1. Introduction

The molecular modulation technique has potential for synthesizing optical pulses as short as a fraction of a fs [1]. This Raman technique, as originally proposed for molecular gases, shows promise for highly efficient production of such ultrashort pulses in the near-visible spectral region, where such pulses inevitably express single-cycle nature and may allow non-sinusoidal field synthesis [2].

We have extended the molecular modulation method to Raman-active crystals, which has the potential of generating fewer ultrashort pulses per train [3, 4]. Controlling the phases of the Raman sidebands is an essential step in synthesizing ultrashort pulses [5]. In [6] we showed precise phase control of a broadband coherent Raman spectral comb. We also did a cross-correlation-like measurement (instead of time, phase was varied), confirming the short pulse nature when the spectral phase is adjusted to be flat.

Diagnostics and optimization of relative phases of sidebands which have spectrally well-separated peaks—“holey” spectrum—is not trivial [7]. Retrieval of a directly measured second harmonic generation (SHG) frequency resolved optical gating (FROG) trace may not converge and therefore may not give reasonable pulse duration and phase information. On the other hand, a nonlinear process such as sum frequency generation (SFG) can be employed in order to compare the phases of waves with different frequencies, as has been pointed out by Hänsch more than 20 years ago [8]. Hsieh et. al. used this method and showed full interpulse phase locking of a Raman generated optical frequency comb. However, their comb consisted of narrower spectral lines as gas was used as a medium and the pump laser had ns pulse duration [9]. In this article, we show a step by step procedure to optimize the phase of a “holey” spectrum. The spectrum is comb-like but each comb line (Raman sideband) has a broad bandwidth. We also show the phase coherence between the sidebands by investigating the interference between them in groups of three. Moreover, we show how coherent beating among the sidebands varies as we scan different sideband phases and how a slight phase slip of one sideband affects the coherent beating at the central fringe.

2. Experimental setup and Raman sideband generation

The schematics of the experimental setup is the same as the one used in [6]. It is shown in Fig. 1(a). Two 50 fs, 1 kHz repetition rate pump pulses have center wavelengths at 1200 nm (Stokes) and 1035 nm (pump) respectively [Fig. 1(b)]. The pump and Stokes beams are focused and crossed at an angle of 3.7° into a 1-mm thick synthetic single-crystal diamond. The power of the pump and the Stokes beam (loosely focused with a 100 cm focal length lens) before the diamond crystal is 3.2 and 23.0 mW respectively. After the crystal, the power of the Stokes beam is 15.6 mW, mainly due to linear losses. The pump beam power after the crystal is 2.1 mW. It reduces to 1.7 mW when it overlaps both in time and space with the Stokes beam and Raman sidebands are generated. In this experiment, the focused intensity of the input beams is mainly limited by parasitic nonlinear processes, such as self phase modulation and white light generation within the diamond. More power can be applied if we focus loosely or use a pair of linearly chirped pulses [10]. The total power of the generated sidebands is about 0.2 mW; the observed conversion efficiency is therefore quite reasonable.

 

Fig. 1 (a) Schematics of the experimental setup. (b) Spectrum of the pump beams and the first 7 sidebands; (c) Peak frequency of one pump pulse and 6 sidebands as a function of the external output angles. Insert: combined sidebands beam profile measured at 1m after the prism.

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For proof of principle, we focus on the first 5 orders of Raman sidebands. These anti-Stokes sidebands are referred to by their orders (AS1, AS2, etc.). The center wavelengths (from AS1 to AS5) are 930 nm, 840 nm, 780 nm, 720 nm, and 680 nm respectively. The spectra of the sidebands are shown in Fig. 1(b), which is taken close to the point where the sidebands are re-focused by the mirror.

We combine the sidebands to a single beam by using a spherical mirror and a fused silica prism [11]. The peak frequencies of the sidebands vs angle dependence has a slope of 524 cm⁻¹/degree [Fig. 1(c)], while the dispersed beam frequency as a function of the angle for the prism has a slope of 11271 cm⁻¹/degree. Therefore, to combine the beams, we need the spherical mirror to provide a 21.5× magnification. This consideration determines the placement of the mirror. Subsequently, fine adjustment of the spherical mirror position and the prism angle are used to improve the combined beam profile. We achieve a uniform and round beam profile for the combined beam, which is measured 1m after the combining prism [Fig. 1(c) insert].

The combined beam comes out from the prism close to the Brewster angle at 56°. It is then sent into a programmable pulse shaper for fine-tuning. The pulse shaper (Phazzler, FastLite) is based on an acousto-optic programmable dispersive filter, which allows compact, in-line and high efficiency pulse shaping and pulse characterization [12]. Specifically, as shown in Fig. 1(a), there is a 20 μm-thick BBO crystal after the Dazzler, the pulse shaping unit, which enables collinear SHG FROG measurement for pulses down to 20 fs. However, this pulse shaper has a fundamental limitation for broadband pulses [13]. The temporal shaping window is limited by the crystal length and its birefringence. In our case, the temporal shaping window is around 7 ps. Each sideband is delayed by about 2 ps with respect to the adjacent order after propagating through the 25 mm thick TeO₂ crystal. To shape all 5 sidebands, we pre-compensate the delays among the sidebands by adding different amounts of glass in each sideband beam path [Fig. 1(a)]. For example, 1 mm glass in AS1 will compensate the delay difference between AS1 and AS2. As a result, they will arrive within 1 ps at the BBO crystal. This smaller difference can be compensated by the pulse shaper. We essentially help the programmable pulse shaper manually, and make fine-tuning the overall pulse shape feasible.

3. Step-by-step phase optimization of the “holey” spectrum

We start by compensating linear chirp (second-order) phase for each sideband, similarly to our recent work [6]. Here as an example, in Fig. 2(a) we show the spectrum of SHG of AS2 as a function of the applied second-order phase. After we integrate the intensity over the spectral width, we find that it has a maximum at −1250 fs². We then apply a 1250 fs² chirp to AS2 to compensate this second-order phase distortion. The higher-order phase distortions are relatively small so no compensation is required.

 

Fig. 2 (a) SHG of AS2 spectrum as a function of applied second-order phase on AS2 sideband. (b) SFG of AS1 and AS5 as a function of applied second-order phase on AS5. (c) SFG of AS1 and AS3. Zero delay between the sidebands is defined to coincide with the strongest SFG position.

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The compensation of AS5 second-order phase distortion is not so straightforward, as the pulse energy of AS5 is about 3 orders of magnitude smaller than that of AS1 [14]. It is too weak to measure the SHG of AS5 individually, with the sensitivity of our spectrometer. To circumvent this problem, we use the SFG between AS1 and AS5: we compensate the second-order dispersion by maximizing the SFG signal, as shown in Fig. 2(b). This SFG between sidebands is also used to find the temporal overlap between the sidebands. One example is given in Fig. 2(c), where we show the SFG between AS1 and AS3. Zero delay is defined at the point where maximum SFG signal is generated.

As explained by Walker et al., there is one well-defined relative phase for three sidebands which are equally separated in frequency [15]. For 5 sidebands we have 3 well-defined phases: 2φ₂ – φ₁ – φ₃, 2φ₃ – φ₂ – φ₄ and 2φ₄ – φ₃ – φ₅, where φ_n are the spectral phases of sideband ASn. These relative phases can be measured from the coherent interference between the SHG/SFG of the sidebands at the BBO crystal, as explained next. First we record the spectrum of SHG/SFG of the AS1, AS2 and AS3 sidebands as a function of AS3 phase, as shown in Fig. 3(a). There is a sinusoidal change of the peak intensity at the SHG of AS2, as a result of the beating between SHG of AS2 and SFG of AS1 and AS3. It is proportional to cos(2φ₂ – φ₁ – φ₃). The intensity has a maximum (constructive interference) when we apply 0.79π phase to AS3. We then assign −0.79π as the zero phase for AS3. After that we record the SHG/SFG of the AS2, AS3 and AS4 sidebands as a function of AS4 phase, with AS3 phase set to 0.79π. The result is shown in Fig. 3(b). Similarly, when φ₄=0.79π, we see the maximum intensity at the position of SHG of AS3. At last in Fig. 3(c) we show the SHG/SFG of the AS3, AS4 and AS5 sidebands as a function of AS5 phase, with φ₃=0.79π, and φ₄=0.79π. (Please note it is just a coincidence that both phases are 0.79π). As we mentioned earlier, SHG of AS5 is not strong enough to be seen with the sensitivity of our spectrometer. Therefore the coherent high visibility fringes at the position of AS4 SHG is a demonstration of the mutual coherence among the sidebands. With AS5 phase set to produce maximum intensity at SHG of AS4, we believe we have achieved flat phase across the 5 sidebands, which is sufficient to produce Fourier transform limited pulses [9].

 

Fig. 3 SHG/SFG spectrum generated by (a) AS1, AS2 and AS3 sidebands in the BBO crystal, as we vary the AS3 phase; (c) AS2, AS3 and AS4 sidebands in the BBO crystal, as we vary the AS4 phase, and (d) AS3, AS4 and AS5 sidebands in the BBO crystal, as we vary the AS5 phase. Parts (b), (d) and (f) show the coherent beating across the dotted lines in (a), (c) and (e).

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The flat phase can also be verified by scanning AS3 phase, as shown in Fig. 4(a). Compared to the SHG/SFG spectra of the sidebands as we vary AS2 phase, which is shown in [6], we see all the sidebands beat in unison and interfere constructively at the multiple integers of 2π. As mentioned above, the pulse energy of AS5 is much lower than that of AS1. Question may arise whether AS5 contributes to the center fringe. Due to the following three reasons, we believe that the center fringe is a result of interference from 3 channels. First of all, although AS5 is weak, AS1 is several magnitudes stronger. As a result, the SFG of AS1 and AS5 has comparable amplitude to that of SHG of AS3. Secondly, the presence of the beating on AS4 can only be due to the existence of AS5, which can be clearly seen in Fig. 3(c), as well as in Fig. 4(a) subband G. At last, in Fig. 4(b) we show the spectral components which contribute to the center fringe, i.e., the SHG of AS3, SFG of AS1 and AS5, and SFG of AS2 and AS4. They are not perfectly overlapped in spectrum. But from the above demonstration of the mutual coherence among the sidebands in the groups of three, we believe that all three terms contribute to the center fringe. If AS5 is not in phase with the other 4 sidebands, the beating at the SHG of AS3 will have a shift, as will be shown later in Fig. 5(d).

 

Fig. 4 (a) The SHG/SFG spectrum of the 5 sidebands after the BBO crystal as we vary the AS phase from 0 to 10π (with a step of 0.05π), after we adjust the relative phase among all sidebands to be equal. Subband A, C, G: SHG of AS1, AS2 and AS4, respectively; B: SFG of AS1&AS2; D: SFG of AS1&AS4, and SFG of AS2&AS3; E: SHG of AS3, SFG of AS1&AS5, and AS2&AS4; F: SFG of AS3&AS4. (b) The spectra of the frequency components from 3 different channels which are contributed to the center fringe.

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Fig. 5 The peak intensity of the SHG/SFG subbands as we vary the AS2 phase (a), AS3 phase (b) and AS1 phase (c) from 0 to 10π, after we adjust the relative phases among all sidebands to be equal. (d) The peak intensity of the subbands as we vary the AS2 phase from 0 to 10π when the sideband phases are not perfectly flat. Step size is 0.01 π in (d) and 0.05 π in (a), (b) and (c).

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In Table 1 we show the parameters which are used for pulse-shaping.

Table 1. Parameters used to adjust the phases of the 5 Raman sidebands to be flat

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4. Coherent beating when varying different sideband phases

We explore how the coherent beating of the subbands varies as we scan AS1, AS2 and AS3 phases. When all 5 sidebands are in phase, the coherent beating for each subband as a function of the AS3 phase [Fig. 4(a)] is similar to a situation when AS2 phase is scanned, as shown in Fig. 5(a). The main difference is that, compared to the other subbands, the fringes have a doubled frequency for SHG of AS2 when AS2 phase is scanned, while the fringes have a doubled frequency for SHG of AS3 when AS3 phase is scanned. The SHG/SFG subband intensities vs AS1 phase are shown in Fig. 5(c). Due to shorter integration time of 500 ms (in all other scans, we do two scans with integration time of both 500 ms and 5000 ms) we can not distinguish the beating for SHG of AS1. However, we observe that the sidebands beat in unison and produce a perfect coincidence of the central 4 subband patterns (C, D, E, and F).

We note that a slight sideband phase shift (with respect to flat phase) can lead to a large shift of the center fringe pattern (subband E). Specifically, when the phase is not flat, patterns at SHG of AS3 are shifted by a phase which is not only determined by the phase difference among sidebands, but also the relative amplitudes of the sidebands. Therefore, it is crucial to scan up to 10π radians of phase change with a step of at most 0.05π when one obtains the relative phase between the sidebands, as shown in Fig. 2(c) and (d). Then, with the sinusoidal fitting, an accurate relative phase between the sidebands can be determined. One example of the shift of the peaks due to this non-flat spectral phase is shown in Fig. 5(d). Large phase error occurred when we determined the relative phase between the sidebands using only 2 cycles (phase scan up to π rad). The phase shift on subband E is close to π in this example, where we estimate we have up to 0.2π phase error on each sideband.

5. Conclusion

We demonstrate the mutual coherence of the spectral sidebands generated through molecular modulation in diamond. We show in detail how we control the phases of a broad “holey” spectrum by pre-chirping the pulse and thus extending the effective temporal pulse shaping window. We show how the sidebands’ SHG and SFG vary as we vary different sideband phases. We find that it is crucial to adjust the relative phases of the sideband to a high precision to achieve flat phase across the 5 sidebands and therefore observe the constructive interference at multiples of 2π when we vary different sideband phases. This point is important when one wants to control the phases of the sidebands to synthesize arbitrary optical waveforms [16].