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Abstract
Coherent beam combination (CBC) holds promise for scaling the output power of the laser system while maintaining good beam quality. Owing to the thermal effect and mechanics instability, piston and tilt distortions always exist and affect the performance of the combined beam. To ensure the constructive interference in the far-field, dynamic correction of the distortions is highly required. Here, we propose an approach for the simultaneous correction of piston and tilt distortions in CBC systems. Based on the position and interval information of the near-field interference fringes, the theoretical expressions for the relative piston phase and tilt errors of each array element are derived, indicating that dynamic distortions in CBC systems can be directly calculated and then corrected by employing phase control servos. To demonstrate the feasibility of the proposed approach, Monte-Carlo Simulations have been carried out for different perturbative environments. Our results indicate that both piston phase and tilt errors can be calculated and compensated accurately (λ/25 and 0.11μrad) by the proposed approach even in 169 beamlets, which also has high tolerance for defocus errors. This work could provide valuable reference on the practical implementation of high-power, phase-locked fiber laser array systems.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
There is a strong demand for high-power fiber lasers in industrial processing, free-space optical communication, and scientific research fields [1,2]. Coherent beam combination (CBC) of fiber lasers is an efficient way to scale the output power while maintaining near diffraction-limited beam quality, which can avoid nonlinear effects and thermal effects in a single laser [3–9]. In CBC systems, the optical fields of multiple laser beams are coherently overlapped, and constructive interference would be formed in the far-field. Owing to the thermal and environmental fluctuations, dynamic distortions always occur in CBC systems, which cause the degradation of combining efficiency and seriously affect the performance of the combined beam. Therefore, accurate compensation of dynamic distortions, especially the piston phase and tilt errors of laser beams, is highly required and challenging. To compensate the distortion in piston phase, ongoing effort has been made to develop the phase control methods in the past decades, which can be divided into two categories according to the basic physical principles. One is the passive phase control method, a straightforward method that realizes self-phase locking based on pure optical principle, such as energy coupling mechanisms [10–14] and nonlinear effects [15]. This method has a simple and effective structure since it does not require an electrical circuit to calculate the piston phase errors. However, the optical structure of the entire system would become complex and sensitive when the number of combining elements increases, and the maximum number of combining elements was reported as 25 [13]. The other one is the active phase control method, which involves an electrical circuit for the calculation and compensation of piston phase errors [16,17], such as the dithering techniques [18–21], optimization algorithms [22–32], and interference techniques [33–39].
Based on the active phase control method, various notable results have been reported in CBC systems. In the year of 2020, efficient phase locking for more than 100 channels was realized [40], and 16 kW output power of CBC system was reported [41]. Despite the remarkable progress in piston phase control methods, simultaneous compensation of piston phase and tilt distortions is essential for CBC systems while still remains a critical challenge. In fact, salient features of CBC systems including spatially beam shaping, all-electronic beam steering, and atmospheric turbulence induced wavefront distortions compensating are enabled by the flexible control of both piston and tilt wavefronts. Accurately manipulating the piston and tilt wavefronts of combining elements at the same time is an important step towards the future implementation of CBC outside the laboratory. Furthermore, additional elements are required for tilt correction in piston-only correction systems. When the system is operated in the perturbative environments with intensive tilt errors, the coupling between piston and tilt distortions would cause the significant degradation of the compensation accuracy. To date, serial and parallel performed optimization algorithms [24,42,43] and four-quadrant detection approach [44] have been attempted to solve this issue. Recently, it has been reported that one can directly calculate the piston and tilt distortions at the same time [45], and an additional grating is required to be employed. Here, we have the motivation to directly calculate the piston and tilt distortions at the same time based on the Mach-Zehnder interferometer, thus the calculated distortions are expected to be compensated without iterations of algorithms and the introduction of additional detectors or gratings.
In this paper, we put forward and validate an approach for simultaneous correction of piston and tilt distortions in CBC systems by using the information of near-field interference fringes. The theoretical model of the near-field interference fringe based active control for CBC has been constructed. Based on the theoretical model, the promising prospect for number scalability of array elements is demonstrated, and the efficiency of phase locking when the system is operated in different perturbative environments can be analyzed. The necessity for the tilt errors compensation is verified by numerical simulation, and furthermore, the utility of our proposed method for accurate calculation and direct compensation of piston and tilt distortions is demonstrated. This work could offer suggestive instructions for the development, design and implementation of large-scale CBC systems.
2. Theoretical analysis
The simultaneous piston and tilt distortions correction system mainly consists of two parts: a reference beamlet and an optical beam reducer, as shown in Fig. 1. The seed is amplified by the pre-amplifier. The amplified laser is split to two beams. One is the reference beam. The other one is the seed laser for coherent beam combination. For coherent beam combination, the seed is split to N beamlets. Each beamlet goes through a phase modulator(PM), an amplifier(AMP) and a collimator. The phase modulator is drived by the controller to change the piston phase of each beamlet, in order to make the piston phase differences of each beamlet divisible by ${2 \pi }$. The AMPs amplifies the power of each beamlet for the high-power output laser. The collimator is adaptive fiber-optics collimator(AFOC) which can change the tilt phase of each beamlet with a compact space structure. The high-power laser is emitted after the emitting plane formed by the AFOCs. There is a sampler to sample the high-power lasers for phase-locking. An optical beam reducer is set to reduce the size of the phased-array to adapt the size of the high-speed camera. The pre-amplifier of the reference beam should be adjusted to obtain a laser output that matches the power density of the sampling beam on the sensor plane of the high-speed camera. Then, there are clear interference fringes from the image plane of the high-speed camera after the half mirror caused by the reduced beam and the reference beam.
Fig. 1. Schematic of the simultaneous piston and tilt distortions correction system for hexagonally distributed seven-beam CBC. PM, phase modulator; AMPs, amplifiers; AFOC, adaptive fiber-optics collimator; CCD, high-speed camera.
When N beams are emitted by a fiber phased-array, the output light field at the emitting plane is expressed as:
After passing through the optical beam reducer, the light field is similar with the light field at the emitting plane. The influence of the reducer is the change of energy distribution. The amplitude of each sub-beam will increase to $\kappa /s$ times than it in the emitting plane, where $\kappa$ and $1/s$ are the beam reducing ratio of the reducer and the sampling rate of the sampler.
The interference fringe is caused by the interference of the reduced beam array and the reference beam. The intensity distribution at the image space can be obtained by:
Hence, the tilt phase error of the j$^{th}$ sub-beam in the x direction can be expressed as:
And the tilt phase error of the j$^{th}$ sub-beam in the y direction can also be expressed as: where ${\delta _{y,j}}$ is the interval information in the y direction of the interference fringe in the area of the j$^{th}$ sub-beam.According to the calculated tilt phase error in the x direction and the tilt phase error in the y direction of the jth sub-beam, combined with the position coordinates of the interference fringe extreme value, the piston phase error of the j$^{th}$ sub-beam can be calculated:
Based on the equations (6)– (8), one can simultaneously calculate and correct the piston and tilt distortions of each beamlet by using the near-field interference fringes, and therefore, the combining efficiency of the CBC system with the compensation of piston and tilt distortions is expected to be enhanced.
3. Numerical simulation and discussion
The scheme of our simulation has been given in Fig. 1. The parameter values are as follows. Our simulation system chooses a seed laser with a center wavelength at 1064 nm. The diameters of the collimators are 22.5 mm. The collimators are designed for the truncated Gaussian beamlets, which has been discussed in Ref. [46] and Ref. [47]. The Gaussian beam width ${\omega _0}$ is 8.4mm in the simulation for the truncation design. The distance between the centers of the two collimators is 25.0 mm. The reflectivity of sampler is 1$\%$. As for the high-speed camera, the number of active pixels is 2400(H)${\times }$1350(V) in the simulation. The output of the camera is 10-bit data, and the intensity would be changed to 0-1023 for the different intensity. The size of the pixels is $8\ \mu m \ \times \ 8\ \mu m$. The optical format of the sensor is 4/3"(19.2 mm ${\times }$ 10.8 mm). With the size limit of the high-speed camera’s sensor, the beam reducer should reduce the output beamlets by 45 times. The parameter of the beam reducer is the same throughout all of the simulations, and the size of the reduced beamlets is the same no matter how the number of elements changes. The peak power of the reference beam is twice that of the beamlets, which is determined by the Equation (3) presented in the Theoretical Analysis section. The accuracy of piston and tilt calculations depends on the number of the formed interference fringes. Without loss of generality, to ensure the high accuracy of piston and tilt calculations at the same time, the angle between the reference beam and the optical axis is selected to be 8.5 mrad, as a typical value, to form five interference fringes. For the sake of piston and tilt calculations accuracy, the value of the angle for the following simulations is chosen to be 8.5 mrad throughout this work, as shown in Fig. 2(a) and Fig. 2(b).
Fig. 2. The small errors in accuracy caused by the discretization of the high-speed camera. (a) The piston phase differences make the stripe shift, (b) the detailed 2D distributions for different piston phase errors, (c) the data of fringes recorded by the high-speed camera in simulation and (d) the calculation accuracy with 63 data points for one beamlet.
The discretization of the high-speed camera makes the equations (6)– (8) not accurate strictly because the value of ${x_{\max }}$ and ${y_{\max }}$ are not rigorously accurate. The influence on the calculation accuracy is extremely required caused by the discretization of the high-speed camera at the very beginning. Hence, the simulation is carried out when the beamlet’s piston phase changes at the array [0, ${\pi /4}$, ${\pi /2}$, ${3\pi /4}$, ${\pi }$, ${5\pi /4}$, ${3\pi /2}$, ${7\pi /4}$, ${2\pi }$], and we can calculate the phase errors according to the interferogram based on the high-speed camera, shown in Fig. 2(c). We assume that the camera is a linear system without extra noise throughout the simulation. As shown in Fig. 2(d), the discretization value of the high-speed camera sensor introduces the calculation error of the piston phase. In the simulation, we simulate the imaging process of the camera, especially the sampling and imaging process of its pixel unit. The pixel size is $8\ \mu m \ \times \ 8\ \mu m$ for the camera. As for the number of pixels allocated to a beamlet, in the simulation, the sampling points are 63 ${\times }$ 63 for the reduced beamlet with the length of 0.504 m. We choose this value in our simulation as a uniform value. The sampling and imaging process of its pixel unit change the data from Fig. 2(a) to arrays with the length of 63 and the value of unsigned integer between 0 and 1023, as shown in Fig. 2(c). When the sampling rate is 63 ${\times }$ 63 for one beamlet in the high-speed camera’s sensor, the performance of calculation accuracy is quite satisfactory. The maximum calculation error is less than 0.1 rad (${\sim \lambda /60}$) when the beamlet’s piston phase changes at the array [0, ${\pi /4}$, ${\pi /2}$, ${3\pi /4}$, ${\pi }$, ${5\pi /4}$, ${3\pi /2}$, ${7\pi /4}$, ${2\pi }$], which would reduce combining efficiency by less than 1$\%$ [48].
As for the length of time required from the sensor of the camera to the correction of the PM, we give the flow diagram of the signal in Fig. 3. Trough the temporal analysis of the system, the length of time required is given by the following equation.
where ${\tau _e}$, ${\tau _t}$, ${\tau _p}$ and ${\tau _c}$ are the time costs for the exposure time of the camera, the transfer for the image from the camera to the controller, the image processing and the signal transform of the controller after receiving the image and the operating of the PM and the AFOC, respectively. The exposure time of the camera, ${\tau _e}$, is set to 5 ${\mu s}$ in Eq. (9). The frame rate is 2000 frames per second. And the time cost for transferring an image, ${\tau _t}$, is 500 $\mu s$. As for the value of ${\tau _p}$, the image processing is parallel with the transfer of the image in the controller. Hence, the value of ${\tau _p}$ is just the time cost for the signal transform from the processing result to the voltage of the driver, which is only a few clock periods. The clock period is 10 ns since the clock frequency is 100 MHz for the controller. The value of ${\tau _p}$ is set to 50 ns. The value of ${\tau _c}$ is the responsive time of the PM and the AFOC and its The maximum is the responsive time of the AFOC, which is 2 $\mu s$. According to the above-mentioned parameters, the length of time required from sensing the phase error to implement the phase correction is calculated to be less than 510 $\mu s$. The characteristic frequency of piston phase noise of the fiber amplifiers is much lower than 1 kHz. Hence, the system can compensate for the piston noise.In order to more systematically characterize the control accuracy in this CBC system, we conduct a more detailed simulation based on this method.
3.1 Environment with only piston phase errors
In previous study [38], a 61-channel CBC system by using the near-field interference fringes based phase control technique, and the piston distortions are extracted from the position information of the interference fringes. To develop the technique, investigating the further scalability of array elements in the CBC system is a significant issue. Here, we consider the environment with only piston phase errors and illustrate whether the residual piston phase (after compensation) increases as the number of array elements increase. In the CBC system without tilt and defocus errors, the fringes only consist of the piston phase noise value in the near field interferogram. The Root-Mean-Square(RMS) value of the residual errors should be taken into consideration in the CBC system. We investigate the control ability in a CBC system with 7 beamlets. The beamlets are are applied with random piston phase errors. And the controller calculates the piston phase error through the image from the high-speed camera. As time goes by, the control will calculate 100 different piston phase distributions in this system. And all the residual errors will be recorded to calculate the RMS value in order to represent the control accuracy.
In the simulation, the controller controls seven channels (see Fig. 4(a)) at the same time. The piston phase noise distribution is given by Fig. 4(b). The interferogram is recorded by the high-speed camera with a single pattern (see Fig. 4 (c)), and the combined beam is shown in Fig. 4(d) with the piston phase noise distribution shown in Fig. 4(b). Then, the controller makes calculation based the interferogram shown in Fig. 4(c). The calculation is shown in Fig. 4e). After the piston phase control, the distribution is shown in Fig. 4(f), where the residual errors are around 0. Up to now, the controller will stop changing the control voltage of the phase modulator with the interferogram shown in Fig. 4(g). The combined beam is shown in Fig. 4(h) after the control.
Fig. 4. Performance of the phase compensation based on the near field interferogram in a CBC system composed of 7 beamlets. (a) Intensity profile. (b), (c) and (d) are the phase map, intensity profile of the near field interferogram, and intensity profile of combined beam with random phase errors, respectively. (e) the calculation of controller. (f), (g) and (h) are the phase map, intensity profile of the near field interferogram, and intensity profile of combined beam after the controller’s compensation, respectively.
For the RMS value of the residual errors, We conduct the simulation by 100 different initial random piston phase distributions such as the Fig. 4(b). The different initial random piston phase distributions are generated by a random generator with the magnitude of ${2\pi }$. And we calculate the interference pattern on the camera and simulate the camera output, shown in Fig. 4(c) as an example. Then, we simulate the image processing by the interference pattern to calculate the piston phase distributions. The random phase distributions and the calculated phase distributions are all known in our simulation. Hence, we can obtain the histogram shown in Fig. 4(e). We record the residual errors when the controller calculate the piston phase for 100 different phase distributions. All residual errors are presented in Fig. 5. To present the maximum of the calculation errors, the maximum value of the absolute value of the calculation errors is 0.16 rad (${\sim \lambda /40}$). Among all simulation results, the controller can calculate the piston phase accurately with an average RMS value of 0.08 rad (${\sim \lambda /80}$).
To illustrate the scalability of array elements, we extend the number of elements in the laser array to 169 with a hexagonal arrangement of 7 circles, as shown in Fig. 6(a). To initialize, the 169 beamlets are placed piston phase distortion as shown in Fig. 6(b). The size of the interferogram, shown in Fig. 6(c), is smaller than that of the high-speed camera sensor. The controller can divide the pattern by the position of each beam through only one exposure. The performance of the controller is presented in Fig. 6 for an initial piston phase distribution. Figures 6(b), (c) and (d) are the phase map of the beams, the intensity profiles of the near field interferogram and intensity profile of combined beam with the initial piston phase distribution, respectively. Figure 6(e) presents the computed ability of the controller, which shows the relationship of piston phase noises and the calculations of 8 beamlets masked in Fig. 6(b). Figures 6(f), (g) and (h) present the computed results for the controller, which are the residual errors, intensity profile of the near field interferogram after control, and intensity profile of combined beam after the controller’s compensation. In order to make the intensity profile of the combined beam with random phase errors clearer, the maximum of the intensity scale in Fig. 6(d) is equal to 0.1 of the maximum of the intensity scale in Fig. 6(h).
Fig. 6. Performance of the phase compensation based on the near field interferogram. (a) Intensity profile of 169 beamlets. (b), (c) and (d) are the phase map, intensity profile of the near field interferogram, and intensity profile of combined beam with random phase errors, respectively. (e) the calculation of controller. (f), (g) and (h) are the phase map, intensity profile of the near field interferogram, and intensity profile of combined beam after the controller’s compensation, respectively.
For the RMS value of the residual errors, we conduct the simulation which is similiar with the data processing for Fig. 5. Analyzing these data (A matrix with a dimension of 169${\times }$100) shown in Fig. 7, the maximum value of the calculated errors is 0.26 rad (${\sim \lambda /20}$) among the 16900 values. It can be observed that the average RMS value is also 0.08 rad (${\sim \lambda /80}$), which does not increase with the expansion of the number of beamlets.
From the above data, it can be seen that the interferogram recorded by the high-speed camera can achieve high accuracy of the piston phase calculation, which is supported by the introduction of reference light to form a Mach-Zehnder interferometer. Since the input of the controller is a 2-D image which is is divided according to the number of beams, the controller can present all the phase errors of the laser array based on the sub-image of every beamlet. The controller can provide the phase errors of the whole beamlets when the controller is running, which is the basis for real-time monitoring of the phase noise distributions of the lasers.
To conclude, the residual phase error increases little along with the number of beamlets increases, indicating that the near-field interference fringes based phase control technique exhibits salient feature in terms of the high array elements scalability. However, the CBC of fiber lasers is sometimes implemented in more unstable environments, thus the system would suffer from the tilt and defocus distortions. These distortions may cause the deformation of the stripes and the corresponding loss of the accuracy. Therefore, further investigation for the tolerance of the control technique in the environment with tilt and defocus errors is highly required.
3.2 Challenges for the environment with tilt and defocus errors
It is worth noting that the phase control technique is only based on the position information of the fringes which are presented in Section 2, ${({x_{\max }},{y_{\max }})}$. However, the tilt and defocus errors would also change the distributions of the interference fringe. The position information of the fringes is different with the same piston phase since the tilt and defocus distortions exist. Hence, the performance of the piston phase calculaiton without tilt compensation needs to be studied based on the position information of the fringes only.
In this section, we study the CBC systems that are operated in the more complex environment with tilt and defocus distortions, which are the higher-order terms of the Zernike polynomials. To elucidate the influences of these errors, the tilt and defocus errors are added by changing the amplitude of the corresponding term of the Zernike polynomials, and the optical field of the laser array at the emitting plane can now be expressed as
Based on these added errors, we calculate the residual errors for the piston phase compensation. In the CBC system of 7 beamlets, the changes of the interferogram are presented after the tilt and defocus errors are introduced, as shown in Fig. 8. When we calculate the piston errors in this system, the results become inaccurate since the fringes has shifted according to the tilt and defocus errors.
Fig. 8. Influence on the fringes of the tilt and defocus errors. (a) Intensity profile of fringes without tilt and defocus errors. (b) Intensity profile of the near field interferogram when the beamlets include tilt phase errors. (c) Intensity profile of the near field interferogram when the beamlets include defocus phase errors.
Based on the interferograms of the CBC system at the high-speed camera plane shown in Figs. 8 (b) and (c), the computational accuracy could be investigated by modulating the amplitude of the tilt and defocus errors as is shown in Fig. 9.
Fig. 9. The relationship between the computational accuracy with (a) tilt phase errors or (b) defocus errors.
The tilt and defocus errors are is gradually strengthened by increasing the value of the Zernike polynomial coefficient. To make the parameters easier to understand in the CBC system, the tilt phase errors are transformed to $\frac {{\Delta \theta }}{{{\theta _0}}}$, where $\Delta \theta$ and $\theta _0$ are the maximum value of the tilt random noises and the divergence angle of Gaussian beam, respectively. [48] The derivation process is shown in Eq. (11).
The results are from a Monte-Carlo Simulation to find the relationship between the computational accuracy with tilt or defocus errors, which are shown in Figs. 9(a) and 9(b), respectively. The controller calculates the piston phase by the interferogram recorded by the high-speed camera through the position information of the fringes, which ignores the influence of the tilt and defocus errors. The calculated errors of piston phase increases with the applied tilt and defocus errors. However, the slope of increase is quite different. For the influence of tilt errors, as shown in Fig. 9(a), the calculated errors increase rapidly with the increase of the tilt errors. Hence, the tilt errors have a major impact on the piston phase’s calculaiton. In contrast, the results for the influence of defocus errors are quite different. As shown in Fig. 9 (b), the RMS of calculated errors is 0.11 rad (${\sim \lambda /60}$) when the ${Z_{defocus}}$ is 9. Hence, the influence of the defocus errors can be ignored.
In a nutshell, the tolerance of the previous piston phasing technique, which is only based on the position information of the fringes, on the tilt and defocus errors has been studied. From the analysis above, the interferogram including the tilt errors can not give the piston phase directly and accurately. As the amplitude of these errors grow, the piston phase calculation becomes inaccurate. The tilt errors of beamlets play a major role. The defocus errors are negligible for the piston phase calculation. However, in the practical implementation of CBC, especially operating at high power and requires far distance propagation, tilt and defocus errors would be generated and seriously impact the efficiency of the CBC system. Therefore, it is a vital issue to reduce the tilt errors or to calculate the tilt phase errors for compensation of the piston phase calculation errors.
3.3 Simultaneous piston and tilt distortion correction
According to the results from Section 3.2, we have demonstrated that only using the position information of the fringes is robust towards the defocus errors, whereas in the environment with dynamic tilt errors, the estimation of piston phase would deviate without the prior knowledge of the tilt distortion information. Hence, new methods are required to compensate for tilt control in the systems with tilt errors. As mentioned in the theoretical analysis in Section 2, our approach is presented to compensate for tilt control using the interval information of near-field interference fringes. The proposed method would calculate the tilt errors through the interval information at the very beginning. Then the controller calculate the piston phase errors. The tilt errors and the piston phase errors are compensated by only one interference pattern through the multiplexing of the position and interval information of the fringes. In this section, we show the utility of this proposed method for simultaneous piston phase and tilt correction in the perturbative environment with both piston and tilt distortions. The approach calculates the tilt errors through the interval information in the x and y directions, ${\delta _{y,j}}$ and ${\delta _{y,j}}$. In general, we choose the beamlet in the bottom right corner, with the sequence number of 169 in our simulation, to illustrate the control flow for the simultaneous correction of piston and tilt distortions, as shown in Fig. 10. Figure 10(a) presents how we extract ${{\delta _{x,j}}}$ and ${{\delta _{y,j}}}$, the interval information in the x and y directions, and the ${x_{\max }}$ and ${y_{\max }}$ are given based on the position information of the maximum point. Then, the piston phase can be calculated by Eqs. (6)– (8). In order to illustrate the calculation process under certain conditions, we give the calculation details of interference fringes in a single direction, as presented in Fig. 10(b). The tilt angle of the reference beam only exists on the y axis. Hence, ${\mu _{ref} = 0 \ mrad}$ under this condition. Equation (8) can be changed to a more concise form.
Fig. 10. The control flow for the simultaneous correction of piston and tilt distortions. (a) and (b) are the interferogram of the actual and the ideal environment, respectively.
To test the effectiveness of this method, we carry out the experiment by this method in the above system. In Section 3.2, the RMS value of the piston calculaiton is inaccurate (${\sim 1.6 rad}$) when the fractional tilt errors is greater than 0.2, shown in Fig. 9 (a). It has been presented that the tilt error is within ${ 7\ \mu rad}$ when the fiber amplifier operates at 1.8 kW in Ref. [49]. The value of $\frac {{\Delta \theta }}{{{\theta _0}}}$ in Ref. [49] is less than 0.1. In the simulation, we choose 0.5 as the value of $\frac {{\Delta \theta }}{{{\theta _0}}}$ to demonstrate the effectiveness of the compensation of the tilt errors. The tilt errors are generated by a random generator with the magnitude between - 0.5 to + 0.5 for the value of $\frac {{\Delta \theta }}{{{\theta _0}}}$, and the piston phase noises are generated by a random generator with the magnitude of ${2\pi }$. The simulated controller calculate the interference pattern to obtain the value of the piston phase, the tilt distortion in the x direction, and the tilt distortion in the y direction. Subsequently, we record the distributions of the piston phase errors, the piston phase calculations, the tilt phase errors in the x direction, its calculations in the x direction, the tilt phase errors in the y direction, and its calculations in the y direction. The results are shown in Fig. 11. From the data of the 100 times Monte-Carlo simulations, the piston and tilt phase errors can be calculated by only one frame. The RMS values of piston and tilt phase errors are 0.25 rad (${\sim \lambda /25}$) and 0.11 ${\mu }$rad ($\frac {{\theta _{RMS}}}{{{\theta _0}}}$ = 0.00289) respectively.
Fig. 11. The calculation performance of the controller with the tilt and piston phase errors. (a1) - (a3) are the phase distribution with tilt and piston phase noises, the compensation, and ideal calculations, respectively. (b1) - (b3) The intensity profile of the interferogram with the phase distributions of (a1) - (a3). (c1) - (c3) The intensity profile of the combined beam with the phase distributions of (a1) - (a3).
To summarize, our approach that extracts the optical field information from both the position and interval information of near-field interference fringes ensures the high-accuracy calculation of piston distortions based on the compensation of the tilt phase errors. As a result, with the efficient correction of the piston and tilt distortions, the performance of the CBC system can be improved.
4. Conclusion
In this work, we propose a scheme for simultaneous compensation of piston and tilt distortions in CBC systems by using the position and interval information of near-field interference fringes. The theoretical model is constructed and numerical simulations have been performed to verify the extensibility and robustness of the near-field interference fringes based phase control method. The results of comparison between CBC systems that consist of 7 and 169 array elements indicate that the phase locking method holds great promise for the further scaling of array elements. Moreover, the robustness of phase locking in different perturbative environments have been studied, and we found that uncompensated tilt errors would seriously affect the performance of piston phase control, indicating that the simultaneous correction of piston and tilt distortions for the CBC systems with near-field interference fringes based control is highly required. Furthermore, the feasibility of our proposed scheme in terms of the accurate compensation of piston and tilt distortions is demonstrated. This work could provide a valuable reference on the upgradation, design, and future implementation of large-scale CBC systems.
Appendix
The general code is listed in Algorithm 1 to construct piston and tilt errors in CBC systems. The feature of the errors could be adjusted by the parameters of the amplitude and noise function. The piston and tilt errors used in this work is generated by the parameters listed in Algorithm 1.
Algorithm 1. A script to generate pistion and tilt errors in CBC system.
To process the data in the simulation, a code is required to read the results, add them to a vector, and calculate the RMS value, which is listed in Algorithm 2.
Algorithm 2. A script to read the simulation results, add them to a vector, and calculate the RMS value.
Funding
National Natural Science Foundation of China (62075242); Innovative Research Groups of Hunan Province (2019JJ10005); Training Program for Excellent Young Innovators of Changsha (KQ1905051).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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