1 Introduction

Ultrashort optical pulses are used in studies of ultrafast phenomena in science and technology [1,2,3,4]. Sub-cycle optical pulses have been generated in the visible region and used in spectroscopic applications [5]. Further extension to the deep ultraviolet (DUV) and to the near-infrared (NIR) regions poses challenges for use in a variety of applications. For example, ultrashort optical pulses in the DUV region are useful for non-resonant two-photon ionization in mass spectrometry for observing a molecular ion to improve reliability in forensic science [6, 7]. In contrast, ultrashort optical pulses in the NIR region can be employed for efficient field ionization, in which a Keldysh parameter should be less than 1, and then a high-intensity laser emitting at longer wavelengths is required [8]. Until now, several methods have been proposed to generate and characterize ultrashort optical pulses in a wide spectral region from the DUV to the NIR. Indeed, ultrashort optical pulses are generated by filamentation in neon gas [9] and by self-phase and cross-phase modulations [10]. Four-wave Raman mixing in hydrogen gas is another option to generate octave-spanning multi-color emissions and to generate ultrashort pulses by Fourier synthesis of the emissions [11,12,13]. It is possible to generate intense sub-2-fs pulses in the DUV region as well as sub-cycle pulses in the NIR region.

To prove the generation of such ultrashort optical pulses, it is necessary to develop an advanced and simple tool to evaluate laser pulses. In past decades, several techniques have been developed and used for this purpose, e.g., autocorrelation (AC) [14] and spectral phase interferometry for direct electric-field reconstruction (SPIDER) [15]. Sub-4-fs pulses have been measured by spatially encoded arrangement SPIDER [16] and by the dispersion scan (D-scan) [17]. Another technique is to reconstruct the absolute phase of optical pulses, as exploited in an attosecond streak camera [18]. Frequency-resolved optical gating (FROG) has been widely used to characterize the temporal waveform and the spectral phase [19]. In particular, cross-correlation FROG (XFROG) has a broad bandwidth to characterize an ultrashort optical pulse [20]. Such optical pulses generated by four-wave mixing in a bulk medium [21] and by noncollinear optical parametric amplification (NOPA) [22] have been successfully measured by this technique. A method combining XFROG and electro-optic sampling [23] has been used for the measurement of the absolute phase of the pulse [24]. XFROG has recently been employed for the measurement of ultrashort optical pulses generated by four-wave Raman mixing in a gas medium. It should be noted that the bandwidth usable for phase matching in XFROG is very wide, supporting a measurement of a 1.2-fs optical pulse when a thin nonlinear optical medium such as a 5-μm beta-barium borate (β-BBO) is used [25]. Thus, XFROG has potential for use in the characterization of an ultrashort optical pulse train generated by Fourier-synthesizing several vibrational Raman emissions.

The technique of XFROG requires a reference pulse. However, conventional XFROG using a single-color reference pulse with duration longer than the test pulse cannot be used for characterizing the relative spectral phase of the multi-color laser emissions [25]. For example, although XFROG has been applied for the measurement of a pulse train, no discussion has been reported about the relative phase, and the temporal waveform has been calculated by assuming the relative phase of the pulse [21, 26]. For this reason, it is desirable to employ a reference pulse whose duration is comparable to the test pulse [27]. However, it is difficult to generate and use a reference pulse of a few femtoseconds in the measurement of a pulse train. Relative phases among several spectral components with nanosecond pulse widths have been measured using a two-color beam as a reference pulse [28]. This technique referred to as modified SPIDER has been used to Fourier-synthesize the ultrashort pulses in a nanosecond envelope.

In this study, we developed a diagnostic tool based on XFROG using an optical beat comprised two spectral components in the test pulse as a reference pulse to characterize a train of a few-femtosecond optical pulses in an envelope of duration of several tens of femtoseconds. When a single-color spectral component was used as a reference pulse, the temporal profile of an ultrashort pulse train was determined with the following parameters: (a) frequency separation, (b) temporal separation, (c) relative intensities, and (d) spectral profile. However, it was difficult to characterize the relative phase among the spectral components in the test pulse as reported in Ref. [25]. The present technique using the optical beat provided interference fringes in the XFROG trace and was useful for evaluating the relative phase of the components. Since the temporal profile of an ultrashort pulse train can be changed considerably by the relative phase, this technique provides a useful means for evaluation of the pulse train. In addition, Raman emissions generated by four-wave Raman mixing in hydrogen were verified to be phase locked.

2 Experimental setup

The experimental setup (Fig. 1) consists of the fundamental beam of a Ti:sapphire laser (35 fs, 800 nm, 1 kHz, Solstice Ace, Spectra-Physics, USA) and a beam from an optical parametric amplifier (35 fs, 1200 nm, TOPAS, Spectra-Physics, USA) pumped by the Ti:sapphire laser being superimposed using a dichroic mirror. A portion of the combined beam (optical beat) was reflected using a 1-mm-thick fused silica plate for use as a reference pulse. To generate a multi-color laser emission by four-wave Raman mixing, the two-color beam was focused into a gas cell filled with hydrogen (1.3 atm) using an aluminum-coated concave mirror. Three spectral components emitting at 600, 800, and 1200 nm were separated by dichroic mirrors, and were recombined using the same dichroic mirrors after suitable delays and beam reflections using two silver mirrors mounted on delay stages. A 0.5-mm-thick fused silica plate was inserted into the beam emitting at 600 nm and was rotated to change the relative phase among these spectral components.

Fig. 1
figure 1

Block diagram of experimental apparatus. DM dichroic mirror, CM concave mirror, FS fused-silica plate, FROG frequency-resolved optical gating

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In the setup for the XFROG measurement (Fig. 2), the reference pulse was divided into two beams using a beam splitter, one of which was after a delay combined with the other. A pulse train was generated by superimposing the three spectral components emitting at 600, 800, and 1200 nm. The reference pulse comprises two components, i.e., the emissions at 800 and 1200 nm. The reference and test pulses were non-collinearly focused onto a type-I β-BBO crystal (θ = 45°, thickness 5 μm). While changing the temporal delay, second harmonic (SH) and sum-frequency (SF) signals were measured simultaneously. An XFROG trace was recorded by changing the delay.

Fig. 2
figure 2

Block diagram of FROG system. BBO β-barium borate crystal, CM concave mirror, SH second harmonic, SF sum-frequency, BS beam splitter, MS multi-channel spectrometer

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3 Results and discussion

3.1 Observed XFROG trace

Figure 3 shows an XFROG trace observed using a two-color reference pulse (optical beat) emitting at 800 and 1200 nm. There are four signal components at 342, 400, 480, and 600 nm, and interference fringes are seen for the components at 400 and 480 nm. They are in striking contrasts to an XFROG trace measured using a single-color reference pulse emitting at 800 nm; three spectral components were observed at 342, 400, and 480 nm and no interference fringes were seen in any components. The spectral component observed at 480 nm in Fig. 3 consists of two sum-frequency signals of the emission at 800 nm in the test pulse (reference pulse) and the emission at 1200 nm in the reference pulse (test pulse). These two emissions interfere to form the fringe pattern at 480 nm. In a similar manner, the component at 400 nm consists of sum-frequency signals of the following three components: the emissions at 800 nm in the test and reference pulses, the emissions at 1200 nm in the reference pulse, and the emissions at 600 nm in the test pulse. As will be discussed later, information concerning the relative phase among these emissions can be extracted from the FROG trace.

Fig. 3
figure 3

XFROG trace observed using an optical beat as a reference pulse

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An XFROG trace was measured by changing the angle of the fused-silica plate inserted in the path of the beam emitting at 600 nm. As expected, the fringe pattern at 480 nm remained unchanged because this component appears by sum-frequency generation of the beams at 1200 and 800 nm. The angular dependence of the shift in the fringe pattern relative to that observed at 480 nm (Fig. 4) shows that the observed data are in good agreement with the results of the calculation. This suggests that the Raman sidebands generated by four-wave Raman mixing are phase locked and that the fringe pattern can be used to control the relative phase in Fourier synthesis to generate a specified temporal waveform.

Fig. 4
figure 4

Dependence of the angle of the fused-silica plate on the phase shift in the fringe pattern observed at 400 nm. Solid line calculated data, red circles observed data

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3.2 Reconstruction of XFROG trace

The structure of the XFROG trace is complicated and is difficult to retrieve the observed data. Hence, an XFROG trace was calculated employing several conditions to fit the observed data to interpret the structure of the trace and know the direction to optimize the spectral parameters in Fourier synthesis to generate ultrashort optical pulses.

3.2.1 Interference fringe pattern

An XFROG trace was calculated in a computer simulation assuming an optical beat with a pulse width of 35 fs emitting at 800 and 1200 nm as a reference pulse, and a pulse train with a pulse width of 35 fs emitting at 600, 800, and 1200 nm was used as a test pulse. In theory, a pulse train consists of several 2.4-fs pulses in a 35-fs envelope. Figure 5a, b show two XFROG traces calculated using a single-color beam and a two-color beam (optical beat) as a reference pulse, respectively. They are apparently different from each other, and the main features are identical to the observed data (see Sect. 3.1). With a single-color reference pulse, only three spectral components with no interference fringe appear in the trace. In contrast, when an optical beat was used as a reference pulse, interference fringes are clearly observed in the components at 400 and 480 nm. The interference fringe at 480 nm is the superposition of the spectral component arising from the sum-frequency generation of the emissions at 800 nm in the test pulse and at 1200 nm in the reference pulse, and of the spectral component arising from the sum-frequency generation of the emission at 1200 nm in the test pulse and at 800 nm in the reference pulse. These two spectral components interfere with each other to form the interference fringe at 480 nm in Fig. 5b. In a similar manner, the interference fringe at 400 nm results from the superposition of the spectral component arising from the sum-frequency generation of the emissions at 600 nm in the test pulse and at 1200 nm in the reference pulse, and the spectral component arising from second harmonic generation of the emission at 800 nm in the reference pulse and at 800 nm in the test pulse. These two spectral components interfere with each other to form the interference pattern at 400 nm. Hence, the interference fringe at 400 nm contains information concerning the relative phase among the three spectral components at 600, 800, and 1200 nm.

Fig. 5
figure 5

XFROG traces calculated using a single-color beam (800 nm) and b two-color beam (800 and 1200 nm) as a reference pulse

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Figure 6 shows the expanded views calculated under the two conditions: (a) all the emissions at 600, 800, and 1200 nm are in phase, whereas (b) the phase of the emission at 600 nm is shifted by π rad (out of phase) against the phases of the other two emissions. The calculated patterns of the two are very different. It should be noted that such information is not obtainable using conventional XFROG with a signal-color reference pulse. Alternatively, the interference patterns were calculated by changing the relative phase of the reference pulse from zero (in phase) to π/2 rad for the 1200-nm emission. However, no difference was observed, suggesting that changing the relative phase in the reference pulse is pointless about characterizing the test pulse.

Fig. 6
figure 6

Expanded view of XFROG traces calculated for two cases: a all the emissions are in phase; b the emission at 600 nm is out of phase from the other two emissions

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3.2.2 Stretch and tilt of the signal components

It is well known that the transform-limited pulse is chirped by passing the beam through dispersive media such as solid optics or even ambient air. The signal components (Fig. 3) are stretched and are tilted down toward the right. These effects would arise from group delay dispersion (GDD) of the emissions, as the pulse width can be increased and sum-frequency generation with two chirped pulses provide a tilted signal component in the XFROG trace. To verify this, an XFROG trace was calculated assuming the emissions at 800 and 1200 nm in the test pulse are positively chirped by GDD = 1000 fs2. As shown in Fig. 7a, the positive GDD leads to signal components tilted down toward the right. In contrast, when the emissions at 800 and 1200 nm were negatively chirped by GDD = −1000 fs2, the components were stretched and were tilted down toward the left. Hence, the direction and angle of tilt in the signal component depends on the GDD in the test pulse.

Fig. 7
figure 7

Calculated XFROG traces. All the spectral components in the test pulse were assumed to be chirped: a GDD = 1000 fs2, b TOD = 10,000 fs3

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When the pulse is extensively deteriorated, the third-order dispersion (TOD) in the pulse cannot be neglected. Figure 7b shows an XFROG trace calculated assuming all emissions at 600, 800, and 1200 nm in the test pulse are positively chirped by TOD = 10,000 fs3. It is interesting to note that the signal component is asymmetrical along the delay axis and has a triangular shape. Such an asymmetry has been observed for the signal components in Fig. 3, suggesting that the effect of TOD is non-negligible for the test pulse used in this study.

3.2.3 Shift of the signal components from zero delay

The signal components in the observed data shown in Fig. 3 shift from the zero delay defined by the center of gravity in the trace. From Fig. 7, the positions of the signal components do not shift even when the GDD (or TOD) of the emissions in the test pulses was changed. Moreover, the situation remained unchanged even when the GDD of the reference pulse was changed. To explain the shift of the components from the zero delay, the effect of group delay (GD) of the reference and test pulses was examined. Figure 8a shows an XFROG trace, in which the emission at 800 nm in the reference pulse was shifted by GD = 50 fs. The components occurring from the 800-nm emission (480, 400, 342 nm) are apparently shifted by a change in the GD. Similarly, the effect of the GD in the test pulse was also examined. When the emission at 600 nm in the test pulse was shifted by GD = 50 fs, the XFROG trace in Fig. 8b shows only shifts in the 400- and 342-nm components because the 600-nm emission is used for the generation of these components. The components related to the delayed emissions are shifted accordingly in the FROG trace. It is interesting to note that the interference fringes concerning the shift are tilted. In addition, the fringe pattern becomes unclear on both sides, probably because of an imbalance in the intensity of the emissions superimposed. Thus, the GD of the emission in the reference and test pulses induces a significant effect in the XFROG trace.

Fig. 8
figure 8

Calculated XFROG traces. A group delay of 50 fs was added only a to the 800-nm emission in the reference pulse and b to the 600-nm emission in the test pulse

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3.2.4 Change in the tilt angle of the fringe pattern

The tilt angle of the interference fringes observed at 480 nm in Fig. 3 is changed in the middle of the signal component. The dispersion of a material is well known to depend on the wavelength and changes the temporal waveform of the pulse. To examine this effect, only the 800-nm emission was chirped by GDD = 3000 fs2. From the calculated trace (Fig. 9), the tilt angle changes in the middle of the interference pattern.

Fig. 9
figure 9

Calculated XFROG trace. The 800-nm emission in the test pulse is chirped by GDD = 3000 fs2

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3.2.5 Inhomogeneous signal components

The XFROG trace shown in Fig. 3 consists of spectral components with irregular structures. This cannot be explained straightforwardly by simple combinations of transform-limited and chirped pulses. Occasionally, the spectral and temporal shapes of the laser pulse are deteriorated by self-phase modulation (SPM) and self-focusing (SF), which are further followed by degradation of beam quality, e.g., deformation of the wave front. In this study, this effect was examined by assuming a laser pulse generated by superimposing a few optical pulses. The result is shown in Fig. 10, where the emission at 1200 nm was assumed to consist of two spectral components shifted by 0 and 700 cm−1. This figure shows the splitting or broadening of the signal components (600 and 480 nm) relating to these emissions, as expected. This also shows the complex nature of the laser pulse used as a test pulse in this study.

Fig. 10
figure 10

Calculated XFROG trace. Data were calculated assuming that the NIR pulse consists of two optical components shifted by 0 and 700 cm−1

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3.2.6 Reconstruction of the observed XFROG trace

To reconstruct the observed data, an XFROG trace was calculated using a reference pulse (60 fs) consisting of an emission at 1200 nm (GD = 0 fs, GDD = 0 fs2, TOD = 0 fs3) and an emission at 800 nm (GD = 30 fs, GDD = 0 fs2, TOD = 0 fs3) and a test pulse (20 fs) consisting of emissions at 1200 nm (GD = 10 fs, GDD = 200 fs2, TOD = 0 fs3, and GD = −20 fs, GDD = 300 fs2, TOD = 0 fs3 and the center frequency is shifted by 700 cm−1), emissions at 800 nm (GD = 60 fs, GDD = 100 fs2, TOD = 0 fs3 and GD = −30 fs, GDD = 800 fs2, TOD = 0 fs3) and an emission at 600 nm (GD = 30 fs, GDD = 0 fs2, TOD = 3000 fs3). The calculated XFROG trace (Fig. 11) has reproduced the following four features appearing in the observed data shown in Fig. 3; (1) a fringe pattern induced by phase lock of the emissions; (2) lengthening of the spectral component by GDD; (3) tilt of the component by GDD; (4) asymmetry of the pattern for the component by TOD; (5) tilt of the fringe pattern by GD; (6) change of the tilt angle in the middle of the signal component; and (7) splitting (broadening) of the component by degradation of the beam (SPM/SF). The reasonable agreement of the calculated trace with the observed data suggests a sufficient performance of XFROG for interpreting the complex structure in a test pulse using a two-color reference pulse.

Fig. 11
figure 11

Calculated XFROG trace. See text for details of calculation conditions

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4 Conclusion

We have developed a new type of diagnostic tool based on XFROG to measure a train of ultrashort optical pulses generated by four-wave Raman mixing in hydrogen gas. Using a two-color beam as a reference pulse, whose frequency separation was identical to that in the test pulse, information about the relative phase of the spectral components was extracted from the interference fringe pattern in the XFROG trace. This is in striking contrast to the method using a single-color reference pulse. The observed fringe patterns suggest phase locking in the process of four-wave Raman mixing that was used for multi-color generation in this study. Because the phase matching bandwidth of XFROG is very wide, XFROG has the potential for use as a powerful tool for the measurement of sub-cycle optical pulses in the NIR region.