|
High fidelity laser beam shaping using liquid crystal on silicon spatial light modulators as diffractive neural networks
|
1. Introduction .
In laser materials processing, application-adapted intensity distributions gain importance, as they enable an increase in productivity and/or the quality of the processing result [ 1 – 5 ]. Such intensity distributions can range from simple homogeneous distributions (so-called top-hat distributions) [ 6 ] or donut shapes [ 7 ] to process-specific distributions, which are derived by solving the inverse heat-conduction problem [ 8 ]. Such application-adapted distributions can be realized with freeform optics [ 9 , 10 ] or via diffractive optical elements (DOEs) [ 11 , 12 ]. In recent years, spatial light modulators based on liquid crystal on silicon [ 13 , 14 ] have gained popularity because they offer the optical functionality of a diffractive element while being dynamically switchable so that they can be flexibly adapted for the respective application.
While LCoS-SLMs provide great flexibility in laser beam shaping and are constantly improved to be usable at ever increasing power [ 14 , 15 ], the created field distributions exhibit deviations from the nominal design acquired with common simulation approaches. A prevalent deviation occurs when shaping a distribution in the focal plane of a lens: the profile exhibits an undesired bright spot in the center of the distribution that is often called the ’0-th diffraction order’. This can be attributed to mostly two dominant factors, an unmodulated, direct reflection at the surface of the SLM and an electronic interaction between neighboring pixels, conventionally called pixel crosstalk or fringing effect [ 16 , 17 ]. In the common approach for designing phase masks for application-adapted beam shaping, the Gerchberg-Saxton algorithm or iterative Fourier transform algorithm [ 11 ], crosstalk is not considered in the optimization. For beam shaping, the conjugate gradient method and iterative feedback between experiment and simulation has been used to reduce crosstalk in the focal plane of a lens [ 18 ]. For beam splitting, there exists extensive work on addressing crosstalk, either by individually optimizing each pixel [ 19 ] or by methods that are based on correcting phase masks after designing. They are able to compensate the effect partially utilizing precise measurements of the SLM’s behavior [ 16 , 17 , 20 ] and using higher phase modulation than $2\pi$ which is possible with conventional LCoS-SLMs [ 21 ].
Diffractive neural networks (DNNs) are a physical equivalent to artificial neural networks. They use light as the information carrier and phase masks as the layers. These networks have already demonstrated to solve classical image processing tasks at the speed of light [ 22 ]. We have demonstrated recently, that they can also be used as a design tool for optical systems consisting out of one or multiple phase masks to enable precise laser beam shaping with several useful features including robustness against deviations, simultaneous phase and amplitude manipulation and optimization for multiple target distributions for effective three-dimensional beam shaping [ 23 ]. DNNs also allow for efficient phase mask optimization with GPUs using standard frameworks for artifical neural networks. The layers of DNNs can be implemented with SLMs.
In this work we demonstrate that our method also allows to consider and compensate both crosstalk and unmodulated reflection already during phase mask design using DNNs for arbitrary beam shaping. This significantly reduces the resulting deteriorations without the need for experimental feedback. Additionally, we will also address the native astigmatism that stems from the different angles of incidence in x- and y-direction on the SLM surface, as otherwise it becomes the dominant deviation.
2. Modeling and implementation .
For all experiments and simulations, we use a setup (see Fig.? 1 ) with a linearly polarized HeNe laser with 632.8?nm wavelength and $5.3$ ?mm Gaussian beam diameter. The SLM is a Hamamatsu X15213 LCoS with a pixel pitch of $ {12.5}\;\mathrm{\mu}\textrm{m}$ and 1024 x 1272 pixels. While we only present results with this specific SLM and laser, the method is applicable to all LCoS-SLMs even though some of the parameters for the deteriorating effects will vary. As a focusing lens we use a biconvex lens with a focal length of $f= {200}\;\textrm{mm}$ . The distance between SLM and lens is $ {150}\;\textrm{mm}$ and the respective target distances are measured behind the lens. This is just an example setup, the here described effects and solution can be applied to any optical system containing LCoS-SLMs.
? Fig. 1. On the left: the experimental setup with the movable target plane. On the right: the used SLM on top of a 5-axis stage for positioning.
Download Full Size PDF
In order to compensate for the effects of astigmatism, pixel crosstalk and the unmodulated, direct reflection on the beam shaping results, we will briefly describe these inherent properties of LCoS-SLMs and show how they can be modeled and implemented in a DNN training architecture.
2.1 Optical aberrations by oblique incidence .
For non-parallel light rays, the incoming rays enters the cover glass and the liquid crystal layer at different angles with respect to the x-axis ( $\approx 5^{\circ }-10^{\circ }$ ) and the y-axis ( $0^{\circ }$ ). This means that the effective pixel size is different in x- and y-direction resulting in a form of astigmatism. One way to compensate the astigmatism is to add an additional asymmetric refractive power to the system - either by a cylindrical lens or a cylindrical phase mask. This has previously been done for steep angles where the effect is very pronounced [ 24 ]. In the optical model, the SLM has no thickness but is modelled by a plane at which the respective phase mask is applied to the field. To consider the aberration due the astigmatism in our model, we add an additional cylindrical phase to the field at this plane that leads to the same astigmatism we observe in the experiment. Inclusion of this effect in the optical model enables the DNN to correct the astigmatism during training. In conventional phase mask design, this is not necessary and the cylindrical phase can be directly added to the beam shaping phase mask after design. However, since we also consider crosstalk in the optimization (cf. Section 2.2 ), modifying the phase mask by adding a cylindrical lens phase would lead to phase values that are larger than the phase range of the SLM. Usually, one would still be able to display this new phase mask by applying a modulo function as in conventional masks a phase shift of $0$ is equivalent to a phase shift of $2\pi$ . This is not the case when dealing with crosstalk and thus renders the approach of adding phase masks after training impossible. The effect of astigmatism for a top-hat intensity distribution in the focal plane of a lens can be seen in Fig.? 2 . In contrast to the edges along the x-axis, only the edges along the y-axis are sharply resolved.
? Fig. 2. Simulation of the individual effects of astigmatism, pixel crosstalk and the unmodulated reflection on the beam shaping result for a square top-hat distribution in the focal plane of a lens.
Download Full Size PDF
2.2 Pixel crosstalk .
The causes of pixel crosstalk in LCoS-SLMs and approaches to its modelling are described in high detail in [ 21 , 25 ]. Fundamentally, pixel crosstalk boils down to interactions between neighboring pixels. The voltage that is applied to a single pixel in an SLM cannot be entirely isolated to this pixel. Because of this, controlling a single pixel will also affect all neighboring pixels and especially also the transition region between pixels. A schematic of this effect is depicted in Fig.? 3 .
? Fig. 3. Illustration of the crosstalk effect. Here, $x$ refers to the voltage which is applied to rotate the crystal orientations and induce the phase modulation. The left side shows the ideal case where the applied voltage induces a sharp phase modulation profile. In contrast, on the right side, the neighboring pixels are influenced by the applied voltage in the center pixel, which leads to a smoothed out phase profile.
Download Full Size PDF
In order to model pixel crosstalk properly in the DNN training architecture, modifications have to be implemented: as mentioned before, in conventional phase masks, a phase shift of $0$ will be equivalent to a phase shift of $2\pi$ due to the periodic nature of the phase of a electromagnetic field. Thus, having phase shifts which are multiples of $2\pi$ will behave exactly as phase shifts forced to the interval $[0, 2\pi )$ through a modulo operation. This is not the case anymore when dealing with crosstalk where effectively a shift of $0$ will ’pull down’ all neighboring values and a shift of $2\pi$ will raise them. This means, when training with unconstrained weights and using a modulo operation for staying in the phase range of the respective SLM, that there will be an non-differentiable point between $2\pi$ and $0$ leading to unstable training. This is addressed by applying constraints to the weights, by either clamping to limits or applying an appropriate function like $\tanh (x)$ or $\sin (x)$ , which take an arbitrary input but limit the output to a fixed range of values. Training results have shown that clamping the weights after every training step leads to stable and successful training results. Thus, when naming the unconstrained training parameter $\phi _p$ , the phase shift $\phi$ of a diffractive layer before passing through the crosstalk convolution layer is
(1) $$\phi = \textrm{clamp}(\phi_p, [0, f_{\pi}\cdot \pi]).$$
Here, the factor $f_{\pi }$ is conventionally $2$ but since common SLMs often have a modulation range larger than $2\pi$ , as e.g. utilized in [ 21 ], $f_{\pi }$ can be chosen to be the modulation range of the respective SLM for utilizing its full range.
We model the effect of crosstalk by a convolution of the nominal phase mask $\phi$ with a super-gaussian kernel [ 20 ]
(2) $$\phi_{CT}(x, y) = \phi(x, y) \ast G(x, y)$$
with the kernel $G(x, y)$ defined as (3) $$G(x, y) = \exp \left[-\left(\frac{x^2}{2\sigma_x^2}+\frac{y^2}{2\sigma_y^2} \right)^\gamma \right].$$
We chose this model, because it is straight-forward to implement, differentiable and describes the effect sufficiently as demonstrated in the following examples. More complex models for the crosstalk, as for instance described in [ 21 ], can also be implemented but may require additional measurements of the defining parameters. In the simpler convolution approach, only the values of $\sigma _x$ , $\sigma _y$ and $\gamma$ have to be determined for the specific SLM in use. Strategies for measuring these values are described in [ 25 ]. We have determined these parameters by iterative comparison of measurements with simulations (including crosstalk) (cf. Table? 1 ).
Table 1. Parameters for the crosstalk model and the direct reflection on the SLMs cover glass determined for the setup by comparison of experimental results with simulations.
View Table
Adding the convolution for the crosstalk model to the training process can be done in a few steps: to the existing pool of modules like free-space propagation, scalar phase mask, lenses etc. we define a new module specifically for implementing the super-gaussian convolution. This means that in the forward pass through the network we pass the phase modulation of a diffractive layer into this new convolution layer. It is important to note that this should not be confused with convolutional layers in convolutional neural networks. The effect of crosstalk on the top-hat intensity distribution in the focal plane can be again seen in Fig.? 2 . Both fringes and a ’0-th diffraction’ order are very apparent.
2.3 Reflection at cover glass .
We use a simplified model for the unmodulated reflection. As the SLMs work with linearly polarized light only, distinction of the reflection coefficient for different polarizations is neglected. The reflectivity of the anti-reflection coating of the specific SLM is denoted as $r$ and the transmittance as $t$ . These are solely numerical parameters and not identical to the Fresnel coefficients which would require the knowledge of the refractive index of the cover glass. The directly reflected part of the field undergoes a constant phase shift $\phi _c$ relative to the modulated field which can then be expressed as
(4) $$U_{\textrm{out}}(x,y) = t \cdot U_{\textrm{in}}(x,y) \cdot \exp(i \phi_{CT}) + r \cdot U_{\textrm{in}}(x,y) \exp(i \phi_{\textrm{c}}).$$
Here, $U_{\textrm {in}}(x,y)$ denotes the incoming field and $U_{\textrm {out}}(x,y)$ the outgoing field.
The effect of unmodulated reflection with $\phi _c=0$ on the top-hat is shown in Fig.? 2 . It only results in a bright spot in the center and otherwise does not affect the distribution. It is important to note that this model assumes the cover glass of the SLM to be completely planar. Since the mirror of the LCoS-SLM is also slightly curved, this might not be the case and we will discuss the implications in Section 3 . Additionally, it was reported, that the assumption of only a single unmodulated reflection might be insufficient [ 26 ] and that these multiple reflections result in Fabry-Perot interference. An adaption of a more diligent model is planned for future work.
2.4 Calibration and evaluation .
Besides crosstalk, unmodulated reflection and astigmatism, another influencing factor is the curvature of the SLM’s mirror which we have not described before. The dielectric mirrors in LCoS-SLMs from Hamamatsu are slightly curved due to the manufacturing process. Hamamatsu provides correction phase masks for each individual SLM that can be added to calculated phase masks in order to compensate this curvature. However, similarly to the astigmatism correction, it would not be expedient to add this correction to a phase mask containing a correction for crosstalk because the crosstalk effects before addition would be different than those after the addition, since the addition of the phase masks would require a new modulo operation. We employ the same solution as for the astigmatism by taking the negative of this correction phase mask into the physical model that the DNN is optimizing. Then, the DNN can find a proper solution for this effect while still considering crosstalk. The phase masks of both the curvature and the cylindrical lens are positioned virtually in the same plane as the SLM.
The parameters determining the effects of the described deviations that represent our setup (see Fig.? 1 ) were estimated by varying the model parameters in simulations and then comparing the propagation results to single measurements. The determined values are summarized in Table? 1 . The modulation range (provided by the manufacturer) of the SLM for 632.8?nm is $f_{\pi } = 2.3$ . It should be emphasized that the relative constant phase shift $\phi _c$ between the modulated and the unmodulated field is a sensitive model parameter. It depends on both the specific SLM and the angle of incidence on the SLM due to the optical path length variation that occurs inside the liquid crystal layer itself. This is especially relevant as in the case when working with a completely erroneous assumption for $\phi _c$ (which would be an error of $\pi$ ), this would mean the difference between constructive and destructive interference and lead to even stronger deviations compared to the case where the effect is than not considered at all. Additionally, as the parameter $\phi _c$ is sensitive to alignment, it has to be recalibrated for a change in the system - for instance after alignment. On the other hand, it is possible to manually set an arbitrary $\phi _c$ in the experimental setup, by slightly adjusting the alignment. This can result in intensity distributions with similar quality for any assumed $\phi _c$ . That is, why we simply set $\phi _c=0$ and manually changed the alignment until the distribution is optimal. An alternate approach is also possible: we trained 20 DNNs simultaneously and assume a different $\phi _c$ ranging from $0$ to $2\pi$ in steps of $0.1 \pi$ for the computations. Then the most suitable $\phi _c$ is determined by testing out all phase masks. This procedure is preferable if one does not want to modify the alignment at all.
Comparison between simulations and experiments confirm our approach for including the deteriorating effects in the model and their respective parametrization (Fig.? 4 ). A clear agreement between simulation and experiment can be seen for a top-hat distribution, once in the focal plane ( $d= f = {200}\;\textrm{mm}$ ) and once with 10?mm defocus ( $d = f - {10}\;\textrm{mm} = {190}\;\textrm{mm}$ ). In the focal plane, in simulation and measurement the ’0-th diffraction order’ is observed, which is identified as the combined pattern of crosstalk and unmodulated reflection (Fig.? 2 ). In the case of the target plane outside of the focus, the ’0-th diffraction order’ appears as pronounced fringes.
? Fig. 4. Comparison of simulated and measured intensities for a top-hat distribution at two different distances behind the focusing lens. In the left column the simulated intensity distributions are shown that result from a model which includes an ideal phase mask but no deteriorating effects. The center column shows the results including the effects of direct reflection on the cover glass, astigmatism due to oblique incidence, and crosstalk in the model. In the right column the corresponding measurements are depicted. There is a good agreement between the computed and measured distributions which demonstrates that the model is capable to produce the dominant deteriorating effects for beam shaping with the SLM.
Download Full Size PDF
3. Results .
3.1 Case study .
Two different example target intensity distributions are considered, a square top-hat and a homogeneous donut distribution, see Fig.? 5 . We have demonstrated in previous work that much more complex distributions can be realized successfully with DNNs built from SLMs [ 27 ], but here we focus on simple distributions as they are easier to quantify visually. The distributions are quite distinct in size to show how different sizes affect the beam shaping results. The color bar is always limited to 1.5 of the respective maximum target intensity which we call $I_{0_{\textrm {max}}}$ . This is done to have a consistent scale where small deviations from the target intensity are still visible which would otherwise have very low contrast if the intensity scale was large enough to not crop the intensity of the highest peak. This refers mostly to the ’0-th diffraction order’ that dominates the uncorrected distributions. For all intensity distributions that we optimized, the highest measured peaks are significantly lower than in the cases that are not optimized. This normalization also means, that since the donut distribution is much larger, given the same total power, $I_{0_{\textrm {max}}, {\textrm {donut}}} < I_{0_{\textrm {max}},{\textrm {top-hat}}}$ . For brevity, we will from now on in all figures write $I_{0_{\textrm {max}}}$ where it will refer to the respective target distribution that is depicted.
? Fig. 5. The two example target intensity distributions used in this work. The length and intensity scale is adapted for each distribution respectively to address the difference in size.
Download Full Size PDF
We choose three different target planes - with distance $d$ measured from the focusing lens - to highlight different design aspects and guidelines:
1. $d= {200}\;\textrm{mm}$ : the focal plane as conventionally used with SLMs because the propagation is performed with a Fourier transform [ 28 ] and is fast and straight forward to compute.
2. $d= {190}\;\textrm{mm}$ : to demonstrate the effect of defocusing on the ’0-th diffraction order’ and the impact of the compensation
3. $d= {163}\;\textrm{mm}$ / $d= {150}\;\textrm{mm}$ : these are the respective distances for the top-hat/donut distribution, where the phase masks requires the least amount of modulation. We determined these distances by setting the SLM phase mask to one constant value during training and transforming the target distance into a trainable parameter. This way the optimizer automatically finds the distance where the training loss is lowest without changing the beam shape which means that the beam size is approximately consistent with the target distribution. This is similar to estimating the geometrical beam size in various distances compared to the target intensity distribution size but trivial to include in our training architecture.
We trained DNNs for each distance and target distribution and cumulatively consider more effects during training for compensation. These are the four cases that are considered:
? Case 1 - Ideal or default system: besides the curvature of the dielectric mirror, the SLM is treated as an ideal phase mask. Deviations are not compensated.
? Case 2 - Compensation of astigmatism: the effect of the tilted angle is also modeled as a virtual cylindrical lens during training.
? Case 3 - Astigmatism and crosstalk compensated: additionally, we consider the neighbouring pixel interactions through the convolution.
? Case 4 - Astigmatism, crosstalk and unmodulated reflection compensated: here, we consider the combination of all three discussed effects.
The free-space propagation is implemented as the band-limited angular spectrum method [ 29 ] which is an accurate propagation method for all considered distances. The SLM has $1024 \times 1272$ pixels and in the numerical model a grid size of $2048\times 2544$ pixels is used. On this grid, the convolution for the crosstalk and the optical propagation is calculated.
3.2 Training parameters and procedure .
As described in [ 23 ], the training infrastructure is implemented in PyTorch Lightning [ 30 , 31 ]. Training is performed with a single Gaussian input beam since robustness training as originally proposed in [ 32 ] does not benefit beam shaping results significantly, when using only one phase mask. The ADAM optimizer is chosen [ 33 ] due to its flexibility. We use the mean-squared error loss function of target and propagated intensity. The ideal systems are trained with 200 epochs to ensure that the loss converges. These ideal phase masks are then used as the initial guess for the following design studies, because it results in phase masks that look more similar to each other and thus makes identifying subtle differences in the corresponding intensity distributions easier.
For the distances that are not the respective target plane where the phase masks require the least amount of modulation (so $d= {163}\;\textrm{mm}$ / $d= {150}\;\textrm{mm}$ respectively), during the first training steps we do not treat the pixels as individual trainable parameters but instead we use the phase mask of an ideal lens with its refractive power $1/f$ as the sole trainable parameter. This results in a macroscopic phase mask that can already establish a beam size close to the target beam size - similar to setting the target distance as a trainable parameter to find an optimal distance. Even though it does not necessarily impact the loss significantly, the phase masks will be more structured and continuous, resulting in reduced noise in the experiment. Additionally, it is not necessary to choose an initial guess in this case. When enabling all pixels to be trainable parameters, the lens refractive power is also a trainable parameter, so the effective phase mask is a superposition of the trainable lens and a mask with trainable pixels. This results in faster convergence of the network as the macroscopic scale of the phase masks can be optimized simultaneously as the individual pixels.
Generally, when considering pixel crosstalk, due to the more complex model and the limitation of the weights to the range $[0, f_{\pi }\cdot \pi ]$ , the training is more likely to get stuck in local minima, resulting in a higher loss when working only with the trainable lens for generating an initial guess. Usually we solve this by training the first twenty (including the training steps with only the curvature of a lens as a parameter) epochs without considering crosstalk and only start to simulate and correct crosstalk after these first epochs. In this work of course, we actually used the ideal phase masks as initial guesses.
For evaluation of the measurement results we use a normalized root-mean-square deviation (RMSD) from the target intensity distribution defined as
(5) $$\textrm{RMSD}(I) = \frac{1}{n} \sum_{i=0}^{n-1} \frac{\sqrt{(I_i-I_{0_i})^2}}{ I_{0_{max}}}$$
with the number of data points $n$ , the measured/simulated intensity distribution $I$ , the target distribution $I_0$ and the maximal target intensity $I_{0_{max}}$ in a single data point. The RMSD is one possible metric to quantify the deviations and there are alternative metrics that could be more relevant for certain applications. We also use target distributions that have perfectly sharp edges which are not achievable due to the diffraction limit. This means that the RMSD can never reach zero but more important than the absolute value of the RMSD is its evolution when looking at the compensation of the various effects. We normalize all measured intensity distributions by dividing it by the sum of all values to not lose too much information to the diffraction efficiency of the SLM which is constant for all measurements. Additional differences of the efficiency of the phase masks are smaller than the power fluctuations of our laser source. 3.3 Experimental validation .
The measurements show that the RMSD can be reduced for all distances and all effects for both the top-hat distribution (Fig.? 6 ) and the donut distribution (Fig.? 7 ). In the focal plane, the ’0-th diffraction order’ in the top-hat distributions vanishes nearly completely while in the donut distribution it is at least significantly reduced. Also at 190?mm, the imhomogeneities that occur in the ’default’ intensity distributions can be diminished drastically. At 163/150?mm respectively, the differences are more subtle but the RMSD values clearly show an improvement with each additional considered and corrected effect. When examining the development of the RMSD, one can see that the fully compensated phase masks for both distributions achieve a lower RMSD than the uncompensated masks at all distances. This means that there is always a benefit to compensating the deviations which cannot be achieved by solely shifting the target plane. From these measurements alone, one cannot infer a generalized rule which distance is optimal for arbitrary beam shaping. We will discuss this hereafter. While the correction of unmodulated reflection generally has the smallest impact on the RMSD, it is vital for correcting the ’0-th diffraction order’ in the focal plane for the top-hat distribution.
? Fig. 6. All measurement results for the sequential compensation of astigmatism, crosstalk and unmodulated reflection for the top-hat distribution. In the left most column the results with a default phase mask are shown where no deviation is compensated besides the curved mirror of the SLM. With each subsequent column the compensation of an additional effect is depicted, so that in the right most column, ’Reflection’ all three effects are compensated simultaneously according to the cases 1)-4). The different rows correspond to the different target distances. In the bottom row, we depict the normalized root-square-deviation ( $\textrm {RSD} = \sqrt {(I-I_0)^2}/ I_{0_{\textrm {max}}}$ ) for the focal plane.
Download Full Size PDF
? Fig. 7. All measurement results for the combined compensation of astigmatism, crosstalk and unmodulated reflection for the donut distribution. In the bottom row the root square deviation for the focal plane is shown.
Download Full Size PDF
A key lesson from these results is that it is possible to reduce the RMSD of arbitrary target distributions at arbitrary distances using our method and all three considered effects contribute to this. There are limits to the model and method which are discussed in the following section.
4. Simulative analysis and discussion .
While the experimental results confirm the viability of the approach, the theoretical expectations of the RMSD reduction are more significant (see Fig.? 8 for the focal plane as an example) than observed in the experiment. We can use the simulative results to evaluate general guidelines for laser beam shaping with LCoS-SLMs. When looking at the intensity distributions of the fully optimized phase masks (Fig.? 9 ), one can see that the RMSD gets reduced for distances closer to the lens and further away from the focal plane. This can be justified, because target distances closer to the focal plane require smaller structure size (and thus higher variation) on the phase mask (Fig.? 10 ) which are more difficult to resolve when considering crosstalk.
? Fig. 8. Simulation correction results for the focal plane. These simulations correspond to the respective measurements in the third row in Fig.? 6 and 7 . The effects are again considered cumulatively with each additional column and propagated with the model including all deteriorating effects.
Download Full Size PDF
? Fig. 9. Simulation results for the respective target distances with all effects compensated.
Download Full Size PDF
? Fig. 10. The phase masks for the respective distances and for the distribution that compensates for all deviations. The visible asymmetry stems from the asymmetrically curved mirror of our SLM that needs to be corrected.
Download Full Size PDF
Two main aspects can be made responsible for the remaining discrepancies: model and physical limitations. We will discuss them in the following:
4.1 Model limitations .
We only chose fairly simple models for modeling the three different effects and did not determine the corresponding parameters with sophisticated methods. If the models are not fully appropriate, the correction can also not work optimally. It can be seen in the right most column in Fig.? 8 that the crosstalk fringes could theoretically be corrected with higher accuracy than seen in the experiments. This most likely means that the parameters for the convolution were not determined with enough precision as slight deviations of $\sigma _{x}$ and $\sigma _{y}$ (both positive and negative) will result in a different fringe pattern and we did not adapt $\gamma$ at all. When comparing the most bottom right top-hat in Fig.? 6 to the same top-hat in Fig.? 8 one can also see that in the measurement instead of a bright ’0-th diffraction order’ there is an intensity dip that is more pronounced than in the simulation. A probable explanation is that the glass cover of the SLM is not planar but slightly curved, as discussed in Section 2.3 . A slight curvature of the glass would mean that in the focal plane, the ’0-th diffraction order’ would vary in shape and position. Assuming the cover to be flat would lead to a mismatch between the corrected ’0-th diffraction order’ and the actual effect. For example, locally the intensity could become even lower than the target because the position of the ’0-th diffraction order’ is not modeled correctly. Furthermore, we neglected the multiple reflections described in [ 26 ]. The implications of the curvature and the multiple reflections for target planes beyond the focal plane will be investigated in the future. For the donut distribution, the RMSD for the default case is significantly larger than in the measurement since the magnitude of the ’0-th diffraction order’ is overestimated and has an intensity peak that is more than $16$ times larger than the target intensity $I_{0_{\textrm {max}}}$ which is not visible in the shown images due to the cutoff of the intensity scale to $1.5\cdot I_{0_{\textrm {max}}}$ . In the corresponding experiment, the ’0-th diffraction order’ reaches only a maximum of roughly $6\cdot I_{0_{\textrm {max}}}$ . This can be mostly attributed to crosstalk, which is apparent in the significant drop of the RMSD when looking at the third column. This further motivates using a more sophisticated crosstalk model. In general, only astigmatism, crosstalk and unmodulated reflection are considered in this work. There are more possible discrepancies between simulation and experiment for example the assumption of an ideal lens neglecting higher-order optical aberrations or neglecting the gaps between pixels of the SLM.
4.2 Physical limitations .
While astigmatism correction does not negatively impact the training results at least for conventional angles of SLMs, crosstalk and unmodulated reflection put hard physical constraints on the achievable phase masks. Crosstalk especially limits the achievable edge sharpness between neighboring pixels. This means that any phase mask that requires large phase jumps, for instances for lenses with $f \leq 1$ ?m (in our configuration), cannot be fully realized without exhibiting significant crosstalk effects. This is most notable when considering the focal plane, where a Gaussian beam (without any phase modulation) will have a size close to its diffraction limit. Here, achieving an intensity distribution that is significantly larger than the focal spot requires enough ’refraction power’ that corresponds to a large gradient between adjacent pixels which cannot necessarily be achieved with crosstalk. We selected the respective target distributions to highlight this effect: the top-hat distribution with a size of 750??m can barely be fully corrected in the focal plane while the donut distribution is already too large with a size of 1200??m, which is apparent due to the still remaining ’0-th diffraction order’. When working with smaller distributions (in our setup roughly smaller than 500??m) even in the focal plane the difference between target and simulation result can be made negligible. In practice, the realizable distribution size in the focal plane depends mostly on the pixel size of the SLM, the crosstalk parameters and the wavelength. Additionally, a larger modulation range $f_{\pi }$ (higher than 2) of the SLM leads to better compensation of the crosstalk.
In certain configurations, the effect of the unmodulated reflection in the focal plane becomes more prominent. While for the combination of target distributions and experimental setup presented in this work crosstalk is the dominant effect for the ’0-th diffraction order’, this is not always the case. For smaller target distributions in the focal plane the unmodulated reflection generally gains more significance. Furthermore, as discussed in 2.4 , the constant phase shift $\phi _c$ strongly depends on the specific alignment and since the effect of unmodulated reflection is most notable in the focal plane, this means that here the beam shaping result is highly dependent on accurate alignment. The model currently neglects additional internal reflections and the resulting Fabry-Perot interference as described in [ 26 ]. Further improvements for the beam shaping results are expected when this effect is considered additionally.
5. Conclusion .
In this work we demonstrate that phase mask design methods based on DNNs improves the beam shaping capabilities of SLMs. We consider three dominant deteriorating effects, namely astigmatism, pixel crosstalk, and direct unmodulated reflection at the cover glass to compensate them in the phase mask optimization. A significant benefit of our proposed method is the resulting reduction of the ’0-th diffraction order’ that is a commonly known effect when using LCoS-SLMs. The phase mask calculation can be done at arbitrary target planes and with arbitrary target distributions. It was confirmed experimentally, that for various target distances the consideration of each effect respectively results in a reduction of the RMSD. The results show that it is not always sensible to choose the focal plane of a lens as the target plane. The method is robust, does not require an initial guess, and optimizes both the phase masks and additional parameters of the optical system (for instance focal lengths or target distance) simultaneously. Calculating a phase mask with a resolution of 2048 x 2544 pixels takes around 30?s on an NVIDIA Tesla-100 32GB GPU.
Going forward, we plan to implement more complex models and automatically fit the corresponding parameters from experimental results. This will enable the method to be easily applicable to arbitrary SLMs without having to determine the parameters manually. It is also planned to analyze the effects and their correction on DNNs with multiple SLMs.
Funding .
Deutsche Forschungsgemeinschaft (EXC-2023 Internet of Production - 390621612).
Disclosures .
The authors declare no conflicts of interest.
Data availability .
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References .
1. A. B. González and J. Pozo, “Optical beam shaping: Unmet needs in laser materials processing,” Optik & Photonik (2017).
2. D. Flamm, D. G. Grossmann, M. Jenne, et al. , “Beam shaping for ultrafast materials processing,” Proc. SPIE 10904 , 109041G (2019). [ CrossRef ] ?
3. S. Vogt, A. V?ll, S. Wollgarten, et al. , “Local laser softening of high-strength steel with an adapted intensity,” J. Laser Appl. 31 (1), 012007 (2019). [ CrossRef ] ?
4. J. Rauschenberger, D. Vogler, C. Raab, et al. , “Diffractive beam shaping for enhanced laser polymer welding,” Proc. SPIE 9351 , 935110 (2015). [ CrossRef ] ?
5. C. Bischoff, F. V?lklein, J. Schmitt, et al. , “Design and manufacturing method of fundamental beam mode shaper for adapted laser beam profile in laser material processing,” Materials 12 (14), 2254 (2019). [ CrossRef ] ?
6. F. M. Dickey, L. S. Weichman, and R. N. Shagam, “Laser beam shaping techniques,” in High-Power Laser Ablation III, vol. 4065C. R. Phipps, ed., International Society for Optics and Photonics (SPIE, 2000), pp. 338–348.
7. O. Pütsch, J. Stollenwerk, M. Kogel-Hollacher, et al. , “Annular BEAM SHAPING SYSTEM FOR ADVANCED 3D LASer brazing,” Adv. Opt. Technol. 1 (5), 397–402 (2012). [ CrossRef ] ?
8. A. V?ll, S. Vogt, R. Wester, et al. , “Application specific intensity distributions for laser materials processing: Tailoring the induced temperature profile,” Opt. Laser Technol. 108 , 583–591 (2018). [ CrossRef ] ?
9. Z. Feng, L. Huang, M. Gong, et al. , “Beam shaping system design using double freeform optical surfaces,” Opt. Express 21 (12), 14728–14735 (2013). [ CrossRef ] ?
10. A. V?ll, R. Wester, M. Berens, et al. , “Accounting for laser beam characteristics in the design of freeform optics for laser material processing,” Adv. Opt. Technol. 8 (3-4), 279–287 (2019). [ CrossRef ] ?
11. J. S. Liu and M. R. Taghizadeh, “Iterative algorithm for the design of diffractive phase elements for laser beam shaping,” Opt. Lett. 27 (16), 1463–1465 (2002). [ CrossRef ] ?
12. D. C. O’Shea, T. J. Suleski, A. D. Kathman, et al. , Diffractive Optics: Design, Fabrication, and Test (SPIE Vol. TT62) (SPIE- International Society for Optical Engineering, 2003).
13. T. Inoue, H. Tanaka, N. Fukuchi, et al. , “Lcos spatial light modulator controlled by 12-bit signals for optical phase-only modulation,” in Emerging Liquid Crystal Technologies II, vol. 6487L.-C. Chien, ed., International Society for Optics and Photonics (SPIE, 2007), pp. 212–222.
14. G. Zhu, D. Whitehead, W. Perrie, et al. , “Thermal and optical performance characteristics of a spatial light modulator with high average power picosecond laser exposure applied to materials processing applications,” Proc. CIRP 74 , 594–597 (2018). 10th CIRP Conference on Photonic Technologies [LANE 2018]. [ CrossRef ] ?
15. J. Harriman, S. Serati, and J. Stockley, “Comparison of transmissive and reflective spatial light modulators for optical manipulation applications,” Proc. SPIE 5930 , 59302D (2005). [ CrossRef ] ?
16. E. Ronzitti, M. Guillon, V. de Sars, et al. , “Lcos nematic slm characterization and modeling for diffraction efficiency optimization, zero and ghost orders suppression,” Opt. Express 20 (16), 17843–17855 (2012). [ CrossRef ] ?
17. J. Liang, S.-Y. Wu, F. K. Fatemi, et al. , “Suppression of the zero-order diffracted beam from a pixelated spatial light modulator by phase compression,” Appl. Opt. 51 (16), 3294–3304 (2012). [ CrossRef ] ?
18. P. Schroff, A. L. Rooij, E. Haller, et al. , “Accurate holographic light potentials using pixel crosstalk modelling,” Sci. Rep. 13 (1), 17670 (2023). [ CrossRef ] ?
19. G. Milewski, D. Engstr?m, and J. Bengtsson, “Diffractive optical elements designed for highly precise far-field generation in the presence of artifacts typical for pixelated spatial light modulators,” Appl. Opt. 46 (1), 95–105 (2007). [ CrossRef ] ?
20. M. Persson, D. Engstr?m, and M. Goks?r, “Reducing the effect of pixel crosstalk in phase only spatial light modulators,” Opt. Express 20 (20), 22334–22343 (2012). [ CrossRef ] ?
21. S. Moser, M. Ritsch-Marte, and G. Thalhammer, “Model-based compensation of pixel crosstalk in liquid crystal spatial light modulators,” Opt. Express 27 (18), 25046–25063 (2019). [ CrossRef ] ?
22. X. Lin, Y. Rivenson, N. T. Yardimci, et al. , “All-optical machine learning using diffractive deep neural networks,” Science 361 (6406), 1004–1008 (2018). [ CrossRef ] ?
23. P. Buske, A. V?ll, M. Eisebitt, et al. , “Advanced beam shaping for laser materials processing based on diffractive neural networks,” Opt. Express 30 (13), 22798–22816 (2022). [ CrossRef ] ?
24. J. Starobrat, P. Wilczynska, and M. Makowski, “Aberration-corrected holographic projection on a two-dimensionally highly tilted spatial light modulator,” Opt. Express 27 (14), 19270–19281 (2019). [ CrossRef ] ?
25. M. Guesmi and K. ?ídek, “Calibration of the pixel crosstalk in spatial light modulators for 4f pulse shaping,” Appl. Opt. 60 (25), 7648–7652 (2021). [ CrossRef ] ?
26. J. L. Martínez, I. Moreno, M. del Mar Sánchez-López, et al. , “Analysis of multiple internal reflections in a parallel aligned liquid crystal on silicon slm,” Opt. Express 22 (21), 25866–25879 (2014). [ CrossRef ] ?
27. P. Buske, A. V?ll, J. Stollenwerk, et al. , “Diffractive neural networks built from spatial light modulators for laser beam shaping in the NIR and VIS,” Proc. SPIE 12798 , 127981Q (2023). [ CrossRef ] ?
28. J. W. Goodman, Introduction to Fourier optics (McGraw Hill, 1996). 2nd edition.
29. K. Matsushima and T. Shimobaba, “Band-limited angular spectrum method for numerical simulation of free-space propagation in far and near fields,” Opt. Express 17 (22), 19662–19673 (2009). [ CrossRef ] ?
30. A. Paszke, S. Gross, F. Massa, et al. , “Pytorch: An imperative style, high-performance deep learning library,” in Advances in Neural Information Processing Systems 32, H. Wallach, H. Larochelle, A. Beygelzimer, et al. , eds. (Curran Associates, Inc., 2019), pp. 8024–8035.
31. W. Falcon and The PyTorch Lightning team, “PyTorch Lightning,” (2019).
32. J. Shi, D. Wei, C. Hu, et al. , “Robust light beam diffractive shaping based on a kind of compact all-optical neural network,” Opt. Express 29 (5), 7084–7099 (2021). [ CrossRef ] ?
33. D. P. Kingma and J. Ba, “Adam: A method for stochastic optimization,” (2017). .
|
|
From：
|
|
|
系统抽取主题
|
(1)
|
|
(1)
|
|
(1)
|
|
(1)
|
|
(1)
|
|
(1)
|
|
(1)
|
|
(1)
|
|
(1)
|
|
(1)
|
|
(1)
|
|
(1)
|
|
(1)
|
(1)
|
|