Diffractive deep neural network based adaptive optics scheme for vortex beam in oceanic turbulence

Haichao Zhan; Yixiang Peng; Bing Chen; Le Wang; Wennai Wang; Shengmei Zhao; Shengmei Zhao

doi:10.1364/OE.462241

1. Introduction

In recent years, underwater wireless optical communication (UWOC) has attracted significant attention due to its high data rate, high security, high bandwidth, low latency, and low cost [1–4]. UWOC has many applications in the rapidly growing human underwater activities, such as submarines, autonomous underwater vehicles and unmanned underwater vehicles [5,6]. In order to meet the increasing demand for data transmission in military, civilian and commercial, orbital angular momentum (OAM) is introduced into UWOC [7]. In 1992, Allen et.al. proved that OAM mode can be characterized by a spatial wave-function with a helical phase $\text {exp}\left ( i\ell \theta \right )$ [8], where $\ell$ is an arbitrary integer named topological charge, representing the number of $2\pi$ phase shifts across the beam, and $\theta$ corresponds to the azimuthal angle. The different OAM modes carried by vortex beam are orthogonal, which provides a new degree of freedom for the multiplexing of OAM beams, and increases transmission capacity and spectral efficiency [9]. Unfortunately, the spiral wavefront structure of OAM mode is easily distorted by oceanic turbulence (OT), which aggravates crosstalk between OAM modes and affects the communication quality of the UWOC system [10]. Therefore, adaptive optics (AO), as a technique that can reduce the distortion caused by turbulence, is applied to the UWOC system [11].

Generally, Shack-Hartmann (SH) wavefront correction method needs an SH wavefront sensor (WFS) to obtain distortion information, and a deformable mirror to correct wavefront distortion. But, due to the existence of phase singularities, the SH wavefront correction method cannot well reconstruct the spiral wavefront of the OAM mode [12]. Accordingly, the AO method without WFS has received a lot of attentions, such as, Gerchberg-Saxton (GS) algorithm [13,14], stochastic parallel gradient descent (SPGD) algorithm [15], hybrid input-output algorithm [16], genetic algorithm [17]. However, these methods are either time-consuming and prone to local optima troubles, or are expensive in experimental consumables. Recently, the AO method without WFS combined with deep learning [18] has become a research hotspot. In 2020, Zhai et.al. proposed an AO method based on convolutional neural network (CNN) [19]. The CNN trained the corresponding relationship between the distortion intensity recorded by a charge-coupled device (CCD) camera and the first 20 Zernike coefficients representing the turbulence phase, then output the predicted Zernike coefficients to compensate for distortion. Later, Lu et.al. designed a jointly-trained CNN-based OAM recognition method, which implicitly used an upsample model as a substitute for a one-shot AO system [20]. Upsampling and recognition share the same CNN backbone, and then the transposed convolution is utilized to restore the distortion intensity image. In addition, compared with CNN, generative adversarial network (GAN) can generate more realistic samples through adversarial training methods in limited datasets, and realize image-to-image translations, which has been applied to the AO system. Zhan et.al. proposed a GAN based AO scheme to compensate for the distortion caused by OT, where the GAN was trained by numerical simulations to obtain the mapping between the distortion intensity distribution and its corresponding phase screen representing OT [21]. The effects of different OT strengths on the compensation performance of the GAN-based AO scheme were compared through experimental analysis.

In 2018, Lin et.al. introduced an all-optical deep learning network framework in which the neural work is composed of multiple layers of diffractive surfaces, which is called a diffractive deep neural network (DDNN) [22]. DDNN can realize various functions based on deep learning and efficiently operate at the speed of light using passive components and the all-optical diffraction layers, which has the advantage that it can be easily scaled up to use a variety of high-throughput and large-area 3D fabrication methods. So it has attracted a lot of interests. Zhao et.al. proposed an OAM mode detection method based on DDNN, and tested the recognition performance of three types of networks: amplitude-only, phase-only, and hybrid type [23]. Wang et.al. utilized DDNN to realize the logic operation of OAM mode, which improved the parallel processing capability and enhanced the logic robustness due to the infinity and orthogonality of OAM modes [24]. Huang et.al. used DDNN’s light field control ability to realize the signal processing of the vortex beam, including OAM-shift-keying (OAM-SK), OAM multiplexing and demultiplexing, and OAM-mode switching [25]. These researches provide references for the realization of AO based on DDNN.

In this paper, we propose a DDNN-based AO compensation scheme, where DDNN is trained to learn the relationship between the intensity pattern of the vortex beam and the phase screen simulating OT by numerical simulations. The diffractive layers of the DDNN are solidified, fabricated and loaded into the spatial light modulator (SLM), and the CCD is utilized to record the intensity information of the modulated distorted light field and the intensity distribution before and after compensation. We demonstrate the reliability of the proposed AO scheme, and compare the compensation performance with the previous CNN-based AO schemes and the conventional GS algorithm under different OT strengths.

2. Theory

2.1 DDNN-based AO scheme

The concept of the AO compensation scheme based on DDNN is sketched in Fig. 1. In Fig. 1(a), a vortex beam passes through a phase screen that simulating OT, its wavefront is distorted, and its perfect doughnut-like intensity distribution is also destroyed. The distorted intensity distribution image is input to the trained DDNN to obtain the phase screen representating the distortion. Then, the phase screen output by DDNN is reversed to form a compensation phase screen. The distorted vortex beam is corrected after passing through the compensation phase screen, and the intensity distribution is also converged into a perfect doughnut. Figure 1(b), Fig. 1(c) and Fig. 1(d) are the mode purity of the vortex beam before passing through OT, after passing through OT and after being compensated, respectively.

Fig. 1. Schematic diagram of the AO compensation scheme based on DDNN. (a) The prediction and compensation of AO based on DDNN. (b) Mode purity of the original vortex beam. (c) Mode purity of the vortex beam after passing through OT. (d) Mode purity after the distorted vortex beam is compensated.

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In order to simulate the effect of OT on the beam propagation, the power spectrum inversion method is used to generate separate phase thin sheet at regular intervals [26]. The phase thin sheet is also called phase screen, which can adjust the phase of the beam. On the beam propagation axis, the phase disturbance caused by the fluctuation of the refractive index can be expressed as the multiplication of the input field and the phase exponential function (a phase screen). And the phase screen is independent of spatial statistics matching the index of refraction fluctuations. The spectrum of refractive index fluctuations of homogeneous and isotropic OT [27] can be expressed as:

(1)$$\begin{aligned} {{\Phi }_{ot}}( \kappa) &= 0.388C_{n}^{2}{{\kappa }^{{-}11/3}}[1+2.35{{( \kappa \eta)}^{2/3}}][ \exp( -{{A}_{T}}\delta)\\ &- 2{{\tau }^{{-}1}}\exp( -{{A}_{TS}}\delta )+{{\tau }^{{-}2}}\exp ( -{{A}_{S}}\delta )], \end{aligned}$$

where $C_{n}^{2}={{10}^{-8}}{{\varepsilon }^{-1/3}}{{\chi }_{T}}$ is a measure of the strength of ocean fluctuations. $\varepsilon$, ${{\chi }_{T}}$, and $\tau$ are the kinetic energy dissipation rate per unit mass of fluid, the dissipation rate of the mean square temperature, and the balance parameter that means the relative strength of temperature and salinity fluctuations, respectively, which ranges from ${{10}^{-10}}{{\text {m}}^{2}}/{{\text {s}}^{3}}$ to ${{10}^{-1}}{{\text {m}}^{2}}/{{\text {s}}^{3}}$, from ${{10}^{-10}}{{\text {K}}^{2}}/\text {s}$ to ${{10}^{-4}}{{\text {K}}^{2}}/\text {s}$, and from −5 to 0. $\kappa$ and $\eta$ are the spatial frequency and the Kolmogorov microscale (inner scale), respectively. ${{A}_{T}}=1.863\times {{10}^{-2}}$, ${{A}_{S}}=1.9\times {{10}^{-4}}$, ${{A}_{TS}}=9.41\times {{10}^{-3}}$, and $\delta =8.284{{\left ( \kappa \eta \right )}^{4/3}}+12.978{{\left ( \kappa \eta \right )}^{2}}$. The phase spectrum can be expressed as:

(2)$$\Phi \left( \kappa \right)=2\pi k_{0}^{2}Z{{\Phi }_{ot}}\left( \kappa \right),$$

where ${{k}_{0}}$ denotes the wave number of the incident beam, $Z$ is the propagation distance. So, the phase screen can be obtained by the Fourier transform:

(3)$$\varphi \left( x,y \right)=F\left[ C\left( \frac{2\pi }{N\Delta x} \right)\sqrt{\Phi \left( \kappa \right)} \right],$$

where $F$ is the Fourier transformation, $C$ represents an $N\times N$ array of complex random numbers with 0 mean and variance 1, and $\Delta x$ is the grid spacing.

2.2 Network structure

DDNN, with learning, memory and light field processing capabilities, is composed of a series of transmission layers or reflective diffraction layers. Figure 2 shows the framework of the DDNN used to compensate for wavefront distortion. Each sensor node on the diffraction layer corresponds to a neuron in the neural network layer, and each node is fully connected to all sensor nodes in the next diffraction layer. Each diffractive layer is filled with $200\times 200$ sensor nodes, and each sensor node has a pixel size of $2\mu m$. Following the Rayleigh-Sommerfeld diffraction equation [28], a single sensor node of DDNN can be considered as a secondary source of a wave, which is composed of the following optical mode:

(4)$$\omega _{i}^{N}\left( x,y,z \right)=\frac{z-{{z}_{i}}}{{{r}^{2}}}\left( \frac{1}{j\lambda }+\frac{1}{2\pi r} \right)\exp \left( \frac{j2\pi r}{\lambda } \right),$$

where $N$ denotes the $N^{th}$ layer of the DDNN, $i$ represents the sensor node located at $\left ( {{x}_{i}},{{y}_{i}},{{z}_{i}} \right )$ of the $N^{th}$ layer. $r=\sqrt {{{\left ( x-{{x}_{i}} \right )}^{2}}+{{\left ( y-{{y}_{i}} \right )}^{2}}+{{\left ( z-{{z}_{i}} \right )}^{2}}}$, is the distance from the source to the $i^{th}$ sensor node in the $N^{th}$ layer, $j=\sqrt {-1}$, and $\lambda$ is the wavelength. The amplitude and phase of the secondary wave are determined by the input wave to the sensor nodes and its transmission coefficient $t_{i}^{N}\left ( {{x}_{i}},{{y}_{i}},{{z}_{i}} \right )$. So, the output function of the $i^{th}$ sensor node in the $N^{th}$ layer can be expressed as:

(5)$$s_{i}^{N}\left( x,y,z \right)=\omega _{i}^{N}\left( x,y,z \right)\times t_{i}^{N}\left( {{x}_{i}},{{y}_{i}},{{z}_{i}} \right) \times \sum\nolimits_{k}{s_{k}^{N-1}\left( {{x}_{i}},{{y}_{i}},{{z}_{i}} \right)},$$

where $\sum \nolimits _{k}{s_{k}^{N-1}\left ( {{x}_{i}},{{y}_{i}},{{z}_{i}} \right )}$ is the input wave superposition of all nodes of the $\left ( N-1 \right )^{th}$ layer to the $i^{th}$ sensor node of the $N^{th}$ layer. The transmission coefficient of the sensor node is composed of amplitude and phase, $t_{i}^{N}\left ( {{x}_{i}},{{y}_{i}},{{z}_{i}} \right )=a_{i}^{N}\left ( {{x}_{i}},{{y}_{i}},{{z}_{i}} \right )\exp \left ( j\phi _{i}^{N}\left ( {{x}_{i}},{{y}_{i}},{{z}_{i}} \right ) \right )$. During DDNN training, the network model is optimized through error back-propagation and gradient descent, so that DDNN reaches the optimal state, and then each diffraction layer can be solidified and fabricated. According to the variable amounts in the transmission coefficient, DDNN can be divided into three types: phase modulation only, amplitude modulation only and hybrid modulation. The secondary wave of the sensor node in the previous layer is diffracted to the next layer, and to the output layer finally.

Fig. 2. The framework of the DDNN used to compensate for wavefront distortion.

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The forward propagation process of vortex beam between DDNN diffractive layers can be simulated by Fresnel diffraction theory [25]. The frequency domain distribution of the incident beam on the $\left ( N-1 \right )^{th}$ layer can be obtained by the Fourier transform of the light field in the spatial domain, as shown in the following equation:

(6)$$U_{{{z}_{N-1}}}^{N-1}\left( {{f}_{x}},{{f}_{y}} \right)=F\left( u_{{{z}_{N-1}}}^{N-1}\left( {{x}_{N-1}},{{y}_{N-1}} \right) \right),$$

where ${{z}_{N-1}}$ is the observation plane on the propagation axis, $F$ is the Fourier transform. Therefore, the output of the $\left ( N-1 \right )^{th}$ diffraction layer can be expressed as:

(7)$$U_{{{z}_{N-1}}}^{N-1}\left( {{f}_{x}},{{f}_{y}} \right)=F[ u_{{{z}_{N-1}}}^{N-1}\left( {{x}_{N-1}},{{y}_{N-1}} \right) \times t_{{{z}_{N-1}}}^{N-1}\left( {{x}_{N-1}},{{y}_{N-1}} \right)].$$

After the beam propagates to the $N^{th}$ diffraction layer, based on the Fresnel diffraction theory, the light field in the spatial domain can be expressed as:

(8)$$u_{{{z}_{N}}}^{N}\left( {{x}_{N}},{{y}_{N}} \right)={{F}^{{-}1}}\left( U_{{{z}_{N-1}}}^{N-1}\left( {{f}_{x}},{{f}_{y}} \right)\times H\left( {{f}_{x}},{{f}_{y}} \right) \right),$$

where ${{F}^{-1}}$ represents the reverse Fourier transform. $H\left ( {{f}_{x}},{{f}_{y}} \right )$ is the transform function, $H\left ( {{f}_{x}},{{f}_{y}} \right )=\exp \left [ jk\left ( {{z}_{N}}-{{z}_{N-1}} \right ) \right ]\exp \left [ -j\pi \lambda \left ( {{z}_{N}}-{{z}_{N-1}} \right )\left ( f_{x}^{2}+f_{y}^{2} \right ) \right ]$, where $k=2\pi /\lambda$ is the wave number, and $\left ( {{z}_{N}}-{{z}_{N-1}} \right )$ represents the separation distance between the diffractive layers.

In order to ensure the consistency of the network output results and the target output (ground truth), the error backward propagation algorithm and Adam gradient descent optimizer [29] are used to train the DDNN. The loss function is defined as:

(9)$$loss={{\left( {{\varphi }_{o}}(x,y)-{{\varphi }_{gt}}(x,y) \right)}^{2}},$$

where ${{\varphi }_{o}}\left ( x,y \right )$ and ${{\varphi }_{gt}}\left ( x,y \right )$ represent the actual output of DDNN and ground truth, respectively.

3. Performance evaluation

In order to evaluate the compensation performance of the AO scheme based on DDNN, we utilize LabVIEW 2012 to simulate and acquire the OT phase screen and the distorted vortex beam intensity distribution images. We choose phase screens with different OT strength levels and corresponding vortex beam intensity distributions, where $C_{n}^{2}=1\times {{10}^{-15}}{{\text {K}}^{\text {2}}}{{\text {m}}^{\text {-2/3}}}$ represents weak turbulence, $C_{n}^{2}=1\times {{10}^{-14}}{{\text {K}}^{\text {2}}}{{\text {m}}^{\text {-2/3}}}$ and $C_{n}^{2}=1\times {{10}^{-13}}{{\text {K}}^{\text {2}}}{{\text {m}}^{\text {-2/3}}}$ represent medium turbulence, and $C_{n}^{2}=1\times {{10}^{-12}}{{\text {K}}^{\text {2}}}{{\text {m}}^{\text {-2/3}}}$ represents strong turbulence. The beam propagation distance $Z$ at each turbulence strength level is 30m. Each OT strength level has 12,000 random phase screens, which are marked as ground truth. At the same time, the corresponding 12,000 vortex beam intensity distribution images are also recorded. For each turbulence strength level, the training datasets include 10,000 ground truth and 10,000 distortion intensity images, and the remaining 2,000 sets of images are used as testing datasets. The original images obtained from LabVIEW 2012 are resized to 256$\times$256, converted to gray scale images, and normalized. The training of DDNN model is implemented with the Tensorflow framework on four NVIDIA TITAN RTX GPUs. Furthermore, the distance between diffractive layers is 0.1 m, and the DDNN model is trained for 50 epochs using the Adam optimizer with a learning rate of 0.01.

Figure 3 shows the experimental setup of AO compensation scheme based on DDNN. The laser (Thorlabs, HNL050R) emits a Gaussian beam with beam waist of 0.35mm and wavelength of 633nm. The Gaussian beam passes through a polarization beam splitter (PBS) (Thorlabs, CM1-PBS252) and half-wave plate (HWP) (Thorlabs, RSP1C/M) to obtain linearly polarized light beam that matches the SLM (Hamamatsu, X10468-07). Then, the Gaussian beam is enlarged by a beam expander composed of two lenses, L1 and L2, and the focal lengths of the two lenses are 50mm and 100mm, respectively. The expanded Gaussian beam is propagated through four SLMs. L3 and L4 are Fourier lens (Thorlabs, LMR1/M, AC254-100-A), which are used to focus the light intensity distribution to CCD camera (Thorlabs, BC106N-VIS/M) in Fourier plane. Firstly, an OAM phase mask ($\ell =-3$ in the Fig. 3(a)) is imprinted to the SLM1. At this time, the intensity pattern of the vortex beam is a perfect doughnut structure. Then, a phase screen, which simulates OT, as shown in Fig. 3(b), is loaded into the SLM2. After the vortex beam passes through the phase screen, its spiral wavefront is distorted. At this moment, the intensity pattern is as shown in Fig. 3(c). The SLM3 is divided into 5 regions, which are used to load the five diffractive layers that are solidified and fabricated of the trained DDNN. The distribution of the five diffractive layers loaded on the SLM3 is obtained by simulation. The distorted vortex beam passes through the SLM3 and the mirror array, and is modulated by the diffraction layer of the DDNN. At this time, the light field intensity information of the modulated distorted vortex beam is recorded by the CCD1. Finally, the phase screen predicted by the DDNN model through simulation, as shown in Fig. 3(d), is reversed to obtain the compensated phase screen. The compensation phase screen is uploaded to the SLM4 to compensate for the distortion caused by OT. As shown in Fig. 3(e), the intensity pattern of the compensated vortex beam is captured by the CCD2. Mode purity, also called transmission probability [7], can be used to evaluate the purity of OAM mode, which can be expressed as:

(10)$$M{{P}_{m}}=\frac{{{I}_{m}}}{\sum\limits_{\ell }{{{I}_{\ell }}}},$$

where ${{I}_{m}}$ is the light intensity of the $\ell =m$. ${{I}_{\ell }}$ denotes the light intensity of all modes at the receiver. The quality of the predicted phase screen can be quantitatively evaluated by the peak signal-to-noise ratio (PSNR) [30].

(11)$$PSNR=10{{\log }_{10}}\left( \frac{Max_{Val}^{2}}{MSE} \right),$$

where $Max_{Val}^{2}$ represents the maximum possible pixel value of the predicted phase screen. The $MSE$ can be expressed as:

(12)$$MSE=\frac{1}{N}\sum_{x,y}{{{\left( {{T}^{'}}\left( x,y \right)-T\left( x,y \right) \right)}^{2}}},$$

where $N$ is the number of pixels of the image. ${{T}^{'}}\left ( x,y \right )$ and $T\left ( x,y \right )$ denote the predicted phase screen and the ground truth, respectively.

Fig. 3. The experimental setup of AO compensation scheme based on DDNN. PBS: polarization beam splitter; HWP: half-wave plate; L1, L2: beam expander; L3, L4: Fourier lens; SLM: spatial light modulator; BS: beam splitter; CCD: charge coupled device camera. (a) The OAM phase mask. (b) The phase screen simulating OT. (c) The distorted intensity pattern of vortex beam. (d) The predicted phase screen output by DDNN. (e) The intensity pattern of the vortex beam after compensation.

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We construct a five-layer DDNN to establish the mapping relationship between the distortion intensity pattern of vortex beam and the phase screen. With training, the difference between the predicted phase screen and ground truth is gradually eliminated. Figure 4 shows the predicted phase and amplitude distribution of each diffraction layer after the training is completed. For the hybrid modulation type DDNN, each diffraction layer contains the predicted phase and amplitude distribution.

Fig. 4. The predicted phase and amplitude distribution of each diffraction layer.

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Figure 5 shows the ground truth at different OT strengths, the predicted phase screen output by the DDNN, and the experimental results of the intensity pattern before and after the vortex beam is compensated. When the vortex beam is not interfered by OT, its MP is 1. The greater the OT strength, the more severe the distortion of the vortex beam. The MP under strong turbulence is only 0.156, while under weak turbulence it is 0.513. The phase screen predicted by DDNN is reversed to obtain the compensated phase screen, and then loaded into SLM4, the distortion of the vortex beam is restored. The intensity pattern of the vortex beam is also restored to the doughnut structure, even under strong OT, the MP reaches 0.980. In addition, comparing the PSNR between the predicted phase screen and ground truth under different OT strengths, it can be found that the predictive ability of DDNN is not affected by the strength of OT, which provides important support for AO compensation in strong OT strength.

Fig. 5. The prediction results of DDNN under different OT strengths and the experimental results of the vortex beam intensity patterns before and after compensation.

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We investigate other OAM mode and multiplexed OAM modes, as shown in Fig. 6. The intensity patterns of OAM-1, OAM+7 and multiplexed OAM modes ($\ell =+6,-6$) are more distorted with increasing OT, but their perfect intensity distribution can be recovered after compensation, even for strong OT. We increase the optical power at the receiving end under strong OT and record the intensity patterns after different OAM modes are compensated, as shown in Fig. 7. The results show that the distortion of the OAM mode is compensated, and the ring-like feature of its intensity pattern is more obvious and easier to distinguish. In addition, it can be seen from Fig. 6 and Fig. 7 that the multiplexed OAM modes have stronger anti-turbulence properties than a single OAM mode.

Fig. 6. The intensity patterns of OAM-1, OAM+7 and multiplexed OAM modes ($\ell =+6,-6$) before and after compensation by the DDNN-based AO scheme.

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Fig. 7. The intensity patterns of the OAM modes after increasing the received optical power.

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To further evaluate the performance of the DDNN-based AO compensation scheme under strong OT, we compare the influence of different epoch numbers on the predicted phase screen output by DDNN, as shown in Fig. 8. Under strong OT, with the increase of epoch, the predicted phase screen is gradually consistent with ground truth. The PSNR reaches a maximum of 37.209 around 50 epochs, which means that the predicted phase screen is almost equivalent to the ground truth.

Fig. 8. The predicted phase screen of DDNN output under different epochs.

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We use the predicted phase screen output by the DDNN under different epochs to compensate for the distortion of the vortex beam, and obtain its MP, which is recorded in Table 1. The change trend of MP with epoch corresponds to Fig. 8. When the epoch is 50, the MP reaches the maximum value of 0.980, which shows that DDNN can excellently compensate the distortion of the vortex beam even under strong OT.

Table 1. The MP after the vortex beam is compensated under different epochs.

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The CNN-based AO compensation schemes are basically divided into two types, one outputs the coefficients of the Zernike polynomial representing the turbulence phase, and the other outputs the phase screen directly. We upload the compensation phase screen generated by various AO methods to SLM4, record the intensity pattern of the vortex beam through CCD2, and calculate the MP. Then, we compare the MP after the vortex beam is compensated by various AO methods under different OT strengths, as shown in Fig. 9, CCN1 represents the output of Zernike coefficients, CNN2 represents the output of phase screen. Under weak OT strength, DDNN, CNN1, CNN2 and GS algorithm achieve excellent compensation performance, while under strong OT strength, only DDNN can well compensate the distortion of the vortex beam, MP is close to 1.

Fig. 9. The MP of vortex beam compensated by AO compensation scheme based on DDNN, CNN, and GS algorithm.

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We record the response time of various AO methods from processing distortion information to outputting prediction results. Figure 10 shows the response time of various AO methods under different OT strengths on the same computer. As the OT strength increases, the response time of the GS algorithm that needs to be iterated also increases. Under strong OT, it takes 20 seconds for the GS algorithm to complete 100 iterations, and may fall into the trouble of local optimization. We use the trained DDNN and CNN models to record the time required for prediction, so the time for datasets preparation and training is not taken into account. It can be seen from Fig. 10 that the response time of DDNN and CNN models does not change with the increase of OT strength. The response time of the DDNN model is the least, only 0.977s. Compared with the 2.156s of CNN2, CNN1 needs to convert the predicted Zernike coefficients into the compensation phase, so its response time is 3.823s.

Fig. 10. The response time of AO compensation scheme based on DDNN, CNN, and GS algorithm.

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The training loss function is used to evaluate the closeness of the prediction results and the output results. Figure 11 shows the training loss of the AO scheme based on DDNN and CNN within 50 epochs. The results show that the training loss of the DDNN model has a faster convergence trend than the CNN1 and CNN2 models, and the final stable value is also lower, which means that the AO compensation scheme based on DDNN can achieve excellent performances.

Fig. 11. The training loss of the AO compensation scheme based on DDNN and CNN within 50 epochs.

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4. Conclusion

Several works have utilized the deep neural networks to modulate and manipulate vortex beams. In [19], a CNN-based turbulence aberration correction model was proposed to recover the phase distribution, mode purity and intensity distribution of vector vortex beams (VVB). The CNN model contains multiple convolutional layers, which are able to learn the mapping between the intensity pattern of the distorted VVB and the turbulent phase produced by the first 20-order Zernike coefficients through a large number of experimental examples. The results show that the mode purity of the corrected VVB is significantly improved and the intensity pattern is recovered. Compared with CNN-based turbulence aberration correction method, the proposed scheme utilizes DDNN to learn the mapping between the intensity pattern of the distorted vortex beam and the phase screen representing the OT. The innovation is that DDNN can implement an image-to-image transformation task, which avoids the generation of phase screens by Zernike coefficients. In addition, the diffractive layers of DDNN can be solidified, fabricated and loaded onto the SLM for optical actuation, which is quite different from conventional electrically actuated CNN. In [25], a DDNN-based all-optical signal processing platform was proposed to manipulate multiple vortex beams. DDNN model was trained to implement different functions including hybrid OAM modes generation, recognition and transformation. Furthermore, the trained DDNN model implemented all-optical signal processing communications, such as OAM shift keying, OAM multiplexing and demultiplexing, and OAM mode switching. The results show that the DDNN model has a strong ability to manipulate light fields. Compared with DDNN-based all-optical signal processing method for vortex beams, the proposed scheme also exploits the capability of DDNN for OAM light field manipulation. The innovation is that the proposed scheme alleviates the interference of OAM mode propagation in UWOC system by OT. In addition, the proposed scheme exhibits excellent compensation performance for the distortion of OAM caused by OT in experiments.

In this work, we have proposed a DDNN-based AO compensation scheme to recover the distortion on the vortex beam caused by OT. The DDNN model has been trained to obtain the corresponding relationship between the intensity pattern of the distorted vortex beam and the representing OT phase screen. When the intensity pattern of the distorted vortex beam is input, the trained DDNN model outputs the predicted phase screen. The predicted phase screen is reversed to represent compensation phase screen, so that the distortion of the vortex beam can be compensated. We have verified the ability of the DDNN-based AO scheme to compensate for the distorted vortex beam through experiments, and compared it with the CNN-based AO compensation scheme and the GS algorithm. The results show that the AO scheme based on DDNN can compensate the distortion on the vortex beam caused by OT, especially in the strong turbulence environment. The mode purity of the vortex beam after compensation is improved to 0.980 for the strong OT with $C_{n}^{2}=1\times {{10}^{-12}}{{\text {K}}^{\text {2}}}{{\text {m}}^{\text {-2/3}}}$. Also, the proposed scheme has the advantages of low response time and fast convergence of the loss function. In short, this scheme demonstrates the combination of diffractive deep neural network and adaptive optics, which can compensate for the distortion of the vortex beam caused by OT in time, and provide help for the development of the UWOC system in the future.

Funding

National Natural Science Foundation of China (61871234, 62001249); Postgraduate Research and Practice Innovation Program of Jiangsu Province (KYCX200718).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Diffractive deep neural network based adaptive optics scheme for vortex beam in oceanic turbulence

Abstract

1. Introduction

2. Theory

2.1 DDNN-based AO scheme

2.2 Network structure

3. Performance evaluation

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (11)

Tables (1)

Equations (12)

Optics Express