Supercontinuum （SC） generation in nonlinear materials excited by ultrashort pulses of peak power has raised great research interest， due to its wide applications in various fields ^［1］. This is especially evident in mid-infrared （MIR） SC sources， which have been used in sensing and detection of gas molecules， due to most of the gas molecules possessing intrinsic vibration absorption in the MIR ^［2-5］. In recent years， a wide variety of numerical and experimental investigations on mid-infrared supercontinuum generation were reported in fluoride glass ^［6］， GaAs crystal ^［7］， ZBLAN fiber ^［8］， SiN waveguide ^［9］， chalcogenide （ChG） planar waveguides ^［10-11］ and fibers ^［12-13］. Wide transparency， high laser damage threshold，high optical nonlinearity of the materials， and suitable engineered dispersion in the waveguide or fiber structure are key factors to achieve broad SC spectra. Generally， SC generation requires the waveguide or fiber structure to be pumped by a laser with a wavelength near its zero-dispersion wavelength （ZDW） ^［14］. The ZDW of the fiber and planar waveguide structure can be finely tuned via adjusting the refractive index of the materials or structural parameters of the waveguide or optical fiber ^［15-17］. For example， in a 10-μm core step-index As₂Se₃ fiber， As₂Se₃ shows a material ZDW around 7 μm， which can be tailored to ∼5 μm ^［15-16］. By tapering the As₂Se₃ fiber down to a diameter of 1.28 μm， the ZDW can also be further shifted to 1.73 μm ^［17］. Otherwise， by adjusting the structural parameters and cladding materials of Ge_11.5As₂₄Se_64.5 in the chalcogenide-glass channel waveguides， the zero dispersion point can be also shifted to different wavelengths ^［5］.

Except for the adjustable ZDW in optical fiber， planar waveguide structures are also compatible with the well-developed semiconductor processing， which have better scalability and low fabrication cost. It has been demonstrated that supercontinuum spectra can be excited in ChG planar waveguides using a high peak power of few kW ^［18-20］. It is well known that high pump power is beneficial to broadening of supercontinuum spectrum since optical nonlinearity of the materials would be maximally excited under the high pump power. However， due to weak mechanical property and low laser damage threshold， ChG glasses are easily damaged under high pump power illumination， and this in turn limits the output power of SC generation ^［21-22］. ZnSe has a larger laser damage threshold around 1 300 mJ/cm2^［23］， which is nearly one or two orders of magnitude larger than the typical chalcogenide glasses ^{［22， 24］}. The large damage threshold indicates that higher peak power up to tens of kW is possible for supercontinuum generation in ZnSe waveguide. ZnSe with a broad transmission range from 0.5 to 22 µm has been widely used for the fabrication of mid-infrared optical components， such as optical lenses and windows ^［25］. Moreover， it has a reasonable high optical nonlinearity of 1.1×10-18m2/W^［26］ at 1.5 µm compared with a value of 2.9×10-18m2/W for typical As₂S₃ glass ^［27］ at 1.5 µm，^{which is suitable for supercontinuum generation. Although SC generation from ZnSe bulk crystals has been reported ［28-31］ the ZnSe waveguides have been used in mid-infrared sensing ［32-33］， there is no reports on SC generation from a dispersion-engineered ZnSe waveguide as far as we know.}

In this paper， we used ZnSe as core materials and simulated the dispersion， optical field distribution， the nonlinear coefficient， and effective mode area in the waveguide consisting of bottom and cladding layers with different thicknesses and refractive index contrast. Different cladding materials and waveguide structures affect the zero-dispersion wavelength. Furthermore， we numerically simulated SC generation in a 5-cm-long dispersion-engineered ZnSe rib waveguide pumped at different wavelength from 3 to 4.5 μm and different peak power from 100 W to 20 kW， and demonstrated a broadband SC spectrum from 3.0 to 12.2 μm in a 4-μm-width waveguide.

The schematic diagram of the designed ZnSe rib waveguide is shown in Fig. 1 （a）. The width ‘w’ of the waveguide is 4 µm or 8 µm. H1 and H2 represents ZnSe rib and ZnSe slab height， respectively. The refractive index of ZnSe ^［25］ at a wavelength range from 2 to 10 μm is shown in Fig.1 （b）. The refractive index changes from 2.446 3 at 2 μm to 2.406 5 at 10 μm gradually. Previous experimental result has identified that Ga₂As₃₀S₆₈ has a refractive index of 2.25， while Ge₅As₁₀S₈₅ has a refractive index of 2.05 at 2 μm ^［34］. Therefore， it is much easier to design the cladding layer to achieve a refractive index contrast up to 0.4. For example by tuning the composition of S in cladding materials of GaAsS or GeAsS glasses during film deposition.

The Sellmeier equation used in the simulation of the wavelength-dependent linear refractive index n over the entire wavelength range of ZnSe， Ga₂As₃₀S₆₈， and Ge₅As₁₀S₈₅ glass is given by Equation（1）， where λ is the wavelength in micrometers， Ai and λi2（i = 1， 2， 3） are material related constants， listed in Table1.

The waveguide structures were optimized by using commercial software （COMSOL）. With numerical analysis， the effective refractive index can be calculated by the Finite Element Modeling solver. Subsequently， the effective index was used for calculating the dispersion parameter as well as all other higher-order dispersion parameters.

The dispersion parameter curves of the modes can be calculated using：

here λ is the wavelength in micrometers， D is the dispersion parameter of the transmission mode， neff is the effective refractive index of the fundamental mode and c is the light speed in vacuum. We investigated the dispersion of the waveguide， in which the top and bottom cladding layers were fixed at 2 µm， the total thickness of ZnSe film was 3 µm， and the ratio of H1 to H2 was variable.

Dispersion of the waveguide plays a significant role in SC generation. Ideally， dispersion near the pump wavelength should be anomalous as well as relatively flat^［5］. Figure 2 （a） shows the dispersion of the fundamental quasi- transverse electric （TE） polarization in the waveguides with different structural parameters where the core-cladding refractive index contrast is kept around 0.2 （employing Ga₂As₃₀S₆₈ glass for both the upper and lower claddings）. It can be seen that， with the changing ratio of H1 to H2and waveguide width， ZDW is always located at a longer wavelength more than 9 μm. It seems unlikely to shift ZDW to a shorter wavelength by employing such a lower refractive index contrast cladding material such as Ga₂As₃₀S₆₈ glass. To realize anomalous dispersion around the pump wavelength， increasing the refractive index contrast is essential. Figures. 2 （b-c） show the mapping of the calculated group velocity dispersion for TE polarization as functions of wavelength and thickness of H1 in a waveguide with a width of 4 and 8 μm， respectively. In both cases， the core-cladding refractive index contrast is kept around 0.4 （employing Ge₅As₁₀S₈₅ glass for both the upper and lower claddings）， and the ZDWs are tunable via changing parameter H_1. As H1 increases， the overall dispersion gradually increases， and the anomalous dispersion appears when H₁ reaches 1 μm in Fig. 2 （b） and 0.5 μm in Fig. 2 （c）， respectively.

Therefore， all simulations in the rest part of the paper were performed in the waveguide with a refractive index contrast of 0.4， since the larger refractive index contrast can tune the ZDW to a shorter wavelength as shown in Figs 2 （b） and （c）. Moreover， we concentrated on the waveguides with the two typical structural parameters （w=4 μm， H1=2 μm，H2=1 μm and w=8 μm， H1=2 μm，H2=1 μm）. The ZDWs can be seen as the cross-points in the solid and dash lines in Fig. 2 （b） and （c）， respectively. Both of them have the first ZDW around 3.0 μm， and the second ZDW at ~5 μm for the 4-μm-width waveguide and ~ 8.5 μm for the 8-μm-width waveguide.

Figures 3 （a-f） show the optical field distribution of the quasi-TE mode at different wavelengths in the waveguide with a width of 4 μm and 8 μm， respectively. It can be seen that， for the 4-μm-width waveguide， the light is well confined within the core as shown in Fig. 3 （a）， slightly leaked to the cladding as shown in Fig. 3 （b）， and considerably leaked out in Fig. 3 （c）. For the 8-μm-width waveguide， the optical field distribution exhibits a similar feature with increasing wavelength， as shown in Figs. 3 （d- f）， respectively. Comparing Fig. 3 （b） with （e）， we can see that the light is better confined with increasing waveguide width since the waveguide width in Fig. 3 （b） is less than the wavelength of 6 μm. However， in both cases， the optical field is hardly leaked into the substrate layer even at 10 μm.

The nonlinear coefficient （Kerr effect）， γ， which is determined by the effective mode area of the waveguide and the nonlinear refractive index of the material， can be calculated using the following formula，

where n2 is the nonlinear refractive index of ZnSe ^［26］， Aeff is the effective area of the propagating mode， given by：

where E is the electric field’s transverse component propagating inside the waveguide.

The effective mode area and Kerr nonlinearity coefficient were calculated and shown in Fig. 4 （a-b）， for the waveguide with a width of 4 μm and 8 μm， H1=2 μm and H2=1 μm， respectively. A general tendency is that as the nonlinear coefficient decreases， the effective area increases with increasing wavelength. However， the nonlinear coefficient is lower， but an effective area is larger in 8 μm-width waveguide compared with that in 4 μm-width one. This indicates that more power in the guided mode could enter the cladding as the wavelength increases， and thus the nonlinear coefficient gradually decreases ^［35］， which is in agreement with the optical field distribution in Fig.4.

We performed the simulations of SC generation by using a generalized nonlinear Schro¨dinger equation （GNLSE） with a chirp-free Gaussian-shaped pump pulse as the initial condition ^［36-37］，

where A is the electrical field amplitude， A（z，T） is the electric field wave amplitude as a function of propagation distance and time， T=t-zvg is the retarded time frame moving at the group velocity v_g = 1β1， α is the linear propagation loss of the waveguide including a wavelength-independent propagation loss of 0.6 dB/cm ^［32］ for our 5-cm-long rib waveguides， βkω=dkβdωk|ω=ω0(k≥2) is the kth-order dispersion parameter. The 10-order dispersion parameters are obtained by calculating the effective mode index with the finite-element method. And the nonlinear coefficient is γ = n2ω0cAeff(ω0) ， where n₂ is the nonlinear refractive index ^［26］ and c is the speed of light in vacuum， Aeff(ω0) is the effective area of the mode at the pump frequency ω₀， and α₂ = 5.5×10^-14 m/W is the two-photon absorption coefficient ^［38］. Finally， the material response function includes both the instantaneous electronic response （Kerr type） and the delayed Raman response and has the form：

where the delayed Raman contribution hRt is given by：

We calculated the Raman gain from the data available， and τ₁ and τ₂ from the linewidth of Raman spectrum of ZnSe， the response function coefficient fR is 0.08， τ₁ is 21.06 fs， and τ₂ is 4.4 ps for ZnSe waveguide ^［39］.

The GNLSE for the fundamental quasi-TE mode of the waveguides was calculated by using commercial software （MATLAB） to simulate SC generation. For numerical analysis of SC generation in a novel 5-cm-long dispersion engineered ZnSe rib waveguide， sub-femtosecond pulses with 150 fs duration and a repetition rate of 1 kHz were used as an exciting source.

Figures. 5-6 summarize the simulations of SC spectra with four different pump wavelengths for the two waveguides at different pump power. In Figs. 5 （a） and （c）， when the pump wavelength is 3.0 μm， which is close to the ZDWs of two waveguides， the change of the waveguide size has almost no effect on the spectral broadening. The SC bandwidth below -30 dB generally has a width from 2.6 μm （2.7 μm） to 3.4 μm （3.2 μm） in 4-μm-width waveguide （in 8-μm-width waveguide） at 100 W， and slightly increases to a width from 2.3 μm （2.3 μm） to 4.6 μm （4.3 μm） at 20 kW. A significant difference can be found in Figs. 5（b） and （d） where the pump wavelength is 4.5 μm. SC spectrum generally broadens with increasing peak power. With a peak power of 20 kW， a bandwidth of 4.6 μm in 8-μm-width waveguide can be found compared with that of 9.2 μm in 4-μm-width waveguide below -30 dB.

For SC generation， self-phase modulation （SPM） alters the broadening rate imposed on the pulse by the group-velocity dispersion （GVD）， and this has a correlation with the optical solitons in the abnormal dispersion region of the waveguide. The group velocity dispersion （β₂） in Fig.4（d） is derived from the dispersion parameter in Fig.4（c）. From Fig.4（d）， since the value of β₂ is negative， the solitons are perturbed by high order dispersion and intrapulse Raman scattering and fission， and changed into multiple， much narrower， fundamental solitons， leading to the asymmetrical broadening of the spectrum. The present simulations demonstrated that the shift of the frequency of the solitons generated in abnormal dispersion region increases with increasing pump wavelength， and the edge of the solitons shifts to longer wavelength when the pump wavelength is from 3.0 to 4.5 μm as shown in Figs. 6（a-d）. For sub-femtosecond pulses with a duration of 150 fs， its spectrum width is very wide， so that the blue-shifted spectrum component of the pulse can be used as a pump to effectively amplify the red-shifted component of the same pulse through Raman gain. Such a continuous process in the waveguide results in a constant energy transformation from the blue to the red component， as represented by the redshift of the soliton spectrum ^［40］.

The spectral evolution corresponding to two curves in Fig. 6 （d） are shown in Fig. 7. For the 4-μm-width waveguide， it can be observed from Fig. 7 （a） that the SC extends over 9.2 μm covering a wavelength range from 3.0 μm to around 12.2 μm above 2 octaves. For the waveguide width is 8 μm， the SC extends over 4.6 μm covering a range from 3.1 μm to 7.7 μm above 1.3 octaves as shown in Fig. 7 （b）. For the waveguides with different width of 4 μm and 8μm， the dispersion parameter D are13.724 ps/nm/km and 22.863 ps/nm/km at 4.5 μm， respectively. The second-order dispersion β2=-Dλ22πc = -0.1483 ps²/m and -0.240 6 ps²/m， and this leads to a dispersion length LD=τP2/β2 = 0.152 m and 0.093 m whereτp is the laser pulse duration. Using the waveguide nonlinear parameter γ of 0.151W-1m-1 and 0.0965W-1m-1 for these two waveguides as shown in Fig.4 （a） and （b）， we obtained a nonlinear length L_NL at 20 kW of coupled power via LNL=1/γP， being 0.331 mm and 0.520 mm， respectively. The soliton order nsol=LD/LNL = 21.4 and 13.3.

When the dispersion length is comparable to the length of the waveguide， both GVD and SPM contribute to the formation of the solitons in the abnormal dispersion region ^［40］. When nsol exceeds 1.5， the light pulses are transmitted as the second or higher-order solitons， and such the solitons undergo a splitting process if they are disturbed by higher-order dispersion and intrapulse Raman scattering. nsol-order solitons can produce N fundamental solitons and the frequencies of all the solitons are shorter than the original input pulse， and the shortest pulse width of the soliton is 1/（2N-1） of the input pulse width ^［40］. The pulse width of the soliton is around 3.6 fs and 6 fs， in 4 μm and 8μm-width waveguide， respectively. For an ultra-short pulse， its pulse width is opposite to its spectral width. With the assistance of Raman scattering process， the pulse spectrum toward longer wavelengths. A spectral shift can occur even in the normal-GVD regime of the waveguide where the solitons are not formed ^［40］. In the normal dispersion region， the pulse broadens rapidly while its spectrum is broadened through SPM. In contrast， in the anomalous dispersion region， the pulse slows down because the group velocity of a pulse is lower at longer wavelengths， and the SPM decreases the broadening rate ^［40］_. Therefore， when the pump wavelength is 4.5 μm， the SC spectrum generated from 4-μm-width waveguide is broader than that from 8-μm-width waveguide， and the solitonic traces can be observed in the temporal density plots in Fig. 7（c-d）， respectively.

We have numerically analyzed dispersion parameters， optical field distribution， nonlinear coefficient， and SC generation in ZnSe rib waveguide. It was found that ZDW can be shifted to shorter wavelength with increasing refractive index contrast between the core and cladding layer in the waveguide， and the optical field distribution can be well confined in 4- and 8-μm-width waveguide employing Ge₅As₁₀S₈₅ glass as both the upper and lower claddings. With increasing pump wavelength from 3.0 μm to 4.5 μm， SC spectrum broadens， and an ultrabroad SC spectrum can be obtained up to 9.2 μm （> 2 octaves） in a waveguide pumped by a peak power of 20 kW and a wavelength of 4.5 μm. Furthermore， the simulations confirm that the SC generation is initiated by self-phase modulation， followed by soliton dynamics and soliton self-frequency shift， both of which increase with increasing pump wavelength.