Femtosecond electron beam probe of ultrafast electronics

The need for ever-faster information processing requires exceptionally small devices that operate at frequencies approaching the terahertz and petahertz regimes. For the diagnostics of such devices, researchers need a spatiotemporal tool that surpasses the device under test in speed and spatial resolution. Consequently, such a tool cannot be provided by electronics itself. Here we show how ultrafast electron beam probe with terahertz-compressed electron pulses can directly sense local electro-magnetic fields in electronic devices with femtosecond, micrometre and millivolt resolution under normal operation conditions. We analyse the dynamical response of a coplanar waveguide circuit and reveal the impulse response, signal reflections, attenuation and waveguide dispersion directly in the time domain. The demonstrated measurement bandwidth reaches 10 THz and the sensitivity to electric potentials is tens of millivolts or −20 dBm. Femtosecond time resolution and the capability to directly integrate our technique into existing electron-beam inspection devices in semiconductor industry makes our femtosecond electron beam probe a promising tool for research and development of next-generation electronics at unprecedented speed and size.

The need for ever-faster information processing requires extremely small devices operating at frequencies approaching the terahertz and petahertz regimes.For the diagnostics of such devices, researchers need a spatiotemporal tool that surpasses the device under test in speed and spatial resolution and therefore, such a tool cannot be provided by electronics itself.
Here we show how ultrafast electron beam probe with terahertz-compressed electron pulses can directly sense local electro-magnetic fields in electronic devices with femtosecond, micrometre and millivolt resolution under normal operation conditions.We analyse the dynamical response of a coplanar waveguide circuit and reveal the impulse response, signal reflections, attenuation and waveguide dispersion directly in the time domain.The demonstrated measurement bandwidth reaches 10 THz and the sensitivity to electric fields is tens of millivolts or -20 dBm.Femtosecond time resolution and the potential to directly integrate our technique into existing electron-beam inspection devices in semiconductor industry makes our femtosecond electron beam probe a promising tool for research and development of next-generation electronics at unprecedented speed and size.
The demand for ultimate speed in modern information processing and data transfer has prompted extensive research on high-speed electronics at maximum frequencies and shortest switching times [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] .While most electronic devices currently operate at gigahertz frequencies, the leading edge of electronics verges into the terahertz 1,2 and even petahertz domain [3][4][5][6][7][8] .For example, terahertz technology and millimetre waves 9 are the basis of the forthcoming sixth-generation telecommunication standard (6G) 1 , and high-speed transistors [10][11][12][13] , hybrid photonic platforms 14,15 or terahertz metadevices 2 begin to merge the electronic and optical domains.Numerical modelling above 100 GHz is difficult because the skin depth approaches the surface roughness 16 , radiation losses strongly increase 17 and the concept of carrier mobility breaks down as the motion becomes ballistic 18 .Therefore, the experimental characterization of local electromagnetic fields and their dynamics in such future devices so far remains an open challenge, because the required bandwidth and time resolution can obviously not be provided by electronics itself.Characterizing recordbreaking devices requires a conceptionally different approach.
Modern research offers some creative diagnostic solutions to deal with the small size of devices or their high speed, but not at the same time.For example, sub-nanometre resolution can be provided by scanning tips 19 or electron beam probing (eBeam) 20 , but the maximum bandwidth is limited to several gigahertz [21][22][23][24] , far below the terahertz domain.Up-conversion of gigahertz signals with high-frequency transistors 11 and microwave analysis 25 provide frequency resolution of up to 1.1 THz 26 , but these approaches cannot resolve the local fields inside the functional parts of a device.If optical techniques are combined with scanning probe tips for electro-optic 27 or photoconductive 28 sampling, scanning tunnelling microscopy 29 or electric force microscopy 30 , a probe tip must be placed in close physical proximity to the device and inevitably disturbs the local fields.Femtosecond point-projection microscopy [31][32][33] is a promising development but the specimen is not in a field-free region and the low electron energies require free-standing materials.

Principle of femtosecond electron beam probe
Here, we show how an ultrafast electron microscope 34 with femtosecond electron beams [35][36][37][38][39][40][41][42] under the control of optical fields of laser light 36 can probe the position-dependent and time-dependent local electric and magnetic fields of ultrafast electronic circuits while operating in the terahertz regime.Figure 1 shows the concept and fundamentals of our proof-of-principle experiment.A broadband impulse response of a device under test (DUT) is triggered by a laser-excited photoconductive switch 43,44 .The produced ultrashort voltage pulse propagates through the DUT and triggers its functionality in space and time.Femtosecond electron pulses for probing purposes are created by laser-driven two-photon photoemission 45 and subsequent acceleration to tens of keV 42 .If necessary for time resolution, these electron pulses are compressed in time with an optically generated terahertz field 36 to obtain sub-100-fs duration.Using magnetic lens systems, the femtosecond electron pulses are focused onto interesting parts of the DUT.There, the local electromagnetic fields in the DUT are probed directly in the time domain via deflection by timefrozen Lorentz forces 46,47 .The resulting beam deflections are observed on a screen and reveal the magnitude and vectorial direction of the local and time-frozen in-plane fields.Electric and magnetic components can be distinguished by their different dependencies on electron energy 46 .
A variation of the delay between the DUT trigger and the femtosecond electron pulses by means of moveable mirrors and scanning the electron beam across the DUT then provide a stroboscopic movie of the local electro-magnetic field vectors in space and time.Figure 2 shows an example of such beam deflection, taken at -2 ps and 1.5 ps after trigger, respectively.

Experiments and results
For our proof-of-principle experiments, we investigate one of the key components for terahertz electronics, that is, a transmission line in form of coplanar waveguide on insulating substrate, one of the most common ways for signal transport.Femtosecond electron pulses at a beam energy of 70 keV are produced by femtosecond laser photoemission 36,37 and subsequently compressed in time by terahertz radiation 36 .Magnetic lenses without temporal distortions 48 are used to focus and steer the beam onto the specimen for pump-probe experiments.A schematic of our device under test (DUT) is shown in Fig. 3a.Our circuit consists of (left to right) a bias pad, a photoconductive switch, a coplanar waveguide and a terminating rectangular pad with a 50- shunt resistor.
Undoped GaAs is used both as the insulating substrate of the coplanar waveguide and the active gap material in the photoconductive switch.The terahertz circuit is manufactured with ultraviolet lithography and wet etching, following established designs 49 .The central conductor of the coplanar waveguide has width of 30 μm and is separated by 20-μm gaps from two adjacent ground plates.
The width of the photoconductive gap is 10 μm. Figure 3b shows a microscopic image of the fabricated structure and Extended Data Fig. 1 shows additional results.The photoconductive switch is biased with a direct-current voltage source (Keithley 6517B, Tektronix) at a range of −15 V to +15 V. Femtosecond triggering is achieved by 80-fs long laser pulse with a wavelength of 515 nm and a pulse energy of ~100 nJ, focused to a 10-m spot at the photoconductive gap.
The electric fields of the generated pulses are mostly pointing along the y direction (blue and magenta arrows) and therefore deflect the electron beam while the magnetic fields mostly point in electron beam direction and therefore provide no substantial effect.At two locations, marked with blue and magenta ellipses in Fig. 3b, we thin the specimen for electron transparency by laser-drilling two 20-m holes.The first probe region allows us to measure the voltage between the central transmission line and the ground.The second probe region enables measuring the voltage between the bias channel and the ground.The deflected beam electrons are detected on a fluorescent screen (TemCam-F416, TVIPS) and the beam positions are determined by a series of Gaussian fits (see Fig. 2).Time-zero of the apparatus is calibrated by alloptical streaking 36 or alternatively by measuring the relative time delay between two probing positions at given distances from the gap.In a first experiment, we investigate the ultrafast dynamics of our DUT with uncompressed electron pulses at a duration of ~800 fs 36 .Figure 3c shows the measured ultrafast voltage dynamics in the two probe regions (blue and magenta circles in Fig. 3b).We can detect electric potentials as low as 30 mV or powers of -20 dBm. Figure 3d shows a zoom into the earliest part of the response.We mark the most distinct features of our data with time labels t1-t8.In the first probe region (magenta), the initial response of the circuit (t1) is a rapid voltage change from 0 V to 5 V with a measured 2-ps rise time (10% to 90%), eventually broadened by the 800-fs electron pulse duration.
There is a delay of 3.6 ps with respect to the trigger of the photoconductive switch at 0 ps.After this impulsive rise, the voltage pulse decreases to 50% of its maximum value within ~45 ps (t2).
Two weak additional voltage peaks are observed at 80 ps and 87 ps (t3), and a stronger peak emerges at 130 ps (t4).In the second probe region (blue), the initial response (t5) is also a strong but this time negative impulsive voltage change from 0 V to -6 V.The fall time is 2 ps (10% to 90%).There is also a delay with respect to the trigger of the photoelectric switch, this time 0.9 ps (t5).During the subsequent rise of the voltage back to ground, there are again several additional voltage peaks, for example at 25 ps (t6), 65 ps (t7) and 120 ps (t8).
When the photoconductive switch is triggered by the laser pulses, the optical carrier-hole pairs make the gap conductive within femtosecond times and the bias voltage is shortened to ground.
Consequently, there emerges an electric field in form of approximately a Heaviside function that subsequently propagates as a positive or negative electric field into the left and the right side of our coplanar waveguide.This dynamics and the time that is needed to reach the points of measurement (blue, magenta) explains the strong and ultrafast voltage changes in our two measurements at opposite polarity and different delay (t1 and t5).In the 1-4 picoseconds that are needed to propagate from the source to the measurement, the laser-generated voltage pulse already attenuates and disperses to picosecond duration 47 .The measured rise time in Fig. 3d is ~2 ps and corresponds, if associated with a quarter wave period, to a frequency of ~0.12 THz which is well below our sampling frequency of about 1/(800 fs) = 1.2 THz.The measured rise time is therefore not the time resolution of our experiment but rather the time-resolved dynamics of the waveform in our transmission line.Experiments with time-compressed electron pulses confirm this assessment; see below.The delays of the two waveforms in Fig. 3d are caused by propagation of the voltage pulse from the photoconductive switch to the probe region.The observed values of t1 = 3.6 ps and t5 = 0.9 ps together with the physical distances of the two probe locations at 410 µm and 105 m away from the photoconductive gap reveal a group velocity of 0.38 c.This speed corresponds well to a simple estimate of pulse propagation speed in our coplanar waveguide, given by  √( + 1)/2 ⁄ = 0.38 c for a dielectric constant for GaAs of  = 13 (Ref. 50).The recovery time of the positive pulse propagating to the right (~45 ps) is longer than the recovery time of the negative pulse propagating to the left (~10 ps).We attribute this observation to the different effective capacities and ohmic resistances in the left and the right parts of the coplanar waveguide.Additional contributions may also arise from a potentially asymmetric position of the laser beam profile on the gap as well as heating effects 51 or differences in the electron and hole mobilities via the photo-Dember effect.
The additional voltage peaks in both recovery signals (t3, t4, t6, t7, t8) are time-domain signal reflections from geometry changes in our transmission line with partial impedance mismatch 51 (see arrows in Fig. 3a).The two minor voltage peaks in the right-propagating pulse (magenta) at t3 = 80 ps and 87 ps correspond to reflections at the interconnection to the shunt pad (see Fig. 3a), first into a bare Goubau line without adjacent ground lines and then into the macroscopic rectangular bond area.The stronger reflection observed at t4 = 130 ps originates from the end of the shunt pad at a total distance of 11.06 mm.These two values reveal a slower average group velocity of 0.294 c as compared to 0.38 c in the transmission line, because the shunt pad has a different size and distances to ground, producing a lower cutoff frequency and large dispersion.
The wirebond connections to the 50- resistor are too thin to be relevant for our picosecond dynamics.On the left side of the coplanar waveguide (blue data), the voltage pulse in the negative voltage region at t6 = 30 ps originates from impedance mismatch when the waveguide opens in a triangular way to the DC bias pad.The voltage pulse at t7 = 65 ps is a reflection from the end of the bias pad.The latest measured voltage pulse in our scan range at t8 = 125 ps is a double reflection within the DC bias pad (see Fig. 3a).All secondary voltage pulses at later times are increasingly broadened by dispersion and also increasingly attenuated by ohmic resistances and total propagation time.
Figures 3e and 3f show the power spectral density (PSD) of the measured signals at the left and right side of the waveguide, evaluated by Fourier transformation from the time-domain results.We see broadband curves with cutoff frequencies of ~0.5 THz, where the response of our waveguide merges with the noise.This value agrees well with the cutoff frequencies observed in similar coplanar waveguides with electro-optical sampling 52 .The frequency spectrum of Fig. 3e-f is therefore the response of the device and not limited by our time resolution.The multiple reflections from the left chip boundary (t5, t7, t8) come in quasi-periodically and consequently show up in Fig. 3f as frequency peaks at ~16 GHz and its second harmonic at ~32 GHz.A pure step function, like initially created in the photoconductive gap, should have a 1/ spectrum, where  is frequency, but our data fits better with a pink noise with  −2.7 dependency (grey curves), because we measure the waveform after substantial dispersion from the femtosecond to the picosecond domain.
In a second experiment, we demonstrate the additional diagnostic capability that is gained by compressing the electron pulses from 800 fs to 100 fs with laser-generated terahertz fields 36 .The duration of the laser pulses (Fig. 1a, red) that trigger the photoconductive switch is still ~80 fs.In order to measure the electron pulse duration at the location of the DUT and thereby the time resolution of our ultrafast voltage probe experiment, we invoke an all-optical streak camera that is driven by the cycles of optically generated far-infrared light 36 .A bow-tie resonator is used to mediate the electron-photon interaction by providing energy-momentum conservation.Electrons are deflected sideways (y direction in Fig. 3b) as a function of their arrival time with respect to one optical cycle of the reference wave.Consequently, long electron pulses produce a blurry streak while short ones produce a point and therefore sample the optical field cycle in an almost classical way 36 .Figure 4a shows the measured electron streaking trace 36 at the location of the DUT.We see a sinusoidal deflection of the beam electrons as a function of delay.The streaking speed around the zero crossing is 1.4 mrad/fs (dashed line).The width of the electron beam there, deconvoluted with the unstreaked electron beam profile 37 , reveals an electron pulse duration of 100±10 fs.The corresponding measurement bandwidth is ~10 THz.
Figure 4b shows the time-resolved dynamics of the electric field between the central transmission line and the ground at the right side of the gap (magenta ellipse in Fig. 3b).The data point separation is 80 fs and the Nyquist frequency is 6 THz.In this experiment, we apply a positive bias voltage of +15 V (magenta dots) and a negative bias of -15 V (magenta diamonds) to the bias pad.The time-resolved voltage dynamics in both cases (magenta dots and magenta diamonds) are mirror images of each other, demonstrating the unidirectional response of our photoconductive switch.Like in the lower-resolution experiment of Fig. 3d, we can see an ultrafast rise or fall time of ~2 ps, but this time the compressed electron pulses allow to resolve the shape of this response and conclude on the broadening mechanisms in the time domain.
Extended data Fig. 2 shows fitting attempts with a broadened Heaviside function or an exponential growth.The residuals in both traces are substantial and these models therefore do not capture the physics of our experiment.In order to fit the measured data in a proper way (black lines), we consider our 100-fs apparatus function in the time domain, an exponential rise time due to the finite capacity of our photoconductive gap and a dispersive pulse elongation from propagation in our transmission line.Consequently, we fit our data () by where ℋ is the Heaviside function and ⨂ is a convolution in time.The fit parameters are the signal amplitude , the time zero  0 , the time constant   of the photoconductive gap and the pulse broadening   due to propagation.The instrumental function of our electron beam probe   is fixed to 100 fs (Fig. 4a).Signal decay due to electron-hole recombination or damping in the transmission line can be ignored on this timescale, because it takes tens of picoseconds, at least.
The best global fit for both bias cases (black lines) is obtained with an RC-limited rise time of RC = 0.72 ps and a dispersive broadening of disp = 0.42 ps.Fitting attempts with simpler models do not yield good results; see extended data Fig. 2. We assign the measured RC time constant to the electromagnetic response directly in the photoconductive gap, defined by the capacity of the structure and the effective mobility of the hot charge carriers after laser excitation, while the dispersion time reveals the effects of the first picoseconds of propagation within our transmission line.Although the gap as well as the waveguide can on their own support a femtosecond response, the accumulated effects add up to an effective rise time of ~2 ps at the location of the probe.These proof-of-principle results demonstrate the value of using femtosecond time resolution and oversampling of picosecond signals to extract and discern the underlying spatiotemporal broadening mechanisms for optimizing future devices towards desired functionalities.

Resolution and speed limits
Given that electron pulses in electron microscopes can be made as short as attoseconds 39,40 , what are the fundamental resolution limits of our approach?Since the local electromagnetic fields in a DUT have always a finite spatial extent in the direction of the probing electron beam, the maximum detectable frequency is limited by the time of flight of the electrons through such a field.This aspect is known as the transit time effect in secondary-electron based voltage contrast in a scanning electron microscope (SEM) 53 .However, the electron beam in our experiment travels at almost half of the speed of light and therefore needs hundred to thousand times less time to pass through a material than the low-energy secondary electrons in a conventional SEM.Also, detection of the direct beam after transmission preserves the emittance and therefore allows to detect positions and deflections at the same time 46 .For ultimate space-time resolution, the electron time-of-flight through the local fields with characteristic longitudinal dimensions  in electron beam direction must be shorter than half a cycle period of the highest frequency  max to be determined, in order to not smear out the temporal response.At these conditions, the electron gains via the Lorentz forces   a sideways momentum change of Δ ⊥ ≈   • 0.5 •   −1 .The factor 0.5 accounts for our desire to resolve half a cycle of   .Fields in features of lateral size d in a DUT (y-axis in Fig. 3b) roughly penetrate into free space by Λ ≈ d.For a voltage amplitude Δ to be resolved, the electron deflection angle ∆θ is given by ∆ = .Interestingly, this speed limit does not depend on the size d of the structures in a DUT because a smaller structure produces higher electric fields for deflection but also requires a more divergent electron beam to be resolved in space.In our proof-of-concept experiment, the beam emittance at the specimen is  ≈ 100 pmrad and a signal with 1-V features in a 20-µm gap (see Fig. 3b) can be resolved up to   ≈ 5 THz or at a temporal resolution of ~200 fs (compare Fig. 4).However, modern pulsed electron beams from better sources than ours, for example from Schottky field emitters, have a beam emittance of ~5 pmrad 54,55 which allows one, assuming a beam energy of 200 keV, to resolve DUT frequencies as high as 60 THz or features as short as 20 fs at 1 V signal strength.
Static electron beam testing is ubiquitous in semiconductor industry and inevitable for research and development, quality control or failure analysis.The special electron microscopes in such eBeam testing facilities work typically with high-brightness field-emission sources similar to those in transmission electron microscopy.The only changes to convert such devices into an ultrashort fs-eBeam diagnostics are a laser-triggered electron source and a DUT-triggering by a photoconductive switch.Alternatively, a terahertz function generator can be synchronized to a femtosecond laser at 10-fs precision 56 .

Conclusion and Outlook
These results and considerations show that electron beam probe with ultrashort electron pulses (fs-eBeam) can resolve the electromagnetic response of working electronic devices in space and time.
Conventional scanning electron beam probe 57,58 , a cornerstone technology of modern electronics research 20 , is therefore advanced to terahertz bandwidth and femtosecond time resolution.No physical contact or proximity of a sensing object is required to achieve these resolutions, and the method is therefore non-distortive and impedance-free.In order to provide the necessary electron transparency, substrates can be thinned or precision-cut with ion mills or a focused ion beam.For example, large membranes of almost arbitrary material can be prepared by a dimple grinder and ion beam 59 .Alternatively, a non-invasive probing of circuit surfaces could be realized by detecting the energy or angular distribution of backscattered high-energy electrons in scanning electron microscopy.In principle, the spatial resolution in our fs-eBeam technique is only limited by the ability to focus an electron beam to a tiny spot, typically 0.2 nm in electron microscopy.The highest frequency and lowest electric fields that we can detect are only limited by the brightness of the electron beam and the averaging time.In contrast to near-field probes 19,27,29,30 , the electrons in our experiment do not influence the dynamics in the DUT, and measurements can therefore be made on running devices under normal operation conditions.In principle, electric and magnetic field components can be distinguished by repeating the experiment with different electron velocities 46 and electron beam accelerations or decelerations by longitudinal electric fields can be revealed by applying an electron energy filter 55 .
Detecting the most basic quantity of electronics, electric and magnetic fields in space and time, at nanometre, femtosecond and millivolt resolutions should therefore help researchers to discover hidden mechanisms and design novel ultrafast electronics beyond the state-of-the-art.

Methods
The terahertz circuit is produced on a 625-m thick, undoped, double-side polished GaAs wafer (University Wafers) by ultraviolet lithography (Smartprint, Smartforce technologies) with a 1-m thick photoresist layer (AZ MIR 701, Merck).The design of the structure is based on a 50 impedance line and the dimensions are chosen to minimise losses 27 .After developing (MIF, Merck), the terahertz structure is deposited by thermal evaporation of 300 nm of gold.A 5-nm thick layer of chromium is used for adhesion.Lift-off is done by dissolving the sacrificial resist layer in acetone.The holes for the electrons are then drilled by using femtosecond laser pulses focused to 10 µm diameter.The chip is then contacted with the bias source with silver paste, and a 50  resistor between ground and signal is contacted with wire bonds.Extended data Fig. 1 shows optical and scanning electron microscopy images of our circuitry.
Femtosecond electron pulses are produced by two-photon photoemission from a back-illuminated gold cathode at 70 keV 36,37 by frequency-doubled femtosecond laser pulses of 270 fs pulse duration (Pharos, Light Conversion).The centre wavelength for photoemission is 515 nm and the repetition rate is 50 kHz.The average number of electrons per pulse is kept below ten to avoid dispersion by space-charge effects 42 .The emittance of the electron beam at the source is  ≈ 2 nmrad 61 but later improved to ~100 pmrad by using a 20-µm aperture in a 100-µm beam.
The electron pulses are compressed to ~100-fs duration by velocity-matched transmission through a metal membrane 37 under illumination with terahertz radiation that is produced by Cherenkov radiation a LiNbO3 slab 62 .All magnetic lenses for beam control are aligned for minimum temporal distortions 48 .The detector at 1.4 m distance from the specimen has a pixel size of 15.6 μm.The electron pulse duration at the specimen and time-zero are characterized by all-optical streaking at a bow-tie resonator 36 .The pump laser pulses for triggering the DUT are compressed from 270 fs to 80 fs by cascaded  (2) interactions 60 and then frequency-doubled to 515 nm wavelength for efficient carrier-hole pair creation in the photoconductive switch.The relation between measured electron beam deflections and absolute voltages are calibrated with help of the bias voltage; we find a linear response for the beam deflection with bias voltage in the entire range from −15 V to +15 V.The fitting error is computed as the residuals quadrature and minimized with a derivativefree Nelder-Mead simplex algorithm.By varying the initial guess parameters, we ensure consistent convergence of the fit and emergence of a global minimum (see extended Data Fig. 2).
Extended Data Fig. 2. Various fits and their residuals.

Figure 1 .
Figure 1.Femtosecond electron beam probe of terahertz electronics.A laser pump pulse with femtosecond duration (red) creates a terahertz voltage pulse (magenta) by closing a photoconductive switch.The terahertz voltage pulse then travels into the device under test (DUT) and triggers its operation.Femtosecond electron pulses (grey) probe the local electric and magnetic field vectors (magenta) by means of time-frozen Lorentz forces and electron beam deflections.Changing the delay between activation of the switch and the probing electron pulses (dashed lines) provides a measurement with femtosecond time resolution and terahertz bandwidth.

Fig. 2 .
Fig. 2. Electron beam spots on the detector.(a) Measured electron beam profiles at -2 ps and 1.5 ps after waveform generation.The dashed line is the equilibrium beam.(b) Gaussian fits (solid lines) reveal the position of the electron beam in y direction (integrated over x).Black, reference beam; blue, beam at 1.5 ps.

Fig. 3 .
Fig. 3.In operando measurement of propagating voltage pulses in a terahertz device.(a) Schematics of the device under test (DUT) in our experiment.Light blue, metal layer; white, isolating substrate; red circle, origin of the laser-triggered terahertz dynamics.The blue and magenta pulses show the positive and negative voltage pulses propagating along the chip.Labels t1-t8 indicate direct

Fig. 4 .
Fig. 4. Time resolution of our experiment and femtosecond dynamics of the DUT.(a) Electron streaking data reveals an electron pulse duration of ~100 fs.Dashed line, streaking speed; Nel, number of electrons on the detector.(b) Measured femtosecond dynamics of our DUT.Red curve, shape of the triggering laser pulse, measured with frequency-resolved optical gating before frequencydoubling.Black curve, 100-fs electron pulses.The measured device dynamics for positive (magenta dots) and negative bias (magenta diamonds) is fitted with a model (black lines) that accounts for the 100-fs instrumental function, a 720-fs exponential voltage buildup at the photoconductive gap and a 420-fs pulse elongation by waveguide dispersion.
(a), (b) Fits of the experimental data (black dots and diamonds) with a model assuming RC-dynamics and the finite electron pulse duration but no voltage pulse dispersion (blue), and another model, assuming only dispersion but no RC-dynamics (magenta).(c), (d) Fit residuals corresponding to panels (a), (b).For comparison, the black squares show the residuals obtained with the model from eq. (1).(e) Residual fit error for different initial conditions.Orange and magenta, unphysical initial guesses with close-to-zero dispersive broadening time.Blue, cyan, realistic initial guesses.(f) Map of fitting errors for positive voltage pulses as a function of   and   .(g) Map of fitting error for negative pulses.The smallest errors (white colour) show a single global minimum in both maps.
≈ 276 keV/c is the longitudinal electron momentum and e is the elementary charge.If we focus an electron beam of emittance  into a spot size of diameter d as a local probe, we obtain a minimum beam divergence angle of Δ  ≈ /.This angle should not be much larger than the sample-induced deflection Δ in order to provide a clear determination of beam shifts.Note that beams with Δ  > Δ may still be resolved at sufficient ratio of signal to noise.Using Δ  < Δ as a worst-case criterion, we obtain a maximum measurable frequency of   ≈