Simultaneously sorting vector vortex beams of 120 modes

Polarization (P), angular index (l), and radius index (p) are three independent degrees of freedom (DoFs) of vector vortex beams, which have been widely used in optical communications, quantum optics, information processing, etc. Although the sorting of one DoF can be achieved efficiently, it is still a great challenge to sort all these DoFs simultaneously in a compact and efficient way. Here, we propose a beam sorter to deal with all these three DoFs simultaneously by using a diffractive deep neural network (D$^2$NN) and experimentally demonstrated the robust sorting of 120 Laguerre-Gaussian (LG) modes using a compact D$^2$NN formed by one spatial light modulator and one mirror only. The proposed beam sorter demonstrates the great potential of D$^2$NN in optical field manipulation and will benefit the diverse applications of vector vortex beams.


INTRODUCTION
Since 1992, Allen et al. discovered that a light beam with the helical phase structured of exp(il ϕ) carries an orbital angular momentum (OAM) of lh per photon (l, named the topological charge, which is usually an integer number) [1].Unlike spin angular momentum (SAM), OAM, in principle, has infinite orthogonal eigenstates indicated by the integer l.OAM beams have been widely adopted in various applications with unprecedented performances due to the merit of the helical phase structure, including optical trapping [2][3][4][5], resolution enhanced microscopy [6][7][8], nonlinear optics [9,10], high-dimensional quantum states [11][12][13][14], and high-capacity optical communications [15][16][17][18][19][20][21].As the most representative OAM mode, the Laguerre-Gaussian (LG) mode has not only the topological charge l, but also the radius index of p (a non-negative integer), which is also very important both in theory and application [13,[22][23][24][25].As a vector field, except for l and p, the polarization P of a light beam is also a vital freedom, which cannot be ignored in practice.
The sorting of vector vortex beams (VVBs) according to the indexes of l, p, and P becomes a prerequisite operation in the various applications mentioned above.When only one of the indexes is considered, it is not difficult to sort.For example, the polarization P can be separated efficiently using a polarization beam splitter (PBS).Recently, the sorting of angular index l has attracted much attention, such as using optical geometric transformation [26][27][28][29][30][31], which only uses less than two phase modulators.It is worth noting that using metasurface can sort both the polarization P and angular index l simultaneously [30].However, it is not satisfactory for the radius index p, because p is mainly reflected in the intensity.The Gouy phase of LG mode is related to l, and p, which provides a potential method to sort l and p simultaneously [32][33][34].Unfortunately, however, the Gouy phase is degenerate for the angular index of ±l, which means this method can not sort l and −l directly.Alternatively, using the combination of Dove prisms and PBSs can sort l also [35], which generally requires a system cascaded by many units.This means that the system's complexity increases rapidly with its sorting capability, making it unsuitable for a larger mode number.Therefore, up to now, it is still a great challenge to sort all three indexes with high efficiency, low crosstalk, and a compact system.
In such a circumstance, based on the diffractive deep neural network (D 2 NN), we theoretically propose a vector vortex beam sorter and experimentally demonstrate the sorting of 120 modes (i.e., l from −7 to 7, p from 0 to 3, and P for |H and |V ).Our key idea is to transform a polarized diffractive deep neural network into two scalar diffractive deep neural networks.First, we use a calcite crystal to split the orthogonal polarization (|H and |V ).And then, after the half-wave plate, |V changes into |H .Last, we use two 5-layer D 2 NNs consisting of a spatial light modulator and mirror to sort LG modes from two channels.We demonstrated the output results of several vector structured beams, tested the robustness in atmospheric turbulence, and obtained satisfactory results in the experiment.We believe this sorter benefits the applications of vector structured beams.

METHODS
In 2018, Lin et al. proposed the diffractive deep neural network (D 2 NN) [36].After that, D 2 NN has made great progress in image processing [37][38][39][40][41][42][43][44][45][46].Besides, due to the photon owned more degrees of freedom (i.e., polarization, wavelength, mode), these DoFs provide a new way of thinking [47][48][49][50][51][52][53][54][55][56][57].Normally, D 2 NN theoretically has the ability to process polarization information.However, corresponding polarization control units are required in practice.Using metasurfaces seems like a good option, which modulates both polarizations and enables a system-on-chip [31,56].Unfortunately, in the experiment, it is a huge challenge to align the adjacent layer, and the transmittance is also regrettable for multi-metasurface in the visible spectrum.Fortunately, we have a more feasible method for vector structured beams using the spatial light modulator to achieve this sorter.Without loss of generality, the vector structured beam can be described as, where α LG p l is the Laguerre-Gaussian mode with circular symmetry, which are the earliest reported vortex beams carrying OAM and described as [1], LG where C p l is the normalization constant, k is the wave number, l is the angular quantum number, and p is the radial quantum number.ω(z), ψ(z) and z R are where λ is the wavelength and ω 0 is the basement membrane waist radius.Notoriously, the |H and |V are orthogonal.Thus, we can separate |H and |V first, which is easy to achieve by polarizing elements (i.e., polarizing beam splitter and calcite crystal).Then, we sort LG modes from two channels.This way, we transform the polarization diffractive deep neural network into two scalar diffractive deep neural networks.
For the scalar diffractive deep neural network, according to the angular spectrum theory, the propagation of an optical field can be described as [58] where F is the operator for the 2D Fourier transform, and correspondingly, F −1 represents the inverse 2D Fourier transform.
Here, k is the angular wavenumber of the fields, k x and k y are spatial frequencies.Moreover, the backward propagating and gradient descent algorithms are used to train the diffractive deep neural networks (more details shown in Supplement).

RESULTS
For the vector structured beam sorter, we first choose the LG modes and Gaussian spots as the input train set and the target output train set of the LG mode sorter.For the train LG set, angular quantum index l is from −7 to 7, and radius quantum index p is from 0 to 3 (total of 60 modes).For D 2 NN, there are a total of 5 neural layers, and each layer has 310 × 420 optical neurons with the neuron size of 8 × 8 µm (matched the pixel size of SLM in the experiment), and the wavelength λ is 532 nm.Moreover, the distance of each layer d is 3.51 cm, and the distance from the last layer to the output plane d 1 is 9.3 cm.After training, the result is shown in Fig. 1.As shown in Fig. 1, the D 2 NN can sort the LG mode after training.Furthermore, we show the more detailed propagation of the LG 3  4 , and the propagation distance of the adjacent frame is about 3.2 mm in Video1.Apart from Video1, we also show the in Video2 and Video3.These videos show that the D 2 NN recognizes the feature of LG modes and outputs it at the corresponding position with low crosstalk in simulation.For the VSB sorter, we use two LG mode sorters to process LG modes from two polarization channels |H and |V .In order to verify the performance of the vector structured beam sorter in practice, we built a corresponding experimental system.Fig. 2 shows the schematic setup of the VSB sorter used in our experiment.The incident Gaussian laser beam with a wavelength of 532 nm is expanded and collimated by Lens 1 , Lens 2 , and Iris 1 .Followed by the polarizer P, the polarization of the incident beam is changed into |H .Then, two forked phase hologram is loaded on SLM 1 (Holoeye PLUTO-2) to generate two exiting LG beams with various l and p, after the pinhole filtering (Iris 2 ) and 4-f system (Lens 3 and Lens 4 ).Through the HWP, we transformed the polarization of one of the beams from Fig. 2. Experimental setup of vector structured beams sorting using the D 2 NN.The Gaussian laser beam with a wavelength of 532 nm is expanded and collimated by Lens 1 and Lens 2 and followed by a polarizer P. Then SLM 1 converts the Gaussian beam into two beams, corresponding to two orthogonal polarization states.After 4-f system, using HWP changes the polarization of one beam from |H to |V .Then using a calcite crystal, combines two orthogonal polarization beams into one.Last, the input vector structured beams are sorted by D 2 NN, consisting of a calcite crystal, HWP, SLM 2 , and M. In the blue dotted line, the input vector structured beam is measured after the linear polarizer aligned at 0 • , 90 • , 45 • , and 135 • .In the red dotted line, the output result is the channel |H and |V successively.List of abbreviations: non-polarizing beam splitter (NPBS); spatial light modulator (SLM); half-wave plate (HWP); mirror (M); and polarizer (P).
|H to |V .Finally, we combined the two orthogonally polar- ized LG beams into one using a calcite crystal and obtained the corresponding VSB.As we pointed out earlier, we use a calcite crystal and HWP to separate the two orthogonal polarization (|H , |V ) and transform |V into |H , corresponding to the upper and lower D 2 NNs.Here, transforming |V into |H is because the SLM only modulator the |H .Then, we use SLM 2 and a plane mirror M to compose two 5-layer D 2 NNs by dividing SLM 2 into 10 regions (as shown in Supplement).The numerically solved phase patterns are loaded in the 5 upper parts of SLM 2 .Moreover, the 5 lower parts of SLM 2 are the same.Using this configuration, one SLM is enough to realize the two 5-layer D 2 NNs shown in Fig. 1, which is very convenient in the experiment.
Fig. 3 shows this system's vector structured beam sorting experimental results.In the upper two lines of Fig. 3, we show the input VSBs measured after the linear polarizer aligned at 0 • , 90 • , 45 • , and 135 • .Furthermore, in the lower line, we show the sorting results.The output channels of polarization are the |H and |V channels from top to bottom.Furthermore, for the output channels of LG mode, the channels of angular quantum index l (from bottom to top) are from −7 to 7, and the channels of radial quantum index p (from left to right) are from 0 to 3. In Figs. 3. (a 1 -a 5 ), we show the input and output result of VSB |ψ in = LG 1  3 |H + LG 2 −2 |V .The vector structured beam sorter can separate the different VSBs in the designed positions, and the crosstalk of different channels is low.Moreover, we also test the system's performances for incident multi-mode (up to 8 modes) and get satisfactory results, as shown in Figures.3.

DISCUSSION
Besides multi-mode results, the crosstalks between different channels are also critical in practice.Compared with the forked grating and Dammann grating, our method only gets the output quasi-Gaussian beam without other outputs.In Fig. 4, we show all the normalized energies W l ,p ;l,p from two polarizations, which is the measured energy of LG p l mode normalized by the incident energy of LG p l mode.All the other parameters used in Fig. 4 are the same as those in Fig. 3. Fig. 4 shows that the energy efficiencies of the sorting are high, and the average value of E l,p;l,p is about 99.43%.The average crosstalk of W l ,p ;l,p with l = l and p = p is near 0%.The maximum crosstalk does not exceed −12dB.It is important for the applications of VBS.It is noted that the crosstalk is mainly due to the aberrations accumulated when passing through the SLMs.In principle, these errors can be compensated using several methods or the Zernike function to generate the compensated phase [39,40].
In practice, it is a challenge to sort the distorted LG beams [59,60].Here, we use the distorted VBSs to test the sorter's accuracy (more details shown in Supplement).Compared with ordinary VSB, the crosstalk of the system affects its use.The system's error rate is above 23%, and most errors occur in l.For p, the error rate is about 5%.Due to atmospheric turbulence, the intensity and phase have been distorted, which changes the mode of VBS.Although the system cannot obtain satisfactory results for distorted beams, which can use adaptive optics to compensate for the distorted wavefront [59,60].
We presented a vector structured beam sorter based on the diffractive deep neural network (120 modes in total).Compared with optical geometric transformation, our method can simultaneously sort polarization P, the angular quantum index l, and the radius quantum index p.The vector structured beam sorter can be separated into two LG mode sorters.After training, D 2 NN can recognize the feature of LG modes (l and p) and output it at the corresponding position with low crosstalk in simulation.
To achieve this sorter in the experiment, we firstly use a polarizing splitter system to separate two orthogonal polarizations (|V , |V ) and change |V to |H , consisting of a calcite crystal and HWP.Furthermore, we use a spatial light modulator and mirror to form two 5-layer D 2 NNs in the visible spectrum of 532 nm.Moreover, the crosstalk of the system is less than −12dB.We believe our vector structural beam sorter would because of a beneficial component in practical multi-dimensional multi-

p l and β p l
are normalization coefficient.|H and |V mean horizontal polarization and vertical polarization, respectively.

Fig. 1 .
Fig. 1.Schematic for the sorting of LG beams with angular quantum index l and radius quantum index p realized by 5 phase planes of Layer 1,2,3,4,5 , the distance of each layer d is 3.51 cm, and the distance from the last layer to the output plane d 1 is 9.3 cm.For clarity, the red round shows the enlarged view of the output beam.

Fig. 4 .
Fig. 4. Normalized energy W l ,p ;l,p in the sorting of the 60 modes for each position, which is the measured energy of channel LG p l mode normalized by the total energy of output channels for incident mode LG p l .Here l and l change from −7 to 7, and p and p change from 0 to 3. All the parameters are the same as those in FIG. 3. (a) is measured in |H polarization.(b) is measured in |V polarization.