1. Introduction

Ultrafast time-resolved measurements in physics, chemistry, biology and material science, e.g. the study of small molecules and their clusters in the gas phase, require a technique for the generation of tunable femtosecond pulses in the vacuum ultraviolet (VUV, λ < 185 nm) with sufficiently high energy (> 0.1 μJ) and high repetition rates (∼ 1 kHz) [1]. The most efficient techniques for generation of ultrashort VUV pulses are based on phase-matched nonlinear optical frequency up-conversion of powerful femtosecond pulses in visible and NIR. By using χ⁽²⁾ sum-frequency generation in nonlinear optical crystals it is possible to generate femtosecond pulses with 0.2-μJ energy near 185 nm at 1 kHz repetition rate [2]. The shortest wavelength of femtosecond pulses generated by phase-matched sum frequency mixing in commercially available crystals is 166 nm. Frequency mixing in gases makes it possible to extend the wavelength range for high-energy femtosecond pulses deeper in the VUV. 300-fs pulses were generated in a gas cell at 155 nm with energies up to a few 100 nJ at 10-Hz repetition rate [3]. However, despite the improved optical transparency and enhanced near-resonant gas nonlinearity, high-energy excimer amplifiers were required, which possess drawbacks such as long pulses and low repetition rates [4]. The shortest achievable pulse durations in these techniques are limited by the narrow spectral acceptance of the phase-matching close to the absorption band of the crystal or to the resonance of the gas [2–4].

Collinear phase-matching using off-resonant χ⁽³⁾ processes in gases is possible only in hollow waveguides [5] and allows generation of ultrashort and broadly tunable pulses in various spectral ranges. Third harmonic generation of Ti:sapphire laser yielding 8-fs pulses with 1 μJ [5] and 25-fs pulses with 8.5 μJ [6] energies were demonstrated at 1 kHz by four-wave difference-frequency mixing (FWM) of the fundamental and the second harmonic in argon-filled capillary. By chirped-pulse FWM 125-fs pulses with energies of up to 65 μJ were achieved at 10 Hz repetition rate [7].

In this paper we demonstrate the generation of high-energy VUV femtosecond pulses at 161 nm and 1 kHz repetition rate by using direct FWM of the fundamental and the third harmonic (TH) of Ti:sapphire laser in argon-filled hollow waveguides. We show that by exciting higher-order transverse modes and coating of the inner surface of the waveguide by aluminum considerable improvement of conversion efficiency is possible. A numerical model, which includes the effects of higher-order modes propagation and the ionization of argon, explains the features of the process.

2. Experimental results

In the FWM scheme two pump photons are mixed with an idler photon to obtain a signal photon: ω_signal = ω_pump1 + ω_pump2 − ω_idler. In the particular case of fundamental (ω) and third harmonic (TH) (3ω) serving as idler and pump photons, respectively, one can generate the fifth harmonic (FH) (5ω = 3ω + 3ω - ω) at 161 nm in the VUV for the fundamental at 805 nm. The experimental setup is presented in Fig. 1. Since the conversion efficiency is quadratically dependent on the TH pump intensity and only linearly dependent on the idler intensity, we have developed a 12-mJ 1-kHz Ti:sapphire CPA laser system at 805 nm optimized for efficient third harmonic generation (THG) [8]. Typical pulse duration was about 135 fs with a time bandwidth product of 0.62. Input pulses with an energy of 0.35 mJ (268 nm, pump) and 0.25-mJ (idler) were coupled into an argon-filled fused-silica hollow waveguide with a circular cross-section having an inner diameter of 100 μm and length of 25 cm (see Fig. 1). The generated VUV signal pulses were separated with a dichroic mirror from the pump and the idler pulses and directed to a commercial energy meter calibrated for the VUV at 157 and 193 nm (Startech). The remaining pump and idler were blocked by a VUV interference bandpass filter at 160 nm contributing to a negligible background in the VUV energy measurements. The measured VUV energy was verified to a precision within 10% by calibrated diamond and Pt-Si photodiodes [9].

 

Fig. 1. Schematic of the experimental setup. THG: third-harmonic generation, D1: delay stage to temporally overlap the pump and the idler pulses, D2: delay stage for autocorrelator, CaF2: 1-mm thick calcium fluoride plate, VUV Det.: VUV energy meter.

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For temporal pulse characterization in the VUV we applied noncollinear autocorrelation using non-resonant two-photon absorption (TPA) in bulk crystals as intensity-dependent broadband nonlinear detector. The coherent FH beam was divided into two parts by a spatially split mirror, one of the pulses was delayed in time by the D2 delay line, and then both parts were focused into a two-photon absorbing crystal plate of CaF₂ with a thickness of 1 mm (see Fig. 1) [10]. In order to keep the dependence of the absorption on intensity close to quadratic, we adjusted the time-dependent drop in transmission to be no more than 8%, resulting in a systematic error of the pulse duration less than 1%. For an easier handling (no vacuum) of the generated FH pulses, we used a commercial argon-filled glove-box chamber equipped with a closed-loop gas purification system (M. Braun GmbH).

 

Fig. 2. Pressure dependence of the fifth harmonic pulse energy with EH₁₁ modes excitation: the length of the hollow waveguide is 25 cm and the diameter 100 μm. The dotted curve with squares is the experimental data, and the solid curve is the result of the theoretical calculations.

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In the case when the pump and the idler were both coupled nearly perfectly into the EH₁₁ mode with an input spot size of about 65 μm (see Ref. [11]), the typical pressure dependence of the generated FH is presented in the Fig. 2. A pronounced peak at about 28 Torr can be observed. The very low pulse energy at the FH is mainly attributed to the low pressure. The theoretical phase-matching pressure for FWM with the given pump, idler, and signal pulses in the EH₁₁ mode is 32 Torr, as illustrated by the solid curve, which results in a low nonlinearity at such low pressure and consequently low efficiency. The maximum FH energy was only about 50 nJ which is not sufficient for most applications.

In the small-signal regime at phase-matching the VUV intensity scales quadratic with the gas pressure. On the other hand, the propagation losses for higher-order transverse modes, which are phase-matched at higher pressures, are much larger than the losses for the fundamental mode [12]. Therefore, in order to increase the VUV pulse energy the fraction of the input beam energy that is coupled into higher-order waveguide modes has to be optimized. We increased the energy coupled into higher-order modes by reducing the input spot sizes at the fiber entrance. Additionally, we have developed a metal-organic chemical vapor deposition reactor for coating the inner surface of the hollow fiber with diameters smaller than 200 μm with pure aluminum using the precursors (TiCl₄ and dimethylethylamine alane) described in Ref. [13]. The linear polarizations of pump and idler waves were preserved during their propagation in coated and uncoated hollow fibers suggesting that only hybrid electric EH_1n transverse modes [11] were excited in both cases.

 

Fig. 3. Pressure dependence of the fifth harmonic pulse energy when three waves are optimized for higher transverse modes: (a) for the uncoated and (b) for the coated hollow waveguide with the length 25 cm and diameter 100 μm. The black curves with circles are the experimental data, and the red curves with triangles are the theoretical calculations.

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The dependence of the fifth-harmonic energy on argon pressure is shown in Fig. 3 for an uncoated (a) and a coated (b) waveguide. The observed peaks are in the range from 80 to 200 Torr, which covers several phase-matching pressures for combinations of different spatial modes of pump, signal and idler waves. Using a coated waveguide, pulse energies up to 600 nJ were obtained at a gas pressure of 150 Torr, which corresponds to the phase-matching pressure with all waves propagating in the EH₁₂ mode. When using an uncoated waveguide, the maximum pulse energy was only 250 nJ and was achieved at 125 Torr and at 95 Torr. The pulse spectrum obtained using an uncoated waveguide at 115 Torr is shown in Fig. 4. The central wavelength was 161 nm and the FWHM spectral width was about 0.55 nm, corresponding to the transfer-limited pulse duration of 70 fs, if a Gaussian pulse is assumed. We have to note that the measured bandwidth was close to the spectral resolution of the spectrograph (about 0.3 nm).

 

Fig. 4. The measured (black curve with circles) and calculated (red curve) spectra at 115 Torr. The calculated spectral phase is given by the dotted blue curve.

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A typical autocorrelation of the fifth harmonic measured by two photon absorption is shown in Fig. 5. The FWHM is about 230 fs, corresponding to a pulse duration of 160 fs and a time bandwidth product of 1 (for Gaussian pulses), which is close to the length of the pump and idler pulses. The propagation through the 0.4-mm-thick MgF₂ entrance window to the glove box and of the 1-mm CaF₂ crystal plate in the TPA autocorrelator, used in the experiments, was found to be not completely negligible at 160 nm. Based on experimentally estimated deteriorations of the pulses at the FH by comparable windows of the same material [10], we calculate approximately a sub-140-fs VUV pulse duration at the exit of the fiber comparable to the idler pulse duration and more than two times the calculated pump-idler group velocity mismatch.

 

Fig. 5. Experimental autocorrelation trace using non-resonant two-photon absorption in 1-mm CaF₂ plate (dots) and the corresponding fit (solid curves) assuming a Gaussian pulse shape.

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3. Theoretical description and discussion

The numerical simulations were performed using a model based on a generalized version for the forward Maxwell equation [14] with inclusion of the transverse higher-order mode structure and without the assumption of slowly varying envelope approximation. Our model incorporates the following physical effects: generation of all possible frequencies from the input fields, dispersive properties and the transverse mode structure of the hollow waveguide calculated by using the Marcatili-Schmeltzer theory [11], multimode light propagation including nonlinear coupling between the transverse modes, influence of plasma and photoionization calculated by the two-color generalization [15] of Faisal-Reis-Keldysh model, self- and cross-phase modulations, and other higher-order nonlinear effects for all spectral components. A detailed description of the model will be presented elsewhere.

The parameters for numerical simulations correspond to the experimental ones (see the captions of Figs. 2 and 3). We have checked numerically that high-order transverse modes EH_1n with n more than 5 do not influence the process due to very high losses, therefore we have considered only 5 first transverse modes. In the experiments, the formation of plasma before the waveguide entrance causes an irregular redistribution of the energy among the transverse modes. Since the exact information about this redistribution is not available, we have accounted for it by assuming equal energies at the input for all five transverse modes. The results of the numerical simulations for the pressure dependence of the FH energy are presented in Figs. 2 and 3 by the red curves with open triangles. A good agreement with the experimental results is achieved, reproducing both the peaks for lower pressures as well as plateau for higher pressures, without any additional fit or renormalization. The resulting pulse has a spectrum with FWHM of 0.4 nm as presented in Fig. 4 (red curve), a smooth spectral phase (dotted blue curve), and a duration of 120 fs also in reasonable agreement with the experiment. Additionally, we have checked the validity of our assumption about the input energy distribution by assuming that input energy is concentrated only in the two lowest modes. In this case we obtained almost no difference between coated or uncoated waveguides, and the pressure dependencies contradicted the experimental data.

The crucial role of higher-order transverse modes in the FH generation suggested by the experiment and also confirmed by the model can be understood as follows. Efficient generation of the FH is possible only for pressures fulfilling the phase-matching condition. For higher transverse modes the phase-matched pressure is much higher, and the efficiency increases quadratically with the nonlinearity which is proportional to pressure. For a waveguide diameter of 100 μm there exist many phase-matched pressures for different combinations of modes, which are located in the region from 30 to 600 Torr. Combinations of EH₁₁, EH₁₂ and EH₁₃ modes have phase-matching pressures between 80 and 200 Torr. This explains the shape of the pressure dependence curves (see Fig. 3) with maxima between 80 and 200 Torr and the plateau for higher pressures.

Coating the inner surface of the fiber with aluminum strongly reduces the losses of the TH and FH radiation because aluminum is characterized by a relatively large magnitude of the dielectric function and small losses for UV frequencies. This effect is even stronger pronounced for higher-order transverse modes. This explains the increase of the efficiency in the case of the coated fibers.

Despite the efficiency of the FH generation is higher when using higher-order EH1n (n≥2) modes in comparison to fundamental EH₁₁ mode, the overall conversion efficiency is still considerably lower than the reported ones in the literature for similar experiments at longer wavelengths [5–7]. Our simulations show, that the major reasons for the low efficiency are (i) higher linear losses for the higher-order modes and (ii) plasma formation dominating at tighter focusing of pump and idler at the hollow fiber tip. The only published quantitative estimation of the yield of the FH generation by FWM in hollow waveguides is 0.1% of the output energy of the TH using cascaded phase-matching in the fundamental mode at much lower intensities [16], which is an order of magnitude lower efficiency than in our results. In spite of the low FWM efficiency the presented fifth harmonic generation setup delivers sub-200-fs VUV pulses with unprecedented average power of up to 0.6 mW at 1 kHz repetition rate. This is more than two orders of magnitude higher than the reported results in the literature so far to our knowledge by using nonlinear optical processes in gases.

4. Summary and Conclusion

In conclusion we demonstrated the generation of fifth-harmonic pulses at 161 nm from a Ti:sapphire laser by four-wave difference-frequency mixing in an argon-filled hollow waveguide. The efficiency was greatly improved by coupling to higher waveguide modes, as well as by coating the inner surface of the waveguide with aluminum. A numerical model of the process yields a basic understanding of the role of higher-order transverse modes and the coating. Further calculations (not shown) predict that the acceptance bandwidth of FWM process allows the generation of sub-10-fs VUV pulses using correspondingly shorter input pulses.