Nanometric probing with a femtosecond, intra-cavity standing wave

Tobias Heldt; Jan-Hendrik Oelmann; Lennart Guth; Nick Lackmann; Lukas Matt; Thomas Pfeifer; José R. Crespo López-Urrutia

doi:10.1515/nanoph-2024-0332

Article Open Access

Nanometric probing with a femtosecond, intra-cavity standing wave

Tobias Heldt , Jan-Hendrik Oelmann , Lennart Guth , Nick Lackmann , Lukas Matt , Thomas Pfeifer and José R. Crespo López-Urrutia

Published/Copyright: November 28, 2024

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From the journal Nanophotonics Volume 13 Issue 25

Abstract

Optical standing waves are intrinsically nanometric, spatially fixed interference-field patterns. At a commensurate scale, metallic nanotips serve as coherent, atomic-sized electron sources. Here, we explore the localized photofield emission from a tungsten nanotip with a transient standing wave. It is generated within an optical cavity with counter-propagating femtosecond pulses from a near-infrared, 100-MHz frequency comb. Shifting the phase of the standing wave at the tip reveals its nodes and anti-nodes through a strong periodic modulation of the emission current. We find the emission angles to be distinct from those of a traveling wave, and attribute this to the ensuing localization of emission from different crystallographic planes. Supported by a simulation, we find that the angle of maximum field enhancement is controlled by the phase of the standing wave. Intra-cavity nanotip interaction not only provides higher intensities than in free-space propagation, but also allows for structuring the light field even in the transverse direction by selection of high-order cavity modes.

Keywords: nanotip; strong-field; electron emission; ultrafast; standing wave; cavity

1 Introduction

Controlling electrons on the natural femtosecond time scale of light has many promising applications [1], [2]. Strong optical fields induce nonlinear emission processes confined to this or even shorter time scales [3]. Their spatial localization by nanometric structures can cause both geometric as well as plasmonic field enhancements [4], [5], [6] making field emission much stronger. A sharp metallic tip features up to one order of magnitude higher field strength [7] with a corresponding increase of coherent electron emission from its apex [8]. Detailed studies of this process have elucidated both a multiphoton and a strong-field regime by combining optical near-field measurements with electron spectrometry [9], [10], [11], [12], [13], [14], [15], [16], [17]. Using few-cycle lasers, such electron pulses reached into the attosecond regime [18], [19], [20], [21], [22], [23], [24], [25]. Recently, both ultrafast scanning and transmission electron microscopes have achieved simultaneous nanometric spatial and femtosecond temporal resolution by steadily improving control of the free-electron phase [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37]. Phase measurements on slow, tip-emitted electrons for studying optical near-fields can be performed, e.g., in point-projection electron microscopes [38], [39], [40], [41], [42], [43], [44]. Such applications require good understanding of the coherence properties of the electrons and nanoscopic localization of their source. One example is when several electrons are emitted per pulse leading to correlated electron-number states due to their Coulomb interaction [45], [46]. Another one is work on tungsten tips demonstrating how apex crystallography and the angle of illumination govern the emission position [47], [48].

Here, we show how combining a standing light wave, i.e. light with a nanometric optical field distribution, with a nanotip allows us to probe the electron-emission sites on a sub-wavelength scale. Our finite-difference time-domain (FDTD) simulations strongly support that tuning the phase of the standing wave, and thus the near-field at the tip, even controls the crystallographic emission locus.

Figure 1a shows the optical setup and Figure 1b depicts the focal region, where two counterpropagating near-infrared (NIR) laser pulses sharing the same cavity mode interfere to form the standing wave. Its phase ϕ relative to the nanotip apex with radius r < 50 nm is controlled by the delay Δx between the laser pulses.

Figure 1:

Principle of the experiment. (a) An amplified frequency comb is split into two branches resulting in two coherent NIR pulses counter-propagating in a cavity. (b) In the focus the pulses interfere with a relative delay of Δx above a nanometric tungsten tip. Electrons emitted there are guided by a velocity-map imaging system to a microchannel-plate detector. (c) The relative delay determines the distance between the tip apex and the next anti-node of the standing wave. (d) Multiphoton absorption results in a nonequilibrium electron distribution f(E) in the tungsten tip out of which electrons can tunnel through the surface barrier. The barrier height depends on the work function of the crystallographic plane.

2 Results

We position a nanometric tungsten tip into the standing wave, as illustrated in Figure 1c. First, we adjusted the intensity ratio of the two counterpropagating pulses until the electron-emission pattern on the detector becomes independent of the relative delay Δx. This implied that for the given settings only one emission site was active on the tip. By changing Δx, we shift the nodes and anti-nodes with respect to the tip position while counting the emitted electrons. As shown in Figure 2, the respective electron yield exhibits a strong periodic modulation, vanishing at the nodes. The electrons have to overcome a potential barrier W eff = W − e 3 F ( 4 π ϵ 0 ) − 1 , given by the work function of tungsten W reduced by the applied electric field F because of the Schottky effect [49]. Under the present conditions, multiphoton absorption raises electrons to states above the Fermi level, from which they can tunnel through a reduced potential barrier [47]. Figure 1d illustrates this emission process for two exemplary crystal planes with the corresponding nonequilibrium electron distribution [48], [50]. Due to the exponential suppression of the tunneling step, emission from the highest electronic states and from planes with the lowest work function are strongly favored. The multiphoton absorption necessary to populate these states is nonlinearly dependent on the local electric field, which is in the present study determined by the phase of the standing wave at the tip. Additionally, we observe a nearly linear drift of ∼50 nm/min on successive measurements, which we attribute to thermal effects on the > 2 m-long, separate beam branches from the beam splitter separating the two pulses until the first cavity mirror.

Figure 2:

Measured electron yield from a tip near a standing light wave versus delay between the two interfering laser pulses. (Upper panel) Gray shaded area: laser-intensity at the tip. (Lower panel) Electron counts in 20 consecutive scans after correction for a phase drift (see inset) and amplitude normalization. Blue curve: approximation of an assumed effective third-order power dependence on the laser intensity.

In the following, we show that the standing wave can also be used to investigate the electron emission sites at the tip. We set the intensity ratio of the two counterpropagating pulses so that their electron yields (separately measured by blocking either of the two) are equally strong. Figure 3a shows the detected emission patterns resembling crescents oriented in the respective direction of laser propagation. This agrees with previous studies showing that for traveling waves the maximum near-field strength appears slightly behind the apex [48]. The emission pattern in our standing wave seems more complex; an example is given in Figure 3b. It strongly depends on the delay Δx, which we scan three times in a row over four light periods. We apply nonnegative matrix factorization (NMF) [51] on the complete dataset of 93 images to extract a suitable basis set for them. This method enables the separation of distinguishable features, also known as components, that constitute each image. Essential characteristics can often be derived from a small number of components, significantly reducing the dimensionality of the data. Compared to simple averaging, NMF is more robust against residual noise of the delay. Figure 3c shows the five strongest NMF components of the electron-emission dataset. The first two excellently match the patterns for traveling waves. In contrast, Figure 3cⁱⁱⁱ and c^iv show two distinct components with strong emission in the upper and lower center, respectively. The fifth one is already very noisy, which is why we ignore higher orders. Figure 3d depicts the contributions of the individual components for the delay-scan images. A Fourier transform clearly reveals the expected oscillation with the light period in the first four components (see Figure 3e). Figure 3f shows the phase of the oscillation for the different components. The first two components differ in phase by more than π, and the phases of the others appear in between.

Figure 3:

Spatial electron emission characteristics of the nanotip. (a) Observed emission pattern of a traveling wave propagating as the arrows show. (b) Example of pattern of electron emission induced by a standing wave. (c) Main components of a nonnegative matrix factorization (NMF) of all images recorded during three consecutive 4 µm delay scans. See main text for details. (d) Intensity evolution of the calculated components. (e) Power spectrum of each component averaged over all three scans, recovering the optical period at the dotted line. (f) Phase of each component at the scan frequency. Error bars indicate the standard deviation of the three scans. (g) Work-function map (measured in Ref. [52]) of a tungsten-tip apex with symmetry (110). Relevant crystallographic emission planes are circled in magenta.

We attribute these phenomena to emission sites attached to crystallographic surfaces of the tungsten tip apex, which we approximate in Figure 3g with a sphere model with its apex oriented along the (110) direction. The work function of this plane W ₁₁₀ = 5.32(10) eV is higher than those of all other planes, especially (221) and (310) with W ₂₂₁ = 4.30(4) eV and W ₃₁₀ = 4.34(4) eV [52]. According to the Fowler–Nordheim equation [53] with constants K _i, the tunnel current

(1) j ( F , W ) = K 1 | F | 2 W exp − K 2 W 3 / 2 / | F |

is strongly suppressed by a high work function. Photofield emission is thus dominated by those crystal planes with a low one. This points to four different crystal sites with low work functions as the cause of the first four NMF components, each having a strength depending on the phase of the standing wave.

To explain these observations, we model with FDTD simulations the electric field at the tip in a standing wave. Figure 4a and b show the field amplitude at the peak absolute field for the central plane of the tip. At ϕ = 0, the anti-node is centered at the apex, while positive values shift it toward negative z-values. The peak field shifts from the centered apex to the side as the phase changes, as displayed in Figure 4c, where the slices for different phases are stacked and the integrated field of the first 10 nm is projected onto the surface. The peak field shown as an arrow moves continuously across the tip. This indicates nanometer-scale control of the near-field by the phase of the standing wave. Considering the spatial structure of the maximal field across the tip F _max(x, z) and the different work functions of the sphere model W(x, z) an estimate of the local electron current j(x, z) can be calculated from Eq. (1). This quantity, given in Figure 4d and e together with the simulated signal, shows that the phase of the standing wave determines the location of electron emission on a nanometric crystallographic scale; our observations shown in Figure 3 are described well by this model.

Figure 4:

Simulated electric fields and resulting electron emission at the tip versus phase ϕ of the standing wave. (a, b) Peak-field distributions at the symmetry plane for a fixed phase, with tip radius r, opening angle α and electron-emission angle θ. Laser propagation along z with polarization axis y. (c) Full phase scan. Lower part: projection of the central slice on the tip surface as a function of the phase. Arrows mark the position and angle of the peak-field amplitude. Upper part: field amplitude for a subset of phases projected on the electron-emission angle. (d, e) Upper panels: electron emission at two different phases ϕ simulated with the different work functions of the ball model (see Figure 3g). Lower panels: simulated detector image after propagation through the electrode assembly. Note the similarity to the NMF components Figure 3cⁱ and c^iv .

3 Conclusion and outlook

We demonstrated control of electron-photofield emission from a tungsten nanotip by intra-cavity counterpropagating laser pulses forming a standing wave. The nodes and anti-nodes of this transient, spatially standing wave are reflected in a strong modulation of electron emission correlated with the local electric field strength. The emission patterns for different phases of the standing wave were analyzed using nonnegative matrix factorization. Supported by an FDTD simulation, we identify the main NMF components as the emission patterns from localized emission sites on the tungsten tip.

Based on these results, we propose the use of enhancement cavities for laser-assisted electron emission, which offer an intensity boost of a few orders of magnitude at the tip. With intensities of up to I > 10¹³ W/cm², we enter a regime where we expect new applications of the Kapitza–Dirac effect [54], [55], [56], [57] in the context of MHz-rate pulsed electron sources. Furthermore, high-order Hermite–Gaussian cavity modes selected by alignment of the driving laser in-coupling would yield very pure structured light that can be used for electron emission and scattering studies [58]. Using cavity-enhanced nonclassical light is another fascinating prospect, since it imprints its statistical and coherence properties on the emitted electrons [59].

Our demonstration points to future uses of frequency-comb-locked, standing waves strongly confined in three dimensions for controlling nanometric structures. The nonlinearity of electron emission could also allow for studying more complex metasurfaces. Moreover, an improved understanding of electron emission in complex field configurations and manipulation of both electron amplitude and phase with light would be beneficial for applications such as electron microscopy, electron holography, and coherent electron control.

4 Methods

4.1 Experimental setup

We use a bow-tie resonator consisting of five mirrors with a dielectric coating optimized for high damage threshold and reflectivity R _HR = 0.99995. An input coupler with lower reflectivity R _IC = 0.993 closes the cavity. Two of the mirrors are concave with a focal length of f = 125 mm to focus to a waist radius of w ₀ = 25 µm; the others are plane. One of them is glued to a piezoelectric element which allows to change the cavity length for locking it on resonance by the Pound–Drever–Hall technique [60]. The cavity enhances the intra-cavity circulating power by a factor of ∼ 200 .

We use a commercial NIR frequency comb (Menlo Systems, FC1000-250) operating at a repetition rate of 100 MHz centered at 1,040 nm, a home-built rod-type amplifier to increase the average power to 13 W, and a grating compressor to reduce the pulse duration to ≤ 200 fs. These pulses pass a beam splitter of selectable intensity ratio. After it, one beam branch is delayed (Δx) by a motorized retro-reflector. After mode matching, both branches are coupled into the enhancement cavity in a counter-propagating fashion such that the pulses collide at the focus, as explained in Ref. [61].

The focus is positioned at the source point of a velocity-map imaging (VMI) spectrometer mostly used for above-threshold ionization of gaseous targets [62]. For the present study, an additional copper electrode holds a tungsten nanotip pointing toward the VMI detector. We use commercial STM probes (Bruker) made from a polycrystalline tungsten wire etched to form a tip of radius r < 50 nm. This treatment is known to let a (110) surface appear at the apex [63], [64]. The tip is positioned 50–100 µm below the laser focus to keep the cavity mode unblocked. For alignment, the VMI electrode assembly can be moved around the cavity focus in all three directions as a whole. We apply a bias of −1,000 V to the tip and appropriate voltages to the other electrodes to image the emitted electrons on a microchannel-plate detector (MCP) equipped with a phosphor screen. An air-side CCD camera records through a window the image of the electron distribution on the MCP. The cavity, the tip, the electrodes, and the detector are installed in a 2-m long ultra-high vacuum chamber at a pressure of ≈ 1 0 − 8 mbar. Two stages of air suspension decouple the whole laser setup and the chamber from vibrations.

4.2 Data evaluation

The total electron count rate is defined as the integrated CCD camera signal. Consecutive scans are normalized to the mean value of all the cycle maxima. To analyze the electron emission patterns, NMF is employed to reduce the high dimensionality of the camera images. All images are downsampled from 978 × 978 pixels to 100 × 100 (=10,000) pixels to eliminate pixel noise. The 93 images in the dataset are organized into a matrix X of size 93 × 10,000. Using a coordinate-descent solver with ’nonnegative double singular value decomposition’ (NNDSVD) initialization, the data is factorized into 12 components. This results in the best approximation of the data being separated into a coefficient matrix W (93 × 12) and a feature matrix H (12 × 10,000), such that WH ≈ X. The coefficients of the components in the original images are normalized. For a phase analysis at the scan frequency, a drift between scans is corrected, and an offset is chosen so that the phases of the first two components are symmetric with respect to zero.

4.3 Simulation methods

To calculate the field evolution at the tip for different phases of the standing wave, we solve the discretized Maxwell equations in a three-dimensional FDTD simulation using an open-source package [65] on a 175 (propagation direction) × 135 (tip and polarization direction) × 151 grid with 2 nm spacing. We use 20 nm thick perfectly matched layers as absorbing boundaries. The tip is modeled as a hemisphere with radius r = 75 nm and a conical shaft with half-angle α = 25°; the relative permittivity of tungsten is taken as ϵ = −4.03 + 22.4i [66]. We approximate the light field as two plane waves with a phase difference ϕ, since both the laser waist and its Rayleigh length are much larger than the simulation box. For stable numerical propagation according to the Courant–Friedrichs–Lewy condition [67], a time step of Δt = 3.8 as is chosen for 3,000 propagation steps corresponding to three full light periods. We determine the time of maximum field for each phase separately, and use a median filter with the size of three pixels to account for discretization errors of the field amplitude. For estimating the emission probability, the sizes of the median filters in the xz-dimensions were extended to nine pixels. At each position on the tip surface, we approximate the emission by the contribution of the maximum field amplitude obtained during the laser cycle. We smooth the map of maximum field amplitudes by a Gaussian kernel with a size of three pixels, scale the maximum field to 1 GV/m corresponding to an intensity of 2.7 × 10¹¹ W/cm² and use it as the input field |F| in Eq. (1). In this way, for each of the central 64 × 64 pixels a measure for the emission probability is calculated dependent on the local work function. For the spatial structure of the different crystallographic planes on a hemisphere, we use the software VESTA [68]. The orientation of the crystal facets was rotated by 10° to match the experimental results. The respective work functions of all planes, which are used for the calculation, are given in Table 1.

Table 1:

Work functions of tungsten for the modeling of the emission probabilities. Taken from Ref. [52].

Crystal plane	Work function (eV)
(110)	5.32 ± 0.10
(211)	5.12 ± 0.07
(100)	4.93 ± 0.06
(320)	4.58 ± 0.06
(321)	4.50 ± 0.06
(111)	4.45 ± 0.05
(610)	4.43 ± 0.04
(411)	4.42 ± 0.05
(332)	4.41 ± 0.05
(210)	4.36 ± 0.04
(421)	4.35 ± 0.05
(310)	4.34 ± 0.04
(611)	4.32 ± 0.04
(221)	4.30 ± 0.04

A total of 10,000 electron emission sites were sampled for each phase of the standing wave, representing the spatially dependent Fowler–Nordheim emission currents. It was assumed that electrons move perpendicularly to the surface for the first 0.6 mm, after which they were propagated in an electrostatic model of the electrode assembly (grid size 0.2 mm) towards the detector using the software SIMION [69].

Corresponding author: Tobias Heldt, Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany; and Heidelberg Graduate School for Physics, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 226, 69120 Heidelberg, Germany, E-mail: heldt@mpi-hd.mpg.de

Funding source: European Metrology Programme for Innovation and Research

Award Identifier / Grant number: 20FUN01 TSCAC

Funding source: Max-Planck-Gesellschaft

Funding source: Max-Planck-RIKEN-PTB-Center for Time, Constants and Fundamental Symmetries

Funding source: European Metrology Programme for Innovation and Research

Award Identifier / Grant number: 23FUN03 HIOC

Acknowledgment

We thank the fine-mechanics workshop at MPIK lead by Thorsten Spranz for the excellent manufacturing of our experimental apparatus, the divisional mechanical workshop lead by Christian Kaiser for prompt support at the laboratory, and our engineering-design office headed by Frank Müller for their help.

Research funding: This work was funded by the Max-Planck-Gesellschaft, the Max-Planck-RIKEN-PTB-Center for Time, Constants and Fundamental Symmetries. This work has been funded by the “European Metrology Programme for Innovation and Research” (EMPIR) projects 20FUN01 TSCAC and 23FUN03 HIOC. This project has received funding from the EMPIR programme co-financed by the participating states and from the European Union’s Horizon 2020 research and innovation programme.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: Authors state no conflict of interest.
Data availability: The datasets generated during and analyzed during the current study are available from the corresponding author on reasonable request.

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