CN115826161A - Random parallel gradient descent optical fiber coupling method for suppressing target function measurement noise - Google Patents

Random parallel gradient descent optical fiber coupling method for suppressing target function measurement noise Download PDF

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CN115826161A
CN115826161A CN202211447156.5A CN202211447156A CN115826161A CN 115826161 A CN115826161 A CN 115826161A CN 202211447156 A CN202211447156 A CN 202211447156A CN 115826161 A CN115826161 A CN 115826161A
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gradient
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fiber coupling
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亓波
彭锦锦
毛耀
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Institute of Optics and Electronics of CAS
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Abstract

The invention provides a random parallel gradient descent optical fiber coupling method for inhibiting measurement noise of a target function, which is called Kalman random parallel gradient descent algorithm (KSPGD) and is used for improving the optical fiber coupling efficiency of a space laser beam when photoelectric detection noise exists. The traditional Stochastic Parallel Gradient Descent (SPGD) algorithm estimates the gradient of an iteration point by measuring a value of an objective function in real time, and then updates the iteration point in a gradient descent manner. Therefore, when the objective function measurement has large noise, the gradient estimation value of the iteration point can oscillate, thereby influencing the convergence performance of the algorithm. Aiming at the problem, the KSPGD algorithm provided by the invention utilizes the information of the target function model to obtain the optimal weighted estimation of the gradient of the iteration point through the Kalman filtering algorithm, so that the accuracy of the gradient estimation value is improved, and the influence of the measurement noise on the convergence performance of the algorithm is further inhibited. The method is applied to a self-adaptive coupling system, and the system adopting the KSPGD algorithm can stably converge under noises with different sizes.

Description

Random parallel gradient descent optical fiber coupling method for suppressing target function measurement noise
Technical Field
The invention belongs to the field of optimization algorithms and photoelectric application, and particularly relates to a random parallel gradient descent optical fiber coupling method for suppressing objective function measurement noise, which is an optimization control method for suppressing photoelectric detection noise for a self-adaptive optical fiber coupling system.
Background
Optical fiber coupling in space laser beam transmission such as satellite-ground laser communication is an important link, and the optical fiber coupling efficiency directly influences the communication efficiency. The self-adaptive optical fiber coupling technology is a method for improving the optical fiber coupling efficiency of a receiving end. The random parallel gradient descent algorithm (SPGD) is an optimization control algorithm commonly used in the adaptive coupling system. However, the photoelectric detection noise affects the convergence performance of the algorithm, and the algorithm does not even converge under the condition of high noise intensity.
Hu Qinto et al (Hu, qinto, zhen, liangli, mao, yao, zhu, shiwei, zhou, xi, zhou, guozhong.Adaptive stored parallel device gradient apparatus for effective fiber coupling [ J ]. Optics Express,2020,28 (9): 13141-13154.) propose an adaptive fiber coupling system for correcting wavefront tilt aberration using a fast mirror, which controls the fast mirror using an improved SPGD algorithm to suppress the influence of beam shift caused by tilt aberration on coupling efficiency, enabling the fast mirror to converge to a position where the coupling efficiency reaches an extreme value faster. The above method does not take into account the case when the objective function measurement in the algorithm is noisy. The influence of photoelectric detection noise on the closed-loop performance of a single-mode fiber adaptive coupling device is studied by yellow crown and the like (yellow crown, gunn super, lifeng, and the like) [ J ] Physics report, 2021,70 (22): 8.) the influence of the closed-loop performance of a system when the photoelectric detection noise exists in the coupling of adaptive optical fibers by using an SPGD blind optimization algorithm is studied. The result shows that with the increase of the noise degree, the convergence speed of the algorithm and the final closed-loop coupling efficiency after convergence are greatly influenced.
In summary, an adaptive coupling optimization control algorithm for suppressing influence of photodetection noise on system performance is still lacking in the current adaptive fiber coupling technology.
Disclosure of Invention
In order to overcome the defects of the prior art and solve the problem that photoelectric detection noise influences the performance of a self-adaptive optical fiber coupling system, the invention provides an anti-noise self-adaptive coupling blind optimization method for optimizing gradient estimation. Because the gradient estimation value used for updating the iteration point is the optimal weighting of the gradient observation value and the model-based prediction value, the method can effectively inhibit the influence of noise and ensure the stable and rapid convergence of the algorithm.
The invention adopts the following technical scheme that a random parallel gradient descent optical fiber coupling method for inhibiting the measurement noise of a target function comprises the following steps:
step (1) establishing a dynamic equation of gradient change:
Figure BDA0003950828210000021
Figure BDA0003950828210000022
wherein the content of the first and second substances,
Figure BDA0003950828210000023
is u t A gradient of (A) is a unit matrix,. DELTA.u t+1 ＝u t+1 -u t Is the difference between the front and back iteration points, H t Is u t The Hessian matrix of (c).
Figure BDA0003950828210000024
Representing measurements of gradient, C being unit matrix, w t 、v t Representing process noise and measurement noise, respectively, as gaussian noise.
Step (2) estimating an objective function model:
for the application scenario of the present invention, i.e. the adaptive fiber coupling system only considering the low-order tilt aberration correction, the objective function can be regarded as a gaussian function:
Figure BDA0003950828210000025
wherein u is 1 、u 2 Is a coefficient of a 1, 2 order tilt term, AAnd B is a undetermined coefficient. Further, a Hessian matrix in the step (1) can be obtained:
Figure BDA0003950828210000026
step (3) establishing a gradient prediction equation:
Figure BDA0003950828210000027
P t+1|t ＝AP t|t A T +Q, (6)
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0003950828210000028
is a gradient
Figure BDA0003950828210000029
The estimation at the time of the iteration t,
Figure BDA00039508282100000210
is a prediction of the gradient at t + 1. P t|t And P t+1|t Are respectively to the gradient
Figure BDA00039508282100000211
The covariance matrix of the estimated error at iteration t and the covariance matrix of the predicted error at iteration t +1, Q is the covariance matrix of the process noise, here 0.
Step (4) calculating an observation gradient value:
Figure BDA00039508282100000212
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA00039508282100000213
Figure BDA00039508282100000214
the gradient observation is calculated by SPGDIn the method, the gradient is estimated by a parallel disturbance random approximation method, the random disturbance obeys Bernoulli distribution with values of 1 and-1, and | δ u t And | is the disturbance amplitude. Wherein the objective function measurements of both positive and negative perturbations are gaussian noise. And (3) replacing the gradient observation model in the formula (2) with the formula in the algorithm application.
Step (5) establishing a gradient updating equation:
Figure BDA00039508282100000215
P t+1|t+1 ＝P t+1|t -K t+1 CP t+1|t (9)
wherein, K t+1 Representing the Kalman gain:
K t+1 ＝P t+1|t C T (R+CP t+1|t C T ) -1 (10)
where R is the measurement noise variance matrix.
And (6) updating iteration points in a gradient descending manner:
Figure BDA0003950828210000031
where γ is the learning rate. The proposed algorithm (KSPGD) implementation is to loop step (3) to step (6) continuously.
The invention has the following advantages:
(1) The method uses the model information of the target function, adopts the optimal weighting mode to estimate the gradient value at the iteration point, and can effectively inhibit the influence of observation noise on the gradient estimation value, thereby inhibiting the influence of noise on the convergence performance of the algorithm.
(2) The invention improves the algorithm to solve the noise problem without increasing the complexity of system hardware, and simultaneously, the algorithm has the same convergence speed as the SPGD since the most time-consuming operation in the algorithm is the target function measurement and the measurement times of the target function of the algorithm are the same as those of the SPGD. The SPGD algorithm convergence speed is known from the aforementioned documents to be affected when the noise level is large, whereas the KSPGD algorithm of the present invention is less affected due to the effect of suppressing noise.
Drawings
FIG. 1 is a basic schematic diagram of an adaptive fiber coupling system;
FIG. 2 is a flow chart of the adaptive coupling algorithm KSPGD according to the present invention;
FIG. 3 is a schematic diagram of fiber coupling;
FIG. 4 shows the convergence results of the fiber coupling efficiency of the SPGD algorithm and the KSPGD algorithm under different noise levels;
FIG. 5 is a graph of the change in the two directional gradient estimates during convergence for the SPGD and KSPGD algorithms with a noise variance of 0.6.
Detailed Description
The following detailed description of the embodiments of the invention refers to the accompanying drawings.
Example 1:
fig. 1 shows a basic schematic diagram of an adaptive fiber coupling system, which includes a laser, a disturbance fast-reflection mirror, a coupling lens, an energy meter, a single-mode fiber, and a controller, where the light energy coupled into the energy meter is regarded as a function of the position of the coupling fast-reflection mirror, and a blind optimization algorithm in the controller is used to generate a control quantity to make the coupling fast-reflection mirror converge to a position where the light energy received by the energy meter is the maximum. The light energy detected by the energy meter has Gaussian noise. Fig. 2 is a flow chart of the adaptive coupling algorithm KSPGD according to the present invention.
In order to achieve the purpose of the invention, the invention provides a random parallel gradient descent optical fiber coupling method for inhibiting the measurement noise of an objective function, which comprises the following steps:
step (1) establishing a dynamic equation of gradient change:
Figure BDA0003950828210000041
Figure BDA0003950828210000042
wherein the content of the first and second substances,
Figure BDA0003950828210000043
is u t A gradient of (A) is a unit matrix,. DELTA.u t+1 ＝u t+1 -u t Is the difference between the front and back iteration points, H t Is u t The Hessian matrix of (c).
Figure BDA0003950828210000044
Representing measurements of gradient, C being unit matrix, w t 、v t Process noise and measurement noise are respectively represented as gaussian noise.
Step (2) estimating an objective function model:
for the application scenario of the present invention, i.e. the adaptive fiber coupling system only considering the low-order tilt aberration correction, the objective function can be regarded as a gaussian function:
Figure BDA0003950828210000045
wherein u is 1 、u 2 The coefficients of the 1 and 2-order tilt terms are obtained, and A and B are coefficients to be determined. Further, a Hessian matrix in the step (1) can be obtained:
Figure BDA0003950828210000046
step (3) establishing a gradient prediction equation:
Figure BDA0003950828210000047
P t+1|t ＝AP t|t A T +Q, (6)
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0003950828210000048
gradient of gradient
Figure BDA0003950828210000049
The estimation at the time of the iteration t,
Figure BDA00039508282100000410
is a prediction of the gradient at t + 1. P t|t And P t+1|t Are respectively to the gradient
Figure BDA00039508282100000411
The covariance matrix of the estimated error at iteration t and the covariance matrix of the predicted error at iteration t +1, Q is the covariance matrix of the process noise, here 0.
Calculating an observation gradient value:
Figure BDA00039508282100000412
wherein the content of the first and second substances,
Figure BDA00039508282100000413
the gradient observation is to adopt a method of parallel disturbance random approximation in the SPGD algorithm to estimate the gradient, the random disturbance obeys Bernoulli distribution with the values of 1 and-1, | delta u t And | is the disturbance amplitude. Wherein the objective function measurements of both positive and negative perturbations are gaussian noise. And (3) replacing the gradient observation model in the formula (2) with the formula in the algorithm application.
Step (5) establishing a gradient updating equation:
Figure BDA00039508282100000414
P t+1|t+1 ＝P t+1|t -K t+1 CP t+1|t (9)
wherein, K t+1 Representing the Kalman gain:
K t+1 ＝P t+1|t C T (R+CP t+1|t C T ) -1 (10)
where R is the measurement noise variance matrix.
And (6) updating iteration points in a gradient descending mode:
Figure BDA0003950828210000051
where γ is the learning rate. The proposed algorithm (KSPGD) implementation is to loop step (3) to step (6) continuously.
Example 2:
the adaptive fiber coupling system shown in fig. 1 can be modeled as a blind optimization problem as follows:
as shown in FIG. 3, an incident beam with wavelength λ is focused by a coupling lens with effective aperture D and focal length f, and finally enters a mode field with radius w 0 In a single mode optical fiber of (1). The optical field distribution and the fiber mode field distribution of the received incident beam can be approximately regarded as Gaussian distribution. The coupling efficiency can be written as:
Figure BDA0003950828210000052
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0003950828210000053
is the active area of the receive aperture,
Figure BDA0003950828210000054
Figure BDA0003950828210000055
representing the wavefront phase, can be described by a linear combination of zernike polynomials:
Figure BDA0003950828210000056
wherein Z is i Denotes the ith zernike polynomial, a i Represents the i-th order zernike coefficient. Zero order term a 0 Indicating that the piston term does not affect the coupling of the single-mode fibre, Z 1 And Z 2 Representing the tilt aberrations in the x-and y-directions. Coefficient of term a due to wavefront tilt 1 、a 2 And control voltage u of FSM 1 、u 2 Is an approximate linear relation, and directly optimizes the variable coefficient a in simulation 1 、a 2 The value of (c). The present invention takes into account temporarily higher order aberrations.
In the simulation, λ is 1550nm, f is 0.71m, w 0 At 5.2 microns, D is 0.15m, and the objective function is approximated as a gaussian function
Figure BDA0003950828210000057
A =0.81 and b =1.15. Can calculate the Hessian matrix H t . Learning rate gamma and disturbance amplitude | δ u of SPGD and KSPGD t Both 0.1 and 0.5. Initial estimation error covariance matrix of KSPGD is set to
Figure BDA0003950828210000058
Figure BDA0003950828210000059
Random numbers satisfying a positive distribution are added when calculating the objective function measurement.
Fig. 4 (a) shows the convergence of the fiber coupling efficiency of the SPGD algorithm at different noise levels. FIG. 4 (b) shows the results of KSPGD. It can be seen that as the noise variance increases, the accuracy and speed of convergence of the SPGD algorithm is affected, while the accuracy of convergence of the KSPGD is hardly affected, and the speed slightly decreases with the noise enhancement.
Fig. 5 (a) and 5 (b) show changes in the estimated values of the two directional gradients during convergence of SPGD when the noise variance is 0.6. FIGS. 5 (c) and 5 (d) show the results of KSPGD. It can be seen that the gradient value estimated by the SPGD algorithm is oscillating all the time due to the noise effect, while the gradient estimated value of the KSPGD algorithm converges quickly with the iteration. This is why KSPGD is stable in convergence in the presence of measurement noise.

Claims (7)

1. A random parallel gradient descent optical fiber coupling method for suppressing objective function measurement noise is characterized by comprising the following steps:
step (1): establishing a dynamic equation for optimizing gradient change in an iterative process;
step (2): estimating an adaptive optical fiber coupling system target function model;
and (3): establishing a gradient prediction equation;
and (4): calculating an observation gradient value;
and (5): establishing a gradient updating equation;
and (6): and updating the iteration point in a gradient descending mode.
2. The method for random parallel gradient descent fiber coupling for suppressing the measurement noise of the objective function according to claim 1, wherein: the dynamic equation of the gradient change in the step (1) is as follows:
Figure FDA0003950828200000011
Figure FDA0003950828200000012
wherein the content of the first and second substances,
Figure FDA0003950828200000013
is u t A gradient of (A) is a unit matrix,. DELTA.u t+1 ＝u t+1 -u t Is the difference between the front and back iteration points, H t Is u t The Hessian matrix of (a) is,
Figure FDA0003950828200000014
representing measurements of gradient, C being unit matrix, w t 、v t Representing process noise and measurement noise, respectively, as gaussian noise.
3. The method for random parallel gradient descent fiber coupling for suppressing the measurement noise of the objective function as claimed in claim 2, wherein: in the step (2), the objective function model needs to analyze a specific optimization problem, and an application scenario is an adaptive fiber coupling system only considering low-order oblique aberration correction, and the objective function can be regarded as a gaussian function:
Figure FDA0003950828200000015
wherein u is 1 、u 2 And (3) obtaining the Hessian matrix in the step (1) by taking the coefficients of the 1 and 2-order tilt terms and taking A and B as coefficients to be determined:
Figure FDA0003950828200000016
4. the method of claim 3, wherein the method comprises the following steps: the gradient prediction equation in the step (3) is as follows:
Figure FDA0003950828200000017
P t+1|t ＝AP t|t A T +Q, (6)
wherein the content of the first and second substances,
Figure FDA0003950828200000018
is a gradient
Figure FDA0003950828200000019
The estimation at the time of the iteration t,
Figure FDA00039508282000000110
for prediction of the gradient at t +1, P t|t And P t+1|t Are respectively to the gradient
Figure FDA0003950828200000025
The covariance matrix of the estimation error at iteration t and the covariance matrix of the prediction error at iteration t +1, Q being the process noiseThe variance matrix, here 0.
5. The method of claim 4, wherein the method comprises the following steps: the observation gradient value in the step (4) is as follows:
Figure FDA0003950828200000021
wherein the content of the first and second substances,
Figure FDA0003950828200000022
the gradient observation is to adopt a parallel disturbance random approximation method in the SPGD algorithm to estimate the gradient, the random disturbance obeys Bernoulli distribution with values of 1 and-1, and | δ u t And | is disturbance amplitude, wherein the measured values of the target functions of positive and negative disturbances are both provided with Gaussian noise, and the gradient observation model of the formula (2) is replaced by the formula (7) in the application of the algorithm.
6. The method of claim 5, wherein the method comprises the following steps: in step (5), the gradient update equation is:
Figure FDA0003950828200000023
P t+1|t+1 ＝P t+1|t -K t+1 CP t+1|t (9)
wherein, K t+1 Representing the Kalman gain:
K t+1 ＝P t+1|t C T (R+CP t+1|t C T ) -1 (10)
where R is the measurement noise variance matrix.
7. The method of claim 6, wherein the method comprises the following steps: in the step (6), updating the iteration point formula in a gradient descent mode as follows:
Figure FDA0003950828200000024
wherein gamma is the learning rate, and the proposed algorithm (KSPGD) implementation process is to continuously loop the steps (3) to (6).
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