A method for coordinated correction of light field intensity and phase based on multilayer deformable mirrors

Abstract

This invention discloses a method for coordinated correction of optical field intensity and phase based on multi-layer deformable mirrors, belonging to the field of adaptive optics technology. The method involves configuring multiple conjugate deformable mirrors in the optical path. The first deformable mirror is conjugate to the upper turbulent layer and employs a stochastic parallel gradient descent algorithm based on the statistical characteristics of sub-aperture spot intensity, using the variance of sub-aperture spot intensity as the cost function for control. It utilizes the angular spectrum diffraction transmission mechanism to transform upper-layer phase modulation into receiver amplitude modulation, achieving light intensity homogenization. The second deformable mirror is conjugate to the telescope pupil plane, and based on light intensity homogenization, a wavefront reconstruction algorithm is used to perform closed-loop correction of the residual phase. This invention achieves layered decoupled control of light intensity and phase, effectively suppressing strong scintillation effects and expanding the working boundary of adaptive optics systems under harsh atmospheric conditions.

Description

Technical Field

[0001] This invention belongs to the field of adaptive optics technology, specifically relating to a method for coordinated correction of light field intensity and phase based on multilayer deformable mirrors. Background Technology

[0002] During high-resolution imaging with ground-based telescopes, when target light waves pass through atmospheric turbulence, they are affected by the random fluctuations in atmospheric refractive index. This not only causes wavefront phase distortion but also produces significant random fluctuations in light intensity at the pupil surface of the receiving telescope. This phenomenon is known as scintillation. When the turbulence intensity is high or the transmission distance is long, the scintillation effect is particularly severe, resulting in extremely uneven light intensity distribution at the receiving surface, and even local areas with zero light intensity.

[0003] Traditional adaptive optics (AO) systems use Hartmann wavefront sensors as the core detection unit. They calculate the wavefront slope by measuring the centroid shift of the light spot within each sub-aperture, thereby achieving wavefront reconstruction and closed-loop correction. However, the stable operation of Hartmann wavefront sensors relies on sufficient and uniform light intensity within each sub-aperture. When strong scintillation occurs, the light intensity in some sub-apertures falls below the detector's response threshold, directly causing missing wavefront slope data or a sharp drop in the signal-to-noise ratio. This leads to the failure of the wavefront reconstruction algorithm, ultimately preventing the adaptive optics system from achieving closed-loop operation. This problem has become a core bottleneck restricting the stable operation of adaptive optics systems in highly turbulent environments.

[0004] In the prior art, multi-layer conjugate adaptive optics systems have been proposed to expand the correction field of view. For example, patent CN102621687B discloses a solar multi-layer conjugate adaptive optics system, which uses two deformable mirrors conjugate to different turbulent layers and uses a tomographic algorithm to separate wavefront aberrations caused by turbulence in different layers and perform layered phase correction. However, the control objective of this type of system is always focused on the accurate reconstruction and compensation of the wavefront phase, and its wavefront detection still depends on the effective light intensity of each sub-aperture of the Hartmann wavefront sensor, without addressing the active suppression of light intensity scintillation itself. In other words, existing multi-layer conjugate systems only solve the problem of "how to correct the phase of different turbulent layers separately", but fail to solve the fundamental problem of "strong scintillation causing the inability to detect the phase".

[0005] In addition, although traditional wavefront-less adaptive optics systems can directly optimize far-field spot quality using traditional optimization algorithms such as SPGD, their convergence speed is slow and their real-time performance is poor, making it difficult to meet the correction requirements of dynamic turbulence, and they cannot achieve effective coordination with wavefront detection.

[0006] In summary, how to actively suppress light intensity flickering without relying on additional detectors, create uniform light intensity working conditions for Hartmann wavefront sensors, and simultaneously achieve high-precision correction of residual phase has become a pressing technical challenge in this field. Summary of the Invention

[0007] This invention provides a method for coordinated correction of light field intensity and phase based on multilayer deformable mirrors. By utilizing the phase modulation of the higher-layer deformable mirrors, intensity correction is achieved at the receiving surface through angular spectrum diffraction transmission, creating uniform light intensity conditions for the phase correction of the lower-layer deformable mirrors, thereby solving the problem of Hartmann wavefront sensor detection failure in strong scintillation environments.

[0008] To achieve the above objectives, the present invention adopts the following technical solution:

[0009] A method for coordinated correction of optical field intensity and phase based on multilayer deformable mirrors includes:

[0010] Construct an optical path system comprising multiple atmospheric turbulence layers, a telescope, a Hartmann wavefront sensor, and multiple conjugate deformable mirrors, wherein the multiple conjugate deformable mirrors include a first deformable mirror conjugate to the upper turbulence layer and a second deformable mirror conjugate to the telescope pupil plane.

[0011] With the goal of ensuring uniformity of light spot intensity in each sub-aperture of the Hartmann wavefront sensor, a stochastic parallel gradient descent algorithm based on the statistical characteristics of sub-aperture light spot intensity is used to iteratively update the driving voltage of the first deformable mirror to correct the intensity distribution. The correction of the intensity distribution is based on the angular spectrum diffraction transmission theory. During the propagation from the upper turbulent layer to the receiving surface, the phase generated by the first deformable mirror is converted into amplitude modulation of the receiving surface, thereby compensating for the light intensity scintillation effect caused by the upper turbulent layer.

[0012] Based on the uniformity of light spot intensity in each sub-aperture, a stable light spot array is acquired using a Hartmann wavefront sensor. The residual wavefront phase is calculated using a wavefront reconstruction algorithm, which drives the second deformable mirror to perform real-time phase compensation.

[0013] Furthermore, the stochastic parallel gradient descent algorithm based on the statistical characteristics of sub-aperture spot intensity aims at the uniformity of the intensity of each sub-aperture spot. Specifically, it constructs a cost function to quantify the uniformity of spot intensity. The cost function is the average normalized variance of the intensity of all effective sub-aperture spots, which is expressed as the sum of the squares of the differences between the intensity of each sub-aperture spot and its average value divided by the total number of effective sub-apertures. The larger the cost function value, the more severe the light intensity flicker and the more uneven the intensity distribution; conversely, the smaller the value, the more uniform the spot intensity distribution.

[0014] Furthermore, the step of iteratively updating the driving voltage of the first deformable mirror using a stochastic parallel gradient descent algorithm based on the statistical characteristics of the sub-aperture light spot intensity specifically includes: applying a random bidirectional perturbation voltage vector to the current driving voltage vector of the first deformable mirror, obtaining the cost function values ​​corresponding to the positive and negative perturbations respectively; calculating the gradient estimate of the cost function based on the difference between the two cost function values ​​and the perturbation amplitude; and determining the update amount of the driving voltage based on the gradient estimate and the current iteration step size, thereby iteratively updating the driving voltage of the first deformable mirror.

[0015] Furthermore, the iteration step size adopts an adaptive step size decay mechanism, in which the step size decays exponentially with the number of iterations, and is expressed as the initial step size multiplied by a negative exponential function with the number of iterations divided by the decay time constant as the exponent.

[0016] Furthermore, the method also includes a gradient limiting and an early stopping mechanism: the gradient limiting is used to limit the magnitude of the gradient estimate of a single voltage update; the early stopping mechanism calculates the relative improvement rate of the cost function, and when the relative improvement rate of P consecutive iterations is less than a preset threshold, it is determined that the optimal light intensity homogenization has been achieved and the iteration optimization is terminated in advance, where P is a preset positive integer.

[0017] Furthermore, the angular spectrum diffraction transmission theory is specifically used to establish a light field propagation model from the upper turbulent layer where the first deformable mirror is located to the receiving surface. This model describes, through the angular spectrum propagation factor of the light field, how the correction phase distribution introduced by the first deformable mirror changes the complex amplitude distribution of the light field at the receiving surface after diffraction propagation, thereby realizing the transformation from phase distribution to amplitude distribution.

[0018] Furthermore, the calculation of the residual wavefront phase using the wavefront reconstruction algorithm specifically involves: after light intensity homogenization, calculating the centroid shift of the light spot within each sub-aperture of the Hartmann wavefront sensor to obtain the two-dimensional average slope within the sub-aperture; and using the two-dimensional average slope and the wavefront reconstruction matrix, reconstructing the residual wavefront phase of the entire aperture using the mode method or the direct slope method.

[0019] Furthermore, the real-time phase compensation of the second deformable mirror is achieved by using a proportional-integral controller to convert the reconstructed residual wavefront phase into the driving voltage of the second deformable mirror; and by using integral gain and cumulative error gain, real-time closed-loop correction of residual phase distortion is realized.

[0020] In a second aspect, the present invention provides an electronic device, comprising: one or more processors; and a memory for storing one or more programs; wherein, when the one or more programs are executed by the one or more processors, the one or more processors implement the aforementioned method for coordinated correction of light field intensity and phase based on multilayer deformable mirrors.

[0021] Thirdly, the present invention provides a computer-readable storage medium having executable instructions stored thereon, which, when executed by a processor, enable the processor to implement the aforementioned method for coordinated correction of light field intensity and phase based on a multilayer deformable mirror.

[0022] The beneficial effects of this invention are as follows:

[0023] Unlike existing multilayer conjugate adaptive optics systems that only focus on phase layer correction, this invention expands the control objective from "phase compensation" to "intensity compensation" for the first time. Through active phase modulation of the high-level deformable mirror, the phase change is converted into amplitude modulation of the receiving surface using the angular spectrum diffraction transmission mechanism. This actively suppresses light intensity scintillation at the physical level, enabling uniform light intensity distribution across all sub-apertures of the Hartmann wavefront sensor. This method fundamentally solves the problem of wavefront detection failure caused by sub-aperture light deficiency in strong scintillation environments, creating the prerequisite for stable operation of adaptive optics systems under harsh atmospheric conditions.

[0024] This invention clearly defines the functional division of "high-level deformable mirrors responsible for intensity homogenization and low-level deformable mirrors responsible for phase correction," avoiding the control difficulties of coupled intensity distortion and phase distortion processing in traditional methods. The two layers of deformable mirrors have clear division of labor and work together, ensuring effective detection by the Hartmann wavefront sensor and achieving high-precision correction of the residual phase, thereby improving the overall correction efficiency of the system.

[0025] This invention, without adding additional detection hardware, enables traditional adaptive optics systems to be applied to extreme scenarios such as strong turbulence and long-distance transmission through innovative algorithms and control strategies, significantly expanding the working range and engineering applicability of adaptive optics technology. Attached Figure Description

[0026] Figure 1 This is a schematic diagram of the optical path transmission of a system based on a multilayer deformable mirror-based optical field intensity and phase co-correction method according to the present invention;

[0027] Figure 2 This is a far-field spot intensity distribution diagram in an embodiment of the present invention without using a multi-layer deformable mirror system for closed-loop collaborative correction;

[0028] Figure 3 This is a diagram showing the far-field light spot intensity distribution after closed-loop collaborative correction using a multi-layer deformable mirror system in an embodiment of the present invention. Detailed Implementation

[0029] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0030] This invention proposes a method for coordinated correction of optical field intensity and phase based on multi-layer deformable mirrors. The method is based on constructing a multi-layer atmospheric turbulence and layered transmission model, and sequentially placing a first deformable mirror DM1 conjugate to the upper layer and a second deformable mirror DM2 conjugate to the receiving surface in the optical path. To address scintillation caused by upper-layer turbulence, the first deformable mirror DM1 employs a stochastic parallel gradient descent (SISC-SPGD) control strategy based on the statistical characteristics of sub-aperture spot intensity. Specifically, this involves constructing a cost function characterizing the uniformity and concentration of intensity of each sub-aperture spot. , Defined as the average normalized variance of the intensity data of all effective sub-aperture spots, i.e. ,in The light spot intensity is the i-th sub-aperture, and N is the total number of effective sub-apertures. The average intensity of the light spot across all sub-apertures is given when light intensity flicker causes the light spot to break apart or disperse. The value increases significantly; conversely, when the light spot intensity is uniform and the focus is good, The value tends to be extremely small. The SISC-SPGD algorithm is used to measure the voltage of the first deformable mirror DM1. Optimization is performed. Through this process, the first deformable mirror DM1 actively modulates the phase, and after angular spectrum transmission, compensates for the intensity fluctuation effect at the receiving surface. For residual aberrations, the second deformable mirror DM2 adopts a PI closed-loop control strategy based on wavefront reconstruction: the cost function of the first deformable mirror DM1... After optimization to the convergence region, the Hartmann wavefront sensor acquires stable spot images. The residual wavefront phase is reconstructed using the mode method. The voltage of the second deformable mirror DM2 is updated via the PI controller. This enables real-time compensation for residual phase distortion.

[0031] like Figure 1 The diagram shows the principle of the optical system used in this invention. The target light passes through the atmosphere to the primary mirror, converges, is reflected by the secondary mirror to the third mirror, and finally reflected by the third mirror to the Nyquist focus, reaching the optical path platform. For multi-layered turbulent atmospheric transmission systems, the propagation of light waves follows Fresnel diffraction theory. Phase distortion at higher levels causes amplitude (light intensity) fluctuations during long-distance transmission. Let the wavelength of the light wave be... , wave number In the direction perpendicular to the propagation direction On the plane, the complex amplitude distribution of the light field is as follows x, y, z represent arbitrary spatial positions (x, y, z).

[0032] This invention utilizes angular spectral transmission theory to describe the transmission of light waves from upper-level turbulence (the height at which the first deformable mirror DM1 is conjugate to the upper-level turbulent layer). (The second deformable mirror DM2 is conjugate to the receiver plane at the height) The propagation process of ). Let ) The angular spectrum of the light field modulated by the first deformable mirror DM1. Let fx and fy be the Fourier transform of the modulated light field, representing the spatial frequencies in the x and y directions, respectively. Then the propagation distance is... The subsequent angular spectrum relationship of the light field is:

[0033] (1)

[0034] By performing an inverse Fourier transform on the result of formula (1), the optical field distribution at the receiver can be obtained. Due to the effect of the transmission factor, the phase introduced by the first deformable mirror DM1 Will change Intensity distribution The core of this invention lies in controlling... , making The apertures are evenly distributed within each sub-aperture of the Hartmann wavefront sensor.

[0035] Due to the complex nonlinear coupling between amplitude and phase, it is difficult to establish an analytical model. Therefore, this invention employs an improved Stochastic Parallel Gradient Descent (SISC-SPGD) algorithm based on the statistical characteristics of sub-aperture spot intensity to control the first deformable mirror DM1. The spot intensity of the i-th sub-aperture of the Hartmann wavefront sensor is defined as... To characterize the intensity uniformity and effectiveness of the light spot, a cost function is constructed. The average normalized variance of the intensity data for all effective sub-aperture spots:

[0036] (2)

[0037] In the formula, This represents the total number of effective sub-apertures. When light intensity flicker causes spot divergence or measurement noise is high, The value is relatively large; when the light spot intensity is uniform and convergent, The value decreases. To optimize... The control voltage vector to the first deformable mirror DM1 Apply two-way random perturbation Measure the positive disturbance separately. and negative disturbance The corresponding cost function value and Then the gradient estimate of the cost function It can be represented as:

[0038] (3)

[0039] in, This represents the disturbance amplitude.

[0040] To address the slow convergence of the traditional SPGD algorithm in dynamic turbulence, this invention introduces an adaptive exponential decay step size strategy. Update step size of the next iteration The calculation formula is:

[0041] (4)

[0042] in, The initial step size, The decay time constant is set to (e.g., half the maximum number of iterations). The control voltage vector update rule for the first deformable mirror DM1 is:

[0043] (5)

[0044] In the formula, This is a gradient limiting function to prevent excessively large single update values ​​from compromising system stability. This represents the control voltage vector of the first deformable mirror DM1 at time t+1. Let t represent the control voltage vector of the first deformable mirror DM1 at time t.

[0045] After the light intensity is homogenized by the first deformable mirror DM1, the Hartmann wavefront sensor can acquire a high signal-to-noise ratio light spot image. At this point, the second deformable mirror DM2 is activated for phase correction. Using the direct slope method, the slope vector... Calculate the control voltage of the second deformable mirror DM2. Phase update is performed using PI control:

[0046] (6)

[0047] in, For integral gain, This represents the cumulative error gain, with the superscript t indicating time. This represents the control voltage of the second deformable mirror DM2 at time t. This represents the control voltage of the second deformable mirror DM2 at time t-1.

[0048] Example:

[0049] The simulation system parameters are set as follows: telescope aperture D = 1.8m, operating wavelength... The atmospheric turbulence model is set to two layers, with the first layer having a height of... Second floor height During the simulation initialization phase, system parameters and the DM influence function matrix are loaded. A multi-layered dynamic phase screen is generated that evolves over time, including the upper atmospheric turbulence intensity. Stronger winds with higher speeds; weaker low-level turbulence.

[0050] Entering the closed-loop calibration cycle:

[0051] 1. Light field generation and transmission:

[0052] The incident light wave is modulated by an atmospheric phase screen at 10 km, and superimposed with the current corrected phase of DM1. Using the angular spectral transmission theory described in formula (1), the complex amplitude distribution after propagation for 10 km is calculated, and the atmospheric phase screen at 0 km and the corrected phase of DM2 are superimposed to form the final residual light field, as shown below. Figure 2 As shown.

[0053] 2. DM1 SPGD Optimization:

[0054] Generate random voltage disturbances .

[0055] The optical field reaching the receiving surface after long-distance transmission is calculated separately after applying positive and negative perturbations.

[0056] Simulate Hartmann wavefront sensor imaging to calculate the average normalized variance of effective sub-aperture spot intensity data. and .

[0057] The gradient is calculated and the DM1 voltage is updated using formulas (3) to (5).

[0058] Introducing an early stopping mechanism: Calculating the relative improvement rate of the cost function. If there are P consecutive iterations (e.g., P=50) If the light intensity is less than the threshold, it is determined that DM1 has achieved optimal light intensity uniformity, and the optimization process for this frame ends prematurely. This represents the optimal cost function value. This represents the current cost function value.

[0059] 3. DM2 Closed-Loop Calibration:

[0060] Based on the iterative optimization of DM1, the Hartmann wavefront sensor detects the residual light field.

[0061] The corrected phase of DM2 is calculated using formula (6) to eliminate residual phase distortion, such as Figure 3 As shown.

[0062] In summary, this invention provides a method for coordinated correction of optical field intensity and phase based on multilayer deformable mirrors. This method effectively suppresses scintillation caused by strong turbulence using higher-layer deformable mirrors without adding an additional detector, creating conditions for the normal operation of Hartmann wavefront sensors. Lower-layer deformable mirrors correct residual aberrations, thereby achieving full-field correction. This invention provides a new technical approach to solving the challenges of adaptive optics imaging in highly turbulent environments.

[0063] In a second aspect, the present invention provides an electronic device, comprising: one or more processors; and a memory for storing one or more programs; wherein, when the one or more programs are executed by the one or more processors, the one or more processors implement the aforementioned method for coordinated correction of light field intensity and phase based on multilayer deformable mirrors.

[0064] Thirdly, the present invention provides a computer-readable storage medium having executable instructions stored thereon, which, when executed by a processor, enable the processor to implement the aforementioned method for coordinated correction of light field intensity and phase based on a multilayer deformable mirror.

[0065] The specific embodiments described above further illustrate the purpose, technical solution, and beneficial effects of the present invention. It should be understood that the above descriptions are merely specific embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for coordinated correction of optical field intensity and phase based on multilayer deformable mirrors, characterized in that, include: An optical path system is constructed, comprising a multi-layer atmospheric turbulence layer, a telescope, a Hartmann wavefront sensor, and a multi-layer conjugate deformable mirror. The multi-layer conjugate deformable mirror includes a first deformable mirror conjugate to the upper turbulence layer and a second deformable mirror conjugate to the telescope pupil plane. With the goal of ensuring uniformity of light spot intensity in each sub-aperture of the Hartmann wavefront sensor, a stochastic parallel gradient descent algorithm based on the statistical characteristics of sub-aperture light spot intensity is used to iteratively update the driving voltage of the first deformable mirror to correct the intensity distribution. The correction of the intensity distribution is based on the angular spectrum diffraction transmission theory. During the propagation from the upper turbulent layer to the receiving surface, the phase generated by the first deformable mirror is converted into amplitude modulation of the receiving surface, thereby compensating for the light intensity scintillation effect caused by the upper turbulent layer. Based on the uniformity of light spot intensity in each sub-aperture, a stable light spot array is acquired using a Hartmann wavefront sensor. The residual wavefront phase is calculated using a wavefront reconstruction algorithm, which drives the second deformable mirror to perform real-time phase compensation.

2. The method for coordinated correction of optical field intensity and phase based on multilayer deformable mirrors according to claim 1, characterized in that, The stochastic parallel gradient descent algorithm based on the statistical characteristics of sub-aperture spot intensity aims at the uniformity of the spot intensity of each sub-aperture. Specifically, it constructs a cost function to quantify the uniformity of spot intensity. The cost function is the average normalized variance of the spot intensity of all effective sub-apertures, which is expressed as the sum of the squares of the differences between the spot intensity of each sub-aperture and its average value, divided by the total number of effective sub-apertures. The larger the cost function value, the more severe the light intensity flicker and the more uneven the intensity distribution; conversely, the smaller the value, the more uniform the spot intensity distribution.

3. The method for coordinated correction of optical field intensity and phase based on a multilayer deformable mirror according to claim 2, characterized in that, The method of iteratively updating the driving voltage of the first deformable mirror using a stochastic parallel gradient descent algorithm based on the statistical characteristics of the sub-aperture light spot intensity specifically includes: applying a random bidirectional perturbation voltage vector to the current driving voltage vector of the first deformable mirror, obtaining the cost function values ​​corresponding to the positive and negative perturbations respectively; calculating the gradient estimate of the cost function based on the difference between the two cost function values ​​and the perturbation amplitude; and determining the update amount of the driving voltage based on the gradient estimate and the current iteration step size, thereby iteratively updating the driving voltage of the first deformable mirror.

4. The method for coordinated correction of optical field intensity and phase based on multilayer deformable mirrors according to claim 3, characterized in that, The iteration step size adopts an adaptive step size decay mechanism. The step size decays exponentially with the number of iterations, and is expressed as the initial step size multiplied by a negative exponential function with the number of iterations divided by the decay time constant as the exponent.

5. The method for coordinated correction of optical field intensity and phase based on a multilayer deformable mirror according to claim 3, characterized in that, The method also includes a gradient limiting and an early stopping mechanism: the gradient limiting is used to limit the magnitude of the gradient estimate of a single voltage update; the early stopping mechanism calculates the relative improvement rate of the cost function, and when the relative improvement rate of P consecutive iterations is less than a preset threshold, it is determined that the optimal light intensity homogenization has been achieved and the iteration optimization is terminated in advance, where P is a preset positive integer.

6. The method for coordinated correction of optical field intensity and phase based on a multilayer deformable mirror according to claim 1, characterized in that, The angular spectrum diffraction transmission theory is specifically used to establish a light field propagation model from the upper turbulent layer where the first deformable mirror is located to the receiving surface. This model describes how the correction phase distribution introduced by the first deformable mirror changes the complex amplitude distribution of the light field at the receiving surface after diffraction propagation through the angular spectrum propagation factor of the light field, thereby realizing the transformation from phase distribution to amplitude distribution.

7. The method for coordinated correction of optical field intensity and phase based on multilayer deformable mirrors according to claim 1, characterized in that, The calculation of the residual wavefront phase using the wavefront reconstruction algorithm specifically involves: after light intensity homogenization, calculating the centroid shift of the light spot within each sub-aperture of the Hartmann wavefront sensor to obtain the two-dimensional average slope within the sub-aperture; and using the two-dimensional average slope and the wavefront reconstruction matrix, reconstructing the residual wavefront phase of the entire aperture using the mode method or the direct slope method.

8. The method for coordinated correction of optical field intensity and phase based on a multilayer deformable mirror according to claim 7, characterized in that, The method for driving the second deformable mirror to perform real-time phase compensation is as follows: a proportional-integral controller is used to convert the reconstructed residual wavefront phase into the driving voltage of the second deformable mirror; and real-time closed-loop correction of residual phase distortion is achieved through integral gain and cumulative error gain.

9. An electronic device, characterized in that, include: One or more processors; Memory, used to store one or more programs; When one or more programs are executed by the one or more processors, the one or more processors implement the optical field intensity and phase co-correction method based on a multilayer deformable mirror as described in any one of claims 1-8.

10. A computer-readable storage medium, characterized in that, It stores executable instructions that, when executed by a processor, enable the processor to implement the optical field intensity and phase co-correction method based on a multilayer deformable mirror as described in any one of claims 1-8.

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