Turbulence aberration correction for vector vortex beams using deep neural networks on experimental data

Yanwang Zhai; Yanwang Zhai; Shiyao Fu; Shiyao Fu; Jianqiang Zhang; Jianqiang Zhang; Xueting Liu; Xueting Liu; Heng Zhou; Heng Zhou; Chunqing Gao; Chunqing Gao

doi:10.1364/OE.388526

1. Introduction

The scalar vortex modes with a helical term of exp(ilφ) carrying orbital angular momentum (OAM), have drawn increasingly attention since pioneered by Allen and associates [1], where l is referred to as the topological charge and describes the number of the 2π helical phase winding in one wavelength. The integer topological charge l of OAM provides a degree of freedom for a large research fields with a myriad of applications in optical communication [2], quantum optics and information [3], optical tweezers and micromechanics [4], remote sensing [5], and so on (see [6–8] and reference therein). Furthermore, a similar but distinguishing mode, the vector vortex beams (VVB) [9] with non-separable states of light in which polarization and OAM are coupled, have also been studied in various disciplines. Similar to the scalar vortex beams with a phase singularity and a helical wave-front, the polarization singularity exists in the axis of the VVBs with an inhomogeneous polarization distribution. The vector beams especially cylindrical vector vortex (CVV) beams can be applied to sharp focus for high-resolution imaging [10], increased channel capacity for optical communication [11–14], rotating body remote sensing detection [15] and so on (see [16] and reference therein). Furthermore, it has also been inferred numerically that VVB analogous to partially coherent beams, are more resilient in turbulence [17].

In practice, nonhomogeneous media such as atmospheric turbulence, oceanic turbulence, biological tissue, strongly effects the optical transmission. Non-uniformity of medium refractive index will distort the helical wave-front and decrease the energy of received light field, resulting in the crosstalk between different modes. Amounts of studies have been done in assessing the atmospheric turbulence effects on vortex beams [18–22] and vector beams [23–28]. The irradiance pattern, scintillation index, degree of polarization, and OAM spectrum of the beams in an atmospheric turbulence with different turbulence strength have been studied numerically and experimentally. In order to improve the robustness of the vortex beams against the non-uniform media, many methods have been employed. The adaptive optics (AO) system is an effective approach, and common algorithms are Gerchberg-Saxton (GS) [29], stochastic-parallel-gradient-descent (SPGD) algorithm [30]. However, existing algorithms generally belong to search algorithms taking lots of iterations to acquire global optimal results. Hence, reducing the latency and improving the correction ability in strong turbulence are still challenges. As well as, to our best knowledge, the aberration correction for vector beams on experimental data have not been investigated.

As the powerful interdisciplinary science combined with the mathematics, computer and biology science, machine learning, including deep learning, has become an important field. Deep Neural Network (DNN) [31] uses multi-layered artificial neural networks implemented with a computer to perform advanced tasks, comparable to or even superior than the performance of human experts. Recently deep learning has made major advances in medical image analysis [32], speech recognition [33], image classification [34], and inverse imaging problems [35–41]. In addition, deep learning methods have potential applications in phase retrieval in image [42] and communication [43–44].

In this scenario, we propose and investigate a turbulence aberration correction convolutional neural network (TACCNN) model for compensation of distorted VVB in terms of intensity profile and spatial modes purity, which respectively corresponding to the polarization distribution and polarization topological charge [45–46]. To compensate the distortion for the vector beams, the distorted intensity captured by the CCD camera and the first 20 Zernike coefficients generating turbulence phase, are regarded as the studying samples. After supervised training, the TACCNN model can be trained to generate compensation phase screen according to the intensity distribution. After compensation, the phase distortion is decreased significantly in different atmosphere turbulences and the mode purity of the distorted vector beams can be improved. And then the intensity profiles get close to the original distribution. Meanwhile, experimental results shown that the TACCNN model can well experimentally predict the compensation phase and have advantages compared with previous work.

2. The vector vortex beams passing through nonhomogeneous media

Light beams can carry total angular momentum (TAM) consist of intrinsic spin angular momentum (SAM) associated with polarization, and OAM associated with the spatial profile of the phase. The vector vortex beams are a linear combination of orthogonal circular polarization scalar vortex optical beams of opposite topological charge [44], where the constituent components are eigenstates of TAM. In the paraxial approximation, a monochromatic generalized VBB has a unique spatial polarization, which can be represented as:

(1)$$|{{\psi_l}} \rangle = \psi _R^l|{{R_l}} \rangle + \psi _L^l|{{L_l}} \rangle ,$$

with respect to the orthonormal circular polarization basis {R_l, L_l} such that

(2a)$$|{{R_l}} \rangle = \exp ( - il\varphi )(x + iy)/\sqrt 2 ,$$

(2b)$$|{{L_l}} \rangle = \exp ( + il\varphi )(x - iy)/\sqrt 2.$$

Eqs. (2a)–(2b) represent a right circular polarized (RCP) and left circular polarized (LCP) optical vortex of topological charge -l and + l, respectively. When the magnitude of $\psi _R^l$ is equal to that of $\psi _l^l$, Eq. (1) can be transformed as follows:

(3)$$|{{\psi_{p,{\varphi_0}}}} \rangle = \vec{E}({r,\varphi } )= A(r )\cdot \left[ {\begin{array}{{c}} {\cos (p\varphi + {\varphi_0})}\\ {\sin (p\varphi + {\varphi_0})} \end{array}} \right]$$

with A(r) the amplitude distribution, φ the azimuthal angle, and φ₀ the initial orientation. p is the index of polarization topological charge (PTC) equal to (l₁-l₂)/2, where l₁ and l₂ are the OAM values of the two basic modes respectively. The space-dependent polarization distribution associated with the SAM and OAM, can be characterized by PTC. The PTC is defined as the repetition number of polarization state change along the azimuthal axis, while its sign stands for the rotating direction of the polarization. Due to the vector beam modes with different PTCs are orthogonal to each other, they have also potential to improve the capacity of optical communication.

The atmosphere is largely non-birefringent [46]. As such, the polarization is unaffected during propagation. Let us consider vector modes behind the turbulence described by the φ_Turb, which state can be expressed as follows:

(4)$$|{{\psi_{p,{\varphi_0}}}} \rangle_{Turb} = \vec{E}({r,\varphi } )\cdot \exp (i{\varphi _{Turb}}) = A(r )\cdot \left[ {\begin{array}{c} {\cos (p\varphi + {\varphi_0})}\\ {\sin (p\varphi + {\varphi_0})} \end{array}} \right] \cdot \exp (i{\varphi _{Turb}})$$

The Fourier expansion form of phase modulation function exp(iφ_Turb) is equal to $\sum {{B_n}} (\theta )\exp (in\varphi )$, and then Eq. (4) can be written as:

(5)$$\vec{E}({r,{\varphi_{Turb}}} )= A(r )\cdot \sum {{B_n}(\theta )} \left[ {\begin{array}{c} {\exp (i((p + n)\varphi + {\varphi_0})) + \exp ( - i((p - n)\varphi + {\varphi_0}))}\\ {i(\exp (i((p + n)\varphi + {\varphi_0})) - i\exp ( - i((p - n)\varphi + {\varphi_0}))} \end{array}} \right]$$

From Eq. (5), one can see that the distorted VBB are composed of many modes with different orders n, which corresponding to different PTCs. Thus the PTC is dispersed, reducing mode purity. Then, we can acquire the distortion intensity pattern in the far field

(6)$$I{(z)_{Turb}} = {f_z}(\vec{E}{({r,\varphi } )_{turb}}),$$

where f is the Fresnel propagation function for distance z. As shown in Fig. 1, the intensity patterns in vertical polarization and horizontal polarization directions behind turbulent atmosphere are distorted. Meanwhile the polarization distribution is also distorted. Both of the turbulence effect on the intensity and polarization distribution lead to poor performance in optical communication, super resolution imaging and so on. Thus, turbulence aberration correction is essential in practice.

Fig. 1. (a) The intensity profile of CVV beams at the waist in different polarized direction. The polarization topological charges equal to 1. (b) Atmospheric turbulence phase screen using Zernike polynomials with D / r₀ value respectively in 1.0, 2.3, 3.74, 5.28, 6.9. (c) The intensity patterns in different polarized directions respectively after turbulent atmosphere.

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Then the distorted phases caused by the atmospheric turbulence can be described as follows:

(7)$${\varphi _{Turb}} = F({I{{(z)}_{Turb}}} ),$$

where F is the inverse transform of the Fresnel propagation function f. According to Eqs. (6)–(7), provided that the mapping F and the distorted intensity I_Turb, the compensate screen can be predicted and parameters of the CVV beams can be corrected. But F is hardly solvable, which belongs to an inverse problem of laser transmission without analytic solution in mathematics. Here, we have trained a TACCNN model to match the relationship F. Through learning plentiful experimental samples, TACCNN model optimizes its parameters to approximate the objective function and finally predicts the distorted phase once distorted intensity I_Turb is input.

A turbulence phase obeys the Kolmogorov turbulence theory. The atmosphere phase screen here can be generated by the Zernike polynomials following Kolmogorov energy distribution [43], which can be written as:

(8)$${\varphi _{Turb}} = \sum {{a_j}} {Z_j}$$

where coefficient a_j and the j-th Zernike polynomial Z_j. Z_j defined is expressed as [47]:

(9)$${Z_{evenj}} = \sqrt {2(n + 1)} R_n^m(r )\cos (m\theta ),m \ne 0,$$

(10)$${Z_{oddj}} = \sqrt {2(n + 1)} R_n^m(r )\sin (m\theta ),m \ne 0,$$

(11)$${Z_j} = \sqrt {n + 1} R_n^0(r ),m = 0.$$

The analytical definition of radial function $R_n^m(r )$ is given by:

(12)$$R_n^m(r )= \sum\limits_{s = 0}^{(n - m)/2} {\frac{{{{( - 1)}^s}(n - s)!}}{{s![{(n + m)/2 - s} ]![{(n - m)/2 - s} ]!}}} {r^{n - 2s}}$$

where polar coordinates (r, θ) in the pupil plane, the radial degrees n and azimuthal frequency m. The value of n and m are integral and satisfy the following condition: m ≤ n, n - |m| = even. The index j represents both n and m, the relationship described in Table 1.

Table 1. The relationship of the index j, m and n.

View Table

The Zernike coefficients a_j characterizing the atmospheric turbulence are not irrelevant. The covariance between two Zernike polynomials z_j and z_j_’, with amplitudes a_j and a_j_’ are satisfying [48]:

(13)$$E({a_j},{a_{j^{\prime}}}) = \frac{{Kzz^{\prime}{\delta _z}\Gamma [{({n + n^{\prime} - 5/3} )/2} ]{{({D/{r_0}} )}^{5/3}}}}{{\Gamma [{({n - n^{\prime} + 17/3} )/2} ]\Gamma [{({n - n^{\prime} + 17/3} )/2} ]\Gamma [{({n - n^{\prime} + 23/3} )/2} ]}}$$

where δ_z is a logical Kronecker symbol depending upon azimuthal frequencies (m, m’) and K_zz is a factor depending upon the frequency characteristics (n, n’, m) of z_j and z_j’. D is the system aperture diameter and r₀ is the Fired parameter. Turbulence effect is proportional to the value of D/r₀. In the process, a diagonal matrix S and a unitary matrix U are acquired by the SVD (singular value decomposition) of the covariance E(a_j,a_j’). Gaussian random variables B with zero mean and variance given by S is generated, as the coefficients of the Karhunen-Loeve functions. Then the Zernike coefficients can be obtained by U^T multiplying B. Figure 1(b) shows the simulated atmospheric phase screen with different D/r₀ ratio. Here, only first 1-20 order Zernike coefficient are used, because higher-order Zernike polynomials over 20 take up relatively small components in turbulence atmosphere.

3. Aberration correction using TACCNN

The TACCNN model is constructed based on convolutional neural network (CNN) using the TensorFlow framework. As shown in Fig. 2, the model is composed of 8 convolution layers and a global average pooling layer. The convolution filters have 32, 64, 128, 256 kernels of size 5×5 respectively. The value of filter in each convolution layer is randomly initialized and the convolution layers (from 2^th to 6^th) are contained with Maxpooling. Meanwhile, each of the first 8th layers combined with an activation function named ReLu. The mean square error (MSE) between the estimated value and actual value of the first 20 Zernike polynomial coefficients is selected as the loss function as follow:

(14)$$L = {\sum\limits_{i = 0}^{20} {({a_{turb}^i - a_{est}^i} )} ^2}/20$$

where aⁱ_turb is the actual Zernike coefficients for the i-th sample, aⁱ_est is the estimated Zernike coefficients for the i-th sample. Then the weights of the neural network w can be optimized by

(15)$$w = \arg \mathop {\min }\limits_w L = \arg \mathop {\min }\limits_w {\sum\limits_{i = 0}^{20} {({a_{turb}^i - a_{est}^i} )} ^2}/20$$

Then, w are trained to bring lower value of the loss function, until reach the most appropriate values.

Fig. 2. Specific TACCNN framework based on Alexnet. The input is a series of pictures with 224×224×3 pixels and the output is 1×12 Zernike coefficients vector. ReLu, Rectified Linear Unit, a kind of activation function. Maxpooling, a kind of down-sampling method.

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In TACCNN model, the input is the distorted intensity profile captured by the CCD camera. The output is the Zernike coefficient vector of size 1×20, which are used to characterize the atmospheric turbulence. 42000 pairs of distorted images and their corresponding Zernike coefficients vectors are generated in experiment, which are used as the training set. The training set consists of five kinds of turbulence strengths (D/r₀ equal to 1, 1.52, 3.74 5.28, 6.90), each kind has 8000 experimental samples. During the training progress, 1000 pairs of data are used as validation data, which is employed to check the training effect. If the turbulence phase predicted by CNN is different from the ideal turbulence phase, we stop the training and re-adjust the parameters or the model’s structure. Besides, the validation data are used to test the generalization ability of the model and whether overfitting existence. As well as the filters learn how to extract features to match the relationship between the distorted intensity profile and the phase distortion. After training, additional 1,000 pairs of data are employed to assess the performance of the network.

Figure 3(a) presents the crucial loss function in the training. With the increase of the number of iterations, the loss value becomes progressively smaller and then tends to the minimum value. The values of five points A, B, C, D, E on the loss curve reflects trends intuitively. When the iterations increase to 25000, the loss value is already close to 0, which indicates that the predicted value gets close to actual value. This curve implies that TACCNN model can well extract turbulence effect information via learning samples.

Fig. 3. (a) Loss function curve along with the number of training progress. (b) The MSE between the Zernike coefficients with and without the compensation in different turbulence strength on test data. The dotted line represents the MSE before compensation, and the solid line represents the MSE after compensation. Different colors denote different turbulence strengths, and different turbulence realization means different random complex matrixes when computing the turbulence mask [29]. (c) The phase distributions in three kinds of turbulence strengths (D/r₀ equal to 3.74, 5.28, 6.9) before and after the turbulence aberration correction.

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Figure 3(b) shows the compensation results with the test sets in three kinds of turbulence strength (D/r₀ equal to 3.74, 5.28, 6.9). The MSE of the Zernike coefficients are zero without the atmospheric turbulence. However, it can be seen that the MSE becomes larger due to the nonuniform of the atmospheric turbulence in practice before compensation. The larger value of D/r₀, the larger value of MSE, which means more serious distorted phase. But, after compensation, the MSE is lower, which is even close to the original zero, and the phase distortion is decreased significantly in different atmosphere turbulences, which means that the proposed method is effective in changing atmospheric turbulences.

Figure 3(c) presents the phase distribution before and after turbulence aberration correction. We can see that the phase distribution is disorderly with a large fluctuation. The bigger the turbulence strength, the greater the phase fluctuation. However, the phase distribution after compensation is almost the same as the real phase screen. Especially, it can remove the larger phase fluctuations and return back a relatively smooth phase distribution. This larger phase fluctuation is difficult for traditional algorithms to recover, due to the limitations of iterative algorithms, which may fall into local optimality. This means that the TACCNN model has a better correction ability than the traditional algorithm in term of the larger phase fluctuations. These results indicate that the TACCNN model we proposed has high correction ability in term of large phase fluctuation.

With the help of this model, only one iteration is used to predict the turbulence aberration. For each prediction, the time consumption is ∼100ms when using a workstation computer(Intel(R) Xeon(R) CPU @ 2.30GHz, Kingston 64GB DDR4, NVIDIA Quadro P4000). We believe that when a higher-performance computer and more learning samples or iterations are implemented, the speed can be further improved.

4. Experimental configuration and results analysis

Figure 4 shows that the experimental setup used for TACCNN for VBB. It consisted of three main stages: generation of CVV modes, turbulence using a Spatial Light Modulator (SLM) and finally, correction and detection. In the first stage, Gaussian modes whose wavelength is 1.55µm are generated by a laser diode and collimated into free space with a diameter of 3mm. The CVV mode is generated using a Q-plate in conjunction with the polarized beam splitter (PBS1). Secondly, a SLM (Holoeye, PLUTO-TELCO-013-C) is used to simulate the atmospheric turbulence. Since the SLM only modulates horizontally polarized beams, a polarization in variant arrangement was implemented, as used in [24]. Finally, after propagation, the perturbed CCV modes in a horizontal direction were collected using a lens L1 and CCD1 (Xenics, Bobcat-320-star) is used to capture intensity distributions, which are used as the input of TACCNN model to predict phase distortion. The CCD2 are used to record the perturbed intensity distribution in different polarization direction (with PBS), or the total light field (without PBS). Then the mode purity can be measured by Q-plate and CCD3.

Fig. 4. Experimental configuration. PBS1& PBS2& PBS3& PBS4, polarized beams splitter. Q1, Q-plate. NPBS1& NPBS2& NPBS3, Non-polarization beams splitter. R1, reflector. HWP, half wave plate. SLM, liquid-crystal spatial light modulator. L1&L2, lenses with focal length f = 100 mm. L3, lens with focal length 200 mm. CCD, infrared CCD camera. TACCNN, the well-trained turbulence aberration correction CNN model.

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Nothing is added on SLM firstly to record the initial intensity distribution. Then, a turbulence phase screen obeying the Kolmogorov turbulence theory is uploaded on SLM. A turbulence phase screen characterized by the Fried coherence length r₀ and the structure constant of the refractive index C_n². The turbulence strength is imposed on the beams, which can be described by D/r₀, with the beam diameter at the turbulence D. By this time, the pattern of distorted CVV beams recorded by CCD1 as the input of well-trained TACCNN model, to predict 20 Zernike coefficients corresponding to the turbulence phase screen. Then the compensation screen is added on SLM. We can measure corrective results in terms of the distorted, corrected light pattern and mode purity, through the CCD2 and CCD3.

The far-field intensity distribution without and with turbulence, as well as after compensation, are shown in Fig. 5. The intensities in different polarization directions of VBB with polarization topological charge +1 and +2 are shown respectively. The intensity distributions are distorted after propagating through the turbulence. The tip-tilt aberration arising from atmospheric turbulence and mechanical movement is one of the largest concerns. This form of atmospheric aberration results in a change of the beam propagation direction, which can be seen that the intensity distributions go outside the center white ring. Once the compensation phase is loaded, the turbulence effects will decrease a lot. The intensity distribution after compensation are close to the original intensity. Especially it can degrade position drift of the centroid induced by the tip and tilt mode of Zernike coefficients. The intensity distributions in different polarization directions nearly go inside the white circles. Thus, the well trained TACCNN model can correct the turbulence aberration in terms of the intensity distribution in different polarization directions of the vector beams.

Fig. 5. The distorted intensities of vector vortex beam without and with turbulence, as well as after compensation, in different polarization directions in (Fried parameters r₀ = 1mm, and D/ r₀ = 5.28) turbulence. (a) - (c) with polarization topological charge +1, (d) - (f) with polarization topological charge +2. The white ring is the central ring which is used to compare the offset of the spot.

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Mode purity is an important parameter describing the cross talk between different spatial mode. The inverse process (exploiting the reciprocity of light) with the Q plate adding the light path before CCD3. Consequently, the CVV beams without atmospheric turbulence are back converted into a Gaussian-like beam (a bright spot in the beam center) and then recorded by CCD3 as shown in the first line of the Fig. 6(a). The normalized intensities, which can be regarded as the mode purity, without compensation, for different turbulence strengths, are sketched in the second line. It can be seen that the bright spot is weak and drift with shape distortion due to the turbulence. And then the third line shows the bright spot after the turbulence correction. The bright spot becomes more bright and center with a round shape.

Fig. 6. (a) The intensities of mode purity without and with turbulence, as well as after compensation. (b) Detected mode purity before and after compensation in various turbulence atmospheres (D/r₀ within the scope of a value in 0 - 6.29). The light blue bar represents the mode purity before compensation, and the prink bar represents the incremental mode purity after compensation.

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For quantitative analysis of the correction effect in terms of mode purity, we compare the power of the bright spot before and after phase compensation, at the original position of the light spot without the turbulence. As shown in Fig. 6(b), the blue bar represents the distorted mode purity while the pink bar of each group stands for the value after the correction. Similar to the MSE change of the Zernike coefficient in terms of the phase distribution, mode purity is lower because of the turbulence. The stronger turbulence, the lower mode purity. After correction, the mode purity becomes higher in different turbulence strength. When D/r₀ equals to 3.74, the mode purity increases from 0.12 to 0.82. When D/r₀ equals to 5.28, the mode purity increases from 0.19 to 0.70. When D/r₀ equals to 6.9, the mode purity increases from 0.05 to 0.35. The results show that the mode purity all increase significantly, which means the TACCNN model has great generalization ability to resist the larger turbulence strength.

5. Discussion

In fact, the TACCNN model containing multiple convolutional layers learns the mapping relationship of intensity profile of the distorted vector vortex modes and the turbulence phase. After supervised learning, the model can acquire the first 20 Zernike coefficients without iterative operation, compared with the traditional algorithm. In order to make clear the response latency of two kinds of phase retrieval algorithms, we compared the computing times of the CNN model and parallel gradient descent (SPGD) algorithm. As shown in Fig. 7, SPGD takes much longer time to complete hundreds of iterations while TACCNN model only requires one iteration. The latency is calculated on a workstation computer. Furthermore, the latency at different turbulence strengths is also the same (100ms). However, the SPGD algorithm approximately costs about 4s to complete 100 iterations. When using the same time about 100ms, SPGD completes so a few iterations that a well performance is hard to acquire. Moreover, the stronger the turbulence, the more iteration times are needed. Thus the TACCNN model has advantage in calibration time in spite of a large amount of training time. In another aspect, not only the TACCNN model has an advantage in terms of computing time, but also we find that it has strong aberration correction capability, especially tilt and offset issues of the intensity distribution, where traditional algorithms perform poorly in these situations due to interference of local optimality. Thus, deep learning can greatly promotes the update of adaptive optics in terms of calibration time and calibration capability.

Fig. 7. The latency of TACCNN model compared with the SPGD under different turbulence realization. The front row is the latency for CNN and the back row is the latency for SPGD algorithm.

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Despite all above, we think we just do the preliminary work about the performance of deep learning in adaptive optics system for turbulence aberration correction. As well as we believe the TACCNN model can be further optimized by some ways. Firstly, a lot work should be paid to study the performance of other CNNs model (such as VGG-Net, GoogleNet) in adaptive optics, and the performances of different CNN models in terms of calculation time remain to be discussed. Secondly hyperparameters in the network model (including learning rate, batch size and so on) should be further optimized for a lower MSE and a shorter calculation time. Finally, the 224 × 224 × 3 RGB images as intensity measurements can be replaced by the 224 × 224 × 1 grayscale images as TACCNN inputs for smaller data volumes. By these approaches, we believe the computation time will be greatly decreased.

6. Conclusion

We have demonstrated a deep learning based adaptive optics system to recover the turbulence aberrations of the VVB in terms of phase distribution, mode purity as well as intensity distribution. The turbulence aberration correction convolutional neural network (TACCNN) model, containing multiple convolutional layers, have learned the relationship between the intensity profile and the turbulence phase through lots of experimental examples. The turbulence compensation screen can be quickly and accurately acquired once the distorted intensity profile is input. The experimental configuration for TACCNN-based turbulence aberration correction is built up. The experimental results show that the corrected optical mode profile at the receiver are found to be nearly identical to the desired profiles, and the mode purity increase significantly (i.e. from 0.19 to 0.70, when D/r₀=5.28). Furthermore, the method can deal with different turbulence strength, and the calculated time can be reduced to 100ms without iterations compared with the traditional algorithms. Thus the TACCNN model has advantage in calibration time and calibration capability. In conclusion, this scheme shows that the present results combining the fields of deep learning and adaptive optics will be great helpful for the structure beams in communication and imaging.

Funding

National Natural Science Foundation of China (11834001, 61905012); CETC Joint Research Foundation (6141B08231125); National Postdoctoral Program for Innovative Talents (BX20190036); China Postdoctoral Science Foundation (2019M650015).

Acknowledgments

The authors acknowledge Dr. Bing Dong, Chun Liu and Boyang, Xing for the helpful discussions.

Disclosures

The authors declare no conﬂicts of interest.

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n, m	0,0	1,1	1,-1	2,0	2,-2	2,2	3,-1	3,1	3,-3	3,3
j	1	2	3	4	5	6	7	8	9	10
n, m	4,0	4,2	4,-2	4,4	4,-4	5,1	5,-1	5,3	5,-3	5,5
j	11	12	13	14	15	16	17	18	19	20

Turbulence aberration correction for vector vortex beams using deep neural networks on experimental data

Abstract

1. Introduction

2. The vector vortex beams passing through nonhomogeneous media

3. Aberration correction using TACCNN

4. Experimental configuration and results analysis

5. Discussion

6. Conclusion

Funding

Acknowledgments

Disclosures

References

Cited By

Figures (7)

Tables (1)

Equations (16)

Optics Express