1. Introduction

Coherent beam combining (CBC) of actively phase-locked laser arrays is a widely used method for scaling laser brightness beyond the limitations of the underlying sources [1]. CBC imposes tight tolerances on individual beam alignments to ensure fully constructive interference [2]. For CBC of fiber lasers, the individual fiber tips must be co-aligned to within a small fraction of their mode field diameter (MFD) to maintain high efficiency. For typical MFDs of tens of microns, this leads to mechanical attachment tolerances of only a few microns. This is quite challenging to achieve and maintain for high power lasers due to assembly tolerance stack-ups, dynamic changes in the beam parameters, thermal expansion of mechanical fixtures due to stray light absorption, and platform deformations [3]. These problems are particularly acute for systems deployed outside a controlled laboratory environment. One solution is to implement active beam pointing and/or position controls.

For large channel count arrays, active controls can be cumbersome owing to the difficulty of sensing beam parameters for each of a set of N lasers. In principle, misalignments can be diagnosed using arrays of N sampling optics and position-sensitive detectors, one for each input laser. Large parts count and optomechanical complexity make this approach unattractive for fieldable systems. Another disadvantage of multi-detector sensing is that misalignments can be quite subtle and difficult to detect. For example, ± 1-μm tip displacements can lead to 1% combining loss for a 20-μm MFD fiber [2]. At such levels of precision, miniscule changes in the relative position or responses of the different detectors can easily be misinterpreted as changes in the laser array, thus reducing control fidelity.

For these reasons, single-detector methods of sensing coherent array misalignments provide an attractive alternative to detector arrays. Such single-detector methods for piston phase control are well-understood, and fall into two general categories: synchronous multi-dither [4], of which the Locking of Optical Coherence by Single-detector Electronic-frequency Tagging (LOCSET) technique is the best-known implementation [5]; and hill-climbing methods, of which the Stochastic Parallel Gradient Descent (SPGD) technique is the most widely used [6]. An earlier demonstration of single-detector coherent position control applied the SPGD algorithm directly to control fiber tip positioning [7]. However, this required mechanical dithering of the fiber tips to generate an error signal, which is unattractive since the mechanical dithering unavoidably reduces the control precision, control bandwidth, and final combining efficiency.

In this work, we demonstrate automated, diffraction-limited spatial co-alignment of 5 coherently combined fiber channels. The demonstration uses a novel and simple control method that leverages phase dithers that are already imposed by the active phasing controls, eliminating the need for additional mechanical dithering. We also show that proper filtering of the detected light can extend the control range to align non-overlapping beams.

2. Principle of operation

The method of sensing alignment errors is based on the Fourier shift theorem and is illustrated in Fig. 1 . A far field amplitude profile that is shifted by S transforms into a wavefront tilt in the near field(and vice-versa): E(s – S) ↔ e^-iSxE(x), where E(x) is the Fourier transform of the laser far field profile E(s). Hence, relative beam misalignments (either near field or far field offsets) can be transduced into relative piston phase errors by spatially filtering the combined beam at an appropriate Fourier plane before detection. Phase errors can be effectively demodulated from the sensed beam by utilizing a duplicate set of the existing active phase controls. Closed-loop optimization is then achieved by applying feedback to appropriate actuators to null the sensed phase errors.

 

Fig. 1 Principle of operation for transducing spatial alignment errors into piston phase errors. The principle applies equivalently for misalignments in either the near field (beam overlap) or far field (beam pointing).

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In Fig. 1, light transmitted through the bottom spatial filter aperture is detected and used to lock all the lasers in-phase at that location. Any wavefront tilt errors between the beams will be transduced into piston phase offsets upon transmission through the aperture at the top of the beam. A detector senses the combined beat signal from all lasers, and standard phasing techniques isolate the error signals from individual lasers. Adjusting the relative beam positions to nullify these error signals corresponds to perfect co-alignment of the beams.

Conceptually, this approach is identical to one that we recently demonstrated for closed-loop stabilization of group delay mismatches between beams [8]. In the present work, the optical filter is applied in the spatial Fourier domain rather than in the temporal Fourier domain (i.e., the spectral domain).

3. Error signal analysis

To understand the approach more quantitatively, consider an array of N beams that are mutually phase-locked at x = 0 in the near field, corresponding to the bottom spatial filter at the detection plane in Fig. 1 (for simplicity we confine our attention to a single transverse dimension x; extension to 2D is straightforward). The beams are assumed to be identical with the exception of relative displacement errors at the actuation plane (far field).A Fourier optic of focal length f will transform a fiber tip displacement s_n of the n^th beam into a near-field wavefront tilt at the detection plane:

Multi-channel, single-detector phasing algorithms, whether based on LOCSET or on SPGD, impose sets of small, time-dependent phase dithers δφ(t) = {δϕ₁(t), δϕ₂(t), …, δϕ_N(t)} on the set of input beams in order to develop error signals from the resulting time-dependence of V:

Taking the small dither approximation exp[iδϕ_k(t)] ≈1 + iδϕ_k(t), and eliminating terms in the double summation that are identically 0 due to permutation symmetry of the indices, Eq. (3) can be rewritten as:

The exact nature of the dithers δφ(t) and error signal demodulation depends on the phase-locking algorithm. The dithers for LOCSET are sinusoids at a unique RF frequency for each channel [5], while SPGD dithers are step-wise changes applied at the identical dither rate on all channels [6]. Despite the apparent dissimilarity, both methods share two common time-averaged properties that allow development of unique accumulated error signals from the product of V and δφ(t). The first property is that the time-averaged dither on each channel is zero-mean, i.e.,

These properties allow construction of accumulated (time-averaged) error signals ε_n for feedback to all position actuators from a single detector signal. We assume that the (locked) piston phases and tip positions remain essentially stationary over the accumulation period T, which is generally true for slowly varying drifts and for an adiabatic separation of timescales between the phase and position control loops. The error signal is the time-averaged product of the photodetector signal Vw ith the normalized dithers, δϕ_n(t)/<δϕ>:

For LOCSET, Eq. (7) describes the function of an RF mixer and low-pass filter to isolate oscillations that are synchronous with the dither frequency applied to the n^th channel. For SPGD, the integrand of Eq. (7) is the error signal that is typically applied to control phase upon each loop iteration [6], and the accumulation is performed using a digital computer or microprocessor. In either case, inserting Eq. (4) into Eq. (7) and applying the zero-mean property of the dithers to eliminate the cosine term in V yields:

From Eq. (9) one can see the error signal for the n^th beam depends on the pairwise position errorsbetween that beam and the other k ≠ n beams in the array, intensity-weighted over the transmitting aperture A. Once the phase errors due to relative beam misalignments become a substantial fraction of a wave across A, the error signal amplitude will decrease as visibility is lost due to fringe formation on the detector.

For the simplifying case in which all but the m^th beam are perfectly co-aligned, s_k = s_mδ_km, and the error signals become

4. Experimental configuration

The closed loop concept was experimentally tested with N = 5 fibers using the layout shown schematically in Fig. 2 . Light from a single-frequency,λ = 1064 nm master oscillator (MO) was split among five single-mode, polarization maintaining (PM) fibers (ThorlabsPM980). Each fiber path contained an electro-optic modulator (EOM) for active phase control, and each cleaved fiber tip was bonded to a custom-made compact flexure stage (AOA Xinetics). The flexure stages provided ± 25 μm of stroke in both transverse (X, Y) dimensions by controlling the voltage applied to electrostrictive lead-magnesium-niobate (PMN) actuators. One of the five fibers was disconnected electrically to serve as both a phase and tip position reference. The light emitted from each fiber tip had a calculated MFD of 7.1 μm based on the PM980 fiber parameters. Each fiber was collimated by an 8-mm focal length lens to ~1.5-mm Gaussian beam diameter. These collimating lenses serve as the Fourier optic shown in Fig. 1. The collimated beams were coherently polarization combined in a filled aperture using a series of half-waveplates (HWPs) and polarizing beam splitters (PBSs) as shown in Fig. 2 [9].

 

Fig. 2 Experimental schematic for 5-fiber active positioning control demonstration.

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The near field was defined to be in collimated beam space approximately midway between the combining elements, and the far field was then defined as the exit plane at each fiber tip. Owing to the small difference in propagation distances ( ± 10 cm) between the various fiber tips and the defined near field plane, there was some small coupling of transverse fiber tip displacements into near field beam displacements. This coupling is ignored in our analysis but served to ultimately limit the dynamic range over which tip motions could be controlled. Variations in the collimating lenses, PBSs, and tip-collimator misalignments resulted in some mode mismatch between beams as can be seen from the slight differences in beam profiles as shown in Fig. 5 ; these too are ignored in the analysis.

 

Fig. 5 Individual and coherently combined far fields, before and after engaging closed-loop tip control. The coherently combined 5-beam far field images are single-frame excerpts from a video recording over the time sequence shown in Fig. 4 (Media 1).

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Following CBC, the combined output beam was reimaged onto cameras ata far-field image plane and several near-field image planes. The far field camera served as the primary diagnostic of fiber tip alignments and coherent combining efficiency. At each of the three near field image planes (for X, Y, and phase), the combined beam was spatially filtered and split between a camera and a photodetector. The spatial filters for X and Y detectors were initially set to block approximately half the near field beam footprint as shown by the beam profiles in Fig. 2. The phase detector was unfiltered.

The three photodetector signals were acquired by a field-programmable gate array (FPGA) and processed into sets of (N – 1) = 4 error signals each using an SPGD algorithm as described in Section 3 to drive the appropriate actuator sets. The FPGA loop rate for phase control was 100 kHz. Error signals for position control were accumulated at the same 100 kHz loop rate and were read out by a PC for application to the tip actuators at a 20 Hz loop rate.

5. Results and discussion

All five beams were initially co-aligned manually and locked in-phase using the FPGA. Feedback to the position actuators was disabled. Both sets of position error signals for X and Y axis tip positions were recorded as one of the beams was deliberately misaligned along the Y-axis.

From Fig. 3 , one can see that the error signals follow the accumulated, intensity-weighted phase errors on the spatially filtered beams as predicted by Eq. (9). For the X error signals, this results in near-0 error regardless of Y misalignments. For the Y error signals, the misaligned beam sees a strong error signal that is linear for small misalignments. For larger misalignments, the error signal drops off due to both its sinusoidal nature as well as due to the loss of coherence as wavefront tilts cause fringes to appear across the filtered aperture. As predicted by Eq. (10) and as shown by the solid curve construct in Fig. 3(a), the Y error signals for the aligned beams are approximately–1/(N– 1) = –1/4as large as those for the misaligned beam. The nonzero error signals for the aligned beams are a response to the shift in the average tip location across the array due to the misaligned beam.

 

Fig. 3 Position error signals derived from the X and Y photodetectors upon misaligning a single beam (labeled J2) in the Y-axis.The insets show the spatially filtered near-field beam profiles incident on the detectors. The solid curve in (a) is –4 × the average of the three aligned error signals.

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Closed loop position control was demonstrated by applying the position error signals in a proportional feedback loop per the usual SPGD algorithm [6]. The results are shown in Fig. 4 , starting with all N = 5 beams unphased and misaligned and with both phase and position controls disengaged. The unfiltered piston phase detector signal was proportional to the CBC power [Fig. 4(a)], and was initially low amplitude and time-varying as the relative phases drifted. The initial position error signals also drifted randomly as a consequence of the piston phase drifts [Fig. 4(b)]. The coordinates of each of the N – 1 = 4 active fiber tips were recorded by calibrating the drive signals to each actuator [Fig. 4(c)]. Starting at time 5.5 sec, the phase control was engaged, stabilizing the CBC power and resulting in well-defined error signals (the apparent noise on the error signals in Fig. 4(b) is an aliasing artifact arising from the asynchronous data recording rate and is not transmitted to the actuators). At time 10.5 sec, the position control loop was engaged and the fiber tips were brought into co-alignment within ~500 ms (90% convergence), maximizing the CBC power and nulling both the error signals and the tip misalignments.

 

Fig. 4 Closed loop demonstration of active phase and fiber tip position control. (a) Piston detector signal vs. time, which is proportional to the total CBC power. (b) Extracted error signals for all 8 actuators under control. (c) Fiber tip alignment errors inferred from the calibrated drive commands to the position actuators. The beams are labeled J1…J5, with beam J3 serving as the reference and hence not plotted.

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Figure 5 shows still images of the individual fiber tips recorded sequentially by blocking all but one of the beams, before and after engaging (and then freezing) the closed loop, as well as a video of the coherently combined far field over the time sequence graphed in Fig. 4. The accuracy of the fiber tip control was quantified by numerically calculating the RMS centroid errors of the 5 beams from these images. The initial RMS error of 6.3 μm was reduced to 0.26 μm after freezing the closed loop. Given the fiber MFD of 7.1 μm, this residual tip positioning error would lead to an expected CBC loss of ~(0.26/3.55)² = 0.5% [2].

Dynamic tip positioning errors after the loop had fully settled (time period 13-20 seconds) were also calculated from the frames of the combined far field and from the individual drives to each of the 8 position actuators under control[Fig. 4(c)]. Both methods indicated dynamic RMS tip motions were less than 0.05 μm for any fiber in either axis.

The 90% convergence time of 500 ms suggests closed-loop bandwidth capability on the order of single Hertz. This was limited by both the 20 Hz software loop speed, as well as the low current drive electronics used to charge the PMN tip actuators. Implementing both the tip control as well as phase control fully in the FPGA, along with commercially available higher current drivers (AOA Xinetics) would lead to expected control bandwidths in the multi-100 Hz regime, ultimately limited by the physical response of the actuators which have their first mechanical acoustic resonance at ~700 Hz. The control method intrinsically supports high speed tip positioning, as there is no need to dither the tip positions to generate error signals.

Since the error signals are based on the detected piston phases, the control range is limited by the requirement to maintain coherent visibility over the integrated near field aperture A. For large apertures such as shown in Fig. 3, the control range is limited to approximately one beam diameter (7.1 μm), beyond which the visibility drops to near-0 as interference fringes appear on the detector.

Figure 6 demonstrates that the control range can be arbitrarily increased by more aggressively spatially filtering the beam incident on the detectors. As the beam misalignments are increased, the corresponding increase in near field wavefront tilts lead to complex near field speckle patterns[Fig. 6(c)]. However, coherence can be recovered by reducing the size of the spatial filter to pass only a fraction of a single speckle spot, corresponding to less than 1 wave of variation in optical path differences across the filtered aperture A. With such a spatial filter, initial beam misalignments exceeding five beam diameters could be corrected upon engaging the closed loop (Fig. 7 ), even starting with almost no far-field beam overlap.

 

Fig. 6 Combined 5-beam near-field intensity profiles on the X-detector with varying levels of tip positioning errors between the beams. (a) Well-aligned beams. (b) Beams misaligned as shown in Fig. 5 prior to engaging closed loop. (c) Beams misaligned by up to 5 MFDs (>35 μm) as shown in Fig. 7.

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Fig. 7 Coherently combined 5-beam far fields (a) before and (b) after engaging closed-loop tip control, starting with misalignments corresponding to the near fields shown in Fig. 6(c). Images are single-frame excerpts from a video recording of the loop engagement sequence (Media 2), with intensities renormalized between (a) and (b).

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The action of the near field spatial filter can be understood intuitively as forcing an increase in the spot sizes of the diffraction-limited, filtered far fields to recover interferometric visibility between beams. This is analogous to prior work in the time domain demonstrating coherent control of broadband laser group delays over multiple coherence lengths by applying a narrow spectral filter to increase the coherence time of the detected light [8].

While this demonstration was performed with a filled-aperture beam combiner, the method appears readily extensible to tiled-aperture phase-locked arrays by replacing the near-field spatial filter by an appropriate mask array to sample identical portions of each beam’s footprint. Since the error signals behave identically regardless of the phasing algorithm, another potential extension would be to replace the SPGD phase controls with the LOCSET phasing approach(as was shown in [8]), which has been demonstrated to scale to N = 32 channels without impacting locking fidelity or closed-loop bandwidth [10].

6. Summary

We have demonstrated automated, high precision co-phasing and spatial co-alignment of a 5-fiber, coherently combined array using a Fourier-domain filtering concept to generate single-detector error signals for each beam parameter for the entire array. By leveraging phase dithers that are already imposed by the active phasing controls, beam misalignments are sensed and automatically nulled without requiring additional dithering. Proper filtering of the detected light was shown to extend the control range to align non-overlapping beams despite the reliance on mutual interference. This control method is expected to substantially ease integration of high power fiber lasers into large coherent arrays by eliminating the need for precision passive optomechanical alignment of the fiber tips.