1. Introduction

Coherent multi-dimensional spectroscopy (CMDS) is a powerful tool for probing the fastest dynamics in materials science. CMDS has already clarified fundamental questions about electron-phonon coupling in covalent II-VI semiconducting nanocrystals [1–4] and ionic perovskite nanostructures [5,6]. Other notable measures are the excited structure through double quantum spectroscopy [7–9] and electronic coherence. Electronic coherences have been detected between the 1S_e1S_3/2 and the 1S_e2S_3/2 excitonic states in single quantum dots [10] and between excitons on distinct quantum dots [9].

A common approach for broadband pulses in CMDS is to use a non-collinear optical parametric amplifier (NOPA) [11–13]. These devices are large and involve several frequency conversions and mixing. Their proper alignment for maximum bandwidth, low noise, and high power operation can be complicated. To switch a NOPA from visible to IR operation requires different crystal cuts and re-optimization of crossing angles, or construction of a completely separate NOPA.

Hollow-core fibers (HCFs) for continuum generation are a general approach for pulse broadening, and have been used to generate multiple-octave spanning spectra [14,15]. These approaches use very high peak field strengths and high pressures to generate the super-broad spectra, and are often highly unstable, due to the several orders of nonlinearity in pulse propagation. In regards to precision spectroscopy, stability in phase, spectrum and power are all very important and these characteristics should be prioritized over octave spanning pulses. For spectroscopy, unstructured fibers driven by weaker pulses have been used [16–20]. HCFs have emerged as a simple source for CMDS with pulse characteristics comparable to NOPAs. Some implementations can produce pulses shorter than 10 fs, spanning the visible spectrum [19,20]. One issue with HCFs is the inability to tailor the central wavelength of the laser spectrum to a particular sample; the experimentalist is limited to samples whose transitions of interest lie in the fixed spectrum provided by the light source. In response to the problems presented by HCFs driven by the fundamental laser wavelength, we present an OPA-driven HCF as a tunable, broadband, low noise source for CMDS.

The block diagram of the tunable continuum source is shown in Fig. 1(a). A Ti:Sapphire based amplifier system producing 8 mJ, 100 fs pulses (Coherent Legend Elite Duo) is used. 2 mJ of the amplifier output drives a commercial OPA (TOPAS). The output of the OPA is in turn used to drive a 2.57 m long, 400 µm inner diameter hollow-core fiber filled with Argon [17,21]. The OPA pulses are typically somewhat shorter than the 800 nm driving pulses (∼ 90 fs vs. 100 fs), and have central wavelength spanning 300-3000 nm and energies from 20-200 μJ. For our interest we used the OPA in the range 500-750 nm, as this is roughly the bandwidth of the DAZZLER acoustic pulse shapers and GRISMs used in our CMDS experiment. Using the direct signal and idler beams, from 1200-2600 nm, should produce more energetic pulses and provide broader spectra than seen here. To characterize the white light generation process, transient-grating frequency resolved optical gating (TG-FROG) signals were measured at 3 locations: Before the entrance window to the fiber, after the exit window of the fiber, and after pulse compression, as shown in Fig. 1(b)-(d). In order to compress the fiber output, dispersion in the DAZZLER is pre-compensated by a GRISM pair and the DAZZLER applies a fourth-order polynomial phase mask to compress the pulse. The phase mask is varied to minimize the FWHM of the spectrally-integrated TG-FROG trace. The white light generation process was found to work across the visible spectrum; the results of driving the fiber with central wavelengths spanning 500-750 nm is shown in Fig. 1(e). Approximately 200 meV of bandwidth is generated over this entire range. The inset of Fig. 1(e) shows an image of the collimated beam after transmission through the fiber. The spatial structure is well-described by a gaussian curve along both axes.

 

Fig. 1. Overview of HCF driven by an OPA for visible CMDS. (a) Optical layout of the setup. (b)-(d) TG-FROG traces of the pulse at the (b) fiber entrance, (c) fiber exit, and (d) after compression. The spectrally-integrated trace in (d) has a FWHM of 13 fs (12 fs for transform limited pulses). (e) Output spectra (solid lines) as a function of input central wavelength (Filled curves). inset: Image of collimated beam profile, with sums along respective axes in black.

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An important metric for CMDS is the stability of the source in terms of spectrum, phase, and power. As almost all CMDS experiments rely on the subtraction of subsequent shots, any shot to shot variation in the spectral width or amplitude of the driving fields will appear as signal. By driving the fiber with much less energetic pulses, plasma generation is avoided and stability is improved when compared to using the 800 nm fundamental for white light generation. The excellent stability of the source is shown in Fig. 2. Here, the DAZZLER pulse shapers are used to diffract two pulses with a delay 100 fs. Over the course of ∼ 6 hours, there is no drift in interpulse phase or delay. Furthermore, spectral bandwidth and power are stable over a similar time period, as shown in Fig. 2(a). Figure 2(b) shows the width of the diffracted spectrum over time, defined as the square root of the second central moment of the intensity spectrum. The overall standard error in the bandwidth is 2.8%. To show the precision of delay and phase control, a TG-FROG trace of the pulse pair is measured, as shown in Fig. 2(c). Here the pulse copies are evident at ${\pm} $100 fs, as is the spectral interference pattern at $t = 0$. By changing the phase between the two pulses, the phase of this interference is shifted. In Fig. 2(d) the interference patterns for $\Delta {\phi _{12}} = 0$ and $\pi $ are shown. As these TG-FROG patterns are recorded over 1000s of laser shots, the interference patterns would be washed out if there were drifts in either delay or interpulse-phase.

 

Fig. 2. Overview of fiber stability. (a) Stable spectral interferograms show stability of phase, delay, and spectrum over the course of ∼ 6 hours. (b) Spectral width over time, defined as square root of second central moment of the data shown in panel (a). (c) TG-FROG trace of two pulses delayed by 100 fs. (d) time-integrated trace of the central peak in (c) showing spectral interferograms for $\Delta {\phi _{12}} = 0$ (red) and $\pi $ (blue).

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2. Modelling

The spectral broadening and phase evolution is in large part the balance of two processes: Self-phase modulation (SPM) and dispersion. SPM will create new frequencies while leaving the temporal pulse envelope unchanged. Dispersion serves to add chirp and lengthen the pulse, decreasing peak intensities and SPM. At high peak powers, close to resonance, or at high pressures, plasma generation becomes an issue as well [21]. Using a split-step Fourier method [17,22] and taking into account Kerr coefficients up to 11th order, the performance of this system has been modelled by solving the nonlinear optical Schrödinger's equation.

Here $\tilde{\varepsilon }$ is the Fourier transform of the slowly varying, time-domain electric field envelope $\varepsilon (r,t,z)$. The right hand side of Eq. (2.1) describes spectral dispersion;

With $k(\omega )$ the frequency-dependent wavevector, and ${k_0}$ and ${k_1}$ the first terms in a Taylor expansion of $k(\omega )$.

A self-steepening term T;

Simulation results are summarized in Fig. 3. The simulated output spectra for the input pulses and fiber conditions used in Fig. 1(e) are shown in Fig. 3(a). The numerical simulations match experimental results very well in terms of pressure dependence, performance across the visible spectrum, and spectral structure. Increased broadening is predicted at shorter wavelengths, however this was not observed, possibly due to the anti-reflection windows at the fiber exit, which are coated for a center wavelength of 800 nm. Fiber pressure is another important consideration for fiber operation; Here numerical and experimental results are in excellent agreement, as shown in Fig. 3(b). Here black is an experimental output spectrum, red is simulated, and grey is the common input spectrum. Spectral width increases almost linearly with pressure in the range explored. To improve performance of our HCF, increasing pressure is a good avenue to explore. Due to the absence of plasma generation in our OPA-driven HCF, greater fiber pressures compared to 800 nm driving fields is possible. Figure 3(c) shows the simulated evolution of the spectral pulse envelope as it propagates down the length of the fiber. The appearance of spectral fringes as the pulse propagates is a classic sign of SPM.

 

Fig. 3. Simulated fiber performance. (a) Broadening across the visible spectrum is replicated. Input and output spectra are the same as in Fig. 1(a). (b) Pressure dependence is accurately reproduced. Simulated spectra (red lines) reproduce experimental data (black lines). (c) Evolution of the spectral envelope as the pulse propagates through the fiber.

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3. Results

To demonstrate the quality of experiment enabled by this light source, a CMDS dataset was acquired on CdSe quantum dots purchased from NNLabs. For experimental details on the CMDS spectrometer, see the work presented in [21] and [24]. Results are shown in Fig. 4. Figure 4(a) shows the sample linear absorption in black and the laser spectrum in red. The linear absorption spectrum was modelled as a sum of gaussian peaks corresponding to excitonic resonances, in addition to a polynomial background [25]. The dotted lines in Fig. 4(a) and (b) correspond to the two lowest excitonic resonances. The laser spectrum is broad enough to completely cover both excitonic peaks, as well as any sub-resonant features associated with multiexcitonic or emissive states [26]. Figure 4(b) shows a CMDS spectrum acquired at T₂= 250 fs. The well resolved diagonal peaks are evidence of the good spectral power at higher energies. To demonstrate stability of the system, the 2DE spectra were acquired at many population times over the course of several hours. Traces along the population time at three coordinates of $({E_1},{E_3})$ are shown in Fig. 4(c). At all three locations a beating in the CMDS signal is apparent at the LO phonon frequency of 208 cm^-1. Also evident is a phase change between the blue and green curves, which would be expected for coherent electron-phonon coupling [3]. To isolate the beating signal, each transient was modelled with a decaying exponential and the residuals were Fourier transformed. The results of this analysis are shown in Fig. 4(d). For reference, the LO phonon frequency of CdSe is shown as a vertical dashed line. The clarity of vibrational coherence, and the obvious phase shifts with movement along either the ${E_1}$ or ${E_3}$ axis, demonstrates the suitability of this light source for CMDS.

 

Fig. 4. CMDS experimental results on CdSe quantum dots. (a) Linear absorption of the sample (black solid line) and laser spectrum (red filled area) (b) A 2D spectrum obtained at T₂ = 250 fs. Dotted vertical and horizontal lines in (a) and (b) indicate excitonic resonances of the sample. (c) Traces of the CMDS spectrum along T₂ (solid) and exponential models (dashed) used to fit them. (d) Fourier transform of the fit residuals. Dashed line shows the LO mode of CdSe at 208 cm^-1. Colors in (c) and (d) correspond to rectangular patches in (b).

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4. Conclusion

In conclusion, we have demonstrated the feasibility and strengths of using an OPA to drive tunable supercontinuum generation in Ar-filled hollow-core fibers. This concept allows CMDS experiments to be performed wherever an OPA can be tuned, roughly 0.5-2.5 eV. Notably, the results presented in Fig. 4 could not have been obtained by driving our HCF with 800 nm pulses. This OPA-driven HCF would work very well in the near to mid IR, where the signal and idler wavelengths provide more pulse energy and the central wavelength is further from plasma generation. Developing light sources is an important endeavor for the overall development of CMDS, as the excitation source is fundamentally linked to the samples that can be probed and the artifacts present in the data. The work presented here demonstrates an improvement in the dynamic range available to CMDS experiments.