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A laboratory demonstration of an adaptive optics system using a point-diffraction interferometer and a continuous MEMs mirror is presented. The dynamic performance of the system was investigated experimentally using a holographic optical aberration generator. The system was tested both in conditions corresponding to weak phase only aberrations and for horizontal propagation through uniform turbulence giving scintillation and optical vortices. The system was shown to work well in weak turbulence and gave correction for the strong turbulence regime up to the highest scintillation strength tested, σ R 2 = 3.3.
©2007 Optical Society of America
1. Introduction
The propagation of light through atmospheric turbulence results in random phase and amplitude fluctuations in the wave [1, 2, 3]. The phase fluctuations severely reduce the resolution of ground based imaging systems [4]. As the propagation path or turbulence strength increases strong scintillation develops and vortices appear in the phase [5]. This is especially problematic for atmospheric laser beam propagation applications such as line of sight communications [6, 7]. Adaptive optics (AO) has been successful in correcting for these dynamic aberrations when the amplitude fluctuations are small [3]. However strong turbulence and extended paths give rise to optical vortices, which are extremely detrimental to the operation of AO systems based on gradient wavefront sensors and conventional least-squares reconstructors [6, 8, 9, 10, 11]. To avoid the reconstruction problem associated with gradient wavefront sensors in strong scintillation, wavefront sensors based on self-referencing interferometers (SRIs) have been proposed as a promising alternative [12, 13, 9, 14, 15, 16]. SRIs measure the local phase differences from differential intensity measurements making them much less sensitive to the presence of vortices (by avoiding the gradient reconstruction problem), and scintillation across the aperture [9]. This paper demonstrates a laboratory adaptive optics systems using a point-diffraction interferometer (PDI) [17, 18, 19, 20, 21, 22, 23] with a local quadrature re-constructor and a continuous 140 actuator MEMs mirror for correction in both weak and strong turbulence conditions.
2. System description
Fig. 1. Optical layout of the closed-loop AO system. Key: L1-15, lenses; PBS,
polarizing beam-splitters; BS, non-polarizing beam-splitters; M1-8, mirrors
and QW1-3, quarter-wave plates. The line colours in the quadrature PDI
depict polarization (
Figure 1 shows the optical layout of the laboratory adaptive optics test bed.
The wavefront sensor (WFS) is a phase-shifting PDI. The PDI is built around a Mach-Zehnder geometry with a pinhole in one arm to generate a reference wave by spatially filtering in the input [17]. For this system a pinhole with a diameter of 1.6λ/D was used, where f is the focal length of the lens L10, λ is the wavelength, and D is the diameter of the input aperture. A polarizing beam splitter at the input is used to produce orthogonally polarized object and reference beams. The phase-shifting PDI uses further polarizing components to acquire four π/2 phase-stepped interferograms simultaneously side by side on a single image sensor. The operation of the polarization phase-stepper is described in detail in reference [24]. The four interference patterns were recorded on an 8 bit, 512x640 Pixelfly CCD camera operated using 2x2 hardware binning giving a total resolution of 256x320 pixels. The spacing between the mirror actuators re-imaged on the WFS camera was approximately 10 pixels. Each wavefront sensor signal was then taken as the average intensity over a 5x5 pixel array centred on the mirror actuator coordinates.
Dynamic optical aberrations in the pupil plane were generated using a 512x512 pixel ferroelectric spatial light modulator operating as a first-order diffractive element [25]. Lee encoded binary holograms were used that were capable of generating optical aberrations with both phase and amplitude fluctuations [26]. Through simulations using a Fourier reconstruction of generated holograms it was found that encoding errors resulted in an average noise to signal ratio of less than 0.02, and a RMS phase error of 0.05 for amplitude and phase fluctuations corresponding to atmospheric propagation with scintillation strengths characterized by the Rytov number σ R 2 = 3.3, and an aperture size , where L is the propagation distance, and λ the wavelength. The noise is considered to be the component of the generated wave that is orthogonal to the original wave. Note that the largest aperture size used in these experiments was D = 4√λL, so this generation technique should give a good representation of the desired optical aberrations.
The reconstruction algorithm for four phase-shifted interference patterns is given by [27]
where (x,y) are the coordinates, and δϕ is the phase difference between the reference and object waves. An advantage of this reconstructor is that it is local: computationally, it scales linearly with the number of phase sample points N, in contrast to conventional linear least-squares reconstructors (from gradients) that scale as N 2, a fact that is important for high-order systems.
Open-loop systems based on a stationary PDI are sensitive to wavefront tilts that result in decreased visibility in the interference pattern [14]. This PDI was designed for closed-loop operation where the beam propagating through the PDI is assumed to be partially, or fully, corrected so should not suffer from this problem once closed-loop is achieved. The mirror used for correction was a square, 3.3x3.3mm, 140 actuator continuous face-sheet MEMs spatial light modulator manufactured by Boston Micromachines [28].
3. Closed-loop control system
The control matrix C defines a linear mapping of the vector of wavefront signals s onto a vector of control signals x and can be written as
An important property of a local reconstructor is that since it includes no cross terms between different actuators, phase wraps between actuators have no impact on the reconstruction. Therefore we choose to use a diagonal control matrix C with
where gmn = dϕn/dxm is the response of the nth sensor channel to the mth actuator control signal. The response of the wavefront sensor signal channels to the corresponding actuators are shown in Fig. 2. In fact there are small coupling terms between adjacent actuators in the forward system matrix, but these are not included in the reconstructor in order to keep the reconstructor local. Note a number of gaps in the response plots that corresponded to unresponsive actuators within the pupil for our particular mirror. This inevitably degrades the correction ability of the mirror, however, since the unresponsive actuators make up about one tenth of the pupil, the overall impact is expected to be small.
The system is run in closed-loop using a simple integration control. For the nth iteration the mirror control signals are
where α is an integrator factor, chosen to ensure stability in the presence of latencies in the different system components (image capture, reconstruction, and mirror response). For the results presented here α = 0.5 was used.
The WFS uses a local reconstructor and returns phase differences wrapped in the interval [-π, π] which are used to control the mirror directly without unwrapping. Although it is simple to see how this might work naturally with a segmented mirror—the mirror is able to implement sharp phase wraps between adjacent segments—it is less obvious for a smoothly varying continuous surface mirror, where any phase wraps will not be sharp. However in the presence of vortices it may prove useful to be able to introduce wraps onto the mirror.
Fig. 2. Plots of the response of the wavefront sensor signal channels to the corresponding actuators.
Fig. 3. Control wrapping schemes. Left, method 1; signals wrapped between -π and π. Right, method 2; using full stroke and wrapping by 2nπ where n = 1, or 2, or 3,...
The stroke of the mirror is larger than a single wrapped phase cycle and several different control signals can give rise to the same wrapped phase at the mirror surface. The mirror control is therefore not uniquely defined by the wavefront sensor signal. We consider two possibilities for wrapping the control signals (illustrated in Fig. 3):
- Confine the control signals to the interval [-π,π], which corresponds to limiting the stroke of the mirror to a range of 2π.
- Allow the full stroke of the mirror to be used and wrap the control signals (by adding or subtracting 2π) only when they would otherwise move outside the stroke of the mirror.
The first method has a single threshold and is susceptible to noise. A noisy input signal hovering around ±π would lead to the mirror jumping between its two extreme positions to the detriment of the correction. This problem is reduced with the second method which operates in a fashion analogous to the Schmitt trigger in electronics: the control signal thresholds are different for positive and negative phase wraps, making the system less sensitive to dithering on noise. A dual threshold also implies the system has some memory, i.e. hysteresis, in that the shape of the mirror is dependent on the correction history.
Fig. 4. Illustration of two possible mirror solutions to a plane wave, (a) and (b).
A possible consequence of allowing the full mirror stroke with a wrapped phase WFS is the presence of so called ‘waffle modes’—modes of the mirror that the wavefront sensor cannot detect, but which nevertheless have an impact on the correction (see Fig. 4). Note however, that these are quite different to the waffle modes in a linear adaptive optics system. In contrast to those, the unsensed mirror modes due to wrapping are “quantized” in the sense that only discrete 2nπ phase jumps are possible. Although we note the possibility for such modes in the system, for the results presented here, no attempt has been made to deal with waffle modes.
Fig. 5. On-axis intensity v. time for the AO system using wrapping schemes 1 and 2 (-π to π, and full stroke ±2π) for MEMs mirror on a static aberration (D/r 0 = 6). τ 0/τS = 1/10, and gain= 0.5.
A preliminary investigation was conducted for the two different wrapping schemes on a static aberration. The results of the on-axis intensity in the far-field against time for one run are shown in Fig. 5. Wrapping method (2) gives a higher on-axis intensity and more stable solution.
4. Adaptive optics system performance
4.1. Weak turbulence approximation - uniform intensity aberrations
The performance of the AO system correcting for a single layer of Kolmogorov turbulence with different strengths and wind speeds was investigated. The strength of turbulence, and the wind speed are characterized by the quantities D/r 0 and v/r 0, where D is the input aperture diameter, r 0 is the Fried parameter and v is the velocity the turbulence layer moves across the aperture. The AO system’s maximum frame rate of 24 fps is far short of that required to correct for real atmospheric turbulence. Consequently the wind speeds in these experiments need to be considered relative to the system frame rate. Therefore the wind speed is expressed using the dimensionless ratio of the frequency of the turbulence (τ 0 = v/r 0) over the system frame rate (τS). Short exposure images recorded by the far-field camera were used to calculate the Strehl ratio.
Fig. 6. Far-field image of point-source with AO on and off for two different turbulence strengths, D/r 0 = 6, and D/r 0 = 10, τ 0/τS = 1/10, and gain= 0.5 (average of 100 short exposures).
The dynamic wavefronts used corresponded to a single layer of turbulence with quasi-Kolmogorov spatial statistics translated across the input aperture according to Taylor’s frozen-flow hypothesis [1, 2]. (Note: the phase screens were spatially periodic, with period 4 times the aperture width and were generated by a discrete Fourier transform of the sampled Kolmogorov spectrum. This results in them having a finite outer scale L 0 that has the effect of suppressing low spatial frequencies [29, 30]. The periodic phase screens allow the dynamic aberrations to run continuously.)
Figure 6 illustrates the improvement in the imaging of the point source with the AO system on for two different turbulence strengths of D/r 0 = 6, and D/r 0 = 10. Figure 7 plots examples of the short-exposure on-axis intensity against time both with and without adaptive correction. The plots show a clear improvement with the AO system. The temporal periodicity of the phase screen’s evolution is evident.
Figure 8 shows the performance of the AO system for increasing strengths of turbulence for 0 ≤ D/r 0 ≤ 10. At D/r 0 = 10, r 0 is approximately equal to the actuator spacing. The Strehl ratio values were calculated as an average of the short exposure values. The plot shows that even at turbulence strengths of D/r 0 = 10 there is a considerable improvement in the average measured Strehl ratio.
Figure 9 shows the results of varying the normalized wind speed for two different turbulence strengths (D/r 0 = 6, and D/r 0 = 10). As before, the Strehl ratios plotted are the average of the short exposure Strehl ratios. Note there is still an improvement in the Strehl ratio up to the maximum wind speed tested (τ 0/τS = 1/4).
4.2. Strong scintillation regime - phase and amplitude aberrations
The performance of the AO system was investigated for dynamic aberrations corresponding to the propagation of plane waves over an extended path through uniform Kolmogorov turbulence. The turbulence generator was used to produce waves with both amplitude and phase fluctuations, which were calculated using wave propagation simulations through multiple phase screens using the angular spectrum of waves [31]. Periodic phase screens (with period four times the optical aperture diameter) were used resulting in spatially periodic amplitude and phase fluctuations. We emphasise that although we use a single spatial light modulator, the waves generated still correspond to propagation over an extended path. An example of a generated pupil plane optical aberration (both phase and intensity) for a scintillation strength σ R 2 = 3.3 is shown in Fig. 10.
Fig. 7. Plots of on-axis intensity v. time for uniform intensity aberrations with AO on and off. Plots: (a) D/r 0 = 6 and (b) D/r 0 = 10. For all plots τ 0/τS = 1/16, and gain= 0.5. Note the traces are not synchronized.
Fig. 8. Strehl v. D/r 0 for AO on and off in the weak (phase-only) regime (τ 0/τS = 1/16, and gain= 0.5).
The scintillation strength, wind speed, and aperture size were characterized by the quantities the Rytov number σ R 2, v/r 0, and the Fresnel length Lf given by , where λ is the wavelength, and L is the propagation distance. In weak scintillation the Rytov number σ R 2 and Fried parameter r 0 are given by [6, 3]
and
where k = 2π/λ, and Cn 2 is the refractive index structure constant.
Fig. 10. Phase and intensity distributions for an example wave, with σ R 2 = 3.3. The circle indicates the aperture size (4Lf).
At a scintillation strength of σ R 2 = 1.5 a number of vortices will be present within the pupil. Figure 11 illustrates the improvement in imaging with the adaptive correction on for two different aperture sizes (3Lf and 4Lf) with σ R 2 = 1.5. Even in the presence of vortices, the system still gives correction at both aperture sizes. Figure 12 shows an example of the reconstructed residual phase from the wavefront sensor, and a histogram of the corresponding phase, with the AO system running with σ R 2 = 2.7. Note that vortices are still present (at the points where all colours meet in the phase map). It is not possible for the continuous surface mirror to remove them. However, the regions of poor correction are mainly confined to relatively narrow bands in the pupil, approximately the width of the actuator spacing, connecting pairs of vortices. This gives what are in effect smoothed branch cuts. The resulting phase distribution (histogram) shows that the phase is well corrected over most but not all of the pupil. Thus, although the system does not remove vortices, it still gives good correction in their presence.
Figures 13 and 14 show the performance of the system against time for the two aperture sizes for two different scintillation strengths. In all cases the AO system gives considerable improvement in the Strehl ratio. At the higher scintillation strengths, especially with the larger aperture size, there is significant variation in the corrected Strehl ratio and there are a number of points where the correction is poor.
Fig. 11. Far-field image of point-source with AO on and off for 2 different aperture sizes (D = 3Lf, and 4Lf), with σ R 2 = 1.5, τ 0/τS = 1/24, and gain= 0.5 (average of 100 short exposures).
Fig. 12. Example of reconstructed residual phase and corresponding phase distribution from closed-loop AO system (D = 3Lf, and σ R 2 = 2.7).
The long-exposure results are plotted in Fig. 15. Even at the maximum scintillation strength tested, σ R 2 = 3.3, when the phase distribution will contain many vortices the system gives some improvement in the Strehl ratio for both aperture sizes.
Finally, the performance of the system was tested against increasing wind speeds for the two aperture sizes with a scintillation strength of σ R 2 = 1.5. Figure 16 shows how the correction performance varies with wind speed. Note that there is still an improvement in the Strehl ratio up to the highest wind speed tested, τ 0/τS = 1/4.
The results show that an AO system using a PDI wavefront sensor combined with a continuous mirror can make considerable improvements to the measured Strehl well into the strong scintillation regime.
Fig. 13. Strehl v. time for D = 3Lf(τ 0/τS = 1/24, and gain= 0.5). Note the traces are not synchronized.
Fig. 14. Strehl v. time for D = 4Lf(τ 0/τS = 1/24, and gain= 0.5). Note the traces are not synchronized.
5. Conclusion
A laboratory adaptive optics system using a point-diffraction interferometer with a local quadrature reconstructor, and a continuous MEMs mirror has been demonstrated in both weak and strong scintillation conditions.
The performance of the system was investigated using a dynamic turbulence generator to produce optical aberrations in the pupil plane corresponding to conditions of the weak turbulence approximation and to the scintillation regime for horizontal propagation through uniform turbulence. The system was shown to give correction under both types of condition. With the AO system on considerable improvements in the Strehl ratio (S ≥ 0.3) were observed for turbulence strengths up to D/r 0 = 10, and with wind speeds up to τ 0/τS = 1/6. For amplitude and phase aberrations the system showed significant levels of correction well into the saturation regime (σ R 2 = 3.3) for an aperture size D = 4Lf. The results show that the performance of the PDI is resilient to increasing scintillation and optical vortices. They also show that a continuous mirror, driven with wrapped control signals and without non-local phase reconstruction, can be used to make significant correction with scintillation and vortices. Driving the mirror with wrapped phase control signals can give rise to phase wraps in the mirror surface. Those that are associated with optical vortices are unavoidable. However, phase wraps can also result from non-vortex components of the wavefront such as Zernike tip and tilt. These phase wraps will still degrade the correction performance, particularly with a continuous surface mirror. In the system presented here, we did not include any separate tip-tilt correcting device. However, we note that it may be beneficial to include such correction, say as part of a separate independent tip-tilt tracking module prior to the PDI system, to reduce the impact of this avoidable phase wrapping.
In summary, we have demonstrated that the PDI and continuous MEMs mirrors are viable options for atmospheric adaptive optics systems for applications in the strong scintillation regime.
Acknowledgments
This research was supported by PPARC. Carl Paterson is funded by the Royal Society.
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and 
are orthogonal linear
polarizations).
are the WFS sampling
points.