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The rigorous solution of a generic impulsive planet-to-planet transfer by means of a Taylor-model-based global optimizer is presented. Although a planet-to-planet transfer represents the simplest case of interplanetary transfer, its formulation and solution is a challenging task when the rigorous global optimum is sought. A customized ephemeris function is derived from JPL DE405 to allow the Taylor-model evaluation of planets positions and velocities. Furthermore, the validated solution of Lambert's problem is addressed for the rigorous computation of transfer fuel consumption. The optimization problem, which consists in finding the optimal launch and transfer time to minimize the required fuel mass, is complex due to the abundance of local minima and relatively high search-space dimension. Its rigorous solution by means of the Taylor-model-based global optimizer COSY-GO is presented considering Earth-Mars and Earth-Venus transfers as test cases.

Frequently the design of schemes for correction of aberrations or the determination of possible operating ranges for beamlines and cells in synchrotrons exhibit multitudes of possibilities for their correction, usually appearing in disconnected regions of parameter space which cannot be directly qualified by analytical means. In such cases, frequently an abundance of optimization runs are carried out, each of which determines a local minimum depending on the specific chosen initial conditions. Practical solutions are then obtained through an often extended interplay of experienced manual adjustment of certain suitable parameters and local searches by varying other parameters. However, in a formal sense this problem can be viewed as a global optimization problem, i.e. the determination of all solutions within a certain range of parameters that lead to a specific optimum. For example, it may be of interest to find all possible settings of multiple quadrupoles that can achieve imaging; or to find ahead of time all possible settings that achieve a particular tune; or to find all possible manners to adjust nonlinear parameters to achieve correction of high order aberrations. These tasks can easily be phrased in terms of such an optimization problem; but while mathematically this formulation is often straightforward, it has been common belief that it is of limited practical value since the resulting optimization problem cannot usually be solved. However, recent significant advances in modern methods of rigorous global optimization make these methods feasible for optics design for the first time. The key ideas of the method lie in an interplay of rigorous local underestimators of the objective functions, and by using the underestimators to rigorously iteratively eliminate regions that lie above already known upper bounds of the minima, in what is commonly known as a branch-and-bound approach. Recent enhancements of the Differential Algebraic methods used in particle optics for the computation of aberrations allow the determination of particularly sharp underestimators for large regions. As a consequence, the subsequent progressive pruning of the allowed search space as part of the optimization progresses is carried out particularly effectively. The end result is the rigorous determination of the single or multiple optimal solutions of the parameter optimization, regardless of their location, their number, and the starting values of optimization. The methods are particularly powerful if executed in interplay with genetic optimizers generating their new populations within the currently active unpruned space. Their current best guess provides rigorous upper bounds of the minima, which can then beneficially be used for better pruning. Examples of the method and its performance will be presented, including the determination of all operating points of desired tunes or chromaticities, etc. in storage ring lattices.

A method is presented that allows the computation of space charge effects of arbitrary and large distributions of particles in an efficient and accurate way based on a variant of the Fast Multipole Method (FMM). It relies on an automatic multigrid-based decomposition of charges in near and far regions and the use of high-order differential algebra methods to obtain decompositions of far fields that lead to an error that scales with a high power of the order. Given an ensemble of N particles, the method allows the computation of the self-fields of all particles on each other with a computational expense that scales as O(N). Using remainder-enhanced DA methods, it is also possible to obtain rigorous estimates of the errors of the methods. Furthermore, the method allows the computation of all high-order multipoles of the space charge fields that are necessary for the computation of high-order transfer maps and all resulting aberrations.

The precise determination of the dynamics in accelerators with complicated field arrangements such as Fixed Field Alternating Gradient accelerators (FFAG) depends critically on the ability to describe the appearing magnetic fields in full 3D. However, frequently measurements or models of FFAG fields postulate their behavior in the midplane only, and rely on the fact that this midplane field and its derivatives determine the field in all of space. The detailed knowledge of the resulting out-of-plane fields is critical for a careful assessment of the vertical dynamics. We describe a method based on the differential algebraic (DA) approach to obtain the resulting out-of-plane expansions to any order in an order-independent, straightforward fashion. In particular, the resulting fields satisfy Maxwell's equations to the order of the expansion up to machine precision errors, and without any inaccuracies that can arise from conventional divided difference or finite element schemes for the computation of out-of-plane fields. The method relies on re-writing the underlying PDE as a fixed point problem involving DA operations, and in particular the differential algebraic integration operator. We illustrate the performance of the method for a variety of practical examples, and obtain estimates for the orders necessary to describe the fields to a prescribed accuracy.

Accelerators are playing increasingly important roles in basic science, technology, and medicine. Ultra high-intensity and high-energy (GeV) proton drivers are a critical technology for accelerator-driven sub-critical reactors (ADS) and many HEP programs (Muon Collider) but remain particularly challenging, encountering duty cycle and space-charge limits in the synchrotron and machine size concerns in the weaker-focusing cyclotrons; a 10–20 MW proton driver is not presently considered technically achievable with conventional re-circulating accelerators. One, as-yet, unexplored re-circulating accelerator, the Fixed-field Alternating Gradient or FFAG, is an attractive alternative to the other approaches to a high-power beam source. Its strong focusing optics can mitigate space charge effects and achieve higher bunch charges than are possible in a cyclotron, and a recent innovation in design has coupled stable tunes with isochronous orbits, making the FFAG capable of fixed-frequency, CW acceleration, as in the classical cyclotron but beyond their energy reach, well into the relativistic regime. This new concept has been advanced in non-scaling nonlinear FFAGs using powerful new methodologies developed for FFAG accelerator design and simulation. The machine described here has the high average current advantage and duty cycle of the cyclotron (without using broadband RF frequencies) in combination with the strong focusing, smaller losses, and energy variability that are more typical of the synchrotron. The current industrial and medical standard is a cyclotron, but a competing CW FFAG could promote a shift in this baseline. This paper reports on these new advances in FFAG accelerator technology and presents advanced modeling tools for fixed-field accelerators unique to the code COSY INFINITY.1

Muon ionization cooling involves passing particles through solid or liquid absorbers. Careful simulations are required to design muon cooling channels. New features have been developed for inclusion in the transfer map code COSY Infinity to follow the distribution of charged particles through matter. To study the passage of muons through material, the transfer map approach alone is not sufficient. The interplay of beam optics and atomic processes must be studied by a hybrid transfer map-Monte-Carlo approach in which transfer map methods describe the deterministic behavior of the particles, and Monte-Carlo methods are used to provide corrections accounting for the stochastic nature of scattering and straggling of particles. The advantage of the new approach is that the vast majority of the dynamics are represented by fast application of the high-order transfer map of an entire element and accumulated stochastic effects. The gains in speed are expected to simplify the optimization of cooling channels which is usually computationally demanding. Progress on the development of the required algorithms and their application to modeling muon ionization cooling channels is reported.

Knowledge about stable and unstable manifolds of hyperbolic fixed points of certain maps is desirable in many fields of research, both in pure mathematics as well as in applications, ranging from forced oscillations to celestial mechanics and space mission design. We present a technique to find highly accurate polynomial approximations of local invariant manifolds for sufficiently smooth planar maps and rigorously enclose them with sharp interval remainder bounds using Taylor model techniques. Iteratively, significant portions of the global manifold tangle can be enclosed with high accuracy. Numerical examples are provided.

In this paper a differential algebra version of the gravity assist space pruning algorithm is presented. The use of differential algebraic techniques is proposed to overcome the two main drawbacks of the existing algorithm, i.e., the steep increase of the number of function evaluations with the number of planets involved in the transfer, and the use of a bounding procedure that relies on Lipschitzian tolerances. Differential algebra allows us to process boxes in place of grid points, and to substitute pointwise evaluations of the constraint functions with their Taylor expansions. Thanks to the particular instance of multi-gravity assist problems dealt with, all the planet-to-planet legs can be treated independently, and forward and backward constraining can be applied. The proposed method is applied to preprocess the search space of sample interplanetary transfers and it also serves as a stepping stone towards a fully rigorous treatment of the pruning process based on Taylor models.

The Hefei light source (HLS) is a second generation synchrotron radiation light source, in which a superconducting wiggler is installed and operating. The effects of the wiggler on the beam dynamics on the HLS storage ring are studied, in order to make sure the wiggler can operate properly when the ring is working in the high brilliance mode. We generate a model of the magnetic field in the midplane of the wiggler. The 3D magnetic field model is also builded up by COSY infinity 9.0. Both the linear and nonlinear effects of the wiggler are discussed. The vertical tune is changed from 2.535 to 2.567 and the vertical beta function is heavily distorted, while a symplectic tracking study shows the dynamic aperture is only slightly affected by the wiggler. And the wiggler should be able to run on the high brilliance mode after the linear effects get compensated.

The calculation of the nonlinear tune shift with amplitude based on the results of measurements and the linear lattice information is discussed. The tune shift is calculated based on a set of specific measurements and some extra information which is usually available, namely that about the size and particle distribution in the beam and the linear optics effect on the particles. The method to solve this problem uses the technique of normal form transformation. The proposed model for the nonlinear tune shift calculation is compared to both the numerical results for the nonlinear model of the Tevatron accelerator and the independent approximate formula for the tune shift by Meller et al. The proposed model shows a discrepancy of about 2%.

For charged particle beams that are wider in the dispersive plane compared to the transverse plane it is cost efficient to utilize magnets that accept beams with elliptic cross section. In this paper we presents the conceptual design of a quadrupole magnet with elliptic cross section and with tunable higher order multipoles. The design consists of 18 superconducting race-track coils placed on two hollow concentric rhombic prism support structures. To compute the magnetic field for the proposed design a new method of calculating 2D and 3D fields for the air core magnets based on differential algebra (DA) techniques is developed. We will present the new method and discuss its implementation of new numerical tools based on this method in the code COSY Infinity.

In this paper, we describe newly developed tools for the study and analysis of the dynamics in FFAG accelerators based on transfer map methods unique to the code COSY INFINITY. With these new tools, closed orbits, transverse amplitude dependencies and dynamic aperture are determined inclusive of full nonlinear fields and kinematics to arbitrary order. The dynamics are studied at discrete energies, via a high-order energy-dependent transfer map.The order-dependent convergence in the calculated maps allows precise determination of dynamic aperture and detailed particle dynamics. Using normal form methods, and minimal impact symplectic tracking, amplitude- and energy-dependent tune shifts and resonance strengths are extracted. Optimization by constrained global optimization methods further refine and promote robust machine attributes.Various methods of describing the fields will be presented, including representation of fields in radius-dependent Fourier modes, which include complex magnet edge contours and superimposed fringe fields, as well as the capability to interject calculated or measured field data from a magnet design code or actual components, respectively.

The Beam Dynamics simulation package in COSY INFINITY is built upon a differential algebra data type. With it, it is possible to compute transfer maps or arbitrary systems to arbitrary order. However, this data type is limited by the precision of the underlying floating point number model provided by the computer processor. We will present a method to extend the effective precision of the calculations based purely on standard floating point operations. Those algorithms are then integrated into the differential algebra data type to efficiently extend the available precision, without unnecessarily affecting overall efficiency. To that effect, the precision of each coefficient is adjusted automatically during the calculation. We will then proceed to show the effectiveness of our implementation by calculating high precision maps of combinations of homogeneous dipole segments, for which the exact results are known, and comparing the high precision coefficients with the results produced by the traditional COSY beam physics package.

Space trajectory design always requires the solution of an optimal control problem in order to maximize the payload launch-mass ratio while achieving the primary mission goals. A certain level of approximation always characterizes the dynamical models adopted to perform the design process. Furthermore the state identification is usually affected by navigation errors. Thus, after the nominal optimal solution is computed, a control strategy that assures the execution of mission goals in the real scenario must be implemented. In this frame differential algebraic techniques are here proposed as an effective alternative tool to design the guidance law. By using differential algebra the final state dependency on initial conditions, environmental and control parameters is represented by high order Taylor series expansions. The mission constraints can then be solved to high order using a so-called high order partial inversion of the polynomial relationship for every admissible uncertainty. The control strategy is eventually reduced to a simple function evaluation. The performances of the proposed methods are assessed by two examples of space mission trajectory design: a continuous propelled Earth-Mars transfer and an aerocapture maneuver at Mars.

A new experiment is described to detect a permanent electric dipole moment of the proton with a sensitivity of $10^{-29}e\cdot$cm by using polarized "magic" momentum $0.7$~GeV/c protons in an all-electric storage ring. Systematic errors relevant to the experiment are discussed and techniques to address them are presented. The measurement is sensitive to new physics beyond the Standard Model at the scale of 3000~TeV., Comment: 8 pages, 3 figures

Current approaches to uncertainty propagation in astrodynamics mainly refer to linearized models or Monte Carlo simulations. Naive linear methods fail in nonlinear dynamics, whereas Monte Carlo simulations tend to be computationally intensive. Differential algebra has already proven to be an efficient compromise by replacing thousands of pointwise integrations of Monte Carlo runs with the fast evaluation of the arbitrary order Taylor expansion of the flow of the dynamics. However, the current implementation of the DA-based high-order uncertainty propagator fails when the non-linearities of the dynamics prohibit good convergence of the Taylor expansion in one or more directions. We solve this issue by introducing automatic domain splitting. During propagation, the polynomial expansion of the current state is split into two polynomials whenever its truncation error reaches a predefined threshold. The resulting set of polynomials accurately tracks uncertainties, even in highly nonlinear dynamics. The method is tested on the propagation of (99942) Apophis post-encounter motion. © 2015, Springer Science+Business Media Dordrecht.

A new method to determine the spin tune is described and tested. In an ideal planar magnetic ring, the spin tune - defined as the number of spin precessions per turn - is given by $\nu_s = \gamma G$ (gamma is the Lorentz factor, $G$ the magnetic anomaly). For 970 MeV/c deuterons coherently precessing with a frequency of ~120 kHz in the Cooler Synchrotron COSY, the spin tune is deduced from the up-down asymmetry of deuteron carbon scattering. In a time interval of 2.6 s, the spin tune was determined with a precision of the order $10^{-8}$, and to $1 \cdot 10^{-10}$ for a continuous 100 s accelerator cycle. This renders the presented method a new precision tool for accelerator physics: controlling the spin motion of particles to high precision is mandatory, in particular, for the measurement of electric dipole moments of charged particles in a storage ring., Comment: 6 pages, 6 figures

The tools used to compute high-order transfer maps based on differential algebraic (DA) methods have recently been augmented by methods that also allow a rigorous computation of an interval bound for the remainder. In this paper we will show how such methods can also be used to determine rigorous bounds for the global extrema of functions in an efficient way. The method is used for the bounding of normal form defect functions, which allows rigorous stability estimates for repetitive particle accelerator. However, the method is also applicable to general lattice design problems and can enhance the commonly used local optimization with heuristic successive starting point modification. The global optimization approach studied rests on the ability of the method to suppress the so-called dependency problem common to validated computations, as well as effective polynomial bounding techniques. We review the linear dominated bounder (LDB) and the quadratic fast bounder (QFB) and study their performance for various example problems in global optimization. We observe that the method is superior to other global optimization approaches and can prove stability times similar to what is desired, without any need for expensive long-term tracking and in a fully rigorous way.

Increasing demands for luminosity in existing and future colliders have made lattice design and error tolerance and correction critical to achieving performance goals. The current state of the Tevatron collider is an example, with a strong skew quadrupole error present in the operational lattice. This work studies the high-order performance of the Tevatron and the strong nonlinear behavior introduced when a significant skew quadrupole error is combined with conventional sextupole correction, a behavior still clearly evident after optimal tuning of available skew quadrupole circuits. An optimization study is performed using different skew quadrupole families, and, importantly, local and global correction of the linear skew terms in maps generated by the code COSY INFINITY [M. Berz, COSY INFINITY version 8.1 user's guide and reference manual, Department of Physics and Astronomy MSUHEP-20704, Michigan State University (2002). URL http://cosy.pa.msu.edu/cosymanu/index.html]. Two correction schemes with one family locally correcting each arc and eight independent correctors in the straight sections for global correction are proposed and shown to dramatically improve linearity and performance of the baseline Tevatron lattice.

This paper provides a computational method to model a three-dimensional static electromagnetic field within a finite source free volume starting from discrete field information on its surface. The method uses the Helmholtz vector decomposition theorem and the differential algebraic framework of COSY INFINITY to determine a solution to the Laplace equation. The solution is locally expressed as a Taylor expansion of the field which can be computed to arbitrary order. It provides a natural multipole decomposition of the field which is required for the computation of transfer maps, and also allows to obtain very accurate finite element representations with very small numbers of cells.

In this paper, we review the features in the newly released version of COSY INFINITY, which currently has a base of more than 1000 registered users, focusing on the topics which are new and some topics which became available after the first release of the previous versions 8 and 8.1. The recent main enhancements of the code are devoted to reliability and efficiency of the computation, to verified integration, and to rigorous global optimization. There are various data types available in COSY INFINITY to support these goals, and the paper also reviews the feature and usage of those data types.

In the US scenario for a Neutrino Factory presented in “A feasibility study of a neutrino source based on a muon storage ring”, N. Holtkamp (Ed.), D. Finley (Ed.), Fermilab, April 15th, 2000), a large percentage of the cost is related to an induction linac for phase-energy rotation and bunching of the muon beam collected after the production target and decay channel. A more cost-effective adiabatic buncher and phase-energy rotator has been proposed to replace the induction linac system (D. Neuffer, A. Van Ginneken, High-frequency bunching and ( φ - δ E ) rotation for a muon source, Proceedings of the 2001 Particle Accelerators Conference, Chicago, 2001, p. 2029). The new method uses consecutive RF cavities with differing frequencies. The frequencies are changed to enable bunching and phase-energy rotation. In this paper, the theoretical concept is developed and demonstrated with simulation results obtained with the map code COSY Infinity ( http://cosy.pa.msu.edu ). An optimization strategy is also explored.

All schemes to produce intense sources of high-energy muons—Neutrino factories, beta beams, Colliders—require collection, RF capture, and transport of particle beams with unprecedented emittances, both longitudinally and transversely. These large initial emittances must be reduced or “cooled” both in size and in energy spread before the muons can be efficiently accelerated to multi-GeV energies. The acceleration stage becomes critical in formulating and optimizing muon beams; individual stages are strongly interlinked and not independent as is the case in most conventional acceleration systems. Most importantly, the degree of cooling, or cooling channel, depends on the choice of acceleration. In the current US baseline scenario, the cooling required for acceleration is about a factor of 10 in transverse emittance per plane. Longitudinal cooling is also required. In the proposed Japanese scenario, using an alternative acceleration scheme, no cooling is presumed. This work discusses two basic, but different approaches to a Neutrino Factory and how the optimal strategy depends on beam parameters and method of acceleration.

Schemes to produce intense sources of high-energy muons, Neutrino Factories, beta beams, and colliders, require collection, rf capture, and transport of particle beams with unprecedented emittances, both longitudinally and transversely. These large initial emittances must be reduced or cooled both in size and in energy spread before the muons can be efficiently accelerated to multi-GeV energies. The acceleration stage becomes critical in formulating and optimizing muon beams; individual stages are strongly interlinked and not independent as is the case in most conventional acceleration systems. Most importantly, the degree of cooling, or cooling channel, depends on the choice of acceleration. This work discusses two basic, but different approaches to a Neutrino Factory and how the optimal strategy depends on beam parameters and method of acceleration.

A detailed study of power series on the Levi-Civita elds is presented. After reviewing two types of convergence on those elds, including convergence criteria for power series, we study some analytical properties of power series. We show that within their domain of convergence, power series are innitely often dieren tiable and reexpandable around any point within the radius of convergence from the origin. Then we study a large class of functions that are given locally by power series and contain all the continuations of real power series. We show that these functions have similar properties as real analytic functions. In particular, they are closed under arithmetic operations and composition and they are innitely often dieren tiable.

High-order transfer maps of particle systems play an important role in the design and optimization of particle optical systems, both for satisfying the basic design needs as well as for the correction of aberrations and non-linear effects, and the differential algebraic (DA) method has proved useful for this problem. Since the high-order maps represent an approximation of the motion, in particular in strongly nonlinear cases in which convergence of the maps may be slow, it is important to know the quality of the approximation. Recent work has shown that it is in principle possible to not only propagate the conventional differential algebraic high-order objects, but also adjoint remainder terms that rigorously account for any errors made by the approximation by the Taylor expansion over the domain of interest. In this paper we describe various recent enhancements of the original method of computations with remainder bounds that allow the control of the errors made both by the integrator scheme and any possible inaccuracy of the description of the system. Using suitable extensions of the DA and Taylor model operators used in the Schauder fixed point theorem formulation of the ODE problem leads to a very transparent approach for the calculation of enclosures for the intergration errors. Under the presence of a scheme for effective treatment of sparsity in the DA vectors, such as the method available in the code COSY INFINITY, the additional resources necessary for this algorithm are very modest.

In a scenario for either a Neutrino Factory or Muon Collider, the anticipated transverse beam emittance subsequent to capture and phase rotation is so large that it permits a relaxation of the requirements on beam spot size in the early stages relative to the final stages of ionization cooling. Staging the cooling process according to initial emittances, coupled with modest cooling factors, permits more optimal and efficient cooling channel designs and avoids much of the difficulty encountered with channels which attempt to maintain strong focusing (large, 300–500 mrad, divergences) across ultra-large momentum ranges (⩾±20% δ p / p ). Relaxation of spot size at the absorber, especially in the “precooling” stage, allowed development of an efficient transverse cooling channel based simply on a quadrupole FODO cell. This work describes the design of such a cooling channel and its application as an upstream stage of beam cooling. Being a linear channel with no bends, it serves to reduce the large transverse beam size delivered from muon-beam capture and bunching before entering more restricted optical structures such as transverse plus longitudinal cooling channels or accelerators.

Various modern systems for the transport and manipulation of large acceptance beams of rare and short-lived particles require the treatment of nonlinear optics of acceleration, absorption, and focusing in a combined approach. We describe a differential algebraic method for the treatment of such nonlinear dynamics via high-order transfer maps. We include the processes of scattering and straggling through absorbing material, which are inherently non-deterministic and hence not representable in the map formalism, in a split operator approach. Some examples of simulations of muon beam ionization cooling channels are provided.

A muon ionization cooling channel consists of three major components: the magnet optics, an acceleration cavity, and an energy absorber. The absorber of liquid hydrogen contained by thin aluminum windows is the only component which introduces stochastic processes into the otherwise deterministic acceleration system. The scattering dynamics of the transverse coordinates is described by Gaussian distributions. The asymmetric energy loss function is represented by the Vavilov distribution characterized by the minimum number of collisions necessary for a particle undergoing loss of the energy distribution average resulting from the Bethe–Bloch formula. Examples of the interplay between stochastic processes and deterministic beam dynamics are given.

We are witnessing tremendous opportunities in ultrafast sciences with the development of extremely bright radiation sources to investigate the structure and spectroscopy of matter with atomistic space and femtosecond time resolution. While generally a strong focus has been on X-ray sources—notably free electron laser (FEL) sources—the use of femtosecond electron pulses has also shown enormous promise in the last decade, especially in the investigation of materials from the sub-micrometer down to the angstrom scale, facilitated by the high sensitivity of electron scattering and the relative ease in designing electron optics for imaging and diffraction from nanomaterials. Moreover, important innovations have been achieved by incorporating ultrafast photoemission sources into various electron microscope setups. Most recently, a new trend of integrating the FEL high-brightness electron beam concept into the ultrafast electron diffraction and microscope system design is likely to open up new prospects and applications of femtosecond diffraction, imaging, and spectroscopy with high throughput. The conference on Femtosecond Electron Imaging and Spectroscopy (FEIS 2013) was held on December 9–12, 2013 in Key West, Florida. FEIS 2013 built on the potential synergy between related technology developments and various emerging scientific opportunities and brought together leaders engaged in cutting-edge development of high-brightness electron and X-ray beam systems and their applications to frontier science problems. FEIS 2013, the first in this series, was organized with the goal of initiating conversation between different communities with the following objectives in mind: (1) to review the current state-of-the-art development and open issues of ultrafast electron imaging technologies; (2) to discuss emerging scientific opportunities enabled by ultrafast imaging and spectroscopy; (3) to identify the key technical challenges in the design and applications of ultrafast electron imaging systems; and (4) to forge cross-fertilization between the electron microscopy, accelerator and beam physics, and ultrafast communities, and to have experimentalists and theorists address common challenges and promote synergistic developments.

A method has been developed to calculate accurate high-order ion-optical maps for electrostatic and radio-frequency accelerating elements. The method has been incorporated into the arbitrary-order map-based beam optics code COSY Infinity. The treatment is restricted to the case of negligible magnetic fields, as is typical of heavy-ion accelerating cavities, and does not include space charge. For validation purposes, the beam dynamics calculated for these elements is compared against ray tracing for typical beam and cavity parameters. Different from the ray-tracing approach, parameter changes of individual components typically require only recalculation of the maps of the particular components and not the entire system, and thus the method is particularly suitable for optimization. The approach developed for accurate analytical representation of the on- and off-axis electric fields of cylindrically symmetric electrostatic lenses and radio-frequency cavities is described. Some of the many possible applications for using accurate high-order map representations of Einzel lenses, electrostatic accelerating gaps, and radio-frequency accelerating structures are discussed.

The coexistence of various electronic and structural phases that are close in free-energy is a hallmark in strongly correlated electron systems with emergent properties, such as metal-insulator transition, colossal magnetoresistance, and high-temperature superconductivity. The cooperative phase transitions from one functional state to another can involve entanglements between the electronically and structurally ordered states, hence deciphering the fundamental mechanisms is generally difficult and remains very active in condensed matter physics and functional materials research. We outline the recent ultrafast characterizations of 2D charge-density wave materials, including the nonequilibrium electron dynamics unveiled by ultrafast optical spectroscopy-based techniques sensitive to the electronic order parameter. We also describe the most recent findings from ultrafast electron crystallography, which provide structural aspects to correlate lattice dynamics with electronic evolutions to address the two sides of a coin in the ultrafast switching of a cooperative state. Combining these results brings forth new perspectives and a fuller picture in understanding lightmatter interactions and various switching mechanisms in cooperative systems with many potential applications. We also discuss the prospects of implementing new ultrafast electron imaging as a local probe incorporated with femtosecond select-area diffraction, imaging and spectroscopy to provide a full scope of resolution to tackle the more challenging complex phase transitions on the femtosecond-nanometer scale all at once based on a recent understanding of the spacespace- charge-driven emittance limitation on the ultimate performance of these devices. The projection shows promising parameter space for conducting ultrafast electron micordiffraction at close to single-shot level, which is supported by the latest experimental characterization of such a system.

In this paper, we study various fringe field effects. Previously, we showed the large impact that fringe fields can have on certain lattice scenarios of the proposed Neutrino Factory. Besides the linear design of the lattice, the effects depend strongly on the details of the field fall off. Various scenarios are compared. Furthermore, in the absence of detailed information, we study the effects for the LHC, a case where the fringe fields are known, and try to draw some conclusions for Neutrino Factory lattices.