## Abstract

Fabrication of multiple arbitrary diffractive optical elements (DOEs) on multiple curved surfaces is always a challenge; here we propose an effective optimization method to fabricate complicated DOEs on several curved surfaces at the same time. First we design phase distribution to modulate complicated three-dimensional (3D) intensity distribution on multiple curved surfaces simultaneously, and then by exposure, the intensity distribution is transferred into the pure-phase or depth distribution. Numerical simulations and optical fabrication are performed for different intensity distributions: 3D binary patterns and 3D gray level patterns, on two or three curved surfaces, and both are in nice agreement. Since multiple DOEs are fabricated on curved surfaces simultaneously, the collimation of different curved surfaces is avoided, and it could improve the fabrication efficiency. It is expected that this proposed method would be employed in various precision 3D optical fabrication and processing in the future.

© 2015 Optical Society of America

## 1. Introduction

There are various applications of the diffractive optical elements (DOEs) on curved surfaces, such as thin-film transistors [1], artificial compound eye [2] and so on. It has been a challenge to fabricate arbitrary DOEs on multiple planar or curved surfaces simultaneously. In recent years, many methods, such as: the laser direct writing [3–6 ], the ion beam proximity lithography [7], the soft lithography [8,9 ], the nano-imprint lithography [10], and the interference lithography [11–16 ], are employed to fabricate micro/nano-structures on planar or curved surface. Most methods are time-consuming and expensive while interference method is simple and fast. Shi et al. have designed and fabricated arbitrary DOEs on planar surface by interference [17], and Zhao et al. improved his method and further fabricated arbitrary DOEs on one curved surface [18], where the precise alignment of two laser beam is required. To solve this problem, Wang et al. proposed a convenient method to fabricate DOEs on curved surface using the computer generated hologram which also adopted for three-dimensional display in [19,20 ], where only one laser beam modulated by spatial light modulator (SLM) is employed [21]. However, all the methods above can only fabricate one DOE at a time and the depth of field is also small. If multiple arbitrary DOEs on multiple curved surfaces are required to fabricate, the common method need multiple exposures on different surfaces, which will lead to the difficulties of the collimation of different curved surfaces, and also it’s time-consuming. Stimulated by those challenges, in this paper, we propose a method to fabricate multiple arbitrary DOEs on multiple curved surfaces simultaneously.

In this paper, we present an effective method to design and fabricate multiple arbitrary intensity distribution of DOEs on multiple curved surfaces. The proposed design method is combined the Fidoc method [22] (improved GS algorithm) and the Huygens diffraction [23] in scalar diffraction domain. 3D binary patterns and 3D gray level patterns are simulated and fabricated on multiple curved surfaces, which verifies the feasibility of this method. The efficiency of the fabrication is improved and the multiple collimation of different curved surfaces is avoided since it requires only one optimized phase (loaded in SLM) to produce multiple desired profiles of DOEs. In addition, the depth of field is larger. It is believed that it is an effective method to fabricate multiple arbitrary complicated DOEs on curved surfaces simultaneously, and it could be useful in various optical fields.

## 2. Basic principles

The basic principle is shown in Fig. 1 . ${P}_{1}$, ${P}_{2}$denote the curved surfaces on which the desired DOEs will be fabricated.$H$denotes the plane that we place a phase distribution. ${P}_{1}\left({{\rm X}}_{1}\right)$,${P}_{2}\left({{\rm X}}_{2}\right)$and $H\left(\Xi \right)$ denote the complex amplitude distributions the on surface${P}_{1}$, ${P}_{2}$and $H$respectively, where${X}_{1}$,${X}_{2}$ and $\Xi $ represent the Cartesian coordinate systems${x}_{1}{y}_{1}{z}_{1}$, ${x}_{2}{y}_{2}{z}_{2}$and $\xi \eta z$respectively. Since the DOEs are fabricated by exposure, the profiles of DOEs can transfer into the 3D intensity distributions $I({X}_{i})={\left|P({X}_{i})\right|}^{2}$ on corresponding surfaces. Then, based on the desired intensity distributions $I({X}_{i})$, the phase distribution $H(\Xi )$ on plane $H$ will be optimized by the following method.

At first, it is necessary to clarify that the proposed method to fabricate multiple DOEs on multiple curved surfaces simultaneously is based on the principle of independently propagating and superposition of light. Then, the Huygens diffraction and the inverse Huygens diffraction [24,25
] in scalar diffraction domain (considering the paraxial approximation) is adopted to denote the process of light propagation. And the positive propagation direction is from $H$ to curved surfaces. So, the complex amplitude distributions on curved surfaces${P}_{1}$, ${P}_{2}$ propagated from plane *H* can be expressed as Eq. (1)

*H*propagated from curved surfaces ${P}_{1}$ and ${P}_{2}$can be expressed as Eq. (2)

*d*, and

_{0}*d*is the distance between surface ${P}_{1}$ and ${P}_{2}$. The diffraction distance between two points on$H$and curved surface ${P}_{1}\left({P}_{2}\right)$ is ${r}_{1,2}=\sqrt{{({x}_{1,2}-\xi )}^{2}+{({y}_{1,2}-\eta )}^{2}+l{({z}_{1,2})}^{2}}$, where $l\left({z}_{1,2}\right)$ is the z-component of distance which is variable for different points on curved surfaces.

Then, combined the Huygens diffraction we adopted above with the Fidoc method [22], Fig. 2
shows the whole process of the method we propose to design the pure-phase distribution to fabricate multiple arbitrary DOEs on multiple curved surfaces. The intensity distributions of DOEs we fabricate combined with random phases are adopted as the initial complex amplitude distributions ${P}_{10}\left({X}_{1}\right)$and $\times $on the curved surfaces, respectively. When the $j$th iteration is performed, at first, the complex amplitude distributions ${P}_{1j}\left({X}_{1}\right)$and ${P}_{\text{2}j}\left({X}_{\text{2}}\right)$on the curved surfaces conduct the inverse Huygens diffraction (denoted by *HuF ^{-}*

^{1}{…}). And according to Eq. (2), the distribution ${H}_{j}(\Xi )$ on plane $H$is obtained. After that, the phase of ${H}_{j}(\Xi )$ is kept and the amplitude is imposed to unity, and the distribution ${\overline{H}}_{j}(\Xi )$ is obtained. Then, according to Eq. (1), the Huygens diffraction (denoted by

*HuF*{…}) of ${\overline{H}}_{j}(\Xi )$ from plane $H$to curved surfaces is computed separately to acquire the complex amplitude distributions ${\tilde{P}}_{1j}\left({X}_{1}\right)$ on ${P}_{1}$ and ${\tilde{P}}_{\text{2}j}\left({X}_{\text{2}}\right)$ on ${P}_{2}$. Then, the phase is kept, while the amplitude is both replaced with${A}_{i,j+1}$,

_{where}the feedback parameter

*k*adds negative feedback to each pixel value to force it to converge faster,

*M*is a function of zero at the positions of the zero padding and unity at the image’s pixels, and the noise suppression parameter $\gamma $,combined with

*M*, affect the noise of the reconstructed patterns [22]. And the obtained complex amplitude distributions ${P}_{i,j+1}\left({X}_{i}\right)$ is adopted to update ${P}_{i,j}\left({X}_{i}\right)$ for next iteration. The iteration will continue until the number of iteration reaches the maximum or the reconstructed intensity distributions from ${\overline{H}}_{j}(\Xi )$ have acceptable and convergent relative errors from the ideal intensity distributions. If feedback parameter

*k*and the noise suppression parameter $\gamma $ are reasonable, the algorithm will converge within tens of iterations and the errors are stably reduced so that the noise is stable and acceptable. Finally, the optimized pure-phase distribution ${\phi}_{j}$ can be obtained which will be loaded in the phase-only SLM, and 3D intensity distributions of DOEs on multiple curved surfaces can be acquired. After exposure once, the multiple desired DOEs can be fabricated.

## 3. Numerical simulations

In this section, we will conduct numerical simulations to evaluate our proposed approach. At first, we should establish the mathematical model of the curved surfaces. Considering the practicality of the subsequent fabrication, we choose two cylindrical surfaces, one of which is concave and the other one is convex, as shown in Fig. 3
. Figure 3(a) shows the top view, and the pink part is the cylindrical surfaces we established which sampled into a series of grids with equal interval as shown in Fig. 3(b). The subscript 1 and 2 means the two cylindrical surfaces respectively, *L* is the side length of cylindrical surface, *R* is the radius of the curvature, *2α* is the field angle, and *d* is the distance of two surfaces. Then, we simulate the process of the proposed method to calculate the pure-phase distribution on a plane to modulate the 3D intensity distributions on the curved surfaces simultaneously.

We apply a 3D grey level pattern and a 3D binary pattern on the two cylindrical surfaces respectively to conduct the numerical simulation. The initial intensity distributions on the two cylindrical surface is shown in Fig. 4(a)
: the left one on concave cylindrical surface ${P}_{1}$ is a 3D binary pattern, while on the right convex surface ${P}_{2}$, a 3D grey level badge pattern is adopted. Both two patterns with pixel of 256 $\times $256 are expended to the patterns with pixels of 400$\times $400 by zero padding. The parameters for the simulation are as follows: *L*
_{1,2}
*=* 12 *mm, R*
_{1,2}
*=* 51.852 *mm, d =* 2 *mm,$\lambda $*
*=* 532 *nm.* The hologram is comprised of 800 $\times $800 pixels, each having square size of $8\mu m$. And the distance between the plane $H$and the cylindrical surface ${P}_{1}$ is *d*
_{0}
*= 361 mm* which needs to satisfy the sampling theorem. We choose the noise suppression parameter $\gamma =1$, the feedback parameter *k =* 0.9. The relative error (*RE*), defined as$RE={R}_{e}/{R}_{i}\times 100\%$, is selected to evaluate the quality of the reconstruction, where ${R}_{e}={\displaystyle \sum _{m=1}^{M}{\displaystyle \sum _{n=1}^{N}{(\left|I\text{'}\left(m,n\right)\right|-\left|I\left(m,n\right)\right|)}^{2}}}$is the error item,${R}_{i}={\displaystyle \sum _{m=1}^{M}{\displaystyle \sum _{n=1}^{N}{\left|I\left(m,n\right)\right|}^{2}}}$ represents the ideal distribution item, $I\text{'}$ denotes the reconstructed intensity distribution, $I$ denotes the desired intensity distribution, *M* and *N* represents the number of row and column of the sampling points. So the smaller value of *RE* implies the higher quality of reconstruction. After 30 iterations, the reconstructed intensity distributions on curved surfaces is shown in Fig. 4(b). Figure 4(c) shows the relationship between the values of *RE* and the number of iteration. The values of *RE* both decrease with the increase of iterations, and after *35* iterations, the *RE* reduce to 3.28*%* and 2.67*%* respectively which represent the high quality of the reconstructed patterns.

The mathematical model above is inerratic. So in order to better demonstrate the availability of the method we proposed, we design a more complicated mathematical model to conduct the simulation, which can be described by the formula: $z=a\mathrm{sin}(bx)\times \mathrm{sin}(cy)$, where *a =* 0.2, *b = c =* 0.5. The ideal patterns we selected is the same as the preceding patterns in first case with 256 $\times $256 pixels, and they are expended to 400 $\times $400 pixels by zero padding. Other parameters are also the same as those used in the first case. The results of the numerical simulation are shown in Fig. 5
. After 30 iterations, the reconstructed patterns is shown in Fig. 5(b) and the value of *RE* decrease to only 3.15*%* and 2.99*%* as shown in Fig. 5(c). So it is proved furthermore that this method is effective to implement the intensity modulation on various sorts of curved surfaces.

For the purpose to better demonstrate the feasibility of the method we proposed to modulate 3D intensity distributions on multiple curved surface, we construct a more complicated mathematical models which contains three curved surface (${P}_{1}$ and ${P}_{3}$ are cylindrical surfaces, ${P}_{2}$ is the same as the second case where *a =* 0.3, *b = c =* 0.5) to conduct the numerical simulation. The ideal patterns on surfaces ${P}_{1}$ and ${P}_{3}$ are the same as the preceding patterns, and the ideal pattern on surface ${P}_{2}$ is a new gray level pattern as show in Fig. 6(a)
. All the patterns with 128 $\times $128 pixels are expended to 256 $\times $256 pixels by zero padding. The distance between the plane $H$and the surface ${P}_{1}$ is *d _{0} =* 1015

*mm*, the distance between these three curved surfaces are both 2

*mm*, the pixels of hologram is 768 $\times $768, the noise suppression parameter$\gamma =1$, the feedback parameter$k=1$and other parameters are also the same as those used in the first case. After 20 iterations, the reconstructed patterns is shown in Fig. 6(b) and the value of

*RE*decrease to only 1.39%, 1.58

*%*and 1.36

*%*respectively as shown in Fig. 6(c). According to the results, the method is proved its feasibility to realize the 3D intensity modulation on multiple arbitrary curved surfaces simultaneously.

## 4. Experiment and discussion

The method we adopt to conduct the experiment is the holographic projection technique. Figure 7
shows the schematic view of optical experimental setup. The laser with $\lambda $
*=* 532 *nm* (Oxxius 532-300-COL-PP-LAS-01462) is adopted as light source. A laser beam is spatial-filtered and collimated and then illuminates the SLM (HOLOEYE PLUTO), in which the pure-phase hologram is loaded to modulate the incident collimating laser beam. The 4-f system and a high-pass filter are adopted to remove the strong zero-order noise in the modulated light which reflected from the SLM. Through these setups, the desired intensity distributions of DOEs are generated and then expose the photopolymer on the surfaces of two cylindrical lens located at their corresponding position. The parameters of the photopolymer we use is as follows: the refractive index modulation is $\Delta n$ = 0.04, the thickness is $15\mu m$ and the exposure is 15 *mJ/cm ^{2}*. In the experiment, according to the power of exposing light, the exposing time we chose is 3 seconds. Finally, the desired DOEs will be fabricated on two pieces of photopolymer.

The two initial patterns we selected is the same as the patterns of the first case in section 3 with 256$\times $256 pixels and expended to 400$\times $400 pixels by zero padding. And the parameters in this experiment are as follows: *L*
_{1,2}
*=* 12 *mm, R*
_{1,2}
*=* 51.852 *mm,$\lambda $*
*=* 532 *nm*, the distance between two cylindrical surfaces is *d =* 2 *mm*, and the diffraction distances between cylindrical surface ${P}_{1}$and the modulation plane is *d _{0}* = 361

*mm*. The pixels of the SLM we adopt is 800$\times $800 in the middle. The focal length of Fourier lens in the 4-f system is 500

*mm*, so the total distance between SLM and cylindrical surface ${P}_{1}$is 2361

*mm*.

The results of the modulation on cylindrical surface ${P}_{1}$and ${P}_{2}$ are shown in Fig. 8 & Fig. 9 . They are both captured by Canon EOS 5D Mark II. Figure 8(b) and Fig. 9(b) shows the partial enlarged images. The proportional dimension is also shown at the bottom of the images. According to these images of experimental results, the two patterns are effectively fabricated on two different curved surfaces simultaneously. So the feasibility of the method we proposed is demonstrated.

However, some details of the produced profiles are not very perfect. The reasons might be as follows. At first, the dust in air cannot avoid during the experiments, which would reduce the quality of the pattern projected on photopolymer. Then, the non-uniformity and the unevenness of the photopolymer will influence the fabrication results. If the environment of experiment is improved, the results could be better.

## 5. Conclusion

In brief, we propose an effective method based on phase optimization to design the pure-phase distribution for fabricating multiple arbitrary DOEs on multiple curved surfaces. Since the proposed method can be applied to fabricate complex 3D patterns on multiple curved surfaces simultaneously, the multiple collimation of different curved surfaces in the system is avoided and the efficiency of DOEs’ fabrication is significantly improved. Moreover, the depth of field is larger in the method we propose. Both the numerical simulations and the optical fabrications are performed and they are in good agreement. And the curved surfaces can be more complicated according to the simulating results. The proposed method is easy and efficient. It is believed that it is an effective method to fabricate multiple arbitrary complicated DOEs on curved surfaces simultaneously, and it could be useful in various optical fields such as optical communication, optical interconnects, optical design, and so on.

## Acknowledgments

This work was supported by the National Natural Science Founding of China (NSFC) (61575024, 61235002 and 61420106014), the National High Technology Research and Development Program of China (863 Program Grant No. 2015AA015905), and the National Basic Research Program of China (973 Program Grant No. 2013CB328801 and No. 2013CB328806).

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