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All-fiber coherent supercontinuum generation in a
cascade of silica, fluoride, and chalcogenide fibers

To cite this article: Md Hosne Mobarok Shamim et al 2024 *J. Phys. Photonics* **6** 045018

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PAPER

All-fiber coherent supercontinuum generation in a cascade of silica, fluoride, and chalcogenide fibers

Md Hosne Mobarak Shamim¹,* , Laurent Brillard², Radwan Chahal³, Johann Troles² and Martin Rochette¹

¹ Department of Electrical and Computer Engineering, McGill University, 3480 University Street, Montréal H3A 0E9, Canada

² CNRS, ISCR-UMR 6226, Université Rennes, F-35000 Rennes, France

* Author to whom any correspondence should be addressed.

E-mail: hosne.shamim@mail.mcgill.ca

**Keywords:** nonlinear optics, supercontinuum generation, mid infrared photonics, photonic crystal fibers

Abstract

We present an all-fiber coherent supercontinuum spanning the spectral range of 1.7–5.0 µm from a cascade of silica, ZBLAN, and chalcogenide (ChG) nonlinear fibers (NLFs). Coherence is maintained by the combined use of femtosecond pump pulses as well as by allowing deterministic spectral broadening mechanism at every stage of the cascade. The use of femtosecond pump pulses enables avoiding modulation instability (MI) at the onset of the supercontinuum generation process and thus prevent subsequent MI-seeded random noise. Once in the NLF cascade, the pump pulse is instead converted into a soliton of order maintained at *N* < 6 in the silica and ZBLAN NLFs, ensuring soliton fission followed by self-frequency shift of a few solitons. Finally, in the ChG NLF, spectral broadening is facilitated through self-phase modulation and dispersive wave generation. The deterministic nature of these nonlinear phenomena results in the generation of a coherent supercontinuum. The supercontinuum delivers an average power of 54 mW from an average pump power of 300 mW, yielding a power conversion efficiency of 18%. The experimental results closely align with numerical simulations, from which coherence is estimated. Such a coherent supercontinuum with a megahertz repetition rate is essential for spectroscopic systems based on optical frequency combs and applications in high-precision optical coherence tomography.

**1. Introduction**

All-fiber supercontinuum sources operating at mid-infrared (MIR) wavelengths (2–20 µm) are reaching
practical maturity thanks to their compact design, mechanical stability, and compatibility with harsh
environments [1, 2]. Of particular interest are the broadband supercontinuum sources that maintain a high
degree of coherence. A coherent supercontinuum plays a pivotal role in enabling spectroscopic systems based
on self-referenced optical frequency combs [3] as well as facilitating high-precision optical coherence
tomography [4].

Recent designs of all-fiber MIR supercontinuum sources make use of a cascade of nonlinear fibers (NLFs) of various materials, including silica, fluoride, and chalcogenide (ChG) for stepwise spectral extension towards longer wavelengths [5–12]. Cascaded supercontinuum generation typically begins with a commercially available kHz repetition rate pump laser operating in the C-band and emitting pulses with a duration ranging from picosecond to nanosecond and energy ranging from tens of nanojoule to tens of microjoules. The first NLF of the cascade provides anomalous dispersion combined with nonlinearity, where a pump pulse experiences noise-seeded modulation instability (MI) and decomposes into a broad collection of short and high peak power fundamental solitons [13]. This collection of solitons propagates to subsequent NLFs with increased nonlinearity and extended transparency toward long wavelengths. Depending on the dispersion regime, solitons entering the second and the subsequent NLFs undergo self-phase modulation (SPM), soliton fission, soliton self-frequency shift (SSFS), dispersive wave (DW) generation, and/or four-wave mixing. These nonlinear effects enrich the spectral content and combine into a supercontinuum.
Despite resulting in a broad supercontinuum, a limitation of this approach is the presence of intrinsic noise mainly seeded from MI that occurs in the first NLF [14]. MI induces random fluctuations in the amplitude and duration of solitons from one pump pulse to the next, which subsequently manifest as shot-to-shot wavelength, amplitude, and phase fluctuations. This ultimately leads to a noisy and incoherent supercontinuum.

With cascaded supercontinuum generation, attempts have been made to mitigate relative intensity noise (RIN) induced by MI. Hansen *et al* demonstrated a reduction in RIN by terminating a cascade of silica and ZBLAN NLFs with a short piece of normal dispersion ChG fiber [15]. The ChG fiber forced the spectrally distributed solitons to undergo SPM, resulting in spectral broadening and overlap of solitons to average out the noise. In another report, Kwarkye *et al* added a thulium-doped gain fiber into a similar cascade of silica and ZBLAN NLFs to minimize noise [16]. The interplay between the absorption-emission process and nonlinear soliton dynamics in the gain fiber contributes to noise reduction in the supercontinuum. In these approaches, MI remains the primary mechanism for initiating spectral broadening and continues to be a significant source of noise. Moreover, reducing the RIN via normal dispersion or gain medium to flatten the supercontinuum spectrum does not necessarily imply coherence improvement because of the random phase relation among solitons triggered by MI. The cascaded supercontinuum source should be designed without MI to achieve a high degree of coherence.

The most common approach for generating a coherent supercontinuum involves pumping an all-normal dispersion NLF with a femtosecond laser [17–20]. In this approach, spectral broadening primarily relies on deterministic and noise-free SPM and optical wave breaking. For instance, Hudson *et al* utilized 230 fs pump pulse from a mode-locked laser at a wavelength of 2.9 µm to generate a coherent supercontinuum spanning from 1.8–9.5 µm in an As₂Se–As₂S₃ tapered fiber [19]. In another example, Yuan *et al* employed a 150 fs pump pulse from an optical parametric amplifier at 5.0 µm to achieve a coherent supercontinuum spanning from 2.0–13.2 µm in a Ge₂₀As₂₀Se₂₀Te₄₀-As₂S₃ photonic crystal fiber (PCF) [20]. Alternatively, in an anomalous dispersion regime, coherence can be preserved through deterministic processes such as soliton fission, as opposed to random noise-seeded MI [21, 22]. This requires ensuring that the soliton fission length is shorter than the characteristic length of MI. In contrast with nanosecond-picosecond pulses, sub-picosecond pulses in an NLF typically result in lower soliton orders and fission lengths much shorter than the MI characteristic length. It has been observed that supercontinuum generation from solitons with an order of N < 10 maintains a high degree of coherence [23]. For example, Salem *et al* generated a coherent supercontinuum from 1.25–4.60 µm by launching a 150 fs pulse from an SSFS-based wavelength converter at 2.12 µm to an InF₃ fiber [21]. Coherent supercontinuum demonstrations in the MIR using these approaches rely on pump sources involving bulk optics and free space setups, which are less practical for applications outside of the laboratory environment.

In this paper, we present the first demonstration of a simplified all-fiber coherent supercontinuum in a cascade of NLFs. The cascade includes silica, ZBLAN, and ChG NLFs. Coherence is preserved in the NLFs through deterministic and noise free nonlinear processes such as soliton fission followed by SSFS in anomalous silica and ZBLAN NLFs as well as SPM and DW generation in the ChG NLF. The cascade is seeded by a femtosecond fiber laser that operates at a wavelength of 1.94 µm and delivers pulses with a repetition rate of 30 MHz. The output of the cascade is a coherent supercontinuum that spans over the wavelength range of 1.7 µm to 5.0 µm, achieving an output power of 54 mW and power conversion efficiency of 18%. A comprehensive numerical study provides insights about the supercontinuum evolution and estimates the coherence at each stage. Simulations match the experiment and predict a maximum average coherence of 0.97 across the supercontinuum spectrum.

## 2. Experimental section

Figure 1(a) shows the experimental setup used to demonstrate coherent supercontinuum generation in cascaded NLFs. Pulses from a mode-locked fiber laser (AdValue) at a wavelength of 1.94 µm, duration of 900 fs, repetition rate of 30 MHz, and average power of 7 mW, serve as seed for supercontinuum generation. The pulses pass through an inline polarization controller (Thorlabs) to maximize nonlinear interaction in the NLFs. An 8 m long normal dispersion fiber (Nufern, UHNA7) with group velocity dispersion (GVD) of β₂ = 0.043 ps²m⁻¹ at the pump wavelength compensates the chirp and pulse broadening induced by the anomalous standard single-mode fibers (SSMF) used prior to reaching the NLFs [24]. At the pump wavelength, the SSMF has a GVD of β₂ = −0.076 ps²m⁻¹ and waveguide nonlinearity of γ = 0.0008 W⁻¹m⁻¹. Pulses are then amplified by a thulium-doped fiber amplifier (TDFA) to reach an average power of 300 mW. The TDFA includes a 32 cm long TDF (Coractive DCF-TM-6/125) connected with wavelength division multiplexers (WDMs) at both ends. The amplifying fiber is bi-directionally
<figure><img src="image_3.png" /><figcaption>**Figure 1.** (a) Schematic of the supercontinuum generation setup, including a cascade of silica, ZBLAN, and chalcogenide NLFs. Arrows A, B, and C are probing points where spectra are analyzed. PC: Polarization controller, DCF: Dispersion compensating fiber, TDFA: Thulium doped fiber amplifier, SSMF: Standard single mode fiber, T: transmittance. (b) Geometry of the chalcogenide NLF. (c) Schematic of the tapered chalcogenide NLF.</figcaption></figure>

pumped with 2.2 W of C-band light from two erbium-doped fiber amplifiers through the WDMs. The second WDM transfers the amplified pulses to the cascade of NLFs.

The first NLF of the cascade is a 0.45 m long silica fiber (Coractive) with a high numerical aperture (NA) of 0.23 and core/cladding diameters of 6.5/125 µm. The silica NLF is fusion spliced with the SSMF with a transmittance of $T_1 = 85\%$. The second NLF is a 0.65 m long ZBLAN fiber (Le Verre Fluoré) with NA = 0.23 and core/cladding diameters of 6.5/125 µm. The silica and ZBLAN NLFs are aligned using a 5-axis translation stage (Luminos Photonics, I5000) and then permanently butt-coupled using index-matching gel and UV-cured epoxy (Thorlabs NOA81) with a transmittance of $T_2 = 91\%$. The butt-coupling process is presented in detail in [25]. The high transmittance is attributed to the identical mode field diameters of the NLFs and their low refractive index mismatch. The third NLF is a Ge₁₀As₂₂Se₆₈ ChG PCF (SelenOptics) with core/cladding diameters of 11/125 µm. Figure 1(b) shows the cross-section of the PCF, where the core is surrounded by air holes with a pitch (Λ) = 7.2 µm and a hole diameter (d) = 3.38 µm. The $d/Λ = 0.47$ allows single mode operation in the MIR [26]. The ChG NLF is tapered down to a diameter of 2.3 µm using a well-known heat-brush technique [27]. Figure 1(c) shows a schematic of the tapered ChG NLF, comprising two fiber sections with a length of $L_f = 3.0$ cm, surrounded by two transition sections of length $L_t = 2.0$ cm, symmetrically positioned at both ends of a wire section with a length of $L_w = 1$ cm. The ZBLAN and the ChG NLFs are also butt-coupled, leading to a transmittance of $T_3 = 80\%$. For ChG PCF, the epoxy penetrates only about 2 mm into the air holes before curing due to its high viscosity (300 times that of water), causing a negligible impact on light transmission. The positioning of the NLFs in the cascade allows stepwise spectral broadening towards longer wavelengths. The spectra are analyzed at probing locations A, B, and C, as indicated in figure 1(a), with a wavelength-insensitive thermal power sensor (Ophir Photonics) and an optical spectrum analyzer (OSA, Yokogawa AQ6377 and AQ6375). A 0.5 m long multimode As₂Se₃ fiber with negligible variation of transmittance in the desired spectral span is used for the measurement at probing point C.

Figure 2 shows the experimental spectra sampled at probing points A, B, and C. In the silica NLF, soliton fission occurs, resulting in a fundamental soliton that experiences SSFS towards longer wavelengths. As observed in figure 2(a), the fundamental soliton is red shifted and reaches a wavelength of 2.06 µm at the output of the silica NLF. The measured average power at this position (probing point A) is 217 mW, including the soliton that contains 85 mW. This soliton serves as the pump for the ZBLAN NLF, where soliton fission occurs again, followed by SSFS that extends to a wavelength of 2.77 µm, as shown in figure 2(b). At probing point B, the output of the ZBLAN NLF carries an average power of 143 mW, including the red shifted fundamental soliton that contains 45 mW. Upon injection into the ChG NLF, nonlinear interactions including SPM and DW generation unfold across the fiber, transition, and wire sections. Next section provides insights into the spectral broadening dynamics in the NLFs. At probing point C, a supercontinuum spanning the wavelength range of 1.7–5.0 µm at a –30 dB level is recorded. The supercontinuum exhibits an average power of 54 mW, resulting in a pump to supercontinuum power conversion efficiency of 18%. Due to the single mode operation of the ChG NLF, the supercontinuum is expected to provide good beam quality.

A preliminary stability test of the supercontinuum is performed where the supercontinuum spectra are recorded for 60 min, with measurements taken at 10 min intervals. Recording began 15 min after the system
<figure><img src="image_4.png" /><figcaption>Figure 2. Experimental spectra at the output of (a) silica NLF, (b) ZBLAN NLF, and (c) chalcogenide NLF.</figcaption></figure>

was powered on to allow it to stabilize and reach thermal equilibrium. The supercontinuum remains stable without any sign of degradation, showing that the fiber cascade made of silica, ZBLAN, and ChG (Ge₁₀As₂₂Se₆₈) makes a promising source design. However, it should be noted that the cascade does not use polarization maintaining fibers, which may lead to power fluctuations due to temperature-dependent polarization variations when tested over longer durations.

### 3. Numerical simulations

Numerical simulations are performed to gain insight into the dynamics of supercontinuum generation in the NLF cascade. For this purpose, the generalized nonlinear Schrodinger equation (GNLSE) is solved using split-step Fourier method (SSFM) with adaptive step size [28]. The GNLSE models the evolution of a slowly varying pulse envelope A(z, t) as it propagates in an optical fiber with a frame of reference moving with the group velocity of the pulse. The GNLSE is expressed as [29],

$$ \frac{\partial A}{\partial z} = -\frac{\alpha}{2}A + i \sum_{n=2}^{18} \frac{i^n \beta_n}{n!} \frac{\partial^n}{\partial t^n} A + i \left( \gamma_0 + \sum_{n=1}^{9} \frac{i^n \gamma_n}{n!} \frac{\partial^n}{\partial t^n} \right) \left( A \int_{0}^{\infty} R(t') |A(z, t-t')|^2 dt' \right). \quad (1) $$

Here, $\alpha$ is the wavelength dependent propagation loss. $\beta_n = \partial^n \beta / \partial \omega^n$ and $\gamma_n = \partial^n \gamma / \partial \omega^n$ are the Taylor coefficients of the wavelength dependent propagation constant ($\beta$) and waveguide nonlinearity ($\gamma$), respectively, evaluated at the center frequency $\omega_0$. $R(t) = (1-f_R)\delta(t) + f_R h_R(t)$ is the Raman response function, where $f_R$ is the fractional contribution to the delayed Raman response, $\delta(t)$ is the Dirac delta function, and $h_R(t)$ is the delayed Raman response function. $R(t)$ is normalized such that $\int_{-\infty}^{\infty} R(t) = 1$.

For SSFM, step size $h$ is crucial to avoid erroneous simulation results and must be much smaller than both dispersion and nonlinear lengths. In a nonlinear process like supercontinuum generation or soliton fission, a high-order pump pulse fissions into several short pulses with high peak power and short pulse duration. During the fission process, sudden changes in dispersion and nonlinear lengths occur. To ensure accurate results, an adaptive step size $h$ is necessary. The simulations indicate that the nonlinear length is a much more limiting factor than the dispersion length for the step size due to the high peak power attained within the solitons [8]. For a step size $h$, the maximum nonlinear phase shift is given by $\phi_{NLmax} = \gamma P_{0max} h$, where $P_{0max}$ represents the maximum peak power of the pulse. To ensure convergence to the solution, $h$ in each iteration is calculated as $h(i) = \phi_{NLmax} / \gamma P_{0max}(i)$. The optimal value for $\phi_{NLmax}$ is 1/50 rad, offering both accuracy and reduced computation time [8].

Figure 3(a) shows wavelength dependent transmission loss of silica, ZBLAN, and (un-tapered) ChG NLFs. For silica NLF, the transmission loss is directly measured whereas for the ZBLAN and ChG NLFs, the transmission loss is reproduced from the manufacturer's datasheet. Although the ChG NLF transmits well beyond 5 µm, the confinement loss in the wire section becomes significant after tapering, as shown in figure 3(b). The confinement loss is numerically determined from its effective refractive index [30]. The wavelength dependent effective refractive index is calculated through modal analysis of the fiber geometry using the finite element method with COMSOL Multiphysics software. Figure 3(c) shows GVD ($\beta_2$) of the silica and ZBLAN NLFs, calculated by solving the characteristic equation of the fundamental mode in a step-index optical fiber [31]. Zero-dispersion wavelength (ZDW) of silica and ZBLAN NLFs are at 1.20 µm
<figure><img src="image_5.png" /><figcaption>Figure 3. (a) Wavelength dependent transmission loss of silica, ZBLAN, and chalcogenide NLFs. (b) Wavelength dependent confinement loss of chalcogenide NLF wire section. (c) Wavelength dependent group velocity dispersion ($\beta_2$) of silica and ZBLAN NLFs. (d) Diameter and wavelength dependent $\beta_2$ of chalcogenide NLF. (e) Wavelength dependent nonlinearity ($\gamma$) of silica and ZBLAN NLFs. (f) Diameter and wavelength dependent $\gamma$ of chalcogenide NLF.</figcaption></figure>

and 1.90 µm, respectively. Figure 3(d) shows diameter and wavelength dependent GVD of the ChG NLF, calculated from the effective refractive index [18]. The fiber section of the ChG NLF exhibits a ZDW at 4.49 µm. In the transition section, the ZDW reduces gradually from 4.9 µm to 2.8 µm with associated diameter reduction from 11.0 µm to 3.2 µm. A second ZDW emerges at longer wavelengths when the diameter further reduces from 3.2 µm down to 2.4 µm. The wire section with a diameter of 2.3 µm exhibits all-normal dispersion across the spectrum with a minimum dispersion of 0.032 ps²m⁻¹ at a wavelength of 2.80 µm. Figure 3(e) shows the wavelength dependent nonlinearity of silica and ZBLAN NLFs. Figure 3(f) shows the diameter and wavelength dependent nonlinearity of the ChG NLF. While the nonlinearities of the silica and ZBLAN NLFs are comparable, the ChG NLF exhibits a nonlinearity that is three orders of magnitude higher.

The Raman response function, $h_R(t)$ of silica NLF is analytically modeled using single damped oscillator model [32, 33],

$$h_R(t) = (\tau_1^{-2} + \tau_2^{-2}) \tau_1 \exp\left(-\frac{t}{\tau_2}\right) \sin\left(\frac{t}{\tau_1}\right) \quad (2)$$

where, $\tau_1 = 12.2$ fs is the Raman delay and $\tau_2 = 32$ fs represents the damping time of silica molecules. The fractional Raman contribution for silica NLF is $f_R = 0.18$. As the Raman gain spectra for ZBLAN and ChG NLFs contain two dominant peaks, the intermediate broadening model is adapted for better accuracy [34, 35]. According to the intermediate broadening model, the Raman response function is expressed by [34].

$$h_R(t) = \sum_{j=1}^{m} A'_j \exp(-\sigma_j t) \exp\left(-\frac{\rho_j^2 t^2}{4}\right) \sin(\omega_{v,j} t). \quad (3)$$

Here, $A'_j$ is the peak amplitude, $\omega_{v,j}$ is the center frequency, $\sigma_j$ is the full width at half maximum (FWHM) of the Lorentzian, and $\rho_j$ is the FWHM of Gaussian for mode j. $j$ varies from 1 to m. For ZBLAN NLF, $f_R = 0.24$, and the parameters related to $h_R$ are tabulated in [35]. For ChG NLF, the Raman gain spectrum is reproduced from experimental results of [36] using eight ($m=8$) modes. The parameters for each mode are tabulated in table 1. For ChG NLF, $f_R = 0.03$ [36]. Figure 4 presents a good agreement in between the modeled Raman gain spectrum and the Raman gain spectrum obtained experimentally.

Figures 5(a) and (b) show the results of the simulated temporal and spectral supercontinuum evolution in the cascaded NLFs, respectively. In line with the experimental condition, a pump pulse with a FWHM duration of 900 fs and peak power of 10 kW is launched into the SSMF prior to propagation through the
<figure><img src="image_6.png" /><figcaption>**Figure 4.** Experimentally obtained Raman gain spectrum of Ge₁₀As₂₂Se₆₈ from [36] compared with numerically reproduced Raman gain spectrum using intermediate broadening model.</figcaption></figure>

<table><caption>**Table 1.** Parameters used in intermediate-broadening model<sup>a</sup>.</caption><thead><tr><th>Mode number, j</th><th>Component position (cm<sup>-1</sup>)</th><th>Peak intensity, A<sub>j</sub></th><th>Gaussian FWHM (cm<sup>-1</sup>)</th><th>Lorentzian FWHM (cm<sup>-1</sup>)</th></tr></thead><tbody><tr><td>1</td><td>135.80</td><td>7.62</td><td>40.91</td><td>16.36</td></tr><tr><td>2</td><td>191.30</td><td>9.20</td><td>20.99</td><td>8.40</td></tr><tr><td>3</td><td>199.30</td><td>13.50</td><td>14.42</td><td>5.77</td></tr><tr><td>4</td><td>212.84</td><td>1.62</td><td>10.45</td><td>4.18</td></tr><tr><td>5</td><td>220.79</td><td>5.43</td><td>19.30</td><td>7.72</td></tr><tr><td>6</td><td>236.40</td><td>35.22</td><td>34.86</td><td>13.94</td></tr><tr><td>7</td><td>264.13</td><td>20.90</td><td>29.34</td><td>11.74</td></tr><tr><td>8</td><td>299.58</td><td>2.53</td><td>31.04</td><td>12.42</td></tr></tbody></table>

NLF cascade. In SSMF, the input pulse exhibits a soliton order $N_{SSMF} = 5.7$. High-order soliton ($N > 1$) experiences temporal compression and spectral broadening because of the soliton compression effect [37]. After 0.50 m of propagation, the pulse reaches maximum compression and results in a duration of 178 fs and peak power of 43.2 kW. The temporal and spectral conditions at the output of the SSMF significantly influence the supercontinuum evolution in the subsequent NLFs [38, 39]. When coupled into a 45 cm long silica NLF, the pulse takes a soliton order $N_{silica} = 2.8$. In silica NLF, fission occurs after a propagation distance of 5 cm, giving rise to a fundamental soliton ($FS_{silica}$) that undergoes SSFS up to a wavelength of 2.06 µm, attaining a duration of 100 fs and a peak power of 43.8 kW. Upon coupling to the ZBLAN NLF, $FS_{silica}$ becomes a high order soliton with $N_{ZBLAN} = 5.5$. A second fission occurs in the ZBLAN NLF after a propagation distance of 4 cm and $FS_{ZBLAN}$ is generated. Under the influence of SSFS, $FS_{ZBLAN}$ extends up to a wavelength of 2.77 µm and attains a duration and peak power of 61 fs and 27.5 kW, respectively.

In the ChG NLF, supercontinuum evolution is divided into three sections: fiber section ($L_f = 3.0$ cm), transition section ($L_t = 2.0$ cm), and wire section ($L_w = 1$ cm). The fiber section with a core diameter of 11 µm exhibits a normal dispersion of $0.32 \text{ ps}^2\text{m}^{-1}$ at the $FS_{ZBLAN}$ wavelength of 2.77 µm. Upon entering the ChG fiber section, the $FS_{ZBLAN}$ experiences a nonlinear length of $L_{NL} = 1/\gamma P = 0.25$ mm. The submillimeter-scale nonlinear length enables strong SPM over a short propagation distance. The spectral broadening caused by SPM is asymmetric up to 0.15 cm, after which the spectral energy is distributed across multiple spectral bands. To identify these spectral bands, a separate numerical analysis is performed, isolating $FS_{ZBLAN}$ from the rest of the spectrum and propagating it through the fiber section of the ChG NLF. The simulation reveals that the energy of the $FS_{ZBLAN}$ is distributed across wavelength bands at 1.65 µm, 2.15 µm, and 3.50 µm. The spectral band at 3.50 µm, labeled 'A' in the figure, is the strongest with its wing extending

<footnote><sup>a</sup> A<sub>j</sub> = A'<sub>j</sub> / ω<sub>v,j</sub>, ρ<sub>j</sub> = πc× Gaussian FWHM, σ<sub>j</sub> = πc× Lorentzian FWHM, ω<sub>v,j</sub> = 2πc× Component position, and c= speed of light.</footnote>
<figure><img src="image_7.png" /><figcaption>**Figure 5.** Simulation results showing the evolution of the supercontinuum generation along the propagation distance of the NLFs. (a) Temporal evolution and (b) spectral evolution in silica, ZBLAN, and ChG NLFs. FS: fiber section; TS: transition section; WS: wire section; ZDW: zero-dispersion wavelength.</figcaption></figure>

up to 4.8 µm. In the temporal domain, this corresponds to ‘A’, which exhibits an oscillatory structure with deep modulations. This behavior is commonly observed in SPM, influenced by higher-order dispersion [40].

The transition section is simulated by discretizing its total length into 600 segments, each 33.3 µm long, and solving equation (1) independently for each segment. In this section, pulses experience varying dispersion and nonlinearity due to changes in fiber diameter along the propagation distance. The ZDW decreases as the propagation distance increases. At the beginning of the transition section, $A'$ undergoes temporal broadening, causing $A$ to lose its spectral wing. This results in a spectrum confined to wavelengths below 4.0 µm. As it progresses toward the middle of the transition section, $A'$ encounters anomalous dispersion and experiences temporal compression. At a propagation distance of 1.3 cm into the transition section, $A'$ reaches maximum compression and emits DWs in the 4.0–4.5 µm wavelength range.

The wire section exhibits all-normal dispersion, with its primary function being to level the power spectral density through SPM, resulting in a relatively flat supercontinuum. The fiber and transition sections at the output end of the ChG NLF contribute negligibly to spectral broadening due to their reduced nonlinearity compared to the wire section. Overall, the extension of the spectrum to longer wavelengths is limited by increasing confinement losses in the wire section (see figure 3(b)).

Figures 6(a)–(c) show a close match between simulation and experimental spectra from silica, ZBLAN, and ChG NLFs, respectively. Simulations also confirm the high level of power conversion efficiency demonstrated in the experiment. There are several factors associated with high efficiency. Firstly, the use of a femtosecond pump laser ensures a limited number of fundamental solitons in silica and ZBLAN NLFs, ensuring efficient power conversion. Secondly, exceptional coupling efficiencies between the NLFs foster a good power conversion process. Thirdly, the efficiency is not compromised by the need for a spectral filter, typically used to block content below 2 µm wavelength to save the ChG NLF from damage, thanks to the low power used for the supercontinuum generation. Lastly, the lengths of the NLFs are kept short to minimize power attenuation.
<figure><img src="image_8.png" /><figcaption>Figure 6. Experimental and numerical spectra of the supercontinuum output from (a) silica, (b) ZBLAN, and (c) ChG NLFs. Coherence spectra of the supercontinua from (d) silica, (e) ZBLAN, and (f) ChG NLFs.</figcaption></figure>

## 4. Coherence

Coherence of the supercontinuum is numerically estimated by adding quantum noise of one photon per mode (OPPM), $\delta_{QN}$ to the input pulse $A(0, t)$. $\delta_{QN}$ is modeled in the frequency domain by adding one photon of energy $\hbar\omega_m$ with random phase $\phi_m$ to each frequency bin with angular frequency $\omega_m$ and bin size $\Delta\Omega$ [41].

$$ \delta_{QN} = \sqrt{\frac{\hbar\omega_m}{\Delta\Omega}} \exp(i2\pi\phi_m). \qquad (4) $$

With OPPM, the initial pulse envelope becomes,

$$ A(0, t) = \sqrt{P_0 \text{sech}\left(\frac{t}{T_0}\right)} + F^{-1}\{\delta_{QN}\} \qquad (5) $$

where, $F^{-1}$ represents the inverse Fourier transform. As the output of the first NLF becomes the input of the second NLF, added noise in the initial pulse has a ripple effect on the coherence in the subsequent NLFs. Wavelength dependent coherence of the supercontinuum is determined by calculating the modulus of the first order coherence [42].

$$ |g_{12}^{(1)}(\lambda)| = \left| \frac{\langle E_1^*(\lambda) E_2(\lambda) \rangle}{\sqrt{\langle |E_1(\lambda)|^2 \rangle \langle |E_2(\lambda)|^2 \rangle}} \right|. \qquad (6) $$

Here, the angular brackets denote an ensemble average of a large set of, in this case, 150 independently generated supercontinuum spectra. $E_1$ and $E_2$ are complex spectral electric field envelopes of the supercontinuum pairs. Coherence is quantified from $0 < |g_{12}^{(1)}| < 1$, where $|g_{12}^{(1)}| = 1$ is the highest level of coherence and $|g_{12}^{(1)}| = 0$ is incoherent.

During supercontinuum evolution in the NLFs, coherence degradation primarily occurs due to noise seeded MI. Modulation sidebands amplify random quantum noise, causing non-deterministic fission of the
pump pulse and generating fundamental solitons with randomly fluctuating amplitudes and durations from one shot of pulse to the next. Subsequently, these random fluctuations manifest into a supercontinuum with spectral features slightly differing from one shot to another. On the contrary, soliton fission is a deterministic process that leads to the generation of shot-to-shot identical fundamental solitons, ensuring high coherence. The competition between the occurrence of these two nonlinear phenomena in the NLF can be analytically predicted based on their respective characteristic lengths, with the shorter characteristic length determining the dominant fission mechanism of the pump pulse. The soliton fission length is estimated from $L_{fiss} \sim L_D/N$, whereas the MI length is estimated from $L_{MI} \sim 16L_{NL}$, where $L_D = T_0^2/|\beta_2|$ [22]. Since $L_{fiss} \propto T_0^2$, femtosecond pulses enable soliton fission rather than MI. In this experiment, $L_{fiss} = 0.6$ m and $L_{MI} = 1.7$ m for SSMF. Consequently, in figures 5(a) and (b), soliton compression is evident, and modulation sidebands are suppressed in the SSMF. Figures 6(d)–(f) show the coherence after silica, ZBLAN, and ChG NLFs, calculated by adding $\delta_{QN}$ to the input pulse. In silica and ZBLAN NLFs, high coherence is maintained with $|g_{12}^{(1)}| > 0.99$ in the entire spectra. The coherence spectrum of the ChG NLF shows points of reduced coherence due to large variations of dispersion, nonlinearity, and transmission loss at different segments of the fiber taper. The overall average coherence of the supercontinuum is $\langle|g_{12}^{(1)}|\rangle = \int|g_{12}^{(1)}(\lambda)|\langle|E_1(\lambda)|^2\rangle d\lambda/\int|E_1(\lambda)|^2\rangle d\lambda = 0.97$.

Decoherence of a supercontinuum is also influenced by external factors such as shot-to-shot amplitude fluctuation of the pump pulse, commonly referred to as rms noise ($\delta_{AN}$) of the pump laser. To test the robustness of the proposed cascade design, $\delta_{AN} = 0.01\%$, $0.05\%$, and $0.1\%$ is added to the simulation. The amplitude fluctuations due to $\delta_{AN}$ also lead to duration fluctuations, which is estimated using equation (7) [43]

$$ \delta t_0 = -0.8(\delta_{AN} - 1). \qquad (7) $$

With both OPPM and rms noises included, the initial condition of the pulse envelope becomes,

$$ A(0, t) = \sqrt{P_0}\delta_{AN}\mathrm{sech}\left(\frac{t}{T_0(1 - \delta t_0)}\right) + F^{-1}\{\delta_{QN}\}. \qquad (8) $$

Figures 6(d)–(f) also show the coherence of the supercontinuum at the output of silica, ZBLAN, and ChG NLF, calculated with the addition of rms noise to the quantum noise ($\delta_{QN} + \delta_{AN}$). Nonlinear processes like SSFS, SPM, and DW generations are deterministic and have minimal impact on coherence under OPPM. However, their influence becomes more pronounced when rms noise is present. Fluctuations in peak power and pulse duration of the pump due to rms noise lead to fluctuations in the fundamental solitons through SSFS in silica and ZBLAN NLF. The impact of this directly influences SPM and DW generation in the ChG NLF, leading to coherence degradation. In the silica and ZBLAN NLFs, reduced coherence is noticeable where fine spectral features emerge, particularly at points where the fundamental solitons separate from the main pulse. In the ChG NLF, increasing rms noise deteriorates the coherence, especially at the longer wavelength spectral edge, due to the strong confinement loss in the ChG wire section [41]. Overall, the proposed cascaded design achieves an average coherence of 0.95, 0.82, and 0.67 when subjected to quantum noise and rms noise of 0.01%, 0.05%, and 0.10%, respectively.

The impact of the analytical Raman model on coherence has been thoroughly examined numerically. The numerical investigations indicate no significant difference in coherence properties whether the Raman gain profile is modeled using a single damped oscillator model or an intermediate broadening model. Coherence can be further enhanced by reducing the length of the NLFs while concurrently increasing the pump power to preserve the overall supercontinuum bandwidth [41].

## 5. Conclusion

In summary, we have demonstrated the first all-fiber coherent supercontinuum source generated in a cascade of silica, ZBLAN, and ChG NLFs, spanning over an octave in the wavelength range of 1.7–5.0 µm. The supercontinuum delivers an average power of 54 mW from a pump power of 300 mW, yielding a power conversion efficiency of 18%. The coherence of the supercontinuum is maintained across the three NLFs by suppressing MI at the onset of supercontinuum evolution, thanks to the femtosecond pump laser used in the experiment. With OPPM noise, the estimated overall average coherence of the supercontinuum reaches a value of 0.97. Inclusion of rms noise of 0.01%, 0.05%, and 0.10% into the simulation results in an average coherence of 0.95, 0.82, and 0.67. The high degree of coherence and compact design of this supercontinuum source are desirable for portable frequency comb and ultrafast spectroscopic systems.
## Data availability statement

The data cannot be made publicly available upon publication because they are not available in a format that is sufficiently accessible or reusable by other researchers. The data that support the findings of this study are available upon reasonable request from the authors.

## Acknowledgment

The authors are thankful to Coractive for the silica fiber, Le Verre Fluoré for ZBLAN fiber utilized in this work. We also thank Yokogawa and Electro-Meters for generously lending the AQ6377 optical spectrum analyzer. We acknowledge the funding from Natural Sciences and Engineering Research Council of Canada.

## ORCID iD

Md Hosne Mobarok Shamim https://orcid.org/0000-0002-5728-2665

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