2.1 Morphology dependent resonance

Transcription

1 Literature Review 2.1 Morphology dependent resonance Optical cavities of two or more mirrors are utilised in all branches of modern linear and nonlinear optics. Practical usage of such cavities is technically restricted, especially when high performance is important. The operating performance of a cavity can be measured in terms of quality factor, mode stability and restricted environments. The macroscopic fabrication of good optical mirrors, their alignment and binding is rather expensive and difficult task which is echoed in the price modern commercial laser systems. The exploitation of the intrinsic microcavity within the confines of a microsphere via morphology dependent resonances (MDR) provides high mode stability and quality factors. To this end, the examination of MDR, sometimes referred to as whispering gallery modes [9], has been of considerable importance to many field in optical science. Fabrication of these microcavities is also comparatively simple and inexpensive. MDR arises due to the constructive interference of light rays at near-glancing angles by total internal reflection within a microcavity (see Fig. 2.1). A dielectric 13

2 sphere possesses natural internal modes of oscillation at characteristic frequencies corresponding to the specific ratio of size to wavelength. Fig. 2.1: Schematic diagram of a long and short wavelength light ray undergoing total internal reflection, the superposition of which is constructive for a round trip, forming separate MDR modes within a microcavity. Fluorescence via electron transitions within an atom or a molecule from an excited state to a ground state by the emission of a photon into the continuum of radiation modes is an example of weak coupling. If the atom is placed in a microcavity and strong coupling conditions are satisfied the atom can interact coherently with a cavity mode for a meaningful time. Strongly coupled systems are those in which the Rabi dynamic can exist [10], even if only briefly, despite the reality of dissipation and loss. Strong coupling occurs when the atom-field coupling strength (which is half the Rabi frequency) is faster than 14

3 any dissipative rate and larger than 1/T where T is the interaction time. Strong coupling results when an atom initially in the excited state of a dipole transition enters a microcavity of volume (V ). If a single mode of the cavity is in its ground state and is resonant with the atomic transition, the atom and vacuum field couple. This results in a quanta of energy shifting back and forth between the atom and the cavity mode at the Rabi frequency. The interaction strength of the atom and cavity mode is linear in the field and hence smaller cavity volumes concentrate the vacuum field of the mode producing larger Rabi frequencies. In reality the cavity will have a finite photon lifetime and a finite quality factor (Q) that will limit and sometimes prevent Rabi oscillations by allowing energy to leak irreversibly into the continuum. Likewise the atomic transition will couple to continuum radiation modes and thereby experience spontaneous decay of its population as well as polarisation dephasing. Weak coupling within a cavity results when dissipation overwhelms the fundamental Rabi dynamic. In this regime the enhancement and suppression of spontaneous emission via the Purcell effect is possible within microcavities [11]. Because all cavities exhibit some degree of loss or dissipation, a cavity mode is proportional to a continuum density of modes function [12]. A two-level atom will decay spontaneously (fluoresce) by the interaction with a vacuum continuum at a rate proportional to the spectral density of modes per unit volume evaluated at the transition frequency. Within a cavity the density of modes is modified and large changes are possible. Cavity modes in the presence of dissipation are more accurately described as quasi modes [12]. The maximal density of modes occurs at the quasi mode resonant frequency and can greatly exceed the corresponding free space density. Purcell [11] noted that a single quasi mode occupied a spectral bandwidth ν/q within a cavity volume (V ). The resultant normalised cavity enhanced mode density per unit volume to the mode density of free space gives the Purcell factor, 15

4 P = 3λ3 4π 2 Q V. (2.1) An atom whose transition falls within the mode linewidth will experience an enhancement to its spontaneous decay rate. More significantly, because the enhancement comes from the coupling to only those continuum modes that make up the corresponding quasi mode of the cavity, the spontaneous emission is directed into this quasi mode. Fig. 2.2: Atom/molecule in free space (a). Purcell enhancement of spontaneous emission (b). Weak coupling to a cavity mode enhances the spontaneous emission rate by increasing the local density of modes compared to that of free space. Fermi s Golden Rule, given in Eq. 2.2, states how emission rates can be modified by the geometrical structure of the matrix in which an atom or a molecule is situated. The transition rate from a state i to j may be expressed as [10], A i j = 1 2 i H ij j 2 ρ(ν), (2.2) 16

5 where i represents the excited state with no photon present, j is the ground state with one photon present, is Plank s constant h 2π, i H ij j is the volume normalised Hamiltonian matrix element representing the atom field interaction and ρ(ν) is the density of final photon states. Placing the emitter inside an optical cavity whose dimension is of the same order as the transition wavelength causes the emitted light to be coupled into discrete cavity modes instead of the continuum of vacuum states in free space. Since the density of states is large when ν corresponds to a cavity mode and small when ν is nonresonant, the emission rate will be enhanced or inhibited depending on whether the emission frequency corresponds to a particular cavity mode or not [11]. The golden rule which underpins the vast majority of discussion of optical phenomena is inadequate in the strong coupling regime for a two-level atom. golden rule suggests that for a transition exactly tuned to the cavity resonance the spontaneous decay rate should be proportional to Q as pointed out by Purcell [11]. This proportionality breaks down when Q is large; the fluorescence lifetime decreases with Q and eventually goes as 1/Q [13]. This means the fluorescence lifetime in the strong coupling regime shows unusual features: non exponential decay, maximum decay rate for intermediate Q values, non additivity of partial rates and the possible reduction of the total decay rate upon addition of an extra decay channel [13]. The strong coupling limit is experimentally accessible, for example a single transition between two levels in general shows up as double peaks in the photon spectra. The wavelengths at which these MDR occur can be calculated by Lorenz-Mie theory [14, 15], since the MDR is dependent on the boundary conditions including the refractive index mismatch, shape and size of the particle in a medium and not on the actual scattering processes either elastic or inelastic, coherent or incoherent [16]. For a given spherical microcavity, resonances occur at specific values on q m,l. Here q is the size parameter given in Eq The 17

6 q m,l = 2πa λ( E m,l ), (2.3) where a is the radius of the spherical cavity, λ( E m,l ) is the emission wavelength and m and l are integers. The mode number m indicates the order of the spherical Bessel and Hankel functions (ζ) describing the radial field distribution and the order l indicates the number of maxima in the radial dependence of the internal field distribution ( E m,l which is a function of ζ m,l ) [17]. m and l indicate that both discrete transverse electric (TE) and transverse magnetic (TM) resonances exist [9, 18]. The emission from dielectric microspheres [19 21] and fibres [22] containing fluorescent dyes show sharp resonance structures superimposed on the normal broadband fluorescence emission. Each mode has dramatically different spatial properties, cavity Q factor and consequently varying degrees of spectral resonant enhancement. In addition contributions to the total fluorescence from other portions of the microsphere not affected by the cavity complicate MDR measurement by providing an intense broad fluorescence background Q-factor and lasing Microparticles that can sustain MDR provide a cavity environment in which optical feedback can occur. The introduction of an active medium, like a fluorescent dye, in the presence of feedback can lead to optical gain within the cavity over time. Atoms and molecules undergo a variety of radiative processes as they interact with cavity photons. These processes are readily described by, and known as the Einstein A and B coefficients [10]. The first of these Einstein coefficients describes spontaneous emission (A 21 ), in which energy from an atom in an excited state relaxes back to a ground state in the form of a photon. Conversely, a photon can be absorbed by an atom raising it from 18

7 the ground state to an excited state, defined as absorption (B 12 ). The final coefficient, stimulated emission (B 21 ), is the most pertinent in the induction of lasing oscillation. It can be regards as reverse absorption in that an atom in an excited state can be triggered by an incoming photon to release its energy in the form of another photon identical to the trigger photon. These two photons have an identical frequency, polarisation state, direction and phase, the backbone characteristics resultant in laser radiation. In the presence of a cavity the balance between these three Einstein coefficients can be manipulated in a way that stimulated emission dominates over spontaneous emission and laser oscillation can be sustained [23]. For a spherical cavity rearrangement and manipulation of the size parameter given in Eq. 2.3 yields a mode spacing given by [9] λ = λ2 2πa arctan n 2 1 n2 1, (2.4) where n is the refractive index of the spherical microcavity. An ideal cavity would confine light indefinitely without loss and have resonant frequencies at precise values given for a microparticle in Eq Deviation from this ideal situation is described by the cavity Q factor which is proportional to the light confinement time in units of the optical period. For a spherical microparticle a higher Q factor at selected wavelengths is provided by MDR. The strength of cavity resonances is a result of the cavity Q factor which can be estimated from elastic-scattering linewidths (ν), calculated from Lorenz-Mie theory [21, 24, 25]. Q = ν ν. (2.5) 19

8 The total cavity quality factor Q is a measure of the feedback of the mode. This is governed by the amount of light that is leaked from the cavity and the amount absorbed by the cavity. Leakage accounts for the deviation of a real microparticle from a homogeneous sphere due to shape and index perturbations. Excellent surface finish is crucial for maximising Q. Laser emission is expected when the round trip gain is larger than the round trip loss resulting from optical absorption and radiation leakage from the particle. The quality factor of MDR modes in microspheres is ultimately limited by the optical attenuation of the resonator material [26]. In resonators of relatively large dimensions, a 500, all other loss mechanisms such as radiation due to curvature and λ scattering on residual molecular-size surface inhomogeneities do not prevent realisation of material limited Q at the level of Q Cavities with very high values of Q quickly degrade in normal laboratory conditions by a factor of approximately three to five times within about a ten-minute time scale after fabrication. The mechanism responsible for this degradation has been identified as a microsphere surface hydration through chemical absorption of atmospheric water [26]. This conclusion was backed by the partial restoration of Q upon baking of the sphere and also by direct tracking of a MDR mode frequency in the corresponding time scale. This monitoring confirmed deposition of a dielectric layer with the thickness of about 1 monolayer (0.2 nm), in agreement with the current data on water chemical absorption on silica surface [26]. The preservation of ultimate material limited Q in a longer time scale can be obtained in evacuated or a dry gas environment. Containing microspheres in a waterfree chamber may also be a method to seek for ever larger values of Q in the near infrared where quick tests of Q in within minutes from fabrication is hardly feasible due to alignment complications. (The maximum expected value of fused silica microsphere is Q = near 1.55 µm). The degradation rate of Q inside the dry box is strongly decreased, allowing preservation of Q for up to two 20

9 hours after sphere fabrication, by contrast to five minutes in normal conditions [26]. MDR Q values are substantially reduced for cavities in an aqueous environment Methods of generation morphology dependent resonance To excite high Q MDR, light needs to be coupled into the microcavity beyond the critical angle. This cannot be done efficiently by illumination of a plane wave. A more efficient method of coupling to MDR can be achieved by using resonant frustrated total internal reflection. This is achieved by placing the microcavity in the evanescent field produced by the total internal reflection of a focused laser beam at the surface of a high refractive index prism [27, 28]. Enhanced MDR for the purpose of optical probing and high signal output requires phased matched coupling to MDR [29], which is typically achieved using total internal reflection from the back face of a prism. Other coupling methods have also been demonstrated including tapered optical fibres and obstructed high numerical aperture objectives [30, 31]. The ability of prisms to induce evanescent fields at their surface is one of simplest and well understood methods of inducing localised fields. A prism allows, with inherent ease, the ability to confine optical fields at angles greater than the critical angle governed by Snell s law. The addition of a cavity that encompasses the prism input and a total internal reflected beam allows increased enhancement of the localised field at the prisms surface. The addition of a cavity by external mirrors and by metallic coating of the prism on two sides has been shown to induce enhancement to the localised field [32, 33] and applied to near-field imaging. This technique achieved only moderate enhancement due to the low cavity Q factor. MDR with greater Q factors are more readily induced in objects with at least one 21

10 degree of circular symmetry due to the inherent ability of energy to complete round trips. Optical fibres have such a symmetry which allows MDR to be excited in the plane perpendicular to that of the normal guiding length of a drawn fibre [22]. Optical fibres themselves rely on total internal reflection between the core and the cladding to guide light along the fibre. Evanescent coupling between two lengths of a fibre stripped of their cladding is now a readily accepted technology [34]. This same technology can be utilised to evanescently couple into MDR. Crossing the same two lengths of the striped optical fiber perpendicular to each other has seen MDR excitation with large Q values [35]. The MDR can be observed in the transmission spectrum of the excitation fibre and by probing the resonant fibre with an external applied fibre waveguide or tapered fibre tip. The excitation of a microcavity by frustrated total internal reflection can be achieved by use of an eroded optical fibre. The transverse dimension of the guided mode in a single mode optical fibre is typically 3 5 µm for visible to infrared wavelengths, close to the transverse confinement of MDR modes in microcavities that can range in size from tens to hundreds of micrometers. Eroding a small length of cladding from such an optical fibre and placing its exposed core in close proximity to the microcavity enables coupling between them. The evanescent field of the propagating mode in the exposed core can be utilised for MDR excitation in the same fashion as that of the total internal reflection from the surface of a high refractive index prism. The construction of such an exposed fibre is analogous to that of half a fibre coupler, as placing two of these exposed regions together would result in the construction of the more familiar bidirectional fibre coupler. Analogous coupling of an evanescent field to a microcavity can be achieved from a optical fibre taper. An optical fibre taper is formed by heating and pulling the fibre, resulting in a relocalisation of energy from the core to the outer edge of the fibre [35]. A single fibre taper can be used as an efficient means of guiding pump laser excitation to the MDR of the microcavity, resonantly coupling to the cavity as well as collecting the 22

11 resulting cavity emission. The fibre taper not only provides a means of efficient input and output coupling, but can be tailored as to only excite a single or multiple MDR modes of the cavity. Naturally one or more fibres can be coupled to a given microcavity. The use of two tapered fibres to act as input and output couplers respectively has been used to achieve almost complete optical power transfer ( 99.8%) to MDR modes. The low intrinsic loss of the cavity and the symmetric dual-coupling structure are critical for obtaining such high efficiencies [36]. It should be pointed out that MDR is not limited to microcavities with spherical geometries, although they have proven the most efficient structure to date. MDR can be induced and sustained in asymmetric microcavities and square microcavities. The simplest mode that can be formed in a square microcavity is one in which all the possible rectangular orbits within the cavity have the same perimeter (see schematic of a square microcavity in Table 2.1). It has also been possible to observe multimode resonance from a square microcavity, a result that is not immediately intuitive [37]. The Q factor of a square etched microcavity has also been shown to be sufficiently large enough to sustain lasing [38]. Asymmetric microcavities cavities made up of oblate spheres are readily observed in nature in the form of water droplets. MDR within these droplets manifests itself in the observation of rainbows due to varying spectral leakage from a combination of Fresnel refraction and total internal reflection depending on the individual light rays interaction with the asymmetric boundary of the droplet [52]. The study of MDR within laboratory asymmetric microcavities in the form of eggshaped, deformed droplets and microspheres has been of interest due to the ability of an asymmetric cavity to directionally concentrate their emission. These asymmetric particles have also been shown to lase, with variation in the output coupling via tuning of the degree of asymmetry. The sizing of asymmetric microparticles by time dependent measurement of MDR spectra has also been demonstrated [53 57]. 23

12 Table 2.1: Different microcavities and their measure of MDR properties. Low Q: , Intermediate Q: , High Q: and Ultrahigh Q: Cavity Quality factor Q Schematic Prism [32, 33] Q 8 Metallic coating Evanescent fie Fabry-Perot [1, 39] Q 2000 Metallic coating Mirror Beam splitter Square microcavity [37, 38] Q > 10 5 Evanescent fie Microdisc [40, 41] Q 10 4 Metallic coating Mirror Beam splitter Evanescent fi Metallic coating Mirror Be spl Microcavity chip [42, 43] Metallic coating Mirror Beam splitter Microcavity Fabry-Perot [44, 45] Q 10 7 Metallic coating Mirror Beam splitter Photonic crystal [46 48] Q > 10 5 Metallic coating Mirror Beam splitter Microsphere [19, 21, 26, 49 51] Q < Metallic Beam Evanescent coating field Mirror splitter Metallic coating Toroid [43] Q 10 8 Mirror Beam splitter The induction of MDR via an obstructed high numerical aperture objective operates in a method analogous to that of total internal reflection microscopy (TIRM) [31]. TIRM utilises an appropriately apodised optical field distribution at the back aperture of a high numerical aperture objective in such a way that the propagating components of the field in the focal region are prohibited. The simplest form of apodisation for TIRM is a coaxially centred opaque disc at the back aperture of the objective, which 24

13 has a diameter large enough to block all incoming excitation that would otherwise be focused into the sample at an angle less than the critical angle. The employment of a doughnut beam is an alternate method of confining the incident illumination to angles grater than the critical angle [58]. Various methods of coupling a thin beam at the very outer edge of the objective lens aperture have also been employed to achieve excitation beyond the critical angle [31, 59]. Dielectric microcavities with graded refractive index in the radial direction have been proposed as a method of making microcavity modes equally spaced with precision corresponding to a fraction of the resonance bandwidth [60]. A dielectric microcavity with equidistant spectra is analogous to the Fabry-Perot resonator. A dielectric microcavity having equidistant spectral resonances has applications as frequency comb generators, optical pulse generators and broadband energy storage circuits for electrooptical devices [61, 62]. They are similarly applicable to all applications in which conventional optical cavities are used. The different coupling properties of individual microcavities enable performance and the range of applications based on MDR microcavities to be significantly expanded. Providing a critical step in MDR microcavity development before becoming a customary building block of modern photonics Application of morphology dependent resonance Optical microcavities with MDR form the core of many evolving photonics applications from high stability narrow linewidth microlasers, high resolution spectroscopy, remote sensing and optoelectronic oscillators to optical memory devices, true-time optical delay lines and optical switching devices [63 65]. Although the MDR effect and its subsequent applications, such as microcavity lasing, have been investigated for microdisk, microring, microtorus, microsphere, microparticles, microdroplets and other microobjects [22, 43, 66 70] and various 25

14 materials such as silica [71], polystyrene [51] and semiconductors [40], MDR has also been examined in square shaped microcavities [37, 38]. High Q MDR cavities have become a useful tool which has been applied across multiple fields within optical science. The strong MDR features of these cavities are helping examine fundamental issues pertaining to particle cavity interaction. The application of MDR and the use of quantum dots as artificial atoms for cavity quantum electrodynamics, as well as a source of single and entangled photons for quantum information processing has driven microcavity development [70, 72]. Optical pumping of a microcavity has seen the onset of cavity lasing due to the high quality factors and corresponding optical feedback of cavity modes. The amount of pump excitation required for the onset of microcavity lasing can be quite low when compared to its macroscopic cavity laser counter parts. The low threshold of microcavity lasing combined with the long photon storage time and small mode volumes leads to high circulation intensities of MDR within the microcavity [73, 74]. These large intensities enable the observation and exploitation of nonlinear optical processes such as second harmonic generation, stimulated Raman scattering and parametric interactions [75 77]. The use of an optical fibre to address an optical microcavity has been an effective way to couple radiation to and from it. The positioning of a tapered optical fibre close to the perimeter of a microcavity allows evanescent coupling between the two. In the taper region of a single mode optical fibre, the fibre mode (or modes in a multimode fibre) is converted into air-cladding taper modes that extend into the environment surrounding the taper. Proper taper design ensures that coupling is primarily into the fundamental taper mode. Power transfer to the microcavity occurs in the region in which the taper is in proximity of the microcavities MDR modes. In the same fashion, reversing the process enables energy from the mirocavity to couple back to the fibre. Engineering of a microcavity in order to increase the cavity Q factor and MDR has 26

15 both advantages and disadvantages for various applications. Like the aforementioned case of a radially graded index profile across the cavity. It results in regularly spaced modes and increased Q values due to dispersion compensation within the cavity. This has advantages in fundamental studies, drop in filters and lab on a chip applications where coupling and cavity dynamics are of critical importance. The disadvantage is that the modes propagate closer to the centre of the cavity, reducing the strength of the evanescent field at the cavities surface. MDR modes can be made to travel closer to the cavity surface, increasing the evanescent field, when the index of refraction of the cavity close to its surface exceeds that of the internal index of refraction [78]. In this regime the Q factor is normally reduced due to the increasing tolerance placed on the cavities surface finish, though the extenuated evanescent field at the surface is advantageous when the cavity is to be used as an optical sensor. One of the more underutilised applications of MDR is sensing [79 82]. Evanescent wave spectroscopy methods based on the use of an integrated optical waveguide provide a powerful means for biosensing and sensing applications. Macroscopically, resonant techniques have been employed as an improved source of the evanescent field via the use of waveguides and coated prisms most notably applied to near-field imaging. Only a modest improvement is achieved, limited by the inherently low cavity quality factor. Waveguide spectroscopy is naturally suitable for detecting biochemical processes at surfaces with high signal to background efficiency due to the high transverse and lateral confinement of the propagating mode. The detection of Raman spectra of a mono-layer and femtomole concentrations of fluorescence molecules have been reported by elaborately designed waveguides [81]. The trade-off in achieving these challenging sensitivities is that a relatively large sensing length (at least several millimeters) of the waveguide is required. This is sometimes impractical and incompatible with the biological material under examination. Decreasing the sensing device to the order micrometers is advantageous in biological and single molecule detection. 27

16 The spectroscopic signal generated from a sample per unit of the waveguide sensing area can be dramatically increased with resonant optical techniques. MDR excitation provides a local enhancement of the optical field immediately surrounding the circumference of a microsphere making it an ideal candidate for spectroscopic sensing. 2.2 Laser trapping Laser trapping is one of the most attractive methods for the remote, non invasive manipulation of matter. The principle of laser trapping is to utilise a laser beam focused by a high numerical aperture microscope objective to induce a confining force on a dielectric or metallic particle in the focal region [83 85]. The application of this laser technology has been shown to manipulate substances on scales from a single atom to tens of micrometers. In the pioneering work of Ashkin [83,86], it was observed that transparent particles, such as latex spheres and air bubbles freely suspended in liquids or gasses, could be accelerated and levitated under the radiation pressure induced by a focused single or multiple laser beam. The radiation force, in the order of piconewtons, is strong enough to levitate and move microparticles against gravity and against the viscous drag of the solution in which the particles are suspended. This technique, sometimes termed laser tweezers, has led to a strong impact on physical [87] and biological research [8, 85, 88, 89]. This is exemplified in studies of the detection of localised light scattering [90], investigation of energy transfer in polymers [91, 92], selflasing [93] and fluorescence imaging [94]. Laser trapping has been applied to the manipulation and sorting of biological samples such as viruses and bacteria [85], blood cells [95], kinesin [8], DNA molecules [88] and human erythrocyte membranes [96]. For a dielectric particle, three-dimensional trapping can be achieved in the focal region regardless of particle size. 28

17 2.2.1 Trapping force The force induced by light on a particle depends on the size of the particle to be trapped. Ashkin was one of the first people to devise a method for the calculation of forces on dielectric particles due to the radiation pressure of light [97]. A single beam gradient optical trap was originally designed for Rayleigh particles, i.e. the size of a particle is much less than the wavelength of the trapping light. This technique was also found to be effective on the larger Mie sized particles, where the diameter of the particle is large compared with the wavelength of the incident light. If the size of the particle is large compared with the wavelength of the light then the force can be described by geometrical ray optics [97]. This method describes the force on the particle as being due to the momentum exchange between light and matter caused by absorption, reflection, refraction and scattering. Both the magnitude and direction of force are determined according to the law of momentum conservation. The effectiveness of a single beam gradient trap acting on a dielectric particle is a result of the gradient force pulling the particle toward the intensity maxima at the focus of the beam, as opposed to the scattering force trying to push the particle away from the focus in the direction of the incident light. The trapping force on a dielectric particle is predominantly generated by the refraction of the incident light through the particle. The strength of scattering and gradient forces is dependent on the angle of incidence of the ray with respect to the particle. The trap consists of an incident parallel beam of arbitrary mode structures and polarisation, which enters a high numerical aperture microscope objective and is focused to a point f, as illustrated in Fig 2.3 [97]. The angle of convergence Φ of a ray is related to the angle ψ by the relation f = 90 ψ. A ray of power P hitting a dielectric sphere of refractive index n at an angle of incidence Θ with incident momentum per second of np/c exerts a force on the sphere. The total force on the sphere is the sum of the contributions due to the reflected ray of 29

18 Fig. 2.3: Ray optics model schematic at an arbitrary trap location with relevant angles for the projection of trapping force. ˆn is the normal to the sphere, S and S are the Y and Z positions of the focus f with respect to the origin O at the centre of the sphere. power P R and the infinite number of emergent refracted rays of successively decreasing power P T 2, P T 2 R,... P T 2 R n is shown in Fig The quantities R and T are the Fresnel reflection and transmission coefficients [17] at the surface at angle Θ. The net force acting through the centre of the particle can be broken down into scattering (F s ) and gradient (F g ) components given by: F s = np c F g = np c { 1 + R cos 2Θ T 2 [cos(2θ 2r a ) + R cos 2Θ] 1 + R 2 + 2R cos 2r a { R sin 2Θ T 2 [sin(2θ 2r a ) + R sin 2Θ] 1 + R 2 + 2R cos 2r a }, (2.6) }, (2.7) where r a and Θ are the angles of refraction and incidence, respectively. These formulae 30

19 P PR r a + PT 2 PT 2 R 2 O Y Z PT 2 R Fig. 2.4: Force contributions due to refraction through a dielectric sphere. sum over all scattered rays and are therefore exact. F s and F g are the force components parallel and perpendicular to the direction of the incident ray, respectively. The scattering and gradient forces F s and F g are polarisation dependent since the Fresnel coefficients R and T are different for rays polarised perpendicular or parallel to the plane of incidence. For linearly polarised light perpendicular to the Y -axis the fractions of the input power contributing to the force in terms of p- and s-components is given by [97]: 31

20 f p = (cos β sin µ sin β cos µ) 2, (2.8) f s = (cos β cos µ sin β sin µ) 2. (2.9) If the incident polarisation is parallel to the X-axis then the expressions for f p and f s reverse. The gradient and scattering force components for p- and s-polarisation states are calculated separately using the appropriate formulae for F s and F g. The force acting on the particle from an individual ray can be resolved into its three components X, Y and Z: F Z = F s sin ψ + F g cos µ cos ψ, (2.10) F Y = F s cos ψ cos β + F g cos µ sin ψ cos β + F g sin µ sin β, (2.11) F X = F s cos ψ sin β + F g cos µ sin ψ sin β + F g sin µ cos β. (2.12) Here the force in the Z-direction, F (Z), is referred to as the axial force and the force in the Y -direction, F (Y ), is referred to as the transverse force. The strength of the trapping force, and its components can be expressed by a dimensionless trapping efficiency Q, which is related to the force by [97]: Q = F c np, (2.13) where c is the speed of light in vacuum, n is the refractive index of the water medium inside the sample cell, and P is the trapping power in the focus of the trapping objective. 32

21 The total trapping efficiency for a particle is evaluated by integrating the contribution of each ray from the paraxial rays out to maximum angle of convergence, Φ max and around the beam axis with respect to β, which is given by: Q = 2π 0 rmax 0 2π 0 Q2 g + Q 2 s I(r) rdrdβ rmax I(r) rdrdβ 0 (2.14) where I(r) is the intensity distribution over the aperture of the trapping objective. The intensity distribution of a Gaussian beam is given by: I(r) = I 0 e ( 2r 2 ω 0 2 ), (2.15) where I 0 is the intensity at the centre of the beam, r is the radius over the back aperture of the trapping objective and ω 0 is the beam radius. If ω 0 is large, the beam distribution over the aperture becomes uniform, I(r) = 1. The maximum transverse trapping force F on a trapped particle can be measured by the Stokes law [98]. The Stokes law is a measure of the viscous drag force and is given by Eq. 2.16: F = 6πrvµ, (2.16) where r is the radius of a trapped particle, v is the maximum translation speed and µ is the viscosity of the surrounding medium. 33

22 2.2.2 Laser trapping with a spatial light modulator Manipulation of a laser trapped particle in the focal plane of the trapping objective is an important feature in practical applications of laser tweezers, in terms of tailoring the trapping potential to a transparent or absorptive particle and applications such as particle cell sorting and micro optical devices. Dielectric and metallic particles are examples of transparent and absorptive particles respectively. The transmission properties of the particle dictate which trapping conditions, refraction or scattering, are best suited [97, ]. In the case of dielectric particles, the particle is attracted to the high intensity of the focal region within the trapping beam. The exchange of momentum of the light refracting within and through the particle in the focal region exerts force on the particle. Apodisation of the trapping beam intensity and phase provides direct control over the incident illumination distribution within the focal region and hence the force that can be applied to a trapped particle. The trapping potential can be arbitrarily manipulated by appropriate apodisation of the incident trapping beam. A spatial light modulator is an excellent candidate for the manipulation of incident trapping illumination. In practice, arbitrary beam profiles can be generated by a computer-generated hologram (CGH) or by a liquid crystal phase modulator [104, 105]. A CGH works efficiently for a specific trapping application via the insertion of a pre-fabricated high resolution lithographically recorded optical element into the trapping beam path. However, this limits the manipulation of the trapping region to a predetermined static state. A liquid crystal spatial light modulator (SLM) enables more flexible and dynamic manipulation of the trapping region [106] Trapping beam aperture apodisation Different beam-profiles can be used in trapping experiments in order to manipulate the particles interaction with the trapping field in the focal region. The most common 34

23 beam profiles employed in trapping experiments are Gaussian, uniform, obstructed and doughnut beams. These beam profiles are illustrated in Fig. 2.5 as incident at the back aperture of the trapping objective. Each has its own distinct trapping characteristics. Gaussian beam Uniform & Obstructed beams Doughnut beam intensity intensity intensity 0 radial direction 0 radial direction 0 radial direction Fig. 2.5: Beam profiles in the back aperture of a trapping objective. At the back aperture of the trapping objective, obstructed and doughnut beam profiles have similar appearance (see Fig. 2.5). However, this is where the similarities between these two beam profiles end. Both obstructed and doughnut beams provide distinctly different enhancements to the trapping efficiency in the focal plane. In the focus these beam types have different intensity distributions in the radial direction as illustrated in Fig. 2.6 due to the transformation of the incident illumination to the focus by the trapping objective lens [107]. Gaussian distribution Obstructed distribution Doughnut distribution intensity 0 radial direction intensity 0 radial direction intensity 0 radial direction Fig. 2.6: Beam profiles in the focal region of a trapping objective. A Gaussian beam, i.e. a TEM 00 mode beam, is most suited to laser trapping of dielectric particles in applications that require a large transverse trapping efficiency [86, 35

24 90,97]. It is also used in other trapping regimes, despite being less efficient, because it is the easiest and most common laser beam to attain. A Gaussian beam can also be used for the transverse trapping of metallic particles [102, 103]. Greatly expanding a Gaussian beam results in the intensity distribution becoming uniform across the back aperture of the focusing lens. A doughnut beam, i.e. a TEM 01 mode beam, is most suited to the axial laser trapping of dielectric particles and is appealing for both the transverse and axial trapping of absorptive particles, such as metallic particles. The advantages for the use of a doughnut beam for laser trapping is best derived in comparison with a Gaussian and uniform beams. Firstly, the force acting on a dielectric particle in the axial direction produced by a Gaussian beam is less than that for a uniform beam [97]. This is because the intensity of the outer portion of the beam is smaller than that of a uniform beam. It is this outer portion of the beam, which contributes mainly to the axial force. The paraxial rays are the main contributors to the transverse force. An increase in the beam radius, even with the same laser power, produces a large axial force. Since the increase of the beam radius is equivalent to that of the numerical aperture of the lens, the use of a high numerical aperture objective is recommended for axial optical trapping [97, 103, 104]. By the same reasoning, a TEM 01 (doughnut mode) beam produces an axial force more efficiently than either Gaussian or uniform beams because a doughnut mode has a strong intensity around, but not on the beam axis in the trapping focus [97]. The central intensity minimum of a doughnut beam in the focus is advantageous for trapping metallic particles. A doughnut beam also has a potential for three-dimensional trapping of metallic particles [108]. A doughnut beam also has the ability to rotate a trapped particle by the transfer of angular momentum inherent within the beam [107]. 36

25 Spatial light modulation Reflective binary spatial light modulators typically consist of a thin layer of birefringent ferroelectric liquid crystal (FLC) material sandwiched between a metal conductor and a glass window coated with a transparent conductive layer such as indium tin oxide (ITO). A voltage is applied across the FLC layer and depending on the polarity of the applied voltage the fast axis of the FLC material is forced into one of two possible states. Both operation states are separated from each other by approximately 45. Thus, when viewed normal to the conductive surface the device can be thought of as a waveplate with an electrically switchable fast axis orientation. Increasing the number of addressable states allows for graded ramp style modulation. Variations of the sandwich layer and its constituents also allow operation in transmission [109]. Spatial light modulators operate in two common configurations, amplitude modulation and phase modulation. The simplest binary amplitude modulation system using a reflective FLC spatial light modulator is to place a linear polariser in front of it so that the transmission axis of the polariser is parallel to the fast axis of the FLC in one of its two possible states. Binary phase modulation is achieved by orientating the FLC device so that the polarisation direction of the incident light bisects the angle formed by the two possible fast axis orientations of the device. In one fast axis orientation the polarisation vector of the incident light is rotated by 45 clockwise after passing through the FLC cell. While in the other fast axis orientation the incident polarisation vector is rotated 45 counter clockwise. Upon passing through the analyser both states produce light polarised in the same plane 180 out of phase with each other. A doughnut beam can be created by the interaction of an incident plane wave illumination with the simulated interference pattern displayed of the face of the phase modulator of the desired doughnut beam: 37

26 Π = cos(kx mϕ) (2.17) where k is the wavenumber, x is the propagation distance, m is the topological charge and ϕ is the phase. The binary patterns displayed on the liquid crystal display alter the phase of the reflected beam. Increasing the topological charge increases the number of dislocations in the fringe interferogram. This is evident in the calculation of binary phase patterns of doughnut beam profiles in Fig (a) (b) (c) (d) (e) (f) (g) (h) (i) Fig. 2.7: Calculated binary phase pattern to produce doughnut beam profiles: Topological charges 1 to 9, (a) to (i), respectively. 38

27 A doughnut beam has been produced using a liquid crystal phase modulator [110] (Displaytec). The experimental layout for doughnut beam creation using a liquid crystal phase modulator is shown in the schematic depicted in Fig Polarisers HeNe laser = 633 nm Beam expansion LCD SLM Screen Reference arm Mirror Fig. 2.8: Schematic diagram of the experimental setup for doughnut beam generation via a liquid crystal spatial light modulator and beam interferometry. When the calculated interferograms are loaded to the spatial light modulators display and it is incident with a plane wave the reflected beam acquires the corresponding doughnut beam characteristics. The experimentally generated beams from the fringe patterns corresponding to doughnut beams of topological charges 1 to 9 are shown in Fig Due to the diffractive nature of the spatial light modulator and its pixel array, higher order diffracted derivatives of the doughnut beams are evident. The doughnut beams charge can be determined by the on-axis interference of the doughnut beam and a plane wave. Calculation of the on-axis interference of a plane wave and doughnut beams of topological charges 1 to 4 are given in Fig. 2.10, the number of breaks in the pattern correspond to the charge of the doughnut beam. The measured on-axis interference patterns in Fig prove the generated beams are actual doughnut beams. On-axis interferograms produce fan shaped fringes characteristic of the topological charge of the doughnut beam due to slight beam divergence between the interfering beams. These interference patterns can be 39

28 Fig. 2.9: Experimentally generated doughnut beam profiles from LCD-SLM: Dotted arrow for zero orders, solid arrows for topological charges 1 to 9, (a) to (i) respectively, and dashed arrows for high order diffracted beams. themselves used for controlled rotation of a laser trapped particle Laser trapping for microscopy Near-field scanning optical microscopy is a powerful tool for obtaining high spatial frequency information from a sample. This information is gathered by placing a probe in the immediate proximity of the object to be studied and measuring its interaction with an evanescent field. The problem lies in extracting a measurable quantity of this inherently weak field. Several techniques have been employed to examine this field. Including, a fibre optic probe tapered down into a very fine point (in the order of tens of nanometers) which is then immersed into the evanescent field and signal collected. 40

29 Chapter 2 Fig. 2.10: Calculated on-axis doughnut beam interferograms: Topological charges 1 to 4, (a) to (d), respectively. Current probes include etched apertures, protrusions [111], fluorescent tips [112] and pulled fiber tips [113]. The positioning of these probes requires mechanical access to a sample object, meaning that samples with rough surfaces or intervening membranes (in the case of biological specimens) are not always accessible for examination. This technique requires extremely sensitive positioning of the fragile fibre probe so not to cause impact between it and the surface under inspection. Any impact is likely to reduce its efficiency and in the worst case irreversible damage to the probe and sample occurs. A laser trapped particle as a near-field probe eliminates this mechanical access 41

30 Fig. 2.11: Experimentally generated doughnut beam interferograms: Topological charges 1 to 4, (a) to (d), respectively. restriction by the use of a non-intrusive optical trap to position and scan the optical probe [114]. In addition, the elastic nature of a probe held within a laser trap permits the sample and the probe to collide without damage. A laser trapped particle is an ideal tool for probing and relaying near-field information to the far-field. It has been demonstrated for a small laser trapped particle, acting as a near-field probe, can scatter an evanescent wave produced by total internal reflection of an externally applied laser beam. Scanning the trapped probe through this evanescent field enables an image of the scattered field to be built up [103]. Laser trapping scanning near-field optical microscopy (SNOM) utilises a trapped microparticle as a scanning probe for near-field imaging and has some advantages over conventional SNOM which uses a tapered fiber tip as a scanning probe. One of the most significant advantages is that external mechanical control of the distance between a probe and a substrate is not required in laser trapping SNOM [87,115]. In addition, 42

31 the concern associated with a fragile tapered fibre probe is not an issue in laser trapping SNOM. However, due to the low signal strength that plagues all near-field microscopic studies and subsequent degradation in image contrast, laser trapping SNOM requires a significant improvement. This method is also limited to surface studies of thin samples which enable total internal reflection over a large area. This is impractical for many samples whose morphology and refractive index make total internal reflection difficult to impossible, especially if the sample is of biological origins. Thin samples can be back illuminated by a field undergoing total internal reflection, producing an evanescent field at the substrates surface. This method is not suitable for thick or absorptive samples and it is difficult to control all the parameters for the generation of strong evanescent field. Control of parameters like the sample refractive index and the confinement of the incident illumination angle to greater than the critical angle is not always possible. Even when total internal reflection is possible the detectable scattered signal is weak. To efficiently generate an evanescent field from a prism it needs to have a high surface quality and high refractive index. This means that the prisms can be quite expensive, especially when the sample is mounted immediately on top of the prism to the detriment of the prism surface. This prism technique can also introduce difficulties in applications where the sample is scanned relative to a fixed probe. Variations on this prism technique have been attempted in a bid to increase the strength of the evanescent field. Thin film coating of the prism surface is one method that has been quite successfully employed to enhance the evanescent field [116]. Thin film coating of a high index prism is a tedious and costly affair, particularly if the sample is mounted on a coated surface. This thin film coating technique is normally reserved for specialist applications. The addition of an external cavity incorporating the prism is another approach that successfully strengthens the evanescent field [3]. This method gives an enhancement of the evanescent field and is ultimately limited by the low quality factor of the external cavity. 43

32 In order to increase the detectable signal level a dopant included within the trapped particle can be employed as a small localised light source. This removes the need for external back illumination and enables probing of a wider range of samples. Most importantly it increases the detectable signal levels due to the strong field enhancement generated by MDR within the trapped particle. MDR and its characteristic spectral structure enables an increase to the modality of the detection system. The onset of MDR particle lasing has enabled the measurement of very fine displacements and chemical sensing via the monitoring of cavity loss processes [117, 118]. MDR [9,19 21,67] provides a useful tool for sensing and imaging in laser trapping applications. Increased modality and versatility are obtained buy using a trapped resonant probe over conventional probes. The utilisation of MDR [19 21, 67, 118] in a trapped microsphere for SNOM is investigated in this thesis. Laser trapping with a continuous wave (CW) laser source is a well established and characterised phenomena [87, 119, 120]. A high power CW laser source has also been used to induce two-photon fluorescence within a laser trapping system [121]. This technique employs a small suitably doped particle totally immersed in the trapping field. The high flux associated with CW two-photon excitation gives very rigid traps with high lateral force constants which inhibits cavity effects and limits the modality, reducing the ability to resolve small details in the scanning plane. One of the solutions to this problem is based on the use of a femtosecond pulsed laser for two-photon excitation, which will be investigated in Chapters Two-photon excitation The energy of a molecule can be raised from its ground state to an excited state through the absorption of energy (E) from a photon. This process can be described by the equation 44