The quasi-continuous-wave (QCW) pulse generated from an Nd:YAG laser has a wide range of applications in many fields. In industrial processing, the QCW laser pulse can form a unique melting and solidification process in the aimed material due to distinguishing thermal effects, which is different from a continuous-wave laser [1–3]. It is also pretty valuable for generating QCW laser at wavelengths of 532 nm, 355 nm, and 589 nm by frequency conversion [4, 5]. Additionally, it is a powerful tool in laser medicine: a long-pulsed Nd:YAG laser can be applied in the treatment of onychomycosis [6, 7], for instance. Therefore, the high-power QCW Nd:YAG laser is highly anticipated due to its wide applications in many areas, and many studies on QCW Nd:YAG lasers have been carried out [8–13]. However, there is an issue that is yet to be solved about QCW: the QCW pulse is usually accompanied by chaotic relaxation spikes that are detrimental for certain applications [14]. As a result, spike suppression is essential for QCW generation in an Nd:YAG laser.

In order to remove the inherent spikes, both active and passive suppression methods have been developed in solid-state lasers. High-speed negative feedback is needed to add to the intracavity elements in active suppression. In passive suppression, an intracavity nonlinear crystal, such as potassium titanyl phosphate (KTP) and lithium triborate (LBO), is needed to effectively suppress spikes in the output by converting high-intensity laser spikes into second harmonic radiation [15–17]. The intracavity nonlinear crystal provides an intensity-dependent nonlinear loss in the cavity, which effectively suppresses the spikes in the output. The problem is, however, that the cavity design is usually sophisticated both in active and passive suppression.

In previous work, we have numerically indicated that chaotic spiking phenomenon in pulse-pumped fiber lasers can be regulated by adopting the bias pumping technique [18–20], in which the pumping power of a fiber laser consists of both a pulsed power and a continuous-wave (CW) power with the same wavelength. We also recently demonstrated experimentally that a duration-controllable mid-infrared pulse can be produced from a bias-pumped Er:ZBLAN fiber laser [21]. In this paper, we explore whether the bias-pumped method can be employed in the QCW operation of a solid-state laser to suppress the inherent spiking, which has rarely been reported before, and do numerical research. Specifically, we investigate the temporal characteristics of output pulse under different pump durations, repetition rates and peak powers in a bias-pumped Nd:YAG laser. The results show that bias pumping is an alternative approach to suppress chaotic spiking in the QCW operation of a solid-state laser, and a stable output QCW pulse can be generated from a bias-pumped Nd:YAG laser.

A schematic diagram of the bias-pumped Nd:YAG laser is shown in Fig. 1. It consists of a CW-laser diode and a pulsed laser diode serving as a pumping source. The lengths of the resonator and Nd:YAG are denoted by L and l_a, respectively. The optical length of the resonator can be calculated by the expression L_e = L + (n − 1)l_a, where n represents the refractive index of Nd:YAG. The resonator mirror M1 is fully reflective at 1.064 µm, while M2 is an output coupler mirror. The power reflectivities (transmissions) of mirrors M1 and M2 are denoted by R1 and R2 (T1 and T2), respectively. The logarithmic losses per pass due to the mirror transmission are given by the expressions γ₁ = −ln(1 − T₁) and γ₂ = −ln(1 − T₂), respectively. In the cavity, an internal loss denoted by δ is assumed, and hence the logarithmic internal loss per pass can be expressed by γ_i = −ln(1 − δ). With those logarithmic loss notations, a total logarithmic loss per pass can be defined as γ = γ_i + (γ₁ + γ₂) / 2. And the lifetime of a cavity photon is given by ${\tau }_c=\frac{L_e}{\gamma c}$, where c is the speed of light in vacuum.

Figure 1. Schematic representation of the bias-pumped Nd:YAG laser. F1 and F2, focus lens; M1, input mirror; M2, output mirror.

For simplicity, we assume the laser oscillates in the active medium in a single mode with the beam waist denoted by ω0. According to the notations above and laser theory, the rate equations of the Nd:YAG laser are as follows [22, 23]:

(1)$$\frac{\partial N\left(t\right)}{\partial t}=R_p\left(t\right)-c\sigma \varphi \left(t\right)N\left(t\right)-\frac{N\left(t\right)}{\tau }$$

(2)$$\frac{\partial \varphi \left(t\right)}{\partial t}=\frac{l_a}{L_e}c\sigma N\left(t\right)\varphi \left(t\right)-\frac{\varphi \left(t\right)}{{\tau }_c}+s$$

where N(t) is the population inversion density of the gain medium, and N₀ is the total concentration of the active ions. ϕ represents the photon density inside the laser cavity, and s is the spontaneous emission rate. σ and τ are the stimulated emission cross section of the laser crystal and the lifetime of the upper laser level, respectively. The quantity R_p (t) is the volumetric pump rate into the upper laser level and is proportional to the total pump power P_cw + P_pulse (t), where P_cw and P_pulse (t) are the CW pump power and the pulsed pump power, respectively. The expression of R_p (t) can be given by

(3)$$R_{\text{p}}\left(t\right)={\eta }_p\left[\frac{P_{\text{cw}}+P_{\text{pulse}}\left(t\right)}{h{\nu }_P}\right]\frac{2}{\pi {\omega }_P^2{l}_a}\left[1-\text{exp}\left(-\alpha l_a\right)\right]$$

where, η_p is the pump quantum efficiency, and hν_P is the pump photon energy. Both the CW pump and the pulsed pump are assumed to be a fundamental mode with spot size denoted by ω_P at the entrance of the Nd:YAG crystal. α represents the absorption coefficient of the pump laser in the gain medium. The output power through mirror M2 is given by $P_{\text{out}}=\left(\frac{{\gamma }_{2}c}{2L_e}\right)\left(h\nu \right)\varphi \left(\frac{\pi {\omega }_0^2}{2}\right)L_e$, where hν is the laser photon energy.

In the following, we numerically simulate Eq. (1), Eq. (2), and Eq. (3) to analyze the QCW pulses generated in the bias-pumped Nd:YAG laser. The main parameters used in the simulation are shown in Table 1.

TABLE 1. Main parameters used in the simulations.

Parameter	Value
λP	808 nm
λs	1064 nm
N0	1.38 × 1026 m−3
la	10 mm
ω0	150 μm
σ	2.8 × 10−23 m2
τ	230 μs
α	σN0
L	20 mm
n	1.82

The long-pulse-pumped Nd:YAG laser will generally operate in a QCW mode with relaxation oscillation spikes in the output pulse. We first investigate the temporal characteristics of QCW pulses from the Nd:YAG laser pumped only by pulsed power with a Gaussian profile. In the QCW operation, the output laser is built up from noise and goes through a series of relaxation oscillations, as seen in Figs. 2 and 3. The simulation results in Fig. 2 indicate that the temporal profiles of output QCW pulses vary with the duration of pump pulses. Also, the peak values of the relaxation oscillations vary with the pump repetition rate, indicating that the pump repetition rate can also affect the temporal characteristics of the QCW outputs, as illustrated in Fig. 3. Therefore, the QCW output from the Nd:YAG laser pumped only by pulsed power is usually accompanied by a series of chaotic spikes, and the temporal characteristics of those spikes are sensitive to the pump pulse parameters.

Figure 2. Temporal characteristics of output pulses under pulsed pump durations of (a) 50 µs and (b) 70 µs, fixing the pump repetition rate at 2 kHz.

Figure 3. Temporal characteristics of output pulses under pump repetition rates of (a) 1 kHz and (b) 5 kHz, fixing the pump duration at 50 µs.

It is expected that the chaotic spikes in the output QCW pulses can be eliminated by adding a CW pump power. Meanwhile, the CW pump power should less than the threshold power of an Nd:YAG laser to prevent CW laser generation. The threshold power of an Nd:YAG laser can be estimated by the expression $P_{th}=\left(\frac{\gamma }{{\eta }_p}\right)\left(\frac{h{\nu }_p}{\tau }\right)\left[\frac{\pi \left({\omega }_0^2+{\omega }_p^2\right)}{2\sigma }\right]$. In the following simulation, the CW pump power (40 mW) is kept slightly below the threshold power (~ 50 mW) of the Nd:YAG laser.

When the Nd:YAG laser is pumped by both a CW power and a pulsed power simultaneously, the simulation results, as shown in Figs. 4 and 5, are different from those in Figs. 2 and 3. In Figs. 4 and 5, owing to the additional CW pump power (40 mW), the relaxation spikes in the output pulse disappear, and the output QCW pulse maintains the same temporal shape as that of the pump under those two different pump durations (depicted in Fig. 4) and different repetition rates (shown in Fig. 5). That is to say, the output QCW pulse becomes stable with bias pumping. Therefore, adopting the bias pumping method for an Nd:YAG laser is very effective to generate a stable QCW laser.

Figure 4. Temporal characteristics of output pulses under pump pulse durations of (a) 50 µs and (b) 70 µs, when the Nd:YAG laser is pumped by both a pulsed power and a continuous-wave (CW) power bias of 40 mW, fixing the pump repetition rate at 2 kHz.

Figure 5. Temporal characteristics of output pulses under different pump repetition rates of (a) 1 kHz and (b) 5 kHz, when the Nd:YAG laser is pumped by both a pulsed power and a continuous-wave (CW) power bias of 40 mW, fixing the pump duration at 50 µs.

As mentioned in our previous work [19], the temporal characteristics of an output laser have much to do with the temporal shape of the pump pulse. In general, chaotic spikes can be caused by steepening the temporal profile of the pump pulse. Whether a stable output can be created in a bias-pumped Nd:YAG laser with a steeper pump profile is an issue worth studying. Accordingly, we simulate the bias-pumped Nd:YAG laser pumped by a six-order super-Gaussian pulse with duration unchanged (50 µs), and the results are shown in Fig. 6. The output pulse stays stable under the six-order super-Gaussian pump pulse with a 2 kHz repetition rate, as demonstrated in Fig. 6(a). When the pump repetition rate increases to 5 kHz, the stable temporal shape of the output laser remains unchanged, as seen in Fig. 6(b), implying that the pump repetition rate has no influence on the temporal shape of a stable output pulse from a bias-pumped Nd:YAG laser. Also, the blue line in Fig. 6 representing population inversion density shows that N(t) is still clamped to its steady-state value during output pulse generation. Thus, the bias-pumped Nd:YAG laser supports pulse generation of several tens-microsecond with a more practical pump pulse shape.

Figure 6. Temporal characteristics of the output laser when the Nd:YAG laser is pumped by both a continuous-wave (CW) power bias of 40 mW and a pulsed power, which has a six-order super-Gaussian profile, duration of 50 µs, repetition rates of (a) 2 kHz and (b) 5 kHz.

The peak power of the pump pulse is also a key factor that influences the temporal characteristics of the output pulse from the Nd:YAG laser. Keeping the six-order super-Gaussian pumping temporal shape and duration of 50 µs unchanged, the QCW pulses from the bias-pumped Nd:YAG laser are investigated under different peak powers of pump pulse. As seen in Fig. 7, no random spike occurs as the pumping peak power increases with a certain power (40 mW) bias. The output QCW pulse remains stable with a low CW power bias, even though the pumping peak power increases to a high value (200 W), as exhibited in Fig. 7. It indicates that increasing the peak power of the pump pulse does not lead to a chaotic spiking phenomenon of output QCW pulse from the bias-pumped Nd:YAG laser. The peak power of the output QCW pulse from the bias-pumped Nd:YAG laser increases linearly with that of the pump pulse at an arbitrary repetition rate, as shown in Fig. 8. Consequently, a stable QCW pulse with a high peak power can be produced in the bias-pumped Nd:YAG laser by increasing the peak power of the pump pulse.

Figure 7. With a 40 mW continuous-wave (CW) pump power bias, the temporal characteristics of output quasi-continuous-wave (QCW) pulses under the pumping pulses with peak powers of (a) 100 W and (b) 200 W, fixing the pump duration at 50 µs and repetition rate at 2 kHz.

Figure 8. Variation of laser peak power with the pump peak power under 40 mW bias power at an arbitrary repetition rate.

We numerically demonstrate a bias-pumped Nd:YAG laser that can generate QCW pulses without a spiking phenomenon. With a certain pumping power bias, relaxation spikes, which are ubiquitous in the QCW operation of an Nd:YAG laser, can be suppressed effectively. After spiking suppression, the output QCW pulses from the bias-pumped Nd:YAG laser turn out to be very stable: their temporal shapes are immune to the pump repetition rate and peak power. In addition, the QCW pulse produced in the bias-pumped Nd:YAG laser can maintain nearly the same temporal shape as that of a pump pulse with a square-like profile and high peak power. Therefore, our study implies that bias pumping is an alternative method of spiking suppression in the QCW operation of a solid-state laser, and adopting the bias pumping approach may extend the application areas of solid-state lasers in QCW mode due to output pulses with stable temporal profiles.

The authors declare no conflicts of interest.

No data were generated or analyzed in the presented research.

The author(s) received no financial support for the research, authorship, and/or publication of this article.