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SECTION I

## INTRODUCTION

Generation of coherent radiation in the 2–20 $\mu\hbox{m}$ wavelength range holds great promises for various application fields including spectroscopy/environment, surgery/medicine as well as countermeasures. Fiber lasers are potential candidates for such applications where high power, superior beam quality and ruggedness are generally required. However, the long-wavelength limited transparency of standard silica glass fibers restricts their wavelength coverage to less than 2.1 $\mu\hbox{m}$. The use of low phonon glass hosts (i.e., fluoride and chalcogenide glasses) possessing an extended transmission window is an interesting alternative to that problem. However, low-phonon energy glass fiber lasers doped with rare-earth ions are currently limited to fluoride glass fibers and actually offer only a few emission bands, which become increasingly scarce at wavelengths longer than 3 $\mu\hbox{m}$. Raman Fiber Lasers (RFLs), which can be implemented from either fluoride or chalcogenide fibers, provide an efficient solution to fill the wavelength gaps where no other fiber laser can operate. Moreover, the Raman fiber laser overall assembly, including the pump laser cavity and the Raman cavity itself, can be made entirely monolithic once fiber Bragg gratings (FBGs) inscription and fusion splices are mastered.

In recent years, it was demonstrated that radiation in the vicinity of 2.1–2.3 $\mu\hbox{m}$ could be generated using a fluoride glass based Raman fiber laser consisting of a pair of FBGs directly written inside a low-loss fluoride fiber and pumped by a Tm:silica fiber laser [1], [2]. More recently, a first-order $\hbox{As}_{2}\hbox{S}_{3}$ Raman fiber laser was demonstrated for the first time, delivering a 600 mW peak power at 3340 nm [3]. This demonstration relied on recent progresses in the development of powerful and reliable erbium doped fluoride glass fiber lasers operating near 3 $\mu\hbox{m}$ [4].

The aim of this paper is to demonstrate that the entire spectrum from 3 to 4 $\mu\hbox{m}$ could be covered using the combination of first- and second-order chalcogenide $(\hbox{As}_{2}\hbox{S}_{3})$ based RFLs. First, the details of the RFL numerical model are presented. Then, we estimate the Raman gain and Stokes attenuation coefficient using experimental results obtained from a specially designed Raman fiber laser cavity. Finally, we show that high efficiencies can be expected when the fiber length and output coupler reflectivity are properly optimized.

SECTION II

## RAMAN FIBER LASER MODEL

Our numerical study of the chalcogenide RFL is based on the standard power coupled equations for Raman lasers, as described in [5]. In this equation set, we treat the pump and Stokes (1st and 2nd) as monochromatic signals an suppose the power transfer occurs only between two successive orders. We also neglect polarization effects and the spontaneous Raman scattering contribution. Two equations are used for each wavelength to describe forward (+) and backward (−) propagation, respectively. For a second-order Raman fiber laser, the equation set can be written as: TeX Source \eqalignno{{dP_{0} (z)^{\pm} \over dz} = &\, \mp \alpha_{0}P_{0} (z)^{\pm} \mp {\lambda_{1} \over \lambda_{0}}{g_{R0} \over A_{{\rm eff}0}}\, x_{0 - 1}\left(P_{1} (z)^{+} + P_{1} (z)^{-}\right)P_{0} (z)^{\pm}\cr {dP_{1} (z)^{\pm} \over dz} = & \, \mp \alpha_{1}P_{1} (z)^{\pm} \pm {g_{R0} \over A_{{\rm eff}0}}\, x_{0 - 1}\left(P_{0} (z)^{+} + P_{0} (z)^{-} \right)P_{1} (z)^{\pm}\cr& \mp {g_{R1} \over A_{{\rm eff}1}}\, x_{1 - 2}{\lambda_{2} \over \lambda_{1}} \left(P_{2} (z)^{+} + P_{2} (z)^{-}\right)P_{1} (z)^{\pm}\cr {dP_{2} (z)^{\pm} \over dz} = &\, \mp \alpha_{2}P_{2} (z)^{\pm} \pm {g_{R1} \over A_{{\rm eff}1}}\, x_{1 - 2}\left(P_{1} (z)^{+} + P_{1} (z)^{-} \right)P_{2} (z)^{\pm}&\hbox{(1)}} with boundary conditions given by TeX Source \eqalignno{P_{0} (0)^{+} = &\, P_{\rm launched}\cr P_{0} (L)^{-} = &\, R_{{\rm out}0}P_{0} (L)^{+}\cr P_{1} (0)^{+} = &\, R_{{\rm in}1}P_{1} (0)^{-}\cr P_{1} (L)^{-} = &\, R_{{\rm out}1}P_{1} (L)^{+}\cr P_{2} (0)^{+} = &\, R_{{\rm in}2}P_{2} (0)^{-}\cr P_{2} (L)^{-} = & \, R_{{\rm out}2}P_{2} (L)^{+}&\hbox{(2)}} where the indices 0, 1, and 2 refer to the pump, first-order Stokes, and second-order Stokes respectively, $g_{R, i}$ is the Raman gain coefficient, $A_{{\rm eff}, i}$ is the effective area of the $\hbox{LP}_{01}$ mode of the fiber, $x_{i - j}$ is the correction from the modal overlap (between $\hbox{LP}_{01}$ modes at $\lambda_{i}$ and $\lambda_{j}$ wavelengths) and $\alpha_{i}$ is the attenuation coefficient. In addition, $R_{{\rm in}, i}$ and $R_{{\rm out}, i}$ correspond to the reflectivity of the input and output fiber Bragg gratings (FBGs). Note that first-order equations can be found by setting $P_{2}(z)^{\pm} = 0$. The coupled equations are solved in Matlab using a finite difference code that implements a three-stage Lobatto IIIa formula which provide a fourth-order accuracy.

It is well known that spectral broadening inside a RFL cavity has a significant impact on laser performances [6], [7], [8], [9], [10]. In fact, it reduces the FBG effective reflectivity by causing a portion of the intracavity power to leak outside the cavity. These effects are not readily accounted for in Eqs. (1) and (2). However, in some cases where experimental data on the FBG and laser spectra are available (as a function of the output power), the impact on the reflectivity can be implemented by calculating the effective reflectivity $R_{\rm eff}$ with the following integrals: TeX Source $$R_{\rm eff} = {P_{\rm reflected} \over P_{\rm incident}} = {\int {P_{\rm out}(\lambda) \over 1 - R_{\rm out}(\lambda)}R_{\rm out}(\lambda)d\lambda \over \int {P_{\rm out}(\lambda) \over 1 - R_{\rm out}(\lambda)}d\lambda}.\eqno{\hbox{(3)}}$$

Note that it is not always possible to perform this calculation for every FBGs involved. For instance, we do not have access to the spectrum leaking from the input grating as it would require a setup that is specifically designed for this purpose. Moreover, even when the effective reflectivity calculation is performed on a specific Raman cavity; the results cannot be readily transposed to different cavity parameters (e.g., fiber length, output coupler reflectivity, FBG bandwidth, etc.). Therefore, we chose not to include the effective reflectivity effects in the last section of this paper, where predictions are made for a wide range of first- and second-order RFLs parameters. Such approximation is justified for situations where the power leakage is minimized with the use of broadband (i.e., chirped) FBGs [9].

SECTION III

## VALIDATION OF THE MODEL USING EXPERIMENTAL RESULTS

Reliable numerical predictions of chalcogenide RFLs critically rely on accurate values of both the fiber Raman gain and attenuation coefficients. While the attenuation of our single-mode fiber was previously measured in a cutback procedure [11], a significant amount of diffusion points (i.e., point defects) were later revealed when the residual fiber segment was screened using an InSb IR camera (Telops TEL-3306). The losses associated to these point defects were found to be responsible for a significant part of the overall (average) attenuation coefficient. For the present experiment, we have carefully selected a fiber segment without defects and thus, a significantly lower attenuation than previously reported (100 dB/km) is expected.

The experimental setup used for the model validation experiments consists of a 1.8 meter defect-free $\hbox{As}_{2}\hbox{S}_{3}$ fiber in which two first-order Stokes FBGs are written (shown in Fig. 1). The output FBG has an initial value (right after inscription) of about 90%. To obtain a better estimation of the Raman gain and Stokes attenuation, the output FBG was subsequently heated to lower its reflectivity from 90% to 85%, to 80% and finally to 75%, corresponding to four distinct Raman cavities. The FBGs spectral responses are shown in Fig. 1(b). The chalcogenide fiber has a 4/150 $\mu\hbox{m}$ (core/clad) diameter and a numerical aperture of 0.36, leading to a cutoff wavelength of about 1900 nm. A 3005 nm Er:ZBLAN fiber laser operated in quasi-CW regime is used to pump the Raman cavity (see Ref. [12] for details). Note that this wavelength is the longest ever reported from an erbium-doped fiber laser. This laser delivers 5 ms pulses at a repetition rate of 20 Hz which are sufficiently long for the RFL to reach its steady state. The difference between the pump and Stokes wavelengths (i.e., 3005 and 3337 nm) corresponds to a frequency shift of 331 $\hbox{cm}^{-1}$.

Fig. 1. (a) Experimental setup of the $\hbox{As}_{2} \hbox{S}_{3}$ first-order RFL. (b) Spectrum of the FBGs.

The average power is measured with a thermopile detector (Gentec XLP12-3S-H2) and the corresponding peak power is inferred from the pulse precise temporal shape [3]. To separate the Stokes signal from the pump, an in-house fabricated long pass filter is placed before the detector. The Stokes spectrum as well as the FBG transmission spectra are measured using a grating-based scanning spectrometer (Digikrom DK480) coupled to a nitrogen-cooled InSb detector (Judson, J10D-M204-R01M-60).

Numerical calculations based on Eqs. (1) and (2) are performed to fit the fiber attenuation and Raman gain coefficients. For each simulation, we first calculate the sum of squared errors between experimental and numerical laser thresholds (for the four laser cavities). To replicate the experimental conditions, the simulations data is corrected to account for the attenuation of the fiber prior and after the laser cavity, for the Fresnel reflection of the fiber tip and for the losses associated with the long pass filter. We notice the existence of a family of parameter sets that give a reasonable fit of the laser thresholds (Fig. 2). Accordingly, the least error is obtained for $\alpha_{1} = 0.025 \ \hbox{dB/m}$ and $g_{R} = 1.65 \ast 10^{-12} \ \hbox{m/W}$ (unpolarized). Since the operating wavelength is slightly shifted from the measured Raman gain peak of our fiber, at 342 $\hbox{cm}^{-1}$ (see Fig. 3), the peak value is estimated at $1.85\ast 10^{-12} \ \hbox{m/W}$. This value is somewhat higher than previous reports spreading from 1.07 and $1.42 \ast 10^{-12} \ \hbox{m/W}$ at the same wavelength (assuming a simple inverse relation with pump wavelength) [13], [14], [15]. Note that a variation of the gain coefficient is possible when slightly different core compositions are used.

Fig. 2. Residual sum of squares between numerical and experimental laser thresholds as a function of the Stokes attenuation and the Raman gain coefficient. Least error is denoted by a white cross.
Fig. 3. Raman gain spectrum of the single-mode $\hbox{As}_{2}\hbox{S}_{3}$ fiber measured by pump-probe experiment, as in [1].

Based on the derived values of $\alpha_{1}$ and $g_{R}$, we then computed the lasing curves. First note that in order to simultaneously fit the laser thresholds and the slope efficiencies, a power leak of up to 7% (due to spectral broadening) was introduced in the numerical simulation. This was accounted for by fitting the input FBG's effective reflectivity as a function of Stokes output power. Accordingly, Fig. 4 shows that one can reproduce quite accurately both the pump and Stokes experimental results when including the effect of spectral broadening on the FBGs effective reflectivities. As expected, Fig. 4(d) curves indicate that a higher output coupler reflectivity leads to increased power leakage through the input FBG.

Fig. 4. (a) Stokes output peak power and (b) residual pump peak power with respect to the launched pump peak power (the x's represent experimental data points). (c) Calculated effective reflectivity associated to the output FBG [obtained using Eq. (3)]. (d) Fitted input FBG's effective reflectivity.
SECTION IV

## OPTIMIZED FIRST- AND SECOND-ORDER RFLs EMITTING BETWEEN 3 AND 4 $\mu$ m

Numerical simulations of optimized first- and second-order RFLs are presented in this section. Based on the actual gain/loss parameters derived in the previous section, we show the 3–4 $\mu\hbox{m}$ spectral range could be covered provided the pump wavelength can be varied from 2700 to 3050 nm (i.e., the expected limits of an Er:ZBLAN fiber laser). For each pump wavelength, first and second Stokes order wavelengths are chosen to match the peak Raman gain coefficient at 342 $\hbox{cm}^{-1}$. Then, the attenuation coefficients are adjusted as well as the effective area and the Raman gain itself, which is scaled using an inverse relationship between the gain and pump wavelength. For Stokes wavelengths extending beyond the expected long wavelength limit of an Er:ZBLAN fiber laser pump (at 3050 nm), we set the pump wavelength at 3050 nm and lower the Raman gain according to the $\hbox{As}_{2}\hbox{S}_{3}$ Raman gain spectrum (Fig. 3).

The modeled laser cavity (see Fig. 5) consists of two pairs of FBGs, one located at the first-order Stokes wavelength and the other at the second Stokes order (where applicable). A highly reflective FBG (99%) is included to recycle the residual pump and thus improve the Raman conversion efficiency.

Fig. 5. Schematic diagram of the modeled $\hbox{As}_{2} \hbox{S}_{3}$ Raman cavity.

For all the simulations, the spectral broadening effects are neglected based on the assumption that broadband FBGs could be written in $\hbox{As}_{2}\hbox{S}_{3}$ fibers. The value of the peak Raman gain coefficient is set to $1.85 \ast 10^{-12} \ \hbox{m/W}$ at 3005 nm (reference wavelength) and the background attenuation of the fiber is scaled to fit our value at 3337 nm (i.e., $\alpha_{1} = 0.025 \ \hbox{dB/m}$) from the previous section. The OH absorption peak around 2.9 $\mu \hbox{m}$ is uncoupled (numerically) from the rest of the attenuation curve and is scaled separately to have 1.5 dB/m at 3005 nm, a value obtained from a cutback experiment. The resulting attenuation curve is shown in Fig. 6(a) for the wavelength range of interest (i.e., from 2.7 to 4 $\mu\hbox{m}$).

Fig. 6. (a) Spectral attenuation of the single-mode $\hbox{As}_{2}\hbox{S}_{3}$ fiber showing the fitted background losses and OH absorption peak. (b) First-order and (c) second-order Stokes output power as a function of wavelength and fiber length $(P_{\rm launched} = 5 \ \hbox{W})$.

Fig. 6(b) and (c) presents simulations performed for different fiber lengths at a launched pump power of 5 W. For each point, the output FBG reflectivity was simultaneously optimized to obtain the maximum output power, as shown in Fig. 7. One should note that for long fiber segments, the power drop observed (e.g., near 3200 and 3600 nm) could be reduced using a pump wavelength shorter than the Raman peak shift, where the attenuation is lower. At this power level, the optimal length varies between 0.5 and 4 meters, depending primarily on the fiber attenuation. These lengths are significantly shorter than typical Raman cavity lengths, which are usually of the order of a few hundred meters to a kilometer [16]. This can be explained by the fact that both the Raman gain and attenuation of our chalcogenide fiber are significantly higher than for standard silicate based fibers. We also note an important variation of the optimal output coupler's reflectivity as the pump wavelength is moved between 2700 and 3050 nm. For first-order RFLs, a higher output reflectivity is needed to reach the laser threshold when the pump is near the OH absorption peak. On the other hand, second-order RFLs require higher finesse cavities for shorter pump wavelengths because the first Stokes order falls within the OH absorption peak. An increase of the optimal reflectivity is also observed outside the Er:ZBLAN pump wavelength range where the Raman gain coefficient decrease according to the gain spectrum.

Fig. 7. Optimized output coupler's reflectivity for (a) a first-order and (b) a second-order RFL as a function of wavelength and fiber length $(P_{\rm launched} = 5 \ \hbox{W})$.

Fig. 8 presents Stokes power curves with respect to the launched pump power using cavity parameters that provide the highest output power (i.e., optimized length and output coupler reflectivity). Four pump wavelengths were chosen: 2750, 2850, 2950, and 3050 nm. The slope efficiency of the laser is maximal for pump wavelengths beyond the OH peak and can reach a value of 66% for first-order and 46% for second-order cavities. On the other hand, the worst performances are obtained when pumping on the short wavelength side of the OH peak (i.e., for a 2750 nm pump). This can be explained by the fact that the corresponding first-order Stokes wave is subjected to a significantly higher attenuation. In general, the efficiencies can be improved even more by lowering the output FBG reflectivity. However, as the threshold will increase as well, a stronger pump will be required to extract a higher Stokes output power. As for the effect of the FBG used as pump reflector, it was found to play a more critical role for pump wavelengths lying between 2700 nm and 2850 nm than for those lying within the OH peak (i.e., between 2850 nm and 3000 nm) where the pump is more strongly absorbed.

Fig. 8. Stokes output power of (a) a first-order and (b) a second-order RFL as a function of the launched pump power, using optimized length and output FBG reflectivity.

A few general guidelines can be drawn from our simulations. Evidently, with a fiber having a lower attenuation, we can expect a longer optimal length and an enhanced efficiency as both pump and Stokes waves can propagate a longer distance before suffering from significant attenuation. To achieve higher output powers, one might consider the use of a chalcogenide fiber having a larger effective mode area. While this would reduce the effective Raman gain, it would also facilitate the coupling of the pump into the $\hbox{As}_{2}\hbox{S}_{3}$ fiber (i.e., increase the pump launch efficiency) and increase the power damage threshold substantially. For such a fiber, a higher Raman laser threshold is expected; nevertheless, the efficiency should be comparable to those of Fig. 8 if sufficient pump power is provided.

These results reveal the great potential of RFLs as compact laser sources operating in the mid-infrared. Nevertheless, to reach experimental performances comparable to the numerical ones, a few improvements are required. First, we note that the discrepancies observed between experimental and numerical efficiencies are primarily caused by significant power leaks through the FBGs. By improving our techniques to write broadband chirped FBGs, we believe the power leakage could be minimized and even become negligible [9]. A reduction of OH impurity content in these fibers would also be highly beneficial as it could raise the conversion efficiency dramatically, particularly in the low spectral range of the pump (2700–2950 nm). Calculations reveal that the thresholds could be lowered by as much as a factor of 2 if the OH absorption peak was removed.

We also note that even if our study was limited to output wavelengths between 3 and 4 $\mu\hbox{m}$, longer wavelengths could be reached using third order Raman cavities or higher. With currently available fibers, the main limiting factor would be the rising material attenuation near 4 $\mu\hbox{m}$ due to the proximity with the S-H bond main absorption peak [17]. In addition, it would also be possible to optimize the coupler reflectivity ($R_{{\rm out}1}$ and $R_{{\rm out}2}$) to promote multiwavelength operation of the Raman cavity, as it was done previously with more standard RFLs [18]. The simultaneous emission of 2 or more wavelengths in the mid-IR would be useful for the generation of terahertz waves by the difference frequency in a nonlinear media [19].

SECTION V

## CONCLUSION

This paper is bringing clear evidence that chalcogenide based RFL can be designed for efficient operation on any wavelength between 3 and 4 $\mu\hbox{m}$. Current bottlenecks of $\hbox{As}_{2}\hbox{S}_{3}$ Raman fiber lasers include the limited control over the FBGs bandwidth, the use of free-space components to launch the 3 $\mu\hbox{m}$ pump power and the presence of OH impurities that increase the attenuation between 2850–3000 nm. In the near future, the FBGs inscription process based on chirped phase mask will be optimized to provide better experimental results. In addition, the replacement of bulk components by an all-fiber solution will be studied to achieve higher power levels.

## Footnotes

This work was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC), by the Canada Foundation for Innovation (CFI), by the Fonds de recherche du Québec—Nature et technologies (FRQ.NT), by Canada Excellence Research Chairs (CERC), and by CorActive High-Tech, Inc. Corresponding author: V. Fortin (e-mail: vincent.fortin.5@ulaval.ca).

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