Abstract

Arbitrary shaping of the on-axis intensity of Bessel beams requires spatial modulation of both amplitude and phase. We develop a non-iterative direct space beam shaping method to generate Bessel beams with high energy throughput from direct space with a single phase-only spatial light modulator. For this purpose, we generalize the approach of Bolduc et al. to non-uniform input beams. We point out the physical limitations imposed on the on-axis intensity profile for unidirectional beams. Analytical, numerical and experimental results are provided.

© 2016 Optical Society of America

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References

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2016 (1)

F. Courvoisier, R. Stoian, and A. Couairon, “Ultrafast laser micro and nano processing with nondiffracting and curved beams,” J. Opt. Laser Technol. 80, 125–137 (2016).
[Crossref]

2015 (2)

M. Zamboni-Rached and M. Mojahedi, “Shaping finite-energy diffraction- and attenuation-resistant beams through Bessel-Gauss–beam superposition,” Phys. Rev. A 92(4), 043839 (2015).
[Crossref]

T. A. Vieira, M. R. R. Gesualdi, M. Zamboni-Rached, and E. Recami, “Production of dynamic frozen waves: controlling shape, location (and speed) of diffraction-resistant beams,” Opt. Lett. 40(24), 5834–5837 (2015).
[Crossref] [PubMed]

2014 (7)

2013 (4)

L. Li, N. Eckerskorn, R. A. Kirian, J. Küpper, D. P. DePonte, W. Krolikowski, W. M. Lee, H. N. Chapman, and A. V. Rode, “quasi-Bessel hollow beam as optical guide for micro-particles,” Proc. SPIE 8810, 88100N (2013).
[Crossref]

F. Courvoisier, J. Zhang, M. K. Bhuyan, M. Jacquot, and J. M. Dudley, “Applications of femtosecond Bessel beams to laser ablation,” App. Phys. A 112(1), 29–34 (2013).
[Crossref]

E. Bolduc, N. Bent, E. Santamato, E. Karimi, and R. W. Boyd, “Exact solution to simultaneous intensity and phase encryption with a single phase-only hologram,” Opt. Lett. 38(18), 3546–3549 (2013).
[Crossref] [PubMed]

C. G. Durfee, J. Gemmer, and J. V. Moloney, “Phase-only shaping algorithm for Gaussian-apodized Bessel beams,” Opt. Express 21(13), 15777–15786 (2013).
[Crossref] [PubMed]

2012 (3)

2010 (1)

2009 (1)

2008 (1)

P. Polesana, M. Franco, A. Couairon, D. Faccio, and P. Di Trapani, “Filamentation in Kerr media from pulsed Bessel beams,” Phys. Rev. A 77(4), 043814 (2008).
[Crossref]

2007 (2)

2006 (2)

C. A. Dartora, K. Z. Nobrega, A. Dartora, G. A. Viana, and H. T. S. Filho, “A general theory for the frozen waves and their realization through finite apertures,” Opt. Commun. 265(2), 481–487 (2006).
[Crossref]

M. Zamboni-Rached, “Diffraction-Attenuation resistant beams in absorbing media,” Opt. Express 14(5), 1804–1809 (2006).
[Crossref] [PubMed]

2005 (4)

2004 (2)

M. A. Porras, A. Parola, D. Faccio, A. Dubietis, and P. D. Trapani, “Nonlinear unbalanced bessel beams: stationary conical waves supported by nonlinear losses,” Phys. Rev. Lett. 93(15), 153902 (2004).
[Crossref] [PubMed]

M. Zamboni-Rached, “Stationary optical wave fields with arbitrary longitudinal shape by superposing equal frequency Bessel beams: Frozen Waves,” Opt. Express 12(17), 4001–4006 (2004).
[Crossref] [PubMed]

1999 (1)

1996 (1)

S. P. Tewari, H. Huang, and R. W. Boyd, “Theory of third-harmonic generation using Bessel beams, and self-phase-matching,” Phys. Rev. A 54(3), 2314–2325 (1996).
[Crossref] [PubMed]

1993 (1)

Arrizón, V.

Ashok, A.

Bent, N.

Bhuyan, M. K.

F. Courvoisier, J. Zhang, M. K. Bhuyan, M. Jacquot, and J. M. Dudley, “Applications of femtosecond Bessel beams to laser ablation,” App. Phys. A 112(1), 29–34 (2013).
[Crossref]

Bolduc, E.

Boyd, R. W.

Campos, J.

Carrada, R.

Chapman, H. N.

L. Li, N. Eckerskorn, R. A. Kirian, J. Küpper, D. P. DePonte, W. Krolikowski, W. M. Lee, H. N. Chapman, and A. V. Rode, “quasi-Bessel hollow beam as optical guide for micro-particles,” Proc. SPIE 8810, 88100N (2013).
[Crossref]

Chichkov, B.

Cižmár, T.

Cottrell, D. M.

Couairon, A.

F. Courvoisier, R. Stoian, and A. Couairon, “Ultrafast laser micro and nano processing with nondiffracting and curved beams,” J. Opt. Laser Technol. 80, 125–137 (2016).
[Crossref]

P. Polesana, M. Franco, A. Couairon, D. Faccio, and P. Di Trapani, “Filamentation in Kerr media from pulsed Bessel beams,” Phys. Rev. A 77(4), 043814 (2008).
[Crossref]

Courtial, J.

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005).
[Crossref]

Courvoisier, F.

F. Courvoisier, R. Stoian, and A. Couairon, “Ultrafast laser micro and nano processing with nondiffracting and curved beams,” J. Opt. Laser Technol. 80, 125–137 (2016).
[Crossref]

L. Froehly, M. Jacquot, P. A. Lacourt, J. M. Dudley, and F. Courvoisier, “Spatiotemporal structure of femtosecond Bessel beams from spatial light modulators,” J. Opt. Soc. Am. A 31(4), 790–793 (2014).
[Crossref] [PubMed]

F. Courvoisier, J. Zhang, M. K. Bhuyan, M. Jacquot, and J. M. Dudley, “Applications of femtosecond Bessel beams to laser ablation,” App. Phys. A 112(1), 29–34 (2013).
[Crossref]

Dartora, A.

C. A. Dartora, K. Z. Nobrega, A. Dartora, G. A. Viana, and H. T. S. Filho, “A general theory for the frozen waves and their realization through finite apertures,” Opt. Commun. 265(2), 481–487 (2006).
[Crossref]

Dartora, C. A.

C. A. Dartora, K. Z. Nobrega, A. Dartora, G. A. Viana, and H. T. S. Filho, “A general theory for the frozen waves and their realization through finite apertures,” Opt. Commun. 265(2), 481–487 (2006).
[Crossref]

Davis, J. A.

Dennis, M. R.

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005).
[Crossref]

DePonte, D. P.

L. Li, N. Eckerskorn, R. A. Kirian, J. Küpper, D. P. DePonte, W. Krolikowski, W. M. Lee, H. N. Chapman, and A. V. Rode, “quasi-Bessel hollow beam as optical guide for micro-particles,” Proc. SPIE 8810, 88100N (2013).
[Crossref]

Dholakia, K.

T. Čižmár and K. Dholakia, “Tunable Bessel light modes: engineering the axial propagation,” Opt. Express 17(18), 15558–15570 (2009).
[Crossref] [PubMed]

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46(1), 15–28 (2005).
[Crossref]

Di Trapani, P.

Dubietis, A.

A. Dubietis, P. Polesana, G. Valiulis, A. Stabinis, P. Di Trapani, and A. Piskarskas, “Axial emission and spectral broadening in self-focusing of femtosecond Bessel beams,” Opt. Express 15(7), 4168–4175 (2007).
[Crossref] [PubMed]

M. A. Porras, A. Parola, D. Faccio, A. Dubietis, and P. D. Trapani, “Nonlinear unbalanced bessel beams: stationary conical waves supported by nonlinear losses,” Phys. Rev. Lett. 93(15), 153902 (2004).
[Crossref] [PubMed]

Dudley, J. M.

L. Froehly, M. Jacquot, P. A. Lacourt, J. M. Dudley, and F. Courvoisier, “Spatiotemporal structure of femtosecond Bessel beams from spatial light modulators,” J. Opt. Soc. Am. A 31(4), 790–793 (2014).
[Crossref] [PubMed]

F. Courvoisier, J. Zhang, M. K. Bhuyan, M. Jacquot, and J. M. Dudley, “Applications of femtosecond Bessel beams to laser ablation,” App. Phys. A 112(1), 29–34 (2013).
[Crossref]

Durfee, C. G.

Eckerskorn, N.

L. Li, N. Eckerskorn, R. A. Kirian, J. Küpper, D. P. DePonte, W. Krolikowski, W. M. Lee, H. N. Chapman, and A. V. Rode, “quasi-Bessel hollow beam as optical guide for micro-particles,” Proc. SPIE 8810, 88100N (2013).
[Crossref]

Faccio, D.

P. Polesana, M. Franco, A. Couairon, D. Faccio, and P. Di Trapani, “Filamentation in Kerr media from pulsed Bessel beams,” Phys. Rev. A 77(4), 043814 (2008).
[Crossref]

M. A. Porras, A. Parola, D. Faccio, A. Dubietis, and P. D. Trapani, “Nonlinear unbalanced bessel beams: stationary conical waves supported by nonlinear losses,” Phys. Rev. Lett. 93(15), 153902 (2004).
[Crossref] [PubMed]

Filho, H. T. S.

C. A. Dartora, K. Z. Nobrega, A. Dartora, G. A. Viana, and H. T. S. Filho, “A general theory for the frozen waves and their realization through finite apertures,” Opt. Commun. 265(2), 481–487 (2006).
[Crossref]

Franco, M.

P. Polesana, M. Franco, A. Couairon, D. Faccio, and P. Di Trapani, “Filamentation in Kerr media from pulsed Bessel beams,” Phys. Rev. A 77(4), 043814 (2008).
[Crossref]

Froehly, L.

Gao, P.

Gemmer, J.

Gesualdi, M. R. R.

González, L. A.

Hernández-Figueroa, H. E.

Huang, H.

S. P. Tewari, H. Huang, and R. W. Boyd, “Theory of third-harmonic generation using Bessel beams, and self-phase-matching,” Phys. Rev. A 54(3), 2314–2325 (1996).
[Crossref] [PubMed]

Jacquot, M.

L. Froehly, M. Jacquot, P. A. Lacourt, J. M. Dudley, and F. Courvoisier, “Spatiotemporal structure of femtosecond Bessel beams from spatial light modulators,” J. Opt. Soc. Am. A 31(4), 790–793 (2014).
[Crossref] [PubMed]

F. Courvoisier, J. Zhang, M. K. Bhuyan, M. Jacquot, and J. M. Dudley, “Applications of femtosecond Bessel beams to laser ablation,” App. Phys. A 112(1), 29–34 (2013).
[Crossref]

Jaroszewicz, Z.

Kachalov, D.

Karimi, E.

Kirian, R. A.

L. Li, N. Eckerskorn, R. A. Kirian, J. Küpper, D. P. DePonte, W. Krolikowski, W. M. Lee, H. N. Chapman, and A. V. Rode, “quasi-Bessel hollow beam as optical guide for micro-particles,” Proc. SPIE 8810, 88100N (2013).
[Crossref]

Kolodziejczyk, A.

Krolikowski, W.

L. Li, W. M. Lee, X. Xie, W. Krolikowski, A. V. Rode, and J. Zhou, “Shaping self-imaging bottle beams with modified quasi-Bessel beams,” Opt. Lett. 39(8), 2278–2281 (2014).
[Crossref] [PubMed]

L. Li, N. Eckerskorn, R. A. Kirian, J. Küpper, D. P. DePonte, W. Krolikowski, W. M. Lee, H. N. Chapman, and A. V. Rode, “quasi-Bessel hollow beam as optical guide for micro-particles,” Proc. SPIE 8810, 88100N (2013).
[Crossref]

Küpper, J.

L. Li, N. Eckerskorn, R. A. Kirian, J. Küpper, D. P. DePonte, W. Krolikowski, W. M. Lee, H. N. Chapman, and A. V. Rode, “quasi-Bessel hollow beam as optical guide for micro-particles,” Proc. SPIE 8810, 88100N (2013).
[Crossref]

Lacourt, P. A.

Lancis, J.

Leach, J.

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005).
[Crossref]

Lee, W. M.

L. Li, W. M. Lee, X. Xie, W. Krolikowski, A. V. Rode, and J. Zhou, “Shaping self-imaging bottle beams with modified quasi-Bessel beams,” Opt. Lett. 39(8), 2278–2281 (2014).
[Crossref] [PubMed]

L. Li, N. Eckerskorn, R. A. Kirian, J. Küpper, D. P. DePonte, W. Krolikowski, W. M. Lee, H. N. Chapman, and A. V. Rode, “quasi-Bessel hollow beam as optical guide for micro-particles,” Proc. SPIE 8810, 88100N (2013).
[Crossref]

Lei, M.

Li, L.

L. Li, W. M. Lee, X. Xie, W. Krolikowski, A. V. Rode, and J. Zhou, “Shaping self-imaging bottle beams with modified quasi-Bessel beams,” Opt. Lett. 39(8), 2278–2281 (2014).
[Crossref] [PubMed]

L. Li, N. Eckerskorn, R. A. Kirian, J. Küpper, D. P. DePonte, W. Krolikowski, W. M. Lee, H. N. Chapman, and A. V. Rode, “quasi-Bessel hollow beam as optical guide for micro-particles,” Proc. SPIE 8810, 88100N (2013).
[Crossref]

Li, Y.

Liang, R.

Magaña-Loaiza, O. S.

McGloin, D.

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46(1), 15–28 (2005).
[Crossref]

Mehta, S.

Méndez, G.

Mendoza-Yero, O.

Mínguez-Vega, G.

Mirhosseini, M.

Mojahedi, M.

M. Zamboni-Rached and M. Mojahedi, “Shaping finite-energy diffraction- and attenuation-resistant beams through Bessel-Gauss–beam superposition,” Phys. Rev. A 92(4), 043839 (2015).
[Crossref]

Moloney, J. V.

Moreno, I.

Nobrega, K. Z.

C. A. Dartora, K. Z. Nobrega, A. Dartora, G. A. Viana, and H. T. S. Filho, “A general theory for the frozen waves and their realization through finite apertures,” Opt. Commun. 265(2), 481–487 (2006).
[Crossref]

Osipov, V.

Padgett, M. J.

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005).
[Crossref]

Parola, A.

M. A. Porras, A. Parola, D. Faccio, A. Dubietis, and P. D. Trapani, “Nonlinear unbalanced bessel beams: stationary conical waves supported by nonlinear losses,” Phys. Rev. Lett. 93(15), 153902 (2004).
[Crossref] [PubMed]

Pavelyev, V.

Peng, L.

Piskarskas, A.

Polesana, P.

Porras, M. A.

M. A. Porras, A. Parola, D. Faccio, A. Dubietis, and P. D. Trapani, “Nonlinear unbalanced bessel beams: stationary conical waves supported by nonlinear losses,” Phys. Rev. Lett. 93(15), 153902 (2004).
[Crossref] [PubMed]

Recami, E.

Rode, A. V.

L. Li, W. M. Lee, X. Xie, W. Krolikowski, A. V. Rode, and J. Zhou, “Shaping self-imaging bottle beams with modified quasi-Bessel beams,” Opt. Lett. 39(8), 2278–2281 (2014).
[Crossref] [PubMed]

L. Li, N. Eckerskorn, R. A. Kirian, J. Küpper, D. P. DePonte, W. Krolikowski, W. M. Lee, H. N. Chapman, and A. V. Rode, “quasi-Bessel hollow beam as optical guide for micro-particles,” Proc. SPIE 8810, 88100N (2013).
[Crossref]

Rodenburg, B.

Ruiz, U.

Sánchez-de-La-Llave, D.

Santamato, E.

Sheppard, C. J. R.

Sochacki, J.

Stabinis, A.

Staronski, L. R.

Stoian, R.

F. Courvoisier, R. Stoian, and A. Couairon, “Ultrafast laser micro and nano processing with nondiffracting and curved beams,” J. Opt. Laser Technol. 80, 125–137 (2016).
[Crossref]

Tewari, S. P.

S. P. Tewari, H. Huang, and R. W. Boyd, “Theory of third-harmonic generation using Bessel beams, and self-phase-matching,” Phys. Rev. A 54(3), 2314–2325 (1996).
[Crossref] [PubMed]

Trapani, P. D.

M. A. Porras, A. Parola, D. Faccio, A. Dubietis, and P. D. Trapani, “Nonlinear unbalanced bessel beams: stationary conical waves supported by nonlinear losses,” Phys. Rev. Lett. 93(15), 153902 (2004).
[Crossref] [PubMed]

Valiulis, G.

Viana, G. A.

C. A. Dartora, K. Z. Nobrega, A. Dartora, G. A. Viana, and H. T. S. Filho, “A general theory for the frozen waves and their realization through finite apertures,” Opt. Commun. 265(2), 481–487 (2006).
[Crossref]

Vieira, T. A.

Wang, J.

L. Zhu and J. Wang, “Arbitrary manipulation of spatial amplitude and phase using phase-only spatial light modulators,” Sci. Rep. 4, 7441 (2014).
[Crossref] [PubMed]

Xie, X.

Yang, Y.

Yao, B.

Ye, T.

Yzuel, M. J.

Zamboni-Rached, M.

Zhang, H.

Zhang, J.

F. Courvoisier, J. Zhang, M. K. Bhuyan, M. Jacquot, and J. M. Dudley, “Applications of femtosecond Bessel beams to laser ablation,” App. Phys. A 112(1), 29–34 (2013).
[Crossref]

Zhao, M.

Zheng, J.

Zhou, J.

Zhou, W.

Zhu, L.

L. Zhu and J. Wang, “Arbitrary manipulation of spatial amplitude and phase using phase-only spatial light modulators,” Sci. Rep. 4, 7441 (2014).
[Crossref] [PubMed]

Zukauskas, A.

App. Phys. A (1)

F. Courvoisier, J. Zhang, M. K. Bhuyan, M. Jacquot, and J. M. Dudley, “Applications of femtosecond Bessel beams to laser ablation,” App. Phys. A 112(1), 29–34 (2013).
[Crossref]

Appl. Opt. (2)

Biomed. Opt. Express (1)

Contemp. Phys. (1)

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46(1), 15–28 (2005).
[Crossref]

J. Opt. Laser Technol. (1)

F. Courvoisier, R. Stoian, and A. Couairon, “Ultrafast laser micro and nano processing with nondiffracting and curved beams,” J. Opt. Laser Technol. 80, 125–137 (2016).
[Crossref]

J. Opt. Soc. Am. A (3)

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

New J. Phys. (1)

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005).
[Crossref]

Opt. Commun. (2)

T. A. Vieira, M. Zamboni-Rached, and M. R. R. Gesualdi, “Modeling the spatial shape of nondiffracting beams: Experimental generation of Frozen Waves via holographic method,” Opt. Commun. 315, 374–380 (2014).
[Crossref]

C. A. Dartora, K. Z. Nobrega, A. Dartora, G. A. Viana, and H. T. S. Filho, “A general theory for the frozen waves and their realization through finite apertures,” Opt. Commun. 265(2), 481–487 (2006).
[Crossref]

Opt. Express (8)

T. Čižmár and K. Dholakia, “Tunable Bessel light modes: engineering the axial propagation,” Opt. Express 17(18), 15558–15570 (2009).
[Crossref] [PubMed]

M. Zamboni-Rached, “Diffraction-Attenuation resistant beams in absorbing media,” Opt. Express 14(5), 1804–1809 (2006).
[Crossref] [PubMed]

A. Dubietis, P. Polesana, G. Valiulis, A. Stabinis, P. Di Trapani, and A. Piskarskas, “Axial emission and spectral broadening in self-focusing of femtosecond Bessel beams,” Opt. Express 15(7), 4168–4175 (2007).
[Crossref] [PubMed]

V. Arrizón, G. Méndez, and D. Sánchez-de-La-Llave, “Accurate encoding of arbitrary complex fields with amplitude-only liquid crystal spatial light modulators,” Opt. Express 13(20), 7913–7927 (2005).
[Crossref] [PubMed]

C. J. R. Sheppard and S. Mehta, “Three-level filter for increased depth of focus and Bessel beam generation,” Opt. Express 20(25), 27212–27221 (2012).
[Crossref] [PubMed]

V. Osipov, V. Pavelyev, D. Kachalov, A. Zukauskas, and B. Chichkov, “Realization of binary radial diffractive optical elements by two-photon polymerization technique,” Opt. Express 18(25), 25808–25814 (2010).
[Crossref] [PubMed]

M. Zamboni-Rached, “Stationary optical wave fields with arbitrary longitudinal shape by superposing equal frequency Bessel beams: Frozen Waves,” Opt. Express 12(17), 4001–4006 (2004).
[Crossref] [PubMed]

C. G. Durfee, J. Gemmer, and J. V. Moloney, “Phase-only shaping algorithm for Gaussian-apodized Bessel beams,” Opt. Express 21(13), 15777–15786 (2013).
[Crossref] [PubMed]

Opt. Lett. (6)

Phys. Rev. A (3)

M. Zamboni-Rached and M. Mojahedi, “Shaping finite-energy diffraction- and attenuation-resistant beams through Bessel-Gauss–beam superposition,” Phys. Rev. A 92(4), 043839 (2015).
[Crossref]

S. P. Tewari, H. Huang, and R. W. Boyd, “Theory of third-harmonic generation using Bessel beams, and self-phase-matching,” Phys. Rev. A 54(3), 2314–2325 (1996).
[Crossref] [PubMed]

P. Polesana, M. Franco, A. Couairon, D. Faccio, and P. Di Trapani, “Filamentation in Kerr media from pulsed Bessel beams,” Phys. Rev. A 77(4), 043814 (2008).
[Crossref]

Phys. Rev. Lett. (1)

M. A. Porras, A. Parola, D. Faccio, A. Dubietis, and P. D. Trapani, “Nonlinear unbalanced bessel beams: stationary conical waves supported by nonlinear losses,” Phys. Rev. Lett. 93(15), 153902 (2004).
[Crossref] [PubMed]

Proc. SPIE (1)

L. Li, N. Eckerskorn, R. A. Kirian, J. Küpper, D. P. DePonte, W. Krolikowski, W. M. Lee, H. N. Chapman, and A. V. Rode, “quasi-Bessel hollow beam as optical guide for micro-particles,” Proc. SPIE 8810, 88100N (2013).
[Crossref]

Sci. Rep. (1)

L. Zhu and J. Wang, “Arbitrary manipulation of spatial amplitude and phase using phase-only spatial light modulators,” Sci. Rep. 4, 7441 (2014).
[Crossref] [PubMed]

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Figures (5)

Fig. 1
Fig. 1 (a) Experimental setup. Ainc is the incident laser beam, U(m,n) is the beam amplitude reflected on the SLM, S(kr,z = 0) is the amplitude spatial spectrum and E(r,z = 0) is the reconstructed Bessel beam envelope (b) Example of the phase mask applied on the SLM; (c) Experimentally measured Bessel beam intensity distribution.
Fig. 2
Fig. 2 Numerical comparison between the target on-axis intensity distribution of a Bessel beam with flat on-axis intensity profile and the ones retrieved using (triangles) the approximation to Eq. (6) proposed in the Ref [31], and (circles) our implementation of the Eq. (6) to build the phase mask.
Fig. 3
Fig. 3 (a) Comparison of the spatial spectra for abrupt and smooth on-axis intensity variations showing the 2 truncations of the spectra at kz = k0 and kz = kzmin. The spectrum is expressed in the SLM space (cone angle reduced by the magnification factor f2/f1) (b) For a target with abrupt variations, the retrieved on-axis intensity shows significant deviations with the target. (c) For a target with smooth intensity variations, i.e. with typical variation length greater than ΔH = 54 µm for our simulation conditions, the retrieved on-axis intensity fits quasi-perfectly with the target field.
Fig. 4
Fig. 4 (a) image of the incident beam at the SLM plane, (b) phase mask corresponding to an incident plane wave, (c) phase mask corresponding to the incident laser beam.
Fig. 5
Fig. 5 numerical and experimental data of the intensity distribution along the propagation axis of Bessel beams with (left) uniformly growing and (right) uniform on-axis intensity profiles.

Equations (13)

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E( r,z=0 )= 1 2π 0 + S( k r ,z=0 ) J 0 ( k r r ) k r d k r
S( k 0 2 k z 2 ,z=0 )= 1 k z 0 + I( z ) exp[ i( k z0 k z )z ]dz
ψ( m,n )=M( m,n )mod[ F( m,n )+ ϕ ref ( m,n ),2π ]
U( m,n )= A inc ( m,n ).exp[ iψ( m,n ) ]
U 1 ( m,n )= A inc ( m,n )sinc( πM( m,n )π )exp[ i( F( m,n )+πM( m,n ) ) ]
{ M( m,n )=1+ sinc 1 ( A( m,n )/ A inc ( m,n ) ) π F( m,n )=ϕ( m,n )πM( m,n )
H( z )=( k 0 k z min )sinc( k 0 k z min 2 z ) e i ( k 0 + k z min 2 z )
I( z )={ I max z/ z 1 if z z 1 0 if z z 1
I( z )={ I max z/ z 1 if z z 1 I max ( 1 z z 1 z 2 z 1 ) 2 if z 1 z z 2 0 if z z 2
S( k 2 k r 2 ,z=0 )= I max k z [ 0 z 1 z z 4 e i( k z k z0 )z dz S 1 + z 1 z 2 ( 1 z z 4 z 5 z 4 ) e i( k z k z0 )z dz S 2 ]
S 1 = 1 z 1 { z 1 a [ sin( a z 1 )icos( a z 1 ) ] π/2 a 3/2 [ S( 2a z 1 /π )iC( 2a z 1 /π ) ] }
S 2 = z 2 z 2 z 1 e ia z 2 e ia z 1 a + e ia z 2 ( ia z 2 1 ) e ia z 1 ( ia z 1 1 ) ( z 2 z 1 ) a 2
S( x )= 0 x sin( π 2 t 2 )dt and C( x )= 0 x cos( π 2 t 2 )dt

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