1 Introduction

An ultrashort optical pulse has been successfully utilized in studies of nonlinear optical phenomena. For example, an attosecond pulse can be produced based on high-order harmonic generation in the soft X-ray region and has been applied to the study of ultrafast phenomena [1]. Another approach would be the generation of a multi-frequency laser emission in the near-ultraviolet (NUV)–visible (VIS) region, based on non-resonant four-wave mixing (FWM). In fact, ultrashort optical pulses as short as 2.2 fs have been produced by cascaded four-wave mixing (CFWM) and a subsequent Fourier synthesis of the emission lines [2].

The observation of high-order Raman sidebands in a wide frequency region was first reported a few decades ago [3]. In this approach based on resonant four-wave Raman mixing (FWRM), a laser emitting at two different frequencies, the spacing of which corresponds to a rotational Raman shift frequency (587 cm−1) for a transition from (v, J) = (0, 1) to (v′, J′) = (0, 3), was focused into hydrogen gas to originate numerous rotational Raman sidebands from the deep-ultraviolet (DUV) to the near-infrared (NIR) region [4]. Due to the extremely wide spectral region, it is suggested that this phenomenon could be used for the generation of an ultrashort optical pulse [5]. A two-color beam yields an optical beat (18 THz or a spacing of 57 fs), the frequency of which is determined by the reciprocal value of the Raman shift frequency, and coherently modulates hydrogen molecules. A subsequent pulse, referred to as a probe pulse, is split into multiple emission lines separated by a Raman shift frequency. This technique, which is based on molecular phase modulation for the generation of rotational Raman sidebands, can be used to produce an ultrashort optical pulse. In fact, an 11-fs optical pulse has been generated in the NUV region [6–8]. On the other hand, a different type of approach, referred to as an impulsive Raman technique, has been used. In this approach, the first pulse, which is shorter than the period of molecular rotation, impulsively excites the coherent molecular rotation and the second pulse serves as a probe pulse to provide high-order rotational Raman sidebands [9–12]. A 3.8-fs optical pulse has been produced using a 30-fs pump pulse.

The vibrational Raman shift frequency for a transition from (v, J) = (0, 1) to (v′, J′) = (1, 1) is larger than the rotational Raman shift frequency. As a result, the modulation frequency can be increased, permitting the production of a shorter optical pulse. A train of highly repetitive optical pulses (250 THz or a spacing of 8 fs) has occurred in a ns envelope, with a pulse width of 1.6 fs being achieved [13, 14]. Recently, Raman sidebands were generated from the NUV to the NIR region based on FWRM using 2-ns lasers emitting at wavelengths of 2,406 and 1,203 nm [15]. These wavelengths, integral multiples of the Raman shift frequency, were employed to control the carrier-envelope-offset frequency (CEO), providing a constant carrier-envelope phase (CEP) for each pulse during the pulse train [16, 17]. In addition, the use of lasers emitting at longer wavelengths prevents the occurrence of Stokes Raman sidebands by cascade-stimulated Raman scattering (CSRS), thus increasing the efficiency in the generation of anti-Stokes Raman sidebands through FWRM. In order to obtain a single ultrashort optical pulse, it is necessary to use two femtosecond lasers (<8 fs) serving as pump and Stokes pulses. A picosecond or sub-picosecond VIS laser, e.g., 250 fs at 600 nm and 70 fs at 800 nm, has been utilized to produce six vibrational sidebands in molecular hydrogen [18–21]. In a previous study, we reported on the observation of ten multiple Raman sidebands extending from the DUV to the NIR region (184–1,203 nm) using a femtosecond NUV probe pulse (35 fs, 401 nm) in addition to femtosecond NIR pump and Stokes pulses (35 fs, 802 and 1,203 nm) produced by a Ti:sapphire laser and the signal beam of a parametric optical amplifier (OPA) pumped by the same Ti:sapphire laser. In theory, the spectral bandwidth achieved yields a few 0.9-fs optical pulses with a constant CEP, as recognized by a Fourier synthesis [22]. To originate a single ultrashort optical pulse, it is necessary to use shorter laser pulses and to compensate for dispersion arising from the optical components including the air in the beam path, since such dispersion is not negligible especially in the DUV region.

In coherent anti-Stokes Raman scattering (CARS) spectroscopy, a pair of chirp pulses has been utilized as pump and Stokes beams to observe anti-Stokes Raman emission. This technique, referred to as “spectral focusing,” improves the spectral resolution [23], expands the spectral region to lower frequencies [24], and suppresses the background signal arising from non-resonant effects [25]. In order to efficiently generate high-order Raman sidebands in a solid material such as PbWO4 or diamond, linearly chirped pump and Stokes pulses have been employed to suppress the peak intensity and then to avoid parasitic nonlinear effects such as non-resonant FWM and self-phase modulation (SPM) [26, 27]. This technique provides numerous Raman sidebands up to 40 anti-Stokes and 5 Stokes sidebands with a spacing of 190 cm−1, which covers a range of 12,000 cm−1. However, the temporal property of the Raman emission such as a group delay dispersion (GDD) has not been measured in the above studies.

In this study, we changed the chirp of the pump (802 nm) and Stokes (1,203 nm) pulses (GDD = 0, ±1,000 fs2) and examined the chirp of the Raman sideband (602 nm) using a frequency-resolved optical gating (FROG) system. This approach based on chirped-pulse four-wave Raman mixing (CPFWRM) is useful for controlling the chirp of Raman sidebands, which is difficult, especially in the DUV region, e.g., the use of a pair of chirp mirrors. Then, this technique of chirp control can be used for compression of the pulse to generate an ultrashort optical pulse in the future by a Fourier synthesis of the Raman sidebands.

2 Experimental apparatus

The experimental setup used in this study is shown in Fig. 1. The signal beam (1,203 nm, 700 μJ) of the OPA (OPerASolo, Coherent Inc.) pumped by the fundamental beam of a Ti:sapphire laser (802 nm, 35 fs, 3.8 W, 1 kHz, Legend, Coherent Inc.) was utilized as a Stokes pulse for CPFWRM. The output of the fundamental beam (800 μJ) of the Ti:sapphire laser, which is designed for use in subsequent frequency mixing in OPA, was taken out using a mirror as a pump pulse in CPFWRM. The output of the Stokes beam was introduced into a telescope (not shown in the figure) for beam collimation and a neutral density (ND) filter, to avoid SPM in the prism (SF10) used for dispersion control. The typical pulse energy of the Stokes beam was 310 μJ after the ND filter, 230 μJ after the prism pair, and 200 μJ before the Raman cell, respectively. The energy of the pump beam emitting at 802 nm was 220, 190, and 170 μJ at the above locations, respectively. One of the prisms in the pairs was mounted on the translational stage placed on a rail to permit the distance between the pair of prisms to be changed and thus to control the dispersion of the laser pulse. The mirror was placed on the same stage, on which the prism was mounted, to produce a constant optical path length. A FROG system based on second harmonic generation (SHG) was constructed in our laboratory. A spectrometer (HR-4000, Ocean Optics Inc.) was used to measure the spectrum of the SHG beam. This second-harmonic-generation frequency-resolved optical gating (SHG-FROG) system was applied to both the pump and Stokes pulses to evaluate the parameters, including the spectrum, the pulse width, and the GDD, at different distances between the prisms. These two pulses were spatially collimated and synchronized in time by changing the position of the reflecting mirrors. The beams were focused into a Raman cell with 0.5-mm-thick windows (fused silica) using a parabolic mirror with a focal length of 65 cm. The cell was filled with hydrogen gas (1 atm). The Raman sidebands generated through FWRM were measured using the HR-4000 spectrometer. A self-diffraction frequency-resolved optical gating (SD-FROG) system was constructed in our laboratory and was used to evaluate the parameters for the first anti-Stokes Raman pulse emitting at 602 nm.

Fig. 1
figure 1

Experimental setup. DL delay line, PM parabolic mirror, ND neutral density filter, DM dichroic mirror

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3 Chirp pulse basics

The phase of the signal pulse in the OPA is shifted by a constant factor of π/2 against the fundamental pulse [28], suggesting that the phase of the signal pulse is fixed (not random). In the process of FWRM, the phase in the time domain for the anti-Stokes pulse can be expressed as ϕ AS = 2ϕ P − (ϕ S + π/2), where ϕ P and ϕ S are the phases of the pump (fundamental pulse in OPA) and Stokes (signal pulse in OPA) pulses emitting at 802 and 1,203 nm, respectively. As a result, multiline Raman emissions are coherently phased and provide an ultrashort optical pulse when their phases are appropriately controlled. A basic operation principle of the CPFWRM is similar to that of chirped-pulse coherent anti-Stokes Raman scattering (CPCARS) spectroscopy and is reported elsewhere [25, 26]. The parameters for a chirped pulse can be briefly summarized as follows [29, 30]. The electric field in the time (t) domain can be written as \(\widetilde{E} = E_{0} e^{{ - at^{2} }} e^{i(\omega t -\phi )}\) when it is assumed to have a Gaussian profile, where E 0 is the amplitude, Δt =\(\sqrt {2\ln 2} /\sqrt a\) is the pulse width, ω is a carrier frequency, and ϕ = −bt 2 is the phase and b is the chirp parameter. Then, the chirp parameter of the anti-Stokes pulse, b AS, can be calculated by

$$b_{\text{AS}} = { 2}b_{\text{P}} {-}b_{\text{S}}$$
(1)

where b P and b S are the chirp parameters of the pump and Stokes pulses, respectively. The chirp parameter can be written as (Δω)2 ϕ 2/2(Δt)2, where Δω is the spectral width and ϕ 2 is referred to as the GDD, which is a coefficient for the second term of a Taylor series for the phase, ϕ(ω). When the ratio of the spectral width and the pulse width is the same for the pump, Stokes, and anti-Stokes pulses, Eq. (1) can be simplified as

$$\phi_{{ 2 {\text{AS}}}} = { 2}\phi_{{ 2 {\text{P}}}} - \phi_{{2{\text{S}}}}.$$
(2)

In the experiment, the GDD values of the pump and Stokes pulses, i.e., ϕ 2P and ϕ 2S, were changed by translating the positions of the prisms either to 0 or ±1,000 fs2. The instantaneous frequency separation is 4,155 cm−1, when the pump and Stokes pulses have the same sign of GDD. When the sign is not the same, the instantaneous frequency separation changes during the pulse.

4 Results and discussion

The pulse width, spectral width, and GDD measured for the pump and Stokes pulses at different prism positions are shown in Fig. 2. The minimal pulse widths were 60 and 35 fs, which were obtained at a prism separation of 320 mm for both the pump and Stokes pulses emitting at 802 and 1,203 nm, respectively, as shown in Fig. 2a. The spectral width increased at distances shorter than 270 mm, as shown in Fig. 2b, which can be attributed to SPM due to a small beam diameter at the position of the first prism. The spectral widths approach minimum values, i.e., 16 nm (250 cm−1) and 100 nm (700 cm−1), for pump and Stokes pulses emitting at 802 and 1,203 nm, respectively. The value of time-bandwidth product (TBP) can be calculated to be 0.45 and 0.73 for the pump and Stokes pulses, respectively, suggesting that they are transform-limited (0.44) and nearly transform-limited (0.73 > 0.44), respectively. As shown in Fig. 2c, the value of GDD is proportional to the distance between the prism pairs as expected from a theory and the chirp of the pulse can be adjusted using this calibration curve.

Fig. 2
figure 2

Characteristics of the pump and Stokes pulses measured at different distances between the prism pairs: a pulse width b spectral width c GDD of pump and Stokes pulses emitting at (a) 802 (b) 1,203 nm, respectively

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When the transform-limited pulses (GDD = 0) were employed as pump and Stokes pulses, numerous Raman sidebands were generated from the DUV to the NIR region [22]. In the present study, the efficiency in the generation of the first anti-Stokes emission was investigated by changing the chirp of the pump and Stokes pulses. As a result, the output pulse energy was 11 μJ, when the pump and Stokes pulses were both transform-limited, positively chirped, and negatively chirped. On the other hand, the energy decreased to 8 μJ, when a positively chirped pump pulse and a negatively chirped Stokes pulse were employed. These results suggest that the anti-Stokes emission occurs efficiently only when the pump and Stokes pulses have the GDD with the same sign and the requirements in phase matching and energy conservation are satisfied. The intensity of the second anti-Stokes emission was ca. 10 % of the first anti-Stokes emission, and a similar trend was observed for the second anti-Stokes emission.

The chirp of the first anti-Stokes pulse was measured using the SD-FROG system. When the GDDs of the pump and Stokes pulses were zero, the pulse width and the spectral width were 62 fs and 11 nm (TBP = 0.60), respectively, as shown in Fig. 3. A simple pattern was observed in the FROG trace, and the value of GDD was calculated to be +500 fs2, which is larger than the value of 0 fs2 theoretically predicted from both Eqs. (1) and (2). This suggests that the GDD would become positive because of the dispersion arising from the optics such as the window materials and the air. A transform-limited anti-Stokes pulse would be obtained using a pair of mirrors with negative chirps for pulse compression.

Fig. 3
figure 3

Characteristics of the first anti-Stokes pulse obtained using transform-limited pump and Stokes pulses (GDD = 0 at 802 nm, GDD = 0 at 1,203 nm). a original FROG trace b retrieved trace (FROG error, 0.8 %); c temporal d spectral profiles calculated from the trace; the phase is indicated by an arrow; a measured spectrum using a spectrometer is indicated by a dotted arrow. The pulse width and the spectral width calculated from the FROG trace are shown in the figure

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The results obtained using the positively chirped pump and Stokes pulses (+1,000 fs2) are shown in Fig. 4. The pulse width and the spectral width of the anti-Stokes pulse were 56 fs and 16 nm (440 cm−1) (TBP = 0.76), respectively, and the GDD observed was +1,000 fs2. This value was identical to the value of 2(+1,000) − (+1,000) = +1,000 fs2 using Eq. (2). The chirp parameters can be calculated to be as follows.

$$b_{\text{AS}} = + \left( { 2\pi } \right)^{ 2} \left( { 4 40} \right)^{ 2} ( 3\times 10^{ 10} )^{ 2} 500/ 2\times \left( { 5 6} \right)^{ 2} = 0. 5 5\times 10^{ 2 7} {\text{s}}^{ - 2}$$
(3)
$$b_{\text{P}} = \, + \left( { 2\pi } \right)^{ 2} \left( { 7 70} \right)^{ 2} ( 3\times 10^{ 10} )^{ 2} 10^{ 3} / 2\times \left( { 100} \right)^{ 2} = { 1}. 1\times 10^{ 2 7} {\text{s}}^{ - 2}$$
(4)
$$b_{\text{S}} = \, + \left( { 2\pi } \right)^{ 2} \left( { 1 200} \right)^{ 2} ( 3\times 10^{ 10} )^{ 2} 10^{ 3} / 2\times \left( { 2 50} \right)^{ 2} = 0. 4 1\times 10^{ 2 7} {\text{s}}^{ - 2}.$$
(5)

In the calculation, the values of the spectral width, pulse width, and GDD for the anti-Stokes pulse (440 cm−1, 56 fs, 500 fs2), pump pulse (770 cm−1, 100 fs, 1,000 fs2), and Stokes pulse (1,200 cm−1, 250 fs, 1,000 fs2) were used. The SPM arising from the first prism increased the spectral width and then the pulse width, which would induce a mismatching between the pump and Stokes pulses.

Fig. 4
figure 4

Characteristics of the first anti-Stokes pulse obtained using chirped pump and Stokes pulses (GDD = +1,000 fs2 at 802 nm, GDD = +1,000 fs2 at 1,203 nm). a original FROG trace b retrieved trace (FROG error, 0.5 %); c temporal d spectral profiles calculated from the trace; the phase is indicated by an arrow; a measured spectrum using a spectrometer is indicated by a dotted arrow. The pulse width and the spectral width calculated from the FROG trace are shown in the figure

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As shown in Fig. 5, when negatively chirped pump and Stokes pulses (−1,000 fs2) were employed, the anti-Stokes pulse was slightly negatively chirped (−250 fs2). The pulse width and the spectral width were 60 fs and 6.4 nm (180 cm−1) (TBP = 0.32), respectively. By inserting a fused silica plate giving a positive dispersion [thickness, 250 fs2/(56 fs2/mm at 602 nm) = 4.5 mm], this negatively chirped pulse would be compressed to a transform-limited pulse. It has been reported that SPM reduces the spectral width and then increases the pulse width for a negatively chirped pulse [31]. Therefore, SPM induced by the optical components should be minimal. The observed value of GDD was in reasonable agreement with the value of 2(−1,000) − (−1,000) = −1,000 fs2 when the dispersion (+500 fs2) arising from the optics in the beam path was added to this value (−1,000 + 500 = −500 fs2). The chirp parameters calculated were as follows.

$$b_{\text{AS}} = \, - \left( { 2\pi } \right)^{ 2} \left( { 1 80} \right)^{ 2} ( 3\times 10^{ 10} )^{ 2} 2 50/ 2\times \left( { 60} \right)^{ 2} = \, - 0.0 40 \times 10^{ 2 7} {\text{s}}^{ - 2}$$
(6)
$$b_{\text{P}} = \, - \left( { 2\pi } \right)^{ 2} \left( { 2 50} \right)^{ 2} ( 3\times 10^{ 10} )^{ 2} 10^{ 3} / 2\times \left( { 80} \right)^{ 2} = \, - 0. 1 7\times 10^{ 2 7} {\text{s}}^{ - 2}$$
(7)
$$b_{\text{S}} = \, - \left( { 2\pi } \right)^{ 2} \left( { 6 90} \right)^{ 2} ( 3\times 10^{ 10} )^{ 2} 10^{ 3} / 2\times \left( { 80} \right)^{ 2} = \, - 1. 3\times 10^{ 2 7} {\text{s}}^{ - 2}.$$
(8)

In these calculations, the experimentally evaluated values of the spectral width, pulse width, and GDD for the anti-Stokes pulse (180 cm−1, 60 fs, −250 fs2), pump pulse (250 cm−1, 80 fs, −1,000 fs2), and Stokes pulse (690 cm−1, 80 fs, −1,000 fs2) were employed. A major discrepancy between the chirp parameters obtained experimentally from Eq. (6) and theoretically from Eq. (1) can be attributed to insufficient accuracy in the estimation of the spectral width for the Stokes pulse emitting at 1,203 nm (690 cm−1), since this value was indirectly evaluated from the SHG-FROG trace. In fact, the values predicted from Eq. (2), which is derived by assuming the same values for the pulse width and the spectral width, were in reasonably good agreement with the experimentally obtained values.

Fig. 5
figure 5

Characteristics of the first anti-Stokes pulse obtained using chirped pump pulses (GDD = −1,000 fs2 at 802 nm, GDD = −1,000 fs2 at 1,203 nm). a original FROG trace b retrieved trace (FROG error, 0.5 %); c temporal d spectral profiles calculated from the trace; the phase is indicated by an arrow; a measured spectrum using a spectrometer is indicated by a dotted arrow. The pulse width and the spectral width calculated from the FROG trace are shown in the figure

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Figure 6 shows the results obtained when a positively chirped (+1,000 fs2) pump pulse and a negatively chirped (−1,000 fs2) Stokes pulse were used. The pulse width and the spectral width of the anti-Stokes pulse were 51 fs and 6.1 nm (170 cm−1), respectively. The TBP was 0.27, which appears to be smaller than the transform-limited value of 0.44. The unexpected result would be explained by the fact that the spectral width was calculated using the FROG program only for a largest peak (see Fig. 6d); a significant discrepancy between measured and retrieved spectra would arise from an appreciable background noise in the FROG data. The anti-Stokes pulse was positively chirped (+800 fs2). The calculated value for the chirp in FWRM would be +800 − (+500) = +300 fs2, when the dispersion in the beam path is taken into account. The result deviates greatly from the value of 2(+1,000) − (−1,000) = +3,000 fs2 calculated using Eq. (2). In addition, the FROG trace is split and has a complicated structure. The chirp parameters calculated were as follows.

$$b_{\text{AS}} = \, + \left( { 2\pi } \right)^{ 2} \left( { 1 70} \right)^{ 2} ( 3\times 10^{ 10} )^{ 2} 800/ 2\times \left( { 5 1} \right)^{ 2} = 0. 1 6\times 10^{ 2 7} {\text{s}}^{ - 2}$$
(9)
$$b_{\text{P}} = \, + \left( { 2\pi } \right)^{ 2} \left( { 7 70} \right)^{ 2} ( 3\times 10^{ 10} )^{ 2} 10^{ 3} / 2\times \left( { 100} \right)^{ 2} = { 1}. 1\times 10^{ 2 7} {\text{s}}^{ - 2}$$
(10)
$$b_{\text{S}} = \, - \left( { 2\pi } \right)^{ 2} \left( { 6 90} \right)^{ 2} ( 3\times 10^{ 10} )^{ 2} 10^{ 3} / 2\times \left( { 80} \right)^{ 2} = \, - 1. 3\times 10^{ 2 7} {\text{s}}^{ - 2}.$$
(11)

The values of the spectral width, pulse width, and GDD for the anti-Stokes pulse (170 cm−1, 51 fs, 800 fs2), pump pulse (770 cm−1, 100 fs, 1,000 fs2), and Stokes pulse (690 cm−1, 80 fs, −1,000 fs2) were utilized. A serious discrepancy would arise from the resonance effect in FWRM. In fact, only the portion, at which the instantaneous frequencies of the pump and Stokes pulses are spaced by the Raman shift frequency (4,155 cm−1), can be converted into a Raman pulse, which is in contrast to non-resonant FWM demonstrated using positively and negatively chirped pump pulses in an optical glass [32, 33]. For this reason, the efficiency of the generation of the anti-Stokes pulse is decreased when the signs of the chirp for the pump and Stokes pulses are different from each other. As a result, a nearly transform-limited pulse could be obtained using pump and Stokes pulses with different GDDs at the expense of conversion efficiency. It would be necessary to use a theory that takes account the resonance effect for more quantitative discussion.

Fig. 6
figure 6

Characteristics of the first anti-Stokes pulse obtained using chirped pump and Stokes pulses (GDD = +1,000 fs2 at 802 nm, GDD = −1,000 fs2 at 1,203 nm). a original FROG trace b retrieved trace (FROG error, 0.9 %); c temporal d spectral profiles calculated from the trace; the phase is indicated by an arrow; a measured spectrum using a spectrometer is indicated by a dotted arrow. The pulse width and the spectral width calculated from the FROG trace are shown in the figure

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5 Conclusion

We report herein on the use of CPFWRM for generating vibrational anti-Stokes pulse in molecular hydrogen. When the sign of the chirp was the same as each other, an anti-Stokes pulse was generated efficiently. The chirp of the anti-Stokes pulse was controlled by changing the chirp of the pump and Stokes pulses. In fact, a positively (or negatively) chirped anti-Stokes pulse was obtained using positively (or negatively) chirped pump and Stokes pulses. Thus, it is possible to generate a negatively chirped pulse, which can readily be converted into a transform-limited pulse by inserting a fused silica plate with an optimal thickness in the beam path. In a similar manner, the chirps of the high-order Raman sidebands would be controlled, providing ultrashort transform-limited Raman pulses at different frequencies in the spectral region extending from the DUV to the NIR. Such a multi-color emission would be useful in a variety of applications, e.g., multi-color multi-photon ionization in mass spectrometry and for generating an ultrashort optical pulse approaching 1 fs by a Fourier synthesis of the emission lines.