1. Introduction

Supercontinuum (SC) generation operating in the mid-infrared (mid-IR) has attracted considerable attention because of its applications in medicine, food production, molecular fingerprint spectroscopy, optical coherent tomography, and optical sensing. Recently, broadband sources in the mid-IR (>2.5 µm) have received a lot of interest for on-chip sensing of molecules. In the spectral range, molecules have fundamental aspects of rotational-vibrational spectra, which are important for health and environmental sensing. Sensing devices with a mid-IR source can detect them with high sensitivity, owing to the strong photon-molecule interaction. The SC in the mid-IR has become a focus for research in performance, footprint, and low cost for integrated optical on-chip solutions [1–3]. The efficient SC relies on both high nonlinearity materials and micron-waveguide geometries that provide strong dispersion engineering. Recently, the chalcogenide glass (ChG) family has played a significant role in SC generation in the mid-IR regime because these materials can provide transparency by transmitting beyond 8.5 µm [2,4,5] and have an optical nonlinearity of more than several hundred times that of silica. This high nonlinearity can generate SC with low input power [6,7]. Recently, several experimental and theoretical SCs were reported in both channel waveguides and microstructured fibers. A 1-cm-long 4H-SiC waveguide could generate SC spectra extending from ∼1.00 µm to ∼3.56 µm using a 100-fs pulse with a 2000 W peak power at a wavelength of 1.55 µm [8]. A 1.2-cm-long silicon-on-silica (SOI) waveguide could theoretically generate SC spectra extending from 1.3 µm to 1.7 µm using a 50-fs pulse with a 25 W peak power at a wavelength of 1.55 µm [9]. A 1.8-cm-long GeAsSe ChG rid waveguide could experimentally generate SC spectra covering from 2.2 µm to 10.2 µm pumping a 330-fs pulse with a 4500 W peak power at a wavelength of 4.184 µm [10]. A 3-mm-long As₂S₃-silica hybrid waveguide could experimentally generate SC spectra covering from 1.1 µm to 2.5 µm, pumped by a thulium-doped all-fiber femtosecond laser and amplifier system at a wavelength of 1.9 µm [11]. A 1-cm-long As₂S₃ ChG waveguide could experimentally generate SC spectra covering from 1.5 µm to 2.2 µm, pumped by all-fiber structured dual-femtosecond solitons [12]. A 60-mm-long As₂S₃ ChG planar waveguide could generate broadband SC with a 30-dB bandwidth of 750 nm, using a 610-fs pulse with a 68 W peak power at a wavelength of 1.55 µm, employing an inorganic polymer glass coating and silica substrate [6]. A 6.6-cm-long As₂S₃ ChG rib waveguide could generate SC spectra spanning from 2.9 µm to 4.2 µm using a 7.5 ps pulse with a 2 kW peak power at a wavelength of 3.26 µm, employing Teflon coating and thermally oxidized silicon substrate [13]. A 4.7-cm-long As₂S₃ ChG rib waveguide could generate SC up to 4.7 µm, using a 7.5-ps pulse with a 1 kW peak power at a wavelength of 3.26 µm, employing MgF₂ ChG as a substrate [14]. As₂Se₃ ChG rib waveguides were designed and fabricated with low optical losses [15]. As₂S₃ ChG ridge waveguides were fabricated with controllable dispersion and high nonlinearity for SC generation [12]. As₂-S₃-silica hybrid waveguides could generate SC spectra extending from 1.1 µm to 2.5 µm, using a pump wavelength of 1.9 µm [11]. For microstructured fiber based on sulphides, an 18-cm-long, suspended core of As₃₈S₆₂ microstructured optical fiber (MOF) could generate SC spectra extending from 1.7 µm to 7.5 µm, using a 320-ps pulse with a 5.2 kW peak power at a wavelength of 4.4 µm [16]. A 72-mm-long As₂S₃ fiber could generate an SC spectrum extending from 1.6 µm to 5.9 µm, using a 67-fs pulse with a 520 kW peak power at a wavelength of 3.1 µm [17]. As₂S₅ ChG optical fibers could generate broadband flat SC spectra from 1.030 µm to 3.441 µm using a pump wavelength of 2 µm [18]. A four-hole As₂S₅ ChG hybrid microstructured optical fiber (HMOF) was fabricated by using a rod-in-tube method with the zero-dispersion wavelength (ZDW) of 2.28 µm. The HMOF was pumped in the normal dispersion at wavelength of 2.0 µm and in anomalous dispersion at wavelengths of 2.3 µm and 2.5 µm. The widest SC could be extended from 1.37 µm to 5.65 µm with HMOF 4.8 cm long, pumped in the anomalous dispersion waveguide with a 1.5 kW peak power at a wavelength of 2.3 µm [19]. These results demonstrate that a pump wavelength in the normal and anomalous dispersion region can generate broadband SCs in the mid-IR regime. In our previous work, a dispersion-engineered SOI waveguide was designed to fall in the anomalous dispersion. By varying peak power inputs of 100, 75, 50, 25, 10, and 3 W with a 50 fs pulses at the pump wavelength of 1.55 µm, a 1-cm-long SOI waveguide can generate SC spectra bandwidth of 532, 493, 445, 366, 325, and 257 nm and the peak power output of 4.3, 4.8, 4.9, 5.8, 4.4, and 1.94 W, respectively [20]. Although the SC spectra output is extended by increasing the peak power input, the peak power output is independent of the peak power input. Recently, Gebhardt et al. [21] reported nonlinear pulse compression of ultrashort pulses, reducing the pulse duration and increasing the pulse peak power. An ultrashort-pulse thulium-based fiber was based on a second-order dispersion of (β₂) of -4.4 × 10⁻³ ps³/m, a third-order dispersion (β₃) of -7 × 10⁻³ ps³/m, a nonlinear refractive index (n₂) of -9 × 10⁻²² m²/W and a pulse duration input of about 400 fs. The simulated output of an ultrashort-pule laser was a compressed full width at half maximum (FWHM) pulse duration shorter than 70 fs with 200 MW pulse peak power, in good agreement with the measurements. Consequently, we focus on the negative third-order dispersion to create a nonlinear compression of mid-IR SC generation in waveguides.

In this paper, we demonstrate numerically that nonlinear compression of mid-IR SC generation can be generated by using As₂S₅ ChG ridge waveguides. The As₂S₅ ChG material is considered because of its mid-IR transparency from ∼0.5 to >10 µm, high nonlinearity, and low two-photon absorption [6,19,22,23]. The dispersion-engineered As₂S₅ rectangular core waveguide is designed with air on the top and MgF₂ ChG as the lower cladding. By varying the waveguide dimensions of the width (W) and height (H) for realizing the near zero-dispersion wavelength (ZDW) and the negative slope of the dispersion, the dispersion engineering of the two waveguides can be tailored to exhibit the normal and anomalous dispersion regime, because this regime has low linear propagation losses and low multiphoton absorption [1]. The designed waveguides in the normal and anomalous dispersions, numerically calculated using the finite element method (FEM) mode-solver [24–26], have achieved the dispersions of −10.547 ps/nm/km and +5.314 ps/nm/km for the fundamental TE mode at the pump wavelength of 2.5 µm, with W = 1.9 µm and H = 0.6 µm, and W = 1.8 µm and H = 0.6 µm, respectively. These negative slopes of the dispersion led to the negative third-order dispersion, which are applied to generate the nonlinear compression of SC generation. The SC evolution along the waveguide length is performed by the generalized nonlinear Schrödinger equation with the split-step Fourier method [2,3,9,20,27,28]. Using a 50-fs pulse with different peak powers of 500 W, 1000 W, 1500 W, and 2000 W at the pump wavelength of 2.5 µm, the As₂S₅ ChG ridge waveguide in normal dispersion could generate the compressed pulse peak power outputs of 593 W, 2580 W, 4780 W, and 6900 W and the widest SC spectra extending from 1.56 µm to 4.85 µm, 1.33 µm to 7.94 µm, 1.25 µm to 10.35 µm, and 1.20 µm to 12.96 µm at the waveguide output, with 4.7-mm, 2.1-mm, 1.2-mm, and 0.85-mm lengths, respectively. Another waveguide in anomalous dispersion could generate the compressed pulse peak power outputs of 496 W, 3020 W, 4010 W, and 6360 W and the widest SC spectra extending from 1.55 µm to 5.34 µm, 1.28 µm to 8.61 µm, 1.30 µm to 11.21 µm, and 1.25 µm to >13 µm at the waveguide output, with 3.2-mm, 1.7-mm, 1.0-mm, and 1.0-mm lengths, respectively. The widest efficient mid-IR SC generation with short waveguides less than 1 mm long can be extended over 11.75 µm and compressed to about 6360 W of pulse peak power with short waveguides less than 1 mm long. The key process behind the SC formation is related to self-phase modulation (SPM), four-wave mixing (FWM), and nonlinear compression [3,6,9,22].

2. Theory

The schematic diagram of the As₂S₅ ChG ridge waveguide used in our simulations is shown in Fig. 1. Its core is made of the As₂S₅ ChG, with air as the top and MgF₂ as the bottom cladding. The wavelength-dependent linear refractive indices for the core and cladding over the wide wavelength range are given by using the Sellmeier equation [2,22,23],

 

Fig. 1. As₂S₅ ChG ridge waveguide geometry.

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Here, λ is the wavelength in micrometers and m is the integer. The Sellmeier coefficients are obtained by fitting the Sellmeier formula with the refractive index data of the As₂S₅ [22] and the MgF₂ [2,23] ChGs, as shown in Table 1. The high refractive index difference between the core and cladding of waveguides can increase the light confinement ability and decrease the confinement loss.

Table 1. Sellmeier coefficients of As₂S₅ and MgF₂ ChGs

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The dispersion (D) parameter and nonlinear coefficient (γ) play an important role in determining the spectral broadening of SC generation. The dispersion of waveguides is a combination of both the material properties and waveguide geometry. The core regime of the dimensions, with the width and height, can be tailored to exhibit the zero-dispersion wavelength (ZDW) near the pump wavelength. The finite element method (FEM)-based mode-solver is used to obtain the propagation constant β(ω) of the waveguide for the fundamental mode over a wide range of frequencies [23–25]. The effective index is given, n_eff= λβ(ω)/2π, which is used to calculate the group-velocity dispersion (GVD) parameter, D(λ) = −(λ/c)(d²n_eff/dλ²), and all other higher-order dispersions [2,9]. To study the SC generation by launching a femtosecond polarization pulse or soliton, it resonates and propagates in the form of a high-order soliton. SC evolution along waveguides is governed by using a generalized nonlinear Schrödinger equation (GNLSE) for the slowly varying envelope approximation of the pulse [2,3,9,27–31], which is given by

Here, A is the electrical field amplitude, α is the linear propagation loss of the waveguide, α₂ is the two-photon absorption coefficient, T = t − z/v_g is the retarded frame moving at the group velocity v_g = 1/β₁(ω₀) at the pump frequency (ω₀), and β_m(ω₀) = d^mβ/dω^m |_ω=ω0 (m≥2) is the m^th order dispersion parameter. The nonlinear coefficient is γ = 2πn₂/(λ₀A_eff), where n₂ is the nonlinear refractive index, A_eff is the effective mode area at the pump wavelength (λ₀), obtained by using the FEM mode-solver. The material response function is given by R(t)= (1 − f_R)δ(t)+f_Rh_R(t), where δ(t) is the instantaneous Kerr response and ${h_R}(t) = [(\tau _1^2 + \tau _2^2)/({\tau _1}\tau _2^2)\textrm{exp}( - t/{\tau _2})\sin ( - t/{\tau _1})$ is the delayed Raman response, where τ₁ and τ₂ relate to the phonon oscillation frequency and the characteristic damping time in the network of vibrating atoms, respectively, and f_R represents the fraction contribution of the delayed Raman response [2,9]. The numerical simulation of the SC generation for the As₂S₅ ChG waveguides in the dispersion regime is carried out using GNLSE with the split-step Fourier method (SSFM) [2,3,9,27]. A soliton input pulse, A(0,t)=$\sqrt {{P_0}} $ sech(t/T₀), is considered, where P₀ is the peak power of the pulse’s duration, T_FWHM ≈1.76T₀ is the pulse full width at the half-maximum (FWHM) duration. The soliton propagation into the waveguide is in the form of N order soliton, ${N^2} = Re (\gamma ){P_0}T_0^2{\beta _2}$, and the soliton fission occurs with the soliton fission length L_fiss= L_D/N, where ${L_D} = T_0^2/|{\beta _2}|$ is the dispersion length [2,3,9,27,32]. The parameters of these SC simulations are α = 0.6 dB/cm [6], n₂ = 3 × 10⁻¹⁸ m²/W [19,22,30], f_R = 0.11, τ₁ = 15.2 fs, and τ₂ = 230.5 fs [30].

3. Numerical results

As the SC generation in the mid-IR regime relates to materials being transparent along their entire wavelength and the high nonlinearity of materials, the As₂S₅ and MgF₂ ChG glass materials are considered to create As₂S₅ ChG ridge waveguides with strong mode confinement, whose refractive index of 2.22 and 1.36 at the wavelength of 2.5 µm, respectively, provides a high index contrast of 0.46. The As₂S₅ core waveguide is designed with air on top and MgF₂ as the lower cladding, as shown in Fig. 1. By varying the waveguide dimensions of the width and height for realizing ZDW and the negative slope of the dispersion, the dispersion engineering is tailored to exhibit the normal or anomalous dispersion regime at the pump wavelength, because this regime has low linear propagation losses and multiphoton absorption [1]. The four different structures were calculated by the dispersion as a function of wavelength for the fundamental TE mode using the FEM mode-solver, as shown in Fig. 2(a). It can be observed this figure that the waveguide of W = 4.4 µm, and H = 1.0 µm, shown by a back dotted curve, exhibits the anomalous dispersions with a positive slope of the dispersion at the pump wavelength of 2.5 µm. As the dimension of the waveguides is reduced, the dispersion becomes lager with a negative slope of the dispersion at the pump wavelength of 2.5 µm, in comparison with the dispersion, with W = 2.4 µm and H = 0.6 µm, shown by a black dashed curve. The two waveguides in the normal and anomalous dispersions achieved W = 1.9 µm, and H = 0.6 µm, and W = 1.8 µm, and H = 0.6 µm, resulting in the dispersion of −10.547 ps.nm⁻¹.km⁻¹ and +5.314 ps.nm⁻¹.km⁻¹, respectively, for the fundamental TE mode. These negative slopes of the dispersion at the pump wavelength of 2.5 µm, shown by the color curves, led to the negative third-order dispersion, which are applied to generate the nonlinear compression of SC generation. Their mode field distribution indicates that the two waveguides can support the propagation of the fundamental TE mode, as shown in Figs. 2(b) and 2(c). The calculated effective mode area is A_eff = 1.29 µm² and A_eff = 1.24 µm², resulting in the nonlinear parameters of γ = 5.8 W⁻¹.m⁻¹ and γ = 6.0 W⁻¹.m⁻¹, respectively. These waveguides also support a second mode; however, their GVD is quite different from that of the fundamental mode. Consequently, coupling between both modes has a negligible effect on the soliton formation [1,9].

 

Fig. 2. (a) Calculated dispersion for the two As₂S₅ ChG ridge waveguides, exhibiting the normal (red curve) and anomalous dispersions (blue curve) with W = 1.9 µm and H = 0.6 µm, and W = 1.8 µm and H = 0.6 µm, respectively, and the mode field distribution of the waveguides in (b) the normal, and (c) the anomalous dispersions for the fundamental TE mode at a pump wavelength of 2.5 µm. The vertical dotted line shows the pump wavelength.

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We considered the SC generation in the normal dispersion for our optimized As₂S₅ ChG ridge waveguide with W = 1.9 µm and H = 0.6 µm. The parameter D curve, related to the propagation constant β(ω), can calculate the second-order and third-order dispersion, which obtained the values of β₂ = 3.3832 × 10⁻² ps².m⁻¹ and β₃ = −2.2027 × 10⁻³ ps³.m⁻¹, respectively, for the fundamental TE mode at a pump wavelength of 2.5 µm. The SC simulation was carried out using GNLSE with SSFM in Eq. (2), and a 50-fs pulse with a 1500 W peak power at a pump wavelength of 2.5 µm. The SC results of the waveguide lengths of 0.8 mm, 1.0 mm, 2.0 mm, and 3.0 mm produced the compressed pulse peak power outputs of 3300 W, 4370 W, 1560 W, and 1370 W and the SC spectra with a −40 dB bandwidth of 5.0 µm, 9.1 µm, 10.0 µm, and 10.0 µm, covering the wavelength ranges of 1.2 µm to 6.2 µm, 1.2 µm to 10.3 µm, 1.3 µm to 11.3 µm, and 1.4 µm to 11.4 µm, respectively, as shown in Fig. 3(a) and 3(b). Its temporal and spectrum evolution along the waveguide length of 3 mm is shown in Fig. 3(c) and 3(d), corresponding to Fig. 3(a) and 3(b). It is observed from Fig. 3(c) and 3(d) that the temporal and spectra evolution are initially simply broadened spectra (Fig. 3(d)) by self-phase modulation (SPM), which can be seen after 0.8 mm. Later, the FWM process occurs about 1 mm, the SC spectra are broadened by the appearance about 3.1 µm idler terms and 1.7 µm signal terms, with the apparent asymmetry in the wavelength domain (or the apparent symmetry in the frequency domain) [1,22]. The compressed pulses at the observed output of L = 1 mm consist of the many short pulses with a dominant peak power of 4370 W and FWHM duration of about 5.5 fs, as shown in the temporal profiles in Fig. 3(a). Then, the pulses evolve along the waveguide length in the time domain that the red part propagation of the pulse lags the blue part due to the negative third-order dispersion, as shown in the temporal profiles at the output of L = 2 mm and 3 mm in Fig. 3(a), corresponding to temporal evolution along the waveguide length in Fig. 3(c). These pulses introduce the spectra toward longer wavelengths, as shown in the spectra profiles at the output of L = 2 mm and 3 mm in Fig. 3(b), corresponding to spectra evolution along the waveguide length in in Fig. 3(d). The compressed pulses are the result of nonlinear compression, which depends on the value of the negative third-order dispersion. This is consistent with the experiment of Gebhardt et al. [21]. The numerical nonlinear parameters indicate that the soliton fission occurred L_fiss= 1.6 mm, and the high-order soliton propagation was equal to N = 14. On the other hand, the soliton formation comprises the compressed pulses and propagates in the form of the high-order soliton along the waveguide length, which introduces a shift of energy toward longer wavelengths. However, the widest SC generation occurs at the waveguide length of about 1.2 mm before the soliton fission. The key process behind this SC formation relates to SPM, FWM, nonlinear compression, and soliton fission.

 

Fig. 3. Simulated SC generation in the normal dispersion for the As₂S₅ ChG ridge waveguide. (Solid curves) (a) Temporal and (b) spectra profiles at the output of 0.8, 1.2, 2.0, and 3.0 mm long waveguides (the dashed curve shows input pulse temporal and spectra for comparison). (c) Temporal and (b) spectra evolution along the waveguide length of 3 mm, using a 50-fs pulse with a 1500 W peak power at a pump wavelength of 2.5 µm.

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Figure 4 shows the SC spectra for the As₂S₅ ChG ridge waveguide in the anomalous dispersion with W = 1.8 µm and H = 0.6 µm, using the obtained dispersion, β₂ = −1.9052 × 10⁻² ps².m⁻¹and β₃ = −2.4075 × 10⁻³ ps³.m⁻¹, for the fundamental TE mode at a pump wavelength. By using the same soliton input, The SC results of the waveguide lengths of 0.8 mm, 1.0 mm, 2.0 mm, and 3.0 mm produced the compressed pulse peak power outputs of 4340 W, 5220 W, 2550 W, and 2470 W and the spectra from 1.2 µm to 6.7 µm, 1.3 µm to 11.2 µm, 1.3 µm to12.4 µm, and 1.3 µm to 12.4 µm, producing a −40 dB bandwidth of 5.5 µm, 9.9 µm, 11.1 µm, and 11.1 µm, respectively, as shown in Fig. 4(a) and 4(b). These SCs are broader because of higher nonlinear effects. Figure 4(c) and 4(d) show the temporal and spectra evolution along the waveguide length of 3 mm, corresponding to Fig. 4(a) and 4(b). It is observed from Fig. 4(c) and 4(d). However, the widest SC output occurs at the waveguide length of about 1.0 mm before the soliton fission of L_fiss = 2.2 mm. The soliton formation has similar features as the previous case, related to SPM, FWM, nonlinear compression, and soliton fission.

 

Fig. 4. Simulated SC generation in the anomalous dispersion for the As₂S₅ ChG ridge waveguide. (Solid curves) (a) Temporal and (b) spectra profiles at the output of 0.8, 1.2, 2.0, and 3.0 mm long waveguides (the dashed curve shows input pulse temporal and spectra for comparison). (c) Temporal and (b) spectra evolution along the waveguide length of 3 mm, using a 50-fs pulse with a 1500 W peak power at a pump wavelength of 2.5 µm.

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To see what role the negative third-order dispersion plays in the nonlinear compression of SC generation, we have repeated the simulation of the positive slope of the anomalous dispersion with the positive third-order dispersion for the As₂S₅ ChG ridge waveguide, W = 4.4 µm and H = 1.0 µm, as shown in Fig. 2(a) (black dotted curve). The obtained parameters were β₂ = −9.3090 × 10⁻² ps².m⁻¹, β₃ = 3.5690 × 10⁻³ ps³.m⁻¹, A_eff = 3.40 µm², and γ = 2.2 W⁻¹.m⁻¹ for the fundamental TE mode at a pump wavelength of 2.5 µm. The SC output of 3 mm produced the peak power outputs of 1500 W, and the spectra from 1.6 µm to 3.6 µm with a −40 dB bandwidth of 2.0 µm, as shown in Fig. 5(a) and 5(b), respectively. The soliton fission for N = 5 occurs within L_fiss= 1.6 mm after the pulse is pumped, and the SC is formed by the SPM, FWM, and soliton fission. However, the compressed pulses are not apparent as shown in Fig. 5(c) and 5(d). This SC spectrum is much narrower, compared to the previous SC spectrum. With the particular design is effective, ChG waveguides in the negative slope of dispersion with a negative third-order dispersion can create the high peak power and the widest spectra of SC generation.

 

Fig. 5. Simulated SC generation in the anomalous dispersion with the positive third-order dispersion for the As₂S₅ ChG ridge waveguide of W = 4.4 µm and H = 1.0 µm. (solid curve) (a) Temporal and (b) spectral profiles at the output of 3.0 mm long waveguide (the dashed curve shows input pulse temporal and spectra for comparison), and (c) its temporal and (d) spectral evolution along the waveguide length of 3 mm, using a 50-fs pulse with a 1500 W peak power at a pump wavelength of 2.5 µm.

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The widest SC generation is developed by using a 50-fs pulse with different pump peak powers at a pump wavelength of 2.5 µm for our optimized As₂S₅ ChG ridge waveguides in the normal and anomalous dispersions. By increasing pump peak powers of 500 W, 1000 W, 1500 W, and 2000 W, the widest SC spectra outputs in the normal dispersion waveguide could be extended from 1.56 µm to 4.85 µm, 1.33 µm to 7.94 µm, 1.25 µm to 10.35 µm, and 1.20 µm to 12.96 µm, producing a -40 dB bandwidth of 3.29 µm, 6.61 µm, 9.1 µm, and 11.76 µm, with the waveguides 4.7 mm long, 2.1 mm long, 1.2 mm long, and 0.85 mm long, respectively, as shown in Fig. 6(b). These spectra corresponded to the compressed pulse peak power of 593 W, 2580 W, 4780 W, and 6900 W, as shown in Fig. 6(a). Using the same power level in the anomalous dispersion waveguide, the widest SC spectra could be extended from 1.55 µm to 5.34 µm, 1.28 µm to 8.61 µm, 1.30 µm to 11.21 µm, and 1.25 µm to >13 µm, producing a -40 dB bandwidth of 3.79 µm, 7.33 µm, 9.91 µm, and beyond 11.75 µm, with the waveguide 3.2 mm long, 1.7 mm long, 1.0 mm long, and 1.0 mm long, respectively, as shown in Fig. 6(d). These spectra corresponded to the compressed pulse peak power of 496 W, 3020 W, 4010 W, and 6360 W, as shown in Fig. 6(c). We performed numerical calculations at the power level of 500 W, 1000 W, 1500 W, and 2000 W. In the normal dispersion waveguide, the soliton fission was L_fiss= 2.9 mm, 2.0 mm, 1.6 mm, and 1.4 mm, and the order soliton of propagation was N = 8, 11, 14, and 16, respectively. In the anomalous dispersion waveguide, the soliton fission was L_fiss= 3.7 mm, 2.6 mm, 2.2 mm, and 1.9 mm, and the order of soliton propagation was N = 11, 16, 19, and 22, respectively. These results can be interpreted as revealing that the widest SC generation occurs before the soliton fission for the peak powers of 1000 W, 1500 W, and 2000 W. Consequently, the key process behind the SC formation relates to SPM and FWM. However, at the peak power of 500 W, the key process behind the widest SC generation relates to SPM, FWM, and soliton fission. By increasing the power level up to 2000 W, the two waveguides can generate the widest SC over 11.75 µm (>3.6 octaves), with short waveguides less than 1 mm long. Although the broadband SC is the same, these lengths are less than 85 octaves by comparison with those achieved using 85-mm-long As₄₀S₆₀ chalcogenide step-index fibre with coupled maximum peak power of 2.29 MW [33].

 

Fig. 6. (solid curves) The widest SC output of the As₂S₅ ChG ridge waveguides with different pump peak powers of 500 W, 1000 W, 1500 W, and 2000 W for (a) temporal and (b) spectral profiles in the normal dispersion and (c) temporal and (d) spectral profiles in the anomalous dispersion, using a 50-fs pulse at a pump wavelength of 2.5 µm. The dashed curve shows the input pulse spectra for comparison.

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Recently, waveguides fabricated using MgF₂ ChG for the lower cladding may be at a disadvantage because of the fragility of MgF₂ glass, but this problem can be managed for short waveguides around 10 mm long [2,14]. The surface contamination of ChG materials by absorbing water and hydrocarbons can prevent the use of a thin 10-nm fluoro-polymer protective coating layer [2,34]. Cheng et al. [22] have reported that, experimentally, an AsSe₂-As₂S₅ HMOF was able to generate SC spectra extending from 1256 nm to 6385 nm. The transparent As₂S₅ ChG material is stable and transmits 50%, covering the wavelength range from 1.747 nm to 12.043 nm. In this paper, our waveguides have been numerically tested for a protective coating of inorganic polymer glass (IPG) material with a refractive index of ≈1.51 [29] by placing IPG on the top of the MgF₂ ChG waveguides. The results of the two waveguides in normal and anomalous dispersions were β₂ = 2.3756 × 10⁻² ps².m⁻¹, and β₃ = −2.0447 × 10⁻³ ps³.m⁻¹ (β₂ = 3.3832 × 10⁻² ps².m⁻¹, and β₃ = −2.2027 × 10⁻³ ps³.m⁻¹ without coating) and β₂ = −3.1358 × 10⁻² ps².m⁻¹, and β₃ = −2.2209 × 10⁻³ ps³.m⁻¹ (β₂ = 1.9052 × 10⁻² ps².m⁻¹, and β₃ = −2.4075 × 10⁻³ ps³.m⁻¹ without coating), respectively, for the fundamental TE mode at a pump wavelength. Such dispersion is a small change and does not produce noticeable changes in the SC bandwidth. The two As₂S₅ ChG ridge waveguides were able to generate an efficient mid-IR SC extending over 11.75 µm, with short waveguides less than 1 mm long, using a 50-fs pulse with the 2000 W peak power at the pump wavelength of 2.5 µm. The SC spectra is consistent with the broadest spectral efficiency of the As₂S₅ ChG material transmission. The widest SC generation of our optimized As₂S₅ ChG ridge waveguides can be compared to a maximum of 2.7 octaves, achieved using ∼200 fs pulses in a 2-cm-long four-hole As₂S₅ ChG HMOF [19], as shown in Table 2. Furthermore, one application that has received a lot of interest is a broadband ZrF₂-BaF₂-LaF₃-AlF₃-NaF ChG fiber mid-IR SC source spanning 2 µm to 7.5 µm for the tissue imaging. Using the SC source at the long wavelengths of 5.70 µm, 6.03 µm, 6.45 µm, and 7.30 µm, the results can distinguish between epithelial and surrounding connective tissues within a paraffin tissue section of colon [35]. This work is another important step towards mid-IR SC source in the long-wavelength region, which our ultra-wideband results in Fig. 6 has shown that the significant broad wavelengths from 1.03 µm to 13.0 µm. It also could be applied to broadband light sources for dense wavelengths division multiplexing.

Table 2. Summary of theoretical and experimental mid-IR SC generation.

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4. Conclusions

We have numerically investigated mid-IR SC generation in the As₂S₅ ChG ridge waveguides. Here, nonlinear pulse compression of the SC source for reducing the pulse duration and increasing the pulse peak power was reported, for the first time, the As₂S₅ ChG waveguides in the normal dispersion of −10.547 ps.nm⁻¹.km⁻¹ and anomalous dispersion of +5.314 ps.nm⁻¹.km⁻¹ for the fundamental TE mode at the pump wavelength of 2.5 µm, with the cross-section of 1.9 µm by 0.6 µm and 1.8 µm by 0.6 µm. These waveguides were designed by using the negative slope of the dispersions with the negative third-order dispersion, which are applied to generate the nonlinear compression of SC generation. Using a 50-fs pulse with increasing the power level up to 2000 W, the two waveguides could generate the widest SC spectra, spanning from 1.20 µm to 12.96 µm and from 1.25 µm to >13 µm with very short waveguides 0.85 mm and 1 mm long, respectively, which corresponded to the compressed pulse peak power of 6900 W, and 6360 W. These powers can be compared to a maximum of 3.4 and 3.1 octaves. The key process behind SC formation of such waveguides is related to SPM, FWM, and nonlinear compression [1,6,21,22]. Also, the SC spectra cover the full transparent atmospheric windows of 0.6 to 13 µm for As₂S₅ ChG material [19,22], compared to a maximum of 2.7 octaves, achieved using ∼200 fs pulses in the fabrication of a 2-cm-long four-hole As₂S₅ ChG HMOF [16]. These show that the waveguides are relatively flexible in design, which is concerned with optical design and engineering, and micro-optical devices. Moreover, these spectra SC up to 13 µm correspond to the experimental result of an 85-mm-long As₄₀S₆₀ chalcogenide step-index fibre [30]. With the particular design is effective, ChG waveguides can generate the high peak power and the widest spectra of SC generation. As the obtained ultra-wideband mid-IR SC, high peak power, very short waveguides, and low-energy pulses (<1 pJ) are important for on-chip mid-IR SC sources on wearable technology sensors, the proposed work would offer the greatest benefits in performance, footprint, and cost.