On emittance and optics calculation from the tracking data in periodic lattices
Malte Titze
TL;DR
This paper explores advanced mathematical methods to improve the calculation of beam emittance and optics in periodic lattices, linking established techniques with new decompositions for better analysis.
Contribution
It introduces the use of Iwasawa decomposition for beam optics parameterization and connects it with sigma-matrix methods and Williamson's Theorem in a linear setting.
Findings
Iwasawa decomposition offers a natural framework for beam optics analysis.
The sigma-matrix method is connected to Williamson's Theorem.
Application to realistic tracking examples demonstrates the approach's effectiveness.
Abstract
In this work we examine the interplay between normal form and matched particle distributions in a linear setting. We first outline the connection between the established sigma-matrix method and Williamson's Theorem. Then we show that the Iwasawa decomposition provides a natural framework for a description of beam optics parameters. Along the way we will apply these methods to a realistic tracking example, as well as provide additional examples, including the connection to the parameterization of Courant-Synder.
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Taxonomy
TopicsAtomic and Molecular Physics · Advanced Chemical Physics Studies · Particle accelerators and beam dynamics
On emittance and optics calculation from the tracking data in periodic lattices
Malte Titze
CERN, Geneva, Switzerland
Abstract
In this work we examine the interplay between normal form and matched particle distributions in a linear setting. We first outline the connection between the established -matrix method and Williamson’s Theorem. Then we show that the Iwasawa decomposition provides a natural framework for a description of beam optics parameters. Along the way we will apply these methods to a realistic tracking example, as well as provide additional examples, including the connection to the parameterization of Courant-Synder.
I Introduction
In a particle simulation involving a periodic lattice, it is usually desired to generate particles in a matched state, which means that the shape of the distribution should not change after one passage through the lattice. In fact, if a matched distribution can be found, one often has already accomplished a great deal in the understanding of the simulation. Additionally, there are circumstances in which the knowledge of the effective emittance and optics parameters is required but difficult to compute, for example in the case of the MAD-X space charge module for the CERN PS lattice (without poleface windings) near the integer resonance , where the search for a closed-orbit can break down.
In recent years, efforts have been made with success in using covariance matrices to compute emittances and beam optics parameters. This was demonstrated in Ref. bib:lebedev1999 in the 4D case and in bib:wolski2006 ; bib:nash2006 for the 6D case. In a later published version of bib:lebedev1999 , i.e. in Ref. bib:lebedev2010 , it was used to examine the relation between the Edwards-Teng bib:edwards1973 (see also Ref. bib:sagan1999 for a summary) and the Mais-Ripken bib:willeke1988 parameterization in a 4D situation with coupled optics. The results were picked up in bib:alexahin2013 in order to compute the emittances from the covariance matrices of 6D tracking data and, in regards of code implementation, recent progress has been made to include some of these techniques into the MAD-X space charge module bib:alex2018 . In Refs. bib:luo2004 ; bib:qin2009 ; bib:qinetal2013 generalizations of the Courant-Snyder parameterization to 4D were examined.
The aim of this work is to continue in this spirit by systematically exploring the connection to linear normal form and established theorems regarding symplectic matrices: By utilizing Williamson’s Theorem we obtain a proof of the remarkable result to obtain emittances by symplectic diagonalization of the given covariance matrix. Such diagonalizations are not unique but, as we shall see in Sec. II, by knowledge of how the underlying freedom enters into the equations we outline how to obtain faithful optics information out of the tracking data.
In particular we found that the Iwasawa decomposition provides a natural framework: Two of their three factors always remain the same, while the third factor can be determined under one additional condition: Namely, that the emittances have to be mutually distinguishable. We are thus led, rather naturally, to a characterization of e.g. the optics -functions. We also discuss an alternative route to obtain -functions by a statistical argument, which was proposed in Ref. bib:wolski2006 , and which we connect and apply to our situation.
Along the way we will provide several tables from a realistic tracking scenario and three examples which illustrate the results of the technical steps. The first example establishes the connection to the well-known (2D) Courant-Snyder parameterization. The second example contains a short way of how to obtain the emittances from a 4D covariance matrix, without the necessity to compute the eigenvectors. The last example also deals with a 4D situation, now with a single coupling parameter. By means of this last example we will demonstrate certain properties of the general decomposition.
For practical purposes we will summarize in Sec. III the techniques of how emittance calculations can be performed in a linear scenario, establish the connection to the familiar emittance of Lapostolle and the single-particle action. For completeness we also discuss the situation of measuring beam sizes at three different locations in the lattice.
II On invariant covariance matrices
II.1 Motivation and preliminaries
Consider a tracking simulation which produces, at every turn, a distribution of particles depending on an initial set of coordinates. We can compute the moments of these distributions in phase space and obtain some sort of measure of the phase space volume occupied. It is of great interest to understand how to set up a distribution in which certain functions of these macroscopic quantities remain unchanged or vary only very slowly in the course of the simulation. In the following we will understand our lattice to be in the form of a ring, but the same reasoning can be applied to a straight periodic lattice.
Let be a canonical transformation from phase space onto itself, which describes the physics of the storage ring in form of a single turn around the machine at a given fixed position in the ring. Such a one-turn or Poincaré map is usually the result of a composition of many elementary maps, which describe the individual elements of the machine.
In this work we will examine the situation in the vicinity of an assumed closed orbit, where linear effects play the dominant role. Therefore we will be focusing on the first derivative of the one-turn map at the closed orbit and do not consider any higher-order effects of the full map . Because of this restriction – and for brevity – we will also call the one-turn map. This map is symplectic since is canonical111Here ’canonical’ means a symplectomorphism..
If denotes the phase space density of a particle distribution, its covariance matrix , consisting of the second-order moments, is given by
[TABLE]
where, for any integrable function , the mean is given by We see that acts by matrix congruence on , where the new covariance matrix is given by
[TABLE]
Such covariance matrices are important, because their entries are the ingredients to compute the emittances of the beam; in the 2D case:
[TABLE]
Moreover, by means of Eq. (2), we have a way to follow the evolution of the moments in the course of the tracking. Because for symplectic maps, the emittance in Eq. (3) is conserved. Note that the emittance is just one example of an invariant. In Ref. bib:rangarajan89 functions of higher-order moments which remain invariant with respect to symplectic matrices were studied.
Here we are focusing on second-order moments and address as our first goal the following question: Given , how can we classifying all ’matched’ cases in which holds? As we shall see in the course of this section, the answer will connect a property used in e.g. Ref. bib:nash2006 to linear normal form.
Definition II.1**.**
We say a matrix is -congruent invariant or, for brevity, -invariant, if holds.
To begin with, we recall an important fact which we will frequently use to identify covariance matrices.
Theorem II.2**.**
* is a covariance matrix if and only if is symmetric and positive semidefinite.*
Proof.
A proof for convenient reference is included in Appendix VI.1. ∎
This means we are interested in -invariant symmetric and positive semidefinite matrices , having in mind that is the symplectic one-turn map of a lattice in a particle accelerator. In particular this means that we can assume that the complexification of can be diagonalized with mutually distinguishable eigenvalues (tunes).
We will now systematically develop important properties of -invariant symmetric matrices . Throughout this section convention VI.1 will hold and all matrices are given with respect to the ordering unless otherwise stated.
II.2 Invariance and linear normal form
In this paragraph we outline the interplay between the technique of transforming a given linear symplectic map with mutually distinguishable eigenvalues into a normal form and invariant covariance matrices. We will use an important lemma which not only helps us to describe the connection, but also in the next paragraph II.3 where we examine the degree of freedom involved in the matrices. For a procedure of how to construct the normal version of a general (higher-order) one-turn map we refer the reader to Refs. bib:forest1990 ; bib:bengtsson1990 . Preliminary tools are given in Appendix VI.1.
Theorem II.3** (Linear normal form).**
Let diagonalizable with mutually distinguishable eigenvalues on the unit circle. Then there exists , so that is orthogonal, leaving the plane for invariant:
[TABLE]
where the phases are related to the eigenvalues of by , and where the ’s correspond to a representation introduced in Dfn. VI.3, so that for the corresponding eigenvectors of , has positive imaginary part.
With the operator introduced in Def. VI.6 we can conveniently write . In the accelerator-physics terminology the phase space in which the one-turn map has the above normal form is also called Floquet-space.
Proof.
A proof can be found in Ref. bib:dragt2018 or in Appendix VI.1. ∎
The matrix in Thm. II.3 is not unique, as can readily be seen if composing the map by additional rotations (which are symplectic) leaving the individual planes invariant:
Remark II.4*.*
By lemma VI.8 we can conclude that the linear normal form map of Thm. II.3 is determined up to an orthosymplectic transformation on the left; namely if , i.e. , then . Because and are symplectic, it follows that and so has the same form as .
Theorem II.5**.**
Let diagonalizable with mutually distinguishable eigenvalues on the unit circle and symmetric. Then if and only if there exists a diagonal matrix with so that , where is given according to Thm. II.3.
Proof.
Let be the orthogonal operator discussed in Rmk. VI.7 and according to Thm. II.3. Then we have , and is -invariant if and only if is -invariant:
[TABLE]
and by lemma VI.8 this the case if and only if has the form , which is equivalent to has the form with . ∎
This result can be connected to a statement used in Ref. bib:nash2006 , and to obtain a familiar expression in the form of eigenvector decompositions, as follows:
Theorem II.6**.**
Every real-valued symmetric invariant of , where is diagonalizable with mutually distinguishable eigenvalues, can be represented as a sum of elementary matrices with , where the are given by , and where is a representation system according to Conv. VI.1.
Proof.
The statement can be found in e.g. Ref. bib:nash2006 in a slightly different version. A proof was included in Appendix VI.2. In VI.2 – and only there – we have changed our notation to . This means, by Conv. VI.1, that the eigenvectors appearing there are proportional to the vectors here. ∎
If we recall that by construction for holds (see the proof of Thm. II.3), then we see that Thm. II.6 is equivalent to Thm. II.5, which was obtained in a rather different manner:
[TABLE]
In some sense Thm. II.5 lays at the heart of computing emittances (i.e. the entries of ) out of covariance matrices using linear normal forms and therefore in answering the questions raised in the introduction of this section. We will now turn our attention to the emittance and optics computation.
II.3 Classification of invariant covariance matrices
The matrix in Thm. II.5 was only assumed to be symmetric. In particular this includes our case, where comes from a covariance matrix of a particle distribution. By our remark in paragraph II.1, these matrices are additionally positive semidefinite. In the typical situation of tracking the distribution through the accelerator, the beam will not be degenerated, i.e. the diagonal entries of in Thm. II.5, which correspond to the emittances, as we shall see in Sec. III, are always positive. This means that we can well assume that is positive definite.
In this case even more can be said about such invariants: Thm. II.5 effectively makes a statement on the conditions by which can be diagonalized by matrix congruence via a symplectic map . Since was constructed via by Thm. II.3, the linear optics of the machine is – up to the tune – effectively contained in . On the other hand, an abstract symplectic diagonalization of without knowledge of the optics is always possible in form of Williamson’s Theorem:
Theorem II.7** (Williamson bib:william1936 ; bib:william1937 ).**
Let be a -dimensional real symmetric positive definite matrix. Then there exist so that
[TABLE]
with and .
Proof.
A concise proof of this theorem can be found in Ref. bib:simon1998 . Because of its relevance in our computations we will sketch the proof here. Since is symmetric and positive definite, it can be diagonalized by an orthogonal matrix and all its eigenvalues are positive. Hence it admits an invertible square root . Now anti-diagonalize the antisymmetric matrix , i.e. find an orthogonal matrix so that
[TABLE]
holds, where is a diagonal -matrix with positive entries. Then set and verify symplecticity and Eq. (5) (with ). ∎
The set of positive real quantities obtained in the above manner is known in the literature as the symplectic spectrum of bib:degosson2006 . As already indicated in the above proof, and in particular by the next theorem, it will become apparent that the symplectic spectrum is -independent. However, the symplectic matrices diagonalizing are not unique, as we shall see by the examples given below. But by the next classification theorem they are not ’too far away’ from each other:
Theorem II.8**.**
Let be a symplectic matrix with mutually distinguishable eigenvalues and , , symmetric and positive definite, so that
[TABLE]
hold. Let be symplectic diagonalizations with . Then it holds .
For the proof of this theorem we have dedicated two smaller paragraphs VI.3 and VI.4 in the appendix. One immediate implication together with Thm. II.5 is the justification of the pretty remarkable result that one can find the emittances out of a covariance matrix alone, without having knowledge of the underlying optics given by the one-turn map. The only assumption on the covariance matrix is that it belongs to a matched distribution with respect to the unknown linear optics: Namely if is given with a decomposition according to Thm. II.5 and another decomposition has been found, for example from Thm. II.7, then we obtain with the orthogonal :
[TABLE]
and since the eigenvalues can not be changed by orthogonal matrix congruence, and must contain the same entries up to a suitable permutation. In Sec. III we will see how the diagonal entries of are connected to the classical emittances by Lapostolle (in the 2D case).
In Tab. 1 we show an example of a covariance matrix coming from a PyOrbit tracking simulation (without space charge) in the CERN SPS. Since is determined under the effect of small imperfections in the lattice, is not perfectly diagonalizing , see Tab. 2. Let us denote by the error between and an ideally matched covariance matrix of the lattice (see Sec. III). We then see that the diagonal entries after diagonalizing by a symplectic matrix according to Thm. II.7 belong to such a , see Tab. 3. Remaining small errors stem from the fact that and are determined by – and involved in – two different procedures.
The check whether the matrix is orthogonal is depicted in Tab. 4, which would be the unit matrix if would perfectly diagonalize . Let us summarize this finding in the following corollary.
Conclusion II.9** (Emittance from covariance matrix by symplectic diagonalization).**
Let be the covariance matrix of a linearly matched particle distribution, then the emittances are given by its symplectic spectrum.
Moreover, by means of Thm. II.8 and the help of the Iwasawa decomposition, we can regain optics functions out of a covariance matrix, by utilizing this natural parameterization. The Iwasawa decomposition reads bib:terras1988 :
Theorem II.10** (Iwasawa decomposition of symplectic matrices).**
Let . Then there exist unique symplectic matrices , and with and the following properties:
[TABLE]
In Thm. II.15 we summarize a result in bib:tinyautam2006 of how to compute such a decomposition. If we have found a symplectic diagonalization of an invariant covariance matrix , we can proceed and determine its decomposition: . On the other hand, a linear normal form , block-diagonalizing , can also be decomposed as . By Thm. II.8 we have with an orthosymplectic , and by uniqueness of the Iwasawa decomposition it therefore follows , and . So we already found two optics factors and only by examination of the covariance matrix .
Our next goal is to understand the nature of the remaining orthosymplectic factor . As a first step note that by , and we have
[TABLE]
and since commutes with it follows
[TABLE]
For the symplectic spectrum we could have used any (positive definite) covariance matrix, but for a determination of the optics functions we will now have to make one additional assumption: Namely that the symplectic spectrum (respectively emittances) is not degenerate, which means that all emittances are mutually distinguishable. By suitable orthosymplectic permutations (their construction is given in Prop. VI.16) on and let us arrange them so that without loss of generality and the entries of are in the following order:
[TABLE]
It follows from Eq. (8), which now reads , that or, on other words, if then , and, by exchanging symbols, also in that case. If we now take a look at the indices and , then we see that for all it holds and , so concerning these rows and columns must have the following form:
[TABLE]
By our assumption on the mutually distinguishable pairs, and because every such group of four entries must be orthosymplectic on its own, we must have as in lemma VI.8 with . By Rmk. II.4 this was the freedom in the choice of , which leaves the individual Floquet-planes invariant. So without loss of generally can be considered as equal to up to a symplectic permutation matrix which is exchanging the order of the symplectic spectrum and up to this freedom. Let us summarize this result:
Conclusion II.11**.**
If is the covariance matrix of a linearly matched particle distribution, and a symplectic diagonalization with mutually distinguishable emittances, then the optics functions are contained in the terms , and of the Iwasawa decomposition . In particular we can regain the linear map to Floquet-space from , block-diagonalizing . Hereby, the factor may contain a freedom of the form with , i.e. of the form , and a suitable symplectic permutation of the components.222In other words: The freedom is a block-rotation as in Thm. II.3.
In analogy to what happens in the 2D case (see Example II.12 below), the term can be understood as a lens- or drift factor and the term can be described as a magnification or squeezing factor. As we shall see in Example II.16, the term may also contain coupling and optic terms in a non-trivial fashion. Under the condition in conclusion II.11, all those three factors, and therefore the linear parameterization to normal form, are thus dependent on the parameters of the covariance matrix only, which are 3, 10 and 21 in the 2, 4 and 6 dimensional cases respectively. The freedom in the coupling term reduces these numbers by , so we can expect to have 2, 8 and 18 independent optics parameters in these cases.
This freedom represents our inability to extract the tunes out of the covariance matrix alone and so we can expect that in repetitive measurements the additional -freedom enters into the covariance matrices statistically. As we will see in Example II.16, is not in general of the form . In cases one wants to obtain the exact coupling terms of an underlying model, this will require a careful analysis in order to disentangle the freedom from these coupling terms. In any case, if the emittances of the given covariance matrix are mutually distinguishable, we obtain a linear normal form map from it which block-diagonalizes (with a certain error), as demonstrated in Tabs. 5 and 6.
As we shall see in examples II.12 and II.16, the three diagonal terms of are directly related to the three classical optics beta functions of the Courant-Snyder parameterization in the uncoupled case. But as they are also appearing in a general coupled situation and are determined by a natural procedure – and also to distinguish them from alternative ’generalized’ -functions discussed below – we call these ’natural’ beta functions. In Tab. 7 we show the matrices determined from the normal form map and the symplectic map (coming from the covariance matrix of our SPS example) and check their relation to a twiss output from MAD-X.
Independently on the problem of finding a faithful parameterization for the optics, one can also introduce ’generalized’ -functions, as proposed in Ref. bib:wolski2006 . The main idea is to use the analogy of the two-dimensional case, in which the -function appears as coefficient in the well-known relation between the emittance and the rms beam size, e.g. . Since the optics functions , and are always the same for beams with non-degenerated spectrum, one can use the symplectic (or of the linear normal form) in order to obtain such relations between the second moments of the matched distribution and the emittances:
[TABLE]
Algebraically speaking they correspond to the coefficients of the parameterization of the -invariant covariance matrices and are by definition related to the in Thm. II.6 via . From the analogy to the 2D case (cf. in example II.12), in which the situation goes over without coupling, one can identify and with and , and with and and and with and respectively.
As was shown in Refs. bib:wolski2006 ; bib:ohkawa2007 ; bib:alexahin2013 , these coefficients have the feature that one can also find expressions for the dispersions. The idea is that in the classical 2D theory the betatron motion and the dispersive part are uncorrelated: Let and for denote the dispersion function with respect to direction , where the 6th component corresponds to the energy offset . Then from it follows , which translates to
[TABLE]
hereby the last equation was assuming a decoupled case, similar to what was previously done to relate the to the classical optics functions. In Tab. 8 we show the general -functions for our SPS example and their agreement with those coming from a MAD-X twiss command. It should therefore be clear how useful these statistical definitions are when it comes to situations where one can not easily determine effective optics functions otherwise, for example in the scenario of a PIC code with space charge.
Having the dispersion parameters at hand, which were determined by including assumptions of the origin of the one-turn map , one can attempt to recover the tunes of the unknown optics – in principle. However, as we shall see, the sensitivity with respect to the dispersion terms is very high. This indicates that such an undertaking, by purely examining covariance matrices, might require more elaborate methods (and probably also better statistics by including more particles):
For the next considerations we change to the ordering and write the one-turn-map into 4 and 2-blocks
[TABLE]
Let for describe a 4D closed-orbit solution of , i.e. , which translates to and so contains the 4 known dispersion terms DX, DPX, DY and DPY in the second column. Although the terms in all columns of can be approximated in an analog fashion as the left-hand side of Eq. (11) if correlations between and the other spatial coordinates are small, and by taking into account only correlations between and , let us assume that the first column is unknown. The relation ( any symplectic map block-diagonalizing , in particular having in mind a map coming from a covariance matrix) reads in this context
[TABLE]
hereby is a block-diagonal rotation matrix, as discussed in Thm. II.3. We can now use the first row to express the unknown by and the rotation matrices and which contain the unknown tunes:
[TABLE]
If we assume that the -map is invertible (which is the case in our SPS example), we can eliminate to obtain
[TABLE]
This corresponds to a system of 8 equations for 7 unknown parameters (3 tunes and the 4 entries of the first column of ). For the given data of our SPS example it turned out, however, that the sensitivity of this problem on the dispersion terms is too high, even if assuming a known -tune, as is summarized in Tab. 9.
We conclude this section with three examples.
Example II.12* (2D).*
For and with consider a linear transport map , see Ref. bib:lee2012 , and a positive definite symmetric
[TABLE]
Then one can show that is symplectic and it holds . Furthermore, the symplectic matrices
[TABLE]
satisfy (in particular, is positive definite). This means that we have all requirements of Thm. II.8 and must be orthogonal. Indeed we have the following Iwasawa decompositions
[TABLE]
and we see and and is orthogonal, as claimed by Thm. II.8. The two independent optics parameters and can be regained by comparison of the above Iwasawa factors with the ones obtained by any matched particle distribution according to conclusion II.11. Since holds, we have and therefore , which means that the associated quadratic form is -invariant. We have or, reversely, . The effects of these two operations are illustrated in Fig. 1 and appear frequently in elementary particle accelerator textbooks.
Example II.13* (4D).*
Let G=\left(\begin{array}[]{cc}A&C\\ C^{tr}&B\end{array}\right) be the covariance matrix of a linearly matched particle distribution in a 4D tracking routine. Then its symplectic spectrum, which are the two emittances, are given by
[TABLE]
where . This formula for the spectrum of a 4D covariance matrix can be found in Ref. bib:pirandola2009 in a different context. From Eq. (32) we regain two familiar symplectic invariants:
[TABLE]
For the next example we need some more machinery.
Lemma II.14**.**
Let be symmetric with so that
[TABLE]
Then the LDL-Cholesky factorization of is given by
[TABLE]
Proof.
The proof is left to the reader. Note that this equation holds also for . ∎
Theorem II.15** (T. Y. Tam).**
Let and with
[TABLE]
Let be the LDL-Cholesky factorization of the positive definite , where is unit upper triangular and positive diagonal. Then the Iwasawa factors and of can be computed by
[TABLE]
Proof.
See Ref. bib:tinyautam2006 . ∎
Example II.16* (4D optics with a single coupling term).*
Similar as in Example II.12 consider for : and with and the matrices . Let be the operator defined in Prop. VI.6. From the properties we have . Let
[TABLE]
It follows from Prop. VI.6 with the symplecticity of that is symplectic and furthermore that is symmetric and . From Prop. VI.6 it also follows that has four complex eigenvalues of the form and by assumption it is guaranteed that no eigenvalue equals . Moreover it follows that is positive definite. can be interpreted as an uncoupled lattice. Let us now introduce a basic coupling term; for set , and as in Example II.12, Eqs. (30c), (30f), with using and .
[TABLE]
By the properties of the operator , all four are symplectic. is clearly orthogonal and also symplectic:
[TABLE]
The symplectic satisfies with symmetric and positive definite :
[TABLE]
The interpretation of is that it describes the covariance matrix of a linearly matched particle distribution in this coupled optics. From Example II.12 we know that must hold. Therefore the symplectic are diagonalizing as in Williamson’s Theorem: . Let us now use lemma II.14 and Thm. II.15 to compute the Iwasawa decomposition exemplary in the case of . By construction:
[TABLE]
with diagonal block-matrices . It follows
[TABLE]
Thm. II.15 tells us that we have to compute the LDL-Cholesky factorization of the positive definite
[TABLE]
By lemma II.14, using , we obtain with
[TABLE]
By Eq. (36a) we see that the determinant of the -factor of a given covariance matrix yields . Moreover, we can recover the term and by Eq. (36d) the . Eq. (36a) - (36e) constitute a system of equations for the five parameters , , , and of this model and more such equations may be obtained by computing the other decompositions of . In this way we can attempt to recover the entire set of parameters of this model from a given covariance matrix.
Since is symplectic, we have , i.e.
[TABLE]
Using this equation, we can then compute the orthosymplectic via . After some steps we find the block-diagonal form
[TABLE]
Hereby we attached the indices on to indicate that it emerges out of the map . Note that , so if , then this term is not in the form . This example therefore completes the statement in Conclusion II.11. What happens for the other cases , and ? Since we expect a symmetric result for let us investigate the case. As we have already computed the Iwasawa factors and , we do not need to recompute them again. has the form:
[TABLE]
After further computations we arrive at the following orthosymplectic :
[TABLE]
We see that also turns into the standard form if we have no coupling. Moreover, there has to be a term with which transforms into . In fact we find with and . This corresponds to a rotation in the second plane by an angle of . In general, any rotation in these planes can lead to valid ’s. Therefore it requires a careful analysis of the covariance matrices involved in order to disentangle the sought coupling terms from that freedom.
III Emittances from simulation and experiment
For large-scale simulations with many particles, the particle positions are usually not stored turn by turn, because this will generate an enormous amount of data. What can be stored without generating too much data, however, are the turn-by-turn covariance matrices of the distribution. In this section we will outline the connection to the well-established formula of Lapostolle. For completeness we will also include practical formulae to obtain emittances from experimental data.
III.1 Simulation case
By means of the map , given by Thm. II.3 or II.11 (from a covariance matrix) and utilized in Thm. II.5, we can parameterize all symmetric invariants of . As motivated in paragraph II.3, one may have the task to find, for a given covariance matrix an -invariant covariance matrix which is closest to , so that we can apply Thm. II.5. Hereby we understand the distance between and as given by the Frobenius norm. This can be formulated in a precise fashion as follows:
Let be the canonical basis of . Introduce for the matrices
[TABLE]
i.e. consists of zeros besides its and components, which are one. Denote for brevity with the notations of Thm. II.5. Then the problem stated above corresponds to the task of finding so that
[TABLE]
is minimized. The ’s then correspond to the emittances, because the covariance matrix in Floquet-space has zero off-diagonal elements for independent variables and the determinant in the individual Floquet-planes are therefore just (see also example III.1 below). We remark that an expression as the sum in Eq. (42) also appears in Ref. bib:wolski2006 . Since is extremal at a given point if and only if is extremal at that point, we obtain, by using the the symmetry of and :
[TABLE]
Hence, is minimal if and only if
[TABLE]
This process provides us with a closest symmetric invariant . Since was assumed to be positive (semi)definite, there exist so that , i.e. with , so is again positive (semi)definite.
Example III.1*.*
In the 2-dimensional case we obtain the classical emittance definition by Lapostolle (cf. bib:lawsonetal1973 ) as follows: Assume that is given and set according to Eq. (44). With we get
[TABLE]
On the other hand, by Eq. (43) and we have
[TABLE]
hereby
[TABLE]
Now combining Eqs. (45), (46) and (47) we obtain
[TABLE]
and since we have , therefore we regain the emittance of Lapostolle up to the Frobenius distance between and the -invariant :
[TABLE]
Example III.2*.*
In the special case of a single particle, i.e. if has the form with , then (compare above) and so we get with the action as a ’single-particle emittance’:
[TABLE]
We recall the standard 2D example of a linear transport map of Example II.12 (or found e.g. in Ref. bib:lee2012 ):
[TABLE]
where , and . Then one can show that with
[TABLE]
we have and
[TABLE]
and with , , we obtain for the action (50):
[TABLE]
III.2 Measurement case
In this last paragraph we will summarize of how to obtain emittances from measured profile data in this general linear setting. We assume here that the coordinates are arranged in the form . Denote for the symplectic transport maps to the location of the scanners which measure our profiles (which are usually two wirescanners and a wall-current monitor) and by the map from Floquet-space to ordinary phase space, which diagonalize the one-turn map by according to Thm. II.3.
Assume that is the covariance matrix belonging to a matched distribution. By Thm. II.5 we have with and , i.e. can be interpreted as the image of a covariance matrix of a distribution in Floquet-space, in which the individual planes are uncorrelated, transported by the map to ordinary phase space. For consider the maps , where denotes the unit vector having a one in the th position, so they project onto the spaces belonging to the , and directions at the corresponding scanner locations. Now consider the linear map given by , so its matrix entries are . Since covariance matrices transport under linear maps in form of matrix congruence (see paragraph II.1), the image of this map can be identified with the second moments of the distribution at the corresponding scanner positions: , and . They are known from our experiments, hence provides us with the emittances of the distribution.
IV Conclusion
We have examined in detail the close connection between linear normal form and covariance matrices belonging to a matched particle distribution. In fact, a linear normal form is contained in such a matrix: If the emittances are mutually distinguishable, then up to an -freedom (which can be understood as a rotation part related to the tunes) the entries of the normal form are uniquely determined. Furthermore, by means of the Iwasawa decomposition, we obtain a natural generalization of the optics -functions and coupling terms, which complement, together with their relation (10) to the embedding coefficients , our picture in this linear scenario. In addition, we have provided the connection to the Courant-Snyder parameterization and the Lapostolle-emittance and summarized useful formulae for practical applications regarding simulations and experiments.
V Acknowledgments
The author want to thank Prof. Dr. M. de Gosson and Dr. F. Schmidt for inspiring comments. This work has been sponsored by the Wolfgang Gentner Programme of the German Federal Ministry of Education and Research (grant no. 05E15CHA)
VI Appendix
VI.1 Preliminaries
This part of the appendix is intended as a convenient reference of properties and notations which we used in the main text. Some of these facts are known in the literature but often scattered or not easy to find in a concise and self-contained fashion. We will begin with the proof of Thm. II.2.
* is a covariance matrix if and only if is symmetric and positive semidefinite.*
Proof.
’’ Symmetry is a consequence of . Positive semidefiniteness follows with . ’’ Since is symmetric, we can find, by Sylvester’s law of inertia, an orthogonal matrix and a diagonal matrix so that hold. Since is positive semidefinite, the diagonal entries are non-negative. Set by taking the square root of these diagonal elements, so that we obtain a Cholesky decomposition of by with . Now take independent random variables , i.e. for . Set . It follows , so is a covariance matrix. ∎
Let us make a notation convention:
Convention VI.1**.**
The Symbol means either or . denotes the symplectic structure
[TABLE]
where denotes the identity matrix in . The upper letter ’’ on a matrix means transposition and complex conjugation. For we understand as the sesquilinear form if nothing else is stated. We will sometimes use the abbreviation for . If is a vector space, we denote its complexification by and for , we sometimes denote its action onto by . However, this notion will be dropped whenever the context is clear.
Let be diagonalizable with mutually distinguishable eigenvalues. We denote by a fixed basis of eigenvectors, where belongs to the eigenvalue . Because is real, is the eigenvalue of the eigenvector of . We have for all :
[TABLE]
so we conclude, since all eigenvalues are mutually distinguishable, that if , then and are orthogonal. Because of this orthogonality, the fact that is a basis of and is non-degenerate, it must hold that , and these values are purely imaginary, which follows by with . Set with . Since , we can choose a representation system of the equivalence relation introduced in Dfn. VI.3, so that hold.
The eigenvalues of coincide with , and if we set , then
[TABLE]
i.e. is an eigenvector or with respect to . Let us summarize:
If we speak of an eigenvector of we will always understand for a given (and fixed) system of eigenvectors of . 2. 2.
From the properties of we always have . 3. 3.
There is a subset so that with holds.
Proofs of the next statements are straightforward. They are required in the proof of the linear normal form Thm. II.3.
Proposition VI.2**.**
Let be a -vector space and linear. Then all eigenvectors of belonging to mutually distinguishable eigenvalues are linearly independent.
Definition VI.3**.**
Let be an -vector space and linear. Assume that has mutually distinguishable eigenvalues . Since is real, it also admits the complex conjugate eigenvalues. So we can introduce on the equivalence relation . We denote the equivalence class of by . They constitute of pairs of indices.
Proposition VI.4**.**
Let be a real vector space and linear. Assume that has mutually distinguishable eigenvalues with corresponding eigenvectors so that . Then are linearly independent in for every representation system of the equivalence relation introduced in Dfn. VI.3.
Proposition VI.5**.**
Let be a real vector space and linear. Let , be an eigenvalue of with eigenvector so that . Then the -vector space spanned by and in is -invariant and it holds
[TABLE]
where and are the real and imaginary parts of .
The next map emerged rather often in our programs as well as in some formulae, so that we found it useful to write it down as reference. It appears whenever we had to switch between the phase-space notation to a block notation of the form . But we also used this (non-symplectic) isomorphism in Thm. II.5 to transport a statement regarding matrices to a statement regarding block matrices and where it is useful to keep track that changing the notation does not have any effect on the symplecticity of the result.
Definition and Proposition VI.6**.**
Let and be two block-matrices of the form
[TABLE]
with and respectively. Define as
[TABLE]
It holds
* is bilinear.* 2. 2.
If and , then . 3. 3.
For , and associativity holds: . 4. 4.
. 5. 5.
. 6. 6.
.
Remark VI.7*.*
For convenience we may want to recast for , , in block-diagonal form, which we will denote by the symbol as . For this purpose we introduce the following orthogonal operator (and also provide its inverse) on the canonical basis of :
[TABLE]
Then it holds .
A proof of Thm. II.3 can be found for example in Ref. bib:dragt2018 , but because of its importance and also because of certain details in the construction of the map we will recall it here:
Proof.
Let us rescale the ’s by , where is given according to Conv. VI.1. So without loss of generality we consider eigenvectors so that hold. By Conv. VI.1 there is a representation system so that holds. According to Prop. VI.4, we obtain a corresponding real basis of with . We thus have by construction :
[TABLE]
Therefore :
[TABLE]
and so the linear map defined on the canonical basis of via
[TABLE]
is symplectic. By Prop. VI.5 we know that for the planes are -invariant; i.e. if denote the vector , then with :
[TABLE]
By assumption has eigenvalues on the unit circle, , so has the properties as claimed. ∎
Lemma VI.8**.**
Let be block diagonal with
[TABLE]
with mutually disjoint , where . Let be given. Then
[TABLE]
*where are diagonal matrices and of the form . *
Proof.
Since the ’s are orthogonal and commute with , the ’’ direction is clear, so let us prove the ’’ direction. Decompose into a symmetric and an antisymmetric part . Since is antisymmetric, it follows
[TABLE]
By adding and subtracting Eqs. (62a) and (62b) we obtain the equivalent conditions and . So let us assume for a moment that is (anti)symmetric.
Condition in block indices reads: . In particular, by exchanging indices, we can write down the following two equations for every and :
[TABLE]
From these two equations we obtain
[TABLE]
Since is (anti)symmetric, , and therefore with the positive semidefinite symmetric , Eq. (64) reads
[TABLE]
Now write in terms of a symmetric matrix
[TABLE]
For brevity denote and . Then Eq. (65) reads
[TABLE]
Since it follows from the or -component that . Inserting this into e.g. the -component we see that must hold and from , it follows that . We now attach the indices and on . We conclude that , so if , must be orthogonal.
In the case that there are now two options: Either or . In the first case, and therefore commutes with and we get together with Eq. (63a)
[TABLE]
Now the second case. By multiplication of with the inverse of , also hold. In this version, Eq. (63a) reads
[TABLE]
Let W:=\left(\begin{array}[]{cc}0&1\\ 1&0\end{array}\right). Then and therefore commutes with , so we get
[TABLE]
Hence in both cases a equation of the form follows, so and therefore, by construction, this is only possible if and so . Conversely we have shown that if , then , so must be block-diagonal and its diagonal blocks must have positive determinant.
Recall that we assumed that was (anti)symmetric. In the symmetric case, since , and at the same time is orthogonal (if ), the individual off-diagonal elements must vanish and therefore has a diagonal form as claimed. In the antisymmetric case , so its diagonal entries vanish and it has the form . ∎
VI.2 A basis for symmetric invariant matrices
Symmetric matrices which are -congruent invariant, where is diagonalizable with mutually distinguishable eigenvalues, can be given a basis built out of the eigenvalues of itself. This very useful result, as stated in Cor. II.6, is used in Ref. bib:nash2006 in order to find matched distributions near coupled synchrobetatron resonances. As we shall see at the end of the next paragraph, this decomposition is linked to the diagonalization of invariant covariance matrices by linear normal form, which is used in some of the other references mentioned in the introduction. For convenience, we change our notation to only in this paragraph VI.2, as otherwise we would have to attach many minus signs on the maps. Assume that is a symmetric invariant and define . Since is symplectic, , condition II.1 can be recast as
[TABLE]
and so
[TABLE]
Denote by the Lie-algebra of . One can show that can be characterized as . The elements of this semisimple Lie-algebra are called Hamiltonian matrices. Now observe that since is symmetric,
[TABLE]
so together with Eq. (69) we have the characterization with . If , then conversely is symmetric: . Let . is a Lie-subalgebra of since enters in both entries. Condition II.1 therefore essentially means that we are considering elements of the Lie-subalgebra of and so that may in addition be positive semidefinite.
Lemma VI.9**.**
Let be a diagonalizable real or complex -matrix with eigenvalues . Then has the eigenvalues for and the corresponding eigenmatrices to have the form
[TABLE]
where and are eigenvectors of and with respect to and .
Proof.
Since has the eigenvalues , let and . It follows
[TABLE]
∎
Proposition VI.10**.**
Let be diagonalizable with mutually distinguishable eigenvalues and denote by the Lie-algebra of and . Then is abelian.
Proof.
By application of lemma VI.9 to the real matrix we obtain the eigenmatrices of . By Conv. VI.1, the entry is zero if or and otherwise not, and so these eigenmatrices are linearly independent. Therefore is spanned by the matrices , which are the eigenmatrices belonging to the eigenvalue [math]. By Conv. VI.1 we have, since the eigenvalues are mutually distinguishable, for : . It follows if , showing that is abelian, especially its sub Lie-algebras . ∎
Proposition VI.11**.**
There exist a basis of linearly independent and -unitary vectors of with respect to the hermitian bilinear form , i.e.
[TABLE]
Proof.
Let be a representation system according to Conv. VI.1 and rescale the ’s so that holds. Set and use the notation for the other element in the equivalence class . Correspondingly let . By the proof of Prop. VI.10 constitute a basis of . We have
[TABLE]
We see that by an additional rescaling by of the ’s we obtain -unitarity. ∎
Corollary VI.12** (See Ref. bib:nash2006 ).**
The set of all real symmetric invariants of , where is diagonalizable with mutually distinguishable eigenvalues, is spanned by the linearly independent matrices , where is a representation system according to Conv. VI.1.
Proof.
From Prop. VI.11 we saw that admits a -unitary basis . Let be given with real- and imaginary parts , . Then
[TABLE]
It follows if is real
[TABLE]
and so is real if and only if it can be represented as a sum of the real matrices and Now observe that is symmetric, while is antisymmetric. So if and only if is represented as a sum of the -orthogonal for . ∎
As we shall see, the result II.6 can equivalent be obtained by means of linear normal form, which will be discussed now.
VI.3 Proof of Thm. II.8 Part 1
For the proof Thm. II.8 it is more convenient to change the notation in its claim to (as otherwise we get many minus signs in the exponents). A similar assertion can be found in Ref. bib:degosson2006 , however the proof unfortunately contained a mistake bib:degosson_pc . We did not found an alternative proof. For the next part we will drop the indices 1 and 2 for convenience.
Since is positive definite, there exist a Cholesky-decomposition of in the form , with invertible (in fact, is real but for convenience we keep the complex notation). Then the invariance condition II.1 can be rewritten as
[TABLE]
i.e. is unitary. Since , must have the same eigenvalues as . Let be a basis of eigenvectors of with respect to the eigenvalue , i.e. .
We have and therefore for every and :
[TABLE]
in particular () it follows that for every , i.e. all eigenvalues lay on the unit circle. Furthermore if , then by assumption . Consequently we must have in this case.
Let be the eigenvectors of with respect to and the eigenvectors of with respect to . Since the eigenvalues are mutually distinguishable by our assumption, they are connected to the orthogonal eigenvectors of (see above) as follows:
[TABLE]
with dependent on (and therefore unspecified yet). It follows
[TABLE]
If we fix an index , then for every : , thus
[TABLE]
and so , i.e. are the eigenvalues of with respect to the eigenvectors . Hence we make the following
Convention VI.13**.**
Let be a basis of eigenvectors of . We will assume that this system is fixed and relabeled in such a way that for it holds with . 2. 2.
The eigenvalue of for with respect to the eigenvector of (labeled according to point 1) is denoted by .
Since , Eq. (79b) provides us with the following relation between the norm of , and :
[TABLE]
Remark VI.14*.*
By Eq. (80) it follows in particular, since , that the eigenvalues must be purely imaginary.
By assumption it holds , so and therefore has the same eigenvalues as . A unitary basis of eigenvectors of is given as follows:
Definition and Proposition VI.15**.**
Denote by the canonical basis of . For define
[TABLE]
Then it holds:
* span a unitary basis of .* 2. 2.
Let block-diagonal with diagonal -matrices . Then
[TABLE] 3. 3.
* for and else.* 4. 4.
.
Proof.
If , then ; similarly is the case . If and , then . 2. 2.
[TABLE] 3. 3.
This follows from Eq. (81) by setting . 4. 4.
.
∎
Definition and Proposition VI.16**.**
For define an orthosymplectic transposition as follows:
[TABLE]
Then it holds
* is orthogonal and symplectic.* 2. 2.
If then has the same form of where is exchanged with .
Proof.
is orthogonal, since is a combination of permutation and a reflection which are orthogonal. The symplecticity and the second property can be seen as follows: Without loss of generality we consider only indices . Since is orthogonal, . Then
[TABLE]
∎
Now attach on the matrices the index 1 and 2. Since the eigenvalues of are the ’s and the ’s are positive, there must exist, by Conv. VI.13, a permutation so that for . Note that this implies automatically for , since the complex conjugated belongs to the eigenvector of , which in turn equals by our Conv. VI.13 and therefore it is related to the eigenvalue .
By Prop. VI.16, we can assign to a suitable composition of symplectic permutation matrices so that the indices of the diagonal entries of now coincide with the indices of , with respect to our fixed eigensystem . These diagonal matrices belong to a similar problem than the original one, now formulated with the symplectic matrices (and Prop. VI.15 holds also for these new block-diagonal matrices). Since is orthogonal, is orthogonal iff is orthogonal. The important fact of this consideration is that we treat both cases simultaneously (if we would have looked at only one case, we could have simply relabeled the ’s). So we conclude:
Corollary VI.17**.**
Without loss of generality we can assume that for all it holds and , i.e.
[TABLE]
in particular , by which we denote the eigenspace of with respect to , and is a unitary basis of eigenvectors of .
VI.4 Proof of Thm. II.8 Part 2
Proposition VI.18**.**
By assumption we have , therefore we can apply the results of paragraph VI.3, using in particular . Then the corresponding orthogonal eigenvectors of (with this particular ) are also eigenvectors of and satisfy:
[TABLE]
Proof.
By Eq. (79b), using , we have with corresponding and values (they are not yet specified)
[TABLE]
Since the ’s constitute a basis, the claim follows. ∎
By Prop. VI.18 and paragraph VI.3 we thus have for each an orthogonal basis of eigenvectors of , satisfying for not yet specified complex numbers . Let us now choose so that are normalized to one, i.e. they describe a unitary basis. By Eq. (80) this is fulfilled if and only if
[TABLE]
leaving an freedom in the choice of the ’s. In particular we obtain
[TABLE]
Now let be the unitary transformation sending to . Let us drop the index for the next lemma.
Lemma VI.19**.**
The unitary map satisfies
. 2. 2.
. 3. 3.
. 4. 4.
.
Proof.
Since by Eq. (83), we have by Eq. (82): . 2. 2.
Let be the eigenspace of with respect to and the equivalence class of indices with . Since , and is a basis of , it follows that there exist so that
[TABLE]
Then
[TABLE] 3. 3.
Follows immediately from 2. 4. 4.
By 1 and 3: .
∎
Now we are ready to prove the original claim. Set , i.e. . It follows
[TABLE]
By lemma VI.19 it holds and , and therefore with Eq. (85)
[TABLE]
Since multiplication of a unitary basis with complex phases is a unitary operation, can entirely be described on as a unitary operation. And because and itself are real, we conclude that must be orthogonal.
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