Abstract
We study an impedance due to coherent synchrotron radiation (CSR) generated by a short bunch of charged particles passing through a dipole magnet of finite length in a vacuum chamber of a given cross section. In our method we decompose the electromagnetic field of the beam over the eigenmodes of the toroidal chamber and derive a system of equations for the expansion coefficients in the series. The general method is further specialized for a toroidal vacuum chamber of a rectangular cross section where the eigenmodes can be computed analytically. We also develop a computer code that calculates the CSR impedance for a toroid of rectangular cross section. Numerical results obtained with the code are presented in the paper.
I. INTRODUCTION
Synchrotron radiation of a relativistic beam moving in a toroidal chamber with conducting walls has been extensively studied in the past (see, e.g., [1] [2] [3] ). The effect of the coherent radiation on beam dynamics is usually formulated in terms of a so-called coherent synchrotron radiation (CSR) impedance or wakefield. Many important features of the CSR impedance have been analyzed based on an exact solution of Maxwell's equations for rectangular cross section of the chamber. In the limit when the characteristic transverse size of the vacuum chamber L is much smaller than the toroid radius R, L ( R, this solution can be simplified, and the wakefield and impedance can be computed. A special case of a beam propagating between parallel conducting plates can be treated using a simpler method of image charges [4] . The results of the parallel plates model are formulated in a relatively simple analytical expressions, and are widely used in applications.
I. INTRODUCTION
One of the drawbacks of the previous approaches [1] [2] [3] [4] is that they are only applicable to a very specific geometry of the toroid, for which Maxwell's equations allow separation of variables. A different and a more general approach to the problem was proposed by the authors in Ref. [5] . It uses the smallness of the parameter ¼ ffiffiffiffiffiffiffiffiffi ffi L=R p to simplify Maxwell's equations, keeping only terms of the lowest order in . It turns out that in this approximation the transverse components of the electric field satisfy a so-called parabolic equation which was proposed more than 50 years ago in the diffraction theory [6] , and has been widely used since that time. It is also a standard approximation in the FEL theory [7] , and more recently was applied to the beam radiation problems in free space [8] and to the theory of high frequency impedance [9] . In a subsequent development of the method [10, 11] , Agoh and Yokoya added a source term to the parabolic equation due to the beam current and developed an algorithm for its numerical solution. Important practical results for various accelerators were obtained using the code in [12, 13] .
I. INTRODUCTION
In this paper we continue developing the approach of Ref. [5] and apply it to the case of the coherent synchrotron radiation of a beam that passes through a bend magnet of finite length. The geometry of the problem is shown in Fig. 1 : it includes a toroidal segment and two long straight sections connected to the toroidal one. The first, entrance straight section is connected to the toroidal one at point A, and the second, exit section is connected at point B. The cross section of every pipe is assumed to be the same. The length of the toroidal segment is equal to l as measured along the central line of the toroid between points A and B. This central line is assumed to be a plane curve. The coordinate along the axis of the toroid is s; it is chosen in such a way that the beam propagates in the positive direction. We also use x and y coordinates in directions transverse to s: the coordinate y is perpendicular to the plane of the central line, and the coordinate x is measured along the radius of the curvature of the central line. The origin of the coordinate system x, y is located on the central line of the pipe.
I. INTRODUCTION
We note that the wakefield of a point charge for the trajectory shown in Fig. 1 in free space, in the absence of metallic walls, was calculated in Ref. [14] .
I. INTRODUCTION
We assume that a bunch of ultrarelativistic charged particles travels through the bend with the velocity v close to the speed of light and use the approximation v ¼ c. We also assume that the bunch length z is much smaller than the characteristic transverse size L-in the opposite limit the frequencies associated with the bunch are below the cutoff, and the bunch does not radiate. Although not required in the general method, we make several additional simplifying assumptions: (a) the transverse size of the beam is small and can be neglected, and (b) the beam orbit coincides with the central line of the pipe. It follows also from these assumptions that the longitudinal distribution of the beam (relative to its central point) does not change with time-it is ''frozen'' by the requirements of v ¼ c and the zero transverse extension of the beam.
I. INTRODUCTION
Our approach to the general problem is split into several steps. In Sec. II we first present and analyze a parabolic equation that describes excitation of the electromagnetic field by moving charges inside of a toroidal section of the waveguide of radius R. Of course, this equation is also valid for a straight cylindrical pipe, where formally R ¼ 1. Eigenmodes in a toroidal waveguide and the general method of finding the electromagnetic field of the beam by means of expanding it over the modes are described in Sec. III. We then consider, in Sec. IV, the specific problem of a beam entering the toroid from a straight pipe, and, in Sec. V, a beam exiting from the toroid to the straight waveguide. General formulas obtained for the arbitrary shape of the cross section of the waveguides and arbitrary transverse profile of the beam density are then applied to the case of a rectangular cross section and a narrow beam in Sec. VI. Finally, in Sec. VII, we present numerical results for the longitudinal CSR wakefield for several different geometries of the toroid.
I. INTRODUCTION
Throughout the paper we assume a perfect conductivity of the walls.
II. PARABOLIC EQUATION FOR THE FIELD
In this section we present a parabolic equation for the electromagnetic field generated by a beam of charged particles in a toroidal waveguide with a given cross section. The detailed derivation of the equation can be found in [5, 10] ; here we only present the equation and discuss its conditions of validity.
II. PARABOLIC EQUATION FOR THE FIELD
We consider a section of a smooth toroidal vacuum chamber of radius R and arbitrary cross section as the one shown in Fig. 1 between points A and B. The parabolic equation is formulated for the electric field and the current in the curvilinear coordinate system x, y, and s shown in Fig. 1 A mathematical assumption that leads to the parabolic equation is a slow dependence of the functionsÊ ? andĵ s versus s, such that @=@s ( k. It means that we are interested in components of the field propagating in the positive direction of s at small angles to the axis of the toroid. In particular, we neglect a part of the field propagating in the negative direction of s. As was shown in Refs. [5, 10] using this approximation for the electric fieldÊ ? generated by the beam inside the toroid leads to the following parabolic equation:
II. PARABOLIC EQUATION FOR THE FIELD
where r ? ¼ ð@=@x; @=@yÞ. The longitudinal electric field can be expressed through the transverse one and the current
II. PARABOLIC EQUATION FOR THE FIELD
Boundary conditions for Eq. (2) are imposed on the metal wall by the requirement that the tangential components of the electric field vanish on the surface of the toroid. If n is a normal vector to the surface, we havê
II. PARABOLIC EQUATION FOR THE FIELD
Note that an axisymmetric version of Eq. (2) for a straight pipe (R ¼ 1) was used in [9] for a study of the longitudinal impedance in the limit of high frequencies.
III. EIGENMODES AND BEAM FIELD EXPANSION
Within the framework of the parabolic equation a toroidal waveguide possesses a set of eigenmodes which are special solutions of Eq. (2) 
III. EIGENMODES AND BEAM FIELD EXPANSION
with k mp ð!Þ ¼ q mp ð!Þ þ k. Note that the validity condition for the parabolic equation @=@s ( k requires jq mp j ( k, which means that the frequency of the mode is much larger than the cutoff frequency (at which k mp ¼ 0). The transverse structure of the mode and the eigenvalue q mp ð!Þ are determined from the homogeneous parabolic equation:
III. EIGENMODES AND BEAM FIELD EXPANSION
which is obtained by putting (5) into Eq. (2) and settinĝ j s ¼ 0. The longitudinal component of the field can be expressed through the perpendicular components using Eq. (3):
III. EIGENMODES AND BEAM FIELD EXPANSION
As we will see below in the analysis of eigenmodes of a rectangular waveguide [Eq. (59)], the parabolic equation correctly describes modes at frequencies much higher than their cutoff frequencies. Such modes make a dominant contribution to the impedance of a bunch with a length much shorter than the characteristic transverse size of the waveguide.
III. EIGENMODES AND BEAM FIELD EXPANSION
One can prove that the modes E mp;? compose a complete orthogonal set, which can always be normalized in such a way that the following orthogonality condition
III. EIGENMODES AND BEAM FIELD EXPANSION
holds, where the integration goes over the cross section of the toroid. In the general case, for a given shape of the toroid, Eq. (7) represents a two-dimensional problem which can be solved numerically. An example of such a solution for a round cross section of the toroid is given in Ref. [5] . Eigenmodes for the rectangular cross section can be found analytically [5] . We give a short derivation for this case in Sec. VI A.
III. EIGENMODES AND BEAM FIELD EXPANSION
Following a general method of Ref. [15] , we expand the perpendicular part of the electric fieldÊ ? generated by the currentĵ s into a serieŝ As discussed in the Introduction, we assume that the currentĵ s does not vary with s and integrate (11) over s from s ¼ 0 to the current value of s. The right side of (11) depends on s through the exponential factor e Àiq mp s which makes the integration trivial:
III. EIGENMODES AND BEAM FIELD EXPANSION
Using Eq. (3), we also obtain the longitudinal component of the field:
III. EIGENMODES AND BEAM FIELD EXPANSION
Equations (12) and (13) give a general solution for the longitudinal field generated by the beam inside the toroidal pipe. This solution depends on the initial values of the coefficients C mp ð0Þ which should be found from expansion of the beam field at the entrance to the toroidal segment over the toroidal eigenmodes. We will carry out this expansion in the next section.
IV. BEAM ENTERING A TOROIDAL SEGMENT FROM A STRAIGHT SECTION
When the beam enters the toroidal section at point A (corresponding to s ¼ 0) in Fig. 1 , its field at this point,
IV. BEAM ENTERING A TOROIDAL SEGMENT FROM A STRAIGHT SECTION
Eð0Þ, is equal to the equilibrium steady state Coulomb field that it carries in the long straight pipe. We will use the notation E ss ? for this steady state field. Because we assume that the beam is moving with the speed of light, the longitudinal component of E ss ? is equal to zero. It is therefore equal to zero at the entrance s ¼ 0 to the toroidal segment. The continuity of the electric field at s ¼ 0 allows us to determine the initial values C mp ð0Þ in Eq. (13) .
IV. BEAM ENTERING A TOROIDAL SEGMENT FROM A STRAIGHT SECTION
Indeed, putting simultaneouslyÊ s ¼ 0 and s ¼ 0 in Eq. (13) gives
IV. BEAM ENTERING A TOROIDAL SEGMENT FROM A STRAIGHT SECTION
Findingĵ s from this equation and substituting it back to Eq. (13) giveŝ
IV. BEAM ENTERING A TOROIDAL SEGMENT FROM A STRAIGHT SECTION
The perpendicular electric field of the beam at s ¼ 0, which we denoteÊ ? ð0Þ, being also continuous at this point can be expanded into the series over eigenmodes of the toroid asÊ
IV. BEAM ENTERING A TOROIDAL SEGMENT FROM A STRAIGHT SECTION
Using the orthogonality property (9), we find
IV. BEAM ENTERING A TOROIDAL SEGMENT FROM A STRAIGHT SECTION
Substituting C mp ð0Þ into Eq. (15) giveŝ
IV. BEAM ENTERING A TOROIDAL SEGMENT FROM A STRAIGHT SECTION
Making integration by parts in the first term on the right side and using Eq. (8) yields an alternative form of the last equation:
IV. BEAM ENTERING A TOROIDAL SEGMENT FROM A STRAIGHT SECTION
It is interesting to note here that the second term in Eq. (19) gives a solution to a different problem where a plane electromagnetic wave falls onto an open end of a toroidal pipe in the normal direction. In this case, the source current is zero,ĵ s ¼ 0, and the longitudinal projection of the incoming wave is also zero, E s ð0Þ ¼ 0, as has been assumed in the derivation of Eq. (19). Equation (19) then gives a distribution of the longitudinal field along s trapped inside the toroidal waveguide in the parabolic equation approximation.
IV. BEAM ENTERING A TOROIDAL SEGMENT FROM A STRAIGHT SECTION
As was pointed out above, from field continuity at s ¼ 0 we haveÊ 
IV. BEAM ENTERING A TOROIDAL SEGMENT FROM A STRAIGHT SECTION
As was mentioned at the beginning of this section, the parallel component E s of this field is zero so that Eq. (3) reduces to
IV. BEAM ENTERING A TOROIDAL SEGMENT FROM A STRAIGHT SECTION
Equations (21) and (22) are consistent with the perpendicular field being a potential one, i.e.,
IV. BEAM ENTERING A TOROIDAL SEGMENT FROM A STRAIGHT SECTION
with the potential c ¼ c ðx; yÞ obeying the Poisson equation
IV. BEAM ENTERING A TOROIDAL SEGMENT FROM A STRAIGHT SECTION
The boundary conditionÊ ss ? j w Â n ¼ 0 at the conducting wall of the waveguide means that c has a constant value on the wall, c ¼ const. Without loss of generality, the constant can be set to zero, so that Eq. (24) should be solved with the boundary condition
IV. BEAM ENTERING A TOROIDAL SEGMENT FROM A STRAIGHT SECTION
A general solution of the boundary value problem, Eqs. (24) and (25), for a rectangular cross section, in a form most suitable for our purposes, can be found in Ref. [16] . We will use this solution below in Sec. VI B.
IV. BEAM ENTERING A TOROIDAL SEGMENT FROM A STRAIGHT SECTION
Repeating the calculation that leads to Eq. (19) and using Eqs. (20) and (23) forÊ ? ð0Þ, one finds for the transverse component of the electric field inside the toroid
IV. BEAM ENTERING A TOROIDAL SEGMENT FROM A STRAIGHT SECTION
We will now consider what happens to the electromagnetic field beyond point B, at s > l, after the beam exits from the toroidal section. We assume that the straight exit pipe is long enough so that eventually the electromagnetic field of the beam reaches its steady state, the radiation that came out of the toroidal section gets behind the beam and the interaction between the radiation field and the beam ceases.
IV. BEAM ENTERING A TOROIDAL SEGMENT FROM A STRAIGHT SECTION
With minor modifications, we can repeat the derivation of Sec. II and find the longitudinal electric field of the beam after it exits from the toroid into the straight pipe. We now need to expand the field into a series of eigenmodes in the straight waveguide. They satisfy the same Eq. (7) with R ¼ 1:
IV. BEAM ENTERING A TOROIDAL SEGMENT FROM A STRAIGHT SECTION
where we use the tilde to distinguish various quantities in the straight pipe from their analogs in the toroid. Each mode has a longitudinal component of the field given by Eq. (8) . It is convenient to separate the steady state field given by Eq. (23) from the field in the straight waveguide and expand the difference over the eigenmodes:
IV. BEAM ENTERING A TOROIDAL SEGMENT FROM A STRAIGHT SECTION
Repeating the derivation of Sec. II, it is easy to find that
IV. BEAM ENTERING A TOROIDAL SEGMENT FROM A STRAIGHT SECTION
and henceC mp ðsÞ are equal to their initial values at the transition point s ¼ l,C mp ðsÞ ¼C mp ðlÞ. Equation (29) can be explained as follows. When the beam enters the straight pipe, it ceases to radiate electromagnetic waves, and the amplitudes of the waveguide eigenmodes do not change with distance s. They are determined by the electromagnetic field that comes out from the toroidal segment and is converted into the eigenmodes of the straight waveguide at the transition point s ¼ l.
IV. BEAM ENTERING A TOROIDAL SEGMENT FROM A STRAIGHT SECTION
To find the coefficientsC mp we use the continuity of the field at the transition point B and expand the differencê E ? ÀÊ ss ? ¼Ê ? À r ?ĉ at s ¼ l (that is at the exit from the toroid) over the eigenmodes of the straight waveguide,
IV. BEAM ENTERING A TOROIDAL SEGMENT FROM A STRAIGHT SECTION
Note that
IV. BEAM ENTERING A TOROIDAL SEGMENT FROM A STRAIGHT SECTION
where the integration goes over the cross section of the pipe. Using the orthogonality condition
IV. BEAM ENTERING A TOROIDAL SEGMENT FROM A STRAIGHT SECTION
for the modes of the straight waveguide, and also usinĝ
IV. BEAM ENTERING A TOROIDAL SEGMENT FROM A STRAIGHT SECTION
which follows from Eqs. (26) and (31) at s ¼ l, we find the coefficients
IV. BEAM ENTERING A TOROIDAL SEGMENT FROM A STRAIGHT SECTION
where
IV. BEAM ENTERING A TOROIDAL SEGMENT FROM A STRAIGHT SECTION
stands for the matrix elements in the expansion 
VI. RECTANGULAR CROSS SECTION OF THE VACUUM CHAMBER
We will now apply the general formalism developed in previous sections for an arbitrary geometry of the transverse cross section of the toroid to a particular case of the rectangular cross section. In this case, the parabolic equation can be solved using separation of variables and the eigenmodes are explicitly expressed in terms of the trigonometric and Airy functions.
VI. RECTANGULAR CROSS SECTION OF THE VACUUM CHAMBER
A
VI. RECTANGULAR CROSS SECTION OF THE VACUUM CHAMBER
In such a waveguide, the eigenmodes are separated into two sets. One set comprises the modes polarized along the x axis, and the other set along the y axis.
VI. RECTANGULAR CROSS SECTION OF THE VACUUM CHAMBER
It is convenient to introduce the dimensionless coordinate
VI. RECTANGULAR CROSS SECTION OF THE VACUUM CHAMBER
and
VI. RECTANGULAR CROSS SECTION OF THE VACUUM CHAMBER
Using the variable , the transverse electric field in these modes can be written as [5] 
VI. RECTANGULAR CROSS SECTION OF THE VACUUM CHAMBER
for the E x modes, and
VI. RECTANGULAR CROSS SECTION OF THE VACUUM CHAMBER
for the E y modes, where
VI. RECTANGULAR CROSS SECTION OF THE VACUUM CHAMBER
but different boundary conditions at ¼ AE a =2, namely,
VI. RECTANGULAR CROSS SECTION OF THE VACUUM CHAMBER
The prime in the equations above indicates differentiation with respect to . The notation Ä m in Eqs. (43) stands for
VI. RECTANGULAR CROSS SECTION OF THE VACUUM CHAMBER
where in the last equation we used the smallness q mp ( k.
VI. RECTANGULAR CROSS SECTION OF THE VACUUM CHAMBER
A general solution to Eqs. (43) and (44) involves the Airy functions Ai and Bi, see [5] , and can be written as 
VI. RECTANGULAR CROSS SECTION OF THE VACUUM CHAMBER
where b is defined in (41). We will use the notations q V mp and q U mp to distinguish between the eigenvalues of the functions V m and U m , respectively.
VI. RECTANGULAR CROSS SECTION OF THE VACUUM CHAMBER
Longitudinal fields for the eigenmodes can be found from Eq. (8); they are expressed as
VI. RECTANGULAR CROSS SECTION OF THE VACUUM CHAMBER
for the E x and E y modes, respectively; note that modes with either m ¼ 0 or p ¼ 0 have no longitudinal field. The right side of (50) can vanish at some particular frequencies leading to q mp ¼ 0 and k mp ¼ !=c. Such a frequency corresponds to a resonant mode that has a phase velocity equal to the speed of light and can be excited by an ultrarelativistic beam. Coherent synchrotron radiation due to excitation of such modes was studied in detail in Ref. [5] .
B. Steady state field in straight rectangular waveguide
As explained in Sec. IV, in addition to finding modes in the toroidal waveguides, we need to calculate the steady state field of the bunch in a straight waveguide. For a rectangular waveguide, this field can be found analytically from Eqs. (24) and (25). For our purposes, it is enough to consider only the case when the source current corresponds to a point charge q moving along the geometrical center of the waveguide, i.e., j s ¼ qcðxÞðyÞðs À ctÞ. The Fourier transformation (1) of the current givesĵ s ¼ qðxÞðyÞ:
B. Steady state field in straight rectangular waveguide
It is convenient to consider c in (24) as a function of variables and y and expand it in Fourier series over the variable y: 
B. Steady state field in straight rectangular waveguide
The electric field is then given bŷ 
C. Eigenmodes in straight rectangular waveguide
The eigenmodes of the straight rectangular waveguide in our approximation satisfy the parabolic equation (27) with the longitudinal electric field in the modes given bỹ
C. Eigenmodes in straight rectangular waveguide
(we remind the reader that we use tilde to indicate quantities that refer to a straight waveguide, with toroidal analogs not having the tilde sign). Introducing the functionsṼ m ðÞ andŨ m ðÞ that are related to the field componentsẼ mp;x andẼ mp;y through Eqs. (42), we find that 
C. Eigenmodes in straight rectangular waveguide
The modes with different polarization but the same numbers m and p have equal wave numbers
C. Eigenmodes in straight rectangular waveguide
Note that this expression can be obtained from the exact dispersion relation for eigenmodes in the rectangular waveguide in the limit when the mode frequency is much larger than the cutoff frequency. The quantity divẼ mp;? related to the longitudinal electric field in the mode is equal to
C. Eigenmodes in straight rectangular waveguide
for the E x modes and
C. Eigenmodes in straight rectangular waveguide
for the E y modes; note that the longitudinal field is absent if either m ¼ 0 or p ¼ 0.
C. Eigenmodes in straight rectangular waveguide
Finally, we also note that for large negative values of Ä m , such that jÄ m j ) 
D. Beam entering and exiting rectangular toroidal section
Using the results of the previous sections, we can calculate the longitudinal field (19) generated by a point charge after it enters a toroidal waveguide from a long straight section. These calculations are carried out in Appendix A, and the result is given by Eq. (A8). We reemphasize here that we use the Fourier representation and the right side of (A8) is a function of the frequency !. Note that at frequencies of resonant modes, that is when q U mp ¼ 0 or q V mp ¼ 0, the denominator of the right side of (A8) vanishes. This however does not lead to a singularity of the expression because the corresponding numerator 1 À e iq U mp s or 1 À e iq V mp s vanishes at the same time.
D. Beam entering and exiting rectangular toroidal section
If the toroidal part of the transition is long enough, as the beam propagates away from the entrance to the toroid and s increases, the electromagnetic field inside the toroid will be approaching its limiting steady state value. This steady state toroidal field can be formally obtained as a limit s ! 1 in Eq. (A8). It is derived in Appendix B and is given by Eq. (B11). On the axis ¼ y ¼ 0 it reduces to a relatively simple expression
D. Beam entering and exiting rectangular toroidal section
A similar expression has been derived in Ref. [5] [Eq. (B9) in Appendix B], however, it was incorrectly concluded that from this equation it follows that the field in front of the particle is equal to zero. Note that in contrast to (A8), Eq. (B11) is singular at frequencies of the resonant modes where one of the denominators on the right side vanishes. The terms sgnð!Þi0 in the denominators of (B11) establish the correct way to treat these singularities. According to Eq. (B7), the deltafunction parts of the singular expression contribute to the real part ofÊ s0 ð!Þ, which is an even function of the frequency !. The principal part of the singularity is responsible for the imaginary part ofÊ s0 ð!Þ, and is an odd function of frequency.
D. Beam entering and exiting rectangular toroidal section
After the beam exits the toroidal segment and enters the region s > l of the straight pipe, the electromagnetic field generated inside the toroid travels with the beam for some time. The group velocity of this field, however, is smaller than the speed of light, and eventually it starts to lag behind the beam. Calculation of this field is carried out in Appendix C.
D. Beam entering and exiting rectangular toroidal section
In Appendix D we obtain general expressions for the longitudinal field integrated through the toroidal segment and the exit pipe. This field gives the usual definition of the longitudinal CSR wake caused by the beam passage though the bend magnet.
VII. NUMERICAL RESULTS
To calculate the field generated by a bunch of particles traveling through a toroidal bend shown in Fig. 1 , we wrote a computer program using the Mathematica programming environment [17] . The program assumes a beam moving with the speed of light on the central line of the toroid x ¼ y ¼ 0 and neglects the transverse size of the bunch. The distribution function of the bunch is given by Nfðs À ctÞ, where N is the number of particles in the bunch, and f is normalized by unity. The bunch is characterized by the spectrumfð!Þ:
VII. NUMERICAL RESULTS
For a Gaussian bunch with the rms length z we havê fð!Þ ¼ expðÀ! 2 2 z =2c 2 Þ. The field of the bunch in the space-time domain is given by the inverse Fourier transform:
VII. NUMERICAL RESULTS
We express this field as a function of the variables z ¼ s À ct, which is a coordinate relative to the center of the bunch at a given time, and divide it by the total charge Nq; the resulting quantity is the longitudinal wake of the bunch at location s:
VII. NUMERICAL RESULTS
wðz; sÞ ¼ 1 q
VII. NUMERICAL RESULTS
The code calculates this wake on the beam orbit x ¼ y ¼ 0 for any given value of s. It can also compute the integrated wake
VII. NUMERICAL RESULTS
using Eq. (D5), where the integration takes into account both the wake in the toroid and in the exiting straight pipe. Note that the wake in the straight entrance pipe, where s < 0, is zero, which is reflected in the choice of the lower integration limit in (66). In addition to calculation of the field for the geometry shown in Fig. 1 , our code computes the fieldÊ s using Eq. (62). This steady state field would set up in a long toroidal section when the transition processes caused by the entrance to the toroid can be neglected. As was pointed out in Appendix B, the fieldÊ s has singularities at the frequencies corresponding to the toroidal modes that are resonant with the beam. A special numerical procedure is used to select singularities in (62) and to separately evaluate their contribution to the integral (65). We use this algorithm, in particular, for comparison with a model of two conducting parallel plates widely referred to for evaluation of the shielding effect of the conducting walls.
VII. NUMERICAL RESULTS
To illustrate the capabilities of the code we compute the CSR wake for the following numerical parameters: the rms bunch length z ¼ 0:5 mm and the orbit radius R ¼ 1 m. We vary the aspect ratio of the rectangle A ¼ a=b and the vertical dimension b. The plots below represent three Figs. 4(c) . Similarly, the lowest resonant mode for the toroid with A ¼ 0:5 and b ¼ 2 cm has a wavelength of 2.0 mm, in agreement with the oscillation period in Fig. 5(b) . Figures 6 and 7 show plots of the transient wakefields of the bunch inside the toroid for several distances s from the entrance to the toroid. Figure 6 (a) demonstrates a buildup of the wake with increasing distance from the entrance. Figure 6 (b) shows that even deep inside the toroid, at distances $90 cm, the wakefield behind the beam, z $ 5-10 mm, does not reach an equilibrium value and continues to evolve with increasing s.
VII. NUMERICAL RESULTS
We compared the process of buildup of the wakefield at the entrance to a toroidal segment for the case A ¼ 3 and b ¼ 4 cm with the wakefield at the entrance to a bend in free space (which mathematically corresponds to the limit a, b ! 1 in our method). The wakefield in free space was computed using equations from Ref. [14] , in the limit ! 1. The result of such a comparison is shown in Fig. 8 . The shielding effect in the toroid of this large aperture is relatively weak and this plot shows a rather good agreement with the free-space wakefield. The deviations from the free space case are observed in front of the bunch at larger distances (17 and 22.6 cm) from the entrance to the bend. Figure 9 shows the evolution of the wake after the beam exits from the toroidal section and travels through the straight pipe. Note how with increasing distance from the exit Ás, the wake in the vicinity of the bunch, jzj & 5 mm, gradually disappears, while the wake at large distance behind the bunch, z $ À15 mm, decreases with the distance rather slowly.
VII. NUMERICAL RESULTS
Finally, Fig. 10 shows an integrated wake (66) for three different lengths of the toroidal segments.
VIII. CONCLUSIONS
In this paper we presented a new method for calculation of the CSR wake in a vacuum chamber with perfectly conducting metallic walls. Our general method, in principle, is applicable to arbitrary cross section of the toroidal pipe. It, however, requires knowledge of the toroidal modes for this cross section, which in general can only be found numerically. Note, however, that finding the eigenmodes constitutes a two-dimensional computational problem, and can be relatively easily accomplished with existing numerical codes. Although we focused our analysis on calculation of the longitudinal CSR wakefield, the transverse component of the wake can also be computed in our method.
VIII. CONCLUSIONS
In the case of a rectangular cross section of the pipe, the toroidal eigenmodes can be calculated analytically and summation in the series representation of the electromagnetic field can be carried out. A numerical algorithm that calculates the CSR wake for such a cross section is implemented as a Mathematica computer code. The results of the numerical calculations with the code are presented and compared with the parallel plates model. 
ACKNOWLEDGMENTS


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