## Gouy phase shift in nondiffracting Bessel beams

Optics Express, Vol. 18, Issue 7, pp. 7108-7120 (2010)

http://dx.doi.org/10.1364/OE.18.007108

Acrobat PDF (6990 KB)

### Abstract

The results of a theoretical and experimental investigation of the Gouy effect in Bessel beams are presented. We point out that the peculiar feature of the Bessel beams of being nondiffracting is related to the accumulation of an extra axial phase shift (i.e., the Gouy phase shift) linearly dependent on the propagation distance. The constant spatial rate of variation of the Gouy phase shift is independent of the order of the Bessel beam, while it is a growing function of the transverse component of the angular spectrum wave-vectors, originated by the transverse confinement of the beam. A free-space Mach-Zehnder interferometer has been set-up for measuring the transverse intensity distribution of the interference between holographically-produced finite-aperture Bessel beams of order from zero up to three and a reference Gaussian beam, at a wavelength of 633 nm. The interference patterns have been registered for different propagation distances and show a spatial periodicity, in agreement with the expected period due to the linear increase of the Gouy phase shift of the realized Bessel beams.

© 2010 OSA

## 1. Introduction

*π*when it passes through the focal region [1]. This phase anomaly explains the imaginary factor

*i*in the Huygens’ integral, corresponding to a phase shift of

*π*/2 acquired by the secondary wavelets diverging from each point of the primary incident wave-front [2]. In general the Gouy effect manifests itself as an additional axial phase shift acquired by any diffracting optical beam (hence characterized by a transverse spatial confinement) with respect to a reference on-axis plane wave. It was been also recognized that the Gouy effect can be seen as another example of topological or Berry’s phase [3

3. M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. Lond. A Math. Phys. Sci. **392**(1802), 45–57 (1984). [CrossRef]

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

*m*,

*n*) and (

*p*,

*l*), show a Gouy phase shift equal to

*z*

_{R}is the Rayleigh range [2,6

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

*kz*, corresponding to an on-axis plane wave, and therefore the Gouy effect plays a fundamental role in determining the resonant frequencies of the laser cavities [6

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

8. T. Ackermann, W. Grosse-Nobis, and G. L. Lippi, “The Gouy phase shift, the average phase lag of Fourier components of Hermite-Gaussian modes and their application to resonance conditions in optical cavities,” Opt. Commun. **189**(1-3), 5–14 (2001). [CrossRef]

*π*, and the maximum variation rate is at the beam waist. For higher-order modes the phase shift is higher, due to the moltiplicative factor depending on the mode indexes. Explicit measurements of the Gouy phase shift for Hermite- and Laguerre-Gaussian beams have been only recently reported [9

9. J. H. Chow, G. de Vine, M. B. Gray, and D. E. McClelland, “Measurement of gouy phase evolution by use of spatial mode interference,” Opt. Lett. **29**(20), 2339–2341 (2004). [CrossRef] [PubMed]

10. 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), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-18-8382. [CrossRef] [PubMed]

11. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A **4**(4), 651–654 (1987). [CrossRef]

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

15. J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. **177**(1-6), 297–301 (2000). [CrossRef]

16. 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]

18. 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]

*β*. These beams, although nondiffracting, present a transverse spatial confinement inducing a reduction of

*β*, which is just coincident to the axial component of the angular spectrum wave-vectors, with respect to

*k*. The Gouy extra axial phase shift is then

*z*. We report here the first experimental verification of this linear

*z*-dependence of the Gouy phase shift in Bessel beams up to third order, by using an interferometric technique.

## 2. Theory

*z*, obeys the following general expression

*where the transverse field*

*z*is simply the multiplication of the field by the phase term

*characterized by wave-vectors*

*to the axial component*

*β*with respect to

*k*is simply due to the inclination of the wave-vector

**with respect to the axis**

*k**z*. A change of reference framework, which aligns

**to the axis**

*k**z*, makes

*β*coincident with

*k*.

*z*, hence with a transverse field described bywhere

*l*assumes only integer values for guaranteeing the 2

*π*-periodicity of

*are*

*z*without any diffraction, although they present a transverse spatial confinement, which is tighter as

*α*increases. For instance, the radius of the central spot of the zero-order Bessel beam is

*z*and it is not explicitly dependent on the order of the beam, differently from diffracting Hermite- and Laguerre-Gaussian beams. Moreover the Gouy shift is a growing function of the parameter

*α*, which measures the spatial confinement.

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

*k*, hence giving the Gouy phase shift. The Bessel beams can be obtained as superposition of plane waves with wave-vectors describing a cone around the axis

*z*[22

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

23. M. A. Porras, C. J. Zapata-Rodríguez, and I. Gonzalo, “Gouy wave modes: undistorted pulse focalization in a dispersive medium,” Opt. Lett. **32**(22), 3287–3289 (2007). [CrossRef] [PubMed]

*α*, ruling the transverse field shaping and consequently the Gouy phase shift, is exactly the transverse component of the wave-vectors generating the Bessel beams.

## 3. Holographic generation of finite-aperture Bessel beams

*l*

^{th}-order Bessel beams with finite transverse extent, by using suitable computer generated holograms (CGHs).

13. 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]

24. J. A. Davis, E. Carcole, and D. M. Cottrell, “Intensity and phase measurements of nondiffracting beams generated with a magneto-optic spatial light modulator,” Appl. Opt. **35**(4), 593–598 (1996). [CrossRef] [PubMed]

25. C. Paterson and R. Smith, “Higher-order Bessel waves produced by axicon-type computer-generated holograms,” Opt. Commun. **124**(1-2), 121–130 (1996). [CrossRef]

*D*is the diameter of the pupil aperture and

13. 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]

*ν*(in the

*y*direction) with the phase term of

*ν*, which is distorted by the phase of

*l*

^{th}-order Bessel beam in the transmitted first order of diffraction, when illuminated by a plane wave.

*l*respectively from 0 to 3. In particular, each CGH has been realized by plotting the corresponding bidimensional transmission pattern with a laser printer and photoreducing it on Ilford Pan film. All these holograms have the same parameters

^{−1},

## 4. Experimental verification of the Gouy phase shift in Bessel beams

*l*

^{th}-order Bessel beams, with

*l*varying from zero to three, and a reference Gaussian beam. More in detail, a Gaussian beam with wavelength of 633 nm and spot size of about 0.3 mm is emitted by a He-Ne laser source and then split by a 50/50 cube beam splitter (BS) in two beams propagating respectively in the two different arms of the interferometer. The beam in the first arm is just attenuated, to equalize the optical powers of the interfering beams, and remains Gaussian. The beam in the second arm is expanded of a factor 8 by a two-lens system and goes through the proper CGH for producing the required

*l*

^{th}-order Bessel beam. The unwanted diffraction order

*z*.

*l*

^{th}-order Bessel beam reproduced by the CGH is an approximation of the ideal nondiffracting beam characterized by the fieldaccording to Eqs. (10) and (14). The propagation constant

*β*, from Eq. (9), is given by

*β*as

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

*β*given by Eq. (19), and the Gaussian beam of Eq. (21) is then where the phase difference between the Bessel and Gaussian beams, giving the argument of the cosine in the beating term, is

*z*is mainly due to the Gouy phase shift of the Bessel beam, as we will verify later.

*ζ*due to the two Gaussian phase contributions in Eq. (23), for a variation of the propagation distance

*z*equal to Λ. The initial distance is

*ζ*, since the corresponding variation of the Gouy phase shift of the Bessel beam is 360°. Moreover, the contribution to

*ζ*due to the wave-front curvature of the Gaussian beam is dependent on the radial coordinate

*ρ*. If we evaluate the perturbation of

*l*from 0 to 3 (resulting in

*l*

^{th}-order Bessel beams and the reference Gaussian beam, calculated at different propagation distances

*d*, obtaining the patterns shown in the second and third column, respectively after the increments

*π*and consequently the bright and dark rings of the interference pattern mutually exchange. On the other hand, after a period Λ the Gouy phase shift increment is 2

*π*and the resulting interference pattern is practically identical to the initial pattern.

*l*maxima, which are equally spaced and alternated by

*l*minima. The Gouy phase shift of the

*l*

^{th}-order Bessel beams varies in the propagation along

*z*with constant rate, causing a uniform rotation of the intensity with angular rate

*π*. Actually the phase term due to the wave-front curvature of the Gaussian beam gives a contribution to

*ζ*proportional to the square of the radial distance and consequently the maxima (minima) of the interference pattern acquires an additional angular shift proportional to

*ρ*

^{2}.

*z*

_{0}of the CCD camera by the quantity

*d*, confirms the expected periodical behavior due to the linear accumulation of the Gouy phase shift of the CGH-produced Bessel beams. As shown in Fig. 6 , the registered patterns are in good agreement with the simulated ones, apart from a phase offset, and recur after Λ. In particular, the measured pattern appears circularly symmetric and pulsates cyclically for

*l*bright spots in the inner ring and rotates uniformly for

*l*from 1 to 3, as

*z*increases.

## 5. Conclusion

*β*with respect to the length

*k*of the wave-vector of a plane wave with same direction and wavelength. Therefore the Gouy extra axial phase shift accumulates linearly as the propagation distance increases, with a constant rate equal to the difference between

*k*and

*β*. This rate is independent of the order of the Bessel beam and depends only on the wavelength and on the transverse component

*α*of the angular spectrum wave-vectors, which describe a cone around the propagation axis. In particular the Gouy phase shift is a growing function of

*α*, according to the physical interpretation of the Gouy effect as originated by the transverse confinement of the beam. In fact the angular spectrum spreading, induced by the spatial confinement, causes an increase of

*α*and hence a reduction of

*β*, which is exactly the axial component of the angular spectrum wave-vectors.

*α*. Our experiments show that the patterns actually recur after the expected period, therefore providing a direct observation of the Gouy effect in nondiffracting Bessel beams.

## References and links

1. | L. G. Gouy, “Sur une proprieté nouvelle des ondes lumineuses,” Acad. Sci., Paris, C. R. |

2. | A. E. Siegman, |

3. | M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. Lond. A Math. Phys. Sci. |

4. | R. Simon and N. Mukunda, “Bargmann invariant and the geometry of the Güoy effect,” Phys. Rev. Lett. |

5. | D. Subbarao, “Topological phase in Gaussian beam optics,” Opt. Lett. |

6. | H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. |

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

8. | T. Ackermann, W. Grosse-Nobis, and G. L. Lippi, “The Gouy phase shift, the average phase lag of Fourier components of Hermite-Gaussian modes and their application to resonance conditions in optical cavities,” Opt. Commun. |

9. | J. H. Chow, G. de Vine, M. B. Gray, and D. E. McClelland, “Measurement of gouy phase evolution by use of spatial mode interference,” Opt. Lett. |

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

11. | J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A |

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

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

14. | A. J. Cox and D. C. Dibble, “Nondiffracting beam from a spatially filtered Fabry-Perot resonator,” J. Opt. Soc. Am. A |

15. | J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. |

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

17. | S. A. Tatarkova, W. Sibbett, and K. Dholakia, “Brownian particle in an optical potential of the washboard type,” Phys. Rev. Lett. |

18. | 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. |

19. | G. B. Arfken, and H. J. Weber, |

20. | N. N. Lebedev, |

21. | S. Feng and H. G. Winful, “Physical origin of the Gouy phase shift,” Opt. Lett. |

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

23. | M. A. Porras, C. J. Zapata-Rodríguez, and I. Gonzalo, “Gouy wave modes: undistorted pulse focalization in a dispersive medium,” Opt. Lett. |

24. | J. A. Davis, E. Carcole, and D. M. Cottrell, “Intensity and phase measurements of nondiffracting beams generated with a magneto-optic spatial light modulator,” Appl. Opt. |

25. | C. Paterson and R. Smith, “Higher-order Bessel waves produced by axicon-type computer-generated holograms,” Opt. Commun. |

26. | W.-H. Lee, “Computer-generated holograms,” in |

**OCIS Codes**

(260.1960) Physical optics : Diffraction theory

(350.5030) Other areas of optics : Phase

**ToC Category:**

Physical Optics

**History**

Original Manuscript: December 16, 2009

Manuscript Accepted: February 9, 2010

Published: March 23, 2010

**Citation**

Paolo Martelli, Matteo Tacca, Alberto Gatto, Giorgio Moneta, and Mario Martinelli, "Gouy phase shift in nondiffracting Bessel beams," Opt. Express **18**, 7108-7120 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-7-7108

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### References

- L. G. Gouy, “Sur une proprieté nouvelle des ondes lumineuses,” Acad. Sci., Paris, C. R. 110, 1251–1253 (1890).
- A. E. Siegman, Lasers (University Science Books, Mill Valley, CA, 1986).
- M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. Lond. A Math. Phys. Sci. 392(1802), 45–57 (1984). [CrossRef]
- R. Simon and N. Mukunda, “Bargmann invariant and the geometry of the Güoy effect,” Phys. Rev. Lett. 70(7), 880–883 (1993). [CrossRef] [PubMed]
- D. Subbarao, “Topological phase in Gaussian beam optics,” Opt. Lett. 20(21), 2162–2164 (1995). [CrossRef] [PubMed]
- H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5(10), 1550–1567 (1966). [CrossRef] [PubMed]
- B. E. A. Saleh, and M. C. Teich, Fundamentals of Photonics (John Wiley & Sons, Inc., 1991).
- T. Ackermann, W. Grosse-Nobis, and G. L. Lippi, “The Gouy phase shift, the average phase lag of Fourier components of Hermite-Gaussian modes and their application to resonance conditions in optical cavities,” Opt. Commun. 189(1-3), 5–14 (2001). [CrossRef]
- J. H. Chow, G. de Vine, M. B. Gray, and D. E. McClelland, “Measurement of gouy phase evolution by use of spatial mode interference,” Opt. Lett. 29(20), 2339–2341 (2004). [CrossRef] [PubMed]
- 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), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-18-8382 . [CrossRef] [PubMed]
- J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4(4), 651–654 (1987). [CrossRef]
- J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987). [CrossRef] [PubMed]
- 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]
- A. J. Cox and D. C. Dibble, “Nondiffracting beam from a spatially filtered Fabry-Perot resonator,” J. Opt. Soc. Am. A 9(2), 282–286 (1992). [CrossRef]
- J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177(1-6), 297–301 (2000). [CrossRef]
- 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]
- S. A. Tatarkova, W. Sibbett, and K. Dholakia, “Brownian particle in an optical potential of the washboard type,” Phys. Rev. Lett. 91(3), 038101 (2003). [CrossRef] [PubMed]
- 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]
- G. B. Arfken, and H. J. Weber, Mathematical Methods for Physicists (Harcourt/Academic Press, 5th Ed., 2001).
- N. N. Lebedev, Special Functions and their Applications (Dover Publications, Inc., New York, 1972).
- S. Feng and H. G. Winful, “Physical origin of the Gouy phase shift,” Opt. Lett. 26(8), 485–487 (2001). [CrossRef]
- G. Indebetouw, “Nondiffracting optical fields: some remarks on their analysis and synthesis,” J. Opt. Soc. Am. A 6(1), 150–152 (1989). [CrossRef]
- M. A. Porras, C. J. Zapata-Rodríguez, and I. Gonzalo, “Gouy wave modes: undistorted pulse focalization in a dispersive medium,” Opt. Lett. 32(22), 3287–3289 (2007). [CrossRef] [PubMed]
- J. A. Davis, E. Carcole, and D. M. Cottrell, “Intensity and phase measurements of nondiffracting beams generated with a magneto-optic spatial light modulator,” Appl. Opt. 35(4), 593–598 (1996). [CrossRef] [PubMed]
- C. Paterson and R. Smith, “Higher-order Bessel waves produced by axicon-type computer-generated holograms,” Opt. Commun. 124(1-2), 121–130 (1996). [CrossRef]
- W.-H. Lee, "Computer-generated holograms," in Progress in Optics XVI, E. Wolf, ed., (North-Holland, Amsterdam, 1978).

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