## Abstract

Broadband tunable ultrafast lasers using mid-infrared (MIR) fibers operating at a 3-5 µm atmospheric transmission window are attractive sources because of their numerous applications. Tellurite fibers possess the merits of large linear and nonlinear refractive indices, sufficient chemical stability, and wide transparency range up to ∼5 µm; also, they are highly suitable for high efficiency MIR ultrafast fiber laser sources based on soliton self-frequency shift (SSFS). We numerically simulate SSFS of MIR femtosecond pulses in step-index tellurite optical fibers. A femtosecond erbium-doped fluoride fiber laser at 3.5 µm is employed as the pump source. Parameters including the peak power of the input pulse and nonlinear fiber length are optimized for high efficiency broadband tunable MIR ultrafast laser performance. Our results show that a high-efficiency 3.5-6 µm wavelength-tunable femtosecond laser can be realized by employing SSFS in a 22-cm-long segment of tellurite step-index fiber pumped by femtosecond pulses with 10-300 kW peak powers at 3.5 µm. Ultra-high energy ratios of the most redshifted solitons to the input pulses of >50% are obtained across the 3.5-5 µm tuning range. The presented numerical study provides valuable guidance for SSFS of MIR femtosecond pulses in step-index tellurite fibers and is valuable for future high efficiency wavelength-tunable MIR ultrafast fiber laser development.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Ultrashort fiber lasers emitting in the 3-5 µm mid-infrared (MIR) are in great demand for a wide range of scientific and technological applications, including remote laser sensing [1], optical frequency combs [2], ultrabroadband and ultrasensitive molecular spectroscopy through the nonlinear wavelength conversion [3], minimally invasive human tissue laser surgery [4] and breath analysis [5]. Traditionally, ultrashort pulses in the MIR have been demonstrated either through wavelength down-conversion of near–infrared mode-locked lasers in conjunction with nonlinear crystals [6], based on free space solid-state Cr:ZnSe/S lasers [7], or with microresonator-based Kerr combs [8]. However, these systems suffer from bulky and complexity, which in turn prevent their applications across the fields of science and industry. Fiber lasers possess significant advantages over other laser sources as an effective approach to provide compact, robust, and flexible optical components with excellent heat-dissipating capability and superior beam quality. Thus, the MIR fiber laser represents an appealing alternative. However, the spectral coverage of fiber lasers is limited by the use of silica-based optical fibers, which become opaque beyond 2.2 µm, due to a multiphonon edge. Thanks to the availability of high-purity fluoride glass fibers with transmission wavelength up to 5 µm, the development of fiber lasers operating at wavelengths well above 2.2 µm multiphonon edge of silica has been enabled [9].

With the utilization of nonlinear polarization rotation or saturable absorbers including Fe^{2+}:ZnSe, semiconductor saturable absorber mirror (SESAM), graphene and black phosphorus, mode-locked Ho^{3+}- and Er^{3+}- doped fiber lasers have been demonstrated at 2.7-2.9 µm wavelengths [10–16]. Mode-locked lasers based on nonlinear polarization evolution (NPE) and emitting sub-500 fs pulses at 2.8 µm with peak power up to 37 kW were also reported [17–19]. By using extra-cavity nonlinear pulse compression, ultrashort pulses with 70 fs pulse duration at 2.8 µm have been realized [20]. Very recently, Z. Qin et al realized a 3.5 µm black phosphorus based mode-locked Er^{3+}-doped ZBLAN fiber utilizing dual wavelength pumping lasers [21]. By incorporating a fiber amplifier, a MIR fiber laser system that generates Raman soliton pulses tunable from 2.8 to 3.6 µm has been achieved. Femtosecond laser pulses with a peak power of 200 kW and pulse energy of 37 nJ at 3.4 µm have been obtained [22].

The above mentioned mode-locked fluoride fiber lasers and NPE based MIR fiber lasers, though generate femtosecond/picosecond pulses with high spectral density, their emission wavelengths are limited to specific wavelengths or within limited tuning range. However, new scientific discoveries and practical applications in the MIR region will require versatile light sources with both high spectral density and continuous tunability over a wide spectral range. MIR supercontinuums spanning more than 3 octaves (2-16 µm) was generated in a chalcogenide fiber [23]. However, the power spectral density is naturally low for supercontinuum, and many applications would benefit from widely tunable high-quality intense short pulses. The soliton self-frequency shift (SSFS) in optical fibers provides high-quality femtosecond pulses with wavelengths that cannot be obtained through direct transition between two energy levels of ions [24,25]. In SSFS, long wavelength component is amplified by short wavelength component due to intra-pulse Raman scatting, which causes central wavelength of the soliton to redshift. SSFS has been widely introduced to generate wavelength-tunable short pulse fiber lasers [22,25–27].

Appropriate fiber material and pump wavelength are of crucial importance for shifting soliton pulses further into the MIR by SSFS. In the 3-5 µm MIR region, fluoride glass fibers have intrinsic low loss and generally present anomalous dispersion (*β*_{2}<0), forcing the lasers to operate in the soliton regime, which is instrumental for SSFS [24,26]. However, fluoride has a nonlinear refractive index comparable to silica and hence a long optical fiber and pump pulses with very high peak power have to be used to achieve output pulses with longer output wavelength [27–30]. In addition, the efficiency of the fiber Raman conversion efficiency (defined by the ratio of the energy of the main shifted soliton at the output and the input energy) is naturally low. For instance, Tang et al. obtained soliton shift up to 4.3 µm in an indium fluoride fiber pumped at 1.9 µm [27]. The choice of using fluoride fiber and pump laser systems operating in the near-IR implies the use of complex power amplifier system to enhance the output power of the near-IR laser in order to produce a spectral shift important enough to reach the MIR. In addition, these shifts are generally associated with the generation of dispersive wave or secondary solitons, as well as losses through the SSFS process that reduces significantly the energy transfer into the MIR. Only 10%-20% of the energy coupled into the fundamental mode has been transferred to the most redshifted soliton in a 2 m indium fluoride fiber. Duval et al. obtained watt-level femtosecond soliton pulses tunable from 2.8 to 3.6 µm by pumping a 22-m-long fluoride fiber at 2.8 µm [22]. However, the estimated power ratio of the main shifted soliton to the pump power is less than 10%, and the wavelength of the most redshifted soliton is shorter than 4 µm.

Compared with fluoride glasses, chalcogenide glasses possess much higher nonlinear refractive index of ∼100×10^{−19} m^{2}/W and show excellent IR transmission up to ∼15 µm [28]. However, its zero dispersion wavelength (ZDW) of the bulk is beyond 5 µm. In order to make a chalcogenide fiber with a ZDW below 1.5, 2, 3, or even 3.5 µm, which is the lasing wavelength of the conventional Er^{3+} or Ho^{3+} doped fiber lasers, very large waveguide dispersions need to be introduced, requiring the final fiber core diameter to be submicron [31]. This is disadvantageous for power scaling and laser beam coupling, especially for MIR fiber laser systems where free-space coupling is often required. Moreover, chalcogenide fibers have very low damage threshold, which limits their output power and specific heat dissipation approach is required to reduce thermal effects.

Tellurite glasses have also been extensively used as a low phonon energy oxide glasses in nonlinear photonic devices because of their broad IR transmission and large nonlinearity. Tellurite fibers possess larger nonlinear refractive indices (5.9×10-19 m^{2}/W) than fluoride fibers, broader Raman gain bandwidth (∼300 cm^{−1}), larger Raman shift, higher robustness, stronger corrosion resistance, and better thermal stability than fluoride and chalcogenide fibers [32,33]. Moreover, benefit from their moisture resistance, tellurite glass devices require less protection and can be stored in ambient air without degradation. However, the ZDW for tellurite glass is ∼2.3 µm, conventional near infrared pumping requires microstructured fiber geometry for dispersion management and shifting of ZDW to the range shorter than the pump wavelength. For example, M. Y. Koptev et al. reported SSFS in 0.5 m tellurite fiber [34]. However, due to the short pump wavelength of hybrid Er/Tm fiber system, microstructured fiber was used, and the longest wavelength of the soliton was still below 3 µm. Recently, a black phosphorus enabled 3.5 µm mode-locked Er^{3+}-doped fluoride fiber laser has been developed [21]. The pulse duration is estimated to be >2.5 ps and the pulse energy is ∼1.4 nJ. In Ref. [22], 160 fs laser pulses with an estimated peak power of 200 kW at 3.4 µm have been obtained. Reduction of the pulse duration and significant power scaling can be expected by introducing proper dispersion compensation in the cavity, extra-cavity pulse compression, and multi-stage power amplification systems [20,22]. Besides “real” saturable absorber modulation, NPE can also be an effective technique to realize ultrafast fiber lasers at 3.5 µm. Both aforementioned laser sources are promising pump sources for SSFS in tellurite fibers with large main soliton conversion efficiency since their wavelengths are already in the large anomalous dispersion region where the generation and control of solitons are easier and a relatively low soliton number can be maintained as well. It also means that ultrashort pulses with higher Raman conversion efficiency and longer wavelength can be obtained using conventional step-index tellurite fibers. Therefore, pumping a tellurite fiber with an ultrafast fiber laser at 3.5 µm is a promising approach to obtain a SSFS based high efficiency ultrafast fiber laser source with wide wavelength tuning range in the 3-5 µm atmospheric window.

In this work, we propose pumping the step-index tellurite fiber with femtosecond pulses at 3.5 µm to generate high efficiency widely wavelength-tunable MIR femtosecond laser pulses by SSFS. Our simulation results indicate that widely tunable femtosecond laser pulses with Raman conversion efficiencies above 50% all over the 3.5-5 µm wavelength tuning range can be achieved by employing SSFS in a 22-cm-long tellurite fiber. Raman soliton shift in the range beyond 5 µm for increased pump energy in tellurite fiber is also discussed. The performed numerical study provides useful guidance for the development of 3-5 µm tunable high efficiency ultrafast fiber sources.

## 2. Numerical model

We employed the generalized nonlinear Schrödinger equation (GNLSE) to numerically simulate the pulse evolution in nonlinear fiber. Based on the re-derivation of GNLSE [35], we solved this equation in the frequency domain:

*α*(

*ω*) denotes the frequency-dependent linear loss in the fiber,

*β*(

*ω*) denotes the propagation constant, and ${\beta _1}({\omega _0}) = \partial \beta /\partial \omega$ is evaluated at the carrier frequency of the input pulse

*ω*

_{0}. $\bar{\gamma }(\omega )$ is the frequency-dependent nonlinear coefficient, which is defined as

*n*

_{2}is the nonlinear refractive index,

*n*

_{eff}(

*ω*) is the frequency-dependent refractive index. This definition of nonlinear coefficient is different from the conventional definition of nonlinear coefficient that is defined as

$\bar{A}({z,T} )$ is determined by

*A*

_{eff}(

*ω*) denotes the frequency-dependent effective area, which is given by

*F*(

*x*,

*y*,

*ω*) is the transverse mode distribution in the fiber.

*R*(

*t*) is the nonlinear response function which can be obtained by where

*h*(

_{R}*t*) denotes the delayed Raman response,

*f*denotes the Raman contribution to the total nonlinear response.

_{R}Using Eq. (8) to switch into interaction picture, the stiff dispersive part of the equation can be removed [36]. The frequency dependence of loss and effective area are both considered in this version of GNLSE. The simulations of the MIR wavelength-tunable ultrafast fiber laser based on SSFS were accomplished by using the Runge-Kutta method to numerically solve these equations.

## 3. SSFS in tellurite step-index fibers

Tellurite fibers have large nonlinear refractive index, low propagation loss in 3-5 µm MIR region, broad Raman gain bandwidth, large Raman shift and superior physical and chemical properties [33]. All these advantages make tellurite fibers promising candidates for MIR wavelength-tunable ultrafast fiber lasers based on SSFS. TeO_{2}-Bi_{2}O_{3}-ZnO-Na_{2}O (TBZN) is a standard tellurite glass possessing high thermal stability [37], and is a highly suitable candidate for the investigation of SSFS in tellurite fiber.

#### 3.1 Optical properties of the TBZN step-index fiber

TBZN step-index fiber with a core diameter of 9 µm, a cladding diameter of 125 µm, and a numerical aperture (NA) of 0.3 is used as the nonlinear fiber. Wavelength-dependent refractive index is calculated according to the Sellmeier equation

The Sellmeier coefficients*A*-

*E*of TBZN are the same as those in [38]. Dispersion of the TBZN fiber is calculated by The waveguide dispersion is obtained by the effective refractive index of the fundamental mode and the total dispersion, which is the sum of the material dispersion and the waveguide dispersion, is plotted as the black curve in Fig. 1(a). It can be seen from Fig. 1(a) that the ZDW of TBZN step-index fiber is 2.29 µm. The frequency-dependent loss of TBZN step-index fiber is shown as the red solid curve in Fig. 1(a) [33,39]. The propagation loss is assumed to increase exponentially with wavelength beyond 5 µm and is depicted with red dashed line in Fig. 1(a). TBZN fiber represents low propagation loss (< 0.5 dB/m) at 1-4.5 µm, making it a suitable candidate for a mid-IR ultrafast laser based on SSFS. Although the propagation loss significantly increases beyond 4.5 µm due to the increased probability of multi-phonon decay, our calculations, as shown below, confirm that a 3.5-6 µm wavelength-tunable ultrafast fiber laser can be achieved by using sub-meter length TBZN fibers. The black curve and red curve in Fig. 1(b) show the effective area and the nonlinear coefficient of TBZN fiber, respectively. The calculated effective mode area changes from 38 µm

^{2}for a signal at 1 µm up to 130 µm

^{2}for a signal at 5 µm. The estimated nonlinear coefficient decreases from 97 W

^{−1}·km

^{−1}to 6 W

^{−1}·km

^{−1}with increasing wavelength from 1 µm to 5 µm, assuming

*n*

_{2 }= 5.9×10

^{−19}m

^{2}/W [40]. Incorporating these dependences into the numerical simulation is crucial to accurate modeling of the long-wavelength edges of the spectra. The delayed Raman response of TBZN fiber is modeled according to the intermediate-broadening model for Raman gain spectrum

*ω*,

_{v,i}*γ*, and

_{i}*Γ*denote the Gaussian component position, Lorentzian and Gaussian full-width at half-maximum, respectively. The coefficients of TBZN in this model are the same as those in [37]. The value of

_{i}*f*is 0.51 [37].

_{R}#### 3.2 Numerical investigation of SSFS in a 10 m TBZN step-index fiber pumped at 3.5 µm

To theoretically study the SSFS in TBZN fiber, we first investigated 10 m TBZN step-index fiber pumped at 3.5 µm, which is the typical wavelength of a dual wavelength pumped Er^{3+}-doped fluoride fiber laser. Consider now an input pulse assumed to be a nonchirped hyperbolic secant pulse and having a full width at half-maximum (FWHM) of 100 fs. The SSFS of the fundamental soliton was calculated first. Input pulse remains a fundamental soliton till the peak power of the pump pulses reaches 10 kW according to the relation:

*β*

_{2}is the group velocity dispersion,

*γ*is the nonlinear coefficient defined in Eq. (4),

*P*

_{0}is the peak power, and

*T*

_{0}is the temporal width of the soliton.

The spectral evolution of fundamental soliton in the 10 m TBZN fiber is shown in Fig. 2. It can be seen from Fig. 2 that the central wavelength of the fundamental soliton increases along the fiber and the amount of frequency shift increases with fiber length. At the output of 10 m step-index TBZN fiber, the central wavelength of the soliton shifts to 4 µm.

When we increased the peak power of the pump pulses, SSFS can also be observed. However, the order of the soliton increases with increasing peak power of the pump pulses. Higher-order soliton splits into several fundamental solitons, and then every fundamental soliton undergoes SSFS in the subsequent process [41,42]. Figure 3 shows the wavelength of the most redshifted soliton versus the propagation distance when a 10 m TBZN fiber is pumped by pulses with different peak powers. When the peak power is fixed, the central wavelength of the soliton increases with increasing distance at first and tends to get stabilized with little further redshift after several meters of propagation. When the propagation distance is fixed, the central wavelength increases with increasing peak power of the input pulses. The output wavelength of the most redshifted soliton reaches 5.01 µm at the output of 10 m TBZN fiber when pumped by pulses with 80 kW peak power.

To further investigate the influence of TBZN fiber length on the output performance, we investigated the relationship between pulse parameters (peak power and FWHM) and propagation distance at specific pump power. Figure 4(a) shows the peak power and FWHM of the most redshifted soliton with respect to the propagation distance when TBZN fiber is pumped by pulses with 40 kW peak power. As can be seen in Fig. 4(a), the peak power decreases gradually along the TBZN fiber due to the frequency-dependent loss in the fiber as well as the soliton constraint, whereas the FWHM has an opposite trend owing to dispersion. When soliton redshifts, increasing value of |*β*_{2}| causes FWHM to increase. At the output of 10 m TBZN fiber, FWHM increases to 2580 fs. Such large FWHM is caused by the large dispersion at long wavelength in the TBZN fiber. As the soliton wants to remain a fundamental soliton (N = 1), the increased FWHM also leads to the decrease of the peak power. Figure 4(b) and 4(c) display the normalized spectrum and time domain profiles at different distances when 10 m TBZN fiber is pumped by pulses with 40 kW peak power, respectively. In the wavelength domain, the most redshifted soliton always has the highest peak spectral intensity. However, in the time domain, the pulse at around 0 ps has the highest peak power after the propagation distance of 8 m. This is caused by the increased loss with longer wavelength and soliton constraint as well. Solitons with longer central wavelengths undergo higher losses in the fiber. Thus, the most redshifted soliton experiences higher loss than the soliton remnant.

In order to make a compromise between the central wavelength and the peak power of the most redshifted soliton, and to make sure that the most redshifted soliton always has the highest peak power when TBZN fiber is pumped by high power pulses, length of TBZN fiber needs to be optimized.

#### 3.3 Numerical optimization of the SSFS process in the TBZN step-index fiber

For practical applications, the efficiency of the fiber Raman converter, which is defined by the ratio of the energy of the main shifted soliton at the output and the input energy in this case, is of importance. This energy ratio is strongly affected by fiber propagation loss and the fiber length and needs to be optimized carefully. In order to optimize the Raman converter, we calculated the energy ratio of the most redshifted soliton at 5 µm to the input pulse for different fiber lengths. Figure 5 shows the ratio of energy of the soliton centered at 5 µm to the input energy (denoted as the black line with squares) and the shortest fiber length needed to provide enough wavelength conversion (denoted as the red line with dots) as a function of the input pulse peak power. Although short fiber length is preferred to prevent energy loss caused by fiber propagation loss, input pulses with higher peak power are required for the soliton to redshift to certain wavelength, as can be seen in Fig. 5. As for the energy ratio of the main soliton centered at 5 µm, it increases with increasing peak power of the input pulses at first and tends to stabilize when the peak power reaches 120 kW. When peak power exceeds 140 kW, the energy ratio starts to decrease. To obtain high energy ratio in a TBZN fiber as short as possible when the fiber is pumped with pulses with peak power as low as possible, we chose 0.22 m as an optimized value of fiber length.

Then, we numerically studied the output performance of the 22-cm-long TBZN step-index fiber pumped by 3.5 µm femtosecond pulses, including the wavelength tunability, pulse energy and pulse width of the most redshifted soliton, and energy ratio of the most redshifted soliton. The wavelength of the most redshifted soliton versus the peak power of the input pulses is shown in Fig. 6(a). As is depicted in Fig. 6(a), wavelength of the most redshifted soliton can be continuously tuned from 3.5 µm to 5 µm by varying the incident peak power from 10 kW to 120 kW. The pulse energy and FWHM of the most redshifted soliton are shown in Fig. 6(b). The energy and FWHM both increase with increasing wavelength and reach 7.1 nJ and 194 fs, respectively, when the center wavelength of the most redshifted soliton reaches 5 µm. The ratio of the most redshifted soliton output energy to the input energy as a function of soliton center wavelength is shown in Fig. 6(c). Though the energy ratio decreases with increasing soliton wavelength due to the higher fiber loss for longer wavelength, all of the energy ratios exceed 50% within the whole tuning range of 3.5-5 µm, indicating that most of the input energy is efficiently transferred to the most redshifted soliton.

Spectral and temporal evolution along the 22-cm-long TBZN fiber when pumped by pulses with 120 kW peak power are shown in Fig. 7(b) and 7(d), respectively. A clean and isolated soliton carrying a great amount of energy and continuously shifting toward longer wavelength in the fiber is observed in the evolution diagrams. Normalized spectrum and time domain pulses at the output are shown in Fig. 7(a) and 7(c), respectively. As can be seen in Fig. 7(c), the most redshifted soliton still has the highest peak power. The most redshifted soliton is at 5 µm with 194 fs FWHM and 7.1 nJ energy. Energy transfer efficiency of the soliton at 5 µm is as high as 52.4%.

#### 3.4 Extension of the wavelength of the most redshifted soliton

Although the fiber loss increases dramatically beyond 5 µm, as plotted in Fig. 1(a), soliton with longer output wavelength is expected with higher energy input pulses since only a few centimeters long TBZN fiber can provide sufficient frequency shift. Our calculation shows that the soliton center wavelength reaches 5.5 µm with 11.6 nJ input pulse energy, the Raman converter efficiency can still be beyond 30%. The corresponding spectral output is shown in Fig. 8. Further enhance the pulse energy of the input pulses to 31 nJ, the wavelength of the most redshifted soliton can reach 6 µm. However, the energy transfer efficiency is reduced to 20.1% Shifting to even longer wavelengths is possible. However, the soliton shifting suffers from high propagation loss and large anomalous dispersion, which will eventually decrease the energy ratio of the most redshifted soliton to the input pulses.

## 4. Conclusion

In conclusion, we have presented numerical investigations on SSFS of MIR femtosecond pulses in a short piece of step-index tellurite optical fibers. Our simulation results have shown that femtosecond fiber laser sources tunable from 3.5-6 µm can be achieved by employing SSFS in sub-meter long tellurite fibers pumped by femtosecond Er^{3+}-doped fluoride fiber lasers at 3.5 µm. High energy ratios of the most redshifted solitons to the input pulses of above 50% all over the whole 3.5-5 µm wavelength range can be obtained. The calculation results can provide valuable guidance for the design and development of high efficiency MIR wideband-tunable ultrafast fiber lasers based on SSFS of MIR femtosecond pulses in tellurite fibers.

## Funding

National Natural Science Foundation of China (NSFC) (61875033, 61705147, 61435003, 61775031, 61421002); Chengdu Science and Technology Huimin Project (2016-HM01-00269-SF, 2016-HM01-00265-SF); Fundamental Research Funds for the Central Universities (YJ201654).

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