Spectroscopy harnessing coherent electromagnetic fields of frequency combs enables precise measurements with superfine spectral and temporal resolutions, demonstrated in dual-comb spectroscopy (DCS) and electro-optic sampling (EOS) [1,2]. Frequency-comb sources spanning the near-infrared (NIR) and mid-infrared (MIR) play pivotal roles in molecular spectroscopy across various disciplines including materials science, chemistry [3,4], atmosphere pollutants monitoring [5,6], determination of protein structure [7], combustion analysis [8,9], and medical diagnostics [10]. Therefore, there is a need for broadband frequency combs that are alignment-free, robust, and simple, ensuring out-of-lab applications without sacrificing spectroscopic sensitivity and speed. Soft-glass fiber lasers [11], solid-state lasers [12], and quantum cascade lasers [13] are promising candidates in this field. However, multi-octave spanning frequency combs at a high repetition rate (⩾100MHz) remain environment and alignment sensitive devices. An alternative approach involves extending the near-infrared (NIR) frequency comb from mature fiber oscillators to the MIR region through coherent nonlinear optical processes using fibers or planar waveguides [14–17].

Silica fibers stand out for their exceptional optical performance and mechanical robustness, rendering them ideal candidates for frequency-comb applications in both terrestrial and space-based scenarios [18,19]. While the high phonon energy inherent in silica glass enhances mechanical resilience, it also facilitates non-radiative processes [20–22]. The elevated loss and high-order dispersion characteristics of silica fibers have historically limited coherent nonlinear effects under 2.2 μm [23–26]. Although utilizing the H34−H35 excited-state emission in thulium-doped silica fibers extended the frequency comb to 3 μm, issues such as nanosecond pump duration and spontaneous emission led to incoherent spectra [27–29]. Consequently, alternative approaches, including the use of soft glass optical fibers and integrated waveguides, have been explored for mid-infrared (MIR) frequency-comb generation [17,30–32]. Recent advancement with a 2 μm single-optical-cycle pump promises expanded MIR frequency combs in silica fibers [33]. MIR spectroscopy benefits from a bright and flatter spectrum. Besides, telecom pump lasers can dramatically improve laser compactness and efficiency. In the absence of a comprehensive study, further spectrum optimization and device simplification of MIR silica-fiber frequency combs have become increasingly challenging.

In this paper, we meticulously explore the generation of optical frequency combs in silica fibers. Our experimental setup features an all-silica-fiber optical frequency comb operating at 200 MHz, spanning from 0.8 μm to 3.5 μm. Additionally, we develop a quantitative model that not only pinpoints the optimal “Goldilocks Zone” for pump-fiber combinations but also elucidates the associated pulse dynamics. Figure 1(a) outlines the conceptual framework, showcasing the generation of a multi-octave frequency comb within a silica fiber through the use of a few-cycle pulse, which provides high peak power and a broadband in-phase spectrum. An f-2f interferometer examines the comb coherence and phase noise. Experimentally, we compress the amplified pulses to 25 fs (about four optical cycles) to a highly nonlinear fiber (HNLF). Ultimately, our device emits frequency combs covering the near- to mid-infrared spectrum. Numerical simulations, depicted in Fig. 1(b), track the entire process from the initial condition of the fiber frequency comb to the output of the HNLF. By systematically varying the MIR power with respect to pump power and HNLF length, we identify the region conducive to all-silica-fiber MIR frequency-comb generation. Spectral noise and coherence are rigorously assessed, with a carrier-envelope offset frequency exhibiting a signal-to-noise ratio exceeding 40 dB and a linewidth of 90 kHz. These comprehensive findings not only advance the field of optical frequency-comb spectroscopy in all-silica-fiber systems but also offer valuable insights applicable to other fiber/waveguide platforms, underscoring the broader impact of our research.

Limited by the residual high-order dispersion from the chirped pulse amplification (CPA), our goal is to generate more feasible sub-30-fs pump pulses using an erbium-doped fiber CPA followed by self-compression. To apply self-compression, the net dispersion of the CPA was tuned to normal dispersion. The PM1550 fiber compensated for the residual normal dispersion of the CPA. At the end of the compensation, the chirp of the self-phase modulation balanced the chirp induced by the anomalous dispersion, leading to a few-cycle pulse duration [34,35]. This pump pulse then enters a Sumitomo HNLF with high nonlinearity and low dispersion to synthesize multi-octave frequency combs.

The polarization-maintaining nonlinear amplifying loop mirror (PM NALM) laser architecture is similar to that in our previous work [36] but with better integration by mounting all the space components on aluminum nitride with a size of 50mm×30mm×20mm, omitting any movable parts. The in-loop frequency instabilities of fceo and fbeat are 7.4×10−18τ1/2 and 8.5×10−18τ1/2 for the integration time τ, respectively [36]. The 3 dB bandwidth was approximately 70 nm and centered at 1560 nm. A second-harmonic-generation autocorrelator measured the pulse duration as 83 fs. Due to the optimized intra-cavity dispersion and reduced cavity loss, the seed laser directly outputs a broad spectrum and a short pulse with a flat spectrum.

The seed pulse then enters the Er-doped fiber-chirped pulse amplification (EDF-CPA) system, indicated as Segment 1 in Fig. 1(a), for amplification and net-dispersion manipulation. To induce normal dispersion (positive chirp) of the amplified pulse, a PM small-core EDF (nLight PM-ER-80-4/125) was used for EDF-CPA, which also provides normal dispersion for the CPA. With a 3.5 W pump power at 976 nm, the CPA outputs an average power of 563 mW. We carefully tuned the pre-chirp to minimize the nonlinear effects inside the amplifier. This methodology avoids spectrum components seeded by the ASE, which would severely degrade spectrum coherence. After amplification, the normal dispersion of the pulses was compensated by the anomalous dispersion of PM1550. Near the end of the compensation, the chirp induced by the nonlinear phase shift becomes sufficiently strong to balance the chirp induced by the PM1550 fiber. The self-phase modulation continues to expand the spectrum while the pulse preserves its near-zero chirp in the pulse center. The final output spectrum spanned from 1300 to 1800 nm, with smooth wings and a structured center, implying the best compression [37,38]. Using a second-harmonic-generation autocorrelator, we measured the compressed pulse duration to be 25 fs or approximately four optical cycles, as shown in Fig. 2(b).

The gray traces in Fig. 2 represent simulation results using GNLSE. The initial condition for the simulation was an experimentally recorded MLL pulse. The traces in Figs. 2(a) and 2(b) depict pulse spectral and temporal profiles, respectively. For better comparison of Fig. 2(b), the autocorrelation of the simulated time-domain pulse is plotted. Our experimental and simulated results exhibit good agreement, which is a prerequisite for studying pulse evolution in HNLF for efficient spectrum broadening.

To generate the MIR frequency comb and verify the performance of our fiber laser layout, a piece of HNLF was spliced for the cutback experiment. We recorded two key HNLF lengths: the onset of MIR frequency-comb generation and the optimal length at which the MIR comb reaches the smoothest broadening. The cutback experiment is facilitated by numerical estimation of pulse evolution inside HNLF. To better address this point, we first introduce the methodology for dispersion and nonlinear parameter estimation of the HNLF.

Based on manufacturer data, the Sumitomo HNLF’s GVD, mode field diameter, and estimated γ are 1.3ps2/km, 10.3μm2, and 20(W·km)−1, respectively. Sumitomo calculates the zero-dispersion wavelength (ZDW) to be 1520 nm. A ring of fluoride-doped silica layer is located next to the Ge-doped core, as illustrated in Fig. 3(a). With the Sellmeier equations of Ge-doped [39] and F-doped [40] silica, the distribution of electric field inside the fiber can be calculated using a finite element method code [41] to estimate the effective propagation constant up to 3 μm. By matching the fiber ZDW and group velocity dispersion on the datasheet, it is possible to determine the dispersion profile of our HNLF. The reconstructed HNLF has a core diameter of 3.66 μm and 0.3% (mass fraction) of GeO dopant in the core, resulting in a 3.5% increase of refractive index compared to fused silica in accordance with a past report [42]. The F-doped layer has a thickness of approximately 0.6 μm.

With the dispersion profile, we estimate that the HNLF has ZDW at 1500 nm, GVD of −1.3ps2/km at 1550 nm, and 11μm2 mode field area. Figure 3(a) displays the wavelength-dependent γ parameter of HNLF from 1 μm to 3 μm evaluated using the n2 and effective mode area values. Figure 3(b) shows the GVD of the reconstructed fiber superimposed on Sumitomo data points. The inset of Fig. 3(b) shows the broadband dispersion up to 3 μm. The nonlinear refractive index (n2) is derived from the definition of γ: γ(ω0)=ω0n2cAeff,(1)where ω0 is the angular frequency of the pump. Here, n2 is 5×10−20m2/W, which is approximately five times the nonlinear refractive index of fused silica owing to GeO dopants. The n2 value is assumed constant over the entire spectrum. The γ parameter in Eq. (1) is wavelength-dependent [Fig. 3(a)]. The input pulse-shot noise was modeled as one photon per frequency grid [43]. At this point, the GNLSE simulation was sufficiently precise to estimate the optimal HNLF fiber length for the MIR frequency comb and revealed the underlying nonlinear processes governing the two-octave spectrum.

Simulations with wavelength-dependent γ parameter prove the generation of MIR components starts at around 5 cm in HNLF and reaches the optimal at around 30 cm. Figure 3(c) displays the experimental results alongside a simulated spectrum using constant γ parameter for a fiber length of 3.2 cm rather than 5 cm. For this fiber length, maintaining a constant γ at the pump wavelength better aligned with the experimental results, consistent with a prior report [44]. To enable nonlinear optical processes, electric fields at different frequencies must exhibit spatial and temporal overlap. Consequently, the newly generated frequency components share the same optical mode as the pump wavelength. As new wavelengths emerge, spatial propagation is essential for them to evolve into the corresponding optical mode areas. Thus, at short fiber lengths, a constant nonlinear parameter better estimates the pulse evolution. At longer HNLF length, the soliton self-frequency shifts towards a lower frequency until the shift is constrained by the loss and higher dispersion terms. Subsequent propagation redistributes the power and flattens the spectrum, proving advantageous for broadband frequency-comb spectroscopy [45]. At 30 cm, which is close to the dispersion length LD of HNLF, these new wavelengths can fully evolve into their corresponding optical modes. A wavelength-dependent γ in the simulation better reproduces this process. With predictions from simulations, the cutback starts at 40 cm. The optimal HNLF length was determined to be 33 cm by cutback—very close to the simulated value of 30 cm. At this length, a spectrum covering more than two octaves with high flatness was achieved [bottom of Fig. 3(b)]. In the MIR range of 2 μm to 3.5 μm, the total power exceeds 100 mW, with clear water lines evident around 2.8 μm, validating that the spectrum is not an artefact. An off-axis parabolic mirror can collimate the two-octave spanning spectrum with minimum chromatic dispersion to fully utilize this two-octave spectrum.

In a silica fiber, the soliton self-frequency shift (SSFS) creates the MIR components, while the dispersive-wave (DW) leads to the blue part. Coherent DW and SSFS processes can preserve the comb structure of the seeds. DW components typically manifest around the 1 μm highly transparent band. Moving towards the MIR, the magnitude of SSFS suffers from an increasing linear loss, and its phase experiences larger high-order dispersion. To generalize our results, we conducted a systematic investigation to reveal the underlying pulse dynamics and limiting factors of the system, as depicted in Fig. 1(b). This subsection details our quantitative study of MIR generation in highly nonlinear fibers (HNLFs) to identify the design margins for MIR frequency-comb generation in silica fibers.

Practically, the spectrum from the HNLF is a joint result from all components, including the seed laser, amplifier, and compressor. Despite previous efforts, few studies have addressed the nonlinear pulse dynamics across the entire laser system. The first step in developing a systematic model is to determine the starting point of the simulation or the initial conditions that provide best approximations of experimental parameters. Our 200 MHz frequency comb outputs 83 fs transform-limited seed pulses. The seed pulse enters a fiber-chirped pulse amplifier [CPA; Segment 2 of Fig. 1(b)], and the pulse energy reaches 2.8 nJ. In CPA, nonlinear effects are minimized to avoid extra noise from amplified spontaneous emissions. Thus, a numerical examination that starts after the CPA and concludes at the HNLF can ease the numerical simulation and provides good matching to experiment [Segments 3 and 4 in Fig. 1(b)]. Considering the gain-narrowing effect, the initial pulse condition is approximated to 100 fs duration and 2.8 nJ energy.

Currently, soliton self-compression remains the only all-fiber approach to creating hundreds of femtoseconds down to single-cycle pulses. The slowly varying envelope approximation in a single optical cycle has been theoretically verified [46,47] and experimentally proven [33]. Thus, the time-domain envelope A(z,T) can be modeled using the generalized nonlinear Schrödinger equation (GNLSE) as [43] ∂A∂z+α2A−∑k≥2Nik+1k!βk∂kA∂Tk=iγ(1+iτshock∂∂T)⁢(A(z,t)∫−∞∞R(T′)|A(z,T−T′)|2dT′).(2)

Here, α represents the wavelength-dependent propagation loss extrapolated from our previous measurements [48]. T denotes the co-moving timeframe at speed β1−1. The nonlinear parameter γ is considered as a wavelength-dependent vector and discussed in later sections. τshock=1/ω0 is the shock term, a key parameter in the few-cycle pulse evolution [46]. The function R(t)=(1−fR)δ(t)+fRhR(t) is a nonlinear response, with the Raman contribution fR being 0.18 in silica fibers. The analytical approximation of the Raman response hR(t) includes instantaneous and delayed Raman processes, as detailed in a previous study [49]. βk represents the kth-order Taylor coefficient of the propagation constant about ω0, and N is the maximum order of the coefficient. The sum term of Eq. (2) is numerically replaced in the Fourier domain as follows: ∑k≥2Nβkk!(ω−ω0)k=β(ω)−β(ω0)−β1(ω0)(ω−ω0).(3)

The term β1(ω0) is the group velocity at the center frequency of the few-cycle pulse. Equation (3) requires complete knowledge of the propagation constant β(ω) and group velocity β1(ω) over the wavelength of interest. The propagation constant and group velocity of a waveguide are the combined impacts of waveguide dispersion and material dispersion. This combined impact can be modeled using the silica-fiber profile. Using the fiber profile, we can extract β(ω) and β1(ω) by the finite element method.

Under the aforementioned conditions, Figures 4(a)–4(d) display the self-compressed pulses from 60 fs to 5 fs in PM1550 [Segment 2 of Fig. 1(b)]. High-order dispersion terms cause tailing satellites and pedestals. The pulse central component contributes to nearly all the spectrum broadening process—it is the “peak power” referred to in most literature. Consequently, the degradation of the center pulse energy lowers the “actual” peak power and efficiency of further nonlinear effects. The energy inside the pulse center decreases from 96% to 25% as the pulse narrows. Indeed, a 25 fs pulse [Fig. 4(b)] has nearly the same pulse peak power as a single-cycle pulse [Fig. 4(b)]. Figure 4(e) shows the trend of central power reduction with a fitted curve, which served as a scaling factor for the peak powers in the following simulations.

The compressed pulse then enters the HNLF to create a multi-octave frequency comb, where careful selection of the pump pulse and HNLF length is essential. With a nonlinear parameter of 20(W·km)−1, our HNLF from Sumitomo is expected to be the most suitable HNLF for efficient MIR spectrum generation and was adopted for this work. This HNLF is not a polarization-maintaining (PM) fiber, and its input polarization extinction ratio decreases to approximately 6 dB. Following previous experiment and simulation [50], we expected nearly identical spectra for both polarizations. The detailed parameters of the HNLF are shown in Fig. 3. In this section, we focus on the physics behind the simulation results.

A short HNLF length is beneficial for decreasing the MIR propagation loss, whereas longer fibers could facilitate SSFS. To maximize the MIR power, we simulated the spectrum after a 1 m HNLF with pump pulses compressed to different durations. Figures 4(f) and 4(g) show the simulated power in the region beyond 2.3 μm and 2.5 μm as a function of pump duration and fiber length, respectively. For pumps longer than 40 fs, the power beyond 2.5 μm is close to zero, regardless of HNLF length. HNLFs longer than 1 m are unlikely to output MIR frequency combs due to silica phonon process. HNLFs less than 10 cm in length do not allow the SSFS to shift to the MIR band. The “Goldilocks Zone” for the MIR comb to exist is a small window bonded by of 15 cm to 40 cm and pulse duration from 10 fs to 35 fs. This narrow window requirement explains the lack of previous MIR comb demonstrations using silica fibers. Meanwhile, pump pulses of less than 40 fs have a minimal impact on the spectrum coherence [51], which is confirmed by our later f−2f interferometer. In the following subsections, we describe the experimental implementation, performance, and noise measurements of the MIR frequency comb.

First-order coherence, denoted as g12, is a common parameter employed for the numerical assessment of pulse-to-pulse coherence during the supercontinuum generation process [43]. Alternatively, a more direct experimental approach becomes feasible when dealing with a low-noise frequency comb, utilizing f-2f interferometry. This method translates the carrier-envelope-offset frequency component, fceo, of a frequency comb into the radio-frequency domain. Consequently, the signal-to-noise ratio (SNR) and linewidth of the fceo term directly correlate with the coherence and noise properties of the frequency-comb lines [52]. Utilizing a standard inline f-2f interferometry setup, we recorded a high-quality fceo signal after supercontinuum generation in the highly nonlinear fiber (S-HNLF). The broadband dispersive wave and Raman soliton (Fig. 3) contribute to a simple and high-quality f0 retrieval. Figure 5(a) illustrates the measured fceo with a 100 kHz resolution bandwidth (RBW) and video bandwidth (VBW), displaying an SNR exceeding 40 dB. At a resolution of 3 kHz, the free-running fceo was recorded at 90 kHz with a 3 dB bandwidth. The high quality of the fceo signal attests to the high coherence and low-frequency noise characteristics of the two-octave spectrum in Fig. 3. Once the seed comb is stabilized, all comb lines can be referenced to an atomic clock.

We characterized the relative intensity noise (RIN) of our laser output at various stages, seed laser, pump laser, and MIR frequency comb, using a dynamic signal analyzer (SR785). Figure 6 presents the double-sideband spectra of the laser in the range 1 Hz to 100 kHz. The blue and red traces represent the RIN of the pump pulse and the MIR frequency comb, respectively. Both exhibit nearly identical RIN, particularly at higher frequencies, a crucial factor for frequency-comb heterodyne detection [1,2]. The increased noise in the MIR frequency comb in the low-frequency region may result from thermal fluctuations near the splicing point. Frequency-comb spectroscopy, which transfers optical information to radio frequency (RF) through heterodyne signals, inherently overlays optical frequency noise onto the downconverted signal. Fortunately, majority comb lines occur beyond the tens of kilohertz range, emphasizing the importance of noise reduction in and beyond this range to achieve a high signal-to-noise ratio interferogram for frequency-comb spectroscopy [1].

In conclusion, we implement an all-silica-fiber MIR frequency comb and develop quantitative design guidelines to generate MIR frequency combs in silica fibers. Our configuration produces two octaves from NIR to MIR frequency comb generation, built entirely with step-indexed silica fibers. Such a frequency comb is crucial for simultaneously monitoring multiple molecules at different wavelengths. Spectral coherence was experimentally confirmed using an f-2f interferometer. Numerically, we quantitatively determined the conditions for generating MIR frequency combs in silica fibers. By rigorously estimating fiber parameters and pulse evolution, we deduced scaling rules for pump duration and fiber length. This narrow “Goldilocks Zone” explains the limited success in extending silica-fiber combs to MIR in previous attempts. To the best of our knowledge, this is the first successful attempt to quantitatively realize NIR to MIR frequency-comb generation in an all-silica-fiber configuration. In addition, we anticipate that polarization-maintaining highly nonlinear fibers (PM HNLFs) will support similar MIR frequency combs with only half the nonlinear parameters compared to non-PM versions. Particularly, robust frequency-comb source covering NIR to MIR enables the correction of beam path conditions with reference molecules at NIR (such as O2, CO2, and/or their isotopes) [53], and a similar approach can be adopted in other fields of science. In addition, soft glass optical fibers and integrated waveguides could be fused or coupled with our silica-fiber implementation for enhanced MIR spectrum coverage. Finally, we would like to emphasize that our approach is a universal technique for frequency-comb generation in both fibers and integrated waveguides, fully exploiting their transmission windows.

Acknowledgment. This work was supported by the National Natural Science Foundation of China, the Shenzhen Science and Technology Program, the Natural Science Basic Research Program of Shaanxi, the Strategic Priority Research Program of the Chinese Academy of Sciences, the Mathematical Basic Science Research Project of Shaanxi, the Fundamental Research Funds for the Central Universities, and the Shanghai Pujiang Programs. The authors thank Fuyu Sun and Dnaiel Lesko for useful comments on the paper.