1. Introduction

In spectroscopy, microscopic information is deduced from the optical response induced in the target. Thereby, nonlinear spectroscopy provides access to the nonlinear matter response which generally contains more information than the linear case, e.g. revealing correlations and coherent properties [1]. Among nonlinear methods in the optical regime, coherent time-domain methods, most prominently multidimensional spectroscopy, have in recent years emerged as particular powerful and versatile tools to study complex processes on ultrafast time scales [1–3]. Due to the coherent excitation schemes, phase information is accessible which manifests an important extension to other spectroscopies, for instance allowing to study the coherent nature of biological processes [4], revealing many-body phenomena [5] and in principle enabling a full characterization of quantum pathways [6]. A transfer of these techniques to the extreme ultraviolet (XUV) and X-ray regime would in addition provide attosecond temporal resolution and facilitate site/chemical-selectivity by addressing specific core resonances and/or exploiting their pronounced chemical shifts [7], thus manifesting a fundamentally improved spectroscopic toolkit with unprecedented temporal and spatial resolution.

Coherent time-domain methods rely on the interference of optically induced wave packets (WPs) recorded as a function of the pulse delays, hence reflecting the dynamics of populations and coherences. If the full information content, including electronic coherences, is desired, pulse sequences with a high degree of phase stability have to be applied, i.e. to a fraction of the optical cycle [8, 9]. Numerous phase stabilization methods have been developed to facilitate pump-probe and multidimensional spectroscopy in the infrared (IR), visible (VIS) and ultraviolet (UV) spectral range [10–17]. However, with shorter wavelengths in the vacuum ultraviolet (VUV) to X-ray regime, much higher phase stability is required and commercial optics necessary for the implementation are not available.

Several methods aiming to solve this issue have been developed. Direct pulse manipulation in the XUV is achieved with wavefront division optics [18–22]. However, fundamentally limited by mechanical instabilities, this approach may not provide a general solution at extremely short wavelengths. For free electron lasers (FELs), manipulation of the electron pulse has been exploited. Recently, this lead to the first demonstration of coherent control with FEL light, confirming the ability to generate phase-locked pulses [23]. However, this technique seems intricate and less flexible in view of designing advanced multidimensional schemes. Another approach is based on the driving/seeding laser, transfering its coherent properties onto the XUV pulses as demonstrated for high harmonic generation (HHG) sources as well as for seeded FELs [24, 25]. By applying phase-locked pulse sequences in the driving laser, the generation of phase-stable XUV pulses with HHG sources [26–28] and seeded FELs [29] has been achieved. While pulse manipulation on the fundamental frequency is much easier than in the XUV and phase stability in the driving field can be achieved with very high precision, the stability might still be insufficient for upscaling to extremly short wavelengths. In addition, phase jitter inherent to the conversion process or amplified thereupon, may compromize the phase stability of the XUV pulses.

As an interesting aspect, the latter approach may facilitate the implementation of phase modulation schemes. The phase modulation approach has been first demonstrated for coherent time-domain pump-probe experiments [30] and, later, in a straight forward extension for multi-dimensional spectroscopy [14, 31]. In this scheme, a tracer beam is superimposed in the optical interferometer which tracks all phase fluctuations. Phase jitter is then removed from the signal by applying a lock-in technique with the tracer beam being the reference. This reduces demands on phase stability by orders of magnitude and may represent a general solution for overcoming stability limits in the XUV.

In phase modulation schemes, information about coherent dynamics is gained with incoherent ’action’ signals [14, 30–33]. At the same time, quasi phase-cycling is introduced allowing for rotating frame sampling, pathway selection and stray light suppression while using collinear beam paths. The signal recovery capabilities of lock-in amplification in combination with incoherent observables provide particularly high sensitivity, and the applicability to highly dilute samples has been demonstrated [33–35]. Furthermore, a sensitive detection scheme for multiphoton quantum coherences has been developed [35].

In this paper, we introduce the principle of phase-modulated harmonic light (PMHL) spectroscopy which combines the phase modulation technique with harmonic light generation to perform spectroscopy with harmonics of the phase-modulated fundamental light. We present a proof-of-principle study in which we combine a phase-modulated pump-probe setup with second harmonic generation (SHG) and perform electronic WP interference measurements in the 400 nm wavelength range.

2. Experimental technique

2.1. Phase-modulated pump-probe scheme and harmonic detection

In coherent pump-probe spectroscopy, phase-related pump and probe pulses each excite a WP in the sample, which interfere with each other depending on the relative phase between the laser pulses [36]. Accordingly, constructive and destructive WP interference is reflected as oscillations in the excited state populations which can be monitored via incoherent processes, i.e. an ’action’ signal. In our setup [Fig. 1(a)], a typical Mach-Zehnder interferometer arrangement including a motorized delay stage is used to generate collinear pump-probe pulses of adjustable interpulse delay τ. A spectroscopy cell containing a mixture of alkali-metal vapors is optically excited and the WP interference is recorded in the fluorescence yield. Thereby, two acousto-optical modulators (AOMs), placed inside the interferometer, continuously sweep the phase of transmitted pulses by ϕ_i (t) = Ω_it, where Ω_i (i = 1, 2) denotes the applied radio frequency. Phase-locked driving of the AOMs then imparts a well-defined modulation of the detected signal according to ϕ₂₁(t) = ϕ₂(t) − ϕ₁(t). In phase-modulated time-domain spectroscopy, the signal oscillation therefore depends on two parameters: the pump-probe delay τ and the phase modulation ϕ₂₁(t). Exemplary, for a one-photon transition from initial state |g〉 to excited state |n〉, the signal is

 

Fig. 1 (a) Schematic diagram of the optical setup. Collinear phase-modulated pump-probe pulses of the fundamental frequency are generated in an optical interferometer. A nonlinear crystal (NLC) followed by an optical filter is placed behind the interferometer to generate collinear second harmonic pulses. A monochromator is used to construct the reference signal for harmonic lock-in demodulation of the fluorescence signal. (b) Harmonic lock-in detection scheme using an external reference signal.

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In the proposed concept of PMHL spectroscopy, harmonics of the fundamental light are generated subsequent to the optical interferometer [Fig. 1(a)]. In our example, a nonlinear crystal is used to produce second order harmonics, but in principle HHG and HGHG processes may be incorporated instead, yielding much higher harmonics. Upon harmonic generation of the individual pulses, their phase is shifted to integer multiples ϕ_i (t) = nΩ_it, with n being the number of the harmonic. This relationship has been experimentally confirmed for SHG in nonlinear crystals [37–40], higher harmonics in HHG sources [24] and for the HGHG process in a seeded FEL [25]. Driving the one-photon transition |g〉 → |n〉 with the n’th harmonic light will then yield a signal modulated with ϕ₂₁(t) = n(Ω₂ − Ω₁)t = nΩ₂₁t.

This phase signature allows for the isolation of contributions from different harmonics in the data acquisition through lock-in demodulation. For this purpose, a portion of the phase-modulated pulses is branched off before the HG process and is spectrally filtered in a monochromator [Fig. 1(a)]. At the exit slit, we record the two-pulse interference signal of temporally stretched fundamental pulses:

This signal oscillates with the downshifted frequency $\overline{\omega }={\omega }_{ng}-n{\omega }_{\text{ref}}$. This drastically reduces the amount of required data points. The same applies for phase noise picked up in the optical setup. Any kind of fluctuations of the optical path length inside the interferometer (e.g. vibrations of optics, fluctuation of refractive index) will cause phase/timing jitter in the signal, which scales in conventional pump-probe experiments with the laser frequency, thus becoming an increasing issue for experiments in the UV and below. In the presented concept, this jitter, denoted δτ, is simultaneously imprinted onto the fluorescence and reference signal and is therefore efficiently removed in the lock-in demodulation process. As a result, the demodulated n’th harmonic signal exhibits a residual phase noise of $\delta \varphi =\overline{\omega }\delta \tau$. Phase fluctuations thus scale only with the downshifted frequency $\overline{\omega }$ when going to high harmonics of the fundamental light, manifesting an important advantage over other stabilization strategies.

At the same time, a selection rule is introduced, selecting only signals modulated with nΩ₂₁t and thereby removing background signals such as stray light. This selection procedure can be compared with phase-cycling as commonly used in nonlinear spectroscopy [41]. Furthermore, the phase-synchronous detection recovers amplitude and phase information allowing for the separation of the absorptive and dispersive part of the obtained spectrum or the reconstruction of quantum states [30].

The experimental apparatus employed for the phase modulation has been described elsewhere [33]. To generate the SHG pulses, the laser beam is focused onto a betha barium borate crystal (2 mm thickness) with a f =40 mm lens. Afterwards, the beam is collimated (f =50 mm) and the fundamental light is blocked with two dielectric mirrors (Thorlabs E01-coating) and an optical short pass filter (Edmund Optics 84-712) yielding pulses around 400 nm and 0.2 nJ (pump), 2 nJ (probe) pulse energies. These enter a heated vapor cell (324 K) which contains a mixture of potassium, rubidium and cesium. The fluorescence is detected using a photo-multiplier tube (Hamamatsu H10721-20) combined with a commercial preamplifier. The reference signal is recorded using an avalanche photo diode, band-pass filtered and amplified. Fluorescence and reference signals are split and fed into four digital lock-in amplifiers. Demodulated in-phase (X_n) and in-quadrature (Y_n) signals are acquired simultaneously for four harmonics (n=1,..,4) and the complex-valued signals Z_n = X_n + iY_n are reconstructed. Absorptive and dispersive parts of the Fourier spectrum of Z_n are obtained through the Kramers-Kronig relation as done in [30]. Typically we use Ω₂₁ = 5 kHz yielding higher harmonic contributions modulated with 10, 15 and 20 kHz, respectively. According to the frequency characteristics of employed detection electronics, signal amplitudes and phases were corrected in the data analysis.

2.2. Calculation of signal contributions for linear and nonlinear processes

To study the effects of linear and nonlinear excitation, we focus on systems which support one- and two-photon transitions, such as the model three-level system shown in Fig. 2(a). In this system, excitation pathways may involve superpositions of |g〉 and |e〉 states, termed single quantum coherence (SQC) or of |g〉 and |f〉 states, termed double quantum coherence (DQC). While a SQC signal yields information about the first excited state, DQC signals, or in general multiple-quantum coherence signals, provide information about higher-lying states.

 

Fig. 2 (a) Model three-level system with one- and two-photon transition. (b) Collinear pump-probe pulses induce second harmonic pulses in a nonlinear crystal (NLC). SHG1 and SHG2 denote the SHG pulses of individual incident pump and probe pulses and SFG denotes the SFG pulse occurring during pump-probe overlap. (c) Double-sided Feynman diagrams for SQC and DQC pathways involving interactions with one SHG pump and probe pulse (1 and 2) and for interactions with the SFG and either of the SHG pulses (3 and 4).

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In our scheme, UV pulses are generated by pumping a nonlinear crystal with the phase-modulated fundamental pulses [Fig. 2(b)]. This results in three harmonic contributions: two SHG pulses modulated with ϕ(t′) = 2Ω₁t′ (SHG1) and ϕ(t′) = 2Ω₂t′ (SHG2), respectively and for temporal overlap of the fundamental pulses an additional contribution from sum-frequency generation (SFG) modulated with ϕ(t′) = (Ω₁ + Ω₂)t′ [40].

Exciting the three-level system with this pulse sequence gives rise to a number of signals. We keep track of the possible excitation pathways by employing the double-sided Feynman diagram formalism [1]. Due to the individual phase modulation of the incident pulses their contributions to the detected signal can be isolated with harmonic lock-in demodulation. Disregarding all diagrams which do not yield a modulation Δϕ = nΩ₂₁t′, n ∈ N, four types of processes are identified as major contributions to the signal. Examples are given in Fig. 2(c): diagrams 1 and 2 show SQC and DQC pathways induced upon interactions with a SHG pump and probe pulse, whereas diagrams 3 and 4 show the same processes for interactions with the SFG pulse and either of the two SHG pulses. Note that only representative examples of Feynman diagrams are shown. A complete set of diagrams including different time-ordering during pulse overlap is omitted. Further, processes represented by diagrams 1 and 3 appear in the second harmonic (2H) lock-in demodulation (Δϕ = 2Ω₂₁t′), diagram 2 in the fourth harmonic (4H) (Δϕ = 4Ω₂₁t′) and diagram 3 in the first harmonic (1H) (Δϕ = Ω₂₁t′). Hence, we get a mixing of SQC and DQC processes in the 2H demodulation whenever SFG pulses are involved.

From the Feynman diagrams, we calculate the individual signal contributions using time-dependent perturbation theory and obtain:

The behavior of these signals is intuitive. Equations (5) and (6) represent ordinary phase-modulated SQC and DQC signals, however modulated at twice the phase modulation frequency due to the SHG process prior to the light-matter interaction. Eqs. 7 and 8 describe essentially the same SQC and DQC pathways, however involving interactions with a SFG pulse. Therefore, these signals occur only for a short delay window as apparent from the τ-dependent envelope functions inherited by the SFG pulse. Further, these signals exhibit a τ/2-delay dependence, which is self-evident considering that the delay between SFG and SHG pulses increases only by Δτ/2 for each Δτ increment [Fig. 2(b)]. As a consequence, these signals appear at half the downshifted WP oscillation frequencies.

We will use these expressions for a simple simulation of our measurements. To complete the simulations, we have to include stray light effects. The majority of stray light is efficiently eliminated upon lock-in demodulation since it does not exhibit a well-defined modulation frequency. However, during temporal pump-probe overlap, a portion of stray light reflects the pump-probe pulse interference, i.e. the interferometric quadratic autocorrelation. As modulated contributions we get the low-frequency cross-correlation of SHG and SFG pulses, denoted S_CC and the high-frequency autocorrelation of two SHG pulses, denoted S_AC:

3. Results

3.1. Linear excitation and spectroscopy with multiple harmonic light

At first, we present PMHL spectroscopic data for linear excitation of an alkali gas mixture containing potassium, rubidium and cesium. With this test system, we also demonstrate the ability to perform simultaneous spectroscopy with multiple harmonics of the laser light exciting the sample. The fundamental laser wavelength is tuned to 774.5 nm (12912 cm⁻¹) and the monochromator set to 774.36 nm (12913.9 cm⁻¹). No spectral filtering is done after the nonlinear crystal, hence the sample is excited with portions of the first and second harmonic light simultaneously. In rubidium and potassium, D line transitions are excited with the fundamental light and in cesium the Δn = 2 transition 6S_1/2 → 8P_3/2 is excited with second harmonic light [cf. Fig. 3(a)]. The respective data obtained from a ±30 ps pump-probe scan for 1H and 2H demodulation is shown in Fig. 3(b) to (e). In the time domain, clean high quality interferograms can be observed with oscillation frequencies strongly downshifted due to rotating frame sampling upon lock-in demodulation. This drastically reduces the amount of required data points and the influence of phase noise picked up in the optical setup. Correspondingly, sharp resonances and an excellent signal-to-noise ratio is obtained in the respective Fourier spectra, for excitation with the fundamental as well as second harmonic light.

 

Fig. 3 Obtained signals for first and second harmonic laser pulses exciting an alkali-metal vapor mixture containing potassium (K), rubidium (Rb) and cesium (Cs). (a) shows the relevant NIR and UV transitions in the alkali atoms. (b)-(e) show the first and second harmonic time- and frequency-domain data, denoted 1H, 2H TD and 1H, 2H FD, respectively. Black dashed lines indicate transitions frequencies taken from [42].

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During pump-probe overlap of fundamental pulses in the nonlinear crystal, more UV output is obtained and intuitively a transient exaggeration of the cesium signal around zero pulse delay should be expected, which is in contrast to the measured 2H time domain data [Fig. 3(b)]. However, the increased UV yield during pulse overlap is actually contained in the SFG pulse, which contributes to the 1H signal. Accordingly, a clear signal increase around zero delay is observed in the 1H signal [Fig. 3(d)] and a broad spectral feature appears in the respective Fourier spectrum [Fig. 3(e)] which is centered at half the frequency of the 6S_1/2→8P_3/2 transition in cesium. This is in accordance with the theory discussed above.

The applied phase modulation thus allows for a clear separation of the light-matter response for different harmonics in the light field. An exception occurs during temporal overlap of pulses where due to SFG pulse generation some UV light contribution leaks into the 1H signal channel. As a consequence, pulse overlap effects are minimized in the 2H signal channel and we obtain a constant amplitude there. The UV light contributions to the 1H signal exhibit specific broad line shapes and can be therefore clearly distinguished from other processes whenever pulse durations are shorter than the dynamics in the system.

We point out, that in molecular systems, transitions may be driven either with a one-photon excitation from one of the harmonics of the laser light or with a multi-photon excitation of a respective lower harmonic. Both processes would appear in the same harmonic spectra. Distinguishing these processes must be done on a case by case basis. This scenario, however, does not play a role in the experiments discussed in this paper.

3.2. Nonlinear excitation and stray light effects

As a second example we present data for the 5S_1/2 →6P_3/2,1/2 transitions in rubidium. Here the laser was focused with a f =150 mm lens into the vapor cell and the delay was scanned for 0 – 40 ps. Fundamental light was blocked from entering the vapor cell using an optical filter. Data for the second harmonic pulses tuned to 420.8 nm (20825 cm⁻¹) and the monochromator set to 840.55 nm (11897.0 cm⁻¹) is shown in Fig. 4.

 

Fig. 4 Excitation of the 5S_1/2 → 6P_3/2,1/2 transitions in a rubidium vapor for the laser wavelength 420.8 nm (23764 cm⁻¹). The excitation scheme is shown in (a). (c), (e) show the time domain data for 2H and 1H lock-in demodulation and (b), (d), (f) the discrete Fourier transforms of 4H, 2H and 1H signals. In (b) the collective excitation 5S_1/2 →6P_3/2 and 5S_1/2 → 6P_1/2 is hidden in the noise floor. Black dashed lines show transition frequencies taken from [42].

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In this example, we get signal contributions in the 1H, 2H and 4H demodulation channel. In the 2H demodulation channel, we obtain a high resolution signal for the 5S_1/2 →6P_3/2,1/2 resonances [Fig. 4(c) and (d)]. In the 1H demodulation channel, only processes involving SFG pulses are contributing since the fundamental light is blocked in this experiment. Accordingly, we observe a transient signal in the 1H, decaying on a short time scale determined by the pulse length and a respective broad peak in the Fourier domain [Figs. 4(e) and 4(f)]. In the 4H demodulation channel, we observe clear resonances [Fig. 4(b)] which we attribute to the collective excitation of two rubidium atoms via the 5S_1/2 → 6P_3/2,1/2 transition. The rubidium system thus supports one- and two-photon transitions and therefore serves here as a model system to study nonlinear effects. A corresponding excitation scheme is shown in Fig. 4(a). Similar collective resonances have been previously observed in alkali-metal vapors using two-dimensional spectroscopy [43] and phase-modulated pump-probe spectroscopy [35].

As a unique feature of the 2H signal, we observe a distinct depletion of the pump-probe transient around zero pulse delay [Fig. 4(c)]. To elucidate the nature of this feature, we show measurements for different laser wavelengths in Fig. 5. The 2H time domain data [Fig. 5(a)] indicates that two effects are superimposed at the time origin leading to a signal enhancement if the laser is rather off-resonant and a signal decrease if the laser is more resonant. For the 2H frequency spectra shown in Fig. 5(b), we have Fourier transformed only a small time-domain window in order to make the transient features more pronounced. In these spectra, we observe a positive broad peak centered at the laser wavelength which is identified with the autocorrelation of SHG pulses [Eq. (10)] originating from stray light scattered into our detector. This feature becomes less pronounced when being more resonant to the atomic transitions. Here, broad negative peaks centered at the 5S_1/2 → 6P_3/2,1/2 transition frequencies become evident instead.

 

Fig. 5 Excitation of the 5S_1/2 → 6P_3/2,1/2 transitions in a rubidium vapor for laser center wavelengths between 417.8 and 424.2 nm. (a) shows the second harmonic time-domain (2H TD) data and (b) and (c) show the second (2H FD) and first harmonic (1H FD) frequency-domain spectra. In (a), oscillation frequencies differ among data runs which is due to changes of the monochromator wavelength. In (c), simulated signals are included as discussed in the text. Gray vertical lines indicate the laser center frequency and black dashed lines show transition frequencies taken from [42].

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This effect can be understood by focusing on the 4H spectrum, where we observe the collective excitation of two atoms for excitation with SHG pump-probe pulses. In analogy, the 2H spectrum shows the same process, however for interactions with an SFG and either of the two SHG pulses (cf. Feynman pathways in Fig. 2(c). As discussed above, these signals appear at half the transition frequency and only contribute to the signal during pump-probe overlap. Furthermore, the collective signals in the 4H [Fig. 4(b)] and 2H [Fig. 5(b)] spectra exhibit the same phase signature and appear as negative peaks with respect to the single-atom resonances [44]. This phase signature explains the depletion of the pump-probe signal around zero delay. The depletion depth also matches with predictions derived from the amplitudes of the 4H peaks.

Our assignment for the transient features in the 2H signal is also consistent with our observations in the 1H signal [Fig. 5(c)]. Similar to the 2H signal, two transient processes are superimposing here for small pulse delays. One contribution comes from the the 5S_1/2 → 6P_3/2,1/2 excitation involving a SFG pulse interaction. The other contribution is due to stray light reflecting the cross-correlation of SHG and SFG pulses [Eq. (9)]. Note that both effects produce interference fringes in the time domain, however being not directly apparent in our data since oscillation frequencies are strongly downshifted due to rotating frame detection.

A comparison of the 1H signal with a simulation confirms the discussed assignment for the different signal contributions. For the simulation, we used Eqs. (7) and (9) with the experimentally obtained pulse parameters. The relative amplitudes between the 5S_1/2 → 6P_3/2,1/2 transitions were set to the peak ratios taken from the 2H signal [Fig. 5(b)], whereas the ratio between this signal and the cross-correlation was a free parameter. The Fourier transform of simulated data is shown together with the experimental data in Fig. 5(c). We get good agreement between simulations and experimental data and the two underlying signal contributions can be clearly identified. For simplicity, no spectral chirp was included in the theoretical model and deviations between experiment and simulation are attributed to spectral chirp of the UV pulses due to the relatively long nonlinear crystal used for the frequency conversion.

In conclusion, rubidium served here as a simple model system to study nonlinear electronic excitation with the concept of PMHL spectroscopy. Since our phase modulation method allows for phase-sensitive detection, the weak collective features can be identified by their phase shift with respect to single-atom and stray light contributions. If scanning longer delays, the collective peaks leaking into the 2H spectra as well as the stray light peak are negligible since these effects decay rapidly with increasing pump-probe delay [cf. Fig. 4(d)].

4. Discussion

Our experiments demonstrate the ability to track the coherent time evolution of electronic WPs for linear and nonlinear optical excitation. With nonlinear WP excitation also double or higher-order quantum coherences can be studied. In view of future XUV studies, such signals may yield valuable insight into double-core excitations and related many-body correlations [45]. Furthermore, simultaneous spectroscopy with different harmonics of the light field can be readily performed with our scheme and direct correlation of different harmonic signal contributions is possible. In this sense, XUV experiments with multiple harmonics simultaneously exciting a sample might be a feasible application and optical isolation of specific harmonics normally causing power losses and pulse dispersion could be omitted. Likewise, the presented time-domain spectroscopy approach might be very beneficial to achieve high spectral resolution with broadband high harmonic pulses independent of the detector resolution. In time-domain methods the spectral resolution is solely determined by the Fourier transform of pump-probe traces and thus a resolution in the µeV regime is feasible [33]. This might be further enhanced by constructing the lock-in reference signal with a narrow-bandwidth continuous wave laser superimposed in the optical interferometer, as done in [31].

The employed collinear beam geometry is a significant advantage for the implementation in XUV beam lines. However, in collinear experiments, the typically highly convoluted nonlinear system response is very challenging to disentangle. Therefore, phase-cycling/phase modulation becomes evident to narrow down the signal to the most relevant contributions. As they rely on a precise phase manipulation of the optical pulses, phase-cycling/phase modulation is hard to achieve in the XUV, though. In our approach, a solution to this issue is presented by manipulating the pulses solely on the fundamental frequency. This is combined with high resolution pump-probe scans and phase-sensitive detection, allowing us to disentangle the detected system response containing several overlapping contributions, i.e. from linear and nonlinear excitation of the spectroscopic sample as well as from interaction with different harmonic light including fundamental, SHG and SFG pulses. Thereby, a particular focus lay on possible ambiguities occurring during temporal overlap of pulses. A precise understanding of this part of the experiment may help to reliably extract information about particularly fast dynamics [46]. At short pump-probe delays, we observe an overlap of different signals in the same harmonic demodulation channel due to the generation of SFG pulses. No additional information is contained in the SFG signal contributions. However, they may complicate data interpretation if more advanced systems will be studied.

We point out that the efficiency of our scheme relies on the high purity of acousto-optically induced phase modulation, performed on a shot-to-shot basis. Other modulation techniques based on modulating the pulse delay, e.g. with piezo mirrors, may also operate with high repetition rate lasers on a shot-to-shot basis, but typically introduce pronounced harmonic sidebands. Pulse shaper based setups would allow for a linear phase modulation, but not with high rates and the achievable pulse delay is restricted to the low picosecond regime. In view of applications with higher harmonic light sources, where harmonic distortions may be significantly more critical, we briefly discuss possible harmonic artifacts in our setup. These include rediffraction in the AOMs [38], leaking of the first harmonic signal into higher harmonic channels [35], nonlinearities in the detector such as two-photon absorption [40] or saturation of detection electronics. Effectively, only the latter two phenomena may yield ambiguities in the PMHL scheme, but can be readily minimized by proper design of the experimental setup. We have only observed saturation of electronics in our setup, causing spurious signals in the order of −50 dB, which could be further damped employing optimized detection electronics.

Our lock-in based detection method equalizes mechanical instabilities in the optical setup by comparing the pump-probe transient with a reference which simultaneously tracks the vibrations of optics in the interferometer. As a result demands on phase stability are generally reduced by orders of magnitude. With harmonic lock-in detection, this concept is transferred to the respective harmonic of the fundamental light, therefore allowing electronic WP interference measurements even at very short wavelengths, possibly down to the VUV and XUV regime. Combined with the perfectly integrated lock-in amplification, the optimal stray light suppression and the compatibility with detectors preferable for ultra high vacuum detection, e.g. photoionization or any kind of depletion signal, make our approach ideal for VUV to XUV applications.

The presented approach can also be extended to multidimensional spectroscopy in a straightforward way [14]. Similarly, a pulse shaper based two-dimensional spectroscopy setup has been transferred to shorter wavelengths [38]. Alternatively, a two-dimensional spectroscopy setup was transferred to the UV by placing nonlinear crystals inside each arm of the optical interferometers. This omits the SFG contributions in the signal, albeit being considerably more intricate than our approach, especially in view of HHG or HGHG applications.

5. Conclusion

In this work we introduced the concept of PMHL spectroscopy in a proof-of-principle study based on the combination of phase-modulated nonlinear spectroscopy and second harmonic light generation applied to alkali-metal vapors. We demonstrated simultaneous spectroscopy with different harmonics of the laser light and studied linear as well as nonlinear processes via one- and two-photon excitation of the spectroscopic target. This was accompanied by a thorough analysis of signals with the focus of short delay times where pump and probe pulses overlap.

The presented scheme should be readily transferable to XUV wavelengths via HHG and HGHG processes, thereby reducing demands on phase stability by orders of magnitude. Other advantages are the compatibility with detection methods favorable for ultra high vacuum applications and the fully integrated lock-in amplification being a unique feature among established stabilization methods. The lock-in detection further provides excellent inherent background suppression such as stray light which is often an issue in the VUV regime and below. The collinear beam geometry is a significant advantage for possible implementation in XUV beam lines; however, requires phase-cycling/phase modulation schemes to disentangle the detected signals which are usually highly convoluted in nonlinear spectroscopy. By manipulating the optical phase on the fundamental wavelength, instead of a direct manipulation of the harmonic light, we demonstrated an approach which would make phase modulation feasible in XUV experiments.

Furthermore, our detection provides phase sensitivity, i.e. amplitude and phase information is simultaneously obtained. Combining this with pathway selectivity and by performing high resolution scans, we were able to clearly identify all individual contributions overlapping in our signals. In this sense, our results provide important insights helping to identify ambiguities in future studies on more complex systems and/or with high harmonic light sources. Eventually, all prerequisites for coherent multidimensional spectroscopy are demonstrated. A corresponding extension of our setup can be accomplished in a straightforward way, therefore also providing a feasible scheme for all-XUV multidimensional spectroscopy.