Acceleration Sensing, Feedback Cooling, and Nonlinear Dynamics with Nanoscale Cavity-Optomechanical Devices

Light has long been used for the precise measurement of moving bodies, but the burgeoning field of optomechanics is concerned with the interaction of light and matter in a regime where the typically weak radiation pressure force of light is able to push back on the moving object. This field began with the realization in the late 1960's that the momentum imparted by a recoiling photon on a mirror would place fundamental limits on the smallest measurable displacement of that mirror. This coupling between the frequency of light and the motion of a mechanical object does much more than simply add noise, however. It has been used to cool objects to their quantum ground state, demonstrate electromagnetically-induced-transparency, and modify the damping and spring constant of the resonator. Amazingly, these radiation pressure effects have now been demonstrated in systems ranging 18 orders of magnitude in mass (kg to fg).

In this work we will focus on three diverse experiments in three different optomechanical devices which span the fields of inertial sensors, closed-loop feedback, and nonlinear dynamics. The mechanical elements presented cover 6 orders of magnitude in mass (ng to fg), but they all employ nano-scale photonic crystals to trap light and resonantly enhance the light-matter interaction. In the first experiment we take advantage of the sub-femtometer displacement resolution of our photonic crystals to demonstrate a sensitive chip-scale optical accelerometer with a kHz-frequency mechanical resonator. This sensor has a noise density of approximately 10 micro-g/rt-Hz over a useable bandwidth of approximately 20 kHz and we demonstrate at least 50 dB of linear dynamic sensor range. We also discuss methods to further improve performance of this device by a factor of 10.

In the second experiment, we used a closed-loop measurement and feedback system to damp and cool a room-temperature MHz-frequency mechanical oscillator from a phonon occupation of 6.5 million down to just 66. At the time of the experiment, this represented a world-record result for the laser cooling of a macroscopic mechanical element without the aid of cryogenic pre-cooling. Furthermore, this closed-loop damping yields a high-resolution force sensor with a practical bandwidth of 200 kHZ and the method has applications to other optomechanical sensors.

The final experiment contains results from a GHz-frequency mechanical resonator in a regime where the nonlinearity of the radiation-pressure interaction dominates the system dynamics. In this device we show self-oscillations of the mechanical element that are driven by multi-photon-phonon scattering. Control of the system allows us to initialize the mechanical oscillator into a stable high-amplitude attractor which would otherwise be inaccessible. To provide context, we begin this work by first presenting an intuitive overview of optomechanical systems and then providing an extended discussion of the principles underlying the design and fabrication of our optomechanical devices.

Complex behavior and perception in Drosophila emerges from iterative feedback-regulated reflexes

For a hungry fruit fly, locating and landing on a fermenting fruit where it can feed, find mates, and lay eggs, is an essential and difficult task requiring the integration of both olfactory and visual cues. Understanding how flies accomplish this will help provide a comprehensive ethological context for the expanding knowledge of their neural circuits involved in processing olfaction and vision, as well as inspire novel engineering solutions for control and estimation in computationally limited robotic applications. In this thesis, I use novel high throughput methods to develop a detailed overview of how flies track odor plumes, land, and regulate flight speed. Finally, I provide an example of how these insights can be applied to robotic applications to simplify complicated estimation problems. To localize an odor source, flies exhibit three iterative, reflex-driven behaviors. Upon encountering an attractive plume, flies increase their flight speed and turn upwind using visual cues. After losing the plume, flies begin zigzagging crosswind, again using visual cues to control their heading. After sensing an attractive odor, flies become more attracted to small visual features, which increases their chances of finding the plume source. Their changes in heading are largely controlled by open-loop maneuvers called saccades, which they direct towards and away from visual features. If a fly decides to land on an object, it begins to decelerate so as to maintain a stereotypical ratio of expansion to retinal size. Once they reach a stereotypical distance from the target, flies extend their legs in preparation for touchdown. Although it is unclear what cues they use to trigger this behavior, previous studies have indicated that it is likely under visual control. In Chapter 3, I use a nonlinear control theoretic analysis and robotic testbed to propose a novel and putative mechanism for how a fly might visually estimate distance by actively decelerating according to a visual control law. Throughout these behaviors, a common theme is the visual control of flight speed. Using genetic tools I show that the neuromodulator octopamine plays an important role in regulating flight speed, and propose a neural circuit for how this controller might be implemented in the flies brain. Two general biological and engineering principles are evident across my experiments: (1) complex behaviors, such as foraging, can emerge from the interactions of simple independent sensory-motor modules; (2) flies control their behavior in such a way that simplifies complex estimation problems.

Advancements in jet turbulence and noise modeling: accurate one-way solutions and empirical evaluation of the nonlinear forcing of wavepackets

Jet noise reduction is an important goal within both commercial and military aviation. Although large-scale numerical simulations are now able to simultaneously compute turbulent jets and their radiated sound, lost-cost, physically-motivated models are needed to guide noise-reduction efforts. A particularly promising modeling approach centers around certain large-scale coherent structures, called wavepackets, that are observed in jets and their radiated sound. The typical approach to modeling wavepackets is to approximate them as linear modal solutions of the Euler or Navier-Stokes equations linearized about the long-time mean of the turbulent flow field. The near-field wavepackets obtained from these models show compelling agreement with those educed from experimental and simulation data for both subsonic and supersonic jets, but the acoustic radiation is severely under-predicted in the subsonic case. This thesis contributes to two aspects of these models. First, two new solution methods are developed that can be used to efficiently compute wavepackets and their acoustic radiation, reducing the computational cost of the model by more than an order of magnitude. The new techniques are spatial integration methods and constitute a well-posed, convergent alternative to the frequently used parabolized stability equations. Using concepts related to well-posed boundary conditions, the methods are formulated for general hyperbolic equations and thus have potential applications in many fields of physics and engineering. Second, the nonlinear and stochastic forcing of wavepackets is investigated with the goal of identifying and characterizing the missing dynamics responsible for the under-prediction of acoustic radiation by linear wavepacket models for subsonic jets. Specifically, we use ensembles of large-eddy-simulation flow and force data along with two data decomposition techniques to educe the actual nonlinear forcing experienced by wavepackets in a Mach 0.9 turbulent jet. Modes with high energy are extracted using proper orthogonal decomposition, while high gain modes are identified using a novel technique called empirical resolvent-mode decomposition. In contrast to the flow and acoustic fields, the forcing field is characterized by a lack of energetic coherent structures. Furthermore, the structures that do exist are largely uncorrelated with the acoustic field. Instead, the forces that most efficiently excite an acoustic response appear to take the form of random turbulent fluctuations, implying that direct feedback from nonlinear interactions amongst wavepackets is not an essential noise source mechanism. This suggests that the essential ingredients of sound generation in high Reynolds number jets are contained within the linearized Navier-Stokes operator rather than in the nonlinear forcing terms, a conclusion that has important implications for jet noise modeling.

Generalized modal identification of linear and nonlinear dynamic systems

This dissertation is concerned with the problem of determining the dynamic characteristics of complicated engineering systems and structures from the measurements made during dynamic tests or natural excitations. Particular attention is given to the identification and modeling of the behavior of structural dynamic systems in the nonlinear hysteretic response regime. Once a model for the system has been identified, it is intended to use this model to assess the condition of the system and to predict the response to future excitations.

A new identification methodology based upon a generalization of the method of modal identification for multi-degree-of-freedom dynaimcal systems subjected to base motion is developed. The situation considered herein is that in which only the base input and the response of a small number of degrees-of-freedom of the system are measured. In this method, called the generalized modal identification method, the response is separated into "modes" which are analogous to those of a linear system. Both parametric and nonparametric models can be employed to extract the unknown nature, hysteretic or nonhysteretic, of the generalized restoring force for each mode.

In this study, a simple four-term nonparametric model is used first to provide a nonhysteretic estimate of the nonlinear stiffness and energy dissipation behavior. To extract the hysteretic nature of nonlinear systems, a two-parameter distributed element model is then employed. This model exploits the results of the nonparametric identification as an initial estimate for the model parameters. This approach greatly improves the convergence of the subsequent optimization process.

The capability of the new method is verified using simulated response data from a three-degree-of-freedom system. The new method is also applied to the analysis of response data obtained from the U.S.-Japan cooperative pseudo-dynamic test of a full-scale six-story steel-frame structure.

The new system identification method described has been found to be both accurate and computationally efficient. It is believed that it will provide a useful tool for the analysis of structural response data.

Random response of nonlinear continuous systems

This thesis presents a technique for obtaining the stochastic response of a nonlinear continuous system. First, the general method of nonstationary continuous equivalent linearization is developed. This technique allows replacement of the original nonlinear system with a time-varying linear continuous system. Next, a numerical implementation is described which allows solution of complex problems on a digital computer. In this procedure, the linear replacement system is discretized by the finite element method. Application of this method to systems satisfying the one-dimensional wave equation with two different types of constitutive nonlinearities is described. Results are discussed for nonlinear stress-strain laws of both hardening and softening types.

Exact solutions and transformation properties of nonlinear partial differential equations from general relativity

Various families of exact solutions to the Einstein and Einstein-Maxwell field equations of General Relativity are treated for situations of sufficient symmetry that only two independent variables arise. The mathematical problem then reduces to consideration of sets of two coupled nonlinear differential equations.

The physical situations in which such equations arise include: a) the external gravitational field of an axisymmetric, uncharged steadily rotating body, b) cylindrical gravitational waves with two degrees of freedom, c) colliding plane gravitational waves, d) the external gravitational and electromagnetic fields of a static, charged axisymmetric body, and e) colliding plane electromagnetic and gravitational waves. Through the introduction of suitable potentials and coordinate transformations, a formalism is presented which treats all these problems simultaneously. These transformations and potentials may be used to generate new solutions to the Einstein-Maxwell equations from solutions to the vacuum Einstein equations, and vice-versa.

The calculus of differential forms is used as a tool for generation of similarity solutions and generalized similarity solutions. It is further used to find the invariance group of the equations; this in turn leads to various finite transformations that give new, physically distinct solutions from old. Some of the above results are then generalized to the case of three independent variables.

Nonlinear oscillations in discrete and continuous systems

The first thesis topic is a perturbation method for resonantly coupled nonlinear oscillators. By successive near-identity transformations of the original equations, one obtains new equations with simple structure that describe the long time evolution of the motion. This technique is related to two-timing in that secular terms are suppressed in the transformation equations. The method has some important advantages. Appropriate time scalings are generated naturally by the method, and don't need to be guessed as in two-timing. Furthermore, by continuing the procedure to higher order, one extends (formally) the time scale of valid approximation. Examples illustrate these claims. Using this method, we investigate resonance in conservative, non-conservative and time dependent problems. Each example is chosen to highlight a certain aspect of the method.

The second thesis topic concerns the coupling of nonlinear chemical oscillators. The first problem is the propagation of chemical waves of an oscillating reaction in a diffusive medium. Using two-timing, we derive a nonlinear equation that determines how spatial variations in the phase of the oscillations evolves in time. This result is the key to understanding the propagation of chemical waves. In particular, we use it to account for certain experimental observations on the Belusov-Zhabotinskii reaction.

Next, we analyse the interaction between a pair of coupled chemical oscillators. This time, we derive an equation for the phase shift, which measures how much the oscillators are out of phase. This result is the key to understanding M. Marek's and I. Stuchl's results on coupled reactor systems. In particular, our model accounts for synchronization and its bifurcation into rhythm splitting.

Finally, we analyse large systems of coupled chemical oscillators. Using a continuum approximation, we demonstrate mechanisms that cause auto-synchronization in such systems.

I. Exact solution of some nonlinear evolution equations. II. The similarity solution for the Korteweg-De Vries equation and the related Painleve Transcendent

In Part I, a method for finding solutions of certain diffusive dispersive nonlinear evolution equations is introduced. The method consists of a straightforward iteration procedure, applied to the equation as it stands (in most cases), which can be carried out to all terms, followed by a summation of the resulting infinite series, sometimes directly and other times in terms of traces of inverses of operators in an appropriate space.

We first illustrate our method with Burgers' and Thomas' equations, and show how it quickly leads to the Cole-Hopft transformation, which is known to linearize these equations.

We also apply this method to the Korteweg and de Vries, nonlinear (cubic) Schrödinger, Sine-Gordon, modified KdV and Boussinesq equations. In all these cases the multisoliton solutions are easily obtained and new expressions for some of them follow. More generally we show that the Marcenko integral equations, together with the inverse problem that originates them, follow naturally from our expressions.

Only solutions that are small in some sense (i.e., they tend to zero as the independent variable goes to ∞) are covered by our methods. However, by the study of the effect of writing the initial iterate u_1 = u_(1)(x,t) as a sum u_1 = ^∼/u_1 + ^≈/u_1 when we know the solution which results if u_1 = ^∼/u_1, we are led to expressions that describe the interaction of two arbitrary solutions, only one of which is small. This should not be confused with Backlund transformations and is more in the direction of performing the inverse scattering over an arbitrary “base” solution. Thus we are able to write expressions for the interaction of a cnoidal wave with a multisoliton in the case of the KdV equation; these expressions are somewhat different from the ones obtained by Wahlquist (1976). Similarly, we find multi-dark-pulse solutions and solutions describing the interaction of envelope-solitons with a uniform wave train in the case of the Schrodinger equation.

Other equations tractable by our method are presented. These include the following equations: Self-induced transparency, reduced Maxwell-Bloch, and a two-dimensional nonlinear Schrodinger. Higher order and matrix-valued equations with nonscalar dispersion functions are also presented.

In Part II, the second Painleve transcendent is treated in conjunction with the similarity solutions of the Korteweg-de Vries equat ion and the modified Korteweg-de Vries equation.

Topics in linear and nonlinear dispersive waves

The various singularities and instabilities which arise in the modulation theory of dispersive wavetrains are studied. Primary interest is in the theory of nonlinear waves, but a study of associated questions in linear theory provides background information and is of independent interest.

The full modulation theory is developed in general terms. In the first approximation for slow modulations, the modulation equations are solved. In both the linear and nonlinear theories, singularities and regions of multivalued modulations are predicted. Higher order effects are considered to evaluate this first order theory. An improved approximation is presented which gives the true behavior in the singular regions. For the linear case, the end result can be interpreted as the overlap of elementary wavetrains. In the nonlinear case, it is found that a sufficiently strong nonlinearity prevents this overlap. Transition zones with a predictable structure replace the singular regions.

For linear problems, exact solutions are found by Fourier integrals and other superposition techniques. These show the true behavior when breaking modulations are predicted.

A numerical study is made for the anharmonic lattice to assess the nonlinear theory. This confirms the theoretical predictions of nonlinear group velocities, group splitting, and wavetrain instability, as well as higher order effects in the singular regions.

Existence, uniqueness, and stability of solutions of a class of nonlinear partial differential equations

In this study we investigate the existence, uniqueness and asymptotic stability of solutions of a class of nonlinear integral equations which are representations for some time dependent non- linear partial differential equations. Sufficient conditions are established which allow one to infer the stability of the nonlinear equations from the stability of the linearized equations. Improved estimates of the domain of stability are obtained using a Liapunov Functional approach. These results are applied to some nonlinear partial differential equations governing the behavior of nonlinear continuous dynamical systems.

Some problems in nonlinear elasticity

Two separate problems are discussed: axisymmetric equilibrium configurations of a circular membrane under pressure and subject to thrust along its edge, and the buckling of a circular cylindrical shell.

An ordinary differential equation governing the circular membrane is imbedded in a family of n-dimensional nonlinear equations. Phase plane methods are used to examine the number of solutions corresponding to a parameter which generalizes the thrust, as well as other parameters determining the shape of the nonlinearity and the undeformed shape of the membrane. It is found that in any number of dimensions there exists a value of the generalized thrust for which a countable infinity of solutions exist if some of the remaining parameters are made sufficiently large. Criteria describing the number of solutions in other cases are also given.

Donnell-type equations are used to model a circular cylindrical shell. The static problem of bifurcation of buckled modes from Poisson expansion is analyzed using an iteration scheme and pertubation methods. Analysis shows that although buckling loads are usually simple eigenvalues, they may have arbitrarily large but finite multiplicity when the ratio of the shell's length and circumference is rational. A numerical study of the critical buckling load for simple eigenvalues indicates that the number of waves along the axis of the deformed shell is roughly proportional to the length of the shell, suggesting the possibility of a "characteristic length." Further numerical work indicates that initial post-buckling curves are typically steep, although the load may increase or decrease. It is shown that either a sheet of solutions or two distinct branches bifurcate from a double eigenvalue. Furthermore, a shell may be subject to a uniform torque, even though one is not prescribed at the ends of the shell, through the interaction of two modes with the same number of circumferential waves. Finally, multiple time scale techniques are used to study the dynamic buckling of a rectangular plate as well as a circular cylindrical shell; transition to a new steady state amplitude determined by the nonlinearity is shown. The importance of damping in determining equilibrium configurations independent of initial conditions is illustrated.

Nonlinear dispersive waves in nonlinear optics

A study is made of solutions of the macroscopic Maxwell equations in nonlinear media. Both nonlinear and dispersive terms are responsible for effects that are not taken into account in the geometrical optics approximation. The nonlinear terms can, depending on the nature of the nonlinearity, cause plane waves to focus when the amplitude varies across the wavefront. The dispersive terms prevent the singularities that nonlinearity alone would produce. Solutions are found which de scribe periodic plane waves in fully nonlinear media. Equations describing the evolution of the amplitude, frequency and wave number are generated by means of averaged Lagrangian techniques. The equations are solved for near linear media to produce the form of focusing waves which develop a singularity at the focal point. When higher dispersion is included nonlinear and dispersive effects can balance and one finds amplitude profiles that propagate with straight rays.

Part I. On the breaking of nonlinear dispersive waves. Part II. Variational principles in continuum mechanics

A model equation for water waves has been suggested by Whitham to study, qualitatively at least, the different kinds of breaking. This is an integro-differential equation which combines a typical nonlinear convection term with an integral for the dispersive effects and is of independent mathematical interest. For an approximate kernel of the form e^(-b|x|) it is shown first that solitary waves have a maximum height with sharp crests and secondly that waves which are sufficiently asymmetric break into "bores." The second part applies to a wide class of bounded kernels, but the kernel giving the correct dispersion effects of water waves has a square root singularity and the present argument does not go through. Nevertheless the possibility of the two kinds of breaking in such integro-differential equations is demonstrated.

Difficulties arise in finding variational principles for continuum mechanics problems in the Eulerian (field) description. The reason is found to be that continuum equations in the original field variables lack a mathematical "self-adjointness" property which is necessary for Euler equations. This is a feature of the Eulerian description and occurs in non-dissipative problems which have variational principles for their Lagrangian description. To overcome this difficulty a "potential representation" approach is used which consists of transforming to new (Eulerian) variables whose equations are self-adjoint. The transformations to the velocity potential or stream function in fluids or the scaler and vector potentials in electromagnetism often lead to variational principles in this way. As yet no general procedure is available for finding suitable transformations. Existing variational principles for the inviscid fluid equations in the Eulerian description are reviewed and some ideas on the form of the appropriate transformations and Lagrangians for fluid problems are obtained. These ideas are developed in a series of examples which include finding variational principles for Rossby waves and for the internal waves of a stratified fluid.

I. Singular perturbation problems involving singular points and turning points. II. On the averaged Lagrangian technique for nonlinear dispersive waves

In Part I a class of linear boundary value problems is considered which is a simple model of boundary layer theory. The effect of zeros and singularities of the coefficients of the equations at the point where the boundary layer occurs is considered. The usual boundary layer techniques are still applicable in some cases and are used to derive uniform asymptotic expansions. In other cases it is shown that the inner and outer expansions do not overlap due to the presence of a turning point outside the boundary layer. The region near the turning point is described by a two-variable expansion. In these cases a related initial value problem is solved and then used to show formally that for the boundary value problem either a solution exists, except for a discrete set of eigenvalues, whose asymptotic behaviour is found, or the solution is non-unique. A proof is given of the validity of the two-variable expansion; in a special case this proof also demonstrates the validity of the inner and outer expansions.

Nonlinear dispersive wave equations which are governed by variational principles are considered in Part II. It is shown that the averaged Lagrangian variational principle is in fact exact. This result is used to construct perturbation schemes to enable higher order terms in the equations for the slowly varying quantities to be calculated. A simple scheme applicable to linear or near-linear equations is first derived. The specific form of the first order correction terms is derived for several examples. The stability of constant solutions to these equations is considered and it is shown that the correction terms lead to the instability cut-off found by Benjamin. A general stability criterion is given which explicitly demonstrates the conditions under which this cut-off occurs. The corrected set of equations are nonlinear dispersive equations and their stationary solutions are investigated. A more sophisticated scheme is developed for fully nonlinear equations by using an extension of the Hamiltonian formalism recently introduced by Whitham. Finally the averaged Lagrangian technique is extended to treat slowly varying multiply-periodic solutions. The adiabatic invariants for a separable mechanical system are derived by this method.

Nonlinear dispersive wave problems

The nonlinear partial differential equations for dispersive waves have special solutions representing uniform wavetrains. An expansion procedure is developed for slowly varying wavetrains, in which full nonlinearity is retained but in which the scale of the nonuniformity introduces a small parameter. The first order results agree with the results that Whitham obtained by averaging methods. The perturbation method provides a detailed description and deeper understanding, as well as a consistent development to higher approximations. This method for treating partial differential equations is analogous to the "multiple time scale" methods for ordinary differential equations in nonlinear vibration theory. It may also be regarded as a generalization of geometrical optics to nonlinear problems.

To apply the expansion method to the classical water wave problem, it is crucial to find an appropriate variational principle. It was found in the present investigation that a Lagrangian function equal to the pressure yields the full set of equations of motion for the problem. After this result is derived, the Lagrangian is compared with the more usual expression formed from kinetic minus potential energy. The water wave problem is then examined by means of the expansion procedure.