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Actively stabilized, simultaneous spatial and temporal coherent beam combination is a promising power-scaling technique for ultrafast laser systems. For a temporal combination based on optical delay lines, multiple stable states of operation arise for common stabilization techniques. A time resolved Jones’ calculus is applied to investigate the issue. A mitigation strategy based on a temporally gated error signal acquisition is derived and demonstrated, enabling to stabilize laser systems with arbitrary numbers of amplifier channels and optical delay lines.
© 2016 Optical Society of America
1. Introduction
Today, there is a multitude of ultrafast laser systems able to address a plethora of applications in science, medicine and industry. On the one hand, there are ultrahigh-peak-power large-scale bulk lasers intended for the most demanding scientific applications, which are complex, alignment sensitive, and emit at extremely low repetition rates [1]. On the other hand, there are novel solid-state-laser concepts based on thin-disk [2], slab [3] or fiber [4]. These lasers typically emit pulse energies orders of magnitude lower than conventional bulk lasers, but are superior in virtually any other parameter. Thus, they have become versatile tools for the majority of ultrafast applications, which do not require highest peak powers, but rather both high peak and high average power. Typically, both chirped-pulse amplification (CPA) [5] and large amplifier beam cross sections are employed in order to mitigate nonlinear effects and optically induced damage. Nevertheless, there are applications [6–8] that require or benefit from power levels, which despite of these outstanding advances of the past decades, remain inaccessible for state-of-the-art laser technology.
Spatially separated amplification in parallelized amplifier channels and subsequent coherent beam combination (CBC) allows to circumvent the prevailing limitations of single-aperture laser systems [9–11]. Ideally, peak power and average power can be increased by the number of amplifier channels. The major implementations for beam combination are divided into filled-aperture and tiled-aperture approaches. In the first case, the beams are superposed in both near and far field using a tree-type arrangement of beam splitters or a diffractive optical element, as shown in Figs. 1(a) and (b). The beam splitting is achieved by reverse operation of these schemes. In the second case, a bundle of parallel beams is superposed solely in the far field as depicted in Fig. 1(c). This technique is not suited for beam splitting but is applied successfully for beam combination of high-average power laser systems [12].
Fig. 1 Implementation of filled-aperture coherent beam combination using a) a tree of beam splitters (BS) or b) a diffractive optical element (DOE) and of c) tiled-aperture combination.
Another approach to exceed existing peak-power limitations is temporal multiplexing, referred to as divided-pulse amplification (DPA) [13,14]. Instead of a high-energy pulse, a train of low-energy pulses is amplified to avoid nonlinear effects. Afterwards, the pulse train is temporally combined into a single pulse, ideally increasing the peak power by the number of pulses. The input pulse sequence can be generated either directly from the output pulse train of a master oscillator using amplitude- and phase-shaping techniques or by dividing a single pulse in cascaded optical delay lines. The temporal combination is achieved either again with optical delay lines [15] or with resonant enhancement cavities [16,17], as schematically shown in Fig. 2, aiming for the temporal combination of several tens or more than a hundred pulses, respectively.
Fig. 2 Schematic of temporal pulse stacking with a) optical delay lines or b) resonant cavities. (PBS: polarzing beam splitter, HWP: half-wave-plate, S: beam deflection switch, M: mirror).
Coherent beam combining systems, whether spatial or temporal, are based on interferometric superposition. Thus, optical path length fluctuations due to environmental perturbations have to be compensated for stable operation. This is given automatically in double-pass and Sagnac-interferometer setups [18], where all pulses travel the same path. However, for a scalable architecture typically separated splitting and combing and, therewith, an active phase stabilization is required [19,20].
Evidently, the simultaneous application of both CBC and DPA in actively stabilized architectures is of great interest for future power scaling of ultrafast lasers. Thereby, tiled-aperture CBC is less suited for reasons of combination efficiency and DOEs are also less suited because they require an individual compensation of the angular chirp in each channel [19]. Binary tree-type arrangements of polarizing beam splitters (PBSs) have been demonstrated for spatial combination [21] and have been pushed to record-breaking peak- and average-power levels at high combination efficiency [22]. For temporal combination based on optical delay lines, PBSs allow for high combination efficiencies as well [15,18] and they allow, compared to enhancement cavities, for simpler setups as neither a fast optical switch nor phase shaping is required [16,17].
The concept of such a spatiotemporally multiplexed amplifier is depicted in Fig. 3. First, a seed pulse train of 2N pulses is generated, e.g. by dividing a single pulse with N cascaded optical delay lines. Then, the pulse train is split into M parallel amplifier channels using a tree-type scheme and, afterwards, is recombined in the same manner. Finally, the pulse train is temporally combined using a separate set of optical delay lines. Each interferometric path in the system contains a phase correcting element, e.g. a piezo-mounted mirror, to stabilize the system against environmental perturbations. Stabilization techniques compatible with optical delay lines rely on a single detector at the system output, allowing for dithering techniques and hill-climbing algorithms, such as the stochastic parallel gradient descent (SPGD) [23] or locking of optical coherence via single-detector electronic-frequency tagging (LOCSET) [24].
Fig. 3 Schematic of an actively stabilized spatiotemporally multiplexed amplifier. Phase regulators in a closed-loop control compensate optical path length fluctuations.
In the following and without loss of generality, only LOCSET is considered. In LOCSET, the phase-modulating elements are used to impose a phase dither to each regulator with a distinct radio-frequency (RF) leading to small dithering of the combined output power. This dither is monitored by a photo diode and is used to retrieve error signals for all regulator channels via coherent demodulation. The error signals feed the closed-loop controlled phase modulators.
Recently, pulse splitting, amplification and recombination in such a scalable multidimensional geometry has been demonstrated at low power levels [20]. In this setup, two amplifier channels and two cascaded optical delay stages were stabilized using LOCSET resulting in a high combining efficiency even for both strong saturation of the amplifiers and high B-integral values. However, an undesirable bistable operation behavior of the system can be observed when choosing non-optimized regulation parameters. As depicted in Fig. 4(a), the combined output power switches over the course of minutes between the anticipated maximum output power (state 1) and an about 25% reduced output power (state 2). The temporal characteristic of the output pulse is also affected, as shown by a photo diode trace in Fig. 4(b). The pulse replicas are either correctly combined into a single pulse (state 1) or remain symmetrically distributed around the time where the correctly combined pulse is expected (state 2).
Fig. 4 Bistable behavior of the amplifier system (a) on the long-term scale and (b) the corresponding photo diode traces of the respective pulse trains.
In this contribution, which is an extension to [25], we thoroughly investigate the LOCSET phase stabilization for these multidimensional architectures. The experimental setup, that led to the discovery of the stability issue, is analyzed to find the origin of this behavior. We show that the number of (undesired) stable states increases with the number of temporal divisions. Then, a mitigation strategy is derived and generalized for arbitrary numbers of spatial and temporal combination stages. Finally, the advanced stabilization method is verified experimentally.
2. Experimental setup
Figure 5 depicts the experimental setup used to investigate the phase stabilization of spatiotemporally multiplexed amplification. It is the very same setup as used in [26] consisting of two pairs of optical delay lines for division and combination (I, VI and II, V) and two amplifier channels (III, IV). This architecture can be arbitrarily scaled in terms of temporal divisions via increasing the number of delay lines and in terms of spatial division with a series of cascaded beam splitter cubes. The system is seeded by stretched femtosecond pulses with 1.5 ns duration. In the first temporal division stage (I), a seed pulse is split into two pulse replicas. A half-wave plate (HWP) is applied to rotate the polarization of the linearly polarized seed pulse such that its power is divided equally at the subsequent PBS. The p-polarized component becomes the leading pulse replica as it is transmitted at the PBS while the s-polarized component is reflected into a free-space double-pass delay line. At the top end of the delay line, the s-polarized pulse is reflected at a normal incidence mirror and undergoes a 90° polarization rotation by double passing the quarter-wave plate (QWP) in the beam path. Thus, the returning pulse is p-polarized and is transmitted at the original PBS into the lower part of the delay line. Here, another combination of QWP and normal incidence mirror rotates the polarization again. The returning s-polarized pulse is reflected at the original PBS along the path of the leading p-polarized pulse. Thus, after the delay line, there are two orthogonally polarized pulses with a delay of about 8.6 ns (~2.6 m delay path). Additionally, one of the end mirrors is mounted on a piezo-actuator for the active phase stabilization. Similarly, the pulse replicas are divided again in the second temporal division stage (II) with a delay of 4.3 ns (~1.3 m), leading to a pulse train consisting of four pulses. Then, the pulse train is split onto two amplifier channels (III). Thereby, the s-polarized component is double passing another set of QWP and piezo-mounted mirror to introduce a degree of freedom for the phase stabilization of the two channels. After amplification, the two beams are spatially superposed and the alternating polarization pattern is restored (IV). A HWP rotates this polarization pattern by 90° such that the s-polarized pulses of the train become p-polarized and vice versa. The s-polarized pulses enter the delay path of the first temporal combination stage (V) which path length is matched to the second pulse division stage (II). Thus, the first and the third pulse are superposed with the directly transmitted second and fourth pulse, respectively. The resulting two pulses are orthogonally polarized again but tilted by an angle of 45° and another HWP is applied to rotate them into s- and p-polarization. The second temporal combination stage (VI) reverses the delay introduced by the first temporal division stage (I) in an identical manner. Finally, at the system output (VII), the pulse is transmitted through a HWP and a PBS to separate the correctly combined linearly polarized pulse from any depolarized signal that could arise from relative path mismatch or pulse energy mismatch among the pulse replicas. Finally, a small fraction of the combined output power is redirected towards the photo diode to be used as the input signal for the LOCSET phase stabilization.
Fig. 5 Simplified experimental setup of a spatiotemporally multiplexed amplifier. (HWP: half-wave plate, PBS: polarizing beam splitter, QWP: quarter-wave plate, Amp: amplifier).
3. Theoretical description of the phase stabilization
3.1 Derivation of the error signals
The observation of the two stable output states (cf. Figure 4) indicates that LOCSET retrieves incorrect error signals. A time resolved Jones’ calculus [27] is applied to derive these error signals following functional steps numerated in Fig. 5. All optical elements are assumed to be ideal, which defines the Jones’ matrices ĴR and ĴT for reflection and transmission on a PBS as
where the phase shift upon reflection is omitted. Similarly, the Jones’ matrix ĴHWP(θ) for a half-wave plate is defined aswith θ being the orientation angle of its optical axis with respect to the plane of p-polarization. The pulses are represented by delta functions δ(t) with complex amplitude E as the pulse shape is not of particular interest in this problem. The Jones’ vector of a p-polarized seed pulse at the amplifier input is set asThe seed pulse is split into two replicas of equal power at the first pulse division stage (I) introducing a delay time τ1 between the replicas, which is accounted for by shifting the time of the trailing pulse from t to t-τ1. The double-passed QWPs can be replaced by HWPs in the calculation. Thus, after the first pulse division stage the complex electric field is given byThe time dependence and the delta functions are dropped to simplify the notation. They are replaced by a time stamp to mark the delay. The phase dither imposed to the trailing pulse by the piezo-mounted mirror will be considered upon combination. This simplifies the notation, as the position of the phase shifting element within the splitting or combination stage is free to chose. The second pulse division stage (II) divides the pulses again and introduces a smaller delay τ2 (2τ2 < τ1).Now there is a pulse train consisting of four pulses that are alternatingly p- and s-polarized. Next, the pulse train is divided onto two amplifier channels of the system (III). There is only one polarization in each channel, which allows to treat them as the two dimensions of the Jones’ vectorIn the step (IV), the pulse trains are spatially recombined. Thereby, the p-polarized component is the arbitrarily chosen reference to which the s-polarized component possesses an unknown relative phase Φ3 expressed bywhere β3 is the modulation depth, ω3 is the dithering frequency of the piezo-mounted mirror, ΔΦ3 is the actual phase error of the spatial combination, and Ψ3 is introduced as abbreviation. In case of perfectly matched phases, the linear polarization pattern of the combined pulses is inclined by 45° with respect to p-polarization. The HWP following the spatial combination is set such that its optical axis is inclined by −22.5° with respect to the plane of p-polarization. Thus, the leading pulse becomes s-polarized. Then, the electric field at the output of the functional block (IV) computes towhere the two vector elements regularly represent p- and s-polarization again. In step V, the temporal division of the short delay line is reversed (τ2). Again, there is a relative phase Φ2 between the respective p- and s-polarized pulses, which includes a phase dither with a modulation depth β2, a dithering frequency ω2 of the respective actuator, and an unknown phase error ΔΦ2Afterwards, a HWP is applied with its optical axis set to 22.5°. Thereby, the polarization of the two remaining pulses rotates and the leading pulse is s-polarized and the trailing pulse is p-polarized. Undesired pulses will emerge in the combined pulse train if the previous spatial combination was incorrect, since wrongly polarized pulses will leak into the different paths of the delay stage. The electric field after the first temporal combination step is given byIn step VI, the pulse train enters the second temporal combination stage (τ1). If the previous stages are combining correctly, then there are only two pulses left and the leading pulse enters the delay line to superpose with the trailing pulse. The superposition is affected by the phase dither of the first regulator channel with a modulation depth β1, a dithering frequency ω1, and by an unknown phase error ΔΦ1Afterwards, a HWP set to −22.5° rotates the final pulse into p-polarization. Thus, the field after step VI is given byIf all complex phase terms Ψi become 1, meaning that both phase dither and phase error vanish, the output is a single p-polarized pulse at the time slot τ1 + τ2. The last step (VII) describes the propagation through a last PBS measuring the p-polarized fieldA part of the transmitted light is send onto a photo diode that cannot resolve the combined pulse train but only the slowly varying intensityThe response of the photo diode is assumed to be linear and translates the optical signal in the electrical domain, where the error signals Si are obtained by demodulating at the respective frequency according to [24]For a sufficient integration time T, that is large compared to the beat note of two neighboring modulation frequencies [28], the error signals of the three regulator channels are found as where Ji is the Bessel function of first kind and i-th order. Comparing to the findings for spatially multiplexed amplification [24], there are additional terms occurring herein. The error signals significantly depend on the phase errors of the respective other channels. However, the closed-loop phase stabilization acts according to the sign and the slope of these error signals. Stable operation points, meaning zero crossings of these signals, are found by applying the general stability criteriato Eqs. (16)-(18). As expected, the correctly combined state is reached when all three phase errors ΔΦi are 0 rad. The incorrect state is found for all ΔΦi being π rad, which causes the polarization pattern to be flipped after the spatial combination (IV). Thus, the pulse train is temporally spread even more at the subsequent temporal combination (V) instead of being combined. Then, in the second temporal combination (VI), these pulses are just partially superposed leading to the secondary state depicted in Fig. 4(b). The system can switch between these two states when the momentary phase perturbations on all channels are large enough to change the sign of their error signals. Note, that the same behavior would be observed using non-polarizing beam splitters for pulse splitting and combination, as depicted in Fig. 6. Moreover, the same behavior is obtained for the optically identical case of temporal combination of eight pulse replicas in a single amplifier channel (N = 3, M = 1). This equivalence is because the first combination step, whether temporal or spatial, does not depend on preceding combination steps. This in turn allows separating the theory of LOCSET phase stabilization into purely spatial and purely temporal combination.Fig. 6 Equivalent implementations of spatiotemporal pulse division (left-to-right) and combination (right-to-left) using either a) polarizing or b) non-polarizing beam splitters.
3.2 Mitigation of the bistability
The undesired bistability is found to be intrinsic to temporal beam combination using optical delay lines in combination with LOCSET phase stabilization. This behavior impedes long-term stable operation of the laser system. However, SPGD and other hill-climbing algorithms will yield the same bistability as the combined output intensity, given by Eq. (14), has two local maxima. A passive stabilization of the system, e.g. by applying a housing, is generally advised but certainly not sufficient, as guaranteeing optical path length stability better than a quarter wave length is required at all times (cf. Equations (16)–(18)). However, considering Fig. 4, there are two more approaches to mitigate the stability issue by gathering additional information about the current state of the system. For example, the difference in the output peak-power of the two states could be used to trigger a π rad phase shift on all stages restoring the correctly combined state. A nonlinear optical process, such as SHG, would simplify the distinction between the two stable states. This approach would allow for a self-reset of the closed-loop control in the correct state. However, the signs of the required phase shifts remain unknown, which eventually will lead to a degradation of combining efficiency for a repeatedly reset closed-loop control. This and the need for a defined threshold value make this approach a less desirable solution. The superior mitigation strategy is to alter the error signals themselves, such that the secondary stable state vanishes. This can be achieved by temporally gating of the raw signal acquisition with a window that fits only the correctly combined pulse. This runtime-sensitive approach can be implemented either in the optical domain using a pockels cell or in the electronic domain using a fast electronic switch. Then, new error signals can be retrieved by considering only the term at the time slot τ1 + τ2 of Eq. (13) leading to
with {i,j,k} being permutations of the regulator channels {1,2,3}. Still, the phase errors are cross-influencing in the error signals, but the second stable state is suppressed as becomes evident by applying the stability criteria from Eq. (19).3.3 Generalization to higher channel counts
The analytic derivation of LOCSET error signals for systems comprising M amplifier channels and more than three temporal division stages becomes a time-consuming task. Reducing the effort, no more than two amplifier channels are considered, as they form independent subsystems for which the known solutions of spatially multiplexed LOCSET remain valid. Thus, systems with one or two amplifier channels and up to five temporal pulse combination stages were investigated using a numerical model that resembles the analytic calculation. Thereby, the delay time of the cascaded optical delay stages are discretized in multiples of the shortest delay time. The phase errors space of each channel is the interval [0,2π]. The resulting numbers of stable states are summarized in Table 1. In the evaluated cases, up to five stable states are found, for which the combined output pulse trains are shown in Fig. 7. The phase errors ΔΦi are numerated in the order of appearance of the pulse combination stages. Thus, assuming the two innermost combination stages being perfect (ΔΦ1 = 0, ΔΦ2 = 0) recovers the case of the analytic solution for one spatial combination and two temporal combinations of Figs. 7(a) and 7(b). For a system with two channels and threetemporal combination stages (ΔΦ1 = 0), there are three stable states. When adding another temporal combination stage, there are five stable states.Please note, that all undesired stable states yield a symmetric distribution of pulse energy around the time slot where the correctly combined pulse is expected. Thus, all of them can be mitigated via the gated error-signal acquisition discussed above. The analytic solutions to the error signals for a system with N temporal pulse divisions or N-1 temporal divisions and two amplifier channels are found as
The numeric analysis of systems with even more temporal combination stages was not pursued as all secondary stable states are suppressed using the temporally gated error signal acquisition. Lastly, the solution can be extended to M amplifier channels using the available closed-form solution of the error signals for purely spatial beam combination [24].
Table 1. Stable States for Spatiotemporally Multiplexed Amplifier Systems
Fig. 7 Stable output pulse trains for a five stage temporal beam combining using LOCSET or SPGD. P: normalized combined output power, {ΔΦ1, ΔΦ2, ΔΦ3, ΔΦ4, ΔΦ5}: error phases.
4. Experimental verification of the mitigated bistability
Figure 8 depicts the implementation of the temporally gated error signal acquisition into the existing experimental setup. The originally used RF photo diode (PD 2) at the system output is replaced by a GHz-bandwidth photo diode to provide a raw signal that resolves the temporally divided pulse replicas. The signal is fed into a fast electronic switch (10%-90% transmission in 6 ns), which is transmitting only the electrical pulse that corresponds to the correctly combined optical pulse. The gating window is 4 ns long and is provided by a square-pulse generator, which is triggered by a clock signal that is derived using another photo diode (PD 1) at the system input. This implementation suppresses the bistable behavior as is demonstrated by a three hour long-term measurement of the combined output power shown in Fig. 9(a). The slight power decrease is due to a temporal drift of the alignment.
Fig. 8 Implementation of the temporally gated LOCSET. (HWP: half-wave plate, QWP: quarter-wave plate, PBS: polarizing beam splitter, PD: photo diode, LW: laser window).
Fig. 9 Long-term stability (a) and frequency spectrum of the short-term stability (b) using LOCSET with gated error signal acquisition.
In addition, the noise spectrum of the output power was recorded for open-loop and closed-loop phase stabilization, which is shown in Fig. 9(b). In the open-loop spectrum, more than 99% of the noise power is confined below 100 Hz. The spectrum features also the ever-present AC hum and its harmonics starting from 50 Hz. Lastly, there are the dithering frequency peaks of the three regulator channels between 6 kHz and 7 kHz and a few other high-frequency peaks originating from the control electronics themselves. In closed-loop control, the noise level is decreased to the noise floor of the measurement. Remaining features are the AC hum and a few high frequency resonances that originate from the control electronics. The RMS amplitude of the error signals in closed-loop control indicates that the RMS path length error is less than λ/45 being limited by the measurement accuracy.
5. Conclusion
In this contribution, the phase stabilization of simultaneously temporal and spatial coherent beam combination using binary tree-type division schemes and optical delay lines was investigated. Compatible stabilization methods are LOCSET and SPGD. In the particular case of two amplifier channels and four temporally divided pulse replicas, an undesired bistable behavior was observed during using LOCSET stabilization. Applying a time resolved Jones’ calculus, the issue was found to be intrinsic to temporal combination with optical delay lines and is independent of the stabilization method applied. The theoretical analysis was extended for systems with more temporal division stages and more amplifier channels using numeric calculations. Five stable states were found for systems with up to 32 pulse replicas. Based on the calculation results, temporal gating of the error signal acquisition was proposed as a simple mitigation strategy. Following this approach, the suppression of the observed bistability was proven in the experiment. This result enables to proceed in the peak-power scaling of high-power ultrafast laser systems via spatiotemporally multiplexed amplification.
Acknowledgments
This work has been partly supported by the German Federal Ministry of Education and Research (BMBF) under contract 13N13167 “MEDUSA” and by the European Research Council under the ERC grant agreement no. [617173]. M.M. and T. E. acknowledges financial support by the Carl-Zeiss Stiftung. A. K. and M. K. acknowledge financial support by the Helmholtz-Institute Jena.
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