An intracavity deformable membrane mirror has been successfully used to optimise the brightness of solid-state lasers — a side-pumped Nd3+:YAlO laser where the oscillation of a low-order transverse mode was obtained, and a grazing incidence Nd3+:GdVO4 laser where a brightness increase by an order of magnitude with negligible drop in power was achieved. Several search algorithms were also implemented in the system and compared with respect to intracavity optimisation.
©2008 Optical Society of America
Thermal induced aberrations are the main limitation in solid-state laser when scaling the power . The thermally induced lens degrades the transverse mode profile of the output beam . Thermally induced birefringence can be greatly reduced by careful choice of laser media and only the spherical contribution of this thermal lens can be compensated by careful laser cavity design . Nevertheless, the non-spherical component cannot be compensated in the same manner leading to reduced efficiency and multi-mode oscillation as the heat load increases. Active transverse mode control and optimisation of Nd:YVO4 laser using an intracavity adaptive optics mirror was demonstrated . This optimisation was achieved with a low power end-pumped Nd:YVO4 laser using a hill-climbing algorithm. The use of the latter seriously limits the search power since a hill-climbing algorithm can only ensure the local maximum to be obtained. The use of a more advanced search algorithm is, therefore, required. In this paper, we investigate the transverse mode control of a side pumped Nd:YAlO laser and of a grazing-incidence Nd:GdVO4 laser using an intracavity deformable membrane mirror (DMM) at power level of 15W with a control feedback loop featuring a genetic algorithm. The demonstration of power scaling of this simple and cost-effective system is shown in this paper. First the DMM will be described with an assessment of the feasibility of using it intracavity. Then the optimisation scheme including a description of the different algorithms used will be discussed. Finally, the results will be presented and discussed.
2. Micro-machined deformable membrane mirror
Adaptive optic mirrors have been widely used in astronomical and medical imaging . The mirror used is a cost-effective 15mm diameter OKOTECH  DMM. It comprised 37 electrostatic actuators arranged in a hexagonal pattern. The frequency response of the mirror was 1 kHz. The silicon nitride membrane was coated with a Cr/Ag layer onto which a subsequent 12 layer dielectric mirror was deposited to provide a high reflectivity at 1064nm (>99.8%). With the maximum voltage applied to all the actuators (240V), the maximum stroke was measured to be 5μm (equivalent to a curvature of ~0.18D). The actuators could be addressed individually via a personal computer using a software package. The membrane can only be ‘pulled’ by the actuators, therefore a pre-bias is required to meet the demand of some applications. The mirror was mounted behind an anti-reflection coated window to environmentally shield the fragile mirror surface and reduce the effects of air currents. Similar mirrors have been used successfully in a range of areas: master-oscillator power amplifier (MOPA) configurations , multiphoton microscopy .
3. Assessment of the scalability of the membrane power to high intracavity powers
One of the main issues with using a DMM inside a laser resonator is the stability of the membrane under conditions of high power density . To assess this, a side-pumped grazing incidence Nd:GdVO4 laser shown in Fig. 1 was used and a Michelson interferometer (see Fig. 2) was configured such that one arm of the interferometer included the intracavity DMM — the interference pattern produced is then a sensitive measure of any distortions induced in the membrane by the intracavity laser field. The 1% doped, Nd:GdVO4 slab had dimensions 22mm × 5mm × 2mm - the AR-coated (1064nm @ 10°) end faces were wedged at 5.0 degrees to prevent parasitic oscillation. The slab was bonded on its two larger surfaces to a water-cooled mount using 0.25mm thick indium foil to ensure good thermal contact. The pump power was delivered by two spatially coupled diode laser bars emitting at 808nm giving a total incident power of 50W. The pump beam was focussed to a 7mm × 0.25mm spot on the crystal. A ×3 intracavity telescope was used to provide a 4.5mm diameter spot on the deformable mirror. In this configuration, a R=50% output coupling mirror resulted in 9W of output power.
The Michelson interferometer was used to analyse any thermally-induced distortion of the surface of the DMM. The HeNe probe beam was expanded to cover about 70% (~10.5mm) of the total aperture of the DMM, i.e. much larger than the incident intracavity laser beam. During the experiment, the protective window was removed from the DMM to ensure the distortion was solely due to the membrane surface. With an incident power of 18W, corresponding to power density of 115W/cm2 on the membrane, no significant change in the interference pattern obtained from the Michelson interferometer was observed. At this incident power level no thermally induced mirror deformation occurs. This observation was also independent of the voltages applied to the actuators of the DMM - see Figs. 3(c) and 3(d).
To increase the incident intracavity power on the membrane, the reflectivity of the laser output coupler was increased to R=99%. Whilst this resulted in a reduced output power of 2W, the average intracavity power was increased considerably to 200W. For all cases, the incident intracavity beam diameter remained unchanged, and so, the resultant power density on the DMM was increased to 1.25kW/cm2 - approximately a ×10 increase. Figure 4 details the interference patterns obtained at these elevated power levels for different actuator settings. It is clearly evident here that significant distortion of the membrane occurs at these intracavity power levels - examination of the fringe patterns suggest that >1μm additional deformation is being induced, which, considering the maximum stroke is only a few microns, will cause serious problems in an iterative optimisation scheme [specifically when non-deterministic search algorithms such as the genetic algorithm are deployed — see section 4].
The time constant for thermal equilibrium to be reached by the hot membrane was found to be ~10s, however, a time of >3 minutes were required for the membrane to cool and reestablish its initial shape. During these measurements the beam diameter on the DMM maintained a constant diameter. Even though such low output coupling is inefficient and unlikely to be used in this kind of laser, the data recorded give a useful upper limit to the power handling range of these DMM devices - not limited by damage but by thermally induced membrane distortion.
4. Control loop feedback system
To facilitate distortion compensation, and so, brightness control of the laser a closed-loop feedback system was developed, the components of which will now be described. The core component of the iterative optimisation technique was a peak-location algorithm which was driven by a single, simple measurement on the operating state of the laser system. This input was derived from an assessment of the laser output fitness recorded by a sensor and the control program was then configured to maximise this value. The voltages of the actuators of the DMM, determined by the control program, were then configured via a multi-channel, high-voltage, USB addressed digital-to-analog converter (DAC). A schematic of the complete closed-loop optimisation scheme is depicted in Fig. 5. The laser beam quality sensor and the nature of the control algorithm are, of course, the essential components of the optimisation network and will now be described in more detail.
4.1 Beam quality sensor
An assessment of the quality of the laser output, i.e. its fitness, is an essential input to any optimisation algorithm. In this work two convenient methods were developed: firstly, a laser brightness measure based on a CMOS camera having a software aperture, and secondly, by way of second harmonic generation (SHG).
The software aperture-based sensing method used a simple CMOS camera, as described in , where the modal output from the laser was assessed by integrating the pixel intensity within a selected region of the recorded camera image of the laser beam profile. The dimensions, location and degree of averaging within this aperture were all user settable to promote optimal performance. In addition, beam centroiding was employed such that the aperture could be automatically re-centred with respect to the laser beam image. Possible beam pointing and alignment issues were therefore eliminated in this assessment mode.
An alternative beam fitness assessment method was based on SHG, as it is well known that efficient second harmonic conversion requires a bright pump beam . Furthermore, the SHG efficiency follows a square-law dependence with average laser power. So clearly, the higher the brightness, the greater the SHG. This laser brightness sensor concept based on SHG is shown schematically in Fig. 6.
In practice, the output beam waist on the plane output coupling mirror was relayed onto a 20mm-long KTP crystal using a 45mm focal length lens (see Fig. 7). The infrared light was filtered out allowing the SHG signal to be recorded by a photodiode. Imaging of the beam waist at the flat output mirror ensures consistent SHG values during any laser optimisation procedure. The experimental arrangement of the SHG sensor is shown in Fig. 7 - the interface of the program featuring the user controls for these sensor systems is shown in Fig. 8. This method presented a very straightforward and low-cost way of assessing the brightness of the laser.
4.2 Optimisation algorithms
The DMM has 37 actuators to which voltages between 0 and 200V with 1V steps are applied. Thus, the number of possible membrane shapes is 37200, or ~1085. Optimisation to find the best possible solution of such a multi-dimensional problem is therefore rather challenging. Perhaps the simplest, and most obvious, method to deal with this problem is to employ a hill-climbing algorithm  - here, each actuator is considered in turn, the voltage varied, and the best value set to each actuator. This algorithm is both straightforward and fast (in practice, less than 2 minutes), however, only a local maximum solution is found, which, in general is dependent on the initial conditions of the system. Given the complexity of the search space it is unlikely that the given result would the best possible and consistent. More advanced algorithms are therefore required. In this work, the genetic algorithm (GA) and the simulated annealing (SA) algorithms were integrated within the control program. Two close variants of the simulated annealing algorithm - the adaptive random search (ARS) and random search (RS) - were also developed.
4.2.1 The genetic algorithm
Non-deterministic algorithms such as the GA  have been developed which can efficiently find the global solution from highly complex multi-dimensional search spaces. The GA mimics evolution by way of natural selection (i.e. survival of the fittest) as an approach to effectively sample the search space and reveal the best possible solution. For the DMM optimisation, in the terminology of the genetic algorithm, each actuator is defined as a chromosome and a particular mirror shape (i.e. the combination of 37 chromosomes) as an individual. The algorithm begins by the random generation of N membrane shapes - the fitness of each shape is then assessed, and the ‘best-yet’ individual is stored. A second generation of individuals is then generated through a combination of existing individuals - this breeding process, or cross-over, is influenced by the fitness of the parents. These daughter solutions are then allowed to exhibit a (small) degree of genetic mutation to promote new solutions such that the search space can be more efficiently examined. The mutated daughter solutions then become the new parent population from which the next generation will evolve - this process is then repeated. The number of generations, the cross-over and mutation probabilities are set by the user. The full GA loop is shown schematically in Fig. 9 - note that the function which acts to stop the algorithm is included. This stopping criterion is fairly arbitrary - the one employed here stops the search if the best solution found does not change by a preset percentage (e.g. 2%) over 50 generations. We find this a practical device, especially given that system noise is present.
In practice, the initial conditions and the crossover and mutation rates need to be tuned to optimise the time taken for the algorithm to converge. As is typical, the population size is chosen to be approximately the dimensionality of the search space - for the 37-element mirror, a population size of 40 was chosen. The probability of crossover occurring was set to 85% and the mutation probability used was 2% - again, these are fairly typical values [i.e. crossover - high, mutation - low] for the genetic algorithm. In a typical laser optimisation this whole process takes about 15 minutes to converge to a solution. The algorithm therefore samples only ~1200 different shapes in finding the (approximate) global maximum in such a wide space range. Although the GA is more computationally intensive than the hill-climbing algorithm, finding the global best solution is a significant advantage for this optimisation procedure.
4.2.2 The simulated annealing algorithm
A simulated annealing algorithm was implemented to the laser control program. This algorithm is based on a Monte-Carlo approach  to find solutions of multi-dimensional problems. It is analogous to the physical process of annealing. The algorithm progressively, but slowly, lowers the ‘temperature’ allowing the system to re-configure into a more optimal state. [The concept of temperature in the algorithm is, obviously, abstract, however, it relates to the size of the allowed perturbation, and the probability of a new state being chosen over an existing state.] At ‘high temperatures’ the system is allowed to change by a large extent such that the whole solution space can be searched, whereas, as the temperature lowers, only progressively smaller changes are allowed. The system temperature is also used in calculating the probability of accepting a new state having a lower fitness value. This process continues until the system ‘freezes’ and no further changes occur. The SA, therefore, samples the search space in a completely different way (see the flowchart in Fig. 10) than the GA, and, typically returns the optimum result faster .
4.2.3 Random search algorithms
The RS and ARS algorithms are derivatives of the SA. Here, only positive fitness changes are accepted (see Fig. 10). The ARS involves the notion of temperature in a similar way as the SA, and this has the effect of reducing the magnitude of the search as the algorithm progresses. In the RS algorithm, the maximum perturbation is fixed and made equal to the magnitude of each axis of the search space. Therefore, the RS continues to search the full bounds of the search space for the duration of the optimisation, whereas the ARS continually narrows its focus - a property that can often lead to undesirable results.
4.3 Experimental laser cavity platforms
To fully investigate the use of an intracavity deformable mirror to optimise the modal performance of an all-solid-state laser, several laser cavities were configured: a transversally-pumped Nd:YAlO laser, shown in Fig. 11, and a side-pumped, grazing-incidence Nd:GdVO4 resonator shown in Fig. 12. It is noted that these laser cavities were simply test-beds used to elucidate the central topic to this paper, AO-based laser optimisation techniques.
4.3.1 Side-pumped Nd:YAlO laser
Here, the gain medium used was a Nd:YAlO slab (15mm × 5mm × 1.5mm) with AR-coated end faces at 1079nm. It was pumped by an 8-bar DILAS  diode laser stack delivering up to 100W output power at 805nm. Using focussing optics, a 1.67mm × 0.3mm pumped zone on the side of the crystal was defined. Like previously, the Nd:YAlO slab was coupled to a water-cooled mount via the 15mm × 5mm faces. The absorbed pump power was measured at 35W. A ×10 intracavity telescope was used to ensure a large (~12mm diameter) spot on the deformable mirror. The effect of the deformable mirror on laser cavity operation was therefore maximised as the majority of the actuators could interact with the intracavity beam. When the laser was configured with a R=80% output coupling mirror, the laser output was measured to be 6W.
4.3.2 Grazing-incidence Nd:GdVO4 laser
The grazing-incidence configuration has potential for very efficient, high-power systems. TEM00 operation of 23W with an efficiency of 68% have been demonstrated in a diode-pumped grazing-incidence Nd:YVO4 laser . Crucial to the operation of this laser is a total internal reflection on the pump surface of the Nd:GdVO4 crystal. The bulge of this reflecting surface and high asymmetric thermal lensing produced represents a limiting factor to future power scaling of such lasers. Thus, this set-up is an excellent test-bed for laser brightness optimisation using an intracavity deformable mirror. The laser cavity, the gain medium and the pump configuration have already been described in section 3, however, the magnification of the intracavity telescope was changed. Here, a ×6 intracavity telescope was chosen to promote a larger, ~8mm diameter, spot on the deformable mirror, such that, the intracavity beam better matches the transducer array dimensions. In this arrangement, a R=50% output coupler was found to optimise the laser output power at ~15W.
5. Beam quality optimisation of solid-state lasers
An optimisation routine was performed based on the genetic algorithm on the two test-bed lasers described in section 4. The software aperture technique was used as the beam quality sensor for the optimisation of the Nd:YAlO laser. Initially, the laser was optimally aligned with the DMM actuators at the middle of their voltage range (120V) and the resultant multi-mode output, having a power of 6W, is shown as Fig. 13(a). After optimisation, which lasted ~5 minutes, a near single transverse mode intensity profile (M2<1.3) was recorded (shown in Fig. 13(b)). The laser output power typically changed by less than -5% as a result of the optimisation.
A similar genetic optimisation was performed on the Nd:GdVO4 laser, however, here the SHG-based fitness sensor was used. Again, the initial state of the laser cavity was defined with all the DMM actuators at 120V. In this case the optimisation routine lasted around 15 minutes which, we believe, was due to the substantially higher-order mode content of the initial laser state. The output power was found to slightly decrease from 15W to 14W, however, the beam propagation parameter, M2, in the tangential plane decreased from 27 to 9, whereas in the sagittal plane, it decreased from 10.5 to 3. Near-field output beam profiles before and after optimisation are shown in Fig. 14. Far-field beam profiles taken before and after optimisation are shown in Fig. 15.
The optimisation scheme, therefore, increased the laser brightness by approximately an order of magnitude whereas the laser output power remained essentially constant. It is expected that further improvement in the performance of the optimisation could be induced by using a DMM having a greater maximum stroke.
6. Comparison of optimisation algorithms
A range of different optimisation algorithms were used to enhance the resolution of an adaptive optics confocal microscope . Similarly, for intracavity laser distortion control the nature of the optimisation algorithm used is a critical component. As discussed above we have integrated a range of algorithms into our control software to allow us to find the optimum technique for laser performance optimisation. The following discussion relates to the assessment of these routines with respect to their application to intracavity AO control of lasers.
It was apparent from the optimisation of the 15W Nd:GVO4 laser that the brightness enhancement produced relates to the overlap of the fundamental mode with the pumped volume within the gain medium. The observed modal distribution within the laser rod will configure to efficiently extract the maximum available gain, and so, the basic shape of the pumped region will be mirrored in the oscillating mode profile. If the resonator is configured to give a small fundamental mode area (compared to the pumped area), then substantial higher-order mode content is required to extract the otherwise un-used gain from the laser rod. So, in this case, (without resorting to using apertures which will reduce the modal content of the output but at the expense of total power) brightness enhancement will be limited to that induced by the subtle modal reconfiguration made possible by deformable mirror. Our attention then turned to the other extreme case, i.e. where the fundamental mode is ‘matched’ to the pumped region, to examine the benefits of AO optimisation.
The laser cavity in Fig. 12 had a fundamental mode radius of ~300μm in the tangential plane. Since the majority of the incident pump was absorbed within the first 2mm of the crystal surface, highly multimode oscillation in the tangential plane was to be expected. To enlarge the fundamental mode radius to ~1mm in the tangential plane within the gain medium, the cavity was reconfigured by extending the short cavity arm and adjusting the cylindrical lens to re-establish resonator stability (see Fig. 16). This cavity did indeed exhibit single transverse mode oscillation (see Fig. 17), however, the output power was significantly reduced to ~4W for an incident pump power of 40W.
Results obtained from the optimisation of this low-order mode laser using our algorithm suite are shown below in table 1. The fitness value was measured using the SHG-based method. The GA, RS and ARS converged to the same optimum result, however, the GA was significantly slower by about a factor of 3. The SA required careful tuning of the calibrating Metropolis value , however, after this was achieved, good performance resulted. This calibration procedure would be required for any new search field, and so, would seriously hamper the use of the SA.
The closed-loop, intracavity adaptive optics control system developed was successful in enhancing the brightness of both the laser platforms examined here. The use of the genetic algorithm ensured the maximum performance (i.e. the globally optimal solution) was returned - and so may be thought of as the best technique. However, a few practical issues were identified which limit its general applicability: thermal lag, electrical and optical noise and modal collapse. All require consideration, specifically when working with hot distorted gain media.
7.1 Thermal lag
Thermal lag - the time required for the laser to respond fully to any applied mirror shape change - manifests itself as a limit to the speed of the optimisation procedure. This, of course, has implications to the practical application of this technology but does not, in itself, limit the potential laser performance enhancements that can be induced. In practice, simply, a delay between the application of a mirror change and the measurement of the effect induced is invoked to account for this lag. Control algorithms where sequences of small changes are used to search for the optimal solution (e.g. SA and RS) induce only minimal lag and so can significantly speed up the optimisation.
Fluctuations in output power cause a much more serious problem specifically in the context of the use of non-deterministic algorithms. Simply put, the process is driven by a comparison of a single-valued parameter for each sampled state of the laser; noise can significantly affect this selection process. In our system, signal averaging is employed to reduce these effects, however, such averaging cannot compensate for slow passive optical drift over time. The implication is therefore that the passive laser platform must have a high degree of long-term stability to make certain of good optimisation success. In our development, this was not always possible, so, modifications to the algorithms have been made to account for this problem. Firstly, in the GA, the fitness of the previously best solution is re-tested after each generation such that a fair comparison to new solutions is still valid. Also similarly, in ARS, RS and SA cases the current best mirror shape is re-measured at the end of each cycle. The effect of these modifications is that the search for the optimal solution can still be maintained even if background mechanical or environmental factors cause the average output power to vary.
7.3 Modal collapse
During optimisation, different successive mirror shapes can significantly change the optical properties of the laser cavity (and in some circumstances turn the laser off). This effect becomes particularly problematic in hot lasers where these large changes can significantly modify the thermal loading in the laser rod. This can lead of a situation whereby the laser state becomes non-reversible with respect to the mirror shape - i.e. restoring the previous mirror shape does not restore the previous laser oscillation state. This effect, termed here modal collapse, is of primary significance as it can ruin any attempt to optimise the laser via the techniques developed here (apart maybe, from a true hill-climbing approach). The RS, ARS or SA reduce the probability of modal collapse as the change of mirror shape is small as only one actuator is changed at any one time, however, the GA with its full 37 transducer variations has very little chance of success. At present, there is no obvious solution to mitigate the effects of modal collapse apart from using the ‘gentler’ algorithmic approaches such as SA and RS.
In addition, the possibility of mixing different algorithms during the same search was implemented. A hill-climbing algorithm can be effectively used to terminate a RS, SA or GA optimisation, therefore reduce significantly the optimisation duration - and largely mitigating modal collapse issues. Furthermore, the starting point of the search could be calculated and a sensible starting membrane shape can be applied. The search field could then be reduced enhancing the simplicity and speed of the chosen search algorithm.
Further to these modifications designed to address specific ‘hot’ laser related problems, we have extended the potential for greater degrees of optimisation by incorporating a piezoelectric tip/tilt mirror into the laser resonator. The tip and tilt functions were integrated easily into the control scheme as these were considered simply as two extra actuators in the control procedure. Extension of this tip/tilt function to more than one mirror shows promise for active laser alignment to complement the active optimisation already demonstrated.
We have demonstrated automatic spatial mode control and power optimisation of two diode-side-pumped solid-state lasers using an intracavity deformable membrane mirror. A closed-loop optimisation method was used which featured a sensor, software-based optimisation algorithm and parallel electronic addressing to the actuators of the deformable mirror. Two beam quality assessment techniques were developed: based on using a software-derived aperture, and on direct second harmonic generation. In addition, a genetic algorithm was used in the optimisation process ensuring that the global maximum was obtained.
In both lasers, the output beam mode quality was significantly enhanced: single transverse mode oscillation was obtained without any drop in power for a side-pumped Nd:YAlO laser. In addition, the brightness of the laser output of a grazing-incidence Nd:GdVO4 laser was increased by a factor of 10 at output powers ~15W. Thermally-induced membrane deformations were clearly observed at 1.25kW/cm2, and for higher power lasers (more than 50W output power), the deformation observed will be the limiting factor for successful operation. Moreover, the maximum stroke of the DMM will also limit the range of brightness enhancement. These limitations aside, this work clearly shows that a low-cost intracavity DMM can significantly improve the beam quality of a diode-pumped solid-state laser at power levels around 15W.
In addition, a set of peak-locating algorithms were implemented and tested. Although the genetic algorithm always finds the best solution possible, the time taken and the computer resources required could limit its application. Although the ultimate choice may well be dictated by the user’s needs, we have found that the (simple) random search algorithm is a good compromise between optimisation results and required computer resources in this type of laser application.
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