Theory and particle tracking simulations of a resonant radiofrequency deflection cavity in TM{}_{110} mode for ultrafast electron microscopy

# Theory and particle tracking simulations of a resonant radiofrequency deflection cavity in TM$_{110}$ mode for ultrafast electron microscopy

## Abstract

We present a theoretical description of resonant radiofrequency (RF) deflecting cavities in TM mode as dynamic optical elements for ultrafast electron microscopy. We first derive the optical transfer matrix of an ideal pillbox cavity and use a Courant-Snyder formalism to calculate the 6D phase space propagation of a Gaussian electron distribution through the cavity. We derive closed, analytic expressions for the increase in transverse emittance and energy spread of the electron distribution. We demonstrate that for the special case of a beam focused in the center of the cavity, the low emittance and low energy spread of a high quality beam can be maintained, which allows high-repetition rate, ultrafast electron microscopy with 100 fs temporal resolution combined with the atomic resolution of a high-end TEM. This is confirmed by charged particle tracking simulations using a realistic cavity geometry, including fringe fields at the cavity entrance and exit apertures.

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[cor]Corresponding author

## 1 Introduction

Since the introduction of the Ultrafast (Transmission) Electron Microscope (U(T)EM) by Ahmed Zewail [1], the dynamics of various sorts of material properties have been studied using ultrafast electron techniques such as imaging [2] [3], diffraction [4] and electron energy-loss spectroscopy (EELS) [5] [6] with picosecond to femtosecond temporal resolution. The research described in these references is all based on a pump-probe scheme in which the probing electron pulses are created from a flat photo-cathode using a fs laser system. This causes two limitations: First, the average current of the UTEM is limited by the repetition rate of the fs laser system, although long relaxation times of dynamical processes, or slow thermal diffusion, can also limit the maximal repetition times that can be used. Second, the relatively large area of the flat-photocathode limits the peak brightness of the generated electron pulses, hence the maximally achievable spatial resolution. A significant improvement in the peak brightness of laser-triggered electron sources has been achieved by sideways laser illumination of a nano-tip. The reduced dimensions of the photo-field emitter have resulted in a working UTEM with 200 fs electron pulses with a peak brightness comparable to continuous Schottky sources [7]. This technique has resulted in very impressive results [8] [9] [10].

An alternative approach requiring no laser at all, involves the chopping of a continuous beam of a high-end TEM into ultrashort electron pulses using a fast blanker in combination with a slit. Apart from the lack of need for an amplified laser system to create electron pulses, further advantages are that no alterations are needed to the gun and the fact that it allows easy switching between continuous mode and pulsed mode. The principle of chopping an electron beam has been realized in Scanning Electron Microscopes (SEMs) many years ago [11] [12] in the form of electrostatic blanking capacitors [13] [14] and cavity resonators [15] [16]. More recently, the use of a photo-conducting switch was proposed to create a laser-triggered, electrostatic beam blanker which can be used for pump-probe experiments in a SEM [17]. In parallel, advanced RF-laser synchronization techniques [18] [19] have reduced the timing jitter between electron pulses and laser pulses to levels below 100 fs [20] and even 5 fs [21], making RF cavity-based pulsed beams also suitable for pump-probe experiments.

RF cavities or resonators are specifically tailored metallic structures, in which electromagnetic energy can be stored in standing waves or modes. Because of resonant enhancement, RF cavities can be used to generate EM fields of high amplitudes with relatively low input power. Various types of RF cavities have been important elements of the standard toolbox for particle accelerators for many years. For example, a cavity in TM-mode supports an oscillating, electric field pointing along the beam axis, which is commonly used for the acceleration of relativistic charged particle pulses. Synchronized to a mode-locked laser system, a cavity in TM mode can be used to for the compression of electron pulses in ultrafast electron diffraction experiments, resulting in pulses shorter than 100 fs [22]. A cavity in TM mode supports a magnetic field oscillating perpendicular to the beam axis, transversely deflecting the beam. This mode has been used to chop the continuous beam of a 30 kV SEM into ultrashort pulses [23] and record time-of-flight electron energy loss spectra [24]. Synchronized to a mode-locked laser, a cavity in TM mode can be used for pulse length measurements [22], for example of non-relativistic, ultracold electron pulses extracted from laser cooled gases [25]. Note the same principles of pulse compression and metrology have also been applied with single-cycle THz fields instead of RF cavities [26].

In 2012, Lassise et al. showed that using a miniaturized RF cavity in TM mode, it is possible to chop a 30 kV electron beam while fully maintaining the peak brightness [27]. Moreover he proposed to use this technique to chop the beam of a high-quality beam of a 200 kV TEM, also conserving the peak brightness of the Schottky field emission source [27]. Since then, such an RF cavity-based UTEM has been built at Eindhoven University of Technology (TU/e) and is currently operational [28]. Furthermore, alternative TEM beam chopping schemes involving multiple RF cavities are being investigated elsewhere [29]. For successful implementation of an RF cavity in a charged particle beam line, a thorough understanding of its effect on the beam dynamics is essential. If not used properly, the rapidly oscillating non-uniform and strong EM fields in RF cavities can have a detrimental effect on the beam quality. However, with proper settings of experimental parameters, such as the RF phase and the position of the beam crossover, the quality of the original beam can be fully maintained, essential for applications such as electron microscopy.

In this paper we present the theoretical background of a resonant RF cavity in TM mode as a dynamic optical element for UTEM. In section 2 we explain the principle of deflection and chopping by calculating the trajectories of a charged particle propagating through an ideal TM pillbox cavity. From these trajectories, we derive the optical transfer matrix of the cavity in section 3; and apply this in a Courant-Snyder formalism to calculate the 6D phase space propagation of a Gaussian electron distribution in section 4. In section 5 we apply our findings to study the special case of a focused beam inside a 200 kV TEM column. We derive closed, analytic expressions for the increase in transverse emittance and energy spread. We show that using proper experimental parameter settings, the growth in transverse emittance can be fully eliminated and also the increase in energy spread can be minimized. In section 6 we present charged particle tracking simulations using a realistic cavity geometry, including fringe fields at the cavity entrance and exit apertures, and a realistic electron beam. The simulations confirm our theoretical findings that an RF cavity in TM mode can be used to chop an electron beam with negligible increase in both transverse emittance and energy spread. This property makes it a very interesting alternative for photo-emission based UTEM, especially in the light of the ever increasing brightness of continuous electron sources.

## 2 Charged particle trajectories

### 2.1 Beam brightness and emittance

First, we define a beam as a distribution of charged particles (charge and mass ) with a collective motion along the -axis with average speed , for which each individual particle also has a small relative velocity , with

 vx,vy,δvz≪vz, (1)

We describe the motion of each particle in the distribution by the 6D trace space coordinates , , , , and , in which and are the angles of the particle trajectory with respect to the mean trajectory of the total distribution. Analogously, represents the relative difference in longitudinal velocity of an individual particle with respect to the mean velocity of the beam. Note that we have used the paraxial approximation of equation (1). The 6D trace space volume occupied by this distribution is a measure for the quality of the beam. In accelerator physics, this is often expressed in terms of the root-mean-squared (rms) geometrical emittance in the direction [30]

 ϵj≡√⟨j2⟩⟨j′2⟩−⟨jj′⟩2. (2)

Here the brackets indicate averaging over the entire distribution. The emittance is proportional to the area of the projection of the 6D trace-space density on the ()-plane and is a measure for the focusability of the beam in the -direction, given that the beam energy is fixed. The geometrical emittance is not a Lorentz-invariant quantity and therefore is not conserved during acceleration. To compare beams of different energy, the Lorentz-invariant rms normalized emittance in the direction is defined as

 Missing or unrecognized delimiter for \left (3)

in which is the momentum in the -direction, is the relativistic Lorentz factor and is the normalized speed [30]. In a beam waist, there are no correlations between and , hence the normalized transverse emittance is simply given by

 ϵwaistn,x≡βγσxσx′, (4)

in which and are the rms beam radius and rms semi-divergence angle in the beam waist. The quality of the beam in the -direction is determined by the normalized longitudinal emittance . For an charged particle bunch with no chirp , so the normalized longitudinal emittance can be written as

 ϵn,z=σzσpzmc≈σtσUmc, (5)

in which is the rms pulse duration and is the rms energy spread [30].

Electron microscopists often describe the quality of the beam in terms of the reduced brightness, which is locally defined as

The reduced brightness is proportional to the local current density per unit solid angle . By dividing by the relativistically corrected beam potential with , this quantity is also Lorentz-invariant. To define a measure for the overall quality of a charged particle beam, the practical reduced brightness was introduced by Bronsgeest et al. [31]

 Bpract≡1V∗IApractΩ=1V∗Iπ(d502)2πθ2x. (7)

It defines the amount of current that can be focused into a waist with an area , in which is the full width spot diameter in which 50% of the current is focused. Furthermore, it assumes a uniform angular distribution with semi divergence angle . We can now express the practical brightness in terms of the rms normalized transverse emittance . By assuming a uniform, angular distribution so that , and a Gaussian, spatial distribution in the beam waist so that , the practical reduced brightness can be expressed in terms of the rms, normalized transverse emittance as

 Bpract=qmc2I4ln2⋅π2ϵ2n,x. (8)

### 2.2 Framework and assumptions

Consider an ideal, cylindrical cavity in TM mode of length , aligned along the -axis of a cartesian coordinate system, and with the entrance aperture positioned at . Close to the -axis:

 kx,ky≪1, (9)

with the wavenumber of the RF field, the magnetic field and the electric field of a cavity in TM mode can be approximated by

 B=B0cos(ϕ0+ωt)^yE=−B0ωxsin(ϕ0+ωt)^z}for 0

in which is the magnetic field amplitude, is the cavity resonance frequency and is the phase of the RF field at [27]. The word ’ideal’ refers to the top-hat profile of the magnetic field amplitude as a function of and the lack of fringe fields around the cavity apertures and . The effect of both these non-ideal features are studied using particle tracking simulations in section 6.

The motion of a charged particle described by position vector and velocity vector will be affected by the Lorentz force as the particle travels through the cavity. This is described by the equations of motion

 dpdt =F=−qB0⎛⎜⎝vzcos(ϕ0+ωt)0−vxcos(ϕ0+ωt)+ωxsin(ϕ0+ωt)⎞⎟⎠ (11) drdt =v=pγm, with  γ=1√1−|v|2/c2.

Now consider a 6D charged particle trace space distribution traveling along the -axis with an average velocity . Figure 2 shows the moment at which the center of the distribution enters the cavity, which is indicated by the dashed lines. The black dot in figure 2 indicates a test particle that enters the cavity with trajectories

 v(t) ≡v0(x′i,y′i,1+z′i), (12) r(t) ≡(xi,yi,zi)+v0(x′i,y′i,1+z′i)t.

Here we define (,,) as the position of the test particle with respect to the bunch center, and we define , and as small deviations in propagation angles at in the frame of the traveling bunch center. Furthermore we define the 6D trace space coordinate at .

To obtain an approximate solution of the equations of motion (11) for the test particle of which the original motion is described by expression (12), we use a perturbative approach based on the following two assumptions:

• First, the charged particle gyrates only a small fraction of a full cyclotron orbit during one oscillation period of the EM field, hence

 ωcω≪1, (13)

in which

 ωc≡qB0γ0m (14)

is the cyclotron frequency with the Lorentz factor of the incident beam. This ensures that the charged particles remain close to the -axis and we can use the paraxial approximation throughout the paper.

• Secondly, the distances over which the individual particles move with respect to the bunch center are small compared to the length scales of the collective motion of the bunch, i.e.

 x′i,y′i,z′i,xiL,yiL,ziL≪ωcω. (15)

### 2.3 Transverse trajectories of the bunch center

Based on assumption (15), we first consider the motion of the bunch center, hence . Therefore we substitute and in equation (11) and integrate from to . The momentum of the bunch center calculated to first order in is then given by

 p(1)(t)=⎛⎜ ⎜⎝B0qv0ω(sinϕ0−sin(ϕ0+ωt))0γ0mv0⎞⎟ ⎟⎠. (16)

Equation (16) says that to first order, the bunch center is periodically deflected in the transverse direction while the longitudinal motion is unaffected. Because the transverse deflection is caused by the magnetic field, the kinetic energy remains unchanged, so that

 v(1)(t)=p(1)(t)γ0m=⎛⎜⎝v0ωcω(sinϕ0−sin(ϕ0+ωt))0v0⎞⎟⎠ (17)

and

 r(1)(t)=⎛⎜ ⎜⎝v0ωcω2(−cosϕ0+cos(ϕ0+ωt)+ωtsinϕ0)0v0t⎞⎟ ⎟⎠, (18)

in which we have substituted equation (14). The deflection angle at which the pulse exits the cavity is given by

 α=vxvz=ωcω(sinϕ0−sin(ϕ0+Λ)), (19)

where we have introduced the dimensionless cavity length parameter

 Λ≡ωLv0. (20)

When a small aperture of diameter is placed at a distance behind the cavity, centered along the cavity symmetry axis, only the electrons go through for which , ignoring a small offset in . This condition is satisfied for values of the RF phase close to

 ϕ0=π−Λ2. (21)

In this regime, the deflection of the charged particles by the cavity can be considered as a linear function of the RF phase

 α≈ωcω(ϕ0−π−Λ2)⋅2sinΛ2, (22)

and the acceptance window of the slit in terms of the RF phase is given by

 −ωd4ωclsin(Λ/2)<ϕ0−π−Λ2<ωd4ωclsin(Λ/2). (23)

The range of phases for which equation (23) is satisfied, determines the pulse length of the charged particle bunch behind the slit. Hence, an ideal continuous beam of charged particles is chopped up in temporally uniform pulses of pulse length

 τ=Δϕ0ω=γ0md2lqB0sin(Λ/2). (24)

Equation (24) shows that the pulse length of the resulting bunches for a given magnetic field amplitude can be minimized by choosing the cavity length parameter , or . For such a cavity length the transit time of the charged particle traveling through the cavity equals half an oscillation period of the RF field. For 200 kV electrons traveling through a cavity with a typical resonance frequency of GHz, this corresponds to a cavity length of mm. In combination with a m slit, positioned a distance cm behind the cavity and with a typical magnetic field strength of mT [27], this results in fs pulses.

The first order trajectories of the bunch center described by equations (17) and (18) for this situation are plotted in figure 3 for various values of the RF phase . The red curve in figure 3 shows the charged particles that have experienced RF phase , or more generally . These particles exit the cavity with zero deflection angle, but with a small shift in transverse position . To chop these particles of the beam, the chopping slit would have to be positioned slightly off-axis. However, for easy switching between pulsed mode and continuous mode, in practice the chopping slit is placed on the optical axis. As a result, the created electron pulses that go through the slit have a finite transverse momentum. Standard TEM deflection coils can be used to redirect the pulses back to the optical axis.

As a final remark, note that the spread in RF phase that defines the final pulses, also results in a spread of transverse momentum. Consequently, the transverse emittance is increased which reduces the focusability of the beam. In section 5 it is shown how this can be prevented.

### 2.4 Longitudinal trajectories of the bunch center

Due to the acquired motion in the -direction, the bunch center now also starts to experience a Lorentz force in the -direction. To calculate these second order effects, the first order expressions for , and of equations (17) and (18) are substituted back into the equation of motion (11). The momentum of the bunch center calculated to second order in is then given by

 p(2)(t)=γ0mv0⎛⎜ ⎜ ⎜⎝ωcω(sinϕ0−sin(ϕ0+ωt))01+ω2cω2cos(ϕ0+ωt)(−cosϕ0+cos(ϕ0+ωt)+ωtsinϕ0)⎞⎟ ⎟ ⎟⎠. (25)

The work done by the electric field in the cavity results in a change in Lorentz factor

 Δγ =1mc2∫qE⋅dr (26) =γ0ω2cω2v20c2[1−cosωt+ωtcos(ϕ0+ωt)sinϕ0+cos(2(ϕ0+ωt))−cos2ϕ04],

in which we also have substituted the expression for of equation (18). With this change in Lorentz factor the second order longitudinal velocity and position are given by

 v(2)z(t) =p(2)z(t)m(γ0+Δγ)≈p(2)z(t)γ0m(1−Δγγ0) (27) =v0+v0ω2cω2{cos(ϕ0+ωt)(−cosϕ0+cos(ϕ0+ωt)+ωtsinϕ0) −v20c2[1−cosωt+ωtcos(ϕ0+ωt)sinϕ0−cos2ϕ0−cos(2(ϕ0+ωt))4]}

and

 z(2)(t) =v0t+v0ω2cω3{ωt2−sinωt+ωtsinϕ0sin(ϕ0+ωt)+sin(2(ϕ0+ωt))−sin2ϕ04 (28) −v20c2[ωt+sinϕ0(−cosϕ0+cos(ϕ0+ωt)+ωtsin(ϕ0+ωt)) −ωtcos2ϕ0+(1−cos(2ϕ0+ωt))sinωt4]}

in which we have assumed , and have omitted terms proportional to and higher, based on assumption (13).

Figure 4 shows the longitudinal trajectories of equations (27) and (28) relative to a co-moving frame traveling with the initial velocity of the bunch , for various values of the RF phase . In other words, figure 4a shows the relative change in longitudinal velocity of the bunch as a function of longitudinal position in the cavity. Figure 4b shows the resulting deviation in longitudinal position relative to the moving frame. Again, the red curve shows the charged particles that have experienced RF phase , or more generally . Figure 4 shows that these particles are first decelerated and subsequently accelerated. The spread in RF phase experienced in the bunch results in an increased energy spread. This is also addressed in section 5.

## 3 Optical transfer matrix

To derive the optical transfer matrix of a RF cavity in TM mode we follow the same perturbative approach as in section 2 to calculate the trajectories of particles with , see figure 2. Because of assumptions (13) and (15), the additional coupling between the terms and the fields in the cavity is regarded as a second order effect. Therefore, the first step is to solve the equation of motion for the test particle described by initial trajectories (12), while substituting in the expression for the Lorentz force. Note that now the equations of motions must be integrated from

 tb≡−ziv0(1+z′i) (29)

to , because the test particle no longer enters the cavity at . With this, the first order trajectories of the test particle are given by

 v(1)(t) =⎛⎜ ⎜⎝v0x′i+v0ωcω(sin(ϕ0+ωtb)−sin(ϕ0+ωt))v0y′iv0(1+z′i)⎞⎟ ⎟⎠~{}and (30) r(1)(t) =⎛⎜ ⎜⎝xi+v0x′it+v0ωcω2(cos(ϕ0+ωt)−cos(ϕ0+ωtb)+ω(t−tb)sin(ϕ0+ωtb))yi+v0y′itzi+v0(1+z′i)t⎞⎟ ⎟⎠.

Subsequently, the first order expressions for , and of equation (30) are substituted back in equation (11) to calculate the second order trajectories. This results in lengthy expressions for and with many cross-terms of the various initial particle coordinates . However because of assumptions (13) and (15), only effects that are linearly dependent on initial particle coordinates and up to second order in are taken into account. This allows the definition of the optical transfer matrix via

 xf=M–––cavxi, (31)

as the linear transformation that maps the initial 6D trace space coordinate at onto the final 6D trace space coordinate , defined at the time at which the test particle exits the cavity:

 te=L−ziv0(1+z′i). (32)

More specifically, we define

 ⎛⎜⎝xfyfzf⎞⎟⎠≡⎛⎜ ⎜⎝x(2)(te)y(2)(te)z(2)(te)−v0te⎞⎟ ⎟⎠~{}and~{}⎛⎜ ⎜⎝x′fy′fz′f⎞⎟ ⎟⎠≡1v0⎛⎜ ⎜ ⎜⎝v(2)x(te)v(2)y(te)v(2)z(te)−v0⎞⎟ ⎟ ⎟⎠ (33)

as the position and propagation angle of the test particle at time in the frame of the traveling bunch center. Note that in this step we have assumed the longitudinal velocity of the charged particle remains constant while traversing the cavity.

After calculating the second order trajectories for the test particle, evaluating them at time and expanding them to first order in , we obtain the optical transfer matrix for a TM cavity

 M–––cav=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝1L00C1C20100C3C4001L00000100C5C6001+C9L+C10C7C800C111+C12⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠, (34)

in which the cavity constants through are given by

 C1=−ωcω(Λcosϕ0+sinϕ0−sin(Λ+ϕ0)), (35)
 C2=LΛωcω(cos(Λ+ϕ0)−cosϕ0+Λsin(Λ+ϕ0)), (36)
 C3=−ΛLωcω(cosϕ0−cos(Λ+ϕ0)), (37)
 C4=ωcω(Λcos(Λ+ϕ0)+sinϕ0−sin(Λ+ϕ0)), (38)
 C5=C1γ20, (39)
 C6=LΛωcω[(1−2β20)(cos(Λ+ϕ0)−cosϕ0)−β20Λsinϕ0+Λγ20sin(Λ+ϕ0)], (40)
 C7=C3γ20, (41)
 C8=ωcω(Λγ20cos(Λ+ϕ0)+β20(sin(Λ+ϕ0)−sinϕ0)), (42)
 C9= ω2cω2[(12−54β20)cos(2ϕ0)+(−12+β204)cos(2(Λ+ϕ0)) (43) +β20(cos(Λ+2ϕ0)+Λ2sin(2ϕ0))−Λγ20sin(Λ+2ϕ0)],
 C10= Lω2cω2[12(1−4β20)cosΛ+β204cos2ϕ0+12(1−β202)cos(2(Λ+ϕ0)) (44) +12(1+β20)[cos(Λ+2ϕ0)−1]+(3β20Λ+Λ2γ20)sinΛ +5β204Λsin2ϕ0−β204Λsin(2(Λ+ϕ0))],
 C11= ω2cω2ΛL[sin(2(Λ+ϕ0))−sin(Λ+2ϕ0)−(Λγ20+β20sinΛ)cos(Λ+2ϕ0)] (45)

and

 C12= ω2cω2[Λ{sin(2(Λ+ϕ0))−sin(Λ+2ϕ0)+Λsinϕ0sin(Λ+ϕ0)} (46) −β20(cos(Λ+ϕ0)−cosϕ0+Λsinϕ0)Λsin(Λ+ϕ0) −2β20(1−cosΛ+Λcos(Λ+ϕ0)sinϕ0+14[cos(2(Λ+ϕ0))−cos2ϕ0])],

with

 β0≡v0c (47)

the normalized initial speed of the bunch center.

Note that cavity constants - are proportional to the magnetic field amplitude and therefore scale linearly with . The cavity constants - represent second order effects scaling with . In the limit where the fields are turned off (), all the cavity constants vanish and the matrix simply reduces to a drift over a distance . The determinant of the transfer matrix is unity to first order in , which means that the 6D trace space density of a charged particle distribution is conserved during cavity transit. Furthermore, note that the transfer matrix simplifies significantly for an optimized cavity length of , i.e. .

## 4 Courant-Snyder trace-space transformation

Now we have the optical transfer matrix of the TM cavity, we use the Courant-Snyder formalism [32] to calculate the propagation of an entire 6D trace-space distribution of charged particles. For this purpose, we describe the distribution in terms of its rms ellipsoidal contours in trace space, of which the projections on the planes () are given by the ellipses

 ϵij=^γjj2+2^αjjj′+^βjj2~{}with~{}^βj^γj−^α2j=1. (48)

Figure 5 shows how the Courant-Snyder parameters are related to initial beam properties such as rms radius and rms angular divergence . The rms area of the ellipse is given by , where is the (initial) projected rms emittance in the -plane.

We can write equations (48) in matrix notation

 xTA––−1x=1, (49)

with the real, positive definite, beam matrix at given by

 A––=⎛⎜ ⎜⎝A––xx′0–0–0–A––yy′0–0–0–A––zz′⎞⎟ ⎟⎠~{% }with~{}A––jj′=ϵij(^βj−^αj−^αj^γj). (50)

Here we assume no correlations between spatial degrees of freedom. Note that because of equation (48). We can now propagate this charged particle distribution through an ideal TM cavity. The beam matrix that describes the distribution at the exit of the cavity is given by

 B––=M–––cavA––M–––Tcav≡⎛⎜ ⎜⎝B––xx′0–cross terms0–B––yy′0–cross terms0–B––zz′⎞⎟ ⎟⎠. (51)

The -correlations are introduced by the non-zero, off-diagonal matrix elements of and will cause an exchange between the transverse and longitudinal emittance, hence energy spread. Both will decrease the focusability of the beam.

The final normalized transverse emittance is calculated by

 ϵfn,x=β0γ0√det(B––xx′) (52)

while the rms energy spread of the beam after cavity transit is given by

 σfU=γ30mv0σfv∥, (53)

with the rms spread in velocity along the propagation vector of the deflected beam. To find we consider the 2D beam ellipse that describes the final trace space distribution projected on the -plane

 B––x′z′≡(B(2,2)B(2,6)B(6,2)B(6,6)). (54)

The diagonal matrix elements and are related to the velocity spread of the bunch in the and directions, via and respectively. Next, we consider this beam matrix in the ()-coordinate system as illustrated in figure 6, which is rotated with respect to the -coordinate system about the final propagation angle at which the bunch center exits the cavity

 αf≡ vfxvfz=x′f1+z′f (55) = Missing or unrecognized delimiter for \right −v20c2(1−cosΛ+Λcos(Λ+ϕ0)sinϕ0−sinΛsin(Λ+2ϕ0)2)]}.

In this coordinate system, the and axes are perpendicular and parallel to the final velocity vector of the bunch, respectively.

The final beam matrix in the rotated coordinate system is given by

 B––rotξ′ζ′≡M–––rotB––x′z′M–––Trot~{}with~{}M–––rot≡(1−α2f−αfαf1−α2f), (56)

of which the diagonal matrix elements and are related to the velocity spread parallel and perpendicular to , via and respectively. Now the final energy spread of the bunch after propagating through the cavity is given by

 σfU=γ30mv0σfv∥=γ30mv20√Brot(2,2). (57)

So, using the Courant-Snyder formalism, we can derive analytical expressions for the final normalized transverse emittance (equation (52)) and final energy spread (equation (57)) of the beam after traversing an ideal TM cavity. Moreover these expressions are derived as function of the initial transverse emittance and energy spread of the incident beam. To our knowledge, this is not possible in any other way.

## 5 Application: focused beam in a 200 kV TEM

Now we apply the Courant-Snyder model to the special case of a focused beam in a 200 kV TEM. More specific, we calculate the increase in normalized transverse emittance and energy spread of a 200 kV 6D Gaussian charged particle distribution with a finite initial geometrical emittance and energy spread, focused to a crossover inside a TM cavity. We are aware that Gaussian distributions are not realistic in electron microscopes, but they result in easy calculations for the rms quantities, required for the Courant-Snyder model. Furthermore, the functional dependencies in the final expressions are independent of the shape of distribution, the only difference is a proportionality factor. In section 6, we use charged particle tracking simulations to calculate actual numbers. Furthermore, the Courant-Snyder model obliges us to choose a finite initial pulse length, although in the experiment the initial beam is continuous. However, we are not interested in the electrons that are not part of the final (chopped) pulse. So by choosing an initial pulse length equal to the expected final pulse length we simply leave out the electrons of which we already know they will collide into the chopping aperture. In section 6, we use charged particle tracking simulations to test the validity of this approach.

Figure 7 shows an electron beam with initial geometrical emittance that is focused to a crossover at with rms divergence angle . Therefore the rms beam radius at is given by