Abstract

The Fresnel Zone Plate (FZP) is widely used in a variety of applications. However, the most concerning drawback of the FZP is its low diffraction efficiency limited by fabrication capability. The diffraction efficiency of an FZP with a surface-relief profile can reach 100% at a specified diffracted order, however, this is difficult to engineer in a precise manner. In this paper, a hybrid-level FZP (HLFZP) is proposed to maximize the diffraction efficiency under a certain lithography condition. To theoretically verify this enhancement, the diffracted field of the HLFZP is derived by establishing a diffraction model. According to different machining capabilities, the design criteria for the level number distribution on the HLFZP are presented. Simulated results show significant improvement in imaging quality while comparing the diffraction efficiency of the specifically designed 8-4-2 HLFZP to that of the traditional 2-level FZP. Fabrication processes for the 8-4-2 HLFZP are developed, with experimental results indicating that the diffraction efficiency could be improved by 54% using the proposed HLFZP, compared to the traditional 2-level FZP. The root mean square (RMS) of the wavefront error for the 8-4-2 HLFZP is 1/20λ (λ = 632.8 nm), and imaging results are good as the predicted results. It is concluded that the proposed hybrid-level method is also promising for improving the diffraction efficiency of other diffractive optical elements.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
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References

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2018 (1)

C. Guo, Z. Zhang, D. Xue, L. Li, R. Wang, X. Zhou, F. Zhang, and X. Zhang, “High-performance etching of multilevel phase-type Fresnel zone plates with large apertures,” Opt. Commun. 407, 227–233 (2018).
[Crossref]

2017 (1)

2016 (4)

S. Wang, C. Zhou, Z. Liu, and H. Li, “Design and analysis of broadband diffractive optical element for achromatic focusing,” Proc. SPIE 10022, 100221J (2016).
[Crossref]

J. Loomis, D. Ratnayake, C. McKenna, and K. M. Walsh, “Grayscale lithography—automated mask generation for complex three-dimensional topography,” J. Micro-Nanolith. Mem. 15, 013511 (2016).

I. Mohacsi, I. Vartiainen, M. Guizar-Sicairos, P. Karvinen, V. A. Guzenko, E. Müller, C. M. Kewish, A. Somogyi, and C. David, “Fabrication and characterization of high-efficiency double-sided blazed X-ray optics,” Opt. Lett. 41(2), 281–284 (2016).
[Crossref] [PubMed]

R. Wang, Z. Zhang, C. Guo, D. Xue, and X. Zhang, “Effects of fabrication errors on diffraction efficiency for a diffractive membrane,” Chin. Opt. Lett. 14(12), 120501 (2016).
[Crossref]

2015 (3)

K. Li, M. J. Wojcik, L. E. Ocola, R. Divan, and C. Jacobsen, “Multilayer on-chip stacked Fresnel zone plates: Hard x-ray fabrication and soft X-ray simulations,” J. Vac. Sci. Technol. B 33(6), 06FD04 (2015).
[Crossref]

C. Li, Y. Li, X. Gao, and C. V. Duong, “Ultra-precision machining of Fresnel lens mould by single-point diamond turning based on axis B rotation,” Int. J. Adv. Manuf. Technol. 77(5–8), 907–913 (2015).
[Crossref]

A. Sabatyan and S. Gharbi, “Generation of double line focus and 1D non-diffractive beams using phase shifted linear Fresnel zone plate,” Opt. Laser Technol. 69, 65–70 (2015).
[Crossref]

2014 (4)

2013 (1)

2012 (1)

S. Nakahara, “Fabrication of the multi-level phase type hologram for display using the laser direct write lithography system,” Proc. SPIE 8281, 828116 (2012).

2010 (2)

C. H. Liu, M. Hong, M. C. Lum, H. Flotow, F. Ghadessy, and J. B. Zhang, “Large-area micro/nanostructures fabrication in quartz by laser interference lithography and dry etching,” Appl. Phys., A Mater. Sci. Process. 101(2), 237–241 (2010).
[Crossref]

S. Werner, S. Rehbein, P. Guttmann, S. Heim, and G. Schneider, “Towards high diffraction efficiency zone plates for x-ray microscopy,” Microelectron. Eng. 87(5–8), 1557–1560 (2010).
[Crossref]

2009 (1)

2007 (2)

K. Jefimovs, J. Vila-Comamala, T. Pilvi, J. Raabe, M. Ritala, and C. David, “Zone-doubling technique to produce ultrahigh-resolution x-ray optics,” Phys. Rev. Lett. 99(26), 264801 (2007).
[Crossref] [PubMed]

J. Kim, D. C. Joy, and S. Y. Lee, “Controlling resist thickness and etch depth for fabrication of 3D structures in electron-beam grayscale lithography,” Microelectron. Eng. 84(12), 2859–2864 (2007).
[Crossref]

2006 (1)

W. B. Veldkamp, “Binary Optics and Beyond: Where Do We Go from Here?” Jpn. J. Appl. Phys. 45(8B), 6550–6554 (2006).
[Crossref]

2005 (2)

F. J. Gonzalez, B. Ilic, J. Alda, and G. Boreman, “Antenna-coupled infrared detectors for imaging applications,” IEEE J. Sel. Top. Quantum Electron. 11(1), 117–120 (2005).
[Crossref]

W. Chao, B. D. Harteneck, J. A. Liddle, E. H. Anderson, and D. T. Attwood, “Soft X-ray microscopy at a spatial resolution better than 15 nm,” Nature 435(7046), 1210–1213 (2005).
[Crossref] [PubMed]

2004 (3)

1999 (1)

1997 (1)

1992 (3)

1989 (1)

N. Streibl, “Beam shaping with optical array generators,” J. Mod. Opt. 36(12), 1559–1573 (1989).
[Crossref]

1981 (1)

1979 (1)

J. E. Harvey, “Fourier treatment of near‐field scalar diffraction theory,” Am. J. Phys. 47(11), 974–980 (1979).
[Crossref]

Alda, J.

F. J. Gonzalez, B. Ilic, J. Alda, and G. Boreman, “Antenna-coupled infrared detectors for imaging applications,” IEEE J. Sel. Top. Quantum Electron. 11(1), 117–120 (2005).
[Crossref]

F. J. González, J. Alda, B. Ilic, and G. D. Boreman, “Infrared antennas coupled to lithographic Fresnel zone plate lenses,” Appl. Opt. 43(33), 6067–6073 (2004).
[Crossref] [PubMed]

Anderson, E. H.

W. Chao, J. Kim, S. Rekawa, P. Fischer, and E. H. Anderson, “Demonstration of 12 nm resolution Fresnel zone plate lens based soft x-ray microscopy,” Opt. Express 17(20), 17669–17677 (2009).
[Crossref] [PubMed]

W. Chao, B. D. Harteneck, J. A. Liddle, E. H. Anderson, and D. T. Attwood, “Soft X-ray microscopy at a spatial resolution better than 15 nm,” Nature 435(7046), 1210–1213 (2005).
[Crossref] [PubMed]

Atcheson, P.

P. Atcheson, J. Domber, K. Whiteaker, J. A. Britten, S. N. Dixit, and B. Farmer, “MOIRE: ground demonstration of a large aperture diffractive transmissive telescope,” Proc. SPIE 9143, 91431W (2014).
[Crossref]

Atcheson, P. D.

Attwood, D. T.

W. Chao, B. D. Harteneck, J. A. Liddle, E. H. Anderson, and D. T. Attwood, “Soft X-ray microscopy at a spatial resolution better than 15 nm,” Nature 435(7046), 1210–1213 (2005).
[Crossref] [PubMed]

Baron, R. L.

J. T. Early, R. Hyde, and R. L. Baron, “Twenty-meter space telescope based on diffractive Fresnel lens,” Proc. SPIE 5166, 148–156 (2004).
[Crossref]

Bechtel, M.

Blough, C. G.

Boreman, G.

F. J. Gonzalez, B. Ilic, J. Alda, and G. Boreman, “Antenna-coupled infrared detectors for imaging applications,” IEEE J. Sel. Top. Quantum Electron. 11(1), 117–120 (2005).
[Crossref]

Boreman, G. D.

Britten, J. A.

Buralli, D. A.

Cao, Q.

Chao, W.

W. Chao, J. Kim, S. Rekawa, P. Fischer, and E. H. Anderson, “Demonstration of 12 nm resolution Fresnel zone plate lens based soft x-ray microscopy,” Opt. Express 17(20), 17669–17677 (2009).
[Crossref] [PubMed]

W. Chao, B. D. Harteneck, J. A. Liddle, E. H. Anderson, and D. T. Attwood, “Soft X-ray microscopy at a spatial resolution better than 15 nm,” Nature 435(7046), 1210–1213 (2005).
[Crossref] [PubMed]

David, C.

DeBruyckere, M.

Divan, R.

K. Li, M. J. Wojcik, L. E. Ocola, R. Divan, and C. Jacobsen, “Multilayer on-chip stacked Fresnel zone plates: Hard x-ray fabrication and soft X-ray simulations,” J. Vac. Sci. Technol. B 33(6), 06FD04 (2015).
[Crossref]

Dixit, S. N.

Domber, J.

P. Atcheson, J. Domber, K. Whiteaker, J. A. Britten, S. N. Dixit, and B. Farmer, “MOIRE: ground demonstration of a large aperture diffractive transmissive telescope,” Proc. SPIE 9143, 91431W (2014).
[Crossref]

Domber, J. L.

Duong, C. V.

C. Li, Y. Li, X. Gao, and C. V. Duong, “Ultra-precision machining of Fresnel lens mould by single-point diamond turning based on axis B rotation,” Int. J. Adv. Manuf. Technol. 77(5–8), 907–913 (2015).
[Crossref]

Early, J. T.

J. T. Early, R. Hyde, and R. L. Baron, “Twenty-meter space telescope based on diffractive Fresnel lens,” Proc. SPIE 5166, 148–156 (2004).
[Crossref]

Farmer, B.

Fischer, P.

Flotow, H.

C. H. Liu, M. Hong, M. C. Lum, H. Flotow, F. Ghadessy, and J. B. Zhang, “Large-area micro/nanostructures fabrication in quartz by laser interference lithography and dry etching,” Appl. Phys., A Mater. Sci. Process. 101(2), 237–241 (2010).
[Crossref]

Follath, R.

Gao, X.

C. Li, Y. Li, X. Gao, and C. V. Duong, “Ultra-precision machining of Fresnel lens mould by single-point diamond turning based on axis B rotation,” Int. J. Adv. Manuf. Technol. 77(5–8), 907–913 (2015).
[Crossref]

Ghadessy, F.

C. H. Liu, M. Hong, M. C. Lum, H. Flotow, F. Ghadessy, and J. B. Zhang, “Large-area micro/nanostructures fabrication in quartz by laser interference lithography and dry etching,” Appl. Phys., A Mater. Sci. Process. 101(2), 237–241 (2010).
[Crossref]

Gharbi, S.

A. Sabatyan and S. Gharbi, “Generation of double line focus and 1D non-diffractive beams using phase shifted linear Fresnel zone plate,” Opt. Laser Technol. 69, 65–70 (2015).
[Crossref]

Goering, E.

Gonzalez, F. J.

F. J. Gonzalez, B. Ilic, J. Alda, and G. Boreman, “Antenna-coupled infrared detectors for imaging applications,” IEEE J. Sel. Top. Quantum Electron. 11(1), 117–120 (2005).
[Crossref]

González, F. J.

Grévent, C.

Guizar-Sicairos, M.

Guo, C.

C. Guo, Z. Zhang, D. Xue, L. Li, R. Wang, X. Zhou, F. Zhang, and X. Zhang, “High-performance etching of multilevel phase-type Fresnel zone plates with large apertures,” Opt. Commun. 407, 227–233 (2018).
[Crossref]

R. Wang, Z. Zhang, C. Guo, D. Xue, and X. Zhang, “Effects of fabrication errors on diffraction efficiency for a diffractive membrane,” Chin. Opt. Lett. 14(12), 120501 (2016).
[Crossref]

Guttmann, P.

S. Werner, S. Rehbein, P. Guttmann, S. Heim, and G. Schneider, “Towards high diffraction efficiency zone plates for x-ray microscopy,” Microelectron. Eng. 87(5–8), 1557–1560 (2010).
[Crossref]

Guzenko, V. A.

Hackett, J.

Harteneck, B. D.

W. Chao, B. D. Harteneck, J. A. Liddle, E. H. Anderson, and D. T. Attwood, “Soft X-ray microscopy at a spatial resolution better than 15 nm,” Nature 435(7046), 1210–1213 (2005).
[Crossref] [PubMed]

Harvey, J. E.

J. E. Harvey, “Fourier treatment of near‐field scalar diffraction theory,” Am. J. Phys. 47(11), 974–980 (1979).
[Crossref]

Heim, S.

S. Werner, S. Rehbein, P. Guttmann, S. Heim, and G. Schneider, “Towards high diffraction efficiency zone plates for x-ray microscopy,” Microelectron. Eng. 87(5–8), 1557–1560 (2010).
[Crossref]

Hong, M.

C. H. Liu, M. Hong, M. C. Lum, H. Flotow, F. Ghadessy, and J. B. Zhang, “Large-area micro/nanostructures fabrication in quartz by laser interference lithography and dry etching,” Appl. Phys., A Mater. Sci. Process. 101(2), 237–241 (2010).
[Crossref]

Hrynevych, M.

Hyde, R.

J. T. Early, R. Hyde, and R. L. Baron, “Twenty-meter space telescope based on diffractive Fresnel lens,” Proc. SPIE 5166, 148–156 (2004).
[Crossref]

Hyde, R. A.

Ilic, B.

F. J. Gonzalez, B. Ilic, J. Alda, and G. Boreman, “Antenna-coupled infrared detectors for imaging applications,” IEEE J. Sel. Top. Quantum Electron. 11(1), 117–120 (2005).
[Crossref]

F. J. González, J. Alda, B. Ilic, and G. D. Boreman, “Infrared antennas coupled to lithographic Fresnel zone plate lenses,” Appl. Opt. 43(33), 6067–6073 (2004).
[Crossref] [PubMed]

Jacobsen, C.

K. Li, M. J. Wojcik, L. E. Ocola, R. Divan, and C. Jacobsen, “Multilayer on-chip stacked Fresnel zone plates: Hard x-ray fabrication and soft X-ray simulations,” J. Vac. Sci. Technol. B 33(6), 06FD04 (2015).
[Crossref]

Jahns, J.

Jefimovs, K.

K. Jefimovs, J. Vila-Comamala, T. Pilvi, J. Raabe, M. Ritala, and C. David, “Zone-doubling technique to produce ultrahigh-resolution x-ray optics,” Phys. Rev. Lett. 99(26), 264801 (2007).
[Crossref] [PubMed]

Joy, D. C.

J. Kim, D. C. Joy, and S. Y. Lee, “Controlling resist thickness and etch depth for fabrication of 3D structures in electron-beam grayscale lithography,” Microelectron. Eng. 84(12), 2859–2864 (2007).
[Crossref]

Karvinen, P.

Keskinbora, K.

Kewish, C. M.

Kim, J.

W. Chao, J. Kim, S. Rekawa, P. Fischer, and E. H. Anderson, “Demonstration of 12 nm resolution Fresnel zone plate lens based soft x-ray microscopy,” Opt. Express 17(20), 17669–17677 (2009).
[Crossref] [PubMed]

J. Kim, D. C. Joy, and S. Y. Lee, “Controlling resist thickness and etch depth for fabrication of 3D structures in electron-beam grayscale lithography,” Microelectron. Eng. 84(12), 2859–2864 (2007).
[Crossref]

Lee, S. Y.

J. Kim, D. C. Joy, and S. Y. Lee, “Controlling resist thickness and etch depth for fabrication of 3D structures in electron-beam grayscale lithography,” Microelectron. Eng. 84(12), 2859–2864 (2007).
[Crossref]

Li, C.

C. Li, Y. Li, X. Gao, and C. V. Duong, “Ultra-precision machining of Fresnel lens mould by single-point diamond turning based on axis B rotation,” Int. J. Adv. Manuf. Technol. 77(5–8), 907–913 (2015).
[Crossref]

Li, H.

S. Wang, C. Zhou, Z. Liu, and H. Li, “Design and analysis of broadband diffractive optical element for achromatic focusing,” Proc. SPIE 10022, 100221J (2016).
[Crossref]

Li, K.

K. Li, M. J. Wojcik, L. E. Ocola, R. Divan, and C. Jacobsen, “Multilayer on-chip stacked Fresnel zone plates: Hard x-ray fabrication and soft X-ray simulations,” J. Vac. Sci. Technol. B 33(6), 06FD04 (2015).
[Crossref]

Li, L.

C. Guo, Z. Zhang, D. Xue, L. Li, R. Wang, X. Zhou, F. Zhang, and X. Zhang, “High-performance etching of multilevel phase-type Fresnel zone plates with large apertures,” Opt. Commun. 407, 227–233 (2018).
[Crossref]

Li, Y.

C. Li, Y. Li, X. Gao, and C. V. Duong, “Ultra-precision machining of Fresnel lens mould by single-point diamond turning based on axis B rotation,” Int. J. Adv. Manuf. Technol. 77(5–8), 907–913 (2015).
[Crossref]

Liddle, J. A.

W. Chao, B. D. Harteneck, J. A. Liddle, E. H. Anderson, and D. T. Attwood, “Soft X-ray microscopy at a spatial resolution better than 15 nm,” Nature 435(7046), 1210–1213 (2005).
[Crossref] [PubMed]

Liu, C. H.

C. H. Liu, M. Hong, M. C. Lum, H. Flotow, F. Ghadessy, and J. B. Zhang, “Large-area micro/nanostructures fabrication in quartz by laser interference lithography and dry etching,” Appl. Phys., A Mater. Sci. Process. 101(2), 237–241 (2010).
[Crossref]

Liu, Z.

S. Wang, C. Zhou, Z. Liu, and H. Li, “Design and analysis of broadband diffractive optical element for achromatic focusing,” Proc. SPIE 10022, 100221J (2016).
[Crossref]

Loomis, J.

J. Loomis, D. Ratnayake, C. McKenna, and K. M. Walsh, “Grayscale lithography—automated mask generation for complex three-dimensional topography,” J. Micro-Nanolith. Mem. 15, 013511 (2016).

Lum, M. C.

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K. Jefimovs, J. Vila-Comamala, T. Pilvi, J. Raabe, M. Ritala, and C. David, “Zone-doubling technique to produce ultrahigh-resolution x-ray optics,” Phys. Rev. Lett. 99(26), 264801 (2007).
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J. Loomis, D. Ratnayake, C. McKenna, and K. M. Walsh, “Grayscale lithography—automated mask generation for complex three-dimensional topography,” J. Micro-Nanolith. Mem. 15, 013511 (2016).

Rehbein, S.

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Ritala, M.

K. Jefimovs, J. Vila-Comamala, T. Pilvi, J. Raabe, M. Ritala, and C. David, “Zone-doubling technique to produce ultrahigh-resolution x-ray optics,” Phys. Rev. Lett. 99(26), 264801 (2007).
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Walsh, K. M.

J. Loomis, D. Ratnayake, C. McKenna, and K. M. Walsh, “Grayscale lithography—automated mask generation for complex three-dimensional topography,” J. Micro-Nanolith. Mem. 15, 013511 (2016).

Wang, R.

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K. Li, M. J. Wojcik, L. E. Ocola, R. Divan, and C. Jacobsen, “Multilayer on-chip stacked Fresnel zone plates: Hard x-ray fabrication and soft X-ray simulations,” J. Vac. Sci. Technol. B 33(6), 06FD04 (2015).
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C. Guo, Z. Zhang, D. Xue, L. Li, R. Wang, X. Zhou, F. Zhang, and X. Zhang, “High-performance etching of multilevel phase-type Fresnel zone plates with large apertures,” Opt. Commun. 407, 227–233 (2018).
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[Crossref]

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C. Guo, Z. Zhang, D. Xue, L. Li, R. Wang, X. Zhou, F. Zhang, and X. Zhang, “High-performance etching of multilevel phase-type Fresnel zone plates with large apertures,” Opt. Commun. 407, 227–233 (2018).
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C. Guo, Z. Zhang, D. Xue, L. Li, R. Wang, X. Zhou, F. Zhang, and X. Zhang, “High-performance etching of multilevel phase-type Fresnel zone plates with large apertures,” Opt. Commun. 407, 227–233 (2018).
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S. Wang, C. Zhou, Z. Liu, and H. Li, “Design and analysis of broadband diffractive optical element for achromatic focusing,” Proc. SPIE 10022, 100221J (2016).
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C. Guo, Z. Zhang, D. Xue, L. Li, R. Wang, X. Zhou, F. Zhang, and X. Zhang, “High-performance etching of multilevel phase-type Fresnel zone plates with large apertures,” Opt. Commun. 407, 227–233 (2018).
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J. Loomis, D. Ratnayake, C. McKenna, and K. M. Walsh, “Grayscale lithography—automated mask generation for complex three-dimensional topography,” J. Micro-Nanolith. Mem. 15, 013511 (2016).

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W. B. Veldkamp, “Binary Optics and Beyond: Where Do We Go from Here?” Jpn. J. Appl. Phys. 45(8B), 6550–6554 (2006).
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S. Werner, S. Rehbein, P. Guttmann, S. Heim, and G. Schneider, “Towards high diffraction efficiency zone plates for x-ray microscopy,” Microelectron. Eng. 87(5–8), 1557–1560 (2010).
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C. Guo, Z. Zhang, D. Xue, L. Li, R. Wang, X. Zhou, F. Zhang, and X. Zhang, “High-performance etching of multilevel phase-type Fresnel zone plates with large apertures,” Opt. Commun. 407, 227–233 (2018).
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K. Jefimovs, J. Vila-Comamala, T. Pilvi, J. Raabe, M. Ritala, and C. David, “Zone-doubling technique to produce ultrahigh-resolution x-ray optics,” Phys. Rev. Lett. 99(26), 264801 (2007).
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S. Wang, C. Zhou, Z. Liu, and H. Li, “Design and analysis of broadband diffractive optical element for achromatic focusing,” Proc. SPIE 10022, 100221J (2016).
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[Crossref]

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Q. Huang, Q. Peng, J. Hu, H. Xu, C. Jiang, and Q. Liu, “Design of a high-efficiency and low-cost reflection-type diffractive optical element as the spectrum splitting solar concentrator for lateral multi-junction solar cells architecture,” in Advanced Information Management, Communicates,Electronic and Automation Control Conference (2017).

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Figures (11)

Fig. 1
Fig. 1 Schematic of an FZP with a collimated incident light beam.
Fig. 2
Fig. 2 Profile of area with L phase steps for a HLFZP.
Fig. 3
Fig. 3 Schematic of a HLFZP.
Fig. 4
Fig. 4 Diffraction efficiency of the 8-4-2 HLFZP with varying.
Fig. 5
Fig. 5 Intensity distribution of an 8-4-2 HLFZP with different σL at the focal plane compared to that of a 2-level FZP (σ4 = 0, σ8 = 0).
Fig. 6
Fig. 6 MTF of an 8-4-2 HLFZP with different σL values compared to the MTF of a surface-relief FZP.
Fig. 7
Fig. 7 Fabrication processes for the 8-4-2 HLFZP.
Fig. 8
Fig. 8 Fabricated HLFZP in an aperture of 20 mm; (a) Formal fabricated HLFZP, (b) Cross-section for the 2-level zone and the 4-level zone, (c) Cross-sectional profiles for the 4-level zone and the 8-level zone
Fig. 9
Fig. 9 (a) Testing system for the diffraction efficiency of FZP (HLFZP), (b) Measurement results of intensity of working order (left axis) and diffraction efficiency (right axis).
Fig. 10
Fig. 10 (a) Testing system for the wavefront error of FZP (HLFZP), (b) PV and RMS of the wavefront error for the 2-level FZP, (c) PV and RMS of the wavefront error for the 8-4-2 HLFZP.
Fig. 11
Fig. 11 (a) Resolution testing system, (b) resolution board images of 2-level FZP, and (c) resolution board images of 8-4-2 HLFZP.

Equations (17)

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

U n ( R )= 1 λ 0 2π a n b n f ρ 2 exp( jkρ )rdrdθ
ρ= f 2 + R 2 + r 2 2Rrcos( θφ )
b n = λfn
a n = λf( n1 )
U n ( R )= 2f f n exp[ jk( f n + R 2 2 f n ) ] J 0 ( k r n f n R )sin( k d n 2 f n )
d n =( b n 2 a n 2 )/2
r n =( a n 2 + b n 2 )/2
f n = f 2 + r n 2
h = 1 L λ n s 1
Δ = k ( n s 1 ) ( n L 1 ) h
U n ( R ) HLFZP = 2 f f n exp [ j k ( f n + R 2 2 f n + Δ ) ] J 0 ( k r n f n R ) sin ( k d n 2 f n )
U ( R ) HLFZP = 0 N U n ( R ) HLFZP
Δ r n W = L 2
σ L = 2λ F # WL + LW 4 r 0
F # =f/2 r 0
σ L 2λ F # WL
η= I W I i

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