Modification of relativistic beam fields under the influence of external conducting and ferromagnetic flat boundaries

We derive analytical expressions for external fields of a relativistic bunch of charged particles with a circular and an elliptical cross section under different boundary conditions and interaction of the fields with an accelerator structural elements. The particle density in the bunch is assumed to be uniform as well as non-uniform. At distances far apart from the bunch, in free space the field reduces to the relativistic modified Coulomb form for a pointlike charge and at small distances the expressions reproduce the external fields of a continuous beam. In an ultra-relativistic limit the longitudinal components of the internal and external electric fields of the bunch are strongly suppressed by the Lorentz factor. If the bunch is surrounded by conducting surfaces, the bunch self-fields are modified. Image fields generated by a bunch between two parallel conducting plates are studied in detail. Exact summation of the electric, $E_y$, and magnetic, $B_x$, image field components allows the infinite series to be represented in terms of elementary trigonometric functions. The new expressions for modified fields are applied to study image forces acting on the bunch constituents and the bunch as a whole. The coherent and incoherent tune shifts for an arbitrary bunch displacement from the midplane are calculated in the framework of an improved linear theory, for both infinite and finite parallel flat surfaces. Moreover, the developed method allows us to generalize the Laslett image coefficients $\epsilon_1$, $\epsilon_2$, $\xi_1$, $\xi_2$ to the case of an arbitrary bunch offset and reveal relationships between these coefficients. Appendix C provides a brief historical background of the development of the method of electrical images.

We derive analytical expressions for external fields of a relativistic bunch of charged particles with a circular and an elliptical cross section under different boundary conditions and interaction of the fields with an accelerator structural elements. The particle density in the bunch is assumed to be uniform as well as non-uniform. At distances far apart from the bunch, in free space the field reduces to the relativistic modified Coulomb form for a pointlike charge and at small distances the expressions reproduce the external fields of a continuous beam. In an ultra-relativistic limit the longitudinal components of the internal and external electric fields of the bunch are strongly suppressed by the Lorentz factor. If the bunch is surrounded by conducting surfaces, the bunch self-fields are modified. Image fields generated by a bunch between two parallel conducting plates are studied in detail. Exact summation of the electric, E y , and magnetic, B x , image field components allows the infinite series to be represented in terms of elementary trigonometric functions. The new expressions for modified fields are applied to study image forces acting on the bunch constituents and the bunch as a whole. The coherent and incoherent tune shifts for an arbitrary bunch displacement from the midplane are calculated in the framework of an improved linear theory, for both infinite and finite parallel flat surfaces. Moreover, the developed method allows us to generalize the Laslett image coefficients ǫ 1 , ǫ 2 , ξ 1 , ξ 2 to the case of an arbitrary bunch offset and reveal relationships between these coefficients.
Appendix C provides a brief historical background of the development of the method of electrical images.

Introduction
In an accelerator, the charged beam is influenced by the environment matter (a beam pipe, accelerator gaps, magnets, collimators, etc.), and a high-intensity bunch of particles induces surface charges or currents into this environment. This modifies the electric and magnetic fields around the bunch. There is a relatively simple method to account for the effect of the environment by introducing image charges and currents. The mathematical technique of electrical images was developed by W. Thomson (Lord Kelvin) [1][2][3]. The method of images has found application in various branches of physics, in particular, in hydrodynamics [4][5][6]. Appendix C provides a brief historical background on the subject 1 .
Over fifty years ago Laslett [7,8] analyzed the influence of the transverse space-charge phenomena, due to image forces, on the instability of the coherent transverse motion of an intense beam. Methods of image field summation are described in his paper [7], which presented some field coefficients calculated for infinite parallel plate vacuum chambers, magnet poles, and vacuum chambers with elliptical cross sections and variable aspect ratios. The resulting image fields were calculated only in the linear approximation and depend linearly on the deviationsȳ and y of the bunch center and the position of a test particle, respectively, from the axis (see Fig. 4). They act therefore like a quadrupole causing a coherent tune shift. The approximation used is incorrect if the field observation point y is located far from the bunch or if the bunch centerȳ is close to a conducting wall.
In the present paper we consider this classical problem summation of fields of images once again for a very simple geometry, namely, an ultra-relativistic bunch moving between infinitely wide parallel perfectly conducting plates. The problem is far from being purely academic. In applications, in particular, in the study of dynamics of photoelectrons in the beam transport system [9] and the electron cloud effect intensified by electron field emission in the flat collimator [10], it is important to know the distribution of electromagnetic fields not only in the vicinity of the bunch, but in the whole collimator gap. We have not found publications with attempts to sum up the series (34) (see Sect. 4) in an approximation beyond the linear one. In Sects. 4 and 5 we present exact 1D solutions of the problem for electric and magnetic image fields. The preliminary results were presented in Ref. [11]. 1D solutions are canonically used in the calculation of tune shifts 2 .
Before we solve the problem formulated above, in Sect. 2 we first derive expressions for the external electric and magnetic fields generated by a cylindrical and an elliptical bunch of charged particles. The task is specified as follows.
The external radial electric field E r and azimuthal magnetic induction B φ for a round unbunched relativistic beam of radius a and a uniform charge density are described by [12][13][14] where κ = 1/4πǫ 0 , λ is the linear beam density, q is the charge, β = v/c is a normalized velocity of the beam constituents and c the velocity of light 3 . In many applications, (1) and (2) are used to describe fields of an individual bunch too. However, in the form (1), (2) the bunch fields do not depend on the bunch energy and at large distances do not follow the Coulomb asymptotic. This contrasts sharply with the fields produced (at t = 0) by a rapidly moving single charge q where θ is the angle that the vector r makes with the z-axis. Along the direction of motion the electric field becomes weaker in γ 2 times, while in the transverse direction the electric field is enhanced by the factor γ: Here, γ denotes the particle Lorentz factor. This paper is organized as follows. In the next section, we demonstrate how, when summing up the elementary electromagnetic fields generated by charged relativistic particles, effective external beam fields are formed. We derive expressions for the transverse and longitudinal components of the bunch electric field, where the defects indicated above are rectified, and find the conditions at which the bunch fields are represented by Eqs. (1) and (2). Here we consider bunches shaped as a cylinder with a circular and an elliptical cross section. In Sect. 3 we discuss fields generated by a bunch with an arbitrary linear particle density and make a statement that in the ultra-relativistic limit γ → ∞ the electric field takes a universal form.
Sections 4 and 5 are devoted to the problem of finding exact analytic expressions for the electric and magnet fields generated by a bunch moving between infinitely wide parallel conducting plates and magnet poles.
In Sect. 6 we discuss image forces acting on the bunch constituents and the bunch as a whole and calculate in the framework of an improved linear approximation the coherent and incoherent tune shifts for an arbitrary bunch displacement from the midplane. In Sect 6.3, a practical example is considered as gradients of image fields in a finite-size collimator affecting the betatron frequency of the beam. Conclusions are drawn in Sect. 7. Here we also compare results obtained by different authors with the use of various techniques. Detailed derivations of the obtained results are placed in Appendixes A and B.

Self-fields of a charged cylinder with an elliptical cross section
Let us consider a bunch of charged particles uniformly distributed with a density ρ within a cylinder of length L and an elliptical cross section. The ellipsoid semi-axes in the x − y plane are a and b and the coordinate z-axis is along the bunch axis. Suppose that the bunch is moving along the z-axis with a relativistic velocity v = c β.
To compute the radial electric field of such a rapidly moving bunch, we have to sum up fields of the type (3), generated by the bunch constituents. In this way we get [16] E ⊥ (r, ξ, z) = κργ zI 1 + (L − z)I 2 (5) with 3/29 where σ is the distance in the x − y plane from the z-axis to the elementary charged volume and where e = √ 1 − b 2 /a 2 is the eccentricity of an ellipse and a > b. Equation (5) represents the radial electric field at instant t = 0 as observed at a distance r from the bunch axis, at an angle ξ relative to the x-axis and at a distance z from the bunch tail.
In Ref. [16] integrals I 1 and I 2 were estimated only numerically, because the integrands were taken as they are. However, the integrands are easy to simplify if the bunch is ultrarelativistic, γ ≫ 1, and we would now like to calculate the field in vicinity of the bunch but at distances much larger than the bunch radius, r ≫ a.
To simplify this, we make use of the notations and the integrand of I 1 can be written as Now we expand the above expression in a power series by using A as a small parameter and keeping only terms up to the power A 4 at each step.

Finite circular cylinder with a uniform particle density
For a bunch shaped as a circular cylinder, a = b, we may set ξ = 0. Due to the fact that 2π 0 cos 2k+1 φ dφ = 0, all odd powers of B vanish after integration over φ. This greatly simplifies the series generated from Eq. (9). After lengthy algebraic manipulations with Eq. (9), we get Substituting this expression in Eq. (6), one gets By changing z 2 to (L − z) 2 in (12), we obtain for I 2 the following result where Notice that for particles uniformly distributed in the bunch volume, ρ = qN b /πa 2 n b L = qλ/πa 2 , where N b is the total number of particles in the beam, n b the number of bunches, and λ the linear particle density. Substituting Eqs. (12)−(13) in (5), finally we arrive at This equation describes the transverse component of the electric field produced by a rapidly moving circular bunch. For γ ≫ 1 the correction factors C 1 and C 2 can be 4/29  Table 1).
neglected. In that case at the bunch surface, r = a, Eq. (14) exactly matchs the equation for the internal field [16]. Therefore, the condition r ≫ a used to derive Eq. (14) can be weakened and Eq. (14) is valid in the region r ≥ a.
The field of a relativistic bunch described by Eq. (14) has different behavior at distances far away from the bunch and for r < Lγ/2. Figure 1(a) shows the radial field profile as follows from (14). The parameters of the bunch correspond to the nominal scenario of the LHC proton beam [17]. At very large distances, r ≫ Lγ/2, (14) reduces to the Coulomb form (4) with q replaced by qN b /n b . Calculations show that for a proton bunch at 7 TeV the Coulomb law is restored only at a distance of several kilometers from the bunch. On the other hand, for r ≪ Lγ/2, Eq. (14) simplifies to a form independent of the z-coordinate, which coincides with the external field Eq. (1) of a continuous beam with λ = N b /n b L.
The magnitude of the electric field varies drastically in the head and tail parts of the bunch. For instance, in a very narrow transition region beyond the bunch tail, z < 0, |z| ≪ r/γ, the field strength decreases with z as follows: However, at larger |z|, the suppression of the radial field by the Lorentz factor becomes dominant, This is shown in Fig. 1(b). The field strength decreases to more than three orders of magnitude at the distance of 4 µm beyond the bunch. One finds from Eq. (14) that at z ≥ L the field magnitude tends to zero in a similar way.
The longitudinal part of the electric field reaches the maximum value on the bunch axis, so that on axis the longitudinal electric field is given by [16] At γ ≫ 1, the field magnitude outside the bunch is given by Equation (18) shows that the longitudinal field is independent of the bunch radius and strongly suppressed along the line of motion of the bunch.
The main result of this consideration is that due to features (16) and (18), the space-time distribution of the electric field around an ultra-relativistic circular bunch with a uniform particle density is well approximated by a step-like form: Similarly, we can show that the azimuthal magnetic induction of the bunch is

Finite elliptical cylinder with a uniform particle density
Let us now consider a bunch shaped as an elliptical cylinder. We make use of the same notations as in the previous subsection. In the ultra-relativistic scenario, the correction factors C 1 and C 2 should be neglected from the beginning. By expanding the integrand of I 1 in a power series as above, we get It can be proven that for an even function f n (φ) (22) where k = 0, 1, 2, . . . . In our consideration The integral (6) now can be solved with respect to σ and φ by direct substitution of Eq.
The condition (22) causes all even terms in A to vanish. Here,  For particles uniformly distributed in the elliptical bunch volume, ρ = qλ/πab with λ = N b /n b L the linear particle density. Substituting equations for I 1 and I 2 in Eq. (5), finally we arrive at 2 1 + 1 4 This equation describes the transverse component of the electric field produced by a rapidly moving elliptical bunch of length L. The transverse part is modulated by an angular factor that takes into account an ellipticity of the bunch. The dimensions of the ellipse enters only via the ratio ae/r. The arguments used in deriving Eq. (19) are also applicable here. The electric field of an ultra-relativistic elliptical bunch is therefore well approximated by a step-like form: The representation (27) with such "separation of variables" (r, ξ) is useful in an analytic calculation, including a differentiation and an integration. The number of terms, k max , in Eqs. (24), (27) to be taken into account depends on the required precision. The approximate formula (27) can now be contrasted with an exact expression of the electric field for a uniformly charged elliptical beam. There is a compact formula [18,19] in the complex (x, y) plane, z = x + iy, in the term of the "complex electric field": However, in the real components the formula is more complicated. Outside the beam, the x-component of the field is while the y-component can be obtained from this by exchanging x ↔ y and a ↔ b. Here with x = r cos ξ and y = r sin ξ.
We are now in a position to evaluate the number of terms in Eq. (27) that need to be taken into account in order to get the precision, say, to better than 5%, if compared with the exact formula (30). Figure 2(a) shows the variation of the ratio of Eq. (27) to Eq. (30) with the azimuthal angle ξ in the first quadrant. We observe that at r/ae = 1 (full curve) the desired precision is almost reached at k max = 7, except for the area near ξ = 0. In this area, it is necessary to account for terms with k max > 7 to achieve the required precision. At the same time, at r/ae = 2 and k max = 7 (dashed curve), the accuracy is better than 1% in the entire area of ξ.
The azimuthal field variation is essential only at r ∼ a. For instance, for a flat beam and at r = a, the field is concentrated at ξ = 0 and π. However, at larger r (say, r > 5a) the angular dependence vanishes rapidly and the electric field of an ultra-relativistic elliptical bunch converges to the universal form (19). The last statement is illustrated by Fig. 2

Universality of a bunch external fields
The uniform particle density considered in the last section is an idealization. In reality, the linear particle density, λ(z), varies considerably along the bunch. As an example, Fig. 3(a) shows the current profile of electron bunches in the XFEL accelerator [20]. Electrons of energy 17.5 GeV form bunches with a charge of 1 nC and a peak current of 5 kA. The current distribution is well fitted by a sum of two Gaussian distributions and a polynomial pedestal. Certainly, these bunches are ultra-relativistic since γ = 3.2 × 10 4 . Now the question arises how to calculate the electric field of the bunch for a given distribution of the current density J(z) = qβcλ(z).
We will now argue that in the asymptotic limit γ → ∞ the problem has a simple solution.
The main conclusion that we can draw from the previous consideration is that at a distance of several bunch radii, the electric field of the bunch is independent of the transverse geometry of the bunch. Without loss of generality let consider again a bunch with a circular cross section. The distribution J(z) can be well approximated by a histogram J i = qβcλ i , as shown in Fig. 3(b). Let us imagine that the bunch is a set of layers with a thickness of ∆z i . Suppose that in the transverse direction particles are distributed uniformly and the linear particle density varies from one layer to the next in accordance with 8/29 λ i . Thus, each layer is a cylinder with a uniform particle density acting as an independent field source. The complete field of the bunch is a sum of elementary field sources (19) Here we have used and the sum is replaced by an integration. We obtain the same result (31) even if consider a bunch with a variable cross section and a uniform transverse density in each slice ∆z i . We conclude with a statement that summarizes the obtained results: Theorem: In the ultra-relativistic limit, γ → ∞, the external electric field of a bunch with a linear particle density λ(z) is governed by the universal law It is instructive to compare the strength of the electric field created by a cylindrical bunch with a uniform particle density λ U = N/L and a bunch with the Gaussian particle distribution λ G (z). The ratio of the fields is Thus, if L is equal to the effective length of the Gaussian bunch L eff = √ 2πσ z , the field strength in the both cases are equal at the maximum of λ G (z). Note, however, that in a more general case, as in Fig. 3, that conclusion is not correct, even if L = L eff . For instance, with parameters of a XFEL bunch [20], one find L eff = 0.217 mm and at the maximum of the current density E XFEL /E U = 3.63.
So far, we have considered fields in free space. In an accelerator, the charged beam is influenced by an environment and a high-intensity bunch induces surface charges or currents into this environment. This modifies the electric and magnetic fields around the bunch. There is a relatively simple method to account for the effect of the environment by introducing image charges and currents.

Fields from image charges
"Definition of an electrical image. An electrical image is an electrified point or system of points on one side of a surface which would produce on the other side of that surface the same electrical action which the actual electrification of that surface really does produce" [21].
Following Laslett [7,8], we consider a relativistic bunch between infinitely wide conducting plates placed at y = h and y = −h. Suppose that the constituents of the bunch are positively charged. For full generality, let the bunch be displaced by (0,ȳ, 0) from the midplane (x, 0, z), and the observation point of the field be at (0, y, 0) between conducting parallel plates. The boundary condition for the electric field on perfectly conducting plates is E z (±h) = 0 and is satisfied if the image charges change sign from image to image.
The electric field seen by a particle at location y on the y-axis is generated by the direct source-charge λ 0 and the successive image charges λ ±i [14,22], as shown in Fig. 4. For instance, the image charges λ 1 and λ −1 are generated by λ 0 due to reflection in plates +h and −h, respectively. The image charges λ 2 and λ −2 are generated by λ −1 and λ 1 due to reflection in plates +h and −h, respectively, and so on. With the help of Fig. 4, one can easily calculate the distance between the image charge position and the observation point. So, for odd images, k = 1, 3, 5, ..., the distances between λ ±k and the point y are d ±k = 2kh ∓ y 1 . For even images, m = 2, 4, 6, ..., the distances between λ ±m and the point y are d ±m = 2mh ∓ y 2 . Here y 1 = y +ȳ and y 2 = y −ȳ.
Suppose that the distance between plates is of the order 10a. Thus, the electric field of each image is described by Eq. (32). To calculate the image electric field component E y,im (y) in front of the plate, we add the contributions from all image fields in the infinite series [7,8,13,22]: The representation (34) keeps the same form irrespective of the relative position of the bunch center and the observation point between plates, (ȳ ≥ 0, y ≥ȳ, y <ȳ) or (ȳ < 0, y ≤ȳ, y >ȳ). These image fields must be added to the direct field of the bunch (32) to meet the boundary condition that the electric field enters conducting surfaces perpendicularly.
In the original paper [7] (see also Refs. [13,22]), the series (34) was summed up only in the linear approximation in y andȳ: The electric field seen by a particle at location y on the y-axis is generated by the direct source-charge λ 0 atȳ and the successive image charges λ ±i at locations d ±k and d ±m (see explanation in the text).
The coefficient ǫ 1 = π 2 /48 is known as the Laslett coefficient (or form factor) for infinite parallel plate vacuum chambers. The approximation (35) widespread in textbooks and lectures, is, however, incorrect if the deviation of the bunch center from the axis is large (ȳ ∼ h) or if the field observation point y is located far off the bunch. Therefore, below we present the exact solution of the problem. In Appendix A it is proven that the exact summation of the series (34) gives where the electric image field structure function Λ depends only on scaled variables δ = y/h,δ =ȳ/h in the form We have shown in Appendix A that the truncated linear approximation (36) recovers the part (35) derived by Laslett.
We shall now calculate values of Λ(δ,δ) at several particular points along the y-axis.

11/29
Equation (38) is singular atδ → 1 and shows that the conducting plate attracts the bunch with a force increasing with the bunch displacement from the midplane. The phenomenon, involving the transverse movement of the bunch as a whole, arises from image forces and could lead to a transverse instability of the beam. This is discussed further in Sect. 6. δ = 0, δ = 1: the bunch is in the midplane and the observation point is at the plate y = h The image field (36) must be added to the direct field of the bunch (32) to meet the boundary condition at conducting surfaces. It is interesting to note that the last term in Eq. (37) is opposite in sign to the direct field contribution outside the bunch and cancels it. As a result, the electric field distribution between parallel conducting plates is given by In particular, a bunch moving in the midplane generates the field described by In other words, in the presence of conducting plates the electric field in front of the plate is enhanced by the factor π/2 (Fig. 5). If now we do not assume that the bunch offsetδ is small then the full linear approximation in δ can be derived by means of Eqs. (A.8) and (A.14) from Appendix A. Thus, the 12/29 vertical component of the electric field seen by a test particle in the vicinity of the bunch (|δ −δ| ≪ 1 ) is given by Here we have introduced a generalization of the Laslett electric image coefficient ǫ 1 in the case of an arbitrary offset: This approximation has to be compared with an alternative representation of Eq. (40) in the form (A.8) to show the origin of each term in Eq. (42). The potential function of the field (44) is with E y,tot = −∂U/∂y. In the linear approximation one can obtain the horizontal component of the electric image field directly from with the use of Eqs. (A.14) and (36). Thus, with δ x = x/h, δ y = y/h andδ y =ȳ/h. As follows from (35) and (36), the image fields produce defocusing forces in the y-direction. On the other hand, due to (47), the corresponding forces in the x-direction produce focusing forces. Equations (42) and (44) tell us that with an increase ofδ y , the field strength near the bunch and the field gradient across the bunch, ∂E y,im /∂y ∼ 1/ cos 2 ( π 2δ y ), significantly increase. This is illustrated by Fig. 5, which shows that with an increase ofδ y the field distribution between plates becomes more and more asymmetric. At the opposite ends of the bunch diameter the difference in the value of the field, ∆E y,tot (δ y ), grows as the displacement increases. For the LHC beam one get ∆E y,tot (0.1)= 4360 V/cm, ∆E y,tot (0.5) = 27290 V/cm, and for the ILC beam ∆E y,tot (0.1)= 13640 V/cm and ∆E y,tot (0.5)= 85340 V/cm, respectively. In Sect. 6 we discuss how this effect modifies the tune shifts.

Magnetic images
In the above, we have discussed electric image fields created by an ultra-relativistic bunch. Magnetic images can be treated in much the same way [13], [22]. Let the ferromagnetic boundaries be represented by a pair of infinitely wide parallel plates at y = +g and y = −g. The magnetic field lines must enter the magnet pole faces perpendicularly. For magnetic image fields we distinguish between DC and AC image fields. The DC field penetrates the metallic vacuum chamber and reaches the ferromagnetic poles. In case of bunched beams the AC fields are of rather high frequency, and we assume that they do not penetrate the thick metallic vacuum chamber. The DC Fourier component of a bunched beam current is equal to twice the average beam current J = qcβλB [13], where B is the Laslett bunching factor.
A magnetic field, seen by a particle at location y on the y-axis, is generated by the successive image currents with the same sign as the beam itself. In Appendix B it is proven that the resulting field is described by Here we have made the replacement µ 0 = 1/(ǫ 0 c 2 ) and used the scaled variables η y = y/g andη y =ȳ/g, B = n b L/2πR is the bunching factor, n b the number of bunches, R the average accelerator radius. The structure function H is of the form In the functional sense, H(η y ,η y ) = Λ(η y , η y ), as can be noticed by comparing Eqs. (50) and (37).
In the linear approximation in y andȳ one obtains from the exact formula (50) (see Appendix B for details) where ǫ 2 = π 2 /24 is the Laslett form factor for infinite parallel plate magnet poles 4 . As above for electric images, we define a generalized form of ǫ 2 for an arbitrary offsetη y as follows ǫ 2 (η y ) = π 2 32 1 cos 2 ( π 2η y ) Thus, the complete linear approximations in η x and η y (see Appendix B) are given by and For further applications, we point out that on the bunch axis, η y =η y , from Eq. (B.5) one gets The contribution of the magnetic AC image field due to eddy currents in vacuum chamber walls is similar to electric image fields, and therefore where the factor (1 − B) accounts for the subtraction of the DC component. Thus, at δ x = 0 the net AC field is tangential to the surface. The magnetic image fields must be added to the direct magnetic field (2) to meet the boundary condition at ferromagnetic surfaces. That is, the summary horizontal component of the magnetic field between the conducting plates is The step function θ(1 − δ y ) accounts for the fact that the AC fields do not penetrate the thick metallic vacuum chamber. For a more detailed discussion of the subject see Refs. [23,24].

Image forces and tune shifts
Direct space-charge fields, as well as fields due to image charges and currents shift the betatron frequencies (tunes). We have to distinguish between coherent tune shifts, which express a change of the betatron frequency when the bunch oscillates as a whole, and incoherent tune shifts, which change the single particle tune. In this section we again assume the bunch to have a circular cross section of radius a and a uniform density.
In the next two subsections we restrict our analysis to an idealized case of a vacuum chamber and/or ferromagnetic poles as two infinite parallel plates and the motion of a bunch in the vertical y-direction 5 . Infinite parallel plates are a good approximation for finite-width collimator parallel plates, given the small transverse beam size in modern accelerators. Our equations for tune shifts are valid at an arbitrary bunch offset and presented in notations as given in the textbook [14] and the handbook [25]. This allows us to compare results obtained by different authors. In Sect. 6.3 we consider a more realistic example of a finite-length collimator with parallel conducting jaws.

Coherent motion and tune shift
The motion of the bunch centerδ(s) in the absence of an external focusing force is described by the equation d 2δ (s) where s = βct and the Lorentz force is of the form Here we have applied the results of the previous section; M b = Nm p is the bunch mass, Q b = Nq the bunch charge. On the bunch axis, the electric and magnetic structure Under the influence of the force (60) the bunch is attracted by a conducting plate and at some point s i hits the plate. The actual position of the impact point depends on the initial value constraints, in particular, the bunch offsetδ(0) =δ 0 . Figure 6(a) shows the numerical solutions of Eq. (59) with a set of initial conditions (δ 0 ,δ ′ 0 = 0) atδ 0 = 0.1, 0.5, 0.8 . For example, let protons in the bunch be at the energy 7 TeV and the machine parameters are as in Table 1 [26]. Then the impact points s i (δ 0 ) are located at distances s i (0.1) = 453.3 m, s i (0.5) = 164.2 m and s i (0.8) = 61.0 m, respectively.
It should be noted that Eq. (59) is true in the approximation, when the bunch velocity along the z-axis is constant, and much larger than the drift velocity in the y direction, v z ≃ βc ≫ v y . In this case, the bunch path is a smooth curve, as shown in Fig. 6(a). However, with a more rigorous account of the effect of crossed electric and magnetic fields from the images, it is necessary to solve together a system of two coupled differential equations for movements along the z and y directions. In this case, the bunch trajectory is a cycloid-like curve.
The coherent motion of the bunch in the y-direction is significantly altered in the presence of the linear focusing provided by quadrupoles and described by the equation Here 16/29 where r p = κq 2 /m p c 2 is the classical proton radius and the meaning of the other parameters is explained in Table 1. With the values of the parameters 6 from Table 1, K 2 0 = 2.041 × 10 −4 m −2 and I = 1.73 × 10 −5 m −2 . Thus, for smallδ 0 the linear focusing is a driving force. Figure 6(b) shows numerical solutions of Eq. (62) for the same initial conditions as above. The dashed line shows a solution of the betatron equation (62) in the absence of the image effects, I = 0. A comparison of the two curves atδ 0 = 0.8 demonstrates how big the influence of images on the coherent tune shift is.
To derive an analytical expression for the coherent tune shift for an arbitrary offset we proceed in the standard way [14,22,23]. In the linear theory, we assume that the forces are proportional to the displacement. Therefore, we expand the structure functions Λ and H (61) in a power series in the neighborhood ofδ 0 ,δ =δ 0 + ∆, keeping only terms up to first order in ∆: Here we have introduced generalized Laslett coherent tune shift form factors, for the image fields from the vacuum chamber and the magnet pole. Substituting expressions for Λ(δ 0 , ∆) and H(η 0 , ∆) in (62), we get 7 ∆ν (coh) y One must note that the image coefficients ǫ 1 and ξ 1 , as well as ǫ 2 and ξ 2 , are not independent but are rooted in the same function Λ(δ,δ) and H(η,η), correspondingly. Therefore, in the linear approximation these functions are related via (67)

Incoherent tune shifts
Let us now evaluate the effect of the image forces on the betatron oscillation of the particles in a bunch. The motion of a test particle in a displaced bunch in the presence of the space-charge force and the image fields is described by the equation Here F y,sc = 2κq 2 λy/a 2 γ 2 is the Lorentz force due to the bunch space-charge [13] and F y,im is defined in (60). Inserting the linear approximations (A.14) and (B.7) into Eq. (68), in the The above analysis can be carried out similarly for the x-motion. The result is Equations (66), (69), and (70) generalize the Laslett tune shifts to the case of the arbitrary bunch offset between parallel conducting plates and ferromagnetic poles.
As numerical examples, with machine and beam parameters from Table 1, let us compare the contribution of each term in Eq. (69) at two distinct values ofδ,δ = 0 and Thus, the third term gives the main contribution that increases withδ, and the transverse particle dynamics in a bunch is defined by the influence of electric images.

Collimator of a finite length
The representation of a vacuum chamber and magnetic poles in the form of infinite parallel plates is a very useful mathematical abstraction. However, in real accelerators, all components are finite in size and at the same time some of these components include elements that are structurally designed as parallel conductive and ferromagnetic flat surfaces. In circular accelerators such as LHC [26] and the future HL-LHC [27], flat parallel surfaces are parts of different types of collimators, the normal conducting separator and orbit correction dipole magnets 8 . As a rule, collimator jaws have a length of 600-1400 mm, and their width is about 50 mm. Similarly, the poles of a dipole magnet have a length of 2000-3400 mm and a pole width of 60 mm [27]. With transverse beam sizes as small as 200 µm, the representation of collimators and dipole magnets in the form of infinite parallel plates is a good approximation for these elements and it is legitimate to apply here the results obtained in the previous sections.
Suppose that the accelerator ring only includes one collimator at point s 0 . A collimator is a short straight section of an accelerator and does not have a guide magnetic field. The motion of the bunch centerδ in the collimator is therefore described by Eq. (59), with F y,im from Eq. (60), where we set g = ∞. In the rest part of a circular accelerator the motion of the bunch is described by Eq. (62) with I = 0. Now we would like to derive in the framework of linear theory an analytical expression for the coherent tune shift due to the image effects in a finite-size collimator. For more than a single collimator one would simply add the individual contribution from each collimator to find the total tune shift.
The image fields E y,im and B x,im,AC act along the same line, so we introduce the effective field B e f f and decompose it into dipole-like and quadrupole-like parts with the use of Eqs. (36), (56) and (63), where Image fields act as a perturbation on the betatron oscillation of the bunch. Thus, the equation of motion of the perturbation ∆(s) of the transverse coordinate of the bunch with the offsetδ =δ 0 + ∆ takes the form with As the analysis in Ref. [13], Sect. 12.1.1 shows, dipole terms of the type K D cause a shift in the beam path without affecting the focusing properties of the beam line. In contract, terms that depend linearly on the transverse bunch offset from the orbit will affect focusing and the stability of the transverse motion of the beam, because these perturbations act like quadrupoles.
The longitudinal size of the collimator l c is small in comparison with the wavelength of betatron oscillations, so we consider the collimator as a point source of perturbation. Let us assume that such a perturbation at point (s 0 ,δ 0 ) is created by a thin lens of the focusing strength σ im = g * l c [13,28]. Here g = qG eff /hm p cβγ, with G eff from Eq. (72). The evolution of a bunch through elements of an accelerator is most clearly described by the product of transfer matrices [13,28,29]. The transfer matrix for a full revolution is M = M p M 0 , where for the lens and for the unperturbed part of the ring Here ψ 0 = 2πν 0 is the unperturbed phase advance per turn andβ 0 is the unperturbed betatron function at the location of the perturbation, s = s 0 . On the other hand, the perturbed transfer matrix M can also be written in the form (76) if we replaceβ 0 byβ and ν 0 by ν, where ν is the the betatron tune in the presence of the field gradient, G eff , from images. By equating the traces of the perturbed matrix and the transfer matrix for a full revolution, Tr(M) = Tr(M p M 0 ), with setting ν = ν 0 + ∆ν and |∆ν| ≪ ν 0 /2π, we get the tune shift due to this perturbation, 20/29 The total tune shift is the sum of the individual contribution from each collimator where n c is the number of collimators along the accelerator circumference.
As stated in Ref. [26], p. 100, theβ-functions at the collimators range from 27 m to 360 m and the range of collimator half-gaps h is 4.7 to 11.1 mm for the injection optics and 1.2 to 3.8 mm for the squeezed optics. On evaluating with Eq. (77) the numerical values of ∆ν c (δ 0 ), we set these values ofβ 0 , h = 1.2 mm, l c = 1 m and the machine and beam parameters from Table 1. The results are presented in Fig. 7. Figure 7 shows that atδ 0 < 0.6 the total tune shift due to the quadrupole-type image fields in collimators, ∼ n c · |∆ν im (δ 0 )|, n c ∼ 100, does not exceed the safe value of the order of 0.1. However, if the beam offset accidentally exceedsδ 0 ≈ 0.6 ÷ 0.7, it is possible to develop transverse instability and shift the betatron frequency into the resonance region.

Summary
This paper presents new analytical expressions for electric and magnetic self-fields produced by a bunch shaped as a cylinder with a circular and an elliptical cross section. Calculations are done in the relativistic limit. These expressions show the correct Coulomb asymptotic and in the near-field zone coincide with the external self-fields of a continuous beam. In the ultra-relativistic limit, external fields of a bunch takes the universal form (32) and (20).
We reanalyzed the problem summation of image fields generated by a charged bunch between infinitely wide parallel conducting plates and/or ferromagnetic poles. The exact 1D solutions for resulting electric and magnetic image fields are represented by the structure functions Λ(δ,δ) and H(η,η), respectively.
The new expressions for modified fields are applied to study the coherent and incoherent tune shifts for both infinite and finite parallel flat surfaces and allow within an improved linear approximation generalization of the Laslett image coefficients to the case of an arbitrary bunch offsetδ. These image coefficient functions, ǫ 1 (δ) and ξ 1 (δ), as well as ǫ 2 (η) and ξ 2 (η), are not now independent but are rooted in the functions Λ(δ,δ) and H(η,η), correspondingly. Equation (77) and Fig. 7 allow us to evaluate how the gradient of image fields in a finite-size collimator affects the betatron frequency of the beam. In particular, if the beam offset in collimators accidentally exceedsδ 0 ≈ 0.6 ÷ 0.7, it is possible to develop transverse instability and shift the betatron frequency into the resonance region.
After a first version of the present paper became public [30] the author learned 9 about an old textbook [31] and article [32], where only the scalar potential function of the electric field generated by a line charge between parallel earthed conducting planes is calculated with the use of conformal mapping. In addition, in [32] are calculated incoherent and coherent image coefficients. The results of Ref. [32] for the image coefficients and tune shifts were rederived in Ref. [33] and several misprints are corrected. This allows us to directly compare the results obtained by different methods.
The two parallel infinite plates is a particular case of a rectangular vacuum chamber when the width to height w/h of the rectangle goes to infinity. In Ref. [33] this limit was considered only for a centered beam,δ = 0. The electric and magnetic image coefficients as well as the tune shift coefficients are of the same values as ǫ 1,2 (0) and ξ 1,2 (0), above. The equations for the vertical and horizontal, incoherent and coherent betatron tune shifts [33], after some rearrangements and neglecting the neutralization factor, take the same form as Eqs. (66), (69) and (70) atδ =η = 0.
Tune shifts and Laslett coefficients for a rectangular and a circular beam pipe were also calculated in Refs. [24,34]. Due to the fact that the radial and vertical betatron oscillations are coupled in general, the tune shift and image coefficients accordingly form second-rank tensors. We can state a guess that the relations of type (67) are somehow connected with this fact; see Eq. (15) in Ref. [34]. However, in Ref. [34] explicit formulas have been given only for a square vacuum chamber, w = h; therefore the passage to the limit of parallel plates is impossible.
To conclude, the results presented here can serve as a starting point in applying the method of images to find exact solutions in 2D problems, calculating the field emission current in a collimator and a study of beam-induced multipacting. All these problems are relevant for the High-Limunosity Large Hadron Collider under construction [35].
Using only the linear terms, we recover the part derived by Laslett [7] (see Eq. (35)) An inspection of Eq. (A.11) shows that the contributions of negatively charged images are enhanced by the factor 2 2n − 1, as compared with the contributions from the positively charged images. Equation (A.11) also shows that for y on the bunch axis, i.e. δ 2 = 0, the contributions from the positively charged images vanish. At first glance, Eqs. (A.8) or (A.9) are singular at δ 2 = 0 or δ =δ, respectively. However, this is not the case. Starting once again from Eq. (A.11) with δ 2 = 0 and taking Eq. (A.10) into account, we formally get Knowing the exact form of Λ(δ,δ), one is able to develop a variety of approximation. For instance, to study particle dynamics in a bunch with a significant offset, one needs to decompose Eq. (A.8) assuming δ 2 ≪ 1. The result is (A.14)

B. Magnetic image fields
Let the boundary of magnet pole faces be represented as two parallel plates located at y = ±g. A magnetic field, seen by a particle at location y on the y-axis, is generated by the successive image currents with the same sign as the beam itself [14,22]. Therefore, instead of the series (A.1) we get where Π k and Π m are of the same functional form as Eqs. (A.3) and (A.4) with an interchange of variables h → g, δ 1 → η 1 = y 1 /g, δ 2 → η 2 = y 2 /g. Indexes k and m have odd, k = 1,3,5,..., and even, m = 2,4,6,... values, respectively. Now, if we are to proceed in the same manner as in Appendix A, we obtain from Eq. (B.2) an expression for the structure function of the image magnetic fields For |y| ≪ g and |ȳ| ≪ g, this gives, to first order in η 1 and η 2 The use of Eq. (A.10) in Eq. (B.3) gives the magnetic structure function that we are looking for Now, if we recall the relationships η 1 = (y +ȳ)/g = η +η and η 2 = (y −ȳ)/h = η −η, we obtain With the same reasoning as in Appendix A, the full linear approximation in η 2 is given by where ǫ 2 (η) is defined in Eq. (52).

C. "The principle of electric images"
Throughout his long and fruitful scientific life, William Thomson (later Lord Kelvin) was in active correspondence with his father, the well known mathematician James Thomson as well as with leading scientists in Great Britain and Europe. The topics of the letters were both personal and discussion of the latest scientific news and emerging new ideas.
In addition, at the insistence of his father, W. Thomson kept diaries in mathematics and physics. These letters and diaries have been preserved and published [37,38], allowing us to trace the roots of ideas and the time of their origin. As follows from the mathematical diary of 21-year-old W. Thomson, the story of "the principle of electric images" began in 1845, during his four-month stay in Paris. This trip abroad had several objectives. First, to improve the physical condition and also personally get acquainted with the leading French mathematicians (Liouville, Cauchy, Sturm), physicists (Arago, Biot, Pouillet), chemists (Regnault) and attend their lectures at theÉcole Polytechnique and the Collège de France [38]. Before Thomson had been a month in Paris he sent to Liouville's Journal de Mathématiques (vol. X. p. 137, 1845) a paper, entitled "Démonstration d'un théorème d'analyse", which two years later he expanded in the Cambridge and Dublin Mathematical Journal (vol. II. p. 109, March 1847) under the title: "On certain definite integrals suggested by problems in the theory of electricity". This article allowed Thomson to demonstrate his skills as a mathematician and identified his area of interest as a natural philosopher.
During long conversations and discussions of the results of Faraday's experiments and problems in the development of the mathematical theory of electricity, J. Liouville engaged W. Thomson to write a memoir for the Institute, with his vision of their solution [38]. Common interests in mathematics and physics made friends Thomson  two spheres. I think I can work it out for i spheres ... The image of an exterior point, in a conducting sphere, is a p. in the interior, with opposite electr." In a footnote added in 1872 to one of his 1848 articles, Thomson describes this period as follows (Ref. [3], p. 52): "... A complete exposition of the principle of electrical images (of which a short account was read at the late meeting of the British Association at Oxford) has not yet been published; but an outline of it was communicated by me to M. Liouville in three letters, of which extracts are published in the Journal de Mathématiques (1845 and 1847, vols. X., XII.). A full and elegant exposition of the method indicated, together with some highly interesting applications to problems in geometry not contemplated by me, are given by M. Liouville himself, in an article written with reference to those letters, and published along with the last of them. I cannot neglect the present opportunity of expressing my thanks for the honour which has thus been conferred upon me by so distinguished a mathematician, as well as for the kind manner in which he received those communications, imperfect as they were, and for the favourable mention made of them in his own valuable memoir." Thomson  On June 20, 1847, Thomson writes from Peterhouse to his father about the approaching meeting of the British Association (Ref. [38], p. 204): "...I have been getting out various interesting pieces of work, along with Stokes, connected with some problems in electricity, fluid motion, etc., that I have been thinking on for years, and I am now seeing my way better than I could ever have done by myself, or with any other person than Stokes." In July 1847, Thomson and Stokes attended the meeting of the British Association at Oxford with talks entitled "On Electrical Images" (by W. Thomson) and "On the Resistance of a Fluid to two Oscillating Spheres" (by G. G. Stokes). One year later the Report of the Seventeenth Meeting of the British Association was published; however, the results of Thomson and Stokes were presented by short abstracts. Some extractions from these abstracts follow. W. Thomson [2]: "There is no branch of natural philosophy of which the elementary laws are more simple than those which regulate the distribution of electricity upon conducting bodies; yet its impracticability has always been the reproach of the mathematical theory of electricity. Very few of the varied and interesting problems which it presents have been made subjects for investigation, on account of the apparently extreme complexity of the conditions to be satisfied; and even when results have been forced from it by the analytical skill and energy of a Poisson, the physical interest has been almost lost in the struggle with mathematical difficulties, and the complexity of the solution has eluded that full interpretation without which the mind cannot be satisfied in any analytical operations having for their object the investigation or expression of truth in natural philosophy...
The subject of this communication is 'the principle of electrical images,' which is suggested by Green's elementary propositions, as the proper way of treating a great variety of problems that present themselves with reference to the distribution of electricity on spherical conductors. The effect of a body electrified in any given manner upon an uninsulated sphere is shown to be completely represented by what may be called "the image of the electrified body in the sphere," and a simple geometrical construction is given by which this image may be described. When an electrified body is placed in the neighbourhood of two uninsulated spheres, an inductive effect is produced which may be represented by an infinite series of 'successive images' in each sphere. An algebraic expression of this result leads to solutions, by means of converging series, of the various problems which occur with reference to the distribution of the induced electricity, and the attractions exerted by the two spheres. When a single conductor bounded by segments of two spherical surfaces cutting at an angle which is a submultiple of two right angles is electrified by the influence of a charged body, the effect may be represented by means of a finite number of images disposed in a symmetrical manner in the circumference of a circle passing through the exciting body, and cutting the two spherical surfaces at right angles. The principle of electrical images, as applied in these two cases, may be illustrated by a reference to the successive images of a candle placed between two parallel plane mirrors, and to the symmetrically arranged images which are seen in the kaleidoscope." G. G. Stokes [4]: "The object of this communication was to shew the application of Professor Thomson's method of images to the solution of certain problems in hydrodynamics...
The investigation mentioned in the preceding paper arose out of the communication to me by Sir William Thomson of his beautiful method of electrical images before he had published it. Having myself paid more attention to the motion of fluids than to electricity, I endeavoured to find if it would in any manner apply to the solution of problems in the motion of fluids. I found that what is called above a singular point of the second order 10 had a perfect image in a sphere when its axis was in the direction of a radius, which led to a complete solution of the problem mentioned in the paper when one sphere lay wholly outside or inside the other." Many years later, an extended version of this report was published in Ref. [40]. After this article, the method of images became one of the problem-solving techniques in hydrodynamics [5,6].
It should be noted that the technique of electrical images is described by Thomson himself fragmentary and scattered on the articles of different years. The most consistent presentation of this method with use of the idea of the potential and of equipotential surfaces is given in "A Treatise on Electricity and Magnetism" by Maxwell [21]. This is how Maxwell begins to introduce the reader to the principle of image charges: "... In applying this method to the most elementary case of a sphere under the influence of a single electrified point, we require to expand the potential due to the electrified point in a series of solid harmonics, and to determine a second series of solid harmonics which express the potential, due to the electrification of the sphere, in the space outside. It does not appear that any of these mathematicians observed that this second series expresses the potential due to an imaginary electrified point, which has no physical existence as an electrified point, but which may be called an electrical image, because the action of the surface on external points is the same as that which would be produced by the imaginary electrified point if the spherical surface were removed.
This discovery seems to have been reserved for Sir W. Thomson, who has developed it into a method of great power for the solution of electrical problems, and at the same time capable of being presented in an elementary geometrical form..." [21] Thus, the method of image charges and currents allows us to solve problems on the distribution of real charges and fields not only due to the conductive surface of the simplest geometric form, a plane, but also for conductors of more complex geometric shapes.