Abstract

We show experimentally how diffracting and nondiffracting laser beams can be characterized through their one-dimensional constituent wave. Such a wave stems from an angular decomposition applicable to any cylindrically symmetric laser beam. In our experiment, spatial filtering in a 4-f system is used to generate the constituent wave of each beam under study. Standard one-dimensional root-mean-square (rms) parameters, such as the propagation factor and the generalized Rayleigh range, are then applied to determine the regime of propagation of the beams to characterize.

©2005 Optical Society of America

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References

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  1. A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986).
  2. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [Crossref] [PubMed]
  3. F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–495 (1987).
    [Crossref]
  4. V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Schirripa Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).
  5. G. Rousseau, D. Gay, and M. Piché, “One-dimensional description of cylindrically symmetric laser beams: application to Bessel-type nondiffracting beams,” J. Opt. Soc. Am. A 22, 1274–1287 (2005).
    [Crossref]
  6. A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE 1224, 2–14 (1990).
    [Crossref]
  7. P.-A. Bélanger, “Beam propagation and the ABCD ray matrices,” Opt. Lett. 16, 196–198 (1991).
    [Crossref] [PubMed]
  8. P.-A. Bélanger, Y. Champagnes, and C. Paré, “Beam propagation factor of diffracted laser beams,” Opt. Commun. 105, 233–242 (1994).
    [Crossref]
  9. C. Paré and P.-A. Bélanger, “Propagation law and quasi-invariance properties of the truncated second-order moment of a diffracted laser beam,” Opt. Commun. 123, 679–693 (1996).
    [Crossref]
  10. T. F. Johnston, “Beam propagation (M2) measurement made as easy as it gets: the four-cuts method,” Appl. Opt. 37, 4840–4850 (1998).
    [Crossref]

2005 (1)

1998 (1)

1996 (2)

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Schirripa Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

C. Paré and P.-A. Bélanger, “Propagation law and quasi-invariance properties of the truncated second-order moment of a diffracted laser beam,” Opt. Commun. 123, 679–693 (1996).
[Crossref]

1994 (1)

P.-A. Bélanger, Y. Champagnes, and C. Paré, “Beam propagation factor of diffracted laser beams,” Opt. Commun. 105, 233–242 (1994).
[Crossref]

1991 (1)

1990 (1)

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE 1224, 2–14 (1990).
[Crossref]

1987 (2)

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

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

Bagini, V.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Schirripa Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

Bélanger, P.-A.

C. Paré and P.-A. Bélanger, “Propagation law and quasi-invariance properties of the truncated second-order moment of a diffracted laser beam,” Opt. Commun. 123, 679–693 (1996).
[Crossref]

P.-A. Bélanger, Y. Champagnes, and C. Paré, “Beam propagation factor of diffracted laser beams,” Opt. Commun. 105, 233–242 (1994).
[Crossref]

P.-A. Bélanger, “Beam propagation and the ABCD ray matrices,” Opt. Lett. 16, 196–198 (1991).
[Crossref] [PubMed]

Champagnes, Y.

P.-A. Bélanger, Y. Champagnes, and C. Paré, “Beam propagation factor of diffracted laser beams,” Opt. Commun. 105, 233–242 (1994).
[Crossref]

Durnin, J.

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

Eberly, J. H.

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

Frezza, F.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Schirripa Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

Gay, D.

Gori, F.

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

Guattari, G.

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

Johnston, T. F.

Miceli, J. J.

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

Padovani, C.

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

Paré, C.

C. Paré and P.-A. Bélanger, “Propagation law and quasi-invariance properties of the truncated second-order moment of a diffracted laser beam,” Opt. Commun. 123, 679–693 (1996).
[Crossref]

P.-A. Bélanger, Y. Champagnes, and C. Paré, “Beam propagation factor of diffracted laser beams,” Opt. Commun. 105, 233–242 (1994).
[Crossref]

Piché, M.

Rousseau, G.

Santarsiero, M.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Schirripa Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

Schettini, G.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Schirripa Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

Schirripa Spagnolo, G.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Schirripa Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

Siegman, A. E.

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE 1224, 2–14 (1990).
[Crossref]

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

Appl. Opt. (1)

J. Mod. Opt. (1)

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Schirripa Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

J. Opt. Soc. Am. A (1)

Opt. Commun. (3)

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

P.-A. Bélanger, Y. Champagnes, and C. Paré, “Beam propagation factor of diffracted laser beams,” Opt. Commun. 105, 233–242 (1994).
[Crossref]

C. Paré and P.-A. Bélanger, “Propagation law and quasi-invariance properties of the truncated second-order moment of a diffracted laser beam,” Opt. Commun. 123, 679–693 (1996).
[Crossref]

Opt. Lett. (1)

Phys. Rev. Lett. (1)

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

Proc. SPIE (1)

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE 1224, 2–14 (1990).
[Crossref]

Other (1)

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

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Figures (7)

Fig. 1.
Fig. 1. Schematic diagram of the 4-f system used to produce the resulting beam (without the slit) and the constituent wave (with the slit) of a diffracting beam (with mask m = 0) and a Bessel-type nondiffracting beam (with mask m = 1). The four spatial filters used in the Fourier plane P2 are shown on the right-hand side of the 4-f system.
Fig. 2.
Fig. 2. Positions of the lateral extent limits of the constituent wave defined by (z) ± 2σx (z). The dashed curves, given by (z) ± 2σ x0, represent the limits of the geometrical projection. The axis of propagation of the resulting beam is shown in green. Colored backgrounds represent, respectively, the geometrical interference zone (light grey) and the confocal zone (dark grey). (a) Diffracting and (b) nondiffracting beams obtained, respectively, with the central lobe (m = 0) and the first annular lobe (m = 1) of the Airy diffraction pattern.
Fig. 3.
Fig. 3. Schematic diagram of the experimental setup. CCR stands for corner cube retroreflector.
Fig. 4.
Fig. 4. Measured transverse intensity distributions of the resulting beam and the corresponding constituent wave along the axis of propagation z. (a) Diffracting and (b) nondiffracting beams obtained, respectively, with the central lobe (m = 0) and the first annular lobe (m = 1) of the Airy diffraction pattern. The axis of propagation z is located at the geometric center of each image. Images in (a) and (b) contain, respectively, 480 × 480 pixels (2.23 mm × 2.23 mm) and 960 × 960 pixels (4.46 mm × 4.46 mm).
Fig. 5.
Fig. 5. Intensity profiles of the resulting beam (blue curves) and the constituent wave (red curves) along the axis x (y = 0) at thirteen axial positions. In each case, the green line represents the position of the propagation axis z of the resulting beam. Experimental results are shown for (a) the diffracting and (b) the nondiffracting beams obtained, respectively, with the central lobe (m = 0) and the first annular lobe (m = 1) of the Airy diffraction pattern.
Fig. 6.
Fig. 6. Measured lateral positions of the centroid (red dots) of the constituent wave as a function of the axial position z. The black line is obtained from numerical simulations and the red dashed line is the result of the linear curve fit based on experimental data. The light grey background represents the geometrical interference zone. (a) Diffracting and (b) nondiffracting beams obtained, respectively, with the central lobe (m = 0) and the first annular lobe (m = 1) of the Airy diffraction pattern.
Fig. 7.
Fig. 7. Measured rms widths (red dots) of the constituent wave as a function of the axial position z. The black curve is obtained from numerical simulations and the red dashed curve is the result of the hyperbolic curve fit based on experimental data. Colored backgrounds represent, respectively, the geometrical interference zone (light grey) and the confocal zone (dark grey). (a) Diffracting and (b) nondiffracting beams obtained, respectively, with the central lobe (m = 0) and the first annular lobe (m = 1) of the Airy diffraction pattern.

Tables (2)

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Table 1. Calculated values of the rms parameters of the constituent waves.

Tables Icon

Table 2. Results of the linear and the hyperbolic curve fits. Results expected from numerical simulations (presented in Table 1) are printed over a grey background.

Equations (7)

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U ˜ c s x z H ( s x ) s x U ˜ ( s r = s x , z ) ,
Z max 2 σ x 0 tan Θ 2 σ x 0 λ s ̄ x ,
Q z R , c Z max s ̄ x 2 σ s x ,
U ˜ ( s r , z = 0 ) = { J 1 ( 2 πR s r ) 2 πR s r for s r [ s min , s max [ 0 for s r [ s min , s max [ ,
U ˜ c ( s x , z = 0 ) = { H ( s x ) s x J 1 ( 2 πR s x ) 2 πR s x for s x [ s min , s max [ 0 for s x [ s min , s max [ .
x ̄ ( z ) = Θ ( z δ z l ) ,
σ x ( z ) = σ x 0 [ 1 + ( z δ z h z R , c ) 2 ] 1 2 ,

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