Abstract

Photon sieves are a fairly new class of diffractive lenses that open unprecedented possibilities for high resolution imaging and spectroscopy, especially at short wavelengths such as UV and x-rays. In this paper, we model and analyze the image formation process of photon sieves using Fourier optics. We derive closed-form Fresnel imaging models that relate an input object to the image formed by a photon sieve system, both for coherent and incoherent illumination. These analytical models also provide a closed-form expression for the point-spread function of the system for both in-focus and out-of-focus cases. All the formulas are expressed in terms of Fourier transforms and convolutions, which enable easy interpretation as well as fast computation. The derived analytical models provide a unified framework to effectively develop new imaging modalities enabled by diffractive lenses and analyze their imaging capabilities for different design configurations, prior to physical production. To illustrate their utility and versatility, the derived formulas are applied to several important special cases such as photon sieves with circular holes and pixelated diffractive lenses generated by SLM-type devices. The analytical image formation models presented in this paper provide a generalizable and powerful means for effective analysis and simulation of any imaging system with a diffractive lens, including Fresnel zone plates, Fresnel phase plates, and other modified Fresnel lenses and mask-like patterns such as coded apertures.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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    [Crossref]
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2018 (3)

2017 (7)

A. Vijayakumar, B. Vinoth, I. V. Minin, J. Rosen, O. V. Minin, and C.-J. Cheng, “Experimental demonstration of square Fresnel zone plate with chiral side lobes,” Appl. Opt. 56, F128–F133 (2017).
[Crossref] [PubMed]

W. Sun, Y. Hu, D. G. MacDonnell, H. J. Kim, C. Weimer, and R. R. Baize, “Fully transparent photon sieve,” Opt. Express 25, 17356–17363 (2017).
[Crossref] [PubMed]

M. N. Julian, D. G. MacDonnell, and M. C. Gupta, “Fabrication of photon sieves by laser ablation and optical properties,” Opt. Express 25, 31528–31538 (2017).
[Crossref] [PubMed]

O. Asmolova, G. Andersen, M. Anderson, and M. Cumming, “Photon sieves for creating and identifying orbital angular momentum of light,” Proc. SPIE 10120, 1012009 (2017).
[Crossref]

F. D. Hallada, A. L. Franz, and M. R. Hawks, “Fresnel zone plate light field spectral imaging,” Opt. Eng. 56, 081811 (2017).
[Crossref]

M. Li, W. Li, H. Li, Y. Zhu, and Y. Yu, “Controllable design of super-oscillatory lenses with multiple sub-diffraction-limit foci,” Scientific Reports,  7, 1335 (2017).
[Crossref] [PubMed]

Y. Cheng, J. Zhu, S. Hu, L. Zhao, W. Yan, Y. He, W. Jiang, and J. Liu, “Focusing properties of single-focus photon sieve,” IEEE Photon. Technol. Lett. 29, 275–278 (2017).
[Crossref]

2016 (4)

P. Wang, N. Mohammad, and R. Menon, “Chromatic-aberration-corrected diffractive lenses for ultra-broadband focusing,” Scientific Reports 6, 21545 (2016).
[Crossref] [PubMed]

Y. Shi, X. Zhu, L. Liang, and Y. Yang, “Tunable focusing properties using optofluidic Fresnel zone plates,” Lab on a Chip 16, 4554–4559 (2016).
[Crossref] [PubMed]

G. Andersen, O. Asmolova, M. G. McHarg, T. Quiller, and C. Maldonado, “FalconSAT-7: a membrane space solar telescope,” Proc. SPIE 9904, 99041P (2016).
[Crossref]

E. Arbabi, A. Arbabi, S. M. Kamali, Y. Horie, and A. Faraon, “Multiwavelength polarization-insensitive lenses based on dielectric metasurfaces with meta-molecules,” Optica 3, 628–633 (2016).
[Crossref]

2015 (5)

2014 (2)

T. Roy, E. T. Rogers, G. Yuan, and N. I. Zheludev, “Point spread function of the Optica l needle super-oscillatory lens,” Appl. Phys. Lett. 104, 231109 (2014).
[Crossref]

A. Sabatyan and S. Hoseini, “Diffractive performance of a photon-sieve-based axilens,” Appl. Opt. 53, 7331–7336 (2014).
[Crossref] [PubMed]

2013 (5)

T. Liu, J. Tan, J. Liu, and H. Wang, “Vectorial design of super-oscillatory lens,” Opt. Express 21, 15090–15101 (2013).
[Crossref] [PubMed]

Z. Li, M. Zhang, G. Liang, X. Li, X. Chen, and C. Cheng, “Generation of high-order optical vortices with asymmetrical pinhole plates under plane wave illumination,” Opt. Express 21, 15755–15764 (2013).
[Crossref] [PubMed]

A. Sabatyan and L. Elahi, “FFT-based convolution algorithm for fast and precise numerical evaluating diffracted field by photon sieve,” Optik 124, 4960–4962 (2013).
[Crossref]

H. Ye, C. Qiu, K. Huang, J. Teng, B. Luk’yanchuk, and S. Yeo, “Creation of a longitudinally polarized subwavelength hotspot with an ultra-thin planar lens: vectorial Rayleigh–Sommerfeld method,” Laser Phys. Lett. 10, 065004 (2013).
[Crossref]

Y. Tang, S. Hu, Y. Yang, and Y. He, “Focusing property of high numerical aperture photon sieves based on vector diffraction,” Opt. Comm. 295, 1–4 (2013).
[Crossref]

2012 (2)

2011 (2)

2010 (5)

2009 (2)

2008 (1)

H.-H. Chung, N. Bradman, M. R. Davidson, and P. H. Holloway, “Dual wavelength photon sieves,” Opt. Eng. 47, 118001 (2008).
[Crossref]

2007 (1)

2006 (1)

2005 (3)

R. Menon, D. Gil, G. Barbastathis, and H. I. Smith, “Photon-sieve lithography,” J. Opt. Soc. Am. A 22, 342–345 (2005).
[Crossref]

G. Andersen, “Large optical photon sieve,” Opt. Lett. 30, 2976–2978 (2005).
[Crossref] [PubMed]

P. Gorenstein, J. D. Phillips, and R. D. Reasenberg, “Refractive/diffractive telescope with very high angular resolution for X-ray astronomy,” Proc. SPIE 5900, 590018 (2005).
[Crossref]

2004 (1)

2003 (4)

Q. Cao and J. Jahns, “Nonparaxial model for the focusing of high-numerical-aperture photon sieves,” J. Opt. Soc. Am. A 20, 1005–1012 (2003).
[Crossref]

Q. Cao and J. Jahns, “Modified Fresnel zone plates that produce sharp Gaussian focal spots,” J. Opt. Soc. Am. A 20, 1576–1581 (2003).
[Crossref]

G. E. Artzner, J. P. Delaboudiniere, and X. Song, “Photon sieves as EUV telescopes for solar orbiter,” Proc. of SPIE 4853, 158–161 (2003).
[Crossref]

Y. Wang, W. Yun, and C. Jacobsen, “Achromatic Fresnel optics for wideband extreme-ultraviolet and x-ray imaging,” Nature 424, 50–53 (2003).
[Crossref] [PubMed]

2002 (1)

2001 (1)

L. Kipp, M. Skibowski, R. Johnson, R. Berndt, R. Adelung, S. Harm, and R. Seemann, “Sharper images by focusing soft x-rays with photon sieves,” Nature 414, 184–188 (2001).
[Crossref] [PubMed]

1981 (1)

W. H. Southwell, “Validity of the Fresnel approximation in the near field,” J. Opt. Soc. Am. A 71, 7–14 (1981).
[Crossref]

Adelung, R.

L. Kipp, M. Skibowski, R. Johnson, R. Berndt, R. Adelung, S. Harm, and R. Seemann, “Sharper images by focusing soft x-rays with photon sieves,” Nature 414, 184–188 (2001).
[Crossref] [PubMed]

Alkanat, T.

F. S. Oktem and T. Alkanat, “Fast computation of two-dimensional point-spread functions for photon sieves,” in Imaging and Applied Optics Congress, OSA Technical Digest (online) (Optical Society of America, 2016), paper JT3A.34.

Andersen, G.

O. Asmolova, G. Andersen, M. Anderson, and M. Cumming, “Photon sieves for creating and identifying orbital angular momentum of light,” Proc. SPIE 10120, 1012009 (2017).
[Crossref]

G. Andersen, O. Asmolova, M. G. McHarg, T. Quiller, and C. Maldonado, “FalconSAT-7: a membrane space solar telescope,” Proc. SPIE 9904, 99041P (2016).
[Crossref]

G. Andersen, “Membrane photon sieve telescopes,” Appl. Opt. 49, 6391–6394 (2010).
[Crossref] [PubMed]

G. Andersen and D. Tullson, “Broadband antihole photon sieve telescope,” Appl. Opt. 46, 3706–3708 (2007).
[Crossref] [PubMed]

G. Andersen, “Large optical photon sieve,” Opt. Lett. 30, 2976–2978 (2005).
[Crossref] [PubMed]

Anderson, M.

O. Asmolova, G. Andersen, M. Anderson, and M. Cumming, “Photon sieves for creating and identifying orbital angular momentum of light,” Proc. SPIE 10120, 1012009 (2017).
[Crossref]

Arbabi, A.

Arbabi, E.

Artzner, G. E.

G. E. Artzner, J. P. Delaboudiniere, and X. Song, “Photon sieves as EUV telescopes for solar orbiter,” Proc. of SPIE 4853, 158–161 (2003).
[Crossref]

Asmolova, O.

O. Asmolova, G. Andersen, M. Anderson, and M. Cumming, “Photon sieves for creating and identifying orbital angular momentum of light,” Proc. SPIE 10120, 1012009 (2017).
[Crossref]

G. Andersen, O. Asmolova, M. G. McHarg, T. Quiller, and C. Maldonado, “FalconSAT-7: a membrane space solar telescope,” Proc. SPIE 9904, 99041P (2016).
[Crossref]

Attwood, D.

D. Attwood, Soft X-rays and Extreme Ultraviolet Radiation: Principles and Applications (Cambridge University, 2000).

Baize, R. R.

Barbastathis, G.

Berndt, R.

L. Kipp, M. Skibowski, R. Johnson, R. Berndt, R. Adelung, S. Harm, and R. Seemann, “Sharper images by focusing soft x-rays with photon sieves,” Nature 414, 184–188 (2001).
[Crossref] [PubMed]

Blahut, R. E.

R. E. Blahut, Theory of Remote Image Formation (Cambridge University, 2004).
[Crossref]

Bradman, N.

H.-H. Chung, N. Bradman, M. R. Davidson, and P. H. Holloway, “Dual wavelength photon sieves,” Opt. Eng. 47, 118001 (2008).
[Crossref]

Buck, J.

Cao, Q.

Chen, L.

Chen, X.

Chen, Z.

Cheng, C.

Cheng, C.-J.

Cheng, G.

Cheng, Y.

Y. Cheng, J. Zhu, S. Hu, L. Zhao, W. Yan, Y. He, W. Jiang, and J. Liu, “Focusing properties of single-focus photon sieve,” IEEE Photon. Technol. Lett. 29, 275–278 (2017).
[Crossref]

Chu, D.

Chung, H.-H.

H.-H. Chung, N. Bradman, M. R. Davidson, and P. H. Holloway, “Dual wavelength photon sieves,” Opt. Eng. 47, 118001 (2008).
[Crossref]

Cumming, M.

O. Asmolova, G. Andersen, M. Anderson, and M. Cumming, “Photon sieves for creating and identifying orbital angular momentum of light,” Proc. SPIE 10120, 1012009 (2017).
[Crossref]

Dai, H. T.

Davidson, M. R.

H.-H. Chung, N. Bradman, M. R. Davidson, and P. H. Holloway, “Dual wavelength photon sieves,” Opt. Eng. 47, 118001 (2008).
[Crossref]

Davila, J. M.

J. M. Davila, “High-resolution solar imaging with a photon sieve,” Proc. SPIE 8148, 81480O (2011).
[Crossref]

F. S. Oktem, J. M. Davila, and F. Kamalabadi, “Image formation model for photon sieves,” in Proceedings of IEEE International Conference on Image Processing (IEEE, 2013), pp. 2373–2377.

F. S. Oktem, F. Kamalabadi, and J. M. Davila, “High-resolution computational spectral imaging with photon sieves,” in Proceedings of IEEE International Conference on Image Processing (IEEE, 2014), pp. 5122–5126.

Delaboudiniere, J. P.

G. E. Artzner, J. P. Delaboudiniere, and X. Song, “Photon sieves as EUV telescopes for solar orbiter,” Proc. of SPIE 4853, 158–161 (2003).
[Crossref]

Dong, X.

Du, C.

Dudgeon, D. E.

D. E. Dudgeon and R. M. Mersereau, Multidimensional Digital Signal Processing (Prentice Hall, 1984).

Elahi, L.

A. Sabatyan and L. Elahi, “FFT-based convolution algorithm for fast and precise numerical evaluating diffracted field by photon sieve,” Optik 124, 4960–4962 (2013).
[Crossref]

Faraon, A.

Franz, A. L.

F. D. Hallada, A. L. Franz, and M. R. Hawks, “Fresnel zone plate light field spectral imaging,” Opt. Eng. 56, 081811 (2017).
[Crossref]

Fu, Q.

Furlan, W. D.

Garcia-Vidal, F. J.

K. Huang, L. Hong, F. J. Garcia-Vidal, M. Hong, B. Luk’yanchuk, J. Teng, and C. Qiu, “Ultrahigh-capacity non-periodic photon sieves operating in visible light,” Nature Comm. 6, 7059 (2015).
[Crossref]

Gil, D.

Giménez, F.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (Roberts, 2005).

Gorenstein, P.

P. Gorenstein, J. D. Phillips, and R. D. Reasenberg, “Refractive/diffractive telescope with very high angular resolution for X-ray astronomy,” Proc. SPIE 5900, 590018 (2005).
[Crossref]

Gupta, M. C.

Hallada, F. D.

F. D. Hallada, A. L. Franz, and M. R. Hawks, “Fresnel zone plate light field spectral imaging,” Opt. Eng. 56, 081811 (2017).
[Crossref]

Harm, S.

M. Kalläne, J. Buck, S. Harm, R. Seemann, K. Rossnagel, and L. Kipp, “Focusing light with a reflection photon sieve,” Opt. Lett. 36, 2405–2407 (2011).
[Crossref] [PubMed]

L. Kipp, M. Skibowski, R. Johnson, R. Berndt, R. Adelung, S. Harm, and R. Seemann, “Sharper images by focusing soft x-rays with photon sieves,” Nature 414, 184–188 (2001).
[Crossref] [PubMed]

Hawks, M. R.

F. D. Hallada, A. L. Franz, and M. R. Hawks, “Fresnel zone plate light field spectral imaging,” Opt. Eng. 56, 081811 (2017).
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He, Y.

Y. Cheng, J. Zhu, S. Hu, L. Zhao, W. Yan, Y. He, W. Jiang, and J. Liu, “Focusing properties of single-focus photon sieve,” IEEE Photon. Technol. Lett. 29, 275–278 (2017).
[Crossref]

Y. Tang, S. Hu, Y. Yang, and Y. He, “Focusing property of high numerical aperture photon sieves based on vector diffraction,” Opt. Comm. 295, 1–4 (2013).
[Crossref]

Holloway, P. H.

H.-H. Chung, N. Bradman, M. R. Davidson, and P. H. Holloway, “Dual wavelength photon sieves,” Opt. Eng. 47, 118001 (2008).
[Crossref]

Hong, L.

K. Huang, L. Hong, F. J. Garcia-Vidal, M. Hong, B. Luk’yanchuk, J. Teng, and C. Qiu, “Ultrahigh-capacity non-periodic photon sieves operating in visible light,” Nature Comm. 6, 7059 (2015).
[Crossref]

Hong, M.

K. Huang, L. Hong, F. J. Garcia-Vidal, M. Hong, B. Luk’yanchuk, J. Teng, and C. Qiu, “Ultrahigh-capacity non-periodic photon sieves operating in visible light,” Nature Comm. 6, 7059 (2015).
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Horie, Y.

Hoseini, S.

Hu, C.

Hu, J.

Hu, S.

Y. Cheng, J. Zhu, S. Hu, L. Zhao, W. Yan, Y. He, W. Jiang, and J. Liu, “Focusing properties of single-focus photon sieve,” IEEE Photon. Technol. Lett. 29, 275–278 (2017).
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Y. Tang, S. Hu, Y. Yang, and Y. He, “Focusing property of high numerical aperture photon sieves based on vector diffraction,” Opt. Comm. 295, 1–4 (2013).
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Hua, Y.

Huang, K.

K. Huang, L. Hong, F. J. Garcia-Vidal, M. Hong, B. Luk’yanchuk, J. Teng, and C. Qiu, “Ultrahigh-capacity non-periodic photon sieves operating in visible light,” Nature Comm. 6, 7059 (2015).
[Crossref]

H. Ye, C. Qiu, K. Huang, J. Teng, B. Luk’yanchuk, and S. Yeo, “Creation of a longitudinally polarized subwavelength hotspot with an ultra-thin planar lens: vectorial Rayleigh–Sommerfeld method,” Laser Phys. Lett. 10, 065004 (2013).
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Huang, T. J.

Jacobsen, C.

Y. Wang, W. Yun, and C. Jacobsen, “Achromatic Fresnel optics for wideband extreme-ultraviolet and x-ray imaging,” Nature 424, 50–53 (2003).
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Jahns, J.

Jiang, W.

Y. Cheng, J. Zhu, S. Hu, L. Zhao, W. Yan, Y. He, W. Jiang, and J. Liu, “Focusing properties of single-focus photon sieve,” IEEE Photon. Technol. Lett. 29, 275–278 (2017).
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Johnson, R.

L. Kipp, M. Skibowski, R. Johnson, R. Berndt, R. Adelung, S. Harm, and R. Seemann, “Sharper images by focusing soft x-rays with photon sieves,” Nature 414, 184–188 (2001).
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Julian, M. N.

Kalläne, M.

Kamalabadi, F.

F. S. Oktem, J. M. Davila, and F. Kamalabadi, “Image formation model for photon sieves,” in Proceedings of IEEE International Conference on Image Processing (IEEE, 2013), pp. 2373–2377.

F. S. Oktem, F. Kamalabadi, and J. M. Davila, “High-resolution computational spectral imaging with photon sieves,” in Proceedings of IEEE International Conference on Image Processing (IEEE, 2014), pp. 5122–5126.

Kamali, S. M.

Kim, H. J.

Kipp, L.

M. Kalläne, J. Buck, S. Harm, R. Seemann, K. Rossnagel, and L. Kipp, “Focusing light with a reflection photon sieve,” Opt. Lett. 36, 2405–2407 (2011).
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L. Kipp, M. Skibowski, R. Johnson, R. Berndt, R. Adelung, S. Harm, and R. Seemann, “Sharper images by focusing soft x-rays with photon sieves,” Nature 414, 184–188 (2001).
[Crossref] [PubMed]

Li, F.

Li, H.

Li, J.

Li, M.

M. Li, W. Li, H. Li, Y. Zhu, and Y. Yu, “Controllable design of super-oscillatory lenses with multiple sub-diffraction-limit foci,” Scientific Reports,  7, 1335 (2017).
[Crossref] [PubMed]

Li, W.

M. Li, W. Li, H. Li, Y. Zhu, and Y. Yu, “Controllable design of super-oscillatory lenses with multiple sub-diffraction-limit foci,” Scientific Reports,  7, 1335 (2017).
[Crossref] [PubMed]

Li, X.

Li, Z.

Liang, G.

Liang, L.

Y. Shi, X. Zhu, L. Liang, and Y. Yang, “Tunable focusing properties using optofluidic Fresnel zone plates,” Lab on a Chip 16, 4554–4559 (2016).
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Lin, Z.

Liu, J.

Y. Cheng, J. Zhu, S. Hu, L. Zhao, W. Yan, Y. He, W. Jiang, and J. Liu, “Focusing properties of single-focus photon sieve,” IEEE Photon. Technol. Lett. 29, 275–278 (2017).
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T. Liu, J. Tan, J. Liu, and H. Wang, “Vectorial design of super-oscillatory lens,” Opt. Express 21, 15090–15101 (2013).
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Liu, R.

Liu, T.

Liu, Y. J.

Lu, X.

Luk’yanchuk, B.

K. Huang, L. Hong, F. J. Garcia-Vidal, M. Hong, B. Luk’yanchuk, J. Teng, and C. Qiu, “Ultrahigh-capacity non-periodic photon sieves operating in visible light,” Nature Comm. 6, 7059 (2015).
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H. Ye, C. Qiu, K. Huang, J. Teng, B. Luk’yanchuk, and S. Yeo, “Creation of a longitudinally polarized subwavelength hotspot with an ultra-thin planar lens: vectorial Rayleigh–Sommerfeld method,” Laser Phys. Lett. 10, 065004 (2013).
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Maldonado, C.

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G. Andersen, O. Asmolova, M. G. McHarg, T. Quiller, and C. Maldonado, “FalconSAT-7: a membrane space solar telescope,” Proc. SPIE 9904, 99041P (2016).
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P. Wang, N. Mohammad, and R. Menon, “Chromatic-aberration-corrected diffractive lenses for ultra-broadband focusing,” Scientific Reports 6, 21545 (2016).
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P. Wang, N. Mohammad, and R. Menon, “Chromatic-aberration-corrected diffractive lenses for ultra-broadband focusing,” Scientific Reports 6, 21545 (2016).
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Nawab, S. H.

A. V. Oppenheim, A. S. Willsky, and S. H. Nawab, Signals and Systems (Prentice Hall, 1997).

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F. S. Oktem, F. Kamalabadi, and J. M. Davila, “High-resolution computational spectral imaging with photon sieves,” in Proceedings of IEEE International Conference on Image Processing (IEEE, 2014), pp. 5122–5126.

F. S. Oktem and T. Alkanat, “Fast computation of two-dimensional point-spread functions for photon sieves,” in Imaging and Applied Optics Congress, OSA Technical Digest (online) (Optical Society of America, 2016), paper JT3A.34.

F. S. Oktem, J. M. Davila, and F. Kamalabadi, “Image formation model for photon sieves,” in Proceedings of IEEE International Conference on Image Processing (IEEE, 2013), pp. 2373–2377.

Oppenheim, A. V.

A. V. Oppenheim, A. S. Willsky, and S. H. Nawab, Signals and Systems (Prentice Hall, 1997).

Padgett, M.

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, 1968).

Phillips, D.

Phillips, J. D.

P. Gorenstein, J. D. Phillips, and R. D. Reasenberg, “Refractive/diffractive telescope with very high angular resolution for X-ray astronomy,” Proc. SPIE 5900, 590018 (2005).
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Pons, A.

Pu, D.

Qiu, C.

K. Huang, L. Hong, F. J. Garcia-Vidal, M. Hong, B. Luk’yanchuk, J. Teng, and C. Qiu, “Ultrahigh-capacity non-periodic photon sieves operating in visible light,” Nature Comm. 6, 7059 (2015).
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H. Ye, C. Qiu, K. Huang, J. Teng, B. Luk’yanchuk, and S. Yeo, “Creation of a longitudinally polarized subwavelength hotspot with an ultra-thin planar lens: vectorial Rayleigh–Sommerfeld method,” Laser Phys. Lett. 10, 065004 (2013).
[Crossref]

Qu, H.

Quiller, T.

G. Andersen, O. Asmolova, M. G. McHarg, T. Quiller, and C. Maldonado, “FalconSAT-7: a membrane space solar telescope,” Proc. SPIE 9904, 99041P (2016).
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P. Gorenstein, J. D. Phillips, and R. D. Reasenberg, “Refractive/diffractive telescope with very high angular resolution for X-ray astronomy,” Proc. SPIE 5900, 590018 (2005).
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T. Roy, E. T. Rogers, G. Yuan, and N. I. Zheludev, “Point spread function of the Optica l needle super-oscillatory lens,” Appl. Phys. Lett. 104, 231109 (2014).
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Roy, T.

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Seemann, R.

M. Kalläne, J. Buck, S. Harm, R. Seemann, K. Rossnagel, and L. Kipp, “Focusing light with a reflection photon sieve,” Opt. Lett. 36, 2405–2407 (2011).
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L. Kipp, M. Skibowski, R. Johnson, R. Berndt, R. Adelung, S. Harm, and R. Seemann, “Sharper images by focusing soft x-rays with photon sieves,” Nature 414, 184–188 (2001).
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Shi, L.

Shi, Y.

Y. Shi, X. Zhu, L. Liang, and Y. Yang, “Tunable focusing properties using optofluidic Fresnel zone plates,” Lab on a Chip 16, 4554–4559 (2016).
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Skibowski, M.

L. Kipp, M. Skibowski, R. Johnson, R. Berndt, R. Adelung, S. Harm, and R. Seemann, “Sharper images by focusing soft x-rays with photon sieves,” Nature 414, 184–188 (2001).
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Smith, H. I.

Song, X.

G. E. Artzner, J. P. Delaboudiniere, and X. Song, “Photon sieves as EUV telescopes for solar orbiter,” Proc. of SPIE 4853, 158–161 (2003).
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Y. Tang, S. Hu, Y. Yang, and Y. He, “Focusing property of high numerical aperture photon sieves based on vector diffraction,” Opt. Comm. 295, 1–4 (2013).
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Teng, J.

K. Huang, L. Hong, F. J. Garcia-Vidal, M. Hong, B. Luk’yanchuk, J. Teng, and C. Qiu, “Ultrahigh-capacity non-periodic photon sieves operating in visible light,” Nature Comm. 6, 7059 (2015).
[Crossref]

H. Ye, C. Qiu, K. Huang, J. Teng, B. Luk’yanchuk, and S. Yeo, “Creation of a longitudinally polarized subwavelength hotspot with an ultra-thin planar lens: vectorial Rayleigh–Sommerfeld method,” Laser Phys. Lett. 10, 065004 (2013).
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Y. Wang, W. Yun, and C. Jacobsen, “Achromatic Fresnel optics for wideband extreme-ultraviolet and x-ray imaging,” Nature 424, 50–53 (2003).
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Y. Cheng, J. Zhu, S. Hu, L. Zhao, W. Yan, Y. He, W. Jiang, and J. Liu, “Focusing properties of single-focus photon sieve,” IEEE Photon. Technol. Lett. 29, 275–278 (2017).
[Crossref]

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Y. Shi, X. Zhu, L. Liang, and Y. Yang, “Tunable focusing properties using optofluidic Fresnel zone plates,” Lab on a Chip 16, 4554–4559 (2016).
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Y. Tang, S. Hu, Y. Yang, and Y. He, “Focusing property of high numerical aperture photon sieves based on vector diffraction,” Opt. Comm. 295, 1–4 (2013).
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Ye, H.

H. Ye, C. Qiu, K. Huang, J. Teng, B. Luk’yanchuk, and S. Yeo, “Creation of a longitudinally polarized subwavelength hotspot with an ultra-thin planar lens: vectorial Rayleigh–Sommerfeld method,” Laser Phys. Lett. 10, 065004 (2013).
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H. Ye, C. Qiu, K. Huang, J. Teng, B. Luk’yanchuk, and S. Yeo, “Creation of a longitudinally polarized subwavelength hotspot with an ultra-thin planar lens: vectorial Rayleigh–Sommerfeld method,” Laser Phys. Lett. 10, 065004 (2013).
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Yöntem, A. Ö.

Yu, Y.

M. Li, W. Li, H. Li, Y. Zhu, and Y. Yu, “Controllable design of super-oscillatory lenses with multiple sub-diffraction-limit foci,” Scientific Reports,  7, 1335 (2017).
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Yuan, G.

T. Roy, E. T. Rogers, G. Yuan, and N. I. Zheludev, “Point spread function of the Optica l needle super-oscillatory lens,” Appl. Phys. Lett. 104, 231109 (2014).
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Y. Wang, W. Yun, and C. Jacobsen, “Achromatic Fresnel optics for wideband extreme-ultraviolet and x-ray imaging,” Nature 424, 50–53 (2003).
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Zhang, J.

Zhang, M.

Zhang, X.

Zhao, L.

Y. Cheng, J. Zhu, S. Hu, L. Zhao, W. Yan, Y. He, W. Jiang, and J. Liu, “Focusing properties of single-focus photon sieve,” IEEE Photon. Technol. Lett. 29, 275–278 (2017).
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Zhao, X.

Zheludev, N. I.

T. Roy, E. T. Rogers, G. Yuan, and N. I. Zheludev, “Point spread function of the Optica l needle super-oscillatory lens,” Appl. Phys. Lett. 104, 231109 (2014).
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Zhu, J.

Y. Cheng, J. Zhu, S. Hu, L. Zhao, W. Yan, Y. He, W. Jiang, and J. Liu, “Focusing properties of single-focus photon sieve,” IEEE Photon. Technol. Lett. 29, 275–278 (2017).
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Zhu, Y.

M. Li, W. Li, H. Li, Y. Zhu, and Y. Yu, “Controllable design of super-oscillatory lenses with multiple sub-diffraction-limit foci,” Scientific Reports,  7, 1335 (2017).
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Appl. Opt. (9)

G. Andersen, “Membrane photon sieve telescopes,” Appl. Opt. 49, 6391–6394 (2010).
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G. Andersen and D. Tullson, “Broadband antihole photon sieve telescope,” Appl. Opt. 46, 3706–3708 (2007).
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C. Zhou, X. Dong, L. Shi, C. Wang, and C. Du, “Experimental study of a multiwavelength photon sieve designed by random-area-divided approach,” Appl. Opt. 48, 1619–1623 (2009).
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A. Sabatyan and P. Roshaninejad, “Super-resolving random-Gaussian apodized photon sieve,” Appl. Opt. 51, 6315–6318 (2012).
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A. Sabatyan and S. Hoseini, “Diffractive performance of a photon-sieve-based axilens,” Appl. Opt. 53, 7331–7336 (2014).
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T. Liu, X. Zhang, L. Wang, Y. Wu, J. Zhang, and H. Qu, “Multiregion apodized photon sieve with enhanced efficiency and enlarged pinhole sizes,” Appl. Opt. 54, 7175–7180 (2015).
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A. Vijayakumar, B. Vinoth, I. V. Minin, J. Rosen, O. V. Minin, and C.-J. Cheng, “Experimental demonstration of square Fresnel zone plate with chiral side lobes,” Appl. Opt. 56, F128–F133 (2017).
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T. Liu, L. Wang, J. Zhang, Q. Fu, and X. Zhang, “Numerical simulation and design of an apodized diffractive Optical element composed of open-ring zones and pinholes,” Appl. Opt. 57, 25–32 (2018).
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T. Liu, X. Zhang, L. Wang, Y. Wu, J. Zhang, and H. Qu, “Fast and accurate focusing analysis of large photon sieve using pinhole ring diffraction model,” Appl. Opt. 54, 5327–5331 (2015).
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Appl. Phys. Lett. (1)

T. Roy, E. T. Rogers, G. Yuan, and N. I. Zheludev, “Point spread function of the Optica l needle super-oscillatory lens,” Appl. Phys. Lett. 104, 231109 (2014).
[Crossref]

IEEE Photon. Technol. Lett. (1)

Y. Cheng, J. Zhu, S. Hu, L. Zhao, W. Yan, Y. He, W. Jiang, and J. Liu, “Focusing properties of single-focus photon sieve,” IEEE Photon. Technol. Lett. 29, 275–278 (2017).
[Crossref]

J. Opt. Soc. Am. A (7)

Lab on a Chip (1)

Y. Shi, X. Zhu, L. Liang, and Y. Yang, “Tunable focusing properties using optofluidic Fresnel zone plates,” Lab on a Chip 16, 4554–4559 (2016).
[Crossref] [PubMed]

Laser Phys. Lett. (1)

H. Ye, C. Qiu, K. Huang, J. Teng, B. Luk’yanchuk, and S. Yeo, “Creation of a longitudinally polarized subwavelength hotspot with an ultra-thin planar lens: vectorial Rayleigh–Sommerfeld method,” Laser Phys. Lett. 10, 065004 (2013).
[Crossref]

Nature (2)

L. Kipp, M. Skibowski, R. Johnson, R. Berndt, R. Adelung, S. Harm, and R. Seemann, “Sharper images by focusing soft x-rays with photon sieves,” Nature 414, 184–188 (2001).
[Crossref] [PubMed]

Y. Wang, W. Yun, and C. Jacobsen, “Achromatic Fresnel optics for wideband extreme-ultraviolet and x-ray imaging,” Nature 424, 50–53 (2003).
[Crossref] [PubMed]

Nature Comm. (1)

K. Huang, L. Hong, F. J. Garcia-Vidal, M. Hong, B. Luk’yanchuk, J. Teng, and C. Qiu, “Ultrahigh-capacity non-periodic photon sieves operating in visible light,” Nature Comm. 6, 7059 (2015).
[Crossref]

Opt. Comm. (1)

Y. Tang, S. Hu, Y. Yang, and Y. He, “Focusing property of high numerical aperture photon sieves based on vector diffraction,” Opt. Comm. 295, 1–4 (2013).
[Crossref]

Opt. Eng. (2)

F. D. Hallada, A. L. Franz, and M. R. Hawks, “Fresnel zone plate light field spectral imaging,” Opt. Eng. 56, 081811 (2017).
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H.-H. Chung, N. Bradman, M. R. Davidson, and P. H. Holloway, “Dual wavelength photon sieves,” Opt. Eng. 47, 118001 (2008).
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Opt. Express (9)

X. Zhao, F. Xu, J. Hu, and C. Wang, “Broadband photon sieves imaging with wavefront coding,” Opt. Express 23, 16812–16822 (2015).
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F. Giménez, J. A. Monsoriu, W. D. Furlan, and A. Pons, “Fractal photon sieve,” Opt. Express 14, 11958–11963 (2006).
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Z. Li, M. Zhang, G. Liang, X. Li, X. Chen, and C. Cheng, “Generation of high-order optical vortices with asymmetrical pinhole plates under plane wave illumination,” Opt. Express 21, 15755–15764 (2013).
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Z. Chen, C. Wang, D. Pu, J. Hu, and L. Chen, “Ultra-large multi-region photon sieves,” Opt. Express 18, 16279–16288 (2010).
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M. N. Julian, D. G. MacDonnell, and M. C. Gupta, “Fabrication of photon sieves by laser ablation and optical properties,” Opt. Express 25, 31528–31538 (2017).
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A. Ö. Yöntem, J. Li, and D. Chu, “Imaging through a projection screen using bi-stable switchable diffusive photon sieves,” Opt. Express 26, 10162–10170 (2018).
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Y. J. Liu, H. T. Dai, X. W. Sun, and T. J. Huang, “Electrically switchable phase-type fractal zone plates and fractal photon sieves,” Opt. Express 17, 12418–12423 (2009).
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T. Liu, J. Tan, J. Liu, and H. Wang, “Vectorial design of super-oscillatory lens,” Opt. Express 21, 15090–15101 (2013).
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Opt. Lett. (6)

Optica (2)

Optik (1)

A. Sabatyan and L. Elahi, “FFT-based convolution algorithm for fast and precise numerical evaluating diffracted field by photon sieve,” Optik 124, 4960–4962 (2013).
[Crossref]

Proc. of SPIE (1)

G. E. Artzner, J. P. Delaboudiniere, and X. Song, “Photon sieves as EUV telescopes for solar orbiter,” Proc. of SPIE 4853, 158–161 (2003).
[Crossref]

Proc. SPIE (4)

P. Gorenstein, J. D. Phillips, and R. D. Reasenberg, “Refractive/diffractive telescope with very high angular resolution for X-ray astronomy,” Proc. SPIE 5900, 590018 (2005).
[Crossref]

J. M. Davila, “High-resolution solar imaging with a photon sieve,” Proc. SPIE 8148, 81480O (2011).
[Crossref]

G. Andersen, O. Asmolova, M. G. McHarg, T. Quiller, and C. Maldonado, “FalconSAT-7: a membrane space solar telescope,” Proc. SPIE 9904, 99041P (2016).
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O. Asmolova, G. Andersen, M. Anderson, and M. Cumming, “Photon sieves for creating and identifying orbital angular momentum of light,” Proc. SPIE 10120, 1012009 (2017).
[Crossref]

Scientific Reports (2)

P. Wang, N. Mohammad, and R. Menon, “Chromatic-aberration-corrected diffractive lenses for ultra-broadband focusing,” Scientific Reports 6, 21545 (2016).
[Crossref] [PubMed]

M. Li, W. Li, H. Li, Y. Zhu, and Y. Yu, “Controllable design of super-oscillatory lenses with multiple sub-diffraction-limit foci,” Scientific Reports,  7, 1335 (2017).
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Other (9)

J. W. Goodman, Introduction to Fourier Optics (Roberts, 2005).

F. S. Oktem and T. Alkanat, “Fast computation of two-dimensional point-spread functions for photon sieves,” in Imaging and Applied Optics Congress, OSA Technical Digest (online) (Optical Society of America, 2016), paper JT3A.34.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, 1968).

F. S. Oktem, J. M. Davila, and F. Kamalabadi, “Image formation model for photon sieves,” in Proceedings of IEEE International Conference on Image Processing (IEEE, 2013), pp. 2373–2377.

R. E. Blahut, Theory of Remote Image Formation (Cambridge University, 2004).
[Crossref]

D. E. Dudgeon and R. M. Mersereau, Multidimensional Digital Signal Processing (Prentice Hall, 1984).

A. V. Oppenheim, A. S. Willsky, and S. H. Nawab, Signals and Systems (Prentice Hall, 1997).

F. S. Oktem, F. Kamalabadi, and J. M. Davila, “High-resolution computational spectral imaging with photon sieves,” in Proceedings of IEEE International Conference on Image Processing (IEEE, 2014), pp. 5122–5126.

D. Attwood, Soft X-rays and Extreme Ultraviolet Radiation: Principles and Applications (Cambridge University, 2000).

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Figures (6)

Fig. 1
Fig. 1 Illustration of a photon sieve imaging system. ©[2013] IEEE. Reprinted, with permission, from [48].
Fig. 2
Fig. 2 Aperture functions of different designs. (a) FZP, (b) PS with circular holes, (c) PS with rectangular holes, (d) Pixelated PS.
Fig. 3
Fig. 3 Two-dimensional view of the magnitude of coherent PSFs at focus (after normalization). (a) FZP, (b) PS with circular holes, (c) PS with rectangular holes, (d) Pixelated PS. To better see the relative efficiency of the imaging, line-scans across the PSF centers are also shown in the insets. These PSF slices are normalized to the peak PSF value of the zone-plate.
Fig. 4
Fig. 4 Two-dimensional view of the magnitude of coherent PSFs at three DOF away from focus (after normalization). (a) FZP, (b) PS with circular holes, (c) PS with rectangular holes, (d) Pixelated PS. To better see the relative efficiency of the imaging, line-scans across the PSF centers are also shown in the insets. These PSF slices are normalized to the the peak PSF value of the zone-plate.
Fig. 5
Fig. 5 Two-dimensional view of the incoherent PSFs at focus (after normalization). (a) FZP, (b) PS with circular holes, (c) PS with rectangular holes, (d) Pixelated PS. To better see the relative efficiency of the imaging, line-scans across the PSF centers are also shown in the insets. These PSF slices are normalized to the peak PSF value of the zone-plate.
Fig. 6
Fig. 6 One-dimensional central slice of normalized PSFs. (a) magnitude of coherent PSF at (first-order) focus, (b) magnitude of coherent PSF at three DOF away from focus, (c) incoherent PSF at focus.

Equations (51)

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u ( x , y ) = A ( α , β ) e i 2 π ( α λ x + β λ y ) d α d β ,
A ( α , β ) = 1 / λ 2 U ( α / λ , β / λ ) ,
U ( f x , f y ) = { u ( x , y ) } ( f x , f y ) = u ( x , y ) e i 2 π ( f x x + f y y ) d x d y .
t ( x , y ) = n = 1 N t n ( x , y ) = n = 1 N circ ( x x n d n , y y n d n ) ,
circ ( x , y ) = { 1 , if x 2 + y 2 1 2 0 , otherwise .
r ( x , y ) = e i π x 2 + y 2 λ d i [ u ˜ ( x , y ) * h d i , λ ( x , y ) , ]
u ˜ ( x , y ) = u ( d s d i x , d s d i y ) ( d s d i ) e i π d s ( x 2 + y 2 ) / ( λ d i 2 ) ,
h d i , λ ( x , y ) = i λ Δ e i π x 2 + y 2 Δ λ d i 2 * T ( x λ d i , y λ d i ) .
H d i , λ ( f x , f y ) = ( λ d i ) 4 t ( λ d i f x , λ d i f y ) e i π ( Δ λ d i 2 ) ( f x 2 + f y 2 ) ,
i ( x , y ) = | r ( x , y ) | 2 = | u ˜ ( x , y ) * h d i , λ ( x , y ) | 2 .
h d i , λ ( x , y ) i λ d i e i π x 2 + y 2 λ d i * T ( x λ d i , y λ d i ) ,
H d i , λ ( f x , f y ) ( λ d i ) 4 t ( λ d i f x , λ d i f y ) e i π ( λ d i ) ( f x 2 + f y 2 ) ,
v ( x , y ) = u ( x , y ) * 1 i λ d s e i π x 2 + y 2 λ d s .
v ( x , y ) = 1 i λ d s e i π x 2 + y 2 λ d s { u ( x , y ) e i π x 2 + y 2 λ d s } ( , x λ d s , y λ d s ) ,
s ( x , y ) = t ( x , y ) v ( x , y ) ,
r ( x , y ) = 1 i λ d i e i π x 2 + y 2 λ d i { s ( x , y ) e i π x 2 + y 2 λ d i } ( x λ d i , y λ d i ) .
r ( x , y ) = e i π x 2 + y 2 λ d i [ u ˜ ( x , y ) * i λ Δ e i π x 2 + y 2 λ d i 2 * T ( x λ d i , y λ d i ) ] ,
r ( x , y ) = e i π x 2 + y 2 λ d i [ u ˜ ( x , y ) * Q ( x λ d i , y λ d i ) ] ,
r ( x , y ) = e i π x 2 + y 2 λ d i [ u ˜ ( x , y ) * i λ Δ e i π x 2 + y 2 λ d i 2 * T ( x λ d i , x λ d i ) ] ,
i ( x , y ) = u ˜ i ( x , y ) * g d i , λ ( x , y ) ,
u ˜ i ( x , y ) = ( d s d i ) 2 u i ( d s d i x , d s d i y ) ,
g d i , λ ( x , y ) = | i λ Δ e i π x 2 + y 2 Δ λ d i 2 * T ( x λ d i , y λ d i ) | 2 .
g d i , λ ( x , y ) = | h d i , λ ( x , y ) | 2 .
w ( x , y , t ) = u ( x , y ) ρ ( x , y , t ) .
w ( x , y , t ) w * ( x , y , t ) = | u ( x , y ) | 2 δ ( x x , y y ) = u i ( x , y ) δ ( x x , y y ) ,
| r ( x , y , t ) | 2 = ( d s d i ) 2 w ( d s d i ξ , d s d i η , t ) w * ( d s d i ξ , d s d i η , t ) × h ( x ξ , y η ) h * ( x ξ , y η ) e i π ( ξ 2 + η 2 ξ 2 η 2 ) λ d i 2 / d s d ξ d ξ d η d η , = ( d s d i ) 2 u i ( d s d i ξ , d s d i η ) | h ( x ξ , y η ) | 2 d ξ d η , = ( d s d i ) 2 u i ( d s d i x , d s d i y ) * | h ( x , y ) | 2 ,
h d i , λ = i λ Δ e i π x 2 + y 2 Δ λ d i 2 * ( n = 1 N d n 2 jinc ( d n x λ d i , d n y λ d i ) e i π 2 π Δ λ d i ( x n x + y n y ) ) ,
H d i λ ( f x , f y ) = ( λ d i ) 4 e i π ( Δ λ d i 2 ) ( f x y + f y 2 ) n = 1 N circ ( λ d i f x x n d n , λ d i f y x y d n ) .
jinc ( x , y ) = jinc ( x 2 + y 2 ) = J 1 ( π x 2 + y 2 ) 2 x 2 + y 2 ,
h d i λ ( x , y ) = 2 π ( λ d i ) 2 e i π x 2 + y 2 λ d i n = 1 N e i π R 2 λ d i 0 d n / 2 r e i π r 2 λ d i J 0 ( 2 π λ d i R r ) d r .
t p ( x , y ) = [ t ( x , y ) 1 d x d y comb ( x d x , y d y ) ] * rect ( x a x , y a y ) , = [ t ( x , y ) m n δ ( x m d x , y n d y ) ] * rect ( x a x , y a y ) , = m n t ( m d x , n d y ) rect ( x m d x a x , y n d y a y ) ,
comb ( x , y ) = m = n = δ ( x m , y n ) ,
rect ( x , y ) = { 1 , if | x | 1 2 , | y | 1 2 0 , otherwise .
T p ( f x , f y ) = [ T ( f x , f y ) * c o m b ( d x f x , d y f y ) ] a x a y sinc ( a x f x , a y f y ) , = a x a y d x d y m n T ( f x m d x , f y n d y ) sinc ( a x f x , a y f y )
sinc ( x , y ) = sinc ( x ) sinc ( y ) ,
h d i λ ( x , y ) = i λ a x a y Δ d x d y e i π x 2 + y 2 Δ λ d i 2 m n T ( 1 λ d i ( x m λ d i d x ) 1 λ d i ( y n λ d i d y ) ) sin ( a x λ d i x , a y λ d i y ) ,
H d i λ ( f x , f y ) = ( λ d i ) 4 e i π ( Δ λ d i 2 ) ( f x y + f y 2 ) m n t ( m d x , n d y ) rect( λ d i a x ( f x m d x λ d i ) λ d i a y ( f y m d y λ d i ) )
t ( x , y ) = 1 2 [ 1 + sgn ( cos π λ f ( x 2 + y 2 ) ) ] circ ( x D , y D ) ,
t ( x , y ) = [ m = sin ( π m / 2 ) π m e i m π λ f ( x 2 + y 2 ) ] circ ( x D , y D ) , = [ 1 2 + odd m ( 1 ) | m | 1 2 π m e i π x 2 + y 2 λ f ] circ ( x D , y D ) ,
h d i , λ ( x , y ) = h 0 ( x , y ) + odd m h m ( x , y ) ,
h 0 ( x , y ) = i λ 2 Δ e i π x 2 + y 2 Δ λ d i 2 D 2 jinc ( D λ d i x , D λ d i y ) , h m ( x , y ) = { ( 1 ) | m | 1 2 m π ( λ d i ) 2 D 2 jinc ( D λ d i x , D λ d i y ) , ϵ m = 0 , ( 1 ) | m | 1 2 m π D 2 jinc ( D λ d i x , D λ d i y ) i λ ϵ m e i π x 2 + y 2 ϵ m λ d i 2 , ϵ m 0 .
H d i , λ ( f x , f y ) = H 0 ( f x , f y ) + odd m H m ( f x , f y ) ,
H 0 ( f x , f y ) = ( λ d i ) 4 2 e i π Δ λ d i 2 ( f x 2 + f y 2 ) circ ( λ d i D f x , λ d i D f y ) , H m ( f x , f y ) = ( 1 ) | m | 1 2 m π ( λ d i ) 4 e i π ϵ m λ d i 2 ( f x 2 + f y 2 ) circ ( λ d i D f x , λ d i D f y ) .
η m = { 1 / 4 , if m = 0 , 1 / m 2 π 2 , if m o d d , 0 , if m e v e n .
t 1 ( x , y ) + t 2 ( x , y ) = circ ( x D , y D ) ,
h c i r c ( x , y ) = i λ Δ e i π x 2 + y 2 Δ λ d i 2 * D 2 jinc ( D λ d i x , D λ d i y ) .
h c i r c ( x , y ) = i λ Δ e i π x 2 + y 2 Δ λ d i 2 * ( T 1 ( x λ d i , y λ d i ) + T 2 ( x λ d i , y λ d i ) ) , = i λ Δ e i π x 2 + y 2 Δ λ d i 2 * T 1 ( x λ d i , y λ d i ) + i λ Δ e i π x 2 + y 2 Δ λ d i 2 * T 2 ( x λ d i , y λ d i ) , = h 1 ( x , y ) + h 2 ( x , y ) .
H c i r c ( f x , f y ) = ( λ d i ) 4 e i π ( Δ λ d i 2 ) ( f x 2 + f x 2 ) circ ( λ d i D f x , λ d i D f y ) = H 1 ( f x , f y ) + H 2 ( f x , f y ) ,
h 0 ( x , y ) = i λ 2 Δ e π x 2 + y 2 Δ λ d i 2 * D 2 jinc ( D λ d i x , D λ d i y ) , h m ( x , y ) = { ( 1 ) | m | + 1 2 m π ( λ d i ) 2 D 2 jinc ( D λ d i x , D λ d i y ) , ϵ m = 0 , ( 1 ) | m | + 1 2 m π D 2 jinc ( D λ d i x , D λ d i y ) * i λ ϵ m e i π x 2 + y 2 ϵ m λ d i 2 , ϵ m 0 ,
H 0 ( f x , f y ) = ( λ d i ) 4 2 e i π Δ λ d i 2 ( f x 2 + f y 2 ) circ ( λ d i D f x , λ d i D f y ) , H m ( f x , f y ) = ( 1 ) | m | + 1 2 m π ( λ d i ) 4 e i π ϵ m λ d i 2 ( f x 2 + f y 2 ) circ ( λ d i D f x , λ d i D f y )
z > 1 λ [ ( x ξ ) 2 + ( y η ) 2 ] m a x 2 .

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