Modeling of Cr²⁺-Doped Saturable-Absorber Q-Switched Tm:CaF₂ Lasers

Abstract

We present a model of a Cr²⁺-doped saturable absorber (SA), which is employed in passively Q-switched (PQS) Tm:CaF₂ lasers. The overall round-trip loss, the time evolution of the intracavity photon density, and the effective population inversion density can all be obtained through numerical solutions. Under the mode-matching condition, this model can be used to easily determine the PQS laser’s main output parameters, including the average output power, repetition frequency, peak power, pulse energy, and pulse width. This concept is also applicable to a range of thulium-doped solid-state lasers (SSLs) operating on the transition from the ³F₄ level to the ³H₆ level, which are Q-switched by a Cr²⁺-doped SA. This model is helpful for the design and optimization of this kind of laser.

1. Introduction

High peak power or high energy nonlinear frequency conversion in the mid-infrared wavelength region, lidar, laser communication, medicine, and other fields all make extensive use of 2 μm pulse lasers [1,2,3,4,5]. In recent years, with the emergence of new saturable absorbers, the use of passive Q-switching technology to obtain pulsed lasers has been widely reported [6,7,8,9,10]. Furthermore, PQS lasers are more advantageous than active Q-switched lasers in pulsed laser shrinking due to their low cost and simple, compact designs [11,12]. The right SA and gain medium selection are essential for obtaining high energy or high peak power in a PQS laser.

Bulk materials doped with Cr²⁺ (such as Cr:ZnS and Cr:ZnSe) and two-dimensional layered materials are the primary sources of SA for the 2 μm wavelength range. Compared with two-dimensional materials, the SA with bulk structure can utilize more doped ions, and, thus, obtain a higher initial population inversion density [6,7,8,9,10]. Moreover, Cr²⁺-doped bulk materials also have the characteristics of a high damage threshold, wide absorption spectrum, and large absorption cross-section [13,14]. These characteristics facilitate the production of high peak power or high energy 2 μm pulsed lasers using Cr²⁺-doped SA Q-switched lasers.

Within the wavelength range of 2 μm, Tm³⁺-doped gain media can be directly pumped by commercial semiconductor lasers, while Ho³⁺-doped gain media typically use a two-stage pumping structure (LD → Tm³⁺ → Ho³⁺). Therefore, compared to SSLs doped with Ho³⁺, SSLs doped with Tm³⁺ can have lower production costs and greatly reduce the complexity of the laser system. Apart from the exceptional physical and chemical characteristics of CaF₂ crystals, including their wide transmittance spectrum, increased heat conductivity, low coefficient of thermal expansion, and high damage threshold, their low phonon energy (≤495 cm⁻¹) can also lessen the likelihood of a non-radiative transition [15]. Moreover, Tm³⁺ tends to form cluster structures within the CaF₂ crystal, which can enhance the cross-relaxation effects [16]. These advantages help improve the efficiency of optical-to-optical conversion of Tm:CaF₂ lasers, thereby increasing peak power or output energy. For the first time, in 2013, A. A. Lyapin et al. employed LD as a pumping source for Tm:CaF₂ crystals to produce a continuous-wave laser with a slope efficiency of 26.7% and a wavelength of 1890 nm [17]. In 2018, it was initially shown that a Tm:CaF₂ laser may operate PQS by employing silver nanorods as a SA [18]. With a maximum output power of 385 mW and a minimum pulse width of 3.1 µs, a wavelength of 1935.4 nm was achieved. The associated pulse energy was 41.4 µJ, and the corresponding repetition frequency was 9.3 kHz. In the same year, this research team developed a continuous-wave laser with an output power of 2.71 W and a center wavelength of 1917.0 nm in an LD-pumped Tm:CaF₂ laser, achieving the highest slope efficiency of this class of lasers to date. The associated slope efficiency was 70.3% [19]. Using a 1610 nm fiber laser as the pump source for the Tm:CaF₂ laser, R. Thouroude et al. created a continuous-wave laser in 2020 with a maximum power of 1.25 W at around 1880 nm and a slope efficiency of 55% [20]. Moreover, CaF₂ lasers co-doped with Tm³⁺ and other ions, like Y³⁺ or La³⁺, have been researched to improve the laser’s performance [21,22,23,24,25].

So far, the research of Tm:CaF₂ lasers has mainly focused on continuous-wave operation, and there are few researches on PQS operation, especially since there have been no reports on Tm:CaF₂ lasers which are Q-switched by Cr²⁺-doped saturable absorbers. A model of PQS Tm:CaF₂ lasers with saturable absorbers doped with Cr²⁺ is established in this work. The main output parameters of the PQS laser can be calculated numerically using this model. Furthermore, the effect of resonator parameters on the output characteristics of the PQS laser is investigated. This paper’s paradigm can also be used for various kinds of thulium-doped SSLs (³F₄ → ³H₆) that are Q-switched by saturable absorbers doped with Cr²⁺. This study, in our opinion, has significant reference value for the development and improvement of such lasers.

2. Materials and Methods

The laser-generating process in a Tm³⁺-doped gain medium is shown in Figure 1a [26]. The ³H₆ → ³H₄ transition is where the pump light is absorbed. The Tm³⁺ system contains a potent cross-relaxation (CR) mechanism between two neighboring ions, ³H₄ + ³H₆→³F₄ + ³F₄. By generating two photons at approximately 1.9 μm from a single pump photon at around 0.8 μm, this method achieves a 200% quantum efficiency. Laser action occurs between one sublevel, 5646.5 cm⁻¹, of the ³F₄ level and one sublevel, 402.2 cm⁻¹, of the ³H₆ level, and the corresponding wavelength is ~1907 nm [27,28]. Figure 1b shows the Cr²⁺ energy level diagram in the Cr²⁺-doped SA. The II→I transition is slow, but the III→II and IV→II transitions are quick. The I→III and II→IV transitions can both be absorbed at the laser wavelength, with absorption cross-sections of σ_gs and σ_es, respectively. Therefore, it can be considered only levels I and II are appreciably populated. Furthermore, Cr²⁺-doped crystals show broad absorption bandwidths and a wide range of spectral transparency. These crystals’ wide absorption spectra allow them to modulate lasers across a remarkably wide wavelength range. The emission cross-section of Tm³⁺ is two orders of magnitude lower than the peak absorption and emission cross-sections of Cr²⁺, which are on the scale of 10⁻¹⁸ cm². Consequently, Cr²⁺-doped crystals are appropriate for usage as passive Q-switches in Tm³⁺-doped laser systems in addition to being gain media for widely tunable mid-infrared lasers [13,14]. The common structure of a PQS Tm:CaF₂ laser employing an SA doped with Cr²⁺ is depicted in Figure 2.

Here, we assume immediate thermalization of the laser manifolds and create the rate equations using the effective population inversion density, effective absorption cross-section, and effective emission cross-section, which are more suited for the quasi-three-level system, thus, (1) through (3) provide the rate equations.

$${\displaystyle \frac{d\varphi }{dt}}={\displaystyle \frac{lc{\sigma }_{eff}}{L}}\left[N_2(\varphi +{\displaystyle \frac{1}{L{A}_g}})-{\displaystyle \frac{f_1}{f_2}}{N}_1\varphi \right]-{\displaystyle \frac{l_{s}c\varphi }{L}}\left({\sigma }_{gs}{N}_{gs}+{\sigma }_{es}{N}_{es}\right)-{\displaystyle \frac{\varphi }{{\tau }_c}}$$

(1)

$${\displaystyle \frac{d{N}_2}{dt}}=-c\varphi {\sigma }_{eff}\Delta N-{\displaystyle \frac{N_2}{{\tau }_2}}+W_p$$

(2)

$${\displaystyle \frac{d{N}_{gs}}{dt}}=-{\displaystyle \frac{A_g}{A_s}}{\sigma }_{gs}c\varphi N_{gs}+{\displaystyle \frac{N_{es}}{{\tau }_s}}$$

(3)

A_g is the effective cross-sectional area of the laser beam inside the gain medium, while N₁ and N₂ stand for the lower and higher laser manifold population densities, respectively. Additionally, the equations state that ϕ is the intracavity photon density, l is the gain medium’s length, c is the vacuum speed of light, σ_eff is the gain medium’s effective emission cross-section, and L is the cavity optical length. The Boltzmann factors for the lower and upper laser intensities are denoted by f₁ and f₂, respectively. The effective absorption cross-sections of the ground and excited states are denoted by σ_gs and σ_es, respectively, and l_s is the SA’s length. N_gs and N_es represent the SA ground and excited state population densities, respectively, whereas τ_c represents the cavity photon lifetime and ΔN represents the effective population inversion density. W_p is the pump rate in addition to the effective cross-sectional areas of the SA. The excited-state lifetimes of the SA and the gain metal are denoted by τ_s and τ₂, respectively.

Equation (1) states that the reason for reabsorption in a quasi-three-level laser is “(f₁/f₂) N₁ϕ,” or stimulated absorption of the laser in the lower energy level. A four-level laser inhibits reabsorption since the bottom laser level has no population. The expression “1/LA_g” in Equation (1) denotes spontaneous emission from the upper laser level, which is a prerequisite to producing passive Q-switched pulses. Equation (4) is also satisfied by N₁ and N₂.

$$N_0=N_1+N_2$$

(4)

where N₀ is the dopant concentration in the gain medium. N_gs and N_es satisfy Equation (5)

$$N_{so}=N_{es}+N_{gs}$$

(5)

where N_so is the saturable absorber’s doping concentration. One way to express the effective population inversion density is as

$$\Delta N=N_2-\left({\displaystyle \frac{f_1}{f_2}}\right)N_1$$

(6)

Equation (7) provides N_so, where T₀ is the saturable absorber’s initial transmittance.

$$N_{so}=-{\displaystyle \frac{\ln \left(T_0\right)}{{\sigma }_{gs}{l}_s}}$$

(7)

Equation (8) can be used to obtain the cavity photon lifetime, where R stands for the output coupler reflectivity, the whole travel time is t_r, and δ is the round-trip internal resonator loss.

$${\tau }_c={\displaystyle \frac{t_r}{\ln \left(\frac{1}{R}\right)+\delta }}$$

(8)

The pump beam’s effective cross-sectional area is 0.5πw_p², where w_p is the pump beam’s radius in the gain medium. This is because the pump strength has a Gaussian spatial distribution when the LD end-pumped regime is used. The pump rate can be expressed using Equation (9).

$$W_p={\displaystyle \frac{4P_p}{h{v}_p\pi w_p^{2}l}}\left[1-e^{-\left({\sigma }_p{N}_{1}l\right)}\right]$$

(9)

Due to the presence of cross-relaxation in the Tm³⁺-doped gain medium, the upper laser level will increase by two particles when the gain medium absorbs one pump photon, so the numerator of formula (9) contains the integer 4. In Equation (9), P_p is the pump power, hv_p is the energy of a pump photon, and σ_p is the effective absorption cross-section at the pump wavelength.

Because the SA’s ground and excited state populations change over time when the pulse is established, the total loss also varies with time and it can be expressed as [29].

$$Loss={\displaystyle \frac{2{\sigma }_{gs}{N}_{gs}{l}_s+2{\sigma }_{es}{N}_{es}{l}_s+\ln \left(\frac{1}{R}\right)+\delta }{2{\sigma }_{eff}l}}$$

(10)

Through the output mirror, the cavity’s instantaneous power coupling output can be expressed as [30]:

$$P_{out}(t)={\displaystyle \frac{h{v}_{l}L{A}_g\ln \left(\frac{1}{R}\right)\varphi (t)}{t_r}}$$

(11)

where hv_l is the oscillating light’s photon energy and LA_g is the resonator’s volume that the oscillating light occupies.

Under the influence of the lattice field, the Tm³⁺-doped gain medium’s excited state (³F₄) and ground state (³H₆) will split into several sub-levels. The population of each manifold’s sub-levels satisfies the Boltzmann distribution when the thermal equilibrium is present. Let f_i represent the Boltzmann Fraction (Bolt. Frac.) of the ith level within a given manifold and it can be expressed as:

$$f_i={\displaystyle \frac{e^{-E_i/k_{b}T}}{Z_i}}$$

(12)

where T represents the temperature, k_b the Boltzmann constant, E_i the ith level’s energy, and Z_i is the manifold’s partition function, which can be written as:

$$Z_i={\displaystyle \sum _m{e}^{-E_m/k_{b}T}}$$

(13)

The “m” represents the thermally populated stark level in the manifold. By using Equations (12) and (13) and considering the energy levels in [28], the Bolt. Frac. for each sublevel of the ³H₆ and ³F₄ manifolds can be calculated, as shown in

Table 1. Bolt. Frac. in the Stark energy levels of the Tm:CaF₂ at 280 K.

3H6 (cm−1)	Bolt. Frac.	3F4 (cm−1)	Bolt. Frac.
5.6	0.4048	5646.5	0.5025
94.3	0.2565	5853.4	0.1733
236.9	0.1232	5901.9	0.1350
402.2	0.0526	6070.5	0.0567
465.9	0.0379	6077.5	0.0547
488.9	0.0337	6119.7	0.0440
496.7	0.0324	6171.9	0.0337
555.9	0.0239
562.7	0.0230
690.0	0.0120

.

3. Results and Discussions

Here, the numerical solution approach can be used to determine the intracavity photon density, effective population inversion density, and total round-trip loss. By using these data further, the PQS laser’s main output parameters—peak power, pulse width, pulse energy, repetition frequency, and average output power—can be determined. This research examines in depth how the resonator parameters affect the laser’s output parameters. Here, the SA was chosen to be Cr:ZnS, and a Tm:CaF₂ crystal with a 3at.% thulium doping concentration was used as the gain medium.

Table 2. Values for the rate equation modeling parameters.

Parameter	Value
l	1 cm
c	2.998 × 1010 cm/s
σeff	0.9 × 10−21 cm2 [27]
wp	0.01 cm
wl	0.01 cm
f1	0.0526
f2	0.5025
ls	0.2 cm
σgs	6.97 × 10−19 cm2 [13,14]
σes	3.48 × 10−20 cm2 [13,14]
Lc	10 cm (physics cavity length)
R	0.95
Pp	20 W
T0	0.85
vp	767 nm
vl	1907 nm
τ2	6.16 ms [31]
σp	4.2 × 10−21 cm2 [27]
δ	0.05
τs	4.3 × 10−6 s [13,14]

lists the parameters utilized in the rate equation modeling unless otherwise noted.

Additionally, the PQS laser’s output parameters were acquired with mode matching, or w_p = w_l. Naturally, the laser spot size in the gain medium should be the same as the A_g = A_s laser spot size in the SA. The intracavity photon density, effective population inversion density, and total round-trip loss change curves over time were created using the values from Table 2, as illustrated in Figure 3. Figure 3b displays a partial enlargement of Figure 3a. As seen in Figure 3a, the building time of the first Q-switched pulse is substantially longer than that of the subsequent pulses. This is because the first pulse’s initial population inversion density (negative value) is significantly lower than that of the subsequent pulses. The pulse buildup process is illustrated in Figure 3b. With the increase of intracavity photon density, both total round-trip loss and effective population inversion density gradually decrease, until the effective population inversion density equals the total round-trip loss, the intracavity photon density reaches its maximum value, and then the intracavity photon density rapidly drops to zero.

3.1. The Effect of Incident Pump Power on Laser Characteristics

While Table 2 uses the values of other parameters, Figure 4 and Figure 5 illustrate how the output characteristics change as the incident pump power varies between 6 W and 20 W. As the pump power increases, the pulse width stays almost constant, but the peak power and pulse energy slightly increase, as shown in Figure 4a and Figure 5. This is because the effective population inversion density still somewhat grows under the pump’s excitation during the laser pulse’s creation. The maximum effective population inversion that may be achieved rises with incident pump power, as shown in Figure 6. Approximately 0.6% more effective population inversion occurs at a 20 W pump power than at a 6 W pump power. Both the average output power and the repetition frequency grow monotonically as the incident pump power increases, as seen in Figure 4b. As the incident pump power rises, the pump rate rises as well, increasing the average output power (Figure 6). This raises the frequency of repetition and decreases the time it takes for a pulse to develop.

3.2. The Effect of Reflectivity of Output Coupler on Laser Characteristics

Figure 7 and Figure 8 illustrate how the output coupler’s reflectivity affects the output characteristics. In this case, the output coupler’s reflectivity ranges from 0.65 to 1, and Table 2 lists the values of the other parameters. According to Figure 8, the pulse width grows as the reflectivity rises; in other words, a smaller reflectivity facilitates the achievement of a shorter pulse width. This is because an increase in reflectivity causes the threshold population inversion density to fall (Figure 9) and the intracavity photon lifetime to increase, both of which increase the pulse width. However, as Figure 7a shows, peak power and pulse energy decrease as reflectivity increases. Figure 7b shows that the repetition frequency increases monotonically with reflectivity, while there is an ideal reflectivity at which the average output power reaches its maximum value.

3.3. The Effect of Unsaturated Absorber Transmission on Laser Characteristics

The influence of the unsaturable absorber transmission on the output characteristics is shown in Figure 10 and Figure 11. Here, the unsaturable absorber transmission varies in the range of 0.7~1, and Table 2 makes use of the other parameters’ values. Figure 10a and Figure 11 show that as the unsaturated absorber transmission increases, the pulse width gradually increases while the peak power and pulse energy monotonically decrease. This is because an increase in unsaturable absorber transmission leads to a reduction in the threshold population inversion density, as shown in Figure 12. Figure 10b illustrates that the average output power and repetition frequency increase in tandem with the unsaturable absorber transmission. The unsaturable absorber transmission has a major influence on the repetition frequency and pulse width, particularly in the area with a large unsaturable absorber transmission (around 1).

3.4. The Effect of Cavity Length on Laser Characteristics

The effects of the cavity’s length, which ranges from 10 to 30 cm, on each of the passive Q-switched laser’s output parameters are shown in Figure 13 and Figure 14. The remaining values are used in Table 2. Because Figure 15 shows that the round-trip time (t_r) grows with cavity length, the pulse width in Figure 14 also increases with cavity length. Except for peak power, which decreases as cavity length grows, Figure 13 illustrates that the average output power, repetition frequency, and pulse energy are all almost insensitive to cavity length. The peak power falls as the pulse width rises, but the pulse energy, repetition frequency, and average output power essentially remain constant because the threshold population inversion density is independent of the cavity length (Figure 15).

3.5. The Effect of the Length of Gain Medium on Laser Characteristics

The effects of the gain medium’s length, which ranges from 0.7 to 6 cm, on each of the passive Q-switched laser’s output parameters are shown in Figure 16 and Figure 17. Table 2 uses the remaining values. The pulse width increases as the gain medium lengthens, as seen in Figure 16a, mainly because a longer gain medium results in a longer round-trip time. Both the threshold population inversion density and the pump rate have an impact on the repeat frequency. As the gain medium’s length grows, the pump rate first shows a slight increase and then gradually decreases, while the threshold population inversion density first drops rapidly and then slows down, as seen in Figure 18. According to Figure 16a, these factors cause the repetition frequency to climb quickly initially before tending to remain constant as the gain medium’s length increases. As illustrated in Figure 16b, the average output power will follow a similar trend to the repetition frequency since the pulse energy stays almost constant. But as the length of the gain medium rises, the peak power monotonically decreases, as shown in Figure 17.

4. Conclusions

This work establishes a model of a PQS Tm:CaF₂ laser using bulk material doped with Cr²⁺ as a SA. The theory can be used to simulate the pulsed laser manufacturing process and derive the evolution of photon density, effective population inversion density, and total round-trip loss with time. Furthermore, this model is used to thoroughly examine how the laser parameters affect the output characteristics. The study found that it is easier to create high-energy, short-pulse lasers by reducing the reflectivity of the output coupler and the transmission of unsaturated absorbers. It is advantageous to shorten the cavity length to produce a laser with a small pulse width and high peak power. By raising the pump power, a high average power laser with a high repetition frequency can be achieved. The short length gain medium makes it easier to achieve the low repetition frequency, small pulse width, and high peak power of the pulse laser. There is also an optimal output coupler reflectivity for the average output power and optical-to-optical conversion efficiency. This model can help in the design and optimization of other types of thulium-doped SSLs (³F₄ → ³H₆) which are Q-switched by Cr²⁺-doped saturable absorbers.