## Phase anomalies in Bessel-Gauss beams |

Optics Express, Vol. 20, Issue 27, pp. 28929-28940 (2012)

http://dx.doi.org/10.1364/OE.20.028929

Acrobat PDF (1390 KB)

### Abstract

Bessel-Gauss beams are known as non-diffracting beams. They can be obtained by focusing an annularly shaped collimated laser beam. Here, we report for the first time on the direct measurement of the phase evolution of such beams by relying on longitudinal-differential interferometry. We found that the characteristics of Bessel-Gauss beams cause a continuously increasing phase anomaly in the spatial domain where such beams do not diverge, i.e. there is a larger phase advance of the beam when compared to a referential plane wave. Simulations are in excellent agreement with measurements. We also provide an analytical treatment of the problem that matches both experimental and numerical results and provides an intuitive explanation.

© 2012 OSA

## 1. Introduction

*Gouy phase*or

*phase anomaly*. This peculiar phase behavior plays an essential role in various physical problems and applications thereof. For fundamental physics, first, it intuitively explains the π/2 phase shift of the secondary Huygens’ wavelets emerging from a primary wavefront [2, 3]. Second, in laser cavities the resonance frequencies of different transverse modes are determined by the Gouy phase [2, 3]. Applied physics problems also rely on it. A prime example are optical trapping schemes where the Gouy phase is at the origin of a lateral trapping force [4

4. 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]

5. 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. Express **20**(8), 8317–8328 (2012). [CrossRef] [PubMed]

6. 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]

9. 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]

10. T. Popmintchev, M.-C. Chen, P. Arpin, M. M. Murnane, and H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent X-ray generation,” Nat. Photonics **4**(12), 822–832 (2010). [CrossRef]

11. 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]

12. 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]

13. T. Tyc, “Gouy phase for full-aperture spherical and cylindrical waves,” Opt. Lett. **37**(5), 924–926 (2012). [CrossRef] [PubMed]

14. R. W. Boyd, “Intuitive explanation of the phase anomaly of focused light beams,” J. Opt. Soc. Am. **70**(7), 877–880 (1980). [CrossRef]

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

18. G. F. Brand, “A new millimeter wave geometric phase demonstration,” Int. J. Infrared Millim. Waves **21**(4), 505–518 (2000). [CrossRef]

21. I. G. da Paz, 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**(12), 125005 (2011). [CrossRef]

22. 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]

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

## 2. Various occurrences and quantities of Gouy phase anomalies

*vice versa*, at observation planes just before or after the focus [see Chapter 17 in Ref. 2]. This change of intensity, demonstrating destructive and constructive interferences, indicates a π shift of the converging wave with respect to the phase of the on-axis plane wave. Gouy’s prediction, such as, the occurrence of the effect in any kind of waves, was straightforward and intuitive. As a proof, such phase anomalies have been demonstrated not only in optical waves but also in acoustic waves [24

24. 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]

18. G. F. Brand, “A new millimeter wave geometric phase demonstration,” Int. J. Infrared Millim. Waves **21**(4), 505–518 (2000). [CrossRef]

26. 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]

28. 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]

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

33. J. Hamazaki, Y. Mineta, K. Oka, and R. Morita, “Direct observation of Gouy phase shift in a propagating optical vortex,” Opt. Express **14**(18), 8382–8392 (2006). [CrossRef] [PubMed]

34. 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]

12. 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]

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

36. 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]

37. P. Martelli, M. Tacca, A. Gatto, G. Moneta, and M. Martinelli, “Gouy phase shift in nondiffracting Bessel beams,” Opt. Express **18**(7), 7108–7120 (2010). [CrossRef] [PubMed]

38. W. Zhu, A. Agrawal, and A. Nahata, “Direct measurement of the Gouy phase shift for surface plasmon-polaritons,” Opt. Express **15**(16), 9995–10001 (2007). [CrossRef] [PubMed]

21. I. G. da Paz, 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**(12), 125005 (2011). [CrossRef]

39. 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]

40. 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]

42. 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]

## 3. Bessel beam and its generation

44. F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photonics **4**(11), 780–785 (2010). [CrossRef]

47. B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. **85**(21), 4482–4485 (2000). [CrossRef] [PubMed]

48. V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature **419**(6903), 145–147 (2002). [CrossRef] [PubMed]

51. V. Garcés-Chávez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, “Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle,” Phys. Rev. Lett. **91**(9), 093602 (2003). [CrossRef] [PubMed]

49. E. Mcleod and C. B. Arnold, “Subwavelength direct-write nanopatterning using optically trapped microspheres,” Nat. Nanotechnol. **3**(7), 413–417 (2008). [CrossRef] [PubMed]

52. M. K. Bhuyan, F. Courvoisier, P.-A. Lacourt, M. Jacquot, L. Furfaro, M. J. Withford, and J. M. Dudley, “High aspect ratio taper-free microchannel fabrication using femtosecond Bessel beams,” Opt. Express **18**(2), 566–574 (2010). [CrossRef] [PubMed]

53. X.-F. Li, R. J. Winfield, S. O’Brien, and G. M. Crean, “Application of Bessel beams to 2D microfabrication,” Appl. Surf. Sci. **255**(10), 5146–5149 (2009). [CrossRef]

54. H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics **2**(8), 501–505 (2008). [CrossRef]

55. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. **253**(1274), 358–379 (1959). [CrossRef]

56. J. Durnin, J. J. Miceli Jr, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. **58**(15), 1499–1501 (1987). [CrossRef] [PubMed]

*ik*

_{z}

*z*)·

*J*

_{0}(

*k*

_{r}

*r*), where

*k*

_{r}and

*k*

_{z}are wave vectors in the radial and longitudinal directions, respectively, and

*J*

_{0}is the zeroth order Bessel function of the first kind. Physically, these beams are slightly pathologic since they are infinitely extended in space, each ring of the Bessel beam carries the same amount of energy which adds up to infinity, the phase among adjacent rings differs exactly by π, and the Fourier spectrum of such Bessel beam is an infinitely thin ring. In other words, perfect Bessel beams are inaccessible in real world experiments but can only be approximated. However, nearly non-diffracting beams, so-called Bessel-Gauss beams [57

57. F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. **64**(6), 491–495 (1987). [CrossRef]

58. J. H. McLeod, “The axicon: A new type of optical element,” J. Opt. Soc. Am. **44**(8), 592–597 (1954). [CrossRef]

59. G. Indebetouw, “Nondiffracting optical fields: some remarks in their analysis and synthesis,” J. Opt. Soc. Am. A **6**(1), 150–152 (1989). [CrossRef]

56. J. Durnin, J. J. Miceli Jr, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. **58**(15), 1499–1501 (1987). [CrossRef] [PubMed]

59. G. Indebetouw, “Nondiffracting optical fields: some remarks in their analysis and synthesis,” J. Opt. Soc. Am. A **6**(1), 150–152 (1989). [CrossRef]

60. A. Vasara, J. Turunen, and A. T. Friberg, “Realization of general nondiffracting beams with computer-generated holograms,” J. Opt. Soc. Am. A **6**(11), 1748–1754 (1989). [CrossRef] [PubMed]

61. Q. Huang, S. Coetmellec, F. Duval, A. Louis, H. Leblond, and M. Brunel, “Analytical expressions for diffraction-free beams through an opaque disk,” J. Europ. Opt. Soc. Rap. Public. **6**, 11031 (2011). [CrossRef]

43. 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]

62. C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J. Microwaves, Opt. Acoust. **2**(4), 105–112 (1978). [CrossRef]

## 4. Experimental and theoretical backgrounds

63. 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]

64. 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]

40. 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]

55. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. **253**(1274), 358–379 (1959). [CrossRef]

^{3}in the

*xyz*-axes. This is sufficiently large and matched the experiments.

*x*-axis) collimated beam is incident. In such configuration, the Fourier spectrum at tangential wave vectors smaller than the tangential wave vector corresponding to the inner NA (NA < 0.72, i.e. < 45.7° in air) is blocked and set to zero. As a consequence, the amplitude distribution in the focal plane resembles a zeroth-order Bessel function of the first kind in the radial direction within a finite extent. The focusing causes a Gaussian apodization in the amplitude distribution that can be represented as a product of a Bessel function and a Gaussian profile in real space. Therefore, this type of experimental Bessel beam is known as Bessel-Gauss beam [57

57. F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. **64**(6), 491–495 (1987). [CrossRef]

62. C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J. Microwaves, Opt. Acoust. **2**(4), 105–112 (1978). [CrossRef]

*z*

_{prop}is defined to be the axial diagonal of the diamond-shaped overlapping region in Fig. 1(b). The wave vector representation in Fig. 1(c) facilitates the derivation of the Gouy phase using the tilted wave concept from Ref. 43

43. 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]

*z*-direction, the phase retardation along this axis compared to the on-axis plane wave is given bywith

*z*as the axial distance, the wave number

*k*, and the

*z*component of

*k*as

*k*

_{z}in Fig. 1(c). The term containing the difference of wave vectors can be more generally written with respect to the transverse component of

*k*

_{t}asSince Eq. (2) does not yield analytical expression in most cases, certain approximations are required. While Eq. (2) approximates to -

*k*

_{t}

^{2}/(2

*k*) for the paraxial case, the high NA focusing case (i.e., non-paraxial case) requires higher order approximation and Eq. (2) can be extended up to the second order [43

43. 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]

37. P. Martelli, M. Tacca, A. Gatto, G. Moneta, and M. Martinelli, “Gouy phase shift in nondiffracting Bessel beams,” Opt. Express **18**(7), 7108–7120 (2010). [CrossRef] [PubMed]

_{inner}= 0.72 and NA

_{outer}= 0.9, the angle θ equals 55.1°. Using

*k*

_{z}=

*k*·cosθ and

*z*=

*z*

_{prop}Eq. (1) can be re-written as

## 5. Intensity distributions

*x-z*plane are shown in Fig. 2 , where the incident light propagates along the positive

*z-*axis. The high NA focusing leads to a strong confinement of such a non-diffracting beam in all directions. The typical features, such as relatively strong intensity in the side lobes and the elongated focal spot along the

*z*-axis, are clearly observed in both experiment and simulation. Note that the intensity distribution in the

*x-z*plane, in general, represents the normalized energy density of the total electric field, i.e. |

*E*

_{t}

*|*

^{2}= |

*E*

_{x}

*|*

^{2}

*+ |E*

_{y}

*|*

^{2}

*+ |E*

_{z}

*|*

^{2}. Here,

*E*

_{x},

*E*

_{y}, and

*E*

_{z}are the complex electric field components and |

*E*

_{t}

*|*is usually referred to as the total electric field or the field intensity

^{2}*I*. However, Fig. 2(b) shows only |

*E*

_{x}

*|*

^{2}because

*E*

_{y}is naturally zero at the

*y*= 0 plane and the longitudinal electric field component

*E*

_{z}is not measurable with a far-field measurement system, such as conventional optical microscopes. Since the focusing NA is high, rigorous simulations of the vectorial diffraction problem are necessary to provide correct information of light fields near the focus.

## 6. Phase distributions

_{eff}) along the

*z*-axis of the tilted wave, which is given asNote that this is valid within a simplified 2D (

*x-z*plane) model of the annular geometry of the angular spectrum of the Bessel-Gauss beam. In our case, the tilt angle of the focal cone θ = 55.1° leads to an effective wavelength of approximately 1.1 μm. Due to the abovementioned reasons for non-measurability of

*E*

_{z}and the vanishing of the

*y*-component of electric field in the plane of interest, the phase as shown corresponds to the phase of the

*x*component of the electric field. In general, the Gouy phase of a focused, monochromatic field at an axial point is defined as the difference between the argument (or “phase”) of the object field and that of a referential plane wave of the same frequency [65

65. 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]

*z*

_{prop}is found to be approximately 5 μm, which corresponds to the axial diagonal of the diamond-shape region. The on-axis axial phase shift, i.e. the phase anomaly, is calculated to be 6.7 π using Eq. (4) with λ = 642 nm and θ = 55.1°. The experimental result is found by unwrapping the on-axis LD phase profile of Fig. 3(a). We found experimentally a value of 6.63 π, being in excellent agreement with the analytical estimation. The derived analytical solution Eq. (4) has been obtained without any higher order approximations as given in Eq. (3) [43

**242**(4-6), 351–360 (2004). [CrossRef]

66. E. H. Linfoot and E. Wolf, “Phase distribution near focus in an aberration-free diffraction image,” Proc. Phys. Soc. Lond. **69**(8), 823–832 (1956). [CrossRef]

*z*

_{prop}. This phenomenon can be clearly seen in both experiments and simulations, shown in Figs. 3(b) and 4(b), respectively. There, the phase singularities next to both sides of the optical axes follow linear trajectories. In the entire three-dimensional phase distribution in theory the geometry defined by the trajectories of the phase singularities would correspond to an elliptic cylinder which is a consequence of the vectorial nature of light (it would be a circular cylinder for linearly polarized light in the paraxial and scalar approximation). As the amplitude of the side lobes is relatively strong, the phase singularities between each side lobe are also prominently visible.

## 7. Phase anomalies in Bessel-Gauss beam

_{eff}due to the tilt angle θ [see Eq. (5)] is clearly visible as the period of the 2π modulation and equals approximately 1.1 μm.

*z*

_{prop}and with a factor of (cosθ – 1). The phase anomaly of each lobe can be calculated by Eq. (4), but the π-jump originating from the phase singularity should be considered for each side lobe separately. While the central lobe and the even number of lobes have exactly the same Gouy phase, the odd number of lobes exhibits a Gouy phase with an offset of π [i.e., the result of Eq. (4) + π]. This is due to the nature of the Bessel function in the transverse direction, where each lobe has a π phase difference with respect to its neighboring lobe due to the phase singularity. The cylindrical form of the phase singularity separates each bright lobe within in the non-diffracting region as it propagates. The experimental and numerical data for the central, 1st, and 2nd lobes are extracted from Figs. 3(a) and 4(a). The phase anomalies for the central and 2nd lobes are perfectly overlapping with the analytical result. The phase shift for the 1st lobe has the same slope with the π offset as expected. Analytical and numerical calculations show an excellent agreement with our experimental findings.

55. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. **253**(1274), 358–379 (1959). [CrossRef]

## 8. Conclusions

*z*

_{prop}) with a factor of (cosθ – 1). The transverse field distribution does not vary within

*z*

_{prop}and a phase singularity appears in a cylindrical form. It can be found in the region where the beam does not diverge. The longitudinal-differential phase measurement nicely demonstrates all of the abovementioned phase features of the Bessel-Gauss beam. The numerical simulation using the Richards and Wolf method verifies the measurements and moreover the analytical model [see Eq. (4)]. Such a non-diffracting character influences not only the central lobe but also the side lobes. The growing phase anomalies of each lobe of the Bessel beam up to the 2nd side lobe have been discussed. The Gouy phase of the odd number side lobes is found to have the same amount and slope as the central spot but with an offset of π. Such highly confined Bessel beams are now essential tool for microscopy, meteorology, optical trapping, and micro- and nano-fabrications. Our study provides deeper insight of light fields behavior in such beams.

## Acknowledgment

## References and links

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5. | 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. Express |

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

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

13. | T. Tyc, “Gouy phase for full-aperture spherical and cylindrical waves,” Opt. Lett. |

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22. | T. D. Visser and E. Wolf, “The origin of the Gouy phase anomaly and its generalization to astigmatic wavefields,” Opt. Commun. |

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24. | 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. |

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

34. | 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. |

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

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

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

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

39. | M.-S. Kim, T. Scharf, S. Mühlig, C. Rockstuhl, and H. P. Herzig, “Gouy phase anomaly in photonic nanojets,” Appl. Phys. Lett. |

40. | 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. |

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

42. | 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. |

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

44. | F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photonics |

45. | F. O. Fahrbach and A. Rohrbach, “Propagation stability of self-reconstructing Bessel beams enables contrast-enhanced imaging in thick media,” Nat Commun |

46. | T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods |

47. | B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. |

48. | V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature |

49. | E. Mcleod and C. B. Arnold, “Subwavelength direct-write nanopatterning using optically trapped microspheres,” Nat. Nanotechnol. |

50. | J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. |

51. | V. Garcés-Chávez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, “Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle,” Phys. Rev. Lett. |

52. | M. K. Bhuyan, F. Courvoisier, P.-A. Lacourt, M. Jacquot, L. Furfaro, M. J. Withford, and J. M. Dudley, “High aspect ratio taper-free microchannel fabrication using femtosecond Bessel beams,” Opt. Express |

53. | X.-F. Li, R. J. Winfield, S. O’Brien, and G. M. Crean, “Application of Bessel beams to 2D microfabrication,” Appl. Surf. Sci. |

54. | H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics |

55. | B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. |

56. | J. Durnin, J. J. Miceli Jr, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. |

57. | F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. |

58. | J. H. McLeod, “The axicon: A new type of optical element,” J. Opt. Soc. Am. |

59. | G. Indebetouw, “Nondiffracting optical fields: some remarks in their analysis and synthesis,” J. Opt. Soc. Am. A |

60. | A. Vasara, J. Turunen, and A. T. Friberg, “Realization of general nondiffracting beams with computer-generated holograms,” J. Opt. Soc. Am. A |

61. | Q. Huang, S. Coetmellec, F. Duval, A. Louis, H. Leblond, and M. Brunel, “Analytical expressions for diffraction-free beams through an opaque disk,” J. Europ. Opt. Soc. Rap. Public. |

62. | C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J. Microwaves, Opt. Acoust. |

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

64. | M.-S. Kim, T. Scharf, and H. P. Herzig, “Small-size microlens characterization by multiwavelength high-resolution interference microscopy,” Opt. Express |

65. | 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 |

66. | E. H. Linfoot and E. Wolf, “Phase distribution near focus in an aberration-free diffraction image,” Proc. Phys. Soc. Lond. |

**OCIS Codes**

(050.5080) Diffraction and gratings : Phase shift

(180.3170) Microscopy : Interference microscopy

(260.0260) Physical optics : Physical optics

(100.5088) Image processing : Phase unwrapping

(260.6042) Physical optics : Singular optics

**ToC Category:**

Physical Optics

**History**

Original Manuscript: October 16, 2012

Revised Manuscript: November 30, 2012

Manuscript Accepted: November 30, 2012

Published: December 12, 2012

**Citation**

Myun-Sik Kim, Toralf Scharf, Alberto da Costa Assafrao, Carsten Rockstuhl, Silvania F. Pereira, H. Paul Urbach, and Hans Peter Herzig, "Phase anomalies in Bessel-Gauss beams," Opt. Express **20**, 28929-28940 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-27-28929

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