We demonstrated phase-locking in a laser-diode-array-pumped Nd:YVO4 laser array (15 emitters) using a Talbot cavity. The Nd:YVO4 slab crystal was coated by dielectric material for claddings and formed a planar waveguide for the vertical mode. To stabilize the horizontal array mode, periodical thermal lenses were generated by controlling the heat flow. The phase-locked waveguide array generated 1.65-W output power, while 2.02 W was available in a standard cavity. Two-peak supermode was demonstrated with the Talbot cavity and was converted to a single peak with a spatial light modulator. We also experimentally and numerically analyzed the characteristics of Talbot phase-locking.
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High-power lasers are required for many applications, and there is a continuous desire to scale lasers to higher output powers while maintaining near diffraction-limit beam quality. Unfortunately, as power levels increase, undesirable thermo-optic effects frequently degrade the beam quality. One promising technology to overcome this problem is combining a large number of relatively low-power beams into a single high-power beam, especially coherent beam combining that uses phased beams. To coherently combine beams from an array laser, we must develop a selective feedback mechanism that supports only one specific supermode. The Talbot effect  was studied intensively as a selective feedback mechanism for laser-diode (LD) arrays [2–5] and CO2 lasers [6,7] in the 1990s. More recently, a Talbot cavity with multicore fiber lasers has been studied [8–12] because it has an exciting possibility to extend the power . The Talbot cavity is useful to pursue higher output powers while maintaining near diffraction-limit beam quality. On the other hand, stability and compactness are crucial for industrial use. The Talbot cavity is also useful to develop very compact lasers with high intensity, and so such a laser system is suitable for such industrial products as a laser television .
In our previous work, we demonstrated an LD-pumped Nd:YAG microchip laser array with a Talbot cavity . However, the power extraction efficiency in the Talbot cavity laser was very low due to the low gain of the Nd:YAG laser pumped by a 10-W LD bar. In this paper, we developed a Talbot cavity laser with an LD-array-pumped Nd:YVO4 module. Nd:YVO4 crystal is more suitable as a microchip laser because of its higher stimulated emission cross section than that of Nd:YAG . Our Nd:YVO4 modules have waveguide structures that also improve the spatial mode matching and the power extraction efficiency [17,18]. Waveguide lasers have several advantages, for example, high optical gain and low threshold power because high density excitation can be achieved in a small volume where the cavity mode is also confined . Output powers >250 W have been reported with a planar-Yb:YAG-core waveguide . Therefore, we expect to improve the power extraction efficiency. We also report the characteristics of our Talbot cavity, especially the effect of partial reflective coating on the output-side surface of the Nd:YVO4 crystal. Finally, we investigate the effect on the Talbot cavity of other parameters such as the filling factor.
2. Laser-diode-array-pumped Nd:YVO4 module
The LD-array-pumped Nd:YVO4 module [17,18] is composed of an 808-nm LD array with 15 emitters on a 200-μm pitch, a 40-μm-thick and 1.5-mm-long Nd:YVO4 slab crystal, and a water-cooled heat sink. A schematic of our laser module is shown in Fig. 1. The Nd:YVO4 slab crystal is coated by dielectric material for claddings and forms a planar waveguide for the vertical mode. To stabilize the horizontal mode, we introduced a thermal lens by controlling the heat flow. To induce temperature modulation and form waveguide-like structures due to the thermal lens effect, the heat sink has linear grating structures in a 200-μm pitch that is made of laminated copper and connect with the Nd:YVO4 slab crystal by solder. The heat flow is stable enough around our operating conditions. The pump laser beams are directly launched into these waveguide-like structures from the rear surface of the Nd:YVO4 slab crystal with high-reflection (HR) at 1064-nm coating and anti-reflection (AR) at 808-nm coating. More than 95% of the pump laser beams are coupled to the waveguide-like structure and the propagation loss is negligibly low. We prepared two types of modules with different coating on the output surface of the Nd:YVO4 slab crystal. Module-A is an AR at 1064 nm and HR at 808-nm coating, and Module-B is a 40% partial reflector (PR) at 1064 nm and HR at 808-nm coating. We operated the LD array in a quasi-continuous wave mode of a 178-μs pulse-width and 1680-Hz repetition. The LD array emits 5.2 W at a 20-A LD current in the above conditions, and >99% of output power is absorbed by Nd:YVO4 crystal. Module-A produces 2.02-W output power at a 5.2-W absorbed pump power when a partial reflector of 70% is attached to the output surface of the Nd:YVO4 crystal. Module-B also produces 1.26 W without an external mirror in the same operating condition.
3. Experiment and results
The Talbot effect states that as a periodic optical wave propagates in free space, the wave repeats its initial field pattern after every characteristic distance, which is the so-called Talbot distance Zt, defined as Zt = 2d2/λ. Here, d corresponds to a center-to-center element spacing of an array image and λ corresponds to the wavelength; our module has a Talbot distance of Zt = 75.2 mm. The diffractive coupling of an array laser based on the Talbot effect is useful to discriminate array supermodes. Particularly, the highest order supermode (out-phase mode) is selectively excited when the round trip cavity length corresponds to half of the Talbot distance .
The schematics of our Talbot cavity laser are shown in Fig. 2. We performed three types of experiments. First, we performed a simple Talbot cavity experiment with Module-A (Experiment 1), which is shown in Fig. 2(a). We placed an AR-coated f = 10 mm cylindrical lens in the external cavity to suppress the vertical beam divergence. Second, we used Module-B instead of Module-A to reduce the residual cavity loss (Experiment 2). The main loss in Experiment 1 is the coupling loss between the spatial waveguide modes and the optical field in the free space. We changed the coating on the output surface of the Nd:YVO4 slab crystal from an AR coating to a 40% PR coating for oscillation with a low-reflective output coupler (OC) to decrease the optical energy in the free space. However, according to previous work by Lu et al.  and our numerical simulation in Section 4, the gain of each supermode easily reaches the threshold and starts to oscillate in such a three-mirror cavity, which prevents the Talbot cavity from stable out-phase-locking. Therefore, next we placed another PR mirror instead of the PR coating on Module-B, as Fig. 2(b) (Experiment 3). The cylindrical lens was also replaced by an f = 0.09 mm fast-axis-collimation (FAC) lens due to experimental difficulty. We intended to optimize the main cavity length between the rear surface of the Nd:YVO4 slab crystal and a central partial reflector so that the highest order supermode is stably excited.
Figure 3 compares the power performance of the Talbot cavity laser for the above three kinds of experiments with a non-Talbot condition using an R = 70% OC placed closely to the Nd:YVO4 output surface. To obtain the highest output power, the reflectivity of the OC is R = 70% in Experiment 1, R = 30% in Experiment 2, and R = 20% in Experiment 3. The optimized reflectivity of the OC in Experiment 1 is same as that in the non-Talbot condition because of the low loss in the external cavity. The external cavity lengths were chosen as maximizing output powers (typically ∼24 mm). Output power linearly depends on the absorbed pump power in both the Talbot and non-Talbot conditions. The output powers at a 5.2-W absorbed pump power are 1.65 W in Experiment 1, 1.81 W in Experiment 2, 1.55 W in Experiment 3, and 2.02 W in the non-Talbot condition. Although Experiment 2 successfully suppressed the loss in the Talbot cavity and increased the output power, the output power in Experiment 3 is less than in Experiment 1, probably because the collimated beam by the FAC lens is broadly vertical and the coupling loss between the off-axis waveguide modes and the optical field is not adequately suppressed. The setup in Experiment 3 is unsuitable for developing a high intensity Talbot laser. The slope efficiency in non-Talbot condition is 48%, and with the Talbot cavity condition, the slope efficiencies are both 40% in the Experiment 1 and 3. However, the same efficiency as that in the non-Talbot condition was obtained in the Experiment 2. Since the thermal lens supporting the waveguide in the laser crystal is weak at lower pump power, the slope efficiencies are lower near the laser threshold for all the cases.
We placed an f = 200 mm spherical lens at its focal length from the Nd:YVO4 crystal to form a far-field image of the phased array beam generated from the Nd:YVO4 waveguides. Figure 4(a) corresponds to the far-field image in Experiment 1 with an R = 70% OC, whose transverse pattern has two sharp peaks that indicate out-phase-locking. The distance between these two peaks is 1.07 mm, which is close to the theoretical value of 1.064 mm estimated from fλ/d; here f corresponds to the focal length of the imaging lens. Figures 4(b) and 4(c) correspond to Experiment 2 with an R = 30% OC and Experiment 3 with an R = 20% OC, respectively. The transverse far-field pattern in Fig. 4(b) has an ambiguous two-peak, which is considered that the Talbot cavity oscillates several supermodes including the highest order supermode. The two-peak is clearer since the reflectivity of OC is higher [Fig. 4(d)], which corresponds to Experiment 2 where R = 60% OC. The transverse far-field pattern in Fig. 4(c) has a clearer two-peak than Figs. 4(b) and 4(d), although Fig. 4(a) is sharper. For comparison, Fig. 5(e) shows the far-field image in non-Talbot condition. Even though experiment 3 successfully improved the supermode selectivity from Experiment 2, it cannot surpass Experiment 1 in either supermode selectivity or output power.
As a function of the external cavity length at 20-A LD current (5.2-W absorbed pump power), the output power is shown in Fig. 5. The reflectivity of an OC is R = 70% in Experiment 1 and R = 30% and 60% in Experiment 2. In Experiment 1, an out-phase beam profile like Fig. 4(a) is observed in the whole cavity length ranging in 15–35 mm when OCs R > 50% are used, which is limited by the travel length of a translation stage. In Experiment 2, we plotted in Fig. 5 only the case where a two-peak is observed, like Figs. 4(b) or 4(d). The cylindrical lens in the cavity affects the transverse diffraction like a ∼5.9-mm-thick glass substrate. Therefore, when external cavity length xd is set to 21.8 mm, the round trip cavity length corresponds to half of the Talbot distance. We only observed a two-peak around half of the Talbot distance. In all the cases plotted in Fig. 5, the output power does not vary much with the external cavity length.
Therefore, although Experiment 2 successfully suppressed the coupling loss, phase-locking is difficult. Experiment 1 is suitable for our objective of developing a power scalable laser while maintaining good beam quality. Experiment 1 reduced the laser output power by ∼20% from the non-Talbot condition. We also observed that output power and Talbot phase-locking are insensitive to external cavity length for our module and cavity setup, which we also investigated by numerical calculation in Section 4.
The two-peak-far-field pattern can be converted to a single-peak pattern by a spatial light modulator (SLM). As depicted as Fig. 6(a), an output beam from the Talbot cavity (Experiment 1) passes through a 4- f lens system to form an image of the array. The SLM (Cambridge Research & Instrumentation, SLM-128) has 128 pixels pitched at 100 μm, and two pixels correspond to the array pitch. When π phase modulation is applied to every two arrays, we obtained the conversion of the far-field image from the out-phase array to the in-phase array. The single peak far-field image obtained by this phase modulation is shown in Fig. 6(b). Figure 6(c) compares the transverse profile of Fig. 6(b) with theoretical far-field profiles of the in-phase and incoherent arrays. Although the diffraction width of the experimental in-phase array was 0.4 mrad, which is slightly wider than the 0.31 mrad in the theoretical in-phase profile, the diffraction width is much narrower than the theoretical incoherent array. We measured the beam radius of the central single peak and the corresponding M2 factor was estimated to 1.6, although it is not adequate to describe the non-Gaussian beam propagation with a M2 factor. Therefore, we successfully converted from an out-phase to an in-phase array.
4. Numerical analysis
4.1. Numerical analysis of experiments
We analyzed the mode threshold gains for a fifteen-element array as a function of external cavity length by solving the matrix equation reported by Lu et al. . Although the matrix equation resembles one reported by Mehuys et al. , the nonzero reflectivity of Rf (reflectivity of array’s output surface) is taken into the equation. We calculated both experiments with Module-A and Module-B for comparison. The common parameters in both experiments are set as an array length of l = 1.5 mm, filling factor of 0.5, and an amplitude reflection of the rear mirror of rb = 1. Here, the filling factor is defined as 2ω0/d, where ω0 and d corresponds to the beam radius from each waveguide at the output surface and the center-to-center element spacing of array waveguide, respectively.
Assuming Module-A whose amplitude reflection of the output surface on Nd:YVO4 crystal and OC reflectivity is rf = 0 and , respectively, we obtained the threshold gain of fifteen self-consistent array modes as a function of the round trip cavity length [Fig. 7(a)]. The lowest lasing threshold was obtained for the highest (15th) order supermode when the cavity length was set to half of the Talbot distance of 37.5 mm. The lowest threshold can be obtained for a wide range of cavity lengths around half of the Talbot length. Although Talbot image of the array is blurred when the external cavity length is displaced from half of the Talbot distance, the filling factor of 0.5 is sufficiently high to obtain high coupling efficiency between such a blurred image and waveguides. On the other hand, the threshold difference compared with the next mode (14th order mode) is quite small because we assumed a high filling factor relative to the LD array. The mode discrimination between the highest mode and the next mode decreases as the filling factor increases, as Mehuys et al. reported . In our experiment, the low threshold difference between the 15th and 14th order modes seems ineffective.
When rf is nonzero, the threshold gains start to oscillate in the wavelength scale along the cavity length. Therefore, only the maximum and minimum values of the oscillating threshold gains are plotted in Fig. 7(b), assuming that Module-B is and . Note that we do not include the transverse mode shift in the calculation. Because only 3∼4 transverse modes can oscillate in the 0.96 nm gain width of Nd:YVO4 whose length l = 1.5 mm, the lowest threshold gain supermodes are also changed by the subtle cavity length fluctuation in our experiment. This causes the ambiguous two-peak profile in our experiment, e. g.,Fig. 5(b). To obtain clear two-peak profiles with a nonzero rf arrangement, modifications that decrease rf and lengthen the array length of l are necessary.
4.2. Filling factor dependence
We also analyzed the mode discrimination dependence on all of the parameters, except rf. The matrix equation we used depends on a filling factor of 2ω0/d and a normalized cavity length of L/Zt. Here, L corresponds to the round trip cavity length. If the filling factor and the normalized cavity length are constant, the results do not depend on ω0, d, and L. Therefore, the filling factor is the most significant parameter that determines the threshold gain except rf. We calculated the threshold gain with ω0 = 25 μm and ω0 = 75 μm using the same parameters in Fig. 8. For ω0 = 25 μm, the highest (15th) order supermode obviously has a minimal value around Zt/2 (37.5 mm). But for ω0 = 75 μm, mode discrimination decreases, and the fluctuation of the threshold gain also decreases. Therefore, the lower filling factor increases mode discrimination.
We developed a phase-locked Nd:YVO4 microchip laser array with a Talbot cavity, which is stable and produces 1.65-W output power. This module can produce 2.02 W with the same LD operating conditions; hence, we simultaneously accomplished high efficiency and Talbot phase-locking. The transverse far-field pattern of our Talbot cavity has two sharp peaks, and this two-peak-far-field pattern can be converted to a single-peak pattern by a SLM. We believe that such a microchip laser array with a Talbot cavity could scale the power with more number of array. When a two dimensional waveguide array is realized in a Nd:YVO4 microchip laser, the coherently combined laser output power will linearly increase.
We also prepared a module with partially reflective coating on the output-side surface, and experimentally and numerically analyzed the characteristics. Although the partially reflective coating brings higher efficiency, it degrades the mode discrimination.
We also analyzed the mode discrimination dependence on the filling factor and found that we should decrease it to increase the mode discrimination. This could be an obstacle to develop a more compact Talbot microchip laser array because the array pitch must be decreased to shorten Talbot length Zt. The diameter of the waveguide may have to be small to maintain the filling factor and the mode discrimination in this case.
This research was supported by JSPS KAKENHI GRANT Number 24656055.
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