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A novel approach of tip/tilt control by using divergence cost function in stochastic parallel gradient descent (SPGD) algorithm for coherent beam combining (CBC) is proposed and demonstrated experimentally in a seven-channel 2-W fiber amplifier array with both phase-locking and tip/tilt control, for the first time to our best knowledge. Compared with the conventional power-in-the-bucket (PIB) cost function for SPGD optimization, the tip/tilt control using divergence cost function ensures wider correction range, automatic switching control of program, and freedom of camera’s intensity-saturation. Homemade piezoelectric-ring phase-modulator (PZT PM) and adaptive fiber-optics collimator (AFOC) are developed to correct piston- and tip/tilt-type aberrations, respectively. The PIB cost function is employed for phase-locking via maximization of SPGD optimization, while the divergence cost function is used for tip/tilt control via minimization. An average of 432-μrad of divergence metrics in open loop has decreased to 89-μrad when tip/tilt control implemented. In CBC, the power in the full width at half maximum (FWHM) of the main lobe increases by 32 times, and the phase residual error is less than λ/15.
© 2013 Optical Society of America
1. Introduction
Coherent beam combining (CBC) of fiber amplifiers (FAs) via a master-oscillator-power-amplifier (MOPA) architecture is an outstanding way for brightness scaling with good beam quality [1–13]. Most of the studies concerning CBC have been performed in phase-locking region [3–14]. In recent years, it has been demonstrated that tip/tilt-type phase errors of combined beamlets disturb CBC seriously, even if all the beamlets are phase-locked [2,15–24]. The CBC imposes tight tolerances on individual beam alignments to ensure fully constructive interference [17]. The tip/tilt errors are induced by many factors, like limited precision of assembling, vibration and thermal deformation of mechanism, and atmospheric turbulence effects, etc.
In previous CBC systems, stochastic parallel gradient descent (SPGD) algorithm [26] has been demonstrated to be one of the best control approaches for phase-locking [5,6,9–11,27,28] and even tip/tilt control [2,15,20–24]. In a SPGD-based optimization, the power-in-the-bucket (PIB) cost function, which is acquired from a photodetector (PD) with a pinhole or a high-speed camera with a nominal bucket, is usually employed for intensity-maximization. PIB metrics are fit for phase-locking, but have limited abilities with tip/tilt control if the beamlets are non-overlapping beams or the tip/tilt errors among beamlets are beyond the extent of the bucket. To achieve steady and precise tip/tilt control in CBC, the bucket size of PIB should be smaller than the Airy-disc of a single spot in the far-field [22]. Besides, the tip/tilt control using PIB metrics will compete with and influence phase-locking which also use the same metrics because of the tiny perturbation demand of SPGD itself [19–21]. So, the PIB cost function might be not the best choice of tip/tilt control using SPGD algorithm.
In fact, the common used tip/tilt control devices, like fast-steering mirror (FSM) [18] and adaptive fiber-optics collimator (AFOC) [2,10,15,20–25], have a relatively slow response rate of below several kHz-level, and a high-speed camera with several-kHz frame rate will sufficiently ensure most applications with them. Except for PIB metrics, the images of camera can provide more information of combined spots, like the max-divergence to the target location of beam pointing, which denotes the tip/tilt errors directly and could be employed by tip/tilt control.
In this paper, a novel and effective approach of tip/tilt control by using divergence cost function in SPGD algorithm for CBC is proposed and demonstrated experimentally, for the first time to our best knowledge. Compared with the conventional PIB cost function for SPGD optimization, the tip/tilt control using divergence cost function ensures wider correction range, automatic switching control of program, and freedom of camera’s intensity-saturation. In Section 2, the CBC setup of a seven-channel 2-W fiber amplifier array is introduced, with the homemade piezoelectric-ceramic-ring fiber-optic phase-modulator (PZT PM) and AFOC employed to correct piston- and tip/tilt-type aberrations, respectively. In Section 3, the divergence cost function for tip/tilt control is discussed in detail. In Section 4, the CBC with both phase-locking and tip/tilt control is achieved.
2. Experimental setup
The CBC setup of a seven-channel 2-W fiber amplifier array is illustrated in Fig. 1. The master oscillator (MO) is a linearly polarized single-mode fiber laser (NKT photonics, Koheras Adjustik), generating 11-mW of power at 1064-nm wavelength with a linewidth of 20-kHz. The MO is firstly pre-amplified to 200-mW before split into eight channels. Seven employed channels are coupled into seven homemade PZT PMs, respectively. In this setup, PZT PMs are adopted for phase-locking control, while in previous demonstrated CBC system, LiNbO3 PMs are usually employed. Compared with LiNbO3 PM, the inherent low insert loss and high laser-induced damage threshold of PZT PM make the corresponding CBC system more compact and stable [29]. Each laser beam from the PZT PM is coupled into a fiber amplifier and amplified to 2-W. Laser beams from amplifiers are collimated by the homemade AFOCs with a single beam diameter of about 28-mm. The AFOC incorporates the function of beam-pointing control, was firstly proposed by L. A. Beresnev, et al [25]. Here, the AFOC array aligned in a ‘hexagon’ shape is used to correct tip/tilt-type aberrations among beamlets. Compared with conventional FSM, the AFOC drives fiber tip directly with advantages of precise control, small inertia, high resonance-frequency and convenience for integration. The distance of adjacent beamlets is 42-mm, and thus the fill factor of the near-field laser array is calculated to be 0.67. The collimated output beams emitted from the AFOC array are focused by a transform lens with the focus length of 1.5-m. The reflected beams from folding mirror are divided into three parts by BS1 (beam splitter) and BS2: one is sent to a PD with a pinhole of 20-μm diameter for phase-locking, another is detected by a high-speed CMOS camera for tip/tilt control, and the third is sent to a 10 × micro-objective and a camera for observation. The PD is a PDA36A silicon amplifier detector with a 350 nm-1100 nm response wavelength and 12.5-MHz bandwidth when the gain is at 10-dB, produced by THORLABS Corporation.
Fig. 1 Experimental setup. MO: master oscillator. PZT-PM: piezoelectric-ceramic-ring fiber-optic phase modulator. AFOC: adaptive fiber-optics collimator.
In this setup, SPGD algorithm is employed for cost function optimization. The conventional PIB cost function acquired from PD is chosen for phase-locking, and the divergence cost function acquired from the high-speed CMOS camera is employed for tip/tilt control, which is proposed for the first time to our knowledge and will be discussed in Section 3 for details. The SPGD algorithm generates seven phase-locking signals and fourteen tip/tilt control signals and the steps of SPGD algorithm optimizing cost function are:
- 1) Generate a group of random voltage perturbations, which obey the Bernoulli probability distribution with zero mean.
- 2) Apply the signalson compensators and get the corresponding cost function; then apply the signalson compensators and get the cost function.
- 3) Update the control-voltage signals on the compensators to maximize the PIB cost function for phase-locking:
or update the control-voltage signals on the compensators to minimize the divergence cost function for tip/tilt control:
whereandare the gain coefficients of SPGD algorithm.
- 4) Go to step 1 and continue the process, until the control procedure is stopped manually.
Two kinds of aberration correction devices mentioned above, PZT PM and AFOC, are homemade in the Key Laboratory on Adaptive Optics, Chinese Academy of Sciences, and their structural schematic diagrams are described in Figs. 2(a) and 2(b), respectively. The PZT PM has a half-wave voltage of 3.1-V and a frequency response of about 90-kHz. Figure 3 shows the performances of AFOCs used here. As plotted in Fig. 3(a), the deflection angles of collimated beams are in range ± 0.5-mrad when the driving voltages applied on AFOCs are in range ± 400-V. The first resonance-frequency of AFOC is about 1.85-kHz, as depicted in Fig. 3(b).
Fig. 3 The performances of AFOC. (a) Deflection angle of the AFOC as the function of applied driving voltage. (b) Frequency response curve.
3. Divergence cost function for tip/tilt control
In previous CBC or incoherent beam combining (IBC) experiments with tip/tilt control, the conventional PIB cost function was often chosen with a fixed bucket-size in SPGD-based optimization, where a PD or a high-speed camera was used as the sensors. Since the response rate of common used tip/tilt control devices, like FSM and AFOC, is below several kHz-level, the high-speed camera with several-kHz frame rate might sufficiently ensure most applications with them. Except for PIB metrics, the images of camera can provide more information of combined spots, like the max-divergence to the target location of beam pointing. The steps of calculating max-divergence via the images of camera can be described as:
- 1) Acquire an image from the camera, and then set threshold value to each pixel of the image. If the gray-scale value of the pixel is greater than or equal to the threshold, substitute the gray value to 1; else substitute the gray value to 0. It is obvious that the divergence metrics will be independent to the intensity saturation of sensors.
- 2) As shown in Fig. 4, (x0, y0) stands for the coordinates of the target location in the image, and (x, y) is the coordinates of any one pixel with value of 1. Calculate the pixel number n(x, y) between every available pixel and the target location:
- 3) Use the maximum value nmax of n(x, y) and the max-divergence θmax of combined spots to the target location is:
where l is the length of the pixel and f is the focal length of the transform lens.
- 4) The θmax will be used as the divergence cost function for minimization of SPGD optimization, which will also been employed as a sign for automatic switching of tip/tilt control.
The feasibility of divergence cost function for tip/tilt control has been investigated via the seven channel CBC setup. The iteration rate of tip/tilt control using SPGD algorithm is about 300-Hz. Figure 5 shows two snapshots of the far-field images acquired by the tip/tilt control CMOS camera before and after closed loop achieved. The gray-scale values of image’s pixels have been compared with the threshold and substituted to 1 or 0, so the divergence cost function is independent to the intensity-saturation of camera. When using the PIB cost function based on intensity-optimization, the intensity-saturation of camera might cause the incorrect evolution of PIB metrics, which will affect the stability and accuracy of tip/tilt control system [22]. As depicted in Fig. 5(a), in open loop, seven spots are visibly and absolutely separate in the far-field, where the maximum divergence to the target location (the center of the pattern) is about fourfold of a single spot’s diameter. The maximum distance between spots (spot-1 to spot-7) is about sixfold of a single spot’s diameter. While in references [2,20–24] where the PIB cost function was chosen for tip/tilt control, the maximum distance between spots is generally no larger than twice of a single spot’s diameter. As depicted in Fig. 5(b), the dispersive spots overlap well, which indicates the tip/tilt aberrations of beamlets are corrected and closed loop is achieved. In this experiment, the use of divergence cost function ensures wider tip/tilt correction range. The spreading of combined spots compared with the single spot is because of the threshold setting.
Fig. 5 Two snapshots of the far-field images acquired by the tip/tilt control CMOS camera (a) before and (b) after closed loop achieved.
Figure 6 plots the evolution curves of divergence metrics when tip/tilt control off and on, respectively. In open loop, the average of the divergence metrics is calculated to be 432-μrad with the MSE (mean square error) of 2-μrad.The tip/tilt closed loop is achieved after about 700-iterations of SPGD min-optimization, where the average of the divergence metrics decreases to 89-μrad with the MSE of 2-μrad as well.
Fig. 6 The evolution curves of divergence metrics as the function of iteration number when tip/tilt control off and on, respectively.
Another advantage of using divergence cost function is that, the cost function itself denotes tip/tilt errors directly which could be used as a sign for automatic switching of tip/tilt control program. If the divergence metric is smaller than required precision of the system, the control loop will be stopped automatically to eliminate the adverse influence to phase-locking course which uses SPGD algorithm as well. Once the divergence metric is detected to be larger than the required precision, the control loop will be turned on immediately. This function has been implemented in Section 4. When using the intensity-based PIB cost function for tip/tilt control, only the indirect intensity-sign could be employed as the program switch, which might have less accuracy and has not been reported yet.
4. CBC experiment
After demonstrating the divergence-based tip/tilt control approach, the CBC with correcting both piston- and tip/tilt-type phase errors simultaneously is investigated. The iteration rate of phase-locking using SPGD algorithm is 30-kHz, and the rate of tip/tilt control is about 300-Hz. Figure 7 shows the normalized PIB metrics acquired from PD as the function of time during three stages, that is, open-loop stage, tip/tilt control stage, and phase-locking & tip/tilt control stage. In Stage I, the average of PIB metrics is only 0.03, which is mainly caused by the background noises of the PD itself. The MSE of metrics is only 0.004. In Stage II, the voltage signal from PD fluctuates between 0.3 and 0.9 randomly with an average of 0.56 and MSE of 0.152. In Stage III, the voltage signals are locked steadily at its highest value with an average of 0.97 and MSE of 0.01, which is only 1/15 of that in Stage II. The power in the full width at half maximum (FWHM) of the main lobe (collected by a pinhole of 20-μm diameter) increases by 32 times compared with Stage I. Here, the divergence cost function has been employed as a sign for automatic switching of tip/tilt control, as mentioned in Section 3. The insert drawing in Stage III depicts the initial evolution curve of normalized PIB as the function of iteration number in CBC. The closed loop of phase-locking is achieved after about 150-iterations of SPGD max-optimization. A residual phase error is evaluated to be less than λ/15 using the expression [7,8]:
where J(t) is the cost function evolution when CBC achieved.The 37-second long-exposure far-field intensity distributions acquired by the 10 × micro-objective and the observation camera are shown in Fig. 8. The target location of beam pointing is at the center of the image. In open loop, seven spots are visibly separate in the far-field, as shown in Fig. 8(a). When tip/tilt control is implemented, the dispersive spots overlap well, but the corresponding long-exposure pattern is an incoherent one with a fringe visibility of nearly zero, as depicted in Fig. 8(b). Figures 8(c) and 8(d) describe the realized CBC results with intensity maximum and minimum in the pinhole, respectively. Figures 8(e) and 8(f) are partial enlarged views of Figs. 8(c) and 8(d) in the red square area. Compared with the theoretical results in Figs. 8(g) and 8(h), it can be concluded that good CBC effects have been achieved. Figures 8(i) and 8(j) are the one-dimensional intensity distributions of Figs. 8(e) and 8(f) along the central lines, respectively. As depicted in Fig. 8(i), the FWHM of the main lobe takes up about 23-pixels on the observation camera, where the length of each pixel is 10.8-μm. So, the FWHM of the main lobe is calculated to be 248-μm. Considering the decade amplification of the micro-objective, the non-amplified FWHM value is 24.8-μm, which is quite fit for the 20-μm diameter pinhole before the PD for PIB confirmation. The actual evolution curves of PIB metrics acquired from PD before and after phase control (depicted in Figs. 8(b) and 8(c), respectively) are corresponding to the Stage II and Stage III of Fig. 7, respectively.
Fig. 8 The 37-second long-exposure far-field intensity distributions acquired by the observation camera. (a) Open loop. (b) Tip/tilt control. (c) and (d) CBC with intensity maximum and minimum in the pinhole, respectively. (e) and (f) Partial enlarged views of (c) and (d) in the red square area, respectively. (g) and (h) Theoretical results of (e) and (f), respectively. (i) and (j) The one-dimensional intensity distributions of (e) and (f) along the central lines, respectively.
The corrected tip/tilt errors of beamlets can be estimated by the use of driving voltages applied on the AFOCs via tip/tilt control. In this way, the absolute values of initial tip/tilt errors of seven beamlets in Fig. 8(a) are calculated to be (362, 241) μrad, (207, 35) μrad, (98, 194) μrad, (153, 78) μrad, (177, 181) μrad, (40, 221) μrad, and (171, 172) μrad, respectively, with an average of 166.4-μrad, as shown in Fig. 9.
Figure 10 is the spectral density of energy collected by the pinhole of 20-μm diameter as the function of frequency. The calculated data remarked in blue solid curve and red dash curve are from the Stage II and Stage III of Fig. 7, respectively. The SPGD iteration rate for phase-locking is 30-kHz and corresponding closed loop is achieved after about 150-iterations via SPGD max-optimization. The bandwidth of PD from THORLABS Corporation is 12.5-MHz. So, the actual control bandwidth of the phase-locking control system is about 30-kHz / 150 = 200-Hz. It can be concluded from Fig. 10 that phase noises below 20-Hz have been noticeably compensated and two peaks at 80-Hz and 100-Hz have been depressed.
Fig. 10 Spectral density of energy collected by the pinhole of 20-μm diameter as the function of frequency.
5. Conclusion
We have demonstrated the feasibility of the new-proposed divergence cost function for tip/tilt control via the CBC of a seven-channel 2-W fiber amplifier array. Compared with the conventional PIB cost function for SPGD optimization, the tip/tilt control using divergence cost function ensures wider correction range, automatic switching control, and freedom of camera’s intensity-saturation. An average of 432-μrad of divergence metrics in open loop has decreased to 89-μrad when tip/tilt control implemented. In CBC, the power in the FWHM of the main lobe increases by 32 times, and the phase residual error is less than λ/15. The outdoor CBC application of this tip/tilt control method will be performed in the near future based on a target-in-the-loop architecture.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (Grant No. 61205069 and 61138007).
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