Abstract

A kind of compact all-optical learning-based neural network has been constructed and characterized for efficiently performing a robust layered diffractive shaping of laser beams. The data-driven control lightwave strategy demonstrates some particular advantages such as smart or intelligent light beam manipulation, optical data statistical inference and incident beam generalization. Based on the proposed method, several typical aberrated light fields can be effectively modulated into the desired fashion including the featured flat-top beams, an arrayed sub-beam arrangement and complex annular fringes compared with conventional GS-based DOEs. An actual THz laser is utilized to evaluate the effectiveness of the method developed.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Generally, the undesirable spatial intensity distribution of laser beams demonstrates several adverse results in many applications. To typical high-energy laser processing systems, the non-uniform or even distorted beams will obviously decrease the focusing and manufacturing efficiency by generating unexpected or detrimental thermal effects and therefore remarkable micro-structural damage of workpieces, so as to seriously affect the fabrication quality [1]. Currently, the imperfect Gaussian beams out from common terahertz (THz) lasers for performing THz super-resolution imaging detection [2] are still indispensable. Other main applications such as adaptive optical imaging [3], laser radars [4], optical communication [5], laser direct writing [6], biomedical imaging detection [7], and near-field optical storage [8], all exhibit their individual requirement for constructing particular functional lightwave mode based on needed beam spatial distribution with relatively high uniformity, for example, special wavefronts, designated light field morphology, patterned light intensity or amplitude arrangement, and so on. Considering the situation of the typical beam deformation or even distortion originated from the imperfect optical sub-systems or components or atmosphere turbulence, several beam shaping approaches such as the aspheric refractive lens group [9], diffractive or refractive micro-lens arrays [10], functioned metal micro-waveguides or micro- cavities [11], or conventional diffractive optical elements (DOEs) [1214], are often utilized for correcting or even modulating incident light beams into demand appearance. The common DOEs that incorporate the computer-based optimized algorithm for efficiently configuring wavelet retardance, can be used in a relatively wide visible or infrared wavelength range to effectively perform wavefront transformation using integrated phase arrangement. And it will lead to a unique lightwave amplitude or intensity dispersion attributing to a relatively portable phase arrangement solution. The operation above means that an efficient target pattern emission for imaging detection or beam shaping can be achieved by only employing one or more smart phase plates corresponding to a given light field or a laser beam fashion.

Nevertheless, the light intensity distribution over the observation plane corresponding to a common laser source will show an aberrated Gaussian formation resulting from several factors such as the variance of the physical conditions or circumstance situation, the equipment or component vibration, the misaligned optical axis caused by artificial factor or electromagnetic disturbance. And the aberrated Gaussian will make the passive conventional DOEs with fixed phase structure invalid to a certain extent. It should be indicated that the conventional DOEs, as a kind of integrated phase assembly for basically retarding the incident wavefronts and therefore nicely adjusting or modulating the far field beam distribution or formation or propagation according to the scheduled light field pattern or beam mode, can be modeled by a set of appropriate scalar formulation obtained through mature iterative Fourier transformation algorithm (IFTA), for instance, the typical Gerchberg-Saxton (GS) algorithm [15] or the stochastic search-based global convergence algorithm [16]. Usually, they are used to resist disturbance by active methods [1719]. However, they still exhibit little ability to deal with some aberrated lightwave inputs by conventional passive DOEs. Furthermore, there are some effective optical strategies or configurations for beam shaping [2031], which can achieve high-quality imaging, but they are also not robust for beam disturbances. Currently, a new kind of optical diffractive deep neural network (D2NN), as an optical machine learning framework for performing a fast and adaptive or even intelligent optical process in several applications including optical logic operations [32], optical pattern recognition [33], salience detection [34], pulse generation [35], and so on, has been introduced for solving the problems mentioned above. We also demonstrate an all-optical weight-noise-injection network framework (SRNN) for remarkably improving the noise resistance of diffractive optical classifier by augmenting the robustness of layered diffractive phase assembly [36].

In this paper, a compact passive light beam shaping strategy based on a kind of all-optical neural network (AONN) for an advanced THz imaging detection significantly improves the beam shaping capacity to unplanned or randomly distorted or off-centered Gaussian light fields compared with conventional IFTA, is proposed. Taking advantage of optical diffractive networks, the proposed passive robust beam shaping system is different from conventional adaptive optical systems, which realize the robust response to a variety of distortion input light fields by the fixed phase structures not to be adjusted. Following the training operation with respect to aberrated incident beams and the corresponding output light fields desired, the AONN presents a capability of being immune to the multiple aberrated inputs to a certain extent using deep learning methods, for instance, the stochastic gradient descent and error-backpropagation, and thus refrained from over-fitting like IFTA in virtue of only iterating on a single light field sample with constant modulus assumed. As a consequence, a relatively compact but more robust beam shaping architecture is exhibited, which can be practically constructed by common methods such as 3D-printing [37], femtosecond photon direct writing [38] or conventional KOH wet corrosion. Especially, a two-layer optical diffractive network, which has been proven to significantly improve the beam shaping robustness compared to conventional GS-based DOEs, can be achieved by a single sheet of substrate through respectively profiling a single diffractive layer on each side of the substrate, so as to shape a matched dual-cascaded control lightwave phase assembly with relatively high transmittance. And considering the situation of above process already limiting the fabrication precision, both the number of neurons and the area of each layer can not be set arbitrarily. In addition to ordinary aberrated Gaussian beams, the fundamental design approach suggested here can be readily adapted to diversely-distributed Gaussian random noise generated individually as well as the extreme pathological inputs, for example, a drastic shift of the light path caused by violent shaking, through the deeper diffractive neural network with superior robustness. Furthermore, we use the trainable diffractive layers in tandem that form a small footprint and a compact projection system to achieve a large-angle anti-noise Gaussian lightwave splitter. Similarly, it presents a relatively high signal-to-noise-ratio or signal-to-clutter-ratio and a beam shaping robustness compared to conventional GS-based DOEs.

2. Proposed method

To construct a kind of all-optical learnable diffractive component set in tandem, we have established a multilayer optical neural network framework for obviously improving the beam shaping robustness to aberrated incident light fields. Both architectures corresponding to a conventional IFTA and a diffractive neural network training strategy have been demonstrated in Fig. 1, respectively. The spatial spectrum method that is an approximation of the scalar diffraction theory expressed by Rayleigh-Sommerfeld diffraction formula to encode Fourier-domain inputs for calculating the complex amplitude distribution, which can well support a fast parallel operation of Graphics Processing Unit (GPU), are utilized in both methods above. The conventional iterative Fourier transformation strategy through multiple iterations of the forward Fourier transformation and the inversion processing with a pre-defined input light field for obtaining the phase details of the DOE is shown in Fig. 1(a). An optical diffractive neural network is established to collectively implement a layered lightwave diffraction transformation for achieving an advanced data-driven beam shaping based on adaptive datum estimation (Adam) and error-backpropagation, as shown in Fig. 1(b). The basic features are as follows.

 

Fig. 1. Operation principles of the IFTA and the AONN

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A hundred third-order super-Gaussian beams with varying levels of distortion and offset centering are generated randomly, which are utilized as training data patterned by different incident beam appearance according to the profile and spatial location. Usually the hyperparameters of the AONN are closely related to the experience acquired. Considering the fabrication difficulty and the connection degree between the layers, the number of neurons in each layer is set to 200×200 with a spacing of 3 cm, and also a 400 GHz laser source is used. In Fig. 1, several AONNs including the two-layer and the three-layer and the five-layer for 1500 epochs using the same data set with similar Epoch-Loss variance trend are trained, respectively. Where one epoch means that all samples in the training set are used to train the network once, the loss refers to the loss between the predicted light field and the target light field calculated by the loss function [37]. The training time of the AONNs above is 35 min, 48 min, and 73 min, respectively. The networks present a gradually improved property, as indicated by both green inserts and the Epoch-Loss curves shown in part-b of the figure. All in all, the beam shaping operation based on the AONN exhibits a robust phase modulation process compatible with multiple types of distorted incident light fields, as shown by patterned training data. Furthermore, the proposed method benefits from the learnable attributes of the neural networks, and remarkably ameliorates the distress that the conventional IFTA is only suitable to a single goal light field and then implement a decentralized modulation for more types of incidents. The diffractive neural network is implemented by using Python 3.85 and the framework of TensorFlow (r2.30 version) on a PC with Intel Core i7-9700H CPU (3.00GHz), 48 GB of RAM, and the GeForce RTX 2070 (NVIDIA).

2.1 Principle

To the optical diffractive neural network, the layered propagation of lightwaves can be described by the relations [37] of

$$m_i^l = {e^{\textrm{j}{\varphi _2}}},$$
$$w_i^l = \frac{{{z^l} - {z^{l - 1}}}}{{{r^2}}}\left( {\frac{1}{{2\mathrm{\pi }r}} + \frac{1}{{\textrm{j}\lambda }}} \right)\textrm{exp} \left( {\frac{{\textrm{j}2\mathrm{\pi }r}}{\lambda }} \right),$$
$$r = \sqrt {{{(x_i^l - x_j^{l - 1})}^2} + {{(y_i^l - y_j^{l - 1})}^2} + {{(z_i^l - z_j^{l - 1})}^2}} ,$$
$$output_i^l = w_i^l \times \sum\nolimits_{k} output_k^{l - 1} \times m_i^l = w_i^l \times |A |{e^{\textrm{j}{\varphi _1}}} \times {e^{\textrm{j}{\varphi _2}}} = |{{A_w}} |{e^{\textrm{j}\Delta \varphi }},$$
where $m_i^l$ is the modulation of the i-th neuron of the layer l, and ${\varphi _\textrm{2}}$ represents the phase change of the wavefront caused by the phase plate, and $w_i^l$ is the Rayleigh-Sommerfeld diffraction between the neurons in both the layer l and l-1, and zl denotes the coordinate of the l-th layer on the axis-z, and r denotes the distance between the i-th neuron of the layer l and the j-th neuron of the layer l-1, and $output_i^l$ is the output lightwave of the i-th neuron of the layer l, and ${\varphi _\textrm{1}}$ represents the phase change of the transportation wavefront due to lightwave diffraction, and both A and Aw are the amplitude of the output lightwaves corresponding to the layer l-1 and l, respectively.

2.2 Loss function and evaluation index

Generally, the loss function of the optical neural network in the training period has three basic components including the fundamental loss term ${\overline \Sigma _m}({RMS{E_m}} )$ used to penalize the mismatch between the desired light field and the network output, and the uniformity loss term and the aberrated factor (AF) for evaluating the aberrated degree of incident light fields. They satisfy a common relation of

$$Loss\textrm{ = }\frac{1}{{MF}}(\alpha {\overline \Sigma _m}({RMS{E_m}} )\textrm{ + }\beta {\overline \Sigma _m}(N{U_m})),$$
where both α and β are the hyper-parameters used to adjust the proportion of each loss term mentioned, so as to make the beam shaping network pay more attention to the uniformity of the output in some special occasions, where the uniformity is full priority.

Furthermore, several related parameters including RMSEm and NUm and DEm, where m means the m-th light intensity distribution, have also been employed as the evaluation index for evaluating the beam shaping efficiency based on the layered lightwave diffractive processing, which is defined as [35]

$$RMS{E_m} = \sqrt {{\Sigma _p}{{({I_m} - {I_0})}^2}\textrm{/}{\Sigma _p}I_0^2} ,$$
$$N{U_m} = \sqrt {\mathop{\sum}\nolimits_{p \in W}{{[({I_m} - \overline {{I_m}} )/\overline {{I_m}} ]}^2}} ,$$
$$D{E_m} = \frac{{\sum_{p \in W} {I_m}}}{{\sum_{p} {I_m}}},$$
where Im indicates the estimated the m-th light intensity distribution reconstructed by the diffractive network, and I0 is the 12-order super-Gaussian field intensity distribution of the target beams selected, and RMSEm represents the m-th root mean square error between both Im and I0, and p expresses the number of the imaging pixel, and NU means the non-uniformity of the flat-top area W of the modulated light field, and $\overline {{I_m}} $ denotes an average light intensity over the flat-top associated closely with each Im, and DE is the diffraction efficiency. The flat-top area of the 12-order super-Gaussian beam I0 of the target beams is defined as
$${I_\textrm{0}}\textrm{ = }{I_n}\textrm{exp} [ - 2{\left( {\frac{{{x^2}}}{{{R_0}^2}}} \right)^{\textrm{12}}} - 2{\left( {\frac{{{y^2}}}{{{R_0}^2}}} \right)^{\textrm{12}}}],$$
where In and R0 are the normalized light intensity and the beam waist radius, respectively. Through adjusting the order of the super-Gaussian beams, the super-Gaussian function can be varied from common Gaussian mode to the flat-top mode discussed above. The definition of the third-order super-Gaussian equation is as follows, where the parameters of a, b, x0 and y0, are variable, and n denotes the added noise. By randomly changing the above parameters within a limited range, we can generate a training set about incident Gaussian beams with off-axis and contour distortion and noisy.
$$I\textrm{ = }{I_n}\textrm{exp} \left[ { - 2{{\left( {\frac{{{{(ax - {x_0})}^2}}}{{{R_0}^2}}} \right)}^\textrm{3}} - 2{{\left( {\frac{{{{(by - {y_0})}^2}}}{{{R_0}^2}}} \right)}^\textrm{3}}} \right] + n,$$

2.3 Aberrated factor

The constructed AONN in this work has no nonlinear component and economized the network layers to improve its transmittance and decrease the physical system error such as the misalignment error and the layer spacing error. Therefore, its expression ability is relatively weak compared to the deeper diffractive neural networks. So, we defined an aberrated factor (AF) by calculating the degree of the distortion as well as the misalignment of the input beams to optimize the training process. Every Gaussian lightwave sample generated by modifying the parameters of the super-Gaussian function independently, will demonstrate a different level of the beam deformation labelled by AF. It is set in computer-based training, which is used to weight the loss of every lightwave sample in the process of the gradient descent and error backpropagation.

The equation we defined is shown in the relation of

$$A{F_m} = \frac{\textrm{1}}{{{c_m}}} + {d_m},$$
where the m-th aberrated factor AFm is constituted by the circularity ${c_m} = 4\mathrm{\pi }S/{L^2}$($c \in (0,1)$) of the m-th input light field as well as the offset distance ${d_m} = \sqrt {{{(x - {x_i})}^2} + {{(y - {y_i})}^2}} $, for example, the distance between the center $(x,y)$ of the misalignment light field and an ideal center $({x_i},{y_i})$ of the m-th Gaussian bema sample. In training stage, we force the network to reduce the weight of the serious problematic Gaussian light field corresponding to a larger aberrated factor, and then pay more attention to trifling distorted and off-center Gaussian lightwaves corresponding to a smaller aberrated factor by using a two-layer linear network to deal with complex shaping problems.

Through utilizing additional adjustable parameters above, the optical neural network can be numerically achieved by an average RMSE of $0.092 \pm 0.034$ and a NU of $9.27 \pm 2.36\%$ over the beam reformation corresponding to a aberrated Gaussian light field generated randomly, so as to prevent extremely aberrated but rare Gaussian samples from dragging down the network learning of other samples with mild symptoms compared to the average RMSE of $0.102 \pm 0.015$ and the NU of $10.89 \pm 0.88\%$ of non-factor training. In other words, the evident standard deviation according to the AF-based training results demonstrates a discrimination treatment of improved AONN to distinct aberrated incident beams.

3. Results and discussions

To fully exhibiting beam shaping efficiency and basic characters through the optical neural network constructed by us, an experimental approach is established, as shown in Fig. 2. According to the experiment layout, the beams of 400 GHz Gaussian lightwave of 40 mW and 100 mW are generated by a Coherent SIFIR 50 THz laser. They first go through a diaphragm and then project upon a GS-based phase plate, which is fabricated over a silicon wafer by a single mask UV-photolithography and a dual-step KOH etching (see Supplement 1 S1) [39,40]. And then a diffractive light field, which is already modulated by the phase plate, is formed over the observation plane and finally imagined by a Spiricon's Pyrocam III ultra-wide spectrum camera, as shown by Fig. 2(a).

 

Fig. 2. THz beam characters and their profile details corresponding to two types of optical diffractive architectures. (a) Light path for performing THz pattern emission using a single GS-based phase mask and a two-layer AONN. (b) Typical THz light fields of 40 mW and 100 mW out from the same THz laser source and then beam appearances including a 3×3 sub-beams and an inverted triangle, and three sub-beam pattern and digital 1 based on a single GS-based DOE manufactured by common KOH wet corrosion. (c) Laser beams shaped by a GS-based single phase plate and a two-layer diffractive neural network, where the output light field of the first layer mask is indicated by a black dashed frame. (d) Comparison between the flat-top beams shaped by a GS-based DOE indicated by the left photography and the constructed AONN according to the simulation demonstrated by the right photography.

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The incident light fields of 40 mW and 100 mW from the same terahertz source demonstrate different brightness as well as different distortion profile of the output THz beam, as shown by both the top and bottom left photographs in Fig. 2(b). By using conventional GS-based DOEs, whose default input is an ideal 3-order super Gaussian beam, the 40mw light field is converted to a 3×3 sub-beam array and a basic inverted triangle, and also the 100mw light field to three sub-beams and digital 1. Apparently, the GS-based phase plate cannot be used to handle those noisy elongated Gaussian light fields disturbed by physical condition variance and artificial factors, as shown in Fig. 2(b). The obvious far field beam deformation generated as above can be attributed to a relatively large change of the THz lightwaves power, which also increases the difficulty of efficiently performing fine phase selection and wavefront modulation for acquiring ideal THz beams. The given solutions through different diffraction architectures, shown in Fig. 2(c), illustrate a basic consideration about the featured phase configuration and thus light field modulation based on the GS-based strategy and a two-layer optical diffractive network, respectively. The phase fluctuation range is restricted from 0 to 2π, and then the lightwaves amplitude from 0 to 1 by conducting data normalization for more clear comparison.

Based on both the GS-based phase plate and the layered diffractive optical assembly already being optimized and trained for 1500 epochs, individually, a 400 GHz monochromatic aberrated Gaussian beam is used to evaluate the beam shaping strategies mentioned above. As shown in Fig. 2(c), a typical shifted and squashed incident light field, which neither appear in the GS algorithm iteration routing nor the diffractive neural network training stage is the incident field. According to the beam shaping processes demonstrated in Fig. 2(d), the conventional GS-based DOE will suffer a relatively severe disturbance or light signal shock where a sparse light field shown by the bottom left subplot, exhibits a district aberrated appearance leading to an intolerable loss of the lightwave intensity or amplitude distribution upon the observation screen. Due to the GS algorithm's intrinsic mechanism, the GS-based DOEs are usually used to recover the phase validity to relatively fixed input lightwaves for constructing a pre-supposed output light field. And instead, through taking advantage of excellent generalization and robustness of the optical neural networks, the RMSE of 0.075 and the NU of 3.27% can be achieved by the trained diffractive optical network. It should be credited to the layer-to-layer cooperation that the first layer cleverly disperses the wrongly concentrated light field in the central region and neighborhood, and then being rebuilt by the next functioned layer, which is much similar to an encoding and decoding process.

Our simulation results based on a conventional IFTA-based single phase mask and a two-layer AONN for efficiently shaping multi-species THz beams with distinct aberrated light field components involving several typical lightwave imperfections such as boundary distorted Gaussian lightwaves and off-centered Gaussian lightwaves and noisy distorted Gaussian lightwaves, are shown in part a of Fig. 3. After passing the IFTA-based single phase mask and a two-layer AONN, the multiple incident light of the first column on the left are shaped into the corresponding light intensity distribution by two fixed physical diffractive structure. Where the results for GS-based DOE are marked by (a-1) to (e-1), the results for optical network are marked by (a-2) to (e-2), and the small inserts are the output light field of the first layer of network. The left color bar is a normalized light intensity indicator. The corresponding light intensity distribution along a cross-sectional curve going through the geometric center of modulated results are given in part b of Fig. 3. The red and blue curves demonstrate the beam shaping results of the GS-based DOE and the two-layer AONN, respectively. The average level of the top small region of the blue intensity distribution with an obvious fluctuation is marked by a gray dotted line. Even though lacking of the beam shaping generalization, the GS-based DOE can still be effectively utilized to perform an almost perfect adjustment or modulation to some given incident beams by stimulating the phase mask being rebuilt, so as to iteratively undergo an optimized process shown by the subplot set (a), and also the final light field illustrated by (a-1), where the incident lightwave is default during running GS algorithm. Almost similar situation can also be viewed according to the beam appearances and profiles shown by (a-2), but already presents a slight beam deterioration because of the average beam intensity or normalized height of 0.84 being slightly lower than that only achieved by the GS-based DOE, and then appears a weak fluctuation over the flat top of the light intensity curve and a slight increase of both the parameters of RMSE and NU. It should be noted that there is almost no high-order diffraction loss of the final beams achieved by the AONN. While the beam shaping is immediately reversed with respect to the subplot set (b) to (e), which are more prone to come out in optical experiments because of existing internal and external disturbances compared with the ideal case shown by the subplot set (a). In addition, the measurement results have been numerically validated once the optical axis shifting (off-center) degree of the perfect Gaussian light field shown in the subplot set (a) being greater than 4%, which means that the limited advantage of the GS-based DOE will lose. Similar situations will also occur when the circularity of the beams is less than 0.73, where the RMSE and the NU of both the approaches are (0.059,0.065) and (7.9%,7.1%), respectively.

 

Fig. 3. Beam shaping efficiency comparison of the IFTA-based DOE and the two-layer AONN. a Comparison of intensity distribution for multiple input. (a) The ideal 3-order super-Gaussian beam, and (b) to (e) the typical problematic Gaussian beams with different aberrated characters, for example, optical axis shifting (off-center) and random noise importing. The evaluation parameters including the RMSE and the NU and the DE are inserted in the upper left corner of each beam shaping diagram. b Cross sections of corresponding modulated intensity distribution. Where both the red and blue curves demonstrate the beam shaping results of the GS-based DOE and the two-layer AONN, respectively.

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According to the light field characters shown by Figs. 3(b-1) to 3(e-1), the beam appearances and profiles shaped by the GS-based DOE already present a gradually deteriorated trend, when the different aberrated incident beams are guided into beam shaping operation. As shown by the subplot set (b) to (d), the conventional GS-based DOE attached by an average RMSE of $\textrm{0}\textrm{.119} \pm \textrm{0}\textrm{.076}$ and an average NU of $\textrm{11}\textrm{.65} \pm {7.26\%}$, arduously deals with the distorted incident beams and thus the evaluation index will also deteriorate sharply. An extreme case without any effective phase adjustment is demonstrated by the second diagram from the left of the subplot set (e). While the AONN can be easily used to realize an average RMSE of $\textrm{0}\textrm{.095} \pm \textrm{0}\textrm{.015}$ and an average NU of $\textrm{6}\textrm{.27} \pm {1.54\%}$, so as to show a particular meliority that a robust data-driven phase or wavefront modulation strategy for efficiently constructing an ideal monochromatic light field is suitable to inconstant or changeful incident beams. The above manipulation can be easily realized by an advanced optical neural network, which actually benefits from the statistical inference and generalization capabilities of existing computer-based neural networks. The first layer of the designed AONN has fully adjusted the offset center of the incident beams to the geometric center and remarkably reduced the beam inclining distortion generated, as shown by small inserts attached to each diagram at (a-2) to (e-2). And then the output light field from the first layer is continuously modulated by the second layer so as to lead to a demand beam appearance and profile. With the aberrated level of incident beams increasing as shown from the subplot set (b) to (e), the beam shaping efficiency of the GS-based DOE is decreased gradually and even lost finally. But the AONN always exports the needed beams in spite of the average beam intensity being decreased apparently from the maximum value of 0.84 to 0.56, as indicated by gray dotted lines in part b of Fig. 3. As mentioned above, the two-layer AONN also conducts a relatively weak shaping to severe aberrated light fields as shown by the diagram in the subplot set (e). We can expect that the operation of introducing more layers will be a promising solution. However, it cannot be ignored that the increase of the number of phase plates in tandem also leads to an increase in background noise to a certain extent, as shown by the curves in part b of Fig. 3, which might be alleviated by improving training strategy [36]. And it should be noted that a two-layer AONN output through a trained fixed phase structure already reveals clearly an insensitivity to aberrated input beams, and then a collective control can thus be conducted due to a passive diffractive layer assembly.

So, the detailed beam shaping processes of the linear AONN with different number of diffractive layer or functioned depth, which defines its statistical inference and generalization capability, are illustrated in Fig. 4. As shown in this figure, the adjustment or modulation manipulations of incident lightwaves that are carried out by the AONN with two-layered and three-layered and five-layered numerical forward model of optical networks, are labeled by the course-A and -B and -C, respectively. During performing diffractive wavefront control along different path mentioned above, the AONN is loaded by the same extremely aberrated incident beams. Three subplot sets labeled by the sign-A and -B and -C corresponding to the different manipulation course, also present the layered shaping intermediate results and the final output beam appearance and light intensity characters according to the sign sequence such as from B-1 to B-3. As shown, the diagrams indicated by the sign A-3 and B-4 and C-6, exhibit the typical three-dimensional beam appearances, and also sign A-4 and B-5 and C-7 the cross-sectional curves going through the geometric center of the final beam intensity distribution shaped.

 

Fig. 4. The detailed beam shaping processes of the linear AONN with different number of diffractive layer or functioned depth, which defines its statistical inference and generalization capability. The diagram set A to C are the two-layer and three-layer and five-layer AONN respectively, where the adjusted or modulated light fields by each functioned layer are graphicly displayed based on the beam profile and intensity distribution.

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According to the subplot set (B) and (C) shown in Fig. 4, the additional functioned layers encourage the constructed optical network to process input beams in a softer modulation manner; for example, the gradient of the phase change of each functioned layer is no longer drastic compared to the networks with rare layer such as only two functioned layers shown by course-A above. We think that this soft or relatively slow processing may be the main reason of already enhancing the robustness of multilayer optical networks, which also be represented by the variance trend of the beam profiles outfrom each functioned layer, for instance, the light intensity distribution fashion indicated by the diagram B-1 to B-3 and also the diagram C-1 to C-5, where the former only concerning three functioned layers for adjusting or modulating the intensity distribution deviation to a desired similar circular appearance and adjusted location even though the output beams still being far away the final profile obtained.

Compared with three-layered AONN, the five-layer optical network tends to more strongly disperse the incident lightwaves and further present a relatively low beam intensity distribution over a larger spatial region completely covering the next entire functional layers so as to make full use of the mapping ability of each neuron in the optical neural network forward propagation processing and then integrated into the target area at the output port, and result in a more uniform and well-defined flat-top beam. To sum up, the acquired shaping diagram A-2 and B-3 and C-6 highlight the control lightwave operation leading to the desired beam appearance and profile outfrom initial distorted or aberrated incident beams through properly increasing the functioned depth. To further improve the performances, we have tried to use different spacing for each layer of the five-layer AONN, for example the spacing increasing with the number of the layer from 1-4 cm, but the results are not much different from the current results. On the other hand, it should be note that the introducing nonlinearity activation function, for example, using the photorefractive materials [41], will be able to further enhance the deductive ability of the AONN for realizing greater robustness. As well known, an arbitrary linear calculation can be compressed into a single matrix. But this does not conflict with the principle of the deeper AONN with more powerful expression capacity. Because the modulation process of a few phase plate is unable to represent arbitrary matrices, in other words, the phase plates need to be connected in series to represent some optimal matrix [42]. Despite its excellent performance compared to the thin optical network, the five-layer network or even deeper network, cannot be used in occasions of requiring relatively high transmittance coefficient due to relatively greater energy reflection and absorption of the manufacturing material, for instance, a five-layer silicon wafer (single layer transmittance coefficient of ∼61.88%) being resulting in a ∼90.93% energy loss while a two-layer network losing ∼61.71%. Compared with conventional optical diffractive components such as a single GS-based phase mask, the two-layer AONN with accomplishing a relatively robust light field adjustment or modulation can handle most of the aberrated incident light fields at a cost of ∼23.59% or more energy loss. Furthermore, too many layers will also amplify the input error layer by layer. For example, when the frequency of the incident light field is shifted by 10Ghz, the diffraction efficiency of the above AONN with different layers will decrease on an average by 5.1%, 8.6%, or 14.2%, individually. It should be noted that a two-layer AONN output through a trained phase- only optical network already reveals clearly an insensitivity to aberrated incident light beams, and then efficiently conducts a collective control lightwave operation through a passive layered diffractive assembly.

In order to further illustrate that the robust beam adjustment or modulation capability of the designed optical network is valid widely, a 2-layer AONN is continuously employed to split an incident beam with non-ideal appearance due to relatively lousy circumstance factors or optical misalignment generated, as shown in Fig. 5. We present a quantitative comparison using the histograms of the normalized peak intensity of 9 sub-beams corresponding to the incident beams of the subplot set (a) to (d) in Fig. 3. As shown in Fig. 5(a), the sub-beam NU with respect of the GS-based DOE and the two-layer AONN are 0.125 and 0.069, respectively. As indicated, the deep learning-based approach has been achieved a 44.8% improvement especially for those light field samples distorted severely. For more clearly exhibiting the distributing and morphological characters of the sub-beams constructed by the shaping architectures mentioned, the arrayed beam appearances and the detailed profiles obtained along the central dotted line so as to comprehensively compare the light field particulars of the above-mentioned two methods based on the incident lightwave shown in the sub plot (c) in Fig. 3, are given by Fig. 5(b). We found that even though considering the intensity loss of 9% caused by the additional functioned layer, the two-layered diffractive network still complete a relatively high average beam intensity of 0.72 compared with that of the traditional single-layered DOE of 0.69, and then output a more precise Gaussian mode of the sub-beam with a higher signal-to-clutter-ratio by jointly modulating two functioned layers, which presents a different phase selection and arrangement behavior according to the final arrayed beam configuration, as shown in Fig. 5(c).

 

Fig. 5. Performance comparison of the GS-based DOE and the two-layer AONN for two dimensional wide-angle beam splitting. a Histogram of the normalized peak intensity of the arrayed sub-beams, where the label-a, -b, -c and -d denote the incident lightwave sample corresponding to that shown in Fig. 3, and the cold and warm colors represent that obtained by the GS-based DOE and the two-layer AONN, respectively. b Profiles of the beam intensity distribution indicated by both the two-dimensional appearances and the curves obtained along the central dotted lines with respect of the lightwave sample c in Fig. 3. c Layered phase plates of the trained two-layer AONN for 2-dimensional beam splitting.

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For further investigate the property of the proposed method to relatively complex light fields with a sharp peak beam intensity and contrast. The two-layer AONN trained with a series of aberrated Gaussian as well as the target angular beam fringes and the optimized single DOE (iteration with an ideal Gaussian input and target angular beam fringes) based on the IFTA strategy, are used to present the featured beam shaping manipulation leading to highly efficient forming axial symmetric light beams, as shown in Fig. 6. Firstly, a Gaussian beam formed by mixing that of the subplot set (c) and (d) shown in Fig. 3 is modulated into a concentric annular light field used as a target image composed of two half semi-circles fully filled by annular beam fringes with different line width and distribution density, and thus employed to evaluate the beam shaping efficiency of the optical network architectures constructed, as shown by the left subgraph in the middle row. The beam shaping results by both the optical structures mentioned, are depicted by other subgraphs also in the middle row including the second and third subgraphs indicating the shaped light fields outfrom the first and the second functioned layers of the two-layer AONN, respectively, where an insert indicating the detailed fine angular fringe distribution with a clear and sharp edge and a relatively high beam concentration. But the fourth and the right subgraphs of the output light fields from the GS-based DOE, which demonstrate relatively bad shaping results such as non-uniform light intensity distribution compared with that of the two-layer AONN even through expressing a similar beam fringe profile, which is also indicated by an insert giving the similar fine angular fringe with relatively fuzzy edges and low and non-uniform beam concentration compared with that of the left insert. The corresponding beam intensity curves measured along the black diameter are also exhibited by the five bottom subgraphs, where the given gray dashed line marks the normalized beam intensity of ∼8.3% and ∼70.6% associated with the blue background and the annular beam fringes for the target image.

 

Fig. 6. Typical beam appearance and intensity distribution before and after performing beam shaping manipulation by both the GS-based DOE and the two-layer AONN for acquiring desired far field annular beam fringes shown by the middle row subgraphs and the bottom beam intensity testing curves. Both top inserts present the detailed the edge and light distribution uniformity of the annular fringes formed.

Download Full Size | PPT Slide | PDF

As seen, the bottom beam intensity testing curves over selected cross sections also demonstrate the typical shortcomings of the conventional GS-based iteration method to cope with seriously distorted or aberrated incident lightwaves. For example, the light intensity distribution of the target beam is already deformed due to the optical axis offset of the incident beams, which also brings about a significant drop of the actual central area beam intensity. While the trained two-layer AONN has already established a light field very closed to the target beam even though obviously reducing the overall beam energy or intensity level to deal with more aberrated light fields with larger background noise or more remarkable contour distortion or off-centering. Simultaneously, the proposed method also apparently improves the overall beam uniformity according to the patterned light field texture, as shown by the enlarged inserts. All in all, this novel cascaded diffractive control light strategy can be used to establish a more entire and uniform patterned light field so as to highlight the beam shaping manipulation leading to complex patterned beam appearance relied on the more representative training data including typical profile and morphology distortion and off- centering offset, which means to realize a robust adaptive adjustment or modulation to various input lightwaves.

The situation that an actual THz Gaussian output beam from the Coherent SIFIR 50 THz laser source already presenting a relatively severe off-centering offset is manipulated by both the GS-based DOE and the optical neural network leading to a uniform circular THz flat-top beam, respectively, is shown in Fig. 7. Where the experimentally or intermediately measured light fields are not informed to the approaches in their iterative or network training stages. The left subgraph demonstrates a typical output THz beam from the Coherent SIFIR 50. The second subgraph gives a typical shaping result corresponding to the left THz beam only by the GS-based DOE. Both the right subgraphs already give the desired appearances and profiles of the output THz beams from the two-layered and three-layered optical neural network, respectively.

 

Fig. 7. An actual THz Gaussian output beam from the Coherent SIFIR 50 THz laser source is used to perform beam shaping based on a conventional GS-based DOE and an AONN with different optical functioned layer constructed, respectively.

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It can be seen that the single layered DOE (iteration with an ideal Gaussian input and an uniform flat-top target beam) already drives incident lightwaves onto several wrong regions with different size and brightness, and then produces a main optical facula with a relatively rough and irregular and fuzzy edge and a non-uniform intensity distribution according to the color bar used, since the input lightwave is an experimentally aberrated light field rather than that used during phase recovery stage according to the diffractive computation by the AONN. Although the incident light field corresponding to the left subgraph is not in the training set of the AONN, the two-layered AONN still reconstructs a relatively complete and uniform flat- top beam corresponding to an unfamiliar input light field, which maybe different with the ideal Gaussian beam as shown by the left subgraph. Furthermore, a modulated light field outfrom the three-layered AONN is exhibited by the right subgraph, which already shows a nearly perfect flat-top beam with negligible laser speckle caused by cascaded or continuous wavefront diffractive operations.

To sum up, the optical neural network designed by us can be efficiently utilized to implement a robust phase-only beam adjustment or modulation, which benefits from some remarkable features including a kind of data-driven intelligent lightwave manipulation and the statistical inference and incident light field generalization based on a layered lightwave diffractive adjustment or modulation operation, so as to result in a kind of intelligent beam shaping strategy and architecture to aberrated incident light fields during a dynamic or changeable course.

4. Conclusion

So far, several typical patterned target light fields with aberrated Gaussian beam distribution characters caused by physical condition variance including the key device or component configuration errors, the equipment or setup vibration, radiation disturbance, or the optical axis misalignment or off-centering according to artificial factors, still remains the most challenging problem. In this paper, a data-driven and phase-only all-optical neural network framework is proposed and verified by simulation and practical manipulating a typical THz laser beam, so as to realize a robust beam shaping strategy. The method highlights the further development of the beam shaping techniques by loading intelligent manipulation based on existing adaptive optical diffractive approaches in related areas such as 3D laser fabrication, lidar, biomedicine detection, and so on.

Funding

National Natural Science Foundation of China (61176052, 61432007).

Disclosures

The authors declare that there are no conflicts of interest.

Supplemental document

See Supplement 1 for supporting content.

References

1. S. Li, Y. Wang, and Z. Lu, “High-quality near-field beam achieved in a high-power laser based on SLM adaptive beam-shaping system,” Opt. Express 23(2), 681 (2015). [CrossRef]  

2. L. Xu, W. Fan, and J. Liu, “High-resolution reconstruction for terahertz imaging,” Appl. Opt. 53(33), 7891 (2014). [CrossRef]  

3. P. S. Salter and M. J. Booth, “Adaptive optics in laser processing,” Light: Sci. Appl. 8(1), 110 (2019). [CrossRef]  

4. P. Ghelfi, F. Laghezza, and F. Scotti, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014). [CrossRef]  

5. S Poole, “Application Specific Optical Fibres And Fibre Devices For Optical Fibre Sensors,” Optical Fiber Sensors (Optical Society of America). 1983–1997 (1992).

6. C. Mauclair, D. Pietroy, and Y. Maieo, “Ultrafast laser micro-cutting of stainless steel and PZT using a modulated line of multiple foci formed by spatial beam shaping,” Opt. Lasers Eng. 67(3), 212–217 (2015). [CrossRef]  

7. N Feng, Y. Fu, and T Wang, “Photoacoustic Elasticity Imaging for Soft Tissue-mimicking Phantom,” International Conference on Photonics & Imaging in Biology & Medicine. (2017).

8. L. Criante, F. Vita, R. Castagna, D. E. Lucchetta, and F. Simoni, “High resolution optical data storage in composite polymeric blue sensitive materials,” Europe and IQEC 2007 Conference. (2007).

9. B. R. Frieden, “Lossless Conversion of a Plane Laser Wave to a Plane Wave of Uniform Irradiance,” Appl. Opt. 4(11), 1400–1403 (1965). [CrossRef]  

10. CR Phipps, “High-Power Laser Ablation VI,” Proceedings of SPIE - The International Society for Optical EngineeringC (2003).

11. R. Degl’Innocenti, Y. Shah, and Y. Ren, “Hollow metallic waveguides for beam shaping THz QCL,” International Conference on Infrared. (2014).

12. F. Yong-Qi, N. Bryan, and O. Shing, “Diffractive optical elements with continuous relief fabricated by focused ion beam for mono-mode fiber coupling,” Opt. Express 7(3), 141 (2000). [CrossRef]  

13. S. Noach, Y. Arieli, and N. Eisenberg, “Wavefront control and aberration correction with a diffractive optical element,” Opt. Lett. 24(5), 333 (1999). [CrossRef]  

14. J. Jiang and G. Nordin, “A rigorous unidirectional method for designing finite aperture diffractive optical elements,” Opt. Express 7(6), 237–242 (2000). [CrossRef]  

15. R. W. Gerchberg, “Phase determination from image and diffraction plane pictures in the electron microscope,” Optik., 35 (1971).

16. S. Kirkpatrick and M Vecchi, “Optimization by Simulated Annealing,” Spin Glass Theory and Beyond: An Introduction to the Replica Method and Its Applications. (1987).

17. J. Rosen, U. Mahlab, and J. Shamir, “Adaptive learning with joint transform correlators,” Opt. Eng. 29(9), 1101–1106 (1990). [CrossRef]  

18. I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett. 32(16), 2309–2311 (2007). [CrossRef]  

19. O. Katz, E. Small, and Y. Bromberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Photonics 5(6), 372–377 (2011). [CrossRef]  

20. M. Herrera-Fernandez J and M. Sanchez-Brea L, “Double diffractive optical element system for near-field shaping,” Appl. Opt. 50(23), 4587 (2011). [CrossRef]  

21. O. Hartmut, “Applications of the tandem component: an element with optimum light efficiency,” Appl. Opt. 24(22), 3811–3816 (1985). [CrossRef]  

22. A. Jesacher, C. Maurer, and A. Schwaighofer, “Full phase and amplitude control of holographic optical tweezers with high efficiency,” Opt. Express 16(7), 4479–4486 (2008). [CrossRef]  

23. H. Lee, J. Park, and K. Oh, “Recent Progress in All-Fiber Non-Gaussian Optical Beam Shaping Technologies,” J. Lightwave Technol. 37(11), 2590–2597 (2019). [CrossRef]  

24. A. Forbes, “Structured light from lasers,” Laser Photonics Rev. 13(11), 1970043 (2019). [CrossRef]  

25. A. Forbes, A. Dudley, and M. McLaren, “Creation and detection of optical modes with spatial light modulators,” Adv. Opt. Photonics 8(2), 200–227 (2016). [CrossRef]  

26. C. Rosales-Guzmán and A. Forbes, How to Shape Light with Spatial Light Modulators. (SPIE, 2017).

27. A. Litvin I, G. King, and H. Strauss, “Beam shaping laser with controllable gain,” Appl. Phys. B 123(6), 174 (2017). [CrossRef]  

28. S. Scholes and A. Forbes, “Improving the beam quality factor (M2) by phase-only reshaping of structured light,” Opt. Lett. 45(13), 3753–3756 (2020). [CrossRef]  

29. H. Yang, J. Meng, and W. Chen, “High efficiency and high-energy intra-cavity beam shaping laser,” Laser Phys. 25(9), 095005 (2015). [CrossRef]  

30. S. Tao and W. Yu, “Beam shaping of complex amplitude with separate constraints on the output beam,” Opt. Express 23(2), 1052 (2015). [CrossRef]  

31. I. A. Litvin and A. Forbes, “Intra-cavity flat-top beam generation,” Opt. Express 17(18), 15891–15903 (2009). [CrossRef]  

32. C. Qian, X. Lin, and X. Lin, “Performing optical logic operations by a diffractive neural network,” Light: Sci. Appl. 9(1), 59 (2020). [CrossRef]  

33. H. Dou, Y. Deng, and T. Yan, “Residual D2NN: Training diffractive deep neural networks via learnable light shortcuts,” Opt. Lett. 45(10), 2688–2691 (2020). [CrossRef]  

34. T. Yan, J. Wu, and T. Zhou, “Fourier-space Diffractive Deep Neural Network,” Phys. Rev. Lett. 123(2), 023901 (2019). [CrossRef]  

35. V Muhammed, M Deniz, T Nezih, L Yi, L Jingxi, Yair Ri, J Mona, and O Aydogan, “Terahertz Pulse Shaping Using Diffractive Legos, “ arXiv:2006.16599

36. S. Jiashuo, C. Mingce, W. Dong, H. Chai, L. Jun, W. Haiwei, Z. Xinyu, and X. Changsheng, “Anti-noise diffractive neural network for constructing an intelligent imaging detector array,” Opt. Express 28(25), 37686–37699 (2020). [CrossRef]  

37. X. Lin, Y. Rivenson, and N. T. Yardimei, “All-Optical Machine Learning Using Diffractive Deep Neural Networks,” Science 361(6406), 1004–1008 (2018). [CrossRef]  

38. T. Gissibl, S. Thiele, and A. Herkommer, “Two-photon direct laser writing of ultracompact multi-lens objectives,” Nat. Photonics 10(8), 554–560 (2016). [CrossRef]  

39. B. Dr, R. Martin, G. Thomas, and A. Marwan, “Trimming method for a high-yield manufacturing of high- efficiency diffraction gratings,” Opt. Lett. 43(16), 4017–4020 (2018). [CrossRef]  

40. P Cormont, M Pfiffer, E Fargin, B Bousquet, M Dussauze, S. Lambert, and J Néaruport, “Wet chemical etching with KOH solutions for fused silica optics manufacturing,” Optical Society of America (2017).

41. L. Waller, G. Situ, and J. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6(7), 474–479 (2012). [CrossRef]  

42. M Deniz, L Yi, R Yair, L Xing, V. Muhammed, and O Aydogan, “Response to Comment on All-optical machine learning using diffractive deep neural networks,” arXiv:1810.04384

References

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  • |
  • |

  1. S. Li, Y. Wang, and Z. Lu, “High-quality near-field beam achieved in a high-power laser based on SLM adaptive beam-shaping system,” Opt. Express 23(2), 681 (2015).
    [Crossref]
  2. L. Xu, W. Fan, and J. Liu, “High-resolution reconstruction for terahertz imaging,” Appl. Opt. 53(33), 7891 (2014).
    [Crossref]
  3. P. S. Salter and M. J. Booth, “Adaptive optics in laser processing,” Light: Sci. Appl. 8(1), 110 (2019).
    [Crossref]
  4. P. Ghelfi, F. Laghezza, and F. Scotti, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
    [Crossref]
  5. S Poole, “Application Specific Optical Fibres And Fibre Devices For Optical Fibre Sensors,” Optical Fiber Sensors (Optical Society of America). 1983–1997 (1992).
  6. C. Mauclair, D. Pietroy, and Y. Maieo, “Ultrafast laser micro-cutting of stainless steel and PZT using a modulated line of multiple foci formed by spatial beam shaping,” Opt. Lasers Eng. 67(3), 212–217 (2015).
    [Crossref]
  7. N Feng, Y. Fu, and T Wang, “Photoacoustic Elasticity Imaging for Soft Tissue-mimicking Phantom,” International Conference on Photonics & Imaging in Biology & Medicine. (2017).
  8. L. Criante, F. Vita, R. Castagna, D. E. Lucchetta, and F. Simoni, “High resolution optical data storage in composite polymeric blue sensitive materials,” Europe and IQEC 2007 Conference. (2007).
  9. B. R. Frieden, “Lossless Conversion of a Plane Laser Wave to a Plane Wave of Uniform Irradiance,” Appl. Opt. 4(11), 1400–1403 (1965).
    [Crossref]
  10. CR Phipps, “High-Power Laser Ablation VI,” Proceedings of SPIE - The International Society for Optical EngineeringC (2003).
  11. R. Degl’Innocenti, Y. Shah, and Y. Ren, “Hollow metallic waveguides for beam shaping THz QCL,” International Conference on Infrared. (2014).
  12. F. Yong-Qi, N. Bryan, and O. Shing, “Diffractive optical elements with continuous relief fabricated by focused ion beam for mono-mode fiber coupling,” Opt. Express 7(3), 141 (2000).
    [Crossref]
  13. S. Noach, Y. Arieli, and N. Eisenberg, “Wavefront control and aberration correction with a diffractive optical element,” Opt. Lett. 24(5), 333 (1999).
    [Crossref]
  14. J. Jiang and G. Nordin, “A rigorous unidirectional method for designing finite aperture diffractive optical elements,” Opt. Express 7(6), 237–242 (2000).
    [Crossref]
  15. R. W. Gerchberg, “Phase determination from image and diffraction plane pictures in the electron microscope,” Optik., 35 (1971).
  16. S. Kirkpatrick and M Vecchi, “Optimization by Simulated Annealing,” Spin Glass Theory and Beyond: An Introduction to the Replica Method and Its Applications. (1987).
  17. J. Rosen, U. Mahlab, and J. Shamir, “Adaptive learning with joint transform correlators,” Opt. Eng. 29(9), 1101–1106 (1990).
    [Crossref]
  18. I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett. 32(16), 2309–2311 (2007).
    [Crossref]
  19. O. Katz, E. Small, and Y. Bromberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Photonics 5(6), 372–377 (2011).
    [Crossref]
  20. M. Herrera-Fernandez J and M. Sanchez-Brea L, “Double diffractive optical element system for near-field shaping,” Appl. Opt. 50(23), 4587 (2011).
    [Crossref]
  21. O. Hartmut, “Applications of the tandem component: an element with optimum light efficiency,” Appl. Opt. 24(22), 3811–3816 (1985).
    [Crossref]
  22. A. Jesacher, C. Maurer, and A. Schwaighofer, “Full phase and amplitude control of holographic optical tweezers with high efficiency,” Opt. Express 16(7), 4479–4486 (2008).
    [Crossref]
  23. H. Lee, J. Park, and K. Oh, “Recent Progress in All-Fiber Non-Gaussian Optical Beam Shaping Technologies,” J. Lightwave Technol. 37(11), 2590–2597 (2019).
    [Crossref]
  24. A. Forbes, “Structured light from lasers,” Laser Photonics Rev. 13(11), 1970043 (2019).
    [Crossref]
  25. A. Forbes, A. Dudley, and M. McLaren, “Creation and detection of optical modes with spatial light modulators,” Adv. Opt. Photonics 8(2), 200–227 (2016).
    [Crossref]
  26. C. Rosales-Guzmán and A. Forbes, How to Shape Light with Spatial Light Modulators. (SPIE, 2017).
  27. A. Litvin I, G. King, and H. Strauss, “Beam shaping laser with controllable gain,” Appl. Phys. B 123(6), 174 (2017).
    [Crossref]
  28. S. Scholes and A. Forbes, “Improving the beam quality factor (M2) by phase-only reshaping of structured light,” Opt. Lett. 45(13), 3753–3756 (2020).
    [Crossref]
  29. H. Yang, J. Meng, and W. Chen, “High efficiency and high-energy intra-cavity beam shaping laser,” Laser Phys. 25(9), 095005 (2015).
    [Crossref]
  30. S. Tao and W. Yu, “Beam shaping of complex amplitude with separate constraints on the output beam,” Opt. Express 23(2), 1052 (2015).
    [Crossref]
  31. I. A. Litvin and A. Forbes, “Intra-cavity flat-top beam generation,” Opt. Express 17(18), 15891–15903 (2009).
    [Crossref]
  32. C. Qian, X. Lin, and X. Lin, “Performing optical logic operations by a diffractive neural network,” Light: Sci. Appl. 9(1), 59 (2020).
    [Crossref]
  33. H. Dou, Y. Deng, and T. Yan, “Residual D2NN: Training diffractive deep neural networks via learnable light shortcuts,” Opt. Lett. 45(10), 2688–2691 (2020).
    [Crossref]
  34. T. Yan, J. Wu, and T. Zhou, “Fourier-space Diffractive Deep Neural Network,” Phys. Rev. Lett. 123(2), 023901 (2019).
    [Crossref]
  35. V Muhammed, M Deniz, T Nezih, L Yi, L Jingxi, Yair Ri, J Mona, and O Aydogan, “Terahertz Pulse Shaping Using Diffractive Legos, “ arXiv:2006.16599
  36. S. Jiashuo, C. Mingce, W. Dong, H. Chai, L. Jun, W. Haiwei, Z. Xinyu, and X. Changsheng, “Anti-noise diffractive neural network for constructing an intelligent imaging detector array,” Opt. Express 28(25), 37686–37699 (2020).
    [Crossref]
  37. X. Lin, Y. Rivenson, and N. T. Yardimei, “All-Optical Machine Learning Using Diffractive Deep Neural Networks,” Science 361(6406), 1004–1008 (2018).
    [Crossref]
  38. T. Gissibl, S. Thiele, and A. Herkommer, “Two-photon direct laser writing of ultracompact multi-lens objectives,” Nat. Photonics 10(8), 554–560 (2016).
    [Crossref]
  39. B. Dr, R. Martin, G. Thomas, and A. Marwan, “Trimming method for a high-yield manufacturing of high- efficiency diffraction gratings,” Opt. Lett. 43(16), 4017–4020 (2018).
    [Crossref]
  40. P Cormont, M Pfiffer, E Fargin, B Bousquet, M Dussauze, S. Lambert, and J Néaruport, “Wet chemical etching with KOH solutions for fused silica optics manufacturing,” Optical Society of America (2017).
  41. L. Waller, G. Situ, and J. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6(7), 474–479 (2012).
    [Crossref]
  42. M Deniz, L Yi, R Yair, L Xing, V. Muhammed, and O Aydogan, “Response to Comment on All-optical machine learning using diffractive deep neural networks,” arXiv:1810.04384

2020 (4)

2019 (4)

T. Yan, J. Wu, and T. Zhou, “Fourier-space Diffractive Deep Neural Network,” Phys. Rev. Lett. 123(2), 023901 (2019).
[Crossref]

P. S. Salter and M. J. Booth, “Adaptive optics in laser processing,” Light: Sci. Appl. 8(1), 110 (2019).
[Crossref]

H. Lee, J. Park, and K. Oh, “Recent Progress in All-Fiber Non-Gaussian Optical Beam Shaping Technologies,” J. Lightwave Technol. 37(11), 2590–2597 (2019).
[Crossref]

A. Forbes, “Structured light from lasers,” Laser Photonics Rev. 13(11), 1970043 (2019).
[Crossref]

2018 (2)

X. Lin, Y. Rivenson, and N. T. Yardimei, “All-Optical Machine Learning Using Diffractive Deep Neural Networks,” Science 361(6406), 1004–1008 (2018).
[Crossref]

B. Dr, R. Martin, G. Thomas, and A. Marwan, “Trimming method for a high-yield manufacturing of high- efficiency diffraction gratings,” Opt. Lett. 43(16), 4017–4020 (2018).
[Crossref]

2017 (1)

A. Litvin I, G. King, and H. Strauss, “Beam shaping laser with controllable gain,” Appl. Phys. B 123(6), 174 (2017).
[Crossref]

2016 (2)

T. Gissibl, S. Thiele, and A. Herkommer, “Two-photon direct laser writing of ultracompact multi-lens objectives,” Nat. Photonics 10(8), 554–560 (2016).
[Crossref]

A. Forbes, A. Dudley, and M. McLaren, “Creation and detection of optical modes with spatial light modulators,” Adv. Opt. Photonics 8(2), 200–227 (2016).
[Crossref]

2015 (4)

S. Li, Y. Wang, and Z. Lu, “High-quality near-field beam achieved in a high-power laser based on SLM adaptive beam-shaping system,” Opt. Express 23(2), 681 (2015).
[Crossref]

C. Mauclair, D. Pietroy, and Y. Maieo, “Ultrafast laser micro-cutting of stainless steel and PZT using a modulated line of multiple foci formed by spatial beam shaping,” Opt. Lasers Eng. 67(3), 212–217 (2015).
[Crossref]

H. Yang, J. Meng, and W. Chen, “High efficiency and high-energy intra-cavity beam shaping laser,” Laser Phys. 25(9), 095005 (2015).
[Crossref]

S. Tao and W. Yu, “Beam shaping of complex amplitude with separate constraints on the output beam,” Opt. Express 23(2), 1052 (2015).
[Crossref]

2014 (2)

L. Xu, W. Fan, and J. Liu, “High-resolution reconstruction for terahertz imaging,” Appl. Opt. 53(33), 7891 (2014).
[Crossref]

P. Ghelfi, F. Laghezza, and F. Scotti, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref]

2012 (1)

L. Waller, G. Situ, and J. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6(7), 474–479 (2012).
[Crossref]

2011 (2)

O. Katz, E. Small, and Y. Bromberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Photonics 5(6), 372–377 (2011).
[Crossref]

M. Herrera-Fernandez J and M. Sanchez-Brea L, “Double diffractive optical element system for near-field shaping,” Appl. Opt. 50(23), 4587 (2011).
[Crossref]

2009 (1)

2008 (1)

2007 (1)

2000 (2)

1999 (1)

1990 (1)

J. Rosen, U. Mahlab, and J. Shamir, “Adaptive learning with joint transform correlators,” Opt. Eng. 29(9), 1101–1106 (1990).
[Crossref]

1985 (1)

1965 (1)

Arieli, Y.

Aydogan, O

V Muhammed, M Deniz, T Nezih, L Yi, L Jingxi, Yair Ri, J Mona, and O Aydogan, “Terahertz Pulse Shaping Using Diffractive Legos, “ arXiv:2006.16599

M Deniz, L Yi, R Yair, L Xing, V. Muhammed, and O Aydogan, “Response to Comment on All-optical machine learning using diffractive deep neural networks,” arXiv:1810.04384

Booth, M. J.

P. S. Salter and M. J. Booth, “Adaptive optics in laser processing,” Light: Sci. Appl. 8(1), 110 (2019).
[Crossref]

Bousquet, B

P Cormont, M Pfiffer, E Fargin, B Bousquet, M Dussauze, S. Lambert, and J Néaruport, “Wet chemical etching with KOH solutions for fused silica optics manufacturing,” Optical Society of America (2017).

Bromberg, Y.

O. Katz, E. Small, and Y. Bromberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Photonics 5(6), 372–377 (2011).
[Crossref]

Bryan, N.

Castagna, R.

L. Criante, F. Vita, R. Castagna, D. E. Lucchetta, and F. Simoni, “High resolution optical data storage in composite polymeric blue sensitive materials,” Europe and IQEC 2007 Conference. (2007).

Chai, H.

Changsheng, X.

Chen, W.

H. Yang, J. Meng, and W. Chen, “High efficiency and high-energy intra-cavity beam shaping laser,” Laser Phys. 25(9), 095005 (2015).
[Crossref]

Cormont, P

P Cormont, M Pfiffer, E Fargin, B Bousquet, M Dussauze, S. Lambert, and J Néaruport, “Wet chemical etching with KOH solutions for fused silica optics manufacturing,” Optical Society of America (2017).

Criante, L.

L. Criante, F. Vita, R. Castagna, D. E. Lucchetta, and F. Simoni, “High resolution optical data storage in composite polymeric blue sensitive materials,” Europe and IQEC 2007 Conference. (2007).

Degl’Innocenti, R.

R. Degl’Innocenti, Y. Shah, and Y. Ren, “Hollow metallic waveguides for beam shaping THz QCL,” International Conference on Infrared. (2014).

Deng, Y.

Deniz, M

V Muhammed, M Deniz, T Nezih, L Yi, L Jingxi, Yair Ri, J Mona, and O Aydogan, “Terahertz Pulse Shaping Using Diffractive Legos, “ arXiv:2006.16599

M Deniz, L Yi, R Yair, L Xing, V. Muhammed, and O Aydogan, “Response to Comment on All-optical machine learning using diffractive deep neural networks,” arXiv:1810.04384

Dong, W.

Dou, H.

Dr, B.

Dudley, A.

A. Forbes, A. Dudley, and M. McLaren, “Creation and detection of optical modes with spatial light modulators,” Adv. Opt. Photonics 8(2), 200–227 (2016).
[Crossref]

Dussauze, M

P Cormont, M Pfiffer, E Fargin, B Bousquet, M Dussauze, S. Lambert, and J Néaruport, “Wet chemical etching with KOH solutions for fused silica optics manufacturing,” Optical Society of America (2017).

Eisenberg, N.

Fan, W.

Fargin, E

P Cormont, M Pfiffer, E Fargin, B Bousquet, M Dussauze, S. Lambert, and J Néaruport, “Wet chemical etching with KOH solutions for fused silica optics manufacturing,” Optical Society of America (2017).

Feng, N

N Feng, Y. Fu, and T Wang, “Photoacoustic Elasticity Imaging for Soft Tissue-mimicking Phantom,” International Conference on Photonics & Imaging in Biology & Medicine. (2017).

Fleischer, J.

L. Waller, G. Situ, and J. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6(7), 474–479 (2012).
[Crossref]

Forbes, A.

S. Scholes and A. Forbes, “Improving the beam quality factor (M2) by phase-only reshaping of structured light,” Opt. Lett. 45(13), 3753–3756 (2020).
[Crossref]

A. Forbes, “Structured light from lasers,” Laser Photonics Rev. 13(11), 1970043 (2019).
[Crossref]

A. Forbes, A. Dudley, and M. McLaren, “Creation and detection of optical modes with spatial light modulators,” Adv. Opt. Photonics 8(2), 200–227 (2016).
[Crossref]

I. A. Litvin and A. Forbes, “Intra-cavity flat-top beam generation,” Opt. Express 17(18), 15891–15903 (2009).
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C. Rosales-Guzmán and A. Forbes, How to Shape Light with Spatial Light Modulators. (SPIE, 2017).

Frieden, B. R.

Fu, Y.

N Feng, Y. Fu, and T Wang, “Photoacoustic Elasticity Imaging for Soft Tissue-mimicking Phantom,” International Conference on Photonics & Imaging in Biology & Medicine. (2017).

Gerchberg, R. W.

R. W. Gerchberg, “Phase determination from image and diffraction plane pictures in the electron microscope,” Optik., 35 (1971).

Ghelfi, P.

P. Ghelfi, F. Laghezza, and F. Scotti, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref]

Gissibl, T.

T. Gissibl, S. Thiele, and A. Herkommer, “Two-photon direct laser writing of ultracompact multi-lens objectives,” Nat. Photonics 10(8), 554–560 (2016).
[Crossref]

Haiwei, W.

Hartmut, O.

Herkommer, A.

T. Gissibl, S. Thiele, and A. Herkommer, “Two-photon direct laser writing of ultracompact multi-lens objectives,” Nat. Photonics 10(8), 554–560 (2016).
[Crossref]

Herrera-Fernandez J, M.

Jesacher, A.

Jiang, J.

Jiashuo, S.

Jingxi, L

V Muhammed, M Deniz, T Nezih, L Yi, L Jingxi, Yair Ri, J Mona, and O Aydogan, “Terahertz Pulse Shaping Using Diffractive Legos, “ arXiv:2006.16599

Jun, L.

Katz, O.

O. Katz, E. Small, and Y. Bromberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Photonics 5(6), 372–377 (2011).
[Crossref]

King, G.

A. Litvin I, G. King, and H. Strauss, “Beam shaping laser with controllable gain,” Appl. Phys. B 123(6), 174 (2017).
[Crossref]

Kirkpatrick, S.

S. Kirkpatrick and M Vecchi, “Optimization by Simulated Annealing,” Spin Glass Theory and Beyond: An Introduction to the Replica Method and Its Applications. (1987).

Laghezza, F.

P. Ghelfi, F. Laghezza, and F. Scotti, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref]

Lambert, S.

P Cormont, M Pfiffer, E Fargin, B Bousquet, M Dussauze, S. Lambert, and J Néaruport, “Wet chemical etching with KOH solutions for fused silica optics manufacturing,” Optical Society of America (2017).

Lee, H.

Li, S.

Lin, X.

C. Qian, X. Lin, and X. Lin, “Performing optical logic operations by a diffractive neural network,” Light: Sci. Appl. 9(1), 59 (2020).
[Crossref]

C. Qian, X. Lin, and X. Lin, “Performing optical logic operations by a diffractive neural network,” Light: Sci. Appl. 9(1), 59 (2020).
[Crossref]

X. Lin, Y. Rivenson, and N. T. Yardimei, “All-Optical Machine Learning Using Diffractive Deep Neural Networks,” Science 361(6406), 1004–1008 (2018).
[Crossref]

Litvin, I. A.

Litvin I, A.

A. Litvin I, G. King, and H. Strauss, “Beam shaping laser with controllable gain,” Appl. Phys. B 123(6), 174 (2017).
[Crossref]

Liu, J.

Lu, Z.

Lucchetta, D. E.

L. Criante, F. Vita, R. Castagna, D. E. Lucchetta, and F. Simoni, “High resolution optical data storage in composite polymeric blue sensitive materials,” Europe and IQEC 2007 Conference. (2007).

Mahlab, U.

J. Rosen, U. Mahlab, and J. Shamir, “Adaptive learning with joint transform correlators,” Opt. Eng. 29(9), 1101–1106 (1990).
[Crossref]

Maieo, Y.

C. Mauclair, D. Pietroy, and Y. Maieo, “Ultrafast laser micro-cutting of stainless steel and PZT using a modulated line of multiple foci formed by spatial beam shaping,” Opt. Lasers Eng. 67(3), 212–217 (2015).
[Crossref]

Martin, R.

Marwan, A.

Mauclair, C.

C. Mauclair, D. Pietroy, and Y. Maieo, “Ultrafast laser micro-cutting of stainless steel and PZT using a modulated line of multiple foci formed by spatial beam shaping,” Opt. Lasers Eng. 67(3), 212–217 (2015).
[Crossref]

Maurer, C.

McLaren, M.

A. Forbes, A. Dudley, and M. McLaren, “Creation and detection of optical modes with spatial light modulators,” Adv. Opt. Photonics 8(2), 200–227 (2016).
[Crossref]

Meng, J.

H. Yang, J. Meng, and W. Chen, “High efficiency and high-energy intra-cavity beam shaping laser,” Laser Phys. 25(9), 095005 (2015).
[Crossref]

Mingce, C.

Mona, J

V Muhammed, M Deniz, T Nezih, L Yi, L Jingxi, Yair Ri, J Mona, and O Aydogan, “Terahertz Pulse Shaping Using Diffractive Legos, “ arXiv:2006.16599

Mosk, A. P.

Muhammed, V

V Muhammed, M Deniz, T Nezih, L Yi, L Jingxi, Yair Ri, J Mona, and O Aydogan, “Terahertz Pulse Shaping Using Diffractive Legos, “ arXiv:2006.16599

Muhammed, V.

M Deniz, L Yi, R Yair, L Xing, V. Muhammed, and O Aydogan, “Response to Comment on All-optical machine learning using diffractive deep neural networks,” arXiv:1810.04384

Néaruport, J

P Cormont, M Pfiffer, E Fargin, B Bousquet, M Dussauze, S. Lambert, and J Néaruport, “Wet chemical etching with KOH solutions for fused silica optics manufacturing,” Optical Society of America (2017).

Nezih, T

V Muhammed, M Deniz, T Nezih, L Yi, L Jingxi, Yair Ri, J Mona, and O Aydogan, “Terahertz Pulse Shaping Using Diffractive Legos, “ arXiv:2006.16599

Noach, S.

Nordin, G.

Oh, K.

Park, J.

Pfiffer, M

P Cormont, M Pfiffer, E Fargin, B Bousquet, M Dussauze, S. Lambert, and J Néaruport, “Wet chemical etching with KOH solutions for fused silica optics manufacturing,” Optical Society of America (2017).

Phipps, CR

CR Phipps, “High-Power Laser Ablation VI,” Proceedings of SPIE - The International Society for Optical EngineeringC (2003).

Pietroy, D.

C. Mauclair, D. Pietroy, and Y. Maieo, “Ultrafast laser micro-cutting of stainless steel and PZT using a modulated line of multiple foci formed by spatial beam shaping,” Opt. Lasers Eng. 67(3), 212–217 (2015).
[Crossref]

Poole, S

S Poole, “Application Specific Optical Fibres And Fibre Devices For Optical Fibre Sensors,” Optical Fiber Sensors (Optical Society of America). 1983–1997 (1992).

Qian, C.

C. Qian, X. Lin, and X. Lin, “Performing optical logic operations by a diffractive neural network,” Light: Sci. Appl. 9(1), 59 (2020).
[Crossref]

Ren, Y.

R. Degl’Innocenti, Y. Shah, and Y. Ren, “Hollow metallic waveguides for beam shaping THz QCL,” International Conference on Infrared. (2014).

Ri, Yair

V Muhammed, M Deniz, T Nezih, L Yi, L Jingxi, Yair Ri, J Mona, and O Aydogan, “Terahertz Pulse Shaping Using Diffractive Legos, “ arXiv:2006.16599

Rivenson, Y.

X. Lin, Y. Rivenson, and N. T. Yardimei, “All-Optical Machine Learning Using Diffractive Deep Neural Networks,” Science 361(6406), 1004–1008 (2018).
[Crossref]

Rosales-Guzmán, C.

C. Rosales-Guzmán and A. Forbes, How to Shape Light with Spatial Light Modulators. (SPIE, 2017).

Rosen, J.

J. Rosen, U. Mahlab, and J. Shamir, “Adaptive learning with joint transform correlators,” Opt. Eng. 29(9), 1101–1106 (1990).
[Crossref]

Salter, P. S.

P. S. Salter and M. J. Booth, “Adaptive optics in laser processing,” Light: Sci. Appl. 8(1), 110 (2019).
[Crossref]

Sanchez-Brea L, M.

Scholes, S.

Schwaighofer, A.

Scotti, F.

P. Ghelfi, F. Laghezza, and F. Scotti, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref]

Shah, Y.

R. Degl’Innocenti, Y. Shah, and Y. Ren, “Hollow metallic waveguides for beam shaping THz QCL,” International Conference on Infrared. (2014).

Shamir, J.

J. Rosen, U. Mahlab, and J. Shamir, “Adaptive learning with joint transform correlators,” Opt. Eng. 29(9), 1101–1106 (1990).
[Crossref]

Shing, O.

Simoni, F.

L. Criante, F. Vita, R. Castagna, D. E. Lucchetta, and F. Simoni, “High resolution optical data storage in composite polymeric blue sensitive materials,” Europe and IQEC 2007 Conference. (2007).

Situ, G.

L. Waller, G. Situ, and J. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6(7), 474–479 (2012).
[Crossref]

Small, E.

O. Katz, E. Small, and Y. Bromberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Photonics 5(6), 372–377 (2011).
[Crossref]

Strauss, H.

A. Litvin I, G. King, and H. Strauss, “Beam shaping laser with controllable gain,” Appl. Phys. B 123(6), 174 (2017).
[Crossref]

Tao, S.

Thiele, S.

T. Gissibl, S. Thiele, and A. Herkommer, “Two-photon direct laser writing of ultracompact multi-lens objectives,” Nat. Photonics 10(8), 554–560 (2016).
[Crossref]

Thomas, G.

Vecchi, M

S. Kirkpatrick and M Vecchi, “Optimization by Simulated Annealing,” Spin Glass Theory and Beyond: An Introduction to the Replica Method and Its Applications. (1987).

Vellekoop, I. M.

Vita, F.

L. Criante, F. Vita, R. Castagna, D. E. Lucchetta, and F. Simoni, “High resolution optical data storage in composite polymeric blue sensitive materials,” Europe and IQEC 2007 Conference. (2007).

Waller, L.

L. Waller, G. Situ, and J. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6(7), 474–479 (2012).
[Crossref]

Wang, T

N Feng, Y. Fu, and T Wang, “Photoacoustic Elasticity Imaging for Soft Tissue-mimicking Phantom,” International Conference on Photonics & Imaging in Biology & Medicine. (2017).

Wang, Y.

Wu, J.

T. Yan, J. Wu, and T. Zhou, “Fourier-space Diffractive Deep Neural Network,” Phys. Rev. Lett. 123(2), 023901 (2019).
[Crossref]

Xing, L

M Deniz, L Yi, R Yair, L Xing, V. Muhammed, and O Aydogan, “Response to Comment on All-optical machine learning using diffractive deep neural networks,” arXiv:1810.04384

Xinyu, Z.

Xu, L.

Yair, R

M Deniz, L Yi, R Yair, L Xing, V. Muhammed, and O Aydogan, “Response to Comment on All-optical machine learning using diffractive deep neural networks,” arXiv:1810.04384

Yan, T.

H. Dou, Y. Deng, and T. Yan, “Residual D2NN: Training diffractive deep neural networks via learnable light shortcuts,” Opt. Lett. 45(10), 2688–2691 (2020).
[Crossref]

T. Yan, J. Wu, and T. Zhou, “Fourier-space Diffractive Deep Neural Network,” Phys. Rev. Lett. 123(2), 023901 (2019).
[Crossref]

Yang, H.

H. Yang, J. Meng, and W. Chen, “High efficiency and high-energy intra-cavity beam shaping laser,” Laser Phys. 25(9), 095005 (2015).
[Crossref]

Yardimei, N. T.

X. Lin, Y. Rivenson, and N. T. Yardimei, “All-Optical Machine Learning Using Diffractive Deep Neural Networks,” Science 361(6406), 1004–1008 (2018).
[Crossref]

Yi, L

V Muhammed, M Deniz, T Nezih, L Yi, L Jingxi, Yair Ri, J Mona, and O Aydogan, “Terahertz Pulse Shaping Using Diffractive Legos, “ arXiv:2006.16599

M Deniz, L Yi, R Yair, L Xing, V. Muhammed, and O Aydogan, “Response to Comment on All-optical machine learning using diffractive deep neural networks,” arXiv:1810.04384

Yong-Qi, F.

Yu, W.

Zhou, T.

T. Yan, J. Wu, and T. Zhou, “Fourier-space Diffractive Deep Neural Network,” Phys. Rev. Lett. 123(2), 023901 (2019).
[Crossref]

Adv. Opt. Photonics (1)

A. Forbes, A. Dudley, and M. McLaren, “Creation and detection of optical modes with spatial light modulators,” Adv. Opt. Photonics 8(2), 200–227 (2016).
[Crossref]

Appl. Opt. (4)

Appl. Phys. B (1)

A. Litvin I, G. King, and H. Strauss, “Beam shaping laser with controllable gain,” Appl. Phys. B 123(6), 174 (2017).
[Crossref]

J. Lightwave Technol. (1)

Laser Photonics Rev. (1)

A. Forbes, “Structured light from lasers,” Laser Photonics Rev. 13(11), 1970043 (2019).
[Crossref]

Laser Phys. (1)

H. Yang, J. Meng, and W. Chen, “High efficiency and high-energy intra-cavity beam shaping laser,” Laser Phys. 25(9), 095005 (2015).
[Crossref]

Light: Sci. Appl. (2)

C. Qian, X. Lin, and X. Lin, “Performing optical logic operations by a diffractive neural network,” Light: Sci. Appl. 9(1), 59 (2020).
[Crossref]

P. S. Salter and M. J. Booth, “Adaptive optics in laser processing,” Light: Sci. Appl. 8(1), 110 (2019).
[Crossref]

Nat. Photonics (3)

O. Katz, E. Small, and Y. Bromberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Photonics 5(6), 372–377 (2011).
[Crossref]

T. Gissibl, S. Thiele, and A. Herkommer, “Two-photon direct laser writing of ultracompact multi-lens objectives,” Nat. Photonics 10(8), 554–560 (2016).
[Crossref]

L. Waller, G. Situ, and J. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6(7), 474–479 (2012).
[Crossref]

Nature (1)

P. Ghelfi, F. Laghezza, and F. Scotti, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref]

Opt. Eng. (1)

J. Rosen, U. Mahlab, and J. Shamir, “Adaptive learning with joint transform correlators,” Opt. Eng. 29(9), 1101–1106 (1990).
[Crossref]

Opt. Express (7)

Opt. Lasers Eng. (1)

C. Mauclair, D. Pietroy, and Y. Maieo, “Ultrafast laser micro-cutting of stainless steel and PZT using a modulated line of multiple foci formed by spatial beam shaping,” Opt. Lasers Eng. 67(3), 212–217 (2015).
[Crossref]

Opt. Lett. (5)

Phys. Rev. Lett. (1)

T. Yan, J. Wu, and T. Zhou, “Fourier-space Diffractive Deep Neural Network,” Phys. Rev. Lett. 123(2), 023901 (2019).
[Crossref]

Science (1)

X. Lin, Y. Rivenson, and N. T. Yardimei, “All-Optical Machine Learning Using Diffractive Deep Neural Networks,” Science 361(6406), 1004–1008 (2018).
[Crossref]

Other (11)

V Muhammed, M Deniz, T Nezih, L Yi, L Jingxi, Yair Ri, J Mona, and O Aydogan, “Terahertz Pulse Shaping Using Diffractive Legos, “ arXiv:2006.16599

C. Rosales-Guzmán and A. Forbes, How to Shape Light with Spatial Light Modulators. (SPIE, 2017).

R. W. Gerchberg, “Phase determination from image and diffraction plane pictures in the electron microscope,” Optik., 35 (1971).

S. Kirkpatrick and M Vecchi, “Optimization by Simulated Annealing,” Spin Glass Theory and Beyond: An Introduction to the Replica Method and Its Applications. (1987).

N Feng, Y. Fu, and T Wang, “Photoacoustic Elasticity Imaging for Soft Tissue-mimicking Phantom,” International Conference on Photonics & Imaging in Biology & Medicine. (2017).

L. Criante, F. Vita, R. Castagna, D. E. Lucchetta, and F. Simoni, “High resolution optical data storage in composite polymeric blue sensitive materials,” Europe and IQEC 2007 Conference. (2007).

CR Phipps, “High-Power Laser Ablation VI,” Proceedings of SPIE - The International Society for Optical EngineeringC (2003).

R. Degl’Innocenti, Y. Shah, and Y. Ren, “Hollow metallic waveguides for beam shaping THz QCL,” International Conference on Infrared. (2014).

S Poole, “Application Specific Optical Fibres And Fibre Devices For Optical Fibre Sensors,” Optical Fiber Sensors (Optical Society of America). 1983–1997 (1992).

P Cormont, M Pfiffer, E Fargin, B Bousquet, M Dussauze, S. Lambert, and J Néaruport, “Wet chemical etching with KOH solutions for fused silica optics manufacturing,” Optical Society of America (2017).

M Deniz, L Yi, R Yair, L Xing, V. Muhammed, and O Aydogan, “Response to Comment on All-optical machine learning using diffractive deep neural networks,” arXiv:1810.04384

Supplementary Material (1)

NameDescription
» Supplement 1       details of the fabrication for GS-based phase plate

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Figures (7)

Fig. 1.
Fig. 1. Operation principles of the IFTA and the AONN
Fig. 2.
Fig. 2. THz beam characters and their profile details corresponding to two types of optical diffractive architectures. (a) Light path for performing THz pattern emission using a single GS-based phase mask and a two-layer AONN. (b) Typical THz light fields of 40 mW and 100 mW out from the same THz laser source and then beam appearances including a 3×3 sub-beams and an inverted triangle, and three sub-beam pattern and digital 1 based on a single GS-based DOE manufactured by common KOH wet corrosion. (c) Laser beams shaped by a GS-based single phase plate and a two-layer diffractive neural network, where the output light field of the first layer mask is indicated by a black dashed frame. (d) Comparison between the flat-top beams shaped by a GS-based DOE indicated by the left photography and the constructed AONN according to the simulation demonstrated by the right photography.
Fig. 3.
Fig. 3. Beam shaping efficiency comparison of the IFTA-based DOE and the two-layer AONN. a Comparison of intensity distribution for multiple input. (a) The ideal 3-order super-Gaussian beam, and (b) to (e) the typical problematic Gaussian beams with different aberrated characters, for example, optical axis shifting (off-center) and random noise importing. The evaluation parameters including the RMSE and the NU and the DE are inserted in the upper left corner of each beam shaping diagram. b Cross sections of corresponding modulated intensity distribution. Where both the red and blue curves demonstrate the beam shaping results of the GS-based DOE and the two-layer AONN, respectively.
Fig. 4.
Fig. 4. The detailed beam shaping processes of the linear AONN with different number of diffractive layer or functioned depth, which defines its statistical inference and generalization capability. The diagram set A to C are the two-layer and three-layer and five-layer AONN respectively, where the adjusted or modulated light fields by each functioned layer are graphicly displayed based on the beam profile and intensity distribution.
Fig. 5.
Fig. 5. Performance comparison of the GS-based DOE and the two-layer AONN for two dimensional wide-angle beam splitting. a Histogram of the normalized peak intensity of the arrayed sub-beams, where the label-a, -b, -c and -d denote the incident lightwave sample corresponding to that shown in Fig. 3, and the cold and warm colors represent that obtained by the GS-based DOE and the two-layer AONN, respectively. b Profiles of the beam intensity distribution indicated by both the two-dimensional appearances and the curves obtained along the central dotted lines with respect of the lightwave sample c in Fig. 3. c Layered phase plates of the trained two-layer AONN for 2-dimensional beam splitting.
Fig. 6.
Fig. 6. Typical beam appearance and intensity distribution before and after performing beam shaping manipulation by both the GS-based DOE and the two-layer AONN for acquiring desired far field annular beam fringes shown by the middle row subgraphs and the bottom beam intensity testing curves. Both top inserts present the detailed the edge and light distribution uniformity of the annular fringes formed.
Fig. 7.
Fig. 7. An actual THz Gaussian output beam from the Coherent SIFIR 50 THz laser source is used to perform beam shaping based on a conventional GS-based DOE and an AONN with different optical functioned layer constructed, respectively.

Equations (11)

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m i l = e j φ 2 ,
w i l = z l z l 1 r 2 ( 1 2 π r + 1 j λ ) exp ( j 2 π r λ ) ,
r = ( x i l x j l 1 ) 2 + ( y i l y j l 1 ) 2 + ( z i l z j l 1 ) 2 ,
o u t p u t i l = w i l × k o u t p u t k l 1 × m i l = w i l × | A | e j φ 1 × e j φ 2 = | A w | e j Δ φ ,
L o s s  =  1 M F ( α Σ ¯ m ( R M S E m )  +  β Σ ¯ m ( N U m ) ) ,
R M S E m = Σ p ( I m I 0 ) 2 / Σ p I 0 2 ,
N U m = p W [ ( I m I m ¯ ) / I m ¯ ] 2 ,
D E m = p W I m p I m ,
I 0  =  I n exp [ 2 ( x 2 R 0 2 ) 12 2 ( y 2 R 0 2 ) 12 ] ,
I  =  I n exp [ 2 ( ( a x x 0 ) 2 R 0 2 ) 3 2 ( ( b y y 0 ) 2 R 0 2 ) 3 ] + n ,
A F m = 1 c m + d m ,