Current Search: Solitons (x)


Title

DISCRETE SURFACE SOLITONS.

Creator

Suntsov, Sergiy, Stegeman, George, University of Central Florida

Abstract / Description

Surface waves exist along the interfaces between two different media and are known to display properties that have no analogue in continuous systems. In years past, they have been the subject of many studies in a diverse collection of scientific disciplines. In optics, one of the mechanisms through which optical surface waves can exist is material nonlinearity. Until recently, most of the activity in this area was focused on interfaces between continuous media but no successful experiments...
Show moreSurface waves exist along the interfaces between two different media and are known to display properties that have no analogue in continuous systems. In years past, they have been the subject of many studies in a diverse collection of scientific disciplines. In optics, one of the mechanisms through which optical surface waves can exist is material nonlinearity. Until recently, most of the activity in this area was focused on interfaces between continuous media but no successful experiments have been reported. However, the growing interest that nonlinear discrete optics has attracted in the last two decades has raised the question of whether nonlinear surface waves can exist in discrete optical systems. In this work, a detailed experimental study of linear and nonlinear optical wave propagation at the interface between a discrete onedimensional Kerrnonlinear system and a continuous medium (slab waveguide) as well as at the interface between two dissimilar waveguide lattices is presented. The major part of this dissertation is devoted to the first experimental observation of discrete surface solitons in AlGaAs Kerrnonlinear arrays of weakly coupled waveguides. These nonlinear surface waves are found to localize in the channels at and near the boundary of the waveguide array. The key unique property of discrete surface solitons, namely the existence of a power threshold, is investigated in detail. The second part of this work deals with the linear light propagation properties at the interface between two dissimilar waveguide arrays (socalled waveguide array heterojunction). The possibility of three different types of linear interface modes is theoretically predicted and the existence of one of them, namely the staggered/staggered mode, is confirmed experimentally. The last part of the dissertation is dedicated to the investigation of the nonlinear properties of AlGaAs waveguide array heterojunctions. The predicted three different types of discrete hybrid surface solitons are analyzed theoretically. The experimental results on observation of inphase/inphase hybrid surface solitons localized at channels on either side of the interface are presented and different nature of their formation is discussed.
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Date Issued

2007

Identifier

CFE0001989, ucf:47426

Format

Document (PDF)

PURL

http://purl.flvc.org/ucf/fd/CFE0001989


Title

OPTICAL WAVE PROPAGATION IN DISCRETE WAVEGUIDE ARRAYS.

Creator

Hudock, Jared, Christodoulides, Demetrios, University of Central Florida

Abstract / Description

The propagation dynamics of light in optical waveguide arrays is characteristic of that encountered in discrete systems. As a result, it is possible to engineer the diffraction properties of such structures, which leads to the ability to control the flow of light in ways that are impossible in continuous media. In this work, a detailed theoretical investigation of both linear and nonlinear optical wave propagation in one and twodimensional waveguide lattices is presented. The ability to...
Show moreThe propagation dynamics of light in optical waveguide arrays is characteristic of that encountered in discrete systems. As a result, it is possible to engineer the diffraction properties of such structures, which leads to the ability to control the flow of light in ways that are impossible in continuous media. In this work, a detailed theoretical investigation of both linear and nonlinear optical wave propagation in one and twodimensional waveguide lattices is presented. The ability to completely overcome the effects of discrete diffraction through the mutual trapping of two orthogonally polarized coherent beams interacting in Kerr nonlinear arrays of birefringent waveguides is discussed. The existence and stability of such highly localized vector discrete solitons is analyzed and compared to similar scenarios in a single birefringent waveguide. This mutual trapping is also shown to occur within the first few waveguides of a semiinfinite array leading to the existence of vector discrete surface waves. Interfaces between two detuned semiinfinite waveguide arrays or waveguide array heterojunctions and their possible applications are also considered. It is shown that the detuning between the two arrays shifts the dispersion relation of one array with respect to the other. Consequently, these systems provide spatial filtering functions that may prove useful in future alloptical networks. In addition by exploiting the unique diffraction properties of discrete arrays, diffraction compensation can be achieved in a way analogous to dispersion compensation in dispersion managed optical fiber systems. Finally, it is demonstrated that both the linear (diffraction) and nonlinear dynamics of twodimensional waveguide arrays are significantly more complex and considerably more versatile than their onedimensional counterparts. As is the case in onedimensional arrays, the discrete diffraction properties of these twodimensional lattices can be effectively altered depending on the propagation Bloch kvector within the first Brillouin zone. In general, this diffraction behavior is anisotropic and as a result, allows the existence of a new class of discrete elliptic solitons in the nonlinear regime. Moreover, such arrays support twodimensional vector soliton states, and their existence and stability are also thoroughly explored in this work.
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Date Issued

2005

Identifier

CFE0000833, ucf:46687

Format

Document (PDF)

PURL

http://purl.flvc.org/ucf/fd/CFE0000833


Title

OPTICAL SOLITONS IN PERIODIC STRUCTURES.

Creator

Makris, Konstantinos, Christodoulides, Demetrios, University of Central Florida

Abstract / Description

By nature discrete solitons represent selftrapped wavepackets in nonlinear periodic structures and result from the interplay between lattice diffraction (or dispersion) and material nonlinearity. In optics, this class of selflocalized states has been successfully observed in both oneand twodimensional nonlinear waveguide arrays. In recent years such lattice structures have been implemented or induced in a variety of material systems including those with cubic (Kerr), quadratic,...
Show moreBy nature discrete solitons represent selftrapped wavepackets in nonlinear periodic structures and result from the interplay between lattice diffraction (or dispersion) and material nonlinearity. In optics, this class of selflocalized states has been successfully observed in both oneand twodimensional nonlinear waveguide arrays. In recent years such lattice structures have been implemented or induced in a variety of material systems including those with cubic (Kerr), quadratic, photorefractive, and liquidcrystal nonlinearities. In all cases the underlying periodicity or discreteness leads to new families of optical solitons that have no counterpart whatsoever in continuous systems. In the first part of this dissertation, a theoretical investigation of linear and nonlinear optical wave propagation in semiinfinite waveguide arrays is presented. In particular, the properties and the stability of surface solitons at the edge of Kerr (AlGaAs) and quadratic (LiNbO3) lattices are examined. Heterostructures of two dissimilar semiinfinite arrays are also considered. The existence of hybrid solitons in these latter types of structures is demonstrated. Rabitype optical transitions in zmodulated waveguide arrays are theoretically demonstrated. The corresponding coupled mode equations, that govern the energy oscillations between two different transmission bands, are derived. The results are compared with direct beam propagation simulations and are found to be in excellent agreement with coupled mode theory formulations. In the second part of this thesis, the concept of paritytimesymmetry is introduced in the context of optics. More specifically, periodic potentials associated with PTsymmetric Hamiltonians are numerically explored. These new optical structures are found to exhibit surprising characteristics. These include the possibility of abrupt phase transitions, band merging, nonorthogonality, nonreciprocity, double refraction, secondary emissions, as well as power oscillations. Even though gain/loss is present in this class of periodic potentials, the propagation eigenvalues are entirely real. This is a direct outcome of the PTsymmetry. Finally, discrete solitons in PTsymmetric optical lattices are examined in detail.
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Date Issued

2008

Identifier

CFE0002013, ucf:47610

Format

Document (PDF)

PURL

http://purl.flvc.org/ucf/fd/CFE0002013


Title

SOLITON SOLUTIONS OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS USING VARIATIONAL APPROXIMATIONS AND INVERSE SCATTERING TECHNIQUES.

Creator

Vogel, Thomas, Kaup, David, University of Central Florida

Abstract / Description

Throughout the last several decades many techniques have been developed in establishing solutions to nonlinear partial differential equations (NPDE). These techniques are characterized by their limited reach in solving large classes of NPDE. This body of work will study the analysis of NPDE using two of the most ubiquitous techniques developed in the last century. In this body of work, the analysis and techniques herein are applied to unsolved physical problems in both the fields of...
Show moreThroughout the last several decades many techniques have been developed in establishing solutions to nonlinear partial differential equations (NPDE). These techniques are characterized by their limited reach in solving large classes of NPDE. This body of work will study the analysis of NPDE using two of the most ubiquitous techniques developed in the last century. In this body of work, the analysis and techniques herein are applied to unsolved physical problems in both the fields of variational approximations and inverse scattering transform. Additionally, a new technique for estimating the error of a variational approximation is established. Note that the material in chapter 2, "Quantitative Measurements of Variational Approximations" has recently been published. Variational problems have long been used to mathematically model physical systems. Their advantage has been the simplicity of the model as well as the ability to deduce information concerning the functional dependence of the system on various parameters embedded in the variational trial functions. However, the only method in use for estimating the error in a variational approximation has been to compare the variational result to the exact solution. In this work, it is demonstrated that one can computationally obtain estimates of the errors in a onedimensional variational approximation, without any a priori knowledge of the exact solution. Additionally, this analysis can be done by using only linear techniques. The extension of this method to multidimensional problems is clearly possible, although one could expect that additional difficulties would arise. One condition for the existence of a localized soliton is that the propagation constant does not fall into the continuous spectrum of radiation modes. For a higher order dispersive systems, the linear dispersion relation exhibits a multiple branch structure. It could be the case that in a certain parameter region for which one of the components of the solution has oscillations (i.e., is in the continuous spectrum), there exists a discrete value of the propagation constant, k(ES), for which the oscillations have zero amplitude. The associated solution is referred to as an embedded soliton (ES). This work examines the ES solutions in a CHI(2):CHI(3), type II system. The method employed in searching for the ES solutions is a variational method recently developed by Kaup and Malomed [Phys. D 184, 15361 (2003)] to locate ES solutions in a SHG system. The variational results are validated by numerical integration of the governing system. A model used for the 1D longitudinal wave propagation in microstructured solids is a KdVtype equation with third and fifth order dispersions as well as first and third order nonlinearities. Recent work by Ilison and Salupere (2004) has identified certain types of soliton solutions in the aforementioned model. The present work expands the known family of soliton solutions in the model to include embedded solitons. The existence of embedded solitons with respect to the dispersion parameters is determined by a variational approximation. The variational results are validated with selected numerical solutions.
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Date Issued

2007

Identifier

CFE0001800, ucf:47379

Format

Document (PDF)

PURL

http://purl.flvc.org/ucf/fd/CFE0001800


Title

DISCRETE NONLINEAR WAVE PROPAGATION IN KERR NONLINEAR MEDIA.

Creator

Meier, Joachim, Stegeman, George, University of Central Florida

Abstract / Description

Discrete optical systems are a subgroup of periodic structures in which the evolution of a continuous electromagnetic field can be described by a discrete model. In this model, the total field is the sum of localized, discrete modes. Weakly coupled arrays of single mode channel waveguides have been known to fall into this class of systems since the late 1960's. Nonlinear discrete optics has received a considerable amount of interest in the last few years, triggered by the experimental...
Show moreDiscrete optical systems are a subgroup of periodic structures in which the evolution of a continuous electromagnetic field can be described by a discrete model. In this model, the total field is the sum of localized, discrete modes. Weakly coupled arrays of single mode channel waveguides have been known to fall into this class of systems since the late 1960's. Nonlinear discrete optics has received a considerable amount of interest in the last few years, triggered by the experimental realization of discrete solitons in a Kerr nonlinear AlGaAs waveguide array by H. Eisenberg and coworkers in 1998. In this work a detailed experimental investigation of discrete nonlinear wave propagation and the interactions between beams, including discrete solitons, in discrete systems is reported for the case of a strong Kerr nonlinearity. The possibility to completely overcome "discrete" diffraction and create highly localized solitons, in a scalar or vector geometry, as well as the limiting factors in the formation of such nonlinear waves is discussed. The reversal of the sign of diffraction over a range of propagation angles leads to the stability of plane waves in a material with positive nonlinearity. This behavior can not be found in continuous selffocusing materials where plane waves are unstable against perturbations. The stability of plane waves in the anomalous diffraction region, even at highest powers, has been experimentally verified. The interaction of high power beams and discrete solitons in arrays has been studied in detail. Of particular interest is the experimental verification of a theoretically predicted unique, all optical switching scheme, based on the interaction of a so called "blocker" soliton with a second beam. This switching method has been experimentally realized for both the coherent and incoherent case. Limitations of such schemes due to nonlinear losses at the required high powers are shown.
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Date Issued

2004

Identifier

CFE0000186, ucf:46176

Format

Document (PDF)

PURL

http://purl.flvc.org/ucf/fd/CFE0000186


Title

QUADRATIC SPATIAL SOLITON INTERACTIONS.

Creator

Jankovic, Ladislav, Stegeman, George I., University of Central Florida

Abstract / Description

Quadratic spatial soliton interactions were investigated in this Dissertation. The first part deals with characterizing the principal features of multisoliton generation and soliton selfreflection. The second deals with two beam processes leading to soliton interactions and collisions. These subjects were investigated both theoretically and experimentally. The experiments were performed by using potassium niobate (KNBO3) and periodically poled potassium titanyl phosphate (KTP) crystals....
Show moreQuadratic spatial soliton interactions were investigated in this Dissertation. The first part deals with characterizing the principal features of multisoliton generation and soliton selfreflection. The second deals with two beam processes leading to soliton interactions and collisions. These subjects were investigated both theoretically and experimentally. The experiments were performed by using potassium niobate (KNBO3) and periodically poled potassium titanyl phosphate (KTP) crystals. These particular crystals were desirable for these experiments because of their large nonlinear coefficients and, more importantly, because the experiments could be performed under noncriticalphasematching (NCPM) conditions. The single soliton generation measurements, performed on KNBO3 by launching the fundamental component only, showed a broad angular acceptance bandwidth which was important for the soliton collisions performed later. Furthermore, at high input intensities multisoliton generation was observed for the first time. The influence on the multisoliton patterns generated of the input intensity and beam symmetry was investigated. The combined experimental and theoretical efforts indicated that spatial and temporal noise on the input laser beam induced multisoliton patterns. Another research direction pursued was intensity dependent soliton routing by using of a specially engineered quadratically nonlinear interface within a periodically poled KTP sample. This was the first time demonstration of the selfreflection phenomenon in a system with a quadratic nonlinearity. The feature investigated is believed to have a great potential for soliton routing and manipulation by engineered structures. A detailed investigation was conducted on two soliton interaction and collision processes. Birth of an additional soliton resulting from a two soliton collision was observed and characterized for the special case of a nonplanar geometry. A small amount of spiraling, up to 30 degrees rotation, was measured in the experiments performed. The parameters relevant for characterizing soliton collision processes were also studied in detail. Measurements were performed for various collision angles (from 0.2 to 4 degrees), phase mismatch, relative phase between the solitons and the distance to the collision point within the sample (which affects soliton formation). Both the individual and combined effects of these collision variables were investigated. Based on the research conducted, several alloptical switching scenarios were proposed.
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Date Issued

2004

Identifier

CFE0000090, ucf:46135

Format

Document (PDF)

PURL

http://purl.flvc.org/ucf/fd/CFE0000090


Title

STABLE SPATIAL SOLITONS IN SEMICONDUCTOROPTICAL AMPLIFIERS.

Creator

ultanir, erdem ahmet, Stegeman, George I., University of Central Florida

Abstract / Description

A spatial soliton is a shape invariant self guided beam of light or a self induced waveguide.Spatial solitons appear as a result of the balance of diffraction and nonlinear focusing in asystem. They have been observed in many different conservative media in the last couple ofyears. Solitons are ubiquitous, because of the probability of using their interactions in opticaldata processing, communications etc. Up to now due to the power required to generate thesolitons, and the response times of...
Show moreA spatial soliton is a shape invariant self guided beam of light or a self induced waveguide.Spatial solitons appear as a result of the balance of diffraction and nonlinear focusing in asystem. They have been observed in many different conservative media in the last couple ofyears. Solitons are ubiquitous, because of the probability of using their interactions in opticaldata processing, communications etc. Up to now due to the power required to generate thesolitons, and the response times of the soliton supporting media, these special waves of naturecould not penetrate the applications arena. Semiconductors, with their resonant nonlinearities, arethought to be ideal candidates for fast switching, low power spatial solitons.In this dissertation it is shown theoretically and experimentally that it is possible toobserve stable spatial solitons in a periodically patterned semiconductor optical amplifier(PPSOA). The solitons have unique beam profiles that change only with system parameters, likepumping current, etc. Their coherent and incoherent interactions which could lead to all opticaldevices have been investigated experimentally and theoretically. The formation of filaments ormodulational instability has been studied theoretically and yielded analytical formulae forevaluating the filament gain and the maximum spatial frequencies in PPSOA devices.Furthermore, discrete array amplifiers have been analyzed numerically for discrete solitons, andthe prospect of using multi peak discrete solitons as laser amplifiers is discussed.
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Date Issued

2004

Identifier

CFE0000142, ucf:46153

Format

Document (PDF)

PURL

http://purl.flvc.org/ucf/fd/CFE0000142


Title

OPTICAL NONLINEAR INTERACTIONS IN DIELECTRIC NANOSUSPENSIONS.

Creator

ElGanainy, Ramy, Christodoulides, Demetrios, University of Central Florida

Abstract / Description

This work is divided into two main parts. In the first part (chapters 27) we consider the nonlinear response of nanoparticle colloidal systems. Starting from the NernstPlanck and Smoluchowski equations, we demonstrate that in these arrangements the underlying nonlinearities as well as the nonlinear Rayleigh losses depend exponentially on optical intensity. Two different nonlinear regimes are identified depending on the refractive index contrast of the nanoparticles involved and the...
Show moreThis work is divided into two main parts. In the first part (chapters 27) we consider the nonlinear response of nanoparticle colloidal systems. Starting from the NernstPlanck and Smoluchowski equations, we demonstrate that in these arrangements the underlying nonlinearities as well as the nonlinear Rayleigh losses depend exponentially on optical intensity. Two different nonlinear regimes are identified depending on the refractive index contrast of the nanoparticles involved and the interesting prospect of selfinduced transparency is demonstrated. Soliton stability is systematically analyzed for both 1D and 2D configurations and their propagation dynamics in the presence of Rayleigh losses is examined. We also investigate the modulation instability of plane waves and the transverse instabilities of soliton stripe beams propagating in nonlinear nanosuspensions. We show that in these systems, the process of modulational instability depends on the boundary conditions. On the other hand, the transverse instability of soliton stripes can exhibit new features as a result of 1D collapse caused by the exponential nonlinearity. Manybody effects on the systems' nonlinear response are also examined. Mayer cluster expansions are used in order to investigate particleparticle interactions. We show that the optical nonlinearity of these nanosuspensions can range anywhere from exponential to polynomial depending on the initial concentration and the chemistry of the electrolyte solution. The consequence of these interparticle interactions on the soliton dynamics and their stability properties are also studied. The second part deals with linear and nonlinear properties of optical nanowires and the coupled mode formalism of paritytime (PT) symmetric waveguides. Dispersion properties of AlGaAs nanowires are studied and it is shown that the group velocity dispersion in such waveguides can be negative, thus enabling temporal solitons. We have also studied power flow in nanowaveguides and we have shown that under certain conditions, optical pulses propagating in such structures will exhibit power circulations. Finally PT symmetric waveguides were investigated and a suitable coupled mode theory to describe these systems was developed.
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Date Issued

2009

Identifier

CFE0002847, ucf:48538

Format

Document (PDF)

PURL

http://purl.flvc.org/ucf/fd/CFE0002847


Title

ACCELERATING OPTICAL AIRY BEAMS.

Creator

Siviloglou, Georgios, CHRISTODOULIDES, DEMETRIOS, University of Central Florida

Abstract / Description

Over the years, nonspreading or nondiffracting wave configurations have been systematically investigated in optics. Perhaps the best known example of a diffractionfree optical wave is the socalled Bessel beam, first suggested and observed by Durnin et al. This work sparked considerable theoretical and experimental activity and paved the way toward the discovery of other interesting nondiffracting solutions. In 1979 Berry and Balazs made an important observation within the context of...
Show moreOver the years, nonspreading or nondiffracting wave configurations have been systematically investigated in optics. Perhaps the best known example of a diffractionfree optical wave is the socalled Bessel beam, first suggested and observed by Durnin et al. This work sparked considerable theoretical and experimental activity and paved the way toward the discovery of other interesting nondiffracting solutions. In 1979 Berry and Balazs made an important observation within the context of quantum mechanics: they theoretically demonstrated that the SchrÃƒÂ¶dinger equation describing a free particle can exhibit a nonspreading Airy wavepacket solution. This work remained largely unnoticed in the literaturepartly because such wavepackets cannot be readily synthesized in quantum mechanics. In this dissertation we investigate both theoretically and experimentally the acceleration dynamics of nonspreading optical Airy beams in both one and twodimensional configurations. We show that this class of finite energy waves can retain their intensity features over several diffraction lengths. The possibility of other physical realizations involving spatiotemporal Airy wavepackets is also considered. As demonstrated in our experiments, these Airy beams can exhibit unusual features such as the ability to remain quasidiffractionfree over long distances while their intensity features tend to freely accelerate during propagation. We have demonstrated experimentally that optical Airy beams propagating in free space can perform ballistic dynamics akin to those of projectiles moving under the action of gravity. The parabolic trajectories of these beams as well as the motion of their center of gravity were observed in good agreement with theory. Another remarkable property of optical Airy beams is their resilience in amplitude and phase perturbations. We show that this class of waves tends to reform during propagation in spite of the severity of the imposed perturbations. In all occasions the reconstruction of these beams is interpreted through their internal transverse power flow. The robustness of these optical beams in scattering and turbulent environments was also studied. The experimental observation of selftrapped Airy beams in unbiased nonlinear photorefractive media is also reported. This new class of nonlocal selflocalized beams owes its existence to carrier diffusion effects as opposed to selffocusing. These finite energy Airy states exhibit a highly asymmetric intensity profile that is determined by the inherent properties of the nonlinear crystal. In addition, these wavepackets selfbend during propagation at an acceleration rate that is independent of the thermal energy associated with twowave mixing diffusion photorefractive nonlinearity.
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Date Issued

2010

Identifier

CFE0003193, ucf:48569

Format

Document (PDF)

PURL

http://purl.flvc.org/ucf/fd/CFE0003193


Title

DISCRETE WAVE PROPAGATION IN QUADRATICALLY NONLINEAR MEDIA.

Creator

Iwanow, Robert, Stegeman, George, University of Central Florida

Abstract / Description

Discrete models are used in describing various microscopic phenomena in many branches of science, ranging from biology through chemistry to physics. Arrays of evanescently coupled, equally spaced, identical waveguides are prime examples of optical structures in which discrete dynamics can be easily observed and investigated. As a result of discretization, these structures exhibit unique diffraction properties with no analogy in continuous systems. Recently nonlinear discrete optics has...
Show moreDiscrete models are used in describing various microscopic phenomena in many branches of science, ranging from biology through chemistry to physics. Arrays of evanescently coupled, equally spaced, identical waveguides are prime examples of optical structures in which discrete dynamics can be easily observed and investigated. As a result of discretization, these structures exhibit unique diffraction properties with no analogy in continuous systems. Recently nonlinear discrete optics has attracted a growing interest, triggered by the observation of discrete solitons in AlGaAs waveguide arrays reported by Eisenberg et al. in 1998. So far, the following experiments involved systems with third order nonlinearities. In this work, an experimental investigation of discrete nonlinear wave propagation in a second order nonlinear medium is presented. This system deserves particular attention because the nonlinear process involves two or three components at different frequencies mutually locked by a quadratic nonlinearity, and new degrees of freedom enter the dynamical process. In the first part of dissertation, observation of the discrete Talbot effect is reported. In contrast to continuous systems, where Talbot selfimaging effect occurs irrespective of the pattern period, in discrete configurations this process is only possible for a specific set of periodicities. The major part of the dissertation is devoted to the investigation of soliton formation in lithium niobate waveguide arrays with a tunable cascaded quadratic nonlinearity. Soliton species with different topology (unstaggered – all channels inphase, and staggered – neighboring channels with a pi relative phase difference) are identified in the same array. The stability of the discrete solitons and plane waves (modulational instability) are experimentally investigated. In the last part of the dissertation, a phaseinsensitive, ultrafast, alloptical spatial switching and frequency conversion device based on quadratic waveguide array is demonstrated. Spatial routing and wavelength conversion of milliwatt signals is achieved without pulse distortions.
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Date Issued

2005

Identifier

CFE0000420, ucf:46382

Format

Document (PDF)

PURL

http://purl.flvc.org/ucf/fd/CFE0000420


Title

SOLITARY WAVE FAMILIES IN TWO NONINTEGRABLE MODELS USING REVERSIBLE SYSTEMS THEORY.

Creator

Leto, Jonathan, Choudhury, S. Roy, University of Central Florida

Abstract / Description

In this thesis, we apply a recently developed technique to comprehensively categorize all possible families of solitary wave solutions in two models of topical interest. The models considered are: a) the Generalized PochhammerChree Equations, which govern the propagation of longitudinal waves in elastic rods, and b) a generalized microstructure PDE. Limited analytic results exist for the occurrence of one family of solitary wave solutions for each of these equations. Since, as mentioned...
Show moreIn this thesis, we apply a recently developed technique to comprehensively categorize all possible families of solitary wave solutions in two models of topical interest. The models considered are: a) the Generalized PochhammerChree Equations, which govern the propagation of longitudinal waves in elastic rods, and b) a generalized microstructure PDE. Limited analytic results exist for the occurrence of one family of solitary wave solutions for each of these equations. Since, as mentioned above, solitary wave solutions often play a central role in the longtime evolution of an initial disturbance, we consider such solutions of both models here (via the normal form approach) within the framework of reversible systems theory. Besides confirming the existence of the known family of solitary waves for each model, we find a continuum of delocalized solitary waves (or homoclinics to smallamplitude periodic orbits). On isolated curves in the relevant parameter region, the delocalized waves reduce to genuine embedded solitons. For the microstructure equation, the new family of solutions occur in regions of parameter space distinct from the known solitary wave solutions and are thus entirely new. Directions for future work, including the dynamics of each family of solitary waves using exponential asymptotics techniques, are also mentioned.
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Date Issued

2008

Identifier

CFE0002151, ucf:47930

Format

Document (PDF)

PURL

http://purl.flvc.org/ucf/fd/CFE0002151


Title

VARIATIONAL EMBEDDED SOLITONS, AND TRAVELING WAVETRAINS GENERATED BY GENERALIZED HOPF BIFURCATIONS, IN SOME NLPDE SYSTEMS.

Creator

Smith, Todd, Choudhury, Roy, University of Central Florida

Abstract / Description

In this Ph.D. thesis, we study regular and embedded solitons and generalized and degenerate Hopf bifurcations. These two areas of work are seperate and independent from each other. First, variational methods are employed to generate families of both regular and embedded solitary wave solutions for a generalized Pochhammer PDE and a generalized microstructure PDE that are currently of great interest. The technique for obtaining the embedded solitons incorporates several recent generalizations...
Show moreIn this Ph.D. thesis, we study regular and embedded solitons and generalized and degenerate Hopf bifurcations. These two areas of work are seperate and independent from each other. First, variational methods are employed to generate families of both regular and embedded solitary wave solutions for a generalized Pochhammer PDE and a generalized microstructure PDE that are currently of great interest. The technique for obtaining the embedded solitons incorporates several recent generalizations of the usual variational technique and is thus topical in itself. One unusual feature of the solitary waves derived here is that we are able to obtain them in analytical form (within the family of the trial functions). Thus, the residual is calculated, showing the accuracy of the resulting solitary waves. Given the importance of solitary wave solutions in wave dynamics and information propagation in nonlinear PDEs, as well as the fact that only the parameter regimes for the existence of solitary waves had previously been analyzed for the microstructure PDE considered here, the results obtained here are both new and timely. Second, we consider generalized and degenerate Hopf bifurcations in three different models: i. a predatorprey model with general predator death rate and prey birth rate terms, ii. a laserdiode system, and iii. travelingwave solutions of twospecies predatorprey/reactiondiusion equations with arbitrary nonlinear/reaction terms. For speci c choices of the nonlinear terms, the quasiperiodic orbit in the postbifurcation regime is constructed for each system using the method of multiple scales, and its stability is analyzed via the corresponding normal form obtained by reducing the system down to the center manifold. The resulting predictions for the postbifurcation dynamics provide an organizing framework for the variety of possible behaviors. These predictions are veri ed and supplemented by numerical simulations, including the computation of power spectra, autocorrelation functions, and fractal dimensions as appropriate for the periodic and quasiperiodic attractors, attractors at in nity, as well as bounded chaotic attractors obtained in various cases. The dynamics obtained in the three systems is contrasted and explained on the basis of the bifurcations occurring in each. For instance, while the two predatorprey models yield a variety of behaviors in the postbifurcation regime, the laserdiode evinces extremely stable quasiperiodic solutions over a wide range of parameters, which is very desirable for robust operation of the system in oscillator mode.
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Date Issued

2011

Identifier

CFE0003634, ucf:48887

Format

Document (PDF)

PURL

http://purl.flvc.org/ucf/fd/CFE0003634


Title

DISSIPATIVE SOLITONS IN THE CUBIC–QUINTIC COMPLEX GINZBURG–LANDAU EQUATION:BIFURCATIONS AND SPATIOTEMPORAL STRUCTURE.

Creator

Mancas, Ciprian, Choudhury, Roy S., University of Central Florida

Abstract / Description

Comprehensive numerical simulations (reviewed in Dissipative Solitons, Akhmediev and Ankiewicz (Eds.), Springer, Berlin, 2005) of pulse solutions of the cubicquintic GinzburgLandau equation (CGLE), a canonical equation governing the weakly nonlinear behavior of dissipative systems in a wide variety of disciplines, reveal various intriguing and entirely novel classes of solutions. In particular, there are five new classes of pulse or solitary waves solutions, viz. pulsating, creeping,...
Show moreComprehensive numerical simulations (reviewed in Dissipative Solitons, Akhmediev and Ankiewicz (Eds.), Springer, Berlin, 2005) of pulse solutions of the cubicquintic GinzburgLandau equation (CGLE), a canonical equation governing the weakly nonlinear behavior of dissipative systems in a wide variety of disciplines, reveal various intriguing and entirely novel classes of solutions. In particular, there are five new classes of pulse or solitary waves solutions, viz. pulsating, creeping, snake, erupting, and chaotic solitons. In contrast to the regular solitary waves investigated in numerous integrable and nonintegrable systems over the last three decades, these dissipative solitons are not stationary in time. Rather, they are spatially confined pulsetype structures whose envelopes exhibit complicated temporal dynamics. The numerical simulations also reveal very interesting bifurcations sequences of these pulses as the parameters of the CGLE are varied. In this dissertation, we develop a theoretical framework for these novel classes of solutions. In the first part, we use a traveling wave reduction or a socalled spatial approximation to comprehensively investigate the bifurcations of plane wave and periodic solutions of the CGLE. The primary tools used here are Singularity Theory and Hopf bifurcation theory respectively. Generalized and degenerate Hopf bifurcations have also been considered to track the emergence of global structure such as homoclinic orbits. However, these results appear difficult to correlate to the numerical bifurcation sequences of the dissipative solitons. In the second part of this dissertation, we shift gears to focus on the issues of central interest in the area, i.e., the conditions for the occurrence of the five categories of dissipative solitons, as well the dependence of both their shape and their stability on the various parameters of the CGLE, viz. the nonlinearity, dispersion, linear and nonlinear gain, loss and spectral filtering parameters. Our predictions on the variation of the soliton amplitudes, widths and periods with the CGLE parameters agree with simulation results. For this part, we develop and discuss a variational formalism within which to explore the various classes of dissipative solitons. Given the complex dynamics of the various dissipative solutions, this formulation is, of necessity, significantly generalized over all earlier approaches in several crucial ways. Firstly, the two alternative starting formulations for the Lagrangian are recent and not well explored. Also, after extensive discussions with David Kaup, the trial functions have been generalized considerably over conventional ones to keep the shape relatively simple (and the trial function integrable!) while allowing arbitrary temporal variation of the amplitude, width, position, speed and phase of the pulses. In addition, the resulting EulerLagrange equations are treated in a completely novel way. Rather than consider the stable fixed points which correspond to the wellknown stationary solitons or plain pulses, we use dynamical systems theory to focus on more complex attractors viz. periodic, quasiperiodic, and chaotic ones. Periodic evolution of the trial function parameters on stable periodic attractors constructed via the method of multiple scales yield solitons whose amplitudes are nonstationary or time dependent. In particular, pulsating, snake (and, less easily, creeping) dissipative solitons may be treated in this manner. Detailed results are presented here for the pulsating solitary waves  their regimes of occurrence, bifurcations, and the parameter dependences of the amplitudes, widths, and periods agree with simulation results. Finally, we elucidate the Hopf bifurcation mechanism responsible for the various pulsating solitary waves, as well as its absence in Hamiltonian and integrable systems where such structures are absent. Results will be presented for the pulsating and snake soliton cases. Chaotic evolution of the trial function parameters in chaotic regimes identified using dynamical systems analysis would yield chaotic solitary waves. The method also holds promise for detailed modeling of chaotic solitons as well. This overall approach fails only to address the fifth class of dissipative solitons, viz. the exploding or erupting solitons.
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Date Issued

2007

Identifier

CFE0001571, ucf:47116

Format

Document (PDF)

PURL

http://purl.flvc.org/ucf/fd/CFE0001571


Title

Comparing the Variational Approximation and Exact Solutions of the Straight Unstaggered and Twisted Staggered Discrete Solitons.

Creator

Marulanda, Daniel, Kaup, David, Moore, Brian, Vajravelu, Kuppalapalle, University of Central Florida

Abstract / Description

Discrete nonlinear Schr(&)#246;dinger equations (DNSL) have been used to provide models of a variety of physical settings. An application of DNSL equations is provided by BoseEinstein condensates which are trapped in deep opticallattice potentials. These potentials effectively splits the condensate into a set of droplets held in local potential wells, which are linearly coupled across the potential barriers between them [3]. In previous works, DNLS systems have also been used for symmetric...
Show moreDiscrete nonlinear Schr(&)#246;dinger equations (DNSL) have been used to provide models of a variety of physical settings. An application of DNSL equations is provided by BoseEinstein condensates which are trapped in deep opticallattice potentials. These potentials effectively splits the condensate into a set of droplets held in local potential wells, which are linearly coupled across the potential barriers between them [3]. In previous works, DNLS systems have also been used for symmetric onsitecentered solitons [11]. A few works have constructed different discrete solitons via the variational approximation (VA) and have explored their regions for their solutions [11, 12]. Exact solutions for straight unstaggeredtwisted staggered (SUTS) discrete solitons have been found using the shooting method [12].In this work, we will use Newton's method, which converges to the exact solutions of SUTS discrete solitons. The VA has been used to create starting points. There are two distinct types of solutions for the soliton's waveform: SUTS discrete solitons and straight unstaggered discrete solitons, where the twisted component is zero in the latter soliton. We determine the range of parameters for which each type of solution exists. We also compare the regions for the VA solutions and the exact solutions in certain selected cases. Then, we graphically and numerically compare examples of the VA solutions with their corresponding exact solutions. We also find that the VA provides reasonable approximations to the exact solutions.
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Date Issued

2016

Identifier

CFE0006350, ucf:51570

Format

Document (PDF)

PURL

http://purl.flvc.org/ucf/fd/CFE0006350


Title

Particle Manipulation Via Optical Forces and Engineering SoftMatter Systems With Tunable Nonlinearities.

Creator

Fardad, Shima, Christodoulides, Demetrios, Hagan, David, Amezcua Correa, Rodrigo, Likamwa, Patrick, Chen, Zhigang, University of Central Florida

Abstract / Description

One of the most intriguing properties of lightmatter interaction is the ability of an electromagnetic field to exert mechanical forces on polarizable objects. This phenomenon is a direct consequence of the fact that light carries momentum, which in turn can be transferred to matter. Mediated by scattering, this interaction usually manifests itself as a (")pushing force(") in the direction of beam propagation. However, it is possible to judiciously engineer these optical forces, either by...
Show moreOne of the most intriguing properties of lightmatter interaction is the ability of an electromagnetic field to exert mechanical forces on polarizable objects. This phenomenon is a direct consequence of the fact that light carries momentum, which in turn can be transferred to matter. Mediated by scattering, this interaction usually manifests itself as a (")pushing force(") in the direction of beam propagation. However, it is possible to judiciously engineer these optical forces, either by tailoring particle polarizability, and/or by structuring the incident light field. As a simple example, a tightly focused laser beam demonstrates strong gradient forces, which may attract and even trap particles with positive polarizability in the focal volume. The opposite occurs in the regime of negative polarizability, where particles are expelled from the regions of highest intensity. Based on this fundamental principle, one can actively shape the beam using spatial light modulators to manipulate individual objects as well as ensembles of particles suspended in a liquid. In the latter case, a modulation of the local particle concentration is associated with changes of the effective refractive index. The result is an artificial nonlinear medium, whose Kerrtype response can be readily tuned by the parameters of its constituent particles.In the course of this work, we introduce a new class of synthetic colloidal suspensions exhibiting negative polarizabilities, and observe for the first time robust propagation and enhanced transmission of selftrapped light over long distances. Such light penetration in strongly scattering environments is enabled by the interplay between optical forces and selfactivated transparency effects. We explore various approaches to the design of negativepolarizability arrangements, including purely dielectric as well as metallic and hybrid nanoparticles. In particular, we find that plasmonic resonances allow for extremely high and spectrally tunable polarizabilities, leading to unique nonlinear lightmatter interactions. Here, for the first time we were able to observe plasmonic resonant solitons over more than 25 diffraction lengths, in colloidal nanosuspensions.
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Date Issued

2014

Identifier

CFE0005610, ucf:50239

Format

Document (PDF)

PURL

http://purl.flvc.org/ucf/fd/CFE0005610


Title

Nonlinear Dynamics in Multimode Optical Fibers.

Creator

Eftekhar, Mohammad Amin, Christodoulides, Demetrios, Amezcua Correa, Rodrigo, Li, Guifang, Kaup, David, University of Central Florida

Abstract / Description

Multimode optical fibers have recently reemerged as a viable platform for addressing a number of longstanding issues associated with information bandwidth requirements and powerhandling capabilities. The complex nature of heavily multimoded systems can be effectively exploited to observe altogether novel physical effects arising from spatiotemporal and intermodal linear and nonlinear processes. Here, we have studied nonlinear dynamics in multimode optical fibers (MMFs) in both the normal...
Show moreMultimode optical fibers have recently reemerged as a viable platform for addressing a number of longstanding issues associated with information bandwidth requirements and powerhandling capabilities. The complex nature of heavily multimoded systems can be effectively exploited to observe altogether novel physical effects arising from spatiotemporal and intermodal linear and nonlinear processes. Here, we have studied nonlinear dynamics in multimode optical fibers (MMFs) in both the normal and anomalous dispersion regimes. In the anomalous dispersion regime, the nonlinearity leads to a formation of spatiotemporal 3D solitons. Unlike in singlemode fibers, these solitons are not unique and their properties can be modified through the additional degrees of freedom offered by these multimoded settings. In addition, soliton related processes such as soliton fission and dispersive wave generation will be also drastically altered in such multimode systems. Our theoretical work unravels some of the complexities of the underlying dynamics and helps us better understand these effects. The nonlinear dynamics in such multimode systems can be accelerated through a judicious fiber design. A cancelation of Raman selffrequency shifts and Blueshifting multimode solitons were observed in such settings as a result of an acceleration of intermodal oscillations. Spatiotemporal instabilities in parabolicindex multimode fibers will also be discussed. In the normal dispersion regime, this effect can be exploited to generate an ultrabroad and uniform supercontinuum that extends more than 2.5 octaves. To do so, the unstable spectral regions are pushed away from the pump, thus sweeping the entire spectrum. Multimode parabolic pulses were also predicted and observed in passive normally dispersive tapered MMFs. These setting can obviate the harsh bandwidth limitation present in singlemode system imposed by gain medium and be effectively used for realizing high power multimode fiber lasers. Finally, an instant and efficient secondharmonic generation was observed in the multimode optical fibers. Through a modification of initial conditions, the efficiency of this process could be enhanced to a record high of %6.5.
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Date Issued

2018

Identifier

CFE0007399, ucf:52063

Format

Document (PDF)

PURL

http://purl.flvc.org/ucf/fd/CFE0007399