Abstract

The nature of light, extending from the optical to the x-ray regime, is reviewed from a diffraction point of view by comparing field-based statistical optics and photon-based quantum optics approaches. The topic is introduced by comparing historical diffraction concepts based on wave interference, Dirac’s notion of photon self-interference, Feynman’s interference of space–time photon probability amplitudes, and Glauber’s formulation of coherence functions based on photon detection. The concepts are elucidated by a review of how the semiclassical combination of the disparate photon and wave concepts have been used to describe light creation, diffraction, and detection. The origin of the fundamental diffraction limit is then discussed in both wave and photon pictures. By use of Feynman’s concept of probability amplitudes associated with independent photons, we show that quantum electrodynamics, the complete theory of light, reduces in lowest order to the conventional wave formalism of diffraction. As an introduction to multi-photon effects, we then review fundamental one- and two-photon experiments and detection schemes, in particular the seminal Hanbury Brown–Twiss experiment. The formal discourse of the paper starts with a treatment of first-order coherence theory. In first order, the statistical optics and quantum optics formulations of coherence are shown to be equivalent. This is elucidated by a discussion of Zernike’s powerful theorem of partial coherence propagation, a cornerstone of statistical optics, followed by its quantum derivation based on the interference of single-photon probability amplitudes. The treatment is then extended to second-order coherence theory, where the equivalence of wave and particle descriptions is shown to break down. This is illustrated by considering two photons whose space–time probability amplitudes are correlated through nonlinear birth processes, resulting in entanglement or cloning. In both cases, the two-photon diffraction patterns are shown to exhibit resolution below the conventional diffraction limit, defined by the one-photon diffraction patterns. The origin of the reduction is shown to arise from the interference of two-photon probability amplitudes. By comparing first- and second-order diffraction, it is shown that the conventional first-order concept of partial coherence with its limits of chaoticity and first-order coherence has the second-order analogue of partial entanglement, with its limits corresponding to two entangled photons (“entangled biphotons”) and two cloned photons (“cloned biphotons”), the latter being second-order coherent. The concept of cloned biphotons is extended to the case of n cloned photons, resulting in a 1/n reduction of the diffraction limit. In the limit of nth-order coherence, all photons within the nth-order collective state are shown to propagate on particle like trajectories, reproducing the 0th-order ray-optics picture. These results are discussed in terms of the linearity of quantum mechanics and Heisenberg’s space–momentum uncertainty principle. A general concept of coherence based on photon density is developed that in first order is equivalent to the conventional wave-based picture.

© 2019 Optical Society of America

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2018 (7)

R. Schneider, T. Mehringer, G. Mercurio, L. Wenthaus, A. Classen, G. Brenner, O. Gorobtsov, A. Benz, D. Bhatti, L. Bocklage, B. Fischer, S. Lazarev, Y. Obukhov, K. Schlage, P. Skopintsev, J. Wagner, F. Waldmann, S. Willing, I. Zaluzhnyy, W. Wurth, I. A. Vartanyants, R. Röhlsberger, and J. von Zanthier, “Quantum imaging with incoherently scattered light from a free-electron laser,” Nat. Phys. 14, 126–129 (2018).
[Crossref]

O. Yu. Gorobtsov, N. Mukharamova, S. Lazarev, M. Chollet, D. Zhu, Y. Feng, R. P. Kurta, J.-M. Meijer, G. Williams, M. Sikorski, S. Song, D. Dzhigaev, S. Serkez, A. Singer, and A. V. Petukhov, and I. A. Vartanyants, “Diffraction based Hanbury Brown and Twiss interferometry at a hard x-ray free-electron laser,” Sci. Rep. 8, 2219 (2018).
[Crossref]

O. Yu. Gorobtsov, N. Mukharamova, S. Lazarev, M. Chollet, D. Zhu, Y. Feng, R. P. Kurta, J.-M. Meijer, G. Williams, M. Sikorski, S. Song, D. Dzhigaev, S. Serkez, A. Singer, and A. V. Petukhov, and I. A. Vartanyants, “Diffraction based Hanbury Brown and Twiss interferometry at a hard x-ray free-electron laser,” Sci. Rep. 8, 2219 (2018).
[Crossref]

Z. Li, N. Medvedev, H. N. Chapman, and Y. Shih, “Radiation damage free ghost diffraction with atomic resolution,” J. Phys. B 51, 025503 (2018).
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Z. Chen, D. J. Higley, M. Beye, M. Hantschmann, V. Mehta, O. Hellwig, A. Mitra, S. Bonetti, M. Bucher, S. Carron, T. Chase, E. Jal, R. Kukreja, T. Liu, A. H. Reid, G. L. Dakovski, A. Föhlisch, W. F. Schlotter, H. A. Dürr, and J. Stöhr, “Ultrafast self-induced x-ray transparency and loss of magnetic diffraction,” Phys. Rev. Lett. 121, 137403 (2018).
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T. Kroll, C. Weninger, R. Alonso-Mori, D. Sokaras, D. Zhu, L. Mercadier, V. P. Majety, A. Marinelli, A. Lutman, M. W. Guetg, F.-J. Decker, S. Boutet, A. Aquila, J. Koglin, J. Koralek, D. P. DePonte, J. Kern, F. D. Fuller, E. Pastor, T. Fransson, Y. Zhang, J. Yano, V. K. Yachandra, N. Rohringer, and U. Bergmann, “Stimulated x-ray emission spectroscopy in transition metal complexes,” Phys. Rev. Lett. 120, 133203 (2018).
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O. Yu. Gorobtsov, G. Mercurio, F. Capotondi, P. Skopintsev, S. Lazarev, I. A. Zaluzhnyy, M. B. Danailov, M. Dell’Angela, M. Manfredda, E. Pedersoli, L. Giannessi, M. Kiskinova, K. C. Prince, W. Wurth, and I. A. Vartanyants, “Seeded x-ray free-electron laser generating radiation with laser statistical properties,” Nat. Commun. 9, 4498 (2018).
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A. Paul and T. Qureshi, “Biphoton interference in a double-slit experiment,” Quanta 7, 1 (2018).
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2017 (8)

I. E. Zadeh, J. W. N. Los, R. B. M. Gourgues, V. Steinmetz, G. Bulgarini, S. M. Dobrovolskiy, V. Zwiller, and S. N. Dorenbos, “Single-photon detectors combining high efficiency, high detection rates, and ultra-high timing resolution,” APL Photon. 2, 111301 (2017).
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C. Milne, T. Schietinger, M. Aiba, A. Alarcon, J. Alex, A. Anghel, V. Arsov, C. Beard, P. Beaud, S. Bettoni, M. Bopp, H. Brands, M. Brönnimann, I. Brunnenkant, M. Calvi, A. Citterio, P. Craievich, M. C. Divall, M. Dällenbach, M. D’Amico, A. Dax, Y. Deng, A. Dietrich, R. Dinapoli, E. Divall, S. Dordevic, S. Ebner, C. Erny, H. Fitze, U. Flechsig, R. Follath, F. Frei, F. Gärtner, R. Ganter, T. Garvey, Z. Geng, I. Gorgisyan, C. Gough, A. Hauff, C. P. Gauri, N. Hiller, T. Humar, S. Hunziker, G. Ingold, R. Ischebeck, M. Janousch, P. Juranić, M. Jurcevic, M. Kaiser, B. Kalantari, R. Kalt, B. Keil, C. Kittel, G. Knopp, W. Koprek, H. T. Lemke, T. Lippuner, D. Llorente Sancho, F. Löhl, C. Lopez-Cuenca, F. Märki, F. Marcellini, G. Marinkovic, I. Martiel, R. Menzel, A. Mozzanica, K. Nass, G. L. Orlandi, C. O. Loch, E. Panepucci, M. Paraliev, B. Patterson, B. Pedrini, M. Pedrozzi, P. Pollet, C. Pradervand, E. Prat, P. Radi, J.-Y. Raguin, S. Redford, J. Rehanek, J. Réhault, S. Reiche, M. Ringele, J. Rittmann, L. Rivkin, A. Romann, M. Ruat, C. Ruder, L. Sala, L. Schebacher, T. Schilcher, V. Schlott, T. Schmidt, B. Schmitt, X. Shi, M. Stadler, L. Stingelin, W. Sturzenegger, J. Szlachetko, D. Thattil, D. M. Treyer, A. Trisorio, W. Tron, S. Vetter, C. Vicario, D. Voulot, M. Wang, T. Zamofing, C. Zellweger, R. Zennaro, E. Zimoch, R. Abela, L. Patthey, and H. H. Braun, “SwissFEL: The Swiss x-ray free electron laser,” Appl. Sci. 7, 720 (2017).
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J. Stöhr, “Two-photon x-ray diffraction,” Phys. Rev. Lett. 118, 024801 (2017).
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J. Liu, H. Chen, Y. Zhou, H. Zheng, F. Li, K. He, F. Li, and Z. Xu, “Second-order fermionic interference with independent photons,” J. Opt. Soc. Am. B 34, 1215–1222 (2017).
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D.-J. Zhang, S. Wu, H.-G. Li, H.-B. Wang, J. Xiong, and K. Wang, “Young’s double-slit interference with two-color biphotons,” Sci. Rep. 7, 17372 (2017).
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S.-K. Liao, W.-Q. Cai, W.-Y. Liu, L. Zhang, Y. Li, J.-G. Ren, J. Yin, Q. Shen, Y. Cao, Z.-P. Li, F.-Z. Li, X.-W. Chen, L.-H. Sun, J.-J. Jia, J.-C. Wu, X.-J. Jiang, J.-F. Wang, Y.-M. Huang, Q. Wang, Y.-L. Zhou, L. Deng, T. Xi, L. Ma, T. Hu, Q. Zhang, Y.-A. Chen, N.-L. Liu, X.-B. Wang, Z.-C. Zhu, C.-Y. Lu, R. Shu, C.-Z. Peng, and J.-Y. Wang, and J.-W. Pan, “Satellite-to-ground quantum key distribution,” Nature 549, 43–47 (2017).
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S.-K. Liao, W.-Q. Cai, W.-Y. Liu, L. Zhang, Y. Li, J.-G. Ren, J. Yin, Q. Shen, Y. Cao, Z.-P. Li, F.-Z. Li, X.-W. Chen, L.-H. Sun, J.-J. Jia, J.-C. Wu, X.-J. Jiang, J.-F. Wang, Y.-M. Huang, Q. Wang, Y.-L. Zhou, L. Deng, T. Xi, L. Ma, T. Hu, Q. Zhang, Y.-A. Chen, N.-L. Liu, X.-B. Wang, Z.-C. Zhu, C.-Y. Lu, R. Shu, C.-Z. Peng, and J.-Y. Wang, and J.-W. Pan, “Satellite-to-ground quantum key distribution,” Nature 549, 43–47 (2017).
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O. Yu. Gorobtsov, G. Mercurio, G. Brenner, U. Lorenz, N. Gerasimova, R. P. Kurta, F. Hieke, P. Skopintsev, I. Zaluzhnyy, S. Lazarev, D. Dzhigaev, M. Rose, A. Singer, W. Wurth, and I. A. Vartanyants, “Statistical properties of a free-electron laser revealed by Hanbury Brown–Twiss interferometry,” Phys. Rev. A 95, 023843 (2017).
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A. Classen, K. Ayyer, H. N. Chapman, R. Röhlsberger, and J. von Zanthier, “Incoherent diffractive imaging via intensity correlations of hard x rays,” Phys. Rev. Lett. 119, 053401 (2017).
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2016 (5)

C. Bostedt, S. Boutet, D. M. Fritz, Z. Huang, H. J. Lee, H. T. Lemke, A. Robert, W. F. Schlotter, J. J. Turner, and G. J. Williams, “Linac Coherent Light Source: The first five years,” Rev. Mod. Phys. 88, 015007 (2016).
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P. Chen and H. Kleinert, “Deficiencies of Bohm trajectories in view of basic quantum principles,” Electron. J. Theor. Phys. 13, 1 (2016).

H. Yu, R. Lu, S. Han, H. Xie, G. Du, T. Xiao, and D. Zhu, “Fourier-transform ghost imaging with hard x rays,” Phys. Rev. Lett. 117, 113901 (2016).
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D. Pelliccia, A. Rack, M. Scheel, V. Cantelli, and D. M. Paganin, “Experimental x-ray ghost imaging,” Phys. Rev. Lett. 117, 113902 (2016).
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B. Wu, T. Wang, C. E. Graves, D. Zhu, W. F. Schlotter, J. J. Turner, O. Hellwig, Z. Chen, H. A. Dürr, A. Scherz, and J. Stöhr, “Elimination of x-ray diffraction through stimulated x-ray transmission,” Phys. Rev. Lett. 117, 027401 (2016).
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2015 (2)

H. Yoneda, Y. Inubushi, K. Nagamine, Y. Michine, H. Ohashi, H. Yumoto, K. Yamauchi, H. Mimura, H. Kitamura, T. Katayama, and T. Ishikawa, and M. Yabashi, “Atomic inner-shell laser at 1.5-Ångström wavelength pumped by an x-ray free-electron laser,” Nature 524, 446–449 (2015).
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H. Yoneda, Y. Inubushi, K. Nagamine, Y. Michine, H. Ohashi, H. Yumoto, K. Yamauchi, H. Mimura, H. Kitamura, T. Katayama, and T. Ishikawa, and M. Yabashi, “Atomic inner-shell laser at 1.5-Ångström wavelength pumped by an x-ray free-electron laser,” Nature 524, 446–449 (2015).
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J. Stöhr and A. Scherz, “Creation of x-ray transparency of matter by stimulated elastic forward scattering,” Phys. Rev. Lett. 115, 107402 (2015).
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2014 (2)

A. Singer, U. Lorenz, A. Marras, A. Klyuev, J. Becker, K. Schlage, P. Skopintsev, O. Gorobtsov, A. Shabalin, H.-C. Wille, H. Franz, H. Graafsma, and I. A. Vartanyants, “Intensity interferometry of single x-ray pulses from a synchrotron storage ring,” Phys. Rev. Lett. 113, 064801 (2014).
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C. J. Chunnilall, I. P. Degiovanni, S. Kück, I. Müller, and A. G. Sinclair, “Metrology of single-photon sources and detectors: a review,” Opt. Eng. 53, 081910 (2014).
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2013 (8)

J. Liu, Y. Zhou, W. Wang, R. Liu, K. He, F. Li, and Z. Xu, “Spatial second-order interference of pseudothermal light in a Hong-Ou-Mandel interferometer,” Opt. Express 21, 19209–19218 (2013).
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M. Beye, S. Schreck, F. Sorgenfrei, C. Trabant, N. Pontius, C. Schüßler-Langeheine, W. Wurth, and A. Föhlisch, “Stimulated x-ray emission for materials science,” Nature 501, 191–194 (2013).
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A. Singer, U. Lorenz, F. Sorgenfrei, N. Gerasimova, J. Gulden, O. M. Yefanov, R. P. Kurta, A. Shabalin, R. Dronyak, R. Treusch, V. Kocharyan, E. Weckert, W. Wurth, and I. A. Vartanyants, “Hanbury Brown–Twiss interferometry at a free-electron laser,” Phys. Rev. Lett. 111, 034802 (2013).
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J. D. Biggs, Y. Zhang, D. Healion, and S. Mukamel, “Watching energy transfer in metalloporphyrin heterodimers using stimulated x-ray Raman spectroscopy,” Proc. Nat. Acad. Sci. USA 110, 15597–15601 (2013).
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C. Weninger, M. Purvis, D. Ryan, R. A. London, J. D. Bozek, C. Bostedt, A. Graf, G. Brown, J. J. Rocca, and N. Rohringer, “Stimulated electronic x-ray Raman scattering,” Phys. Rev. Lett. 111, 233902 (2013).
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K. Jung, “Is the de Broglie-Bohm interpretation of quantum mechanics really plausible?” J. Phys. Conf. Ser. 442, 012060 (2013).
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S. Eibenberger, S. Gerlich, M. Arndt, M. Mayor, and J. Tüxen, “Matter–wave interference of particles selected from a molecular library with masses exceeding 10,000 amu,” Phys. Chem. Chem. Phys. 15, 14696 (2013).
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R. Bach, D. Pope, S.-H. Liou, and H. Batelaan, “Controlled double-slit electron diffraction,” New J. Phys. 15, 033018 (2013).
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2012 (5)

J.-W. Pan, Z.-B. Chen, C.-Y. Lu, H. Weinfurter, A. Zeilinger, and M. Żukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84, 777–838 (2012).
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N. Rohringer, D. Ryan, R. A. London, M. Purvis, F. Albert, J. Dunn, J. D. Bozek, C. Bostedt, A. Graf, R. Hill, and S. P. Hau-Riege, and J. J. Rocca, “Atomic inner-shell x-ray laser at 1.46 nanometres pumped by an x-ray free-electron laser,” Nature 481, 488–491 (2012).
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N. Rohringer, D. Ryan, R. A. London, M. Purvis, F. Albert, J. Dunn, J. D. Bozek, C. Bostedt, A. Graf, R. Hill, and S. P. Hau-Riege, and J. J. Rocca, “Atomic inner-shell x-ray laser at 1.46 nanometres pumped by an x-ray free-electron laser,” Nature 481, 488–491 (2012).
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T. Wang, D. Zhu, B. Wu, C. Graves, S. Schaffert, T. Rander, L. Müller, B. Vodungbo, C. Baumier, D. P. Bernstein, B. Bräuer, V. Cros, S. De Jong, R. Delaunay, A. Fognini, R. Kukreja, S. Lee, V. López-Flores, J. Mohanty, B. Pfau, H. Popescu, M. Sacchi, A. B. Sardinha, F. Sirotti, P. Zeitoun, M. Messerschmidt, J. J. Turner, W. F. Schlotter, O. Hellwig, R. Mattana, N. Jaouen, F. Fortuna, Y. Acremann, C. Gutt, H. A. Dürr, E. Beaurepaire, C. Boeglin, S. Eisebitt, G. Grübel, J. Lüning, and J. Stöhr, and A. O. Scherz, “Femtosecond single-shot imaging of nanoscale ferromagnetic order in Co/Pd multilayers using resonant x-ray holography,” Phys. Rev. Lett. 108, 267403 (2012).
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T. Wang, D. Zhu, B. Wu, C. Graves, S. Schaffert, T. Rander, L. Müller, B. Vodungbo, C. Baumier, D. P. Bernstein, B. Bräuer, V. Cros, S. De Jong, R. Delaunay, A. Fognini, R. Kukreja, S. Lee, V. López-Flores, J. Mohanty, B. Pfau, H. Popescu, M. Sacchi, A. B. Sardinha, F. Sirotti, P. Zeitoun, M. Messerschmidt, J. J. Turner, W. F. Schlotter, O. Hellwig, R. Mattana, N. Jaouen, F. Fortuna, Y. Acremann, C. Gutt, H. A. Dürr, E. Beaurepaire, C. Boeglin, S. Eisebitt, G. Grübel, J. Lüning, and J. Stöhr, and A. O. Scherz, “Femtosecond single-shot imaging of nanoscale ferromagnetic order in Co/Pd multilayers using resonant x-ray holography,” Phys. Rev. Lett. 108, 267403 (2012).
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S. Shwartz, R. N. Coffee, J. M. Feldkamp, Y. Feng, J. B. Hastings, G. Y. Yin, and S. E. Harris, “X-ray parametric down-conversion in the Langevin regime,” Phys. Rev. Lett. 109, 013602 (2012).
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O. Thomas, Z. L. Yuan, and A. Shields, “Practical photon number detection with electric field-modulated silicon avalanche photodiodes,” Nat. Commun. 3, 644 (2012).

2011 (4)

M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, “Single-photon sources and detectors,” Rev. Sci. Instrum. 82, 071101 (2011).
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I. A. Vartanyants, A. Singer, A. P. Mancuso, O. M. Yefanov, A. Sakdinawat, Y. Liu, E. Bang, G. J. Williams, G. Cadenazzi, B. Abbey, H. Sinn, D. Attwood, K. A. Nugent, E. Weckert, T. Wang, D. Zhu, B. Wu, C. Graves, A. Scherz, J. J. Turner, W. F. Schlotter, M. Messerschmidt, J. Lüning, Y. Acremann, P. Heimann, D. C. Mancini, V. Joshi, J. Krzywinski, R. Soufli, M. Fernandez-Perea, S. Hau-Riege, A. G. Peele, Y. Feng, O. Krupin, S. Moeller, and W. Wurth, “Coherence properties of individual femtosecond pulses of an x-ray free-electron laser,” Phys. Rev. Lett. 107, 144801 (2011).
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K. Tamasaku, K. Sawada, E. Nishibori, and T. Ishikawa, “Visualizing the local optical response to extreme-ultraviolet radiation with a resolution of λ/380,” Nat. Phys. 7, 705–708 (2011).
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S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the average trajectories of single photons in a two-slit interferometer,” Science 332, 1170–1173 (2011).
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2010 (5)

D. Zhu, M. Guizar-Sicairos, B. Wu, A. Scherz, Y. Acremann, T. Tyliszczak, P. Fischer, N. Friedenberger, K. Ollefs, M. Farle, J. R. Fienup, and J. Stöhr, “High-resolution x-ray lensless imaging by differential holographic encoding,” Phys. Rev. Lett. 105, 043901 (2010).
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H. N. Chapman and K. A. Nugent, “Coherent lensless x-ray imaging,” Nat. Photonics 4, 833–839 (2010).
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M. Bei, M. Borland, Y. Cai, P. Elleaume, R. Gerig, K. Harkay, L. Emery, A. Hutton, R. Hettel, R. Nagaoka, D. Robin, and C. Steier, “The potential of an ultimate storage ring for future light sources,” Nucl. Instrum. Methods Phys. Res. A 622, 518–535 (2010).
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J. Liu and G. Zhang, “Unified interpretation for second-order subwavelength interference based on Feynman’s path-integral theory,” Phys. Rev. A 82, 013822 (2010).
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I. A. Vartanyants and A. Singer, “Coherence properties of hard x-ray synchrotron sources and x-ray free-electron lasers,” New J. Phys. 12, 035004 (2010).
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2009 (4)

R. Hadfield, “Single-photon detectors for optical quantum information applications,” Nat. Photonics 3, 696–705 (2009).
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B. Henrich, A. Bergamaschi, C. Broennimann, R. Dinapoli, E. Eikenberry, I. Johnson, M. Kobas, P. Kraft, A. Mozzanica, and B. Schmitt, “PILATUS: A single photon counting pixel detector for x-ray applications,” Nucl. Instrum. Meth. A 607, 247–249 (2009).
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R. Rick, A. Scherz, W. F. Schlotter, D. Zhu, J. Lüning, and J. Stöhr, “Optimal signal-to-noise ratios for soft x-ray lensless imaging,” Opt. Lett. 34, 650 (2009).
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R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81, 865–942 (2009).
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2008 (2)

T. L. Dimitrova and A. Weis, “The wave-particle duality of light: a demonstration experiment,” Am. J. Phys. 76, 137–142 (2008).
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A. Scherz, D. Zhu, R. Rick, W. F. Schlotter, S. Roy, J. Lüning, and J. Stöhr, “Nanoscale imaging with resonant coherent x rays: extension of multiple-wavelength anomalous diffraction to nonperiodic structures,” Phys. Rev. Lett. 101, 076101 (2008).
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2007 (5)

A. Scherz, W. F. Schlotter, K. Chen, R. Rick, J. Stöhr, J. Lüning, I. McNulty, C. Günther, F. Radu, W. Eberhardt, O. Hellwig, and S. Eisebitt, “Phase imaging of magnetic nanostructures using resonant soft x-ray holography,” Phys. Rev. B 76, 214410 (2007).
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B. J. Smith and M. G. Raymer, “Photon wave functions, wave-packet quantization of light, and coherence theory,” New J. Phys. 9, 414 (2007).
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F. W. Sun, B. H. Liu, Y. X. Gong, Y. F. Huang, Z. Y. Ou, and G. C. Guo, “Stimulated emission as a result of multiphoton interference,” Phys. Rev. Lett. 99, 043601 (2007).
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K. W. Chan, J. P. Torres, and J. H. Eberly, “Transverse entanglement migration in Hilbert space,” Phys. Rev. A 75, 050101 (2007).
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T. Jeltes, J. M. McNamara, W. Hogervorst, W. Vassen, V. Krachmalnicoff, M. Schellekens, A. Perrin, H. Chang, D. Boiron, and A. Aspect, and C. I. Westbrook, “Comparison of the Hanbury Brown–Twiss effect for bosons and fermions,” Nature 445, 402–405 (2007).
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T. Jeltes, J. M. McNamara, W. Hogervorst, W. Vassen, V. Krachmalnicoff, M. Schellekens, A. Perrin, H. Chang, D. Boiron, and A. Aspect, and C. I. Westbrook, “Comparison of the Hanbury Brown–Twiss effect for bosons and fermions,” Nature 445, 402–405 (2007).
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2006 (3)

R. Shimizu, K. Edamatsu, and T. Itoh, “Quantum diffraction and interference of spatially correlated photon pairs and its Fourier-optical analysis,” Phys. Rev. A 74, 013801 (2006).
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R. J. Glauber, “Nobel Lecture: One hundred years of light quanta,” Rev. Mod. Phys. 78, 1267–1278 (2006).
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A. Gatti, M. Bache, D. Magatti, E. Brambilla, F. Ferri, and L. A. Lugiato, “Coherent imaging with pseudo-thermal incoherent light,” J. Mod. Opt. 53, 739–760 (2006).
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2005 (6)

A. Valencia, G. Scarcelli, M. D’Angelo, and Y. Shih, “Two-photon imaging with thermal light,” Phys. Rev. Lett. 94, 063601 (2005).
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M. D’Angelo and Y. Shih, “Quantum imaging,” Laser Phys. Lett. 2, 567–596 (2005).
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V. Scarani, S. Iblisdir, N. Gisin, and A. Acín, “Quantum cloning,” Rev. Mod. Phys. 77, 1225–1256 (2005).
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J. Du, T. Durt, P. Zou, H. Li, L. C. Kwek, C. H. Lai, C. H. Oh, and A. Ekert, “Experimental quantum cloning with prior partial information,” Phys. Rev. Lett. 94, 040505 (2005).
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T. Qureshi, “Understanding Popper’s experiment,” Am. J. Phys. 73, 541–544 (2005).
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D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46, 15–28 (2005).
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2004 (2)

P. Emma, K. Bane, M. Cornacchia, Z. Huang, H. Schlarb, G. Stupakov, and D. Walz, “Femtosecond and subfemtosecond x-ray pulses from a self-amplified spontaneous-emission–based free-electron laser,” Phys. Rev. Lett. 92, 074801 (2004).
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S. Eisebitt, J. Lüning, and W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt, and J. Stöhr, “Lensless imaging of magnetic nanostructures by x-ray spectro-holography,” Nature 432, 885–888 (2004).
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S. Eisebitt, J. Lüning, and W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt, and J. Stöhr, “Lensless imaging of magnetic nanostructures by x-ray spectro-holography,” Nature 432, 885–888 (2004).
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2003 (5)

G. Di Giuseppe, M. Atatüre, M. D. Shaw, A. V. Sergienko, B. E. A. Saleh, M. C. Teich, A. J. Miller, S. W. Nam, and J. Martinis, “Direct observation of photon pairs at a single output port of a beam-splitter interferometer,” Phys. Rev. A 68, 063817 (2003).
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Y. H. Shih, “Entangled biphoton source–property and preparation,” Rep. Prog. Phys. 66, 1009–1044 (2003).
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S. Eisebitt, M. Lörgen, W. Eberhardt, J. Lüning, J. Stöhr, C. T. Rettner, O. Hellwig, E. E. Fullerton, and G. Denbeaux, “Polarization effects in coherent scattering from magnetic specimen: Implications for x-ray holography, lensless imaging, and correlation spectroscopy,” Phys. Rev. B 68, 104419 (2003).
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R. Shimizu, K. Edamatsu, and T. Itoh, “Quantum diffraction and interference of spatially correlated photon pairs generated by spontaneous parametric down-conversion,” Phys. Rev. A 67, 041805 (2003).
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W. J. Mullin and G. Blaylock, “Quantum statistics: Is there an effective fermion repulsion or boson attraction?” Am. J. Phys. 71, 1223–1231 (2003).
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2002 (4)

C. H. Holbrow, E. Galvez, and M. E. Parks, “Photon quantum mechanics and beam splitters,” Am. J. Phys. 70, 260–265 (2002).
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R. S. Bennink, S. J. Bentley, and R. W. Boyd, “‘Two-photon coincidence imaging with a classical source,” Phys. Rev. Lett. 89, 113601 (2002).
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A. Lamas-Linares, C. Simon, J. C. Howell, and D. Bouwmeester, “Experimental quantum cloning of single photons,” Science 296, 712–714 (2002).
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A. S. Sanz, F. Borondo, and S. Miret-Artés, “Particle diffraction studied using quantum trajectories,” J. Phys. Condens. Matter 14, 6109–6145 (2002).
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2001 (4)

B. Lengeler, “Coherence in x-ray physics,” Naturwissenschaften 88, 249–260 (2001).
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A. Abouraddy, M. Nasr, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Demonstration of the complementarity of one- and two-photon interference,” Phys. Rev. A 63, 063803 (2001).
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J. M. Raimond, M. Brune, and S. Haroche, “Manipulating quantum entanglement with atoms and photons in a cavity,” Rev. Mod. Phys. 73, 565–582 (2001).
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M. D’Angelo, M. V. Chekhova, and Y. Shih, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 87, 013602 (2001).
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2000 (3)

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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1999 (3)

L. Mandel, “Quantum effects in one-photon and two-photon interference,” Rev. Mod. Phys. 71, S274 (1999).
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1998 (3)

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1997 (1)

D. T. Pegg and S. M. Barnett, “Quantum optical phase,” J. Mod. Opt. 44, 225–264 (1997).

1996 (1)

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D. Attwood, K. Halbach, and K.-J. Kim, “Tunable coherent x-rays,” Science 228, 1265–1272 (1985).
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1983 (1)

L. Mandel, “Is a photon amplifier always polarization dependent?” Nature 304, 188 (1983).
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1981 (1)

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1980 (1)

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1978 (1)

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1977 (4)

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1976 (2)

J. T. Foley and M. S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26, 297–300 (1976).
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1974 (1)

J. F. Clauser, “Experimental distinction between the quantum and classical field—theoretic predictions for the photoelectric effect,” Phys. Rev. D 9, 853–860 (1974).
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1972 (1)

S. J. Freedman and J. F. Clauser, “Experimental test of local hidden-variable theories,” Phys. Rev. Lett. 28, 938–941 (1972).
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1971 (1)

P. Eisenberger and S. L. McCall, “X-ray parametric conversion,” Phys. Rev. Lett. 26, 684–688 (1971).
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1970 (3)

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1968 (1)

1967 (3)

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D. Magde and H. Mahr, “Study in ammonium dihydrogen phosphate of spontaneous parametric interaction tunable from 4400 to 16,000 Å,” Phys. Rev. Lett. 18, 905–907 (1967).
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1966 (2)

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1965 (2)

L. Mandel and E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231–287 (1965).
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1964 (2)

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1962 (1)

I. S. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Inf. Theory 8, 194–195 (1962).
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1961 (4)

L. Mandel, “Photon degeneracy in light from optical maser and other sources,” J. Opt. Soc. Am. 51, 797–798 (1961).
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1958 (2)

R. Hanbury Brown and R. Q. Twiss, “Interferometry of the intensity fluctuations in light. II. An experimental test of the theory for partially coherent light,” Proc. R. Soc. London Ser. A 243, 291–319 (1958).
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1957 (1)

R. Hanbury Brown and R. Q. Twiss, “Interferometry of the intensity fluctuations in light. I. Basic theory: the correlation between photons in coherent beams of radiation,” Proc. R. Soc. London Ser. A 242, 300–324 (1957).
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1956 (3)

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1955 (1)

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1952 (2)

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1951 (1)

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1949 (1)

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1948 (1)

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Abbey, B.

I. A. Vartanyants, A. Singer, A. P. Mancuso, O. M. Yefanov, A. Sakdinawat, Y. Liu, E. Bang, G. J. Williams, G. Cadenazzi, B. Abbey, H. Sinn, D. Attwood, K. A. Nugent, E. Weckert, T. Wang, D. Zhu, B. Wu, C. Graves, A. Scherz, J. J. Turner, W. F. Schlotter, M. Messerschmidt, J. Lüning, Y. Acremann, P. Heimann, D. C. Mancini, V. Joshi, J. Krzywinski, R. Soufli, M. Fernandez-Perea, S. Hau-Riege, A. G. Peele, Y. Feng, O. Krupin, S. Moeller, and W. Wurth, “Coherence properties of individual femtosecond pulses of an x-ray free-electron laser,” Phys. Rev. Lett. 107, 144801 (2011).
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Abrams, D. S.

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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T. Wang, D. Zhu, B. Wu, C. Graves, S. Schaffert, T. Rander, L. Müller, B. Vodungbo, C. Baumier, D. P. Bernstein, B. Bräuer, V. Cros, S. De Jong, R. Delaunay, A. Fognini, R. Kukreja, S. Lee, V. López-Flores, J. Mohanty, B. Pfau, H. Popescu, M. Sacchi, A. B. Sardinha, F. Sirotti, P. Zeitoun, M. Messerschmidt, J. J. Turner, W. F. Schlotter, O. Hellwig, R. Mattana, N. Jaouen, F. Fortuna, Y. Acremann, C. Gutt, H. A. Dürr, E. Beaurepaire, C. Boeglin, S. Eisebitt, G. Grübel, J. Lüning, and J. Stöhr, and A. O. Scherz, “Femtosecond single-shot imaging of nanoscale ferromagnetic order in Co/Pd multilayers using resonant x-ray holography,” Phys. Rev. Lett. 108, 267403 (2012).
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