1. Introduction

Supercontinuum generation (SCG) is intensively studied within current photonics research, especially from the advent of photonic crystal fiber (PCF) in 2000 [1–3], which allowed an unprecedented control of dispersion by tuning geometry [4] and/or material [5]. Currently supercontinuum (SC) sources are applied in a multitude of fields ranging from for instance metrology to advanced optical coherence tomography. In particular, SC sources with high degree of temporal coherence have revolutionized the field of time-resolved spectroscopy [6], frequency comb metrology [7], tomography [8] and biophotonic imaging [9] etc.

Particular in case of ultrashort optical pulses efficient SCG demands precise dispersion engineering of the underlying waveguide, allowing tailoring the light generation process via the properties of the waveguide. Here the novel concept of dispersion control via geometrically induced resonances was recently reported in gas-filled anti-resonant hollow-core fibers (AR-HCFs) [10], experimentally demonstrating multi-octave coherent SCG covering the ultraviolet, visible and near infrared spectral domains. This demonstration brought a new insight to the beneficial influence of resonances on SCG and thus can principally break through current dispersion management limitations of state-of-the-art waveguide designs. Before this demonstration, the majority of research focused on nonlinear pulse propagation in various transmission bands of all-solid photonic band gap structures [11], while the pump wavelengths were located relatively far away from the resonances, diminishing their influence on SCG.

Recently several studies on how resonances impact broadband light generation [12–14] revealed that resonances lead to new phase-matching (PM) opportunities for four-wave-mixing (FWM) and dispersive wave (DW) emission in the SCG process [12]. In particular, the resonances introduce dramatically changed dispersion within a very small frequency range resulting in a multitude of dispersion regimes with different sign of group velocity dispersion ${\beta _2}$. Due to this special dispersion property, two-colour DW emission in the ultraviolet (UV) region and at the resonance frequency was demonstrated in an argon-filled hypocycloid-core kagome fiber [13]. Moreover, a high level of control on FWM processes was presented in a gas-filled hollow-core PCF thanks to the specific dispersion profile [14].

These great advancements define a clear scientific case for understanding the fundamental physics of the impact of resonances on SCG. The goal of this work is, from simulation and modeling perspective, to present the relevant dependencies about how resonances affect modal dispersion and SCG by varying parameters such as spectral distance between pump and resonance wavelength or resonance properties. We organize this paper in the following way. Firstly, the numerical model used to analyze resonance-enhanced SCG is introduced. Specifically, a resonance is incorporated into the generalized nonlinear Schrödinger equation (GNLSE) by including a Lorentzian-type dispersion term. The impact of resonance on SCG is revealed by comparing the power-spectral pulse evolutions of a capillary waveguide (without resonance) with the counterpart that includes a resonance. Afterwards, to understand the fundamental physics behind the resonance-enhanced SCG, the PM conditions of DW generation and FWM are discussed in great detail. Lastly, the influence of the spectral distance between pump and resonance wavelength, ${\lambda _p}$ and ${\lambda _R}$, (defined as ${\Delta }\lambda = {\lambda _R} - {\lambda _p}$) and the impact of the resonance amplitude on SCG are discussed together with an approximate model that includes analytic equations for DW phase-mismatch and thus DW frequency.

2. Modeling of pulse propagation

2.1 Dispersion model

The effect of resonances is incorporated into the GNLSE by re-designing the dispersion term. Since most investigations on resonance-enhanced SCG experiments are conducted in gas-filled AR-HCFs, we consider the gas-filled hollow-capillary waveguide model (Fig. 1(a)) [15,16] in our simulation. To account for the strongly dispersive behavior of the resonances anticipated, a Lorentzian-type response function [12] (last term on the right-handed side of Eq. (1)) is introduced, which strongly modifies the effective index of the waveguide mode.

 

Fig. 1. (a) Sketch of hollow-capillary waveguide. Spectral distribution of (b) phase indices and (c) group velocity dispersion in case the waveguide includes (solid red) or excludes (dashed blue) one resonance (simulation parameters are given in the main text). Colour bar represents different dispersion regimes (gray: normal dispersion; orange: anomalous dispersion). The two green dots in (c) represent the zero-dispersion frequencies of AR-HCF model.

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Here ${\textrm{A}_\textrm{q}}$ and ${\textrm{B}_\textrm{q}}$ are the amplitude and damping of the ${\textrm{q}^{\textrm{th}}}$-resonance centered at the resonance frequency ${{\omega }_\textrm{q}}$, ${\textrm{n}_\textrm{g}}$ is the pressure- and temperature-dependent refractive index of the gas, ${\omega }$ is the frequency, $\textrm{R}$ is the core radius, ${\textrm{u}_{11}}$ is the first root of the zero-order Bessel function (${\textrm{u}_{11}} \approx 2.405$) and $\textrm{c}$ is the vacuum speed of light. The real part of Eq. (1) represents the resonance-induced modification of effective refractive index, and the imaginary part embodies the loss/gain introduced by the resonance. The employment of a Lorentzian-type function relates to the fact that leaky modes show sharp peaks in the modal attenuation spectrum at each resonance frequency. According to the Kramers-Kronig relation [17] such maxima in the loss distribution are related to characteristic shapes of the dispersion curves, which mathematically can be described by a Lorentzian-type function of the form of the right-handed term of Eq. (1). In the work of M. Bache et al. [18] the authors showed that using a Lorentzian-type function is a valid approximation for the dispersion of the fundamental mode of single-ring anti-resonant hollow core fiber. Specifically, the authors compared Lorentzian-type dispersion curves with results from full numerical modelling, leading to an overall good agreement across the entire spectral domain of interest. It should be noted that Lorentzian-type response functions can also be applied to other fibers and waveguides that include resonances such as all-solid band gap fibers [19], light cages [20,21], hollow core band gap fibers [22], directional mode coupling fibers (modified step index fiber) [23] etc. Technically, Eq. (1) can be fitted to the spectral distribution of the effective index which is obtained from numerical simulations in order to extract the resonance parameters. Due to its correlation to the gas-filled hollow core fibers mentioned above, we refer the model that is associated with Eq. (1) as AR-HCF model in the following.

To clearly demonstrate the impact of resonances on fiber dispersion, the spectral distributions of phase index ${n_p}$ and group velocity dispersion ${\beta _2}$ using capillary waveguide (dashed blue) and AR-HCF model (solid red) for one example configuration are plotted in Figs. 1(b) and 1(c). Here, capillary and AR-HCF have a core radius of $R = 25$ µm and are filled with Krypton at a pressure of $p = 6\; \textrm{bar}$. The refractive index of Krypton is calculated using the Sellmeier-type equation [24]. Here we assume one resonance centered at ${\omega _1} = 2\pi \cdot 0.32\; \textrm{PHz}$ ($\lambda = 930\; \textrm{nm}$) with an amplitude of ${A_1} = 2.5\; \textrm{THz}$ and a damping coefficient of ${B_1} = 10\; \textrm{THz}$. The influence of the resonance is already highly visible for the phase index showing distinct change of ${n_p}$ at the resonance frequency (Fig. 1(b)). This phase modification causes a dramatically changed dispersion around the resonance frequency, which yields in total two normal-dispersion (ND, grey bar) and two anomalous-dispersion (AD) regimes (orange bar in Fig. 1) together with two zero-dispersion frequencies locating both sides of the resonance frequency, in contrast to one ND and one AD regimes and one zero-dispersion frequency for the resonance-free case. Note that for the latter case having zero ${\beta _2}$ is a result of the material dispersion of the gas in the core. It is worth mentioning that this dispersion property is totally different from that of the resonance-free fiber (red solid and blue dashed lines in Figs. 1(b) and 1(c)) leading to new and distinctive characteristics in SCG process. We would like to highlight that for unambiguously unlocking the underlying physics of resonance-enhanced SCG, only one resonance is included in simulation, despite there are always several resonance regions in the transmission spectrum of AR-HCFs (spectral distance between resonance is mainly determined by the thickness of the glass strands surrounding the central core [25,26]). Preliminary simulations of SCG in a system with two spectrally well-separated resonances (${\omega _1} = 2\pi \cdot 0.32\; \textrm{PHz}$ (930 nm), ${A_1} = 2.5\; \textrm{THz}$, ${B_1} = 10\; \textrm{THz}$ and ${\omega _2} = 2\pi \cdot 0.60\; \textrm{PHz}$ (500 nm), ${A_2} = 1.8\; \textrm{THz}$, ${B_2} = 10\; \textrm{THz}$) shows that there is no significant difference from SCG with one resonance with the parameters (${\omega _1}$, ${A_1}$, ${B_1}$) mentioned above in case the 80 fs pump pulse is centered at 800 nm (pulse energy: 12 µJ).

2.2 Nonlinear pulse propagation equation

The optical pulse propagation in the resonance-enhanced fiber is numerically calculated using the GNLSE that accounts for the full dispersion at any frequency $\beta = \beta (\omega )$ defined in Eq. (1), leading to

3. Results and discussion

3.1 Impact of resonance dispersion on SCG

To demonstrate the impact of a resonance on SCG, Fig. 2 compares the energy/spectral evolutions with (Figs. 2(a) and 2(b)) and without resonance (Figs. 2(c) and 2(d)) at two relative pump wavelengths ${\Delta }\lambda $ (Figs. 2(a) and 2(c): 230 nm, Figs. 2(b) and 2(d): 80 nm). The parameters of the resonance are the same as Fig. 1. The pump pulse has a “sech” shape with full width at half maximum (FWHM) duration of 80 fs. The spectral evolutions clearly suggest that the resonance modifies the nonlinear pulse evolution substantially and in particular leads to an increased spectral bandwidth at highest pulse energy. As visible in Figs. 2(a) and 2(b), regardless of the pump pulse being located in the AD or ND regime, blue- and red-shifting radiation at the wings of SC spectra that is not present in the resonance-free situation is observed, which is a result of the multiple PM opportunities regarding FWM and DW emission introduced by the resonance. In contrast, the capillary case shows only slight self-phase modulation (SPM) related broadening when the pump pulse is located in the ND regime (Fig. 2(c)). A larger broadening is observed when pumping in the AD domain (Fig. 2(d)), while compared to the resonance-enhanced case (Fig. 2(b)) this broadening is significantly smaller (incl. res.: 1720 nm, no res.: 370 nm bandwidth at power level −20 dB, shown in Table 1) and is observed only at massively higher soliton numbers (SC onset soliton numbers: incl. res.: 4, no res.: 40). As shown in the following the larger spectral broadening in case the resonance is introduced can be attributed to the strongly changed dispersion landscape imposing (i) new DWs at different spectral locations, (ii) large spectral distances between DWs and solitons and (iii) new phase-matching opportunities for FWM.

 

Fig. 2. Simulated energy/spectral evolution of SCG process with ((a) and (b)) and without ((c) and (d)) resonance. The left column refers to a pump wavelength of $700\;\textrm{nm}$, the right to $850\; \textrm{nm}$. The black dotted lines represent the zero-dispersion frequencies, the magenta dashed lines the spectral location of the resonance. Corresponding dispersion domains are indicated on the top of each plot. The two plots on the right include the respective soliton numbers as additional $\textrm{y}$-axis scale.

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Table 1. Parameters and results of the configuration simulated in Fig. 2

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3.2 Mechanism analysis

The comparably low soliton numbers involved in the resonance-enhanced situation presented in Fig. 2(b) suggest that the SCG processes relevant here is coherent (coherence criterion $N < 16$ defined by Dudley et al. [28]) and is a result of the fission of a higher-order soliton into its fundamental counterparts, which is associated with the emission of DWs. In contrast the resonance-free case shown in Fig. 2(d) involves substantially higher soliton numbers beyond the mentioned limit, suggesting that noise-driven modulation instability is the origin of the process. DWs are most efficiently generated at those wavelengths the associated PM condition is fulfilled, allowing the generated fundamental solitons to release excess energy to these wavelengths. The DW phase mismatch is given by [29]:

Another important parametric process that is additionally relevant here is FWM. For degenerated FWM case, in which two pump photons at frequency ${\omega _0}$ generate a frequency-upshifted (signal, ${\omega _s}$) and a frequency-downshifted (idler, ${\omega _i}$) photon satisfies $\Delta \omega = 2{\omega _0} - {\omega _s} - {\omega _i} = 0$ and

To uncover the fundamental physical mechanism of resonance-enhanced SCG, Fig. 3 shows the spectral pulse evolution along the fiber length at three different soliton numbers ($N = 7,\; 10$ and $15$, all within the coherence limit ($N < 16$)) together with the phase-mismatching curves of DW (red) and degenerated FWM (black) calculated using Eqs. (3)–(4). In these simulations, a pump pulse with 80 fs duration at 800 nm ($\omega = 2\pi \cdot 0.37\; \textrm{PHz}$) and a fiber of fixed length (50 nm) with the same parameters as Fig. 1 are considered. The value of ${\beta _2}$ at the pump frequency is $- 5.6\; \textrm{p}{\textrm{s}^2}/\textrm{cm}$, meaning that this configuration represents the AD situation.

 

Fig. 3. Simulated pulse propagation (bottom row) and phase-mismatching rate (red: DW, black: FWM) for different soliton numbers ((a)${\textrm{N}_\textrm{s}} = 7$, (b) ${\textrm{N}_\textrm{s}} = 10$, (c) ${\textrm{N}_\textrm{s}} = 15$). The 80 fs pump pulse is centered at 800 nm and the resonance at 930 nm. In the bottom plots, the black dotted lines represent the zero-dispersion frequencies, the magenta dashed lines the resonance frequencies.

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As the pump pulse is located in the AD region, soliton-related propagation effects dominate the SCG process. The fission lengths are 55cm, 37 cm and 24 cm for the three cases, calculated by the approximate equation ${L_{fission}} \approx {L_D}/N$ (dispersion length ${L_D} = \; T_0^2/|{{\beta_2}} |$) [29] which roughly matches the lengths where DWs are found in simulations (50 cm, 35 cm and 22 cm, respectively). Note that even for soliton number $N = 7$ (Fig. 3(a)), the overall fiber length (50 cm) is shorter than the approximate soliton fission length (55 cm) and a weak DW located at $\omega = 2\pi \cdot 0.7\; \textrm{PHz}$ is still seen at maximal propagation length. This DW strongly blue shifts with increasing soliton number as a result of the increasing contribution of the nonlinear phase to the phase-mismatch condition (Eq. (3)). Besides, we can see there are three DWs which satisfy the DW PM condition: one locates in the vicinity of the resonance frequency and the other two locate at both sides of the resonance frequency (ultraviolet and near-infrared regions). In particular, one DW locates in the AD domain. The energy transferred to this DW actually forms a new soliton, an effect being referred as spectral tunneling effect [31]. It is particularly interesting to see the correlation between soliton number and frequencies of DWs (Fig. 4) showing an opposite spectral tuning behavior of the high- and low-frequency DWs, which we attribute to the special dispersion profile. In contrast, the DW locating in the vicinity of the resonance frequency is quite stably located at the edge of the fundamental transmission band regardless of increasing input pulse energy [13].

 

Fig. 4. Dependence of frequencies of the generated DWs as functions of soliton number, determined from the energy/spectral evolutions using the example configurations shown in Fig. 3. The black dotted lines represent the zero-dispersion frequencies, the magenta dashed line the resonance frequency and the red dash-dotted line the pump frequency.

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Apart from DW emission, FWM plays a vital part in the SCG process. For soliton number ${N_s} = 7$ (Fig. 3(a)), broadening of the spectra due to SPM effectively provides energy at a signal frequency (0.44 PHz in Fig. 3(a)), leading to energy transfer of the pump to the idler (0.29 PHz in Fig. 3(a)). Because of the unique dispersion profile, another pair of signal (0.6 PHz in Fig. 3(a)) and idler (0.19 PHz in Fig. 3(a)) is generated, which is consistent with the phase-mismatching curve of FWM (black line in Fig. 3(a)). Subsequently, cascaded FWM contributes to the middle part (0.2-0.6 PHz) of the broadband spectrum in the form of multiple peaks separated by ${\Delta }\omega = 2\pi \cdot 60\; \textrm{THz}$, which can be seen in Fig. 3(c) (soliton number ${N_s} = 15$) before the DW generation (up to a length of $z = 15\textrm{cm}$).

To visualize the nonlinear dynamics, spectrograms at different locations in the fiber (Fig. 5) are plotted using cross-correlation frequency-resolved optical gating (16 fs Gaussian gate pulse, conditions as those of Fig. 3(c)). Up to a distance of 7 cm, only SPM-induced spectral broadening occurs causing slight temporal pulse compression and a positive nonlinear chirp across the pulse. At a length of 10 cm, one of the spectral lobes reaches the ND regime (particularly 0.44 PHz), acting as the signal of FWM, an idler is generated on the low-frequency (red) side of the pulse spectrum at around 0.29 PHz. Consequently, more and more spectral lobes appear at the blue side of the pump with increasing propagation distance due to cascaded FWM exhibiting a constant frequency interval ${\Delta }\omega = 2\pi \cdot 60\; \textrm{THz}$.

 

Fig. 5. Spectrograms at selected propagation distances inside the resonance-enhanced fiber calculated for the conditions of Fig. 3(c) (black dotted lines: zero-dispersion frequencies, magenta dashed line: resonance). The parabolic black dashed curve shows changes in group index with frequency (known as $\textrm{c}/{\textrm{v}_\textrm{g}}$).

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The most striking feature of Fig. 5 is three-color DW emission generated during the nonlinear pulse propagation as a result of the unique dispersion profile introduced by the resonance. At the length of 15 cm, the spectrum is greatly broadened thanks to cascaded FWM and extends to the frequencies that satisfy the DW PM condition [32]. At a distance of 22 cm, which corresponds to the theoretically approximated soliton fission length (24 cm), the higher-order soliton undergoes fission due to the perturbations of higher-order dispersion terms, leading to the sudden emergence of new spectral peaks (DWs) on the high-frequency (1 PHz) and low-frequency (0.2 PHz) sides of the pulse spectrum and at the resonance frequency (0.32 PHz). The DW PM condition (Eq. (3)) yields the spectral locations of the DWs (Fig. 3(c), 0.2 PHz, 0.32 PHz, and 1 PHz), which are in good agreement with those of the presented spectrograms. As mentioned above the DW at 0.2 PHz is located within the AD regime, leading to the formation of a soliton via spectral tunneling [31]. As the low-frequency DW is located spectrally closer to the initial soliton and has larger spectral seed energy, it experiences higher energy-transfer efficiency than its high-frequency counterparts [33].

Note that even though solitons and DWs are spectrally separated, they overlap in the time domain after the fission process (propagation distance 22 cm, Fig. 5). This temporal overlap allows solitons and associated DWs to interact through cross-phase modulation (XPM) [34] and FWM [35], with the two excitations being locked together as a result of the more-intense excitation (which in the present case is the soliton) trapping its counterpart [29]. After trapping, both excitations propagate together at the same group velocity (dashed black line, propagation distance 50 cm, Fig. 5).

3.3 Impact of pump wavelength on SCG

It is worth noting that the spectral distance between pump and resonance ${\Delta \lambda}$ plays a crucial role in the resonance-enhanced SCG process. This behavior is immediately evident from Fig. 6, which shows the simulated energy/spectral SC evolution for different values of ${\Delta }\lambda $, while the resonance wavelength is fixed to 930 nm. When the pump wavelength gradually comes closer to the resonance wavelength (which also means pumping further away from the zero-dispersion wavelength (ZDW)), the spectral broadening gradually increases, yet until a certain value of ${\Delta \lambda }$ at which it dramatically decreases and eventually stops. This behavior is unique and entirely different from resonance-free fibers where an increasing spectral distance between pump and ZDW imposes the generated SC spectrum to spectrally narrow.

 

Fig. 6. Simulated energy/spectral SC evolution for six different situations of relative spectral distance between pump and resonance wavelength (resonance wavelength: 930 nm, ${\textrm{A}_1} = 2.5\; \textrm{THz}$, ${\textrm{B}_1} = 10\; \textrm{THz}$, input pulse: 80 fs). The right axis shows the initial soliton number associated with the input pulse energy on the respective left axis. The black dotted lines represent the zero-dispersion frequencies, the magenta dashed lines the resonance frequencies.

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The dramatic impact of the spectral distance on SC is particular evident when plotting the generated spectral bandwidth as function of ${\Delta }\lambda $ (Fig. 7) showing a clear maximum, which in the present case is ${\Delta }\lambda = 40\; \textrm{nm}$, whereas the corresponding minimum of the SC onset energy is located at a different value (${\Delta }\lambda = 130\; \textrm{nm}$). This plot essentially reveals that the resonance has a sophisticated and non-obvious impact on nonlinear pulse propagation, thus motivating the study performed in the present work.

 

Fig. 7. Spectral SC bandwidth (at −20 dB level, pump energy 12 µJ) and SC onset energy versus spectral distance between the pump and the resonance wavelength (resonance wavelength: 930 nm, ${\textrm{A}_1} = 2.5\; \textrm{THz}$, ${\textrm{B}_1} = 10\; \textrm{THz}$). The input pulse is located at 800 nm (pulse duration: 80 fs).

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To interpret the observed phenomena, some key nonlinear and linear parameters of all the cases considered in Fig. 6 are shown in Table 2. It is evident that the value of ${\beta _2}$ becomes much larger when the pump wavelength increases (i.e., gets closer to the resonance wavelength) as a result of the impact of the resonance. This dramatic increase of ${\beta _2}$ by orders of magnitude within a spectral range of only 20 nm (Figs. 6(d)–6(f)) yields a rapid reduction of the soliton number, which explains why the broadening decreases when the pump wavelength gets too close to the resonance. On the other hand, the initial increase of SC bandwidth (Figs. 6(a)–6(c)) can be explained by DW PM that is highly impacted by the dispersion of the resonance.

Table 2. Various parameters of the simulations shown in Fig. 6 (resonance wavelength: 930 nm). The third order dispersion length is given by ${\boldsymbol L}_{\boldsymbol D}^{\boldsymbol \prime} = {\boldsymbol T}_0^3/|{{{\boldsymbol \beta }_3}} |$

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Moreover, it is important to mention that for increasing pump wavelengths (i.e. reducing ${\Delta }\lambda $), the coherence of the generated SC spectra improves as a result of the decreasing soliton number (coherence limit $N < 16$ as defined in [28]). Specifically, the soliton number reduces below 10 for the pump wavelength $> 850\; \textrm{nm}$ (Fig. 6(c)), which ensures a high level of SC coherence [28] where the soliton fission length is significantly smaller compared to the distance over which modulation instability has amplified any input noise to a significant level.

As the resonance-enhanced fiber discussed here shows two AD and two ND regimes at opposite sides of the resonance, it is worth comparing the soliton dynamics for pumping within either of the AD domains. Figure 8 shows the SC evolution with pump energy when pumping at 800 nm (Fig. 8(a), ${\Delta }\lambda > 0$) and 1430 nm (Fig. 8(b), ${\Delta \lambda } < 0$) while the resonance wavelength is fixed at 930 nm with amplitude ${A_1} = 2.5\; \textrm{THz}$ and damping ${B_1} = 10\; \textrm{THz}$. Very different nonlinear dynamics are observed: A strongly blue-shifted DW emission in UV region is observed for the situation of ${\Delta }\lambda < 0$ (Fig. 8(b)) together with a spectral gap of low spectral density between the resonance and the UV DW. The complementary situation (${\Delta }\lambda > 0$, Fig. 8(a)) which is addressed in this work, shows a more homogenous energy distribution particularly at those wavelengths the ${\Delta }\lambda < 0$ case exhibits low spectral densities with the help of cascaded FWM. Together with those of Fig. 6, these results therefore clearly emphasize the importance of choosing the correct configuration to achieve sufficient spectral densities in the desired spectral domain.

 

Fig. 8. Simulated energy/spectral SC evolution for the resonance-enhanced fiber situation when pumping within (a) the short-wavelength (${{\lambda }_\textrm{p}} = 800\; \textrm{nm}$, ${\Delta \lambda } > 0$) and (b) the long-wavelength (${{\lambda }_\textrm{p}} = 1430\textrm{nm}$, ${\Delta \lambda } < 0$) anomalous-dispersion region. The resonance is fixed to $930\; \textrm{nm}$ with amplitude ${\textrm{A}_1} = 2.5\; \textrm{THz}$ and damping ${\textrm{B}_1} = 10\; \textrm{THz}$. The input pulse has a FWHM of $80\; \textrm{fs}$. The black dotted lines represent the zero-dispersion frequencies, the magenta dashed lines the resonance frequencies.

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3.4 Impact of resonance amplitude on SCG

Since the resonance term in the modal dispersion (Eq. (1)) is highly influenced by the parameters of the resonance anticipated, we calculated the group velocity dispersion (Fig. 9(a)) and the energy/spectral evolution of the SC process for three different resonance amplitude cases (Figs. 9(b)–9(d)). It is worth noting that not only the value of ${{\beta }_2}$ at the resonance frequency is greatly enhanced (inset of Fig. 9(a)), but the zero-dispersion frequencies shift away from the resonance through increasing the value of the resonance amplitude in Eq. (1). Regardless of resonance amplitude, the pulse qualitatively undergoes the same physical processes as demonstrated in Sec. 3.2, including SPM, cascaded FWM, soliton fission and DW emission, while the respective parameter combination (Table 3) leads to differently shaped output spectra. From our perspective the most prominent feature that changes for the different resonance amplitudes is that the UV DW emission gets more blue-shifted when the resonance amplitude is increased.

 

Fig. 9. Impact of resonance amplitude ${\textrm{A}_1}$ on SCG. Calculated spectral distribution of the group velocity dispersion (a) and simulated energy/spectral SC evolution for three different resonance amplitude cases ((b-d), indicated by the label in the lower right corner of each plot) with the same damping ($\textrm{B}\, = \,10\; \textrm{THz}$)). Inset in (a) shows the ${{\beta }_2}$/${\omega }$ dependence on a larger scale. The black dotted lines in (b)-(d) represent the zero-dispersion frequencies, the magenta dashed lines the resonance frequencies.

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Table 3. Parameters of the three configurations in Fig. 9 with different resonance amplitudes (pump: 800 nm)

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The impact of the resonance damping parameter B on the SCG process has also been studied in simulation (not shown here). Compared to the resonance amplitude, however, the impact of B on the dispersion properties is rather insignificant, meaning that the damping of a resonance has less impact on SCG. For instance, in case the damping parameter ${B_1}$ is changed from $5\; \textrm{THz}$ to $18\; \textrm{THz}$, while keeping the resonance amplitude ${A_1}$ fixed to $2.5\; \textrm{THz}$ (resonance wavelength: $930\; \textrm{nm}$, pump pulse: $80\; \textrm{fs}$, center pulse wavelength: $800\; \textrm{nm}$, pulse energy: 12 µJ.

4. Analytic model to approximate DW frequency

The mathematical form of Eq. (1) suggests that it is basically possible to derive an analytical expression for the spectral location of DW in case the gas dispersion is given by an analytic expression. To qualitatively reveal the impact of one resonance on the DW frequency in a resonance-enhanced waveguide within a first order approximation, we assume that the gas has a constant refractive index of unity (${\textrm{n}_\textrm{g}} = 1$) and that no nonlinear phase is present (${\textrm{P}_0} = 0$). Inserting the relation for the phase index (Eq. (1)) into the DW phase mismatch equation (Eq. (3)) and dropping one insignificant resonance-type term yields a fully analytic DW phase-mismatch equation:

 

Fig. 10. (a) Spectral distribution of the DW phase mismatch within the scope of the approximation defined in the main text (green: numerical solution incl. a non-zero damping parameter, dashed blue: analytic approximation using Eq. (5)). The curves refer to the following parameters: ${\textrm{n}_\textrm{g}} = 1$, ${{\lambda }_{\textrm{res}}} = 0.93$ µm, ${{\lambda }_{\textrm{sol}}} = 0.8$ µm, ${\textrm{B}_1} = 10\; \textrm{THz}$. The different brightness of the colour of the curves refer to different values of ${\textrm{A}_1}$ (from bright to dark: $0.5\; \textrm{THz}$, $1\; \textrm{THz}$, $1.5\; \textrm{THz}$, $2\; \textrm{THz}$, $2.5\; \textrm{THz}$, $3\; \textrm{THz}$). The yellow dots indicate the spectral location of the corresponding DW calculated using Eq. (6). (b) Normalized DW frequency as a function of resonance amplitude (using Eq. (6)) for various spectral separations of soliton and resonance ${\Delta }{{\lambda }_{\textrm{rs}}} = {{\lambda }_{\textrm{sol}}} - {{\lambda }_{\textrm{res}}}$ (from blue to red: $- 330\; \textrm{nm}$, $- 280\; \textrm{nm}$, $- 230\; \textrm{nm}$, $- 180\; \textrm{nm}$, $- 130\; \textrm{nm}$, $- 80\; \textrm{nm}$, $- 30\; \textrm{nm}$).

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Using the PM condition (${\Delta }{\beta _{DW}} = 0$) the normalized DW-frequency $\omega _{DW}^{norm} = {\omega _{DW}}/{\omega _1}$ can be calculated by finding the root of the right handed term of Eq. (5), leading to

An interesting scientific question is whether the inclusion of gases with Raman response will lead to qualitatively different nonlinear dynamics. Here we would like to highlight that the main characteristics of the spectra shown here can be attributed to the unique dispersion landscape provided by the introduction of a strong resonance. Therefore a substantial change of the spectral locations of DWs and FWM peaks can at least to a first order approximation be neglected in case the Raman effect is included, as the corresponding phase-matching conditions (Eqs. (3)–(4)) remain unchanged. Here it is important to note that in contrast to glass systems, the Raman response of gases is relatively weak and the corresponding spectrum consists of a cascade of narrow lines well distant from the excitation source. Thus, coherent excitation of the Raman lines is therefore not straightforward, suggesting that the specific supercontinuum characteristics simulated here will not change drastically.

5. Conclusions

Soliton-based ultrafast SCG represents one of the most efficient pathways towards distributing electromagnetic energy across desired spectral domains and highly depends on the dispersion of the underlying waveguide. On the example of a Krypton-filled resonance-enhanced anti-resonant hollow core fiber, we reveal the fundamental dependencies of the temporal/spectral evolution of femtosecond pulses and in particular of the SCG process in case the dispersion of the fiber is modified by a strongly dispersive resonance. A unique dispersion landscape is obtained by including a Lorentzian-type dispersion term leading to novel soliton dynamics that are presented here by solving the generalized nonlinear Schrödinger equation. Using phase-matching considerations and parameter studies, features such as three-color DW emission on both sides of the pump and at the resonance frequency are revealed, whereas in the vicinity of the resonance cascaded FWM is identified to significantly contribute to the flatness of the SC spectra. In-depth studies of the impact of the resonance parameters on the SCG process show that the spectral distance between pump and resonance is the essential parameter to obtain broadband spectra with spectral bandwidths greatly exceeding those of the corresponding non-resonant fibers. Moreover, we develop an approximate model that yields analytic equations for both DW phase-matching condition and DW frequency allowing for qualitatively unlocking the fundamental dependencies of DW formation in resonance-enhanced waveguides.

The inclusion of a resonance into modal dispersion has appealing implications: Towards the resonance group velocity dispersion increases by orders of magnitude, imposing the corresponding soliton numbers to drop below the coherence limit, with the consequence that the nonlinear spectral broadening processes shows improved pulse-to-pulse stability. Our simulations clearly indicate that at exceptionally low soliton numbers comparably flat and broadband output spectra can be achieved, which will be the target of a future investigation. Moreover, in case the resonance properties can be modified by means of an external influence, a real-time tunable supercontinuum source can be straightforwardly anticipated. Due to the generic character of the dispersion equation, our findings are not restricted to a particular fiber or waveguide geometry or a specific spectral domain but it is applicable to any waveguide system with resonant elements, making the concept of resonance-induced dispersion tuning a general scheme for broadband light generation.