## Abstract

We demonstrate that Arbitrary-Detuning ASynchronous OPtical Sampling (AD-ASOPS) makes possible multiscale pump-probe spectroscopy with time delays spanning from picosecond to millisecond. The implementation on pre-existing femtosecond amplifiers seeded by independent free-running oscillators is shown to be straightforward. The accuracy of the method is determined by comparison with spectral interferometry, providing a distribution with a standard deviation ranging from 0.31 to 1.7 ps depending on experimental conditions and on the method used to compute the AD-ASOPS delays.

© 2015 Optical Society of America

## 1. Introduction

Femtosecond pump-probe spectroscopy is a powerful and widespread method for time resolving a system response with a resolution limited only by the duration of the pump and probe pulses, rather than by the response time of electronic photodetectors. By varying the pump-probe time delay using a mechanical delay line, it is thus readily possible to reach the ultrafast regime. However, this method of controlling the delay becomes unpractical for values exceeding a few nanoseconds due to issues such as vibrations, defocussing and slow scanning frequencies [1]. Hence, for investigating systems presenting a broad distribution of timescales, often encountered e.g. in biomolecules [2, 3], it is desirable to design alternative methods which do not rely on a variable optical path for producing the pump-probe delay.

One such method is ASynchronous Optical Sampling (ASOPS), a stroboscopic technique exploiting a slight repetition-rate difference between two femtosecond oscillators in order to generate a train of pump-probe pulses with a linearly-increasing pump-probe delay [4–8]. ASOPS is now a well-established technique that has found many applications, as the lack of moving parts allows to achieve very high scanning frequencies [9]. However, its time range remains limited by the inverse repetition rate of the pump femtosecond oscillator, typically of the order of 10 ns.

As demonstrated by Bredenbeck *et al.* [10], far greater values of the pump-probe delay can be achieved by use of two femtosecond amplifiers whose trigger signals are independently controlled. While fine tuning relies on phase shifting the two phase-locked oscillators seeding the amplifiers, coarse tuning is controlled by electronically setting the delay between the trigger signals firing the two amplifiers [10, 11]. There is virtually no upper limit in the achievable pump-probe delay, since the repetition rate of the amplifier producing the pump pulses can be easily decreased as much as required. However, the method does require that the two oscillators have identical repetition rates and are phase locked, in order to avoid the timing jitter of about 10 ns that would otherwise result when seeding the amplifiers with two independent oscillators. This requirement prevents a straightforward implementation with pre-existing laser systems, e.g. when the oscillators have different repetition rates or when installing a servo loop on one of the two oscillators is considered unpractical. Apart from this drawback, the two-amplifier method is a nearly-ideal approach for performing pump-probe spectroscopy over a broad range of time scales.

We have recently demonstrated [12] a new ASOPS variant, coined AD-ASOPS for Arbitrary-Detuning ASOPS. As its name implies, AD-ASOPS can be applied to any pair of free-running femtosecond oscillators, even when their repetition rates differ by a great amount. AD-ASOPS relies on an accurate determination of the relative delays between all pulses produced by the two oscillators, so that the acquired data can be *a posteriori* sorted and averaged in the appropriate time bin for each probe pulse. The lack of requirement on the repetition rates greatly extends the variety of experiments that can be performed, as exemplified by our measurement of the time response of the reaction center from *Rhodobadcter sphaeroides* over a time domain ranging up to 200 ns with a subpicosecond time resolution [13].

In this article, we demonstrate that AD-ASOPS can be readily applied to the two-amplifier approach. Hereafter, we will refer to this new method as kHz AD-ASOPS whereas our previous implementation of AD-ASOPS on femtosecond oscillators will now be referred to as MHz AD-ASOPS. As in the two-amplifier method, kHz AD-ASOPS relies on the use of two femtosecond amplifiers independently triggered so that coarse tuning of the delay can be controlled electronically. However, in contrast with the standard two-amplifier method, the oscillators seeding the two amplifiers now have arbitrary repetition rates and are free running, so that there is a timing jitter of typically 10 ns. Yet, the exact time delays between the two free-running oscillators are continuously tracked by use of AD-ASOPS, so that the actual delays between amplified pump and probe pulses are determined *a posteriori* allowing to average the data in the appropriate time bin. We thus combine the advantages of AD-ASOPS with those of the two-amplifier method, resulting in a straightforward implementation with free-running pre-existing seeding oscillators.

In the following, we first detail the principle of operation of kHz AD-ASOPS. We proceed in demonstrating a more accurate – but time-range limited – measurement of the time delays by use of Fourier-Transform Spectral Interferometry (FTSI). Finally we compare the results of the two methods over the time range where they overlap, thus demonstrating the validity and accuracy of kHz AD-ASOPS.

## 2. Determination of the time delays by kHz AD-ASOPS

The experimental setup for implementing kHz AD-ASOPS is shown in Fig. 1. Two femtosecond amplifiers, hereafter called laser 1 (delivering the pump pulses) and laser 2 (delivering the probe pulses), provide the amplified femtosecond pulses needed in the pump-probe experiment. These two amplifiers are seeded by two independent free-running oscillators of periods respectively *T*_{1} and *T*_{2}. An electronic system, hereafter referred to as the AD-ASOPS device, delivers the trigger signals for both amplifiers. The coarse time delay is thus controlled by shifting the electronic signal triggering the pump amplifier (laser 1) with respect to that triggering the probe amplifier (laser 2). The AD-ASOPS device additionally continuously monitors the time delay between oscillator pulse pairs and determines the actual pump-probe delay between amplified pulses. For each amplified probe pulse, this information is then transferred to the computer in charge of data acquisition so that each data point (or spectrum) can be averaged in the appropriate time bin for reconstruction of the pump-probe dynamics. Hence, a subpicosecond time resolution can be retrieved despite the timing jitter between the two independent free-running oscillators.

Let us now discuss how the AD-ASOPS device is able to accurately monitor the time delay between all oscillator pulse pairs. As previously demonstrated [12], the AD-ASOPS method relies on the very good clock stability of femtosecond oscillators, even when they are free running. This clock stability allows to approximate with a linear law the time delay between the most-recent pulse delivered by oscillator 1 and the *n*^{th} pulse delivered by oscillator 2:

*n*is an integer number,

*a*

_{0}and

*a*

_{1}are real parameters, and [

*T*

_{1}] stands for the modulo-

*T*

_{1}operation, resulting in folding the delay in an interval of width

*T*

_{1}. Parameter

*a*

_{1}is simply the incremental delay whose value car be deduced from the difference in oscillator periods:

Slow drift of the free-running oscillator cavity lengths implies that parameters *a*_{0} and *a*_{1} must be regularly updated based on experimental measurements. Although different methods can be envisioned for performing this measurement, we rely here on coincidence detection as in our previous work [12]. Indeed, determination of the offset parameter *a*_{0} is straightforward from the detection of a coincidence event between the two oscillator pulse trains. Furthermore, if we count the number of pulses *N*_{1} (resp. *N*_{2}) delivered by oscillator 1 (resp. 2) between two consecutive coincidence events, we can write the elapsed time as *N*_{1}*T*_{1} = *N*_{2}*T*_{2}, making possible a determination of *a*_{1} as a function of *T*_{1} only:

This allows the determination of the time delays in units of *T*_{1}. Finally, a simple measurement of oscillator-1 repetition rate using a frequency counter allows calibration of the time axis. As shown in Fig. 1, coincidence events are detected here by use of a fiber-based interferometer. A departure of the differential signal from zero by an amount greater than predefined thresholds indicates constructive or destructive interferences between oscillators 1 and 2 and hence a time delay smaller than the coherence time, which corresponds to a coincidence.

In our experiment, we use two commercial Titanium:Sapphire compact femtosecond amplifiers running at a repetition rate of 1 kHz : the Hurricane (Spectra-Physics) as laser 1 and the Libra-HE (Coherent) as laser 2. The integrated oscillators seeding these amplifiers are respectively the Mai Tai (Spectra-Physics, repetition rate 1*/T*_{1} = 79.9 MHz), and Vitesse (Coherent, repetition rate 1*/T*_{2} = 80.1 MHz). Their spectra are centered at 800 nm, and are conditioned for better spectral overlap by bandpass interference filters (FWHM 3 nm) before entering the interferometer. The AD-ASOPS device consists of a VHDL (VHSIC-Very High Speed Integrated Circuit - Hardware Description Language) designed system embedded on a commercial development card including a Spartan 6 FPGA SP601 (Xilinx) associated with a fast homemade I/O card. However, as compared to our previous work on MHz ADASOPS [13], there is no requirement here for a fast ADC card or for fast data processing at a high throughput, which significantly simplifies the demands on the hardware.

Figure 2 shows 100 consecutive delays between amplified pulses determined by kHz AD-ASOPS for a constant coarse delay. As expected, the retrieved AD-ASOPS time delays spread over a window of width *T*_{1} *≈* 12.5ns, due to the timing jitter between the two free-running oscillators. As a constant number of oscillator pulses separates each measurement, note that the folded linear variation of the time delay is still visible. In order to validate this measurement, we now turn to an independent determination of the time delay between amplified pulses by use of spectral interferometry.

## 3. Determination of the time delays by spectral interferometry

In order to provide an independent measurement of the time delay *τ* between amplified pulses, we insert the fiber-based spectrally-resolved interferometer shown in Fig. 3 in place of the pump-probe arrangement shown as the ”Experiment” box in Fig. 1. Thanks to spectral interferometry [14], the recombined beam will produce spectral fringes of period inversely proportional to *τ*, enabling determination of the delay. However, the finite spectral resolution of the spectrometer results in a useful measurement only when the time delay is small enough so that spectral fringes can be actually resolved. In practice, our 0.5-m SpectraPro 2500i spectrometer (Acton, Princeton Instruments) associated with a 1200 grooves/mm diffraction grating results in a spectral resolution of 1.1 cm^{−1}, limiting the practical measurement window to |*τ*| < 25 ps.

We set the electronically-controlled coarse time delay to zero, which by definition is adjusted by a constant offset so that the amplified pulses overlap in the fiber coupler. The actual delays between amplified pulses will then be randomly distributed in a 12.5-ns time window due to the timing jitter between the two oscillators. Figure 4 shows a series of 500 consecutive spectra thus obtained. As expected, in most cases the time delay is too large and the spectral fringes remain unresolved. The measured spectrum is then simply the sum of the spectra of the two lasers. However, for some spectra indicated by arrows, spectral fringes are indeed resolved, indicating a time delay |*τ*| smaller than 25 ps.

The measured spectrum reads:

*E*

_{1}(

*ω*) and

*E*

_{2}(

*ω*) are the fields associated with beams 1 and 2, while ∆

*φ*(

*ω*) is the difference in spectral phase between beam number 2 and beam number 1. Thanks to the use of a strongly unbalanced interferometer, we know that ∆

*φ*(

*ω*) must present a strictly-positive curvature, as the additional 5-m of optical fiber on the probe beam results in a group delay dispersion far greater than the small deviation from transfom-limited pump and probe pulses. It will thus be possible to determine the sign of

*τ*, despite the fact that we measure only the cosine of the phase difference.

Figure 5(a) shows an example of measured spectral interferogram. Due to the strong dispersion in the additional propagation through 5 m of optical fiber, the fringe spacing is not uniform but decreases when frequency increases, indicating that the absolute value of the group delay increases with frequency. Considering the positive dispersion of the unbalanced interferometer, this is the signature of a positive time delay *τ*. In order to process the experimental data, we make use of Fourier-Transform Spectral Interferometry (FTSI) which allows through a simple processing in Fourier space to retrieve the difference in spectral phase, ∆*φ*(*ω*)+*ωτ*, as long as the two pulses do not overlap in time [15, 16]. Figure 5(b) shows the resulting spectral amplitude and phase. Among the two possible choices of sign in the retrieved spectral phase, Fig. 5(b) shows the only one with a positive curvature, resulting in a positive slope and hence a positive time delay. Figure 6 shows a different example where the fringe spacing increases when frequency increases. Keeping as above the spectral phase with a positive curvature now results in a negative slope and hence a negative delay. A simple polynomial fit then yields the time delay associated with the linear term. Note that in practice we use one of the experimentally-measured spectral phases as a reference subtracted to subsequent measurements, so that the relative time delay can be more accurately determined from a linear fit on the phase difference.

We are then able to retrieve independently the delays between pump and probe pulses, as long as the spectral fringes are resolved by the spectrometer (|*τ*| < 25 ps). Another noteworthy limitation arises for short delays, since FTSI processing in Fourier space requires non-overlapping correlation functions [15, 16]. This condition corresponds here to |*τ*| > 3 ps, due to the strong differential dispersion acquired in the additional 5-m optical fiber. Delays measured by FTSI are thus limited here to a window [−25 ps, −3 ps] ∪ [3 ps, 25 ps], in contrast to delays measured by AD-ASOPS which cover a continuous time window of much broader extent.

## 4. Results and discussion

Let us now compare the time delays retrieved by AD-ASOPS and FTSI. Note that, in principle, there is a constant offset between these two measurements due to the difference in optical paths between the two regenerative amplifiers as well as in the different propagation length from the amplifiers to the spectral interferometer. As mentioned above, we thus define the AD-ASOPS delay as the gross value minus this constant offset so that a zero delay corresponds to the actual zero delay in the spectral-interferometry measurement.

Figure 7(a) shows a comparison between AD-ASOPS delays, on the horizontal axis, and corresponding FTSI delays, on the vertical axis, recorded during an acquisition time of 15 minutes. Blue dots correspond to the entire measurement set consisting of data points where FTSI processing is suitable, resulting as discussed above in a 50-ps time window with a 6-ps gap around zero delay. As expected, data points are spread in the vicinity of a straight line with a slope equal to one. Figure 7(b) shows the histogram of the difference between FTSI and AD-ASOPS delays, evidencing a narrow peak that demonstrates the validity of kHz AD-ASOPS. However, a few data points can be observed with an error greater than a few picoseconds, resulting in a distribution whose standard deviation is 1.7 ps. Although suitable for many pump-probe experiments, we note that this time resolution is not as good as in our previous work on MHz AD-ASOPS [12]. This is attributed to the fact that the variation of oscillator cavity length is more important in the perturbed environment of a femtosecond amplifier, resulting in the need for updating the *a*_{0} and *a*_{1} parameters of Eq. (1) at a higher rate. This can be achieved by selecting data points between coincidence pairs separated by less than 0.2 ms – red diamonds in Fig. 7(a) –, resulting in a narrower distribution with a standard deviation of only 0.31 ps – shown in red on Fig. 7(b). The cost is that this selection results in keeping only 23% of all laser shots, so that one must eventually compromise between time resolution and acquisition time [13].

Another option to better keep track of a varying cavity length is to use a quadratic fit instead of the linear law used up to now. We thus replace Eq. (1) with the second-order function

where real parameters*a*

_{0},

*a*

_{1}and

*a*

_{2}are now determined so that the quadratic law interpolates three coincidence events, taking into account the associated number of pulses in between. This higher-order approach results in the histogram shown in Fig. 7(c). Although the FWHM is not significantly reduced, the distribution pedestal vanishes so that the standard deviation is greatly improved, down to 0.75 ps.

In order to explore a different region of the broad time window achievable using kHz AD-ASOPS, we insert a 25-m optical fiber in beam 2 of the spectral interferometer, bringing the total propagation difference to 30 m. Spectral fringes can be retrieved by shifting accordingly the electronically-controlled coarse time delay between the two femtosecond amplifiers. By repeating the previous experiments, we obtain the result shown in Fig. 8, now centered on an AD-ASOPS time delay of 112350 ps. Note that the greater gap around zero delay can be attributed to the greater value of the second-order dispersion, resulting in longer correlation functions. As above, we observe an excellent agreement between the two measurements, with a standard deviation of 0.74 ps thanks to the use of a quadratic fit for calculating the AD-ASOPS delay.

## 5. Conclusion and perspectives

To summarize, we have demonstrated a new multiscale pump-probe spectroscopy method based on the two-amplifier approach combined with AD-ASOPS. The accuracy of the method has been demonstrated by comparing the AD-ASOPS time delays with an independent measurement using spectral interferometry. The error distribution shows a standard deviation ranging from 0.31 to 1.7 ps depending on the degree of data selection and on the method used to compute the AD-ASOPS delays. This opens the way to easy multiscale pump-probe spectroscopy ranging from sub-picosecond to millisecond time scales in a single experiment based on preexisting laser amplifiers seeded by independent free-running oscillators.

One drawback of kHz AD-ASOPS as compared to the standard two-amplifier method is the lack of active control on the short-timescale delay, which is uniformly distributed in a 12.5-ns window. For example, when studying the short-time component of a multiscale dynamics, it would be desirable to concentrate some of the actual data points closer to zero delay. However, it should be emphasized that this current AD-ASOPS limitation can be overcome by taking advantage of the fact that most kHz amplifiers tolerate a small shot-to-shot variation in repetition period. For example, if one allows a 10-*µ*s variation in the exact delay between amplified pulses, it is possible to choose each amplified pulse among 800 candidates delivered by the oscillator. Assuming the two oscillator repetition rates differ by more than 0.1 MHz – so that the entire period is scanned at least once in 10 *µ*s – it is thus possible to choose the actual delay in the desired time range. The width of the resulting delay distribution is then reduced by the same factor of 800, corresponding here to a 16-ps time window where all amplified laser shots will be concentrated. However, in order to concentrate the delay distribution in the desired time region, much greater demands on the AD-ASOPS device will have to be met. Indeed, only fast-throughput data processing at the oscillator repetition rate will allow to compute the delays of all candidate pulses fast enough for making the appropriate decision in due time. This requirement involves specific VHDL developments currently under progress, although beyond the scope of the present article.

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