We first investigate the polarization coexistence and switching between the orthogonal π- and σ-polarization states in a free running optical bistability Tm,Ho:LLF laser. The output performances of the optical bistability Tm,Ho:LLF laser is numerically simulated, based on a new rate equation theory model taking into account the influences of polarization-dependent gain and losses. The simulation results show the polarization coexistence and switching of output laser in the process of decreasing pump power. The physical mechanism of polarization coexistence and switching is clarified by analyzing the polarization-dependent net-gain coefficients. Furthermore, the polarization coexistence and switching are experimentally realized in the optical bistability Tm,Ho:LLF laser, which validates the theoretical analysis. The theoretical model can be applied to analyze the polarization coexistence and switching in other kinds of (quasi-) three-level optical bistable lasers.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Solid state lasers emitting in the 2 μm eye-safe region have attracted much interest in recent years due to their wide applications in optical communication, coherent Doppler wind lidar, differential absorption lidar, range-finding, photo-medicine, and nolinear frequency conversion [1–4]. Benefiting from large emission cross-section, long upper level lifetime, and high quantum efficiency, Tm-Ho codoped laser materials are excellent candidates for generating 2 μm laser. Among them, Tm,Ho:LuLiF4 (LLF) is a kind of excellent laser crystal because of high natural birefringence, low upconversion effect, and large energy spread of the manifolds [5,6]. In the past years, continuous wave (CW), Q-switched, and mode-locked Tm,Ho:LLF lasers were widely researched [7–9]. Besides, the optical bistability of Tm,Ho:LLF laser in CW scheme as an interesting phenomenon has also been experimentally investigated . The optical bistability is a behavior in which the system exhibits two output intensities for the same input intensity. Owing to this specific feature, the optical bistability has many potential applications in optical communication, optical computing, optical logic, optical switching, and optical cooling [11–15]. The optical bistability has been realized in CO2 laser , quantum cascade laser , and two-section quantum dot laser , etc. The optical bistability was also obtained in some quasi-three-level solid state lasers. Liu et al reported the optical bistability in Yb:LuVO4, Yb:YGdVO4, and Yb:GdVO4 lasers respectively [19–21]. The optical bistability phenomenon was also observed in Tm,Ho:YLF laser, moreover the influence of temperature on the optical bistability was investigated . Recently, the optical bistability was realized in the Tm,Ho:LLF laser, furthermore the intensity switching based on optical bistability was achieved successfully by our group . At the same time, the theoretical investigation on the optical bistability in a quasi-three-level solid state laser has been performed [24,25]. It is worth mentioning that the optical bistability achieved in some quasi-three-level solid state lasers was accompanied with polarization coexistence and switching (PCS) of the orthogonal π- and σ-polarization states [19,20]. The PCS in solid state lasers, without inserting any polarization-dependent optical elements to introduce different losses, have attracted growing attention. X. Mateos et al reported the PCS of Tm:KLu(WO4)2 lasers [26–28], and the PCS was also realized in Yb:KGW laser . The polarization switching of Tm:KLu(WO4)2 and Yb:KGW lasers occurred when the pump power was increased to a certain range, which was attributed to the effect of polarization-dependent thermal lens [28,29]. The PCS in the quasi-three-level optical bistability lasers were relatively different, which happened in the process of decreasing pump power from a relative high level [19,20]. In the process of increasing pump power, the lasers operated in pure π-polarization state. The PCS has been observed in several kinds of Yb doped optical bistability quasi-three-level solid state lasers. However, to the best of our knowledge, the detailed theoretical analysis and physical mechanism of PCS in the quasi-three-level optical bistability solid state laser has not been given so far.
In this paper, we theoretically and experimentally investigate the PCS of π- and σ-polarization states in a free running optical bistability Tm,Ho:LLF laser. The new coupled rate equations are established, considering the polarization-dependent gain and losses. Based on the coupled rate equations, the PCS are numerically investigated. The physical mechanism of PCS is clarified by the analysis of polarization-dependent net gain coefficients. Furthermore the PCS in a free-running optical stability Tm,Ho:LLF laser is realized. The experimental results are in agreement with the theoretical analysis, and the theoretical model is proved to be reasonable.
2. Gain spectra analysis
First, we analyze the polarization-dependent gain spectra of quasi-three-level Tm,Ho:LLF, and the dimensionless gain G under steady-state condition can be expressed as Eq. (1) and the following parameters : l = 2.5 mm, NHo = 5.59 × 1019 cm−3, T = 283 K, Zexc = 11.4, Zgnd = 9, EZL = 5153 cm−1, the π- and σ-polarization gain spectra of Tm,Ho:LLF are calculated. The gain spectra for several inversion parameters β = 0.2, 0.25, 0.3, and 0.4 are shown in Fig. 1. It is found from Fig. 1(a) that the maximum gain is around 2065 nm when the inversion rate β is less than 0.3 for the π-polarization. With the increase of β value, the maximum gain will appear at 2053 nm short wavelength. However, for σ-polarization, the maximum gain is always near 2065 nm, regardless of the change of β value, as shown in Fig. 1(b). It is worth noting that the gain values for π- and σ-polarization states around 2065 nm can be equivalent between β = 0.2 and 0.25, which means that the PCS near 2065 nm can be achieved by properly adjusting the polarization-dependent cavity losses.
3. Theoretical investigations
For the singly end-pumped Tm,Ho:LLF laser, the distribution of pump power is nonuniform along the axis of the crystal. Therefore, the laser crystal in the laser mode can be divided into two parts: gain region (high pump intensity region) and absorption region (low pump intensity region) . The energy transfer between the Tm3+ 3H4 and Ho3+ 5I7 manifolds is much fast compared to their upper state lifetimes, so they can be treated as a coupled level under CW pumping. Under considering the polarization characteristic, ground state reabsorption (GSR), and energy transfer upconversion (ETU), the rate equations can be expressed as [25,31]32]32]Eqs. (10)-(12) and the following parameters [6,31]: dnc/dT = −3.6 × 10−6 K−1 (c-axis), dna/dT = −6 × 10−6 K−1 (a-axis), Kcc = 6.3 WK−1m−1 (c-axis), Kca = 5 WK−1m−1 (a-axis), α = 5.4 cm−1, l = 2.5 mm, λ = 2065 nm, ξ = 0.4, rb = 2.5 mm, ωl = 180 μm and ωp0 = 85 μm, the diffraction losses for the π- and σ-polarization states as a function of pump power are calculated, as shown in Fig. 2. It can be seen from Fig. 2, that the higher pump power, the larger the thermally induced diffraction losses become for a given mode-to-pump ratio. Furthermore, the thermally induced diffraction loss of σ-polarization is larger and grows faster than the one of π-polarization state owing to the smaller thermal conductivity Kc and the bigger temperature-dependent index change dn/dT.
With Eqs. (2)-(9), the results of Fig. 2, and the following parameters : λp = 792 nm, λ = 2065 nm, ηp = 1.57, τ = 11.4 ms, Q = 5 × 10−18 cm3s−1, σπ = 1.2 × 10−20 cm2, σσ = 0.89 × 10−20 cm2, fuπ = 0.089, fuσ = 0.176, fHo = 0.65, flπ = flσ = 0.026, c = 3 × 108 m/s, NHo = 5.59 × 1019cm−3, δfπ = 0.03, δfσ = 0.025, lcav = 55 mm, n = 1.4, TOC = 0.05, the output power as a function of pump power for the free-running Tm,Ho:LLF laser is calculated, as shown in Fig. 3. For demonstrating the phenomenon more clearly, here we plot the numerical results in two figures for increasing and decreasing pump power, respectively. Figure 3(a) depicts the relationship between the output power and the increasing pump power. When the pump power is increased from zero, the laser does not oscillate until a critical point of pump power, referred to as on-threshold, is reached at Pin = Pon = 1.54 W, at which the output power jumps from zero to a substantial level of 58 mW. Above this point the output power increases linearly with the pump power, furthermore the output laser is purely π-polarized. For the quasi-three-level Tm,Ho:LLF laser, the optical bistability belongs to absorption type because of GSR of Ho3+ 5I8 lower level to 2 μm laser. Before the laser oscillates, the absorption region is just like the role of saturable absorber in the passively Q-switched operation, which leads to a high inversion population density. In this case, the emission cross section becomes the key factor, and oscillating will occur on the transition with the larger emission cross section [7,8]. Moreover, the thermal induced diffraction loss of the σ-polarization state is larger than that of the π-polarization state. Hence, the laser is π-polarized at the point of on-threshold due to the larger emission cross section and smaller diffraction loss compared with those of the σ-polarization state, though the net gain coefficient of the σ-polarization state also satisfies the oscillation condition. At the same time, the 2053 nm π-polarized laser cannot start oscillation, due to the lower gain, compared with the π- and σ-polarization states at 2065 nm for a relative low β value, as shown in Fig. 1.
In the process of decreasing pump power, the output power as a function of pump power is described in Fig. 3(b). Decreasing the pump power from 2 to 1.65 W, the output laser is only π-polarized around 2065 nm. When the pump power is further decreased to 1.65 W, the σ-polarized laser around 2065 nm starts to oscillate. With the decreasing of pump power from 1.65 to 1.48 W, the output power of σ-polarized laser increases from zero to 54 mW, however the output power of π-polarized laser decreases from 64 mW to zero. In the range of 1.65-1.48 W in terms of decreasing pump power, the Tm,Ho:LLF laser not only shows the coexistence of the orthogonal π- and σ-polarization states, but also completes the polarization switching from π-polarization to σ-polarization. At the pump power of 1.48 W, the π-polarized laser ceases, leaving only the σ-polarized laser. Reducing the pump power further to the off-threshold Poff = 1.25 W, the output power of σ-polarized laser drops from 24 mW to zero. According to the numerical simulation results, a hysteresis loop in the dependence of total output power on pump power is presented in Fig. 3. For total output power, in the range of pump power Poff<Pin<Pon, the operation behavior of the Tm,Ho:LLF laser is bistable.
Apart from the output power characteristic, the PCS between π- and σ-polarization states in the optical bistability Tm,Ho:LLF laser can be illustrated more intuitively from the perspective of the polarization-dependent net-gain. According to Eq. (4), the total gain coefficient GTotal, and the π-polarization gain coefficient Gπ, and the σ-polarization gain coefficient Gσ can be respectively expressed asEqs. (13)-(18), the net-gain coefficients for total, π- and σ-polarization states can be respectively given byFig. 4. It can be seen from Fig. 4(a) that the total net-gain coefficient is smaller than zero and the laser will not oscillate, before pump power is increased to the on-threshold pump power Pon = 1.54 W from zero. Increasing the pump power to Pon = 1.54 W, the total net- gain coefficient is equal to zero and the laser will begin to oscillate. In the process of decreasing pump power from a high level (more than Pon = 1.54 W), the total net-gain coefficient is equal to zero until the pump power is decreased to the off-threshold pump power Poff = 1.25 W. Reducing the pump power further, the total net-gain coefficient becomes smaller than zero and the laser ceases. When the pump power is in the bistability region (Poff<Pin<Pon), there are two possible total net-gains, depending on the route (increasing or decreasing) by which the pump power goes into the bistability region.
Figure 4(b) and 4(c) demonstrate the net-gain coefficients of the π- and σ-polarization states, respectively. Before increasing pump power to Pon = 1.54 W, the net-gain coefficients of π- and σ-polarization states are both less than zero, and the laser does not oscillate. The net-gain coefficients of π- and σ-polarization states both reach zero at the on-threshold pump power of 1.54 W, however only the π-polarized laser begins to oscillate due to a larger emission cross section as previously mentioned. As the pump power is increased to over 1.65 W, compared with the π-polarization state, the higher σ-polarization diffraction loss makes the σ-polarization net-gain coefficient become smaller than zero. However the π-polarization net-gain coefficient is still kept to be zero. As a consequence, the output laser is always π-polarized in the process of increasing pump power. During the pump power is decreased from 2 to 1.65 W, the π-polarization net-gain is always equal to zero, the σ-polarization net gain coefficient increases from a negative value to zero, and the laser still operates in purely π-polarization state. In the coexistence region between 1.48 and 1.65 W, the net-gain coefficients of the orthogonally π- and σ-polarized laser are both equal to zero. The equalized net-gain coefficients allow both π- and σ-polarized laser to oscillate and coexist. As the pump power is decreased to below 1.48 W, the π-polarization net-gain coefficient becomes smaller than zero and the σ-polarization net-gain coefficient keeps to be zero, which makes the σ-polarized laser replace the π-polarized laser. Therefore, a complete switching between the two orthogonal π- and σ-polarization states is accomplished, leaving only the σ-polarization state laser. Further decreasing the pump power to the off-threshold pump power Poff = 1.25 W, the σ-polarization net-gain coefficient drops from zero to a negative value, which leads to the ceasing of σ-polarized laser.
Comparing the simulation results shown in Fig. 3 and Fig. 4, it can been found that the polarization-dependent net-gain coefficients explain the output performances of the free-running optical bistability Tm,Ho:LLF laser very well. Based on the above analysis, it can be concluded that the polarization-dependent diffraction losses can balance the net gain coefficients of π- and σ-polarization states by appropriate selecting the mode-to-pump ratio and nondiffraction losses. The PCS in the optical bistability Tm,Ho:LLF laser mainly result from the regulation of polarization-dependent losses and the competition of net gain between π- and σ-polarization states.
4. Experimental results
The experimental setup is shown in Fig. 5, and a simple plane-concave cavity is employed. The pump source is a 792 nm fiber-coupled diode with a maximum output power of 3 W. The radius and numerical aperture of the fiber core are 50 μm and 0.22, respectively. A coupling optics system is used to focus the pump beam, and the pump spot radius in the crystal is about 85 μm. The a-cut Tm,Ho:LLF laser crystal has dopant concentrations of 5% Tm, 0.5% Ho with dimensions of 5 mm × 5 mm × 2.5 mm. The crystal is wrapped with indium foil and held in a brass heat sink, and the temperature of the heat sink is kept at 283 K. A dichromic coating on the front face of the crystal is high transmition at 792 nm, but is totally reflection at 2 µm. The other face is antireflection coated at 792 nm and 2 µm. The curvature radius and transmittance of the output coupler are 103 mm and 5%, respectively. The resonator is formed between the planar crystal front face and the output coupler, and the cavity length of is 55 mm.
Figures 6(a) and 6(b) show the output power characteristics of the Tm,Ho:LLF laser with two processes of increasing pump power and decreasing pump power, respectively. For increasing pump power, the output power as a function of pump power is shown in Fig. 6(a). When the pump power is increased to the on-threshold pump power Pon = 1.8 W from zero, the output power jumps from zero to a substantial level of 59 mW. Above this point, the output power increases linearly with the pump power, and the laser is purely π-polarized at 2069 nm, as shown in Fig. 7(a). In the process of decreasing pump power, the output power as a function of pump power is shown in Fig. 6(b). Decreasing the pump power from 2.2 to 1.96 W, the output power decreases with nearly the same slope and the laser is also purely π-polarized at 2069 nm. When pump power is decreased to 1.96 W, the σ-polarized laser starts oscillation. The measured optical spectrum for the σ-polarized laser is shown in Fig. 7(b), and the central wavelength is 2066 nm. Decreasing pump power from 1.96 to 1.75 W, the output power of σ-polarized laser increases from zero to 47 mW, however the output power of π-polarized laser decreases from 83mW to zero. The two orthogonal π- and σ-polarization states can coexist and complete the polarization switching in the range of 1.96-1.75 W. Further reducing the pump power to the off-threshold Poff = 1.6 W, the output power of σ-polarized laser drops from 17 mW to zero. So the PCS is experimentally realized in the free-running optical bistability Tm,Ho:LLF laser. The experimental results are in qualitative accordance with the numerical simulation results shown in Fig. 3, and the present theoretical analysis is confirmed to be reasonable.
In summary, the polarization coexistence and switching between the orthogonal π- and σ-polarization states in the free-running optical bistability Tm,Ho:LLF laser is first theoretically and experimentally investigated. Based on the gain spectra and the coupled rate equations, the output power as a function of pump power is numerically simulated. The theoretical results predict that, in the process of decreasing pump power, the polarization coexistence and switching in the optical bistability Tm,Ho:LLF laser can be realized by selecting suitable mode-to-pump ratio and nondiffraction losses. The physical mechanism of polarization coexistence and switching is clarified, and it mainly resulted from the competition of polarization-dependent net gain between π- and σ-polarization states. The polarization coexistence and switching in the optical bistability Tm,Ho:LLF laser is experimentally realized, and the experimental results validate the present rate equation model. The theory model can also be used to other kinds of (quasi-) three-level optical bistable lasers with polarization coexistence and switching.
National Natural Science Foundation of China (61775166 and 61275138); 111 Project to the Harbin Engineering University (B13015); Program for Innovatative Research Team in University of Tianjin (TD13-5035).
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