Abstract

Beginning with a beam coherence polarization (BCP) matrix, we obtain an analytical intensity expression for radially polarized ultrashort pulsed laser beams that pass through an apertureless aplanatic lens. We also investigate the intensity distribution of radially polarized beams in the vicinity of the focus. The focal shift of these beams is studied in detail. The focal shift depends strongly on ZF that coincides with π times the Fresnel number.

© 2007 Optical Society of America

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References

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    [CrossRef]
  14. C. Varin and M. Piché, "Acceleration of ultra-relativistic electrons using high-intensity TM01 laser beams," Appl. Phys. B 74, S83-S88 (2002).
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  18. A. A. Tovar, "Production and propagation of cylindrically polarized Laguerre-Gaussian laser beams," J. Opt. Soc. Am. A 15, 2705-2711 (1998).
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2006 (1)

2005 (1)

C. H. Niu, B. Y. Gu, B. Z. Dong, and Y. Zhang, "A new method for generating axially symmetric and radially polarized beams," J. Phys. D 38, 827-832 (2005).
[CrossRef]

2002 (1)

C. Varin and M. Piché, "Acceleration of ultra-relativistic electrons using high-intensity TM01 laser beams," Appl. Phys. B 74, S83-S88 (2002).
[CrossRef]

2001 (1)

2000 (3)

1999 (3)

1998 (4)

A. A. Tovar, "Production and propagation of cylindrically polarized Laguerre-Gaussian laser beams," J. Opt. Soc. Am. A 15, 2705-2711 (1998).
[CrossRef]

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

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, "Beam coherence-polarization matrix," Pure Appl. Opt. 7, 941-951 (1998).
[CrossRef]

C. Palma, G. Cincotti, and G. Guattari, "Spectral shift of a Gaussian Schell-model beam beyond a thin lens," IEEE J. Quantum Electron. 34, 378-383 (1998).
[CrossRef]

1993 (1)

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1993).

1990 (1)

1984 (2)

1982 (2)

1981 (1)

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

Agrawal, G. P.

Borghi, R.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, "Beam coherence-polarization matrix," Pure Appl. Opt. 7, 941-951 (1998).
[CrossRef]

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

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1993).

Carter, W. H.

Cincotti, G.

C. Palma, G. Cincotti, and G. Guattari, "Spectral shift of a Gaussian Schell-model beam beyond a thin lens," IEEE J. Quantum Electron. 34, 378-383 (1998).
[CrossRef]

Deng, D.

Dong, B. Z.

C. H. Niu, B. Y. Gu, B. Z. Dong, and Y. Zhang, "A new method for generating axially symmetric and radially polarized beams," J. Phys. D 38, 827-832 (2005).
[CrossRef]

Dorn, R.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, "Focusing light to a tighter spot," Opt. Commun. 179, 1-7 (2000).
[CrossRef]

Eberler, M.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, "Focusing light to a tighter spot," Opt. Commun. 179, 1-7 (2000).
[CrossRef]

Ford, D. H.

Gahagan, K. T.

Glöckl, O.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, "Focusing light to a tighter spot," Opt. Commun. 179, 1-7 (2000).
[CrossRef]

Gori, F.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, "Beam coherence-polarization matrix," Pure Appl. Opt. 7, 941-951 (1998).
[CrossRef]

Greene, P. L.

Gu, B. Y.

C. H. Niu, B. Y. Gu, B. Z. Dong, and Y. Zhang, "A new method for generating axially symmetric and radially polarized beams," J. Phys. D 38, 827-832 (2005).
[CrossRef]

Guattari, G.

C. Palma, G. Cincotti, and G. Guattari, "Spectral shift of a Gaussian Schell-model beam beyond a thin lens," IEEE J. Quantum Electron. 34, 378-383 (1998).
[CrossRef]

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, "Beam coherence-polarization matrix," Pure Appl. Opt. 7, 941-951 (1998).
[CrossRef]

Hall, D. G.

Kimura, W. D.

Larkin, K. G.

Leuchs, G.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, "Focusing light to a tighter spot," Opt. Commun. 179, 1-7 (2000).
[CrossRef]

Li, Y.

Lü, B.

Nesterov, A. V.

V. G. Niziev and A. V. Nesterov, "Influence of beam polarization on laser cutting efficient," J. Phys. D 32, 1455-1461 (1999).
[CrossRef]

Niu, C. H.

C. H. Niu, B. Y. Gu, B. Z. Dong, and Y. Zhang, "A new method for generating axially symmetric and radially polarized beams," J. Phys. D 38, 827-832 (2005).
[CrossRef]

Niziev, V. G.

V. G. Niziev and A. V. Nesterov, "Influence of beam polarization on laser cutting efficient," J. Phys. D 32, 1455-1461 (1999).
[CrossRef]

Palma, C.

C. Palma, G. Cincotti, and G. Guattari, "Spectral shift of a Gaussian Schell-model beam beyond a thin lens," IEEE J. Quantum Electron. 34, 378-383 (1998).
[CrossRef]

Piché, M.

C. Varin and M. Piché, "Acceleration of ultra-relativistic electrons using high-intensity TM01 laser beams," Appl. Phys. B 74, S83-S88 (2002).
[CrossRef]

Pu, J.

Quabis, S.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, "Focusing light to a tighter spot," Opt. Commun. 179, 1-7 (2000).
[CrossRef]

Santarsiero, M.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, "Beam coherence-polarization matrix," Pure Appl. Opt. 7, 941-951 (1998).
[CrossRef]

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

Sheppard, C. J. R.

Sucha, G. D.

Swartzlander, G. A.

Tidwell, S. C.

Tovar, A. A.

Varin, C.

C. Varin and M. Piché, "Acceleration of ultra-relativistic electrons using high-intensity TM01 laser beams," Appl. Phys. B 74, S83-S88 (2002).
[CrossRef]

Vicalvi, S.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, "Beam coherence-polarization matrix," Pure Appl. Opt. 7, 941-951 (1998).
[CrossRef]

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

Wolf, E.

Zhang, Y.

C. H. Niu, B. Y. Gu, B. Z. Dong, and Y. Zhang, "A new method for generating axially symmetric and radially polarized beams," J. Phys. D 38, 827-832 (2005).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. B (1)

C. Varin and M. Piché, "Acceleration of ultra-relativistic electrons using high-intensity TM01 laser beams," Appl. Phys. B 74, S83-S88 (2002).
[CrossRef]

IEEE J. Quantum Electron. (1)

C. Palma, G. Cincotti, and G. Guattari, "Spectral shift of a Gaussian Schell-model beam beyond a thin lens," IEEE J. Quantum Electron. 34, 378-383 (1998).
[CrossRef]

J. Opt. Soc. Am. (1)

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

J. Opt. Soc. Am. B (2)

J. Phys. D (2)

V. G. Niziev and A. V. Nesterov, "Influence of beam polarization on laser cutting efficient," J. Phys. D 32, 1455-1461 (1999).
[CrossRef]

C. H. Niu, B. Y. Gu, B. Z. Dong, and Y. Zhang, "A new method for generating axially symmetric and radially polarized beams," J. Phys. D 38, 827-832 (2005).
[CrossRef]

Opt. Commun. (3)

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, "Focusing light to a tighter spot," Opt. Commun. 179, 1-7 (2000).
[CrossRef]

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

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

Opt. Express (1)

Pure Appl. Opt. (1)

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, "Beam coherence-polarization matrix," Pure Appl. Opt. 7, 941-951 (1998).
[CrossRef]

Other (1)

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1993).

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Figures (5)

Fig. 1
Fig. 1

Notation relating to the focused radially polarized ultrashort pulsed laser beams.

Fig. 2
Fig. 2

Three-dimensional intensity distribution in the rz plane near the geometric focus for different mode numbers: (a) p = 1 , Z F = 3 ; (b) p = 1 , Z F = 5 ; (c) p = 3 , Z F = 3 ; (d) p = 3 , Z F = 5 .

Fig. 3
Fig. 3

Normalized intensity distribution at the focal plane for different mode numbers and Z F : (a) p = 1 and (b) p = 3 .

Fig. 4
Fig. 4

Beam width versus relative propagation distance z / f for different mode numbers.

Fig. 5
Fig. 5

Relative focal shift Δ z / f varies with Z F for mode number p = 2 .

Equations (16)

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E p 1 ( x , y , 0 ) = 2 E 0 w 0 L p 1 [ 2 ( x 2 + y 2 ) w 0 2 ] × exp [ x 2 + y 2 w 0 2 ] [ x e ^ x + y e ^ y ] = E p 1 x ( x , y , 0 ) e ^ x + E p 1 y ( x , y , 0 ) e ^ y ,
J ^ ( r 1 , r 2 , z ) = [ J x x ( r 1 , r 2 , z ) J x y ( r 1 , r 2 , z ) J y x ( r 1 , r 2 , z ) J y y ( r 1 , r 2 , z ) ] ,
J x x ( r 1 , r 2 , z ) = E x * ( r 1 , z ) E x ( r 2 , z ) ,
J y y ( r 1 , r 2 , z ) = E y * ( r 1 , z ) E y ( r 2 , z ) ,
J x y ( r 1 , r 2 , z ) = γ 0 E x * ( r 1 , z ) E y ( r 2 , z ) = J y x * ( r 1 , r 2 , z ) .
I ( r , z ) = J x x ( r , r , z ) + J y y ( r , r , z ) = E x * ( r , z ) E x ( r , z ) + E y * ( r , z ) E y ( r , z ) .
J α β ( r 1 , r 2 , z ) = J α β ( ρ 1 , ρ 2 ) K * ( r 1 , ρ 1 , z ) × K ( r 2 , ρ 2 , z ) d 2 ρ 1 d 2 ρ 2 ,
α , β = x , y ,
E α ( r , z ) = 0 2 π 0 E α ( ρ , 0 ) K ( r , ρ , z ) ρ d ρ d θ ,
α = x , y .
K ( r , ρ , z ) = k   exp ( i k z ) 2 π i z   exp [ i k 2 z ( r ρ ) 2 ] exp [ i k ρ 2 2 f ] .
E p 1 x ( r , z ) = ( 1 ) p + 1 2 E 0 Z R 2 x w 0 z 2 S 2 ( S * S ) p × exp [ i k z + i k r 2 2 z Z R 2 r 2 z 2 w 0 2 S ] L p 1 [ 2 Z R 2 r 2 w 0 2 z 2 | S | 2 ] ,
E p 1 y ( r , z ) = ( 1 ) p + 1 2 E 0 Z R 2 y w 0 z 2 S 2 ( S * S ) p × exp [ i k z + i k r 2 2 z Z R 2 r 2 z 2 w 0 2 S ] L p 1 [ 2 Z R 2 r 2 w 0 2 z 2 | S | 2 ] ,
S = 1 + i Z F i ( Z R / z ) .
I ( r , z ) = 2 E 0 2 Z R 4 r 2 w 0 2 z 4 | S | 4   exp [ 2 Z R 2 r 2 z 2 w 0 2 | S | 2 ] ( L P 1 [ 2 Z R 2 r 2 w 0 2 z 2 | S | 2 ] ) 2 .
0 2 π 0 r 0 I ( ρ , θ , z 0 ) ρ d ρ d θ 0 2 π 0 I ( ρ , θ , z 0 ) ρ d ρ d θ = 0.8 .

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