Mid-Infrared Supercontinuum and Frequency Comb Generations by Different Optical Modes in a Multimode Chalcogenide Strip Waveguide

Supercontinuum (SC) with broad bandwidth and high coherence is crucial in the SC-based frequency comb source generation. In this paper, we numerically investigate the mid-infrared (MIR) SC generations with the three optical modes (TE<sub>00</sub>, TE<sub>10</sub>, and TE<sub>20</sub>) in a multimode chalcogenide (As<sub>2</sub>Se<sub>3</sub>) strip waveguide. The waveguide structure is carefully engineered to ensure that the pump pulses are propagated in the normal dispersion regions of the considered three optical modes. Highly coherent and octave-spanning MIR SCs are generated when the optimized pump pulse with 80-fs pulse duration, 3-kW peak power, and 3-<inline-formula> <tex-math notation="LaTeX">$\mu \text{m}$ </tex-math></inline-formula> center wavelength is used. Moreover, the nonlinear dynamics of the SC generation are numerically analyzed. Finally, the SC-based frequency combs are numerically demonstrated when a pulse train with a repetition rate of 50 MHz is used as the pump source and launched into the multimode As<sub>2</sub>Se<sub>3</sub>-based strip waveguide. It is believed that the generated MIR SC and SC-based frequency comb sources have important applications in biophotonics, metrology, and sensing.


I. INTRODUCTION
Generation of optical frequency comb has attracted great research interests due to the broad applications in optical communication, metrology, molecular detection, etc [1], [2]. For example, the wavelength-division multiplexing as one of the key technologies in optical communication systems requires a large number of discrete-wavelength laser sources. They can be replaced by one frequency comb source produced through a single seed laser. Thus, the complexity and cost of the system are greatly reduced [3], [4]. In general, frequency comb sources can be obtained from The associate editor coordinating the review of this manuscript and approving it for publication was Nicola Andriolli . mode-locked lasers, micro-resonator based Kerr comb, or supercontinuum (SC) generated in nonlinear fiber or waveguides [5]- [10]. However, direct generation of self-referenced frequency comb in mode locked lasers is currently impossible since no gain medium could cover the span of an octave. In contrast, octave spanning frequency combs are achievable by exciting Kerr comb in micro-resonators or SC in nonlinear optical medium. Kerr comb in micro-resonators is the most promising technique to fabricate on-chip comb sources [11]. However, there are still problems that impede the application of Kerr frequency comb sources. Due to the high quality factor (Q-factor), the intrinsic thermal effect will cause instability of the source. Another problem that limits the application is the challenging fabrication of the high Q-factor micro-resonators [12]. Compared with Kerr comb sources, the generation of SC-based frequency comb is much easier to launch a pump pulse train into a nonlinear fiber or waveguide. Before the thermal and fabrication problems of Kerr comb sources are solved, it is expected that SC-based frequency comb generation will serve as the preferred technique to be adopted in applications.
The dynamics that contribute to the SC generation including modulation instability (MI), soliton fission, four wave mixing (FWM), dispersive wave generation, and Raman scattering have been systematically investigated [13]. The group velocity dispersion at the pump launching wavelength and higher-order dispersion coefficients over the whole spectral range are critical effects that determine the quality of the SC generated [13], [14]. The SC generation pumped at anomalous dispersion region has been studied a lot owing to the large bandwidth benefiting from the combined contribution of all effects mentioned above. However, the perturbation sensitivity of MI greatly degrades the coherence of SC generated with pump pulses accompanied by noise [15]. A method of generating the coherent SC is to design nonlinear media with all-normal dispersion (ANDi) profile within the wavelength of interest [16], [17]. The incoherent processes of MI will be well suppressed. Recently, Milan et al. experimentally reported the coherent SC generation through a simple post-process technique to control the waveguide dispersion to obtain ANDi profile in a hybrid chalcogenide/silicongermanium system [18]. Yuan et al. experimentally demonstrated a coherent SC by using ANDi chalcogenide all-solid microstructured fiber [19]. Fang et al. numerically achieved a three-octave coherent SC using ANDi Si 3 N 4 slot waveguide [20]. When the pump pulses are launched and propagate into a normal dispersion waveguide, the spectral broadening is mainly induced by the self-phase modulation (SPM) and optical wave breaking (OWB) [21]. The SPM and OWB effects are self-seeded processes, which are coherent processes, degradation of the coherence of the generated SC will be avoided [22], [23]. However, there are still some limitations for coherent SC generation in ANDi region when the pump pulse duration or the propagation length is increasing [24], [25]. The reasonable pulse duration and propagation length need to be carefully selected. Besides the dispersion effect, the nonlinearity of waveguide is also important for SC generation. Adoption of materials with high Kerr nonlinearity and low nonlinear loss can enhance the nonlinear processes, which are beneficial to the SC generation.
Chalcogenide glasses are considered as one of the promising materials for mid-infrared (MIR) SC generations [26]- [29]. For example, the transparency window of As 2 Se 3 is in the wavelength range from 0.85 to 17.5 µm [28]. However, the research works are mainly focusing on the MIR SC generation by the fundamental mode of chalcogenide waveguide [29]- [32]. Obviously if the pump light is coupled into a higher-order mode of the waveguide, the nonlinear dynamics of the MIR SC generation will be different to that with fundamental mode because of the remarkable difference in the dispersion characteristics [33], [34]. In this paper, we will investigate the MIR SC generations by different optical modes, which have potential applications in mode-division multiplexing and multimode sensing systems. Highly coherent and octave-spanning MIR SCs are generated in a multimode chalcogenide (As 2 Se 3 ) strip waveguide by using three optical modes (TE 00 , TE 10 , and TE 20 ), which are designed with ANDi profiles within the wavelength of interest. SC-based frequency combs are achieved when a pulse train with 50 MHz repetition rate is used as the pump source. The paper is organized as follows: In Section II, we design a multimode As 2 Se 3 -based strip waveguide, and introduce the theoretical model. In Section III, the MIR SC generations with the three optical modes are numerically investigated with variation of pump parameters. Generation of octave-spanning MIR frequency combs are demonstrated with the pulse train. Conclusions are drawn in Section IV. Fig. 1(a) shows the cross-section of the designed As 2 Se 3 strip waveguide on a MgF 2 substrate. The As 2 Se 3 waveguide has a width W of 14 µm and a height H of 0.8 µm. The thickness of the MgF 2 substrate used is 5.0 µm in the simulation. The fabrication of such a waveguide has been developed [35], [36]. It can be fabricated through the film deposition with thermal evaporation [37], photolithography [38], inductively coupled plasma etching with the CF 4 /O 2 gas mixture or CHF 3 gas [39], [40], and photoresist removing by wet chemical stripping [29]. The refractive indices of As 2 Se 3 and MgF 2 are calculated by the Sellmeier equation

II. WAVEGUIDE DESIGN AND THEORETICAL MODEL
where λ is the wavelength. The coefficients of the Sellmeier equation for As 2 Se 3 and MgF 2 are given in Table 1.
With the full-vector finite element method (FEM), the mode field distributions of the three quasi-TE modes (TE 00 , TE 10 , and TE 20 ) calculated at wavelengths 3 and 6 µm are shown in Fig. 1(b). From Fig. 1(b), most of the mode field energy is well confined in the waveguide.
The effective refractive index N eff and effective mode field area A eff at different wavelengths for the three optical modes  are shown in Fig. 2(a), which are also calculated by the fullvector FEM. From Fig. 2(a), the differences of N eff and A eff for different modes are enlarged as the wavelength increases. The group-velocity dispersion D is derived from N eff as [42] where c is the velocity of light in vacuum. Fig. 2(b) shows the D(λ) curves for the three optical modes. From Fig. 2  where F(x, y) is the distribution of the mode fields and n 2 = 2.4 × 10 −17 m 2 /W is the nonlinear refractive index of As 2 Se 3 [41]. The γ (λ) curves of the three optical modes are shown in Fig. 2(b). From Fig. 2(b), the three curves are mostly overlapped, which means they have almost same nonlinearity. This is mainly because the small deviation of A eff between the three optical modes. The evolution dynamics of the short pulse propagating in the designed waveguide are modelled by a generalized nonlinear Schrödinger equation [31], [32] ∂A ∂z + where A is the slowly varying envelope in a retarded frame T , β m (m = 2, 3, . . ., and 13) is the mth-order dispersion coefficient calculated from a Taylor expansion of the propagation constant at the carrier frequency of the pump pulse, and α is the linear loss coefficient. R is the nonlinear response function as following where f R , δ(t), and h R (t) are the fractional contribution of the Raman response, instantaneous electronic response, and delayed Raman response function, respectively. h R (t) is described by the Green's function of a damped harmonic oscillator, which is expressed as where τ 1 is the Raman period corresponding to the phonon oscillation frequency and τ 2 is the Raman gain spectral bandwidth. Some parameters used in the simulations are listed in Table 2 [40], [41], [43]. The coherence of SC is a key indicator to evaluate the quality of the SC generated. To quantify the coherence of SC, three important factors are calculated as following [44], [45] where g 1,2 is the first-order degree of coherence, which indicates the correlation of the signals from shot-to-shot at wavelength λ. A k (λ) is the spectral amplitude of the SC generated with the k-th shot pump pulse with random noise. The angular brackets denote the ensemble average among the independent pairs of the generated spectra. The weighted degree of coherence R in (8) measures the averaged coherence in the whole spectrum, where P(λ) = |A(λ)| 2 denotes the ensemble average power spectrum of the generated SC. In (9), K is used to enlarge the detail of R when it is close to 1 for highly coherent SC.

III. SIMULATION RESULTS AND DISCUSSION
As the base of frequency comb, generation of SC with large bandwidth and high coherence is first investigated. By using the Runge-Kutta method with adaptive step-size to solve (4), the evolution dynamics of the SC generation can be characterized [46]. In the following simulation, the number of the frequency bins is chosen as 2 15 . And the up to 13th-order dispersion coefficients are considered. A hyperbolic secant pulse with a complex amplitude of P 1/2 0 sech(t/T 0 ) are used as the pump and coupled into the three optical modes TE 00 , TE 10 , and TE 20 of the designed waveguide with a length of 1 cm, respectively. The coupling efficiency can be improved through designing the tapered couplers at both ends of the waveguide [47]. And the other two higher-order modes can be efficiently excited by the TE 00 mode based on the tapered directional coupler scheme [48], [49]. In this section, we will vary the launching wavelength, peak power and pulse duration respectively to investigate the impact of these parameters and seek for appropriate parameter combination that should be used to optimize the performance of the SC generated.
We first investigate the impact of launching wavelength of the pump pulse. The pulse duration T 0 = 240 fs and peak power P 0 = 2 kW, which are moderate values to model the typical trends of dynamics along the variation of λ 0 = 2.5, 3, and 3.5 µm. Such laser sources centered at the three wavelengths are available from the previous works [50]- [52]. By using the dispersion and nonlinearity parameters of TE 00 , TE 10 , and TE 20 modes, respectively, the spectral and temporal profiles of the SCs generated at the output point of the waveguide are shown in Fig. 3. In Figs. 3(a) and 3(b), when a pump pulse at λ 0 = 2.5 µm is coupled into the TE 00 mode, the −40 dB bandwidth of the generated SC spans from 1.6 to 4.0 µm, covering 1.26 octaves. As λ 0 is increased to 3 and 3.5 µm, the SC is also shifted towards longer wavelength with simultaneous increase of the bandwidth. The −40 dB bandwidths of the generated SCs with 3 and 3.5 µm pump pulses span from 1.7 to 4.5 µm and 2.0 to 5.0 µm, respectively, covering 1.49 and 1.59 octaves. Besides, the spectral and temporal shapes of the generated SCs are asymmetric, and some burrs appear in the temporal profile due to the asymmetric dispersion profile. For the SC generation, because the pump pulses are launched at wavelengths with normal dispersions, SPM is the dominating nonlinear process at the initial stage. Then, the OWB, which is caused by the third-order dispersion and self-steepening, further broadens the optical spectra.  . For the TE 10 mode, the −40 dB bandwidths of the SCs cover 1.43, 1.6, and 1.72 octaves, respectively. When the pump pulse is coupled into the TE 20 mode, the spectral and temporal profiles of the generated SCs show notable differences as shown in Figs. 3(e) and 3(f). The SCs cover 1.33, 1.83, and 2.07 octaves with pump pulses at 2.5, 3, and 3.5 µm, respectively. The oscillation structures on the spectra mainly caused by enhanced SPM effect compared to TE 00 and TE 10 modes. Its dispersion is reduced since the dispersion curve of the TE 20 mode is much closer to zero than TE 00 and TE 10 modes in most spectral region considered, which is beneficial to the nonlinear spectral broadening. In addition, it is worth noting that when λ 0 is further shifted from 3 to 3.5 µm, the bandwidth increments of the SCs are not obvious except for the TE 20 mode. The long wavelength part of the SC generation of TE 20 mode is enhanced, but the pump sources with center wavelength above 3 µm are difficult to obtain. Therefore, λ 0 is chosen as 3 µm in the following discussion.
Except for the launching wavelength λ 0 , the peak power P 0 also plays an important role in the SC generation. In Fig. 4, we show the comparison of the generated SC with λ 0 = 3 µm and T 0 = 240 fs while the peak power is varied as P 0 = 1, 2, and 3 kW. The three rows of Fig. 4 show the results with TE 00 , TE 10 , and TE 20 modes, respectively. When the pump pulse is launched into TE 00 mode as shown in Figs. 4(a) and 4(b), the optical spectra extend on both sides due to the SPM effect and reach 1.10, 1.47, and 1.69 octaves for P 0 = 1, 2, and 3 kW, respectively. Figs. 4(c) and 4(d) show the results obtained with the TE 10 mode. The −40 dB bandwidths of the generated SCs cover 1.21, 1.59, and 1.79 octaves, respectively, which are slightly larger than those of the TE 00 mode. The −40 dB bandwidths of the SCs are further extended to 1.24, 1.83, and 2.13 octaves with the TE 20 mode. Compared with the results shown in Figs. 4(a) to 4(d), the spectral and temporal profiles of the SCs with TE 20 mode are asymmetric and show some oscillating structures. According to the results shown in Figs. 4(a) to 4(f), the optimized P 0 = 3 kW is used in the following discussions.
The influence of T 0 on the SC generation is also investigated. Figs. 5(a) to 5(f) show the spectral and temporal profiles of the SCs when pump pulses with λ 0 = 3 µm and P 0 = 3 kW are operated at T 0 = 80, 240, and 480 fs and coupled into the TE 00 , TE 10 , and TE 20 modes, respectively. In Figs. 5(a) and 5(b), when the pump pulses with T 0 = 80, 240, and 480 fs are coupled into the TE 00 mode, the −40 dB bandwidths of the SCs cover 1.82, 1.69, and 1.57 octaves, respectively. When the TE 10 mode is chosen, the −40 dB bandwidths of the SCs are slightly increased to 1.93, 1.79, and 1.66 octaves, respectively, as shown in Figs. 5(c) and 5(d). The performance of the SCs with the TE 20 mode are some different, the −40 dB bandwidths of the SCs cover 2.33, 2.13, and 1.48 octaves, respectively. The bandwidths of SCs pumped with 80 and 240 fs pulses are significantly enhanced comparing with the other two modes. However, the SC bandwidth pumped with 480 fs pulse is decreased. Comparing the results shown in Fig. 5, it is obvious that the SCs with the 80-fs pump pulse have the best performance in optimizing the bandwidth and flatness. The main reason can be explained as following. The pulse energy will be increased when P 0 is fixed and T 0 increases. At the same time, its initial bandwidth gradually becomes narrower. Each wavelength part will have more energy. The nonlinear effect will be enhanced. Many oscillations appear when the pulse with duration of 480 fs is used. The flatness of SC becomes poor. Besides, the narrow initial bandwidth limits the spectral broadening because the number of frequency components participated in the initial SPM and OWB stages might be less than the shorter duration pulse which has larger initial bandwidth. Therefore, compared with the 240-fs and 480-fs pump pulses, the 80-fs pump pulse has the best performance for generating the flat and broadband SC.
Based on the above investigation, the optimized parameter combination λ 0 = 3 µm, P 0 = 3 kW, and T 0 = 80 fs will be used to study the evolutions of the SC signal. Fig. 6 shows the evolutions of the spectral and temporal profiles along the propagation of the pump pulse for the TE 00 , TE 10 , and TE 20 modes. In Figs. 6(a) and 6(b), the pump pulses with λ 0 = 3 µm, P 0 = 3 kW, and T 0 = 80 fs are coupled into the TE 00 mode. At the initial stage, the SPM plays an important role. The optical spectrum of the pump pulse broadens symmetrically and shows some oscillations. After the propagation of ∼0.05 cm, the OWB effect becomes obvious, and sidelobes emerge on both sides of the optical spectrum. New frequency components are generated at the leading and trailing edges of the pump pulses through the four-wave mixing effect. The sidelobe on the short wavelength side emerges earlier. It is mainly due to the self-steepening effect [23]. With further propagation, the broadening of spectrum on blue side is terminated because of the large dispersion in the spectral region of <2 µm. The shape of the generated SC becomes asymmetric gradually because of the continuous broadening on red side. In the time domain, the pulse duration is broadened monotonously with the increase of the propagation length. The evolutions of SC in the TE 10 mode are similar with those of the TE 00 mode. However, the evolutions with the TE 20 mode shown in Figs. 6(e) and 6(f) are significantly different to those with the TE 00 and TE 10 modes. Deep oscillations appear on the spectrum during the spectral broadening at very early stage, and gradually disappear after the propagation in a long distance. The oscillations decrease a lot near z = 1 cm. The SC in the TE 20 mode extends on the long wavelength side to much longer wavelength than those of the TE 00 and TE 10 modes. And the increment in bandwidth does not change obviously near z = 1 cm. That is why we chose 1 cm as the length of waveguide. Finally, the SCs generated with the TE 00 , TE 10 , and TE 20 modes reach 1.82, 1.93, and 2.33 octaves, respectively.
Besides the spectral bandwidth, the coherence of SCs is an important feature that determines the quality of frequency comb that based on the SC generation. To investigate the coherence of the generated SCs, the first-order degree of coherence g (1) 1,2 is used. Figs. 7(a) to 7(d) show the generated VOLUME 8, 2020 SCs and g (1) 1,2 calculated with 50 shot pump pulses with the one-photon-per-mode quantum noise [13], [44]. The random noise is individually generated for each shot by n = ηN exp(i2π U ), where η is the noise factor denoting the amplitude of the noise relative to the input pulse amplitude, N is a random variable with the standard normal distribution, and U is a random variable with the uniform distribution between 0 to 1 [44]. A noise level of η = 0.01 is used in the simulation. In Figs. 7(a) to 7(c), the grey plots are the overlapped SC spectra of the 50 shots for the TE 00 , TE 10 , and TE 20 modes, respectively. The red, blue, and olive curves in Figs. 7(a)-7(c) represent the averaged spectra of each 50 shots. All the generated spectra show slight variations from shot to shot. The calculated g (1) 1,2 of the three modes are shown in Fig. 7(d). From Fig. 7 1,2 is almost 1 in the whole wavelength ranges considered, which indicates a high coherence of the SCs.
To quantify the coherence and the stability of the SCs, we simulated the SC generation with different noise levels and calculate the weighted degree of coherence R. Fig. 8(a) shows the relationship between R and lg(η) in the whole spectral range for the three modes considered. The values of R for the TE 00 , TE 10 , and TE 20 mode are 0.52, 0.40, and 0.32 at lg(η = 0.2). It implies that the TE 00 mode may have slightly better noise tolerance than the TE 10 and TE 20 modes when η is high. For all the three modes, the value of R increases as lg(η) decreases, and it is very close to 1 when lg(η) ≤ −2.
The variable K is also plotted in Fig. 8(b) since R is close to 1 when lg(η) ≤ −2. It can be seen from Fig. 8(b) that the three curves are very close, and the difference between the values of K for the modes are not obvious when lg(η) ≤ −2. Thus, the SCs generated by the considered three modes are highly coherent when η ≤ 0.01.
In order to demonstrate the generation of octave-spanning SC-based frequency comb, we launch a train of 50 pulses with 50 MHz repetition rate into the designed waveguide. The frequency combs generated by the TE 00 , TE 10 , and TE 20 modes are shown in Figs. 9(a), 9(b), and 9(c), respectively. The shaded color regions represent the highly dense frequency combs. The zoom-in views shown as the insets clearly illustrate the fine frequency comb lines with an interval of 50 MHz which equals to the repetition rate of the seed pulse train.

IV. CONCLUSION
In summary, we have designed a multimode As 2 Se 3 -based strip waveguide and numerically investigate the MIR SC generations when pump pulses launched in the normal dispersion regions of the TE 00 , TE 10 , and TE 20 modes under different wavelengths of pump pulse, peak powers and pulse duration. Highly coherent multi-octave spanning SCs are generated with each of the three modes. When the optimized pump pulse parameters of λ 0 = 3 µm, P 0 = 3 kW, and T 0 = 80 fs are used, the SCs generated with the TE 00 , TE 10 , and TE 20 modes reach 1.82, 1.93, and 2.33 octaves, respectively. All of the SCs generated by the considered three modes are highly coherent even at the noise amplitude η is 0.01. Finally, based on the generated MIR SCs, the octave-spanning frequency comb sources are achieved when a train of 50 pulses at 50 MHz repetition rate is launched into the designed waveguide. It is believed that the generated MIR SC and SC-based frequency comb sources have important applications in biophotonics, metrology, and sensing. And the exploration of higher-order mode with unique dispersion characteristics to generate the special SCs in one single waveguide for some MIR applications would be of great interest.   CHONGXIU YU graduated from the Beijing University of Posts and Telecommunications (BUPT), Beijing, China, in 1969. She is currently a Professor with BUPT. She is also engaged in university education and research work and has been the Principal Investigator of many projects supported by the China 863 plan, the National Natural Science Foundation, the National Ministry of Science Technology, and so on. Until now, she has published more than 300 papers. Her research interests are optical fiber communication, photonic switching, and optoelectronics technology and its applications. She is a member of the Chinese Institute of Communication, the Committee of Fiber Optics and Integral Optics, and the Chinese Optical Society.