Courant-Snyder Theory Research Articles

Equivalence of Gouy and Courant-Snyder phase

The generation of electron vortex beams and the conversion of these beams into beams without angular momentum by means of astigmatic optical systems, or vice versa, has been pursued in the optical and the electron microscopy community, but also in the accelerator community in the past decades. Despite different conceptual approaches similar results have been achieved. By adapting the Courant-Snyder theory, which was originally developed for the description of optical properties of accelerators, to the description of laser modes it is shown, that identical mode converters have been developed for charged particle and for light beams, and that the Courant-Snyder phase and the Gouy phase are equivalent.

Decoupling a transversely-coupled beam based on symplectic transformation theory and its application

Transverse projected emittances of injected beams are strictly limited for synchrotrons with multi-loop injection. Horizontal and vertical emittances easily grow up when beam phase spaces in the xx′ and yy′ planes are coupled. Coupling in beam motion can be induced by tilted quadrupoles and solenoids. Various formalisms have been investigated to describe the beam coupling, which include the matrix and Hamiltonian formalism. In the matrix formalism, the transverse beam motion is parametrized by 4 × 4 beam matrices and the transfer matrices of the elements. Several methods or theories are used to obtain the decoupling matrix, including the equations solving, eigensystem analysis, geometrical method, Courant–Snyder theory and Hamiltonian perturbation theory. We propose an approach to obtain the decoupling matrix based on linear symplectic transformation theory. By adopting several consecutive tilted quadrupoles, the philosophy, as well as the numerical code for solving the decoupling and matching problem, are developed and checked for the general case. The transversely-coupled beam due to a 45∘-rotated radio frequency quadrupole (RFQ) can be decoupled in the downstream drift tube linac with only five tilted fixed-gradient permanent-magnet quadrupoles (PMQs), which is suggested to be the minimal number of PMQs for decoupling and matching. The transverse emittance growth is verified to be suppressed below 5% by multi-particle simulation. The error study shows that the transmission rate can reach 98% with a probability of 99%.

Generalized parametrization methods for centroid and envelope dynamics of charged particle beams in coupled lattices

For almost 60 years, the well-known Courant-Snyder (CS) theory has been employed as the standard method to describe the uncoupled dynamics of charged particle beams in electromagnetic focusing lattices. Meanwhile, the generalization of the CS theory to coupled dynamics with two or more degrees of freedom has been attempted in numerous directions. The parametrization method developed by Qin and Davidson is particularly noteworthy, because their method enables the treatment of complicated coupled beam dynamics using a remarkably similar mathematical structure to that of the original CS theory. In this paper, we revisit the Qin-Davidson parametrization method and extend it to include beam centroid motions. The linear terms in the quadratic Hamiltonian for the coupled dynamics are handled by introducing a special time-dependent canonical transformation. In this manner, we show that the centroid dynamics is decoupled from the envelope dynamics, even for the cases of coupled lattice, and all formulations of the Qin-Davidson method can be applied in a straightforward manner.

Spectral and structural stability properties of charged particle dynamics in coupled latticesa)

It has been realized in recent years that coupled focusing lattices in accelerators and storage rings have significant advantages over conventional uncoupled focusing lattices, especially for high-intensity charged particle beams. A theoretical framework and associated tools for analyzing the spectral and structural stability properties of coupled lattices are formulated in this paper, based on the recently developed generalized Courant-Snyder theory for coupled lattices. It is shown that for periodic coupled lattices that are spectrally and structurally stable, the matrix envelope equation must admit matched solutions. Using the technique of normal form and pre-Iwasawa decomposition, a new method is developed to replace the (inefficient) shooting method for finding matched solutions for the matrix envelope equation. Stability properties of a continuously rotating quadrupole lattice are investigated. The Krein collision process for destabilization of the lattice is demonstrated.

Beam envelope calculations in general linear coupled lattices

The envelope equations and Twiss parameters (β and α) provide important bases for uncoupled linear beam dynamics. For sophisticated beam manipulations, however, coupling elements between two transverse planes are intentionally introduced. The recently developed generalized Courant-Snyder theory offers an effective way of describing the linear beam dynamics in such coupled systems with a remarkably similar mathematical structure to the original Courant-Snyder theory. In this work, we present numerical solutions to the symmetrized matrix envelope equation for β which removes the gauge freedom in the matrix envelope equation for w. Furthermore, we construct the transfer and beam matrices in terms of the generalized Twiss parameters, which enables calculation of the beam envelopes in arbitrary linear coupled systems.

Analytical methods for describing charged particle dynamics in general focusing lattices using generalized Courant-Snyder theory

The dynamics of charged particles in general linear focusing lattices with quadrupole, skew-quadrupole, dipole, and solenoidal components, as well as torsion of the fiducial orbit and variation of beam energy is parametrized using a generalized Courant-Snyder (CS) theory, which extends the original CS theory for one degree of freedom to higher dimensions. The envelope function is generalized into an envelope matrix, and the phase advance is generalized into a 4D symplectic rotation, or a $U(2)$ element. The 1D envelope equation, also known as the Ermakov-Milne-Pinney equation in quantum mechanics, is generalized to an envelope matrix equation in higher dimensions. Other components of the original CS theory, such as the transfer matrix, Twiss functions, and CS invariant (also known as the Lewis invariant) all have their counterparts, with remarkably similar expressions, in the generalized theory. The gauge group structure of the generalized theory is analyzed. By fixing the gauge freedom with a desired symmetry, the generalized CS parametrization assumes the form of the modified Iwasawa decomposition, whose importance in phase space optics and phase space quantum mechanics has been recently realized. This gauge fixing also symmetrizes the generalized envelope equation and expresses the theory using only the generalized Twiss function $\ensuremath{\beta}$. The generalized phase advance completely determines the spectral and structural stability properties of a general focusing lattice. For structural stability, the generalized CS theory enables application of the Krein-Moser theory to greatly simplify the stability analysis. The generalized CS theory provides an effective tool to study coupled dynamics and to discover more optimized lattice designs in the larger parameter space of general focusing lattices.

Generalized Courant-Snyder theory for charged-particle dynamics in general focusing lattices.

The Courant-Snyder (CS) theory for one degree of freedom is generalized to the case of coupled transverse dynamics in general linear focusing lattices with quadrupole, skew-quadrupole, dipole, and solenoidal components, as well as torsion of the fiducial orbit and variation of beam energy. The envelope function is generalized into an envelope matrix, and the phase advance is generalized into a 4D sympletic rotation. The envelope equation, the transfer matrix, and the CS invariant of the original CS theory all have their counterparts, with remarkably similar expressions, in the generalized theory.

Analysis of continuously rotating quadrupole focusing channels using generalized Courant-Snyder theory

By extending the recently developed generalized Courant-Snyder theory for coupled transverse beam dynamics, we have constructed the Gaussian beam distribution and its projections with arbitrary mode emittance ratios. The new formulation has been applied to a continuously rotating quadrupole focusing channel because the basic properties of this channel are known theoretically and could also be investigated experimentally in a compact setup such as the linear Paul trap configuration. The new formulation retains a remarkably similar mathematical structure to the original Courant-Snyder theory, and thus, provides a powerful theoretical tool to investigate coupled transverse beam dynamics in general and more complex linear focusing channels.

Geometrical method of decoupling

The computation of tunes and matched beam distributions are essential steps in the analysis of circular accelerators. If certain symmetries - like midplane symmetrie - are present, then it is possible to treat the betatron motion in the horizontal, the vertical plane and (under certain circumstances) the longitudinal motion separately using the well-known Courant-Snyder theory, or to apply transformations that have been described previously as for instance the method of Teng and Edwards. In a preceeding paper it has been shown that this method requires a modification for the treatment of isochronous cyclotrons with non-negligible space charge forces. Unfortunately the modification was numerically not as stable as desired and it was still unclear, if the extension would work for all thinkable cases. Hence a systematic derivation of a more general treatment seemed advisable. In a second paper the author suggested the use of real Dirac matrices as basic tools to coupled linear optics and gave a straightforward recipe to decouple positive definite Hamiltonians with imaginary eigenvalues. In this article this method is generalized and simplified in order to formulate a straightforward method to decouple Hamiltonian matrices with eigenvalues on the real and the imaginary axis. It is shown that this algebraic decoupling is closely related to a geometric "decoupling" by the orthogonalization of the vectors $\vec E$, $\vec B$ and $\vec P$, that were introduced with the so-called "electromechanical equivalence". We present a structure-preserving block-diagonalization of symplectic or Hamiltonian matrices, respectively. When used iteratively, the decoupling algorithm can also be applied to n-dimensional systems and requires ${\cal O}(n^2)$ iterations to converge to a given precision.

De Moivre's formula: are Sands' H-functions the same as Chao's?

It has often been said that there is no clear extension of the Courant-Snyder theory to coupled dynamics. In particular, one never sees an extension of the symplectic de Moivre type formula beyond one degree of freedom. In the process of analysing the existence and meaning of such a formula, I discovered quite accidentally that Sands' formalism [1], if expressed in terms of Ripken-like lattice functions germane to the full three dimensional oscillator, gives the exact same result as the more accurate Chao theory [2]. This semi-serious paper displays this elegant connection. It is based in the lattice functions of Ripken [3] as extended by Forest [4]. We review Forest's derivation from the point of view de Moivre's formula extended to three degrees of freedom.

Use of real Dirac matrices in two-dimensional coupled linear optics

The Courant-Snyder theory for two-dimensional coupled linear optics is presented, based on the systematic use of the real representation of the Dirac matrices. Since any real $4\ifmmode\times\else\texttimes\fi{}4$ matrix can be expressed as a linear combination of these matrices, the presented ansatz allows for a comprehensive and complete treatment of two-dimensional linear coupling. A survey of symplectic transformations in two dimensions is presented. A subset of these transformations is shown to be identical to rotations and Lorentz boosts in Minkowski space-time. The transformation properties of the classical state vector are formulated and found to be analog to those of a Dirac spinor. The equations of motion for a relativistic charged particle---the Lorentz force equations---are shown to be isomorph to envelope equations of two-dimensional linear coupled optics. A universal and straightforward method to decouple two-dimensional harmonic oscillators with constant coefficients by symplectic transformations is presented, which is based on this isomorphism. The method yields the eigenvalues (i.e., tunes) and eigenvectors and can be applied to a one-turn transfer matrix or directly to the coefficient matrix of the linear differential equation.

Twiss parameters and beam matrix formulation of generalized Courant–Snyder theory for coupled transverse beam dynamics

Courant–Snyder (CS) theory for one degree of freedom has recently been generalized by Qin and Davidson to the case of coupled transverse dynamics with two degrees of freedom. The generalized theory has four basic components of the original CS theory, i.e., the envelope equation, phase advance, transfer matrix, and the CS invariant, all of which have their counterparts in the original CS theory with remarkably similar expressions and physical meanings. In this brief communication, we further extend this remarkable similarity between the original and generalized CS theories and construct the Twiss parameters and beam matrix in generalized forms for the case of a strong coupling system.

Generalized Courant-Snyder theory for coupled transverse dynamics of charged particles in electromagnetic focusing lattices

The Courant-Snyder theory gives a complete description of the uncoupled transverse dynamics of charged particles in electromagnetic focusing lattices. In this paper, the Courant-Snyder theory is generalized to the case of coupled transverse dynamics with two degrees of freedom. The generalized theory has the same structure as the original Courant-Snyder theory for one degree of freedom. The four basic components of the original Courant-Snyder theory, i.e., the envelope equation, phase advance, transfer matrix, and the Courant-Snyder invariant, all have their counterparts, with remarkably similar expressions, in the generalized theory presented here. In the generalized theory, the envelope function is generalized into an envelope matrix, and the envelope equation becomes a matrix envelope equation with matrix operations that are noncommutative. The generalized theory gives a new parametrization of the 4D symplectic transfer matrix that has the same structure as the parametrization of the 2D symplectic transfer matrix in the original Courant-Snyder theory. All of the parameters used in the generalized Courant-Snyder theory correspond to physical quantities of importance, and this parametrization can provide a valuable framework for accelerator design and particle simulation studies. A time-dependent canonical transformation is used to develop the generalized Courant-Snyder theory. Applications of the new theory to strongly and weakly coupled dynamics are given. It is shown that the stability of coupled dynamics can be determined by the generalized phase advance developed. Two stability criteria are given, which recover the known results about sum and difference resonances in the weakly coupled limit.

A physical parametrization of coupled transverse dynamics based on generalized Courant–Snyder theory and its applications

A physical parametrization of coupled transverse dynamics is developed by generalizing the Courant–Snyder (CS) theory for one degree of freedom to the case of coupled transverse dynamics with two degrees of freedom. The four basic components of the original CS theory, i.e., the envelope equation, phase advance, transfer matrix, and CS invariant, all have their counterparts with remarkably similar expressions in the generalized theory. Applications of the new theory are given. It is discovered that the stability of coupled dynamics is completely determined by the generalized phase advance.

Linear beam optics in solenoidal channels

In this paper we study beam transport through a straight solenoidal channel using the single-particle and linear optics approach. We derive the single-particle invariants and show their use in an extended Courant-Snyder theory for a solenoidal coupled system. Matching between solenoidal channels and between solenoidal and quadrupolar channels is discussed. We give envelope solutions and illustrate them with some numerical examples.

A new derivation of the generalized Courant-Snyder theory with coupling

The theory of charged beam propagation in magnetic structures, including the transverse motion coupling, is developed using a formalism close to the original Courant-Snyder formulation of uncoupled beam transport. We introduce three 2×2 matrices (β̂, α̂ and γ̂) which are the generalizations of the usual Twiss parameters and study their properties. The physical meaning of the above matrices is discussed and their role, in specifying e.g. the spatial beam distribution, elucidated. The usefulness of such a point of view is discussed within the context of a specific example.

Coupled transverse motion in charged-beam transport dynamics and generalized twiss coefficients

In this paper we discuss a generalized Courant-Snyder theory forn-dimensional quadratic Hamiltonians. We stress the usefulness of the obtained results both for transverse-mode coupling in accelerators and in quantum mechanics, for the generation of squeezed states.