Exploring the influence of pump beam quality on designing millijoule diode-end-pumped passively Q-switched lasers

Chun-Yu Cho; Xin-Cheng Lin; Je-Hau Shr

doi:10.1364/OE.462323

1. Introduction

Q-switched lasers have been widely utilized in industry, medical applications, and academic researches. Recently, the application of gas detecting that utilize nanosecond high energy mid-infrared wavelength tunable lasers has become an important topic for environmental protection [1,2]. Consequently, using high quality and high pulse energy passively Q-switched (PQS) lasers for converting near infrared laser via optical parametric oscillation (OPO) is the most efficient method [3–5], especially for the consideration of cost limitation and compactness. To generate high pulse energy PQS laser while maintaining high beam quality, using master oscillation power amplifier (MOPA) with high quality seed laser is a common method [4–9]. However, in comparison with seed laser oscillator, MOPA will suffer from the low extraction efficiency and beam quality degradation after few stages of amplifiers [7–9]. Since some of the requirements of nanosecond output energy for OPO conversion is in a range from 0.5 to 2 mJ [2], it is more efficient to obtain such an output from a single-stage energy-scaled seed PQS laser instead of using MOPA. As a result, it is important to explore design criteria of the millijoule end-pumped PQS laser.

The theoretical analyses for PQS laser output parameters, such as pulse energy, peak power, pulse width, and repetition rate have been thoroughly investigated [10–16]. Plenty of practical models for different laser designs have been proposed and verified via complete experiments. Recently, the influence of transverse pump to cavity mode overlapping has been deeply investigated for more precisely predicting of PQS output parameters [14–16]. Even for different design parameters of laser resonator, the output parameters variation among different models were quite small, which will not affect the prediction of output energy design. However, it was found that the threshold pump powers of the PQS laser were nearly not discussed in these theoretical models. Plenty of these works were featured in the output parameters and verified the laser properties by using a pump diode with enough pump power to eliminate the effect of threshold power restriction. Moreover, some of these researches have relatively low pulse energy and hence will reduce the influence of pump power requirement. Nevertheless, when designing a millijoule PQS laser, the variation of threshold pump power will significantly affect the selection of pump diode. Consequently, it is highly necessary for the practically industrial application to precisely predict the required pump power level.

In this work, we explored the precise theoretical model for millijoule PQS laser, especially for threshold pump power. It is obvious that the spatial dependence of pump to cavity mode volume must be taken into consideration for better prediction. In Ref. [14], an analytical formula of threshold pump power with transversal dependence was derived along with other output parameters. In that research, both cavity and pump mode size along longitudinal direction were assumed to be invariant inside the gain medium. However, this assumption only valid when the gain medium was shorter than the Rayleigh range of the refocusing pump beam. By using this threshold pump power model and the parameters given in previous works, we found that the theoretical pump powers were underestimated in comparison with experimental results [13,14]. The reason that it did not influence the experiment might due to the fact that the power difference was nearly 1 to 2 W for such low energy lasers. As a result, not only the transverse overlapping but also the longitudinal spatial dependence must be considered [17]. The divergence of pump profile should be included, especially when operating in tight focusing region, since the beam quality of pump diodes were usually worse. Here, we derived theoretical model of PQS laser with M² pump beam quality factor. The influence of M² were first analyzed with various design parameters. The threshold pump powers revealed up to 5 times difference when the M² factor was up to 80, which resulted in >10 W power difference at high pump power level. The output pulse energy with longitudinal mode size overlapping were also discussed. It was found that the difference was less than 15%, which revealed the fact that the transverse dependent PQS models proposed in past researches were still good enough for other output parameters especially for operating at low energy level [13,14].

The theoretical model was verified experimentally by the Nd:YAG/Cr⁴⁺:YAG PQS laser. We used two pump diodes with M² factors of 60 and 80 combined with four different output coupler reflectivity in the experiment. The experimental results showed excellent agreement with the proposed model. We also compared the modified model with top-hat distribution of the pump diode. The result of output pulse energy fits well with the experimental results. However, the theoretical threshold pump power using top-hat pump distribution were still much lower than the experimental results. This comparison indicated the importance of considering the influence of pump beam quality for threshold pump power. After verification of the theory, we further discussed the cavity design and the influence of thermal lensing for our laser resonator. We found that it was necessary to match the cavity mode size to be slightly larger than pump mode size for avoiding temporal satellite pulse and pattern distortion. Finally, we scaled the PQS laser output energy to 1.07 mJ based on our model. The pulse width was 6 ns and the peak power was 178 kW. The results demonstrated in this work can be practically useful for designing a millijoule PQS laser.

2. Theoretical analysis

The PQS model with both transversely and longitudinally spatial dependence must be derived for further investigation. It can be derived from PQS coupled rate equations [10–16]. After considering the spatial dependence, the pump threshold, P_th, and output pulse energy, E, can be given as the following [14–16]:

(1)$${P_{th}} = \frac{{h{\nu _p}}}{{2\sigma {\tau _f}}}\frac{{{V_{eff}}}}{{{l_{cav}}}}\left[ {\ln \left( {\frac{1}{{T_0^2}}} \right) + \ln \left( {\frac{1}{R}} \right) + L} \right]\textrm{,}$$

(2)$$E = \frac{{h{\nu _l}}}{{2\sigma }}\frac{{S{V_{eff}}}}{{{l_{cav}}}}({1 - R} )\left\{ {1.55\left( {\frac{{{\alpha^2} - 1}}{{{\alpha^2}}}} \right){{\left[ {\chi - \frac{\alpha }{{\alpha - 1}}} \right]}^{0.85}}} \right\}\textrm{,}$$

where h was the Plank constant, ν_p was the pump frequency, σ was the emission cross section of the gain medium, τ_f was the upper-level life time of the gain medium, l_cav was the cavity length, T₀ was the initial transmission of the saturable absorber, R was the output coupler reflectivity, L was the round-trip cavity loss, ν_l was the laser emission frequency. For the output pulse energy in Eq. (2), we utilized (1-R) instead of ln(1/R) for more precise modeling of low output reflectivity design [15]. A fitting function was utilized here for easier calculation of the output energy. The detail discussion of this fitting function was revealed in Ref. [14] to [16]. χ and α were given as the following:

(3)$$\chi = {{\left[ {\ln \left( {\frac{1}{{T_0^2}}} \right) + \ln \left( {\frac{1}{R}} \right) + L} \right]} / {\left[ {\beta \ln \left( {\frac{1}{{T_0^2}}} \right) + \ln \left( {\frac{1}{R}} \right) + L} \right]}}\textrm{,}$$

(4)$$\alpha = {{A{\sigma _{gs}}} / {{A_s}{\sigma _{es}}}}\textrm{,}$$

(5)$$\beta = {{{\sigma _{es}}} / {{\sigma _{gs}}}}\textrm{,}$$

where A was the mode area on gain medium, A_s was the mode area on the saturable absorber, σ_gs and σ_es were the ground state and excited state emission cross sections of the saturable absorber, respectively. Notice that the final loss of the saturable absorber, which was related to (1-β)ln(1/T₀²) [11], was already included in the fitting function. V_eff and S were the spatial overlapping factors that can be given by:

(6)$${V_{eff}} = {\left[ {\int\!\!\!\int\!\!\!\int {\phi ({x,y,z} ){r_p}({x,y,z} )dv} } \right]^{ - 1}}\textrm{,}$$

(7)$$S = \frac{{{{\left( {\int\!\!\!\int\!\!\!\int {\phi ({x,y,z} ){r_p}({x,y,z} )dv} } \right)}^2}}}{{\int\!\!\!\int\!\!\!\int {{\phi ^2}({x,y,z} ){r_p}({x,y,z} )dv} }}\textrm{,}$$

where ϕ was the normalized laser intracavity photon density distribution and r_p was the normalized pump density distribution. By considering pump and mode size transverse profiles to be both Gaussian distributions, ϕ and r_p can be derived as:

(8)$$\phi ({x,y,z} )= \frac{2}{{\pi \omega _l^2(z ){l_{cav}}}}\textrm{exp} \left[ { - 2\frac{{{x^2} + {y^2}}}{{\omega_l^2(z )}}} \right]\textrm{,}$$

(9)$${r_p}({x,y,z} )= \frac{{2\kappa }}{{\pi \omega _p^2(z )[{1 - \textrm{exp} ({ - \kappa {l_{med}}} )} ]}}\textrm{exp} \left[ { - 2\frac{{{x^2} + {y^2}}}{{\omega_p^2(z )}} - \kappa z} \right]\textrm{,}$$

where κ and l_med were the absorption coefficient and the length of the gain medium. ω_l(z) and ω_p(z) were the cavity mode size and pump mode size with respect to the longitudinal axis. Since the gain medium length was relatively short, the divergence of the cavity mode size can be ignored. Hence, ω_l(z) can be considered to be invariant:

(10)$${\omega _l}(z )\approx {\omega _{l0}}\textrm{.}$$

On the other hand, according to our hypothesis, the pump mode size cannot be treated as constant since it’s beam quality was relatively worse. Hence, the pump mode size should be given as the following:

(11)$${\omega _p}(z )= {\omega _{p0}} + \frac{{{M^2}{\lambda _p}}}{{\pi {\omega _{p0}}}}|{z - {z_0}} |\textrm{,}$$

where ω_p0 was the refocusing beam waist, M² was the pump beam quality, λ_p was the pump wavelength, and z₀ was the location of the pump beam waist. By substituting Eq. (8) to (11) into Eq. (6) and (7), we found that both V_eff and S can be derived with a same integral equation [17]:

(12)$${V_{eff}} = \frac{1}{{F({{M^2},{\omega_{l0}},{\omega_{p0}},{z_0}} )}}\frac{{\pi \omega _{l0}^2}}{2}{l_{cav}}\textrm{,}$$

(13)$$S = \frac{{F{{({{M^2},{\omega_{l0}},{\omega_{p0}},{z_0}} )}^2}}}{{F\left( {{M^2},{{{\omega_{l0}}} / {\sqrt 2 }},{\omega_{p0}},{z_0}} \right)}}\textrm{,}$$

where F was shown as the following:

(14)$$F({{M^2},{\omega_{l0}},{\omega_{p0}},{z_0}} )= \frac{1}{{1 - \textrm{exp} ({ - \kappa {l_{med}}} )}}\int_0^{{l_{med}}} {\frac{{\kappa \omega _{l0}^2\textrm{exp} ({ - \kappa z} )}}{{\omega _{l0}^\textrm{2} + {{\left( {{\omega_{p0}} + \frac{{{M^2}{\lambda_p}}}{{\pi {\omega_{p0}}}}|{z - {z_0}} |} \right)}^2}}}dz} \textrm{.}$$

Here, F was a factor with a value from 0 to 1 that can determine the influence of spatial dependence on both longitudinal and transverse directions. In Ref. [17], the optimal pump beam waist location, z₀, for the lowest pump threshold at continuous-wave operation was thoroughly discussed. By letting dF/dz approaching to zero and including the fact that the pump absorption efficiency approaching to 100%, that is l_med >> κ, F with optimized z₀ can be fitted to be [17]:

(15)$$F({{M^2},{\omega_{l0}},{\omega_{p0}}} )= {{\omega _{l0}^2} / {\left\{ {\omega_{l0}^2 + {{\left[ {{\omega_{p0}} + \frac{{{M^2}{\lambda_p}}}{{\kappa \pi {\omega_{p0}}}}B({{M^2},{\omega_{l0}}} )} \right]}^2}} \right\}}}\textrm{,}$$

where

(16)$$B({{M^2},{\omega_{l0}}} )= {\left[ {{{\left( {\frac{{{M^2}{\lambda_p}}}{{\kappa \pi }}} \right)}^5} + 0.23\omega_{l0}^{\textrm{0}\textrm{.2}}} \right]^{{{{{\left( {\frac{{{M^2}{\lambda_p}}}{{\kappa \pi }}} \right)}^{0.25}}} / {({{\omega_{l0}}\textrm{ + 0}\textrm{.72}} )}}}}\textrm{.}$$

For the experiment of PQS laser, this step indicated the free running operation must be optimized first and the pump beam waist location cannot be adjusted during the Q-switching operation. This approach will directly affect our experiment result due to the fact that the PQS output energy will be different as long as the pump beam waist was adjusted. By using Eq. (15), the PQS threshold pump power and output pulse energy with both transversely and longitudinally spatial dependence can be obtained as the following:

(17)$${P_{th}}({{M^2}} )= \frac{{h{\nu _p}}}{{2\sigma {\tau _f}}}\frac{1}{{F({{M^2},{\omega_{l0}},{\omega_{p0}}} )}}\frac{{\pi \omega _{l0}^2}}{2}\left[ {\ln \left( {\frac{1}{{T_0^2}}} \right) + \ln \left( {\frac{1}{R}} \right) + L} \right]\textrm{,}$$

(18)$$E({{M^2}} )= \frac{{h{\nu _l}}}{{2\sigma }}\frac{{F({{M^2},{\omega_{l0}},{\omega_{p0}}} )}}{{F\left( {{M^2},{{{\omega_{l0}}} / {\sqrt 2 }},{\omega_{p0}}} \right)}}\frac{{\pi \omega _{l0}^2}}{2}({1 - R} )\left\{ {1.55\left( {\frac{{{\alpha^2} - 1}}{{{\alpha^2}}}} \right){{\left[ {\chi - \frac{\alpha }{{\alpha - 1}}} \right]}^{0.85}}} \right\}\textrm{.}$$

From Eqs. (17) and (18), it can be found that the main difference when longitudinal dependence was taking into consideration was the pump beam quality factor, M². The common fiber-coupled laser diodes have M² factors in a range from 40 to 100. Here, if we numerically let the M² equals to 0, the pump mode size in Eq. (11) will become invariant, ω_p(z)=ω_p0. The theoretical analysis will lead to the model without longitudinal dependence:

(19)$$F({{\omega_p}(z) = {\omega_{p0}}\textrm{,}{\omega_{l0}},{\omega_{p0}},{z_{0,opt}}} )= {{\omega _{l0}^2} / {({\omega_{l0}^2 + \omega_{p0}^2} )}}\textrm{.}$$

Then, spatial dependence factors V_eff and S will become the widely known equations [14–16]:

(20)$${V_{eff}}({{\omega_p}(z) = {\omega_{p0}}} )= {{\pi ({\omega_{l0}^2 + \omega_{p0}^2} ){l_{cav}}} / \textrm{2}}\textrm{,}$$

(21)$$S({{\omega_p}(z) = {\omega_{p0}}} )= {{\omega _{l0}^2({\omega_{l0}^2 + \textrm{2}\omega_{p0}^2} )} / {{{({\omega_{l0}^2 + \omega_{p0}^2} )}^\textrm{2}}}}\textrm{.}$$

Notice that M² = 0 was not a real laser parameter. It was only utilized in this discussion to represent the assumption case that pump mode size was invariant along optical axis. Using Eqs. (20) and (21), we can obtain the threshold power and output energy for ω_p(z)=ω_p0 that is utilized in former research [14]:

(22)$${P_{th}}({{\omega_p}(z) = {\omega_{p0}}} )= \frac{{h{\nu _p}}}{{2\sigma {\tau _f}}}\frac{{\pi ({\omega_{l0}^2 + \omega_{p0}^2} )}}{2}\left[ {\ln \left( {\frac{1}{{T_0^2}}} \right) + \ln \left( {\frac{1}{R}} \right) + L} \right]\textrm{,}$$

(23)$$E({{\omega_p}(z) = {\omega_{p0}}} )= \frac{{h{\nu _l}}}{{2\sigma }}\frac{{\pi \omega _{l0}^2({\omega_{l0}^2 + \textrm{2}\omega_{p0}^2} )}}{{2({\omega_{l0}^2 + \omega_{p0}^2} )}}({1 - R} )\left\{ {1.55\left( {\frac{{{\alpha^2} - 1}}{{{\alpha^2}}}} \right){{\left[ {\chi - \frac{\alpha }{{\alpha - 1}}} \right]}^{0.85}}} \right\}.$$

After obtaining the spatially dependent PQS model, we can investigate the output difference for various pump diode beam quality in the Nd:YAG/Cr⁴⁺:YAG laser. The optical parameters utilized in the analysis were shown as the following [13,14]: τ_f = 230 µs, σ = 2.35 × 10⁻¹⁹ cm², σ_gs = 8.7 × 10⁻¹⁹ cm², σ_es = 2.2 × 10⁻¹⁹ cm², L = 0.01, and κ = 3.6 cm⁻¹. The mode area on the saturable absorber was set to be slightly smaller than the mode area on gain medium. Figure 1(a) to 1(c) illustrate threshold pump powers with respect to designed pump mode size at several cavity parameters: R = 30% & T₀= 30%, R = 50% & T₀= 50%, and R = 80% & T₀= 80%. The R and T₀ were set to be equal for the optimal output pulse energy design [11]. It is worth to mention that an optimal pump to mode size ratio for the continuous-wave operation with lowest threshold power can be obtained by calculation [17]. Normally, the cavity mode size will be slightly larger than the pump mode size for lower threshold pump power. However, larger mode size can lead to higher output pulse energy for PQS laser. Hence, we simply let cavity mode size equal to 1.2 times larger than the pump beam waist, ω_l0= 1.2ω_p0, for easier analysis in this discussion. Using Fig. 1, we can predict the designed pump mode size at different pump level. From Fig. 1(a) to 1(c), it can be observed that the threshold pump power was significantly larger especially for M²>30, which was the common pump diode beam quality. For pump mode size equaled to 100 µm, the threshold power difference became even larger, which indicated that the pump divergence was much larger at such tight focusing status. The trends for different beam qualities were the same for all three cavity designs because they all depended on the match function F. In fact, the ratio between threshold pump power without and with longitudinal dependence, P_th(ω_p(z)=ω_p0)/P_th(M²) can be derived to F(M²)/F(ω_p(z)=ω_p0) from Eq. (15), which is shown in Fig. 1(d). From Fig. 1(d), we can observe that the threshold power ratio can be less than 0.2, which indicated that the threshold pump power can be up to 5 times larger when we considered the pump beam quality. In addition, it can be seen that the actual pump power difference was only few watts for low energy design in Fig. 1(c) and become up to 10 W for high energy design in Fig. 1(a). This result verified our point of view that when energy scaling, the pump diode beam quality factor must be taken into consideration for precisely modeling the PQS threshold pump power. Our experimental verification condition in this work were denoted in Fig. 1(b), which will be further discussed in experimental results.

Fig. 1. Theoretical results of spatial dependent passively Q-switched laser threshold pump powers for different pump beam quality factors when operated at output reflectivity and saturable absorber initial transmission both equal to (a) 30%, (b) 50%, and (c) 80%. (d) The ratio of pump power without longitudinal overlapping divided by that with overlapping for different pump beam quality factors.

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Theoretical output pulse energies for three cavity designs are shown in Fig. 2(a) to 2(c). We found that the output pulse energy difference was relatively small in comparison with threshold pump power even for M ²= 80. As a result, we only illustrated the result of M²= 80. From the theoretical results, we can verify that the pump beam quality dose not really affect the output pulse energy. From the ratio of E(ω_p(z)=ω_p0)/E(M²= 80) shown in Fig. 2(d), we can see that the smallest ratio was only 0.8 and it is operated at small output energy level. This result verifies the fact that past researches did not need to consider pump beam quality factor for the calculation of output pulse energies [13,14]. The reason that the influence of pump beam quality was not significant for output energy model is due to the fact that the output energy depended on the multiple of overlapping factors SV_eff. From Eqs. (7) and (8), we can find that SV_eff will be dominant by the square of laser photon density, ϕ, on its denominator, which was related to invariant cavity mode size in the model. As a result, the pump beam quality only showed negligible effect on the output pulse energy.

Fig. 2. Theoretical results of spatial dependent passively Q-switched laser output pulse energies for different pump beam quality factors when operated at output reflectivity and saturable absorber initial transmission both equal to (a) 30%, (b) 50%, and (c) 80%. (d) The ratio of output energy without longitudinal overlapping divided by that with overlapping for different pump beam quality factors.

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Since the pump diode beam profile was normally considered as top-hat distribution for multimode fiber output, we further investigated the difference between Gaussian distribution and top-hat distribution numerically. The normalized top-hat pump distribution can be given as the following with a Heaviside step function, H [16]:

(24)$${r_p}({x,y,z} )= \frac{{2\kappa }}{{\pi \omega _p^2(z )[{1 - \textrm{exp} ({ - \kappa {l_{med}}} )} ]}}\textrm{exp} [{ - \kappa z} ]H({\omega_p^2(z )- {x^2} + {y^2}} ).$$

Figure 3 depicts the ratio of Gaussian distribution to top-hat distribution for threshold pump power and output pulse energy when pump mode size was invariant. It can be found that when operated with ω_l0= 1.2ω_p0, the ratios were both approximately 0.9. The small difference for threshold pump power indicated that the influence of pump beam quality was necessary to be considered. Since the numerical analysis increased difficulty for analysis, it will be more practical to utilize the Gaussian-fit analytical solution proposed in this work.

Fig. 3. The ratio between utilizing Gaussian and top-hat pump distributions for the threshold pump power and output pulse energy at different cavity to pump mode size ratios.

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3. Experimental results

After analyzing the PQS model with the influence of pump beam quality, we demonstrated experiment to verify the theoretical model. The experimental setup of the PQS laser is depicted in Fig. 4. Two 808-nm fiber-coupled diodes with different beam qualities were utilized for verification. LD1 has a maximum output power of 25 W and a core diameter of 200 µm. LD2 has a maximum output power of 40 W and a core diameter of 400 µm. We measured the beam quality M² factors of pump diodes by using a focusing lens and verified them to be approximately 60 and 80 for LD1 and LD2, respectively. The pump light was collimated and refocusing into the gain medium through a lens pair to form a pump mode radius of 250 µm. The ratios of focal length for the lens pair were 2.5 and 1.25 for LD1 and LD2 experiments, respectively. The pump mode size was determined to scale the output pulse energy to approximately 0.5 mJ for <20 W threshold pump power according to Fig. 2(b). The front mirror of the laser cavity was a plane-parallel mirror with AR (R < 0.1%) coating at 808 nm on the first surface and HR (R > 99.9%) coating at 1064 nm as well as HT (T > 98%) coating at 808 nm on second surface. The actual incident pump power was then measured for precise comparison after the front mirror. To verify the theoretical analysis, we used four plane-parallel output couplers with reflectivity, R, of 50%, 60%, 70%, and 80% at 1064 nm. The laser cavity lengths were set to be approximately 90 mm and 80 mm when using LD1 and LD2, respectively, so that the cavity mode size on the gain medium can be 1.2 times larger than pump beam waist. The cavity design will be discussed after demonstrating experimental results. The gain medium was a 1.1-at.-% Nd:YAG crystal with an aperture of 3 × 3 mm² and a length of 10 mm. Both end facets of the gain medium were coated with AR (R < 0.1%) coating at 808 nm and 1064 nm. A Cr⁴⁺:YAG crystal with initial transmission, T₀, of 50% was utilized as the saturable absorber. Both end surfaces of the saturable absorber were coated with AR (R < 0.1%) coating at 1064 nm. Both gain medium and saturable absorber were wrapped with indium foil and mounted into water-cooled copper holders with a cooling temperature of 25°C. The laser temporal behavior was recorded by a digital oscilloscope (Lecroy, SDA5000A, bandwidth 5GHz, sampling rate 20GSa/s) and a high-speed photodiode (Thorlabs, DET08C/M, bandwidth 5GHz). The transverse distribution of the laser was recorded by an infrared CCD camera.

Fig. 4. Experimental setup of the passively Q-switched laser.

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At first, the free running operation was demonstrated to find the pump position, z₀, by power optimization. Figure 5 illustrates output powers with respected to incident pump powers of the continuous-wave operation for R = 70%. For LD1, the threshold pump power was 3.3 W. The maximum output power was 9.7 W at a pump power of 21.7 W. The conversion efficiency was 44.7% and the slope efficiency was 52.7%. For LD2, the threshold pump power was 4.4 W. The maximum output power was 9.5 W at a pump power of 24.3 W. The conversion efficiency was 39.1% and the slope efficiency was 48.4%. The free running output indicated the fact that better beam quality can lead to better performances [17].

Fig. 5. Output powers using LD1 and LD2 versus incident pump powers for laser operated at free running with output coupler reflectivity of 70%.

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After optimizing the free running operation for both pump diodes, we simply inserted the saturable absorber and changed the output coupler for the PQS laser without adjusting pump location, front mirror, and the gain medium. To avoid serious thermal effect and for more easier measurement, we operated pump diode at quasi-continuous-wave mode with a repetition rate of 1 kHz and 30% duty. Figure 6(a) demonstrates the experimental results and the theoretical analysis of the threshold pump powers. The experimental threshold pump powers by using R = 50%, 60%, 70%, 80% were 15.1 W, 13.9 W, 12.8 W, 11.6 W for LD1 and 18.3 W, 16.5 W, 14.7 W, 14 W for LD2, respectively. The red dash line and blue solid line shown in Fig. 6(a) were theoretical threshold pump powers using Eq. (17) with M²= 60 and M²= 80, respectively. The cavity mode size utilized in the analysis were approximately 290 µm and 300 µm for LD1 and LD2, respectively, which were determined by considering the thermal lensing effect on the gain medium. The green dash-dot line was the theoretical result with pump mode size invariant calculated from Eq. (22). We can verify from Fig. 6(a) that the theoretical analyses with pump beam quality were in good agreement with the experimental results. The slight difference might come from the thermal lensing effect which resulted in slight cavity mode size difference when operated at different pump level. Without considering pump beam quality, the theoretical analysis will be underestimated. When using R = 50%, the theoretical threshold power for LD1 and LD2 were 16.1 W and 17.9 W, respectively. For using invariant pump size with Gaussian distribution, the threshold pump power was 10.9 W. The power difference can be up to 5.2 W or 32% difference for LD1. We further showed the top-hat approach with orange dot line in Fig. 6(a) by dividing Eq. (22) with a factor of 0.91 that obtained in Fig. 3. Using the top-hat distribution, the threshold pump power was 12 W, which was still much smaller than the experimental results. It can be seen that it is important to include pump beam quality for threshold pump power calculation.

Fig. 6. The comparison between experimental and theoretical analyses for (a) threshold pump power and (b) output energy, where dot are the experimental results, blue solid line and red dash line were the theoretical results with pump beam quality of 80 and 60, orange dot line and green dash-dot line are the theoretical results without pump beam quality using top-hat and Gaussian pump distribution, respectively.

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Figure 6(b) shows experimental results and the theoretical analysis of the output pulse energies. We obtain output pulse energies of 450 µJ, 436 µJ, 432 µJ, 366 µJ for LD1 and 455 µJ, 450 µJ, 428 µJ, 400 µJ for LD2 when using R = 50%, 60%, 70%, 80%, respectively. The pulse width was found to be 5 ns and will slightly increase to 5.5 ns when R increased from 50% to 80%. The corresponding peak power was in a range from 66 kW to 91 kW. The red dash line and blue solid line shown in Fig. 6(b) were the theoretical pulse energies using Eq. (18) with M²= 60 and M²= 80, respectively. There was nearly no difference between using different beam quality diodes for the output pulse energies. The experimental results showed good agreement with theoretical analysis within pump beam quality except for R = 80%. It might come from multi-transverse mode generation at such high resonance condition. The green dash-dot line was the result with pump mode size invariant calculated from Eq. (23). Although the theoretical model without pump beam quality will still be underestimated, the difference was only 11% for maximum energy. The difference might come from actual cavity mode size difference caused by the thermal lensing effect. Moreover, the PQS laser will normally operate at a pump power slightly above threshold for stable output. It will be difficult to precisely calculate cavity mode sizes for each case. When using R = 50%, the theoretical pulse energies for LD1 and LD2 were 434 µJ and 440 µJ, respectively. For using invariant pump size with Gaussian distribution, the pulse energy was 389 µJ. The energy difference was 45 µJ or 11% difference for LD1. It is worth to mentioned that using top-hat distribution for output pulse energy calculation can match the experimental results. The top-hat approach with orange dot line shown in Fig. 6(b) was calculated by using Eq. (23) dividing by a factor of 0.88 that obtained in Fig. 3. This result support our point of view that the past researches can have precise prediction on output pulse energy when using transverse pump to mode size matching only. Slight modification the model to top-hat pump distribution can also increase the accuracy of output energy calculation. However, it is still necessary to include pump beam quality for threshold pump power calculation.

Figure 7 depicts the design criteria and the analysis of our resonator. The designed cavity mode size should be approximately 1.2 times larger than the pump mode size of 250 µm. Since there existed thermal lensing effect on the gain medium, the cavity mode size on the gain medium for plane-parallel resonator can be obtained using following equation [14]:

(25)$${\omega _{l0}} = \sqrt {\frac{{{\lambda _l}{l_{eff}}}}{\pi }\frac{{g_2^\ast }}{{g_1^\ast ({1 - g_1^\ast g_2^\ast } )}}\left[ {1 + {{\left( {\frac{{{d_1}}}{{{l_{eff}}}}} \right)}^2}\frac{{g_1^\ast ({1 - g_1^\ast g_2^\ast } )}}{{g_2^\ast }}} \right]} ,$$

(26)$$g_i^\ast{=} 1 - \frac{1}{{{f_{th}}}}{d_j},\textrm{ }i,j = 1,2,$$

(27)$${l_{eff}} = {l_{opt}} - \frac{1}{{{f_{th}}}}{d_1}{d_2},$$

where λ_l =1064 nm was the laser wavelength, d₁ and d₂ were the distance from front mirror and from output coupler to the gain medium, respectively, l_opt was the optical cavity length. f_th was the focal length caused by thermal lensing of the gain medium, which can be obtained by [14,18]:

(28)$${f_{th}} = \frac{{\pi {K_c}\omega _{p0}^2}}{{{P_{in}}\xi \left( {\frac{{dn}}{{dt}}} \right)}}\frac{\textrm{1}}{{1 - \textrm{exp} ({ - \kappa {l_{med}}} )}}\textrm{,}$$

where K_c was the thermal conductivity of the gain medium, P_in was the incident pump power, ζ was the thermal fraction of pump power, and dn/dt was the thermo-optics coefficient of the gain medium. For Nd:YAG crystal at room temperature, K_c was 0.13 W/cm-K and dn/dt was 7.3×10⁻⁶ K⁻¹ [18]. ζ was 0.24 for 808-nm pumped 1064-nm laser. By considering the refractive index of the gain medium, the optical cavity length utilized in Eq. (27) will be slightly different from real cavity length, l_cav:

(29)$${l_{opt}} = {l_{cav}} - {l_{med}} + \frac{{{l_{med}}}}{n}\textrm{,}$$

where n = 1.82 was the refractive index of the Nd:YAG crystal. It is worth to mention that the thermal lensing effect was difficult to express by single formula since there will be additional thermally induced effects such as stress. Moreover, the pump spatial and temporal profile might also lead to different thermal lensing effect. In fact, it will be more suitable to fit the focal length value based on each experimental condition [14]. However, we verified that the calculated focal length value using Eq. (28) was close to our experiment. Here, we used the maximum stable cavity length in our experiment to find the approximate focal length of the thermal lensing effect. Consequently, we believed using Eq. (28) is good enough to analyze our experimental results. Figure 7 illustrates calculated focal lengths of the thermal lensing effect at two pump mode sizes and the corresponding experimental results.

Fig. 7. Calculated focal length of thermal lensing effect for the Nd:YAG crystal in this experiment.

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Figure 8(a) shows cavity mode size on gain medium for different cavity length when operated at 15.5 W and 18.5 W. d₁ was set to be 1 mm since the gain medium was very close to the front mirror. The pump powers were determined to be slightly larger than PQS threshold pump powers that we obtained experimentally when utilizing LD1 and LD2. We can find that by using cavity length of 91 mm and 80 mm for LD1 and LD2, the cavity mode sizes were similar to the values we utilized for theoretical analysis, 290 µm and 300 µm. It is worth to mention that when we decreased the cavity length to be approximately 50 mm, a satellite pulse will appear. Figures 8(b) and 8(c) depict the pulse shape obtained with cavity length of 80 mm and 50 mm when using LD2 with R = 50%. We believed the satellite pulse came from the high-order transverse mode emission when pump mode size was larger than the cavity mode size. On the other hand, when we increased cavity length to larger than 85 mm, the laser transverse pattern will be distortion on its outer ring. Figures 8(d) and 8(e) show the transverse patterns obtained with cavity lengths of 80 mm and 85 mm when using LD2 with R = 50%. The distortion might be caused by the thermal lensing effect since the effective area of the thermal lensing effect was restricted by the pump area. When the cavity mode size was much larger than pump mode size, only the central part will be focused and there will appear ring on the edge. As a consequence, it is important to match the cavity mode size when designing a PQS laser.

Fig. 8. (a) The theoretical cavity mode size on gain medium with thermal lensing effect versus cavity length when operated at incident power of 15.5 W and 18.5 W. The temporal pulse shapes for using LD2 at cavity lengths of (b) 80 mm and (c) 50 mm. The transverse patterns for using LD2 at cavity lengths of (d) 80 mm and (e) 85 mm.

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Finally, we demonstrated the energy scaling for the PQS laser. To increase the output pulse energy, the mode size should be increased. According to the result in Figs. 1(b) and 2(b), the theoretical threshold pump power and output pulse energy were 35.4 W and 1.05 mJ when pump mode size increased to 400 µm. By changing the refocusing lens pair with a focal length ratio of 2, we can obtain millijoule pulse energy output using LD2. After carefully adjusted the cavity length to be approximately 105 mm, the satellite pulse can be suppressed. The threshold pump power was 38.2 W and output pulse energy was up to 1.07 mJ. Figure 9(a) shows the current cavity mode size calculation using Eq. (25) and we found that the cavity mode size on the gain medium will be approximately 500 µm, which was 1.25 times larger than pump mode size. This might be the reason that theoretical analysis was slightly smaller. The temporal pulse shape and pulse train of the PQS laser were shown in Figs. 9(b) and 9(c). The pulse width was approximately 6 ns and the peak power reached 178 kW. The peak-to-peak stability was less than 3%. It is worth to mention that the theoretical threshold pump power without considering pump beam quality was only 30 W. If we utilized this underestimated power to design the laser, it is highly possible that an inadequate laser diode will be considered for this experiment.

Fig. 9. (a) The theoretical cavity mode size on gain medium with thermal lensing effect versus cavity length when operated at incident power of 38 W and pump size of 400 µm. (b)The temporal pulse shape and (c) pulse train of the PQS laser with 1.07-mJ output pulse energy.

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4. Conclusions

In conclusions, we have investigated the influence of pump diode beam quality on the diode-end-pumped PQS laser output, which can be practically utilized for designing millijoule high-beam-quality PQS laser. The transverse cavity to pump mode size matching has already been widely discussed that can help modeling the output parameters of PQS laser in past researches. In our theoretical analysis, we further include the longitudinal mode size dependence to express the influence of pump diode beam quality. An analytical model is derived that allow us to vary the beam quality factor M² to clarify the effects of pump diode beam quality. Theoretically, it is found that the threshold pump power can be larger up to 5 times in comparison with modeling result without longitudinal mode size matching. This will dramatically affect the practical design of PQS laser especially at high pump power level. On the other hand, our theoretical model also shows that the pulse energy has less than 15% difference in comparison with model without longitudinal mode size matching. This result indicates that the output energy is dominant by the cavity mode size and the previous transverse spatial dependent model is good enough for analysis. The influence of using top-hat pump distribution is also discussed and the result is the same. Experimentally, we verify the theoretically analysis by using two diodes with different M² factors to demonstrate the PQS lasers. The results are in good agreement with the theoretical analysis. We further discussed the cavity design with the thermal lensing effect, which indicated the importance of mode size matching for avoiding satellite pulse and pattern distortion. Finally, we successfully scale the output pulse energy to approximately 1.07 mJ at a pump power of 38.2 W with 6 ns pulse width and 178 kW peak power based on our theoretical analysis and cavity design. The analytical model derived in this work can practically help the design of millijoule diode-end-pumped PQS laser.

Funding

Ministry of Science and Technology, Taiwan (MOST-109-2112-M-239-004 -MY2).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are available in Refs. [13–18]

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Exploring the influence of pump beam quality on designing millijoule diode-end-pumped passively Q-switched lasers

Abstract

1. Introduction

2. Theoretical analysis

3. Experimental results

4. Conclusions

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (9)

Equations (29)

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