Abstract

The far-field intensity pattern of laser beams diffracted by axicons is extensively characterized both theoretically and experimentally. The regular structure of the pattern, consisting of high-contrast fringes, is explained. The experimental results have been interpreted by representing the diffracted field as generated by an extended virtual source shaped as a circle centered on the optical axis of the incident laser beam. The simulations include modifications to the diffraction pattern arising from the laser radiation diffraction limit at the axicon tip, and they reproduce well the measured intensity profile at different distances from the axicon.

© 2004 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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2002 (2)

C. Altucci, R. Bruzzese, C. de Lisio, A. Porzio, S. Solimeno, V. Tosa, “Diffractionless beams and their use for harmonic generation,” Opt. Lasers Eng. 37, 565–575 (2002).
[CrossRef]

B. Dépret, Ph. Verkerk, D. Hennequin, “Characterization and modeling of the hollow beam produced by a real conical lens,” Opt. Commun. 211, 31–38 (2002).
[CrossRef]

2001 (1)

L. Cacciapuoti, M. de Angelis, G. Pierattini, L. Ricci, G. M. Tino, “Single-beam optical bottle for cold atoms using conical lenses,” Eur. Phys. J. D 14, 373–376 (2001).
[CrossRef]

2000 (3)

1998 (1)

I. Manek, Yu. B. Ovchinnikov, R. Grimm, “Generation of a hollow laser beam for atom trapping using an axicon,” Opt. Commun. 147, 67–70 (1998).
[CrossRef]

1997 (1)

V. E. Peet, R. V. Tsubin, “Third-harmonic generation and multiphoton ionization in Bessel beams,” Phys. Rev. A 56, 1613–1620 (1997).
[CrossRef]

1996 (1)

V. E. Peet, “Resonantly enhanced multiphoton ionization of xenon in Bessel beams,” Phys. Rev. A 53, 3679–3682 (1996).
[CrossRef] [PubMed]

1991 (1)

1989 (1)

1987 (1)

J. Durnin, J. Miceli, J. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

1986 (1)

M. V. Pérez, C. Gómez-Reino, J. M. Cuadrando, “Diffraction patterns and zone plates produced by thin linear axicon,” Opt. Acta 33, 1161–1176 (1986).
[CrossRef]

1980 (1)

1978 (1)

1974 (1)

L. W. Casperson, M. S. Shekhani, “Air breakdown in a radial-mode focusing element,” Appl. Opt. 19, 104–108 (1974).
[CrossRef]

1970 (1)

1960 (1)

1954 (1)

Altucci, C.

C. Altucci, R. Bruzzese, C. de Lisio, A. Porzio, S. Solimeno, V. Tosa, “Diffractionless beams and their use for harmonic generation,” Opt. Lasers Eng. 37, 565–575 (2002).
[CrossRef]

C. Altucci, R. Bruzzese, D. D’Antuoni, C. de Lisio, S. Solimeno, “Harmonic generation in gases by use of Bessel-Gauss laser beams,” J. Opt. Soc. Am. B 17, 34–42 (2000).
[CrossRef]

Bélanger, P.-A.

Born, M.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, London, 1970).

Bruzzese, R.

C. Altucci, R. Bruzzese, C. de Lisio, A. Porzio, S. Solimeno, V. Tosa, “Diffractionless beams and their use for harmonic generation,” Opt. Lasers Eng. 37, 565–575 (2002).
[CrossRef]

C. Altucci, R. Bruzzese, D. D’Antuoni, C. de Lisio, S. Solimeno, “Harmonic generation in gases by use of Bessel-Gauss laser beams,” J. Opt. Soc. Am. B 17, 34–42 (2000).
[CrossRef]

Butkus, R.

Cacciapuoti, L.

L. Cacciapuoti, M. de Angelis, G. Pierattini, L. Ricci, G. M. Tino, “Single-beam optical bottle for cold atoms using conical lenses,” Eur. Phys. J. D 14, 373–376 (2001).
[CrossRef]

Casperson, L. W.

L. W. Casperson, M. S. Shekhani, “Air breakdown in a radial-mode focusing element,” Appl. Opt. 19, 104–108 (1974).
[CrossRef]

Collins, G.

Cuadrando, J. M.

M. V. Pérez, C. Gómez-Reino, J. M. Cuadrando, “Diffraction patterns and zone plates produced by thin linear axicon,” Opt. Acta 33, 1161–1176 (1986).
[CrossRef]

D’Antuoni, D.

de Angelis, M.

L. Cacciapuoti, M. de Angelis, G. Pierattini, L. Ricci, G. M. Tino, “Single-beam optical bottle for cold atoms using conical lenses,” Eur. Phys. J. D 14, 373–376 (2001).
[CrossRef]

de Lisio, C.

C. Altucci, R. Bruzzese, C. de Lisio, A. Porzio, S. Solimeno, V. Tosa, “Diffractionless beams and their use for harmonic generation,” Opt. Lasers Eng. 37, 565–575 (2002).
[CrossRef]

C. Altucci, R. Bruzzese, D. D’Antuoni, C. de Lisio, S. Solimeno, “Harmonic generation in gases by use of Bessel-Gauss laser beams,” J. Opt. Soc. Am. B 17, 34–42 (2000).
[CrossRef]

Dépret, B.

B. Dépret, Ph. Verkerk, D. Hennequin, “Characterization and modeling of the hollow beam produced by a real conical lens,” Opt. Commun. 211, 31–38 (2002).
[CrossRef]

Durnin, J.

J. Durnin, J. Miceli, J. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Eberly, J.

J. Durnin, J. Miceli, J. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Elsaesser, T.

Friberg, A. T.

Gómez-Reino, C.

M. V. Pérez, C. Gómez-Reino, J. M. Cuadrando, “Diffraction patterns and zone plates produced by thin linear axicon,” Opt. Acta 33, 1161–1176 (1986).
[CrossRef]

Griebner, U.

Grimm, R.

I. Manek, Yu. B. Ovchinnikov, R. Grimm, “Generation of a hollow laser beam for atom trapping using an axicon,” Opt. Commun. 147, 67–70 (1998).
[CrossRef]

Grunwald, R.

Hartmann, H.-J.

Hennequin, D.

B. Dépret, Ph. Verkerk, D. Hennequin, “Characterization and modeling of the hollow beam produced by a real conical lens,” Opt. Commun. 211, 31–38 (2002).
[CrossRef]

Herman, R. M.

Jnibbering, T.

Juptner, W.

Karlsson, H.

Kebbel, V.

Khoo, I.

Laurell, F.

Leith, E. N.

Lit, J. W. Y.

Manek, I.

I. Manek, Yu. B. Ovchinnikov, R. Grimm, “Generation of a hollow laser beam for atom trapping using an axicon,” Opt. Commun. 147, 67–70 (1998).
[CrossRef]

McLeod, J. H.

Miceli, J.

J. Durnin, J. Miceli, J. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Ovchinnikov, Yu. B.

I. Manek, Yu. B. Ovchinnikov, R. Grimm, “Generation of a hollow laser beam for atom trapping using an axicon,” Opt. Commun. 147, 67–70 (1998).
[CrossRef]

Pasiskevicius, V.

Peet, V. E.

V. E. Peet, R. V. Tsubin, “Third-harmonic generation and multiphoton ionization in Bessel beams,” Phys. Rev. A 56, 1613–1620 (1997).
[CrossRef]

V. E. Peet, “Resonantly enhanced multiphoton ionization of xenon in Bessel beams,” Phys. Rev. A 53, 3679–3682 (1996).
[CrossRef] [PubMed]

Pérez, M. V.

M. V. Pérez, C. Gómez-Reino, J. M. Cuadrando, “Diffraction patterns and zone plates produced by thin linear axicon,” Opt. Acta 33, 1161–1176 (1986).
[CrossRef]

Pierattini, G.

L. Cacciapuoti, M. de Angelis, G. Pierattini, L. Ricci, G. M. Tino, “Single-beam optical bottle for cold atoms using conical lenses,” Eur. Phys. J. D 14, 373–376 (2001).
[CrossRef]

Piskarskas, A.

Porzio, A.

C. Altucci, R. Bruzzese, C. de Lisio, A. Porzio, S. Solimeno, V. Tosa, “Diffractionless beams and their use for harmonic generation,” Opt. Lasers Eng. 37, 565–575 (2002).
[CrossRef]

Ricci, L.

L. Cacciapuoti, M. de Angelis, G. Pierattini, L. Ricci, G. M. Tino, “Single-beam optical bottle for cold atoms using conical lenses,” Eur. Phys. J. D 14, 373–376 (2001).
[CrossRef]

Rioux, M.

Schirschwitz, F. T.

Shekhani, M. S.

L. W. Casperson, M. S. Shekhani, “Air breakdown in a radial-mode focusing element,” Appl. Opt. 19, 104–108 (1974).
[CrossRef]

Smilgevicius, V.

Solimeno, S.

C. Altucci, R. Bruzzese, C. de Lisio, A. Porzio, S. Solimeno, V. Tosa, “Diffractionless beams and their use for harmonic generation,” Opt. Lasers Eng. 37, 565–575 (2002).
[CrossRef]

C. Altucci, R. Bruzzese, D. D’Antuoni, C. de Lisio, S. Solimeno, “Harmonic generation in gases by use of Bessel-Gauss laser beams,” J. Opt. Soc. Am. B 17, 34–42 (2000).
[CrossRef]

Stabinis, A.

Tellefsen, J. A.

Tino, G. M.

L. Cacciapuoti, M. de Angelis, G. Pierattini, L. Ricci, G. M. Tino, “Single-beam optical bottle for cold atoms using conical lenses,” Eur. Phys. J. D 14, 373–376 (2001).
[CrossRef]

Tosa, V.

C. Altucci, R. Bruzzese, C. de Lisio, A. Porzio, S. Solimeno, V. Tosa, “Diffractionless beams and their use for harmonic generation,” Opt. Lasers Eng. 37, 565–575 (2002).
[CrossRef]

Tsubin, R. V.

V. E. Peet, R. V. Tsubin, “Third-harmonic generation and multiphoton ionization in Bessel beams,” Phys. Rev. A 56, 1613–1620 (1997).
[CrossRef]

Turunen, J.

Vasura, A.

Verkerk, Ph.

B. Dépret, Ph. Verkerk, D. Hennequin, “Characterization and modeling of the hollow beam produced by a real conical lens,” Opt. Commun. 211, 31–38 (2002).
[CrossRef]

Wiggins, T. A.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, London, 1970).

Wynn, T.

Appl. Opt. (2)

L. W. Casperson, M. S. Shekhani, “Air breakdown in a radial-mode focusing element,” Appl. Opt. 19, 104–108 (1974).
[CrossRef]

P.-A. Bélanger, M. Rioux, “Ring pattern of a lens-axicon doublet illuminated by a Gaussian beam,” Appl. Opt. 17, 1080–1086 (1978).
[CrossRef] [PubMed]

Eur. Phys. J. D (1)

L. Cacciapuoti, M. de Angelis, G. Pierattini, L. Ricci, G. M. Tino, “Single-beam optical bottle for cold atoms using conical lenses,” Eur. Phys. J. D 14, 373–376 (2001).
[CrossRef]

J. Opt. Soc. Am. (4)

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

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

Opt. Acta (1)

M. V. Pérez, C. Gómez-Reino, J. M. Cuadrando, “Diffraction patterns and zone plates produced by thin linear axicon,” Opt. Acta 33, 1161–1176 (1986).
[CrossRef]

Opt. Commun. (2)

B. Dépret, Ph. Verkerk, D. Hennequin, “Characterization and modeling of the hollow beam produced by a real conical lens,” Opt. Commun. 211, 31–38 (2002).
[CrossRef]

I. Manek, Yu. B. Ovchinnikov, R. Grimm, “Generation of a hollow laser beam for atom trapping using an axicon,” Opt. Commun. 147, 67–70 (1998).
[CrossRef]

Opt. Lasers Eng. (1)

C. Altucci, R. Bruzzese, C. de Lisio, A. Porzio, S. Solimeno, V. Tosa, “Diffractionless beams and their use for harmonic generation,” Opt. Lasers Eng. 37, 565–575 (2002).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. A (2)

V. E. Peet, “Resonantly enhanced multiphoton ionization of xenon in Bessel beams,” Phys. Rev. A 53, 3679–3682 (1996).
[CrossRef] [PubMed]

V. E. Peet, R. V. Tsubin, “Third-harmonic generation and multiphoton ionization in Bessel beams,” Phys. Rev. A 56, 1613–1620 (1997).
[CrossRef]

Phys. Rev. Lett. (1)

J. Durnin, J. Miceli, J. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Other (1)

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, London, 1970).

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Figures (8)

Fig. 1
Fig. 1

(a) Schematic of the experimental set up. (b) Details of propagation of an input laser beam through an axicon.

Fig. 2
Fig. 2

(a) Far-field intensity profile of the diffracted He–Ne beam versus the radial coordinate that represents (b) a section of the corresponding overall image obtained by the CCD for OO ¯ = 360   mm and a distance between the laser and the axicon of 3 m. The distance between adjacent peaks is 0.42, 0.3, 0.24, 0.22, and 0.22 mm, respectively, going from the highest to the lowest intensity peak value. The intensity radial envelope represented in (a) by the dashed curve is not Gaussian.

Fig. 3
Fig. 3

Description of the geometry projected onto one of the equivalent planes passing through the optical axis of the axicon. The points S and S are the intersections of this projection with the extended virtual source, shaped as a circle having its center on the optical axis, that replaces the incident laser beam. The y axis represents in this projection the observation plane where the CCD element is placed. OO is perpendicular to the x and y axes. Note that the length scales of the figure do not correspond to the real case; in fact, all the radial distances such as OX ¯ and OY 0 ¯ are approximately three orders of magnitude shorter than the distances from the virtual source to the axicon tip and from the axicon tip to the CCD.

Fig. 4
Fig. 4

(a) Complete 3D representation of the surface generated by the envelope of all the straight lines tangent to the diffracted phase front at point O. The resulting conical surface with the tip at the point O coincident with the tip of the axicon can be viewed as the diffraction surface. Γ stands for the section of the diffraction surface with plane normal to the optical axis ζ and including the point X . Γ is thus a circle with center at the point M (M lies in the plane generated by x and y axes) and with radius r = MX ¯ . Note that all the points except K lie in the xy plane. (b) Illustration of the infinitesimal area dσ on the diffraction surface having sides with length r d α in the tangential direction and d x along the x axis.

Fig. 5
Fig. 5

Approximate geometry of the phase front of the diffracted beam around O. Because of the laser beam diffraction limit, this part of the phase front is seriously distorted from the conical shape around O and becomes much smoother there. The bold lines d and d set the integration limits from 0 to α for the angular variable α of the diffraction integral.

Fig. 6
Fig. 6

General radial behavior of the amplitude factor generated on the CCD by the infinitesimal areas of the diffraction surface, with ( A + , solid curve) and without (A, dashed curve) considering the diffraction limit.

Fig. 7
Fig. 7

Comparison between measured intensity pattern (solid curve) of the diffracted field and the corresponding numerical simulation (dashed curve) for three values of the distance between the axicon tip and the CCD, namely, (a) 240 mm, (b) 300 mm, and (c) 450 mm. The other parameters used throughout the simulations are a = 1.5 (field amplitude, mm), b = 0.1   mm , α = 0.05 rad. The distance from the He–Ne laser and the axicon tip is 1 m, which corresponds to a distance between the virtual source and the axicon tip of ∼0.6 m. Note that the CCD gain has been set so as to evidence the low-intensity fringe; consequently, the most intense fringe saturates the detector.

Fig. 8
Fig. 8

Detail of the 3D configuration with Γ and the point Y 0 in evidence. The point Y 0 does not belong to the perpendicular to Γ passing through M; therefore, the represented conical surface is elliptical. The section of the elliptical cone passing through M and perpendicular to Y 0 M is also drawn. Note that in this configuration QQ MX .

Equations (16)

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γ = arcsin ( n   sin   δ ) - δ ,
OZ ¯ = ζ ( ρ ) = ρ 1 tan   γ - 1 tan   θ .
d A ( x ) = A ( x ) d α d x = A 0 cos   γ sz   x exp [ - ( x 2 / w 2 ) ] d α d x ,
A ( Y 0 ) = A 0 sz cos   γ 0 d x 0 2 π d α x × exp [ - ( x 2 / w 2 ) ] × cos 2 π λ   ( s 2 + x 2 + l ( x ,   α ) ) ,
A ( Y 0 ) = A 0 sz cos   γ 0 d x - α / 2 α / 2 d α x × exp [ - ( x 2 / w 2 ) ] × cos { 2 π / λ [ ( s 2 + x 2 ) 1 / 2 + l ( x ,   α ) ] } .
d A + ( x ) = A + ( x ) d α d x = cos   γ sz x A 0 exp - x 2 w 2 + a   exp - x 2 b 2 d α d x ,
A + ( x ) = A ( x ) + cos   γ sz   a   exp - x 2 b 2 .
OM ¯ = x sin   γ ,
r = MX ¯ = x cos   γ ,
O O ˆ Y 0 = arctan ( y 0 / z ) y 0 / z , y 0 z .
MY 0 ¯ = z = [ z 2 + y 0 2 + x 2 sin 2   γ - 2 z 2 + y 0 2 x sin   γ   cos ( γ - y 0 / z ) ] 1 / 2 ,
X Y 0 ¯ = z 2 + ( y 0 - x ) 2 ,
QY 0 ¯ = z 2 + r 2 ,
l = KY 0 ¯ = KL ¯ + LY 0 ¯ KL ¯ + QY 0 ¯ ,
KL ¯ = ( X Y 0 ¯ - QY 0 ¯ ) cos   α .
l ( x ,   α ) = z 2 + r 2 + ( z 2 + ( y 0 - x ) 2 - z 2 + r 2 ) cos   α .

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