Abstract

Lens-axicon doublets have been used to produce Bessel-Gaussian beams, a narrow non-diffracting beam of relatively constant width. One problem of using Bessel-Gaussian beams is that there is a compromise between achieving a long effective focal length with a small central core radius and distributing the beam intensity between the central core and the off-axis rings. Here, we explore the advantage of tuning the lens-axicon separation, which allows us to have an additional degree of freedom to tailor the beam profile. Moreover, the separation between the lens and the axicon reduces the spherical aberrations in the beam profile, which can then be modeled within the paraxial regime. We study the detrimental effects of the spherical aberrations and provide several options to minimize them. We examine both sharp and shallow axicons used in combination with different converging lenses. We perform a series of detailed experiments to image the structure of the beam through the Bessel region. The spatial light distribution of the lens-axicon system is analyzed by using high dynamic range imaging and complemented with consistent theoretical calculations within the paraxial regime.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (1)

2017 (2)

2016 (1)

2013 (1)

G. S. Sokolovskii, V. V. Dyudelev, S. N. Losev, M. Butkus, K. K. Soboleva, A. I. Sobolev, A. G. Deryagin, V. I. Kuchinskii, V. Sibbet, and E. U. Rafailov, “Influence of the axicon characteristics and beam propagation parameter M2 on the formation of bessel beams from semiconductor lasers,” Quantum Electron. 43, 423–427 (2013).
[Crossref]

2012 (1)

2010 (2)

2009 (1)

L. V. Dao, K. B. Dinh, and P. Hannaford, “Generation of extreme ultraviolet radiation with a bessel-gaussian beam,” Appl. Phys. Lett. 95, 131114 (2009).
[Crossref]

2008 (3)

2006 (1)

A. Stockham and J. G. Smith, “Optical design for generating bessel beams for micromanipulation,” Proc. SPIE 6326, 63261D (2006).

2005 (1)

D. McGloin and K. Dholakia, “Bessel beams: Diffraction in a new light,” Contemp. Phys. 46, 15–28 (2005).
[Crossref]

2004 (2)

2003 (2)

M. de Angelis, L. Cacciapuoti, G. Pierattini, and G. Tino, “Axially symmetric hollow beams using refractive conical lenses,” Opt. Lasers Eng. 39, 283 – 291 (2003).
[Crossref]

V. Vaičaitis and Š. Paulikas, “Formation of bessel beams with continuously variable cone angle,” Opt. Quantum Electron. 35, 1065–1071 (2003).
[Crossref]

1999 (1)

C. A. McQueen, J. Arlt, and K. Dholakia, “An experiment to study a “nondiffracting” light beam,” Am. J. Phys. 67, 912–915 (1999).
[Crossref]

1997 (2)

C. Parigger, Y. Tang, D. H. Plemmons, and J. W. L. Lewis, “Spherical aberration effects in lens–axicon doublets: theoretical study,” Appl. Opt. 36, 8214–8221 (1997).
[Crossref]

S. Sogomonian, S. Klewitz, and S. Herminghaus, “Self-reconstruction of a bessel beam in a nonlinear medium,” Opt. Commun. 139, 313 –319 (1997).
[Crossref]

1991 (1)

1988 (1)

1987 (3)

J. Durnin, “Exact solutions for nondiffracting beams. i. the scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[Crossref]

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

F. Gori, G. Guattari, and C. Padovani, “Bessel-gauss beams,” Opt. Commun. 64, 491–495 (1987).
[Crossref]

1978 (1)

1976 (1)

P.-A. Bélanger and M. Rioux, “Diffraction ring pattern at the focal plane of a spherical lens - axicon doublet,” Can. J. Phys. 54, 1774–1780 (1976).
[Crossref]

Arlt, J.

C. A. McQueen, J. Arlt, and K. Dholakia, “An experiment to study a “nondiffracting” light beam,” Am. J. Phys. 67, 912–915 (1999).
[Crossref]

Asuncion, A. J.

Bélanger, P.-A.

P.-A. Bélanger and M. Rioux, “Ring pattern of a lens-axicon doublet illuminated by a gaussian beam,” Appl. Opt. 17, 1080–1088 (1978).
[Crossref] [PubMed]

P.-A. Bélanger and M. Rioux, “Diffraction ring pattern at the focal plane of a spherical lens - axicon doublet,” Can. J. Phys. 54, 1774–1780 (1976).
[Crossref]

Belyi, V.

Belyi, V. N.

Bock, M.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, London, 1975).

Brzobohatý, O.

Butkus, M.

G. S. Sokolovskii, V. V. Dyudelev, S. N. Losev, M. Butkus, K. K. Soboleva, A. I. Sobolev, A. G. Deryagin, V. I. Kuchinskii, V. Sibbet, and E. U. Rafailov, “Influence of the axicon characteristics and beam propagation parameter M2 on the formation of bessel beams from semiconductor lasers,” Quantum Electron. 43, 423–427 (2013).
[Crossref]

Cacciapuoti, L.

M. de Angelis, L. Cacciapuoti, G. Pierattini, and G. Tino, “Axially symmetric hollow beams using refractive conical lenses,” Opt. Lasers Eng. 39, 283 – 291 (2003).
[Crossref]

Chávez-Cerda, S.

Cizmar, T.

T. Cizmar, V. Garces-Chavez, K. Dholakia, and P. Zemanek, “Optical trapping in counter-propagating bessel beams,” Proc. SPIE 5514, 643–651 (2004).
[Crossref]

Cižmár, T.

Dao, L. V.

L. V. Dao, K. B. Dinh, and P. Hannaford, “Generation of extreme ultraviolet radiation with a bessel-gaussian beam,” Appl. Phys. Lett. 95, 131114 (2009).
[Crossref]

de Angelis, M.

M. de Angelis, L. Cacciapuoti, G. Pierattini, and G. Tino, “Axially symmetric hollow beams using refractive conical lenses,” Opt. Lasers Eng. 39, 283 – 291 (2003).
[Crossref]

Deryagin, A. G.

G. S. Sokolovskii, V. V. Dyudelev, S. N. Losev, M. Butkus, K. K. Soboleva, A. I. Sobolev, A. G. Deryagin, V. I. Kuchinskii, V. Sibbet, and E. U. Rafailov, “Influence of the axicon characteristics and beam propagation parameter M2 on the formation of bessel beams from semiconductor lasers,” Quantum Electron. 43, 423–427 (2013).
[Crossref]

Dholakia, K.

D. McGloin and K. Dholakia, “Bessel beams: Diffraction in a new light,” Contemp. Phys. 46, 15–28 (2005).
[Crossref]

T. Cizmar, V. Garces-Chavez, K. Dholakia, and P. Zemanek, “Optical trapping in counter-propagating bessel beams,” Proc. SPIE 5514, 643–651 (2004).
[Crossref]

K. Volke-Sepúlveda, S. Chávez-Cerda, V. Garcés-Chávez, and K. Dholakia, “Three-dimensional optical forces and transfer of orbital angular momentum from multiringed light beams to spherical microparticles,” J. Opt. Soc. Am. B 21, 1749–1757 (2004).
[Crossref]

C. A. McQueen, J. Arlt, and K. Dholakia, “An experiment to study a “nondiffracting” light beam,” Am. J. Phys. 67, 912–915 (1999).
[Crossref]

Dinh, K. B.

L. V. Dao, K. B. Dinh, and P. Hannaford, “Generation of extreme ultraviolet radiation with a bessel-gaussian beam,” Appl. Phys. Lett. 95, 131114 (2009).
[Crossref]

Dudutis, J.

Durnin, J.

Dyudelev, V. V.

G. S. Sokolovskii, V. V. Dyudelev, S. N. Losev, M. Butkus, K. K. Soboleva, A. I. Sobolev, A. G. Deryagin, V. I. Kuchinskii, V. Sibbet, and E. U. Rafailov, “Influence of the axicon characteristics and beam propagation parameter M2 on the formation of bessel beams from semiconductor lasers,” Quantum Electron. 43, 423–427 (2013).
[Crossref]

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Comparison of bessel and gaussian beams,” Opt. Lett. 13, 79–80 (1988).
[Crossref] [PubMed]

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

Forbes, A.

Garces-Chavez, V.

T. Cizmar, V. Garces-Chavez, K. Dholakia, and P. Zemanek, “Optical trapping in counter-propagating bessel beams,” Proc. SPIE 5514, 643–651 (2004).
[Crossref]

Garcés-Chávez, V.

Gecys, P.

Gori, F.

F. Gori, G. Guattari, and C. Padovani, “Bessel-gauss beams,” Opt. Commun. 64, 491–495 (1987).
[Crossref]

Grunwald, R.

Guattari, G.

F. Gori, G. Guattari, and C. Padovani, “Bessel-gauss beams,” Opt. Commun. 64, 491–495 (1987).
[Crossref]

Guerrero, R. A.

Hannaford, P.

L. V. Dao, K. B. Dinh, and P. Hannaford, “Generation of extreme ultraviolet radiation with a bessel-gaussian beam,” Appl. Phys. Lett. 95, 131114 (2009).
[Crossref]

Herman, R. M.

Herminghaus, S.

S. Sogomonian, S. Klewitz, and S. Herminghaus, “Self-reconstruction of a bessel beam in a nonlinear medium,” Opt. Commun. 139, 313 –319 (1997).
[Crossref]

Hong, W.

Huferath, S.

Jiang, L.

Kazak, N.

Kebbel, V.

Khilo, N.

Khilo, N. A.

Klewitz, S.

S. Sogomonian, S. Klewitz, and S. Herminghaus, “Self-reconstruction of a bessel beam in a nonlinear medium,” Opt. Commun. 139, 313 –319 (1997).
[Crossref]

Kuchinskii, V. I.

G. S. Sokolovskii, V. V. Dyudelev, S. N. Losev, M. Butkus, K. K. Soboleva, A. I. Sobolev, A. G. Deryagin, V. I. Kuchinskii, V. Sibbet, and E. U. Rafailov, “Influence of the axicon characteristics and beam propagation parameter M2 on the formation of bessel beams from semiconductor lasers,” Quantum Electron. 43, 423–427 (2013).
[Crossref]

Lei, S.

Lewis, J. W. L.

Li, M.

Li, X.

Li, Y.

Litvin, I. A.

Losev, S. N.

G. S. Sokolovskii, V. V. Dyudelev, S. N. Losev, M. Butkus, K. K. Soboleva, A. I. Sobolev, A. G. Deryagin, V. I. Kuchinskii, V. Sibbet, and E. U. Rafailov, “Influence of the axicon characteristics and beam propagation parameter M2 on the formation of bessel beams from semiconductor lasers,” Quantum Electron. 43, 423–427 (2013).
[Crossref]

Lu, P.

Lu, Y.

McGloin, D.

D. McGloin and K. Dholakia, “Bessel beams: Diffraction in a new light,” Contemp. Phys. 46, 15–28 (2005).
[Crossref]

McQueen, C. A.

C. A. McQueen, J. Arlt, and K. Dholakia, “An experiment to study a “nondiffracting” light beam,” Am. J. Phys. 67, 912–915 (1999).
[Crossref]

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Comparison of bessel and gaussian beams,” Opt. Lett. 13, 79–80 (1988).
[Crossref] [PubMed]

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

Nelson, J.

Néron, J.-L.

Neumann, U.

Nussbaum, A.

A. Nussbaum and R. Phillips, Contemporary optics for scientists and engineers, (Prentice-Hall, 1976).

Padovani, C.

F. Gori, G. Guattari, and C. Padovani, “Bessel-gauss beams,” Opt. Commun. 64, 491–495 (1987).
[Crossref]

Parigger, C.

Paulikas, Š.

V. Vaičaitis and Š. Paulikas, “Formation of bessel beams with continuously variable cone angle,” Opt. Quantum Electron. 35, 1065–1071 (2003).
[Crossref]

Phillips, R.

A. Nussbaum and R. Phillips, Contemporary optics for scientists and engineers, (Prentice-Hall, 1976).

Piché, M.

Pierattini, G.

M. de Angelis, L. Cacciapuoti, G. Pierattini, and G. Tino, “Axially symmetric hollow beams using refractive conical lenses,” Opt. Lasers Eng. 39, 283 – 291 (2003).
[Crossref]

Plemmons, D. H.

Raciukaitis, G.

Rafailov, E. U.

G. S. Sokolovskii, V. V. Dyudelev, S. N. Losev, M. Butkus, K. K. Soboleva, A. I. Sobolev, A. G. Deryagin, V. I. Kuchinskii, V. Sibbet, and E. U. Rafailov, “Influence of the axicon characteristics and beam propagation parameter M2 on the formation of bessel beams from semiconductor lasers,” Quantum Electron. 43, 423–427 (2013).
[Crossref]

Raoul, M.

Rioux, M.

P.-A. Bélanger and M. Rioux, “Ring pattern of a lens-axicon doublet illuminated by a gaussian beam,” Appl. Opt. 17, 1080–1088 (1978).
[Crossref] [PubMed]

P.-A. Bélanger and M. Rioux, “Diffraction ring pattern at the focal plane of a spherical lens - axicon doublet,” Can. J. Phys. 54, 1774–1780 (1976).
[Crossref]

Ropot, P.

Sibbet, V.

G. S. Sokolovskii, V. V. Dyudelev, S. N. Losev, M. Butkus, K. K. Soboleva, A. I. Sobolev, A. G. Deryagin, V. I. Kuchinskii, V. Sibbet, and E. U. Rafailov, “Influence of the axicon characteristics and beam propagation parameter M2 on the formation of bessel beams from semiconductor lasers,” Quantum Electron. 43, 423–427 (2013).
[Crossref]

Siegman, A. E.

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

Smith, J. G.

A. Stockham and J. G. Smith, “Optical design for generating bessel beams for micromanipulation,” Proc. SPIE 6326, 63261D (2006).

Sobolev, A. I.

G. S. Sokolovskii, V. V. Dyudelev, S. N. Losev, M. Butkus, K. K. Soboleva, A. I. Sobolev, A. G. Deryagin, V. I. Kuchinskii, V. Sibbet, and E. U. Rafailov, “Influence of the axicon characteristics and beam propagation parameter M2 on the formation of bessel beams from semiconductor lasers,” Quantum Electron. 43, 423–427 (2013).
[Crossref]

Soboleva, K. K.

G. S. Sokolovskii, V. V. Dyudelev, S. N. Losev, M. Butkus, K. K. Soboleva, A. I. Sobolev, A. G. Deryagin, V. I. Kuchinskii, V. Sibbet, and E. U. Rafailov, “Influence of the axicon characteristics and beam propagation parameter M2 on the formation of bessel beams from semiconductor lasers,” Quantum Electron. 43, 423–427 (2013).
[Crossref]

Sogomonian, S.

S. Sogomonian, S. Klewitz, and S. Herminghaus, “Self-reconstruction of a bessel beam in a nonlinear medium,” Opt. Commun. 139, 313 –319 (1997).
[Crossref]

Sokolovskii, G. S.

G. S. Sokolovskii, V. V. Dyudelev, S. N. Losev, M. Butkus, K. K. Soboleva, A. I. Sobolev, A. G. Deryagin, V. I. Kuchinskii, V. Sibbet, and E. U. Rafailov, “Influence of the axicon characteristics and beam propagation parameter M2 on the formation of bessel beams from semiconductor lasers,” Quantum Electron. 43, 423–427 (2013).
[Crossref]

Steinmeyer, G.

Stibenz, G.

Stockham, A.

A. Stockham and J. G. Smith, “Optical design for generating bessel beams for micromanipulation,” Proc. SPIE 6326, 63261D (2006).

Summers, A. M.

Tang, Y.

Tino, G.

M. de Angelis, L. Cacciapuoti, G. Pierattini, and G. Tino, “Axially symmetric hollow beams using refractive conical lenses,” Opt. Lasers Eng. 39, 283 – 291 (2003).
[Crossref]

Todd, D.

Trallero-Herrero, C. A.

Vaicaitis, V.

V. Vaičaitis and Š. Paulikas, “Formation of bessel beams with continuously variable cone angle,” Opt. Quantum Electron. 35, 1065–1071 (2003).
[Crossref]

Volke-Sepúlveda, K.

Wang, A.

Wang, S.

Wang, X.

Wang, Z.

Wiggins, T. A.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, London, 1975).

Wu, F.

X. Zeng and F. Wu, “Effect of elliptical manufacture error of an axicon on the diffraction-free beam patterns,” Opt. Eng. 47, 08340 (2008).

Yao, Z.

Yu, X.

Zemanek, P.

T. Cizmar, V. Garces-Chavez, K. Dholakia, and P. Zemanek, “Optical trapping in counter-propagating bessel beams,” Proc. SPIE 5514, 643–651 (2004).
[Crossref]

Zemánek, P.

Zeng, X.

X. Zeng and F. Wu, “Effect of elliptical manufacture error of an axicon on the diffraction-free beam patterns,” Opt. Eng. 47, 08340 (2008).

Zhang, Q.

Zigo, S.

Am. J. Phys. (1)

C. A. McQueen, J. Arlt, and K. Dholakia, “An experiment to study a “nondiffracting” light beam,” Am. J. Phys. 67, 912–915 (1999).
[Crossref]

Appl. Opt. (3)

Appl. Phys. Lett. (1)

L. V. Dao, K. B. Dinh, and P. Hannaford, “Generation of extreme ultraviolet radiation with a bessel-gaussian beam,” Appl. Phys. Lett. 95, 131114 (2009).
[Crossref]

Can. J. Phys. (1)

P.-A. Bélanger and M. Rioux, “Diffraction ring pattern at the focal plane of a spherical lens - axicon doublet,” Can. J. Phys. 54, 1774–1780 (1976).
[Crossref]

Contemp. Phys. (1)

D. McGloin and K. Dholakia, “Bessel beams: Diffraction in a new light,” Contemp. Phys. 46, 15–28 (2005).
[Crossref]

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

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

Opt. Commun. (2)

S. Sogomonian, S. Klewitz, and S. Herminghaus, “Self-reconstruction of a bessel beam in a nonlinear medium,” Opt. Commun. 139, 313 –319 (1997).
[Crossref]

F. Gori, G. Guattari, and C. Padovani, “Bessel-gauss beams,” Opt. Commun. 64, 491–495 (1987).
[Crossref]

Opt. Eng. (1)

X. Zeng and F. Wu, “Effect of elliptical manufacture error of an axicon on the diffraction-free beam patterns,” Opt. Eng. 47, 08340 (2008).

Opt. Express (8)

A. M. Summers, X. Yu, X. Wang, M. Raoul, J. Nelson, D. Todd, S. Zigo, S. Lei, and C. A. Trallero-Herrero, “Spatial characterization of bessel-like beams for strong-field physics,” Opt. Express 25, 1646–1655 (2017).
[Crossref]

Y. Li, Q. Zhang, W. Hong, S. Wang, Z. Wang, and P. Lu, “Efficient generation of high beam-quality attosecond pulse with polarization-gating bessel-gauss beam from highly-ionized media,” Opt. Express 20, 15427–15439 (2012).

O. Brzobohatý, T. Čižmár, and P. Zemánek, “High quality quasi-bessel beam generated by round-tip axicon,” Opt. Express 16, 12688–12700 (2008).
[Crossref] [PubMed]

J. Dudutis, P. Gečys, and G. Račiukaitis, “Non-ideal axicon-generated bessel beam application for intra-volume glass modification,” Opt. Express 24, 28433–28443 (2016).
[Crossref] [PubMed]

Z. Yao, L. Jiang, X. Li, A. Wang, Z. Wang, M. Li, and Y. Lu, “Non-diffraction-length, tunable, bessel-like beams generation by spatially shaping a femtosecond laser beam for high-aspect-ratio micro-hole drilling,” Opt. Express 26, 21960–21968 (2018).
[Crossref] [PubMed]

R. Grunwald, M. Bock, V. Kebbel, S. Huferath, U. Neumann, G. Steinmeyer, G. Stibenz, J.-L. Néron, and M. Piché, “Ultrashort-pulsed truncated polychromatic bessel-gauss beams,” Opt. Express 16, 1077–1089 (2008).
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[Crossref] [PubMed]

I. A. Litvin, N. A. Khilo, A. Forbes, and V. N. Belyi, “Intra–cavity generation of bessel–like beams with longitudinally dependent cone angles,” Opt. Express 18, 4701–4708 (2010).
[Crossref] [PubMed]

Opt. Lasers Eng. (1)

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[Crossref]

Opt. Lett. (1)

Opt. Quantum Electron. (1)

V. Vaičaitis and Š. Paulikas, “Formation of bessel beams with continuously variable cone angle,” Opt. Quantum Electron. 35, 1065–1071 (2003).
[Crossref]

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[Crossref]

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Figures (9)

Fig. 1
Fig. 1 From left to right through z-axis a collimated Gaussian beam incident on the lens with aperture radius a1 and focal length f. The beam is then propagated through the lens, adding the phase factor, by distance d and integrated over ρ1. Then, we have a cylindrically symmetric intensity profile at distance d (axicon plane with radius aperture a2 and apex angle α) from the lens and at a distance ρ2 from the optical axis. Finally, we have a cylindrically symmetric intensity profile from the tip of the axicon along the axial distance and at distance r from the optical axis. The final ellipse represents out of plane region of the far field ring.
Fig. 2
Fig. 2 Numerical calculation of the beam profile after crossing a 30mm-5° lens-axicon combination with six different distances (a) 4mm, (b) 8mm, (c) 12mm, (d) 16mm, (e) 20mm and (f) 24mm. All intensity distributions are normalized to the maximum of (f) with the exception of its inset, which reproduces (a) normalized to its maximum intensity. Notice that in the selected region the number of rings is the same in all cases.
Fig. 3
Fig. 3 Numerical calculation of the intensity through the central axis of propagation for 30mm-5° lens-axicon with six different values of d: 4mm (red), 8mm (green), 12mm (blue), 16mm (cyan), 20mm (magenta) and 24mm (yellow). Insets; Comparison with only a 20° axicon with the combination 30mm-5° for d 4mm (red) and 24mm (yellow), The incident beam radius W on the single 20° axicon is equal the beam size after crossing the 30mm lens and propagate the distance d: Solid black for equivalent W as d = 4mm and dashed black for d=24mm (a) On-axis the core intensity profile. (b) Radial intensity distribution at the peak intensity. (c) Central core radius through the axis of propagation. Central core radius from peak intensity to 1/e2. Red dashed line indicates the central core radius at the peak intensity.
Fig. 4
Fig. 4 Numerical calculations (a) and experiment (b) of the normalized intensity beam profile generated with a 50mm-5° lens-axicon. Comparison of the simulation (black line) with the experimental data (red line) for (c) the intensity distribution of the central core normalized at their maximum intensity and (d) the central core radius through the axial direction.
Fig. 5
Fig. 5 Numerical calculation (top row) and experiment (bottom row) cross sections normalized to the peak intensity (fourth column) of the beam profile shown in Fig. 4 at 3.5mm, 4.7mm, 5.9mm, 7.1mm and 8.2mm from the tip of the axicon.
Fig. 6
Fig. 6 Numerical calculation (top row) and experiment (bottom row) of the intensity beam profile generated with a 30mm-5° lens-axicon (left column) and a 30mm-20° lens-axicon (right column). All plots normalized at their maximum intensity.
Fig. 7
Fig. 7 Observation of spherical aberrations modifying the ideal Bessel zone. (a) Three experimental transverse view images of the Bessel-Gaussian beam after the 30mm-20° lens-axicon combination in different positions inside the Bessel zone. From left to right 1.38mm, 1.43mm and 1.46mm from the tip of the axicon. (b) Example of radial intensity of the central cross-section. (c) and (d) are zoomed axial intensity distributions showing the sharp peaks in the central core region for the 30mm-20° and the 30mm-5° combinations, respectively. Red dots in (c) correspond to the center core of the images in (a).
Fig. 8
Fig. 8 Numerical calculation (a) and experiment (b) of a normalized intensity beam profile generated with a 32mm-20° aspheric lens-axicon combination. Comparison of the simulation (black line) with the experimental data (red line) for (c) the intensity distribution of the central core normalized at their maximum intensity and (d) the central core radius through the axial direction.
Fig. 9
Fig. 9 Numerical calculation (a) and experiment (b) of a normalized intensity beam profile generated with a 32mm-5° aspheric lens-axicon combination. Comparison of the simulation (black line) with the experimental data (red line) for (c) the intensity distribution of the central core normalized at their maximum intensity and (d) the central core radius through the axial direction.

Tables (1)

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Table 1 Focal spot size (FWHM) of an axicon or a lens-axicon combination for a 532nm incoming Gaussian beam with a beam radius W = 2.5mm.

Equations (5)

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N z max 2 Δ r , P core P total N .
u ( r , z ) = k z e i k r 2 2 z 0 a A ( ρ ) e i k ϕ ( ρ ) e i k ρ 2 2 z J 0 ( k r ρ z ) ρ d ρ ,
u 1 ( ρ 2 , d ) = k d e i k ρ 2 2 2 d 0 a 1 A ( ρ 1 ) e i k ϕ l e n s ( ρ 1 ) e i k ρ 1 2 2 d J 0 ( k ρ 2 ρ 1 d ) ρ 1 d ρ 1 ,
u 2 ( r , z ) = k z e i k r 2 2 z 0 a 2 u 1 ( ρ 2 , d ) e i k ϕ a x i c o n ( ρ 2 ) e i k ρ 2 2 2 z J 0 ( k r ρ 2 z ) ρ 2 d ρ 2 ,
u 1 ( ρ 2 , d ) = I 0 ( f f d ) e ρ 2 2 W 2 ( f d f ) 2 e i k ρ 2 2 2 ( f d ) .

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