Design of a Mid-IR Laser Based on a Ho:Nd-codoped Fluoroindate Fiber

In this work, a novel mid-infrared continuous wave laser, based on a fluoroindate fiber co-doped with holmium and neodymium, is designed to emit at <inline-formula><tex-math notation="LaTeX">${\lambda }_s = \ 3.92\ \mu {\rm m}$</tex-math></inline-formula>, when pumped at <inline-formula><tex-math notation="LaTeX">${\lambda }_p = \ 808\ {\rm nm}$</tex-math></inline-formula>. The laser is modeled considering a nine-level system, by taking into account experimental spectroscopical parameters. Since the energy transfer coefficients are unknown, they have been evaluated starting from the measured emission spectra of the bulk glass, reported in literature, and comparing their ratio with respect to the ratio between the simulated signal gain coefficients. The designed laser promises higher slope efficiency and power threshold lower than those obtainable with a holmium-heavily-doped fiber, having same fiber section geometry, same refractive indices and pumped at <inline-formula><tex-math notation="LaTeX">${\lambda }_p = \ 888\ {\rm nm}$</tex-math></inline-formula>. Slope efficiency <inline-formula><tex-math notation="LaTeX">$\eta \ = \ 16.67{\boldsymbol{\% }}$</tex-math></inline-formula> and input power threshold <inline-formula><tex-math notation="LaTeX">${P}_{th} = \ 0.2\ W$</tex-math></inline-formula> are obtained for the fiber length <inline-formula><tex-math notation="LaTeX">${L}_{fiber} = \ 0.4\ {\rm m}$</tex-math></inline-formula>, dopants concentrations <inline-formula><tex-math notation="LaTeX">${N}_{Ho} = \ 8 \times {10}^{26}\ {\rm ions}/{\rm m}^3$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">${N}_{Nd} = \ 1 \times {10}^{26}\ {\rm ions}/{\rm m}^3$</tex-math></inline-formula>, and output mirror reflectivity <inline-formula><tex-math notation="LaTeX">${R}_{\rm out} = \ 60{\boldsymbol{\% }}$</tex-math></inline-formula>. This result encourages the fabrication of a continuous wave laser based on a Ho:Nd-codoped fluoroindate fiber.

In this work, for the first time to the best of our knowledge, the design of a novel continuous wave (CW) laser, pumped at λ p = 808 nm and emitting at λ s = 3.92 μm, based on a fluoroindate fiber co-doped with holmium and neodymium, is proposed. A model based on a nine-level system and taking into account the experimental spectroscopic parameters of the rare earth doped bulk glass, is developed [29], [33], [34], [35], [36], [37], [38]. The unknown energy transfer coefficients allowing to match the model with measured emission spectra are identified. The laser behavior is optimized by using a homemade numerical solver. The designed laser is very interesting, allowing simulated slope efficiency and input power threshold improved with respect to those obtained with a holmium-heavily-doped fiber, having the same fiber section geometry, same refractive indices, with a cavity length optimized at the pump wavelength λ p = 888 nm.

II. RATE-EQUATION MODEL
The Ho:Nd-codoped fluoroindate fiber stimulated emission at λ s = 3.92 μm can be modeled by considering a nine-levels system, pumped at λ p = 808 nm. including all the significant ion interactions, is reported in Fig. 1. These are the pump absorption, the stimulated emission, the radiative and nonradiative decays, and the energy transfers (ET) between Ho 3+ and Nd 3+ ions [29].
By following the rate equations approach [12], [14], [24], the energy level populations N 1 , . . . , N 5 of neodymium can be written by the nonlinear system (1a)-(1j) below: whereas the energy level populations N 6 , . . . , N 9 of holmium can be written as τ Ri are the radiative decay rates; β i,j are the branching ratios; τ Ri are the i-th level lifetimes; K ET 1 and K ET 2 are the ET coefficients; W NR are the non-radiative decay rates. The emission/absorption transition rate W i,j for the i → j transition is 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 (1-5, 5-1 and 8-9, 9-8) transitions; λ p/s is the pump/signal wavelength; h is the Plank constant; c 0 is the light speed in vacuum; P ± p is the forward/backward pump power; P ± s is the forward/backward signal power; Γ p /Γ s are the overlap coefficients between the pump/signal beam and the doped area A d . The conditions The power propagation for pump and signal beams is governed by the following partial differential equations are the gain coefficients for the pump and the signal, respectively, and α is the glass optical background loss. To solve (3), the following boundary conditions are imposed.
where z = 0 and z = L represent the ends of the laser cavity, P in 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. ENERGY TRANSFER COEFFICIENTS RECOVERING
The considered fiber is a step-index double-cladding fluoroindate (InF 3 ) glass fiber, doped with Ho 3+ and Nd 3+ ions. Its transverse section is shown in Fig. 2. The core diameter is d co = 16 μm. The cladding is 2-D shaped, obtained with circular diameter d cl1 = 100 μm truncated by two parallel planes at a distance d = 90 μm, to enhance cladding pump absorption. The second cladding, made of low index resin, has diameter d cl2 = 155 μm. The inner and outer numerical apertures are N A 1 = 0.2 and N A 2 = 0.5, respectively. The optical losses are conservatively considered α = 0.2 dB/m for both pump and signal wavelengths, according to the measurement reported in [21]. This kind of double cladding fiber doped with holmium is produced by Le Verre Fluoré [21], [22]. In the following, the co-doping with Ho 3+ and Nd 3+ ions is supposed. The pump and signal wavelengths are λ p = 808 nm and λ s = 3920 nm, respectively. The pump wavelength is feasible, since obtainable with commercial pigtailed semiconductor lasers.
The electromagnetic investigation, performed with a commercial Finite Element Method (FEM) solver, has shown that the fiber is slightly multimode at the normalized frequency number V = 2.56 of the signal wavelength. However, the second order mode can be neglected in the laser operation since its overlapping coefficient Γ s is less than a half of the one of the fundamental mode; its contribution in the laser operation is not considered without significant error, as confirmed by preliminary simulation performed without any approximation [24]. The spectroscopic parameters used in the simulations are listed in Table I. They are taken from [29], [33], [34], [35], [36], [37], [38]. The ion rate equations and the power propagation equations are implemented in a home-made computer code to simulate the optical gain and the laser behavior. Since ET coefficients K ET 1 and K ET 2 are not available from literature, they have been evaluated starting from measured emission spectra from [29] of the bulk glass and comparing their ratio with respect to the gain coefficient ratio simulated for the fiber laser, as reported below. This approach was proposed in a previous work [39]. The aforesaid comparison is feasible in the linear region of the laser characteristic. Fig. 3 shows the measured emission spectra intensities s sn (λ) for different Ho 3+ concentrations [29], normalized with respect to s s1 (λ) for a better reading; the Nd 3+ concentration is set to N Nd = 2 × 10 26 ions/m 3 (1 mol.%). In particular, s s1 (λ) is the normalized emission spectrum intensity for N Ho = 2 × 10 26 ions/m 3 (1 mol.%) (blue curve), s s2 (λ) for N Ho = 1 × 10 26 ions/m 3 (0.5 mol.%) (red curve), and s s3 (λ) for N Ho = 4 × 10 25 ions/m 3 (0.2 mol.%) (yellow curve). The ratios R n between measured emission spectra at λ s = 3920 nm are defined as in Table II   The ratios RG n are defined as follows R G 1 = g s1 (λ s )/g s3 (λ s ); R G 2 = g s1 (λ s )/g s2 (λ s ); R G 3 = g s2 (λ s )/g s3 (λ s ). Fig. 4(a)-(c) show the colormaps of the percentage difference between the ratios (R n − RG n )/R n as a function of the trial energy transfer coefficients K ET 1 and K ET 2 . These three percentage differences must be contemporarily minimized. This condition is obtained for K ET 1 = 4 × 10 −22 m 3 ions −1 s −1 and K ET 2 = 6 × 10 −21 m 3 ions −1 s −1 , for which (R 1 − RG 1 )/R 1 = 8%, (R 2 − RG 2 )/R 2 = 1%, and (R 3 − RG 3 )/R 3 = 0.7%. Table III reports  To validate these results, the global Ho : I 5 level lifetime τ R9 and the global Ho : I 6 level lifetime τ R8 have been simulated and compared with the experimental ones taken from literature [29], for different holmium concentrations N Ho . The lifetimes have been simulated solving the rate equations (1a)-(1j) as a function of time, pumping the system until the ion populations steady-state condition. Then, the pump power is turned off, setting P p = 0 W , and the simulated population exponential decays are observed. The level lifetimes are calculated as the time constants of the obtained exponential curves. Fig. 5 shows the Ho : I 5 level lifetime τ R9 and the Ho : I 6 level lifetime τ R8 simulated (blue) and measured (red) [29] as a function of the holmium concentration N Ho . The good accordance confirms that the recovered energy transfer coefficients K ET 1 and K ET 2 are correct.

IV. LASER DESIGN
In the design, a deep investigation of the laser output signal power P s versus: i) the laser fiber length L f iber ; ii) the dopants concentration N Nd and N Ho ; iii) the output mirror reflectivity R 2 is carried out, in order to identify the configuration allowing the maximum slope efficiency η and the minimum threshold power P th . Fig. 6 shows the output power P s as a function of the input pump power P p , for different values of the fiber length L f iber . The characteristics show a discontinuity with a sawtooth shape in all cases. The reason behind this behavior will be deeply investigated in Section V. The discontinuity shifts towards higher pump power as the fiber length L f iber increases. Moreover, the slope efficiency η slightly decreases, while the input pump threshold is close to P th = 0.5 W in all cases. It is worthy to observe that experiments in literature suggest avoiding input power larger than P p = 6 W in typical fluoroindate fibers [22]. In a lightly doped fiber higher pumping levels could be potentially employed. Therefore, a different cavity optimization could be required. A good trade-off length is L f iber = 0.4 m, for which the discontinuity occurs beyond the realistic range of power, i.e., for P p = 8 W , and for which the efficiency η = 8.47% is obtained. The slope efficiency is calculated after the threshold, between P p = 1 W and P p = 1.5 W . Fig. 7 shows the output power P s as a function of the input pump power P p , for different values of the holmium concentration N Ho . As the concentration increases, the slope efficiency η also increases and the discontinuity shifts to higher input pump powers. The input pump threshold decreases to P th = 0.1 W . The optimal holmium concentration is the maximum considered in the simulations N Ho = 8 × 10 26 ions/m 3 (4 mol.%) (purple curve). Generally, higher holmium concentrations are not used in practice to avoid second order phenomena, such as cross-relaxation or up-conversion.        9 shows the output power P s as a function of the input pump power P p , for different values of output mirror reflectivity R out . The slope efficiency η increases as the output mirror reflectivity R out decreases until R out = 60%, while the input pump threshold P th decreases from P th = 0.3 W to P th = 0.05 W .
The optimal laser configuration is obtained for fiber length L f iber = 0.4 m, holmium concentration N Ho = 8 × 10 26 ions/m 3 , neodymium concentration N Nd = 1 × 10 26 ions/m 3 , and output mirror reflectivity R out = 60%,  allowing the slope efficiency η = 16.67% and the input pump threshold P th = 0.2 W . These simulated performances are better than those typical of CW lasers obtained with heavilyholmium-doped fiber with the same geometry, pumped at λ p = 888 nm, showing η = 10.2% and input pump threshold P th = 4.3 W [22].

V. RESULTS DISCUSSION
By the inspection of the energy level diagram of Fig. 1, for each ion couple involved in the ET2 transition an energy loss equal to the energy difference ΔE = 1.7 × 10 3 cm −1 [29] between level 8 and level 3 occurs, due to the 1-3 and 8-6 transitions. This energy leakage could be the cause of the sawtooth. Indeed, simulating the system without ET2 effect, by putting null K ET 2 , the laser shows the typical characteristic with a pump threshold P th = 0.6 W , a slope efficiency η = 0.2%, and a saturation power P ss = 4.3 W , without any sawtooth. Fig. 10 shows the output power P s (blue curve) and the output residual pump power P res at the end of the fiber (red curve) as a function of the input pump power P p . The output residual pump power P res steeply increases when the output signal P s shows the discontinuity, close to P p = 7.5 W . For larger values of the input pump power P p , a large amount is not absorbed, reaching about the 50% for P p = 12 W . This is plausibly caused by a too large depopulation of level 1 (Nd : I 9/2 ), reducing ET2 effect. By simulation, the N 1 ion population at the end of the fiber steeply decreases for pump power higher than P p = 7.5 W . Therefore, few ions can be promoted from level 1 to level 3 (Nd : I 15/2 ) and the related transition 8-6 does not occur efficiently. Accordingly with this phenomenon, for pump power larger than P p = 7.5 W , the N 8 ion population steeply increases affecting the laser population inversion. This could be a further cause of the sawtooth.

VI. CONCLUSION
For the first time, a CW laser emitting at λ s = 3.92 μm based on a holmium and neodymium co-doped fluoroindate glass fiber is accurately designed, by using measured and recovered spectroscopic parameters. By employing an input pump power at the wavelength λ p = 808 nm, a fiber length L f iber = 0.4 m, an holmium concentration N Ho = 8 × 10 26 ions/m 3 (4 mol.%), a neodymium concentration N Nd = 1 × 10 26 ions/m 3 (0.5 mol.%), and an output mirror reflectivity R out = 60%, a slope efficiency η = 16.67% and an input pump threshold P th = 0.2 W can be obtained. This result is interesting if compared with the efficiency obtainable with holmium doped fiber.