Generation of isolated attosecond extreme ultraviolet pulses employing nanoplasmonic field enhancement: optimization of coupled ellipsoids

The production of extreme ultraviolet (XUV) radiation via nanoplasmonic field-enhanced high-harmonic generation (HHG) in gold nanostructures at MHz repetition rates is investigated theoretically in this paper. Analytical and numerical calculations are employed and compared in order to determine the plasmonic fields in gold ellipsoidal nanoparticles. The comparison indicates that numerical calculations can accurately predict the field enhancement and plasmonic decay, but may encounter difficulties when attempting to predict the oscillatory behavior of the plasmonic field. Numerical calculations for coupled symmetric and asymmetric ellipsoids for different carrier-envelope phases (CEPs) of the driving laser field are combined with time-dependent Schrödinger equation simulations to predict the resulting HHG spectra. The studies reveal that the plasmonic field oscillations, which are controlled by the CEP of the driving laser field, play a more important role than the nanostructure configuration in finding the optimal conditions for the generation of isolated attosecond XUV pulses via nanoplasmonic field enhancement.

3 energies, they will coherently interfere, leading to pronounced modulations within the corresponding spectral domain. Conversely, a smooth continuous spectrum is indicative of the emission of an isolated XUV pulse [18]. Isolated, attosecond XUV pulses can be obtained via HHG by a number of well-established techniques (see e.g. [19]) including the application of few-cycle laser pulses with well-defined electric fields in combination with spectral filtering of the XUV light from the cut-off region [18]. This technique has led to the generation of the shortest isolated XUV pulses to date of only 80 as duration [20].
Nanoplasmonic field-enhanced HHG, pioneered by Kim et al [5], offers a promising route for the generation of (attosecond) XUV light pulses at high repetition rates. In Kim et al's approach, an array of metallic bow-tie antennas is seeded with Ar gas, and a 75 MHz Ti : Sa oscillator is focused to a peak intensity of 10 11 W cm −2 [5], yielding sufficiently high local fields via plasmonic field enhancement within the gaps of the bow-tie antennas for HHG in Ar. This initial work has prompted the present studies on the potential generation of isolated attosecond XUV pulses at MHz repetition rates by employing nanoplasmonic field enhancement. Such light sources could be used to e.g. achieve extremely high spatial and temporal resolutions in experiments on nanostructured surfaces in the angstrom-(sub)nanometer range, while avoiding space-charge problems [3]. Here, we theoretically investigate the potential generation of isolated attosecond XUV pulses in coupled symmetric and asymmetric ellipsoid nanoparticles. Although nanoplasmonic enhancement is expected to be lower for ellipsoid nanostructures than for bow-tie nanostructures, the reason for choosing ellipsoid nanostructures in this work is twofold: (i) for single ellipsoids, the Maxwell equations can be solved analytically and the solution compared with numerical approaches such as finite-difference-time-domain (FDTD) calculations to validate the accuracy of the theoretical results; (ii) ellipsoids can be produced in large quantities by chemical synthesis [21] and can also be produced in larger aligned arrays on surfaces (see e.g. [22]), so that the fabrication of large arrays with small gap sizes between the ellipsoids can be achieved more easily than using lithography techniques, which are typically used to produce bow-tie nanostructures.
The scheme for HHG by nanoplasmonic field enhancement in coupled ellipsoids is shown in figure 1. Coupled symmetric or asymmetric ellipsoid nanostructures are assembled on a thin sapphire substrate. The structures are back-illuminated by a few-cycle Ti : Sa oscillator pulse with linear polarization along the z-axis. The laser is focused to peak intensities up to 10 11 W cm −2 . Note that higher intensities are likely to damage the nanostructures [23]. The nanostructures are seeded with Ar gas. Nanoplasmonic field enhancement within the gaps between coupled ellipsoids results in local fields reaching intensities >10 14 W cm −2 , sufficient for HHG in Ar. In contrast to conventional HHG, where the XUV light generation is directly driven by the laser electric field waveform, nanoplasmonic field-enhanced HHG is based on the local plasmonic fields. This adds an additional degree of complexity to the HHG process, which opens the opportunity to optimize not only the laser parameters but also the nanostructure in order to shape the resulting XUV pulses.
For laser pulses with durations in the few-cycle regime, i.e. <10 fs for visible light (where 1 fs = 10 −15 s), the electric field waveform, which is given by with pulse envelope E 0 (t), frequency ω and carrier-envelope phase (CEP) φ, can be controlled by the CEP. The CEP is an important degree of control over the response of the local plasmonic fields that are induced by few-cycle pulses [24,25]. Although the influence of the CEP on Geometry of a coupled ellipsoid dimer that is back-illuminated by the driving laser field. Laser polarization is along the z-axis, and maximum field enhancement is located within the gap in the dimer as shown. For elements with a major axis length of 100 nm, separated by 5 nm, this field enhancement (with respect to the normalized laser field) is approximately 40 in the center of the gap.
the HHG process in the vicinity of nanostructures has not been studied to date, a considerable influence of the CEP including the potential generation of isolated attosecond light pulses is expected. This work theoretically investigates the role of both the nanostructure configuration and the CEP of the driving few-cycle laser field in the nanoplasmonic field-enhanced HHG in order to identify the optimal conditions for the generation of isolated attosecond XUV pulses. While analytical solutions for the plasmonic response of spherical nanoparticles are straightforward, we have derived here an analytical solution of the Maxwell equations for a single nanoellipsoid (see section 2) and we compare the response with numerical results obtained from FDTD calculations [26] to validate the accuracy of the theoretical results obtained from such numerical calculations. The influence of the coupled nanoparticle configuration, where symmetric and asymmetric ellipsoid dimers have been tested, and of the CEP of the excitation pulse on the induced plasmonic field is evaluated in section 3. The corresponding XUV spectra are determined by solving the time-dependent Schrödinger equation within the single-active-electron approximation [27]. Filtering of the cut-off region, similar to the conventional scheme used for producing isolated attosecond pulses, is applied to determine whether isolated attosecond pulses can be obtained for various structural and CEP parameters. Section 4 gives a summary of the results and their implications.

Analytical solution for ellipsoid nanoparticle in a uniform external field
Consider a sub-wavelength-size metal spheroid subjected to an optical wave. In this case, the problem becomes quasi-static, i.e. the wave can be replaced by a uniform electric field. An analytical solution of this problem for a monochromatic field with a given frequency ω is well known [28]. Here, for the sake of completeness, we will briefly summarize it. We will apply this analytical solution to our problem of the temporal dynamics of the ultrafast local fields and the HHG processes induced by the local fields in the vicinity of such a metal spheroid.
We consider an ellipsoidal nanoparticle with semiaxes a b c oriented along the x, y and z axes respectively, of the coordinate system. This nanoparticle consists of a metal with permittivity (dielectric function) ε m (ω), which is embedded into a dielectric medium with permittivity ε d . It is subjected to an external uniform electric field E 0ω = E 0ω e −iωtê i , where i = x, y, z andê i is a unit vector along the corresponding axis.
Consider the external (excitation) field with polarization in the z-direction. Then the scalar potential ω on the z-axis is given by [28] where 0ω = −E 0ω e −iωt z is an external optical-field potential oscillating with optical frequency ω, is Bergman's spectral parameter, s z is an eigenvalue of the problem (see the definition below after equation (4)), and ξ is an elliptical coordinate satisfying −c 2 < ξ < ∞ (on the z-axis, ξ = z 2 − c 2 ). The surfaces ξ = const are confocal ellipsoids, and the ellipsoid corresponding to ξ = 0 is the surface of the nanoparticle itself.
The functions L i (ξ ) are generally defined as where a x = a, a y = b and a z = c. We introduce the eigenvalues of the problem corresponding to the three principal axes as s i = L i (0). Note that the spectral parameter s(ω) contains information about the material properties of the system, while the eigenvalues s i depend only on its geometry [29,30], which is a general property of the quasi-static problems. The complex frequencies of surface plasmons (SPs) can be found from the equation where ω i is the corresponding real frequency and τ i = (2γ i ) −1 is the SP lifetime [31]. In good plasmonic metals (gold, silver, etc), the SP resonance quality factor Q = ω i /γ i is high, Q 1. Then the resonance frequencies and spectral widths are determined by the following expressions [32] (see also figure 2(a)): Note that SP lifetimes [31] are geometry independent and are determined only by the material properties. Consider a prolate spheroid for which a = b. In such a case, the SP eigenvalues depend only on the spheroid aspect ratio R = a/c 1 (see also figure 2(b)),  (7) and (8): note that integral and algebraic representations match).
For the convention used here, the polarization of the excitation field is directed along the z-axis (the major axis) of the ellipsoid, and the total electric field corresponding to the potential of equations (2) and (3) is According to equations (9)and (10), the total field at the surface of the sharp tip of this spheroid is given by where it is taken into account that ξ | tip = 2aê z and L z = −1/(2a 2 ). The analytical expressions for fields mentioned above in this section are given for monochromatic waves as functions of ω. Later in this paper, we use ultrashort pulses with the corresponding choice of the excitation-field amplitude E 0ω as a function of ω. The local fields are computed in the time domain by the conventional Fourier transform,

Analytical calculations and numerical simulations for a single gold ellipsoid
In order to test the model, the plasmonic field enhancement for a single, solid gold nanoellipsoid was determined analytically using the equations described in the previous section and compared with the results obtained numerically by the FDTD method [26] employing XFDTD (version 7, Remcom). The real and imaginary parts of the wavelength-dependent dielectric function for gold for the calculations were determined using the modified Debye model [33]. The nanoparticles are surrounded by vacuum (index of refraction, n = 1). We calculated a total  Configuration of (a) single and (b) coupled symmetric and (c) asymmetric gold ellipsoids with semi-axes c (major) and a (minor); g is the distance between the coupled ellipsoids. In the asymmetric case, the minor axis of the first ellipsoid (2a) is kept constant, while that of the second ellipsoid (2a 2 ) is varied. In all cases, the length of the major axis is 2c = 100 nm. The excitation laser field is polarized parallel to the symmetry axis of the nanostructures (i.e. in the directionê z ).
volume of 1 × 1 × 1 µm 3 and used a perfect matched layer boundary with a thickness of 500 nm to terminate the volume [34]. To ensure an accurate description of the near field of the particles, the three-dimensional mesh in the calculations had a unit size of 0.5 × 0.5 × 0.5 nm 3 within the gap between the ellipsoids and up to 1.0 × 1.0 × 1.0 nm 3 otherwise. By employing variable mesh parameters, it was ensured that the results shown here are not distorted by the mesh. The polarization direction of the excitation field is horizontal (along the major axes of the ellipsoids).
A gold nano-ellipsoid with the full lengths of the minor and major axes of 2a = 16.7 nm and 2c = 100 nm, respectively, was chosen as shown in figure 3(a). These dimensions are chosen such that the structure is near resonant with the 800 nm excitation laser field. In all calculations the peak intensity of the laser is 1 × 10 11 W cm −2 (corresponding to a field of 6.14 × 10 8 V m −1 ), which lies below the typical damage threshold of Au nanoparticles [23]. Because a recent work has demonstrated the generation of <4 fs pulses from Ti : Sa oscillators [35], laser pulses with a duration of 3.5 fs and a central wavelength of 800 nm were employed in the present theoretical study.
The enhanced plasmon fields calculated 2.5 nm from the apex of the structure with both the introduced analytical and numerical FDTD methods are shown in figure 4. There is good agreement between the analytical and numerical results with near-identical overall exponential field decays showing similar field enhancements. There is a phase mismatch between the two plasmon fields as a result of the assumptions made in the two theoretical approaches. In the numerical approach, the wavelength dependence of the metal's permittivity is determined using the modified Debye model (see above), whereas the analytical approach uses experimental values for this parameter [36] and can therefore be assumed to provide more accurate results. The phase of the plasmonic field is expected to be of importance for the potential generation of single-or multiple attosecond light pulses, similar to the role of the CEP in conventional HHG [18].
The maximum field enhancement arising from the single ellipsoid is calculated to be 13 with respect to the normalized laser field, corresponding to an overall local intensity of 1.7 × 10 13 W cm −2 assuming a peak laser intensity of 1 × 10 11 W cm −2 . As this is an insufficient intensity for HHG, the following section considers coupled ellipsoids, which will be shown to result in significantly higher field enhancements [37].

Nanoplasmonic field-enhanced high-harmonic generation in coupled ellipsoids
In this section, the plasmonic field-enhanced HHG is investigated for coupled ellipsoids. Symmetric and asymmetric nano-ellipsoid configurations were tested, and the CEP of the laser pulse was varied to explore the influence of these parameters on the resulting HHG spectra and potential XUV pulse durations obtained from filtering out a portion of the cut-off region of these spectra. The configurations of the coupled ellipsoids are shown in figure 3.
The symmetric configuration, figure 3(b), consists of two ellipsoids with the same minor and major axis lengths, 2a = 16.7 nm and 2c = 100 nm, respectively. The aspect ratio of these particles is R = R 2 = a/c = 0.17. Asymmetry is introduced into the system by varying the minor axis of one of the ellipsoids; see figure 3(c). We introduced asymmetries by choosing aspect ratios for the second ellipsoid of R 2 = 0.25 and R 2 = 0.33. The separation between the coupled ellipsoids was set to g = 5 nm. The enhanced plasmonic fields were determined at the center of the gaps between the coupled ellipsoids.
The near-field amplitude spectra for the three coupled nanoellipsoid configurations that were calculated via the FDTD method are shown in figure 5. All these configurations are slightly off-resonant with respect to the driving laser field to facilitate ultrafast responses. The spectra exhibit one dominant peak at 882 and 850 nm in the symmetric and asymmetric configurations, respectively. For the asymmetric configurations, the reduced spatial symmetry leads to the appearance of an additional peak, which is blue-shifted in wavelength by λ = 50-100 nm; there is also an appreciable blue shift of the original resonance by λ = 30 nm.
This symmetry effect has a transparent physical interpretation. For the center-symmetric systems, there are two longitudinal SP modes, i.e. modes with the dipole of the individual ellipsoids directed along the symmetry axis. These can be classified by parity as symmetric and antisymmetric (or even and odd with respect to the z-component of the electric field). The even mode, where the dipoles of both the ellipsoids oscillate in phase, is bright (dipole-type) and strongly red-shifted from the SPs of the constituent nanoellipsoids, because the opposite polarization charges face each other across the gap between the nanoellipsoids, which reduces the electrostatic energy and leads to the red shift of the mode. In contrast, the antisymmetric mode possesses the like charges separated by this gap and is therefore blue-shifted. Due to the fact that the dipoles on the two nanoellipsoids oscillate completely out of phase (corresponding to the quadrupolar symmetry), this antisymmetric mode is (in the dipole approximation) dark and does not show itself in the far-field optical spectra.
For the present slightly asymmetric systems, there are still two longitudinal modes that originate from the symmetric and asymmetric SPs of the symmetric system. They are still approximately (although not precisely) symmetric and antisymmetric: the individual dipoles oscillate almost in phase or out of phase. However, these dipoles are not equal and do not cancel each other out exactly in the antisymmetric case, resulting in a nonzero combined dipole of the nearly antisymmetric mode that is not dark now and manifests itself in the spectra shown in figure 5. This mechanism of broken symmetry is well known in plasmonics; see e.g. [38].
The corresponding plasmonic field enhancements for the three configurations were determined for two CEP values of φ = 0 and φ = π/2 and are shown in figures 6(a) and (b), respectively.
Maximum field enhancements for the three aspect ratios of R 2 = 0.17, R 2 = 0.25 and R 2 = 0.33 are found, with 43.5, 43.3 and 38.5, respectively, corresponding to intensities of 1.89 × 10 14 , 1.88 × 10 14 and 1.48 × 10 14 W cm −2 , assuming an incident laser intensity of 1 × 10 11 W cm −2 . These plasmonic field intensities are sufficient for HHG in argon. The plasmon field for the symmetric configuration and laser-pulse CEPs of 0 and π/2 (figures 6(a) and (b), respectively) undergoes a rapid increase to a maximum of approximately 43. As there is essentially only one bright plasmon mode within this configuration, the first few cycles of the enhanced field show a direct imprint of the driving laser field followed by a smooth exponential decay with a time constant of 10.2 fs. The enhanced fields for the asymmetric configurations also display a rapid increase to their maximum fields, but in contrast to the symmetric configuration the successive dynamics of the local field amplitude shows a strongly modulated exponential decay. The modulation arises from the beating pattern of the two different plasmon modes of these structures. The dependence of the plasmon fields on the laser CEP is more subtle, but close inspection reveals that a shift in the laser CEP leads to a shift in the phase of the plasmonic field oscillations.
The dependence of the plasmon fields on the shape of the nanostructures and on the CEP of the driving laser field has a direct impact on the high-harmonic spectra. The number of peaks and cycles of the enhanced field with sufficient strength to drive the HHG process, and in particular sufficient strength to produce photons near the cutoff of the XUV spectrum, affects the number of attosecond pulses that are produced when a spectral window from the cutoff of the spectra is selected. In laser-driven XUV sources [20], isolated attosecond XUV pulses can be generated if HHG within a (sufficiently broad) spectral region can be restricted to originate from a single ionization and recollision event. In this respect, the asymmetric nanostructures may exhibit an advantage with respect to symmetric structures as the field oscillations can be further modified and controlled by the temporal beating between different modes.
The plasmonic fields were subsequently used to determine the corresponding highharmonic spectra by solving the time-dependent Schrödinger equation within the single-activeelectron model [27]. There are a number of features that are common to all of the spectra shown in figure 7. These spectra show a series of peaks corresponding to the odd harmonics of the driving laser field. The large peak centered at 1.55 eV is the photon energy of the laser with the two subsequent peaks at 4.65 and 7.75 eV corresponding to the third and fifth harmonics. An extended region where the high-order harmonics are of similar intensity (the plateau region) ranges from about 17 to 58 eV for the symmetric structure with R 2 = 0.17 and the asymmetric structure with R 2 = 0.25 and ranges from about 17 to 53 eV for the asymmetric structure with R 2 = 0.33. Within the plateau regions, the XUV spectra exhibit almost equal spectral intensities for the three structures and two different CEP values. Beyond the plateau region, the XUV spectral intensities fall off rapidly with photon energy (cut-off region) and are strongly dependent on the CEP and nanostructure configuration. The cutoff with the highestenergy XUV photon E max is classically given by

12
where I p is the ionization potential of the atom and U p is the ponderomotive potential of an electron in the field given by where e and m are the elementary electronic charge and mass, E is the effective field and ω is the angular frequency. The high-harmonic spectra for the symmetric structure with R 2 = 0.17 and the asymmetric structure with R 2 = 0.25 exhibit approximately the same cut-off energy with E max ≈ 68 eV. A lower cut-off energy of E max ≈ 60 eV is observed for the asymmetric structure with R 2 = 0.33, which arises from the significantly lower maximum field enhancement factor of 38 corresponding to a lower effective laser peak intensity.
In order to determine the feasibility of producing an isolated attosecond pulse via nanoplasmonic field enhancement from the studied nanostructures, a portion of the cutoff of the spectra was selected by imposing a 5 eV full-width-at-half-maximum (FWHM) Gaussian filter within the cut-off region. For the symmetric structure with R 2 = 0.17 and the asymmetric structure with R 2 = 0.25, this filter was centered at 60 eV, whereas for the asymmetric structure with R 2 = 0.33, it was centered at 53 eV. The filtered spectral data were Fourier-transformed to obtain the temporal XUV pulse profiles shown in figure 8. It is evident that the CEP of the laser pulse has a much stronger impact on the produced attosecond pulses than does the nanostructure configuration. As outlined above, the CEP of the laser field determines the absolute phase of the plasmonic field, which can be shifted via a CEP shift. The phase of the plasmonic field in turn determines whether a single ionization-recollision event or many such events produce an XUV photon of a certain energy. More generally, a modulated XUV spectrum indicates the contribution from many such events, while a smooth continuum is related to an isolated ionization-recollision event [18]. Accordingly, selecting a spectral window from an unmodulated, smooth cutoff leads to in the generation of isolated attosecond pulses (achieved here for e.g. R 2 = 0.17 and φ = 0 and for R 2 = 0.25 and φ = π/2). It should be noted that the phase of the field oscillations obtained via FDTD calculations is influenced by the particular model chosen to describe the wavelength-dependent permittivity (see section 2). Calculated CEP values for the production of isolated attosecond pulses might thus differ from experimental ones, but can be compensated for by a CEP-shift of the laser field. A more accurate prediction of the absolute phase of nanoplasmonic fields requires systematic future theoretical work applying various numerical models, the results of which should be compared with experimental data, where high-harmonic spectra from nanoplasmonic enhancement were obtained for various laser CEPs. The implementation of such experimental studies for various nanostructures is currently being pursued in our laboratory.

Conclusions
HHG via nanoplasmonic field enhancement has been theoretically investigated for coupled gold ellipsoid nanostructures. The feasibility of generating an isolated attosecond pulse by selecting a portion of the cutoff of the XUV spectrum through spectral filtering has been studied for coupled symmetric and asymmetric nano-ellipsoid configurations and for different laser CEPs using a 3.5 fs excitation pulse. We have analytically derived the field enhancement for a single ellipsoid and compared the result to FDTD calculations, showing excellent agreement in the obtained field amplitudes, but deviations in the absolute phase of the resulting plasmon field oscillations. Sufficient field enhancement for the HHG process while avoiding laser damage of the gold nanostructures is obtained for the coupled symmetric and asymmetric ellipsoids. While the nanostructure configuration does not significantly influence the ability to find the conditions for generating an isolated attosecond pulse, the absolute phase of the enhanced plasmonic field is an important parameter that for a given nanostructure determines whether an isolated attosecond pulse or multiple pulses are generated. Note that these conclusions are drawn on the basis of the current modeling of the HHG process, which did not include spatially dependent fields during the electron excursion. Such computations are subject to future work. Similarly, the accurate prediction of the absolute phases of plasmonic fields remains a challenge for future theoretical work, which shall be compared with experimental studies expected to emerge in the near future. For a realistic comparison between experiment and theory, it should be considered that a measured high-harmonic spectrum will be the sum of the contributions from multiple emitters in an array. The HHG spectra and the visibility of CEP effects might be affected by imperfections in the fabrication process of the nanostructures.
Although we have studied the feasibility of the production of isolated attosecond pulses via nanoplasmonic field enhancement for coupled ellipsoid nanostructures, we expect the results to be of importance also for other coupled nanostructures such as coupled symmetric and asymmetric spheres, cubes or the bow-tie-type nanostructures that have been used in the pioneering experimental work of Kim et al [5]. This is supported by recent work on symmetric and asymmetric bow-tie nanostructures [39]. Our studies indicate the possibility of generating single-attosecond XUV pulses at low laser intensities and MHz repetition rates. Such novel laser sources would find many applications including measurements on spatio-time-resolved electron dynamics on nanostructured surfaces using photoelectron emission microscopy [3] or ultramicroscopy [40].