1. Introduction

Fiber-optic Cherenkov radiation (FOCR), also known as dispersive wave generation or non-solitonic radiation, describes the radiation from a fiber-optic soliton when perturbed by higher-order fiber dispersion [1–5]. First theoretically studied in 1986, FOCR has attracted renewed research interest in company with the advent of photonic crystal fibers (PCFs) whose properties can be flexibly engineered [6]. For example, controlling a PCF’s hole-size and inter-hole spacing results in two zero-dispersion wavelengths (ZDWs) which in turn give rise to two FOCR bands [7]. As a matter of fact, deliberate designs allow the presence of three ZDWs [8]; such a PCF would accommodate a variety of novel phenomena arising from phase-matched interactions between soliton and the resulting FOCR [9]. To date, most of FOCR research in PCFs is carried out in the context of suppercontinuum generation since FOCR extends the spectrum towards shorter wavelength [10–18]. Supercontinuum generation requires that the pump pulse propagate in the anomalous dispersion regime and form a higher-order soliton. To achieve a SC without substantial spectral gaps, the pump’s center wavelength needs to be close to the fiber’s ZDW so that the generated FOCR merges with the spectral components building up from other nonlinear effects. As this wavelength separation increases, FOCR manifests as an isolated spectrum that corresponds to a wavelength up-conversion or down-conversion of the pump depending on the sign of the third-order dispersion [7,19–22]. The up-conversion FOCR is of particular importance due to its capability of converting near-infrared (NIR) ultrafast lasers (e.g., Ti:Sapphire laser) to their counterparts in the visible wavelength range. The resulting ultrafast sources are desired in many applications, such as multi-photon microscopy, fluorescence spectroscopy, and optical coherence tomography. It is widely believed that FOCR generates resonant, narrowband (i.e., ~10 nm in the visible wavelength range) spectrum with relatively low conversion efficiency (~10%). These two drawbacks have hampered many real-world applications that demand high photon flux with a broadband coverage. Recently, we have explored the dependence of these two quantities (i.e., bandwidth and conversion efficiency) on the NIR pump-pulse’s parameters (such as duration and pulse energy), and have demonstrated FOCR featuring high efficiency (>40%), broadband spectrum (>50 nm), and low threshold (<100 pJ for pulse energy) [23]. Such a dramatic improvement results from using few-cycle pump pulses to drive the radiation.

Generation of few-cycle laser pulses and their applications for investigating ultrafast processes in physics and chemistry have received a great deal of research attention [24]. While FOCR excited by few-cycle pulses has been pursued for different purposes (e.g., investigation of soliton-trapped FOCR [25] and synthesis of single-cycle pulses [26]), a thorough and detailed study of FOCR emphasizing bandwidth and conversion efficiency in this new regime is yet absent.

In this paper, we both theoretically and experimentally investigate FOCR in the few-cycle regime. In Ref. [23], we have identified three characteristic propagation scales: 1) initial buildup stage in which FOCR acquires most (>90%) of its energy; 2) quasi-independent propagation with minimal interaction with its host soliton; and 3) strong interaction with the host soliton that is decelerated by stimulated Raman scattering (SRS). Since the FOCR spectrum splitting and trapping that take place in the 3rd stage has been well studied, we rather focus on the first two stages.

2. Underlying physics in the initial buildup stage: phase-matching revisited by introducing coherence length

Propagation of an ultrashort pulse inside an optical fiber is well modeled by the generalized nonlinear Schrödinger (GNLS) equation [27]

Typical values of $f_R$, $f_a$, $f_b$, and $f_c$are 0.245, 0.75, 0.21, and 0.04, respectively [28]. Isotropic Raman response$h_a(t)$and anisotropic Raman response $h_b(t)$ are defined as $h_a(t)=\Theta (t)({\tau }_1^2+{\tau }_2^2)/({\tau }_1{\tau }_2^2)\exp (-t/{\tau }_2)\sin (t/{\tau }_1)$ and $h_b(t)=\Theta (t)(2{\tau }_b-t)/{\tau }_b^2\exp (-t/{\tau }_b)$, where ${\tau }_1$, ${\tau }_2$, and${\tau }_b$ have values of 12 fs, 32 fs, and 96 fs, respectively. $\Theta (t)$ is the Heaviside step function. If higher-order dispersion (i.e. ${\beta }_n$,$n>2$), self-steepening, and SRS are absent Eq. (1) under ${\beta }_2<0$ supports fundamental soliton solution with propagation constant${\beta }_s(\omega )={\beta }_0+{\beta }_1(\omega -{\omega }_0)+(1-f_R)\gamma P_0/2$. The peak power $P_0$ and full-width-half-maximum (FWHM) duration $T_0$are related by the soliton area theorem, i.e., $(1-f_R)\gamma P_0=3.1\left|{\beta }_2\right|/T_0^2$.

Optical soliton—a non-dispersive pulse with the propagation constant ${\beta }_s(\omega )$—is a consequence of interplay between anomalous dispersion and self-phase modulation (SPM). Higher-order dispersion exerts perturbation onto a soliton and enforces it to shed away energy carried out by a dispersive pulse; the pulse travels with a propagation constant which coincides with the fiber’s dispersion curve $\beta (\omega )$. Evidently, two sources contribute to the FOCR at position $z+\Delta z$: 1) the FOCR at z, which propagates $\Delta z$distance and acquires an extra phase due to fiber dispersion; and 2) the new radiation emitted by the soliton as it moves from z to$z+\Delta z$. Depending on their phase difference, these two fields add up constructively or destructively. If the phase-matching condition, i.e. $\beta (\omega )={\beta }_s(\omega )$, is satisfied for any frequency, the two fields stay in-phase all the time and the generation of FOCR is maximized. In reality, however, the phase-matching condition only holds at a certain discrete frequencies. This is owing to the fact that ${\beta }_s(\omega )$ is a linear function of frequency (that is why soliton is non-dispersive) while $\beta (\omega )$ could be highly oscillatory for specially engineered dispersion. These phase-matched frequencies grow up monotonically along the propagation due to constructive interference. In contrast, non-phase-matched frequencies experience destructive interference and thus are strongly suppressed given enough propagation distance. Such a phase-sensitive mechanism explains why FOCR is widely regarded as a resonance process and forms a narrowband spectrum—a popular image one bears in mind. To describe phase-mismatch caused spectral narrowing, we introduce a new quantity $L_c$—coherence length—defined as $L_c(\omega )=\pi /\left|\beta (\omega )-{\beta }_s(\omega )\right|$. Plugging in the expression of the soliton’s propagation constant, after a little math, the coherence length can be expressed as a function of fiber’s dispersion and input soliton peak power,

It is noteworthy that the physical picture behind FOCR is quite universal in nonlinear optics, especially in illustrating ${\chi }^{(2)}$-initiated nonlinear phenomena, such as second-harmonic generation, sum-frequency generation, difference-frequency generation, etc. For example, when driven by femtosecond pulses, optical rectification—a degenerate difference-frequency generation—gives rise to generation of broadband spectrum located at terahertz regime. In fact, we have borrowed the concept of coherence length from the terahertz community. In the context of ultrafast terahertz generation, coherence length quantifies the matching between the terahertz pulse’s phase-velocity and the optical pulse’s group velocity [29]. This velocity-matching is intrinsically equivalent to phase-matching required for efficient three-wave (i.e. terahertz frequency and two optical frequencies for optical rectification) mixing in a ${\chi }^{(2)}$nonlinear process.

For a ${\chi }^{(2)}$nonlinear process, it is the thickness of the nonlinear crystal that largely determines phase-matching bandwidth. Thus a thinner crystal is desired to achieve broader bandwidth at the cost of a decrease in conversion efficiency. Since FOCR shares a similar physical mechanism, one may ask whether such a tradeoff between bandwidth and conversion efficiency also exists for FOCR. Answering this question necessitates numerically solving the GNLS equation.

3. Continuum generation in the few-cycle regime

To investigate the effect of coherence length, we perform simulations for a fundamental soliton propagating in a PCF. The mode-field diameter of the PCF is 1.2 µm. The group-velocity dispersion shown in Fig. 1(a) indicates that the PCF has a ZDW at 0.71 µm. Also plotted in the same figure are the coherence lengths for three fundamental solitons with different FWHM duration (i.e. $T_0$) or center wavelength (i.e. ${\lambda }_0$) which are specified in the legend of Fig. 1(a). These three curves share similar features:

 

Fig. 1 (a) the PCF’s group velocity dispersion and calculated coherent length for three fundamental solitons with different peak power or center wavelength (see the legend). $L_{con}$labels the continuum length for the blue curve; (b)-(d) spectrum evolution along propagation distance for fundamental solitons with different FWHM duration or different center wavelength (as indicated in each figure). The spectrum intensity is shown in logarithm scale. The double-arrow line marks the continuum that connects the phase-matching wavelength and the soliton center wavelength.

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The broad phase-matching bandwidth around the continuum length strongly suggests that a continuum might build up spanning between the phase-matching wavelength and the soliton center wavelength. To verify such a prediction, we take each of the three fundamental solitons as the input and numerically solve the GNLS equation. Figures 1(b)–1(d) record the resulting spectral evolution for these three solitons propagating up to 10 cm; the spectra normalized to their peak power at each position are shown in logarithm scale. Despite calculated for different inputs, the corresponding spectra evolve following a similar pattern:

Generation of continuum highly depends on input soliton’s duration. A comparison between Fig. 1(b) and Fig. 1(c) reveals that using a shorter, few-cycle pulse as the input favors continuum generation accompanied by higher power conversion efficiency and better spectral flatness. For most reported FOCR experiments, which exploit input pulses of ~100-fs duration, the continuum generation is extremely weak and thus not observed.

4. Higher conversion efficiency in the few-cycle pulse regime

Results in Fig. 1 confirm the physical picture described in section 2; that is, FOCR at a propagation distance $z+\Delta z$includes two parts: the radiation accumulated before z and new radiation as the soliton travels from z to $z+\Delta z$. Note that the new added radiation $\Delta E_{CR}(\omega )$ is proportional to $\Delta z$ as well as to the soliton’s spectral amplitude at ω. Therefore, a soliton with shorter duration extends more in the spectral domain which enhances FOCR generation, leading to a higher conversion efficiency as indicated by comparing Fig. 1(c) with Fig. 1(b).

The simulation performed in last section has assumed fundamental solitons as the input, which possess low pulse energy. For example, the soliton centered at 0.75 µm with 20-fs duration (i.e. the simulation input for generating Fig. 1(b)) is of only 12 pJ. The typical pulse energy in FOCR experiments is normally at ~1 nJ level. In this scenario, the input hyperbolic secant pulse becomes a Nth-order soliton rather than a fundamental soliton; the soliton order N is given by

From above analysis, one might speculate that, with input pulse energy fixed, conversion efficiency may monotonically increase as the input pulse duration decreases. As a matter of fact, this is not true. It has been well known that, if $T_0$ is so small that the resulting soliton number is much less than 1, fiber dispersion overtakes the nonlinearity and no soliton can form. As a result, FOCR is absent. Therefore there must be an optimum pulse duration that maximizes conversion efficiency for a given input energy. Figure 2 plots dependency of FOCR conversion efficiency on input pulse duration for different pulse energy. Evidently, an optimum duration in maximizing conversion efficiency exists for a given input pulse energy, and it becomes shorter with increasing pulse energy. More specifically, the optimum duration decreases from 18 fs to 12 fs as the input pulse duration increases from 80 pJ (red line with diamond marker) to 150 pJ (green line with square marker); the conversion efficiency at the optimum duration goes up from 10% to 24%. For a given duration, increasing input pulse energy always improves conversion efficiency. The dashed, orange line marks the duration corresponding to 10 carrier oscillation cycles at 0.8 µm. The fact that the optimum duration is less than 10 optical cycles even for a moderate energy level (80 pJ - 150 pJ) indicates that FOCR in the few-cycle regime is highly desirable to achieve higher conversion efficiency and stronger FOCR pulse. The soliton order ranges 1.5 – 2 at the optimum duration.

 

Fig. 2 Dependency of FOCR conversion efficiency on input pulse duration for different pulse energy. An optimum duration in maximizing conversion efficiency exists for a given input pulse energy; the optimum duration becomes shorter with increasing pulse energy. The orange dashed line indicates the duration that corresponds to a pulse of 10 cycles centered at 0.8 µm.

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5. Broader bandwidth in the few-cycle regime

In the physical picture that qualitatively describes FOCR generation and evolution, the emitted FOCR linearly propagates inside the fiber, only experiencing fiber dispersion. Such a model is valid when the conversion efficiency is low so that the resulting weak FOCR pulse creates negligible nonlinear effects. Apparently this is not the case for FOCR in the few-cycle regime featuring high conversion efficiency. In this new regime, FOCR pulse may carry ~40% of the total input energy [23], strong enough to cause nonlinear effects (e.g., SPM) which in turn reshape the FOCR pulse and broaden its optical spectrum.

Figure 3 plots the evolution of FOCR conversion efficiency and spectral bandwidth along propagation distance for an input 10-fs pulse of hyperbolic secant shape centered at 0.8 µm with 300-pJ pulse energy. The spectral bandwidth is FWHM of the optical spectrum located in the range of 0.4-0.7 µm. The spectra at 0.8 mm, 4 mm and 20 mm are presented as inset (a)-(c), respectively. A comparison of FOCR spectral bandwidth reveals a three-stage evolution, which normally does not coincide with the aforementioned three-stage process in terms of conversion efficiency:

 

Fig. 3 Evolution of FOCR conversion efficiency and spectral bandwidth along propagation distance for a 10-fs hyperbolic secant pulse centered at 0.8 µm with 300-pJ pulse energy. Insets (a) and (b) plot spectra corresponding to the maximum and minimum bandwidth. Further propagation of spectrum recorded in (b) up to 2 cm results in the FOCR spectrum shown as the blue, solid line in inset (c). If we propagate only the FOCR spectrum (i.e., spectrum in 0.4-0.6 µm), the corresponding spectrum at 2-cm distance is denoted by the red, dashed curve. See the text for details. Note that the spectra are shown in linear scale.

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Note that the concept of coherence length explains the buildup of the FOCR in the first two stages. In the third stage in which CR emission stops, coherence length is not meaningful since interaction between CR and the pump pulse is absent. As for further propagation beyond 20 mm, part of the residual pump develops into a soliton (hereafter, we refer it as Raman soliton) which continuously shifts towards longer wavelength driven by SRS, and as a result, slows down. Eventually the FOCR pulse catches up with the decelerating Raman soliton; their strong nonlinear interaction splits the FOCR spectrum into two well-separated parts. The shorter wavelength part will be captured by the Raman soliton. Detailed studies on FOCR spectral splitting and trapping have been presented in Refs. [23] and [25].

6. Experimental results

Using a home-built 85-MHz Ti:Sapphire laser emitting ~10-fs pulses to pump a commercially available PCF, we have experimentally investigated FOCR in the few-cycle regime. The PCF has a ZDW at 710 nm with a dispersion curve shown in Fig. 1(a). A detailed description of the whole setup can be found in Ref. [23].

Figure 4(a) records the FOCR spectra generated by three PCFs with different lengths: 2 mm, 4 mm, and 2 cm; the input pulse energy coupled into these fibers is fixed at 300 pJ. While the coupled spectrum [inset in Fig. 4(a)] is not of hyperbolic shape, the experimental results well reproduce the main features of the three-stage evolution discussed in session 5 and depicted in Fig. 3:

 

Fig. 4 (a) Evolution of FOCR spectrum with increased PCF length, labeled as 2 mm, 4 mm, and 2 cm, respectively. The input pulse energy is fixed at 300 pJ. Inset shows the laser spectrum coupled into the fiber (i.e., the optical spectrum at the beginning of the PCF). (b) Evolution of FOCR spectrum with increased input pulse energy for the 2-mm long PCF.

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The dependence of continuum generation on input pulse energy is also explored using the 2-mm PCF. Figure 4(b) plots the resulting spectra as we vary input pulse energy. Similar to the spectral evolution with respect to increased PCF length, a continuum appears with the input pulse energy increased; further increase results in an isolated FOCR spectrum growing up at the continuum’s short wavelength side. Such an evolution trend also indicates that the continuum length becomes shorter with increased input pulse energy.

As the results in Fig. 2 show, using few-cycle pulses to excite FOCR enhances its conversion efficiency, leading to a stronger FOCR pulse; the corresponding stronger nonlinearity eventually leads to a broader FOCR spectrum in the visible wavelength range. In other words, FOCR excited by few-cycle pulses allows achieving broadband visible-wavelength spectra from relatively low-energy ultrafast sources. For example, scaling up the repetition rate of a Ti:sapphire femtosecond oscillator beyond 1 GHz is inevitably accompanied by low pulse energy (<1 nJ). Such high repetition-rate sources are desired in many applications, such as frequency metrology, optical arbitrary waveform generation, and high speed A/D conversion, to name a few. To demonstrate that FOCR in the few-cycle regime constitutes a powerful wavelength up-conversion tool to efficiently convert a NIR, GHz laser source into its broadband counterpart in the visible wavelength range, we switch to a home-built, 1 GHz Ti:sapphire oscillator as the pump source, centered at 0.83 µm with ~140 nm bandwidth and ~800 mW average power. Figure 5 presents the FOCR spectra generated by 10-cm PCFs with different ZDW, i.e., 710 nm, 735 nm, and 750 nm. The maximum powers coupled into these fibers are 210 mW, 250 mW, and 190 mW, respectively [see the figure legend that specifies each fiber’s ZDW and coupled power]. The relatively low coupling efficiencies into these PCFs are due to chromatic dispersion caused by the singlet coupling lenses. Higher coupled powers are expected if achromatic coupling lenses are employed. The FOCR spectrum, its center wavelength mainly determined by the phase-matching condition, shifts toward longer wavelength along with increasing the PCF’s ZDW. Indeed, 10-cm length of PCF is enough to substantially slow down the Raman soliton (emerging from the residual pump) so that it has a considerable overlap with the FOCR pulse; the overlap-introduced XPM manifests as amplitude modulation onto the FOCR spectra shown in Fig. 5. With 20-30% conversion efficiency, the resulting FOCR spectra, locating in the visible wavelength range, exhibit a bandwidth of 50-70 nm.

 

Fig. 5 FOCR spectra generated by 10-cm PCFs with different ZDWs, i.e., 710 nm (red curve), 735 nm (green curve), and 750 nm (blue curve). The one with 750-nm ZDW is a polarization maintaining (PM) PCF.

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Recent years have seen growing research interest in optimizing femtosecond-laser based frequency combs for astrophysical spectrograph calibration (“astro-combs”) [32–40]. Through precision radial velocity (PRV) observation, astro-combs hold the promise to enable the search for Earth-like extra-solar planets (exoplanets), direct observation of cosmological deceleration, and the study of temporal variation of fundamental constants. Astro-combs in the visible wavelength range are of particular importance because, in this wavelength region (400-600 nm), emission from Sun-like stars provides the largest photon flux as well as high-quality spectral features most suited for PRV observation [41]. While astro-combs in the blue and green have been demonstrated via second harmonic generation of NIR source combs, they exhibit narrow bandwidth (2 nm for the green astro-comb [37] and 15 nm for the blue [40]). In this scenario, the broadband, highly efficient, GHz FOCR in the few-cycle pulse regime emerges as a powerful wavelength up-conversion tool to implement a broadband astro-comb in the visible wavelength range. As a proof-of-concept, we have demonstrated a broadband green astro-comb using this method [42].

7. Discussion and conclusion

In this paper, we have both theoretically and experimentally studied FOCR in the few-cycle regime, with a focus on its evolution before spectral splitting due to the decelerated Raman soliton. Although FOCR originates from ${\chi }^{(3)}$ nonlinear susceptibility of an optical fiber, its buildup and initial evolution resembles well-known ${\chi }^{(2)}$nonlinear effects, such as second-harmonic generation, sum-frequency generation, difference-frequency generation, and optical rectification. A universal feature of these phenomena is that they all require phase-matching to achieve efficient power conversion. At the initial evolution stage of FOCR, its bandwidth is determined by phase-matching condition in a manner similar to ${\chi }^{(2)}$ nonlinear process; that is longer propagation leads to narrower bandwidth. Inspired by such a resemblance, we have introduced the coherence length for continuum generation—a characteristic phenomenon only evident for FOCR in the few-cycle regime. Coherence length also elucidates the physical mechanism behind the formation of an isolated FOCR spectrum that builds up at the short wavelength side of the continuum and experiences spectral narrowing. Inside a ${\chi }^{(2)}$medium, normally the effect of ${\chi }^{(3)}$ is negligible and spectral broadening due to SPM is absent. Therefore spectral bandwidth is nearly exclusively determined by the phase-matching condition. In contrast, after the spectral narrowing due to limited phase-matching bandwidth, FOCR is broadened as a consequence of nonlinear propagation. Such a nonlinear spectral broadening process is enabled by the high conversion efficiency achieved by exciting FOCR with few-cycle pulses.

In summary, we have demonstrated (both theoretically and experimentally) that FOCR in the few-cycle regime exhibits three unique features that are absent when pumped with often-used, long pulses: (1) continuum generation (may span one octave in connection with the pump spectrum), (2) high conversion efficiency (up to 40%), and (3) isolated FOCR spectra with broadband (70 nm experimentally obtained) coverage. Dependence of coherence length on fiber dispersion [see Eq. (3)] suggests that these three features can be engineered by tailoring the fiber’s dispersion—a manufacturing flexibility readily provided by PCFs. For example, during the nonlinear propagation in which FOCR acquires more bandwidth, large normal dispersion quickly stretches the FOCR pulse up to hundreds of fs within a couple of centimeters, which dramatically reduces its peak power, weakens the nonlinear effects (e.g. SPM), and thus diminish the spectral broadening. Apparently, such a pulse-stretching effect can be suppressed by reducing the amount of normal dispersion in the FOCR spectral region. Consequently, the strengthened nonlinear effects will enhance the spectral broadening process. Furthermore, a reduced normal dispersion increases the coherence length, implying that constructive addition between the existing FOCR and newly emitted radiation takes place within a longer propagation distance. As a result, more input energy transfers to the FOCR, giving rise to an even higher conversion efficiency, which in turn benefits FOCR’s spectral broadening.