Abstract

This paper proposes a method for the generation of high-contrast localized sinusoidal fringes with spatially noncoherent illumination and relatively high light throughput. The method, somehow similar to the classical Lau effect, is based on the use of a Fresnel biprism. It has some advantages over previous methods for the noncoherent production of interference fringes. One is the flexibility of the method, which allows the control of the fringe period by means of a simple axial shift of the biprism. Second is the rapid axial fall-off in visibility around the high-contrast fringe planes. And third is the possibility of creating fringes with increasing or with constant period as the light beam propagates. Experimental verifications of the theoretical statements are also provided.

© 2013 Optical Society of America

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    [CrossRef]

2010

A. F. Isakovic, A. Stein, J. B. Warren, A. R. Sandy, S. Narayanan, M. Sprung, J. M. Ablett, D. P. Siddons, M. Metzler, and K. Evans-Lutterodt, “A bi-prism interferometer for hard x-ray photons,” J. Synchrotron Rad. 17, 451–455 (2010).
[CrossRef]

2009

M. Martinez-Corral and G. Saavedra, “The resolution challenge in 3D optical microscopy,” Prog. Opt. 53, 1–67 (2009).
[CrossRef]

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

V. Torres-Company, C. R. Fernández-Pousa, and L. R. Chen, “Temporal Lau effect: a multiwavelength self-imaging phenomenon,” Opt. Lett. 34, 1885–1887 (2009).
[CrossRef]

2008

V. Jacques, N. D. Lai, A. Dréau, D. Zheng, D. Chauvat, F. Treussart, P. Grangier, and J.-F. Roch, “Illustration of quantum complementarity using single photons interfering on a grating,” New J. Phys. 10, 123009 (2008).
[CrossRef]

2007

2006

J. Lancis, C. M. Gómez-Sarabia, J. Ojeda-Castañeda, C. Fernández-Pouza, and P. Andrés, “Temporal Lau effect: noncoherent reconstruction of periodic pulse trains,” J. Eur. Opt. Soc. Rapid Pub. 1, 6018 (2006).
[CrossRef]

2005

V. Jacques, E. Wu, T. Toury, F. Treussart, A. Aspect, P. Grangier, and J.-F. Roch, “Single-photon wavefront-splitting interference: an illustration of the light quantum in action,” Eur. Phys. J. D 35, 561–565 (2005).
[CrossRef]

2002

2001

M. Berry, I. Marzoli, and W. Schleich, “Quantum carpets, carpets of light,” Phys. World 14, 39–46 (2001).

P. O’Shea, M. Kimmel, X. Gu, and R. Trebino, “Highly simplified device for ultrashort-pulse measurement,” Opt. Lett. 26, 932–934 (2001).
[CrossRef]

2000

M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198, 82–87 (2000).
[CrossRef]

1999

A. R. Lang and A. P. W. Makepeace, “Production of synchrotron x-ray biprism interference patterns with control of fringe spacing,” J. Synchrotron Rad. 6, 59–61 (1999).
[CrossRef]

1997

1995

M. S. Chapman, C. R. Ekstrom, T. D. Hammond, J. Schmiedmayer, B. E. Tannian, S. Wehinger, and D. E. Pritchard, “Near-field imaging of atom diffraction gratings: the atomic Talbot effect,” Phys. Rev. A 51, R14–R17 (1995).
[CrossRef]

1993

E. Noponen and J. Turunen, “Electromagnetic theory of Talbot imaging,” Opt. Commun. 98, 132–140 (1993).
[CrossRef]

J. C. Barreiro and J. Ojeda-Castañeda, “Degree of coherence: a lensless measuring technique,” Opt. Lett. 18, 302–304 (1993).
[CrossRef]

1992

E. Tepichin, E. Sanchez-Aguilera, and J. Ojeda-Castañeda, “Polarization sensitive Young interferometer,” Opt. Commun. 92, 165–169 (1992).
[CrossRef]

1991

J. C. Barreiro, P. Andrés, J. Ojeda-Castañeda, and J. Lancis, “Multiple incoherent 2D optical correlator,” Opt. Commun. 84, 237–241 (1991).
[CrossRef]

1990

G. Indebetouw, “Necessary condition for temporal self-imaging in a linear dispersive medium,” J. Mod. Opt. 37, 1439–1451 (1990).
[CrossRef]

1989

1988

J. Ojeda-Castañeda, P. Andrés, and J. Ibarra, “Lensless theta decoder with high throughput,” Opt. Commun. 67, 256–260 (1988).
[CrossRef]

1983

A. W. Lohmann and J. Ojeda-Castañeda, “Spatial periodicities in partially coherent fields,” Opt. Acta 30, 475–479 (1983).
[CrossRef]

A. W. Lohmann, J. Ojeda-Castañeda, and N. Streibl, “Spatial periodicities in coherent and partially coherent fields,” Opt. Acta 30, 1259–1266 (1983).
[CrossRef]

K.-H. Brenner, A. W. Lohmann, and J. Ojeda-Castañeda, “Lau effect: OTF theory,” Opt. Commun. 46, 14–17 (1983).
[CrossRef]

1981

G. F. Missiroli, G. Pozzi, and U. Valdri, “Electron interferometry and interference electron microscopy,” J. Phys. E Sci. Instrum. 14, 649–671 (1981).
[CrossRef]

1975

1967

1950

Ablett, J. M.

A. F. Isakovic, A. Stein, J. B. Warren, A. R. Sandy, S. Narayanan, M. Sprung, J. M. Ablett, D. P. Siddons, M. Metzler, and K. Evans-Lutterodt, “A bi-prism interferometer for hard x-ray photons,” J. Synchrotron Rad. 17, 451–455 (2010).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, 1972), Chap. 7.

Alferness, R.

Ambrosini, D.

G. S. Spagnolo, D. Ambrosini, and D. Paoletti, “Displacement measurement using the Talbot effect with a Ronchi grating,” J. Opt. A 4, 376–380 (2002).
[CrossRef]

Andrés, P.

J. Lancis, C. M. Gómez-Sarabia, J. Ojeda-Castañeda, C. Fernández-Pouza, and P. Andrés, “Temporal Lau effect: noncoherent reconstruction of periodic pulse trains,” J. Eur. Opt. Soc. Rapid Pub. 1, 6018 (2006).
[CrossRef]

J. C. Barreiro, P. Andrés, J. Ojeda-Castañeda, and J. Lancis, “Multiple incoherent 2D optical correlator,” Opt. Commun. 84, 237–241 (1991).
[CrossRef]

J. Ojeda-Castañeda, P. Andrés, and J. Ibarra, “Lensless theta decoder with high throughput,” Opt. Commun. 67, 256–260 (1988).
[CrossRef]

Aspect, A.

V. Jacques, E. Wu, T. Toury, F. Treussart, A. Aspect, P. Grangier, and J.-F. Roch, “Single-photon wavefront-splitting interference: an illustration of the light quantum in action,” Eur. Phys. J. D 35, 561–565 (2005).
[CrossRef]

Barreiro, J. C.

J. C. Barreiro and J. Ojeda-Castañeda, “Degree of coherence: a lensless measuring technique,” Opt. Lett. 18, 302–304 (1993).
[CrossRef]

J. C. Barreiro, P. Andrés, J. Ojeda-Castañeda, and J. Lancis, “Multiple incoherent 2D optical correlator,” Opt. Commun. 84, 237–241 (1991).
[CrossRef]

Berry, M.

M. Berry, I. Marzoli, and W. Schleich, “Quantum carpets, carpets of light,” Phys. World 14, 39–46 (2001).

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), Chap. 10.

Brenner, K.-H.

K.-H. Brenner, A. W. Lohmann, and J. Ojeda-Castañeda, “Lau effect: OTF theory,” Opt. Commun. 46, 14–17 (1983).
[CrossRef]

Chang, B. J.

Chapman, M. S.

M. S. Chapman, C. R. Ekstrom, T. D. Hammond, J. Schmiedmayer, B. E. Tannian, S. Wehinger, and D. E. Pritchard, “Near-field imaging of atom diffraction gratings: the atomic Talbot effect,” Phys. Rev. A 51, R14–R17 (1995).
[CrossRef]

Chauvat, D.

V. Jacques, N. D. Lai, A. Dréau, D. Zheng, D. Chauvat, F. Treussart, P. Grangier, and J.-F. Roch, “Illustration of quantum complementarity using single photons interfering on a grating,” New J. Phys. 10, 123009 (2008).
[CrossRef]

Chen, J.

Chen, L. R.

Cronin, A. D.

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

Dennis, M.

Dréau, A.

V. Jacques, N. D. Lai, A. Dréau, D. Zheng, D. Chauvat, F. Treussart, P. Grangier, and J.-F. Roch, “Illustration of quantum complementarity using single photons interfering on a grating,” New J. Phys. 10, 123009 (2008).
[CrossRef]

Ekstrom, C. R.

M. S. Chapman, C. R. Ekstrom, T. D. Hammond, J. Schmiedmayer, B. E. Tannian, S. Wehinger, and D. E. Pritchard, “Near-field imaging of atom diffraction gratings: the atomic Talbot effect,” Phys. Rev. A 51, R14–R17 (1995).
[CrossRef]

Endo, J.

Evans-Lutterodt, K.

A. F. Isakovic, A. Stein, J. B. Warren, A. R. Sandy, S. Narayanan, M. Sprung, J. M. Ablett, D. P. Siddons, M. Metzler, and K. Evans-Lutterodt, “A bi-prism interferometer for hard x-ray photons,” J. Synchrotron Rad. 17, 451–455 (2010).
[CrossRef]

Fernández-Pousa, C. R.

Fernández-Pouza, C.

J. Lancis, C. M. Gómez-Sarabia, J. Ojeda-Castañeda, C. Fernández-Pouza, and P. Andrés, “Temporal Lau effect: noncoherent reconstruction of periodic pulse trains,” J. Eur. Opt. Soc. Rapid Pub. 1, 6018 (2006).
[CrossRef]

Fujita, H.

García de Abajo, F.

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms and Optics (Wiley, 1978), Chap. 3.

Gómez-Sarabia, C. M.

J. Lancis, C. M. Gómez-Sarabia, J. Ojeda-Castañeda, C. Fernández-Pouza, and P. Andrés, “Temporal Lau effect: noncoherent reconstruction of periodic pulse trains,” J. Eur. Opt. Soc. Rapid Pub. 1, 6018 (2006).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996), Chap. 3.

Grangier, P.

V. Jacques, N. D. Lai, A. Dréau, D. Zheng, D. Chauvat, F. Treussart, P. Grangier, and J.-F. Roch, “Illustration of quantum complementarity using single photons interfering on a grating,” New J. Phys. 10, 123009 (2008).
[CrossRef]

V. Jacques, E. Wu, T. Toury, F. Treussart, A. Aspect, P. Grangier, and J.-F. Roch, “Single-photon wavefront-splitting interference: an illustration of the light quantum in action,” Eur. Phys. J. D 35, 561–565 (2005).
[CrossRef]

Gu, X.

Gustafsson, M. G. L.

M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198, 82–87 (2000).
[CrossRef]

Hammond, T. D.

M. S. Chapman, C. R. Ekstrom, T. D. Hammond, J. Schmiedmayer, B. E. Tannian, S. Wehinger, and D. E. Pritchard, “Near-field imaging of atom diffraction gratings: the atomic Talbot effect,” Phys. Rev. A 51, R14–R17 (1995).
[CrossRef]

Ibarra, J.

J. Ojeda-Castañeda, P. Andrés, and J. Ibarra, “Lensless theta decoder with high throughput,” Opt. Commun. 67, 256–260 (1988).
[CrossRef]

Indebetouw, G.

G. Indebetouw, “Necessary condition for temporal self-imaging in a linear dispersive medium,” J. Mod. Opt. 37, 1439–1451 (1990).
[CrossRef]

Isakovic, A. F.

A. F. Isakovic, A. Stein, J. B. Warren, A. R. Sandy, S. Narayanan, M. Sprung, J. M. Ablett, D. P. Siddons, M. Metzler, and K. Evans-Lutterodt, “A bi-prism interferometer for hard x-ray photons,” J. Synchrotron Rad. 17, 451–455 (2010).
[CrossRef]

Jacques, V.

V. Jacques, N. D. Lai, A. Dréau, D. Zheng, D. Chauvat, F. Treussart, P. Grangier, and J.-F. Roch, “Illustration of quantum complementarity using single photons interfering on a grating,” New J. Phys. 10, 123009 (2008).
[CrossRef]

V. Jacques, E. Wu, T. Toury, F. Treussart, A. Aspect, P. Grangier, and J.-F. Roch, “Single-photon wavefront-splitting interference: an illustration of the light quantum in action,” Eur. Phys. J. D 35, 561–565 (2005).
[CrossRef]

Jenkins, F. A.

F. A. Jenkins and H. E. White, Fundamental Optics (McGraw-Hill, 1957), Chap. 13.

Juskaitis, R.

Kimmel, M.

Kobayashi, D.

Lai, N. D.

V. Jacques, N. D. Lai, A. Dréau, D. Zheng, D. Chauvat, F. Treussart, P. Grangier, and J.-F. Roch, “Illustration of quantum complementarity using single photons interfering on a grating,” New J. Phys. 10, 123009 (2008).
[CrossRef]

Lancis, J.

J. Lancis, C. M. Gómez-Sarabia, J. Ojeda-Castañeda, C. Fernández-Pouza, and P. Andrés, “Temporal Lau effect: noncoherent reconstruction of periodic pulse trains,” J. Eur. Opt. Soc. Rapid Pub. 1, 6018 (2006).
[CrossRef]

J. C. Barreiro, P. Andrés, J. Ojeda-Castañeda, and J. Lancis, “Multiple incoherent 2D optical correlator,” Opt. Commun. 84, 237–241 (1991).
[CrossRef]

Lang, A. R.

A. R. Lang and A. P. W. Makepeace, “Production of synchrotron x-ray biprism interference patterns with control of fringe spacing,” J. Synchrotron Rad. 6, 59–61 (1999).
[CrossRef]

Leith, E. N.

Lohmann, A. W.

A. W. Lohmann and J. Ojeda-Castañeda, “Spatial periodicities in partially coherent fields,” Opt. Acta 30, 475–479 (1983).
[CrossRef]

A. W. Lohmann, J. Ojeda-Castañeda, and N. Streibl, “Spatial periodicities in coherent and partially coherent fields,” Opt. Acta 30, 1259–1266 (1983).
[CrossRef]

K.-H. Brenner, A. W. Lohmann, and J. Ojeda-Castañeda, “Lau effect: OTF theory,” Opt. Commun. 46, 14–17 (1983).
[CrossRef]

A. W. Lohmann, Optical Information Processing (Universitätsverlag Ilmenau, 2006), Chap. 20.

Makepeace, A. P. W.

A. R. Lang and A. P. W. Makepeace, “Production of synchrotron x-ray biprism interference patterns with control of fringe spacing,” J. Synchrotron Rad. 6, 59–61 (1999).
[CrossRef]

Martinez-Corral, M.

M. Martinez-Corral and G. Saavedra, “The resolution challenge in 3D optical microscopy,” Prog. Opt. 53, 1–67 (2009).
[CrossRef]

Marzoli, I.

M. Berry, I. Marzoli, and W. Schleich, “Quantum carpets, carpets of light,” Phys. World 14, 39–46 (2001).

McMorran, B. J.

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

Metzler, M.

A. F. Isakovic, A. Stein, J. B. Warren, A. R. Sandy, S. Narayanan, M. Sprung, J. M. Ablett, D. P. Siddons, M. Metzler, and K. Evans-Lutterodt, “A bi-prism interferometer for hard x-ray photons,” J. Synchrotron Rad. 17, 451–455 (2010).
[CrossRef]

Missiroli, G. F.

G. F. Missiroli, G. Pozzi, and U. Valdri, “Electron interferometry and interference electron microscopy,” J. Phys. E Sci. Instrum. 14, 649–671 (1981).
[CrossRef]

Montgomery, W.

Narayanan, S.

A. F. Isakovic, A. Stein, J. B. Warren, A. R. Sandy, S. Narayanan, M. Sprung, J. M. Ablett, D. P. Siddons, M. Metzler, and K. Evans-Lutterodt, “A bi-prism interferometer for hard x-ray photons,” J. Synchrotron Rad. 17, 451–455 (2010).
[CrossRef]

Neil, M. A. A.

Noponen, E.

E. Noponen and J. Turunen, “Electromagnetic theory of Talbot imaging,” Opt. Commun. 98, 132–140 (1993).
[CrossRef]

O’Shea, P.

Ojeda-Castañeda, J.

J. Lancis, C. M. Gómez-Sarabia, J. Ojeda-Castañeda, C. Fernández-Pouza, and P. Andrés, “Temporal Lau effect: noncoherent reconstruction of periodic pulse trains,” J. Eur. Opt. Soc. Rapid Pub. 1, 6018 (2006).
[CrossRef]

J. C. Barreiro and J. Ojeda-Castañeda, “Degree of coherence: a lensless measuring technique,” Opt. Lett. 18, 302–304 (1993).
[CrossRef]

E. Tepichin, E. Sanchez-Aguilera, and J. Ojeda-Castañeda, “Polarization sensitive Young interferometer,” Opt. Commun. 92, 165–169 (1992).
[CrossRef]

J. C. Barreiro, P. Andrés, J. Ojeda-Castañeda, and J. Lancis, “Multiple incoherent 2D optical correlator,” Opt. Commun. 84, 237–241 (1991).
[CrossRef]

E. Tepichin and J. Ojeda-Castañeda, “Talbot interferometer with simultaneous dark and bright fields,” Appl. Opt. 28, 1517–1520 (1989).
[CrossRef]

J. Ojeda-Castañeda, P. Andrés, and J. Ibarra, “Lensless theta decoder with high throughput,” Opt. Commun. 67, 256–260 (1988).
[CrossRef]

A. W. Lohmann and J. Ojeda-Castañeda, “Spatial periodicities in partially coherent fields,” Opt. Acta 30, 475–479 (1983).
[CrossRef]

A. W. Lohmann, J. Ojeda-Castañeda, and N. Streibl, “Spatial periodicities in coherent and partially coherent fields,” Opt. Acta 30, 1259–1266 (1983).
[CrossRef]

K.-H. Brenner, A. W. Lohmann, and J. Ojeda-Castañeda, “Lau effect: OTF theory,” Opt. Commun. 46, 14–17 (1983).
[CrossRef]

Paoletti, D.

G. S. Spagnolo, D. Ambrosini, and D. Paoletti, “Displacement measurement using the Talbot effect with a Ronchi grating,” J. Opt. A 4, 376–380 (2002).
[CrossRef]

Patorski, K.

K. Patorski, “The self-imaging phenomenon and its applications,” Prog. Opt. 27, 1–108 (1989).
[CrossRef]

Pozzi, G.

G. F. Missiroli, G. Pozzi, and U. Valdri, “Electron interferometry and interference electron microscopy,” J. Phys. E Sci. Instrum. 14, 649–671 (1981).
[CrossRef]

Pritchard, D. E.

M. S. Chapman, C. R. Ekstrom, T. D. Hammond, J. Schmiedmayer, B. E. Tannian, S. Wehinger, and D. E. Pritchard, “Near-field imaging of atom diffraction gratings: the atomic Talbot effect,” Phys. Rev. A 51, R14–R17 (1995).
[CrossRef]

Roch, J.-F.

V. Jacques, N. D. Lai, A. Dréau, D. Zheng, D. Chauvat, F. Treussart, P. Grangier, and J.-F. Roch, “Illustration of quantum complementarity using single photons interfering on a grating,” New J. Phys. 10, 123009 (2008).
[CrossRef]

V. Jacques, E. Wu, T. Toury, F. Treussart, A. Aspect, P. Grangier, and J.-F. Roch, “Single-photon wavefront-splitting interference: an illustration of the light quantum in action,” Eur. Phys. J. D 35, 561–565 (2005).
[CrossRef]

Saavedra, G.

M. Martinez-Corral and G. Saavedra, “The resolution challenge in 3D optical microscopy,” Prog. Opt. 53, 1–67 (2009).
[CrossRef]

Sanchez-Aguilera, E.

E. Tepichin, E. Sanchez-Aguilera, and J. Ojeda-Castañeda, “Polarization sensitive Young interferometer,” Opt. Commun. 92, 165–169 (1992).
[CrossRef]

Sandy, A. R.

A. F. Isakovic, A. Stein, J. B. Warren, A. R. Sandy, S. Narayanan, M. Sprung, J. M. Ablett, D. P. Siddons, M. Metzler, and K. Evans-Lutterodt, “A bi-prism interferometer for hard x-ray photons,” J. Synchrotron Rad. 17, 451–455 (2010).
[CrossRef]

Schleich, W.

M. Berry, I. Marzoli, and W. Schleich, “Quantum carpets, carpets of light,” Phys. World 14, 39–46 (2001).

Schmiedmayer, J.

M. S. Chapman, C. R. Ekstrom, T. D. Hammond, J. Schmiedmayer, B. E. Tannian, S. Wehinger, and D. E. Pritchard, “Near-field imaging of atom diffraction gratings: the atomic Talbot effect,” Phys. Rev. A 51, R14–R17 (1995).
[CrossRef]

Siddons, D. P.

A. F. Isakovic, A. Stein, J. B. Warren, A. R. Sandy, S. Narayanan, M. Sprung, J. M. Ablett, D. P. Siddons, M. Metzler, and K. Evans-Lutterodt, “A bi-prism interferometer for hard x-ray photons,” J. Synchrotron Rad. 17, 451–455 (2010).
[CrossRef]

Spagnolo, G. S.

G. S. Spagnolo, D. Ambrosini, and D. Paoletti, “Displacement measurement using the Talbot effect with a Ronchi grating,” J. Opt. A 4, 376–380 (2002).
[CrossRef]

Sprung, M.

A. F. Isakovic, A. Stein, J. B. Warren, A. R. Sandy, S. Narayanan, M. Sprung, J. M. Ablett, D. P. Siddons, M. Metzler, and K. Evans-Lutterodt, “A bi-prism interferometer for hard x-ray photons,” J. Synchrotron Rad. 17, 451–455 (2010).
[CrossRef]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, 1972), Chap. 7.

Stein, A.

A. F. Isakovic, A. Stein, J. B. Warren, A. R. Sandy, S. Narayanan, M. Sprung, J. M. Ablett, D. P. Siddons, M. Metzler, and K. Evans-Lutterodt, “A bi-prism interferometer for hard x-ray photons,” J. Synchrotron Rad. 17, 451–455 (2010).
[CrossRef]

Streibl, N.

A. W. Lohmann, J. Ojeda-Castañeda, and N. Streibl, “Spatial periodicities in coherent and partially coherent fields,” Opt. Acta 30, 1259–1266 (1983).
[CrossRef]

Tannian, B. E.

M. S. Chapman, C. R. Ekstrom, T. D. Hammond, J. Schmiedmayer, B. E. Tannian, S. Wehinger, and D. E. Pritchard, “Near-field imaging of atom diffraction gratings: the atomic Talbot effect,” Phys. Rev. A 51, R14–R17 (1995).
[CrossRef]

Tepichin, E.

E. Tepichin, E. Sanchez-Aguilera, and J. Ojeda-Castañeda, “Polarization sensitive Young interferometer,” Opt. Commun. 92, 165–169 (1992).
[CrossRef]

E. Tepichin and J. Ojeda-Castañeda, “Talbot interferometer with simultaneous dark and bright fields,” Appl. Opt. 28, 1517–1520 (1989).
[CrossRef]

Toraldo di Francia, G.

Torres-Company, V.

Toury, T.

V. Jacques, E. Wu, T. Toury, F. Treussart, A. Aspect, P. Grangier, and J.-F. Roch, “Single-photon wavefront-splitting interference: an illustration of the light quantum in action,” Eur. Phys. J. D 35, 561–565 (2005).
[CrossRef]

Trebino, R.

Treussart, F.

V. Jacques, N. D. Lai, A. Dréau, D. Zheng, D. Chauvat, F. Treussart, P. Grangier, and J.-F. Roch, “Illustration of quantum complementarity using single photons interfering on a grating,” New J. Phys. 10, 123009 (2008).
[CrossRef]

V. Jacques, E. Wu, T. Toury, F. Treussart, A. Aspect, P. Grangier, and J.-F. Roch, “Single-photon wavefront-splitting interference: an illustration of the light quantum in action,” Eur. Phys. J. D 35, 561–565 (2005).
[CrossRef]

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E. Noponen and J. Turunen, “Electromagnetic theory of Talbot imaging,” Opt. Commun. 98, 132–140 (1993).
[CrossRef]

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G. F. Missiroli, G. Pozzi, and U. Valdri, “Electron interferometry and interference electron microscopy,” J. Phys. E Sci. Instrum. 14, 649–671 (1981).
[CrossRef]

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Warren, J. B.

A. F. Isakovic, A. Stein, J. B. Warren, A. R. Sandy, S. Narayanan, M. Sprung, J. M. Ablett, D. P. Siddons, M. Metzler, and K. Evans-Lutterodt, “A bi-prism interferometer for hard x-ray photons,” J. Synchrotron Rad. 17, 451–455 (2010).
[CrossRef]

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M. S. Chapman, C. R. Ekstrom, T. D. Hammond, J. Schmiedmayer, B. E. Tannian, S. Wehinger, and D. E. Pritchard, “Near-field imaging of atom diffraction gratings: the atomic Talbot effect,” Phys. Rev. A 51, R14–R17 (1995).
[CrossRef]

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F. A. Jenkins and H. E. White, Fundamental Optics (McGraw-Hill, 1957), Chap. 13.

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Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), Chap. 10.

Wood, R. W.

R. W. Wood, Physical Optics (Optical Society of America, 1988), Chap. 6.

Wu, E.

V. Jacques, E. Wu, T. Toury, F. Treussart, A. Aspect, P. Grangier, and J.-F. Roch, “Single-photon wavefront-splitting interference: an illustration of the light quantum in action,” Eur. Phys. J. D 35, 561–565 (2005).
[CrossRef]

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Zheng, D.

V. Jacques, N. D. Lai, A. Dréau, D. Zheng, D. Chauvat, F. Treussart, P. Grangier, and J.-F. Roch, “Illustration of quantum complementarity using single photons interfering on a grating,” New J. Phys. 10, 123009 (2008).
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Eur. Phys. J. D

V. Jacques, E. Wu, T. Toury, F. Treussart, A. Aspect, P. Grangier, and J.-F. Roch, “Single-photon wavefront-splitting interference: an illustration of the light quantum in action,” Eur. Phys. J. D 35, 561–565 (2005).
[CrossRef]

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J. Lancis, C. M. Gómez-Sarabia, J. Ojeda-Castañeda, C. Fernández-Pouza, and P. Andrés, “Temporal Lau effect: noncoherent reconstruction of periodic pulse trains,” J. Eur. Opt. Soc. Rapid Pub. 1, 6018 (2006).
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G. S. Spagnolo, D. Ambrosini, and D. Paoletti, “Displacement measurement using the Talbot effect with a Ronchi grating,” J. Opt. A 4, 376–380 (2002).
[CrossRef]

J. Opt. Soc. Am.

J. Phys. E Sci. Instrum.

G. F. Missiroli, G. Pozzi, and U. Valdri, “Electron interferometry and interference electron microscopy,” J. Phys. E Sci. Instrum. 14, 649–671 (1981).
[CrossRef]

J. Synchrotron Rad.

A. R. Lang and A. P. W. Makepeace, “Production of synchrotron x-ray biprism interference patterns with control of fringe spacing,” J. Synchrotron Rad. 6, 59–61 (1999).
[CrossRef]

A. F. Isakovic, A. Stein, J. B. Warren, A. R. Sandy, S. Narayanan, M. Sprung, J. M. Ablett, D. P. Siddons, M. Metzler, and K. Evans-Lutterodt, “A bi-prism interferometer for hard x-ray photons,” J. Synchrotron Rad. 17, 451–455 (2010).
[CrossRef]

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V. Jacques, N. D. Lai, A. Dréau, D. Zheng, D. Chauvat, F. Treussart, P. Grangier, and J.-F. Roch, “Illustration of quantum complementarity using single photons interfering on a grating,” New J. Phys. 10, 123009 (2008).
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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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A. W. Lohmann and J. Ojeda-Castañeda, “Spatial periodicities in partially coherent fields,” Opt. Acta 30, 475–479 (1983).
[CrossRef]

A. W. Lohmann, J. Ojeda-Castañeda, and N. Streibl, “Spatial periodicities in coherent and partially coherent fields,” Opt. Acta 30, 1259–1266 (1983).
[CrossRef]

Opt. Commun.

J. Ojeda-Castañeda, P. Andrés, and J. Ibarra, “Lensless theta decoder with high throughput,” Opt. Commun. 67, 256–260 (1988).
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J. C. Barreiro, P. Andrés, J. Ojeda-Castañeda, and J. Lancis, “Multiple incoherent 2D optical correlator,” Opt. Commun. 84, 237–241 (1991).
[CrossRef]

E. Noponen and J. Turunen, “Electromagnetic theory of Talbot imaging,” Opt. Commun. 98, 132–140 (1993).
[CrossRef]

E. Tepichin, E. Sanchez-Aguilera, and J. Ojeda-Castañeda, “Polarization sensitive Young interferometer,” Opt. Commun. 92, 165–169 (1992).
[CrossRef]

K.-H. Brenner, A. W. Lohmann, and J. Ojeda-Castañeda, “Lau effect: OTF theory,” Opt. Commun. 46, 14–17 (1983).
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Phys. Rev. A

M. S. Chapman, C. R. Ekstrom, T. D. Hammond, J. Schmiedmayer, B. E. Tannian, S. Wehinger, and D. E. Pritchard, “Near-field imaging of atom diffraction gratings: the atomic Talbot effect,” Phys. Rev. A 51, R14–R17 (1995).
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M. Berry, I. Marzoli, and W. Schleich, “Quantum carpets, carpets of light,” Phys. World 14, 39–46 (2001).

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K. Patorski, “The self-imaging phenomenon and its applications,” Prog. Opt. 27, 1–108 (1989).
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M. Martinez-Corral and G. Saavedra, “The resolution challenge in 3D optical microscopy,” Prog. Opt. 53, 1–67 (2009).
[CrossRef]

Other

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), Chap. 10.

R. W. Wood, Physical Optics (Optical Society of America, 1988), Chap. 6.

F. A. Jenkins and H. E. White, Fundamental Optics (McGraw-Hill, 1957), Chap. 13.

See Ref. [17], Chap. 7.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996), Chap. 3.

J. D. Gaskill, Linear Systems, Fourier Transforms and Optics (Wiley, 1978), Chap. 3.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, 1972), Chap. 7.

A. W. Lohmann, Optical Information Processing (Universitätsverlag Ilmenau, 2006), Chap. 20.

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Figures (16)

Fig. 1.
Fig. 1.

Geometric scheme to study the diffracted field by a Fresnel biprism illuminated by a spatially incoherent quasi-monochromatic planar source.

Fig. 2.
Fig. 2.

Absolute value of the visibility of the interference patterns obtained in planes behind the biprism. For the calculation of these curves we assumed a specific setup in which n=1.5, δ=0.51°, λ=0.634μm, x0=400μm, and Δ=60μm.

Fig. 3.
Fig. 3.

Scheme for the production of far-field periodic resonant patterns.

Fig. 4.
Fig. 4.

Visibility of fringe patterns behind the biprism when the array of linear sources is placed at infinity by using a converging lens. For the calculation of these curves we assumed a particular case: n=1.5, δ=0.51°, λ=0.634μm, f=200mm, x0=400μm, and Δ=60μm.

Fig. 5.
Fig. 5.

Numerically evaluated envelope functions V(x⃗,z;η) and env(x⃗,z;η) for (a) z=116mm and (b) z=261mm. The dashed lines define the region of interference superposition in the parageometrical approximation. For the calculation we set n=1.5, δ=0.51°, λ=0.634μm, and η=207.8mm.

Fig. 6.
Fig. 6.

Numerically evaluated fringe patterns located behind the biprism for different numbers of incoherent point sources. For the calculation we set n=1.5, δ=0.51°, λ=0.634μm, η=207.8mm, x0=400μm, and Δ=60μm.

Fig. 7.
Fig. 7.

Numerically evaluated fringe profile in the planes (a) z=105mm (maximum visibility) and (b) z=160mm (minimum visibility), for the case N=5 of point sources.

Fig. 8.
Fig. 8.

Numerically evaluated fringe patterns located behind the biprism for different numbers of incoherent point sources located at infinity. For the calculation we set n=1.5, δ=0.51°, λ=0.634μm, f=200mm, x0=400μm, and Δ=60μm.

Fig. 9.
Fig. 9.

Schematic diagram of the experimental setup used for the production of resonant superposition of interference fringe.

Fig. 10.
Fig. 10.

Interference patterns obtained for N=5 at different distances from the Fresnel biprism. The patterns shown in the right column correspond to the first, second, and third planes of resonance, respectively.

Fig. 11.
Fig. 11.

Experimental irradiance distribution along the axial direction for different numbers of finite-width sources.

Fig. 12.
Fig. 12.

Experimentally measured fringe patterns corresponding to N=5 point sources and a plane of (a) maximum visibility (z=105mm) and (b) minimum visibility (z=160mm).

Fig. 13.
Fig. 13.

Numerically evaluated (triangles) and experimentally measured (asterisks) values of the visibility of the fringes for different numbers of incoherent finite-width sources.

Fig. 14.
Fig. 14.

Schematic diagram for generating interference patterns behind a Fresnel biprism when finite-width sources are located at an infinite distance.

Fig. 15.
Fig. 15.

Experimental irradiance distribution along the axial direction N=5 for sources when the grating is located at the front focal plane of a converging lens.

Fig. 16.
Fig. 16.

Numerically evaluated (triangles) and experimentally measured (dots) values of the visibility of the fringes for N=5.

Equations (22)

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I(x⃗,z;η)=1MS2IS(x⃗MS)2I0(x⃗,z;η),
I0(x⃗,z;η)=1+cos(2πxp).
p=M2u0,
I(x⃗,z;η)=I˜S(0⃗)MS2[1+V(z)cos(2πxp+Φ(z))],
I˜S(u⃗)=|I˜S(u⃗)|exp(jΦ(u⃗)),
V(z)=|I˜S(MSp,0)|I˜S(0⃗)=|I˜S(2u0zz+η,0)|I˜S(0⃗).
IS(x⃗)=I=1NIPδ(xx1,y),
x1=(N+12I)x0I=1,,N,
I(x⃗,z;η)=IPI=1NI0(xMSx1,y,z;η).
V(z)=sin(πNMSx0p)Nsin(πMSx0p).
zm=mλη2x0(n1)δmλ
V(z)=sin(πNMSx0p)Nsin(πMSx0p)sinc(ΔMSp).
I(x⃗,z)1+V(z)cos(2πxp),
p=12u0
V(z)=sin(πNMSx0p)Nsin(πMSx0p)sinc(ΔMSp),
T=f2u0x0.
t(x⃗)=exp(j2πu0x)step(x⃗),
U0=12Mexp[jπλzM(x+λzu0)2]{1+j2+Fres[2λzM(x+λzu0)]}+12Mexp[jπλzM(xλzu0)2]{1+j2Fres[2λzM(xλzu0)]},
Fres[α]=0αexp(jπ2x2)dx=C(α)+jS[α].
I0(x⃗,z;η)=env(x⃗,z;η)2M[1+V(x⃗,z;η)cos(2πxp)],
env(x⃗,z;η)=1+C[2λzM(x+λzu0)]2+S[2λzM(x+λzu0)]2+C[2λzM(x+λzu0)]+S[2λzM(x+λzu0)]+C[2λzM(xλzu0)]2+S[2λzM(xλzu0)]2C[2λzM(xλzu0)]S[2λzM(xλzu0)],
V(x⃗,z;η)=1env(x⃗,z;η){1+2C[2λzM(x+λzu0)]C[2λzM(xλzu0)]2S[2λzM(x+λzu0)]S[2λzM(xλzu0)]+C[2λzM(x+λzu0)]+S[2λzM(x+λzu0)]C[2λzM(xλzu0)]S[2λzM(xλzu0)]}.

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