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Over 50 W all-fiber mid-infrared supercontinuum laser

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Abstract

Broadband supercontinuum laser sources in the mid-infrared region have attracted enormous interest and found significant applications in spectroscopy, imaging, sensing, defense, and security. Despite recent advances in mid-infrared supercontinuum laser sources using infrared fibers, the average power of those laser sources is limited to 10-watt-level, and further power scaling to over 50 W (or hundred-watt-level) remains a significant technological challenge. Here, we report an over 50 W all-fiber mid-infrared supercontinuum laser source with a spectral range from 1220 to 3740 nm, by using low loss (<0.1 dB/m) fluorotellurite fibers we developed as the nonlinear medium and a tilted fusion splicing method for reducing the reflection from the fluorotellurite-silica fiber joint. Furthermore, the scalability of all-fiber mid-infrared supercontinuum laser sources using fluorotellurite fibers is analyzed by considering thermal effects and optical damage, which verifies its potential of power scaling to hundred-watt-level. Our results pave the way for realizing all-fiber hundred-watt-level mid-infrared lasers for real applications.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Mid-infrared supercontinuum (SC) laser sources have attracted enormous interest and found significant applications in frequency metrology, molecular spectroscopy, biomedicine, remote sensing, hyperspectral imaging, defense, and security [111]. Several types of infrared fibers (including fluoride [1215], tellurite [1618], and chalcogenide fibers [1923]) have been developed for improving the spectral bandwidth and the output power of mid-infrared SC laser sources. Broadband SC laser sources covering the mid-infrared region (up to 18 µm) have been realized by using chalcogenide fibers as the nonlinear media [21]. However, the output power of mid-infrared SC laser sources using chalcogenide fibers is limited to watt-level owing to the limitation of thermal properties and optical damage. Recently, many efforts have been made to improve the output power of mid-infrared SC laser sources for meeting the requirement of various applications, such as long-distance remote sensing, hyper-spectral imaging, defense, and security. Among infrared fibers, fluoride fibers have been used as the nonlinear media for constructing all-fiber high power (>10 W) mid-infrared SC lasers with a spectral range from near infrared to over 4 µm. In 2009, Xia et al. reported an all-fiber SC laser source with a spectral range from 0.8 to 4 µm and an average output power of ∼ 10.5 W, in which 53ZrF4-20BaF2-4LaF3-3AlF3-20NaF (ZBLAN) fibers are used as the nonlinear media and connected to the output silica fiber of the pump source by a butt-coupling method [13]. In 2019, Wu et al. reported an 11.3 W all-fiber mid-infrared SC laser source with a spectral range from 0.8 to 4.7 µm by using fluoroindate fibers as the nonlinear media [14]. In 2020, Yang et al. demonstrated a 20.6 W all-fiber SC laser source with a spectral range from 1.9 to 4.3 µm by using ZBLAN fibers as the nonlinear media, in which a direct fusion splicing method is used to connect ZBLAN fibers to silica fibers [15]. In 2021, G A Newburgh and M Dubinskii demonstrated the potential of ZBLAN fiber for constructing mid-infrared fiber lasers with an output power of ∼70 W [24]. However, according to recent investigations on high power mid-infrared fiber lasers using ZBLAN or fluoroindate fibers, OH diffusion in the fluoride fiber tip causes the damage of the ZBLAN or fluoroindate endcap quickly, which prevents the long-term operation of high power mid-infrared SC laser sources using ZBLAN or fluoroindate fibers [25]. Although several types of protective endcaps are developed for preventing OH diffusion [26], the long-term stability of high power mid-infrared SC laser sources needs to be improved for real applications owing to poor chemical and thermal stability of ZBLAN or fluoroindate fibers, and further power scaling to 50 W (or hundred-watt-level) remains a significant technological challenge.

Here, we experimentally demonstrate an over 50 W all-fiber mid-infrared SC laser source with a spectral range from 1220 to 3740 nm, in which low loss (<0.1 dB/m) fluorotellurite fibers with a broadband transmission window of 0.4∼6.0 µm, good water resistance and high glass transition temperature (Tg) were developed for using as the nonlinear media, and a tilted fusion splicing method was used to connect fluorotellurite fibers to silica fibers for reducing the reflection from the fluorotellurite-silica fiber joint. Furthermore, the scalability of all-fiber mid-infrared supercontinuum laser sources using fluorotellurite fibers is analyzed by considering thermal effects and optical damage. This analysis verifies its potential of power scaling to hundred-watt-level.

2. Experiments and results

The core and cladding materials for fluorotellurite fibers we selected are 70TeO2-20BaF2-10Y2O3 (70TBY) and 65TeO2-25BaF2-10Y2O3 (65TBY) glasses, respectively, for the following reasons. First, the above fluorotellurite (TBY) glasses have a broadband transmission window of 0.4∼6.0 µm, which enables TBY fibers to become a promising nonlinear medium for generating broadband SC laser sources with a spectral range from visible to 5.5 µm [27]. Second, TBY glasses have stable thermal and chemical properties compared to fluoride glasses. The Tg and onset crystallization temperatures (Tx) of 70TBY and 65TBY glasses are about 424 ± 1 °C, 422 ± 1 °C and 527 ± 1 °C, 520 ± 1 °C, respectively [28,29], which is higher than that of fluoride glasses (Tg: ∼252°C, Tx: ∼343°C for ZBLAN glasses. [30,31]). The values of parameter △T = Tx-Tg which was normally used to evaluate the glass stability for fiber fabrication [28], were 103 ± 2 °C and 98 ± 2 °C, respectively, larger than that of ZBLAN glass (∼91 °C). The figure-of-merit parameter for characterizing the thermal mechanical properties of a laser material is about 0.215 W/m1/2 for TBY glass, which is 1.5 times larger than that of ZBLAN glasses [28]. Especially, TBY glasses have good water resistance, which makes sure of the prevention of OH diffusion in the fiber tip (or end face). The above properties of TBY glasses indicate that TBY fibers might be promising nonlinear media for further power scaling of all-fiber mid-infrared SC laser sources. Third, by slightly varying the composition of TBY glasses, the refractive index of the 70TBY glass is a little larger than that of the 65TBY glass (see Supplement 1 for more details), and the former is used as the core material and the latter as the cladding material. In addition, since the thermal properties of the 70TBY glass is nearly same as that of the 65TBY glass, the fabricated fluorotellurite fibers based on 70TBY and 65TBY glasses might have low transmission loss, which is required for power scaling of all-fiber mid-infrared SC laser sources.

Based on 70TBY and 65TBY glasses, fluorotellurite fibers are fabricated by using a rod-in-tube method, in which a suction method is used to make the fiber preforms [32]. The inset of Fig. 1 shows the scanning electron micrograph image of the cross-section of the fabricated fiber. The fiber has a step-index structure and the core diameter is about 14 µm. The numerical aperture (NA) at 2 µm of the fiber was about 0.28. The group velocity dispersion (GVD) profiles of the fiber were calculated by using commercial software MODE solutions (Lumerical Solutions, Inc.) with the full vectorial finite difference method. The calculated GVD profile of the fundamental propagation mode LP01 in the fiber was shown in Fig. 1. The zero-dispersion wavelength (ZDW) of the fiber was located at 1994nm. The transmission loss of the above fiber was measured to be ∼0.09 dB/m at 2 µm by using a cutback method (see Supplement 1 for more details). The nonlinear coefficient at 2 µm of the fiber was calculated to be about 11.5 km-1 W-1 by using a nonlinear refractive index of 3.5 × 10−19 m2 W-1 for fluorotellurite glasses [33].

 figure: Fig. 1.

Fig. 1. Calculated GVD curve of the fundamental propagation mode for fluorotellurite fiber with 14 µm core diameters. Inset: scanning electron micrograph of the fluorotellurite fiber.

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In the all-fiber mid-infrared SC laser source system, a tilted fusing splicing method was used to reduce the reflection from the fluorotellurite-silica fiber joint. The used above fluorotellurite fiber has a cladding diameter of 220 µm, and the used silica fiber has a core/cladding diameter of 10/130 µm with a core NA of 0.15. The fluorotellurite fiber and the silica fiber were fusion spliced with each other by a CO2 laser fiber fusion processing workstation (Fujikura, LZM-100). Being different from the direct fusion splicing method as introduced in Ref. [34], the silica fiber was specially prepared before the splicing. First, the silica fiber was cleaned and cleaved at 7° by using a fiber cleaver (Fujikura, CT106). Then, both the cleaved fluorotellurite and silica fiber were put into the CO2 laser fiber fusion processing workstation for fusing splicing. The parameter of the splicer was optimized by adjusting the power of CO2 laser, setting the fiber end gap and the overlap amount of splicing to 5 and 15 µm, respectively. The heater was located at the side of fluorotellurite fiber. When the temperature at the cleaved end of fluorotellurite fiber exceeded the Tg of the fluorotellurite glass, the two fibers were pushed toward each other immediately and fused together. Finally, the temperature around the fusion splicing point was kept at the Tg of the fluorotellurite fiber for 660 ms and then gradually cooled to the room temperature to enhance the strength of the fusion splicing point. Figure 2(b) shows the microscope photo of connection point between silica and fluorotellurite fiber before and after fusion splicing. Because of the larger Tg of silica fiber, the fused fluorotellurite fiber wrapped onto the unfused silica fiber and kept the connection tilted, which was the key to prevent the harm to the system from the Fresnel reflection between two fibers. A continuous-wave fiber laser at 2000nm was used to test the splicing results. About 90% of the launched light was output from the 48 cm fluorotellurite fiber, corresponding to a fusing splicing loss of ∼0.31 dB after taking into account the fiber propagation loss of 0.09 dB/m and the mode mismatch loss of 0.104 dB. To show the reproducibility of the tilted fusion splicing technique, we repeated the splicing process for six times. The corresponding losses were measured to be about 0.28, 0.36, 0.40, 0.28, 0.32 and 0.31 dB at 2000nm, respectively.

 figure: Fig. 2.

Fig. 2. (a) Experimental setup of the 50-W mid-infrared SC laser source. (b) Microscope photo of connection point between silica and fluorotellurite fiber before and after fusion splicing. (c) Output spectrum of the pump laser with maximal launched pump power. (HNLF, highly nonlinear optical fiber; DCF, dispersion compensating fiber).

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Figure 2(a) shows the experimental setup for all-fiber mid-infrared SC laser source. The whole SC laser is comprised of a high-power Tm3+-doped fiber amplifier (TDFA) with a broadband output spectrum spanning from 1.9 to 2.5 µm as a pump laser and a piece of 48 cm long fluorotellurite fiber as the nonlinear medium. The pump laser was seeded with a 2000nm Raman soliton fiber laser (repetition rate: 50 MHz) and has the maximum available output power of ∼80 W. Figure 2(c) shows the output spectrum of the TDFA with maximal launched pump power. The spectral broadening on the red side was mainly caused by the generation of Raman soliton self-frequency shift. A high-power circulator is set between the pre-amplifier and main-amplifier of TDFA to detect the return light power during the experiment. The pigtail of the TDFA is a piece of 2 m long passive double cladding silica fiber with a core/cladding diameter of 10/130 µm and an effective core/cladding NA of 0.15/0.46. The fluorotellurite fiber was tilted fusion spliced to the pigtail of the TDFA with above method and the joint was placed on an aluminum plate water-cooled at 12°C for efficient heat dissipation. The output end of the fluorotellurite fiber was angle-cleaved. The output signals were monitored by using an optical spectrum analyzer with a measurement range of 1200-2400 or 1900-5500 nm (Yokogawa, AQ6375 or AQ6377). The output power of the mid-infrared SC laser source was measured by using a power meter.

Figure 3(a) shows spectrum evolution of the generated SC laser under different output powers from the above fluorotellurite fiber. As the launched average pump power was increased to 23 W, the average output power of the generated SC laser was gradually increased to 17 W, large spectral broadening around the pump light occurred and an emission peak at 2260 nm appeared. Since the peak wavelength of the pump laser was located at the anomalous dispersion region of the above fluorotellurite fiber, the mechanisms of large spectral broadening for a launched average pump power of ≥ 23 W was self-phase modulation, stimulated Raman scattering, higher-order soliton compression, soliton fission, Raman soliton self-frequency shift, and the generation of blueshifted dispersive waves. The emission peak at 1655 nm was attributed to the blueshifted dispersive wave, and with further increasing of the launched average pump power to 73.35 W, the short wave-length edge of SC spectrum was expanded to 1.22 µm. The emission peak at 2260 nm was attributed to the stimulated Raman scattering, and the further spectral broadening on the red side was attributed to the Raman soliton self-frequency shift. Finally, the long wave-length edge of SC spectrum was expanded to 3.74 µm. As the pump peak around 2 µm was neglected, the 10-dB bandwidth of the generated SC laser was 1824nm, corresponding to a spectral range of 1404-3228 nm. Figure 3(b) shows the dependence of the average output power of the generate SC laser in the fluorotellurite fiber on the launched average power of the pump laser. With increasing the launched average pump power to 73.35 W, the average output power of the generated SC laser was gradually increased to 50.22 W, and the corresponding optical-to-optical conversion efficiency was about 68.47%. The measured maximum return power was about 20 mW, which was much lower than that under non-tilted fusion splicing (see Supplement 1 for more details). In the experiments, we monitored the output power and spectrum of the generated SC laser source at the maximal output power of 50 W for 12 hours, and no obvious changes in the output power and spectrum of the SC laser source were observed. Moreover, no obvious damage at the output fiber end or the fusion point was observed. The effect of nonlinear fiber length on the output power and spectra of the generated SC laser source could be seen in Supplement 1. To the best of our knowledge, the results obtained (over 50 W power) represent a new record for all-fiber high power mid-infrared SC lasers. In the future, we will try to further promote the output power and spectral bandwidth by optimizing the parameters of the fluorotellurite glass fibers and the pump lasers.

 figure: Fig. 3.

Fig. 3. (a) Mid-infrared SC spectrum evolution under different output powers from the fluorotellurite fiber. (b) Dependence of the SC average power output from fluorotellurite fiber on the launched pump power. Inset: The power meter photograph during the mid-infrared SC laser operating at the output power of 50.22 W.

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The numerical simulations on spectral broadening of the above SC laser were also performed by solving the generalized Schrödinger equations. In our simulations, the initial pump laser has an operating wavelength of 2000nm, a pulse width of 3 ps, and a repetition rate of 50 MHz. A 2.0 m long double cladding single-mode silica fiber and a 0.48 m long fluorotellurite fiber were used for SC laser generation. First, the SC laser with a spectral region of 1.9-2.5 µm was generated from the silica fiber. The nonlinear coefficient of the silica fiber we used was 1 km-1 W-1. The Raman response function, the GVD curve and the loss of silica fiber was derived from Refs. [3537], respectively. By using 1.9-2.5 µm SC laser as pump laser, a high-power mid-infrared SC laser was generated from the fluorotellurite fiber. The parameters including nonlinear coefficient, GVD curve and transmission loss of fluorotellurite fibers were mentioned above. The Raman response function was derived from the Raman gain spectrum of fluorotellurite glass.

Figure 4(a) and 4(b) show the simulated SC spectra (the black curves) and measured SC spectra (the red curves) output from the silica fiber of TDFA and the above fluorotellurite fiber for a launched average pump power of 73.35 W. The simulated results agreed with the measured one, which indicated that the parameters used in the simulations were appropriate. Figure 4(c) shows the spectral evolution of SC generation in the 2.0 m silica fiber and the 0.48 m fluorotellurite fiber for a launched average pump power of 73.35 W. The spectral broadening inside the silica fiber was due to the self-phase modulation, modulation instability, high-order soliton compression, soliton fission, and Raman soliton self-frequency shift. And the large spectral broadening inside the fluorotellurite fiber was mainly caused by the Raman soliton self-frequency shift and the generation of blue-shifted dispersive waves. The corresponding temporal evolution of SC generation was shown in Fig. 4(d), which confirmed the above interpretation.

 figure: Fig. 4.

Fig. 4. (a) and (b) The simulated (the black curves) and measured (the red curves) SC spectra output from the silica fiber and fluorotellurite fiber for the launched average pump power of 73.35 W, respectively. (c) and (d) Spectral and temporal evolution of SC generation in concatenated 2.0 m silica and 0.48 m fluorotellurite fiber for a launched average pump power of 73.35 W, respectively.

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3. Power scaling and limits of mid-infrared SC lasers

In recent years, the average power of mid-infrared SC laser sources based on fluorotellurite fibers continues to increase with the development of low loss fluorotellurite fibers and the fusion splicing techniques [28,33,34]. However, the power scaling limit has not yet been investigated. We considered this issue and calculated the maximum extractable powers of mid-infrared SC laser sources based on fluorotellurite fibers. The primary limits relevant to the power scaling of fiber lasers are imposed by thermal effect, optical damage and finite pump brightness. Next, we investigate the power scaling of mid-infrared SC lasers based on fluorotellurite fibers by considering those limiting factors.

3.1 Thermal limitations

The primary thermal effects for optical fibers are thermal fracture, melting of the core, melting of the coat and thermal lens [38,39]. The thermal load of the fiber is mainly caused by loss. The maximum extractable powers at the limits of thermal fracture, melting of the core, melting of the coating and thermal lens are given by the following equations [40,41]:

$${P_{TF}} = \frac{{4{\eta _{laser}}\pi {R_m}L}}{{{\eta _{heat}}\left( {1 - \frac{{{a^2}}}{{2{b^2}}}} \right)}}$$
$${P_{MCE}} = \frac{{4{\eta _{laser}}\pi {k_{core}}({{T_{mce}} - {T_c}} )L}}{{{\eta _{heat}}\left( {1 + \frac{{2{k_{core}}}}{{c \cdot h}} + 2\ln (\frac{b}{a}) + 2\ln (\frac{c}{a})} \right)}}$$
$${P_{MCT}} = \frac{{2{\eta _{laser}}\pi {k_{coat}}({{T_{mct}} - {T_c}} )L}}{{{\eta _{heat}}\left( {\frac{{{k_{coat}}}}{{c \cdot h}} + \ln (\frac{b}{c})} \right)}}$$
$${P_{TL}} = \frac{{{\eta _{laser}}\pi k_{{co}re}{\lambda ^2}L}}{{2{\eta _{heat}}\frac{{dn}}{{dT}}{a^2}}}$$
where the subscript TF, MCE, MCT and TL refer to thermal fracture, melting of the core, melting of the coating and thermal lens. L is the length of fiber, a is the core radius, b is the cladding radius and c is the coating radius. The relation ηlaser + ηheat =1, ηlaser is optical-optical conversion efficiency, while ηheat is the faction of pump light converted to heat. Ideally, ηlaser = exp(-α·L), limited only by transmission loss. In Eq. (1), Rm denotes the rupture modulus; in Eq. (2) and (3) kcore and kcoat are thermal conductivity of the core and coating material, respectively. Tmce is the melt temperature of the core and Tmct is the maximum continuous operating temperature of the coating material. h is the convective film coefficient and Tc is the coolant temperature; in Eq. (4) λ is the operating wavelength and dn/dT denotes the thermo-optic coefficient of the core material. Since fluorotellurite glass has a negative thermo-optic coefficient of -3.69 × 10−6 K-1 (see Supplement 1 for more details), the influence of thermal lens on the power limit might be neglected.

3.2 Optical damage

Optical damage is often observed in pulsed fiber lasers as it is easy for them to achieve high peak power. The pulsed laser damage threshold is mainly determined by the self-focusing effect in the fiber. The critical power imposed by self-focusing is given by [42]

$${P_{SF}} = \frac{{0.148{\lambda ^2}}}{{n{n_2}}}$$
where λ is the operating wavelength, n is the refractive index, and n2 is the nonlinear refractive index. The peak power at which self-focusing occurs at fluorotellurite fibers was calculated to be 0.909 MW at 2 µm.

3.3 Pump limitation

Pump limitation might be a problem for conventional diode pump sources as they can not meet high output power and good beam quality simultaneously. However, since mid-infrared SC laser source is pumped by fiber laser with good beam and low-loss fusion splicing could be realized between gain medium and pump laser source, the pump brightness limitation is minor.

Table 1 lists the parameters, symbols and values used in the calculation. The fluorotellurite fiber has a cladding diameter of 125 µm and a coating diameter of 200 µm. The coating material is polyetherimide (PEI), its thermal conductivity is 0.24 W m-1 K-1, and the maximum continuous operating temperature is 443 K [43,44], as shown in Table 1. The pump light we used has an operating wavelength of 2 µm, a pulse width of 3 ps and a repetition rate of 50 MHz, which is regarded as a quasi-continuous light source. Figure 5 shows the maximum extractable powers of mid-infrared SC laser sources limited by thermal fracture, melting of the core and melting of the coating versus the transmission loss of 0.48 m long fluorotellurite fibers with the core diameter of 14 µm. As the total transmission loss of the fluorotellurite fiber decreases from 0.48 to 0.048 dB (corresponding to 1 to 0.1 dB/m), the maximum extractable power increases from 38 to 401 W. The maximum extractable power of fluorotellurite fiber with a total transmission loss of 0.048 dB could reach to 401 W, limited by the melting temperature of the coating material. And higher output power in hundred-watt-level is expected to be achieved by further reducing fiber loss.

 figure: Fig. 5.

Fig. 5. Maximum extractable powers of mid-infrared SC laser sources limited by thermal fracture, melting of the core and melting of the coating versus the total transmission loss of 0.48 m long fluorotellurite fibers with the core diameter of 14 µm.

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Tables Icon

Table 1. List of parameters, symbols and values in calculation as well as units and references

4. Conclusions

In summary, we demonstrated an over 50 W mid-infrared SC laser source with full spectral coverage of 1220-3740 nm based on low loss fluorotellurite fibers in all-fiber configuration. Tilted fusion splicing method was used to reduce the reflection from the fluorotellurite-silica fiber joint. Furthermore, the calculations show that power scaling to hundred-watt-level of SC laser sources could be realized by considering thermal effects and optical damage. To the best of our knowledge, our work represents a new record for all-fiber high power mid-infrared SC lasers based on fluorotellurite fibers to date, and our results pave the way for realizing all-fiber hundred-watt-level mid-infrared lasers for real applications.

Funding

National Natural Science Foundation of China (62090063, 62075082, U20A20210, 61827821, U22A2085, 62235014, 62205121); the Opened Fund of the State Key Laboratory of Integrated Optoelectronics.

Disclosures

The authors declare no conflict of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Supplemental document

See Supplement 1 for supporting content.

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Supplementary Material (1)

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Supplement 1       Supplemental Document

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (5)

Fig. 1.
Fig. 1. Calculated GVD curve of the fundamental propagation mode for fluorotellurite fiber with 14 µm core diameters. Inset: scanning electron micrograph of the fluorotellurite fiber.
Fig. 2.
Fig. 2. (a) Experimental setup of the 50-W mid-infrared SC laser source. (b) Microscope photo of connection point between silica and fluorotellurite fiber before and after fusion splicing. (c) Output spectrum of the pump laser with maximal launched pump power. (HNLF, highly nonlinear optical fiber; DCF, dispersion compensating fiber).
Fig. 3.
Fig. 3. (a) Mid-infrared SC spectrum evolution under different output powers from the fluorotellurite fiber. (b) Dependence of the SC average power output from fluorotellurite fiber on the launched pump power. Inset: The power meter photograph during the mid-infrared SC laser operating at the output power of 50.22 W.
Fig. 4.
Fig. 4. (a) and (b) The simulated (the black curves) and measured (the red curves) SC spectra output from the silica fiber and fluorotellurite fiber for the launched average pump power of 73.35 W, respectively. (c) and (d) Spectral and temporal evolution of SC generation in concatenated 2.0 m silica and 0.48 m fluorotellurite fiber for a launched average pump power of 73.35 W, respectively.
Fig. 5.
Fig. 5. Maximum extractable powers of mid-infrared SC laser sources limited by thermal fracture, melting of the core and melting of the coating versus the total transmission loss of 0.48 m long fluorotellurite fibers with the core diameter of 14 µm.

Tables (1)

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Table 1. List of parameters, symbols and values in calculation as well as units and references

Equations (5)

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P T F = 4 η l a s e r π R m L η h e a t ( 1 a 2 2 b 2 )
P M C E = 4 η l a s e r π k c o r e ( T m c e T c ) L η h e a t ( 1 + 2 k c o r e c h + 2 ln ( b a ) + 2 ln ( c a ) )
P M C T = 2 η l a s e r π k c o a t ( T m c t T c ) L η h e a t ( k c o a t c h + ln ( b c ) )
P T L = η l a s e r π k c o r e λ 2 L 2 η h e a t d n d T a 2
P S F = 0.148 λ 2 n n 2
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