Class of cooperative stochastic models: Exact and approximate solutions, simulations, and experiments using ionic self-assembly of nanoparticles.

We present exact and approximate results for a class of cooperative sequential adsorption models using matrix theory, mean-field theory, and computer simulations. We validate our models with two customized experiments using ionically self-assembled nanoparticles on glass slides. We also address the limitations of our models and their range of applicability. The exact results obtained using matrix theory can be applied to a variety of two-state systems with cooperative effects.

Cooperative sequential-adsorption model in two dimensions with experimental applications for ionic self-assembly of nanoparticles.

Self-assembly of nanoparticles is an important tool in nanotechnology, with numerous applications, including thin films, electronics, and drug delivery. We study the deposition of ionic nanoparticles on a glass substrate both experimentally and theoretically. Our theoretical model consists of a stochastic cooperative adsorption and evaporation process on a two-dimensional lattice. By exploring the relationship between the initial concentration of nanoparticles in the colloidal solution and the density of particles deposited on the substrate, we relate the deposition rate of our theoretical model to the concentration.

Fabrication of 200 nanometer period centimeter area hard x-ray absorption gratings by multilayer deposition.

We describe the design and fabrication trials of x-ray absorption gratings of 200 nm period and up to 100:1 depth-to-period ratios for full-field hard x-ray imaging applications. Hard x-ray phase-contrast imaging relies on gratings of ultra-small periods and sufficient depth to achieve high sensitivity. Current grating designs utilize lithographic processes to produce periodic vertical structures, where grating periods below 2.0 µm are difficult due to the extreme aspect ratios of the structures. In our design, multiple bilayers of x-ray transparent and opaque materials are deposited on a staircase substrate, and mostly on the floor surfaces of the steps only. When illuminated by an x-ray beam horizontally, the multilayer stack on each step functions as a micro-grating whose grating period is the thickness of a bilayer. The array of micro-gratings over the length of the staircase works as a single grating over a large area when continuity conditions are met. Since the layers can be nanometers thick and many microns wide, this design allows sub-micron grating periods and sufficient grating depth to modulate hard x-rays. We present the details of the fabrication process and diffraction profiles and contact radiography images showing successful intensity modulation of a 25 keV x-ray beam.

Computer aided minimally invasive cardiac procedures.

Minimally invasive cardiac procedures have been investigated to reduce the risks associated with open heart surgery. With the assistance of improvements in engineering technologies such as medical imaging, surgical navigation, and robotic devices, more cardiac surgeries can be performed in a minimally invasive fashion. We have surveyed these state-of-the-art engineering technologies and the minimally invasive cardiac procedures that are benefited from these technologies.

Collisions between counter-rotating solitary vortices in the three-dimensional Ginzburg-Landau equation.

We report results of collisions between coaxial vortex solitons with topological charges +/-S in the complex cubic-quintic Ginzburg-Landau equation. With the increase of the collision momentum, merger of the vortices into one or two dipole or quadrupole clusters of fundamental solitons (for S=1 and 2, respectively) is followed by the appearance of pairs of counter-rotating "unfinished vortices," in combination with a soliton cluster or without it. Finally, the collisions become elastic. The clusters generated by the collisions are very robust, while the "unfinished vortices," eventually split into soliton pairs.

Stable three-dimensional optical solitons supported by competing quadratic and self-focusing cubic nonlinearities.

We show that the quadratic (chi(2)) interaction of fundamental and second harmonics in a bulk dispersive medium, combined with self-focusing cubic (chi(3)) nonlinearity, give rise to stable three-dimensional spatiotemporal solitons (STSs), despite the possibility of the supercritical collapse, induced by the chi(3) nonlinearity. At exact phase matching (beta = 0) , the STSs are stable for energies from zero up to a certain maximum value, while for beta not equal 0 the solitons are stable in energy intervals between finite limits.

Stable vortex tori in the three-dimensional cubic-quintic Ginzburg-Landau equation.

We demonstrate the existence of stable toroidal dissipative solitons with the inner phase field in the form of rotating spirals, corresponding to vorticity S=0, 1, and 2, in the complex Ginzburg-Landau equation with the cubic-quintic nonlinearity. The stable solitons easily self-trap from pulses with embedded vorticity. The stability is corroborated by accurate computation of growth rates for perturbation eigenmodes. The results provide the first example of stable vortex tori in a 3D dissipative medium, as well as the first example of higher-order tori (with S=2) in any nonlinear medium. It is found that all stable vortical solitons coexist in a large domain of the parameter space; in smaller regions, there coexist stable solitons with either S=0 and S=1, or S=1 and S=2.

Three-dimensional spatiotemporal optical solitons in nonlocal nonlinear media.

We demonstrate the existence of stable three-dimensional spatiotemporal solitons (STSs) in media with a nonlocal cubic nonlinearity. Fundamental (nonspinning) STSs forming one-parameter families are stable if their propagation constant exceeds a certain critical value that is inversely proportional to the range of nonlocality of nonlinear response. All spinning three-dimensional STSs are found to be unstable.

Stable solitons of even and odd parities supported by competing nonlocal nonlinearities.

We introduce a one-dimensional phenomenological model of a nonlocal medium featuring focusing cubic and defocusing quintic nonlocal optical nonlinearities. By means of numerical methods, we find families of solitons of two types, even-parity (fundamental) and dipole-mode (odd-parity) ones. Stability of the solitons is explored by means of computation of eigenvalues associated with modes of small perturbations, and tested in direct simulations. We find that the stability of the fundamental solitons strictly follows the Vakhitov-Kolokolov criterion, whereas the dipole solitons can be destabilized through a Hamiltonian-Hopf bifurcation. The solitons of both types may be stable in the nonlocal model with only quintic self-attractive nonlinearity, in contrast with the instability of all solitons in the local version of the quintic model.

Stable spatiotemporal solitons in bessel optical lattices.

We investigate the existence and stability of three-dimensional solitons supported by cylindrical Bessel lattices in self-focusing media. If the lattice strength exceeds a threshold value, we show numerically, and using the variational approximation, that the solitons are stable within one or two intervals of values of their norm. In the latter case, the Hamiltonian versus norm diagram has a swallowtail shape with three cuspidal points. The model applies to Bose-Einstein condensates and to optical media with saturable nonlinearity, suggesting new ways of making stable three-dimensional solitons and "light bullets" of an arbitrary size.

Parametric light bullets supported by quasi-phase-matched quadratically nonlinear crystals.

We present a comprehensive analysis of the dynamics of three-dimensional spatiotemporal nonspinning and spinning solitons in quasi-phased-matched (QPM) gratings. By employing an averaging approach based on perturbation theory, we show that the soliton's stability is strongly affected by the QPM-induced third-order nonlinearity (which is always of a mixed type, with opposite signs in front of the corresponding self-phase and cross-phase modulation terms). We study the dependence of the stability of the spatiotemporal soliton (STS) on its energy, spin, the wave-vector mismatch between the fundamental and second harmonics, and the relative strength of the intrinsic quadratic and QPM-induced cubic nonlinearities. In particular, all the spinning solitons are unstable against fragmentation, while zero-spin STS's have their stability regions on the system's parameter space.

Two-dimensional solitons with hidden and explicit vorticity in bimodal cubic-quintic media.

We demonstrate that two-dimensional two-component bright solitons of an annular shape, carrying vorticities (m,+/-m) in the components, may be stable in media with the cubic-quintic nonlinearity, including the hidden-vorticity (HV) solitons of the type (m,-m) , whose net vorticity is zero. Stability regions for the vortices of both (m,+/-m) types are identified for m=1 , 2, and 3, by dint of the calculation of stability eigenvalues, and in direct simulations. In addition to the well-known symmetry-breaking (external) instability, which splits the ring soliton into a set of fragments flying away in tangential directions, we report two new scenarios of the development of weak instabilities specific to the HV solitons. One features charge flipping, with the two components exchanging angular momentum and periodically reversing the sign of their spins. The composite soliton does not directly split in this case; therefore, we identify such instability as an intrinsic one. Eventually, the soliton splits, as weak radiation loss drives it across the border of the ordinary strong (external) instability. Another scenario proceeds through separation of the vortex cores in the two components, each individual core moving toward the outer edge of the annular soliton. After expulsion of the cores, there remains a zero-vorticity breather with persistent internal vibrations.

A computer assisted method for guide-wore and catheter evaluation.

We developed a method for testing guide wires and catheters that realistically evaluates the forces applied to anatomical structures by these instruments during urological procedures. The placement of guide wires and catheters to gain access to the upper urinary tract can induce undesirable stress on the tissue. Previous studies have characterized wire/catheter performances base on their physical properties, such as stiffness and friction coefficient. However, the results of these studies do not directly quantify their effect on the tissues. Additionally, individual physical properties do not entirely characterize the behavior of the wire/catheter ensemble. Our model utilizes a computer-controlled test stand that simulates the urological environment by including a tortuous path and a stone obstruction. Experimental results indicate that the method shows significant promise in reflecting wire/catheter performance data in congruence with reliable real-life measures of stress upon relevant anatomical structures. Furthermore, due to the modularity of the approach, the model can be easily reconfigured to simulate environments from other medical fields.

Precision placement of instruments for minimally invasive procedures using a "needle driver" robot.

Medical practice continues to move toward less invasive procedures. Many of these procedures require the precision placement of a needle in the anatomy. Over the past several years, our research team has been investigating the use of a robotic needle driver to assist the physician in this task. This paper summarizes our work in this area. The robotic system is briefly described, followed by a description of a clinical trial in spinal nerve blockade. The robot was used under joystick control to place a 22 gauge needle in the spines of 10 patients using fluoroscopic imaging. The results were equivalent to the current manual procedure. We next describe our follow-up clinical application in lung biopsy for lung cancer screening under CT fluoroscopy. The system concept is discussed and the results of a phantom study are presented. A start-up company named ImageGuide has recently been formed to commercialize the robot. Their revised robot design is presented, along with plans to install a ceiling-mounted version of the robot in the CT fluoroscopy suite at Georgetown University.

Robotic assisted radio-frequency ablation of liver tumors--randomized patient study.

The minimally invasive treatment of liver tumors represents an alternative to the open surgery approach. Radio-frequency ablation destroys a tumor by delivering radio-frequency energy through a needle probe. Traditionally, the probe is placed manually using imaging feedback. New approaches use robotic devices to accurately place the instrument at the target. The authors developed an image-guided robotic system for percutaneous interventions using computed tomography. The paper presents a randomized patient study comparing the manual versus robotic needle placement for radio-frequency ablation procedures of liver tumors. The results of this study show that in our case robotic interventions were a very viable solution. Several treatment parameters such as radiation exposures and procedure-times were found to be significantly improved in the robotic case.

Stable three-dimensional spatiotemporal solitons in a two-dimensional photonic lattice.

We investigate the existence and stability of three-dimensional spatiotemporal solitons in self-focusing cubic Kerr-type optical media with an imprinted two-dimensional harmonic transverse modulation of the refractive index. We demonstrate that two-dimensional photonic Kerr-type nonlinear lattices can support stable one-parameter families of three-dimensional spatiotemporal solitons provided that their energy is within a certain interval and the strength of the lattice potential, which is proportional to the refractive index modulation depth, is above a certain threshold value.

Stable vortex solitons supported by competing quadratic and cubic nonlinearities.

We address the stability problem for vortex solitons in two-dimensional media combining quadratic and self-defocusing cubic [chi(2):chi(3)- ] nonlinearities. We consider the propagation of spatial beams with intrinsic vorticity S in such bulk optical media. It was earlier found that the S=1 and S=2 solitons can be stable, provided that their power (i.e., transverse size) is large enough, and it was conjectured that all the higher-order vortices with S> or =3 are always unstable. On the other hand, it was recently shown that vortex solitons with S>2 and very large transverse size may be stable in media combining cubic self-focusing and quintic self-defocusing nonlinearities. Here, we demonstrate that the same is true in the chi(2):chi(3)- model, the vortices with S=3 and S=4 being stable in regions occupying, respectively, approximately 3% and 1.5% of their existence domain. The vortex solitons with S>4 are also stable in tiny regions. The results are obtained through computation of stability eigenvalues, and are then checked in direct simulations, with a conclusion that the stable vortices are truly robust ones, easily self-trapping from initial beams with embedded vorticity. The dependence of the stability region on the chi(2) phase-mismatch parameter is specially investigated. We thus conclude that the stability of higher-order two-dimensional vortex solitons in narrow regions is a generic feature of optical media featuring the competition between self-focusing and self-defocusing nonlinearities. A qualitative analytical explanation to this feature is proposed.

Robust soliton clusters in media with competing cubic and quintic nonlinearities.

Systematic results are reported for dynamics of circular patterns ("necklaces"), composed of fundamental solitons and carrying orbital angular momentum, in the two-dimensional model, which describes the propagation of light beams in bulk media combining self-focusing cubic and self-defocusing quintic nonlinearities. Semianalytical predictions for the existence of quasistable necklace structures are obtained on the basis of an effective interaction potential. Then, direct simulations are run. In the case when the initial pattern is far from an equilibrium size predicted by the potential, it cannot maintain its shape. However, a necklace with the initial shape close to the predicted equilibrium survives very long evolution, featuring persistent oscillations. The quasistable evolution is not essentially disturbed by a large noise component added to the initial configuration. Basic conclusions concerning the necklace dynamics in this model are qualitatively the same as in a recently studied one which combines quadratic and self-defocusing cubic nonlinearities. Thus, we infer that a combination of competing self-focusing and self-defocusing nonlinearities enhances the robustness not only of vortex solitons but also of vorticity-carrying necklace patterns.

Two-dimensional solitons in quasi-phase-matched quadratic crystals.

We study the existence and dynamics of two-dimensional spatial solitons in crystals that exhibit a periodic modulation of both the refractive index and the second-order susceptibility for achieving quasi-phase-matching. Far from resonances between the domain length of the periodic crystal and the diffraction length of the beams, it is demonstrated that the properties of the solitons in this quasi-phase-matched geometry are strongly influenced by the induced third-order nonlinearities. The stability properties of the two-dimensional solitons are analyzed as a function of the total power, the effective wave-vector mismatch between the first and second harmonics, and the relative strength between the induced third-order nonlinearity and the effective second-order nonlinearity. Finally, the formation of two-dimensional solitons from a Gaussian beam excitation is investigated numerically.

Stable spatiotemporal spinning solitons in a bimodal cubic-quintic medium.

We investigate the formation of stable spatiotemporal three-dimensional (3D) solitons ("light bullets") with internal vorticity ("spin") in a bimodal system described by coupled cubic-quintic nonlinear Schrödinger equations. Two relevant versions of the model, for the linear and circular polarizations, are considered. In the former case, an important ingredient of the model are four-wave-mixing terms, which give rise to a phase-sensitive nonlinear coupling between two polarization components. Thresholds for the formation of both spinning and nonspinning 3D solitons are found. Instability growth rates of perturbation eigenmodes with different azimuthal indices are calculated as functions of the solitons' propagation constant. As a result, stability domains in the model's parameter plane are identified for solitons with the values of the spins of their components s=0 and s=1, while all the solitons with s> or =2 are unstable. The solitons with s=1 are stable only if their energy exceeds a certain critical value, so that, in typical cases, the stability region occupies approximately 25% of their existence domain. Direct simulations of the full system produce results that are in perfect agreement with the linear-stability analysis: stable 3D spinning solitons readily self-trap from initial Gaussian pulses with embedded vorticity, and easily heal themselves if strong perturbations are imposed, while unstable spinning solitons quickly split into a set of separating zero-spin fragments whose number is exactly equal to the azimuthal index of the strongest unstable perturbation eigenmode.