Laser atmospheric transmission efficient simulation method based on quantum computing

Abstract

The invention relates to a laser atmospheric transmission efficient simulation method based on quantum computing, and belongs to the crossing field of quantum computing and atmospheric optics. Under the paraxial approximation condition, the transmission process of laser in atmospheric turbulence is mapped into a quantum calculation framework, and efficient simulation is achieved through a quantum circuit and a quantum algorithm. The method specifically comprises the following steps: determining the number of required quantum bits, and initializing a quantum wave function of a light field; generating a phase screen according to the atmospheric turbulence model, and loading phase modulation into a quantum state; quantum Fourier transform is executed, and a spatial domain is converted into a frequency domain; applying a transfer function to simulate propagation of light in a free space; returning to a spatial domain through inverse quantum Fourier transform; and repeating the process to simulate multi-phase screen transmission, and finally obtaining the intensity distribution of the light field. According to the method, the simulation efficiency and precision are greatly improved, and the method is suitable for high-precision and large-scale laser transmission simulation requirements and has a wide application prospect.

Description

Technical Field

[0001] The present invention belongs to the cross - field of quantum computing and atmospheric optics, and specifically relates to an efficient simulation method for laser atmospheric transmission based on quantum computing. Background Art

[0002] The research on the transmission of laser in atmospheric turbulence has important application value in the fields of laser communication, laser imaging, and laser directed energy. Atmospheric turbulence can cause random changes in the phase and amplitude of the laser beam, affecting the performance of the laser system. Therefore, accurately simulating the transmission process of laser in atmospheric turbulence is crucial for designing high - performance laser systems.

[0003] Currently, classical laser atmospheric transmission simulation is mainly based on the multi - phase screen transmission method. This method approximates the transmission process of laser in the atmosphere by decomposing the atmospheric turbulence model into a series of phase screens and simulating the interaction of the laser beam on each phase screen. However, this method requires a large number of Fourier transforms during the calculation process, and the computational complexity is O(NlogN), where N is the number of sampling points. When the simulation accuracy requirement is high and the number of sampling points increases, the amount of calculation increases exponentially, resulting in low computational efficiency and difficulty in meeting the actual application requirements.

[0004] With the rapid development of quantum computing technology, quantum computers, with their parallel computing capabilities and unique quantum algorithms, provide new possibilities for solving complex computational problems, especially in simulating quantum physical systems and solving problems with high complexity. However, the current quantum simulation research on laser atmospheric transmission is still in its infancy, lacking effective quantum algorithms and implementation schemes.

[0005] Therefore, how to utilize the advantages of quantum computing to develop an efficient quantum simulation method for laser atmospheric transmission, reduce the computational complexity, and improve the simulation accuracy has become a technical problem to be solved urgently. Summary of the Invention

[0006] (1) Technical Problems to be Solved

[0007] The technical problem to be solved by the present invention is how to provide an efficient simulation method for laser atmospheric transmission based on quantum computing to solve the problems of high computational complexity and low simulation efficiency in the existing laser atmospheric transmission technology.

[0008] (2) Technical Solutions

[0009] To solve the above - mentioned technical problems, the present invention proposes an efficient simulation method for laser atmospheric transmission based on quantum computing, which includes the following steps:

[0010] S1. Quantum bit selection: Determine the required number of quantum bits \(n = \log_2N\), where \(N\) is the number of sampling points; construct a wave function \(|\psi\rangle\) using \(n\) quantum bits, where the eigenstate of each quantum bit corresponds to the spatial or spectral sampling point of the laser, represented as the computational basis state \(|\gamma\rangle\), where the value range of \(\gamma\) is Negative values are represented in two's complement;

[0011] S2. Initialize the wave function: Encode the initial optical field distribution \(U (0) (\gamma)\) into the quantum state ;

[0012] S3. Phase modulation of turbulence: According to the refractive index power spectrum of atmospheric turbulence, use the power spectrum inversion method to generate a series of phase screens to simulate the influence of atmospheric turbulence; use the selective phase shift operator to load the phase screen information onto the quantum state to achieve phase modulation;

[0013] S4. Fourier domain conversion: Perform a quantum Fourier transform (QFT) on the phase-modulated quantum state to convert the spatial domain to the frequency domain; use Hadamard gates and controlled phase rotation gates to construct a QFT circuit to achieve an efficient Fourier transform;

[0014] S5. Second-order propagation processing: In the frequency domain, apply the free space transfer function to each frequency component to simulate the propagation of light in free space; through the combination of phase gates and controlled phase gates, implement the phase operation of the transfer function in the quantum circuit;

[0015] S6. Inverse Fourier transform: Apply the inverse quantum Fourier transform (IQFT) to convert the quantum state in the frequency domain back to the spatial domain to obtain the propagated quantum state;

[0016] S7. Repeated phase screen processing: For the multi-phase screen transmission process, repeat the processes of phase modulation, Fourier transform, transfer function application, and inverse Fourier transform in S3 - S6, sequentially load the phase modulation of each phase screen, simulate the multiple interactions of the laser in atmospheric turbulence, and obtain the final quantum state and the optical field intensity distribution.

[0017] (III) Beneficial effects

[0018] The present invention proposes an efficient simulation method for laser atmospheric transmission based on quantum computing. The present invention provides an efficient simulation method for laser atmospheric transmission based on quantum computing, which significantly reduces the computational complexity and improves the simulation accuracy and efficiency. Compared with traditional classical algorithms, this method utilizes the parallelism of quantum computing and the efficiency of quantum algorithms to reduce the computational complexity from \(O(N\log N)\) to \(O((\log N) 2 ), achieving an exponential performance improvement.

[0019] Under the paraxial approximation condition, the method of the present invention models atmospheric turbulence through the multi-phase screen method, and combines quantum Fourier transform and inverse Fourier transform to achieve efficient propagation simulation of optical fields in a turbulent environment. This method not only improves the computational efficiency but also enhances the storage efficiency, making it possible to process a large number of sampling points.

[0020] In addition, the present invention can handle the dispersion relations of arbitrary-order polynomials, not limited to the linear dispersion model, and has wide applicability. By adjusting the number of qubits and the parameters of the phase screen, it can flexibly adapt to different laser transmission conditions and atmospheric environments, and has good scalability and adaptability.

[0021] The present invention applies quantum computing technology to the transmission simulation of lasers in atmospheric turbulence, expands the application of quantum computing in the field of optical simulation, and provides a theoretical and technical basis for future research on light propagation in more complex environments.

[0022] Through the above steps, the method of the present invention reduces the computational complexity of the classical algorithm from O(NlogN) to O((logN) 2 ), greatly improving the simulation accuracy and efficiency. This method can handle the dispersion relations of arbitrary-order polynomials, is applicable to a wider range of light propagation problems, and has good generality and application prospects. BRIEF DESCRIPTION OF THE DRAWINGS

[0023] Figure 1 is a flow chart of the method of the present invention;

[0024] Figure 2 is an example of a quantum circuit for transmission based on 8 qubits and three-layer phase screens;

[0025] Figure 3 is Figure 2 the probability distribution diagram of the transmission measurement results of the quantum circuit shown. DETAILED DESCRIPTION OF THE INVENTION

[0026] To make the objectives, contents, and advantages of the present invention clearer, the following further describes the specific implementation manners of the present invention in detail with reference to the drawings and embodiments.

[0027] The present invention relates to the cross field of quantum computing and atmospheric optics, and specifically relates to a method for efficiently simulating the transmission of lasers in atmospheric turbulence using quantum computing technology, belonging to the technical field of quantum computing simulation applications.

[0028] To solve the problems of high computational complexity and low simulation efficiency in the prior art, the present invention proposes an efficient simulation method for laser atmospheric transmission based on quantum computing. Under the paraxial approximation condition, this method maps the transmission process of laser in atmospheric turbulence into a quantum computing framework and uses quantum circuits and quantum algorithms to achieve efficient simulation.

[0029] Under the paraxial approximation condition, the present invention maps the transmission process of laser in atmospheric turbulence into a quantum computing framework and uses quantum circuits and quantum algorithms to achieve efficient simulation. The specific method includes: determining the required number of qubits, and initializing the quantum wave function of the optical field; generating a phase screen according to the atmospheric turbulence model and loading the phase modulation into the quantum state; performing a quantum Fourier transform to convert the spatial domain into the frequency domain; applying a transfer function to simulate the propagation of light in free space; then returning to the spatial domain through an inverse quantum Fourier transform; repeating the above process to simulate multi-phase screen transmission, and finally obtaining the intensity distribution of the optical field. The calculation process is as Figure 1 shown. This method reduces the computational complexity from O(NlogN) of the classical algorithm to O((logN) 2 ), greatly improving the simulation efficiency and accuracy, being applicable to the simulation requirements of high-precision and large-scale laser transmission, and having broad application prospects.

[0030] The method of the present invention includes the following steps:

[0031] S1. Qubit selection: Determine the required number of qubits n = log2N, where N is the number of sampling points. Use n qubits to construct the wave function |ψ>, where the eigenstate of each qubit corresponds to the spatial or spectral sampling point of the laser, expressed as the computational basis state |γ>, where the value range of γ is Negative values are represented by two's complement to improve memory efficiency.

[0032] S2. Initializing the wave function: Encode the initial optical field distribution U (0) (γ) into the quantum state . For a specific light beam (such as a Gaussian beam), the Kitaev-Webb algorithm can be used for efficient initialization.

[0033] S3. Phase modulation of turbulence: According to the refractive index power spectrum of atmospheric turbulence (such as the Kolmogorov spectrum or the Von Karman spectrum), use the power spectrum inversion method to generate a series of phase screens to simulate the influence of atmospheric turbulence. Use the selective phase shift operator to load the phase screen information onto the quantum state to achieve phase modulation.

[0034] S4. Fourier domain conversion: Perform a quantum Fourier transform (QFT) on the phase-modulated quantum state to convert the spatial domain to the frequency domain. Use Hadamard gates and controlled-phase rotation gates to construct a QFT circuit to achieve an efficient Fourier transform.

[0035] S5. Second-order propagation processing: In the frequency domain, apply the free-space transfer function to each frequency component to simulate the propagation of light in free space. Through the combination of phase gates and controlled-phase gates, implement the phase operation of the transfer function in the quantum circuit.

[0036] S6. Inverse Fourier transform: Apply the inverse quantum Fourier transform (IQFT) to convert the quantum state in the frequency domain back to the spatial domain to obtain the propagated quantum state.

[0037] S7. Repeated phase screen processing: For the multi-phase screen transmission process, repeat the processes of phase modulation, Fourier transform, transfer function application, and inverse Fourier transform in S3 - S6, sequentially load the phase modulation of each phase screen, simulate the multiple interactions of the laser in atmospheric turbulence, and obtain the final quantum state and the light field intensity distribution.

[0038] Embodiment 1:

[0039] An efficient simulation method for laser atmospheric transmission based on quantum computing, comprising the following steps:

[0040] (1) Quantum bit selection: Determine the number of sampling points N, and construct a quantum wave function |ψ> using n = log2N quantum bits. The eigenstate of each quantum bit corresponds to the spatial or spectral sampling point of the laser, denoted as the computational basis state |γ>, and negative values are represented in two's complement form;

[0041] (2) Initialize the wave function: Use the Kitaev-Webb algorithm to efficiently load the initial state of the Gaussian beam into the quantum register to obtain the initial quantum state where U (0) (γ) is the complex amplitude of the initial light field;

[0042] (3) Phase modulation of turbulence: Use the power spectrum inversion method to generate a sequence of phase screens of atmospheric turbulence, and use the selective phase shift operator to load the phase information φ i (γ) of the i-th phase screen onto the quantum state to achieve phase screen loading and obtain the phase-modulated quantum state

[0043] (4) Quantum Fourier transform: Perform a quantum Fourier transform (QFT) on the phase-modulated quantum state |ψ (i) > to convert the spatial domain to the frequency domain and obtain the frequency-domain quantum state and obtain the complex amplitude in the frequency domain after the i-th phase screen

[0044] (5) Transfer function application: In the frequency domain, apply the transfer function to the quantum state through phase gates and controlled-phase gates, and perform phase rotation on each frequency component to obtain the complex amplitude in the frequency domain where φ prop is the phase factor of the transfer function, related to Δz, where Δz is the spacing between two phase screens, and represents the complex amplitude in the frequency domain after the i-th phase screen;

[0045] (6) Inverse quantum Fourier transform: Perform the inverse quantum Fourier transform (IQFT) on the quantum state processed by the transfer function to convert the frequency domain back to the spatial domain, obtaining the updated quantum state |ψ (i+1) >;

[0046] (7) Repeated phase screen processing: Repeat steps (3) to (6) to process all phase screens in sequence, simulating multiple interactions of the laser in atmospheric turbulence;

[0047] (8) Measurement and result acquisition: Measure the final quantum state to obtain the optical field intensity distribution I(γ) = |U(γ,z)| 2 where N screen represents transmission through N phase screens, and U(γ,z) represents the spatial complex amplitude after transmission over a distance z.

[0048] Furthermore, in the initialization step of the quantum wave function, the Kitaev-Webb algorithm is used to efficiently load the initial state of the Gaussian beam into the quantum register by constructing a series of controlled rotation gates and single-qubit rotation gates.

[0049] Furthermore, in the selective phase shift operator φ i (γ) is generated from the refractive index power spectrum of atmospheric turbulence and calculated using the power spectrum inversion method or the Zernike polynomial expansion method.

[0050] Furthermore, the quantum Fourier transform (QFT) and the inverse quantum Fourier transform (IQFT) are constructed using Hadamard gates and controlled-phase rotation gates to achieve efficient transformation of the quantum state.

[0051] Furthermore, the phase factor φ of the transfer function prop is where Δz is the propagation distance increment, λ is the laser wavelength, and Δx is the spatial sampling interval.

[0052] Furthermore, the method can handle the dispersion relations of arbitrary-order polynomials, not limited to the linear dispersion model, and is applicable to a wider range of light propagation problems.

[0053] Furthermore, the negative value is represented in two's complement form to adapt to the coding method of quantum computing and improve memory efficiency.

[0054] Furthermore, by adjusting the number of qubits and the parameters of the phase screen, it can flexibly adapt to different laser transmission conditions and atmospheric environments, and has good scalability and adaptability.

[0055] Furthermore, under the paraxial approximation condition, the quantum computing simulation method models atmospheric turbulence through the multi-phase screen method and combines quantum algorithms to achieve efficient simulation.

[0056] Furthermore, the method can be implemented on different types of quantum hardware platforms, including superconducting qubits, ion trap quantum computers, or photonic quantum computers.

[0057] Example 2:

[0058] For the simulation of laser atmospheric turbulence transmission based on quantum computing, the calculation process is as Figure 1 shown. The schematic diagram of the simplified 8-qubit quantum circuit is as Figure 2 shown, and the probability distribution of the output quantum state of the simplified schematic diagram is as Figure 3 shown.

[0059] Step 1: Qubit selection

[0060] 1. Determine the number of sampling points N and the number of qubits n:

[0061] Select the number of sampling points N = 1024×1024 = 2 20 , to meet the requirements of high-precision simulation.

[0062] The corresponding number of qubits is n = log2N = 20.

[0063] 2. Quantum state representation:

[0064] The ground state of the quantum state is represented as |γ>, where

[0065] the negative part is represented in two's complement form.

[0066] Step 2: Initialize the wave function

[0067] 1. Set the initial light field distribution U (0) (γ):

[0068] Select a Gaussian beam as the initial light field, and the amplitude distribution is:

[0069]

[0070] Among them, the waist radius w0 = 0.3 m, the position x = γΔx, and Δx is the sampling interval.

[0071] 2. Initialization of wave function:

[0072] Encode the initial optical field distribution into the quantum state in.

[0073] 3. Use the Kitaev-Webb algorithm for efficient initialization:

[0074] Use the Kitaev-Webb algorithm to efficiently load the initial state of the Gaussian beam into the quantum register.

[0075] Step 3: Phase modulation of turbulence

[0076] 1. Select the atmospheric turbulence model:

[0077] Adopt the Vn Karman turbulence power spectrum model, and the expression is:

[0078]

[0079] Among them:

[0080] ■ is the turbulence structure constant.

[0081] ■ The outer scale of turbulence L0 = 10 m, corresponding wave number

[0082] ■ The inner scale of turbulence l0 = 2 mm, corresponding wave number

[0083] 2. Generation of phase screen:

[0084] Use the power spectrum inversion method to generate 100 phase screens, with an interval of 500 meters for each phase screen and a total transmission distance of 50 kilometers.

[0085] 3. Realization of phase modulation:

[0086] In the quantum circuit, use the selective phase shift operator to load the phase information of the phase screen onto the quantum state to achieve phase modulation:

[0087]

[0088] Among them, φ i (γ) is the phase value of the i-th phase screen at the position γ.

[0089] Step 4: Fourier domain conversion

[0090] 1. Quantum Fourier Transform (QFT):

[0091] Perform a quantum Fourier transform on the phase-modulated quantum state to convert the spatial domain to the frequency domain:

[0092]

[0093] 2. Circuit implementation of QFT:

[0094] Construct a QFT circuit using Hadamard gates and controlled-phase rotation gates to optimize the circuit depth and the number of gates.

[0095] Step Five: Second-order propagation processing

[0096] 1. Apply the transfer function:

[0097] In the frequency domain, apply the free-space transfer function to each frequency component:

[0098]

[0099] where

[0100] 2. Quantum circuit implementation:

[0101] Implement the phase operation of the transfer function in the quantum circuit through phase gates and controlled-phase gates.

[0102] Step Six: Inverse Fourier transform

[0103] 1. Inverse Quantum Fourier Transform (IQFT):

[0104] Perform an inverse quantum Fourier transform on the quantum state processed by the transfer function to obtain the propagated quantum state:

[0105]

[0106] Step Seven: Repeated phase screen processing

[0107] ● Multi-phase screen transmission:

[0108] Repeat the processes of phase modulation, Fourier transform, transfer function application, and inverse Fourier transform, and sequentially process each phase screen to simulate multiple interactions of the laser in atmospheric turbulence.

[0109] Step Eight: Measurement and result acquisition

[0110] 1. Quantum state measurement:

[0111] Measure the final quantum state to obtain the optical field amplitude at each position γ.

[0112] 2. Result analysis:

[0113] Calculate the light intensity distribution I(γ) = |U(γ,z)| 2 , and analyze the change of the optical field after the laser propagates through atmospheric turbulence.

[0114] The beneficial effects of the present invention include:

[0115] The present invention provides an efficient simulation method for laser atmospheric transmission based on quantum computing, which significantly reduces the computational complexity and improves the simulation accuracy and efficiency. Compared with the traditional classical algorithm, this method utilizes the parallelism of quantum computing and the high efficiency of quantum algorithms to reduce the computational complexity from O(NlogN) to O((logN) 2 ), achieving an exponential performance improvement.

[0116] Under the paraxial approximation condition, the method of the present invention models atmospheric turbulence through the multi-phase screen method, and combines quantum Fourier transform and inverse Fourier transform to realize the efficient propagation simulation of the optical field in a turbulent environment. This method not only improves the computational efficiency, but also realizes the improvement in storage efficiency, making it possible to process a large number of sampling points.

[0117] In addition, the present invention can handle the dispersion relation of any-order polynomial, not limited to the linear dispersion model, and has wide applicability. By adjusting the number of qubits and the parameters of the phase screen, it can flexibly adapt to different laser transmission conditions and atmospheric environments, and has good scalability and adaptability.

[0118] The present invention applies quantum computing technology to the transmission simulation of lasers in atmospheric turbulence, expands the application of quantum computing in the field of optical simulation, and provides a theoretical and technical basis for future optical propagation research in more complex environments

[0119] Through the above steps, the method of the present invention reduces the computational complexity of the classical algorithm from O(NlogN) to O((logN) 2 ), greatly improving the simulation accuracy and efficiency. This method can handle the dispersion relation of any-order polynomial, is applicable to a wider range of optical propagation problems, and has good generality and application prospects.

[0120] The above are only the preferred embodiments of the present invention. It should be noted that for those of ordinary skill in the art, without departing from the technical principle of the present invention, several improvements and deformations can be made, and these improvements and deformations should also be regarded as the protection scope of the present invention.

Claims

1. An efficient simulation method for laser atmospheric transmission based on quantum computing, characterized in that The method includes the following steps: S1. Qubit selection: Determine the required number of qubits \(n = \log_2 N\), where \(N\) is the number of sampling points; construct a wave function \(|\psi\rangle\) using \(n\) qubits, where the eigenstate of each qubit corresponds to a spatial or spectral sampling point of the laser, denoted as the computational basis state \(|\gamma\rangle\), where the value range of \(\gamma\) is Negative values are represented in two's complement; S2. Initialize the wave function: Encode the initial optical field distribution U (0) (γ) into the quantum state |ψ0> = ∑ γ U (0) (γ)|γ>. S3. Phase modulation of turbulence: According to the refractive index power spectrum of atmospheric turbulence, a series of phase screens are generated by using the power spectrum inversion method to simulate the influence of atmospheric turbulence; the phase screen information is loaded onto the quantum state by using the selective phase shift operator to achieve phase modulation; S4. Fourier domain conversion: Perform a quantum Fourier transform (QFT) on the phase-modulated quantum state to convert the spatial domain to the frequency domain; construct a QFT circuit by using Hadamard gates and controlled phase rotation gates to achieve an efficient Fourier transform; S5. Second-order propagation processing: In the frequency domain, apply a free-space transfer function to each frequency component to simulate the propagation of light in free space; implement the phase operation of the transfer function in the quantum circuit through a combination of phase gates and controlled phase gates; S6. Inverse Fourier transform: Apply an inverse quantum Fourier transform (IQFT) to convert the quantum state in the frequency domain back to the spatial domain to obtain the propagated quantum state; S7. Repeated phase screen processing: For the multi-phase screen transmission process, repeat the processes of phase modulation, Fourier transform, transfer function application, and inverse Fourier transform in S3 - S6, sequentially load the phase modulation of each phase screen, simulate the multiple interactions of the laser in atmospheric turbulence, and obtain the final quantum state and the optical field intensity distribution.

2. The high-efficiency simulation method for laser atmospheric transmission based on quantum computing according to claim 1, wherein, The S2 includes: adopting the Kitaev-Webb algorithm to efficiently load the initial state of the Gaussian beam into the quantum register, obtaining the initial quantum state |ψ0> = ∑ γ U (0) (γ)|γ>, where U (0) (γ) is the complex amplitude of the initial optical field.

3. The high-efficiency simulation method for laser atmospheric transmission based on quantum computing according to claim 2, wherein In the above S3, the refractive index power spectrum of atmospheric turbulence is the Kolmogorov spectrum or the Von Karman spectrum.

4. The high-efficiency simulation method for laser atmospheric transmission based on quantum computing according to claim 2, characterized in that, The S3 includes: generating a phase screen sequence of atmospheric turbulence by using a power spectrum inversion method, and loading the phase information φ i (γ) of the i-th phase screen onto the quantum state to achieve phase screen loading and obtain the phase-modulated quantum state 5. The high-efficiency simulation method for laser atmospheric transmission based on quantum computing according to claim 4, wherein The selective phase shift operator is where φ i (γ) is generated by the refractive index power spectrum of atmospheric turbulence and calculated by the power spectrum inversion method or the Zernike polynomial expansion method.

6. The high-efficiency simulation method for laser atmospheric transmission based on quantum computing according to claim 4, wherein The S4 includes: performing a quantum Fourier transform (QFT) on the phase-modulated quantum state |ψ (i) >, converting the spatial domain to the frequency domain to obtain a frequency-domain quantum state and obtaining the complex amplitude in the frequency domain after the i-th phase screen 7. The high-efficiency simulation method for laser atmospheric transmission based on quantum computing according to claim 6, wherein, The S5 includes: in the frequency domain, for the quantum state applying a transfer function, implemented by a phase gate and a controlled-phase gate, to each frequency component to perform phase rotation and obtain the complex amplitude in the frequency domain where φ prop is the phase factor of the transfer function, related to Δz, where Δz is the spacing between two phase screens, denotes the complex amplitude in the frequency domain after the i-th phase screen.

8. The high-efficiency simulation method for laser atmospheric transmission based on quantum computing according to claim 7, characterized in that The phase factor φ of the transfer function prop is where Δz is the propagation distance increment, λ is the laser wavelength, and Δx is the spatial sampling interval.

9. The high-efficiency simulation method for laser atmospheric transmission based on quantum computing according to claim 7, wherein The S6 includes: performing an inverse quantum Fourier transform (IQFT) on the quantum state processed by the transfer function, converting the frequency domain back to the spatial domain, and obtaining the updated quantum state |ψ ( i +1) >.

10. The high-efficiency simulation method for laser atmospheric transmission based on quantum computing according to claim 9, characterized in that, The obtaining of the final quantum state sum and the optical field intensity distribution in S7 includes: for the final quantum state perform measurement to obtain the optical field intensity distribution I(γ) = |U(γ, z)| at each position γ 2 , where N screen represents the transmission through N phase screens, and U(γ, z) represents the spatial complex amplitude after transmission of distance z.

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