Abstract

Lensless coherent diffractive imaging usually requires iterative phase-retrieval for recovering the missing phase information. Holographic techniques, such as Fourier-transform holography (FTH) or holography with extended references (HERALDO), directly provide this phase information and thus allow for a direct non-iterative reconstruction of the sample. In this paper, we analyze the effect of detector noise on the reconstruction for FTH and HERALDO with linear and rectangular references. We find that HERALDO is more sensitive to this type of noise than FTH, especially if rectangular references are employed. This excessive noise, caused by the necessary differentiation step(s) during reconstruction in case of HERALDO, additionally depends on the numerical implementation. When considering both shot-noise and detector noise, we find that FTH provides a better signal-to-noise ratio (SNR) than HERALDO if the available photon flux from the light source is low. In contrast, at high photon flux HERALDO provides better SNR and resolution than FTH. Our findings will help in designing optimum holographic imaging experiments particularly in the photon-flux-limited regime where most ultrafast experiments operate.

© 2016 Optical Society of America

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2016 (1)

2015 (1)

J. Miao, T. Ishikawa, I. K. Robinson, and M. M. Murnane, “Beyond crystallography: diffractive imaging using coherent x-ray light sources,” Science 348, 530–535 (2015).
[Crossref] [PubMed]

2014 (5)

C. von KorffSchmising, B. Pfau, M. Schneider, C. M. Günther, M. Giovannella, J. Perron, B. Vodungbo, L. Müller, F. Capotondi, E. Pedersoli, N. Mahne, J. Lüning, and S. Eisebitt, “Imaging ultrafast demagnetization dynamics after a spatially localized optical excitation,” Phys. Rev. Lett. 112, 217203 (2014).
[Crossref]

J. Rothhardt, S. Hädrich, A. Klenke, S. Demmler, A. Hoffmann, T. Gotschall, T. Eidam, M. Krebs, J. Limpert, and A. Tünnermann, “53 W average power few-cycle fiber laser system generating soft X rays up to the water window,” Opt. Lett. 39(17), 5224–5227 (2014).
[Crossref] [PubMed]

S. Hädrich, A. Klenke, J. Rothhardt, M. Krebs, A. Hoffmann, O. Pronin, V. Pervak, J. Limpert, and A. Tünnermann, “High photon flux table-top coherent extreme-ultraviolet source,” Nat. Photonics 8, 779–783 (2014).
[Crossref]

M. Zürch, J. Rothhardt, S. Hädrich, S. Demmler, M. Krebs, J. Limpert, A. Tünnermann, A. Guggenmos, U. Kleineberg, and C. Spielmann, “Real-time and subwavelength ultrafast coherent diffraction imaging in the extreme ultraviolet,” Sci. Rep. 4, 7356 (2014).
[Crossref]

V. T. Tenner, K. S. E. Eikema, and S. Witte, “Fourier transform holography with extended references using a coherent ultra-broadband light source,” Opt. Express 22(21), 25397–25409 (2014).
[Crossref] [PubMed]

2013 (1)

A. Wirth, R. Santra, and E. Goulielmakis, “Real time tracing of valence-shell electronic coherences with attosecond transient absorption spectroscopy,” Chem. Phys. 414, 149–159 (2013).
[Crossref]

2012 (1)

2011 (1)

2010 (1)

D. Gauthier, M. Guizar-Sicairos, X. Ge, W. Boutu, B. Carré, J. R. Fienup, and H. Merdji, “Single-shot femtosecond X-ray holography using extended references,” Phys. Rev. Lett. 105, 093901 (2010).
[Crossref] [PubMed]

2009 (2)

A. Ravasio, D. Gauthier, F. R. N. C. Maia, M. Billon, J-P. Caumes, D. Garzella, M. Géléoc, O. Gobert, J-F. Hergott, A-M. Pena, H. Perez, B. Carré, E. Bourhis, J. Gierak, A. Madouri, D. Mailly, B. Schiedt, M. Fajardo, J. Gautier, P. Zeitoun, P. H. Bucksbaum, J. Hajdu, and H. Merdji, “Single-shot diffractive imaging with a table-top femtosecond soft x-ray laser-harmonics source,” Phys Rev Lett. 103, 028104 (2009).
[Crossref] [PubMed]

R. L. Sandberg, D. A. Raymondson, C. La-o-vorakiat, A. Paul, K. S. Raines, J. Miao, M. M. Murnane, H. C. Kapteyn, and W. F. Schlotter, “Tabletop soft-x-ray Fourier transform holography with 50 nm resolution,” Opt. Lett. 34(11), 1618–1620 (2009).
[Crossref] [PubMed]

2008 (1)

2007 (2)

2006 (1)

W. F. Schlotter, R. Rick, K. Chen, A. Scherz, J. Stöhr, J. Lüning, S. Eisebitt, Ch. Günther, W. Eberhardt, O. Hellwig, and I. McNulty, “Multiple reference Fourier transform holography with soft X-rays,” Appl. Phys. Lett. 89, 163112 (2006).
[Crossref]

2005 (1)

K. Holldack, T. Kachel, S. Khan, R. Mitzner, and T. Quast, “Characterization of laser-electron interaction at the BESSY II femtoslicing source,” Phys. Rev. Spec. Top-Ac. 8, 040704 (2005).

2004 (1)

H. He, U. Weierstall, J. C. H. Spence, M. Howells, H. A. Padmore, S. Marchesini, and H. N. Chapman, “Use of extended and prepared reference objects in experimental Fourier transform X-ray holography,” Appl. Phys. Lett. 85(13), 2454–2456 (2004).
[Crossref]

2001 (1)

M. Howells, C. Jacobsen, S. Marchesini, S. Miller, J. Spence, and U. Weierstall, “Toward a practical X-ray Fourier holography at high resolution,” Nucl. Instrum. Meth. A 467–468, 864–867 (2001).
[Crossref]

1992 (1)

I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells, and D. P. Kern, “High-resolution imaging by Fourier transform X-ray holography,” Science 256(5059), 1009–1012 (1992).
[Crossref] [PubMed]

1965 (2)

G. W. Stroke, “Lensless Fourier transform method for optical holography,” App. Phys. Lett. 6(10), 201–203 (1965).
[Crossref]

J. Winthrop and C. Worthington, “X-ray microscopy by successive Fourier transformation,” Phys. Lett. 15(2), 124–126 (1965).
[Crossref]

1964 (1)

G. Stroke and D. Falconer, “Attainment of high resolutions in wavefront reconstruction imaging,” Phys. Lett. 13(4), 306–309 (1964).
[Crossref]

Adams, D. E.

Anderson, E. H.

I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells, and D. P. Kern, “High-resolution imaging by Fourier transform X-ray holography,” Science 256(5059), 1009–1012 (1992).
[Crossref] [PubMed]

Barthelmess, M.

Bendat, J. S.

J. S. Bendat and A. G. Piersol, Random Data: Analysis and Measurement Procedures, 4th ed. (Wiley, 2010).
[Crossref]

Billon, M.

A. Ravasio, D. Gauthier, F. R. N. C. Maia, M. Billon, J-P. Caumes, D. Garzella, M. Géléoc, O. Gobert, J-F. Hergott, A-M. Pena, H. Perez, B. Carré, E. Bourhis, J. Gierak, A. Madouri, D. Mailly, B. Schiedt, M. Fajardo, J. Gautier, P. Zeitoun, P. H. Bucksbaum, J. Hajdu, and H. Merdji, “Single-shot diffractive imaging with a table-top femtosecond soft x-ray laser-harmonics source,” Phys Rev Lett. 103, 028104 (2009).
[Crossref] [PubMed]

Bourhis, E.

A. Ravasio, D. Gauthier, F. R. N. C. Maia, M. Billon, J-P. Caumes, D. Garzella, M. Géléoc, O. Gobert, J-F. Hergott, A-M. Pena, H. Perez, B. Carré, E. Bourhis, J. Gierak, A. Madouri, D. Mailly, B. Schiedt, M. Fajardo, J. Gautier, P. Zeitoun, P. H. Bucksbaum, J. Hajdu, and H. Merdji, “Single-shot diffractive imaging with a table-top femtosecond soft x-ray laser-harmonics source,” Phys Rev Lett. 103, 028104 (2009).
[Crossref] [PubMed]

Boutu, W.

W. Boutu, D. Gauthier, X. Ge, R. Cassin, M. Ducousso, A.I. Gonzalez, B. Iwan, J. Samaan, F. Wang, M. Kovačev, and H. Merdji, “Impact of noise in holography with extended references in the low signal regime,” Opt. Express 24(6), 6318–6327 (2016).
[Crossref] [PubMed]

D. Gauthier, M. Guizar-Sicairos, X. Ge, W. Boutu, B. Carré, J. R. Fienup, and H. Merdji, “Single-shot femtosecond X-ray holography using extended references,” Phys. Rev. Lett. 105, 093901 (2010).
[Crossref] [PubMed]

Bucksbaum, P. H.

A. Ravasio, D. Gauthier, F. R. N. C. Maia, M. Billon, J-P. Caumes, D. Garzella, M. Géléoc, O. Gobert, J-F. Hergott, A-M. Pena, H. Perez, B. Carré, E. Bourhis, J. Gierak, A. Madouri, D. Mailly, B. Schiedt, M. Fajardo, J. Gautier, P. Zeitoun, P. H. Bucksbaum, J. Hajdu, and H. Merdji, “Single-shot diffractive imaging with a table-top femtosecond soft x-ray laser-harmonics source,” Phys Rev Lett. 103, 028104 (2009).
[Crossref] [PubMed]

Capotondi, F.

C. von KorffSchmising, B. Pfau, M. Schneider, C. M. Günther, M. Giovannella, J. Perron, B. Vodungbo, L. Müller, F. Capotondi, E. Pedersoli, N. Mahne, J. Lüning, and S. Eisebitt, “Imaging ultrafast demagnetization dynamics after a spatially localized optical excitation,” Phys. Rev. Lett. 112, 217203 (2014).
[Crossref]

F. Capotondi, E. Pedersoli, M. Kiskinova, A.V. Martin, M. Barthelmess, and H. N. Chapman, “A scheme for lensless X-ray microscopy combining coherent diffraction imaging and differential corner holography,” Opt. Express 20(22), 25152–25160 (2012).
[Crossref] [PubMed]

Carré, B.

D. Gauthier, M. Guizar-Sicairos, X. Ge, W. Boutu, B. Carré, J. R. Fienup, and H. Merdji, “Single-shot femtosecond X-ray holography using extended references,” Phys. Rev. Lett. 105, 093901 (2010).
[Crossref] [PubMed]

A. Ravasio, D. Gauthier, F. R. N. C. Maia, M. Billon, J-P. Caumes, D. Garzella, M. Géléoc, O. Gobert, J-F. Hergott, A-M. Pena, H. Perez, B. Carré, E. Bourhis, J. Gierak, A. Madouri, D. Mailly, B. Schiedt, M. Fajardo, J. Gautier, P. Zeitoun, P. H. Bucksbaum, J. Hajdu, and H. Merdji, “Single-shot diffractive imaging with a table-top femtosecond soft x-ray laser-harmonics source,” Phys Rev Lett. 103, 028104 (2009).
[Crossref] [PubMed]

Cassin, R.

Caumes, J-P.

A. Ravasio, D. Gauthier, F. R. N. C. Maia, M. Billon, J-P. Caumes, D. Garzella, M. Géléoc, O. Gobert, J-F. Hergott, A-M. Pena, H. Perez, B. Carré, E. Bourhis, J. Gierak, A. Madouri, D. Mailly, B. Schiedt, M. Fajardo, J. Gautier, P. Zeitoun, P. H. Bucksbaum, J. Hajdu, and H. Merdji, “Single-shot diffractive imaging with a table-top femtosecond soft x-ray laser-harmonics source,” Phys Rev Lett. 103, 028104 (2009).
[Crossref] [PubMed]

Chapman, H. N.

F. Capotondi, E. Pedersoli, M. Kiskinova, A.V. Martin, M. Barthelmess, and H. N. Chapman, “A scheme for lensless X-ray microscopy combining coherent diffraction imaging and differential corner holography,” Opt. Express 20(22), 25152–25160 (2012).
[Crossref] [PubMed]

H. He, U. Weierstall, J. C. H. Spence, M. Howells, H. A. Padmore, S. Marchesini, and H. N. Chapman, “Use of extended and prepared reference objects in experimental Fourier transform X-ray holography,” Appl. Phys. Lett. 85(13), 2454–2456 (2004).
[Crossref]

Chen, C.-C.

Chen, K.

W. F. Schlotter, R. Rick, K. Chen, A. Scherz, J. Stöhr, J. Lüning, S. Eisebitt, Ch. Günther, W. Eberhardt, O. Hellwig, and I. McNulty, “Multiple reference Fourier transform holography with soft X-rays,” Appl. Phys. Lett. 89, 163112 (2006).
[Crossref]

Demmler, S.

M. Zürch, J. Rothhardt, S. Hädrich, S. Demmler, M. Krebs, J. Limpert, A. Tünnermann, A. Guggenmos, U. Kleineberg, and C. Spielmann, “Real-time and subwavelength ultrafast coherent diffraction imaging in the extreme ultraviolet,” Sci. Rep. 4, 7356 (2014).
[Crossref]

J. Rothhardt, S. Hädrich, A. Klenke, S. Demmler, A. Hoffmann, T. Gotschall, T. Eidam, M. Krebs, J. Limpert, and A. Tünnermann, “53 W average power few-cycle fiber laser system generating soft X rays up to the water window,” Opt. Lett. 39(17), 5224–5227 (2014).
[Crossref] [PubMed]

Ducousso, M.

Eberhardt, W.

W. F. Schlotter, R. Rick, K. Chen, A. Scherz, J. Stöhr, J. Lüning, S. Eisebitt, Ch. Günther, W. Eberhardt, O. Hellwig, and I. McNulty, “Multiple reference Fourier transform holography with soft X-rays,” Appl. Phys. Lett. 89, 163112 (2006).
[Crossref]

Eidam, T.

Eikema, K. S. E.

Eisebitt, S.

C. von KorffSchmising, B. Pfau, M. Schneider, C. M. Günther, M. Giovannella, J. Perron, B. Vodungbo, L. Müller, F. Capotondi, E. Pedersoli, N. Mahne, J. Lüning, and S. Eisebitt, “Imaging ultrafast demagnetization dynamics after a spatially localized optical excitation,” Phys. Rev. Lett. 112, 217203 (2014).
[Crossref]

W. F. Schlotter, R. Rick, K. Chen, A. Scherz, J. Stöhr, J. Lüning, S. Eisebitt, Ch. Günther, W. Eberhardt, O. Hellwig, and I. McNulty, “Multiple reference Fourier transform holography with soft X-rays,” Appl. Phys. Lett. 89, 163112 (2006).
[Crossref]

Fajardo, M.

A. Ravasio, D. Gauthier, F. R. N. C. Maia, M. Billon, J-P. Caumes, D. Garzella, M. Géléoc, O. Gobert, J-F. Hergott, A-M. Pena, H. Perez, B. Carré, E. Bourhis, J. Gierak, A. Madouri, D. Mailly, B. Schiedt, M. Fajardo, J. Gautier, P. Zeitoun, P. H. Bucksbaum, J. Hajdu, and H. Merdji, “Single-shot diffractive imaging with a table-top femtosecond soft x-ray laser-harmonics source,” Phys Rev Lett. 103, 028104 (2009).
[Crossref] [PubMed]

Falconer, D.

G. Stroke and D. Falconer, “Attainment of high resolutions in wavefront reconstruction imaging,” Phys. Lett. 13(4), 306–309 (1964).
[Crossref]

Fienup, J. R.

Garzella, D.

A. Ravasio, D. Gauthier, F. R. N. C. Maia, M. Billon, J-P. Caumes, D. Garzella, M. Géléoc, O. Gobert, J-F. Hergott, A-M. Pena, H. Perez, B. Carré, E. Bourhis, J. Gierak, A. Madouri, D. Mailly, B. Schiedt, M. Fajardo, J. Gautier, P. Zeitoun, P. H. Bucksbaum, J. Hajdu, and H. Merdji, “Single-shot diffractive imaging with a table-top femtosecond soft x-ray laser-harmonics source,” Phys Rev Lett. 103, 028104 (2009).
[Crossref] [PubMed]

Gauthier, D.

W. Boutu, D. Gauthier, X. Ge, R. Cassin, M. Ducousso, A.I. Gonzalez, B. Iwan, J. Samaan, F. Wang, M. Kovačev, and H. Merdji, “Impact of noise in holography with extended references in the low signal regime,” Opt. Express 24(6), 6318–6327 (2016).
[Crossref] [PubMed]

D. Gauthier, M. Guizar-Sicairos, X. Ge, W. Boutu, B. Carré, J. R. Fienup, and H. Merdji, “Single-shot femtosecond X-ray holography using extended references,” Phys. Rev. Lett. 105, 093901 (2010).
[Crossref] [PubMed]

A. Ravasio, D. Gauthier, F. R. N. C. Maia, M. Billon, J-P. Caumes, D. Garzella, M. Géléoc, O. Gobert, J-F. Hergott, A-M. Pena, H. Perez, B. Carré, E. Bourhis, J. Gierak, A. Madouri, D. Mailly, B. Schiedt, M. Fajardo, J. Gautier, P. Zeitoun, P. H. Bucksbaum, J. Hajdu, and H. Merdji, “Single-shot diffractive imaging with a table-top femtosecond soft x-ray laser-harmonics source,” Phys Rev Lett. 103, 028104 (2009).
[Crossref] [PubMed]

Gautier, J.

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J. Rothhardt, S. Hädrich, A. Klenke, S. Demmler, A. Hoffmann, T. Gotschall, T. Eidam, M. Krebs, J. Limpert, and A. Tünnermann, “53 W average power few-cycle fiber laser system generating soft X rays up to the water window,” Opt. Lett. 39(17), 5224–5227 (2014).
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I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells, and D. P. Kern, “High-resolution imaging by Fourier transform X-ray holography,” Science 256(5059), 1009–1012 (1992).
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Kern, D. P.

I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells, and D. P. Kern, “High-resolution imaging by Fourier transform X-ray holography,” Science 256(5059), 1009–1012 (1992).
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K. Holldack, T. Kachel, S. Khan, R. Mitzner, and T. Quast, “Characterization of laser-electron interaction at the BESSY II femtoslicing source,” Phys. Rev. Spec. Top-Ac. 8, 040704 (2005).

Kirz, J.

I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells, and D. P. Kern, “High-resolution imaging by Fourier transform X-ray holography,” Science 256(5059), 1009–1012 (1992).
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S. Hädrich, A. Klenke, J. Rothhardt, M. Krebs, A. Hoffmann, O. Pronin, V. Pervak, J. Limpert, and A. Tünnermann, “High photon flux table-top coherent extreme-ultraviolet source,” Nat. Photonics 8, 779–783 (2014).
[Crossref]

J. Rothhardt, S. Hädrich, A. Klenke, S. Demmler, A. Hoffmann, T. Gotschall, T. Eidam, M. Krebs, J. Limpert, and A. Tünnermann, “53 W average power few-cycle fiber laser system generating soft X rays up to the water window,” Opt. Lett. 39(17), 5224–5227 (2014).
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Krebs, M.

J. Rothhardt, S. Hädrich, A. Klenke, S. Demmler, A. Hoffmann, T. Gotschall, T. Eidam, M. Krebs, J. Limpert, and A. Tünnermann, “53 W average power few-cycle fiber laser system generating soft X rays up to the water window,” Opt. Lett. 39(17), 5224–5227 (2014).
[Crossref] [PubMed]

S. Hädrich, A. Klenke, J. Rothhardt, M. Krebs, A. Hoffmann, O. Pronin, V. Pervak, J. Limpert, and A. Tünnermann, “High photon flux table-top coherent extreme-ultraviolet source,” Nat. Photonics 8, 779–783 (2014).
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M. Zürch, J. Rothhardt, S. Hädrich, S. Demmler, M. Krebs, J. Limpert, A. Tünnermann, A. Guggenmos, U. Kleineberg, and C. Spielmann, “Real-time and subwavelength ultrafast coherent diffraction imaging in the extreme ultraviolet,” Sci. Rep. 4, 7356 (2014).
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Limpert, J.

M. Zürch, J. Rothhardt, S. Hädrich, S. Demmler, M. Krebs, J. Limpert, A. Tünnermann, A. Guggenmos, U. Kleineberg, and C. Spielmann, “Real-time and subwavelength ultrafast coherent diffraction imaging in the extreme ultraviolet,” Sci. Rep. 4, 7356 (2014).
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S. Hädrich, A. Klenke, J. Rothhardt, M. Krebs, A. Hoffmann, O. Pronin, V. Pervak, J. Limpert, and A. Tünnermann, “High photon flux table-top coherent extreme-ultraviolet source,” Nat. Photonics 8, 779–783 (2014).
[Crossref]

J. Rothhardt, S. Hädrich, A. Klenke, S. Demmler, A. Hoffmann, T. Gotschall, T. Eidam, M. Krebs, J. Limpert, and A. Tünnermann, “53 W average power few-cycle fiber laser system generating soft X rays up to the water window,” Opt. Lett. 39(17), 5224–5227 (2014).
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Lüning, J.

C. von KorffSchmising, B. Pfau, M. Schneider, C. M. Günther, M. Giovannella, J. Perron, B. Vodungbo, L. Müller, F. Capotondi, E. Pedersoli, N. Mahne, J. Lüning, and S. Eisebitt, “Imaging ultrafast demagnetization dynamics after a spatially localized optical excitation,” Phys. Rev. Lett. 112, 217203 (2014).
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A. Ravasio, D. Gauthier, F. R. N. C. Maia, M. Billon, J-P. Caumes, D. Garzella, M. Géléoc, O. Gobert, J-F. Hergott, A-M. Pena, H. Perez, B. Carré, E. Bourhis, J. Gierak, A. Madouri, D. Mailly, B. Schiedt, M. Fajardo, J. Gautier, P. Zeitoun, P. H. Bucksbaum, J. Hajdu, and H. Merdji, “Single-shot diffractive imaging with a table-top femtosecond soft x-ray laser-harmonics source,” Phys Rev Lett. 103, 028104 (2009).
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A. Ravasio, D. Gauthier, F. R. N. C. Maia, M. Billon, J-P. Caumes, D. Garzella, M. Géléoc, O. Gobert, J-F. Hergott, A-M. Pena, H. Perez, B. Carré, E. Bourhis, J. Gierak, A. Madouri, D. Mailly, B. Schiedt, M. Fajardo, J. Gautier, P. Zeitoun, P. H. Bucksbaum, J. Hajdu, and H. Merdji, “Single-shot diffractive imaging with a table-top femtosecond soft x-ray laser-harmonics source,” Phys Rev Lett. 103, 028104 (2009).
[Crossref] [PubMed]

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H. He, U. Weierstall, J. C. H. Spence, M. Howells, H. A. Padmore, S. Marchesini, and H. N. Chapman, “Use of extended and prepared reference objects in experimental Fourier transform X-ray holography,” Appl. Phys. Lett. 85(13), 2454–2456 (2004).
[Crossref]

M. Howells, C. Jacobsen, S. Marchesini, S. Miller, J. Spence, and U. Weierstall, “Toward a practical X-ray Fourier holography at high resolution,” Nucl. Instrum. Meth. A 467–468, 864–867 (2001).
[Crossref]

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McNulty, I.

W. F. Schlotter, R. Rick, K. Chen, A. Scherz, J. Stöhr, J. Lüning, S. Eisebitt, Ch. Günther, W. Eberhardt, O. Hellwig, and I. McNulty, “Multiple reference Fourier transform holography with soft X-rays,” Appl. Phys. Lett. 89, 163112 (2006).
[Crossref]

I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells, and D. P. Kern, “High-resolution imaging by Fourier transform X-ray holography,” Science 256(5059), 1009–1012 (1992).
[Crossref] [PubMed]

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Merdji, H.

W. Boutu, D. Gauthier, X. Ge, R. Cassin, M. Ducousso, A.I. Gonzalez, B. Iwan, J. Samaan, F. Wang, M. Kovačev, and H. Merdji, “Impact of noise in holography with extended references in the low signal regime,” Opt. Express 24(6), 6318–6327 (2016).
[Crossref] [PubMed]

D. Gauthier, M. Guizar-Sicairos, X. Ge, W. Boutu, B. Carré, J. R. Fienup, and H. Merdji, “Single-shot femtosecond X-ray holography using extended references,” Phys. Rev. Lett. 105, 093901 (2010).
[Crossref] [PubMed]

A. Ravasio, D. Gauthier, F. R. N. C. Maia, M. Billon, J-P. Caumes, D. Garzella, M. Géléoc, O. Gobert, J-F. Hergott, A-M. Pena, H. Perez, B. Carré, E. Bourhis, J. Gierak, A. Madouri, D. Mailly, B. Schiedt, M. Fajardo, J. Gautier, P. Zeitoun, P. H. Bucksbaum, J. Hajdu, and H. Merdji, “Single-shot diffractive imaging with a table-top femtosecond soft x-ray laser-harmonics source,” Phys Rev Lett. 103, 028104 (2009).
[Crossref] [PubMed]

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Miller, S.

M. Howells, C. Jacobsen, S. Marchesini, S. Miller, J. Spence, and U. Weierstall, “Toward a practical X-ray Fourier holography at high resolution,” Nucl. Instrum. Meth. A 467–468, 864–867 (2001).
[Crossref]

Mitzner, R.

K. Holldack, T. Kachel, S. Khan, R. Mitzner, and T. Quast, “Characterization of laser-electron interaction at the BESSY II femtoslicing source,” Phys. Rev. Spec. Top-Ac. 8, 040704 (2005).

Müller, L.

C. von KorffSchmising, B. Pfau, M. Schneider, C. M. Günther, M. Giovannella, J. Perron, B. Vodungbo, L. Müller, F. Capotondi, E. Pedersoli, N. Mahne, J. Lüning, and S. Eisebitt, “Imaging ultrafast demagnetization dynamics after a spatially localized optical excitation,” Phys. Rev. Lett. 112, 217203 (2014).
[Crossref]

Murnane, M. M.

Padmore, H. A.

H. He, U. Weierstall, J. C. H. Spence, M. Howells, H. A. Padmore, S. Marchesini, and H. N. Chapman, “Use of extended and prepared reference objects in experimental Fourier transform X-ray holography,” Appl. Phys. Lett. 85(13), 2454–2456 (2004).
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Pedersoli, E.

C. von KorffSchmising, B. Pfau, M. Schneider, C. M. Günther, M. Giovannella, J. Perron, B. Vodungbo, L. Müller, F. Capotondi, E. Pedersoli, N. Mahne, J. Lüning, and S. Eisebitt, “Imaging ultrafast demagnetization dynamics after a spatially localized optical excitation,” Phys. Rev. Lett. 112, 217203 (2014).
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A. Ravasio, D. Gauthier, F. R. N. C. Maia, M. Billon, J-P. Caumes, D. Garzella, M. Géléoc, O. Gobert, J-F. Hergott, A-M. Pena, H. Perez, B. Carré, E. Bourhis, J. Gierak, A. Madouri, D. Mailly, B. Schiedt, M. Fajardo, J. Gautier, P. Zeitoun, P. H. Bucksbaum, J. Hajdu, and H. Merdji, “Single-shot diffractive imaging with a table-top femtosecond soft x-ray laser-harmonics source,” Phys Rev Lett. 103, 028104 (2009).
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Perez, H.

A. Ravasio, D. Gauthier, F. R. N. C. Maia, M. Billon, J-P. Caumes, D. Garzella, M. Géléoc, O. Gobert, J-F. Hergott, A-M. Pena, H. Perez, B. Carré, E. Bourhis, J. Gierak, A. Madouri, D. Mailly, B. Schiedt, M. Fajardo, J. Gautier, P. Zeitoun, P. H. Bucksbaum, J. Hajdu, and H. Merdji, “Single-shot diffractive imaging with a table-top femtosecond soft x-ray laser-harmonics source,” Phys Rev Lett. 103, 028104 (2009).
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Perron, J.

C. von KorffSchmising, B. Pfau, M. Schneider, C. M. Günther, M. Giovannella, J. Perron, B. Vodungbo, L. Müller, F. Capotondi, E. Pedersoli, N. Mahne, J. Lüning, and S. Eisebitt, “Imaging ultrafast demagnetization dynamics after a spatially localized optical excitation,” Phys. Rev. Lett. 112, 217203 (2014).
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Pervak, V.

S. Hädrich, A. Klenke, J. Rothhardt, M. Krebs, A. Hoffmann, O. Pronin, V. Pervak, J. Limpert, and A. Tünnermann, “High photon flux table-top coherent extreme-ultraviolet source,” Nat. Photonics 8, 779–783 (2014).
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C. von KorffSchmising, B. Pfau, M. Schneider, C. M. Günther, M. Giovannella, J. Perron, B. Vodungbo, L. Müller, F. Capotondi, E. Pedersoli, N. Mahne, J. Lüning, and S. Eisebitt, “Imaging ultrafast demagnetization dynamics after a spatially localized optical excitation,” Phys. Rev. Lett. 112, 217203 (2014).
[Crossref]

Piersol, A. G.

J. S. Bendat and A. G. Piersol, Random Data: Analysis and Measurement Procedures, 4th ed. (Wiley, 2010).
[Crossref]

Podorov, S. G.

Pronin, O.

S. Hädrich, A. Klenke, J. Rothhardt, M. Krebs, A. Hoffmann, O. Pronin, V. Pervak, J. Limpert, and A. Tünnermann, “High photon flux table-top coherent extreme-ultraviolet source,” Nat. Photonics 8, 779–783 (2014).
[Crossref]

Quast, T.

K. Holldack, T. Kachel, S. Khan, R. Mitzner, and T. Quast, “Characterization of laser-electron interaction at the BESSY II femtoslicing source,” Phys. Rev. Spec. Top-Ac. 8, 040704 (2005).

Raines, K. S.

Ravasio, A.

A. Ravasio, D. Gauthier, F. R. N. C. Maia, M. Billon, J-P. Caumes, D. Garzella, M. Géléoc, O. Gobert, J-F. Hergott, A-M. Pena, H. Perez, B. Carré, E. Bourhis, J. Gierak, A. Madouri, D. Mailly, B. Schiedt, M. Fajardo, J. Gautier, P. Zeitoun, P. H. Bucksbaum, J. Hajdu, and H. Merdji, “Single-shot diffractive imaging with a table-top femtosecond soft x-ray laser-harmonics source,” Phys Rev Lett. 103, 028104 (2009).
[Crossref] [PubMed]

Raymondson, D. A.

Rhyzik, I. M.

I. S. Gradstheyn and I. M. Rhyzik, Table of Integrals, Series, and Products, Corrected and Enlarged Edition (Academic Press, 1980), Chap. 0.

Rick, R.

W. F. Schlotter, R. Rick, K. Chen, A. Scherz, J. Stöhr, J. Lüning, S. Eisebitt, Ch. Günther, W. Eberhardt, O. Hellwig, and I. McNulty, “Multiple reference Fourier transform holography with soft X-rays,” Appl. Phys. Lett. 89, 163112 (2006).
[Crossref]

Robinson, I. K.

J. Miao, T. Ishikawa, I. K. Robinson, and M. M. Murnane, “Beyond crystallography: diffractive imaging using coherent x-ray light sources,” Science 348, 530–535 (2015).
[Crossref] [PubMed]

Rong, L.

Rothhardt, J.

M. Zürch, J. Rothhardt, S. Hädrich, S. Demmler, M. Krebs, J. Limpert, A. Tünnermann, A. Guggenmos, U. Kleineberg, and C. Spielmann, “Real-time and subwavelength ultrafast coherent diffraction imaging in the extreme ultraviolet,” Sci. Rep. 4, 7356 (2014).
[Crossref]

S. Hädrich, A. Klenke, J. Rothhardt, M. Krebs, A. Hoffmann, O. Pronin, V. Pervak, J. Limpert, and A. Tünnermann, “High photon flux table-top coherent extreme-ultraviolet source,” Nat. Photonics 8, 779–783 (2014).
[Crossref]

J. Rothhardt, S. Hädrich, A. Klenke, S. Demmler, A. Hoffmann, T. Gotschall, T. Eidam, M. Krebs, J. Limpert, and A. Tünnermann, “53 W average power few-cycle fiber laser system generating soft X rays up to the water window,” Opt. Lett. 39(17), 5224–5227 (2014).
[Crossref] [PubMed]

Samaan, J.

Sandberg, R. L.

Santra, R.

A. Wirth, R. Santra, and E. Goulielmakis, “Real time tracing of valence-shell electronic coherences with attosecond transient absorption spectroscopy,” Chem. Phys. 414, 149–159 (2013).
[Crossref]

Scherz, A.

W. F. Schlotter, R. Rick, K. Chen, A. Scherz, J. Stöhr, J. Lüning, S. Eisebitt, Ch. Günther, W. Eberhardt, O. Hellwig, and I. McNulty, “Multiple reference Fourier transform holography with soft X-rays,” Appl. Phys. Lett. 89, 163112 (2006).
[Crossref]

Schiedt, B.

A. Ravasio, D. Gauthier, F. R. N. C. Maia, M. Billon, J-P. Caumes, D. Garzella, M. Géléoc, O. Gobert, J-F. Hergott, A-M. Pena, H. Perez, B. Carré, E. Bourhis, J. Gierak, A. Madouri, D. Mailly, B. Schiedt, M. Fajardo, J. Gautier, P. Zeitoun, P. H. Bucksbaum, J. Hajdu, and H. Merdji, “Single-shot diffractive imaging with a table-top femtosecond soft x-ray laser-harmonics source,” Phys Rev Lett. 103, 028104 (2009).
[Crossref] [PubMed]

Schlotter, W. F.

Schneider, M.

C. von KorffSchmising, B. Pfau, M. Schneider, C. M. Günther, M. Giovannella, J. Perron, B. Vodungbo, L. Müller, F. Capotondi, E. Pedersoli, N. Mahne, J. Lüning, and S. Eisebitt, “Imaging ultrafast demagnetization dynamics after a spatially localized optical excitation,” Phys. Rev. Lett. 112, 217203 (2014).
[Crossref]

Seaberg, M. D.

Spence, J.

M. Howells, C. Jacobsen, S. Marchesini, S. Miller, J. Spence, and U. Weierstall, “Toward a practical X-ray Fourier holography at high resolution,” Nucl. Instrum. Meth. A 467–468, 864–867 (2001).
[Crossref]

Spence, J. C. H.

H. He, U. Weierstall, J. C. H. Spence, M. Howells, H. A. Padmore, S. Marchesini, and H. N. Chapman, “Use of extended and prepared reference objects in experimental Fourier transform X-ray holography,” Appl. Phys. Lett. 85(13), 2454–2456 (2004).
[Crossref]

Spielmann, C.

M. Zürch, J. Rothhardt, S. Hädrich, S. Demmler, M. Krebs, J. Limpert, A. Tünnermann, A. Guggenmos, U. Kleineberg, and C. Spielmann, “Real-time and subwavelength ultrafast coherent diffraction imaging in the extreme ultraviolet,” Sci. Rep. 4, 7356 (2014).
[Crossref]

Stöhr, J.

W. F. Schlotter, R. Rick, K. Chen, A. Scherz, J. Stöhr, J. Lüning, S. Eisebitt, Ch. Günther, W. Eberhardt, O. Hellwig, and I. McNulty, “Multiple reference Fourier transform holography with soft X-rays,” Appl. Phys. Lett. 89, 163112 (2006).
[Crossref]

Stroke, G.

G. Stroke and D. Falconer, “Attainment of high resolutions in wavefront reconstruction imaging,” Phys. Lett. 13(4), 306–309 (1964).
[Crossref]

Stroke, G. W.

G. W. Stroke, “Lensless Fourier transform method for optical holography,” App. Phys. Lett. 6(10), 201–203 (1965).
[Crossref]

Tenner, V. T.

Townsend, E. L.

Tünnermann, A.

M. Zürch, J. Rothhardt, S. Hädrich, S. Demmler, M. Krebs, J. Limpert, A. Tünnermann, A. Guggenmos, U. Kleineberg, and C. Spielmann, “Real-time and subwavelength ultrafast coherent diffraction imaging in the extreme ultraviolet,” Sci. Rep. 4, 7356 (2014).
[Crossref]

S. Hädrich, A. Klenke, J. Rothhardt, M. Krebs, A. Hoffmann, O. Pronin, V. Pervak, J. Limpert, and A. Tünnermann, “High photon flux table-top coherent extreme-ultraviolet source,” Nat. Photonics 8, 779–783 (2014).
[Crossref]

J. Rothhardt, S. Hädrich, A. Klenke, S. Demmler, A. Hoffmann, T. Gotschall, T. Eidam, M. Krebs, J. Limpert, and A. Tünnermann, “53 W average power few-cycle fiber laser system generating soft X rays up to the water window,” Opt. Lett. 39(17), 5224–5227 (2014).
[Crossref] [PubMed]

Vodungbo, B.

C. von KorffSchmising, B. Pfau, M. Schneider, C. M. Günther, M. Giovannella, J. Perron, B. Vodungbo, L. Müller, F. Capotondi, E. Pedersoli, N. Mahne, J. Lüning, and S. Eisebitt, “Imaging ultrafast demagnetization dynamics after a spatially localized optical excitation,” Phys. Rev. Lett. 112, 217203 (2014).
[Crossref]

von KorffSchmising, C.

C. von KorffSchmising, B. Pfau, M. Schneider, C. M. Günther, M. Giovannella, J. Perron, B. Vodungbo, L. Müller, F. Capotondi, E. Pedersoli, N. Mahne, J. Lüning, and S. Eisebitt, “Imaging ultrafast demagnetization dynamics after a spatially localized optical excitation,” Phys. Rev. Lett. 112, 217203 (2014).
[Crossref]

Wang, F.

Weierstall, U.

H. He, U. Weierstall, J. C. H. Spence, M. Howells, H. A. Padmore, S. Marchesini, and H. N. Chapman, “Use of extended and prepared reference objects in experimental Fourier transform X-ray holography,” Appl. Phys. Lett. 85(13), 2454–2456 (2004).
[Crossref]

M. Howells, C. Jacobsen, S. Marchesini, S. Miller, J. Spence, and U. Weierstall, “Toward a practical X-ray Fourier holography at high resolution,” Nucl. Instrum. Meth. A 467–468, 864–867 (2001).
[Crossref]

Winthrop, J.

J. Winthrop and C. Worthington, “X-ray microscopy by successive Fourier transformation,” Phys. Lett. 15(2), 124–126 (1965).
[Crossref]

Wirth, A.

A. Wirth, R. Santra, and E. Goulielmakis, “Real time tracing of valence-shell electronic coherences with attosecond transient absorption spectroscopy,” Chem. Phys. 414, 149–159 (2013).
[Crossref]

Witte, S.

Worthington, C.

J. Winthrop and C. Worthington, “X-ray microscopy by successive Fourier transformation,” Phys. Lett. 15(2), 124–126 (1965).
[Crossref]

Zeitoun, P.

A. Ravasio, D. Gauthier, F. R. N. C. Maia, M. Billon, J-P. Caumes, D. Garzella, M. Géléoc, O. Gobert, J-F. Hergott, A-M. Pena, H. Perez, B. Carré, E. Bourhis, J. Gierak, A. Madouri, D. Mailly, B. Schiedt, M. Fajardo, J. Gautier, P. Zeitoun, P. H. Bucksbaum, J. Hajdu, and H. Merdji, “Single-shot diffractive imaging with a table-top femtosecond soft x-ray laser-harmonics source,” Phys Rev Lett. 103, 028104 (2009).
[Crossref] [PubMed]

Zürch, M.

M. Zürch, J. Rothhardt, S. Hädrich, S. Demmler, M. Krebs, J. Limpert, A. Tünnermann, A. Guggenmos, U. Kleineberg, and C. Spielmann, “Real-time and subwavelength ultrafast coherent diffraction imaging in the extreme ultraviolet,” Sci. Rep. 4, 7356 (2014).
[Crossref]

App. Phys. Lett. (1)

G. W. Stroke, “Lensless Fourier transform method for optical holography,” App. Phys. Lett. 6(10), 201–203 (1965).
[Crossref]

Appl. Phys. Lett. (2)

W. F. Schlotter, R. Rick, K. Chen, A. Scherz, J. Stöhr, J. Lüning, S. Eisebitt, Ch. Günther, W. Eberhardt, O. Hellwig, and I. McNulty, “Multiple reference Fourier transform holography with soft X-rays,” Appl. Phys. Lett. 89, 163112 (2006).
[Crossref]

H. He, U. Weierstall, J. C. H. Spence, M. Howells, H. A. Padmore, S. Marchesini, and H. N. Chapman, “Use of extended and prepared reference objects in experimental Fourier transform X-ray holography,” Appl. Phys. Lett. 85(13), 2454–2456 (2004).
[Crossref]

Chem. Phys. (1)

A. Wirth, R. Santra, and E. Goulielmakis, “Real time tracing of valence-shell electronic coherences with attosecond transient absorption spectroscopy,” Chem. Phys. 414, 149–159 (2013).
[Crossref]

Nat. Photonics (1)

S. Hädrich, A. Klenke, J. Rothhardt, M. Krebs, A. Hoffmann, O. Pronin, V. Pervak, J. Limpert, and A. Tünnermann, “High photon flux table-top coherent extreme-ultraviolet source,” Nat. Photonics 8, 779–783 (2014).
[Crossref]

Nucl. Instrum. Meth. A (1)

M. Howells, C. Jacobsen, S. Marchesini, S. Miller, J. Spence, and U. Weierstall, “Toward a practical X-ray Fourier holography at high resolution,” Nucl. Instrum. Meth. A 467–468, 864–867 (2001).
[Crossref]

Opt. Express (6)

S. G. Podorov, K. M. Pavlov, and D. M. Paganin, “A non-iterative reconstruction method for direct and unambiguous coherent diffractive imaging,” Opt. Express 15(16), 9954–9962 (2007).
[Crossref] [PubMed]

M. Guizar-Sicairos and J. R. Fienup, “Holography with extended reference by autocorrelation linear differential operation,” Opt. Express 15(26), 17592–17612 (2007).
[Crossref] [PubMed]

M. D. Seaberg, D. E. Adams, E. L. Townsend, D. A. Raymondson, W. F. Schlotter, Y. Liu, C. S. Menoni, L. Rong, C.-C. Chen, J. Miao, H. C. Kapteyn, and M. M. Murnane, “Ultrahigh 22 nm resolution coherent diffractive imaging using a desktop 13 nm high harmonic source,” Opt. Express 19(23), 22470–22479 (2011).
[Crossref] [PubMed]

F. Capotondi, E. Pedersoli, M. Kiskinova, A.V. Martin, M. Barthelmess, and H. N. Chapman, “A scheme for lensless X-ray microscopy combining coherent diffraction imaging and differential corner holography,” Opt. Express 20(22), 25152–25160 (2012).
[Crossref] [PubMed]

W. Boutu, D. Gauthier, X. Ge, R. Cassin, M. Ducousso, A.I. Gonzalez, B. Iwan, J. Samaan, F. Wang, M. Kovačev, and H. Merdji, “Impact of noise in holography with extended references in the low signal regime,” Opt. Express 24(6), 6318–6327 (2016).
[Crossref] [PubMed]

V. T. Tenner, K. S. E. Eikema, and S. Witte, “Fourier transform holography with extended references using a coherent ultra-broadband light source,” Opt. Express 22(21), 25397–25409 (2014).
[Crossref] [PubMed]

Opt. Lett. (3)

Phys Rev Lett. (1)

A. Ravasio, D. Gauthier, F. R. N. C. Maia, M. Billon, J-P. Caumes, D. Garzella, M. Géléoc, O. Gobert, J-F. Hergott, A-M. Pena, H. Perez, B. Carré, E. Bourhis, J. Gierak, A. Madouri, D. Mailly, B. Schiedt, M. Fajardo, J. Gautier, P. Zeitoun, P. H. Bucksbaum, J. Hajdu, and H. Merdji, “Single-shot diffractive imaging with a table-top femtosecond soft x-ray laser-harmonics source,” Phys Rev Lett. 103, 028104 (2009).
[Crossref] [PubMed]

Phys. Lett. (2)

G. Stroke and D. Falconer, “Attainment of high resolutions in wavefront reconstruction imaging,” Phys. Lett. 13(4), 306–309 (1964).
[Crossref]

J. Winthrop and C. Worthington, “X-ray microscopy by successive Fourier transformation,” Phys. Lett. 15(2), 124–126 (1965).
[Crossref]

Phys. Rev. Lett. (2)

D. Gauthier, M. Guizar-Sicairos, X. Ge, W. Boutu, B. Carré, J. R. Fienup, and H. Merdji, “Single-shot femtosecond X-ray holography using extended references,” Phys. Rev. Lett. 105, 093901 (2010).
[Crossref] [PubMed]

C. von KorffSchmising, B. Pfau, M. Schneider, C. M. Günther, M. Giovannella, J. Perron, B. Vodungbo, L. Müller, F. Capotondi, E. Pedersoli, N. Mahne, J. Lüning, and S. Eisebitt, “Imaging ultrafast demagnetization dynamics after a spatially localized optical excitation,” Phys. Rev. Lett. 112, 217203 (2014).
[Crossref]

Phys. Rev. Spec. Top-Ac. (1)

K. Holldack, T. Kachel, S. Khan, R. Mitzner, and T. Quast, “Characterization of laser-electron interaction at the BESSY II femtoslicing source,” Phys. Rev. Spec. Top-Ac. 8, 040704 (2005).

Sci. Rep. (1)

M. Zürch, J. Rothhardt, S. Hädrich, S. Demmler, M. Krebs, J. Limpert, A. Tünnermann, A. Guggenmos, U. Kleineberg, and C. Spielmann, “Real-time and subwavelength ultrafast coherent diffraction imaging in the extreme ultraviolet,” Sci. Rep. 4, 7356 (2014).
[Crossref]

Science (2)

J. Miao, T. Ishikawa, I. K. Robinson, and M. M. Murnane, “Beyond crystallography: diffractive imaging using coherent x-ray light sources,” Science 348, 530–535 (2015).
[Crossref] [PubMed]

I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells, and D. P. Kern, “High-resolution imaging by Fourier transform X-ray holography,” Science 256(5059), 1009–1012 (1992).
[Crossref] [PubMed]

Other (5)

M. Guizar-Sicairos, “Methods for Coherent Lensless Imaging and X-Ray Wavefront Measurement,” Ph.D. dissertation, University of Rochester (2010).

J. R. Janesick, Scientific Charge-Coupled Devices (SPIE Press, 2001).
[Crossref]

J. S. Bendat and A. G. Piersol, Random Data: Analysis and Measurement Procedures, 4th ed. (Wiley, 2010).
[Crossref]

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, 1965).

I. S. Gradstheyn and I. M. Rhyzik, Table of Integrals, Series, and Products, Corrected and Enlarged Edition (Academic Press, 1980), Chap. 0.

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Figures (4)

Fig. 1
Fig. 1 Numerical comparison of the response of FTH and HERALDO to a uniform noise in the detector plane. The first column shows the object for (a) FTH (d) HERALDO with line reference and (g) HERALDO with rectangular reference. The second column shows the diffraction patterns, after adding noise, from the corresponding objects. The corresponding reconstructions are shown in the third column.

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Fig. 2
Fig. 2 Reconstruction of HERALDO sample by differentiation of the autocorrelation. (a) Reconstruction from the diffraction pattern in Fig. 1(e). (b) Reconstruction from the diffraction pattern in Fig. 1(h). A significant improvement of the SNR can be seen when comparing Fig. 2(b) with Fig. 1(i).

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Fig. 3
Fig. 3 The SNR as a function of (a) variance and (b) the incoming intensity for FTH (Eq. (20)), HERALDO line reference with polynomial multiplication (Eq. (21)), and HERALDO line reference with approximate derivative of the autocorrelation (Eq. (18)).

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Fig. 4
Fig. 4 The evolution of the resolution and SNR as (a) the pinhole radius in FTH, (b) the slit length in HERALDO line reference, and (c) the length of the rectangle in HERALDO rectangular reference are increased. In (a)–(c), the vertical and horizontal resolutions are denoted by the × and + markers, respectively (a∗ marker is actually an overlapping + and ×). (d) the threshold resolution as a function of the incoming field amplitude ( I).

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Tables (1)

Tables Icon

Table 1 Comparison of the mean and variance of noise in the reconstruction in the case of FTH and HERALDO line reference.

View Table

Equations (37)

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

I [ j , k ] = I 0 [ j , k ] + η [ j , k ]
η [ j , k ] = η 0 [ j , k ] + η m
R [ m , n ] = IFFT [ I [ j , k ] ] = IFFT [ I 0 [ j , k ] + η [ j , k ] ] = R 0 [ m , n ] + 1 M 2 ∑ j k M − 1 η [ j , k ] exp ( 2 π i M ( j m + k n ) )
R [ m , n ] ⇒ exp ( − i 2 π M ( j c n + k c m ) ) R [ m , n ] = R 0 [ m , n ] + 1 M 2 ∑ j k M − 1 η [ j , k ] exp ( 2 π i M ( J m + K n ) )
ℜ [ R [ m , n ] ] = ℜ [ R 0 [ m , n ] ] + 1 M 2 ∑ j k M − 1 η [ j , k ] cos ( 2 π M ( J m + K n ) ) = ℜ [ R 0 [ m , n ] ] + ℜ [ E [ m , n ] ]
C A B = 〈 A B 〉 − 〈 A 〉 〈 B 〉
〈 ℜ [ E [ m , n ] ] 〉 = η m M 2 ∑ j k M − 1 cos ( 2 π M ( J m + K n ) ) = 0
〈 ℜ [ E [ m , n ] ] 2 〉 = 〈 ( ∑ j k M − 1 η [ j , k ] M 2 cos ( 2 π M ( J m + K n ) ) ) 2 〉 = σ I 2 M 4 ∑ j k M − 1 cos 2 ( 2 π M ( J m + K n ) )
σ ℜ R 2 = σ ℜ E 2 = 〈 ℜ [ E [ m , n ] ] 2 〉 − 〈 ℜ [ E [ m , n ] ] 〉 2 = σ I 2 2 M 2
R [ m , n ] = IFFT [ i Δ K I [ j , k ] ] = R 0 [ m , n ] + i Δ M 2 ∑ j k M − 1 K η [ j , k ] exp ( 2 π i M ( J m + K n ) )
ℜ [ R [ m , n ] ] = ℜ [ R 0 [ m , n ] ] − Δ M 2 ∑ j k M − 1 K η [ j , k ] sin ( 2 π i M ( J m + K n ) )
σ ℜ R 2 = 〈 ℜ [ E [ m , n ] ] 2 〉 − 〈 ℜ [ E [ m , n ] ] 〉 2 = π 2 σ I 2 6 M 4 ( M 2 − 1 ) ≈ π 2 σ I 2 6 M 2
A [ m , n ] = IFFT [ I [ j , k ] ] = IFFT [ I 0 [ j , k ] + η [ j , k ] ] = A 0 [ m , n ] + 1 M 2 ∑ j k M − 1 η [ j , k ] exp ( 2 π i M ( J m + K n ) )
R [ m , n ] = A [ m , n + 1 ] − A [ m , n ]
σ ℜ R 2 = σ I 2 2 M 2 + σ I 2 2 M 2 = σ I 2 M 2
I [ j , k ] = I 0 [ j , k ] + η [ j , k ] + ϒ [ j , k ]
〈 | R [ m , n ] | 2 〉 = 〈 ℜ [ R [ m , n ] ] 2 〉 + 〈 ℑ [ R [ m , n ] ] 2 〉 = ( σ ℜ R 2 + 〈 ℜ [ R [ m , n ] ] 〉 2 ) + ( σ ℑ R 2 + 〈 ℑ [ R [ m , n ] ] 〉 2 )
σ ℜ R 2 + σ ℑ R 2 = 1 M 2 ( 2 σ l 2 + ∑ m n M − 1 | O 0 [ m , n + 1 ] − O 0 [ m , n ] | 2 )
S N R = ∑ ( P Q ) | S [ m , n ] | 2 ∑ ( P Q ) ( σ ℜ R 2 + σ ℑ R 2 ) = M 2 P Q ∑ ( P Q ) | S [ m , n ] | 2 2 σ l 2 + ∑ m n M − 1 | O 0 [ m , n + 1 ] − O 0 [ m , n ] | 2
2 σ l 2 → σ l 2
| O 0 [ m , n + 1 ] − O 0 [ m , n ] | 2 → | O 0 [ m , n ] | 2
S N R = M 2 P Q ∑ ( P Q ) | S [ m , n ] | 2 2 σ l 2 + ∑ m n M − 1 | O 0 [ m , n ] | 2
S N R = M 2 P Q ∑ ( P Q ) | S [ m , n ] | 2 ( π 2 2 σ I 2 ) / 3 + ∑ m n M − 1 | O 0 [ m , n + 1 ] − O 0 [ m , n ] | 2
∑ k = 0 M − 1 exp ( i 2 π n M ( k − k c ) )
∑ k = 0 M − 1 exp ( i 2 π n M ( k − k c ) ) = exp ( − i 2 π n M k c ) M ( exp ( 2 π n ) − 1 ) exp ( 2 π n / M ) − 1 = 0
〈 ℜ [ E [ m , n ] ] 〉 = − η m Δ M 2 ∑ j k M − 1 K sin ( 2 π M ( J m + K n ) ) = 0
〈 ℜ [ E [ m , n ] ] 2 〉 = 〈 ( − Δ M 2 ∑ j k M − 1 K η [ j , k ] sin ( 2 π M ( J m + K n ) ) ) 2 〉 = − Δ 2 σ I 2 2 M 4 ∑ j k M − 1 K 2
∑ k = 1 M k 2 = 1 6 M ( M + 1 ) ( 2 M + 1 )
∑ k = 1 M k = 1 2 M ( M + 1 )
R [ m , n ] = R 0 [ m , n ] + 1 M 2 ∑ j k M − 1 ∑ p = 1 ∞ η [ j , k ] exp ( 2 π i M ( K n + J m ) ) ( 2 π i K / M ) p p !
R 0 [ m , n ] = 1 M 2 ∑ j k M − 1 ∑ p = 1 ∞ I 0 [ j , k ] exp ( 2 π i M ( K n + J m ) ) ( 2 π i K / M ) p p !
exp ( 2 π i M ( K n + K + J m ) ) − exp ( 2 π i M ( K n + J m ) ) = exp ( 2 π i M ( K n + J m ) ) ∑ p = 1 ∞ ( 2 π i K / M ) p p !
R 0 [ m , n ] ≈ 1 M 2 2 π i M ∑ j k M − 1 K I 0 [ j , k ] exp ( 2 π i M ( K n + J m ) )
σ ℜ R 2 + σ ℑ R 2 = σ ℜ R 2 ( η ) + σ ℜ R 2 ( ϒ ) + σ ℑ R 2 ( η ) + σ ℑ R 2 ( ϒ )
σ ℜ R 2 ( ϒ ) + σ ℑ R 2 ( ϒ ) = Δ 2 ∑ j k M − 1 K 2 I 0 [ j , k ] = Δ 2 ∑ j k M − 1 | K U 0 [ j , k ] | 2
σ ℜ R 2 ( ϒ ) + σ ℑ R 2 ( ϒ ) = 1 M 2 ∑ m n M − 1 | O 0 [ m , n + 1 ] − O 0 [ m , n ] | 2
σ ℜ R 2 + σ ℑ R 2 = 1 M 2 ( 2 σ l 2 + ∑ m n M − 1 | O 0 [ m , n + 1 ] − O 0 [ m , n ] | 2 )

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