INTRACAVITY OPTICAL TRAPPING WITH
FIBER LASER
a dissertation submitted to
the graduate school of engineering and science
of bilkent university
in partial fulfillment of the requirements for
the degree of
doctor of philosophy
in
physics
By
Fatemeh Kalantarifard
June 2019
Intracavity optical trapping with fiber laser By Fatemeh Kalantarifard
June 2019
We certify that we have read this dissertation and that in our opinion it is fully adequate, in scope and in quality, as a dissertation for the degree of Doctor of Philosophy.
Giovanni Volpe(Advisor)
Fatih ¨Omer ˙Ilday
Alpan Bek
Mehmet Bur¸cin ¨Unl¨u
Seymur Jahangirov
Approved for the Graduate School of Engineering and Science:
Ezhan Kara¸san
ABSTRACT
INTRACAVITY OPTICAL TRAPPING WITH FIBER
LASER
Fatemeh Kalantarifard Ph.D. in Physics Advisor: Giovanni Volpe
June 2019
After Ashkin’s seminal works, optical trapping has been a powerful technique for capturing and manipulating sub micro particles not only in physics research fields but also in biology and photonics. Standard optical tweezers consists of a single beam with Gaussian or profile which focused by a high numerical aperture (NA) water or oil immersion microscope objective. Typically, objective with NA>1.2 is used to provide strong enough gradient forces being able to overcome Brownian fluctuations and gravity and trap the particle stably. On the other hand, compare with high NA, trapping with low NA, has its own advantage and among all the advantages, low local heating of the sample has a particular interest in molecular biology and manipulating living cells. The main concern is that the interaction of trapping laser beam and biological object induces a damage on the specimen which is mainly due to light absorption of the sample. It is, therefore, recom-mended to use NIR (near infrared ) wavelength due to its minimal absorption by water and biological objects. Other important factors that must be considered, to secure the viability of the cell, are spot size of the focused beam and laser power at the sample plane. Thus, it deserves an effort to look for new configurations with low NA with the capability of creating 3D confinement. Standard optical tweezers rely on optical forces that arise when a focused laser beam interacts with a microscopic particle: scattering forces, which push the particle along the beam direction, and gradient forces, which attract it towards the high-intensity focal spot. Importantly, the incoming laser beam is not affected by the particle posi-tion because the particle is outside the laser cavity. Here, we demonstrate that intracavity nonlinear feedback forces emerge when the particle is placed inside the optical cavity, resulting in orders-of-magnitude higher confinement per unit laser intensity on the sample. We first present a toy model that intuitively explains how the microparticle position and the laser power become nonlinearly coupled: The loss of the laser cavity depends on the particle position due to scattering, so the laser intensity grows whenever the particle tries to escape. We describe a
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simple toy model to clarify how the nonlinear feedback forces emerge as a result of the interplay between the particle’s motion and the laser’s dynamics. It also quantifies how and to what extent this scheme reduces the average laser power to which a trapped particle is exposed. In this model, the power and hence trap-ping force are considered to be zero for small particle displacements. However, in reality they have small values that do operate the trap even when the particle is near the equilibrium position. Thus, we need an accurate description of the coupling between the laser and the trapped particle thermal dynamics at equilib-rium to compare with experiments. In particular, accurate simulations can help to associate an effective harmonic potential to the optical trap for small displace-ments from the equilibrium position, and hence to define a meaningful stiffness using the standard calibration methods based on the thermal fluctuations of a trapped particle. We therefore present a series of numerical simulations based on an extended theoretical model, including highly realistic descriptions of the laser dynamics, optical losses incurred by the particle, and the particle’s Brownian mo-tion in order to gain a quantitative understanding of the dynamics of intracavity optical trapping and to guide the experiments. Finally, guided by the simulation results, we have built an experimental setup to prove the operational principle of intracavity optical trapping and experimentally realize this concept by opti-cally trapping microscopic polystyrene and silica particles inside the ring cavity of a fiber laser. One of the major advantages of the intracavity optical trapping scheme is that it can operate with very low-NA lenses, with a consequent large field-of-view, and at very low average power, resulting in about two orders of magnitude reduction in exposure to laser intensity compared to standard optical tweezers. When compared to other low-NA optical trapping schemes, positive and negative aspects can be considered, such as in terms of trap stiffness and av-erage irradiance of the sample. These features can yield advantages when dealing with biological samples. Ultra-low intensity at our wavelength can grant a safe, temperature controlled environment, away from surfaces for microfluidics manip-ulation of biosamples. Accurate studies on Saccharomices cerevisiae yeast cells in near-infrared counterpropagating traps and standard optical tweezers have found no evidence for a lower power threshold for phototoxicity. We observed that we can 3D trap single yeast cells with about 0.47 mW, corresponding to an inten-sity of 0.036 mW µm−2, that is more than a tenfold less intensity than standard techniques.
v
¨
OZET
F˙IBER LAZER˙I ˙ILE OYUK-˙IC
¸ ˙I OPT˙IK YAKALAMA
Fatemeh Kalantarifard Fizik, Doktora
Tez Danı¸smanı: Giovanni Volpe Haziran 2019
Arthur Ashkin’in ufuk a¸cıcı ¸calı¸smalarından bu yana, optik yakalama tekni˘gi, mikropar¸cacıkların yakalanması ve manip¨ulasyonunda g¨u¸cl¨u bir teknik olarak kar¸sımıza ¸cıkarak, yalnızca fizik de˘gil, biyoloji ve fotonik alanlarında da kullanılan ¨onemli ve yaygın bir metod olagelmi¸stir. Standart optik cımbızlar, Gauss’yan ¨ozellik ta¸sıyan sade ı¸sık demetinin odaklanmasıyla olu¸sturulurlar. Bu ı¸sın demeti, mercek a¸cıklı˘gı y¨uksek olan objektiflerle su ya da ya˘g i¸cerisine batırılarak, ¨orne˘ge odaklanır ve optik cımbız olu¸sturulur. Genellikle, optik a¸cıklı˘gı NA>1.2 ¨ozelli˘gini ta¸sıyan objektifler tercih edilir. Bunun sebebi de, Brownian titre¸simleri ve yer ¸cekimi kuvvetine baskın gelerek, cımbızlanmanın ger¸cekle¸sebilmesi i¸cin g¨u¸cl¨u en-lemsel kuvvetler sa˘glanmasının gereklili˘gidir. Y¨uksek optik a¸cıklıklı objektif kul-lanmanın yanıba¸sında, d¨u¸s¨uk optik a¸cıklıklı objektif kullanmanın da bazı avan-tajları vardır. Ozellikle, molek¨¨ uler biyolojide ve ya¸sayan h¨ucrelerin manip¨ule edilmesinde b¨olgesel ısınmanın azaltılması i¸cin d¨u¸s¨uk optik a¸cıklı˘gı olan objek-tifler tercih edilmektedir. Burada ana problem, ı¸sı˘gın so˘gurulmasına sebebiyle ¨orne˘gin ısınması neticesinde, yakalayıcı lazer demeti ile biyolojik objenin etk-ile¸siminin, ¨ornek ¨uzerinde hasara yol a¸cmasıdır. Bu y¨uzden genelde kırmızıya yakın dalga boyundaki ı¸sı˘gın kullanılması tavsiye edilir ki, su ve biyolojik objeler tarafından en az so˘gurulan rejim, kırmızıya yakın dalgaboylu rejimdir. H¨ucrenin ya¸samsal faaliyetlerinin g¨uvence altına alınması i¸cin di˘ger bazı ¨onemli fakt¨orler de, odaklanmı¸s ı¸sı˘gın spot geni¸sli˘gi ve ¨ornek ¨uzerindeki lazer g¨uc¨ud¨ur. Bu y¨uzden, d¨u¸s¨uk optik a¸cıklı˘ga sahip, 3 boyutlu yakalama yapabilecek yeni d¨uzenekler ara¸stırmaya de˘gerdir. Standart optik cımbızlar, odaklanmı¸s lazer demetinin mikropar¸cacıklarla etkile¸siminden do˘gan optik kuvvetlerin (sa¸cılan y¨ondeki ve demet y¨on¨undeki kuvvetler) kullanımı ¨uzerine in¸sa edilegelmi¸stir. Buradaki ¨onemli nokta, gelen lazer demeti, par¸cacı˘gın pozisyonuna ba˘glı de˘gildir ¸c¨unk¨u par¸cacık lazer hapsinin i¸cerisinde de˘gildir. Burada par¸cacı˘gın optik kavitenin i¸cine yerle¸stirilmesi durumunda ortaya ¸cıkan i¸cten kaviteli lineer olmayan geri besleme kuvvetlerini g¨osteriyoruz, bu kuvvetler birim lazer ¸siddeti ile b¨uy¨ukl¨uk
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mertebesi birka¸c kez daha y¨uksek optik hapsedilme sa˘glamaktadır. Oncelikle¨ mikropar¸cacık pozisyonu ile lazer g¨uc¨un¨un nasıl lineer olmayan ¸sekilde birbirine ba˘glı oldu˘gunu g¨osteren bir model sunuyoruz: Lazer kavitesindeki kayıp par.acı˘gın pozisyonuna sa¸cılma sebebi ile ba˘glıdır, bu nedenle lazer ¸siddeti par¸cacık her ka¸cmaya ¸calı¸stı˘gında y¨ukselir. Bu model ile par¸cacık pozisyonu ve lazer dinamik-lerinin kuvvetdinamik-lerinin kar¸sılıklı etkile¸simi sonucu a¸cı˘ga ¸cıkan lineer olmayan geri besleme kuvvetlerini a¸cıklıyoruz. Ayrıca bu metodun par¸cacıkları tuzaklamak i¸cin gerekli ortalama lazer g¨uc¨un¨un nasıl ne ne derece azalttı˘gını ¨ol¸c¨uyoruz. Bu modele g¨ore lazer g¨uc¨u ve dolayısıyla kuvvet par¸cacı˘gın k¨u¸c¨uk yer de˘gi¸stirmeleri i¸cin ihmal edilebilir derecede k¨u¸c¨ukt¨ur ve sıfır kabul edilir. Ancak ger¸cek hay-atta k¨u¸c¨uk kuvvetler par¸cacık denge noktasının etrafındayken de etki eder. Bu nedenle lazer ve tuzaklanan par¸cacı˘gın denge halindeki termal dinamikleri ve lazer arasındaki etkile¸simin hassas olarak tanımlanması gerekmektedir. ¨Ozellikle par¸cacı˘gın optik tuzak i¸cindeki k¨u¸c¨uk yer de˘gi¸stirmelerinde etkili olan efektif har-monik potansiyelin hassas ¸sekilde simule edilmesi ve standart metotlarla kalibre edilmesi gerekmektedir. Bu nedenle bir seri simulasyon ile var olan teorik modeli ileri g¨ot¨urerek, lazer dinamiklerini par¸cacı˘gın Brown hareketi ve optik kayıplar da hesaba katılarak, i¸cten kaviteli optik tuzaklamanın dinamiklerini anlamak ve deneyleri y¨onetmek amacıyla nicel olarak tanımlayan bir model sunuyoruz. Son olarak, simulasyon sonu¸cları ı¸sı˘gında i¸cten kaviteli optik tuzaklama y¨ontemini deneysel olarak realize ediyoruz ve bunu silicon ve polistiren mikropar¸cacıklarını fiber lazerin halka kavitesinin i¸cinde tuzaklayarak g¨osteriyoruz. ˙I¸cten kaviteli optik tuzaklamanın ba¸slıca avantajları d¨u¸s¨uk n¨umerik a¸cıklıklı objeltifler ile ve dolayısıyla b¨uy¨uk g¨or¨unt¨uleme alanlı sistemlerle kurulabilmesi ve ¸cok d¨u¸s¨uk orta-lama lazer g¨u¸clerinde ¸calı¸sabilmesidir, bu sayede optik tuzaklama standart optik cımbızlara g¨ore iki mertebe daha d¨u¸s¨uk lazer ¸siddeti ile sa˘glanmaktadır. Di˘ger d¨u¸s¨uk n¨umerik a¸cıklı˘ga sahip optik tuzaklama sistemleriyle kıyaslandı˘gında tuza-klama ¸siddeti ve ortalama rasyasyon gibi belli ba¸slı farklılıklar biyolojik ¨orneklerle ¸calı¸sırken ¨onemli avantajlar sa˘glamaktadır. Kullanıcan dalga boyunda ¸cok d¨u¸s¨uk lazer g¨uc¨unde ve sıcaklık kontrol¨undeki ortam mikrobiyolojik ¨orneklerin g¨uvenli bir ¸sekilde ve y¨uzeyden izole olarak tuzaklanmasını sa˘glamaktadır. Hassas ¸calı¸smalar Saccharomice ceverevisiae maya h¨ucrelerinin kar¸sılıklı ilerleyen lazer tuzaklar ile tutulan par¸cacıkların fototoksik olarak lazer g¨uc¨unde bir alt limitinin olmadı˘gını g¨ostermektedir. Bizler bu ¸calı¸smada maya h¨ucrelerini 0.47 mW g¨u¸c ve 0.036 mW µm−2 enerji yo˘gunlu˘gu ile standart tekniklere kıyasla on kattan daha
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Anahtar s¨ozc¨ukler: Optical trapping, Optical tweezers, Fiber laser, Feedback, Cellulose fibers, Raman tweezers.
Acknowledgement
The research work presented in this thesis is the result of a scientific and per-sonal growth path started several years ago, a period during which I had the opportunity and the privilege to meet extraordinary people and get in touch with different scientific realities. Each person and place I met during my PhD gave me something or left me a sign which inevitably has contributed to get me this far with this work, which I wish were as a springboard to the future. First and foremost, I would like to thank my supervisor and mentor Dr. Giovanni Volpe for giving me the beautiful opportunity to join his research group, to travel and enter in contact with several scientific realities. He has all my respect for having created a scientific environment where I had the chance to manage and organize my work and studies. I am thankful to Dr. Parviz Elahi for my training and the time spent together in the laboratory performing experiments and building new setups. Thanks to him for showing me the experimental life in a laboratory and the patience needed to carefully build and align the several OT setups. A special thanks to Dr. Ghaith Makey for being a wonderful person and scientist, for his abilities for image processing and for always having new ideas when one was failed. Special thanks to Dr. F. ¨Omer Ilday for his knowledge, experience, for his scientific advice and ideas and for his careful, detailed revision of our article that improved it to a remarkably higher level. Thanks to Dr. Agnese Callegari for teaching me the Geometrical optics toolbox that is the base of my numerical simulations. I am very grateful to my other group mates in Advanced Research Laboratories (ARL) of Bilkent university, for scientific discussions and for their amazing friendship, interesting conversation and the great time spent together, especially to S. Maasoumeh Mousavi, Sabareesh K. P. Velu, Aykut Ar-gun, Alessandro Magazz`u, Tu˘gba Anda¸c and Jalpa Soni. My PhD experience was greatly enhanced by the opportunities to collaborate with excellent researchers from other institutions thanks to COST project (MPNS COST Action 1305), which financially supported my Short-Term Scienti c Missions at CNR-IPCF, Messina, Italy. I would especially like to thank Dr. Onofrio M. Marag´o first for realizing the amazing potential of intracavity optical trapping approach and, in addition, for giving me the great opportunity to work on different studies at
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his laboratory. Thanks to him for giving me the chance to learn about optical trapping of nanofibers, Raman spectroscopy and Raman tweezers and also for his positive thoughts and encouragements. Thanks also to Maria G. Donato and her help and scientific discussions during my visit at CNR-IPCF. A special thanks to Antonella for her beautiful friendship, help and support and also to all the mem-bers of the extraordinary research group for their help and support, especially to Antonino Foti, P. G. Gucciardi, et al. I also extend my thanks to Dr. Alpan Bek for the valuable discussion, and to Dr. Burcin ¨Unl¨u and Dr. Seymour Jahangirov for reading the thesis and helpful feedback. At last but not least, thanks to all the people who supported and stood by me, my parents, my siblings and my sisters- in- law and especially to my husband and my son for their precious help and their patience.
Contents
1 Introduction 1
1.1 A historical overview on optical tweezers . . . 1
1.2 Optical trapping forces in different ranges . . . 3
1.2.1 Geometrical optics approximation for small particles . . . 5
1.2.2 Dipole approximation for very small objects . . . 9
1.2.3 Electromagnetic theory and intermediate regime . . . 14
1.3 Application of optical tweezers . . . 15
1.3.1 Biophysics . . . 15 1.3.2 Cellular biology . . . 17 1.3.3 Optofluidics . . . 19 1.3.4 Spectroscopy . . . 20 1.3.5 Plasmonics . . . 21 1.3.6 Colloid science . . . 21
2 Standard optical tweezers 22 2.1 Trapping . . . 23 2.1.1 Laser beam . . . 24 2.1.2 Objective . . . 25 2.1.3 Sample . . . 27 2.2 Imaging . . . 27 2.3 Position detection . . . 28
2.3.1 Digital video microscopy . . . 28
2.3.2 Interferometry . . . 29
2.4 Calibration techniques . . . 31
CONTENTS xii
2.4.2 Power spectrum analysis . . . 35
2.4.3 Potential analysis . . . 37
2.4.4 Mean square displacement analysis . . . 39
3 Intracavity Optical Tweezers 41 3.1 Introduction . . . 41
3.2 Working principle . . . 44
3.3 Toy model . . . 44
3.3.1 Model for laser dynamics. . . 45
3.3.2 Stationary probability density of the particle position. . . . 46
3.3.3 Analytical expressions for σ2 r and Pave. . . 47
3.3.4 Rate equations for the laser dynamics . . . 53
4 Intra-cavity optical trapping (Simulation) 55 4.1 Introduction . . . 55
4.2 Simulation . . . 56
4.2.1 Rate equations for the laser dynamics . . . 58
4.2.2 Trapping equations . . . 58
4.3 Results and discussion . . . 59
4.3.1 Force and power versus displacement of a particle without thermal fluctuations . . . 59
4.3.2 Trapped particle in the intracavity optical tweezers . . . . 60
4.3.3 Analysis of the optical potential generated by the intracav-ity optical trap . . . 63
4.3.4 More detailed analysis and more particles . . . 64
4.4 Comparison . . . 66
4.4.1 Intracavity OT versus standard OT with low NA lens . . . 69
4.4.2 Intracavity OT versus standard OT with high NA objective 70 4.4.3 Intracavity optical trapping for small particles . . . 72
5 Intra-cavity optical trapping (Experiment) 76 5.1 Experiment . . . 76
5.2 3D digital video microscopy . . . 79
CONTENTS xiii
5.4 Experimental results . . . 84 5.4.1 Analysis of the optical potential generated by the
intracav-ity optical trap . . . 84 5.4.2 Temporal dynamics of fiber laser, intracavity trapping, and
Brownian fluctuations . . . 87 5.4.3 More detailed and more particles . . . 88 5.5 Comparison . . . 91
5.5.1 Comparison of intracavity optical trapping with standard OTs with high NA . . . 91 5.5.2 Comparison of intracavity optical trapping with other
op-tical trapping schemes . . . 92 5.6 Yeast cell . . . 96
List of Figures
1.1 Comet Hale-Boop was one of the most shiny comet in the sky in 20th century. The diffusive tail is due to the sun’s light force and the blue tail is from the solar wind and composed of charged particles. Credits: E. Kolmhofer, H. Raab; Johannes-Kepler-Observatory, Linz, Austria. . . 2 1.2 Optical trapping ranges. Different sizes objects that are trapped
in typical optical trapping experiments. Trapping wavelength is usally assumed to be visible or near infra red (NIR). Reproduced from ref. [10] . . . 4 1.3 Multiple scattering of a light ray on a spherical particle. . . 5 1.4 Optical forces in OT. Each ray which is scattered by the particle
will have a change in the momentum and the momentum change is the cause of the reaction force on the particle towards the beam center. The net force on the particle, F has two components: The component in the direction of beam propagation is scattering force, Fs which pushes the particle away from the center of trap.
The component perpendicular to the direction of incoming beam is called gradient force, Fg and pulls the particle towards the optical
LIST OF FIGURES xv
1.5 Optical forces in OT for the case np > n. (a) The particle is not
centered with the laser beam, changes in the momentum of the rays is causing a net reaction force towards the beam center. Total force, Ft, which in this case is the total radial gradient force pulls
the particle to the center of the beam. (b) when the particle is lo-cated below the focal point, along the beam propagation, the axial gradient produces a net restoring force towards the focal point. (c) Also in case of a particle beyond the focal point the axial gradient force is towards the center of focus point. . . 7 1.6 Probing DNA thermal fluctuation. (a) Experimental configuration
and (b) theoretical model of thermal fluctuation in a single DNA molecule. Reproduced from ref. [34]. . . 16 1.7 Bacterial adhesion force. The bead which is trapped by optical
tweezers is coated with receptor molecules and is brought near a bacterium stuck on a large bead’s surface. By moving the stage by a distance xlb and as a result displacement of the small bead,
xsb, the bond is measured by deflection of the beam. Reproduced
from ref. [38]. . . 17 1.8 Neuronal growth. Flat (left column) and tubular (right column)
neuronal cones. the position of the laser spot is shown with a circle. Frames are taken at intervals of 10 minutes (left column) and 5 minutes (right column). Reproduced from ref. [39]. . . 18 1.9 Raman spectra of red blood cells that are optically trapped. The
spectra of normal and β-thalassemic RBCs are compared. Dashed lines indicate the observed energy shifts and arrows highlight spec-tral features influenced by intensity changes. Reproduced from ref. [50]. . . 20
2.1 Basic optical tweezers setup. Digital video microscopy. . . 23 2.2 Basic optical tweezers setup. Interferometry. . . 24
LIST OF FIGURES xvi
2.3 Autocorrelation functions (ACFs) corresponding to a silica 4 µm bead trapped in three dimensions. The symbols are the experimen-tal ACFs and the solid lines are the theoretical fitted parameters. The lateral trap in (a) x and (b) y is stronger than axial trap, (c) z as ACF in x and y directions decay faster than in z direction. . 34 2.4 Power spectral densities (PSDs) corresponding to a silica 4 µm
bead optically trapped. The symbols are the experimental PSDs and the solid lines are the theoretical fitted parameters in the di-rections x (a), y (b) and z (c). . . 37 2.5 Histograms corresponding to a silica 4 µm bead optically trapped.
The symbols are the experimental histograms and the solid lines are the theoretical fitted parameters in the directions x (a), y (b) and z (c). . . 37 2.6 Mean square displacements (MSDs) corresponding to a silica 4 µm
bead optically trapped. The symbols are the experimental MSDs and the solid lines are the theoretical fitted parameters.The lateral trap in (a) x and (b) y is stronger than axial trap, (c) z as the value of plateau in x and y directions are smaller than in z direction. 40
LIST OF FIGURES xvii
3.1 Intracavity optical trapping. The trapping optics (collimators C1
and C2, lenses L1 and L2) are placed within the cavity of a ring
fiber laser so that the position of the particle can influence the cavity loss. (a) When the particle is not in the trap region, the optical loss of the cavity is low, the intracavity laser power P is high, and consequently the particle is attracted towards the center of the trap. The laser power scaling curve (solid line) shows that the pump power Ppump (vertical dashed line) is above the lasing
threshold. (b) When the particle is at the center of the trap region, cavity losses due to scattering of light out of the cavity by the par-ticle are maximum. The power scaling curve is right-shifted and the laser is below or barely above threshold for the same Ppump.
The particle is not strongly trapped. (c) When thermal fluctua-tions displace the particle away from the trap region, the optical loss of the cavity decreases, P increases, and the particle is pulled back towards the center of the trap. . . 43 3.2 Toy model of the dependence of the laser power on the particle
position. (a) Laser power P (r) as a function of particle position r (Eq. 3.6) employed in the toy model (dashed line) and the the actual laser power including saturation (solid line). (b) Corre-sponding probability density of the particle position (Eq. 3.9, with parameters P0 = 3 mW, rL = 0.5 µm, κP = 0.1 pNµm−1mW−1,
and T = 300 K). The solid line represents the the probability density of the particle position obtained with a standard optical tweezers employing the same average power. . . 49
LIST OF FIGURES xviii
4.1 Block diagram of the simulation. The simulation consists of three major software packages: RR performs the simulation of the ring-resonator laser; S performs the simulation of the optical forces and scattering of the particle at the focus; and BM performs the simu-lation of the Brownian motion of the particle. The three packages share information as shown by the arrows: RP receives in input the information on the optical losses from S; S receives in input the information on the optical power from RR and on the position of the particle from BM; BM receive in input the information on the optical forces from S. . . 56 4.2 Force and power versus displacement.(a-b) Simulations of the
ra-dial and axial force, and (c-d) corresponding intracavity power for a 4.9-µm-diameter polystyrene particle displaced from its equilib-rium position. The solid lines are linear fits that show the linearity of the force for small displacements, i.e., Fx ≈ −kxx. For large
dis-placements the nonlinear behaviour dictated by the increased op-tical feedback related to the reduction of scattered light is clearly visible. . . 59 4.3 Simulation results. (a) Ray optics diagrams of the propagation of
a focused beam through an optically trapped particle (i) when the particle is at the center of the trap (equilibrium position in the trap that takes into account also the effective gravitational force acting on the particle), (ii) when it is displaced in the radial direction, (iii) when it is displaced along the axial direction downwards and (iv) upwards. (b) Radial (r) and axial (z) particle position, and corresponding laser power (P ) obtained from the simulation of the motion of a 4.9-µm polystyrene particle trapped in the intracavity optical trap. (c) Dependence of the laser power on the radial and axial position of the particle. The points (i)-(iv) correspond to the configurations in (a) and the dashed lines correspond to the insets graphs the dependence on z and r on the left and bottom respectively. . . 62
LIST OF FIGURES xix
4.4 Analysis of the simulation results. (a) Probability distribution, (b) position autocorrelation functions (ACFs), and (c) power spectral densities (PSDs) along the transverse x-direction for a simulated trajectory of a 4.9-µm-diameter polystyrene particle held in the in-tracavity optical trap (corresponding to the results shown in Figure 3). (d-e) Probability distribution, ACF and PSD along the axial z-direction. The solid lines are the best fit to the data from which we calculate the trap stiffnesses of the simulated intracavity trap. These analyses are consistent with a Hookean force profile for small displacements (r < 1 µm) and short times (t < 1 s), which per-mits us to perform a meaningful calibration of the optical trapping forces using standard techniques. . . 63 4.5 Simulation results for 4.9 µm polystyrene particle in intracavity
OT. Autocorrelation functions of r (a) and z (b) positions. Sym-metric or summation cross correlation functions of radial position and power (c) and axial position and power (d). . . 65 4.6 Simulation results for 6.24 µm polystyrene particle in intracavity
OT . . . 66 4.7 Simulation results for a silica 2.8 µm particle in intracavity OT . 67 4.8 Simulation results for a silica 4 µm particle in intracavity OT . . 67 4.9 Simulation results for a silica 4.8 µm particle in intracavity OT . 68 4.10 Comparison of the axial forces for intracavity and low NA standard
trapping. Total axial force calculated for a 4 µm silica particle in intracavity OT (filled circles) and standard OT with low NA lens (open circles) . . . 69
LIST OF FIGURES xx
4.11 Uncorrelated power-position map in standard OTs using high NA objective. Simulations of a high NA trap operating at a power of about 0.12 mW corresponding to the intracavity average trap power (a) and at power of 3 mW that is close to the trapping threshold in experiments (b). The trap power is decoupled by the particle fluctuations and hence the color map is constant at the specified value. The positional fluctuations follow the Gaussian distribution typical of an optical tweezers that for small displace-ments is well approximated by a harmonic potential. Note that in both cases the intensity of laser light at sample is 2 order of magnitude more than that of the intracavity trap. Moreover, the actual threshold trapping power in (b) is much higher than the intracavity case. . . 70 4.12 Comparison of intracavity with high NA standard trapping.
Com-parison of the simulation results of inverse radial and axial trap confinement (σ−2
r and σz−2, respectively) per unit intensity at the
sample for an intracavity optical trap (circles) and a standard high-NA optical tweezers (squares) for polystyrene (a-b) and silica (c-d) particles of various diameter 2R. . . 71 4.13 Intracavity optical loss for small particles. (a) Optical loss as a
function of displacement calculated for our experimental param-eters (NA=0.12, λ0 = 1030 nm) in the dipole approximation for
polystyrene particles with radii of 300, 400, 500, and 600 nm (ma-genta). The optical loss profile follows the Gaussian intensity pro-file of the incident laser beam. (b) Size scaling of the optical loss in the dipole approximation for polystyrene particles. Since the particles are non-absorbing the scaling rapidly decreases as R6 for
small radii. This clearly shows that, for our experimental parame-ters, the intracavity feedback trapping is efficient at the microscale while it reduces to a standard single-beam optical trapping at the nanoscale. . . 73
LIST OF FIGURES xxi
5.1 Experimental setup. (a) The setup comprises a diode-pumped Yb-doped fiber laser, the trapping optics, and the digital video micro-scope. (b) Measured power scaling with a trapped 4.9-µm-radius polystyrene particle (orange squares) and without the trapped par-ticle (red circles). At a pump power of 66 mW (dashed verti-cal line), the laser is below threshold with the particle (orange squares), but above threshold without the particle (red circles). . 77 5.2 Detection of axial displacements. A sequence of calibration a)
raw b) digitally processed and enhanced images of a stuck particle recorded at different depths by moving the sample away from the lens with 1 µm step. Each image in b) is translated, azimuthally averaged around the detected center and cropped. Notice that this image processing is applied for every frame of experimental data continued by comparing them to the calibration images in b by calculating normalized cross correlation. . . 81 5.3 An example of output result of the MATLAB code for tracking of
the particle in the intracavity optical tweezers using digital video microscopy technique plus our algorithm for axial direction. The code takes a recorded frame (a), after translating, averaging and cutting (b), compares it with the calibration images and finds the one that has the closest shape (c) and finally in (d) it plots the axial position (green), radial position (blue) and power (red). . . 82 5.4 Effects of the trapped particle on the laser threshold. Feedback
power in the absence (blue curves) and presence (red curves) of a stuck a) silica 4 µm and b) polystyrene 4.9 µm particle versus input current. A polystyrene particle (b) shows more reduction in the power compared to that of the silica bead in (a). This is because refractive index of polystyrene (1.59) is higher than refractive index of silica (1.42), scatters more light and applies more loss to the cavity. . . 83
LIST OF FIGURES xxii
5.5 Experimental results. (a) Radial (r) and axial (z) particle position, and corresponding laser power (P ) for a 4.9-µm-radius polystyrene held in the intracavity optical trap. (b) Dependence of the laser power on the radial and axial position of the particle. The posi-tions indicated with i, ii, iii and iv correspond to particle position and laser power configurations similar to those illustrated in the diagrams shown in Fig. 4.3a. The points (i)-(iv) correspond to the configurations shown in Fig. Fig3a, and the dashed lines cor-respond to the insets graphs the dependence on z and r on the left and bottom respectively. . . 85 5.6 Analysis of experimental results. (a) Probability distribution, (b)
position autocorrelation functions (ACFs), and (c) power spectral densities (PSDs) along the transverse x-direction for an experi-mental trajectory of a 4.9-µm-diameter polystyrene particle held in the intracavity optical trap (corresponding to the results shown in 5.5). (d-e) Probability distribution, ACF and PSD along the axial z-direction. The solid lines are the best fit to the data from which we calculate the trap stiffnesses of the simulated intracavity trap. These results are in agreement to those shown in Supple-mentary Figure 4.4. These analyses are consistent with a Hookean force profile for small displacements (r < 1 µm) and short times (t < 1 s), which permits us to perform a meaningful calibration of the optical trapping forces using standard techniques. . . 86
LIST OF FIGURES xxiii
5.7 Temporal dynamics of intracavity optical trapping. Temporal dy-namics of a 4.9-µm-diameter polystyrene particle during intracav-ity trapping. The particle position is tracked (a) while the particle reaches its equilibrium position. As the particle is trapped the ring-laser dynamics changes (b) and intracavity power drops to the low-level value corresponding to the increased optical losses. The slopes of both the particle position and power drop corre-spond to about 100 ms, clearly showing how the particle position directly affect the laser operation. The insets in (a) and (b) repre-sent the position and corresponding power for a shorter time scale where the particle is falling into the trap. (c) Measured response of the laser cavity to a dynamical loss produced using an AOM (acousto-optic modulator) inside the cavity. The inset represents the zoomed in data where the signal drops. As shown in the inset, 10-90% response time of the laser is about 20 ns. . . 87 5.8 Experimental results for 4.9 µm polystyrene particle in intracavity
OT. Autocorrelation functions of r (a) and z (b) positions. Sym-metric or summation cross correlation functions of radial position and power (c) and axial position and power (d). . . 89 5.9 Simulation results for 6.24 µm polystyrene particle in intracavity
OT . . . 90 5.10 Experimental results for a silica 2.8 µm particle in intracavity OT 91 5.11 Experimental results for a silica 4 µm particle in intracavity OT . 92 5.12 Experimental results for a silica 4.8 µm particle in intracavity OT 93 5.13 Comparison of intracavity with high NA standard optical trapping.
Comparison of the experimentally measured inverse radial and ax-ial trap confinement (σ−2
r and σz−2, respectively) per unit intensity
at the sample for an intracavity optical trap (circles) and a stan-dard high-NA optical tweezers (squares) for polystyrene (a-b) and silica (c-d) particles of various diameter 2R. The dashed lines are the corresponding results from numerical simulations. . . 94
LIST OF FIGURES xxiv
5.14 Comparison of different optical trapping schemes. We compare our intracavity nonlinear-feedback trapping scheme to standard optical tweezers (OT) using a high numerical aperture objective (NA=1.3) and trapping the same polystyrene particle with a diameter of 4.9 µm as well as to other optical trapping schemes proposed in the literature, namely counter-propagating beam trapping (CPOT) ( dual-beam optical trapping [104], mirror optical trapping (MT) [96], and SIBA trapping [63] in the main text). For a quantitative comparison we consider various parameters, such as intensity, I, at the sample and trap stiffness per unit intensity, κ/I, for polystyrene particles of similar size (when possible). Note how the intracavity optical trapping scheme has the lowest intensity at the sample with the highest stiffness per unit intensity. . . 95 5.15 Trapping yeast cell by means of intracavity optical tweezers. A
yeast cell in the intracavity optical tweezers at laser off (a) and on (b). Average power on the trapped yeast cell is about 0.47 mW and the intensity of laser light at sample is about 0.036 µm−2. . . 97
Chapter 1
Introduction
1.1
A historical overview on optical tweezers
The idea of that light can exert forces by radiation pressure was known in 17th century when Johannes Kepler announced that (in his work Decometis) a comet tail is, in fact, due to the interaction of Sun’s rays and the particles around the body of comet. As seen in the picture (figure 1.1), there are two different tails, formed by dust and ions. The former depends on the radiation pressure of the solar light and the latter is based on solar wind. Two centuries later, in 1873, Maxwell equations explained the radiation pressure within electromagnetic theory clarifying that light has the ability of exerting force on matter due to momentum exchange. This momentum exchange is so small that only in 1901 its existence was showed in the experimental demonstration [1, 2]. Later on, first experimental observation of optical torque on a macroscopic object was reported by Beth in 1936 [3]. Since optical forces are extremely small, only after the invention of laser [4] by using coherent light, it has been possible to observe the effect of them on the motion of microscopic objects. This observation in the early 1970s, in fact, led to the invention of optical tweezers by ArturAshkin. In the early 1970s, Artur Ashkin and his co-workers showed that the motion of neutral micro-spheres [5] and atoms [6] could be affected by laser-induced optical
Figure 1.1: Comet Hale-Boop was one of the most shiny comet in the sky in 20th century. The diffusive tail is due to the sun’s light force and the blue tail is from the solar wind and composed of charged particles. Credits: E. Kolmhofer, H. Raab; Johannes-Kepler-Observatory, Linz, Austria.
forces . He observed an unexpected phenomena that transparent neutral micro-particles were attracted perpendicular to the direction of beam propagation while pushed in the axial direction [7]. This effect identifies two kinds of forces by light: gradient f orce caused by the focalization of laser beam [7] and scattering f orce that comes from radiation pressure. After discovering of gradient force, Ashkin demonstrated an optical trap using two counter propagated laser beams to get a stable trap by balancing the radiation pressure forces. He could move the trapped dielectric particle along the beam propagation direction by changing the power of one laser. Then, in 1971, he made first levitation trap assisted by gravity in order to balance the radiation pressure and trap the particle [8]. In 1986, Ashkin and his colleagues reported the first demonstration of an optical tweezers using a highly focused laser beam by means of a high numerical aperture objective [9]. In this configuration, the focal spot is so tight that is possible to create a gradient force along the propagation direction too. This force is always towards the focal spot and capable of trapping and manipulating of dielectric microscopic particles and atoms.
1.2
Optical trapping forces in different ranges
Typically objects ranging from tens of nanometers to tens of micrometers, such as cells, micron-sized dielectric particles or nano-sized metallic beads, are optically manipulated. Optical trapping force is a result of conservation of electromagnetic momentum in the interaction between light and matter. Considering the ratio between the characteristic dimension L of the object and the wavelength λ of the trapping light, three different trapping regimes can be defined. In figure 1.2 [10] an overview of the kind of objects belonging to each of these regimes is presented, assuming that the trapping wavelength is in the visible or NIR spectral region [11]. In any of these regimes, the electromagnetic equations can be solved to evaluate the force acting on the object. However, this can be a cumbersome task to use a full electromagnetic theory based on Maxwell’s equations [12] to study the scattering process involving non-spherical or non-homogeneous particles for instance. For the Rayleigh regime and geometrical optics regime approximated
Figure 1.2: Optical trapping ranges. Different sizes objects that are trapped in typical optical trapping experiments. Trapping wavelength is usally assumed to be visible or near infra red (NIR). Reproduced from ref. [10]
models have been developed, which allow one to gain insights into the physics of optical trapping. However, most of the objects that are normally trapped in are placed in the intermediate regime region, where the Rayleigh or geometrical optics approximation cannot be used. For a better and easier understanding of optical forces, simpler approximations have been made:
kL = 2πnm λ0
L (1.1)
Where k = 2πnλ
0 is the light wave number in the medium , L is the characteristic
dimension of the object, λ0 is the wavelength of the trapping laser in the vacuum
and nm is the refractive index of the medium, usually water (nm=1.33).
For the particles bigger than the laser wavelength, where kL >> 1, to calculate optical forces created by optical tweezers, remarkable simplifications, geometrical optics approximation or Ray optics regime, has been applied. The accuracy of this approximation increases with the size parameter and exact theories are
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%,)Figure 1.3: Multiple scattering of a light ray on a spherical particle. not useful any more because as the size of particle increases, the computational complexity rises up as well. When the size of particle is comparable with the wavelength of the laser light (kL ≈ 1), intermediate regime, a complete wave optics model for light-particle interaction is needed to obtain the optical forces. For the particles at size much smaller than the wavelength, (kL << 1), we can apply Rayleigh regime which the object is approximated with a dipole [13] or a number of dipoles [14].1.2.1
Geometrical optics approximation for small
parti-cles
In the geometrical optics approximation, the optical field is described as a set of N rays which carry a certain portion, Pi, of the whole incident power, P = NPiPi.
The incident power P is defined as the number of photons per unit of time multiplied by the energy of the photon: P = N v, where v is the energy of photon, v = λhc
0. Thus, the linear momentum of a ray per second can be obtained
as: nPi
c . To a better understanding how trapping occurs, we consider an incident
ray carries N photons incoming on a dielectric spherical particle with a refractive index higher than the refractive index of medium. When the ray hits the surface
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b
Figure 1.4: Optical forces in OT. Each ray which is scattered by the particle will have a change in the momentum and the momentum change is the cause of the reaction force on the particle towards the beam center. The net force on the particle, F has two components: The component in the direction of beam propagation is scattering force, Fs which pushes the particle away from the center
of trap. The component perpendicular to the direction of incoming beam is called gradient force, Fg and pulls the particle towards the optical axis when np > n.
of the particle, it is reflected back and also refracted inside the particle. The refracted ray leads to a change of direction and this, in turn, changes the linear momentum associated to the ray. This change in the momentum will consequently cause a force on the center of mass of the particle as seen in the figure 1.4. In fact, when a ray Ri impinges on the particle, its power splits into two rays, reflected
ray, Rr(0) and transmitted ray, Rt(0). However, most of the power is carried by
the transmitted ray that passes through the particle until it hits the opposite surface which it will be reflected and transmitted again. A large portion of power is delivered to the ray transmitted outside the particle, Rt(1) and this process
will continue until all light is scattered outside particle (figure 1.3). Here we can calculate the optical force inserted to the particle by the reflected and refracted rays and changes in the momentum: [15, 16]
FR = nPi c Rˆi− nPr c Rˆ (0) r − ∞ X j=1 nPt(j) c Rˆ (j) t (1.2)
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centered with the laser beam, changes in the momentum of the rays is causing a net reaction force towards the beam center. Total force, Ft, which in this case is
the total radial gradient force pulls the particle to the center of the beam. (b) when the particle is located below the focal point, along the beam propagation, the axial gradient produces a net restoring force towards the focal point. (c) Also in case of a particle beyond the focal point the axial gradient force is towards the center of focus point.
Where ˆRi, ˆR (0)
r and ˆR(j)t are the unit vectors represent the direction of the
incident ray, reflected 0-ray and jth transmitted rays respectively. Pi, Prand Pt(j)
are the power of incident, reflected and transmitted rays obtained from Fresnel’s coefficients [16]. The optical force presented in equation 1.2 has components in the plane of incident only because all reflected and refracted rays are placed in this plane. The ray force FR can be split into two components that are perpendicular.
One in the direction of incident ray representing optical scattering f orce and the another component perpendicular to the incoming ray represents gradient f orce. If np > n the scattering force pushes the particle towards the direction of
incoming ray and gradient force pulls it to the optical axis (high intensity focal region) (figure 1.5). Instead, when np < n, (for microbubbles for instance [17]),
the particle will be pushed away from the optical axis.
FR = FR,scat+ FR,grad = FR,scatRˆi+ FR,gradRˆ⊥. (1.3)
Here we can define a dimensionless quantity, trapping ef f iciency that is ob-tained by dividing the force components by nPi
c : QR,scat = c nPi FR,scat, (1.4) QR,grad= c nPi FR,grad. (1.5)
A. Ashkin obtained the expressions for scattering and gradient efficiency of a circularly polarized ray on a spherical particle based on Fresnel reflection and transmission coefficients (R and T ) and the angle of incidence and transmission (θiand θt) [15]. To model an optical trapping we need to consider a highly focused
laser beam not only a single ray. Therefore, the total force acting on the particle, in geometrical approximation, is the sum of all contributions of each ray that forms the beam:
Fgo=
X
s
A highly focused single beam composed by focused rays will be able to produce a restoring force proportional to the displacement of the particle. Thus, for very small displacements (about particle radius), in one direction, x, the force will be as a harmonic response:
Fx≈ −κx(x − xequilibrium) (1.7)
Where κx is the trap stiffness in x direction and by measuring or calculating
it, we can calibrate the optical trap. For a circularly polarized beam κx and κy
in the plane perpendicular to the beam propagation direction are equal but the axial spring constant, κz is different and typically smaller. Notice that while F
is independent of the particle size, the stiffness is inversely proportional to the particle radius because it is given by the force divided by the displacement. The geometrical optics approach can be also applied for non spherical particles such as cylindrical objects by considering the optical torque and transverse radiation force [18]. As an example, the effect of torque due to rays is to align a cylindri-cal object along its opticylindri-cal axis. The transverse radiation force that comes from the an-isotropic shape of non spherical particles, causes an optical lift effect and creates a motion transversely to the beam propagation direction. As the size of particle increases, exact electromagnetic theories become unpractical due to the computational complexity, whereas, the accuracy of geometrical optics approx-imation increases. Thus, ray optics becomes a key technique to model optical trapping of large particles [19].
1.2.2
Dipole approximation for very small objects
For the particles whose size is smaller than the wavelength, optical forces calcu-lation can be approximated according to Reyliegh approximation. In this regime particle is considered as a small induced dipole in an electromagnetic field E(r, t), oscillating at frequency ν, and it is homogeneous inside the particle [13]. If the external field is strong, the induced dipole moment, p(r, t), can be calculated
based on the polarizability of the particle: [20]
p(r, t) = αpE(r, t) (1.8)
Where αp is the relative complex polarizability of the particle to the medium
given as: [21] αp = α0 1 −iα0k30 6π0 (1.9)
And α0 is the static Clausius-Mossatti polarizability:
α0 = 4π0n2mr3(
m2− 1
m2+ 2) (1.10)
Where r is radius of the particle, 0 and k0 are the vacuum permitivity, and
m is the ratio of np
n. An object such as a particle or a dipole illuminated by
an electromagnetic wave, not only scatters the wave but also absorbs the power from the wave. The rate of removing the energy from the electromagnetic wave, σext is equal to scattering cross section, σscat, multiplied by the power density of
the incoming wave plus absorption cross section, σabs, multiplied by the power
density of the incoming wave, σext = σscat+ σabs. The power per unit energy of
an electric dipole illuminated by a plane electromagnetic field is: Ii =
1
20c|Ei|
2, (1.11)
and the total power removed from the electromagnetic wave by the dipole, using Poynting’s theorem can be obtained from:
Pext =
ω
2Im[pd.E
∗
i(rd)] (1.12)
which is the work done by electric field on the charge distribution of the dipole. Therefore, the extinction cross section, the ratio between Ii and Pext will be
calculated as:
σext =
k0
0
To calculate σscat, we similarly obtain the rate of energy dissipates from the work
done by the dipole due to scattering at the position of the dipole: Pscat= −
ω
2Im[pd.E
∗
d(rd)], (1.14)
where Edis the radiated electric field of a dipole placed at the center of integrated
throughout the sphere: Ed= 1 3[−1 + (k0a) 2+ 2i 3(k0a)3 ]P, (1.15)
where P is the polarization of the sphere which is uniform and is related to the dipole moment, pd= PV , and then the scattering cross section will be obtained
as: σscat = k40 6π2 0 |αd|2. (1.16)
The absorption cross section then will be found from the difference between the extinction and scattered cross sections:
σabs = k0 0 Im[αd] − k4 0 6π2 0 |αd|2. (1.17)
For small metallic particles, with imaginary parts of refractive index, the ab-sorption is so larger that dominates the scattering: σabs ≈ σext. However, small
dielectric particle scatter most of the light with no absorption and so: σscat ≈ σext.
Now we are going to calculate the optical forces exerted on an electric dipole in the presence of time-varying electric and magnetic fields. Here we consider an electric dipole formed by two particles of m and charges ±q placed at r± with a distance ∆r = r+− r−:
pd = q∆r (1.18)
If the angle between pd and the incident electrical field is not zero, the forces,
F+ and F− on the charges ±q are oppositely directed which means there will be
a torque that tends to align the dipole with the electrical field:
In case of a non-uniform electric field, an additive force which is due to its gra-dient, must be considered. In addition, electrical field and electrostatic potential of the dipole are cylindrically symmetric around the dipole axis. In this case, the electric field can be expressed as the gradient of the electrostatic potential. To obtain the optical forces in this approximation, assuming that |∆r| << λ0, we
first write the equations of motion of two particles in the presence of electric and magnetic fields that vary with time, Ei(r, t), Bi(r, t):
md2r+ dt2 = +q[Ei(r+, t) + dr+ dt × Bi(r+, t)] − ∇Uint(∆r, t) md2r−
dt2 = −q[Ei(r−, t) +drdt− × Bi(r−, t)] + ∇Uint(∆r, t)
(1.20)
Here, the Lorentz forces, a combination of electric and magnetic forces, are in the square bracket, and Uint is the potential keeping two particles together.
Expanding in Taylor series, the fields Ei(r, t) and Bi(r, t) around the center of
mass of the dipole and inserting the new expressions in the above force equations, we obtain an important formula:
< Fd>=
X
j=x,y,z
pd,j∇Ei,j(rd, t) (1.21)
After doing some algebra, using Maxwell’s equations and introducing σext = k0α00p
0 as the extinction cross-section, we can express the time averaged optical force
exerted on a small particle when illuminated by time-varying electromagnetic field as: [13, 22, 23] < Fd >= 1 4α 00 p∇|E|2+ σext c < S > − 1 2σextc∇× < Lspin> (1.22) Where < S >= 1
2Re(E × H∗) is the time-averaged Poynting vector of the
incoming wave and < Lspin > is the time -averaged spin angular momentum
density [22, 23]. The first term in the equation represents gradient force which is responsible for the confinement of the particle:
Fd,grad(r) = 1 2 α0 p c0 ∇I(rd) (1.23)
Where I(rd) is the intensity of the electric field and rd is the position of the
dipole center. This shows that the gradient force, which comes from the potential energy of the dipole immersed in the electric field, is conservative so its work is independent of the path taken. Particles with the refractive index higher than medium, having a positive α0p, will be attracted towards the center of laser spot [9]. By contrast, for the case np < n , the polarizability is negative and the
particle is repelled by the high intensity region. Here we can obtain the gradient force for a focused Gaussian intensity distribution: Ii(ρ) = I0e
−2ρ2
W 20 , where ρ is
the radial coordinate in the transverse plane, I0 is the maximum intensity and
W0 is the laser beam waist, and for ρ ≈ 0, we can have:
Ii(ρ) ≈ I0(1 − 2
ρ2
W2 0
) (1.24)
Substituting the derivation of 1.24 eq. into 1.23 eq. and recalling Fgrad(ρ) =
−κρρ, we find the elastic force with radial spring constant:
κρ=
2α0I 0
c0W02
(1.25) And the corresponding radial potential :
U (ρ) = 1 2κρρ
2 (1.26)
The scattering force, responsible for radiation pressure is as the second term in 1.22 eq.:
Fd,scat(r) =
σext
Where is non-conservative because momentum transferred from the electric field to the dipole is a result of both absorption and scattering. The direction of this force as known before is in the direction of beam propagation [5]. Third term in the equation 1.22 , spin-curl force [22]:
Fspin(r) = −
1
2σextc∇× < Lspin> (1.28)
Spin-curl force is due to the polarization gradients in the electromagnetic field and non-conservative. This force does not play an important role in the trap-ping as it is very small compared to the other terms. However, when considering optical trapping with the beams of higher order with in-homogeneous polariza-tion, cylindrical vector beams for instant [24, 25], it may become significantly important.
1.2.3
Electromagnetic theory and intermediate regime
In this regime, the particle size is comparable with the wavelength, so neither ray optics nor dipole approximation are not applicable. Therefore, full electro-magnetic (wave optics) theory is necessary to model the light-matter interaction and to obtain the optical force and torque. In the electromagnetic theory and using the conservation of linear momentum, the time averaged optical force on a particle, in near field approximation is as: [26, 27, 28, 29]
Frad =
I
s
ˆ
n· < TM > dS (1.29)
Where ˆn is the surface outer normal unit vector , S is the surrounding medium and < TM > is the averaged Maxwell stress tensor that is related to the
mechan-ical interaction of light and matter:
< TM >= 2Re E ⊗ E∗+ c 2 n2B ⊗ B ∗− 1 2(|E| 2+ c2 n2|B| 2)I (1.30)
Where E and B are total electric and magnetic fields superposed of the incident and scattered fields: E = Ei+ Es, B = Bi+ Bs. Considering the conservation of
angular momentum, in the similar way, we can express the radiation torque as:
Γrad = −
I
s
(< TM > ×r) · ˆndS (1.31)
Where r is the position of surface area element.
1.3
Application of optical tweezers
Since the invention of optical tweezers (OT), they have been used in many differ-ent research fields due to the relatively cheap setups. Typically, optical tweezers setup can be built from readily available components. In fact, optical tweezers have revolutionized the study of microscopic system capable of manipulating and assembling of biological objects, cells, single atoms and nanostructures [30, 31]. It is possible to investigate and manipulate the living cells inside their native environment without any physical contact since the trapped objects are usually suspended in aqueous solutions. Also, OT allow to measure the red blood cell elasticity in order to investigate the variation of the cells due to diseases and drugs [32]. Here we give a brief overview of few subjects of interest.
1.3.1
Biophysics
Optical trapping is one of the techniques suited to the study of physical prop-erties of biological molecules due to their size (nano to micro meters) and force scale (from femtonewtons to piconewtons). Optical tweezers act as a transducer and although bio-molecules, most of time, are not trapped directly, micro beads act as handles and the DNA, for instance, is attached to the surface of one or two micro beads. Using optical tweezers single-molecule assay technique, it is possible to determine the force-extension curve and , therefore, the mechanical
Figure 1.6: Probing DNA thermal fluctuation. (a) Experimental configuration and (b) theoretical model of thermal fluctuation in a single DNA molecule. Re-produced from ref. [34].
and elastic properties of a DNA. Thermal fluctuation of the micro particles used as handles causes the resolution of this technique to be limited. Recently [33] , another method, probing thermal fluctuation, has been applying to overcome this limitation by using two separate optical tweezers which trap two beads and a DNA molecule held between them (figure 1.6) [34]. In addition to the mechani-cal properties of bio-molecules, the dynamics of molecular motors can be studied by optical tweezers due to the dynamic range and sensitivity of them. Myosin molecule is an important example of a molecule motor driven by chemical energy released from the hydrolysis of adenosine triphosfate (ATP). Using a motility assay based on a dual optical tweezers, the mechanics of this molecular motor was probed [35] and to study the dynamic, silica beads, coated with skeletal muscle heavy meromyosin (HMM), were attached to the cover slip. The actin filament then was brought close to the silica bead in order to interact with HMM molecules on the silica bead surface and to prob the molecular motor dynamics by measuring its displacement versus time.
Figure 1.7: Bacterial adhesion force. The bead which is trapped by optical tweez-ers is coated with receptor molecules and is brought near a bacterium stuck on a large bead’s surface. By moving the stage by a distance xlb and as a result
displacement of the small bead, xsb, the bond is measured by deflection of the
beam. Reproduced from ref. [38].
1.3.2
Cellular biology
Optical tweezers have also played an important role in the study of complex cel-lular processes as with single biological molecules which we discussed in previous section. Optical tweezers, as force transducers, have been used to prob mechan-ical interactions between cells or between cells and their environments and cell biology forces such as adhesion forces in a minimum invasive technique. In addi-tion, Optical tweezers have been altered to control the behavior of cells such as the direction and rate of growth of neurons. As for examples of applications of OT in cellular biology, we can name: 1) Measurement of cellular adhesion forces: Cellular bonds are dynamic and environmental factors such as mechanical stress may affect the associate and disassociate rates of the bond. This is important in cellular dynamics where, for example, the force exerted by a cell on a substrate depends on this bond. OT have been employed to measure these forces by trap-ping the cells directly or by traptrap-ping microscopic prob beads [36, 37]. 2) Probing the structure by binding bacteria to surfaces: Bacteria adhesion to tissues medi-ated by pili , which are hair like structures, occurs during the colonization and
Figure 1.8: Neuronal growth. Flat (left column) and tubular (right column) neuronal cones. the position of the laser spot is shown with a circle. Frames are taken at intervals of 10 minutes (left column) and 5 minutes (right column). Reproduced from ref. [39].
infection processes. An optical tweezers force transducer system was built [38] to measure the strength of Escherchia coli bacteria adhesion force. The bac-teria, E.coli which consist of a short flexible fibrillum with a long helical rod, were attached to the surface of large beads that were immobilized on a cover slip. Smaller beads functionalised with galaboise which binds to the P pilus adhesion, trapped by means of optical tweezers, were brought close to the bacterium so that the pilus could bind. By moving the large bead, pulling the bacterium away from the optical tweezers and measuring the displacements of the small bead, the binding forces were measured (figure 1.7). 3) Guiding the growth of neurons: The mechanics of the membrane plays an important role in the motility and growth of cells, so the mechanical properties of cell membrane have become the subjects of interests in biology research field. Using an optical tweezers [39] positioned near the leading edge of a neuronal growth cone, the direction of growth is guided towards the laser light, as shown in figure 1.8.
1.3.3
Optofluidics
Optofluidics research, which integrates photonics and microfluidics, filed provides a platform that improves optical manipulation and sensing of various nature sam-ples. In this research area there are specific application such as optically sorting and delivering of cells and particles by size or refractive index using scattering force [40, 41] and using random potential by speckle fields [42]. Optical forces capable of trapping, manipulating, sorting and characterizing samples without mechanical contacts, broaden the vision of microfluidics to lab − on − a − chip concept,i.e., the scaling down of a lab fit on a chip with physical, biological and chemical capabilities. Due to development of contactless techniques micro and nano-machines can be fabricated within microfluidics chips [43, 44]. More-over, these machines such as rotors and pumps, which work based on spin angu-lar momentum transfer from circuangu-larly poangu-larized beams to birefringent materials [45, 46, 47, 48, 49], can be driven and monitored by optical tweezers.
Figure 1.9: Raman spectra of red blood cells that are optically trapped. The spectra of normal and β-thalassemic RBCs are compared. Dashed lines indicate the observed energy shifts and arrows highlight spectral features influenced by intensity changes. Reproduced from ref. [50].
1.3.4
Spectroscopy
One of the most powerful analytical techniques to characterize materials and biological objects is spectroscopy. The combination of spectroscopic function-alities with optical tweezers leads to the new powerful technique, spectroscopic optical tweezers that permits one to manipulate and analyze a wide range of particles particularly biological entities such as living cells, bacteria, viruses and organelles in their natural environment [50] (figure 1.9). Raman tweezers, as the integration of optical tweezers with Raman spectroscopy, allows one to analyze the chemical and physical properties of an individual trapped objects, e.g., bun-dle of carbon nanotubes (CNT), graphene flakes and silicon nanowires (SiNW) [31, 33, 51, 52, 53, 54] without the need of fixing them on a substrate.
1.3.5
Plasmonics
From the 1990s to 2000s different groups suggested to enhance the optical forces by plasmonic resonance associated to metal nano-particles (MNPs), to obtain a stable trap at low power [55]. There are two different approaches that can be employed to enhance optical force exploiting plasmons. The first is based on trapping MNPs due to localized surface plasmons produced by the particles themselves [55, 56]. The second is using the surface plasmon polaritons (SPPs) generated by nanostructures on a substrate like pads [57, 58], nanoantennas [59, 60] and nanohole [61, 62, 63]. Volpe et al., experimentally demonstrated that plasmonic enhancement can also be obtained at interface between a non-pattern golden layer and dielectric particles in solution and in the presence of a resonant illumination [64].
1.3.6
Colloid science
Colloidal particles, with a wide range of size, from nanometer to several microm-eters, are usually dispersed in a medium which is typically gas or liquid. The interactions between particles in a colloidal suspension are divided in 3 terms: hydrodynamic interactions, electrostatic forces and depletion forces. These inter-actions depend on the suspending solvent and they can rise to complex behaviors. Using Optical tweezers, as a useful tool, it is possible to prob these colloidal in-teractions and measure the coupling between the particles in suspension. To determine the hydrodynamic, two colloidal spheres are held in independent opti-cal tweezers and their thermal fluctuations are tracked [65]. While electrostatic interaction is Coulomb interaction between charged colloids [66], depletion inter-action comes purely from the solvent [67].
Chapter 2
Standard optical tweezers
As mentioned before, an optical tweezers setup is an ideal tool to capture and manipulate micro and nanoparticles in liquids and to measure forces to the pi-conewton scale [68]. A typical optical tweezers setup comprises an optical trap to hold a probe, a dielectric or metallic micron size particle, and a position sensing system. In the case of biophysical applications the probe is a small dielectric bead tethered to the cell or molecule under study. The particle randomly moves due to Brownian motion in the potential well formed by the optical trap. The refracted rays differ over the volume of the particle and exert a piconewton scale force on it, drawing it towards the region of highest light intensity. The optical trap is harmonic near the focal point: the force acting on a colloidal particle positioned at x within a trap centered at x0 is Fopt = −k(x − x0), where k is
the trap stiffness. Several characteristics influence the performance of an optical trap: (1) the trap stiffness increases with the power in the light beam. (2) for a given laser power, the trap stiffness increases with a decrease in the size of the focused spot. (3) the trap is weakest in the direction of the bream propagation where the intensity gradient is weakest and the force associated with reflection is opposed to the restoring one depending on how much light is reflected at the bead surface, the force due to reflection can easily push the sphere out of a weak trap. Furthermore, in presence of damping, if the trap depth is not significantly greater than the kinetic energy characteristic of the Brownian fluctuations, the
Camera Illumination PC Sample Laser Objective
Figure 2.1: Basic optical tweezers setup. Digital video microscopy.
particle will easily escape. Figure 2.2 and 2.1 show the three parts of a typical setup that accomplish these roles: the trapping optics, the imaging optics and the detection optics.
2.1
Trapping
In the simplest configuration, illustrated in figures 2.2 and 2.1, experimentally optical trapping is generally achieved by focusing a laser beam using a high numerical aperture objective. Here, the trapping beam is directed upwards and the sample is placed above the objective. Using this inverted microscope, it is possible to get a more stable trap because gravity is in the opposite direction of the radiation pressure.
Figure 2.2: Basic optical tweezers setup. Interferometry.
2.1.1
Laser beam
An optical trap is generated by strongly focusing a laser beam. The crucial step is to choose an adequate laser depending on the applications. There are important factors such as Laser power, Wavelength, beam quality and polarization that must be considered.
Power Typically, to generate a single optical trap for manipulation of micro-scopic particles, a laser with a power up to maximum 100 mW is more than sufficient. One should note that since the objective looses a large amount of power of the beam, the laser power at focus on the sample might be significantly lower than the original power.
Wavelength In most current applications the trapping beam is generated by a near infrared (NIR) laser. This is mainly due to the fact that NIR light is less damaging to biological cells and molecules than visible and ultra-violet (UV) light. The NRI wavelength falls close to a local minimum of water absorption spectrum, minimizing heating effects on biological samples.
