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.

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  1. L. G. Gouy, “Sur une proprieté nouvelle des ondes lumineuses,” Acad. Sci., Paris, C. R. 110, 1251–1253 (1890).
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  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).
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    [CrossRef] [PubMed]
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    [CrossRef]
  12. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [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]
  17. 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]
  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]
  19. G. B. Arfken, and H. J. Weber, Mathematical Methods for Physicists (Harcourt/Academic Press, 5th Ed., 2001).
  20. N. N. Lebedev, Special Functions and their Applications (Dover Publications, Inc., New York, 1972).
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    [CrossRef]
  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]
  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]
  26. W.-H. Lee, "Computer-generated holograms," in Progress in Optics XVI, E. Wolf, ed., (North-Holland, Amsterdam, 1978).

2007

2006

2004

2003

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]

2001

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

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. 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]

2000

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]

1996

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]

1995

1993

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

1992

1989

1987

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]

1984

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

1966

1890

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

Ackermann, T.

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]

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]

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]

Berry, M. V.

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

Carcole, E.

Chow, J. H.

Cottrell, D. M.

Cox, A. J.

Davis, J. A.

de Vine, G.

Dholakia, K.

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]

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]

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]

Dibble, D. C.

Dultz, W.

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]

Durnin, J.

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]

Eberly, J. H.

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

Feng, S.

Friberg, A. T.

Garces-Chavez, V.

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]

Garcés-Chávez, V.

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]

Gonzalo, I.

Gouy, L. G.

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

Gray, M. B.

Grosse-Nobis, W.

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]

Hamazaki, J.

Indebetouw, G.

Kogelnik, H.

Li, T.

Lippi, G. L.

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]

McClelland, D. E.

McGloin, D.

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]

Miceli, J. J.

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

Mineta, Y.

Morita, R.

Mukunda, N.

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

Oka, K.

Padgett, M. J.

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]

Paterson, C.

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]

Porras, M. A.

Schmitzer, H.

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]

Sibbett, W.

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]

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]

Simon, R.

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

Smith, R.

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]

Subbarao, D.

Tatarkova, S. A.

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]

Turunen, J.

Vasara, A.

Winful, H. G.

Zapata-Rodríguez, C. J.

Acad. Sci., Paris, C. R.

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

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Commun.

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]

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]

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]

Opt. Express

Opt. Lett.

Phys. Rev. Lett.

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

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]

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

Proc. R. Soc. Lond. A Math. Phys. Sci.

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

Other

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

B. E. A. Saleh, and M. C. Teich, Fundamentals of Photonics (John Wiley & Sons, Inc., 1991).

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).

W.-H. Lee, "Computer-generated holograms," in Progress in Optics XVI, E. Wolf, ed., (North-Holland, Amsterdam, 1978).

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Figures (6)

Fig. 1
Fig. 1

Cone of the angular spectrum wave-vectors generating a Bessel beam.

Fig. 2
Fig. 2

Transmission functions of the binary-amplitude off-axis CGHs for obtaining finite-aperture l th-order Bessel beams: (a) l = 0; (b) l = 1; (c) l = 2; (d) l = 3. The black region corresponds to a transmission equal to zero.

Fig. 3
Fig. 3

Measured intensity distribution of the finite-aperture l th-order Bessel beams reproduced by the realized binary-amplitude off-axis CGHs: (a) l = 0; (b) l = 1; (c) l = 2; (d) l = 3.

Fig. 4
Fig. 4

Scheme of the interferometric set-up for observing the Gouy phase shift of Bessel beams.

Fig. 5
Fig. 5

Transverse intensity distributions of the interference between l th-order Bessel beams (with l from 0 to 3) and a reference Gaussian beam, calculated at the initial position z 0 (first column), at z 0 + Λ/2 (second column), and at z 0 + Λ (third column).

Fig. 6
Fig. 6

Transverse intensity distributions of the interference between l th-order Bessel beams (with l from 0 to 3) and a reference Gaussian beam, acquired by a CCD camera at the initial position z 0 (first column), at z 0 + Λ/2 (second column), and at z 0 + Λ (third column).

Equations (25)

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ψ ( x , y , z , t ) = u t ( x , y )   exp (- i β z exp ( i ω t ),
t 2 u t ( x , y ) + ( k 2 β 2 )   u t ( x , y ) = 0 ,
ψ ( r , t ) = A exp [ - i ( k r ω t ) ] = A   exp ( i k x x )   exp ( i k y y )   exp ( i k z z )   exp ( i ω t )
u t ( x , y ) = A   exp (- i k x x exp (- i k y y )
ψ ( ρ , ϑ , z , t ) = u t ( ρ , ϑ )   exp (- i β z exp ( i ω t ),
2 u t ρ 2 + 1 ρ u t ρ + 1 ρ 2 2 u t ϑ 2 + ( k 2 β 2 )   u t = 0
u t ( ρ , ϑ ) = F ( ρ )   exp ( i l ϑ ),
d 2 F d ρ 2 + 1 ρ d F d ρ + ( k 2 β 2 l 2 ρ 2 )   F = 0 ,
α = k 2 β 2 > 0.
ψ ( r , t ) = A   J l ( α ρ )   exp ( i l ϑ exp (- i β z exp ( i ω t ) .
ϕ Gouy ( z ) = ( k β ) z = α 2 k + k 2 α 2 z .
k t = k 2 k z 2 = k 2 β 2 = α ,
t l ( ρ , ϑ ) = {    exp ( i l ϑ )   exp ( i   2 π ρ ρ 0 )                   if    ρ D    0                                                         if   ρ > D ,
α = 2 π ρ 0
T l ( ρ , ϑ ) = {     1 2   [ 1 + cos ( 2 π ν ρ sin ϑ + l ϑ 2 π ρ ρ 0 ) ]                   if    ρ D     0                                                                         if   ρ > D
T l (B) ( ρ , ϑ ) = {     1          if    1 2 1 2   [ 1 + cos ( 2 π ν ρ sin ϑ + l ϑ 2 π ρ ρ 0 ) ] 1    and    ρ D   0          if    0 1 2   [ 1 + cos ( 2 π ν ρ sin ϑ + l ϑ 2 π ρ ρ 0 ) ] 1 2    or    ρ > D .
u Bessel ( ρ , ϑ , z ) = A B   J l ( 2 π ρ ρ 0 )   exp ( i l ϑ exp (- i β z ),
β = k 2 α 2 = k 1 ( α / k ) 2 = k 1 ( λ / ρ 0 ) 2 .
β k ( 1 α 2 2 k 2 ) = k α 2 2 k = k π λ ρ 0 2 .
ϕ Gouy ( z ) α 2 2 k z = π λ ρ 0 2 z .
u Gauss ( ρ , z ) = A G w 0 w ( z )   exp [ ρ 2 w 2 ( z ) ]   exp [ i k z + i arctan ( z z R ) i k ρ 2 2 R ( z ) ] ,
I ( ρ , ϑ , z ) = | A B | 2 J l 2 ( 2 π ρ ρ 0 ) + | A G | 2 w 0 2 w 2 ( z )   exp [ 2 ρ 2 w 2 ( z ) ] +
+ 2 | A B A G J l ( 2 π ρ ρ 0 ) | w 0 w ( z )   exp [ ρ 2 w 2 ( z ) ]   cos [ ζ ( ρ , ϑ , z ) ] ,
ζ ( ρ , ϑ , z ) = l ϑ + π λ ρ 0 2 z arctan ( z z R ) + k ρ 2 2 R ( z ) + ϕ 0 .
Λ = 2 π   ( d ϕ Gouy d z ) 1 8 π 2 α 2 λ = 2 ρ 0 2 λ .

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