OSA's Digital Library

Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 1 — Jan. 14, 2013
  • pp: 1287–1300

Phase anomalies in Talbot light carpets of self-images

Myun-Sik Kim, Toralf Scharf, Christoph Menzel, Carsten Rockstuhl, and Hans Peter Herzig  »View Author Affiliations

Optics Express, Vol. 21, Issue 1, pp. 1287-1300 (2013)

View Full Text Article

Enhanced HTML    Acrobat PDF (5201 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



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

OCIS Codes
(050.1950) Diffraction and gratings : Diffraction gratings
(050.5080) Diffraction and gratings : Phase shift
(070.6760) Fourier optics and signal processing : Talbot and self-imaging effects
(180.3170) Microscopy : Interference microscopy
(260.1960) Physical optics : Diffraction theory

ToC Category:
Diffraction and Gratings

Original Manuscript: November 5, 2012
Revised Manuscript: December 21, 2012
Manuscript Accepted: December 26, 2012
Published: January 11, 2013

Myun-Sik Kim, Toralf Scharf, Christoph Menzel, Carsten Rockstuhl, and Hans Peter Herzig, "Phase anomalies in Talbot light carpets of self-images," Opt. Express 21, 1287-1300 (2013)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. F. Talbot, “Facts relating to optical science. No. IV,” Philos. Mag.9, 401–407 (1836).
  2. L. Rayleigh, “On copying diffraction gratings and some phenomena connected therewith,” Philos. Mag.11(67), 196–205 (1881). [CrossRef]
  3. J. C. Bhattacharya, “Measurement of the refractive index using the Talbot effect and a moire technique,” Appl. Opt.28(13), 2600–2604 (1989). [CrossRef] [PubMed]
  4. 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]
  5. 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]
  6. J. R. Leger, “Lateral mode control of an AlGaAs laser array in a Talbot cavity,” Appl. Phys. Lett.55(4), 334–336 (1989). [CrossRef]
  7. L. Stuerzebecher, T. Harzendorf, U. Vogler, U. D. Zeitner, and R. Voelkel, “Advanced mask aligner lithography: fabrication of periodic patterns using pinhole array mask and Talbot effect,” Opt. Express18(19), 19485–19494 (2010). [CrossRef] [PubMed]
  8. 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]
  9. 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]
  10. O. Bryngdahl, “Image formation using self-imaging techniques,” J. Opt. Soc. Am.63(4), 416–419 (1973). [CrossRef]
  11. E. Bonet, J. Ojeda-Castañeda, and A. Pons, “Imagesynthesis using the Laueffect,” Opt. Commun.81(5), 285–290 (1991). [CrossRef]
  12. 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]
  13. R. F. Edgar, “The Fresnel diffraction images of periodic structures,” J. Mod. Opt.16, 281–287 (1969).
  14. J. T. Winthrop and C. R. Worthington, “Theory of Fresnel images. I. Plane periodic objects in monochromatic light,” J. Opt. Soc. Am.55(4), 373–381 (1965). [CrossRef]
  15. A. Kołodziejczyk, “Realization of Fourier images without using a lens by sampling the optical object,” J. Mod. Opt.32, 741–746 (1985).
  16. Y.-S. Cheng and R.-C. Chang, “Theory of image formation using the Talbot effect,” Appl. Opt.33(10), 1863–1874 (1994). [CrossRef] [PubMed]
  17. S. Teng, Y. Tan, and C. Cheng, “Quasi-Talbot effect of the high-density grating in near field,” J. Opt. Soc. Am. A25(12), 2945–2951 (2008). [CrossRef] [PubMed]
  18. M. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt.43(10), 2139–2164 (1996). [CrossRef]
  19. E. Noponen and J. Turunen, “Electromagnetic theory of Talbot imaging,” Opt. Commun.98(1-3), 132–140 (1993). [CrossRef]
  20. Y. Lu, C. Zhou, and H. Luo, “Talbot effect of a grating with different kinds of flaws,” J. Opt. Soc. Am. A22(12), 2662–2667 (2005). [CrossRef] [PubMed]
  21. 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]
  22. M. Berry, I. Marzoli, and W. Schleich, “Quantum carpets, carpets of light,” Physics World June, 39–46 (2001).
  23. 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]
  24. 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]
  25. S. Nowak, Ch. Kurtsiefer, T. Pfau, and C. David, “High-order Talbot fringes for atomic matter waves,” Opt. Lett.22(18), 1430–1432 (1997). [CrossRef] [PubMed]
  26. P. Cloetens, J. P. Guigay, C. De Martino, J. Baruchel, and M. Schlenker, “Fractional Talbot imaging of phase gratings with hard x rays,” Opt. Lett.22(14), 1059–1061 (1997). [CrossRef] [PubMed]
  27. B. J. McMorran and A. D. Cronin, “An electron Talbot interferometer,” New J. Phys.11(3), 033021 (2009). [CrossRef]
  28. M. R. Dennis, N. I. Zheludev, and F. J. García de Abajo, “The plasmon Talbot effect,” Opt. Express15(15), 9692–9700 (2007). [CrossRef] [PubMed]
  29. 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]
  30. 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]
  31. 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]
  32. 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]
  33. L. G. Gouy, “Sur une propriété nouvelle des ondes lumineuses,” C. R. Acad. Sci. Paris110, 1251–1253 (1890).
  34. R. W. Boyd, “Intuitive explanation of the phase anomaly of focused light beams,” J. Opt. Soc. Am.70(7), 877–880 (1980). [CrossRef]
  35. D. Subbarao, “Topological phase in Gaussian beam optics,” Opt. Lett.20(21), 2162–2164 (1995). [CrossRef] [PubMed]
  36. R. Simon and N. Mukunda, “Bargmann invariant and the geometry of the Güoy effect,” Phys. Rev. Lett.70(7), 880–883 (1993). [CrossRef] [PubMed]
  37. S. J. M. Habraken and G. Nienhuis, “Geometric phases in astigmatic optical modes of arbitrary order,” J. Math. Phys.51(8), 082702 (2010). [CrossRef]
  38. G. F. Brand, “A new millimeter wave geometric phase demonstration,” Int. J. Infrared Millim. Waves21(4), 505–518 (2000). [CrossRef]
  39. P. Hariharan and P. A. Robinson, “The Gouy phase shift as a geometrical quantum effect,” J. Mod. Opt.43, 219–221 (1996).
  40. S. Feng and H. G. Winful, “Physical origin of the Gouy phase shift,” Opt. Lett.26(8), 485–487 (2001). [CrossRef] [PubMed]
  41. 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).
  42. T. D. Visser and E. Wolf, “The origin of the Gouy phase anomaly and its generalization to astigmatic wavefields,” Opt. Commun.283(18), 3371–3375 (2010). [CrossRef]
  43. 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]
  44. A. E. Siegman, Lasers (University Science Books, Stanford, 1986).
  45. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, Cambridge, 1999).
  46. F. Gittes and C. F. Schmidt, “Interference model for back-focal-plane displacement detection in optical tweezers,” Opt. Lett.23(1), 7–9 (1998). [CrossRef] [PubMed]
  47. 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]
  48. 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]
  49. R. W. Boyd, Nonlinear Optics 2nd ed. (Academic Press, San Diego, 1992).
  50. S. Carrasco, B. E. A. Saleh, M. C. Teich, and J. T. Fourkas, “Second- and third-harmonic generation with vector Gaussian beams,” J. Opt. Soc. Am. B23(10), 2134–2141 (2006). [CrossRef]
  51. C. Zhang, Y.-Q. Qin, and Y.-Y. Zhu, “Perfect quasi-phase matching for the third-harmonic generation using focused Gaussian beams,” Opt. Lett.33(7), 720–722 (2008). [CrossRef] [PubMed]
  52. 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]
  53. C. R. Carpenter, “Gouy phase advance with microwaves,” Am. J. Phys.27, 98–100 (1959).
  54. 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]
  55. 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]
  56. 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]
  57. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (John Wiley & Sons, Inc., 1991).
  58. H. Kogelnik and T. Li, “Laser Beams and Resonators,” Appl. Opt.5(10), 1550–1567 (1966). [CrossRef] [PubMed]
  59. 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]
  60. J. Courtial, “Self-imaging beams and the Guoy effect,” Opt. Commun.151(1-3), 1–4 (1998). [CrossRef]
  61. J. Hamazaki, Y. Mineta, K. Oka, and R. Morita, “Direct observation of Gouy phase shift in a propagating optical vortex,” Opt. Express14(18), 8382–8392 (2006). [CrossRef] [PubMed]
  62. 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]
  63. 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]
  64. X. Pang, G. Gbur, and T. D. Visser, “The Gouy phase of Airy beams,” Opt. Lett.36(13), 2492–2494 (2011). [CrossRef] [PubMed]
  65. 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]
  66. P. Martelli, M. Tacca, A. Gatto, G. Moneta, and M. Martinelli, “Gouy phase shift in nondiffracting Bessel beams,” Opt. Express18(7), 7108–7120 (2010). [CrossRef] [PubMed]
  67. 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]
  68. 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]
  69. 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]
  70. L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A14(10), 2758–2767 (1997). [CrossRef]
  71. M.-S. Kim, T. Scharf, S. Mühlig, C. Rockstuhl, and H. P. Herzig, “Engineering photonic nanojets,” Opt. Express19(11), 10206–10220 (2011). [CrossRef] [PubMed]
  72. M.-S. Kim, T. Scharf, and H. P. Herzig, “Small-size microlens characterization by Multiwavelength High-Resolution Interference Microscopy,” Opt. Express18(14), 14319–14329 (2010). [CrossRef] [PubMed]
  73. E. Lau, “Beugungserscheinungen an Doppelrastern,” Ann. Phys.437(7-8), 417–423 (1948). [CrossRef]
  74. A. W. Lohmann and A. S. Marathay, “About periodicities in 3-D wavefields,” Appl. Opt.28(20), 4419–4423 (1989). [CrossRef] [PubMed]
  75. A. W. Lohmann and J. Ojeda-Castaneda, “Spatial periodicities in partially coherent fields,” Opt. Acta: Int. J. Opt.30(4), 475–479 (1983). [CrossRef]
  76. 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]
  77. 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]
  78. 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]
  79. J. T. Foley and E. Wolf, “Wave-front spacing in the focal region of high-numerical-aperture systems,” Opt. Lett.30(11), 1312–1314 (2005). [CrossRef] [PubMed]
  80. T. D. Visser and J. T. Foley, “On the wavefront spacing of focused, radially polarized beams,” J. Opt. Soc. Am. A22(11), 2527–2531 (2005). [CrossRef] [PubMed]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited