1 Introduction

Efficient, stable, thulium lasers operating at wavelengths near 2 μm are highly desirable for a variety of applications, including surgery [1], as pump sources for holmium-doped lasers [2, 3] and Cr2+:ZnSe lasers [4], for the production of 3–5-μm light using OPOs, and for eye-safe LIDAR. For many of these applications, it is vital that the frequency and output power of the 2 μm laser is stable.

The Tm:YAlO3 (or Tm:YAP) gain medium is a good candidate for these applications as it is highly efficient [3, 5], and 795-nm pump laser diodes are readily available. It also has thermo-mechanical properties that are similar to those of YAG. Further, its natural birefringence facilitates the production of polarized laser beams and minimizes the impact of stress-induced birefringence. However, continuous-wave (CW) Tm:YAlO3 lasers often exhibit strong self-pulsing [6–8]. The associated frequency instability, along with the risk of damage due to the high peak power of these pulses, renders these lasers unsuitable for use as master lasers and for use in many of the applications listed above.

Various mechanisms have been proposed to explain the self-pulsing in Tm:YAlO3 lasers, including pump power fluctuations [9] and mechanical instabilities in the laser [10]. Intra-cavity saturable absorption can also produce self-pulsing [11], and various authors have proposed that self-pulsing in Tm:YAlO3 lasers is due to saturable absorption within the gain medium caused by up-conversion and excited state absorption [4, 6, 12].

Razdobreev and Shestakov [6] observed that stable output could be obtained at low normalized pump rates for a monolithic laser, but that a regular self-pulsing occurred at higher pump powers. They also reported that a rate equation model that incorporated phonon-assisted excited state absorption (ESA) could explain this behavior. Subsequently, Šulc et al. [10] compared the behavior of monolithic and external resonator Tm:YAlO3 lasers that used identical gain media and pump source, and exhibited the same threshold and slope efficiency. They observed that the output of the monolithic laser was stable over a broad range of normalized pump rates, but the beam quality of the output was degraded significantly for output powers >1 W. The output of the external resonator laser pulsed irregularly. It is unclear why the two monolithic Tm:YAlO3 lasers behaved differently. However, one conclusion that can be made is that resonator mechanical instabilities can lead to irregular self-pulsing and that the regular pulsing observed by Razdobreev et al. is due to a different effect, one perhaps specific to their gain medium.

Nevertheless, as observed by Šulc et al. [10], it is not possible to produce output powers greater than a few watts with good beam quality using a monolithic resonator. Thus, a technique that will reliably produce stable outputs in Tm:YAlO3 lasers with good beam quality for all power levels is required. This can be achieved by actively suppressing the self-pulsing via feedback. Intra-cavity acousto-optic modulators (AOMs) have been used to stabilize the output via loss modulation [7, 8]. However, this method introduces additional cavity losses and complication.

Current feedback to the pump diode laser has been used to eliminate low modulation depth intensity noise in other dopants and hosts, such as Nd:YAG [13, 14] and co-doped Tm–Ho:YAG [15] lasers. However, the self-pulsing in Tm:YAlO3 lasers results in almost 100 % modulation of the output power, which is much larger than the modulation observed in those lasers [16]. Pulse suppression of such strong modulation has only been shown at very low power in a four-level Nd-doped fiber laser [17].

In this paper, we show for the first time that the self-pulsing in Tm:YAlO3 can be suppressed reliably by using proportional current feedback to the pump diode laser only. The feedback can be implemented while the laser is operating at full power and exhibiting 100 % modulation. It achieves complete suppression, up to the maximum output power of 6.5 W, which was limited by the pump source.

This type of feedback eliminates the need for additional intra-cavity components, making it suitable for both monolithic lasers and external resonator lasers. It also ensures CW operation regardless of operating power and should be scalable to high power. We also present the first measurement of the Tm:YAlO3 laser transfer function and show that the self-pulsing can be modeled as a dynamically unstable relaxation oscillation.

2 The Tm:YAlO3 laser system

The Tm:YAlO3 laser configuration used in these experiments is shown in Fig. 1. The gain medium is a 4 at.% Tm:YAlO3 crystalline cuboid with dimensions 4.55 mm (a axis) × 5.02 mm (b axis) × 5.54 mm (c axis) (Pnma basis). A 790-nm fiber-coupled diode, with a core diameter of 100 μm and an N.A. of 0.22, was used to pump the gain medium. The output from the fiber was imaged into the crystal via a 2× magnifying lens system, resulting in a 200-μm-diameter spot within the crystal.

Fig. 1
figure 1

Schematic of the Tm:YAlO3 laser

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The laser resonator consists of a flat mirror that has high reflectivity (>99.6 %) at 1.94 μm and high transmission at 790 nm, the gain medium, an intra-cavity lens to compensate for thermal lensing, and a flat output coupler with a reflectivity of 95 % at 1.94 μm. The fundamental mode waist radius of the resonator was estimated to be 100 μm.

The ends of the gain medium are anti-reflection (AR) coated for both 790 nm and 1.94 μm, while the intra-cavity lens is AR-coated for 1.94 μm. The laser gain medium absorbs 60 % of the incident pump light. The laser output and the unabsorbed pump light are separated using a dichroic mirror. An extended range InGaAs photodetector (PD) that has a 25-ns rise/fall time is used as the sensor for the feedback loop.

3 Free-running laser output

The average output power of the free-running Tm:YAlO3 laser is plotted versus the absorbed pump power in Fig. 2. The laser threshold of 1.5 W and slope efficiency of 56 % are comparable to the best published values [8, 18]. As expected [5], the output of the laser was c axis polarized with a wavelength of about 1.94 μm. The output had a measured beam quality, M 2 = 4, which remains approximately constant with power.

Fig. 2
figure 2

Average output power of the free-running laser

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Large amplitude self-pulsing of the output power was observed, as shown in Fig. 3. It consists of an exponentially growing burst of an oscillation that has a frequency of about 100 kHz. The frequency of the oscillation was measured at different laser output powers for different cavity lengths; the results are plotted in Fig. 4. These data show that the frequency is proportional to the square root of the laser output power, as expected for a stable resonant relaxation oscillation [19], with a shift to higher frequencies for shorter cavity lengths.

Fig. 3
figure 3

Top A plot of the self-pulsing of the laser output. Bottom A plot of a single ‘burst’ with improved temporal resolution, showing the high-frequency oscillation within the pulse envelope, and an exponential fit to the envelope

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Fig. 4
figure 4

Plot of the oscillation frequency versus the square root of the output power, for two resonator lengths

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Thus, it appears as though the self-pulsing is due regenerative amplification of small fluctuations at the frequency of the relaxation oscillation of the laser, which grows until it dominates the output and is quenched.

4 Suppression of self-pulsing

A schematic of the feedback loop used to suppress the self-pulsing is shown in Fig. 5. The photodetector shown in Fig. 1 is used as the sensor for the loop. The signal from the photodetector is amplified by the AC-coupled pre-amplifier and then converted into a current, which is injected directly into the pump diode. Note that the electronic components were all non-inverting, due to the phase delay resultant from the cross-relaxation lifetime in Tm:YAlO3.

Fig. 5
figure 5

Schematic of the feedback system

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A comparison between the open-loop (free-running) and closed-loop output of the laser is shown in Fig. 6. In closed-loop operation, the fluctuations in the output power were similar to the photodetector noise floor. Stable CW output was achieved for all operating powers by adjusting the gain of the feedback loop. Stability was insensitive to the exact value of the gain, as further increase in the feedback gain beyond the threshold for suppression did not result in any instability. The average output power, beam quality, and the output spectrum of the laser were unchanged by the application of feedback.

Fig. 6
figure 6

Laser output for the Tm:YAlO3 laser when free-running (left) and when feedback is active (right)

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5 Transfer function

The transfer function of the Tm:YAlO3 laser was measured by injecting a small electrical oscillation into the pump diode while the self-pulsing was suppressed, and comparing the optical outputs of the laser and pump diode. The result is plotted in Fig. 7; it shows a significant resonant response near 90 kHz, as expected for a relaxation oscillation, but with a phase lead of ~180°. This type of transfer function is characteristic of an unstable oscillator, which has a negative damping coefficient. In the time domain, this results in exponential growth of small perturbations at the eigen frequency, as observed in Fig. 3.

Fig. 7
figure 7

Measured transfer function of the Tm:YAlO3 laser and that predicted by Eq. 1, at 2-W laser output power

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The laser transfer function, \( G_{\text{laser}} (s) \), is therefore described by a model that includes two components: a low-pass filter corresponding to the time constant of the cross-relaxation into the upper lasing level (3H4 + 3H6 → 3F4 + 3F4) and a negatively damped oscillator corresponding to the unstable relaxation oscillation:

$$ G_{\text{laser}} (s) = \frac{{\omega_{\text{CR}} }}{{s + \omega_{\text{CR}} }} \cdot \frac{{G\omega_{\nu }^{2} }}{{s^{2} + 2\zeta \omega_{\nu } s + \omega_{\nu }^{2} }} $$
(1)

where ω CR = 2π/τ, τ is the cross-relaxation time constant, ω ν and ζ are the natural angular frequency and damping constant of the relaxation oscillation, respectively, and G is the low-frequency gain. The values of G, ω v , and ζ were estimated from the measured data, giving G = 0.7, ω v  = 2π(87,000 Hz), and ζ = 0.025, while τ was estimated to be 3 μs [20]. The transfer function in Eq. 1 is plotted in Fig. 7, showing general agreement with the measured transfer function.

As indicated above, only proportional feedback is required to damp the self-pulsing. The s-plane plot of the poles of the closed-loop transfer function of the model at various feedback gains is shown in Fig. 8. The pole locations for a near-zero cross-relaxation lifetime and for a loop that includes an additional low-pass filter at 200 kHz are also shown. At low gain, the poles are approximately those of the open-loop transfer function. If the cross-relaxation is nearly instantaneous, the poles remain on the right-hand side of the s-plane and the system remains unstable for all feedback gains. However, for a 3-μs cross-relaxation lifetime, increasing the gain results in the poles moving into the left-hand half of the s-plane. Thus, the nonzero cross-relaxation lifetime ensures that simple proportional feedback can be used to damp the relaxation oscillation, as observed. Introducing an additional low-pass filter into the feedback loop would further assist the damping, as shown in Fig. 8, but is not necessary.

Fig. 8
figure 8

s-plane plot of the relaxation oscillation poles of the closed-loop transfer function of the model at different feedback gains for three different cases: a near-zero cross-relaxation lifetime, a 3-μs cross-relaxation lifetime [16], and a 3-μs cross-relaxation lifetime with an additional low-pass filter at 200 kHz. The gains used to calculate the locations of the poles ranged from 0 at the open-loop pole location through 0.1, 0.2… to 0.99

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6 Conclusion

We have described an efficient Tm:YAlO3 laser that had a threshold of 1.5 W and a slope efficiency of 56 % and exhibited self-pulsing, similar to that observed by other authors. We have shown, for the first time, that the irregular self-pulsing in this laser can be suppressed by proportional feedback to the pump diode laser. While the use of a monolithic resonator results in a stable output, that architecture is not suitable for high-power, diffraction-limited lasers. Our current feedback approach, by contrast, can be applied to all architectures, does not require additional intra-resonator optics, and is easy to implement.