Abstract

Based on the Collins diffraction integral formula and the complex Gaussian expansion of the aperture function, an analytical expression for a Lorentz–Gauss beam focused by an optical system with a thin lens and a circular aperture has been derived. The focal shift of the focused truncated Lorentz–Gauss beam is investigated with numerical examples, and the dependence of the focal shift on the different parameters of the focused truncated Lorentz–Gauss beam is discussed in detail. This research is useful to the applications of highly divergent laser beams.

© 2008 Optical Society of America

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References

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  1. Y. Li, “Dependence of the focal shift on Fresnel number and f number,” J. Opt. Soc. Am. 72, 770-775 (1982).
    [CrossRef]
  2. 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]
  3. Y. Li and E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211-215 (1981).
    [CrossRef]
  4. M. P. Givens, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 41, 145-148 (1982).
    [CrossRef]
  5. W. H. Carter, “Focal shift and concept of effective Fresnel number for a Gaussian laser beam,” Appl. Opt. 21, 1989-1994 (1982).
    [CrossRef] [PubMed]
  6. 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]
  7. Y. Li, “Focal shift formula for focused, apertured Gaussian beams,” J. Mod. Opt. 39, 1761-1764 (1992).
    [CrossRef]
  8. Y. Li, “Oscillations and discontinuity in the focal shift of Gaussian laser beams,” J. Opt. Soc. Am. A 3, 1761-1765 (1986).
    [CrossRef]
  9. X. Du and D. Zhao, “Focal shift and focal switch of focused truncated elliptical Gaussian beams,” Opt. Commun. 275, 301-304 (2007).
    [CrossRef]
  10. W. H. Carter and M. F. Aburdene, “Focal shift in Laguerre-Gaussian beams,” J. Opt. Soc. Am. A 4, 1949-1952 (1987).
    [CrossRef]
  11. P. L. Greene and D. G. Hall, “Focal shift in vector beams,” Opt. Express 4, 411-419 (1999).
    [CrossRef] [PubMed]
  12. R. I. Hernandez-Aranda and J. C. Gutiérrez-Vega, “Focal shift in vector Mathieu-Gauss Beams,” Opt. Express 16, 5838-5848 (2008).
    [CrossRef] [PubMed]
  13. M. Martínez-Corral, C. J. Zapata-Rodríguez, P. Andrés, 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]
  14. O. E. Gawhary and S. Severini, “Lorentz beams and symmetry properties in paraxial optics,” J. Opt. A, Pure Appl. Opt. 8, 409-414 (2006).
    [CrossRef]
  15. A. Naqwi and F. Durst, “Focus of diode laser beams: a simple mathematical model,” Appl. Opt. 29, 1780-1785 (1990).
    [CrossRef] [PubMed]
  16. W. P. Dumke, “The angular beam divergence in double-heterojunction lasers with very thin active regions,” IEEE J. Quantum Electron. QE-11, 400-402 (1975).
    [CrossRef]
  17. S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60, 1168-1177 (1970).
    [CrossRef]
  18. J. J. Wen and M. A. Breazeale, “Computer optimization of the Gaussian beam description of an ultrasonic field,” in Computational Acoustics, D.Lee, A.Cakmak, and R.Vichnevetsky, eds. (Elsevier, 1990).
  19. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1980), p. 1147.
  20. Y. Li and E. Wolf, “Focal shift in focused truncated Gaussian beams,” Opt. Commun. 42, 151-156 (1982).
    [CrossRef]

2008 (1)

2007 (1)

X. Du and D. Zhao, “Focal shift and focal switch of focused truncated elliptical Gaussian beams,” Opt. Commun. 275, 301-304 (2007).
[CrossRef]

2006 (1)

O. E. Gawhary and S. Severini, “Lorentz beams and symmetry properties in paraxial optics,” J. Opt. A, Pure Appl. Opt. 8, 409-414 (2006).
[CrossRef]

1999 (1)

1998 (1)

1992 (1)

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

1990 (1)

1989 (1)

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]

1987 (1)

1986 (1)

1984 (1)

1982 (4)

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

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, “Focal shift in focused truncated Gaussian beams,” Opt. Commun. 42, 151-156 (1982).
[CrossRef]

1981 (1)

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

1975 (1)

W. P. Dumke, “The angular beam divergence in double-heterojunction lasers with very thin active regions,” IEEE J. Quantum Electron. QE-11, 400-402 (1975).
[CrossRef]

1970 (1)

Aburdene, M. F.

Andrés, P.

Breazeale, M. A.

J. J. Wen and M. A. Breazeale, “Computer optimization of the Gaussian beam description of an ultrasonic field,” in Computational Acoustics, D.Lee, A.Cakmak, and R.Vichnevetsky, eds. (Elsevier, 1990).

Carter, W. H.

Collins, S. A.

Du, X.

X. Du and D. Zhao, “Focal shift and focal switch of focused truncated elliptical Gaussian beams,” Opt. Commun. 275, 301-304 (2007).
[CrossRef]

Dumke, W. P.

W. P. Dumke, “The angular beam divergence in double-heterojunction lasers with very thin active regions,” IEEE J. Quantum Electron. QE-11, 400-402 (1975).
[CrossRef]

Durst, F.

Gawhary, O. E.

O. E. Gawhary and S. Severini, “Lorentz beams and symmetry properties in paraxial optics,” J. Opt. A, Pure Appl. Opt. 8, 409-414 (2006).
[CrossRef]

Givens, M. P.

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

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1980), p. 1147.

Greene, P. L.

Gutiérrez-Vega, J. C.

Hall, D. G.

Hernandez-Aranda, R. I.

Li, Y.

Y. Li, “Focal shift formula for focused, apertured Gaussian beams,” J. Mod. Opt. 39, 1761-1764 (1992).
[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, “Oscillations and discontinuity in the focal shift of Gaussian laser beams,” J. Opt. Soc. Am. A 3, 1761-1765 (1986).
[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]

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, “Focal shift in focused truncated Gaussian beams,” Opt. Commun. 42, 151-156 (1982).
[CrossRef]

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

Martínez-Corral, M.

Naqwi, A.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1980), p. 1147.

Severini, S.

O. E. Gawhary and S. Severini, “Lorentz beams and symmetry properties in paraxial optics,” J. Opt. A, Pure Appl. Opt. 8, 409-414 (2006).
[CrossRef]

Silvestre, E.

Wen, J. J.

J. J. Wen and M. A. Breazeale, “Computer optimization of the Gaussian beam description of an ultrasonic field,” in Computational Acoustics, D.Lee, A.Cakmak, and R.Vichnevetsky, eds. (Elsevier, 1990).

Wolf, E.

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]

Y. Li and E. Wolf, “Focal shift in focused truncated Gaussian beams,” Opt. Commun. 42, 151-156 (1982).
[CrossRef]

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

Yu, F. T. S.

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]

Zapata-Rodríguez, C. J.

Zhao, D.

X. Du and D. Zhao, “Focal shift and focal switch of focused truncated elliptical Gaussian beams,” Opt. Commun. 275, 301-304 (2007).
[CrossRef]

Appl. Opt. (2)

IEEE J. Quantum Electron. (1)

W. P. Dumke, “The angular beam divergence in double-heterojunction lasers with very thin active regions,” IEEE J. Quantum Electron. QE-11, 400-402 (1975).
[CrossRef]

J. Mod. Opt. (1)

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

J. Opt. A, Pure Appl. Opt. (1)

O. E. Gawhary and S. Severini, “Lorentz beams and symmetry properties in paraxial optics,” J. Opt. A, Pure Appl. Opt. 8, 409-414 (2006).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (4)

Opt. Commun. (5)

X. Du and D. Zhao, “Focal shift and focal switch of focused truncated elliptical Gaussian beams,” Opt. Commun. 275, 301-304 (2007).
[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 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]

Y. Li and E. Wolf, “Focal shift in focused truncated Gaussian beams,” Opt. Commun. 42, 151-156 (1982).
[CrossRef]

Opt. Express (2)

Other (2)

J. J. Wen and M. A. Breazeale, “Computer optimization of the Gaussian beam description of an ultrasonic field,” in Computational Acoustics, D.Lee, A.Cakmak, and R.Vichnevetsky, eds. (Elsevier, 1990).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1980), p. 1147.

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Figures (7)

Fig. 1
Fig. 1

Schematic diagram of an optical system with a thin lens and a circular aperture. The waist of the Lorentz–Gauss beam is assumed to be located on the plane of the aperture.

Fig. 2
Fig. 2

Axial normalized intensity distribution of a focused truncated Lorentz–Gauss beam. λ = 0.8 μ m , f = 200 mm , and w 0 x = w 0 y = 0.8 mm . (a) G = 4 . (b) G = 8 . The solid, long-dashed, and short-dashed curves correspond to N = 1 , 4, and 9, respectively.

Fig. 3
Fig. 3

Axial normalized intensity distribution of a focused truncated Lorentz–Gauss beam. Parameters as in Fig. 2, with N = 1 . The solid, long-dashed, and short-dashed curves correspond to (a) G = 0.25 , 0.5, and 4, respectively, and to (b) G = 4 , 8, and 16, respectively.

Fig. 4
Fig. 4

Axial normalized intensity distribution of a focused truncated Lorentz–Gauss beam. Parameters as in Fig. 2, with N = 4 . (a) The solid, long-dashed, and short-dashed curves correspond to (a) G = 0.25 , 0.5, and 4, respectively, and to (b) G = 4 , 8, and 16, respectively.

Fig. 5
Fig. 5

Axial normalized intensity distribution of a focused truncated Lorentz–Gauss beam. λ = 0.8 μ m , f = 200 mm , w 0 y = 0.8 mm , and N = 1 . The solid, long-dashed, and short-dashed curves correspond to w 0 x = 0.2 , 0.4, and 1.6 mm , respectively, with (a) G = 1 , (b) G = 4 .

Fig. 6
Fig. 6

Axial normalized intensity distribution of a focused truncated Lorentz–Gauss beam. f = 200 mm , w 0 x = w 0 y = 0.8 mm , and w 0 = 0.8 mm . The solid, long-dashed, and short-dashed curves correspond to λ = 0.2 , 0.4, and 0.8 μ m , respectively, with (a) R = 0.4 mm , (b) R = 0.8 mm .

Fig. 7
Fig. 7

Axial normalized intensity distribution of a focused truncated Lorentz–Gauss beam. λ = 0.8 μ m , w 0 x = w 0 y = 0.8 mm , and w 0 = 0.8 mm . The solid, long-dashed, and short-dashed curves correspond to f = 100 , 200, and 400 mm , respectively, with (a) R = 0.4 mm , (b) R = 0.8 mm .

Equations (24)

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[ A B C D ] = [ Δ z f f + Δ z 1 f 1 ] .
E ( x 0 , y 0 , 0 ) = E 0 w 0 x w 0 y 1 [ 1 + ( x 0 w 0 x ) 2 ] [ 1 + ( y 0 w 0 y ) 2 ] exp ( x 0 2 + y 0 2 w 0 2 ) ,
E ( x , y , z ) = i exp ( i k z ) λ B E ( x 0 , y 0 , 0 ) circ ( ζ ) exp { i k 2 B [ A ( x 0 2 + y 0 2 ) 2 ( x 0 x + y y 0 ) + D ( x 2 + y 2 ) ] } d x 0 d y 0 ,
circ ( ζ ) = { 1 0 ζ < 1 0 ζ 1 } .
circ ( ζ ) = n = 1 15 α n exp ( β n ζ 2 ) ζ [ 0 , ) ,
E ( x , y , z ) = i E 0 exp ( i k z ) λ B n = 1 15 α n E n ( x , z ) E n ( y , z ) ,
E n ( j , z ) = 1 w 0 j [ 1 + ( j 0 w 0 j ) 2 ] exp [ i k 2 B ( A n j 0 2 2 j 0 j + D j 2 ) ] d j 0 ,
E n ( j , z ) = w 0 j exp ( i k D j 2 2 B ) exp ( i k j 2 2 A n B ) [ f 1 ( j A n ) f 2 ( j A n ) ] ,
f 1 ( τ ) = 1 w 0 j 2 + τ 2 ,
f 2 ( τ ) = exp ( k A n 2 B τ 2 ) .
E n ( j , z ) = π 2 exp ( i k C n 2 A n j 2 ) [ E j + ( j , z ) + E j ( j , z ) ] ,
E j ± ( j , z ) = exp [ i k A n 2 B ( w 0 j ± i j A n ) 2 ] { 1 erf [ i k A n 2 B ( w 0 j ± i j A n ) ] } ,
erf ( x ) = 2 π 0 x exp ( s 2 ) d s
E ( x , y , z ) = i π 2 E 0 exp ( i k z ) 4 λ B n = 1 15 α n exp ( i k C n 2 A n ρ 2 ) [ E x + ( x , z ) + E x ( x , z ) ] [ E y + ( y , z ) + E y ( y , z ) ] ,
E ( 0 , 0 , z ) = i π 2 E 0 exp ( i k z ) λ B n = 1 15 α n exp [ i k A n 2 B ( w 0 x 2 + w 0 y 2 ) ] { 1 erf [ i k A n 2 B w 0 x ] } { 1 erf [ i k A n 2 B w 0 y ] } .
I ( 0 , 0 , z ) = E ( 0 , 0 , z ) 2 .
erf [ i k A n 2 B w 0 x ] = 1 + i k A n 2 B w 0 x π exp ( i k A n 2 B w 0 x 2 ) m = 1 ( 1 ) m Γ ( m 1 2 ) ( i k A n 2 B w 0 x 2 ) m ,
erf [ i k A n 2 B w 0 y ] = 1 + i k A n 2 B w 0 y π exp ( i k A n 2 B w 0 y 2 ) m = 1 ( 1 ) m Γ ( m 1 2 ) ( i k A n 2 B w 0 y 2 ) m ,
E ( 0 , 0 , z ) = exp ( i k z ) n = 1 15 α n Δ z f i 2 ( f + Δ z ) ( β n k R 2 + 1 k w 0 2 ) .
E ( 0 , 0 , z ) = exp ( i k z ) Δ z f i 2 ( f + Δ z ) k w 0 2 .
d I ( 0 , 0 , z ) d Δ z = 2 Δ z f 2 + 4 ( Δ z f + 1 ) f k 2 w 0 4 [ ( Δ z f ) 2 + 4 ( f + Δ z ) 2 k 2 w 0 4 ] 2 = 0 ,
Δ z max f = 1 ( 1 + π 2 w 0 4 λ 2 f 2 ) .
G = w 0 2 λ f ,
N = R 2 λ f .

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