The feasibility and performance of the pyramid wavefront sensor without modulation used in closed-loop adaptive optics system is investigated in this paper. The theory concepts and some simulation results are given to describe the detection trend and the linearity range of such a sensor with the aim to better understand its properties, and then a laboratory setup of the adaptive optics system based on this sensor and the liquid-crystal spatial light modulator is built. The correction results for the individual Zernike aberrations and the Kolmogorov phase screens are presented to demonstrate that the pyramid wavefront sensor without modulation can work as expected for closed-loop adaptive optics system.
© 2011 OSA
The pyramid wavefront sensor (PWFS) is an innovative device proposed by Ragazzoni  with the special characteristics of variable gain and adjustable sampling in real time to enable an optimum match of the system performance, which make it an attractive option for next generation adaptive optics system compared with the Shack-Hartmann [2,3]. The Telescopio Nazionale Galileo is the first astronomical telescope that implements the PWFS in its adaptive optics system  and the Large Binocular Telescope also used this sensor as a key component . Moreover, the PWFS which operates at near-infrared wavelengths is designed on the 3.5m telescope on the Calar Alto observatory in southern Spain .Furthermore, the PWFS is acted as a new and promising approach in the MCAO (multi-layer conjugate adaptive optics) to solve the problems related with the limited field of view , and it has also demonstrated that the PWFS has the ability to phase and align segmented mirrors .
At present most of the PWFS are used with modulation based on oscillating optical component in order to give a linear measurement of the local tilt, and it has been studied in the laboratory and successfully operated on the sky based on the modulation mode. Modulation plays a central role in the geometrical optical approximation to improve the linear range of the PWFS, which is very small in the nonmodulated case. Burvall discussed the approximate linear relationship between the wavefront derivative and the sensor response with modulation , and Vérinaud’s analysis explained the effects of modulation according to the diffraction optics theory . However, the sensitivity of the PWFS will decrease with the modulation amplitude and the moving parts increase the complexity of the system. There is an attractive idea that rise with the use of the PWFS is how to work without any dynamic modulation, which would greatly simplify the optical and mechanical design of the adaptive optics system and also give highest sensitivity as expected to be achieved. Costa presented a study showing that the higher order uncompensated residuals of the atmospheric turbulence are equivalent to a modulation for the lower compensated modes [11,12]. Ragazzoni introduced a static modulation method based on a light diffusing element, which causes a blur of the spot on the pyramid pin, in a similar way to the dynamic modulation , and then LeDue tested this idea using a holographic diffuser . Korkiakoski has found an iterative reconstruction method that can be used to evaluate the significance of the nonlinearities in the nonmodulated PWFS .
The aim of this paper is to clearly present the choices that can be made to design the PWFS without modulation using in the closed-loop adaptive optics system. Firstly, the special behavior of the PWFS without modulation is described with theoretical analysis and numerical simulations. The detection trend of the PWFS is studied to illustrate that the correction orientation of the nonmodulated PWFS is right to have the successive iteration process for the closed-loop adaptive optics system. (According to this paper, the detection trend is the normalized multiplication of the wavefront gradient with the PWFS signal. If the signal of the detection trend is a positive number, the polarity of the PWFS signal is accord with the wavefront slope and the closed-loop correction along this orientation will be effective.) At the same time, the behavior of the small linear range for the PWFS without modulation is considered in this paper. Finally the laboratory experimental setup based on this sensor and the liquid-crystal spatial light modulator (LC SLM) has been developed and the experimental results demonstrate the feasibility of the PWFS without modulation used in the closed-loop adaptive optics system.
2. Theory and simulation of the PWFS without modulation
The principle of the PWFS has been well described in many other papers so that we just refer the conclusions important to this work. When the PWFS is used as a wavefront sensor, four images of the pupil are created on the detector and the detection signals of the PWFS are calculated with the following formula :.
If no modulation is applied, the PWFS only yields the sign of the wavefront slope based on the knife-edge test, that is to say the linear range of the signal is very small according to the geometrical optics theory. Using the diffraction theory of the Foucault knife-edge test, the relationship between the wavefront phase and the PWFS signal is given by the following expression  (here the PWFS is simply replaced by a roof prism and interferences between the two pupils are neglected, and the same formula to the PWFS signal )
It is well known that the system can achieve successful closed-loop operation if the system satisfies two important conditions. The first one is that the detection trend of signal must be right during the closed-loop process, which means that the estimated correction is in the right direction to cause the residual variance of the closed-loop error to decrease continuously by successive iterations. The other one is that a small linear range is exist when the loop is closed successfully, which makes the closed-loop system to retain such a status in the closed-loop process. Due to the complexity of the theoretical formula (especially considering other impact factors, for example the interferences between the four pupils of the PWFS, the theoretical formula could become more complicated and hard to analyze), hence we adopt simulation method in the following discussion and present the preliminary simulation results to explain that it is possible to use the PWFS without modulation in the closed-loop adaptive system.
Firstly, to make a qualitative description of the detection trend of the PWFS without modulation, we derive a modal for the non-modulated PWFS measurement signal using diffraction optics and the elements for our simulation is consistent with the laboratory setup in the following text. Figure 1 shows the schematic diagram of this modal. The incoming wavefront is , and a Fourier transform (labeled in the figure) related the to the in the focal plane, where it is masked by the function representing the pyramid (in this case is the spatial filter effect introduced by the i-th PWFS facet in the focal plane). Finally an optical relay lens projects four pupil images on the detector, that is to say the intensity distribution ,, and can be accepted, and then the PWFS measurement signal and are formed from these four intensity patterns. The signal of represents the normalized multiplication of the wavefront local tilt with the PWFS signal (the same to). Here the signal is a quantitative representation of the detection trend and it has been normalized according to the maximum value in the simulation signal (because the detection signal amplitude is different for the different aberrations and we mainly want to explain that the actual slope and the PWFS signal have the same trend, we conducted a normalized processing to obtain the images with certain unity and intuitive). If the signal is bigger than zero, this is in agreement with the fact that the detection trend of the phase difference is right when using as the local wave-front derivative and the system will have the successive iteration process.
The simulated figures (shown from Fig. 2 –9 ) of PWFS without modulation for a series of Zernike modes (from Z3 to Z10, with RMS = 0.3λ) are obtained according to the above simulation modal. The Zernike order used is the Noll ordering without the piston mode . As shown in these figures, (i = 1,2,3,4) corresponds to the normalized intensity map created through the i-th quadrant of the PWFS, and denotes the normalized local derivatives on the wavefront (where ), and indicates the signal of the PWFS given by formula (1), and represents the normalized multiplication of the wavefront local tilt with the PWFS signal (the same to , and ). With the qualitative point of view, some characteristics are found from these figures: firstly, the figure has saturation effect due to the small linear range, but the outline of image is identical with the wavefront difference ; secondly, for the signal Trend and , this is in agreement with the comment previously stated that the detection trend of the phase difference is right when using and as the local wavefront derivative. To study the general behavior of these characteristics, the simulated images for the Kolmogorov phase screen with RMS = 0.47λ (according to D/r0 = 12, where D is the input aperture diameter, and r0 is the Fried parameter) composed of 3 to 40 Zernike modes are shown in Fig. 10 , and the same results can be obtained.
Secondly, the performance of the small linear range for the PWFS without modulation is considered. The approach of modal reconstruction has been used in the design of most existing adaptive optics systems and the goal of the reconstruction process is to find the modal coefficients of the modal polynomial, for example the Zernike polynomial. Therefore the following simulation is based on the modal reconstruction with the implement process as following: (1) a set of independent Zernike modes is introduced as the incoming wavefront and the corresponding signal of PWFS is recorded to get the interaction matrix; (2) with the computational technique of singular value decomposition, the reconstruction matrix is obtained; (3) the detection signal is calculated from the four images of the PWFS according to the formula provided above; (4) the mode coefficient is obtained using the multiplication of the reconstruction matrix with the detection signal of PWFS. It is clear that the linear range of the interaction matrix is important to make the system to have exact measurements, so the calibration amplitude of the Zernike modes to obtain the interaction matrix should be considered. To study the characteristic of the interaction matrices with different calibration amplitude, the following simulation is made: firstly four interaction matrices are done for 42 Zernike polynomials based on four different calibration amplitudes of the modes (these four calibration amplitudes corresponding to the modes with different RMS 0.05λ, 0.1λ, 0.2λ and 0.5λ respectively), and then with these matrices the constructed amplitude for the input Z3 polynomial is obtained (the constructed amplitude is plotted as a function of the applied amplitude, as can be seen in Fig. 11 .). We also get the simulated results for an aberration formed by Z3 with amplitude 0.06 and Z43 with amplitude 0.02 based on these four interaction matrices, which is illustrated in Fig. 12 . As can be seen from the Fig. 11 and Fig. 12, the calibration amplitude should be smaller than 0.1 to make the interaction matrix inside the linear range. For the calibration amplitude greater than 0.1, the constructed amplitude has an overestimation of the input aberration (which is consistent with the conclusions ) and the estimated modes are leaked out due to the nonlinearity of the interaction matrix, so it is impossible to make exact measurements.
3. Experiment results
The closed-loop optical layout of the adaptive optics system based on the PWFS without modulation and the LC SLM in the laboratory setup is sketched in Fig. 13 . The pyramid optical part is the double refractive pyramid that behaves as a single pyramid having a base angle equal to the difference between the two base angles of these two pyramids. The first pyramid component is made of BAK6 glass, and the other one is made of ZK9 glass. A polarized He-Ne laser with 632.8 nm wavelength is used as the light source for the experiment. After passing through the lens and the pinhole, the collimated plane laser output illuminates the LC SLM, and then the beam (with an aperture diameter of 4.61mm) is reflected from the SLM and incidences into the PWFS. By using the signal from the images of the PWFS, the reconstructed wavefront is loaded onto the SLM and the wave aberration is corrected. In this configuration the LC SLM has two functions. First it is used to introduce the static aberration into the system and afterwards to correct the distorted wavefront as the deformable mirror. The lens 1 forms an F/217 beam that is focused on the tip of the glass pyramid, and the choice of the F-ratio leads to a spot size of approximately 330μm on the pyramid tip, which reduces the influence of the light loss due to the pyramid edges and the roof-shaped. Then the four beams formed by the four facets of the pyramid are re-imaged on the CCD 1 through the lens 2. Each of the four pupils has an optically defined diameter of 0.277mm. The lens 3 and the CCD 2 form the far-field imaging optical layout, so that the actual shape of the far-field spot can be seen at any time.
Firstly, using a singular value decomposition algorithm we determined a modal least-square reconstruction matrix. Secondly, after introducing a given aberration by the SLM, the optical layout allows simultaneous recording of the PWFS signal. Thirdly, with the reconstruction matrix and the detecting signals, we can obtain an estimate of the Zernike modes of the aberration. Fourthly, the SLM produces the estimated Zernike modes correctly to compensate the wavefront distortion. Finally, the measurement continues repeated till the loop is stabilized.
The plots in Fig. 14 represent the iteration process in closed-loop compensation for the defocus, astigmatism, and coma aberration, with four different initial values of RMS (0.1λ, 0.2λ, 0.3λ and 0.4λ) respectively. The results shown are for 20 correction loops, and it can be seen that after 15 loops the RMS (root-mean-square) and SR (strehl ratio) of the surface measurements have a good convergence property.
The results of the laboratory measurements using the PWFS without modulation for the Kolmogorov turbulence phase screens (D/r0 = 40 and D/r0 = 80, without global slope) are obtained in Fig. 15 and Fig. 16 respectively. According to these figures, the corrected wavefront has a RMS of about 0.05λ in each case. Referring to initial RMS values (1.300λ and 2.750λ) of these two turbulence phase screens, it is clear that the wavefront correction using the PWFS without modulation is feasible.
Two points must be emphasized about the experimental results. Firstly, the LC SLM is used as the deformable mirror with the main advantage to generate the modes with high precision at small amplitudes, so the interaction matrix that connects the wave-space to the PWFS is determined inside the linear range. Additionally, the aberration applied to the SLC is static without change in real-time like the practice atmospheric turbulence, namely the experiment is just a “static closed-loop” so the temporal characteristics of atmospheric turbulence are not considered in the calculation of the data. It is planned in future to complement the laboratory system with simulated turbulence phase screen generated by the Hot-Air Turbulence Generator.
4. More considerations for the PWFS used in the practical astronomical AO system
In the above described content an important point must be illustrated that the regime of correction in which the PWFS is working in the experiment detailed above is not properly representing the conditions of the PWFS working in an astronomical AO system. The experimental system is able to produce a residual error of less than 0.1λ, and in this range the PWFS is found linear and providing good wavefront estimates. However this limit is rarely achieved in an astronomical AO system. In order to study the characteristic of the PWFS in the practical astronomical AO system, we have made a computer simulation of the PWFS using in the closed-loop AO system. On the circular entrance pupil we generate a matrix with the distribution of phases that is described by Zernike polynomials with 128 × 128 points. With the Fourier Transform we obtain the field in the focal plane, and then based on the PWFS model the four intensity images is obtained and the signal of the PWFS is calculated. The simulation allows us to get the interaction matrix of the PWFS system, and through SVD the reconstruction matrix is obtained. Firstly we have computed two reconstruction matrices (these two calibration amplitudes corresponding to the modes with different RMS 0.1λ and 0.3λ respectively). Secondly the signal is achieved by the PWFS modal and the aberrations estimated by the sensor through multiplication by the reconstruction matrix are then used to calculate an estimated wavefront, which is subtracted from the next phase screens. To stabilize the loop and study the performance a gain factor is multiplied with the estimated modes before applying the corrections to the deformable mirror (which is the gain in the following figures).
The result in Fig. 17 is according to the calibration matrix with amplitude 0.1λ, and the gain factor is changed from 0.5 to 3. With the aim to represent the real astronomical condition, one way to achieve this is by adding some phase perturbation of the proper amplitude to each of the loop iteration, so we add the random phase perturbation to each of the loop iteration with the RMS about 0.1λ. It is clear with the proper gain it is possible to close the loop without modulation. The result with the random phase perturbation about 0.15λ is illustrated in Fig. 18 and the same conclusion is obtained.
We know that the calibration amplitude plays an important role for the closed loop performance. Especially for a non-modulated PWFS this amplitude is critical because of the small linear regime, so we do the same simulation but with the calibration amplitude 0.3 (the results is shown in Fig. 19 and Fig. 20 ). It can be seen from the figures that although the calibration amplitude is out of the linearity range of the sensor, with the proper gain factor the system can also be closed.
The modulation linearizes the PWFS behavior and reduces the number of iterations required to converge to the proper estimate of the wavefront, and the PWFS without modulation will take more iterations to close the loop. This is because the PWFS is saturated in the beginning and therefore the correction applied to the deformable mirror is too small, so it takes more iterations to “pull” the PWFS out of saturation and to adapt for changes in the atmosphere. The simulation is analyzed according to different modulation amplitude and is shown in Fig. 21 (the modulation amplitudes are 1 times diffraction limit width, 2 times, 3 times and 4 times, respectively). As can be seen from this figure, along with the increase of the modulation amplitude the number of iterations required to achieve the certain closed-loop effect will gradually decrease and apparently the condition without modulation needs maximum number of iterations.
5. Conclusions and future work
The detection trend and the linearity range of the PWFS without modulation are explained in this paper and an optical system has been built in the laboratory to analyze the characteristics of the PWFS without modulation. For the defocus, astigmatism, coma and the atmospheric turbulence phase screens, the corresponding results are obtained during the closed-loop process. The experimental results demonstrate the feasibility of PWFS without modulation used in the adaptive optics system. The final stage of the experiment will be developed to work with the dynamic turbulence phase screen and the other deformable mirror. The actual astronomical observation AO system must consider a series of other issues, for example the atmospheric turbulence seeing, the turbulence time feature, the detector photon noise and read noise. So the PWFS without modulation applied in practical astronomical observation still needs a lot of work to do.
The project is supported by the National Natural Science Foundation of China (Grant No 61008038), and is also supported by West Light Foundation of The Chinese Academy of Sciences.
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