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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 4 — Feb. 25, 2013
  • pp: 5140–5148
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Modulation of optical intensity on curved surfaces and its application to fabricate DOEs with arbitrary profile by interference

Haozhi Zhao, Juan Liu, Ru Xiao, Xin Li, Rui Shi, Peng Liu, Haizheng Zhong, Bingsuo Zou, and Yongtian Wang  »View Author Affiliations


Optics Express, Vol. 21, Issue 4, pp. 5140-5148 (2013)
http://dx.doi.org/10.1364/OE.21.005140


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Abstract

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

1. Introduction

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 [1

1. H. C. Ko, M. P. Stoykovich, J. Song, V. Malyarchuk, W. M. Choi, C. J. Yu, J. B. Geddes 3rd, J. Xiao, S. Wang, Y. Huang, and J. A. Rogers, “A hemispherical electronic eye camera based on compressible silicon optoelectronics,” Nature 454(7205), 748–753 (2008). [CrossRef] [PubMed]

] and artificial compound eyes [2

2. D. Radtke, J. Duparré, U. D. Zeitner, and A. Tünnermann, “Laser lithographic fabrication and characterization of a spherical artificial compound eye,” Opt. Express 15(6), 3067–3077 (2007). [CrossRef] [PubMed]

]. Over the past few years, several techniques including ruling engine [3

3. T. Kita and T. Harada, “Ruling engine using a piezoelectric device for large and high-groove density gratings,” Appl. Opt. 31(10), 1399–1406 (1992). [CrossRef] [PubMed]

], ion beam proximity lithography [4

4. P. Ruchhoeft, M. Colburn, B. Choi, H. Nounu, S. Johnson, T. Bailey, S. Damle, M. Stewart, J. Ekerdt, S. V. Sreenivasan, J. C. Wolfe, and C. G. Willson, “Patterning curved surfaces: template generation by ion beam proximity lithography and relief transfer by step and flash imprint lithography,” J. Vac. Sci. Technol. B 17(6), 2965–2969 (1999). [CrossRef]

] and laser direct writing [5

5. Y. Xie, Z. Lu, F. Li, J. Zhao, and Z. Weng, “Lithographic fabrication of large diffractive optical elements on a concave lens surface,” Opt. Express 10(20), 1043–1047 (2002). [CrossRef] [PubMed]

8

8. T. Wang, W. Yu, D. Zhang, C. Li, H. Zhang, W. Xu, Z. Xu, H. Liu, Q. Sun, and Z. Lu, “Lithographic fabrication of diffractive optical elements in hybrid sol-gel glass on 3-D curved surfaces,” Opt. Express 18(24), 25102–25107 (2010). [CrossRef] [PubMed]

] 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

9. R. J. Jackman, S. T. Brittain, A. Adams, M. G. Prentiss, and G. M. Whitesides, “Design and fabrication of topologically complex, three-dimensional microstructures,” Science 280(5372), 2089–2091 (1998). [CrossRef] [PubMed]

, 10

10. J. G. Kim, N. Takama, B. J. Kim, and H. Fujita, “Optical-softlithographic technology for patterning on curved surfaces,” J. Micromech. Microeng. 19(5), 055017 (2009). [CrossRef]

] and nanoimprint lithography [11

11. T. Senn, J. P. Esquivel, N. Sabate, and B. Lochel, “Fabrication of high aspect ratio nanostructures on 3D surfaces,” Microelectron. Eng. 88(9), 3043–3048 (2011). [CrossRef]

] 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

12. T. A. Savas, S. N. Shah, M. L. Schattenburg, J. M. Carter, and H. I. Smith, “Achromatic interferometric lithography for 100-nm-period gratings and grids,” J. Vac. Sci. Technol. B 13(6), 2732–2735 (1995). [CrossRef]

,13

13. M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404(6773), 53–56 (2000). [CrossRef] [PubMed]

]. 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. [14

14. K. M. Baker, “Highly corrected close-packed microlens arrays and moth-eye structuring on curved surfaces,” Appl. Opt. 38(2), 352–356 (1999). [CrossRef] [PubMed]

] 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

15. J. Nishii, “Glass-imprinting for optical device fabrication,” in Advances in Optical Materials, OSA Technical Digest (CD) (Optical Society of America, 2009), paper AThC1.

17

17. A. Mizutani, S. Takahira, and H. Kikuta, “Two-spherical-wave ultraviolet interferometer for making an antireflective subwavelength periodic pattern on a curved surface,” Appl. Opt. 49(32), 6268–6275 (2010). [CrossRef] [PubMed]

]. In 2010, Mizutani et al. [17

17. A. Mizutani, S. Takahira, and H. Kikuta, “Two-spherical-wave ultraviolet interferometer for making an antireflective subwavelength periodic pattern on a curved surface,” Appl. Opt. 49(32), 6268–6275 (2010). [CrossRef] [PubMed]

] 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. [18

18. R. Shi, J. Liu, J. Xu, D. Liu, Y. Pan, J. Xie, and Y. Wang, “Designing and fabricating diffractive optical elements with a complex profile by interference,” Opt. Lett. 36(20), 4053–4055 (2011). [CrossRef] [PubMed]

] 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

19. L. Ahrenberg, P. Benzie, M. Magnor, and J. Watson, “Computer generated holograms from three dimensional meshes using an analytic light transport model,” Appl. Opt. 47(10), 1567–1574 (2008). [CrossRef] [PubMed]

,20

20. Y. Z. Liu, J. W. Dong, Y. Y. Pu, B. C. Chen, H. X. He, and H. Z. Wang, “High-speed full analytical holographic computations for true-life scenes,” Opt. Express 18(4), 3345–3351 (2010). [CrossRef] [PubMed]

] 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. [21

21. T. Shimobaba, N. Masuda, and T. Ito, “Arbitrary shape surface Fresnel diffraction,” Opt. Express 20(8), 9335–9340 (2012). [CrossRef] [PubMed]

] 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 [22

22. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).

] can be expressed as
U(P2)=1jλU(P1)exp(jkr)rcosθdσ,
(1)
and its inverse form, which is called the inverse Huygens diffraction [23

23. J. R. Shewell and E. Wolf, “Inverse diffraction and a new reciprocity theorem,” J. Opt. Soc. Am. 58(12), 1596–1603 (1968). [CrossRef]

,24

24. S. Yan, Design of Diffractive Micro-optics (National Defense Industry Press, Beijing, 2011), pp. 25, 70, 284 (in Chinese).

], can also be expressed as
U(P1)=jλU(P2)exp(jkr)rcosθdσ,
(2)
where U(P1) and U(P2) are complex light-field distributions on the source surface (SS) and destination surface (DS), respectively, λ and k are the wavelength and wave number of light wave, respectively, coordinates ξηz and xyz are both Cartesian coordinate systems. The diffraction distance between two points from the source surface and destination surface isr=(xξ)2+(yη)2+d(z)2, where d(z) is the distance along the z-axis. If the source surface is planar, d(z) equals d, which is the distance between two parallel planes. While the source surface is curved, d(z) is variable for different points on the curved surface. cosθ is the direction factor, andθ denotes the angle between r12 and n(the normal vector of micro area dσ), as shown in Fig. 1
Fig. 1 Schematic view of the light wave propagation from a source surface to a destination surface. SS-source surface, DS-destination surface.
. When the paraxial approximation condition is well matched, cosθ1is assumed. When it doesn’t meet the paraxial approximation condition, cosθ should be considered for those micro areas, respectively. Please note that in the actual calculation process as shown in Fig. 2(a)
Fig. 2 (a) Calculation process of the desired light wave from CS to the input plane, (b) Michelson interferometer system for realizing the modulation of light intensity distribution.
, optical propagation direction is from surface P1 to surfaceP2, and coordinates ξηz and xyz are both Cartesian coordinate systems. Therefore U(B)=Beiφon surface P1 is obtained by inverse Huygens diffraction and can be expressed as: U(B)=HuF1{U(A)}, where U(A)=Aeiα denotes the original complex distribution on CS and HuF1{...}represents the inverse Huygens diffraction.

As shown in Fig. 2(a) the light distribution of U(B) can be modulated by two phase modulators M1andM2, that is U(B)=eiφ1+eiφ2 and the phase distributions can be analytically obtained as [18

18. R. Shi, J. Liu, J. Xu, D. Liu, Y. Pan, J. Xie, and Y. Wang, “Designing and fabricating diffractive optical elements with a complex profile by interference,” Opt. Lett. 36(20), 4053–4055 (2011). [CrossRef] [PubMed]

,25

25. Y. Zhang and B. Wang, “Optical image encryption based on interference,” Opt. Lett. 33(21), 2443–2445 (2008). [CrossRef] [PubMed]

]
φ1=arg(D)±cos1(abs(D)/2),φ2=arg(Deiφ1),
(3)
where D = U(B), arg(...) and abs(...) represent the phase value and modulus value, respectively. Therefore, U(A) can be expressed as
U(A)=HuF{eiφ1}+HuF{eiφ2},
(4)
where HuF{...} represents the Huygens diffraction. It is indicated that a complex light-field distribution on CS can be formed by interference of the fields generated by two phase-only distributions.

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 U(B)=Beiφ is achieved; step 2, for any complex distribution Beiφ can be decomposed into two pure phases analytically, that isBeiφ=eiφ1+eiφ2; 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 M1and M2 will modulate the wavefronts into eiφ1and eiφ2, 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)
Fig. 3 (a) Side view of a convex lens surface, (b) Schematic view of the modeling of a convex lens surface.
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 dσwhich 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)
Fig. 4 Intensity distributions of (a) an ideal pattern, and the reconstructed patterns of (b) case 1, (c) case 2.
, 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 U(A)=Aeiα on the lens surface is defined as follows: the amplitude part A equals I, while the phase part αis set to be the zero distribution for simplicity, which means that ideal complex distribution becomes U(A)=A. 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 SNR=10log10(Ps/Pn), wherePs=m=1Mn=1NI(m,n)represents the signal item, Pn=m=1Mn=1N|I(m,n)I(m,n)|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)
Fig. 5 Intensity distributions of (a) the ideal NPCR pattern, and (b) the reconstructed NPCR pattern.
. 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
Fig. 6 Dependence of SNR on the diffraction distance between the input plane and the curved surface.
. 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 M1and M2 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
Fig. 7 Schematic view of experimental setup.
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

26. P. Kumar, J. Joseph, and K. Singh, “Optical image encryption using a jigsaw transform for silhouette removal in interference-based methods and decryption with a single spatial light modulator,” Appl. Opt. 50(13), 1805–1811 (2011). [CrossRef] [PubMed]

,27

27. X. Wang and D. Zhao, “Optical image hiding with silhouette removal based on the optical interference principle,” Appl. Opt. 51(6), 686–691 (2012). [CrossRef] [PubMed]

]. 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 [22

22. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).

] 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)
Fig. 8 (a) Photograph of the pattern fabricated on the convex lens. (b) Enlarged picture of the fabricated pattern.
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.

5. Conclusions

Acknowledgments

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).

References and links

1.

H. C. Ko, M. P. Stoykovich, J. Song, V. Malyarchuk, W. M. Choi, C. J. Yu, J. B. Geddes 3rd, J. Xiao, S. Wang, Y. Huang, and J. A. Rogers, “A hemispherical electronic eye camera based on compressible silicon optoelectronics,” Nature 454(7205), 748–753 (2008). [CrossRef] [PubMed]

2.

D. Radtke, J. Duparré, U. D. Zeitner, and A. Tünnermann, “Laser lithographic fabrication and characterization of a spherical artificial compound eye,” Opt. Express 15(6), 3067–3077 (2007). [CrossRef] [PubMed]

3.

T. Kita and T. Harada, “Ruling engine using a piezoelectric device for large and high-groove density gratings,” Appl. Opt. 31(10), 1399–1406 (1992). [CrossRef] [PubMed]

4.

P. Ruchhoeft, M. Colburn, B. Choi, H. Nounu, S. Johnson, T. Bailey, S. Damle, M. Stewart, J. Ekerdt, S. V. Sreenivasan, J. C. Wolfe, and C. G. Willson, “Patterning curved surfaces: template generation by ion beam proximity lithography and relief transfer by step and flash imprint lithography,” J. Vac. Sci. Technol. B 17(6), 2965–2969 (1999). [CrossRef]

5.

Y. Xie, Z. Lu, F. Li, J. Zhao, and Z. Weng, “Lithographic fabrication of large diffractive optical elements on a concave lens surface,” Opt. Express 10(20), 1043–1047 (2002). [CrossRef] [PubMed]

6.

Y. Xie, Z. Lu, and F. Li, “Lithographic fabrication of large curved hologram by laser writer,” Opt. Express 12(9), 1810–1814 (2004). [CrossRef] [PubMed]

7.

D. Radtke and U. D. Zeitner, “Laser-lithography on non-planar surfaces,” Opt. Express 15(3), 1167–1174 (2007). [CrossRef] [PubMed]

8.

T. Wang, W. Yu, D. Zhang, C. Li, H. Zhang, W. Xu, Z. Xu, H. Liu, Q. Sun, and Z. Lu, “Lithographic fabrication of diffractive optical elements in hybrid sol-gel glass on 3-D curved surfaces,” Opt. Express 18(24), 25102–25107 (2010). [CrossRef] [PubMed]

9.

R. J. Jackman, S. T. Brittain, A. Adams, M. G. Prentiss, and G. M. Whitesides, “Design and fabrication of topologically complex, three-dimensional microstructures,” Science 280(5372), 2089–2091 (1998). [CrossRef] [PubMed]

10.

J. G. Kim, N. Takama, B. J. Kim, and H. Fujita, “Optical-softlithographic technology for patterning on curved surfaces,” J. Micromech. Microeng. 19(5), 055017 (2009). [CrossRef]

11.

T. Senn, J. P. Esquivel, N. Sabate, and B. Lochel, “Fabrication of high aspect ratio nanostructures on 3D surfaces,” Microelectron. Eng. 88(9), 3043–3048 (2011). [CrossRef]

12.

T. A. Savas, S. N. Shah, M. L. Schattenburg, J. M. Carter, and H. I. Smith, “Achromatic interferometric lithography for 100-nm-period gratings and grids,” J. Vac. Sci. Technol. B 13(6), 2732–2735 (1995). [CrossRef]

13.

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404(6773), 53–56 (2000). [CrossRef] [PubMed]

14.

K. M. Baker, “Highly corrected close-packed microlens arrays and moth-eye structuring on curved surfaces,” Appl. Opt. 38(2), 352–356 (1999). [CrossRef] [PubMed]

15.

J. Nishii, “Glass-imprinting for optical device fabrication,” in Advances in Optical Materials, OSA Technical Digest (CD) (Optical Society of America, 2009), paper AThC1.

16.

K. Kintaka, J. Nishii, and N. Tohge, “Diffraction gratings of photosensitive ZrO2 gel films fabricated with the two-ultraviolet-beam interference method,” Appl. Opt. 39(4), 489–493 (2000). [CrossRef] [PubMed]

17.

A. Mizutani, S. Takahira, and H. Kikuta, “Two-spherical-wave ultraviolet interferometer for making an antireflective subwavelength periodic pattern on a curved surface,” Appl. Opt. 49(32), 6268–6275 (2010). [CrossRef] [PubMed]

18.

R. Shi, J. Liu, J. Xu, D. Liu, Y. Pan, J. Xie, and Y. Wang, “Designing and fabricating diffractive optical elements with a complex profile by interference,” Opt. Lett. 36(20), 4053–4055 (2011). [CrossRef] [PubMed]

19.

L. Ahrenberg, P. Benzie, M. Magnor, and J. Watson, “Computer generated holograms from three dimensional meshes using an analytic light transport model,” Appl. Opt. 47(10), 1567–1574 (2008). [CrossRef] [PubMed]

20.

Y. Z. Liu, J. W. Dong, Y. Y. Pu, B. C. Chen, H. X. He, and H. Z. Wang, “High-speed full analytical holographic computations for true-life scenes,” Opt. Express 18(4), 3345–3351 (2010). [CrossRef] [PubMed]

21.

T. Shimobaba, N. Masuda, and T. Ito, “Arbitrary shape surface Fresnel diffraction,” Opt. Express 20(8), 9335–9340 (2012). [CrossRef] [PubMed]

22.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).

23.

J. R. Shewell and E. Wolf, “Inverse diffraction and a new reciprocity theorem,” J. Opt. Soc. Am. 58(12), 1596–1603 (1968). [CrossRef]

24.

S. Yan, Design of Diffractive Micro-optics (National Defense Industry Press, Beijing, 2011), pp. 25, 70, 284 (in Chinese).

25.

Y. Zhang and B. Wang, “Optical image encryption based on interference,” Opt. Lett. 33(21), 2443–2445 (2008). [CrossRef] [PubMed]

26.

P. Kumar, J. Joseph, and K. Singh, “Optical image encryption using a jigsaw transform for silhouette removal in interference-based methods and decryption with a single spatial light modulator,” Appl. Opt. 50(13), 1805–1811 (2011). [CrossRef] [PubMed]

27.

X. Wang and D. Zhao, “Optical image hiding with silhouette removal based on the optical interference principle,” Appl. Opt. 51(6), 686–691 (2012). [CrossRef] [PubMed]

OCIS Codes
(050.0050) Diffraction and gratings : Diffraction and gratings
(090.0090) Holography : Holography
(090.1760) Holography : Computer holography
(220.0220) Optical design and fabrication : Optical design and fabrication
(220.4000) Optical design and fabrication : Microstructure fabrication
(050.6875) Diffraction and gratings : Three-dimensional fabrication

ToC Category:
Diffraction and Gratings

History
Original Manuscript: November 28, 2012
Revised Manuscript: January 16, 2013
Manuscript Accepted: January 21, 2013
Published: February 22, 2013

Citation
Haozhi Zhao, Juan Liu, Ru Xiao, Xin Li, Rui Shi, Peng Liu, Haizheng Zhong, Bingsuo Zou, and Yongtian Wang, "Modulation of optical intensity on curved surfaces and its application to fabricate DOEs with arbitrary profile by interference," Opt. Express 21, 5140-5148 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-4-5140


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References

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