1. Introduction

One of the remaining challenges of microlaser research is the achievement of a giant pulse (over megawatt) ultraviolet (UV) laser with low timing jitter σ (below 100 ns) at high repetition rates. This problem has resisted solution since the first demonstration of giant pulsed laser from ruby in 1962 [1]. Passively Q-switched high-power microlaser and giant pulse outputs were made thereafter by using end-pumped composite Nd³⁺:YAG [2] and intense pulse pumping [3]. Recent attempts to achieve giant pulse microlaser by combining [100]-cut Nd³⁺:YAG crystal as gain material with [110]-cut Cr⁴⁺:YAG crystal as saturable absorber succeeded in demonstrating intrinsically reduced depolarization ratio [4]. However, serious problems still remain to be overcome, mainly having to do with thermal effects at high repetition rate (1 kHz), before a compact flange mounted passively Q-switched microlaser system could be achieved.

Timing jitter is the other key parameter in many applications such as terahertz pulse generation from a passively Q-switched laser [5]. The origin of timing jitter comes from spontaneous emission fluctuations during pulse generation. Although timing jitter around microsecond (μs) level from a traditional passively Q-switched system is suitable for engine ignition [6], it encounters challenges from other applications such as Laser Ultrasonic Visualizing Inspection (LUVI) and Laser Ignition Breakdown Spectroscopy (LIBS). Several methods were used to reduce timing jitter. The first method is Cr⁴⁺:YAG bleaching, where additional diode laser source is used to modulate the saturation of absorber crystal and thus reduce pulse fluctuations [7,8]. Cole et al. could suppress standard deviation of timing jitter from 241 ns to 20 ns in passively Q-switched Nd³⁺:YAG/Cr⁴⁺:YAG laser [8]. Although this value is small, additional modulator increases the system size and it is no longer compact. The second approach to reduce timing jitter is hybrid active/passive Q-switch [9,10]. In this case active modulator is additionally inserted into the cavity. Although by this approach the timing jitter could be reduced down to 65 ps [9], the achieved peak power was not enough for our applications due to increased cavity length and longer pulse duration. The third approach to reduce timing jitter requires modulation of pump power. Khurgin et al. used composite pump pulse, which was made out of long low-intensity pulse followed by short high-intensity pulse [11]. By using this method it was possible to reduce timing jitter one order of magnitude, down to 500 ns.

In this article, Intense and Fast Pulse Pump (IFPP) technique is proposed to solve above mentioned issues of thermal effects and timing jitter, which is mainly based on higher pumping efficiency at higher pulse pump power density. Firstly, pulsed pumping, which could also be called as rapid pumping, could alleviate thermal effects in comparison to the continuous wave (cw) pumping. Secondly, higher pumping efficiency could be achieved through reduced pump pulse duration at higher pump peak power since the required pump energy remains at the same level for a fixed passively Q-switch laser cavity. With intense pulse pumped microlaser, one could make compact lens-less UV laser system with strikingly new capabilities, such as giant pulse UV laser over megawatt (MW) peak power and reduced timing jitter σ below 100 ns. This capability also leads to the design of a giant-pulse, flange-mounted UV microlaser operating at 1 kHz repetition rate [12].

2. Theory model

The purpose of this theoretical simulation is to explain the phenomenon of thermal load and timing jitter at different repetition rates and pump power densities. Higher repetition rate (1kHz) and low timing jitter (below 100 ns) is required in many applications including mass spectroscopy imaging (MSI). Bhandari et al. reported a microlaser operating at repetition rate 500 Hz [13]. With increased repetition rates, thermal management becomes critical in high power and high-energy laser system. Hence, IFPP originating from pumping efficiency η_p is introduced in this work as a solution for reducing thermal load and timing jitter. Insights into η_p date back to the 1980s which could be gained by rate equation [14]. The equation for pumping efficiency at different normalized pump durations is given in Eq. (15) as illustrated in Appendix A. Together with the understanding of the mechanism of a passively Q-switched laser, pump power density efficiency η_D could be defined as a function of pump power density D as shown in Eq. (1).

D_τf is the pump power density used for laser operation when the pulse duration is at the value of fluorescence lifetime τ_f. The pump energy to start Q-switch will decrease if keeping pump beam size the same but increasing the peak power (diode laser current goes up). Accordingly, the pump pulse duration T_p is reduced. This could be expressed as T_p·D<τ_f⋅D_τ for T_p<τ_f. Hence the mathematical deduction of T_p/ τ_f<D_τf/D could be obtained. From the combined considerations of Eq. (1), Eq. (15) and Fig. 7, η_p>η_D could then be deduced for T_p<τ_f. In other words, shortened pump pulse duration could result in higher pumping efficiency at higher pulsed pump intensity.

The thermal load as a function of pump power density P_D could then be derived in Eq. (2) following the definition of thermal load which is determined by fractional thermal load η_h as discussed in Appendix B. Here, A = (P_in·T_p·f)·η_abs and B = η_q·η_r·λ_p/λ_f, where P_in is incident pump peak power, f is pump repetition rate, η_abs is absorption efficiency, η_q is pump quantum efficiency, η_r is radiative quantum efficiency, λ_p is pump wavelength (808 nm) and λ_f is average fluorescence wavelength (1038 nm). Thus P_h<P_D could be obtained for T_p<τ_f.

Figure 1 simply shows that the thermal load is reduced at higher pump power density following Eq. (2). Thermal load could be alleviated at higher pump power densities D through enhanced pumping efficiency when the cavity parameters remain the same. Bhandari et al. reported 266 nm UV microlaser with peak power 0.5 MW at 1 kHz by suppressing the thermal birefringence from a diode laser with peak power 100 W [15]. Based on above analysis, it is operable to further suppress thermal load through IFPP approach.

 

Fig. 1 Simple diagram of thermal load P_h and jitter δ as a function of pump power density.

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Another important topic in laser research is the suppression of timing jitter for applications such as MSI, LIBS or LUVI, where synchronization of a laser pulse with other measurement equipment is crucial. For clarification, the symbols δ and σ will be used in the article to express timing jitter, where δ is the definition of a jitter and σ is statistically measured value. The mechanism of timing jitter in a passively Q-switched laser is illustrated in Appendix C. Timing jitter δt_s as a function of pump power density is indicated in Eq. (3a). and Eq. (17). α is the ratio of pump population when Q-switch starts as shown in Eq. (3b) and Eq. (16). N₀ stands for total atom number and N_s is inversion population when Q-switch starts. Figure 1 also shows a simple diagram of timing jitter as a function of pump power density. Timing jitter δt_s is inversely proportional to the pump power density. To reach the same N_s in a pulse pumped passively Q-switched laser, shorter pump pulse durations are required at higher pump power densities.

Since current work is aiming for Mass Spectrometry Imaging (MSI), we have to suppress the standard deviation of timing jitter from several µs of traditional passively Q-switched laser to less than 100 ns. Based on above analysis, we have to increase the pump power density in order to suppress timing jitter to one order magnitude. Hence, we took IFPP technology to improve the giant pulse microlaser. The design of IFPP is to use high pump power diode laser for high repetition rate based on high radiative quantum efficiency close to unity and high pumping efficiency at shortened normalized lifetime T_p/τ_f, which could be superior to longer normalized lifetime [15] and beneficial for reducing thermal load in the gain material. This will reduce thermo-optical aberration, thermal lens effect, birefringence and timing jitter through suppressing the spontaneous emission in gain material.

Figure 2 shows the block diagram of timing jitter measurement which includes electrical jitter δt₃ from detector, analogue triggered jitter δt₄ from diode laser driver, as well as optical jitter δt₂ from microlaser, δt₁ from diode laser and δt₅ from oscilloscope. The complete system jitter is then given by . For details of timing jitter in a passively Q-switched laser, please refer to Appendix C. The timing jitter from oscilloscope is 22 ps which is three orders of magnitude less than that of master jitter, hence the former could be ignored. The rise time of the detector is less than 25 ps and the timing jitter δt₃ also could be omitted. It makes sense that the system jitter mainly comes from diode laser δt₁, diode laser driver δt₄ and intrinsic microlaser δt₂. In other words, the intrinsic microlaser jitter could be determined by . It is worthwhile to mention here that the jitter from diode laser driver δt₄ is triggered by computer control process. The influence of electric circuit on timing jitter from diode laser driver requires further improvement.

 

Fig. 2 Block diagram of timing jitter measurement.

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3. Experiment

One of the key issues of the IFPP technique is to select a suitable pump power for different pump beam size in Nd³⁺:YAG crystal. The saturated pump power P_sat = I_sat × Area was calculated from saturation intensity I_sat = hν/(σ_abs·τ_f), where h is a Plank constant, ν is pump laser frequency and σ_abs is absorption cross-section of a gain crystal. Accordingly, fiber coupled quasi-cw diode laser with maximum pump peak power 400 W and pulse duration 200 μs at repetition rate 1 kHz was used in this work.

Figure 3 shows the configuration of giant pulse 1064 nm microlaser at 1 kHz. A [100]-cut YAG / 1.1 at.% Nd³⁺:YAG composite crystal (HG Optronics Inc., China) with aperture 3 × 3 mm² and thickness 5 mm was used as gain material. The input mirror was coated on YAG surface with high transmission (HT) at 808 nm and high reflection (HR) at 1064 nm. Thermal diffusion bonding technique was used for the uncoated interfaces bonding between YAG and Nd³⁺:YAG. Undoped YAG crystal worked as a heat sink. The output surface of Nd³⁺:YAG crystal was coated with HR at 808 nm and HT at 1064 nm. The saturable absorber was a [110]-cut Cr⁴⁺:YAG crystal (Scientific Materials Co., USA) with initial transmission T₀ = 30%. The depolarization loss in intense pump solid-state laser could be reduced by using a [100]-cut Cr⁴⁺:YAG crystal originating from polarization effects of distorted tetrahedral site of Cr⁴⁺ ions in cubic garnet [4,16]. The output surface of Cr⁴⁺:YAG crystal was coated with partial reflection (PR) 50% at 1064 nm working as an output coupler (OC) mirror.

 

Fig. 3 Scheme of intense pulse pumped UV microlaser. A [100]-cut YAG / 1.1 at.% Nd³⁺:YAG composite crystal was used as a gain material with aperture 3 × 3 mm² and thickness 5 mm (1mm for YAG and 4mm for Nd³⁺:YAG). [100]-cut Cr⁴⁺:YAG crystal with initial transmission T₀ = 30% was used as saturable absorber. The total module length was 63 mm.

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The Q-switched pulse was measured with detector ET 3500 (bandwidth over 15 GHz, Electro-Optics Technology, Inc.) and oscilloscope (DPO71604C, 16 GHz, Tektronix). For timing jitter measurements, the diode laser current signal was used as a reference to start Q-switch pulse measurements. Histogram data was collected for approximately 15 s, from which standard deviation of timing jitter was defined.

After the 1064 nm microlaser was fixed, LBO crystal (TYPE I, θ = 90°, ϕ = 11.4°, 5 × 5 × 15 mm³, CASTECH China) was used for generating 532 nm laser. β-BBO crystal (TYPE I, 5 × 5 × 6 mm³, CASTECH China) was used for fourth harmonic generation (FHG) of 266 nm laser. The effects of different pump beam size on nonlinear crystals will be explained in the next section.

4. Results and discussion

As shown in Appendix B, thermal load P_h = (P_in⋅T_p⋅f)⋅η_abs⋅η_h is a function of absorption efficiency η_abs and fractional thermal load η_h = (1-η_q⋅η_r⋅η_p⋅λ_p/λ_f), which is determined by pump quantum efficiency η_q, radiative quantum efficiency η_r and pumping efficiency η_p. The coating on Nd³⁺:YAG crystal was designed to reflect the pump beam back into the gain crystal and to increase the absorption efficiency η_abs = 1-exp(−2α_a·l), where α_a is absorption coefficient and l is the length of gain crystal. The measured absorption efficiency η_abs was 0.98. The radiative quantum efficiency η_r was 0.764 for 1.1 at.% Nd³⁺:YAG crystal [17].

According to Eq. (15) as illustrated in Appendix A, thermal load is a function of η_p. Meanwhile, η_p is the function of normalized pulse duration T_p/τ_f. To better understand thermal load, the relationship between P_h and T_p/τ_f could be deduced in the equation P_h = α × {1-β × [1-exp(-x)]/x}/{[1-exp(-x)]/x} and depicted in Fig. 4, where x≡T_p/τ_f is defined as horizontal axis. The absorbed factor α = 17.86 is determined by the absorbed pump power at fixed frequency and varies at different pump conditions. By combining Eq. (2) and Eq. (13) (Appendix A), the thermal factor β = 0.56 could be determined by pump rate W_pτ_f and varies at different fractional thermal load. Simulation on thermal load was carried out by heat analysis program after constructing geometry model of the cavity elements. The dependence of P_h at different pump power densities was then obtained. Thermal load was 12.12 W when the normalized pulse duration T_p/τ_f was 0.6 and the pump peak power was 170 W. By increasing the pump peak power to 400 W, T_p/τ_f was reduced to 0.2 and the thermal load P_h became 8.8 W. Accordingly, thermal load could be reduced by 27% when changing peak power of diode laser from 170 W to 400W at 1 kHz.

 

Fig. 4 Thermal load P_h at different normalized pulse duration T_p/τ_f. Thermal load was 12.12 W when the normalized pulse duration T_p/τ_f was 0.6 and the pump peak power was 170 W. By increasing the pump peak power to 400 W, T_p/τ_f was reduced to 0.2 and the thermal load P_h became 8.8 W. Accordingly, P_h was reduced by 27% when changing diode laser power from 170 W to 400W at 1 kHz.

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Pumping efficiency η_p was measured at different pump peak powers and corresponding pump pulse durations to keep the output pump energy at the same level in the 1064 nm passively Q-switched microlaser at 1 kHz. The pump intensity was kept around 200 W/cm² when estimating the temperature on the gain material. η_p was enhanced from 58% to 91% by increasing pump peak power from 100 W to 400 W. Higher pump power densities could result in shortened T_p leading to a higher η_p and lower thermal loads. For details, please refer to Appendix A.

Figure 5 depicts optical efficiency ratio as a function of pump power density ratio D/D_τf following the equation η/η_τf = 1.95 × [1-exp(−1/x)]/(1/x), where x = D/D_τf. η is optical pump efficiency and η_τf is the optical pump efficiency when the pump pulse duration is set at the lifetime value. The value below D/D_τf<1 was simulated from curves of pump energy and jitter. It could be obtained that η/η_τf at different pulse duration ratio T_p/τ_f follows the equation η/η_τf = γ × [1-exp(-T_p/τ_f)]/(T_p/τ_f) = γ × η_p. The efficiency factor γ = 1.581 was obtained at the condition of T_p = τ_f. The optical efficiency got 50% rising when pump power was increased from 170 W to 400 W.

 

Fig. 5 Optical efficiency ratio η/η_τf and timing jitter ratio δ/δ_τf as a function of pump power density ratio D/D_τf at 1 kHz. The optical efficiency got 50% rising when pump power increased by 2.3 times. By remaining the output energy at the same level, the timing jitter could be reduced by 6 times when increasing the pump power density up to 5 times higher than that required for a laser operation at pulse duration value of lifetime.

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Figure 5 also shows the timing jitter ratio δ/δ_τf at different pump density ratio D/D_τf in passively Q-switched 1064 nm microlaser following the equation δ/δ_τf = ζ/x. Here, the jitter factor ζ = 0.85 was determined both by the pump power density and the ratio of pump power to the threshold pump power as illustrated in Eq. (3a) and Eq. (3b). The threshold was influenced by the cavity alignment. The evolution of horizontal delay time over the cause of a single pump pulse is derived. When the optical signal appeared closer to the start of electrical signal from the pump current signal, the measured timing jitter became smaller which could be attributed to less spontaneous emission noise and better spatial modes match inside laser cavity. In this work, the IFPP technique is successfully applied in timing jitter reduction by employing high intense pulsed pump power. When increasing the pump power density up to 5 times higher than that required for the operation at lifetime value, the timing jitter could be reduced to 6 times lower. In the kHz system, standard deviation of timing jitter was 37 ns at pump power 363 W and 209 ns at pump power 73 W, respectively. It fits well with the prediction of IFPP technology that higher pump power density is beneficial for reducing timing jitter. The effective pumping rate of the upper laser level is related to electrical power input, pumping efficiency, operation frequency and laser mode volume. Here, pumping efficiency is affected both by diode laser efficiency and optical coupling efficiency. For more details, please refer to rate equation as presented in Appendix A and the block diagram of timing jitter measurement as shown in Fig. 2. The jitter in microlaser cavity is actually lower than 37 ns since the measured jitter is a sum of optical jitter and electrical jitter as explained in theory model part.

Finally, the fundamental laser output at 1 kHz was obtained with peak power 2.2 MW, pulse energy 1.1 mJ and pulse width 484 ps. The measured standard deviation of timing jitter was 37 ns at the condition of η_p = 94%. The M² factor for the fundamental wavelength was 5.6 and 4.6 in horizontal and vertical directions, respectively.

The fundamental laser was directly guided into the LBO crystal for second harmonic generation (SHG) of green laser at 1 kHz. No focus lens was used. The SHG green laser was obtained with peak power 1.45 MW, pulse energy 500 μJ and pulse width 344 ps. The M² factor at 532 nm was 2.2 and 2.6 in horizontal and vertical directions, respectively. The conversion efficiency from fundamental laser to green laser was 64%.

The fourth harmonic generation (FHG) UV laser at 1 kHz was achieved with peak power 1.01 MW, pulse energy 221 μJ and pulse width 218 ps. The conversion efficiency from fundamental laser to UV laser was 44%. The green beam was directly guided into the β-BBO crystal for generating UV laser without using focus lens. The M² factor of UV beam was 2.6 and 2.8 in horizontal and vertical directions, respectively. Four dichroic mirrors with average reflection of 88.56% at 266 nm were used to separate the 266 nm beam from mixed 1064 nm / 532 nm / 266 nm beam. The stability for calibrated energy of 227 μJ was tested over one hour with coefficient of variation 2.4%. The stability over 10 hours was done for calibrated pulse energy of 146 μJ with coefficient of variation 1.1%.

By increasing pump beam diameter, output energy 2.5 mJ, pulse duration 567 ps and peak power 4.4 MW at 1 kHz could be achieved in 1064 nm passively Q-switched laser. But multi-modes also appear due to the thermal effects. The gain crystal and saturable absorber also suffered from coating damage which might be caused by a sudden burst at high peak power as a result of multi-modes competition. As one example of nonlinear conversion, the green laser was obtained with pulse energy 659 µJ, pulse duration 385 ps and peak power 1.7 MW. Multi-modes behavior resulted in low SHG conversion efficiency of 38.4%. Green laser with same output energy level could be both obtained either by using a lens with focus length 50 mm before LBO crystal or without it but placing the LBO crystal close (within 10 mm) to the output coupler. The UV 266 nm laser was then obtained with energy 229 µJ, pulse duration 215 ps and peak power 1.06 MW by using lens with focus length of 38.1 mm. After operation for some period (e.g. one month), LBO crystal could suffer from damages either from coating or internal crystal. β-BBO crystal would also suffer from coating and fracture damages in the frequency harmonic generation setup using focus lens.

5. Conclusion

By using Intense and Fast Pulse Pump (IFPP) technique proposed in this work, over megawatt lens-free UV microlaser was realized in kHz system. IFPP is effective in suppressing thermal load and timing jitter through enhancement of pumping efficiency. Thermal load was reduced by 27% when increasing the peak power of diode laser from 170 W to 400W at 1 kHz. Timing jitter was reduced to 37 ns at pump power of 363 W at frequency 1 kHz with Q-switched output energy around 1 mJ. Finally, IFPP technique was used for the fourth harmonic generation of UV (266 nm) laser at 1 kHz and over 1 MW peak power was achieved with calibrated pulse energy 221 μJ, pulse width 218 ps from lens-free passively Q-switched microlaser at pumping efficiency 94%.

Appendix A Pumping efficiency in pulse pumped laser system

By starting with rate equations that describing photon density and population inversion in the steady-state case of a laser cavity, one could model the pumping efficiency in a pulse pumped laser system.

Figure 6 gives a simplified model for laser pumping in Nd³⁺:YAG gain crystal. Atoms are pumped from the ground level ⁴I_9/2 into upper level ⁴F_5/2 and relaxed down into upper laser level ⁴F_3/2, from where the atoms make stimulated laser transitions down to the lower laser level ⁴I_11/2 and then back to the ground level. W_p is pumping transition probability (probability per atom per second). W_pN₀ is the pumping rate (atoms per second, per unit volume) being lifted up out of the ground level, which is proportional to the pump light intensity in an optically pumped laser. The number of atoms per second reaching the upper laser level is then given by an effective pumping rate R_p=η_q·W_p·N_o, where η_q represents the quantum efficiency for pump excitation into this upper laser. The rate equation for upper laser level ⁴F_3/2 with population density N₂ is given in Eq. (4) and Eq. (5). τ_f is lifetime of the upper laser level ⁴F_3/2.

 

Fig. 6 Simplified model for laser pumping.

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(4)$$\frac{d{N}_2}{dt}=W_P\cdot \left(N_0-N_2\right)-\frac{N_2}{{\tau }_f}$$

(5)$$\frac{d{N}_2}{dt}+\left(\frac{1}{{\tau }_f}+W_P\right)\cdot N_2=W_P\cdot N_0$$

It is necessary to define the following parameters.

(6)$$\frac{1}{{\tau }_f}+W_P\equiv \frac{1}{T}=P\left(x\right)$$

(7)$$W_P\cdot N_0=W\cdot N=Q\left(x\right)$$

(8)$$\frac{d{N}_2}{dt}+\frac{1}{T}\cdot N_2=W\cdot N$$

By substituting Eq. (6), Eq. (7) and Eq. (8) into Eq. (5), we obtain Eq. (9) and Eq. (10). The steady-state population on upper laser level is then given in Eq. (11). When the maximum upper-level population N₂=0, constant C is obtained with C=-WNT. The steady-state solution of the maximum upper-level population at the end of pump pulse is then given by Eq. (12).

(9)$$\frac{dy}{dx}+P\left(x\right)\cdot y=Q\left(x\right)$$

(10)$$N_2=e^{-\frac{t}{T}}\cdot \left(W\cdot N\cdot T\cdot e^{\frac{t}{T}}+C\right)$$

(11)$$y=e^{-{\displaystyle \int P\left(x\right)}dx}\left[{\displaystyle \int Q\left(x\right)e^{{\displaystyle \int P\left(x\right)dx}}dx}+C\right]$$

(12)$$N_2=W\cdot N\cdot T\cdot \left(1-e^{-\frac{t}{T}}\right)=\frac{W_P\cdot {\tau }_f\cdot N_0}{W_P\cdot {\tau }_f+1}\cdot \left[1-e^{-\left(W_P\cdot {\tau }_f+1\right)\cdot \frac{t}{{\tau }_f}}\right]$$

In a pulse pumped Q-switching laser, it is of little use to continue the pump pulse with longer than about two times of upper-level lifetimes, since the upper-level population no longer increases at further pumping beyond that point. The pumping efficiency η_p is a ratio of the maximum number of atoms stored in the upper level at the end of the pumping pulse, to the total number of pump atoms lifted up during the pump pulse. The total number of atoms lifted up during the pump pulse is W_pN₀T_p. By assuming t=T_p, the pumping efficiency η_p could be obtained in Eq. (13).

(13)$${\eta }_P=\frac{\frac{W_P\cdot {\tau }_f\cdot N_0}{W_P\cdot {\tau }_f+1}\cdot \left[1-e^{-\left(W_P\cdot {\tau }_f+1\right)\cdot \frac{T_P}{{\tau }_f}}\right]}{W_P\cdot T_P\cdot N_0}=\frac{1}{W_P\cdot {\tau }_f+1}\cdot \frac{1-e^{-\left(W_P\cdot {\tau }_f+1\right)\cdot \frac{T_P}{{\tau }_f}}}{T_P/{\tau }_f}$$

In the case of 1.1 at.% Nd³⁺:YAG gain material, τ_f is 219 μs. The absorption coefficient α_a was measured at 5 cm⁻¹ in 1.1 at. % Nd³⁺:YAG crystal. The coating on Nd³⁺:YAG crystal was designed for double pass through the crystal to enhance the absorption efficiency. N_o is the total atom number 1.527×10²⁰ cm⁻³ in 1.1 at. % Nd³⁺:YAG. Absorption cross section σ_abs is 3.27 × 10⁻²⁰ cm² from σ_abs=α_a/N₀. Planck constant h is 6.63 × 10⁻³⁴ m²·kg·s⁻¹. Light speed c is 3 × 10⁸ m·s⁻¹. Finally the population of the upper laser level in Nd³⁺:YAG crystal is given by Eq. (14). The pumping efficiency is given in Eq. (15). Figure 7 shows the pumping efficiency at different normalized pump duration.

 

Fig. 7 Pumping efficiency η_p at different normalized pump duration T_p/τ_f. When the value of pulse duration T_p is 1.2 times higher than that of lifetime τ_f, pumping efficiency is 58%. By shortening the pump pulse duration value to T_p/τ_f = 0.12, the pumping efficiency reach 94%.

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(14)$$N_2\equiv W_P\cdot {\tau }_f\cdot N_0\cdot \left[1-\exp \left(-T_P/{\tau }_f\right)\right]$$

(15)$${\eta }_P\equiv \frac{1-\exp \left(-T_P/{\tau }_f\right)}{T_P/{\tau }_f}$$

Appendix B Fractional thermal load in pulse pumped laser system

The thermal load in Nd³⁺:YAG crystal was performed under condition of non-lasing extraction and measured by Infrared Camera together with simulation from heat analysis program (ANSYS Workbench finite element analysis software). The rectangular Nd³⁺:YAG sample was pumped by pulse laser.

The fractional thermal load η_h was written as η_h=1-η_q⋅[(1-η_l) ⋅η_r⋅ (λ_p/λ_f)+η_l⋅ (λ_p/λ_f)]. Here pump quantum efficiency η_q is estimated at 0.9 due to concentration quenching, η_r is radiative quantum efficiency, λ_p is pump wavelength, λ_f is average fluorescence wavelength, η_l is the fraction of ions excited to the upper metastable level that are extracted by stimulated emission. In case of non-lasing extraction, η_l equals to 0. Hence η_h could be written as η_h=1-η_q⋅η_r⋅ (λ_p/λ_f). By measuring the temperature on Nd³⁺:YAG during non-lasing condition and simulating the thermal load by ANSYS Workbench finite element analysis software, a revised equation P_h=(P_in·T_p·f)·η_abs·η_h =(P_in·T_p·f)·η_abs·(1-η_q·η_r·η_p·λ_p/λ_f) is used to calculate the thermal load in pulse pump condition, where η_p is considered for η_h. Table 1 shows the thermal load in 1 kHz microlaser. η_p was enhanced from 58% to 91% and P_h was reduced by 27% when increasing pump peak power from 100 W to 400 W. The tolerance between measured and simulated temperature was within 12%.

Table 1. Thermal load P_h in 1 kHz microlaser as a function of fractional thermal load η_h by considering pumping efficiency η_p. η_p was enhanced from 76% to 90% and P_h was reduced by 27%when increasing pump peak power from 170 W to 400 W. The measured and simulated temperature tolerance was within 12%.

View Table

Appendix C Timing jitter in a Q-switched laser

Figure 8 shows the mechanism of a passively Q-switched laser operating at two different pump powers. The pump power density dependent timing jitter could be illustrated as follows. From Fig. 8(a), higher pump power intensity D requires shorter pump pulse duration t_p=t₁-t₂ to reach the same population inversion δN_s as shown in Fig. 8(b). Accordingly, delay time t=t_s,1-t₁ in gain material becomes shorter at higher pump power, where t_s stands for the starting point of Q-switch laser as given in Fig. 8(b). As shown in Fig. 8(c), timing jitter value δt_s could be limited in shorter period, while peak to peak deviation is also mitigated as indicated Figure 8(d). In unsaturated state, saturable absorber Cr⁴⁺:YAG introduces a high optical loss. The laser gain must overcome that loss before lasing could start. When the laser radiation becomes stronger, it eventually surmounts the losses. From this point on, the laser power rises rapidly until the gain is also saturated. Q-switch build-up time t_s is shown in Fig. 8(b) and Eq. (16), where τ_f is lifetime of upper laser level. W_p is pump rate. N_s is inversion population when Q switch starts. α is the ratio of pump population to those when Q-switch starts.

 

Fig. 8 The mechanism of timing jitter in a passively Q-switched laser. (a): Higher pump power intensity D requires shorter pump pulse duration t_p = t₁-t₂ to reach the same population inversion δN_s. (b): Delay time t = t_s,1-t₁ in gain material becomes shorter at higher pump power, where t_s stands for the starting point of Q-switch laser. (c): Timing jitter value δt_s could be limited in shorter period. (d): Peak to peak deviation could be mitigated in shorter period.

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(16)$$t_s={\tau }_f\cdot \ln \left(\frac{\alpha }{\alpha -1}\right)={\tau }_f\cdot \ln \left(\frac{W_P\cdot {\tau }_f\cdot N_0/N_s}{W_P\cdot {\tau }_f\cdot N_0/N_s-1}\right),\begin{array}{cc}& \alpha =W_P\cdot {\tau }_f\cdot N_0/N_s\end{array}$$

(17)$$\delta t_s=\frac{{\tau }_f}{1-\alpha }\cdot \frac{\delta \alpha }{\alpha }=\frac{{\tau }_f}{1-\alpha }\cdot \frac{\delta P}{P}=\frac{{\tau }_f}{1-\alpha }\cdot \frac{\delta D}{D}=W_P\cdot {\tau }_f\cdot \frac{\delta D}{D}$$

Timing jitter δt_s is illustrated in Fig. 8(d) and obtained by differentiating Eq. (17). P is pump power and D is pump power density. From Eq. (17), a small change in pump power density could cause an obvious variation in the timing jitter σt_s. As indicated in Fig. 8, shorter rise time t_s,1 could give faster slope speed to reach the inversion population and result in faster pump with reduced timing jitter δt_s,1.

Funding

Japan Science and Technical Agency (JST) (SENTAN); ImPACT Program of Council for Science, Technology and Innovation (Cabinet Office, Government of Japan).

Acknowledgments

The author would like to thank Dr. R. Bhandari from Institute for Molecular Science, Dr. Y. Furukawa and S. Makio from Oxide Co. for their skilled technical assistance.

References and links

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