## Abstract

We present results of the isoplanatic performance of an astronomical adaptive optics system in the laboratory, by using a dual layer turbulence simulator. We describe how the performance of adaptive correction degrades with off-axis angle. These experiments demonstrate that it is now possible to produce quantifiable multi-layer turbulence in the laboratory as a precursor to constructing multi-conjugate adaptive optics.

©2000 Optical Society of America

## 1 Introduction

The limitations of conventional adaptive optics (AO) in correcting large areas of the sky have been well-documented [1, 2, 3, 4]. Light from a natural or laser guide star is well-corrected by an AO system. However, imaging of a nearby object degrades with increasing angle from the guide star, because light from the object does not pass through the same upper layers of the atmosphere as the guide star. Acceptable correction is obtained for a limited field of view around the guide star only, defined by the isoplanatic angle, *θ*
_{0}.

Multi-conjugate adaptive optics will enable wide field (~1 arcminute) imaging with large ground—based telescopes (see e.g.[1, 2, 3, 4]), by using a series of deformable mirrors conjugate to the dominant layers of turbulence in the atmosphere. To—date, however, no multi—conjugate AO systems have been constructed. We are developing a laboratory prototype system based on liquid crystal spatial light modulators (LC-SLMs) as the wavefront correctors. In such laboratory systems it is important to be able to quantifiably generate turbulence with the correct statistics. Neil *et al*.[5] describe a technique using a ferroelectric LC-SLM to produce high bandwidth (both spatial and temporal) turbulence. Here we present results using an extension of this technique to produce dual-layer turbulence, which we then use to feed a single—conjugate adaptive optics system. The turbulent layers are illuminated by both an on—axis beam and an off—axis beam with variable angular separation. In this way we can quantify the effects of the off—axis performance of the AO system.

## 2 The turbulence generator

The turbulence generator described in Ref [5] used a holographic technique to produce an analogue phase from a binary SLM. The method allows an analogue wavefront to be produced from a binary element using optical Fourier transformation and spatial filtering. Some higher frequencies may be removed by the spatial filter, but Ref [5] showed acceptable reproduction of the desired wavefront for atmospheric turbulence up to *D*/*r*
_{0}=30, where *D* is the telescope pupil diameter, and r0 is the Fried parameter. The optical throughput of the system is also limited by the holographic technique (40.5% theoretical maximum per layer, or a few % in reality), but this is not important in the laboratory.

We used two LC-SLMs in series in order to generate dual—layer turbulence. The optical setup is shown in Fig 1. Each of the ferroelectric LC SLMs (Displaytech Inc.) are binary reflective devices, consisting of 256×256 pixels with a pixel size of 15*µ*m×15*µ*m. The LC SLMs can operate at speeds up to 2.5kHz, allowing turbulence generation at rates similar to atmospheric wind speeds, and significantly faster than analogue nematic LC SLMs. To simulate an AO system which uses a natural guide star to correct an off—axis object, a second light source is introduced into the optical setup via a pellicle beamsplitter BS and 2 mirrors M1 and M2. The imaging between the two turbulent layers is controlled by lenses L1 and L2, and is arranged so that emerging light from both the on— and off—axis beams overlap at the lower SLM whereas they are displaced on the upper SLM with the correct pupil shear. The amount of shear, or the equivalent height of the upper SLM, is controlled by the distance of the lower SLM from the image of the upper SLM. If this distance is given by Δ*z* in real space, then the spacing of the equivalent layers in the (virtual) atmosphere is given by Δ*z* (*D*/*d*)^{2}, where *D* is the telescope pupil size, and *d* is the SLM size (=3.84mm). We initially designed the system to model a 1m telescope, with the lower turbulent layer at the telescope pupil and the upper turbulent layer at an altitude of 3.4km. It is very difficult to reproduce accurately the optics of a large telescope using small optics, because the angle of the off—axis beam is magnified by a factor of *D*/*d*, which means that the off axis angles in the system become unfeasible (e.g. 35° for an 8m aperture and a 1arcmin off—axis angle). However, when simulating a system it is the off—axis angle normalised by the isoplanatic patch size which is of interest. Therefore we can model the performance of large apertures by effectively assuming that that the turbulence is at a very high altitude.

To model anisoplanatic effects, the maximum simulated off—axis angle in the sky produced by the turbulence generator should be compared with the isoplanatic angle *θ*
_{0}. Simple Kolmogorov theory states the size of the isoplanatic patch is given by *θ*
_{0}, where

and *ħ* is the average height of turbulence. As we have only 2 layers, one of which is effectively at the telescope pupil, we can replace (*ħ*) with *h*, as long as *r*
_{0} is replaced with the *r*
_{0} of only the upper layer. Alternatively the total *r*
_{0} can be used,

where *r _{i}* is calculated from the

*D*/

*r*

_{0}values for each layer, and the

*ħ*can be calculated using

In addition, modifications to the single plane wavefront generator need to ensure that the turbulence generator accurately represents the displacement of the on and off axis beams in the upper atmosphere. The upper turbulent layer should be oversized with respect to the lower layer so that the on and off—axis beams have the correct pupil shear in the upper layer, but experience the same turbulence in the lower layer. Hence the angle of the off—axis beam at the upper SLM, *α*, will be magnified with respect to the angle at the lower SLM, *β*. At the same time, the off—axis beam should not be vignetted by the SLMs, spatial filters, lens apertures or other optical components. The aperture size of the lenses L1 and L2 place an additional constraint on the maximum values of *α* and *β*, as does the effective separation of the SLMs, Δ*z*. Therefore the lenses must be chosen carefully for focal length, aperture size, magnification and *f*/#.

## 3 Experiment

The optical setup is shown in Fig 1. L1 has a focal length of 50 mm and an aperture of 25 mm. L2 has a focal length of 125 mm and an aperture of 25 mm, giving a magnification *M*=2.5. Lenses L3 and L4 are identical 100 mm focal length lenses with apertures of 25 mm. The off—axis beam is produced via the pellicle beamsplitter BS and the two mirrors M1 and M2. The spatial filters were slits, rather than pinholes, to allow both the on— and off—axis beams to propagate through the system. The off—axis angle can be varied by adjusting either M1 or M2.

Time-evolving phase screens were generated corresponding to Kolmogorov turbulence, using an algorithm described in Ref [6]. Separate phase screens were produced for the two layers, which were temporally and spatially uncorrelated and had different values of *r*
_{0}. The data had the wavefront slopes removed before they were applied to the SLMs. In principle, the system could produce data with wavefront slopes, as long as the spatial filters were adjusted so as not to significantly vignette the beams. The system could be used with either one or both SLMs switched on, to create either single or dual layer turbulence. Single layer turbulence at the telescope pupil was generated by applying a flat phase screen to the upper SLM.

The LC wavefront generator was used to inject turbulence degraded wavefronts into a single deformable mirror (DM) AO system (Electra[7]). The segmented DM has 76 segments with three piezoactuators on each segment, and therefore has 228 degrees of freedom. Hysteresis was corrected by a feedback control system using strain gauges on the piezoactuators. Relay optics were used to input the aberrated wavefront to the AO system. The wavefront sensor was a Shack-Hartmann array with 10×10 lenslets.

The DM was placed in the pupil plane of the telescope, the lower layer was placed at an altitude of 100m (which effectively is the same as the pupil plane), and the upper layer was placed at an altitude of 3.6km (for a *D*=1 m telescope). The turbulence applied to the upper SLM was actually 2.5 times the desired turbulence, to account for the magnification of the upper layer with respect to the lower layer. The parameters of turbulence generated were

1. single layer turbulence, *D*/*r*
_{0}=6

2. single layer turbulence, *D*/*r*
_{0}=7.7.

3. dual layer turbulence, lower layer with *D*/*r*
_{0}=6, and the upper layer with *D*/*r*
_{0}=4, which is equivalent to a total turbulence of *D*/*r*
_{0}=7.7.

4. dual layer turbulence, both upper and lower layer with *D*/*r*
_{0}=6 (equivalent to a total turbulence *D*/*r*
_{0}=9.1)

We could not increase the maximum simulated off-angle angle (in the sky) significantly beyond 13 arcsecs, as it was limited by the resolution of the SBIG CCD camera used to capture the images. The range of angles (compared with *θ*
_{0}) could be increased by using a higher resolution camera, or by changing the turbulence to decrease *θ*
_{0}. The minimum allowable off—axis angle we could simulate was determined by leakage of light from the off—axis beam into the wavefront sensor.

## 4 Results

The Strehl ratios of the corrected and uncorrected on and off—axis beams for varying off—axis angles were calculated by comparing the peak intensities of the point spread functions (PSFs) with the peak intensity of a reference beam. For the reference we used a “flat” deformable mirror configuration which was obtained by correcting the static aberrations in the system with a simplex “hill climbing” algorithm. The simplex algorithm optimised the on—axis PSF by iterative sampling of a merit function calculated from the PSF while manipulating the deformable mirror[8]. For each angle, Strehl ratios were calculated from the average intensity of three exposures of the PSF. Due to fluctuations in the on—axis correction by the AO system, the off—axis correction was normalised by the on—axis improvement in Strehl for each of the three exposures. Fig 2 shows surface plots of PSFs scaled to Strehl ratio for uncorrected (top) and corrected (bottom) single layer turbulence, *D*/*r*
_{0}=6. The left column is the PSF of the on—axis beam, while the right column is the off—axis beam. Fig 3 shows similar plots for dual layer turbulence, both upper and lower layers with *D*/*r*
_{0}=6. For both Figs 2 and 3, the off—axis angle is approximately 13 arcsecs for a *D*=1m telescope. In both cases, the on axis beam is well-corrected, but the off—axis correction is much better for the single layer turbulence than for the dual-layer turbulence.

The results, which show Strehl ratio versus angle are shown Fig. 4. We have plotted the x—axis in normalised units of *R*/*h* where *R* is the telescope aperture radius and *h* is the height of the turbulence. For a 1m telescope, and turbulence at 3.4km, this scale would have a range from 0 to 20 arcseconds. The experimental results show that …

1. for single layer turbulence, the off—axis corrected Strehl shows only a slight drop-off

2. the on—axis (zero angle) corrected Strehl for Turbulence 2 (single layer) and 3 (dual layer) coincide as expected, because the effective turbulence is the same for both.

3. for dual layer turbulence, the corrected Strehl degrades with increasing angle.

Using simple Kolmogorov theory and Eqns 1, 2 and 3, the Strehl ratio, *S* should then show the following dependence with off—axis angle, *θ*, using the Maréchal approximation (*S*=exp(-*σ*
^{2})),

However, as described by Roddier [9], this is only true for perfect adaptive correction. If only a finite number of wavefront modes are corrected, then the angular fall off is slower because the lower order modes are more strongly correlated with angle. Roddier states that for low order correction, then the wavefront variance should follow a power law with an exponent close to 2.

We performed a curve fitting of the experimental data points of the form *y*=*A*exp(*Bx ^{C}*) +

*D*(similarly to Eqn 4) for turbulence 3 and 4. The exponent (C) for turbulence 3 was 1.37±0.05 and for turbulence 4 it was 1.63±0.09, indicating that the experimental data more closely follow an exponent of 5/3 rather than an exponent of 2. However, the experimental data are sparse for small off—axis angles because of light leakage into the wavefront sensor as explained above. Our results are also complicated by the fact that our turbulence had no tip—tilt included. We are currently performing a more complete theoretical analysis of our system to include these effects.

## 5 Conclusion

We have described a dual-conjugate turbulence generator which we have used to investigate off—axis effects in conventional AO systems. Experimental results show initial agreement with theoretical predictions for both single and dual-layer turbulence. An in-depth theoretical simulation of our system, including the effects of tip-tilt removal and partial correction of Zernike modes is the subject of current research. We propose to use the turbulence generator in the future with a laboratory dual-conjugate AO system.

## References and links

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