## 1 Introduction  
Coherent light sources in the mid-infrared spectral region (MIR, 3-20 \(\mu\)m) are highly important for studies in molecular science since a number of molecular vibrations have resonance in this wavelengthregion. The light source can be applied to various advanced molecular spectroscopies, such as frequency comb spectroscopy for the molecular fingerprint region [1], pump-probe spectroscopy to trace ultrafast structural dynamics [2], and control of photodissociation by selective excitation of vibrational states [3]. On the other hand, intense few-cycle MIR sources are recently getting much attention from the strong-field research community. By using a long wavelength driver laser pulse, it becomes possible to generate shorter wavelength high harmonics due to the ponderomotive energy scaling with square of wavelength [4]. It may result in shorter attosecond pulse generation via high harmonic generation. Recently, the scaling of the harmonic cutoff was clearly demonstrated with high energy MIR pulses [5]. A typical MIR femtosecond pulsed laser is based on combination of an optical parametric amplifier (OPA) and difference frequency generation. The OPA with a \(\beta\)-barium borate (BBO) crystal pumped by the fundamental of a Ti:sapphire amplifier generates a signal and an idler wave in the wavelength range between 1 and 2.5 \(\mu\)m, and the difference frequency generation between the signal and the idler in a AgGaS\({}_{2}\) crystal produces an MIR pulse [6,7]. Such a system has already been commercially available (TOPAS, light conversion). This provides the temporal width and the output energy of about 100 fs and several micro joules, respectively. The reason for using the sequential down-conversion process is that the wavelength difference between the pump pulse (800 nm) and the MIR pulse is so large that few crystals are transparent in the both wavelength regions. As an alternative to the MIR pulse generation with the second-order nonlinear processes, third-order nonlinear processes were considered [8]. Although third-order processes are much less efficient than second-order processes, a variety of nonlinear media are available since isotropic media can be used for third-order processes. In [8], the OPA signal (1.2-1.5 \(\mu\)m) and the fundamental of a Ti:sapphire laser were mixed in a calcium fluoride plate or a barium fluoride plate, which are transparent in the wavelength range from 200 nm to 10 \(\mu\)m. Tunable (2.4-7.7 \(\mu\)m) MIR pulses were produced by four-wave mixing (FWM) in fluoride plates [8]. On the other hand, a broadband spectrum cannot be generated in the MIR region with this scheme, because of a narrow phase-matching bandwidth. The output energy is also limited to a few hundred nano joule level due to the damage threshold of the fluoride plates. Using gas media instead of solid media would be effective in terms of phase matching bandwidth and damage threshold. However, the nonlinearity of gas media is much smaller than that of bulk media. To overcome the low efficiency, filamentation [9, 10, 11] in gases can be used. The balance between self-focusing and plasma self-defocusing makes the pulse propagate much longer than the Rayleigh range with a very high intensity. It results in a dramatic enhancement of nonlinear processes occurring in the filamentation zone. The enhanced nonlinear-optical processes in laser-induced filaments suggest a new strategy for frequency conversion of femtosecond pulses. In contrast to the third harmonic generation process, FWM does not suffer from phase mismatch caused by Gouy phase shift [12]. This means that the conversion efficiency of FWM can be significantly improved with a filamentation effect. Wavelength conversion of Ti:sapphire laser output by using FWM through filament to ultraviolet [13, 14, 15, 16], visible [17], infrared [18, 19, 20, 21, 22], and terahertz (THz) [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36] regions were demonstrated so far.