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We present a programmable beam-shaping method based on the combination of a deformable mirror membrane mirror and a simulated annealing algorithm. The algorithm iteratively adjusts the control voltages of 37 independent electrodes to reduce the variance between the chosen shape and the actual beam shape. The experimental results show that the system is capable of adaptively creating, on demand, Gaussian and super-Gaussian beam profiles that closely match the desired target parameters.
©2005 Optical Society of America
1. Introduction
Electrostatic deformable mirrors (DMs) are amongst the most commonly used adaptive optics for beam shaping and for the correction of wavefront aberrations. DMs are a popular choice for laser beam shaping because of their reflective nature and broad bandwidth, which distinguishes them from competing devices such as liquid-crystal modulators (LCMs) [1]. Unlike LCMs, deformable mirrors are not naturally programmable devices that can give arbitrary shapes. The deformation is linear in applied force but the force varies with the square of the applied voltage and inversely with the membrane-electrode separation. Although open-loop wavefront or beam-shape control can in principle be achieved using knowledge of a DM’s influence matrix [2], its nonlinear characteristics make implementation of a wide range of predeformed shapes problematic without the use of feedback through a wavefront sensor or some other form of closed-loop control. The approaches used to date can be separated into those which make full use of an exact measurement of the incident wavefront using, for example, a Shack-Hartmann wavefront sensor [3], or those which employ an iterative strategy to optimise the mirror surface based on a single experimentally measurable variable, for example the transmission of the focused beam through an optical fiber [4]. Compared with exact methods, the latter category of beam shaper (the so-called “power-in-the-bucket” method) is characterised by its ease of implementation but is commonly limited to correcting the beam for a flat wavefront because this normally corresponds to optimal focusing through a pinhole. In this paper we demonstrate a modification of this familiar strategy which allows, within the mechanical constraints of the deformable mirror, the creation of arbitrary beam profiles focused at a chosen plane. This approach is motivated by the fact that in some applications, for example laser machining, it is the prescription of the shape of the beam-profile focused at a target plane, rather than the output wavefront, that is the ultimate experimental objective.
The simulated-annealing algorithm [5] is one of a group of stochastic optimization algorithms which is well-suited to finding a global minimum (or maximum) of some objective error function. In the context of adaptive-optical control, the simulated annealing algorithm is well-suited to the task because of its ability to independently optimise many variables at once. In the case of a deformable mirror, each actuator voltage represents one independently adjustable variable. All stochastic algorithms work by assessing the quality of any proposed solution by defining an error function - normally a single number - whose value indicates how close any solution is to the target. In this way, deformable mirrors have been used together with evolutionary algorithms to compensate for the effects of group-delay dispersion on ultrashort laser pulses (using a one-dimensional deformable mirror) [6–7] and to optimise the laser output power from a solid-state laser in the presence of a thermal lens [8]. In both of the examples given here, the optical measurement was made using a simple point detector. In the first cases the detector was a frequency-doubling crystal followed by a photodiode and in the second case it was a physical pinhole and photodiode or a variable synthetic pinhole achieved using a CCD camera. Point detectors are a natural choice because they yield a single scalar output that can be used directly as a measure of the fitness of the solution. In our present work we have used a camera rather than a point detector to record the exact beam profile focused by a lens and have derived an error value by taking the root-mean square (RMS) difference between the beam intensity profile measured by a camera and a stored two-dimensional target profile.
2. Experimental configuration
The adaptive optical system is shown in Fig. 1 and consisted of a micro-machined deformable mirror fabricated by OKO technology [9]. The 15 mm diameter mirror comprised a Au-coated reflective membrane supported by 37 hexagonal actuators arranged in hexagonal pattern. The mirror was computer controlled directly from MATLAB and was driven with a custom-designed 12-bit multi-channel high-voltage digital-to-analogue interface circuit. This degree of precision is necessary because the deflection produced in electrostatic-based deformable mirrors varies quadratic ally with the applied voltage and therefore leads to coarse steps in the deflection at large applied voltages if only 8-bit resolution is used to drive the mirror. The maximum actuator voltage was limited to 225 V which is close the maximum permitted voltage on the mirror actuators. At maximum voltage the deflection of the mirror surface was specified by the manufacturer to be 7 μm.
Fig. 1. Experimental configuration of the beam-shaping system showing the feedback loop formed by the deformable mirror, the camera and the control computer.
The laser source for the experiments was a low-power He-Ne laser whose beam was expanded and collimated using a 1:12 telescope before being incident on the deformable mirror. After the mirror the beam was focused by a 250mm plano-convex lens to form a spot on the surface of a CCD camera [10]. The camera was directly interfaced using a IEEE 1394 (Firewire) connection and was addressed under MATLAB control using the same program that also controlled the mirror actuator voltages. The MATLAB program was capable of monitoring and processing images from the camera at a typical rate of 10 Hz. The control system used the camera measurements to adjust the surface of the deformable mirror to produce a target beam profile and the following section presents details of the feedback algorithm used.
3. Simulated annealing control algorithm
The adaptive mirror was controlled using a simulated annealing algorithm that formed part of a closed control loop between the beam profile recorded on the camera and the set of voltages written to the mirror actuators. The control process is depicted by the flowchart in Fig. 2. The mirror voltages were represented by a 37-element vector, v = [v 1, v 2,…,v 37], which was initially defined with every voltage set at some constant and mid-range value, typically around 100V. At the beginning of the procedure a target beam profile was defined according to,
where a and b determine the beam width in directions x and y, p and q are integers that specify the steepness of the beam sides (the degree to which the beam is represented by a super-Gaussian) and xo and yo are the pixel co-ordinates of the target beam centre.
The algorithm began by acquiring the camera image, I camera of the focused spot that was produced by the initial voltages. The background of the acquired camera image was removed by averaging the pixel intensity in the four corner regions of the image then subtracting this value from the entire image data.
The beam centre position within the camera image, (xo, yo), was found by calculating its centre of mass using a fifth-order moment method. Using this information about the beam centre, the target function was then re-centred on this point and the RMS error between the camera and target beam profiles was calculated according to:
In the above calculation the target and camera images were both normalised to have the same maximum pixel intensity. For the purposes of efficiency, only the small region of the camera data that contained the focused spot image was compared with the target image (approximately a 50 × 50 pixel field). At this point the simulated annealing strategy begins by testing if the change in the error is higher than an acceptable value according to,
where P is the probability that the proposed set of voltages will be accepted and T is the algorithm “temperature” which is slowly reduced with each iteration of the algorithm. By only accepting voltage sets, v, that resulted in P > 0.5, the simulated-annealing strategy was used to minimize the RMS error between the target and the actual shape. On each iteration, v was randomly perturbed by up to 10 % and the new voltages accepted or rejected on the basis of the implied change to the error. During the process the system ran freely, and continuously updated and stored the set of voltages and the camera image giving the closest match to the target shape. Depending on the type of target profile demanded, the algorithm converged to give a beam that matched the target in between 100 – 2000 iterations (typically ~ 5 – 100 minutes).
4. Experimental results and discussion
The beam-shaping system was tested for two sets of target profiles, the first comprising radially-symmetric Gaussian beams of three different diameters and the second taking the form of three super-Gaussian profiled beams, each with different diameters. Table 1 lists the parameters of the target profiles used in all six cases.
Table 1. Parameters of the Gaussian and super-Gaussian target beams focused at the camera plane
Fig. 3. Beam shaping results based on three different Gaussian target profiles (see Table 1). Row 1: target profiles; Row 2: experimentally obtained beam profiles; Row 3: horizontal cross-section through the centre of the camera image (solid line) and comparison with a cross-section through the target profile (symbols); Row 4: evolution of the RMS error during algorithm execution.
Figure 3 shows the results obtained when the algorithm was used to create a Gaussian-profiled beam of a specific diameter. The top row corresponds to the three Gaussian target profiles and the second row shows the actual beam profiles obtained after the beam-shaping algorithm had been executed. The third row compares a horizontal cross-section through the centre of the camera image (solid line) with a cross-section through the target (symbols) and the fourth row shows the evolution of the RMS error as the algorithm proceeded. For comparison, Fig. 4 shows the actual beam shape captured by the CCD camera before the algorithm was run and this corresponds to the beam produced by the initial set of uniform actuator voltages. Figure 5 shows a similar set of results for the super-Gaussian target beam.
Fig. 4. Measured beam profile produced by the initial voltage set and used as the starting condition in all cases
Fig. 5. Beam shaping results based on three different super-Gaussian target profiles (see Table 1). For layout description, see Fig. 3 caption.
In all cases good agreement was achieved between the target and the experimentally measured beam shape, despite the target in some cases being defined by only a small number of non-zero pixels. The number of iterations required to achieve good convergence depended on the size of the target spot but in many cases there was only limited improvement after 100 iterations. We generally observed that smaller beam shapes were less demanding to produce, resulting in smaller RMS values and being obtained in a fewer number of iterations than larger spot sizes. We attribute this to the geometrical configuration of the camera and focusing lens which were separated by one focal length and therefore enabled a small spot to be obtained by using a relatively undistorted mirror surface. For larger spot sizes the mirror surface had to be significantly deformed in order to produce the required beam diameters on the camera surface. The comparison between the Gaussian and super-Gaussian beam shapes indicated that, within the limits of diffraction and the possible mirror deformations, it was possible to produce beams whose shapes and cross-sectional intensity profiles matched the target data, eg. sharply falling sides were more evident in the beams shaped using a super-Gaussian target.
5. Conclusion
In summary, we have demonstrated a new and simple method for beam shaping based on the combination of a deformable membrane mirror and the simulated annealing algorithm. The technique is appropriate for the correction of aberrations where the quality of the result is measured in terms of the beam profile on a target, rather than in terms of the wavefront quality. The method will be particularly relevant to the correction of aberrations in beams used for laser machining and we intend to apply the technique in this area in the near future.
Acknowledgments
The authors are grateful to the Leverhulme Trust for supporting this work.
References and links
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