Abstract

Three-dimensional intensity and phase distributions generated by microaxicons are evaluated in the low-Fresnel-number regime. Apertured and nonapertured conical wavefronts may generate transverse patterns with notable deviations from the expected nondiffracting Bessel beam. First-order analytical expressions are proposed for the evaluation of the wave field produced by axicons of different Fresnel number in the focal region.

© 2006 Optical Society of America

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    [CrossRef]
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  25. C. J. Zapata-Rodríguez, "Paraxial waves in the far-field region," Optik (Stuttgart) 113, 361-365 (2002).
  26. M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, 1999), Chap XIII.
  27. K. Reivelt and P. Saari, "Experimental demonstration of realizability of optical focus wave modes," Phys. Rev. E 66, 056611 (2002).
    [CrossRef]
  28. G. Rousseau, D. Gay, and M. Piché, "One-dimensional description of cylindrically symmetric laser beams: application to Bessel-type nondiffracting beams," J. Opt. Soc. Am. A 22, 1274-1287 (2005).
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  29. M. A. Porras, G. Valiulis, and P. D. Trapani, "Unified description of Bessel X waves with cone dispersion and tilted pulses," Phys. Rev. E 68, 016613 (2003).
    [CrossRef]

2005

2004

2003

J. Amako, D. Sawaki, and E. Fujii, "Microstructuring transparent materials by use of nondiffracting ultrashort pulse beams generated by diffractive optics," J. Opt. Soc. Am. B 20, 2562-2568 (2003).
[CrossRef]

M. A. Porras, G. Valiulis, and P. D. Trapani, "Unified description of Bessel X waves with cone dispersion and tilted pulses," Phys. Rev. E 68, 016613 (2003).
[CrossRef]

2002

C. J. Zapata-Rodríguez, "Paraxial waves in the far-field region," Optik (Stuttgart) 113, 361-365 (2002).

K. Reivelt and P. Saari, "Experimental demonstration of realizability of optical focus wave modes," Phys. Rev. E 66, 056611 (2002).
[CrossRef]

2001

J. R. Jiménez and E. Hita, "Babinet's principle in scalar theory of diffraction," Opt. Rev. 8, 495-497 (2001).
[CrossRef]

2000

1998

1997

1996

1995

I. Golub, "Superluminal-source-induced emission," Opt. Lett. 15, 1847-1849 (1995).
[CrossRef]

1993

T. Wulle and S. Herminghaus, "Nonlinear optics of Bessel beams," Phys. Rev. Lett. 70, 1401-1404 (1993).
[CrossRef] [PubMed]

1984

1981

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

1976

A. Arimoto, "Intensity distribution of aberration-free diffraction patterns due to circular apertures in large F-number optical systems," Appl. Opt. 23, 245-250 (1976).

1972

1962

1954

Amako, J.

Andrés, P.

Arimoto, A.

A. Arimoto, "Intensity distribution of aberration-free diffraction patterns due to circular apertures in large F-number optical systems," Appl. Opt. 23, 245-250 (1976).

Avizonis, P. V.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, 1999), Chap XIII.

Brinkmann, S.

Dresel, T.

Elsaesser, T.

R. Grunwald, U. Griebner, F. Tschirschwitz, E. T. J. Nibbering, T. Elsaesser, V. Kebbel, H.-J. Hartmann, and W. Jüptner, "Generation of femtosecond Bessel beams with microaxicon arrays," Opt. Lett. 25, 981-983 (2000).
[CrossRef]

R. Grunwald, U. Griebner, T. Elsaesser, V. Kebbel, H.-J. Hartmann, and W. Jüptner, "Femtosecond interference experiments with thin-film micro-optical components," in Fringe. 2001, W.Osten and W.Jüptner, eds. (Elsevier, 2001), pp. 33-40.

Friberg, A. T.

Fujii, E.

Fujiwara, S.

Gay, D.

Golub, I.

I. Golub, "Superluminal-source-induced emission," Opt. Lett. 15, 1847-1849 (1995).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), Chaps. 4 and 5.

Griebner, U.

R. Grunwald, U. Griebner, F. Tschirschwitz, E. T. J. Nibbering, T. Elsaesser, V. Kebbel, H.-J. Hartmann, and W. Jüptner, "Generation of femtosecond Bessel beams with microaxicon arrays," Opt. Lett. 25, 981-983 (2000).
[CrossRef]

R. Grunwald, U. Griebner, T. Elsaesser, V. Kebbel, H.-J. Hartmann, and W. Jüptner, "Femtosecond interference experiments with thin-film micro-optical components," in Fringe. 2001, W.Osten and W.Jüptner, eds. (Elsevier, 2001), pp. 33-40.

Grunwald, R.

Hartmann, H.-J.

R. Grunwald, U. Griebner, F. Tschirschwitz, E. T. J. Nibbering, T. Elsaesser, V. Kebbel, H.-J. Hartmann, and W. Jüptner, "Generation of femtosecond Bessel beams with microaxicon arrays," Opt. Lett. 25, 981-983 (2000).
[CrossRef]

R. Grunwald, U. Griebner, T. Elsaesser, V. Kebbel, H.-J. Hartmann, and W. Jüptner, "Femtosecond interference experiments with thin-film micro-optical components," in Fringe. 2001, W.Osten and W.Jüptner, eds. (Elsevier, 2001), pp. 33-40.

Herminghaus, S.

T. Wulle and S. Herminghaus, "Nonlinear optics of Bessel beams," Phys. Rev. Lett. 70, 1401-1404 (1993).
[CrossRef] [PubMed]

Hernández, F. E.

C. J. Zapata-Rodríguez and F. E. Hernández, "Focal squeeze in axicons," Opt. Commun. 254, 3-9 (2005).
[CrossRef]

Hita, E.

J. R. Jiménez and E. Hita, "Babinet's principle in scalar theory of diffraction," Opt. Rev. 8, 495-497 (2001).
[CrossRef]

Holmes, D. A.

Jhe, W.

Jiménez, J. R.

J. R. Jiménez and E. Hita, "Babinet's principle in scalar theory of diffraction," Opt. Rev. 8, 495-497 (2001).
[CrossRef]

Jüptner, W.

R. Grunwald, U. Griebner, F. Tschirschwitz, E. T. J. Nibbering, T. Elsaesser, V. Kebbel, H.-J. Hartmann, and W. Jüptner, "Generation of femtosecond Bessel beams with microaxicon arrays," Opt. Lett. 25, 981-983 (2000).
[CrossRef]

R. Grunwald, U. Griebner, T. Elsaesser, V. Kebbel, H.-J. Hartmann, and W. Jüptner, "Femtosecond interference experiments with thin-film micro-optical components," in Fringe. 2001, W.Osten and W.Jüptner, eds. (Elsevier, 2001), pp. 33-40.

Kebbel, V.

Kim, J.

Korka, J. E.

Kühn, H.-J.

Lee, K.

Leinhos, U.

Li, Y.

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

Lindlein, N.

Mann, K.

Martínez-Corral, M.

McLeod, J. H.

Mischke, H.

Miyamoto, K.

Muñoz-Escrivá, L.

Neumann, U.

Nibbering, E. T. J.

Noh, H.

Ohtsu, M.

Papoulis, A.

A. Papoulis, Systems and Transformations with Applications in Optics (McGraw-Hill, 1968), Chap. 7.

Piché, M.

Porras, M. A.

M. A. Porras, G. Valiulis, and P. D. Trapani, "Unified description of Bessel X waves with cone dispersion and tilted pulses," Phys. Rev. E 68, 016613 (2003).
[CrossRef]

Reivelt, K.

K. Reivelt and P. Saari, "Experimental demonstration of realizability of optical focus wave modes," Phys. Rev. E 66, 056611 (2002).
[CrossRef]

Rousseau, G.

Saari, P.

K. Reivelt and P. Saari, "Experimental demonstration of realizability of optical focus wave modes," Phys. Rev. E 66, 056611 (2002).
[CrossRef]

Sawaki, D.

Schreiner, R.

Schwider, J.

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, 1986), Chap. 16.

Silvestre, E.

Trapani, P. D.

M. A. Porras, G. Valiulis, and P. D. Trapani, "Unified description of Bessel X waves with cone dispersion and tilted pulses," Phys. Rev. E 68, 016613 (2003).
[CrossRef]

Tschirschwitz, F.

Valiulis, G.

M. A. Porras, G. Valiulis, and P. D. Trapani, "Unified description of Bessel X waves with cone dispersion and tilted pulses," Phys. Rev. E 68, 016613 (2003).
[CrossRef]

Wolf, E.

Wulff-Molder, D.

Wulle, T.

T. Wulle and S. Herminghaus, "Nonlinear optics of Bessel beams," Phys. Rev. Lett. 70, 1401-1404 (1993).
[CrossRef] [PubMed]

Zapata-Rodríguez, C. J.

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Commun.

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

C. J. Zapata-Rodríguez and F. E. Hernández, "Focal squeeze in axicons," Opt. Commun. 254, 3-9 (2005).
[CrossRef]

Opt. Lett.

Opt. Rev.

J. R. Jiménez and E. Hita, "Babinet's principle in scalar theory of diffraction," Opt. Rev. 8, 495-497 (2001).
[CrossRef]

Optik (Stuttgart)

C. J. Zapata-Rodríguez, "Paraxial waves in the far-field region," Optik (Stuttgart) 113, 361-365 (2002).

Phys. Rev. E

K. Reivelt and P. Saari, "Experimental demonstration of realizability of optical focus wave modes," Phys. Rev. E 66, 056611 (2002).
[CrossRef]

M. A. Porras, G. Valiulis, and P. D. Trapani, "Unified description of Bessel X waves with cone dispersion and tilted pulses," Phys. Rev. E 68, 016613 (2003).
[CrossRef]

Phys. Rev. Lett.

T. Wulle and S. Herminghaus, "Nonlinear optics of Bessel beams," Phys. Rev. Lett. 70, 1401-1404 (1993).
[CrossRef] [PubMed]

Other

A. Papoulis, Systems and Transformations with Applications in Optics (McGraw-Hill, 1968), Chap. 7.

R. Grunwald, U. Griebner, T. Elsaesser, V. Kebbel, H.-J. Hartmann, and W. Jüptner, "Femtosecond interference experiments with thin-film micro-optical components," in Fringe. 2001, W.Osten and W.Jüptner, eds. (Elsevier, 2001), pp. 33-40.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), Chaps. 4 and 5.

A. E. Siegman, Lasers (University Science Books, 1986), Chap. 16.

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, 1999), Chap XIII.

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Figures (11)

Fig. 1
Fig. 1

Schematic diagram of the focusing conical lens. Also, we show a ray incident on the substrate of the axicon that is deviated on the conical surface in a direction toward the optical axis at angle β.

Fig. 2
Fig. 2

Angular dependence of the intensity pattern F 2 normalized to its maximum value for different Fresnel numbers: N = 10 3 (solid curve), N = 1 (short-dashed curve line), N = 10 3 (long-dashed curve). The data have been numerically obtained from Eq. (17).

Fig. 3
Fig. 3

Three-dimensional intensity distributions of an apertured axicon U ̃ A 2 for (a) N = 0.1 , (b) N = 1 , (c) N = 10 , and field patterns for a nonapertured axicon U ̃ NA 2 also for N = 0.1 , (e) N = 1 , (f) N = 10 . The maximum intensity in a subfigure is normalized to unity. The transverse spatial coordinate is normalized as N ρ , and the axial coordinate is ζ.

Fig. 4
Fig. 4

Axial intensity distributions for focusing geometries of Fresnel number N = 0.1 and N = 10 in the cases of (a) apertured and (b) nonapertured axicons.

Fig. 5
Fig. 5

Intensity of an apertured axicon of (a) N = 10 at different transversal planes ζ = 0.2 (solid curve), ζ = 0.5 (dashed curve), ζ = 0.9 (dotted curve), normalized to the value at the origin. When (b) N = 1 we find transverse intensity patterns deviating considerably from the expected Bessel distribution at ζ = 0.8 (solid curve), ζ = 0.15 (dashed curve), ζ = 0.27 (dotted curve).

Fig. 6
Fig. 6

FWHM of the transverse central lobe for an apertured axicon. The Fresnel numbers are N = 10 (solid curve), N = 1 (dashed curve), N = 0.1 (dotted curve). Inset: 3D intensity distribution for N = 1 in a focal region spanning from ζ = 0.2 to ζ = 0.4 , where the focal beam generates a doughnut structure and the FWHM is a meaningless parameter.

Fig. 7
Fig. 7

Dependence of the intensity distribution at the transverse plane ζ = 1 2 versus the Fresnel number for (a) an apertured axicon, (b) a nonapertured axicon.

Fig. 8
Fig. 8

On-axis intensity at ζ = 1 2 versus the Fresnel number of the focusing geometry.

Fig. 9
Fig. 9

Phase distribution (normalized to 2 π ) in the focal region of a nonapertured axicon of Fresnel number N = 10 in different transverse planes: ζ = 0.1 (long-dashed curve), ζ = 0.5 (solid curve), and ζ = 0.9 (short-dashed curve).

Fig. 10
Fig. 10

Phase of the complex field U NA along the optical axis for different Fresnel numbers. The phase is normalized to the quantity 2 π N .

Fig. 11
Fig. 11

3-D phase distributions of an apertured axicon for (a) N = 0.1 , (b) N = 1 , and (c) N = 10 , and phase patterns for a nonapertured axicon also for (d) N = , 0.1 , (e) N = 1 , and (f) N = 10 . Again, the transverse spatial coordinate is normalized as N ρ , and the axial coordinate is ζ.

Equations (28)

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ϕ ( r ) = k n s e + k n Δ ( r ) + k [ Δ 0 Δ ( r ) ] ,
t ( r ) = P ( r ) exp [ j k ( n 1 ) Δ 0 ] exp ( j k β r ) + [ 1 P ( r ) ] .
t A ( r ) = exp [ j k ( n 1 ) Δ 0 ] exp ( j k β r ) P ( r ) ,
U NA ( r , z ) = j k z exp ( j k 2 z r 2 ) 0 t ( r 0 ) exp ( j k 2 z r 0 2 ) J 0 ( k r z r 0 ) r 0 d r 0 ,
( r 2 + r 1 r + 2 j k z ) U NA ( r , z ) = 0 ,
U NA ( r , z ) = U A ( r , z ) + U PW ( z ) U C ( r , z ) .
N = a 2 λ L = a λ β .
U A ( r , z ) = 2 π exp [ j χ 1 ( r , z ) + j χ 2 ( z ) ] N z L 0 L z exp [ j π N z L ( ρ 0 1 ) 2 ] J 0 ( 2 π N r a ρ 0 ) ρ 0 d ρ 0 ,
U C ( r , z ) = 2 π exp [ j χ 1 ( r , z ) ] N z L 0 L z exp ( j π N z L ρ 0 2 ) J 0 ( 2 π N r a ρ 0 ) ρ 0 d ρ 0 ,
0 ζ 1 exp ( j π N ζ ρ 0 2 ) J 0 ( 2 π N ρ ρ 0 ) ρ 0 d ρ 0 = [ exp ( j π N ζ ρ 0 2 ) j 2 π N ζ J 0 ( 2 π N ρ ρ 0 ) ] ρ 0 = 0 ρ 0 = ζ 1 j ρ ζ 0 ζ 1 exp ( j π N ζ ρ 0 2 ) J 1 ( 2 π N ρ ρ 0 ) d ρ 0 .
U ̃ C ( ρ , ζ ) exp ( j π N ρ 2 ζ ) [ 1 exp ( j π N ζ ) J 0 ( 2 π N ρ ζ ) ] .
0 1 ζ exp [ j π N ζ ( ρ 0 1 ) 2 ] J 0 ( 2 π N ρ ρ 0 ) ρ 0 d ρ 0 J 0 ( 2 π N ρ ) exp [ j π N ζ ( ρ 0 1 ) 2 ] d ρ 0 = exp ( j π 4 ) N ζ J 0 ( 2 π N ρ ) ,
U ̃ A ( ρ , ζ ) exp ( j π N ρ 2 ζ ) { exp ( j π N ζ ) 1 ζ J 0 ( 2 π N ρ ζ ) + 2 π N ζ exp ( j π 4 ) exp [ j π N ( 2 ζ ) ] J 0 ( 2 π N ρ ) }
U ̃ A ( ρ , ζ ) j exp ( j π N ρ 2 ζ ) J 1 ( 2 π N ρ ζ ) ρ ,
U ̃ NA ( ρ , ζ ) = 1 + j 2 π N U ̃ A ( ρ , ζ ) 4 π 2 N 2 3 ζ exp ( j π ρ 2 ζ ) 1 F 2 ( 3 2 ; 1 , 5 2 ; π 2 N 2 ρ 2 ζ 2 ) ,
U ( r , z ) = 1 z exp ( j k 2 z r 2 ) F ( θ ) ,
F ( θ ) = j k 0 t ( r 0 ) J 0 ( k θ r 0 ) r 0 d r 0
F ( θ ) = j 2 π N L exp ( j 2 π N ) 0 1 exp ( j 2 π N ρ 0 ) J 0 ( 2 π N ρ 0 θ β ) ρ 0 d ρ 0 .
J 0 ( 2 π N ρ 0 θ β ) 1 2 π ( N ρ 0 θ β ) 1 2 exp ( j π 4 ) exp ( j 2 π N ρ 0 θ β ) .
F ( θ ) = L 2 N 8 π exp ( j 2 π N ) { 2 ( 1 j ) x 1 exp ( j 2 π x ) + j x 3 2 Erfi [ ( 1 + j ) π x ] } ,
x = N ( θ β 1 ) = N ( ρ ζ 1 )
F ( θ ) = j N L J 1 ( 2 π x ) x ,
Δ ζ = 1 1 + N ,
U ̃ NA ( ρ = 0 , ζ = 1 2 ) 2 1 + 8 3 π 2 N 2 ,
ϕ = π 4 + 2 π N π N ζ ,
ϕ B ( z ) = k z cos β ϕ 0 = 2 k z sin 2 ( β 2 ) ,
v p = ω ( k + z ϕ ) 1 c ( 1 k 1 z ϕ ) ,
v p = c ( 1 + 1 2 β 2 ) ,

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