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Abstract
Diffractive optical elements play a crucial role in the miniaturization of the optical systems, especially in correcting achromatic aberration. Considering the rapidity and validity of the design method, we propose a fast method for designing broadband achromatic diffractive optical elements. Based on the direct binary search algorithm, some improvements have been made including the selection of the initial height map to mitigate the uncertainty, the reduction of the variations to accelerate the optimization and the increase of sampling rate to deal with the large operation bandwidth. The initial height map is calculated instead of random initial value. Due to different regions of the height map contributing to point spread functions differently, the variations are reduced to speed up the optimization. The large operation bandwidth is solved by increasing the sampling rate at unfitted wavelengths instead of setting weighting coefficients. We demonstrate via simulations that our method is effective through several examples. The design of broadband achromatic diffractive optical elements can be quickly achieved by our method.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
With the rapid development of miniaturization and integration of optical systems in recent years, diffractive optical element (DOE) has become more essential, especially in broadband achromatic focusing [1–10]. In contrast to conventional optical systems which require a combination of multiple lenses to correct chromatic aberration [11–13], DOE can achieve the same effect with a single element, allowing optical systems to be much smaller and lighter. In recent years, metalenses have developed rapidly in the field of achromatic focusing. However, metalenses have more strict fabrication requirements than diffractive optical elements in terms of both resolution and precision. Furthermore, DOEs are polarization insensitive [1]. Conventional diffractive lenses (such as zone plates) also contain chromatic aberration, broadband achromatic DOEs require special design and optimization.
There are two types of design methods for broadband achromatic DOEs, which are the end-to-end design method based on deep learning and the traditional iterative design method. The applications of diffractive optical elements designed by the end-to-end optimization methods include single-shot HDR imaging [14], achromatic extended depth of field imaging [2], and full-spectrum computational imaging [7]. This type of methods generally needs to cooperate with the postprocessing algorithms, which leads to certain limitations. Direct binary search (DBS) algorithm is the representative traditional iterative algorithm. The wave bands of the achromatic DOE achieved using this type of optimization methods include visible [1], near-infrared [8], longwave infrared [4], terahertz [3] and broad bandwidth from visible to longwave infrared [9]. Although the discrete changes limit the optimization, this type of methods takes the resolution limitation of the lithography tool into account and it is more concerned with the optimization of the PSFs (point spread functions). Our method proposed in this paper is also based on this kind of methods.
In recent years, many pioneering efforts have attempted to optimize high-quality broadband imaging with DOE. A diffractive achromat (DA) that combines optimization in both diffractive optical design and post-capture image reconstruction is introduced in [15] firstly. In order to better adapt the deconvolution algorithm, the spectral PSFs are designed to be nearly wavelength independent. And they only consider rotationally symmetric patterns to optimize a 1D problem which is also applied by us. However, due to the special designed PSFs, significant image blurring is observed, which requires powerful post-processing to obtain sharp images. Moreover, it may not be the optimal design paradigm from the perspective of computational imaging. [7,16] improve on the design approach based on [15]. They realize the joint learning of a DA and an image recovery neural network in an end-to-end manner across the full visible spectrum. Higher full-spectrum image quality can be achieved without the prior knowledge of the optimal OTF (optical transfer function) distribution. However, The DOE is highly bound to the recovery algorithm, and the deep learning algorithm is updated quickly. For a stronger recovery network, they need to jointly redesign the DOE instead of directly using the original DOE. We design achromatic DOEs by maximizing the optimization function, and our approach is able to maintain the quality of the images comparable to that achievable with more complex systems of lenses. Post-processing algorithms also improve our imaging performance. In addition, there are also designs of harmonic diffractive elements to achieve broadband imaging [6,17]. The design of the harmonic diffractive elements is non-iterative and does not need to consider the initial value. [17] even realizes a multi-color diffractive lens with adjustable focal length. The disadvantage of the harmonic diffraction element is also obvious. It realizes the imaging of multi-wavelength through high-order diffraction, which is suitable for multiple bands, but not suitable for continuous spectrum. Compared with other methods that apply DBS algorithm to design DOEs [3,4,8,9,18], our method uses less time and is robust.
The DBS algorithm is an iterative algorithm that is applied to search for the optimal solution of the height map. After a random initial height map, the goal of this algorithm is to maximize the optimization function (or figure-of-merit, FOM) by adding positive or negative perturbations (step height). Termination conditions should ensure convergence, and the optimization is completed when termination condition is reached [1]. The DBS algorithm can be modified to GDABS algorithm by applying gradient descent assistance [3]. However, the selections of initial height values and parameters in the DBS algorithm are random and complicated. Moreover, the complexity and difficulty of DOE designs increases with the widening of the operating bandwidth. It is still a problem to achieve good and robust results while reducing the optimization time. The conversion of one-dimensional DOE and two-dimensional DOE is also less studied.
In this study, we propose a fast method for designing and optimizing broadband achromatic aberration DOEs called the CPBS (central part binary search) algorithm. In addition, we use a simple method to convert 1D DOE and 2D DOE. Finally, we demonstrate via simulations that our method is effective, and the design of broadband achromatic diffractive optical elements can be quickly achieved.
2. Principle and design
The basic flow of the design of broadband achromatic DOEs in Ref. [1] is shown in Fig. 1. First, a series of required parameters such as operation bandwidth, focal length are selected, and the random height map $h_{initial}$ is obtained. The initial value of FOM is calculated by bringing $h_{initial}$ into the optimization function. Then use the GDABS algorithm [3] to optimize the height map of the DOE. When the termination condition is reached, the optimization ends, and the final height map is obtained.
2.1 Parameters and initial height map
The selection of initial height value is random in the GDABS algorithm. In order to reach the termination condition of the optimization function faster, we propose a universal approach to the initial height map. For the broadband achromatic DOE over a large bandwidth of $\lambda _{min}$ to $\lambda _{max}$, the initial height map can be achieved as the height map designed for the single wavelength $\lambda _{0}=\frac {\lambda _{min}+\lambda _{max}}{2}$, we call it middle height map. The advantages of our initial height map will be presented in the next section. And $h_{initial}$ can be expressed as
where $z$ is the distance of Fresnel propagation, $n_{\lambda }$ is the refractive index of the DOE depending on the wavelength $\lambda$, $r$ is the distance from the optical axis, and $m$ is an integer. Meanwhile, other parameters of the DOE are given. To ensure that the phase difference can be modulated from $0$ to $2\pi$ when $\lambda =\lambda _{max}$, we get the maximum height $h_{max}$ of the DOE is $\frac {\lambda _{max}}{n_{\lambda _{max}}-1}$ and the step height $h_{step}$ depends on fabrication technology. Determined by the FWHM (full-width-at-half-maximum) of the PSF $W_{i}$, the step width (or the feature size) should be less than $\frac {\lambda _{min}}{2NA}$ [4] where $NA$ is the number aperture. When operating wave bands are different, DOEs need to change materials, including fused silica, silicon, and photoresist. The refractive indexes of these materials vary slightly with wavelengths, as well as slight absorption. For the sake of simplicity, we do not consider the impact of absorption.2.2 FOM
FOM is proposed to make the optimized PSFs closer to target functions, so it plays a key role in the optimization algorithm. Here, target functions described as the approximation of ideal PSFs are defined as Gaussian functions centered at $\left (x'_{min}+x'_{max}\right )/2$ with FWHM $W_{i}$ determined by the far-field diffraction limit [1], $x'_{min}$ and $x'_{max}$ limit the range of the DOE. The target function for the $i$th wavelength sample can be expressed as
It has been shown that when the operation waveband becomes wider, the design of broadband DOE becomes more difficult. The design for several discrete wavelengths is much simpler than that for continuous wave band. For example, imaging in visible light, we can consider the PSFs of three wavelengths blue, green, and red ($460nm, 540nm, 620nm$), or the full visible spectrum ($450-700nm$). The number of wavelength samples $N$ is $3$ in the former condition, in the latter condition $N$ which depending on the sampling interval is about $50$. If we apply the former results to directly verify the latter, we can find that at some wavelengths in the middle of the interval, the PSFs do not fit the ideal well in Fig. 2. Due to the difficulty in selecting $\omega _{i}$, we balance the contributions from different wavelength samples in another way. Our method is to directly shorten the sampling interval on the unfitted wave bands. The results of our method are shown in the next section.
Fig. 2. The performance of the PSFs obtained by optimizing only three wavelengths ($460nm, 540nm, 620nm$) in the entire visible band.
2.3 Optimization algorithm
The GDABS algorithm is an iterative algorithm that is applied to search for the optimal solution of the height map in the previous articles. Here we demonstrate a method called the CPBS (central part binary search) algorithm to further accelerate the iteration based on the GDABS algorithm.
First we analyze the wave field propagation of diffraction before designing the achromatic DOE. In simple terms, after a phase shift occurs by the DOE, the incident wave goes through Fresnel diffraction and reaches the image plane. To obtain ideal PSFs on the image plane, the phase difference between the ray on the optical axis and the ray with a distance $r$ from the axis in Fig. 3 can be expressed as
where $\phi _{h}$ is the phase difference that occurs by the DOE, $\phi _{g}$ is the phase difference that occurs by the geometrical paths. $\phi _{h}$ and $\phi _{g}$ are represented asWhen adding a perturbation $h_{step}$ to the height map, we can get
Intuitively, when the small unit is close to the center of the DOE (considering the 2D DOE), the phase difference caused by the propagation distance is very small, and the phase difference mainly depends on the height difference of the DOE. Then when a perturbation occurs, it has a great influence. When the small unit is far away from the center of the DOE, the phase difference caused by the propagation distance is large, and the influence of the phase difference caused by the height of the DOE is reduced. We also do some simulation experiments to support this inference in the following section. Choosing the same initial height map, in one case, we only optimize half of the small elements which are close to the center, and in the other case, we only optimize half of the small elements which are close to the edge. We find that optimizing the central half tends to get better optimization results (almost as good as optimizing all units). In this way, we can choose to optimize only a certain proportion of small units close to the central part according to the time consumption we can accept. This is also the reason why we called it the CPBS algorithm.
2.4 Conversion of 1D DOE and 2D DOE
We present a trick to implement the conversion of 1D DOE and 2D DOE. To simplify the structure of the entire DOE, we assume that the DOE is symmetric. Then the variations of 1D DOE are only half number of the width. We can get the height map of the DOE $h_{1D}$ by
where $h_{half}$ is a half part of the height map of the DOE, $flip$ is the flip operation. By applying the rotationally symmetric parameterization on the height map of 1D DOE, we can get 2D DOE and reduce the number of optimization parameters as well in Fig. 4. This rotationally symmetric structure has been used in the design of end-to-end diffractive optical elements and computational imaging with diffractive optics [7,15] to reduce consumption and memory. We also calculate the focusing efficiency of 1D DOE and 2D DOE in Section 3.1. Compared with 1D DOE, the focusing efficiency of 2D DOE decreases.2.5 Termination condition
If the slope of the growth of FOM is less than $0$, we consider the termination condition is reached and the optimization ends. After these analyses, we can get the flow of our CPBS algorithm in Fig. 5.
3. Simulation experiments and results
In this section, we would introduce three DOEs that worked in different wavebands to verify the effectiveness and speed of our method. The matlab version is R2022a, the CPU is i5-9400F, and the memory size is 16G.
3.1 Dual-wavelength DOE for near-infrared imaging
First, we designed a diffractive optical element that could focus pairs of wavelength ($1.064\mu m$ and $1.55\mu m$) to the same point introduced in [8]. As we analyzed, the achromatic DOE for several discrete wavelengths were easier to achieve. According to [8], we chose the same parameters of lens, including an aperture of $560\mu m$, a focal length of $10mm$, a feature size of $2\mu m$, a step height of $0.1\mu m$, and a refractive index of $1.482$. As mentioned before, the maximum height $h_{max}$ of the DOE using our method was $\frac {\lambda _{max}}{n_{\lambda _{max}}-1}$, so we chose $33$ steps instead of $30$. We verified the validity of our initial height map. From Table 1, using our middle height map the optimization converged faster and yielded better results. Random initialization had strong uncertainty. Since the initial height map had a great influence on the optimization, even if the method was known, the result might not be reproduced. Our middle height map could avoid this problem and use less time. As shown in Fig. 6, the initial psfs obtained by our method were more concentrated. Although the final PSFs by some random height map might better than our final PSFs as shown in Fig. 7(c) and (d), it had a large randomness.
Fig. 6. (a), (b) are the initial PSFs by our initial height map. (c), (d), (e), (f) are the initial PSFs by random height maps.
Fig. 7. (a), (b) are the final psfs by our initial height map. (c), (d), (e), (f) are the final psfs by random height maps.
Table 1. Optimization performance using different initial height map
Then applying the conversion from the previous section, we could get 2D DOE and the results demonstrated in Fig. 8. Compared with other conversion methods, the advantage of our method was that only one optimization was performed, and two results of 1D DOE and 2D DOE could be obtained at the same time. In comparison with the results in [8], our method also gave good results in Fig. 8. Furthermore, the focusing efficiencies of 1D DOE and 2D DOE were considered. We took the definition of the focusing efficiency as the ratio of the power in the ideal Airy disk to the total power on the image plane. And the focusing efficiencies of 1D DOE were calculated to be $95.74\%$ and $89.32\%$ at $1.064\mu m$ and $1.55\mu m$, respectively. Compared with 1D DOE, the focusing efficiencies of 2D DOE dropped to $74.15\%$ and $54.17\%$.
Fig. 8. (a), (b) are the normalized intensity distributions at two wavelengths. (c), (d), are the simulation results on the focal plane at two wavelengths. (e), (f) are the PSFs at two wavelengths.
3.2 Broadband DOE for longwave-infrared imaging
Apart from the near-infrared imaging, we also designed DOE for longwave-infrared ($8-12\mu m$) imaging. The lens had an aperture of $8 mm$ and a focal length of $8 mm$. It was comprised of concentric rings of width = $8\mu m$ with the height of each ring being varied between 0 and a maximum value of $5\mu m$ (a total of $50$ levels with a difference of $0.1\mu m$ between each level), and a refractive index of $3.42$. The number of steps was $50$ which was less than the number in [4].
Using our initial value selection, when the wavelength $\lambda _{0}$ was $10\mu m$, after $5$ iterations, the result could be obtained, the PSFs did not perform well at $11\mu m$ and $12\mu m$. We could adjust the choice of initial wavelength a bit based on the performances of the PSFs, when $\lambda _{0}=11\mu m$, the problem was solved. At this time, it was also $5$ iterations, and the FOM of $0.1763$ was greater than the previous $0.1233$. As shown in the Table 2, despite using the GDABS algorithm and optimizing the PSFs for only $5$ wavelengths, the time for the entire iterative process exceeded $1500s$. We could apply the CPBS method we proposed to accelerate the iteration. We only optimized middle half variations of the DOE. Under this condition that the FOM (=$0.1710$) did not reduce much, it only took about half the time to get the results. The final PSFs in the Fig. 9 also demonstrated the effectiveness of our CPBS algorithm. To show the effect of reducing variations on FOM, we gave the relationship between optimization rate and FOM in this example in Fig. 10. The decline of FOM at optimization rate$=0.7$ was actually caused by the gradient descent of FOM in the algorithm. At optimization rate$=0.7$, only $5$ cycles were iterated, and at optimization rate $=0.6$, $6$ cycles were iterated. But it could still be shown that when the optimization rate exceeded $0.5$, FOM was very close to the final result, and our method was effective.
Fig. 9. (a) are the final psfs by optimizing all variations, (b) are the final psfs by optimizing middle half variations.
Table 2. Optimization performance using different initial height map
3.3 Broadband DOE for the entire visible band
Finally, we designed 2D diffractive optical elements that only considering the RGB tricolor and the entire visible band, respectively. For three discrete wavelengths ($460nm, 540nm, 620nm$) we chose the parameters of lens, including an aperture of $0.84mm$, a focal length of $10mm$, a feature size of $3\mu m$, a step height of $50 nm$, and we utilized photoresist SC1813 as the material which is similar to SC1827. The number of steps was $21$ which was enough for $\lambda _{max}$ modulated to $2\pi$. It was easy to obtain good results of these three wavelengths using our initial height map as shown in Fig. 11. However, for full visible spectrum this diffractive element was far from good in Fig. 2, especially at about $575nm$ and $675nm$. For the continuous broadband across the visible spectrum ($450nn-700nm$) it became more complicated, but in our example the optimization time was controlled to approximately 600s. The parameters remained the same except focal length of $20mm$.
First step, we needed to change the wavelength sampling to $25 nm$ during the design. We accelerated the iteration by our CPBS algorithm and the optimization rate was $0.5$. It was only $3$ iterations to finish the optimization, and the optimization time was about $215$s in Fig. 12(a) and we got the height map as the initial height map of the second step in Fig. 13(a). From the Fig. 11(c) and (d), it showed that the PSFs during $520nm$ to $540nm$ performed poor. So we added $520nm$ and $530nm$ in the wavelength sampling to improve the performance of these wavelengths. Here we had to optimize the whole height map. The reason for optimizing the whole height map was that we made some minor adjustments in the second step. And the optimization time of the second step was about $350$s because of the added wavelength samples and the more optimization object. As shown in Fig. 12(b), the PSFs in the full visible spectrum performed well though the PSFs during $620nm$ to $660nm$ were wider. As shown inFig. 12(e) and (f). the performance of $520nm$ and $530nm$ became better. We could obtain the final height map in Fig. 13(c), the optimization of step 2 could be visualized from the cross section of the height map in Fig. 13(b) and (d). This example has demonstrated the effectiveness and rapidity of our methods.
Fig. 12. (a) The simulated PSFs of as function of incident wavelength in step 1. (b) The simulated PSFs of as function of incident wavelength in step 2. (c) The simulated PSF at $520nm$ in step 1, (d) the simulated PSF at $530nm$ in step 1, (e) the simulated PSF at $520nm$ in step 2, (f) the simulated PSF at $530nm$ in step 2.
Fig. 13. (a) The height map calculated by Step 1. (c) The final height map for the entire visible band.
4. Discussion
For DOE with small feature size, there may be millions of small units and it takes a long time to run an iteration. Moreover, if the random initial height map is not good, it is possible that you are wasting much time on unfitted results. In this situation, our algorithm exhibits some advantages in saving time and robustness. Compared to traditional refractive optical systems, the point spread functions we get are not good enough, and the focusing efficiency is not high enough. This problem can be effectively solved by powerful deblurring algorithms. We sacrifice better results for the efficiency and robustness of the algorithm. This is acceptable when the DOE is complex. Another problem is that there are no experimental results. Later, we need to fabricate the designed DOE to further verify the effectiveness of our algorithm.
5. Conclusions
In this paper we propose a fast method of designing broadband achromatic diffractive optical elements considering the rapidity and validity. We carefully analyze and optimize every step of the entire optimization process. Furthermore, we mitigate the uncertainty of the initial height maps and provide the solution of the large operation bandwidth. We illustrate that our method is effective through several examples including near-infrared imaging, longwave-infrared imaging and visible imaging. Our method can achieve almost the same results as other broadband diffractive element designs in less time and the method is novice friendly. However, we have not yet fabricated the diffractive optical elements designed by the simulation and experimentally verified its effectiveness. And this will also be a topic we will do later.
Funding
Key Research Project of Zhejiang Laboratory (2021MH0AC01).
Disclosures
The authors declare no conflicts of interest.
Data availability
Codes can be obtained from the authors upon reasonable request.
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