where Qx and Qy are the tunes which one would expect without coupling. A similar formula, with the same fractional values of QI,II , describes the coupled tunes in the horizontal plane. Equation (2.87) shows that the measured fractional parts of QI and QII are never exactly equal, but can only approach each other up to a distance |κ−|. Figure 2.30 illustrates this with an example from the PEP-II High Energy Ring (HER), which is consistent with our simple example in (2.79) and (2.80). A common technique for correcting the betatron coupling in a storage ring is to minimize the distance of closest approach using at least two skew quadrupole magnets. It is often the only correction necessary, especially if the tunes are close to the difference resonance Qx − Qy + q = 0.

Fig. 2.30. Closest tune approach in the PEP-II HER before final correction [73]. Shown are the measured fractional tunes as a function of the horizontal tune 'knob' (which would only change Qx if the accelerator were uncoupled), in dimensionless units. The minimum tune distance is equal to the driving term |κ−| of the difference resonance (Courtesy Y. Cai, 1998)

### 2.8.5 Compensating the Sum Resonance

In the vicinity of the difference resonance, while there is a continuous energy exchange between the two transverse planes, the beam or particle motion remains bounded. In contrast, close to the sum resonance for which |Qx + Qy + q| < |κ+|, the motion is unstable. The total width of the stop band around the sum resonance is equal to twice the driving term |κ+| of (2.81) [67]. In practice, while perhaps more difficult, than determining the driving term |κ−| for the difference resonance using the closest-tune approach, the driving term for the sum resonance |κ+| can be compensated by adjusting two skew quadrupoles so as to minimize the stop band width.

In order to optimize the vertical emittance in an electron damping ring, in general it is necessary to correct both driving terms, |κ−| and |κ+|. A minimum of six skew quadrupole correctors are needed to correct both coupling driving terms and the the vertical dispersion function since one sine-like and one cosine-like corrector are required for correction type of each aberration.

Even after perfect global coupling compensation, there can be a residual contribution of local coupling to the vertical emittance. Equation (2.83) suggests that local correction might be necessary. It is indeed employed at several light sources, the KEK ATF, and the B-factories.

### 2.8.6 Emittance near Difference Resonance

A third approach to determine the coupling parameter |κ−| is to measure the horizontal (or vertical) emittance as a function of the distance from the difference resonance [71]. Equation (2.83) does not apply close to the resonance. Instead, near the difference resonance, the horizontal emittance is described by [74]

$$
\epsilon\_x = \epsilon\_{x0} \frac{(\Delta Q\_-)^2 + |\kappa\_-|^2/2}{(\Delta Q\_-)^2 + |\kappa\_-|^2}, \tag{2.88}
$$

where x0 is the unperturbed horizontal emittance without coupling or far from the resonance. The tune difference ΔQ− = (Qx − Qy − n) function is related to the measured tune difference ΔQI,II = |QII − QI | via

$$
\Delta Q\_{-} = \sqrt{\Delta Q\_{I,II}^{2} - |\kappa\_{-}|^{2}}.\tag{2.89}
$$

After inserting this into (2.88), the latter becomes

$$
\epsilon\_x = \epsilon\_{x0} \left( 1 - \frac{|\kappa\_-|^2}{2\Delta Q\_{I,II}^2} \right) \,. \tag{2.90}
$$

A more recent and different derivation [75] describes the horizontal and vertical emittances near the difference resonance in the following modified form:

$$\epsilon\_x = \epsilon\_{x0} - (\epsilon\_{x0} - \epsilon\_{y0}) \frac{|\kappa\_-|^2/2}{(\Delta Q\_-)^2 + |\kappa\_-|^2 + \Delta Q\_- \sqrt{(\Delta Q\_-)^2 + |\kappa\_-|^2}}, \quad \text{(2.91)}$$

$$\epsilon\_y = \epsilon\_{y0} + (\epsilon\_{x0} - \epsilon\_{y0}) \frac{|\kappa\_-|^2/2}{(\Delta Q\_-)^2 + |\kappa\_-|^2 + \Delta Q\_- \sqrt{(\Delta Q\_-)^2 + |\kappa\_-|^2}}. \tag{2.92}$$

In an experiment at the KEK ATF, the horizontal emittance x was inferred from the spot size σx measured with the ATF interferometric monitor [76], using the formula x = (σ2 x−(Dxδ)2)/βx, where the design beta function (βx = 0.33 m) and dispersion (Dx = 4 cm) at the point of light emission, and the calculated energy spread of δ = 7.14 × 10−4 were assumed.

Performing a nonlinear fit to (2.90) of the emittance x obtained for various tune differences ΔQI,II , the driving term |κ−| can be extracted [71]. Figure 2.31 shows the measured horizontal emittance as a function of tune separation ΔQI,II . The result of the nonlinear fit is also depicted. The coupling strength of |κ−| ≈ 0.037 inferred from the fit [71] agrees well with the coupling strength |κ−| ≈ 0.042 obtained from a measurement of the closest tune approach under the same conditions [77].

Fig. 2.31. Horizontal emittance as a function of the tune separation ΔQI,II at the ATF Damping Ring; the measured data [77] as well as the result of a nonlinear fit to (2.90) are shown; the fitted coupling strength of |κ−| ≈ 0.037 is consistent with the value |κ−| ≈ 0.042 inferred from a simultaneous measurement of the closest tune approach [77, 71]

### 2.8.7 Emittance near Sum Resonance

The driving term of the sum resonance κ+ may also be inferred by measuring the emittance in the vicinity of this resonance [78]. However, we caution the reader that experimental experience with this scheme is scarce, and that there appear to be some uncertainties in the predictions by different theories and computer simulations.

Using the formulae in [74] and [79], the vertical emittance should depend on the measured distance to the sum resonance, ΔQI,II,+ ≡ |QI + QII − p| (where p is the integer which minimizes the expression) as [78]

$$
\epsilon\_y \approx \epsilon\_{x0} \frac{2|\kappa\_+|^2}{5|\kappa\_+|^2 + \Delta Q\_{I,II,+}^2},
\tag{2.93}
$$

while the horizontal emittance should vary as

$$
\epsilon\_x \approx \epsilon\_{x0} \frac{3|\kappa\_+|^2 + \Delta Q\_{I,II,+}^2}{5|\kappa\_+|^2 + \Delta Q\_{I,II,+}^2}. \tag{2.94}
$$

In deriving (2.93) and (2.94), we have used that the actual tune distance to the sum resonance, ΔQ+ = (Qx+Qy−n), is related to the measured distance ΔQI,II,+ = |QII + QI − p| via

$$
\Delta Q\_{+} = \sqrt{\Delta Q\_{I,II,+}^{2} + |\kappa\_{+}|^{2}}.\tag{2.95}
$$

### 2.8.8 Coupling Transfer Function

A different method of measuring the coupling is through the 'coupling transfer function' [80]. Here, the beam is excited horizontally, while detecting the resulting vertical coherent motion. Such a technique was used to continually monitor and correct the coupling strength during collisions in the CERN ISR [80].

### 2.8.9 Excursion: Flat Versus Round Beams

Frequently the question is asked why flat beams are used in electron-positron colliders. One answer is that the beams naturally become flat due to synchrotron radiation, as will be discussed in a later chapter. However, as we have seen, one could make these beams round, by operating close to the linear difference resonance Qx = (Qy + k), where k is an integer.

In general, whether round or flat beam collisions would give higher luminosity is not easy to answer. The luminosity is defined as the reaction rate divided by the cross section. Aside from the beam energy, it is the most important performance parameter of a high-energy collider. For a ring collider, assuming Gaussian beam distributions, the luminosity, typically measured in units of cm−2 s−1, can be expressed as (compare Chap. 1)

$$L = \frac{N\_b^2 n\_b f\_{\text{rev}}}{4\pi\sigma\_x\sigma\_y} \,, \tag{2.96}$$

where Nb is the number of particles per bunch, nb the number of bunches per beam, and frev the revolution frequency. The beam sizes at the interaction point (IP), σx,y, can be expressed in terms of the IP beta function and the transverse emittances as

$$
\sigma\_x = \sqrt{\beta\_x \epsilon\_x} \,, \tag{2.97}
$$

$$
\sigma\_y = \sqrt{\beta\_y \epsilon\_y} \,. \tag{2.98}
$$

The sum of the emittances may be held constant, equal to x0, while the ratio of the horizontal and vertical emittance can be varied via the betatron coupling according to

$$
\epsilon\_x = \frac{\epsilon\_{x0}}{1+\kappa},
\tag{2.99}
$$

$$
\epsilon\_y = \kappa \frac{\epsilon\_{x0}}{1+\kappa}.\tag{2.100}
$$

An important constraint of a collider is the beam-beam tune shift, i.e., the tune shift and tune spread due to the additional focusing at the collision point by the field of the opposing beam. The beam-beam tune shift is characterized by the parameters ξx,y,

$$
\xi\_{x,y} = \frac{N\_b r\_e}{2\pi\gamma} \frac{\beta\_x}{\sigma\_{x,y}(\sigma\_x + \sigma\_y)} \,, \tag{2.101}
$$

where re is the classical electron radius, and γ the relativistic factor. Best performance is achieved if the vertical and horizontal are both at the maximum possible value. If we assume that the maximum tune shift parameter is approximately equal for the two planes, ξx = ξy, it follows immediately that

$$\frac{\epsilon\_x}{\epsilon\_y} = \frac{\beta\_x}{\beta\_y} = \frac{1}{\kappa} \,. \tag{2.102}$$

Inserting this into (2.96), one can rewrite the luminosity as

$$L = \frac{1+\kappa}{2} \left. \gamma \xi\_{x,y} \right. \frac{1}{r\_e} \frac{N\_b f\_{\text{rev}} n\_b}{\beta\_y} \,. \tag{2.103}$$

This expression shows that round beams may give two times higher luminosity, thanks to the factor (1 + κ), but only if the horizontal beta function can be made as small as the vertical, βx = βy/κ = βy.

Recently a transformation was proposed [81] which can convert a round beam created in a solenoid field into a flat beam outside of the solenoid or vice versa. As discussed in Chap. 4, the transformation which achieves the conversion can be realized by a combination of skew and regular quadrupoles. As a practical example, using this scheme the round beam from an rf photoinjector can be transformed into a beam with a flat emittance ratio y/ x 1, which is more suitable for a linear collider [82]. A first experiment at the Fermilab A0 line has demonstrated the viability of this scheme [83] (see Chap. 5). Another application is the matching of a flat electron beam to a round proton beam, e.g., in an electron cooling device.

# Exercises

### 2.1 Schottky Signals

The longitudinal spectrum of a single particle, denoted by k, of an unbunched beam circulating in a storage ring, considering positive frequencies only, is given [84] by

$$i\_k(t) = ef\_{\text{rev}} + 2ef\_{\text{rev}} \sum\_{n=1}^{\infty} \cos(n\omega\_{\text{rev}}t + \phi) \,\,,\tag{2.104}$$

where n represents the turn number and ωrev = 2πfrev is the angular revolution frequency of this particle.

a) Sketch the current spectrum for this particle in both the time and frequency domains.

b) Show that summing (2.104) over N particles and averaging in time gives the dc beam current Idc = !N k=1ik " t = Nefrev, assuming that different particles (index k) have slightly different angular revolution frequencies ωrev,k and different initial phases φk, and that frev denotes the average revolution frequency.

c) The power spectrum one would measure with a spectrum analyzer is proportional to the rms current Irms = ( k ik)2t 1 2 . Show that

Irms = 2efrev#N 2 . (2.105)

### 2.2 Betatron Tunes

a) From Fig. 2.5, what was the synchrotron tune in LEP?

b) Suppose that when the quadrupole family for the horizontal tune is increased in strength, the horizontal tune line is observed to move to the right in similar measurements. What is the fractional horizontal betatron tune?

c) After how many turns would a particle with the tunes found in a) and b) return to exactly the same place in the longitudinal or horizontal phase space?

### 2.3 Application of Multipole Field Expansion

A useful formula for fields produced by a magnet of order n (n = 0 represents a dipole) is2

$$B\_y + \mathbf{i}B\_x = B\_0 \sum\_{n=0} (b\_n + \mathbf{i}a\_n)(x + \mathbf{i}y)^n,\tag{2.106}$$

where for upright, or 'normal' magnets bn = 0 and an = 0 while for magnets rotated by 45◦, so-called 'skew' magnets, bn = 0 and an = 0.

a) For the case of a upright, or 'normal', sextupole (b2 = 0, a2 = 0), show that the Lorentz force seen by the particle is

$$F\_x \propto (x^2 - y^2) \quad \text{and} \quad F\_y \propto xy \,. \tag{2.107}$$

b) For a nondispersive orbit (x = xco + xβ and y = yco + yβ, where the subscripts co and β refer to the closed orbit and to the betatron orbit), show that a beam off-axis in x experiences a normal quadrupole field while a beam off-axis in y experiences a skew quadrupole field.

c) What transverse-to-longitudinal coupling terms arise if the dispersive contribution to the orbit (xδ = Dxδ and yδ = Dyδ) are also included?

2 We note that (2.106) refers to the American convention, and that in Europe the exponent of (x+iy) is chosen as (n−1), so that a sextupole field would be given by b3 and not by b2.

### 2.4 Beta-Beat

Consider a quadrupole error. We split the quadrupole into two halves each of which is parametrized by the usual matrix for a quadrupole

$$R\_q = \begin{pmatrix} 1 & 0 \\ -1/(2f) & 1 \end{pmatrix} \,. \tag{2.108}$$

where 1/f ≡ ΔK in a thin-lens approximation, ΔK being the (integrated) quadrupole strength error. Let the one turn map (OTM) with the field error be given by R = RqR0Rq (where for algebraic convenience we have taken as a reference point the center of the quad), where R0 is the OTM for the ideal ring:

$$M = \begin{pmatrix} \cos \phi\_0 & \beta\_0 \sin \phi\_0 \\ -(1/\beta\_0) \sin \phi\_0 & \cos \phi\_0 \end{pmatrix} \,. \tag{2.109}$$

a) Show that the perturbed phase advance φ is given in terms of the unperturbed phase advance φ0 by

$$
\cos \phi = \cos \phi\_0 - \frac{\beta\_0}{2f} \sin \phi\_0 \,. \tag{2.110}
$$

b) From the measurement of phase advance errors in Fig. 2.9, estimate the amplitude of the modulation on the beam size.

### 2.5 Quadrupole with a Shorted Coil

So-called difference orbit measurements are taken by saving a BPM reference orbit under nominal conditions, perturbing the beam (for example, by powering a single corrector) and forming the difference orbit consisting of measurements of the perturbed orbit minus that of the nominal orbit. This difference orbit may be easily compared with model expectation. Suppose such lattice diagnostic measurements are made using a negatively charged beam and reveal that the beam is errantly deflected up and outboard (away from ring center) at a focussing quadrupole.

a) Based on these observations, which of the quadrupoles' coils is suspect of a turn-to-turn short?

b) With a position resolution of 10 μm at a BPM located l = 1 m downstream of the quadrupole, how well can the relative change in the current through the offending coil be determined assuming K = Blq/((Bρ)a) = 1.0 m−1 with a pole tip radius of a = 1 cm?

### 2.6 Quadrupole Gradient Errors

a) Design a closed π-bump spanning a focussing quadrupole of strength K assuming perfect calibration of the corrector dipoles.

b) Using this bump, suppose that the orbit is observed not to close with a residual amplitude A measured using a BPM. Find an expression relating the measured leakage A to the gradient error ΔK.

### 2.7 Multiknobs

A storage ring employs two families of quadrupoles for varying the tranverse focussing. As is common, each quadrupole family has influence on both the horizontal and vertical betatron tunes, Qx and Qy respectively. Design a multiknob that allows independent control of each of the betatron tunes; that is, find strength coefficients for each of the quadrupole families for the cases (i) ΔQx = 0, ΔQy = 0 and (ii) ΔQx = 0, ΔQy = 0.

# 3 Orbit Measurement and Correction

In practice, there are many uncertainties whose presence must be appreciated when correcting the beam orbit in both linear and circular accelerators. Such uncertainties include the variations in the electronic and/or mechanical centers of the beam position monitors (BPMs), in the magnetic center of the quadrupoles (inside which the position monitors are often mounted), or in the electromagnetic center of accelerating structures. Consider the case illustrated in Fig. 3.1. In this case, the absolute beam position, with respect to a reference axis, is given by

$$x = x\_d + x\_b + x\_m \,, \tag{3.1}$$

where xd represents the quadrupole offset from the reference axis, xb gives the offset of the electronic center of the BPM relative to the quadrupoles, and xm denotes the measurement from the BPM. The reference axis may be chosen to minimize emittance dilutions in a global sense. In the ideal case for which the reference axis is straight and in the center of all quadrupoles, a typical most simple steering algorithm aimed towards zeroing the BPM readings would

Fig. 3.1. Sketch showing relative positions of the BPM, the quadrupole, and the beam position measurement from an ideal reference axis (Courtesy C. Adolphsen, 1999)

This chapter has been made Open Access under a CC BY 4.0 license. For details on rights and licenses please read the Correction https://doi.org/10.1007/978-3-662-08581-3\_13

© The Author(s) 2003 M. G. Minty et al., *Measurement and Control of Charged Particle Beams,* https://doi.org/10.1007/978-3-662-08581-3\_3

still place the beam off-axis in the quadrupole if xb = 0. For accelerators with tight emittance budgets and correspondingly tight alignment tolerances, clearly more sophisticated steering algorithms are needed.

If the orbit is off-center in a quadrupole magnet, dispersion is generated, and, in a ring, also the beam energy may be changed or the polarization may decrease. An orbit that is off-center in a sextupole induces skew coupling and/or beta beating. Thus it is very important to center the orbit in these magnets. The standard tools for correcting the orbit are corrector dipoles. Of course, such an orbit correction will never be perfect. Figure 3.2 shows a typical absolute orbit reading from the PEP-II HER, after moderate orbit correction during commissioning.

Fig. 3.2. Typical commissioning orbit in the PEP-II HER: (top) horizontal orbit in mm; (center ) vertical orbit in mm; (bottom) intensity in 1010 (Courtesy U. Wienands, J. Seeman et al, 1998)

We will see that if the BPM offsets are not known, and possibly larger than the alignment errors, a better strategy for optimal emittance preservation is to reduce the rms strength of the steering correctors, and to pay less attention to the absolute orbit reading. In several cases, at the SLC and at the ATF, this second approach significantly reduced the magnitude of the residual vertical dispersion [1]. Sometimes other constraints are imposed on the orbit. For example, a certain orbit amplitude or a certain angle may be desired near the injection or extraction points, or near a synchrotron light beamline. In such cases, a constant orbit must be maintained at the adjacent BPMs.

Increased beam emittances may arise from beam-to-magnet or beam-tostructure position deviations as shown conceptually in Fig. 3.3. The beam passing through a single misplaced quadrupole experiences the next lowerorder field namely a dipole field. The misalignment therefore generates a betatron oscillation and dispersion as higher energy particles are less deflected by the dipole field. Also in the case of a displaced structure a betatron os-

Fig. 3.3. Conceptual drawing illustrating orbit perturbations due to misaligned quadrupoles (a) or structures (b)

cillation is induced, due to the transverse wake field. The deflection caused by the wake field changes linearly with the beam displacement and with the bunch charge. In either case, the ensuing orbit is such that further emittance dilutions may result downstream of the perturbation due to the initial errors.

In this chapter we begin by reviewing beam-based alignment of single components. Such methods are particularly useful for those components with tight alignment tolerances, for example, near interaction regions. The techniques presented may be applied to the alignment of any accelerator component in general. For large accelerators, for which component-by-component alignment is too time-consuming or for which the beam trajectories must be routinely monitored, global orbit measurement and correction techniques are required. Many such algorithms will be presented in order of increasing complexity. So-called 'one-to-one' steering and R-matrix reconstruction are typically used not only to correct the orbit, but also to identify optical and/or instrumental errors. While such simple checks are absolutely mandatory, to achieve design beam parameters, more advanced tuning algorithms may be required. Common algorithms will be described including global beam-based alignment, singular value decomposition (SVD), application of wake field bumps, and dispersion-free steering.

# 3.1 Beam-Based Alignment

In many modern accelerators, the alignment tolerances on quadrupole and sextupole magnets are so tight that they cannot be achieved by state-of-the art surveying and installation methods. Typical residual alignment errors are in the range of 100–200 μm while the alignment tolerances may be even below 10 μm. The standard approach to meet and maintain tight tolerances is beam-based alignment.

Beam-based alignment determines the relative offset between magnet centers and nearby BPMs. If these offsets are sufficiently stable, a simple orbit correction (steering) can maintain a well-centered orbit until the optical alignment measurement is repeated at a later time (e.g., after several months).

### 3.1.1 Quadrupole Excitation

If the beam is not centered in a quadrupole magnet, and the strength of this quadrupole is varied, the beam receives a kick. This causes a change in the beam trajectory for single-turn measurements or a change in the closed orbit for measurements on a stored beam.

For a single-pass measurement, the dipole kick θ can easily be inferred by fitting the difference trajectory to a betatron oscillation including one additional kick at the location of the quadrupole. The dipole kick θ obtained from the fit is proportional to the quadrupole misalignment xq and the change in the integrated quadrupole strength ΔK:

$$
\theta = \Delta K \ x\_q \,. \tag{3.2}
$$

Note that the beam is offset from the center of the quadrupole by −xq, so that there is no minus sign in (3.2) assuming K > 0 refers to a horizontally focusing quadrupole.

If beam-based alignment is performed using a stored beam, the additional kick of the closed orbit induced by the change in quadrupole strength is given by the sum of two components: the change in field strength and the change in the closed-orbit offset at the quadrupole. To lowest order [2],

$$
\theta \approx \Delta K \ x\_q - K \ \Delta x \,, \tag{3.3}
$$

where xq is the original quadrupole offset, Δx the change in closed-orbit position, K the integrated quadrupole gradient in units of m−1, and a secondorder term (ΔK Δx) has been neglected. Applying the formula for the closed orbit distortion at the location of the dipole kick, (2.34), with s = s0. Neglecting here the possible small contribution from the dispersion Dx, then

$$
\Delta x = \begin{pmatrix} \Delta K \ x\_q - K \ \Delta x \end{pmatrix} \begin{pmatrix} \beta \\ \frac{\pi}{2 \tan \pi Q} \end{pmatrix}, \tag{3.4}
$$

which may be solved for Δx:

$$
\Delta x = \Delta K \ x\_q \left( \frac{\beta/(2\tan\pi Q)}{1 + K\beta/(2\tan\pi Q)} \right) \,. \tag{3.5}
$$

Inserting this back into (3.3) gives the closed-orbit kick induced by a gradient change ΔK:

$$\theta = \Delta K \, x\_q \left( \frac{1}{1 + K\beta/(2\tan\pi Q)} \right). \tag{3.6}$$

This is the stored-beam equivalent of (3.2).

The precision of this method is much improved by taking difference orbits for several quadrupole-to-beam off-sets, Δxq, varied with a local bump [3]. One can also define a merit function

$$f(\Delta x\_q) = \frac{1}{N\_{\rm BPM}} \sum\_{i=1}^{N\_{\rm BPM}} \left[ x\_i(\Delta K) - x\_i(-\Delta K) \right]^2,\tag{3.7}$$

where NBPM is the total number of BPMs in the ring and xi(ΔK) are the BPM readings for a quadrupole strength change of ΔK. The quadrupole offset may then determined by minimizing f(Δxq) as a function of the bump amplitude Δxq, using a least-squares parabolic fit. At the Advanced Light Source (ALS), this procedure allows determination of the center of the quadrupoles to within ±5 μm [2] (in case of the ALS, the orbit at the quadrupole is varied with a single corrector and not by a closed bump).

This type of measurement does not require an independent power supply for each quadrupole to be aligned, but, for several magnets in series, a simple switchable shunt resistor across each magnet will suffice. Simultaneously, such shunt resistors allow a measurement of the local beta function, according to (2.28) or (2.29) of the previous chapter.

Figure 3.4 illustrates the application of this technique at the storage ring SPEAR. The left figure shows the circuit diagram for a magnet with shunt resistor, and the right figure presents a typical alignment measurement for a SPEAR quadrupole. Plotted in the right figure is the orbit shift induced by

Fig. 3.4. Beam-based alignment with quadrupole shunts at SPEAR [4]: (top) electric circuit with shunt resistor; (bottom) shunt-induced orbit shift at two downstream BPMs as a function of the beam-position read back at the BPM nearest to the quadrupole being varied (Courtesy J. Corbett, 1998)

the shunt at two downstream BPMs as a function of the orbit at the shunted quadrupole, which is varied by a local bump. The orbit is centered in the quadrupole when no orbit shift is induced by the shunt (the intersection of the two lines).

If the number of BPMs is small and only groups of quadrupoles can be changed simultaneously, it is still possible to determine the quadrupole misalignments, by applying a statistical fit to a sufficiently large number of trajectories taken for different quadrupole-group excitations, different incoming conditions and different corrector settings. An interesting example of such an analysis can be found in [5].

Fig. 3.5. Overview of the TTF linear accelerator including 2 accelerating modules (ACC1 and ACC2) and a bunch compressor chicane (BC2)

Another application of this method is illustrated for the Tesla Test Facility (TTF). An overview of part of the linear accelerator is shown in Fig. 3.5. In this measurement a horizontal corrector located just upstream of the bunch compressor chicane (BC2) was varied in steps for different settings of quadrupole settings. In the data shown in Fig. 3.6 the quadrupole of interest was the most upstream quadrupole in the triplet following the chicane. The next downstream beam position monitor was used for measurement of

Fig. 3.6. Beam-based alignment of a single quadrupole at the TTF (Courtesy P. Castro, 2000)

the horizontal displacements. The orbit was centered when variation of the quadrupole field produced no change in the measured beam position (at the intersection of the lines).

### 3.1.2 Quadrupole Gradient Modulation

A scheme which allows continuous monitoring of quadrupole alignment and BPM offsets was implemented at LEP; see for example [6]. Here the strength of several quadrupoles was modulated at different frequencies in the range 0.8–15.6 Hz, and the induced oscillation amplitude, of the order of 1 μm was detected. Figure 3.7 shows the FFT over 4096 data points of this detector signal, at a time when four quadrupoles were modulated. Clearly visible are 4 peaks in the frequency sprectrum, corresponding to the four different modulation frequencies. The amplitude of the peak is proportional to the beam offset in that quadrupole.

Fig. 3.7. FFT spectra with 4 modulated quadrupoles in LEP [6]. The amplitude of the peaks is proportional to the beam displacement in the 4 quadrupoles (Courtesy I. Reichel, 1998)

Using this 'K-modulation' technique, one could infer the BPM offsets from the naturally occurring beam-orbit jitter and orbit variation. This is illustrated in Fig. 3.8. The left figure shows a BPM orbit reading in LEP during several hours of a luminosity run. The reasons for the slow changes are not fully understood; the fast steps reflect corrections of the closed orbit. Making use of this natural orbit variation, one can plot the amplitude of the beam response to the quadrupole modulation as a function of the BPM

Fig. 3.8. Determination of BPM offsets using k modulation and natural orbit variation in LEP [6]: (top) natural orbit drifts and corrections during a LEP luminosity run at one quadrupole; (bottom) amplitude of beam response to k modulation vs. BPM orbit reading for the modulated quadrupole. The minimum of this plot gives the BPM offset. These data were taken continuously during 5 hours of luminosity run (Courtesy I. Reichel, 1998)

reading for the corresponding quadrupole. The result is a 'V plot', as shown in the right figure. The minimum in this plot determines the BPM reading at which the beam is centered in the quadrupole.

### 3.1.3 Sextupole Excitation

In present-day storage rings, it is often assumed that the sextupoles are well enough aligned with respect to the quadrupoles that only the quadrupole alignment has to be verified. An orbit off center in a sextupole will result in vertical dispersion, betatron coupling, or beta beating. Although, in principle, also the sextupoles in a storage ring can be aligned by changing their strength and measuring the induced orbit shift (which is a quadratic function of the excitation) there is little experience with such a scheme. To reach the same sensitivity as for the equivalent quadrupole alignment, the change in the sextupole gradient ΔKs would have to be equal to

$$
\Delta K\_s = \frac{\Delta K\_q}{2x\_s},
\tag{3.8}
$$

where xs is the horizontal orbit offset at the sextupole, and ΔKq the corresponding change in quadrupole gradient. A different approach, which was tested at KEK [7], is to equip the sextupole magnets with additional quadrupole trim windings for beam-based alignment. This is based on the assumption that the magnetic centers of quadrupole trim coil and sextupole will coincide. Sextupole alignment with a precision better than 50 μm was demonstrated [7].

Local orbit bumps across single sextupoles have been used for the purpose of sextupole alignment at KEK [8, 9] and DESY [10]. The strengths of all sextupoles are changed together by ΔKs and the induced orbit change is measured. Then the measurement is repeated for a different bump amplitude. The horizontal deflection depends quadratically on the horizontal bump amplitude, while the vertical deflection is a linear function:

$$
\Delta\theta\_x = -0.5 \,\,\Delta K\_s (x\_{\text{bump}} - x\_s)^2,\tag{3.9}
$$

$$
\Delta\theta\_y = \Delta K\_s (x\_{\text{bump}} - x\_s) y\_s \,, \tag{3.10}
$$

where xbump is the amplitude of the bump, and xs, ys are the sextupole misalignments. The advantage of this method is that it does not require individual power supplies for the sextupoles.

The vertical sextupole alignment can be determined in a similar way, but in this case a (preferably large) horizontal orbit offset x0 at the sextupole must first be introduced, to enhance the sensitivity. The vertical deflection angle then varies linearly with the amplitude of a vertical orbit bump at the sextupole as follows:

$$
\Delta\theta\_y = \Delta K\_s (x\_0 - x\_s)(y\_{\text{bump}} - y\_s) \,, \tag{3.11}
$$

where we have also included a residual horizontal misalignment xs. After measuring the vertical misalignment ys, the original horizontal orbit can be restored.

Alternative approaches are conceivable: one could vary multiple sextupoles at once, and fit for multiple kicks. Also, one could vary the sextupole strength and measure the induced tune variation or the tune separation near the difference resonance [11].

In the final-focus systems of linear colliders, sextupole alignment is essential. At the SLC final focus, the orbit in the sextupoles was frequently measured and adjusted to maintain high luminosity. The SLC sextupole alignment procedure was based on varying the sextupole strength and detecting the induced optics (not orbit) change [12]. If the orbit is off center, the first order effect of the sextupole excitation is a waist shift (change in the beta function), skew coupling, or dispersion at the interaction point. These optics changes can be quantified easily by reoptimizing the spot-size at the collision point, after a change in the sextupole strength. This reoptimization is done by scanning a group of quadrupole and skew quadrupole magnets excited together so that they only affect one optical parameter, e.g., either dispersion, skew coupling or waist shift. Such combinations of changes in magnet settings are called multiknobs; see the discussion in Chap. 2. For each setting of the multiknob, the IP spot size is remeasured with beam-beam deflection scans, and the magnets are finally set to the value for which the beam size is minimum. The change in the optimum settings of the multiknobs controlling waist, dispersion, etc., as a function of the sextupole excitation is proportional to the orbit offset at the sextupole. The measured offsets are corrected by means of closed bumps.

An interesting feature of the SLC final focus is that it comprises 2 pairs of interleaved sextupoles. The sextupoles in each pair, connected to the same power supply, are separated by an optical −I transform. Thus, the alignment procedure actually consists in generating symmetric or antisymmetric orbit bumps for each sextupole pair, in response to the amount of waist motion or dispersion etc., induced by a change in the sextupole-pair strength [13].

### 3.1.4 Sextupole Movement

It is also possible to align the sextupole magnets by detecting the second-order effect of the sextupole excitation: the induced orbit kick. This method works well when the sextupoles are installed on precision movers, which can be used

Fig. 3.9. Sextupole alignment in the Final Focus Test Beam (FFTB) [14]. The downstream orbit variation is measured as a function of sextupole mover position; the sextupole is aligned at the minimum of the parabola (Courtesy P. Tenenbaum, 1998)

for both the measurement and the alignment. The measurement principle is straightforward. Measuring the orbit change downstream as a function of horizontal or vertical sextupole-mover position results in a parabolic curve. The sextupole is aligned when the mover position is set to the minimum of this curve. A sample measurement from the FFTB [14] is displayed in Fig. 3.9.

### 3.1.5 Structure Alignment Using Beam-Induced Signals

For future high-gradient linear accelerators operating at even higher accelerating frequencies it becomes even more essential to center the beam orbit in the accelerating structures, thus minimizing the effect of transverse wake fields on the beam. Alignment techniques have been studied on test structures for the Next Linear Collider, which were installed in the SLAC linac as part of the ASSET experiment [15]. These studies demonstrated that the beam-induced dipole-mode signals can be used to center the beam to the level of 40 μm [16]. The result in Fig. 3.10 shows the amplitude and phase

Fig. 3.10. Amplitude (top) and phase (bottom) of the beam-induced dipole mode signal in an X-band accelerating structure versus the nominal beam position (arbitrary zero), which was varied by steering correctors [16] (Courtesy M. Seidel, 1998)

(with respect to a reference phase derived from a BPM signal) of a 15-MHz wide slice of the beam-induced dipole mode signal, centered near 15 GHz, as a function of the nominal beam position. The beam position was varied with dipole steering magnets. Clearly visible is a minimum in the amplitude along with a 180 degree phase jump. Steering the beam to the position with minimum signal successfully centered the orbit in the structure as was verified by detecting the deflection experienced by a subsequent witness bunch.

We now turn our attention to global steering algorithms designed to efficiently align the beam relative to offset magnets and accelerating structures relying heavily on computational techniques. The algorithms are presented in order of increased complexity. Not by chance, this order coincides to a large degree with the order of implementation as an accelerator is being commissioned and the beam properties are refined.

# 3.2 One-to-One Steering

This algorithm aims to steer the beam so that the transverse displacements measured by beam position monitors (BPMs) are minimized. The BPMs are typically mounted near the center of quadrupoles since their sensitivity is highest at large β-function. A conceptual orbit steered one-to-one is shown in Fig. 3.11. The beam is successfully deflected to pass through the magnet center and, assuming that the BPM is not offset with respect to the quadrupole, the BPM would show zero displacement. Notice however that one-to-one steering generates dispersion will contribute to emittance dilutions.

Fig. 3.11. Conceptual illustration of a closed bump that would minimize the BPM reading after one-to-one steering

In a transport line the beam centroid position measured downstream at location s = j obeys

$$x\_j = \sum\_{i=0}^j \sqrt{\beta\_i \beta\_j} \theta\_i \sin(\theta\_j - \theta\_i) \,\tag{3.12}$$

which has contributions from each dipole kick θi and depends on the β-functions at the location of the initial disturbance (i) and at the observation point (j). The corrector magnet fields to be applied to minimize the BPM readings will be solved for assuming linear transport; that is, assuming that there are no nonlinear magnetic fields and that the measurements are made at low bunch current so that nonlinear wake field effects may be ignored.

In matrix form

$$x = M\theta \,, \tag{3.13}$$

where x is the set of measurements from m BPMs, θ is the set of deflection angles to be applied by n correctors, and M contains the transfer matrix elements between the correctors and the BPMs:

$$x^{t} \;= \begin{pmatrix} x\_0, x\_1, \dots, x\_m \end{pmatrix} \;,\tag{3.14}$$

$$\theta^t \quad = \left(\theta\_0, \theta\_1, \dots, \theta\_n\right), \tag{3.15}$$

$$M\_{ij} = \sqrt{\beta\_i \beta\_j} \sin(\phi\_j - \phi\_i) \,. \tag{3.16}$$

Solving (3.13), the kick angles to be applied for minimizing the BPM readings are obtained:

$$M^t \mathbf{x} = M^t M \boldsymbol{\theta} \quad \text{or} \quad \boldsymbol{\theta} = (M^t M)^{-1} M^t \mathbf{x} \,. \tag{3.17}$$

If the number of correctors equals the number of BPMs, then M is a square matrix. In this case, (3.17) reduces to simply θ = M−1x. Otherwise the general form is taken. If m>n the matrix is overdetermined. For m<n the number of unknown corrector strengths exceeds the number of measurements, so that an independent measurement should be made after changing some parameter, for example, the beam energy. In a linear accelerator, (3.12) must be modified [17] to include the energy scaling factor -Ei/Ej , which reflects the transverse damping due to the longitudinal acceleration. This introduces n additional unknown variables, so that additional measurements are required to constrain the solution.

As motivation for the algorithms to be used below in the discussion of beam-based alignment and dispersion-free steering, the solution (3.17) can be equivalently formulated in terms of a minimization procedure, which is well adapted to computational evaluation. The function to be minimized, given by (3.13), is

$$\sum\_{j} \left[ x\_j - \sum\_{i} M\_{ji} \theta\_i \right]^2,\tag{3.18}$$

where xj again represents the BPM measurements and the fitting function i Mjiθi contains the unknowns θi. The minimization procedure demands

$$\begin{split} 0 &= \frac{\partial}{\partial \theta\_k} \Big[ \sum\_j (x\_j - \sum\_i M\_{ji}\theta\_i)^2 \Big] \\ &= 2 \sum\_j \Big[ x\_j - \sum\_i M\_{ji}\theta\_i \Big] M\_{jk} \, , \end{split} \tag{3.19}$$

or

$$\sum\_{j} M\_{jk} x\_j = \sum\_{j} \sum\_{i} M\_{ji} M\_{jk} \theta\_i \,, \tag{3.20}$$

which is identical to (3.17).

One-to-one steering is used routinely during initial commissioning of an accelerator as it is by far the simplest of all steering algorithms. The procedure relies on having reasonably well functioning beam position monitors and may diverge in the presence of either large monitor errors or, possibly, with multiple smaller errors. A complementary algorithm, presented next, is useful to help identify BPM errors (polarity, gain, offset, etc.) and to help localize optical errors as well.

# 3.3 Lattice Diagnostics and R Matrix Reconstruction

Consider a beam line, without any elements coupling horizontal and vertical motion, as shown in Fig. 3.12 consisting of dipole and quadrupole magnets, BPMs and corrector magnet dipoles. The point-to-point transfer map between any two points (1) and (2) is given by

$$
\begin{pmatrix} x \\ x' \end{pmatrix}\_2 = \begin{pmatrix} R\_{11} & R\_{12} \\ R\_{21} & R\_{22} \end{pmatrix} \begin{pmatrix} x \\ x' \end{pmatrix}\_1. \tag{3.21}
$$

$$
\begin{pmatrix} \cdot \\ \end{pmatrix} \overbrace{\nabla \cdot \square \square \square \square}^{\perp} \begin{pmatrix} \cdot \\ \end{pmatrix}\_2 \overbrace{\Delta \square \square \square \square \square}^{\perp} \begin{pmatrix} \cdot \\ \end{pmatrix}\_2
$$

Fig. 3.12. Simple FODO lattice. Shown are the focussing quadrupoles (QF), the defocussing quadrupoles (QD), and dipoles (rectangles). The BPMs are usually mounted within the quadrupoles. Corrector magnets are denoted by triangles

Let the initial point (1) be at a corrector and the final point (2) be at a BPM. Two measurements are required to determine R12: xBPM with the nominal beam trajectory and the nominal initial values (x, x )COR, and with xBPM for the initial position and slope (x, x + θ)COR, i.e., after the beam is kicked by an angle θ. The difference in xBPM between these two measurements is R12 = Δx/θ. In practice to decrease sensitivity to measurement error, one introduces a series of large betatron oscillations by varying the corrector in steps. The response of the BPM readings to these perturbations [18] is then measured. The resulting dependence of position on the kick angle θ is fitted with a straight line as shown in Fig. 3.13.

Shown in Fig. 3.14 is an example of a beam trajectory after excitation by a corrector magnet and an amplitude fit using many downstream BPMs. For good viewing conditions the measured trajectory is plotted for the case of a maximum kick angle θ = θmax. The solid line connects the measurements from each BPM. The dashed line represents the fitted positions xi = Rθ→i 12 θ + xo,i evaluated at θmax where xo,i is an offset at the ith BPM.

Fig. 3.13. Example of an R12 measurement: transverse position x versus B dl, where the kick angle, in terms of the magnetic rigidity Bρ, is θ = B dl/(Bρ)

Fig. 3.14. Comparison of measured data (solid curve) and amplitude fit (dashed curve). Plotted are the horizontal and vertical beam positions as measured as a function of distance along the transport line. See also [19]

in the linear fit. A single value of θ and the values of m BPM offsets xo,i are fitted, so as to minimize the difference between the measured BPM readings and the model prediction, i.e., i(xBPM,i −Rθ→i 12 θ −xo,i)2. Here Rθ→i 12 is the (1,2) transport-matrix element between the corrector and the ith BPM computed by the optics model. To probe all magnetic elements in the beamline, a second measurement is required using a second corrector dipole separated by about 90◦ in betatron phase advance. Discrepancies between the meas-

urement, xBPM,i and the fit, xi, are used to reveal phase errors, which could result from a shorted quadrupole magnet, and/or "bad" BPMs for example. Identification and exclusion of bad BPMs is required for good convergence of steering algorithms.

Assuming the linear transport matrices between the different correctors and the transport matrices between adjacent BPMs are known with sufficient accuracy, the R matrix between the correctors and the BPMs can be determined by a simple least squares fit [18]. Including the additional constraint that the R matrix has to be symplectic eliminates several degrees of freedom, but then the problem must be solved by non-linear regression [18]. Reference [18] describes how a rigorous error analysis allows an estimate of the unknown systematic errors.

Suppose the agreement between the measured data and fit is unsatisfactory. In the absence of hardware errors, this may result from systematic measurement errors or from an incomplete model. Accuracy of the model is vital for basic optics checkout and requires, for example, accurate representations of magnetic field strengths.

In a linear accelerator, it may be necessary to take into account the energy dependence of the point-to-point transfer matrix elements. The change in the betatron phase Δψ relative to the expected phase ψ is given by

$$\frac{\Delta\psi}{\psi} = \delta\xi \,, \tag{3.22}$$

where δ is the relative energy deviation and ξ is the chromaticity (in a storage ring ξ is related to Q via ξ ≡ Q /Q, where Q denotes the betatron tune). With ξ = −1/π for a 90◦ FODO cell, then a 1% energy error corresponds to a change in phase advance of Δψ = −0.003ψ per cell.

To take into account the energy dependence of the transport matrix, the matrix elements may be expanded in a Taylor series. Keeping the linear term only, the transport matrix from the entrance ('1') to the end of the ('2') of the linac can be written as

$$
\begin{pmatrix} x \\ x' \end{pmatrix}\_2 = \prod\_{k=1}^{N\_s} \begin{pmatrix} R\_{1,1\_k} + \frac{\mathrm{d}R\_{1,1\_k}}{\mathrm{d}E\_k} \Delta E\_k & R\_{1,2\_k} + \frac{\mathrm{d}R\_{1,2\_k}}{\mathrm{d}E\_k} \Delta E\_k \\ R\_{2,1\_k} + \frac{\mathrm{d}R\_{2,1\_k}}{\mathrm{d}E\_k} \Delta E\_k & R\_{2,2\_k} + \frac{\mathrm{d}R\_{2,2\_k}}{\mathrm{d}E\_k} \Delta E\_k \end{pmatrix} \begin{pmatrix} x\_0 \\ x'\_0 \end{pmatrix}\_1,\tag{3.23}
$$

where Ns is the number of regions ('sectors', typically taken to be one sector per power source) into which the accelerator has been subdivided, and the total R matrix is expressed as a product of the R matrices for the different sectors. A possible procedure now consists of

$$\chi^2 = \sum (x\_m - x\_c)^2,\tag{3.24}$$

where the sum is over all the BPMs used in the measurement, and 4. iterating steps 2 and 3 to minimize the χ2.

An example of this approach is shown in Fig. 3.15 for the SLAC linear accelerator [20]. In the top plot an amplitude fit was used without taking into account any energy deviations (as in Fig. 3.14). In the bottom plot, also the values of Ek were fitted. The minimum χ2 corresponded to an energy error of about 30%, which far exceeded the estimated uncertainty in the energy. Possible reasons for the discrepancy might include calibration errors of the quadrupole strengths, random errors in the BPM gains, or wake fields, which all are not included in the model. Nonetheless, the dynamics were made more predictable by incorporating the fitted energy errors into the accelerator model, despite the fact that the actual source of the error (perhaps wake fields) was different from the assumed one (energy errors).

Fig. 3.15. Comparison of measured data and fit to betatron amplitude only (top) or fit to both betatron amplitude and energy in the different linac sections (bottom). Plotted is the horizontal beam position as a function of distance along the linear accelerator (Courtesy T. Himel, 1999)

# 3.4 Global Beam-Based Steering

We next present a more global approach to beam-based alignment than that discussed previously. We consider beam-based steering algorithms as ones which provide information on magnet, BPM, or structure misalignments using measurements with the beam. With this definition, one-to-one steering may also be considered a beam-based alignment algorithm since the applied kicks θ are related to the quadrupole (or BPM) displacements Δx by θ = KΔx, where K is the integrated quadrupole focussing field.

More generally, we take in this example into account that the electrical zero of the BPMs may not be coincident with the magnetic center of the quadrupoles and that the quadrupoles themselves may be displaced with respect to the reference axis. The coordinate system used is sketched in Fig. 3.16. Here the beam position x measured with respect to some reference axis, which is common to all magnets, is given as a sum of the quadrupole displacement xq, the difference in location of the electrical center of the BPM and the magnetic center of the quadrupole xbpm, and the measured BPM value xm.

Fig. 3.16. Coordinate system used in the example of beam-based alignment (Courtesy C. Adolphsen, 2000)

The beam position xk and angle xk at quadrupole k, defined with respect to the reference axis, are given by [21]

$$
\begin{pmatrix} x\_k \\ x\_{k'} \\ 1 \end{pmatrix} = \sum\_{j=1}^{k-1} R\_{j+1,k} \left\{ R\_{j,j+1} \left[ \begin{pmatrix} x \\ x' \\ 1 \end{pmatrix}\_j + \begin{pmatrix} -x\_{q,j} \\ 0 \\ 0 \end{pmatrix} \right] + \begin{pmatrix} x\_{q,j} \\ 0 \\ 0 \end{pmatrix} \right\},\tag{3.25}
$$

where ()j gives the beam position and angle with respect to the quad center, the term in [] is the beam position with respect to the reference axis, Rj,j+1[] is the beam position with respect to the reference axis transported between quad j and quad j + 1, and the term in {} is the beam position and angle with respect to the quad center transported between quads j and j + 1. Rearranging terms gives

$$
\begin{pmatrix} x\_k \\ x\_k' \\ 1 \end{pmatrix} = R\_{0,k} \begin{pmatrix} x \\ x' \\ 1 \end{pmatrix}\_0 + \sum\_{j=1}^{k-1} (R\_{j+1,k} - R\_{j,k}) \begin{pmatrix} x\_{q,j} \\ 0 \\ 0 \end{pmatrix},\tag{3.26}
$$

where the sum is taken over upstream quadrupoles. The function to be minimized is then

$$\sum\_{k} \left[ x\_m - (x\_k - x\_q - x\_{\text{bpm}}) \right]^2,\tag{3.27}$$

where xm are the measurements and (xk − xq − xbpm) is the fitting function with unknowns xq, xbpm and the initial position and angle x0 and x0 .

The number of measurements is about half the number of unknowns. So the system is underconstrained. To constrain the solution, two independent measurements are required. An independent set of data may be obtained by scaling all the quadrupoles and correctors by a common factor and repeating the measurements. Multiple such scalings may be used to overdetermine the system which reduces the sensitivity of the solution to statistical errors.

# 3.5 Singular Value Decomposition

A common situation is that the BPM offsets are known fairly well and the orbit already fulfills a number of constraints, but many of the corrector magnets are strongly excited with some of them 'fighting' (compensating) each other. Fortunately, there exists a very powerful technique to reduce the rms strength of the orbit correctors, while maintaining a set of constraints. This technique is sometimes called 'corrector ironing' [22] and it is based on the 'singular value decomposition' (SVD) [23].

The linear equation to be solved is

$$
\Delta \mathbf{x} = \mathbf{A} \cdot \boldsymbol{\theta} \,, \tag{3.28}
$$

where the vector Δx = (Δx1, . . . Δxm) may describe the desired correction (or constraint) at m BPMs, and θ = (θ1,...,θn) are the excitation strengths of n correctors, that we want to determine. If m ≥ n, we can decompose the matrix A as

$$\mathbf{A} = \mathbf{U} \cdot \begin{pmatrix} w\_1 & 0 & \dots & 0 \\ 0 & w\_2 & \dots & 0 \\ & \dots & \dots & \\ 0 & 0 & \dots & w\_n \end{pmatrix} \cdot \mathbf{V}^t. \tag{3.29}$$

The column vectors of the m × n matrix U and the n × n matrix V are orthonormal,

$$\mathbf{U}^t \cdot \mathbf{U} = \mathbf{I}\_n \,, \tag{3.30}$$

$$\mathbf{V}^t \cdot \mathbf{V} = \mathbf{I}\_n \,, \tag{3.31}$$

where In denotes the n × n unity matrix. The decomposition of (3.29) can be performed, for example, using the FORTRAN subroutine described in [23]. An SVD decomposition is also provided in a convenient form by many mathematical analysis packages, such as MATLAB [24].

We now consider three different cases: First, we suppose the number of correctors is equal to the number of BPMs. In this case the matrix A is square. The formal solution is given by

$$\boldsymbol{\theta} = \mathbf{A}^{-1} \cdot \Delta \mathbf{x} = \mathbf{V} \cdot \begin{pmatrix} 1/w\_1 & 0 & \dots & 0 \\ 0 & 1/w\_2 & \dots & 0 \\ & \dots & \dots \\ 0 & 0 & \dots & 1/w\_n \end{pmatrix} \cdot \mathbf{U}^t \cdot \Delta \mathbf{x} . \tag{3.32}$$

If none of the wi is zero, this is the unique solution to the problem. If one or more of the wi are zero, the equation may not have an exact solution, but for these wi one can simply replace 1/wi by 0, and with this replacement (3.32) still gives the solution in a least squares sense. This means it minimizes the distance r = |A·θ−Δx|. Furthermore, the solution vector θ so obtained is the (either least-squares or exact) solution with the smallest possible length |θ| 2. In other words, the solution derived from the SVD decomposition also minimizes the rms strength of the correctors.

In addition, it is worthwhile to note that the columns of U whose same numbered wi are nonzero form an orthonormal set of basis vectors that span the range of the matrix A while the columns of V whose same-numbered elements wj are zero form an orthonormal set for the nullspace of A.

Next, we consider the case that there are fewer equations than correctors. In this case, we can simply add rows with zeroes to the vectors and matrices of (3.28) until the matrix is square, and then apply the SVD formalism, as described above. In this case, there is (at least) one zero eigenvalue wj for every row of zeroes added.

Finally, in the case of more BPM constraints than unknown correctors (m>n), SVD works just as well. In general the wj will not be zero, and the SVD solution will agree with the result of a least-square fit. If there are still some small values wj , these indicate a degeneracy in A and the corresponding 1/wj should be set to zero as before. The corresponding column in V deserves attention, since it describes a linear combination of corrector excitations, which does not affect the constraints.

The SVD steering algorithm has been used successfully at many accelerators, for example, at the synchrotron light source SPEAR [25], throughout the SLC, and most recently at LEP. Applications include not only the minimization of corrector strengths and the identification of strongly excited correctors 'fighting' each other, but also orbit feedback, dispersion-free steering, and the computation of optical tuning knobs, e.g., for the beta function, dispersion, or coupling. Multiknobs were discussed in Sect. 2.5.

An Example of SVD. As a more concrete illustration, let us consider two steering correctors, separated by a betatron phase advance of π, which are followed by two BPMs at locations 1 and 2, which are also a phase of π apart from each other.

After applying a convenient normalization, the matrix equation relating the corrector strengths θ1 and θ2 with the BPM readings x1 and x2 is

$$
\begin{pmatrix} x\_1 \\ x\_2 \end{pmatrix} = \begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix} \begin{pmatrix} \theta\_1 \\ \theta\_2 \end{pmatrix} . \tag{3.33}
$$

The 2 × 2 matrix A can be decomposed as follows:

$$\mathbf{A} \equiv \begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix} \tag{3.34}$$

$$= \begin{pmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}}\\ -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{pmatrix} \begin{pmatrix} 2 \ 0\\ 0 \ 0 \end{pmatrix} \begin{pmatrix} \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{pmatrix} \tag{3.35}$$

$$\equiv \mathbf{U} \mathbf{W} \mathbf{V}^t,\tag{3.36}$$

where W denotes the diagonal matrix. Suppose now we want to steer to xt,1 = 1 and xt,2 = 0. An exact solution does not exist, since there is no corrector between the two BPMs and the latter are a phase advance of π apart. However, there is an SVD solution:

$$
\begin{pmatrix} \theta\_1 \\ \theta\_2 \end{pmatrix} = \mathbf{V} \mathbf{W}^{-1} \mathbf{U}^t \begin{pmatrix} 1 \\ 0 \end{pmatrix} \tag{3.37}
$$

$$= \begin{pmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}}\\ -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{pmatrix} \begin{pmatrix} \frac{1}{2} \, 0\\ 0 \, 0 \end{pmatrix} \begin{pmatrix} \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{pmatrix} = \begin{pmatrix} \frac{1}{4} \\ -\frac{1}{4} \end{pmatrix} . \tag{3.38}$$

Hence, the SVD solution equals θ1 = 1/4 and θ2 = −1/4. For these corrector settings one obtains the BPM readings x1 = 1/2 and x2 = −1/2, where indeed the quadratic distance to the target values, i(xt,i − xi)2, assumes a minimum. Not only does SVD determine a pair of corrector settings for which the BPM readings are closest to the solution looked for, but in addition, among all possible such solutions (adding any constant to the above values of θ1 and θ2 would give the same minimum value of i(xt,i − xi)2), it finds that one which also minimizes the corrector strengths, namely the sum of the squares θ2 i .

# 3.6 'Wake Field Bumps'

Through the early 1990's emittance dilutions in the SLC linac were controlled by imposing tight tolerances on injection errors as a precursor to BNS damping [26, 27], steering using both one-to-one correction and localized beam-based alignment [21, 28], and by invoking BNS damping. The name BNS damping refers to V. Balakin, A. Novokhatsky, and V. Smirnov, who in the early 1980s proposed this type of scheme, in which the tail of the bunch is focussed more weakly than the bunch head in order to compensate for the defocusing effect of the single-bunch wake field [29]. At the SLC the variation of focusing across the bunch was achieved by introducing an energy difference. More details will be discussed in Chap. 4.

As the beam currents were increased, a more localized emittance preservation technique was developed in which empirically determined trajectory oscillations ('bumps') were used to cancel emittance dilutions from transverse wake fields and dispersive errors. While the origins of the disturbances could not be easily localized longitudinally along the linac, the accumulated effects could be cancelled using such bumps and the emittance dilution could be reduced by a factor of almost ten [30]. The effect on the beam was determined by emittance measurements near the end of the linac.

Two trajectories in both x and y are shown [30] in Fig. 3.17. Both trajectories produced about the same small emittance measured near the end of the linac. Notice the vertical scale which shows excursions of nearly 750 μm peakto-peak. While wake field bumps were used for many years, it became clear as the currents were increased that this technique was inherently unstable; small (e.g., thermal) changes in the reference line phase, for example, changed the phase advance over the bump range so that even this more localized correction scheme was not sufficiently local to be stable against realistic variations in the accelerator. Both trajectories in Fig. 3.17 resulted in about the same

Fig. 3.17. Two measured orbits with empirically determined coherent betatron oscillations used to cancel accumulated wake field and dispersion errors (Courtesy J. Seeman, 2000)

final beam emittances indicating that the procedure did not converge to a unique solution.

Physical insights were gained by simulations carried out using the program LIAR [31]. The model included representative amplitudes of the wake field bumps which minimized the relative emittance growth at the locations of the measurements. The simulated relative growth in normalized emittance [32, 33] is shown in Fig. 3.18 as a function of position along the linac. Several important conclusions were drawn from this and other simulation results:

Fig. 3.18. Simulated emittance growth as a function of position along the linac. The locations of the feedback loops, which controlled the amplitude of the wake field bumps, are shown along with the locations of the emittance measurements (Courtesy R. Assmann, 2000)

# 3.7 Dispersion-Free Steering

So far we have described one-to-one steering which is a first step in orbit optimization but is imperfect as minimization of the BPM reading in a displaced quadrupole generates dispersive errors, beam-based alignment of quadrupole displacements which works beautifully at low beam currents where there is no wake field-generated dispersion, and wake field bumps which while more local than BNS damping is highly sensitive to small perturbations in the electromagnetic optics. With perfect implementation of either procedure, dispersive emittance dilutions may still result. As an example, consider a closed trajectory bump of the kind illustrated in Fig. 3.3. It has been shown [38] using LIAR and realistic optical parameters of the SLC linac that a closed 100 μm π-bump at a quadrupole located early in the linac generates nearly 0.5 mm dispersion at the end of the linac. Naively, about 6 such bumps acting independently would produce a dispersive emittance contribution equal to the final emittances typically achieved at the SLC.

Dispersion-free steering [35, 36, 38] is an algorithm which corrects even more locally the dispersive errors from misaligned quadrupoles and the dispersive errors arising from transverse wake fields. For mostly technical reasons (e.g., data acquisition and processing time) implementation was unduly delayed at the SLC. Dispersion-free steering (and rf phase stability [39, 40, 41]) proved crucial for maintaining stable linac emittances at the SLC.

The centroid trajectory1 is given by

$$x\_j = \sum\_{i=1}^{j-1} \sqrt{\frac{E\_i}{E\_j}} \sqrt{\beta\_i \beta\_j} \theta\_i \sin(\phi\_j - \phi\_i) \tag{3.39}$$

$$=\sum\_{i=1}^{j-1} R\_{12}^{ij} \theta\_i \,, \tag{3.40}$$

where the damping factor -Ei/Ej has been included. To constrain the system, one can equivalently change the beam energy (which may be difficult in practice) or scale the strength of the lattice, i.e., the strengths of the quadrupoles. Then the orbit difference is measured for a deflection applied in the original lattice and in the scaled lattice. This orbit difference amounts to

$$\begin{split} \Delta x\_{j} &\approx \sum\_{i=1}^{j-1} \left[ R\_{12}^{ij} - \sqrt{\frac{E\_{i}}{E\_{j}}} \sqrt{\beta\_{i,\kappa} \beta\_{j,\kappa}} \sin(\phi\_{j,\kappa} - \phi\_{i,\kappa}) \right] \theta\_{i} \\ &= \sum\_{i=1}^{j-1} R\_{12,\kappa}^{ij} \theta\_{i} \,, \tag{3.41} \end{split} \tag{3.41}$$

where βi,κ and φi,κ denote the beta function and betatron phase in the scaled lattice, and the change in lattice focussing is expressed by the parameter

$$
\kappa = \frac{\Delta K}{K} + 1\,,\tag{3.42}
$$

where K is the quadrupole strength.

1 a warning: intrabunch position-energy correlations, when projected, may result in measured centroid displacements which underestimate the contributions from off-axis bunch tails

The function to be minimized is

$$\sum\_{j} \left[ x\_j - \sum\_{i} M\_{ji} \theta\_i \right]^2,\tag{3.43}$$

where xj is an m × 1 vector containing the difference measurements and the fitting function is given by i Mjiθi where θi is an n×1 vector of unknowns. In the simplest case, the coefficients Mji represent an m×n matrix containing the transfer matrix elements Mji = Rij 12.

In practice it is not difficult to minimize not only the absolute orbit, but simultaneously several (k) difference orbits Δx. In this case xj is a (k + 1) × 1 vector containing the difference measurements and the absolute orbit, Mji is a (k + 1) × n matrix and θ remains an n × 1 matrix. This approach was used at the SLC where in addition, to overconstrain the solution and minimize systematic errors arising from magnet hysteresis, the measurements were performed for k = 4 or 5 values of κ corresponding to energy variations of +5% to −30%. In later years, the problem of magnet hysteresis was eliminated and the application became noninvasive as two independent measurements could be obtained without changing the lattice by measuring independently the orbits of the electrons and positrons which passed through the same lattice. The opposite charge of electrons and positrons was equivalent to a lattice scaling by −200%.

Shown in Fig. 3.19 are absolute (κ = 1.0) and difference trajectories (κ = 0.9, 0.8, and 0.7) measured after trajectory steering of the SLC linac [38] using 20-pulse BPM averaging. With an equivalent energy change of 30% (κ = 0.7), a difference trajectory of up to 1.5 mm was observed. Similar measurements made after iteration of dispersion-free steering [38] are given in Fig. 3.20. Iteration proved useful to reduce sensitivity to errors in the

Fig. 3.19. Absolute and difference vertical trajectories measured after trajectory steering before dispersion-free steering of the SLC linac (Courtesy R. Assmann, 2000)

Fig. 3.20. Absolute and difference vertical trajectories measured after dispersionfree steering in the SLC linac (Courtesy R. Assmann, 2000)

assumed optics even though experience showed that the first iteration yielded the largest improvements. With κ = 0.3, neglecting the errant point due possibly to a bad BPM near sector 20, the maximum orbit difference after dispersion-free steering was reduced from 1.5 mm to less than 200 μm. Notice that the rms of the measurements of the absolute orbit are actually larger following dispersion-free steering. This suggests that significant BPM and/or quadrupole misalignment errors were still present.

# 3.8 Errors

For simplicity of expression, measurement errors have been neglected up to now. At the SLC, error sources and their typical rms contributions included BPM resolution errors σ(xj ) < 10 μm, BPM misalignments σbpm ∼ 100 μm, and systematic errors arising from beam jitter and/or slow drifts σsys ∼ 20 μm. To propagate the measurement errors used in the minimization procedures, a weighting function may be defined as

$$w\_j = \frac{1}{\sum\_j \sigma\_{m,j}^2},\tag{3.44}$$

where the subscripts m give the different error sources and j is a sum over the BPM measurements. The functions to be minimized then are (c.f. (3.18), (3.27), and (3.43))

$$\sum\_{j} \qquad \left[\frac{x\_j - \sum\_{i} M\_{ij}\theta\_i}{\sum\_{m} \sigma\_{m,j}^2}\right]^2 \qquad \text{(one-to-one starting)},$$

$$\sum\_{k} \left[\frac{x\_m - \left(x\_k - x\_q - x\_{\text{bpm}}\right)}{\sum\_{m} \sigma\_{m,j}^2}\right]^2 \text{ (beam} - \text{based alignment)},$$

$$\sum\_{j} \quad \left[\frac{x\_j - \sum\_{i} M\_{ij}\theta\_i}{\sum\_{m} \sigma\_{m,j}^2}\right]^2 \quad \text{ (dispersion-free shearing)}\,. \tag{3.45}$$

A goodness of fit parameter, or χ−squared may be correspondingly constructed. In the dispersion-free steering example given above for which both the trajectory and the trajectory differences were to be simultaneously minimized,

$$\chi^2 = \sum\_{j} \left[ \frac{x\_j^{\cdot 2}}{\sigma\_{\text{bpm}}^{2}} + \sum\_{\kappa} \frac{\Delta x\_{j,\kappa}^{\cdot}^{\cdot}}{\sigma\_{\text{sys}}^{\cdot 2}} \right],\tag{3.46}$$

where the second summation over κ corresponds to the difference orbits for the different energy scalings under which the measurements were made. The errors from BPM resolution were assumed to be negligible and the summation over errors has been simplified to reflect the dominating errors; that is, the systematic errors contribute less than the alignment errors in the measurements of the absolute trajectories, while in the measurements of difference trajectories the BPM misalignments cancel and the associated errors are therefore set to zero.

# 3.9 Orbit Feedback

Automated feedback systems that continually stabilize the beam orbit are becoming more common in accelerators. A comprehensive overview of orbit feedback system design can be found in [42]. A simple orbit feedback maintains a constant orbit by adjusting the strength of 2 or 4 steering correctors based on BPM readings. Many orbit feedback systems employ an SVD algorithm which flattens the orbit while at the same time minimizing the strength of the correctors.

Slightly more complicated feedback loops are designed so that they maintain both the beam orbit and the beam energy. Orbit and energy can be separated using BPMs at dispersive locations. The orbit is corrected via steering correctors; the beam energy by adjustments to some upstream rf phase.

The effectiveness of a feedback can be tested by measuring its response to a step change. An example in Fig. 3.21 shows the response of an SLC feedback loop to a sudden step change in energy. The picture illustrates the improvement achieved by increasing the number of feedback BPMs to better constrain the fit. Specifically, 2 BPMs separated by −I were included so that the sum of the (difference) signals from the two BPMs gave the dispersive contribution to the particle orbit independent of the betatron component. In addition to having a well understood accelerator model, for accelerators with high repetition frequency (with intra-pulse spacing comparable to the the response time of the correction magnets) it was found necessary to carefully match the response times of correction magnets [43].

Fig. 3.21. Response of the orbit and energy feedback in the ring-to-linac transfer line of the SLC to a fast step change in energy [43]: (left) before and (right) after additional BPMs were included in the feedback loop

There are different techniques to calibrate the local transport matrices between correctors and BPMs within each feedback loop, which are used to continually compute the excitation of the feedback steering correctors. For example, the induced change in orbit position and angle can be measured as a function of the individual feedback corrector strengths.

If there are successive feedback loops on a beam line, these loops could interfere with each other, and cause unwanted orbit oscillations due to "doublecompensation". This interference can be avoided by one of 4 different approaches [42]: (1) orthogonality of correction, (2) different feedback response time, (3) inter-loop communication (feedback "cascades") and (4) integration of loops into one global loop. The orbit feedbacks in the SLAC linac are connected by a so-called adaptive cascade, where each feedback passes its information to the loop downstream, in order to avoid multiple corrections of the same perturbation. The linear transport matrix between successive loops is monitored and updated continuously using the naturally occurring beam-orbit variation.

# 3.10 Excursion – AC Dipole

Not always is there a well defined closed orbit in a storage ring which is stable from turn to turn. As an example of a case without regular closed orbit, we consider a dipole field that is modulated at a frequency near the fractional part of the betatron tune. We assume that the field changes on successive turns, but has a sinusoidal variation, so that it repeats after a larger number of turns. In this case there is a new dynamic 'closed orbit' which returns to its starting point after a full modulation period. The amplitude of this periodic orbit is [44]

$$
\hat{x} = \frac{\beta\_x (B\_{\text{mod}} l\_{\text{dip}})}{4\pi (B\rho)\Delta Q},
\tag{3.47}
$$

where Bmod is the amplitude of the dipole-field modulation, ldip the length of the dipole, (Bρ) the magnetic rigidity, ΔQ = |Qx −Qm −k|, Qx the betatron tune, Qm the modulation tune, and k an integer which minimizes ΔQ. The amplitude of the dynamic orbit increases inversely with the frequency difference of modulation and betatron oscillations. Such a dynamic closed orbit may be used to increase the strength of spin resonances in order to preserve the polarization during acceleration [45] (see also the discussion of polarization in Chap. 9) or it can aid in measuring lattice nonlinearities [44] without diluting the transverse emittance. First tests were successfully performed at the BNL AGS [44, 45] and the CERN SPS [46].

# Exercises

### 3.1 Design of an Orbit Feedback Loop

Write an algorithm for orbit correction in one plane assuming uncoupled, linear transport. Let the beam position and angle be detected using two BPMs and use two fast corrector dipoles for implementing the desired deflections. Introduce (assumed known) relative phase advances and beta functions as needed to take into account phase differences between the correctors, between the BPMs, and between the correctors and BPMs. Comment on the optimum phase advances between the correctors and BPMs and comment on the optimum beta functions at the locations of the measurements and corrections. Consider

### 3.2 Linac Dispersion and Orbit Correction

a) Dispersion for a free betatron oscillation. Consider a beam deflected by an angle θ at s = 0. In a smooth approximation, and ignoring wake field effects, the betatron motion of a single particle with relative momentum deviation δ1 is

$$x\_1''(s) + \frac{k\_\beta^2}{1 + \delta\_1} x\_1(s) = \frac{\theta}{1 + \delta\_1} \delta(s) \,, \tag{3.48}$$

where the last δ(s) is the Dirac delta function, indicating a single deflection of strength θ at location s = 0, and kβ = √ K is the wave number of the betatron oscillation. Equation (3.48) describes the motion of an individual offmomentum particle in the bunch, as well as of the bunch centroid if the latter experiences a momentum offset δ1. The equality of the particle and centroid motion (for small bunch charges) is an advantage, since the single-particle dispersion can be measured by observing the response of the centroid motion to an energy error. Solve (3.48), linearize the solution in δ1 and determine the dispersion for kβsδ1 1.

b) Dispersion behind a π bump. A one-to-one orbit correction can be thought of as a superposition of π bumps. Calculate the dispersion generated by a single π bump represented by two kicks θ, at s1 = 0 and s2 = π/kβ.

# 4 Transverse Beam Emittance Measurement and Control

The beam emittance xyz represents the volume of the beam occupied in the six dimensional phase space (x, x , y, y , φ, δ), where x and y are the transverse positions, x and y are the transverse angles, φ is the time-like variable representing the relative phase of the beam, and δ is the relative beam momentum error. Using the notation of the beam matrix Σbeam introduced in Chap. 1, the 6-dimensional emittance is

$$
\epsilon\_{xyz} = \det \,\, \Sigma\_{\text{beam}}^{xyz}.\tag{4.1}
$$

Considering now only the horizontal plane, the corresponding 2-dimensional horizontal emittance is obtained from

$$
\epsilon\_x = \sqrt{\langle x^2 \rangle \langle x'^2 \rangle - \langle xx' \rangle^2}, \tag{4.2}
$$

where the first moments have been subtracted, and the average (...) is taken over the distribution function of the beam; recall also (1.27–1.29). An analoguous expression holds for the vertical plane. For a coupled system, the general form of (4.1) must be taken.

Control of the beam emittance and prevention of emittance dilutions are mandatory for achieving high brilliance in light sources and high luminosity in colliding beam accelerators. Some sources of emittance dilution, e.g., beam-gas scattering, are inevitable and can be reduced only via hardware improvements. In the same vane, processes involving space-charge dilutions, which constitute a predominant limitation for low-energy ion or proton beams, have after-theobservation been treated by fundamental changes in the acceleration optics, as illustrated, e.g., by the FNAL linac upgrade, and a new booster ring at BNL. Another class of dilutions involving man-made sources, e.g., component vibration, ground motion, power supply regulation, etc., may be curable using sophisticated measurement devices and feedback or feedforward schemes. In this case, the crucial ingredient leading to improved performance is the detection of the offending presence, e.g., using model-independent analysis, as discussed in Chap. 2, or more commonly analyses in the frequency domain.

An interesting example of the latter case is shown in Fig. 4.1. Plotted is the Fourier transform of pulse-by-pulse BPM measurements from the SLC linac. The data evidence a strong component at about 1 Hz which seemed

This chapter has been made Open Access under a CC BY 4.0 license. For details on rights and licenses please read the Correction https://doi.org/10.1007/978-3-662-08581-3\_13

to partly explain the 1 Hz 'wave' in the backgrounds recorded by the SLD detector. However, at this time, the data acquisition was limited to 30 Hz (with a beam repetition frequency of 120 Hz). As a result, the sampled data were aliased; that is, the Fourier line appearing at 1 Hz corresponded actually to 59 Hz as verified later with a faster (120 Hz) data acquisition speed. The source of these excitations was eventually traced back to asynchronous (to the beam) operation of the cooling water pumps for the linac quadrupoles. Once the disturbance was identified1, corresponding measures were implemented – in this case, damping of the vibrations resulting from the turbulent water flow and modifications to the pump impellers – and the impact on the beam was correspondingly reduced. Identification of the error sources evidenced by the remaining peaks in Fig. 4.1 was also partially successful [1].

Fig. 4.1. Frequency spectrum of beam motion in the SLC linac, revealing the contribution of water pumps operating at 59 Hz [1]

Other common situations which lead to emittance dilutions arise from poor setting of the accelerator including poor "matching" between accelerator subsystems, residual betatron coupling (which may also be caused by unknown field errors), or spurious dispersion which arises either directly from magnet misalignments or from imperfect attempts to correct for alignment errors by one-to-one steering. The latter may arise from insufficient knowledge of the accelerator optics (as limited by the BPM resolution, for example) or

1 which proved difficult since in fact there actually was at that time a 1 Hz component on the beam introduced incorrectly by the orbit feedback loops as these also reacted on the aliased frequency

may arise intrinsically from the alignment procedure itself (as is the case, e.g., of using the one-to-one steering algorithm).

Finally, there are processes which are in practice hard to predict that also lead to emittance dilutions. Such processes, predominant especially with high beam intensities, include, at high single-bunch charge both transverse and longitudinal wakefields effects leading to beam instabilities in circular accelerators and to beam centroid excursions in linear accelerators, and at high single-bunch charge and high total current also beam instabilities arising from ions or electron clouds.

In all the above cases, the diagnosis is precedented by an accurate measurement of the beam emittance. In this chapter, standard emittance measurement techniques will be outlined in a step-by-step manner. Then some methods used to minimize the beam emittance by changing the underlying accelerator optics will be reviewed.

# 4.1 Beam Emittance Measurements

In this section we describe not single particle transport, but transport of the beam as a whole. The beam quality will be characterized by the beam emittance which is often measured with reference to a particular plane of interest; i.e., the horizontal, vertical, or longitudinal emittance. Methods for measuring the beam emittance and for parametrizing the degree of mismatch will be outlined.

We note that many novel measurement techniques have recently been developed for measuring very small beam sizes including interferometric methods, applied in the beamline at the SLAC final focus test facility [2] or in the ATF damping ring [3], or tomographic phase space reconstruction applied at the TTF linac [4]. A recent review of similar topics is given in [5]. In this section we restrict the discussion to emittance measurement techniques, which are based on commonly used and relatively simple hardware. For illustration we describe measurements and present experimental data obtained using wire scanners although the measurement principles have been similarly applied with the use of fluorescent screens.

### 4.1.1 Single Wire Measurement

The beam emittance can be measured by varying the field strength of a quadrupole located upstream of a single wire or screen. In general this will lead to trajectory and beam-size changes downstream, and, hence, is referred to as an "invasive" measurement. The measurement could be made less invasive by simultaneously adjusting another quadrupole downstream of the wire scanner, so as to compensate for the change in the beta function induced by the first quadrupole. This has been rarely done in the past, but is a proposed scheme for future linear colliders.

#### 102 4 Transverse Beam Emittance Measurement and Control

The total transfer matrix of interest here is R = SQ, where S denotes the known transfer matrix between the quadrupole and the wire, and Q is the transfer matrix of the quadrupole:

$$Q = \begin{pmatrix} 1 & 0 \\ K & 1 \end{pmatrix},\tag{4.3}$$

where we have invoked the thin-lens approximation, which is valid if the length of the quadrupole is short compared to its focal length f = 1/K. After matrix multiplication, one obtains

$$R = \begin{pmatrix} S\_{11} + KS\_{12} & S\_{12} \\ S\_{21} + KS\_{22} & S\_{22} \end{pmatrix},\tag{4.4}$$

where the coefficients Sij are the components of the matrix S. Expanding the matrix product for the 2 × 2 beam matrix Σx beam = (SQ)Σx beam,0(SQ)T and equating the (1,1) elements on both sides, the square of the horizontal beam size follows as

$$\begin{aligned} \Sigma\_{11}(=\langle x^2 \rangle) &= (S\_{11} \, ^2 \Sigma\_{11\_0} + 2S\_{11} S\_{12} \Sigma\_{12\_0} + S\_{12} \, ^2 \Sigma\_{22\_0}) \\ &+ (2S\_{11} S\_{12} \Sigma\_{11\_0} + 2S\_{12} \, ^2 \Sigma\_{12\_0}) K + S\_{12} \, ^2 \Sigma\_{11} K^2, \end{aligned} \tag{4.5}$$

which is quadratic in the field parameter K.

To make use of these results in an emittance measurement, the following procedure is often employed:

1. For each value of quadrupole field strength K, the wire is scanned and the amplitude of the response measured by a detector is obtained as a function of wire position.

2. For each wire scan at fixed K, the distribution is fitted with a Gaussian of the form

$$f(x) = f\_0 + f\_{\text{max}} e^{-\frac{(x-\langle x\rangle)^2}{2\langle x^2\rangle}},\tag{4.6}$$

where f0 is the baseline level offset and fmax is the peak value of the Gaussian distribution.

3. The fitted beam size x2 is plotted as a function of K.

4. The result is fitted with a parabola. One parametrization for the fit [6] is

$$\begin{split} \Sigma\_{11} &= A(K - B)^2 + C \\ &= AK^2 - 2ABK + (C + AB^2) \end{split} \tag{4.7}$$

5. The Σ matrix is reconstructed by equating coefficients of (4.5) and (4.7):

$$A = S\_{12}^2 \Sigma\_{11} \,, \tag{4.8}$$

$$-2AB = 2S\_{11}S\_{12}
\Sigma\_{11} + 2S\_{12}^2
\Sigma\_{12},\tag{4.9}$$

$$C + AB^2 = S\_{11} \, ^2\Sigma\_{11} + 2S\_{11}S\_{12}\Sigma\_{12} + S\_{12} \, ^2\Sigma\_{22} \,, \tag{4.10}$$

and solving for Σ11, Σ12 (= Σ21), and Σ22. The results are

$$
\Sigma\_{11} = A / S\_{12}\,^2,\tag{4.11}
$$

$$
\Sigma\_{12} = -\frac{A}{S\_{12}} \left( B + \frac{S\_{11}}{S\_{12}} \right) \,, \tag{4.12}
$$

$$\Sigma\_{22} = \frac{1}{S\_{12}} \left[ (AB^2 + C) + 2AB \left( \frac{S\_{11}}{S\_{12}} \right) + A \left( \frac{S\_{11}}{S\_{12}} \right)^2 \right]. \tag{4.13}$$

6. The beam emittance is then calculated from the determinant of the beam matrix x = det Σx beam and the errors are propagated:

$$\det \,\Sigma\_{\text{beam}}^x = \Sigma\_{11}\Sigma\_{22} - \Sigma\_{12}^{\ \ 2} \tag{4.14}$$

$$=AC/S\_{12}^{4}\,,\tag{4.15}$$

so that

$$
\epsilon\_x = \sqrt{AC}/S\_{12}^2 \,. \tag{4.16}
$$

Fig. 4.2. Example transverse beam emittance measurements from the SLC prior to injection into the main linac using a single wire. The fit parameters for the horizontal emittance (top) were A = 3494 ± 52, B = −118.8 ± 0.03, C = 3.2 × 104 ± 297 with χ2/dof = 1.5 giving x = 12.9± 0.2 nm-rad or γx = 30.1 ± 0.4 μmrad. In the vertical plane (bottom), A = 158.5 ± 3.5, B = −129.3 ± 0.08, and C = 5.1 × 103 ± 103 with χ2/dof = 4.6 giving y = 1.71 ± 0.02 nm-rad or γx = 3.98 ± 0.04 μm-rad

#### 104 4 Transverse Beam Emittance Measurement and Control

The above results also give the beam-ellipse parameters αx, βx, and γx:

$$
\beta\_x = \frac{\Sigma\_{11}}{\epsilon} = \sqrt{\frac{A}{C}},\tag{4.17}
$$

$$\alpha\_x = -\frac{\Sigma\_{12}}{\epsilon} = \sqrt{\frac{A}{C}} \left( B + \frac{S\_{11}}{S\_{12}} \right) \,, \tag{4.18}$$

$$\gamma\_x = \frac{S\_{12}}{\sqrt{AC}} \left[ (AB^2 + C) + 2AB \left( \frac{S\_{11}}{S\_{12}} \right) + A \left( \frac{S\_{11}}{S\_{12}} \right)^2 \right]. \tag{4.19}$$

As a useful check, the beam-ellipse parameters should satisfy (βxγx−1) = α2 x.

An example of emittance measurements in the two transverse planes x and y is shown in Fig. 4.2. Notice that the optics at the wire scanners has been optimally chosen to allow simultaneous measurement of the beam waists in both transverse planes.

### 4.1.2 Multiple Wire Measurement

The beam emittance in a transport line or linac may be measured (in many applications noninvasively) using multiple wire scanners. Here, the quadrupole gradients are fixed, but the R matrices between s0 and the different wire scanners (or other beam-size monitors) are different. If there are no coupling elements, three measurements using three wire scanners are required. With coupling, four wire scanners are needed (in this case each wire scanner should be equipped with several wires oriented at different angles in the transverse plane, e.g., a horizontal, a vertical and a wire oriented at 45◦ in the case of 'round beams' with equal horizontal and vertical beam size). The optimum wire locations for maximum sensitivity (without coupling) are such that the separation between wires corresponds to a difference in betatron phase advance Δμ of 90◦/Nw, where Nw is the number of wires used in the measurement.

The matrix equation to be solved is

$$
\begin{pmatrix}
(\sigma\_x^{(1)})^2\\ (\sigma\_x^{(2)})^2\\ (\sigma\_x^{(3)})^2\\ \dots\\ (\sigma\_x^{(n)})^2
\end{pmatrix} = \begin{pmatrix}
(R\_{11}^{(1)})^2 \, 2R\_{11}^{(1)} R\_{12}^{(1)} \, (R\_{12}^{(1)})^2\\ (R\_{11}^{(2)})^2 \, 2R\_{11}^{(2)} R\_{12}^{(2)} \, (R\_{12}^{(2)})^2\\ (R\_{11}^{(3)})^2 \, 2R\_{11}^{(3)} R\_{12}^{(3)} \, (R\_{12}^{(3)})^2\\ \dots\\ (R\_{11}^{(n)})^2 \, 2R\_{11}^{(n)} R\_{12}^{(n)} \, (R\_{12}^{(n)})^2
\end{pmatrix} \begin{pmatrix}
\beta(s\_0)\epsilon\\ -\alpha(s\_0)\epsilon\\ \gamma(s\_0)\epsilon
\end{pmatrix} . \tag{4.20}
$$

This equation is applicable for both a multiple wire measurement or for a quadrupole scan. The superindex within parenthesis refers to the different different measurements; i.e., it either corresponds to the setting of some quadrupole magnet, in the case of a quadrupole scan, or to a different wire scanner or monitor, in the case of a multi-wire emittance measurement. The superindex '2' is meant to denote the square of the quantity. At least 3 measurements are required in order to solve for the three independent parameters , β(s0) and α(s0).

To simplify the notation, let us denote the n × 3 matrix on the righthand side of (4.20) as B, the n-component vector on the left side by Σx = (σ(1)2 x , ..., σ(n)2 x ), and the 3-component vector on the far right by

$$\mathbf{o} = \left(\beta(s\_0)\epsilon, -\alpha(s\_0)\epsilon, \gamma(s\_0)\epsilon\right). \tag{4.21}$$

The equation is then

$$
\Sigma\_x = \mathbf{B} \cdot \mathbf{o} \,. \tag{4.22}
$$

The problem of determining the elements of the vector o can be solved by a simple least-squares fit. We have to minimize the sum

$$\chi^2 = \sum\_{l=1}^n \frac{1}{\sigma\_{\Sigma\_x^{(l)}}^2} \left( \Sigma\_x^{(l)} - \sum\_{i=1}^3 B\_{li} o\_i \right)^2,\tag{4.23}$$

where σΣ(l) x denotes the rms error of Σ(l) x = σ(l)2 x . This error is obtained from the fit to the lth wire scan which determines the rms beam size σ(l) x .

We find it convenient to normalize the coordinates Σ(l) so that the rms error is 1, introducing

$$
\hat{\Sigma}\_x^{(l)} = \frac{\Sigma\_x^{(l)}}{\sigma\_{\Sigma\_x^{(l)}}},\tag{4.24}
$$

and

$$
\hat{B}\_{li} = \frac{B\_{li}}{\sigma\_{\Sigma\_x^{(l)}}}.\tag{4.25}
$$

Forming a symmetric n × n covariance matrix

$$\mathbf{T} = (\hat{\mathbf{B}}^t \cdot \hat{\mathbf{B}})^{-1},\tag{4.26}$$

the least-squares solution to (4.22) is

$$\mathbf{o} = \mathbf{T} \cdot \hat{\mathbf{B}}^t \cdot \hat{\boldsymbol{\Sigma}}\_x \,, \tag{4.27}$$

and the error of any scalar function f(o) is given by

$$\left(\sigma(f)\right)^{2} = \left(\nabla\_{\mathbf{o}}f\right)^{t} \cdot \mathbf{T} \cdot \left(\nabla\_{\mathbf{o}}f\right). \tag{4.28}$$

In particular, the errors of the parameters o themselves are

$$
\sigma\_{o\_i} = \sqrt{T\_{ii}}\,. \tag{4.29}
$$

Once the components of o are known, we still need to perform a simple nonlinear transformation to infer , β, and α:

106 4 Transverse Beam Emittance Measurement and Control

$$
\epsilon \, \, = \sqrt{o\_1 o\_3 - o\_2^2} \,, \tag{4.30}
$$

$$
\beta = o\_1/\epsilon \text{, and } \tag{4.31}
$$

$$
\alpha = -\mathbf{o}\_2/\epsilon.\tag{4.32}
$$

The error propagation is straightforward, using (4.28).

A possible procedure for the multiple wire emittance measurement is as follows:

1. Each wire is scanned to obtain detector counts versus wire position x.

2. For each wire scan, the distribution is fit with a Gaussian using (4.6).

3. The Σ matrix is reconstructed using (4.20), the transfer matrix elements Ri from the model, and the σi from the measurements.

4. The emittance is calculated = √det Σbeam.

5. The ellipse parameters α = −Σ12/ , β = Σ11/ , and γ = Σ22/ are calculated, if desired.

### 4.1.3 Graphics

Increased operational efficiency may be obtained from a meaningful graphical representation of the experimental data. In the multiple wire emittance measurement it is useful to project the measurements to a single point along the accelerator and to plot the normalized phase space. The emittance , multiplied by π, corresponds to the area of the ellipse parametrized by

$$
\epsilon = \gamma\_x x^2 + 2\alpha\_x x x' + \beta\_x x'^2. \tag{4.33}
$$

Since βγ =1+ α2,

$$\epsilon = \frac{1}{\beta\_x} [x^2 + (\alpha\_x x + \beta\_x x')^2] = \frac{1}{\beta\_x} (x^2 + p\_x^{-2}) \,, \tag{4.34}$$

where px = αxx + βxx is the coordinate orthogonal to x.

A succinct representation of the measured beam emittances is obtained by the following procedure (applied at the SLC), which displays the data in the normalized phase space, so that deviations from the design values are immediately obvious. The wire orientations are also plotted to indicate the phase space coverage provided by the wires:

1. Plot the design rms ellipse in the coordinates

$$\left(\frac{x}{\sqrt{\beta\_x}}, \frac{\alpha\_x x + \beta\_x x'}{\sqrt{\beta\_x}}\right) \tag{4.35}$$

at some reference point s along the trajectory. This results in a circle. Normalize the design ellipse to unit radius.

2. Plot also the ellipse obtained from the measurements of the ellipse parameters transposed to the reference point. Apply the same normalization as in step 1.

3. Using the lattice model, project each beam-size measurement back to the reference point and add the result as a line to the figure; that is, for each point (σx,w, x w), where σx,w is the beam size measured at the wire, apply an inverse mapping to the reference point:

$$
\begin{pmatrix} x \\ x' \end{pmatrix}\_{\text{ref point}} = R^{-1} \begin{pmatrix} \sigma\_{x,w} \\ x'\_w \end{pmatrix}, \tag{4.36}
$$

where R is the 2×2 transport matrix from the reference point to the location of the wire. Here x w represents the undetermined angle variable (divergence) at the wire which parametrizes the location along the straight line in the phase space defined by (4.35). The slope of this line is related to the phase advance between the reference point and the wire.

An accompanying display of numbers should summarize the measurements which might include the measured and expected beam widths at each of the wires, the measured and design beam emittances, as well as the beam intensity. In addition, a measure of the degree of "mismatch" is useful. This will be further discussed in the next section.

Fig. 4.3. Graphics output of multiple-wire transverse beam emittance measurement in the injector linac at the SLC

An example of this graphics is shown in Fig. 4.3. The corresponding raw data are given in Fig. 4.4. From Fig. 4.3 it is immediately obvious that while the measured ellipse has roughly the same emittance as the design circle (the horizontal emittance is 208.8 ± 9.9 [mm-mrad] compared to the design of 200 [mm-mrad], the vertical emittance is 323 ± 26.7 [mm-mrad] compared to the design of 200 [mm-mrad]), the ellipse orientation is incorrect. As will be shown in the next section, if this beam were allowed to propagate uncorrected,

Fig. 4.4. Raw data showing individual wire scans used in the summary display of Fig. 4.3 and the "asymmetric Gaussian" fits of (4.37)

the final emittance titled Bmag would be 390.0 ± 10.2 [mm-mrad] in x and 543.3 ± 13.2 [mm-mrad] in y. The emittance dilution factor Bmag represents the degree of the mismatch. From Fig. 4.3 can be deduced immediately the degree of phase space coverage spanned by the wires. In the horizontal plane, for example, the wire orientations are about 0◦, −45◦, −22.5◦, and −67.5◦, which is ideal for a 4-wire emittance measurement.

The "measured ellipse", that is the ellipse that was reconstructed from the beam widths obtained by Gaussian fits to the individual wire scans, does not always represent the true rms emittance of a Gaussian beam distribution. This can be seen by inspecting the raw data shown in Fig. 4.4. For more complex beam distributions, a somewhat better characterization is achieved by applying an "asymmetric Gaussian" fit to the wire-scan measurement, in which the left and right hand sides of the measured beam profile are approximated independently by two separate Gaussians. For example, the fitting function used at the SLC [7] was

$$f(x) = f\_0 + f\_{\text{max}} \exp\left[ -\frac{(x - \langle x \rangle)^2}{2\langle x^2 \rangle (1 + \alpha [\text{sign}(x - \langle x \rangle)])} \right],\tag{4.37}$$

where α represents an asymmetry factor and is zero for a perfect Gaussian2. The σ for the left and right hand sides of the fitted distribution are σ = x2(1 ± α). For the ellipse reconstruction the average σ was used. When large tails are present in the raw data this more accurately represents the beam distribution. Based on the raw data it is clear, however, that even with the more sophisticated fitting algorithm, the fit only marginally represents the actual distributions.

For reasonably well "matched" beams, the graphical summary display is most useful for quick evaluation of the beam. In this example the deviations between the design and measured ellipse in the graphics suggest that a closer inspection of the raw data may be warranted. The "double-humps" in the single-wire measurements are characteristic of an upstream error; a beam, if kicked transversely, will filament, i.e., it loses coherency due to the natural spread in the phase advance between particles, resulting in an increased emittance and the characteristic double humps.

If a wire is mounted at 45◦ with respect to x and y (a "u-plane" wire), then it is also possible to measure the coupling between the horizontal and vertical plane. The (4×4) Σ-matrix is

$$
\Sigma\_{\text{beam}}^{xy} = \begin{pmatrix}
\Sigma\_{11} & \Sigma\_{12} & \Sigma\_{13} & \Sigma\_{14} \\
\Sigma\_{21} & \Sigma\_{22} & \Sigma\_{23} & \Sigma\_{24} \\
\Sigma\_{31} & \Sigma\_{32} & \Sigma\_{33} & \Sigma\_{34} \\
\Sigma\_{41} & \Sigma\_{42} & \Sigma\_{43} & \Sigma\_{44}
\end{pmatrix},
\tag{4.38}
$$

where, for example, Σ14 represents the correlation between x and y . Notice that Σ14 = Σ23. Whereas for the single plane uncoupled beam matrix reconstruction a minimum of 3 measurements are required, to fully reconstruct the coupled beam matrix a total of 10 measurements is needed. This includes the 3 measurements in the x plane, 3 in the y plane, and 4 in the u plane. The raw data used in such a coupled emittance measurement is presented in Figs. 4.5–4.7. In this case the raw data are well fitted using Gaussian fits. From these fits, using the uncoupled analysis (with only the x and y wires) presented previously, γ x = (2.20 + / − 0.01) × 10−5 m-r and

2 with high resolution scanners this parameter may also prove useful for characterizing 'banana' beams in future linear colliders

Fig. 4.5. Raw data in the x-plane for emittance measurement with x−y coupling

Fig. 4.6. Raw data in the y-plane for emittance measurement with x−y coupling

γ y = (1.89 +/−0.02)×10−5 m-r. Using the complete data set, and projecting the distributions onto the eigenplanes of the tilted beam ellipses, the fits yielded γ 1 = 2.13×10−5 m-r and γ 2 = 1.71×10−5 m-r, which is in reasonable agreement with the results obtained excluding coupling effects indicating that the coupling is small. However, the error bars were not fully propagated. Recent experience at the ATF has shown that proper wire orientations are

Fig. 4.7. Raw data in the u-plane for emittance measurement with x−y coupling

critical for maximizing sensitivity particularly for flat beam measurements. Moreover, measurement errors can strongly influence the value of the emittance [8] interpreted from the approach given above. More robust tuning procedures for minimizing the linear coupling are given in [8].

### 4.1.4 Emittance Mismatch

In this section we begin by explicitly computing, in two dimensions, the transverse position and angle using the general form of the beam transfer matrix for a periodic lattice. This result is used to calculate the individual elements of the beam transfer matrix and to derive an expression for the mismatch parameter Bmag [9, 10].

The parameter Bmag has an important physical meaning. If a beam is injected into a ring or linac with a mismatch, the beam will filament until its distribution approaches a shape that is matched to the ring or the linac lattice. However, the filamentation causes the beam emittance to increase, such that, after complete filamentation, the emittance is given by the product of Bmag and the initial value of . The mismatch parameter is well suited for analysis in circular machines for which the periodicity is implicit. We will see that the same formalism is useful in describing emittance transport in linear accelerators and transport lines as well.

Derivation of Beam Matrix Elements. According to (1.27) and after subtracting the mean values from all coordinates, the beam matrix is

$$
\Sigma^x\_{\text{beam}} = \begin{pmatrix}
\langle x^2 \rangle & \langle xx' \rangle \\
\langle xx' \rangle & \langle x'^2 \rangle
\end{pmatrix}.
\tag{4.39}
$$

#### 112 4 Transverse Beam Emittance Measurement and Control

From (1.25), the point-to-point transfer matrix is

$$R\_{fi} = \begin{pmatrix} \sqrt{\frac{\beta\_f}{\beta\_i}}(\cos\phi\_{fi} + \alpha\_i \sin\phi\_{fi}) & \sqrt{\beta\_f\beta\_i}\sin\phi\_{fi} \\ \frac{\alpha\_i - \alpha\_f}{\sqrt{\beta\_f\beta\_i}}\cos\phi\_{fi} - \frac{1 + \alpha\_f\alpha\_i}{\sqrt{\beta\_f\beta\_i}}\sin\phi\_{fi} & \sqrt{\frac{\beta\_i}{\beta\_f}}(\cos\phi\_{fi} - \alpha\_f\sin\phi\_{fi}) \end{pmatrix},\tag{4.40}$$

where αf and βf are the ellipse parameters at a (final) observation point f downstream of a reference point, which is denoted by the subscript i. Here φf i is the phase advance between the reference point and the observation point and is equal to

$$
\phi\_{fi} = \int\_{s\_i}^{s\_f} \frac{\mathrm{d}s}{\beta} \,. \tag{4.41}
$$

For a periodic lattice for which α = α0 and β = β0, the periodic, one turn map, transfer matrix Rotm is given by (1.26):

$$R\_{\rm atm} = \begin{pmatrix} \cos \mu + \alpha \sin \mu & \beta \sin \mu \\ -\gamma \sin \mu & \cos \mu - \alpha \sin \mu \end{pmatrix} \,. \tag{4.42}$$

The beam matrix elements after 1 iteration through the periodic lattice are

$$\begin{aligned} \langle x^2 \rangle\_n &= \langle x^2 \rangle\_0 R\_{11}{}^2 + 2 \langle xx' \rangle\_0 R\_{11} R\_{12} + \langle x'^2 \rangle\_0 R\_{12}{}^2, \\ \langle xx' \rangle\_n &= \langle x^2 \rangle\_0 R\_{11} R\_{21} + \langle xx' \rangle\_0 [R\_{11} R\_{22} + R\_{12} R\_{21}] \end{aligned} \tag{4.43}$$

$$\begin{aligned} \langle \rangle\_n &= \langle x^2 \rangle\_0 R\_{11} R\_{21} + \langle xx' \rangle\_0 [R\_{11} R\_{22} + R\_{12} R\_{21}] \\ &+ \langle x'^2 \rangle\_0 R\_{12} R\_{22} \,, \end{aligned} \tag{4.44}$$

$$
\langle x'^2 \rangle\_n = \langle x^2 \rangle\_0 R\_{21}{}^2 + 2 \langle xx' \rangle\_0 R\_{21} R\_{22} + \langle x'^2 \rangle\_0 R\_{22}{}^2. \tag{4.45}
$$

After substitution of the matrix elements of (4.42) into (4.44–4.45), and using cos 2ψ = cos2 ψ − sin2 ψ and sin 2ψ = 2 sin ψ cos ψ,

$$\begin{aligned} \langle x'^2 \rangle\_n &= \frac{1}{2} [\langle x^2 \rangle\_0 + \langle (\alpha x\_0 + \beta x'\_0)^2 \rangle] \\ &- \frac{1}{2} [\langle x^2 \rangle\_0 \left(\alpha^2 - 1\right) + 2\beta \alpha \langle xx' \rangle\_0 + \beta^2 \langle x'^2 \rangle\_0] \cos 2\psi \\ &+ [\alpha \langle x^2 \rangle\_0 + \beta \langle xx' \rangle\_0] \sin 2\psi, \\ \langle x'^2 \rangle\_n &= \frac{1}{2} [\langle x'^2 \rangle\_0 + \langle (\alpha x' + \gamma x)^2 \rangle\_0] \end{aligned} \tag{4.46}$$

$$\begin{aligned} \langle \langle x'^2 \rangle\_n &= \frac{1}{2} [\langle x'^2 \rangle\_0 + \langle (\alpha x' + \gamma x)^2 \rangle\_0] \\ &+ \frac{1}{2} [\langle x'^2 \rangle\_0 - \langle (\alpha x' + \gamma x)^2 \rangle\_0] \cos 2\psi \\ &- [\alpha \langle x'^2 \rangle\_0 + \gamma \langle x x' \rangle\_0] \sin 2\psi \,, \end{aligned} \tag{4.47}$$

$$\begin{split} \langle xx' \rangle\_{n} &= \frac{1}{2} [-\alpha \gamma \langle x^{2} \rangle\_{0} - 2\alpha \langle xx' \rangle\_{0} - \alpha \beta \langle x'^{2} \rangle\_{0}] \\ &+ \left[ \frac{\alpha \gamma}{2} \langle x^{2} \rangle\_{0} + (1 + \alpha^{2}) \langle xx' \rangle\_{0} + \frac{\alpha \beta}{2} \langle x'^{2} \rangle\_{0} \right] \cos 2\psi \\ &+ \frac{1}{2} [-\gamma \langle x^{2} \rangle\_{0} + \beta \langle x'^{2} \rangle\_{0}] \sin 2\psi . \end{split} \tag{4.48}$$

Next, let

$$a = \frac{\beta}{2} [\gamma \langle x^2 \rangle\_0 + 2\alpha \langle xx' \rangle\_0 + \beta \langle x'^2 \rangle\_0],\tag{4.49}$$

$$b = \frac{\gamma}{\beta}a,\tag{4.50}$$

$$c = -\frac{\alpha}{\beta}a,\tag{4.51}$$

and use

$$c\_1 \cos 2\psi + c\_2 \sin 2\psi = \sqrt{c\_1^2 + c\_2^2} \cos(2\psi - \chi), \text{ with } \chi = \tan^{-1}\left(\frac{c\_2}{c\_1}\right). \tag{4.52}$$

Then,

$$\langle x^2 \rangle\_n = a + \sqrt{a^2 - \beta^2 (\langle x^2 \rangle\_0 \langle x'^2 \rangle\_0 - \langle xx' \rangle\_0^2)} \cos(2\psi + \chi\_{\{x^2\}\_0}),\tag{4.53}$$

$$\langle \left\langle x'^2 \right\rangle\_n = b + \sqrt{b^2 - \gamma^2(\langle x^2 \rangle\_0 \langle x'^2 \rangle\_0 - \langle xx' \rangle\_0^{\;2})} \cos(2\psi + \chi\_{\langle x'^2 \rangle\_0}), \tag{4.54}$$

and

$$<\langle xx'\rangle\_n = c + \sqrt{(\langle xx'\rangle\_0 - c)^2 + \left(-\frac{\gamma}{2}\langle x^2\rangle\_0 + \frac{\beta}{2}\langle x'^2\rangle\_0\right)^2} \cos(2\psi + \chi\_{\{xx'\}\_0}),\tag{4.55}$$

where the angles χx2 0 , χx-2 0 , and χxx- 0 , follow from (4.52), (4.46), (4.47), and (4.48), respectively. Note that since = √ det σ is an invariant in the absence of filamentation,

$$
\langle \langle x^2 \rangle\_n \langle x'^2 \rangle\_n - \langle xx' \rangle\_n^2 = \langle x^2 \rangle\_0 \langle x'^2 \rangle\_0 - \langle xx' \rangle\_0^2. \tag{4.56}
$$

The Mismatch Parameter [9] Bmag. Dividing both sides of (4.53) by β 0 = β x20x20 − xx 0, we have

$$\begin{split} \frac{\langle x^2 \rangle\_n}{\beta \epsilon\_0} &= \frac{a/\beta}{\sqrt{\langle x^2 \rangle\_0 \langle x'^2 \rangle\_0 - \langle xx' \rangle\_0^2}} \\ &+ \sqrt{\left(\frac{a/\beta}{\sqrt{\langle x^2 \rangle\_0 \langle x'^2 \rangle\_0 - \langle xx' \rangle\_0^2}}\right)^2 - 1} \left[\cos(2\psi - \chi\_{\{x^2\}\_0})\right] \\ &= \mathbf{B\_{mag}} + \sqrt{\mathbf{B\_{mag}^2 - 1} \cos(2\psi - \chi\_{\{x^2\}\_0})}, \end{split} \tag{4.57}$$

where the mismatch parameter Bmag is defined as

$$\mathbf{B\_{mag}} = \frac{a/\beta}{\sqrt{\langle x^2 \rangle\_0 \langle x'^2 \rangle\_0 - \langle xx' \rangle\_0}}.\tag{4.58}$$

With the ellipse parameters α, β, and γ referring to the steady-state or equilibrium beam distribution, we have

$$\mathbf{B}\_{\rm mag} = \frac{1}{2} \frac{[\gamma \langle x^2 \rangle\_0 + 2\alpha \langle xx' \rangle\_0 + \beta \langle x'^2 \rangle\_0]}{\sqrt{\langle x^2 \rangle\_0 \langle x'^2 \rangle\_0 - \langle xx' \rangle\_0}}. \tag{4.59}$$

Thus, Bmag is the ratio of the area of the decohered beam to the area of the injected beam. The factor of 2 results from the numerator representing an rms area.

Examples of Emittance Dilution due to Mismatch. Emittance dilution results if Bmag = 1 due to the difference in the transverse phase advance of the particles within the bunch. There are multiple sources of such phase advance variations. The two most commonly considered sources depend on the chromaticity or on the amplitude of the betatron oscillations. The chromaticity ξ (≡ Q /Q)=(Δψ/ψ)/δ characterizes the energy dependence of the phase advance where Δψ is the difference in the phase advance of a particle from the mean phase advance of the bunch and δ is the relative energy deviation of that particle compared to the mean energy of the bunch. The amplitude dependence of the phase advance due to sextupole or octupolar magnetic fields is approximately described by

$$2\pi\psi = 2\pi\psi\_0 - \mu a^2 \,, \tag{4.60}$$

where ψ0 is the phase advance for a reference particle on the closed orbit, μ characterizes the strength of the sextupolar or octupolar fields, and a is the betatron oscillation amplitude of the particle. Less commonly considered sources for phase advance variations include wakefield focussing or spacecharge defocussing for high current beams, focussing due to ions or electron clouds, and focussing due to the beam-beam tune shift in colliding beam accelerators.

### a) Periodic Lattice

Let the length of the lattice period be L. Then, as shown in Fig. 4.8, for Bmag = 1, the beam always fills the same area in phase space after each lattice period. The rms area of the ellipse after n turns is

$$
\langle x^2 \rangle\_n = \beta \epsilon\_0 \,, \tag{4.61}
$$

and the beam is said to be matched. Under these conditions, no emittance dilution will occur. In particular, since x2n is independent of the phase advance ψ, the phase space area is unchanged even if the phase of each of the particles in the beam advances differently.

For Bmag > 1, then

$$\begin{split} \epsilon\_n &= \frac{\langle x^2 \rangle\_n}{\beta} \\ &= \epsilon\_0 \left[ \mathbf{B}\_{\text{mag}} + \sqrt{\mathbf{B}\_{\text{mag}}^2 - 1} \cos(2\psi - \chi\_{\langle x^2 \rangle\_0}) \right], \end{split} \tag{4.62}$$

Fig. 4.8. Horizontal phase space for a matched beam in a periodic lattice. The emittance is preserved even if the phase advance is different for different particles (denoted here by the numbers 1 and 2)

as shown in Fig. 4.9. The solid, small ellipse represents the matched ellipse for which Bmag = 1. The shaded ellipse represents the (1σ) rms distribution of the mismatched beam. During the first few traversals of identical lattice segments, the phase advance variations of the different particles may not be obvious. As n approaches infinity, however, the phase advance variations lead to a smearing in the transverse phase space resulting in a larger emittance. This is represented by the area occupied by the hatched ellipse as given by (4.57).

Fig. 4.9. Schematic of the horizontal phase space for a mismatched beam in a periodic linear lattice after 1, 2, 3 and N 1 periods. The emittance is not preserved: the dilution is given by the ratio of the areas of the hatched ellipse to the design ellipse (for N → ∞)

### b) Circular Lattice

For a circular accelerator, the periodicity is usually taken to be not the superperiodicity of the machine (i.e., the number of identical lattice sections), but the revolution period. The index n therefore represents the turn number. The mismatch Bmag most often arises from improper orientation of the beam ellipse at injection. Neglecting the constant phase offset χx2 0 in (4.57), the

#### 116 4 Transverse Beam Emittance Measurement and Control

equilibrium emittance is

$$\epsilon\_n = \frac{\langle x^2 \rangle\_n}{\beta} = \epsilon\_0 \left[ \mathbf{B}\_{\text{mag}} + \sqrt{\mathbf{B}\_{\text{mag}}^{\prime}^2 - 1} \cos(4\pi\nu) \right],\tag{4.63}$$

where ν is the phase advance per turn. Shown in Fig. 4.10 is the evolution of the transverse phase space for Bmag along with the projections onto the horizontal axis. With a turn-by-turn beam size monitor, the mismatch can be measured directly by detecting the beam size changes at every turn. An example is given in Sect. 9.6.

Fig. 4.10. Horizontal phase space and x-projection for a mismatched beam in a circular accelerator

# 4.2 Beta Matching in a Transport Line or Linac

The beam size (squared) at the location s can be expressed in terms of the α and β functions and the emittance at an upstream location s0 as

$$
\langle x^2(s) \rangle = R\_{11}^2 \beta(s\_0) \epsilon - 2R\_{12}R\_{11}\alpha(s\_0)\epsilon + R\_{12}^2 \gamma(s\_0)\epsilon \,. \tag{4.64}
$$

In a quadrupole scan, the transfer matrix elements R11 and R12 are varied, by changing the strength of a quadrupole between s0 and s. Beam-size measurements for at least 3 different quadrupole settings are required in order to solve for the three independent unknown parameters: , β(s0) and α(s0). The fourth parameter, γ(s0) is not free, but determined by α(s0) and β(s0): γ = (1 + α2)/β.

The deviation of the β, α, and γ from the design parameters βD, αD and γD is often characterized in terms of the 'Bmag' (β matching) parameter [9, 10] of (4.59), which can also be written as

$$\mathbf{B}\_{\rm mag} = \frac{1}{2} \left( \beta \gamma\_D - 2\alpha \alpha\_D + \gamma \beta\_D \right) \,. \tag{4.65}$$

Once the values of β and α are known, quadrupole magnets can be adjusted so as to match the optical functions at a selected point to their design value, which is equivalent to Bmag = 1.

The SLC had more than 10 multi-wire emittance measurement stations, which monitored the beam emittances in various parts of the machine in hourly intervals, and were indispensable for emittance control and tuning. As will be shown in Sect. 4.4.3, in the SLC linac transverse orbit bumps were intentionally induced as a global correction which cancelled the accumulated local effects of dispersion or wakefields. The bumps were optimized by minimizing the emittance downstream, as calculated by this measurement technique.

Example. To illustrate the beta matching method, Fig. 4.11 shows an example from the KEK/ATF beam transport line (BT), connecting the S-band linac and the ATF damping ring. The top picture shows the result of a typical quadrupole scan at the end of the BT. Plotted is the square of the vertical beam size versus the strength of an upstream quadrupole, as well as a quadratic fit to the data. The Twiss parameters deduced from such a fit can be propagated through the BT, using a model derived from the actual or the design magnet settings. The bottom picture displays the inferred beta functions compared to the design optics.

# 4.3 Equilibrium Emittance

We now discuss different methods for changing and controlling the equilibrium emittance in electron or positron storage rings. In these rings, synchrotron radiation gives rise to an equilibrium beam size, which is independent of the beam emittance at injection. While at high beam currents, collective effects and intrabeam scattering may be important as well, at low beam intensity the value of the equilibrium emittance is determined solely by the ring optics and the beam energy.

The discreteness and the random character of the synchrotron radiation increases the beam emittance. The expression for the transverse emittance growth in the plane u (u = x or y) due to quantum excitation is [12]:

$$\frac{d\epsilon\_u}{dt} = cC\_Q E^5 \left< \frac{\mathcal{H}\_u}{\rho^3} \right> \,, \tag{4.66}$$

where the function Hu, introduced by Sands [13], is

$$\mathcal{H}\_{u}(s) = \frac{1}{\beta\_{u}} \left\{ D\_{u}^{2} + \left( \beta\_{u} D\_{u}^{\prime} + \alpha\_{u} D\_{u} \right)^{2} \right\},\tag{4.67}$$

Fig. 4.11. Beta matching in the KEK/ATF beam transport line (BT) [11]: (top) quadrupole-scan emittance measurement; shown is the square of the vertical beam size measured using a fluorescent screen vs. the strength of an upstream quadrupole; (bottom) the vertical beta function obtained by propagating the measured Twiss parameters (solid) through the actual BT optics is compared with the beta function expected for the design optics (dashed)

the coefficient CQ is

$$C\_Q = \frac{55}{48\sqrt{3}} \frac{r\_e \hbar c}{(m\_e c^2)^6} \approx 2 \times 10^{-11} \text{ m}^2 \text{ GeV}^{-5}.\tag{4.68}$$

and ρ is the bending radius. The angular brackets denote an average over the ring.

On the other hand, the average energy loss due to the synchrotron radiation usually leads to damping in all three degrees of freedom. The emittance decrease due to radiation damping is described by

$$\frac{d\epsilon\_u}{dt} = -2\epsilon\_u C\_d J\_u E^3 \left< \frac{1}{\rho^2} \right> ,\tag{4.69}$$

where is the beam emittance,

$$C\_d = \frac{c}{3} \frac{r\_e}{(m\_e c^2)^3} = 2.1 \times 10^3 \text{ m}^2 \text{ GeV}^{-3} \text{ s}^{-1},\tag{4.70}$$

and Ju is the damping partition number.

The equilibrium emittance is reached when the quantum excitation and the damping are equal. It is

$$
\epsilon\_{u,\infty} = C\_q \frac{\gamma^2}{J\_u} \frac{\left<\mathcal{H}\_u/\rho^3\right>}{\left<1/\rho^2\right>},\tag{4.71}
$$

where the new constant Cq is defined by

$$C\_q = \frac{55}{32\sqrt{3}} \frac{\hbar c}{m\_e c^2} = 3.84 \times 10^{-13} \text{ m} \,. \tag{4.72}$$

Thus u is inversely proportional to the transverse damping partition number Ju. Similarly, the longitudinal emittance is inversely proportional to Jz.

The exponential amplitude-damping time τu is obtained from the equation

$$\frac{1}{\tau\_u} \equiv \frac{1}{2\epsilon\_u} \frac{d\epsilon\_u}{dt} \,. \tag{4.73}$$

Including also the longitudinal degree of freedom, the exponential damping times for all three oscillation modes can be written as [13]

$$\tau\_i = \frac{2E\_0}{\langle P\_\gamma \rangle J\_i},\tag{4.74}$$

where E0 is the nominal energy, and Pγ is the average rate of energy loss. The latter is given by

$$
\langle P\_{\gamma} \rangle = \frac{cC\_{\gamma}}{2\pi} E\_0^4 \left\langle \frac{1}{\rho^2} \right\rangle \,, \tag{4.75}
$$

where yet another constant is introduced, namely

$$C\_{\gamma} = \frac{4\pi r\_e}{3(m\_e c^2)^3} \approx 8.877 \times 10^{-5} \text{ m GeV}^{-3},\tag{4.76}$$

as defined by Sands [13].

The factor Ji in (4.74) is the damping partition number for the ith degree of freedom. A general theorem by Robinson [14] states that the sum of the three partition numbers is a constant:

$$J\_x + J\_y + J\_z = 4\,\text{.}\tag{4.77}$$

A general proof of this theorem is given in Chap. 11. If the ring only contains horizontal, but no vertical bending magnets, then Jy = 1 and the partition numbers in the other two planes are related by a term D [13]

$$J\_x = 1 - \mathcal{D},\tag{4.78}$$

$$J\_z = 2 + \mathcal{D},\tag{4.79}$$

120 4 Transverse Beam Emittance Measurement and Control

where

$$\mathcal{D} = \frac{\oint D\_x/\rho \ \left(1/\rho^2 + 2k\right) \,\mathrm{d}s}{\oint 1/\rho^2 \ \mathrm{d}s}.\tag{4.80}$$

For separated function magnets k/ρ = 0 and the value of D is typically much smaller than 1.

It is often desirable to increase one of the damping rates or to vary the horizontal emittance. For example, in linear collider applications, a fast horizontal damping and a small horizontal emittance are advantageous, whereas the longitudinal emittance is of less concern. In storage-ring colliders one may instead want to increase the horizontal emittance near the beam-beam limit3.

The damping rate and the equilibrium emittance can be changed by adjusting the value of D and/or by adding wiggler magnets. Depending on the application, there are various possibilities to do so. In the following, we describe the effect of a change in the ring circumference, an almost equivalent application in which the accelerating frequency is changed, and two different applications of wigglers.

### 4.3.1 Circumference Change

If the geometric circumference of the ring is changed by moving the magnet centers outwards by a step Δxmag while holding the ring rf frequency fixed (so as to maintain synchronization with other systems) the quantity D changes by

$$
\Delta \mathcal{D} \approx -\left(\sum\_{q} k\_{q}^{2} D\_{x,q} L\_{q}\right) \frac{2\rho^{2}}{C} \,\Delta x^{\text{mag}},\tag{4.82}
$$

where kq is the non-integrated quadrupole gradient, Lq the quadrupole length, Dx,q the dispersion function at the quadrupole, C the ring circumference, and ρ the bending radius of the dipole magnets. The minus sign arises because the orbit moves inwards with respect to the quadrupole magnets (the orbit shift is opposite to the displacement: Δx = −Δxmag). Note that the contributions from focusing and defocusing magnets add, because the effect is quadratic in k1,q. Note that −2 < ΔD < 1 else beam loss may occur at

$$
\xi\_x = \xi\_y = \frac{r\_e}{2\pi\gamma} \frac{N\_b}{\epsilon\_x (1+\kappa)}\tag{4.81}
$$

3 Near the beam-beam limit, experimental observations [15] have shown that attempts to raise the luminosity by increasing the bunch charge leads to an increase in the beam size such that the beam-beam tune shift, from (2.101),

(with κ = y/x the emittance ratio), remains approximately constant. However, since L ∝ Nb 2/, such an increase in bunch charge and emittance still can result in a higher luminosity

the damping poles. The maximum tolerable shift Δx is determined by the available aperture, and by the beam size at injection.

In 1992, the magnet support girders of the SLC North Damping Ring were pulled outwards by about Δxmag = 1.5 mm, increasing the geometric ring circumference by 9 mm. As a result the measured horizontal damping time decreased [16] from 4.11 ± 0.11 ms to 3.41 ± 0.09 ms, consistent with predictions.

### 4.3.2 RF Frequency Change

An equivalent change in D can be achieved with a shift of the rf frequency by

$$\frac{\Delta f\_{\rm rf}}{f\_{\rm rf}} = \frac{2\pi \Delta x^{\rm mag}}{C} \,. \tag{4.83}$$

More accurately, the orbit shift in the quadrupole is proportional to the local dispersion function

$$
\Delta x(s) = -\frac{D\_x}{\alpha\_x} \frac{\Delta f\_{\rm rf}}{f\_{\rm rf}},
\tag{4.84}
$$

with αc the momentum compaction factor. The change in the partition number is

$$
\Delta \mathcal{D} \approx \frac{\oint 2k^2 D\_x^2 \,\mathrm{d}s}{\oint \mathrm{d}s/\rho^2} \,\frac{\Delta p}{p} \equiv C\_0 \,\,\frac{\Delta p}{p} \,\,. \tag{4.85}
$$

However, in practice the rf frequency must be locked to the rf of the injection (or extraction) system. Therefore, at the SLC damping rings in addition to the static circumference change a dynamic rf frequency shift was implemented [17]. The dynamic rf frequency shift by up to 100 kHz started about 1.33 ms after injection, and was stopped 200 μs before extraction, in order to stabilize the injected beam and to minimize emittance and extraction jitter, respectively [18]. The total store time was 8.33 ms, equal to about 2.5 nominal damping times. For a dynamic frequency shift of 62.5 kHz the normalized emittance of the extracted beam decreased from 3.30 ± 0.07 m to 2.66 ± 0.06 m. The 20% reduction agreed with SAD calculations [17]. This example is discussed further in Sect. 8.10 after describing the influence of heavy beam loading on the rf system.

Emittance control via the accelerating rf has been used already before [19]. More recently it was applied at LEP [20] and in the HERA electron ring [21]. At HERA the associated increase in beam energy spread was compensated by a larger rf bucket height. With limited rf power, this was achieved by increasing the transverse focussing thereby reducing the dispersion and hence the momentum compaction factor.

We note an interesting side effect, remarked by Wiedemann [12]. From (4.85) the partition number changes with the particle momentum. If a particle performs synchrotron oscillations

122 4 Transverse Beam Emittance Measurement and Control

$$\frac{\Delta p}{p} = \delta\_{\text{max}} \sin \Omega\_s t \,, \tag{4.86}$$

its horizontal partition number and damping time vary with the synchrotron period:

$$\frac{1}{\tau} = \frac{1}{\tau\_0} (1 - C\_0 \delta\_{\text{max}} \sin \Omega\_s t) \,, \tag{4.87}$$

The equation for the horizontal equilibrium emittance of such particles is then time-dependent according to [12]

$$\epsilon\_x(t) = \epsilon\_{x,\infty} \, \exp\left[\frac{2\delta\_{\text{max}}C\_0}{\Omega \tau\_0} (\cos \Omega t - 1)\right] \,. \tag{4.88}$$

The effect is largest for particles with large synchrotron oscillations.

### 4.3.3 Wigglers

A wiggler magnet generates additional synchrotron radiation and, thus, can enhance the radiation damping and/or change the equilibrium emittance. The damping time is modified as

$$\tau\_{u,w} = \tau\_{u,0} \frac{1}{1 + \langle 1/\rho^2 \rangle\_w / \langle 1/\rho^2 \rangle\_0},\tag{4.89}$$

where τu,x is the damping time in the plane u including the effect of the wiggler, τu,0 on the right is the damping time for the ring proper, while on the right-hand side of the equality, the subindex 0 indicates an average over the ring without wiggler magnets, while the subindex w indicates the contribution from the wiggler magnets. On the left, τu,x is the damping time in the plane u, including the effect of the wiggler, and τu,0 on the right is the damping time for the ring proper.

Similarily, the relative emittance increase due to the presence of the wiggler is

$$\frac{\epsilon\_{u,w}}{\epsilon\_{u,0}} = \frac{1 + \left< \mathcal{H}\_u / \rho^3 \right>\_w / \left< \mathcal{H}\_u / \rho^3 \right>\_0}{1 + \left< 1 / \rho^2 \right>\_w / \left< 1 / \rho^2 \right>\_0},\tag{4.90}$$

where the averages are given, for example, by

$$
\left\langle \frac{1}{\rho^2} \right\rangle\_w = \frac{1}{C} \oint \frac{1}{\rho\_w^2} \,\mathrm{d}s \,, \tag{4.91}
$$

with C the circumference, and ρw the bending radius in the wiggler.

In addition to changing the emittance, wigglers also affect the energy spread [12]:

$$\frac{\sigma\_{\delta,w}^2}{\sigma\_{\delta,0}^2} = \frac{1 + \left<1/\rho^3\right>\_w / \left<1/\rho^3\right>\_0}{1 + \left<1/\rho^2\right>\_w / \left<1/\rho^2\right>\_0}. \tag{4.92}$$

Damping Wigglers. If we place a wiggler in a region with no dispersion, Dx = 0, we might expect that the equilibrium emittance decreases according to (4.90) with Hw = 0. However this is not completely correct, because the wiggler itself generates dispersion. As an example, we consider a sinusoidal wiggler with field

$$B(z) = B\_w \cos k\_p z \,, \tag{4.93}$$

where kp = 2π/λp and λp denotes the length of the wiggler period. The differential equation for the dispersion function reads:

$$D\_{u}^{\prime\prime}(z) = \frac{1}{\rho\_{w}} \cos k\_{p} z \,, \tag{4.94}$$

which, assuming Du(0) = D u(0) = 0, can be solved as

$$D\_u(z) = \frac{1}{k\_p^2 \rho\_w} (1 - \cos k\_p z) \,. \tag{4.95}$$

Using

$$\frac{1}{\rho} = \frac{1}{\rho\_w} \left| \cos k\_p z \right|,\tag{4.96}$$

for each half period of the wiggler we find [12]

$$\int\_0^{\lambda\_p/2} \frac{\mathcal{H}\_u}{|\rho|^3} \,\mathrm{d}z = \frac{36}{15} \frac{1}{\beta\_u} \frac{1}{k\_p^5 \rho\_w^5} + \frac{4}{15} \frac{\beta\_u}{k\_p^3 \rho\_w^5} \approx \frac{4}{15} \,\frac{\beta\_u}{k\_p^3 \rho\_w^5} \,, \tag{4.97}$$

where β is the beta function and we assumed that λp β. Introducing the deflection angle per wiggler pole θw = 1/(ρwkp), and the number of wiggler periods Nw, we can rewrite (4.97) as

$$\int\_{w} \frac{\mathcal{H}\_{u}}{\rho^{3}} \,\mathrm{d}z \approx N\_{w} \frac{8}{15} \frac{\beta}{\rho\_{w}^{2}} \,\theta\_{w}^{3} \tag{4.98}$$

Similarily, we find