Abstract

An interesting feature of light fields is a phase anomaly, which occurs on the optical axis when light is converging as in a focal spot. Since in Talbot images the light is periodically confined in both transverse and axial directions, it remains an open question whether at all and to which extent the phase in the Talbot images sustains an analogous phase anomaly. Here, we investigate experimentally and theoretically the anomalous phase behavior of Talbot images that emerge from a 1D amplitude grating with a period only slightly larger than the illumination wavelength. Talbot light carpets are observed close to the grating. We concisely show that the phase in each of the Talbot images possesses an anomalous axial shift. We show that this phase shift is analogous to a Gouy phase of a converging wave and occurs due to the periodic light confinement caused by the interference of various diffraction orders. Longitudinal-differential interferometry is used to directly demonstrate the axial phase shifts by comparing Talbot images phase maps to a plane wave. Supporting simulations based on rigorous diffraction theory are used to explore the effect numerically. Numerical and experimental results are in excellent agreement. We discover that the phase anomaly, i.e., the difference of the phase of the field behind the grating to the phase of a referential plane wave, is an increasing function with respect to the propagation distance. We also observe within one Talbot length an irregular wavefront spacing that causes a deviation from the linear slope of the phase anomaly. We complement our work by providing an analytical model that explains these features of the axial phase shift.

© 2013 OSA

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2012 (8)

M.-S. Kim, T. Scharf, C. Menzel, C. Rockstuhl, and H. P. Herzig, “Talbot images of wavelength-scale amplitude gratings,” Opt. Express20(5), 4903–4920 (2012).
[CrossRef] [PubMed]

M.-S. Kim, T. Scharf, C. Etrich, C. Rockstuhl, and H. H. Peter, “Longitudinal-differential interferometry: Direct imaging of axial superluminal phase propagation,” Opt. Lett.37(3), 305–307 (2012).
[CrossRef] [PubMed]

J. P. Rolland, K. P. Thompson, K.-S. Lee, J. Tamkin, T. Schmid, and E. Wolf, “Observation of the Gouy phase anomaly in astigmatic beams,” Appl. Opt.51(15), 2902–2908 (2012).
[CrossRef] [PubMed]

B. Roy, S. B. Pal, A. Haldar, R. K. Gupta, N. Ghosh, and A. Banerjee, “Probing the dynamics of an optically trapped particle by phase sensitive back focal plane interferometry,” Opt. Express20(8), 8317–8328 (2012).
[CrossRef] [PubMed]

L. Friedrich and A. Rohrbach, “Tuning the detection sensitivity: a model for axial backfocal plane interferometric tracking,” Opt. Lett.37(11), 2109–2111 (2012).
[CrossRef] [PubMed]

H. He and X.-C. Zhang, “Analysis of Gouy phase shift for optimizing terahertz air-biased-coherent-detection,” Appl. Phys. Lett.100(6), 061105 (2012).
[CrossRef]

H. X. Cui, X. L. Wang, B. Gu, Y. N. Li, J. Chen, and H. T. Wang, “Angular diffraction of an optical vortex induced by the Gouy phase,” J. Opt.14(5), 055707 (2012).
[CrossRef]

X. Pang, D. G. Fischer, and T. D. Visser, “Generalized Gouy phase for focused partially coherent light and its implications for interferometry,” J. Opt. Soc. Am. A29(6), 989–993 (2012).
[CrossRef] [PubMed]

2011 (6)

X. Pang, G. Gbur, and T. D. Visser, “The Gouy phase of Airy beams,” Opt. Lett.36(13), 2492–2494 (2011).
[CrossRef] [PubMed]

M.-S. Kim, T. Scharf, S. Mühlig, C. Rockstuhl, and H. P. Herzig, “Engineering photonic nanojets,” Opt. Express19(11), 10206–10220 (2011).
[CrossRef] [PubMed]

I. G. Da Pazl, P. L. Saldanha, M. C. Nemes, and J. G. Peixoto de Faria, “Experimental proposal for measuring the Gouy phase of matter waves,” New J. Phys.13, 125005 (2011).

M.-S. Kim, T. Scharf, S. Mühlig, C. Rockstuhl, and H. P. Herzig, “Gouy phase anomaly in photonic nanojets,” Appl. Phys. Lett.98(19), 191114 (2011).
[CrossRef]

F. J. Torcal-Milla, L. M. Sanchez-Brea, and J. Vargas, “Effect of aberrations on the self-imaging phenomenon,” J. Lightwave Technol.29(7), 1051–1057 (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(3), 033902 (2011).
[CrossRef] [PubMed]

2010 (5)

2009 (4)

J. F. Federici, R. L. Wample, D. Rodriguez, and S. Mukherjee, “Application of terahertz Gouy phase shift from curved surfaces for estimation of crop yield,” Appl. Opt.48(7), 1382–1388 (2009).
[CrossRef] [PubMed]

P. Maddaloni, M. Paturzo, P. Ferraro, P. Malara, P. De Natale, M. Gioffrè, G. Coppola, and M. Iodice, “Mid-infrared tunable two-dimensional Talbot array illuminator,” Appl. Phys. Lett.94(12), 121105 (2009).
[CrossRef]

B. J. McMorran and A. D. Cronin, “An electron Talbot interferometer,” New J. Phys.11(3), 033021 (2009).
[CrossRef]

S. Cherukulappurath, D. Heinis, J. Cesario, N. F. van Hulst, S. Enoch, and R. Quidant, “Local observation of plasmon focusingin Talbot carpets,” Opt. Express17(26), 23772–23784 (2009).
[CrossRef] [PubMed]

2008 (2)

2007 (6)

W. Zhu, A. Agrawal, and A. Nahata, “Direct measurement of the Gouy phase shift for surface plasmon-polaritons,” Opt. Express15(16), 9995–10001 (2007).
[CrossRef] [PubMed]

H. Chen, Q. Zhan, Y. Zhang, and Y.-P. Li, “The Gouy phase shift of the highly focused radially polarized beam,” Phys. Lett. A371(3), 259–261 (2007).
[CrossRef]

M. Vasnetsov, V. Pas’ko, A. Khoroshun, V. Slyusar, and M. Soskin, “Observation of superluminal wave-front propagation at the shadow area behind an opaque disk,” Opt. Lett.32(13), 1830–1832 (2007).
[CrossRef] [PubMed]

M. Thiel, M. Hermatschweiler, M. Wegener, and G. von Freymann, “Thin-film polarizer based on a one-dimensional–three-dimensional–one-dimensional photonic crystal heterostructure,” Appl. Phys. Lett.91(12), 123515 (2007).
[CrossRef]

M. R. Dennis, N. I. Zheludev, and F. J. García de Abajo, “The plasmon Talbot effect,” Opt. Express15(15), 9692–9700 (2007).
[CrossRef] [PubMed]

F. M. Huang, N. Zheludev, Y. Chen, and F. Javier Garcia de Abajo, “Focusing of light by a nanohole array,” Appl. Phys. Lett.90(9), 091119 (2007).
[CrossRef]

2006 (2)

2005 (3)

2004 (1)

Q. Zhan, “Second-order tilted wave interpretation of the Gouy phase shift under high numerical aperture uniform illumination,” Opt. Commun.242(4-6), 351–360 (2004).
[CrossRef]

2003 (1)

N. C. R. Holme, B. C. Daly, M. T. Myaing, and T. B. Norris, “Gouy phase shift of single-cycle picosecond acoustic pulses,” Appl. Phys. Lett.83(2), 392–394 (2003).
[CrossRef]

2002 (2)

D. Chauvat, O. Emile, M. Brunel, and A. Le Floch, “Direct measurement of the central fringe velocity in Young-type experiments,” Phys. Lett. A295(2-3), 78–80 (2002).
[CrossRef]

G. Spagnolo, D. Ambrosini, and D. Paoletti, “Displacement measurement using the Talbot effect with a Ronchi grating,” J. Opt. Soc. A4(6), S376–S380 (2002).
[CrossRef]

2001 (3)

R. Gadonas, V. Jarutis, R. Paškauskas, V. Smilgevičius, A. Stabinis, and V. Vaičaitis, “Self-action of Bessel beam in nonlinear medium,” Opt. Commun.196(1-6), 309–316 (2001).
[CrossRef]

T. Ackemann, W. Grosse-Nobis, and G. L. Lippi, “The Gouy phase shift, the average phase lag of Fourier components of Hermite-Gaussian modes and their application to resonance conditions in optical cavities,” Opt. Commun.189(1-3), 5–14 (2001).
[CrossRef]

S. Feng and H. G. Winful, “Physical origin of the Gouy phase shift,” Opt. Lett.26(8), 485–487 (2001).
[CrossRef] [PubMed]

2000 (1)

G. F. Brand, “A new millimeter wave geometric phase demonstration,” Int. J. Infrared Millim. Waves21(4), 505–518 (2000).
[CrossRef]

1999 (1)

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, “Direct observation of the Gouy phase shift with single-cycle terahertz pulse,” Phys. Rev. Lett.83(17), 3410–3413 (1999).
[CrossRef]

1998 (2)

1997 (3)

1996 (2)

M. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt.43(10), 2139–2164 (1996).
[CrossRef]

P. Hariharan and P. A. Robinson, “The Gouy phase shift as a geometrical quantum effect,” J. Mod. Opt.43, 219–221 (1996).

1995 (2)

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. A51(1), R14–R17 (1995).
[CrossRef] [PubMed]

D. Subbarao, “Topological phase in Gaussian beam optics,” Opt. Lett.20(21), 2162–2164 (1995).
[CrossRef] [PubMed]

1994 (1)

1993 (2)

R. Simon and N. Mukunda, “Bargmann invariant and the geometry of the Güoy effect,” Phys. Rev. Lett.70(7), 880–883 (1993).
[CrossRef] [PubMed]

E. Noponen and J. Turunen, “Electromagnetic theory of Talbot imaging,” Opt. Commun.98(1-3), 132–140 (1993).
[CrossRef]

1991 (2)

E. Bonet, J. Ojeda-Castañeda, and A. Pons, “Imagesynthesis using the Laueffect,” Opt. Commun.81(5), 285–290 (1991).
[CrossRef]

J. C. Barreiro, P. Andrés, J. Ojeda-Castañeda, and J. Lancis, “Multiple incoherent 2D optical correlator,” Opt. Commun.84(5-6), 237–241 (1991).
[CrossRef]

1989 (3)

1988 (1)

J. R. Leger, M. L. Scott, and W. B. Veldkamp, “Coherent addition of AlGaAs lasers using microlenses and diffractive coupling,” Appl. Phys. Lett.52(21), 1771–1773 (1988).
[CrossRef]

1985 (1)

A. Kołodziejczyk, “Realization of Fourier images without using a lens by sampling the optical object,” J. Mod. Opt.32, 741–746 (1985).

1983 (1)

A. W. Lohmann and J. Ojeda-Castaneda, “Spatial periodicities in partially coherent fields,” Opt. Acta: Int. J. Opt.30(4), 475–479 (1983).
[CrossRef]

1980 (1)

1973 (1)

1969 (1)

R. F. Edgar, “The Fresnel diffraction images of periodic structures,” J. Mod. Opt.16, 281–287 (1969).

1966 (1)

1965 (1)

1959 (1)

C. R. Carpenter, “Gouy phase advance with microwaves,” Am. J. Phys.27, 98–100 (1959).

1948 (1)

E. Lau, “Beugungserscheinungen an Doppelrastern,” Ann. Phys.437(7-8), 417–423 (1948).
[CrossRef]

1890 (1)

L. G. Gouy, “Sur une propriété nouvelle des ondes lumineuses,” C. R. Acad. Sci. Paris110, 1251–1253 (1890).

1881 (1)

L. Rayleigh, “On copying diffraction gratings and some phenomena connected therewith,” Philos. Mag.11(67), 196–205 (1881).
[CrossRef]

1836 (1)

F. Talbot, “Facts relating to optical science. No. IV,” Philos. Mag.9, 401–407 (1836).

Ackemann, T.

T. Ackemann, W. Grosse-Nobis, and G. L. Lippi, “The Gouy phase shift, the average phase lag of Fourier components of Hermite-Gaussian modes and their application to resonance conditions in optical cavities,” Opt. Commun.189(1-3), 5–14 (2001).
[CrossRef]

Agrawal, A.

Ambrosini, D.

G. Spagnolo, D. Ambrosini, and D. Paoletti, “Displacement measurement using the Talbot effect with a Ronchi grating,” J. Opt. Soc. A4(6), S376–S380 (2002).
[CrossRef]

Andrés, P.

J. C. Barreiro, P. Andrés, J. Ojeda-Castañeda, and J. Lancis, “Multiple incoherent 2D optical correlator,” Opt. Commun.84(5-6), 237–241 (1991).
[CrossRef]

Banerjee, A.

Barreiro, J. C.

J. C. Barreiro, P. Andrés, J. Ojeda-Castañeda, and J. Lancis, “Multiple incoherent 2D optical correlator,” Opt. Commun.84(5-6), 237–241 (1991).
[CrossRef]

Baruchel, J.

Berry, M.

M. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt.43(10), 2139–2164 (1996).
[CrossRef]

Bhattacharya, J. C.

Bonet, E.

E. Bonet, J. Ojeda-Castañeda, and A. Pons, “Imagesynthesis using the Laueffect,” Opt. Commun.81(5), 285–290 (1991).
[CrossRef]

Boyd, R. W.

Brand, G. F.

G. F. Brand, “A new millimeter wave geometric phase demonstration,” Int. J. Infrared Millim. Waves21(4), 505–518 (2000).
[CrossRef]

Brunel, M.

D. Chauvat, O. Emile, M. Brunel, and A. Le Floch, “Direct measurement of the central fringe velocity in Young-type experiments,” Phys. Lett. A295(2-3), 78–80 (2002).
[CrossRef]

Bryngdahl, O.

Carpenter, C. R.

C. R. Carpenter, “Gouy phase advance with microwaves,” Am. J. Phys.27, 98–100 (1959).

Carrasco, S.

Cesario, J.

Chang, R.-C.

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. A51(1), R14–R17 (1995).
[CrossRef] [PubMed]

Chauvat, D.

D. Chauvat, O. Emile, M. Brunel, and A. Le Floch, “Direct measurement of the central fringe velocity in Young-type experiments,” Phys. Lett. A295(2-3), 78–80 (2002).
[CrossRef]

Chen, H.

H. Chen, Q. Zhan, Y. Zhang, and Y.-P. Li, “The Gouy phase shift of the highly focused radially polarized beam,” Phys. Lett. A371(3), 259–261 (2007).
[CrossRef]

Chen, J.

H. X. Cui, X. L. Wang, B. Gu, Y. N. Li, J. Chen, and H. T. Wang, “Angular diffraction of an optical vortex induced by the Gouy phase,” J. Opt.14(5), 055707 (2012).
[CrossRef]

Chen, Y.

F. M. Huang, N. Zheludev, Y. Chen, and F. Javier Garcia de Abajo, “Focusing of light by a nanohole array,” Appl. Phys. Lett.90(9), 091119 (2007).
[CrossRef]

Cheng, C.

Cheng, Y.-S.

Cherukulappurath, S.

Cloetens, P.

Coppola, G.

P. Maddaloni, M. Paturzo, P. Ferraro, P. Malara, P. De Natale, M. Gioffrè, G. Coppola, and M. Iodice, “Mid-infrared tunable two-dimensional Talbot array illuminator,” Appl. Phys. Lett.94(12), 121105 (2009).
[CrossRef]

Courtial, J.

J. Courtial, “Self-imaging beams and the Guoy effect,” Opt. Commun.151(1-3), 1–4 (1998).
[CrossRef]

Cronin, A. D.

B. J. McMorran and A. D. Cronin, “An electron Talbot interferometer,” New J. Phys.11(3), 033021 (2009).
[CrossRef]

Cui, H. X.

H. X. Cui, X. L. Wang, B. Gu, Y. N. Li, J. Chen, and H. T. Wang, “Angular diffraction of an optical vortex induced by the Gouy phase,” J. Opt.14(5), 055707 (2012).
[CrossRef]

Da Pazl, I. G.

I. G. Da Pazl, P. L. Saldanha, M. C. Nemes, and J. G. Peixoto de Faria, “Experimental proposal for measuring the Gouy phase of matter waves,” New J. Phys.13, 125005 (2011).

Daly, B. C.

N. C. R. Holme, B. C. Daly, M. T. Myaing, and T. B. Norris, “Gouy phase shift of single-cycle picosecond acoustic pulses,” Appl. Phys. Lett.83(2), 392–394 (2003).
[CrossRef]

David, C.

De Martino, C.

De Natale, P.

P. Maddaloni, M. Paturzo, P. Ferraro, P. Malara, P. De Natale, M. Gioffrè, G. Coppola, and M. Iodice, “Mid-infrared tunable two-dimensional Talbot array illuminator,” Appl. Phys. Lett.94(12), 121105 (2009).
[CrossRef]

Dennis, M. R.

Edgar, R. F.

R. F. Edgar, “The Fresnel diffraction images of periodic structures,” J. Mod. Opt.16, 281–287 (1969).

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. A51(1), R14–R17 (1995).
[CrossRef] [PubMed]

Emile, O.

D. Chauvat, O. Emile, M. Brunel, and A. Le Floch, “Direct measurement of the central fringe velocity in Young-type experiments,” Phys. Lett. A295(2-3), 78–80 (2002).
[CrossRef]

Enoch, S.

Etrich, C.

Federici, J. F.

Feng, S.

S. Feng and H. G. Winful, “Physical origin of the Gouy phase shift,” Opt. Lett.26(8), 485–487 (2001).
[CrossRef] [PubMed]

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, “Direct observation of the Gouy phase shift with single-cycle terahertz pulse,” Phys. Rev. Lett.83(17), 3410–3413 (1999).
[CrossRef]

Ferraro, P.

P. Maddaloni, M. Paturzo, P. Ferraro, P. Malara, P. De Natale, M. Gioffrè, G. Coppola, and M. Iodice, “Mid-infrared tunable two-dimensional Talbot array illuminator,” Appl. Phys. Lett.94(12), 121105 (2009).
[CrossRef]

Fischer, D. G.

Foley, J. T.

Fourkas, J. T.

Friedrich, L.

Gadonas, R.

R. Gadonas, V. Jarutis, R. Paškauskas, V. Smilgevičius, A. Stabinis, and V. Vaičaitis, “Self-action of Bessel beam in nonlinear medium,” Opt. Commun.196(1-6), 309–316 (2001).
[CrossRef]

García de Abajo, F. J.

Gatto, A.

Gbur, G.

Ghosh, N.

Gioffrè, M.

P. Maddaloni, M. Paturzo, P. Ferraro, P. Malara, P. De Natale, M. Gioffrè, G. Coppola, and M. Iodice, “Mid-infrared tunable two-dimensional Talbot array illuminator,” Appl. Phys. Lett.94(12), 121105 (2009).
[CrossRef]

Gittes, F.

Gouy, L. G.

L. G. Gouy, “Sur une propriété nouvelle des ondes lumineuses,” C. R. Acad. Sci. Paris110, 1251–1253 (1890).

Grosse-Nobis, W.

T. Ackemann, W. Grosse-Nobis, and G. L. Lippi, “The Gouy phase shift, the average phase lag of Fourier components of Hermite-Gaussian modes and their application to resonance conditions in optical cavities,” Opt. Commun.189(1-3), 5–14 (2001).
[CrossRef]

Gu, B.

H. X. Cui, X. L. Wang, B. Gu, Y. N. Li, J. Chen, and H. T. Wang, “Angular diffraction of an optical vortex induced by the Gouy phase,” J. Opt.14(5), 055707 (2012).
[CrossRef]

Guigay, J. P.

Gupta, R. K.

Habraken, S. J. M.

S. J. M. Habraken and G. Nienhuis, “Geometric phases in astigmatic optical modes of arbitrary order,” J. Math. Phys.51(8), 082702 (2010).
[CrossRef]

Haldar, A.

Hamazaki, J.

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. A51(1), R14–R17 (1995).
[CrossRef] [PubMed]

Hariharan, P.

P. Hariharan and P. A. Robinson, “The Gouy phase shift as a geometrical quantum effect,” J. Mod. Opt.43, 219–221 (1996).

Harzendorf, T.

He, H.

H. He and X.-C. Zhang, “Analysis of Gouy phase shift for optimizing terahertz air-biased-coherent-detection,” Appl. Phys. Lett.100(6), 061105 (2012).
[CrossRef]

Heinis, D.

Hermatschweiler, M.

M. Thiel, M. Hermatschweiler, M. Wegener, and G. von Freymann, “Thin-film polarizer based on a one-dimensional–three-dimensional–one-dimensional photonic crystal heterostructure,” Appl. Phys. Lett.91(12), 123515 (2007).
[CrossRef]

Herzig, H. P.

Holme, N. C. R.

N. C. R. Holme, B. C. Daly, M. T. Myaing, and T. B. Norris, “Gouy phase shift of single-cycle picosecond acoustic pulses,” Appl. Phys. Lett.83(2), 392–394 (2003).
[CrossRef]

Huang, F. M.

F. M. Huang, N. Zheludev, Y. Chen, and F. Javier Garcia de Abajo, “Focusing of light by a nanohole array,” Appl. Phys. Lett.90(9), 091119 (2007).
[CrossRef]

Iodice, M.

P. Maddaloni, M. Paturzo, P. Ferraro, P. Malara, P. De Natale, M. Gioffrè, G. Coppola, and M. Iodice, “Mid-infrared tunable two-dimensional Talbot array illuminator,” Appl. Phys. Lett.94(12), 121105 (2009).
[CrossRef]

Jarutis, V.

R. Gadonas, V. Jarutis, R. Paškauskas, V. Smilgevičius, A. Stabinis, and V. Vaičaitis, “Self-action of Bessel beam in nonlinear medium,” Opt. Commun.196(1-6), 309–316 (2001).
[CrossRef]

Javier Garcia de Abajo, F.

F. M. Huang, N. Zheludev, Y. Chen, and F. Javier Garcia de Abajo, “Focusing of light by a nanohole array,” Appl. Phys. Lett.90(9), 091119 (2007).
[CrossRef]

Khoroshun, A.

Kim, M.-S.

Klein, S.

M. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt.43(10), 2139–2164 (1996).
[CrossRef]

Kogelnik, H.

Kolodziejczyk, A.

A. Kołodziejczyk, “Realization of Fourier images without using a lens by sampling the optical object,” J. Mod. Opt.32, 741–746 (1985).

Kurtsiefer, Ch.

Lancis, J.

J. C. Barreiro, P. Andrés, J. Ojeda-Castañeda, and J. Lancis, “Multiple incoherent 2D optical correlator,” Opt. Commun.84(5-6), 237–241 (1991).
[CrossRef]

Lau, E.

E. Lau, “Beugungserscheinungen an Doppelrastern,” Ann. Phys.437(7-8), 417–423 (1948).
[CrossRef]

Le Floch, A.

D. Chauvat, O. Emile, M. Brunel, and A. Le Floch, “Direct measurement of the central fringe velocity in Young-type experiments,” Phys. Lett. A295(2-3), 78–80 (2002).
[CrossRef]

Lee, K.-S.

Leger, J. R.

J. R. Leger, “Lateral mode control of an AlGaAs laser array in a Talbot cavity,” Appl. Phys. Lett.55(4), 334–336 (1989).
[CrossRef]

J. R. Leger, M. L. Scott, and W. B. Veldkamp, “Coherent addition of AlGaAs lasers using microlenses and diffractive coupling,” Appl. Phys. Lett.52(21), 1771–1773 (1988).
[CrossRef]

Li, L.

Li, T.

Li, Y. N.

H. X. Cui, X. L. Wang, B. Gu, Y. N. Li, J. Chen, and H. T. Wang, “Angular diffraction of an optical vortex induced by the Gouy phase,” J. Opt.14(5), 055707 (2012).
[CrossRef]

Li, Y.-P.

H. Chen, Q. Zhan, Y. Zhang, and Y.-P. Li, “The Gouy phase shift of the highly focused radially polarized beam,” Phys. Lett. A371(3), 259–261 (2007).
[CrossRef]

Lippi, G. L.

T. Ackemann, W. Grosse-Nobis, and G. L. Lippi, “The Gouy phase shift, the average phase lag of Fourier components of Hermite-Gaussian modes and their application to resonance conditions in optical cavities,” Opt. Commun.189(1-3), 5–14 (2001).
[CrossRef]

Lohmann, A. W.

A. W. Lohmann and A. S. Marathay, “About periodicities in 3-D wavefields,” Appl. Opt.28(20), 4419–4423 (1989).
[CrossRef] [PubMed]

A. W. Lohmann and J. Ojeda-Castaneda, “Spatial periodicities in partially coherent fields,” Opt. Acta: Int. J. Opt.30(4), 475–479 (1983).
[CrossRef]

Lu, Y.

Luo, H.

Luo, K.-H.

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(3), 033902 (2011).
[CrossRef] [PubMed]

Maddaloni, P.

P. Maddaloni, M. Paturzo, P. Ferraro, P. Malara, P. De Natale, M. Gioffrè, G. Coppola, and M. Iodice, “Mid-infrared tunable two-dimensional Talbot array illuminator,” Appl. Phys. Lett.94(12), 121105 (2009).
[CrossRef]

Malara, P.

P. Maddaloni, M. Paturzo, P. Ferraro, P. Malara, P. De Natale, M. Gioffrè, G. Coppola, and M. Iodice, “Mid-infrared tunable two-dimensional Talbot array illuminator,” Appl. Phys. Lett.94(12), 121105 (2009).
[CrossRef]

Marathay, A. S.

Martelli, P.

Martinelli, M.

McMorran, B. J.

B. J. McMorran and A. D. Cronin, “An electron Talbot interferometer,” New J. Phys.11(3), 033021 (2009).
[CrossRef]

Menzel, C.

Mineta, Y.

Moneta, G.

Morita, R.

Mühlig, S.

M.-S. Kim, T. Scharf, S. Mühlig, C. Rockstuhl, and H. P. Herzig, “Engineering photonic nanojets,” Opt. Express19(11), 10206–10220 (2011).
[CrossRef] [PubMed]

M.-S. Kim, T. Scharf, S. Mühlig, C. Rockstuhl, and H. P. Herzig, “Gouy phase anomaly in photonic nanojets,” Appl. Phys. Lett.98(19), 191114 (2011).
[CrossRef]

Mukherjee, S.

Mukunda, N.

R. Simon and N. Mukunda, “Bargmann invariant and the geometry of the Güoy effect,” Phys. Rev. Lett.70(7), 880–883 (1993).
[CrossRef] [PubMed]

Myaing, M. T.

N. C. R. Holme, B. C. Daly, M. T. Myaing, and T. B. Norris, “Gouy phase shift of single-cycle picosecond acoustic pulses,” Appl. Phys. Lett.83(2), 392–394 (2003).
[CrossRef]

Nahata, A.

Nemes, M. C.

I. G. Da Pazl, P. L. Saldanha, M. C. Nemes, and J. G. Peixoto de Faria, “Experimental proposal for measuring the Gouy phase of matter waves,” New J. Phys.13, 125005 (2011).

Nienhuis, G.

S. J. M. Habraken and G. Nienhuis, “Geometric phases in astigmatic optical modes of arbitrary order,” J. Math. Phys.51(8), 082702 (2010).
[CrossRef]

Noponen, E.

E. Noponen and J. Turunen, “Electromagnetic theory of Talbot imaging,” Opt. Commun.98(1-3), 132–140 (1993).
[CrossRef]

Norris, T. B.

N. C. R. Holme, B. C. Daly, M. T. Myaing, and T. B. Norris, “Gouy phase shift of single-cycle picosecond acoustic pulses,” Appl. Phys. Lett.83(2), 392–394 (2003).
[CrossRef]

Nowak, S.

Ojeda-Castaneda, J.

A. W. Lohmann and J. Ojeda-Castaneda, “Spatial periodicities in partially coherent fields,” Opt. Acta: Int. J. Opt.30(4), 475–479 (1983).
[CrossRef]

Ojeda-Castañeda, J.

J. C. Barreiro, P. Andrés, J. Ojeda-Castañeda, and J. Lancis, “Multiple incoherent 2D optical correlator,” Opt. Commun.84(5-6), 237–241 (1991).
[CrossRef]

E. Bonet, J. Ojeda-Castañeda, and A. Pons, “Imagesynthesis using the Laueffect,” Opt. Commun.81(5), 285–290 (1991).
[CrossRef]

Oka, K.

Pal, S. B.

Pang, X.

Paoletti, D.

G. Spagnolo, D. Ambrosini, and D. Paoletti, “Displacement measurement using the Talbot effect with a Ronchi grating,” J. Opt. Soc. A4(6), S376–S380 (2002).
[CrossRef]

Pas’ko, V.

Paškauskas, R.

R. Gadonas, V. Jarutis, R. Paškauskas, V. Smilgevičius, A. Stabinis, and V. Vaičaitis, “Self-action of Bessel beam in nonlinear medium,” Opt. Commun.196(1-6), 309–316 (2001).
[CrossRef]

Paturzo, M.

P. Maddaloni, M. Paturzo, P. Ferraro, P. Malara, P. De Natale, M. Gioffrè, G. Coppola, and M. Iodice, “Mid-infrared tunable two-dimensional Talbot array illuminator,” Appl. Phys. Lett.94(12), 121105 (2009).
[CrossRef]

Peixoto de Faria, J. G.

I. G. Da Pazl, P. L. Saldanha, M. C. Nemes, and J. G. Peixoto de Faria, “Experimental proposal for measuring the Gouy phase of matter waves,” New J. Phys.13, 125005 (2011).

Peter, H. H.

Pfau, T.

Pons, A.

E. Bonet, J. Ojeda-Castañeda, and A. Pons, “Imagesynthesis using the Laueffect,” Opt. Commun.81(5), 285–290 (1991).
[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. A51(1), R14–R17 (1995).
[CrossRef] [PubMed]

Qin, Y.-Q.

Quidant, R.

Rayleigh, L.

L. Rayleigh, “On copying diffraction gratings and some phenomena connected therewith,” Philos. Mag.11(67), 196–205 (1881).
[CrossRef]

Robinson, P. A.

P. Hariharan and P. A. Robinson, “The Gouy phase shift as a geometrical quantum effect,” J. Mod. Opt.43, 219–221 (1996).

Rockstuhl, C.

Rodriguez, D.

Rohrbach, A.

Rolland, J. P.

Roy, B.

Rudd, J. V.

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, “Direct observation of the Gouy phase shift with single-cycle terahertz pulse,” Phys. Rev. Lett.83(17), 3410–3413 (1999).
[CrossRef]

Ruffin, A. B.

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, “Direct observation of the Gouy phase shift with single-cycle terahertz pulse,” Phys. Rev. Lett.83(17), 3410–3413 (1999).
[CrossRef]

Saldanha, P. L.

I. G. Da Pazl, P. L. Saldanha, M. C. Nemes, and J. G. Peixoto de Faria, “Experimental proposal for measuring the Gouy phase of matter waves,” New J. Phys.13, 125005 (2011).

Saleh, B. E. A.

Sanchez-Brea, L. M.

Scharf, T.

Schlenker, M.

Schmid, T.

Schmidt, C. F.

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. A51(1), R14–R17 (1995).
[CrossRef] [PubMed]

Scott, M. L.

J. R. Leger, M. L. Scott, and W. B. Veldkamp, “Coherent addition of AlGaAs lasers using microlenses and diffractive coupling,” Appl. Phys. Lett.52(21), 1771–1773 (1988).
[CrossRef]

Simon, R.

R. Simon and N. Mukunda, “Bargmann invariant and the geometry of the Güoy effect,” Phys. Rev. Lett.70(7), 880–883 (1993).
[CrossRef] [PubMed]

Slyusar, V.

Smilgevicius, V.

R. Gadonas, V. Jarutis, R. Paškauskas, V. Smilgevičius, A. Stabinis, and V. Vaičaitis, “Self-action of Bessel beam in nonlinear medium,” Opt. Commun.196(1-6), 309–316 (2001).
[CrossRef]

Song, X.-B.

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(3), 033902 (2011).
[CrossRef] [PubMed]

Soskin, M.

Spagnolo, G.

G. Spagnolo, D. Ambrosini, and D. Paoletti, “Displacement measurement using the Talbot effect with a Ronchi grating,” J. Opt. Soc. A4(6), S376–S380 (2002).
[CrossRef]

Stabinis, A.

R. Gadonas, V. Jarutis, R. Paškauskas, V. Smilgevičius, A. Stabinis, and V. Vaičaitis, “Self-action of Bessel beam in nonlinear medium,” Opt. Commun.196(1-6), 309–316 (2001).
[CrossRef]

Stuerzebecher, L.

Subbarao, D.

Tacca, M.

Talbot, F.

F. Talbot, “Facts relating to optical science. No. IV,” Philos. Mag.9, 401–407 (1836).

Tamkin, J.

Tan, Y.

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. A51(1), R14–R17 (1995).
[CrossRef] [PubMed]

Teich, M. C.

Teng, S.

Thiel, M.

M. Thiel, M. Hermatschweiler, M. Wegener, and G. von Freymann, “Thin-film polarizer based on a one-dimensional–three-dimensional–one-dimensional photonic crystal heterostructure,” Appl. Phys. Lett.91(12), 123515 (2007).
[CrossRef]

Thompson, K. P.

Torcal-Milla, F. J.

Turunen, J.

E. Noponen and J. Turunen, “Electromagnetic theory of Talbot imaging,” Opt. Commun.98(1-3), 132–140 (1993).
[CrossRef]

Vaicaitis, V.

R. Gadonas, V. Jarutis, R. Paškauskas, V. Smilgevičius, A. Stabinis, and V. Vaičaitis, “Self-action of Bessel beam in nonlinear medium,” Opt. Commun.196(1-6), 309–316 (2001).
[CrossRef]

van Hulst, N. F.

Vargas, J.

Vasnetsov, M.

Veldkamp, W. B.

J. R. Leger, M. L. Scott, and W. B. Veldkamp, “Coherent addition of AlGaAs lasers using microlenses and diffractive coupling,” Appl. Phys. Lett.52(21), 1771–1773 (1988).
[CrossRef]

Visser, T. D.

Voelkel, R.

Vogler, U.

von Freymann, G.

M. Thiel, M. Hermatschweiler, M. Wegener, and G. von Freymann, “Thin-film polarizer based on a one-dimensional–three-dimensional–one-dimensional photonic crystal heterostructure,” Appl. Phys. Lett.91(12), 123515 (2007).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Schematic geometry of the grating structure: the 1D amplitude grating of the period Λ = 1 μm and the duty cycle = 0.5, the 30-μm-wide opaque region, and the 40-μm-wide opening region where the in situ reference plane wave passes. Structures are fabricated on a 1.5-mm-thick glass substrate.

Fig. 2
Fig. 2

The x-z intensity distribution emerging from the grating structure shown in Fig. 1. A plane wave is used for the illumination. (a) Measured and (b) simulated intensity distribution. The region of interest is the central 60-μm part of Fig. 1. Intensities are normalized.

Fig. 3
Fig. 3

Comparison of the exact and approximate Talbot lengths ZT with respect to the grating period Λ. The difference becomes noticeable when the period Λ is below 5λ. The case of Λ = 1.56λ ( = 1 μm) produces approximately 12% difference between two results. When Λ = λ, this relative error reaches up to 100%. The experimental result is in excellent agreement with the exact Talbot length.

Fig. 4
Fig. 4

The measured x-z phase distributions: (a) the longitudinal-differential phase map and (b) the propagation phase map. The reference plane wave passing the opening is shown in x = 1 - 8 μm of both maps in the referential plane at x = 0 µm. The propagation phase is obtained by adding the phase advance of a plane wave in free space to each distance that can be calculated analytically according to kz and unwrapping it with a modulo of 2π.

Fig. 5
Fig. 5

The simulated x-z phase distributions: (a) the longitudinal-differential phase map and (b) the propagation phase map. The reference plane wave passing the opening is shown in x = 1 - 5 μm of both maps. The LD phase distribution is obtained by subtracting the phase of the plane wave from the propagation phase map in the referential plane at x = 0 µm.

Fig. 6
Fig. 6

The axial phase shifts along the center of Talbot images from the experiment (filled square) and simulation (red solid line). The analytical result using Eq. (3) is plotted in dark dashed line. For the single diffraction order, the simulation result is plotted in blue dot-dashed line denoted as FMM single order.

Fig. 7
Fig. 7

Phase profiles within one Talbot length along the center of the Talbot image close to x = 55 μm: (a) the absolute phase from Fig. 5(b) and (b) the unwrapped LD phase from Fig. 5(a). The absolute phase shows the irregular wavefront spacing within one Talbot length. This irregularity leads to the phase anomaly slope deviating from the linear slope of the analytical solution Eq. (3). This repeatedly appears in Fig. 6 as a demonstration for the axial periodicity that represents the Talbot length.

Equations (4)

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

Z T = λ 1 1 ( λ Λ ) 2 .
Z T = 2 Λ 2 λ .
Δϕ =z(cosθ1)k,
Z T = λ 1cosθ ,

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