Abstract

We define the effective Fresnel number of the cylindrical lens illuminated by a plane wave or Schell-model beams. On the basis of the concept of the effective Fresnel number, the focusing properties of the cylindrical lens illuminated by the Schell-model beam are investigated in a simple way. It is shown that the relative focal shift can be evaluated by an analytical formulation, which is expressed as a function of the effective Fresnel number. To evaluate our approach, we make the comparison between the results obtained by our method and the numerical calculation based on the diffraction integral. The results indicate that we can simply and exactly evaluate the focal shifts with our method.

© 2006 Optical Society of America

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  1. V. N. Mahajan, 'Axial irradiance and optimum focusing of laser beams,' Appl. Opt. 22, 3042-3053 (1983).
    [CrossRef] [PubMed]
  2. D. Y. Jiang and J. J. Stamnes, 'Focusing at low Fresnel numbers in the presence of cylindrical or spherical aberration,' Pure Appl. Opt. 6, 85-96 (1997).
    [CrossRef]
  3. E. Wolf and Y. Li, 'Conditions for the validity of the Debye integral representation of focused fields,' Opt. Commun. 39, 205-209 (1981).
    [CrossRef]
  4. Y. Li and E. Wolf, 'Three-dimensional intensity distributions near the focus in systems of different Fresnel numbers,' J. Opt. Soc. Am. A 1, 801-808 (1984).
    [CrossRef]
  5. Y. Li and H. Platzer, 'An experimental investigation of diffraction patterns in low-Fresnel-number focusing systems,' Opt. Acta 30, 1621-1643 (1983).
    [CrossRef]
  6. S. Wang, Q. Lin, L. Yu, and X. Xu, 'Fresnel number of a regular polygon and slit,' Appl. Opt. 39, 3453-3455 (2000).
    [CrossRef]
  7. X. Liu and J. Pu, 'Focal shift and focal switch of partially coherent light in dual-focus systems,' Opt. Commun. 252, 262-267 (2005).
    [CrossRef]
  8. B. Lu and R. Peng, 'Focal shift in Hermite-Gaussian beams based on the encircled-power criterion,' Opt. Laser Technol. 36, 435-440 (2003).
  9. J. Pu, B. Lu, and S. Nemoto, 'Three-dimensional intensity distribution of focused annular non-uniformly polarized beams,' J. Mod. Opt. 49, 1501-1513 (2002).
    [CrossRef]
  10. Y. Li, 'Encircled energy for systems of different Fresnel numbers,' Optik (Stuttgart) 64, 207-218 (1983).
  11. J. Pu and B. Lu, 'Focal shifts in focused nonuniformly polarized beams,' J. Opt. Soc. Am. A 18, 2760-2766 (2001).
    [CrossRef]
  12. 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]
  13. T. D. Visser, G. Gbur, and E. Wolf, 'Effect of the state of coherence on the three-dimensional spectral intensity distribution near focus,' Opt. Commun. 213, 13-19 (2002).
    [CrossRef]
  14. J. Pu and S. Nemoto, 'Spectral changes and 1XN spectral switches in the diffraction of partially coherent light by an aperture,' J. Opt. Soc. Am. A 19, 339-344 (2002).
    [CrossRef]
  15. J. Wu and A. D. Boardman, 'Propagation of a Gaussian-Schell beam through turbulent media,' J. Mod. Opt. 37, 671-684 (1990).
    [CrossRef]
  16. A. Dogariu and S. Amarande, 'Propagation of partially coherent beams: turbulence-induced degradation,' Opt. Lett. 28, 10-12 (2003).
    [CrossRef] [PubMed]
  17. J. Turunen, E. Tervonen, and A. T. Friberg, 'Coherence theoretic algorithm to determine the transverse-mode structure of lasers,' Opt. Lett. 14, 627-629 (1989).
    [CrossRef] [PubMed]
  18. J. Pu, S. Nemoto, H. Zhang, W. Zhang, and W. Zhang, 'Axial intensity distribution of partially coherent light focused by a lens with spherical aberration,' J. Mod. Opt. 47, 605-612 (2000).

2005 (1)

X. Liu and J. Pu, 'Focal shift and focal switch of partially coherent light in dual-focus systems,' Opt. Commun. 252, 262-267 (2005).
[CrossRef]

2003 (2)

B. Lu and R. Peng, 'Focal shift in Hermite-Gaussian beams based on the encircled-power criterion,' Opt. Laser Technol. 36, 435-440 (2003).

A. Dogariu and S. Amarande, 'Propagation of partially coherent beams: turbulence-induced degradation,' Opt. Lett. 28, 10-12 (2003).
[CrossRef] [PubMed]

2002 (3)

T. D. Visser, G. Gbur, and E. Wolf, 'Effect of the state of coherence on the three-dimensional spectral intensity distribution near focus,' Opt. Commun. 213, 13-19 (2002).
[CrossRef]

J. Pu and S. Nemoto, 'Spectral changes and 1XN spectral switches in the diffraction of partially coherent light by an aperture,' J. Opt. Soc. Am. A 19, 339-344 (2002).
[CrossRef]

J. Pu, B. Lu, and S. Nemoto, 'Three-dimensional intensity distribution of focused annular non-uniformly polarized beams,' J. Mod. Opt. 49, 1501-1513 (2002).
[CrossRef]

2001 (1)

2000 (2)

S. Wang, Q. Lin, L. Yu, and X. Xu, 'Fresnel number of a regular polygon and slit,' Appl. Opt. 39, 3453-3455 (2000).
[CrossRef]

J. Pu, S. Nemoto, H. Zhang, W. Zhang, and W. Zhang, 'Axial intensity distribution of partially coherent light focused by a lens with spherical aberration,' J. Mod. Opt. 47, 605-612 (2000).

1998 (1)

1997 (1)

D. Y. Jiang and J. J. Stamnes, 'Focusing at low Fresnel numbers in the presence of cylindrical or spherical aberration,' Pure Appl. Opt. 6, 85-96 (1997).
[CrossRef]

1990 (1)

J. Wu and A. D. Boardman, 'Propagation of a Gaussian-Schell beam through turbulent media,' J. Mod. Opt. 37, 671-684 (1990).
[CrossRef]

1989 (1)

1984 (1)

1983 (3)

Y. Li and H. Platzer, 'An experimental investigation of diffraction patterns in low-Fresnel-number focusing systems,' Opt. Acta 30, 1621-1643 (1983).
[CrossRef]

Y. Li, 'Encircled energy for systems of different Fresnel numbers,' Optik (Stuttgart) 64, 207-218 (1983).

V. N. Mahajan, 'Axial irradiance and optimum focusing of laser beams,' Appl. Opt. 22, 3042-3053 (1983).
[CrossRef] [PubMed]

1981 (1)

E. Wolf and Y. Li, 'Conditions for the validity of the Debye integral representation of focused fields,' Opt. Commun. 39, 205-209 (1981).
[CrossRef]

Amarande, S.

Andrés, P.

Boardman, A. D.

J. Wu and A. D. Boardman, 'Propagation of a Gaussian-Schell beam through turbulent media,' J. Mod. Opt. 37, 671-684 (1990).
[CrossRef]

Dogariu, A.

Friberg, A. T.

Gbur, G.

T. D. Visser, G. Gbur, and E. Wolf, 'Effect of the state of coherence on the three-dimensional spectral intensity distribution near focus,' Opt. Commun. 213, 13-19 (2002).
[CrossRef]

Jiang, D. Y.

D. Y. Jiang and J. J. Stamnes, 'Focusing at low Fresnel numbers in the presence of cylindrical or spherical aberration,' Pure Appl. Opt. 6, 85-96 (1997).
[CrossRef]

Li, Y.

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

Y. Li and H. Platzer, 'An experimental investigation of diffraction patterns in low-Fresnel-number focusing systems,' Opt. Acta 30, 1621-1643 (1983).
[CrossRef]

Y. Li, 'Encircled energy for systems of different Fresnel numbers,' Optik (Stuttgart) 64, 207-218 (1983).

E. Wolf and Y. Li, 'Conditions for the validity of the Debye integral representation of focused fields,' Opt. Commun. 39, 205-209 (1981).
[CrossRef]

Lin, Q.

Liu, X.

X. Liu and J. Pu, 'Focal shift and focal switch of partially coherent light in dual-focus systems,' Opt. Commun. 252, 262-267 (2005).
[CrossRef]

Lu, B.

B. Lu and R. Peng, 'Focal shift in Hermite-Gaussian beams based on the encircled-power criterion,' Opt. Laser Technol. 36, 435-440 (2003).

J. Pu, B. Lu, and S. Nemoto, 'Three-dimensional intensity distribution of focused annular non-uniformly polarized beams,' J. Mod. Opt. 49, 1501-1513 (2002).
[CrossRef]

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

Mahajan, V. N.

Martínez-Corral, M.

Nemoto, S.

J. Pu and S. Nemoto, 'Spectral changes and 1XN spectral switches in the diffraction of partially coherent light by an aperture,' J. Opt. Soc. Am. A 19, 339-344 (2002).
[CrossRef]

J. Pu, B. Lu, and S. Nemoto, 'Three-dimensional intensity distribution of focused annular non-uniformly polarized beams,' J. Mod. Opt. 49, 1501-1513 (2002).
[CrossRef]

J. Pu, S. Nemoto, H. Zhang, W. Zhang, and W. Zhang, 'Axial intensity distribution of partially coherent light focused by a lens with spherical aberration,' J. Mod. Opt. 47, 605-612 (2000).

Peng, R.

B. Lu and R. Peng, 'Focal shift in Hermite-Gaussian beams based on the encircled-power criterion,' Opt. Laser Technol. 36, 435-440 (2003).

Platzer, H.

Y. Li and H. Platzer, 'An experimental investigation of diffraction patterns in low-Fresnel-number focusing systems,' Opt. Acta 30, 1621-1643 (1983).
[CrossRef]

Pu, J.

X. Liu and J. Pu, 'Focal shift and focal switch of partially coherent light in dual-focus systems,' Opt. Commun. 252, 262-267 (2005).
[CrossRef]

J. Pu, B. Lu, and S. Nemoto, 'Three-dimensional intensity distribution of focused annular non-uniformly polarized beams,' J. Mod. Opt. 49, 1501-1513 (2002).
[CrossRef]

J. Pu and S. Nemoto, 'Spectral changes and 1XN spectral switches in the diffraction of partially coherent light by an aperture,' J. Opt. Soc. Am. A 19, 339-344 (2002).
[CrossRef]

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

J. Pu, S. Nemoto, H. Zhang, W. Zhang, and W. Zhang, 'Axial intensity distribution of partially coherent light focused by a lens with spherical aberration,' J. Mod. Opt. 47, 605-612 (2000).

Silvestre, E.

Stamnes, J. J.

D. Y. Jiang and J. J. Stamnes, 'Focusing at low Fresnel numbers in the presence of cylindrical or spherical aberration,' Pure Appl. Opt. 6, 85-96 (1997).
[CrossRef]

Tervonen, E.

Turunen, J.

Visser, T. D.

T. D. Visser, G. Gbur, and E. Wolf, 'Effect of the state of coherence on the three-dimensional spectral intensity distribution near focus,' Opt. Commun. 213, 13-19 (2002).
[CrossRef]

Wang, S.

Wolf, E.

T. D. Visser, G. Gbur, and E. Wolf, 'Effect of the state of coherence on the three-dimensional spectral intensity distribution near focus,' Opt. Commun. 213, 13-19 (2002).
[CrossRef]

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

E. Wolf and Y. Li, 'Conditions for the validity of the Debye integral representation of focused fields,' Opt. Commun. 39, 205-209 (1981).
[CrossRef]

Wu, J.

J. Wu and A. D. Boardman, 'Propagation of a Gaussian-Schell beam through turbulent media,' J. Mod. Opt. 37, 671-684 (1990).
[CrossRef]

Xu, X.

Yu, L.

Zapata-Rodríguez, C. J.

Zhang, H.

J. Pu, S. Nemoto, H. Zhang, W. Zhang, and W. Zhang, 'Axial intensity distribution of partially coherent light focused by a lens with spherical aberration,' J. Mod. Opt. 47, 605-612 (2000).

Zhang, W.

J. Pu, S. Nemoto, H. Zhang, W. Zhang, and W. Zhang, 'Axial intensity distribution of partially coherent light focused by a lens with spherical aberration,' J. Mod. Opt. 47, 605-612 (2000).

J. Pu, S. Nemoto, H. Zhang, W. Zhang, and W. Zhang, 'Axial intensity distribution of partially coherent light focused by a lens with spherical aberration,' J. Mod. Opt. 47, 605-612 (2000).

Appl. Opt. (2)

J. Mod. Opt. (3)

J. Pu, B. Lu, and S. Nemoto, 'Three-dimensional intensity distribution of focused annular non-uniformly polarized beams,' J. Mod. Opt. 49, 1501-1513 (2002).
[CrossRef]

J. Wu and A. D. Boardman, 'Propagation of a Gaussian-Schell beam through turbulent media,' J. Mod. Opt. 37, 671-684 (1990).
[CrossRef]

J. Pu, S. Nemoto, H. Zhang, W. Zhang, and W. Zhang, 'Axial intensity distribution of partially coherent light focused by a lens with spherical aberration,' J. Mod. Opt. 47, 605-612 (2000).

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

Opt. Acta (1)

Y. Li and H. Platzer, 'An experimental investigation of diffraction patterns in low-Fresnel-number focusing systems,' Opt. Acta 30, 1621-1643 (1983).
[CrossRef]

Opt. Commun. (3)

X. Liu and J. Pu, 'Focal shift and focal switch of partially coherent light in dual-focus systems,' Opt. Commun. 252, 262-267 (2005).
[CrossRef]

E. Wolf and Y. Li, 'Conditions for the validity of the Debye integral representation of focused fields,' Opt. Commun. 39, 205-209 (1981).
[CrossRef]

T. D. Visser, G. Gbur, and E. Wolf, 'Effect of the state of coherence on the three-dimensional spectral intensity distribution near focus,' Opt. Commun. 213, 13-19 (2002).
[CrossRef]

Opt. Laser Technol. (1)

B. Lu and R. Peng, 'Focal shift in Hermite-Gaussian beams based on the encircled-power criterion,' Opt. Laser Technol. 36, 435-440 (2003).

Opt. Lett. (2)

Optik (Stuttgart) (1)

Y. Li, 'Encircled energy for systems of different Fresnel numbers,' Optik (Stuttgart) 64, 207-218 (1983).

Pure Appl. Opt. (1)

D. Y. Jiang and J. J. Stamnes, 'Focusing at low Fresnel numbers in the presence of cylindrical or spherical aberration,' Pure Appl. Opt. 6, 85-96 (1997).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Effective Fresnel number N eff p and N eff s ( Δ ) as functions of the radius of the lens a, where λ = 1.06 μ m , f = 1 m . The solid and dashed curves correspond to the effective Fresnel number of the cylindrical lens illuminated by a plane wave and Schell-model beams ( Δ ) , respectively..

Fig. 2
Fig. 2

Relative focal shift versus the Fresnel number, where λ = 1.06 μ m , f = 1 m . The dotted, dashed, and solid curves correspond to Δ = 5 , 0.2 , 0.05 , respectively.

Fig. 3
Fig. 3

Relative focal shift versus the effective Fresnel number, where λ = 1.06 μ m , f = 1 m . The dotted, dashed, and solid curves correspond to the case of Δ = 5 , 0.2 , 0.05 , respectively.

Fig. 4
Fig. 4

Relative focal shift versus the relative coherent length, where λ = 1.06 μ m , f = 1 m . The dotted, dashed, and solid curves correspond to N a = 20 , 10 , 5 , respectively.

Fig. 5
Fig. 5

Relative focal shift versus the effective Fresnel number, where λ = 1.06 μ m , f = 1 m . The dotted, dashed, and solid curves correspond to N a = 20 , 10 , 5 , respectively.

Fig. 6
Fig. 6

Relative focal shift versus the effective Fresnel number.

Fig. 7
Fig. 7

Relative focal shift as a function of the effective Fresnel number given by numerical calculation of the diffraction integral as well as by our method (a) Δ = 0.05 and (b) N a = 5 .

Equations (36)

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E ( x ) = exp ( i k x 2 2 f ) ,
E ( z ) = exp [ i k ( z + f ) ] i λ ( z + f ) a a exp [ i π x 2 z λ f ( z + f ) ] d x ,
J N ( z ) = E ( z ) E ( z = 0 ) 2 = f z + f a a exp [ i π x 2 z λ f ( z + f ) ] d x a a 1 d x 2 ,
J N ( u ) = ( 1 2 λ f u ) a a exp ( i 2 π u x 2 ) d x a a 1 d x 2 ,
u = z 2 λ f ( z + f ) .
I N ( u ) = J N ( 0 ) + J N ( 0 ) u + J N ( 0 ) 2 u 2 .
I N ( u ) = 1 2 λ f u 4 π 2 2 u 2 ,
2 = m 2 m 0 ( m 1 m 0 ) 2 ,
m n = a a ( x 2 ) n d x .
u max p = 1 4 λ f 1 π 2 ζ 2 ,
z max p f = 1 1 + 2 π 2 ζ 2 ,
ζ = λ f .
N eff p = 1 π 1 + 2 π 2 ζ 2 ,
z max f = 1 π 2 N eff p 2 .
N eff p = 8 45 a 4 π 2 + λ 2 f 2 π λ f .
W ( 0 ) ( x 1 , x 2 , ω ) = [ I ( 0 ) ( x 1 , ω ) I ( 0 ) ( x 2 , ω ) ] 1 2 μ ( 0 ) ( x 2 x 1 , ω ) = I 0 exp [ ( x 2 x 1 ) 2 2 σ 0 2 ] ,
W ( x 1 , x 2 , ω ) = U * ( x 1 ) U ( x 2 ) ,
{ U ( x ) } = { U 0 ( x ) exp ( i k 2 f x 2 ) } .
W ( x 1 , x 2 , ω ) = I 0 exp [ ( x 2 x 1 ) 2 2 σ 0 2 ] exp [ i k 2 f ( x 1 2 x 2 2 ) ] .
W ( x 1 , x 2 , z ) = I 0 k 2 π ( z + f ) a a a a exp [ ( x 2 x 1 ) 2 2 σ 0 2 ] cos φ d x 1 d x 2 ,
φ = k 2 f ( x 1 2 x 2 2 ) k 2 f ( f + z ) [ ( x 1 x 1 ) 2 ( x 2 x 2 ) 2 ] .
H ( x , z ) = I 0 k 2 π ( z + f ) a a a a exp [ ( x 2 x 1 ) 2 2 σ 0 2 ] cos φ d x 1 d x 2 .
H ( z ) = I 0 k a 2 2 π ( z + f ) 1 1 1 1 exp [ ( x 2 x 1 ) 2 2 Δ 2 ] cos [ 2 π z a 2 ( x 1 2 x 2 2 ) 2 λ f ( f + z ) ] d x 1 d x 2 ,
H ( u ) = I 0 a 2 [ 1 λ f ( 1 2 λ f u ) ] 1 1 1 1 exp [ ( x 2 x 1 ) 2 2 Δ 2 ] cos [ 2 π u a 2 ( x 1 2 x 2 2 ) ] d x 1 d x 2 ,
u = z 2 λ f ( z + f ) .
J N ( u ) = H ( u ) H ( u = 0 ) = 1 1 1 1 q ( x 1 , x 2 ) cos [ 2 π u a 2 ( x 1 2 x 2 2 ) ] d x 1 d x 2 1 1 1 1 q ( x 1 , x 2 ) d x 1 d x 2 ,
q ( x 1 , x 2 ) = exp [ ( x 2 x 1 ) 2 2 Δ 2 ] .
I N ( u ) J N ( 0 ) + J N ( 0 ) u + J N ( 0 ) 2 u 2 .
I N ( u ) 1 2 λ f u 2 π 2 2 u 2 ,
2 = m 2 m 0 ,
m n = 1 1 1 1 q ( x 1 , x 2 ) [ a 2 ( x 1 2 x 2 2 ) ] n d x 1 d x 2 .
u max s = 1 2 λ f 1 π 2 ζ 2 .
z max f = 1 1 + π 2 ζ 2 .
ζ = λ f .
N eff s = 1 π 1 + π 2 ζ 2 .
z max f = 1 π 2 N eff s 2 .

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