Improving temporal resolution of ultrafast electron diffraction by eliminating arrival time jitter induced by radiofrequency bunch compression cavities

The temporal resolution of sub-relativistic ultrafast electron diffraction (UED) is generally limited by the radio frequency (RF) phase and amplitude jitter of the RF lenses that are used to compress the electron pulses. We theoretically show how to circumvent this limitation by using a combination of several RF compression cavities. We show that if powered by the same RF source and with a proper choice of RF field strengths, RF phases, and distances between the cavities, the combined arrival time jitter due to RF phase jitter of the cavities is cancelled at the compression point. We also show that the effect of RF amplitude jitter on the temporal resolution is negligible when passing through the cavity at a RF phase optimal for (de)compression. This will allow improvement of the temporal resolution in UED experiments to well below 100 fs.


I. INTRODUCTION
A successful method to improve the temporal resolution in sub-relativistic pump-probe ultrafast electron diffraction (UED) experiments is the use of a resonant radio frequency (RF) cavity in the TM 010 mode 1-5 to compress electron pulses to the 100 fs range. In this way, single-shot UED has been demonstrated with 100 fs electron bunches. 6,7 To achieve this, the phase of the oscillating electro-magnetic field is synchronized 8 to both the pump and photoemission laser.
However, RF phase instabilities in the synchronization system lead to variations in the arrival time of the electron bunches, thus limiting the temporal resolution of UED experiments to a few 100 fs. [1][2][3][4][5] In addition, RF amplitude instabilities may lead to further degradation of the temporal resolution. 10 This paper theoretically describes how to eliminate the RF phase jitter using two or three TM 010 cavities, depending on the velocity chirp of the incoming electron beam. If powered by the same RF source and with a proper choice of RF field strengths, phases, and distances between the cavities, the combined phase jitter is cancelled at the compression point. The effect of amplitude instabilities can be minimized by operating the compression cavity at a RF phase for optimal (de)compression. In this way, the temporal resolution can be improved substantially.
This paper is organized as follows: First (Sec. II), we will introduce the concept of using a compression cavity as a longitudinal lens and derive its corresponding focal length. Hereafter (Sec. II A), we will show how RF phase and amplitude fluctuations result in arrival time jitter and how this is connected to the focal length of the longitudinal lens. Next (Secs. II B and II C), a) Electronic mail: o.j.luiten@tue.nl 2329-7778/2017/4(4)/044026/10 V C Author(s) 2017. 4, 044026-1 we show how to use two or three cavities to effectively cancel the arrival time jitter at the compression point. Hereafter (Sec. III), we will present detailed charged particle tracking simulation results that perfectly agree with the derived analytical theory. We thus show that it is possible to create a longitudinal focus that is inherently insensitive to both phase and amplitude fluctuations of the RF field in the compression cavities. Finally (Sec. IV), we discuss the limitations.

II. THEORY
The principle of using resonant RF cavities as longitudinal lenses for sub-relativistic UED is an established technique which is described in Refs. 6 and 9. The on-axis oscillating electric field inside the RF cavity is given byẼ ¼ EðzÞ cos ðxt þ /Þẑ with E(z) the on-axis longitudinal electric field amplitude, x the angular frequency, and / the RF phase. The change in longitudinal momentum Dp z an electron acquires by traveling through an RF cavity is given by 6,9 Dp with e the electron charge, d c ¼ Ð 1

À1
EðzÞ E 0 cos xz v z À Á dz the effective cavity length, E 0 ¼ E(0) the electric field strength at the center of the cavity, v z the average speed of the electron bunch, and f z À v z t the longitudinal electron coordinate with respect to the center of the bunch; / is chosen as the RF phase at the moment the center of the electron bunch passes through the center of the cavity. The longitudinal focal length f of a such a cavity is given by 6 with m the electron mass and c ¼ 1= ffiffiffiffiffiffiffiffiffiffiffi ffi 1 À v 2 c 2 q the Lorentz factor with v % v z . Equation (1) shows that the average momentum change Dp z of the electron pulse passing through the cavity is zero if the center of the bunch passes through the center of the cavity when the RF electric field goes through zero, i.e., / ¼ 6 p 2 . Operating the cavity at a phase of / ¼ p 2 will result in bunch compression: the electrons in the front part of the bunch will be decelerated while the electrons in the back will be accelerated. Operating the cavity at / ¼ À p 2 will result in decompression; the electrons in the front part are accelerated and the ones in the back are decelerated.
RF phase variations d/ and electric field amplitude fluctuations e DE E 0 will result in a net acceleration or deceleration of the electron bunch depending on the sign of d/, e, and the focal length of the lens. This leads to arrival time fluctuations dt at a distance d from the cavity, 9 given by Equation (3) shows that the arrival time depends on the focal length of the lens, so choosing two lenses with opposite focal lengths will allow us to cancel the arrival time fluctuations due to both RF phase and amplitude fluctuations at some point behind the two cavities. For optimal (de-)compression, i.e., / ¼ 6 p 2 , the latter equation reduces to showing that the arrival time fluctuations due to amplitude fluctuations e are a second order effect. We can illustrate this with a numerical example: state-of-the-art synchronization by an RF phase locked loop system has a typical residual phase RF phase jitter d/ ¼ 2 mrad. 8 Assuming a typical angular frequency x ¼ 2pÁ3 GHz and f ¼ d, we find dt phase % 110 fs. Solid state RF amplifiers are commercially available with a RF amplitude stability of dP ¼ 5 Â 10 À4 , which results in an electric field amplitude stability of e ¼ dP 2 ¼ 2:5 Â 10 À4 and thus to additional arrival time jitter on the order of dt amp ¼ dt phase Áe % 28 as. Clearly, the amplitude contribution to the arrival time jitter is negligible, owing to it being a second order effect. The validity of Eq. (4) is confirmed by charged particle simulations which will be presented in Sec. III.
RF amplitude fluctuations also cause the longitudinal focus to shift position, thus resulting in bunch length fluctuations at the nominal (e ¼ 0) position of the waist. The Courant-Snyder b parameter 12 in the longitudinal waist is given bŷ with s w the pulse length at the longitudinal waist and z the normalized longitudinal emittance. The Courant-Snyder parameterb waist is equivalent to the Raighley length in optics. We want b waist to be much larger than the shift of the focal position to ensure that the pulse length at the nominal focus is not affected by RF amplitude instabilities. This means that the shift in focal position should be much smaller thanb waist , i.e., f 1 À 1 1þe <b waist , which is equivalent to For 100 keV electrons, f ¼ 0.5 m, s w ¼ 20 fs, and a normalized root-mean-squared (rms) longitudinal emittance z ¼ 350 fs Á eV, this results in the condition e < 0.1, which is easily achievable.

A. Longitudinal focussing
We will now first derive how to longitudinally compress an electron pulse by using a two lens focussing system, as is illustrated in Fig. 1. We will use geometrical optics to describe the longitudinal focussing system, i.e., the paraxial beam approximation and thin and weak-lens approximations. 9 The first lens is a negative lens with focal length f 1 < 0. This lens stretches the electron bunch; the second lens is a positive lens with a focal length f 2 > 0. This lens is used to compress the electron bunch, as illustrated in Fig. 1. The distance between the lenses is given by d lens . The longitudinal divergence of the incoming electron beam is parameterized by the length d 0 , which is the distance of the focal point with respect to the position of the first lens if corresponds to a converging beam which is longitudinally focused a distance d 0 behind the first cavity, as is schematically indicated in Fig. 1. d 0 < 0 represents an diverging electron beam which originates from a beam waist a distance d 0 before the first lens.
FIG. 1. Schematic representation of a two-lens longitudinal focussing system. The first lens is a negative lens with a focal length f 1 < 0, the second lens is a positive lens with a focal length f 2 > 0. The combination of the two lenses compresses the beam at a distance L focus behind the second lens.
with d lens , f 2 > 0, f 1 < 0, and e the electric field amplitude jitter which modulates the focal length of the lens. In the case that the incoming beam is parallel the latter equation reduces to In the case that the first lens collimates the incoming converging beam (f 1 ¼ -d 0 ) the position of the focal point becomes

B. Jitter correction
We assume that both cavities have the exact same phase and amplitude variations since they are driven by the same RF amplified signal. From Eq. (4), it then follows that for optimal (de)compression the arrival time jitter of the electron pulse at the second cavity (lens 2) due to the first cavity (lens 1) is given by Similarly, the arrival time jitter at a distance L jitter behind the second cavity (lens 2) due to the first cavity (lens 1) is given by The arrival time jitter at a distance L jitter behind the second cavity (lens 2) due to the second cavity is given by There will be no arrival time jitter at a distance L jitter behind the second cavity (lens 2) when dt 1L þ dt 2L ¼ 0 which shows that both phase d/ and amplitude e variations cancel in first order. The point where there is no jitter is given by with d lens ; f 2 > 0; f 1 < 0 and f 2 < d lens À f 1 since L jitter > 0.
To improve the temporal resolution of UED experiments, the no-jitter point L jitter has to overlap with the longitudinal focal point L focus . These points overlap when the following equation holds: with d 0 > d lens since Eq. (13) requires f 2 < d lensf 1 . The point where the pulse length is independent of RF phase fluctuations for all electrons inside a bunch is then given by This means that overlapping the focal point and the no-jitter point is only possible for beams with negative energy chirp (d 0 > d lens > 0). This means that in order to cancel the phase jitter in an RF focussing system with two cavities, we need an already focussing electron beam. This is the case for an electron beam which is extracted from a longitudinally extended source such as a laser cooled gas. 13,14 An electron beam extracted from a photo-emission gun can be negatively chirped using magnetic compression schemes. An RF photo-gun operated at the right phase can also produce longitudinally converging bunches.
RF amplitude variations lead to arrival time variations dt foc at the position of the jitter correction point given by Here, we see that RF amplitude fluctuations in both RF compression cavities result in only small deviations in arrival time at the focus due to the second order nature of the contribution. In addition, amplitude fluctuations lead to the shifts of the focal position given by As shown in Sec. II, jdL foc j <b waist which results in the following condition: which is easily achievable in practice.

C. Three lens jitter correction
In the previous section (Sec. II B), we have shown that it is possible to create a jitter free focus using a set of two RF cavities if the incoming beam is longitudinally converging. If the incoming beam is longitudinally diverging, a set of minimally three RF cavities is required to create a jitter free focus. The derivation is similar to the one described in the previous section (Sec. II B) and yields a jitter free focal point with the focal length of the third lens and with f 1 < 0; f 2 > 0; d l1 > d l2 f 2 d l2 Àf 2 > 0; d l2 > f 2 > 0 because L focus > 0. Here, the first and the second lens are separated by a distance d l1 and the second and the third lens by a distance d l2 .

III. PARTICLE TRACKING SIMULATIONS
We have performed particle tracking simulations to test our concept in a realistic setting. The General Particle Tracer software package 15 was used for calculating the particle trajectories. The full 3D electromagnetic fields inside the RF cavities were calculated by a field expansion 9 using the on-axis normalized field distribution EðzÞ E 0 shown in Fig. 2. In all simulations, we used an electron beam with an average beam energy of 100 keV, a rms transverse emittance of ? ¼ 30 pm Á rad, and a normalized rms longitudinal emittance of z ¼ 2 ps Á eV. Space-charge effects have not been taken into account. First, we simulated the arrival time jitter of a conventional single-cavity focussing system. The electron bunch was longitudinally compressed at distance L focus % f % 450 mm behind the cavity. Figure 3 shows the arrival time of such an electron bunch at the position of the focus for various RF phase offsets d/. The simulated arrival time is indicated by the circles. The solid line represents the theoretical arrival time jitter [Eq. (4)] and perfectly agrees with the simulations. The arrival time jitter follows the linear behavior even beyond 20 mrad of phase jitter.
Next, we simulated the arrival time dependence on relative electric field amplitude variations e. Figure 4 shows the arrival time difference for various phase offsets, from d/ ¼ À20 mrad to d/ ¼ 20 mrad in steps of 10 mrad. The circles indicate the simulation results and the solid lines are calculated using Eq. (4). Again the theory perfectly describes the simulations, showing that the amplitude fluctuations are indeed a second order effect.
Subsequently, we simulated the elimination of the arrival time jitter in the longitudinal focus by using two RF cavities. According to theory (Sec. II B), we can eliminate the RF phase jitter of an already focussing electron bunch (i.e., d 0 > 0) with a two lens focussing system, as schematically indicated in Fig. 1. In the simulations d 0 ¼ 700 mm, d lens ¼ 200 mm, and f 1 ¼ À1000 mm. Figure 5 shows the pulse length of the electron bunch as a function of longitudinal position for various focal lengths f 2 , ranging from f 2 ¼ 750 mm to f 2 ¼ 1000 mm in steps of 50 mm. The dashed lines indicate the positions of the first and second cavities. The figure shows that we enter the first cavity with a negatively chirped bunch. The first cavity defocusses (f 1 < 0) the electron bunch; the front gets accelerated and the back decelerated. The second cavity focusses (f 2 > 0) the electron beam. Figure 6 shows  Figure 7 shows both L jitter (circles) and L focus (squares) as a function of the focal length f 2 . The solid black curve was calculated using Eq. (13) with e ¼ 0. The solid grey curve was calculated using Eq. (7) with e ¼ 0. The theoretical curves perfectly describe the simulations. At the position where the longitudinal focus and the zero jitter point intersect, we find a longitudinal waist that is insensitive to arrival time jitter due to RF phase fluctuations. Cavity 1 decompresses (f 1 < 0) the electron pulse and cavity 2 compresses the electron pulse (f 2 > 0). and the solid black curve was calculated using Eq. (16). The figure shows that the simulation results are perfectly described by theory, even for electric field amplitude fluctuations up to jej ¼ 2%. The figure also shows that the change in arrival time at the zero jitter point is below half a femtosecond for jej < 2%, which is due to the second order nature of the amplitude fluctuations.
Finally, Fig. 9 shows the shift of the focal point position as a function of relative electric field variations e. The circles represent the simulation results and the solid black line was chirp, a set of minimally two RF cavities is required. When the incoming bunch is longitudinally diverging, i.e., with a positive chirp, a set of minimally three cavities is required. The analytical theory results are confirmed by charged particle simulations. This means that we can improve the temporal resolution of UED experiments to well below 100 fs by creating a jitter free longitudinal focus allowing both phase and amplitude variations.