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Abstract
Phase locking of coupled lasers is severely hindered by the spread in their natural lasing frequencies. We present an intra-cavity adaptive optics method that reduces the frequency spread and thereby improves phase locking. Using an intra-cavity spatial light modulator and an iterative optimization algorithm, we demonstrate a fourfold enhancement of phase locking 450 coupled lasers, as quantified by the peak intensity and the inverse participation ratio of the far-field output distributions. We further show that the improvement is long-lasting, and suitable for phase locking of weakly coupled lasers.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Phase locking of coupled laser arrays continues to play a prominent role in many possible applications and basic scientific investigations. Passive phase locking of many coupled lasers has been demonstrated in coherent beam combining, for obtaining high output intensities and high brightness [1–3]. Coupled laser arrays have been exploited to investigate intriguing and complex physical phenomena such as geometrical frustration [4], topological states [5–7], and non-Hermitian dynamics [8,9]. Moreover, coupled laser arrays can be used as spin simulators and solvers of complex optimization problems [10–16]. In all these, phase locking was found to be extremely sensitive to frequency detuning between the lasers [17] which require complicated experimental arrangements and very precise alignment, thereby limiting its widespread use.
Adaptive optics was first suggested in 1953 [18] as a technique to compensate for atmospheric aberrations in astrophysical observations [19] and has since been extensively used in additional fields such as microscopy [20], and laser processing [21]. In the context of coupled lasers, adaptive optics techniques can be used to reduce frequency detuning between different lasers and thus enhance their phase locking. However, this is challenging since the degrees of freedom of the coupled lasers system are interrelated because of the inherent nonlinearity of the lasers so they cannot be independently controlled. For example, a change in the resonant frequency of one laser can change the lasing frequencies of the lasers coupled to it, which will in turn change its lasing frequency in a nonlinear manner [22]. The resulting complex landscape cannot be treated by addressing one degree of freedom at a time.
In this work, we present an intra-cavity adaptive optics scheme to improve the phase locking of hundreds of coupled lasers by reducing their relative frequency detuning. We resort to a gradient descent inspired optimization algorithm to change the cavity parameters according to feedback from the laser output. As the sensitivity to disorder depends on the strength of the coupling between the lasers [22–24] we introduce a tunable coupling scheme that allows us to study adaptive optics over a wide range of coupling strengths.
Our results show significant improvement of phase locking, even in weakly coupled laser arrays, which lasts for much longer than the optimization process itself. Intra-cavity adaptive optics can allow for optimized passive coherent beam combining systems, with significantly higher peak output intensity as compared to the approach with no adaptive optics, as well as reveal new phenomena in weakly coupled oscillators that were so far obscured by noise and disorder.
2. Experimental arrangement
The experimental arrangement for forming an array of coupled lasers and intra-cavity adaptive optics is schematically presented in Fig. 1. It is comprised of two branches. The first branch is a digital degenerate cavity laser (DDCL) [12,14,25] that includes a 4f telescope with 2 plano-convex lenses ($f_1=40cm, f_2=20cm$), an 99% reflective mirror, a Nd:YAG side-pumped gain medium, and a reflective spatial light modulator (SLM) with 792x600 pixels and a 20$\mu m$ pixel pitch. The SLM is used to generate an amplitude mask with which we formed 450 independent lasers in a square lattice array. Each laser has a diameter of $160\mu m$, and the distance between neighboring lasers is $d_{lat}=200\mu m$. The $\lambda /2$-waveplate near the SLM (labeled (A) in Fig. 1) together with the adjacent polarizing beam splitter (PBS) (A) are used to ensure that light incident on the SLM is polarized in the appropriate linear polarization, as the effects of the SLM are polarization dependent. Therefore, the angle of waveplate (A) is aligned before the beginning of the experiment and remains fixed throughout. Part of the light reflected by the SLM (2%) is deflected by PBS (A) towards an imaging system, which detects both the SLM (the near-field) and far-field intensity distributions.
Fig. 1. Experimental arrangement for forming an array of coupled lasers and intra-cavity adaptive optics. The coupling strength between neighboring lasers is tuned by rotating $\lambda /2$-waveplate (B); it determines the amount of light that experiences strong negative coupling via Talbot coupling.
The second branch includes a mirror displaced from the telescope by a quarter Talbot length of the square lattice, $\frac {1}{4}Z_{talbot}=\frac {d_{lat}^2}{2\lambda }$ such that the cavity roundtrip is increased by $\frac {1}{2}Z_{talbot}$. Displacing the mirror in this fashion provides a strong negative Talbot coupling between nearest-neighbor lasers [26,27]. Tunable coupling between lasers is achieved by inserting an additional waveplate and PBS (B) which deflects the light towards the second branch, where the amount of light propagating towards the Talbot coupling branch is set by the angular rotation $\theta$ of half wave-plate (B). We define the coupling as a normalized overlap integral [28,29],
where $m$ and $n$ are the laser indices, and $i$ is the roundtrip number. The coupling coefficient $K_{mn}$ is normalized by the self coupling such that it depends only on the system geometry.The interaction between lasers is given by a term of the form $\frac{\textrm{d}}{{\textrm{d}t}}E_m\propto K_{mn}E_n$ [5,30]. For Talbot coupling and our lattice parameters, the maximal nearest neighbor (NN) coupling value is $K_{\text {NN max}}\approx -0.45$. As the uncoupled branch of the system is amplified by the roundtrip gain G, the actual coupling parameter of our system is
where we define $\theta =0^o$ as that of zero coupling with no light in the coupling branch and $\theta =45^o$ as that of all the light in the branch. Note that in the experimental arrangement the maximal coupling strength is limited, because the Talbot branch has no gain, and for high coupling values the cavity losses are higher than the available gain of the system. For our laser system, the available $K_{\text {NN}}$ range is $0 < |{K_{\text {NN}}}|\overset {\sim }{<}0.2$. To allow for maximal sensitivity to detuning between lasers, we performed the experiments with the smallest coupling strength that showed some degree of phase locking, i.e., $K_{\text {NN}}\approx -0.02$ ($\theta =8^o$). The negative sign of the coupling causes the laser to phase lock in an out-of-phase configuration, where neighboring lasers lock with a $\pi$ phase difference.It is worth noting that our two-branch coupling scheme effectively creates an intra-cavity etalon, which can reduce the number of longitudinal modes in the cavity [31]. However, for the parameters used here, this has a negligible effect on phase locking [32,33].
In addition to acting as an amplitude mask that defines the geometry of the lasers array, the intra-cavity SLM also acts as a phase mask that controls the effective cavity length of each laser with an accuracy of $\frac {\lambda }{256}$. Optimizing the phase distribution of the SLM so as to minimize the frequency detuning between the lasers, requires many degrees of freedom - one per each pixel of the SLM. To simplify the optimization, we consider only smooth detuning patterns. This is reasonable when the detuning is dominated by misalignment, aberrations, and fabrication errors that are generally smooth, and can be effectively represented by low-order Zernike polynomials [34].
3. Results and discussion
We use an iterative optimization procedure that involves a modified gradient descent algorithm [35]. At each step, the algorithm alters the SLM phase pattern by small increments in the Zernike basis, measures the difference in phase locking quality, and takes a step based on these measurements.
A complete evaluation of the number of phase locked lasers for a given far field (FF) intensity distribution requires analyzing the two-point phase correlation function in the near field (NF) and the far field distributions [36]. However, a simplified evaluation can be made by determining the ratio between the width of the diffraction peaks to the distance between them in the FF intensity distribution. As an example consider a simple one dimensional array of $N$ lasers with period $d$, and each laser output is a Gaussian beam with waist $w_0$. The entire NF distribution is modulated by a rectangular envelope with full width $2W$, such that the array is finite and the lasers have roughly equal intensity. The NF can be written as,
where $N=\frac {W}{d}+1$. The FF intensity distribution is the absolute value squared of the Fourier transform of the NF distribution, and the distance between far field peaks, $\Delta k$, is $\Delta k=\frac {2\pi }{d}$ which is the inverse array constant. The width of each FF diffraction peak is inversely related to the width of the $rect$ function, as $W_{\text {FF peak}}=\frac {4\pi }{W}$. Accordingly, the ratio of the distance between far field peaks to their width isEquation (4) shows that the number of phase locked lasers can be estimated from the far field intensity distribution. This result can be generalized to two dimensions. We note that the derivation assumes that the phase locked lasers have roughly equal intensities as in our experiments, and that they occupy neighboring lattice sites which is consistent with a short-range coupling scheme such as Talbot coupling. It is also important to emphasize that the exact proportionality constant between the ratio and the number of phase locked lasers depends on the exact shapes of the lasers and the NF envelope.
In practice, using the above method to estimate the number of phase locked lasers requires fitting the FF intensity distribution to a bimodal distribution function. In cases where the number of phase locked lasers is small and their FF peak intensity is weak compared to the background, fitting can be prone to errors, and so we elected to use an alternative method for purposes of optimization. Instead of directly estimating the number of phase locked lasers, phase locking quality is determined from the measured FF distribution of the lasers array by two criteria - the full width half max (FWHM) area of the diffraction peaks and its inverse participation ratio (IPR) [37]. The FWHM area of the FF diffraction peaks scales as $\propto \frac {1}{N}$ where N is the number of phase-locked lasers [38,39]. The IPR is a commonly used criterion for measuring the localization of a field [37], defined as
The IPR is maximal for a highly localized distribution and sensitive to the presence of a background field. It serves as an additional useful tool to measure the degree of phase locking of the lasers, when their FF has a bimodal distribution [4,38,40].
The FWHM peak area and IPR of the FF distribution provide complementary evaluation of the phase locking quality, and are highly correlated with the amount of phase locked lasers (see Supplement 1 for supporting numerical simulations). Our optimization algorithm performs best when it first aims to minimize the FWHM area, then switches to maximize the IPR, and stops when a steady state is reached. This is because at the start of the optimization process, when phase locking is poor, it is difficult to measure gradients in the IPR, due to the strong background in the FF distribution.
Some typical results are presented in Figs. 2–4. Figure 2 shows the effect of the adaptive optics corrections using Zernike polynomials of the first and second order (5 polynomials in total), compared to a weakly coupled cavity aligned manually. As evident, there is a significant narrowing of the FF peaks and improved uniformity in the near field. Figure 3 shows typical normalized FWHM and IPR values during and experimental sequence with intra-cavity adaptive optics. As evident, the adaptive optics correction reduced the FWHM area by a factor of 5, and increased the IPR by 2.3. This improvement of phase locking quality remained stable within 5% for 72 hours (right side of Fig. 3).
Next, we investigated adaptive optics correction when tuning the coupling strength between the lasers while keeping $\frac {P_{\text {Pump}}}{P_{\text {th}}}$ constant by calibrating the pump power to compensate for the losses of the increased coupling. Figure 4 compares the measured mean FF peak intensity and the extracted number of phase-locked lasers with and without adaptive optics correction as the coupling strength is increased from $|{K_{\text {NN}}}|=0$ to $|{K_{\text {NN}}}|=0.25$. As evident, the improvement of phase locking by adaptive optics increases with the coupling strength. For even higher coupling strength the improvement by adaptive optics becomes less pronounced and for the maximal coupling strength of $|{K_{\text {NN}}}|\approx 0.45$, we observed almost no improvement. This indicates that the system is limited by detuning noise that is not corrected by our algorithm, such as rapidly changing detuning noise which is not properly approximated by low order Zernike polynomials. Theoretical studies of locally coupled oscillators suggest that such noise accumulates along the phase locked laser lattice, limiting the maximal number of phase locked lasers [24]. In addition, our algorithm seems to be ineffective when using very strong coupling, as small changes in detuning hardly affect the measurable criteria with which we determine phase locking quality.
Fig. 2. Adaptive optics correction applied on 450 lasers on a square array with nearest neighbor coupling $K_{\text {NN}}\approx -0.02$ using Zernike polynomials up to the second order. (a),(b) Near-field intensity distributions, and (c),(d) Far-field intensity distributions after manual alignment of the cavity and after adaptive optics correction, respectively. All pictures were normalized to best display the changes in peak sharpness and uniformity. The bottom plots show the far-field cross sections. The four diffraction peaks are due to the lasers locking in out-of-phase configuration caused by the negative coupling.
Fig. 3. Typical normalized FWHM and IPR values of the far field distribution for an experimental sequence of intra-cavity adaptive optics. Left: The FF IPR (red) and FWHM (blue) normalized values measured at each step. The first vertical line signifies the point where the FWHM measurements reached a steady state, and so the algorithm shifts to optimizing the IPR instead. Right: Measuring the FF IPR and FWHM values over time after the optimization ended. The steady state value is stable to within $5\%$ for 72 hours after the optimization procedure was over.
Fig. 4. Far field peaks mean intensity (top), estimated number of phase locked lasers (middle), and far field IPR (bottom) as a function of the coupling strength $K_{\text {NN}}$ (controlled by rotating the $\frac {\lambda }{2}$-waveplate), with (red) and without (blue) adaptive optics correction using low-order Zernike polynomials.
By evaluating the phase compensation masks resulting from our algorithm, we learn that they typically contain mostly tilts (Zernike polynomials $Z(\pm 1,1)$) and defocusing ($Z(0,2)$), with additional 10-20% contribution (in relative magnitude) of higher order Zernike polynomials. We repeated the experiments using different sets of Zernike polynomials to evaluate the performance of our algorithm. First, we repeated the experiment using Zernike polynomials up to the fourth order, increasing the number of degrees of freedom from 5 to 14, and obtained similar results but with significantly longer run times. Additionally, when using only a subset of Zernike polynomials relevant to common optical aberrations in 4f optical systems such as tilt, defocusing, and spherical aberrations, we found a significantly smaller improvement in phase locking. In numerical simulations of the system we observed that the presence of a weak background of detuning noise (such as that caused by uncorrected aberrations) distort the optimization landscape, creating local minima and obscuring the global minimum (these results are presented in Supplement 1 section 3 and Fig. S3). As a result, choosing specific polynomials might shorten the runtime of our algorithm, but at the cost of reducing the improvement in phase locking.
Finally, we use our system with the adaptive optics correction to study the sensitivity to various well-controlled perturbations. For example, the effects of phase differences between nearest neighbor lasers on the FF IPR. The phase differences are obtained by mimicking a tilt of the end mirror in our experimental arrangement (Fig. 1) with a linear phase gradient on the intra-cavity SLM. The results are shown in Fig. 5. For weak coupling between neighboring lasers of $|{K_{\text {NN}}}|=0.017$, the IPR drops remarkably fast and reduces to half its value at $\Delta \phi _{NN}=0.04\pi$ between lasers, corresponding to an effective mirror tilt by $0.05 mrad$. For a stronger coupling strength of $|{K_{\text {NN}}}|=0.037$, the IPR decays slower, reducing to half its value at $\Delta \phi _{NN}=0.07\pi$. The other criteria of phase locking quality showed similar behavior. We note that within our signal-to-noise and system stability, we can determine and maintain the optimal tilt with an accuracy of $\Delta \phi _{NN}=2.5\cdot 10^{-6}\pi$ corresponding to an effective mirror tilt of $3.5 \mu rad$, testifying the remarkable control provided by the intra-cavity SLM.
Fig. 5. Far-field IPR as a function of the applied phase difference between nearest neighboring lasers by the intra-cavity SLM. Increasing the coupling strength yields a slower decay of the IPR.
Theoretical studies of coupled oscillators dynamics predicted that phase locking with short-range coupling depends on the accumulated detuning between all participating oscillators [24,41]. A linear phase gradient causes detuning to accumulate linearly between lasers, and thus rapidly diminishes the possibility for distant lasers to phase lock with one another, as confirmed by our results.
4. Concluding remarks
We demonstrated that intra-cavity adaptive optics can significantly improve the phase locking of weakly coupled laser arrays. The adaptive optics involves an iterative optimization procedure that is based on a gradient descent algorithm. Our procedure provides a long-lasting improvement in phase locking, measurable by the FWHM and the IPR of the output FF distribution. Indeed our results indicate significant improvement for passive beam combining with over a threefold increase in the number of phase locked lasers. The procedure enables investigations of interesting phenomena at the weak coupling regime, which hitherto were not accessible [24,41]. Our approach could be improved in the future by further optimizing the parameters of the algorithm, or by resorting to other optimization algorithms.
Funding
Minerva Foundation; Israel Science Foundation (1520-22, 3652-21).
Acknowledgments
We thank Eran Bernstein, Alon Goldring, and Neta Shaul for valuable help and discussions.
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.
Supplemental document
See Supplement 1 for supporting content.
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