1. INTRODUCTION

Frequency conversion extends the spectral region accessible by lasers from the mid-infrared toward soft x rays. To obtain short wavelength in the vacuum-ultraviolet (VUV) or extreme ultraviolet (XUV) regime, usually gases are applied as nonlinear media. Compared to bulk media, they offer a broader transparency window toward short wavelength and can be exposed without damage to higher intensities (e.g., in the case of rare gases). However, gases have typically rather small nonlinear susceptibilities and, hence, yield only small conversion efficiencies, even at larger pressures on the order of one atmosphere. It is an obvious idea to increase the conversion efficiency by increasing the interaction length, i.e., by confining both the atomic medium and the laser beam in a long, hollow-core optical waveguide [1]. The simplest type of dielectric waveguide is the glass capillary, used since the 1990s in pulse compression [2] and phase-matched harmonic generation [3] in rare gases. Here, the waveguide dispersion can be cancelled by the pressure-tunable dispersion of the gaseous medium to yield a phase-matched generation of (high) harmonics. In recent years, hollow-core photonic fibers have also paved the way toward even higher intensities and strong pulse compression, providing negligible transmission losses and negative dispersion [1]. We note that apart from the dispersion properties, the transversal mode structure and propagation properties are quite similar in conventional and photonic waveguides with cylindrical symmetry [4].

Another straightforward approach to increase nonlinear susceptibilities is to tune the driving laser to atomic (multiphoton) resonances. This has been well known for decades and was extensively applied in nonlinear optics at moderate intensities [5,6], e.g., driven with rather long laser pulses in the nanosecond (ns) domain or ultrafast pulses at low energies. However, there is still only a limited amount of work on resonance enhancements, applied for high-intensity frequency conversion. The role of resonances and atomic structure in strong field interactions is still “scarcely explored” [7]. Early experiments on high harmonic generation (HHG) in xenon [8] and argon [9] revealed some evidence for such resonance structures. The strength of the effect depended upon the laser intensity. It was found theoretically that the resonances occur when a highly excited state is dynamically shifted by the driving laser, such that it becomes multiphoton resonant with an individual harmonic from the HHG spectrum [10,11]. Related experiments on HHG in ionic targets from laser ablation on solid surfaces revealed quite clear proof for resonance enhancements of individual harmonics [12]. Recently, we performed the first experiments on resonance enhancement of lower-order harmonics via a multiphoton resonance in an atomic jet, already yielding large enhancements by an order of magnitude [13].

The following work deals with a combination of both approaches to increase the conversion efficiency of harmonic generation in a gas, i.e., tuning to a multiphoton resonance and confining light and medium in an optical waveguide. In particular, we will demonstrate optimal phase-matching conditions in a gas-filled capillary also at inevitable dynamic level shifts at high intensities. Moreover, we discuss the significant contribution of higher-order transversal modes to the total conversion efficiency, and the contribution of cascaded frequency conversion processes (which were already applied for efficient frequency conversion in waveguides [14], but required pumping with a bicolor field). Finally, we show additional signal enhancements by buffer gas admixtures, i.e., another standard strategy to enhance conversion efficiencies in gases, which was recently also used in HHG to control ionization [15]. The experimental data are confirmed by extended numerical simulations.

2. EXPERIMENT

The experiments are conducted in argon atoms, confined in a UV fused silica (UVFS) capillary. In our experimental setup (Fig. 1), ultrashort (picosecond) laser pulses with tunable fundamental center wavelength ${\lambda }_F=506\dots 520\mathrm{nm}$ drive the atoms close to a five-photon transition between the ground state $3p^6$ (${}^1{S}_0$) and the excited state $3p^{5}4s{}^2{[1/2]}_1$ of argon, yielding resonantly enhanced fifth-harmonic VUV radiation at ${\lambda }_{5H}=101\dots 104\mathrm{nm}$ (see Fig. 2). The unperturbed five-photon resonance is expected at 524 nm. The (picosecond) laser pulses are generated from a mode-locked Ti:Sa oscillator (MIRA 900P, Coherent), driving an optical parametric oscillator (OPO) (OPO FAN, APE) with intracavity frequency doubling. The output of the OPO is amplified in a homemade four-stage dye amplifier to yield pulses with a typical pulse energy of several 100 μJ. For the following experiments, we tuned the laser system with pulse duration of 1.2 ps (FWHM) and bandwidth of 0.86 nm (FWHM) at a repetition rate of 20 Hz. Due to spectral gain differences, the pulse length and bandwidth exhibit a statistical spread of 0.2 ps and 0.12 nm, respectively. For details on the laser setup, see [13]. We characterize the temporal pulse profile in a homemade polarization-gated frequency-resolved optical gating (FROG) setup by 2D phase retrieval. Multistage spatial filtering is applied to obtain a laser beam profile with $M^2<1.2(1)$, which is important for efficient coupling into the fundamental spatial mode of the waveguide. (Whenever we state values in parentheses, we give the (combined) experimental standard uncertainty, unless otherwise noted.)

 

Fig. 1. Experimental setup for phase-matched harmonic generation inside the 55 mm long waveguide (blue tube) with two holes for gas inlet each 1 cm from the waveguide end. The filter F1 as well as the mirrors M1 and M2 are inserted to image the laser focus on a camera chip (CMOS1) and the waveguide end onto CMOS2. The inserted plot shows the calculated static core gas pressure versus the chamber pressure $p_c$.

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Fig. 2. Level scheme of argon with the relevant atomic levels and the tuning range (gray area) of the fundamental with respect to the $4s$ energy levels.

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The waveguide is a UVFS capillary with a length of 55 mm and a nominal bore radius of 50 μm. The capillary is processed by selective laser-induced etching [16] to provide two gas supply channels of 30(1) μm radius, each 10 mm from its extends. This enables a long section of constant pressure (see inset in Fig. 1) for phase matching in the capillary and small perturbation of wave guiding. After the etching, we find the bore to be slightly elliptic with an average radius $a=52(2)\mathrm{\mu m}$ and an aspect ratio of 1.06(3) (as indicated in the upper images of Fig. 3 by a white dashed line). The capillary is airtightly mounted, yielding a chamber of static, tunable gas pressure, measured by a piezo gauge (Pfeiffer APR250) or a Pirani gauge (Edwards PR10-K) for pressures below 1 mbar. We note that the gas flow in the tight waveguide cannot be modeled by a continuum flow approach, but requires the Burnett equation [17] to calculate the static pressure (i.e., the particle density) inside the waveguide along the $z$ axis. We use the approach of Yang and Garimella [18] and account for the dynamic pressure at the inlet and outlet of the capillary to compute the static pressure versus longitudinal position, as displayed in Fig. 1. We employ differential vacuum pumping at either end of the capillary to avoid self-focusing/defocusing and reabsorption of generated VUV radiation.

 

Fig. 3. Beam intensity profiles of the fundamental laser pulses at ${\lambda }_f=512\mathrm{nm}$, before (top left) and after the waveguide (top right). Red color indicates large intensity, purple, low intensity. Distances are normalized to the average bore radius $a=52\mathrm{\mu m}$. The white dashed lines show the capillary aperture. The lower plots show cuts in horizontal (gray, solid dots) and vertical (black, hollow squares) direction through the 2D intensity profiles. The colored lines show simulations with the mode amplitudes as deduced from the experimental data (blue line), or with the slightly corrected mode amplitudes (green line); compare Table 1. Due to the slight ellipticity of the waveguide, the horizontal intensity distribution at the exit is shallower, which enhances the peak intensity compared to the simulation without taking the ellipticity into account.

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We focus the fundamental pump beam with a plano-convex lens system of focal length 230 mm into the waveguide, yielding an almost perfectly round beam profile with a $1/e^2$ radius of $w=32(2)\mathrm{\mu m}$ at the entrance of the waveguide (see Fig. 3, top left). This yields a ratio $w/a=0.61$, very close to the optimum of $w_{\text{opt}}/a\approx 0.64$, which is expected to give the highest coupling efficiency into the lowest-order fundamental hybrid modes ${EH}_{1m}$ of the waveguide [19] (see Appendix A, Section 1 for the definition). Figure 3, top right, shows the beam profile at the exit of the waveguide. As comparison with the input beam profile (Fig. 3, top left) shows, the profile broadens and changes shape. We find that the beam profile at the exit also varies with the pump wavelength, which we attribute to wavelength-dependent mode beating in the waveguide. Hence, there are some contributions of higher-order modes $E{H}_{1m}$, with $m>1$, to the beam propagation in the capillary. From the measured beam profiles at the entrance, we determine the amplitudes of the $E{H}_{1m}$ modes using the overlap integral Eq. (A7), derived in Appendix A, Section 2, assuming an electric field with real part only (i.e., a plane phase front) at the focal plane. Table 1 lists the relative mode contribution in the input pump beam profile, given by fractions of the power coupled into the modes versus total power. While almost 95% are in the lowest mode $E{H}_{11}$, less than 2% are in $E{H}_{12}$, less than 1% in $E{H}_{13}$, and far less than 1% in $E{H}_{14}$. The asymmetric error bars include the dispersion of the calculated power decompositions for all wavelengths used, as well as the effect of self-focusing in the 5 mm thick window (BK7 glass) in the beam path (see Fig. 1), possibly shifting the waist away from the waveguide entrance and enhancing the amplitude of the $E{H}_{12}$ mode versus $E{H}_{11}$ and $E{H}_{13}$.

Table 1. Modal Power Decomposition of the Input Pump Beam Profile^a

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In our simulations of harmonic generation in the waveguide, we take modes up to $E{H}_{14}$ into account. As we will discuss below, even these small contributions of higher-order waveguide modes in the driving pump beam result in a (on first glance) surprisingly large contribution to the generated harmonics. We note that, for the simulations, we slightly varied the mode amplitudes, as derived from the measured beam profile, to better match the simulations with the experimental data discussed in the following section. However, as Table 1 and the overlapping blue and green curve in Fig. 3 show, the required corrections are tiny and well within the range of the possible self-focusing. We estimated the effect and found that it can well justify the small required corrections of the mode contributions.

The focused fundamental beam drives harmonic generation in the argon-filled hollow-core waveguide. A homemade VUV spectrometer with a 1200 l/mm concave grating (Horiba 52200250) serves to separate the generated fifth harmonic from the fundamental radiation and detect it on an electron multiplier tube (Hamamatsu R595). We monitor the fundamental pulse energy on a fast, calibrated photodiode (Becker&Hickl PDI-400) and process only data points in an intensity window of $\pm 3\%$ around an average fundamental intensity.

3. NUMERICAL SIMULATION

We briefly summarize now our approach for the numerical simulations of multimode frequency conversion in the capillary. For details we refer the reader to Appendix A. We numerically solve the set of differential equations for frequency conversion in the waveguide, for the lowest five ($m=1-5$) modal amplitudes $E_{j,m}$ at the third and fifth harmonic $j=3,5$, as well as the four lowest modes of the fundamental for all positions in $z$. In the equations, we use the pressure-dependent nonlinear susceptibilities and propagation constants, computed for a given input peak intensity and fundamental pump frequency ${\omega }_P$.

To account for the temporal shape of the laser pulse, we employ an adaptive step-size temporal integration with at least five base points in fundamental intensity to yield the pulse energy of a quasi-cw laser pulse.

We determine the error of the temporal integration to be well below 10% when compared with a temporal integration with more than 60 base points in fundamental intensity. To limit the computation time, we neglect pulse propagation effects. This is well justified for the major part of our parameter range (see Appendix A, Section 5 for details).

To account for the spectral width of the fundamental pulses with a residual chirp of about 1 THz/ps, as determined by our FROG setup, we calculate a set of “quasi-cw” pulse energies at five fundamental wavelengths, centered at ${\lambda }_P$, with a step size of 0.25 nm and average the data to obtain the simulation results for pulsed excitation at the central wavelength ${\lambda }_P$. As the spectral components are temporally separated, their interaction is expected to be small, as long as the detuning to the resonance is bigger than the wavelength step size.

4. RESULTS

A. Pressure and Intensity Dependence of Harmonic Generation

The efficiency of frequency conversion in a nonlinear optical medium is limited by phase mismatch, which is due to dispersion, i.e., different phase velocities of the fundamental and generated harmonic radiation. As is well known, gas-filled capillaries permit balancing pressure-dependent (typically negative) gas dispersion and (positive) waveguide dispersion [20] to enable perfect phase matching. Hence, the frequency conversion efficiency will reach a maximum for an appropriately chosen (gas) pressure (which we will term “phase-matching pressure” in the following).

Figure 4 shows the results of measurements in which we varied the gas pressure and monitored the generated fifth-harmonic pulse energy. We took data sets at three different fundamental pump intensities. For any pump intensity, the harmonic signal reaches a maximum at a certain range of gas pressures, providing phase-matched conditions. However, the phase-matching pressure varies with laser intensity: At higher intensity, the phase-matching pressure is substantially lower (i.e., with increasing intensity, the signal maxima shift to the left in Fig. 4).

 

Fig. 4. Fifth-harmonic pulse energy versus chamber gas pressure at a fundamental pump peak intensity of $4\mathrm{TW}/{\mathrm{cm}}^2$ (blue dots), $7\mathrm{TW}/{\mathrm{cm}}^2$ (red dots), and $8.3\mathrm{TW}/{\mathrm{cm}}^2$ (green dots). The fundamental wavelength is ${\lambda }_F=512\mathrm{nm}$. Solid lines show numerical simulations averaged over 1 nm spectral width. Data and simulations are normalized to the peak harmonic yield at $7\mathrm{TW}/{\mathrm{cm}}^2$.

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Indeed, if we would consider harmonic generation far off resonances, the variation of the phase-matching pressure with the driving fundamental intensity is negligible, at least until strong ionization promotes plasma dispersion. However, in the vicinity of a resonance, we have a steep dispersion, i.e., the refractive index changes quickly with the laser frequency. Strong dispersion is also possible when the laser frequency remains fixed, but the atomic resonance frequency shifts.

Such variations of the atomic resonance frequency are inevitable at intense light-matter interaction, which drives large AC Stark shifts even under conditions of small ionization. Stark shifts $\mathrm{\Delta }W$ increase linearly with the pump intensity $I_P$. In principle, the total shift could be calculated by adding up nonresonant contributions of all bound and continuum states [21], which is a challenging task. As an approximation, the Stark shifts of highly excited atomic states (i.e., weakly bound electrons) can be estimated by the ponderomotive potential $\mathrm{\Delta }W\approx {\mathrm{\varphi }}_P=\frac{e^2}{4m_e{\omega }^2}{I}_P$ of a free electron in a laser field [22]. At a peak intensity of $8\mathrm{TW}/{\mathrm{cm}}^2$, we get a shift in the range of ${\mathrm{\varphi }}_P\approx 47\mathrm{THz}$. In an earlier experiment involving the $4s$ states in argon, we already confirmed this range [13]. Fitting our numerical simulation to the data in Fig. 4, we determine a net Stark shift (averaged over the transversal intensity profile) $\mathrm{\Delta }W\approx 0.85$ ${\mathrm{\varphi }}_P$ across the fundamental beam diameter, i.e., very close to the simple approximation.

In our resonantly enhanced harmonic generation scheme, the Stark shift drives the $4s$ level in argon toward higher energies. As a consequence, the refractive index of argon decreases for the generated, single-photon resonant VUV radiation at ${\lambda }_{5H}=102.4\mathrm{nm}$, while it remains almost constant at the single-photon far off-resonant pump wavelength ${\lambda }_F=512\mathrm{nm}$. To compensate the (positive) waveguide dispersion, we require smaller gas pressure of the (negative dispersive) argon, compared to frequency conversion at lower intensities and smaller Stark shift. This explains the shifts in the phase-matching pressure in Fig. 4.

As an important consequence of the Stark shifts, at high laser intensity, the optimal particle density is smaller compared to the low intensity case. Thus, increasing the pump intensity does not increase the fifth-harmonic signal as much as it would if we could keep the gas pressure fixed. For a rough estimation, let us assume frequency conversion in the perturbative regime. The fifth-harmonic intensity depends then upon the square of the particle density and the fifth power of the driving pump intensity, i.e., $I_{5H}\sim N^2{I}_P^5$. Doubling the fundamental pump intensity from $4\mathrm{TW}/{\mathrm{cm}}^2$ to $8\mathrm{TW}/{\mathrm{cm}}^2$ should yield a fifth-harmonic gain by a factor of $2^5=32$, if we neglect Stark shifts and would be permitted to maintain constant pressure. However, the phase-matching pressure of 12 mbar at an intensity of $8\mathrm{TW}/{\mathrm{cm}}^2$ is only 60% of the optimal pressure of 20 mbar at an intensity of $4\mathrm{TW}/{\mathrm{cm}}^2$. Hence, the lower particle density yields a signal reduction by a factor of ${0.6}^2=0.36$. We get a total signal gain of $0.36·32\approx 12$. This rough estimation is confirmed by comparison of the maxima of the green and blue data points in Fig. 4. This already shows that resonant multiphoton excitation enhances the conversion efficiency in harmonic generation, but Stark shifts limit the potential gain.

B. Contributions of Higher Waveguide Modes to the Harmonic Yield

We will now have a closer look into the pressure dependence of the harmonic yield. As the data in Fig. 4 show, the signal maxima are rather broad, with a slow drop toward larger pressures, also exhibiting some residual oscillations. To study details, Fig. 5 again depicts the harmonic signal for a fundamental intensity of $7\mathrm{TW}/{\mathrm{cm}}^2$ (see black dots in Fig. 5), along with numerical simulations of contributions from different waveguide modes to the harmonic yield (see Appendix A, Section 1). We recall that our fundamental beam profile is very close to the lowest-order waveguide mode $E{H}_{11}$ (see Table 1), with roughly 3% contributions only from higher modes. Thus, on first glance, we would expect negligible contributions of the higher modes to the harmonic yield. Simple theory, based on a single-waveguide mode $E{H}_{11}$, would predict a smooth dependence $I_{5H}\sim p^2\sin {\mathrm{c}}^2(p-p_{PM})$, with the deviation $(p-p_{PM})$ from the phase-matching pressure. It would yield a rather narrow peak around a low phase-matching pressure of 8 mbar, and quick drop of the harmonic yield at higher pressures (see blue, dashed line in Fig. 5). While the simple model describes the rise of the harmonic yield quite well for small pressures below 10 mbar, it does not fit at large pressures. Let us consider now the harmonics generated in waveguide modes up to $E{H}_{15}$ driven by fundamental radiation guided in higher hybrid waveguide modes up to $E{H}_{14}$ (contributions given in Table 1). The calculation includes the Stark shift, as well as different phase-matching pressures for the different waveguide modes. We note that the phase-matching pressures (with respect to the fundamental radiation guided in $E{H}_{11}$) for the higher-order modes are smaller compared to the $E{H}_{11}$ mode. Fifth-harmonic generation toward the $E{H}_{12}$ and $E{H}_{13}$ mode is phase matched at 6 and 4 mbar chamber pressure, respectively, (at $7\mathrm{TW}/{\mathrm{cm}}^2$). This results in the local maximum of the cyan line near 7 mbar in Fig. 5 and the maximum of the green line near 4.5 mbar (barely visible) for pulsed excitation.

 

Fig. 5. Fifth-harmonic pulse energy versus pressure. Pump intensity $I_P=7\mathrm{TW}/{\mathrm{cm}}^2$, wavelength ${\lambda }_P=512\mathrm{nm}$, pulse bandwidth $\mathrm{\Delta }{\lambda }_P=1\mathrm{nm}$. Experimental data (black dots), full simulation with contributions from all waveguide modes (thick, solid red line), simulation assuming the full fundamental pulse energy to be in the lowest mode $E{H}_{11}$ (thick, dashed blue line). Thin colored lines show the distribution of the generated fifth harmonic (H5) into the waveguide modes.

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The red line shows the calculated total harmonic signal. The simulation describes the experimental data pretty well. The first maximum resulting from the phase matching for the fundamental field in the $E{H}_{11}$ mode around 8 mbar is still visible. The simulation also yields the larger maximum around 15 mbar, which comes from the contributions of the higher modes, although those are significantly phase mismatched with respect to the driving field at this pressure. If we would allow in the simulation for a slightly larger contribution of the $E{H}_{12}$ mode, the ratio of the two maxima at 15 mbar and 8 mbar would further increase and even better fit the experimental data.

As an important (and surprising) finding, the weak modes $E{H}_{12},E{H}_{13},E{H}_{14}$, which guide only roughly 3% of the total power, have a quite strong effect upon the total harmonic yield. To understand this feature, in our simulations we investigate propagation along the waveguide. We find that interference between the higher modes of the fundamental pump field strongly modulates the radial intensity distribution as well as the peak intensity at the waveguide center. This increases the overlap of the nonlinear polarization to higher modes in the fifth-harmonic field, which modulates the overall harmonic gain, because most of the harmonic signal is generated in the center of the waveguide at maximal intensity.

We illustrate the effect in a simplified example and simulation, i.e., a waveguide with constant argon pressure, driven by cw radiation at $7\mathrm{TW}/{\mathrm{cm}}^2$ and the modal power decomposition of Table 1. Figure 6(b) shows the spatial variation of the fundamental intensity due to the different phase velocity of the four fundamental modes, along the propagation distance $z$. Three distinct regions of maximal fundamental intensity appear near $z=0$, 18, and 36 mm. For the $E{H}_{11}$ mode of the harmonic, frequency conversion is maximal at the phase-matching pressure of 8 mbar. Thus, the accumulated phase difference between fundamental and harmonic remains close to zero along the waveguide [see dashed orange line in Fig. 6(a)]. However, the generated harmonic power of the $E{H}_{11}$ mode does not simply rise exponentially [solid orange line in Fig. 6(a)], as there are three regions of maximal fundamental intensity. The effect is even more pronounced for the harmonics guided in $E{H}_{12}$, mode at a phase-matching pressure of 6 mbar.

 

Fig. 6. Model simulation of fifth-harmonic generation in a waveguide at constant argon pressure. (a) Solid lines show the fifth harmonic generated in mode ${EH}_{11}$ at $p_c=8\mathrm{mbar}$ (orange) and ${EH}_{12}$ (cyan) at $p_c=6\mathrm{mbar}$. Dashed and dotted lines indicate the phase differences between the fundamental and the harmonic in ${EH}_{11}$ (orange, dotted) and ${EH}_{12}$ (cyan, dashed); (b) fundamental intensity variation by mode beating in the waveguide; (c) phase differences and fifth-harmonic power, generated in a QPM scheme for the ${EH}_{11}$ mode at $p_c=16\mathrm{mbar}$ or the ${EH}_{12}$ mode at $p_c=13\mathrm{mb}\mathrm{a}\mathrm{r}$.

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We return now to the experimentally observed and numerically confirmed efficiency maximum near a pressure of 15 mbar (see Fig. 5). At a pressure of 14 mbar, we estimate a phase difference $\mathrm{\Delta }\mathrm{\varphi }\ge 5\pi$ between the fifth harmonic and the fundamental field. Nevertheless, the conversion efficiency at 14 mbar remains high, comparable to the phase-matched case. We attribute this to the modulation period of the pump peak intensity $\mathrm{\Lambda }=2\pi /\mathrm{\Delta }k$, fulfilling a quasi-phase-matching (QPM) condition by polarization beating [23]. The intensity of the fundamental is low where back conversion occurs, reducing the loss of the harmonic significantly at $z=9$ and 27 mm [see orange line in Fig. 6(c)]. The higher particle density leads to an increased nonlinear gain and compensates for the nonzero back conversion. The effect is even more pronounced for the fifth harmonic guided in the $E{H}_{12}$ mode [see cyan line in Fig. 6(c)]. At 13 mbar, the harmonic yield is more than twice the yield of the phase-matched case at 6 mbar [see Fig. 6(a)], because the intensity modulation in the waveguide center enhances the nonlinear gain for this mode. In this case, the ratio of the pressures would allow for more than 4 times stronger signal, but the full length of the medium is not efficiently applicable due to back conversion. Thus, when a laser with a beam waist (even only slightly) different from the ratio $w/a=0.64$ is coupled to a hollow-core waveguide, QPM can increase the harmonic yield. We note that if the beam exhibits an $M^2$ value bigger than unity, which leads to wavefront distortion and imperfect coupling to the $E{H}_{11}$ mode, the phase-matching behavior can significantly change. The effect of QPM was already demonstrated in high-harmonic generation [23–26]. Our numerical simulation predicts an increase by a factor of 5 in the conversion efficiency for coupling the waveguide with a beam waist of $w_0/a=0.5$, instead of $w_0/a=0.64$. However, as a drawback, the amplitudes for the first two waveguide modes are equal then, which causes strong interference to degrade the spatial harmonic beam profile.

C. Wavelength Dependence of the Harmonic Yield

We investigate now the harmonic yield versus the wavelength of the driving pump laser, tuned in the vicinity of a five-photon resonance in argon. The aim is to resonantly enhance the harmonic signal. As we have already seen in Section 4.A above, the phase-matching condition changes with resonance shifts. The same will occur for variations in the excitation frequency. Thus, we performed systematic measurements to monitor the harmonic signal versus both the excitation wavelength as well as the pressure (see Fig. 7). The data show that, to every fundamental wavelength below 518 nm, we can match a certain pressure to reach a maximum efficiency. The appropriate pressure decreases from 30 mbar at a fundamental wavelength of 506 nm to zero at a wavelength of 518 nm. The behavior is mainly due to the strong dispersion of the refractive index of argon in the VUV near the $4s$ energy levels. The phase-matching pressures (i.e., normal and QPM) are determined by the resonance position of the $4s$ level and vary toward higher pressure for larger five-photon detuning.

 

Fig. 7. Fifth-harmonic pulse energy (experimental data) versus pump fundamental wavelength and argon pressure. The pump intensity was $I_P=7\mathrm{TW}/{\mathrm{cm}}^2$. We note that the pulse duration from our laser system varied slightly with the wavelength. To calibrate for the effect, at each wavelength we deduced the pulse duration from a FROG trace. We compensate the variation in the pulse duration by slightly increased pulse power to acquire data at constant pump peak intensity, and normalize with respect to pulse area.

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Assuming a spatially averaged Stark shift of $\mathrm{\Delta }\mathrm{W}=0.85{\mathrm{\varphi }}_P$, at a pump intensity of $7\mathrm{TW}/{\mathrm{cm}}^2$ the single-photon transition wavelength shifts from 104.8 to 103.5 nm, resulting in a five-photon resonance moving to 517.5 nm. The shifted resonance position corresponds well with the strong decrease in the signal yield between 516 and 518 nm in Fig. 7. At fundamental wavelengths longer than 517.3 nm, the refractive index at the fifth harmonic flips from values smaller than unity toward normal dispersive behavior with values bigger than unity, thus preventing phase matching. Nevertheless, at lower intensities (i.e., in the wings of the pump beam profile) phase matching is still possible for quite small pressures, as the resonance has not shifted much. The data confirm this expectation, as Fig. 7 still shows some nonzero signal (i.e., above the noise background) around ${\lambda }_P\approx 518\mathrm{nm}$.

To verify these experimental findings, we compare the data in Fig. 7 with a numerical simulation (see Fig. 8). For details on the calculation parameters, see Appendix A, Section 4. The simulation fits very well with the experimental data. In particular, the experimentally obtained VUV maxima (see black dots in Fig. 8) versus pressure and wavelength are well described. We obtained these data by fitting to the areas of large signal in Fig. 8.

 

Fig. 8. Fifth-harmonic pulse energy (numerical simulation) versus pump fundamental wavelength and argon pressure. We assumed a fundamental peak intensity of $7.1\mathrm{TW}/{\mathrm{cm}}^2$ and averaged over a laser bandwidth of 1 nm. Black dots with error bars depict the pressures of maximal efficiency from Fig. 7. The red dashed line represents the pressure of maximal efficiency, neglecting the resonance shift. We note that due to the pronounced Stark shifts, at peak intensity the dispersion for the fifth harmonic increases. This results in a lower group velocity at the peak intensity and thus modifies the pulse shape of the fifth harmonic. The shaded, parabolic area indicates the region where parts of the fifth-harmonic pulse experience a group delay larger than half the pump pulse duration.

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Figure 8 also shows a simulation of the VUV maxima ignoring Stark shifts. The incomplete simulation deviates from the experimental data. This confirms the important effect of Stark shifts upon resonance enhancements and phase-matching pressures.

We note the fan-like substructures with four branches in the numerical simulation (see labels 1–4 in Fig. 8). These cannot be observed with great detail in the experimental data, as the signal “sidebands” are rather weak. Nevertheless, the experimental data clearly reveal the strong branch (2), and there is evidence for branches (3) and (4) in the broader spreading of the data points at wavelengths below 508 nm and the local maxima at pressures above 40 mbar. The first maximum of the signal [branch (1)] with increasing argon pressure is only evident in the simulation results as the actual modal amplitudes might exhibit even higher $E{H}_{12}$ coupling than supported from the focal spot images, leading to stronger QPM and thus a more pronounced second maximum. At fundamental pump wavelengths above 514 nm, the pulse bandwidth approaches the phase-matching bandwidth. This results in a convergence of the branches toward a single maximum at 517 nm, where the phase-matching bandwidth of 0.15 nm is much smaller than the pulse bandwidth, and a strong spectral averaging (and spectral narrowing of the generated harmonic) occurs. For small detuning from the resonance, the dispersion at the fifth harmonic also gives rise to a group delay comparable to or bigger than the driving pulse length. We indicate the parabolic region where the group delay at the peak intensity is larger than half the pump pulse length by a gray shading in Fig. 8. In this region, the actual harmonic yield might be overestimated by our simulation model, as we neglect pulse propagation. For a detailed analysis, please refer to Appendix A, Section 5. Anyway, this imposes no serious limit to the comparison with the experimental data, as in the shaded area we did not observe large signal or specific features (compare Fig. 7), and the simulation still seems to fit well also in the shaded area.

We will now have a closer look into the data and simulation. In Fig. 9(b) we plot the fifth-harmonic signal maxima from Figs. 7 and 8 versus fundamental wavelength. The VUV yield increases with decreasing detuning from the Stark-shifted resonance till it reaches a broad maximum around 512 nm and quickly drops for larger wavelengths afterwards. As in Fig. 7, the possible resonance enhancement is visible. The full numerical simulation [see green line in Fig. 9(b)] shows in principle a similar behavior, i.e., a smooth increase toward a broad maximum and a sharp drop for long wavelengths. However, compared to the experimental data, the wavelength position of the steep falling slope is shifted toward longer wavelength in the simulation.

 

Fig. 9. (a) Dependence of ${\chi }^{(5)}$ versus wavelength for a low, constant argon pressure (gray dashed line). Dependence of ${\chi }^{(5)}$ versus wavelength at the corresponding QPM argon pressure (green solid line). (b) Fifth-harmonic signal maxima versus wavelength (black dots); experimental data taken from Fig. 8. The green thick line shows the results of the full numerical simulation, taken from Fig. 9. Thin lines show numerical simulations of normalized dependencies for negative values of ${\chi }^{(5)}$ (dashed) as well as for a lower positive value (orange). The blue line shows the results without the contribution of cascading conversion processes, i.e., setting ${\chi }^{(3)}=0$. The shaded area indicates the wavelength region, where pulse propagation effects may occur, which are neglected in the simulations (compare Fig. 8).

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Our numerical simulation permits us to study the effect of different contributions to the spectral dependence in Fig. 9. We start with a calculation of the fifth-order susceptibility for fifth-harmonic generation ${\chi }_{\mathrm{FHG}}^{(5)}$ at low constant pressure, i.e., neglecting phase-matching issues [see gray dotted line in Fig. 9(a)]. For details on the calculation, see Appendix A, Section 4. The susceptibility shows a typical resonance line profile, as expected. We compare this with the spectral behavior of ${\chi }_{\mathrm{FHG}}^{(5)}$ at the corresponding QPM pressures [see green solid line in Fig. 9(a)]. The choice of the QPM pressure, as determined by the driving fundamental wavelength, automatically includes phase matching. Therefore, the behavior of ${\chi }_{\mathrm{FHG}}^{(5)}$ looks now very different from the low-pressure case. This consideration of the susceptibility ${\chi }_{\mathrm{FHG}}^{(5)}$ only already reveals a qualitatively similar behavior as the experimental data in Fig. 9(b). We observe that, for the QPM case, the higher possible phase-matching pressure (caused by the drop of the linear index of refraction for the fifth harmonic, or the linear susceptibility ${\chi }^{(1)}({\omega }_5)$ rp.) almost compensates the decrease of ${\chi }_{\mathrm{FHG}}^{(5)}$ when the detuning from resonance increases. Nevertheless, still substantial resonance enhancement in the fifth-harmonic yield remains.

To understand the envelope of the maximal harmonic intensity with respect to fundamental wavelength in more detail, we turn now to the phase-matching bandwidth, which must be sufficiently large to permit efficient frequency conversion. In our case, for a 55 mm long capillary with a constant pressure section of 35 mm, the phase-matching bandwidth (FWHM) drops from $\mathrm{\Delta }{\lambda }_{PM}=3.5\mathrm{nm}$ at 500 nm to $\mathrm{\Delta }{\lambda }_{PM}=1\mathrm{nm}$ at 514.5 nm. For longer wavelengths, the conversion efficiency suffers from spectral narrowing. Thus, the simulation [see green line in Fig. 9(b)] shows a strong drop of the VUV yield beyond wavelengths of 515 nm. The experimental data confirm these considerations, i.e., the data reveal a similar shape. However, the VUV signal already starts to drop at shorter wavelengths beyond 513 nm. This is most probably caused by the group velocity mismatch, stretching the harmonic pulse and temporally shifting it away from the pump pulse, thus reducing the conversion efficiency.

We must not fail to mention that a bandwidth of $\mathrm{\Delta }{\lambda }_{PM}=3.5\mathrm{nm}$ at 500 nm corresponds to a laser pulse duration in the regime of 100 fs. Thus, though we implemented our experiment with picosecond pulses, resonance enhancements in argon are possible also for shorter (femtosecond) laser pulses.

We note that there are always two processes that contribute to the total fifth-harmonic yield: (1) a direct fifth-harmonic generation process with ${\omega }_5=5{\omega }_P$, adding up five fundamental photons in a single step; (2) a cascade conversion process by third-harmonic generation (THG) ${\omega }_3=3{\omega }_P$ followed by sum-frequency mixing (SFM) ${\omega }_5={\omega }_3+2{\omega }_P$ with two additional fundamental photons. In our numerical simulations, we included both contributions to the fifth-harmonic yield. The exact simulation results depend upon the ratio of the two conversion channels. Our simple modeling of the fifth-order nonlinearity via a generalized Miller’s formula [27] should resemble the functional behavior near the five-photon resonance quite well. However, the sign and absolute magnitude of ${\chi }^{(5)}$ at infrared and mid-infrared wavelength are still a matter of debate in the literature [28–32]. As our model is normalized to these values, we analyze results for different ${\chi }^{(5)}$ at constant ${\chi }^{(3)}$.

The third-harmonic power scales quadratic with the argon pressure, so at high phase-matching pressures (lower fundamental wavelength) the contribution of the cascaded process is bigger. The relative sign between ${\chi }_{\mathrm{FHG}}^{(5)}$ and ${\chi }_{\mathrm{SFM}}^{(3)}$ determines whether this contribution adds constructively (different sign) or destructively (same sign) to the direct fifth-order process. In Fig. 9(b) we show the resulting maximal harmonic pulse energies for constructive (lines) and destructive (dashes) interference, as well as a simulation without cascaded mixing (${\chi }^{(3)}=0$, blue line).

If we reduce the magnitude of ${\chi }^{(5)}$ by a factor of 4 (i.e., a ratio of $\left|\frac{{\chi }_{\mathrm{FHG}}^{(5)}}{{\chi }_{\mathrm{SFG}}^{(3)}}\right|<350\frac{p{m}^2}{{\mathrm{V}}^2}$), the simulation yields a significant change of the envelope at maximum VUV energy [compare orange dashed and solid lines in Fig. 9(b)], which in both cases reduces the consistency with the experimental data. Furthermore, we calculate a second increase in VUV yield near 120 mbar argon pressure due to SFM that could not be observed experimentally.

No matter which sign for ${\chi }^{(5)}$ we choose, the shape of the experimental data can be already well described by direct fifth-harmonic generation [compare green solid and dashed line versus blue line in Fig. 9(b)]. Hence, for all calculations above, we used ${\chi }^{(5)}$ with a positive sign. We conclude that in this case the contribution of cascaded processes to the VUV power is in the range of 10% at the QPM pressure. We want to stress that this finding cannot be generalized toward gas jet or filamentation experiments. Inside the waveguide, the third-harmonic field accumulates a phase of more than $3\pi$ with respect to the fundamental, so the relative third-harmonic intensity is quite limited, compared to a dilute gas with negligible phase mismatch for all harmonics.

In summary, the numerical simulation confirms the experimentally observed strong spectral variation of the conversion efficiency. The data in Figs. 7 and 9 clearly demonstrate resonantly enhanced harmonic generation, even when the multiphoton transition is affected by strong Stark shifts.

The possible VUV signal enhancements are limited by the rather low phase-matching pressures in the regime of 10 mbar. In principle, it is possible to further increase the signal yield by using higher intensities in tighter waveguides, increasing the waveguide dispersion and group delay proportional to $1/a^2$. However, the signal yield is limited by damping in the waveguide, scaling with $1/a^3$ (which could be overcome in photonic crystal fibers [1]), additional plasma dispersion, and very strong Stark shifts at high intensities. Therefore, it is convenient to fill a buffer gas in the waveguide to tailor the refractive index of the medium in favor of higher phase-matching pressures. Figure 10 shows the VUV signal in our waveguide, using a mixture of 13(3)% argon and 87(3)% neon. Neon offers positive dispersion to compensate the negative dispersion of argon in our wavelength regime. The total phase matching shifts to 250 mbar, corresponding to a partial pressure of 32(7) mbar in argon. This permits a further signal gain of roughly 50%.

 

Fig. 10. Fifth-harmonic pulse energy versus pressure in a mixture of argon with neon (blue data points), compared to pure argon (orange data points). Data taken at a fundamental peak intensity of $7.5\mathrm{TW}/{\mathrm{cm}}^2$ and a wavelength of 512 nm.

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5. SUMMARY AND OUTLOOK

We presented experimental data, accompanied by extended numerical simulations, on phase-matched fifth-harmonic generation of (picosecond) laser pulses in a hollow-core waveguide with a bore radius of roughly 50 μm, filled with argon. The center frequency of fundamental pump pulses at ${\lambda }_f=506\dots 520\mathrm{nm}$ was tuned in the vicinity of a five-photon resonance between the ground state $3p^6$ (${}^1{S}_0$) and the excited state $3p^{5}4s{}^2{[1/2]}_1$ of argon, yielding resonantly enhanced fifth-harmonic VUV radiation at ${\lambda }_{5H}=101\dots 104\mathrm{nm}$. We systematically investigated the dependence of the signal yield versus driving wavelength, argon pressure, and laser intensity. The five-photon resonance was shifted up to by a strong AC Stark effect at peak laser intensities of several $\mathrm{TW}/{\mathrm{cm}}^2$. We found the AC Stark shift of the excited state to be well approximated by the ponderomotive energy. Tuning close to the Stark-shifted resonance, and setting the argon pressure in the waveguide to the (wavelength-dependent) phase-matching pressure, we observed significant resonance enhancements of the fifth-harmonic yield. The phase-matching bandwidth in the regime of a few nanometers should permit resonance enhancements of laser pulses with durations down to 100 fs.

Through comparison of the experimental data with numerical simulations, we determined the significant contribution of higher transversal waveguide modes to the total conversion efficiency–even if the driving fundamental beam profile deviated only very little from the lowest-order waveguide mode. The contribution of the higher waveguide modes is clearly visible in a broadening and oscillatory substructure of the signal yield versus pressure and the advent of a second maximum in conversion efficiency, yielding up to a factor of 2 more efficiency. We explain this by a QPM effect, based upon interference between the different waveguide modes of the fundamental pump field, which modulates the radial intensity distribution as well as the peak intensity at the waveguide center.

We note that such a QPM scheme was already proposed [33] and demonstrated [23] for conventional (i.e., off-resonant) high-harmonic generation. We showed that it is also applicable to enhancing the yield of harmonic generation close to resonances. Furthermore, our numerical simulations indicate enhancements of up to a factor of 5 for purely Gaussian input beams. Shaping the spatial input beam could further support mode beating, reduce back conversion, and thus enhance the efficiency even without the need to increase the total input power.

A closer inspection of the experimental data also revealed a possible contribution of a cascading process to generate fifth-harmonic radiation by (off-resonant) THG followed by SFM with two additional fundamental photons. Finally, we demonstrated the applicability of neon buffer gas to push the otherwise rather low phase-matching pressure toward higher values, which further increased the signal yield by 50%.

The data indicate the potential of multiphoton resonance enhancements for harmonic generation of ultrashort laser pulses.

APPENDIX A

In the following sections, we summarize our theoretical treatment for the numerical simulation of fifth-harmonic generation in a gas-filled waveguide. The approach takes higher transversal waveguide modes in the fundamental and fifth-harmonic fields into account, as well as cascading fifth-harmonic generation by THG followed by SFM. We also describe the calculation of nonlinear susceptibilities required in the calculation.

1. PROPAGATION AND PHASE MATCHING IN HOLLOW-CORE WAVEGUIDES

The propagation (in $z$ direction) of initially linearly polarized light at angular frequency ${\omega }_j$ inside a circular, dielectric step index waveguide of radius $a$ can be described on the basis of the simplified hybrid $E{H}_{1m}$ modes [34]. The field guided in the $m$th mode,

(A1)$${\overrightarrow{E}}_{j,m}(r,z,t)={\overline{E}}_{j,m}(r,z)·e^{-i({\omega }_{j}t)}·{\overrightarrow{e}}_T,$$

is also linearly polarized along the unitary vector ${\overrightarrow{e}}_T$ in the transversal direction of the input polarization. These modes neglect the component of the field in the propagation direction as well as other corrections of the same order ($\lambda /a$).

The spatial dependence of the electric field is given in the radial direction by the Bessel function of first kind in lowest order, $J_0$ and in propagation direction determined by the propagation constant ${\gamma }_{j,m}$ (of mode $m$ at harmonic frequency $j{\omega }_1$). The complex amplitude of each mode is given by $E_{j,m}$:

(A2)$${\overline{E}}_{j,m}=E_{j,m}·J_0\left(u_{1m}\frac{r}{a}\right)\exp (i{\int }_0^z{\gamma }_{j,m}(z^{\prime })\mathrm{d}{z}^{\prime }).$$

The complex phase of these linearly polarized, radially symmetric modes evolves with the propagation constant

(A3)$${\gamma }_{j,m}=k_j-\frac{1}{2k_j}{\left(\frac{u_{1m}}{a}\right)}^2-i\frac{1}{2k_j}{\left(\frac{u_{1m}}{a}\right)}^2·\frac{2{\nu }_{1,j}}{k_{j}a},$$

where $u_{1m}$ is the $m$th root of the Bessel function of first kind. For waveguides with a radius larger than the wavelength of the light wave, the propagation is dominated by the complex plane wave propagation constant $k_j={\omega }_{j}n({\omega }_j,p)c_0^{-1}$, with the pressure-dependent complex refractive index of the gas $n({\omega }_j,p)$ and the speed of light in vacuum $c_0$.

The waveguide radius $a$ determines the waveguide dispersion (see second term in), which can be modified by tuning the gas density and thereby modifying $k_j$.

For harmonic frequencies within the bound states of a rare gas, absorption is low when the laser is not tuned to multiphoton resonances. Therefore, the damping [last term in Eq. (A3)] of the electromagnetic waves is dominated by

(A4)$${\nu }_{1,j}=({\tilde{n}}_e^2({\omega }_j)+1)/\left(2\sqrt{{\tilde{n}}_e^2({\omega }_j)-1}\right),$$

where ${\tilde{n}}_e({\omega }_j)$ is the ratio of the refractive index of the capillary material to that of the medium inside the bore [3]. In the waveguide in our experiment, damping is below 5% for a capillary length of 55 mm.

In contrast to plane wave phase matching of the propagation constants $\mathrm{\Delta }{k}_j=k_j-j·k_1$ at fundamental frequency ${\omega }_1$ and harmonic frequency ${\omega }_j=j\omega$, the propagation inside the capillary enables multiple phase-matching conditions, $\mathrm{\Delta }{\gamma }_{j,(m,m^{\prime })}=j·{\gamma }_{1,m}-{\gamma }_{j,m^{\prime }}$ for the generation of a harmonic frequency ${\omega }_j$ of the fundamental frequency via the combination of mode numbers $(m,m^{\prime })$ as shown by Durfee et al. [3]. For combinations of different modes, the overlap between the nonlinear polarization and the guiding mode field is reduced, reducing the overall efficiency of the process. Therefore, a low difference in mode number is preferable. The phase mismatch,

(A5)$$\mathrm{\Delta }{\gamma }_{m{m}^{\prime }}^{(j)}=j{\gamma }_{1,m}-{\gamma }_{j,m^{\prime }},$$

between the fundamental and the $j$th harmonic guided in mode $m^{\prime }$ can be nullified by tuning the pressure $p$ of the gas inside the waveguide such that the gas dispersion cancels the waveguide dispersion for the chosen combination of modes.

Which mode combinations can be phase matched depends on the dispersion of the gas. Harmonics above a single-photon resonance, where the refractive index is smaller than in the visible regime, support the most mode combinations. Harmonic generation energetically below bound states usually demands the harmonic to be guided in a higher mode compared to the fundamental beam.

2. FIFTH-HARMONIC GENERATION IN A WAVEGUIDE

To calculate the buildup of the harmonic pulse energy inside the waveguide, we follow an approach similar to Tani et al. [35] and decompose the measured intensity distribution at the input of the waveguide in terms of the waveguide modes to satisfy

(A6)$${\overrightarrow{I}}_{\mathrm{in}}(r,z=0)=\frac{\pi }{Z}{|\sum _m{E}_{1,m}(r,0)+\text{c.c.}|}^2,$$

where $Z$ is the electromagnetic impedance of the medium and c.c. represents the complex conjugate of the previous terms, with the coupling relation

(A7)$$E_{1,m}=\frac{{\int }_0^a{J}_0(u_{1m}r·a^{-1})·F_s(r)r\mathrm{d}r}{{\int }_0^a{J}_0^2(u_{1m}r·a^{-1})r\mathrm{d}r}.$$

This represents the energy flow in the $z$ direction [36]. Here, $F_s(r)$ is the source term. Assuming a plane wavefront (in the focal plane of the Gaussian input beam) at the entrance of the dielectric waveguide, the source is the electric field at the input $F_s={[\frac{Z}{\pi }·I_{\text{in}}(r,\varphi ,z=0)]}^{\frac{1}{2}}$.

We note that this modal expansion reproduces only intensity profiles with circular symmetric shape, as we neglect the higher modes in the azimuthal dimension.

We assign a $z$ dependence to the amplitudes $E_{j,m}$ and numerically solve the one-dimensional nonlinear wave equation in slowly varying envelope approximation:

(A8)$$\begin{gathered} \frac{\partial E_{j,m}}{\partial z}=\frac{i}{2{\gamma }_{j,m}}[(k_{j,m}^2-{\gamma }_{j,m}^2)E_{j,m} \\ +\frac{{\omega }_j^2}{c_0^2{\epsilon }_0}{\tilde{P}}_{j,m}^{NL}(z)·e^{-i{\int }_0^z{\gamma }_{j,m}(z^{\prime })\mathrm{d}{z}^{\prime }}], \end{gathered}$$

with the initial electric field amplitudes $E_{1,m}$ from Eq. (A7) at $z=0$ and no electric field for the harmonics $j=3$, 5.

This is done for each frequency $j=1$, 3, 5 for the five lowest waveguide modes $m=1\dots 5$. Here the complex nonlinear polarization amplitude ${\tilde{P}}_{j,m}^{NL}(z)$ of the respective frequency and mode (calculated in Appendix A, Section 4) is the source term for harmonic generation. We note that the phase integral ${\int }_0^z{\gamma }_{jm}(z^{\prime })\mathrm{d}{z}^{\prime }$ is also required, as the refractive index of the gas in our case of a ported waveguide with pressure ramps toward vacuum at the ends is a function of the propagation coordinate $z$.

3. CALCULATION OF THE INDEX OF REFRACTION

Our simulation requires the (linear) index of refraction in argon, in the wavelength regime $101\dots 104\mathrm{nm}$. The Sellmeier equation for refractive indices in argon by Bideau-Mehu [37], covering most of the ultraviolet spectral region, is applicable up to 140 nm only. The equation involves two resonance terms only, corresponding to the levels of lowest energy, $4s{}^2{[3/2]}_1^0$ at a transition wavelength of 104.8 nm and $4s{}^2{[1/2]}_1^0$ at 106.6, as well as an empiric term for the ionization continuum. The accuracy of the model above the two resonances (i.e., at wavelength shorter than 104.8 nm) has not been proven experimentally and is expected to exhibit increasing error with decreasing wavelength. Therefore, we extended the refractive index model, taking higher states into account. We start by computing the atomic polarizability [38],

(A9)$$\begin{gathered} {\alpha }_b(\omega )=\frac{e^2}{2{ϵ}_{0}m}\sum _n\frac{f_{na}}{{\omega }_{na}} \\ [{({\omega }_{na}-\omega -\frac{i\mathrm{\Gamma }}{2})}^{-1}+{({\omega }_{na}+\omega +\frac{i\mathrm{\Gamma }}{2})}^{-1}], \end{gathered}$$

for all dipole-allowed bound states up to 5d (i.e., 81.62 nm), where $e$ is the electron charge, ${\epsilon }_0$ the vacuum permittivity, $m_e$ the electron mass, $\omega$ the light frequency, $\mathrm{\Gamma }$ the decay rate, and ${\omega }_{na}$ the resonance frequencies. We take the oscillator strengths $f_{na}$ from the work of Chan [39], and average them with the data of Xu and Wu [40], with relative weights determined by the specified errors.

At lower wavelength, the available data sets do not resolve the spectral lines of the highly excited states. From these unresolved lines, together with the absorption cross section ${\sigma }_c$ of the ionization continuum [39], we calculate the imaginary part of the atomic polarizability,

(A10)$$\text{Im}[{\alpha }_c]={\sigma }_c\frac{c_0}{\omega },$$

and calculate the real part from the Kramers–Kronig relation [41]. We then obtain the pressure-dependent refractive index from the atomic polarizability,

(A11)$$\alpha (\omega )={\alpha }_b(\omega )+{\alpha }_c(\omega ),$$

(A12)$$n(\omega ,p)={(1+N(p)·\alpha (\omega ))}^{1/2},$$

by assuming the ideal gas law for the number density $N(p)=p/(k_{B}T)$, where $k_b$ is the Boltzmann constant.

Our above calculation yields values of the index of refraction $(n(\omega ,p)-1)$ from the NIR toward the UV spectral regime. The values differ less than 1% from the Sellmeier equation approach within its valid spectral regime [37].

4. CALCULATION OF THE NONLINEAR POLARIZATION AND SUSCEPTIBILITIES

The nonlinear polarization ${\overline{P}}_{jm}^{NL}$, oscillating at frequency ${\omega }_j$, coupled to the $m$th mode is calculated from the full fields

(A13)$${\overline{\overline{E}}}_j=\sum _m{\overline{E}}_{j,m}(r,z)$$

at frequencies ${\omega }_j$, including the different modal propagation constants ${\gamma }_{j,m}$. We reduce the nonlinear polarization to the terms at highest amplitude (i.e., the highest powers of the fundamental field $E_{1,m}$). We checked the simplified model (as presented below) versus the full polarization model and found the difference to be less than 10% for each data point.

For the fifth-harmonic frequency, we consider direct fifth-harmonic generation via ${\chi }^{(5)}$ as well as cascaded generation of the fifth harmonic from a field at ${\omega }_3=3{\omega }_1$, generated in the capillary at shorter propagation distances. As a result, we obtain the nonlinear polarizations ${\overline{P}}_{jm}^{NL}$ at the three frequencies by setting the source in the coupling relation to

(A14)$${\overline{\overline{P}}}_1^{NL}(r,z)={ϵ}_0(3{\chi }_{(\omega ,-\omega ,\omega )}^{(3)}{\overline{\overline{E}}}_1{|{\overline{\overline{E}}}_1|}^2+10{\chi }_{(\omega ,-\omega ,\omega ,-\omega ,\omega )}^{(5)}{\overline{\overline{E}}}_1{|{\overline{\overline{E}}}_1|}^4)$$

for the fundamental radiation, only taking self-phase modulation into account,

(A15)$${\overline{\overline{P}}}_3^{NL}(r,z)={ϵ}_0(3{\chi }_{(\omega ,\omega ,\omega )}^{(3)}{\overline{\overline{E}}}_1^3+5{\chi }_{(\omega ,-\omega ,\omega ,\omega ,\omega )}^{(5)}{\overline{\overline{E}}}_1^3{|{\overline{\overline{E}}}_1|}^2),$$

for the third harmonic, also including cross-phase modulation.

The fifth-harmonic nonlinear polarization is simplified to

(A16)$${\overline{\overline{P}}}_5^{NL}(r,z)={ϵ}_0(3{\chi }_{(\omega ,\omega ,\omega )}^{(3)}{\overline{\overline{E}}}_1^2{\overline{\overline{E}}}_3+{\chi }_{(\omega ,\omega ,\omega ,\omega ,\omega )}^{(5)}{\overline{\overline{E}}}_1^5),$$

considering fifth-harmonic generation and SFM only.

From these radially dependent polarizations, we finally calculate the complex nonlinear polarization amplitude ${\tilde{P}}_{j,m}^{NL}(z)$ to insert in Eq. (A8) by the overlap integral,

(A17)$${\tilde{P}}_{j,m}^{NL}(z)=\frac{{\int }_0^a{J}_0(u_{1m}r·a^{-1})·{\overline{\overline{P}}}_j^{NL}(r,z)r\mathrm{d}r}{{\int }_0^a{J}_0^2(u_{1m}r·a^{-1})r\mathrm{d}r},$$

in analogy to Eq. (A7).

We model the required nonlinear susceptibilities from the single-photon transition probabilities by the generalized Miller formulas [27], i.e., as a product of linear polarizations at the relevant frequencies. Thereby we neglect two-photon and four-photon resonances that would appear in the full quantum mechanical calculation. We expect these resonances to only contribute a nearly frequency-independent factor (accounted for by a pre-factor $A^{(p)}$), as the detuning to the lowest two-photon-allowed transition is more than three fundamental photon energies.

We calculate the pre-factor $A_p^{(3)}$ for the third-order nonlinear susceptibility using literature values of the nonlinear refractive index of argon at 800 nm $n_2=10·{10}^{-20}\frac{{\mathrm{cm}}^2}{\mathrm{W}}$ [42–44] as a reference. We obtain the nonlinear susceptibilities of third order by

(A18)$${\chi }_{({\omega }_1,{\omega }_2,{\omega }_3,{\omega }_4)}^{(3)}=A_p^{(3)}·N(p)·\prod _{i=1}^4\alpha ({\omega }_i),$$

which involves the product of the pressure-dependent, dynamically shifted linear susceptibilities, with $\alpha (\omega )$ from Eq. (A15).

The same approach is used for the fifth-order nonlinear susceptibilities, normalizing the model function

(A19)$${\chi }_{({\omega }_1,{\omega }_2,{\omega }_3,{\omega }_4,{\omega }_5,{\omega }_6)}^{(5)}=A_p^{(5)}·N(p)·\prod _{i=1}^{6}a({\omega }_i)$$

to a literature value of $n_4$ for the second-order nonlinear refractive index. Using the most recent value $n_4=1·{10}^{-33}\frac{{\mathrm{cm}}^4}{\mathrm{W}}$ of Tarazkar [28] we obtain ${\chi }_{({\omega }_1,{\omega }_1,{\omega }_1,{\omega }_1,{\omega }_1,{\omega }_5)}^{(5)}\approx -1·{10}^2{pm}^4{\mathrm{V}}^{-4}$ for the fifth-harmonic generation.

In contrast to $n_2$, which has been measured in good agreement by numerous groups, there are also data sets with negative sign of $n_4$ available [29,30,42] that we initially used to calculate the ${\chi }^{(5)}$ listed in Table 2 for the simulation. We show and discuss the potential variations in the magnitude and sign of ${\chi }^{(5)}$ in Fig. 9. The obtained values for the nonlinear susceptibilities are given in Table 2.

We note that, in our experiment, the relevant nonlinear susceptibilities for fifth-harmonic generation ${\chi }_{({\omega }_1,{\omega }_1,{\omega }_1,{\omega }_1,{\omega }_1,{\omega }_5)}^{(5)}$ and cascading via SFM ${\chi }_{({\omega }_1,{\omega }_1,{\omega }_3,{\omega }_5)}^{(3)}$ involve additional (four- three- and two-photon) resonances. Miller’s generalized approach is not precise in this case. However, we can use it as an order-of-magnitude estimation of the susceptibility.

Table 2. Nonlinear Susceptibilities Applied in Our Numerical Simulation

View Table | View all tables in this article

5. PULSE PROPAGATION EFFECTS

The bandwidth of the fundamental and harmonic pulses is 1 THz only, yielding a group delay below 4 as in a given waveguide mode. The nonlinear refractive index at an intensity of $7\mathrm{TW}/{\mathrm{cm}}^2$ and the maximum considered pressure of 60 mbar in our experiments at 512 nm fundamental wavelength yield a phase shift below 0.02 rad only, which does not alter the pulse envelope. Also, the waveguide modal dispersion produces a group delay around 0.04 ps only (compared to a pulse duration in the range of 1 ps).

The largest group delay appears between the fifth harmonic and the fundamental pulse, due to the nearby resonance at 102.4 nm. This group delay is strongly intensity dependent due to the Stark shift of the excited state and therefore changes in radial as well as temporal dimension of the Gaussian laser pulse. The group delay reaches its maximum at the peak intensity, when the five-photon detuning to the $4s$ level is minimal. Only a more advanced model (e.g., [35]) could account for this, but would require very much longer calculation time.

In our simpler model, we monitor the maximal (worst case) group delay. For our investigations at 512 nm fundamental wavelength, we estimate a maximal group delay of 0.21 ps at the largest argon pressure used in our measurements. This is much less than half the pump pulse length and therefore imposes no severe limit on the numerical results.

Funding

Horizon 2020 Framework Programme (H2020) (Marie Sklodowska-Curie, 641272).

Acknowledgment

We acknowledge fruitful discussions with L. Yatsenko and B. W. Shore. Furthermore, we thank LightFab GmbH for the preparation of the waveguide.

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