Pickup concepts for ultra-low charged short bunches in X-Ray Free-Electron Lasers
Bernhard E. J. Scheible, Stefano Mattiello, Marie K. Czwalinna, Holger Schlarb, Wolfgang Ackermann, Herbert De Gersem, Andreas Penirschke
PPickup concepts for ultra-low charged short bunches in X-Ray Free-Electron Lasers
B. E. J. Scheible, ∗ S. Mattiello, and A. Penirschke
Technische Hochschule Mittelhessen, Wilhelm-Leuschner-Straße 13, 61169 Friedberg, Germany.
M. K. Czwalinna and H. Schlarb
Deutsches Elektronen-Synchrotron DESY, Notkestrasse 85, 22607 Hamburg, Germany.
W. Ackermann and H. De Gersem
Technische Universit¨at Darmstadt, Schloßgartenstr. 8, 64289 Darmstadt, Germany. (Dated: February 3, 2021)The all-optical synchronization systems used in various X-ray free-electron lasers (XFEL) suchas the European XFEL observe the transient fields of passing electron bunches coupled into one ormore pickups in the Bunch Arrival Time Monitors (BAM). The extracted signal is then amplitudemodulated on reference laser pulses in a Mach-Zehnder type electro-optical modulator. With theemerging demand for future experiments with ultra-short FEL shots, fs precision is required forthe synchronization systems even with 1 pC bunches. Since the sensitivity of the BAM depends inparticular on the slope of the bipolar signal at the zero-crossing and thus, also on the bunch charge,a redesign with the aim of a significant increase by optimized geometry and bandwidth is inevitable.In this contribution the theoretical foundations of the pickup signal are aggregated and treated witha focus on ultra-short bunches as well as a general formulation. A possible new pickup concept issimulated and its performance is compared to the previous concept. A significant improvement ofslope and voltage is found. The improvement is mainly achieved by the reduced distance to thebeam and a higher bandwidth.
I. INTRODUCTION
Free-electron lasers (FEL) became an important lightsource for experiments in various fields since they pro-vide ultra-short pulses with extreme brilliance in atomiclength and time scales [1]. FEL are well suited for appli-cations in pump-probe experiments [1], where the timingjitter is specifically critical [2], as well as for capturingimage sequences with atomic resolution on fs-time-scales,even below the FEL repetition rate [1, 3, 4].For the generation of ultra-short X-ray pulses, FELwith short and ultra-low charge electron bunches ( ≤ .
02 nC to 1 nC [7] with a possible bunch length below3 fs in the undulator section [8]. Moreover, a decrease toultra-low charges of 1 pC is targeted.The application in time-resolved experiments entailstight requirements for the overall machine synchroniza-tion in order to reduce the timing jitter [2]. The timinginformation is also used for post-processing experimentaldata [1]. The synchronization concerns all critical subsys-tems, specifically in the injector and if present the seedingand the pump laser [1]. Furthermore, the instrumenta-tion must be suited for a broad spectrum of operation ∗ [email protected]; Also at Technische Universit¨atDarmstadt, Schloßgartenstr. 8, 64289 Darmstadt, Germany. modes with different bunch properties even in a singlebunch train [9, 10]. Besides, bunch arrival time monitors(BAM) are installed throughout the whole facility, thusexperiencing different bunch properties.A tremendous improvement in synchronization, ex-ceeding RF techniques, and reduction of arrival time aswell as energy jitter was achieved by the implementa-tion of an all-optical synchronization system with twodifferent feedback loops [11]. Though some updates havebeen introduced [12, 13], the basic scheme in use by the Deutsches Elektronen-Synchrotron (DESY) remained un-changed.In this contribution the all-optical synchronization sys-tem will be introduced with special attention on thestate-of-the-art cone-shaped pickups. The schematicBAM description is followed by an in-depth analyticaldiscussion of the voltage signal, regarding different lim-iting cases as well as a general solution, to identify prin-cipal parameters determining the BAM resolution andtheir effect on the signal shape. For design purposes anadditional numerical treatment is introduced. These con-siderations are the basis for three designs, which are pre-sented at the end of this paper.
II. ALL-OPTICAL SYNCHRONIZATIONSYSTEM
The all-optical synchronization system, as successfullytested at the Free-Electron Laser in Hamburg (FLASH)by L¨ohl [11], mainly comprises of a mode-locked refer-ence laser, length stabilized fiber links and different end-stations for synchronization and arrival time measure- a r X i v : . [ phy s i c s . i n s - d e t ] F e b ment [10–15]. The arrival time is non-destructively mea-sured with respect to the reference laser in the BAM,which include high-bandwidth pickup electrodes in theRF unit, a Mach-Zehnder type electro-optical modulator(EOM) and the data acquisition system (DAQ) [10, 15]. A. Basic working principle of BAM
The transient electric fields of passing electron bunchesare extracted in the RF unit and, if foreseen, initially pro-cessed with analogue components like RF combiners, lim-iters or attenuators [11, 12]. The received bipolar signalis transmitted over radiation hard silicon dioxide coaxialcables to the EOM [12], there it is probed at its zero-crossing by the reference laser [11]. Any temporal devi-ation will lead to an additional voltage, which the EOMturns into an amplitude modulation of the laser pulse.Therefore, the laser amplitude holds the timing informa-tion, which can be retrieved in the DAQ [11]. The signalslope at its zero-crossing strongly influences the BAM’stemporal resolution. The minimum design requirementfor the currently installed BAM was ≥
300 mV ps − with20 pC bunches [12]. B. RF unit and pickups
The RF unit so far comprises four identical pickupsmounted circularly around the beam line. The combina-tion of opposite pickup signals compensates for the orbitdependency [16]. The original pickups used at FLASHwere of button-type and designed for a 10 GHz band-width [17]. This pickup struggled with ringing and strongsignal reflections at the alumina vacuum feedthrough [17]degrading the signal strength and resolution for chargesbelow 150 pC [18]. Hence, a new design was required [18].A novel pickup (Fig. 1, left), similar to those designedfor the CERN linear collider test facility [19], was pro-posed as a solution in FEL applications [20]. The cone-shaped design, finalized in [12], with 40 GHz bandwidthbecame a new standard device used at the EuXFEL andat other FELs.Due to high losses in the RF path a design updatewas necessary (Fig. 1, right). The second generationof cone shaped pickups were optimized towards a max-imum signal voltage at the cost of its slope by increas-ing the active surface and letting the cone slightly pro-trude into the beam pipe [13]. A limit was reached whena larger surface, e.g. more than a few bunch lengths,further decreases the slope without a further increase inpeak-to-peak voltage [13]. The protrusion increases theinductance, which unfavorably deforms the signal shape[13]. Nonetheless, a combination of both modificationsincreased the peak-to-peak voltage sufficiently for 20 pCbunches while maintaining an acceptable slope [13].Recently the performance of the state-of-the-art sys-tem was evaluated at the EuXFEL. The correlation of
Cut -out
Cone
Cone-shapedpickup Modified cone-shaped pickup
FIG. 1. Cross section of the 1 st generation cone-shaped pickup(left) and the modified 2 nd generation cone-shaped pickup forhigh peak-to-peak voltage (right) adapted from [13]. two adjacent monitors with less than 1 m distance wasanalyzed. Examining the measured arrivaltimes for a pe-riod of 1 min gave a timing jitter of approximately 6 fsr.m.s. caused by the BAM resolution and critical partsof the reference laser distribution system [21]. III. RF SIGNAL
The limitations of current pickup structures are evi-dent in the theoretical consideration of the time domainvoltage signal. The image charge on the pickup surfaceis calculated by Q im ( t ) = (cid:90) ∞−∞ λ ( z − c t ) w ( z )2 πr p d z, (1)with the line charge density λ ( t ) of the ultra-relativisticbunch ( v ≈ c ), pickup width w ( z ) and the distance be-tween pickup surface and bunch r p [22]. Because linecharge density and pickup width are real quantities, bysubstitution of z into a moving frame, Eq. (1) can betransformed into the cross-correlation Q im ( t ) = ( λ (cid:63) w ) ( t ) , (2)which is readily solved numerically. This is true for anybunch and pickup form. In the special case of an evencharge density, Eq. (1) is the convolution Q im ( t ) = ( λ even (cid:126) w ) ( t ) . (3)Eqs. (2) and (3) can be utilized in the numerical treat-ment and also for analytical solutions, which are acces-sible for many different cases. Some essential examplesare covered in this section followed by the numerical ap-proach. A. Analytical Model
Idealized representations of bunch and pickup surface,allow to find an analytical solution for Eq. (1). A suitablerepresentation that leads to a substantial simplificationis given by a Gaussian bunch and a rectangular pickupsurface. The stationary charge density of the Gaussianbeam centralized on the z -axis when is ρ ( x, y, z ) = δ ( x ) δ ( y ) λ ( z ) . (4)The line charge density is found by integration of thetransverse area. A Gaussian bunch is given by λ ( z ) = λ exp (cid:18) − z σ (cid:19) = Q b G ( z, σ z ) , (5a)where λ = Q b / (cid:112) πσ is a constant for normalizationby the bunch charge, Q b the total charge of one bunch, σ z the r.m.s. length and G ( τ ) the normalized Gaussiandistribution function G ( τ, σ τ ) = 1 √ πσ τ exp (cid:18) − τ σ τ (cid:19) . (5b)In the scope of this work, we denote λ ( t ) := λ ( z ) | z = c t = Q b c G ( t, σ t ) , (5c)using Eq. (5a), where the speed of light is incorporatedinto the r.m.s length in units of time σ t = σ z c . (6)The rectangular pickup surface width is written as w ( z ) = w Π (cid:96) ( z ) , (7)where (cid:96) is the longitudinal extension of the pickup andΠ (cid:96) ( z ) = (cid:40) | z | ≤ (cid:96) | z | > (cid:96) (8)the rectangular function.
1. Long Bunch Approximation
In many applications, specifically with hadrons, it isreasonable to assume a bunch much longer than thepickup. This is treated in the long bunch approxima-tion (LBA). The pickup form is insignificant in the LBA,hence a more general approach is used. For this purposethe parameterization of a finite pickup profile is w ( z ) = ˜ w ( z ) χ [ a,b ] ( z ) , (9)where χ [ a,b ] ( z ) is the indicator function of the interval[ a, b ], which bounds the pickup in longitudinal direction,and ˜ w ( z ) defines the pickup width at location z . Therectangular pickup in Eq. (7) is a special case of Eq. (9).The longitudinal characteristic dimension of the pickup is defined by (cid:96) = | a − b | and its surface area A p by theintegral A p = (cid:90) ∞−∞ ˜ w ( z ) χ [ a,b ] ( z )d z = (cid:90) ba ˜ w ( z ) d z. (10)The image charge found by introducing Eq. (9) in Eq.(1) is Q im ( t ) = 12 πr p (cid:90) ∞−∞ λ ( z − c t ) ˜ w ( z ) χ [ a,b ] ( z )d z (11a)which becomes Q im ( t ) = λ c πr p b − c t (cid:90) a − c t ˜ λ ( ζ ) ˜ w ( ζ + c t ) d ζ (11b)by substitution of ζ = z − c t and factorizing λ ( ζ ) as theproduct of the constant term λ c , which has the units of aline charge density, and a dimensionless longitudinal dis-tribution function ˜ λ ( ζ ). Despite the explicit dependenceof the bunch form, the following derivation of the longbunch approximation only requires that the distribution˜ λ ( z ) can be written asT˜ λ ( z ; c t ) = ˜ λ ( c t ) + ˜ λ (cid:48) ( c t )( z − c t ) + ... , (12) ∀ z ∈ R apart to a at most countable set of points, whereT˜ λ ( z ; c t ) is the Taylor expansion of ˜ λ ( z ) at c t . Theimage charge can be written as an expansion as well, Q im ( t ) = λ c πr p ∞ (cid:88) n =0 q n ( c t ) , (13a)with q n ( c t ) = ˜ λ ( n ) ( c t ) b − c t (cid:90) a − c t ˜ w ( ζ + c t ) ( ζ − c t ) n d ζ. (13b)For a long bunch, or equivalently a short pickup, it isjustified to consider only the leading term, with n = 0, ifthe reminder term is negligible, which implies the func-tion to be constant in the rolling interval of length (cid:96) . Thisis applicable if the following limit is true | R | = (cid:12)(cid:12)(cid:12) ˜ λ ( z ) − ˜ λ ( c t ) (cid:12)(cid:12)(cid:12) → ∀ z ∈ [ c t − a, c t + b ] (14)and justified for any given function at some point for | a − b | →
0. Therefore, it is useful to introduce a scalingquantity µ for the bunch-pickup system, which controlsthe approximation for any Lipschitz continuous function.For this purpose because the derivative is bounded by M = sup (cid:110)(cid:12)(cid:12)(cid:12) ˜ λ (cid:48) ( t ) (cid:12)(cid:12)(cid:12) : t ∈ R (cid:111) , that is the Lipschitz constantof the distribution ˜ λ ( t ), the following inequality gives anupper limit for the reminder R ˜ λ ( z ) − ˜ λ ( c t ) ≤ M | z − c t | ≤ M | b − a | . (15)Therefore, the scaling is defined as µ = M | b − a | = M (cid:96) (16)and the requirement in Eq. (14) is fullfilled for µ (cid:28) . (17)Applied on a Gaussian bunch according to Eq. (5a) it is µ = 1 √ e lσ z (cid:28) . (18)Under this assumption, the image charge is q and thus Q LBAim ( t ) = λ ( t )2 πr p (cid:90) ba ˜ w ( z ) d z , (19)which by Eq. (10) is Q LBAim ( t ) = A p πr p λ ( t ) . (20)This approximation is widespread and for example foundin [22, 23]. Equation (20) is valid for any pickup whichis short enough to neglect the change of line charge den-sity along the pickup’s length, which is ensured by re-quirement (14), independent of pickup and bunch form.The validity of this situation, i.e. of the long-bunch ap-proximation, is qualitatively controlled by condition (17).If this condition is satisfied, it is ensured that the re-quirement (14) is also met. Nonetheless Eq. (17) is astricter condition and may be untrue for some functionsthat still are sufficiently described by the LBA. Theserequirements are met in many facilities, thus the theoryis well developed and calculations for the image currentand output voltage discussed in various publications onbeam instrumentation and in particular beam positionmonitors as [23].
2. Short Bunch Approximation
In FEL applications as discussed in this work, the re-quirement of the long bunch approximation µ (cid:28) ≥ . µ (cid:29) ⇔ σ z (cid:28) (cid:96) . (21)Under this assumption in the short bunch approximation(SBA), because the normalized Gaussian is a Dirac se-quence, Eq. (5a) is described by a delta distribution withthe total charge Q b concentrated at one point in spacelim σ t → λ ( z − c t ) = Q b δ ( z − c t ) . (22) Since the delta distribution is even, Eq. (1) is treated asa convolution, leading straightforward to Q SBAim ( t ) = Q b w πr p Π t ( t ) , (23)where t = (cid:96) c (24)is the time it takes for the bunch center to pass half thepickup. The image current found by differentiation of Q SBAim ( t ) is given by the following distribution I SBAim ( t ) = Q b w πr p [ δ ( t + t ) − δ ( t − t )] . (25)The output voltage is simply derived for any given trans-fer function H ( ω ) with corresponding impulse responsefunction h ( t ) by the convolution with Eq. (25) U SBAh ( t ) = ( h (cid:126) I im ) ( t ) . (26)Therefore, the voltage signal in time domain is U SBAh ( t ) = Q b w πr p [ h ( t + t ) − h ( t − t )] . (27)The preceding derivation of the short bunch limit is pos-sible for all line charge density distributions in L , thuswith finite total bunch charge Q b . The resulting voltageshape U sh ( t ) is certainly independent of the infinitesimalshort bunch form, but defined by the subtraction of thescaled impulse response function h ( t ) shifted parallel indifferent directions.In the following two idealized transfer functions areused to illustrate the implications of the ultra-shortbunch. The two applied low-pass filters are a rectangularfilter defined by H ( ω ) = R Π c ( ω ) (28a)and a Gaussian filter defined by H ( ω ) = R exp (cid:18) − ln(2) ω (cid:19) . (28b)The cut-off frequency is Ω c and the filters are terminatedwith R, usually 50 Ω. The corresponding time domainresponse functions are h Π ( t ) = R Ω c π si (Ω c t ) (29a)and h G ( t ) = R (cid:101) Ω c √ π exp (cid:32) − (cid:101) Ω t (cid:33) , (29b)where (cid:101) Ω c = Ω c / (cid:112) ln(2) is the cut-off frequency definedby a 1 /e decrease. The voltage signals U sΠ ( t ) respectively − −
100 0 100 200 − − t (ps) U Π ( V ) Ωc = 100 GHzΩc = 40 GHz − − − −
20 0 20 40 60 80 − − t (ps) U G ( V ) Ωc = 100 GHzΩc = 40 GHz
FIG. 2. Voltage signal in the SBA for two idealized filters. A rectangular (left) and a Gaussian filter (right) have been utilized,each at 100 GHz (blue) and 40 GHz (red) cut-off frequency. Bunch and geometry parameters are in a reasonable range forXFEL applications, e.g. σ t = 200 fs, Q b = 20 pC, (cid:96) = w = 4 . r p = 20 mm and R = 50 Ω. U sG ( t ) are found by substitution of the response functionin Eq. (27). The resulting curves are shown in Fig. 2 as-suming dimensions in the range of existing bunch arrivaltime monitors. The characteristic curve, resembling thederivative of a Gaussian, is found at the center of bothexamples. An ideal rectangular filter induces an infiniteripple, which in some cases also reflects on the steep sig-nal slope at the center, while the Gaussian is smooth.The signal slope at the zero-crossing S h , ZC = ˙ U h ( t ZC ) (30)is the figure of merit in BAM applications. The signifi-cant ZC is located at t = 0 for both example filters. Therespective slopes are S SBAΠ , ZC = − Rw π r p Q b Ω c t [cos (Ω c t ) − si (Ω c t )] (31a)and S SBAG , ZC = − RA p πr p Q b √ π (cid:101) Ω c (cid:20) exp (cid:18) − (cid:101) Ω t (cid:19)(cid:21) . (31b)For the rectangular filter the slope is depending on theratio of cut-off frequency and pickup length, with a pe-riodical behavior caused by the bracket term. This termhas local extrema of approximately ±
1. For simplicity itis sufficient to assume exactly ±
1, found at Ω c t = nπ ,with n ∈ N , although the real maximum is located some-what before that.For the Gaussian filter the slope is also depending onthe ratio of cut-off frequency and pickup length. For aninfinite as well as infinitesimal pickup the slope is vanish-ing. The maximum slope is reached at (cid:101) Ω c = 2 c /(cid:96) . Fora lower cut-off frequency the signal width is increased,leading to distortion and partial annihilation of the sig-nal. If the bandwidth is increased further, two distinctive peaks become visible leading to a lower gradient at zero-crossing. The optimum is S SBAG , max = − √ e RA p πr p Q b √ π c (cid:101) Ω . (32)This equals the maximum slope received by the shortpickup and high cut-off frequency approximation, apartfrom the factor 1 / √ e and the replacement of the recipro-cal bunch width by the cut-off frequency, which are thecorresponding decisive widths.The real transfer function is neither rectangular norGaussian. In literature the pickup is usually modeled byan equivalent circuit of a resistor R parallel to the ca-pacity C, with the image current I im serving as a currentsource [23, 24]. The expected low-pass filter then gives H ( ω ) = R iωRC , (33)with cut-off frequency Ω RC = 1 / ( RC ), as transfer func-tion from current source I im to the output voltage U RC .In time domain the impulse response function is h ( t ) = 1 C exp (cid:18) − tRC (cid:19) Θ ( t ) , (34)where Θ ( t ) denotes the Heaviside step function.In order to calculate the output voltage with the equiv-alent circuit’s transfer function the convolution with theimage current in short bunch approximation according toEq. (27) gives U SBARC ( t ) = 1 C Q b w πr p exp (cid:18) − t + t RC (cid:19) × (cid:20) Θ ( t + t ) − exp (cid:18) + 2 t RC (cid:19) Θ ( t − t ) (cid:21) , (35) − −
25 0 25 5000 . . . . t (ps) Q i m ( p C ) LBASBAAnalytical − − − t (ps) Q i m ( p C ) LBASBAAnalytical
FIG. 3. Image charge by a Gaussian bunch on a rectangular pickup surface according to the LBA in Eq. (20) (dashed red),the SBA in Eq. (23) (dashed green) and the analytical solution in Eq. (37) (solid blue) for two different bunch lengths. Oneclose to the LBA regime, with µ ≈ .
65 (left) and one closer to the SBA regime, with µ ≈ . σ t = 0 . σ t = 15 ps, Q b = 20 pC, (cid:96) = w = 4 . r p = 20 mm. In the EuXFEL even shorter bunches occur, further reducing the discrepancy between SBAand analytical solution at the edges. which corresponds to the capacitor experiencing the com-plete electric field of the infinitesimal bunch at once, lead-ing to a compensating current, which is equivalent to thedischarging of a capacitor. The decay curve is observeduntil the coasting beam reaches the end of the pickup.Than the effect of the bunch’s electric field on the pickupsuddenly ends and the initial voltage jump is reversed.Afterwards the charges start flowing back, leading to acontinuation of the exponential voltage decay. Thereforeslope at zero-crossing, found at t = t , is an infinite de-crease.
3. General Solution for the Image Current
In case none of the preceding approximations is suf-ficient a general solution for Eq. (1) is needed. It isfound straightforward for Gaussian bunches and rectan-gular pickups. Applying the shapes defined in Eqs. (5a)and (7) without any approximation leads to Q im ( t ) = w Q b πr p (cid:90) (cid:96)/ − (cid:96)/ G ( z − c t, σ z ) d z. (36)This integral is readily solved, giving Q im ( t ) = 12 w πr p Q b (cid:34) erf (cid:32) (cid:96) − c t √ σ z (cid:33) + erf (cid:32) (cid:96) + c t √ σ z (cid:33)(cid:35) (37)for the image charge induced on the pickup by the coast-ing beam. Figure 3 shows exemplary results of the analytical de-scription in Eq. (37) (blue) compared to the LBA, asin Eq. (20) (dashed red), and the SBA, as in Eq. (23)(dashed green), for two different values of the scaling fac-tor µ . In case of an ultra short bunch, µ ≈ . I im ( t ) = w πr p Q b [ G ( t + t , σ t ) − G ( t − t , σ t )] , (38)with t defined in Eq. (24) and bunch width σ t expressedin units of time according to (6). The first Gaussian rep-resents the incoming charges introduced by the arrivingbunch, the other is caused by the bunch leaving the effec-tive area. This result can be understood as a generaliza-tion to the current found in the ultra short bunch approx-imation. If the delta distributions are replaced by Gaus-sians, the factor differs by the normalization 1 / (cid:112) πσ .Consistently the general solution is readily converted intoEq. (25) by the limit of an ultra short bunch length.The general voltage signal measured at resistor R isfound by the known transfer function based on the pick-ups equivalent circuit, see Eq. (33). It is determined bythe convolution of the general image current, Eq. (38),and the impulse response function, Eq. (34). The result,also mentioned in [24], is − −
10 0 10 20 30 − t (ps) U R C ( V ) ind. (C = 30 fF)cap. (C = 4 pF)intermediate − − − − t (ps) U R C ( V ) ind. (C = 30 fF)cap. (C = 1 pF)intermediate FIG. 4. Voltage signal according to Eq. (39) with dimensions in the range of the LBA with µ ≈ . µ ≈
24 (right). The capacity is varied from almost the inductive limit (red) up to the capacitive case (blue) with threeintermediate steps (dashed). Bunch and geometry parameters are in a reasonable range for XFEL applications, e.g. σ t = 7 ps(left) respectively σ t = 0 . Q b = 20 pC, (cid:96) = 2 . w = 4 . r p = 20 mm. In other applications, whenthe capacitive case is required, the limiting case can be achieved by increasing the terminating resistor R, to counter the 1 /C dependence in Eq. (39), see Fig. 3 (a) in [24]. U RC ( t ) = 12 1 C w πr p Q b exp (cid:18) (cid:16) σ t RC (cid:17) (cid:19) (cid:26) exp (cid:18) − t + t RC (cid:19) (cid:20) (cid:18) t + t √ σ t − σ t √ RC (cid:19)(cid:21) − exp (cid:18) − t − t RC (cid:19) (cid:20) (cid:18) t − t √ σ t − σ t √ RC (cid:19)(cid:21)(cid:27) . (39)The voltage curve shows a very different behavior for dif-ferent parameter choice, which is not immediately acces-sible in Eq. (39). The two introduced limiting cases for µ (cid:28) µ (cid:29) µ , it is possible to findan approximation of Eq. (39) by assuming extreme val-ues for the cut-off frequency or in particular the capacity,as presented in the next section. The limiting cases andthe transitional behavior are exemplified in Fig. 4. Onthe left side bunch and pickup length approach the LBA, µ ≈ .
7. In contrast the right image contains signalsfor a relatively short bunch with µ ≈
24, looking alikethe SBA. In both plots the capacitance is varied start-ing at 30 fF, close to the so called inductive case (red)to 1 pF respectively 4 pF, which is around the capacitivecase (blue). The corresponding cut-off frequencies areΩ RC = 0 .
67 THz and 5 GHz respectively 20 GHz. Thetransition between both cases is indicated with strippedlines.The BAM resolution depends on the signal slope atthe zero-crossing (ZC). The time dependent signal slopefound by derivation of Eq. (39)˙ U RC ( t ) = Ω RC [ RI im ( t ) − U RC ( t )] . (40)At the time of zero-crossing t ZC the voltage U RC ( t ZC )is zero by definition. Therefore, the signal slope at the zero-crossing is only defined by the image current S RC , ZC = ˙ U RC ( t ZC ) = 1 C I im ( t ZC ) . (41)Because I im ( t = 0) is zero, a zero-crossing at t ZC = 0would entail a gradient of zero, but the zero-crossing islocated at t ZC (cid:54) = 0 for any C >
4. Voltage by Frequency Regions
In many theoretical approaches the voltage signal isanalyzed in two special cases defined by frequency re-gions. In the so called capacitive limit, the capacity C islarge, respectively the cut-off frequency approaches zeroΩ RC →
0, hence all exponential functions approach oneand the first term of each error function is predominant.This leads to a great simplification of the output voltage,which is U capRC ( t ) = 1 C Q im ( t ) . (42)Therefore, the signal in the limit of a infinitesimal cut-off frequency resembles a charge source [24]. The limitfor an infinitesimal capacity, the inductive case [24] withΩ RC → ∞ , is straightforward for the frequency domaintransfer function Eq. (33), which becomes the constantR, and thus h ( t ) = Rδ ( t ) in time domain. The voltage isthen U indRC ( t ) = RI im ( t ) , (43)resembling a current source [24]. In this case, the zero-crossing is exactly at t = 0 and for a finite pickup lengththe signal slope is nonzero in a seeming contradiction toEq. (41), but the denominator C is zero as well.
5. Maximum Voltage and Signal Slope
For deployment in a BAM, maximum voltage and sig-nal slope are the key features of the pickup. Moreovera throughout understanding of the decisive parametersis crucial. The general formulation in Eq. (39) confirmsthe proportionality to the bunch charge and pickup widthas well as reciprocal to the distance between bunch andpickup U RC , max ∝ Q b w r p (44)and S RC , max ∝ Q b w r p . (45)The proportionality to the bunch charge was experimen-tally proven for the signal slope of the cone shaped pick-ups at FLASH [16]. The inverse proportionality to r p has to be analyzed in simulations, since these measure-ments are expensive and limited by the facilities designparameters.Another dependency observed in simulations, is on theratio of pickup and bunch lengths. The maximum volt-age approaches its maximum asymptotically and deviatesonly slightly from the value for a pickup longer than a fewbunch lengths [13, 24]. This also effects the signal slope,as the maximum voltage barely changes in this regionwhile both extrema drift apart, leading to a decreasingslope [13]. In Eq. (39) it is readily shown, that the volt-age vanishes for t = 0, which is identical to (cid:96) = 0. Todescribe the behavior for any other ratio, the approxima-tions must be utilized. In case of the pure charge sourcethe maximum voltage found at t max = 0 ismax ( U capRC ) = 1 C w Q b πr p erf (cid:18) (cid:96) √ σ z (cid:19) (46)[24]. For a pure current source t max must adhere to t = t max tanh (cid:18) t σ t t max σ t (cid:19) , (47)which is found by setting ˙ U indRC = 0. Two useful ap-proximations give the positions of extrema for a largeor negligible argument of the hyperbolic tangent, which are t max ≈ ± σ t for the LBA with t (cid:28) σ t respectively t max ≈ ± t for the SBA with t (cid:29) σ t . The correspondingmaximum voltages are for a short pickupmax (cid:16) U LBA , indRC , t (cid:28) σ t (cid:17) ≈ R √ e A p πr p Q b √ πc σ = µ ˆ U ind0 , (48)with ˆ U ind0 = R w πr p Q b √ πσ t , (49)and for a long pickupmax (cid:16) U SBA , indRC , t (cid:29) σ t (cid:17) ≈ ˆ U ind0 (cid:20) − exp (cid:18) − l σ (cid:19)(cid:21) . (50)The first term is the maximum voltage for (cid:96) → ∞ , whichis justified for (cid:96) > σ z with the Gaussian bunch. Inthe cases described by Eqs. (46) and (50) the maximumvoltage saturates for a pickup length of a few σ z , whereasin the realms of LBA, with an infinitesimal pickup, themaximum voltage goes linear with its length.Regarding the BAM resolution a high signal slope isdesired. The main proportionality is already given byEq. (45), but in the inductive limit a general solution ispossible. The slope at the zero-crossing is S indRC (cid:12)(cid:12) t =0 = − RQ b w πr p t σ G ( t , σ t ) . (51)This result is visualized in Fig. 5. For a fixed bunchlength, the optimum is reached at t = σ t , whereas for afixed pickup length the best result is found at t = √ σ t .At t = σ t = 0 is a pole, therefore reduction of both di-mensions is favorable in the case of a pure charge source,but in real applications the bunch length is in a fixedrange specific for the facility and foreseen experiments.
6. Bandwidth Limitation
The measured signal slope is limited by the bandwidthaccording to the uncertainty principle. The product ofa signal’s rise time τ as response to a step function andthe frequency bandwidth ∆ f satisfies τ ∆ f = 1 . (52)This rise time serves as an upper limit and can be ex-pressed by the maximum slope S max and the peak-to-peak voltage U PP with τ = U PP /S max giving S max ≤ U pp ∆ f. (53) Denote that the right-hand side of the related uncertainty in-equalities by K¨upfm¨uller [25] and Gabor [26] depend on the def-initions of bandwidth and duration.
FIG. 5. Absolute signal slope at zero-crossing according toEq. (51) as a function of t and σ t in the interval from 0to 2 ps. The dotted contours mark 1, 10 and 100 V ps − anddashed lines indicate t = σ t and t = √ σ t . This limitation affects the resolution of a transmittedsignal. It determines the highest possible slope allowedin the response function, but if the slope of an incomingsignal is well below this limit, a rise in bandwidth mighteven have a negative effect on the outputs signal slope,as already shown for the idealized Gaussian filter in theSBA.
7. Circular Pickup
In all prior considerations the pickup was assumedrectangular, which is not in accordance with many real-world applications. From Fig. 4 and the formulationof the SBA it is reasonable to assume that the rectan-gular model is sufficient for long bunches, but its shapeis significantly reflected in the voltage signal for shorterbunches. A general analytical solution for the circularsurface of radius r B , defined by w ( z ) = (cid:40) (cid:112) r − z | z | ≤ r B | z | > r B (54)[22], is not available and numerical methods have to beutilized. Nonetheless, some special cases are accessible byanalysis. For the SBA, according to Eq. (22), the imagecharge is proportional to the pickup shape and thus theimage current is I SBAim ( t ) ≈ − c Q b πr p t √ r − ( c t ) | t | < r B c | t | > r B c (55)with two poles at t ±∞ = ± r B /c , which would be soft-ened by a broader bunch. In the inductive limit, interest-ing for synchronization, the voltage is given by Eq. (43), with the zero-crossing exactly at t = 0 and a signal slopeof S SBA , indRC (cid:12)(cid:12)(cid:12) t =0 = − R c Q b πr p r B . (56)The slope is inversely proportional to the radius ofthe button pickup, hence a smaller button is favorable,though this approximation is only valid for a buttonmuch larger than the bunch.
8. Summary
An analytical solution for the voltage signal is readilyfound, which is valid for any bunch or pickup length andfor any value of the lumped elements in the equivalentcircuit representing the physical pickup. For differentlimiting cases, as the SBA, LBA, capacitive as well asinductive limit, appropriate expressions exist, which areeasier to work with.Nonetheless, some evaluations were carried out for thegeneral solution as well. First of all the highly relativisticbunch is assumed Gaussian and centered on the longitu-dinal axis. Usually the pickup was treated as a rectangu-lar surface with the same curvature as the beam pipe, tokeep the radial distance to the beam constant. Fringingfields caused by the gap between pickup surface and thebeam pipe are neglected, but might be considered suffi-ciently by a constant factor depending on the geometry.The equivalent circuit was idealized as RC elements inparallel.The most critical approximation is the rectangularpickup surface. The solution is still useful in the rangeof medium to long bunches, but the form becomes a sig-nificant factor for µ (cid:29)
1. There are numerical methodsfor arbitrary pickup surfaces.
B. Numerical Solution
A numerical solution of Eq. (1) is accessible for anybunch or pickup shape, to study the signal in cases wherethe prior assumptions are not valid. While the Gaussianbunch in Eq. (5a) usually is a good approximation, thebutton pickup has a circular surface with a width givenby Eq. (54). In Fig. 4 the pickup form is apparent in thesignal shape in case of short bunches. An analytical so-lution was only determined in the SBA with an inductivepickup. It is therefore crucial to find a general descriptionconsidering the pickup form, which is possible by numer-ical methods. Acknowledging that for any even chargedistribution, Eq. (1) corresponds to a convolution, theimage charge is readily calculated in frequency domainby the convolution theorem. It is Q im ( ω ) = FFT (cid:20) λ ( c t )2 πr p (cid:21) FFT [ w ( c t )] , (57)0 − −
10 0 10 20 30 40 − − . . t (ps) U R C ( V ) Circular PickupRectangular Pickup − − − − t (ps) U R C ( V ) Circular PickupRectangular Pickup
FIG. 6. Voltage signal of a rectangular (dashed blue) and a round (solid red) pickup surface calculated with Matlab ® accordingto Eqs. (57) to (60). Analog to the analytic approach, a Gaussian bunch and an RC-filter were used. The dimensions arein range of the SBA with µ ≈
49 and at the capacitive limit (left) respectively inductive limit (right). Bunch and geometryparameters, apart from the pickup length, are the same as in Fig. 4 (right), e.g. R = 50 Ω, C = 1 pF (left) respectively C = 30 fF (right), σ t = 0 . Q b = 20 pC, (cid:96) = w = r B = 4 . r p = 20 mm. where FFT is the fast fourier transform. In caseof an uneven charge density distribution it is neces-sary to exchange FFT [ λ ( c t )] by the complex conjugateFFT [ λ ( c t )]. By inverse FFT the time domain imagecharge is Q im ( t ) = FFT − [ Q im ( ω )] (58)[22]. The image current is I im ( t ) = FFT − [ iωQ im ( ω )] (59)and the voltage at the output of a system with transferfunction H ( ω ) U h ( t ) = FFT − [ H ( ω ) I im ( ω )] . (60)This method works well for finite inputs. Residual val-ues, e.g. of the response function, at the end of the timeinterval lead to unphysical effects. Therefore, the timeinterval and sample frequency must be chosen with care.The voltage signal calculated with a parameter set inthe range of the operational parameters at the EuXFELis pictured in Fig. 6. The dashed blue line representsthe numerical result for a rectangular pickup, which isin good agreement with the analytical version in Fig.4 (right). The red line shows the signal by a circularpickup. IV. ULTRA-LOW CHARGE MODE
For new experiments in the EuXFEL, with an ultra-low charge mode ( ≤ nd generation pickups areincapable of providing the required signal for fs resolu-tion. Therefore, the BAM will be upgraded with a novelpickup structure and new EOM. A. Planned signal improvement
New layouts are restrained in each facility by designregulations and previous design choices. For the nextBAM upgrade in the EuXFEL a smaller pipe diameteris now permitted. The possible reduction from 40 . B. Aperture reduction
A straightforward option for improvement is the solereduction of the beam pipe aperture. A diameter of10 mm is possible in combination with the smaller first-generation cone-shaped pickup. The second-generationpickups cannot be installed in a four-pickup configura-tion with this diameter due to their dimensions, as thecut-outs would overlap. A simulation with the wakefieldsolver of CST PARTICLE STUDIO ™ yields a slope of1746 mV ps − with 13 . . . . . . . . . . − − − − t (ns) U ( V ) − − − − − − f (GHz) U ( d B V ) FIG. 7. Simulated signal in time domain (top) and its nor-malized spectrum (bottom) taken at the end of the vacuumfeedthrough of a single pickup with 10 mm minimum distance,adapted from [27].
C. 90-GHz cone shaped pickup
We proposed a concept with pickups scaled to supportup to 90 GHz in 2019 with the dimensions specified inTable I [28]. This pickup was simulated afterwards in aBAM like setup, with four identical pickups equally dis-tributed around a pipe section, omitting the proposedsignal combination. In addition, the former bunch pa-rameters have been initially used. These are a super rel-ativistic ( v = c ) Gaussian bunch with σ z = 1 mm and acharge of 20 pC. Therefore, the voltage can be comparedto the well-described state-of-the-art pickups accordingto Angelovski et al. [13, 16].The bipolar signal pictured in Fig. 8 has a peak-to-peak voltage of 3 .
52 V. The peaks are separated by8 .
01 ps, more than doubling the slope to 722 . − .Though the pipe radius was decreased nearly by a factorof four, the gain is only by 2 .
4. The increased bandwidth
TABLE I. Specifications of the 90 GHz-pickup [28], the mod-ified pickup (2 nd Gen.) [16] and the original (1 st Gen.) [12].Draft’19 2 nd Gen. 1 st Gen.Cut-out dia. (mm) 1 .
00 1 .
62 1 . .
26 13 . . .
45 0 .
70 0 . .
02 6 2 . \ a \ a .
75 4 . . . . . a Not specified in the publication leads to a reduced rise time of 7 . st genera-tion, but a smaller peak-to-peak voltage gives advantagesin machine protection.Reducing the bunch charge to 1 pC accordingly gives aslope of 36 . − , which undershoots the minimumtarget by about a factor of 4. For a further increase thecombination of more than two signals is planned. Bythe combination of 8 pickups, without any phase shiftand 3 dB attenuation at each stage, a factor of 2 . D. Printed circuit board BAM
For ultra-short bunches the transition to a short rect-angular pickup on a printed circuit board (PCB) with atrace thickness still larger than the bunch length appearspossible. This concept possibly allows for a 100 GHzpickup without the drawback of smaller dimensions.Further benefits are the possibility to use well-knowncomponents with precise production methods and well-described materials. The transmission lines (TL) andcombination network may be realized on the PCB re-ducing the RF path, which is specifically important toprevent dispersion effects in broadband quasi-TEM TLs.A microstrip is favorable for its width, but entails disper-sion and is less shielded. Therefore, it is planned to use amicrostrip (MS) for coupling to the field and a stripline(SL) for the combination network.A preliminary simulation, shown in Fig. 9, of a PCB2 . . . . . . . − − − t (ns) U ( V ) − − − − − − f (GHz) U ( d B V ) FIG. 8. The signal in time domain (top) and its normalizedspectrum (center) taken at the end of the vacuum feedthroughof a single pickup as well as the simulation model with 90 GHzcone-shaped pickups (bottom), adapted from [27]. . . . . . . . − − − t (ns) U ( V ) − − − − − − f (GHz) U ( d B V ) FIG. 9. The signal in time domain (top) and its normalizedspectrum (center) taken at the end of the vacuum feedthroughof a single pickup as well as the simulation model with 50 Ωstripline pickups (bottom), adapted from [27]. .
55 mm wide and about15 mm long 50 Ω SL in a (cid:15) r = 4 .
03 substrate disc with10 mm aperture inside. For a 20 pC bunch the simulationreturns a slope of about 1270 mV ps − . The SL pickup isexceeding the current as well as the 90 GHz cone-shapedpickups, but does not achieve the performance of a gener-ation 1 pickup of equal aperture. Furthermore, crosstalkand reflections at the vacuum feedthrough as well as theopen pickup end generate delayed but significant ringing. V. CONCLUSION
A high bandwidth cone-shaped pickup with 10 mmaperture leads to a significant improvement by reduc-tion of the distance and increase of the bandwidth. If amaximum voltage is of no concern, a configuration of 1 st generation pickups in a 10 mm beam pipe is a simple solu-tion estimated sufficient for bunch charges down to 4 pC.In case of ultra-short bunches a PCB-type BAM may be suited to support 100 GHz without the drawback of re-duced dimensions. With the current design restrictions,a signal combination is inevitable. Further studies of anintegrated combination network are required to reducesignal reflections and losses. Furthermore, it is necessaryto investigate the properties of PCB boards regardingvacuum suitability and radiation hardness. Specifically,potential damages caused by beam incidence need to beassessed. ACKNOWLEDGMENTS
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