Dual-conjugate wavefront generation for
adaptive optics
Thu-Lan Kelly† , David F. Buscher‡ , Paul Clark, Colin N.
Dunlop, Gordon D. Love*, Richard M. Myers, Ray M.
Sharples and Andrew Zadrozny
†
Dept of Physics, University of Durham, Durham, DH1 3LE UK
now at the Dept of Physics, University of Adelaide, Australia 5005
‡
now at the Cavendish Laboratory, University of Cambridge,
CB3 0HE, UK
∗
also with the School of Engineering
Corresponding author: Gordon D. Love, email g.d.love@durham.ac.uk
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.
(C) 2000 Optical Society of America
OCIS codes: (010.1080) Adaptive optics; (230.6120) Spatial light
modulators
References and links
1. J. M. Beckers, “Increasing the size of the isoplanatic patch with multi-conjugate adaptive optics,”
in ESO Symposium on Large Telescopes and Their Instrumentation M.-H. Ulrich, ed. (ESO Proc,
Garching) 693-703 (1988)
2. B. Ellerbroek, “First-order performance evaluation of adaptive-optics systems for atmosphericturbulence compensation in extended-field-of-view astronomical telescopes,” J. Opt. Soc. Am. A
11, 783-805 (1994)
3. R. Ragazzoni, E. Marchetti and G. Valente, “Adaptive-optics corrections available for the whole
sky,” Nature (London) 403, 54-56 (2000)
4. D. C. Johnston and B. M. Welsh, “Analysis of multiconjugate adaptive optics,” J. Opt. Soc. Am.
A 11 394-408 (1994)
5. M. A. A. Neil, M. J. Booth, T. Wilson, “Dynamic wave-front generation for the characterization
and testing of optical systems,” Opt. Lett. 23 1849-1851 (1998)
6. R. J. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves
Random Media 2, 209-224 (1992)
7. A. Zadrozny, M.P.J.L. Chang, D.F. Buscher, R.M. Myers, A.P. Doel, C.N. Dunlop, R.M. Sharples,
& R.L. Arnold. In ESO/SPIE Topical Meeting on Astronomy with Adaptive Optics, ed. D. Bonaccini. (1999)
8. N. Doble, G.D. Love, D.F. Buscher, R.M. Myers, and A. Purvis, “The use of image quality metrics
for correction of non-common path errors in the ELECTRA adaptive optics system,” Proc. SPIE
3749 785, ICO 18th Congress (1999)
9. F. Roddier, M.J. Northcott, J.E. Graves, and D.L. McKenna. “One-dimensional spectra of
turbulence-induced Zernike aberrations: time delay and isoplanicity error in partial adaptive compensation,” J. Opt. Soc. Am. A 10 957-965 (1993)
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
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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, θ0 .
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/r0 = 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 X 256 pixels with a pixel size of 15µm×15µm.
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 ∆z in real space, then the spacing of
2
the equivalent layers in the (virtual) atmosphere is given by ∆z (D/d) , 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
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M1
laser light
BS
P1
Image of
SLMupper
offaxis beam
SLMupper
P2
M2
f1
L1 f1 slit
f2
L2
f2
f3
slit
SLMlower
f3
P3
L3
f4
L4
relay optics
To AO
system
Fig. 1. Optical setup for dual-conjugate wavefront generation. P1 to P3 polarizers;
BS, beamsplitter; M1 and M2 mirrors. f1 to f4 are the focal lengths of lenses L1 to
L4.
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
θ0 . Simple Kolmogorov theory states the size of the isoplanatic patch is given by θ0 ,
where
(1)
θ0 = 0.31r0 /h
and h is the average height of turbulence. As we have only 2 layers, one of which is
effectively at the telescope pupil, we can replace (h) with h, as long as r0 is replaced
with the r0 of only the upper layer. Alternatively the total r0 can be used,
r0 =
3
− 53 − 5
(2)
ri
where ri is calculated from the D/r0 values for each layer, and the h can be calculated
using
h=
∞
5
2
3
0 h Cn (h) dh
∞ 2
0 Cn (h) dh
53
(3)
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
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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, ∆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 r0 . 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 X 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/r0 = 6
2. single layer turbulence, D/r0 = 7.7.
3. dual layer turbulence, lower layer with D/r0 = 6, and the upper layer with D/r0 =
4, which is equivalent to a total turbulence of D/r0 = 7.7.
4. dual layer turbulence, both upper and lower layer with D/r0 = 6 (equivalent to a
total turbulence D/r0 = 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 θ0 ) could be increased by
using a higher resolution camera, or by changing the turbulence to decrease θ0 . The
minimum allowable off–axis angle we could simulate was determined by leakage of light
from the off–axis beam into the wavefront sensor.
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Fig. 2. Point spread functions for single layer turbulence with D/r0 = 6. (i) Uncorrected on–axis, (ii) uncorrected off–axis, (iii) corrected on–axis, (iv) corrected
off–axis.
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/r0 = 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/r0 = 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
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Fig. 3. Point spread functions for dual layer turbulence with both layers having
D/r0 = 6. (i) Uncorrected on–axis, (ii) uncorrected off–axis, (iii) corrected on–axis,
(iv) corrected off–axis.
with increasing angle.
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 )),
5/3
θ
S ∝ exp −
.
(4)
θ0
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(BxC ) + 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
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Fig. 4. Corrected Strehl ratio as a function of off–axis angle. Turbulence 1 (diamonds) and 2 (squares) - single layer turbulence. Turbulence 3 (crosses) and 4 (triangles) - dual layer turbulence. The turbulence strength for each curve is described
in the text.
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.
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