1. Introduction

Spectroscopy in the vacuum and extreme ultraviolet (VUV-XUV) regions of the spectrum has been recently made more accessible by the use of Ramsey-type techniques in combination with high-order laser harmonic sources [1–4]. This approach, in its different variants, has resulted in a valid table-top alternative to synchrotron radiation sources. In all cases, the method consists in generating the two identical time-delayed and accurately phase-locked harmonic pulses for spectroscopy starting from two delayed pump pulses in the near infrared.

One possible variant, based on frequency combs, generates the pump pulse pair by amplifying two successive pulses from a mode-locked train [5]. Varying the laser repetition rate allows one to finely control the pulse separation for delays in the 10 ns range, or longer.

On the other hand, the split-pulse approach relies on splitting a single amplified pump pulse and delaying the two resulting replicas in a Michelson interferometer. It has been shown [3] that a path difference of just a few millimeters in a Ramsey-Michelson setup is sufficient to achieve a resolving power of 10⁵, at the same level of current state-of-the-art synchrotron monochromators [6] and about one order of magnitude below that of recent Fourier-transform synchrotron spectrometers [7]. Our group has also recently demonstrated an evolution of such a scheme, combining measurements only in a limited subset of randomly chosen time-sampling intervals [8, 9], which led us to perform the first high-resolution XUV spectroscopy of an atomic transition with this simple split-pulse technique [10].

Scaling up the demonstrated performances of the split-pulse approach towards higher spectral resolutions requires moving to larger delays, in the range of several nanoseconds (corresponding to meters of arm displacements). Going much further is probably not necessary, as these delays can be well covered by the frequency-comb technique, which, on the contrary, cannot analyze energy levels with a short lifetime.

The realization of a Michelson interferometer with a large and accurately controlled delay may exploit some of the several technical solutions developed during the evolution of conventional Fourier Transform Spectroscopy (FTS). However, the system requirements for the application to Ramsey-type spectroscopy in the XUV are much more demanding. In fact, the interferometric control of the delay has to be guaranteed at the harmonic wavelength, which implies positioning uncertainties at the nanometer scale, or better. Moreover, due to the slow acquisition, typically dictated by the need to accumulate a sufficient number of ion/electron counts for each value of the delay, such a precise positioning has to be accurately stabilized over arbitrarily long times.

In addition to the issues typical of conventional FTS, the short-wave nature of the harmonic radiation gives further constraints connected to possible wavefront differences in the two pulses, which have to be kept small on the scale of the wavelength to obtain a good fringe contrast, whose degradation would make the ideal instrumental resolution set by the pulse delay useless. Moreover, using XUV harmonics for Ramsey-type spectroscopy poses additional challenges connected to the highly nonlinear generation process, which, besides amplifying the residual, unwanted, phase and intensity differences between the IR pump pulses by the harmonic order, might also introduce further distortions between the XUV pulses. Although these have been generally proved negligible at the intensities and harmonic orders investigated so far by more than a decade of experiments by our and other groups, moving to much shorter wavelengths might not be straightforward.

In the following we describe the realization and characterization of a large-beam Michelson interferometer suitable to be employed in a deeply unbalanced arm configuration for splitting-pulse XUV spectroscopy.

2. The interferometer

The interferometer is assembled on a vacuum compatible 1.5-m long optical breadboard, with a standard hole pattern that allows one to flexibly adapt the optical setup to different configurations. The breadboard is suspended by means of four springs (placed at a distance from the breadboard angles calculated to minimize the elastic deflections) inside a 160×50×30-cm vacuum chamber equipped with ports for pumping and for electrical connections, as well as with windows for optical access. Vacuum is important in this context both for minimizing dispersion effects in the long path traveled by the beams, and for efficient acoustical noise suppression.

The long path difference between the two arms of the interferometer requires the use of a large beam diameter to minimize the detrimental effects of diffraction and of non-ideal translation of the mirrors during the scan. The maximum optical path difference in the simplest configuration (no additional mirror) of our interferometer is ∆z=225 cm, corresponding to a maximum delay of 7.5 ns. The IR femtosecond pump beam, at λ ≈780 nm, has a diameter of 2w₀ ≈ 15 mm, and a corresponding Rayleigh length of z₀ = kw²/2≈227 m, which results in a very small $\mathrm{\Delta }w/w_0=\mathrm{\Delta }{z}^2/2z_0^2=5\cdot {10}^{-}{}^5$ relative variation of the beam width and a correspondingly negligible fringe contrast diminution even at the maximum delay. Also the condition ∆x ≪ w, where ∆x is the displacement of the mirror perpendicular to the delay axis during the scan, can be readily fulfilled in our setup. The large beam diameter also limits possible nonlinear effects and damage due to the high peak power of the laser beam, but has the drawback of requiring optics with a certified high quality over a sufficiently large area. Our system uses two hollow retro reflector cubes from PLX (of nominal quality λ/10 over 60 mm and a beam parallelism error of 0.5 arcsec) and a beam-splitter and compensator fused silica plates from Layertec (quality λ/20 over 60 mm). We performed a careful analysis of the optical system by measuring the angular displacement between the two arms of the interferometer with a long focal-length lens and a CCD camera. We estimated a combined effect of the cube front parallelism and of the beam-splitter surface error around 1.1 arcsec, while the cube contribution alone was compatible with the specifications.

Suspending the whole interferometer in a vacuum environment is the first solution to the problem of interferometer stability and insulation from vibrations. Coupling the breadboard of the Michelson interferometer to the environment through a system of springs constitutes a passive resonant system characterized by a low resonant frequency ν_o and a modest quality factor Q. Elementary considerations [11] for a system consisting in a mass hanging from an ideal spring give a vertical oscillation frequency ${\nu }_{ov}={(2\pi )}^{-}{}^1\sqrt{g/\mathrm{\Delta }L}$, being ∆L the equilibrium elongation of the spring, and an horizontal pendulum oscillation frequency ${\nu }_{oh}={(2\pi )}^{-}{}^1\sqrt{g/L}$, being L the total length of the spring. For our system ∆L≈270 mm, and L≈480 mm, giving resonant frequencies of 0.96 and 0.72 Hz, respectively.

In the simplest case of a single resonance, the transfer function for displacements at frequency ν is given by [12]

Far from resonance, the environmental noise is suppressed differently depending on the magnitude of the quality factor: in a first region (ν < Qν_o) we have |T(ν)| ≈ (ν_o/ν)², while |T(ν)| ≈ (ν_o/Qν) for ν > Qν_o. Noise insulation can thus improve at a rate of 40 or 20 dB/decade at higher frequencies, but the amplitude of oscillations around ν_o is increased approximately by a factor Q. Therefore, while passive stabilization obtained by simply suspending the interferometer in a vacuum chamber can effectively make the system immune to external high-frequency perturbations, some other solution has to be introduced to compensate for the low-frequency ones.

In fact, an active stabilization is required for accurately locking the interferometer path difference during each step of a Ramsey scan. Since data acquisition for each step can last from seconds to minutes, the passive stabilization alone, working for ν ≫ 1 Hz, is certainly not effective. Possible approaches to actively modulate the optical path in one arm of the Michelson interferometer could be based on the transmission through a movable glass wedge or on the direct piezoelectric translation of one of the cube retro reflector mirrors. We chose the latter not to introduce different dispersion between the pulses traveling the two interferometer paths.

A backlash-free translator is realized via a brass cantilever, suspending the mirror and giving the necessary mechanical load to the piezoelectric assembly (PI, mod. P-885.11, resonance frequency 135 KHz). Due to the mirror weight (around 150 g), the resonance frequency of the system results around 1 kHz. Therefore, the feedback loop cannot be very fast, but this is not a great limitation since passive stabilization at high frequencies is already granted. The whole mirror setup is mounted on a stepping motor translation stage for coarse adjustments of the path difference.

A 45-degrees linearly-polarized beam from a metrological-grade He-Ne laser at 633 nm (SIOS-SL03), following nearly the same path of the IR beam in the interferometer, provides the reference signal for locking the path difference (see Fig. 1 for a scheme of the experimental setup). A λ/4 waveplate is placed in the path of the He-Ne beam in the fixed interferometer arms. After transmission through the wave-plate and recombination with the beam coming from the other arm, the two orthogonal polarization components are separated by a polarizing beam-splitter. Upon detection by a home-made differential photodiode pair (electronic noise $\approx 20\mathrm{nV}/\sqrt{Hz}$), these provide the A–B quadrature fringe signal used by a PID amplifier for controlling the PZT movement. An additional integration stage is also added in cascade to the home-made PID amplifier to further reduce the low–frequency noise.

 

Fig. 1 Simplified scheme of the experimental set-up. PID is a home-made proportional-integral-derivative feedback controller; BS is a 50% beam-splitter; λ/2 and λ/4 are a quarter- and half-waveplates, respectively.

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3. Stability analysis

The power spectral density of the He-Ne error signal, as measured by a spectrum analyzer (Tektronix, Mod. RSA-3303A), is shown in Fig. 2 for the two experimental conditions of open (PID OFF) and closed loop (PID ON). For comparison, we also report in the same picture the PSD obtained by measuring a signal of the same intensity bypassing the interferometer. In this case the curve (which has been re-scaled in path-difference scale for comparison purposes) only includes electronic noise contributions and intensity fluctuations from the He-Ne laser. When the feedback loop is closed (PID ON) the electronic noise limit is approximately reached at the highest and at the lowest frequencies. However, the feedback loop is responsible for the introduction of additional noise in the range 30 Hz÷2 kHz, due to the mechanical resonance of the retro-reflector cube mirror mount. The integrated noise in this range of frequencies results in a standard deviation of about 0.2 nm on the path difference.

 

Fig. 2 Square root of the power spectral density (PSD) of the interferometer optical path difference, measured as a function of the frequency. Red: open-loop (PID OFF) signal; blue: closed-loop (PID ON) signal; gray: equivalent signal corresponding to pure He-Ne intensity fluctuations and electronic noise. The standard deviation of the path difference in the 1.5 Hz–3 KHz frequency range $(\sigma =\sqrt{\int PSD(\nu )d\nu })$ is 116 pm, 170 pm, and 1.2 pm, for the three cases, respectively.

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For actual experiments with high-order harmonic sources, where acquisition times of the order of 60 s are often needed, one has to characterize the interferometer in the low-frequency region (1 Hz and below). In order to obtain this information, and to perform some preliminary tests of its application in conjunction with harmonic sources, we made a simple stability check based on the observation of spectral interference fringes of the third harmonic generated by focusing the intense IR pulses in air. The 1 mJ, 30 fs-long, pulses at 780 nm are produced by the amplified Ti:sapphire laser operating at a repetition rate of 1 kHz that is normally used for high-order harmonic generation. We also used the same normal incidence vacuum monochromator but, differently from the case of harmonics produced in a gas jet, we did not evacuate the chamber and just observed the fluorescence pattern of the air-produced third harmonic on a screen placed in the exit focal plane.

Delaying the two IR pulses of about 200 fs by means of the translation stage in the interferometer, the combined spectrum of the resulting delayed third-harmonic pulses acquires a sinusoidal modulation that is then mapped to a spatial intensity modulation at the exit focal plane of the monochromator. Such spectra were imaged on a CCD camera and fitted to extract the fringe shifts from one acquisition (lasting about 0.5s) to another (spaced by about 2s), as shown in Figs. 3(a) and 3(b).

 

Fig. 3 Spectral interference fringes for time-delayed third harmonic pulses. a) CCD image of the monochromator focal plane. (b) Integrated lineout of spectral interference fringes (normalized to the total image counts), superimposed to a sinusoidally modulated Gaussian fit function (two CCD acquisitions, corresponding to the extrema of the series). c) Values of the fitted delays τ for a run of 47 acquisitions, each corresponding to an integration time of 0.5 s (sampling frequency f_s=0.5 Hz). The standard deviation is σ_τ =2.2 as on a mean value τ =191.8942 fs. (d) Histogram of the equivalent path differences. ∆τ = 2.2 as corresponds to a path standard deviation of approximately 0.7 nm.

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Any variation in the relative phase between the two pulses at 266 nm (either coming from some path difference fluctuation for the IR pump pulses in the Michelson interferometer, or caused by possible perturbations in the generation process) can be revealed as a shift of the interference pattern along the frequency axis: a shift equal to the distance between consecutive fringe peaks corresponds to an optical path difference of one wavelength for the 266 nm light. To get rid of artifacts due to intensity fluctuations, we normalized each spectrum to the total number of counts in the image. This measurement is particularly interesting, as it gives a direct estimation of the phase jitter for a realistic Ramsey-type spectroscopy experiment in the VUV-XUV made with high-order harmonics and a split pulse spectrometer [8]. In fact, by comparing the fitted UV fringes taken in different acquisitions (see Figs. 3(c) and 3(d)), we find a standard deviation of σ ≈ 0.7 nm for the equivalent variations in the optical path difference in the actively stabilized interferometer. This measured standard deviation can be attributed to the integrated path difference PSD in the low-frequency region (approximately comprised between the inverse duration of the measurement 1/T, where T = 94 s, and the sampling frequency f_s = 0.5 Hz), which was not measurable with the spectrum-analyzer approach described above. Note that this value also includes effects of mechanical fluctuations in the detection setup and therefore is only valid as an upper limit for the interferometer instabilities.

If we consider the entire frequency interval of interest, comprised between the inverse of the typical duration time of a measurement and the kHz-range repetition rate of the laser, we can estimate an upper limit for relative path fluctuations in the two arms of the actively-stabilized interferometer of about 1 nm.

Such a good stability and small uncertainty in the absolute positioning of the interferometer arms should allow one to measure Ramsey fringes of sufficient visibility for XUV radiation down to the < 10 nm wavelength range provided that eventual intensity and wavefront distortions are maintained identical in the generation of the two XUV pulses. This capability, combined with the long path unbalancing available in our new setup, and in conjunction with the random-sampling approach [10], which simplifies the measurement procedure by reducing the acquisition intervals, opens the path to split-pulse XUV Ramsey-type spectroscopy with resolving powers in excess of 10⁸.

Also considering lower-order harmonics, like the ninth harmonic of our laser, at 88 nm, the exceptional precision and stability of the interferometer path unbalancing up to the meter range should guarantee a resolving power larger than 10⁷, better than any state-of-the-art system based on synchrotron radiation.

4. Conclusions

In conclusion, we have developed an ultra-stable and accurately controllable Michelson interferometer for applications to XUV Ramsey-type spectroscopy with high-order laser harmonics. Thanks to the passive and active stabilization systems implemented in the setup, time delays of the order of ten nanoseconds (corresponding to meters of optical path differences) can be achieved with a stability of attoseconds (equivalent to spatial instabilities in the nanometer range). The possibility to accurately monitor and control the large path difference between the two arms of the interferometer with such a high precision will allow us to produce the pump pulses for generating precisely delayed pairs of XUV harmonic pulses. Combined with the already demonstrated capabilities of split-pulse Ramsey-type spectroscopy, and provided that an adequate long-term stable reference for the absolute interferometer positioning is used [13], this new tool has the potential to significantly improve the achievable spectral resolution and the precision of absolute frequency measurements in the XUV.