Simulator of Wave Front Phase Distortions
The term “adaptive optics” summarizes the wide range of tasks and optical devices proper which allow the reduction of negative influence of atmospheric or other inhomogeneities of refractive index on the efficiency of optical systems. In respect to the atmospheric tasks, the purpose for adaptive systems development is to minimize the distortions of beams and images. The principle of adaptive system operation consists in the control of wave front form of received or transmitted radiation. This control is performed with the help of active elements such as flexible mirrors or integrated controlled mirrors, controlled phase transparencies etc.
The design of modern systems of adaptive optics is indeed technically complicated task because practically all parameters reach their limit. In this regard the numerical simulation as the tool for the selection of optimal parameters is required at the stage of system design as well at the stage of functioning. The issues connected with the system adjustment refer to the group of specific issues. It is known that AOS adjustment is complicated scientific and engineering task. It is connected with the indeterminacy of system parameters as well as fluctuations of characteristics of optical radiation propagation medium (atmosphere, active medium of lasers etc.).
In reality it is quite difficult to divide these factors. AOS adjustment under the conditions of full-scale experiment results in large financial and time expenditures. Therefore, the actual and reasonable procedure consists in the division of the task of AOS adjustment in the previous stage of preparation and tuning of the system under laboratory conditions and the following stage – operation under the conditions of full-scale phase perturbations. It is necessary to make the special tool for such division. First of all, it is to reproduce the determined amplitude-phase distortions of wave front for AOS adjustment as the classical system of automatic control. Secondly, it should be capable of stimulation of these distortions by the substitution of various models of atmosphere and active medium of laser. This article is devoted to the practical aspects of such tool design.
SIMULATION OF PHASE DISTORTIONS OF OPTICAL WAVES
During the period from 1991 to 1995 we developed the first four-dimensional dynamic computer code in the world for the simulation of atmospheric adaptive optical systems [1-3]. The structure of such computer program is given in Fig. 1 in the form of two fragments: for the systems of beam formation and systems of image formation including the astronomic systems.
Particularly, using this program the calculations for the project of the first adaptive compound 10-meter telescope AST-10 were carried out in 1994 [4-5]. In 1997 the numerical analysis of parameters of large-size stellar interferometers was performed taking into account the influence of interferometer base orientation, distribution of the wind speed vector in atmosphere and models of external scale of turbulence . Later, in the early 2000s the analysis of the project of 50-meter astronomic telescope with adaptive optics was performed .
On the basis of modern technologies of concurrent programming the software for computer-based study of algorithms and layouts of adaptive optics in atmospheric laser applications was designed. The scalar parabolic equation for monochromatic wave can be solved numerically using the fast Fourier transform method for homogeneous media and splitting method for inhomogeneous media.
The model of adaptive optical system (AOS) includes geometry of radiating aperture and beam propagation path, vertical profiles of atmosphere parameters, concurrent splitting algorithm – Fourier transforms for the solution of diffraction and wave propagation equations, dynamic model of the “frozen” atmospheric turbulence with wide range of scales, model of wave front sensor and controlled deformable mirror. We suggest the appearance of structural layout (Fig. 2) of typical AOS which can be suitable for the simulation of adaptive systems too. Of course, the key element is the flexible mirror control unit which can be used during design activities as well as adjustment.
In order to perform the calculations on large initial data matrix the technologies of concurrent calculations are applied:
•INTEL Math Kernel Library (MKL) v.10.3,
•INTEL Integrated Performance Primitives (IPP) v.7.0,
•NVIDIA CUDA Toolkit 4.01.
Basic functions of MKL library:
•Vector and matrix algebra,
•Static sensor and random number generator,
•Fast Fourier Transform (1-D, 2-D, 3-D)
Main IPP libraries:
•Signal processing (convolution, correlation),
•Image processing (filtration, digital transform, coding and decoding).
It is known that the turbulent inhomogeneities of atmosphere have wide range of scales (relation of external scale to internal scale is more than 4 orders or L0/l0 > 10000) . For their correct presentation (Fig. 3) during computer simulation the two-dimensional computational meshes with large size and fast algorithms-generators were used including the basic program generator of uncorrelated pseudo-random numbers. Also this generator must have the maximum period because the standard library linear congruent sensors have the period of the order of computer word length or ~231–1 which is obviously insufficient for the matrices with large dimensions. Generators of Intel MKL and NVIDIA CUDA Toolkit libraries have the required properties – long period, excellent statistics and fast operation. SIMD-oriented fast Mersenne Twister generator SFMT19937 (Intel MKL) with the period = 219937 which is the best generator as of today was selected as the main generator.
In the program-generator of random processes and fields with set correlation properties the spectral method was used with the application of concurrent versions of two-dimensional fast Fourier transform (FFT) algorithms DFTi and CUFFT. In order to describe the time variations of the field of refractive index caused by wind-induced motion the algorithm of “infinite” dynamic random inhomogeneous medium simulation was developed. It is based on the hypothesis of “frozen” turbulence which consists in the further generalization of spectral-phase method (Lukin, Fortes, 1999) for the simulation of two-dimensional random fields with the known time spectrum. The model of the first-order autoregression with the moving average is its basis; it can be described through the discrete difference equation:
f (nT) = a1 f (n –1) T) + z (nT) = b1r ((n – 1)T) + r (nT).
In order to simulate the process of turbulence wind drift [9, 10] the method of spectral multiplication was used.
Comparison of frame sequence shows the capabilities to transfer the dynamic character of processes. In general, comparatively small variations (Fig. 4) cause the significant changes of the picture in time. This computer program can be effectively applied for the computer simulation of random processes and fields varying in time. Its implementation is possible with the use of concurrent algorithms. Every turbulent layer (phase screen) in this model has independent speed of movement which is transverse to the path. It is known that the discrete Fourier transform has periodicity property (the phase in boundaries of rectangular mesh is continuous). Therefore, the dynamic model formed with the help of FFT algorithm has the cyclic property and becomes infinitely continuous in case of assigning the drift which is multiple of the distance between mesh points during movement.
We consider AOS as the dynamic system with feedback and system with constant time lag [11-14] when the adaptive mirror reproduces the phase surface which corresponds to the wave front on system entrance aperture with considerable time lag.
In the paper  the adaptive correction with forecasting was suggested for the first time. On the basis of various schemes of statistic forecasting of random phase distortions the calculations of errors were carried out for the evaluation of medium field and average field intensity in AOS with constant time lag. The evaluation for enhancement of correction quality was obtained for the system with forecasting in comparison with the system having constant time lag. It was shown that it is possible to improve the time dynamic characteristics of adaptive system and reduce the influence of residual distortions caused by the time lag in feedback loop. For this purpose it is necessary to estimate the time correlation functions of the fluctuations of phase mode components preliminarily. Nowadays, the solution of wave front forecasting problem is proved by the positive results of numerical [11-14] as well as full-scale experiments.
Application of the new technological elements and devices for data processing in adaptive systems – high-speed video cameras, improved models of correctors and high-power computers – gives better operation speed and higher accuracy for the control of laser radiation, however, it does not solve the problems connected with AOS constant time lag. This problem can be solved by the only method: reduce the time lag implementing the adaptive correction with lead. In other words, it is necessary to send the control actions to the mirror calculated relative to the wave front which is being measured at the moment. In our software product the options of system operation with the transition from forecasting of modal components [12-14] to leading adaptive correction on a real time basis are suggested.
Besides, it is necessary to take into account that as a rule the AO intended for the correction of turbulent distortions of system laser radiation represents the dynamic system with constant time lag. It consists of the time for data processing, time for calculation of control actions to adaptive mirror, time for mirror adjustment including the lag and transitional processes of mechanical construction of this mirror. Thus, the correcting surface of adaptive mirror is formed from the surface of wave front existing on the adaptive system entrance aperture which intentionally has the lag equal to the value of system lag time. We solved this problem in our papers using the methods of leading adaptive correction. However, many times these methods do not take into account the error of wave front measurement carried out by Shack-Hartmann sensor and wave front forecasting error. Taking into consideration the necessity of forecasting for the increase of AO performance the Kalman filter is suggested to be used for the synthesis of statistically optimal algorithms of deformable mirror control. It should be noted that the studies connected with the phase forecasting issue for the short time intervals using the Kalman filter are reflected in our earlier papers [11-12].
The algorithm based on application of Kalman filter was qualitatively assessed in the task of wave front correction on the basis of measurements carried out by Shack-Hartmann wave front sensor. Analysis data was received by the simulation of radiation transmission through turbulent screen . The mean square deviations of the wave front calculated with the help of forecasting unit of Kalman filter from the wave front existing on AOS entrance aperture with the various drifts of turbulent screen are specified in Table 1.
Analysis of the data shown in Table 1 indicates that the application of Kalman filter (in comparison with the method of leading correction) gives the results of considerably higher quality. Carried out numerical experiments on the evaluation of wave front correction quality using the Kalman filter showed that this algorithm is relatively efficient and at the expense of its evaluation properties surpasses the algorithm of leading correction represented in the paper  by its accuracy.
Thus, we suggested the physical simulator of optical wave phase distortions which represents the active bimorph mirror with the system of high-voltage amplifiers controlled by computer. Simulator layout is given in Fig. 5. Practically, the simulated phase distribution of optical field occurs in the optical radiation reflected from active mirror. Under the influence of control signals the phase of reflected radiation can change in space and time.
Simulator consists of:
•Personal computer (PC);
•Program Simulator of WF distortions controlling the N-channel digital-to-analog converter (D/A converter);
•N-channel D/A converter controlling the high-voltage amplifiers;
•Set of high-voltage amplifiers loaded on bimorph actuators of deformable mirror;
•Deformable mirror of simulator located on the path of “standard” radiation source.
Computer with the special software sends the digital code to 16-channel digital-to-analog converter (D/A converter) which is the part of computer in the form of interface board. Analog signals with the amplitude up to ±10 V arrive from the D/S converter to the input of high-voltage amplifier. Amplified signals are sent to the actuators of deformable mirror (DM). Amplification coefficient of high-voltage amplifier (HVA) is ~ 30, signal amplitude is ~ ±250 V. Upon the voltage supply to actuator the DM reflecting surface becomes deformed in the relevant spot; it bulges upon the positive-going signal and sags upon the negative-going signal. Thus, the wave front of the light beam reflected from mirror obtains the phase distortions. The software allows setting of different phase distortions (static and dynamic distortions) in order to determine the properties of the adaptive system under test. The main technical characteristics of the simulator of phase distortions are specified in Table 2.
The simulator software has interface (see Fig. 6 for main window) including:
1.diagrammatic view of the controlled deformable mirror with numeration of controlled actuators;
2.four buttons for the selection of WF distortion type: stochastic mode, rotation, defocusing, general mode;
3.toolbar where the simulator operation is displayed in graphic form, particularly the behavior of actuator voltages (in the form of diagrams and animation);
4.buttons “Start” and “Stop”;
5.toolbar where the parameters of simulator operation are displayed.
The program contains four types of wave front distortions.
Stochastic mode. In this mode the following parameters are set: peak-to-peak voltage Umax which is the same for all actuators (this value is limited to ); typical frequency of voltage variation f which is the same for all actuators (this value is limited to ). At every point of time the actuator voltages are calculated according to the formula:
Ui = Umax . sin (2 π f + ϕi),
where ϕi are the random variables which are evenly distributed on the intercept (0, 2 π).
Rotation mode. In this mode the following parameters are set: peak-to-peak voltage Umax which is the same for all actuators (this value is limited to ); typical frequency of rotation f which is the same for all actuators (this value is limited to ). At every point of time the actuator voltages are calculated according to the formula:
Ui = Umax sin (2 π f t + ϕi),
where ϕi are the variables which ensure the rotation of mirror deformation picture (Table 3).
Defocusing mode. In this mode the voltages U which are the same for all actuators are set (this value is limited to ).
General mode. In this mode 3 variables are set for each ith actuator: peak-to-peak voltage Uimax (this value is limited to ); typical frequency of voltage variation fi (this value is limited to ); initial phase ϕi (0, 2 π). At every point of time the actuator voltages are calculated according to the formula:
Ui = Uimax sin (2 πt + ϕi).
The constant values of voltages can be set as special case (fi = ϕi = 0). The values U, Umax, Uimax, f , fi are entered in the relevant windows of the program main window.
The values and are entered in ini-file of the program. In this mode user can change the peak-to-peak voltage, typical frequency of voltages and initial phases. All these parameters are specified independently for each actuator.
Visualization toolbar is intended for the graphic representation of the processes occurring on the output of D/A converter. There are three variants of data view:
•Animation: information is displayed in the form of time-dependent cylindrical objects simulating the actuators which are located in the same order as on mirror. The algorithm of variation of graphic “actuators” height visually reflects the values of parameters set by user.
•Histogram: varying voltages of each actuator are displayed in the form of histogram.
•Diagram: information is displayed for each actuator separately (actuator number is selected in cell) in the form of the diagram of voltage dependence on time.
Hardware (active mirrors of bimorph type) of phase distortion simulator is shown in two fragments, Fig. 7. The sets of electronic amplifiers are shown in Fig. 8, 9. These elements are designed using the best world components [15, 16].
Developed simulator has the analog – phase simulator produced by the European Southern Observatory which represents the rotating retarder. However, in comparison with this analog the tool which was developed by us has a number of significant advantages :
•it allows the simulation of turbulence with the widest range of scales;
•computer program can be effectively applied for the computer simulation of time-dependent random processes and fields (the capability of its implementation is provided with the help of concurrent algorithms);
•the tool makes it possible to generate the turbulences of various types including the simulation of non-Kolmogorov turbulence;
•coherent turbulence simulation is provided;
•simulation of non-Kolmogorov turbulence with finite external and internal scales;
•the tool suggests several methods of calculation of control actions on the active bimorph mirror;
•it has the program estimation of required amount of polynomials in order to display the relevant phase distortions;
•capability of simulation of medium time variation, time evolution of scales.