Static telescope aberration measurement using lucky imaging techniques

DOI 10.1007/s10686-012-9291-4 ORIGINAL ARTICLE Static telescope aberration measurement using lucky imaging techniques Marcos López-Marrero Luis Fernando Rodríguez-Ramos José Gil Marichal-Hernández José Manuel Rodríguez-Ramos Received: 15 September 2011 / Accepted: 9 February 2012 Springer Science+Business Media B.V. 2012 Abstract A procedure has been developed to compute static aberrations once the telescope PSF has been measured with the lucky imaging technique, using a nearby star close to the object of interest as the point source to probe the optical system. This PSF is iteratively turned into a phase map at the pupil using the Gerchberg Saxton algorithm and then converted to the appropriate actuation information for a deformable mirror having low actuator number but large stroke capability. The main advantage of this procedure is related with the capability of correcting static aberration at the specific pointing direction and without the need of a wavefront sensor. Keywords Static aberration Lucky imaging Phase retrieval Gerchberg Saxton 1 Introduction Most present-day telescopes have been designed bearing in mind the seeing statistics as the reference for the error budget in static aberrations. It was M. López-Marrero (B) J. G. Marichal-Hernández J. M. Rodríguez-Ramos University of La Laguna, La Laguna, Spain e-mail: mlmarr@ull.es J. G. Marichal-Hernández e-mail: jmariher@ull.es J. M. Rodríguez-Ramos e-mail: jmramos@ull.es L. F. Rodríguez-Ramos Instituto de Astrofísica de Canarias, Tenerife, Canary Island, Spain e-mail: lrr@iac.es

expected for the telescope mirror to support flexures and not to introduce significant aberration when pointing to the different directions on sky, in comparison with mean seeing figures. New observing techniques like Lucky Imaging (FastCam) [1, 2] have demonstrated their capability to routinely provide sharp images in the I-Band, at telescopes in the range from 1 to 4 m and even to obtain Airy rings at the 1.5 m TCS at Observatorio del Teide (Tenerife, Canary Is., Spain). Once the effect of the atmospheric turbulence is minimized using this Lucky Imaging technique, the remaining aberration due to mirror misalignment is the limiting factor in spatial resolution, assuming that the shape of the primary mirror has been adequately adjusted, whenever feasible. This aberration depends on the pointing direction, mostly due to the effect of gravity on the mechanical structure. A procedure has been developed to compute these static aberrations after the Point Spread Function (PSF) is measured with the lucky imaging technique, using a nearby star as the point source to probe the optical system. This reference star is not expected to be present in the same field of the object of interest, but will produce similar gravity effects in the telescope primary mirror shape. This PSF is iteratively turned into a phase map at the pupil using the Gerchberg Saxton algorithm [3] and will be converted in the future to the adequate actuation information for a deformable mirror having low actuator number but large stroke capability. The main advantage of this procedure is related with the capability of correcting static aberration at the specific pointing direction and without the need of a wavefront sensor. This paper describes the PSF measuring and phase recovery that have been accomplished and tested at the 1.5 TCS at Obs. del Teide (Canary Is.) The algorithm has been developed and tested using numerical software but the aim of the project is to use FPGAs (Field Programmable Gate Array) as computing platform to execute the algorithm in real time. 2 Static aberration recovery The static aberration recovery algorithm will be described in this chapter. This method uses high resolution (almost diffraction limited) star images where the atmospheric turbulence has been removed. Recovering the static aberration due to mirror misalignment and deformations can be referred as recovering the star radiation wavefront at the telescope pupil. 2.1 Wavefront at the telescope pupil Let 1 (x) be the complex amplitude of the electromagnetic field of an extended object at the celestial sphere. Assuming a distance near to infinite

between the celestial sphere and the atmosphere, the complex amplitude at this point will be: 2 (u) = F { 1 (x)} =A(u)e iφ(u) (1) This ideal radiation scheme is distorted by the atmosphere and the telescope optics, both in amplitude and phase. The complex amplitude obtained after these two distortions can be written as: 3 (u) =[A(u) + a(u)]e i[φ(u)+θ(u)], (2) where a(u) is the scintillation (amplitude distortion) and θ(u) is the atmospherical and telescope wavefront phase (phase distortion). The lucky imaging technique [1] consist of selecting the best frames and displacement correcting before co-adding and virtually removes the effects of the atmosphere to a very high degree, especially in the low-order Zernike modes, by averaging out the turbulence in time scales of several minutes. It is then reasonable to associate any remaining static wavefront distortion to the telescope optics, mainly the primary mirror shape, and assume that all the effects are related to phase distortion, neglecting any amplitude distortion in the telescope optical path because reflectivity non-uniformities will be much smaller than shape effects. Then, neglecting scintillation, the radiation at the telescope focal plane will be: 4 (x) = F 1 { 3 (u)} (3) This is a complex amplitude with module and phase. Only the intensity of light (square module) is sensed to get the image of the celestial object, so the phase information is lost and the information we get is: I(x) = 4 (x) 4 (x) (4) Assuming that the object was a natural reference point star and that all the atmospheric turbulence has been removed, this intensity should be the Point Spread Function (PSF) of the system where the aberrations are only due to the telescope. The proposed algorithm should obtain the phase of the complex amplitude at the telescope pupil φ(u) from the PSF or the intensity of the radiation at the focal plane I(x). More generally: the phase of a complex object Fourier transform should be recovered from this object square module. 2.2 Wavefront recovery algorithm The static aberration recovery algorithm is based in a variation of the iterative Gerchberg Saxton [3] algorithm by J. R. Fienup [4] that aims to rebuild an object from the modulus of its Fourier transform. The Fienup algorithm will be introduced first and our variation will be presented afterwards.

Fig. 1 Block diagram of the Fienup algorithm Let f (x) be an object and its Fourier transform F(u) = F(u) e iφ(u) = F { f (x)} = f (x)e i2πu x dx, (5) where the vector position x represents a two-dimensional spatial coordinate and u the spatial frequency. For sky objects, f (x) is a real, non-negative function. The problem is to find an object that is consistent with all the known constraints: that it should be non-negative and that the modulus of its Fourier transform equals the measured modulus, F(u). The problem is solved with an error reduction system as shown in Fig. 1.At the k th iteration, g k (x), an estimate of the object, is Fourier transformed; the Fourier transform is made to conform to the known modulus; and the result is inverse-fourier transformed, giving the image g k (x). Then, the iteration is completed by forming a new estimate of the object that conforms to the objectdomain constraints: { g k (x), x γ g k+1 (x) = (6) 0 x γ where the region γ includes all points at which g k (x) violates the constraints. The principal constraint is that the object should be non-negative. The iterations can be started by using a sequence of random numbers for g 1 (x). This approach reconstructs an object from the modulus of its Fourier transform, but our goal is to recover the phase of a complex object Fourier transform from the modulus of the object. With some changes, the Fienup algorithm can be adapted to this other purpose. The variation, shown in Fig. 2, is described as follows. The iterative algorithm begins in the upper-left corner of the block diagram, where θ(u) is the estimated wavefront that will converge to the recovered aberration. This estimation is initially constant and its shape may have certain Fig. 2 Block diagram of the Fienup variation algorithm. The initial data (PSF) is introduced. It should be noticed that in this variation the objects are complex on both domains

influence in the final recovered wavefront. This influence is explained in the results chapter. The telescope itself constitutes a physical system that defines the pupil plane constraints by nature, so applying these constraints means finding the product between the complex amplitude -whose phase is the estimated phaseand the annular mask of the telescope. Besides, another pupil plane constrain is considered. If the scintillation is neglected, the modulus of the complex amplitude which phase is the estimation is made one. On the other side, the focal plane constrains are those that make the modulus of the object approach to the actual measured object. This way, and taking into account that what is sensed in the CCD is the intensity of the electromagnetic field (see (4)), the modulus is just changed by the PSF square root. Using the expression shown in Fig. 2, the constraints in both planes can be summarized like this: Focal plane constraints: g (u) = PSF (7) { G (u) =1 Pupil plane constraints: G(u) = G (u) AM(u), (8) where AM(u) is the annular mask corresponding to the telescope used to get the PSF (Fig. 3). At this point the whole algorithm is described. Figure 4 shows the algorithm with constraints in both planes and example images for modulus and phase. 2.3 Error measurement An error computation is done at each iteration, to evaluate the convergence of the algorithm. The difference between the original PSF and the PSF obtained from the estimation of θ(u) is computed. If the estimation is meaningful, the Fig. 3 Annular mask representing the pupil of a cassegrain telescope. The value is zero in the shadowed area and unity in the clear area

Fig. 4 Diagram for Fienup variation algorithm with example images for modulus and phase. The first iteration begins at 1, where modulus and phase are both unity inside the annular mask. Then, a inverse Fourier transform is done 2. The focal plane constraints are applied by replacing the modulus obtained by the PSF (initial data) 3. The result is Fourier transformed 4 and the pupil plane constrains are applied. The loop restarts with modulus one and phase θ (u) with the annular mask superposed 5 difference should converge to a small value. At k th iteration, an average of the differences is computed this way: PSF g(x) Avg k = x N, (9) where N is the number of pixels of the image and the summation in x refers to the sum of all the pixels of the image. Then, a variance is computed this way: Var(x) k = Var(x) k 1 + ( PSF g(x) Avg k ) 2 (10)

Finally, the accumulated energy of the image until this iteration is computed like: E k = 1 Var(x) k (11) N k x In every case E k tends to zero, so we can say the approximation of θ(u) is good. 3 Simulations and results Before using real PSF images, the system has been tested using synthetic PSF created from known wavefronts. Those wavefronts have been constructed combining Zernike polynomials at different levels. First of all, PSFs made from a single Zernike polynomial were used in the algorithm and the results were nearly perfect for the 15 first Zernike polynomials. Then, simulated static aberration wavefronts were used and finally, real FastCam images. The results of the recovered wavefronts are shown in this section. 3.1 Zernike polynomials recovery Some examples of recovered wavefronts are presented in this section. The procedure is the following: A Zernike figure is taken and used as the phase of a complex map, then the annular mask is superposed and the result is inverse Fourier transformed. The resulting PSF is the input data for the iterative algorithm. The output data can be compared with the Zerinke figure in order to prove that the result is the expected. Fig. 5 Results for single Zernike polynomials Z 4 and Z 7 wavefronts recoveries. First column: Zernike figure used to get the PSF. Second column: Recovered wavefront. Third column: Evolution of E k during the execution. Fourth column: Section of the original wavefront (continuous line) and the recovered one (dashed line)

In Fig. 5 the following images and graphics are presented: the Zernike figure used to generate the synthetic PSF, the recovered wavefront, the evolution of E k during the execution of the algorithm and a section of the original wavefront and the recovered one. In some cases the wavefront is inverse recovered. In this cases, E k does converge to zero and this happens because the original wavefront and the inverted one have both the same inverse Fourier transform, that is, the same PSF. Despite of the pedestal that appears in the recovered wavefront, it can be appreciated that the shape of both wavefronts are nearly identical. 3.2 Simulated static aberration phase recovery The equivalent procedure is used to test the algorithm in this case. Now, instead of single Zernike polynomial figures, simulated wavefronts are used to create the PSF that is used as input data for the algorithm. These wavefronts have been created with a combination of many Zernike polynomials randomly weighted following Kolmogorov turbulence statistics and resulting in a wavefront far more complex than the typical static aberration. It has been considered that the magnitude of the expected static aberration will always be less than λ/4, which is the widely accepted value for the primary mirror polishing quality. Fig. 6 Result for simulated static aberration wavefront recovery. Two sections are shown to detail the recovery information. Original (continuous line) and recovered (dashed line)

The same scheme is used in Fig. 6 to show the results of the recovered wavefronts. Again, some wavefronts are recovered inverted or rotated, due to the non-univoque character of the Fourier transform but the recovering is as good as in the Zernike figure simulation. 3.3 Real static aberration phase recovery (FastCam) FastCam [1] employs the Lucky Imaging technique [2] to improve the resolution of the images and freeze out the atmospheric turbulence. The principle of FastCam is based in the use of the turbulence statistics. A small number of sharp images can be selected in every set of them. These images are identified, aligned and co-added, producing a sharp image. Theoretically, some of the selected frames are diffraction limited, or nearly, but as we are not able to measure the fraction of flux in the wings of the PSF from a single exposure, we will say there is a diffraction limited core in our FastCam image. The described procedure removes nearly all atmospheric distortion, as can be seen in Fig. 7, so if one of those images is used in the presented algorithm, the recovered distortion should be caused by static aberrations and misalignment of the telescope. An important issue should be addressed here: the main drawback of Lucy Imaging methods is the presence of broad wings surrounding the PSF in the resulting image. This wings are the product of re-centering and co-adding the selected speckle images where atmospheric distortion is present and will be considered as a residue. In fact, the wings have to be erased in order to guarantee a coherent recovering of the static aberration so, in practice, only the brightest pixels of the image are used, where the information about static aberrations should be contained. A normalization and an apodization are applied and the resulting edges are smoothened. Now we can say the FastCam image should have a corresponding wavefront that can be recovered using the proposed algorithm. A consideration has to be done here: erasing PSF residual atmospherical wings means losing information about spherical aberration in the future wavefront recovery, as spherical aberration is contained in the area of the image that is being discounted. Simulations are coherent with this issue and Fig. 7 Speckle image and FastCam processed images of ADS8446 at the TCS telescope. 500 images out of 10,000 were used to produce the result. Notice that it is not possible to identify the binary star in the speckle image, but it is clearly visible after processing

Fig. 8 Central pixels for FastCam image (left)of FK209 and the area used to recover the static aberration (right) where the Airy disc is visible spherical static aberration could not be recovered from a simulated FastCam PSF where a static spherical aberration was present. This is not considered a significant problem because spherical static aberration is usually stable during observations and can be compensated during operation changing the telescope focus. Using the right image shown in Fig. 8 once apodization is applied, the obtained result is a wavefront corresponding to the static aberration in the pupil of the telescope at the moment of the observation. Due to the arctangent operation used when computing the phase, some jumps usually appear in the recovered phase map, but this phase wrapping can be corrected easily without using any additional unwrapping algorithm because of the simplicity of the phase jumps. The results are shown in Fig. 9. In order to evaluate the results, a wavefront sensor (Shack Hartmann) should be used to physically determine the shape of the wavefront that reaches the telescope pupil. This task should be identified as a calibration of the proposed algorithm and will be undertaken in the next future. The recovered wavefront can be decomposed in Zernike polynomials in order to clarify the obtained results. The decomposition for the first ten polynomials of the unwrapped wavefront is shown in Fig. 9 using the ISO arrangement. Low-order polynomials (piston, tip and tilt) are not relevant in this context, while astigmatism and coma will usually be the most significant contributions (Table 1). Fig. 9 Wrapped (left) and unwrapped (right) recovered phase maps. The unwrapped one shows the static aberration

Table 1 Zernike decomposition of the recovered wavefront Zernike polynomial Magnitude (rad) Z(0) Piston 0.8641 Z(1) Tip 1.0546 Z(2) Tilt 1.1291 Z(3) Defocus 0.1547 Z(4) Astigmatism 0.5071 Z(5) Astigmatism 0.4409 Z(6) Coma 0.2592 Z(7) Coma 0.4359 Z(8) Defocus, 2nd order 0.1368 Z(9) Trefoil 0.2915 4 Conclusions and future work As far as the astronomical application of this system is concerned, the possibility of measuring aberrations of a telescope using only its high resolution images, without invading the mechanical system, is indeed a significant improvement in the astronomical data acquisition area. The simulations were done taking into account the real magnitude of the static aberration, always below λ/4. All the results were obtained working within this limits. The influence of apodization has been considered and do not mean a significant drawback of the method. Our next task will be calibrating the algorithm. Controlled deformations can be mechanically created in the telescope pupil and then measured to find coincidences and calibration errors. Future works include the implementation of this algorithm in FPGA, focusing in real time execution. Using an optical beam splitter and a deformable mirror the aberration can be immediately removed, even if the telescope is following a celestial object during a whole observation night. This supposes the telescope mechanics will slightly bend or suffer misalignments during this period. Despite being pending for calibration and practical verification, the algorithm has shown to be as viable in simulations as in data obtained on real observations. Acknowledgements This work has been funded by Programa Nacional I+D+i (Project AYA 2009-13075) of the Ministerio de Educación y Ciencia, by ACIISI Camera 3DTV project and by the European Regional Development Fund (ERDF). References 1. Law, N.M., Mackay, C.D., Baldwin, J.E.: Lucky imaging: high angular resolution imaging in the visible from the ground. Astron. Astrophys. 446, 739 745 (2006) 2. Oscoz, A., et al.: FastCam: A New Lucky Imaging Instrument for Medium-Sized Telescopes. SPIE, Bellingham (2008) 3. Gerchberg, R.W., Saxton, W.O.: A practical algorithm for the determination of the phase from image and diffraction plane pictures. Optik 35, 237 (1972) 4. Fienup, J.R.: Reconstruction of an object from the modulus of its Fourier transform. Opt. Lett. 3(1), 27 29 (1978)