## 1 Introduction  
The newest light sources, extreme ultraviolet (XUV) and X-ray free electron lasers (FELs), are extending laboratory laser experiments to shorter wavelengths, adding element and chemical state specificity by exciting and probing electronic transitions from core levels. The high pulse energies available ensure that spectroscopies belonging to table top laser, such as, for instance, non-linear optics, can cross the frontier into this new wavelength range. Since the first short wavelength FEL FLASH began operation in Hamburg [1], development has been very rapid. The Linac Coherent Light Source (LCLS) was specified to produce 200 fs pulses, but soon produced few fs pulses [2]. Schemes have been developed to create multi pulse and polychromatic radiation, and FEL light has been used to pump atomic lasers in the X-ray region [3; 4]. The XUV/soft X-ray laser FERMI is the first fully coherent FEL facility, as the light possesses the full longitudinal coherence lacking in a pure SASE (Self-Amplified Stimulated Emission) configuration [5]. Coherence is achieved by initiating the FEL process with an external seed. The coherence properties of the seed are indeed transferred to the electron modulation, leading to coherent emission at the undulator resonance and at its harmonics. A similar configuration is adopted in the new facility in China; the Dalian Coherent Light Source (DCLS, Dalian, China) covers the vacuum-ultraviolet (VUV) photon energy range [6]. Advanced schemes of self seeding, both in the soft and hard X-rays, were adopted at LCLS [7] to improve coherence of a SASE FEL. With the development of coherent FEL radiation sources, a new era of X-ray spectroscopy commenced, which may have a comparable impact to that of lasers in optics and spectroscopy [8]. In non-linear ultrafast time-resolved techniques, state-specific information is often provided through multiphoton resonances with combinations of sequential photons. Theoretically, combinations of multiple X-ray photons resonant with core transitions can also characterize different excitation processes due to specific sequences of light-matter interaction. Thus,particular sub-processes can be enhanced by matching the pulse frequencies to transitions between molecular Eigen-states. This provides high selectivity and flexibility due to momentum and energy conservation of the interacting photons with the material. The different non-linear processes can typically be ordered by the number of involved photons: sum-frequency generation, two-photon absorption and stimulated emission with two photons, followed by other non-linear X-ray phenomena, time resolved transient gratings (XUV-TG) or four wave mixing (XUV-FWM) spectroscopy (see [8] and references therein). Fluorescent decays in the soft X-ray region allow unique access to the structure of the occupied valence states, while keeping the element selectivity and chemical state specificity of soft X-ray spectroscopies [9]. Unfortunately, the probability for a fluorescent decay in the soft X-ray range is below 1%. With a typical spectrometer acceptance of less than \(10^{-5}\) of the full solid angle into which the fluorescence is emitted, soft X-ray emission spectroscopy (XES) or resonant inelastic X-ray scattering (RIXS) are experiments where single photons have to be counted for hours in order to obtain useful information. Using stimulated emission spectroscopy combined to high intensity, small bandwidth, tunable soft X-ray beams of different colors, a stimulated beam containing the same information as a fluorescence spectrum can be formed--mitigating the low acceptance angle of the spectrometers, and suppressing the dominating Auger decays that also create electronic damage to the sample.