Abstract

We present a method to design the pure-phase distribution based on phase optimization for realizing the three-dimensional (3D) intensity modulation on curved surfaces (CS) and apply it to fabricate desired 3D patterns on CS. 3D intensity patterns are reconstructed numerically as well as fabricated experimentally on CS with high quality, which demonstrates the validity of the method. Since the arbitrary phase profile of diffractive optical elements (DOEs) on CS can be mapped into the 3D optical intensity distribution on CS, the method can be directly applied to fabricate any desired DOEs on CS. As far as we know, it is the first time to design the pure-phase distribution for realizing the 3D intensity modulation on CS and apply it to fabricate arbitrary patterns on CS.

© 2014 Optical Society of America

1. Introduction

The methods to fabricate micro/nano-structures on curved surfaces (CS) can be applied to produce many useful devices such as artificial compound eyes [1] and thin-film transistors [2]. In recent years, there have been several methods to fabricate micro/nano-structures on CS including the ruling engine [3], the ion beam proximity lithography [4], the laser direct writing [5–8]. However, these methods require expensive equipment and the fabricating process is time-consuming. Soft lithography [9,10] and nanoimprint lithography [11] are proposed to solve the problem of low-throughput while the fabricating accuracy is limited to micrometer dimension. The interference lithograph is a simple method to fabricate micro patterns in large areas [12–17]. The modulation of the three-dimensional (3D) optical intensity with arbitrary distribution on CS is a key problem for fabricating 3D desired patterns on CS. Shi et al. have proposed an approach to fabricate diffractive optical elements (DOEs) with arbitrary profile by interference on planar surfaces [18], and Zhao et al. have proposed an improved method to fabricate DOEs with arbitrary profile and large area on CS [19]. Though the approach proposed by Zhao et al. can realize the arbitrary 3D intensity modulation on CS by interference theoretically, the precise alignment of two phase-only spatial light modulators (SLMs) within micrometer precision is required. Since it is difficult to align two SLMs precisely for achieving the desired patterns on CS, actually, they employed one phase (loaded in SLM) to fabricate the pattern on the curved surface for verifying the proposed theory, and the experimental result is approximate. Thus it is necessary to explore a convenient and practical method to fabricate micro patterns on CS.

In this paper, we present a method to design the pure-phase distribution on the plane based on phase optimization for realizing the 3D intensity modulation on CS, and apply the method of arbitrary 3D intensity modulation on CS to produce corresponding patterns on CS. The experimental fabrication process is convenient since only one optimized phase (loaded in SLM) is required. 3D intensity patterns are designed and reconstructed numerically on complicated CS. Both a 3D binary pattern and a 3D gray level pattern are reconstructed numerically as well as fabricated experimentally on cylindrical surfaces with high quality. Since the arbitrary phase profile of DOEs on CS can be mapped into the 3D optical intensity distribution on CS, the method can be directly applied to fabricate any desired DOEs on CS.

2. Basic principles

In scalar diffraction domain, Huygens diffraction [20] can be expressed as

U˜(Χ2)=1jλU(Χ1)exp(jkr)rcosθdσ
and its inverse form called inverse Huygens diffraction [21,22] can be expressed as
U(Χ1)=jλ'U˜(Χ2)exp(jkr')r'cosθ'dσ'
where U˜(Χ2) and U(Χ1) are complex light-field distributions on curved surface and P1 (coordinate plane ξηz), respectively, as shown in Fig. 1. Χ1 and Χ2 represent coordinates ξηz and xyz, respectively. Coordinates ξηzand xyz are both Cartesian coordinate systems. λ and k are the wavelength and wave number of light wave in free space, respectively. The distance between P1 and P2(coordinate planexyz) is d. The diffraction distance between two points from P1 and the curved surface is r=(xξ)2+(yη)2+l(z)2, where l(z) is the distance along the z-axis, which is variable for different points on the curved surface. is the plane while ' is the curved surface. cos(θ) and cos(θ') are the direction factors, θ denotes the angle between r12 and n (the normal vector of the micro area dσ on P1), θ' denotes the angle between r21' and n' (the normal vector of the micro area dσ' on curved surface). The positive optical propagation direction is from P1 to P2, thus complex light-field distribution U˜(Χ2)=Aeiα on the curved surface is obtained by Huygens diffraction of U(Χ1)=Beiφ on the P1, which can be expressed as U˜(Χ2)=HuF{U(Χ1)}, where HuF{...} represents the Huygens diffraction, as described in Eq. (1). On the contrary U(Χ1)=Beiφ is obtained by U(Χ1)=HuF1{U˜(Χ2)}, where HuF1{...} represents the inverse Huygens diffraction, as described in Eq. (2).

 figure: Fig. 1

Fig. 1 Schematic view of the light wave propagation from the plane P1 to the curved surface.

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The phase retrieval method is introduced to design the pure-phase distribution for realizing the 3D intensity modulation on CS. As shown in Fig. 2, the whole process of the phase retrieval method can be described as follows: Firstly, the initial complex amplitude distribution U˜0(Χ2) on the curved surface is assigned with the ideal intensity A0 and a random phase. When the jth iteration is performed, the complex amplitude distribution Uj(Χ1) on P1 is computed by using the inverse Huygens diffraction of the distribution on the curved surface. As to the complex amplitude distribution Uj(Χ1), amplitude of unity is imposed and the phase is kept, then the distribution U¯j(Χ1) is obtained. After that, the Huygens diffraction of U¯j(Χ1) is computed to form the complex amplitude distribution U¯˜j(Χ2) on the curved surface. The phase is kept while the amplitude is replaced withAj+1. M is a function of zero at the positions of the zero padding and unity at the image’s pixels, the parameters γ and kwith the value between 0 and 1 are the noise suppression parameter and the feedback parameter, respectively [23]. Subsequently, with replacement of the new complex amplitude distribution as the initial distribution for the next iteration, the procedure is repeated until the distribution U¯j(Χ1) converges into a value that forms the target distribution with acceptable relative error or the process reaches the maximum number of iterations. Finally, the output distribution U¯j(Χ1)on P1 is obtained. Thus the pure-phase distribution φj can be acquired and then loaded in the phase-only SLM in order to modulate ideal 3D intensity distribution on CS.

 figure: Fig. 2

Fig. 2 Flowchart of the phase retrieval method.

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3. Numerical simulations

Numerical simulations are performed to demonstrate the validity of the phase retrieval method for realizing the 3D intensity modulation on CS. First of all, the mathematical modeling method is established. The simplest curved surface is a cylindrical surface which is only curved in one dimension with constant curvature. The sampling pixel of the cylindrical surface is shown in Fig. 3, the top view is shown in Fig. 3(a), and the center green part is divided into many grids with equal areas as shown in Fig. 3(b). Here L is side length of the simulated pattern, R is the radius of the surface curvature, and 2α is the field angle. The area of the single grid will become extremely small if the number of the grids is large enough, which can be achieved through sampling more points for the simulated pattern. Thus every tiny grid can be regarded as a micro area. Next, we design the pure-phase distribution on the plane for modulating 3D intensity distribution on the cylindrical surface. Numerical simulations are performed subsequently for both a 3D binary pattern and a 3D gray level pattern.

 figure: Fig. 3

Fig. 3 (a) Top view and (b) Sampling pixel of cylindrical surface.

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In order to demonstrate the viability of the phase retrieval method, the actual calculation of a 3D binary pattern is conducted numerically. As shown in Fig. 4(a), the ideal 3D intensity distribution I is a four-Chinese-character (meaning: the curved surface processing) pattern on cylindrical surface, and the color bar on the right denotes the range of intensity distribution. The designed results are shown in Figs. 4 (b) and (c). The binary pattern with pixel of 128 × 128 is expanded to a pattern with pixels of 200 × 200 by zero padding. The number of the sampling points is 200 × 200. The parameters used for the simulation of binary pattern are: L = 12mm, R = 51.852mm, λ = 532nm, d = 433mm, and the size of SLM is 7.68mm × 7.68mm. The size of the pattern without zero padding is 7.68mm × 7.68mm and after zero padding the size is 12mm × 12mm. We choose the noise suppression parameter γ=1 and the feedback parameter k=1 for the pattern after zero padding. The ideal complex distribution on the cylindrical surface is U˜(Χ2)=Aeiα, where the amplitude part A is imposed as I. Finally, the reconstructed intensity distribution I' with 35 iterations is shown in Fig. 4(b). The relative error (RE) is introduced to evaluate the reconstruction quality, which can be defined asRE=R1/R2×100%, whereR1=m=1Mn=1N(|I'(m,n)||I(m,n)|)2represents the error item,R2=m=1Mn=1N|I(m,n)|2 denotes the ideal pattern item, and M and N in the formulas represent the number of grids of the pattern. As shown in Fig. 4(c), the RE can reduce to 2.91% after 35 iterations.

 figure: Fig. 4

Fig. 4 Numerical simulation for 3D binary pattern: intensity distributions of (a) an ideal pattern and (b) the reconstructed pattern, (c) relative error as a function of iteration.

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For the purpose of better illuminating the validity of this method, we study on a 3D gray level pattern subsequently. As shown in Fig. 5(a), the 3D gray level badge pattern of Beijing Institute of Technology is used as the ideal pattern. The gray level pattern with pixels of 256 × 256 is expanded to a pattern with pixels of 400 × 400 by zero padding and the number of the sample points is 400 × 400. The other parameters and processing procedure of the pattern are the same as those used for 3D binary pattern mentioned above. As shown in Figs. 5(b) and 5(c), the reconstructed pattern after 35 iterations is acquired with high quality and the RE reduces to 0.66% after 35 iterations, which again demonstrates the availability of the method.

 figure: Fig. 5

Fig. 5 Numerical simulation for 3D gray level badge pattern: intensity distributions of (a) an ideal pattern, (b) the reconstructed pattern, (c) relative error as a function of iteration.

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To better prove the validity of this method for realizing the 3D intensity modulation on various CS, we then perform numerical simulations on more complicated CS with different curvatures. The curved surface can be described by the mathematical formula which is expressed as z=asin(bx)*sin(cy), where parameters a, b and c are constant. The first case is selected as the curved surface with the parameters a = 1/10, b = 1, c = 1. The radius of curvature of the surface is variable along both Cartesian coordinates. The gray level pattern “cameraman” is selected as the desired pattern, as shown in Fig. 6(a). The size of the pattern without zero padding is 4.92mm × 4.92mm and after zero padding the size is 7.68mm × 7.68mm. The other parameters and processing procedure of the pattern are the same as those used for 3D binary pattern mentioned above. Then the numerical simulations of the 3D gray level pattern are performed, and the results are shown in Figs. 6(b) and 6(c). It is seen that the reconstructed pattern is achieved with high quality and the RE reduces to 1.02% after 20 iterations.

 figure: Fig. 6

Fig. 6 Numerical simulation for 3D gray level pattern “cameraman” (a = 1/10, b = 1, c = 1): intensity distributions of (a) an ideal pattern, (b) the reconstructed pattern, and (c) relative error as a function of iteration.

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To study the influence of the surface’s curvature on the reconstruction quality, we then increase the curvature of the CS by setting the parameter a = 1 and keeping other parameters unchanged. The ideal pattern on the curved surface is shown in Figs. 7(a), where the curved surface is steeper with the increase of the curvature. The simulated results are shown in Figs. 7(b) and 7(c). The corresponding RE reduces to 1.71% after 20 iterations. Though a slightly quality reduction of reconstructed pattern is found, the validity of this method for realizing the 3D intensity modulation on complicated CS with different curvatures is demonstrated.

 figure: Fig. 7

Fig. 7 Numerical simulation for 3D gray level pattern “cameraman” (a = 1, b = 1, c = 1): intensity distributions of (a) an ideal pattern, (b) the reconstructed pattern, and (c) relative error as a function of iteration.

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4. Experiment results and discussion

We employ the holographic projection technique to perform the experiment. The simplest curved surface (cylindrical surface) is chosen to perform the optical experiments because the actual fabrication can be conveniently controlled. The schematic view of experimental setup is shown in Fig. 8. In the actual experiment, the light source is a laser with wavelength of 532nm (Oxxius 532-300-COL-PP-LAS-01462), the SLM (BNS XY series, 512 × 512 pixels, the active area is 7.68mm × 7.68mm) is illuminated by the spatial-filtered and collimated laser beam produced by the light source. The pure-phase distribution we designed is subsequently loaded in the SLM to modulate the incident laser beam. A very strong zero-order noise of the modulated light reflected by SLM can be eliminated by a combination of a 4-f system and a high-pass filter. It is necessary to adjust the central position and axial distance of cylindrical lens precisely. The modulated optical intensity is produced. Finally, the photopolymer on the cylindrical lens surface is exposed by the modulated light for recording the modulated 3D intensity distribution.

 figure: Fig. 8

Fig. 8 Schematic view of experimental setup.

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For both experiments, the sizes of patterns are selected in accordance with the size of SLM’s active area (7.68mm × 7.68mm). The parameters referred in the experiment are as follows: L = 12mm, R = 51.852mm, λ = 532nm, the number of grids is 200 × 200 for the binary pattern, and 400 × 400 for the gray level badge pattern, respectively. The diffraction distances between the modulators plane and the cylindrical surface are both 433mm for two cases. The whole length is 2633mm from SLM to cylindrical lens in the actual optical system.

The experimental results are shown in Fig. 9 and Fig. 10, where the 3D binary pattern with four Chinese characters and the 3D gray level badge pattern of Beijing Institute of Technology are used as the ideal patterns. Figure 9 and Fig. 10 (a) are captured by 500X Series Digital Microscope, while the magnified partial image in Fig. 10(b) is captured by OLYMPUS BX51 optical microscope. There is a proportional dimension at the lower right corner of the image. As we can see, the two patterns are fabricated successfully, which demonstrates the validity of the proposed method.

 figure: Fig. 9

Fig. 9 (a) Photograph of the 3D binary pattern fabricated on the cylindrical lens. (b) Enlarged picture of the fabricated pattern.

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 figure: Fig. 10

Fig. 10 (a) Photograph of the 3D gray level badge pattern fabricated on the cylindrical lens. (b) Enlarged picture of the fabricated pattern.

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Although the two patterns are formed, the resolution is not very well in the following several possible reasons: first, the smoothness and uniformity of the photopolymer used is not good; second, the filter system blocks brings about additional noise when it is used to eliminate the low-frequency noise; third, the dust is inevitable during the image acquisition experiments, which also reduces the captured image quality. In brief, ideal experimental environment will guarantee the good results. It proves to be an effective method to modulate 3D intensity distribution on CS.

This method could be applied to fabricate the DOEs with arbitrary phase distribution on CS. When a DOE with arbitrary phase profile on CS needs to be fabricated, since the chemical development of the photoresist on CS is linearly related to the 3D intensity of the light wave, we can convert the required phase distribution into the optical intensity of the light wave. When the optical intensity of the light wave on CS is modulated, the fabrication of DOEs with arbitrary phase distribution on CS can be achieved.

5. Conclusion

The method based on phase optimization is proposed to design the pure-phase distribution on the plane for realizing the 3D intensity modulation on CS. Both the numerical simulations and the experimental verification are performed with high quality. The method simplifies the experimental fabrication process by using a single SLM. It can be applied to fabricate the non-periodic and more complex 3D patterns on CS, such as any desired DOEs on CS. It is worth to mention that the CS can be even more complicated though we use the cylindrical surface in the experiments. Because a single SLM is required, this method is convenient and it could have broader application prospects of various fields such as image processing, 3D display, and optical manipulation on CS.

Acknowledgments

This work was supported by the National Basic Research Program of China (973 Program Grant Nos. 2013CB328801 and 2013CBA01702), the National Natural Science Foundation of China (61235002).

References and links

1. 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]  

2. G. Yoo, H. Lee, D. Radtke, M. Stumpf, U. Zeitner, and J. Kanicki, “A maskless laser-write lithography processing of thin-film transistors on a hemispherical surface,” Microelectron. Eng. 87(1), 83–87 (2010). [CrossRef]  

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. H. Zhao, J. Liu, R. Xiao, X. Li, R. Shi, P. Liu, H. Zhong, B. Zou, and Y. Wang, “Modulation of optical intensity on curved surfaces and its application to fabricate DOEs with arbitrary profile by interference,” Opt. Express 21(4), 5140–5148 (2013). [CrossRef]   [PubMed]  

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

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

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

23. A. Georgiou, J. Christmas, N. Collings, J. Moore, and W. A. Crossland, “Aspects of hologram calculation for video frames,” J. Opt. A, Pure Appl. Opt. 10(3), 035302 (2008). [CrossRef]  

References

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  1. 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]
  2. G. Yoo, H. Lee, D. Radtke, M. Stumpf, U. Zeitner, and J. Kanicki, “A maskless laser-write lithography processing of thin-film transistors on a hemispherical surface,” Microelectron. Eng. 87(1), 83–87 (2010).
    [Crossref]
  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. H. Zhao, J. Liu, R. Xiao, X. Li, R. Shi, P. Liu, H. Zhong, B. Zou, and Y. Wang, “Modulation of optical intensity on curved surfaces and its application to fabricate DOEs with arbitrary profile by interference,” Opt. Express 21(4), 5140–5148 (2013).
    [Crossref] [PubMed]
  20. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
  21. J. R. Shewell and E. Wolf, “Inverse diffraction and a new reciprocity theorem,” J. Opt. Soc. Am. 58(12), 1596–1603 (1968).
    [Crossref]
  22. S. Yan, Design of Diffractive Micro-optics (National Defense Industry, 2011), pp. 25, 70, 284 (in Chinese).
  23. A. Georgiou, J. Christmas, N. Collings, J. Moore, and W. A. Crossland, “Aspects of hologram calculation for video frames,” J. Opt. A, Pure Appl. Opt. 10(3), 035302 (2008).
    [Crossref]

2013 (1)

2011 (2)

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]

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]

2010 (3)

2009 (1)

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]

2008 (1)

A. Georgiou, J. Christmas, N. Collings, J. Moore, and W. A. Crossland, “Aspects of hologram calculation for video frames,” J. Opt. A, Pure Appl. Opt. 10(3), 035302 (2008).
[Crossref]

2007 (2)

2004 (1)

2002 (1)

2000 (2)

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]

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]

1999 (2)

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]

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]

1998 (1)

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]

1995 (1)

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]

1992 (1)

1968 (1)

Adams, A.

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]

Bailey, T.

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]

Baker, K. M.

Brittain, S. T.

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]

Campbell, M.

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]

Carter, J. M.

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]

Choi, B.

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]

Christmas, J.

A. Georgiou, J. Christmas, N. Collings, J. Moore, and W. A. Crossland, “Aspects of hologram calculation for video frames,” J. Opt. A, Pure Appl. Opt. 10(3), 035302 (2008).
[Crossref]

Colburn, M.

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]

Collings, N.

A. Georgiou, J. Christmas, N. Collings, J. Moore, and W. A. Crossland, “Aspects of hologram calculation for video frames,” J. Opt. A, Pure Appl. Opt. 10(3), 035302 (2008).
[Crossref]

Crossland, W. A.

A. Georgiou, J. Christmas, N. Collings, J. Moore, and W. A. Crossland, “Aspects of hologram calculation for video frames,” J. Opt. A, Pure Appl. Opt. 10(3), 035302 (2008).
[Crossref]

Damle, S.

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]

Denning, R. G.

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]

Duparré, J.

Ekerdt, J.

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]

Esquivel, J. P.

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]

Fujita, H.

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]

Georgiou, A.

A. Georgiou, J. Christmas, N. Collings, J. Moore, and W. A. Crossland, “Aspects of hologram calculation for video frames,” J. Opt. A, Pure Appl. Opt. 10(3), 035302 (2008).
[Crossref]

Harada, T.

Harrison, M. T.

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]

Jackman, R. J.

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]

Johnson, S.

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]

Kanicki, J.

G. Yoo, H. Lee, D. Radtke, M. Stumpf, U. Zeitner, and J. Kanicki, “A maskless laser-write lithography processing of thin-film transistors on a hemispherical surface,” Microelectron. Eng. 87(1), 83–87 (2010).
[Crossref]

Kikuta, H.

Kim, B. J.

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]

Kim, J. G.

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]

Kintaka, K.

Kita, T.

Lee, H.

G. Yoo, H. Lee, D. Radtke, M. Stumpf, U. Zeitner, and J. Kanicki, “A maskless laser-write lithography processing of thin-film transistors on a hemispherical surface,” Microelectron. Eng. 87(1), 83–87 (2010).
[Crossref]

Li, C.

Li, F.

Li, X.

Liu, D.

Liu, H.

Liu, J.

Liu, P.

Lochel, B.

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]

Lu, Z.

Mizutani, A.

Moore, J.

A. Georgiou, J. Christmas, N. Collings, J. Moore, and W. A. Crossland, “Aspects of hologram calculation for video frames,” J. Opt. A, Pure Appl. Opt. 10(3), 035302 (2008).
[Crossref]

Nishii, J.

Nounu, H.

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]

Pan, Y.

Prentiss, M. G.

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]

Radtke, D.

Ruchhoeft, P.

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]

Sabate, N.

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]

Savas, T. A.

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]

Schattenburg, M. L.

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]

Senn, T.

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]

Shah, S. N.

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]

Sharp, D. N.

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]

Shewell, J. R.

Shi, R.

Smith, H. I.

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]

Sreenivasan, S. V.

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]

Stewart, M.

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]

Stumpf, M.

G. Yoo, H. Lee, D. Radtke, M. Stumpf, U. Zeitner, and J. Kanicki, “A maskless laser-write lithography processing of thin-film transistors on a hemispherical surface,” Microelectron. Eng. 87(1), 83–87 (2010).
[Crossref]

Sun, Q.

Takahira, S.

Takama, N.

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]

Tohge, N.

Tünnermann, A.

Turberfield, A. J.

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]

Wang, T.

Wang, Y.

Weng, Z.

Whitesides, G. M.

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]

Willson, C. G.

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]

Wolf, E.

Wolfe, J. C.

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]

Xiao, R.

Xie, J.

Xie, Y.

Xu, J.

Xu, W.

Xu, Z.

Yoo, G.

G. Yoo, H. Lee, D. Radtke, M. Stumpf, U. Zeitner, and J. Kanicki, “A maskless laser-write lithography processing of thin-film transistors on a hemispherical surface,” Microelectron. Eng. 87(1), 83–87 (2010).
[Crossref]

Yu, W.

Zeitner, U.

G. Yoo, H. Lee, D. Radtke, M. Stumpf, U. Zeitner, and J. Kanicki, “A maskless laser-write lithography processing of thin-film transistors on a hemispherical surface,” Microelectron. Eng. 87(1), 83–87 (2010).
[Crossref]

Zeitner, U. D.

Zhang, D.

Zhang, H.

Zhao, H.

Zhao, J.

Zhong, H.

Zou, B.

Appl. Opt. (4)

J. Micromech. Microeng. (1)

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]

J. Opt. A, Pure Appl. Opt. (1)

A. Georgiou, J. Christmas, N. Collings, J. Moore, and W. A. Crossland, “Aspects of hologram calculation for video frames,” J. Opt. A, Pure Appl. Opt. 10(3), 035302 (2008).
[Crossref]

J. Opt. Soc. Am. (1)

J. Vac. Sci. Technol. B (2)

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]

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]

Microelectron. Eng. (2)

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]

G. Yoo, H. Lee, D. Radtke, M. Stumpf, U. Zeitner, and J. Kanicki, “A maskless laser-write lithography processing of thin-film transistors on a hemispherical surface,” Microelectron. Eng. 87(1), 83–87 (2010).
[Crossref]

Nature (1)

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]

Opt. Express (6)

Opt. Lett. (1)

Science (1)

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]

Other (3)

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

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

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

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Figures (10)

Fig. 1
Fig. 1 Schematic view of the light wave propagation from the plane P 1 to the curved surface.
Fig. 2
Fig. 2 Flowchart of the phase retrieval method.
Fig. 3
Fig. 3 (a) Top view and (b) Sampling pixel of cylindrical surface.
Fig. 4
Fig. 4 Numerical simulation for 3D binary pattern: intensity distributions of (a) an ideal pattern and (b) the reconstructed pattern, (c) relative error as a function of iteration.
Fig. 5
Fig. 5 Numerical simulation for 3D gray level badge pattern: intensity distributions of (a) an ideal pattern, (b) the reconstructed pattern, (c) relative error as a function of iteration.
Fig. 6
Fig. 6 Numerical simulation for 3D gray level pattern “cameraman” (a = 1/10, b = 1, c = 1): intensity distributions of (a) an ideal pattern, (b) the reconstructed pattern, and (c) relative error as a function of iteration.
Fig. 7
Fig. 7 Numerical simulation for 3D gray level pattern “cameraman” (a = 1, b = 1, c = 1): intensity distributions of (a) an ideal pattern, (b) the reconstructed pattern, and (c) relative error as a function of iteration.
Fig. 8
Fig. 8 Schematic view of experimental setup.
Fig. 9
Fig. 9 (a) Photograph of the 3D binary pattern fabricated on the cylindrical lens. (b) Enlarged picture of the fabricated pattern.
Fig. 10
Fig. 10 (a) Photograph of the 3D gray level badge pattern fabricated on the cylindrical lens. (b) Enlarged picture of the fabricated pattern.

Equations (2)

Equations on this page are rendered with MathJax. Learn more.

U ˜ ( Χ 2 )= 1 jλ U( Χ 1 ) exp(jkr) r cosθ dσ
U( Χ 1 )= j λ ' U ˜ ( Χ 2 ) exp(jkr') r' cos θ ' dσ'

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