Abstract

We examine the three-dimensional intensity distribution of vector Bessel–Gauss beams with general polarization near the focus of a nonaperturing thin lens. Recently reported linearly and azimuthally polarized Bessel–Gauss beams are members of this family. We define the width and focal plane of such a vector beam using an encircled-energy criterion and calculate numerically that, as for scalar beams, the true focus occurs not at the geometric focus of the lens but rather somewhat closer to the lens. This focal shift depends on the mode number and system parameters and is largest for a narrow beam, long lens focal length, and large wavelength.

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References

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  1. W. H. Carter, "Focal shift and concept of effective Fresnel number for a Gaussian laser beam," Appl. Opt. 21, 1989-1994 (1982).
    [CrossRef] [PubMed]
  2. Y. Li, "Dependence of the focal shift on Fresnel number and f number," J. Opt. Soc. Am. 72, 770-775 (1982).
    [CrossRef]
  3. Y. Li and E. Wolf, "Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers," J. Opt. Soc. Am. A 1, 801-808 (1984).
    [CrossRef]
  4. M. Mart�nez-Corral, C. J. Zapata-Rodr�guez, P. Andr�s and E. Silvestre, "Effective Fresnel-number concept for evaluating the relative focal shift in focused beams," J. Opt. Soc. Am. A 15, 449-455 (1998).
    [CrossRef]
  5. A. E. Siegman, Lasers, (University Science Books, Mill Valley, Calif., 1986), 769-773.
  6. Y. Li and E. Wolf, "Focal shifts in diffracted converging spherical waves," Opt. Commun. 39, 211-215 (1981).
    [CrossRef]
  7. M. P. Givens, "Focal shifts in diffracted converging spherical waves," Opt. Commun. 41, 145-148 (1982).
    [CrossRef]
  8. T.-C. Poon, "Focal shift in focused annular beams," Opt. Commun. 65, 401-406 (1988).
    [CrossRef]
  9. M. Mart�nez-Corral, C. J. Zapata-Rodr�guez, P. Andr�s and M. Kowalczyk, "Analytical formula for calculating the focal shift in apodized systems," J. Mod. Opt. 45, 1671-1679 (1998).
    [CrossRef]
  10. Y. Li and F. T. S. Yu, "Intensity distribution near the focus of an apertured focused Gaussian beam," Opt. Commun. 70, 1-7 (1989).
    [CrossRef]
  11. Y. Li, "Focal shift formula for focused, apertured Gaussian beams," J. Mod. Opt. 39, 1761-1764 (1992).
    [CrossRef]
  12. Y. Li, "Oscillations and discontinuity in the focal shift of Gaussian laser beams," J. Opt. Soc. Am. A 3, 1761- 1765 (1986).
    [CrossRef]
  13. Y. Li, "Axial intensity and the focal shift of a focused non-truncated elliptical Gaussian beam," Optik 80, 121- 123 (1988).
  14. W. H. Carter and M. F. Aburdene, "Focal shift in Laguerre-Gaussian beams," J. Opt. Soc. Am. A 4, 1949-1952 (1987).
    [CrossRef]
  15. S. De Nicola, "On-axis focal shift effects in focused truncated J 0 Bessel beams," Pure Appl. Opt. 5, 827-831 (1996).
    [CrossRef]
  16. B. L� and W. Huang, "Focal shift in unapertured Bessel-Gauss beams," Opt. Commun. 109, 43-46 (1994).
    [CrossRef]
  17. B. L� and W. Huang, "Three-dimensional intensity distribution of focused Bessel-Gauss beams," J. Mod. Opt. 43, 509-515 (1996).
    [CrossRef]
  18. R. Borghi, M. Santarsiero and S. Vicalvi, "Focal shift of focused flat-topped beams," Opt. Commun. 154, 243- 248 (1998).
    [CrossRef]
  19. F. Gori, G. Guattari and C. Padovani, "Bessel-Gauss beams," Opt. Commun. 64, 491-495 (1987).
    [CrossRef]
  20. R. H. Jordan and D. G. Hall, "Free-space azimuthal paraxial wave equation: the azimuthal Bessel-Gauss beam solution," Opt. Lett. 19, 427-429 (1994).
    [CrossRef] [PubMed]
  21. D. G. Hall, "Vector-beam solutions of Maxwells wave equation," Opt. Lett. 21, 9-11 (1996).
    [CrossRef] [PubMed]
  22. P. L. Greene and D. G. Hall, "Properties and diffraction of vector Bessel-Gauss beams," J. Opt. Soc. Am. A 15, 3020-3027 (1998).
    [CrossRef]
  23. L. W. Casperson, D. G. Hall and A. A. Tovar, "Sinusoidal-Gaussian beams in complex optical systems," J. Opt. Soc. Am. A 14, 3341-3348 (1997).
    [CrossRef]
  24. See, for example, J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 59-60.

Other (24)

W. H. Carter, "Focal shift and concept of effective Fresnel number for a Gaussian laser beam," Appl. Opt. 21, 1989-1994 (1982).
[CrossRef] [PubMed]

Y. Li, "Dependence of the focal shift on Fresnel number and f number," J. Opt. Soc. Am. 72, 770-775 (1982).
[CrossRef]

Y. Li and E. Wolf, "Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers," J. Opt. Soc. Am. A 1, 801-808 (1984).
[CrossRef]

M. Mart�nez-Corral, C. J. Zapata-Rodr�guez, P. Andr�s and E. Silvestre, "Effective Fresnel-number concept for evaluating the relative focal shift in focused beams," J. Opt. Soc. Am. A 15, 449-455 (1998).
[CrossRef]

A. E. Siegman, Lasers, (University Science Books, Mill Valley, Calif., 1986), 769-773.

Y. Li and E. Wolf, "Focal shifts in diffracted converging spherical waves," Opt. Commun. 39, 211-215 (1981).
[CrossRef]

M. P. Givens, "Focal shifts in diffracted converging spherical waves," Opt. Commun. 41, 145-148 (1982).
[CrossRef]

T.-C. Poon, "Focal shift in focused annular beams," Opt. Commun. 65, 401-406 (1988).
[CrossRef]

M. Mart�nez-Corral, C. J. Zapata-Rodr�guez, P. Andr�s and M. Kowalczyk, "Analytical formula for calculating the focal shift in apodized systems," J. Mod. Opt. 45, 1671-1679 (1998).
[CrossRef]

Y. Li and F. T. S. Yu, "Intensity distribution near the focus of an apertured focused Gaussian beam," Opt. Commun. 70, 1-7 (1989).
[CrossRef]

Y. Li, "Focal shift formula for focused, apertured Gaussian beams," J. Mod. Opt. 39, 1761-1764 (1992).
[CrossRef]

Y. Li, "Oscillations and discontinuity in the focal shift of Gaussian laser beams," J. Opt. Soc. Am. A 3, 1761- 1765 (1986).
[CrossRef]

Y. Li, "Axial intensity and the focal shift of a focused non-truncated elliptical Gaussian beam," Optik 80, 121- 123 (1988).

W. H. Carter and M. F. Aburdene, "Focal shift in Laguerre-Gaussian beams," J. Opt. Soc. Am. A 4, 1949-1952 (1987).
[CrossRef]

S. De Nicola, "On-axis focal shift effects in focused truncated J 0 Bessel beams," Pure Appl. Opt. 5, 827-831 (1996).
[CrossRef]

B. L� and W. Huang, "Focal shift in unapertured Bessel-Gauss beams," Opt. Commun. 109, 43-46 (1994).
[CrossRef]

B. L� and W. Huang, "Three-dimensional intensity distribution of focused Bessel-Gauss beams," J. Mod. Opt. 43, 509-515 (1996).
[CrossRef]

R. Borghi, M. Santarsiero and S. Vicalvi, "Focal shift of focused flat-topped beams," Opt. Commun. 154, 243- 248 (1998).
[CrossRef]

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

R. H. Jordan and D. G. Hall, "Free-space azimuthal paraxial wave equation: the azimuthal Bessel-Gauss beam solution," Opt. Lett. 19, 427-429 (1994).
[CrossRef] [PubMed]

D. G. Hall, "Vector-beam solutions of Maxwells wave equation," Opt. Lett. 21, 9-11 (1996).
[CrossRef] [PubMed]

P. L. Greene and D. G. Hall, "Properties and diffraction of vector Bessel-Gauss beams," J. Opt. Soc. Am. A 15, 3020-3027 (1998).
[CrossRef]

L. W. Casperson, D. G. Hall and A. A. Tovar, "Sinusoidal-Gaussian beams in complex optical systems," J. Opt. Soc. Am. A 14, 3341-3348 (1997).
[CrossRef]

See, for example, J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 59-60.

Supplementary Material (4)

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Figures (7)

Fig. 1.
Fig. 1.

Schematic diagram of the geometry under consideration, showing the notation used. The blue dashed lines represent the geometric focus of the lens, and the red curves the profile of the focused beam.

Fig. 2.
Fig. 2.

(1 MB each) Snapshots from animations showing vector beam intensity patterns as the viewer travels along the direction of propagation, near the geometric focus. Images (a)–(d) show modes m = 0, 1, 2, and 7, respectively. The beam parameters f = 80 mm and β ≃ 0.006 μm-1 correspond to Fresnel numbers Nw ≃ 20 and Nβ ≃ 0.5 (m + 2)2. Each mode has been independently normalized, but the individual images within each animation are not separately scaled. The white lines have been added to aid in the interpretation of Fig. 3. [Media 1] [Media 2] [Media 3] [Media 4]

Fig. 3.
Fig. 3.

Intensity plotted in an ρ-z plane slicing longitudinally through the three-dimensional beam, for (a) m = 0, ϕ = 0 (circularly symmetric beam), (b) m = 1, ϕ = π/2, (c) m = 2, ϕ = π/4, and (d) m = 7, ϕ = π/14. Beam parameters are as in Fig. 2. In images (b) and (c), note that the beam does not vanish entirely in the region surrounding z = f, but the energy is no longer found at this particular azimuth ϕ.

Fig. 4.
Fig. 4.

Beam width in micrometers plotted against normalized propagation distance z/f for several mode numbers m. The beam parameters are as in Fig. 2. The minimum width for each mode occurs slightly before z/f = 1, indicating a positive focal shift ∆z that decreases slowly with mode number for m ≥ 1.

Fig. 5.
Fig. 5.

Relative focal shift ∆z/f plotted against the Gaussian Fresnel number Nw , varied by changing w 0, for modes m = 0 (blue line) and 1 (red line). The Bessel Fresnel number is Nβ ≃ 2 for each beam, corresponding to β ≃ 0.006 μm-1 and β ≃ 0.0095 μm-1, respectively.

Fig. 6.
Fig. 6.

Relative focal shift ∆z/f plotted against the Bessel Fresnel number Nβ , varied by changing β, for modes m = 0 (blue line) and 1 (red line). The Gaussian Fresnel number is Nw ≃ 20. Differences between the two modes arise primarily from the (m + 2)2 factor in Nβ , as defined in Eq. (11); when the focal shift is plotted against β itself, the lines are nearly identical.

Fig. 7.
Fig. 7.

Relative focal shift plotted against the Gaussian Fresnel number Nw , for m = 0 and several values of the Bessel Fresnel number Nβ . For a single mode, the largest focal shifts are seen for small Nw and Nβ . This situation corresponds to either small w 0 and large β, which together produce a very narrow beam, or a large wavelength λ or focal length f

Equations (14)

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E = U ( ρ , ϕ , z ) exp [ i ( kz ωt ) ] .
U ( ρ , ϕ , z ) = g ( ρ , z ) f ( ρ , ϕ , z )
g ( ρ , z ) = w 0 w ( z ) exp [ i Φ ( z ) ] exp ( ρ 2 / w 0 2 1 + iz / L ) ,
f { ρ , ϕ } ( ρ , ϕ , z ) = a m Q ( z ) [ J m 1 ( u ) ± J m + 1 ( u ) ] Θ { ρ , ϕ } ( ϕ ) .
Q ( z ) = exp [ i β 2 z / ( 2 k ) 1 + iz / L ] ,
u = βρ 1 + iz / L ,
[ Θ ρ ( ϕ ) Θ ϕ ( ϕ ) ] = [ cos ( ) sin ( ) ] or [ sin ( ) cos ( ) ] ,
U { ρ , ϕ } ( ρ , ϕ , z ) = a m ki m 2 Fz exp [ ik ( z + ρ 2 2 z ) ] exp [ 1 4 F ( β 2 + k 2 ρ 2 z 2 ) ]
× [ I m 1 ( βkρ 2 Fz ) I m + 1 ( βkρ 2 Fz ) ] Θ { ρ , ϕ } ( ϕ ) ,
F = 1 w 0 2 + ik 2 ( 1 z 1 f ) ,
N w = w 0 2 / ( λf ) ,
N β = ( m + 2 ) 2 / ( λf β 2 ) .
I ( ρ , ϕ , z ) = ( 1 / 2 ) Re { E ( ρ , ϕ , z ) × H * ( ρ , ϕ , z ) z }
0 2 π 0 ρ 0 I ( ρ ' , ϕ ' , z 0 ) ρ ' d ρ ' ' 0 2 π 0 I ( ρ ' , ϕ ' , z 0 ) ρ ' d ρ ' ' = 0.8 .

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