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

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2012 (9)

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]

F. O. Fahrbach and A. Rohrbach, “Propagation stability of self-reconstructing Bessel beams enables contrast-enhanced imaging in thick media,” Nat Commun3, 632 (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]

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

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]

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]

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]

2011 (6)

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

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

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. Methods8(5), 417–423 (2011).
[CrossRef] [PubMed]

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]

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]

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]

2010 (7)

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]

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. Photonics4(12), 822–832 (2010).
[CrossRef]

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

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photonics4(11), 780–785 (2010).
[CrossRef]

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. Express18(2), 566–574 (2010).
[CrossRef] [PubMed]

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]

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]

2009 (2)

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]

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]

2008 (3)

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. Photonics2(8), 501–505 (2008).
[CrossRef]

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

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]

2007 (3)

2006 (2)

2005 (1)

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 (2)

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]

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]

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature419(6903), 145–147 (2002).
[CrossRef] [PubMed]

2001 (4)

J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun.197(4-6), 239–245 (2001).
[CrossRef]

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. Ackemanna, W. Grosse-Nobis, and G. L. Lippia, “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 (2)

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

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]

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)

1996 (1)

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

1995 (1)

1993 (1)

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

1989 (2)

1987 (2)

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

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

1980 (1)

1978 (1)

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]

1966 (1)

1959 (2)

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]

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

1956 (1)

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]

1954 (1)

1890 (1)

L. G. Gouy, “Sur une propriété nouvelle des ondes lumineuses,” C. R. Acad. Sci.110, 1251–1253 (1890).

Ackemanna, T.

T. Ackemanna, W. Grosse-Nobis, and G. L. Lippia, “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.

Arlt, J.

J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun.197(4-6), 239–245 (2001).
[CrossRef]

Arnold, C. B.

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

Arpin, P.

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. Photonics4(12), 822–832 (2010).
[CrossRef]

Banerjee, A.

Betzig, E.

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. Methods8(5), 417–423 (2011).
[CrossRef] [PubMed]

Bhuyan, M. K.

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.

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Figures (6)

Fig. 1
Fig. 1

(a) Schematic of the generation of the Bessel-Gauss beam by focusing the annular shaped collimated incidence. (b) Magnified view of the central spatial domain with a diamond shape: the overlapping region that corresponds to the propagation distance (zprop). (c) The wave vector representation of the Bessel beam: k is the incident wave vector, kz = k·cosθ the longitudinal component of k, and kt = k·sinθ the transverse component of k.

Fig. 2
Fig. 2

The measured and simulated amplitude distributions in the x-z plane: (a) experimental and (b) numerical results (only Ex is accounted). The intensities are normalized. The image size is 5 x 5 μm2.

Fig. 3
Fig. 3

(Color online) The measured phase distributions in the x-z plane: (a) the longitudinal-differential phase and (b) the propagating phase. The phase is displayed in radian [from -π to π]. The image size is 5 x 5 μm2.

Fig. 4
Fig. 4

The simulated phase distributions in the x-z plane: (a) the longitudinal-differential phase and (b) the propagating phase. The phase is displayed in radian [from -π to π]. The image size is 5 x 5 μm2.

Fig. 5
Fig. 5

On-axis phase profiles from Figs. 3(b) and 4(b). The solid line represents the experiment. The dashed line represents the simulation. The period of 2π modulation defines the effective wavelength, here λeff = 1.1 μm [see Eq. (5)].

Fig. 6
Fig. 6

The overall phase anomalies within the propagation distance (zprop): the solid line represents for the analytical solutions, Eq. (4) for the central and 2nd lobes and its π offset of for the 1st lobe, the dashed lines for the simulations, and the markers for the experiments. Experimental and numerical data are obtained by unwrapping the LD phase profiles from Figs. 3(a) and 4(a). The odd number side lobes have the same Gouy phase with a π offset due to the phase singularity. The initial axial phase shifts for the central and the 2nd lobes are set to be zero for the easy comparison.

Equations (5)

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

Δϕ=z( k z -k),
k z k= k 2 k t 2 k.
k 2 k t 2 k k t 2 2k k t 4 8 k 3 .
Δϕ= z prop k(cosθ1).
λ eff = λ cosθ .

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