1 Introduction

Numerous ultrashort laser applications use laser pulses originating from separate synchronized laser sources. In particular, this is an especially important and challenging task with free-electron lasers (FELs) producing short pulses of X-ray light: Both for the FEL operation and for the conduction of pump–probe experiments with FELs, different external lasers need to be kept tightly synchronized for long periods of time [1–3]. In addition, in many pump–probe experiments with table-top ultrafast lasers, pulses of different wavelengths are generated and/or amplified using regenerative, multipass or parametric amplifier systems and then have to be recombined on the sample with a precisely determined timing (see, e.g., [4]). Thus, there is a strong motivation for continuous improvement and development of techniques related to the measurement and subsequent stabilization of the relative arrival time of pulses produced by different laser sources or originating from the same source and consequently undergoing amplification or frequency conversion in separated channels. It should be noted that also in the latter case, the process of amplification (both in regenerative and parametric amplifiers) exhibits time jitter and drifts that are not negligible on a 10-fs timescale [5–8], so the exact timing synchronization of all pulses is not automatically granted. The relative pulse timing is typically controlled by using optical cross-correlators, where the two pulses under investigation interact in a medium and produce a signal which depends on the degree of their temporal overlap [5, 7, 8]. When both pulses are at optical wavelengths, a second-order nonlinear process in a crystal is typically used and the frequency-mixing signal (FMS) is detected. In this case, the intensity of the FMS is proportional not only to the relative timing, but also to the intensity of both pulses, while, ideally, the pulse timing determination requires a signal which depends only on the pulse delay and is immune to the changes of the pulse(s) intensity. An elegant method allowing to reach this condition has been proposed in [9], and it uses a balanced scheme where the studied collinearly propagating pulses are split in two channels and flipped in time in the second one. In this way, the difference between the FMS obtained in the two channels, measured by a balanced photodiode, depends only on the time delay between the two pulses. The idea has been implemented in different versions of the balanced cross-correlator (BCC) and is often used for the phase-locking and timing control of ultrafast oscillators and fiber links [10–12]. An alternative and potentially simpler approach could be the use of a single-shot type scheme, where the two pulses under study meet in the nonlinear medium at an angle, which leads to a dependence of the pulse relative arrival time along the transverse direction. In the single-shot geometry, the relative arrival time of the pulses at each shot is coded in the transverse distribution of the generated FMS so it can be used for pulse duration and for timing diagnostics if combined with a position sensitive detector [13]. The single-shot scheme relies on the position of the peak of the signal, and it could be expected that it would be intrinsically insensitive to pulse energy changes. In the standard non-collinear single-shot scheme, however, the maximum angle between the pulse fronts, and therefore the observable time window, is typically limited by the phase-matching and group velocity conditions imposed by the nonlinear crystal [14]. Single-shot collinear cross-correlator scheme that may overcome this limitation have been proposed in the literature [15, 16]. It is based on collinearly propagating pulses having pulse front(s) tilted with respect to the propagation direction by using the dispersion introduced by a diffraction grating. The above-mentioned works proposing this geometry are referring mostly to pulse measurement applications, and to our knowledge, the aspects related to pulse timing applications have not been studied in the literature. In this paper, we propose and characterize a very simple and compact geometry of collinear, tilted-front, single-shot cross-correlator applied to measurement of relative timing between pulses. The scheme is based on the different action of a diffraction grating on the two pulses impinging on it at a different angle and can be used with pulses of equal or different wavelengths. We present experimental data demonstrating that the scheme is capable of measuring the relative time delay of low-energy (<0.5 nJ) pulses derived from a fiber oscillator with high precision and in a very large time window, making it very suitable for pulse timing applications.

2 Basic scheme

As depicted in Fig. 1, in the proposed scheme, the two pulses to be cross-correlated impinge on a diffraction grating at different angles that are chosen so that one of the beams (P1) is diffracted (in first order) in the same direction in which the second beam (P2) is transmitted in zero order. In this way, the first pulse has a pulse-front tilt caused by the grating’s angular dispersion [17, 18], while the second one has the pulse front unaffected (i.e., orthogonal to the direction of propagation). After passing the grating, the two pulses propagate collinearly in a nonlinear crystal cut for a phase-matching–frequency-mixing interaction. Because of the pulse-front tilt of the pulse P1, the intensity distribution of the sum-frequency signal contains information on the pulse duration of the two pulses, and the transverse position of its peak, measured by a position sensitive detector, depends on their relative arrival time. As it is known, upon propagation, the dispersed pulse broadens both spatially and temporarily, so it is important to first determine the propagation range in which this effect does not affect the measurement and can be neglected. For this purpose, we will use some of the formulae related to pulse-front tilt derived by Martinez [16]. Following the notation in [17], the tilt angle ψ of the pulse with central wavelength λ diffracted on a grating at an angle θ is given by:

$${ \tan }\,\psi = kc\beta ,$$
(1)

where k is the wavenumber and β is a coefficient characterizing the dispersion of the grating given by:

$$\beta = - m\lambda^{2} /(2\pi cs\,{ \cos }\theta ).$$
(2)

Above, m is the diffraction order and s is the grating groove spacing. As it can be seen, for a given wavelength, the pulse-front tilt can easily be adjusted to be in the range 20°–60° by properly choosing the diffraction grating and the geometry of incidence. We note that both transmission and reflection gratings can be used; however, in reflection geometry, there may be some limitations on the usable angle of incidence/diffraction imposed by the nonlinear crystal size. Use of higher diffraction order could in principle give access to higher tilt angle and therefore larger time window, provided gratings optimized for such regime are available. Assuming that the nonlinear crystal is positioned in very close vicinity with the grating exit surface and that the delay of the two pulses is adjusted so that the zero delay coincides with the center of the two beams, the relation between relative time delay Δt and horizontal displacement Δx measured from the center is given by:

$$\Delta t = \frac{{\Delta x\,{ \tan }\psi }}{c} =\Delta xk\beta ,$$
(3)

The temporal window in which the frequency-mixing signal can be observed will be limited by the beam size of the smaller of the two beams or by the crystal aperture. Thus, to obtain a window in the ps range, the minimum beam and crystal size in this plane should be on the order of a millimeter. The relative pulse timing is visualized by the transverse position of the peak (or center of mass) of the signal image, so the precision of the time measurement will be determined by the accuracy of determination of the peak or center of mass of the image. It can be expected that, when the beams propagate away from the grating, the conditions for this deteriorate because of the pulse lengthening of the dispersed pulse and the corresponding image spreading. As it is known, the pulse broadening after diffraction on a grating is caused by group delay dispersion and by the lateral walk-off of the spectral components. The analysis in [17] quantifies these two contributions and in addition shows that there is also some modification of the initial pulse-front tilt upon propagation. Here we will directly use some of the formula derived in [17]. The pulse duration is governed by the following equation:

$$\tau^{2} = \tau_{0}^{2} \left[ {\left( {1 + u} \right) + \frac{{\left( {2k\beta^{2} z} \right)^{2} }}{{\left( {1 + u} \right)\tau_{0}^{4} }}} \right],$$
(4)

where u is a parameter related to the ratio of the lateral walk-off caused by dispersion to the beam size and is defined as:

$$u = \left[ {2\beta z/\left( {w_{0} \tau_{0}^{2} } \right)} \right]^{2} .$$
(5)

For the scheme under study, as it was mentioned above, the beam size in the plane of dispersion need to be kept large in order to insure a large time window, so in typical conditions, u ≪ 1 and the main term causing the pulse lengthening will be the one related to group delay dispersion. For keeping the latter low and avoiding excessive pulse duration increase, the distance of the nonlinear crystal from the grating has to be kept in the sub-5 mm, if a femtosecond-level precision is required. As mentioned above, it has been shown in [17] that the propagation also leads to a deviation of the pulse-front tilt with respect to the value given by (3); however for the propagation length and crystal thickness of interest for this scheme, this deviation is negligible and the pulse-front tilt can be considered constant and defined by (3).

Fig. 1
figure 1

Basic scheme

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In Fig. 1, the two pulses are assumed to have two different wavelengths, and the polarization of the dispersed (P1) pulse to be in the plane where the grating efficiency is maximized. The polarization of the other pulse (P2) can be chosen depending on the type of the frequency-mixing process used, and it can be in the same plane (type 1 mixing), or orthogonal (type 2 mixing). When two pulses of the same wavelength are to be cross-correlated, their polarization can be made orthogonal and a type 2 second-harmonic generation process will be used, in order to provide a background free cross-correlation signal. We note that there is some additional flexibility on choosing the incidence angles in a way that the transmitted/reflected and diffracted pulses are not exactly collinear, which may offer advantage under certain experimental conditions (for example, if one of the pulses has an order of magnitude higher energy, this can be used to screen it in order to avoid leakage through the filter).

3 Experimental setup and performance

We tested the two versions of the scheme shown in Fig. 2, using a commercial femtosecond Er-doped fiber laser (Menlo Systems C-Fiber) as a source. Furthermore, we will discuss in details only the results obtained with the version, where the two pulses have different wavelengths, because it can be considered more general. The performance obtained with the single-wavelength scheme was very similar. The laser oscillator delivers a train of pulses with duration of about 95 fs (assuming a Gaussian shape) at 110 MHz repetition rate, having a rather broad and modulated spectrum around 1550 nm. The output of the laser is divided in two beams by a 70/30 beam splitter (BS). For obtaining the second wavelength pulse needed to test the scheme, the reflected (80 mW) beam is frequency doubled in a type 1 PPLN crystal, generating about 30 mW of second-harmonic light centered at 775 nm and having bandwidth of about 8-nm FWHM. The frequency-doubling process in this crystal is accompanied also with an unwanted third-harmonic light around 520 nm, which is cut by a color glass filter (F1). The second-harmonic beam passes through a coarse delay line and two steering mirrors (where also the residual fundamental beam is removed) and then impinges on the surface of the grating at about 29°. The path of the fundamental beam contains a delay line, incorporating both a micrometer-driven delay and a piezo-driven delay stage (PI P-625.1CD), allowing fine tuning of the optical path difference of the two pulses arriving at the grating. The choice of transmission grating for the two-wavelength version and a reflective grating for the single wavelength one was dictated only by the available gratings. The transmission grating used has a groove density of 1250 l/mm and an efficiency of c-ca 80 % in the chosen Littrow geometry (angles of incidence and diffraction about 29°). At this angle of incidence, the pulse-front tilt calculated from (1) is about 48°. Care is taken that the diffracted 775-nm beam (about 25 mW) and the transmitted 1550 mm beam (about 17 mW) propagate collinearly after the grating. The two beams are independently focused in the vertical plane by the cylindrical lenses CL1 and CL2 of focal distances 100 and 80 mm, respectively. As an alternative, a common long focal length (1000 mm) cylindrical lens positioned in front of BS was also tested and gave a sufficient signal and similar performance. The nonlinear crystal, which was positioned about 3 mm from the grating, was a 4 × 4 × 4 mm BIBO cut in the YZ plane for type 1 frequency doubling of 1064 nm (θ = 169.6°, φ = 90°). It can be seen that this crystal allows to obtain phase matching also for the process we need, i.e., type 1 frequency mixing of 1550 and 775 nm with a rather high effective nonlinearity of 2.87 pm/V (θ = 171°, φ = 90°)Footnote 1. With the polarization orientation used in our experiment, the phase matching was obtained by tilting the crystal by about 2° in the vertical plane (i.e., orthogonal to the plane of grating dispersion). The group velocity mismatch of the two infrared pulses propagating along the phase-matching direction can be calculated to be about 117 fs/mm. The generated sum-frequency signal was collected by a doublet lens (f = 60 mm) and imaged onto the surface of CCD camera (Basler Scout scA1300-32gm) with a magnification 3× to match roughly the beam size in the crystal with the CCD horizontal dimension (and in this way, the available time window). A thick color glass filter was used for cutting the two infrared beams (F2). The cross-correlator was calibrated by applying a known shift on the PZT delay line.

Fig. 2
figure 2

a The two-wavelength experimental setup using transmission grating and b the single-wavelength version using reflective grating. BS: beam splitter; WP: half-wave plate; L1, L2, L: spherical lenses; CL1, CL2: cylindrical lenses; NLC1, NLC2, NLC: nonlinear crystals; DG: diffraction grating; F1, F2, F: color glass filters; DL: delay line; PZT DL: piezo translation stage delay line; P: polarizing cube beam splitter

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A typical signal image, taken close to the center of the time window, is shown in Fig. 3. The width of the image represents the real size of the CCD, i.e., the time window of our cross-correlator. A CCD exposure time of 20 ms was used. The signal has a temporal width of about 450 fs FWHM. The dominating mechanism causing this spreading is the group velocity mismatch in the crystal, which might also be the cause of the observed slight asymmetry of the cross-correlation profile. We note that a shorter crystal (e.g., of 2 mm length) would give a shorter signal pulse (and therefore an image which is narrower in the horizontal direction) and may allow a temporal resolution even higher than the one demonstrated below. The crystal was positioned rather close to the grating (at about 3 mm) so the contribution of the 780-nm pulse lengthening due to the angular dispersion is smaller than the one of the crystal length (about 160 fs according to Eq. 4).

Fig. 3
figure 3

A typical cross-correlation signal image

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A typical short-term (30 s) trace is shown in Fig. 4a. It shows, at a fixed translator position, the pulse relative timing position calculated by the center of mass of the image. As it can be noticed, it exhibits fast fluctuations (seen on 1 s or shorter timescale) that are mainly due to the intrinsic noise of the system. On the basis of this measurement, we can estimate an upper limit of about 70-fs RMS for the intrinsic noise of this version of the setup which can be attributed in part to the noise of the detector causing an uncertainty of the determination of the beam position and in part to real fluctuations of the beam position caused by pointing instability of the laser. In order to understand better this point, we measured the increase in noise when the signal level was deliberately decreased (by attenuating the coupled laser power using a waveplate and keeping the CCD exposure time unchanged) by a factor of 2 and by a factor of 5 and we found that the cross-correlator timing noise increased by 40 % and by 300 %, respectively. This indicates that the performance obtained in optimum conditions in our case (generated sum-frequency signal in the 5 μW level) is already partially limited by the noise of the detector. We note that a very similar timing noise level was observed in the balanced cross-correlator described in [12] which we built previously for laser oscillator optical locking following the scheme proposed in [9]. In addition to short-term noise, the long-term stability of the time delay measured by the cross-correlator is very important, especially in view of the foreseen applications of the proposed scheme. So, we also performed 8-h-long measurements; keeping the translation stage fixed, a typical trace is shown in Fig. 5. As it can be seen, the measured timing exhibits a well-pronounced modulation with a period of about 10 min and a peak-to-peak amplitude of about 1.7 fs. This behavior was found to exactly correlate with the periodic switching of the air-flux of the air-conditioning system which regulates the air temperature of the room to within ±0.5°. In fact, with the air-conditioning OFF, the modulation completely disappeared; however, room temperature was rising fast and was impossible to perform a long-term measurement. It is important to note that the observed timing fluctuations may have two main sources: The real time delay changes due to the fact that the optical paths of the two pulses change in a different way with temperature, and the error in the timing measurement caused by the beam pointing shift in the horizontal plane. The exact measurement of the weight of these two contributions would require a reference cross-correlator and goes beyond the scope of this paper. At present, we will (conservatively) accept that without beam pointing stabilization, the latter factor is dominant and state that in this particular geometry (i.e., time-to-space calibration factor), the proposed cross-correlator has an upper limit of the accuracy of long-term timing delay measurement of below 2 fs.

Fig. 4
figure 4

a Measured short-term stability of the cross-correlation signal and b its histogram

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Fig. 5
figure 5

Measured long-term stability of the cross-correlation signal with a fixed time delay

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The other parameter important for timing applications is the sensitivity of detection of relative time changes. To test it, we applied a series of small steps to the PZT. Figure 6 shows the data for steps of 1, 0.5 and 0.25 fs.

Fig. 6
figure 6

a Measured position of the cross-correlation signal in response of consecutive pulse delay steps introduced by the PZT. b Linear fits based on the average positions (each one obtained from 32 acquisitions of 2 s) performed after every step

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It can be seen that a delay change of 0.5 fs is well above the noise level and is reliably detected by this version of the setup. We note that one part of the fluctuations that makes the interpretation of the 0.25-fs shifts not so clear is caused by the slow drift of the cross-correlator discussed above.

Next, we demonstrate that the total relative pulse time delay window in which a good quality timing measurement could be done (without changing any parameter of the detection) is indeed very large, and for this particular case, it is about 3.3 ps (see Fig. 7a). On Fig. 7b, we plot the deviation of the measured versus the real value of the delay change from the best linear fit. This deviation is reproducible within below a fs if the measurement is done fast and therefore can be corrected for; it can be attributed to a slight signal image shape distortion along the long-range scans. For the sake of comparison, it is important to note here that the above-mentioned balanced cross-correlators [9–12] have typical time windows of few hundred femtoseconds and the linear region is only a fraction of it.

Fig. 7
figure 7

a Full-range linearity of the cross-correlator response. Each point represents an average of 32 measurements, done in 2 s. The line is a linear fit. b Difference between the measured positions and the linear fit

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As it was mentioned in the introduction, an important feature of the tilted-front cross-correlation scheme is that it is expected to have a rather low sensitivity to changes of the energy of one of the two cross-correlated pulses. To explore this point, a zero-order λ/2 waveplate has been inserted in front of the frequency-doubling stage. By rotating the waveplate, the generated second-harmonic power could be smoothly changed. A curve showing the pulse timing position indicated by the cross-correlator as a function of the normalized integrated intensity of the signal, when it changes by about 12 % is shown in Fig. 8. We note that the observed pulse timing change of about 0.3 fs is real and is due to the delay change introduced by the modified effective refractive index, and therefore optical path, of the pulse inside the zero-order waveplate [19]. In typical conditions, the energy stability of the pulses delivered by oscillators, regenerative amplifiers and even parametric amplifiers is better than a few percent, so on the basis of the above measurement, we can safely state that in the tested configuration, the scheme can be considered insensitive to the monitored pulse intensity.

Fig. 8
figure 8

Three consecutive measurements of time delay measured by the cross-correlator where energy of one of the pulses (775 nm) is changed by 12 %

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4 Conclusions

The results presented above show in a convincing way that the studied single-shot geometry may be applied with success for precise laser timing applications. The reported long-term cross-correlator stability measurement indicates that a good temperature stabilization and well-isolated environment need to be provided in order to achieve the potential of combining sub-fs timing accuracy with a few picosecond wide instantaneous time window. We have used low-energy, high-repetition rate pulses delivered by a single-fiber oscillator for this demonstration. While the full timing information is contained in every single pulse of the generated sum-frequency signal, an exposure time of 20 ms (i.e., >106 pulses) was needed with a standard CCD sensor. The use of nonlinear crystals with higher nonlinearity (e.g., PPLN or PPKTP) as in all recently reported BCC schemes [10–12] can increase the signal in the same conditions by an order of magnitude and allow further reducing the exposure time if needed. It can be foreseen that the method could easily be adapted and find applications in the detection and stabilization of timing of regenerative or parametric amplifiers to their seed or pump pulses. In such case, the cross-correlator setup would still be very similar and the main difference with respect to the reported scheme could be that a slightly non-collinear geometry may need to be applied in order to allow sending only the signal pulse to detector and remove potential background for the high-energy fundamental pulses. In case the scheme is applied for optical synchronization of separated laser oscillators (this may include also a synchronization of a laser oscillator to an optical reference signal delivered by a fiber link), there will be the need that the repetition rate of the slave oscillator is coarsely locked to the master one, e.g., by a standard RF locking (see, e.g., [12] and refs therein). Then, the error signal generated by the single-shot cross-correlator could be used to fully compensate the slow timing drifts and to improve the timing jitter performance in a frequency range that will be mostly limited by the speed of the position sensitive detector. While a standard (matrix type) CCD, like the one used here, would introduce a limit of 50–100 Hz on the detection speed, we note that one could implement a line-scan CCD (e.g., BASLER RACER series) allowing detection bandwidths exceeding several kHz and that even higher speed of image position detection are feasible with position sensitive photodiodes.