1. Introduction

Construction of ultrashort pulses requires compensation for spectral phase dispersion, in which the second-order term (group velocity dispersion; GVD) is usually dominant. Therefore, eliminating GVD is the most important aspect of ultrashort pulse generation. Because almost all materials have positive GVD, the GVD only increases as pulses pass through materials. Therefore, to construct ultrashort pulses, various methods providing “negative GVD”, such as prism pairs [1], grating pairs [2], chirped mirrors [3], and a spatial light modulator (SLM) [4] have so far been developed to remove the aroused positive GVD.

In contrast to these methods, when a spectrum is substantially discrete, there is another approach for generating ultrashort pulses that involves adding further positive GVD to compensate for the presence of positive GVD. In fact, ultrashort pulse shaping on a nanosecond time scale has been demonstrated by utilizing a high dispersion around a sharp resonance in gaseous atom [5]. Furthermore, mainly in relation to multiplying the pulse repetition rate, various related phenomena have been extensively studied in the context of the temporal Talbot effect [6]. Recently, by using this temporal Talbot effect as a basis, picosecond ultrashort pulses have been generated by passing them through fiber Bragg gratings or fiber/or waveguide splitters/combiners [7].

Here, we demonstrate the generation of a train of femtosecond ultrashort pulses by applying this idea to a coherent spectrum with a frequency spacing exceeding 10 THz (Raman sidebands). The train of femtosecond ultrashort pulses is shaped simply by passing the Raman sidebands through a dispersive material with a thickness of a few tens of mm. The method has attractive practical advantages, including simplicity, robustness, and applicability to both spectra with various GVDs and high power pulses.

2. Theory

We consider a discrete spectrum with a center frequency of ω₀ and a frequency spacing of Δω. The spectral phase of such spectrum is generally expressed by

However, if we take the discreteness of the spectrum into account, we find FTL conditions other than ϕ₂ = 0. Equation (2) describes such conditions.

We apply this scheme to highly-discrete broad Raman sidebands with a bandwidth of 20 THz and a frequency spacing of 10.6 THz. Figure 1(a) shows a typical spectrum generated in reality from the rotational coherence between J = 2 and 0 in parahydrogen [8]. The curves in Fig. 1(a) show spectral phases corresponding to n = –1 (blue), 0 (black), 1 (red), and 2 (green) in Eq. (2). Horizontal dotted lines indicate the values of the spectral phases where spectral lines exist; all the values are integer multiples of ‘π’. The temporal profiles calculated from these four different spectral phases surely provide the same FTL pulse train as that shown in Fig. 1(b). (The time origin for each temporal profile is adjusted accordingly.) Here, the difference in ϕ₂ between the neighboring spectral-phase curves amounts to 1,400 fs². This implies that the FTL condition is recovered every 1,400 fs² when we add GVDs to the Raman sidebands.

 

Fig. 1 Basic concept of the phase-dispersion compensation method. (a) Red lines show a discrete spectrum with a frequency spacing of 10.6 THz corresponding to the rotational transition of J = 0 to 2 in parahydrogen. The envelope has Gaussian shape with a width of 20 THz at FWHM. The curves show spectral phases with the factors n = –1 (blue), 0 (black), 1 (red), and 2 (green) in Eq. (2). All the curves cross the horizontal lines (integer multiples of 2π) at the dots where spectral lines exist. (b) The temporal intensity waveform reconstructed from the four different spectral phases in (a).

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3. Numerical experiment

In order to add positive GVD, one may simply insert a dispersing material. The amount of positive GVD can be adjusted by changing the material type and thickness. We calculated the dispersion-adding phase compensation for the discrete spectrum in Fig. 1(a) by employing three kinds of material: fused quartz, borosilicate crown glass (BK7), and sapphire crystal. The refractive indices of such materials were known from Sellmeier equations. Note that the Sellmeier equations naturally include higher-order dispersions, ϕ^h, in addition to the second-order (GVD) term. Figures 2(a) , 2(b), and 2(c) show the peak intensity variations of the constructed waveforms from the spectrum shown in Fig. 1(a) as functions of the thicknesses of the inserted dispersing materials. The initial phase of the spectrum was set to the FTL condition of ϕ₂ = 0. The FTL pulses were then distorted as the inserted material thickness increased, but they periodically recovered their shapes at specific thicknesses corresponding to the condition of ϕ₂ = 2π/Δω² (n = 1). For a discrete spectrum with a frequency spacing of 10.6 THz, we found these specific thicknesses for fused quartz, BK7, and sapphire crystal to be 38.5, 31.2, and 24.0 mm, respectively, all of which are experimentally achievable.

 

Fig. 2 Peak intensity variations of the intensity waveforms constructed from the power spectrum and spectral phases in Fig. 1(a), shown as functions of the added thicknesses of dispersing materials made of (a) fused quartz, (b) borosilicate crown glass (BK7), and (c) sapphire crystal. The discrete spectrum has a Gaussian shape with a width of 20 THz at FWHM, and its initial spectral phase is set to the FTL condition.

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Here, we should note that, with this method, there is a limitation due to the higher-order dispersion of the material, ϕ^h, because the higher-order dispersions merely accumulate and never subtract the deviations from the conditions of Eq. (2). As seen in Figs. 2(a), 2(b), and 2(c), the peaks of the reconstructed intensity waveforms around the FTL conditions declined as the insertion thicknesses increased (i.e. as n increased). This is due to the higher-order dispersions. In other words, these higher-order dispersions limit the applicable bandwidth in terms of the number of spectral lines to be phase-dispersion compensated. In this case, where we employed a discrete spectrum with a frequency spacing of 10.6 THz, the shortest pulse duration that could be produced by the dispersion-adding compensation was 15 fs, corresponding to a bandwidth of 30 THz. If we employ different dispersive materials and combine them to compensate for phase dispersion, in principle we can extend this technique to include higher-order dispersion. This will constitute future study.

4. Experimental

To achieve spectral phase-dispersion compensation by the concept explained above, we designed a device (dispersion controller) that continuously varied the thickness (i.e. the GVD) [Fig. 3(a) ]. The device was composed of two glass wedges that were identically shaped and moved along their hypotenuses relative to each other. We used BK7 wedges; thereby the variable thickness range was designed so that it exceeded 31.2 mm [see Fig. 2(b)]. The vertex angles of the wedges were 30°, and the hypotenuses were 80 (for the large wedge) and 40 mm (for the small wedge). By sliding the large wedge along its hypotenuse, the total thickness of the wedge pair continuously varied from 10 to 50 mm. The large wedge was mounted on a rack-and-pinion gear stage with a resolution of 0.1 mm. Therefore, the thickness changed with a resolution of 0.05 mm. The air gap between the wedges was set to be less than 1 mm, in order to make the displacement among the spectral components negligible.

 

Fig. 3 (a) Dispersion controller composed of a pair of BK7 wedges. The thickness is continuously varied from 10 to 50 mm. (b) Whole experimental setup. Intense, two-color laser pulses are produced from the dual-frequency injection-locked Ti:sapphire laser. The dispersion controller is placed after the collimating lens. The generated sidebands are passed through the dispersion controller, split into two and sent to the measurement systems on the spectral phase and the second harmonic intensity, respectively.

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Figure 3(b) shows the whole experimental setup. We used a dual-frequency injection-locked Ti:sapphire laser [9,10] to adiabatically drive [11–14] a high rotational-Raman-coherence in parahydrogen (77 K, 3 × 10²⁰ cm^–3 [8]). This Raman process simultaneously generated a discrete spectrum with a frequency spacing of 10. 6 THz, called Raman sidebands (details: see Refs. 8,15). The generated spectrum was measured by an optical multichannel analyzer. A typical spectrum is shown at top right in Fig. 3(b). We passed the generated Raman sidebands through the dispersion controller to add positive GVDs and then introduced them to the measurement systems.

The compensation of the spectral-phase-dispersion in the Raman sidebands was confirmed directly by spectral phase measurement. The measurement was based on a recently developed spectral interferometry method (spectral interferometry for direct electric-field reconstruction for a discrete spectrum) [16]. We also simultaneously measured the integrated intensities of the second harmonic (SH) from a β-BBO crystal, which was generated by the Raman sidebands with a constant energy. Here, losses of surface reflections were independent of the positions of the wedges; thereby, we could estimate the effect of the dispersion compensation by monitoring this SH intensity.

5. Results and discussion

In Fig. 4(a) , the solid, dashed and shaded lines respectively show the initial spectral phase of the discrete spectrum just before the dispersion controller, the FTL conditions calculated for the cases with n = 0, 1, and 2 in Eq. (2), and the observed spectral phases for the device thicknesses every 5 mm from 10 to 40 mm. The initial spectral phase includes both the intrinsic spectral phase aroused in the process of generation of the Raman sidebands and the additional dispersions by the windows of the cell and the cryostat and by the collimating lens. By varying the thickness of the dispersion controller, the spectral phase was continuously changed and nearly matched the FTL conditions twice [Fig. 4(a)]. The pulse shapes in Fig. 4(b) are temporal intensity waveforms corresponding to the spectral phases in Fig. 4(a). The shortest pulse was observed at a thickness of 39 mm, as shown by the black solid curve in Fig. 4(b). The pulse duration was 22 fs at full width at half maximum (FWHM), which matched an FTL duration of 22.05 fs.

 

Fig. 4 (a) Spectral phases and (b) temporal intensity waveforms reconstructed with the spectral phases in (a). Solid, dashed and shaded curves respectively indicate initial, FTL, and observed spectral phases and the corresponding temporal intensity waveforms. The initial spectral phase (solid) was measured without the dispersion controller. The three indicated FTL conditions (dashed) in (a) correspond to n of 0, 1, and 2 in Eq. (2). The observed spectral phases in (a) and the intensity waveforms (shaded) in (b) are shown for the inserted thicknesses of the dispersion controller every 5 mm from 10 to 40 mm.

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In addition to these direct spectral-phase measurements, we also measured the integrated intensity of the second harmonic produced by the pulse train. Figure 5(a) shows the SH intensity variation as a function of the thickness of the dispersion controller. We found that peaks appear around thicknesses of 10 and 40 mm in this dispersion-control range. This is consistent with the results of Figs. 4(a) and 4(b). In Fig. 5(b) we also estimated the SH intensity variation expected from the intensity waveforms in Fig. 4(b). Moreover, from the intensity spectrum and the initial spectral phase before the dispersion controller, we calculated the SH intensity as a function of the thickness of the dispersion controller [Fig. 5(c)]. These three results [Figs. 5(a), 5(b) and 5(c)] are in good agreement with each other, especially in terms of the peak positions, the minimum to maximum ratio, and the existence of a sub-peak appearing at a thickness of 25 mm. Therefore, we confirmed the efficacy of our method of ultrashort pulse generation.

 

Fig. 5 Integrated intensities of the second harmonic generated by the Raman sidebands as a function of the inserted thickness of the dispersion controller. (a) Observed experimentally; (b) calculated with the reconstructed temporal intensity waveforms in Fig. 4(b); and (c) estimated numerically with the initial spectral phase and the refractive indices of BK7 known from the Sellmeier equation.

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Although our dispersion-adding phase-compensation method is applicable only to a highly-discrete spectrum, it has three advantages. First, the setup is robust and simple, because it does not require angular dispersion and spatial decomposition for each spectral component. Second, GVD can be controlled continuously and precisely without any change in the pulse propagation direction. Third, it is applicable to high power pulses up to the damage threshold of the dispersing material (generally higher than those of dielectric- or metal-coated optics). The method is attractive for generating ultrashort pulses for various practical uses.

6. Conclusion

In conclusion, we demonstrated femtosecond ultrashort pulse generation by adding a positive material GVD to compensate for the presence of a positive GVD. The broad Raman sidebands with a frequency spacing of 10.6 THz, which were produced by an adiabatic Raman process in gaseous parahydrogen, were compressed to a train of FTL pulses by passing them through a device made of thickness-variable dispersive-material. Compensation of the GVD and generation of the FTL ultrashort pulses were verified by both direct measurement of the spectral phase and measurement of the second harmonic intensity produced by the Raman sidebands.

When we employ a gaseous medium in ultrashort pulse generation, creation of additional positive GVDs by the windows of the sample cell is inevitable. If one takes into account this dispersion and designs the thickness of the window on the basis of the concept in this paper, ultrashort pulses are produced immediately after the sample cell without any other phase-dispersion compensation, even though the spectral phase is positively dispersed in the cell. This will make ultrashort pulse generation simpler and more practical.