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The Talbot cavity is one promising method to synchronize the phase of a laser array. However, it does not achieve the lowest array mode with the same phase but the highest array mode with the anti-phase between every two adjacent lasers, which is called out-phase locking. Consequently, their far-field images exhibit 2-peak profiles. We propose intra-Talbot-cavity frequency-doubling. By placing a nonlinear crystal in a Talbot cavity, the Talbot cavity generates an out-phased fundamental wave array, which is converted into an in-phase-locked second harmonic wave array at the nonlinear crystal. We demonstrate numerical calculations and experiments on intra-Talbot-cavity frequency-doubling and obtain an in-phase-locked second harmonic wave array for a Nd:YVO4 array laser.
© 2015 Optical Society of America
1. Introduction
Waveguide laser arrays are now widely developed and commercialized as a small and low-cost laser module [1]. Laser arrays are scalable in total output power, and waveguide structure brings low cavity loss and higher laser gains [2]. Consequently, we can reduce the size of laser systems and maintain high total output power by waveguide laser arrays. However, the output laser beam of laser arrays cannot be focused in a small spot without phase locking between the lasers in the array. Therefore, phase locking schemes between lasers are necessary to develop high brightness laser devises using multiple lasers [3]. To lock the laser array’s phase, one promising technique is the Talbot cavity [4], which is an external cavity that contains a laser-active medium array. Its only necessary condition is that the external cavity length must be a certain length.
The Talbot effect [5] is self-imaging of the periodical wave-front profile at a certain distance known as the Talbot length. An image of a periodic array laser is reproduced at its Talbot length. Only when the relative phase between every two adjacent lasers is kept at π, the image is reproduced at its half Talbot length. Such an array mode oscillates discriminately when the round-trip external-cavity-length is its half Talbot length [6]. This phase locking denotes out-phase locking in this paper.
The Talbot cavity, which does not need an electrical feedback system, is quite useful for the development of compact, low-cost, and literally scalable high intensity laser modules. However, since the lasers in the array synchronize with the anti-phase between two adjacent lasers, its far-field profile always exhibits a two-peak profile.
To convert such an out-phase-locked laser array into an in-phase-locked laser array, we propose intra-Talbot-cavity frequency-doubling, which is a Talbot cavity that contains a nonlinear crystal for second harmonic generation (SHG). Although the Talbot cavity has been studied with laser-diode (LD) arrays [4,7–13], CO2 lasers [14,15], multicore fiber lasers [15–20], and solid-state lasers [21,22], we believe that wave-guided solid-state laser arrays are most suitable to intra-Talbot-cavity frequency-doubling because the waveguide structure brings higher intensity and higher SHG conversion efficiency.
In this paper, we performed numerical analysis and experimented using a laser-diode-array-pumped Nd:YVO4 waveguide laser and a periodically-poled LiNbO3 (PPLN) waveguide. Although a combination between the Talbot effect and second order nonlinearity was reported in research of super resolution [23,24], to the best of our knowledge, our work is the first report on SHG with a Talbot cavity.
2. Principle and numerical analysis
In the principle of intra-Talbot-cavity frequency-doubling, we start from an out-phase-locked array of the fundamental wave produced by Talbot-cavity phase locking. Then part of the fundamental wave array is converted to a second harmonic (SH) wave at the nonlinear crystal. Because the SH wave is generated proportionally to the square of the fundamental wave, its phase is double that of the fundamental wave’s. Since the π phase difference between two adjacent fundamental wavelength laser beam becomes zero in SH wave, the Talbot cavity produces an in-phase-locked SH wave array.
The above interpretation does present some concerns. A nonlinear crystal for SHG has nonzero length, and the Talbot length of the SH wave is twice as long as the fundamental wave’s. That is why the SH wave might not form the same phase profile as that of the fundamental wave. Another concern is that the fundamental wave might be disturbed by the interaction with the SH wave, negatively influencing the self-imaging in the Talbot cavity. Therefore, we numerically analyzed the standing wave propagation of the array laser beams in the half-Talbot-length planar waveguide in a LiNbO3, where part of the LiNbO3 waveguide was periodically poled.
For the numerical calculations, the wave equations, including diffraction and χ(2) nonlinear interaction, are depicted as follows [25,26]:
where j is the imaginary unit and variable ε is the complex electric field. n and k correspond to the refractive index and the wave number of the waveguide, respectively. Coefficient deff is the effective nonlinear coefficient. Physical constant c is the speed of light. Subscripts denote that 1 is a fundamental wave and 2 is a SH wave. Phase mismatching Δk is defined as Δk = k2 − 2k1.Because planar waveguide propagation is assumed, we reduced the y-axis from the above wave equations. Although the estimation is rough, we assume that both ε1 and ε2 have Gaussian profiles on the y-axis with identical width everywhere. Under this assumption, the diffraction terms on the y-axis can be eliminated. After the elimination, we substitute ε(x,0,z) f (y) for ε(x,y,z), where f (y) is defined as , and wy is beam width on y-axis. This equation is usually not valid because the nonlinear term is a square of the Gaussian profile on the y-axis. Assuming the nonlinear terms have an averaged influence on the complex electric field, which has Gaussian profile on the y-axis, we can consider only the integral of the equations over the y-axis. Because , the nonlinear term should be reduced by . Therefore, we used the following wave equations for the numerical simulations:
Since we set the fundamental wave to 1064 nm, n1 and n2 correspond to 2.156 and 2.234. Effective nonlinear coefficient deff flips its sign for every poled period of PPLN, and its absolute value is 34.4 pm/V. PPLN’s ideal period is 3.4103 μm for maximum conversion efficiency. Since initial value ε2(x,0) is zero, ε1(x,0) is a 15-laser array of the highest order array mode [27], defined as follows:
where a and d are the beam width of a single beam on the x-axis and the center-to-center interval of the laser array on it, respectively. We set these values to 50 and 200 μm based on our experiment. The Talbot length is defined by Zt = 2d2/λ, so Zt = 75.2 mm in this condition. Because Talbot effects are based on Fresnel diffraction, the Talbot length gets longer and becomes nZt in a medium with the refractive index of n. Therefore, this calculation brings the 1/2 Talbot length of 81.1 mm. To achieve this 1/2 Talbot length in round trip, we placed the output coupler at the 1/4 Talbot length. ε11 is calculated from the total optical power for the simulation.First, we assumed that the end facet of the LiNbO3 waveguide is coated for high reflection (HR) at 1064 nm and anti-reflection (AR) at 532 nm, and the LiNbO3 contains 5-mm-long periodically-poled part at the beginning. Although the SH wave is generated at two parts of the PPLNs, we calculated these SH waves individually by assuming non reflection. The model, which is stated above, is shown in Fig. 1(a), and we calculated the propagation using the unfold model as Fig. 1(b). To simulate the SH wave emitted to both directions, the SH wave is once reset after the 40.5-mm propagation.
The PPLN period is set to approx. 3.41 μm, which is ideal. The total input power is set to 100 W, and ∼40 W is converted to the SH wave. Figure 2(a) shows the far-field image of the fundamental wave on the x-axis after propagation. Even though ∼40% of the optical power is converted to the SH wave, the fundamental wave’s far field image does not change, which means the nonlinear interaction does not disturb the fundamental wave array in this condition. The far-field images of the SH wave are shown in Fig. 2(b). The SH wave is generated at both ends of the LiNbO3 waveguide, and both far-field images have a sharp peak in the center.
Fig. 2 (a) Far-field profile of fundamental wave with 100-W input power and ideal 3.41-μm PPLN period. (b) Far-field profiles of SH wave generated in first and second PPLNs, respectively.
Second, we calculated SH wave generation under the same condition as above except the PPLN period. If the PPLN period is not optimized, the SH wave is converted to the fundamental wave again. The 5-mm-long PPLN is sufficiently long to cause back-conversion at 0.1%∼0.2% variation of PPLN’s period. Back-conversion might disturb the fundamental wave in both the amplitude and phase profiles. We set the PPLN period to 3.4171 μm (0.2% variation), and Fig. 3(a) shows the total optical power change of the SH wave in the first PPLN part of the waveguide. The SH wave is entirely back-converted to the fundamental wave and returns to 1.5 W. The far-field profiles of the fundamental and SH waves are shown in Figs. 3(b) and 3(c). Their far-field profiles have almost the same shape as the ideal PPLN period.
Fig. 3 Calculation results from 100 W input and 0.2% variation from ideal PPLN period: (a) Output power of SH wave from first PPLN. (b) Far-field profile of fundamental wave. (c) Far-field profiles of SH wave.
From these results, we observed no disturbance of the Talbot cavity for the fundamental wave, and the far-field profile of the SH waves match the in-phase array’s well, probably because the Rayleigh length of the individual fundamental wave beams is ∼16 mm in PPLN, which is much longer than PPLN’s length. Therefore, the diffraction of both the fundamental and SH waves barely influences the amplitude and phase profiles of the laser array.
3. Experiment and results
The schematics of our intra-Talbot-cavity frequency-doubling are shown in Fig. 4. The 15-emitter LD array pumped Nd:YVO4 waveguide laser has the same structure as the one in our previous work [22], and its details were previously described [1,28]. The Nd:YVO4 waveguide is a 40-μm-thick and 1.5-mm-long Nd:YVO4 slab crystal. It has dielectric claddings to confine light for the vertical direction and exploits thermal lensing effect, which is induced by a periodically structured heat sink, to build waveguide array in the horizontal direction. The emitters have a 200-μm center-to-center pitch; its Talbot length is 75.2 mm.
Fig. 4 Bird’s eye view (left), top view (upper-right), and side view (lower-right) of experimental setup
In this paper, we used two kinds of laser modules. One is a unit consisting of a LD array, an Nd:YVO4 waveguide, and a PPLN waveguide. The other does not contain a PPLN waveguide. We placed a PPLN waveguide near the Nd:YVO4 waveguide and carefully adjusted its position. For the first module, all the components were integrated on a metal heat sink [1], and the PPLN waveguide was immobilized. Although we cannot adjust the PPLN waveguide’s angle, this module has a heater and a thermistor very close to the PPLN waveguide. Therefore, PPLN’s temperature can be controlled very precisely. Since the second module needs a mechanical holder and a temperature controller for the PPLN waveguide, it is hard to control the PPLN’s temperature, but its angle is adjustable. The thermal response of this mechanical holder is very slow because the heater is placed away from the PPLN waveguide due to geometrical limitations. The poor thermal control made the cavity alignment difficult because the waveguide generates heat that depends on the optical power in the cavity. For this reason, during the most of experiments in the following description, we used the first module, although later we used the second one to achieve more precise alignment. The 40-μm-thick, 5-mm-long PPLN waveguide was placed very close to the Nd:YVO4 waveguide. Because LiNbO3 exhibits a refractive index that resembles Nd:YVO4’s (ne = 2.17 at 1064 nm), we attained good coupling efficiency between the two waveguides, ∼97% for the first module with a PPLN waveguide. For stable SHG conversion, the temperature of the PPLN waveguide was precisely controlled to ∼110°C. After the PPLN waveguide, the laser beam array was collimated on the y-axis by an f = 10 mm cylindrical lens. The output coupler (OC) was placed around the 1/4 Talbot length. The coatings on S1–S4, as seen from Fig. 4, were as follows: S1: HR at 1064 nm and AR at 808 nm, S2: AR at 1064 nm and HR at 808 nm, S3: AR at 1064 nm and HR at 532 nm, S4: AR at 1064 nm and AR at 532 nm, and OC: HR at 1064 nm and AR at 532 nm. The LD array was operated in a quasi-continuous wave mode with 500-μs pulse-width at 100-Hz repetition.
To form a far-field image of the phased array beam, an f = 200 mm spherical lens was placed at its focal length from the PPLN waveguide. Figure 5 shows examples of the far-field profile from intra-Talbot-cavity frequency-doubling. We used a unit module containing a PPLN in Figs 5(a) and 5(b). The length between the PPLN and the OC was L = 22 mm, and the absorbed power in the Nd:YVO4 was ∼880 mW. Figures 5(a) and 5(b) correspond to the fundamental wave that leaked from the OC and SH waves. To discriminate the fundamental wave from the SH wave (or vice versa) we used a long-pass (short-pass) filter before a CCD camera. A two-peak-far-field profile appeared on the transverse profile in Fig. 5(a), which indicates out-phase locking. A sharp single peak (peak 1) is shown in Fig. 5(b) between the two peaks in Fig. 5(a) on the transverse axis, indicating that a SH wave formed in-phase locking. Figure 5(b) has another peak (peak 2) at the lower right of peak 1. We speculate that peak 1 is generated while the fundamental wave propagates forward in the PPLN and peak 2 is generated during backward propagation toward the Nd:YVO4. Peak 2 is reflected at surface S3, where S3’s angle is not adjustable in this experiment.
Fig. 5 Far-field pattern of (a) fundamental wave and (b) SH wave. (c) Far-field pattern when adjusting S3 angle.
We also performed the same experiment using a module without a PPLN waveguide to adjust S3’s angle. The two peaks aligned on the longitudinal axis and overlapped on the transverse axis (Fig. 5(c)). However, the output power decreased from 92 to 55 mW. This power drop was mainly caused by excess thermal load. Although the PPLN does not significantly absorb fundamental light, the bonding material on it might absorb uncoupled light. Therefore the cavity alignment changes the temperature of the PPLN waveguide, and that variation is sufficiently high to reduce the conversion efficiency. To precisely control the PPLN waveguide’s temperature, the heater and the thermistor must be placed as close to the PPLN waveguide as possible. This procedure is currently difficult with our mechanical holder and temperature controller.
Figures 6(a) and 6(b) show the dependence of the transverse profiles on the cavity length for the fundamental and SH waves. These transverse profiles are the summation over the longitudinal axis of the 2D profile. We measured the profile only from L = 12 to L = 26 mm due to geometrical limitation in the experiment. The shape of the SH wave in Fig. 6(b) is not symmetric due to peak 2, which was discussed before. Although the geometrical 1/4 Talbot length is ∼18.8 mm, L ∼19.8 mm corresponds to the 1/4 Talbot length after taking into account the length of a PPLN waveguide and the refractive index of the cylindrical lens. Figure 6(a) shows a clear two-peak profile for cavity lengths over 22 mm, which is slightly longer than the 1/4 Talbot length. One possible reason is that the beam waist of the fundamental wave array should be located outside of the Nd:YVO4 waveguide because it exploits the thermal lensing effect to stabilize the transverse mode [1,22,28]. In fact, we observed a similar result, as described in our previous report [22], where we used the module with the same structure. A longer cavity forms a Talbot phase locking well. On the SH wave, the single peak also becomes clearer for cavity lengths over 22 mm. Since Talbot phase locking can be seen from 22 to 26 mm of the external cavity length. It has been known that the ambiguity of the Talbot effect is quite large.
Fig. 6 Transverse far-field profile of (a) fundamental wave and (b) SH wave with various external cavity lengths.
The dependence of the output powers on the absorbed powers is shown in Fig. 7. The absorbed powers are estimated by injection currents to the LD array, data of the laser power, and absorbance of the Nd:YVO4 crystal. The cavity length was set at 22 mm. The slope efficiency was 14%, which is quite low because of such internal cavity loss as the coupling loss between the PPLN waveguide and the free space. To increase output power, the conversion efficiency of SHG must be increased and the internal loss must be decreased.
Our experimental results suggest the following two problems. One is that two peaks appeared in the far-field images of the SH wave. An intense peak is generated by the fundamental wave traveling toward the OC. The second peak corresponds to the SH generated by the backward propagation of the fundamental wave, which is reflected at the PPLN surface (S3). Therefore, even a small angle misalignment causes a spatial shift from the other peak. We cannot adjust the angle in the unit module. On the other hand, we successfully adjusted these two peaks to make them coincident at the output with the laser module with the separated PPLN. However, for the above reason, the temperature fluctuation was not completed, and the SH output power degraded. A module that can manage both the temperature control and PPLN’s angle adjustment will solve this problem.
The second issue is the low slope efficiency of 14%. Using a similar unit-type array laser module but without a Talbot self-imaging mechanism, optical conversion efficiency of ∼40% was reported at a total output power of 10.8 W [1]. We expect that an all-waveguide Talbot cavity will significantly reduce the intra cavity loss and improve the output performance.
4. Conclusion
To the best of our knowledge, we developed the first intra-Talbot-cavity frequency-doubled laser. Our numerical calculation and experimental results revealed that an in-phase-locked laser array can be obtained by combining the Talbot cavity and SHG. For an external cavity length of 22–26 mm, which is slightly longer than our estimation, Talbot phase locking was observed. The slope efficiency is currently quite low, mainly due to the diffraction loss at the free-space Talbot length. We are now preparing all waveguide-type structures that will significantly reduce the loss and improve the efficiency. We believe this technique will contribute to the development of a small, low-cost, and scalable high intensity coherent laser array.
Acknowledgments
This research was supported by JSPS KAKENHI GRANT Number 24656055.
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