Abstract

The three dimensional distribution of focused vector Mathieu-Gauss beams (vMG) is studied in the vicinity of the geometrical focus of an unapertured thin lens. We adopt two different intensity based criteria for defining the actual focus. Our analysis confirms the existence of a focal shift towards the lens for this type of beams. The dependence of the focal shift on the different parameters of the beams is discussed in detail. Beams with different states of polarization are studied as well, and it is shown that the focal shift is independent of the polarization.

© 2008 Optical Society of America

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References

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    [CrossRef] [PubMed]
  6. V. N. Mahajan, "Axial irradiance and optimum focusing of laser beams," Appl. Opt. 19, 3042-3053 (1983).
    [CrossRef]
  7. M. Martinez-Corral, C. J. Zapata Rodriguez, P. Andres, 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]
  8. C. J. R. Sheppard and P. Torok, "Dependence of focal shift on Fresnel number and angular aperture," Opt. Lett. 23, 1803-1804 (1998).
    [CrossRef]
  9. S. De Nicola, D. Anderson, and M. Lisak, "Focal shift effects in diffracted focused beams," Pure Appl. Opt. 7, 1249-1259 (1998).
    [CrossRef]
  10. T. C. Poon, "Focal shift in focused annular beams," Opt. Commun. 65, 401-406 (1988).
    [CrossRef]
  11. W. H. Carter and M. F. Aburdene, "Focal shift in Laguerre-Gaussian beams," J. Opt. Soc. Am. A 10, 1949-1952 (1987).
    [CrossRef]
  12. B. Lu and W. Huang, "Focal shift in unapertured Bessel-Gauss beams," Opt. Commun. 109, 43-46 (1994).
    [CrossRef]
  13. S. De Nicola, "On axis focal shift effects in focused truncated J0 Bessel beams," Pure Appl. Opt. 5, 827-831 (1996).
    [CrossRef]
  14. Z. Hricha, L. Dalil-Essakali, and A. Belafhal, "Axial intensity distribution and focal shifts of focused partially coherent conical Bessel-Gauss beams," Opt. Quantum. Electron. 35, 101-110 (2003).
    [CrossRef]
  15. M. Dong and J. Pu, "Effective Fresnel number and the focal shifts of focused partially coherent beams," J. Opt. Soc. Am. A 24, 192-196 (2007).
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  16. P. L. Greene and D. G. Hall, "Focal shift in vector beams," Opt. Express 4, 411-419 (1999).
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  17. J. Pu and B. Lu, "Focal shifts in focused nonuniformly polarized beams," J. Opt. Soc. Am. A 18, 2760-2766 (2001).
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  19. Y. V. Kartashov, A. A. Egorov, V. A. Vysloukh, and L. Torner, "Shaping soliton properties in Mathieu lattices," Opt. Lett. 31, 238-240 (2006).
    [CrossRef] [PubMed]
  20. C. Lopez-Mariscal, J. C. Gutierrez-Vega, G. Milne, and K. Dholakia, "Orbital angular momentum transfer in helical Mathieu beams," Opt. Express. 14, 4182-4187 (2006).
    [CrossRef] [PubMed]
  21. C. A. Dartora and H. E. Hernandez-Figueroa, "Properties of a localized Mathieu pulse," J. Opt. Soc. Am. A 21, 662-667 (2004).
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  22. J. Davila-Rodriguez and J. C. Gutierrez-Vega, "Helical Mathieu and parabolic localized pulses," J. Opt. Soc. Am. A 24, 3449-3455 (2007).
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  23. M. A. Bandres and J. C. Guti’errez-Vega, "Vector Helmholtz-Gauss and vector Laplace-Gauss beams," Opt. Lett. 30, 2155-2157 (2005).
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  24. R. I. Hernandez-Aranda, M. A. Bandres, and J. C. Gutierrez Vega, "Propagation dynamics of vector Mathieu- Gauss beams," in Laser Beam Shaping VII, Fred M. Dickey and David L. Shealy, eds., Proc. SPIE 6290, 629011 (2006).
    [CrossRef]
  25. R. I. Hernandez-Aranda, J. C. Gutierrez-Vega, M. Guizar-Sicairos, and M. A. Bandres, "Propagation of generalized vector Helmholtz-Gauss beams through paraxial optical systems," Opt. Express. 14, 8974-8988 (2006).
    [CrossRef] [PubMed]
  26. A. Chafiq, Z. Hricha, and A. Belafhal, "Propagation properties of vector Mathieu-Gauss beams," Opt. Commun. 275, 165-169 (2007).
    [CrossRef]
  27. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  28. M. Lax, W. H. Louisell, and W. B. McKnight, "From Maxwell to paraxial wave optics," Phys. Rev. A 11, 1365- 1370 (1975).
    [CrossRef]
  29. J. C. Gutierrez-Vega and M. A. Bandres, "Helmholtz-Gauss waves," J. Opt. Soc. Am. A 22, 289-298 (2005).
    [CrossRef]
  30. B. Lu and R. Peng, "Focal shift in Hermite-Gaussian beams based on the encircled-power criterion," Opt. Laser Technol. 35, 435-440 (2003).
    [CrossRef]
  31. L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, "Longitudinal Field Modes Probed by Single Molecules," Phys. Rev. Lett. 86, 5251-5254 (2001).
    [CrossRef] [PubMed]
  32. R. Dorn, S. Quabis, and G. Leuchs, "Sharper Focus for a Radially Polarized Light Beam," Phys. Rev. Lett. 91, 233901 (2003).
    [CrossRef] [PubMed]
  33. J. Alda, "Laser and Gaussian beam propagation and transformation," in Encyclopedia of Optical Engineering, (Marcel Dekker, New York, 2003).

2007

2006

Y. V. Kartashov, A. A. Egorov, V. A. Vysloukh, and L. Torner, "Shaping soliton properties in Mathieu lattices," Opt. Lett. 31, 238-240 (2006).
[CrossRef] [PubMed]

C. Lopez-Mariscal, J. C. Gutierrez-Vega, G. Milne, and K. Dholakia, "Orbital angular momentum transfer in helical Mathieu beams," Opt. Express. 14, 4182-4187 (2006).
[CrossRef] [PubMed]

R. I. Hernandez-Aranda, J. C. Gutierrez-Vega, M. Guizar-Sicairos, and M. A. Bandres, "Propagation of generalized vector Helmholtz-Gauss beams through paraxial optical systems," Opt. Express. 14, 8974-8988 (2006).
[CrossRef] [PubMed]

2005

2004

2003

B. Lu and R. Peng, "Focal shift in Hermite-Gaussian beams based on the encircled-power criterion," Opt. Laser Technol. 35, 435-440 (2003).
[CrossRef]

R. Dorn, S. Quabis, and G. Leuchs, "Sharper Focus for a Radially Polarized Light Beam," Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

Z. Hricha, L. Dalil-Essakali, and A. Belafhal, "Axial intensity distribution and focal shifts of focused partially coherent conical Bessel-Gauss beams," Opt. Quantum. Electron. 35, 101-110 (2003).
[CrossRef]

2001

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, "Longitudinal Field Modes Probed by Single Molecules," Phys. Rev. Lett. 86, 5251-5254 (2001).
[CrossRef] [PubMed]

J. Pu and B. Lu, "Focal shifts in focused nonuniformly polarized beams," J. Opt. Soc. Am. A 18, 2760-2766 (2001).
[CrossRef]

2000

1999

1998

M. Martinez-Corral, C. J. Zapata Rodriguez, P. Andres, 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]

C. J. R. Sheppard and P. Torok, "Dependence of focal shift on Fresnel number and angular aperture," Opt. Lett. 23, 1803-1804 (1998).
[CrossRef]

S. De Nicola, D. Anderson, and M. Lisak, "Focal shift effects in diffracted focused beams," Pure Appl. Opt. 7, 1249-1259 (1998).
[CrossRef]

1996

S. De Nicola, "On axis focal shift effects in focused truncated J0 Bessel beams," Pure Appl. Opt. 5, 827-831 (1996).
[CrossRef]

1994

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

1988

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

1987

W. H. Carter and M. F. Aburdene, "Focal shift in Laguerre-Gaussian beams," J. Opt. Soc. Am. A 10, 1949-1952 (1987).
[CrossRef]

1983

V. N. Mahajan, "Axial irradiance and optimum focusing of laser beams," Appl. Opt. 19, 3042-3053 (1983).
[CrossRef]

1982

1981

J. J. Stamnes and B. Spjelkavik, "Focusing at small angular apertures in the Debye and Kirchhoff approximations," Opt. Commun. 40, 81-85 (1981).
[CrossRef]

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

1980

1975

M. Lax, W. H. Louisell, and W. B. McKnight, "From Maxwell to paraxial wave optics," Phys. Rev. A 11, 1365- 1370 (1975).
[CrossRef]

Appl. Opt.

V. N. Mahajan, "Axial irradiance and optimum focusing of laser beams," Appl. Opt. 19, 3042-3053 (1983).
[CrossRef]

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

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. A.

M. Martinez-Corral, C. J. Zapata Rodriguez, P. Andres, 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]

Opt. Commun.

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

J. J. Stamnes and B. Spjelkavik, "Focusing at small angular apertures in the Debye and Kirchhoff approximations," Opt. Commun. 40, 81-85 (1981).
[CrossRef]

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

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

A. Chafiq, Z. Hricha, and A. Belafhal, "Propagation properties of vector Mathieu-Gauss beams," Opt. Commun. 275, 165-169 (2007).
[CrossRef]

Opt. Express

Opt. Express.

R. I. Hernandez-Aranda, J. C. Gutierrez-Vega, M. Guizar-Sicairos, and M. A. Bandres, "Propagation of generalized vector Helmholtz-Gauss beams through paraxial optical systems," Opt. Express. 14, 8974-8988 (2006).
[CrossRef] [PubMed]

C. Lopez-Mariscal, J. C. Gutierrez-Vega, G. Milne, and K. Dholakia, "Orbital angular momentum transfer in helical Mathieu beams," Opt. Express. 14, 4182-4187 (2006).
[CrossRef] [PubMed]

Opt. Laser Technol.

B. Lu and R. Peng, "Focal shift in Hermite-Gaussian beams based on the encircled-power criterion," Opt. Laser Technol. 35, 435-440 (2003).
[CrossRef]

Opt. Lett.

Opt. Quantum. Electron.

Z. Hricha, L. Dalil-Essakali, and A. Belafhal, "Axial intensity distribution and focal shifts of focused partially coherent conical Bessel-Gauss beams," Opt. Quantum. Electron. 35, 101-110 (2003).
[CrossRef]

Phys. Rev. A

M. Lax, W. H. Louisell, and W. B. McKnight, "From Maxwell to paraxial wave optics," Phys. Rev. A 11, 1365- 1370 (1975).
[CrossRef]

Phys. Rev. Lett.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, "Longitudinal Field Modes Probed by Single Molecules," Phys. Rev. Lett. 86, 5251-5254 (2001).
[CrossRef] [PubMed]

R. Dorn, S. Quabis, and G. Leuchs, "Sharper Focus for a Radially Polarized Light Beam," Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

Pure Appl. Opt.

S. De Nicola, D. Anderson, and M. Lisak, "Focal shift effects in diffracted focused beams," Pure Appl. Opt. 7, 1249-1259 (1998).
[CrossRef]

S. De Nicola, "On axis focal shift effects in focused truncated J0 Bessel beams," Pure Appl. Opt. 5, 827-831 (1996).
[CrossRef]

Other

R. I. Hernandez-Aranda, M. A. Bandres, and J. C. Gutierrez Vega, "Propagation dynamics of vector Mathieu- Gauss beams," in Laser Beam Shaping VII, Fred M. Dickey and David L. Shealy, eds., Proc. SPIE 6290, 629011 (2006).
[CrossRef]

J. Alda, "Laser and Gaussian beam propagation and transformation," in Encyclopedia of Optical Engineering, (Marcel Dekker, New York, 2003).

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

Supplementary Material (8)

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Figures (5)

Fig. 1.
Fig. 1.

Focusing system under consideration. The lens is assumed to be thin and unapertured, f defines the lens focal distance, i.e. geometrical focus, z 0 the actual focus, and Δz the focal shift.

Fig. 2.
Fig. 2.

Transverse intensity profiles of a TM type focused vMG beam with seed function W=We m (ξ,η;q), q=5 and beam orders (a)m=0, (b) m=1, (c) m=2, and (d) m=7. Plots in the top row correspond to initial profiles at plane z=0, bottom row images are snapshots for animations showing the three dimensional evolution in the focal region from z/f=0.95 to 1.05. (Movie files: (a) 1.68 MB, (b) 2.07 MB, (c) 1.94 MB, and (d) 2.31 MB). [Media 1][Media 2][Media 3][Media 4]

Fig. 3.
Fig. 3.

Transverse intensity profiles of a TM type focused vMG beam with seed function W=We m (ξ,η;q)+iWo m (ξ,η;q), q=5 and beam orders (a) m=4, and (b) m=7. Plots in (c) and (d) correspond to a right-circularly polarized vMG beam with seed function W=We m (ξ,η;q), q=5, and beam orders m=0 and m=7 respectively. Plots in the top row show initial profiles at plane z=0, bottom row images are snapshots for animations showing the three dimensional evolution in the focal region from z/f=0.95 to 1.05. (Movie files: (a) 2.12 MB, (b) 2.33 MB, (c) 1.65 MB, and (d).2.25 MB) [Media 5][Media 6][Media 7][Media 8]

Fig. 4.
Fig. 4.

Beam waist plotted against the normalized propagation distance z/f, for q=[0,5,25] and mode orders m=0 through 7. (a) Calculation of beam waist using the encircled energy criteria and (b) using the mean squared radius of the intensity distribution. The minimum beam waist is achieved just before the geometric focus, i.e. z/f=1, indicating a focal shift Δz towards the lens.

Fig. 5.
Fig. 5.

Relative focal shift versus Gaussian-Fresnel number Nw of the beam, for q=5 and beam orders m=[0,1,7]. A change in Nw was achieved by changing the Gaussian waist w 0 from 600 µm to 2000 µm. All simulation parameters are the same as in Fig. 2. Notice how narrower beams present larger shifts.

Equations (20)

Equations on this page are rendered with MathJax. Learn more.

E t = Z ( ζ ) G ( r , ζ ) U ( X , Y ) ,
G ( r , ζ ) = ζ 1 exp ( r 2 w 0 2 ζ ) .
U ( 1 ) = T W ( X , Y ) , U ( 2 ) = z ̂ × U ( 1 ) .
E t = Z ( ζ ) G ( r , ζ ) T W ( X , Y ) ,
H t = ε 0 μ 0 Z ( ζ ) G ( r ) ( z ̂ × T W ( X , Y ) ) .
E z = i k T · E t , H z = i k T · H t .
W m e ( ξ , η ; q ) = Je m ( ξ , q ) ce m ( η , q ) ,
W m o ( ξ , η ; q ) = Jo m ( ξ , q ) se m ( η , q ) ,
T W ( ξ , η ) = ( 1 h ξ ξ ξ ̂ + 1 h η η η ̂ ) W ( ξ , η ) ,
T W ( ξ , η ) = X W ( ξ , η ) x ̂ Y W ( ξ , η ) y ̂ ,
X W ( ξ , η ) = h sinh ξ cos η h ξ 2 ξ W ( ξ , η ) h cos ξ sin η h η 2 η W ( ξ , η ) ,
Y W ( ξ , η ) = h cosh ξ sin η h ξ 2 ξ W ( ξ , η ) + h sinh ξ cos η h η 2 η W ( ξ , η ) ,
E t ( ξ ˜ , η ˜ , z ) = k i 2 F z exp [ ik ( z + r 2 2 z ) ] exp [ 1 4 F ( k t 2 + k 2 r 2 z 2 ) ] ˜ t W ( ξ ˜ , η ˜ ) ,
F = 1 w 0 2 ik 2 ( 1 z 1 f ) ,
x = 2 iFzh k cosh ξ ˜ cos η ˜ , y = 2 iFzh k sinh ξ ˜ sin η ˜ .
P = S · z ̂ dA = S z dA .
s z = 1 2 ( ε 0 μ 0 ) 1 2 ( E t · E t * ) ,
0 2 π 0 r w s z r d r d θ = 0.8 P ,
r m 2 ( z ) = 1 P [ r 2 s z dA m r ] ,
N w = w 0 2 λ f ,

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