Abstract

High-resolution lens-coupled indirect X-ray scintillator imagers are required by many imaging applications. However, the severe weakening of image details prevents its further performance improvement. Through our research, this image degradation is attributed to the broadband loss of the high-spatial-frequency information caused by the high refractive index. A technique known as high-spatial-frequency spectrum enhanced reconstruction is thus proposed to retrieve this information. A two-dimensional high-density array is covered on the scintillator’s exit surface and operates as an encoder based on which high-frequency information can be shifted to the low-frequency region to improve the signal-to-noise ratio. The experimental results show that the middle-high-frequency signal intensities can be increased by an order of magnitude or more, up to 50 times. Therefore, the image details can be effectively enhanced to break through the performance bottleneck of such widely used X-ray imagers for synchrotron radiation facilities or tabletop X-ray tubes.

© 2020 Chinese Laser Press

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  1. P. A. Douissard, A. Cecilia, X. Rochet, X. Chapel, T. Martin, T. van de Kamp, L. Helfen, T. Baumbach, L. Luquot, and X. Xiao, “A versatile indirect detector design for hard X-ray microimaging,” J. Instrum. 7, P09016 (2012).
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    [Crossref]
  3. M. Moszynski, M. Kapusta, M. Mayhugh, and D. Wolski, “Absolute light output of scintillators,” IEEE Trans. Nucl. Sci. 44, 1052–1061 (1997).
    [Crossref]
  4. M. Kronberger, E. Auffray, and P. Lecoq, “Probing the concepts of photonic crystals on scintillating materials,” IEEE Trans. Nucl. Sci. 55, 1102–1106 (2008).
    [Crossref]
  5. Z. Zhu, S. Wu, C. Xue, J. Zhao, L. Wang, Y. Wu, and R. Tai, “Enhanced light extraction of scintillator using large-area photonic crystal structures fabricated by soft-X-ray interference lithography,” Appl. Phys. Lett. 106, 241901 (2015).
    [Crossref]
  6. M. Chen, H. Yu, J. Zhao, L. Wang, H. Xie, Y. Wu, and R. Tai, “Effect of photonic crystal preparation methods on imaging resolution of synchrotron radiation scintillation detector,” Nucl. Technol. 38, 5–10 (2015).
    [Crossref]
  7. M. G. L. Gustafsson, “Nonlinear structured-illumination microscopy: wide-field fluorescence imaging with theoretically unlimited resolution,” Proc. Natl. Acad. Sci. USA 102, 13081–13086 (2005).
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  11. Y. C. Kim, K. H. Kim, D. Y. Son, D. N. Jeong, J. Y. Seo, Y. S. Choi, and N. G. Park, “Printable organometallic perovskite enables large-area, low-dose X-ray imaging,” Nature 550, 87–91 (2017).
    [Crossref]
  12. S. Ohtsuka, Y. Sugishita, T. Takeda, Y. Itai, K. Hyodo, and M. Ando, “Dynamic intravenous coronary arteriography using synchrotron radiation and its application to the measurement of coronary blood flow,” Jpn. Circ. J. 61, 432–440 (1997).
    [Crossref]
  13. C. Tötzke, N. Kardjilov, I. Manke, and S. E. Oswald, “Capturing 3D water flow in rooted soil by ultra-fast neutron tomography,” Sci. Rep. 7, 6192 (2017).
    [Crossref]
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  16. Y. Ben-Aryeh, “Transmission enhancement by conversion of evanescent waves into propagating waves,” Appl. Phys. B 91, 157–165 (2008).
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  17. G. Cristóbal, B. Javidi, and S. Vallmitjana, Information Optics (Springer, 2009).
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    [Crossref]
  19. S. Yang, J. Zhao, L. Wang, F. Zhu, C. Xue, H. Liu, and R. Tai, “The influence of symmetry and duty cycle on the pattern generation in achromatic Talbot lithography,” J. Vac. Sci. Technol. B 35, 021601 (2017).
    [Crossref]
  20. S. Yang, J. Zhao, L. Wang, F. Zhu, C. Xue, H. Liu, and R. Tai, “Developments at SSRF in soft X-ray interference lithography,” Nucl. Sci. Tech. 26, 010101 (2015).
    [Crossref]
  21. R. C. Gonzalez and R. E. Woods, Digital Image Processing (Prentice Hall, 2002).
  22. Q. Meng, J. Su, and W. Li, Comparative Anatomy of Fishes (Science Press, 1987).

2018 (1)

H. M. Thirimanne, K. D. G. I. Jayawardena, A. J. Parnell, R. M. I. Bandara, A. Karalasingam, S. Pani, J. E. Huerdler, D. G. Lidzey, S. F. Tedde, and S. R. P. Silva, “High sensitivity organic inorganic hybrid X-ray detectors with direct transduction and broadband response,” Nat. Commun. 9, 2926 (2018).
[Crossref]

2017 (4)

C. Tötzke, N. Kardjilov, I. Manke, and S. E. Oswald, “Capturing 3D water flow in rooted soil by ultra-fast neutron tomography,” Sci. Rep. 7, 6192 (2017).
[Crossref]

Y. C. Kim, K. H. Kim, D. Y. Son, D. N. Jeong, J. Y. Seo, Y. S. Choi, and N. G. Park, “Printable organometallic perovskite enables large-area, low-dose X-ray imaging,” Nature 550, 87–91 (2017).
[Crossref]

Z. Zhu, B. Liu, C. Cheng, H. Zhang, H. Chen, M. Gu, and Y. Wu, “Enhancement of directional broadband luminescence from a scintillation film via guided-mode resonance in a photonic crystal structure,” Appl. Phys. Lett. 110, 051901 (2017).
[Crossref]

S. Yang, J. Zhao, L. Wang, F. Zhu, C. Xue, H. Liu, and R. Tai, “The influence of symmetry and duty cycle on the pattern generation in achromatic Talbot lithography,” J. Vac. Sci. Technol. B 35, 021601 (2017).
[Crossref]

2016 (1)

P. Cloetens, R. Barrett, J. Baruchel, J. P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard X-ray imaging,” J. Phys. D 9, 133–146 (2016).
[Crossref]

2015 (3)

S. Yang, J. Zhao, L. Wang, F. Zhu, C. Xue, H. Liu, and R. Tai, “Developments at SSRF in soft X-ray interference lithography,” Nucl. Sci. Tech. 26, 010101 (2015).
[Crossref]

Z. Zhu, S. Wu, C. Xue, J. Zhao, L. Wang, Y. Wu, and R. Tai, “Enhanced light extraction of scintillator using large-area photonic crystal structures fabricated by soft-X-ray interference lithography,” Appl. Phys. Lett. 106, 241901 (2015).
[Crossref]

M. Chen, H. Yu, J. Zhao, L. Wang, H. Xie, Y. Wu, and R. Tai, “Effect of photonic crystal preparation methods on imaging resolution of synchrotron radiation scintillation detector,” Nucl. Technol. 38, 5–10 (2015).
[Crossref]

2012 (1)

P. A. Douissard, A. Cecilia, X. Rochet, X. Chapel, T. Martin, T. van de Kamp, L. Helfen, T. Baumbach, L. Luquot, and X. Xiao, “A versatile indirect detector design for hard X-ray microimaging,” J. Instrum. 7, P09016 (2012).
[Crossref]

2008 (3)

M. Kronberger, E. Auffray, and P. Lecoq, “Probing the concepts of photonic crystals on scintillating materials,” IEEE Trans. Nucl. Sci. 55, 1102–1106 (2008).
[Crossref]

M. W. Westneat, J. J. Socha, and W. K. Lee, “Advances in biological structure, function, and physiology using synchrotron X-ray imaging,” Annu. Rev. Physiol. 70, 119–142 (2008).
[Crossref]

Y. Ben-Aryeh, “Transmission enhancement by conversion of evanescent waves into propagating waves,” Appl. Phys. B 91, 157–165 (2008).
[Crossref]

2007 (2)

Z. Liu, S. Durant, H. Lee, Y. Pikus, N. Fang, Y. Xiong, and X. Zhang, “Far-field optical superlens,” Nano Lett. 7, 403–408 (2007).
[Crossref]

M. J. Kitchen, G. A. Buckley, T. E. Gureyev, M. J. Wallace, N. Andres-Thio, and K. Uesugi, “CT dose reduction factors in the thousands using X-ray phase contrast,” Sci. Rep. 7, 15953 (2007).
[Crossref]

2005 (1)

M. G. L. Gustafsson, “Nonlinear structured-illumination microscopy: wide-field fluorescence imaging with theoretically unlimited resolution,” Proc. Natl. Acad. Sci. USA 102, 13081–13086 (2005).
[Crossref]

2004 (1)

1997 (2)

M. Moszynski, M. Kapusta, M. Mayhugh, and D. Wolski, “Absolute light output of scintillators,” IEEE Trans. Nucl. Sci. 44, 1052–1061 (1997).
[Crossref]

S. Ohtsuka, Y. Sugishita, T. Takeda, Y. Itai, K. Hyodo, and M. Ando, “Dynamic intravenous coronary arteriography using synchrotron radiation and its application to the measurement of coronary blood flow,” Jpn. Circ. J. 61, 432–440 (1997).
[Crossref]

Ando, M.

S. Ohtsuka, Y. Sugishita, T. Takeda, Y. Itai, K. Hyodo, and M. Ando, “Dynamic intravenous coronary arteriography using synchrotron radiation and its application to the measurement of coronary blood flow,” Jpn. Circ. J. 61, 432–440 (1997).
[Crossref]

Andres-Thio, N.

M. J. Kitchen, G. A. Buckley, T. E. Gureyev, M. J. Wallace, N. Andres-Thio, and K. Uesugi, “CT dose reduction factors in the thousands using X-ray phase contrast,” Sci. Rep. 7, 15953 (2007).
[Crossref]

Auffray, E.

M. Kronberger, E. Auffray, and P. Lecoq, “Probing the concepts of photonic crystals on scintillating materials,” IEEE Trans. Nucl. Sci. 55, 1102–1106 (2008).
[Crossref]

Bandara, R. M. I.

H. M. Thirimanne, K. D. G. I. Jayawardena, A. J. Parnell, R. M. I. Bandara, A. Karalasingam, S. Pani, J. E. Huerdler, D. G. Lidzey, S. F. Tedde, and S. R. P. Silva, “High sensitivity organic inorganic hybrid X-ray detectors with direct transduction and broadband response,” Nat. Commun. 9, 2926 (2018).
[Crossref]

Barrett, R.

P. Cloetens, R. Barrett, J. Baruchel, J. P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard X-ray imaging,” J. Phys. D 9, 133–146 (2016).
[Crossref]

Baruchel, J.

P. Cloetens, R. Barrett, J. Baruchel, J. P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard X-ray imaging,” J. Phys. D 9, 133–146 (2016).
[Crossref]

Baumbach, T.

P. A. Douissard, A. Cecilia, X. Rochet, X. Chapel, T. Martin, T. van de Kamp, L. Helfen, T. Baumbach, L. Luquot, and X. Xiao, “A versatile indirect detector design for hard X-ray microimaging,” J. Instrum. 7, P09016 (2012).
[Crossref]

Belkebir, K.

Ben-Aryeh, Y.

Y. Ben-Aryeh, “Transmission enhancement by conversion of evanescent waves into propagating waves,” Appl. Phys. B 91, 157–165 (2008).
[Crossref]

Buckley, G. A.

M. J. Kitchen, G. A. Buckley, T. E. Gureyev, M. J. Wallace, N. Andres-Thio, and K. Uesugi, “CT dose reduction factors in the thousands using X-ray phase contrast,” Sci. Rep. 7, 15953 (2007).
[Crossref]

Cecilia, A.

P. A. Douissard, A. Cecilia, X. Rochet, X. Chapel, T. Martin, T. van de Kamp, L. Helfen, T. Baumbach, L. Luquot, and X. Xiao, “A versatile indirect detector design for hard X-ray microimaging,” J. Instrum. 7, P09016 (2012).
[Crossref]

Chapel, X.

P. A. Douissard, A. Cecilia, X. Rochet, X. Chapel, T. Martin, T. van de Kamp, L. Helfen, T. Baumbach, L. Luquot, and X. Xiao, “A versatile indirect detector design for hard X-ray microimaging,” J. Instrum. 7, P09016 (2012).
[Crossref]

Chaumet, P. C.

Chen, H.

Z. Zhu, B. Liu, C. Cheng, H. Zhang, H. Chen, M. Gu, and Y. Wu, “Enhancement of directional broadband luminescence from a scintillation film via guided-mode resonance in a photonic crystal structure,” Appl. Phys. Lett. 110, 051901 (2017).
[Crossref]

Chen, M.

M. Chen, H. Yu, J. Zhao, L. Wang, H. Xie, Y. Wu, and R. Tai, “Effect of photonic crystal preparation methods on imaging resolution of synchrotron radiation scintillation detector,” Nucl. Technol. 38, 5–10 (2015).
[Crossref]

Cheng, C.

Z. Zhu, B. Liu, C. Cheng, H. Zhang, H. Chen, M. Gu, and Y. Wu, “Enhancement of directional broadband luminescence from a scintillation film via guided-mode resonance in a photonic crystal structure,” Appl. Phys. Lett. 110, 051901 (2017).
[Crossref]

Choi, Y. S.

Y. C. Kim, K. H. Kim, D. Y. Son, D. N. Jeong, J. Y. Seo, Y. S. Choi, and N. G. Park, “Printable organometallic perovskite enables large-area, low-dose X-ray imaging,” Nature 550, 87–91 (2017).
[Crossref]

Cloetens, P.

P. Cloetens, R. Barrett, J. Baruchel, J. P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard X-ray imaging,” J. Phys. D 9, 133–146 (2016).
[Crossref]

Cristóbal, G.

G. Cristóbal, B. Javidi, and S. Vallmitjana, Information Optics (Springer, 2009).

Douissard, P. A.

P. A. Douissard, A. Cecilia, X. Rochet, X. Chapel, T. Martin, T. van de Kamp, L. Helfen, T. Baumbach, L. Luquot, and X. Xiao, “A versatile indirect detector design for hard X-ray microimaging,” J. Instrum. 7, P09016 (2012).
[Crossref]

Durant, S.

Z. Liu, S. Durant, H. Lee, Y. Pikus, N. Fang, Y. Xiong, and X. Zhang, “Far-field optical superlens,” Nano Lett. 7, 403–408 (2007).
[Crossref]

Fang, N.

Z. Liu, S. Durant, H. Lee, Y. Pikus, N. Fang, Y. Xiong, and X. Zhang, “Far-field optical superlens,” Nano Lett. 7, 403–408 (2007).
[Crossref]

Gonzalez, R. C.

R. C. Gonzalez and R. E. Woods, Digital Image Processing (Prentice Hall, 2002).

Gu, M.

Z. Zhu, B. Liu, C. Cheng, H. Zhang, H. Chen, M. Gu, and Y. Wu, “Enhancement of directional broadband luminescence from a scintillation film via guided-mode resonance in a photonic crystal structure,” Appl. Phys. Lett. 110, 051901 (2017).
[Crossref]

Guigay, J. P.

P. Cloetens, R. Barrett, J. Baruchel, J. P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard X-ray imaging,” J. Phys. D 9, 133–146 (2016).
[Crossref]

Gureyev, T. E.

M. J. Kitchen, G. A. Buckley, T. E. Gureyev, M. J. Wallace, N. Andres-Thio, and K. Uesugi, “CT dose reduction factors in the thousands using X-ray phase contrast,” Sci. Rep. 7, 15953 (2007).
[Crossref]

Gustafsson, M. G. L.

M. G. L. Gustafsson, “Nonlinear structured-illumination microscopy: wide-field fluorescence imaging with theoretically unlimited resolution,” Proc. Natl. Acad. Sci. USA 102, 13081–13086 (2005).
[Crossref]

Helfen, L.

P. A. Douissard, A. Cecilia, X. Rochet, X. Chapel, T. Martin, T. van de Kamp, L. Helfen, T. Baumbach, L. Luquot, and X. Xiao, “A versatile indirect detector design for hard X-ray microimaging,” J. Instrum. 7, P09016 (2012).
[Crossref]

Huerdler, J. E.

H. M. Thirimanne, K. D. G. I. Jayawardena, A. J. Parnell, R. M. I. Bandara, A. Karalasingam, S. Pani, J. E. Huerdler, D. G. Lidzey, S. F. Tedde, and S. R. P. Silva, “High sensitivity organic inorganic hybrid X-ray detectors with direct transduction and broadband response,” Nat. Commun. 9, 2926 (2018).
[Crossref]

Hyodo, K.

S. Ohtsuka, Y. Sugishita, T. Takeda, Y. Itai, K. Hyodo, and M. Ando, “Dynamic intravenous coronary arteriography using synchrotron radiation and its application to the measurement of coronary blood flow,” Jpn. Circ. J. 61, 432–440 (1997).
[Crossref]

Itai, Y.

S. Ohtsuka, Y. Sugishita, T. Takeda, Y. Itai, K. Hyodo, and M. Ando, “Dynamic intravenous coronary arteriography using synchrotron radiation and its application to the measurement of coronary blood flow,” Jpn. Circ. J. 61, 432–440 (1997).
[Crossref]

Javidi, B.

G. Cristóbal, B. Javidi, and S. Vallmitjana, Information Optics (Springer, 2009).

Jayawardena, K. D. G. I.

H. M. Thirimanne, K. D. G. I. Jayawardena, A. J. Parnell, R. M. I. Bandara, A. Karalasingam, S. Pani, J. E. Huerdler, D. G. Lidzey, S. F. Tedde, and S. R. P. Silva, “High sensitivity organic inorganic hybrid X-ray detectors with direct transduction and broadband response,” Nat. Commun. 9, 2926 (2018).
[Crossref]

Jeong, D. N.

Y. C. Kim, K. H. Kim, D. Y. Son, D. N. Jeong, J. Y. Seo, Y. S. Choi, and N. G. Park, “Printable organometallic perovskite enables large-area, low-dose X-ray imaging,” Nature 550, 87–91 (2017).
[Crossref]

Kapusta, M.

M. Moszynski, M. Kapusta, M. Mayhugh, and D. Wolski, “Absolute light output of scintillators,” IEEE Trans. Nucl. Sci. 44, 1052–1061 (1997).
[Crossref]

Karalasingam, A.

H. M. Thirimanne, K. D. G. I. Jayawardena, A. J. Parnell, R. M. I. Bandara, A. Karalasingam, S. Pani, J. E. Huerdler, D. G. Lidzey, S. F. Tedde, and S. R. P. Silva, “High sensitivity organic inorganic hybrid X-ray detectors with direct transduction and broadband response,” Nat. Commun. 9, 2926 (2018).
[Crossref]

Kardjilov, N.

C. Tötzke, N. Kardjilov, I. Manke, and S. E. Oswald, “Capturing 3D water flow in rooted soil by ultra-fast neutron tomography,” Sci. Rep. 7, 6192 (2017).
[Crossref]

Kim, K. H.

Y. C. Kim, K. H. Kim, D. Y. Son, D. N. Jeong, J. Y. Seo, Y. S. Choi, and N. G. Park, “Printable organometallic perovskite enables large-area, low-dose X-ray imaging,” Nature 550, 87–91 (2017).
[Crossref]

Kim, Y. C.

Y. C. Kim, K. H. Kim, D. Y. Son, D. N. Jeong, J. Y. Seo, Y. S. Choi, and N. G. Park, “Printable organometallic perovskite enables large-area, low-dose X-ray imaging,” Nature 550, 87–91 (2017).
[Crossref]

Kitchen, M. J.

M. J. Kitchen, G. A. Buckley, T. E. Gureyev, M. J. Wallace, N. Andres-Thio, and K. Uesugi, “CT dose reduction factors in the thousands using X-ray phase contrast,” Sci. Rep. 7, 15953 (2007).
[Crossref]

Kronberger, M.

M. Kronberger, E. Auffray, and P. Lecoq, “Probing the concepts of photonic crystals on scintillating materials,” IEEE Trans. Nucl. Sci. 55, 1102–1106 (2008).
[Crossref]

Lecoq, P.

M. Kronberger, E. Auffray, and P. Lecoq, “Probing the concepts of photonic crystals on scintillating materials,” IEEE Trans. Nucl. Sci. 55, 1102–1106 (2008).
[Crossref]

Lee, H.

Z. Liu, S. Durant, H. Lee, Y. Pikus, N. Fang, Y. Xiong, and X. Zhang, “Far-field optical superlens,” Nano Lett. 7, 403–408 (2007).
[Crossref]

Lee, W. K.

M. W. Westneat, J. J. Socha, and W. K. Lee, “Advances in biological structure, function, and physiology using synchrotron X-ray imaging,” Annu. Rev. Physiol. 70, 119–142 (2008).
[Crossref]

Li, W.

Q. Meng, J. Su, and W. Li, Comparative Anatomy of Fishes (Science Press, 1987).

Lidzey, D. G.

H. M. Thirimanne, K. D. G. I. Jayawardena, A. J. Parnell, R. M. I. Bandara, A. Karalasingam, S. Pani, J. E. Huerdler, D. G. Lidzey, S. F. Tedde, and S. R. P. Silva, “High sensitivity organic inorganic hybrid X-ray detectors with direct transduction and broadband response,” Nat. Commun. 9, 2926 (2018).
[Crossref]

Liu, B.

Z. Zhu, B. Liu, C. Cheng, H. Zhang, H. Chen, M. Gu, and Y. Wu, “Enhancement of directional broadband luminescence from a scintillation film via guided-mode resonance in a photonic crystal structure,” Appl. Phys. Lett. 110, 051901 (2017).
[Crossref]

Liu, H.

S. Yang, J. Zhao, L. Wang, F. Zhu, C. Xue, H. Liu, and R. Tai, “The influence of symmetry and duty cycle on the pattern generation in achromatic Talbot lithography,” J. Vac. Sci. Technol. B 35, 021601 (2017).
[Crossref]

S. Yang, J. Zhao, L. Wang, F. Zhu, C. Xue, H. Liu, and R. Tai, “Developments at SSRF in soft X-ray interference lithography,” Nucl. Sci. Tech. 26, 010101 (2015).
[Crossref]

Liu, Z.

Z. Liu, S. Durant, H. Lee, Y. Pikus, N. Fang, Y. Xiong, and X. Zhang, “Far-field optical superlens,” Nano Lett. 7, 403–408 (2007).
[Crossref]

Luquot, L.

P. A. Douissard, A. Cecilia, X. Rochet, X. Chapel, T. Martin, T. van de Kamp, L. Helfen, T. Baumbach, L. Luquot, and X. Xiao, “A versatile indirect detector design for hard X-ray microimaging,” J. Instrum. 7, P09016 (2012).
[Crossref]

Manke, I.

C. Tötzke, N. Kardjilov, I. Manke, and S. E. Oswald, “Capturing 3D water flow in rooted soil by ultra-fast neutron tomography,” Sci. Rep. 7, 6192 (2017).
[Crossref]

Martin, T.

P. A. Douissard, A. Cecilia, X. Rochet, X. Chapel, T. Martin, T. van de Kamp, L. Helfen, T. Baumbach, L. Luquot, and X. Xiao, “A versatile indirect detector design for hard X-ray microimaging,” J. Instrum. 7, P09016 (2012).
[Crossref]

Mayhugh, M.

M. Moszynski, M. Kapusta, M. Mayhugh, and D. Wolski, “Absolute light output of scintillators,” IEEE Trans. Nucl. Sci. 44, 1052–1061 (1997).
[Crossref]

Meng, Q.

Q. Meng, J. Su, and W. Li, Comparative Anatomy of Fishes (Science Press, 1987).

Moszynski, M.

M. Moszynski, M. Kapusta, M. Mayhugh, and D. Wolski, “Absolute light output of scintillators,” IEEE Trans. Nucl. Sci. 44, 1052–1061 (1997).
[Crossref]

Ohtsuka, S.

S. Ohtsuka, Y. Sugishita, T. Takeda, Y. Itai, K. Hyodo, and M. Ando, “Dynamic intravenous coronary arteriography using synchrotron radiation and its application to the measurement of coronary blood flow,” Jpn. Circ. J. 61, 432–440 (1997).
[Crossref]

Oswald, S. E.

C. Tötzke, N. Kardjilov, I. Manke, and S. E. Oswald, “Capturing 3D water flow in rooted soil by ultra-fast neutron tomography,” Sci. Rep. 7, 6192 (2017).
[Crossref]

Pani, S.

H. M. Thirimanne, K. D. G. I. Jayawardena, A. J. Parnell, R. M. I. Bandara, A. Karalasingam, S. Pani, J. E. Huerdler, D. G. Lidzey, S. F. Tedde, and S. R. P. Silva, “High sensitivity organic inorganic hybrid X-ray detectors with direct transduction and broadband response,” Nat. Commun. 9, 2926 (2018).
[Crossref]

Park, N. G.

Y. C. Kim, K. H. Kim, D. Y. Son, D. N. Jeong, J. Y. Seo, Y. S. Choi, and N. G. Park, “Printable organometallic perovskite enables large-area, low-dose X-ray imaging,” Nature 550, 87–91 (2017).
[Crossref]

Parnell, A. J.

H. M. Thirimanne, K. D. G. I. Jayawardena, A. J. Parnell, R. M. I. Bandara, A. Karalasingam, S. Pani, J. E. Huerdler, D. G. Lidzey, S. F. Tedde, and S. R. P. Silva, “High sensitivity organic inorganic hybrid X-ray detectors with direct transduction and broadband response,” Nat. Commun. 9, 2926 (2018).
[Crossref]

Pikus, Y.

Z. Liu, S. Durant, H. Lee, Y. Pikus, N. Fang, Y. Xiong, and X. Zhang, “Far-field optical superlens,” Nano Lett. 7, 403–408 (2007).
[Crossref]

Rochet, X.

P. A. Douissard, A. Cecilia, X. Rochet, X. Chapel, T. Martin, T. van de Kamp, L. Helfen, T. Baumbach, L. Luquot, and X. Xiao, “A versatile indirect detector design for hard X-ray microimaging,” J. Instrum. 7, P09016 (2012).
[Crossref]

Schlenker, M.

P. Cloetens, R. Barrett, J. Baruchel, J. P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard X-ray imaging,” J. Phys. D 9, 133–146 (2016).
[Crossref]

Sentenac, A.

Seo, J. Y.

Y. C. Kim, K. H. Kim, D. Y. Son, D. N. Jeong, J. Y. Seo, Y. S. Choi, and N. G. Park, “Printable organometallic perovskite enables large-area, low-dose X-ray imaging,” Nature 550, 87–91 (2017).
[Crossref]

Silva, S. R. P.

H. M. Thirimanne, K. D. G. I. Jayawardena, A. J. Parnell, R. M. I. Bandara, A. Karalasingam, S. Pani, J. E. Huerdler, D. G. Lidzey, S. F. Tedde, and S. R. P. Silva, “High sensitivity organic inorganic hybrid X-ray detectors with direct transduction and broadband response,” Nat. Commun. 9, 2926 (2018).
[Crossref]

Socha, J. J.

M. W. Westneat, J. J. Socha, and W. K. Lee, “Advances in biological structure, function, and physiology using synchrotron X-ray imaging,” Annu. Rev. Physiol. 70, 119–142 (2008).
[Crossref]

Son, D. Y.

Y. C. Kim, K. H. Kim, D. Y. Son, D. N. Jeong, J. Y. Seo, Y. S. Choi, and N. G. Park, “Printable organometallic perovskite enables large-area, low-dose X-ray imaging,” Nature 550, 87–91 (2017).
[Crossref]

Su, J.

Q. Meng, J. Su, and W. Li, Comparative Anatomy of Fishes (Science Press, 1987).

Sugishita, Y.

S. Ohtsuka, Y. Sugishita, T. Takeda, Y. Itai, K. Hyodo, and M. Ando, “Dynamic intravenous coronary arteriography using synchrotron radiation and its application to the measurement of coronary blood flow,” Jpn. Circ. J. 61, 432–440 (1997).
[Crossref]

Tai, R.

S. Yang, J. Zhao, L. Wang, F. Zhu, C. Xue, H. Liu, and R. Tai, “The influence of symmetry and duty cycle on the pattern generation in achromatic Talbot lithography,” J. Vac. Sci. Technol. B 35, 021601 (2017).
[Crossref]

S. Yang, J. Zhao, L. Wang, F. Zhu, C. Xue, H. Liu, and R. Tai, “Developments at SSRF in soft X-ray interference lithography,” Nucl. Sci. Tech. 26, 010101 (2015).
[Crossref]

M. Chen, H. Yu, J. Zhao, L. Wang, H. Xie, Y. Wu, and R. Tai, “Effect of photonic crystal preparation methods on imaging resolution of synchrotron radiation scintillation detector,” Nucl. Technol. 38, 5–10 (2015).
[Crossref]

Z. Zhu, S. Wu, C. Xue, J. Zhao, L. Wang, Y. Wu, and R. Tai, “Enhanced light extraction of scintillator using large-area photonic crystal structures fabricated by soft-X-ray interference lithography,” Appl. Phys. Lett. 106, 241901 (2015).
[Crossref]

Takeda, T.

S. Ohtsuka, Y. Sugishita, T. Takeda, Y. Itai, K. Hyodo, and M. Ando, “Dynamic intravenous coronary arteriography using synchrotron radiation and its application to the measurement of coronary blood flow,” Jpn. Circ. J. 61, 432–440 (1997).
[Crossref]

Tedde, S. F.

H. M. Thirimanne, K. D. G. I. Jayawardena, A. J. Parnell, R. M. I. Bandara, A. Karalasingam, S. Pani, J. E. Huerdler, D. G. Lidzey, S. F. Tedde, and S. R. P. Silva, “High sensitivity organic inorganic hybrid X-ray detectors with direct transduction and broadband response,” Nat. Commun. 9, 2926 (2018).
[Crossref]

Thirimanne, H. M.

H. M. Thirimanne, K. D. G. I. Jayawardena, A. J. Parnell, R. M. I. Bandara, A. Karalasingam, S. Pani, J. E. Huerdler, D. G. Lidzey, S. F. Tedde, and S. R. P. Silva, “High sensitivity organic inorganic hybrid X-ray detectors with direct transduction and broadband response,” Nat. Commun. 9, 2926 (2018).
[Crossref]

Tötzke, C.

C. Tötzke, N. Kardjilov, I. Manke, and S. E. Oswald, “Capturing 3D water flow in rooted soil by ultra-fast neutron tomography,” Sci. Rep. 7, 6192 (2017).
[Crossref]

Uesugi, K.

M. J. Kitchen, G. A. Buckley, T. E. Gureyev, M. J. Wallace, N. Andres-Thio, and K. Uesugi, “CT dose reduction factors in the thousands using X-ray phase contrast,” Sci. Rep. 7, 15953 (2007).
[Crossref]

Vallmitjana, S.

G. Cristóbal, B. Javidi, and S. Vallmitjana, Information Optics (Springer, 2009).

van de Kamp, T.

P. A. Douissard, A. Cecilia, X. Rochet, X. Chapel, T. Martin, T. van de Kamp, L. Helfen, T. Baumbach, L. Luquot, and X. Xiao, “A versatile indirect detector design for hard X-ray microimaging,” J. Instrum. 7, P09016 (2012).
[Crossref]

Wallace, M. J.

M. J. Kitchen, G. A. Buckley, T. E. Gureyev, M. J. Wallace, N. Andres-Thio, and K. Uesugi, “CT dose reduction factors in the thousands using X-ray phase contrast,” Sci. Rep. 7, 15953 (2007).
[Crossref]

Wang, L.

S. Yang, J. Zhao, L. Wang, F. Zhu, C. Xue, H. Liu, and R. Tai, “The influence of symmetry and duty cycle on the pattern generation in achromatic Talbot lithography,” J. Vac. Sci. Technol. B 35, 021601 (2017).
[Crossref]

S. Yang, J. Zhao, L. Wang, F. Zhu, C. Xue, H. Liu, and R. Tai, “Developments at SSRF in soft X-ray interference lithography,” Nucl. Sci. Tech. 26, 010101 (2015).
[Crossref]

Z. Zhu, S. Wu, C. Xue, J. Zhao, L. Wang, Y. Wu, and R. Tai, “Enhanced light extraction of scintillator using large-area photonic crystal structures fabricated by soft-X-ray interference lithography,” Appl. Phys. Lett. 106, 241901 (2015).
[Crossref]

M. Chen, H. Yu, J. Zhao, L. Wang, H. Xie, Y. Wu, and R. Tai, “Effect of photonic crystal preparation methods on imaging resolution of synchrotron radiation scintillation detector,” Nucl. Technol. 38, 5–10 (2015).
[Crossref]

Westneat, M. W.

M. W. Westneat, J. J. Socha, and W. K. Lee, “Advances in biological structure, function, and physiology using synchrotron X-ray imaging,” Annu. Rev. Physiol. 70, 119–142 (2008).
[Crossref]

Wolski, D.

M. Moszynski, M. Kapusta, M. Mayhugh, and D. Wolski, “Absolute light output of scintillators,” IEEE Trans. Nucl. Sci. 44, 1052–1061 (1997).
[Crossref]

Woods, R. E.

R. C. Gonzalez and R. E. Woods, Digital Image Processing (Prentice Hall, 2002).

Wu, S.

Z. Zhu, S. Wu, C. Xue, J. Zhao, L. Wang, Y. Wu, and R. Tai, “Enhanced light extraction of scintillator using large-area photonic crystal structures fabricated by soft-X-ray interference lithography,” Appl. Phys. Lett. 106, 241901 (2015).
[Crossref]

Wu, Y.

Z. Zhu, B. Liu, C. Cheng, H. Zhang, H. Chen, M. Gu, and Y. Wu, “Enhancement of directional broadband luminescence from a scintillation film via guided-mode resonance in a photonic crystal structure,” Appl. Phys. Lett. 110, 051901 (2017).
[Crossref]

Z. Zhu, S. Wu, C. Xue, J. Zhao, L. Wang, Y. Wu, and R. Tai, “Enhanced light extraction of scintillator using large-area photonic crystal structures fabricated by soft-X-ray interference lithography,” Appl. Phys. Lett. 106, 241901 (2015).
[Crossref]

M. Chen, H. Yu, J. Zhao, L. Wang, H. Xie, Y. Wu, and R. Tai, “Effect of photonic crystal preparation methods on imaging resolution of synchrotron radiation scintillation detector,” Nucl. Technol. 38, 5–10 (2015).
[Crossref]

Xiao, X.

P. A. Douissard, A. Cecilia, X. Rochet, X. Chapel, T. Martin, T. van de Kamp, L. Helfen, T. Baumbach, L. Luquot, and X. Xiao, “A versatile indirect detector design for hard X-ray microimaging,” J. Instrum. 7, P09016 (2012).
[Crossref]

Xie, H.

M. Chen, H. Yu, J. Zhao, L. Wang, H. Xie, Y. Wu, and R. Tai, “Effect of photonic crystal preparation methods on imaging resolution of synchrotron radiation scintillation detector,” Nucl. Technol. 38, 5–10 (2015).
[Crossref]

Xiong, Y.

Z. Liu, S. Durant, H. Lee, Y. Pikus, N. Fang, Y. Xiong, and X. Zhang, “Far-field optical superlens,” Nano Lett. 7, 403–408 (2007).
[Crossref]

Xue, C.

S. Yang, J. Zhao, L. Wang, F. Zhu, C. Xue, H. Liu, and R. Tai, “The influence of symmetry and duty cycle on the pattern generation in achromatic Talbot lithography,” J. Vac. Sci. Technol. B 35, 021601 (2017).
[Crossref]

S. Yang, J. Zhao, L. Wang, F. Zhu, C. Xue, H. Liu, and R. Tai, “Developments at SSRF in soft X-ray interference lithography,” Nucl. Sci. Tech. 26, 010101 (2015).
[Crossref]

Z. Zhu, S. Wu, C. Xue, J. Zhao, L. Wang, Y. Wu, and R. Tai, “Enhanced light extraction of scintillator using large-area photonic crystal structures fabricated by soft-X-ray interference lithography,” Appl. Phys. Lett. 106, 241901 (2015).
[Crossref]

Yang, S.

S. Yang, J. Zhao, L. Wang, F. Zhu, C. Xue, H. Liu, and R. Tai, “The influence of symmetry and duty cycle on the pattern generation in achromatic Talbot lithography,” J. Vac. Sci. Technol. B 35, 021601 (2017).
[Crossref]

S. Yang, J. Zhao, L. Wang, F. Zhu, C. Xue, H. Liu, and R. Tai, “Developments at SSRF in soft X-ray interference lithography,” Nucl. Sci. Tech. 26, 010101 (2015).
[Crossref]

Yu, H.

M. Chen, H. Yu, J. Zhao, L. Wang, H. Xie, Y. Wu, and R. Tai, “Effect of photonic crystal preparation methods on imaging resolution of synchrotron radiation scintillation detector,” Nucl. Technol. 38, 5–10 (2015).
[Crossref]

Zhang, H.

Z. Zhu, B. Liu, C. Cheng, H. Zhang, H. Chen, M. Gu, and Y. Wu, “Enhancement of directional broadband luminescence from a scintillation film via guided-mode resonance in a photonic crystal structure,” Appl. Phys. Lett. 110, 051901 (2017).
[Crossref]

Zhang, X.

Z. Liu, S. Durant, H. Lee, Y. Pikus, N. Fang, Y. Xiong, and X. Zhang, “Far-field optical superlens,” Nano Lett. 7, 403–408 (2007).
[Crossref]

Zhao, J.

S. Yang, J. Zhao, L. Wang, F. Zhu, C. Xue, H. Liu, and R. Tai, “The influence of symmetry and duty cycle on the pattern generation in achromatic Talbot lithography,” J. Vac. Sci. Technol. B 35, 021601 (2017).
[Crossref]

S. Yang, J. Zhao, L. Wang, F. Zhu, C. Xue, H. Liu, and R. Tai, “Developments at SSRF in soft X-ray interference lithography,” Nucl. Sci. Tech. 26, 010101 (2015).
[Crossref]

M. Chen, H. Yu, J. Zhao, L. Wang, H. Xie, Y. Wu, and R. Tai, “Effect of photonic crystal preparation methods on imaging resolution of synchrotron radiation scintillation detector,” Nucl. Technol. 38, 5–10 (2015).
[Crossref]

Z. Zhu, S. Wu, C. Xue, J. Zhao, L. Wang, Y. Wu, and R. Tai, “Enhanced light extraction of scintillator using large-area photonic crystal structures fabricated by soft-X-ray interference lithography,” Appl. Phys. Lett. 106, 241901 (2015).
[Crossref]

Zhu, F.

S. Yang, J. Zhao, L. Wang, F. Zhu, C. Xue, H. Liu, and R. Tai, “The influence of symmetry and duty cycle on the pattern generation in achromatic Talbot lithography,” J. Vac. Sci. Technol. B 35, 021601 (2017).
[Crossref]

S. Yang, J. Zhao, L. Wang, F. Zhu, C. Xue, H. Liu, and R. Tai, “Developments at SSRF in soft X-ray interference lithography,” Nucl. Sci. Tech. 26, 010101 (2015).
[Crossref]

Zhu, Z.

Z. Zhu, B. Liu, C. Cheng, H. Zhang, H. Chen, M. Gu, and Y. Wu, “Enhancement of directional broadband luminescence from a scintillation film via guided-mode resonance in a photonic crystal structure,” Appl. Phys. Lett. 110, 051901 (2017).
[Crossref]

Z. Zhu, S. Wu, C. Xue, J. Zhao, L. Wang, Y. Wu, and R. Tai, “Enhanced light extraction of scintillator using large-area photonic crystal structures fabricated by soft-X-ray interference lithography,” Appl. Phys. Lett. 106, 241901 (2015).
[Crossref]

Annu. Rev. Physiol. (1)

M. W. Westneat, J. J. Socha, and W. K. Lee, “Advances in biological structure, function, and physiology using synchrotron X-ray imaging,” Annu. Rev. Physiol. 70, 119–142 (2008).
[Crossref]

Appl. Phys. B (1)

Y. Ben-Aryeh, “Transmission enhancement by conversion of evanescent waves into propagating waves,” Appl. Phys. B 91, 157–165 (2008).
[Crossref]

Appl. Phys. Lett. (2)

Z. Zhu, S. Wu, C. Xue, J. Zhao, L. Wang, Y. Wu, and R. Tai, “Enhanced light extraction of scintillator using large-area photonic crystal structures fabricated by soft-X-ray interference lithography,” Appl. Phys. Lett. 106, 241901 (2015).
[Crossref]

Z. Zhu, B. Liu, C. Cheng, H. Zhang, H. Chen, M. Gu, and Y. Wu, “Enhancement of directional broadband luminescence from a scintillation film via guided-mode resonance in a photonic crystal structure,” Appl. Phys. Lett. 110, 051901 (2017).
[Crossref]

IEEE Trans. Nucl. Sci. (2)

M. Moszynski, M. Kapusta, M. Mayhugh, and D. Wolski, “Absolute light output of scintillators,” IEEE Trans. Nucl. Sci. 44, 1052–1061 (1997).
[Crossref]

M. Kronberger, E. Auffray, and P. Lecoq, “Probing the concepts of photonic crystals on scintillating materials,” IEEE Trans. Nucl. Sci. 55, 1102–1106 (2008).
[Crossref]

J. Instrum. (1)

P. A. Douissard, A. Cecilia, X. Rochet, X. Chapel, T. Martin, T. van de Kamp, L. Helfen, T. Baumbach, L. Luquot, and X. Xiao, “A versatile indirect detector design for hard X-ray microimaging,” J. Instrum. 7, P09016 (2012).
[Crossref]

J. Phys. D (1)

P. Cloetens, R. Barrett, J. Baruchel, J. P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard X-ray imaging,” J. Phys. D 9, 133–146 (2016).
[Crossref]

J. Vac. Sci. Technol. B (1)

S. Yang, J. Zhao, L. Wang, F. Zhu, C. Xue, H. Liu, and R. Tai, “The influence of symmetry and duty cycle on the pattern generation in achromatic Talbot lithography,” J. Vac. Sci. Technol. B 35, 021601 (2017).
[Crossref]

Jpn. Circ. J. (1)

S. Ohtsuka, Y. Sugishita, T. Takeda, Y. Itai, K. Hyodo, and M. Ando, “Dynamic intravenous coronary arteriography using synchrotron radiation and its application to the measurement of coronary blood flow,” Jpn. Circ. J. 61, 432–440 (1997).
[Crossref]

Nano Lett. (1)

Z. Liu, S. Durant, H. Lee, Y. Pikus, N. Fang, Y. Xiong, and X. Zhang, “Far-field optical superlens,” Nano Lett. 7, 403–408 (2007).
[Crossref]

Nat. Commun. (1)

H. M. Thirimanne, K. D. G. I. Jayawardena, A. J. Parnell, R. M. I. Bandara, A. Karalasingam, S. Pani, J. E. Huerdler, D. G. Lidzey, S. F. Tedde, and S. R. P. Silva, “High sensitivity organic inorganic hybrid X-ray detectors with direct transduction and broadband response,” Nat. Commun. 9, 2926 (2018).
[Crossref]

Nature (1)

Y. C. Kim, K. H. Kim, D. Y. Son, D. N. Jeong, J. Y. Seo, Y. S. Choi, and N. G. Park, “Printable organometallic perovskite enables large-area, low-dose X-ray imaging,” Nature 550, 87–91 (2017).
[Crossref]

Nucl. Sci. Tech. (1)

S. Yang, J. Zhao, L. Wang, F. Zhu, C. Xue, H. Liu, and R. Tai, “Developments at SSRF in soft X-ray interference lithography,” Nucl. Sci. Tech. 26, 010101 (2015).
[Crossref]

Nucl. Technol. (1)

M. Chen, H. Yu, J. Zhao, L. Wang, H. Xie, Y. Wu, and R. Tai, “Effect of photonic crystal preparation methods on imaging resolution of synchrotron radiation scintillation detector,” Nucl. Technol. 38, 5–10 (2015).
[Crossref]

Opt. Lett. (1)

Proc. Natl. Acad. Sci. USA (1)

M. G. L. Gustafsson, “Nonlinear structured-illumination microscopy: wide-field fluorescence imaging with theoretically unlimited resolution,” Proc. Natl. Acad. Sci. USA 102, 13081–13086 (2005).
[Crossref]

Sci. Rep. (2)

M. J. Kitchen, G. A. Buckley, T. E. Gureyev, M. J. Wallace, N. Andres-Thio, and K. Uesugi, “CT dose reduction factors in the thousands using X-ray phase contrast,” Sci. Rep. 7, 15953 (2007).
[Crossref]

C. Tötzke, N. Kardjilov, I. Manke, and S. E. Oswald, “Capturing 3D water flow in rooted soil by ultra-fast neutron tomography,” Sci. Rep. 7, 6192 (2017).
[Crossref]

Other (3)

R. C. Gonzalez and R. E. Woods, Digital Image Processing (Prentice Hall, 2002).

Q. Meng, J. Su, and W. Li, Comparative Anatomy of Fishes (Science Press, 1987).

G. Cristóbal, B. Javidi, and S. Vallmitjana, Information Optics (Springer, 2009).

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Figures (6)

Fig. 1.
Fig. 1. Schematic of an X-ray scintillator imager based on the use of the proposed high-spatial-frequency spectrum enhanced reconstruction (HSFER) method. A two-dimensional (2D) encoder is used to extract the middle-high-frequency and high-frequency components of the image generated in the scintillator. The image is decoded by a PSF/OTF of the encoder.
Fig. 2.
Fig. 2. (a) Areas below the dashed, dotted, and solid curves represent the parameter zones that allow the +1st-order diffraction to occur for the two-dimensional (2D) encoder periods of 600, 450, and 300 nm, respectively. This shows that the encoder with a 300 nm period can yield a maximum suppression of the +1-order diffraction compared with the other two encoders. In fact, no +1st-order diffraction occurs with a 300 nm period in FDTD simulations. (b) The green area shows the parameter zone that allows the 1st-order diffracted waves to be accepted by the optical system, whereby the encoder with the 300 nm period is selected. The angle 33.0° represents the total reflection angle. The FDTD simulation results of the transverse electric (TE) and transverse magnetic (TM) modes are also shown here. (c) MTFs are used for the evaluation of the performance of this system. The thick curve represents the MTF of the case with the 2D encoder, while the thin one represents the case without the 2D encoder. Obviously, the spectrum is extended considerably by the encoder. The fc value is the cutoff frequency determined by the equivalent camera pixel size, which is equal to (1.3  μm)1 in our experiment. The green area indicates the extra high-frequency information received from the encoder.
Fig. 3.
Fig. 3. Experimental setup with the HSFER imager. The fluorescent pattern produced by the X rays through the sample was first encoded by a 2D encoder, then imaged by the camera, and finally decoded by an iteration method. The inset upper-left subfigure shows the scanning electron micrograph of the 2D encoder: a YAG:Ce film covered with a 2D SiNx array.
Fig. 4.
Fig. 4. Radiographs of the resolution chart obtained with a 3 s exposure (the dose is 4×108  phs/mm2). (a), (b) Imaging results for a resolution chart without and with the HSFER method, respectively. (c), (d) Magnified views of the regions of interest in (a) and (b), respectively. (e) PSDs of the images in (a) and (b), whereby the red and the blue curves represent the cases with and without the HSFER, respectively, and fc2 is the cutoff frequency for the thin curve. (f) Normalized PSD with respect to the one without HSFER, thus showing that the cutoff frequency is extended to fc1. The fn is the spatial frequency at which the average effect of the deconvolution algorithm begins to work. The dashed line describes the denoising effect for frequencies much higher than fn.
Fig. 5.
Fig. 5. Radiographs of the resolution chart obtained with a 0.5 s exposure (the dose is 6×107  phs/mm2). (a), (b) Imaging results without and with the HSFER method, respectively. (c), (d) Magnified views of the regions of interest in (a) and (b), respectively. (e) PSDs of the images in (a) and (b), whereby the red and blue curves represent the cases with and without the HSFER, respectively. (f) PSDs with respect to the case at which the HSFER was not used.
Fig. 6.
Fig. 6. Radiographs of the gill of a dehydrated zebrafish following an exposure of 1 s (the dose is 1.2×108  phs/mm2). More detailed structures in the regions of interest can be found in (b) the HSFER image compared with that in (a) the conventional image.

Equations (13)

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

w0=wsh,
wI0=|ws|2|h|2.
w=(ws·g)h.
wI=|ws·g|2|h|2.
g=1+η0·ei(kgxx+Øx)·ei(kgyy+Øy),
w={ws·[1+η0ei(kgxx+Øx)·ei(kgyy+Øy)]}h.
wI{|ws|2·[1+η·ei(kgxxkgyy)+η*·ei(kgxx+kgyy)]}|h|2,
W˜I(k)[W˜s(k)+η·W˜s(kkg)+η*·W˜s(k+kg)]·H˜(k,λ),
W˜I(k)W˜s(k)·[(1/Δλ)·H(k,λ)dλ]+W˜s(kkg)·[(1/Δλ)·η(k,λ)H(k,λ)dλ],
W˜I(k)W˜s(k)·H˜(k)¯·{1+[W˜s(kkg)·η(k,λ)H˜(k)¯]/[W˜s(k)·H˜(k)¯]}.
H˜(k)=1+[W˜s(kkg)·η(k,λ)·H˜(k)¯]/[W˜s·H˜(k)¯],
|nsc·(2π/λ)·sinθm+m·G0|<2π/λ,
1<sinθm<(m/p+sinγ)/nsc.

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