We numerically study chirped four-wave mixing for VUV pulse generation in hollow waveguides filled with a noble gas. Taking into account ionization effects we predict the generation of signal pulses at 160 nm with shortest durations up to 6.5 fs, highest pulse energy up to the mJ level and maximum energy efficiency of about 30% by broadband chirped idler pulses at 800 nm and narrow-band pump pulses at 270 nm. Using cascaded processes sub-10-fs pulses in the spectral range from 90 to 140 nm can also be generated.
©2008 Optical Society of America
In recent years large progress has been achieved in the generation of few-cycle pulses in the near IR with pulse energy in the mJ range. Even shorter attosecond pulses can be produced by using high-order harmonic generation (HHG) in the soft X-ray range. In contrast in the UV and VUV spectral region the generation of shortest pulses with relatively high energy is not standard and in general not yet so impressive. However, ultrashort pulse sources in the UV and VUV spectral range are essential tools in physics, chemistry, biology and material sciences requiring further progress in the development of VUV fs pulse generation methods.
Sub-10 fs pulses in the UV range can be generated by optical parametric amplification in nonlinear crystals . Frequency mixing in gases enables to extend the wavelengths deeper into the VUV. At 155 nm a pulse duration of 300 fs has been achieved by near-resonant four-wave mixing . In Re.  13 fs pulses have been produced using chirp compensation in the fifth-harmonic generation in the HHG process. Recently 11 fs pulses at 162 nm with 4 nJ pulse energy has been achieved directly by pumping with 12 fs pulses at 810 nm .An alternative method for UV and VUV fs pulse generation with a possible much higher pulse energy is the use of non-resonant four wave mixing (FWM) in hollow waveguides filled with a noble gas. Generation of 8 fs pulses with 1µJ at 270 nm were reported with this method by pumping with the fundamental and second harmonic of a Ti:sapphire laser . This method has been extended into the VUV range generating 600 nJ, 160 fs pulses at 160 nm using the fundamental and third harmonic of a Ti:sapphire laser as idler and pump, respectively: ω S=ω P+ω P-ω I with ω P=3ω I and λ I=800 nm . Recently we studied numerically  the potential of this method for sub-5 fs VUV pulse generation and predicted the generation of 2.5 fs pulses at 160 nm by 10 fs, 800 nm idler and narrow-band pump pulses at 267 nm. However, in this case the VUV pulse energy is limited to the nJ range. Therefore a modification of this method for sub-10 fs VUV pulse generation is desirable which allow a significant increase of the output pulse energy. An elemental option for this purpose is chirped four-wave mixing in hollow waveguides.
In the present paper we study the generation of VUV pulses by chirped FWM with broadband stretched, positively chirped near-IR idler pulses and narrow-band UV pump pulses. Assuming idler pulses at 800 nm and its third harmonic as pump pulses with an energy in the range of 1 mJ high-energy VUV signal pulses can be generated which can be compressed to the sub-10 fs level by a layer of MgF2 with normal dispersion. The FWM process has a maximum efficiency of about 30% of the pump pulse at 276 nm converted to 160 nm. The shortest predicted VUV pulse duration is 7 fs with 200 µJ energy, but a pulse energy of 1.4 mJ and 20 fs duration is achievable for an input energy of J I=J P=20 mJ and stretching of the input pulses to 10 ps to avoid ionization. We show that by cascaded FWM sub-10 fs pulses with smaller wavelengths in the range from 90 to 140 nm with a pulse energy in the range of 1µJ for an input energy of J I=J P=2:6 mJ can also be generated.
For the numerical simulations, we used a model based on a generalization of the forward Maxwell equation  taking into account ionization effects. This first-order equation is derived from the wave equation neglecting the back-reflected wave but without the use of the slowly-varying-envelope approximation. The complex-valued Fourier transform E(x, y, z,ω) of the real-valued optical electric field E(x, y, z, t) can be written as E(x, y, z,ω)= E(z,ω)F(x, y,ω), where F(x, y,ω) is the fundamental transverse mode profile, z is the coordinate along the waveguides, x and y are the transverse coordinates. The evolution of the Fourier transform of the field E≡E(z,ω) along the z coordinate is then governed by the following equation:
Here β(ω) is the propagation constant for the fundamental transverse mode, v is the velocity of the moving coordinate frame. The nonlinear polarization P nl(z, t)=ε 0 χ (3) E 3(z, t)+P pl includes the Kerr nonlinearity and the polarization P pl due to ionization. Ppl is calculated using the multiphoton generalization  of the Keldysh-Faisal-Reiss model . The details of the model are presented elsewhere .
The propagation constant β(ω) includes the contribution of the bulk gas and the waveguide one: β(ω)=β gas(ω)+β wg(ω). The contribution of the gas is calculated from the Sell-meier formula . The waveguide contribution is given by the Marcatili-Schmelzer theory . The loss (ω) is determined only by the waveguide contribution. The contribution β wg(ω) is anomalous, therefore it leads to an increasing range of negative group velocity dispersion (d 2 β/dω 2<0). In addition, this contribution also influences the wave vector mismatch for the FWM process δk=2β(ω P)-β(ω I)-β(ω S). This is illustrated in Fig. 1(a)–(c) for a hollow waveguide filled with argon, where the group velocity dispersion (GVD), δk and the loss α are shown for different waveguide diameters. One can see, that with decreasing waveguide diameter a stronger waveguide contribution to dispersion leads to a shift of the zero-dispersion wavelength to shorter wavelengths [Fig. 1(a)]. For d=100 µm at 1 atm the GVD is anomalous (d 2 β/d ω 2<0) for λ>600 nm. A larger waveguide contribution to dispersion enables to fulfill the phase matching condition δk=0 for a higher pressure [Fig. 1(b)]. For d=100 µm the phase matching pressure is 27 Torr. On the other hand, the loss is higher for smaller diameters [Fig. 1(c)], but it is still acceptable for a waveguide length less than 1 m.
We preformed numerical simulations of Eq. (1) using the split-step method. We consider hollow waveguides filled with argon and an inner wall made from fused silica. To study high-energy VUV pulse generation by chirped FWM we consider a broadband idler input pulses at 800 nm stretched to the duration of the pump and a narrow band third-harmonic pump of the same energy (J P=J I, τ I=τ P). In particular, we assume an input idler broadband pulse stretched from 3 fs to the pump pulse duration by propagation through a piece of dispersive MgF2 glass. For the pump at 273 nm unchirped pulses were assumed with duration of 300 fs, 1 ps and 10 ps.
In Fig. 2(a) one can see the results for a waveguide diameter d=100 µm after z=20 cm propagation with input pulse energy J I=J P=0:65 mJ and a duration of 300 fs. The signal pulse energy in this case is 200 µJ and the spectrum in Fig. 2(a) (red curve) reaches from 150 nm to 170 nm, the spectral phase (green curve) shows a negative chirp opposite to the chirp of the idler. It allows compression of the output signal pulse [Fig. 2(b)] to the duration of 6.8 fs by a 1 mm layer of MgF2.
In Fig. 2(a) the input intensity at the center of the waveguide is 120 TW/cm2. This intensity should not be increased significantly to avoid a damage of the waveguide wall. However, the energy of the pulse can be increased by stretching the pulse to a larger pulse duration. In Fig. 2(c) the signal spectrum and in Fig. 2(d) the pulse shape after compression are shown for an input pulse energy of idler and pump of J I=J P=2:6 mJ, a duration of 1 ps and 120 TW/cm2 peak intensity. The VUV pulse energy is 860 µJ and the pulse duration (FWHM) after compression is 11 fs. However, due to incomplete chirp compensation by the layer of MgF2 in Fig. 2(d), a strong tail arises after compression. As seen in the inset of Fig. 2(d), an ideal compensation of the spectral phase yields a compressed pulse with a FWHM of 4.7 fs without tail. In Fig 2 (e), (f) the signal output pulse is shown for an input pulse energy J I=J P=20 mJ and a pulse duration of 10 ps. In this case, the compressed pulse has a FWHM of 20.6 fs obtained after propagation through 23 mm of MgF2. The VUV pulse energy in this case is J=1.4 mJ with about 0.4 mJ in the main part of the pulse. An ideal phase compensation yields a FWHM of 8.9 fs without pulse tail. The tail carries a significant part of the pulse energy [around 60 % for Fig. 2 (a), (b), 80 % for Fig. 2 (c), (d) and 70 % for Fig. 2 (e), (f)]. Note that we assumed the simplest method for the production and compensation of the chirp, but more sophisticated methods should improve the pulse quality.
Output energies and pulse durations of the VUV pulses in dependence on the input energies are presented in Fig. 3 for waveguides with diameters d=100, 200 and 300 µm and corresponding phase-matching pressures 28, 7, and 3 Torrs, respectively. In Fig. 3 (a), (b) the signal characteristics are show for varying input energies J I=J P but fixed input idler and pump pulse durations τ I=τ P=300 fs. For the waveguide with the smallest diameter d=100 µm the energy efficiency reaches a level of about 30%. With increasing the diameter the energy efficiency of the chirped FWM process decreases. For lower input energy J I=J P the VUV output energy J S grows with J 3 P. For higher input energies the energy transfer to the signal pulse is saturated due to the influence of photoionization in Ar. In Fig. 3(c), (d) signal pulse energy and duration are presented in dependence on the input pulse durations τ P=τ I for d=100 µm while keeping the input pump and idler intensity constant: I I=I P=120 TW/cm2. One can see that the energy of the signal output pulse grows almost linearly up to the range of 1 mJ. The duration of the signal pulse (black curve) is below 10 fs for input durations up to 1 ps.
A simplified description of the above-described dynamics can be obtained by an analytical solution for the signal pulse using the slowly-varying envelope and small signal approximation. Neglecting the di erence in the group velocities, the influence of GVD and the linear losses and assuming the phase-matching pressure Eq. 1 can be reduced to the equation for the complex amplitude of the VUV signal pulse . Here n 2 is the nonlinear refractive index of Ar per 1 atm; n 2=9.3×10-7 cm2 TW-1 atm-1, g=∫a 0 r J 4 0 dr/∫a 0 r J 2 0 dr≈0.566 is the nonlinear reduction factor (J 0(r) is the Bessel function of zero oder), p 0 is the phase-matching pressure and c is the speed of light in vacuum. Corresponding the Marcatili- Schmelzer theory  the phase-matching pressure depends on the waveguide diameter as p 0~d -2. For Gaussian shapes of the pump (with the FWHM duration τ P) and idler (with the FWHM duration τ I≡τ I 0 r obtained by r times stretching the idler) the solution of this equation is
where L is the propagation distance. One can see, that the signal amplitude is proportional to the conjugate of the idler one and therefore has an opposite sign of the chirp [see Fig. 2 (a), (c), (e)].
Performing the Fourier transformation of this solution and assuming ideal chirp compensation one obtains after back transformation for τ d=τ P a Gaussian pulse shape with FWHM which goes to √3τ I0 for r≫1 [see Fig. 3 (d), red line]. It is in reasonable agreement with the ideal chirp compensation for the numerically obtained results [see Fig. 3 (d), blue line]. The energy of the signal pulse is given by J S=(γL/τ Pd 2)2 J 2 P J I for τ P=τ I, where γ≈0.0078n 2 ω S cm3sTW-1atm-1. This analytical formula is in a good agreement with the numerical simulations in Fig. 3 below the saturation and reproduces the dependence on the energy of the idler and pump as well as the dependence on the waveguide diameter J 5~d -8.
The high efficiency of the FWM leads to the generation of frequency-shifted components below 150 nm due to cascaded processes. To study these processes we choose a different idler wavelegth λ I=570 nm than in Fig. 2, 3 to avoid the competition of processes which lead to the same signal frequency (as it appears for λ I=λ P/3). The pump wavelength was selected, as in the previous case, at λ P=267 nm and the pressure is 22.7 Torr (phase-matching for the FWM process ω S=2×ω P-ω I). The short-wavelength part of the spectrum for these input frequencies after 20 cm of propagation and with input parameters J I=J P=2.6 mJ, τ P=τ I=1.2 ps is shown in Fig. 4. It arises from different processes with different sign of the chirp, as seen in Fig. 4(a) (green curve). The cascaded processes ω S 1=ω S+ω P-ω I (129 nm) and ω S 2=2×ω S-ω I(102 nm) lead to a negative chirp, therefore these parts of the spectrum can be compressed by a normal-dispersive medium, here assuming argon at 1 atm. In Fig. 4(b), the pulse shape is shown, which is obtained by selecting a spectral range from 117 to 135 nm by a frequency filter and compressing the resulting pulse by 20 cm propagation in argon. The duration (FWH) of the pulse is 8.6 fs and its energy is 1 µJ, but the pulse shape has a leading edge. As seen in the inset an ideal compression yealds a 3.7 fs pulse without edges. The analogous procedure for the range from 90 to 104 nm results in a pulse of 5.9 fs duration and 43 nJ energy compressed by propagation in a 7 mm argon-filled waveguide [see Fig. 4(c)]. The ideal compression yealds 3.2 fs in this case. Note that the spectral part below 100 nm lies already above the one-photon resonance transition of argon but the loss coefficient α=4×10-2 cm-1 for 30 Torr leads only to a reduction of the signal by about 60%.
In conclusion, we studied numerically chirped four-wave mixing for VUV pulse generation in hollow waveguides filled with argon with unprecedented short pulse durations in the sub-10 fs range and high energies up the mJ level. In particular, we predict that by using a broadband chirped Ti:sapphire laser pulse as idler and a narrowband third harmonic as pump 6.8 fs VUV signal pulses at 160 nm with 200 µJ energy can be generated with a maximum efficiency of 30% from the pump for waveguide diameters of 100 µm. Ionization limits the maximum pump and idler intensity to the range of ~150 TW/cm, but the signal energy can be scaled by using longer pump and idler pulses. The generation of pulses with still shorter wavelength by cascadedFWM with a pulse energy in the range of 1 µJ is also predicted. By using this method tunable sub-10 fs VUV pulses in the range of 100 to 200 nm can be generated by using a frequency-tunable idler from an optical parametric amplifier (OPA) and a pump in the UV range.
We acknowledge financial support from the Deutsche Forschungsgemeinschaft (project He 2083/13-1). We are grateful to A. Husakou for useful discussions.
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