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A kind of optical diffractive element named photon sieve, which is essentially Fresnel zone plate in which the transmissive rings are replaced with a large number of randomly distributed isolated pinholes, can be used to focus X-ray and extreme ultraviolet lithography spectrum into spots with sizes smaller than the diameter of the smallest circular pinhole. However, both the traditional photon sieves and Fibonacci sieves have no more than two axial foci. In order to break this limitation, the Lucas sequence is introduced into the design of photon sieves, and thus producing four axial foci. With respect to the previous Fibonacci sequence, Lucas sequence has the same recursion relation as well as the same eigenvalue of golden mean =(1 + 5)/2. The only difference between them is the first two initial seeds. Based on Fresnel-Kichhoff diffraction theory, the simulation results show that there exist four focal spots with approximately equal intensity along the optical axis on condition that the hole diameters are set to be 1.16 times the underlying Fresnel zone width. Then in order to verify the validity of our proposed model, a Lucas sieve of diameter 12.11 mm and referred focal length 180 mm is fabricated by photolithography and its focusing properties are precisely measured by the in-line phase-shifting digital holography. In experiment, a quarter wave plate is used to realize two-step phase-shift interferences, and obtain the quad-focal length by auto-focusing algorithm in holography. Meanwhile, the quad-focal spots can also be calculated through the diffraction propagation of reconstructed object wave. Compared with the theoretical values, the measurement results indicate that the maximum deviation of quad-focal lengths is less than 0.9%, and the relative errors of the full width at half maximum of four Airy spots are all less than 5%. The experimental results agree well with the theoretical analysis results. Owing to the advantages of small volume, little weight and easy processing, Lucas sieve has great potential in X-ray microscopy, array imaging for living biological cell and especially in the next generation of synchrotron light sources.
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Keywords:
- diffractive optical elements /
- multifocal imaging /
- digital holography /
- X-ray optics
[1] Kirz J 1974 J. Opt. Soc. Am. 64 301
[2] Suzuki Y, Takeuchi A, Takano H, Takenaka H 2005 Jpn. J. Appl. Phys. 44 1994
[3] Kyuragi H, Urisu T 1985 Appl. Opt. 24 1139
[4] Sun J A, Cai A 1991 J. Opt. Soc. Am. A 8 33
[5] Kipp L, Skibowski M, Johnson R L, Berndt R, Adelung R, Harm S, Seemann R 2001 Nature 414 184
[6] Xie C Q, Zhu X L, Shi L N, Liu M 2010 Opt. Lett. 35 1765
[7] Chung H H, Bradman N M 2008 Opt. Engng. 47 118001
[8] Jia J, Xie C Q 2009 Chin. Phys. B 18 183
[9] Gimenez F, Monsoriu J A, Furlan W D, Pons A 2006 Opt. Express 14 11958
[10] Gimenez F, Furlan W D, Monsoriu J A 2007 Opt. Commun. 277 1
[11] Kallane M, Buck J, Harm S, Seemann R, Rossnagel K, Kipp L 2011 Opt. Lett. 36 2405
[12] Xie C Q, Zhu X L, Li H L, Shi L N, Wang Y H 2010 Opt. Lett. 35 4048
[13] Cheng G X, Hu C 2011 Acta Phys. Sin. 60 080703 (in Chinese)[程冠晓, 胡超 2011 物理学报 60 080703]
[14] Cheng G X, Xing T W, Lin W M, Zhou J M, Qiu C K, Liao Z J, Yang Y, Hong L, Ma J L 2007 Proc. SPIE 6517 651736
[15] Cheng Y G, Tong J M, Zhu J P, Liu J B, Hu S, He Y 2016 Opt. Lasers Eng. 77 18
[16] Kincade K 2004 Laser Focus World 40 34
[17] Andersen G, Asmolova O, McHarq M G, Quiller T, Maldonado C 2016 Proc. SPIE 9904 99041P
[18] Zhang J Y 2015 Opt. Express 23 30308
[19] Ke J, Zhang J Y 2015 Acta Opt. Sin. 35 0923001 (in Chinese)[柯杰, 张军勇 2015 光学学报 35 0923001]
[20] Ke J, Zhang J Y 2015 Appl. Opt. 54 7278
[21] Ke J, Zhang J Y 2016 Opt. Commum. 368 34
[22] Takaki Y, Kawai H, Ohzu H 1999 Appl. Opt. 38 4990
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[1] Kirz J 1974 J. Opt. Soc. Am. 64 301
[2] Suzuki Y, Takeuchi A, Takano H, Takenaka H 2005 Jpn. J. Appl. Phys. 44 1994
[3] Kyuragi H, Urisu T 1985 Appl. Opt. 24 1139
[4] Sun J A, Cai A 1991 J. Opt. Soc. Am. A 8 33
[5] Kipp L, Skibowski M, Johnson R L, Berndt R, Adelung R, Harm S, Seemann R 2001 Nature 414 184
[6] Xie C Q, Zhu X L, Shi L N, Liu M 2010 Opt. Lett. 35 1765
[7] Chung H H, Bradman N M 2008 Opt. Engng. 47 118001
[8] Jia J, Xie C Q 2009 Chin. Phys. B 18 183
[9] Gimenez F, Monsoriu J A, Furlan W D, Pons A 2006 Opt. Express 14 11958
[10] Gimenez F, Furlan W D, Monsoriu J A 2007 Opt. Commun. 277 1
[11] Kallane M, Buck J, Harm S, Seemann R, Rossnagel K, Kipp L 2011 Opt. Lett. 36 2405
[12] Xie C Q, Zhu X L, Li H L, Shi L N, Wang Y H 2010 Opt. Lett. 35 4048
[13] Cheng G X, Hu C 2011 Acta Phys. Sin. 60 080703 (in Chinese)[程冠晓, 胡超 2011 物理学报 60 080703]
[14] Cheng G X, Xing T W, Lin W M, Zhou J M, Qiu C K, Liao Z J, Yang Y, Hong L, Ma J L 2007 Proc. SPIE 6517 651736
[15] Cheng Y G, Tong J M, Zhu J P, Liu J B, Hu S, He Y 2016 Opt. Lasers Eng. 77 18
[16] Kincade K 2004 Laser Focus World 40 34
[17] Andersen G, Asmolova O, McHarq M G, Quiller T, Maldonado C 2016 Proc. SPIE 9904 99041P
[18] Zhang J Y 2015 Opt. Express 23 30308
[19] Ke J, Zhang J Y 2015 Acta Opt. Sin. 35 0923001 (in Chinese)[柯杰, 张军勇 2015 光学学报 35 0923001]
[20] Ke J, Zhang J Y 2015 Appl. Opt. 54 7278
[21] Ke J, Zhang J Y 2016 Opt. Commum. 368 34
[22] Takaki Y, Kawai H, Ohzu H 1999 Appl. Opt. 38 4990
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