CN114815436B - Optical phased array element mutual coupling compensation method based on neighborhood sampling principal component analysis - Google Patents

Optical phased array element mutual coupling compensation method based on neighborhood sampling principal component analysis Download PDF

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CN114815436B
CN114815436B CN202210437250.6A CN202210437250A CN114815436B CN 114815436 B CN114815436 B CN 114815436B CN 202210437250 A CN202210437250 A CN 202210437250A CN 114815436 B CN114815436 B CN 114815436B
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CN114815436A (en
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汪相如
黄彦威
王康哲
严倩盈
谭庆贵
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University of Electronic Science and Technology of China
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    • G—PHYSICS
    • G02—OPTICS
    • G02F—OPTICAL DEVICES OR ARRANGEMENTS FOR THE CONTROL OF LIGHT BY MODIFICATION OF THE OPTICAL PROPERTIES OF THE MEDIA OF THE ELEMENTS INVOLVED THEREIN; NON-LINEAR OPTICS; FREQUENCY-CHANGING OF LIGHT; OPTICAL LOGIC ELEMENTS; OPTICAL ANALOGUE/DIGITAL CONVERTERS
    • G02F1/00—Devices or arrangements for the control of the intensity, colour, phase, polarisation or direction of light arriving from an independent light source, e.g. switching, gating or modulating; Non-linear optics
    • G02F1/29—Devices or arrangements for the control of the intensity, colour, phase, polarisation or direction of light arriving from an independent light source, e.g. switching, gating or modulating; Non-linear optics for the control of the position or the direction of light beams, i.e. deflection
    • G02F1/292—Devices or arrangements for the control of the intensity, colour, phase, polarisation or direction of light arriving from an independent light source, e.g. switching, gating or modulating; Non-linear optics for the control of the position or the direction of light beams, i.e. deflection by controlled diffraction or phased-array beam steering
    • G—PHYSICS
    • G02—OPTICS
    • G02F—OPTICAL DEVICES OR ARRANGEMENTS FOR THE CONTROL OF LIGHT BY MODIFICATION OF THE OPTICAL PROPERTIES OF THE MEDIA OF THE ELEMENTS INVOLVED THEREIN; NON-LINEAR OPTICS; FREQUENCY-CHANGING OF LIGHT; OPTICAL LOGIC ELEMENTS; OPTICAL ANALOGUE/DIGITAL CONVERTERS
    • G02F2203/00—Function characteristic
    • G02F2203/18—Function characteristic adaptive optics, e.g. wavefront correction

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Abstract

The invention discloses an optical phased array element mutual coupling compensation method based on neighborhood sampling principal component analysis, which comprises the steps of dimension reduction and iterationThe dimension reduction comprises the following steps of: s1, calculating a driving voltage corresponding to each array element of the phased array; s2, obtaining a neighborhood sampling matrix X; s3, calculating a covariance matrix C of the X; s4, calculating the characteristic value and the characteristic vector of the C, and splicing the characteristic vector according to the row to obtain a matrix U; s5, sorting the eigenvalues from large to small, and sorting eigenvectors in the U matrix correspondingly to obtain a space transformation matrix P; s6, multiplying the voltage vector by the P matrix to obtain a new vector
Figure DDA0003607336830000011
The iteration comprises the following steps: s7, obtaining an updated voltage vector and loading the updated voltage vector onto array elements of the phased array; s8, acquiring a value J of the evaluation function, and according to the change delta J of the evaluation function
Figure DDA0003607336830000012
Is updated. The invention can greatly reduce the iteration dimension of the optimization process, improve the convergence speed and improve the robustness of the system operation.

Description

Optical phased array element mutual coupling compensation method based on neighborhood sampling principal component analysis
Technical Field
Belongs to the technical field of optical phased array control and self-adaptive optical optimization algorithms, and particularly relates to an optical phased array element mutual coupling compensation method based on neighborhood sampling principal component analysis.
Background
The optical phased array is an ideal method for realizing non-mechanical beam deflection, and is applied to various fields such as optical detection and ranging, free space communication, target tracking, remote sensing and the like. However, many optical phased arrays suffer from the problem of mutual coupling between array elements, such as lateral electric fields of liquid crystal phased arrays, thermal crosstalk of silicon-based phased arrays, and the like, which can lead to phase errors of near-field wavefronts, thereby causing deterioration of far-field deflected beam quality. These coupling problems caused by the device structure are difficult to solve by merely improving the manufacturing process or materials. Therefore, a common approach is to compensate for the phase difference by some iterative based adaptive optimization algorithm, such as genetic algorithm, particle swarm algorithm, random parallel gradient descent algorithm, etc. However, the rate of iterative convergence of these algorithms decreases significantly as the number of array elements increases. Since these iterative algorithms independently optimize each array element of the phased array, the dimension of the iterative variable is equal to the number of array elements, and finding a globally optimal solution in a high-dimensional space through an iterative method is very difficult, and may even fall into a locally optimal solution, so that convergence cannot be completed. Therefore, the iteration convergence speed is low, and the method becomes a main obstacle for restricting the application of the optimization algorithm in an actual system.
In order to improve the convergence rate of the optimization algorithm, three main methods exist at present. The first method is decoupling of array elements, namely modeling the coupling relation among the array elements by a numerical calculation method, so as to strip out the influence of a single array element on an evaluation function. However, in practical systems, it is very difficult to accurately model the coupling relation between the array elements, so that most of the method stays at the theoretical level, and has few applications in practical engineering. The second method is phase difference modeling, namely modeling the phase difference of a system according to a phase difference theory, establishing a relation between the phase difference and an evaluation function, and then carrying out targeted optimization on the reason for the generation of the phase difference. The method can realize the optimization speed of the system, but the built model can only be applied to specific scenes, different models are needed for different scenes, and the accurate modeling of the phase difference is difficult in certain scenes. The third method is machine learning modeling, which builds an error model of the phased array device through a large number of sample training, and then optimizes the device. However, this method requires a large number of samples for pre-training the model, and the model is not generic between different devices.
In general, a practical algorithm still lacks at present, and can perform self-adaptive iterative optimization and phase compensation aiming at phase errors caused by the coupling problem among optical phased array elements, so that the convergence speed is high, and the universality is ensured while the online operation is supported.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provides an optical phased array element mutual coupling compensation method based on neighborhood sampling principal component analysis, which can greatly reduce iteration dimension of an optimization process, avoid falling into a local optimal solution, improve convergence speed and improve system operation robustness.
The aim of the invention is realized by the following technical scheme: the optical phased array element mutual coupling compensation method based on neighborhood sampling principal component analysis comprises two processes of dimension reduction and iteration, wherein the dimension reduction process comprises the following steps of:
s1, calculating driving voltage corresponding to each array element of a phased array according to a target angle of deflection through a phased array formula and a voltage-phase relation, and recording the driving voltage of all the array elements as an N-dimensional column vector
Figure BDA0003607336810000021
Wherein N is the total number of array elements of the phased array;
s2, setting K-1 sampling points in one adjacent area of the target angle, and calculating a voltage vector corresponding to each sampling point
Figure BDA0003607336810000022
They are combined with->
Figure BDA0003607336810000023
Splicing according to columns to obtain a neighborhood sampling matrix +.>
Figure BDA0003607336810000024
Denoted as X, where K is the total number of voltage vectors;
s3, calculating covariance matrix C=XX of neighborhood sampling matrix X T ；
S4, calculating a characteristic value of the covariance matrix C and a characteristic vector corresponding to the characteristic value, and splicing the characteristic vector according to rows to obtain a matrix U;
s5, sorting the eigenvalues of the covariance matrix C from large to small, and sorting eigenvectors in the U matrix correspondingly to obtain a space transformation matrix P;
s6, using the P matrix to multiply the voltage toMeasuring amount
Figure BDA0003607336810000025
Spatially transforming it to obtain a new vector +.>
Figure BDA0003607336810000026
The iterative process comprises the following steps:
s7, pairing
Figure BDA0003607336810000027
Random perturbation is applied to the first K dimensions of (2)>
Figure BDA0003607336810000028
For updated->
Figure BDA0003607336810000029
Inverse transformation is performed using the inverse of the spatial variation matrix P to obtain an updated voltage vector +.>
Figure BDA00036073368100000210
And will->
Figure BDA00036073368100000211
Loading the array elements of the phased array;
s8, acquiring a value J of the evaluation function, and according to the change delta J of the evaluation function
Figure BDA00036073368100000212
The first K values of (a) are updated by the following update formula: />
Figure BDA00036073368100000213
Wherein->
Figure BDA00036073368100000214
Data representing the nth iteration, γ being the iteration step.
Further, the specific implementation method of the step S1 is as follows: the phased array formula refers to a relation between the phase shift quantity delta phi between adjacent phased array elements and a target angle theta, namely delta phi = 2 pi/lambda dsin theta, wherein lambda is the wavelength of incident laser, and d is the center-to-center distance of the phased array elements; the voltage-phase relation is a relation curve between phased array phase shift quantity delta phi and driving voltage, and is measured by experiments and used for mapping the phase shift quantity delta phi into a voltage value so as to obtain the driving voltage required by each array element.
Further, a neighborhood of the target angle in the step S2 refers to an angle range located in [ θ - δθ, θ+δθ ] with the target angle θ as a center, where δθ satisfies: and (2) sin (theta+delta theta) -sin (theta) | < lambda/(Nd), wherein lambda is the wavelength of incident laser, and d is the center-to-center distance of the array elements of the phased array.
The beneficial effects of the invention are as follows: the invention extracts the structural information of the optical phased array through principal component analysis, approximates the structural information by a small number of dimensions, and realizes the compensation of phase difference brought by mutual coupling of array elements by adaptively optimizing the driving voltage of the array elements of the optical phased array in a low-dimensional space; on the basis, coupling information in different dimensions is obtained through multiple sampling in the neighborhood of the target angle, and the accuracy of linear approximation is improved. The iteration dimension of the optimization process can be greatly reduced, the fall into a local optimal solution is avoided, the convergence speed is improved, and the robustness of the system operation is improved; meanwhile, the algorithm is independent of any specific device structure or application scene, can be applied to any phased array system with the problem of coupling among array elements, and has extremely strong universality.
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Fig. 1 is a flowchart of an optical phased array element mutual coupling compensation method based on neighborhood sampling principal component analysis.
Detailed Description
The technical scheme of the invention is further described below with reference to the accompanying drawings.
As shown in FIG. 1, the optical phased array element mutual coupling compensation method based on neighborhood sampling principal component analysis comprises two processes of dimension reduction and iteration, wherein the dimension reduction process utilizes a principal component analysis method to extract structural information of an optical phased array, and a space transformation matrix is generated through eigenvalue and eigenvector sequencing, and is approximated by a few dimensions in a new space, so that dimension reduction is carried out on high-dimension data; on the basis, coupling information in different dimensions is obtained through multiple sampling in the neighborhood of the target angle, so that the accuracy of linear approximation is improved; and the iterative process adopts a random parallel gradient descent algorithm to carry out iterative optimization on the data subjected to dimension reduction, so that phase errors caused by mutual coupling of the optical phased array elements are compensated.
The dimension reduction process is optimized for each target angle only once, and comprises the following steps:
s1, calculating driving voltage corresponding to each array element of a phased array according to a target angle of deflection through a phased array formula and a voltage-phase relation, and recording the driving voltage of all the array elements as an N-dimensional column vector
Figure BDA0003607336810000031
Figure BDA0003607336810000032
Wherein N is the total number of array elements of the phased array;
the specific implementation method comprises the following steps: the phased array formula refers to a relation between the phase shift quantity delta phi between adjacent phased array elements and a target angle theta, namely delta phi = 2 pi/lambda dsin theta, wherein lambda is the wavelength of incident laser, and d is the center-to-center distance of the phased array elements; the voltage-phase relation is a relation curve between phased array phase shift quantity delta phi and driving voltage, and is measured by experiments and used for mapping the phase shift quantity delta phi into a voltage value so as to obtain the driving voltage required by each array element.
S2, setting K-1 sampling points in one adjacent area of the target angle, and calculating a voltage vector corresponding to each sampling point
Figure BDA0003607336810000033
They are combined with->
Figure BDA0003607336810000034
Splicing according to columns to obtain a neighborhood sampling matrix +.>
Figure BDA0003607336810000035
Denoted as X, where K is the total number of voltage vectors; a neighborhood of the target angle is centered around the target angle θ and is located at [ θ - δθ, θ+δθ ]]Wherein δθ satisfies: and (2) sin (theta+delta theta) -sin (theta) | < lambda/(Nd), wherein lambda is the wavelength of incident laser, and d is the center-to-center distance of the array elements of the phased array.
S3, calculating covariance matrix C=XX of neighborhood sampling matrix X T The superscript T indicates transposition, the obtained covariance matrix C is an N-dimensional real symmetric matrix, diagonal elements of the covariance matrix C are variances of all dimensions of X, and non-diagonal elements of the covariance matrix C are covariance;
s4, calculating eigenvalues { lambda } of covariance matrix C 1 ，λ 2 ...λ N Characteristic vector xi corresponding to characteristic value 1 ，ξ 2 ...ξ N Splicing the feature vectors according to rows to obtain a matrix U; from the nature of the real symmetric matrix, C satisfies:
D＝UCU T
wherein U is a matrix formed by splicing characteristic vectors of C as row vectors; d is a diagonal matrix, and the diagonal elements are each eigenvalue of C in turn, namely:
Figure BDA0003607336810000041
Figure BDA0003607336810000042
s5, eigenvalue { lambda } of covariance matrix C 1 ，λ 2 ...λ N Sequencing from large to small, and sequencing feature vectors in the U matrix correspondingly to obtain a space transformation matrix P, so that feature vectors corresponding to feature values with larger values are arranged in the front row in the P matrix;
s6, using the P matrix to multiply the voltage vector
Figure BDA0003607336810000043
Spatially transforming it to obtain a new vector +.>
Figure BDA0003607336810000044
/>
The dimension reduction process principle of the S1-S6 is as follows: the eigenvectors form a new set of orthogonal bases, S6, the original voltage vectors are multiplied by the matrix P
Figure BDA0003607336810000045
Corresponds to defining a spatial transformation of +.>
Figure BDA0003607336810000046
Mapping each element of (2) onto the new set of substrates, the value in each dimension of the new substrates corresponding to +.>
Figure BDA0003607336810000047
Is included in the set of elements. According to the definition of the covariance matrix C, the larger the eigenvalue corresponding to the eigenvector, the larger the variance of the data representing the dimension, and the larger the information amount covered by the data after space transformation. Therefore, by sequencing the eigenvalues, the dimension corresponding to the eigenvector with larger information quantity is arranged at the front position, so that the linear approximation of the phased array structure information is realized, namely, main information is reserved; only optimizing the dimensions in the iterative process, compared with directly optimizing the N-dimensional space of the original voltage vector, the dimension reduction is realized; the other dimensions correspond to smaller characteristic values, so that the contained information quantity is smaller, the information is relatively minor information, and the optimization efficiency is improved by neglecting.
The neighborhood sampling is introduced in the S2, because for the same optical phased array, different angles correspond to different voltage vectors, and the mutual coupling degree among array elements is also different. Thus, each additional voltage vector introduces new dimensional information for principal component analysis.
The covariance matrix C is an N-dimensional real symmetric square matrix, and the rank of the covariance matrix C is equal to the number K of neighborhood sampling points and the number of non-zero eigenvalues; since the space transformation matrix is formed by the eigenvectors, the larger the sampling point number K is, the more eigenvectors corresponding to the nonzero eigenvalues can be obtained, the more accurate the approximation to the phased array structure information is, but the slower the calculation speed is. In practical applications, the number of sampling points should be reasonably selected, and a balance between accuracy and speed should be sought.
The iterative process needs to be operated repeatedly until reaching a preset convergence condition, namely, the optimization is completed, and the method comprises the following steps:
s7, pairing
Figure BDA0003607336810000051
Random perturbation is applied to the first K dimensions of (2)>
Figure BDA0003607336810000052
For updated
Figure BDA0003607336810000053
Inverse transformation is performed using the inverse of the spatial variation matrix P to obtain an updated voltage vector +.>
Figure BDA0003607336810000054
And will->
Figure BDA0003607336810000055
Loading the array elements of the phased array;
s8, acquiring a value J of the evaluation function, and according to the change delta J of the evaluation function
Figure BDA0003607336810000058
The first K values of (a) are updated by the following update formula: />
Figure BDA0003607336810000056
Wherein->
Figure BDA0003607336810000057
Data representing the nth iteration, γ being the iteration step. An evaluation function, which refers to the current far fieldThe common evaluation function comprises deflection efficiency, side mode rejection ratio, full width half maximum of main lobe and the like, and can be selected according to actual requirements.
And stopping iteration until a preset convergence condition is reached, such as a desired convergence value or a set iteration number is reached, or an evaluation function change amount caused by perturbation reaches a set value, and the like.
Those of ordinary skill in the art will recognize that the embodiments described herein are for the purpose of aiding the reader in understanding the principles of the present invention and should be understood that the scope of the invention is not limited to such specific statements and embodiments. Those of ordinary skill in the art can make various other specific modifications and combinations from the teachings of the present disclosure without departing from the spirit thereof, and such modifications and combinations remain within the scope of the present disclosure.

Claims (2)

1. The optical phased array element mutual coupling compensation method based on neighborhood sampling principal component analysis is characterized by comprising two processes of dimension reduction and iteration, wherein the dimension reduction process comprises the following steps of:
s1, calculating driving voltage corresponding to each array element of a phased array according to a target angle of deflection through a phased array formula and a voltage-phase relation, and recording the driving voltage of all the array elements as an N-dimensional column vector
Figure FDA0004178014930000011
Wherein N is the total number of array elements of the phased array; the specific implementation method comprises the following steps: the phased array formula refers to a relation between the phase shift quantity delta phi between adjacent phased array elements and a target angle theta, namely delta phi = 2 pi/lambda dsin theta, wherein lambda is the wavelength of incident laser, and d is the center-to-center distance of the phased array elements; the voltage-phase relation is a relation curve between phased array phase shift quantity delta phi and driving voltage, and is measured by experiments and used for mapping the phase shift quantity delta phi into a voltage value so as to obtain driving voltage required by each array element;
s2, setting K-1 sampling points in one adjacent area of the target angle, and calculating each sampling point pairVoltage vector of the application
Figure FDA0004178014930000012
They are combined with->
Figure FDA0004178014930000013
Splicing according to columns to obtain a neighborhood sampling matrix +.>
Figure FDA0004178014930000014
Denoted as X, where K is the total number of voltage vectors;
s3, calculating covariance matrix C=XX of neighborhood sampling matrix X T ；
S4, calculating a characteristic value of the covariance matrix C and a characteristic vector corresponding to the characteristic value, and splicing the characteristic vector according to rows to obtain a matrix U;
s5, sorting the eigenvalues of the covariance matrix C from large to small, and sorting eigenvectors in the U matrix correspondingly to obtain a space transformation matrix P;
s6, using the P matrix to multiply the voltage vector
Figure FDA0004178014930000015
Spatially transforming it to obtain a new vector +.>
Figure FDA0004178014930000016
Figure FDA0004178014930000017
The iterative process comprises the following steps:
s7, pairing
Figure FDA0004178014930000018
Random perturbation is applied to the first K dimensions of (2)>
Figure FDA0004178014930000019
For updated->
Figure FDA00041780149300000110
Inverse transformation is performed using the inverse of the spatial variation matrix P to obtain an updated voltage vector +.>
Figure FDA00041780149300000111
And will->
Figure FDA00041780149300000112
Loading the array elements of the phased array;
s8, acquiring a value J of the evaluation function, and according to the change delta J of the evaluation function
Figure FDA00041780149300000113
The first K values of (a) are updated by the following update formula: />
Figure FDA00041780149300000114
Wherein->
Figure FDA00041780149300000115
Data representing the nth iteration, γ being the iteration step.
2. The optical phased array element mutual coupling compensation method based on the neighborhood sampling principal component analysis according to claim 1, wherein one neighborhood of the target angle in the step S2 is an angle range centered on the target angle θ and located in [ θ - δθ, θ+δθ ], where δθ satisfies: and (2) sin (theta+delta theta) -sin (theta) | < lambda/(Nd), wherein lambda is the wavelength of incident laser, and d is the center-to-center distance of the array elements of the phased array.
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