A method for measuring atmospheric turbulence aberration wavefront and orbital angular momentum of a vortex beam

Abstract

The present application relates to a kind of methods for measuring vortex light beam atmospheric turbulence aberration wavefront and orbital angular momentum, belong to optical measurement field.The present application uses fan micro-lens array as wavefront sensor, the radial and angular decomposition of the displacement vector of sub-spot is carried out under polar coordinates, and the thought of relevant algorithm is used to calculate the aberration wavefront of vortex light beam generated by atmospheric turbulence, then the angular displacement vector information of residual wavefront spot array is used to calculate the orbital angular momentum information of vortex light beam.Relative to the method for measuring vortex light beam atmospheric turbulence aberration using traditional probe beam or deep learning, the method does not need probe beam, does not need a large number of calculations, can simultaneously calculate the aberration wavefront and orbital angular momentum of vortex light beam after atmospheric transmission.The method is simple and efficient, and the advantage is obvious.

Description

Technical Field

[0001] This invention belongs to the field of optical measurement, specifically relating to a method for measuring the wavefront and orbital angular momentum of a vortex beam subjected to atmospheric turbulence distortion. Background Technology

[0002] The angular momentum of light is divided into spin angular momentum and orbital angular momentum. Spin angular momentum is manifested through the polarization of light, while orbital angular momentum was only known in 1992. A beam carrying orbital angular momentum (OAM) is also called a vortex beam, which is a beam with a continuous helical phase and a ring-shaped optical field distribution. The magnitude of the orbital angular momentum is... ,in This is called the topological load number or pattern number. It is the reduced Planck constant. Furthermore, the modes of a vortex beam are mutually orthogonal, forming a complete orthogonal basis. Therefore, orbital angular momentum becomes another fundamental dimension of light, with applications in various fields. In the field of space optical communication, since the number of modes of a vortex beam can be infinite, it theoretically possesses infinite coding capabilities. This endows it with infinite potential for high-capacity communication, and is expected to break through the capacity limitations of traditional optical communication.

[0003] In the field of OAM (Optical Aperture AM) space optical communication, two main problems are faced: 1. The orbital angular momentum diffusion of vortex beams after atmospheric transmission causes mode crosstalk; 2. The distortion wavefront of vortex beams after atmospheric transmission is difficult to detect. Currently, commonly used methods include probe beams and neural networks to address these issues. The probe beam method involves coaxially transmitting a Gaussian beam and a vortex beam, and detecting the wavefront distortion of the Gaussian beam at the receiving end to compensate for the wavefront distortion caused by the vortex beam. The neural network method mainly utilizes convolutional neural networks or other improved network models to establish a correspondence between the light intensity distribution at the receiving end and the turbulent phase, thereby detecting the distortion wavefront of the vortex beam during atmospheric transmission.

[0004] The methods described above cannot simultaneously detect both orbital angular momentum and the distorted wavefront of atmospheric turbulence. Furthermore, the use of probe beams increases system complexity. Methods based on neural networks and deep learning increase computational load. Additionally, the limited number of training samples for the network may cause the model to fail in real-world applications due to the variable atmospheric turbulence across different regions, times, and climates. Therefore, both of these methods are still some distance from practical application. Thus, a more efficient method is needed to measure the distorted wavefront of atmospheric turbulence and orbital angular momentum of vortex beams, which will accelerate the development of vortex beam space communication applications. Summary of the Invention

[0005] The purpose of this invention is to address the shortcomings of the aforementioned detection methods by proposing a method that can simultaneously detect the atmospheric turbulence wavefront distortion and orbital angular momentum of a vortex beam. After the vortex beam propagates through the atmosphere, it is received by a Hartmann wavefront sensor. The centroid displacement information of its beam array is contributed by the helical phase and turbulence phase of the vortex beam. Utilizing the phase slope distribution characteristics of the helical phase in polar coordinates, combined with relevant algorithmic ideas, the distorted wavefront of the atmospheric turbulence of the vortex beam can be obtained from the centroid displacement information of the beam. Then, through the mapping relationship between orbital angular momentum and centroid displacement, the magnitude of the orbital angular momentum can be obtained.

[0006] The technical solution adopted in this invention is: a method for measuring the wavefront and orbital angular momentum of a vortex beam subjected to atmospheric turbulence distortion, the method comprising the following steps:

[0007] Step 1: Use a fan-shaped array of microlenses to receive signals to obtain a ring-shaped array of light spots;

[0008] Step 2: In the polar coordinate system, calculate the radial and angular centroid displacement of each sub-aperture spot in the spot array. Then, select a reference sub-aperture on each ring and calculate the column vector G of the relative centroid displacement of other sub-aperture spots on the ring relative to the reference sub-aperture spot.

[0009] Step 3: Calculate the Zernike coefficient A of the turbulent distortion wavefront using the relative centroid displacement column vector G;

[0010] Step 4: Calculate the radial and angular displacements of the sub-aperture spot towards the centroid of the restored wavefront;

[0011] Step 5: Combine the radial component of the residual wavefront centroid displacement to obtain the Zernike coefficients for defocus and spherical aberration;

[0012] Step 6: Combine the angular component of the residual wavefront centroid displacement to obtain the orbital angular momentum of the vortex beam.

[0013] The present invention has the following beneficial effects:

[0014] The present invention provides a method for measuring the turbulent distortion wavefront and orbital angular momentum of a vortex beam. This method does not require a probe beam or a complex neural network structure. It only uses a fan-shaped microlens array and obtains the turbulent distortion wavefront and orbital angular momentum information simultaneously through a single frame of Hartmann spot array information. The method is simple in structure and highly efficient in system. Attached Figure Description

[0015] Figure 1 This is a flowchart of a method for measuring the wavefront and orbital angular momentum of a vortex beam under atmospheric turbulence distortion according to the present invention.

[0016] Figure 2 (a) shows the phase slope distribution of the spiral phase;

[0017] Figure 2 (b) Phase slope distribution of the defocused phase;

[0018] Figure 2 (c) Phase slope distribution of primary spherical aberration;

[0019] Figure 2 (d) Phase slope distribution of second-order spherical aberration;

[0020] Figure 3(a) is a schematic diagram of the arrangement of the fan-shaped microlens array;

[0021] Figure 3(b) is a diagram of the annular light spot array on the focal plane of the fan-shaped microlens array;

[0022] Figure 3 (c) Vector diagram of centroid displacement of the decomposed sub-spots in the radial and angular directions;

[0023] Figure 4(a) is =5 spiral phase diagram;

[0024] Figure 4(b) is a turbulence phase diagram;

[0025] Figure 4 (c) Phase diagram of spiral distortion;

[0026] Figure 5(a) is the restored Zernike coefficient diagram, excluding defocus and spherical aberration;

[0027] Figure 5(b) is the restored Zernike coefficient diagram, including defocus and spherical aberration;

[0028] Figure 5(c) is the measured orbital angular momentum diagram. Detailed Implementation

[0029] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention. Furthermore, the technical features involved in the various embodiments of this invention described below can be combined with each other as long as they do not conflict with each other. To achieve the above objectives, this invention adopts the following technical solution.

[0030] The following, with reference to the accompanying drawings, briefly describes the algorithm of this invention for extending wavefront detection of targets in adaptive optics. In a Cartesian coordinate system, since the contribution of tilt aberration to the centroid displacement of each sub-aperture is the same, the algorithm will subtract the tilt aberration information of the wavefront as a whole, resulting in the detection of only aberrations other than tilt. Based on this idea, this invention analyzes and studies the phase slope distribution of the helical phase of a vortex beam, as follows:

[0031] (1)

[0032] In the above formula, λ is the topological charge of the vortex beam, θ is the eigenvalue of OAM, (r, θ) are polar coordinates, and λ is the wavelength. The spiral phase slope distribution is shown in Figure 2(a), where the black arrows indicate the slope direction. It can be seen that, in the polar coordinate system, the wavefront slope of the vortex beam is independent of the angular coordinate θ, and only depends on the radial coordinates r and θ. The wavefront slope is circularly symmetric, meaning the slope is the same along the same radius. Based on this characteristic, and using relevant algorithms in polar coordinates, the spiral phase in the spiral distortion wavefront of the vortex beam can be subtracted to obtain the turbulent distortion wavefront.

[0033] This invention employs a fan-shaped microlens array, making it more suitable for signal processing in polar coordinates. Its arrangement is shown in Figure 3(a), and a ring-shaped array of light spots can be obtained on the focal plane, as shown in Figure 3(b). Since the wavefront information of the received light field includes turbulent phase and spiral phase, the centroid displacement information of the Hartmann light spot array also includes spiral phase and turbulent phase information. In polar coordinates, the radial d-axis displacement of the light spot centroid of each sub-aperture is calculated. r and angular direction d θ The decomposition, as shown in Figure 3(c), is expressed as follows:

[0034] (2)

[0035] In the above formula, (x n,m y n,m ) is the coordinate of the (n, m)th pixel within the corresponding sub-aperture on the camera, I n,m This represents the intensity information acquired by the (n, m)th pixel, where (X(i, j), Y(i, j)) are the Cartesian coordinates of the centroid of the j-th sub-aperture on the i-th ring. i represents the i-th ring, and j represents the index of the sub-aperture on each ring. These are the radial and angular displacements of the j-th sub-aperture on the i-th ring, respectively, and θ is the angle between the center of the sub-aperture and the origin of the coordinate system and the x-coordinate.

[0036] Then, the centroid information of the reference sub-aperture of each ring is subtracted from the centroid information of the sub-aperture on each ring to obtain the relative centroid displacement column vector G, which can be expressed as follows:

[0037] (3)

[0038] In the above formula, This represents the relative centroid displacement of the j-th sub-aperture of the i-th ring. Let the first sub-aperture of the i-th ring be taken as a reference. The relative displacement of each sub-aperture corresponding to each Zernike polynomial can be expressed as follows:

[0039] (4)

[0040] In the above formula, Let A and B represent the radial and angular centroid displacements of the k-th Zernike aberration within the j-th sub-aperture of the i-th ring, respectively. Therefore, the Zernike coefficient A of the turbulent distortion wavefront can be obtained from the following equation:

[0041] (5)

[0042] In the above formula, It is the restoration matrix, which is the inverse matrix of D. Since the phase slopes of defocus and spherical aberration are also circularly symmetric, as shown in Figures 2(b), 2(c), and 2(d), the Zernike coefficients obtained by the calculations of formulas (2) to (5) do not include defocus and spherical aberration. At this time, the restored wavefront can be expressed as:

[0043] (6)

[0044] in, The i-th term of the Zernike coefficient A on the turbulent distortion wavefront. Representing the i-th order Zernike aberration, the restored wavefront matrix... The corresponding sub-aperture displacement is expressed as:

[0045] (7)

[0046] In the above formula, S is the region where the j-th sub-aperture of the i-th ring is located. Indicates Fourier transform, This represents the centroid displacement of each sub-aperture of the restored wavefront in rectangular coordinates. Formula (2) yields the radial and angular centroid displacements of each sub-aperture. ).

[0047] Subtracting the centroid displacement introduced by the restored wavefront from the detected initial spot centroid information, the centroid displacement of the residual wavefront can be obtained, as follows:

[0048] (8)

[0049] As shown by the black arrows in Figures 2(a)-(d), the phase slopes of defocus and spherical aberration are radially distributed, while the helical phase slope is angularly distributed, and they are orthogonal to each other. Therefore, the angular component of the residual wavefront's spot displacement vector is provided by the helical phase, and the radial component is provided by defocus and spherical aberration. Thus, by extracting the radial component of the residual wavefront's centroid displacement vector, the Zernike coefficients of defocus and spherical aberration can be obtained, as expressed below:

[0050] (9)

[0051] In the above formula, It is the radial component of the residual wavefront centroid displacement. It is a restoration matrix containing only defocus and spherical aberration, which can be obtained by formula (4), by replacing the Zernike aberration in the matrix with defocus and spherical aberration. Combining formulas (5) and (9), the Zernike coefficient of the turbulent distortion wavefront of the vortex beam after atmospheric transmission can be detected.

[0052] According to formula (1), the phase slope of the vortex beam is related to the topological charge number. The topological charge is an eigenvalue of the orbital angular momentum; therefore, when mapped onto the displacement of the center of mass, the topological charge... It can be represented as:

[0053] (10)

[0054] In the above formula, d represents the beam spot displacement. For the entire beam spot array, the topological charge of the vortex beam... It can be obtained from the following formula:

[0055] (11)

[0056] In the above formula, It is the angular component of the residual wavefront centroid displacement. It is a restored matrix containing only the spiral phase, which can be obtained by formula (4), by replacing the Zernike aberration in the matrix with A spiral phase of 1 is sufficient.

[0057] In summary, by using a fan-shaped microlens array in polar coordinates and combining relevant algorithms, single-frame data can be used to detect the turbulent phase information and orbital angular momentum information of a vortex beam after it has been transmitted through the atmosphere.

[0058] Example 1: Topological Charge of Vortex Beam =5, atmospheric coherence length r0=10cm, receiver diameter 600mm, blocking ratio 0.286.

[0059] In this embodiment, the topological charge of the emitted vortex beam =5, and its spiral phase is shown in Figure 4(a). The atmospheric coherence length r0 = 10 cm, and the turbulent phase is shown in Figure 4(b). A 600 mm aperture telescope with an obstruction ratio of 0.286 is used as the receiving telescope. After the vortex beam passes through the atmosphere, the phase distribution at the receiving telescope is shown in Figure 4(c). Compared with the initial spiral phase, the spiral phase is distorted. Since the wavefront at the receiving telescope is a spirally distorted wavefront, the Hartmann wavefront sensor calibrated with a plane wavefront cannot detect the turbulently distorted wavefront.

[0060] The algorithm flowchart of this invention is as follows: Figure 1 As shown, firstly, the centroid displacement of the annular spot is calculated. Then, a reference sub-aperture is selected for each ring, and the relative displacement of the other sub-aperture spots on the ring is calculated. The Zernike coefficient of the turbulent distortion wavefront is calculated using formula (4). Then, the centroid displacement of the sub-aperture spot on the restored wavefront is calculated, and the difference between the centroid displacement and the initial spot centroid displacement is calculated to obtain the centroid displacement of the residual wavefront. According to formula (5), combined with the radial component of the centroid displacement of the residual wavefront, the Zernike coefficients of defocus and spherical aberration are obtained. According to formula (7), combined with the angular component of the centroid displacement of the residual wavefront, the orbital angular momentum of the vortex beam is obtained.

[0061] The calculation results are shown in Figures 5(a)-(c), showing the Zernike coefficients of the first 35 orders. Figure 5(a) shows the Zernike coefficients obtained by formula (4), where the 3rd, 10th, and 21st orders are defocus, primary spherical aberration, and secondary spherical aberration, respectively, and the restored Zernike coefficients are zero. Combining formula (5), the Zernike coefficients for defocus and spherical aberration can be calculated, as shown in Figure 5(b). The restored results are basically consistent with the initial Zernike coefficients. The measured orbital angular momentum results are shown in Figure 5(c), which are completely consistent with the orbital angular momentum of the emitted vortex beam. Since this invention separates the turbulent distortion wavefront and the spiral phase, the eigenmodes of the vortex beam can be directly detected without considering the influence of mode crosstalk caused by atmospheric turbulence.

[0062] The above embodiments demonstrate that the invention can detect the distorted wavefront and orbital angular momentum of a vortex beam after it has been transmitted through the atmosphere without requiring a complex structure or probe beam.

[0063] This invention is not limited to the specific embodiments described above. Any modifications and alterations to this invention that are within the spirit and principles of this invention should also fall within the scope of protection of the claims of this invention.

Claims

1. A method for measuring the wavefront and orbital angular momentum of a vortex beam subjected to atmospheric turbulence distortion, characterized in that, The method includes the following steps: Step 1: Use a fan-shaped array of microlenses to receive signals to obtain a ring-shaped array of light spots; Step 2: In the polar coordinate system, calculate the radial and angular centroid displacement of each sub-aperture spot in the spot array. Then, select a reference sub-aperture on each ring and calculate the column vector G of the relative centroid displacement of other sub-aperture spots on the ring relative to the reference sub-aperture spot. Step 3: Calculate the Zernike coefficient matrix A of the turbulent distortion wavefront using the relative centroid displacement column vector G; Step 4: Calculate the centroid displacement of the sub-aperture spot on the restored wavefront, calculate the difference between the centroid displacement and the initial spot centroid displacement, obtain the centroid displacement of the residual wavefront, and further obtain the radial and angular components of the centroid displacement of the residual wavefront. Step 5: Combine the radial component of the residual wavefront centroid displacement to obtain the Zernike coefficients for defocus and spherical aberration; Step 6: Combine the angular component of the residual wavefront centroid displacement to obtain the orbital angular momentum of the vortex beam. In step 5, the Zernike coefficients for defocus and spherical aberration are obtained by combining the following formula with the radial component of the residual wavefront centroid displacement. (9) In the above formula, It is the radial component of the residual wavefront centroid displacement. It is a restoration matrix that only contains defocus and spherical aberration, obtained by restoring matrix D. + The Zernike aberration in the image is obtained by replacing defocus and spherical aberration; Step 6: Based on the following formula, and combined with the angular component of the residual wavefront centroid displacement, obtain the orbital angular momentum of the vortex beam. (11) In the above formula, l is the topological charge number, which is the eigenvalue of the orbital angular momentum. It is the angular component of the residual wavefront centroid displacement. It is a restoration matrix containing only spiral phase, obtained by replacing the Zernike aberration in matrix D+ with a spiral phase of l=1.

2. The method according to claim 1, characterized in that, In step 2, the method for calculating the relative centroid displacement column vector G in polar coordinates is as follows: In polar coordinates, the centroid displacement of each sub-aperture spot is decomposed radially and angularly, as expressed below: (2) In the above formula, (x n,m y n,m ) is the coordinate of the (n, m)th pixel within the corresponding sub-aperture on the camera, I n,m This represents the intensity information acquired by the (n, m)th pixel. (X(i, j), Y(i, j)) are the Cartesian coordinates of the centroid of the j-th sub-aperture on the i-th ring, where i represents the i-th ring and j represents the sequence number of the sub-aperture on each ring. , These are the radial and angular displacements of the j-th sub-aperture spot on the i-th ring, respectively, and θ is the angle between the center of the sub-aperture and the origin of the coordinate system and the x-axis. Then, the column vector G of the relative centroid displacement of the other sub-aperture spots on the ring relative to the reference sub-aperture is expressed as follows: (3) In the above formula, This represents the centroid displacement of the j-th sub-aperture on the i-th ring. This represents the centroid displacement of the first sub-aperture of the i-th ring, which is used as the reference sub-aperture. , These represent the radial and angular components of the displacement of the light spot of the j-th sub-aperture on the i-th ring relative to the centroid, respectively.

3. The method according to claim 2, characterized in that, In step 3, the relative centroid displacement matrix D corresponding to each sub-aperture for each order Zernike aberration is represented as follows: (4) in, Let A represent the radial and angular centroid displacements of the k-th Zernike aberration within the j-th sub-aperture of the i-th ring, respectively. Then, the Zernike coefficient matrix A of the turbulent distortion wavefront is calculated using the following formula. (5) In the above formula, It is the restored matrix, which is the inverse matrix of D.

4. The method according to claim 3, characterized in that, In step 4, the wavefront matrix is ​​restored. Represented as: (6) in, The i-th term of the Zernike coefficient A on the turbulent distortion wavefront. Representing the i-th order Zernike aberration, the restored wavefront matrix... The corresponding sub-aperture displacement is expressed as: (7) In the above formula, S is the region where the j-th sub-aperture of the i-th ring is located. Indicates Fourier transform, , It is the centroid displacement of each sub-aperture of the restored wavefront in rectangular coordinates. According to formula (2), the radial displacement of each sub-aperture is obtained. and angular displacement ; Then, the difference between the centroid displacement and the initial spot centroid displacement is calculated to obtain the centroid displacement of the residual wavefront. The radial and angular components of the centroid displacement of the residual wavefront are then obtained, as shown in the following equation: (8)。

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