© The Author(s), published by EDP Sciences, 2025

1 Introduction

Spectrally tunable laser sources providing Gigahertz-level (GHz) linewidths at high energies and average power are valuable for applications such as Impulsive Stimulated Raman Scattering [1], and orientational control of molecules [2]. The pulse durations corresponding to GHz linewidths lie in the range of picoseconds to nanoseconds. These parameters can be achieved using injection-seeded Q-Switched Solid-State lasers [3] achieving millijoule energies with limited spectral tunability, injection-seeded Optical Parametric Oscillators [4], or nonlinear spectral compression [6, 7]. An alternative approach involves the generation of pulse bursts with pulse spacings comparable to the pulse duration. By this, interpulse spectral interference leads to an energy redistribution from a continuous distribution over the whole spectral bandwidth into a few narrowband spectral interference peaks [8]. In this context, the optical bandwidth of the burst pulses acts as an optical-frequency reservoir allowing resonance tuning. However, a drawback of this method is the existence of multiple equidistant spectral peaks rather than a single, isolated tunable line. In this work, we demonstrate the generation of a broadband spectrum consisting of only a few narrow peaks by picosecond burst formation, combined with spectral filtering of an isolated peak within this spectrum. Due to the Linear Time-Invariant (LTI)-nature of the burst-shaping process, we perform spectral filtering prior to and after burst formation. This commutability is of high relevance for high-peak-intensity amplifier systems, as it directly affects system efficiency and damage-threshold management.

2 Theory and simulations

In the following, we outline the LTI nature of burst-forming techniques. We demonstrate that both approaches – the transformation of an isolated pulse into a burst and the Vernier-type accumulation of multiple subsequent pulses into a single burst – can be identified as LTI processes. As such, they can be formulated as$${\tilde{E}}_{\mathrm{out}}\left(\omega \right)={\tilde{H}}_{\mathrm{burst}}\left(\omega \right)\bullet {\tilde{E}}_{\mathrm{in}}\left(\omega \right)$$(1)

with the filter response function ${\tilde{H}}_{\mathrm{burst}}\left(\omega \right)$, which describes burst formation, and the input/output electric fields ${\tilde{E}}_{\mathrm{in}}\left(\omega \right)/{\tilde{E}}_{\mathrm{out}}\left(\omega \right)$. Within a broad bandwidth, the interpulse interference results in a coherent redistribution of the total optical energy into a few equidistant spectral peaks, spaced by $f_{\mathrm{burst}}=1/t_{\mathrm{burst}}$ that is determined by the pulse spacing t_burst. The LTI-condition of the burst-forming process has direct implications for its combination with other pulse-shaping techniques. Specifically, if the process is LTI, it commutes with any other LTI pulse-shaping operation, such as spectral windowing $\tilde{\mathcal{W}}\left(\omega \right)$ achieved by applying a bandpass filter$${\tilde{H}}_{\mathrm{burst}}\left(\omega \right)\bullet \tilde{\mathcal{W}}\left(\omega \right)=\tilde{\mathcal{W}}\left(\omega \right)\bullet {\tilde{H}}_{\mathrm{burst}}\left(\omega \right).$$(2)

We hereby define spectral windowing as $\tilde{\mathcal{W}}\left(\omega \right)={\mathcal{W}}_{f,0}$ for $\left|\omega -{\omega }_f\right|<{∆\omega }_f/2$, and $\tilde{\mathcal{W}}\left(\omega \right)=0$, otherwise. In this case, ${\mathcal{W}}_{f,0}$ is the filter amplitude, ω_f the filter central frequency and ∆ω_f the filter bandwidth.

First, we will discuss the transformation of an isolated pulse into a burst. This process is commonly implemented either at the front-end of high-energy laser systems – by spectrally filtering seed pulses – or at the back-end, after amplification, through temporal pulse splitting with interferometric precision [9, 10]. Generating a pulse train from a single pulse via spectral filtering can be formulated as an LTI operation. In this case, ${\tilde{H}}_{\mathrm{burst}}\left(\omega \right)$ corresponds to the frequency-periodic spectral filter response [11]. Similarly, temporal pulse splitting can be expressed as an LTI operation, that represents a comb of N pulses, by$$E_{\mathrm{out}}\left(t\right)= \sum _{n=0}^{N-1}{k}_n{E}_p(t-{\tau }_n).$$(3a) $${\tilde{E}}_{\mathrm{out}}\left(\omega \right)=\left(\sum _{n=0}^{N-1}{k}_n{\exp }^{(-\mathrm{i}\omega {\tau }_n)}\right)\bullet {\tilde{E}}_p\left(\omega \right)={\tilde{H}}_{\mathrm{comb}}\left(\omega \right)\bullet {\tilde{E}}_p\left(\omega \right)$$(3b)

Recently, Vernier-type techniques applied to burst-mode amplifier systems have attracted attention [12–15], which transform widely spaced seed pulses into a closely packed burst. The transit time t_loop of the optical loop is precisely mismatched from the intial pulse separation t_seed. By directing subsequent pulses through a slightly shorter loop, each pulse ends up close to the preceding one with a temporal spacing of t_burst = t_seed−t_loop. After N round-trips, the intracavity electric field E_loop (t) at the input/output coupler is given by$$E_{\mathrm{loop}}\left(t\right)= \sum _{n=0}^{N-1}\mathrm{\Pi }\left(t-\frac{t_{\mathrm{loop}}}{2}\right)\bullet e^{\mathrm{in}{\Phi }_s}\bullet E_p(t-n\bullet t_{\mathrm{loop}})= \mathrm{\Pi }\left(t-\frac{t_{\mathrm{loop}}}{2}\right)\bullet \sum _{n=0}^{N-1}{e^{\mathrm{in}{\Phi }_s}\bullet E}_p(t-n\bullet t_{\mathrm{loop}}).$$(4)

In this expression, Φ_s is the pulse-to-pulse phase slip, and Π(t) is the rectangular function, defined as Π(t/a) = 1 for $\left|t\right|<\frac{a}{2}$ , and Π(t/a) = 0, otherwise. We assume identical pulse shapes, except for the time delays n∙t_loop and the phase slip Φ_s, which is imprinted on the burst during Vernier-type burst forming processes – a reasonable approximation for mode-locked oscillator pulse trains. The similarity between equations (3a) and (4) becomes evident by identifying τ_n with n · t_loop and k_n with $e^{\mathrm{in}{\mathrm{\Phi }}_s}$. Thus, Vernier-type techniques can also be classified as LTI operations.

To further demonstrate the equivalence of spectral filtering before and after burst formation, we calculate the burst properties for two cases see Fig. 1):

• The $\mathcal{W}H$ case, which is defined as burst formation $\tilde{H}\left(\omega \right)$ followed by spectral windowing $\tilde{\mathcal{W}}\left(\omega \right)$.

• The $H\mathcal{W}$ case, which is defined as spectral filtering followed by burst formation of the spectrally filtered pulses.

Fig. 1 Comparison of the cases WH and HW. a) Dependence of the final burst properties on the pulse number. Filt. Energy: Filtered burst energy, normalized to the total energy of the initial unfiltered pulses. Peak Intensity: Burst spectral peak intensity normalized to the spectral peak intensity at N = 100. FWFN bandwidth: Full-Width-at-Full-Null width of the center spectral peak. (b), (c) Exemplary spectra at N = 10 pulses of the (b) WH case. (c) HW case.

The final burst energy (Filt. Energy in Fig. 1) is normalized to the energy of the incoherent sum ${ϵ}_{\mathrm{sum},\mathrm{nonf}}^{\left(\mathrm{incoh}\right)}=\sum {ϵ}_{n,\mathrm{nonf}}$ of the N burst-pulse energies without spectral filtering ${ϵ}_{n,\mathrm{nonf}}$. To maintain a constant ${ϵ}_{\mathrm{sum},\mathrm{nonf}}^{\left(\mathrm{incoh}\right)}$, we scale the pulse amplitudes by $1/\surd N$ when varying the pulse number. Due to this normalization, the filtered energy shown in Figure 1 reflects the efficiency of broadband energy conversion into the spectral windowing range. The spectral window limits were chosen to be [−∆f_FWHM, ∆f_FWHM], with ±∆f_FWHM being the Full-Width-at-Half-Maximum (FWHM) frequencies of the single-pulse spectrum. This choice yields a single-pulse filter throughput of 76.09%. In our calculations, the spectral-peak spacing f_burst is set to the FWHM range (f_burst = 2∆f_FWHM).

In the $\mathcal{W}H$ case, spectral interference redistributes the energy into three spectral peaks (accompanied by much weaker side-lobes), with the central peak being far more intense than the others (see Fig. 1). The peak intensity scales N-proportionally (due to the constant burst energy scaling), while the Full-Width-at-Full-Null (FWFN) of the high-intensity peak scales inversely proportional to N. Due to the spectral interference, the remaining normalized energy after spectral windowing exceeds the single-pulse filter efficiency, saturating at 84.05%. The saturation behaviour occurs as the side-lobes become negligible compared to the strong interference peaks.

In the $H\mathcal{W}$ case, the pulses are first spectrally filtered by the windowing function, restricting the single-pulse envelope to the FWHM range. Surprisingly, the coherent-sum energy of the resulting $H\mathcal{W}$-burst exceeds the single-pulse filter efficiency and matches the $\mathcal{W}H$-burst energy, a consequence of the LTI nature of burst formation. This can be further understood by considering the pulse interference in the time-domain: Spectral windowing causes the burst pulses to overlap temporally, introducing a non-negligible interference term ϵ_interf > 0 in the coherent-energy sum of the complex pulse amplitudes A_n $${ϵ}_{\mathrm{sum}}^{\left(\mathrm{coh}\right)}\propto {\left|\sum _n{A}_n\right|}^2=\sum _n{\left|A_n\right|}^2+\sum _{\left(l,m\right)}{A}_l{A}_m^{*}={ϵ}_{\mathrm{sum}}^{\left(\mathrm{incoh}\right)}+{ϵ}^{\left(\mathrm{interf}\right)}.$$(5Please note that equations were not sequential order in the text, and have been renumbered. Please check, and correct if necessary.)

We note that in both cases, the shaping efficiency (see Fig. 1a, Filt. Energy) saturates at about 88.8%, being 12.7% higher than the spectral filtering of an isolated pulse.

3 Experiment

We experimentally investigate the differences between the $\mathcal{W}H$ and $H\mathcal{W}$ cases using a Vernier-type burst-mode amplifier. We couple 10 pulses at a 76 Megahertz repetition rate from a mode-locked oscillator (Light Conversion, FLINT) into a home-built regenerative amplifier (RA) and accumulate the pulses in the amplifier to form a burst. Prior to incoupling, the pulses are stretched to 300 ps duration and selected from the oscillator pulse train by an acousto-optical modulator to improve the pulse contrast. A more detailed description of the laser system is given in [8, 12].

The RA’s round-trip time is slightly detuned from the oscillator repetition time, resulting in an intraburst pulse spacing of about 0.52 ps, such that only one peak is dominant within the single-pulse spectrum, which spans a 6 nm FWHM bandwidth (see Fig. 2a). Upon burst-formation, we amplify the pulses from few picojoules to approximately 100 μJ. Due to an etalon effect caused by the waveplates in the amplifier, a parasitic modulation with a ~20% modulation depth appears in the single-pulse spectra. We employ a 3 nm bandpass filter (Thorlabs FLH1030-3) to isolate one of the spectral interference peaks, which are spaced by 7 nm due to their 0.52 ps temporal burst-pulse spacing. In chirped-pulse amplification, the spectral filter also reduces the chirped pulse duration. As usual for Vernier-type burst formation, a pulse-to-pulse phase slip ${\mathrm{\Phi }}_s$ is imprinted on the burst. This phase-slip determines the frequency offset f_offset of the filtered burst spectral peak by$$f_{\mathrm{offset}}= \frac{{\mathrm{\Phi }}_s}{2\pi t_{\mathrm{burst}}}.$$(6)

Fig. 2 a) Single-pulse spectrum (lightblue-solid), the filtered single-pulse spectrum (lightblue-dotted), the burst spectrum (darkblue-solid), and the filtered burst spectrum (orange-filled). (b)–(d) Measured energy (top) and spectra (bottom) dependent on the phase-slip offset in burst-mode (10 pulses) at 0.52 ps intraburst pulse spacing. (b) Without spectral filtering. (c), (d) With spectral filtering (c) after and (d) prior to amplification. For validation, the spectral sum $\sum _{i}S({\mathrm{\Phi }}_{s,m},{\lambda }_i)\Delta \lambda$ at each corresponding phase slip Φs,m (blue line) is compared with the energies acquired from the powermeter (purple-dashed line). For the spectrally filtered cases, the filtered single-pulse spectrum is also included.

4 Discussion and conclusion

Figures 2b–2d show the measured spectra depending on the phase slip for three cases: burst formation without filtering (Fig. 2b), burst formation followed by spectral filtering ($\mathcal{W}H$ case, Fig. 2c), and spectral filtering followed by burst formation ($H\mathcal{W}$ case, Fig. 2d). For validation, we compare the burst energy – measured with a power meter (Coherent FieldMaxII-TO) – with the spectral integral $\sum _{i}S({\mathrm{\Phi }}_{s,m},{\lambda }_i)\Delta \lambda$ for each corresponding phase slip Φ_s,m. The two quantities show excellent agreement in all three cases. In the non-filtered case, the burst energy remains largely independent of the phase slip, except for minor fluctuations on the order of 3% caused by parasitic modulation in the single-pulse spectrum. Free-running burst operation with an unstable phase-slip, thus shows an increased RMS noise in comparison to the measured 0.6% RMS noise in single-pulse mode. For the filtered cases ($\mathcal{W}H$ and $H\mathcal{W}$), the spectrum supports only the filtered spectral peak, as intended. We also include the filtered single-pulse spectrum from Figure 2a, but converted the wavelength axis first to frequencies, and then to the corresponding phase-offsets using equation (6). Surprisingly, the spectral intensity extends over a larger range than the filtered single pulse, which is advantageous for achieving broader tunability. The 2D spectral structure practically remains unchanged for the $H\mathcal{W}$ and $\mathcal{W}H$ cases because it is characterized by coherent inter-pulse interference, to which the LTI nature of burst formation applies. However, amplification does relate the seed waveform with the amplifier output in a linear way only when the amplifier is operated without saturation and with a sufficiently strong seed energy, such that there is no competition with parasitic Q-Switching. Filtering and thereby reducing the seed-pulse energies prior to amplification changes the nonlinear gain dynamics by reducing the energy extraction efficiency from the amplifier. This leads to a non-negligible Q-Switched background radiation driven by Amplified Spontaneous Emission (ASE), as is evident from the phase-slip independent ≈13 μJ background energy observed for ${\mathrm{\Phi }}_s<-\pi$ and ${\mathrm{\Phi }}_s>1.5\pi$. This is consistent with the $H\mathcal{W}$ case (Fig. 2d) appearing more smeared out than the unfiltered case (Fig. 2b) and the $\mathcal{W}H$ case (Fig. 2c) showing a well-defined modulation pattern. The Q-switched background adds a broadband spectral component, reducing the contrast of the interference structure.

As conclusion, although burst formation and spectral filtering are theoretically commutable, the nonlinear gain dynamics of amplification disrupt this equivalence in the case of strong amplification. Implementing spectral filtering at the front-end requires careful seed-energy management to maintain efficient gain and avoid parasitic effects. These considerations are critical for developing future high-energy, wavelength-tunable sources.

Funding

This research was funded in whole or in part by the Austrian Science Fund (FWF) [10.55776/F1004, 10.55776/I5590]. For open access purposes, the author has applied a CC BY public copyright license to any author accepted manuscript version arising from this submission.

Conflicts of interest

The authors have nothing to disclose.

Data availability statement

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Author contribution statement

Conceptualization, V. Stummer and A. Baltuška; Methodology, V. Stummer; Software, V. Stummer; Validation; V. Stummer, A. Pugžlys, and A. Baltuška; Formal Analysis, V. Stummer; Investigation, V. Stummer; Resources, E. Kaksis, and A. Baltuška; Data Curation, V. Stummer; Writing – Original Draft Preparation, V. Stummer; Writing – Review & Editing, V. Stummer, A. Pugžlys, and A. Baltuška; Visualization, V. Stummer; Supervision, A. Baltuška; Project Administration, A. Baltuška; Funding Acquisition, A. Baltuška.

References

All Figures

Fig. 1 Comparison of the cases WH and HW. a) Dependence of the final burst properties on the pulse number. Filt. Energy: Filtered burst energy, normalized to the total energy of the initial unfiltered pulses. Peak Intensity: Burst spectral peak intensity normalized to the spectral peak intensity at N = 100. FWFN bandwidth: Full-Width-at-Full-Null width of the center spectral peak. (b), (c) Exemplary spectra at N = 10 pulses of the (b) WH case. (c) HW case.

In the text

Fig. 2 a) Single-pulse spectrum (lightblue-solid), the filtered single-pulse spectrum (lightblue-dotted), the burst spectrum (darkblue-solid), and the filtered burst spectrum (orange-filled). (b)–(d) Measured energy (top) and spectra (bottom) dependent on the phase-slip offset in burst-mode (10 pulses) at 0.52 ps intraburst pulse spacing. (b) Without spectral filtering. (c), (d) With spectral filtering (c) after and (d) prior to amplification. For validation, the spectral sum $\sum _{i}S({\mathrm{\Phi }}_{s,m},{\lambda }_i)\Delta \lambda$ at each corresponding phase slip Φs,m (blue line) is compared with the energies acquired from the powermeter (purple-dashed line). For the spectrally filtered cases, the filtered single-pulse spectrum is also included.

In the text