1. Introduction

Optical frequency combs have become an indispensable tool for optical frequency metrology [1]. A frequency comb consists of an array of continuous wave (cw) laser modes that are regularly arranged in the frequency domain and can be phase-coherently referenced to an atomic clock. Not only does this allow for absolute optical frequency calibration of cw lasers, but the individual comb lines themselves may also be used for direct frequency comb spectroscopy (see e.g. [2, 3]). While the application of narrow band cw lasers required for precision spectroscopy is limited to wavelengths up to the near ultraviolet, frequency combs can be extended to the extreme ultraviolet (XUV) spectral region using high harmonic generation (HHG) in a jet of noble gases. This gives access for high resolution laser spectroscopy to this hitherto unexplored spectral region where nearly all elements show primary resonances. To allow the absorbing atom, ion or molecule to single out the modes from the comb, the pulse repetition rate, which sets the mode spacing of the frequency comb, must be larger than the observed line width. For many years the standard method to reach the high intensities required for high harmonic generation was to reduce the repetition rate of the laser system typically to the kHz regime in order to concentrate the available average power in only a few pulses per unit of time. Recently, HHG inside an external enhancement cavity, very much like resonantly enhanced second harmonic generation, has been introduced which can operate at the full repetition rate of the oscillator [4, 5]. While the initial experiments did not generate sufficient XUV power for spectroscopy of interesting charged hydrogen-like systems such as He⁺, Li⁺⁺, significant improvements have been made so that µW’s of power can now be generated [6]. High harmonics generated collinearly to the fundamental light, as in most experiments to date, are difficult to extract from the generating cavity because no material exists that is transparent in the XUV. Therefore the technical challenge is to design an output coupling mechanism with minimum losses in the infrared (IR).

In previous experiments, a thin intracavity plate was installed at Brewster’s angle for the infrared, and total external reflection of the XUV radiation [4, 5, 6] was used for output coupling. However, this Brewster’s plate output coupler brings about several problems: First, it is typically useful in a limited wavelength region only since its reflectivity is determined by material properties. For the sapphire plates used in previous experiments, the reflectance decreases drastically at wavelengths shorter than 40 nm, thereby limiting the order of harmonics that can be coupled out. Second, with enhanced intracavity pulse energies and peak intensities [7], nonlinear effects arising from intracavity materials can no longer be neglected [6, 8]. Third, in contrast to cw cavities, intracavity group delay dispersion (GDD) has to be taken into account when coupling a broadband oscillator comb into a femtosecond enhancement cavity. So far, the use of chirped mirrors has been necessary to exactly compensate for the large contributions of the Brewster’s plate output coupler to the intracavity GDD. However, a reliable compensation with chirped mirrors is difficult for high intracavity peak intensities due to the intensity dependence of the GDD arising from the nonlinear effects of the intracavity material, let alone the technical challenges of manufacturing precisely chirped mirrors.

Several alternative ideas for XUV output couplers have been proposed to address these problems. Yost and co-workers report on an XUV grating that is etched into the surface of one of the dielectric mirrors of the cavity [9]. Another proposed possibility that does not require to introduce material inside the enhancement cavity, is to generate high harmonics that are emitted in a direction which is non-collinear with the driving IR beams [10, 11]. Thus, when employing carefully designed low dispersion high reflectors, dispersion compensated enhancement cavities without chirped mirrors become possible. The direction and the efficiency of the high harmonic emission crucially depend on the phase matching between the generated harmonics and the driving beam. For conventional high harmonic generation from a single fundamental beam, the phase matching condition is satisfied only for the collinear direction. However, for two driving beams crossing at a common focus, numerical calculations suggest that phase matching is also possible non-collinearly depending on the initial atomic densities in the interaction region, the crossing angle between the fundamental beams, and the driving intensity [11, 12].

 

Fig. 1. Schematic of the experimental setup: BS, beamsplitter; DS, delay stage; PAM, piezo-actuated mirror; MW, moveable wedge plate

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Besides a favorable power scaling compared to the Brewster’s plate method, NCHHG also has the advantage of spatially separating the high harmonic radiation and the IR. This is of particular interest with regard to spectroscopy in the XUV, because it might render the use of additional filters for the infrared unnecessary. These are required for setups based on Brewster’s plate output couplers in order to block the residual IR reflection which is typically orders of magnitude stronger than the XUV output. Therefore, since the filters are only partially transmissive in the XUV as well, NCHHG will result in about one order of magnitude higher absolute XUV power available for spectroscopy. In addition, the restriction of XUV outcoupling to wavelength regions, where appropriate materials for Brewster’s plates exist, is lifted by NCHHG, thus making it a promising outcoupling method in the short wavelength region of higher order harmonics.

Thus far, to our knowledge, there has been no experimental demonstration of NCHHG in a setup that allows to take advantage of all the benefits mentioned above and that is therefore directly applicable to cavity-enhanced HHG [12]. Here we demonstrate that a collimated XUV beam can be emitted to the bisector of the crossing angle of two IR driving beams in such a way that it can be separated from the IR in the farfield. We show that the loss introduced by such a beam separation is low enough to allow the use in a high Q enhancement resonator without any additional intracavity material besides a gas jet. For this proof of principle, we do not use an enhancement resonator yet but investigate possible geometries with amplified pulses from a chirped pulse amplification system.

2. Experimental setup

The schematic of our setup for non-collinear high harmonic generation is shown in Fig. 1. A chirped pulse amplification (CPA) system (Spectra-Physics Spitfire) that is seeded by the output of a mode-locked Ti:Sapph oscillator (Femtolasers Femtosource 20) provides 150-fs-duration pulses with an average pulse energy of 1 mJ at a repetition rate of 1 kHz and at a wavelength centered about 790 nm. The pulses from the CPA system are propagated along two different paths of equal length and then recombined non-collinearly inside the vacuum chamber with two identical spherical mirrors (f=300mm). Whereas a piezo-actuated mirror is added into one path for precise spatial alignment, the other path contains a pair of wedge plates of which one is mounted on a piezo-actuated stage and a manual delay stage in order to precisely control the delay between the pulses from each path. The two beams are focused into a Xe gas target inside a metal tube (inner diameter: 1 mm) through a transverse bore hole of 200µm in diameter. During the experiments the gas flow supplied to the nozzle is controlled by a regulating valve (Pfeiffer Vacuum EVR 116) and the nozzle itself can be precisely aligned for maximum non-collinear high harmonic emission with a three-dimensional translation stage. The waists of the driving beams are w ₀=24µm resulting in an intensity of 2.6×10¹⁴W/cm² at the focus for each beam. An XUV sensitive camera with a back-illuminated charge-coupled device (CCD) without anti-reflection coating (Princeton Instruments PIXIS XO-100B) allows to monitor and record the spatial profile of the high harmonics at a distance of ~750 mm from the focus. The scattered laser light and high harmonics of orders q<11 are blocked by two aluminum filters (200 µm thickness, Lebow 0.2Al-0-L3.0).

 

Fig. 2. Two-dimensional beam profiles of high harmonics obtained with a CCD camera. Top, bottom: Collinear high harmonics observed with only one driving beam, respectively. Center: non-collinear high harmonics observed with both fundamental beams open. Note that the collinear harmonics vanish in that case.

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3. Discussion of results

Typical images obtained from the CCD camera at a Xe gas flow of 6×10^-2 mbar l/s and a backing pressure of about 10^-2 mbar are presented in Fig. 2. The top and bottom pictures show the collinearly generated high harmonics of both fundamental beams for the case when the respective other driving beam is blocked. When both fundamental beams overlap at the common focus, collimated XUV harmonic radiation can be observed in the bisection direction. Adjusting the beam overlapping, the delay and the nozzle position for maximum non-collinear HHG, the collinear emission vanishes so that the full XUV power becomes available at the bisectrix (center picture in Fig. 2).

The horizontal beam profiles of both collinear and non-collinear high harmonics shown in Fig. 3(a) are obtained by integrating the images in the vertical direction. For comparison, the calculated beam profiles of the driving IR beams have been added. The full non-collinear angle is determined to be 30 mrad from the images. The high harmonics generated non-collinearly exhibit a collimation that is comparable to that of the collinear high harmonics and show a broad pedestal. Although we currently do not have a quantitative model to explain the latter observation, one might potentially attribute the pedestal to three different effects: Since high harmonics of different orders show a different angular divergence, the pedestal might be a result of different harmonic orders (q>9) overlapping. At the same time, high harmonics of the same order are known to show an inherently different behavior in terms of divergence resulting from the short and long trajectories of the electrons until recombination, respectively [13]. In addition, the angular emission pattern will also be affected by the spatial intensity distribution of the two interfering IR pulses at the focus, which in turn depends on their timing and phasing.

 

Fig. 3. (a) Horizontal beam profiles of collinear (dashed/green, dotted/green) and non-collinear high harmonics (thick solid/blue) obtained from vertical integration of the recorded images. The non-collinear angle is set to 30 mrad. To illustrate the potential as an outcoupling method, the estimated beam profiles of the fundamental infrared beams are also shown (thin solid/red). (b) Same as (a) for a larger non-collinear angle (45 mrad). The same intensity scale is used for both (a) and (b). Note that for this angle, the intensity ratio between collinear and non-collinear high harmonics stays similar.

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The overlap between the driving infrared and the non-collinear high harmonics along the bisector of the two fundamental beams found in Fig. 3(a) suggests an investigation of NCHHG at larger non-collinear angles. Figure 3(b) shows the results of the same experiment as described above with an increased non-collinear angle of 45 mrad which is the maximum achievable angle in our current experimental apparatus. For this measurement, the distance between the focus and the CCD camera has to be decreased to 525 mm to monitor the full two-dimensional beam profiles. After careful alignment of the two IR beams, collimated non-collinear high harmonics are observed close to the bisectrix and with a more pronounced pedestal. Figure 3 shows that the intensity ratio between the collinear and non-collinear high harmonics does not deteriorate at 45 mrad. At the same time, however, the spatial overlap between the IR and the high harmonic radiation is reduced. This behavior is of crucial importance for efficient output coupling of the XUV from an enhancement cavity.

 

Fig. 4. Possible configuration of non-collinear high harmoncis generation in an enhancement cavity. With a small outcoupling hole between the folding mirrors (inset), the generated XUV beam can be efficiently outcoupled without introducing any additional material into the cavity (see also [10]).

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As shown in Fig. 3, the angles at which we observe the maximum non-collinear high harmonic intensity might deviate from the bisectrix of the two IR beams. We find that the emission direction depends on the relative intensities of the two beams and their precise timing. When attenuating one of the beams, the direction of the XUV emission changes continuously towards the un-attenuated IR beam until only collinear emission remains. Even though these sensitivities can lead to increased beam pointing instabilities, we believe that a stable focus position can still be obtained with appropriate optics. The optimum conditions for NCHHG are also found to be dependent on the shape of the gas nozzle. When changing the inner diameter of the metal tube or the size of the holes, the intensity and the beam profile of the generated non-collinear harmonics are observed to vary. We plan to systematically investigate the dependence of NCHHG on the parameters mentioned above in the near future.

To gain some insight in the underlying physics, we have set up a code based on a first crude model: Our simulation determines the intensity dependent amplitude and phase of the high harmonic dipole moments of the atoms in three dimensions. By propagating the emitted waves, we obtain the far field distribution that qualitatively agrees with the observed patterns shown in Fig 2. However, it remains to be tested whether and to what extent this simplified description is justified, because our current model completely neglects effects such as the time variation of the refractive index due to the generated charges. The latter effect was taken into account in more sophisticated numerical simulations that Fomichev et al. and Wu and Zeng [11, 12] have performed based on the so-called Lewenstein model [13] and the integration of the time-dependent Schrödinger equation, respectively.

One possible implementation of NCHHG as an out-coupling method in femtosecond enhancement cavities is to use two separate Fabry-Perot cavities arranged at a small crossing angle such that they share a common focus (Fig. 4)[10]. In this scheme, the non-collinear harmonics can be outcoupled through an opening between the two adjacent folding mirrors which is assumed to be spherical here (see inset of Fig. 4). Efficient outcoupling without introducing severe diffraction loss into the cavities requires the size of the hole to be carefully chosen. Since the eigenmode of the buildup cavities will be modified slightly to avoid high intensities at the outcoupling hole, a ray-tracing analysis of a cavity round trip is required to precisely evaluate the diffraction loss introduced by the hole. In this brief report, we estimate the diffraction loss from the intensity of an unmodified Gaussian beam hitting the outcoupling hole. Figure 5 shows the diffraction loss calculated in this way as a function of the outcoupling efficiency which is defined here as the ratio between the outcoupled XUV power and the collinear high harmonic power obtained with a single driving beam. At the larger non-collinear angle, more than 20 % of outcoupling efficiency can be obtained by introducing only 4×10^-5 of loss into the cavity. These values are far superior to those of a sapphire Brewster’s plate. For higher order harmonics, which are expected to be more collimated so that the diameter of the outcoupling hole can be reduced, the outcoupling efficiency should be even larger.

 

Fig. 5. Estimation of intracavity loss as a function of outcoupling efficiency for noncollinear angles of 30 mrad (solid (blue)) and 45 mrad (dashed (red)). Here, the outcoupling efficiency is defined as the power-ratio between the outcoupled XUV and the collinear harmonics.

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4. Conclusion

To conclude, we experimentally demonstrated non-collinear high harmonic generation with two crossing beams from a chirped pulse amplification system. Our results show that NCHHG is feasible for outcoupling XUV radiation from an enhancement cavity. This novel outcoupling method does not introduce any material into the cavity and is thus expected to be ideal for the next generation of cavity-assisted HHG with much higher intracavity pulse energies.