Highly Nonlinear and Ultrafast Optical Phenomena in Metallic Nanostructures

Abstract

This Chapter presents recent findings on nonlinear ionization and photoemission processes at metallic nanostructures. A particular emphasis is placed on processes which—due to the localized excitation in optical near-fields—exhibit different properties and scalings from their counterparts in the gas phase or at planar surfaces. The Chapter is structured in two parts. The first part discusses various regimes in highly nonlinear photoelectron emission from metallic nanotips, including field-driven photoemission at near- and mid-infrared frequencies, and the control of localized photoemission using intense terahertz transients. In the second part, multiphoton and strong-field ionization of atomic gases in plasmonic antennas and waveguides is presented. It is demonstrated that local ionization is enabled by optical field enhancements in various types of structures. At the same time, fundamental physical limitations preventing efficient high-harmonic generation in nanostructures are discussed.

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1 Introduction

Throughout recent years, significant progress was made in the study of highly nonlinear, ultrafast ionization and photoemission processes in nanoscale matter by employing the enhancement of ultrashort laser pulses in tailored metallic nanostructures such as sharp needles, nanoparticles, resonant antennas or waveguides. Two important optical features of these metallic nanostructures are their capability to localize electromagnetic radiation at sub-wavelength scales and, at the same time, to enhance incident light intensities in nanometric hot spots by several orders of magnitude. The associated sub-wavelength localization allows for a detailed analysis and deeper understanding of ultrafast ionization processes driven by highly inhomogeneous fields.

In this context, nonlinear photoemission from metallic and dielectric structures [6–12] and nanostructure-enhanced atomic gas ionization causing extreme-ultraviolet (EUV) light generation [13–19] were shown to exhibit scalings strongly deviating from bulk behavior. In this Chapter, we discuss a set of our recent studies in this field on photoelectron emission and atomic ionization in nanostructures [11, 12, 15, 17, 18].

In the first Section, experiments on nonlinear photoemission from sharp metallic tips are presented. The emission and acceleration of photoelectrons with ultrashort, intense laser pulses are examined for a variety of experimental conditions. Ultrashort near- and mid-infrared laser pulses are used to induce photoemission, and electron kinetic energy spectra are analyzed. In particular, a novel regime of strong-field photoemission is found, in which the electrons are accelerated in the rapidly decaying near-field within a fraction of the optical half-cycle. Furthermore, streaking-type two-color experiments are carried out by additionally coupling delayed single-cycle terahertz (THz) transients onto the tip, enabling a control of the photoemission current and nanoscale electron trajectories.

The second Section covers recent findings about EUV light generation in plasmonic nanostructures, induced by the plasmonic enhancement of atomic gas ionization. Specifically, the feasibility of coherent high-harmonic generation in resonant nanoantennas and tapered hollow waveguides, as reported previously [13, 14], is critically revisited [15, 17, 18].

2 Photoelectron Dynamics at Sharp Metal Nanotips

In this Section, we investigate the effects of field-enhancement and localization on photoelectron emission in optical near-fields. Sharp metal nanotips are employed to locally enhance the incident optical fields at the tip apex. The excitation wavelengths in our experiments span from near-infrared (NIR) to terahertz (THz) frequencies. As demonstrated in the following, strong-field excitation at long wavelengths facilitates access to sub-cycle electron dynamics in spatially confined electric fields [11, 20].

2.1 Nonlinear Photoemission

Conventional photoemission, involving photon energies exceeding the electron binding energy (or work function) $\varPhi$, exhibits photoemission yields proportional to the incident light intensity on a surface [21]. In contrast, nonlinear photoemission is governed by the simultaneous absorption of multiple photons of an energy below the work function, leading to the emission of a single electron in the perturbative process of multiphoton photoemission. For higher intensities and at long wavelengths, in a different conceptual picture, the optical electric field can bend the surface potential to a barrier which allows for quantum-mechanical tunneling of electrons (optical field emission). The transition between the multiphoton photoemission and optical field emission regimes is characterized by the dimensionless Keldysh parameter , i.e., the square root of the ratio of the work function and the ponderomotive energy $U_p$ [22]:

$$\begin{aligned} \gamma =\sqrt{\frac{\varPhi }{2U_p}}. \end{aligned}$$

(10.1)

Here, the ponderomotive energy  $U_p=\frac{e^2F^2}{4m\omega ^2}$ is the average kinetic energy of an electron with mass m and charge e oscillating in an electric field of amplitude F with the angular frequency $\omega$. For moderate intensities or high frequencies, the ponderomotive energy is negligible compared to the metal work function ($\gamma \gg 1$), and the photoemission process can be described by multiphoton photoemission , as depicted in Fig. 10.1a. At metallic nanotips, multiphoton photoemission was reported, e.g., in [6, 8, 23]. In this regime, the photocurrent J scales with a power law in the intensity I, corresponding to the photon number n: $J\propto I^{n}$.

Fig. 10.1

Nonlinear photoemission processes. a Multiphoton photoemission (top) leads to a slightly modified photoelectric law for the electron kinetic energies (bottom). b Optically-induced tunneling (top) and subsequent field acceleration exhibits drastically different kinetic energy scalings. c Components of the two-step photoemission model for one optical half-cycle: the photocurrent (gray area) increases nonlinearly with the sinusoidal electric force (solid line). For direct (not rescattered) electrons, the kinetic energy gain (dashed line) in the electric field reaches a maximum value of 2 $U_p$. Rescattered electrons reach maximum energies up to 10$\,U_p$

Under such conditions ($\gamma >1$), multiphoton transitions of the lowest possible photon order $n_0$ dominate [24–26]. At ponderomotive energies comparable to the photon energy (for higher intensities or lower frequencies), the absorption of excess photons ($n>n_0$) results in additional spectral peaks at multiples of the photon energy (Above threshold photoemission [27, 28]), which was observed for metal nanotips in [7]. For increasing intensity, the growing kinetic energy contained in the quiver motion of the final state of the electron after photoemission (Volkov state) eventually leads to the closing of photoemission channels of low order [29].

At even higher intensities and/or longer wavelengths ($\gamma \ll 1$), the emission dynamics can be described by a field-driven instead of a photon-driven process. In this strong-field regime , the emission is viewed as optically-induced tunneling through a potential barrier, adiabatically following the applied electric field (see Fig. 10.1b). The actual barrier height is lowered to an effective work function $\varPhi _{eff}$ due to the Schottky effect (see, e.g. [30, 31]). Electrons are liberated via tunneling through the potential barrier around the peak of the laser field (cf. Fig. 10.1b). In this adiabatic regime, the current density j is described using the Fowler-Nordheim equation from static tunneling [32], which in its simplest version takes the form:

$$\begin{aligned} j(F)\propto \varTheta (-F)\cdot F^2\cdot \exp \left( -\frac{4}{3}\sqrt{\frac{2m}{\hbar ^2}}\frac{\varPhi ^{3/2}}{{e^1}{F^1}}\right) . \end{aligned}$$

(10.2)

Here, ${\hbar }$ is planck’s constant, and $\varTheta (-F)$ is the Heaviside step function. Subsequent to photoemission, the electrons are exposed to the unscreened optical electric field, which accelerates them to kinetic energies determined by the ponderomotive energy [33]. The kinetic energy distributions resulting from this process can be calculated within a modified simple man’s model , as previously employed to describe scalings in strong-field atomic ionization and high-harmonic generation [34, 35]. In a first step, electrons are generated by Fowler-Nordheim tunneling, and in a second step, their respective kinetic energy is calculated via classical propagation of point charges in the optical field. The resulting electron energy depends on the phase of the electromagnetic wave at the moment of photoemission (see Fig. 10.1c), and overall spectra are a result of weighting with the final kinetic energy after acceleration the instantaneous current density. Due to the rectifying nature of a metal-vacuum interface, only half-cycles driving electrons out of the surface contribute to the emission. For electrons emitted early in the optical half-cycle (increasing electrical force on electrons), the final kinetic energy reaches a maximum value of $2\,U_p$, while photoelectrons generated after the peak of the force curve are accelerated back, with a fraction elastically rescattered at the metal surface and reaching final kinetic energies of up to $10\,U_p$ in the case of spatially homogeneous fields [34, 36, 37].

Fig. 10.2

Different regimes in the field-driven acceleration of photoelectrons. a In the quiver regime, the optical pulse ends, before the electron has significantly moved out of the optical focus. b For more confined fields, the effective interaction time of the electron with the field is limited to few optical cycles. c In the sub-cycle regime, the electron escapes the highly confined field within less than one optical half-cycle

2.2 Sub-cycle Electron Dynamics in Highly Localized Electric Fields

The field-driven photoemission in localized fields, e.g., induced at the apex of metallic nanotips, exhibits characteristic electron dynamics, which may strongly deviate from the situation in diffraction-limited optical foci. In a far-field optical focus, photoemitted electrons interact with the entire remainder of the oscillatory field after the instance of emission. In contrast, this electron quiver motion may be suppressed in confined near-fields, as the typical spatial extension of the enhanced near-field (in the range of few to few tens of nanometers) can approach the dimensions of an electron’s excursion path.

Generally, the kinetic energy change $\varDelta E$ of a photoemitted electron in a space- and time-dependent electric field F is described by the integral of the force acting along its trajectory x(t), starting at the time of emission $t_e$:

$$\begin{aligned} \varDelta E=-\int \nolimits _{t_e}^\infty e F(x(t), t)dt. \end{aligned}$$

(10.3)

Figure 10.2 schematically depicts electron trajectories for a sequence of increasing confinement in the driving field. When the photoelectrons are accelerated out of the high-field region by multiple cycles of a pulse with long pulse duration, as depicted in Fig. 10.2b, the electrons experience a focal intensity gradient which leads to a ponderomotive force. As a result, the electrons gain an energy offset of 1$\,U_p$ in addition to the kinetic energy distribution in spatially homogeneneous fields (compare Sect. 10.2.1) [38].

In photoemission experiments with very short pulses and diffraction limited foci, the quiver amplitude is much smaller than the decay length of the field, and the photoelectrons do not experience the spatial inhomogeneity during the pulse duration (cf. Fig. 10.2a). Thus, their momentum change $\varDelta p$ is given by the temporal integral of the time-dependent electric field, which is proportional to the vector potential A [39]:

$$\begin{aligned} \varDelta p=-\int \nolimits _{t_e}^\infty e F(t)dt=eA(t_e). \end{aligned}$$

(10.4)

In this case, the photoelectron energy is shifted compared to the long pulse limit, a phenomenon first observed in above threshold ionization spectra from atoms [40].

Photoemission in highly localized, sub-wavelength fields allows for a third regime, in which electrons rapidly leave the driving field, potentially within a fraction of an optical half-cycle (see Fig. 10.2c). In the extreme, quasi-stationary limit of this sub-cycle acceleration regime, the electron kinetic energy reflects the momentary electrical field at the instant of emission:

$$\begin{aligned} \varDelta E=-\int \nolimits _{x_e}^\infty e F({x_1}{t_e})dx, \end{aligned}$$

(10.5)

Where $x_e$ is the position of emission. Sub-cycle electron dynamics can be achieved at long wavelengths, high intensities (corresponding to high ponderomotive energies) and very short decay lengths of the field. The quiver regime (see (10.4)) and the quasi-static regime (see (10.5)) represent two extreme cases of electron propagation in temporally and spatially varying electric fields. The transition between these scenarios can be characterized by a dimensionless spatial adiabaticity parameter  $\delta$ , which compares the corresponding length scales, namely the quiver amplitude $l_q=\frac{eF}{m\omega ^2}$ of the photoelectrons and the decay length $l_F$ of the electrical field [11]:

$$\begin{aligned} \delta =\frac{l_F}{l_q}=\frac{l_Fm\omega ^2}{eF}. \end{aligned}$$

(10.6)

This adiabaticity parameter for the acceleration (step two in the above-mentioned simplified model) takes on a similar role as the Keldysh parameter does for the emission (step one), as both parameters distinguish between cycle-integrated and sub-cycle dynamics. For low spatial adiabaticity parameters $\delta \ll 1$, the corresponding quiver amplitude in a homogeneous field would be much larger than the field decay length, leading to sub-cycle acceleration. Similar to sub-cycle emission conditions, this acceleration regime can be reached at high intensities and long wavelengths. However, the Keldysh parameter has a different frequency dependency as the $\delta$-parameter and also depends on the work function of the material, while the spatial adiabaticity parameter is a function of the field decay length. Thus, both parameters are independently adjustable, and strong-field dynamics at nanostructures facilitate ultrafast emission adiabatically following the electric field at $\gamma \ll 1$, with quasi-instantaneous acceleration of the charges at $\delta \ll 1$.

2.3 Photoemission from Gold Nanotips Induced by Near- and Mid-infrared Femtosecond Pulses

In order to study strong-field electron dynamics in highly confined fields, we analyze photoemission from single metal nanotips in the near- and mid-infrared range, investigating the effects of field-driven electron dynamics on the photoelectron energy spectra [11]. The experimental setup is depicted in Fig. 10.3a. A Ti:Sapphire laser amplifier system provides 30-fs pulses at 800 nm wavelength, and tunable optical pulses from the near- to the mid-infrared are produced via optical parametric amplification and difference frequency generation, resulting in an accessible wavelength range between 0.8 and $10\,\upmu$m. This broad tuning range enables a detailed study of the transition from ponderomotive electron acceleration to field-driven electron dynamics. The laser pulses are focused onto an electrochemically etched gold nanotips with radii of curvature of few tens of nanometers. Photoemission is induced in the enhanced field at the tip apex for polarization parallel to the tip axis. The kinetic energy spectra of the photoelectrons are measured using a multichannel-plate (MCP) detector combined with a retarding field analyzer.

Figure 10.3b shows examplary spectra for increasing laser intensities at a fixed excitation wavelength of 3.8 $\upmu$m for two tips with different radii of curvature. The tips are moderately biased to $-40$ V to optimize the detection efficiency. In Fig. 10.3b, the additional bias potential is subtracted from the final kinetic energy. For growing intensities, we observe a strong increase in the maximum electron kinetic energy up to several hundred electron volts. Corresponding to the absorption of about one thousand photon energies by a single electron, it is clear that a description in terms of multiphoton orders is no longer practical for this field-driven process.

Fig. 10.3

a Scheme of the experimental setup. b Electron energy spectra for increasing laser intensity recorded at 3.8 $\upmu$m (adapted from [11]). The measured spectra from two tips with different radii (tip radii: 12 nm (solid circles) and 22 nm (open circles)) were reproduced by a modified two-step model (solid lines)

2.3.1 Wavelength-Scaling of the Near-Field Electron Acceleration

We investigate the wavelength-dependence of the photoelectron dynamics by recording kinetic energy spectra for varying excitation wavelengths at sub-100 fs pulse durations as measured with interferometric photocurrent autocorrelations at the nanotip (see Fig. 10.4b). In these measurements, an identical pair of pulses is interfered on the tip, and the photoemission yield is recorded as a function of the relative pulse delay.

Figure 10.4a displays a set of electron energy spectra for excitation wavelengths between 0.8 and 8.0 $\upmu$m at constant emitted charge per pulse. In adiabatic strong-field emission, the photocurrent is determined by the field strength, and a constant emitted charge at comparable pulse durations is an indication for nearly constant local electric field strengths. In comparing these measurements, we find a saturation in the maximum kinetic energy for long wavelengths, see Fig. 10.4a—in contrast to the quadratic increase of kinetic energy as expected from pure ponderomotive energy scalings (compare Sect. 10.2.1). This observation can be directly attributed to sub-cycle electron acceleration in the optical near-field. According to (10.6), the quiver amplitude of electrons in a homogeneous electromagnetic field increases with the wavelength due to the longer acceleration periods. In the sub-cycle regime (see Sect. 10.2.2), the quiver amplitude begins to exceed the decay length of the near-field at the tip apex, which in a first approximation can be taken as a weak function of wavelength. The resulting saturation of the maximum kinetic energy indicates electron dynamics driven by the momentary electric field.

Fig. 10.4

a Electron energy spectra for varying wavelengths. The cutoff energy saturates for wavelengths above 3.8 $\upmu$m. b Photocurrent autocorrelation measurements for several wavelengths (adapted from [11])

Fig. 10.5

High energy cutoff in a spatially decaying electric field. a Experimental cutoff energies for varying wavelength at two different local intensities ($I_1=5.4$ TWcm$^{-2}$ and $I_2=40$ TWcm$^{-2}$). The gray areas are limited by simulations of the cutoff energy for complete rescattering (upper bound) and without rescattering (lower bound). b $\delta$-Parameter associated to the experimental data. c Electron trajectories in a spatially homogeneous field. d Electron trajectories in an electric field with a decay length in the order of the electron quiver amplitude. e Simulated maximum energy of the directly emitted photoelectrons as a function of the $\delta$-parameter (solid black line). Dashed line Homogeneous case ($\delta \rightarrow \infty$) shown for comparison (a–d adapted from [11])

For a quantitative analysis of the wavelength-scaling, we evaluate the spectral high-energy cutoff , represented by the energy exceed by the fastest 1 % of the electron distribution. Figure 10.5a shows the cutoff energies as a function of wavelength for two different intensities. At higher intensity, the transition to sub-cycle acceleration dynamics is found to be steeper as a result of the increased quiver amplitude for high intensities. The observations are reproduced by a modified two-step model for the calculation of kinetic energy distributions, and the numerical cutoff energies are presented in Fig. 10.5a. The gray shaded area is formed by the cutoff energies of direct (lower curve) and of elastically rescattered (upper curve, 100 % rescattering probability) electrons. The experimental data are found within this corridor.

In homogenous fields, the cutoff energy of rescattered electrons typically exceeds the cutoff of direct electron by a factor of 5 (compare Sect. 10.2.1). Here, we find that with increasing wavelength, the cutoff energies of direct and rescattered electrons converge as a result of reduced back-acceleration towards the surface in the sub-cycle regime. Figure 10.5c, d illustrate the characteristic electron trajectories in spatially homogeneous and highly inhomogeneous fields, respectively. Whereas in Fig. 10.5c, a large fraction of trajectories shows backscattering at the metal surface, the same emission times result in trajectories for which the photoelectrons directly leave the field in Fig. 10.5d.

Figure 10.5e presents a calculation of the maximum energy of directly emitted electrons as a function of the spatial adiabaticity parameter $\delta$. Approaching the transition at $\delta =1$, the energy cutoff exceeds $2\,U_p$ due to the additional ponderomotive energy from the gradient force in the near-field. In the limit of sub-cycle field-driven acceleration ($\delta \ll 1$), the energy cutoff decreases below $2\,U_p$ and is finally proportional to the electric field [20]. This scaling may be employed, e.g., to directly map the time-dependent electric near-field at a nanostructure in a streaking scheme, as demonstrated below.

2.4 Nanostructure Streaking with Ultrashort THz Pulses

In order to phase-resolve field-driven electron dynamics, we conduct streaking-type experiments with phase-stable single-cycle THz transients and femtosecond NIR pulses. All-optical streaking allows for the mapping of temporal information on, e.g., the kinetic energy of photoelectrons [41, 42]. Numerous schemes and applications for nanostructure-based near-field streaking have been theoretically studied (i.e., [43–46]), discussing the characterization of plasmonic near-fields with attosecond XUV pulses. In these schemes, the attosecond pulses generate photoelectrons, which experience a phase-dependent acceleration in the streaking near-field.

Fig. 10.6

a Experimental setup for a streaking experiment at a single nanotip using NIR and THz pulses. b Close-up of the streaking scheme (adapted from [12])

Here, we transfer this concept to longer wavelengths and present near-field streaking of NIR-induced photoelectrons at the nanotip using THz frequencies [12]. The experimental setup is sketched in Fig. 10.6. Both NIR and the THz pulses are focused collinearly onto single gold nanotips (see Fig. 10.6b). The near-infrared pulses of 50-fs duration and 800 nm centre wavelength generate photoelectrons, which are subsequently streaked in the tip-enhanced THz electric field.

The THz transients are produced with the AC-bias method in a light-induced air plasma [47, 48]. This method provides phase-stable THz pulses and the control of pulse energy and carrier-envelope-phase. The air plasma is generated with infrared pulses of up to 2 mJ pulse energy, a small fraction of which is frequency-doubled and overlapped with the fundamental beam. The photoelectron spectra are detected with a time-of-flight electron spectrometer or a retarding field analyzer. A series of kinetic energy spectra, recorded as a function of the relative time delay between both pulses, forms a spectrogram as shown, for example, in Figs. 10.7a and 10.8a. The tip is moderately biased to draw the photoelectrons to the detector. We characterize the incident THz transient in-situ at the position of the nanotip using electro-optic sampling in a ZnTe crystal.

Fig. 10.7

Near-field streaking at a very sharp nanotip (radius of curvature 10 nm) (adapted from [12]). a Spectrogram. b Photocurrent extracted from spectrogram (a). c Comparison between the incident electric field and assembled waveforms from the delay-dependent photocurrent and electron kinetic energy. d Scanning electron microscope image of the nanotip employed in the measurements

Fig. 10.8

a Spectrogram with broad initial energy spectrum and an effective decay length of around 40 nm. b Scheme for spectral compression and c broadening (adapted from [12])

2.4.1 Experimental Observation of Field-Driven Streaking

A streaking spectrogram at a very sharp gold tip with a radius of curvature of 10 nm is displayed in Fig. 10.7. This spectrogram shows the impact of the THz transient on the electron kinetic energy: The THz electric field accelerates the electrons by 50 eV, resulting in the final energy of 80 eV (bias tip voltage: $-30$ V). At large negative delays, the 800 nm pulse precedes the THz transient, and the photoelectrons are not affected by the THz field. Thus, the electron energy is determined only by the static bias voltage. Due to the rectifying effect of the field emission geometry, the photocurrent is suppressed in the half-cycles of the THz electric force pointing towards the tip.

The spectrally integrated photocurrent as a function of the delay is plotted in Fig. 10.7b and shows the current suppression and enhancement depending on the phase of the THz transient, with the enhancement induced by the impact of the THz field on the effective work function (Schottky effect) . The peak of the photocurrent is in-phase with the maximum of the kinetic energy. This observation represents the limiting case of acceleration by a momentary electric field. Electrons emitted at the peak field are quasi-statically accelerated ($\delta \ll 1$) and leave the enhanced near-field with minimal temporal integration.

Due to the suppression of the photocurrent in half-cycles of negative electric force, the spectrogram contains only information about the accelerating half-cycles of the THz near-field. Thus, the THz transient with opposite polarity is measured in addition, in order to characterize the entire continuous THz waveform . In the experiment, the polarity of the THz transient can be inverted by a rotation of the BBO crystal to obtain the complementary spectrogram [12]. The energy expectation value of the assembled waveform¹ is shown in Fig. 10.7c. This curve (black) represents the local THz waveform. A comparison with the incident electric field, characterized via electro-optic sampling (blue), reveals the time-domain response of the nanostructure. We find subtle waveform modulations and a slight phase shift. These effects can be attributed to the electromagnetic response of the nanotip and are reproduced in an antenna circuit model [49, 50].

2.4.2 Controlling Electron Trajectories in Time-Varying Near-Fields

At the transition to the sub-cycle acceleration regime, the electron dynamics sensitively depend on the initial electron energy and the temporal slope of the electric field. Figure 10.8a shows a spectrogram at a nanotip with a decay length longer than in the previous spectrograms, and with a broader initial energy spectrum induced by the NIR pulse. Here, the width of the kinetic energy spectra is modulated by the slope of the THz field. We observe that streaking by a temporally decreasing electric field leads to a broadening of the kinetic energy spectra, whereas an increasing streaking field causes spectral compression .

The initial energy of the photoelectrons at the instant of emission determines the interaction time with the streaking field. Electrons with low kinetic energy are accelerated over longer time intervals as compared to faster electrons. Therefore, slow electrons are more sensitive to the temporal evolution of the streaking field. For increasing electric fields, initially slower electrons gain more kinetic energy than their faster counterparts. As a result, the kinetic energy distribution can be compressed, cf. Fig. 10.8b. Accordingly, decreasing streaking fields (see Fig. 10.8c) result in a broadening of the initial spectra.

The dependence of electron trajectories on the slope of the streaking field enables the control of electron spectra and electron pulse durations via the relative delay between NIR and THz pulses. The presented spectral compression could be employed for the generation of optimized electron spectra for ultrafast optical-pump/electron-probe experiments.

2.4.3 Modeling THz Streaking at a Nanotip

The streaking spectrograms are simulated by classical particle propagation in the temporally and spatially varying THz streaking field as shown in Fig. 10.9. Due to the different pulse durations of NIR and THz transients, quasi-instantaneously generated electron spectra are propagated in the THz field as described in Sect. 10.2.3. The electron trajectories in the temporally- and spatially-varying streaking near-field are numerically integrated for different initial electron velocities. This model enables a detailed reproduction of the main features of the spectrograms, especially the modulation of the spectral width.

Fig. 10.9

a Numerical Simulation of a spectrogram with broad initial energy distribution and a with a decay length of around 40 nm (adapted from [12]). b Spectral reshaping computed with analytical model. Increasing and decreasing field strength lead to compression (red, $E_{final, 1}$) and broadening (blue, $E_{final, 2}$) of the initial energy spectrum (black). The spectra are shifted to a common peak energy for comparison

The basic principle of such spectral reshaping can be analytically described with a model presented in the following. We derive the final kinetic energy $E_{final}$ of a photoelectron propagating in an exponentially decaying near-field $F(x,t)=F_0(t)\exp (-x/l_F)$ (field decay length $l_F$) and in the linear temporal slope of the streaking transient. The final energy $E_{final}$ for a electron with the initial energy $E_0$ at $x_e=0$ is given by the path integral over the spatially varying time-dependent field:

$$\begin{aligned} E_{final}=E_0-\int \nolimits _0^\infty eF(t,x)dx. \end{aligned}$$

(10.7)

For the integration, we parametrize the time as a function of space:

$$\begin{aligned} t(x)=t_e+\int \nolimits _0^x\frac{dx'}{v(x')}, \end{aligned}$$

(10.8)

where $t_e$ is the time of photoemission. In the first order, the velocity v(x) is determined by the kinetic energy in the static electric field $F(x)=F_0(t_e)\exp (-x/l_F)$:

$$\begin{aligned} v(x)=\sqrt{\frac{2}{m}\left[ E_0-el_FF_0(t_e)+el_FF_0(t_e)\exp \left( -\frac{x}{l_F}\right) \right] }. \end{aligned}$$

(10.9)

The temporal evolution of the electric field is introduced by the Taylor expansion of linear order facilitating the integration:

$$\begin{aligned} F(t,x)=\left( F_0(t_e)+ (t-t_e)\cdot \frac{\partial F_0}{\partial t}\bigg |_{t=t_e}\right) \exp \left( -\frac{x}{l_F}\right) . \end{aligned}$$

(10.10)

This expression enables the analytical solution of the integral (10.7), yielding the following term for the final kinetic energy:

$$\begin{aligned} E_{final}=E_0-eF_0(t_e)l_F-e\frac{\partial F}{\partial t}\bigg |_{t=t_e}\frac{\sqrt{2m}l_F^2}{\sqrt{E_0-eF_0(t_e)l_F}+\sqrt{E_0}}. \end{aligned}$$

(10.11)

The average velocity of the electron in the near-field is $v=\left( \sqrt{E_0-eF_0(t_e)l_F}\,+\right. \left. \sqrt{E_0}\right) /\sqrt{2m}$. The energy gain depends on the initial energy and the width of an energy distribution is—under the present assumptions—proportional to the slope of the THz transient, as illustrated in Fig. 10.9.

In this section, we discussed strong-field photoemission in the vicinity of nanostructures and identified sub-cycle electron dynamics at near- and mid-infrared as well as THz frequencies. The extreme case of field-driven electron acceleration is characterized by a spatial adiabaticity parameter $\delta \ll 1$. The field confinement affects all relevant scalings, including wavelength- and intensity-dependent cutoff behaviors and the spectral reshaping of photoelectron spectra in streaking configurations. Some of these features may be employed in the ultrafast and high-resolution characterization of optical near-fields, while others, such as the THz trajectory control, may be utilized in the temporal compression of ultrashort electron pulses for time-resolved electron imaging and diffraction.

3 Extreme-Ultraviolet Light Generation in Plasmonic Nanostructures

In recent years, nanoscale plasmonic light confinement in metallic nanostructures has been extensively used to study numerous linear and nonlinear optical phenomena in strongly-enhanced, inhomogeneous fields [51–53]. Specifically, high intensity enhancements in tailored structures, such as sharp needles, resonant antennas or waveguides, have enabled the investigation of highly nonlinear effects, including strong-field photoemission and acceleration (see Sect. 10.2), in nanostructure-based implementations.

Fig. 10.10

Schematic illustration of plasmon-enhanced gas excitation and EUV light generation by using a bow-tie nanoantennas and b tapered hollow waveguides

Another prominent example for this development is the generation of extreme-ultraviolet (EUV) light in gaseous media by utilizing bow-tie antennas and tapered hollow waveguides [13–19]. Figure 10.10 schematically illustrates the concept of such approaches. In particular, ultrashort low-energy laser pulses are enhanced in nanometric hot spots leading to orders of magnitude higher local intensities, either between the triangular tips (see Fig. 10.10a) or at the very end of the conical taper (see Fig. 10.10b). Noble gas atoms can be injected into these hot spots and get excited in the strong near-fields, resulting in the emission of EUV photons.

The very first reports about a successful realization of these experimental schemes [13, 14] suggested that the produced radiation originates from high-order harmonic generation (HHG) [34, 54, 55] and, therefore, raised high expectations that such compact sources could be harnessed for an eased and efficient generation of attosecond EUV pulses [56–61]. If correct, this scheme would enable the integration of attosecond science into the domain of nonlinear nano-optics, with potential benefits for ultrafast spectroscopy [62, 63] or coherent nanoscale photonic imaging [64].

However, despite particular efforts to reproduce plasmon-enhanced HHG [15, 17–19], an independent experimental verification of this effects was not reported up to date. Recently, our studies revealed that the concept of HHG in nanostructures is fundamentally flawed [15, 17, 18]. Specifically, the tiny nanostructure volume displaying field enhancement is found to prohibit an efficient generation of coherent radiation. Whereas local intensities high enough for the nonlinear excitation and ionization of atoms are reached, we could show that the dominant radiation process is not in fact HHG, but incoherent atomic and ionic fluorescence.

In the following, we present the main results of our comprehensive study on EUV light generation in plasmonic nanostructures. Our experiments and spectral analysis clearly evidence that the EUV emission seen under the reported conditions is predominantly caused by incoherent fluorescence stemming from multiphoton and strong-field gas excitation and ionization.

3.1 Strong-Field EUV Light Generation from Gas Atoms

Currently, the key approach to obtain access to EUV and soft X-ray wavelengths on laboratory scales is the conversion of intense visible or infrared laser radiation in gaseous media. Two distinct excitation processes are responsible for the generation of EUV light in gas atoms. In particular, ultrashort high-energy laser pulses can cause the emission of highly-energetic photons via nonlinear excitation of gas atoms accompanied by either high-order harmonic generation or fluorescence.

3.1.1 High Harmonic Generation

High-order harmonic generation—first realized in the late 1980’ [54, 55]—is the frequency up-conversion of intense laser pulses in gaseous media and has paved the way to lab-scale sources enabling coherent attosecond light pulses at EUV and soft-X-ray wavelengths. In short, the physical mechanism of HHG in a single atom picture can be summarized in a semi-classical simple man’s model [34]: (1) A linearly polarized, ultrashort (pico- to femtosecond) laser pulse with sufficient peak intensity (typically in excess of $10^{\mathrm {13}}$ W/cm$^{\mathrm {2}}$ for visible or infrared wavelengths) ionizes a gas atom, creating a quasi-free electron. A part of the electrons’ wavefunction stays in the remaining ion. (2) The electron is accelerated in the oscillating strong laser field (with frequency $\omega _0$ and intensity I), gaining additional ponderomotive energy $U_p \propto I/\omega _0^2$. (3) The electron returns to its parent ion and causes the emission of high-energetic photons upon re-collision. As a result of this coherent process only radiation at odd multiples (harmonics) of the driving frequencies $\omega _0$ is generated in each half cycle of the laser pulse [65]. The HHG process retains the optical and temporal properties of the driving laser pulses and, thus, has the potential to enable EUV attosecond pulses [66, 67].

Beyond these unique characteristics, it has to be noted that the HHG process is rather inefficient due to a low re-collision probability of the electrons with their parent ions [68]. In general, sufficient photon fluxes are only obtainable by considering the coherent buildup of the harmonic radiation over a large propagation distance in a mesoscopic (tens to hundreds of micrometers or even millimeters) generation volume. Specifically, the high harmonic output scales quadratically with the pressure-length product [55]. However, such a coherent radiation buildup is only obtainable under ideal—phase-matched—conditions that are influenced by propagation effects like dephasing between the driving and the generated radiation or absorption and defocusing [65].

To date, state-of-the-art experimental concepts for efficient HHG typically rely on exposing gas filled capillaries or hollow waveguides to ultrashort, intense laser pulses, either by employing chirped-pulse amplification [69, 70] or enhancement cavities [71, 72] at kHz and MHz repetition rates, respectively.

3.1.2 EUV Fluorescence

In addition to high harmonic radiation, the excitation of gas atoms in intense light fields can also cause fluorescent light emission due to electronic transitions stemming from multiphoton absorption, strong-field ionization and plasma formation [73–75]. The characteristic energy of a fluorescent photon is given by the transitions between different atomic or ionic states. At EUV wavelengths, the photon energies are on the order of several tens of electron volts (eV), which implies that, in the case of excitation with infrared light frequencies (photon energies below 2 eV) multiple photon absorption is required. Hence, the process exhibits an intensity threshold of about 10$^{13}$ W/cm$^2$, which corresponds to the intensities required for HHG.

In contrast to HHG, the emission characteristics of fluorescence are governed by the intrinsically incoherent nature of the generation process with long spontaneous lifetimes of the excited energy levels. Therefore, the properties of the driving laser pulses are lost, and the fluorescent radiation is temporally and spatially incoherent.

3.2 Experimental Methods

The experiments conducted in this study are based on the excitation of rare gas atoms by using plasmonic enhancement of ultrashort low-energy laser pulses in tailored metallic nanostructures (see Fig. 10.10). The particular nanostructures used are resonant bow-tie antennas on sapphire substrates and tapered hollow waveguides milled into gold bulk supports (cf. scanning electron microscope images in Fig. 10.11). Both types of structures are fabricated via focused ion beam (FIB) lithography. Compared to other nano-fabrication techniques, such as electron beam lithography, the used FIB milling represents a convenient method to produce prototype-like structures with very high control during the fabrication process. Therefore, by using the FIB technique and starting with high quality supports, i.e. smooth gold films on sapphire for the antennas and smooth gold platforms for the waveguides, it is possible to optimize the quality of the structures with respect to a high field-enhancement [56] and thermal stability upon laser irradiation [76].

Fig. 10.11

Scanning electron microscope images of the used nanostructures. a An array of bow-tie nanoantennas fabricated on the smooth gold film of 100 nm thickness deposited on a sapphire plate. Inset shows a close-up view. b Tapered waveguides milled into a 10 $\upmu$m thick gold plateau at the end of a conically etched gold wire. Inset shows a top-view of an individual waveguide with elliptical cross section

The experimental setup used to investigate the EUV light generation in the nanostructures is schematically depicted in Fig. 10.12. It consists of a vacuum generation chamber and a flat-field EUV spectrometer. Optical illumination is provided with a 78 MHz Ti:sapphire laser oscillator delivering 8-fs, nJ pulses at a centre wavelength of 800 nm. The generated EUV radiation is collected within a solid angle of $\pm$1.2$^{\circ }$ and spectrally resolved with a flat-field spectrometer. Detection of the EUV photons is realized with an imaging microchannel-plate (MCP) detector in combination with a CCD camera. The inset of Fig. 10.12a shows an exemplary flat-field image for a measurement with xenon gas.

The excitation schemes for the measurements using arrays of bow-tie antennas and single tapered hollow waveguides are depicted in Fig. 10.12b, c, respectively. In both cases, the low-energy pulses are tightly focused on the respective specimen, enabling incident intensities in excess of 0.1 TW/cm$^{2}$. Noble gas atoms are exposed to the plasmon-enhanced near-fields of the particular nanostructures and are efficiently excited (see Fig. 10.10). In the measurements with the bow-tie antennas, the gas atoms are supplied via a nozzle facing the structures, whereas the hollow waveguides are placed in a purged gas cell with a small exit aperture (200 $\upmu$m) for the collection of the generated EUV radiation. Backing gas pressures up to 500 mbar were applied in both scenarios.

Fig. 10.12

Experimental setup for plasmon-enhanced EUV light generation. a Schematic of the vacuum setup consisting of a generation chamber and an EUV flat-field spectrometer. The spectra (xenon fluorescence spectrum in the inset) are detected with a microchannel-plate detector (MCP) in combination with a CCD camera. b, c Excitation schemes for the measurements using b resonant bow-tie antennas and c tapered waveguides

3.3 Results and Discussion

Figure 10.13 shows an exemplary plasmon-enhanced EUV spectrum (solid black curve) obtained from argon using an array of bow-tie antennas (structure iv in Fig. 10.14a). The most prominent spectral features (higher spectral resolution is provided in the second grating diffraction order; red curve) are identified to be fluorescence from neutral and singly ionized argon atoms, with a signal-to-noise ratio larger than 10$^3$. The exact wavelength positions of expected atomic and ionic fluorescence lines are indicated by the triangles [77]. Although conditions for strong-field ionization are clearly reached, evidenced by significant fluorescence signals from singly ionized atoms, no coherent HHG signals were detectable at harmonic orders (H9-H19 wavelengths indicated in the upper x-axis of Fig. 10.13) of the driving laser frequency.

Fig. 10.13

Extreme-ultraviolet spectrum from argon gas atoms excited via plasmonic field-enhancement in an array of bow-tie nanoantennas (see Fig. 10.14a structure iv). Solid black/red curves represent spectra taken in the frist/second diffraction orders of the used spectrometer grating. Fluorescence line positions for neutral/singly ionized argon atoms are indicated by filled/open triangles. Upper scale indicates the positions for expected high-harmonic orders (H9–H19). Graph adapted from [15]

Fig. 10.14

Plasmon-enhanced EUV spectra from argon and xenon with bow-tie antennas. a Scanning electron microscope images of individual bow-tie antennas. Color-coded scale bars 200 nm. b Dark-field scattering spectra of two antenna arrays with different bow-tie arm lengths. c, d EUV spectra recorded with different bow-tie antennas (indicated by the color code see scale bars in a) for c argon and d xenon gas. Graphs and images adapted from [17]

Fig. 10.15

Plasmon-enhanced EUV spectra from noble gases (argon, xenon and neon) using a tapered hollow waveguide (cf. Fig. 10.11b). The argon and neon spectra are up-shifted for better visibility. Dotted gray lines indicates the noise floor at 0.5 count/s/nm. The wavelength positions of the expected fluorescence lines from neutral and singly ionized atoms are indicated by the triangles. Graph adapted from [18]

Figure 10.14 shows an extended set of similar measurements for argon and xenon gas, using several bow-tie geometries (see Fig. 10.14a) with different plasmonic resonances. The resonances are determined by analyzing dark-field scattering spectra, which are shown in Fig. 10.14b for the shortest and longest bow-tie antennas. In agreement with the previous measurements, all spectra shown in Fig. 10.14c, d exclusively originate from fluorescent emission without any signature of coherent high harmonic radiation. Due to the characteristic energy transitions of xenon, the spectral features shown in Fig. 10.14d are completely different compared to those of the argon spectra. Whereas the overall spectral shape is qualitatively comparable for measurements with the same gas species, the strength of the fluorescence signal is governed by the plasmon resonances and the structural quality of the different bow-tie antennas.

Further experiments with tapered hollow waveguides instead of bow-tie antennas yield results which also indicate the dominance of fluorescent over high-harmonic emission in plasmon-enhanced implementations. Figure 10.15 presents the EUV spectra from argon, xenon and neon gas, which were all obtained by using the waveguide structure shown in the inset of Fig. 10.11b. Notably, here, the recorded signal levels are about one order of magnitude larger than those in the bow-tie measurements. This is mainly caused by the approximately 10 times larger field-enhanced generation volume in the waveguide. Furthermore, even clear fluorescent signals from neutral and singly ionized neon atoms are detectable, which is an indicator for very high local field strengths, considering the ionization threshold of 21.5 eV (corresponding to the energy of 13 infrared photons) of neutral neon.

The above results show that strong-field EUV light generation in plasmonic nanostructures is generally possible under the examined conditions. However, the spectra also clearly identify the origin of the EUV emission as incoherent fluorescence stemming from multiphoton excitation and strong-field ionization of the gas atoms. Previously reported high-harmonic emission in such implementations [13, 14, 16] could not be observed, although the experiments were reproduced multiple times using different nanostructure geometries, gas pressures and incident laser intensities. Accurate detection alignment with respect to an efficient collection of coherent signals was proven by surface-enhanced low-order harmonic generation [17, 18, 78] in the bare nanostructures (without gas). Special care was also taken to rule out insufficient local intensities as a possible reason for the absence of HHG signals. A novel means to gauge local fields by employing intensity-dependent relative fluorescence line strengths in the measured spectra was used to determine the intensity enhancements within the nanostructures [17]. Therefore, local intensities of up to 60 TW/cm$^2$, generally sufficient for HHG, were confirmed. Furthermore, the observation of significant fluorescence signals, in particular from singly ionized gas atoms, over several hours of operation evidence that the structures remain intact upon incident laser intensities up to 0.15 TW/cm$^2$ [17].

A physical reason for the lack of HHG in these experiments can be found by considering the nanometric generation volume in the used nanostructures. Specifically, in implementations based on localized surface plasmons, the very small excited gas volume prohibits an efficient coherent build up of the harmonic radiation. In other words, the quadratic scaling of the HHG output with the number of emitting dipoles—or more generally with the pressure-length product—is diminishing the likelihood to observe any significant coherent contributions in the generated EUV emission.

Taking this attribute of the coherent HHG process into account, estimates have shown that the high-harmonic signal levels in these plasmon-enhanced implementations are expected to be several orders of magnitude below that of the simultaneously excited incoherent fluorescence [17, 79]. Thus, it will be very difficult to discriminate potential coherent signals from the dominant fluorescence without employing special detection procedures.

The findings in this Section constitute a clear setback for plasmon-assisted HHG, but they also illustrate that other highly nonlinear effects, such as strong-field gas ionization and fluorescence, or low-order harmonic generation, are present. As a further example for such an effect, we have found a new kind of optical bistability [80–82] in the waveguide experiments. In particular, an intensity-dependent hysteresis in the fluorescence signal strength indicates the formation of a nano-plasma in the waveguides [18]. All of these effects warrant deeper investigation and could help to determine the ultimate frontiers of highly nonlinear strong-field nano-optics. However, the severe limitations for HHG in such implementation calls for alternative approaches to re-open this important research field with respect to coherent EUV signal generation.