OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 27 — Dec. 17, 2012
  • pp: 28929–28940
« Show journal navigation

Phase anomalies in Bessel-Gauss beams

Myun-Sik Kim, Toralf Scharf, Alberto da Costa Assafrao, Carsten Rockstuhl, Silvania F. Pereira, H. Paul Urbach, and Hans Peter Herzig  »View Author Affiliations


Optics Express, Vol. 20, Issue 27, pp. 28929-28940 (2012)
http://dx.doi.org/10.1364/OE.20.028929


View Full Text Article

Acrobat PDF (1390 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

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

Anomalous axial phase behavior of optical beams has been drawing attention since Gouy’s discovery in 1890 [1

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

] and is called after him 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

2. A. E. Siegman, Lasers (University Science Books, 1986).

, 3

3. M. Born and E. Wolf, Principles of Optics, 7th ed (Cambridge University Press, 1999).

]. Second, in laser cavities the resonance frequencies of different transverse modes are determined by the Gouy phase [2

2. A. E. Siegman, Lasers (University Science Books, 1986).

, 3

3. M. Born and E. Wolf, Principles of Optics, 7th ed (Cambridge University Press, 1999).

]. 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]

] and where it can provide a tracking mechanism of trapped particles [5

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

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]

]. Moreover, the generation of higher harmonics employs phase-matching techniques by considering this phase anomaly; not only in nonlinear optics [7

7. R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic Press, San Diego, 1992)

9

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]

] but also in attosecond science [10

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]

].

Although, Gouy’s discovery took place more than hundred years ago, curiosity about the origin and physical meanings of this phenomenon continually induces discussions which are developed from different theoretical perspectives. The wavefront spacing, which is defined as the smallest distance between surfaces of constant phase on which the values differ by 2π [11

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]

], can demonstrate how such on-axis phase shifts occur in focused waves compared to the plane wave. Note that for monochromatic plane wave fields this wavefront spacing equals the wavelength but it may differ significantly for spatially inhomogeneous wave fields. In this context, both analytical and numerical studies for radially polarized beams have been reported [12

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]

]. The 2D and 3D focusing cases have been analyzed applying mathematical techniques that directly demonstrate irregular wavefront spacing [13

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

]. Other theories were considered as well. For example, the geometric properties of Gaussian beams [14

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

], Berry’s geometrical phase [15

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

18

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

], and even quantum mechanics [19

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

21

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]

] have been considered to give more insights. Recently, it has also been explored in the context of deviating wave fields, i.e. astigmatic wave fields [22

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

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]

].

The aim of this study is to investigate the Gouy phase anomaly in non-diffracting Bessel-Gauss beam, where the quantity of the axial phase shift is expected to be distinct from that of the focused fundamental Gaussian beam. We apply experimental and numerical methods, and then an analytical treatment is provided for an intuitive explanation. The remainder of this work is organized as follows. We start with the varieties of occurrences and quantities of the Gouy phase anomalies (section 2) and discuss the non-diffracting Bessel beam and its generation (section 3). In section 4, the details of the experimental theoretical backgrounds are explained. Next, in sections 5 and 6, the intensity and phase distributions of the focused Bessel-Gauss beam are presented, respectively. In section 7, the overall discussions for the Gouy phase anomalies in the Bessel-Gauss beam follow. The conclusions of this study are given in section 8.

2. Various occurrences and quantities of Gouy phase anomalies

In Gouy’s original experiment [1

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

], a light diverging from a point source was reflected from a planar and a concave mirror. These two light beams were overlapped to cause circular interference fringes in a plane of observation. The appearance of the central fringe was shifted from dark to bright, or vice versa, at observation planes just before or after the focus [see Chapter 17 in Ref. 2

2. A. E. Siegman, Lasers (University Science Books, 1986).

]. 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]

], microwaves [18

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

, 25

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

], and terahertz waves [26

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

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]

]. In optical waves, various classes of beams exhibit the Gouy phase, e.g., general higher Gaussian modes like Hermite-Gaussian and Laguerre-Gaussian beams [2

2. A. E. Siegman, Lasers (University Science Books, 1986).

, 29

29. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).

32

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

], more specifically a vortex beam [33

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

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]

], a radially polarized beam [12

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]

], the Airy beam [35

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

], and the Bessel beam [36

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

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]

]. In addition to such optical beams, surface plasmon-polaritons [38

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]

], matter waves [21

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]

], scattered hotspots (i.e. a photonic nanojet) [39

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]

], and diffracted hotspots (i.e. the spot of Arago) [40

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

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]

] also exhibit axial phase shifts. The amount of such axial phase shifts differs depending on the type of beams and the confinement situations. The exact determination of this amount is of major importance.

3. Bessel beam and its generation

Bessel beams itself are of major importance. The remarkable features of the Bessel beam are of great interest in applications requiring a large depth of focus and a self-healing capacity, especially for imaging microscopy [44

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

47

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]

], optical manipulations [48

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

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]

] or micro- and nano-fabrication [49

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

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

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]

]. Moreover, increasing demands of the resolution in nano-science pushes the size of Bessel beams down to sub-wavelength range and the combination with the radially polarized light is a great attraction for Raman spectroscopy, fluorescent imaging, particle acceleration, and second harmonic generation [54

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]

and the references therein].

To shed new light on the peculiarities of Bessel beams while concentrating on phase phenomena, we investigate here the field behavior of a submicron-size non-diffracting beam by experimental and theoretical means. We use a highly confined Bessel-Gauss beam that is generated by focusing an annular shaped collimated illumination using a high NA objective. Specifically, the longitudinal phase field of such a beam is investigated to demonstrate the Gouy phase and the emergence of phase singularities. A method developed by Richards and Wolf [55

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]

] allows us to calculate the vector light fields near the focus of an aplanatic lens with a high NA. Filtering out the inner NA of the illuminating beam simulates the experimental situations. This allows us to simulate in an elegant way the light propagation in agreement to the experimental situation. Therefore, simulation results can be compared to our measurements.

Generally, the non-diffracting Bessel beam [56

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

] is an exact solution to Maxwell’s equations where the radial amplitude distribution corresponds to a Bessel function. The complex field amplitude is denoted as exp(ikzzJ0(krr), where kr and kz are wave vectors in the radial and longitudinal directions, respectively, and J0 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]

], with finite power can be realized that propagate over a comparably long distance without significant divergence. This kind of quasi-Bessel beams, whose transverse field distribution imitates the Bessel function and the non-diverging distance is significantly extended compared to that of the focused Gaussian beam, can be generated by using an axicon lens [58

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

, 59

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

], an annular slit at the back focal plane of the focusing lens [56

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

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

], computer generated holograms [60

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]

], and diffraction of Gaussian beam by an opaque disk [61

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]

]. Another practical way is to focus an annularly shaped parallel illumination [3

3. M. Born and E. Wolf, Principles of Optics, 7th ed (Cambridge University Press, 1999).

, 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]

, 62

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]

]. This method allows an easy adjustment of the beam size along the radial and longitudinal directions by changing the numerical aperture (NA) of the focusing lens. Moreover, the size of the inner blocking disc of the annularly incident beam can be used to vary the amount of the axial phase shift that find important roles in third harmonic generation microscopy and coherent anti-Stokes Raman scattering microscopy [43 and the references therein].

4. Experimental and theoretical backgrounds

To study experimentally the complex field distribution in space, we employ a high-resolution interference microscope (HRIM) to measure amplitude and phase in the entire 3D space. Details of the experimental setup are reported elsewhere [63

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

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]

]. Longitudinal-differential (LD) interferometry [40

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]

], which is a particular measurement mode of the HRIM, allows to directly measure the axial phase shift of a beam of interest with reference to a plane wave. All experimental and theoretical investigations were performed at a single wavelength of 642 nm (CrystaLaser: DL640-050-3). The achievable spatial resolution for the amplitude field is subject to the diffraction limit of the observing objective, a 100X/NA0.9 HC PL FLUOTAR from Leica Microsystems. The Richards and Wolf method [55

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]

] was employed to rigorously simulate the field distributions near the focus of an NA = 0.9 aplanatic lens in air from a linearly polarized (along the x-axis) plane wave. The computational domain is set to be 5 x 5 x 5 μm3 in the xyz-axes. This is sufficiently large and matched the experiments.

The Bessel-Gauss beam is generated by focusing an annular shaped collimated beam, whose inner NA corresponds to 0.72 (half angle = 45.7°). In practice, an annular central disc blocks approximately 50% of the entrance pupil of the focusing objective (Leica Microsystem, 50X/NA0.9 HXC PL APO) on which a linearly polarized (in the 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

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]

].

Figure 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.
shows the schematic of the experimental arrangement and the wave vector representation for the angular spectrum of the Bessel beam. The half angle of the inner focal cone θ corresponds to the inner NA of the annular illumination. For simplicity, the focused annular beam is illustrated as two parallel beams overlapping near the focal point of the focusing lens. It will be shown that this basic concept of two interfering plane waves properly describes all the phenomena that will be observed. The propagation distance zprop 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]

. When a tilted plane wave propagates at a direction that has an angle θ with respect to the optical axis, in our case the z-direction, the phase retardation along this axis compared to the on-axis plane wave is given by
Δϕ=z(kz-k),
(1)
with z as the axial distance, the wave number k, and the z component of k as kz in Fig. 1(c). The term containing the difference of wave vectors can be more generally written with respect to the transverse component of kt as
kzk=k2kt2k.
(2)
Since Eq. (2) does not yield analytical expression in most cases, certain approximations are required. While Eq. (2) approximates to -kt2/(2k) 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]

] as
k2kt2kkt22kkt48k3.
(3)
In our scenario the inner NA of 0.72 would usually suggest the consideration of such higher order approximation. However, it will be shown that such higher orders do not need to be taken into account and Eq. (1) will be shown to provide an adequate measure for the Gouy phase of the Bessel beam [37

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]

]. We approximate the tilt angle θ as the average value of the inner and outer NAs. In our example with NAinner = 0.72 and NAouter = 0.9, the angle θ equals 55.1°. Using kz = k·cosθ and z = zprop Eq. (1) can be re-written as

Δϕ=zpropk(cosθ1).
(4)

5. Intensity distributions

The measured and simulated intensity distributions of the generated Bessel-Gauss beam in the x-z plane are shown in 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.
, 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. |Et|2 = |Ex|2 + |Ey|2 + |Ez|2. Here, Ex, Ey, and Ez are the complex electric field components and |Et|2 is usually referred to as the total electric field or the field intensity I. However, Fig. 2(b) shows only |Ex|2 because Ey is naturally zero at the y = 0 plane and the longitudinal electric field component Ez 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

The LD phase mode of the HRIM directly obtains the phase difference between the object wave and a reference plane wave in each measurement plane. The longitudinal slice, crossing the optical axis, of such a phase map is called the LD phase distribution and visualizes directly the phase anomaly. Furthermore, a 2D propagating phase map, which provides a glimpse on the actual phase evolution in space, can be easily reconstructed by wrapping the LD phase with a known value, for instance, the effective wavelength (λeff) along the z-axis of the tilted wave, which is given as
λeff=λcosθ.
(5)
Note 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 Ez 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]

]. We would like to stress that the term Guoy phase in a strict sense is used to assess the phase evolution of a wave near its focus. There, the Guoy phase describes the deviation of the phase relative to that of a plane wave. Although we investigate here the phase evolution of a Bessel-Gauss beam and discuss its inherent properties along the entire propagation direction and not just in a spatial region that can be understood as the focus, we would like to interpret this here as a Guoy phase for simplicity as well. We understand this as a reasonable nomenclature since, at least for the ideal Bessel beam, a focus cannot be identified since the beam is non-diffracting, i.e., it entirely preserves its shape everywhere in space.

In our study, the difference of the phase of the plane wave from that of the object wave is directly provided by the LD phase distribution. Figure 3(a)
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.
shows the measured LD phase distribution of the Bessel-Gauss beam. The propagating phase map is reconstructed by wrapping the LD phase map with λeff = 1.1 μm as shown in Fig. 3(b). The planar wavefronts emerging from the left and right hand side corners in the bottom of the Fig. 3(b) perfectly corresponds to the proposed 2D model where two tilted plane waves propagate towards each other. They are characterized by a tangential wave vector component with opposite sign but they do share the same longitudinal wave vector component. Here, the tilt angle of these planar wavefront is the same (55.1°) with respect to the positive z-axis. When two planar wavefronts overlap, as shown in the diamond shaped region of Fig. 1(b), the resulting field pattern corresponds to that of a two-beam interference pattern. Such a diamond shaped overlapping region can also be found in the measurement in Fig. 3(b).

The results of numerical simulations by the Richards and Wolf method are shown in Figs. 4(a)
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.
and 4(b) for the LD phase and the propagating phase, respectively. Note that the natural outcome from the simulation is the propagating phase (i.e. absolute phase) as shown in Fig. 4(b). The LD phase as shown in Fig. 4(a) is reconstructed by subtracting the calculated phase in Fig. 4(b) from the phase of a referential plane wave of the same frequency in each transverse plane. The construction in simulation, therefore, is exactly opposite as in the measurements. Nonetheless, the simulation is in perfect agreement with the experiments, which are shown in Fig. 3.

As observed in Figs. 3 and 4, the non-diffracting character of the Bessel beam causes the Gouy phase to grow along the propagation direction [see Eq. (4)]. In other words, the phase anomaly exceeds the ordinary Gouy phase of π and it continues to grow across the spatial domain where the Bessel-Gauss beam possesses a non-diffracting shape, i.e. within the spatial domain that was called the overlap region. In Fig. 3(b), the propagation distance of the non-diffracting Bessel-Gauss beam zprop 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

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]

]. Therefore, to neglect higher order approximations seems to be justified by the experimental results on both a quantitative and qualitative level.

7. Phase anomalies in Bessel-Gauss beam

To quantify the results to an even larger extent, we concentrate on the axial phase not only for the center of the beam (the optical axis) but also for the side lobes. For such non-diffracting beams the on-axis fields within the central lobe are of utmost importance since many applications exploit it. In 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)].
we provide a comparison of the on-axis propagating phase profiles obtained from Figs. 3(b) and 4(b): the solid line for the experiment and the dashed line for the simulation. As expected, the larger effective wavelength λeff 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.

The overall axial phase shift is obtained by unwrapping the extracted longitudinal-differential phase data from Figs. 3(a) and 4(a). Since the non-diffracting transverse field distribution influences fields not only in the central lobe but also in the side lobes, it is interesting to compare the phase anomalies in the central, 1st, and 2nd side lobes. Figure 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.
plots the Gouy phase of each lobe obtained from experiment and simulation together with the analytical result. For convenience, the initial axial phase shifts of the central and the 2nd lobes are set to be zero. The solid line represents the analytical result, the dashed lines the simulations, and the markers the experiments. As it can be anticipated by Eq. (4), the anomalous axial phase shift grows linearly with the propagation distance zprop 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.

The non-diffracting character of Bessel Gauss beams causes two prominent features in the phase distributions. First, the amount of the phase shift continuously grows with respect to the propagation distance and eventually becomes larger than π (i.e. general Gouy phase for a 3D converging wave). Second, the phase singularity that surrounds the central bright spot with a dark ring (i.e. zero amplitude) extends along the entire spatial domain where diffraction is suppressed. Both aspects are shown in the experiments. The rigorous simulations using the Richards and Wolf method [55

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]

] allow verifying the experimental situation by filtering out the inner NA of the annular beam and the numerical results confirm our experimental findings.

8. Conclusions

We experimentally and theoretically investigated the Gouy phase anomaly in a Bessel-Gauss beam generated by focusing an annular collimated beam with a high NA objective. The generated beam shows typical features of a Bessel beam, such as relatively strong side lobes and a long propagation distance without a significant divergence. The non-diffracting character makes the Gouy phase to grow proportionally to the propagation distance (zprop) with a factor of (cosθ – 1). The transverse field distribution does not vary within zprop 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

The research leading to these results has received funding from the Swiss National Science Foundation under Project No. 200021_125177/1 and the European Community's Seventh Framework Programme FP7-ICT-2007-2 under grant agreement no. 224226.

References and links

1.

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

2.

A. E. Siegman, Lasers (University Science Books, 1986).

3.

M. Born and E. Wolf, Principles of Optics, 7th ed (Cambridge University Press, 1999).

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]

7.

R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic Press, San Diego, 1992)

8.

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. B 23(10), 2134–2141 (2006). [CrossRef]

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]

16.

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

17.

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

18.

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

19.

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

20.

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

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]

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]

25.

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

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]

27.

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]

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]

29.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).

30.

H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5(10), 1550–1567 (1966). [CrossRef] [PubMed]

31.

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]

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]

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]

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]

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 295(2-3), 78–80 (2002). [CrossRef]

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]

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]

44.

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

45.

F. O. Fahrbach and A. Rohrbach, “Propagation stability of self-reconstructing Bessel beams enables contrast-enhanced imaging in thick media,” Nat Commun 3, 632 (2012). [CrossRef] [PubMed]

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 8(5), 417–423 (2011). [CrossRef] [PubMed]

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]

49.

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

50.

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]

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]

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]

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]

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]

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]

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]

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]

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]

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


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. L. G. Gouy, “Sur une propriété nouvelle des ondes lumineuses,” C. R. Acad. Sci.110, 1251–1253 (1890).
  2. A. E. Siegman, Lasers (University Science Books, 1986).
  3. M. Born and E. Wolf, Principles of Optics, 7th ed (Cambridge University Press, 1999).
  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. Express20(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]
  7. R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic Press, San Diego, 1992)
  8. 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]
  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. Photonics4(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. A371(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]
  16. R. Simon and N. Mukunda, “Bargmann invariant and the geometry of the Güoy effect,” Phys. Rev. Lett.70(7), 880–883 (1993). [CrossRef] [PubMed]
  17. S. J. M. Habraken and G. Nienhuis, “Geometric phases in astigmatic optical modes of arbitrary order,” J. Math. Phys.51(8), 082702 (2010). [CrossRef]
  18. G. F. Brand, “A new millimeter wave geometric phase demonstration,” Int. J. Infrared Millim. Waves21(4), 505–518 (2000). [CrossRef]
  19. P. Hariharan and P. A. Robinson, “The Gouy phase shift as a geometrical quantum effect,” J. Mod. Opt.43, 219–221 (1996).
  20. S. Feng and H. G. Winful, “Physical origin of the Gouy phase shift,” Opt. Lett.26(8), 485–487 (2001). [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]
  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, T. Schmid, and E. Wolf, “Observation of the Gouy phase anomaly in astigmatic beams,” Appl. Opt.51(15), 2902–2908 (2012). [CrossRef] [PubMed]
  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]
  25. C. R. Carpenter, “Gouy phase advance with microwaves,” Am. J. Phys.27, 98–100 (1959).
  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]
  27. 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]
  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]
  29. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
  30. H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt.5(10), 1550–1567 (1966). [CrossRef] [PubMed]
  31. 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]
  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. Express14(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]
  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. Express18(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. Express15(16), 9995–10001 (2007). [CrossRef] [PubMed]
  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]
  41. 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]
  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]
  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]
  44. F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photonics4(11), 780–785 (2010). [CrossRef]
  45. 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]
  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. Methods8(5), 417–423 (2011). [CrossRef] [PubMed]
  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,” Nature419(6903), 145–147 (2002). [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]
  50. 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]
  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]
  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. Express18(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. Photonics2(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, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett.58(15), 1499–1501 (1987). [CrossRef] [PubMed]
  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. A6(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. A6(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]
  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]
  63. M.-S. Kim, T. Scharf, S. Mühlig, C. Rockstuhl, and H. P. Herzig, “Engineering photonic nanojets,” Opt. Express19(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. Express18(14), 14319–14329 (2010). [CrossRef] [PubMed]
  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. A29(6), 989–993 (2012). [CrossRef] [PubMed]
  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]

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