1. Introduction

Co-doped Tm, Ho solid-state lasers present significant interest for a number of advanced applications. 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 [1–22]. However, many related physical effects which are non-linearly coupled with each other continue to remain unclear. For instance, notwithstanding the importance attributed to thermal effects [23], they are often treated in an excessively simplistic way, which does not allow a correct interpretation of experimental data as well as an adequate laser simulation and optimization. In this Communication we develop a coupled themo-optoical computational model in which specific non-steady-state thermal effects are rigorously coupled with population dynamics and spectroscopic processes involved in the energy transitions, and lasing in Tm,Ho solid state lasers. These effects are found to play a very significant role in high-power high- frequency Q-switched laser operation required for development of coherent-detection lidar systems.

2. Optics model

There are several models describing the electron population dynamics of co-doped Tm,Ho solid state lasers using the main levels involved in the pumping and laser generation shown in Fig. 1. These models have included various effects such as end and side pumping, ground state depletion, energy transfer between Tm³⁺ and Ho³⁺ ions, and also different types of up-conversion processes decreasing the upper laser level population. Several studies have used simplifications allowing model reduction to two-rate equations describing the electron density at the excited levels ³F₄ and ⁵I₇, and ground-state levels ³H₆ and ⁵I₈. In our study we endeavor to retain all the terms in the rate dynamics model, allowing us to reveal several specific effects. In particular, our analysis is based on computational non-steady state thermo-optical model using 8-level rate equations and related data describing the population dynamics of Tm,Ho lasers from Walsh et al. [17]:

For the upper laser level (⁵I₇):

For the lower laser level (⁵I₈):

where n _i(t,r) are the level concentrations, p _ij are the probabilities of the optical transitions, τ _i are the level lifetimes, R _p(t) is the pumping source, ϕ(t,r) is the local laser photon density, σ_se is the stimulated emission cross-section, f _i(t,r) are the Boltzmann level population factors and η is the refractive index of the crystal.

 

Fig. 1. Energy transfer processes in co-doped Tm,Ho materials and energy differences used in Eq. (16).

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All optical transition probabilities and level lifetimes, including characteristic radiative times, have been considered in detail in Ref. [17]. Although that study neglects some of the possible radiation decays from the upper manifolds which could easily be taken into account in present study, we have not extended the original model of Ref. [17] in view of the good agreement with the known experimental data on Q-switched pulse operation [6].

The local laser photon density ϕ(t,r) is represented by the product of (i) the total number of photons inside the oscillator cavity, Φ₀(t), depending on t and (ii) the normalized space distribution function, ϕ ₀(r). The resulting equation for Φ₀(t) is given by a differential equation including integration of the stimulated and spontaneous radiation over the crystal volume [17, 23, 24]:

where V _cr is the crystal volume, and ε≈10^-7-10^-8 is a factor taking into account the proportion of photons spontaneously emitted within the solid angle of the mirrors, and τ _c is the cavity lifetime given 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 _l is the back mirror reflectance, T _out is the output mirror transmittance and β is the parameter used in our simulations for the optical loss associated with the active Q-switching: β=0 for the open resonator and β≫-ln R ₁(1-T _out) for the closed resonator. For the acousto-optic Q-switch, if the fraction of the main beam diffracted out of the resonator is 0.9, β=-ln(1-0.9)=2.3. We neglect here additional reflectance and scattering loss on crystal and Q-switch. However, these factors can also be included into the round trip optical loss in Eq. (10).

For the case of 100–-500 ns pulse generation considered here the cavity length L _cav≫L _cr and the spatial photon distribution inside the operating crystal can be described by TEM₀₀ fundamental mode as:

where w ₀ is the beam waist radius of TEM₀₀ mode defined by the resonator parameters.

The solution of the rates equations together with the main oscillator Eq. (9) gives the radial distribution of the output power density (W/m²) at the output mirror as:

where w*₀ is the modified beam radius outside the resonator (for instance, for the case of a TEM₀₀ Gaussian beam inside the confocal spherical resonator one has $w_0=\sqrt{L_{\mathrm{cav}}{\lambda }_l⁄2\pi }$ and w*₀=√2w ₀ at the output mirror).

In this paper we consider a particular case of Tm (6%), Ho(0.4%):YLF operation side pumped by 785 nm LD radiation. For 6 % Tm doped YLF crystal one finds for the absorption coefficient α=σ _a N _Tm≈2.8 cm ^-1 [17]. Thus, a 2 mm diameter YLF crystal is able to absorb a (1-exp(-2αd))≈0.67 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. In the simulation we assume that the laser rod axis is directed along the c-axis of the YLF crystal and we neglect anisotropy in absorption of the polarized beams of the LD bars. We assume that this anisotropy is not significant for the side-pumped configuration in which three LD bars are arranged around the crystal in threefold symmetry and the internal surface of the tube used for water cooling has a high diffusive reflection [25]. In fact, high incident fluxes are able to deplete the ³H₆-level in Tm³⁺ and to reduce significantly absorption [23]. However, in this study the concentration of the ³H₆-level does not fall below 0.9 of the Tm-concentration, the related variations of α do not exceed 5 %, and in the simulations we use:

where η _a=(1-ρ)[1-exp(-2 αd)] is the absorption efficiency of pumping, ρ is the reflection factor of the pumping radiation into laser material, Q _p is the pumping pulse energy, Δt _p is the pumping pulse duration and η _p is quantum efficiency.

In Fig. 2 we show a simulated giant pulse (G-pulse) generated by a Tm,Ho:YLF laser producing ≈0.1 J pulses of ≈150 ns duration. In particular, we simulate an active Q-switched laser side-pumped by 0.5 ms LD pulses of 785 nm wavelength for a crystal 2 cm long and 2 mm in diameter placed inside a 1 m long cavity (T _out=0.05 and R _l=0.98) with a 0.85 mm radius waist in the TEM₀₀ laser beam distribution. The Q-switch is open after a 0.5 ms pumping period with a delay of 0.7 ms to ensure that the G-pulse generation starts after achieving the maximal possible gain. This delay is associated with the delay of excitation transfer from ³H₄ to ³F₄, and finally towards the lasing ⁵I₇ level [22].

 

Fig. 2. Simulation of G-pulse generation: pulse power versus time.

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3. Thermal model

The heat absorbed inside the crystal leads to a temperature increase over the crystal volume. For high power operation this temperature shift is able to change the local values of the Boltzmann population factors of the upper and lower lasing levels:

where k _B is the Boltzmann constant, g _i is the degeneracy of the i-level, and T (t,r) is the local temperature.

Generally, the operating crystal is heated via lattice vibrations due to non-radiative decay of electrons from all levels involved in the excitations. The local heat source is defined by:

where ΔE _i is the energy difference between the i-manifold and the next lower manifold into which the electron makes the transition (Fig. 1) and τ _inr are the non-radiative times inversely proportional to the non-radiative transition probabilities.

In order to avoid difficulties in defining the probabilities of non-radiative transitions, an estimate of the heat source can be made via the difference between the pumped energy and the energy of stimulated and spontaneous radiation leaving the crystal [12]. This approach is mainly used for the CW mode or as an averaged estimate for high-repetition pulsed mode. However, we use this approach for normal or Q-switched mode operation by introducing a modification which takes into account the rate, $\sum _{i=2}^7\Delta E_i^{*}{\mathrm{dn}}_i⁄\mathrm{dt}$, at which the pumped energy is stored inside Tm³⁺ and Ho³⁺ ions as:

where in addition to ΔE _i we introduce the energy difference between the i-manifold and the ground state ΔE*_i (Fig. 1), and then τ _ir are the corresponding radiative times [17].

The calculation of Eq. (16) for the Tm,Ho:YLF laser reveals several effects significant for energy extraction by lasing pulse. First, Fig. 3(a) shows the energy balance integrated over the crystal volume versus time. It reveals a very significant extension of the heat release period as compared with the pumping period. Fig. 3(a) shows that the heat is released inside the crystal over a period of ≈10 ms, whereas the pumping period is 0.5 ms during which only ≈30 % of heat is released. A two-time lower resulting temperature increase is achieved in the crystal prior to G-pulse generation (1.2 ms) as follows from an estimate neglecting the thermal conductivity effect:

 

Fig. 3. Energy balance versus time during laser operation: (a) energy pumping and release and (b) optical loss by spontaneous radiation from different levels.

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Second, Fig. 3(a) also shows that in the final energy balance ≈0.12 J corresponds to the G-pulse energy, ≈0.75 J corresponds to the heat released inside the crystal and ≈0.84 J corresponds to the energy lost by spontaneous emission. Thus, about 43 % of the pumped energy is directly converted into heat. We should note that the estimates of heat release based on 2-level rate equations treat this value as the difference between the pumped energy and the optical energy of the laser pulse and the spontaneous emission from two levels, ³F₄ and ⁵I₇ [12]. The energy spontaneously emitted by other levels, i.e. ³H₅, ³H₄, ⁵I₅ and ⁵I₆, are implicitly included into the heat released inside the crystal [12]. The 8-level model used here shows that the contribution of the ⁵I₅ and ⁵I₆ levels into the spontaneous emission loss is negligibly small, whereas the contribution of ³H₄ and ³H₅ appears to be quite significant, ≈0.1 and ≈0.4 J, respectively (see Fig. 3(b)). Adding these values to the heat release of ≈0.75 J gives ≈70 %, similar to the result from the 2-level model [12].

Thus, only ≈43 % of the pumped energy is released directly as heat inside the crystal, whereas ≈45 % is spontaneously emitted radiation from the crystal at wavelengths: λ₂=1.93 µm, λ ₃=4.32 µm, λ ₄=2.46 µm and λ ₇=2.07 µm. These wavelengths are within the transparency range of the crystal and are therefore able to leave the crystal. Fig. 3(b) shows the values of the total optical loss, $\sum _{i=2}^7\Delta E_i{n}_i⁄{\tau }_{\mathrm{ir}}$, and also the losses emitted from all levels, ΔE _i n _i/τ _ir integrated over the crystal volume. These radiation fluxes leaving the crystal are absorbed by the water flow typically used for crystal cooling. The water absorption coefficients for these wavelengths are [26]: α ₂=124 cm^-1, α ₃=300 cm^-1, α ₄=63.5 cm^-1 and α ₇=31 cm^-1. That is, the spontaneously emitted fluxes leaving the crystal are absorbed within lengths of ≈α ^-1 _i, i.e. within 80, 33, 157 and 320 µm from the surface, respectively. The absorption of these fluxes in the vicinity of the crystal surface can significantly inhibit heat dissipation from the crystal. The heat transfer to the water flow depends on the Reynolds number, Re, defining the level of the flow turbulency dependent on the water flow rate through the channel inside which the operating crystal is set up. Numerical estimates show that for the typical coaxial crystal in a tube water channel geometry and typical flow rates, the value of the heat transfer coefficient is h=10³-10⁵ W/m² K [27]. The main thermal resistance to the heat flow from the crystal surface is due to the thermal boundary layer, δ _T, within which the heat conductance dominates over the convective transport. The estimate of δ _T follows from the equivalency of -k _cr∂T _cr/∂r|_sur=-k _w∂T _w/∂r|_sur=h(T _cr|_sur-T _w∞), where k _cr≈6 and k _w≈0.6 W/m K are the thermal conductivity of crystal and water, respectively. That is, using ∂T _w/∂r|_sur≈-(T _cr|_sur-T _w∞)/δ _T one finally obtains for h=10³-10⁵ W/m²K:

Thus, the spontaneous IR fluxes are absorbed by water within a distance where the heat transfer is dominated by the thermal conductivity. Hence, the absorption of these fluxes is able to significantly inhibit the heat dissipation from the crystal. In order to consider the thermal effect we simulate the complex heat transfer non-steady state, two-dimensional problem by coupling the above optical model with the heat generation and heat transport through the operating crystal, and the water boundary layer inside which the absorption of spontaneously emitted IR radiation takes place. The radially symmetric temperature distribution inside the cylindrical crystal, T _cr(t, r), and the thermal boundary layer in water, T _w(t, r), are defined by:

for crystal (i=cr) and water (i=w) with the boundary condition T _w=T _w∞ at r=R ₀+δ _T, where δ _T=R ₀[exp(k _w/R ₀ h)-1] takes into account the radial curvature.

Heat source density inside the crystal is defined by Eq. (16) whereas the heat source density due to the absorption of spontaneously emitted IR fluxes in water is defined by:

where J _0i(t) are the IR flux densities isotropically leaving the crystal given by:

The effect of IR radiation absorption is negligibly small for h>10⁵ W/m²K and δ _T<6 µm, when δ _T≪α ^-1 _i. However, for h≈10⁴ W/m²K (δ _T≈60 µm) this effect is very significant, and can lead to the onset of an inverted temperature distribution inside the crystal when the temperature inside the boundary layer is higher than that inside the crystal. This effect is shown to take place in a coupled thermo-optical simulation performed using a conservative 50x100 conservative finite-difference approximation. The main results of this simulation given in Fig. 4 at three times show that a significant temperature increase has occured (≈2 K) by the start of G-pulse generation (1.2 ms), which leads to a pulse energy decrease due to the decrease in f ₇ n ₇-f ₈ n ₈ over the crystal volume. This figure also shows that an inverted temperature distribution inside the crystal is present at period of time of ≈10 ms.

 

Fig. 4. Temperature distribution inside the operating Tm,Ho:YLF crystal and thermal boundary layer for single G-pulse generation for h=10⁴ W/m²K (δ _T≈60 µm, water temperature T _w∞=290 K).

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This effect, associated with the strong absorption of emitted IR radiation in water, produces a different result in high repetition mode. In particular, the simulation of 20 and 50 Hz G-pulse repetition mode given in Fig. 5 shows that in contrast with first few pulses after pulsed operation stabilization the temperature inside the crystal becomes higher than that at the crystal surface due to the onset of a quasi-steady state gradient. Finally, Fig. 5(a) shows that this thermal effect leads to ≈10–25 % reduction of G-pulse energy combined with a strong thermal lensing effect known to be detrimental for laser beam quality.

4. Summary

A complex thermo-optical model for Tm,Ho solid state lasers has been developed based on an 8-level rate dynamics model for the excitation transfer to Ho³⁺ ions from LD pumped Tm³⁺ ions integrated together with the equation for the total number of stimulated photons inside the cavity. This model is also coupled with a two-dimensional time dependent heat transfer model including absorption, heat release and heat transfer inside the operating crystal as well as the absorption and the thermal effect of infrared radiation fluxes spontaneously emitted by the operating crystal. In the case of water cooled laser operation the thermal effect is shown to be split into two simultaneously occurring processes: (i) direct heat release inside the crystal and (ii) infrared spontaneously emitted radiation fully absorbed in water over a distance of several hundreds of microns, which corresponds to a typical value of boundary layer thickness. In particular, the simulations show that only ≈43 % of the pumped energy is transformed into heat directly inside the crystal, whereas ≈45 % is IR radiation spontaneously emitted by ³H₄, ³H₅, ³F₄ and ⁵I₇ levels and absorbed in the vicinity of the crystal surface. The absorption taking place within the boundary layer provides an additional strong thermal effect, inhibiting the dissipation of the heat from the crystal and significantly increasing crystal temperature. The resulting temperature increase is shown to reduce significantly G-pulse energy.

 

Fig. 5. 20 and 50 Hz G-pulse laser operation: (a) G-pulse power modification with time and (b) temperature increase in the operating crystal versus time for h=10⁴ W/m² K for crystal axis and surface.

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