Transverse to Longitudinal Emittance Exchange *

A scheme is proposed to exchange the transverse and longitudinal emittances of an electron bunch. A general analysis is presented and a specific beamline is used as an example where the emittance exchange is achieved by placing a transverse deflecting mode radio-frequency cavity in a magnetic chicane. In addition to reducing the transverse emittance, the bunch length is also simultaneously compressed. The scheme has the potential to introduce an added flexibility to the control of electron beams and to provide some contingency for the achievement of emittance and peak-current goals in free-electron lasers.


Introduction
The main challenge of the Linac Coherent Light Source [1] and other free-electron lasers (FEL) that are currently planned or under design, remains the achievement of a bright electron beam in the transverse plane.Although the FEL also constrains the longitudinal emittance, it appears to be more easily obtainable than that of the transverse plane.In fact, the predictions are that the incoherent momentum spread originating from the photo-injector is too small to effectively damp the micro-bunching instability induced by coherent synchrotron radiation (CSR) [2,3,4].A motivation therefore exists to reduce the transverse emittance and increase the longitudinal, since this may lead to SASE (self-amplified spontaneous emission) lasing in a shorter undulator length and simultaneously less CSR micro-bunching in the compressor.
We show that, under certain conditions, a transfer of emittance from the transverse to the longitudinal plane (or the reverse) is possible and not impractical.Our implementation uses a radiofrequency cavity in a dispersive region of a four dipole-magnet chicane.The cavity operates in the dipole mode, having a longitudinal electric field with gradient such that its strength varies linearly with transverse distance from the axis.A time dependent magnetic deflecting field is also present.A complete emittance analysis is presented and a specific example is studied.

The Dipole-Mode Cavity
Occasionally, an application arises of an RF cavity operating in a dipole mode, where the longitudinal electric field varies linearly with transverse distance from the axis.The earliest mention of such cavities, to the authors' knowledge, appeared in Ref. [5].The hope of using such cavities to change the damping of the three modes of oscillation of particles in an electron circular accelerator was dashed by Robinson's famous paper [6] that shows that the partition numbers cannot be changed with an RF field.A discussion of the physical mechanism of this general principle as it applies to a dipole mode cavity was presented in Ref. [7].Cylindrical cavities operating in the TM 210 mode (thus with a quadratic dependence of the longitudinal electric field on the distance from the axis) to couple the longitudinal and transverse motion to enhance laser cooling of ions in a storage ring [8], or to establish a correlation between betatron amplitude and momentum deviation to condition an FEL electron beam [9], have also been proposed.For the system under consideration we use a rectangular cavity having a longitudinal electric field which varies linearly with transverse distance, x, from the axis, as shown in Fig. 1.
In the neighborhood of the axis (x << a) we have an accelerating field for an electron crossing the cavity at time t, ( ) where z is the longitudinal axis of the reference trajectory, x the horizontal axis, y the vertical, ω the frequency of the cavity oscillations, and a is a constant characteristic of the cavity dimensions.The peak field is E 0 = V 0 /l, where V 0 is the peak RF voltage and l the cavity length.The vertical motion is neglected in this analysis, since, to first order, no force of the cavity acts in the vertical plane.
The associated magnetic field is obtained from Maxwell's equation: The small relative energy change, δ (≡ ∆γ /γ << 1), of an electron traversing the cavity at a distance x from the axis is ( ) where E is the nominal electron energy.We phase the cavity such that the center of the bunch (the reference particle) passes through the cavity at time t = 0, when the electric field gradient is at its maximum and the magnetic field passes through zero.We consider a bunch length that is much smaller than the RF wavelength (i.e., |ω t| << 1).Thus, to first order, 0 0 The horizontal deflection angle due to the vertical magnetic field of the cavity is and z is the longitudinal distance from the center of this ultra-relativistic bunch.

Emittance Exchange
We now analyze the emittance exchange concept and return to the cavity implementation later.The following is a general 4-dimensional linear beam transport analysis [10] in the x-z plane (or x-y plane).The initial uncoupled 4×4 beam covariance matrix, V 0 , can be written as [11] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 where α x,z, β x,z , and γ x,z (≡ {1+α x,z 2 }/β x,z ) are the beam envelope functions, and ε x 0 and ε z 0 are the initial uncoupled beam emittances in the horizontal and longitudinal planes.The rms beam sizes (horizontal and longitudinal) are related to the respective rms emittances by the relations 0 0 , The bunch 'chirp', or linear energy slope along the bunch length, is related to the longitudinal parameters by with the total rms relative energy spread, σ δ , given by 0 2 2 2 (1 ) Here σ δ u and σ δ c are the time-uncorrelated and time-correlated relative energy spread components, respectively, which add in quadrature.The normalized longitudinal emittance is with γ (= E/mc 2 ) the beam energy in units of electron rest mass.In the simple case, with no timecorrelated energy spread (i.e., 〈zδ 〉 = 0), the longitudinal emittance is ( ) where σ E is the absolute rms energy spread.Now propagate the beam through a 4×4 beamline transfer matrix, R, starting from an initial beam V 0 , with R T as the transpose of the matrix R.
Since R is symplectic and therefore det(R) ≡ |R| = 1, the 4-D emittance (= ε x 0 ε z 0 ) of V 0 is unchanged by R. The 4×4 matrix R is constructed from four 2×2 blocks, A, B, C, and D [12], as and it follows from ( 12) that the new beam, after beamline R, is with V x and V z the 2×2 block matrices of the x and z planes as shown in (6).The squares of the projected rms x and z emittances are the determinants of the 2×2 on-diagonal blocks.
We recall that the determinant of the sum of 2×2 matrices can be expressed using the trace (tr) as where X a is the adjoint of X (used here to avoid inverting A, B, C, or D which may be singular).
Applying the above matrix property, (16) becomes where |A|, |B|, |C|, and |D|, are the determinants of the 2×2 blocks of the net transfer matrix R.
Using an alternate form for the initial uncoupled beam, V x and V z , .
We now use the symplectic condition with S [13] as the 4×4 form of J, and find the following relations between the sub-matrix determinants 1, , The U and V matrices of ( 22) are shown to be related by using ( ) and from the off-diagonal 2×2 block of (23), C T JD = −A T JB, so that ( ) and therefore which is simply the sum of the squares of the normalized coupling block of the transfer matrix, and is positive.
The emittances at the exit of the beamline are now related to the emittances at the start of the beamline (subscript "0") by ( ) ( ) From (30), if ε x 0 = ε z 0 , then ε x = ε z .That is, equal initial uncoupled emittances will always remain equal through a symplectic map.Additionally, if λ 2 is insignificant, which it can be, then setting |A| = 0 will produce a complete x-to z-plane emittance exchange.Note that λ 2 ≠ 0 unless all A ij = 0, or the trivial case of no coupling at all, where all B ij = C ij = 0.

An Emittance Exchanger Beamline
We then apply this derivation to the chicane and dipole-mode cavity system shown in Fig. 2. A magnetic chicane sets up a dispersive region at its center, where the cavity is located.The chicane, of full length L, is made of four bending magnets and no quadrupole magnets.

Schematic diagram of the chicane and transverse cavity.
From ( 4) and ( 5), the transfer matrix of the 'thin-lens' cavity is which is similar to a thin-lens skew quadrupole transfer matrix, but in x, x′, z, δ space, rather than x, x′, y, y′ space.(The effects of a thick-lens are discussed in section 6.)The transfer matrix across the entire chicane is where R 1 and R 2 are the transfer matrices of the first and second section of the chicane, respectively (see Fig. 2), and ξ is the momentum compaction (ξ ≡ R 56 ) of the full chicane (ξ > 0 for chosen coordinates with bunch head at z > 0).
The matrix of the full chicane and cavity system is The expression for λ 2 has four terms and is quite long and awkward, even for this system.

( )( )
Thus the emittances at the end of the chicane can be exchanged up to a cross term which is related to the rms divergence, σ x′ , and energy spread, σ δ , of the initial beam, or It should be recalled that the parameters, β x,z , α x,z , ε x 0 ,z 0 , σ x′ , and σ δ , all describe the beam at entrance to the chicane.As demonstrated in the example below, the cross-term coefficient, λ 2 , can be made insignificantly small for reasonable beam parameters allowing almost complete emittance exchange.

Numerical Example
For an example, we take for the four-dipole chicane shown in Fig. 2 with an X-band RF deflecting cavity (ω /2π ≈ 11.4 GHz. a ≈ 1.3 cm) at its center.The beamline and beam parameters at the start of the chicane are listed in Table 1.Here we use ε x 0 > ε z 0 , which is a required condition for the reduction of the transverse emittance, and one which may not be trivially realized.With these parameters used in (38), we have and have completely exchanged the emittance levels.These results are verified with the computer tracking code TURTLE [14] up to 2 nd -order.The tracking output is shown in Fig. 3.The system described here, with k = 1/η, leaves the x and z planes insignificantly correlated (i.e., 〈xz〉 ≈ 0, 〈xδ 〉 ≈ 0, 〈x′z〉 ≈ 0, 〈x′δ 〉 ≈ 0).Note that the bunch length is also compressed by a factor of five at chicane exit (σ z = 100 µm → 19 µm), which is a very desirable feature for an FEL requiring a high peak current.The final bunch length, σ z f , and energy spread, σ δ f , for k = 1/η, and The energy spread has also increased to 0.24%, a level that is sensitive to the choice of η (= 1/k) and also β x at chicane entrance.In addition, the final β x and α x functions are greatly magnified by the transverse deflecting field (in this case: β x = 2.6 m → 520 m, α x = 0 → −400).In this example the initial energy-time correlation, α z , was set to zero.In fact a reasonable tolerance on this condition is acceptable.If the initial energy spread is ~3-times larger due to a linear time-correlation (α z ≈ 2.6), the final horizontal emittance is increased by ~10% in this case, as given by (38).A non-linear initial energy-time correlation, such as induced by space-charge forces or longitudinal wakefields prior to the chicane, will generate a non-linear position-angle correlation in transverse phase space after the chicane.The longitudinal and transverse emittances, ε x 0 and ε z 0 , should therefore be considered as projected emittances, which may be increased by nonlinear correlations with their conjugate variables.This presents a practical limitation for the exchange process, where the initial beam may need to be cleaned of aberrations prior to emittance exchange.
Finally, the exchanger beamline has some strange properties, which may be surprising on first observation.For example, betatron centroid oscillations initiated prior to the chicane will nearly disappear after the chicane (when scaled to local beam size), instead generating energy and timing shifts to the electron bunch (〈x 0 〉 = 1⋅σ x → 〈z〉 ≈ 1⋅σ z ).On the other hand, bunch arrival time variations upstream of the chicane will not change the bunch arrival time after the chicane, instead generating betatron oscillations in the horizontal plane.This may be an advantage over standard compressors since it effectively absorbs electron gun-timing variations and keeps them from becoming final bunch length and final energy jitter.This behavior is evident in (34) with ηk = 1.

Thick-Lens and Second-Order Effects
Second-order optical aberrations can become significant, due mostly to the second-order dispersion from cavity to end of chicane, if the final energy spread becomes too large.This can be controlled by decreasing the initial beta function, β x , or increasing the chicane dispersion, η (which reduces the cavity voltage).The relative emittance increase above the linear calculation of (38), which is due to second-order dispersion, is approximately given by 0 2 0 2 2 1 2 where σ z and β x are the initial beam parameters at chicane entrance.In the above case, secondorder aberrations are insignificant, but a choice of η = 50 mm (rather than 100 mm) and β x = 10 m (rather than 2.6 m) causes a factor of three final horizontal emittance increase above the linear expectations of (38), and a final rms energy spread of 0.8% (rather than 0.2%).This has been verified using TURTLE tracking.The values for β x and η should be chosen carefully with (42) as a guide.
The emittance exchanger beamline described above uses a thin-lens model of a transverse deflecting RF cavity to demonstrate the concept.Of course, the cavity will have some length, especially to produce many mega-volts.A modification of (31), (34), (37), and (38) is necessary to include this.The matrix of the thick-lens transverse cavity is where l is the cavity length.Using this in (32), and continuing with the case ηk = 1, produces a modified R with A and B blocks which are then used in (36) to calculate λ 2 . ( With l = 0 this reduces to (37), but otherwise can be a much more significant limitation in the emittance exchange.If this is now minimized with respect to α z , it becomes ( )( ) at a value of α z (related by ( 8) to the initial energy-time chirp in the bunch), which is given by 2 The expression for λ 2 in (45) will reduce to that of (37) with α z = 0 if k 2 lξ/24 << 1.For an X-band cavity with ~50 MV/m, the level of 20 MV is achieved with l = 0.4 m.From Table 1, the values of k and ξ give k 2 lξ/24 ≈ 0.03.Therefore, the emittance exchanger for a thick-lens works almost exactly like the thin-lens as long as α z (i.e., the incoming energy chirp) is given by (46) (i.e., α z ≈ 9.4 in this case).From (9), this means an initial correlated energy spread of 0.03%.
Particle tracking is repeated in Fig. 4 with a thick-lens cavity (l = 0.4 m) and α z set according to (46).The initial energy chirp is evident in the upper right plot.The emittances are again completely exchanged in this more general, and more realistic case.The emittance exchange relations of (38) are now modified for the thick-lens case (l ≠ 0), and are given (with ηk = 1) by, ( )( ) ( )( ) In (47), the initial energy chirp, α z , is not a free parameter and must follow (46).Otherwise (44) must be substituted into (30) for the more general case.Note in (30) that if ηk ≠ 1 and k ≠ 0, both 'projected' emittances can simultaneously increase to very large levels, both much larger than the largest initial emittance.This is because the beams become highly coupled in this case and the single-plane projected emittances do not reflect the intrinsic beam emittances, but are simply the quantities measured in those particular planes.The full 4D phase space volume is, of course, preserved since always |R| = 1.

Applications
We have shown that, under appropriate conditions, it is possible to transfer the transverse emittance into the longitudinal plane, and the reverse.In a practical design, two systems might be used, one chicane-cavity system to reduce the horizontal emittance, and the other might be a similar concept, but using skew quadrupoles rather than the chicane-cavity system, to produce equal x and y emittances by exchanging some of the larger ε y into the smaller ε x .Two chicanecavity systems, the second rotated by 90°, will not work because the first one increases the zemittance above the transverse goal, and therefore inhibits the next y-emittance exchange.
The advantages of the emittance transfer scheme proposed here are a reduced dependence on the photoinjector to meet the transverse emittance goals, thus adding a considerable safety margin to the design.The chicane also compresses the bunch by acting on the amplified betatron oscillations of the RF-cavity, thereby adding another useful function to the scheme.The compression takes place entirely in the last bend.Coherent synchrotron radiation or longitudinal wakefields in the first two bends may present a severe limitation if the energy spread is increased significantly.Vacuum chamber shielding or low charge levels may be necessary depending on bend magnet and beam parameters.An additional bonus is that the reduction of the transverse emittance is accompanied by an increase of the local energy spread, a desirable requisite for the control of the CSR microbunching instability.Finally, the system allows a degree of control over bunch length, energy spread, and emittance and may add to the flexibility in manipulating electron beam parameters.

Figure 1 .
Figure 1.Electric field (top-left) in a dipole-mode cavity at synchronous time (t = 0), and the magnetic field (top-right) one-quarter oscillation later.Longitudinal electric and vertical magnetic fields around t = 0 (bottom).

Figure 3 .
Figure 3. Initial (top) and final (bottom) phase space tracking plots.The horizontal and longitudinal emittances are completely exchanged, as predicted by (38).

Figure 4 .
Figure 4. Initial (top) and final (bottom) phase space tracking plots with thick-lens cavity and α z set according to (46).The emittances are still completely exchanged.

Table 1 .
Beam and system parameters as an example for emittance exchange.