## Abstract

A new approach is proposed for the adaptive phase-locking of a set of parallel laser beams. It is based on an optical conversion of phase differences in the array into an intensity pattern which feeds an optimization algorithm for iterated adjustments of the phase modulators. A numerical analysis and proof of principle experiment support the method and demonstrate its speed.

© 2015 Optical Society of America

## 1. Introduction

For a long time laser beam coherent combining has been considered as a way to produce laser radiations at powers which are difficult to reach with a single laser chain while preserving a good beam quality. Earlier projects concerned continuous wave (CW) or quasi CW laser sources but other more recent studies dealt with the combination of ultrashort femtosecond pulses to get high intensity beam at high average powers [1,2]. Coherent combining is not only a means to get record breaking laser systems it is also a laser scheme which is relevant because of the high flexibility offered by arrays of identical lasers. Although the power record was obtained by combining bulk solid-state lasers [3], most of the current research on that matter concerns Ytterbium doped and Thulium doped fiber lasers [4,5]. Numerous techniques have been proposed and implemented to achieve phase-locking of laser beam array by means of active phase control [1]. A standard architecture starts with a master oscillator that feeds an array of parallel amplifiers. Electro-optics phase modulators are included at the input of each amplifying arm and serve to fix the output phase of each elementary beam to a common value thanks to a feedback signal. In most cases, the amplified laser fields are arranged to make a tiled aperture beam array [4]. There are then two major types of methods to lock the phase in the beam network. The first type is based on the measurement of the different phases across the array which directly yields the error signal to apply to the individual modulators. The laser fields’ phase can be obtained by heterodyning with a frequency shifted sample of the master laser [3]. Alternatively, the differential phase pattern in the beam array can be obtained by simple interference with a reference plane wave [6] or by shearing interferometry, without the need of a reference [7]. In the second type, a fraction of all the elementary beams must overlap on a single photodetector. The phase of the different beams is weakly disturbed through a specific command added to the main phase offset control. In one option, the perturbation is periodic and steady, the different beams being tagged by different frequencies [8,9]. The photocurrent is further filtered out to give access to the perturbed beam phase. In another option, there is no phase offset measurement and the phase perturbations as well as the photocurrent serve for an optimization loop which maximizes the phase uniformity in the array. Stochastic parallel gradient descent (SPGD) is the most frequently encountered optimization routine in that domain [10,11]. All techniques mentioned above (see [1] for a more complete review) have proved to be effective for the co-phasing of various sets of laser beams.

In this article, we propose a new phasing method that cannot be put in any of the previous categories. There is no phase measurement and there is no added phase perturbation although the process is driven by an iterative optimization approach. The novelty comes from the use of a phase intensity mapping (PIM) device in combination with a dedicated optimization algorithm that are described in the following sections.

## 2. Principle of the method

Let us first assume that the set of beams concerned is free of practical imperfections and that the phase piston only needs to be adjusted to make the array deliver a discretized plane wave. The proposed laser scheme is schematically depicted in Fig. 1. The first half of the process is analogic and performed by the transformation of optical waves. The second half of the process is computational with electrical inputs and outputs. A weak fraction of the laser field is taken from the laser output by the beam splitter (BS). It is then launched into a passive device which maps in intensity the relative phase of the beamlets. The intensity of the light exiting the phase intensity mapping (PIM) device is converted into photocurrents by an array of photodiodes (PDs). Each photodiode is associated with one beam of the array. The photodiode currents are further processed by the optimization routine that generates the command voltages for the phase modulators (PMs). Before entering into the details of the operation we will discuss the two crucial parts of the servo-control.

#### Phase Intensity Mapping device

Various components and set-ups can be chosen for performing the phase to intensity conversion [12–14]. More clearly PIM means here that the local intensity in the output pattern at the exit of the device is a function of the phase of the input laser field at the same point in the array. It is worth emphasizing that there is no need for the phase mapping to be represented by a simple function, just that the function be linear with respect to the fields in the array. All that is required here is a prior knowledge of the transmission matrix connecting the input field vectors to the output field vectors of the PIM device. For example a standard implementation of PIM can be based on a wavevector filtering which offers the advantage that the transmission matrix can be represented analytically. Some appropriate filters can be found among the ones employed for phase contrast microscopy [15]. More specifically in the following we consider a filtering that we previously exploited in a laser cavity to get self-organized phase-locking of a ring fiber laser array [16]. The high spatial frequencies are attenuated while the low spatial frequencies are just delayed by λ/4 where λ denotes the laser wavelength. A plate with appropriate coatings can perform this type of filtering once introduced in the Fourier plane of a unit magnification telescope. For a square array of NxN laser beams, the transfer matrix **T** of such a PIM device has NxNxNxN dimension. Its elements can be expressed as:

**T**matrix. T

_{klmn}is the coefficient for multiplication of input field at the crossing of row “m” and column “n” in the array to get the linear contribution to the output field in row “k” and column “l”. The parameter β represents the filter transmission for the high spatial frequencies. γ denotes the normalized (square) aperture width for the transmission of low spatial frequencies, w is the radius at 1/e

^{2}in intensity of the beamlets assumed to be of Gaussian profile, and p stands for the array pitch. Phase intensity mapping results from the interference of a transmitted fraction of the beams with a background formed by the coherent summation of a diffracted fraction of all the beams in the array. At the output of the PIM device the photodiode array delivers electric signals which are proportional to the intensity of the filtered image of the laser pattern.

#### Optimization loop

The optimization approach we have chosen can be considered as an extension of the generalized projection methods [17]. It consists of iterated sequences of measurement followed by projections of the measured value, and of computation, followed by projections of the computed value. Projections are performed in the two sets of the measured and computed targeted data (constraints). In the more general case, the two data sets are not convex. On each optimization round, there is a reset of the inputs because the phase modulators modify the laser field array. That makes the approach very specific and different from standard projection methods. The goal of the routine is to minimize the distance between measurements and target values through management of the phase modulators. There is no requirement that the PIM transfer function be a linear isometry. We assume the power carried by the different beams is known, so a uniform power distribution in the array is not required. The practical goal is that in a given reference cross-section, the different elementary waves all get in phase synchrony, i.e. that the field array corresponds to a discretized plane wave. It requires that the targeted laser field array [F] be represented by pure real positive values (a real two-dimensional matrix). A preliminary task of the optimization is the derivation of the data expected at the exit of the PIM device for the phase-locked array. They are computed from the knowledge of transmission matrix **T** and of [F] (derived from the beamlet power): [E] = **T**. [F]. Since measurements will only provide data on the filtered field intensity, we store Arg[E]. Therefore, the two sets of the targeted data are given by [F] on one side and by Arg[E] on the other side. The loop begins with an unknown phase state for the beamlets that we can assume to be randomly distributed on the phase circle. Operation of the optimization loop is schematically pictured in Fig. 2.

The photodiodes give a first set of measured data from which we can obtain the modulus of the measured filtered fields: [|E_{m}|]. Projection on the solutions’ set gives a new field: [E_{s}] = [|E_{m}|] . exp(j.Arg[E]). We then compute the laser field [F_{s}] that would lead to such a filtered field using: [F_{s}] = **T ^{−1}**. [E

_{s}]. Projection of the result on the second solutions’ set means:

- (i) [|F
_{s}|] is replaced by [F], automatically achieved in practice, and there is no action on the laser beams’ amplitude, since the power of each beam is kept unchanged, and - (ii) the phase lag Arg[F
_{s}] is compensated by a reverse sign phase modulation to restore a pure real field.

A command is therefore sent to the phase modulator array to alter the phase pattern by the computed values - Arg[F_{s}]. Consequently, the measured intensity at the PIM device output is modified and the process enters a new iteration loop. This is a kind of error reduction process where the error ${e}_{k,l}{}^{(n)}$ at iteration n for the beam at position k,l can be written: ${e}_{k,l}{}^{(n)}={\displaystyle \sum \left|{F}_{k,l}\right|}-\left|{\displaystyle \sum {F}_{k,l}^{(n)}}\right|$ . ${F}_{k,l}^{(n)}$ denotes the laser field at iteration n, at position k,l in the array, at the exit of the PMs matrix. In the laser beam combining frame, it is a more common practice to compute a phasing parameter (also known as an “order parameter” in the oscillator synchronization frame), $\eta ={\left(\frac{\left|{\displaystyle \sum _{k,l}{F}_{k,l}^{(n)}}\right|}{{\displaystyle \sum _{k,l}\left|{F}_{k,l}\right|}}\right)}^{2}$ that permits to monitor the efficiency and the dynamics of the proposed scheme. In the numerical results reported in the next chapter, one can see that the feedback loop quickly and continuously reduces the error and improves the phase uniformity until perfect phase locking is reached.

## 3. Numerical analysis

To assess the efficiency of the phase-locking system, we numerically simulated the operation of the device. The PIM device transmission matrix **T** and its inverse **T ^{−1}** were first computed from Eq. (1). Laser beam arrays with random phase distributions uniformly distributed on [0, 2π] were used as initial conditions. Then, the evolution of the laser fields was computed based on the above mentioned process and the parameter η was deduced for each iteration step. We first set the beam array with a uniform intensity.

Figure 3 shows the phasing dynamics for a 2D array of 25 beams under various initial conditions. Convergence was gradual, continuous and fast. It is worth emphasizing that there was no added perturbation on the phase pattern during an iteration step, contrary to alternativeapproaches [11]. We next studied the evolution of the co-phasing dynamics and speed according to the size of the beam array. The results are reported in Fig. 4 where the data have been averaged on 100 different initial conditions. The number of iterations required to reach almost complete phase-locking (η = 0.99, standard phase deviation${\sigma}_{\phi}\le 0.1rad$) evolves very slowly when the number of laser beams to control increases. This is illustrated on Fig. 5. The behavior is significantly different from SPGD whose number of steps (in its basic form) is higher and scales linearly with the number of lasers [11].

The impact of potential electrical noise on the signals delivered by the PDs array has been assessed. In the simulation of the phase-locking set-up we added some randomly chosen values (from the range [-V_{no}, V_{no} ] where 2V_{no} denotes the peak to peak maximum noise voltage) to the PDs data . These random values were all different in the signal array and varied at each round trip in the loop to mimic the most disturbing situation. We observed that the noise on the measured data had almost no impact on the number of iterations required to achieve phase-locking, whatever the beam array size.

The dynamics of the phasing were also unchanged. However, as expected, the noise level had an impact on the uniformity of the phase among the beams once a stationary state was reached. A signal to noise ratio as low as 10 was sufficient to preserve a phase variance below λ/60 (0.1 rad). That proves that the proposed scheme is highly robust with respect to noise. As an example, Fig. 6 compares for a given set of random initial phases (case of 25 laser beams), the convergence of the proposed scheme for a noiseless detection (blue lines) and for detection with a SNR of 10 (red lines) and 7 (green lines). Since at steady state the detected signal varied smoothly across the array, the signal to noise ratio indicated here corresponds to the ratio of the PDs average voltage and the uniform noise effective voltage.

We also considered the fact that the beams may slightly vary in intensity according to time, i.e. during the process of reaching a uniform phase. The main source of intensity noise for high power amplifiers coming from the pump laser diodes in the relevant bandwidth, we assumed in our computations that the noises were uncorrelated across the array. Again to consider the worst situation, the random intensity noises were reset at each round in the loop. It appeared that the concept is extremely robust to the intensity uncorrelated noises. Numerical simulations have shown that the phase-locking speed as well as the steady-state phase locking level did not changed even with up to 10% rms power fluctuations. That property is preserved when the array size is extended. We have also considered the case where the detectors have different responses across the PD’s array. If the non-uniformity of the photodiodes is known, the perturbation can be compensated in the algorithm by a simple product of the measured data by the appropriate coefficients. Even in case it is unknown, a 10% rms random response deviation among the PDs is tolerable and moderately alters the phase-locking level (few %). Moreover, there is no need to perform some fine characterization of the **T** matrix and in the experiments reported below we assumed the PIM was ideal.

## 4. Experiments

For a proof of concept we implemented an experimental bench with three liquid crystal spatial light modulators (SLMs) which is schematically descripted in Fig. 7.

The light source was an amplified laser diode at 1064nm. The first SLM1 served for the generation of an array of 4x4 parallel beams with a desired relative phase distribution that mimicked the random distortions to compensate. The beamlets were of nearly top hat profile with a width of 800 μm for an array pitch of 850 μm. The second SLM2 was used as the phase modulator array of the adaptive synchronization system. The third one SLM3 was the amplitude and phase filtering element of the PIM based on a 4f set-up. A plate with appropriate coatings would have been sufficient but we chose a SLM because it offered a large flexibility in the filter design. Therefore parameters of the filter in the PIM have been varied in the range [10%, 80%] for β^{2} and in the range [0.3, 0.7] for γ. An array of 4x4 photodiodes (Thorlabs FDS100) delivered the electric signals required for the optimization engine after analog to digital conversion. The data computed using the approach described in section 2, corresponding to a simple matrix product, were then sent to command the SLM2. Other imaging set-ups and cameras complemented the bench and let us record the near field and far field patterns. One example of the initial plane wave spectrum of the array of 16 beams with random phases is given on Fig. 8(a). After the automated system started, the pattern quickly evolved towards one of a phase-locked array as illustrated in Fig. 8(b), demonstrating that the proposed scheme worked as expected.

We also measured the evolution of the phase-locking level on each iteration of the locking servo. A typical plot is given in Fig. 9 (dark dashed line). The 4x4 array was phase-locked in approximately 15 iterations demonstrating the fast convergence speed of the process.

Convergence speed is given here in terms of number of iterations. The time needed for one iteration mostly depends on the time required for computation and the associated ADC/DAC conversions since detectors and phase modulators are usually very fast and since their signals are processed in parallel. However that was not the case in our experiments based on a liquid crystal SLM for phase compensation. They are known to be slow devices with time response in the tens of milliseconds range and they fixed here the time for one iteration to nearly the same time. They are not suited to a practical phase-locking system but they are convenient for a proof of concept experiment like the one reported here.

The fast oscillations close to steady state and at steady state that can be observed on Fig. 9 have nothing to do with the synchronization process itself. In fact, these disturbances were connected to the chaining of liquid crystal SLMs. Beatings due to their independent driving electronics produced artefacts and parasitic intensity modulations of the optical waves. For comparison, we obtained from a numerical simulation with the parameters of the experimental bench, the theoretical trace in the red solid line shown in Fig. 9. Although the dynamics are slightly different in theory with respect to experimental data, the convergence speeds are relatively close. The agreement between simulation and practice was rather good, so we can be confident about the outcomes of the previous numerical analysis. Convergence to a phase-locked array was obtained in the whole range of parameters chosen for the PIM filter. The phasing speed however significantly depended on the choice of β and γ, requiring between 12 and 63 iterations. Again that was in good agreement with our numerical study.

## 5. Conclusion

We have proposed a new method for the control of the phase in an array of laser beams in order to obtain their coherent combining. The investigated process belongs to the iterative approaches. Its operation does not require to add phase modulations or phase perturbations in striking difference with most of the other iterative processes. There is no need of a reference wave. The proposed scheme is characterized by two specific elements. The first one is an optical device for the conversion in intensity of a beamlet phase-shift. It can be realized in various ways. The second one is a specific optimization routine derived from projection methods. Numerical simulations have demonstrated that the phase-locking system is efficient with a smooth convergence in a small number of iterations. A significant advantage of the scheme is that the convergence speed is not very dependent on the number of laser beams to control. This property comes in part from the fact that the number of detectors is identical to the number of beams. Parallel processing of the data would keep short the time needed for one iteration. Robustness of the method has been assessed and our simulations have demonstrated that the performances remain high, even in the presence of significant optical or electrical intensity noises. A practical implementation of the phase-locking system has been carried out with a 4x4 laser beam array at 1030 nm. A liquid crystal spatial light modulator was used to control the beamlet phase. In this proof of principle, we demonstrated that the proposed scheme actually performed phase synchronization of the light fields in the array. Furthermore the measured number of iterations required by the adaptive system to reach phase-locking was low and consistent with the numerical analysis. Phase-locking servo based on the proposed scheme should run with bandwidth above 1 kHz with the use of current fast converters and processing units like FPGA [4].

## Acknowledgments

The authors acknowledge CILAS Company for their financial support.

## References and links

**1. **A. Brignon, *Coherent Laser Beam Combining* (Wiley-VCH, 2013).

**2. **E. Seise, A. Klenke, S. Breitkopf, J. Limpert, and A. Tünnermann, “88 W 0.5 mJ femtosecond laser pulses from two coherently combined fiber amplifiers,” Opt. Lett. **36**(19), 3858–3860 (2011). [CrossRef] [PubMed]

**3. **S. McNaught, C. Asman, H. Injeyan, A. Jankevics, A. Johnson, G. Jones, H. Komine, J. Machan, J. Marmo, M. McClellan, R. Simpson, J. Sollee, M. Valley, M. Webera, and S. Weiss, “100kW coherently combined Nd:YAG MOPA laser array,” in *Frontiers in Optics* (OSA, 2009), paper FthD2.

**4. **C. X. Yu, S. J. Augst, S. M. Redmond, K. C. Goldizen, D. V. Murphy, A. Sanchez, and T. Y. Fan, “Coherent combining of a 4 kW, eight-element fiber amplifier array,” Opt. Lett. **36**(14), 2686–2688 (2011). [CrossRef] [PubMed]

**5. **P. Honzatko, Y. Baravets, F. Todorov, P. Peterka, and M. Becker, “Coherently combined power of 20 W at 2000 nm from a pair of thulium-doped fiber lasers,” Laser Phys. Lett. **10**(9), 095104 (2013). [CrossRef]

**6. **H. Chosrowjan, H. Furuse, M. Fujita, Y. Izawa, J. Kawanaka, N. Miyanaga, K. Hamamoto, and T. Yamada, “Interferometric phase shift compensation technique for high-power, tiled-aperture coherent beam combination,” Opt. Lett. **38**(8), 1277–1279 (2013). [CrossRef] [PubMed]

**7. **C. Bellanger, B. Toulon, J. Primot, L. Lombard, J. Bourderionnet, and A. Brignon, “Collective phase measurement of an array of fiber lasers by quadriwave lateral shearing interferometry for coherent beam combining,” Opt. Lett. **35**(23), 3931–3933 (2010). [CrossRef] [PubMed]

**8. **T. M. Shay, V. Benham, J. T. Baker, B. Ward, A. D. Sanchez, M. A. Culpepper, D. Pilkington, J. Spring, D. J. Nelson, and C. A. Lu, “First experimental demonstration of self-synchronous phase locking of an optical array,” Opt. Express **14**(25), 12015–12021 (2006). [CrossRef] [PubMed]

**9. **T. M. Shay, “Theory of electronically phased coherent beam combination without a reference beam,” Opt. Express **14**(25), 12188–12195 (2006). [CrossRef] [PubMed]

**10. **M. A. Vorontsov, G. W. Carhart, and J. C. Ricklin, “Adaptive phase-distortion correction based on parallel gradient-descent optimization,” Opt. Lett. **22**(12), 907–909 (1997). [CrossRef] [PubMed]

**11. **H. Yang and X. Li, “Comparison of several stochastic parallel optimization algorithms for adaptive optics system without a wavefront sensor,” Opt. Laser Technol. **43**(3), 630–635 (2011). [CrossRef]

**12. **T. Kim and G. Popescu, “Laplace field microscopy for label-free imaging of dynamic biological structures,” Opt. Lett. **36**(23), 4704–4706 (2011). [CrossRef] [PubMed]

**13. **S. Bernet, A. Jesacher, S. Fürhapter, C. Maurer, and M. Ritsch-Marte, “Quantitative imaging of complex samples by spiral phase contrast microscopy,” Opt. Express **14**(9), 3792–3805 (2006). [CrossRef] [PubMed]

**14. **R. Horisaki, Y. Ogura, M. Aino, and J. Tanida, “Single-shot phase imaging with a coded aperture,” Opt. Lett. **39**(22), 6466–6469 (2014). [CrossRef] [PubMed]

**15. **F. Zernike, “Phase contrast, a new method for the observation of transparent objects,” Physica **7**, 686–698 (1942).

**16. **F. Jeux, A. Desfarges-Berthelemot, V. Kermene, and A. Barthelemy, “Efficient passive phasing of an array of 20 ring fiber lasers,” Laser Phys. Lett. **11**(9), 095003 (2014). [CrossRef]

**17. **A. Levi and H. Stark, “Image restoration by the method of generalized projections with application to restoration from magnitude,” J. Opt. Soc. Am. A **1**(9), 932–943 (1984). [CrossRef]