Chalcogenide glass photonic crystal fibers (PCFs) have aroused the particular interest of researchers due to high nonlinearities and lower transmission losses in the mid-infrared (MIR) regime.^[1] Among them, two glass PCFs based on As₂Se₃ and As₂S₃ chalcogenide glass offer the most superior optical properties and show quite a wide range of applications in the medical, military, sensing areas^[2] and optical signal procession.^[3] The chalcogenide As₂Se₃ and As₂S₃ glass are transparent at MIR wavelengths as long as 10 μm,^[4] and they potentially allow the generation of significant supercontinuum (SC) emission beyond 6 μm, which is high desirable for the variant advanced MIR spectral and optical applications.^[5]

As far as the nonlinear effects are concerned, high Raman response plays an essential role in a four-wave mixing (FWM) process, SC generation, and wavelength conversion scheme based on cross-phase modulation (XPM), and can be used to develop Raman amplifier and Raman laser.^[6,7] Besides, its high nonlinearity (n₂ is almost 100–1000 times as high as that of silica at 1.55 μm^[8]) has led to the demonstrations of high speed, all optical signal processing in a compact, planar waveguide geometry for next generation optical communications systems.^[9] From our previous work,^[10–12] one can know that modulation instability (MI) gain depends on Raman effect and nonlinearity of fiber, while the As₂Se₃ and As₂S₃ chalcogenide glass PCFs have high Raman gains and high nonlinearities. Therefore, the effects of high Raman scattering and high nonlinearity on MI gain in the As₂Se₃ and As₂S₃ glass PCFs are a worthwhile research topic.

In this paper, with the obtained MI gain expression from the scalar nonlinear Schrödinger equation (NLSE), we will give a detailed description for MI gain characteristics including the dependences on pump wavelength and power in the As₂Se₃ and As₂S₃ glass PCFs, and mainly discuss their gain bandwidths and gain amplitudes.

For non-birefringence photonic crystal fiber, the light propagation characteristic can be described with the scalar NLSE. Considering all nonlinear phenomena such as fiber dispersion, self-phase modulation (SPM), self-steepening (SS), stimulated Raman scattering (SRS), and four-wave mixing (FWM), the NLSE is generally expressed as^[13]

Using the undepleted-dump approximation method, MI can be studied by adding small perturbations into the steady solution U_st(z) = (P_p)^1/2exp(iγP_pz) of Eq. (1), where P_p is the pump power. Thus, the real solution of Eq. (1) can be expressed by U(z) = U_st(z) + u_s(z)exp(iΩt)exp(iγP_pz) + u_as(z)exp(–iΩt)exp(iγP_pz), where u_s(z) and u_as(z) are the small signal amplitudes of the Stokes and anti-Stokes waves at detuning frequency ±Ω, respectively, and Ω = ω_p – ω_s = ω_as – ω_p(Ω > 0). Inserting U(z) into Eq. (1), two coupled linear ordinary differential equations on the perturbations u_s(z) and u_as(z) are obtained by picking out the Stokes term and the complex conjugate of the anti-Stokes term, and making them linearized. They can be written as

Fig. 1.

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	Fig. 1. Raman susceptibilities and normalized Raman gain spectral of As2S3 and As2Se3 glass.

For analyzing the MI gain property of the chalcogenide glass PCF, we first design PCFs each with a hexagonal geometry and five layers of air holes for optimal studies. The transverse structures and fundamental mode energy distributions of As₂Se₃ and As₂S₃ PCFs are shown in Fig. 2(a), respectively. Their air-hole diameter d = 1.2 μm and the air-hole pitch Λ = 3 μm, which are the two important parameters that control the dispersions and the nonlinear coefficients of the designed PCFs. The ratio of the hole diameter to pitch is set to be d/Λ = 0.4, so that the fundamental mode can be found in the MIR (2 μm–10 μm). Background refractive index is defined by their Sellemier formula,^[14,16] whose values are different for the As₂Se₃ and As₂S₃ PCFs. Based on the full-vector solver COMSOL, by using the perfect-electric-conductor boundary condition and dividing the fiber cross-sectional surface into over 10⁵ non-uniform mesh grids, the effective refractive index of the fundamental mode of fiber is first calculated. Then, we can use the effective index to obtain their total dispersions in the spectral range from 2 μm to 7 μm (see Fig. 2(b)), which include the waveguide dispersion and material dispersion. It is easy to see from Fig. 2(b) that, the zero dispersion wavelengths of As₂Se₃ and As₂S₃PCFs are 3722 nm and 2908 nm, respectively. Moreover, their nonlinear coefficients are also gained from the mode field distribution (see Fig. 2(c)), which are higher than that of fused silica PCF.

Fig. 2.

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	Fig. 2. Parameters of the designed As2Se3 PCF and As2S3 PCF: (a) transverse structure and field distribution; (b) dispersion variation with wavelength; (c) effective mode area and nonlinear coefficient versus wavelength.

Figure 3 shows the MI gain variations with angular detuning frequency Ω in the abnormal dispersion regimes of As₂S₃ and As₂Se₃ PCFs when pump wavelength is increased and pump power P_p = 100 W. It can be seen from Fig. 3 that for the As₂Se₃ PCF, there are two gain spectral bands in the anti-Stokes region (ω_as > ω_p) or the Stokes region (ω_s < ω_p) (see Figs. 3(a) and 3(c)), while only a gain bandwidth appears in the Stokes or anti-Stokes region for the As₂S₃ PCF (see Figs. 3(b) and 3(d)). Moreover, two gain bands gradually move closer to each other in the Stokes or anti-Stokes region for the As₂Se₃ PCF when the pump wavelength varies from 3724 nm to 3748 nm (see Figs. 3(a) and 3(c)), while the single gain peak moves away from the zero angular detuning frequency point (Ω = 0) for the As₂S₃ PCF (see Figs. 3(b) and 3(d)) by changing pump wavelength from 2910 nm to 2934 nm. By further analyzing Fig. 3, one can also find that, when pump wavelength moves away from their zero dispersion wavelengths (3722 nm for As₂Se₃ PCF, 2908 nm for As₂S₃ PCF) and Raman effect is considered, their gain spectral peaks gradually move away from pump wavelength. Moreover, comparing Fig. 3(a) and 3(b) with Figs. 3(c) and 3(d) respectively, one can see that the gain magnitude per meter fiber length is smaller with considering the Raman effect than without considering the Raman effect. Also, the gain magnitude per meter fiber length remains constant (see Figs. 3(a) and 3(b)) when the Raman effect is not considered. It indicates that Raman scattering makes part of the pump energy transferred into that of Raman frequency component. In fact, the phenomenon can also be obtained from the variation of the small spectral peak close to the zero detuning frequency (see the green (dotted line) boxes in Fig. 3(c) and 3(d)) because the detuning frequencies corresponding to these small peaks just lie in the Raman gain band ranges of As₂Se₃ and As₂S₃ glass (see Fig. 1(b)). Besides those results mentioned above, it should also be noted that although the MI gain peaks move with the pump wavelength variation, their MI gain bandwidths are almost equal. Therefore, the variation of pump wavelength almost has little or no influence on the MI gain bandwidth when pump power is a constant. However, the smooth degrees of these gain spectra mainly depend on the pump wavelength, and they gradually become smoother with the pump wavelength decreasing, and the reason is that the Raman gain effect disappears when pump wavelength is decreased to a value beyond the Raman gain regions of As₂Se₃ and As₂S₃ material.

Fig. 3.

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	Fig. 3. MI gain variations with detuning frequency Ω in the abnormal dispersion regimes of As2Se3 and As2S3 PCF when pump wavelength is increased and pump power Pp = 100 W without ((a) and (b)) and with ((c) and (d)) considering Raman effect.

Although MI gain bandwidth does not quite depend on the pump wavelength in the abnormal dispersion region for each of the As₂Se₃ and As₂Se₃ PCF, pump power may have an important effect on the MI gain bandwidth. Figure 4 shows the variations of sideband wavelength with pump power. To better express the MI gain magnitude, a color bar is used to represent the normalized MI gain value. Analyzing Fig. 4, we find that the influence of pump power on MI gain bandwidth is very obvious. For the As₂Se₃ PCF, the MI gain bandwidth gradually increases when pump power P_p < 1000 W, while it gradually decreases for P_p > 1000 W (see Fig. 4(a)), which indicates that there is an optimal pump power. For the As₂S₃ PCF, the MI gain bandwidth gradually increases with the increase of pump power (see Fig. 4(b)). Moreover, their gain bandwidths in the Stokes sideband are much larger than in the anti-Stokes sideband for the As₂Se₃ and As₂S₃ PCF. Further comparing Fig. 4(a) with Fig. 4(b), it is easy to see that the total MI gain bandwidth is much larger in the As₂Se₃ PCF than in the As₂S₃ PCF when pump power P_p < 1600 W, which may be related to the high nonlinearity of the As₂Se₃ PCF. Especially for the As₂Se₃ PCF, if P_p < 600 W, a two-peak gain profile appears in the Stokes sideband. When P_p > 600 W, the two gain peaks merge together into an ultra-broadband gain profile in the Stokes sideband (see Fig. 4(a)). Similar results cannot be observed in the As₂S₃ PCF.

Fig. 4.

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	Fig. 4. Variations of sideband wavelength with pump power when pump wavelength is 3724 nm for the As2Se3 PCF (a), and 2910 nm for the As2S3 PCF (b).

Based on the above analysis, in order to achieve the smooth and ultra-broadband MI gain, pump wavelengths should be chosen to be close to their zero dispersion wavelengths (2910 nm for the As₂S₃ PCF, and 3724 nm for the As₂Se₃ PCF), while pump powers may be set to be about 700 W for the As₂Se₃ PCF and 2000 W for the As₂S₃ PCF respectively. Figure 5 shows the obtained ultra-broadband MI gain spectra in the chalcogenide glass PCFs. For the As₂Se₃ PCF, the optimal MI gain bandwidth reaches 2738 nm (498 nm + 2240 nm, see Fig. 5(a)), while it reaches 1961 nm (400 nm + 1561 nm, see Fig. 5(b)) for the As₂S₃ PCF. As far as MI gain bandwidth is concerned, the ultra-broadband gain is first gained, which is larger than that in the fused silica PCF.^[11,12] Moreover, for the As₂Se₃ PCF, the gain amplitude is 500 m^–1, while it reaches 1600 m^–1 for the As₂S₃ PCF. They are much larger in the chalcogenide glass PCFs. These results show that the chalcogenide glass PCFs can be used to amplify the continuous spectra based on the MI gain principle.

Fig. 5.

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	Fig. 5. Optimal ultra-broadband MI gains in the chalcogenide glass As2Se3 and As2S3 PCF.

In this work, we design the chalcogenide glass As₂Se₃ and As₂S₃ PCF, and calculate their MI gain spectra. The results indicate that it is possible to gain an ultra-broadband gain by using the chalcogenide glass As₂Se₃ and As₂S₃ PCF with the high nonlinearity and high Raman effect. By designing some special structure PCFs and choosing appropriate pump wavelength and power, the MI gain bandwidths of the As₂Se₃ and As₂S₃ PCF reach 2738 nm and 1961 nm respectively, which is the first time to use MI characteristic to obtain a ultra-broadband spectral gain structure. These findings may be used in the communication and optical amplification fields in the future.