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Photonics
, Volume 7 (3) – Jul 3, 2020

/lp/multidisciplinary-digital-publishing-institute/arsenic-sulfide-suspended-core-fiber-simulation-with-three-parabolic-gNS4RmVDVu

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- Multidisciplinary Digital Publishing Institute
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- 2304-6732
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- 10.3390/photonics7030046
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hv photonics Article Arsenic Sulﬁde Suspended-core Fiber Simulation with Three Parabolic Air Holes for Supercontinuum Generation 1 , 2 , 2 2 Tao Peng *, Xunsi Wang and Tiefeng Xu College of Electrical Engineering, Zhejiang University of Water Resources and Electric Power, Hangzhou 310018, China Laboratory of Infrared Material and Devices, The Research Institute of Advanced Technologies, College of Information Science and Engineering, Ningbo University, Ningbo 315211, China; wangxunsi@nbu.edu.cn (X.W.); xutiefeng@nbu.edu.cn (T.X.) * Correspondence: pengt@zjweu.edu.cn Received: 17 May 2020; Accepted: 2 July 2020; Published: 3 July 2020 Abstract: Highly nonlinear suspended-core ﬁbers (SCFs) with tunable dispersion have attracted much attention in the ﬁelds of Raman ampliﬁcation, optical frequency combs, broadband and ﬂat supercontinuum generation (SCG). To address the limitation of applications due to its fragile suspension arms, this study proposes the design of a ﬁber structure with three parabolic air holes. Numerical simulations are performed to optimize an arsenic sulﬁde SCF in terms of dispersion management and SCG in the wavelength range from 0.6 m to 11.6 m. Results show that the proposed SCF has dual zero-dispersion wavelengths (ZDWs) that can be shifted by adjusting the parabolic coecient of the air-hole and the equivalent diameter of the suspended core. By means of structural optimization, an SCF with 1 m equivalent diameter and a parabolic coecient of 0.18 m is proposed. The ﬁrst ZDW of the SCF is blue-shifted to 1.541 m, which makes it possible to use a commercial light source with a cheaper price, more mature technology and smaller volume as the pump source. SCG is studied by solving the generalized nonlinear Schrödinger equation using the split-step Fourier method, and a 0.6–5.0 m supercontinuum spectrum is obtained at a pump source peak power of 40 kW. Keywords: suspended-core ﬁber; parabolic air hole; characteristics analysis; nonlinear optics; supercontinuum 1. Introduction Because of its wide spectrum, high brightness and high coherence, supercontinuum (SC) has wide applications in photometry, optical coherent imaging, spectroscopy, etc. [1–3]. Supercontinuum generation (SCG) is a kind of spectrum broadening caused by the pump pulse generating new frequency components under the inﬂuence of dispersion and various nonlinear eects in medium [4–6]. As a special microstructured ﬁber, SCF has an important characteristic of adjustable dispersion [7,8]. In addition, it has a smaller core, so a higher nonlinear coecient can be obtained [9,10]. By optimizing the structure parameters of the SCF, it can obtain a higher nonlinear coecient and more reasonable dispersion distribution, which is of great help to the SCG. Silica-based glass has proven to be highly appropriate for the preparation of SCF. Because of the high viscosity, higher pressures are required during the extrusion process, which leads to the deformation of the mold during the drawing process [11]. The nonlinear application of silica-based SCF in the midinfrared region (MIR) is limited due to the infrared cut-o wavelength and lower nonlinear coecient. In comparison to silica-based glass, the higher optical nonlinearity and broader Photonics 2020, 7, 46; doi:10.3390/photonics7030046 www.mdpi.com/journal/photonics Photonics 2020, 7, 46 2 of 16 transmission window of chalcogenide glass enable the generation of a wider SC spectrum in the Photonics 2020, 7, x FOR PEER REVIEW 2 of 17 MIR [12]. Arsenic sulﬁde glass is an ideal candidate for the fabrication of SCF, as it has a more mature preparation process, lower loss and better mechanical properties than other chalcogenide [12]. Arsenic sulfide glass is an ideal candidate for the fabrication of SCF, as it has a more mature p glass repar forms ation p [13 ro,c14 es]. s, lAn owe SC r losour ss an ce d b with ettera m wavelength echanical prrange opertie fr som tha0.6 n oth m er c to ha 4.1 lco ge m niwas de glgenerated ass forms in a 2 cm long three-hole As S chalcogenide SCF [15]. Gao et al. demonstrated an SCG spanning [13,14]. An SC source with a wavelength range from 0.6 µm to 4.1 µm was generated in a 2 cm long 2 3 th a r 1.37–5.65 ee-hole A sm 2S3 wavelength chalcogenide range SCF in [15a ]. four Gao-hole et al.As dem S on chalcogenide strated an SC SCF G sp [16 an]. nin A g midinfrar a 1.37–5.6ed 5 µSC m 2 5 w spanning avelengtthe h ra 2.5–5.5 nge in a m four spectral -hole A region s2S5 ch was alcodemonstrated genide SCF [1in 6].a A 25 m mm idinlong frared thr See-hole C spann suspended-cor ing the 2.5–5.5 e µ As m s Spe chalcogenide ctral region w SCF as de [17 mo ].nXue strate etdal. in r aepo 25 rted mm lan ong SC thspectr ree-ho um le sus spanning pended- fr com ore A 2.05 s2S 3 c m ha to lco 6.95 gen id m, e 2 3 S which CF [17was ]. Xue generated et al. repo in rte ad19-cm an SC long spectr four um-hole spann SCF ing, fwhich rom 2.0 combined 5 µm to 6.9 a5chalcogeni µm, whichde wa SCF s gen with erate an d iAs n a S 19- center cm lon cor g fe our [18 -h ].ole SCF, which combined a chalcogenide SCF with an As2S3 center core [18]. 2 3 R Reducing educing the thecor coe re size sizis e among is amothe ng most the m e oective st effemethods ctive meto thobtain ods toa o higher btain nonlinear a higher coe non licient. near c At oef pr fic eisent, ent. A ther t pe rear see ntwo t, the types re areof tw SCF o ty , p as es shown of SCF in , aFigur s shoe w1 n . iThe n Fig cor ure e part 1. Th of e c one ore type part of of optical one typﬁber e of o is pti determined cal fiber is by dete the rmsuspended-arm. ined by the suspeIt nd is ed only -arm r.ealized It is onby ly r re educing alized bthe y rewidth ducingof ththe e wsuspension idth of the sarm, uspen as sio shown n arm, a in s s Figur hown e i1 na, Fiwhich gure 1acauses , whichsigniﬁcant causes signdi ifi caculty nt diff in icu the lty ipr n eparation the prepara of tio the n ofﬁber the . fThe iber.As ThS e A thr s2ee-hole S3 threeSCF -hole pr S oposed CF proby posMouawad ed by Moua et w al. ad [17 e]t and al. [the 17] As and S th four e A -hole s2S5 fSCF our-h pr ooposed le SCF 2 3 2 5 p by roG po ao seet d b al. y G [19 ao ] belong et al. [1to 9] this belokind ng toof th SCF is k.in Mor d oeover f SCF,. various Moreovpr eroperties , variousof pr the ope SCF rtiesar oe f signiﬁcantly the SCF are s ai gected nifican by tly br aeakage ffected of by the bre suspension akage of tharm e sus during pensiouse. n arm Consequently during use., another Conseque type ntly of , a SCF noth str eructur typee ois f S pr Coposed F structto ure absolve is prop the osedependence d to absolve on the the dearm. pende Although nce on ththe e arcor m. eA is lth no oug longer h thedetermined core is no lby ong the er d suspension etermined arm, by thits e sus size pecannot nsion abe rmr,e iduced ts size to can obtain not be a rlar edger ucenonlinear d to obtain coe a l arcient ger n(Figur onlineear 1 b), coe and fficithis ent (str Fig uctur ure 1 ebis ), a mor nd e thvulnerable is structurethan is mo the re v pr ul evious nerable one. than The thesilicate previous SCF onwith e. The thr siee lica holes te SCpr F w oposed ith thre by e h Eo bendor les pro p-Heidepriem osed by Eben et do al. rff[-20 He ] iand depr the iem As et S al. SCF [20] with and th four e Aholes s2S3 SC pr Foposed with foby ur W ho ang les p et ro al. po[s 21 ed ] ar by e 2 3 Wa this nkind g et a of l. [ SCF 21] .a Ther re th efor is ke, inthe d of design SCF. T of he are mor fore e, stable the de str sig uctur n of e a with more lower stabled e sdependence tructure with of lo the wecor red e d on epthe endsuspension ence of thearm core is o highly n the s desir uspe ed nsi [o 22 n]. aNevertheless, rm is highly d ther esir eear d e [2 few 2]. Ne studies verthaddr elessessing , therethis are issue few s at tud pr iesent. es addressing this issue at present. Figure 1. Two dierent types of suspended-core ﬁbers (SCFs). (a) SCF with non-independent core Figure 1. Two different types of suspended-core fibers (SCFs). (a) SCF with non-independent core (b) SCF with non-independent core. (b) SCF with non-independent core. Moreover, since the zero-dispersion wavelength (ZDW) of the arsenic sulﬁde bulk glass is ~4 m, Moreover, since the zero-dispersion wavelength (ZDW) of the arsenic sulfide bulk glass is ~4 it will produce a certain blue-shift after being prepared into optical ﬁber. We present in Table 1 a µm, it will produce a certain blue-shift after being prepared into optical fiber. We present in Table 1 brief overview of pioneering work in this area. As observed in Table 1, the ZDWs of these ﬁbers a brief overview of pioneering work in this area. As observed in Table 1, the ZDWs of these fibers are are distributed between 2.3 and 4.5 m due to structural design. However, the pump source in this distributed between 2.3 and 4.5 µm due to structural design. However, the pump source in this wavelength range is huge and expensive, which hinders the commercial use of the SC spectrum. wavelength range is huge and expensive, which hinders the commercial use of the SC spectrum. The The current light source with a wavelength of 1.55 m has the most mature technology, cheapest price current light source with a wavelength of 1.55 µm has the most mature technology, cheapest price and smallest volume. If the ZDW of arsenic sulfide SCF is adjusted to this wavelength by structural optimization, the feasibility of the commercial production of SCG devices will be greatly improved. Photonics 2020, 7, 46 3 of 16 and smallest volume. If the ZDW of arsenic sulﬁde SCF is adjusted to this wavelength by structural optimization, the feasibility of the commercial production of SCG devices will be greatly improved. Table 1. Overview of previously published works. Pump Pump Peak Spectral Year/Ref. Glass Components Structure Length ZDW FWHM Wavelength Power Bandwidth Unit cm m m kW fs m 2013/[16] As S 3-hole 1.3/2.4 2.52 2.6 0.24–1.32 ~200 1.520–4.610 2 3 2014/[15] As S 3-hole 2 2.5 2.5 1.25–4.86 200 0.6–4.1 2 3 2014/[23] As S 4-hole 4.8 2.28 2.3 0.22–1.55 200 1.370–5.650 2 5 2014/[24] AsSe -As S 4-hole 2 3.38 3.389 1.356 ~200 1.256–5.400 2 2 5 2016/[17] As S 3-hole 2.5 2.65 3.5 0.015 300 2.5–5.5 2 3 As S -Ge As Se Te 4-hole 19 3.93 4.5 66 150 2.06–6.95 2018/[24] 2 3 20 20 15 45 In this paper, an As S nonindependent SCF with three parabolic air holes is designed. The eects 2 3 of the structural parameters of an SCF on its eective refractive index (n ), nonlinear coecient, and dispersion can be obtained by simulation. Through optimization of the parameters, the most suitable ﬁber structure is therefore proposed to produce an SC spectrum. Finally, the generalized nonlinear Schrödinger equation (GNLSE) is adopted in order to obtain the corresponding SC spectrum by adjusting the parameters of the pump source and the SCF, and the basic reasons for its generation are analyzed. 2. Structure Design Figure 1a,b shows that suspension arms of the SCF are traditionally constructed by the parallel edges of adjacent air holes. Thereby, a smaller core can be obtained by reducing the suspension arm width. In our previous study [25], we found that most of the energy in the fundamental mode (FM) was transmitted in the core; however, a small part leaked to the junction of the arm and core. To obtain a higher nonlinear coecient, the suspension arm has to be narrowed as much as possible, which makes the preparation of SCF more dicult and signiﬁcantly reduces the mechanical strength. However, this problem is not solvable by employing a circular air-hole [16,26,27]. In this study, a new type of SCF with a parabolic air-hole is designed. The suspension core is composed of the top of the parabolic air-hole, and the width of the arm can be determined by the parabolic function, instead of by simple parallel lines. The cross-sectional view of the proposed SCF is shown in Figure 2, where the gray shading depicts As S glass, and the three white holes depict air. 2 3 The diameter of the SCF is 125 m. The inner edge of the air-hole is a parabolic structure, whose speciﬁc function is y = a x , where a is a variable structural parameter, whose value is selected between 0.02 and 0.3 m . When parameter a increases, the top of the air holes becomes narrow, which can make the core smaller. The suspension core is measured by a circle, whose diameter is parameter d [28]. The diameter d determines the size of the suspension core, and it is varied to be 1, 3, 5 and 9 m. The outer edge of the air-hole is a circular structure with diameter d1. In past research, we found that the diameter of the air holes in the SCF has a limited eect on the FM. Hence, d1 is set to a ﬁxed value of 60 m, which facilitates the fabrication of the ﬁber. Photonics 2020, 7, x FOR PEER REVIEW 3 of 17 Table 1. Overview of previously published works. Glass Pump Pump Peak Spectral Year/Ref. Structure Length ZDW FWHM Components Wavelength Power Bandwidth Unit cm µm µm kW fs µm 2013/[16] As2S3 3-hole 1.3/2.4 2.52 2.6 0.24–1.32 ~200 1.520–4.610 2014/[15] As2S3 3-hole 2 2.5 2.5 1.25–4.86 200 0.6–4.1 2014/[23] As2S5 4-hole 4.8 2.28 2.3 0.22–1.55 200 1.370–5.650 2014/[24] AsSe2-As2S5 4-hole 2 3.38 3.389 1.356 ~200 1.256–5.400 2016/[17] As2S3 3-hole 2.5 2.65 3.5 0.015 300 2.5–5.5 As2S3- 2018/[25] 4-hole 19 3.93 4.5 66 150 2.06–6.95 Ge20As20Se15Te45 In this paper, an As2S3 nonindependent SCF with three parabolic air holes is designed. The effects of the structural parameters of an SCF on its effective refractive index (neff), nonlinear coefficient, and dispersion can be obtained by simulation. Through optimization of the parameters, the most suitable fiber structure is therefore proposed to produce an SC spectrum. Finally, the generalized nonlinear Schrödinger equation (GNLSE) is adopted in order to obtain the corresponding SC spectrum by adjusting the parameters of the pump source and the SCF, and the basic reasons for its generation are analyzed. 2. Structure Design Figure 1a,b shows that suspension arms of the SCF are traditionally constructed by the parallel edges of adjacent air holes. Thereby, a smaller core can be obtained by reducing the suspension arm width. In our previous study [26], we found that most of the energy in the fundamental mode (FM) was transmitted in the core; however, a small part leaked to the junction of the arm and core. To obtain a higher nonlinear coefficient, the suspension arm has to be narrowed as much as possible, which makes the preparation of SCF more difficult and significantly reduces the mechanical strength. However, this problem is not solvable by employing a circular air-hole [16,27,28]. In this study, a new type of SCF with a parabolic air-hole is designed. The suspension core is composed of the top of the parabolic air-hole, and the width of the arm can be determined by the parabolic function, instead of by simple parallel lines. The cross-sectional view of the proposed SCF is shown in Figure 2, where the gray shading depicts As2S3 glass, and the three white holes depict air. The diameter of the SCF is 125 µm. The inner edge of the air-hole is a parabolic structure, whose specific function is y = a x , where a is a variable structural parameter, whose value is selected between −1 0.02 and 0.3 µm . When parameter a increases, the top of the air holes becomes narrow, which can make the core smaller. The suspension core is measured by a circle, whose diameter is parameter d [29]. The diameter d determines the size of the suspension core, and it is varied to be 1, 3, 5 and 9 µm. The outer edge of the air-hole is a circular structure with diameter d1. In past research, we found that Photonics 2020, 7, 46 4 of 16 the diameter of the air holes in the SCF has a limited effect on the FM. Hence, d1 is set to a fixed value Figure 2. Design views of the SCF geometrical formation. Figure 2. Design views of the SCF geometrical formation. Photonics 2020, 7, x FOR PEER REVIEW 4 of 17 With the decrease in a, the opening of the air-hole becomes larger, the thickness of the suspension With the decrease in a, the opening of the air-hole becomes larger, the thickness of the suspension arm becomes thinner and the area of the suspension core decreases, facilitating the binding of the FM arm becomes thinner and the area of the suspension core decreases, facilitating the binding of the FM in the core. Figure 3a shows the FM ﬁeld distribution of the structure (d = 3 m, a = 0.10−1m ) at in the core. Figure 3a shows the FM field distribution of the structure (d = 3 µm, a = 0.10 µm ) at 0.6 0.6 m wavelength. The LP mode with two vertical degenerate modules, indicated with the red µm wavelength. The LP01 mode with two vertical degenerate modules, indicated with the red arrows arrows in Figure 3b,c is conﬁned tightly in the suspension core demonstrated by the simulation using in Figure 3b,c is confined tightly in the suspension core demonstrated by the simulation using the the commercial software COMSOL. commercial software COMSOL. Figure 3. Simulated 3D plot of fundamental mode (FM) (a) and intensity profile of (b) LP01-x and (c) Figure 3. Simulated 3D plot of fundamental mode (FM) (a) and intensity proﬁle of (b) LP and 01-x −1 LP01-y (d = 3 µm, a = 0.10 µm , λ =1 0.6 µm). (c) LP (d = 3 m, a = 0.10 m , = 0.6 m). 01-y The shape of parabolic air-hole can be generated by means of extrusion instead of stacking. The shape of parabolic air-hole can be generated by means of extrusion instead of stacking. Owing to the pressure, temperature and deformation of the mold, the shape of the air holes may not Owing to the pressure, temperature and deformation of the mold, the shape of the air holes may not be be maintained during the hot drawing process, resulting in significant deformation of the edge of the maintained during the hot drawing process, resulting in signiﬁcant deformation of the edge of the air air holes. Figure 4 shows that even if the air-hole has a significant deformation, the distribution of the holes. Figure 4 shows that even if the air-hole has a signiﬁcant deformation, the distribution of the FM FM is not affected, as long as the top of the air-hole remains unchanged. Because the propagation is not aected, as long as the top of the air-hole remains unchanged. Because the propagation mode is mode is not affected by deformation, the neff of the SCF maintains its original value. not aected by deformation, the n of the SCF maintains its original value. −1 Figure 4. Influence of air-hole (d = 1µm, a = 0.2 µm ) with (a) perfect boundary (b) large defects on SCF’s the electric field distribution of FM and neff. The traditional suspension arm adopts a parallel structure, which drains part of the energy of the FM, whereas the parabolic structure effectively limits the FM to the core. Assuming that the Photonics 2020, 7, x FOR PEER REVIEW 4 of 17 With the decrease in a, the opening of the air-hole becomes larger, the thickness of the suspension arm becomes thinner and the area of the suspension core decreases, facilitating the binding of the FM −1 in the core. Figure 3a shows the FM field distribution of the structure (d = 3 µm, a = 0.10 µm ) at 0.6 µm wavelength. The LP01 mode with two vertical degenerate modules, indicated with the red arrows in Figure 3b,c is confined tightly in the suspension core demonstrated by the simulation using the commercial software COMSOL. Figure 3. Simulated 3D plot of fundamental mode (FM) (a) and intensity profile of (b) LP01-x and (c) −1 LP01-y (d = 3 µm, a = 0.10 µm , λ = 0.6 µm). The shape of parabolic air-hole can be generated by means of extrusion instead of stacking. Owing to the pressure, temperature and deformation of the mold, the shape of the air holes may not be maintained during the hot drawing process, resulting in significant deformation of the edge of the air holes. Figure 4 shows that even if the air-hole has a significant deformation, the distribution of the FM is not affected, as long as the top of the air-hole remains unchanged. Because the propagation Photonics 2020, 7, 46 5 of 16 mode is not affected by deformation, the neff of the SCF maintains its original value. Figure 4. Inﬂuence of air-hole (d = 1m, a = 0.2 m ) with (a) perfect boundary (b) large defects on −1 Figure 4. Influence of air-hole (d = 1µm, a = 0.2 µm ) with (a) perfect boundary (b) large defects on SCF’s the electric ﬁeld distribution of FM and n . SCF’s the electric field distribution of FM and neff. Photonics 2020, 7, x FOR PEER REVIEW 5 of 17 The traditional suspension arm adopts a parallel structure, which drains part of the energy of the The traditional suspension arm adopts a parallel structure, which drains part of the energy of FM, whereas the parabolic structure eectively limits the FM to the core. Assuming that the suspension suspension arm is broken, as shown in Figure 5b,d, the neff of the parabolic structure does not change, the FM, whereas the parabolic structure effectively limits the FM to the core. Assuming that the arm is broken, as shown in Figure 5b,d, the n of the parabolic structure does not change, whereas eff whereas for the traditional structure it decreases from 1.9072 to 1.9067 due to the influence of the for the traditional structure it decreases from 1.9072 to 1.9067 due to the inﬂuence of the propagation propagation mode. Since the nonlinear coefficient, dispersion, and SC spectrum of the fiber are all mode. Since the nonlinear coecient, dispersion, and SC spectrum of the ﬁber are all calculated based calculated based on neff, even a slight variation can cause a dramatic change in these parameters. on n , even a slight variation can cause a dramatic change in these parameters. Figure 5. Inﬂuence of suspension arm defects on two kinds of SCFs. (a) Proposed SCF without Figure 5. Influence of suspension arm defects on two kinds of SCFs. (a) Proposed SCF without structural defects; (b) proposed SCF with structural defects; (c) traditional SCF without structural structural defects; (b) proposed SCF with structural defects; (c) traditional SCF without structural defects; (d) traditional SCF with structural defects. defects; (d) traditional SCF with structural defects. 3. Characteristics Analysis 3. Characteristics Analysis The characteristics of the ﬁber include the n , nonlinear coecient, dispersion, SC, etc. The study The characteristics of the fiber include the neff, nonlinear coefficient, dispersion, SC, etc. The of other characteristics is highly important, such as four-wave mixing (FWM), soliton and the SCG study of other characteristics is highly important, such as four-wave mixing (FWM), soliton and the based on these parameters [19]. These characteristics are not the same in block glass and ﬁber, as they SCG based on these parameters [19]. These characteristics are not the same in block glass and fiber, comprise large dierences in their various structures of ﬁbers. Hence, it is necessary to study the as they comprise large differences in their various structures of fibers. Hence, it is necessary to study inﬂuence of dierent structural parameters on the properties of the SCFs. the influence of different structural parameters on the properties of the SCFs. 3.1. Effective Refractive Index First, we investigate the influence of a on the neff of the fiber with different d values using COMSOL. The neff of the fiber is obtained by the refractive index of As2S3, which is calculated by the Sellmeier formula [30]: B 2 i n ()1 (1) 2 ' C i1 where B and C in (1) are the parameters related to materials. For As2S3 block glass, they are i i 1.8983678, 1.9222979, 0.8765134, 0.1188704, 0.9569903, 0.0225, 0.0625, 0.1225, 0.2025 and 750, respectively. In the simulation, different input wavelengths have different solutions that correspond to different mode fields. Figure 6a shows the neff curves as a function of the wavelength at d = 1 µm. It can be observed from the figure that the effect of a on the neff is not particularly apparent, and we found that the function change of the neff at d = 3, 5 and 9 µm is basically consistent with the law at d = 1 µm; therefore, the introduction of similar data is omitted. We observe that all curves decrease monotonically with the increasing wavelengths, as the fiber structure effectively limits the FM to the suspended core. As shown in Figure 6b, when the value of a is fixed, the neff of the fiber decreases with the increase of d. Further, this law intensifies with increasing wavelength. Photonics 2020, 7, 46 6 of 16 3.1. Eective Refractive Index First, we investigate the inﬂuence of a on the n of the ﬁber with dierent d values using COMSOL. The n of the ﬁber is obtained by the refractive index of As S , which is calculated by the Sellmeier e 2 3 formula [29]: X 2 2 i n () = 1 + (1) 2 0 i=1 where B and C in (1) are the parameters related to materials. For As S block glass, they are i 2 3 1.8983678, 1.9222979, 0.8765134, 0.1188704, 0.9569903, 0.0225, 0.0625, 0.1225, 0.2025 and 750, respectively. In the simulation, dierent input wavelengths have dierent solutions that correspond to dierent mode ﬁelds. Figure 6a shows the n curves as a function of the wavelength at d = 1 m. It can be observed from the ﬁgure that the eect of a on the n is not particularly apparent, and we found that the function change of the n at d = 3, 5 and 9 m is basically consistent with the law at d = 1 m; therefore, the introduction of similar data is omitted. We observe that all curves decrease monotonically with the increasing wavelengths, as the ﬁber structure eectively limits the FM to the suspended core. As shown in Figure 6b, when the value of a is ﬁxed, the n of the ﬁber decreases with the increase of d. Photonics 2020, 7, x FOR PEER REVIEW 6 of 17 Further, this law intensiﬁes with increasing wavelength. Figure 6. Impact of parameters (a) a, (b) d on n of FM. Figure 6. Impact of parameters (a) a, (b) d on neff of FM. It is found that with the increase of d, the FM appears at dierent wavelengths, regardless of a. It is found that with the increase of d, the FM appears at different wavelengths, regardless of a. With the decrease in a and d, the size of the suspension core decreases accordingly. When the operating With the decrease in a and d, the size of the suspension core decreases accordingly. When the wavelength is higher than that in the suspended core, the FM can no longer propagate in the core, and operating wavelength is higher than that in the suspended core, the FM can no longer propagate in there is no corresponding n . The simulation indicates that the n of the ﬁber increases with the e e the core, and there is no corresponding neff. The simulation indicates that the neff of the fiber increases increase of d. This is because a larger area of the suspended core indicates a greater inﬂuence of glass with the increase of d. This is because a larger area of the suspended core indicates a greater influence on the FM in comparison to the ﬁber structure. of glass on the FM in comparison to the fiber structure. Since the n is an important aspect of the dispersion calculation and other nonlinear parameters Since the neff is an important aspect of the dispersion calculation and other nonlinear parameters of the SCF, the functional relationship between the n and the operating wavelength needs to be of the SCF, the functional relationship between the neff and the operating wavelength needs to be accurately determined. In particular, the calculation of SCG is strongly dependent on the dispersion accurately determined. In particular, the calculation of SCG is strongly dependent on the dispersion curve, so the ﬁtting eect and ﬁtting error of the function have a great inﬂuence on the calculation curve, so the fitting effect and fitting error of the function have a great influence on the calculation of of these parameters, which makes the selection of ﬁtting function extremely important. However, these parameters, which makes the selection of fitting function extremely important. However, the the traditional dierence method has unsatisfactory results in cases where the amount of discrete data traditional difference method has unsatisfactory results in cases where the amount of discrete data is is large. A large error arises, particularly in the second derivative. This leads to an inaccuracy in the large. A large error arises, particularly in the second derivative. This leads to an inaccuracy in the dispersion value. Numerous types of functions can be employed for ﬁtting. To reduce the systematic dispersion value. Numerous types of functions can be employed for fitting. To reduce the systematic error caused by the ﬁtting process, the same function type must be chosen. An excessively low order error caused by the fitting process, the same function type must be chosen. An excessively low order of the ﬁtting function yields an R-square value that is minuscule, which cannot accurately express the of the fitting function yields an R-square value that is minuscule, which cannot accurately express functional relationship. In contrast, if the order of the function is excessively high, although R-square the functional relationship. In contrast, if the order of the function is excessively high, although R- square approaches the value of one, the function generates an extreme value in the second derivation of dispersion, which is not in line with the actual situation of dispersion distribution. Since neff is based on the Sellmeier formula, we find that the third-order Gaussian function is the most optimal fitting function through fitting comparisons. xb f (x) a exp[( ) ] i (2) i1 i Using the function fitting tool in the MATLAB software, the preliminary function fitting results can be obtained. Because the fitting results given by the software cannot meet the requirements, some parameters need to be adjusted slightly. It is found that fixing other parameters while changing the c3 value can not only improve the fitting effect, but also ensure that the series of fitting functions have similar change rules. By slightly adjusting c3 (range-accuracy is 0.0001) in the fitting process, the average R-square and the sum of squares due to error of the fitting function reach values up to −5 0.999933314 and 8.24 × 10 , which can accurately reflect the relationship between the operating wavelength and the neff. 3.2. Nonlinear Coefficient The nonlinear coefficient of the fiber is calculated by [31]: Photonics 2020, 7, 46 7 of 16 approaches the value of one, the function generates an extreme value in the second derivation of dispersion, which is not in line with the actual situation of dispersion distribution. Since n is based on the Sellmeier formula, we ﬁnd that the third-order Gaussian function is the most optimal ﬁtting function through ﬁtting comparisons. X 2 x b f(x) = a exp[( ) ] (2) i=1 Using the function ﬁtting tool in the MATLAB software, the preliminary function ﬁtting results can be obtained. Because the ﬁtting results given by the software cannot meet the requirements, some parameters need to be adjusted slightly. It is found that ﬁxing other parameters while changing the c value can not only improve the ﬁtting eect, but also ensure that the series of ﬁtting functions have similar change rules. By slightly adjusting c (range-accuracy is 0.0001) in the ﬁtting process, the average R-square and the sum of squares due to error of the ﬁtting function reach values up to 0.999933314 and 8.24 10 , which can accurately reﬂect the relationship between the operating wavelength and the n . 3.2. Nonlinear Coecient The nonlinear coecient of the ﬁber is calculated by [30]: 2n () = (3) A () e f f 19 2 where n is the nonlinear refractive index of the ﬁber material (for As S , n = 2.92 10 m /W [31]); 2 2 3 2 is the operating wavelength; and A () is the eective area of the FM, which can be obtained by the following expression: ( E(x, y,) dxdy) A () = (4) e f f E(x, y,) dxdy where E(x, y, ) is the electric ﬁeld transverse distribution of the FM, which can be determined by simulation. The corresponding A () can be obtained after postdata processing. Equation (3) indicates that three approaches, including the selection of glass with larger n , blue shifting of the operating wavelength and reduction of A (), are eective in terms of increasing the nonlinear coecient of the SCF. Because the shape of the air-hole and the diameter of the core are determined by a and d respectively, A () and () can be resized by adjusting the two structural parameters. Figure 7 shows that the nonlinear coecient of the SCF has a signiﬁcant inverse proportional relationship with the wavelength. Although the nonlinear coecient decreases with increasing a, this eect is almost negligible compared to its variation with d. When = 0.6 m, the maximum nonlinear 1 1 coecient can reach 49.26965 m W at d = 1 m, which is more than 70 times of that at d = 9 m. Hence, to obtain a higher nonlinear coecient, the core should be reduced to the greatest possible degree, which renders the preparation of the SCF more dicult. However, the large loss coecient of the ﬁber cannot be ignored due to the limited mode ﬁeld diameter. Therefore, the operating wavelength should also be considered as another important factor that signiﬁcantly aects the nonlinear coecient. From Equation (3) and Figure 7, we can clearly deduce that the operational wavelength is inversely proportional to the nonlinear coecient. Notably, the absorption peaks of arsenic sulﬁde glass are mainly concentrated at ~3 m (H2O) and ~4.3 m (H-S) in the MIR. As there is no obvious absorption peak in the near-infrared region, the loss coecient of As S SCF is not excessively large. The blue-shift 2 3 of the operating wavelength is another eective approach to improve the nonlinear coecient. Photonics 2020, 7, x FOR PEER REVIEW 7 of 17 2n ( ) (3) A ( ) eff −19 2 where n2 is the nonlinear refractive index of the fiber material (for As2S3, n2 = 2.92 × 10 m /W [32]); λ is the operating wavelength; and Aeff (λ) is the effective area of the FM, which can be obtained by the following expression: ( E (x,y,) dxdy) A () (4) eff E (x,y,) dxdy where E(x, y, λ) is the electric field transverse distribution of the FM, which can be determined by simulation. The corresponding Aeff (λ) can be obtained after postdata processing. Equation (3) indicates that three approaches, including the selection of glass with larger n2, blue shifting of the operating wavelength and reduction of Aeff (λ), are effective in terms of increasing the nonlinear coefficient of the SCF. Because the shape of the air-hole and the diameter of the core are determined by a and d respectively, Aeff (λ) and γ (λ) can be resized by adjusting the two structural parameters. Figure 7 shows that the nonlinear coefficient of the SCF has a significant inverse proportional relationship with the wavelength. Although the nonlinear coefficient decreases with increasing a, this effect is almost negligible compared to its variation with d. When λ = 0.6 µm, the maximum nonlinear −1 −1 coefficient can reach 49.26965 m W at d = 1 µm, which is more than 70 times of that at d = 9 µm. Hence, to obtain a higher nonlinear coefficient, the core should be reduced to the greatest possible degree, which renders the preparation of the SCF more difficult. However, the large loss coefficient of the fiber cannot be ignored due to the limited mode field diameter. Therefore, the operating wavelength should also be considered as another important factor that significantly affects the nonlinear coefficient. From Equation (3) and Figure 7, we can clearly deduce that the operational wavelength is inversely proportional to the nonlinear coefficient. Notably, the absorption peaks of arsenic sulfide glass are mainly concentrated at ~3 µm (H2O) and ~4.3 µm (H-S) in the MIR. As there is no obvious absorption peak in the near-infrared region, the loss coefficient of As2S3 SCF is not Photonics 2020, 7, 46 8 of 16 excessively large. The blue-shift of the operating wavelength is another effective approach to improve the nonlinear coefficient. Figure 7. Impact of parameter (a) a, (b) d on the nonlinear coecient of FM. Figure 7. Impact of parameter (a) a, (b) d on the nonlinear coefficient of FM. 3.3. Chromatic Dispersion 3.3. Chromatic Dispersion Chromatic dispersion in the ﬁber is predominantly determined by the material and waveguide Chromatic dispersion in the fiber is predominantly determined by the material and waveguide dispersions. In the case of a large ﬁber core, chromatic dispersion is mainly determined by material dispersions. In the case of a large fiber core, chromatic dispersion is mainly determined by material dispersion, whereas waveguide dispersion plays an important role in a narrow core. Due to the small dispersion, whereas waveguide dispersion plays an important role in a narrow core. Due to the small core diameter, the main contribution is waveguide dispersion which can be calculated according to the following equation [32]: @ Re[n ()] e f f D() = (5) where Re[n ] is the real part of n , and c is the velocity of light. e e With the decrease in d, it is increasingly dicult for the FM of long wavelengths to transmit within the suspension core. The cut-o wavelength of the FM is red-shifted to 4–5 m at d = 1 m. Thus, the cut-o wavelength will red-shift further, as d gradually decreases. Simultaneously, with the increase in d, a lower maximum value of dispersion leads to a ﬂatter dispersion curve. This is mainly because the size of the suspended core becomes larger with d, and the n of the SCF is therefore increasingly closer to the value of As S block glass. The dispersion in the ﬁber is predominantly 2 3 determined by the material, such that the dispersion curve and zero-dispersion point (ZDP) are increasingly coincident with the block glass. In contrast, waveguide dispersion plays an important role only when the suspension core is small. Figure 8a shows the maximum dispersion value of the ﬁber gradually decreasing from 418.37724 ps/(kmnm) to 154.59883 ps/(kmnm) as a increases from 0.18 to 0.30 m at d = 1 m, and the wavelength of the maximum red-shifts from 2.481 m to 2.184 m. The dispersion curve tends to ﬂatten with increasing d, as shown in Figure 8b, such that the maximum dispersion decreases gradually. Therefore, in order to obtain a wider SC spectrum, it is preferable choosing a ﬁber structure with a small d. This is mainly because a larger air-hole opening, as shown in Figure 9, when a increases from 0.16 to 0.3 m , leads to more of the light ﬁeld distribution of the FM overﬂowing from the suspension arm. This increases the eective mode ﬁeld area and decreases the nonlinear coecient. Because the nonlinear coecient is inversely proportional to the wavelength, as indicated in Equation (3), this is more obvious at long wavelengths. Figure 7 shows that the inﬂuence of a on the nonlinear coecient is not signiﬁcant, whereas its inﬂuence on the dispersion is evident. Figure 8 shows that a has a more signiﬁcant inﬂuence on the dispersion with the decrease in d. Hence, smaller a values lead to higher maximum dispersion values, and a larger slope of dispersion. Photonics 2020, 7, x FOR PEER REVIEW 8 of 17 core diameter, the main contribution is waveguide dispersion which can be calculated according to the following equation [33]: Re[n ()] eff (5) D() c where Re[neff] is the real part of neff, and c is the velocity of light. With the decrease in d, it is increasingly difficult for the FM of long wavelengths to transmit within the suspension core. The cut-off wavelength of the FM is red-shifted to 4–5 µm at d = 1 µm. Thus, the cut-off wavelength will red-shift further, as d gradually decreases. Simultaneously, with the increase in d, a lower maximum value of dispersion leads to a flatter dispersion curve. This is mainly because the size of the suspended core becomes larger with d, and the neff of the SCF is therefore increasingly closer to the value of As2S3 block glass. The dispersion in the fiber is predominantly determined by the material, such that the dispersion curve and zero-dispersion point (ZDP) are increasingly coincident with the block glass. In contrast, waveguide dispersion plays an important role only when the suspension core is small. Figure 8a shows the maximum dispersion value of the fiber gradually decreasing from 418.37724 −1 ps/(km∙nm) to 154.59883 ps/(km∙nm) as a increases from 0.18 to 0.30 µm at d = 1 µm, and the wavelength of the maximum red-shifts from 2.481 µm to 2.184 µm. The dispersion curve tends to flatten with increasing d, as shown in Figure 8b, such that the maximum dispersion decreases gradually. Therefore, in order to obtain a wider SC spectrum, it is preferable choosing a fiber structure with a small d. This is mainly because a larger air-hole opening, as shown in Figure 9, when a increases −1 from 0.16 to 0.3 µm , leads to more of the light field distribution of the FM overflowing from the suspension arm. This increases the effective mode field area and decreases the nonlinear coefficient. Because the nonlinear coefficient is inversely proportional to the wavelength, as indicated in Equation (3), this is more obvious at long wavelengths. Figure 7 shows that the influence of a on the nonlinear coefficient is not significant, whereas its influence on the dispersion is evident. Figure 8 shows that a has a more significant influence on the dispersion with the decrease in d. Hence, smaller a values lead Photonics 2020, 7, 46 9 of 16 to higher maximum dispersion values, and a larger slope of dispersion. Photonics 2020, 7, x FOR PEER REVIEW 9 of 17 Figure 8. Impact of parameter (a) a, (b) d on the dispersion of FM. Figure 8. Impact of parameter (a) a, (b) d on the dispersion of FM. Figure 9. Electric ﬁeld distribution of FM at = 1.5 m for SCFs with d = 1 m and (a) a = 0.18 m , −1 Figure 9. Electric field distribution of FM at λ = 1.5 µm for SCFs with d = 1 µm and (a) a = 0.18 µm , (b) a = 0.3 m . −1 (b) a = 0.3 µm . The dispersion of As S block glass, as denoted by the black dotted line in Figure 8, is proportional 2 3 The dispersion of As2S3 block glass, as denoted by the black dotted line in Figure 8, is to the wavelength, such that there is only one ZDP of ~4.9 m. It was found that almost all the SCFs proportional to the wavelength, such that there is only one ZDP of ~4.9 µm. It was found that almost designed in this study exhibit dual-ZDW when d 9 m. Their ﬁrst ZDPs are more concentrated, all the SCFs designed in this study exhibit dual-ZDW when d ≤ 9 µm. Their first ZDPs are more whereas the second ZDPs are more dispersed. Table 2 indicates that the standard deviation is very concentrated, whereas the second ZDPs are more dispersed. Table 2 indicates that the standard small, regardless of d. Therefore, parameter a has little inﬂuence on the ﬁrst ZDPs. deviation is very small, regardless of d. Therefore, parameter a has little influence on the first ZDPs. Table 2. Mean and standard deviation of ﬁrst zero-dispersion wavelength (ZDW). Table 2. Mean and standard deviation of first zero-dispersion wavelength (ZDW). d = 1 m d = 3 m d = 5 m d = 9 m d = 1 μm d = 3 μm d = 5 μm d = 9 μm Mean (m) 1.5336 2.0814 2.6860 4.5493 Mean (μm) 1.5336 2.0814 2.6860 4.5493 Standard Deviation 0.0243 0.0579 0.0687 0.1218 Standard Deviation 0.0243 0.0579 0.0687 0.1218 The inﬂuence of a on the second ZDP is more evident. With the increase in a, a clear blue-shift The influence of a on the second ZDP is more evident. With the increase in a, a clear blue-shift occurs and becomes more signiﬁcant as d increases. The second ZDP can be blue-shifted from 10.442 to occurs and becomes more significant as d increases. The second ZDP can be blue-shifted from 10.442 11.334 m by increasing the value of a when d = 9 m. Furthermore, the diameter d likewise aects the to 11.334 µm by increasing the value of a when d = 9 µm. Furthermore, the diameter d likewise affects trend of the ZDPs, both of which have the tendency to red-shift as d increases. It was found that the the trend of the ZDPs, both of which have the tendency to red-shift as d increases. It was found that second ZDP has a more obvious red-shift than the ﬁrst one. The second ZDP red-shifts from ~3 to the second ZDP has a more obvious red-shift than the first one. The second ZDP red-shifts from ~3 ~11 m as d increases from 1 to 9 m. to ~11 µm as d increases from 1 to 9 µm. 3.4. Supercontinuum As a general numerical approach to study SCG, the pulse evolution inside As2S3 SCFs was calculated by solving the GNLSE [34]: n+1 n n A A A A i A 2 i A i A A t A (6) n R z 2 n! t t t n2 0 SPM Loss SRS Dispersion SS where A = A(z, t) is the electric field envelope of FM; α is the loss coefficient of the SCF, the terms βn depict various dispersion coefficients in the Taylor series expansion of the propagation constant β at the central frequency ω0; tR is the Raman response function, which is usually expressed as: 2 2 t t 1 2 t (1 f )(t ) f exp( ) sin( ) (7) R R R 1 2 2 1 where the fractional contribution of the delayed Raman response is fR = 0.11 [35], the Raman period is τ1 = 15.5 fs and the lifetime is τ2 = 230.5 fs for As2S3 [36]. Photonics 2020, 7, 46 10 of 16 3.4. Supercontinuum As a general numerical approach to study SCG, the pulse evolution inside As S SCFs was 2 3 calculated by solving the GNLSE [33]: 2 3 6 7 6 7 6 7 6 7 6 7 X 2 n+1 n 6 @ jAj A 7 @A i @ A i @ A 6 j j 7 6 2 7 6 7 + A = i jAj A + t A (6) n 6 R 7 6 7 @z 2 n! @t 6 ! @t @t 7 |{z} 6 7 n2 |{z} 6 | {z }7 6 | {z } 7 4 5 | {z } SPM Loss SRS SS Dispersion where A = A(z, t) is the electric ﬁeld envelope of FM; is the loss coecient of the SCF, the terms depict various dispersion coecients in the Taylor series expansion of the propagation constant at the central frequency ! ; t is the Raman response function, which is usually expressed as: 2 2 t t 1 2 t = (1 f )(t) + f exp sin (7) R R R 2 1 where the fractional contribution of the delayed Raman response is f = 0.11 [34], the Raman period is = 15.5 fs and the lifetime is = 230.5 fs for As S [35]. 2 3 1 2 In this study, the split-step Fourier method (SSFM) is employed to calculate the GNLSE. The formula indicates that the dispersion expression on the left has a signiﬁcant inﬂuence on all three nonlinear eects, which are stimulated Raman scattering (SRS), self-steepening (SS) and self-phase modulation (SPM), on the right side of the formula. Therefore, the structural design of the SCF is crucial to obtain a wider SC. The parameters of the pump source and the structure of the SCF both have a signiﬁcant inﬂuence on SCG. When d = 1 m and a = 0.18 m , the nonlinear coecient of the ﬁber is relatively large. More importantly, its ﬁrst ZDW is 1.541 m, which is very close to 1.550 m. Presently, the pump source with a 1.550 m wavelength is widely used in communication owing to the maturity of the technology, low cost and high power. As shown in Figure 8a, the second ZDW is 3.543 m, and the anomalous dispersion region occurs between the two ZDWs [36]. We ﬁxed the wavelength of the pump source to 1.541 m and the pulse width to 200 fs. SCG is studied by adjusting the pump source peak power. Because of the high nonlinear coecient of the As S material, a short SCF can 2 3 achieve the saturation of SC, such that its length can be selected as 0.01 m, which can reduce the calculation complexity by omitting the loss factor of the ﬁber. The nonlinear coecient of the ﬁber is 1 1 13.84129 m W at 1.541 m. Based on the accurate dispersion function obtained in the previous section, the dispersion coecients are calculated at high accuracy (to improve the accuracy of SC calculation, the tenth-order dispersion coecient is used in simulation). The speciﬁc parameters are shown in Table 3. Table 3. Values of 1–10 order of dispersion coecients of SCF with d = 1 m and a = 0.18 m at 1.541 m. 2 3 4 5 6 7 8 9 10 (fs/mm) 1 2 3 4 5 6 7 8 9 10 (fs /mm) (fs /mm) (fs /mm) (fs /mm) (fs /mm) (fs /mm) (fs /mm) (fs /mm) (fs /mm) 6.73833 1.31008 4.6199 2.39585 4.6505 2.43177 5.22843 5.21097 7.83179 5.3381 10 16 27 42 56 71 85 99 112 124 10 10 10 10 10 10 10 10 10 Results show that the SCG can be divided into four stages. As shown in Figure 10, the spectrum of the pump source is symmetrically extended to both long and short wavelengths simultaneously by SPM, which results in the basic broadening of the SC. If the SC needs to be extended to the short wavelength, the pump wavelength is of particular importance [37]. The SCF designed in this work adjusts the ZDW to ~1.55 m, which can use more common pump sources and extend the SC to 1 m. Photonics 2020, 7, x FOR PEER REVIEW 10 of 17 In this study, the split-step Fourier method (SSFM) is employed to calculate the GNLSE. The formula indicates that the dispersion expression on the left has a significant influence on all three nonlinear effects, which are stimulated Raman scattering (SRS), self-steepening (SS) and self-phase modulation (SPM), on the right side of the formula. Therefore, the structural design of the SCF is crucial to obtain a wider SC. The parameters of the pump source and the structure of the SCF both have a significant influence −1 on SCG. When d = 1 µm and a = 0.18 µm , the nonlinear coefficient of the fiber is relatively large. More importantly, its first ZDW is 1.541 µm, which is very close to 1.550 µm. Presently, the pump source with a 1.550 µm wavelength is widely used in communication owing to the maturity of the technology, low cost and high power. As shown in Figure 8a, the second ZDW is 3.543 µm, and the anomalous dispersion region occurs between the two ZDWs [37]. We fixed the wavelength of the pump source to 1.541 µm and the pulse width to 200 fs. SCG is studied by adjusting the pump source peak power. Because of the high nonlinear coefficient of the As2S3 material, a short SCF can achieve the saturation of SC, such that its length can be selected as 0.01 m, which can reduce the calculation complexity by omitting the loss factor of the fiber. The nonlinear coefficient of the fiber is 13.84129 −1 −1 m W at 1.541 µm. Based on the accurate dispersion function obtained in the previous section, the dispersion coefficients βn are calculated at high accuracy (to improve the accuracy of SC calculation, the tenth-order dispersion coefficient is used in simulation). The specific parameters are shown in Table 3. −1 Table 3. Values of 1–10 order of dispersion coefficients of SCF with d = 1 µm and a = 0.18 µm at 1.541 µm. β1 β2 β3 β4 β5 β6 β7 β8 β9 β10 2 3 4 5 6 7 8 9 10 (fs/mm) (fs /mm) (fs /mm) (fs /mm) (fs /mm) (fs /mm) (fs /mm) (fs /mm) (fs /mm) (fs /mm) 5.3381× −6.73833× 1.31008× −4.6199× 2.39585× −4.6505× −2.43177× 5.22843× 5.21097× −7.83179× 5 −16 −27 −42 −56 −71 −85 −99 −112 −124 10 10 10 10 10 10 10 10 10 10 Results show that the SCG can be divided into four stages. As shown in Figure 10, the spectrum of the pump source is symmetrically extended to both long and short wavelengths simultaneously Photonics 2020, 7, 46 11 of 16 by SPM, which results in the basic broadening of the SC. If the SC needs to be extended to the short wavelength, the pump wavelength is of particular importance [38]. The SCF designed in this work adjusts the ZDW to ~1.55 µm, which can use more common pump sources and extend the SC to 1 The spectra show that the broadening width and saturation length of the SC in the early stage is mainly µm. The spectra show that the broadening width and saturation length of the SC in the early stage is determined by the characteristics of the pump source and optical ﬁber material. Because of the large mainly determined by the characteristics of the pump source and optical fiber material. Because of nonlinear coecient of arsenic sulﬁde glass, as shown in Figure 11a, the ﬁber only needs to be ~4 mm the large nonlinear coefficient of arsenic sulfide glass, as shown in Figure 11a, the fiber only needs to in length to achieve the saturated output of the SC at P = 1 kW. With the increase in peak power, be ~4 mm in length to achieve the saturated output of the SC at P = 1 kW. With the increase in peak the length is even smaller. power, the length is even smaller. Photonics 2020, 7, x FOR PEER REVIEW 11 of 17 Figure 10. Spectrum spreading at the early stage of supercontinuum generation (SCG) at different peak powers. In the second stage, the SS effect renders the pulse asymmetric with an increase in the distance or power. This is because when the peak power reaches the Raman threshold, the SRS will selectively increase the spectral width to the long-wavelength measurement, such that the SC exhibits a red-shift [39]. As it can no longer generate frequency components to the short wavelength side, as shown in Figure 10. Spectrum spreading at the early stage of supercontinuum generation (SCG) at dierent Figure 11a, the short wavelength will cease to expand after reaching 1 µm at the initial stage. peak powers. Figure 11. Simulated SCG at peak powers of (a) 1 kW, (b) 10 kW, (c) 20 kW, (d) 30 kW, (e) 40 kW and Figure 11. Simulated SCG at peak powers of (a) 1 kW, (b) 10 kW, (c) 20 kW, (d) 30 kW, (e) 40 kW and (f) 50 kW. (f) 50 kW. In the second stage, the SS eect renders the pulse asymmetric with an increase in the distance Subsequently, SC continues to be distorted under the influence of high-order dispersion and or power. This is because when the peak power reaches the Raman threshold, the SRS will nonlinearity of the SCF. Figure 11c shows that when P = 20 kW, the higher-order soliton splits into selectively increase the spectral width to the long-wavelength measurement, such that the SC exhibits four Raman solitons due to the anomalous dispersion region, and the pulse wavelength of the soliton a red-shift [38]. As it can no longer generate frequency components to the short wavelength side, becomes longer through the soliton self-frequency shift [40]. Moreover, Raman solitons and the as shown in Figure 11a, the short wavelength will cease to expand after reaching 1 m at the initial stage. dispersive waves emitted by them generate new frequency components through the cross-phase Subsequently, SC continues to be distorted under the inﬂuence of high-order dispersion and modulation (XPM) and FWM effect, which further broadens the SC. nonlinearity of the SCF. Figure 11c shows that when P = 20 kW, the higher-order soliton splits into In the last stage, when the Raman solitons red-shift in the anomalous dispersion region, the four Raman solitons due to the anomalous dispersion region, and the pulse wavelength of the soliton dispersive wave quickly fills the energy gap between the solitons. When the wavelength exceeds the becomes longer through the soliton self-frequency shift [39]. Moreover, Raman solitons and the second ZDW, the tendency of the spectral red-shift is greatly reduced with the loss of the soliton. dispersive waves emitted by them generate new frequency components through the cross-phase Although the dispersive wave and other nonlinear effects can still support the continued broadening modulation (XPM) and FWM eect, which further broadens the SC. of the SC, the effect will not be obvious. Even with a further increase in the pump energy, the SC cannIn ot c the ontilast nue stage, to red-when shift dthe ue to Raman saturasolitons tion [41,4 r2 ed-shift ]. Figure in 12 the sho anomalous ws the evodispersion lution of thr eegion, SC the dispersive wave quickly ﬁlls the energy gap between the solitons. When the wavelength exceeds spectrum with increasing pump power (P) from 10 kW to 50 kW. As the peak power reaches 40 kW, the the second SC will ZDW not r,ed the -shtendency ift after re of acthe hing spectral 5.0 µm,r ed-shift which m is ea gr ns eatly that rth educed e spectr with um the cann loss ot b of e e the xtesoliton. nded Although further. Th the e tw dispersive o ZDPs m wave ust nand ot be other too fnonlinear ar apart, ae s in ects tha can t castill se th support e energy the gacontinued p betweenbr th oadening e solitonsof cannot be filled by the dispersive wave. Therefore, the fiber structure with two ZDPs, exhibiting flat and low dispersion, is an important factor in the design of ultrawide SC. Photonics 2020, 7, 46 12 of 16 the SC, the eect will not be obvious. Even with a further increase in the pump energy, the SC cannot continue to red-shift due to saturation [40,41]. Figure 12 shows the evolution of the SC spectrum with increasing pump power (P) from 10 kW to 50 kW. As the peak power reaches 40 kW, the SC will not red-shift after reaching 5.0 m, which means that the spectrum cannot be extended further. The two ZDPs must not be too far apart, as in that case the energy gap between the solitons cannot be ﬁlled by the dispersive wave. Therefore, the ﬁber structure with two ZDPs, exhibiting ﬂat and low dispersion, Photonics 2020, 7, x FOR PEER REVIEW 12 of 17 is an important factor in the design of ultrawide SC. Figure 12. Simulated evolution of SCG pumped at 1.541 m wavelength at dierent peak powers. Figure 12. Simulated evolution of SCG pumped at 1.541 µm wavelength at different peak powers. In addition to the peak power, we also study the inﬂuence of pulse duration and the central In addition to the peak power, we also study the influence of pulse duration and the central wavelength of the pump source on the SC spectrum. Figure 13 depicts the evolution of the SC spectrum wavelength of the pump source on the SC spectrum. Figure 13 depicts the evolution of the SC with increasing pulse duration from 50 fs to 200 fs at peak powers of 40 kW. When the femtosecond pulse spectrum with increasing pulse duration from 50 fs to 200 fs at peak powers of 40 kW. When the width is wider, the soliton number N is larger and the fundamental soliton splits into N higher-order femtosecond pulse width is wider, the soliton number N is larger and the fundamental soliton splits soliton pulses with dierent red-shifted central frequencies. The closer the split fundamental soliton into N higher-order soliton pulses with different red-shifted central frequencies. The closer the split approaches the second ZDW, the greater the eect of third-order dispersion [42]. At this time, the phase fundamental soliton approaches the second ZDW, the greater the effect of third-order dispersion [43]. matching is easier to achieve, which enhances the FWM eect. Finally, the spectrum is broadened At this time, the phase matching is easier to achieve, which enhances the FWM effect. Finally, the to the long wavelength by FWM, third-order dispersion, and other nonlinear eects [18]. Moreover, spectrum is broadened to the long wavelength by FWM, third-order dispersion, and other nonlinear with the increase of the pulse duration, the multi peak oscillation appears in the direction of the long effects [25]. Moreover, with the increase of the pulse duration, the multi peak oscillation appears in wave, which may be aected by the higher-order dispersion. the direction of the long wave, which may be affected by the higher-order dispersion. When d = 1 m and a = 0.18 m , as shown in Figure 8a, two ZDWs are 1.540 m and 3.544 m. We have studied the expansion of the SC spectrum of two near-ZDWs (1.541 m and 3.543 m) and their midpoint (2.542 m) in the anomalous dispersion region. Figure 14 illustrates that the SC spectrum expands in both long and short wave directions with dierent pump wavelengths. Due to the cut-o of the ﬁrst ZDW, the pump pulse with the wavelength of 1.541 m is squeezed by the normal dispersion region, so it is dicult to move to the short wavelength. In the anomalous dispersion region, the dispersive wave makes the band gap between high-order solitons easier to be ﬁlled and the SC spectrum red-shifts more easily [43]. Therefore, the ability to extend to the long wavelength is stronger than others, and the spectrum is ﬂatter. Figure 13. Simulated evolution of SCG pumped at different pulse durations with a pump power of 40 kW. Photonics 2020, 7, x FOR PEER REVIEW 12 of 17 Figure 12. Simulated evolution of SCG pumped at 1.541 µm wavelength at different peak powers. In addition to the peak power, we also study the influence of pulse duration and the central wavelength of the pump source on the SC spectrum. Figure 13 depicts the evolution of the SC spectrum with increasing pulse duration from 50 fs to 200 fs at peak powers of 40 kW. When the femtosecond pulse width is wider, the soliton number N is larger and the fundamental soliton splits into N higher-order soliton pulses with different red-shifted central frequencies. The closer the split fundamental soliton approaches the second ZDW, the greater the effect of third-order dispersion [43]. At this time, the phase matching is easier to achieve, which enhances the FWM effect. Finally, the spectrum is broadened to the long wavelength by FWM, third-order dispersion, and other nonlinear Photonics 2020, 7, 46 13 of 16 effects [25]. Moreover, with the increase of the pulse duration, the multi peak oscillation appears in the direction of the long wave, which may be affected by the higher-order dispersion. Photonics 2020, 7, x FOR PEER REVIEW 13 of 17 −1 When d = 1 µm and a = 0.18 µm , as shown in Figure 8a, two ZDWs are 1.540 µm and 3.544 µm. We have studied the expansion of the SC spectrum of two near-ZDWs (1.541 µm and 3.543 µm) and their midpoint (2.542 µm) in the anomalous dispersion region. Figure 14 illustrates that the SC spectrum expands in both long and short wave directions with different pump wavelengths. Due to the cut-off of the first ZDW, the pump pulse with the wavelength of 1.541 µm is squeezed by the normal dispersion region, so it is difficult to move to the short wavelength. In the anomalous dispersion region, the dispersive wave makes the band gap between high-order solitons easier to be filled and the SC spectrum red-shifts more easily [44]. Therefore, the ability to extend to the long F Figure igure 1 13. 3. S Simulated imulated ev evo ollution ution o of f S SCG CG p pumped umped a at t d di iffer eren ent t p pulse ulse d durat uratiions ons w with ith a a p pump ump p power ower o of f wavelength is stronger than others, and the spectrum is flatter. 4 40 0 k kW W. . Figure 14. Simulated evolution of SCG with a pump power of 30 kW at dierent operation wavelengths. Figure 14. Simulated evolution of SCG with a pump power of 30 kW at different operation wavelengths. In brief, the peak power and pulse duration of the pump source have a great inﬂuence on the In brief, the peak power and pulse duration of the pump source have a great influence on the red-shift of the SC spectrum, and the central wavelength determines the cut-o of the blue-shift. For the red-shift of the SC spectrum, and the central wavelength determines the cut-off of the blue-shift. For SCF, the dispersion curve can be controlled by adjusting the structural parameters of the ﬁber, so as to the SCF, the dispersion curve can be controlled by adjusting the structural parameters of the fiber, so adjust the width and ﬂatness of the SC spectrum. as to adjust the width and flatness of the SC spectrum. 4. Conclusions 4. Conclusions To achieve a smaller core, the traditional SCF must reduce the thickness of its suspension To achieve a smaller core, the traditional SCF must reduce the thickness of its suspension arm, arm, which causes considerable diculties in the preparation of the SCF and moreover reduces the which causes considerable difficulties in the preparation of the SCF and moreover reduces the mechanical strength of the ﬁber. Because of the fragile suspension arm, the entire SCF is easily damaged mechanical strength of the fiber. Because of the fragile suspension arm, the entire SCF is easily during operation. In this study, a special As S SCF with three parabolic air holes, allowing for both a 2 3 damaged during operation. In this study, a special As2S3 SCF with three parabolic air holes, allowing for both a very small core size and a more robust suspension arm, was designed. We carried out a comprehensive analysis of the impact of structural parameters (a and d) on the neff, nonlinear coefficient, and chromatic dispersion within the wavelength range from 0.6 µm to 11.6 µm using COMSOL. The simulation results indicate that the two structural parameters are both inversely proportional to the neff, nonlinear coefficient, and chromatic dispersion. The size of the suspension core is mainly determined by d, which consequently assumes a greater impact than parameter a on the SCF. The higher nonlinear coefficient is mainly achieved by reducing d. By this approach, the maximum dispersion is increased, and the flat dispersion curve is more difficult to obtain. By appropriately increasing parameter a, the nonlinear coefficient is reduced accordingly. However, the flatness of the dispersion curve is significantly improved. The SCF with a flat dispersion and high nonlinear coefficient can be obtained by properly reducing d while increasing a. Moreover, the Photonics 2020, 7, 46 14 of 16 very small core size and a more robust suspension arm, was designed. We carried out a comprehensive analysis of the impact of structural parameters (a and d) on the n , nonlinear coecient, and chromatic dispersion within the wavelength range from 0.6 m to 11.6 m using COMSOL. The simulation results indicate that the two structural parameters are both inversely proportional to the n , nonlinear coecient, and chromatic dispersion. The size of the suspension core is mainly determined by d, which consequently assumes a greater impact than parameter a on the SCF. The higher nonlinear coecient is mainly achieved by reducing d. By this approach, the maximum dispersion is increased, and the ﬂat dispersion curve is more dicult to obtain. By appropriately increasing parameter a, the nonlinear coecient is reduced accordingly. However, the ﬂatness of the dispersion curve is signiﬁcantly improved. The SCF with a ﬂat dispersion and high nonlinear coecient can be obtained by properly reducing d while increasing a. Moreover, the designed SCFs have dual-ZDWs, both of which red-shift with the increase in d. The second ZDP is likewise aected by a, which in contrast to the trend with d is blue-shifted as a increases. By adjusting a and d, the ﬁrst ZDP can be red-shifted from 1.509 m to 4.712 m, and the second ZDP is 2.909–11.565 m. In particular, at d = 1 m, the ﬁrst ZDW is ~1.53 m, which enables the generation of the SC by pumping of the SCF by low cost and commercial lasers. According to the dispersion characteristics, the SCF (d = 1 m and a = 0.18 m ) can obtain 0.6–5.0 m SC at the peak power of 40 kW. Author Contributions: Conceptualization, T.X.; software, T.P.; investigation, T.P.; data curation, T.P.; writing—original draft preparation, T.P.; writing—review and editing, X.W.; visualization, T.P.; supervision, X.W.; project administration, X.W.; funding acquisition, T.X. and X.W. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the National Science Foundation of China under grant number 61875097, 61705091, 61627815, 61775109 and Zhejiang Province Public Welfare Technology Application Research Project under grant number LGF20F010004. Acknowledgments: The authors gratefully appreciate the support from College of Electrical Engineering, Zhejiang University of Water Resources and Electric Power and Ningbo University that provided oce and laboratory. Conﬂicts of Interest: The authors declare no conﬂict of interest. 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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**Published: ** Jul 3, 2020

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