We demonstrate a novel method for the modulation of the optical intensity on curved surfaces (CS) by interference and apply it to fabricate diffractive optical elements (DOEs) with arbitrary profile and large area on CS. The intensity on CS is modulated accurately by two phase distributions. Both a binary pattern and a gray pattern are reconstructed numerically on the lens surfaces with big curvatures in large areas, while a binary and non-periodic pattern is produced experimentally on a lens surface with a radius of curvature in 25.8 mm. The simulations together with the experiment demonstrate the validity of the method. To our knowledge, it is the first time to present an approach for fabricating DOEs with arbitrary profile and large area on CS by interference.
© 2013 OSA
Nowadays, the technique to fabricate micro/nano-structures on curved surfaces (CS) can be applied to produce many useful devices such as electronic eye camera  and artificial compound eyes . Over the past few years, several techniques including ruling engine , ion beam proximity lithography  and laser direct writing [5–8] have been widely investigated for patterning on CS. However, these methods require a set of expensive equipment and the fabricating process for large area is time-consuming. Soft lithography [9, 10] and nanoimprint lithography  might solve the problem of low-throughput but the fabricating accuracy is limited to micrometer dimension when applied to curved substrates with very large curvature due to the properties of the soft and flat mold.
Interference lithography is a low-cost and high-efficient technique to fabricate micro patterns in large areas [12,13]. This method doesn’t require complicated and expensive equipment, besides, the patterning resolution can reach subwavelength of the incident light. Early in the year 1999, Baker et al.  made subwavelength periodic patterns on CS by interference exposure method. After that, this method has been widely utilized to make periodic patterns on CS [15–17]. In 2010, Mizutani et al.  developed a two-spherical-wave ultraviolet interferometer to fabricate patterns, and the most distinguish advantage of this method over the two-plane-wave interferometer is that the variation of fringe period on CS can be highly restrained. However, since the interference fringes are simply generated by basic wave fronts, such as planar waves and spherical waves, it can fabricate only periodic line patterns or lattice patterns. The modulation of the optical intensity with arbitrary distribution on CS is a key problem for patterning on the curved surface. In order to make non-periodic patterns on CS, one needs to control the wave fronts with complex distributions to interfere with each other. Though Shi et al.  proposed an approach to fabricate DOEs with arbitrary profile by interference, it can only be used for fabricating DOEs on the planar surfaces. In this paper, we propose an approach to realize the arbitrary intensity modulation on CS by interference, and apply it to fabricate DOEs with arbitrary profile and large area on CS. Both binary pattern and gray pattern are reconstructed on convex lens surfaces in large areas numerically. Due to the limitation of condition in our laboratory, a binary and non-periodic pattern is made on a convex lens surface experimentally, which demonstrates the validity of this approach.
2. Basic principles
To fabricate patterns on a curved surface the intensity distribution on this surface should be modulated as precisely as possible. Regard to this problem, the polygon-based method [19,20] has been proposed to compute diffraction from a curved surface to a planar surface, where the curved surface is approximated with many non-parallel planar surfaces and the diffractions of them are calculated respectively. Very recently, Tomoyoshi Shimobaba et al.  proposed a fast algorithm which is capable of calculating Fresnel diffraction from a curved surface by integral. These methods are very useful for calculating a computer generated hologram (CGH) from a three-dimensional object composed of multiple polygons or arbitrary shape surfaces. However, both methods employ the approximation condition to speed up the calculation, so the accuracy is limited. To precisely fabricate patterns on CS, more accurate calculation method should be employed.
As is well known, Huygens-Fresnel principle can be expressed in the form of formula. In this article, we name it as Huygens diffraction for simplification. Considering the precision requirement, Huygens diffraction is more suitable than its approximate forms. Huygens diffraction  can be expressed as23,24], can also be expressed asFig. 1. When the paraxial approximation condition is well matched, is assumed. When it doesn’t meet the paraxial approximation condition, should be considered for those micro areas, respectively. Please note that in the actual calculation process as shown in Fig. 2(a), optical propagation direction is from surface to surface, and coordinates ξηz and xyz are both Cartesian coordinate systems. Therefore on surface is obtained by inverse Huygens diffraction and can be expressed as: , where denotes the original complex distribution on CS and represents the inverse Huygens diffraction.
In summary, the basic principle to modulate arbitrary intensity on CS can be described in three steps: step 1, the complex light wave with desired intensity distribution on CS propagates to the input plane P1 according to the inverse Huygens diffraction, and the complex light wave is achieved; step 2, for any complex distribution can be decomposed into two pure phases analytically, that is; step 3, the combination of two pure phases can be realized by the Michelson interferometer.
The schematic view of the interference lithography system for fabricating DOEs on CS is shown in Fig. 2(b). The uniform plane waves with wavelength of λ illuminate the input planes Q1 and Q2, respectively, where the phase-only modulators and will modulate the wavefronts into and , and they are combined by the beam splitter (BS). Finally the complex distribution U(A) is reconstructed and the desired intensity |U(A)|2 is obtained on the target CS.
3. Simulations and analyses
To demonstrate the validity of this method, numerical simulations are performed. Firstly, the modeling method and calculation principles are introduced. As shown in Fig. 3(a) the green part (color on line, and gray on print page) is the side view of a convex lens surface, where R is the radius of curvature of the surface, r is the radius of the circle, and 2α is the field angle, where r = R sin α, and coordinate xz is a part of the xyz coordinate system. To calculate the diffraction of the light wave on a curved surface, we divide the main center part into many grids with equal areas as shown in Fig. 3(b). As the number of the grids is large enough, the grid size becomes extremely small. Thus every tiny grid can be regarded as a micro area which is mentioned in Fig. 1 and Eq. (2). Therefore, the inverse Huygens diffraction can be applied to compute the diffraction from the curved surface to the planar surface.
Then, the actual calculation is conducted numerically. The ideal intensity distribution I is a symbol ‘BIT’ on a convex lens surface, as shown in Fig. 4(a), and the color bar on the right denotes the range of intensity distribution. As discussed in the modeling process, the curved surface is simplified to a main center area (an inscribed square) of the surface and the square consists of 256 × 256 grids. The parameters used for the first case (case 1) are: r = 2.828 mm, R = 5.656 mm, 2α = 60°, λ = 532 nm, the sizes of two phase modulators are both 4 mm × 4 mm, and the distance between the modulators plane and the curved surface is chosen as 118 mm. The ideal complex distribution on the lens surface is defined as follows: the amplitude part A equals , while the phase part αis set to be the zero distribution for simplicity, which means that ideal complex distribution becomes . The inverse Huygens diffraction is computed according to Eq. (2), and the phase distributions φ1 and φ2 are calculated by Eq. (2). Then the reconstructed intensity distribution I′ is shown in Fig. 4(b). To evaluate the reconstruction quality, we define SNR as , whererepresents the signal item, denotes the noise item, and M and N in the formulas represent the number of grids of the pattern. For the sake of comparison, we calculate the SNR of the pattern reconstructed on a planar surface by this formula, and it is about 20. The SNR of this pattern on the convex lens surface is 19. It can be seen that the pattern is reconstructed with very high quality, which verifies the validity of the proposed approach.
To study the influence of the surface’s curvature on the reconstruction quality, we increase the curvature by reducing R of the surface to 4 mm, and corresponding 2α = 90°, and keep other parameters unchanged (case 2). After the complete calculation process like case 1, the pattern is reconstructed as shown in Fig. 4(c). The SNR is just over 16, a little bit lower than that of case 1, and it is clear that the reconstructed pattern is still at a high quality. In brief, a binary and non-periodic pattern is reconstructed successfully, which demonstrates that the arbitrariness of fabricating can be in x-direction and y-direction.
Furthermore, to demonstrate the validity of arbitrariness in z-direction, we select a non-periodic concentric ring (NPCR) with the intensity distribution of 256 gray-level as the ideal pattern. The parameters used are the same as the first case except for the propagation distance is change to 118.3 mm, and the ring number of the NPCR is 5. The ideal NPCR pattern is shown in Fig. 5(a). After the same calculation process like case 1, the pattern is reconstructed as shown in Fig. 5(b). The SNR is 18.5, similar to that of case 1, and it can be seen that the quality of the reconstructed pattern is still at a high level. The result indicates that the arbitrary pattern in z-direction can also be modulated successfully on CS.
Before the calculation process, firstly, we consider that the sampling frequency cannot be too large, otherwise it will cost the computing time; secondly, we set the size and sampling frequency of the output surface according to the size and resolution of the desired pattern; thirdly, we set the size and sampling frequency of the input plane according to the size and resolution of the actual phase modulator. So when the sizes and sampling frequencies of both the pattern and the input plane are fixed, in accordance with the Nyquist Sampling Principle the propagation distance should be bigger than a minimum, or the reconstructed quality will be reduced gradually. On the other hand, longer distance leads to the loss of light energy, especially the light wave with high frequency. Therefore, there should be an optimal diffraction distance for numerical simulations. To investigate the dependence of reconstruction quality on the propagation distance (d represents the diffraction distance between the input plane and the curved surface), we study the two cases for binary pattern with different curvatures, where the SNRs are calculated as the function of distances d. The simulation results are plotted in Fig. 6. It is clear that there is an optimal propagation distance for both cases, just as our prediction. We can also see that the reconstructed pattern on surface with a larger curvature has a lower SNR due to the loss of the high frequency of the light wave during propagation and the marginal distortions of the grids during sphere modeling.
4. Experiment results and discussion
Basically, the fabrication process can be performed by the interference lithography (as shown in Fig. 2(b)), and the phase-only modulators and can be replaced by the spatial light modulators (SLMs). Actually since the precise alignment of two SLMs is required and the accuracy of the alignment should be within one micrometer, about one tenth of the dimension of a SLM’s pixel, it is difficult to achieve such precise alignment under present condition of our laboratory. Here we employ the holographic projection technique to fabricate a non-periodic and millimeter-size pattern on a concave lens surface for demonstrating the validity of this proposed approach. Figure 7 shows the schematic of the preliminary experiment system. It is worth saying that only one pure-phase SLM is employed and the phase loaded on the SLM is the summation of two phase distributions φ1 and φ2 [26,27]. The laser with wavelength of 532 nm (Oxxius 532-300-COL-PP-LAS-01462) is employed as the light source. The laser beam is spatial-filtered and collimated, then illuminates the SLM (BNS XY series, 512 × 512 pixels, the active area is 7.68 mm × 7.68 mm). It should be pointed out that the light reflected by SLM has a very strong zero-order noise due to dead areas of SLM, which largely reduces the quality of reconstructed pattern. Since the zero-order noise is at very low frequency, a high-pass filter setup is required to reduce or eliminate it. In the actual experiment as shown in Fig. 7, a 4-f system  consisting of two Fourier lenses with the focal length in 500 mm is employed, and its total length is 2000 mm.
To make it convenient to design and calculate the pattern, the center area of the lens surface is selected in accordance with the size of SLM’s active area (7.68 mm × 7.68 mm). The ‘BIT’ is used as the idea pattern, and parameters of the pattern’s surface are as follows: R = 25.8 mm, r = 5.43 mm, 2α = 24.2° and the number of grids is 128 × 128. Actually the size of pattern can be designed without the restrict of SLM, but the resolution of the SLM indeed affects the quality of reconstructed pattern. In this case, the optimal diffraction distance between the modulators plane and the curved surface is 433 mm. Therefore in the actual optical system, the convex lens is located at 2433 mm from the SLM, which is the sum of optimal diffraction distance (433 mm) and the length of 4-f system (2000 mm). After precisely adjusting the central position and axial distance of convex lens, the filtered light wave illuminates the convex lens (R = 25.8 mm) directly, as shown in Fig. 7, and the modulated optical intensity then is produced. Finally the photopolymer coated on the lens surface is exposed for recording the modulated intensity distribution.
The experimental result is shown in Figs. 8(a) and 8(b). Both pictures are captured by Canon EOS 5D Mark II with the lens of Canon EF 28-300 mm f/3.5-5.6L IS USM. Figure 8(a) shows a photograph of the ‘BIT’ pattern fabricated on the lens surface, while Fig. 8(b) is a magnified picture of the pattern. It can be observed that the sign‘BIT’is patterned, which indicates that the proposed method works well. The size of ‘BIT’ is measured with the width in 6 mm and height in 3 mm, which is well matched with the size of the original designed pattern. However, among the pattern, there is much noise. There are four main reasons which contribute to it: (1) The photopolymer is homemade in our lab and coated by hand, so the smoothness and uniformity of the material is not good. The fluctuation of the surface is at the range of 10-100 μm, which is quite big for such fine fabrication; (2) The curved surface raises the hardship to coat, and it further degrades the uniformity of the photopolymer. (3) The unsteady chemical development of photopolymer causes the random noise and marginal noise. (4)The filter system blocks some of the useful information and brings about additional noise when it is used to eliminate the low-frequency noise. It is noted that limited by the condition of our lab, we cannot align two SLMs precisely to realize the interference lithography directly and we demonstrate this proposed method by holography projection experiment. That is why the feature size of the pattern is so large. The feature size of DOEs on CS can be at the range of subwavelength if the two SLMs are aligned precisely and an optical microscopy system is used, and the quality of the DOEs can be very high. In brief, the numerical simulations and experimental results show that arbitrary intensity distribution with large area on convex lens surface can be modulated by this proposed method properly.
In summary, a novel approach for modulating the optical intensity on CS and fabricating DOEs with arbitrary profile and large area on CS by interference is proposed. The numerical simulations and the experimental verification are performed. A binary pattern and a 256 gray-level pattern are reconstructed numerically. A non-periodic and binary pattern is fabricated experimentally on a convex lens with a large radius of curvature. Nice results are achieved, and this demonstrates the validity of the proposed method. Compared with the previous methods which can only make periodic patterns on CS by interference, this method can fabricate non-periodic and more complex patterns. To our knowledge, it is the first time of realizing the intensity modulation with arbitrary distribution on CS by interference, and we apply it to fabricate DOEs. Inheriting the advantages of interference lithography, this approach is also an efficient and convenient way to realize mass production of DOEs on CS. If one wants to actually pattern on CS with very large curvature, the whole areas with resist can be divided into several proper areas, and each area can be exposed perpendicularly and respectively by rotating the substrate to face directly to the exposure system, that is multiple-exposure technique. The fabrication system is designed to be established by two SLMs or two phase modulators if a high-precision mechanical system can achieve the precise alignment of the devices, which will improve the quality of reconstructed pattern remarkably. It is believed that with the aid of an optical microscopy system and multiple-exposure technique, micro- or nano-optical elements with smaller feature sizes and larger curvatures can be fabricated. Furthermore, this proposed approach can be used for modulating the optical intensity with arbitrary distribution on preset curved surface in the field of optical manipulating.
This work was supported by the National Basic Research Program of China (973 Program Grant No. 2013CB328801 and 2011CB32801), the National Natural Science Founding of China (61235002 and 61077007), and the National High Technology Research and Development Program of China (863 Program Grant No. 2009AA01Z3091).
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