Underwater vortex beam distortion wavefront correction method based on improved GS algorithm

Abstract

The invention discloses an underwater vortex beam distortion wavefront correction method based on an improved GS algorithm, and the method specifically comprises the steps: 1, generating an ocean turbulence random phase screen, obtaining the field intensity distribution of a distorted Laguerre-Gaussian vortex beam after the Laguerre-Gaussian vortex beam passes through a turbulence distance z and the light field amplitude U '(x, y) of the Laguerre-Gaussian vortex beam subjected to wavefront distortion by adopting a phase screen transmission method; 2, according to the U '(x, y) and the initial amplitude and phase of the Laguerre Gaussian vortex beam, an improved GS algorithm is adopted to obtain a pre-compensation phase; and step 3, obtaining the field intensity distribution of the corrected Laguerre Gaussian vortex beam by adopting a phase screen transmission method according to the pre-compensation phase. According to the invention, the light field distribution before and after the transmission of the Laguerre Gaussian vortex beam is used as the input of the improved GS algorithm to obtain the pre-compensation phase screen, the pre-compensation phase is loaded on the transmission beam, the vortex beam is pre-corrected, and the corrected vortex beam is high in purity and strong in correction capability.

Description

Technical Field

[0001] The present invention belongs to the technical field of wireless optical communication methods, and relates to an underwater vortex beam aberration wavefront correction method based on an improved GS algorithm. Background Art

[0002] With the continuous development of underwater operations such as environmental monitoring, underwater exploration, submarine communication, and offshore oilfield exploration, the scope of marine activities is becoming increasingly extensive, and underwater wireless communication technology has become one of the important communication technologies that major countries in the world are competing to develop. Due to the huge demand for data transmission from underwater sensor networks, unmanned underwater vehicles, autonomous underwater vehicle submarines, etc., high-capacity underwater communication technology has received increasing attention. Compared with traditional underwater acoustic communication and electromagnetic wave communication, underwater wireless optical communication (UWOC) can cover a larger bandwidth and transmit higher data rates. Moreover, the propagation speed of light in water is five orders of magnitude higher than the transmission speed of sound waves. Therefore, information is transmitted more quickly, safely, and effectively in UWOC systems. With the development of technologies such as low-cost and energy-saving light-emitting diodes (EDs) or laser diodes (LDs), UWOC can utilize the blue-green light low-attenuation window to achieve transmission at the Gbit / s level in short-to-medium-distance transmission with the characteristics of low-cost, high-security, and low-latency communication.

[0003] As a brand-new degree of freedom, orbital angular momentum (OAM) can greatly increase the channel capacity of communication as each photon carries orbital angular momentum. A promising method to increase the transmission capacity of UWOC links is to introduce this dimension of OAM. In recent years, OAM multiplexing technology has been widely applied in the field of underwater blue-green light communication. The vortex beam carrying OAM can be used for encoding or as a data carrier, thereby increasing the channel capacity and spectral efficiency.

[0004] However, when the vortex beam carrying orbital angular momentum propagates underwater for communication, it is vulnerable to the interference of ocean turbulence. Ocean turbulence is a common phenomenon underwater, which is a random and unstable energy transfer in space and time. When light propagates underwater, it is affected by factors such as fluid motion, temperature, salinity, and bubbles, resulting in random changes in the refractive index of light at different positions underwater. Ocean turbulence can cause beam expansion, drift, phase fluctuations, intensity scintillation, fading, etc. The beam phase distortion caused by turbulence then leads to the dispersion of the OAM beam mode, crosstalk between different OAM modes, and a decrease in the quality of the OAM beam at the receiving end, which will seriously affect underwater optical communication. To mitigate the impact of turbulence on the OAM beam, in recent years, researchers have proposed the use of adaptive optics. Adaptive optics (AO) is a relatively good compensation method. This method is divided into the WFS-based AO system and the WFS-free AO system according to whether there is a wavefront sensor (WFS) for measuring the distorted phase. Among them, the WFS-based AO system generally uses the Shack-Hartmann wavefront sensor as the detector and the deformable mirror as the wavefront corrector. However, due to the helical wavefront phase structure of the OAM beam, the detection of the OAM beam is difficult, and there are disadvantages such as high cost and difficult installation and maintenance of equipment.

[0005] The GS (Gerchberg-Saxton) algorithm is a classical algorithm for wavefront phase recovery. When using GS for wavefront phase recovery, since the error of the GS algorithm itself does not decrease with the number of iterations and is prone to falling into local optimization, it will lead to poor correction effects. Summary of the Invention

[0006] The purpose of the present invention is to provide an underwater vortex beam distorted wavefront correction method based on an improved GS algorithm. The light field distributions before and after the propagation of the Laguerre-Gaussian vortex beam are used as the input of the improved GS algorithm to obtain a pre-compensation phase screen, and the pre-compensation phase is loaded onto the transmitted beam to achieve pre-correction of the vortex beam, with high purity and strong correction ability after correction.

[0007] The technical solution adopted by the present invention is an underwater vortex beam distorted wavefront correction method based on an improved GS algorithm, which is specifically implemented according to the following steps:

[0008] Step 1, use the power spectrum inversion method combined with the sub-harmonic compensation method to generate an ocean turbulence random phase screen to simulate the phase perturbation caused by ocean turbulence, and use the phase screen transmission method to obtain the field strength distribution of the distorted Laguerre-Gaussian vortex beam after the Laguerre-Gaussian vortex beam passes through a turbulence distance of z and the light field amplitude U'(x, y) of the Laguerre-Gaussian vortex beam with wavefront distortion.

[0009] Step 2, using the improved GS algorithm to obtain the pre-compensation phase according to U'(x, y) and the initial amplitude and phase of the Laguerre-Gaussian vortex beam;

[0010] Step 3, using the phase screen transmission method according to the pre-compensation phase to obtain the field intensity distribution of the corrected Laguerre-Gaussian vortex beam.

[0011] The present invention is also characterized in that:

[0012] In step 1, the ocean turbulence random phase screen is generated by combining the power spectrum inversion method with the subharmonic compensation method to simulate the phase disturbance caused by ocean turbulence. Specifically:

[0013] Step 1.1, obtain the spatial distribution of the phase screen, specifically:

[0014] Calculate the phase spectrum of seawater on any slice perpendicular to the beam propagation direction, that is, the z-axis direction:

[0015] F Φ (k x ,k y )=2πk 2 ΔzΦ(k x ,k y )

[0016] Where Δz is the thickness of the turbulent layer or the propagation distance of the light beam, k = 2π / λ is the wave number of the light beam, λ is the wavelength, Φ(k x ,k y ) is the ocean turbulence refractive index power spectrum;

[0017] Then use F Φ (k x ,k y ) for the Gaussian random complex matrix h(k x ,k y ) is filtered, and then the inverse Fourier transform is performed to obtain the spatial distribution of the phase screen.

[0018]

[0019] Where i is the imaginary unit, C is the constant factor controlling the phase screen variance, C = (Δk x Δk y ) 1 / 2 In the discretized space domain, x and y are the space domain coordinates, x = mΔx, y = nΔy, Δx and Δy are the sampling intervals of the space domain in the x and y directions respectively, m and n are integers, let Δx = Δy, in the wave number domain, k x =m′Δk x , k y =n′Δk y , Δkx , Δk y are the sampling intervals in the x and y directions in the wavenumber domain, respectively. Δk x = 2π / (NΔx), Δk y = 2π / (NΔy), where m' and n' are integers, and N is the number of grids of the phase screen. Then the spatial distribution of the phase screen is:

[0020]

[0021] Let Then the expression for the spatial distribution of the phase screen is:

[0022]

[0023] Step 1.2: Obtain the low-frequency phase screen

[0024] Step 1.3: Obtain the final phase screen

[0025] Step 1.2 is as follows:

[0026] Obtain the low-frequency harmonics F sub (k x , k y ) = (3) -2p F(k lx , k ly );

[0027] where p is the order of the low-frequency harmonics, and (3) -2p is the weight of the low-frequency harmonics. k lx = (3) -2p k x , k ly = (3) -2p k y are the low-frequency harmonic frequencies in the x and y directions in the spatial domain, respectively. After discretizing the spatial distribution of the phase screen obtained in Step 1, the corresponding low-frequency phase screen is obtained as:

[0028]

[0029] where N x and N y are the number of sampling points in the x and y directions in the spatial domain, respectively. h(m', n') is a complex Gaussian random number with zero mean and unit variance. p is the order of the low-frequency harmonics, and N p is the maximum value of the order of the low-frequency harmonics.

[0030] In Step 1.1, the ocean turbulence refractive index power spectrum Φ(k x , k y) The expression is:

[0031]

[0032] where δ = 8.284(κη) 4 / 3 + 12.978(κη) 2 ; A T = 1.863×10 -2 ; A S = 1.9×10 -4 ;

[0033] A TS = 9.41×10 -3 ;

[0034] ε is the kinetic energy dissipation rate, representing the conversion rate of the kinetic energy of ocean turbulence into heat energy, and also indicating the kinetic energy lost by unit mass of seawater per unit time. ε ∈ [10 -10 , 10 -1 m 2 / s 3 , where ε takes 10 -10 representing the deep - sea area, and ε takes 10 -1 representing the turbulent active area; η is the inner - scale factor, η ∈ [6×10 -5 , 0.01] m, χ T is the mean - square temperature dissipation rate, representing the amount of turbulence acting on the temperature field per unit time, and its value range is χ T ∈ [10 -10 , 10 -4 K 2 / s.

[0035] In step 1, the phase - screen propagation method is used to obtain the field - strength distribution of the distorted Laguerre - Gaussian vortex beam after passing through a turbulent distance z and the optical - field amplitude U'(x, y) of the Laguerre - Gaussian vortex beam with wave - front distortion, specifically as follows:

[0036] Initialize a Laguerre - Gaussian vortex beam with a topological charge number l, and its field - strength distribution at the light source is

[0037]

[0038] where ω0 is the beam - waist radius of the zero - order Gaussian beam, ρ is the radius coordinate in the polar coordinate system, is the angular coordinate, is the generalized Laguerre polynomial, l is the topological charge number, and m represents the radial mode number;

[0039] Convert to the expression E0(x, y) in the rectangular coordinate system:

[0040]

[0041] Among them, U0(x, y) is the optical field amplitude of the ideal Laguerre-Gaussian vortex beam without wavefront distortion, is the ideal phase of the Laguerre-Gaussian vortex beam;

[0042] The phase screen transmission method is used to obtain the field strength distribution E(x, y, z) of the distorted Laguerre-Gaussian vortex beam after the Laguerre-Gaussian vortex beam has passed through a turbulent distance of Z:

[0043]

[0044] Among them, f x and f y are the spatial frequencies on the x-axis and y-axis, k = 2π / λ is the wave number of the beam, and λ is the wavelength. is the phase screen after finally passing through sub-harmonic compensation; F(·) represents the Fourier transform, and F -1 represents the inverse Fourier transform;

[0045] Then, in MATLAB, abs(E(x, y, z)) is used to obtain the optical field amplitude U'(x, y) of the Laguerre-Gaussian vortex beam with wavefront distortion.

[0046] Step 2 is specifically as follows:

[0047] Step 2.1: Select the optical field amplitude U0(x, y) of the ideal Laguerre-Gaussian vortex beam without wavefront distortion as the amplitude of the input optical field, and the ideal phase of the Laguerre-Gaussian vortex beam as the initial random phase, and combine U0(x, y) and to obtain the input optical field distribution E0(x, y) as the input for diffraction calculation:

[0048]

[0049] Step 2.2: Perform Fresnel diffraction transmission operation on the input optical field distribution E0(x, y) to obtain the diffracted optical field distribution F k (x, y);

[0050] Step 2.3: Use U'(x, y) to construct the input F k+1 (x, y) for the next iteration of the Fresnel inverse diffraction part:

[0051]

[0052] Among them, a is the weighting coefficient, and U'(x, y) is the optical field amplitude of the Laguerre-Gaussian vortex beam with wavefront distortion; is F k The phase of (x, y);

[0053] Step 2.4, perform the inverse Fresnel diffraction operation on F k+1 (x, y) to obtain the optical field distribution E k (x, y);

[0054] Step 2.5, construct the optical field distribution E k (x, y) for the next iteration of the Fresnel diffraction part: k+1 (x, y):

[0055]

[0056] where γ represents the restricted region in the spatial domain, β is the feedback coefficient, and x, y are the spatial domain coordinates;

[0057] Step 2.6, determine whether k is greater than N. If so, exit the iterative process; if not, return to Step 2.2 and use the optical field distribution E k+1 (x, y) obtained in Step 2.5 as the input to execute Steps 2.2 - 2.5 again until the defined maximum number of iterations is reached, at which point the calculation terminates, and the phase corresponding to E k (x, y) obtained in the last iteration step 2.4 is used as the reconstructed beam distortion phase, where N is the maximum number of iterations;

[0058] Step 2.7, obtain the final pre - compensation phase D(x, y) according to the reconstructed beam distortion phase :

[0059]

[0060] In Step 2.2, F k (x i , y i ) is specifically:

[0061]

[0062] where g k (x, y) and are respectively the amplitude and phase obtained by performing the Fresnel diffraction transmission operation on the input optical field distribution E0(x, y).

[0063] In Step 2.4, E k (x, y) is specifically:

[0064]

[0065] where E k (x, y) represents the result of performing an operation on F k+1(x, y) performs the Fresnel inverse diffraction transmission operation to obtain the optical field distribution, where U k (x, y) is the amplitude of E k (x, y), and k is the phase of E

[0066] The field strength distribution of the Laguerre-Gaussian vortex beam after correction in step 3 is:

[0067]

[0068] The beneficial effects of the present invention are:

[0069] (1) The present invention improves the GS algorithm and combines it with the Fresnel diffraction theorem. The optical field distributions before and after the transmission of the Laguerre-Gaussian vortex beam are used as the input of the improved GS algorithm, and iterative calculations are performed on it to calculate the phase distortion caused by ocean turbulence, and a pre-compensation phase screen can be obtained more effectively, so as to achieve the purpose of correcting the distorted vortex beam. Compared with the traditional GS algorithm, the mode purity of OAM after correction using this algorithm is higher and the correction ability is stronger.

[0070] (2) The present invention does not require a wavefront sensor. At the receiving end, only a CCD camera is needed to record the intensity distribution of the distorted vortex beam, and wavefront detection can be realized, greatly reducing the complexity and cost of the adaptive optical system. Description of the Drawings

[0071] Figure 1 is the schematic diagram of the underwater vortex beam distorted wavefront correction method based on the improved GS algorithm of the present invention;

[0072] Figure 2 is the correction flowchart of the underwater vortex beam distorted wavefront correction of the improved GS algorithm in the underwater vortex beam distorted wavefront correction method based on the improved GS algorithm of the present invention;

[0073] Figure 3(a) is the light intensity diagram of the distorted vortex beam without wavefront correction in Embodiment 7 of the present invention;

[0074] Figure 3(b) is the light intensity diagram of the vortex beam after correction using the traditional GS algorithm in Embodiment 7 of the present invention;

[0075] Figure 3(c) is the light intensity diagram of the vortex beam after correction using the improved GS algorithm in Embodiment 7 of the present invention;

[0076] Figure 4(a) is the phase diagram of the distorted vortex beam without wavefront correction in Embodiment 7 of the present invention;

[0077] Figure 4(b) is the phase diagram of the vortex beam after correction using the traditional GS algorithm in Embodiment 7 of the present invention;

[0078] Figure 4(c) is the phase diagram of the vortex beam after correction using the improved GS algorithm in Embodiment 7 of the present invention;

[0079] Figure 5(a) is the histogram of the mode purity of the distorted vortex beam without wavefront correction in Embodiment 7 of the present invention;

[0080] Figure 5(b) is the histogram of the mode purity of the vortex beam after correction using the traditional GS algorithm in Embodiment 7 of the present invention;

[0081] Figure 5(c) is the histogram of the mode purity of the vortex beam after correction using the improved GS algorithm in Embodiment 7 of the present invention. Specific implementation manners

[0082] The following is a detailed description in combination with specific implementation manners.

[0083] Embodiment 1

[0084] The method for correcting the distorted wavefront of the underwater vortex beam based on the improved GS algorithm of the present invention has the principle as Figure 1 shown, and is specifically implemented according to the following steps:

[0085] Step 1, use the power spectrum inversion method combined with the sub-harmonic compensation method to generate an ocean turbulence random phase screen to simulate the phase perturbation caused by ocean turbulence, and use the phase screen propagation method to obtain the field strength distribution of the distorted Laguerre-Gaussian vortex beam after the Laguerre-Gaussian vortex beam passes through the turbulence distance z and the optical field amplitude U'(x, y) of the Laguerre-Gaussian vortex beam with wavefront distortion;

[0086] Step 2, obtain the pre-compensation phase using the improved GS algorithm according to U'(x, y) and the initial amplitude and phase of the Laguerre-Gaussian vortex beam;

[0087] Step 3, obtain the field strength distribution of the Laguerre-Gaussian vortex beam after correction using the phase screen propagation method according to the pre-compensation phase.

[0088] Embodiment 2

[0089] On the basis of Embodiment 1,

[0090] The specific implementation of using the power spectrum inversion method combined with the sub-harmonic compensation method to generate an ocean turbulence random phase screen to simulate the phase perturbation caused by ocean turbulence in Step 1 is as follows:

[0091] Step 1.1, obtain the spatial distribution of the phase screen, specifically:

[0092] The randomly distributed phase screen is a random complex domain value reflecting the refractive index variation of ocean turbulence, and can be represented by an N x ×N y random complex matrix, Nx and N y are the sampling points in the x and y directions in the spatial domain respectively, and their statistical characteristics satisfy underwater turbulent fluctuations. Then: Calculate the seawater phase spectrum on any slice perpendicular to the beam propagation direction, that is, the z-axis direction:

[0093] F Φ (k x , k y ) = 2πk 2 ΔzΦ(k x , k y )

[0094] where Δz is the thickness of the turbulent layer or the propagation distance of the beam, k = 2π / λ is the wave number of the beam, λ is the wavelength, and Φ(k x , k y ) is the ocean turbulence refractive index power spectrum;

[0095] Then use F Φ (k x , k y ) to filter the Gaussian random complex matrix h(k x , k y ), and then perform the inverse Fourier transform to obtain the spatial distribution of the phase screen

[0096]

[0097] where i is the imaginary unit, C is a constant factor that controls the variance of the phase screen, C = (Δk x Δk y ), in the discretized spatial domain, x and y are spatial domain coordinates, x = mΔx, y = nΔy, Δx and Δy are the sampling intervals in the x and y directions of the spatial domain respectively, m and n are integers, assuming Δx = Δy, in the wave number domain, k 1 / 2 = m′Δk x , k x = n′Δk y , Δk y and Δk x are the sampling intervals in the x and y directions in the wave number domain respectively, Δk y = 2π / (NΔx), Δk x = 2π / (NΔy), m′ and n′ are integers, and N is the number of grids of the phase screen. Then the spatial distribution of the phase screen is: y Let

[0098]

[0099] Then the expression of the spatial distribution of the phase screen is:

[0100] ​

[0101] Step 1.2, obtaining the low-frequency phase screen Specifically:

[0102] In order to fully sample the low-frequency part of the power spectrum, a (Δk x )×(Δk y ) region centered at the origin is used for low-frequency compensation. At this time, the low-frequency compensation region overlaps with the sampling points of the high-frequency phase screen. Then, the (Δk x )×(Δk y ) region is divided into 3×3 sub-regions. For the first-level sampling, in the outermost sub-region, the remaining sub-regions are used for the next-level sub-harmonic compensation until the set low-frequency sub-harmonic order N p position is reached; then the low-frequency sub-harmonics F sub (k x ,k y ) of the power spectrum are obtained, where F(k -2p ,k lx ,k ly )=(3)

[0103] where p is the low-frequency sub-harmonic order, (3) -2p is the low-frequency sub-harmonic weight, k lx =(3) -2p k x , k ly =(3) -2p k y are the low-frequency sub-harmonic frequencies in the x and y directions in the spatial domain respectively. After discretizing the spatial distribution of the phase screen obtained in Step 1, the corresponding low-frequency phase screen is obtained which is:

[0104]

[0105] where N x and N y are the number of sampling points in the x and y directions in the spatial domain respectively, h(m′, n′) is a complex Gaussian random number with zero mean and unit variance, p is the low-frequency sub-harmonic order, and N p is the maximum value of the low-frequency sub-harmonic order;

[0106] Step 1.3, obtaining the final phase screen

[0107] Example 3

[0108] Based on Example 2, the selected light source is a Laguerre-Gaussian vortex beam. The introduction of the oceanic turbulent phase screen is to more realistically simulate the perturbation of the turbulent environment on the beam. Its simulation implementation is based on the power spectrum inversion method combined with the sub-harmonic compensation method to generate the oceanic turbulent random phase screen.

[0109] Without considering the absorption and scattering effects of seawater particles and assuming that oceanic turbulence is only caused by the perturbation of seawater temperature and salinity, the refractive index fluctuation of clear seawater mainly depends on the fluctuations of seawater temperature and salinity.

[0110] Nikishov proposed that the spatial power spectrum model of the refractive index fluctuation of seawater in isotropic and homogeneous turbulence is the oceanic turbulent refractive index power spectrum, then:

[0111] In step 1.1, the expression of the oceanic turbulent refractive index power spectrum Φ(k x ,k y ) is:

[0112]

[0113] where δ = 8.284(κη) 4 / 3 + 12.978(κη) 2 ; A T = 1.863×10 -2 ; A S = 1.9×10 -4 ;

[0114] A TS = 9.41×10 -3 ;

[0115] ε is the kinetic energy dissipation rate, representing the conversion rate of the kinetic energy of oceanic turbulence into heat energy, and also indicating the kinetic energy lost by unit mass of seawater per unit time. The value range is ε ∈ [10 -10 , 10 -1 m 2 / s 3 , where ε taking 10 -10 represents the deep sea area, and ε taking 10 -1 represents the turbulent active area; η is the inner scale factor (Kolmogorov size), which is comprehensively determined by the molecular viscosity coefficient and the kinetic energy dissipation rate. When the scale is smaller than the Kolmogorov size, it belongs to the molecular dissipation area, and generally this situation is not considered. The value range of η is: η ∈ [6×10 -5 , 0.01] m, and this interval is called the inertial interval; χ T is the mean square temperature dissipation rate, representing the amount of turbulence acting on the temperature field per unit time. The value range is χ T ∈ [10 -10 , 10-4 K 2 / s.

[0116] Example 4

[0117] On the basis of Example 3, in step 1, the phase screen transmission method is used to obtain the field strength distribution of the distorted Laguerre-Gaussian vortex beam after the Laguerre-Gaussian vortex beam passes through a turbulence distance of z, and the optical field amplitude U'(x, y) of the Laguerre-Gaussian vortex beam with wavefront distortion is specifically as follows:

[0118] Initialize a Laguerre-Gaussian vortex beam with a topological charge number of l, and its field strength distribution at the light source is

[0119]

[0120] where ω0 is the waist radius of the zero-order Gaussian light, ρ is the radius coordinate in the polar coordinate system, is the angular coordinate, is the generalized Laguerre polynomial, l is the topological charge number, and m represents the radial mode number;

[0121] Convert to the expression E0(x, y) in the rectangular coordinate system:

[0122]

[0123] where U0(x, y) is the optical field amplitude of the ideal Laguerre-Gaussian vortex beam without wavefront distortion, is the ideal phase of the Laguerre-Gaussian vortex beam;

[0124] Use the phase screen transmission method to obtain the field strength distribution E(x, y, z) of the distorted Laguerre-Gaussian vortex beam after the Laguerre-Gaussian vortex beam passes through a turbulence distance of Z:

[0125]

[0126] where f x and f y are the spatial frequencies on the x-axis and y-axis, k = 2π / λ is the wave number of the light beam, and λ is the wavelength, is the phase screen after final sub-harmonic compensation; F(·) represents the Fourier transform, and F -1 represents the inverse Fourier transform;

[0127] Then, in MATLAB, use abs(E(x, y, z)) to obtain the optical field amplitude U'(x, y) of the Laguerre-Gaussian vortex beam with wavefront distortion.

[0128] Example 5

[0129] Based on Embodiment 3, the process of obtaining the pre-compensation phase by the improved GS algorithm in Step 2 is as Figure 2 shown, specifically:

[0130] Step 2.1, Select the optical field amplitude U0(x,y) of the ideal Laguerre-Gaussian vortex beam without wavefront distortion as the amplitude of the input optical field, and the ideal phase of the Laguerre-Gaussian vortex beam as the initial random phase, and combine U0(x,y) and to obtain the input optical field distribution E0(x,y) as the input for diffraction calculation:

[0131]

[0132] Step 2.2, Perform Fresnel diffraction propagation operation on the input optical field distribution E0(x,y) to obtain the diffracted optical field distribution F k (x,y):

[0133]

[0134] where, g k (x,y) and are the amplitude and phase respectively obtained by performing Fresnel diffraction propagation operation on the input optical field distribution E0(x,y);

[0135] Step 2.3, Use U'(x,y) to construct the input F for the next iteration of the Fresnel inverse diffraction part k+1 (x,y):

[0136]

[0137] where, a is the weighting coefficient, and U'(x,y) is the optical field amplitude of the Laguerre-Gaussian vortex beam with wavefront distortion; is the phase of F k (x,y);

[0138] Step 2.4, Perform Fresnel inverse diffraction operation on F k+1 (x,y) to obtain the optical field distribution E k (x,y):

[0139]

[0140] where, E k (x,y) represents the optical field distribution obtained by performing Fresnel inverse diffraction propagation operation on F k+1 (x,y), where, U k (x,y) is the amplitude of E k (x,y), is the phase of E k(x,y) phase;

[0141] Step 2.5, according to E k construct the optical field distribution E k+1 (x,y) for the next iteration of the Fresnel diffraction part:

[0142]

[0143] where γ represents the restricted area in the spatial domain, β is the feedback coefficient, and x, y are the spatial domain coordinates;

[0144] Step 2.6, determine whether k is greater than N. If so, exit the iterative process; if not, return to Step 2.2, and use the optical field distribution E k+1 (x,y) obtained in Step 2.5 as the input and execute Steps 2.2 - 2.5 again until the defined maximum number of iterations is reached, then the calculation terminates, and the phase corresponding to E k (x,y) obtained in the last iteration Step 2.4 is used as the reconstructed beam distortion phase, where N is the maximum number of iterations;

[0145] Step 2.7, according to the reconstructed beam distortion phase obtain the final pre - compensation phase D(x,y):

[0146]

[0147] Example 6

[0148] Based on Example 5, the field strength distribution of the Laguerre - Gaussian vortex beam after correction in Step 3 is:

[0149]

[0150] Example 7

[0151] Based on Example 6, in this example, the Laguerre - Gaussian vortex beam is simulated passing through an oceanic turbulence phase screen, then the pre - compensation phase screen is obtained by the improved GS algorithm and re - loaded on the Laguerre - Gaussian vortex beam for transmission through the turbulence, so as to realize the correction of the vortex beam. Table 1 shows the specific parameters set in this example:

[0152] Table 1

[0153]

[0154]

[0155] The correction process of the traditional GS algorithm is as follows:

[0156] (1) Select the ideal optical field amplitude U i (x,y) that has not undergone wavefront distortion as the amplitude of the input optical field, and select the ideal phase as the initial random phase. The two together form an optical field as the input optical field for diffraction calculation;

[0157] (2) Perform diffraction propagation calculation on the optical field to obtain its transformed domain amplitude spectrum A(x,y) and phase spectrum θ(x,y);

[0158] (3) Replace A(x,y) with the amplitude spectrum U0(x,y) of the distorted beam to obtain the new complex amplitude of the optical field U0(x,y)exp(iθ(x,y));

[0159] (4) Perform inverse diffraction operation on the optical field U0(x,y)exp(iθ(x,y)) to obtain the spatial domain amplitude spectrum a(x,y) and phase spectrum H(x,y);

[0160] (5) Replace a(x,y) with the amplitude spectrum U i (x,y) of the initial ideal optical field to obtain the initial optical field expression U i (x,y)exp(iH(x,y)) for the next loop iteration. When certain iteration conditions are met or the defined number of loop iterations is reached, the calculation terminates, and the distorted phase H(x,y) of the reconstructed beam can be obtained.

[0161] (6) The corresponding distorted phase of oceanic turbulence is

[0162] As shown in Fig. 3(a), it is the intensity diagram of the distorted vortex beam without wavefront correction in this embodiment. Fig. 3(b) is the intensity diagram of the vortex beam corrected by the traditional GS algorithm in this embodiment. Fig. 3(c) is the intensity diagram of the vortex beam corrected by the improved GS algorithm of the present invention in this embodiment. Fig. 4(a) is the phase diagram of the distorted vortex beam without wavefront correction in this embodiment. Fig. 4(b) is the phase diagram of the vortex beam corrected by the traditional GS algorithm in this embodiment. Fig. 4(c) is the phase diagram of the vortex beam corrected by the improved GS algorithm in this embodiment. Fig. 5(a) is the mode purity histogram of the distorted vortex beam without wavefront correction in this embodiment. Fig. 5(b) is the mode purity histogram of the vortex beam corrected by the traditional GS algorithm in this embodiment. Fig. 5(c) is the mode purity histogram of the vortex beam corrected by the improved GS algorithm in this embodiment.

[0163] It can be found from Fig. 3(a) that after the Laguerre-Gaussian vortex beam passes through oceanic turbulence, the perfect hollow ring structure is deformed and the light intensity is diffused. By comparing Fig. 3(b) and Fig. 3(c), it can be found that both correction algorithms have played an improving role. The hollow structure of the vortex beam has been significantly improved, and the light intensity has become more uniform. Moreover, the correction effect of the improved GS algorithm is significantly better than that of the traditional GS algorithm.

[0164] It can be found from Fig. 4(a) that after the Laguerre-Gaussian vortex beam passes through oceanic turbulence, the isophase lines are distorted and deformed, and the phase singularities are significantly diffused. It is difficult to distinguish the OAM order from the phase diagram in the simulation. By comparing Fig. 3(b) and Fig. 3(c), it can be found that both correction algorithms have played an improving role. The isophase of the vortex beam has been restored, the phase singularities are aggregated, and the OAM order can be determined through the phase. Moreover, the correction effect of the improved GS algorithm is significantly better than that of the traditional GS algorithm.

[0165] It can be found from Fig. 5(a) that most of the energy of the Laguerre-Gaussian vortex beam with a topological charge of 3 has been transferred to the Laguerre-Gaussian vortex beam with a topological charge of 4 after passing through oceanic turbulence. By comparing Fig. 4(b) and Fig. 4(c), it can be found that both correction algorithms have improved the mode purity of the mode with a topological charge of 3. Using the original GS algorithm, the mode purity is increased from 0.1334 to 0.6679. Using the improved GS algorithm, the mode purity is increased from 0.1334 to 0.8321. By comparing the mode purity before and after correction, it can be found that the correction effect of the improved GS algorithm is significantly better than that of the traditional GS algorithm.

[0166] Through the analysis of this embodiment, the effectiveness of the present invention is jointly verified, and the correction effect is significantly better than that of the traditional GS algorithm.

Claims

1. An underwater vortex beam distorted wavefront correction method based on an improved GS algorithm, characterized in that The implementation is carried out according to the following steps: Step 1: Use the power spectrum inversion method combined with the sub-harmonic compensation method to generate an ocean turbulence random phase screen to simulate the phase perturbation caused by ocean turbulence. Use the phase screen propagation method to obtain the field strength distribution of the distorted Laguerre-Gaussian vortex beam after the Laguerre-Gaussian vortex beam has propagated through a turbulence distance of z, and the optical field amplitude U'(x, y) of the Laguerre-Gaussian vortex beam with wavefront distortion; Step 2: Obtain the pre-compensation phase using the improved GS algorithm according to U'(x, y), the initial amplitude, and the phase of the Laguerre-Gaussian vortex beam; Step 3: Use the phase screen propagation method to obtain the field strength distribution of the Laguerre-Gaussian vortex beam after correction according to the pre-compensation phase.

2. The method for correcting the distorted wavefront of an underwater vortex beam based on the improved GS algorithm according to claim 1, wherein The specific method of using the power spectrum inversion method combined with the sub-harmonic compensation method to generate an ocean turbulence random phase screen to simulate the phase perturbation caused by ocean turbulence in Step 1 is as follows: Step 1.1: Obtain the spatial distribution of the phase screen, specifically: Calculate the sea water phase spectrum on any slice perpendicular to the beam propagation direction, that is, the z-axis direction; F Φ (k x ,k y ) = 2πk 2 ΔzΦ(k x ,k y ) where Δz is the thickness of the turbulent layer or the propagation distance of the light beam, k = 2π / λ is the wave number of the light beam, λ is the wavelength, and Φ(k x , k y ) is the refractive index power spectrum of ocean turbulence; Then use F Φ (k x , k y ) to filter the Gaussian random complex matrix h(k x , k y ), and then perform the inverse Fourier transform to obtain the spatial distribution of the phase screen where \(i\) is the imaginary unit, \(C\) is a constant factor controlling the variance of the phase screen, \(C = (\Delta k\) x \(\Delta k\) y ) 1 / 2 . In the discretized spatial domain, \(x\) and \(y\) are spatial domain coordinates, \(x = m\Delta x\), \(y = n\Delta y\), \(\Delta x\) and \(\Delta y\) are the sampling intervals of the spatial domain in the \(x\) and \(y\) directions respectively, \(m\) and \(n\) are integers. Let \(\Delta x=\Delta y\). In the wavenumber domain, \(k\) x \(= m'\Delta k\) x , \(k\) y \(= n'\Delta k\) y , \(\Delta k\) x and \(\Delta k\) y are the sampling intervals of the wavenumber domain in the \(x\) and \(y\) directions respectively, \(\Delta k\) x \(= 2\pi / (N\Delta x)\), \(\Delta k\) y \(= 2\pi / (N\Delta y)\), \(m'\) and \(n'\) are integers, \(N\) is the number of grids of the phase screen. Then the spatial distribution of the phase screen is: Let Then the expression for the spatial distribution of the phase screen is as follows: Step 1.2, obtaining a low-frequency phase screen Step 1.3, obtaining the final phase screen 3. The method for correcting the distorted wavefront of an underwater vortex beam based on the improved GS algorithm according to claim 2, characterized in that Step 1.2 is as follows: Obtain the low-frequency harmonics F of the power spectrum sub (k x ,k y ) = (3) -2p F(k lx ,k ly ); Among them, p is the low-frequency harmonic series, (3) -2p is the low-frequency harmonic weight, k lx =(3) -2p k x , k ly =(3) -2p k y are the low-frequency harmonic frequencies in the x and y directions in the spatial domain respectively. After discretizing the spatial distribution of the phase screen obtained in step 1, the corresponding low-frequency phase screen is obtained is: where N x and N y are the sampling numbers in the x and y directions in the spatial domain respectively, h(m′, n′) is a complex Gaussian random number with zero mean and unit variance, p is the low-frequency harmonic series, and N p is the maximum value of the low-frequency harmonic series.

4. The method for correcting the distorted wavefront of an underwater vortex beam based on the improved GS algorithm according to claim 3, wherein In the above step 1.1, the expression of the ocean turbulence refractive index power spectrum Φ(k x , k y ) is as follows: where δ = 8.284(κη) 4 / 3 + 12.978(κη) 2 ; A T = 1.863×10 -2 ; A S = 1.9×10 -4 ; A TS ＝9.41×10 -3 ； ε is the kinetic energy dissipation rate, representing the conversion rate of the kinetic energy of ocean turbulence into heat energy, and also indicating the kinetic energy lost by seawater per unit mass per unit time. ε ∈ [10 -10 , 10 -1 m 2 / s 3 , where ε taking 10 -10 represents the deep - sea area, and ε taking 10 -1 represents the area with active turbulence; η is the inner - scale factor, η ∈ [6×10 -5 , 0.01] m, χ T is the mean - square temperature dissipation rate, representing the amount of turbulence acting on the temperature field per unit time, and the value range is χ T ∈ [10 -10 , 10 -4 K 2 / s.

5. The method for correcting the distorted wavefront of an underwater vortex beam based on the improved GS algorithm according to claim 4, wherein The specific method of using the phase screen propagation method to obtain the field strength distribution of the distorted Laguerre-Gaussian vortex beam after the Laguerre-Gaussian vortex beam has propagated through a turbulence distance of z, and the optical field amplitude U'(x, y) of the Laguerre-Gaussian vortex beam with wavefront distortion in Step 1 is as follows: Initialize a Laguerre-Gaussian vortex beam with a topological charge of l, and its field strength distribution at the light source is where ω0 is the waist radius of the zero-order Gaussian beam, ρ is the radial coordinate in the polar coordinate system, is the angular coordinate, is the generalized Laguerre polynomial, l is the topological charge number, and m represents the radial mode number; Convert to the expression E0(x,y) in the rectangular coordinate system: where \(U_0(x,y)\) is the optical field amplitude of the ideal Laguerre-Gaussian vortex beam without wavefront distortion, is the ideal phase of the Laguerre-Gaussian vortex beam; Use the phase screen propagation method to obtain the field strength distribution E(x, y, z) of the distorted Laguerre-Gaussian vortex beam after the Laguerre-Gaussian vortex beam has propagated through a turbulence distance of Z; where f x and f y are the spatial frequencies on the x-axis and y-axis, k = 2π / λ is the wave number of the light beam, λ is the wavelength, is the phase screen after final sub-harmonic compensation, F(·) represents the Fourier transform, and F -1 represents the inverse Fourier transform; Then use abs(E(x, y, z)) in MATLAB to obtain the optical field amplitude U'(x, y) of the Laguerre-Gaussian vortex beam with wavefront distortion.

6. The method for correcting the distorted wavefront of an underwater vortex beam based on the improved GS algorithm according to claim 5, wherein Step 2 is specifically as follows: Step 2.1, select the optical field amplitude U0(x,y) of the ideal Laguerre-Gaussian vortex beam without wavefront distortion as the amplitude of the input optical field, and the ideal phase of the Laguerre-Gaussian vortex beam as the initial random phase, and combine U0(x,y) and to obtain the input optical field distribution E0(x,y) as the input for diffraction calculation: Step 2.2, perform Fresnel diffraction propagation operation on the input optical field distribution E0(x, y) to obtain the diffracted optical field distribution F k (x, y); Step 2.3, use U'(x, y) to construct the input F for the next partial iteration of the inverse Fresnel diffraction k+1 (x, y): Among them, a is the weighting coefficient, and U'(x, y) is the optical field amplitude of the Laguerre-Gaussian vortex beam with wavefront distortion; is F k (x, y) phase; Step 2.4, perform the inverse Fresnel diffraction operation on F k+1 (x, y) to obtain the optical field distribution E k (x, y); Step 2.5, according to E k (x, y) construct the optical field distribution E k+1 (x, y) for the next iteration of the Fresnel diffraction part: Among them, γ represents the restricted area in the spatial domain, β is the feedback coefficient, and x, y are the spatial domain coordinates; Step 2.6, determine whether k is greater than N. If so, exit the iterative process; if not, return to Step 2.2, and use the optical field distribution E k+1 (x, y) obtained in Step 2.5 as the input and execute Steps 2.2 - 2.5 again until the calculation terminates when the defined maximum number of iterations is reached. Take the phase corresponding to E k (x, y) obtained in the last iteration of Step 2.4 as the reconstructed beam aberration phase, where N is the maximum number of iterations; Step 2.7, according to the distorted phase of the reconstructed beam obtain the final pre-compensation phase D(x, y):

7. The method for correcting the distorted wavefront of an underwater vortex beam based on the improved GS algorithm according to claim 6, characterized in that, In step 2.2, F k (x i , y i ) is specifically: Among them, g k (x, y) and are the amplitude and phase respectively obtained by performing the Fresnel diffraction transmission operation on the input optical field distribution E0(x, y).

8. The method for correcting the distorted wavefront of an underwater vortex beam based on the improved GS algorithm according to claim 7, characterized in that In step 2.4, E k (x, y) specifically is: Among them, E k (x, y) represents performing the Fresnel inverse diffraction transmission operation on F k+1 (x, y) to obtain the optical field distribution. Among them, U k (x, y) is the amplitude of E k (x, y), is the phase of E k (x, y).

9. The method for correcting the distorted wavefront of an underwater vortex beam based on the improved GS algorithm according to claim 8, wherein The field strength distribution of the Laguerre-Gaussian vortex beam after correction in Step 3 is:

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