Design of a High Performance Mid-IR Fiber Laser Based on Pr3+-Doped Fluoroindate Glass

In this work, a novel continuous wave fiber laser, pumped at <inline-formula><tex-math notation="LaTeX">${\lambda }_p = 1550\ \text{nm}$</tex-math></inline-formula> and emitting at <inline-formula><tex-math notation="LaTeX">${\lambda }_s = 4\ \mu \text{m}$</tex-math></inline-formula>, has been designed and optimized. It is based on a step-index, double-cladding, praseodymium-doped fluoroindate glass fiber, available on market, having dopant concentration <inline-formula><tex-math notation="LaTeX">${N}_{Pr} = 8000\ \text{ppm}$</tex-math></inline-formula>. For a realistic design, measured spectroscopical parameters have been taken into account, writing a five-level rate equation model. The design is carried out by employing a homemade code solver. The best predicted slope efficiency of about <inline-formula><tex-math notation="LaTeX">$\eta = 33{{ \% }}$</tex-math></inline-formula> and pump power threshold <inline-formula><tex-math notation="LaTeX">${P}_{th} = 0.007\ \text{W}$</tex-math></inline-formula> have been obtained for a fiber length <inline-formula><tex-math notation="LaTeX">${L}_{fiber} = 0.4\ \text{m}$</tex-math></inline-formula> and output mirror reflectivity <inline-formula><tex-math notation="LaTeX">${R}_{out} = 30\% $</tex-math></inline-formula>. These values are very interesting with reference to the state of the art and promise the fabrication of high beam quality optical sources in the middle infrared range, by employing conventional erbium-doped fiber pumping lasers, with a potentially easy all-in-fiber integration.

In this work, for the first time to the best of our knowledge, a continuous wave laser based on a Pr 3+ -doped fluoroindate fiber, emitting at λ s = 4 μm when pumped at λ p = 1550 nm, has been designed and optimized, starting from experimental spectroscopical parameters taken from literature [21], [22], [23], [24], [25], [26] and employing a home-made computer code solver [14].

II. RECALL OF THEORY
The praseodymium-doped glass emitting at λ s = 4 μm when pumped at λ p = 1550 nm is modeled with a 5-level scheme, as reported in Fig. 1.It takes into account pumping (bold black arrow), stimulated emission (bold red arrow), radiative and nonradiative emissions, Excited State Absorption (ESA), and crossrelaxation (CR) phenomena.
By considering a rate equation approach, the following nonlinear system (1a)-(1e) can be written to evaluate the ion populations N 1 , . . ., N 5 .
where A i,j = β i,j τ i are the radiative decays, β i,j are the branching ratios, τ i are the i-th level lifetimes, W CR is the cross relaxation rate, and W NR,ij are the non-radiative decay rates.The ion population condition N P r = N 1 + N 2 + N 3 + N 4 + N 5 is considered.The coefficients W ij are the transition rates for i → j transition defined as where σ i,j (λ p/s ) is the emission/absorption cross section at the wavelength λ p/s for the i → j transition, h is the Planck constant, ν p/s is the pump/signal frequency, P p is the pump power, P s is the forward signal power, Γ p and Γ s are the overlap coefficients of pump and signal beams with the doped area A d , respectively.
The power propagation along the fiber, for the pump P p and for the signal P s , is modeled by considering the following equations: where α is the glass attenuation, and g p and g s are the pump and signal gains, respectively, defined as: The following boundaries conditions are imposed: where z = 0 and z = L represent the ends of the laser cavity, P p is the input pump power, R in and R out are the input and output mirror reflectivity, respectively.Initial conditions for level populations are also imposed as follows:

III. LASER DESIGN
The laser has been designed considering a step-index doublecladding fluoroindate fiber doped with praseodymium concentration N P r = 1.6 × 10 26 ions/m 3 = 8000 ppm, by Le Verre Fluoré [17].Fig. 2 I.The fiber has been investigated via a

TABLE I MODELING PARAMETERS
Finite Element Method (FEM) software, in order to calculate the pump and the signal overlap coefficients Γ p = 0.899 and Γ s = 0.312, respectively.The fiber is monomodal at signal wavelength.Table II reports the experimental spectroscopical parameters employed in the design, taken from literature.
The design is carried out via a home-made solver code, the structure of which is based on the rate-equations approach, well validated in a number of cases [9], [11], [14].In the design, several simulations have been carried out to investigate the behavior of the laser output power P s as a function of the input pump power, for different values of: (i) the fiber length L f iber , and (ii) the output mirror reflectivity R out .Moreover, also the behavior of the laser output power P s as a function of (iii) the fiber length L f iber , and (iv) the output mirror reflectivity R out , for different values of the input pump power has been investigated.The input mirror reflectivity is kept fixed to R in = 95%, as a cautionary value to simulate a Fiber Bragg Grating (FBG) in an all-in-fiber set-up.
Fig. 3(a) shows the laser output power P s as a function of the input pump power, for different values of the fiber length L f iber , i.e., laser cavity.The slope efficiency tends to slightly reduce for longer fibers, whereas the saturation pump power P sat increases.Fig. 3(b) shows an enlarged view to better observe the threshold P th .The pump power threshold P th slightly increases as the fiber length L f iber increases.The best value is obtained for L f iber = 0.4 m, P th = 0.003 W, while the saturation pump power is P sat = 1.3 W, corresponding to the laser output power P s = 0.34 W. The slope efficiency is η = 28%.
Fig. 4(a) shows the laser output power P s as a function of the input pump power, for different values of the output mirror reflectivity R out .As the reflectivity decreases, the slope efficiency asymptotically increases reaching η = 32.5%, for R out = 30%, while the saturation pump power P sat remains almost the same in all cases.The maximum laser output power is P s = 0.42 W. The pump power threshold P th slightly increases as the output mirror reflectivity decreases, but it is always below P th < 10 mW, as better illustrated in Fig. 4(b).
Fig. 5 shows the laser output power P s as a function of the output mirror reflectivity R out , for different values of the input Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.Fig. 3. Laser output power P s as a function of the input pump power P p , for different values of the fiber length L fiber , input reflectivity R in 95%, output mirror reflectivity R out = 80%; (b) enlarged view of the pump power threshold.
pump power P p .The laser output power P s slowly increases for lower values of the output mirror reflectivity R out , as also shown in Fig. 4. As the input pump power P p increases, the variation of the output power becomes more evident.
Fig. 6 shows the laser output power P s as a function of the fiber length L f iber , for different values of the input pump power P p .For each value of the input pump power, a saturation of the laser output power can be observed.
Table III reports a comparison among the Pr 3+ -doped fluoroindate fiber laser proposed in this work and other fluoroindate fiber lasers emitting in Mid-IR [8], [9], [10], [12], [13], [14], [18].In particular, the comparison with literature is performed in terms of doping ion, emission wavelength λ s , pump wavelength λ p , pump power threshold P th , and slope efficiency η.All the considered lasers emit between λ s = 3.4 μm and λ s = 4.4 μm, and are pumped in the visible or near-infrared (NIR) range.It is worth noting that in [8], [12], [13], dual-wavelength pumping schemes are proposed to increase the slope efficiency and to reduce the pump power threshold.The proposed laser exhibits the highest slope efficiency and the lowest pump power threshold, with one of the longest emitting wavelengths.Moreover, it can be pumped by employing a commercial erbium-doped fiber laser, to be spliced with the praseodymium-doped fiber, available on the market, thus obtaining an all-in-fiber device, with FGBs employed as cavity mirrors [4].The possibility to employ a single pumping wavelength simplifies the construction scheme of the laser system.Fluoride erbium-doped fiber lasers could be taken into account [7], [8], [9] with a proper design to emit at 1.5 μm.

IV. CONCLUSION
For the first time to the best of our knowledge, a fiber laser based on a praseodymium-doped fluoroindate glass, at λ s = 4 μm, when pumped at λ p = 1550 nm, has been designed and optimized, by considering spectroscopical parameters taken from literature.The predicted slope efficiency η = 33% is promising, along with the low input pump threshold.NIR pumping could be implemented by employing an erbium-doped fiber laser, spliced with the praseodymium fluoroindate fiber cavity.Future developments may consider co-doping with ytterbium, to obtain multi-wavelength emission at both λ s = 3.6 μm and λ s = 4 μm.
shows the fiber cross-section geometry and the HE 11 mode at the signal wavelength.It has core diameter d co = 7.5 μm, inner cladding of diameter d cl1 = 125 μm shaped with a 2-D cut at distance d = 115 μm, and second cladding diameter d cl2 = 180 μm.The parameters employed for modeling are reported in Table

Fig. 2 .
Fig. 2. Fiber cross-section geometry and E-field modulus of the fundamental mode HE 11 at signal wavelength λ s .

Fig. 4 .
Fig. 4. Laser output power P s as a function of the input pump power P p , for different values of the output mirror reflectivity R out , input mirror reflectivity R in = 95%, fiber length L fiber = 0.4 m; (b) enlarged view of the pump power threshold.

Fig. 5 .
Fig. 5. Laser output power P s as a function of the output mirror reflectivity R out , for different values of the input pump power P p , input mirror reflectivity R in = 95%, fiber length L fiber = 0.4 m.

Fig. 6 .
Fig.6.Laser output P s as function of the fiber length L fiber , for different values of the input pump power P p , input mirror reflectivity R in = 95%, output mirror reflectivity R out = 30%.

TABLE II SPECTROSCOPIC
PARAMETERS OF PR 3+ -DOPED FLUOROINDATE GLASS FIBER

TABLE III COMPARISON
OF LASER PERFORMANCE WITH OTHER MID-IR LASERS BASED ON FLUOROINDATE FIBERS