We proposed a new phase-locking technique for multibeam coherent beam combination. By near-field angle modulation and angular spectrum measurement, we obtained the relative phase between each pair of beams with one camera. This method is appropriate for multibeam schemes and possesses the advantages of high accuracy, resistance to energy fluctuation, and simplicity, as shown by the analysis in this study. In a proof-of-principle experiment, we realized the phase-locking of three beams, achieving a Strehl ratio of 89.5%. Our method may supply a scheme for multibeam coherent combining of ultra-intense bulk laser systems.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
In the last few decades, the advent of chirped pulse amplification (CPA)  has revolutionized ultrashort ultrahigh-intensity laser systems, and with it a wide variety of disciplines ranging from advanced attosecond science, laser-plasma electron and ion acceleration to high-energy laboratory astrophysics and even polarization of the vacuum in superstrong fields [2–7]. It has been known for some time that the maximum intensity achieved by a single beam is 2 × 1022 W/cm2. In addition, extensive improvement will be difficult to realize in the near future because of damage threshold and nonlinear effects, etc . As a promising way to avoid the technological limitations of single-beam schemes, coherent beam combination (CBC) is proposed. CBC can be divided into two main methods, the filled-aperture and tiled-aperture approaches. In the former, beams are combined on beam splitters, polarizers or dichroic mirrors, and in the latter, beams are overlapped in the focal plane of the focusing element . To obtain an ultrahigh peak intensity, which is significant in many physical experiments, the tiled-aperture configuration is appropriate, in which the on-axis peak intensity can scale up to N2 times compared to the one of a single beam (where N is the number of combined beams). Owing to the potential capability of CBC, even large-scale facilities such as the Extreme Light Infrastructure and the Exawatt Center for Extreme Light Studies have begun planning for CBC of multiple 10-PW-class beams [10,11].
Several phase-locking techniques are currently employed to stabilize the relative phases among the combined beams, which has proved to be the most important factor in CBC [12,13]. In the fiber regime, the most common methods, such as heterodyne detection , multidithering  and stochastic parallel gradient descent algorithm (SPGD) based phase control , are used to control the relative phase among multiple beams. In the bulk regime, Hansch–Couillaud detectors or nonlinear methods are often used to acquire the relative phase between two beams [13,17,18]. In 2013, Chosrowjan proposed a method using photo-detector pairs to record the fringe shift caused by phase fluctuation . Further, in our previous work, a hybrid method with high accuracy based on near-field interference fringes and single-crystal balanced optical cross-correlation was proposed . These techniques have achieved good performance in femtosecond pulse CBC. However, to achieve ultra-high intensity, CBC of multiple femtosecond pulses based on bulk amplification is urgent needed in recent years. The methods in the bulk regime mentioned above may have a complicated structure because N-1 detectors are needed . Further, the multidithering technique used in the fiber regime using phase modulators cannot be exploited in CBC of multiple femtosecond pulses based on bulk amplification.
We propose a method to realize CBC of multiple pulses propagating in free space, which is based on the near-field angle modulation (NFAM) interference pattern and angular spectrum. The experimental setup is simple, and only one camera is used to record the near-field pattern from which we extract the relative phases among the combined beams. In this paper, we derive the principle and simulate the measurement accuracy and scalability of our method, and then we conduct a proof-of-principle experiment to realize CBC of three beams. Our method can be used to combine four or even more beams, and the experiment with three beams verifies its feasibility.
2. Theoretical Analysis
Interference phase measurement using the Fourier transform method has been widely used in holographic interferometry . As mentioned above, we need to consider this method in multibeam CBC. Thus, a method using near-field angle modulation is proposed and analyzed below.
When N beams are incident on a camera at certain angles, the superposition intensity distribution may be expressed as
Here we consider only the intensity on the x axis for simplicity and do not take the aberration of the wavefront into consideration. , , and represent the amplitude, wave vector on the x axis, and frequency of the nth beam, respectively. is the phase, which varies with the optical length jitter.
In the CBC scheme, all beams are obtained by dividing the same source, thus . Further, can be approximated as , where is the angle between the wave vector and the normal vector of the x axis, satisfying . Thus, the formula can be simplified as
For convenience, the pattern can be rewritten as
When Eq. (5) is Fourier transformed with respect to , it becomes
Now, we consider the experimental conditions. The amplitudes of the N beams are supposed to be the same, which is expressed as
The power captured by the camera can be expressed as
In Eq. (9), the exponential term varies slowly; therefore, the position of the maximum value depends on , and the accuracy of the conventional fringe-finding and tracking method is .
To simply estimate the relative phase measured by NFAM, we assume that (just as plane wave or Super-Gaussian distribution). The Eq. (9) can be deduced to
It is obviously that is the sideband value of the spectrum when . Then we can extract the relative phase from the above formula
According to Eq. (13), the relative phase can be precisely measured, which is mainly affected by the recorded accuracy of the camera. And the measurement accuracy is analyzed by numerical simulation in the next section.
In addition, should not exceed to avoid the sidebands exceeding the DFT window, which means the maximum modulation angle should not exceed . Thus, the number of combined beams is limited by the camera pixel size, which is discussed in the next section in detail.
3. Numerical simulation
According to the analysis in the last section, NFAM possesses a high relative phase measurement accuracy. However, it is also affected by the A/D converter accuracy of the camera in real conditions. Here, we analyze the measurement accuracy of our camera and the resistance to energy fluctuation by a simulation. Several other factors that would influence the accuracy are also considered. Finally, the feasibility of the method for multiple beams is verified by simulation of three-beam CBC measurement and the maximum number of combined beams in our conditions is estimated.
3.1 Analysis of the measurement accuracy
As mentioned in Eq. (13), the measured relative phase is the sum of a constant and the argument of the sideband. The measurement error is mainly resulted by the A/D converter accuracy of the camera. To analyze the measurement accuracy, we simulate the relative phase measurement of two beams according to Eq. (9) and its Fourier transform spectrum without approximation. The parameters are determined in accordance with our experiment, where the camera is an 8-bit camera with an accuracy of 8. The angle between two beams is 3 and the maximum intensity value is set to 255, owing to the use of an 8-bit camera, which would introduce a slight measurement error.
Figures 1(a) and 1(b) show the near-field fringe pattern of two combined beams in one dimension and the angular spectrum (all of the intensity and spectral intensity values in this paper are normalized). A list of phase value is used to simulate the actual phase, while the relative phase acquired from the sideband is the measured phase both in this simulation and the later experiment. The relationship between the measured phase and the actual phase is acquired, and is plotted in Fig. 2(a), from which the root mean square (rms) value of the accuracy is calculated to be 0.6475 .
We also find that the fluctuation of the laser energy has a slight influence on the measurement of the phase when an 8-bit camera is used. We simulate the relationship between the measured and actual phases when the energy of each beam fluctuates in the range of 50%–150% of the unit energy; the result is shown in Fig. 2(b), and the measurement accuracy is 0.9498 .
In addition, changing the A/D converter accuracy of the camera and the angle between the beams in the simulation, we obtain the result shown in Table 1. The measurement accuracy increases with increasing A/D converter accuracy. Further, the modulation angle has little effect on the measurement accuracy, unless the angle is too small for the sideband to be distinguished from the central peak.
3.2 Analysis of the multibeam CBC measurement
We would like to verify the feasibility of multibeam measurement by simulating three beams firstly. We choose one of the three beams to be the reference with an angle , and adjust the angles of the other two beams. Figure 3(a) and 3(b) show the fringe pattern in one dimension of three-beam CBC and the angular spectrum according to Eq. (9).
Here, the angular spectrum consists of three sidebands that represent the angular differences, , and. It can be seen that the distance between the sideband and the center is proportional to the angular difference from Eq. (6). We can obtain the relative phases , , and from the first, second, and third sidebands, respectively. The measurement accuracy is just the same as the result given above.
Further, the maximum number of combined beams is considered. Under the conditions of , it is simulated that the measurement accuracy is 16.3 , 2.8, and 0.808 when the relative angle is 1, 1.1, and 1.2, respectively. Therefore, an appropriate minimum relative angle is chosen as 1.2 for a high accuracy. In addition, as mentioned in section 3, the maximum modulation angle should not exceed , that is 39.5. Thus, it is inferred that at most 32 sidebands without overlap can be found in the single side DFT window. However, from Fig. 3(b), we find that , and should not be arithmetic series to prevent , which would cause an error in phase demodulation. In view of this, an odd number of sequences in these sidebands are chosen to extract relative phase between the rest beams and the reference beam, and the maximum number of combined beams is 17 in theory. Of course, this number varies with the camera pixel size, the wave length, the beam size and so on.
In conclusion, the simulation results in sections 3.1 and 3.2 suggest that NFAM has a high accuracy and a good resistance to energy fluctuation. Further, it can potentially be used for multibeam CBC.
4. Experimental setup and results
To verify the feasibility of our method, a three-beam CBC experimental setup is built, as shown in Fig. 4. Here, a continuous-wave He-Ne laser beam (632.8 nm) is used. The source beam is split into three channels by beam splitters (BS1 and BS2). The setup consists of the main optical path and the measurement path. In the main portion, three beams (B1, B2, and B3) pass through delay lines and are focused by a lens with a focal length of 60 mm, where camera 1 records the focal spot. Then, three beam splitters are used to divide part of the power into the measurement portion, in which three beams are reflected onto camera 2 at different angles, which are on the order of . Here, camera 1 is a 12-bit high-resolution camera, and camera 2 is an 8-bit high-speed camera with a maximum frame rate of 4600 Hz. Further, two piezoelectric transducers (PZT) are used to adjust the path length of B1 and B2, and a negative feedback loop is adopted with a phase-locking bandwidth of 100 Hz.
In the measurement portion, we choose B3 as a reference and then adjust the angles of the other two beams () to obtain a distinct interference fringe. Figure 5(a) shows the fringe pattern recorded by camera 2. As the beams are arranged in a triangle in the main path, the angle modulation of the measurement path is not exactly on the x axis; thus, one of the beams has an angle component on the y axis with the reference beam. However, it does not affect the one-dimensional phase measurement according to Eq. (4). In fact, the phase information is wrapped in both the x and y axes, and the method could be expanded into two dimensions. Here, we just need the angle modulation in one dimension to verify the feasibility. It should be noted that a random phase offset exists in each discrete measurement since the beams will not perfectly overlap and the origin would not be exactly at the centroid of the beams every time. Thus, we need to adjust the delay line of B1 and B2, and compensate for this offset to get a high Strehl ratio. Then we choose the intensity profile on the x axis (the red line in Fig. 5(a)) and use Fourier transformation to extract the phase, as shown in Fig. 5(b) and 5(c). There is background noise in Fig. 5(b), owing to the dark noise of the camera. However, it has little effect on the measurement accuracy because of the Fourier transform.
Figure 6 shows the relative phase between each pair of beams when the feedback loop is off (blue line) and on (red line). We realize phase locking between B1 and B3 with an rms deviation of and between B2 and B3 with an rms deviation of . At the same time, we also record the relative phase between B1 and B2 with an rms deviation of . The differences of the stability between beams may come from the different mechanical structure of the delay line in our experiment. Overall, the phase-locking results exhibit a high accuracy, and the results would be better if the feedback bandwidth is higher, because the high-frequency noise is not controlled under the existing conditions.
The focal spot varies sharply with the fluctuation of the relative phase when the feedback loop is off, as shown in Figs. 7(a) –7(c), whereas it is maintained with a good shape and high Strehl ratio when the feedback loop is on, as shown in Fig. 7(d). The peak intensities of the combined beams when the feedback loop is on and off, are recorded by camera 1 at a frame rate of 7.5 Hz. The data are plotted in Fig. 7(e), from which the mean peak intensity of the combined beam is calculated to be 3852. Further, the peak intensities of each single beam are 359, 370, and 752. The Strehl ratio can be calculated by the following formula.
According to Eq. (14), the Strehl ratio is 89.5%. The Strehl ratio of three beams is lower than that of two beams (which has exceeded 95%) because the relative phases among three beams are more difficult to control, and alignment is harder. However, the result of this proof-of-principle experiment shows that our method would exhibit good performance in multibeam CBC.
In conclusion, we propose a new phase-locking technique for coherent combination of multiple beams. By angle modulation in the near field and angular spectrum analysis, we could detect the relative phase between each pair of beams in multibeam CBC with a single camera. This method may provide a precise and low-cost alternative for ultra-intense bulk laser systems. We analyze the feasibility and precision by simulation and then conduct experimental CBC of three beams to verify our method. We realize phase-locking with rms deviations of , , and between the pairs of beams; the corresponding Strehl ratio is 89.5%. In our next work, we are willing to attempt this method in CBC of an ultra-intense laser.
The authors acknowledge support from the National Natural Science Foundation of China (NSFC) (Grant No. 61775223) and Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDB1603).
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