We describe a concept of detecting the piston and tilt errors with a Shack-Hartmann sensor and correcting them simultaneously via a segmented deformable mirror (DM) in a two-dimensional laser array. The concept in coherent beam combination (CBC) is demonstrated by both simulation and experiment. The simulative results show only two iterations are enough to correct the random initial phase errors in the steady condition. The experimental results confirm that this method works effectively in a dynamic condition. The study represents that the Shack-Hartmann sensor can extract any piston error in –π ~π, which means the concept is an arbitrary phasing technique for a laser array.
©2012 Optical Society of America
The growing needs of high-power lasers with high beam quality and high-brightness could hardly be satisfied by a single laser. There are many objective factors that limit the increasing output power of a single laser, such as nonlinear effects of the gain medium, heavy thermal loading, and low energy transfer efficiency. Coherent combination of a low-power laser array becomes an available method to acquire high-power good-beam-quality lasers because of its’ outstanding abilities in magnifying power and saving beam quality . The approaches of coherent beam combination (CBC) can be mainly classified into active phasing [2–7] and passive phasing [8–11]. Because of the good beam quality and compact size, most of the current CBC systems are based on fiber lasers. And the master oscillator power amplifier (MOPA) configuration [2–7] becomes the most famous system structure. With combination of slab amplifier array, the CBC of MOPA achieves 105kW output , which is the highest output of CBC recently.
Many phase-locking techniques are commonly used in CBC, such as heterodyne [4, 5], zero-order interference (ZOI) detections [13, 14], Self-Synchronous and Self-Reference  and intensity detecting based on SPGD algorithm control [2, 15]. Most of these methods depend on the fiber phase modulators, such as LiNbO3 crystal and fiber-wrapped PZT. They have large advantages in compensating the piston error in fiber laser systems by their high response speed. But in other non-fiber laser systems such as free space laser systems, these elements are not used in compensating piston error. Some other phasing techniques based on the interference pattern of two apertures [16–19] are brought up. They are mostly used to phase the segments in the large segmented telescopes. For example, adaptive optics (AO) is employed to flat the wavefront via the deformable mirror (DM). Chanan introduces the sub-apertures array to detect the piston error between two neighboring segments . This idea of phasing segments by sub-apertures array is then applied to CBC by Yang . Yang describes a Hartmann lens array to measure the piston and tilt errors of a seven-beam laser array and an active segmented mirror (ASM, or named segmented DM) to compensate the measured phase errors. Though, Yang just represents a simulative result of this idea. By introducing AO into CBC, many profits have been brought in. The biggest one is the ASM can compensate both tilt and piston errors of the laser array simultaneously. Another one is the continuous DM can be used to compensate the high-order wavefront distortions of each beam in the large aperture case .
In this paper, we realize the measurement and correction of phase errors (piston and tilt) by the Shack-Hartmann sensor and AO technique. The peak-rate (PR) algorithm proposed by Yang is replaced by the displacement of the brightest point of the interference pattern (DBPIP) algorithm . Because the DBPIP can detect the piston and tilt errors simultaneously by measuring the peak location shift of the diffraction pattern in each sub-aperture. Some optimizations are added in the micro-lens array design. We demonstrate the application of the Shack-Hartmann sensor in CBC in simulation and experiment. In the simulation, we describe the phase errors compensation procedure step by step. In the experiment, we show the coherent combination results on a two-dimensional three-laser array by the method with a Shack-Hartmann sensor and an ASM.
2. Measure the piston and tilt errors by a Shack-Hartmann sensor
The Shack-Hartmann sensor was used to detect the piston error in the large telescopes  and measure the tip-tilt error of the beam array in CBC by Neal . Yang  firstly combined the two applications of Shack-Hartmann sensor in CBC. To optimize the detection method, we have designed the Shack-Hartmann sensor again, as shown in Fig. 1 .
The sub-micro-lens located in the center of the beam measures the tilt error of the corresponding beam. The centroid calibration method commonly used in traditional AO systems [23, 24], is employed to measure the tilt error in our system. Note that the piston error is also measured by the Shack-Hartmann sensor.
The sub-micro-lens which shears the neighboring beams detects their piston error. The piston error is extracted by measuring the shift of peak location of the interference pattern.
Chanan found out the peak location of the two beams’ interference pattern shifts with their piston error. Especially, he pointed out that the peak location shifts linearly with the piston error when the piston error is small . Actually, we find this conclusion is almost suitable for the piston error in the whole interval –π ~π .
For two plane-wave beams with center locations of and the same shape of , their interference intensity pattern at far-field can be calculated by the Fourier optics theory :
Two circular beams with a diameter of 2 mm arranged side by side are shown in Fig. 2 . In this case, the function is:Eq. (1), we can numerically calculate the interference pattern of the two beams having any piston error without loss of generality to look at interference strips only along the x-direction. The other parameters in calculation are λ = 632.8 nm, f = 300 mm, and w = 1 mm. Figure 3 shows the one and two dimensional intensity distributions of the patterns with different piston error. As we see, the peak location shifts as the piston error changing. A series of calculations are done step by step all over the interval –π ~π, and the results are shown in Fig. 4 . We can see that the peak location shifts almost linearly in the whole interval –π ~π. When the piston error exceeds the interval, the measured piston error should be an equivalent value between –π and π which is the exact piston error shifted by several 2π. This character is the 2π ambiguity of the piston detection method.
In the method, the tilt errors of the beams can be expressed as:Fig. 1.
From the conclusion in Fig. 4, the piston error in the laser array can be got by:Fig. 4, and,Fig. 1, each piston error is defined as the difference of the piston phase that of the right side beam subtracts the left one. Here, the piston phase of the first beam is chosen as the reference piston and all the other piston errors are the relative differences referenced to it. We can get the piston phase of each beam by:
As seen in Fig. 4, the piston error in the interval –π ~π corresponds to a unique peak shift. According to the Eq. (10), any phase distribution of the laser array also corresponds to a unique peak shift vector . With accurate compensation, it is feasible to adjust the phase distribution to not only the co-phase statement but also an arbitrary statement.
3. Numerical simulation
A numerical simulation of a seven-laser array with uniform intensity has been done to demonstrate how the Shack-Hartmann sensor works with the method. All the initial piston phase () and tilt phase () of each beam are generated randomly. The field of the ith beam can be described as:
The simulation is in the steady condition. The laser array only has the initial phase errors. The wavelength of each beam is 632.8 nm. The focal length of the micro-lens is 300 mm, and its dimensions are shown in Fig. 1. Figure 5 shows the correction procedure of simulation. The first row shows the wavefront evolution of the laser array. The second row shows the evolution of the image spots created by the Shack-Hartmann micro-lens array. The last row shows the evolution of far-field images. At the beginning (Fig. 5(a)), the randomly distributed phase errors lead to a messy far-field intensity distribution and the asymmetric image spots of the Shack-Hartmann sensor. After the first iteration (Fig. 5(b)), the tilt error is mostly eliminated, and the far-field intensity is more concentrated than that in the initial state. But the asymmetric image spots of the Shack-Hartmann sensor tell us that the piston error still exists. After the second iteration (Fig. 5(c)), both the far-field image and the image spots of the Shack-Hartmann sensor are symmetric, which means the piston error has been eliminated. The far-field image is very close to the diffraction limit.
In Fig. 6, the tilt error is eliminated and can be neglected after the first iteration and the piston error is eliminated after the second iteration. After the second iteration, almost no changes happen to either the piston error or the tilt error in the laser array. It shows that the random static phase errors in the laser array can be eliminated just by two iterations in the steady condition.
Both Fig. 5 and Fig. 6(a) show that one laser’s piston phase is over nearly 2π comparing with the others, it is caused by the 2π ambiguity of the piston detection method, which is described in section 2. Figure 7(a) shows the measured peak shifts of the patterns created by the corresponding sub-micro-lens. The nearly zero peak shifts after the second iteration tell that all the lasers are locked in-phase, which is confirmed by the Strehl ratio as shown in Fig. 7(b). After the second iteration, the Strehl ratio is steadily maintained steadily and closed to the limit. The steep Strehl ratio promotion in the second iteration indicates that the piston error is the main factor which damages the output beam quality.
We could get the conclusion from the simulation results: two iterations are enough to correct the tilt and piston errors in the ideal steady condition. The first iteration mainly compensates the tilt error and the second is mainly for piston error. And the piston error cannot be precisely detected before eliminating the tilt error.
To simplify the verification test, we build up a three-beam CBC experimental setup which is shown in Fig. 8 . A He-Ne laser source produces a line-polarized beam with 632.8 nm wavelength. The laser beam is expanded into a plane wave with Φ50 mm diameter by a beam expander. An aperture screen cuts the plane wave into three beams with Φ14 mm diameter and 16 mm center separation. A beam splitter (BS1) is applied to create the reference laser array which is reflected by a high reflective mirror (HRM). The signal beams passing through the simulated turbulence strike the ASM. The reference laser array is used to calibrate the micro-lens array (MLA). It must be stopped once the calibration finished. Another beam splitter (BS2) is used to split parts of the laser array for phase errors detection. The rest part is for the sample system. A 40 × micro-objective lens is employed to scale the image spot on the focus plane of Lens4 (f = 220 mm). Then a CCD camera records the scaled image. The structure of Shack-Hartmann sensor system (HSS) is shown in Fig. 9 . Two lens (Lens3, f = 440 mm and Lens4, f = 55 mm) compose a 1/8 × beam reducing system. The reduced beams are dissected into a number of small samples by MLA (f = 300 mm), and then focused onto a high-speed CMOS camera. The ASM, Lens3, Lens4 and MLA must be properly adjusted to compose a 4-F Fourier imaging system. The image signal acquired by HSS is sent into calculation and control system where the phase errors are calculated and the control signal is sent to the ASM which is drove by the HVA. The traditional PID control algorithm is adopted in the closed-loop control. Though the CMOS has a frame rate of nearly 560 fps at the resolution of 512 × 512 pixels, the efficient frame rate of the closed-loop is only 330 fps because of the time consumption of the image processing in computer. To simulate a dynamic piston error, a 3 Hz sinusoidal signal with 1V amplitude generated by a signal generator is added to the control signal. It must be noted that the 1V amplitude signal can provide piston errors over 2π and the dynamic limit of the closed-loop is not 3 Hz as we chose in our testbed. Anyhow, it does demonstrate the method works in a dynamic condition. The control bandwidth limitations mainly come from the speed of CMOS and the time consumption of image process. To increase the control bandwidth, higher speed image sensors are needed for detection and DSP modules can be adopted for the local management of tilt and piston errors.
The image spots created by Shack-Hartmann sensor are shown in Fig. 10(a) and Fig. 10(b) which are acquired before and after the correction of phase errors in the experiment. Figure 10(c) shows the measured curve between the peak shift and the piston error. As it shows, the peak shift varies linearly with the piston error in the interval of –λ/2 ~λ/2. When the piston error exceeds this interval, the peak shift curve is folded back into the limited interval of −12 ~12 pixels. It agrees with the 2π ambiguity described in section 2. In Fig. 10(a), the spots in the squares marked with 2, 3 and 5 are not symmetric before the correction. It’s because of the tilt and piston errors. The centroid shifts of the spots in the squares marked with 1, 4 and 6 represent the corresponding tilt error of each beam.
The whole experiment is divided into three phases. Each phase runs 10 seconds. In the first phase, both the piston and tilt control loops are off; in the second phase, the tilt control loop turns on and the piston control loop turns on in the last phase. The curves of tilt and piston errors obtained in the experiment are shown in Fig. 11(a) and Fig. 11(b) respectively. As shown in Fig. 11(a), the tilt error of the third beam in open loop is much bigger than another two beams, because the third beam has a big initial miss alignment. It also can obviously see that the range of tilt error is reduced from 200 μrad in the open loop to 20 μrad in the close loop. The average RMS of the tilt error in close loop is about 5.7 μrad. It is about 1/12 of the diffraction-limited divergence of a single beam in the array which is calculated to be 67 μrad in the experiment. The curve of piston error in Fig. 11(b) is acquired while the tilt control loop is on. After the piston control loop is on, the piston error range decreases from π to 0.15π and the RMS of the piston error is only 0.1π. Commonly in CBC, the tilt error should be less than half of the direction-limited divergence and the piston error should be less than λ/10 (or 0.2π in radian). Figure 12 shows the evolution of the long-exposure far-field images in different phases of the experiment. The image in Fig. 12(a) disperses mainly because of the dynamic tilt error. The peak intensity in Fig. 12(b) is nearly twice of the peak intensity in Fig. 12(a), because the dynamic tilt error is eliminated. The image is blurred, because the dynamic piston error is not been compensated. The shape of the pattern in the close loop (Fig. 12(c)) is very close to the theoretical far-field image shown in Fig. 12(d). The peak intensity rises 3 times compared to Fig. 12(a). The power encircled in the main lobe is 41.3% of the total power of the pattern, comparing to the theoretical value of 57.8% calculated from Fig. 12(d). Figure 13 shows the evolution curve of the normalized power encircled in the area of main lobe. Because the tilt and piston errors are eliminated in the last phase of the experiment, the energy is more concentrated and stable in the main lobe area comparing to the open loop phase.
Compared to the simulative results, the correction efficiency in the experiment is a little lower. Maybe the main reason is the inaccurate execution of the ASM, which means the ASM could not compensate exactly the phase errors which the Shack-Hartmann sensor detected. So, it needs more iterations in the experiment to compensate each perturbation of the phase errors.
In summary, we present a method of detecting and compensating the phase errors (piston and tilt) in a two-dimensional laser array. In this method, a properly designed Shack-Hartmann sensor is applied for the detection of the phase errors and an ASM is adopted for compensation. And a simulation, which demonstrates how the method works in co-phasing a seven-laser array, has been carried out. Its results show how powerful the method is: only two corrective iterations can compensate the random static phase errors. This high efficient correction indicates that the method may have the potential to be used in CBC of a large number of beams. A dynamic experiment based on our verification testbed has been accomplished. Its results accord with the simulative results, which means the method works effectively in co-phasing a two-dimensional laser array. For the Shack-Hartmann sensor can measure the arbitrary piston error in the interval of –π ~π, it also can be used as an arbitrary phasing technique for a two-dimensional laser array.
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