Abstract

The first demonstration of a multi-watt continuous wave fluoride glass Raman fiber laser operating beyond 2.2 μm is reported. A maximum output power of 3.7 W was obtained from a nested cavity setup with a laser slope efficiency of 15% with respect to the launched pump power.

© 2012 OSA

1. Introduction

Raman fiber lasers (RFLs) can be operated at virtually any wavelength as they only require an adequate combination of pump laser and fiber. This unique feature allows access to several interesting emission wavelengths located outside of the lanthanides’ emission bands suitable for lasing. Over the last decade, significant efforts have been deployed to perfect RFLs at specific wavelengths between 1 and 2 μm. For instance, 1480 nm RFLs have been developed and perfected as efficient pumps for Er3+ fibers lasers and amplifiers [13]. However, due to the challenges associated with using non-silica glass fibers, very little work has been done at emission wavelengths exceeding 2 μm. This is unfortunate since wavelengths between 2 and 5 μm are currently of great interest for several applications including defense & security, biomedical and spectroscopy. Of all the oxide materials typically used, GeO2 based fibers were the most appealing candidates for RFLs operating beyond 2µm, due to their high Raman gain coefficient, comparatively low attenuation and good availability [4,5]. The most notable result reported to date was a 2105 nm first order Raman fiber laser emitting a maximum power (CW) of 4.6 W [5]. However, for output wavelengths even further in the mid-IR, new glasses have to be considered. Fluoride and chalcogenide glass fibers are potential candidates for this purpose but they come along with hurdles such as their lower damage threshold, reduced robustness as well as the impracticality – until recently – of writing quality fiber Bragg gratings (FBGs) within their core. In addition, their spectral attenuation (a critical parameter for RFLs) is typically higher than for standard silica fibers. Recently, a fluoride based Raman fiber laser was reported, operating at 2185 nm and delivering a maximum output power of 600 mW [6]. However, the efficiency of this RFL was severely impaired by spectral broadening, thermal shifting of the output FBG, and by the use of a single pass pump configuration.

In this paper, we report on a nested cavity Raman laser based on a fluoride glass fiber producing 3.66 W at 2231 nm, i.e. the highest output power ever reported from such a fiber laser operating beyond 2.2 μm. The RFL operates at a 567 cm−1 Raman spectral shift which is very close to the peak Raman gain of the material (572 cm−1) [6].

2. Experiment

The experimental setup is shown in Fig. 1 . A 36 W, 791 nm laser diode (QPC Lasers BrightLase Ultra-100) is used to pump an 8 m-long Tm3+:silica fiber followed by a 26 m-long undoped fluoride glass (Raman) fiber. The Tm3+-doped double clad fiber (Coractive DCF-TM-6/125) is used to produce the Raman pump wavelength at 1981 nm. It has a 6.2 μm diameter, 0.23 NA core and an estimated Tm3+ ions concentration of 4 wt. %, according to the manufacturer. The fluoride glass fiber from Le Verre Fluoré (model 2818) is almost perfectly mode-matched to the silica fiber with a 6.7 μm diameter core and an NA of 0.23. The two fibers are butt-coupled to each other with the use of high precision mounts. Peltier coolers were added to the mounts to avoid thermally-induced fiber misalignment at high power. At the output of the fluoride fiber, a long pass edge filter at 2050 nm is used to separate the Stokes and Raman pump wavelengths before the monitoring setup. The spectra are recorded using an optical spectrum analyzer (Yokogawa AQ6375) and a thermopile detector (Gentec UP19K-15S-H5) is used to measure the output power.

 

Fig. 1 Experimental setup of the nested cavity Raman fiber laser.

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Two nested pairs of Bragg gratings were written, forming two concurrent laser cavities. FBGs in the fluoride glass fiber were inscribed based on a femtosecond writing method at 800 nm [7] whereas we opted for a 400 nm writing wavelength for the FBG located inside the silica fiber [8]. The first cavity (shown in blue in Fig. 1), acting as the Raman pump, oscillates at 1981 nm and is formed by a highly reflective (HR) chirped Bragg grating (P1) written in a mode-matched double clad undoped silica fiber on the input side and by a HR FBG written at the end of the fluoride fiber (P2). High reflectivity FBGs were used to maximize the intracavity Raman pump power and thus to lower the stimulated Raman scattering threshold.

The second cavity (in red, Fig. 1) is set to operate at the first order Stokes wavelength at 2231.4 nm and is formed by two additional Bragg gratings (S1 and S2) both written in the fluoride glass fiber. Reflectivities of 99.9% and 92% were chosen for the input and output ends, respectively. Figure 2 shows the experimental transmission spectra of these two Stokes FBGs. To ensure their long term stability, the FBGs were thermally annealed at 100 °C for 5 minutes following their inscription.

 

Fig. 2 Transmission spectra of the pair of FBGs forming the Stokes cavity.

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The Stokes FBGs were both written from the same phase mask so that their spectral overlapping did not require any post-writing spectral tuning. However, in order to prevent thermally-induced redshift during laser operation, we glued the Stokes FBGs in passively-cooled copper V-grooves.

3. Experimental results

An output power of 3.66 W was produced at 2231 nm at the maximum launched pump of 36 W. As shown in Fig. 3 , an 8 W Raman threshold and a 15% slope efficiency (down to 11% at high powers) were observed. At the power levels involved, we did not detect any instability from the feedback generated by the butt-coupled junction.

 

Fig. 3 Output power as a function of the launched pump power.

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Spectra of both the Stokes and Raman pump are displayed in Fig. 4 . Spectral broadening is clearly shown for both waves. This effect is commonly observed in high power RFLs [911] and affects the laser behaviour by causing power leaks when the spectrum gets broader than the FBG’s bandwidth. In order to compensate for the reduction in efficiency that would otherwise result from spectral broadening, we purposely used an output Stokes FBG (S2) with a higher than optimal reflectivity so that the effective reflectivity would decrease to its optimal value at high power.

 

Fig. 4 Output spectra for the pump (a) and Stokes (b) wavelengths.

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The power stability of the RFL was also recorded at different output powers (Fig. 5 ). We measured peak to peak fluctuations of less than 0.5% at an output power of 3.3 W, which is only slightly above the peak-to-peak noise level of the thermopile detector.

 

Fig. 5 Power stability for different Stokes output powers from 0.17 W to 3.3 W.

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4. Analysis

4.1 Numerical model

We modeled the nested cavity laser in two steps. First, the Tm:silica fiber laser was simulated using the local population density rate equations along with the steady-state equations for the pump and Raman pump waves [12,13]. According to the notation used in reference [12], we labeled the 3H6, 3F4, 3H5 and 3H4 from 1 to 4, respectively. The laser transition (3F43H6) occurs from level 2 to level 1 and the pump absorption (3H63H4) brings the ions up to level 4. We neglected the population density of level 3 due to its short lifetime compared to those of levels 4 and 2.

dN4dt=σa_QN1(Q(z)hνQAeff_Q)N4τ4k4212N4N1+k2124N22,
dN2dt=2k4212N4N12k2124N22N2τ2+β42N4τ4(P(z)hνPAeff_P)(σe_PN2σa_PN1),
N1=NN2N4,
dQ(z)dz=ηPσa_QN1Q(z),
dP(z)±dz=±(σe_PN2σa_PN1αP)P(z)±,
In Eqs. (1-5), the variables Q and P are used to describe respectively the pump (791 nm) and Raman pump (1981 nm) and the signs in exponent ( ± ) depict forward and backward propagation, σa_Q, σa_P and σe_P are the absorption and emission cross-sections, k4212 and k2124 describe the energy transfer processes, β42 is the branching ratio of the spontaneous transition (4 -> 2), τ2 and τ4 are the spontaneous lifetimes and finally, ηP is the core/clad area ratio.

During the first step, we fitted the value of the coefficient describing the energy transfer process k4212 to reproduce the experimental data of our Tm-doped fiber laser when operated alone (i.e. considering a feedback provided by Fresnel reflection only). The remaining parameters, namely the emission and absorption cross sections, branching ratios, lifetimes and other energy transfer coefficients were taken from previously published values [1214]. We also estimated splice losses lower than 5% between the Tm-doped fiber and the undoped fiber hosting the Raman pump input FBG (P1) and lumped losses from this grating were evaluated at 2% from the measured spectrum.

The second step of our numerical modeling consisted in computing the powers involved in the whole nested cavity laser setup. The Raman cavity was modeled using the following set of steady-state differential equations [15]:

dP(z)±dz=αPP(z)±νPνSgRAeff(S(z)++S(z))P(z)±
dS(z)±dz=αSS(z)±±gRAeff(P(z)++P(z))S(z)±,
with initial conditions
P(0)+=PintracavityP(L)=RP2P(L)+S(0)+=RS1S(0)S(L)=RS2S(L)+,
where S denotes the Stokes power (2231 nm), αP and αS are the attenuation of the fiber for the Raman pump and Stokes wavelengths given by 0.03 and 0.0024 dB/m respectively, Aeff is the effective area of the 1981 nm LP01 mode of the fluoride glass fiber and finally, gR is the unpolarized Raman gain coefficient at 1981 nm, fixed to 3.33*10−14 m/W according to a previous report [6]. We estimated the losses associated to each fluoride glass fiber FBG at 4% based on previous measurements made on similar gratings. Lastly, since we could not measure loss at the butt-coupled junction, it was estimated at 21%, which corresponds to the loss measured in a similar setup using an identical fluoride fiber (without gratings).

In these calculations, we used the value of k4212 derived in step 1 and treated the Raman cavity as an added loss term in Eqs. (4) and (5). For each launched pump power (Q), we initially solved Eqs. (1-5) ignoring Raman conversion to obtain an intracavity pump power (P) estimation. Using this value, we then solved Eqs. (6-8) to calculate the portion of the Raman pump that would be converted to the first order Stokes wavelength by the Raman cavity. This iterative process was carried out until the change in power (P and S) between two successive iterations was negligible (i.e. below 1%).

The effective reflectivity of the output FBGs (due to spectral broadening) was calculated with the simple formula Reff=Preflected/Pincident using the measured laser output spectra and the FBG spectra. We assumed the power leaking from the input Stokes and pump FBGs was negligible; this approximation is reasonable since their bandwidths are substantially larger than their matching output FBGs (i.e. 2.5 and 1.1 nm FWHM widths respectively for S1 and S2 FBGs).

4.2 Discussion

The overall lasing efficiency of 15% is resulting from two wavelength conversion processes. The first conversion (from 791 nm to 1981 nm) is associated with the Raman pump cavity while the second one (from 1981 nm to 2231 nm) is made by the Stokes cavity. Because of the nested cavity nature of the RFL, we do not have access to the 1981 nm – 2231 nm conversion efficiency. Note that it is not relevant since the Raman pump cavity was designed to maximize the intracavity pump power at the expense of the output pump power. Nevertheless, it is possible to show that the measured efficiency is significantly better than for the alternative approach consisting of two independent cavities [6]. In fact, from these previous results, we can deduce an 11% low power efficiency with respect to the 791 nm pump that is reduced to 5% at high powers, the latter being less than half of the efficiency value reported in the present experiment. Aside from the enhanced cavity configuration, we also believe this efficiency increase is attributed to the fact that the overlap between FBGs S1 and S2 was maintained by preventing spectral shifting of either FBGs. Now it should be noted that, in the cavity configuration we used, spectral broadening plays a dominant role in the laser behavior (Fig. 4). This phenomenon is enhanced due to the combination of a low group velocity dispersion (GVD) and the use of highly reflective gratings. In fact, according to simulations based on the formulas derived by Zhang et al [16], our fluoride fiber is showing a plateau region near its zero dispersion wavelength (located at λ = 1.7 μm). Therefore, the GVD is small on an extended spectral range (from 1.7 to 3 μm) and takes an approximate value of 5.5 ps km−1nm−1 at the Stokes wavelength of our fiber laser. This contributes to enhance four-wave mixing (FWM), and thus spectral broadening, due to a larger number of longitudinal modes with quasi phase matching [9]. In addition, the highly reflective FBGs also contribute to raise significantly the intensities inside the cavities, even at low input powers, thus favoring nonlinear effects.

In order to identify bottlenecks and possible improvements for our RFL, two assumptions are made in the numerical modeling. First, we assume the effective reflectivity of the FBG P2 is the same for all sets of cavity parameters. We also assume that the Stokes FBG (S2) reflectivity depends on the total intracavity Stokes power only. This allows us to proceed to an extrapolation for intracavity power values beyond the experimental data range. Simulations were first carried out to analyze the impact of the intracavity fiber butt-coupled losses (Fig. 6 ). By reducing these losses from 21% to 5%, the laser efficiency rose from 15% to 20%. We also believe these losses could explain the slight roll-over of the experimental output Stokes power that was not naturally replicated by the model (Fig. 3). In fact, we observed a maximum experimental output power of 3.7 W compared to the 4 W predicted by the model. This discrepancy could be explained by a thermally-induced shift of the butt-coupled junction alignment which was not taken into account in the model. Because of the high finesse Raman pump cavity, high intensities were passing through this junction and our present setup could not efficiently extract the heat generated. In fact, the model indicated that for additional junction losses of about 4%, the output power predicted would be 3.7 W (Fig. 6(a)), as we measured.

 

Fig. 6 (a) Calculated Stokes output power as a function of the launched pump power for different butt-coupled losses. (b) Calculated Stokes output power as a function of the S2 FBG reflectivity (black curves). The red and blue curves show the output power for a cavity with and without spectral broadening respectively.

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Another set of simulations was performed to determine the optimal Stokes FBG reflectivity (S2) and the results are summarized in Fig. 6(b). In this figure, the black curves show the Stokes output power with respect to the reflectivity of the output Stokes FBG for specific input pump powers (at 791 nm). In addition, the red and blue curves display respectively the output power for a cavity with and without spectral broadening. These simulations revealed that the reflectivity of the Stokes output FBG was already near optimal to obtain a maximum Stokes power (at Q = 36 W). It is interesting to note that the spectral broadening actually improved the efficiency and the maximum power obtainable because, as we increased the pump power, the reflectivity drifted towards the optimal reflectivity (i.e. towards lower reflectivities). This demonstrates the importance to overshoot the reflectivity of the output FBG when designing a Raman cavity subject to strong spectral broadening. Naturally, much higher input powers would require a lower Stokes FBG reflectivity to extract the maximum power possible. For instance, our model predicts a 71% optimal (effective) reflectivity for an input pump power twice as high (Q = 72 W), which would correspond to a (true) reflectivity of about 83% considering the spectral broadening associated with the intracavity Stokes power calculated.

Some improvements could be made to increase the performance of our laser. First, the Raman pump cavity has room for improvement considering some groups achieved higher Tm:silica fiber laser efficiency in the past [17]. Moreover, even better results could be obtained by using a Tm-doped fluoride fiber instead of a silica based fiber, as it was previously shown that a higher efficiency was expected in this case [18]. This would also make a fusion splice between the pump and Raman fibers feasible, therefore lowering intracavity losses (see Fig. 6) and improving the overall robustness of the laser. Secondly, the Raman cavity efficiency depends much on the fluoride fiber used to generate the Raman gain. Although the fiber we had was well adapted for this application, a higher NA and a smaller core diameter would provide a better confinement of the LP01 mode and increase the effective gain coefficient accordingly. This would obviously result in a higher Raman cavity efficiency provided the attenuation coefficient is not changed. Finally, we also believe it would be possible to scale these results to even higher powers by adding more pump diodes since our experiments did not reveal any sign of damage to the fluoride glass fiber at these high power densities. In fact, previous work has already shown that tens of watts of CW power can be launched in fluorozirconate fibers without fiber damage issues [19].

5. Conclusion

In conclusion, we have demonstrated the first fluoride glass Raman fiber laser delivering as much as 3.7 W of output power at a wavelength beyond 2.2 µm. The output wavelength obtained is currently the longest ever produced with a RFL. We believe RFLs have the potential to provide much higher output powers at even longer mid-IR wavelengths with all the benefits of fiber laser technology.

Acknowledgments

This research was supported financially by the Canadian Institute for Photonic Innovations (CIPI), the Fonds québécois de recherche sur la nature et les technologies (FQRNT), the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Canada Foundation for Innovation (CFI).

References and links

1. G. Vareille, O. Audouin, and E. Desurvire, “Numerical optimisation of power conversion efficiency in 1480nm multi-Stokes Raman fibre lasers,” Electron. Lett. 34(7), 675–676 (1998). [CrossRef]  

2. N. S. Kim, M. Prabhu, C. Li, J. Song, and K.-I. Ueda, “1239/1484 nm cascaded phosphosilicate Raman fiber laser with CW output power of 1.36 W at 1484 nm pumped by CW Yb-doped double-clad fiber laser at 1064 nm and spectral continuum generation,” Opt. Commun. 176(1-3), 219–222 (2000). [CrossRef]  

3. N. Kurukitkoson, H. Sugahara, S. K. Turitsyn, O. N. Egorova, A. S. Kurkov, V. M. Paramonov, and E. M. Dianov, “Optimisation of two-stage Raman converter based on phosphosilicate core fibre: modelling and experiment,” Electron. Lett. 37(21), 1281–1283 (2001). [CrossRef]  

4. E. M. Dianov, I. A. Bufetov, V. M. Mashinsky, V. B. Neustruev, O. I. Medvedkov, A. V. Shubin, M. A. Melkumov, A. N. Gur'yanov, V. F. Khopin, and M. V. Yashkov, “Raman fibre lasers emitting at a wavelength above 2 μm,” Quantum Electron. 34(8), 695–697 (2004). [CrossRef]  

5. B. A. Cumberland, S. V. Popov, J. R. Taylor, O. I. Medvedkov, S. A. Vasiliev, and E. M. Dianov, “2.1 microm continuous-wave Raman laser in GeO2 fiber,” Opt. Lett. 32(13), 1848–1850 (2007). [CrossRef]   [PubMed]  

6. V. Fortin, M. Bernier, J. Carrier, and R. Vallée, “Fluoride glass Raman fiber laser at 2185 nm,” Opt. Lett. 36(21), 4152–4154 (2011). [CrossRef]   [PubMed]  

7. M. Bernier, D. Faucher, R. Vallée, A. Saliminia, G. Androz, Y. Sheng, and S. L. Chin, “Bragg gratings photoinduced in ZBLAN fibers by femtosecond pulses at 800 nm,” Opt. Lett. 32(5), 454–456 (2007). [CrossRef]   [PubMed]  

8. M. Bernier, R. Vallée, B. Morasse, C. Desrosiers, A. Saliminia, and Y. Sheng, “Ytterbium fiber laser based on first-order fiber Bragg gratings written with 400 nm femtosecond pulses and a phase-mask,” Opt. Express 17(21), 18887–18893 (2009). [CrossRef]   [PubMed]  

9. J.-C. Bouteiller, “Spectral modeling of Raman fiber lasers,” IEEE Photon. Technol. Lett. 15(12), 1698–1700 (2003). [CrossRef]  

10. P. Suret and S. Randoux, “Influence of spectral broadening on steady characteristics of Raman fiber lasers: from experiments to questions about validity of usual models,” Opt. Commun. 237(1-3), 201–212 (2004). [CrossRef]  

11. D. V. Churkin, S. V. Smirnov, and E. V. Podivilov, “Statistical properties of partially coherent CW fiber lasers,” Opt. Lett. 35(19), 3288–3290 (2010). [CrossRef]   [PubMed]  

12. J. Xu, M. Prabhu, J. Lu, K.-I. Ueda, and D. Xing, “Efficient double-clad thulium-doped fiber laser with a ring cavity,” Appl. Opt. 40(12), 1983–1988 (2001). [CrossRef]   [PubMed]  

13. S. D. Jackson and T. A. King, “Theoretical modeling of Tm-doped silica fiber lasers,” J. Lightwave Technol. 17(5), 948–956 (1999). [CrossRef]  

14. S. D. Agger and J. H. Povlsen, “Emission and absorption cross section of thulium doped silica fibers,” Opt. Express 14(1), 50–57 (2006). [CrossRef]   [PubMed]  

15. M. Rini, I. Cristiani, and V. Degiorgio, “Numerical modeling and optimization of cascaded CW Raman fiber lasers,” IEEE J. Quantum Electron. 36(10), 1117–1122 (2000). [CrossRef]  

16. L. Zhang, F. Gan, and P. Wang, “Evaluation of refractive-index and material dispersion in fluoride glasses,” Appl. Opt. 33(1), 50–56 (1994). [CrossRef]   [PubMed]  

17. P. F. Moulton, G. A. Rines, E. V. Slobodtchikov, K. F. Wall, G. Frith, B. Samson, and A. L. G. Carter, “Tm-doped fiber lasers: fundamentals and power scaling,” IEEE J. Sel. Top. Quantum Electron. 15(1), 85–92 (2009). [CrossRef]  

18. B. M. Walsh and N. P. Barnes, “Comparison of Tm:ZBLAN and Tm: silica fiber lasers; spectroscopy and tunable pulsed laser operation around 1.9 μm,” Appl. Phys. B 78(3-4), 325–333 (2004). [CrossRef]  

19. D. Faucher, M. Bernier, G. Androz, N. Caron, and R. Vallée, “20 W passively cooled single-mode all-fiber laser at 2.8 μm,” Opt. Lett. 36(7), 1104–1106 (2011). [CrossRef]   [PubMed]  

References

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  1. G. Vareille, O. Audouin, and E. Desurvire, “Numerical optimisation of power conversion efficiency in 1480nm multi-Stokes Raman fibre lasers,” Electron. Lett.34(7), 675–676 (1998).
    [CrossRef]
  2. N. S. Kim, M. Prabhu, C. Li, J. Song, and K.-I. Ueda, “1239/1484 nm cascaded phosphosilicate Raman fiber laser with CW output power of 1.36 W at 1484 nm pumped by CW Yb-doped double-clad fiber laser at 1064 nm and spectral continuum generation,” Opt. Commun.176(1-3), 219–222 (2000).
    [CrossRef]
  3. N. Kurukitkoson, H. Sugahara, S. K. Turitsyn, O. N. Egorova, A. S. Kurkov, V. M. Paramonov, and E. M. Dianov, “Optimisation of two-stage Raman converter based on phosphosilicate core fibre: modelling and experiment,” Electron. Lett.37(21), 1281–1283 (2001).
    [CrossRef]
  4. E. M. Dianov, I. A. Bufetov, V. M. Mashinsky, V. B. Neustruev, O. I. Medvedkov, A. V. Shubin, M. A. Melkumov, A. N. Gur'yanov, V. F. Khopin, and M. V. Yashkov, “Raman fibre lasers emitting at a wavelength above 2 μm,” Quantum Electron.34(8), 695–697 (2004).
    [CrossRef]
  5. B. A. Cumberland, S. V. Popov, J. R. Taylor, O. I. Medvedkov, S. A. Vasiliev, and E. M. Dianov, “2.1 microm continuous-wave Raman laser in GeO2 fiber,” Opt. Lett.32(13), 1848–1850 (2007).
    [CrossRef] [PubMed]
  6. V. Fortin, M. Bernier, J. Carrier, and R. Vallée, “Fluoride glass Raman fiber laser at 2185 nm,” Opt. Lett.36(21), 4152–4154 (2011).
    [CrossRef] [PubMed]
  7. M. Bernier, D. Faucher, R. Vallée, A. Saliminia, G. Androz, Y. Sheng, and S. L. Chin, “Bragg gratings photoinduced in ZBLAN fibers by femtosecond pulses at 800 nm,” Opt. Lett.32(5), 454–456 (2007).
    [CrossRef] [PubMed]
  8. M. Bernier, R. Vallée, B. Morasse, C. Desrosiers, A. Saliminia, and Y. Sheng, “Ytterbium fiber laser based on first-order fiber Bragg gratings written with 400 nm femtosecond pulses and a phase-mask,” Opt. Express17(21), 18887–18893 (2009).
    [CrossRef] [PubMed]
  9. J.-C. Bouteiller, “Spectral modeling of Raman fiber lasers,” IEEE Photon. Technol. Lett.15(12), 1698–1700 (2003).
    [CrossRef]
  10. P. Suret and S. Randoux, “Influence of spectral broadening on steady characteristics of Raman fiber lasers: from experiments to questions about validity of usual models,” Opt. Commun.237(1-3), 201–212 (2004).
    [CrossRef]
  11. D. V. Churkin, S. V. Smirnov, and E. V. Podivilov, “Statistical properties of partially coherent CW fiber lasers,” Opt. Lett.35(19), 3288–3290 (2010).
    [CrossRef] [PubMed]
  12. J. Xu, M. Prabhu, J. Lu, K.-I. Ueda, and D. Xing, “Efficient double-clad thulium-doped fiber laser with a ring cavity,” Appl. Opt.40(12), 1983–1988 (2001).
    [CrossRef] [PubMed]
  13. S. D. Jackson and T. A. King, “Theoretical modeling of Tm-doped silica fiber lasers,” J. Lightwave Technol.17(5), 948–956 (1999).
    [CrossRef]
  14. S. D. Agger and J. H. Povlsen, “Emission and absorption cross section of thulium doped silica fibers,” Opt. Express14(1), 50–57 (2006).
    [CrossRef] [PubMed]
  15. M. Rini, I. Cristiani, and V. Degiorgio, “Numerical modeling and optimization of cascaded CW Raman fiber lasers,” IEEE J. Quantum Electron.36(10), 1117–1122 (2000).
    [CrossRef]
  16. L. Zhang, F. Gan, and P. Wang, “Evaluation of refractive-index and material dispersion in fluoride glasses,” Appl. Opt.33(1), 50–56 (1994).
    [CrossRef] [PubMed]
  17. P. F. Moulton, G. A. Rines, E. V. Slobodtchikov, K. F. Wall, G. Frith, B. Samson, and A. L. G. Carter, “Tm-doped fiber lasers: fundamentals and power scaling,” IEEE J. Sel. Top. Quantum Electron.15(1), 85–92 (2009).
    [CrossRef]
  18. B. M. Walsh and N. P. Barnes, “Comparison of Tm:ZBLAN and Tm: silica fiber lasers; spectroscopy and tunable pulsed laser operation around 1.9 μm,” Appl. Phys. B78(3-4), 325–333 (2004).
    [CrossRef]
  19. D. Faucher, M. Bernier, G. Androz, N. Caron, and R. Vallée, “20 W passively cooled single-mode all-fiber laser at 2.8 μm,” Opt. Lett.36(7), 1104–1106 (2011).
    [CrossRef] [PubMed]

2011 (2)

2010 (1)

2009 (2)

M. Bernier, R. Vallée, B. Morasse, C. Desrosiers, A. Saliminia, and Y. Sheng, “Ytterbium fiber laser based on first-order fiber Bragg gratings written with 400 nm femtosecond pulses and a phase-mask,” Opt. Express17(21), 18887–18893 (2009).
[CrossRef] [PubMed]

P. F. Moulton, G. A. Rines, E. V. Slobodtchikov, K. F. Wall, G. Frith, B. Samson, and A. L. G. Carter, “Tm-doped fiber lasers: fundamentals and power scaling,” IEEE J. Sel. Top. Quantum Electron.15(1), 85–92 (2009).
[CrossRef]

2007 (2)

2006 (1)

2004 (3)

B. M. Walsh and N. P. Barnes, “Comparison of Tm:ZBLAN and Tm: silica fiber lasers; spectroscopy and tunable pulsed laser operation around 1.9 μm,” Appl. Phys. B78(3-4), 325–333 (2004).
[CrossRef]

P. Suret and S. Randoux, “Influence of spectral broadening on steady characteristics of Raman fiber lasers: from experiments to questions about validity of usual models,” Opt. Commun.237(1-3), 201–212 (2004).
[CrossRef]

E. M. Dianov, I. A. Bufetov, V. M. Mashinsky, V. B. Neustruev, O. I. Medvedkov, A. V. Shubin, M. A. Melkumov, A. N. Gur'yanov, V. F. Khopin, and M. V. Yashkov, “Raman fibre lasers emitting at a wavelength above 2 μm,” Quantum Electron.34(8), 695–697 (2004).
[CrossRef]

2003 (1)

J.-C. Bouteiller, “Spectral modeling of Raman fiber lasers,” IEEE Photon. Technol. Lett.15(12), 1698–1700 (2003).
[CrossRef]

2001 (2)

N. Kurukitkoson, H. Sugahara, S. K. Turitsyn, O. N. Egorova, A. S. Kurkov, V. M. Paramonov, and E. M. Dianov, “Optimisation of two-stage Raman converter based on phosphosilicate core fibre: modelling and experiment,” Electron. Lett.37(21), 1281–1283 (2001).
[CrossRef]

J. Xu, M. Prabhu, J. Lu, K.-I. Ueda, and D. Xing, “Efficient double-clad thulium-doped fiber laser with a ring cavity,” Appl. Opt.40(12), 1983–1988 (2001).
[CrossRef] [PubMed]

2000 (2)

N. S. Kim, M. Prabhu, C. Li, J. Song, and K.-I. Ueda, “1239/1484 nm cascaded phosphosilicate Raman fiber laser with CW output power of 1.36 W at 1484 nm pumped by CW Yb-doped double-clad fiber laser at 1064 nm and spectral continuum generation,” Opt. Commun.176(1-3), 219–222 (2000).
[CrossRef]

M. Rini, I. Cristiani, and V. Degiorgio, “Numerical modeling and optimization of cascaded CW Raman fiber lasers,” IEEE J. Quantum Electron.36(10), 1117–1122 (2000).
[CrossRef]

1999 (1)

1998 (1)

G. Vareille, O. Audouin, and E. Desurvire, “Numerical optimisation of power conversion efficiency in 1480nm multi-Stokes Raman fibre lasers,” Electron. Lett.34(7), 675–676 (1998).
[CrossRef]

1994 (1)

Agger, S. D.

Androz, G.

Audouin, O.

G. Vareille, O. Audouin, and E. Desurvire, “Numerical optimisation of power conversion efficiency in 1480nm multi-Stokes Raman fibre lasers,” Electron. Lett.34(7), 675–676 (1998).
[CrossRef]

Barnes, N. P.

B. M. Walsh and N. P. Barnes, “Comparison of Tm:ZBLAN and Tm: silica fiber lasers; spectroscopy and tunable pulsed laser operation around 1.9 μm,” Appl. Phys. B78(3-4), 325–333 (2004).
[CrossRef]

Bernier, M.

Bouteiller, J.-C.

J.-C. Bouteiller, “Spectral modeling of Raman fiber lasers,” IEEE Photon. Technol. Lett.15(12), 1698–1700 (2003).
[CrossRef]

Bufetov, I. A.

E. M. Dianov, I. A. Bufetov, V. M. Mashinsky, V. B. Neustruev, O. I. Medvedkov, A. V. Shubin, M. A. Melkumov, A. N. Gur'yanov, V. F. Khopin, and M. V. Yashkov, “Raman fibre lasers emitting at a wavelength above 2 μm,” Quantum Electron.34(8), 695–697 (2004).
[CrossRef]

Caron, N.

Carrier, J.

Carter, A. L. G.

P. F. Moulton, G. A. Rines, E. V. Slobodtchikov, K. F. Wall, G. Frith, B. Samson, and A. L. G. Carter, “Tm-doped fiber lasers: fundamentals and power scaling,” IEEE J. Sel. Top. Quantum Electron.15(1), 85–92 (2009).
[CrossRef]

Chin, S. L.

Churkin, D. V.

Cristiani, I.

M. Rini, I. Cristiani, and V. Degiorgio, “Numerical modeling and optimization of cascaded CW Raman fiber lasers,” IEEE J. Quantum Electron.36(10), 1117–1122 (2000).
[CrossRef]

Cumberland, B. A.

Degiorgio, V.

M. Rini, I. Cristiani, and V. Degiorgio, “Numerical modeling and optimization of cascaded CW Raman fiber lasers,” IEEE J. Quantum Electron.36(10), 1117–1122 (2000).
[CrossRef]

Desrosiers, C.

Desurvire, E.

G. Vareille, O. Audouin, and E. Desurvire, “Numerical optimisation of power conversion efficiency in 1480nm multi-Stokes Raman fibre lasers,” Electron. Lett.34(7), 675–676 (1998).
[CrossRef]

Dianov, E. M.

B. A. Cumberland, S. V. Popov, J. R. Taylor, O. I. Medvedkov, S. A. Vasiliev, and E. M. Dianov, “2.1 microm continuous-wave Raman laser in GeO2 fiber,” Opt. Lett.32(13), 1848–1850 (2007).
[CrossRef] [PubMed]

E. M. Dianov, I. A. Bufetov, V. M. Mashinsky, V. B. Neustruev, O. I. Medvedkov, A. V. Shubin, M. A. Melkumov, A. N. Gur'yanov, V. F. Khopin, and M. V. Yashkov, “Raman fibre lasers emitting at a wavelength above 2 μm,” Quantum Electron.34(8), 695–697 (2004).
[CrossRef]

N. Kurukitkoson, H. Sugahara, S. K. Turitsyn, O. N. Egorova, A. S. Kurkov, V. M. Paramonov, and E. M. Dianov, “Optimisation of two-stage Raman converter based on phosphosilicate core fibre: modelling and experiment,” Electron. Lett.37(21), 1281–1283 (2001).
[CrossRef]

Egorova, O. N.

N. Kurukitkoson, H. Sugahara, S. K. Turitsyn, O. N. Egorova, A. S. Kurkov, V. M. Paramonov, and E. M. Dianov, “Optimisation of two-stage Raman converter based on phosphosilicate core fibre: modelling and experiment,” Electron. Lett.37(21), 1281–1283 (2001).
[CrossRef]

Faucher, D.

Fortin, V.

Frith, G.

P. F. Moulton, G. A. Rines, E. V. Slobodtchikov, K. F. Wall, G. Frith, B. Samson, and A. L. G. Carter, “Tm-doped fiber lasers: fundamentals and power scaling,” IEEE J. Sel. Top. Quantum Electron.15(1), 85–92 (2009).
[CrossRef]

Gan, F.

Gur'yanov, A. N.

E. M. Dianov, I. A. Bufetov, V. M. Mashinsky, V. B. Neustruev, O. I. Medvedkov, A. V. Shubin, M. A. Melkumov, A. N. Gur'yanov, V. F. Khopin, and M. V. Yashkov, “Raman fibre lasers emitting at a wavelength above 2 μm,” Quantum Electron.34(8), 695–697 (2004).
[CrossRef]

Jackson, S. D.

Khopin, V. F.

E. M. Dianov, I. A. Bufetov, V. M. Mashinsky, V. B. Neustruev, O. I. Medvedkov, A. V. Shubin, M. A. Melkumov, A. N. Gur'yanov, V. F. Khopin, and M. V. Yashkov, “Raman fibre lasers emitting at a wavelength above 2 μm,” Quantum Electron.34(8), 695–697 (2004).
[CrossRef]

Kim, N. S.

N. S. Kim, M. Prabhu, C. Li, J. Song, and K.-I. Ueda, “1239/1484 nm cascaded phosphosilicate Raman fiber laser with CW output power of 1.36 W at 1484 nm pumped by CW Yb-doped double-clad fiber laser at 1064 nm and spectral continuum generation,” Opt. Commun.176(1-3), 219–222 (2000).
[CrossRef]

King, T. A.

Kurkov, A. S.

N. Kurukitkoson, H. Sugahara, S. K. Turitsyn, O. N. Egorova, A. S. Kurkov, V. M. Paramonov, and E. M. Dianov, “Optimisation of two-stage Raman converter based on phosphosilicate core fibre: modelling and experiment,” Electron. Lett.37(21), 1281–1283 (2001).
[CrossRef]

Kurukitkoson, N.

N. Kurukitkoson, H. Sugahara, S. K. Turitsyn, O. N. Egorova, A. S. Kurkov, V. M. Paramonov, and E. M. Dianov, “Optimisation of two-stage Raman converter based on phosphosilicate core fibre: modelling and experiment,” Electron. Lett.37(21), 1281–1283 (2001).
[CrossRef]

Li, C.

N. S. Kim, M. Prabhu, C. Li, J. Song, and K.-I. Ueda, “1239/1484 nm cascaded phosphosilicate Raman fiber laser with CW output power of 1.36 W at 1484 nm pumped by CW Yb-doped double-clad fiber laser at 1064 nm and spectral continuum generation,” Opt. Commun.176(1-3), 219–222 (2000).
[CrossRef]

Lu, J.

Mashinsky, V. M.

E. M. Dianov, I. A. Bufetov, V. M. Mashinsky, V. B. Neustruev, O. I. Medvedkov, A. V. Shubin, M. A. Melkumov, A. N. Gur'yanov, V. F. Khopin, and M. V. Yashkov, “Raman fibre lasers emitting at a wavelength above 2 μm,” Quantum Electron.34(8), 695–697 (2004).
[CrossRef]

Medvedkov, O. I.

B. A. Cumberland, S. V. Popov, J. R. Taylor, O. I. Medvedkov, S. A. Vasiliev, and E. M. Dianov, “2.1 microm continuous-wave Raman laser in GeO2 fiber,” Opt. Lett.32(13), 1848–1850 (2007).
[CrossRef] [PubMed]

E. M. Dianov, I. A. Bufetov, V. M. Mashinsky, V. B. Neustruev, O. I. Medvedkov, A. V. Shubin, M. A. Melkumov, A. N. Gur'yanov, V. F. Khopin, and M. V. Yashkov, “Raman fibre lasers emitting at a wavelength above 2 μm,” Quantum Electron.34(8), 695–697 (2004).
[CrossRef]

Melkumov, M. A.

E. M. Dianov, I. A. Bufetov, V. M. Mashinsky, V. B. Neustruev, O. I. Medvedkov, A. V. Shubin, M. A. Melkumov, A. N. Gur'yanov, V. F. Khopin, and M. V. Yashkov, “Raman fibre lasers emitting at a wavelength above 2 μm,” Quantum Electron.34(8), 695–697 (2004).
[CrossRef]

Morasse, B.

Moulton, P. F.

P. F. Moulton, G. A. Rines, E. V. Slobodtchikov, K. F. Wall, G. Frith, B. Samson, and A. L. G. Carter, “Tm-doped fiber lasers: fundamentals and power scaling,” IEEE J. Sel. Top. Quantum Electron.15(1), 85–92 (2009).
[CrossRef]

Neustruev, V. B.

E. M. Dianov, I. A. Bufetov, V. M. Mashinsky, V. B. Neustruev, O. I. Medvedkov, A. V. Shubin, M. A. Melkumov, A. N. Gur'yanov, V. F. Khopin, and M. V. Yashkov, “Raman fibre lasers emitting at a wavelength above 2 μm,” Quantum Electron.34(8), 695–697 (2004).
[CrossRef]

Paramonov, V. M.

N. Kurukitkoson, H. Sugahara, S. K. Turitsyn, O. N. Egorova, A. S. Kurkov, V. M. Paramonov, and E. M. Dianov, “Optimisation of two-stage Raman converter based on phosphosilicate core fibre: modelling and experiment,” Electron. Lett.37(21), 1281–1283 (2001).
[CrossRef]

Podivilov, E. V.

Popov, S. V.

Povlsen, J. H.

Prabhu, M.

J. Xu, M. Prabhu, J. Lu, K.-I. Ueda, and D. Xing, “Efficient double-clad thulium-doped fiber laser with a ring cavity,” Appl. Opt.40(12), 1983–1988 (2001).
[CrossRef] [PubMed]

N. S. Kim, M. Prabhu, C. Li, J. Song, and K.-I. Ueda, “1239/1484 nm cascaded phosphosilicate Raman fiber laser with CW output power of 1.36 W at 1484 nm pumped by CW Yb-doped double-clad fiber laser at 1064 nm and spectral continuum generation,” Opt. Commun.176(1-3), 219–222 (2000).
[CrossRef]

Randoux, S.

P. Suret and S. Randoux, “Influence of spectral broadening on steady characteristics of Raman fiber lasers: from experiments to questions about validity of usual models,” Opt. Commun.237(1-3), 201–212 (2004).
[CrossRef]

Rines, G. A.

P. F. Moulton, G. A. Rines, E. V. Slobodtchikov, K. F. Wall, G. Frith, B. Samson, and A. L. G. Carter, “Tm-doped fiber lasers: fundamentals and power scaling,” IEEE J. Sel. Top. Quantum Electron.15(1), 85–92 (2009).
[CrossRef]

Rini, M.

M. Rini, I. Cristiani, and V. Degiorgio, “Numerical modeling and optimization of cascaded CW Raman fiber lasers,” IEEE J. Quantum Electron.36(10), 1117–1122 (2000).
[CrossRef]

Saliminia, A.

Samson, B.

P. F. Moulton, G. A. Rines, E. V. Slobodtchikov, K. F. Wall, G. Frith, B. Samson, and A. L. G. Carter, “Tm-doped fiber lasers: fundamentals and power scaling,” IEEE J. Sel. Top. Quantum Electron.15(1), 85–92 (2009).
[CrossRef]

Sheng, Y.

Shubin, A. V.

E. M. Dianov, I. A. Bufetov, V. M. Mashinsky, V. B. Neustruev, O. I. Medvedkov, A. V. Shubin, M. A. Melkumov, A. N. Gur'yanov, V. F. Khopin, and M. V. Yashkov, “Raman fibre lasers emitting at a wavelength above 2 μm,” Quantum Electron.34(8), 695–697 (2004).
[CrossRef]

Slobodtchikov, E. V.

P. F. Moulton, G. A. Rines, E. V. Slobodtchikov, K. F. Wall, G. Frith, B. Samson, and A. L. G. Carter, “Tm-doped fiber lasers: fundamentals and power scaling,” IEEE J. Sel. Top. Quantum Electron.15(1), 85–92 (2009).
[CrossRef]

Smirnov, S. V.

Song, J.

N. S. Kim, M. Prabhu, C. Li, J. Song, and K.-I. Ueda, “1239/1484 nm cascaded phosphosilicate Raman fiber laser with CW output power of 1.36 W at 1484 nm pumped by CW Yb-doped double-clad fiber laser at 1064 nm and spectral continuum generation,” Opt. Commun.176(1-3), 219–222 (2000).
[CrossRef]

Sugahara, H.

N. Kurukitkoson, H. Sugahara, S. K. Turitsyn, O. N. Egorova, A. S. Kurkov, V. M. Paramonov, and E. M. Dianov, “Optimisation of two-stage Raman converter based on phosphosilicate core fibre: modelling and experiment,” Electron. Lett.37(21), 1281–1283 (2001).
[CrossRef]

Suret, P.

P. Suret and S. Randoux, “Influence of spectral broadening on steady characteristics of Raman fiber lasers: from experiments to questions about validity of usual models,” Opt. Commun.237(1-3), 201–212 (2004).
[CrossRef]

Taylor, J. R.

Turitsyn, S. K.

N. Kurukitkoson, H. Sugahara, S. K. Turitsyn, O. N. Egorova, A. S. Kurkov, V. M. Paramonov, and E. M. Dianov, “Optimisation of two-stage Raman converter based on phosphosilicate core fibre: modelling and experiment,” Electron. Lett.37(21), 1281–1283 (2001).
[CrossRef]

Ueda, K.-I.

J. Xu, M. Prabhu, J. Lu, K.-I. Ueda, and D. Xing, “Efficient double-clad thulium-doped fiber laser with a ring cavity,” Appl. Opt.40(12), 1983–1988 (2001).
[CrossRef] [PubMed]

N. S. Kim, M. Prabhu, C. Li, J. Song, and K.-I. Ueda, “1239/1484 nm cascaded phosphosilicate Raman fiber laser with CW output power of 1.36 W at 1484 nm pumped by CW Yb-doped double-clad fiber laser at 1064 nm and spectral continuum generation,” Opt. Commun.176(1-3), 219–222 (2000).
[CrossRef]

Vallée, R.

Vareille, G.

G. Vareille, O. Audouin, and E. Desurvire, “Numerical optimisation of power conversion efficiency in 1480nm multi-Stokes Raman fibre lasers,” Electron. Lett.34(7), 675–676 (1998).
[CrossRef]

Vasiliev, S. A.

Wall, K. F.

P. F. Moulton, G. A. Rines, E. V. Slobodtchikov, K. F. Wall, G. Frith, B. Samson, and A. L. G. Carter, “Tm-doped fiber lasers: fundamentals and power scaling,” IEEE J. Sel. Top. Quantum Electron.15(1), 85–92 (2009).
[CrossRef]

Walsh, B. M.

B. M. Walsh and N. P. Barnes, “Comparison of Tm:ZBLAN and Tm: silica fiber lasers; spectroscopy and tunable pulsed laser operation around 1.9 μm,” Appl. Phys. B78(3-4), 325–333 (2004).
[CrossRef]

Wang, P.

Xing, D.

Xu, J.

Yashkov, M. V.

E. M. Dianov, I. A. Bufetov, V. M. Mashinsky, V. B. Neustruev, O. I. Medvedkov, A. V. Shubin, M. A. Melkumov, A. N. Gur'yanov, V. F. Khopin, and M. V. Yashkov, “Raman fibre lasers emitting at a wavelength above 2 μm,” Quantum Electron.34(8), 695–697 (2004).
[CrossRef]

Zhang, L.

Appl. Opt. (2)

Appl. Phys. B (1)

B. M. Walsh and N. P. Barnes, “Comparison of Tm:ZBLAN and Tm: silica fiber lasers; spectroscopy and tunable pulsed laser operation around 1.9 μm,” Appl. Phys. B78(3-4), 325–333 (2004).
[CrossRef]

Electron. Lett. (2)

G. Vareille, O. Audouin, and E. Desurvire, “Numerical optimisation of power conversion efficiency in 1480nm multi-Stokes Raman fibre lasers,” Electron. Lett.34(7), 675–676 (1998).
[CrossRef]

N. Kurukitkoson, H. Sugahara, S. K. Turitsyn, O. N. Egorova, A. S. Kurkov, V. M. Paramonov, and E. M. Dianov, “Optimisation of two-stage Raman converter based on phosphosilicate core fibre: modelling and experiment,” Electron. Lett.37(21), 1281–1283 (2001).
[CrossRef]

IEEE J. Quantum Electron. (1)

M. Rini, I. Cristiani, and V. Degiorgio, “Numerical modeling and optimization of cascaded CW Raman fiber lasers,” IEEE J. Quantum Electron.36(10), 1117–1122 (2000).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

P. F. Moulton, G. A. Rines, E. V. Slobodtchikov, K. F. Wall, G. Frith, B. Samson, and A. L. G. Carter, “Tm-doped fiber lasers: fundamentals and power scaling,” IEEE J. Sel. Top. Quantum Electron.15(1), 85–92 (2009).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

J.-C. Bouteiller, “Spectral modeling of Raman fiber lasers,” IEEE Photon. Technol. Lett.15(12), 1698–1700 (2003).
[CrossRef]

J. Lightwave Technol. (1)

Opt. Commun. (2)

P. Suret and S. Randoux, “Influence of spectral broadening on steady characteristics of Raman fiber lasers: from experiments to questions about validity of usual models,” Opt. Commun.237(1-3), 201–212 (2004).
[CrossRef]

N. S. Kim, M. Prabhu, C. Li, J. Song, and K.-I. Ueda, “1239/1484 nm cascaded phosphosilicate Raman fiber laser with CW output power of 1.36 W at 1484 nm pumped by CW Yb-doped double-clad fiber laser at 1064 nm and spectral continuum generation,” Opt. Commun.176(1-3), 219–222 (2000).
[CrossRef]

Opt. Express (2)

Opt. Lett. (5)

Quantum Electron. (1)

E. M. Dianov, I. A. Bufetov, V. M. Mashinsky, V. B. Neustruev, O. I. Medvedkov, A. V. Shubin, M. A. Melkumov, A. N. Gur'yanov, V. F. Khopin, and M. V. Yashkov, “Raman fibre lasers emitting at a wavelength above 2 μm,” Quantum Electron.34(8), 695–697 (2004).
[CrossRef]

Supplementary Material (6)

» Media 1: JPG (425 KB)     
» Media 2: JPG (445 KB)     
» Media 3: JPG (423 KB)     
» Media 4: JPG (656 KB)     
» Media 5: JPG (488 KB)     
» Media 6: JPG (901 KB)     

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Figures (6)

Fig. 1
Fig. 1

Experimental setup of the nested cavity Raman fiber laser.

Fig. 2
Fig. 2

Transmission spectra of the pair of FBGs forming the Stokes cavity.

Fig. 3
Fig. 3

Output power as a function of the launched pump power.

Fig. 4
Fig. 4

Output spectra for the pump (a) and Stokes (b) wavelengths.

Fig. 5
Fig. 5

Power stability for different Stokes output powers from 0.17 W to 3.3 W.

Fig. 6
Fig. 6

(a) Calculated Stokes output power as a function of the launched pump power for different butt-coupled losses. (b) Calculated Stokes output power as a function of the S2 FBG reflectivity (black curves). The red and blue curves show the output power for a cavity with and without spectral broadening respectively.

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

d N 4 dt = σ a_Q N 1 ( Q(z) h ν Q A eff_Q ) N 4 τ 4 k 4212 N 4 N 1 + k 2124 N 2 2 ,
d N 2 dt =2 k 4212 N 4 N 1 2 k 2124 N 2 2 N 2 τ 2 + β 42 N 4 τ 4 ( P(z) h ν P A eff_P )( σ e_P N 2 σ a_P N 1 ),
N 1 =N N 2 N 4 ,
dQ(z) dz = η P σ a_Q N 1 Q(z),
dP (z) ± dz =±( σ e_P N 2 σ a_P N 1 α P )P (z) ± ,
dP (z) ± dz = α P P (z) ± ν P ν S g R A eff ( S (z) + +S (z) )P (z) ±
dS (z) ± dz = α S S (z) ± ± g R A eff ( P (z) + +P (z) )S (z) ± ,
P (0) + = P intracavity P (L) = R P2 P (L) + S (0) + = R S1 S (0) S (L) = R S2 S (L) + ,

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