Abstract

The focal shift of a focused truncated pulsed-laser beam is investigated. In the case of the Fresnel approximation, the analytic expression of the time-averaged intensity distribution along the axis is derived based on the series expansion. It shows that the focal shift of the pulsed beam can be completely determined by a series of normalized spectrum moments and the central Fresnel number defined according to the central frequency of the pulse. The absolute value of the focal shift of the pulsed beam decreases monotonously and slowly with the normalized spectrum width increasing and the central Fresnel number fixed, and it increases monotonously with the central Fresnel number decreasing and the normalized spectrum width fixed. Besides the central Fresnel number and the normalized spectrum width, the shape of spectral intensity of the pulse affects the focal shift too.

© 2007 Optical Society of America

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References

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  1. Y. Li and E. Wolf, "Focal shifts in diffracted converging spherical waves," Opt. Commun. 39, 211-215 (1981).
    [CrossRef]
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    [CrossRef]
  3. C. J. R. Sheppard and P. Török, "Dependence of focal shift on Fresnel number and angular aperture," Opt. Lett. 23, 1803-1804 (1998).
    [CrossRef]
  4. Y. Li, "Focal shift in small-Fresnel-number focusing systems of different relative aperture," J. Opt. Soc. Am. A 20, 234-239 (2003).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  9. C. Rulliere, Femtosecond Laser Pulses: Principles and Experiments (Springer, 1998).
  10. Zs. Bor and Z. L. Hováth, "Distortion of femtosecond pulses in lenses. Wave optical description," Opt. Commun. 94, 249-258 (1992).
    [CrossRef]
  11. Z. L. Hováth and Zs. Bor, "Focusing of femtosecond pulses having Gaussian spatial distribution," Opt. Commun. 100, 6-12 (1992).
    [CrossRef]
  12. A. Federico and O. Martinez, "Distortion of femtosecond pulses due to chromatic aberration in lenses," Opt. Commun. 91, 104-110 (1992).
    [CrossRef]
  13. M. Kempe, U. Stamm, B. Wilhelmi, and W. Rudolph, "Spatial and temporal transformation of femtosecond laser pulses by lenses and lens systems," J. Opt. Soc. Am. B 9, 1158-1165 (1992).
    [CrossRef]
  14. I. P. Christov, "Propagation of femtosecond light pulses," Opt. Commun. 53, 364-366 (1985).
    [CrossRef]
  15. C. J. R. Sheppard and X. Gan, "Free-space propagation of femtosecond light pulses," Opt. Commun. 133, 1-6 (1997).
    [CrossRef]
  16. R. Ashman and M. Gu, "Effect of ultrashort pulsed illumination on foci caused by a Fresnel zone plate," Appl. Opt. 42, 1852-1855 (2003).
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  17. M. Gu and X. Gan, "Fresnel diffraction by circular and serrated apertures illuminated with an ultrashort pulsed-laser beam," J. Opt. Soc. Am. A 13, 771-778 (1996).
    [CrossRef]

2005 (2)

2003 (2)

1999 (1)

1998 (2)

1997 (1)

C. J. R. Sheppard and X. Gan, "Free-space propagation of femtosecond light pulses," Opt. Commun. 133, 1-6 (1997).
[CrossRef]

1996 (1)

1992 (4)

Zs. Bor and Z. L. Hováth, "Distortion of femtosecond pulses in lenses. Wave optical description," Opt. Commun. 94, 249-258 (1992).
[CrossRef]

Z. L. Hováth and Zs. Bor, "Focusing of femtosecond pulses having Gaussian spatial distribution," Opt. Commun. 100, 6-12 (1992).
[CrossRef]

A. Federico and O. Martinez, "Distortion of femtosecond pulses due to chromatic aberration in lenses," Opt. Commun. 91, 104-110 (1992).
[CrossRef]

M. Kempe, U. Stamm, B. Wilhelmi, and W. Rudolph, "Spatial and temporal transformation of femtosecond laser pulses by lenses and lens systems," J. Opt. Soc. Am. B 9, 1158-1165 (1992).
[CrossRef]

1987 (1)

1985 (1)

I. P. Christov, "Propagation of femtosecond light pulses," Opt. Commun. 53, 364-366 (1985).
[CrossRef]

1982 (1)

1981 (1)

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

Aburdene, M. F.

Ashman, R.

Bor, Zs.

Zs. Bor and Z. L. Hováth, "Distortion of femtosecond pulses in lenses. Wave optical description," Opt. Commun. 94, 249-258 (1992).
[CrossRef]

Z. L. Hováth and Zs. Bor, "Focusing of femtosecond pulses having Gaussian spatial distribution," Opt. Commun. 100, 6-12 (1992).
[CrossRef]

Carter, W. H.

Christov, I. P.

I. P. Christov, "Propagation of femtosecond light pulses," Opt. Commun. 53, 364-366 (1985).
[CrossRef]

Federico, A.

A. Federico and O. Martinez, "Distortion of femtosecond pulses due to chromatic aberration in lenses," Opt. Commun. 91, 104-110 (1992).
[CrossRef]

Gan, X.

Greene, P. L.

Gu, M.

Hall, D. G.

Hováth, Z. L.

Zs. Bor and Z. L. Hováth, "Distortion of femtosecond pulses in lenses. Wave optical description," Opt. Commun. 94, 249-258 (1992).
[CrossRef]

Z. L. Hováth and Zs. Bor, "Focusing of femtosecond pulses having Gaussian spatial distribution," Opt. Commun. 100, 6-12 (1992).
[CrossRef]

Kempe, M.

Li, Y.

Martinez, O.

A. Federico and O. Martinez, "Distortion of femtosecond pulses due to chromatic aberration in lenses," Opt. Commun. 91, 104-110 (1992).
[CrossRef]

Rudolph, W.

Rulliere, C.

C. Rulliere, Femtosecond Laser Pulses: Principles and Experiments (Springer, 1998).

Sheppard, C. J. R.

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

C. J. R. Sheppard and X. Gan, "Free-space propagation of femtosecond light pulses," Opt. Commun. 133, 1-6 (1997).
[CrossRef]

Stamm, U.

Török, P.

Wilhelmi, B.

Wolf, E.

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

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

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

Opt. Commun. (6)

I. P. Christov, "Propagation of femtosecond light pulses," Opt. Commun. 53, 364-366 (1985).
[CrossRef]

C. J. R. Sheppard and X. Gan, "Free-space propagation of femtosecond light pulses," Opt. Commun. 133, 1-6 (1997).
[CrossRef]

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

Zs. Bor and Z. L. Hováth, "Distortion of femtosecond pulses in lenses. Wave optical description," Opt. Commun. 94, 249-258 (1992).
[CrossRef]

Z. L. Hováth and Zs. Bor, "Focusing of femtosecond pulses having Gaussian spatial distribution," Opt. Commun. 100, 6-12 (1992).
[CrossRef]

A. Federico and O. Martinez, "Distortion of femtosecond pulses due to chromatic aberration in lenses," Opt. Commun. 91, 104-110 (1992).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Other (1)

C. Rulliere, Femtosecond Laser Pulses: Principles and Experiments (Springer, 1998).

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Figures (5)

Fig. 1
Fig. 1

Scheme of the focused laser pulse.

Fig. 2
Fig. 2

Relationship between the normalized spectrum width Δ ν ˜ RMS and the temporal width Δt FWHM, with the central frequency ν ¯ corresponding to a wavelength of 800   nm . Curve G is for the pulse with Gaussian spectral intensity and constant spectral phase, and curve R is for the pulse with rectangular spectral intensity and constant spectral phase.

Fig. 3
Fig. 3

Normalized axial intensity distribution I ˜ ( z ) of the focused laser pulse with Gaussian spectral intensity, where Δ ν ˜ RMS = 0.3 , and N ¯ = 5 , 1, 0.5, and 0.1, respectively. Curves I are calculated by integral expressions (1) and (2), and the other curves are calculated by truncated series expressions (9b), (10), and (13) with M = 3 , 5, and 15.

Fig. 4
Fig. 4

Relationship between the focal shift Δ f / f and the central Fresnel number N ¯ with various normalized RMS spectrum widths Δ ν ˜ RMS . Curves G1 and G2 stand for pulses with Gaussian spectral intensities; curves R1 and R2 stand for pulses with rectangular spectral intensities. Curves G1 and R1 correspond to Δ ν ˜ RMS = 0.3 ; curves G2 and R2 correspond to Δ ν ˜ RMS = 0.1 ; and curve M1 corresponds to Δ ν ˜ RMS = 0 , i.e., the monochromatic beam.

Fig. 5
Fig. 5

Relationship between the focal shift Δ f / f and the normalized RMS spectrum width Δ ν ˜ RMS with various central Fresnel numbers N ¯ . Curves with Δ ν ˜ RMS [ 0 , 0.33 ] represent pulses with Gaussian spectral intensities, and curves with Δ ν ˜ RMS [ 0 , 0.58 ] represent pulses with rectangular spectral intensities.

Equations (26)

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U ( ν , z ) = 1 1 z / f exp [ i 2 π ν c ( n Δ + z ) ] × { exp [ i π R 2 ν c ( 1 z 1 f ) ] 1 } ,
I ( z ) = 1 T | U ( ν , z ) | 2 S ( ν ) d ν ,
S ( ν ) = exp [ ( ν ν 0 ) 2 Δ ν 2 ] ,
S ( ν ) = rect ( ν ν 0 Δ ν ) ,
S ( ν ) = δ ( ν ν 0 ) ,
I ˜ ( z ) = 2 f 2 z 2 n = 2 ( 1 z 1 f ) 2 n 4 ( 1 ) n 2 1 ( 2 n 2 ) ! ( π R 2 c ) 2 n 4 × S ( ν ) ν 2 n 2 d ν S ( ν ) ν 2 d ν .
ν ¯ = S ( ν ) ν d ν S ( ν ) d ν ,  
S ( ν ) ν 2 n 2 d ν S ( ν ) ν 2 d ν = ν ¯ 2 n 4 α n ,
α n = S ( ν ˜ ν ¯ ) ν ˜ 2 n 2 d ν ˜ S ( ν ˜ ν ¯ ) ν ˜ 2 d ν ˜ ,
I ˜ ( z ) = 2 ( u + 1 ) 2 n = 2 ( 1 ) n 2 ( 2 n 2 ) ! ( π N ¯ ) 2 n 4 α n u 2 n 4 = α 2 + m = 1 r m u m ,
r m = { 2 ( 1 ) n 2 ( 2 n 2 ) ! ( π N ¯ ) 2 n 4 α n + 2 ( 1 ) n 1 ( 2 n ) ! ( π N ¯ ) 2 n 2 α n + 1 ,   if   m = 2 n 2 , n = 2 , 3 ,   … 4   ( 1 ) n 2 ( 2 n 2 ) ! ( π N ¯ ) 2 n 4 α n ,   if   m = 2 n 3 , n = 2 , 3 ,   …   .
I ˜ ( z ) = α 2 + m = 1 M r m u m .
d I ˜ ( z ) d u = m = 1 M m r m u m 1 = 0 .
ν ¯ = ν 0 .
α n = 1 + m = 1 n 1 [ ( 2 n 2 ) ! / ( 2 n 2 2 m ) ! m ! ] Δ ν ˜ 2 m 2 3 m 1 + ( 1 / 4 ) Δ ν ˜ 2 ,
α n = [ 1 / ( 2 n 1 ) ] k = 0 2 n 2 ( 1 + Δ ν ˜ / 2 ) 2 n 2 k ( 1 Δ ν ˜ / 2 ) k 1 + Δ ν ˜ 2 / 12 ,
α n = 1 ,
Δ ν RMS = [ S ( ν ) ( ν ν ¯ ) 2 d ν S ( ν ) d ν ] 1 / 2
i ( t ) = | S ( ν ) exp ( i 2 π ν t ) d ν | 2 .
S ( ν 1 ) = exp [ ( ν 1 ν 0 ) 2 Δ ν 2 ] = 0.01
α n = d ν ˜ exp [ [ 2 ( ν ˜ 1 ) 2 ] / Δ ν ˜ 2 ] ν ˜ 2 n 2 d ν ˜ exp [ [ 2 ( ν ˜ 1 ) 2 ] / Δ ν ˜ 2 ] ν ˜ 2 ,
d ν ˜ exp [ 2 ( ν ˜ 1 ) 2 Δ ν ˜ 2 ] ν ˜ 2 n 2 = k = 0 2 n 2 ( 2 n 2 ) ! ( 2 ) k 1 k ! ( 2 n 2 k ) ! × Δ ν ˜ k + 1 I k , I k = exp ( σ 2 ) σ k d σ ,
I 2 m = 2 m 1 2 I 2 m 2 = 2 m 1 2 2 m 3 2 I 2 m 4
= = ( 2 m 1 ) ! ! 2 m I 0 ,
α n = rect [ ( ν ˜ 1 ) / Δ ν ˜ ] ν ˜ 2 n 2 d ν ˜ rect [ ( ν ˜ 1 ) / Δ ν ˜ ] ν ˜ 2 d ν ˜ = 1 / ( 2 n 1 ) k = 0 2 n 2 ( 1 + Δ ν ˜ / 2 ) 2 n 2 k ( 1 Δ ν ˜ / 2 ) k 1 + Δ ν ˜ 2 / 12 .
α n = ( 1 / ν ¯ ) δ ( ν ˜ 1 ) ν ˜ 2 n 2 d ν ˜ ( 1 / ν ¯ ) δ ( ν ˜ 1 ) ν ˜ 2 d ν ˜ = 1.

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