基于色散调制硒化锌脊形波导产生超连续谱

方依明; 杨振; 徐培鹏; 严昆仑; 盛燕; 王荣平; FANG Yi-Ming; YANG Zhen; XU Pei-Peng; YAN Kun-Lun; SHENG Yan; WANG Rong-Ping

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Dispersion engineered ZnSe rib waveguide for mid-infrared supercontinuum generation PDF

- ORCID：
FANG Yi-Ming ^1,2
✉
- ORCID：
YANG Zhen ^1,2
- ORCID：
XU Pei-Peng ^1,2
- ORCID：
YAN Kun-Lun ^1,2
- ORCID：
SHENG Yan ^1,2
- ORCID：
WANG Rong-Ping ^1,2,3
✉

1. Laboratory of Infrared Materials and Devices， Ningbo University， Ningbo 315211， China； 2. Key Laboratory of Photoelectric Detection Materials and Devices of Zhejiang Province， Ningbo 315211， China； 3. Laboratory of Silicate Materials Science and Engineering， Wuhan University of Technology， Wuhan 430070， China

Updated：2021-08-25

DOI：10.11972/j.issn.1001-9014.2021.04.010

Abstract

Mid-infrared supercontinuum generation in dispersion-engineered ZnSe rib waveguides was investigated for the first time. Numerical results showed that the zero-dispersion wavelength can be shifted to a shorter wavelength by adjusting structural parameters and refractive index contrast between the core and cladding layers in the waveguide. The optical field can be well confined in the 4- and 8-μm wide waveguides with a 2-μm thick cladding layer of Ge₅As₁₀S₈₅ glass. The effect of waveguide parameters on the bandwidth of the supercontinuum spectrum at a 5-cm-long waveguide was also simulated to understand the effect of the pump wavelength and structure parameters on the supercontinuum generation. Our results showed that supercontinuum output could vary over a wide range depending on structural parameters of the waveguide， the pump power and wavelength. An ultrabroad supercontinuum spectrum from 3.0 up to 12.2 μm （> 2 octaves） was confirmed in a 4 μm-width waveguide with a peak pump power of 20 kW and a pump wavelength of 4.5 μm， which is promising as one of the on-chip supercontinuum light sources for many applications such as biomedical imaging， and environmental and industrial sensing in the mid-infrared.

Keywords

ZnSe rib waveguides; supercontinuum; soliton self-frequency; dispersion-engineered

Introduction

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 ^{［Reference 2-5}2-5］. In recent years， a wide variety of numerical and experimental investigations on mid-infrared supercontinuum generation were reported in fluoride glass^{［Reference 6

Baidu Scholar}6］， GaAs crystal^{［Reference 7

Baidu Scholar}7］， ZBLAN fiber^{［Reference 8

Baidu Scholar}8］， SiN waveguide ^{［Reference 9

Baidu Scholar}9］， chalcogenide （ChG） planar waveguides^{［Reference 10-11}10-11］ and fibers ^{［Reference 12-13}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） ^{［Reference 14

Baidu Scholar}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 ^{［Reference 15-17}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 ^{［Reference 15-16}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 ^{［Reference 17

Baidu Scholar}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 ^{［Reference 5

Baidu Scholar}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 ^{［Reference 21-22}21-22］. ZnSe has a larger laser damage threshold around 1 300

m J / c m^{2}

^{［Reference 23

Baidu Scholar

23］}， which is nearly one or two orders of magnitude larger than the typical chalcogenide glasses ^{［Reference 22

Baidu Scholar}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 ^{［Reference 25

Baidu Scholar}25］. Moreover， it has a reasonable high optical nonlinearity of

1.1 \times 10^{- 18} m^{2} / W

^{［Reference 26

Baidu Scholar}26］ at 1.5 µm compared with a value of

2.9 \times 10^{- 18} m^{2} / W

for typical As₂S₃ glass ^{［Reference 27

Baidu Scholar}27］ at 1.5 µm，which is suitable for supercontinuum generation. Although SC generation from ZnSe bulk crystals has been reported ^{［Reference 28-31}28-31］ the ZnSe waveguides have been used in mid-infrared sensing^{［Reference 32-33}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.

1 Simulation and discussion

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. $H_{1}$ and $H_{2}$ 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 ^{［Reference 34

Baidu Scholar}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.

Fig. 1 (a) ZnSe waveguide structure, (b) refractive index of the core layer ZnSe, the cladding layer Ga₂As₃₀S₆₈ and Ge₅As₁₀S₈₅.

图1 (a) ZnSe波导结构，(b)ZnSe芯层、Ga2As30S68和Ge5As10S85覆盖层折射率

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， $A_{i}$ and $λ_{i}^{2}$ （i = 1， 2， 3） are material related constants， listed in Table1.

n (λ) = \sqrt[]{1 + \sum_{i = 1}^{m} \frac{A_{i} λ^{2}}{λ^{2} - λ_{i}^{2}}}

（1）

Table 1 Refractive coeﬃcient of ZnSe, Ga₂As₃₀S₆₈, and Ge₅As₁₀S₈₅.

表1 ZnSe、Ga2As30S68和Ge5As10S85的折射率系数

Materials	$A_{1}$	$A_{2}$	$A_{3}$	$λ_{1}^{2}$	$λ_{2}^{2}$	$λ_{3}^{2}$
ZnSe	4.925 7	3.247 7 $e^{- 5}$	22.206 3	0.051 3	19.315	16 574
Ga₂As₃₀S₆₈	4.044 5	1.323 6 $e^{- 4}$	1.282 $e^{- 5}$	0.039 5	14.107	29.575
Ge₅As₁₀S₈₅	3.154 6	-3.007 5 $e^{- 4}$	-0.058 1	0.071 9	15.124	-196.7

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：

D = - (\frac{λ}{c}) \cdot (\frac{\partial^{2} n_{e f f}}{\partial λ^{2}})

，

（2）

here $λ$ is the wavelength in micrometers， D is the dispersion parameter of the transmission mode， $n_{e f f}$ 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 $H_{1}$ to $H_{2}$ 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

H_{1}

H_{2}

and 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

H_{1}

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

H_{1}

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.

Fig. 2 Calculated dispersion curves of the fundamental quasi-TE (a) the dispersion parameter curves for the fundamental quasi-TE mode calculated from $n_{e f f}$ for eight waveguide geometries employing Ga₂As₃₀S₆₈ glass for both the upper and lower claddings, (b) and (c) Map of the dispersion parameter of w=4 and 8 μm ZnSe rib waveguides as a function of core thickness and wavelength, respectively, employing Ge₅As₁₀S₈₅ glass for both the upper and lower claddings. The dash lines show the change of the ZDWs

图2 TE基模的色散曲线 (a)采用Ga2As30S68做包层的8种波导结构的有效折射率计算TE基模的色散，(b-c)上下包层为Ge5As10S85 4和8 μm宽不同结构的ZnSe脊波导色散图，虚线表示0色散点位置

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， $H_{1}$ =2 μm， $H_{2}$ =1 μm and w=8 μm， $H_{1}$ =2 μm， $H_{2}$ =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.

Fig. 3 The optical filed distribution for quasi-TE polarization in the waveguide (a-c) for w=4 μm, and (d-f) for w=8 μm waveguide with $H_{1}$ =2 μm and $H_{2}$ =1 μm at a wavelength of 2, 6, and 10 μm, respectively

图3 (a-c)宽4 μm和(d-f)宽8 μm, H1=2 μm和H2=1 μm的波导在2、6和10 μm波长处TE光场分布

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，

γ = \frac{2 π}{λ} \frac{n_{2}}{A_{e f f}}

，

（3）

where $n_{2}$ is the nonlinear refractive index of ZnSe ^［

26］，

A_{e f f}

is the eﬀective area of the propagating mode， given by：

A_{e f f} = \frac{{(\int \int_{- \infty}^{+ \infty} |E^{2}| d x d y)}^{2}}{(\int \int_{- \infty}^{+ \infty} {|E|}^{4} d x d y)}

，

（4）

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， $H_{1}$ =2 μm and $H_{2}$ =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.

Fig. 4 Eﬀective area and nonlinear coeﬃcient of the fundamental mode calculated in the waveguides (a) w = 4 μm, $H_{1}$ =2 μm, and $H_{2}$ =1 μm (b) w = 8μm, $H_{1}$ =2 μm, and $H_{2}$ =1 μm. (c) Dispersion distribution of the waveguides with w = 4 and 8μm. (d) The second-order dispersion of the waveguides with w = 4 and 8 μm.

图4 计算波导基模有效面积和非线性系数(a) 宽4 μm (b) 宽8 μm, H1=2 μm, H2=1 μm。宽4和8 μm波导的(c)色散分布 (d) 二阶色散

We performed the simulations of SC generation by using a generalized nonlinear Schr $\ddot{o}$ dinger equation （GNLSE） with a chirp-free Gaussian-shaped pump pulse as the initial condition ^［

36-37］，

\frac{\partial}{\partial z} A (z, T) = - \frac{α}{2} A + \sum_{k \geq 2} \frac{i^{k + 1}}{k!} β_{k} \frac{\partial^{k} A}{\partial T^{k}} + i (γ + \frac{α_{2}}{2 A_{e f f}}) (1 + \frac{i}{ω 0} \frac{\partial}{\partial T})

\times (A (z, T) \int_{- \infty}^{\infty} R (T) {|A (z, - T^{'})|}^{2} d T^{'})

，

（5）

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- $\frac{z}{v_{g}}$ is the retarded time frame moving at the group velocity v_g = $\frac{1}{β_{1}}$ ， 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，

β_{k} (ω) = \frac{d^{k} β}{d ω^{k}} |_{ω = ω 0} (k \geq 2)

is the

k^{t h}

-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 =

\frac{n_{2} ω_{0}}{c A_{e f f} (ω_{0})}

， where n₂ is the nonlinear refractive index ^{［Reference 26

Baidu Scholar}26］ and c is the speed of light in vacuum，

A_{e f f} (ω_{0})

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 ^{［Reference 38

Baidu Scholar}38］. Finally， the material response function includes both the instantaneous electronic response （Kerr type） and the delayed Raman response and has the form：

R (t) = (1 - f_{R}) δ (t) + f_{R} h_{R} (t)

，

（6）

where the delayed Raman contribution $h_{R} (t)$ is given by：

h_{R} (t) = \frac{τ_{1}^{2} + τ_{2}^{2}}{τ_{1} τ_{2}^{2}} e x p (- \frac{t}{τ_{2}}) s i n (\frac{t}{τ_{1}})

（7）

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 $f_{R}$ 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.

Fig. 5 Simulated SC spectra at a pump wavelength of (a) 3.0 µm, (b) 4.5 µm, (c) 3.0 µm and (d) 4.5 µm for the two waveguides at different peak power up to 20 kW, respectively.

图5 在泵浦波长为(a)3.0 μm (b)4.5 μm (c)3.0 μm和(d)4.5 μm的两个波导在不同峰值功率时的SC仿真光谱

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］.

Fig. 6 Simulated SC spectra at different pump wavelengths of (a) 3.0 µm, (b) 3.5 µm, (c) 4.0 µm, and (d) 4.5 µm for the two waveguides with a peak power of 20 kW, respectively.

图6 两个波导在20 kW峰值功率不同泵浦波长(a)3.0 μm (b)3.5 μm (c)4.0 μm (d)4.5 μm的SC光谱

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 are 13.724 ps/nm/km and 22.863 ps/nm/km at 4.5 μm， respectively. The second-order dispersion $β_{2} = - \frac{D λ^{2}}{2 π c}$ = -0.1483 ps²/m and -0.240 6 ps²/m， and this leads to a dispersion length $L_{D} = τ_{P}^{2} / |β_{2}|$ = 0.152 m and 0.093 m where $τ_{p}$ is the laser pulse duration. Using the waveguide nonlinear parameter γ of $0.151 W^{- 1} m^{- 1}$ and $0.0965 W^{- 1} m^{- 1}$ for these two waveguides as shown in Fig.4 （a） and （b）， we obtained a nonlinear length L_NLat 20 kW of coupled power via $L_{N L} = 1 / γ P$ ， being 0.331 mm and 0.520 mm， respectively. The soliton order $n_{s o l} = \sqrt[]{L_{D} / L_{N L}}$ = 21.4 and 13.3.

Fig. 7 The spectral evolution plots and temporal density plots corresponding to two curves in Fig. 5.2 (d) at a peak power of 20 kW for (a) the spectral evolution plot of 4 μm waveguide, (b) the spectral evolution plot of 8 μm waveguide, (c) the temporal density plot of 4 μm waveguide, (d) the temporal density plot of 8 μm waveguide.

图7 对应图5.2 (d) 20千瓦峰值功率的SC光谱和时间谱,(a)4 μm波导SC光谱,(b)8 μm波导SC光谱,(c)4 μm波导时间谱,(d)8 μm波导时间谱

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

n_{s o l}

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.

n_{s o l}

-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 ^{［Reference 40

Baidu Scholar}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 ^{［Reference 40

Baidu Scholar}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 ^{［Reference 40

Baidu Scholar}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.

2 Conclusion

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.

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