1. Introduction

The current generation of astronomical adaptive optics (AO) systems have ~10²–10³ actuators and results on large telescopes have largely been confined to the infrared, where atmospheric turbulence is relatively benign. Much higher order wavefront correction, known as extreme adaptive optics (XAO) is needed for applications ranging from visible correction on 8m class telescopes and exoplanet detection, up to wavefront correction on future extremely large telescopes. The number of required actuators can be as large as 10⁵–10⁶. There are no currently available wavefront corrections technologies with such specifications, but likely candidates (for systems with a small field of view) are MEMS and liquid crystal (LC) devices.

In an AO system it is advantageous for the control if the wavefront sensor measures the typical wavefront deformations produced by the wavefront corrector, e.g. a bimorph mirror is matched well to a curvature sensor and a segmented deformable mirror is matched well to a Shack Hartmann wavefront sensor. Some MEMS and LC devices typically have piston-only influence functions, and therefore a wavefront sensor which measures phase directly, such as an interferometer is desirable. This has the advantage that it will be possible to produce a “reconstructor-free” control system, whereby the control of each actuator is independent of the other actuators, and therefore the complexity of the control will scale linearly with the number of actuators. Such systems were proposed, in the context of XAO some time ago¹, although there is recent revised interest^2–5 in them as practical advances towards XAO are beginning to be made.

This letter has two aims. First we describe an alternative configuration of a common-path point diffraction interferometer (PDI) which gives simultaneously two phase-shifted fringe patterns, and which also allows for calibration of scintillation. The interferometric arrangements based in refs. 2–4 are based on a Mach-Zhender configuration, and therefore do not have the stability advantages associated with common-path operation described here. LCs have previously been used to produce phase-shifting PDIs^6,7. Here we just use the fact that they are birefringent to produce two phase-shifted fringe patterns simultaneously using incident unpolarized light (and therefore without wasting any light by the use of a polarizer as is often the case when LCs are used). The second aim is to describe the use of the PDI to control a high-speed dual frequency LC cell to produce a single-channel laboratory AO system. The key point is that this system is linearly scalable in terms of the number of actuators.

2. The liquid crystal point diffraction interferometer

The key element in the wavefront sensor is a ferroelectric liquid crystal (FLC) cell (which acts as an electrically rotatable half-wave plate (HWP)) that has small circle, 20µm in diameter, removed from one of the control electrodes. Phase-only modulation with the FLC is achieved⁸ by placing the device between two co-aligned quarter-wave plates (QWPs), as shown in figure 1.

Light which is incident linearly polarized at 45° to the QWP’s crystal axes is phase modulated by an amount 2θ, where θ is the switching angle of the FLC (which is normally twice the tilt angle). When the FLC is turned on, it acts like a phase plate with a small phase discontinuity, where there is no electrode in the center, of magnitude 2θ=90° (since the FLC switching angle is 45°). An orthogonally polarized component experiences a similar phase plate, but with a phase shift of -2θ=-90°. If the wavefront sensor is therefore used with unpolarized light, which can be considered as an incoherent sum of two orthogonally polarized linear states, then two, orthogonally polarized, fringe patterns will be produced. These can be separated by a polarizing beam splitter (the device could also be used in non-astronomical situations using linearly polarized light of the azimuth was set appropriately). The resultant fringe patterns are therefore differing in phase by 180°. When the FLC is in the off state, then the molecules throughout the whole cell are aligned, and no interferogram is observed. The device acts, effectively, like an isotropic plate.

 

Fig. 1. Concept for a common path point diffraction interferometer based on a ferroelectric liquid crystal (FLC) phase shifter. The FLC is essentially a switchable half-wave plate placed between two quarter-wave plates (QWP). The devices works with unpolarized input light to give two simultaneous interferograms which are orthogonally polarized, phase shifted by half a wave, and which are then separated by a polarizing beamsplitter.

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The observed interferograms will be of the form,

where I_A and I_B are the two observed intensity patterns (which will be a function of the position, (x,y)), I₀ is the normalization intensity, γ is the fringe visibility, ϕ(x,y)is the phase of the wavefront, and ϕ̄ is the average phase across the wavefront. This latter term arises because in this kind of PDI the reference wavefronts are produced directly from the test wavefronts. The final $\pm \raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ arises from the phase shifter for each of the outputs, A and B. If the wavefront sensor was to be used for metrology, then it would be usual to record at least 3 fringe patterns in order to determine the three unknowns; I₀, γ, and ϕ(x,y). ϕ̄ is not needed because constant phase shifts over the whole beam can usually be ignored. When controlling an AO system an output which is proportional to the phase, instead of equal to the exact value, is required and therefore fewer interferograms are needed. We therefore calculate, using approximations, a value for the measured phase, ϕ_m, using a number of approximations as described below.

Expanding the cosine term in the above equation gives,

and therefore the required measured phase, ϕ_m≡ϕ(x,y)-ϕ̄ is given by,

For small values of ϕ_m then the approximation sinϕ_m≈ϕ_m can be used, and assuming unit visibility fringes, then,

 

Fig. 2. Simulation results of the residual rms wavefront error from measurements of aberrated wavefronts. The test wavefronts were pure Zernike modes, as indicated on the x-axis and the amplitude of the applied mode is shown on the y-axis. It can be seen that measurement quality decreases with increasing amplitude, as is expected.

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A noiseless computer simulation was produced of the system and aberrated test wavefronts were generated using Zernike polynomials. The wavefronts were reconstructed using the above approximation for ϕ_m and compared to the original wavefronts. The results are shown in Fig. 2, which shows the residual rms wavefront error plotted versus mode number and the amplitude of the mode. The simulation demonstrates the utility of the technique for small amplitude aberrations.

The interferometer also has a “null output” which can be used to measure the intensity of the beam when the phase shifter is not activated. Of course, this could also be achieved in any of the systems described in refs. 2–4, simply by placing a beamsplitter before the wavefront sensor and imaging the pupil onto another detector. However, the system proposed here has the advantage that there are no non-common path errors due to the differences in optics or detectors between the wavefront sensor and the intensity detector. The computer simulation was extended by multiplying the amplitude of the incident wavefronts by a uniformly distributed noise factor, U, varying from 0.5 to 1 (for simplicity). The phase was then reconstructed firstly using equation 1., which ignores the effect of scintillation. Next equation 1. was modified by subtracting the measured random intensity, U, reference frame from each term so that the equation becomes,

${\phi }_m=\frac{\left(I_B-U\right)-\left(I_A-U\right)}{\left(I_B-U\right)+\left(I_A-U\right)}=\frac{I_B-I_A}{I_B+I_A-2U}.$

The results are shown in Fig. 3. for the 8^th Zernike mode (which here is second-order astigmatism). The reconstructed phase was subtracted from the original phase to give the residual error, and the figure shows a slice through this data. The numerical residual errors are shown in Table 1.

 

Fig. 3. Simulation results showing the effect of calibration for scintillation. The red line shows a scan through the residual phase error if no attempt is made to correct for scintillation. The phase was reconstructed using equation 1. The blue line shows the residual phase error when the phase was reconstructed using equation 2.

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Table 1. Simulated residual wavefront errors showing that errors can be reduced by accounting for the scintillation by also measuring the pupil intensity. In the system described here this can be done without non-common path errors by recording data when the FLC is off.

View Table

The description so far, and the simulations, have assumed the use of monochromatic light. In general, broadband light must be used. The QHQ is similar in design to a broadband waveplate (which are made from a series of individual wavplates) and its performance is approximately achromatic⁹. However, how the precise affect of this on the total system performance awaits a full closed loop broadband simulation.

3. Closed loop high speed single channel system

In order to demonstrate the wavefront sensor in closed loop operation a single channel system was constructed as shown in Fig. 4.

 

Fig. 4. Diagram of the closed loop system. Unpolarized light from the laser passes through the turbulence generator, followed by the LC wavefront corrector. The wavefront sensor, formed by the two quarter-wave plates (QWPs) and the FLC half-wave plate (HWP) followed by the polarizing beamsplitter, produces 2 signals on the photodiodes. Pinholes before the photodiodes ensure that the correct pixel on the wavefront corrector is being imaged onto the detectors. The differential signal from the photodiodes (which is the wavefront sensor output) is then compared with a reference DC signal in a comparator, which is used to control an analogue switch which supplies the wavefront corrector with a high voltage low or high frequency voltage.

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A single channel system is obviously not a complete AO system, however it demonstrates the principle of the system, since each channel essentially operates independently of the others. A collimated HeNe (633nm) laser beam passed through a turbulence generator, which was a heat-gun (similar to a hair dryer). The wavefront corrector was a Meadlowlark Optics¹⁰ nematic liquid crystal wavefront corrector¹¹ which had 127 phase-only correction elements. The device operates with unpolarized light by using a QWP between the LC and a mirror¹². The device also contains high speed dual frequency LC material and so it does not suffer from the usual slow response time associated with LCs. Previous work has demonstrated the use of such devices in a closed loop system using a Shack-Hartman wavefront sensor¹³. As the authors of this work point out in their papers, this is not an ideal wavefront sensor for a piston-only wavefront corrector. Here, the output of the wavefront sensor is used in a control system specially designed for operation with dual frequency LCs^14,15. The light then passes to the wavefront sensor, as described in the previous section. The optics were arranged so that just one pixel in the wavefront corrector was imaged onto a pinhole in front of a photodiode in each of the detector arms, in order to give the two required signals, I_A and I_B. A ‘bang-bang’ type control system was implemented. The two signals from the photodetectors are fed to a differential amplifier in order to give a signal proportional to I_A - I_B, denoted as V_det in Fig. 4. The signal V_det is then compared to a set-point voltage, V_ref, using a comparator which operates an analogue switch to controls the signal fed, via a high-voltage (HV) amplifier, to the LC wavefront corrector. If V_det > V_ref then a high frequency (~30KHz) drive signal is selected. If V_det < V_ref then a low frequency (~1KHz) drive signal is selected. Since the comparator used in this implementation had no hysteresis, the operation of this system is such that, if the measured phase differs in either direction from the set-point, then the wavefront corrector applies the maximum compensating signal. This means that the dual frequency LC in the wavefront corrector is always either turning on or turning off.

First, the loop was not closed, the turbulence generator turned off, and the LC wavefront corrector was set to oscillate between phase values of 0 and 2π, and the output of the wavefront sensor (after the differential amplifier) was measured, and is shown in Fig. 5.

 

Fig. 5. Wavefront sensor output, measured after the differential amplifier (in fig. 4.), with the loop not closed, and an alternating signal of 1 wave being applied to the wavefront corrector.

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Next the turbulence generator was turned on, and the loop closed. The reference DC signal was adjusted to optimize the compensation. Results of the output of the wavefront sensor signal (again after the differential amplifier) are shown in Fig. 6. The results have been converted to waves, using the data in Fig. 5 as calibration. It can be seen that the phase variance has been considerably reduced (from 2.44 to 0.40 rad²). The performance of the system is likely limited by the small number of correction pixels: since the turbulence is strong it is unlikely that the phase is constant over to single correction pixel.

The power spectra of these data are shown in Fig. 7. The spectra were smoothed to make the general trend clearer. It can be seen that the crossover frequency, when the system begins to add noise is about 700Hz.

 

Fig. 6. Calibrated wavefront sensor output without (left) and with (right) the loop closed. The turbulence was severe (from a hot air blower). The uncorrected and corrected phase variances are 2.44 rad² 0.40 rad² respectively.

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The noise performance of the system could be improved by potentially including hysteresis in the comparator stage of the control system. In principle, the system could also be improved by adopting a proportional control, rather than a bang-bang type, although the latter was chosen because it particularly lends itself to the control of dual frequency LCs.

 

Fig. 7. Power spectrum of the wavefront sensor output with the loop not closed (red line) and closed (blue line). The system is reducing wavefront aberrations up to a crossover frequency of about 700Hz.

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4. Discussion

An alternative form of point diffraction interferometer has been described, which has the advantage of simultaneously giving phase shifted outputs and common path operation, as well as having an option for reducing the effects of scintillation on the measurements. High speed closed loop operation of a single channel AO system using the wavefront sensor has been demonstrated. This type of system has the advantage that the number of channels can be scaled linearly and therefore is a candidate technique use in a XAO system. As in all interferometric wavefront sensors of this type, a critical issue is ensuring that the intensity of the reference and test beams are equal when the turbulence is strong (before the loop is closed). LC devices may also have a role to play here as it would be possible to make a sensor with an electronically controllable diffracting spot size. A related issue is tracking is as light must pass through the reference hole in the FLC device. When the aberrations are large, the pattern on the FLC may be highly speckled. Normal centroid tracking may place a “dark” spot on the reference hole. Future work should involve a full simulation of such a closed loop system.