LASER-Propulsion of Microparticles in Liquid- Filled ...

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.


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


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


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


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].


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.


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.

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


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


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.


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


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.


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


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:


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


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.


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


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


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


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,


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.


5.3.1 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-


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


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


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,


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.


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


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.


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


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,


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


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


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


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-


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


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.


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.


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


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.


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.


(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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