match the photon momentum
transfer matrix of the particle.
The two lobed LP11 mode exhibits two stable trapping positions for sphere sizes
below 12.0 �m, where particles can be stably trapped in either of the two lobes.
Interestingly, particles between 9 and 12 �m will be pushed against the core wall,
as their stable trapping position lies beyond the white dotted line, indicating the
boundary where a particle collides with the core wall. Even larger spheres cannot
resolve the mode profile and will be trapped on the fiber axis. The bifurcation of
the stable trapping position happens very abruptly in the two lobed mode and is
more gentle in the doughnut shaped profile. The abrupt transition at slightly larger
particle diameter in the LP11 mode is due to the zero intensity on the mirror axis
between the two lobes. The symmetry of the field requires the intensity to vanish
on this axis. Therefore the system changes from the degenerate state to on-axis
trapping instantaneously. Strikingly similar to the doughnut shaped TE01 mode,
the axial force reaches a maximum of (350 pN/W) at 12.3 �m particle diameter.
This is due to the fact that the LP11 mode is a superposition of two doughnut
shaped modes.
In order to better compare the trapping properties of the analyzed modes, axial
force, trapping potential and trap stiffness for particles of different radii on the fiber
axis (figure 4.7, zero displacement) are plotted in figure 4.8.
For particle sizes below the bifurcation diameter of the TE01 mode (10.4 �m),
on-axis trapping is only possible for the fundamental LP01 mode. Only for larger
diameters, the axial force (figure 4.8A) in the TE01 mode (and above 12.0 �m also in
the LP11 mode) exceeds the values found for the fundamental mode. The identical
44 Theory
Figure 4.8: Axial optical force (A), radial trapping potential (B) and trap stiffness (C)for a spherical borosilicate particle of variable radius, located on the fiber axis. Thebifurcation diameters of TE01 and LP11 mode are indicated by vertical dotted lines.
axial force and trapping potential (figure 4.8B) confirm the similarity of TE01 and
LP11 mode for particles trapped on the fiber axis.
The trap stiffness (figure 4.8C) vanishes for both, the TE01 and LP11 mode at
the bifurcation diameter. Not only the first, but also the second radial position
derivative of the potential in figure 4.7 are nought, there. This is an indication for a
broad potential minimum since its curvature also vanishes. In this case particles are
free to move between the stable position on the fiber axis to the trapping positions
in either the two lobes or the ring. However, this is only valid for the doughnut
shaped TE01 mode since in the case of the two lobed LP11 mode, the particle cannot
access the off-axis trapping positions and will crash into the core wall.
4.1 Optical force 45
The strong advantage of the fundamental LP01 mode, however, lies in its trap
stiffness which is positive over all particle sizes. It turns out to be the best choice
for optical force experiments as it exhibits the largest axial force, for radii smaller
than the bifurcation points of TE01 and LP11 mode. In addition, particles of all
sizes will be trapped strongly in the center of the core since the trapping potential
has a minimum and the trap stiffness is maximized there. In order to get a better
quantitative view on the forces on a sphere in the fundamental LP01 mode with a
refractive index of 1.56 in a medium with refractive index 1.33, the axial and radial
force are calculated for 9 different sphere radii. The resulting plots are shown in
figure 4.9.
The physically accessible regions before the particle crashes into the core wall
are indicated by the solid lines, whereas the dashed lines indicate regions which the
particle could only access if no core wall were present. The radial force, which is
depicted in figure 4.9A, increases with particle diameter. This is due to the larger
area of the particle which intersects with the mode profile. Also the axial force
(see figure 4.9B) increases with particle size until a diameter of 10.7 �m since the
interacting area of the particle and the mode is increased. For larger sizes the rim
of the particle, where the photon momentum is transferred most efficiently to the
particle, is located in the low intensity regions of the mode. Thus the axial force is
decreased.
The influence of refractive index on optical propulsion force and on the trap stiff-
ness are examined by running calculations for 5 different particle refractive indices
and variable particle radius. The LP11 mode is used since it offers ideal trapping
and propulsion properties. Figure 4.10 shows the results for particles located on the
fiber axis and normalized to 1 W optical power.
Both the axial optical force in figure 4.10A and the trap stiffness in figure 4.10B
increase globally, as the particle diameter increases. This is due to the stronger
interaction of the optical mode and the particle. Rays are refracted at larger angles,
thus transferring more momentum to the sphere.
For particles small compared to the core size, the axial optical force for each
46 Theory
Figure 4.9: A: radial and B: axial force in the light field of a LP01 mode for a particlerefractive index of 1.56 and a medium index of 1.33. The calculations are performed forsphere diameters from 1�m to 16 �m. The physically accessible regions are indicated bythe solid lines.
refractive index increases quadratically with particle diameter. The intensity of the
optical mode can be approximated to be constant across the particle in this regime.
Thus the axial optical force is then only proportional to the interacting cross-section
of the particle, yielding a parabolic behavior with respect to its diameter.
As the highly efficient scattering regions on the rim of the particle move out of
the high intensity regions of the mode with further increasing size, the propulsion
force reaches a maximum and begins to decrease. This maximum is shifted to
slightly larger diameters with increasing particle refractive index. The origin for this
phenomenon lies in the reflection coefficients Rs and Rp. With increasing refractive
4.1 Optical force 47
Figure 4.10: A: axial optical force and B: trap stiffness versus radius for spherical particlesof different refractive index. The calculations were performed for the LP11 mode and 1 Woptical power and a waveguide medium of refractive index 1.33.
index contrast, both increase more strongly for regions in the center of the particle
compared to its rim. Thus the the momentum transfer efficiency in the central
regions grows relative to the rim, yielding a shift of the maximum possible optical
axial force to larger diameters.
The axial optical force does not vanish, but rather levels off in the large particle
limit. This is due to the fact that the high intensity region of the mode interact with
two nearly flat surfaces. For an infinitely large sphere, the axial force approaches
the force on a flat glass disc where the light is simply reflected at both interfaces.
48 Theory
Figure 4.10B shows the trap stiffness of particles on the fiber axis. In analogy to
to the axial optical force, the trap stiffness increases with refractive index. However,
the relative increase becomes smaller as the refractive index grows. This is an indi-
cation to the resulting increased reflectivity of the particle. The amount of reflected
light, which pushes the particle away from regions of high intensity, increases. Si-
multaneously, the amount of light which is refracted into the particle and pulls the
particle into regions of high field intensities is decreased. Both effects reduce the
trap stiffness, why the relative increase in trap stiffness is smaller for larger refractive
indices.
Particles below ∼3 �m diameter interact with a nearly uniform central part of
the intensity profile and therefore are only weakly trapped. For larger particle
diameters, the trap stiffness increases since the particle rim intersects regions of the
mode where the intensity gradient is larger. A displacement of the particle from
the optical axis results in a restoring force, as the intensity distribution across the
particle becomes non-uniform. The trap stiffness reaches a maximum when the
particle rim coincides with the mode region of highest intensity gradient. Again this
maximum shifts to slightly larger radii for growing refractive index, in analogy to
the axial force case. As the sphere size is increased further, the trap stiffness reduces
since the particle periphery moves to regions of smaller intensity gradient. In the
limit of a sphere much larger than the core, the trap stiffness would vanish. Again
this can be understood by comparing the sphere to a flat glass disc, which can be
translated freely in radial direction.
4.2 Fluidic force
The dynamics of fluidic systems are described exactly by the equations found by
Claude Louis Marie Henri Navier and George Gabriel Stokes in 1822. Although these
equations have been known for almost two centuries, only few geometrical problems
have been solved up to now. Analytical solutions of the Navier-Stokes equations
can only be found for a number of boundary conditions, therefore computational
techniques, involving finite element modeling, are required for most problems.
4.2 Fluidic force 49
4.2.1 Microfluidic theory
For incompressible Newtonian fluids the Navier-Stokes equations simplify to [62]:
ρ
(∂V
∂t+V · ∇V
)=−∇p + μ∇2V+ f
∇ ·V =0,
(4.11)
where ρ is the fluid density, V is the flow velocity, p is the pressure, μ is the fluid
viscosity and f is an external force per unit volume. The LHS of the first equation
includes the unsteady acceleration ∂V∂t
and the convective acceleration V · ∇V. The
first one is zero for a steady flow field, although molecules in the liquid can be
accelerated as their speed changes with position. It describes the change in flow
speed over time for a given position. The second term describes how the flow speed
changes over position, for example as the liquid enters a narrower channel and is
accelerated due to conservation of mass flow. The RHS of the first equation includes
force due to pressure gradient −∇p, viscosity μ∇2V and external forces f. The
second equation conserves the volume as the divergence of velocity, which is equal
to the divergence in mass flow, is zero.
A measure for ratio between inertia and viscous forces in hydrodynamic systems
is given by the Reynolds number Re. For a system of typical dimension L, it is given
by [63, 64]:
Re =ρVL
η(4.12)
The system behaves laminarly if Re is below ∼2100. For microfluidic systems in
water-like liquids (liquid density: ρ = 103 kgm3 , fluid speed: V = 10−3m
s, L = 10−3m,
η = 10−3 Pas) the Reynolds number does not exceed 1. This is why they can be
regarded as purely laminar.
For a spherical particle with radius r in a purely laminar flow environment with-
out any boundary conditions, Stokes derived an analytical formula that describes
the drag force Fdrag:
Fdrag = −6πηrVp, (4.13)
where η is the viscosity of the surrounding medium and Vp the particle speed.
50 Theory
Since the flow in microfluidic channels is purely laminar, equation 4.13 can be
used for the analysis of microparticles propagating in the liquid-filled core of a
PCF. However, the second requirement that the fluid has no boundary conditions
is strongly violated. The vicinity of the core walls to the particle can dramatically
increase the drag force on a particle. Quddus et al. [65] have numerically analyzed
the complex behavior of a spherical object in a liquid-filled cylinder and reviewed
the previous work [66, 67, 68, 69]. The general expression for the total drag force
on a spherical particle with radius r, located on the axis of a cylinder with radius R
is given by:
Fdrag = −6πηr (VpK1 − VmaxK2), (4.14)
where Vp is the particle speed and Vmax is the flow speed of the liquid in the center of
the cylinder, at a point far away from the particle (see figure 4.11). K1 corresponds
to the correction of a moving particle in a steady liquid and K2 to the correction
for a stationary particle in a flow. The dimensionless factors K1 and K2 have to
Figure 4.11: Drag force Fdrag acting on a sphere on the axis of a liquid-filled cylinder.Contributions are due to the movement of the sphere (Vp) and the flow inside the cylinder(Vmax).
be determined numerically and only depend on the ratio of particle to cylinder
radius Γ = r/R. Astonishingly, both correction factors are independent and the
total force on the particle can be simply calculated by a summation of both forces.
A logarithmic plot for K1 and K2 is shown in figure 4.12. The data of Quddus et al.
is fitted with an inverse polynomial fit in order to obtain an analytical expression,
yielding the following values:
4.2 Fluidic force 51
Figure 4.12: Correction factors K1 and K2 calculated by Quddus et al. K1 is the correctionfactor for a particle moving in a steady liquid, where K2 corrects the drag on a stationarysphere in a flow.
K1 =[1− 2.0711 · Γ− 0.37088 · Γ2+
+ 3.6478 · Γ3 − 2.8946 · Γ4 + 0.68845 · Γ5]−1(4.15)
K2 =[1− 2.0413 · Γ + 0.16226 · Γ2+
+ 2.2368 · Γ3 − 1.8409 · Γ4 + 0.48511 · Γ5]−1(4.16)
Both fits match the data simulated by Quddus et al. within 1% up to a Γ value of
0.9, which is beyond all measurements discussed in this thesis.
4.2.2 Microfluidic flow profile in the hollow core
The Reynolds number in a liquid-filled core of 17 �m diameter and maximum flow
speeds in the order of a few mm/s is far below 2100, where turbulences occur.
Therefore the flow is purely laminar and only depends on the size of the core, the
length of the fiber, the pressure difference between both fiber ends and the viscosity
of the fluid. The exact flow-analysis will be discused in the following. The flow
profile inside a cylinder with radius R is parabolic, reaches a maximum velocity
Vmax on the cylinder axis and is zero at the cylinder wall. Given a pressure gradient
dPdz
in the cylinder and viscosity η, Vmax can be calculated from the Hagen-Poiseuille
52 Theory
equation:
Vmax =dP
dz
R2
4η. (4.17)
For not perfectly cylindrical pipes, such as the fiber core, Vmax can be calculated
using the wetted perimeter Pwet. It is defined by the line-integral of the interface
between liquid and solid (here around the fiber core) [70]. The hydraulic diameter
DH for any arbitrarily shaped pipe with cross-sectional area A is given by [71]:
DH =4A
Pwet
=233μm2
53.5μm= 17.4μm. (4.18)
Thus the hydrodynamic flow of the PCF depicted in figure 4.13 is identical to that
in a cylinder with 17.4μm diameter.
Figure 4.13: Scanning electron micrograph of the fiber core. The dotted circle indicatesa cylinder of similar fluidic properties. A slight ellipticity of the core can be seen whencomparing the circle to the fiber structure at the top right and bottom left.
A pressure difference is applied between the two ends of the fiber by setting the
head PH of a D2O reservoir as shown in figure 2.4. The flow resistivity is completely
dominated by the fiber and all other components can be neglected due to their larger
diameter (flow scales with the 4th power of the diameter). Therefore the pressure
gradient along an empty fiber core is well approximated by
dP
dz=
g · PH · ρD2O
Lfiber, (4.19)
where ρD2O = 1.1056 · 103 kgm3 is the density of heavy water, g = 9.81 m
s2is the gravi-
tational field strength and Lfiber is the length of the fiber. Including this expression
4.2 Fluidic force 53
into equation 4.17 and using that ηD2O = 1.25 · 10−3 Pa s for 20◦C and R = DH/2
yields the following expression for Vmax:
Vmax =g · ρD2O · D2
H
16 · ηD2O
PH
Lfiber
= 164.2μm
s· PH
Lfiber
. (4.20)
The flow inside the core only depends on the ratio between the pressure head PH and
the fiber length Lfiber. The fiber intrinsic flow speed of 164.2 μms
can be associated
with the flow due to gravity in the core of a vertical fiber with open ends. Exactly
in this case the identity PH = Lfiber holds. It is trivial but worthwile mentioning,
that this speed does not depend on the total length of the fiber, as long as it
is placed vertically. Another very interesting aspect is the inverse proportionality
of Vmax ∼ η−1 with respect to the liquid viscosity. As the drag force exerted on a
particle due to a flow is proportional to the product of Vmax and η (see equation 4.14),
it is independent of the viscosity. No matter which medium fills the fiber core, the
drag force due to the flow will only depend on the pressure gradient along the fiber.
4.2.3 Particle effects on flow rate
Equation 4.20 only holds for an empty core and the effect of a particle obstructing
it has to be considered. Therefore the case of a stationary sphere with radius r in a
pressure driven flow is investigated using finite element simulations (COMSOL, CFD
module). A normalized pressure is applied to the open ends of a liquid-filled cylinder
of radius R (aspect ratio 5:1) which is obstructed by a sphere of given radius. No
slip boundary conditions are used by fixing the flow speed to nought at the position
of all interfaces. The pressure profile for R = 2 r is depicted in figure 4.14. One
can see that ∼ 50% of the pressure drop along the cylinder axis takes place in the
vicinity of the sphere. It is important that the length of the cylinder is sufficiently
long, so that the flow can relax to its unperturbed profile. The pressure gradient
changes to a purely axial, only about 2 r away from the sphere center, indicating
that the chosen length is sufficient. This is also confirmed by Quddus et al. [65].
In order to estimate how strongly the flow is inhibited by a particle, analogous
calculations are performed using different particle radii (0, 0.25R, 0.5R, 0.75R) and
54 Theory
Figure 4.14: Pressure profile in a cylinder, obstructed by a spherical particle.
a fixed pressure difference between both cylinder ends. The total flow through the
system with a particle is simulated and compared to the empty cylinder. Figure 4.15
shows the flow speed profiles simulated. The transverse flow speed profiles at the
position of the sphere center (A) and the cylinder output (B) are plotted on the left
side of figure 4.15.
Figure 4.15: Left: Flow speed profiles along the sphere center (A) and the cylinder output(B). Right: Simulated flow speed distributions for an empty cylinder and obstructedcylinders with different sphere radii.
4.2 Fluidic force 55
For r/R = 0.75 the flow speed on the fiber axis Vmax is dramatically reduced to
∼ 20% of the value in the free cylinder. However the aspect ratio of the cylinder
is ∼ 104 times smaller compared to that in the experiment. In order to estimate
the impact on the experimental results, the flow can be compared to that in a
longer empty cylinder length. Therefore an effective empty cylinder length Leff is
calculated at which the flow rate (and therefore Vmax) is identical to the obstructed
situation, given the same pressure difference between both ends. This calculation is
very simple since the pressure gradient, and thus Vmax, is inversely proportional to
the cylinder length. Therefore one yields
Leff = L0vmax(free)
vmax(obstructed), (4.21)
where L0 = 10R is the length of the original cylinder. The effective cylinder length
for the simulated examples is shown in figure 4.16. The additional length (Leff − L0,
Figure 4.16: Effective cylinder lengths Leff , having the same flow resistance as a cylinderwith length L0 with a sphere of given radius in its center. L0 = 10R is 87 �m for theinvestigated fiber.
above the dotted line) due to the largest simulated particle, placed in the center of
the cylinder, is ∼ 4 L0 = 40R. Given a 8.7 �m core radius, this means that the flow
in a fiber obstructed by a stationary particle with 13 �m diameter is identical to an
empty fiber which is 350�m longer. Therefore the impact of a stationary particle
56 Theory
on the flow has to be considered for very short fiber lengths of a few millimeters
and particles large compared to the core. All measurements discussed in this thesis
are performed for r < 0.75R, fiber lengths above 10 cm and the error margin in
determining the fiber length is beyond 350 �m. Therefore all effects of microparticles
on the flow rate will be neglected.
4.3 Optofluidic balancing
Knowing the optical and fluidic forces, one can predict how the balancing of both will
be influenced by parameters like refractive index and size of a given microparticle.
On the contrary this allows for real time measurements of refractive index and/or
the size of any kind of particle launched into the hollow core of a PCF. In analogy to
the correction factors K1 and K2 calculated by Quddus et al. two balancing regimes
can be identified. In the first case, corresponding to K1, no flow is present and the
particle moves with a speed Vp along the fiber, balancing optical and drag induced
fluidic force. We define the optical particle mobility μopt,1 by the speed at which a
particle propagates for 1W of optical power in a stationary fluid:
Vp = μopt,1 · 1W. (4.22)
The second case where the particle is stationary and a flow is applied, such that the
drag induced at the particle balances the optical propulsion force, can be identified
with K2. Analogously to μopt,1 we define the optical flow mobility μopt,2 by the flow
speed Vmax necessary to hold a particle stationary against the optical force at 1W
of optical power:
Vmax = μopt,2 · 1W. (4.23)
For a LP11 mode in a waveguide of refractive index 1.33, the optical particle
mobility μopt,1 and the optical flow mobility μopt,2 are calculated by equating the
axial optical force Fz from the ray-optics model with the fluidic force (equation 4.14)
and differentiating over optical power:
4.3 Optofluidic balancing 57
μopt,1 =Vp
dPopt
=dFz
dPopt
1
6πηrK1
,
μopt,2 =Vmax
dPopt
=dFz
dPopt
1
6πηrK2
.
(4.24)
The results are depicted in figure 4.17. Optical particle mobility μopt,1 and optical
flow mobility μopt,2 are very similar for one given refractive index, since for small
radii K1 and K2 are very similar (see figure 4.12). This is due to the fact that small
Figure 4.17: Optical particle mobility μopt,1 and optical flow mobility μopt,2 for sphericalparticles of different refractive index and changing radius. The speed of a particle propa-gating in a stationary liquid, given 1W of optical power corresponds to μopt,1. The flowspeed necessary to hold a particle stationary against the optical force at 1W of powercorresponds to μopt,2.
particles only interact with the central part of the parabolic flow profile where the
flow speed can be approximated to be Vmax in the vicinity of the particle. Therefore
the force on a particle moving in a steady liquid is almost identical to the force due
to the liquid flowing past the stationary particle at the same speed. For increasing
radius, the relative difference between μopt,1 and μopt,2 remains small (within 21%
for n=1.55 and particles below 5 �m radius) although the relative difference of K1
58 Theory
and K2 increases. This is due to the fact that K1 and K2 reach high values and are
located in the denominator of equation 4.17.
For small particle radii μopt,1 and μopt,2 increase linearly with particle radius,
as K1 and K2 can be approximated to be 1 and the optical force increases with
the particle cross-section. Therefore equation 4.24 exhibits linear behavior in this
regime.
Importantly, μopt,1 and μopt,2 exhibit a maximum for particle radii of about 2�m
radius. For larger particles, the drag correction factors increase more strongly than
the optical force. Therefore the particle or flow speed, creating a drag that balances
against 1W of optical power decreases with increasing radius. As the particle reaches
the size of the core, K1 and K2 become infinitely large, causing μopt,1 and μopt,2 to
vanish.
Applications like cell biology monitoring and particle synthesis analysis arise as
both, μopt,1 and μopt,2 exhibit regions which are very sensitive to a change in refractive
index, but insensitive to size fluctuations. These are found for radii around 2 �m,
close to the maxima in figure 4.17. The growth or shrinkage of cells or particles can
be analyzed best for smaller or larger radii, where μopt,1 and μopt,2 change strongly
with particle size.
Chapter 5
Particle guidance in liquid-filledphotonic crystal fibers
The following chapter discusses the trapping and guidance of microparticles made
of different materials and from a wide range of sizes, along liquid-filled hollow core
photonic crystal fibers. Guidance in horizontally and vertically oriented fibers, as
well as the guidance around bends is demonstrated. The optical particle mobility
μopt,1 and the optical flow mobility μopt,2 are determined over a wide range of particle
sizes and compared to theory, giving rise to many interesting applications.
5.1 Particle characterization and launching
As the optical (equation 3.10) and fluidic (equation 4.20) properties inside the hollow
core of the D2O-filled fiber are precisely determined, the propagation of microparti-
cles can be investigated. However, it is important to characterize the particles first.
Additionally one must be able to detect the position of particles along the fiber
over time. Two different measurement schemes will be discussed, where the particle
propagates in a stationary liquid or where it is held sationary against a flow.
In order to test the functionality of the system, particles of exactly known re-
fractive index and size have to be used. Commercially available borosilicate spheres
(Duke Scientific 9002, 9005, 9010) are certified to be monodisperse. Their refractive
index is specified to be 1.56. Although specified monodisperse, their size can vary
over more than one magnitude. Especially the smaller batches frequently show de-
60 Particle guidance in liquid-filled photonic crystal fibers
viations in size of up to 300%. Figure 5.1 shows scanning electron micrographs of
the three different batches, size-matched to an image of the fiber core. Even though
Figure 5.1: Scanning electron micrographs of certified monodisperse borosilicate spheres,in scale with a picture of the fiber used.
the size distribution is very broad, most of the particles are very spherical and show
only little oblaticity or prolaticity. Defects, however, are very common and can be
detected efficiently by an optical microscope where the probing is not limited to the
particle surface.
Therefore it is crucial to image the particles prior to launching and to select one
of desired size, shape and homogenous refractive index distribution. Furthermore a
3D imaging of the sample space is highly desirable in order to pinpoint the position
of the levitated microparticle. Additionally the launched particle has to be imaged
as it travels along the fiber and the mode profile exiting has to be determined
in order to ensure optimized coupling to the fundamental mode. Multiple CCD
cameras are therefore used, as shown in figure 5.2. Using CCD1 and CCD2, a full 3D
reconstruction of the particle position is possible. Additionally the high resolution
tweezers lens allows for excellent imaging of the particles in the sample volume. A
5.1 Particle characterization and launching 61
Figure 5.2: Schematic showing the imaging utilizing several CCD cameras. A: CCD1 andCCD2 allow for 3D imaging of the sample volume. Additionally high resolution imagescan be taken using CCD2 since it images through the high NA 100 x tweezers objective.A movable camera is used to track guided particles along the fiber trajectory (CCD3). B:A window in the D2O cell offers optical access to image the mode profile exiting the fiberon the other end using CCD4.
movable camera (CCD3) is used to track the particle. The mode profile is measured
through a glass window in the D2O cell by imaging the far field onto CCD4.
Once a particle with desired properties is spotted, it can be levitated up to
the fiber core using the laser tweezers setup. A sequence of snapshots during this
process using CCD1 is shown in figure 5.3A-C, looking at the front face of the fiber.
The optically trapped particle can be identified with the red scattering point in the
bottom left corner of figure 5.3A. Figure 5.3D shows a particle with 6 μm diameter,
imaged using CCD2 and the tweezers objective. Its shape, size and contour can be
imaged clearly, even revealing a small defect on its surface. While in this position,
the particle was balanced by a laser beam impinging from the left and a flow coming
out of the fiber core from the right. The laser beam was slightly misaligned, shifting
the particle away from the core axis. In this position the particle could be observed
spinning counter-clockwise due to viscous shear forces. However, the usual loading
procedure is to position the particle exactly in front of the fiber core using the laser
tweezers. The pressure head is set to nought in order to eliminate the flow in the
core. In the next step, the laser beam (coming from the left in figure 5.3D), which
62 Particle guidance in liquid-filled photonic crystal fibers
Figure 5.3: Loading, launching and guidance of a particle (diameter 6 μm). A-C: tweez-ering a particle up to the entrance to the core. D: side-view of the particle held at theentrance to the core by optical forces balanced against counter-flow of liquid from the core.While in this position the particle could be seen to revolve under the action of imbalancedviscous forces. E-H: side-scattering patterns imaged through the cladding of the fiber,photographed at 1 s intervals.
is adjusted (using CCD4) to perfectly couple to the fundamental core mode of the
fiber, is unblocked, kicking the particle out of the trap and pushing it into the fiber
core. As the particle is pushed along the core against the visous drag force, it is
imaged using a flexible camera system (CCD3). A sequence is shown in figure 5.3E-
H where it propagates along the fiber at ∼ 100 μms. The position of the particle can
be determined by light scattered off it at angles which lie not within the bandgap
of the fiber and at which the light can escape from the core.
5.2 Horizontal particle guidance 63
5.2 Horizontal particle guidance
Particles of various sizes are launched into a horizontal fiber piece of 11 cm length
[72, 73] and optically guided to a position convenient for imaging with CCD3. Once
at this position, they can be hindered from further propagating into the D2O cell
by either applying a counterflow that exactly balances the optical force or simply
by blocking the guided beam. In the latter case the particle sinks to the core wall
and remains there stationary.
In the first experiment a particle of known size is recorded while propagating
along a piece of fiber, as depicted in figure 5.3E-H. The optical power at the particle
position Popt is calculated by measuring the output power of the fiber Pout before
launching it and using equation 3.10. In a next step the particle is moved back to its
starting position, using liquid flow drag, in order to maintain identical experimental
conditions. The optical power is changed and the experiment repeated. The propa-
gation speeds are determined from the size-calibrated frames and plotted vs. optical
power at the particle position Popt. Three different spheres of 1.0 �m, 2.0 �m and
3.1 �m radius are launched and investigated for Popt-values between 0 and 180mW,
as shown in figure 5.4. Of course, the optical power exiting the fiber Pout, is mea-
sured only once prior to launching the particle and is calibrated to the power exiting
the laser aperture Papt. By doing so, the particle needs not to be removed from the
fiber core for each measurement point, and Popt can be deduced directly from Papt.
For optical powers below 10mW the particles remain stationary, indicating that
the radial trapping force is not sufficient to lift them off the core wall against gravity.
At increasing Popt the particle speed increases linearly since the optical force scales
linearly with the optical power, and the counteracting drag force scales linearly with
Vp. For optical powers slightly higher than 10mW, Vp is reduced since gravity pulls
the particle slightly below the optical axis, causing the optical force to decrease.
The optical particle mobility μopt,1 is defined as the speed at 1W optical power
(see equation 4.22), and can be identified with the slope of the speed plots dVp
dPoptin
figure 5.4. It is smallest for the 1.0 �m particle and increases for larger particles (see
figure 5.4). Theoretically, the optical particle mobility can be calculated from the
64 Particle guidance in liquid-filled photonic crystal fibers
Figure 5.4: Particle velocity Vp versus launched optical power Popt for three particle sizes(zero liquid flow). The relationship is approximately linear. At low powers the transversetrapping strength is weak, causing gravity to pull the particle closer to the wall, away fromthe center of the optical mode and thus lowering Vp.
force balance at 1W of optical power (as derived in Chapter 4.3).
In a second experiment the optical force on the particles is balanced against
a flow, keeping them in a stationary position. The optical power Popt is varied
and the pressure gradient necessary to balance is monitored. Vmax is derived using
equation 4.20. Figure 5.5 shows the results for 5 different particle radii.
Again for optical powers below 10mW, the particles remain lying on the core
wall and no flow is necessary to balance against the optical force. As the optical
power is increased, the particles exhibit a linear balancing behavior between Popt
and Vmax. This is again due to the proportionality of optical force to Popt and of
fluidic drag force to Vmax. For optical powers slightly larger than 10mW gravity
displaces the particle from the fiber axis and the optical force is decreased, causing
a non-linear behavior between Popt and Vmax. The optical flow mobility μopt,2 which
can be identified with the slopes of the graphs increases with particle size.
The data are compared to the theory for drag and optical forces in chapter 4.3.
5.2 Horizontal particle guidance 65
Figure 5.5: Optical power needed to hold a silica sphere (for five different radii) stationaryagainst the fluid flow driven by the pressure gradient dPH/dz. The righthand axis showsthe velocity Vmax in the center of the flow. Once again the relationship is linear.
A LP11 mode and a D2O-filled core with a refractive index of 1.33 were assumed.
The resulting plots are depicted in figure 5.6 and compared to the experimentally
retrieved data. The theory matches within 30% the experimental values found for
both, μopt,1 and μopt,2, for particles below 3μm radius. For larger particle sizes
however, the experimentally found data scatters within up to 65% compared to
theory. Possible error sources are inhomogeneities in the refractive index and shape
of the particles. The latter dramatically changes the viscous drag [74], especially
for particle sizes close to the core diameter. The error in determining the particle
radius is estimated to be below 0.25μm and cannot explain the strong deviations.
The slopes of the plots in figure 5.4 and 5.5, and thus μopt,1 and μopt,2, depend on
the exact calibration of the optical power Popt at the particle position, as described
in equation 3.10. Therefore a wrong calibration due to additional loss mechanisms
between the particle and the fiber in- or output (e.g. bubbles in the cladding or loss
at glass windows), would yield in erroneous results. If the real optical power is larger
compared to the assumed power, then the real μopt,1 or μopt,2 is lower compared to
the assumed one and vice versa. The error can be estimated to be ∼20%, given the
66 Particle guidance in liquid-filled photonic crystal fibers
Figure 5.6: Experimentally retrieved optical particle mobilities (squares, solid line) andoptical flow mobilities (circles, dashed line) for different borosilicate particles (np = 1.56),compared to the theoretical ray-optics model (lines).
measuring accuracy of the power-meter and the optical elements involved in the path
between the particle and the detector. Related to this, the incoupling conditions can
change over time, exciting higher order modes and reducing the coupling efficiency.
However, this would yield in non-linear characteristics of the plots in figure 5.4 and
5.5 which could not be observed.
In addition, the ray-optics model is limited as it does not include the vectorial
properties of the waveguide mode and excludes effects like Mie resonances [4, 75,
76, 77, 78, 79]. Although the scattering in the data cannot be explained by this, the
overall trend to underestimate μopt,1 and μopt,2 might be an indication that a more
sophisticated theory is necessary to explain the optical phenomena.
5.3 Vertical particle guidance
It is shown that particles with a larger mass density compared to the surrounding
liquid in horizontal fibers are pulled away from the fiber axis, especially for low
5.3 Vertical particle guidance 67
optical powers. In order to prevent particles from leaving the fiber axis, a long piece
of fiber with fixed ends is easily configured such, that it exhibits a vertical section
which is convenient for video imaging. A schematic of the forces on a particle in the
core of a vertical fiber piece is depicted in figure 5.7. In this configuration, light is
Figure 5.7: Schematic of the forces on a particle in the core of a vertical fiber. Due to thesymmetry and the axial direction of the gravitational force the particle will be situatedexactly on the fiber axis.
impinging on the particle from below, pushing it upwards. Gravity pulls the particle
in the opposite direction and acts axially on the particle. Since in this configuration
gravity does not have a radial component, the particle will be located exactly on
the fiber axis. The fluidic force on the particle again consists of the drag due to the
particle movement, combined with the drag due to a flow in the fiber core.
In the experiment a borosilicate microsphere is characterized geometrically in the
optical tweezers setup and launched into the fiber. It is propagated to the vertical
position, convenient for video imaging, using the particle guidance properties. Once
at this position, the optical power Popt is reduced and the flow is carefully adjusted
to keep the particle stationary. In a next step, the optical power is increased and
the flow readjusted in order to keep the particle perfectly stationary again. Both,
68 Particle guidance in liquid-filled photonic crystal fibers
the optical power and the flow necessary to balance against the optical force are
monitored. As the desired amount of data is recorded, the particle is flushed out
of the fiber using the fluid flow, and the initial conditions are restored in order to
launch another particle with different size. Figure 5.8 shows the experimental data
recorded for 9 borosilicate particles of different size. At very low optical powers,
Figure 5.8: Flow necessary to balance borosilicate particles of 9 different radii in thehollow core against the optical force exerted by given optical power and gravity. All curvesstart at negative flow values, indicating that the optical force is smaller than gravity, there.A negative flow speed is necessary to assist the optical force in order to keep the particlesstationary.
the optical force is too small to balance against gravity and an assisting negative
flow is necessary to keep the particle stationary. The slope of the curves decreases
for very large particles, as the drag correction factor becomes large and only little
flow is necessary to induce large viscous forces. The gravitational effects will be
discussed in the following section, before analyzing the gravity independent optical
flow mobility μopt,2 which can be identified with the slopes of the curves.
5.3 Vertical particle guidance 69
5.3.1 Balancing against gravity
Gravitational effects can be investigated for regimes where the optical and drag
forces are comparable to the gravitational force. For a borosilicate particle of 6.5 �m
diameter suspended in D2O, the gravitational force is 2 pN. Given the calculated
optical axial force of 200 pN/W (also see figure 4.7), an optical power in the order
of 10mW is necessary to investigate effects of gravity. Therefore a more precise
analysis of the low power regions in figure 5.8 is presented. A zoom in for the 3.5�m
radius particle (green data) is shown in figure 5.9.
Figure 5.9: Balance flow necessary to hold a borosilicate particle of 3.5 �m radius sta-tionary against optical and gravitational force for powers below ∼10mW. A linear fit isperformed to the measured data. The points where the optical power (blue) or the flow(red) vanish are indicated by circles.
A linear fit is performed to the data retrieved and indicated by the green line.
Circles indicate the points where this line intersects the zero-flow axis (red) and
the zero-power axis (blue). In the first case, the gravitational force of 2.47 pN is
balanced only by optical forces at a power of 8.87mW, meaning that at this power
the optical force is 2.47 pN. The force per Watt can be calculated to be 278 pN/W,
slightly higher than the 212 pN/W predicted by theory. Figure 5.10 shows the the-
70 Particle guidance in liquid-filled photonic crystal fibers
oretical optical power at which the particles are expected to be balanced purely by
light against gravity, compared to the data found experimentally for all 9 measured
particles.
Figure 5.10: Experimentally found optical power, necessary to balance a borosilicatesphere of given radius against gravity in the D2O-filled vertical HC-BGF. The data iscompared to the theoretically calculated power for the fundamental LP01 mode using theray-optics model, indicated by the solid line.
The theoretically predicted optical power necessary to balance the particles
against gravity is larger compared to all data found experimentally. However, the ex-
perimental data scatters strongly. This scattering behaviour can also be observed in
figure 5.11 where the optical power is nought and the gravitational force is balanced
against a flow only.
The theoretically predicted flow is indicated by the solid line. Again, the data
scatters strongly. This is due to the fact that not only the slope of the curves in
figure 5.8 has to be determined, but also the exact position of the reservoir where the
flow inside the fiber core vanishes. A wrong calibration shifts the graphs vertically,
causing the balance flow and the balance power to either both decrease or increase.
By comparing figure 5.11 and 5.8, this is indeed the fact. Large values in figure 5.11
correspond to large values in figure 5.8 and vice versa. More reliable conclusions
5.3 Vertical particle guidance 71
Figure 5.11: Flow in the hollow core, necessary to balance borosilicate particles againstgravity. The theoretically predicted values are indicated by the solid line.
can be drawn straight from the slope, as the pressure difference between both ends
necessary to balance a particle against gravity is only in the order of 10mbar, given
a 1m long piece, about 5 times smaller than typical atmospheric fluctuations.
5.3.2 Optical flow mobility in vertical hollow-core PCFs
Although the graphs in figure 5.8 may be slightly shifted vertically, as the calibra-
tion point for zero-flow conditions might be erroneous, all graphs show very linear
characteristics. The slopes which can be identified with the optical flow mobility
μopt,2 can easily be evaluated and are independent of any constant pressure offsets
between both fiber ends. Figure 5.12 shows the data from figure 5.5 and figure 5.8
compared to theory.
It can be clearly seen that the data scatters only very little and follows a distinct
trend for all 13 measurements taken. It is worthwile mentioning that most of the
measurements were performed at different days, with different particle batches, in
different fiber pieces, using different incoupling conditions. The measured optical
flow mobility reaches a maximum at 2.5 �m radius, in good agreement with the
72 Particle guidance in liquid-filled photonic crystal fibers
Figure 5.12: Optical flow mobility μopt,2 evaluated from figure 5.5 and figure 5.8. The datais compared to the ray-optics model with different refractive indices. The theoreticallypredicted curve for borosilicate (n=1.56) is solid. The shading indicates the particle batchused.
theoretically predicted value of 2.0 �m. The measured flow mobilities μopt,2 show
quantitative agreement within 35% up to the maximum mobility value, although
no fitting parameters were used for neither the theory, nor the experiments. One
possible reason for the deviations is an underestimation of the theoretically calcu-
lated optical force. A more sophisiticated simulation technique, taking into account
the fields of cylindrical waveguide modes and expanding them in terms of eigen-
functions of a sphere (vector spherical wave functions) [80], allows to calculate the
scattering force while including vectorial field components. Preliminary results agree
extremely well with the measured data. Another explanation for the deviation from
theory lies in the quality of the particles. The three different shadings in figure 5.12
indicate the borosilicate batch of the utilized particle. It can be seen that particles
from the 10�m batch give a larger optical mobility, as there seems to be a kink in
the characteristics between the 5 �m and the 10 �m batch. There is indeed evidence,
5.3 Vertical particle guidance 73
that particles from the Duke Scientific 10 �m batch exhibit a rather poor quality.
Although the particle surface does not seem to exhibit major defects as the SEM-
images in figure 5.1 show, defects inside the particles seem to occur frequently. These
can be detected using the optical tweezers microscope. Figure 5.13 shows represen-
tative images of the Duke Scientific 10 �m batch. In the brightfield regime, light is
Figure 5.13: Representative microscope images of the Duke Scientific 10 �m batch. Inthe brightfield image, light is absorbed by the particles, whereas in the darkfield imagethe background is dark and light that is scattered by the particles is detected.
transmitted through the particles. Dark spots indicate regions in the particle where
the light is absorbed or strongly scattered. The background in the darkfield image
is dark and the sample is illuminated. Bright spots indicate that light is scattered
strongly in the particles. Both imaging techniques indicate the poor material qual-
ity. This also explains the increased optical mobility of particles from this batch.
Light is scattered strongly or is absorbed as it hits defect regions, inducing an ex-
tremely large local photon momentum transfer close to 1 (similar to absorption or
uniform scattering in all directions of the photon). This effect increases the optical
force and thus the optical mobility, as observed in the experiments.
74 Particle guidance in liquid-filled photonic crystal fibers
5.4 Particle guidance around bends
It was previously shown in chapter 3.3.2 that light at 1064 nm is efficiently guided
around bends with radii as small as 4mm, offering the great possibility to translate
microparticles along arbitrarily curved trajectories. In order to demonstrate that
also particles of different material than borosilicate can be trapped and launched,
the following experiments are performed using polystyrene beads. The used beads
(Duke Scientific 4205) have a certified refractive index of 1.59 which is close to the
value certified for borosilicate (1.56). However, their mass density ρps = 1.05 · 103 kgm3
is slightly smaller compared to the suspension medium (ρD2O = 1.1056 · 103 kgm3 ),
resulting in buoyancy of the microparticles. Figure 5.14 shows a trapped polystyrene
bead in front of the fiber core. The particle is trapped and ready to be launched. As
Figure 5.14: Polystyrene microparticles floating in D2O. A particle with desired proper-ties is trapped in front of the fiber core. Light from the left which is coupled into the fibercore accidently pushes two random particles against the trapped bead.
5.4 Particle guidance around bends 75
the guidance beam from the left is unblocked, two random particles are accidently
pulled into the beam and pushed against the trapped bead of desired properties.
As the 3 particles form a cluster, the trapping conditions become unstable and they
are pushed out of the trap by the trap beam itself (not shown). A fourth particle
coming from the left can be seen in the last frame, indicating the commonness of
this process. The dark stain on the top part of the fiber facet indicates a particle
stuck to the fiber. As time proceeds, more and more floating beads attach to the
fiber facet or are randomly launched.
Two techniques to circumvent these problems were developed. One is to rinse the
sample volume after a particle is launched. The other one is to keep the trapping
beam active after having launched a desired bead. In the first case, most of the
beads are removed and the residual beads are strongly diluted in the sample volume.
However, this only reduces the probability for accidental launching as the number
of particles is reduced, but not all can be removed. In addition, the incoupling
conditions are probable to change as the fiber experiences strong fluidic drag forces
during rinsing. The second technique works in an interesting way. As a particle is
accidently trapped by the low-NA incoupling beam, it is transported to the tweezers
focus and trapped. As a second particle is accidently transported the tweezers focus,
it forms a cluster with the particle already trapped. After a short rearrangement
time (usually less than a second) both particles are pushed upwards out of the
trap by the tweezers beam as both cannot be trapped stably. Unfortunately the
fiber coupling fluctuates as particles are pulled into the incoupling beam or as they
are located in the optical trap. The guidance around curves is demonstrated by
launching a polystyrene microparticle with 2.5 �m radius into the core of a fiber
coiled twice around a metallic post with 6.25mm radius. Video snapshots of the
experiment are shown in figure 5.15.
The guided light is incident from the left, pushing the particle along the core
against viscous drag. No viscous flow is present in the experiment. The particle
translates by ∼1.2mm in 30 s, yielding a speed of 40 �m/s. Two entire loops can be
absolved by the particle without any problem, proving that the fiber can be used
76 Particle guidance in liquid-filled photonic crystal fibers
Figure 5.15: Polystyrene microparticle with 2.5 �m radius propelled by light along asharply bent (6.25mm bend radius) piece of D2O-filled hollow-core BGF. The particlemoves at a speed of about 40 �m/s.
to robustly guide microparticles optically around sharp bends over macroscopic dis-
tances. Particle guidance along sharp curves is also reproduced with borosilicate
particles, proving that the trap stiffness does not change upon bending the fiber.
However, the buoyancy of the utilized polystyrene particles is an obstacle, as parti-
cles are randomly launched into the fiber core. For this reason all following experi-
ments are performed with non-buoyant microparticles.
Chapter 6
Doppler velocimetry
A Doppler-based velocimetry technique, that makes use of the excellent waveguiding
properties of the PCF, is used to pinpoint the particle position as it propagates along
the fiber core. The principle is explained in chapter 2.3 and depicted schematically
in figure 6.1. As light is back-reflected off a moving particle, its frequency is shifted
Figure 6.1: Schematic of the Doppler velocimetry setup; BS beam splitter; PD photo-diode. A borosilicate microsphere is propelled along the D2O-filled fiber at speed Vp byoptical forces. Backscattered light has a Doppler-shifted frequency νD and is mixed withunshifted light of frequency ν0 at the core entrance.
from the original frequency ν0 to the Doppler-shifted frequency νD. It is then guided
with low loss, back to the entrance of the core, where it mixes with light reflected
at the core entrance. The resulting beating between the two frequencies is picked
up by a photodiode (PD). One beat occurs as the particle is translated by the beat
length LB = λ0/2n, where λ0 is the original wavelength (1064 nm) of the light and
n is the effective refractive mode index (∼1.33). The beat frequency νB is derived
78 Doppler velocimetry
in chapter 2.3 and its proportionality to the particle speed Vp is given by:
νB =Vp
LB(6.1)
6.1 Measurement procedure
A typical signal picked up by the photodiode for a borosilicate particle, propagating
at 240 �m/s, is shown in figure 6.2. Slow fluctuations in the diode signal, due to a
Figure 6.2: Typical photodiode-signal with 20 kHz sampling rate, picked up from aborosilicate particle, travelling at 240 �m/s. Part A shows a 2 s long part of the recordedsignal. The first 20ms, marked in red, are shown in part B. Clearly a periodic varia-tion in the diode current can be observed. One beat corresponds to LB = 400nm particledisplacement.
change of the amount of light reflected by the particle back into the core mode, can
be observed in figure 6.2A. The first 20ms are marked in red and are depicted in
6.2 Doppler based particle tracking 79
figure 6.2B. A clear periodic variation in the backscattered signal with a frequency
of 600Hz can be observed which is due to the beating of the Doppler-shifted and
the original light. Following equation 6.1, this frequency corresponds to a particle
speed of 240 �m/s. This value can also be easily calculated by counting the beats,
each corresponding to a particle displacement of LB = 400 nm, and dividing by the
length of the time interval.
In order to determine the speed of a particle on-the-fly, one can simply calculate
the fast Fourier transform (FFT) of the diode signal, while measuring in the lab-
oratory. Another method of evaluating the data, is to record the diode signal and
process it later on. Therefore a computer code is used which reads the collected
data and analyzes a time window of given length and starting point from the data.
A FFT is performed and saved, analyzing all frequency components in the time
window. Hereby the largest possible frequency component, analyzed by the FFT
is given by twice the inverse time between two subsequent measurement points of
the photodiode, or half the sampling rate. The resolution of the FFT is given by
the inverse time length of the window, meaning that a longer window gives a better
resolution of the speed measurements. However, the longer the time window is, the
more averaged the frequency spectrum is. The FFT of the data in figure 6.2 with a
20 kHz diode sampling rate is shown in figure 6.3. The distinct peak in the Fourier
transformed signal at 600Hz indicates a periodic fluctuation in the reflected light
intensity that the moving particle induces .
The window is shifted in a next step by a given time and the procedure is
repeated, yielding the frequency spectrum at the shifted time. This step is repeated
until the entire data set is analyzed, and the results are saved in a matrix.
6.2 Doppler based particle tracking
The goal of the utilized technique is the exact determination of particle speed and
position, as a particle is propelled along a hollow-core PCF. Therefore, in the fol-
lowing, the position and speed measured with the Doppler based technique are
compared to the values found by a camera measurement.
80 Doppler velocimetry
Figure 6.3: Fast Fourier transform of the recorded photodiode signal in figure 6.2. Adistinct peak at 600Hz can be observed which corresponds to the beating induced by themoving particle.
The evaluated matrix for a particle travelling at ∼250 �m/s and 4 s signal length
is plotted over time in a density plot, as shown in figure 6.4 A. The diode signal
has 20 kHz sampling rate. A sampling interval of 100ms and 50ms time steps are
used. This corresponds to a frequency range of 10 kHz, a frequency resolution of
10Hz, and a time resolution of 50ms. The retrieved matrix is then peak traced,
rescaled to the particle speed, using formula 6.1 and plotted over time as shown in
figure 6.4B. The speed resolution, corresponding to a 50Hz frequency resolution of
the FFT spectrum, is 4 �m/s. In order to determine the exact particle position,
the evaluated particle speed is simply integrated over time, yielding the plot in
figure 6.4C.
The retrieved particle position and speed are compared to data from a video
camera (red dotted lines). However, the video data is retrieved by evaluating the
position first, and then taking the derivative. As the position data is noisy, the
derivative can fluctuate extremely, even exhibiting negative particle speeds. There-
fore the video speed is averaged over ∼0.5 s. Despite the long averaging time in the
video measurement, the velocity data using the Doppler-effect is ∼10 times more
accurate. The particle position from the Doppler setup VD agrees within 20 �m with
6.3 Intermodal beating 81
Figure 6.4: A: FFT-spectrum over time for a sampling interval of 100ms, and 50mstime steps. B: Peak trace of A, rescaled to particle speed, using formula 6.1. C: Particleposition over time, retrieved by integration of B over time. The dotted red lines indicatemeasurements using a CCD camera.
the data retrieved from the video VV (blue line and right axis in figure 6.4C). This
proves that the speed and position can be determined accurately, even in opaque
environments where the particle cannot be imaged, using Doppler based velocimetry.
6.3 Intermodal beating
The Doppler based particle velocimetry allows a direct and precise speed measure-
ment of a particle propelled by optical forces along the hollow core. Figure 6.5A
shows a typical speed measurement for a 6.5 �m borosilicate sphere propagating
along 14mm at an optical power of 270mW. The speed remains constant within
82 Doppler velocimetry
10% which is in agreement with the previous measurements in chapter 5, where
the average speed over a distance of ∼400 �m was determined from a video. How-
ever, the field of view of only ∼400 �m (also see figure 5.3E-H) is too small in order
to detect the fluctuation period of ∼360 �m. Interestingly, the amplitude of the
backscattered light from the particle (see figure 6.5A blue curve and right scale)
also correlates with the particle speed. Fast regions correspond to strong backscat-
tering, whereas slow regions coincide with a weaker backscattered intensity. This is
an indication for the stronger interaction of the particle with the optical mode in
the fast regions, causing stronger scattering and a larger axial optical force. The
opposite is the case in the slower regions. The periodicity of the fluctuations sug-
gests intermodal beating to be a possible explanation for this behavior. In order to
further investigate this hypothesis, the region of figure 6.5A marked in grey is com-
pared to the theoretical model. A reasonable assumption of a 90:10 mode mixture
of LP01 and LP11 mode is simulated, as indicated in the small mode profile on the
top left of figure 6.5B. The beat length between the two modes can be calculated
by evaluating their axial propagation constants. For modes with azimuthal index n
and radial index m these are given by [32]
βn,m =2 π
λ
(1− 1
2
(un−1,m λ
2 πR
)), (6.2)
where λ is the wavelength of the light in the core, taking into account the modal
refractive index, R is the core radius and un,m is the m-th zero of the n-th Bessel
function Jn. The modal refractive index is assumed to be equal to the bulk D2O-
index, neglecting all dispersive effects. The fundamental LP01 mode is, as the name
suggests, linearly polarized and thus a hybrid mode since all transverse electric and
transverse magnetic modes exhibit radial symmetry. It can be identified with the
EH11 hybrid mode and therefore n=m=1 holds for this mode and the propaga-
tion constant β1,1 equals 7.84889 · 106m−1. The LP11 mode is a superposition of
the EH21 and for example the TE01 mode, as discussed earlier. The propagation
constants β0,1 and β2,1 are equal as expected from mode theory and have the value
7.84104 · 106m−1. Given these propagation constants the modal beat length LMB,
6.3 Intermodal beating 83
Figure 6.5: A: particle velocity (top curve) and averaged photodiode signal (bottomcurve). 6.5 �m borosilicate sphere, launched optical power 270mW. B: calculated intensityprofiles of LP01 and LP11 modes (left) in the D2O-filled, 17 �m diameter core. (I-IV)superposition of 90% LP01 and 10% LP11 mode at four positions within one beat period.C: measured (symbol) and calculated (curve) particle velocity versus relative displacement(position relative to 11.5mm).
until both modes are in phase again, can easily be calculated to be
LMB =2 π
β1,1 − β0,1
= 801μm. (6.3)
This agrees well with twice the measured length of 720 �m. The factor 2 can be un-
derstood by looking at the mode profile of the superimposed mode as it propagates
along the fiber (see figure 6.2B). The maximum is located on the axis every 400.5 �m
(II and IV), and thus the speed is identical every half beat period. The particle speed
was calculated for each mode profile, taking into account the changing optical and
84 Doppler velocimetry
fluidic forces [62] at the stable trapping position. Toezeren [81] et al. have numeri-
cally calculated the drag on a sphere placed off-center in a liquid-filled cylinder. The
drag force Fdrag for small deviations from the axis can be approximated to be:
Fdrag = 6π η aVP
(λ0 + λ2 · ε2
), (6.4)
where ε is the displacement from the axis divided by the core radius, λ0 and λ2 are
numerically retrieved correction coefficients. Figure 6.6 shows a plot of the values
found over the ratio of particle to core radius Γ. Counterintuitively, the quadratic
Figure 6.6: Numerically retrieved correction coefficients λ0 and λ2 for the drag force ona particle, slightly off centered in the fiber core plotted versus the ratio of particle to coreradius Γ.
correction factor λ2 is negative, indicating that the drag is not minimized in the
center of the core where the distance to the walls is maximized. As the relative
displacement ε is increased, the drag force decreases. This is due to the fact that
the particle experiences a torque and begins to spin as it leaves the on-axis position.
By doing so, the drag is reduced on the side closer to the wall. On the side further
away, the drag is increased due to the spinning, but therefore the distance from the
wall is larger. In total the drag force decreases for small deviations.
This effect together with the compressed mode profile in the assymetric cases in
figure 6.5B (I and III) leads to an increased particle speed. Interestingly the stable
trapping position is only 2.1 �m from the fiber axis, where the intensity maximum
6.3 Intermodal beating 85
of the mode is further away at 2.3�m. This is due to the fact that the mode
is asymmetric and a larger part of the low intensity region extends towards the
center of the core. The optical force is increased in the asymmetric case by 14%
to 58 pN compared to the on-axis trapping position where it only has a value of
51 pN. Additionally, the drag is reduced by 6% (using λ0 = 3.19 and λ2 = −3.10 by
interpolation of the data in figure 6.6 for Γ = 0.382 as expected for 3.25 �m particle
radius), yielding an increased particle speed of about 21%.
In order to verify the purely optical origin of the fluctuations and exclude other
effects as for example surface charges, particle effects due to spinning or geometrical
inhomogeneities, a number of experiments are performed. Firstly, the experiment is
repeated 3 times in another piece of fiber, as depicted in figure 6.7. A clear corre-
Figure 6.7: Particle propelled along the same piece of fiber 3 times. The fluctuations areclearly correlated to the position in the fiber.
lation between the speed fluctuations and the position in the fiber is found, as fast
and slow speeds are always observed at the same positions in the fiber. Therefore
particle effects can be excluded, as they would occur randomly and not correspond
to a particular position in the fiber. It can be seen that the average speed decreases
86 Doppler velocimetry
for later experiments (the numbers indicate the order of the measurements). Addi-
tionally the amplitude of the fluctuations increases. This is a clear indication that
the incoupling is best in measurement 1 where light is coupled efficiently to the fun-
damental mode and only little higher order modes are excited. For measurement 2
and 3 the coupling is deteriorated, yielding a smaller total coupling efficiency which
corresponds to a smaller average speed. Additionally the coupling to higher order
modes is increased which causes more pronounced fluctuations.
A second experiment is performed where the incoupling is deteriorated deliber-
ately as seen in figure 6.8. A fundamental mode is excited and a particle is trans-
Figure 6.8: Particle speed versus propagation distance in the same piece of fiber fordifferent incoupling displacements.
ported to a defined starting position in the fiber. Once there, the laser beam is
blocked and the particle sinks to the core wall. As it is unblocked, the particle
accelerates and reaches its final speed of about 225 �m/s (see figure 6.8, 0�m). It
can be seen that the small speed fluctuations are random and show no periodicity.
The experiment is then repeated with a detuned fiber incoupling by moving
the incoupling lens by 1 �m and 5 �m. As suggested before, the average speed de-
6.3 Intermodal beating 87
creases and the fluctuation amplitude increases with detuning, as the total coupling
efficiency decreases and higher order modes are excited more efficiently. The fluctu-
ations again occur at distinct positions in the fiber, indicating that they are linked
to modal beating.
Periodically occuring surface charges in the fiber could also explain the fluctua-
tions in speed. These would have less impact for larger axial trap stiffness at higher
guided optical power. Therefore a third experiment is performed where the optical
power at the particle position is strongly increased to ∼700mW, roughly 3 times
larger than in the previous experiment where no fluctuations could be observed for
ideal incoupling conditions. The experiment is repeated 3 times in the same fiber
piece, as shown in figure 6.9. Again fluctuations can be observed at distinct positions
Figure 6.9: Borosilicate particle guided through the same piece of fiber three times at ahigh optical power of ∼700mW.
in the fiber. This proves that only modal effects can explain the speed fluctuations,
as the amplitude of the fluctuations is still in the order of 10%.
A fourth experiment is performed where a different fiber is used that guides
light from a tunable Ti:sapphire laser source of 810 nm in a band gap. Again the
88 Doppler velocimetry
experiment is repeated 3 times as shown in figure 6.10.
Figure 6.10: Borosilicate particle propelled optically through a piece of fiber, guiding at810 nm due to a band gap. The experiment is repeated three times, showing excellentreproducibility of the speed fluctuations.
The fluctuations of ∼500 �m period are more pronounced and vary in amplitude,
showing excellent reproducibility. This is due to the fact that the utilized fiber has
an even larger core of 19 �m diameter and a shorter wavelength is used. Therefore
the excitation of even higher order modes is likely, creating an envelope function
for the beating between the fundamental LP01 and the first higher LP11 mode,
as their propagation constants are different compared to the LP01 and LP11 mode.
The measured beat length of ∼500 �m is confirmed by theory which predicts again a
larger beat length of 650 �m, given a core diameter of 19�m and 810 nm wavelength.
From these experiments we conclude that the observed speed fluctuations can
be purely attributed to intermodal beating in the fiber core. This shows that an
optically propelled particle can be utilized to investigate the mode inside a hollow
core fiber destruction free and continuously over arbitrary lengths.
6.4 Delayed particle lifting 89
6.4 Delayed particle lifting
The excellent position- and speed-resolution of the Doppler based velocimetry tech-
nique can be used to investigate the dynamics of a microparticle, launched off the
core wall of a hollow core PCF. An experiment is performed where the particle is
stopped in the hollow core of a horizontal fiber piece by blocking the laser beam.
After some time when the particle is lying on the core wall due to gravity, the beam
is unblocked and the particle is launched off the fiber core wall. A typical measure-
Figure 6.11: Stop-start velocity measurements, r = 3.25 �m, launched optical power230mW. A: Doppler spectrum. B: velocity measurement. In regions 1 the beam is blockedand the particle lies on the lower core-wall (see inset). In region 2, the beam is switchedon and the particle is slowly moving back to the central position. Exponential fits to thedata have time constants of 0.60 s and 0.61 s. In region 3 the particle is moving close tothe center of the core.
90 Doppler velocimetry
ment is shown in figure 6.11, where the particle speed is evaluated over time and
not position, in order to analyze the temporal dependency of the particle speed.
The particle is stopped twice at different positions where it sinks to the core wall
as indicated in figure 6.11B 1. As the beam is unblocked, the particle is lifted and
pulled into the core center as shown in figure 6.11B 2. The steady state where the
particle is moving continuously in the fiber is depicted in figure 6.11B 3.
Exponential fits to the speed curves were performed, yielding τ values of 0.60 s
and 0.61 s. The origin of this time constant might be due to inertia, therefore the
motion of a particle with radius r and mass m, experiencing a force Fopt in a viscous
medium, is calculated, using the damped equation of motion:
mv = Fopt − βv. (6.5)
The damping term β can be identified with the Stokes drag coefficient 6π η r, where
η is the viscosity of the surrounding medium. Equation 6.5 can be easily solved,
yielding the time dependent speed:
v =Fopt
β
(1− e−
βm·t). (6.6)
The final speed is given by the ratio of the driving force and the Stokes drag. The
typical time constant in the exponential function τ = m/β yields a time of 4.7�s,
using the typical parameters of a borosilicate sphere of 3.25�m radius in D2O. The
assumptions are made that the drag does not have to be corrected and that the
accelerated mass is only the particle mass. The drag is clearly larger in the hollow
core, as discussed in chapter 4.2, yielding a smaller τ . On the other hand, the
accelerated mass m is underestimated as not only the particle has to be accelerated,
but also the fluid that needs to flow around the sphere. This yields in a larger τ .
The calculated time deviates from the experimentally found value by 5 orders
of magnitude. The explanation for this is that, due to the small Reynolds num-
ber, all inertia-effects can be neglected. The particle reaches its terminal velocity
Fopt/β instantaneously (within a few microseconds), which is given by the balance
of optical and viscous forces. Since the axial optical force increases, as the particle
moves radially from the core wall to the fiber axis, where the axial optical force is
6.4 Delayed particle lifting 91
maximized, the particle accelerates. In other words, the time it takes for the particle
to move radially from the core wall to the core center is equal to the time it takes to
reach its terminal speed. As axial and radial speed are decoupled, the radial position
over time is calculated, by using the theoretically obtained optical radial force and
the Stokes drag corrected for a particle moving along the fiber axis. The theoretical
time constant is found to be ∼0.06 s, assuming an optical power of 230mW, gravi-
tational force and a corrected drag coefficient, taking into account wall effects. The
wall effects are estimated by using the analytically obtained correction factor for a
sphere moving normal to an infinite plane, found by Goldman et al. [82]:
Kplane =1
(h/r)
[1− 1
5
h
rln
(h
r
)+ 0.9712
h
r
], (6.7)
where h is the distance between the sphere and the plane surface and r is the
sphere radius. Astonishingly the theory still predicts a 10 times faster launch than
measured experimentally. A possible explanation for this deviation is the Magnus
effect. As the particle translates axially at a speed Vp along the fiber core at an off-
axis position, it experiences a torque due to the microfluidic forces [81]. The drag on
the side of the particle closer to the core wall is larger, compared to the side further
away. The consequential torque causes the particle to spin with an angular speed
ω as shown in figure 6.12. An analytical expression for the Magnus force FMagnus on
Figure 6.12: Forces acting on an optically launched microparticle in the core of a liquid-filled hollow-core PCF. As the particle is launched, it begins to spin due to fluidic forces,resulting in a radial counter-force to the optical gradient force due to the Magnus effect.The gravitational force is not shown explicitly, but taken into account for the calculations.
92 Doppler velocimetry
a spinning sphere in a free medium with mass density ρ is given by Rubinow and
Keller [83]:
FMagnus = πr3ρ (ω ×VP). (6.8)
The angular speed for a particle, translating in a liquid-filled cylinder, depending
on its distance from its axis cannot be determined analytically and needs to be
calculated numerically. The problem is very complex, as the ratio of particle size
to core size also plays a role and increases the parameter dimension. Additionally
the expression for the Magnus force is only valid for a liquid without any boundary
conditions which is strongly violated in a liquid confined by the core. This is why
only the qualitative trend is examined. In general it is correct to assume that the
angular speed ω is linearly proportional to the particle speed Vp, given a certain
particle size and distance from the cylinder axis. This implies that the Magnus
force FMagnus is proportional to V2p and thus also to the the second power of the
optical power Popt. In contrary, the radial optical trapping force only scales linearly
with Popt, indicating that the impact of the Magnus effect increases for larger optical
powers. In any case, the Magnus effect will delay the lifting process off the core wall,
as confirmed experimentally. For extremely high powers one expects the particle to
remain rolling along the core wall, as the Magnus force increases quadratically with
the optical power and eventually overcomes the radial optical force on the particle
given there.
6.5 Multi-particle tracking
The Doppler based velocimetry technique even allows for the tracking of several
particles simultaneously. Doppler shifted light from a leading particle passes the
position of the following particle as it is guided back to the fiber entrance. The
light is scattered by the following particle and again coupled to the core mode. It
is important that light is not Doppler shifted during this forward scattering event,
as the optical path does not change. Therefore the frequency of the light remains
unchanged as it is transmitted through a particle. Only reflected light is Doppler
shifted in frequency, as the optical path is increased for a particle moving away from
6.5 Multi-particle tracking 93
the light source. This could also be observed experimentally for two borosilicate
particles, optically guided in the hollow core of a BGF at 810 nm, as shown in
figure 6.13. Two clear beating signals can be detected by the photo diode, as shown
Figure 6.13: A: Speed-trace of two borosilicate particles guided in the hollow core of aBGF at 810 nm (yellow dots indicate the speed of the following particle). The investigatedpiece is located ∼3 cm away from the fiber end where lossy higher order modes still exist.The resulting extreme speed fluctuations can be observed. B: Video snapshot at 20 s. Theweakly scattering and slow particle is located on the left and indicated by the yellow dot.The leading, fast particle is located to the right.
in figure 6.13A. The signal marked with yellow dots corresponds to the following
particle, whereas the unmarked trace can be identified with the speed of the leading
particle. The speed signals are confirmed by a video measurement, as depicted in
figure 6.13B, showing a snapshot at 20 s. The faster particle scatters more brightly as
it interacts more efficiently with the optical mode. Pronounced and abrupt changes
in speed can be observed in the investigated piece which is located ∼3 cm away from
the fiber end. This close to the core entrance higher order modes can still exist,
despite their large propagation loss. Thus, the mode profile and the optical forces
change very dramatically along the fiber core. Additionally the optical mode for the
leading particle is strongly modified by the following particle which excites higher
94 Doppler velocimetry
order modes, as the light is scattered by it. Also the back-scattered light from the
leading particle might influence the propagation of the following particle. However,
this effect is expected to be weak as only a small fraction of light is scattered in the
backward direction.
Chapter 7
Conclusions and recommendations
7.1 Experimental and theoretical conclusions
The work presented in this thesis demonstrates for the first time that hollow-core
photonic crystal fibers can be used in combination with a laser tweezers setup to
selectively trap and guide microparticles along a liquid-filled fiber in a single, funda-
mental mode [84, 85]. By obeying scaling laws [86], fibers were designed which guide
light due to a photonic band gap when the entire structure is filled by an aqueous
medium. The resulting low loss and single mode guidance, together with a precisely
controlled microfluidic flow, allow the exact investigation of the optical and viscous
forces on microparticles inside the hollow core [73, 87]. The optical particle mobility
and the optical flow mobility were investigated for a range of particle sizes [88, 89],
proving that hollow-core photonic crystal fibers offer an ideal environment to study
the properties of microparticles. Gravitational effects on microparticles were inves-
tigated in vertical fibers where particles were held against gravity, using optical and
viscous forces. It was proven that microparticles can be guided around sharp bends
due to the small bend loss of the utilized fibers, demonstrating their transportation
potential to difficultly accessible places over meter distances. A Doppler based ve-
locimetry technique was used in order to accurately measure the position and speed
of a guided particle as it propagates along the hollow core [90]. Even the tracking
of several particles simultaneously is possible. Small, periodic fluctuations in speed
were revealed by the technique which could be attributed to intermodal beating
96 Conclusions and recommendations
between the fundamental and the first higher order mode. This demonstrates that
microparticles can be used to investigate the mode propagation along a fiber in a
destruction-free manner. Additionally the acceleration of particles, launched off the
core wall in a horizontal piece was investigated, indicating that particle rotation
plays an important role and delays the launch significantly. All data were compared
to a theoretical model that takes into account the optical forces, using a ray-optics
approach, as well as the enhanced fluidic drag inside the hollow core.
7.2 Recommendations and outlook
The theoretical model generally slightly underestimates the optical forces which
were found experimentally. Therefore currently a more sophisticated model of the
optical forces in the fiber [80] is developed. This model uses all components of the
field rather than a ray optics model which is more similar to a plane wave approach.
Preliminary results indeed predict a larger optical force and show good agreement
with the measured results.
Elastic particles can be stretched by two counterpropagating beams [91] which
is of high interest to cancer diagnostics and research where the flexibility of a cell
can be used to diagnose cell diseases such as cancer [92]. This could also be done in
a hollow core PCF by either using two counterpropagating beams or even a single
beam (see figure 7.1). A nonlinear behavior between optical power and cell speed
LASERLASER CELLCELL
Figure 7.1: Schematic of a cell moving along the hollow core of a PCF (by courtesy of S.Unterkofler).
7.2 Recommendations and outlook 97
is expected as the cell is elongated along the fiber axis for increased optical power,
yielding a more hydrodynamic shape and higher speed. For cell sizes close to the
core diameter, small deformations could be measured as the drag force strongly
depends upon a size change of the particle in this regime.
Also particles with two or more different surfaces, the so-called Janus particles
[93, 94, 95, 96] which are named after the double-faced Roman god Janus, could be
launched into HC-PCFs. The optical forces could be investigated or one might use a
Figure 7.2: A: Borosilicate particles from a 5 �m batch, coated with silver patches (Janusparticles) by courtesy of R. N. K. Taylor [97]. B: Laser tweezers power, necessary tolift a certain particle off the microscope slide for 10 uncoated and 10 coated borosilicateparticles from the same batch. The average power needed is ∼4 times smaller for theJanus particles, indicating that trapping in a single beam trap is not possible due to theenhanced axial optical force. C: The first Janus particle, launched into the hollow core ofa PCF. A vertical setup is needed where the particle is launched off the microscope slide,directly into the core, using a low NA lens.
98 Conclusions and recommendations
PCF to synthesize Janus particles, where a layer is deposited on the particle front as
it propagates. Preliminary tests with borosilicate spheres coated with silver patches
by R. N. K. Taylor et al. [97] (see figure 7.2A) were performed, proving that Janus
particles can be guided controllably in a PCF (see figure 7.2C). However, a vertical
launching setup has to be used for these specific particles as the axial scattering
force is increased roughly by a factor of 4 due to the silver patches. This can be
seen in figure 7.2B where the laser power necessary to lift a particle off a microscope
slide was measured for 10 coated and 10 plain borosilicate spheres from the same
batch. The average threshold power where the particle lifts is ∼4 times smaller for
the coated particles. Therefore single beam tweezers trapping is not possible and
the particles have to be launched directly off the microscope slide into the fiber core
using a 10 x low NA objective.
Furthermore, the particle growth in synthesis processes or surface chemistry
where the roughness ond shape of particles could be investigated while they undergo
chemical reactions. The use of higher order modes which can be excited with a
spatial light modulator (SLM) [61] is very useful in this context as their stable
trapping position exhibits a bifurcation diameter (see chapter 4.1.2). Bistability
studies in the micron size regime are possible, where a particle is suddenly lifted off
the core wall and transported away, as it grows. Or the opposite case can be regarded
where it is pushed out of the core center as it shrinks below a critical size and is
deposited at a distinct position in the fiber core. Close to the bifurcation diameter
the axial optical force depends almost digitally on the particle diameter, making
it a precise analysis tool for extremely small size changes, e.g. in cell diagnostics
or particle synthesis. In addition, modes with angular momentum could be excited
which would cause birefringent particles to spin [15] along an axis parallel to the
fiber axis, possibly inhibiting the spinning along other axes which was observed
experimentally.
Chemical reactions where the particle acts as a catalyst, in combination with
photochemistry [98] could be investigated. For example, if a trapped particle is
pushed sideways using a laterally focused laser beam (which can be delivered through
7.2 Recommendations and outlook 99
the cladding [99]), the imbalance of viscous drag on opposite sides will cause it to
spin, enhancing chemical reactions at the particle surface.
As shown in Chapter 6.5, several particles can be tracked simultaneously, allowing
the studies of interactions between particles. A deeper analysis of the observed
phenomena together with counterpropagating beams and the formation of optically
bound particle trains [100, 101, 102, 103, 104] would be highly interesting. These
trains form due to the occurence of standing wave patterns where particles are
trapped in field maxima. The pattern can be translated by changing the optical
path in one of the arms, thus creating an optical conveyor belt [105]. This concept
could be extended to a hollow core PCF where the beat pattern is preserved over
macroscopic distances and a slightly detuned wavelength in one arm could be used
to create a moving fringe pattern and continuously move trains of microparticles or
atoms along the fiber core.
The guidance of atoms, however is only possible in vacuum which leads to the
next possible area, namely trapping in air [33] and vacuum. Preliminary results
Figure 7.3: Silica microparticle suspended in air over several minutes, ready to belaunched into the hollow core of a PCF. The trapping is performed from above wherea laser beam is focused by a high NA microscope lens into an aerosol cloud of silicaparticles, generated by a piezo driven vibrating glass membrane.
100 Conclusions and recommendations
(following the work of Omori et al. [106]) were obtained where a silica bead was
trapped in air by a single beam laser trap as shown in figure 7.3. An extension to
high and ultra high vacuum would strongly reduce the drag force, thus allowing for
extreme particle speeds, theoretically predicted in the order of km/s. Additionally,
the possibility of viscosity measurements in low Reynolds number dilute gases arises.
In an ultra high vacuum (UHV), the particle might even be used as a pump, to
push out residual air molecules, since they behave ballistically in this regime. As
the particle is optically pushed back and forth from one end of the fiber to the other,
all air molecules along its way are pushed out of the fiber core.
Finally, the system could be used as a flexible optofluidic interconnect for trans-
porting particles or cells between microfluidic circuits. So far the optical transport
of microspheres over distances of up to 2m could be achieved.
Chapter 8
Acknowledgements
Although I am the author of this thesis, the work presented is an accomplishment of
a large group of unique people. I dedicate this chapter to everyone who has helped
me in becoming the person who I am today.
Prof. Philip Russell laid the foundation for this work by offering me a PhD-position
in his group. His perpetual scientific enthusiasm and his many ideas and suggestions
were a great inspiration to me. The way he approaches everyday’s scientific issues,
his wonderful Irish humor and his direct character make him a strong leader. Thank
you for all this and for teaching me how to become a better scientist.
Tijmen Euser supported me in many ways. He is incredibly patient when things
don’t work out in the laboratory and always happy to help or discuss scientific
problems. He also introduced me to the great Dutch culture which I previously only
knew from German Autobahns. Many good ideas came up during sauna sessions
after the gym, camping trips, dinners, barbecues or after watching soccer games. I
am also very happy that he met Leyun Zang during my time as a PhD-student and
both got married. I thank both of you for your help and the great time we had. I
wish you all the best and good luck with your driver’s licenses!
My parents have always supported me and guided me on my way through life
in order to make the right choices. They have made sure that I could focus on the
work and offered me the comfort and open-mindedness of a Polish family. Although
scattered across the world, my entire family has supported me in my plans over all
102 Acknowledgements
the years and I have very much enjoyed the few but merry gatherings. Thank you
for being such great parents and thank you to the rest of our entire family!
Anulka’s and my path of life met in 2009 and since then we master our common
trail together. Her outgoing character which has even made her learn the compli-
cated Polish language and her good heart are inspiring. She made sure I could focus
on my PhD-project, was understanding when I worked late or during weekends and
her presence has changed our apartment from a shelter to a warm home. Dziekuje
ci za wszystko, Kochanie!
Jocelyn Chen, Oliver Schmidt and Sarah Unterkofler who are part of the bio-
photonics group accompanied me all along the good and bad episodes in the life of
a PhD-student. Thank you for introducing your very own character to the group,
the fun time we had, the many discussions and the great ideas that came up!
I would also like to thank my office colleagues. Philipp Holzer who is a master
motorist and great dialog partner, did not fully understand the principle of refund,
yet and truly works 24/7. Christine Kreuzer, the passionate small-horse rider who
always took great care of our plants and accompanied me from the first to the last
day of my time as a PhD-student. Howard Lee, the first person I have met who can
make phone calls without making a sound, by only moving his lips and who also
supplied us with great and sometimes exotic sweets. Silke Rammler who introduced
me to ice swimming, is always helpful and ready for a chat and taught me how to
draw fibers. Sebastian Stark with his passion for Australia, sports, funny pictures,
eating animals and good drinks, who is a source for good music and fun to spend
time with. Thank you guys, it was a real pleasure to share an office with all of you!
I am also grateful to everyone who has spent their rare spare time together with
me. I have really enjoyed the camping trip to Alpspitze and many other occasions
with Amir Abdolvand, Anna Butsch and Myeong Soo Kang. Thank you Mohiudeen
Azhar for the fun time with you, cooking wonderful Indian food and inviting me to
your home. We have had many barbecues, nice evenings and other fun events with
Nicolas Joly, Luis Prill Sempere, Benjamin Sprenger, Hemant Tyagi, Patrick Ubel
and Marta Ziemienczuk. I have had a great time together with Martin Butryn,
103
Wonkeun Chang, Ana Cubillas, KaFai Mak, Ana Pinto, Markus Schmidt, John
Travers and Andreas Walser at various occasions. Thank you all for that!
I am also thankful to everyone from the group for their help and the nice time
together. Thank you Fehim Babic, Sadegh Bakhtiarzadeh, Andre Brenn, Claudio
Conti, Stanislaw Dorschner, Zeinab Eskandarian, Michael Frosz, Nicolai Granzow,
Xin Jiang, Ralf Keding, Gunther Kron, Alexander Nazarkin whom I wish all the
best and hope that he will recover soon from his illness, Thang Nguyen, Johannes
Nold, Michael Scharrer, Michael Schmidberger, Francesco Tani, Shailendra Varsh-
ney, Frederick Vinzent, Gordon Wong and Joseph Zyss.
Thank you all for making this work possible and for being who you are.
Martin
104 Acknowledgements
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