1. Introduction

Since the first observation of optical bistability in the 1970s, this phenomenon has attracted substantial attention because of its usefulness for controlling the characteristics of laser radiation and its potential use in all optical circuits and optical information processing [1]. Compared with the mechanism of bistability in a passive system, that in a laser (active) system is complicated, because the condition of population inversion must be satisfied and multiple modes may lase simultaneously. Lugiato et al. theoretically predicted the bistable operation of a laser with an intracavity absorbing medium [2]. Bistability has been experimentally observed in lasers with a saturable absorber [3] and an electro-optical birefringent tuner [4]. Because of applications in optical communications and photonic switching, bistable laser diodes have attracted considerable attention [5]. In solid-state lasers, a bistable laser based on stable–unstable cavity configuration transitions of an active optical resonator was first proposed in a Nd:GGG laser [6]. Recently, bistable laser systems have been observed in Yb-doped vanadate lasers [7,8], Tm,Ho:YLF lasers [9], and Nd-doped vanadate lasers [10].

Among the different bistabilities, polarization bistability is an interesting phenomenon occurring in laser diodes [11–13], associated with a two-mode bistability in orthogonal polarizations and dual wavelengths. Various mechanisms induce different types of bistabilities and bifurcations, and this finding has received considerable attention [12]. Conventional S-shaped bistability and pitchfork bifurcation bistability depend on the values of self- and cross-gain saturation coefficients. Measurements of the nonlinear coupling between orthogonally polarized states in Nd-doped and semiconductor lasers have been reported previously [14–16]. However, dual-wavelength lasing in a solid-state laser is a common phenomenon and can exhibit significantly different wavelengths; for example, the σ- and π-polarization state emissions exhibit a wavelength difference of tens of nanometers [7] while various excited-state emissions (⁴F_3/2→⁴I_11/2 and ⁴F_3/2→⁴I_13/2) exhibit a wavelength difference of hundreds of nanometers [17]. Determining whether polarization bistability exists in these high-wavelength-difference systems warrants further investigation. In a bistable Yb-doped vanadate laser [7], coexistence and switching of polarization states occur when only the pump power is reduced. In a fiber laser reported in a previous study, the residual power used in a pump-bypassed Yb-doped fiber laser for pumping the other Er/Yb co-doped fiber achieved bistability corresponding to two gain media with wavelengths of 1.04 and 1.537 μm; however, polarization was not discussed [18]. To the best of our knowledge, polarization bistability switch in a high-wavelength-difference and single-gain-medium solid-state laser system have not been reported.

In this study, polarization bistability for wavelengths of 1064 and 1342 nm, corresponding to the ⁴F_3/2→⁴I_11/2 and ⁴F_3/2→⁴I_13/2 transitions, respectively, was experimentally demonstrated in a Nd:YVO₄ laser. Because dual wavelengths share the upper level state ⁴F_3/2 population and the difference between the cross-sections for the two wavelengths is high, a variation in the pump power induces complex and fast-varying mode competition, which is not desired for exploring bistability. When the two wavelengths lase simultaneously, the gains of dual-wavelength modes are close to each other. Therefore, the loss at 1064 nm, incurred when using an intra-cavity electro-optic (EO) periodically poled lithium niobate (PPLN) Bragg modulator [19], could enable finely tuned mode competition as a control parameter. Hysteresis enables inversely switching between the 1064 and 1342 nm lasers with orthogonal polarizations by increasing and decreasing the applied voltage of PPLN to modulate the intra-cavity loss at 1064 nm.

2. Experimental setup

Figure 1(a) shows the experimental setup. An 808 nm and 16 W diode laser was collimated by using an optical imaging accessory to yield a laser spot with a diameter of approximately 450 μm in the focal plane, which was used for pumping the Nd:YVO₄ lasers. The c-cut Nd:YVO₄ crystal measured 3×3×8 mm³ with 0.5 at.% Nd³⁺ doping and an antireflection broadband coating from 800 to 1450 nm on both sides. The laser crystal was mounted on a water-cooled cooper block with wrapped indium foil, which was water-cooled at 18 °C. A T-type cavity configuration was designed for exploring the dual-wavelength competition, which could independently vary the overlap integral between the cavity mode and the pump mode for each wavelength to achieve the tunable output power ratio [20]. The mirror M₁ was one of the cavity mirrors for both wavelengths. One side of M₁ had a high-transmission coating at 808 nm, and the other side had a high-reflection coating at 1064 and 1342 nm and a high-transmission coating at 808 nm. A dichroic beam splitter (BS) with an HT coating at 1342 nm and an HR coating at 1064 nm distinguished dual wavelengths and functioned as a T-type cavity. An output coupler, OC₁, of the 1342 nm laser was a spherical concave mirror with a radius of curvature of 200 mm and a reflection of 99%. A plano-concave cavity configuration was formed for the 1342 nm laser. The same plano-concave cavity configuration was formed for the 1064 nm laser; the OC₂ had a radius of curvature of 150 mm and a reflection of 90%. The terms F₁ and F₂ represented 1342 nm and 1064 nm bandpass filters, respectively.

 

Fig. 1 (a) The experimental setup. (b) The transmission of EO PPLN versus the drive voltage at m = 0.

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The PPLN dimensions were 10 mm ($x^{\prime }$)$\times$15 mm ($y^{\prime }$)$\times$2 mm ($z^{\prime }$), where $x^{\prime }$, $y^{\prime }$ and $z^{\prime }$ were the general coordinate axes of an EO crystal, as shown in Fig. 1(a). The PPLN achieved a Bragg grating by cascading several Pockel cells of periodically poled lithium niobate with periodic intervals of 20.3 μm along the $x^{\prime }$ direction and arranging each adjoining cell in the opposite orientation. The z-cut surfaces of the PPLN were coated on the electrodes and a voltage was applied. The EO effect induced a periodic refractive index modulation in the PPLN to form a Bragg grating set. The variation in the refractive index of each Pockel cell is given by$\Delta n_{z^{\prime },x^{\prime }}=-\frac{n_{z^{\prime },x^{\prime }}^3{r}_{33,13}{E}_{z^{\prime }}S(x^{\prime })}{2}$, where$n_{z^{\prime }}$ and $n_{x^{\prime }}$ are the extraordinary and ordinary indices, respectively; $r_{33}$ and $r_{31}$ are the Pockel coefficients of extraordinary and ordinary waves, respectively; $E_{z^{\prime }}$ is the applied electric field; and $S(x^{\prime })=\pm 1$ denotes the sign of the domain orientation of each Pockel cell of the PPLN crystal as a periodic function of $x^{\prime }$. The characteristics of the Bragg grating follow the Bragg condition; in other words, 2Λsinθ_B,_m = mλ₀/n, where m is the diffraction order, λ₀ is the incident laser wavelength, n is the average refractive index of the Bragg grating, and Λ is the grating period. When the PPLN grating period is 20.3 μm and the first diffraction order (m = 1) is adopted, the Bragg angles are 0.7° and 0.67° for the extraordinary ray (e-ray) and ordinary ray (o-ray), respectively, at 1064 nm. Because the e-ray exhibits a high Pockel coefficient in lithium niobate, the Bragg angle was aligned to 0.7° to achieve an increased diffraction loss. Figure 1(b) shows the 1064-nm zeroth order transmission T_p of the EO PPLN Bragg modulator as a function of the applied voltage V_a. The transmission was normalized to that at V_a = 0. The half-wave voltage of this EO PPLN was 658 V. When the PPLN was added in the 1064 nm cavity, controlling V_a for varying the intra-cavity transmission of 1064 nm enabled achieving the tunable intra-cavity loss in the $x^{\prime }$ direction.

3. Experimental results

This study focused on the competition between ⁴F_3/2→⁴I_11/2 and ⁴F_3/2→⁴I_13/2 transitions. Because dual wavelengths share the same upper level state, a dual-wavelength laser was realized by controlling the gain extraction efficiency and intra-cavity loss to balance the competition between the two wavelengths. The extracting efficiency depends on the overlap integral between the pump and cavity distributions; therefore, a T-type cavity independently tuned the dual cavities to and enabled the determination of the power extraction efficiencies for simultaneous emission at 1064 and 1342 nm. For V_a = 0 V, the slope efficiencies were 25.96% and 3.92%, the thresholds were 2.5 and 3.5 W, the maximal levels of output power were 1481.3 and 316.0 mW at the pump power of 10 W, and the cavity lengths were 15.87 and 12.12 cm for 1064 and 1342 nm cavities, respectively. Furthermore, the intra-cavity EO PPLN finely controlled the intra-cavity loss of the emission at 1064 nm and enabled exploring the competition between these two wavelengths.

The intra-cavity loss of the 1064 nm cavity was induced by varying the applied voltage of the PPLN; Fig. 2 shows the dependence of hysteresis on V_a at P = 7.5 W. The hysteresis occurred when the applied voltage of the PPLN varied between 110 and 230 V for dual wavelengths. In this study, the hysteresis depended on the dual wavelengths and was also associated with various polarizations; in other words, polarization bistability, accompanied by two excited-state transitions, occurred. Figures 2(b) and 2(c) show the plots of the output power versus the applied voltage of the PPLN in x and y polarizations, respectively. Hysteresis was observed in the x-polarized 1064 nm and y-polarized 1342 nm lasers.

 

Fig. 2 The output power versus the applied voltage of the PPLN (a) without adding polarizer, (b) with polarizer in x polarization, and (c) with polarizer in y polarization.

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The bipolarization characteristics were further explored. The Fabry–Pérot spectrum at approximately 1064 nm and the patterns at V_a = 0 V are shown in Figs. 3(a) and 3(b), respectively; the corresponding polarizations were verified by adding a polarizer before the Fabry–Pérot interferometer and CCD. In Fig. 3(a), the total cavity length (L) of the 1064 nm light was fixed at approximately 15.87 cm to achieve longitudinal mode spacing at approximately 945 MHz, which was verified with the RF spectrum. A frequency shift of 870 MHz was observed between x- and y-polarized 1064 nm lasers. The frequency shift implies that the orthogonal bipolarization likely resulted from the thermally induced birefringence of a laser crystal, and the index difference induced differences between the spacing of the longitudinal modes in the x and y directions. Because the order of the axial mode was high, a different order of the axial mode in the x and y directions caused frequencies around the maximal gain to lase. In x polarization, the lasing mode was approximately TEM₀₀, as shown in Fig. 3(b). Although, the lasing modes were dominated by two longitudinal modes in y polarizations, the pattern still resembled a central spot. A similar pattern was observed for the y-polarized 1342 nm laser. However, the pattern of the x-polarized 1342 nm laser was dominated by TEM₁₁, and the output power was low. This indicates that the gain was obtained from the residue of the spatial gain distribution in the x polarization. Similar spectra and patterns were observed in the hysteresis region.

 

Fig. 3 (a) The Fabry-Perot spectrum for 1064 nm and (b) the patterns at V_a = 0, in which “pol.” represents polarization.

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At V_a = 0, the 1064 nm light lased in both polarizations, whereas the 1342 nm light mainly lased in the y polarization. Therefore, the gain competition values for the x- and y-polarized 1064 nm lasers and the y-polarized 1342 nm laser were close and greater than that for the x-polarized 1342 nm laser. The output power of the x-polarized 1342 nm laser was under 15 mW for various V_a, and the mode competition inducing variations in the output power could be neglected. This initial condition is easily satisfied in a simultaneously dual-wavelength Nd-doped laser, because the emission cross- section of the 1064 nm light is greater than that of the 1342 nm light. Applying an electric field to the PPLN mainly induced the phase-matching diffraction loss in the y polarization (contributed by the r₃₃ Pockel coefficient) and a slight generation of the nonphase-matching diffraction loss in the x polarization (contributed by the r₃₁ Pockel coefficient) for the 1064 nm laser. On the basis of the mode competition, the output characteristics in Fig. 2 can be divided into three regions depending on V_a, namely V_a ≤ 40 V, 40 V < V_a < 80 V, and V_a ≥ 80 V.

For V_a ≤ 40 V, the PPLN transmissions in both polarizations exceeded 99%, as shown in Fig. 1(b). The competition between the x- and y-polarized 1064 nm lasers dominated the output characteristics, because the loss induced by the PPLN diffraction was not sufficient to support the extraction efficiency superiority at 1342 nm. For the 1064 nm laser, the diffraction loss in the y (two-longitudinal-mode) polarization suppressed one longitudinal mode and equivalently increased the power extraction efficiency of the other longitudinal mode, making it higher than that of the x-polarized laser, resulting in an increase in the output power with increasing V_a. For 40 V < V_a < 80 V, the phase-matching diffraction loss in the y polarization increased. Therefore, the output power of the y-polarized 1064 nm laser decayed quickly with increasing V_a, and the cross-saturation effect [18] enhanced the x-polarized 1064 nm laser. Moreover, the power extraction efficiency of the 1342 nm laser gradually increased compared with that of the 1064-nm laser in the x polarization, because nonphase-matching diffraction loss influenced by the r₃₁ Pockel coefficient of PPLN increased slowly in the x-polarized 1064 nm laser.

For V_a ≥ 80 V, the system gradually became bistable with increasing V_a. The behavior of the dual-wavelength laser might have been governed by the cross-saturation effect; this can be further discussed on the basis of the intensity growth equations of motion [21,22]. The N-mode intensity equations can be expressed as follows:

where I_n is the dimensionless intensity for mode n and n = 1,2,…,N, α_n is a small signal or unsaturated gain minus loss, θ_nm with n = m is the self-saturation coefficient that is also represented as β_n, and θ_nm with n≠m is the cross-saturation coefficient. For two-mode operation, the laser will be a bistable system if the coupling constant $C=({\theta }_{12}{\theta }_{21})/({\beta }_1{\beta }_2)>1$ (strong coupling) and $({\beta }_1{\alpha }_2/{\theta }_{21})<{\alpha }_1<({\theta }_{12}{\alpha }_2/{\beta }_2)$ (unstable condition) by using the linear stability analysis to explore the stability of a steady-state solution [21,22]. The coupling constant between two orthogonal modes can be measured by using methods such as an anisotropic feedback scheme [23] and adding intra-cavity loss scheme [24]. When a plano-concave cavity was used for measuring the coupling constant between two orthogonal 1064 nm modes based on the feedback scheme [23], the coupling constants for a c-cut Nd:YVO₄ gain medium were 0.844 ± 0.026 and 0.858 ± 0.023, associated with x- and y-direction modulation, respectively. These values imply that the polarization bistability was not easily implemented in a two-mode Nd-doped vanadate laser. Indeed, we did not observe polarization bistability of 1064 nm wavelength under loss control in experiments. However, the complicated coupling could induce bistability for 3-mode operation. Based on the experimental results, we could neglect the mode competition of the x-polarized 1342 nm mode because of the high-order mode and low-output-power lasing only. The behavior of the dual-wavelength laser was governed by the cross-saturation effect in the x-polarized 1064 nm, y-polarized 1064 nm, and y-polarized 1342-nm emissions for which the corresponding suffixes in Eq. (1) were labeled as 1, 2, and 3, respectively. Figure 4 shows the steady-state intensity as a function of the y-polarized diffraction loss of the PPLN, L_p. Because adding the PPLN voltage decreased α₁ and α₂ with different weighting, we assumed that ${\alpha }_1=0.5-0.132\times L_p$ and ${\alpha }_2=0.5-L_p$. From Fig. 1, L_p $\propto$(1-T_p) is not a linear function of V_a, and the factor of 0.132 was estimated by the loss ratio at V_a = 500 V and was fixed to simplify the simulation. The other parameters were α₃ = 0.6, θ₁₁ = 0.3, θ₂₂ = 0.3, θ₃₃ = 0.5, θ₁₂ = θ₂₁ = 0.275, θ₁₃ = θ₃₁ = 0.4, and θ₂₃ = θ ₃₃ = 0.4. For L_p < 0.048, the intensity of the y-polarized 1064 nm laser, I₂, decayed to zero with increasing L_p and the cross-saturation effect enhanced the intensity of the x-polarized 1064 nm laser, I₁. A bistable region exists as 0.157 ≤ L_p ≤ 0.375, but $C=({\theta }_{12}{\theta }_{21})/({\beta }_1{\beta }_2)=0.84$ is less than 1. In the three-mode case, varying C by simultaneously tuning θ₁₂ and θ₂₁ will dominate the decay of I₂ for increasing L_p, i.e., for large C I₂ quickly decays to zero with increasing L_p. The evolution of intensity agrees with the experimental results for V_a > 50 V. However, a bistability with a binary 0/1 type may not satisfactorily match the experimental results, which could result from the existence of multiple modes in y polarization, as shown in Fig. 3(a). Based on the spectral hole burning, a relatively high cross-saturation exists when the net gains experienced by the two polarized eigenstates are close. Thus, the other mode in y polarization has a weak cross-saturation relating to other modes and retains some power for forming the base. In practice, three-dimensional cross-saturation coefficients based on the spatial hole burning were further estimated and modified in a solid-state laser [25]. Moreover, conventional S-shape and pitchfork bifurcations, depending on self- and cross-saturation coefficients, were found in laser diodes when the rate equation including gain saturation was further considered [5]. These observations warrant further detailed modeling studies.

 

Fig. 4 The simulation result for the steady-state intensity as a function of the y-polarized diffraction loss of the PPLN, L_p, based on the three-mode operation.

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According to Fig. 2 in which P = 7.5 W, hysteresis starts at V_L = 110 V (T_p = 96.25%) and ends at V_H = 230 V (T_p = 82.75%) for increasing V_a, where the size of hysteresis is defined as V_H − V_L; V_H − V_L = 110 V. Figure 5 shows the size of the hysteresis as a function of P. The size increases with increasing P. When bistability occurs, I₂ approachs zero. According to the aforementioned unstable conditions for a two-mode operation, an increase in P increases the gain and α₂, resulting in an increase in the size of hysteresis.

 

Fig. 5 The size of the hysteresis as a function of the pump power

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4. Conclusion

On the basis of the polarization bistability associated with a two-mode or dual-wavelength bistability in orthogonal polarizations, a scheme controlling the loss of one wavelength was proposed for achieving a high-wavelength-difference polarization bistability. Because the dual wavelengths associated with the ⁴F_3/2→⁴I_11/2 and ⁴F_3/2→⁴I_13/2 transitions share the upper level state, using PPLN to control the loss at 1064 nm enables coupling, varying the mode competition between 1064 and 1342 nm lasers, and achieving polarization bistability in a c-cut Nd:YVO₄ laser. For the pump power of 7.5 W, a hysteresis loop was observed for the applied PPLN voltage between 110 and 230 V and the PPLN transmission between 82.75% and 96.25%. The size of the hysteresis increased with increasing the pump power.