Properties and generation of ultrashort electron pulses
3.1 Phase space description of electron beams
Conventionally, a geometric particle beam refers to the concept of a stream of isolated particles having a common direction of propagation and being laterally confined [101].
Within the description of an electron beam, a natural orthogonal coordinate decoupling par-allel (z-axis, longitudinal) and perpendicular (x-,y-axes, transverse) to the main direction of propagation can be applied, accounting for an order of magnitude larger longitudinal electron momentum (fulfilling paraxial approximation) [101]. By this separation, the
longitudinal beam properties are directly connected the electron beam’s energy distribution and evolving temporal profile (at a fixedz-position).
Electron pulses in classical phase space
In classical mechanics, the state of a single particle in three-dimensional space is given by a set of canonical positionqiand momentum pi coordinates (i=x,y,z), with its propagation described by a trajectory in six-dimensional phase space. Extending this concept, the temporal evolution of an ensemble of electrons may be described by a time-dependent local particle densityn(qi,pi,t). For such a phase space distribution, Liouville’s theorem states that the volume occupied by a given number of non-interacting particles remains constant [101,102]. This is valid in good approximation if single particle-particle interactions can be neglected and electromagnetic forces induced by the electrons are well described by an average scalar potential [101]. Integrating the particle densityn(qi,pi)in phase space yields the total number of electrons
Ne=
˚
n(qi,pi)dqidpi. (3.1) In general, a propagating electron (or ensemble of many thereof) may be described as a classical density functionρin six dimensional phase space with canonical coordinatesqi
andpi(i= x,y,z). The total charge Q=Ne·e=
˚
ρ(qi,pi)dqidpi, (3.2) of such an electron bunch isNetimes the elementary chargeeand obtained by integrating the charge distributionρ(qi,pi)over phase space. To characterize the occupied phase space volume of such an electron bunch, a useful quantity is the normalized rms (root-mean-square) beam emittance, which is independent of the kind of particle, projecting the 6D phase space distribution onto the three orthogonal subspaces(qipi)with (i=x,y,z) [101]
εn,rms,i= 1
=σpi giving the standard deviations of position and momen-tum contained in the beam, while the cross-term
qip0i2
accounts for linear phase space distortions (e.g. elliptical non-parallel topi orqi). At a beam waist, e.g. in a spatial or
transverse pulse properties longitudinal pulse properties
Figure 3.1:Properties of single-electron pulses. The concept of a pulsed electron beam enables a decoupling of the transverse and longitudinal phase space. A quantitative description is given by the ensemble properties of multiple low-charge electron pulses generated at a high repetition rate.
temporal focus of a beam, the cross-term vanishes, reducing Eq.3.3to εn,rms,i= 1
mecσqiσpi. (3.4)
This normalization allows for defining the brightness of an electron bunch in terms of its total charge divided by the occupied phase space volume B¯6D=Q/V6, or explicitly, by integrating the differential brightness
dB6D=(mec)3 d6ρ
dxdydzdpxdpydpz, (3.5) over the entire phase space ((x,y,z)= qx,qy,qz
for clarity). In case of Gaussian shaped distributions ofqi andpi, the peak brightness in the center of the bunch is given by [103]
Bnp6D= 1 8π3
Q
εn,rms,xεn,rms,yεn,rms,z. (3.6) In good approximation, the 6D phase space decouples into a transverse (x,y,px,py) and longitudinal (z,pz) subspace, assuming near-monochromatic (low initial longitudinal mo-mentum spread), non-interacting (free of space charge and scattering) and high energy electron beams of low divergence (pz≈p0) [101]. All of that is fulfilled for single-electron pulses accelerated linearly by a high static potential, as later applied in this thesis (cf. Ch.4) and allows for a separate treatment of the longitudinal and transverse beam properties, as illustrated in Fig.3.1.
Quantum phase space: The Wigner distribution
To account for their simultaneous wave and particle characters, real particles must be described as complex quantum mechanical wave packetsψ(pi,t)↔F ψ(qi,t)(connected by the Fourier transformF). Here, the particles classical canonical coordinates are substituted by the expectation values of the positionqi → hqii and momentum operator pi→ hpii. Additionally, such a matter wave description provides for the particle’s corresponding de Broglie wavelength λe = h/|p0| with Planck constant h and electron momentum in direction of propagationp0. In analogy to the classical phase space density, the Wigner distribution of a pure quantum state is defined by [104]
Wi(qi,pi)= 1 π~
ˆ −∞
+∞
ψ∗ qi+qi0ψ qi−qi0
e−2ipiqi0/~dq0i (3.7) for the conjugate variablesqi andpi, as will be applied for studying the temporal evolution of the longitudinal electron wave function in Ch. 5. Note that the generalized Wigner distribution in case of mixed quantum states is an established tool for describing the transverse ensemble properties of electron beams in electron optical systems, including lenses, aberrations and holography, and is applied for the description of coherent and incoherent sample interactions [105–107].
Heisenberg’s uncertainty principle applies, stating a lower bound for the consecutive measurement of position and linear momentumσqi·σpi ≥~/2, here represented by the standard deviations of the respective operators. In consequence, the single electron wave function always occupies a finite volume in phase space, which results in a lower limit of the normalized rms emittance
εn,rms,i= 1
mecσqiσpi ≥ ~
2mec =ε0=0.26·10−12m (3.8) in all Cartesian projections (identity applies for a Gaussian shaped wave packet). Inter-estingly, one interpretation of Heisenberg’s principle is the quantization of the classical phase space, which yields the volumedV0=(mecε0)3 that a single electron microstate occupies. Hereby, the microcanonical degeneracy of an ensemble of electron pulses in classical phase can be defined by [108]
Ω6D= dV0·Ne
V6 = ε03
εn,rms,xεn,rms,yεn,rms,z. (3.9)
Notably, Pauli’s exclusion principle allows no more than two electron spin states occupying a phase space elementdV0, therefore limiting the degeneracy toΩ6D≤2. Combining both principles yields a theoretical maximum for the brightness of an electron beam,
B6Dnp = 2e dV0 =2e·
2mec
~ 3
=4.45·1019 C
m3. (3.10)
For most applications, an electron beam is sufficiently well described by the classical density function ρ if the lower limit of the emittance ε0 is included, e.g. yielding the well-known diffraction limit in the transverse beam direction.
Transverse beam properties
In a microscope or accelerator, the most easily accessible transverse beam properties are the beam diameter and opening angle of the focused particle beam. Therefore, in analogy to geometrical optics for paraxial approximation (px,ypz), the transverse electron momenta px,pycan be replaced by the slopesx0,y0of the electron beam [101]
x0= px
Pz
y0= py Pz
. (3.11)
The geometrical normalized rms emittance is given by εn,rms,x =βγq
x2 x02
− hx x0i2 εn,rms,y= βγq
y2 y02
− hy y0i2, (3.12) for longitudinal electron momentum pz≈ p0= βγmec, relativistic Lorentz factor γ and β=ve/c withve and c being the electron and light velocities in vacuum, respectively.
In the waist of a Gaussian beam (which corresponds to an axis-aligned elliptical (x,px) distribution), the divergence terms become zero, simplifying Eq.3.12by substitution with the standard deviations of the beam diameterσx and angular spreadσx0
εn,rms,x= βγ·σxσx0 εn,rms,y= βγ·σyσy0. (3.13) Notably, Liouville’s theorem also applies in good approximation to the transverse trace space, allowing for measuring the emittance of an electron source at a later beam waist in an optical system—if the beam was not cut by an aperture. Applying Eq.3.13only yields an upper bound for the emittance, since phase space distortions (like lens aberrations) will
lead to a non-vanishing divergence term.
While the 6D brightness defined by Eq.3.6gives the peak charge density of a Gaussian shaped bunch, an often more practical quantity for characterizing the usability of an electron beam is the 4D brightness [103]
Bnp= 1 4π2
I
εn,rms,x·εn,rms,y, (3.14)
which is a projection into the transverse 4D subspace. Notably, two beams of different longitudinal momentum spread (corresponding to broader energy distribution) but same current and transverse geometrical properties have the same 4D brightness.
A further important property of an electron beam is the transverse degree of coherence, corresponding to the capability of self-interference. For classical electron sources, an electron wavefunction only interferes with itself due to an initial random phase upon generation. Therefore, to observe perfect ensemble interference, all electrons must occupy only one (spin independent) state in transverse phase space, which enables the definition of the transverse degree of coherence by [109]
Kx= ε0
εn,rms,x ≤1. (3.15)
The lateral coherence lengthξc,xis given by the diameter of a single electron wavefunction ψ (twice the standard deviation of the position operator, 2σψ,x) [110]. Consequently, Heisenberg’s uncertainty principle relates the transverse momentum spreadσpx of a beam with its coherence length and substitution with Eq.3.4yields [111]
ξc,x=~/σpx, ξc,x= ~
mec· σx
εn,rms,x. (3.16) Alternatively, the transverse coherence length of a wave can be derived in a more stringent way by considering the Van Cittert–Zernike theorem and calculating the coherence transfer functions [109,112].
Longitudinal beam properties
In longitudinal phase space, an electron beam is most conveniently described in terms of its temporal structure and kinetic energy distribution. The spatial (laboratory-fixed)
coordinatezin beam direction is replaced with the propagation timet=z/veby considering the relativistic electron velocityve. The kinetic energyE0of the electrons after acceleration in a static potentialU0is connected to the longitudinal momentum componentp0≈pz by the relativistic dispersion relation
E0=U0·e=q
p20c2+m2ec4−mec2, (3.17) with slopedE/dp0=ve, yieldingp0= 1cq
E02+2E0mec2solved for the momentum. Substi-tutingdp0=v1edEanddz=vedtin Eq.3.4gives an alternative definition for the longitudinal normalized rms beam emittance
εn,rms,z= 1
mecσtσE ≥ε0= ~
2mec, (3.18)
using the rms bunch durationσtand energy spreadσE. In analogy to Eq.3.16, the temporal coherence length of an electron beam is given by
ξc,t=~ve/σE, ξc,t = ~
mec· σx
εn,rms,x. (3.19) This brief introduction to the theoretical description of electron pulses in phase space will be essential for Ch.4, characterizing the Göttingen UTEM instrument, and for Ch.5using the Wigner function to study the temporal phase space evolution of single-electron pulses.