1. Introduction

The development of solid state 2 μm Tm, Ho lasers has remained a topic of particular interest for many years, due to a number of possible applications such as coherent laser radar, atmospheric sensing, a possible pump source for an optical parametric oscillator operating in the mid-infrared region, and medicine. In recent years significant progress in understanding the basic phenomena underlying Tm,Ho laser operation as well as in developing high power lasers has been achieved [1–21]. Nevertheless significant effort is still necessary for defining optimized parameters and achieving high lasing efficiency.

In this communication we focus on computational simulation and optimization of a side pumped Tm,Ho:YLF laser for producing MW-power giant pulses with durations of several hundred nanoseconds. The results obtained suggest that the operation of such a laser can be significantly enhanced by (i) Ho doping optimization and (ii) adjustment of pumping pulse duration and Q-switch opening time.

2. Pulse generation model

Our study is based on 8-level (see Fig. 1 for excited level scheme) rate dynamics model describing the population dynamics in Tm,Ho lasers by the following set of equations [17]:

For the upper laser state (⁵I₇):

For the lower laser state (⁵I₈):

where n_i are the state concentrations, p _ij are the probabilities of the optical transitions, τ_i are the state lifetimes, R _p(t,r,z) is the pumping source space and time distribution, σ_se is the stimulated emission cross-section, f _i are the Boltzman level populations factors and ϕ(t, r) is the local laser photon density with dimensionality of 1/m³.

The local laser photon density is represented via the product of (i) the total number of photons depending on t and (ii) the normalized space distribution function as ϕ(t,r) = Φ₀(t)ϕ ₀(r,z). The resulting equation for total photon number, Φ₀(t), inside the oscillator cavity is given by the differential equation [11,17, 22, 23]:

which includes the integrals of the stimulated and spontaneous emissions over the crystal volume, V _cr, and where τ _c is the cavity time, and ε≈10^-7-10^-8 is the coefficient which takes into account the proportion of photons spontaneously emitted within the solid angle of resonator mirrors, thereby initiating the development of the laser beam.

 

Fig. 1. Energy transfer processes in Tm, Ho doped materials.

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For the operating resonator the cavity time is given by:

and for the locked resonator by:

where L_opt = L_cav +(η - 1)L_cr is the characteristic optical length, L _cav is the cavity length and L _cr is the crystal length; R ₁=0.98 is the back mirror reflectance and T _out is the output mirror transmittance, α _qs l _qs=10²-10³ is the absorption loss inside the locked Q-switch device.

For the case of ≈500 ns pulse generation the cavity length L _cav ≫ L _cr and the laser beam radius inside the crystal may be considered to be constant. The spatial distribution of photons inside the operating crystal is simplified as:

where w ₀ is the beam waist radius of TEM₀₀ mode defined by the resonator parameters (for instance, for the simplest case of the confocal spherical resonator one has $w_0=\sqrt{\frac{L_{\mathrm{cav}}{\lambda }_l}{2\pi }}).$

The solution of the rates equation together with main oscillator equation gives the value of the output energy fluence, (W), as:

and its radial distribution (W/m²) at the output mirror as:

where the increased beam radius outside of the resonator, w ^* ₀, is taken into account.

We consider Tm,Ho:YLF operation side pumped by 780 nm LD radiation. For 6 % Tm doped YLF one finds for the absorption coefficient α=σ_a N _Tm≈2.8 cm^-1. Thus, a 2 mm diameter YLF crystal is able to absorb (1-exp(-2αd))≈0.67-fraction of the incident beam flux in the case of the double-pass pumping scheme providing high uniformity of the absorbed flux over the crystal volume. High incident fluxes are able, in fact, to deplete the ³H₆-level in Tm³⁺ and to inhibit absorption [11, 22]. However, in the present study the concentration of the ³H₆-level does not fall below 0.9 of the initial Tm-concentration, the related variations of absorption coefficient do not exceed 5 %, and in our simulation we use for R _p:

where η_a = ρ[1 - exp(-2αd)] is the absorption efficiency of the pumping radiation, ρ is the reflection loss of pumping radiation, Q_p is the pumping pulse energy, Δt _p is the pumping pulse duration, η _p≈1.3 is the pumping quantum efficiency for Tm,Ho:YLF.

3. Results and discussion

Model parameters used for Tm,Ho:YLF in the simulations are fully specified in Ref. [17]. Other parameters used in the simulations reported below are: crystal length L _cr=2 cm, crystal diameter d=2 mm, cavity length L _cav=1 m, output transmittance T=0.05, reflectance R ₁=0.98, and beam waist radius w ₀=0.85 mm. The simulations reveal two main effects important for efficiency optimization in developing a laser for producing nanosecond G-pulses.

Figure 2 shows a simulation results for 6 % Tm doping and two values of Ho doping, 0.4 and 1.0 %, for the case of normal pulse operation (without Q-switch) demonstrating a series of relaxation oscillations. The pumping (780 nm laser diode) time for both modes is 0.5 ms and the total absorbed energy η_aQ_P = 1.8 J. Figure 2(a) shows the power oscillations for the cases of Ho = 0.4 and 1.0 % respectively, whereas Fig. 2(b) shows the output pulse energies versus time. The final pulse energy for the 0.4 % Ho case (≈0.9 J), is calculated to be higher than that of 1.0 % Ho case (≈0.8 J). However, Fig. 2(a) shows that the power of the first relaxation spike is higher for the case of 1.0 % Ho. Thus in G-pulse generation the 1.0 % Ho case is able to yield a higher output energy per pulse as compared with the 0.4 % Ho case.

Figure 3 shows calculations of (a) G-pulse powers and (b) energies versus time for 0.4 % and 1.0 % Ho cases. The calculations assume Q-switch operation open by the end of the 0.5 ms pumping period. This figure shows that 1.0 % Ho doping can provide higher energy output per pulse as compared with 0.4 % Ho doping due to significantly larger level of the inverted population achieved after the pumping (See Fig. 4). Figure 5 compares the resulting pulse energy for the normal pulse and for the G-pulse as a function of Ho-concentration for the given 6 % Tm doping. These figures confirm that for the normal pulse 0.4 % Ho doping provides the highest output energy whereas for G-pulse 1.0–1.2 % looks to be optimal.

 

Fig. 2. Pulse generation in normal mode: (a) pulse power versus time and (b) pulse energy versus time for two Tm, Ho concentrations.

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Fig. 3. G-pulse generation: (a) pulse power versus time and (b) pulse energy versus time.

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Fig. 4. Inversion population dynamics in G-pulse generation mode.

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Fig. 5. Comparison of output pulse energy in normal (solid) and G-pulse (dashed) mode.

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Let us now focus on the time optimization of G-pulse operation. The above results on G-pulse generation assumed 0.5 ms pumping followed by immediate Q-switch opening. However, Fig. 4 suggests that such Q-switch opening is premature because the inverted population density does not achieve its maximum at the end of pumping.

Now we provide more details to emphasize the importance of proper time control. In particular, Fig. 6(a) shows the dependence of G-pulse power as function of the time for three characteristic cases: (i) 0.5 ms pumping followed by Q-switch opening (green), giving a pulse energy of ≈0.08 J, (ii) 1.9 ms pumping followed by Q-switch opening (blue) giving a pulse energy of ≈0.11 J and (iii) 0.5 ms pumping followed by Q-switch opening at 1.2 ms (red) giving a pulse energy of ≈0.12 J. Figure 6(b) shows the behavior of Δn for all three cases considered and provides the background information for understanding time optimization issues. That is, in the first case the Q-switch is open prior to the moment when maximal population inversion in the Ho³⁺ is achieved and the resulting output G-pulse energy is small. After the pulse is finished and the Q-switch is closed, the excited Tm³⁺ state ³F₄ continues to transfer excitation towards the Ho^{3+ 5}I₈. In this case a very significant part of the excitation “stored” initially in Tm³⁺ is transfered to Ho³⁺ after the G-pulse generation, and later on uselessly dissipated through spontaneous radiation and non-radiative transitions. In the second case of the significantly increased pumping time 1.9 ms with simultaneous opening of Q-switch a significantly higher generated pulse energy is achieved, 0.042 J, due to significantly higher electronic excitation energy stored on the Ho³⁺⁵I₇ state by the end of the pumping pulse. The third case corresponds to the situation when a relatively short pumping time of 0.5 ms is used together with 0.7 ms delay in opening the Q-switch. This case gives maximal energy output of ≈0.12 J in G-pulse generation because the Q-switch opens exactly at the moment when the concentration on the lasing Ho³⁺⁵I₇ state and the value of Δn reach their maximum, allowing maximal energy extraction by the laser pulse.

 

Fig.6. Optimization of G-pulse generation: (a) G-pulse powers versus time and (b) corresponding population inversion densities versus time for three different scenarios.

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Fig. 7. Optimization of operation times: (a) Q-switch is open by the end of pumping and (b) pumping time is 0.5 ms and Q-switch is open after a delay.

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Figure 7 shows final results on G-pulse energy optimization: (a) for the case of extended LD pumping within the range 0.5 – 10 ms followed by immediate Q-switch opening and (b) 0.5 ms LD pumping with delayed Q-switch opening. Energy maxima exist in both cases: at ≈1.9 ms for (a) and at ≈1.2 ms for (b). However, case (b) provides the higher absolute maximum.

It is worth noting that the delay between the pumping period and Q-switch opening shown above to optimize the energy pulse is associated with the characteristic time of the electronic transfer from the upper exited state ³H₄ to ³F₄ and, later on, towards ⁵I₇. Our simulation based on the above model shows that the excitation transfer from the ³F₄ state towards the ⁵I₇ state happens without any significant deceleration whereas significant delay is associated with the ³H₄ → ³F₄ radiation transfer process, i.e. the rise in concentration at the Tm^{3+ 3}H₄ state, defined by the pumping period used, and the depletion of this level, defined by the two-photon process p ₄₁ n ₄ n ₁ and requiring ≈0.7 ms. In the case of extended pumping of several ms this delay does not play a very significant role, in contrast with short pumping times. The decrease in the delivered G-pulse energy shown in both cases for long times is associated with the increasing role of spontaneous radiation loss from both ³F₄ and ⁵I₇ states which have characteristic lifetimes of 16 and 15 ms respectively.

To finalize the discussion we would like to note that a ≈0.7 ms delay between pumping and achieving the maximal population inversion density in a Tm,Ho:YLF laser found in this study is experimentally supported by the observation of the amplification signal in a Tm,Ho:YLF crystal pumped by a 780 nm radiation pulse of 65 μs duration [12]. Data recorded for the YLF case shows the amplification signal maximum at 0.6–0.7 ms after the start of the pumping. However, the data for Tm,Ho:YAG operation does not show any significant delay between pumping period and maximal amplification signal.

4. Conclusions

This study, based on an 8-level rate equation model, shows two effects significant for giant pulse Tm,Ho:YLF laser optimization. The first one is associated with Ho doping. That is, in generating a G-pulse 1.0 % Ho concentration should provide a higher energy output as compared with normal pulse operation for which 0.4 % Ho concentration is confirmed to yield the highest energy output. The second effect is associated with the optimal time for opening the Q-switch. It is found that a delay of ≈0.7 ms between the 0.5 ms 780 nm LD pumping pulse and opening the Q-switch is able to provide a significant increase in the generated G-pulse energy. This is due to a ≈0.7 ms delay in the energy transfer from the initially excited Tm³⁺ ions to the upper lasing level of Ho³⁺ ions. This effect is captured by the 8-level rate equation model, which takes into account the possible deceleration of the electronic excitation transfer from the Tm³⁺ ions excited by the 780 nm pulse to the Tm³⁺³F₄ state, and later on, towards the lasing Ho³⁺⁵I₇ state.