Abstract

Statistical aspects of Young's double-slit diffraction experiment are analysed quantitatively. It is shown that the signal-to-noise ratio and the spatial resolution in the detected diffraction pattern satisfy a duality relationship which implies that both of them cannot be improved simultaneously beyond a certain limit if the total number of particles forming the image is fixed. As a consequence of this duality, it is possible to estimate the minimal number of particles that have to be detected in order for two slits separated by a given distance to be resolved with a confidence level corresponding to a pre-defined signal-to-noise ratio, e.g. according to the Rose criterion. These results are related to the recently introduced imaging system quality characteristic which combines the spatial resolution and the noise sensitivity, and allows one to estimate the efficiency with which imaging quanta are utilised in a system to deliver maximal amount of information about the imaged object. The presented results can be useful for applications where the imaging quanta are at a premium or where minimization of the radiation dose is important.

© 2015 Optical Society of America

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References

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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
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2014 (3)

T. E. Gureyev, Y. I. Nesterets, F. de Hoog, G. Schmalz, S. C. Mayo, S. Mohammadi, and G. Tromba, “Duality between noise and spatial resolution in linear systems,” Opt. Express 22(8), 9087–9094 (2014).
[Crossref] [PubMed]

F. de Hoog, G. Schmalz, and T. E. Gureyev, “An uncertainty inequality,” Appl. Math. Lett. 38, 84–86 (2014).
[Crossref]

T. E. Gureyev, S. C. Mayo, Y. I. Nesterets, S. Mohammadi, D. Lockie, R. H. Menk, F. Arfelli, K. M. Pavlov, M. J. Kitchen, F. Zanconati, C. Dullin, and G. Tromba, “Investigation of the imaging quality of synchrotron-based phase-contrast mammographic tomography,” J. Phys. D Appl. Phys. 47(36), 365401 (2014).
[Crossref]

2013 (1)

R. Bach, D. Pope, S.-H. Liou, and H. Batelaan, “Controlled double-slit electron diffraction,” New J. Phys. 15(3), 033018 (2013).
[Crossref]

2012 (2)

S. Frabboni, A. Gabrielli, G. Carlo Gazzadi, F. Giorgi, G. Matteucci, G. Pozzi, N. S. Cesari, M. Villa, and A. Zoccoli, “The Young-Feynman two-slits experiment with single electrons: Build-up of the interference pattern and arrival-time distribution using a fast-readout pixel detector,” Ultramicroscopy 116, 73–76 (2012).
[Crossref]

T. Juffmann, A. Milic, M. Müllneritsch, P. Asenbaum, A. Tsukernik, J. Tüxen, M. Mayor, O. Cheshnovsky, and M. Arndt, “Real-time single-molecule imaging of quantum interference,” Nat. Nanotechnol. 7(5), 297–300 (2012).
[Crossref] [PubMed]

2008 (1)

2005 (1)

Y. I. Nesterets, S. W. Wilkins, T. E. Gureyev, A. Pogany, and A. W. Stevenson, “On the optimization of experimental parameters for x-ray in-line phase-contrast imaging,” Rev. Sci. Instrum. 76(9), 093706 (2005).
[Crossref]

2002 (1)

N. Garcia, I. G. Saveliev, and M. Sharonov, “Time-resolved diffraction and interference: Young’s interference with photons of different energy as revealed by time resolution,” Philos. Trans. A Math. Phys. Eng. Sci. 360(1794), 1039–1059 (2002).
[Crossref] [PubMed]

1976 (1)

P. G. Merli, G. F. Missiroli, and G. Pozzi, “On the statistical aspect of electron interference phenomena,” Am. J. Phys. 44(3), 306–307 (1976).
[Crossref]

1963 (1)

J. A. Wheeler, ““No fugitive and cloistered virtue”—A tribute to Niels Bohr,” Phys. Today 16(1), 30–32 (1963).
[Crossref]

1948 (1)

Arfelli, F.

T. E. Gureyev, S. C. Mayo, Y. I. Nesterets, S. Mohammadi, D. Lockie, R. H. Menk, F. Arfelli, K. M. Pavlov, M. J. Kitchen, F. Zanconati, C. Dullin, and G. Tromba, “Investigation of the imaging quality of synchrotron-based phase-contrast mammographic tomography,” J. Phys. D Appl. Phys. 47(36), 365401 (2014).
[Crossref]

Arndt, M.

T. Juffmann, A. Milic, M. Müllneritsch, P. Asenbaum, A. Tsukernik, J. Tüxen, M. Mayor, O. Cheshnovsky, and M. Arndt, “Real-time single-molecule imaging of quantum interference,” Nat. Nanotechnol. 7(5), 297–300 (2012).
[Crossref] [PubMed]

Asenbaum, P.

T. Juffmann, A. Milic, M. Müllneritsch, P. Asenbaum, A. Tsukernik, J. Tüxen, M. Mayor, O. Cheshnovsky, and M. Arndt, “Real-time single-molecule imaging of quantum interference,” Nat. Nanotechnol. 7(5), 297–300 (2012).
[Crossref] [PubMed]

Bach, R.

R. Bach, D. Pope, S.-H. Liou, and H. Batelaan, “Controlled double-slit electron diffraction,” New J. Phys. 15(3), 033018 (2013).
[Crossref]

Batelaan, H.

R. Bach, D. Pope, S.-H. Liou, and H. Batelaan, “Controlled double-slit electron diffraction,” New J. Phys. 15(3), 033018 (2013).
[Crossref]

Carlo Gazzadi, G.

S. Frabboni, A. Gabrielli, G. Carlo Gazzadi, F. Giorgi, G. Matteucci, G. Pozzi, N. S. Cesari, M. Villa, and A. Zoccoli, “The Young-Feynman two-slits experiment with single electrons: Build-up of the interference pattern and arrival-time distribution using a fast-readout pixel detector,” Ultramicroscopy 116, 73–76 (2012).
[Crossref]

Cesari, N. S.

S. Frabboni, A. Gabrielli, G. Carlo Gazzadi, F. Giorgi, G. Matteucci, G. Pozzi, N. S. Cesari, M. Villa, and A. Zoccoli, “The Young-Feynman two-slits experiment with single electrons: Build-up of the interference pattern and arrival-time distribution using a fast-readout pixel detector,” Ultramicroscopy 116, 73–76 (2012).
[Crossref]

Cheshnovsky, O.

T. Juffmann, A. Milic, M. Müllneritsch, P. Asenbaum, A. Tsukernik, J. Tüxen, M. Mayor, O. Cheshnovsky, and M. Arndt, “Real-time single-molecule imaging of quantum interference,” Nat. Nanotechnol. 7(5), 297–300 (2012).
[Crossref] [PubMed]

de Hoog, F.

Dullin, C.

T. E. Gureyev, S. C. Mayo, Y. I. Nesterets, S. Mohammadi, D. Lockie, R. H. Menk, F. Arfelli, K. M. Pavlov, M. J. Kitchen, F. Zanconati, C. Dullin, and G. Tromba, “Investigation of the imaging quality of synchrotron-based phase-contrast mammographic tomography,” J. Phys. D Appl. Phys. 47(36), 365401 (2014).
[Crossref]

Frabboni, S.

S. Frabboni, A. Gabrielli, G. Carlo Gazzadi, F. Giorgi, G. Matteucci, G. Pozzi, N. S. Cesari, M. Villa, and A. Zoccoli, “The Young-Feynman two-slits experiment with single electrons: Build-up of the interference pattern and arrival-time distribution using a fast-readout pixel detector,” Ultramicroscopy 116, 73–76 (2012).
[Crossref]

Gabrielli, A.

S. Frabboni, A. Gabrielli, G. Carlo Gazzadi, F. Giorgi, G. Matteucci, G. Pozzi, N. S. Cesari, M. Villa, and A. Zoccoli, “The Young-Feynman two-slits experiment with single electrons: Build-up of the interference pattern and arrival-time distribution using a fast-readout pixel detector,” Ultramicroscopy 116, 73–76 (2012).
[Crossref]

Garcia, N.

N. Garcia, I. G. Saveliev, and M. Sharonov, “Time-resolved diffraction and interference: Young’s interference with photons of different energy as revealed by time resolution,” Philos. Trans. A Math. Phys. Eng. Sci. 360(1794), 1039–1059 (2002).
[Crossref] [PubMed]

Giorgi, F.

S. Frabboni, A. Gabrielli, G. Carlo Gazzadi, F. Giorgi, G. Matteucci, G. Pozzi, N. S. Cesari, M. Villa, and A. Zoccoli, “The Young-Feynman two-slits experiment with single electrons: Build-up of the interference pattern and arrival-time distribution using a fast-readout pixel detector,” Ultramicroscopy 116, 73–76 (2012).
[Crossref]

Gureyev, T. E.

F. de Hoog, G. Schmalz, and T. E. Gureyev, “An uncertainty inequality,” Appl. Math. Lett. 38, 84–86 (2014).
[Crossref]

T. E. Gureyev, S. C. Mayo, Y. I. Nesterets, S. Mohammadi, D. Lockie, R. H. Menk, F. Arfelli, K. M. Pavlov, M. J. Kitchen, F. Zanconati, C. Dullin, and G. Tromba, “Investigation of the imaging quality of synchrotron-based phase-contrast mammographic tomography,” J. Phys. D Appl. Phys. 47(36), 365401 (2014).
[Crossref]

T. E. Gureyev, Y. I. Nesterets, F. de Hoog, G. Schmalz, S. C. Mayo, S. Mohammadi, and G. Tromba, “Duality between noise and spatial resolution in linear systems,” Opt. Express 22(8), 9087–9094 (2014).
[Crossref] [PubMed]

T. E. Gureyev, Y. I. Nesterets, A. W. Stevenson, P. R. Miller, A. Pogany, and S. W. Wilkins, “Some simple rules for contrast, signal-to-noise and resolution in in-line x-ray phase-contrast imaging,” Opt. Express 16(5), 3223–3241 (2008).
[Crossref] [PubMed]

Y. I. Nesterets, S. W. Wilkins, T. E. Gureyev, A. Pogany, and A. W. Stevenson, “On the optimization of experimental parameters for x-ray in-line phase-contrast imaging,” Rev. Sci. Instrum. 76(9), 093706 (2005).
[Crossref]

Y. I. Nesterets and T. E. Gureyev, in preparation.

Juffmann, T.

T. Juffmann, A. Milic, M. Müllneritsch, P. Asenbaum, A. Tsukernik, J. Tüxen, M. Mayor, O. Cheshnovsky, and M. Arndt, “Real-time single-molecule imaging of quantum interference,” Nat. Nanotechnol. 7(5), 297–300 (2012).
[Crossref] [PubMed]

Kitchen, M. J.

T. E. Gureyev, S. C. Mayo, Y. I. Nesterets, S. Mohammadi, D. Lockie, R. H. Menk, F. Arfelli, K. M. Pavlov, M. J. Kitchen, F. Zanconati, C. Dullin, and G. Tromba, “Investigation of the imaging quality of synchrotron-based phase-contrast mammographic tomography,” J. Phys. D Appl. Phys. 47(36), 365401 (2014).
[Crossref]

Liou, S.-H.

R. Bach, D. Pope, S.-H. Liou, and H. Batelaan, “Controlled double-slit electron diffraction,” New J. Phys. 15(3), 033018 (2013).
[Crossref]

Lockie, D.

T. E. Gureyev, S. C. Mayo, Y. I. Nesterets, S. Mohammadi, D. Lockie, R. H. Menk, F. Arfelli, K. M. Pavlov, M. J. Kitchen, F. Zanconati, C. Dullin, and G. Tromba, “Investigation of the imaging quality of synchrotron-based phase-contrast mammographic tomography,” J. Phys. D Appl. Phys. 47(36), 365401 (2014).
[Crossref]

Matteucci, G.

S. Frabboni, A. Gabrielli, G. Carlo Gazzadi, F. Giorgi, G. Matteucci, G. Pozzi, N. S. Cesari, M. Villa, and A. Zoccoli, “The Young-Feynman two-slits experiment with single electrons: Build-up of the interference pattern and arrival-time distribution using a fast-readout pixel detector,” Ultramicroscopy 116, 73–76 (2012).
[Crossref]

Mayo, S. C.

T. E. Gureyev, S. C. Mayo, Y. I. Nesterets, S. Mohammadi, D. Lockie, R. H. Menk, F. Arfelli, K. M. Pavlov, M. J. Kitchen, F. Zanconati, C. Dullin, and G. Tromba, “Investigation of the imaging quality of synchrotron-based phase-contrast mammographic tomography,” J. Phys. D Appl. Phys. 47(36), 365401 (2014).
[Crossref]

T. E. Gureyev, Y. I. Nesterets, F. de Hoog, G. Schmalz, S. C. Mayo, S. Mohammadi, and G. Tromba, “Duality between noise and spatial resolution in linear systems,” Opt. Express 22(8), 9087–9094 (2014).
[Crossref] [PubMed]

Mayor, M.

T. Juffmann, A. Milic, M. Müllneritsch, P. Asenbaum, A. Tsukernik, J. Tüxen, M. Mayor, O. Cheshnovsky, and M. Arndt, “Real-time single-molecule imaging of quantum interference,” Nat. Nanotechnol. 7(5), 297–300 (2012).
[Crossref] [PubMed]

Menk, R. H.

T. E. Gureyev, S. C. Mayo, Y. I. Nesterets, S. Mohammadi, D. Lockie, R. H. Menk, F. Arfelli, K. M. Pavlov, M. J. Kitchen, F. Zanconati, C. Dullin, and G. Tromba, “Investigation of the imaging quality of synchrotron-based phase-contrast mammographic tomography,” J. Phys. D Appl. Phys. 47(36), 365401 (2014).
[Crossref]

Merli, P. G.

P. G. Merli, G. F. Missiroli, and G. Pozzi, “On the statistical aspect of electron interference phenomena,” Am. J. Phys. 44(3), 306–307 (1976).
[Crossref]

Milic, A.

T. Juffmann, A. Milic, M. Müllneritsch, P. Asenbaum, A. Tsukernik, J. Tüxen, M. Mayor, O. Cheshnovsky, and M. Arndt, “Real-time single-molecule imaging of quantum interference,” Nat. Nanotechnol. 7(5), 297–300 (2012).
[Crossref] [PubMed]

Miller, P. R.

Missiroli, G. F.

P. G. Merli, G. F. Missiroli, and G. Pozzi, “On the statistical aspect of electron interference phenomena,” Am. J. Phys. 44(3), 306–307 (1976).
[Crossref]

Mohammadi, S.

T. E. Gureyev, S. C. Mayo, Y. I. Nesterets, S. Mohammadi, D. Lockie, R. H. Menk, F. Arfelli, K. M. Pavlov, M. J. Kitchen, F. Zanconati, C. Dullin, and G. Tromba, “Investigation of the imaging quality of synchrotron-based phase-contrast mammographic tomography,” J. Phys. D Appl. Phys. 47(36), 365401 (2014).
[Crossref]

T. E. Gureyev, Y. I. Nesterets, F. de Hoog, G. Schmalz, S. C. Mayo, S. Mohammadi, and G. Tromba, “Duality between noise and spatial resolution in linear systems,” Opt. Express 22(8), 9087–9094 (2014).
[Crossref] [PubMed]

Müllneritsch, M.

T. Juffmann, A. Milic, M. Müllneritsch, P. Asenbaum, A. Tsukernik, J. Tüxen, M. Mayor, O. Cheshnovsky, and M. Arndt, “Real-time single-molecule imaging of quantum interference,” Nat. Nanotechnol. 7(5), 297–300 (2012).
[Crossref] [PubMed]

Nesterets, Y. I.

T. E. Gureyev, S. C. Mayo, Y. I. Nesterets, S. Mohammadi, D. Lockie, R. H. Menk, F. Arfelli, K. M. Pavlov, M. J. Kitchen, F. Zanconati, C. Dullin, and G. Tromba, “Investigation of the imaging quality of synchrotron-based phase-contrast mammographic tomography,” J. Phys. D Appl. Phys. 47(36), 365401 (2014).
[Crossref]

T. E. Gureyev, Y. I. Nesterets, F. de Hoog, G. Schmalz, S. C. Mayo, S. Mohammadi, and G. Tromba, “Duality between noise and spatial resolution in linear systems,” Opt. Express 22(8), 9087–9094 (2014).
[Crossref] [PubMed]

T. E. Gureyev, Y. I. Nesterets, A. W. Stevenson, P. R. Miller, A. Pogany, and S. W. Wilkins, “Some simple rules for contrast, signal-to-noise and resolution in in-line x-ray phase-contrast imaging,” Opt. Express 16(5), 3223–3241 (2008).
[Crossref] [PubMed]

Y. I. Nesterets, S. W. Wilkins, T. E. Gureyev, A. Pogany, and A. W. Stevenson, “On the optimization of experimental parameters for x-ray in-line phase-contrast imaging,” Rev. Sci. Instrum. 76(9), 093706 (2005).
[Crossref]

Y. I. Nesterets and T. E. Gureyev, in preparation.

Pavlov, K. M.

T. E. Gureyev, S. C. Mayo, Y. I. Nesterets, S. Mohammadi, D. Lockie, R. H. Menk, F. Arfelli, K. M. Pavlov, M. J. Kitchen, F. Zanconati, C. Dullin, and G. Tromba, “Investigation of the imaging quality of synchrotron-based phase-contrast mammographic tomography,” J. Phys. D Appl. Phys. 47(36), 365401 (2014).
[Crossref]

Pogany, A.

T. E. Gureyev, Y. I. Nesterets, A. W. Stevenson, P. R. Miller, A. Pogany, and S. W. Wilkins, “Some simple rules for contrast, signal-to-noise and resolution in in-line x-ray phase-contrast imaging,” Opt. Express 16(5), 3223–3241 (2008).
[Crossref] [PubMed]

Y. I. Nesterets, S. W. Wilkins, T. E. Gureyev, A. Pogany, and A. W. Stevenson, “On the optimization of experimental parameters for x-ray in-line phase-contrast imaging,” Rev. Sci. Instrum. 76(9), 093706 (2005).
[Crossref]

Pope, D.

R. Bach, D. Pope, S.-H. Liou, and H. Batelaan, “Controlled double-slit electron diffraction,” New J. Phys. 15(3), 033018 (2013).
[Crossref]

Pozzi, G.

S. Frabboni, A. Gabrielli, G. Carlo Gazzadi, F. Giorgi, G. Matteucci, G. Pozzi, N. S. Cesari, M. Villa, and A. Zoccoli, “The Young-Feynman two-slits experiment with single electrons: Build-up of the interference pattern and arrival-time distribution using a fast-readout pixel detector,” Ultramicroscopy 116, 73–76 (2012).
[Crossref]

P. G. Merli, G. F. Missiroli, and G. Pozzi, “On the statistical aspect of electron interference phenomena,” Am. J. Phys. 44(3), 306–307 (1976).
[Crossref]

Rose, A.

Saveliev, I. G.

N. Garcia, I. G. Saveliev, and M. Sharonov, “Time-resolved diffraction and interference: Young’s interference with photons of different energy as revealed by time resolution,” Philos. Trans. A Math. Phys. Eng. Sci. 360(1794), 1039–1059 (2002).
[Crossref] [PubMed]

Schmalz, G.

Sharonov, M.

N. Garcia, I. G. Saveliev, and M. Sharonov, “Time-resolved diffraction and interference: Young’s interference with photons of different energy as revealed by time resolution,” Philos. Trans. A Math. Phys. Eng. Sci. 360(1794), 1039–1059 (2002).
[Crossref] [PubMed]

Stevenson, A. W.

T. E. Gureyev, Y. I. Nesterets, A. W. Stevenson, P. R. Miller, A. Pogany, and S. W. Wilkins, “Some simple rules for contrast, signal-to-noise and resolution in in-line x-ray phase-contrast imaging,” Opt. Express 16(5), 3223–3241 (2008).
[Crossref] [PubMed]

Y. I. Nesterets, S. W. Wilkins, T. E. Gureyev, A. Pogany, and A. W. Stevenson, “On the optimization of experimental parameters for x-ray in-line phase-contrast imaging,” Rev. Sci. Instrum. 76(9), 093706 (2005).
[Crossref]

Tromba, G.

T. E. Gureyev, S. C. Mayo, Y. I. Nesterets, S. Mohammadi, D. Lockie, R. H. Menk, F. Arfelli, K. M. Pavlov, M. J. Kitchen, F. Zanconati, C. Dullin, and G. Tromba, “Investigation of the imaging quality of synchrotron-based phase-contrast mammographic tomography,” J. Phys. D Appl. Phys. 47(36), 365401 (2014).
[Crossref]

T. E. Gureyev, Y. I. Nesterets, F. de Hoog, G. Schmalz, S. C. Mayo, S. Mohammadi, and G. Tromba, “Duality between noise and spatial resolution in linear systems,” Opt. Express 22(8), 9087–9094 (2014).
[Crossref] [PubMed]

Tsukernik, A.

T. Juffmann, A. Milic, M. Müllneritsch, P. Asenbaum, A. Tsukernik, J. Tüxen, M. Mayor, O. Cheshnovsky, and M. Arndt, “Real-time single-molecule imaging of quantum interference,” Nat. Nanotechnol. 7(5), 297–300 (2012).
[Crossref] [PubMed]

Tüxen, J.

T. Juffmann, A. Milic, M. Müllneritsch, P. Asenbaum, A. Tsukernik, J. Tüxen, M. Mayor, O. Cheshnovsky, and M. Arndt, “Real-time single-molecule imaging of quantum interference,” Nat. Nanotechnol. 7(5), 297–300 (2012).
[Crossref] [PubMed]

Villa, M.

S. Frabboni, A. Gabrielli, G. Carlo Gazzadi, F. Giorgi, G. Matteucci, G. Pozzi, N. S. Cesari, M. Villa, and A. Zoccoli, “The Young-Feynman two-slits experiment with single electrons: Build-up of the interference pattern and arrival-time distribution using a fast-readout pixel detector,” Ultramicroscopy 116, 73–76 (2012).
[Crossref]

Wheeler, J. A.

J. A. Wheeler, ““No fugitive and cloistered virtue”—A tribute to Niels Bohr,” Phys. Today 16(1), 30–32 (1963).
[Crossref]

Wilkins, S. W.

T. E. Gureyev, Y. I. Nesterets, A. W. Stevenson, P. R. Miller, A. Pogany, and S. W. Wilkins, “Some simple rules for contrast, signal-to-noise and resolution in in-line x-ray phase-contrast imaging,” Opt. Express 16(5), 3223–3241 (2008).
[Crossref] [PubMed]

Y. I. Nesterets, S. W. Wilkins, T. E. Gureyev, A. Pogany, and A. W. Stevenson, “On the optimization of experimental parameters for x-ray in-line phase-contrast imaging,” Rev. Sci. Instrum. 76(9), 093706 (2005).
[Crossref]

Zanconati, F.

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Figures (4)

Fig. 1
Fig. 1 Schematic diagram of the imaging experiment: two slits (a) versus one slit (b).
Fig. 2
Fig. 2 (a) The ideal observer’s SNR, for the total number of photons equal to 10 (black line), and the minimum required number of photons, for the SNR of the ideal observer equal to 5 (blue line), and (b) ratio SN R I 2 /( N tot d/b) , as functions of the ratio of the distance between the slits to their width, d / b.
Fig. 3
Fig. 3 (a) Examples of the signal S1, described by Eq. (8), for two values of the visibility, and the signal S2 described by Eq. (9); (b) the difference, S1S2, for the same values of the visibility as in Fig. 3(a).
Fig. 4
Fig. 4 (a) The minimum number of photons required to achieve SNRI = 5, and (b) the SNR of the ideal observer in the case of the total number of photons equal to 100, as the functions of the interference pattern visibility; (c) dependences of the visibility of the far-field interference pattern and the minimum number of photons required to achieve SNRI = 5, and (d) the ratio SN R I 2 (d/ l sys )/ N tot , as functions of the ratio of the distance between the slits and the effective transverse coherence length of the imaging system.

Equations (18)

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SN R I 2 (b,d)=2 + dx [ S 1 (x;b,d) S 2 (x;b)] 2 S 1 (x;b,d)+ S 2 (x;b) .
T(x;b)={ 1,|x|b/2, 0,|x|>b/2.
S 1 (x;b,d)=F[T(x+d/2;b)+T(xd/2;b)], S 2 (x;b)=2FT(x;b).
SN R I 2 (b,d;F)=2F + dx [T(x+d/2;b)+T(xd/2;b)2T(x;b)] 2 T(x+d/2;b)+T(xd/2;b)+2T(x;b) .
SN R I 2 (b,d;F)=SN R I 2 (d/b; N tot )= N tot { (4/3)[2(d/b)1],1d/b2, 4,2d/b.
SN R I 2 N tot (d/b) = SN R I 2 2Fd 2,
d 2 /λz<<1.
S 1 (x)=2F{1+Vcos[2π(x x 0 )/Δx]} sinc 2 [(b/d)x/Δx],
S 2 (x)=2F sinc 2 [(b/d)x/Δx].
SN R I,1 2 2 0 Δx dx [ S 1 (x) S 2 (x)] 2 S 1 (x)+ S 2 (x) 4 V 2 ΔxF 0 1 d x cos 2 (2π x ) 2+Vcos(2π x ) =8[2 (4 V 2 ) 1/2 1]ΔxF.
SN R I,1 2 (ΔxF) 1 =8[2 (4 V 2 ) 1/2 1]8(2/ 3 1)=1.2376.
SN R I 2 =4 V 2 ΔxF + d x cos 2 [2π( x x 0 /Δx)] 2+Vcos[2π( x x 0 /Δx)] sinc 2 [(b/d) x ] (d/b)SN R I,1 2 .
N tot = + dx S 2 (x) =2F(d/b)Δx.
SN R I 2 =4[2 (4 V 2 ) 1/2 1] N tot ,
N tot = SN R I 2 4[2 (4 V 2 ) 1/2 1] .
V=exp(2 π 2 σ sys 2 /Δ x 2 ),
2π σ sys /Δx=d/ l sys ,
SN R I 2 N tot ( l sys /d) = 2πSN R I 2 N tot (Δx/ σ sys ) 0.2434,

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