Abstract

The Talbot effect, also referred to as self-imaging or lensless imaging, is of the phenomena manifested by a periodic repetition of planar field distributions in certain types of wave fields. This phenomenon is finding applications not only in optics, but also in a variety of research fields, such as acoustics, electron microscopy, plasmonics, x ray, and Bose–Einstein condensates. In optics, self-imaging is being explored particularly in image processing, in the production of spatial-frequency filters, and in optical metrology. In this article, we give an overview of recent advances on the effect from classical optics to nonlinear optics and quantum optics. Throughout this review article there is an effort to clearly present the physical aspects of the self-imaging phenomenon. Mathematical formulations are reduced to the indispensable ones. Readers who prefer strict mathematical treatments should resort to the extensive list of references. Despite the rapid progress on the subject, new ideas and applications of Talbot self-imaging are still expected in the future.

© 2013 Optical Society of America

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2012 (4)

K. Hornberger, S. Gerlich, P. Haslinger, S. Nimmrichter, and M. Arndt, “Colloquium: quantum interference of clusters and molecules,” Rev. Mod. Phys. 84, 157–173 (2012).
[CrossRef]

H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, “PT-symmetric Talbot effects,” Phys. Rev. Lett. 109, 033902 (2012).
[CrossRef]

Z. Chen, D. Liu, Y. Zhang, J. M. Wen, S. N. Zhu, and M. Xiao, “Fractional second-harmonic Talbot effect,” Opt. Lett. 37, 689–691 (2012).
[CrossRef]

D. Liu, Y. Zhang, Z. Chen, J.-M. Wen, and M. Xiao, “Acousto-optic tunable second-harmonic Talbot effect based on periodically poled LiNbO3 crystals,” J. Opt. Soc. Am. B 29, 3325–3329 (2012).
[CrossRef]

2011 (7)

J.-M. Wen, Y. Zhang, S. N. Zhu, and M. Xiao, “Theory of nonlinear Talbot effect,” J. Opt. Soc. Am. B 28, 275–280 (2011).
[CrossRef]

K.-H. Luo, J.-M. Wen, X.-H. Chen, Q. Liu, M. Xiao, and L.-A. Wu, “Erratum: Second-order Talbot effect with entangled photon pairs,” Phys. Rev. A 83, 029902 (2011).
[CrossRef]

V. Torres-Company, J. Lancis, H. Lajunen, and A. T. Friberg, “Coherence revivals in two-photon frequency combs,” Phys. Rev. A 84, 033830 (2011).
[CrossRef]

E. Poem and Y. Silberberg, “Photon correlations in multimode waveguides,” Phys. Rev. A 84, 041805(R) (2011).
[CrossRef]

X.-B. Song, H.-B. Wang, J. Xiong, K. Wang, X. Zhang, K.-H. Luo, and L.-A. Wu, “Experimental observation of quantum Talbot effects,” Phys. Rev. Lett. 107, 033902 (2011).
[CrossRef]

J.-M. Wen, S. Du, H. Chen, and M. Xiao, “Electromagnetically induced Talbot effect,” Appl. Phys. Lett. 98, 081108 (2011).
[CrossRef]

W. Zhang, X. Huang, and Z. Lu, “Super Talbot effect in indefinite metamaterial,” Opt. Express 19, 15297–15303 (2011).
[CrossRef]

2010 (6)

Y. Wang, K. Zhou, X. Zhang, K. Yang, Y. Wang, Y. Song, and S. Liu, “Discrete plasmonic Talbot effect in subwavelength metal waveguide arrays,” Opt. Lett. 35, 685–687 (2010).
[CrossRef]

B. Erkman and J. H. Shapiro, “Ghost imaging: from quantum to classical to computational,” Adv. Opt. Photon. 2, 405–450 (2010).
[CrossRef]

K.-H. Luo, X.-H. Chen, Q. Liu, and L.-A. Wu, “Nonlocal Talbot self-imaging with incoherent light,” Phys. Rev. A 82, 033803 (2010).
[CrossRef]

X.-B. Song, J. Xiong, X. Zhang, and K. Wang, “Second-order Talbot self-imaging with pseudothermal light,” Phys. Rev. A 82, 033823 (2010).
[CrossRef]

C. H. R. Ooi and B. L. Lan, “Intense nonclassical light: controllable two-photon Talbot effect,” Phys. Rev. A 81, 063832 (2010).
[CrossRef]

Y. Zhang, J.-M. Wen, S. N. Zhu, and M. Xiao, “Nonlinear Talbot effect,” Phys. Rev. Lett. 104, 183901 (2010).
[CrossRef]

2009 (5)

B. J. McMorran and A. D. Cronin, “An electron Talbot interferometer,” New J. Phys. 11, 033021 (2009).
[CrossRef]

K.-H. Luo, J.-M. Wen, X.-H. Chen, Q. Liu, M. Xiao, and L.-A. Wu, “Second-order Talbot effect with entangled photon pairs,” Phys. Rev. A 80, 043820 (2009).
[CrossRef]

A. Maradudin and T. Leskova, “The Talbot effect for a surface plasmon polariton,” New J. Phys. 11, 033004 (2009).
[CrossRef]

W. Zhang, C. Zhao, J. Wang, and J. Zhang, “An experimental study of the plasmonic Talbot effect,” Opt. Express 17, 19757–19762(2009).
[CrossRef]

S. Cherukulappurath, D. Heinis, J. Cesario, N. F. van Hulst, S. Enoch, and R. Quidant, “Local observation of plasmon focusing in Talbot carpets,” Opt. Express 17, 23772–23784 (2009).
[CrossRef]

2008 (5)

G. Niconoff, J. Sanchez-Gil, H. Sanchez, and A. Leija, “Self-imaging and caustics in two-dimensional surface plasmon optics,” Opt. Commun. 281, 2316–2320 (2008).
[CrossRef]

L.-W. Zhu, X. Yin, Z.-P. Hong, and C.-S. Guo, “Reciprocal vector theory for diffractive self-imaging,” J. Opt. Soc. Am. A 25, 203–210(2008).
[CrossRef]

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Bronnimann, C. Grunzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
[CrossRef]

P. Peier, S. Pilz, F. Muller, K. A. Nelson, and T. Feurer, “Analysis of phase contrast imaging of terahertz phonon-polaritons,” J. Opt. Soc. Am. B 25, B70–B75 (2008).
[CrossRef]

W. Yashiro, Y. Takeda, and A. Momose, “Efficiency of capturing a phase image using cone-beam x-ray Talbot interferometry,” J. Opt. Soc. Am. A 25, 2025–2039 (2008).
[CrossRef]

2007 (4)

2006 (5)

2005 (6)

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13, 6296–6304 (2005).
[CrossRef]

T. Weitkamp, B. Nohammer, A. Diaz, C. David, and E. Ziegler, “X-ray wavefront analysis and optics characterization with a grating interferometer,” Appl. Phys. Lett. 86, 054101 (2005).
[CrossRef]

R. Iwanow, D. A. May-Arrioja, D. N. Christodoulides, G. I. Stegeman, Y. Min, and W. Sohler, “Discrete Talbot effect in waveguide arrays,” Phys. Rev. Lett. 95, 053902 (2005).
[CrossRef]

C. J. Corcoran and F. Durville, “Experimental demonstration of a phase-locked laser array using a self-Fourier cavity,” Appl. Phys. Lett. 86, 201118 (2005).
[CrossRef]

J. Azana, “Spectral Talbot phenomena of frequency combs induced by cross-phase modulation in optical fibers,” Opt. Lett. 30, 227–229(2005).
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J. A. Bolger, P. Hu, J. T. Mok, J. L. Blows, and B. J. Eggleton, “Talbot self-imaging and cross-phase modulation for generation of tunable high repetition rate pulse trains,” Opt. Commun. 249, 431–439 (2005).
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2004 (1)

R. W. Robinett, “Quantum wave packet revivals,” Phys. Rep. 392, 1–119 (2004).
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2003 (1)

A. Momose, “Demonstration of x-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42, L866–L868 (2003).
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2002 (4)

C. David, B. Nohammer, H. H. Solak, and E. Ziegler, “Differential x-ray phase contrast imaging using a shearing interferometer,” Appl. Phys. Lett. 81, 3287–3289 (2002).
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P. Xi, C. Zhou, E. Dai, and L. Liu, “Generation of near-field hexagonal array illumination with a phase grating,” Opt. Lett. 27, 228–230 (2002).
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B. Brezger, L. Hackermuller, S. Uttenthaler, J. Petschinka, M. Arndt, and A. Zeilinger, “Matter-wave interferometer for large molecules,” Phys. Rev. Lett. 88, 100404 (2002).
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H. Mack, M. Bienert, F. Haug, M. Freyberger, and W. P. Schleich, “Wave packets can factorize numbers,” Phys. Status Solidi B 233, 408–415 (2002).
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2001 (3)

M. V. Berry, I. Marzoli, and W. P. Schleich, “Quantum carpets, carpets of light,” Phys. World 14(6), 39–44 (2001).

J. Azana and M. A. Muriel, “Temporal self-imaging effects: theory and application for multiplying pulse repetition rates,” IEEE J. Quantum Electron. 7, 728–744 (2001).
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M. D’Angelo, M. V. Chekhova, and Y. H. Shih, “Two-photon diffraction and quantum lithography,” Phys. Rev. Lett. 87, 013602 (2001).
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2000 (2)

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733–2736 (2000).
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D. Wojcik, I. Bialynicki-Birula, and K. Zyczkowski, “Time evolution of quantum fractals,” Phys. Rev. Lett. 85, 5022–5025 (2000).
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1999 (4)

1998 (2)

F. Mitschke and U. Morgner, “The temporal Talbot effect,” Opt. Photon. News 9(6), 45–47 (1998).
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H. Ling, Y. Li, and M. Xiao, “Electromagnetically induced grating: homogeneously broadened medium,” Phys. Rev. A 57, 1338–1344 (1998).
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1997 (3)

1996 (5)

V. Arrizon, J. G. Ibarra, and J. Ojeda-Castaneda, “Matrix formulation of the Fresnel transform of complex transmittance gratings,” J. Opt. Soc. Am. A 13, 2414–2422 (1996).
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M. Testorf and J. Ojeda-Castaneda, “Fractional Talbot effect: analysis in phase space,” J. Opt. Soc. Am. A 13, 119–125 (1996).
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C. Leichtle, I. S. Averbukh, and W. P. Schleich, “Generic structures of multilevel quantum beats,” Phys. Rev. Lett. 77, 3999–4002 (1996).
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M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43, 2139–2164 (1996).
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M. H. Rubin, “Transverse correlation in optical spontaneous parametric down-conversion,” Phys. Rev. A 54, 5349–5360 (1996).
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1995 (4)

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429–R3432 (1995).
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V. Arrizon and J. Ojeda-Castaneda, “Fresnel diffraction of substructured gratings: matrix description,” Opt. Lett. 20, 118–120 (1995).
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M. S. Chapman, C. R. Ekstrom, T. D. Hammond, R. A. Rubenstein, J. Schmiedmayer, S. Wehinger, and D. E. Pritchard, “Optics and interferometer with Na molecules,” Phys. Rev. Lett. 74, 4783–4786 (1995).
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1994 (1)

1991 (1)

1990 (1)

1989 (1)

J. R. Leger, “Lateral mode control of an AlGaAs laser array in a Talbot cavity,” Appl. Phys. Lett. 55, 334–336 (1989).
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1985 (1)

J. Ojeda-Castaneda and E. E. Sicre, “Quasi ray-optical approach to longitudinal periodicities of free and bounded wavefields,” Opt. Acta 32, 17–26 (1985).
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1984 (1)

G. Indebetouw, “Propagation of spatially periodic wavefields,” Opt. Acta 31, 531–539 (1984).
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1983 (1)

A. W. Lohmann and J. Ojeda-Castaneda, “Spatial periodicities in partially coherent fields,” Opt. Acta 30, 475–479 (1983).
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1981 (1)

1975 (2)

R. Ulrich, “Image formation by phase coincidences in optical waveguides,” Opt. Commun. 13, 259–264 (1975).
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R. Ulrich, “Light-propagation and imaging in planar optical waveguides,” Nouv. Rev. Opt. 6, 253–262 (1975).
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1973 (1)

1972 (1)

1971 (1)

J. P. Guigay, “On Fresnel diffraction by one-dimensional periodic objects, with application to structure determination of phase objects,” Opt. Acta 18, 677–682 (1971).

1967 (1)

1965 (1)

1960 (1)

J. M. Cowley and A. F. Moodie, “Fourier images IV. The phase grating,” Proc. Phys. Soc. London 76, 378–384 (1960).
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1957 (3)

J. M. Cowley and A. F. Moodie, “Fourier images I. The point source,” Proc. Phys. Soc. (London) B 70, 486–496 (1957).
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J. M. Cowley and A. F. Moodie, “Fourier images II. The out-of-focus patterns,” Proc. Phys. Soc. (London) B 70, 497–504 (1957).
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J. M. Cowley and A. F. Moodie, “Fourier images III. Finite sources,” Proc. Phys. Soc. (London) B 70, 505–513 (1957).
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1948 (1)

E. Lau, “Beugungserscheinungen an Doppelrastern,” Ann. Phys. (Leipzig) 6, 417–423 (1948).
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1913 (1)

M. Wolfke, “Uber die Abbidung eines Gitters Au erhald der Einstellebene,” Ann. Phys. (Leipzig) 345, 194–200 (1913).
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1910 (1)

H. Weisel, “Uber die nach Fresnelscher Art Beobachteten Beugungserscheninungen der Gitter,” Ann. Phys. (Leipzig) 338, 995–1031(1910).
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A. Winkelmann, “Ubereinige Erscheinungen, die bei der Beugung des Lichtesdurch Gitter Auftreten,” Ann. Phys. (Leipzig) 332, 905–954 (1908).
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1881 (1)

Lord Rayleigh, “On copying diffraction gratings and on some phenomenon connected therewith,” Philos. Mag. 11, 196–205 (1881).

1836 (1)

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A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733–2736 (2000).
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Arndt, M.

K. Hornberger, S. Gerlich, P. Haslinger, S. Nimmrichter, and M. Arndt, “Colloquium: quantum interference of clusters and molecules,” Rev. Mod. Phys. 84, 157–173 (2012).
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B. Brezger, L. Hackermuller, S. Uttenthaler, J. Petschinka, M. Arndt, and A. Zeilinger, “Matter-wave interferometer for large molecules,” Phys. Rev. Lett. 88, 100404 (2002).
[CrossRef]

Arrizon, V.

Averbukh, I. S.

C. Leichtle, I. S. Averbukh, and W. P. Schleich, “Generic structures of multilevel quantum beats,” Phys. Rev. Lett. 77, 3999–4002 (1996).
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Azana, J.

Baruchel, J.

Bech, M.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Bronnimann, C. Grunzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
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Berry, M. V.

M. V. Berry, I. Marzoli, and W. P. Schleich, “Quantum carpets, carpets of light,” Phys. World 14(6), 39–44 (2001).

M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43, 2139–2164 (1996).
[CrossRef]

Bialynicki-Birula, I.

D. Wojcik, I. Bialynicki-Birula, and K. Zyczkowski, “Time evolution of quantum fractals,” Phys. Rev. Lett. 85, 5022–5025 (2000).
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Bienert, M.

H. Mack, M. Bienert, F. Haug, M. Freyberger, and W. P. Schleich, “Wave packets can factorize numbers,” Phys. Status Solidi B 233, 408–415 (2002).
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Blows, J. L.

J. A. Bolger, P. Hu, J. T. Mok, J. L. Blows, and B. J. Eggleton, “Talbot self-imaging and cross-phase modulation for generation of tunable high repetition rate pulse trains,” Opt. Commun. 249, 431–439 (2005).
[CrossRef]

Bolger, J. A.

J. A. Bolger, P. Hu, J. T. Mok, J. L. Blows, and B. J. Eggleton, “Talbot self-imaging and cross-phase modulation for generation of tunable high repetition rate pulse trains,” Opt. Commun. 249, 431–439 (2005).
[CrossRef]

Bortolozzo, U.

Boto, A. N.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733–2736 (2000).
[CrossRef]

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A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733–2736 (2000).
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B. Brezger, L. Hackermuller, S. Uttenthaler, J. Petschinka, M. Arndt, and A. Zeilinger, “Matter-wave interferometer for large molecules,” Phys. Rev. Lett. 88, 100404 (2002).
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F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Bronnimann, C. Grunzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
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Bryngdahl, O.

Bunk, O.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Bronnimann, C. Grunzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
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F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat. Phys. 2, 258–261 (2006).
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Cathey, W. T.

Cesario, J.

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M. S. Chapman, C. R. Ekstrom, T. D. Hammond, R. A. Rubenstein, J. Schmiedmayer, S. Wehinger, and D. E. Pritchard, “Optics and interferometer with Na molecules,” Phys. Rev. Lett. 74, 4783–4786 (1995).
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M. D’Angelo, M. V. Chekhova, and Y. H. Shih, “Two-photon diffraction and quantum lithography,” Phys. Rev. Lett. 87, 013602 (2001).
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J.-M. Wen, S. Du, H. Chen, and M. Xiao, “Electromagnetically induced Talbot effect,” Appl. Phys. Lett. 98, 081108 (2011).
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K.-H. Luo, J.-M. Wen, X.-H. Chen, Q. Liu, M. Xiao, and L.-A. Wu, “Erratum: Second-order Talbot effect with entangled photon pairs,” Phys. Rev. A 83, 029902 (2011).
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K.-H. Luo, X.-H. Chen, Q. Liu, and L.-A. Wu, “Nonlocal Talbot self-imaging with incoherent light,” Phys. Rev. A 82, 033803 (2010).
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K.-H. Luo, J.-M. Wen, X.-H. Chen, Q. Liu, M. Xiao, and L.-A. Wu, “Second-order Talbot effect with entangled photon pairs,” Phys. Rev. A 80, 043820 (2009).
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Chen, Z.

Cherukulappurath, S.

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H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, “PT-symmetric Talbot effects,” Phys. Rev. Lett. 109, 033902 (2012).
[CrossRef]

R. Iwanow, D. A. May-Arrioja, D. N. Christodoulides, G. I. Stegeman, Y. Min, and W. Sohler, “Discrete Talbot effect in waveguide arrays,” Phys. Rev. Lett. 95, 053902 (2005).
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Clark, C. W.

L. Deng, E. W. Hagley, J. Denschlag, J. E. Simsarian, M. Edwards, C. W. Clark, K. Helmerson, S. L. Rolston, and W. D. Phillips, “Temporal, matter-wave-dispersion Talbot effect,” Phys. Rev. Lett. 83, 5407–5411 (1999).
[CrossRef]

Cloetens, P.

Coppola, G.

Corcoran, C. J.

C. J. Corcoran and F. Durville, “Experimental demonstration of a phase-locked laser array using a self-Fourier cavity,” Appl. Phys. Lett. 86, 201118 (2005).
[CrossRef]

Cowley, J. M.

J. M. Cowley and A. F. Moodie, “Fourier images IV. The phase grating,” Proc. Phys. Soc. London 76, 378–384 (1960).
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J. M. Cowley and A. F. Moodie, “Fourier images II. The out-of-focus patterns,” Proc. Phys. Soc. (London) B 70, 497–504 (1957).
[CrossRef]

J. M. Cowley and A. F. Moodie, “Fourier images III. Finite sources,” Proc. Phys. Soc. (London) B 70, 505–513 (1957).
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J. M. Cowley and A. F. Moodie, “Fourier images I. The point source,” Proc. Phys. Soc. (London) B 70, 486–496 (1957).
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J. M. Cowley, Diffraction Physics (North-Holland, 1995).

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B. J. McMorran and A. D. Cronin, “An electron Talbot interferometer,” New J. Phys. 11, 033021 (2009).
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D’Angelo, M.

M. D’Angelo, M. V. Chekhova, and Y. H. Shih, “Two-photon diffraction and quantum lithography,” Phys. Rev. Lett. 87, 013602 (2001).
[CrossRef]

Dai, E.

David, C.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Bronnimann, C. Grunzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
[CrossRef]

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat. Phys. 2, 258–261 (2006).
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T. Weitkamp, B. Nohammer, A. Diaz, C. David, and E. Ziegler, “X-ray wavefront analysis and optics characterization with a grating interferometer,” Appl. Phys. Lett. 86, 054101 (2005).
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T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13, 6296–6304 (2005).
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C. David, B. Nohammer, H. H. Solak, and E. Ziegler, “Differential x-ray phase contrast imaging using a shearing interferometer,” Appl. Phys. Lett. 81, 3287–3289 (2002).
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De Martino, C.

De Natale, P.

De Nicol, S.

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L. Deng, E. W. Hagley, J. Denschlag, J. E. Simsarian, M. Edwards, C. W. Clark, K. Helmerson, S. L. Rolston, and W. D. Phillips, “Temporal, matter-wave-dispersion Talbot effect,” Phys. Rev. Lett. 83, 5407–5411 (1999).
[CrossRef]

Dennis, M. R.

Denschlag, J.

L. Deng, E. W. Hagley, J. Denschlag, J. E. Simsarian, M. Edwards, C. W. Clark, K. Helmerson, S. L. Rolston, and W. D. Phillips, “Temporal, matter-wave-dispersion Talbot effect,” Phys. Rev. Lett. 83, 5407–5411 (1999).
[CrossRef]

Diaz, A.

T. Weitkamp, B. Nohammer, A. Diaz, C. David, and E. Ziegler, “X-ray wavefront analysis and optics characterization with a grating interferometer,” Appl. Phys. Lett. 86, 054101 (2005).
[CrossRef]

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13, 6296–6304 (2005).
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Dowling, J. P.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733–2736 (2000).
[CrossRef]

Du, S.

J.-M. Wen, S. Du, H. Chen, and M. Xiao, “Electromagnetically induced Talbot effect,” Appl. Phys. Lett. 98, 081108 (2011).
[CrossRef]

Durville, F.

C. J. Corcoran and F. Durville, “Experimental demonstration of a phase-locked laser array using a self-Fourier cavity,” Appl. Phys. Lett. 86, 201118 (2005).
[CrossRef]

Eason, R.

Edwards, M.

L. Deng, E. W. Hagley, J. Denschlag, J. E. Simsarian, M. Edwards, C. W. Clark, K. Helmerson, S. L. Rolston, and W. D. Phillips, “Temporal, matter-wave-dispersion Talbot effect,” Phys. Rev. Lett. 83, 5407–5411 (1999).
[CrossRef]

Eggleton, B. J.

J. A. Bolger, P. Hu, J. T. Mok, J. L. Blows, and B. J. Eggleton, “Talbot self-imaging and cross-phase modulation for generation of tunable high repetition rate pulse trains,” Opt. Commun. 249, 431–439 (2005).
[CrossRef]

Eikenberry, E. F.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Bronnimann, C. Grunzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
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M. S. Chapman, C. R. Ekstrom, T. D. Hammond, R. A. Rubenstein, J. Schmiedmayer, S. Wehinger, and D. E. Pritchard, “Optics and interferometer with Na molecules,” Phys. Rev. Lett. 74, 4783–4786 (1995).
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Erkman, B.

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Feurer, T.

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H. Mack, M. Bienert, F. Haug, M. Freyberger, and W. P. Schleich, “Wave packets can factorize numbers,” Phys. Status Solidi B 233, 408–415 (2002).
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V. Torres-Company, J. Lancis, H. Lajunen, and A. T. Friberg, “Coherence revivals in two-photon frequency combs,” Phys. Rev. A 84, 033830 (2011).
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K. Hornberger, S. Gerlich, P. Haslinger, S. Nimmrichter, and M. Arndt, “Colloquium: quantum interference of clusters and molecules,” Rev. Mod. Phys. 84, 157–173 (2012).
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J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

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F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Bronnimann, C. Grunzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
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P. Cloetens, J. P. Guigay, C. De Martino, J. Baruchel, and M. Schlenker, “Fractional Talbot imaging of phase gratings with hard x rays,” Opt. Lett. 22, 1059–1061 (1997).
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J. P. Guigay, “On Fresnel diffraction by one-dimensional periodic objects, with application to structure determination of phase objects,” Opt. Acta 18, 677–682 (1971).

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Hackermuller, L.

B. Brezger, L. Hackermuller, S. Uttenthaler, J. Petschinka, M. Arndt, and A. Zeilinger, “Matter-wave interferometer for large molecules,” Phys. Rev. Lett. 88, 100404 (2002).
[CrossRef]

Hagley, E. W.

L. Deng, E. W. Hagley, J. Denschlag, J. E. Simsarian, M. Edwards, C. W. Clark, K. Helmerson, S. L. Rolston, and W. D. Phillips, “Temporal, matter-wave-dispersion Talbot effect,” Phys. Rev. Lett. 83, 5407–5411 (1999).
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H. Hamam, “Simplified linear formulation of Fresnel diffraction,” Opt. Commun. 144, 89–98 (1997).
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M. S. Chapman, C. R. Ekstrom, T. D. Hammond, R. A. Rubenstein, J. Schmiedmayer, S. Wehinger, and D. E. Pritchard, “Optics and interferometer with Na molecules,” Phys. Rev. Lett. 74, 4783–4786 (1995).
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Haslinger, P.

K. Hornberger, S. Gerlich, P. Haslinger, S. Nimmrichter, and M. Arndt, “Colloquium: quantum interference of clusters and molecules,” Rev. Mod. Phys. 84, 157–173 (2012).
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A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by x-ray Talbot interferometry for biological imaging,” Jpn. J. Appl. Phys. 45, 5254–5262 (2006).
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H. Mack, M. Bienert, F. Haug, M. Freyberger, and W. P. Schleich, “Wave packets can factorize numbers,” Phys. Status Solidi B 233, 408–415 (2002).
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Heinis, D.

Helmerson, K.

L. Deng, E. W. Hagley, J. Denschlag, J. E. Simsarian, M. Edwards, C. W. Clark, K. Helmerson, S. L. Rolston, and W. D. Phillips, “Temporal, matter-wave-dispersion Talbot effect,” Phys. Rev. Lett. 83, 5407–5411 (1999).
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M. Testorf, B. Hennelly, and J. Ojeda-Castaneda, Phase-Space Optics (McGraw Hill, 2010).

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K. Hornberger, S. Gerlich, P. Haslinger, S. Nimmrichter, and M. Arndt, “Colloquium: quantum interference of clusters and molecules,” Rev. Mod. Phys. 84, 157–173 (2012).
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J. A. Bolger, P. Hu, J. T. Mok, J. L. Blows, and B. J. Eggleton, “Talbot self-imaging and cross-phase modulation for generation of tunable high repetition rate pulse trains,” Opt. Commun. 249, 431–439 (2005).
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G. Indebetouw, “Propagation of spatially periodic wavefields,” Opt. Acta 31, 531–539 (1984).
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R. Iwanow, D. A. May-Arrioja, D. N. Christodoulides, G. I. Stegeman, Y. Min, and W. Sohler, “Discrete Talbot effect in waveguide arrays,” Phys. Rev. Lett. 95, 053902 (2005).
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Jalali, B.

Jannson, J.

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M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43, 2139–2164 (1996).
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Kok, P.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733–2736 (2000).
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H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, “PT-symmetric Talbot effects,” Phys. Rev. Lett. 109, 033902 (2012).
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H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, “PT-symmetric Talbot effects,” Phys. Rev. Lett. 109, 033902 (2012).
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F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Bronnimann, C. Grunzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
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Figures (31)

Figure 1
Figure 1

The optical Talbot effect for a monochromatic light, shown as a “Talbot carpet.” On the left of the figure the light can be seen diffracting through a grating, and this exact pattern is reproduced on the right of the picture (one Talbot length away from the grating). Halfway between each edge and the middle, one sees the image shifted to the side, and at regular fractions of the Talbot length the sub-images are clearly seen.

Figure 2
Figure 2

Plasmon Talbot carpets for a metal film drilled by an array of nanoholes with different periods a : (a)  a = λ SP , (b)  a = 5 λ SP , and (c) and (d)  a = 20 λ SP . As one can see, the Talbot effect is not yet developed in (a). For larger periodicities, as in (b)–(d), Talbot self-images are clearly observable. Reproduced with permission, © 2007. Optical Society of America [59].

Figure 3
Figure 3

Plasmon Talbot effect formed by SPPs propagating along a silver surface and scattered by a row of silicon dots with periods of (a)  b = λ SP , (b)  b = 10 λ SP , and (c)  b = 20 λ SP . Reprinted by permission from IOP publishing Ltd: [60], 2009.

Figure 4
Figure 4

(a) Schematic of the SPPLGs. (b) Scanning electron micrograph of the SPPLG with period d = 6 λ SP and α = 1 / 2 . Reproduced with permission, © 2009. Optical Society of America [62].

Figure 5
Figure 5

The first experimental demonstration of plasmon Talbot carpets for the SPPLGs with d = 3 λ SP (a) and d = 6 λ SP (b), respectively. (c) Experimental and theoretical spatial intensity profiles at z t / 2 for the SPPLG with d = 6 λ SP . (d) Theoretical Talbot carpet for the SPPLG with d = 6 λ SP ; white dashed lines from left to right indicate the positions of z t / 2 , d 2 / λ , z t , and 2 d 2 / λ , respectively. Reproduced with permission, © 2009. Optical Society of America [62].

Figure 6
Figure 6

Experimental demonstration of plasmonic Talbot carpets for Au cylindrical particles with periods of (a) 800 nm and (b) 1200 nm. The small white circles indicate the positions of the Au particles. The incident light with a wavelength of 790 nm traverses from left to right. The reflection of the SPP from the nanostructures creates high-intensity regions that saturate the image on the reflection side. Reproduced with permission, © 2009. Optical Society of America [63].

Figure 7
Figure 7

(a) Schematic configuration of the Talbot effect in an indefinite metamaterial. The structure period ( D ) is much smaller than the input wavelength, λ 0 . (b) No Talbot effect happens using a conventional material under the configuration shown in (a). (c) A Talbot carpet appears in an indefinite metamaterial. Reproduced with permission, © 2011. Optical Society of America [64].

Figure 8
Figure 8

Illustration of the Talbot effect in a multimode waveguide for an input field Ψ ( y , 0 ) . Along the propagation, one would observe, respectively, a mirrored single image at 3 L π , a self-image at 2 × 3 L π , and twofold images at 3 L π / 2 and 9 L π / 2 .

Figure 9
Figure 9

Discrete Talbot intensity carpets for different input field patterns: (a)  { 1 , 0 , 1 , 0 , } , (b)  { 1 , 0 , 1 , 0 , } , and (c)  { 1 , 0 , 0 , 1 , 0 , 0 , } . Figure 1 reprinted with permission from R. Iwanow et al., Phys. Rev. Lett. 053902 (2005) [50]. Copyright 2005, the American Physical Society. http://prl.aps.org/abstract/PRL/v95/i5/e053902

Figure 10
Figure 10

Experimental verifications on theoretical discrete Talbot revivals shown in Fig. 9 with input fields of (a)  { 1 , 0 , 1 , 0 , } , (b)  { 1 , 0 , 1 , 0 , } , and (c)  { 1 , 0 , 0 , 1 , 0 , 0 , } . Figure 4 reprinted with permission from R. Iwanow et al., Phys. Rev. Lett. 053902 (2005) [50]. Copyright 2005, the American Physical Society. http://prl.aps.org/abstract/PRL/v95/i5/e053902

Figure 11
Figure 11

Schematic illustration of grating-based hard x-ray interferometer: The beam splitter grating ( G 1 ) splits the incident beam into essentially two diffraction orders, which form a periodic interference pattern in the plane of the analyzer grating ( G 2 ). A phase object in the incident beam causes slight refraction and leads to changes of the locally transmitted intensity through the analyzer. G 1 is usually a phase grating with a period of p 1 , and G 2 is an absorption grating with a period of p 2 . The separation between the two gratings is d . Reproduced with permission, © 2005. Optical Society of America [73].

Figure 12
Figure 12

Three-dimensional density-projection rendering of the reconstructed refractive index of a small spider. The parameters of the interferometer are p 1 = 4 μm , p 2 = 2 μm , and d = 23.2 mm . The phase modulation of the phase grating G 1 is π . Reproduced with permission, © 2005. Optical Society of America [73].

Figure 13
Figure 13

Phase tomogram of a mouse tail observed by an x-ray Talbot interferometer with p 1 = p 2 = 8 μm , λ = 0.04 nm , and d = p 1 2 / 2 λ . In this experiment, G 1 is a π / 2 phase grating. Copyright 2006. The Japan Society of Applied Physics [72].

Figure 14
Figure 14

Configuration of a three-grating x-ray interferometer for hard x-ray phase imaging. G 0 and G 2 are absorption gratings with periods of p 0 and p 2 , respectively. G 1 can be a phase grating or a transmission grating, with period of p 1 . The distance between G 0 and G 1 is l and that between G 1 and G 2 is d . Reprinted by permission from Macmillan Publishers Ltd: Nat. Phys. 2, 258–261 [75], copyright 2006. www.nature.com/nphys/.

Figure 15
Figure 15

X-ray images of a small fish obtained with (a) a conventional transmission image and (b) differential phase-contrast imaging. (c)–(h) Two-times magnified and contrast-optimized parts of the transmission [(c), (e), (g)] and the differential phase-contrast image [(d), (f), (h)]. (c) and (d) show parts of the tail fin, (e) and (f) show the region around the otoliths, and (g) and (h) show the eyes of the fish. Reprinted by permission from Macmillan Publishers Ltd: Nat. Phys. 2, 258–261 [75], copyright 2006. www.nature.com/nphys/.

Figure 16
Figure 16

Observation of a chicken wing (a) in a conventional transmission image and (b) in a dark-field image. The x-ray scattering due to the porous microstructure of the bones and the reflection at internal or external interfaces produce a strong signal in the dark-field image. Reprinted by permission from Macmillan Publishers Ltd: Nat. Mater. 7, 134–137 [76], copyright 2008. http://www.nature.com/nmat/index.html.

Figure 17
Figure 17

Second-harmonic Talbot effect with a 1D PPLT crystal. (a) The domain structure of the 1D PPLT crystal obtained from SEM. (b) and (c) are the SH self-images recorded at the first and the third Talbot planes, respectively. Figure 2 reprinted with permission from Y. Zhang et al., Phys. Rev. Lett. 183901 (2010) [79]. Copyright 2010, the American Physical Society. http://link.aps.org/doi/10.1103/PhysRevLett.104.183901

Figure 18
Figure 18

Second-harmonic Talbot effect with a 2D PPLT crystal. (a) The domain structure. The SH self-images of the 2D PPLT crystal are recorded at the (b) first and (c) third Talbot planes, respectively. Figure 3 reprinted with permission from Y. Zhang et al., Phys. Rev. Lett. 183901 (2010) [79]. Copyright 2010, the American Physical Society. http://link.aps.org/doi/10.1103/PhysRevLett.104.183901

Figure 19
Figure 19

Fractional SH Talbot images experimentally recorded at (a)  1 / 2 , (b)  1 / 3 , and (c)  1 / 4 Talbot lengths, and (d)–(f) their corresponding theoretical simulations. Reproduced with permission, © (2012). Optical Society of America [81].

Figure 20
Figure 20

Schematic diagram of the SH TI, where a driving signal is applied along the y axis of the crystal. The size of the sample is L x × L y × L z . Reproduced with permission, © 2012. Optical Society of America [82].

Figure 21
Figure 21

Second-harmonic TI formed from a 1D PPLN crystal works at different acoustic power. (a) 0.10 W, (b) 0.54 W, (c) 0.64 W, and (d) 1.38 W. Reproduced with permission, © 2012. Optical Society of America [82].

Figure 22
Figure 22

The SH pattern is tuned to 2D when inducing an external acoustic wave (1 W) along the direction perpendicular to the domain walls. Reproduced with permission, © 2012. Optical Society of America [82].

Figure 23
Figure 23

Experimental setup for Talbot-effect-enhanced TWM. Reproduced with permission, © 2006. Optical Society of America [83].

Figure 24
Figure 24

(a) Measured TWM gain as a function of δ . δ is defined by δ = D 25.9 mm , where D is the distance between the two valves. (b) Gain for one valve (open circles) and two valves (dots). β is the intensity ratio between the pump and signal beams. Reproduced with permission, © 2006. Optical Society of America [83].

Figure 25
Figure 25

Simulated Talbot carpets for (a)  N = 1 , (b)  N = 2 , and (c)  N = 3 . Reproduced with permission, © 2006. Optical Society of America [83].

Figure 26
Figure 26

Setup to show the Talbot effect in the configuration of (a) quantum imaging and (b) quantum lithography using SPDC photons. (BS, beam splitter; CC, coincidence counter). Figure 1 reprinted with permission from K.-H. Luo et al., Phys. Rev. A 043820 (2009) [85]. Copyright 2009, the American Physical Society. http://pra.aps.org/abstract/PRA/v80/i4/e043820

Figure 27
Figure 27

Second-order Talbot imaging carpet obtained by scanning the idler detector D i through d i = 0 34 cm along the longitudinal z direction and through x 2 = 0.3 0.3 mm in the transverse x direction while fixing the signal detector D s at position d s 2 = 20 cm and x 1 = 0 , and d s 1 = 11 cm . The color bar denotes the transverse value of the two-photon correlation function. Figure 2 reprinted with permission from K.-H. Luo et al., Phys. Rev. A 043820 (2009) [85]. Copyright 2009, the American Physical Society. http://pra.aps.org/abstract/PRA/v80/i4/e043820

Figure 28
Figure 28

Top: Experimental setup for two-photon second-order Talbot self-imaging in the ghost interference scheme. The open circles in (a)–(c) are the coincidence counts for three different diffraction lengths of (a)  z T / 4 ( z i = 12 cm , z s 1 = 8 cm , z s 2 = 20 cm ) , (b)  z T / 2 ( z i = 25 cm , z s 1 = 15 cm , z s 2 = 40 cm ) , and (c)  z T ( z i = 25 cm , z s 1 = 55 cm , z s 2 = 80 cm ) . The solid lines are theoretical curves. Figure 2 reprinted with permission from X.-B. Song et al., Phys. Rev. Lett. 033902 (2011) [92]. Copyright 2011, the American Physical Society. http://prl.aps.org/abstract/PRL/v107/i3/e033902

Figure 29
Figure 29

Second-order Talbot lithography carpets obtained by (a) scanning both detectors D i and D s along the z direction from d 0 = 0 to 5.7 cm with the transverse position of one detector fixed at x 1 ( x 2 ) = 0 , and the other detector scanned in its transverse x 2 ( x 1 ) direction; (b) moving both detectors simultaneously in the same manner along the transverse x and longitudinal z directions, with transverse scanning range x 1 = x 2 = 0.3 to 0.3 mm. The color bars denote the values of the transverse two-photon correlation functions. Figure 5 reprinted with permission from K.-H. Luo et al., Phys. Rev. A 029902 (2011) [86]. Copyright 2011, the American Physical Society. http://pra.aps.org/abstract/PRA/v83/i2/e029902

Figure 30
Figure 30

Top: Experimental setup for observing two-photon second-order Talbot self-imaging in the quantum lithography configuration. The open circles are experimental data observed for three diffraction lengths of (a)  z s T / 4 , (b)  z s T / 2 , and (c)  z s T . The left and right parts correspond to the entangled two-photon source and the coherent light source, respectively. The solid lines are theoretical curves. Figure 3 reprinted with permission from X.-B. Song et al., Phys. Rev. Lett. 033902 (2011) [92]. Copyright 2011, the American Physical Society. http://prl.aps.org/abstract/PRL/v107/i3/e033902

Figure 31
Figure 31

(a) Closed three-level Λ -type atomic system for electromagnetically induced self-imaging. (b) Configuration of forming an electromagnetically induced grating (EIG). (c) Snapshot of the EIG in (b). Reprinted with permission from Wen et al., Appl. Phys. Lett. 98, 081108 (2011) [93]. Copyright 2011, American Institute of Physics.

Equations (48)

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A ( x ) = n = c n e 2 i π n x d ,
E ( X ) = e 2 i π ( z 1 + z 2 ) λ i λ z 1 z 2 d x s S ( x s ) d x A ( x ) e i π ( x x s ) 2 λ z 1 e i π ( X x ) 2 λ z 2 ,
E ( x ) d x A ( x ) e i 2 π λ ( z + X 2 2 z x X z + x 2 2 z ) .
E ( X ) n = c n e i π λ n 2 z d 2 e i 2 π n X d .
z = 2 m d 2 λ ,
z G = 2 m d 2 λ ( w z w 0 ) 2 ,
M G = ( w z w 0 ) 2 .
z n = λ SP / ( 1 1 ( n λ SP / d ) 2 ) , n = 1 , 2 , 3 , .
A ( x , z ) = n f n e i n q x x e i k z z ,
k z 2 ε x + k x 2 ε z = k 0 2 ,
A ( x , z ) = n f n e i n q x x e i k 0 z ε x ( 1 n 2 ε z ( q x k 0 ) 2 ) .
A ( x , z ) = n f n e i n q x x e i 2 π n z D ε x ε z .
z T D ε z ε x ,
k y v 2 + β v 2 = k 0 2 n r 2 ,
W e v W e 0 = W M + λ 0 π ( n c n r ) 2 σ 1 n r 2 n c 2 ,
β v k 0 n r ( v + 1 ) 2 π λ 0 4 n r W e 0 2 .
L π = π β 0 β 1 4 n r W e 0 2 3 λ 0 ,
β 0 β v v ( v + 2 ) π 3 L π .
Ψ ( y , z ) = v = 0 m 1 c v ψ v ( y ) e i β v z ,
Ψ ( y , z ) = v = 0 m 1 c v ψ v ( y ) e i ( β 0 β v ) z = v = 0 m 1 c v ψ v ( y ) e i v ( v + 2 ) π 3 L π z ,
i d U n d z + κ ( U n + 1 + U n 1 ) = 0 ,
d m = ( m 1 2 ) p 1 2 4 λ , m = 1 , 2 , 3 , .
p 0 = p 2 × l d .
Δ φ m = π λ M 2 2 P s L y L z ,
G = | S out | 2 ( with the pump ) / | S out | 2 ( without the pump ) ,
| ψ = d ω s d ω i d 2 α 1 d 2 α 2 Φ ( ω s , ω i ) δ ( ω s + ω i ω p ) δ ( α⃗ 1 + α⃗ 2 ) | 1 k⃗ s , 1 k⃗ i ,
E j ( + ) ( ρ⃗ j , z j , t j ) = d ω j d 2 α j E j f j ( ω j ) e i ω j t j g j ( α⃗ j , ω j ; ρ⃗ j , z j ) a ( α⃗ j , ω j ) ,
[ a ( α⃗ , ω ) , a ( α⃗ , ω ) ] = δ ( α⃗ α⃗ ) δ ( ω ω ) .
R c c = 1 T 0 T d t 1 0 T d t 2 | Ψ ( ρ⃗ 1 , z 1 , t 1 ; ρ⃗ 2 , z 2 , t 2 ) | 2 ,
Ψ ( ρ⃗ 1 , z 1 , t 1 ; ρ⃗ 2 , z 2 , t 2 ) = 0 | E 1 ( + ) ( ρ⃗ 1 , z 1 , t 1 ) E 2 ( + ) ( ρ⃗ 2 , z 2 , t 2 ) | ψ .
g 1 ( α⃗ 1 , ω s ; ρ⃗ 1 , d s ) = i ω s 2 π c d s 2 e i ω s d s c e i ω s ρ 1 2 2 c d s 2 e i c d s 1 α 1 2 2 ω s d 2 ρ o A o ( ρ⃗ o ) e i ω s ρ o 2 2 c d s 2 e i ρ⃗ o ( α⃗ 1 ω s ρ⃗ 1 c d s 2 ) ,
g 2 ( α⃗ 2 , ω i ; ρ⃗ 2 , d i ) = e i ω i d i c e i c d i α 2 2 2 ω i e i α⃗ 2 ρ⃗ 2 ,
Ψ ( ρ⃗ 1 , z 1 , t 1 ; ρ⃗ 2 , z 2 , t 2 ) = e i ( Ω s τ 1 + Ω i τ 2 ) ψ ( ρ⃗ 1 , τ 1 ; ρ⃗ 2 , τ 2 ) ,
ψ ( ρ⃗ 1 , τ 1 ; ρ⃗ 2 , τ 2 ) = d v s d v i δ ( v s + v i ) e i ( v s τ 1 + v i τ 2 ) f 1 ( Ω s + v s ) f 2 ( Ω i + v i ) W ( ρ⃗ 1 , ρ⃗ 2 ) .
W ( ρ⃗ 1 , ρ⃗ 2 ) = W 0 d 2 ρ o A o ( ρ⃗ o ) e i ω s ρ o 2 2 c d s 2 e i ω s ρ⃗ o ρ⃗ 1 c d s 2 d 2 α 1 e i c α 1 2 2 ( d s 1 ω s + d i ω i ) e i α⃗ 1 ( ρ⃗ o ρ⃗ 2 ) ,
W ( ρ⃗ 1 , ρ⃗ 2 ) = W 0 d 2 ρ o A o ( ρ⃗ o ) e i ω s c ρ⃗ o ( ρ⃗ 1 d s 2 + ρ⃗ 2 d s 1 + ω s d i / ω i ) e i ω s 2 c ρ⃗ o 2 ( 1 d s 2 + 1 d s 1 + ω s d i / ω i ) .
A ( x ) o = n = c n e 2 i π n x a .
W ( x 1 , x 2 ) = W 0 n = c n exp ( i n 2 π a 2 λ s 1 d s 2 + 1 d s 1 + λ i λ s d i ) exp ( i 2 n π a ( x 1 d s 2 + x 2 d s 1 + λ i λ s d i ) 1 d s 2 + 1 d s 1 + λ i λ s d i ) .
1 d s 2 + 1 d s 1 + λ i λ s d i = 1 2 m a 2 / λ s = 1 m z s T ,
W ( x 1 , x 2 ) = W 0 n = c n exp ( i 2 n π a ( d s 1 + λ i λ s d i ) x 1 + d s 2 x 2 d s 1 + d s 2 + λ i λ s d i ) ,
g j ( α⃗ j , ω ; ρ⃗ j , d ) = i λ d e i ω d c e i ω ρ j 2 2 c d d 2 ρ o A o ( ρ⃗ o ) e i ω ρ o 2 2 c d e i ρ⃗ o ( α⃗ j ω ρ⃗ j c d ) .
W ( ρ⃗ 1 , ρ⃗ 2 ) = W 0 d 2 ρ o A o 2 ( ρ⃗ o ) e i ω ρ o 2 c d e i ω c d ρ⃗ o ( ρ⃗ 1 + ρ⃗ 2 ) .
W ( x 1 , x 2 ) = W 0 n = c n e i π n 2 λ 2 a 2 d 0 e i π n a ( x 1 + x 2 ) ,
d 0 = m 4 a 2 λ = m z s T ,
W ( x 1 , x 2 ) = W 0 n = c n e i π n a ( x 1 + x 2 ) .
χ = N | μ | 2 2 ε 0 Δ 2 + i γ b c | Ω c | 2 cos 2 ( π x a ) ( Δ p + i γ a ) ( Δ 2 + i γ b c ) ,
E p ( x , L ) = E p ( x , 0 ) e k p χ L 2 e i k p χ L 2 ,
E p ( x , L ) = n = c n e i 2 π n x a .

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