We demonstrate a new way to analyze stable, multipass optical cavities (Herriott cells), using the linear canonical transform formalism, showing that re-entrant designs reproduce an arbitrary input field at the output, resulting in useful symmetries. We use this analysis to predict the stability of cavities used in interferometric delay lines for temporal pulse addition.

We have generated 81 independently controllable beams using a spatial light modulator and combined them on a diffractive combiner, to characterize the combiner and develop a fast phase error detection scheme. A key parameter of the diffractive combiner is measured in a new way, enabling an efficient combination when programming calibrated phases of each beam. This testbed provides a platform for development of advanced feedback phase control of high channel-count beam combination.

We develop a rapidly converging algorithm for stabilizing a large channel-count diffractive optical coherent beam combination. An 81-beam combiner is controlled by a novel, machine-learning based, iterative method to correct the optical phases, operating on an experimentally calibrated numerical model. A neural-network is trained to detect phase errors based on interference pattern recognition of uncombined beams adjacent to the combined one. Due to the non-uniqueness of solutions in the full space of possible phases, the network is trained within a limited phase perturbation/error range. This also reduces the number of samples needed for training. Simulations have proven that the network can converge in one step for small phase perturbations. When the trained neural-network is applied to a realistic case of 360 degree full range, an iterative scheme exploits random walking at the beginning, with the accuracy of prediction on phase feedback direction, to allow the neural-network to step into the training range for fast convergence. This neural-network-based iterative method of phase detection works tens of times faster than the commonly used stochastic parallel gradient descent approach (SPGD) using a single-detector and random dither when both are tested with random phase perturbations.

We demonstrate, to the best of our knowledge, the first two-dimensional diffractive beam combination for ultrashort pulses-a highly scalable technique capable of using a diffractive optic pair to combine large arrays of ultrashort pulsed beams. A square array of eight 120 fs pulsed beams from eight fiber outputs is coherently combined into one beam using the diffractive combiner. The experimental results show that the combined pulse preserves the input pulse width and shape, and the combining efficiency is measured to be close to the limit of the manufactured diffractive optic. An analysis shows that the combining loss due to uncompensated temporal and spatial dispersions is negligible.

We demonstrate a new method for controlling diffractive, high-power beam combination, sensing phase errors by analyzing the intensity pattern of uncombined side beams at the output. A square array of eight beams is combined with <0.3% stability and 84.6% efficiency. As the channel count is increased, so does the usable information, enabling scaling to large channel counts without significant slowing of control loop response time, an advantage over single-input algorithms.

An 8-beam, diffractive coherent beam combiner is phase controlled by a learning algorithm trained while optical phases drift, using a differential mapping technique. Combined output power is stable to 0.4% with 95% of theoretical maximum efficiency, limited by the diffractive element.

We analyze a new kind of machine learning algorithm designed to feedback stabilize coherently combined lasers. This algorithm learns differential, rather than absolute, values of action in phase space, in order to facilitate learning on initially unstable systems. Experiments have shown that this approach can control small-scale spatial beam combination with high stability. In this paper we analyze the algorithm's performance and limitations in depth, showing that it can continuously learn during operation in order to track changes. Using simulation, we extend the application to temporal combination, and show that it scales to more complex instances by combining 81 beams.

To stabilize the combined pulse energy for coherent temporal pulse stacking using interferometer cavities, we have developed a direct cavity phase measurement method based on analysis of the response to modulated probe pulses. An experiment has demonstrated optical phase control within 50 mrad for four cavities, resulting in a combination of 25 pulses with 1.5% root mean square stability over 30 h.

Coherent pulse stacking (CPS) is a new time-domain coherent addition technique that stacks several optical pulses into a single output pulse, enabling high pulse energy from fiber lasers. We develop a robust, scalable, and distributed digital control system with firmware and software integration for algorithms, to support the CPS application. We model CPS as a digital filter in the Z domain and implement a pulse-pattern-based cavity phase detection algorithm on an field-programmable gate array (FPGA). A two-stage (2+1 cavities) 15-pulse stacking system achieves an 11.0 peak-power enhancement factor. Each optical cavity is fed back at 1.5kHz, and stabilized at an individually-prescribed round-trip phase with 0.7deg and 2.1deg rms phase errors for Stages 1 and 2, respectively. Optical cavity phase control with nanometer accuracy ensures 1.2% intensity stability of the stacked pulse over 12 h. The FPGA-based feedback control system can be scaled to large numbers of optical cavities.