1. Introduction

A laboratory setup of Multi-Conjugate Adaptive Optics (MCAO) system [1] has been built on an optical bench as a test-bed for developing new wavefront sensing concepts [2, 3]. These concepts are intended for sensing the atmosphere-induced wavefront distortion on future giant telescopes with integrated MCAO systems. An effective operation of the MCAO system at any position on the sky requires a number of artificial sources, namely laser guide stars (LGSs), and a wavefront sensing system for adequate probing of the atmospheric turbulence above the telescope.

The ultimate goal of the MCAO test-bed experiments is to resolve LGS imaging problems in multi-mirror optical systems proposed for future Extremely Large Telescopes (ELTs) [4–6]. Due to intrinsic telescope aberrations associated with imaging of LGSs seen as objects at finite distance, the LGS wavefront propagation through the multi-mirror systems to the final telescope focus is hardly feasible. To resolve this problem, one could design a special purpose telescope optical system optimized for imaging both laser and natural guide stars [7] or implement a virtual wavefront sensor (WFS) concept [2].

The virtual wavefront sensing concept employs a primary wavefront sensor (WFS) placed at the first available LGS focus where LGSs images are not significantly aberrated (which is true for typical two-mirror telescopes at Cassegrain or Gregorian focus), and a dedicated test WFS working in the telescope final focus. The latter operates in conjunction with an artificial point source at an intermediate science telescope focus to monitor the effects of the second part of the telescope system containing additional DMs invisible for the primary WFS. The wavefront sensing data obtained from both WFSs are transformed into “virtual” wavefronts being the LGS wavefronts, which would have been measured if LGS wavefronts had passed through the entire telescope optical system.

Although the virtual wavefront sensor comprises two sensors, here we present only the optical design for the primary WFS working as a multi-directional WFS [3] on the laboratory MCAO test-bed. It is our intention to develop and test the primary WFS system first before advancing to the next level of complexity to verify the virtual WFS concept.

2. Tomographic wavefront sensor design

Wavefront sensing of incoming LGS wavefronts provides the knowledge of atmospheric turbulence fluctuations required for optimal control of the deformable mirrors (DMs) to achieve excellent MCAO correction [8]. Since MCAO system corrects over an extended field of view (typically 1–2 arcmin in diameter for correction at the K band with Strehl ratio higher than 0.5 under good atmospheric conditions), to acquire this knowledge, multi-directional wavefront sensing is required. To achieve this, several WFSs are employed at the LGS focus in the telescope. Each WFS has its sensing plane conjugated to DMs or the entrance pupil of the telescope. We shall refer to such scheme as a tomogprahic wavefront sensing, that is sensing the atmospheric turbulence in several directions corresponding to available reference sources near and within the science field.

In the context of MCAO correction, rigorous optical tomography, as a full reconstruction of the 3-D structure of perturbing turbulence, is neither necessary nor possible. As a rule, to derive optimal correcting shapes for DMs, it is sufficient to know projected wavefronts in the telescope pupil and geometrical configuration of the reference sources on the sky. In some cases, as for layer-oriented wavefront sensing [9], correcting shapes for DMs can be obtained iteratively by sensing the wavefronts at the planes conjugated to DMs (one wavefront sensor per DM). In such cases, tomography is carried out with several WFSs collecting wavefronts from all guide stars at several sensing planes. The incoming wavefronts from guide stars will be more overlapped in the sensing plane, the closer it is conjugated to the telescope pupil. The DM shape is controlled based on the wavefront measurements obtained from its conjugated WFS, for which the combined optical effects of the other DMs and turbulence at other conjugates are smeared out. The more correction is achieved on each DM the less contamination from the turbulence at other conjugates will be seen in WFSs.

In principle, one can avoid splitting the light into several channels by sensing all wavefronts at the telescope pupil and then employing a shift and add procedure to obtain wavefronts at the DM conjugation planes [2].

As opposed to optical tomography, the shift and add procedure can be regarded as a digital tomography, for which, one could use a single WFS collecting all wavefronts in a non-overlapped manner. We shall refer to such multi-directional WFS as a tomographic wavefront sensor.

Figure 1 depicts the optical layout of the tomographic WFS incorporating several reference sources and using the same CCD camera for all pupil images. For clarity reasons, the collimated beams from only three sources are shown. These sources (one on-axis and two off-axis) represent natural guide stars (NGS).

 

Fig. 1. Optical layout of the tomographic wavefront sensor: 1 – the telescope entrance pupil; 2 – telescope objective, 3 – field lenslet array; 4 – collimator; 5 – Shack-Hartmann lenslet array; 6 – SH focal plane; 7 – field flattener lens; 8 – reimaging two-doublet system; 9 – CCD camera imaging plane.

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The collimated light from the NGSs passing through the telescope entrance pupil (1) is focused by the telescope, which is represented by a doublet lens (2). The field lenslet array (3) is placed in the telescope focal plane, so each NGS has its own field lens, see Fig. 2. The field lenslets together with a collimator (4) form non-overlapping images of the telescope pupils in the sensing plane (5) of the Shack-Hartmann (SH) lenslet array. A plano-concave lens (7) plays a role of the field flattener for a reimaging system (8), which consists of two identical doublets. The reimaging system provides the proper plate scale on the CCD camera (9) and also resolves the problem of short focal length of the SH lenslets (5). A telecentric pass provided for the chief rays of the probing beams arriving on the CCD helps to keep the central SH spots aligned with the center of their sub-frames while eliminating a defocus error, see a close-up view in Fig. 2.

 

Fig. 2. Magnified optical layout: pupil separation in the tomographic WFS (left side), rays pass near the CCD plane (right side), for legend see Fig. 1 caption.

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3. Laboratory MCAO test-bed

To develop the Virtual WFS concept, an experimental setup was built on an optical bench. Fig. 3 shows the layout of the experimental setup, which consists of three main parts: reference source module (1) incorporating five NGSs, atmospheric turbulence module (2) comprising two phase screens and two DMs, and the telescope system (3) with the tomographic WFS described in the previous section. Two deformable mirrors and atmospheric phase screens are employed to simulate a scaled version of a 10-m adaptive telescope operating in the K band (at wavelength λ=2.2 µm). The real experiment is carried out in the visible in the range λ=0.5-0.65 µm.

Four white light sources separated by 12 mm from an on-axis source are created by using five optical fibers with a white light injected from a halogen lamp. Each optical fiber has a 50 micron core, which is comparable to the Airy disk of the first collimator (2) shown in Fig. 4. By collimating the beam, five NGSs within 2 arcmin field are formed for the following atmospheric module and the telescope system. A close-up view of the first and the second modules is presented in Fig. 4. The collimator (2) images the reference sources to infinity for NGS model. One could also image stars at a finite distance for modeling laser guide star.

 

Fig. 3. Optical layout and actual arrangement of the MCAO test-bed.

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Fig. 4. Optical layout of NGSs and atmospheric turbulence module: 1 – guide star plane; 2 – first collimator; 3 – first beam splitter; 4 – afocal system; 5 – high altitude layer DM2; 6 – second beam splitter; 7 – ground layer DM1 (telescope pupil); 8 – collimated light entering tomographic WFS; PS1 and PS2 are the phase screens.

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In the present, simplified design for our MCAO test-bed, both DMs are located in the atmospheric turbulence module acting as two additional turbulent layers at low (0 km for DM1) and high altitudes (10 km for DM2), similar to the dual-conjugate AO experiment [10]. The atmospheric model incorporates two phase screens PS1 and PS2 shown in Fig. 4 as vertical dashed lines. The PS2 representing turbulent layers at 20 km and the PS1 represents layer at – 6 km altitude. Originally we used phase screens made from CD-case covers with profiles resembling thin atmospheric layers, see [11]. However, for the present experiment both PS1 and PS2 are custom-made phase screens plate with imprinted phase, which has a Von-Karman turbulence spectrum with correlation length r₀=1.5 mm and outer scale L₀=41 mm [12]. The DM1 is a 15-mm 37-channel OKO micro-machined membrane deformable mirror. The DM2 is a 30-mm 59-channel OKO mirror. All optical components used in the MCAO test-bed system are presented in Table 1 in the sequential order (along the light pass). Those components working in a double pass are marked with an asterisk when quoted for the second time. Working diameters are given for 4.6 degree full field of view.

Table 1. Essential parameters for the optical components in the MCAO test-bed system.

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For optimal correction with OKO mirrors we used only their intermediate and central zones, which is about 60–70% of their optical diameter [13]. The telescope pupil size is defined by the 8 mm aperture stop placed near the DM1. An afocal system with a magnification of 1.6 is used to match a larger working area of the DM2 with required 22.8 mm metapupil.

The axial optical distance between DMs is about 80 mm (taking into account the effect of beam splitters), which corresponds to 10 km pass in the parent telescope system. Thus, we have longitudinal scaling S_L=125000. The transversal scaling is S_T=1250, since the telescope pupil diameter is 8 mm, and its parent equivalent is 10 m. The compression factor S_L/S_T=100 implies that field angle in the parent system 100 times smaller than that in the experimental setup. Since the half field of view is 2.3 degrees, we are modeling a 10-m telescope with 83 arcsec field angle. Each DMs provides approximately five actuators across the NGS probing beam, which is equivalent to a 1.6 mm and 2 m actuator pitch projected to the telescope pupil in the setup and parent system respectively. The correlation length r₀ of the phase screen scaled for the parent system is 1.875 m and outer scale L₀=51 m. For a good site, in the K band a typical value of r₀ is about 1 m. Therefore, in the future, we plan to use several screens to increase the turbulence strength.

Figure 5 shows footprints of five NGS beams on DM2, Hartman spots on the CCD plane (Zemax model) and real Hartmann spots as seen on the Dalsa camera CCD. For non-aberrated wavefronts, the distance between adjacent Hartmann spots on the CCD is 12 pixels. Therefore, each sub-aperture covers 12×12 pixels, which corresponds to a 20×20 arcmin field for 12 µm pixel of the CCD. This field is large enough to accommodate spot displacements due to local wavefront tilt and some internal misalignment. The parent equivalent of this field is 12×12 arcsec, which is sufficient for MCAO correction. The five pupils shown in Fig.7(c) do not overlap, however there are no gaps between their bordering sub-apertures.

 

Fig. 5. Full Field Spot Diagrams: (a) metapupil on DM2 (10 km conjugation), (b) Hartmann spots on the CCD (Zemax simulation), and (c) real Hartmann spots in selected boxes on the CCD.

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4. First results: MCAO system correction

Before carrying out a complete test of the entire MCAO system for the virtual WFS concept, we simulated performance of its primary WFS on the MCAO test-bed. For our MCAO system shown in Fig. 3, the total number of actuators is M=96, the number of sub-apertures is 108 for central pupil and 100 for each off-axis pupil leading to the total number of sub-apertures W=508. The DM control is performed using a least squares algorithm based on the linear mapping of the actuator control signals c to the wavefront signals s as follows: s=Ac, where s=[s₁, s₂, …,s_W]^T, c=[c₁, c₂, …,c_M]^T and A is the interaction matrix given by

$\mathbf{A}=\left[\begin{array}{cccc}\frac{\partial s_1}{\partial c_1}& \frac{\partial s_1}{\partial c_2}& \dots & \frac{\partial s_1}{\partial c_M}\\ \frac{\partial s_2}{\partial c_1}& \frac{\partial s_2}{\partial c_2}& \dots & \frac{\partial s_{12}}{\partial c_M}\\ \dots & \dots & \dots & \dots \\ \frac{\partial s_W}{\partial c_1}& \frac{\partial s_W}{\partial c_2}& \dots & \frac{\partial s_W}{\partial c_M}\end{array}\right]$

This matrix is constructed experimentally by poking each actuator of the deformable mirror and measuring the corresponding response with the tomographic wavefront sensor. It is worth noting that partial derivative ∑ s_i /∑ c_j is estimated as [s_i (c _j+) - s_i (c_{j -})]/[(c _j+) - (c _j-)], where s_i (c _j+) and s_i (c _j-) are signals obtained with a j-th actuator poked with positive and negative unit voltage applied with respect to a chosen biased voltage (setting some intermediate DM membrane position). This improves the linearity of the AO system.

The interaction matrix A of W x M dimensions can be written as the product of three matrices,

where: U is a W×M orthogonal matrix with its columns u _i representing a complete set of sensor signal modes, V is an M×M orthogonal matrix, its columns v _i representing a complete set of mirror control modes and ∑ is an M×M diagonal matrix being the singular value decomposition of the interaction matrix A. The diagonal values λ_i are singular values of the matrix A. Each non-zero singular value λ_i relates the orthogonal basis component v _i in actuator control space c to the orthogonal basis component u _i in wavefront sensor signal space s. In other words, the actuator control signal c=v _i results in the corresponding sensor signal s=λ_i u _i. Consequently, the component of the sensor signal s=u _i can be corrected for by applying the corresponding actuator mode c=-${\mathrm{\lambda }}_{\mathrm{i}}^{-1}$ v _i.

We are using the least squares controller for the DMs, for which the control matrix C of M x W dimensions is given by the least-squares inverse (pseudo-inverse) of A,

where, ∑^-1* is pseudo-inverse of ∑ and is formed by transposing ∑ and replacing all non-zero diagonal elements λ_i by their reciprocals ${\mathrm{\lambda }}_{\mathrm{i}}^{-1}$, which represent the gains of the mirror modes in the control matrix. Finally, knowing C we can express our zonal reconstruction operation as c=C s, that provides a vector of actuator control signals c from measured wavefront sensor signals s.

In the simplified version of the experiment, we correct static aberrations of the phase screen using 2 DMs. In the first run, the turbulent layer (a fixed PS) is placed at 20 km altitude, in the second run it is at - 6 km altitude. These rather extreme values are due to limitation in the geometry of the layout shown in Fig. 3, which will be modified in the future by adding a separate atmospheric volume. Figure 6 depicts two locations of the PS and the star field with a target star used for analysis of the dual-conjugate AO correction. The star target image obtained in our acquisition camera at the telescope focus is presented in Fig.7 together with an estimated shape of the star diffraction pattern for perfectly aligned system.

 

Fig. 6. Target star and location of the phase screen during the static aberration correction in dual-conjugate AO: star field with a target under the test, (b) two locations of the turbulent layer, and (c) phase screen.

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Fig. 7. Target star image: original image in the focal plane (left side); Zemax Polycromatic diffraction point spread function normalize to the peak intensity of the unaberrated image (right side), box size is 360 µm.

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It is seen that original image is not a point object. Due to chromatic aberrations at the telescope focus the target star looks like an extended object with 60 µm RMS diameter (λ=0.55-0.65µm) for perfectly centered system. For comparison, the telescope has a 30 µm Airy disk diameter. The real image has even larger size, due to some additional aberration introduced. The AO correction is based on the wavefront estimates for all five stars.

The probing beam footprints on the PS located at 30 km height is depicted in Fig. 8. At 20 km height the beams will be slightly overlapped, whereas for PS at - 6 km height they will have about 50% of common area. Due to limiting field of view in the acquisition camera, only one target star was used for image evaluation after AO correction. The analysis of image quality of the target star for both positions of the PS is presented in Figs. 9 and 10 and Table 2. For majority of the images in Figs.9 and 10, the gray-scale bar indicates relative value of photon counts multiplied by 10⁴. Based on these images, a quantitative analysis of the AO five stars correction has been carried out and presented in Table 2. The en-squared energy (EE) is calculated over 3×3 pixels and normalized to the corresponding value in the original image (24.5×10⁴), see Fig.7. The central intensity ratio (CIR) is calculated as a ratio between the photon counts in the brightest pixel of the image and the maximum value in the original image (6.3×10⁴). The anticipated Strehl ratio (SR) is derived from analytical considerations, see below. Due to pixel binning, the pixel size of the acquisition camera is 24µm leading to low resolution imaging, that is helpful for EE estimation, but less desirable for estimation of CIR, which is in the limiting case of infinitesimal pixels becomes equivalent to SR.

 

Fig. 8. Phase maps for the intersection region of the five probing beams with the phase screen at 30 km height.

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From the interferometeric measurements of the PS working zone for each beam, see Fig. 8, we know that average PTV phase variation is ±λ(λ=0.6328µm) and average RMS is 0.3λ. For Von-Karman turbulence model with ratio D/L₀=0.2 the residual piston- and tilt-removed phase variance can be estimated as Δ₃=0.117 (D/r₀)^5/3 [14], and in terms of SR, for all stars the image degradation due to presence of the PS screen should be exp(-Δ₃)≈0.15. This value is comparable to the CIR values of aberrated images for both PS positions.

The anisoplanatic angle for a single layer is defined as θ₀=r₀/H, where H is a distance from the layer to DM. For the PS placed in both positions, the corresponding anisoplanatic angle is given for each DM in Table 2. Note, that the target star is 83 arcsec away from the field center, which is 1.2–4 times exceeds the estimated θ₀. For correction of a single natural guide star with one DM, the DM fitting error will limit achievable SR at the level exp(-Δ₁₅)≈0.64, where Δ₁₅=0.0279 (D/r₀)^5/3 is a residual phase variance after correction of the first 15 Zernike modes, this formula derived by Noll for Kolmogorov turbulence model [15] is still valid for our case with D/L₀=0.2, for details see [14]. We assume here that DMs are capable of correcting 15 Zernike modes within 8 mm beam. While attaining a good on-axis correction, off-axis star will be suffering from angular anisoplanatism, which, according to Stone et al. [16], after expansion in Zernike modes can be easily analyzed in terms of phase degradation. Instead of general anisoplanatism formula giving the difference between the wavefronts, including even the highest-spatial frequency effects that would never be corrected in a real AO system, one could estimate the phase differences between partially corrected wavefronts. It is possible to eliminate high-frequency problem by limiting correction with Zernike modes at some finite order. Following [16] one can show that for small field angles θ when SR>0.2–0.3, the phase variance due to anisoplanatism is Δ_a=0.08 (θ/θ₀)² (D/r₀)^-1/3 F_n, where F_n is coefficient depending on the correction order n. For the case of 15 Zernike modes, n=4, and F_n=0.7031. The combined effect of DM fitting error and angular anisoplanatism after correction of 15 Zernike modes in the target star wavefront gives SR=exp(-(Δ₁₅+Δ_a)), which is presented in Table2 for both DMs and PS positions. For more discussion on anisoplanatism with DM fitting error see [17].

In a very simplistic approximation [1], correction with N DMs having similar fitting error is equivalent to enlarging anisoplanatic patch by 2N, so for PS1 and PS2 the average angle θ₀=45 and 30 arcsec respectively. After 2 DM correction we expect to obtain four times larger anisoplanatic angles, see Table 2. It means that for MCAO system, the DM fitting errors mainly limits correction. The experimental results show some image improvement for the target star even with a single DM correction as estimated analytically. It is interesting to note that EE values agree much better with predicted SR than CIR values. It could be partly due to the fact that CIR is more sensitive to low-spatial frequency content than EE. Correction achieved over five stars is a trade off between SR at the center and at the edge of the field. Therefore in contrast to a single star correction case, SR will be lower than 0.64 in the field central, while four off-axis stars will share larger amount of the total correction due to their larger influence in the interaction matrix. The correction obtained with 2 DMs leads to a significant improvement in the image quality that agrees with the predicted SR especially EE values, that confirms a noticeable increase of energy concentration in the image.

Table 2. Experimental results for five-star AO correction measured on the target star.

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As we pointed out before, because of restrictions from the opto-mechanical design, in both runs, the phase screen was placed outside the DM conjugation range, which is known as not favorable location for DM correction action [2]. This non-optimal conjugation condition reduced the anisoplanatic angle and made difficult to correct for off-axis stars at the edge of the field. Apart from angular anisoplanatism and DM fitting errors, AO correction might drop as a result of non-efficient operation of the tomographic WFS on extended sources, since chromatic aberration blurs the stars to the extend when they can be resolved in sub-apertures. To reduce this effect, we plant to narrow down the working wavelength range.

 

Fig. 9. Upper layer static aberration correction for conjugations: PS2=20km, OKO37=0 km, OKO59=10 km.

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Fig. 10. Low layer static aberration correction for conjugations: PS1=-5km, OKO37=0 km, OKO59=10 km

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5. Conclusion and future work

A laboratory MCAO test-bed has been designed and completed for testing the primary WFS for the virtual WFS concept. Using least squares algorithm, a successful wavefront reconstruction has been obtained with the tomographic WFS working as the primary WFS. It has been also demonstrated that AO and MCAO systems could correct static aberrations of the single turbulent layer with OKO37 DM and OKO59 DM action. The correction obtained with 2 DMs showed a significant improvement in the image in terms of energy concentration.

It is planned to position several phase screens in the atmosphere module and also move DMs into the telescope system so that MCAO test-bed could be extended for testing Virtual WFS conditions (semi-open loop) with a more realistic turbulence model.