Photonic crystal fiber (PCF) [1, 2], also known as microstructure optical fiber [3], has recently attracted wide interest in scientific research. Typically, a PCF is sa ingle-material-based optical fiber, with air-filled holes surrounding the core along the fiber’s entire length, to provide strong confinement of the light field, long interaction lengths, and customizable wavelength dispersion. Tapered single-mode fibers [4], tapered microstructure fibers [5], and tapered photonic crystal fibers [6, 7] exhibit many novel characteristics, such as mode coupling and high nonlinearity. In the fiber-tapering process, the nonlinearity is enhanced by reducing the core size and modulating dispersion along the fiber’s length. With the rapid development of fiber-optic technology, high-quality ultrashort pulses will play a very important role in modern communication. Recently, compression of an ultrashort pulse has been developed using waveguide techniques [8] or optical fibers [9, 10]. Arnold et al. [8] compared ultrashort laser pulses by nonlinear propagation in gas-filled planar hollow waveguides and pulse compression down to a small-cycle duration with energies of up to 100 mJ. Hadrich et al. [9] used noble-gas-filled hollow fibers for spectral broadening of the optical pulses via self-phase modulation. This pulse-compression scheme can provide a pulse-shortening factor of greater than 10. Using numerical simulations and analyses, Wang et al. [10] proposed a fiber with a multiple-hollow-core structure for optical pulse compression of high-energy ultrashort laser pulses. Martial et al. [11] reported pulse compression in a large mode-area rod-type PCF, and demonstrated the compression of 4-μJ 338-fs pulses from a fiber-chirped pulse-amplification system down to 49 fs. Voronin and Zheltikov [12] and Zheltikov [13] demonstrated the compression of a few single-cycle pulse widths in a highly nonlinear fiber, using the soliton-effect and a soliton compression ratio of up to 50. Jing et al. [14] experimentally and numerically investigated the broadening supercontinuum in an 80-m-long all-normal dispersion PCF, using standard single-mode fiber as a high-order soliton compressor. Chan et al. [15, 16] and Shumin et al. [17] studied soliton-effect pulse compression by the combined action of negative third-order dispersion and Raman self-scattering in optical fibers. Through extensive research, it has been found that dispersion-decreasing optical fibers are suitable for short-pulse compression [18, 19], supercontinuum generation [20], and parametric amplification [21]. Hu et al. [22] studied pulse compression using a tapered microstructure optical fiber, and found the primary limitation on pulse compression to be the loss due to mode leakage. Wen-Wen et al. [23] studied pulse compression in tapered holey fibers, and found that a compression factor of 136.7 can be achieved by pulses with an initial width of 800 fs, propagated through a length of 0.8 m. However, the loss of the tapered holey fiber was not considered in the pulse-propagating process. PCF tapers with large mode area were designed by Li et al. [24], and are suitable for engineering the fiber’s nonlinearity profile γ(z) and dispersion profile β₂(z) along its length. A high-power 1-ps pulse can be compressed self-similarly down to a pulse width of 53.6 fs with a negligible pedestal, by injecting the pulse into a nonlinearity-increasing fiber. Numerical simulations and experimental studies on self-similar amplification of picosecond pulses in a short-gain fiber were reported by Song et al. [25] for obtaining near-100-fs laser pulses with nearly transform-limited temporal quality. The results showed that the picosecond pulse can be compressed and a high-quality femtosecond pulse obtained by controlling fiber parameters such as dispersion and nonlinearity coefficient. In addition, the application of optical parametric amplification based on PCFs to optical communication and optical switching has been studied [26–28].

An infrared laser is an important tool in scientific research [29]. For example, a midinfrared laser was recently shown to cut a variety of tissues effectively, with minimal injury to adjacent structures [30]. Sources of broadband midinfrared light attract considerable attention from many researchers, due to their broad application potential [31] in optical frequency metrology, astronomical spectroscopy, optical tomography, tunable wavelength conversion, and infrared imaging. Compared to other non-silica glasses, chalcogenide glasses—in particular, As₂Se₃ [32, 33] and As₂S₃ [34, 35]—exhibit a larger refractive index and a higher nonlinear index, providing larger mode confinement and higher nonlinearity. Highly nonlinear multimaterial chalcogenide spiral PCF has been prepared by Kalra et al. [36] for supercontinuum generation. Moreover, chalcogenide glasses are transparent at a midinfrared wavelength of about 10 μm ; in the case of As₂Se₃, this can be up to 14 μm. Active mode locking of a Cr²⁺:ZnSe laser was studied by Carrig et al. [37], who showed that the central wavelength of the laser was nearly 2.5 μm ; the laser was able to produce 4.4-ps transform-limited Gaussian-shaped pulses at an output average power of 82 mW.

In this paper, we design a tapered As₂S₃ PCF. The dispersion, nonlinearity coefficient, confinement loss, and other parameters of the tapered As₂S₃ PCF are numerically simulated using the finite-element method. The propagation of a midinfrared pulse in the tapered As₂S₃ PCF is simulated numerically using the adaptive split-step Fourier method. The potential for pulse compression in the tapered fiber is studied, and the influence of fiber confinement loss on pulse compression is analyzed.

In this paper, we design a tapered As₂S₃ PCF with four layers of air holes in a hexagonal array around the core. We assume that the air-hole pitch Λ, air-hole diameter d, and fiber diameter decrease alongside the narrowing PCF. However, the ratio of the air-hole diameter to the pitch d/Λ remains constant along the tapered PCF. A structural diagram of the tapered As₂S₃ PCF is shown in Fig. 1.

Figure 1. Structural diagram of the tapered As₂S₃ PCF.

The air-hole pitch of the tapered PCF with a linearly tapered structure can be expressed as follows [23]:

where Λ(z) is the pitch of the air holes of the tapered PCF at distance z, Λ(0) and Λ(L) are the pitch at the thick and thin ends of the tapered PCF respectively, z is the coordinate along the tapered PCF in the Cartesian coordinate system, and L is the total length of the fiber. The refractive index of the chalcogenide glass As₂S₃ is shown in the equation below [38, 39]:

where B₁ = 1.898367, B₂ = 1.922297, B₃ = 0.87651, B₄ = 0.11887, B₅ = 0.95699, C₁ = 0.15 μm, C₂ = 0.25 μm, C₃ = 0.35 μm, C₄ = 0.45 μm, and C₅ = 27.3861 μm. The nonlinear index coefficient n₂ of As₂S₃ is 2.0 × 10⁻¹⁸ m²/W. According to Eq. (2), the refractive index n of As₂S₃ can be computed as shown in Fig. 2(a), and we obtain the material index n = 2.42 at the wavelength λ = 2.5 μm. The air-hole pitch Λ of the tapered PCF as a function of the distance z is shown in Fig. 2(b).

Figure 2. For the As₂S₃ tapered PCF, L = 1.0 m. (a) Refractive index n of the chalcogenide glass As₂S₃ as a function of the wavelength λ, (b) air-hole pitch of the tapered PCF as a function of the distance z from the thick end of the fiber.

For the As₂S₃-tapered PCF simulated in this paper, we set Λ(0) = 2.26 μm, Λ(L) = 1.5 μm, and L = 1.0 m, while the ratio of the air-hole diameter to the pitch d/Λ isset to 0.4, 0.5, 0.6, 0.7, and 0.8. The vector-effective-index method is used to simulate the dispersion and nonlinearity coefficients of the tapered photonic crystal fibers in this paper, and the corresponding program is written by our research group in the FORTRAN language. At a wavelength of 2.5 μm, dispersion coefficient D of the tapered PCF as a function of distance z is shown in Fig. 3(a), and the nonlinearity coefficient γ of the tapered PCF as a functions of distance z is shown in Fig. 3(b). Fig. 3(a) shows the dispersion coefficient D of the tapered PCF decreasing along the taper from the thick to the thin end, for all d/Λ. Only in the cases of d/Λ = 0.6, 0.7, and 0.8 does the tapered fiber demonstrate anomalous dispersion along the fiber, with the dispersion of the tapered fiber decreasing most rapidly along the z axis when d/Λ= 0.8. Fig. 3(b) shows the nonlinearity coefficient γ of the tapered PCF increasing along the taper from the thick to the thin end, for all d/Λ. The nonlinearity coefficient of the tapered fiber with d/Λ = 0.8 is larger than the other coefficients. A higher nonlinearity coefficient is beneficial in pulse compression.

Figure 3. For the As₂S₃ tapered PCF, we set Λ(0) = 2.26 μm, Λ(L) = 1.5 μm, and L = 1.0 m, while the ratio of the air-hole diameter to the pitch d/Λ is set to 0.4, 0.5, 0.6, 0.7, and 0.8. (a) Dispersion coefficient D of the tapered PCF at a wavelength of 2.5 μm, as a function of distance z, (b) nonlinearity coefficient γ of the tapered PCF at a wavelength of 2.5 μm, as a function of distance z.

To achieve the dispersion-decreasing and nonlinearity-increasing tapered fiber, we select the ratio of the air-hole diameter to the pitch d/Λ to be 0.8. Dispersion D and nonlinearity coefficient γ at a wavelength of 2.5 μm as functions of distance z are given in Fig. 4(a), which shows that D decreases nonlinearity coefficient while γ increases along the taper, from the thick to the thin end. Confinement loss α at a wavelength of 2.5 μm as a function of z is given in Fig. 4(b). Among them, the total loss comprises material-absorption loss and confinement loss. The material absorption loss of As₂S₃ is lower than 0.08 dB/m at 2.5 μm, which is located in the As₂S₃ weak-absorption window of 0.78–5.5 μm [40]. In this paper, the confinement loss of tapered PCF is approximately equal to the total loss.

Figure 4. For the A_s2S₃ tapered PCF, Λ(0) = 2.26 μm, Λ(L) = 1.5 μm, L = 1.0 m, and d/Λ = 0.8. (a) Dispersion D and nonlinearity coefficient γ at a wavelength of 2.5 μm, as functions of distance z, (b) confinement loss α at a wavelength of 2.5 μm, as a function of z.

The variation of dispersion and confinement loss with the structure parameters of a PCF is a complex process affected by multiple parameters, such as the air-hole diameter d, pitch Λ, and the ratio of air-hole diameter to pitch d/Λ. There is some deviation from monotonic variation in dispersion and confinement loss, but the general trend is still close to monotonic, as shown in Figs. 4(a) and 4(b). In addition, the confinement loss is calculated by the multiple scattering of the light wave to the air-holes, and calculation error occurs easily when the loss is small.

To realize some design tolerance, we include Fig. 5 to illustrate that small variations in d/Λ cause variations in the dispersion and nonlinearity coefficient of the tapered PCF. Figures 5(a) and 5(b) show that a slight change of d/Λ changes the dispersion and nonlinearity coefficient of the tapered PCF, but the trend of the dispersion and nonlinearity coefficient does not change along the fiber’s axis. In the tolerance range of 1% of d/Λ, from the thick to the thin end of the tapered PCF, the dispersion of the fiber still keeps decreasing gradually, while the nonlinearity coefficient of the fiber still keeps increasing gradually.

Figure 5. For the As₂S₃ tapered PCF, Λ(0) = 2.26 μm, (L) = 1.5 μm, L = 1.0 m, and d/Λ = 0.79, 0.8, and 0.81. (a) The dispersion D at a wavelength of 2.5 μm, as a function of distance z, (b) nonlinearity coefficient γ at a wavelength of 2.5 μm, as a function of z.

Maintaining the air holes is a difficult problem in the preparation of a photonic crystal fiber. In the process of preparing a PCF, an appropriate pressure is applied to the preform rod to keep the air holes from collapsing, and the adiabatic tapering method is used to keep the air-hole diameter proportional to the tapered region during the tapering process [41]. In addition, as shown in Fig. 5, the performance of the fiber has a certain tolerance to the preparation parameters.

To simulate the transmission of laser pulses in the As₂S₃ tapered PCF, we further calculate the high-order dispersion of the As₂S₃ tapered PCF from β₃ to β₁₅. Second-order dispersion β₂ and third-order dispersion β₃ at a wavelength of 2.5 μm, as functions of distance z, are given in Fig. 6(a), while the fourth-order dispersion β₄ and fifth-order dispersion β₅ are given in Fig. 6(b). Figure 6(a) shows that the third-order dispersion β₃ is negative in the tapered PCF, while the fourth-order dispersion β₄ changes from positive to negative along the tapered fiber.

Figure 6. For the As₂S₃ tapered PCF, Λ(0) = 2.26 μm, Λ(L) = 1.5 μm, L = 1.0 m, and d/Λ = 0.8. (a) Second-order dispersion β₂ and third-order dispersion β₃ at a wavelength of 2.5 μm, as functions of distance z; (b) fourth-order dispersion β₄ and fifth-order dispersion β₅ at a wavelength of 2.5 μm, as functions of z.

The generalized nonlinear Schrödinger equation for pulse propagation in a tapered PCF with nonuniform parameters should be expressed as in Eq. (3) [42]:

where α(z), β_n(z), and γ(z) represent the total loss, n-th-order dispersion, and nonlinearity coefficient of the tapered PCF respectively. These are functions of the distance z along the tapered PCF. Unlike silica, As₂S₃ glass exhibits some special parameters [38], e.g. its material nonlinearity is taken as n₂ = 2.0 × 10⁻¹⁸ m²/W, and its Raman fraction is f_R = 0.2. The nonlinearity coefficient is calculated for the structural parameter in the tapered PCF based on the effective mode area A_eff:

while the nonlinear response function is defined as

For Gaussian-shaped pulses, the input pulses are given by [41] as follows:

where T₀ is the half-width of a pulse, and its relation to the full width at half maximum (FWHM) of the pulse is T_FWHM = 2(ln2)^1/2T₀ ≈ 1.665T₀. For a fundamental soliton propagating in a lossy fiber, when pulses propagate from the thick to the thin end in a tapered PCF, the FWHM of pulses along the tapered PCF can be expressed as follows:

The decrease in dispersion acts as effective amplification, because it precisely compensates for the decrease in soliton energy caused by fiber loss. Equation (7) shows that pulse compression could be achieved during propagation along a fiber with either G > 1 (gain), β₂(z) < β₂(0) (dispersion decreasing), or γ(z) > γ(0) (nonlinearity increasing). The tapered PCF we design can potentially achieve this goal. Figure 4(a) shows the nonlinearity coefficient increasing and the dispersion coefficient D decreasing along the tapered PCF, at a wavelength of 2.5 μm. However, a degree of loss in the fiber always occurs; as a result, G is generally less than 1. As such, the loss of a tapered PCF will seriously affect pulse compression.

Higher-order compression in the dispersion-decreasing fiber is a process of transforming a higher-order soliton into a highly compressed pulse that propagates almost the same as a fundamental soliton. The width of an n^th-order optical soliton is given by the following equation:

where E_S = 2P₀T₀ is the soliton’s energy, P₀ is the peak power, and γ is the fiber’s nonlinearity coefficient. Equation (8) indicates that the tapered PCF designed in this paper is suitable for pulse compression.

In this paper, we simulate pulse transmission by pumping the active mode locking of a Cr²⁺:ZnSe laser [37] into a tapered PCF. The central wavelength of this laser is nearly 2.5 μm, located in the As₂S₃ weak-absorption window of 0.78–5.5 μm, and the absorption wavelength is 2.5 μm, corresponding to minimum material loss for As₂S₃. The laser produces 4.4-ps transform-limited Gaussian-shaped pulses. The average power of the input pulses is 3 mW in the simulation conducted for the current study, the repetition frequency is 81.0 MHz, the corresponding peak pulse power is 12 W, and the energy of a single pulse is approximately 0.037 nJ. In this simulation, an adaptive split-step Fourier method [23] is numerically applied to study pulse propagation in the tapered PCF. We compare without fiber loss with total fiber loss under two operating modes.

The pulse propagation in the tapered As₂S₃ PCF is simulated numerically using the adaptive split-step Fourier method, and the corresponding program is written by our research group in the FORTRAN language. The temporal profile as a function of propagation distance z is shown in Fig. 7. The horizontal ordinate is the normalized time T/T₀, and the half-width of initial pulses T₀ is 2.64 ps, which corresponds to a FWHM of 4.4 ps. Figure 8 shows the spectrum propagating along the tapered PCF. We can see that the laser pulse is effectively compressed, and the corresponding spectrum broadened. When the pulse width gradually becomes narrower, obvious side lobes appear in the compressed pulse.

Figure 7. Temporal profile of a pulse, as a function of propagation distance z. The horizontal ordinate is the normalized time T/T₀. Fiber loss is ignored in this simulation.

Figure 8. The spectrum in the tapered PCF, as a function of propagation distance z. Fiber loss is ignored in this simulation.

To describe pulse compression we use the pulse compression factor, defined as the ratio of the FWHM pulse duration at the beginning and at a given transmission distance in the fiber. Figure 9 shows the pulse width and compression factor as functions of propagation distance z. We find that the pulse FWHM can be changed from 4,400 fs to 56 fs when the pulse propagates from the thick to the thin end. The final compression factor achieved is 78.

Figure 9. Pulse width and compression factor as functions of propagation distance z.

As shown in Fig. 9, the pulse compression mainly occurs over the propagation-distance range of 0–0.7 m, becoming slow after 0.7 m. By analyzing the data in Fig. 9, we can see that the pulse width changed from 252 fs to 56 fs and the compression factor changed from 17.5 to 78.6 when the pulse propagates from 0.7 m to 1.0 m in the tapered PCF. When the transmission distance is greater than 0.7 m, the pulse compression slows because the spectrum is widened and the energy corresponding to the peak wavelength of the pulse decreases, which weakens the nonlinear effect.

A degree of loss in the fiber always exists. In a tapered PCF, the loss coefficient α increases from the thin to the thick end, as shown in Fig. 4(b). The confinement loss is roughly 16 dB/m at the thin end of the tapered PCF. A comparison of the temporal profile with and without loss, as a function of propagation distance z from 0.7 m to 1.0 m, is given in Fig. 10. Due to the loss, the peak power of pulses that propagate inside the tapered PCF alongside this loss is lower than in the tapered PCF without loss. Concurrently, pulse width becomes wider and the compression factor becomes smaller when we considere loss in the tapered PCF. This can be observed in Fig. 11, which shows comparisons of the pulse FWHM (a) and the compression factor (b) with and without loss. When loss is not considered, the pulse width continuously becomes smaller along the tapered fiber. However, when we do consider the loss, the pulse width FWHM reaches a minimum of 72 fs; correspondingly, the compression factor reaches a maximum of 61 at propagation distance z = 0.9 m. Then the pulse width begins to increase and the compression factor to decrease. Due to loss in the tapered fiber, the pulse compression is suppressed.

Figure 10. Comparison of the temporal profile with and without loss, as a function of propagation distance z from 0.7 m to 1.0 m. The ordinate is the normalized time T/T₀.

Figure 11. Comparisons of pulse FWHM and compression factor with and without loss. (a) Pulse FWHM as a function of propagation distance z, (b) compression factor as a function of z.

Compared to silica or Poly methyl methacrylate, As₂S₃ glass has higher refractive index and nonlinear refractive index, the values being n = 2.42 and n₂ = 2.0 × 10⁻¹⁸ m²/W at wavelength λ = 2.5 μm respectively. A PCF made of As₂S₃ glass has a stronger ability to regulate dispersion, and a higher nonlinearity coefficient.

From the numerical simulation results, we find that loss reduces the compression factor when the pulse propagates in the tapered PCF. Pulse compression is a nonlinear optical process related to pulse energy, and the existence of loss will make the pulse energy gradually diminish in the transmission process. As a result, the decrease of pulse energy leads to the decrease of the nonlinear effect, and the compression factor of the pulse is reduced. Equation (7) is the theoretical basis of pulse compression using a tapered PCF. The factor G = e^−az in Eq. (7) is less than 1 for a fiber with loss, which is not conducive to pulse compression. As can be seen from Fig. 4(b), from the thick to the thin end of the tapered fiber the confinement loss gradually increases, which also leads to the gradual decrease of G, but is not conducive to pulse compression. At the thin end of the tapered PCF, the effect of loss on the compression factor is greater than that of dispersion and nonlinearity, so a spike appears at last.

Li et al. [43] studied the phenomenon of wave breaking when pulses propagate from the thick to the thin end along tapered holey fibers. In [43], the central wavelength of 800 nm was located in the normal-dispersion regime, in which second-order dispersion β₂ is positive. With the increase in propagation distance z, the pulse shape becomes broader and the pulse spectrum flattens and is accompanied by oscillatory structures occuring near pulse edges, and also side lobes appear in the pulse spectrum. In this paper, we simulate the midinfrared pulse compression in a tapered As₂S₃ PCF when a pulse propagates from the thick to the thin end. The pulse’s central wavelength of 2.5 μm is located in the anomalous-dispersion region, in which the second-order dispersion β₂ is negative. The results show that in the anomalous-dispersion region, pulses can be efficiently compressed in a dispersion-decreasing and nonlinearity-increasing tapered PCF. When a pulse is transmitted from the thick to the thin end in the tapered fiber, it is found by comparison that wave breaking occurs if the central wavelength is in the normal-dispersion region of the fiber, while compression occurs if the central wavelength is in the abnormal-dispersion region of the fiber.

In this paper, a tapered As₂S₃ PCF with four air-hole layers in a hexagonal array around the core was designed. For structural parameters Λ(0) = 2.26 μm, Λ(L) = 1.5 μm, L = 1.0 m, and d/Λ = 0.8, the linear tapered As₂S₃ PCF presented anomalous dispersion at a wavelength of 2.5 μm, while the dispersion D decreased and the nonlinearity coefficient γ increased along the tapered PCF from the thick to the thin end. Assumed active mode locking of a Cr²⁺:ZnSe laser was pumped into this tapered PCF. When an initially Gaussian input pulse of 4.4 ps, an average input power of 3 mW, and a repetition frequency of 81.0 MHz was employed, we theoretically obtained a pulse duration of 56 fs and a compression factor of 78 when the pulse propagated from the thick to the thin end of the tapered PCF. When considering loss in the tapered PCF, the pulse FWHM reached a minimum of 72 fs; correspondingly, the compression factor reached a maximum of 61. Accordingly, we demonstrated that in the anomalous-dispersion region, midinfrared pulses can be efficiently compressed in a dispersion-decreasing and nonlinearity-increasing tapered As₂S₃ PCF. The efficiency of pulse compression will be suppressed because of loss in the tapered fiber.

This study was supported in part by the Program of the Natural Science Foundation of Hebei Province (Grant No. F2017203193), and in part by Nanjing University of Posts and Telecommunications Foundation under Grants JUH219002, JUH219007, NY215007, and NY215113. This work was supported in part by the Research Center of Optical Communications Engineering & Technology, Jiangsu Province Foundation, under Grant ZXF20170102.