October 15, 2011 / Vol. 36, No. 20 / OPTICS LETTERS
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Designing and fabricating diffractive optical elements with a complex profile by interference Rui Shi,1 Juan Liu,1,2 Jia Xu,1 Dongmei Liu,1 Yijie Pan,1 Jinghui Xie,1 and Yongtian Wang1,3 1
Key Laboratory of Photoelectronic Imaging Technology and System, Ministry of Education of China, School of Optics and Electronics, Beijing Institute of Technology, Beijing 100081, China 2
e-mail:
[email protected] 3
e-mail:
[email protected]
Received June 30, 2011; revised August 19, 2011; accepted September 14, 2011; posted September 15, 2011 (Doc. ID 149972); published October 12, 2011 We demonstrate a novel (to our knowledge) method for the design and the fabrication of diffractive optical elements (DOEs) with an arbitrary complex phase profile based on interference. The DOEs are designed to modulate the complex light wave by the analytical formulas, and an asymmetric holographic DOE with cubic phase modulation is fabricated by a two-step exposure technique. The desired Airy beams are produced experimentally, which demonstrates the validity of this method. It is a simple approach with a low cost for the design and the fabrication of DOEs with a large area and arbitrary phase distribution. © 2011 Optical Society of America OCIS codes: 090.2890, 090.1970, 220.3740, 220.4000, 220.4241, 220.4610.
Diffractive optical elements (DOEs) are widely used in various optical fields, such as beam shaping, holographic projection, and optical encryption. The design of DOEs is an issue of amplitude–phase retrieval. The methods for the design of DOEs are normally based on optimization iteration algorithms, such as the Gerchberg–Saxton algorithm [1,2], the Yang–Gu algorithm [3], and the simulated annealing algorithm [4], where the intensity in the output plane is elaborately achieved regardless of the phase. However, in many optical systems, it is desired that the DOE be designed to modulate the intensity and the phase of the light wave in the output plane simultaneously. DOEs with an arbitrary phase distribution are commonly fabricated by multilevel masks, gray-tone masks, electron beam lithography, focused ion beam lithography, etc. [5], and all these techniques are time consuming and expensive. Holographic or interferometric lithography is a low-cost and efficient technique to fabricate DOEs in large areas [6–10]. However, conventional holographic or interferometric lithography techniques can fabricate either the grating or the simple lens with linear or spherical phase distribution based on the interference of plane waves or spherical waves [11–14]. It is of particular interest to fabricate DOEs with arbitrary phase distribution in large areas. The interferometric phase contrast method [15] is proposed to fabricate arbitrary DOEs, where the phases of ideal DOEs and the ones generated by liquid crystal on silicon are correlated by the wellknown 4-f system. Precise alignment is needed, and the fabricated DOEs can modulate either the amplitude or the phase of the incident light wave in the output plane. For better controlling and operating of the light waves in various optical systems, it is an ideal that the designed and fabricated DOEs can modulate both the amplitude and the phase distributions simultaneously in the output plane. Here we describe an approach for designing DOEs that can modulate both the amplitude and the phase of the light wave simultaneously, and we demonstrate it by fabricating a cubic phase element using the two-step exposure technique. The schematic view of the optical system for designing DOEs based on interference is shown in Fig. 1. The 0146-9592/11/204053-03$15.00/0
uniform plane waves illuminate the input planes P1 and P2, where the pure phase distributions φ1 and φ2 modulate the wavefronts into A01 eiφ1 and A02 eiφ2 (to simplify, A01 ¼ A02 ¼ 1), and they are combined by the beam splitter. The complex amplitude distribution Aeiα in the output plane P3 can be expressed as Aeiα ¼ U 1 þ U 2 , where U 1 ¼ FrTfeiφ1 ; zg, U 2 ¼ FrTfeiφ2 ; zg. FrTf:::g represents the Fresnel or Fraunhofer transform under the paraxial approximation. Then the phase distributions can be obtained as [16,17] jξj φ1 ¼ argðξÞ cos ; 2 −1
φ2 ¼ argðξ − eiφ1 Þ ; ð1Þ
where ξ ¼ FrT−1 fAeiα g. argð:::Þ represents the phase value. Since the phase distributions φ1 and φ2 are obtained analytically, the desired optical function Aeiα can be achieved exactly when both optical waves with pure phases interfere with each other. Therefore, recording these two phases is the key task. It can be achieved by loading two phases in the spatial light modulators (SLMs) placed at the end of the interferometer. The process of fabricating the designed DOEs with arbitrary phase distribution can be described as follows. When a DOE with arbitrary phase profile Pðx; yÞ needs to be fabricated, one first converts the required phase distribution Pðx; yÞ into
Fig. 1. (Color online) Schematic view of the optical interferometric system. BS, beam splitter. © 2011 Optical Society of America
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OPTICS LETTERS / Vol. 36, No. 20 / October 15, 2011
the optical intensity of the light wave I 0 ðx; yÞ, since the chemical development of the photoresist is linearly related to the intensity of p the light ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi wave Pðx; yÞ ¼ cI 0 ðx; yÞ. Then let Aðx; yÞ ¼ I 0 ðx; yÞ, where c is constant. Finally, the fabrication process can be performed by interferometric lithography (as shown in Fig. 1). Since the precise alignment of two SLMs is required and the accuracy of the alignment should be within the scale of 1 μm, about one tenth of a pixel’s dimension, it is difficult to achieve such precision under the present condition of our laboratory. Here we employ holographic lithography and double exposure to fabricate a holographic DOE to demonstrate the validity of this proposed approach where only one SLM is used in a sequence. The holographic lithography system is shown in Fig. 2(a), where an SLM is used for modulating the pure phase distribution and the two-step exposure technique is employed to superpose the two pure phase distributions. The laser beam is split and collimated. One beam is named as the reference wave R, and the other one is modulated by the pure phase SLM as the object wave. The recorded optical intensity distribution can be written as Iðx; yÞ ∝ hOðx; yÞ • R ðx; yÞi;
ð2Þ
where Oðx; yÞ and Rðx; yÞ are the complex amplitudes of the object light wave and the reference light wave, respectively. The superscript “” means the conjugation, and h:::i denotes time average. Rðx;⇀yÞ is assumed to be the uniform ⇀ plane wave Rðx; yÞ ¼ e−i k · r . The complex object light wave Oðx; yÞ can be decomposed into two pure phase distributions, as described by Eq. (1). The fabrication process can be described as follows: first, the pure phase φ1 is loaded onto the SLM and U 1 is frozen in the holographic plate by recording its interference with the reference wave R [I 1 ¼ hjeiφ1 þ Rj2 i]; second, φ2 is loaded onto the SLM and U 2 is frozen in the holographic plate [I 2 ¼ hjeiφ2 þ Rj2 i], and I ¼ I 1 þ I 2 , which is a two-step exposure process. After the chemical development process of the holographic plate, the holographic DOE with the desired complex phase distributions is fabricated. When the reference wave is illuminated into the holographic plate, the reconstructed light wave Eðx; yÞ can be written as Eðx; yÞ ∝ FrTfeiφ1 þ eiφ2 ; zg · R ðx; yÞ · Rðx; yÞ;
ð3Þ
Fig. 2. (Color online) Optical experimental setup: (a) recording system with the double exposure technique, (b) reconstruction of the fabricated holographic DOE. M1–3, mirrors; L1–2, lenses; SF1–2, spatial filters.
Fig. 3. Numerical results of the pure phase distributions (a) φ1 and (b) φ2 .
where the desired light wave Aeiα ¼ FrTfeiφ1 þ eiφ2 ; zg appears in the output plane, as shown in Fig. 2(b). To demonstrate the validity of the approach, we design and fabricate a holographic diffractive optical element (HDOE) with an asymmetric cubic phase distribution that can be used to produce Airy beams with a constant phase [18]. The parameters used are as follows: a He–Ne laser with a wavelength of λ ¼ 632:8 nm and power of 50 mW is employed; a reflective phase-only SLM (Holoeye Pluto) with an active area of 8:64 mm × 8:64 mm, a pitch period of 8 μm × 8 μm, and a pixel size of 1080 × 1080 is used. The recording area on the holographic plate (silver halide plate) is about the same size as the SLM. The distance between the SLM and the recording plane is 200 mm. The tilt angle between the SLM and the recording plane is less than 5°, so the effect caused by the slight tilt can be neglected. The phase distributions φ1 and φ2 that are numerically simulated and then loaded onto the SLM according to the time sequence are shown in Figs. 3(a) and 3(b), respectively. The experiment setup is shown in Fig. 2(a). The modulated beam with the phase distributions φ1 and φ2 in a sequence interferes with the reference plane wave on the recording plane, and then the interferometric patterns are recorded one by one. By carefully controlling the ratio of the two exposure times according to the sensitivity to the light intensity of the holographic material, we can record the interference fringe on the hologram plate. The fabricated HDOE profile is shown in Fig. 4(a), measured by an optical profiler (veeco wyko NT 9100). The pinstripe is modulated by the individual pixel of the SLM, and the main stripe that is observed to be not regular corresponds to the phase distributions of the HDOE. Then we turn to study the optical characteristics of the HDOE. When a plane wave illuminates the obtained pure phase HDOE, as shown in Fig. 2(b), the Airy beams are
Fig. 4. (Color online) Experimental results: (a) portion of the fabricated HDOE, (b) Airy beams produced by the fabricated HDOE.
October 15, 2011 / Vol. 36, No. 20 / OPTICS LETTERS
achieved on the focal plane of the Fourier lens [18], as recorded in Fig. 4(b) by a CCD (Lumenera Infinity 411c). When the beams interfere with a plane wave, it is found that the phase is approximately constant by the phase-shifting method. Further investigation indicates that three extraordinary optical characteristics of Airy beams [18] can be observed when they propagates, which verifies that the fabricated HDOE can realize the modulation of phase and amplitude simultaneously. Figure 4(b) shows the þ1 order of the diffractive beams, that is, the desired Airy beams. The diffraction efficiency of the þ1 order is measured experimentally to be about 22% including the noise caused by the zero order of the SLM [19], where diffraction efficiency of the desired Airy beams is only about 15%. It can be improved by using volume phase materials such as dichromated gelatin. In fact, the twin image also appears as the −1 order, as well as the zero-order beam as shown in Fig. 2(b). The inhomogenous noise in the Airy beams is caused by the zeroorder interruption from the nonfull filled pixlated SLM [19], and the uniform noise is due to the vibration of the hologram plate, the air disturbance, the ratio of the two exposure times, the unsteady chemical process to the hologram plate, etc. The resolution of the fabricated DOE is limited by the pitch period and the resolution of the SLM, which can be further improved by inserting an optical microscope in the fabrication system. In summary, an analytical approach for designing and fabricating DOEs with a complex profile based on interference is proposed. Both a numerical simulation and an optical experiment are performed, and they are in nice agreement, which verifies the validation of the new approach. The whole fabrication system can also be established containing two SLMs if precise alignment can be achieved by a high-precision mechanical system. It is believed that micro- or nano- optical elements with smaller feature sizes can be fabricated by the proposed method when an optical microscopy system is used. It will be a very efficient and convenient way to design and fabricate DOEs with a required complex fine structure. This work is supported by the National Basic Research Program of China (973 Program Grant No. 2011CB301801), the National Natural Science Foundation of China (NSFC) (61077007), the National High
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Technology Research and Development Program of China (863 Program Grant No. 2009AOZ309), and the Innovation Team Development Program of the Chinese Ministry of Education (Grant No. IRT0606). References 1. R. W. Gerchberg and W. O. Saxton, Optik 35, 237 (1972). 2. J. R. Fienup, Opt. Lett. 3, 27 (1978). 3. G. Yang, B. Dong, B. Gu, J. Zhuang, and O. K. Ersoy, Appl. Opt. 33, 209 (1994). 4. S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, Science 220, 671 (1983). 5. Z. Cui, Micro-Nanofabrication Technologies And Applications (Higher Education Press, 2005). 6. M. Farhoud, J. Ferrera, A. J. Lochtefeld, T. E. Murphy, M. L. Schattenburg, J. Carter, C. A. Ross, and H. I. Smith, J. Vac. Sci. Technol. B 17, 3182 (1999). 7. T. A. Savas, S. N. Shah, M. L. Schattenburg, J. M. Carter, and H. I. Smith, J. Vac. Sci. Technol. B 13, 2732 (1995). 8. H. H. Solak, Y. Ekinci, and P. Käser, J. Vac. Sci. Technol. B 25, 91 (2007). 9. A. Fernandez, H. T. Nguyen, J. A. Britten, R. D. Boyd, M. D. Perry, D. R. Kania, and A. M. Hawryluk, J. Vac. Sci. Technol. B 15, 729 (1997). 10. M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, Nature 404, 53 (2000). 11. J. M. Lee, S. H. Oh, C. W. Lee, H. S. Ko, S. Park, K. S. Kim, and M. H. Park, Electrochem. Solid-State Lett. 7, G11 (2004). 12. D. Gil, R. Menon, X. Tang, H. I. Smith, and D. J. D. Carter, J. Vac. Sci. Technol. B 20, 2597 (2002). 13. M. Fritze, T. M. Bloomstein, B. Tyrrell, T. H. Fedynyshyn, N. N. Efremow, D. E. Hardy, S. Cann, D. Lennon, S. Spector, M. Rothschild, and P. Brooker, J. Vac. Sci. Technol. B 23, 2743 (2005). 14. H. Martinsson, T. Sandstrom, A. Bleeker, and J. D. Hintersteiner, J. Microlith. Microfab. Microsyst. 4, 011003 (2005). 15. M. Teschke, R. Heyer, M. Fritzsche, S. Stoebenau, and S. Sinzinger, Appl. Opt. 47, 2550 (2008). 16. Y. Zhang and B. Wang, Opt. Lett. 33, 2443 (2008). 17. N. Zhu, Y. Wang, J. Liu, and J. Xie, Opt. Commun. 283, 4969 (2010). 18. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007). 19. H. Zhang, J. Xie, J. Liu, and Y. Wang, Appl. Opt. 48, 5834 (2009).
