Numerical Design of a Gain-Switched Pulsed Laser at 3.92 μm Wavelength Based on a Ho3+-Doped Fluoroindate Fiber

A gain-switched pulsed laser based on a commercial, heavily holmium-doped fluoroindate glass fiber, is designed to emit in the middle-infrared range, at the wavelength <inline-formula><tex-math notation="LaTeX">$\boldsymbol{\lambda } = 3.92\ {\boldsymbol{\mu } \bf{ m}}$</tex-math></inline-formula>. The laser, pumped at <inline-formula><tex-math notation="LaTeX">${\boldsymbol{\lambda }} = 888{\bf{\ nm}}$</tex-math></inline-formula>, is modeled by a six-level system, by taking into account experimental spectroscopic parameters, to identify a feasible laser configuration. An output signal peak power of about <inline-formula><tex-math notation="LaTeX">${\boldsymbol{P}}_{\boldsymbol{s}}^{{\boldsymbol{peak}}} = 14.62{\bf{\ W}}$</tex-math></inline-formula> with a full width at half maximum (FWHM) pulse duration less than <inline-formula><tex-math notation="LaTeX">${{\boldsymbol{\tau }}_{\boldsymbol{s}}} = 73\ {\boldsymbol{ns}}$</tex-math></inline-formula> and pulse energy <inline-formula><tex-math notation="LaTeX">${{\boldsymbol{E}}_{\boldsymbol{s}}} = 1.214\ {\boldsymbol{\mu } \bf {J}}$</tex-math></inline-formula> is predicted, by considering an input peak power of <inline-formula><tex-math notation="LaTeX">${\boldsymbol{P}}_{\boldsymbol{p}}^{{\boldsymbol{peak}}} = 10{\bf{\ W}}$</tex-math></inline-formula>, and pump repetition rate of <inline-formula><tex-math notation="LaTeX">${\boldsymbol{f}} = 100{\bf{\ kHz}}$</tex-math></inline-formula>, by employing a 8 cm-long fluoroindate fiber with holmium concentration <inline-formula><tex-math notation="LaTeX">${{\boldsymbol{N}}_{{\boldsymbol{Ho}}}} = 100\ 000{\bf{\ ppm}}$</tex-math></inline-formula>. The obtained result encourages the construction of a pulsed laser based on commercially available optical fiber, for applications in different fields as sensing and biomedicine.

at different wavelengths [9]- [13]. Great research interest was focused on Er 3+ -doped fluoride fiber, since emission at about λ = 2.8 μm and λ = 3.5 μm allowed to obtain intriguing laser in continuous wave (CW), gain-switched, and Q-switched regime [14]- [16]. In particular, in [15] a gain-switched fiber laser operating near λ = 3.5 μm via a dual-wavelength pumping scheme was obtained in an erbium-doped fluorozirconate fiber, with stable pulses with repetition rates ranging between 15 and 20 kHz and laser efficiency of η = 4.7%. Fluoroindate optical fibers can exhibit attenuation much smaller with respect to the fluoride ones at longer wavelength, beyond λ = 3.3/3.5 μm, also with reference to available on market products [17]. They exhibit very promising performances in terms of high transparency, with reduced optical attenuation α ≈ 0.2 dB/m from ultra-violet (UV) to Mid-IR range [18]. Therefore, they can enable promising applications in the 3 − 5 μm atmospheric transparency window and in an important part of the molecular fingerprint region. During the last years, rare-earth doped fluoroindate glasses have been spectroscopically investigated with the aim of finding new pumping schemes and operating wavelengths [19]- [24]. As an example, dysprosium and holmium ions allow emission at wavelengths beyond 3 μm [12], [25]. Recently, holmium-doped fluoroindate fibers have been characterized [25]- [29] and CW lasers emitting at λ s = 2.875 μm [30] and λ s = 3.92 μm [31], [32], pumped at λ p = 1120 nm and λ p = 888 nm respectively, have been demonstrated. For that pertaining the pulsed laser operation, emissions at the wavelength λ s = 2.106 μm [33] and in the λ s = 2.95 − 3.015 μm range [34] have been obtained. The literature results reported in [28]- [31], showing the feasibility of CW laser emission in holmium-doped fluoroindate optical fibers, encourage the investigation/ prediction of the pulsed operation with these optical fibers as an alternative to erbium-doped fluoride ones [15], [16].
In this work, for the first time to the best of our knowledge, the design of a heavily holmium-doped fluoroindate fiber, pumped at λ p = 888 nm, is proposed to obtain laser pulses at λ s = 3.92 μm. By using a homemade numerical solver, the pulsed laser behavior is realistically investigated and optimized, employing measured spectroscopic parameter taken from literature [28], [29]. The developed numerical solver is well validated. The employed design approach is similar to that used in [8], where a gain-switched pulsed laser, based on a Dy 3+ :ZBLAN fiber was suggested and then successfully constructed exploiting the same laser transition, even if the pumping wavelength was different [35], [36], with a good agreement between simulation and experimental results. The obtained results are interesting since, until now, only CW laser based on the same commercial Ho 3+ -doped flouroindate fiber, has been designed [32] or fabricated [31]. Moreover, the practical interest of this work lies in the need of fulfilling accurately the optimized laser parameters identified in the proposed design, in order to construct a pulsed laser in the stable single pulse regime based on a commercial fiber. The simulated performance, in term of laser efficiency are comparable with those obtained by considered other promising dopants and glasses, e.g., the erbium doped fluorozirconate fibers reported in [15].

II. RECALL OF THEORY
The Ho 3+ -doped fluoroindate fiber stimulated emission at λ s = 3.92 μm can be modeled by considering a six-levels system [29], pumped at λ p = 888 nm. The complete level scheme, including all the main phenomena, is reported in Fig. 1. The six-level model is employed instead of the five-level model proposed in [28], [31] because it allows an accurate simulation. It is validated by simulating the CW laser presented in [31], obtaining results in good agreement, as reported in the next section, while the five-level model is not suitable and wrong results are obtained in the cases here investigated. Level 5 I 4 and level 5 I 5 degenerate and are considered as single level 4 as reported in [28], [29]. The energy transfer up-conversion phenomenon between level 1 and level 3, starting from level 2, even if included in the model of [29], is here neglected since this approximation does not affect the simulation results. The other light-rare earth interactions taken into account are pump absorption, stimulated emission, radiative and nonradiative decays, excited state absorption (ESA), cross-relaxation (CR), and energy transfer up-conversion (ETU) due to the high Ho 3+ ions concentration that will be simulated.
By following the rate equations approach [11], [13], [37], the energy level populations N 1 , . . . , N 6 can be written as a nonlinear system, τ Ri take into account radiative and non-radiative decays, β i,j are the branching ratios, τ Ri are the i-th level lifetimes, K is the ETU rate, and W CR1 and W CR2 are the cross-relaxation transition rates. The condition where σ i,j (λ p/s ) is the cross section at the wavelength λ p/s for the i → j transition, h is the Plank constant, c 0 is the light speed in vacuum, λ p/s is the pump/signal wavelength, P ± p is the forward/backward pump power, P ± s is the forward/backward signal power, i p and i s are the normalized transverse intensity profiles, i.e., the squared modulus of the electromagnetic field, of pump and signal beams, respectively.
The power propagation for pump and signal beams is considered by the following partial differential equations are the gain coefficients for the pump and the signal, respectively, α is the glass optical loss, v p g and v s g are the group velocity for the pump and the signal, respectively, and B ase is the equivalent noise bandwidth for the Amplified Spontaneous Emission (ASE), n i,p/s are the overlap integrals over the rare earth-doped region Ω d between the i-th level population distribution N i (x, y, z, t) and the pump/signal optical mode intensity i p/s (x, y) and they are defined as follows.
These coefficients allow to take into account the overlapping strength between the spatial distribution of ion populations and the electromagnetic field.
To solve (3), the following initial conditions are imposed.
where z = 0 and z = L represent the ends of the laser cavity, P p0 (t) is the input pump power, R 1 and R 2 are the first and second mirror reflectivity, respectively. Time initial conditions are also considered as follows.
To evaluate the time evolution of the generated pulses, the output power is defined as

III. PULSED LASER DESIGN
The considered fiber is a step-index double-cladding fluoroindate (InF 3 ) glass fiber, doped with 10 mol.% of Ho 3+ ions, commercially available by Le Verre Fluoré [17]. The core diameter is d co = 16 μm and the numerical aperture NA = 0.2.
The cladding is 2-D shaped, obtained with circular diameter d cl1 = 100 μm truncated by two parallel plans at a distance d = 90 μm, to enhance cladding pump absorption. The second cladding has diameter d cl2 = 155 μm. Fig. 2 illustrates the fiber section geometry and the electric field modulus of the fundamental HE 11 mode simulated via a Finite Element Method (FEM) code. The aforesaid fiber is slightly multimode with normalized frequency number V = 2.56 at the signal wavelength. It exhibits a second order mode, which can be neglected in the laser operation since its overlapping coefficient with the spatial distribution of ion populations is less than a half of that pertaining to the fundamental mode. This prevents its contribution in the laser operation.
The rare earth dopant concentration is N Ho = 100 000 ppm = 2 × 10 27 ions/m 3 . The used spectroscopic and optical parameters are taken from [28], [29] and listed in Table I. Since ETU rate K, cross relaxation (CR 1 ) rate W CR1 , and cross relaxation (CR 2 ) rate W CR2 are reported in literature with respect to relative level populations, they were obtained by dividing the values of [28], [29] by the dopant concentration N Ho . The pump and signal wavelengths are λ p = 888 nm and λ s = 3920 nm, respectively. The equivalent ASE noise bandwidth is B ase = 100 nm, while the optical losses are α = 0.2 dB/m for both pump and signal wavelengths as reported in [30]. The simulations are performed via the differential equations by using Finite-Difference Time-Domain (FDTD) method, by considering a time grid with step Δt = 0.1 ns and a space grid with step Δz to obtain twenty space samples along the fiber. The pump excitation waveform is considered as a square wave with amplitude P peak p , repetition rate f , and duty cycle D.

A. Model Validation
The six-level model is validated by considering the experimental data reported in literature [31]. In particular, Fig. 3 shows the comparison between the six-level model and of the five-level model simulated efficiencies, with respect to the measured values, for a CW input pump, i.e., for a duty cycle D = 100%, fiber length L f iber = 23 cm, and second mirror reflectivity R 2 = 84% [31]. A slope efficiency η CW = 8.9% and power threshold P th = 4.2 W are simulated with the six-level model. These values are in good agreement with the experimental ones η CW = 10.2% and P th = 4.3 W . The five-level model provides less accurate simulation results, the slope efficiency being η CW = 2.27% and the power threshold P th = 1.5 W with a significant deviation with respect to the experimental values. The discrepancy between the experimental values and the six-level model simulated parameters could be due to i) the employed attenuation α = 0.2 dB/m for both pump and signal wavelengths, which is probably overestimated and ii) the employed emission and absorption cross-section approximated as coincident at the two considered wavelengths, listed in Table I. However, in absence of further experimental spectroscopic data, we keep this choice which is precautionary for the laser feasibility investigation.

B. Pulsed Laser Results
As an example of the pulsed laser simulation, Fig. 4 shows the unregular output signal pulses P out s (t) (red curve) at the end of the fiber length L f iber and the input pump pulses P in p (t) (black curve), with peak power P peak p = 10 W , as a function of time t; pump repetition rate f = 100 kHz, input pump duty cycle D = 50%, fiber length L f iber = 10 cm, and second mirror reflectivity R 2 = 70%. In the simulation, the considered input pump laser is the effectively coupled in the fiber. The laser of Fig. 4 is not optimized. The emission exhibits multiple output peaks with unstable amplitudes.
A deep investigation about the dependence of the laser output signal peak power P peak s , output signal pulse width τ s , and energy E s , on the laser fiber length L f iber for different pairs of input pump duty cycle D and second mirror reflectivity R 2 , (see Figs. 5 -7) is carried out. Since variations of the input pump duty cycle D and of the second mirror reflectivity R 2 are strictly related, they have been investigated simultaneously. After a high number of simulations, only the cases of practical interest, with stable single pulse operation, are reported. For all investigated  cases, the output residual pump peak power was under the 1% of the input pump peak power. Fig. 5 shows the output signal peak power P peak s as a function of the laser fiber length L f iber , for different pairs of input pump duty cycle D and second mirror reflectivity R 2 ; pump repetition rate f = 100 kHz. As the input pump duty cycle D increases, the second mirror reflectivity R 2 must be reduced to guarantee single pulse output. This induces a strong decrease of the output signal peak power P peak s . The maximum output signal peak power P peak s = 14.76 W is obtained for the fiber length L f iber = 6 cm, with input pump duty cycle D = 25% and second mirror reflectivity R 2 = 85%. Fig. 6 shows the output signal pulse full width at half maximum (FWHM) width τ s as a function of the laser fiber length L f iber , for different pairs of input pump duty cycle D and second mirror reflectivity R 2 ; pump repetition rate f = 100 kHz. The output pulse width τ s increases almost linearly with the fiber length L f iber . It is weakly dependent on the input pump duty cycle D and the second mirror reflectivity R 2 . To shorten the output optical pulse duration, reduced fiber lengths L f iber are more suitable. The shortest output pulse width τ s = 58.9 ns is obtained for input pump duty cycle D = 25% , second mirror reflectivity R 2 = 85%, and fiber length L f iber = 5 cm. This small length value is feasible thank to the very high dopant concentration. Fig. 7 shows the output signal pulse energy E s as a function of the laser fiber length L f iber , for different pairs of input pump duty cycle D and second mirror R 2 ; pump repetition rate f = 100 kHz. It grows almost linearly with the fiber length L f iber and it is quite independent from the input pump duty cycle D and the second mirror reflectivity R 2 . The maximum output signal pulse energy E s = 1.37 μJ is obtained for the input pump duty cycle D = 30%, second mirror reflectivity R 2 = 77%, and fiber length L f iber = 12 cm, leading to an optical-to-optical internal efficiency η = 4.57%. The fiber length L f iber = 5 cm allows the shortest obtained pulse width τ s but leads to the minimum output signal pulse energy E s . To find a tradeoff among the output signal peak power P peak s as high as possible, the output signal pulse width τ s as short as possible, and the output signal pulse energy E s as high as possible, the combination R 2 = 77% and L f iber = 8 cm is chosen for the next investigations.
The investigation is completed by considering the dependence of the laser output signal peak power P peak s , output signal pulse width τ s , and energy E s on the input pump duty cycle D, for different values of the pump repetition rate f (see Figs. 8-10). Fig. 8 shows the output signal peak power P peak s as a function of the input pump duty cycle D, for different values of the pump repetition rate f ; fiber length L f iber = 8 cm; second mirror reflectivity R 2 = 77%, input pump peak power P peak p = 10 W . The domains of correct laser operation are very narrow and strongly discontinuous. For each value of f only a small variation of D is allowed in order to obtain a stable single pulse output. Moreover, as the pump repetition rate f increases, also the input pump duty cycle D must increase to obtain the correct pulsed laser operation with stable single pulse output. The output signal peak power P peak s slightly increases, varying from P peak s = 14.5 W to P peak s = 14.87 W , as f and D increase. Fig. 9 shows the output signal pulse width τ s as a function of the input pump duty cycle D, for different values of the pump   repetition rate f ; fiber length L f iber = 8 cm , second mirror reflectivity R 2 = 77%. The output signal pulse width τ s slightly decreases by increasing the pump repetition rate f and the input pump duty D, changing from τ s = 72.75 ns to τ s = 71.9 ns.  Lastly, the output signal pulses P out s (t) (red curve) and the input pump pulses P in p (t) (black curve), with peak power P peak p = 10 W , as a function of time t, for the optimized cavity is illustrated in Fig. 11 for the pump repetition rate f = 100 kHz, duty cycle D = 30%, fiber length L f iber = 8 cm, output mirror reflectivity R 2 = 77%. A stable single pulse signal at λ s = 3.92 μm with an output signal peak power P peak s = 14.62 W , pulse width τ s = 72.55 ns, signal pulse energy E s = 1.214 μJ and optical-to-optical internal efficiency η = 4.05% is obtained. The time to first pulse is t fp = 15 μs and the stable gain-switched regime is achieved after about t R = 60 μs. The obtained efficiency is consistent with the CW laser one [31].
These results pave the way to fabricate a new pulsed laser, based on a commercially available fluoroindate fiber, with a stable output in a wide range of repetition pump rates f , from f = 50 kHz to beyond 200 kHz. We underline that, due to the narrow domains in which the laser exhibits stable single pulse operation, the optimized laser parameters, identified in the proposed design, provide useful guidelines to be followed in order to obtain a feasible pulsed emission. The interest is also due to the potential optimizations which could be obtained by co-doping the fluoroindate fiber with Ho 3+ and Nd 3+ ions and employing a pumping scheme at λ p = 808 nm [38].

IV. CONCLUSION
For the first time, a pulsed laser emitting at λ s = 3.92 μm based on a commercial double-cladding heavily holmium-doped fluoroindate glass fiber is accurately designed via a validated model, by using measured spectroscopic parameters. By employing an input pump with peak power P peak p = 10 W at the wavelength λ p = 888 nm, repetition rate f = 100 kHz and duty cycle D = 30%, stable output pulses having peak power P peak s = 14.62 W , pulse width FWHM τ s = 72.55 ns and pulse energy E s = 1.214 μJ, are simulated. The proposed gain-switched laser enables stable pulsed output in a wide range of pump repetition rates, from f = 50 kHz to beyond f = 200 kHz. Future development will include different co-doping and pumping scheme solutions.