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We present a closed-loop adaptive optics system based on a holographic sensing method. The system uses a multiplexed holographic recording of the response functions of each actuator in a deformable mirror. By comparing the output intensity measured in a pair of photodiodes, the absolute phase can be measured over each actuator location. From this a feedback correction signal is applied to the input beam without need for a computer. The sensing and correction is applied to each actuator in parallel, so the bandwidth is independent of the number of actuator. We demonstrate a breadboard system using a 32-actuator MEMS deformable mirror capable of operating at over 10kHz without a computer in the loop.
© 2014 Optical Society of America
1. Introduction
Traditional adaptive optics systems use wavefront sensors that require a significant amount of computational overhead [1–3]. This includes the measurements as well as the derivation of phase information from those measurements. Additionally, wavefront phase is usually expressed using some orthogonal basis set such as Zernike polynomials which then has to be translated into particular actuator motions for closed-loop correction. One approach by the author and others [4–12], involves the use of a multiplexed hologram constructed by recording the maximum and minimum expected phase amplitudes of a finite number of Zernike modes. While this method has been shown to work, it suffers from low photometric efficiency as all holograms required for each mode are multiplexed in the same location. Additionally, there is still the large amount of computational overhead in translating the final amplitudes of the Zernike polynomials into individual actuator motions in the phase corrector.
Here we present an alternative solution which uses a multiplexed hologram specially constructed to directly analyze the wavefront error based on actuator response functions. Essentially this modifies the previous holographic adaptive optics approaches by recording “modes” that are based on the actual response functions of each actuator. In operation, the phase information is derived from fast photodetector measurements of stationary reconstructed focal spots. Both the detection, sensing and corrective feedback are achieved with minimal, parallel computations, making it faster than conventional systems while being easily scaled to a large number of actuators without loss of speed.
2. Theory
To best understand the operation of the holographic adaptive optics system it is often instructive to consider the construction of the holographic optical element. For our system we used an optical recording scheme, though with a good knowledge of the response functions of the deformable mirror (DM) a computer generated hologram is an equally viable solution. In our scheme a plane wave is reflected off an actuator in the deformable mirror that has been pushed to its maximum extent to impose some local phase delay (Fig. 1(a)). A hologram is recorded between this object beam and a convergent reference beam focused to some distant point A. Note that the property of a hologram is such that if the initial recording geometry for the actuator and input object beam remained the same, such a hologram would reconstruct a beam focused to this same point A.
Fig. 1 (a) The first recording is made with the actuator pushed to its maximal extent. (b) A second hologram is recorded on the first with the actuator fully pulled and a reference beam focused to B. (c) An input beam with arbitrary phase will reconstruct two focused beams to points A & B. (d) A filled wavefront generates one pair of foci for each actuator.
We now record a second hologram on the first – this time with the actuator pulled to its maximum extent to give the maximum phase delay, and a reference beam focused to a different point B (Fig. 1(b)). If the actuator is now set to some arbitrary value between maximum and minimum phase delay, the object beam created does not match either of the recording cases. As such, the imperfect phase matching results in the reconstruction of two focused beams (Fig. 1(c)). The phase mismatch also imposes a curvature term onto each beam that gives rise to a slight focus to one and defocus to the other. By measuring the power transmitted through pinholes located at points A and B it is possible to derive a simple, direct relationship between focal spot intensity ratio and phase.
The procedure above describes the method for detecting phase over a given actuator location. To create a general wavefront sensor, a separate pair of multiplexed holograms are recorded for each actuator in the same manner. Each new pair, however, can be located in a different spatial location on the holographic medium so as not to reduce the overall efficiency. If we now take an input beam that fills the entire aperture and reflect this off the deformable mirror and on to the hologram, many pairs of focused beams are reconstructed (Fig. 1(d)). Note that the actual spatial arrangement of the focal spots or actuators is not critical but will most likely be determined by the geometry of the detector used.
In essence, where a Shack-Hartmann sensor detects local tilt over a given subaperture, the holographic wavefront sensor is measuring defocus. A key difference, however, is that with global tip/tilt removed, the focal spots remain centered on the pinholes and detectors. The phase is found simply by measuring a response function based on the relative power through each pinhole - something which can be accomplished at bandwidth of 100kHz rates or higher (assuming sufficient light flux) even using inexpensive photodetectors.
As in conventional holography, we can define the recorded hologram by the interference pattern formed from the object beam [13],
whereε is the aberration amplitude, σ is the FWHM associated with the Gaussian-like influence function of the DM actuators and k the wave number. The spherical reference beam E, with an effective focal length f, has the formwith , and the tip and tilt components defined by the angles θx and θy relative to the x and y axes respectively. Note that the reference beam has a propagation vector with a lateral component to that of the object beam essential for the read beam to converge to the correct element of the sensor array. The interference pattern formed from the two beams can be written asThe hologram will have a transmission proportional to the interference intensity, T = βI, where β is a function of the properties of the holographic media. In operation, should the reconstructing object beam, Fin (the beam under investigation) have similar phase components as those reflected from an actuator during the HOE writing (i.e. Fin = O), the resulting beam would beThe first and second terms are just modifications of the object beam which we ignore in this case. The third term is the conjugate of the reference beam, which means that it is diverging in our setup. The last term is the spherical reference beam collected by the detector and proportional to the object beam (used in the writing of the hologram) convolved with the beam under investigation.
Figure 2(a) shows a schematic of the reconstruction geometry for a portion of the beam reflected from a single actuator. With a pair of multiplexed holograms, this single beam will diffract from the hologram to reconstruct two focused beams to two fixed pinholes. In the figures we see the effect of the actuator driven to the two extremes (maximum push and pull) as indicated by the solid and dashed lines respectively. Note that for the desired flat condition, the light transmitted through the pinholes will be equalized, as neither beam will be perfectly in focus.
Fig. 2 (a) A schematic of the reconstruction for a beam reflected off a single actuator, diffracting from the hologram (H) to produce two beams focused on two pinholes (P). The two reconstruction conditions are shown as the actuator is varied from maximum push to pull. A model of the HALOS response function is shown plotted against actuator position (b) and actuator voltage (c).
Rather than using a simple ratio of the pairs of sensor voltages (Va and Vb), we use a fractional energy difference to derive a response function so that E = (Va-Vb)/(Va + Vb). This dimensionless quantity has the advantage that it is invariant under changing input power and largely insensitive to stray background illumination. For an ideal actuator varied from fully pulled to pushed positions, the resultant response function is shown in Fig. 2(b). With the MEMS mirror used in our experiments detailed here, the actuator motion is nonlinear with applied voltage, having an approximately quadratic response. As such, if we plot the modeled response function against voltage we get the asymmetric curve as shown in Fig. 2(c).
The key here is that the response function is always singly determined for a particular actuator voltage. Using this information the actuators can then be manipulated under one-to-one feedback control to correct for any wavefront errors. This avoids the high computational overhead associated with conventional adaptive optics schemes that require translating wavefront phase from one modal basis set to another, so a simple circuit can manage full closed-loop correction up to MHz rates. In principle, since the actuators are controlled in parallel, the system bandwidth is the same for one actuator as it is for thousands or more.
3. Experiment
We constructed a prototype holographic adaptive laser optics system (HALOS) using a 32-element Boston Micromachines Mini-DM. This micro-electromechanical deformable mirror (MEMS-DM) has actuators spaced 300 microns apart on a 6x6 grid (with the corners inactive). This mirror operates by deforming a continuous facesheet by electrostatic attraction and the nominal flat condition is achieved by setting the voltage on all actuators so that their displacement is 50% of their maximum pull. The “push” condition is achieved by reducing the applied voltage to an actuator and allowing the mirror to relax. We achieved 0.36μm stroke of pull and 0.33μm of push (i.e. a total stroke of 0.69μm) using a custom-made driver. Note that the holographic adaptive optics concept cannot be made to work with a segmented mirror as it is insensitive to discontinuities which can lead to a 2π modulus phase shift ambiguity.
We recorded 64 holograms (32 multiplexed pairs) using a cw Nd:YAG laser (λ = 532nm) with the recording and reconstruction layouts as shown in Fig. 3. Motorized stages were used to move an aperture stop and reposition the fiber-fed reference beam to ensure overlap and correct pointing for each exposure. To eliminate diffractive effects, the aperture is imaged onto the DM which is, in turn, imaged onto the plane of the hologram. For larger deformable mirrors, with small distances between the DM and hologram, this may not be necessary.
Fig. 3 (a) Recording. An aperture isolates a particular actuator while the reference beam is formed from a coherent beam directed through a fiber and focusing optics. (b) Replay. The aperture is replaced by a deformable mirror for testing. Light not directed onto the sensor forms the corrected output.
The diameter of the illumination patch for each actuator is approximately 0.5mm (the minimum our variable aperture could achieve). In designing the holographic wavefront sensor it is important to choose an optimum illumination patch isolating the actuator of interest. Too large, and the hologram is unnecessarily recording a redundant extent of flat mirror which will reduce efficiency. Too small, and the hologram will not be recording the full extent of the actuator deformation relative to some global “flat” condition. Our modeling (verified by trial and error) shows that an ideal patch size is 1.2-1.6 times the inter-actuator spacing which is 0.3mm for our DM.
Phase transmission holograms were recorded on dichromated gelatin plate film with 20mW per beam and 50ms exposure per hologram. The angle between recording beams was approximately 20 degrees. Optimizing efficiency was not a primary concern, but we easily managed a total input to output beam diffraction efficiency of 5% - or approximately 100pW per beam for the minimum usable input beam power of 130nW. Here we also note that the system can be adapted to work equally well with transmission, reflection or surface relief holograms, intensity or phase or in any type of medium. There is also the possibility of using holograms recorded at one wavelength for reconstruction at another, thus making it possible for HALOS to operate in the infrared. In such a case, however, allowances have to be made for changes in geometry resulting from dispersion by the grating.
Reconstruction was achieved with a diffraction limited, collimated beam illuminating the entire DM aperture, producing 64 focused spots arranged in a 8x8 grid. We fabricated an array of 350μm pinholes to isolate each focused spot. The light transmitted through each pinhole was collected by 64 elements of a SensL 4p9 avalanche photodiode array. Pairs of these input voltages were then used to generate an error signal for a particular actuator, as defined earlier. By cycling any actuator through its entire range of motion we can measure the response function – an example of which is shown in Fig. 4(a). This data compares well with our model as shown in Fig. 2(c), with the slight difference being due to a minor asymmetry in the mirror shape for the push and pull conditions. Note that both this and the nonlinearity in the actuator response itself pose no limitation to closed-loop correction, so long as the function is singly determined for all actuator voltages across the range of operation.
Fig. 4 (a) The response function measured for a single actuator driven through the full range of motion. (b) The response functions simultaneously recorded for the correct actuator channel, as well as the neighboring channels as the single actuator is driven through its full range.
For an ideal system with no cross-talk, the motion of one actuator will not give rise to a measurable error signal in a neighboring channel. However, since we have a continuous facesheet deformable mirror, the influence function is such that there will always be some inter-actuator cross-talk. Thus, even driving a single actuator in isolation, a change in the response function will not only be measured for the correct channel, but for the neighboring (stationary) elements as well. We measured this cross-talk for one actuator as shown in Fig. 4(b), where we see that moving one actuator through its entire range results in a minimal change in the response function measured in the neighboring 8 actuators.
Note that each actuator channel has its own unique offset which can be any value (so long as there is sufficient dynamic range accommodated by the digitizer) as it is simply recorded as the target value in the calibration. Note that cross-talk is typically small and it does not prevent the system from closing the loop for diffraction limited correction. It may slow the system down to a minor extent initially (as a few more loop cycles may be required to achieve the zero phase target wavefront error), but since the holographic sensing is still orders of magnitude faster than conventional systems, the effect is rarely noticeable.
HALOS uses inexpensive COTS electronics (four XMOS XS1-L8A-64 processors) for sensing and closed-loop control. The processor power consumption is less than 2.2W and closed-loop latency is 8 microseconds. While this is an order of magnitude better than the best adaptive optics systems, it could be further improved by more than 2 orders of magnitude with more exotic components such as FPGAs or dedicated ASICs. An image of the HALOS set-up is in Fig. 5, along with a magnified image of the hologram lit by white light which clearly shows the square print-through of the actuator pads along with the circular illumination patches. The only required equipment not shown in Fig. 5(a) is the driver and power supply.
Fig. 5 (a) An image of HALOS showing the DM (top center), hologram (right) and sensor (lower left) which was uncovered in this photo to show the electronics. (b) An image of the hologram. Note the slight overlap between pairs of holograms (with their circular profile) centered on each actuator location.
The system requires a one-off calibration where a diffraction-limited plane wave is introduced onto the deformable mirror which is set to its flattened condition. The response function values for each actuator are then recorded and stored in the microprocessor as target values. In closed-loop operation, the microprocessor compares the measured response functions at any particular moment to the target values, and from this generates a control signal to drive each actuator to make the appropriate correction. For example, if the target value for the response function is zero, and the measured value is negative, the actuator voltage is increased to obtain a corresponding increase in the response function. The step size of the adjustments can be varied by changing a gain factor, which is adjusted according to the desired sensitivity and response time. A more advanced system could also include a full characterization of each individual response function curve so that faster, more precise correction could be applied in a single step. In most cases, however, this is likely an unnecessary complication, as the moment-to-moment adjustments are quite small.
The above procedure details the case for obtaining a diffraction-limited plane wave output but the system could be equally calibrated to any desired output wavefront shape in the same manner. The target values can even be altered at any time through a GUI if there are any changes required to the final wavefront shape. Also, while a computer is used for the initial upload of the control program and storage of the one-off calibration target values, the closed-loop control is managed entirely by the processor not a computer. In our current prototype the computer connection is used only for realtime monitoring of the system performance.
Finally, the conventional HALOS configuration (and the one used for experiments described here) uses a beamsplitter to direct part of the beam onto the hologram, while the other portion is the corrected output. However, we can also use a modified design in which the beamsplitter is replaced by a mirror, with all the light directed onto the hologram. In that configuration the undiffracted (zero order) light passing through the hologram can form the output port. This arrangement has the benefit of improving the efficiency at the expense of some dynamic range. The reason is that the hologram may introduce some minor residual aberration to the transmitted beam. In that case, the calibration will require the incorporation of an offset aberration into the deformable mirror to ensure a plane wave output – i.e. instead of the target mirror shape being perfectly flat it will have some initial distortion.
4. Results
Upstream of the holographic adaptive optics system we added a second MEMS-DM, programmed to introduce a random, time-varying wavefront error. The deformable mirror was located at the same plane as the aperture used for the original recording and this “aberrator” could be used for testing the system performance. Global tip/tilt is not introduced by the aberrator DM which is essentially a thin distorter, so there is no need for beam-steering pointing correction. The input beam intensity was adjusted such that the power transmitted through each of the pinholes varied between approximately 50nW to 1μW over the full stroke (well within the sensor sensitivity range of 20nW to 1.4μW).
Our first test was to measure the minimum update rate for the entire system, which involved moving an actuator on the aberrator DM and observing the time taken for the system to sense then move the actuator on the corrector DM. We measured this to be 1ms for the full stroke and faster (<100 μsecs) for correcting minor phase fluctuations. It is important to note that this 10kHz bandwidth is simply a result of inertial limitations in the deformable mirror. We measured the actual electronics full loop latency of just 10 microseconds; the minimum time required to make detect a change in phase, make a measurement of the magnitude of this change (using the response function) and send a command to initiate the corrective motion to the deformable mirror. With a sufficiently fast deformable mirror then, we can infer a maximum possible bandwidth of 100kHz.
Next, a time-varying aberration was introduced over the entire aperture, generated by adding a small random stroke to each actuator in the aberrator DM. The corrected output beam could then be tested to evaluate the correction capability. These results are shown in the form of point spread function (Figs. 6(a), 6(b), 6(c), and 6(d)) and wavefront (interferometric) analyses (Figs. 6(c) and 6(d)). Video data is provided as supplemental documentation. An analysis of the fringes in Figs. 6(e) and 6(f) show that 0.43 waves RMS of aberration (Strehl ratio < 0.01) is corrected to 0.16 waves RMS (SR = 0.35). The residual error is limited to the minimum measurable change in the response function as well as the extent of the cross-talk. While the latter is fixed by the nature of the deformable mirror itself, the final wavefront error of our system could be improved by increasing the electronics resolution from its current value of 12 bits as well as decreasing the noise in the sensor, feedback circuit and DM driver.
Fig. 6 Point spread function before (a) and after (b) correction. Surface plots generated from these images are shown in (c) and (d), along with a video (Media 1). Wavefront interferometry before (e) and after (f) correction is shown, along with video (Media 2). The red border indicates the extent of the deformable mirror.
The operation and performance remained unchanged as the room lights were turned on and off. This is partly due to shielding of the sensor but mostly as a direct result of the use of fractional energy difference calculation of the response function which naturally factors out the contribution of a common background signal. Additionally, there is check in the software to identify if the denominator in the response function is zero (most likely because of an obscuration of the beam). In such a case, the actuator is set to its flat condition – neither pushed nor pulled.
5. Conclusion
We have introduced a holographic adaptive optics system that uses a multiplexed hologram to divide an incoming beam into many focused beams. A measure of the ratio of pairs of spot intensities can be directly related to the absolute position of an actuator in the system’s deformable mirror. With a simple circuit, parallel feedback control over individual actuators can be achieved at high speeds. The system has the advantage of being compact, autonomous (computer-free) and scalable to orders more actuators without loss of bandwidth. We have constructed a prototype system consisting of 32 actuators and shown it capable of correcting wavefronts to bandwidths of 10kHz.
Acknowledgments
We wish to acknowledge the support for this project from the United States Air Force Academy, the Air Force Office of Scientific Research as well as the High Energy Laser Joint Technology Office.
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