Complex formalism of the linear beam dynamics
aa r X i v : . [ phy s i c s . acc - ph ] S e p Complex formalism of the linear beam dynamics
J. Lucas ∗ ELYTT Energy S.L., Calle de Orense 11, 28046 Madrid
V. Etxebarria † Depto. de Electricidad y Electr´onica, Universidad del Pa´ıs Vasco - UPV/EHU, 48940 Leioa, Spain (Dated: September 16, 2020)It has long been known that the ellipse normally used to model the phase space extension ofa beam in linear dynamics may be represented by a complex number which can be interpretedsimilarly to a complex impedance in electrical circuits, so that classical electrical methods mightbe used for the design of such beam transport lines. However, this method has never been fullydeveloped, and only the transport transformation of single particular elements, like drift spaces orquadrupoles, has been presented in the past. In this paper, we complete the complex formalismof linear beam dynamics by obtaining a general differential equation and solving it, to show thatthe general transformation of a linear beam line is a complex Moebius transformation. This resultopens the possibility of studying the effect of the beam line on complete regions of the complexplane and not only on a single point. Taking advantage of this capability of the formalism, we alsoobtain an important result in the theory of the transport through a periodic line, proving that theinvariant points of the transformation are only a special case of a more general structure of thesolution, which are the invariant circles of the one-period transformation. Among other advantages,this provides a new description of the betatron functions beating in case of a mismatched injectionin a circular accelerator.
PACS numbers: 29.27.Eg, 41.85.JaKeywords: linear beam dynamics; Moebius transformation; Twiss parameters
I. INTRODUCTION
Complex variable methods have been widely used inthe past to conveniently represent two-dimensional fieldsand phenomena by essentially identifying the compo-nents of a planar vector with the real and imaginary partsof a complex number. This approach has been appliedto most branches of science, but particularly it is worthmentioning that in the very early days of electromagnetictheory, even Maxwell included complex variable meth-ods in a chapter of one of his major works to set uppotential functions satisfying Laplace’s equations in twodimensions [1]. Since then, many complex variable for-mulations elegantly describing a wide variety of electro-magnetic phenomena have been proposed. Interestinglyenough, this complex formulation has been repeatedlyapplied in the past to describe different electromagneticproperties and systems in relation to dynamics, transportand optics of charged particle beams, including electro-static space-charge phenomena [2], [3], two-dimensionalmagnetic fields [4], [5] or phase plane linear beam optics[6]. However, the idea was never very popular, as faras beamline dynamics design is concerned, in compari-son with transport matrices or Twiss parameters, proba-bly because of the lack of a complete complex formalismwhich could boost the method to its full potential as ap-plied to this field. ∗ [email protected] † [email protected] In this paper we extend and develop the idea originallyproposed by Hereward [6] of using a complex number torepresent the phase space ellipse describing the motionin a linear beamline. Instead of just using this formula-tion to treat beam transport lines as electrical circuits byanalogy, as it was intended in principle, here the generaldifferential equation governing the complex representa-tion of the phase plane ellipse is obtained and solved. Asa result, it is demonstrated that the general transforma-tion of a linear beam line is a subgroup of the well-knowncomplex Moebius transformation. Apart from the ele-gance and compactness of the general obtained solutionin complex formulation, the result contributes to improveour fundamental knowledge of linear beamline dynamicsas well as to be able to transform complete regions of thecomplex phase space instead of single points, as it is thecase with the conventional Twiss parameters approach.As a practical example of use of the new formalismproposed, the method is applied to the important caseof beam transport through a periodic line (for instance,through a circular accelerator). By means of the complexformalism we prove the existence of invariant circles un-der such periodic transformation, which generalizes theclassical concept of invariant points of the transforma-tion determined through the Twiss parameters methods,meaning that invariant points in a periodic transportline are indeed invariant circles of null radius. This al-lows us to reinterpret and more accurately describe, pre-dict, quantify and control important effects measured inpractice, such as betatron oscillations and their beating,among other properties.As described in the following, both in theory and illus-trated through example, it becomes clear that the pro-posed complex formalism gives us a deeper insight on lin-ear beam dynamics , generalizes classical concepts andhas clear implications both in theoretical beam opticsand in practical computation and design of present andfuture beam transport lines. II. DEFINITION OF THE COMPLEXPARAMETERS
Although the complex formalism of linear beam dy-namics may be obtained without using the Twiss pa-rameters, here we will refer to them for the sake of com-pleteness and for easier comparison between the standardand the new proposed formulation. The phase space el-lipse is parametrized by Eq. (1), [7][8], where the Twissparameters, α , β and γ define the shape of the ellipse,and the emittance ǫ is related to its area. In addition,because of the symplecticity of the dynamics, the relation βγ − α = 1 holds among the ellipse parameters. γx + 2 αxx ′ + βx ′ = ǫ (1)The relationships between the Twiss parameters andthe shape of the beam space ellipse may be seen in Fig.1. x’ x PSfrag replacements − α p ǫ/β − α p ǫ/γ p ǫ/β √ ǫγ √ ǫβ p ǫ/γφ tan 2 φ = αγ − β FIG. 1. Twiss parameters defining the phase space ellipse
When the beam is transported through a drift space,the point of maximum divergence will keep its divergence γ = γ = γ , while its position will shift according to x ′ and the drift length L . This may be stated as: − α r ǫγ = − α r ǫγ + L √ ǫγ (2) Note that this paper deals only with linear beam dynamics. How-ever a complex formalism might also be established for non-linearbeam dynamics through Birkhoff Normal Forms, which are nat-urally based on complex formalism In all the following we will be referring to the Courant-Snyderlattice function parameters by using the widespread (though notso proper) term Twiss parameters
So, through a drift space, it holds:1 γ = 1 γ (3) − α γ = − α γ + L (4)It can be shown in the same way that through a thinlens of focal length f , the following relationships will fol-low: 1 β = 1 β (5) − α β = − α β − f (6)Now two complex numbers can be defined as: Z = 1 γ − j αγ (7) Y = 1 β + j αβ (8)It may be seen by direct multiplication that ZY = 1.In addition, through a drift space we will have, Z = Z + jL , and through a thin lens Y = Y + j/f .If the Y parameter is used, the size of the beam will beproportional to p ǫ/ Re Y , while the lines passing throughthe origin are of constant α . The upper part of thecomplex plane corresponds to converging beams and thelower one to diverging ones. Fig. 2 shows a qualitativerepresentation of the shape of the phase space ellipsesaccording to their position in the Y plane. a/b 1/b FIG. 2. Shape of the phase space ellipses according to theirlocation in the Y plane. Each ellipse is drawn in a local x - x ′ system. III. THE GENERAL DIFFERENTIALEQUATION OF THE COMPLEX FORM
All the previous material has been studied in the past,starting with the report by Hereward, [6], and it is citedextensively, [9], [10], [11]; but as far as it is known tothe authors, the general differential equation governingthe complex representation of the phase space ellipse hasnever been obtained. In order to obtain this equation,in the following an infinitesimal displacement through alens of strength per unit length k will be analyzed [12].Because the displacement is infinitesimal, the effect ofthe lens, which is straightforward in Y , can be superposedwith the effect of the drift ds , which is straightforward in Z . This superposition may be expressed either in termsof Z or Y , but the latter representation is preferred, be-cause the real part of Y is related to β , which, in turn,is related to the beam size: dY = dY lens + dY drift = jkds + d (cid:18) Z (cid:19) drift = jkds + − dZ drift Z = jkds − Y jds = (cid:0) k − Y (cid:1) jds (9)And finally, this results in the following Riccati differ-ential equation: dYds + jY = jk ( s ) (10)To solve this equation, we start by applying a substi-tution: Y = − j u ′ u (11)which converts the Riccati equation into the -well-known to beamline practitioners- Hill’s second order lin-ear differential equation: u ′′ + k ( s ) u = 0 (12)whose solutions may be expressed, using the superposi-tion principle in linear equations, as a linear combinationof the fundamental functions C ( s ) and S ( s ), which satisfy C (0) = 1, C ′ (0) = 0, S (0) = 0, S ′ (0) = 1, [13]. More-over, two particular solutions of Eq. (10) can be found,which may be written as: Y = − j C ′ C (13) Y = − j S ′ S (14)With one particular solution, for instance the onebased on C , the Riccati equation can be reduced to afirst order linear equation using the substitution: Y = Y + 1 z (15)thus obtaining: z ′ − C ′ C z = j (16)Since a second particular solution of the Riccati equa-tion is already known, a particular solution of Eq. (16)can be found as well: z = 1 Y − Y = jCS (17)where it has been used the fact that CS ′ − C ′ S = 1because of the constancy of the Wronskian of Eq. (12).With this particular solution of Eq. (16), its generalsolution may be obtained as the sum of the general so-lution of the homogeneous equation and the particularsolution: z = AC + jCS (18)where A is an integration constant. By replacingEq. (18) in Eq. (15), the general solution of the Riccatiequation is readily obtained: Y = − j C ′ C + 1 AC + jCS (19)The integration constant can be found by imposing Y (0) = Y , to get A = Y − . After some algebra, thefinal result is obtained: Y = S ′ Y − jC ′ jSY + C (20)and therefore it is concluded that the transformationof the complex parameter is a Moebius transformationof the shape given by Eq. (20). A similar expression isfound in [6], but the proof is restricted to drift spacesand lenses.We should note now that this result is very important,because the Moebius transformations form group, i.e. thecomposition of Moebius transformations is a new trans-formation. Since any general transformation followingEq. (10) may be expressed as a composition of individualtransformations, it may be concluded that if each singlebeamline element is described by a Moebius transform,then also the line will be described by a Moebius trans-form. IV. EXAMPLES OF COMMON BEAM LINEELEMENTS IN COMPLEX FORM
By introducing the well-known transport matrices val-ues of common beam line elements [7] [8] in the gen-eral result of Eq. (20), some illustrative examples of theMoebius transformations associated to these common el-ements can be built.
A. A drift space of length L Y = Y jLY + 1 (21)The effect on the complex plane of the drift space in Z = Y − is a vertical displacement in the upward direc-tion. The complex inverse of a straight line in the com-plex plane is a circle passing through the origin. There-fore the trajectory in the Y plane will be a sector of acircle passing through the origin and tangent to the imag-inary axis. The particle will move clockwise, because theinversion implies a change of sign with respect to themovement as seen from the origin. Fig. 3, shows theeffect of a drift on the Z and Y planes. AB -a/g 1/g A B a/b 1/b
FIG. 3. Effect of a drift on the Z and the Y plane It can also be seen that the length of the drift is mea-sured as the difference in the vertical distance of the twoextreme points defining the drift, A and B . If all thehorizontal lines of the Z plane were labeled by their con-stant pure imaginary coordinate − α/γ , these lines wouldbe transferred to the Y plane as circles passing throughthe origin and tangent to the real plane. Each of thesecircles correspond to a certain − α/γ . B. A thin lens of focal length f Y = Y + jf (22)This represents a vertical displacement upward when thelens is focusing and downward when it is defocusing. C. A thick lens of strength k and thickness l We assume here an ideal horizontally focusing thickquadrupole of length l with constant focusing strength k ( s ) = − k with k positive. Y = cos( √ kl ) Y + j √ k sin( √ kl ) j sin( √ kl ) / √ kY + cos( √ kl ) (23) A similar equation is given in [14] without proof.This transformation is, as well, a circle centered in thereal axis. This may be seen by dividing the numeratorand denominator of Eq. (23) by cos √ kl and considering Y = Y + j √ k tan( √ kl ) j tan( √ kl ) / √ kY + 1 = Y + j √ kuju/ √ kY + 1 (24)as a Moebius transform of u=tan( √ kl ), ie. the realaxis. As it is known, Moebius transformations convertlines to circles (or lines in some cases). In order to obtainthe parameters of the circle, the Riccati equation can bewritten in terms of its real and imaginary components, Y = x + jy . dxds = − xy (25) dyds = k − (cid:0) x − y (cid:1) (26)It is observed in the above equation that when the signof y changes, only the horizontal derivative changes itssign. This is the expected behavior from a circle centeredon the real axis following the equation:( x − x c ) + y = R (27)where x c represents the center coordinate on the realaxis and R is the radius of the circle.The circle parameters can be readily obtained by not-ing that when the vertical derivative Eq. (26) cancels, thiscorresponds to a maximum y , i.e., the point with ( x c , R )coordinates. Thus, the following pair of equations defin-ing the circle passing through the point ( x , y ) and witha focusing strength of k , can be written:( x − x c ) + y = R (28) k − x c + R = 0 (29)so that the desired parameters will be: x c = k + x + y x (30) R = p x c − k (31)It is worth to notice here that, when k >
0, the circleis fully contained in the right part of the complex plane,so that the complex parameter is bounded, while when k <
0, the circle is partly contained on the left part ofthe complex plane.
V. THE COMPLEX PARAMETERS FOR ACIRCULAR ACCELERATOR
In a circular accelerator, or in general a periodic line,the beam is expected to repeat its configuration in phasespace. This is equivalent to operate in one of the fixedpoints of the Moebius transformation of Eq. (20). Sincethe fixed point must have a real part, we will reproducehere the classical condition | C + S ′ | < σ . This parameter is invari-ant through any equivalence transformation, and is ob-tained as: σ = ( a + d ) ad − bc − C + S ′ ) − Elliptic if − ≤ σ < σ > σ ≤ − σ is not real (33)The behavior of an iterative application of the sameMoebius transform with respect to the fixed points isfully described by the transformation type. The mostimportant aspect for the transverse beam dynamics studyis that only for the elliptic transformation does none ofthe invariant point represent an attractor. That is, forall other transformation types, the iterative applicationof the transformation has one of the invariant points asa limit. In addition, the invariant points for the case | C + S ′ | ≥ β → ∞ , which shows that the beam will grow withoutlimit in size for the hyperbolic case. This condition willensure as well that only Moebius transformations of theelliptic type lead to periodic solutions in periodic lattices. A. Structure of the periodic solutions of beamtransport in complex form
The Twiss parameters can readily be expressed as afunction of the fundamental solutions: This section makes use of some mathematical properties of theMoebius transformation, which are summarized in the Appendixfor completeness. In particular in the following we will makeuse of the matrix representation of the Moebius transformation,the fixed points of the transformation and the circle preservingproperties, all of which are discussed in the Appendix. β = 1Re Y + = 2 S q − ( C + S ′ ) (34) α = Im Y + Re Y + = C − S ′ q − ( C + S ′ ) (35)This is, of course, a classical result of the theory ofthe Twiss parameters. We will use now the theory ofthe complex transformation to obtain the structure ofthe solutions of the beam transport on a periodic line.Note that these solutions are not easily obtained from theclassical theory, and that a beautiful and useful structurecan be unveiled for these solutions when seen under thelight of the Moebius transformation.First, we will prove that if there are two circlesinvariant under the Moebius transformation, a one-dimensional set of circles being also invariant can befound. A pencil of circles is formed by the linear combi-nation of two circles: C ( λ , λ ) = λ C + λ C (36)It can be easily proved that if C and C are invariantcircles, all the circles of their pencil are invariant. Thepencil is one-dimensional, because Eq. (36) may be mul-tiplied by a constant without modifying the circle of thepencil.Now we can use the zero radius circles with center inthe fixed points of the transformation as the basis of thepencil of invariant circles. We will call the two fixedpoints Y + and Y − respectively, according to the sign ofthe real part of the fixed points given by Eq. (A.7). Theirrespective Hermitian matrices are: C ± = (cid:20) − Y ± − Y ± Y ± Y ± (cid:21) (37)The parametrization of the invariant circles will be: C λ = (1 + λ ) C + − λ C − (38)This parametrization ensures that the A term of C λ (see Eq. (A.11)) is equal to unity, so that the circle isnormalized, and that λ = 0 corresponds to the invariantpoint with positive real part. It is now possible to obtainthe invariant circle of the invariant pencil that passesthrough any point y , by solving λ from the equation: y H C λ y = 0 (39)The solution of the invariant circle passing throughpoint y will be: C λ = − y H C − yy H C + y − y H C − y C + + y H C + yy H C + y − y H C − y C − (40)The general result of Eq. (40) can be particularized tothe case of the movement around the fixed points givenby Eq. (A.7) for the elliptic case. For this transformation,the two fixed points are symmetric respect to the imag-inary axis, i.e. Y − = − Y + . If we call y the fixed pointwith positive real part (the one with physical meaning),then the matrices of the two zero radius fixed points willbe: C + = (cid:20) − y − y y y (cid:21) C − = (cid:20) y y y y (cid:21) (41)Now, by applying Eq. (40), it is possible to find theinvariant circle that passes through a certain point y .The solution is a circle centered at a point y c given by: y c = | y | + | y | − y ) Im ( y )2Re ( y ) + i Im ( y ) (42)meaning that the invariant circles have the center atthe same horizontal line as the fixed points. The radiusof the invariant circle will then be: R = | y − y | | y + y | y ) (43)With the knowledge of the invariant circle, it is possibleto obtain the maximum beam phase-space limit, whichwill be given by:1 β ≥ | y | + | y | − y ) Im ( y ) − | y − y | | y + y | y ) (44)This result may be very useful to determine the max-imum beam size that may be caused by a mismatch inthe injection of the beam into a periodic line. B. A numerical example
As an example of the use of the theory of invariantcircles on a periodic transport line, let us analyze thestructure of the solutions of the beam when injected, notnecessarily well matched, on a FODO line. In a qualita-tive way, we can show the evolution of the beam along theline for a well matched condition in Fig. 4. The focusingthin lens is the line CD ; the upper circle arc DA is thedrift going to the defocusing lens; AB is the defocusinglens and BC is the drift space going to the focusing lens.It may be worth to notice that the relationship betweenthe Twiss parameters and the complex ones in beam dy-namics is similar to that existing between the Bode andthe Nyquist plots in control theory. In the first one, theamplitude and the phase are expressed in two differentplots, but in exchange the position (frequency) at whichthe gain is expressed is explicitly indicated in the chart. ABCD a/b 1/b
FIG. 4. Trajectory on the Y plane of a beam through a thinlens FODO cell
In the second one, a complex number provides all infor-mation but as a drawback, the position (frequency) atwhich the value is given, must be provided with a labelattached to a certain number of points.To illustrate the complex formulation in a more quan-titative situation, next we will define a thick lens FODOcell with quadrupoles of length 0.2 m and drift spaces2 m long. The strength of the quadrupoles is ± − .Arbitrarily, we will analyze the behavior of the horizontalsolution at 1/3 of the length of the focusing quadrupole.The cell, with the horizontal Twiss parameters for theinjection matched at the invariant point, may be seen inFig. 5. s (m)fodo1 MAD-X 5.02.00 22/02/17 23.04.15 -4.-2.0.02.4.6.8.10. β x α x FIG. 5. Horizontal Twiss parameters in the example cell
The same cell may be described by the movement ofthe beam point in the complex plane. This is shown inFig. 6. The matching parameters at the injection pointare β ≈ m and α ≈ .
1, or the corresponding complexparameter. β α / β FIG. 6. The same cell described in the complex plane
We can now use the theory of invariant circles to studythe behavior of the periodic FODO in case of a mis-matched injection. In this case, the beam will oscillatearound the periodic parameters in such a way that it willbe difficult to find any apparent order. For instance, atFig. 7 the beam has been injected with a β = 5 m and α = 0 .
5. Under this condition, the beam wiggles aroundthe ideal fixed point. Nevertheless, the classical theorydoes not provide any information about the maximumamplitude of the Twiss parameters during the oscillation. s (m)fodo1 MAD-X 5.02.00 22/02/17 23.22.34 -8.-5.-2.0.02.5.8.10.12.15.18. β x α x FIG. 7. Horizontal Twiss parameters in the example cell forthe case of a non-matched injection.
The situation becomes much more clear when the dif-ferent complex points are plotted in the complex planeafter each period. This is shown in Fig. 8. The pointsmust remain in the circle of the invariant pencil of circlesthat passes through the point defined by the injectionparameters. At some passages, the point will be at theright side of the fixed point, corresponding to a smaller beam and at other passages, the point will be at the leftside, which corresponds to a larger beam. Nevertheless,the beam size will be bounded by the left-most side ofthe invariant circle, which is easily obtained through theparameters of the C λ invariant circle of Eq. (40).
20 0.25 0.300.000.050.100.150.200.25 1/ β α / β FIG. 8. The pencil of invariant circles at 1/3 of the lengthof the quadrupole. The position of the beam passes duringseveral periods, as well as the fixed point, are shown
As a final example, for the sake of comparison, we haveincreased the strength of the quadrupoles until the Moe-bius transformation becomes hyperbolic. In this case,the movement of the particles change dramatically bothqualitative and quantitatively, and instead of showing arotation around one of the fixed points, now the beamtrajectory converges to one of the fixed points, which ac-tually is located at the imaginary axis and therefore itrepresents a beam of infinite size, β → ∞ . This situationis shown in Fig. 9. - - / β α / β FIG. 9. The same FODO cell but with unstable parameters.It can be seen that the movement of the beam converges toone of the stable points located at the imaginary axis. Notethe logarithmic scale that is required to show all points as theconvergence towards the fixed point is exponential.
VI. CONCLUSIONS
A general formulation of the complex formalism of onedimensional linear beam dynamics has been presented.Through determining and solving the general differentialequation governing the beam dynamics in complex form,it has been shown that the general complex transforma-tion of a beam line is a subgroup of the Moebius transfor-mation. Using the proposed formalism, the transforma-tion of several common beam line components has beenobtained and graphically represented. It has been shownthat, although the complex formulation is equivalent tothe Twiss parameters approach, it generalizes and com-plements the classical analysis of a beam transport line,it allows to prove some theorems of beam dynamics in asimpler way and it opens the possibility of transformingcomplete regions of the complex phase space instead ofjust single points, as is the case in the classical formalismof beam transport lines.Further, for beam transport through periodic lines, theproposed complex formalism has allowed us to prove theexistence of invariant circles under a periodic transforma-tion. In the classical formalism only the invariant pointsare considered, and they are identified with the actualTwiss parameters for this point location of the transportline. Now, we have seen that the invariant points arenothing but invariant circles of zero radius. In this way,it is possible to obtain a higher bound of the maximumbeam excursion at any point. Through the classical ap-proach, the mismatched injection on a periodic line istreated as if the Twiss parameters in the injected lineare those of the fixed point of the one-period transforma-tion (actually turning an initial conditions problem intoan eigenvalue problem), so that the beam will increase itsemmitance due to higher order effects to accomodate thenew ellipse. Although this may be an acceptable descrip-tion for a circular accelerator, where millions of turns un-der the non-linear field will smear the beam phase space,it may be too pessimistic for a long periodic transfer line.In this case, the description of the present paper, in whichthe transformed point moves in the invariant circle maybe a better description of reality.As a general conclusion, it can be stated that the pro-posed complex formalism provides a deeper complemen-tary understanding of linear beam dynamics as comparedto the classical formalism, it allows to map complete re-gions of the complex phase plane instead of single points,and in general it contributes to improve our fundamentalknowledge of beam transport dynamics.
VII. ACKNOWLEDGEMENTS
The authors are grateful to MINECO and UPV/EHUfor partial support of this work under grants DPI2017-82373-R and GIU18/196, respectively.
Appendix: Some useful properties of the Moebiustransformation
The general result given by Eq. (20) can be further ex-ploited by considering the mathematical properties of theMoebius transformations, which have been extensivelystudied [15],[12],[16]. In this Appendix some results use-ful for the analysis of beam transfer lines as proposed inthis paper are presented and commented.
1. The matrix representation of the Moebiustransformation
A general Moebius transformation, w = az + bcz + d with ad − bc = 0 (A.1)may be represented by a matrix: H = (cid:20) a bc d (cid:21) (A.2)and the composition of Moebius transformations corre-sponds to the matrix multiplication. In our case, takinginto account Eq. (20), the matrix H will be given by themore restricted form: H = (cid:20) S ′ − jC ′ jS C (cid:21) (A.3)which somewhat resembles the transport matrix for asingle particle. However, it should be noted here thatthis case deals with a complex number representing thewhole phase space ellipse of the beam. As | H | = 1, theMoebius transformation is normalized.The matrix representation becomes particularly clearif the complex numbers, w and z are expressed by homo-geneous coordinates. In this case, the Moebius transformmay be written as: w w = a z z + bc z z + d = az + bz cz + dz (A.4)Thus, since it is possible to represent a complex num-ber, z by its column vector of homogeneous coordinates, z , the transformation will be: w = M z (A.5)
2. The fixed points of the transformation
The fixed points are left invariant by the transforma-tion. These points can be obtained by imposing the con-diton that that the complex point is not changed by thetransformation: Y = aY + bcY + d (A.6)Alternatively, the fixed points are defined by the eigen-vectors of Eq. (A.5). The solution for the particular caseof the beam transport is Eq. (A.7): Y ± = j ( C − S ′ ) ± p − ( C + S ′ ) S (A.7)There are three possibilities, if | C + S ′ | <
2, Eq. (A.7)will have two complex solutions symmetrical with respectto the imaginary axis, if | C + S ′ | = 2, there will beonly one double solution in the imaginary axis and ifif | C + S ′ | >
2, there will be two solutions contained inthe imaginary axis.
3. The circle preserving properties
One interesting possibility that opens when consider-ing a beam line transformation as a complex plane trans-formation, is that entire regions of the complex planemay be transformed as conformal mappings [17]. For in-stance, one of the properties of the Moebius transformis that generalized circles are transformed to generalizedcircles. It is possible then to find a set of initial condi-tions, envelope them within a circle and transform thecircle along the beam line. All the initial conditions willremain inside the transformed circle. Because it is pos-sible to analyze the evolution of the radius of the circlealong the transformed planes, it is possible for instanceto know if the solutions converge or not and at whichspeed.Next we will proceed as in [18]. In order to provethe circle preserving property, first the equation of thecircle in the complex plane will be expressed in a moregeneral way. A circle of radius ρ and center at γ , may beexpressed as: | z − γ | = ρ (A.8)or: ( z − γ ) ( z − γ ) = ρ (A.9) zz − zγ − zγ + (cid:0) γγ − ρ (cid:1) = 0 (A.10)Multiplying Eq. (A.10) by an arbitrary factor A , andwriting it in matrix form, it results: (cid:2) z (cid:3) (cid:20) A − Aγ − Aγ A (cid:0) γγ − ρ (cid:1)(cid:21) (cid:20) z (cid:21) = (cid:2) z (cid:3) (cid:20) A BC D (cid:21) (cid:20) z (cid:21) = 0(A.11)In this way, the circle will be represented by the Her-mitian matrix C = [ AB | CD ], and Eq. (A.11) describingthe circle will be compactly expressed as: z H C z = 0 (A.12)where the superindex H represents the transpose con-jugate of a matrix. It can be seen that by construction C must be Hermitian, as the diagonal elements are real andthe non-diagonal elements are conjugate of each other.The arbitrary factor A has been introduced in order toinclude the straight lines as a particular case of the cir-cles. In the projective plane, a line may be considered asa circle with a point at infinite. With the representationof Eq. (A.12), all circles and lines of the complex planemay be represented as the quadratic form of a Hermitianmatrix with respect to the homogeneous coordinates ofthe complex plane.The determinant of the circle matrix C is equal to − Aρ , and it is called the discriminant of the circle. Realcircles will have a negative discriminant. The discrimi-nant will be zero if C represents a line or a zero radiuscircle. A positive discriminant is due to a circle of imagi-nary radius, which cannot be represented in the ordinarycomplex plane.In order to check how the Moebius transform changesa given circle, consider a circle defined at the start of thetransformation by: z H C z = 0 (A.13)If W is the reverse transformation, i.e. the one causing: z = W ω (A.14)then the circle will be transformed in the target planeto: ω H W H C W ω = 0 (A.15)The matrix inside the quadratic form of Eq. (A.15) isHermitian as well, and will represent a new circle in thetransformed plane. C = W H C W (A.16)so it is readily established that the Moebius transfor-mation maps circles into circles in the complex plane.0 [1] J.C. Maxwell. Electricity and Magnetism , volume I Chap-ter XII. Claredon Press, Oxford, 1881.[2] G.B. Walker. Congruent space charge flow.
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