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Abstract
An optofluidic phase shifter can be used to generate virtually arbitrary intensity patterns, but only if the phase shift generated by the controllably deformed fluidic surface can be appropriately defined. To enable this functionality, we present two phase retrieval algorithms based on neural networks and least-squares optimization which are used to determine the necessary phase profile to generate a desired target intensity pattern with high accuracy. We demonstrate the utility of the algorithms by showing experimentally the ability of an optofluidic phase shifter to generate arbitrary complex intensity distributions.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Optical beam forming systems with dynamically programmable intensity profiles, radiance patterns, and beam directions have a wide spectrum of potential applications, ranging from laser processing [1,2] to optical lithography [3,4] to patterned illumination. One promising technology to achieve such functionality is the use of optofluidic components in which a deformable optical surface based on a liquid/liquid interface [5–9] is used to define a spatially-varying phase shift over the entire aperture of a beam. This phase shift gives rise to a controllably deformed wavefront which then in turn defines a two-dimensional intensity distribution in the far-field. However, determining the necessary phase shift to yield a desired intensity pattern remains a difficult inverse problem [10,11] such that having a fast and accurate algorithm for retrieving the required phase from the intended beam profile is an important prerequisite for using tunable optofluidic phase shifters as reconfigurable illumination sources.
Various phase retrieval methods have been proposed for the determination of such freeform surfaces, including the “Monge-Ampère-equation method” [10,12,13] or the “ray mapping method” [10,14,15]. However, both of these approaches are based on solving complex equations and therefore require a high computational effort. Other methods, such as the iterative Fourier transform algorithm [11,16,17] or the Gerchberg-Saxton algorithm [18–22], have also shown that it is possible to extract phase information from target intensity distributions. However, these algorithms are often very slow; frequently find only a local minimum; or have a limited applicability [11,19].
Recently, deep learning based approaches using convolutional neural networks (CNNs) have been demonstrated to be a powerful and computationally efficient tool in numerous optical applications [23–28]. We present here two phase retrieval algorithms based on a CNN which are able to accurately determine the required phase information for generation of a target intensity profile in terms of Zernike coefficients, and show how we use these to generate a desired beam profile using an optofluidic phase-shifting component. The first approach includes a CNN which predicts the Zernike coefficients from a desired intensity distribution. These predicted values form the starting points for a least-squares optimization that further minimizes the error by varying the Zernike coefficients. The second approach is based on a least-squares optimization algorithm that calculates the required phase profile from binary input masks, again to generate intensity profiles that have a defined shape. Using these approaches, we experimentally show that arbitrary beam profiles can be generated by a tunable optofluidic phase shifter, opening up new possibilities for the use of these systems as reconfigurable illumination sources.
2. Optofluidic phase shifter
The optofluidic phase shifter which we will employ to demonstrate the utility of the phase retrieval algorithms proposed consists of a cylindrical liquid-filled tube with 32 azimuthally-distributed actuation electrodes; these are used for controlled deformation of the liquid/liquid interface using electrowetting-on-dielectrics (EWOD) [29,30]. The glass tube is filled with two immiscible liquids with different refractive indices, $n_1$ and $n_2$. By applying defined voltages to the individual electrodes on the inner circumference of the cylinder, the contact angle of the liquid/liquid interface at each electrode can be individually defined, resulting in a precisely deformed optical interface.
In Fig. 1, side views of the transparent glass cylinder showing three different interface configurations are seen; the semitransparent electrodes are the vertical lines and the deformed meniscus between the two liquids is clearly seen. The structure will be discussed in more detail in Section 7.
Fig. 1. Side view of the tubular optofluidic phase shifter used in the experimental section below. By application of appropriate voltages on the azimuthally-distributed electrodes (the semitransparent vertical lines), the liquid/liquid interface can be controllably deformed.
When an input optical field $E_{in}(x',y',0)$ with wavelength $\lambda _0$ passes through this optofluidic phase shifter, it undergoes a two-dimensionally spatially varying phase shift $\phi (x',y')$ at the liquid/liquid interface, which is defined by
where $k_0 = \frac {2 \pi }{\lambda _0}$ is the wavenumber; $n_1$ and $n_2$ are the refractive indices of the two liquids; and $z(x',y')$ is the surface profile of the liquid/liquid interface. The resulting deformed wavefront then defines a 2D intensity distribution $I(x,y,z_s)$ at a screen at a distance $z_s$ from the liquid/liquid interface. This resultant intensity distribution can be calculated using the Fresnel approximation,3. Typical phase retrieval procedure
To generate a desired intensity distribution $I_{target}(x,y,z_s)$ with a phase-shifting component, it is thus necessary to determine the necessary phase shift $\phi (x',y')$, a classic inverse problem. If an applicable phase retrieval algorithm is not available, an estimation of the phase shift from the target intensity distribution is required. The typical approach involves making a guess of the phase from the target intensity distribution; calculating the intensity distribution with Eq. (2); and comparing it with the target intensity distribution. These steps are repeated until an acceptable result is achieved.
Once the phase of the wavefront is determined, this can be used to generate the surface profile $z(x',y')$ of the optofluidic phase shifter. To generate this specific shape of the liquid/liquid interface, it is then necessary to precisely calculate the voltages that must be applied to the individual electrodes. For this step, the desired surface profile $z(x',y')$ is first transformed into cylindrical coordinates, $z(r, \varphi )$, and subsequently expressed in terms of Zernike coefficients $c_{n}^{m}$ as
where $Z_{n}^{m} (x',y')$ is the Zernike polynomial; $m$ and $n$ are integers; $r$ is the radial distance and $\varphi$ is the azimuthal angle. Given $z(r, \varphi )$, we can then calculate the required contact angles $\theta _i$ at each individual electrode usingGiven the required contact angle, the voltage on electrode $i$, $V_i$, is determined from a voltage-contact angle curve measured with the help of a test electrode. For this purpose, a droplet of the aqueous solution is placed on the test electrode surrounded by the oil phase. Several forward and backward actuation cycles are then recorded using a contact angle goniometer (OCA15Pro, DataPhysics Instruments GmbH, Filderstadt, Germany) and the voltages are calculated by evaluating the inverse interpolation of the averaged voltage-contact angle curve. The calculated voltages $V_i$ are then applied to the individual electrodes and the measured intensity distribution is evaluated.
4. New phase retrieval approach
Since such an iterative process of guessing the phase, as described in Section 3, must be repeated until an acceptable result is obtained, it is very time-consuming. To make this procedure considerably more efficient, we introduce a phase retrieval algorithm which represents a powerful and computationally effective tool to calculate the phase profile required to yield arbitrary target intensity distributions and directly drive an optofluidic phase shifter.
Figure 2 shows a schematic representation of this approach. First, the desired intensity distribution $I_{target}(x,y,z_s)$ is defined as input to a CNN that has been trained with pairs of known Zernike coefficients $\vec{c}$ and their corresponding intensity profiles $I (x,y,z_{s})$. The CNN is thus able to predict 14 Zernike coefficients of $Z_{n}^{m} (x',y')$ with $|m|=n<=8$ defining a phase profile which should yield the target intensity distribution. A least-squares optimization algorithm subsequently minimizes the difference between the target intensity distribution and the calculated intensity distribution using this predicted phase profile by varying the Zernike coefficients. From the resulting optimized Zernike coefficients, the contact angles $\vec{\theta }$ at the individual electrodes can then be calculated. In a final step, the voltages $\vec{V}$ to be applied to the electrodes of the optofluidic phase shifter are calculated from the measured voltage-contact angle curve. These calculated voltages are then applied to the electrodes and the measured beam profile is evaluated.
Fig. 2. Schematic representation of the phase retrieval approach. A pretrained neural network and least-squares optimization are used to predict the Zernike coefficients defining the modulated shape of a fluidic interface from an arbitrary intensity profile. The required contact angles at the individual electrodes are calculated from the Zernike coefficients and a measured contact angle curve is used to calculate the voltages. The desired beam profile is then generated by applying the calculated voltages to the individual electrodes of the optofluidic phase shifter.
5. Algorithm 1: convolutional neural network plus optimization
To generate the training data of the CNN, 300,000 pairs of randomly sampled Zernike coefficients $\vec{c}$ and their corresponding generated intensity distributions $I (x,y,z_{s})$ are created. For training, only Zernike coefficients of $Z_{n}^{m} (x',y')$ with $|m|=n<=8$ are used, since only these can be tuned by the optofluidic phase shifter we employ here. The intensity distributions have a size of ${5}\;\textrm{mm}$ x ${5}\;\textrm{mm}$ and, to ensure that this window shows the entire intensity profile, tip and tilt are excluded. The Zernike coefficients $\vec{c}$ are created randomly in a range of $c_{n}^{m} \in [-{100}\;\mathrm{\mu}\textrm{m},{100}\;\mathrm{\mu}\textrm{m}]$, taking into account that the contact angle $\theta$ at all electrodes is limited to the range $\theta _i \in [50.12^\circ,152.76^\circ ]$ due to the possible voltage actuation range $V_i \in [{0}{\textrm{V} _{RMS}},{200}{\textrm{V} _{RMS}}]$ of the optofluidic phase shifter. The corresponding intensity distributions $I (x,y,z)$ are calculated using Eq. (2).
The neural network is designed and trained in MATLAB R2022b. The architecture of the CNN is illustrated in Fig. 3. It consists of a 101 x 101 x 1 input layer, an initial 2D convolution layer and three residual layers, each comprising two blocks of batch normalization, rectified linear unit (ReLU) activation, 2D convolution and max pooling or average pooling, respectively. After the third residual layer, a dropout layer is implemented to avoid overfitting. A fully connected layer, having the same number of outputs as the number of Zernike coefficients to be predicted, ultimately returns the corresponding Zernike coefficients $\vec{c}$.
Fig. 3. Schematic representation of the CNN architecture consisting of the following components: an input layer, an initial 2D convolutional layer, and three residual layers. Each residual layer comprises two blocks of batch normalization, ReLU activation, 2D convolution and pooling layers. After the third residual layer, a dropout layer is implemented, followed by a fully connected layer with the number of outputs equal to the number of Zernike coefficients $\vec{c}$ to be predicted.
The network is trained for 80 epochs and a batch size of 64 on a GPU (NVIDIA Quadro RTX 4000) using the adam optimizer with a learning rate $a_{0}=0.001$. Convergence is reached after training for approximately 70 h, resulting in a CNN that is able to predict the Zernike coefficients from an arbitrary intensity distribution in less than one second. It should be noted that the training time is relatively long, but only needs to be performed once. Unless fundamental parameters, such as the aperture size or the $z_{s}$-distance, are not changed, retraining is not necessary.
For the optimization a trust-region-reflective algorithm is used, which can be described by
After completion of the training and optimization processes, the performance of the phase retrieval algorithm is evaluated using a test set consisting of pairs of Zernike coefficients and intensity distributions not previously used for the training of the CNN. The ability of the phase retrieval algorithm to reconstruct an arbitrary intensity distribution is shown in Fig. 4(a)-(f), where the input, the predicted and the optimized intensity profiles and their corresponding surface profiles are shown. It can be seen that the reconstructed intensity distributions both show good agreement with the target intensity distribution. The error between the retrieved and true surface profile is shown in Fig. 4(g). Since the size of the input beam is limited by an aperture, the error is evaluated within a diameter of $D={2}\;\textrm{mm}$.
Fig. 4. Results of the phase retrieval algorithm. (a) Arbitrary input intensity distribution and the corresponding (known) target surface profile (b); (c) Intensity distribution generated by the predicted surface profile (d); (e) Intensity distribution generated by the optimized surface profile (f); (g) Error of the retrieved surface profile (between true known and optimized profiles) within a diameter of $D={2}\;\textrm{mm}$.
A quantitative evaluation of the performance of the phase retrieval algorithm is given by the root mean square error (RMSE) between the true and retrieved surface profiles. To investigate how reliably the phase retrieval algorithm works, the RMSE of $n = 100$ surface profiles retrieved from arbitrary intensity distributions was determined. The results of this statistical analysis are presented in Fig. 5 in the form of box charts. Each of these charts displays the median, the lower and upper quartiles, the whiskers and any outliers. The median value of the RMSE obtained through the CNN alone is ${0.18}\;\mathrm{\mu}\textrm{m}$. By applying the optimization algorithm, this value could be further reduced to ${0.06}\;\mathrm{\mu}\textrm{m}$ showing the ability of the algorithm to retrieve surface profiles from arbitrary intensity distributions with high accuracy. The small interquartile range also indicates the ability of the algorithm to make precise predictions.
Fig. 5. Root mean square error (RMSE) between true and calculated surface profiles, evaluated for $n = 100$ arbitrary intensity distributions within a diameter of $D={2}\;\textrm{mm}$.
In addition, the error between the retrieved Zernike coefficients and the true values was evaluated. The results show that the larger the Zernike order, the larger the error of the calculated Zernike coefficients. One reason for this is that the Zernike coefficients are calculated over a diameter of $d={5}\;\textrm{mm}$, which is the diameter of the optofluidic phase shifter. However, the diameter of the beam is limited to $D={2}\;\textrm{mm}$ by an aperture that filters out primarily higher Zernike orders, which therefore have only a minimal effect on the intensity distribution. Furthermore, the CNN was trained with randomly chosen Zernike amplitudes, so that the surface profile is in most cases dominated by the low Zernike orders. As a result, the CNN has lower accuracy in predicting the higher order Zernike coefficients, but an increase in accuracy can be achieved by adding training data in which the higher order Zernike coefficients have a stronger influence on the resulting intensity distribution.
Using the proposed algorithm offers advantages in flexibility, generalization and automation. The CNN can adapt to various complex and nonlinear relationships between intensity data and phase profiles, making it suitable for a wide range of beam shaping systems. Once trained to a specific dataset, the CNN can generalize well to unseen intensity data and can retrieve phase profiles in less than one second, making it a computationally efficient phase retrieval technique. Compared to other phase retrieval algorithms [12,14,16], such as the "Monge-Ampère-equation method" or the Gerchberg-Saxton algorithm, the proposed algorithm is more robust against stagnation. The CNN alone already predicts the surface profile with high accuracy and prevents the optimization from converging to a local minimum. Using the proposed algorithm also reduces the need for manual parameter tuning, making the process less dependent on user expertise.
6. Algorithm 2: binary mask optimization
As we saw in Section 5, neural networks can be trained to calculate phase shifts that will result in a desired intensity distribution. However, this is only possible when the desired intensity distribution can be represented as a continuous function or distribution. When the desired intensity profile cannot be represented as a distribution, in the case of a binary intensity profile for example, it becomes impossible for the neural network to determine the required phase information. The reason for this is that the neural network is trained to learn the relationship between phase and a continuous intensity distribution, and it cannot generalize to intensity patterns which have only binary (light/dark) values.
However, the ability to retrieve the phase information from a binary mask can be achieved using a trust-region-reflective algorithm that takes as input a binary mask which has the desired shape of the intensity profile. The algorithm calculates the required Zernike coefficients from the input mask by first calculating an intensity distribution using starting values of the Zernike coefficients and the Fresnel approximation. Then, a thresholding step is performed. The basic idea of this step is to convert the intensity distribution into a binary image by assigning a specific value to each pixel based on whether its intensity exceeds a certain threshold. In our case, pixels with intensities above the threshold of $I/I_0>0.2$ are set to $I/I_0=1$. However, pixels with intensities of $I/I_0<=0.2$ remain unchanged, as a gradient is needed for the optimization. The reason is that the objective function that quantifies the difference between the calculated intensity mask and the input mask, must be a differentiable function. Otherwise gradient-based optimization methods, including the trust-region-reflective algorithm, cannot be applied directly. In a final step, the optimization algorithm minimizes the difference between the calculates intensity mask and the binary input mask by iteratively updating the Zernike coefficients. In this way the optimization algorithm is able to calculate the Zernike coefficients required to generate a desired shaped intensity distribution.
The ability to reconstruct differently shaped intensity profiles with this algorithm using only binary input masks is shown in Fig. 6. In this example, polygonal input masks that have the shape of a triangle, square, pentagon and hexagon are used as shown in Fig. 6(a). For each mask, the Zernike coefficients of $Z_{n}^{m} (x',y')$ with $|m|=n <= 8$ are calculated by the optimization algorithm. The resulting Zernike amplitudes and the retrieved surface profiles are depicted in Fig. 6(b). The intensity profiles are then calculated by substituting the retrieved phase profiles into the Fresnel approximation.
Fig. 6. Results of the binary mask optimization algorithm for differently shaped intensity profiles. (a) Binary target masks that have the shape of a triangle, square, pentagon and hexagon; (b) Optimized Zernike coefficients and the corresponding surface profiles; (c) Resulting intensity distributions calculated with the Fresnel approximation using the optimized Zernike coefficients.
The resulting intensity profiles are shown in Fig. 6(c) and have a good agreement with the input masks. Figure 6(b) shows that the retrieved surface profiles are dominated by the Zernike coefficients of $Z_{n}^{m}$ with $m=n$, where $m$ corresponds to the number of sides of the polygon. Thus, with a total number of 32 electrodes, a polygon with 16 sides could theoretically be generated by the optofluidic phase shifter. More complex distributions can also be generated using multiple electrowetting devices or multiple interfaces within one device.
7. Experimental setup
The optofluidic phase shifter used for the experiments, briefly introduced in Section 2, is based on a technology published previously [5,29–31]. It consists of a flexible polyimide foil with 32 embedded electrodes that is attached to the inner wall of a cylindrical glass tube, creating a three-dimensional phase shifter as depicted in Fig. 7(a). The glass tube has an inner diameter of $d ={5}\;\textrm{mm}$, a length of $l={10}\;\textrm{mm}$ and is mounted onto a Pyrex substrate. After assembly, the tube is filled with Laser Liquid 433 (Cat. 20108, Cargille Laboratories, USA) and a mixture of OHGL (Cat. 19580, Cargille Laboraties, USA) and OHZB (Cat. 19581, Cargille Laboraties, USA). Their refractive indices are $n_{OHGL,OHZB}=1.499$ and $n_{LL433}=1.293$ at $\lambda ={589.3}\;\textrm{nm}$ and ${25}^{\circ}\textrm{C}$. After filling, the tube is completely sealed with a thin adhesive O-ring and a Pyrex substrate chip on which an annular IrO$_{x}$/Ir/IrO$_{x}$ electrode provides the electrical connection to the aqueous phase.
Fig. 7. (a) Side view of the optofluidic phase shifter; (b) Schematic drawing of the experimental setup for measurement of the intensity patterns generated by the phase shifter.
The experimental setup for characterization of the intensity patterns consists of a laser, a circular aperture, the optofluidic phase shifter and two cameras as depicted in Fig. 7(b). The laser ($\lambda _0={532}\;\textrm{nm}$, ${0.9}\;\textrm{mW }$, $CPS532-C2$, Thorlabs, Germany) emits a collimated beam, which is passed through a circular aperture with a diameter of $D={2}\;\textrm{mm}$ to control the diameter and to remove any stray light; the optofluidic phase shifter is placed in the optical path. To observe the behavior of the electrowetting phase shifter two cameras are used: a USB-microscope records the liquid/liquid interface and a camera (NIKON D5300, AF-S Micro-NIKKOR ${60}\;\textrm{mm}$ 1:2.7G ED, Nikon Corporation, Japan) captures the intensity distribution on the screen, which is placed at a distance of $z_{s}={200}\;\textrm{mm}$ from the liquid/liquid interface.
8. Experimental results
Target intensities in the shape of a triangle, square, pentagon and hexagon are used to demonstrate the ability of the optofluidic phase shifter to generate arbitrary intensity distributions. The required phase profiles are calculated from the desired intensity distributions using the phase retrieval algorithm described in Section 6. Figure 8 shows the target intensity distributions and the corresponding measured intensity profiles. Comparing the measured intensity distributions with the target distributions, it is noticeable that although the shape of the distributions matches, the sizes of the distributions differ.
Fig. 8. Experimental results showing the target intensity profiles and the corresponding measured intensity distributions which are generated by applying voltages calculated from the optimized Zernike coefficients to the individual electrodes of the optofluidic phase shifter.
The likely reason for the discrepancy in the size of the intensity distributions is that the optofluidic phase shifter has small gaps between the electrodes. On these ${50}\;\mathrm{\mu}\textrm{m}$ wide sections the contact angle cannot be adjusted: it is permanently at the maximum value of $\theta =152.76^\circ$. Since the gaps are evenly distributed around the circumference, a uniform downward force acts on the contact line. This force mainly causes a change in the Zernike coefficient of $Z_{2}^{0}$, resulting in a defocus and a deviation of the size of the intensity distribution. Another reason could be irregularities of material properties of individual electrodes. Due to the shape of the glass housing, it is not possible to accurately measure the contact angle of each electrode inside the tube. The contact angle curves are recorded using a test electrode that is not part of the foil but fabricated on the same wafer. In this way, irregularities of material properties of individual electrodes and thus their wetting behavior are not taken into account. In addition, other material inconsistencies such as fluid density mismatch or manufacturing and assembly tolerances have a negative impact on the resulting intensity distributions.
The results show that, on the one hand, phase profiles can be calculated from arbitrary intensity distributions and, on the other hand, that these can be generated by the optofluidic phase shifter. The phase shifter thus not only offers the possibility of generating simple intensity profiles, such as the line profile and different spot sizes as shown in [30], but also to generate much more complex profiles. Compared to other freeform optics with a fixed surface, the optofluidic phase shifter thus has the advantage that the shape of the interface can be changed within a few hundred milliseconds [32]. Thus, this component offers a high degree of flexibility and suggests an attractive approach for beam shaping applications.
Nevertheless, there are some limiting factors that affect the performance of the system. One factor is that the voltages to be applied are determined using a measured voltage-contact angle curve. Local changes of the contact angle at individual electrodes are not taken into account, possibly resulting in an error in the reconstructed intensity distribution. However, this could be addressed using a closed-loop control method, where the contact angles at each individual electrode can be recorded and adjusted in real-time. Another possibility would be to use the phase retrieval algorithm in the experimental setup and retrieve and adjust the shape of the liquid/liquid interface by directly measuring the intensity distribution.
9. Conclusion
We have presented two advanced phase retrieval algorithms based on convolutional neural networks and optimization. Leveraging the power and computational efficiency of deep learning and least-squares optimization, our algorithms determine the required phase information to generate a desired intensity profile with high accuracy. Compared to other methods [12,14,16] the proposed algorithms have a reduced complexity and enable the use of optofluidic phase shifters as reconfigurable illumination sources with high precision and flexibility. Using the proposed algorithms, we have experimentally shown that arbitrary beam profiles may be generated using a phase shifter and the ability to generate complex surface profiles shows the great potential of this kind of optofluidic device for beam shaping applications. By using several of these phase shifters in series even more complex intensity distributions could be generated. Collimated beam shaping, for example, which requires two freeform surfaces, could be realized using two interfaces in one optofluidic component. Giving the cutting-edge nature of our methodology and its immediate applicability in the field of optical beam shaping, we believe that our proposed algorithms can also be applied to more complex problems.
Funding
Carl-Zeiss-Stiftung.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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