Simulation of transverse emittance measurements using the single slit method
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1 Simulation of transverse emittance measurements using the single slit method Rudolf Höfler Vienna University of Technology DESY Zeuthen Summer Student Program 007 Abstract Emittance measurements using the single slit scan method was investigated using simulation data for particles from ASTRA [1]. It was shown that neglecting the correlation term leads to an underestimation of the emittance of 14% for a slit width of 50ñm. (15% for 10ñm repsectively). An alternative formula for the transverse emittance is proposed considering the corralation xx. Its convergence is shown for high number of measured beamlets.
2 1 Introduction 1 Introduction Synchrotron radiation was in the first place produced as a by-product in storage rings. Meanwhile a wide range of applications has been developed mostly in material science and medicine. The quality of the synchrotron radiation plays an important role since the wavelengths determines the resolution. Further, most of the beam intensity in the experiments is lost due to several diaphragms, which a beam has to pass before it reaches the sample. The most advanced x-ray radiation source is at the moment the Free Electron Laser (FEL). In a Free Electron Laser electrons are accelerated in a LINAC and injected into the undulator. An undulator is simply a sequence of dipole magnets with alternating polarity. The trajectory of the electrons has then a sinusoidal shape and synchrotron radiation is emitted along the longitudinal direction. The electrons start to interact with their own electromagnetic field. If the radiation emitted from the different segments is coherent one talks about self-amplified spontaneous emission (SASE). The intensity is determined by the phase space density of the electron bunch. The aim of PITZ (Photo Injector Test facility Zeuthen) is to optimise a source capable of such a high quality beam. One demand for future free electron lasers such as XFEL, the European Free Electron Laser is an emittance of ǫ = 1.4 mm mrad at the undulator. This means, since the emittance in a linear accelerator can only increase, that an emittance after the gun of ǫ = 0.9 mm mrad at a bunch charge of 1nC must be reached. An electron beam can be characterised by several parameters such as brilliance and brightness. The latter one is given by B = I/(ǫ x,rms ǫ y,rms ). The denominator represents an additional, especially for electron beams very important parameter - the transverse emittance. It is the product of the emittance in x and in y direction. It is obvious that, for a given beam current I one has to decrease the emittance to increase the brightness. It was shown in [] that a transverse emittance smaller than 1 mm mrad can be reached. The measurement of the beam emittance is a sophisticated task and several different techniques are currently in use or in development respectively. Redundancy in the measurements is one reason for the use of different methods such as phase space tomography and quadrupole scans. This paper treats some aspects of the single slit method using simulated data from ASTRA [1]. Basic concept of emittance and phase space A bunch of N particles 1 in an electron beam can be represented by the phase space density f 6 ( r, p). The number of degrees of freedom here is 6 per particle i.e. {x, y, z, p x, p y, p z }. The number of particles within an infinitesimal Volume is given by: dn = f 6 ( r, p)dv 6 (1) To simplify (1) the interaction between the longitudinal and the transverse components is neglected. The phase space density in then: 1 For the simulation ASTRA was [1] used. This program uses macro particles rather than single particles. The phase space density of N particles is given by f 6N( r, p). Since the treatment of a whole bunch at once is rather difficult one looks at one particle each in a 6 dimensional phase space.
3 Basic concept of emittance and phase space f 6 ( r, p) = f ( x, p x ) f ( y, p y ) f ( z, p z ) () Each factor on the right hand side represent a -dimensional subspace. On the other hand one calculates the d phase space distribution out of the 6 dimensional, simply by tracing out the other degrees of freedom. f ( x, p x ) = 1 dv6 f 6 ( r, p) dz dp z dy dp y f 6 ( r, p) (3) Now the representation of the phase space is changed from {x, y, z, p x, p y, p z } to {x, y, z, x, y, p} where the quantities x = px p z, y = py p Z and p, the total mean momentum are introduced (see figure 1). This representation is convenient because x and y are the transverse divergence angles of the beam which can be measured. Also the total momentum is accessible through measurements. Figure 1: Relation between divergence angle and the components of the momentum The particles occupy a certain area in the reduced trace space which is given by: A x = dxdx = 1 dxdp x (4) p It is assumed that p p << 1 i.e. that p z can be replaced by the total mean momentum. The transverse emittance in x-direction is then defined as the area occupied by the particles i.e ǫ x = A x (5) Another definition of the emittance is the phase space area divided by π. The definition of the emittance is not unique. Here (5) is used. The normalised emittance is given by ǫ n,rms = βγǫ rms, where the two factors are the relativistic Lorentz factors. The normalisation is necessary to make beams with different energies comparable [6]..1 Definition of the emittance using the second central moments of the particle distribution The next step is to find a way to calculate the emittance from measurable quantities. The starting point is a definition of the emittance based on the central moments of the particle distribution 3. ǫ x,rms = x x xx (6) 3 The definition (6) is actually the trace space emittance. It is shown in [3] that phase space and trace space emittance are equivalent. 3
4 Basic concept of emittance and phase space The quantities within the square root are defined by: x = x = xx = dxdx f (x, x )x dxdx f (x, x )x dxdx f (x, x )xx (7) It was assumed that the first central moments x and x equal zero. One can put the moments into the so called sigma matrix and define then the emittance as determinant of the matrix. [ ] x Σ x = xx xx x (8) ǫ x,rms = detσ x (9) It is now obvious that the unit of the emittance is [length angle] - usually [mm mrad]. The rms sizes of the beam are related to the second order moments by: x rms = x rms = Equivalent formulae are obtained for y coordinates. x (10) x (11) The meaning of the correlation term xx can be understood by considering the phase space ellipse. On the left hand side in fig. the correlation term is zero. If it is possible to remove the correlation on the right hand side i.e. to shear the phase space ellipse one can obtain the transverse emittance directly by measuring the rms size and rms divergence angle of the beam. A detailed derivation of the correlation is given in [3]. Figure : Phase space ellipse without correlation on the left hand side and with correlation on the right hand side. The Areas of ellipsis are not exactly the equal but the difference is tiny 4
5 3 Single slit method Figure 3: Scheme of a slit scan. The figure shows distributions of real measured data 3 Single slit method The electron beam at PITZ is highly dominated by the space charges in particular due to the high charge density. The emittance increases due to the space charge effect which can be compensated by a solenoid magnet [4]. The projected emittance can be plotted as a function of the solenoid current and has a minimum near the beam waist at a certain position. A slit is used to select a tiny fraction of the charge such that the beamlet evolution is dominated by the emittance. The measurements are performed by moving a slit with an opening of 50 or 10ñm respectively across the beam fig. 3. The rms size of this so called beamlet is measured at a distance L further downstream. The local beamlet divergence and the local rms size at the slit position are related by: x x n,rms n,rms L (1) where L is the drift distance between the slit position and the screen. The divergence of the whole beam is then the weighted sum of all beamlets. x = N bl ω n x n (13) n=1 The rms normalised emittance writes then as follows: ǫ x,rms = βγ x ω n x n (14) The correlation term in this formula is neglected. The emittance in (14) is given by the square root of the correlated nd moment and the divergence defined in (13). It can be shown that the approximation above works for a phase space ellipse. As it is shown in figure 4 a phase space distribution simulated using ASTRA is fan-shaped. In this case a different approach has to be used. 5
6 3 Single slit method x [mrad] 3 Entries Mean x Mean y e-06 RMS x RMS y x [mm] Figure 4: Transverse phase space distribution in x direction. 3.1 Slit scan under consideration of the correlation term Lets take (7) as starting point. For N slices it is: x = dxdx f,n (x, x )(x x n + x n) (15) n The integrand is modified with the term x n which represents the mean divergence of a beamlet. This can be rewritten as: x = dxdx f,n (x, x )( x n + x n) (16) n Squaring the expression in the parenthesis and carrying out the integral under the assumption that the 1 st moments disappear leads to the final expression: x = n ω n x n + n ω n x n (17) Applying the same ansatz for xx leads to a final expression for the transverse emittance. ǫ x,rms = x rms ( n ω n (x n,rms + x n )) ( n ω n ( x n x n + x n x n)) (18) Here additional terms appear. First the mean divergence of the beamlet which modifies (13). Second a correlation term which is the weighted sum of the product of mean beamlet position the mean beamlet divergence plus the -terms which represents the deviation of a single particle from the mean value of the beamlet. These terms disappear if the sum over all particles within a beamlet is taken. In other words: the mean deviation from the mean value is zero. Formula (18) can be rewritten as: ǫ rms = ǫ x,0 + A B (19) 6
7 4 Simulation and results The formula above is able to reproduce the emittance values that are obtained directly by ASTRA. One has to take care that a sufficient number of particles is taken into account to get the correct values. The crux of the matter is that formula (18) contains terms i.e. the -terms that are not measurable. The first term in (19) on the right hand side is given by (14). A is the additional term ωn x n in (17) and B represents the correlation term. To obtain the correct emittance it is therefore necessary to calculate these quantities correctly. Both corrections A and B can become very large numbers and go into a sum which results a very small quantity. It is therefore essential to determine both values correctly. A deviation in only one of the two values can make the calculation impossible. 4 Simulation and results Before the results are discussed the quantities which appear in (18) are shown as a function of the transverse slit position (see figure 5). Since this is a bunched beam with a flat-top temporal shape the transverse beam profile deviates from the Gaussian one. The distribution is symmetric and centered around zero at the focusing current of 38A x w n vs slit postion weight w n vs. slit postion 0.35 ] [mrad 0.07 weight 0.3 n x x [mm] - slit pos x [mm] x w n vs. slit postion xx w n vs. slit postion [mrad] x n x [mm] [mm mrad] xx n x [mm] Figure 5: Central moments as a function of the transverse slit position. The scan was made for 5 beamlets at 38A. 4.1 Underestimation of the emittance As mentioned in the previous section formula (14) results in emittance values that are lower compared to the direct output of ASTRA (rms projected normalised emittance). Anyway one can clearly see in figure 6(a), that the difference between the curve obtained from ASTRA 7
8 4 Simulation and results and the the curve from the slit scan has a minimum at I focus. The procedure was tested for different number of beamlets and slit widths. Figure 6(b) shows the emittance normalised to the ASTRA emittance as a function of the number of beamlets. It is possible to not take all particles into account i.e. the slit scan has not to cover the whole bunch. Only for a very low number of scanned particles the curve looses its smoothness and statistical errors become larger. The emittance was calculated as the arithmetic mean of ǫ x,rms and ǫ y,rms. Comparison of different emittance curves Emittance normalised to the ASTRA value [mm mrad] ε y ε x Screen at 4.3m ASTRA (direct output) slit scan(50 µm,61 slices) x rms xx /σ x ASTRA ε ε (SW 50 µm, I 38 A): = ε ε sol. Emittance (ASTRA) slit scan measurement I solenoid [A] (a) Number of beamlets (b) Figure 6: Fig. 6(a) Slit scan using formula (14) performed with 61 slices. It shows also the rms size and the correlation term. Fig. 6(b) convergence behaviour - The saturation value is basically reached at 60 beamlets. At 150 beamlets overlapping starts to occur. The shape of the curve strongly depends on the scan sequence. 4. Scanning sequence The scanning sequence was chosen in such a way, that a scan starts in the centre of the bunch and alternates symmetrically to larger values of x. Also a parameter was introduced which gives the possibility to shift all slit positions sligthly along the axis. This results in a oscillatory pattern with a minimum at x = 0. Figure 7 shows the emittance as a function of shift and the number of beamlets. For high number of beamlets the curve approaches a straight horizontal line with a relative deviation of 14% for a slit width of 50ñm. Figure 7: Shift along the x-axis. The equidistant spacing s is not changed. 8
9 4 Simulation and results ASTRA ε ε number of beamlets shift [mm] Figure 8: The value obtained from ASTRA was 1.3 [mm mrad]. The graph shows that the scan pattern only plays a role for a low number of beamlets. The calculated emittance can reach a value close the predicted value by ASTRA but is unusable due to the dependence on the shift. 4.3 Emittance reconstruction considering the correlation The emittance can be reconstructed if formula (18) is used. Since the -terms do not contribute the final equation is: ǫ x,rms = x rms ( n ω n (x n,rms + x n )) ( n ω n ( x n x n)) = ǫ 0,rms + A B (0) Figure 9 shows the equivalent plot to figure 6. Here the emittance from the slit scan fits very well the curve obtained from the ASTRA simulation. In fig. 9(b) the convergence behaviour is plotted. In contrast to figure 6(b) the saturation value is reached at beamlets. This means that one has to take almost all particles into account. The scan range at the focusing current is between 3.5 and 3.8mm. Therefore the step width for 00 beamlets is smaller than the slit width. The relative deviation for the last point i.e. 450 beamlets is 0.7%. At 10 beamlets the curve has a relative deviation of 50%. 9
10 4 Simulation and results Comparison of different emittance curves Emittance normalised to the ASTRA value [mm mrad] ε y 7 6 Screen at 4.3m ASTRA (direct output) slit scan(50 µm,01 slices) ASTRA ε ε (SW 50 µm, I 38 A): ε ε = sol. Emittance (ASTRA) slit scan measurement ε x I solenoid [A] (a) Number of beamlets (b) Figure 9: 9(a) ASTRA curve and curve from the slit scan. 9(b) Convergence is only obtained if the whole bunch is covered by the scan. ε ε Isolenoid Number of beamlets Figure 10: Convergence of the relative deviation Finally the convergence behaviour is shown as a function of I solenoid and the number of beamlets (see figure 10). One can clearly see that ǫ/ǫ approaches zero for high current and high number of beamlets. 10
11 5 Summary and outlook 5 Summary and outlook Single slit scan method was investigated using simulation data for particles from ASTRA. It was shown that the approximation that uses a sheared phase space ellipse underestimates the emittance. An alternative formula for the transverse emittance is proposed considering the correlation xx. Its convergence is shown for high number of measured beamlets. Leaving out more than 1 per mill of the particles leads already to a significant deviation from the simulated emittance values In single slit scan measurements the correlation term must be considered in the calculation. A neglect leads to an underestimation which converges to a relative deviation of 14% for a slit width of 50ñm and 15% for 10ñm. The shape of the convergence curve is strongly influenced by the scan pattern. This dependence disappears when the number of beamlets is sufficient high i.e. the convergence value is reached. The transverse emittance can be reconstructed if the correlation term is taken into account. The emittance converges slower as a function of the number of beamlets. A scan has to cover the whole bunch. To measure the correlation term, one needs to know the mean position and the mean divergence of the beamlets relative to the whole bunch. Since all beamlets are measured on the same screen, it is possible to calculate the mean values of the bunch out of the beamlet mean values and to correct the coordinates accordingly i.e. that the 1 st moments of the bunch are zero. One can also measure the whole bunch without a slit mask and correct afterwards the coordinates accordingly. For further investigations also the possibility of non-uniform scan patterns has to be considered. 11
12 References References [1] K. Flöttman, ASTRA mpyflo [] L. Staykov, Commissioning of an New Emittance Measurement System at PITZ [3] K. Flöttmann, Some basic features of the beam emittance (003) [4] J.-H. Han, Dynamics of Electron Beam and Dark Current in Photo cathode RF Guns [5] K.-J. Kim, Physics Accelerator Physic and Technologies for Linear Colliders (00) [6] V. Miltchev, Investigation on the transverse phase space at a photo injector for minimised emittance (006) [7] K. Wille, Physik der Teilchenbeschleuniger und Synchrotronstrahlungsquellen 1
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