## Phase anomalies in Talbot light carpets of self-images |

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

http://dx.doi.org/10.1364/OE.21.001287

Acrobat PDF (5201 KB)

### 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

## 1. Introduction

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. A **4**(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. Express **18**(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]

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]

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. A **25**(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. A **22**(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]

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. A **51**(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. Express **15**(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. Express **17**(26), 23772–23784 (2009). [CrossRef] [PubMed]

20. Y. Lu, C. Zhou, and H. Luo, “Talbot effect of a grating with different kinds of flaws,” J. Opt. Soc. Am. A **22**(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]

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]

30. M.-S. Kim, T. Scharf, C. Menzel, C. Rockstuhl, and H. P. Herzig, “Talbot images of wavelength-scale amplitude gratings,” Opt. Express **20**(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]

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

*in situ*[31

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]

70. L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A **14**(10), 2758–2767 (1997). [CrossRef]

## 2. Experiment and simulation

71. M.-S. Kim, T. Scharf, S. Mühlig, C. Rockstuhl, and H. P. Herzig, “Engineering photonic nanojets,” Opt. Express **19**(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. Express **18**(14), 14319–14329 (2010). [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]

*in situ*. In this study, all experimental and theoretical investigations were performed at a single wavelength of 642 nm (CrystaLaser: DL640-050-3). In passing we note that the laser source provides perfect coherent light and the light delivering system is based on a collimation using a spatial filtering technique that also assures perfect spatial coherence. The Talbot effect occurs in the amplitude of the wave field only if it is perfectly coherent, as it holds for the present setup. However, even for incoherent light a longitudinal periodicity can be observed that was first described by Lau [73

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]

*in situ*reference plane wave. This provides direct information on the phase evolution of the referential plane wave. To include the referential plane wave in the object space, we design an integrated sample system that includes a wide opening. This opening is distant from the edge of the grating and serves as the passage of the unperturbed illumination, which provides the referential phase. The geometry of the integrated sample system is illustrated in Fig. 1 . It has three distinct regions: An opening for the reference wave, an amplitude stop for spatial separation, and the grating region. The reference field is a 40-μm opening that is located next to a 30-μm opaque region to separate the grating and reference region. The grating is a 1D amplitude grating that has a period of 1 μm and a duty cycle of 0.5. The grating structures and the opaque region are made of 80-nm thick chromium (Cr) coating, which is opaque at the wavelength of interest and a metal coating that used for conventional mask structure of photolithography formed on a 1.5-mm-thick glass substrate (Compugraphics Jena GmbH). Illumination is in TM polarization with respect to the grating.

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]

70. L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A **14**(10), 2758–2767 (1997). [CrossRef]

*n*= 2.039 + i2.879, corresponding to the index of chrome at a wavelength of 642 nm. The entire structure is placed on a substrate with index

*n*= 1.5 and illuminated from the substrate with a TM polarized plane wave. Due to the large super-period, an excessive number of Fourier orders had to be taken into account (

*N*= 4001) for the simulation in order to achieve convergent results. All intensity and phase distributions in the manuscript are shown such that

*z*= 0 µm correspond to the terminating edge of the grating.

## 3. Intensity distributions: Talbot images and Talbot length

*x-z*intensity distributions when the 1D grating structure from Fig. 1 is illuminated by a plane wave of 642 nm wavelength.

*in situ*reference plane wave with minimal perturbations passes through the opening. No light passes through the opaque region,

*x*= 5 - 35 μm in Fig. 2. Only diffracted light emerges at certain angles from both edges of the opaque region. On the right hand side,

*x*= 35 - 60 μm, the self-images of the grating appear, i.e., the Talbot images. Here, one can see a consequence of the finite grating. The diffraction angle of the ± 1st orders defines a geometrical region with an inclined edge where the three diffraction orders superpose and interfere to form the Talbot image. The inclination angle perfectly corresponds to the diffraction angle of the 1st order. For a period of Λ = 1 μm and a wavelength of λ = 642 nm, the 1st order diffraction order propagates at an angle of 39.9°. The region

*x*= 50 - 60 μm shows Talbot images extending up to

*z*= 20 μm. It corresponds to the upper limit of the region of interest. The simulation in Fig. 2(b) shows an excellent agreement with the measurement from Fig. 2(a).

17. S. Teng, Y. Tan, and C. Cheng, “Quasi-Talbot effect of the high-density grating in near field,” J. Opt. Soc. Am. A **25**(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]

*Z*, but we adopt the original definition proposed by Lord Rayleigh [2

_{T}2. L. Rayleigh, “On copying diffraction gratings and some phenomena connected therewith,” Philos. Mag. **11**(67), 196–205 (1881). [CrossRef]

*Z*= 2.8 μm that perfectly matches with the result of Eq. (1) while Eq. (2) would lead to

_{T}*Z*= 3.1 μm.

_{T}## 4. Phase distributions: longitudinal-differential and propagation phase maps

*in situ*reference plane wave that is present in the region of

*x*= 1 - 8 μm and which appears as a constant phase. By wrapping this constant phase of the reference plane wave with a modulo of 2π, a propagation phase map, which is the counterpart of the simulated absolute phase map, is obtained as shown in Fig. 4(b). The corresponding simulations are shown in Fig. 5 . Note that the coordinates of the region of interest are shifted to match with intensity distributions. Therefore, the

*in situ*reference appears now at

*x*= 1 - 5 μm. The simulations are again in excellent agreement with experiments. In simulations, the absolute phase is the natural result that represents the propagation phase map as shown in Fig. 5(b). Subtracting the phase of the plane wave from the propagation map leads to the LD phase map in Fig. 5(a).

*x*= 38 - 60 μm for experiment and

*x*= 35 - 60 μm for simulation), phase Talbot images emerge from the grating surface. In this region, the super-resolution features like phase singularities are found in each self-image plane. Such phase dislocations occur where the destructive interference causes points in space with zero amplitude. Since only the self-images demonstrate such features [30

30. M.-S. Kim, T. Scharf, C. Menzel, C. Rockstuhl, and H. P. Herzig, “Talbot images of wavelength-scale amplitude gratings,” Opt. Express **20**(5), 4903–4920 (2012). [CrossRef] [PubMed]

## 5. Phase anomaly of the Talbot image

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. A **29**(6), 989–993 (2012). [CrossRef] [PubMed]

*in situ*reference plane wave, shown in Fig. 4(a) and 5(a) directly reveals the axial phase shift from the plane wave. Since Talbot images are periodic along the lateral direction, on-axis observation points are defined here as the points connecting the center of each Talbot image. Therefore, the axial profiles of the LD phase maps from the experiment [see Fig. 4(a)] and the simulation [see Fig. 5(a)] are extracted along the center of one Talbot image, i.e., close to

*x*= 55 μm. After unwrapping they are plotted together in Fig. 6 .

*z*-axis. To better understand this phenomenon, an analytical equation is derived by using the tilted wave model [78

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]

*k*being the wavenumber ( = 2π/λ). In this case, the amount of the axial phase shift or phase anomaly Δ

*ϕ*does not has a finite and bound value, as in the case of a converging wave, but it is a growing function with respect to the propagation distance

*z*. The phase anomaly calculated by Eq. (3) is also plotted in Fig. 6 and is compared to the experimental and numerical results. Overall, the analytical results show a very good agreement to the experiment and simulation. The axial phase of the single diffraction order, which is the −1st, has been plotted as well (see FMM single order). This will be discussed later with results of Fig. 7 .

*x*= 40 – 60 μm]. This results in an irregular wavefront spacing within one Talbot length, as shown in the extracted axial phase profile from Fig. 5(b) given in Fig. 7(a). Such irregular wavefront spacing is typically found in focused beams [63

63. H. Chen, Q. Zhan, Y. Zhang, and Y.-P. Li, “The Gouy phase shift of the highly focused radially polarized beam,” Phys. Lett. A **371**(3), 259–261 (2007). [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. A **22**(11), 2527–2531 (2005). [CrossRef] [PubMed]

*z*= 0 - 2.8 μm). Therefore, the periodicity of this deviation in the Talbot image phase map is again equal to the Talbot length.

*k*= 2π/λ·cosθ, which represents exactly a tilted wave. Therefore, the axial phase difference of this isolated diffraction order from the plane wave propagating along the axial direction equals the result of Eq. (3). Now, the axial profile from the simulation from Fig. 5(a) is extracted at

_{z}*x*= 34 μm for the −1st order. Since there is no wavefront deformation and irregular spacing, a periodic deviation from the linear slope is not found. The phase extracted from the simulation along this line is equally shown in Fig. 6 (see FMM single order) and shows clearly a linear slope at no specific periodic deviation. In this way, we can verify that the phase anomaly of the Talbot image is associated with the unified action of all the diffraction orders, for our case, the 0th and ± 1st orders, not the 0th order alone. The tilt angle

*θ*(i.e., the diffraction angle of the higher orders) plays a key role to define the growing slope. But the fine details are clearly dominated by the interference of multiple diffraction orders. For the Talbot images, the wavefront deformation and the irregular wavefront spacing causes periodic deviations from the linear slope that appear along the propagation distance. This periodicity verifies the Talbot length in another way. When the phase difference between higher ( ± 1st) orders and the lowest (0th) order equals 2π, constructive or destructive interference occurs. This suggests that the axial period of the Talbot images, which are the result of the constructive interference, equals the distance

*z*where

*Δϕ*= 2π. Therefore, by letting Eq. (3) = 2π another formulation of the Talbot length can be derived:When applying the grating equation sinθ = λ/Λ, Eq. (1) leads to an identical formula as Eq. (4) that covers up to the non-paraxial case of the diffraction and interference problems.

## 6. Conclusions

*in situ*reference plane wave as required for longitudinal-differential interferometry, particular grating structures were designed that sustain a wide opening far away from the edge of the grating. This opening permits the incidence plane wave to propagate with marginal perturbation so that the wavefront compares very well to that of the initial plane wave. The longitudinal-differential phase distributions, which are the natural outcomes of the experiments, directly demonstrate how much amount of axial phase shifts occur along the Talbot images. By wrapping this LD phase map with 2π modulo referencing the

*in situ*plane wave, the propagation phase map could be obtained as well. Measurements were compared to simulations and showed an excellent agreement. Particular phase features like phase singularities were found in each self-image plane. There, destructive interference among the three propagating orders occurs at both sides of the bright Talbot images. It can be safely anticipated that these features can find application for super-resolution distance measurements, for instance.

## Acknowledgment

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53. | C. R. Carpenter, “Gouy phase advance with microwaves,” Am. J. Phys. |

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. |

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. |

56. | H. He and X.-C. Zhang, “Analysis of Gouy phase shift for optimizing terahertz air-biased-coherent-detection,” Appl. Phys. Lett. |

57. | B. E. A. Saleh and M. C. Teich, |

58. | H. Kogelnik and T. Li, “Laser Beams and Resonators,” Appl. Opt. |

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. |

60. | J. Courtial, “Self-imaging beams and the Guoy effect,” Opt. Commun. |

61. | J. Hamazaki, Y. Mineta, K. Oka, and R. Morita, “Direct observation of Gouy phase shift in a propagating optical vortex,” Opt. Express |

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. |

63. | H. Chen, Q. Zhan, Y. Zhang, and Y.-P. Li, “The Gouy phase shift of the highly focused radially polarized beam,” Phys. Lett. A |

64. | X. Pang, G. Gbur, and T. D. Visser, “The Gouy phase of Airy beams,” Opt. Lett. |

65. | R. Gadonas, V. Jarutis, R. Paškauskas, V. Smilgevičius, A. Stabinis, and V. Vaičaitis, “Self-action of Bessel beam in nonlinear medium,” Opt. Commun. |

66. | P. Martelli, M. Tacca, A. Gatto, G. Moneta, and M. Martinelli, “Gouy phase shift in nondiffracting Bessel beams,” Opt. Express |

67. | W. Zhu, A. Agrawal, and A. Nahata, “Direct measurement of the Gouy phase shift for surface plasmon-polaritons,” Opt. Express |

68. | D. Chauvat, O. Emile, M. Brunel, and A. Le Floch, “Direct measurement of the central fringe velocity in Young-type experiments,” Phys. Lett. A |

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. |

70. | L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A |

71. | M.-S. Kim, T. Scharf, S. Mühlig, C. Rockstuhl, and H. P. Herzig, “Engineering photonic nanojets,” Opt. Express |

72. | M.-S. Kim, T. Scharf, and H. P. Herzig, “Small-size microlens characterization by Multiwavelength High-Resolution Interference Microscopy,” Opt. Express |

73. | E. Lau, “Beugungserscheinungen an Doppelrastern,” Ann. Phys. |

74. | A. W. Lohmann and A. S. Marathay, “About periodicities in 3-D wavefields,” Appl. Opt. |

75. | A. W. Lohmann and J. Ojeda-Castaneda, “Spatial periodicities in partially coherent fields,” Opt. Acta: Int. J. Opt. |

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. |

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. A |

78. | Q. Zhan, “Second-order tilted wave interpretation of the Gouy phase shift under high numerical aperture uniform illumination,” Opt. Commun. |

79. | J. T. Foley and E. Wolf, “Wave-front spacing in the focal region of high-numerical-aperture systems,” Opt. Lett. |

80. | T. D. Visser and J. T. Foley, “On the wavefront spacing of focused, radially polarized beams,” J. Opt. Soc. Am. A |

**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

**History**

Original Manuscript: November 5, 2012

Revised Manuscript: December 21, 2012

Manuscript Accepted: December 26, 2012

Published: January 11, 2013

**Citation**

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)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-1-1287

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