Beam Emittance Measurement with Laser Wire Scanners in the ILC Beam Delivery System

Accurate measurement of the beam phase-space is essential for the next generation of electron accelerators. A scheme for beam optics optimization and beam matrix reconstruction algorithms for the diagnostics section of the beam delivery system of the International Linear Collider based on laser-wire beam profile monitors are discussed. Possible modes of operation of the laser-wire system together with their corresponding performance are presented. Based on these results, prospects for reconstructing the ILC beam emittance from representative laser-wire beam size measurements are evaluated.


I. INTRODUCTION
Future electron machines will need accurate determination and monitoring of their transverse phase space in order to meet their challenging performance specifications. In this paper, prospects for the transverse emittance measurements at the International Linear Collider (ILC) are presented, with special emphasis on the beam delivery system (BDS).
The main parameters of the ILC [1], [2] are presented in Tab. I The ILC luminosity L is given by [3] where the asterix denotes the value at the e + e − interaction point (IP). H D is the disruption parameter due to * Ilya.Agapov@cern.ch † blair@pp.rhul.ac.uk ‡ mdw@slac.stanford.edu the mutual attraction of electrons and positrons in the collision and has value H D 2. Although the beam sizes at the IP depend strongly on the aberrations in the final focus system, accurate measurement of beam parameters upstream of the final focus is required to tune the main linac performance. The information about the transverse beam phase space is gained by putting beam profile scanners at several locations along the beam-line. The vertical beam sizes in the diagnostics section of the beam delivery system of the ILC are of order of 1µm which is too small to measure with a solid wire, so the transverse beam profile measurements will be performed by the laser-wire (LW) system [4], [5]. A similar situation applies to portions of the RTML (ring-to-main-linac) and the main linac sections of the ILC.
In contrast to a ring machine, where an individual bunch can be measured many times as it passes around the ring, the emittance measurement in the ILC BDS will need to be performed on a single pass basis. This will require laser-wire scans that sample across successive bunches within a train, necessarily involving both a projection of any bunch position jitter and an averaging over successive bunches. The analysis described below of the extraction of the emittance from the bunch dimension measurements applies equally to circular and linear machines, although additional allowances for variations between ILC trains may also be necessary, for instance when performing quadrupole scans or measurements of linear dispersion. Throughout this paper, the electron bunch is assumed to be pure Gaussian; an extension of the analysis presented here to more realistic post-linac ILC bunch profiles will be included in a future publication.
In section II the beam matrix reconstruction using a series of beam profile monitors is described. This section deals mostly with an analysis of how the emittance measurement error depends on the precision of the beam profile measurement. The general ideas behind the beam matrix reconstruction method described here are well known ( see e.g. [6], [7], [8], [9] ). After presenting the framework we discuss numerical algorithms for matrix reconstruction and introduce a numerical criterion allowing beam optics optimisation. Simulations of emittance reconstruction using ILC parameters are presented. The methods described in this section are independent of the type of the beam profile monitor or beam sizes.
In section III the laser-wire (LW) beam profile monitor is introduced and its use in the measurement of transverse beam profiles is described. The LW is useful as a non-invasive device to measure electron beam profiles ranging from a few tens of microns down to the micron scale. Issues of Gaussian beam optics that influence the measurement are discussed and quantitative results presented. We conclude by quoting the requirements on the laser-wire system, plus associated laser specifications necessary to achieve the desired emittance measurement precision of a few percent.
Laser-wire specifications are given for the beam sizes relevant to the 500 GeV beam (1 TeV centre of mass) machine upgrade, which is more challenging because the bunches are smaller. Other issues such as beam matrix reconstruction methods are independent of the beam energy and the results shown are normally for the 250 GeV beam.

II. BEAM MATRIX RECONSTRUCTION
In this section we first describe the standard approach to reconstructing the 4d coupled beam matrix with the least-squares fit method [6]. In the presence of coupling the emittance reconstruction precision falls dramatically with the beam size measurement error. The Cholesky decomposition method is analysed as an option to reduce this effect.
We further introduce a criterion which allows numerical optimisation of beam-line lattice parameters to minimise the error of the emittance measurement. The contributions to the beam profile scan from effects such as beam jitter is discussed towards the end of the section. The described methods were used to simulate the emittance reconstruction process with the ILC lattice and with the beam size measurement precision predicted in section III.

A. Beam matrix reconstruction from measured beam sizes
One is generally interested in reconstructing the (x, x , y, y , ∆p p , ∆t) beam phase space. To the first order it is given by correlations like xx , x ∆p p etc. The transverse coordinates r can be represented as the sum of the betatron oscillations r β and dispersive trajectory η ∆p p r = (x, x , y, y ), r β = (x β , x β , y β , y β ) where the dispersion vector is Ideally dispersion should be zero in the diagnostics section. But in reality some residual dispersion can be present. The dispersion at the beginning of the transfer line is defined as [10] is the RMS momentum spread.
The transverse beam envelope matrix is then defined as x 2 x y x y xy x y y 2 yy xy x y yy The equations for dispersion and betatron coordinates are [10] r β + K(s)r β = 0 η + K(s)η = 1 ρ where 1/ρ is the orbit curvature. We assume that in the diagnostics section the beam orbit is first aligned sufficiently close to the magnet axis so that the additional dispersion created there can be neglected. Then one can write down the coordinates at each scanner location i in terms of transfer matrices [11] as and thus This definition of the beam matrix already includes effects of the dispersion.
The phase-space occupied by a ( generally coupled ) beam can be quantified by the intrinsic emittances ε 1,2 [12]. They are recovered from the beam matrix constructed from the betatron coordinates only by bringing it to a diagonal form: In what follows, "vertical" emittance usually denotes the smaller of the intrinsic emittances. The projected (vertical) emittance is defined as 2 y = det y 2 yy yy y 2 It will coincide with the intrinsic emittance if the beam is uncoupled but will be larger if the coupling is present. We can rewrite the beam matrix as and we need at least ten measurements to accomplish the task. At a scanner location in the beam-line it is possible to measure three values, x 2 , y 2 and xy , with the help of a horizontal, a vertical, and a tilted wire scanner. The ten values can be obtained either by changing the optics in a controlled manner at the wire location [8], [6], [7] or by locating the wires at different positions in the beam-line. For the ILC one aims at fast intra-train scanning for which the former method is not possible.
The elements of the beam matrix can be obtained analytically when the coupling elements are neglected and the total number of wire scanners is six (three for each plane, two scanners at each location) that are suitably spaced in the betatron phase [9]. For more general cases it is however convenient to have a numerical procedure for the beam matrix reconstruction, which will be now described. Let the measured values of x 2 , y 2 and xy beσ with i = 1 : N scanners . Let σ k be the elements of the beam matrix at the location where the beam transfer matrices R are evaluated from. Assuming that the transport matrices are uncoupled in the diagnostics section (coupling introduced by misalignment errors is neglected) one obtains by equating coefficients in Eq.2 The problem then reduces to three uncoupled sets of equations: and each set is solved separately by a least squares fit. This may lead to an unphysical result (a non-positive beam matrix) when the measurement is sufficiently noisy. A typical dependency of the fraction of non-positive matrices on the relative measurement error is shown in Fig. 1. Here the 4d diagnostics section with 6 laser wire stations and nominal ILC parameters were assumed. A common relative measurement error is assumed for both the horizontal and vertical dimensions, for reasons outlined below in Sec. II C 4.
A way to avoid non-positive beam matrices is to search for the beam matrix as a Cholesky decomposition [13], [14] σ 0 = GG T where G ∈ R 4×4 is lower (or upper) triangular with positive diagonal entries. From the point of view of numerical errors the lower triangular representation should be used when the horizontal emittance is smaller than the vertical, and the upper in the opposite case. This procedure introduces biasing to the emittance measurement, however it seems to be advantageous when either a small number of measurements is available or when a larger measurement error results in a high rejection fraction (fraction of non-positive matrices). In Figs. 2 and 3 examples of emittance fits using simulated beam size measurement data in the 4d ILC emittance measurement section are shown. A relative error of 35% was introduced to the simulated data. Both methods yield a significantly biased mean emittance. However, the Cholesky decomposition method results in an increase of statistics (i.e. physically meaningful fits) by a factor of three.
When large statistics are available the straightforward method performs satisfactorily. For the ILC one aims at about 1-5% measurement errors within a bunch train. In this range the choice of algorithm is not important during stable operation, however the Cholesky decomposition method will be helpful in certain cases, for in- stance during the measurement tune-up when the errors are large.
In the presence of coupling the intrinsic emittance is smaller then the projected one. With measurement errors the centre of the distribution of reconstructed intrinsic emittances is shifted towards smaller values even if the real beam is uncoupled [12] (also seen in Figs. 2 and 3). The distribution of projected emittances does not shift, so care should be taken if the difference between the projected and the intrinsic emittances is used to evaluate the coupling correction.
When beam position measurements are available and it is possible to vary the beam energy the dispersion functions can be measured at wire locations. Supposing the measurement vector is {η i } the initial dispersion is recovered with a least squares fit from The dispersion model used here might not be adequate since in reality the bunches coming from the linac might have a complicated correlation pattern. Its correction in combination with the emittance measurement is a subject of a separate work. So in what follows we will not analyse the effect of dispersion apart from estimating its influence on the beam profile measurement.

B. Beam optics for the diagnostics section
Concepts of optics for 2d and 4d diagnostics sections were presented in [15]. For optimal performance the diagnostics section lattice should be designed so that the beam sizes and aspect ratios at the wire location are optimal for scanning performance and that the solutions of Eqs 3 are only weakly sensitive to perturbations of the right hand side. For optics analysis it is convenient to express the R-matrices in terms of Twiss parameters [16]. In the uncoupled case where, assuming periodic optics, Assuming further that the wire locations are at the maxima of the β-functions in order to optimise the spotsize resolution, α vanishes and the matrices M X , M Y and M XY will have the form: x sin 2 ∆µ x,1 cos 2 ∆µ x,2 2β x cos ∆µ x,2 sin ∆µ x,2 β 2 x sin 2 ∆µ x,2 cos 2 ∆µ x,3 2β x cos ∆µ x,3 sin ∆µ x,3 β 2 x sin 2 ∆µ x,3 · · · · · · · · ·      cos 2 ∆µ y,1 2β y cos ∆µ y,1 sin ∆µ y,1 β 2 y sin 2 ∆µ y,1 cos 2 ∆µ y,2 2β y cos ∆µ y,2 sin ∆µ y,2 β 2 y sin 2 ∆µ y,2 cos 2 ∆µ y,3 2β y cos ∆µ y,3 sin ∆µ y,3 β 2 y sin 2 ∆µ y,3 · · · · · · · · ·      and One can choose the optimality criterion for the lattice to be the condition numbers of the corresponding matrices [13], [14] Here the norm of a matrix M is defined as the maximum value of ||M x|| over all vectors of unit lengths The condition number is used to quantify the solution error of a linear algebraic system [13], [14]. A small condition number corresponds to well-conditioned systems while a large condition number corresponds to illconditioned systems. This number is hard to evaluate analytically but it can be evaluated numerically for any optics design. Apart from the condition number one has to make sure that both x and x make contributions of the same order of magnitude to the measurements, i.e.(for 2d case), To achieve this one can introduce another optimality criterion to minimise: For the full beam matrix reconstruction one needs to have ten measurements including coupling terms. Beam optics for such a measurement section should be designed so that the condition numbers of M X , M Y and M XY are minimised simultaneously (so-called 4d optics). In many cases one is interested only in correcting the coupling terms rather than measuring them. To do so, a set of skew quadrupoles is introduced upstream of the measurement section. They are used to minimise the projected emittances [12] and no direct measurement of the coupling terms is necessary. In this case six measurements are required and the condition numbers of only M X and M Y need to be minimised (so-called 2d optics).

2d measurement section
For the diagnostic section lattice we choose a FODO channel with a constant phase advance per cell ∆µ 1 = ∆µ 2 = ∆µ. Then κ will depend on two parameters: β and ∆µ. In Fig. 4, 5 these dependencies are shown for different numbers of measurement stations.
The condition number is infinite for 0 • and 90 • phase advance (not seen in the picture). It has a minimum close to 0 • and a second minimum that depends on the number of wire scanners used in the fit. This minimum appears to be at ∆µ = 180 • /N scanners (60 • for 3 scanners). Together with Eq. 6 (see also Fig. 6) this gives 180 • /N scanners as the optimal phase advance in a FODO cell. In Fig. 7 simulated emittance reconstruction error is plotted against the cell phase advance which shows that the error is indeed minimised by following the described optimisation procedure.

4d measurement section
For a 4d diagnostics section we can assume that the lattice is constructed from identical cells of phase advances ∆µ 1 and ∆µ 2 . The matrix M depends on 4 parameters β 1,2 and ∆µ 1,2 . For different values of β x,y the behaviour shows a similar pattern with the matrix being singular for ∆µ 1 = ∆µ 2 and ∆µ 1 = 180 • − ∆µ 2 . Combining the information on plots such as in Fig. 8 with that in Figs. 4,5 one sees that there is no clear optimum for this problem. However, for a good performance one can choose, for instance, a phase advance in one plane of 60 • and in the other plane a phase advance close to 90 • (for 4 wire stations with 3 wires each).
The optics for the ILC diagnostics sections [15] designed for 2d and 4d emittance measurements are shown in Fig. 9. The 2d diagnostics section is sufficient for emit- cell phase advance, 5 wire scanners, 5% error on beam size measurement tance tuning purposes and is shorter; it is thus currently expected to be used at the ILC [1], [2].

C. Machine Contributions to the Transverse Profile Scans
The imperfections in the linac will result in beam jitter, residual dispersion, transverse beam coupling etc. [17], [18]. These may also cause a deviation of the bunch from the Gaussian shape. Estimates of these errors can partially be subtracted from the beam size measurements. Due to imperfections in the diagnostic section the transfer matrices will also not be known precisely, which will introduce additional errors into the reconstruction procedure; this effect is expected to be much smaller than the others and is neglected in this paper.
Neglecting all but the dispersion and beam jitter contributions, the value of σ e extracted from the beam profile scan is: where σ scan is the laser-wire scan after deconvolution of laser effects (discussed in Sec. III A 4) and α J represents the magnitude of the beam orbit jitter, as normalized by the observed beam size, and will be detailed in the next section.
When an effect contributes to the measurement error δσ e /σ e , we define E effect as its contribution to the total relative error, adding in quadrature as: where E scan is the contribution to the error from the raw laser-wire scan; the contributions to this error are discussed in Sec. III A 4. E jitter is the error remaining after subtracting the electron bunch-to-bunch jitter as discussed in Sec. II C 1. The additional effects of any residual dispersion, E η , could in principle also be subtracted explicitly; the error that remains after such a subtraction is estimated below. The significant machine-related errors are now discussed in turn.

Error Contribution from the Jitter of the Beam Location
In the following, the jitter of the location of the bunches within the bunch train at the ILC laser-wire IP locations can be written as σ jitter = α J σ e where σ e is the electron transverse bunch size. The value of α J will depend on the stability of the ILC site and on the additional vibrations arising from beam-line components, from energy and kicker jitter, and from the performance of train-to-train and intra-train feedback. Preliminary studies [19] suggest that values of α J may end up ranging from about 0.01 to about 0.7; in the following the resulting error estimates are normalised relative to a value α J 0.25. The bunch-by-bunch jitter can be determined by local BPMs to within their single-bunch resolution. The performance of cavity BPMs is the subject of ongoing R&D [20], which suggests that single-bunch resolutions of order 20 nm should be obtainable. The error contribution E jitter to δσ e /σ e remaining after subtracting the bunch jitter is given by differentiating Eq 7 to be where δα J is the precision to which the bunch jitter is known; the last approximation assumes that the only bunch jitter measurements are made at the LW location. It is possible that more precise measurements could be made using dedicated machine setups. This equation can be written as

Error Contribution from Residual Dispersion
If the effects of η and η are not subtracted at the location of each LW IP, then the emittance will be overestimated. The error remaining after subtraction of residual vertical dispersion, η, at the laser-wire IP is again given by differentiation of Eq. 7 to be where δη is the precision to which η can be determined. If E η is not to dominate a 1% tranverse bunch size measurement for a typical ILC beam then E η should be less than about 0.5%. If η is measured to δη /η 1%, say, then for σ ey = 1 µm Eq. 10 implies that η must be kept below about 0.5 mm.
A method that could potentially be used to determine η more accurately than 1% is to change the mean energy of the electron beam by a relative amount ∆E/E (a possible choice would be ∆E/E 5 × 10 −3 so as to remain within the energy acceptance of the ILC BDS) and measure any subsequent shift in position of the beam centroid over the N train bunches in a train. If the singlebunch energy resolution of the ILC beam spectrometers is σ E , then: With additional R&D, it should be possible to achieve σ BP M 20 nm and spectrometer resolution σ E 10 −4 [21]. An alternative to explicit subtraction of the dispersion effects is to include them implicitly in the fits to the measured laser-wire distribution; this is the method described above in Sec. II A.

Systematic beam size variations
In general position shifts of individual bunches within a linear collider bunch train can have arbitrary patterns according to the errors of the injection kickers, or effects of the long-range transverse wakefield. Regardless of the shape of the patterns, the individual relative bunch displacements can be subtracted using bunch-by-bunch BPM measurements and so such effects can be absorbed into the treatment of beam position jitter discussed in Sec. II C 1. Ideally the laser-wire fast scanning system will be able to take account of these patterns (for instance by learning the shape from previous trains) so as to maintain an efficient scanning technique across the length of a train.
One possible use of the laser-wire measurements will be to predict the spot-size at the IP and thereby enable a comparison with the spot-size inferred from luminosity measurements via Eq. 7. Systematic variations of the bunch transverse dimension along the train will lead to a bias in such predictions. For example, a distortion of the transverse dimension with maximum value ±s train σ e , which varies linearly along the train, will modify the effective transverse dimension that enters Eq. 1 by In order to correct for such effects, a number of scans will need to be made within a train, requiring ultra-fast scanning methods. Ongoing R&D [22] is aiming at scanning rates of several 10's of kHz using electro-optic tech-niques.
The influence of the beam jitter on the measurement procedure depends strongly on the linac tuning procedure and requires further studies.

Vertical-Horizontal Coupling
One can measure x 2 = σ 2 x , y 2 = σ 2 y and extract xy from an additional measurement of the bunch along the u-axis u 2 = σ 2 u , which is defined to be at an angle φ with respect to the x-axis, as shown in Fig. 10.
When it comes to measurement of σ u with a LW, it will be necessary to take into account the size of the perpendicular dimension, because of Rayleigh-range effects discussed below.
The coupling xy is: The RMS error δ xy on the measurement of the coupling term is then given by: By substituting Eq.13 into Eq. 15 and minimising with respect to φ under the assumption that the relative errors δσ y /σ y and δσ x /σ x are approximately equal, the optimal value for φ is given by Substituting this value of φ into Eq. 15 and using the approximation that xy is small gives the error on the coupling term as: Typical values of interest to the ILC BDS are presented below in Tab. VI, where it can be seen that for the optimal value of φ, σ u ≈ σ y . With this optimal u-wire angle, the relative errors of the vertical and horizontal measurements enter equally in the coupling term in Eq. 17.
This justifies the earlier simplification (Sec.II A), where equal relative errors were assumed for all measurements in the emittance reconstruction simulations. Also, since σ x > σ y for the ILC, the error on the coupling given by Eq. 17 will grow more rapidly than the error on the vertical beam size as the relative measurement errors increase. This explains the fact that the number of unphysical beam matrices grows rapidly with the measurement error, because the coupling terms quickly dominate over those of the vertical part of the beam matrix.

III. THE LASER-WIRE IN BEAM EMITTANCE MEASUREMENT
In this section the laser-wire (LW) beam profile monitor is described and the possible precisions that can be obtained from its use in transverse beam profile measurements are quantified. Ideally the LW will be used at the ILC to measure the electron transverse beam profile at several locations within a bunch train (containing 2625 bunches, Tab I), which will require high power lasers to get sufficient statistics for each laser shot together with ultra-fast laser scanning systems.
The LW is useful for beam profiles ranging from several tens of microns, down to the micron scale. Smaller beam profiles have been measured using laser interferometric techniques [5,23] whereas traditional solid wires or screens can be used for larger profiles (although they are disruptive to the electron beams). Very challenging, low f-number, laser optics are necessary for the LW in order to achieve the required small laser spot-sizes and the subsequent performance is evaluated numerically and described in Sec. III A 3. The laser systems necessary to power the LW are also very challenging and the necessary specifications are derived and discussed in Sec. III D 1.

A. The Laser-wire Beam Profile Monitor
Traditionally the transverse dimensions of an electron beam have been measured by scanning a tungsten or carbon wire across the beam and measuring the resulting backgrounds as a function of relative position of the wire. This method has the disadvantage of being highly disruptive to the electron beam and so it cannot be used during normal luminosity running. At the ILC, the electron beams in the BDS will have vertical transverse size of order 1-few µm; a normal wire scanner would not be able to measure beams of this size, nor would it be able to withstand the energy depositions from such high in-

tensities.
To solve these issues, the solid wire can be replaced by a finely-focused beam of laser light; such a system is called a laser-wire (LW). The Compton collisions between laser photons and beam electrons are detected downstream and the Compton rate as a function of relative positions of electron and laser beams provides the measurement of the electron beam transverse profile. This principle is illustrated in Fig. 11. Two distinct methods have been employed to date. Operating the laser in continuous wave mode together with a Fabry-Perot cavity to enhance the power has been used [24] at the ATF at the KEK laboratory to measure the emittance of the damping ring; this technique would also be applicable to the ILC damping rings. In other parts of the machine, including the BDS, the beam is not circulating so a single-pass method based on high power pulsed lasers is required [25][26][27]. In the following discussion, the latter technique is assumed.

Laser-wire Compton Rates
The Compton cross section decreases as the electron beam energy increases. For an electron beam energy E b and laser photon energy k = hc λ , the Compton cross section is given by σ C (ω) = σ T f (ω) where σ T is the Thomson cross section = 0.665×10 −28 m 2 , ω = kE b m 2 e , and [5,28] f (ω) = 3 4 Values of f (ω) for laser wavelengths and beam energies of typical interest at the ILC are presented in Table II.
In this section, the Compton rate for a set of laser-wire operating conditions is derived as a function of relative horizontal and vertical offsets, ∆ x and ∆ y respectively, between the centroids of the electron bunch and laser beam.
The number N (∆ x , ∆ y ) of Compton photons produced will be proportional to the relevant overlap integral, (∆ x , ∆ y ). In Sec. A 1, (∆ x , ∆ y ) will be evaluated in µm −1 .
P is the instantaneous laser power at the laser-electron IP, and N e is the number of electrons in the bunch. If η det is the detector efficiency then, using realistic numerical values, the number of detected photons is N det (∆ x , ∆ y ), where and ξ = η det 0.05

Conventions for laser-optics
In the following, the conventions used define z along the electron beam direction, y along vertical and x along the laser-beam direction. The light intensity of the laser has the form where λ is the laser wavelength and M 2 is a quality factor for the laser, which effectively increases the wavelength λ → M 2 λ compared to the diffraction limited case; an ideal single-mode laser would have M 2 = 1. x R is the Rayleigh range of the setup and f # is the f -number of the optics, f # = D /F , where D is the diameter of the lens and F is its focal length. As in Ref. [29], 99% of energy in the Gaussian beam profile is required to be contained within the lens aperture. For the TM 00 mode this requirement means: In this case, the opening angle θ = 1/f # between the centre of the diverging Gaussian beam and its e −2 intensity cone is given by: and so, for TM 00 with f 1 optics, σ 0 = λ/π and θ = 1 rad, or 57 • .

Numerical Results
Results for the case of laser-M 2 =1.3 and f 1 final focus optics are shown in Fig. 12 for an electron bunch transverse Gaussian profiles with (a) σ ey = 1 µm, σ ex = 10 µm and (b) σ ey = 1 µm, σ ex = 100 µm; the effect of the Rayleigh-range is very apparent for the larger aspectratio.
Results for the case of operating the laser in TM 01 mode with laser-M 2 =1.3 and f 1 final focus optics are shown in Fig. 13 for electron bunch transverse Gaussian profiles with (a) σ ey = 1 µm, σ ex = 10 µm and (b) σ ey = 5 µm, σ ex = 50 µm; the potential benefit of the TM 01 mode is apparent for the larger vertical spot-size. The relative benefits of the TM 00 and TM 01 modes are presented in Tab. III, where it can be seen that for σ ey > 1−2 µm there is a significant advantage for the statistical power by using the TM 01 mode; this advantage has been demonstrated at the ATF [30]. However the sensitivity to the laser properties (as parameterized by a simple M 2 in these calculations) is greater for the TM 01 mode and, for spot-sizes of order 1 µm, the relative statistical power of the TM 01 to that of the TM 00 mode decreases rapidly, as illustrated in Fig 14. In this study, the laser-spot sizes of order 1 µm are of particular importance for the BDS LW system, so in the following the TM 00 mode is used, while acknowledging that higher-order laser modes may be advantageous in other locations.

Error Contribution from the Laser-Wire Scan
In this section the various contributions to the relative error E scan in Eq. 8 are outlined. If σ scan is the electron beam size after subtracting the laser effects discussed below, then: III: Optimal f -numbers (for Estat) and corresponding statistical precision for electron bunches with aspect ratio σex/σey = 10, λ = 532 nm. The laser M 2 = 1.3 for both TM00 and TM01 modes. The minimum practical f -number is taken as 1.0, even where a better statistical precision could in principle be obtained by smaller f -numbers. The factor of 1.15 of Eq. A11 is included in the TM01 calculations. Also shown is the error E M 2 if there is a 5% error in the value of the laser M 2 .

TM00
TM01 σ scan itself is obtained from subtracting the effects of the laser pointing instability: where σ fit is the raw result of a fit to the laser-wire scan profile, for which the errors are discussed in detail in Sec. III B. σ point is the contribution to the scan profile from the RMS laser pointing angular fluctuations ψ ; σ point = F ψ , where F is the focal length of the lens. As discussed in Sec. III C 1 F ≥ 15 mm. After subtracting σ point as in Eq. 27 the error contribution can thus be written as: where and E point =

B. Laser-wire Performance
In this section the performance of the ILC laser-wire systems will be examined in detail with a view to quantifying the errors that contribute to δσ fit of Eq. 29. The contributions to the raw laser-wire scan can be broken down as follows: where E stat is the statistical error of a 19-point fit to the raw scan curve (Sec. III C 1). In general, N scan laser-shots could be used in a variety of scanning modes. However, as set examples in this paper, all the scans consist of N scan = 19 equally spaced values of ∆ y over a range ±7σ m (as defined in Eq. A4), with ∆ x = 0 and ξ = 1.0, where ξ is as defined in Eq.20. E ξ is the error arising from the shot-by-shot normalisation fluctuations as introduced in Eq. 37. E M 2 is the error on the extraction of σ e introduced by the error on the laser light distribution at the IP; this is characterized here by an error in the M 2 value of the laser and, for a real system, will need to be calculated including the effects of alignment errors etc. in the finalfocus optics. As illustrated in Sec. III C 1 E M 2 can be estimated by fitting the measured profile to σ ey assuming a value of M 2 that is wrong by a factor (1 + δ M 2 ). In the following, the laser M 2 is thus assumed to be determined shot by shot to an accuracy of δ M 2 . Naively, without allowing for Rayleigh range effects, the error on the extracted value of σ e from subtraction of the laser spot-size is Inserting the representative values of M 2 = 1.3 and σ e = 1 µm gives: (33) A full numerical treatment, using the relations given in Appendix A, is presented in Tables VIII-X and shows that this is a good approximation for small δ M 2 1% but is a slight underestimate for larger values.

Fits to TM00 Distributions
The overlap integrals necessary to calculate the number of LW Compton photons are presented in Appendix A. A numerical evaluation of Eq. A2 is now used to simulate laser-wire scans for a variety of situations of interest to the ILC.
Both E stat and E M 2 will depend on the f -number of the laser optics employed. This dependence is illustrated in Fig. 15 for the case of σ ey = 1 µm and σ ex = 25 µm. For each set of σ ex , σ ey , there is an optimal f -number that gives the lowest statistical error for given values of ξ and N scan . However, as can be seen in Fig. 15, the minima are often fairly shallow, which must be contrasted with the difficulty of building low f -number optics. The difficulty is not just in building low f -number alone, but in producing a system that can maintain a small laser spot size approximately ±10 σ off axis, as needed during a scan. For this reason, f -numbers less than 1 are not considered here. In addition to determining the optimal f -number for various electron transverse dimensions, the nominal errors E stat and E M 2 are also determined here for f 1.5 optics, because this is a likely technological goal for a system that will produce a small spot-size both on the optical axis and over an acceptable scan range of order 1 mrad. The beam-pipe in the ILC BDS has inner radius of 12 mm and outer radius of approximately 14 mm. In order to ensure sufficient clearance from the beam halo, the minimum focal length is taken here to be F min = 15 mm. Imposing the requirement of being able to scan ±7 σ m , then the scanning system must be able to scan a range Imposing this condition on the most challenging small spot-size of σ ey = 1 µm, σ = 1 µm (assuming M 2 = 1.3) and so σ m 1.4 µm gives θ scan ≥ 0.7 mrad. The maximum scanning rate will depend on the scan range required so, with this in mind, the scan range should be kept as small as practical. In the following the condition is therefore adopted. For larger spot-sizes and for the horizontal (or skew-angle) scans of the electron bunch, the suitable f -number is determined primarily by the angular scan-range of the final focus lens plus scanning system. The laser optics for these dimensions will probably use diameter D = 5 cm optics (or similar). In this case, assuming again a scan range of 7σ m the practical f # is given by The errors obtained from a laser-wire scan using an f # given by Eq. 36 for the horizontal dimension σ ex are given in Tab. XI. In practice, for very large scan ranges, it may be preferable to use a stepping-motor system to move the final focus lens as opposed to scanning using optical ray deflection, which would enable smaller f # s to be employed. In that case, the scan would have to be very slow compared to the machine repetition rate.

Normalisation Fluctuations
The error on the normalisation factor ξ is given by: where δP is the resolution of the measured laser power, δN e is the bunch charge measurement resolution, and the final term is the contribution from time jitter (or phase error) between the laser and electron beams. The laser power can be measured bunch-by-bunch by a fast photo-diode presumably to a level of order 10 −2 and the bunch charge to a similar level by using dedicated current monitors or by combining measurements from a set of BPMs.
The contribution from time jitter is now addressed. As indicated in Tab. I, the ILC bunch train consists of N train = 2625 bunches with repetition frequency f rep = 5 Hz. Each bunch has length L b = 300 µm with corresponding bunch Gaussian time width of If T is the laser pulse length and τ is the RMS time jitter between the laser and electron pulses then the RMS contribution of this temporal fluctuation to the relative value of the overlap integral of Gaussian laser and electron profiles is given by: or after including typical ILC values: (39) The number of Compton photons (Eq. 18) produced by each laser-shot is proportional to the instantaneous value of ξ and so any shot-by-shot ξ-fluctuations will contribute to the error on the fit to the raw LW scan, as given by Performing the fit with this additional error factor enables its contribution to the total error (Eq. 31) to be determined according to: where values of the coefficient a ξ are given in Tab IV for a range of electron-bunch vertical spot-sizes and aspect ratios σ ex /σ ey . Combining all the terms, the normalisation error as given by Eq. 41 and Eq.37 becomes:

Alternative Scanning Mode
An alternative laser-wire scanning mode can be considered where the laser is kept fixed in space relative to the centre of an accurate BPM. The relative offset ∆ y between the laser and the electron bunch can then be measured by the BPM on a bunch-by-bunch basis to an accuracy given by the BPM resolution σ BPM .
If the bunch charge N e and laser power P are also measured on a bunch-by-bunch basis, then the factor ξ defined in Eq. 20 is known to an accuracy given by Eq. 37. If it is assumed that the electron transverse charge distribution is a pure Gaussian, then Eq. A2 can be inverted bunch-by-bunch.
Given that ∆ y is generated by the bunch jitter with RMS value α J σ ey and assuming σ ey σ σ m / √ 2 1 µm and α J 0.25 then typical values of N det (Eq. 19) are of order

332
(43) giving a statistical error on each measured value of N of order 5%. In addition there will be a contribution to the error from ξ due to the measurements of bunch charge and laser power and from the laser trigger (phase) jitter. In the following, the total error on ξ is taken to be 2%. The expected measurement error on the transverse size σ ey of the electron bunch was then evaluated for a nominal electron spot-size of σ ey = 1 µm by σ ex = 10 µm by inverting the full overlap integral of Eq. A2.
The resulting percentage error on a single-shot measurement of σ ey is presented in Fig 16 as a function of (a) BPM resolution and (b) RMS beam jitter α J . From these plots it can be deduced that the bunch-by-bunch error on σ ey is of order 15% for the chosen realistic parameters. Making such measurements over a whole train would then yield an error on the average spots-size of the train of 0.15/ √ 2625 2.9 × 10 −3 .

D. Summary of Laser-wire Requirements
A LW system for the ILC BDS will involve many sophisticated elements including high power mode-locked lasers, high quality laser final-focus optics, and integrated BPM systems. In this section, the desired performance specifications of the key sub-systems are discussed and the resulting errors on the measured electron transverse spot-size are estimated.

Laser Requirements
The above analysis has discussed various laser requirements that must be met if fast intra-train emittance measurements are to be performed at the ILC. These requirements are gathered here to provide a benchmark for R&D requirements [31] and to act as a base-line for subsequent discussion of the other laser-wire component specifications.
The laser supplying the light to the laser-wire IP will need to match the ILC bunch structure, providing laserpulses each of instantaneous power P 10 MW with sufficient pulse-length to overlap fully with the electron bunch so as to minimise any synchronisation issues. The pulse pattern and synchronisation can be obtained by mode-locking a master laser oscillator to a subharmonic of the machine RF and then by pulse-picking for high-power amplification. Assuming that only the required pulses are amplified, the average power of the laser will be dominated by only those amplified pulses and so can be estimated as: (44) where f rep is the rep-rate of the machine (Tab I). If no pulse-picking for the high-power pulses were applied, then P av would be 108 W, assuming the ILC nominal V: Requirements on laser-system for intra-train laserwire scans at the ILC, with reference to the equation that sets the goal value. Note, an extra factor of 2 has been applied to allow a 50% loss of power due to light transport between the laser and the laser-wire IP, which may involve distances of several hundred metres in the ILC

Parameter
Symbol Goal Value Eqn.   bunch spacing of 369 ns (Tab I), which would make the laser very much more challenging. An extra factor of 2 has been included in the laser power requirements for P , because at the ILC the light transport between laser and IP may be as long as several hundred metres and so will involve substantial power losses en route.

Error Summary
The most challenging laser-wire measurements at the ILC occur in the BDS and some representative values of the bunch dimensions of interest are given in Tab. VI. For illustration in this table, the precisions obtainable on the matrix element xy are listed, assuming that each dimension σ x , σ y , and σ u can be measured to a nominal 1% (which means σ 2 x and σ 2 y are measured to 2%). Some examples of vertical emittance reconstructions for 4d optics using these assumptions are presented in figures 17 and 18.
In order to summarise the ILC laser-wire requirements and to describe the various contribution to the measurement errors, a beam with representative transverse dimensions σ ex = 10 µm × σ ey = 1 µm is now used as a specific example to illustrate the key points. This beam is somewhat more challenging than that expected at the nominal ILC, but points to where additional R&D may be required if specific operating conditions give rise to smaller spot-sizes.
The following laser, optical, and BPM specifications are non-trivial and are themselves subjects of R&D, however it is probable that suitable solutions to them will be found. The assumptions are: • the laser-requirements of Tab. V can be met, FIG. 18: Distribution of reconstructed vertical emittance with 1% (a) and 5% (b) random errors on the beam size measurement for a 4d diagnostics section assuming 50% random mismatch of initial optical functions (statistics corresponding to train length). The true emittance is 0.079 µm · µrad .
• the bunch-by-bunch charge and the laser instantaneous power can both be measured to 10 −2 , • the laser profile at the IP (characterised in this paper by a simple M 2 value) is known to the equivalent of δ M 2 = 0.01, • BPM resolution σ BPM = 20 nm.
• f 1.5 optics can be made to work, together with the scanning system.
All the errors discussed above are summarised in Tab. VII for the most challenging nominal transverse dimensions of interest to the ILC, namely: σ ey = 1 µm, VII: Error terms for σey for an electron bunch whose transverse dimensions are σey = 1 µm and σex = 10 µm, giving σu = 1.41 µm and σv = 9.95 µm. The values were obtained assuming the performance goals of Tab. V, laser δ M 2 = 0.01 , σBPM = 20 nm, αJ = 0.25, The electron bunch charge and laser power are assumed each to be known to 1% and the pointing jitter to 10%. The measurement statistical errors are for a full train (i.e. Nscan 140). No subtraction of residual dispersion has been made for these measurements; instead they are input into the global fit to extract the emittance and dispersion terms together. σ ex = 10 µm, with corresponding skew scans with dimensions: σ u = 1.41 µm and σ v = 9.95 µm.
Note that in order to reduce the error from the laserspot size uncertainty for the 1 µm vertical spot-size, UV laser light (λ = 266 nm) has been used. In this case, additional laser power (perhaps by a factor of approximately 1.5-2) may be required to compensate for the inefficiency of the second laser frequency doubling. Most of this factor has already been included in the extra contingency factor of 2 discussed in Sec. III D 1.

IV. CONCLUSION
The measurement of emittance at the ILC will be essential to maintain the high luminosity performance of the machine. A fast non-invasive scheme to do this has been presented, which involves advanced laser-wire systems and dedicated machine optics. It was shown that the efficiency of emittance determination falls dramatically if the transverse electron-bunch measurements provide an accuracy worse than about 30%. The accuracy of the resulting emittance measurement is directly related to the transverse electron-bunch measurements; it is therefore preferable to achieve accurate profile measurements of order 1-5%. A set of methods for emittance reconstruction were presented to improve the reconstruction efficiency in the event of degraded precision and a general scheme for the optics of the emittance measurement section was discussed.
The most challenging vertical spot-sizes in the ILC BDS will eventually be of order 1 µm. The required LW performance was discussed in detail and a plausible route to obtaining a transverse spot-size measurement with a relative precision of order 1.3% using green laser-light was presented, together with prospects of achieving modestly improved measurements using ultra-violet light. It can be concluded that, while percent-scale measurements on a train-by-train basis seem possible, many sub-systems need significant improvements over the current state-ofart; an ongoing programme of R&D is currently addressing these challenges.

APPENDIX A: LASER-WIRE OVERLAP INTEGRALS
In the following, the electron beam is assumed to have a simple Gaussian charge profile, with σ ex and σ ey be-ing the horizontal and vertical electron spot-sizes respectively. σ z is assumed long compared to the laser spot-size, so the overlap integral in z integrates out trivially.

Scans Using the Laser TM00 Mode
In this section, the full overlap integrals of TM 00 and TM 01 laser modes with a Gaussian electron bunch are presented, building on previous studies [32] by including full effects of Rayleigh range and detailed analysis of the laser final focus optics. The results of the numerical integrals for a range of parameters of interest to ILC laser-wires are presented in Tables VIII-X For the laser TM 00 mode, after performing the zintegral, the remaining transverse overlap integral is: Performing the y-integral gives: In the approximation of an infinite Rayleigh range the equations reduce to the more familiar form with [5] σ m = σ 2 e + σ 2 (A4) and a. Scans Using the Laser TM01 Mode Using the same conventions as in Sec. III A 2 the light intensity of the laser TM 01 mode has the form I (x, y, z) = I 0 2πσ 2 (A6) The necessary overlap integral is now: where, as before, ∆ x and ∆ y are the horizontal and vertical relative displacements of the electron and laser beams.
Performing the y-integral gives: (∆ x , ∆ y ) = I I e 2πσ ex dx σ s (x, ∆ x ) σ ey σ s (x, ∆ x ) The minimum practical f -number is taken as 1.0, even where a better statistical precision could in principle be obtained by smaller fnumbers. The numbers in brackets are the corresponding statistical errors using f1.5 optics; the systematic errors due to δ M 2 = 0.01, 0.05, 0.10 are then 1.09%, 5.68% and 12% respectively and do not depend significantly on the electron bunch aspect ratio. where σ s (x, ∆ x ) is as defined in Eq. A3. For the TM 01 mode, the condition that 99% of the light energy is contained within the lens aperture becomes:  IX: Optimal f -numbers (for Estat) for the measurement of σey and the corresponding statistical precision for σey = 1 µm, λ = 355 nm and M 2 = 1.3. The minimum practical f -number is taken as 1.0, even where a better statistical precision could in principle be obtained by smaller f -numbers. The numbers in brackets are the corresponding statistical errors using f1.5 optics; the systematic errors due to δ M 2 = 0.01, 0.05, 0.10 are then 0.48%, 2.48% and 5.16% respectively and do not depend significantly on the electron bunch aspect ratio.  Optimal f -numbers (for Estat) for the measurement of σey and the corresponding statistical precision for σey = 1 µm, λ = 266 nm and M 2 = 1.3. The minimum practical f -number is taken as 1.0, even where a better statistical precision could in principle be obtained by smaller f -numbers. The numbers in brackets are the corresponding statistical errors using f1.5 optics; the systematic errors due to δ M 2 = 0.01, 0.05, 0.10 are then 0.27%, 1.39% and 2.87% respectively and do not depend significantly on the electron bunch aspect ratio.