1. Introduction

In order to compensate for the detrimental effects of the turbulent atmosphere at astronomical observatories, adaptive optics is used. Conventional adaptive optics systems, using one deformable mirror (DM) and one guide star (GS), are now present at several astronomical observatories. Conventional adaptive optics has its limitations though. The single DM can compensate for the cumulative phase distortion in the direction of the GS, but since atmospheric turbulence is three-dimensional, the corrected field is small. The corrected field, termed isoplanatic patch, is typically a few arc seconds in the visible and NIR [1]. This, together with the fact that wave-front sensing is restricted to bright GSs, implies that the sky coverage for conventional adaptive optics is low.

Beckers [2] introduced the concept of multiconjugate adaptive optics (MCAO), to increase the size of the isoplanatic patch. The principle of MCAO is to achieve this, by optical conjugation of several DMs to different altitudes in the atmosphere. MCAO, together with the possibility to produce synthetic reference beacons or laser guide stars [3], has the potential to overcome the limitations of conventional adaptive optics.

Analytical studies and numerical simulations of MCAO [4–7] have all shown that the corrected field is increased, as compared to conventional AO. As of today, no practical demonstration of this has been achieved. A laboratory AO prototype system emulating a layered atmosphere following Kolmogorov statistics, has demonstrated the effect of anisoplanatism when using conventional adaptive optics [8].

The scope of this paper is to practically demonstrate the increase of the isoplanatic patch by using dual-conjugate adaptive optics (MCAO using two DMs), compared to conventional adaptive optics. In Section 2 the details of the experiment is given, wherein the first subsection contains information about the optical setup, the second subsection presents the wave-front reconstruction alternatives used in the experiment and the last subsection gives the obtained results. The results from the practical setup are compared with results from a Zemax model of the setup in Section 3. Finally, in section 4, some remarks and conclusions are stated, and the main conclusion is that the isoplanatic patch was demonstrated to increase using dual-conjugate adaptive optics.

2. The Experiment

2.1 Optical setup

The experiment is a downscaled version of a 7.5-m aperture telescope equipped with dual-conjugate adaptive optics, a four layer atmosphere with an altitude of 16 km and a GS cross (5 on a dice) with an arm length of 45″. The setup comprises four different parts; a guide star arm, an atmosphere also containing the two DMs, a telescope and finally a wave-front sensing arm. The optical setup is shown in Fig. 1.

 

Fig. 1. Schematic birds eye view of the optical setup, with rays originating from the central GS.

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Broadband light from the halogen lamp (HL, see Fig. 1) is focused by L6, and filtered by the green (λ=550 nm) glass filter (GGF) at a bundle of five optical fibres (FGS). The opposite ends of these fibres constitute the GS cross, with an arm length of 2.5 mm. A mask (GSM) is placed in front of the GS cross in order to choose GS. The diverging light, emanating from the fibres, is collimated by L5 (f _L5=+700 mm). This implies that the stars are placed at infinity, i.e. natural GSs are used in the experiment.

For simplicity, the two DMs are placed directly into the atmosphere, not post-focal at optically conjugating planes, which would be the case for a real telescope. One mirror is placed at ground level and one is placed at an altitude corresponding to 10 km in the parent (scaled-up) version. The DMs used in the experiment are micromachined membrane DMs, manufactured by OKO. Each DM has 37 actuators in a hexagonal pattern with three rings of actuators surrounding the central actuator. The membrane of the mirror is grounded and the deflection of the membrane is proportional to the square of the applied actuator voltage. A null voltage was applied to all actuators in order to allow bi-directional poking of the membrane. The shape of the membrane in this null-position introduces defocus to the wave-front. This was compensated for by shifting L5 for DM2 and L1 for DM1, to eliminate the introduced defocus. The diameter of the DM is 15 mm. In order to allow for a field of view of 1.5′ (in the parent version), and still make use of the entire surfaces of the DMs, different scaling of the lower and upper part of the atmosphere is required. In addition, to fit the setup into the lab, different longitudinal and transversal scaling are used to compress the setup. The different scaling of the upper and lower part of the atmosphere is allowed by an afocal lens combination, i.e. L3 (f _L3=+200 mm) and L4 (f _L4=-100 mm). The scaling factors relating the laboratory setup to the parent version are given in Table 1.

Table 1. Scaling of the atmosphere in the experiment

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The compression of the setup gives that angles in the experiment, α_exp , are related by the compression factor, C, to the angles in the parent version, α_par , according to:

The turbulence in the atmosphere is modeled as static layers, using four phase screens (PS1–PS4). The phase screens are holographic film, exposed by laser speckles and then developed and bleached. The result is a screen with varying optical path length, which introduces the desired rms phase error. Although the spectral content of the phase screen is not known, the Kolmogorov-equivalent r₀ can be calculated from the variance of the angle of arrival [9]:

where λ is the wavelength and D is the diameter of the circular aperture. Since the angular tilt over each subaperture on the Shack-Hartmann sensor and hence ${\sigma }_{\alpha }^{\mathit{2}}$ can be measured, Eq. (2) allows r₀ to be estimated. D is approximated by the width of each square subaperture on the Shack-Hartmann lenset array. Each screen was inserted into the setup one at a time. The average of ten measurements of the average length of the Hartmann-vectors gave the r₀ -values presented in Table 2. Apart from r₀ characterising the high frequency roll-off, a main feature of the phase screen will be the low frequency saturation given by the outer scale L₀ . Fig. 2 gives a sample wave-front plus the structure function of PS2. For this screen L₀ is well-defined given by the cut-off at 5 m. L₀ is in the range 3–6 m for the other phase screens.

The entrance pupil of the telescope (P in Fig. 1) is emulated by an iris diaphragm, placed in front of L2 (f _L2=+350 mm). Wave-front sensing is realised with the collimating lens L1 (f _L1=+100 mm) and the mini-WaveScope Shack-Hartmann sensor. The chosen lenslet array has square lenslets with f _S-H=+18 mm and width D _S-H=0.325 mm, allowing 12 subapertures across the pupil. The CCD array used is 640×480 with a pixel width of 10 µm. The spot size is 2λf _S-H/D _S-H=61 µm. The spot-to-pixel ratio is therefore 6.1. The centroiding algorithm in the mini-WaveScope software, fitting a quadratic surface to the spot area, was used. All relevant parameters of the experiment are given in Table 2. As seen in Table 2 the field of view (FoV) of the Shack-Hartmann sensor is larger than the GS-field. Since a static atmosphere is emulated, the Shack-Hartmann sensor can therefore be used sequentially for all five GSs.

Table 2. Key parameters for the experiment, and scaled to parent version.

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The prime goal of this experiment is to investigate the gain in corrected field that can be obtained by using more than one deformable mirror. Hence, the temporal aspects of AO-correction have not been addressed and the experiment is static.

 

Fig. 2. a) Sample of optical path difference from PS2 given in radians, with distance r scaled to parent version. b) Obtained structure function D_Φ from PS2, averaging over 10 measurements (blue) compared to the Kolmogorov structure function D_Φ=k·r^5/3 (green).

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2.2 Wave-front reconstruction

The actuator commands are given by c=[c₁ , c₂ , …, c_N ] ^T , where N is the number of actuators. The wave-front sensor measurements are given by s=[s₁ , s₂ , …, s_2M ]^T, where M is the number of subapertures. The vector s gives the average x- and y-gradient of the wave-front over each subaperture. These vectors are related according to:

where the Jacobian matrix G=∂s/∂c is the so-called interaction matrix [6]. It is obtained by poking each actuator a unit poke, and collecting the corresponding sensor measurements as the columns in G. This is done with a flat reference wave-front. In this experiment sensor measurements in the directions of the five GSs and actuator commands to the two DMs are used, which give a concatenated version of Eq. (3):

The pseudo-inverse of the interaction matrix, i.e., the least-squares solution, allows the actuator commands to be calculated from sensor measurements on distorted wave-fronts. The pseudo-inverse of the full interaction matrix corresponds to dual-conjugate adaptive optics. Conventional adaptive optics is realised by using only the central GS (GS3) and DM1, i.e. the pseudo-inverse of ∂s₃ /∂c₁ . Additionally, field-averaged conventional adaptive optics is used by using only the left half of the interaction matrix. This method uses only DM1, but information from all five GSs is used in the reconstruction procedure.

2.3 Results

Reference Hartmann spots were obtained sequentially for all five GS-directions, with phase screens excluded from the setup. At the same time poking of the actuators in all directions were executed to obtain the interaction matrix G. After inserting the phase screens the three wave-front reconstruction alternatives were implemented, based on the Hartmann-vectors (tip&tilt excluded). This corresponds to an open-loop system without servo error, and the results presented were obtained after a single iteration. Also, since the tip&tilt in each of the five wave-fronts were excluded, the plate scale modes are excluded from the reconstruction. Point spread functions (PSFs) could be extracted with the software mini-WaveScope (AOA), reflecting the relative improvement after correction with the DM(s). The tip&tilt was excluded from the measured wave-fronts. Thus, the calculated PSFs are centered. The PSFs were extracted in each GS direction for the uncorrected case, after correction with conventional adaptive optics, after correction with field-averaged conventional adaptive optics and after correction with dual-conjugate adaptive optics. To obtain a reasonable statistical basis, 20 measurements were obtained with phase screens transversely shifted between each measurement. The average PSFs were calculated for the cases and directions respectively.

 

Fig. 3. Average PSFs from the experiment. Case a) is uncorrected, b) is corrected with conventional AO, c) is field averaged conventional AO and d) is dual-conjugate AO. The intensity I_N is normalised with the central intensity of the diffraction-limited PSF. The axes of the PSFs are the relative coordinates in arc seconds, scaled to the parent version.

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In order to relate the PSFs to the parent version, they had to be scaled. The obtained PSFs were based upon the size of the pupil as imaged onto the Shack-Hartmann sensor. The angular size of the PSF is proportional to λ/D. Taking care of these two facts, the angular scaling of the PSFs is given by:

where λ_par =2.2 µm, λ_exp =550 nm and D_exp /D_par =1/500. The resulting PSFs, presented in Fig. 3, have been scaled according to this relationship and thus represent the parent version.

The Strehl ratio is equal to the central intensity value normalised with the diffraction-limited intensity. Thus, the Strehl ratio of the PSFs can be read as the maximum I_N-value. The calculated seeing angle is λ/r_0,eff =0.32″. The calculated diffraction limited angle is 1.22λ/D=0.074″. The effects on diffraction, due to scaling between the parent and experimental version, are accounted for by the scaling in Eq. (4). It is different from Eq. (1), which concerns the scaling of geometrical angles. The experiment clearly suffers from more scintillation than the parent version will do, but not to a degree where coherent phase maps could not be calculated. Hence, the change in diffractive effects (Fresnel number) is of no important consequence.

3. Zemax-simulations

In order to confirm the experimental results a Zemax model of the setup was made with atmospheric screens generated by Skylight according to Table 2. Relevant global parameters for the parent atmosphere listed in Table 2 are r_0,eff =1.41 m and θ_i =17.5″ (isoplanatic angle).

 

Fig. 4. Comparison of the experimental results with a Zemax model of the experiment. The black curves depict the Strehl ratio as function of angle, α, from the central guide star in the parent experiment with error intervals marked in green or blue. The red dots are the experimental values including error bars. Case a) is uncorrected, b) is corrected with conventional AO, c) is field averaged conventional AO and d) is dual-conjugate AO. The green families resulted from running the optimization routine a single time, and the blue family resulted from running it 10 times. See text for more explanation.

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Ten stochastically independent sets of four phase screens with appropriate values of r₀ were generated to obtain reasonable statistics. The screens followed Kolmogorov statistics, with infinite L₀ . Adequate wavefront sensing was mimicked by tracing rays in a square pattern comprising 12 rays across the pupil using the “ray aiming on” feature in Zemax. Figure 4 upper left confirms the screen statistics to be adequate. The mirrors were modeled as polynomial phase screens with a certain maximum order, which was restricted to four performing conventional AO and comparing to the experiment (Fig. 4 upper right shows this to be a bit too optimistic). The field averaged conventional AO and the dual-conjugate AO cases used the above ray and mirror formats when performing optimizations varying the polynomial mirror coefficients and weighting the stars according to the relevant case. Only the dual-conjugate case needed more than one iteration to reach saturation.

4. Conclusion

It has been demonstrated that the isoplanatic patch is indeed increased using dual-conjugate adaptive optics. The improved isoplanatism when going from b) (see Fig. 3) to d) is clearly seen. The mean Strehl ratio for the peripheral GSs in b) is 0.11, which is raised to 0.48 in d). The Strehl ratio for the central guide star is also raised, presumably due to the increased number of actuators. It can also be observed in Fig. 4 d) that the Strehl ratio in the field is higher in the direction of the guide stars, already known from [6–7]. In spite of the discrepancy, regarding the outer scale, between the experimental screens and the Skylight screens used in the Zemax model, the agreement between the experimental results and the Zemax model is quite good in particular for dual-conjugate AO. This reflects the fact that the outer scale mainly affects the low frequency content of the fluctuations, which is removed by the mirrors. Finally, it should be noted that the results are based on calculated PSFs, no real images have been obtained.