1. Introduction

Solid-state lasers operating in the 2μm wavelength are the subject of much interest [1–7]. The eye-safe wavelength is attractive in a number of applications, including range finding, coherent laser radar, atmospheric sensing, and several medical applications. Q-switched 2μm lasers are useful as pump sources for optical parametric oscillators operating in the mid-infrared region. In the last decade, room-temperature, diode-pumped 2μm laser (continuous and Q-switched) have been integrated in systems for ground-based or airborne radar measurements [2, 3]. The Tm-Ho co-doped materials, operating on the Ho transition, are usually preferred for high-output energy application, many different hosts and transitions have been reported to laser [6, 7]. Unlike the single Tm-doped materials used in the low output energy lasers, Tm-Ho co-doped materials present a high-emission cross section and long upper state fluorescence lifetimes (~10ms) that makes them suitable for producing high pulse energy under continuous wave pumping condition.

The energy transfer dynamics and the extent of up-conversion losses are often more influential than the threshold pumping level in determining the laser performance. While the presence of Tm reduces the Stokes losses, the resonant Tm-Ho transfer time in YLF is at lease 5μs. Although this is faster than observed in Tm,Ho:YAG, it is longer than the typical pulse buildup time and consequently only that fraction of the excitation energy stored in the Ho ⁵I₇ upper laser level is accessible when Q switching [8]. For a given pump power one can access a greater fraction of the excitation energy in a YLF host which, in conjunction with reportedly reduced up-conversion losses in YLF, would favor Tm,Ho:YLF over co-doped YAG. Furthermore, YLF is chosen as a host crystal because of its long pump integration time, excellent optical damage resistance, lack of thermal induced birefringence, and linearly polarized output. We have conducted some experiments and reported the characteristics of Tm,Ho:YLF microchip lasers [9–13].

In this paper, We give a rate equation theoretical model of continuous wave end-pumped actively Q-switched Tm,Ho:YLF lasers, in which the energy transfer up-conversion (ETU) effect and ground state re-absorption (GSA) are taken into account. Our theoretical analysis shows that the ETU effect reduces the pulse energy and the upper level effective lifetime. The theoretical results are in good agreement with experimental results.

2. Rate equation theory

The energy transfer processes involved in the 2μm laser operation of co-doped Tm-Ho materials have been thoroughly described by several authors [4, 5]. We use the main lines of these analyses with reference to the energy level diagram of Tm and Ho [11]. When the 792nm pump light from the diode laser is absorbed into the ³H₄ manifold, the ³F₄ manifold is efficiently populated through the well-known two-for-one cross-relaxation process. A fraction of the energy stored in the Tm ³F₄ manifold is then transferred to the Ho ⁵I₇ manifold. As the populations of ³F₄ and ⁵I₇ grow, the long lifetimes of the levels are reduced by up-conversion process. The laser emission at 2.06μm is due to a transition between the lowest Stark level in the ⁵I₇ manifold and a high level in the ⁵I₈ ground-state manifold. In such cases, the population density on the lower laser level is not presumed to be zero, but is assumed to have a small thermal population.

For typical doping concentrations (6%Tm, 0.4% Ho), the characteristic energy transfer time from Tm to Ho is between 5 and 20μs. We may treat the ³F₄ and ⁵I₇ levels as one coupled system, since this energy transfer time is much smaller than either of the ³F₄ and ⁵I₇ level lifetimes [13]. We used a space-dependent rate-equation analysis to take into account the influence of ETU and GSA. The quasi-three-level rate equation can be written as [14, 15]

Where N_u is the coupling upper level population density, τ is the lifetime of coupling upper level, ξ is the up-conversion rate. R_p is the average pump intensity and is given by

Where ω_p is the averaged pump size in the active medium, η_p is the pump quantum efficiency, η_a=1-exp(-αl) is the fraction of incident pump power absorbed in a crystal of length l with absorption coefficient α, P_in is the incident pump power, and h is the Planck’s constant. From Eq. (1), the initial coupling upper level population inversion density at the end of the low-Q segment at a pulse repetition frequency f is derived as [14, 15]

Where, N_f is the residual population inversion density of the coupling upper level from the preceding pulse. The resonant time in Tm,Ho:YLF is between 5μs and 20μs, it is longer than the typical pulse buildup time and consequently that fraction of the excitation energy stored in the Ho ⁵I₇ upper laser level is accessible when Q-switch is opened. The initial ⁵I⁷ population inversion density at the end of the low-Q segment at a pulse repetition frequency f is given by

Where N_Ho is the total doping concentrations for Ho³⁺, γ=f_u +f_i is the inversion reduction factor, f_u and f_l are the fractions of the total ⁵I₇ and ⁵I₈ population density residing in laser upper level and lower level, f_Ho represents the fractional population of Ho ions in the coupling upper level. The final inversion density of Ho ⁵I₇ upper laser level is given by

Where, σ_e is the emission cross section, and L is the roundtrip dissipated optical loss. The relationship between the final coupling upper laser level inversion density N_f and the final inversion density of Ho ⁵I₇ upper laser level Δn_f is given by

With the conventional model of coupled rate equations the pulse energy E is given by [16]

Where, hv_l is the output laser photon energy, ${\pi \omega }_l^2$ is the cavity mode area, R is the output mirror reflectivity.

To illustrate the plots of the pulse energy as a function of pulse repetition frequency, we use Eqs. (2)–(9) and the following parameters: f_u =0.095 [4], f_ul=0.024 [4], η_p=1.57, N_Ho =5.59×10¹⁹cm^-3, l=2.5mm, λ_p=792nm, λ_l=2.066μm, α=5.4cm^-1 [4], σ_e=15×10^-20cm² [5], R=98%, ω_p=ω_l=100μm, L=0.03. Fig. 1 shows the variation of pulse energy with pulse repetition frequency for different ETU coefficients when the incident pump power is 3W. From Fig. 1 it can be seen that the pulse energy decreases with increasing ETU coefficient at the same pulse repetition frequency, the influence of ETU effect on pulse energy increase as the pulse repetition decreases, and is more serious and almost maintains at a constant under lower pulse repetition. We also can see from Fig. 1 that the corresponding repetition frequency that the pulse energy begin to fall increases with increasing ETU coefficient. The reciprocal value of the corresponding repetition frequency that the pulse energy begin to fall is approximately equal to the upper effective lifetime. So we can get the conclusion that the ETU effect reduces the pulse energy and the upper level effective lifetime. Fig. 2 shows the plots of the pulse energy as a function of pulse repetition frequency for three different pump powers. It is noted from Fig. 2 that the influence of ETU effect is more serious as pump power increases. The means that the upper level effective lifetime decreases as pump power increases. It is only due to the influence of ETU effect, the increase of incident pump power decreases the upper level effective lifetime.

 

Fig. 1. Pulse energy versus pulse repetition frequency for different energy transfer up-conversion rates

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Fig. 2. Pulse energy versus pulse repetition frequency for different pump powers at the same energy transfer up-conversion rate

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2. Experiments and results

The experimental setup is shown in Fig. 3. The pump laser is Coherent Inc S-79-3000-200-H/L 3W laser diode temperature tuned to 792nm. Its emission is collected with an 8mm focal length collimating lens followed by a cylindrical lens with focal length of 100mm to reshape. This laser beam then is focused onto the Tm,Ho:YLF crystal using a lens with a 50mm focal length. With this arrangement the pumping beam can be focused to a spot size of approximately 100×100μm² at the entrance face of the laser crystal. The total transmission efficiency of the beam-reshaping system is about 91% at 792nm. The Tm,Ho:YLF laser crystal from II -VI corporation has do-pant concentrations of 6%Tm, 0.4%Ho with dimension of 5×5×2.5 mm³. The crystal is oriented with the c axis parallel to the polarization direction of the pump beam to utilize the higher π spectrum absorption because both the pump and laser cross sections are considerably enhanced in the π polarization.

A plane-concave resonator is employed to make the system very simple and compact. The near hemispherical resonant is formed between the planar crystal front face and the output coupler. A dichromatic coating on the front face of the crystal is high transmitting at 792nm but is totally reflecting at 2μm. The other face is only polished and uncoated at both pump and output wavelengths. To efficiently remove the heat generated with pump power from the crystal, the crystal is wrapped with indium foils and held in brass heat sink. Temperature of the heat sink is held at a constant 293K with a thermoelectric cooler. The QSGSU-6Q acoustic-optical modulator (The 26th Electronics Institute, Chinese Ministry of Information Industry), the effective length of which is 44 mm is located between the crystal and the output coupler but near the crystal. The optical polarization and acoustic wave-vector are mutually orthogonal for optimum diffraction. Both ends of the modulator are antireflection coated at 2066nm, and its intrinsic diffraction loss is ~85% which is adequate to prevent lasing action. The modulation repetition rate is tunable from 80Hz to 10 kHz. The laser pulse width and waveforms were observed with a room temperature mercury cadmium telluride photoconductive detector and a TDS3032B digital oscilloscope (Tektronix Inc., USA).

 

Fig. 3. The schematic diagram of LD-pumped Q-switched Tm,Ho:YLF laser

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Output couplers with transmissions of 1.26%, 2.0%, 2.97%, 4.75%, 6.0%, and 10% have been used, with 2.0% giving the best results for the continuous conditions. The following experimental results are obtained with the output with transmission of 2%, and the radius of curvature of the output coupler is 10cm.

The passive loss introduced into the cavity by the Q-switch crystal reduced the CW output power at the pump power of 1.7W from 270mW to 180mW. When pulse repetition frequency was changed from 80 Hz to 10 kHz, the resulting laser pulse width increased almost linearly from 138ns to 226ns. Fig. 4 shows the single pulse energy as a function of the pulse repetition frequency when the incident pump power is 1.7W. It is noted from Fig. 4 that the largest pulse energy of 45μJ can be achieved at the lower pulse repetition frequency of 1kHz, the pulse energy decreases from 45 μJ to 15μJ with the increase of the pulse repetition frequency from 1 kHz to 10 kHz. At sufficiently low pulse repetition frequency the pulse energy will saturate since the pumping time becomes large compared to the upper level effective lifetime, furthermore, the pulse energy slightly decreases with the decrease of pulse repetition frequency due to thermal effect. The pulse-to-pulse amplitude fluctuation of Q-switched pulse train was measured to be less than ±5%.

 

Fig. 4. Pulse energy versus pulse repetition frequency

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From Fig. 4, we find that the repetition frequency related to the maximum output energy is about 1kHz. At frequencies up to 1 kHz, the population inversion between pulses can reach a maximum and the pulse energy is nearly constant. However, beyond this the time between Q-switch pulses is less than the effective upper laser lifetime and the attainable inversion is reduced. The transition region occurs at Q-switch frequencies commensurate with an effective upper laser level lifetime of 1ms, a factor of 14 less than the 14ms fluorescence lifetime because of ETU effect. For comparison we also plot the results computed with and without the ETU effect in the Fig. 4. The values of the parameters used for calculation are ξ=5×10^-18cm³/s, P_in=1.7W, f_u =0.095 [4], f_l =O.024 [4], λ _p=792nm, λ _l=2.066μm, ±=5.4cm^-1 [4], l=2.5mm, σ_e=15×l0^-20cm² [5], R=98%, η_p=1.57, N_Ho =5.59×l0¹⁹cm³, ω_p=ω_ll00μm, L=0.03. It can be seen that the calculated results with the ETU effect are in good agreement with the experimental data. The theoretical results without the ETU effect not only overestimate the output pulse energy but also cannot describe that the pulse repetition frequency that the pulse energy reaches a maximum increases from ~70Hz to ~1kHz.

3. Conclusion

We have built the quasi-three-level rate equation theoretical model of continuous wave end-pumped actively Q-switched Tm,Ho:YLF laser, in which the ETU and GSA effects are included. The theoretical results show that the pulse energy is affected by both pulse repetition frequency and ETU effect. The decrease of pulse energy due to ETU effect increases when the actively Q-switched Tm,Ho:YLF lasers operate at high pump power and low pulse repetition frequency. We find that the ETU effect reduces the upper level effective lifetime of Tm,Ho:YLF laser. The practical example of the diode-end-pumped actively Q-switched Tm,Ho:YLF laser have been used to verify the present theoretical model. We find the influences of the up-conversion effect and ground state re-absorption can be decreased by lowering the temperature of crystal and decreasing the incident intensity. Our theoretical model can be applied to other Tm and Ho co-doped lasers and provides an important guideline of optimizing laser output energy.