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We numerically investigate high-order harmonic generation (HHG) in noble gases in the vicinity of fractal structures of metallic rough surfaces described by the restricted solid-on-solid model. The calculated intensity enhancement factors in the range of 103 enables HHG up to the 50-th order with low pump intensity down to tens of GW/cm2. The increased interaction volume of “hot spots” in the case of grazing incidence of s-polarized pump pulses leads to an efficiency of harmonics in the plateau region of about 10−7.
© 2011 Optical Society of America
High-order harmonic generation (HHG) is an effective method for producing extreme ultraviolet (EUV) radiation and attosecond pulses, and finds applications in a wide range of fields [1]. This process can be explained by the ionization of gaseous atoms in the intense laser field due to tunneling ionization, the acceleration of the freed electrons in the strong electric field and finally the recombination of the electrons with the parent ion, accompanied by the emission of energetic photons. HHG requires intensities higher than 1014 W/cm2 available by complex and expensive femtosecond pulse amplifiers with repetition rates in the range of kHz. Recently a method of high-harmonic generation has been demonstrated that allows a significant reduction of the threshold pump power by using plasmon-assisted field enhancement in bow-tie metallic nanostructures [2]. This arrangement requires only a MHz repetition rate femtosecond oscillator without amplifier. In Ref. [3], a theoretical model for HHG in noble gases near metallic tips or metallic bow-tie-shaped nanostructures has been developed which has shown good agreement with these experimental results.
In the experiment reported in Ref. [2], the achieved field enhancement enabled the generation of up to the 21-th harmonic, but only a rather low conversion efficiency of about 10−9 in the plateau region of the harmonics has been realized by this configuration based on a bow-tie nanostructure. Therefore, the study of other configurations with larger total interaction volume of field enhancement and higher HHG efficiency is desirable. Besides, high effort is demanded for the lithographic manufacturing of nanostructures with curvature radii on the scale of 10 nm as in Ref. [2].
In the present article, we study metallic rough surfaces [4], as an alternative to bow-tie structures, for the realization of plasmonic field enhancement for the HHG process in the vicinity of noble gas atoms. We show that the proposed self-affine fractal structure for a rough surfaces allows intensity enhancement factors larger than 103 and therefore permits corresponding low threshold pump intensities of about 100 GW/cm2 for the generation of harmonics up to the order of 50. For grazing incidence of s-polarized pump pulses, the interaction volume in the HHG process can be increased and the efficiency of HHG in the plateau is in the range of 10−7.
Previously, giant enhancement of local fields on thin metallic rough films of nanometer-sized roughness features in colloidal aggregates or other random nanocomposites has been studied in a large number of papers (see e.g. [4–7]). These results suggest that metallic film surfaces obtained by deposition have self-affine fractal structures. These metallic random surfaces are not characterized by translational invariance but rather by scaling invariance. Therefore surface-plasmon eigenmodes are excited being strongly localized in “hot spots” on subwavelength characteristic roughness scales. The local fields in the “hot spots” on a fractal rough surface are significantly enhanced in comparison with the input fields. Plasmon field-enhancement can be used for a multitude of highly efficient nonlinear processes, e. g. for second and third harmonic generation [8–10], and others. The main advantage of metallic rough surfaces as an inexpensive and strongly field-enhancing element is their applicability in processes which produce very weak signals or have a high threshold.
We describe a random metallic rough surface by the restricted solid-on-solid model [11, 12] which very well describes the self-affine fractal structure typical for metal films produced by thermal evaporation or sputtering of metal onto an insulating substrate. In this model, particles are added one by one on top of the growing surface at random positions. These particles have a form of cubes (so-called voxels) and are added only when the newly created interface does not have steps higher than the size of one voxel.
The enhanced field distribution for a silver fractal rough surface calculated by the discrete-dipole approximation [13,14] is shown in Fig. 1 by red cones together with the 800-nm incident field characterized by the blue cone. The cone directions indicate the polarization and the cone sizes the field enhancement factor. The figure shows that the enhanced field is localized at random positions due to the existence of localized modes on the fractal surface structure. Note that the calculated field has no “hot spots” at the boundaries of the computation domain which could arise by numerical artifacts. The simulation also shows that the physical size of voxels does not affect the maximum enhancement factor and the statistical characteristics of the field distribution.
Fig. 1 (Color online) Structure of the rough surface and enhanced field distribution. The surface structure is illustrated by yellow color and relative magnitudes and directions of enhanced field are shown by red cones. The blue cone shows the polarization of incident field at 800 nm. Voxel size of 1 nm and 50 nm × 50 nm samples were considered.
For the study of HHG in a noble gas in the vicinity of the metallic rough surface, we have to calculate the field enhancement in the spectral range of the pump pulse. Here we consider 10 fs input pulses at central wavelength of 800 nm and calculate the field enhancement distribution A(r,ω) in the spectral range of ∼ 600 nm to ∼ 1000 nm with steps of 12 nm.
In Fig. 2, we show the maximum enhancement factor distribution for wavelengths at 700 nm, 800 nm and 900 nm. One can see that plasmonic hot-spots are formed at different locations depending on the wavelengths which is a typical feature of fractal surfaces [6]. The intensity enhancement factors range up to maximum values of ∼ 5000, and the density of “hot spots” is higher for longer wavelengths, as found also in previous observations [6].
Fig. 2 (Color online) Maximum intensity enhancement factor in different cross-sections for different wavelengths: 700 nm (a)–(c); 800 nm (d)–(f); 900 nm (g)–(i).
Using the wavelength-dependent field enhancement factors A(r,ω) and the pump pulse spectrum E0 (ω), we reconstruct the temporal profile Eenh (t) of the enhanced pulse which is given by the inverse Fourier transform of A(r,ω)E0 (ω). In Fig. 3, we show an example of a pulse shape located at one of the “hot spots”. The figure shows that the enhanced temporal pulse shape is delayed and stretched. This phenomenon is attributed to the dispersion of the plasmonic response on the surface.
Fig. 3 (Color online) Incident pulse (blue dotted line) and enhanced pulse (green line) profiles on the rough Ag surface for one of the realizations of random surface. Maximum field enhancement is more than 40 which corresponds to the instant intensity enhancement of ∼ 2000.
In the calculation of the high harmonics we select only spots with maximum intensity enhancement larger than 100 because other spatial positions do not contribute to HHG. For the calculation of HHG we apply the Lewenstein model [15]. The induced dipole moment for HHG is given by
in the atomic unit system, where is the canonical momentum, is the classical action, A is the magnitude of the vector potential for the enhanced field Eenh at the local positions, and Ip is the ionization energy of the considered atom. For the hydrogen-like atoms, the dipole matrix element is given by d(p) = i23.5α1.25π–1p/(p2 + α), where α is taken to be 2Ip.Next we perform simulations of the HHG spectrum and the efficiency by using the spectral distribution of the field enhancement factors for a 10 fs, 100 GW/cm2 pump pulse with central wavelength at 800 nm and substitute the local fields of this pulse in the “hot spots” into the expression Eq. (1) for the HHG dipole moment of surrounding argon atoms at pressure of 0.18 atm.
The efficiency of harmonic generation is calculated by using the formula η (N) = ω4NT(12πɛ0c3)−1 ∫ |x(ω)|2 dV/Φ, where V is the volume of the space in consideration, N = ω/ω0 is the harmonic order, ω0 is the central frequency of the incident pulse, T is the time window of the numerically performed Fourier transformation, and Φ is the total pulse energy of the incident pulse propagating along the rough surface of the metallic substrate sample. Note that the contributions from the different “hot spots” add up incoherently due to the random nature of the surface, leading to random orientation and phase of the enhanced field. In Fig. 4, we show the spectral distribution of the calculated HHG efficiency for perpendicular incidence of the pump with respect to the base plane of the rough surface as a function of harmonic order N. The spatial beam shape is approximated by a rectangular shape with constant intensity. The efficiency of several elementary samples are presented by the gray curves and the red curve is the averaged efficiency over an ensemble of 20 samples.
Fig. 4 (Color online) The HHG efficiency ηav (N) as a function of harmonic orders N averaged over an ensemble of 20 random samples (thick red line) and for some of the individual samples (thin gray lines).
For harmonics in the plateau the efficiency is in the range of 10−8 ∼ 10−9. In the case of normal incidence of the pump pulses the interaction volume is restricted to the range of the beam spot size. This interaction volume can be significantly increased for grazing incidence of the pump.
In Fig. 5, we show the dependence of the spectral distribution of the efficiency for different incident angles and polarizations on the given fractal surface sample of Ag under the same pump condition as in Fig. 4. For the p-polarized pump pulse with grazing incidence, the main energy is contained in the component perpendicular to the base plane of the sample, leading to low enhancement factors and efficiency [6]. On the other hand, for grazing incident s-polarized pump, the pump field remains parallel to the surface and the interaction length is much longer than in the case of normal incidence. Under this arrangement, the HHG efficiency can be increased because the enhancement factors remain roughly the same for grazing incidence. In the case of s-polarization with incident angle of 89.9 °, the efficiency for harmonics in the plateau is increased by more than a factor of 30 compared with the case of normal incidence and is 1.1 × 10−7 which is approximately 2 orders of magnitude larger than the result in Ref. [2] obtained with a bow-tie gold nanostructure. In Fig. 5(d), the efficiencies are calculated for the harmonic spectrum averaged over the plateau. The figure shows that the efficiency increases with the increase of incident angle for s-polarization and decreases for p-polarization.
Fig. 5 (Color online) The dependence of HHG efficiency on the incident angle for s- (blue curve) and p- (red curve) polarizations of the pump beam. HHG efficiency spectra for incident angles 0 °, 89 ° and 89.9 ° for s-polarization are shown in (a), (b) and (c), respectively. In (d), the HHG efficiencies in the range of plateau are shown.
In conclusion, we have studied plasmon-enhanced HHG in noble gases in the vicinity of metallic rough surfaces with a self-affine fractal structure. The enhanced pulse near the surface is calculated by using the discrete-dipole approximation. The results show that by using these surfaces, it is possible to achieve intensity enhancement factors in the range of 103. We predict that thanks to the corresponding reduction of the HHG threshold, laser pulses with peak intensity of only 100 GW/cm2 and pulse duration of 10 fs can be used for the generation of extreme ultraviolet radiation with harmonic order up to 50. For grazing incidence of s-polarized pump beams, the efficiency for harmonics in the plateau is in the range of 10−7 and 2 orders of magnitude larger than the previous experimental result based on the utilization of bow-tie nanostructures.
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