A Multi-Deformable-Mirror 500 Hz Adaptive Optical System for Atmospheric Turbulence Simulation, Real-Time Reconstruction, and Wavefront Correction Using Bimorph and Tip-Tilt Correctors

Abstract

Atmospheric turbulence introduces distortions to the wavefront of propagating optical radiation. It causes image resolution degradation in astronomical telescopes and significantly reduces the power density of radiation on the target in focusing applications. The impact of turbulence fluctuations on the wavefront can be investigated under laboratory conditions using either a fan heater (roughly tuned), a phase plate, or a deformable mirror (finely tuned) as a turbulence-generation device and a wavefront sensor as a wavefront-distortion measurement device. We designed and developed a software simulator and an experimental setup for the reconstruction of atmospheric turbulence-phase fluctuations as well as an adaptive optical system for the compensation of induced aberrations. Both systems use two 60 mm, 92-channel, bimorph deformable mirrors and two tip-tilt correctors. The wavefront is measured using a high-speed Shack–Hartmann wavefront sensor based on an industrial CMOS camera. The system was able to achieve a 500 Hz correction frame rate, and the amplitude of aberrations decreased from 2.6 μm to 0.3 μm during the correction procedure. The use of the tip-tilt corrector allowed a decrease in the focal spot centroid jitter range of 2–3 times from ±26.5 μm and ±24 μm up to ±11.5 μm and ±5.5 μm.

1. Introduction

As is known, radiation that propagates through an atmosphere is distorted due to turbulent refractive index fluctuations. This can limit telescopes’ resolution as well as decrease laser radiation coherence [1,2,3,4,5,6,7,8,9]. Atmospheric turbulence also affects the wireless transmission of information and energy by means of optical radiation [10,11,12,13]. In particular, it limits optical communication channels in free space [14], wireless optical communication [15,16], and low-Earth-orbit satellites [11]. It also limits the ability to destroy unmanned aerial vehicles or space debris [17,18,19], to create a beam of the desired shape [20,21], to focus a beam inside the aperture for laser communication tasks [22], to increase the radiation power density on a target [23,24] (i.e., laser cutting) and to improve the accuracy of beam positioning for the optical recording of information.

The main reasons for the low efficiency of systems for wireless signal transmission by means of optical radiation are the diffraction of a light beam, radiation scattering via an atmospheric aerosol [25,26,27], and the influence of atmospheric turbulence [28,29,30,31,32,33,34,35,36,37,38,39,40]. The problem of wavefront degradation due to the atmospheric turbulence influence [41] has been studied for more than 50 years, but it is still relevant. And one of the most efficient ways to overcome this issue is to use adaptive optics [42,43].

The problems of increasing a laser radiation propagation range through the atmosphere using adaptive optics methods are being solved by research teams from Russia, Germany, Italy, USA, and the Netherlands [44,45,46]. The Air Force Maui Optical Station (AMOS) [47] in Hawaii, USA, where the ADONIS (Daylight Optical Near-Infrared System) system was built in 1993–1995, initially did not use the adaptive optics on its 1.2 m telescope. After that, the ADONIS system was moved to the 3.6 m EOAR telescope and was equipped with the adaptive optical system. In [48,49], the authors describe a vision system that uses a conventional adaptive optical system and works on the 2.5 km horizontal atmospheric path. Another vision system with adaptive optics was included; produced in the Fraunhofer Institute [50,51], it was developed to run under urban conditions. This adaptive system contains a conventional deformable mirror and works with a frequency of 800 Hz. In [52], the authors describe the use of an adaptive optical system within a 0.35 m telescope on a 20 km slant atmospheric path. In [53], the authors describe an adaptive system with a deformable mirror and a tip-tilt corrector for a 0.12 m telescope on a 3 km horizontal atmospheric path.

Most of the papers described above are devoted either to the compensation of turbulence-phase fluctuations or to turbulence generation using spatial light modulators. The novelty of our research is (1) the use of a bimorph deformable mirror in kinematic tip-tilt mounting as a turbulence-generation device and (2) the implementation of the full research cycle, which includes the simulation, generation, and correction of turbulence fluctuations. We numerically simulate phase-screen sequences using Kolmogorov theory, we experimentally reconstruct/generate a sequence of phase screens using the bimorph mirror in tip-tilt housing, and we compensate for the induced distortions using another bimorph mirror and a standalone tip-tilt corrector. The results show that the whole system works in real time—the turbulence-generation and turbulence-correction modules perform independently. The use of such a system allows for the investigation of the limitations of deformable mirrors for the correction of atmospheric turbulence fluctuations under laboratory conditions.

The sections below describe how we overcome all of these challenges, assembling and investigating a multi-deformable-mirror-adaptive optical system with turbulence-generation and wavefront-correction modules.

2. Materials and Methods

2.1. An Experimental Setup

In our previous experiments, we used a fan heater as a device to generate turbulence-phase fluctuations [54]. Prior to an experiment, we measured the wavefront distortions induced via the fan heater and confirmed that the statistics of phase fluctuations were close to Kolmogorov’s statistics [55]. But the problem of the fan heater is that there is no fine tuning of the turbulence parameters—the operator cannot set the desired wind velocity, aperture diameter, or Fried radius. To overcome this issue, optical phase plates (OPPs) can be used [56,57]. An optical phase plate is a device that can simulate real atmospheric conditions in a lab setting. Another even more flexible approach is to create a completely controllable turbulence generator. To accomplish this, we implemented the model of Kolmogorov’s phase screen simulation, developed the algorithm and software to reconstruct the set of simulated phase screens in real time by means of the response functions of wavefront correctors, and assembled the experimental setup with the bimorph mirror inside tip-tilt kinematic mounting that acts as a turbulence-generation device.

In [54], we used the adaptive system based on the FPGA, which allowed for the achievement of the correction frequency up to 2 kHz. FPGA processed the image bytes coming from the wavefront sensor camera “on the fly” (as they arrive from the camera), which made it possible to calculate the vector of control voltages until the whole image frame is received. This system was successfully assembled and tested, and it demonstrated rather good results—the residual error of phase fluctuations was decreased 10 times during the 2 kHz correction procedure.

However, there are a few disadvantages to systems based on the FPGA. First, a camera with an FPGA is a relatively expensive device. Second, the programming process of an FPGA processor is very time-consuming. Third, systems based on an FPGA are difficult to scale—it is impossible to rapidly implement changes if necessary. Assuming that there are also several applications in which it is necessary to compensate for phase fluctuations with a correction frequency below 1 kHz, the use of video cameras without an FPGA processor can be considered. In this case, the challenge moves to the optimization plane of the execution time of the measurements and correction algorithms as well as to the selection of an appropriate camera sensor that has the capacity to achieve the desired frame rate.

The principal optical scheme and the photo of the adaptive system for the generation and compensation of atmospheric turbulence are presented in Figure 1.

A fiber-coupled diode laser with a wavelength of 0.532 μm was collimated with an achromatic lens. The collimated laser beam is then incident on the first 60 mm bimorph deformable mirror with 92 electrodes in the kinematic tip-tilt mounting—this construction allows us to introduce controllable wavefront distortions that, in our case, are equivalent to those induced by atmospheric turbulence. The distorted beam passes through the conjugating telescope (two lenses that conjugate the planes of two bimorph mirrors) and falls on the second 60 mm bimorph deformable mirror with 92 electrodes, which compensates for the introduced low- and high-order wavefront distortions. The beam then goes to the standalone tip-tilt corrector, passes through the matching telescope and falls on the Shack–Hartmann wavefront sensor based on a CMOS camera Baumer VCXU 13M, Stockach, Germany [58]. Then, part of the beam is reflected off the beam splitter and falls on the CMOS camera The Imaging Source DMK 23UX174, Bremen, Germany [59], which analyzes the intensity distribution of a focal spot in the far field. The sections below provide information about the optical components used in the experimental setup.

2.2. A Bimorph Deformable Mirror

The most significant turbulence-induced aberrations are tilts, defocus, and a few other low-order aberrations. Of course, wavefront aberrated by a turbulence is not limited to such aberrations. There are also high-order aberrations also, but their impact is not as high as that of low-order ones. Bimorph deformable mirrors are modal correctors, and thus they are very efficient in a reconstruction/compensation of low-order large-amplitude wavefront aberrations.

A conventional bimorph deformable mirror consists of a passive glass or silicon substrate coated with reflective coating and two piezoceramic disks glued to it [60]. A common (1st) electrode was applied to the internal piezoceramic disk, which was designed to change the curvature of the reflecting surface of the mirror. A set of control electrodes from the 2nd to the 92nd positions was applied to the external piezoceramic disk [61,62,63,64]. The piezoceramic disk can be divided into sections using laser engraving technology. The wires can be connected to the control electrodes via ultrasonic welding [65]. The reflecting coating can be dielectric, aluminum, silver, or gold [66,67], depending on the application in which the mirror is intended. The principal scheme of the bimorph mirror, the scheme of the electrodes’ layout, and photos of the developed mirror are presented in Figure 2, and the mirror characteristics are given in

Table 1. Parameters of the bimorph deformable mirror in tip-tilt mounting.

Parameter	Value
Substrate aperture	65 mm
Clear aperture	60 mm
Substrate material	Silica
No. of PZT	2
No. of control electrodes	92
Type of actuators	PZT discs
Actuators geometry	sectorial
Maximum input voltage	−300–+500 V
Maximum element diameter for tip-tilt mounting	152 mm
Angle displacement range of tip-tilt mounting	±35 mrad

.

2.3. A Tip-Tilt Corrector

Most of the spectral power of the Kolmogorov spectra phase fluctuations was within the first few low-order wavefront aberrations, i.e., tip-tilt, defocus, astigmatism, and coma, according to Taylor’s hypothesis [68]. Moreover, tip-tilt aberration has a greater than 40% impact on the spectral power, indicating that these aberrations should be considered. One effective device to compensate for tip-tilt aberration is a mirror with magnetostriction [69] or piezoelectric actuators [70,71].

The piezo tilt stage (two-axis motion) was based on a parallel kinematic design with a coplanar axis and moving platform. Four piezo actuators were placed at a 90° angle interval paired differential control distribution. Two pairs of differentially driven actuators provide the highest achievable angular stability over a wide temperature range. The tilt motion is achieved by two pairs of piezo actuators in a push–pull motion controlled by a bridge connection circuit.

Piezoelectric actuators typically consist of stacks of multilayer piezoelectric materials enclosed in a metal housing for preloading [72]. Applying a modulated high-voltage signal to piezoelectric ceramics causes a small deformation of the material. Compared with electromagnetic coils, piezoelectric actuators generate enormous forces in a smaller housing with a much higher frequency response. However, such actuators have a limited range of motion, exhibit hysteresis, and must have a preload mechanism to prevent the piezoelectric stack from delaminating and to overcome external forces. The combination of the high operating frequency, load, and stability of small tilt settings facilitates the use of piezoelectric actuators to drive mirrors to correct wavefront tilts. The design of the tilt corrector, which is based on piezoelectric actuators along with an electronic drive circuit, is shown in Figure 3.

The local deformation of one pair of piezo actuators forms a rotation center (pivot point) and at the same time preloads the piezo actuator. The drives operate electrically on a bridge circuit, which is supplied with a fixed voltage and controlled by an alternating voltage.

The tip-tilt corrector’s linearity was calibrated and tested. The diagram in Figure 4 demonstrates the response linearity of 0.2%.

The main parameters of the tip-tilt corrector are listed in

Table 2. Parameters of the tip-tilt corrector.

Parameter	Value
Clear aperture	56 mm
Substrate material	Silica
No. of control actuators	4
Angle displacement	±0.4 mrad
Voltages range	−0–+180 V
Control frequency	More than 200 Hz
Resolution	0.02 μrad
Unloaded resonant frequency	1.5 kHz
Unloaded step time	2 ms
Electrical capacitance	7.2/axis μF
Operating temperature	−20~80 °C
Material	Aluminum
Closed-loop linearity	0.2% F.S.
Closed-loop repeatability	0.02% F.S.

.

2.4. A Shack–Hartmann Wavefront Sensor

In order to measure the wavefront aberrations, a Shack–Hartmann wavefront sensor [73,74,75,76,77,78] was used. A Shack–Hartmann sensor is a robust, easily calibrated, and widely used device in a large and diverse set of applications, and it is primarily used to measure wavefront aberrations of radiation passing through a turbulent or scattering atmosphere [79], biological tissues [80], etc.

The principle of a conventional Shack–Hartmann sensor can be described as follows. The wavefront of the incident light is divided into a set of subapertures using a microlens array. The microlens array is a thin, flat base with a grid of microlenses etched on it. Each microlens has (usually) a diameter of 100 to 300 µm and a focal length f of 3–8 mm. A field of focal spots is created at the focal plane of the microlens array when the radiation passes through this array (Figure 5).

Because the diameter of each microlens is small, wavefront W is assumed to be flat and only exhibit tip-tilt aberration within a single microlens. In the case of no aberrations (i.e., a wavefront is flat and parallel to the plane of the microlens), the radiation is focused at the center of the corresponding subaperture of the sensor. If the wavefront in a microlens has a non-zero tip-tilt, then the focal spot is displaced (S_x and S_y) from the center of the subaperture in proportion to the tip-tilt value. In other words, if we measure the displacements S_x and S_y of the focal spot per X and Y axis, we can obtain the values of the partial derivatives ∂W/∂x and ∂W/∂y of the wavefront W within each subaperture:

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial W(x,y)}{\partial x}}\\ {\displaystyle \frac{\partial W(x,y)}{\partial y}}\end{array}\right\}={\displaystyle \frac{1}{f}}\left\{\begin{array}{c}{S}_x\\ S_y\end{array}\right\},$$

(1)

where ${\displaystyle \frac{\partial W(x,y)}{\partial x}}$, ${\displaystyle \frac{\partial W(x,y)}{\partial y}}$ are the partial derivatives of wavefront W, f is the focal length of each lenslet, $S_x$, $S_y$ are the displacements of each focal spot within a subaperture per the X and Y axis.

On the other hand, to describe and visualize a wavefront surface analytically, one can use a polynomial approximation, for example, B-splines [81] or Zernike polynomials [82,83,84,85], which are commonly used in optics. Thus, the partial wavefront derivatives ∂W/∂x and ∂W/∂y can be defined analytically using Zernike polynomials:

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial W(x,y)}{\partial x}}\\ {\displaystyle \frac{\partial W(x,y)}{\partial y}}\end{array}\right\}=\left\{\begin{array}{c}{\displaystyle \sum _i^N}{a}_i·{\displaystyle \frac{\partial Z_i(x,y)}{\partial x}}\\ {\displaystyle \sum _i^N}{a}_i·{\displaystyle \frac{\partial Z_i(x,y)}{\partial y}}\end{array}\right\},$$

(2)

where $a_i$ represents a Zernike coefficient representing the aberration value, Z_i represents the i^th Zernike polynomial, N represents the number of Zernike polynomials used.

The values can also be calculated from the measured displacements S_x and S_y of the focal spots on a Shack–Hartmann sensor:

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial W(x,y)}{\partial x}}\\ {\displaystyle \frac{\partial W(x,y)}{\partial y}}\end{array}\right\}=\left\{\begin{array}{c}{\displaystyle \sum _i^N}{a}_i·{\displaystyle \frac{\partial Z_i(x,y)}{\partial x}}\\ {\displaystyle \sum _i^N}{a}_i·{\displaystyle \frac{\partial Z_i(x,y)}{\partial y}}\end{array}\right\}={\displaystyle \frac{1}{f}}\left\{\begin{array}{c}{S}_x\\ S_y\end{array}\right\}.$$

(3)

Finally, we obtain an overdetermined system of linear equations with unknown coefficients $a_i$. By solving the least squares problem [86], we obtain the coefficients $a_i$. The wavefront then be analytically described and analyzed.

For our experimental setup, we developed the Shack-Hartmann wavefront sensor based on a CMOS industrial camera Baumer VCXU 13M [58]. This camera operates at 222 fps at full resolution of 1280 × 1024 pixels (1/2”). By adjusting the region of interest (ROI) and setting it to 480 × 480 pixels, we were able to increase the frame rate up to 1200 fps, which in turn allowed us to achieve the wavefront correction frequency up to 600 Hz (up to 500 Hz stable) in a closed loop. A photo of the developed wavefront sensor and its parameters is presented in Figure 6.

2.5. Algorithm of Phase Screens Simulation and Reconstruction

As is known [87], the simplest and most reliable model of radiation propagation through a turbulent atmosphere is a thin phase-screen model. The aberrations of an optical wave that passes through a set of thin phase screens are similar to the fluctuations of the light field in a continuous randomly inhomogeneous medium. The thin phase screen closely reproduces the influence of large-scale atmospheric inhomogeneities on the characteristics of the light field [88]. The phase screen approach presents a good approximation, and in most cases, it can reproduce the effect of turbulence on the wavefront with acceptable accuracy.

In order to simulate a set of phase screens, we applied the Fast Fourier transform to the Kolmogorov spectrum of the phase fluctuations [89]:

$$p\left(u,v,t+∆t\right)={\iint }_{-\infty }^{\infty }\sqrt{K(x,y)}·f(x,y,t+∆t)·e^{i·\sqrt{x^2+y^2}·V·∆t}dxdy,$$

(4)

where $p\left(u,v,t+∆t\right)$ is the phase screen at $t+∆t$, $(x,y)$ is a spectrum point, $(u,v)$ is a phase screen point, $V$ is the wind velocity, m/s, $t$ is the moment of the previous phase screen generation, $t+∆t$ is the time moment of the new phase screen generation, $∆t$ is the time interval between two phase screens, and $K(x,y)$ is the spectrum of the phase fluctuations.

To calculate the spectrum of the phase fluctuations $K(x,y)$, the following formula was used [90]:

$$K\left(x,y\right)=0.023·{({\displaystyle \frac{2D}{r_0}})}^{{\displaystyle \frac{5}{3}}}·{(x^2+y^2)}^{-{\displaystyle \frac{11}{3}}},$$

(5)

where $D$ is the telescope receiving aperture, $r_0$ is a Fried radius, and $f(x,y,t+∆t)$ is a function defined using the following formula:

$$f\left(x,y,t+∆t\right)=p·f\left(x,y,t\right)+\sqrt{1-p^2}·e^{i·\phi (x,y,t)},$$

(6)

where $p=e^{{\displaystyle \raisebox{1ex}{$-∆t$}\!\left/ \!\raisebox{-1ex}{$\tau $}\right.}}$, $\tau$ is the atmosphere coherence time, and $\phi (x,y,t)$ is a random delta-correlated value in the range of [0; 2π].

When $t=0$, the function $f$ is expressed as shown below:

$$f\left(x,y,t=0\right)=e^{i·\phi (x,y,t=0)}.$$

(7)

The calculated phase values were normalized in accordance with the relation ${\displaystyle \frac{D}{r_0}}$ [46] using the following phase structure function:

$$DN=6.88·{\left({\displaystyle \frac{\sqrt{x^2+y^2}}{r_0}}\right)}^{{\displaystyle \frac{5}{3}}}.$$

(8)

Once the phase screen is generated, it can be approximated using Zernike polynomials. In this way, the simulated and approximated phase maps can be numerically compared, i.e., by calculating the point-to-point phase difference. The resultant phase maps are presented in Figure 7.

After implementation of the phase screen simulation model, we conducted an analysis of the energy distribution of Zernike polynomials within the simulated turbulent wavefront. The diagram in Figure 8 illustrates the influence of each Zernike mode on the resulting wavefront based on a substantial set of simulated phase screens. Additionally, we analyzed the energy distribution during continuous wavefront correction both with and without tip-tilt correction. The results indicate a strong alignment between the observed energy distribution and the theoretical predictions. The corresponding bar diagram is presented in Figure 8.

Once the set of phase screens was simulated, the procedure of phase screens approximation using the response functions of the bimorph mirror [91] started. The response function for a single electrode of the deformable mirror describes how the mirror’s surface deforms in response to the voltage applied to that specific electrode [92]. Response functions are measured using a Shack–Hartmann sensor.

To reconstruct the phase screen with the mirror, the following procedure was used:

• Calculation of the wavefront derivatives in each subaperture of the wavefront sensor based on the known Zernike approximation of the simulated phase screen, as previously described above and illustrated in Figure 7.
• Calculation of the displacements of the focal spots corresponding to the wavefront derivatives calculated in Step 1.
• By knowing the focal spot shifts related to the mirror’s response functions and the focal spot displacements needed to reproduce the wavefront, we were able to solve the overdetermined system of linear equations using the least squares method [93]. This allowed us to calculate the voltage vector that needed to be applied to the mirror electrodes [94].

This procedure was repeated for each simulated phase screen, resulting in a set of voltage vectors. Each vector corresponds to a particular phase screen, meaning that applying this vector of voltages to the mirror electrodes should generate a wavefront that aligns with the simulated phase screen.

3. Results

The principal steps of the experiment are outlined as follows:

• Simulation of phase screens utilizing the Fast Fourier transform to model the Kolmogorov spectrum of phase fluctuations [1,95].
• Determination of the voltage set required for the reconstruction of the phase screens [96].
• Real-time reconstruction of the simulated phase screens employing the bimorph mirror mounted in a tip-tilt mounting.
• Measurement of wavefront distortions in real time using the Shack–Hartmann wavefront sensor along with an analysis of the intensity distribution of the focal spot in the far field.
• Real-time computation of the correction voltages necessary for both the bimorph deformable mirror and the tip-tilt corrector.
• Real-time compensation of the reconstructed phase screens.

The primary parameter characterizing the intensity of atmospheric turbulence is the refractive index structure parameter $C_n^2$ [95]; specifically, a larger $C_n^2$ value indicates stronger turbulence. This parameter can range from ${10}^{-17}{\mathrm{m}}^{-{\displaystyle \frac{2}{3}}}$ for weak turbulence to ${10}^{-12}{\mathrm{m}}^{-{\displaystyle \frac{2}{3}}}$ for very strong atmospheric turbulence. For instance, in Hefei, China, $C_n^2$ has been observed to vary between $6.69\times {10}^{-16}{\mathrm{m}}^{-{\displaystyle \frac{2}{3}}}$ and $9.87\times {10}^{-14}{\mathrm{m}}^{-{\displaystyle \frac{2}{3}}}$ during summer measurements along a horizontal atmospheric path of 1 km [97]. In maritime conditions, $C_n^2$ is approximately ${10}^{-15}{\mathrm{m}}^{-{\displaystyle \frac{2}{3}}}$ for a 10 km path with a coherence radius $r_0$ of $3.8\mathrm{c}\mathrm{m}$ at a laser wavelength $\lambda =0.85\mathsf{\mu }\mathrm{m}$ [98].

For ground-to-space communications between the International Space Station and the Optical Communications Telescope Laboratory (OCTL) in Wrightwood, California, a coherence radius of $r_0=4.5\mathrm{c}\mathrm{m}$ was experimentally determined for a 1200 km path at a zenith angle 75° with input telescope apertures ranging from $10\mathrm{c}\mathrm{m}$ to $100\mathrm{c}\mathrm{m}$ [99]. In terrestrial atmospheric turbulence, $C_n^2$ is approximately ${10}^{-12}{\mathrm{m}}^{-{\displaystyle \frac{2}{3}}}$ for a 1 km path under conditions where the wind velocity is 10 m/s and the telescope aperture is 20 cm [100,101]. In desert environments, $C_n^2$ measures around ${10}^{-13.2}{\mathrm{m}}^{-{\displaystyle \frac{2}{3}}}$ (with an average wind velocity of 6 m/s) along a 1.2 km path at Edward Air Force Base in the Mojave Desert, California, USA [102].

For the atmospheric turbulence measured by our team at the 1.2 km intra-city link, the coherence radius was equal to $r_0=1.6\mathrm{c}\mathrm{m}$ for a laser wavelength $\lambda =0.532\mathsf{\mu }\mathrm{m}$ and $r_0=2.65\mathrm{c}\mathrm{m}$ for a laser wavelength $\lambda =0.81\mathsf{\mu }\mathrm{m}$ with wind velocities ranging from 5 to 10 m/s for a receiving aperture of D = 140 mm.

Furthermore, $C_n^2$ can be derived from the known receiving aperture diameter D and the Fried radius $r_0$. In this study, we set ${\displaystyle \frac{D}{r_0}}=20$ with a wavelength λ = 0.532 μm and a wind velocity v = 6 m/s. Consequently, $C_n^2$ varies approximately from $3.6\times {10}^{-14}{\mathrm{m}}^{-{\displaystyle \frac{2}{3}}}$ to $2.2\times {10}^{-13}{\mathrm{m}}^{-{\displaystyle \frac{2}{3}}}$ for path lengths ranging from 500 m to 3 km.

A comprehensive set of over 1000 phase screens was generated based on the specified parameters. Selected sequential phase screens from this dataset are illustrated in Figure 9.

Several parameters serve as input data for the phase screen simulator, which is based on Kolmogorov’s theory, including aperture diameter, Fried radius, and wind velocity. Given that the model operates in a quasi-static manner—generating only one single phase screen at a time—while turbulence is inherently a volumetric phenomenon (wherein the distortions imposed on the wavefront cannot be accurately represented by a single phase screen), it is necessary to simulate a series of turbulent phase screens. The parameter that governs this temporal effect within the model is wind velocity. The choice of wind velocity influences the degree of difference between sequentially simulated phase screens; as wind velocity increases, the variation between these screens becomes more pronounced, leading to a less similar appearance from one screen to the next. This behavior is illustrated in Figure 9, where the phase appears to “drift” gradually from one screen to another.

Once the turbulence generation module is running, we can simultaneously activate the turbulence correction module. Both modules feature adjustable frequencies, allowing independent control over the generation frequency of turbulence phase screens and the correction frequency. Given that the correction frequency is constrained to 500 Hz by the wavefront sensor, the maximum frequency of turbulence that can be effectively corrected is below 30 Hz [103].

During the continuous correction of induced wavefront distortions, the amplitude of the residual wavefront aberrations ranged from 0.25 to 0.35 μm, whereas the initial amplitude was between 2.5 and 2.8 μm. An example of the wavefront before and after correction is presented in Figure 10.

It is important to emphasize that not only were the common Zernike coefficients—such as defocus, astigmatism, etc.—significantly reduced, but the tip and tilt components were also minimized. Specifically, the tip and tilt values were decreased from −1.62 μm and 2.47 μm to 0.004 μm and 0.04 μm, respectively, due to the use of the standalone tip-tilt corrector.

The significance of the tip-tilt corrector becomes particularly clear from the following Figure 11.

The first row of Figure 11 contains images of the far field focal spot captured by a video camera while the turbulence generator was in operation. It is evident that the focal spot suffers significant degradation with the peak intensity value (expressed in shades of gray) measuring only 47 out of 255.

The second row of Figure 11 displays images of the focal spot when the correction module #1 was activated, utilizing the bimorph mirror to mitigate wavefront distortions while the tip-tilt corrector remained inactive. A notable enhancement in the intensity distribution of the focal spot is observed with the peak intensity increasing sixfold (reaching up to 255 with 3–4 overexposed pixels). Despite this substantial improvement, it is important to note that the focal spot remains unsteady as it shifts within the aperture. This effect is illustrated in the figure, where the focal spot is not centered within the aperture.

The third row of Figure 11 illustrates images of the focal spot when both correction modules were turned on: the bimorph mirror correcting wavefront distortions while the tip-tilt corrector corrects for tip and tilt aberrations. In this scenario, no shifts of the focal spot within the aperture were observed, resulting in a sharp and stable focal spot.

Figure 12 depicts the jitter curves of the centroid of the far field focal spot prior to correction, during correction with the bimorph mirror alone, and with both the bimorph mirror and the tip-tilt corrector operating in tandem. The frequency of the turbulent phase screen reproduction by the turbulence generator was set to 20 Hz.

When the turbulence generator is operational, the amplitude of the centroid displacements of the far field focal spot was measured at 53 μm along the X-axis and 48 μm along the Y-axis. During correction with only the bimorph mirror (without the tip-tilt corrector), the amplitude of the focal spot displacements was reduced to 45 μm along X-axis and 35 μm along the Y-axis. However, when both the bimorph mirror and the tip-tilt corrector were activated, the amplitude of the focal spot displacements decreased by a factor of 2 to 4, yielding values of 23 μm along the X-axis and 11 μm along the Y-axis.

Both the charts and the numerical data indicate that the residual displacements of the focal spot centroid are ±11.5 μm along the X-axis and ±5.5 μm along the Y-axis. Assuming that the pixel size of the camera sensor is 5.86 μm, this suggests that the centroid of the focal spot shifts within a range of approximately 2 pixels during the correction procedure.

There are at least two factors that may contribute to this phenomenon. First, during the experiments, it was observed that the mounting of the standalone tip-tilt corrector requires modification or replacement, as additional vibrations were detected when the operational frequency was increased to 300 Hz during the correction mode. Second, the upper-level program managing the closed-loop system works under the Windows operating system, which must contend with the Windows messaging loop. This can introduce minor delays in the operational procedure. In general, these delays can coincide with the moments when new phase screens are generated by the turbulence generator, leading to shifts in the focal spot.

4. Conclusions

To summarize, in this research, we successfully implemented a model for simulating turbulent phase fluctuations characterized by Kolmogorov spectra. We designed and constructed an experimental setup for real-time turbulence generation as well as an apparatus for wavefront correction. Furthermore, we developed an algorithm and software capable of reconstructing a series of phase screens in real time, utilizing the response functions of a bimorph mirror mounted in a tip-tilt kinematic mounting.

The controllable turbulence generator we created allowed for the generation of phase screens with predefined parameters, including wind velocity, aperture diameter, and Fried radius, in real time. In our experiments, the parameters were set as follows: ${\displaystyle \frac{D}{r_0}}$ was 10, the wavelength was 0.532 μm, and the wind velocity was 6 m/s. The refractive index structure parameter $C_n^2$ ranged approximately from $3.6\times {10}^{-14}{\mathrm{m}}^{-{\displaystyle \frac{2}{3}}}$ to $2.2\times {10}^{-13}{\mathrm{m}}^{-{\displaystyle \frac{2}{3}}}$ for path lengths varying from 500 m to 3000 m. The induced wavefront aberrations exhibited a frequency of 20 Hz and amplitudes ranging from 2.4 to 2.8 μm.

The adaptive optical system functioned at a closed-loop frequency of 500 Hz, successfully reducing wavefront distortion amplitudes to as low as 0.3 μm. During the correction process utilizing both the bimorph mirror and the tip-tilt corrector, the jitter of the far field focal spot centroid was decreased by a factor of 2.5 to 4 compared to the no-correction mode, while the peak intensity of the focal spot was increased sixfold.