Abstract

We experimentally demonstrated Bessel-like beams utilizing digital micromirror device (DMD). DMD with images imitating the equivalent axicon can shape the collimated Gaussian beam into Bessel beam. We reconstructed the 3D spatial field of the generated beam through a stack of measured cross-sectional images. The output beams have the profile of Bessel function after intensity modulation, and the beams extend at least 50 mm while the lateral dimension of the spot remains nearly invariant. Furthermore, the self-healing property has also been investigated, and all the experimental results agree well with simulated results numerically calculated through beam propagation method. Our observations demonstrate that the DMD offers a simple and efficient method to generate Bessel beams with distinct nondiffracting and self-reconstruction behaviors. The generated Bessel beams will potentially expand the applications to the optical manipulation and high-resolution fluorescence imaging owing to the unique nondiffracting property.

© 2013 Optical Society of America

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2013 (2)

2012 (14)

X. Chen, B. Yan, F. Song, Y. Wang, F. Xiao, and K. Alameh, “Diffraction of digital micromirror device gratings and its effect on properties of tunable fiber lasers,” Appl. Opt. 51, 7214–7220 (2012).
[CrossRef]

P. Zhang, Y. Hu, D. Cannan, A. Salandrino, T. Li, R. Morandotti, X. Zhang, and Z. Chen, “Generation of linear and nonlinear nonparaxial accelerating beams,” Opt. Lett. 37, 2820–2822 (2012).
[CrossRef]

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. 109, 193901 (2012).
[CrossRef]

F. O. Fahrbach and A. Rohrbach, “Propagation stability of self-reconstructing Bessel beams enables contrast-enhanced imaging in thick media,” Nat. Commun. 3, 632 (2012).
[CrossRef]

P. Zhu, O. Fajardo, J. Shum, Y. P. Z. Schärer, and R. W. Friedrich, “High-resolution optical control of spatiotemporal neuronal activity patterns in zebrafish using a digital micromirror device,” Nat. Protoc. 7, 1410–1425 (2012).
[CrossRef]

V. Lerner, D. Shwa, Y. Drori, and N. Katz, “Shaping Laguerre–Gaussian laser modes with binary gratings using a digital micromirror device,” Opt. Lett. 37, 4826–4828 (2012).
[CrossRef]

G. Sirinakis, Y. Ren, Y. Gao, Z. Xi, and Y. Zhang, “Combined versatile high-resolution optical tweezers and single-molecule fluorescence microscopy,” Rev. Sci. Instrum. 83, 093708 (2012).
[CrossRef]

S. K. Tiwari, S. R. Mishra, S. P. Ram, and H. S. Rawat, “Generation of a Bessel beam of variable spot size,” Appl. Opt. 51, 3718–3725 (2012).
[CrossRef]

J. Liang, S. Y. Wu, R. N. Kohn, M. F. Becker, and D. J. Heinzen, “Bandwidth-limited laser image projection using a DMD-based beam shaper,” Proc. SPIE 8254, 82540M (2012).
[CrossRef]

L. Gao, L. Shao, C. D. Higgins, J. S. Poulton, M. Peifer, M. W. Davidson, X. Wu, B. Goldstein, and E. Betzig, “Noninvasive imaging beyond the diffraction limit of 3D dynamics in thickly fluorescent specimens,” Cell 151, 1370–1385 (2012).
[CrossRef]

S. B. Purnapatra, S. Bera, and P. P. Mondal, “Spatial filter based Bessel-like beam for improved penetration depth imaging in fluorescence microscopy,” Sci. Rep. 2, 692 (2012).
[CrossRef]

D. B. Ruffner and D. G. Grier, “Optical conveyors: a class of active tractor beams,” Phys. Rev. Lett. 109, 163903 (2012).
[CrossRef]

H. A. Rendall, R. F. Marchington, B. B. Praveen, G. Bergmann, Y. Arita, A. Heisterkamp, F. J. Gunn-Moore, and K. Dholakia, “High-throughput optical injection of mammalian cells using a Bessel light beam,” Lab Chip 12, 4816–4820 (2012).
[CrossRef]

M. Duocastella and C. B. Arnold, “Bessel and annular beams for materials processing,” Laser Photon. Rev. 6, 607–621 (2012).
[CrossRef]

2011 (5)

M. J. Comstock, T. Ha, and Y. R. Chemla, “Ultrahigh-resolution optical trap with single-fluorophore sensitivity,” Nature Methods 8, 335–340 (2011).
[CrossRef]

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8, 417–423 (2011).
[CrossRef]

D. Brousseau, J. Drapeau, M. Piché, and E. F. Borra, “Generation of Bessel beams using a magnetic liquid deformable mirror,” Appl. Opt. 50, 4005–4010 (2011).
[CrossRef]

R. Bowman, N. Muller, X. Zambrana-Puyalto, O. Jedrkiewicz, P. di Trapani, and M. Padgett, “Efficient generation of Bessel beam arrays by means of an SLM,” Eur. Phys. J. Special Topics 199, 159–166 (2011).
[CrossRef]

H. Ryoo, D. W. Kang, and J. W. Hahn, “Analysis of the effective reflectance of digital micromirror devices and process parameters for maskless photolithography,” Microelectron. Eng. 88, 235–239 (2011).
[CrossRef]

2010 (2)

Y. X. Ren, M. Li, K. Huang, J. G. Wu, H. F. Gao, Z. Q. Wang, and Y. M. Li, “Experimental generation of Laguerre–Gaussian beam using digital micromirror device,” Appl. Opt. 49, 1838–1844 (2010).
[CrossRef]

F. Courvoisier, M. Jacquot, P. A. Lacourt, M. Bhuyan, L. Furfaro, R. Ferrière, and J. Dudley, “Generation of ultrafast Bessel micro-beams and applications to laser surface nanoprocessing,” Proc. SPIE 7728, 77281W (2010).

2009 (7)

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1, 1–57 (2009).
[CrossRef]

J. P. Rice, J. E. Neira, M. Kehoe, and R. Swanson, “DMD diffraction measurements to support design of projectors for test and evaluation of multispectral and hyperspectral imaging sensors,” Proc. SPIE 7210, 72100D (2009).

F. Courvoisier, P. A. Lacourt, M. Jacquot, M. Bhuyan, L. Furfaro, and J. Dudley, “Surface nanoprocessing with nondiffracting femtosecond Bessel beams,” Opt. Lett. 34, 3163–3165 (2009).
[CrossRef]

S. Akturk, C. L. Arnold, B. Prade, and A. Mysyrowicz, “Generation of high quality tunable Bessel beams using a liquid-immersion axicon,” Opt. Commun. 282, 3206–3209 (2009).
[CrossRef]

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

J. M. D. Kowalczyk, S. N. Smith, and E. B. Szarmes, “Generation of Bessel beams using a 4-f spatial filtering system,” Am. J. Phys. 77, 229–236 (2009).
[CrossRef]

X. F. Li, R. Winfield, S. O’Brien, and G. Crean, “Application of Bessel beams to 2D microfabrication,” Appl. Surf. Sci. 255, 5146–5149 (2009).
[CrossRef]

2008 (3)

P. Polynkin, M. Kolesik, A. Roberts, D. Faccio, P. di Trapani, and J. Moloney, “Generation of extended plasma channels in air using femtosecond Bessel beams,” Opt. Express 16, 15733–15740 (2008).
[CrossRef]

G. Milne, G. D. M. Jeffries, and D. T. Chiu, “Tunable generation of Bessel beams with a fluidic axicon,” Appl. Phys. Lett. 92, 261101 (2008).
[CrossRef]

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[CrossRef]

2007 (2)

A. V. Novitsky and D. V. Novitsky, “Negative propagation of vector Bessel beams,” J. Opt. Soc. Am. A 24, 2844–2849 (2007).
[CrossRef]

V. Kollarova, T. Medrik, R. Celechovsky, Z. Bouchal, O. Wilfert, and Z. Kolka, “Application of nondiffracting beams to wireless optical communications,” Proc. SPIE 6736, 67361C (2007).
[CrossRef]

2006 (3)

P. Polesana, A. Dubietis, M. Porras, E. Kučinskas, D. Faccio, A. Couairon, and P. Di Trapani, “Near-field dynamics of ultrashort pulsed Bessel beams in media with Kerr nonlinearity,” Phys. Rev. E 73, 056612 (2006).
[CrossRef]

T. Tsai, E. McLeod, and C. B. Arnold, “Generating Bessel beams with a tunable acoustic gradient index of refraction lens,” Proc. SPIE 6326, 63261F (2006).
[CrossRef]

A. Stockham and J. G. Smith, “Optical design for generating Bessel beams for micromanipulation,” Proc. SPIE 63211, 63261D (2006).
[CrossRef]

2005 (1)

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

2004 (2)

V. Garcés-Chávez, D. Roskey, M. Summers, H. Melville, D. McGloin, E. Wright, and K. Dholakia, “Optical levitation in a Bessel light beam,” Appl. Phys. Lett. 85, 4001–4003 (2004).
[CrossRef]

B. Paredes, A. Widera, V. Murg, O. Mandel, S. Fölling, I. Cirac, G. V. Shlyapnikov, T. W. Hänsch, and I. Bloch, “Tonks–Girardeau gas of ultracold atoms in an optical lattice,” Nature 429, 277–281 (2004).
[CrossRef]

2003 (3)

D. McGloin, V. Garcés-Chávez, and K. Dholakia, “Interfering Bessel beams for optical micromanipulation,” Opt. Lett. 28, 657–659 (2003).
[CrossRef]

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[CrossRef]

D. Dudley, W. Duncan, and J. Slaughter, “Emerging digital micromirror device (DMD) applications,” Proc. SPIE 4985, 14 (2003).
[CrossRef]

2002 (2)

J. H. Lee, S. J. Lee, C. S. Kyong, J. B. Song, Y. W. Lee, and C. H. Kwak, “Propagation characteristics of Bessel beam using phase type CGH,” Proc. SPIE 4924, 302–310 (2002).

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[CrossRef]

2001 (1)

J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197, 239–245 (2001).
[CrossRef]

1999 (1)

J. Arlt, K. Dholakia, L. Allen, and M. Padgett, “Efficiency of second-harmonic generation with Bessel beams,” Phys. Rev. A 60, 2438–2441 (1999).
[CrossRef]

1992 (1)

G. Scott and N. McArdle, “Efficient generation of nearly diffraction-free beams using an axicon,” Opt. Eng. 31, 2640–2643 (1992).
[CrossRef]

1991 (1)

1987 (2)

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

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

Akturk, S.

S. Akturk, C. L. Arnold, B. Prade, and A. Mysyrowicz, “Generation of high quality tunable Bessel beams using a liquid-immersion axicon,” Opt. Commun. 282, 3206–3209 (2009).
[CrossRef]

Alameh, K.

Allen, L.

J. Arlt, K. Dholakia, L. Allen, and M. Padgett, “Efficiency of second-harmonic generation with Bessel beams,” Phys. Rev. A 60, 2438–2441 (1999).
[CrossRef]

Arita, Y.

H. A. Rendall, R. F. Marchington, B. B. Praveen, G. Bergmann, Y. Arita, A. Heisterkamp, F. J. Gunn-Moore, and K. Dholakia, “High-throughput optical injection of mammalian cells using a Bessel light beam,” Lab Chip 12, 4816–4820 (2012).
[CrossRef]

Arlt, J.

J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197, 239–245 (2001).
[CrossRef]

J. Arlt, K. Dholakia, L. Allen, and M. Padgett, “Efficiency of second-harmonic generation with Bessel beams,” Phys. Rev. A 60, 2438–2441 (1999).
[CrossRef]

Arnold, C. B.

M. Duocastella and C. B. Arnold, “Bessel and annular beams for materials processing,” Laser Photon. Rev. 6, 607–621 (2012).
[CrossRef]

T. Tsai, E. McLeod, and C. B. Arnold, “Generating Bessel beams with a tunable acoustic gradient index of refraction lens,” Proc. SPIE 6326, 63261F (2006).
[CrossRef]

Arnold, C. L.

S. Akturk, C. L. Arnold, B. Prade, and A. Mysyrowicz, “Generation of high quality tunable Bessel beams using a liquid-immersion axicon,” Opt. Commun. 282, 3206–3209 (2009).
[CrossRef]

Becker, M. F.

J. Liang, S. Y. Wu, R. N. Kohn, M. F. Becker, and D. J. Heinzen, “Bandwidth-limited laser image projection using a DMD-based beam shaper,” Proc. SPIE 8254, 82540M (2012).
[CrossRef]

Bera, S.

S. B. Purnapatra, S. Bera, and P. P. Mondal, “Spatial filter based Bessel-like beam for improved penetration depth imaging in fluorescence microscopy,” Sci. Rep. 2, 692 (2012).
[CrossRef]

Bergmann, G.

H. A. Rendall, R. F. Marchington, B. B. Praveen, G. Bergmann, Y. Arita, A. Heisterkamp, F. J. Gunn-Moore, and K. Dholakia, “High-throughput optical injection of mammalian cells using a Bessel light beam,” Lab Chip 12, 4816–4820 (2012).
[CrossRef]

Betzig, E.

L. Gao, L. Shao, C. D. Higgins, J. S. Poulton, M. Peifer, M. W. Davidson, X. Wu, B. Goldstein, and E. Betzig, “Noninvasive imaging beyond the diffraction limit of 3D dynamics in thickly fluorescent specimens,” Cell 151, 1370–1385 (2012).
[CrossRef]

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8, 417–423 (2011).
[CrossRef]

Bhuyan, M.

F. Courvoisier, M. Jacquot, P. A. Lacourt, M. Bhuyan, L. Furfaro, R. Ferrière, and J. Dudley, “Generation of ultrafast Bessel micro-beams and applications to laser surface nanoprocessing,” Proc. SPIE 7728, 77281W (2010).

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V. Garcés-Chávez, D. Roskey, M. Summers, H. Melville, D. McGloin, E. Wright, and K. Dholakia, “Optical levitation in a Bessel light beam,” Appl. Phys. Lett. 85, 4001–4003 (2004).
[CrossRef]

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[CrossRef]

Miceli, J.

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

Milkie, D. E.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8, 417–423 (2011).
[CrossRef]

Milne, G.

G. Milne, G. D. M. Jeffries, and D. T. Chiu, “Tunable generation of Bessel beams with a fluidic axicon,” Appl. Phys. Lett. 92, 261101 (2008).
[CrossRef]

Mishra, S. R.

Moloney, J.

Mondal, P. P.

S. B. Purnapatra, S. Bera, and P. P. Mondal, “Spatial filter based Bessel-like beam for improved penetration depth imaging in fluorescence microscopy,” Sci. Rep. 2, 692 (2012).
[CrossRef]

Morandotti, R.

P. Zhang, Y. Hu, D. Cannan, A. Salandrino, T. Li, R. Morandotti, X. Zhang, and Z. Chen, “Generation of linear and nonlinear nonparaxial accelerating beams,” Opt. Lett. 37, 2820–2822 (2012).
[CrossRef]

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. 109, 193901 (2012).
[CrossRef]

Muller, N.

R. Bowman, N. Muller, X. Zambrana-Puyalto, O. Jedrkiewicz, P. di Trapani, and M. Padgett, “Efficient generation of Bessel beam arrays by means of an SLM,” Eur. Phys. J. Special Topics 199, 159–166 (2011).
[CrossRef]

Murg, V.

B. Paredes, A. Widera, V. Murg, O. Mandel, S. Fölling, I. Cirac, G. V. Shlyapnikov, T. W. Hänsch, and I. Bloch, “Tonks–Girardeau gas of ultracold atoms in an optical lattice,” Nature 429, 277–281 (2004).
[CrossRef]

Mysyrowicz, A.

S. Akturk, C. L. Arnold, B. Prade, and A. Mysyrowicz, “Generation of high quality tunable Bessel beams using a liquid-immersion axicon,” Opt. Commun. 282, 3206–3209 (2009).
[CrossRef]

Neira, J. E.

J. P. Rice, J. E. Neira, M. Kehoe, and R. Swanson, “DMD diffraction measurements to support design of projectors for test and evaluation of multispectral and hyperspectral imaging sensors,” Proc. SPIE 7210, 72100D (2009).

Novitsky, A. V.

Novitsky, D. V.

O’Brien, S.

X. F. Li, R. Winfield, S. O’Brien, and G. Crean, “Application of Bessel beams to 2D microfabrication,” Appl. Surf. Sci. 255, 5146–5149 (2009).
[CrossRef]

Overfelt, P. L.

Padgett, M.

R. Bowman, N. Muller, X. Zambrana-Puyalto, O. Jedrkiewicz, P. di Trapani, and M. Padgett, “Efficient generation of Bessel beam arrays by means of an SLM,” Eur. Phys. J. Special Topics 199, 159–166 (2011).
[CrossRef]

J. Arlt, K. Dholakia, L. Allen, and M. Padgett, “Efficiency of second-harmonic generation with Bessel beams,” Phys. Rev. A 60, 2438–2441 (1999).
[CrossRef]

Paredes, B.

B. Paredes, A. Widera, V. Murg, O. Mandel, S. Fölling, I. Cirac, G. V. Shlyapnikov, T. W. Hänsch, and I. Bloch, “Tonks–Girardeau gas of ultracold atoms in an optical lattice,” Nature 429, 277–281 (2004).
[CrossRef]

Peifer, M.

L. Gao, L. Shao, C. D. Higgins, J. S. Poulton, M. Peifer, M. W. Davidson, X. Wu, B. Goldstein, and E. Betzig, “Noninvasive imaging beyond the diffraction limit of 3D dynamics in thickly fluorescent specimens,” Cell 151, 1370–1385 (2012).
[CrossRef]

Piché, M.

Planchon, T. A.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8, 417–423 (2011).
[CrossRef]

Polesana, P.

P. Polesana, A. Dubietis, M. Porras, E. Kučinskas, D. Faccio, A. Couairon, and P. Di Trapani, “Near-field dynamics of ultrashort pulsed Bessel beams in media with Kerr nonlinearity,” Phys. Rev. E 73, 056612 (2006).
[CrossRef]

Polynkin, P.

Porras, M.

P. Polesana, A. Dubietis, M. Porras, E. Kučinskas, D. Faccio, A. Couairon, and P. Di Trapani, “Near-field dynamics of ultrashort pulsed Bessel beams in media with Kerr nonlinearity,” Phys. Rev. E 73, 056612 (2006).
[CrossRef]

Poulton, J. S.

L. Gao, L. Shao, C. D. Higgins, J. S. Poulton, M. Peifer, M. W. Davidson, X. Wu, B. Goldstein, and E. Betzig, “Noninvasive imaging beyond the diffraction limit of 3D dynamics in thickly fluorescent specimens,” Cell 151, 1370–1385 (2012).
[CrossRef]

Prade, B.

S. Akturk, C. L. Arnold, B. Prade, and A. Mysyrowicz, “Generation of high quality tunable Bessel beams using a liquid-immersion axicon,” Opt. Commun. 282, 3206–3209 (2009).
[CrossRef]

Praveen, B. B.

H. A. Rendall, R. F. Marchington, B. B. Praveen, G. Bergmann, Y. Arita, A. Heisterkamp, F. J. Gunn-Moore, and K. Dholakia, “High-throughput optical injection of mammalian cells using a Bessel light beam,” Lab Chip 12, 4816–4820 (2012).
[CrossRef]

Purnapatra, S. B.

S. B. Purnapatra, S. Bera, and P. P. Mondal, “Spatial filter based Bessel-like beam for improved penetration depth imaging in fluorescence microscopy,” Sci. Rep. 2, 692 (2012).
[CrossRef]

Ram, S. P.

Rawat, H. S.

Ren, Y.

G. Sirinakis, Y. Ren, Y. Gao, Z. Xi, and Y. Zhang, “Combined versatile high-resolution optical tweezers and single-molecule fluorescence microscopy,” Rev. Sci. Instrum. 83, 093708 (2012).
[CrossRef]

Ren, Y. X.

Rendall, H. A.

H. A. Rendall, R. F. Marchington, B. B. Praveen, G. Bergmann, Y. Arita, A. Heisterkamp, F. J. Gunn-Moore, and K. Dholakia, “High-throughput optical injection of mammalian cells using a Bessel light beam,” Lab Chip 12, 4816–4820 (2012).
[CrossRef]

Rice, J. P.

J. P. Rice, J. E. Neira, M. Kehoe, and R. Swanson, “DMD diffraction measurements to support design of projectors for test and evaluation of multispectral and hyperspectral imaging sensors,” Proc. SPIE 7210, 72100D (2009).

Roberts, A.

Rohrbach, A.

F. O. Fahrbach and A. Rohrbach, “Propagation stability of self-reconstructing Bessel beams enables contrast-enhanced imaging in thick media,” Nat. Commun. 3, 632 (2012).
[CrossRef]

Roskey, D.

V. Garcés-Chávez, D. Roskey, M. Summers, H. Melville, D. McGloin, E. Wright, and K. Dholakia, “Optical levitation in a Bessel light beam,” Appl. Phys. Lett. 85, 4001–4003 (2004).
[CrossRef]

Ruffner, D. B.

D. B. Ruffner and D. G. Grier, “Optical conveyors: a class of active tractor beams,” Phys. Rev. Lett. 109, 163903 (2012).
[CrossRef]

Ryoo, H.

H. Ryoo, D. W. Kang, and J. W. Hahn, “Analysis of the effective reflectance of digital micromirror devices and process parameters for maskless photolithography,” Microelectron. Eng. 88, 235–239 (2011).
[CrossRef]

Salandrino, A.

Schärer, Y. P. Z.

P. Zhu, O. Fajardo, J. Shum, Y. P. Z. Schärer, and R. W. Friedrich, “High-resolution optical control of spatiotemporal neuronal activity patterns in zebrafish using a digital micromirror device,” Nat. Protoc. 7, 1410–1425 (2012).
[CrossRef]

Scott, G.

G. Scott and N. McArdle, “Efficient generation of nearly diffraction-free beams using an axicon,” Opt. Eng. 31, 2640–2643 (1992).
[CrossRef]

Shalaby, M. Y.

Shao, L.

L. Gao, L. Shao, C. D. Higgins, J. S. Poulton, M. Peifer, M. W. Davidson, X. Wu, B. Goldstein, and E. Betzig, “Noninvasive imaging beyond the diffraction limit of 3D dynamics in thickly fluorescent specimens,” Cell 151, 1370–1385 (2012).
[CrossRef]

Sheppard, C.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[CrossRef]

Shi, L.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[CrossRef]

Shlyapnikov, G. V.

B. Paredes, A. Widera, V. Murg, O. Mandel, S. Fölling, I. Cirac, G. V. Shlyapnikov, T. W. Hänsch, and I. Bloch, “Tonks–Girardeau gas of ultracold atoms in an optical lattice,” Nature 429, 277–281 (2004).
[CrossRef]

Shum, J.

P. Zhu, O. Fajardo, J. Shum, Y. P. Z. Schärer, and R. W. Friedrich, “High-resolution optical control of spatiotemporal neuronal activity patterns in zebrafish using a digital micromirror device,” Nat. Protoc. 7, 1410–1425 (2012).
[CrossRef]

Shwa, D.

Sibbett, W.

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[CrossRef]

J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197, 239–245 (2001).
[CrossRef]

Sirinakis, G.

G. Sirinakis, Y. Ren, Y. Gao, Z. Xi, and Y. Zhang, “Combined versatile high-resolution optical tweezers and single-molecule fluorescence microscopy,” Rev. Sci. Instrum. 83, 093708 (2012).
[CrossRef]

Slaughter, J.

D. Dudley, W. Duncan, and J. Slaughter, “Emerging digital micromirror device (DMD) applications,” Proc. SPIE 4985, 14 (2003).
[CrossRef]

Smith, J. G.

A. Stockham and J. G. Smith, “Optical design for generating Bessel beams for micromanipulation,” Proc. SPIE 63211, 63261D (2006).
[CrossRef]

Smith, S. N.

J. M. D. Kowalczyk, S. N. Smith, and E. B. Szarmes, “Generation of Bessel beams using a 4-f spatial filtering system,” Am. J. Phys. 77, 229–236 (2009).
[CrossRef]

Song, F.

Song, J. B.

J. H. Lee, S. J. Lee, C. S. Kyong, J. B. Song, Y. W. Lee, and C. H. Kwak, “Propagation characteristics of Bessel beam using phase type CGH,” Proc. SPIE 4924, 302–310 (2002).

Stockham, A.

A. Stockham and J. G. Smith, “Optical design for generating Bessel beams for micromanipulation,” Proc. SPIE 63211, 63261D (2006).
[CrossRef]

Summers, M.

V. Garcés-Chávez, D. Roskey, M. Summers, H. Melville, D. McGloin, E. Wright, and K. Dholakia, “Optical levitation in a Bessel light beam,” Appl. Phys. Lett. 85, 4001–4003 (2004).
[CrossRef]

Swanson, R.

J. P. Rice, J. E. Neira, M. Kehoe, and R. Swanson, “DMD diffraction measurements to support design of projectors for test and evaluation of multispectral and hyperspectral imaging sensors,” Proc. SPIE 7210, 72100D (2009).

Szarmes, E. B.

J. M. D. Kowalczyk, S. N. Smith, and E. B. Szarmes, “Generation of Bessel beams using a 4-f spatial filtering system,” Am. J. Phys. 77, 229–236 (2009).
[CrossRef]

Tiwari, S. K.

Tsai, T.

T. Tsai, E. McLeod, and C. B. Arnold, “Generating Bessel beams with a tunable acoustic gradient index of refraction lens,” Proc. SPIE 6326, 63261F (2006).
[CrossRef]

Wang, H.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[CrossRef]

Wang, Y.

Wang, Z. Q.

Widera, A.

B. Paredes, A. Widera, V. Murg, O. Mandel, S. Fölling, I. Cirac, G. V. Shlyapnikov, T. W. Hänsch, and I. Bloch, “Tonks–Girardeau gas of ultracold atoms in an optical lattice,” Nature 429, 277–281 (2004).
[CrossRef]

Wilfert, O.

V. Kollarova, T. Medrik, R. Celechovsky, Z. Bouchal, O. Wilfert, and Z. Kolka, “Application of nondiffracting beams to wireless optical communications,” Proc. SPIE 6736, 67361C (2007).
[CrossRef]

Winfield, R.

X. F. Li, R. Winfield, S. O’Brien, and G. Crean, “Application of Bessel beams to 2D microfabrication,” Appl. Surf. Sci. 255, 5146–5149 (2009).
[CrossRef]

Wright, E.

V. Garcés-Chávez, D. Roskey, M. Summers, H. Melville, D. McGloin, E. Wright, and K. Dholakia, “Optical levitation in a Bessel light beam,” Appl. Phys. Lett. 85, 4001–4003 (2004).
[CrossRef]

Wu, J. G.

Wu, S. Y.

J. Liang, S. Y. Wu, R. N. Kohn, M. F. Becker, and D. J. Heinzen, “Bandwidth-limited laser image projection using a DMD-based beam shaper,” Proc. SPIE 8254, 82540M (2012).
[CrossRef]

Wu, X.

L. Gao, L. Shao, C. D. Higgins, J. S. Poulton, M. Peifer, M. W. Davidson, X. Wu, B. Goldstein, and E. Betzig, “Noninvasive imaging beyond the diffraction limit of 3D dynamics in thickly fluorescent specimens,” Cell 151, 1370–1385 (2012).
[CrossRef]

Xi, Z.

G. Sirinakis, Y. Ren, Y. Gao, Z. Xi, and Y. Zhang, “Combined versatile high-resolution optical tweezers and single-molecule fluorescence microscopy,” Rev. Sci. Instrum. 83, 093708 (2012).
[CrossRef]

Xiao, F.

Yan, B.

Yin, X.

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. 109, 193901 (2012).
[CrossRef]

Younse, J. M.

J. M. Younse, “Projection display systems based on the digital micromirror device (DMD),” in Micromachining and Microfabrication (International Society for Optics and Photonics, 1995), pp. 64–75.

Zambrana-Puyalto, X.

R. Bowman, N. Muller, X. Zambrana-Puyalto, O. Jedrkiewicz, P. di Trapani, and M. Padgett, “Efficient generation of Bessel beam arrays by means of an SLM,” Eur. Phys. J. Special Topics 199, 159–166 (2011).
[CrossRef]

Zhan, Q.

Zhang, P.

Zhang, X.

P. Zhang, Y. Hu, D. Cannan, A. Salandrino, T. Li, R. Morandotti, X. Zhang, and Z. Chen, “Generation of linear and nonlinear nonparaxial accelerating beams,” Opt. Lett. 37, 2820–2822 (2012).
[CrossRef]

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. 109, 193901 (2012).
[CrossRef]

Zhang, Y.

G. Sirinakis, Y. Ren, Y. Gao, Z. Xi, and Y. Zhang, “Combined versatile high-resolution optical tweezers and single-molecule fluorescence microscopy,” Rev. Sci. Instrum. 83, 093708 (2012).
[CrossRef]

Zhao, J.

Zhu, P.

P. Zhu, O. Fajardo, J. Shum, Y. P. Z. Schärer, and R. W. Friedrich, “High-resolution optical control of spatiotemporal neuronal activity patterns in zebrafish using a digital micromirror device,” Nat. Protoc. 7, 1410–1425 (2012).
[CrossRef]

Adv. Opt. Photon. (1)

Am. J. Phys. (1)

J. M. D. Kowalczyk, S. N. Smith, and E. B. Szarmes, “Generation of Bessel beams using a 4-f spatial filtering system,” Am. J. Phys. 77, 229–236 (2009).
[CrossRef]

Appl. Opt. (5)

Appl. Phys. Lett. (2)

G. Milne, G. D. M. Jeffries, and D. T. Chiu, “Tunable generation of Bessel beams with a fluidic axicon,” Appl. Phys. Lett. 92, 261101 (2008).
[CrossRef]

V. Garcés-Chávez, D. Roskey, M. Summers, H. Melville, D. McGloin, E. Wright, and K. Dholakia, “Optical levitation in a Bessel light beam,” Appl. Phys. Lett. 85, 4001–4003 (2004).
[CrossRef]

Appl. Surf. Sci. (1)

X. F. Li, R. Winfield, S. O’Brien, and G. Crean, “Application of Bessel beams to 2D microfabrication,” Appl. Surf. Sci. 255, 5146–5149 (2009).
[CrossRef]

Cell (1)

L. Gao, L. Shao, C. D. Higgins, J. S. Poulton, M. Peifer, M. W. Davidson, X. Wu, B. Goldstein, and E. Betzig, “Noninvasive imaging beyond the diffraction limit of 3D dynamics in thickly fluorescent specimens,” Cell 151, 1370–1385 (2012).
[CrossRef]

Cont. Phys. (1)

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

Eur. Phys. J. Special Topics (1)

R. Bowman, N. Muller, X. Zambrana-Puyalto, O. Jedrkiewicz, P. di Trapani, and M. Padgett, “Efficient generation of Bessel beam arrays by means of an SLM,” Eur. Phys. J. Special Topics 199, 159–166 (2011).
[CrossRef]

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

Lab Chip (1)

H. A. Rendall, R. F. Marchington, B. B. Praveen, G. Bergmann, Y. Arita, A. Heisterkamp, F. J. Gunn-Moore, and K. Dholakia, “High-throughput optical injection of mammalian cells using a Bessel light beam,” Lab Chip 12, 4816–4820 (2012).
[CrossRef]

Laser Photon. Rev. (1)

M. Duocastella and C. B. Arnold, “Bessel and annular beams for materials processing,” Laser Photon. Rev. 6, 607–621 (2012).
[CrossRef]

Microelectron. Eng. (1)

H. Ryoo, D. W. Kang, and J. W. Hahn, “Analysis of the effective reflectance of digital micromirror devices and process parameters for maskless photolithography,” Microelectron. Eng. 88, 235–239 (2011).
[CrossRef]

Nat. Commun. (1)

F. O. Fahrbach and A. Rohrbach, “Propagation stability of self-reconstructing Bessel beams enables contrast-enhanced imaging in thick media,” Nat. Commun. 3, 632 (2012).
[CrossRef]

Nat. Methods (1)

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8, 417–423 (2011).
[CrossRef]

Nat. Photonics (1)

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[CrossRef]

Nat. Protoc. (1)

P. Zhu, O. Fajardo, J. Shum, Y. P. Z. Schärer, and R. W. Friedrich, “High-resolution optical control of spatiotemporal neuronal activity patterns in zebrafish using a digital micromirror device,” Nat. Protoc. 7, 1410–1425 (2012).
[CrossRef]

Nature (3)

B. Paredes, A. Widera, V. Murg, O. Mandel, S. Fölling, I. Cirac, G. V. Shlyapnikov, T. W. Hänsch, and I. Bloch, “Tonks–Girardeau gas of ultracold atoms in an optical lattice,” Nature 429, 277–281 (2004).
[CrossRef]

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[CrossRef]

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[CrossRef]

Nature Methods (1)

M. J. Comstock, T. Ha, and Y. R. Chemla, “Ultrahigh-resolution optical trap with single-fluorophore sensitivity,” Nature Methods 8, 335–340 (2011).
[CrossRef]

Opt. Commun. (2)

S. Akturk, C. L. Arnold, B. Prade, and A. Mysyrowicz, “Generation of high quality tunable Bessel beams using a liquid-immersion axicon,” Opt. Commun. 282, 3206–3209 (2009).
[CrossRef]

J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197, 239–245 (2001).
[CrossRef]

Opt. Eng. (1)

G. Scott and N. McArdle, “Efficient generation of nearly diffraction-free beams using an axicon,” Opt. Eng. 31, 2640–2643 (1992).
[CrossRef]

Opt. Express (1)

Opt. Lett. (5)

Optics Express (1)

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

Phys. Rev. A (1)

J. Arlt, K. Dholakia, L. Allen, and M. Padgett, “Efficiency of second-harmonic generation with Bessel beams,” Phys. Rev. A 60, 2438–2441 (1999).
[CrossRef]

Phys. Rev. E (1)

P. Polesana, A. Dubietis, M. Porras, E. Kučinskas, D. Faccio, A. Couairon, and P. Di Trapani, “Near-field dynamics of ultrashort pulsed Bessel beams in media with Kerr nonlinearity,” Phys. Rev. E 73, 056612 (2006).
[CrossRef]

Phys. Rev. Lett. (3)

D. B. Ruffner and D. G. Grier, “Optical conveyors: a class of active tractor beams,” Phys. Rev. Lett. 109, 163903 (2012).
[CrossRef]

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

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. 109, 193901 (2012).
[CrossRef]

Proc. SPIE (8)

J. P. Rice, J. E. Neira, M. Kehoe, and R. Swanson, “DMD diffraction measurements to support design of projectors for test and evaluation of multispectral and hyperspectral imaging sensors,” Proc. SPIE 7210, 72100D (2009).

A. Stockham and J. G. Smith, “Optical design for generating Bessel beams for micromanipulation,” Proc. SPIE 63211, 63261D (2006).
[CrossRef]

D. Dudley, W. Duncan, and J. Slaughter, “Emerging digital micromirror device (DMD) applications,” Proc. SPIE 4985, 14 (2003).
[CrossRef]

F. Courvoisier, M. Jacquot, P. A. Lacourt, M. Bhuyan, L. Furfaro, R. Ferrière, and J. Dudley, “Generation of ultrafast Bessel micro-beams and applications to laser surface nanoprocessing,” Proc. SPIE 7728, 77281W (2010).

J. Liang, S. Y. Wu, R. N. Kohn, M. F. Becker, and D. J. Heinzen, “Bandwidth-limited laser image projection using a DMD-based beam shaper,” Proc. SPIE 8254, 82540M (2012).
[CrossRef]

V. Kollarova, T. Medrik, R. Celechovsky, Z. Bouchal, O. Wilfert, and Z. Kolka, “Application of nondiffracting beams to wireless optical communications,” Proc. SPIE 6736, 67361C (2007).
[CrossRef]

T. Tsai, E. McLeod, and C. B. Arnold, “Generating Bessel beams with a tunable acoustic gradient index of refraction lens,” Proc. SPIE 6326, 63261F (2006).
[CrossRef]

J. H. Lee, S. J. Lee, C. S. Kyong, J. B. Song, Y. W. Lee, and C. H. Kwak, “Propagation characteristics of Bessel beam using phase type CGH,” Proc. SPIE 4924, 302–310 (2002).

Rev. Sci. Instrum. (1)

G. Sirinakis, Y. Ren, Y. Gao, Z. Xi, and Y. Zhang, “Combined versatile high-resolution optical tweezers and single-molecule fluorescence microscopy,” Rev. Sci. Instrum. 83, 093708 (2012).
[CrossRef]

Sci. Rep. (1)

S. B. Purnapatra, S. Bera, and P. P. Mondal, “Spatial filter based Bessel-like beam for improved penetration depth imaging in fluorescence microscopy,” Sci. Rep. 2, 692 (2012).
[CrossRef]

Other (1)

J. M. Younse, “Projection display systems based on the digital micromirror device (DMD),” in Micromachining and Microfabrication (International Society for Optics and Photonics, 1995), pp. 64–75.

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Figures (5)

Fig. 1.
Fig. 1.

Experiment layout and gamma curve correction of DMD. (a) Schematic of the experimental setup. The laser utilized is a He–Ne laser with wavelength 632.8 nm. A telescope (f1=30mm, f2=150mm) is employed to expand the beam to 10 mm in diameter. In order to improve the beam quality, a pinhole P1 with diameter 500 μm is adopted to filter the laser beam. The expanded and collimated laser beam is then steered to uniformly illuminate the surface of DMD with precise incident angle of 24° through adjustment of mirrors M1, M2. The DMD modulates the laser profile through projection of gray scale holograms and the incident Gaussian beam was transformed to Bessel beam through intensity modulation exerted by DMD. Then convex lens L3 (f=250mm) is employed to collect the modulated light. A pinhole P2 placed near the back focal plane of the lens selectively passes the first diffraction order of beam. The optical intensity profile is digitalized and recorded by a CCD camera (MINTRON MTV-1881EX 795×596) mounted on a guide rail for multiplane image collection. When performing the energy measurements, the camera is replaced by a power meter (Thorlabs, PM100D). (b) Image sequence with gray value from 0 to 255 used to test the intensity response property of DMD. The gray scale changes gradually from black to white when increasing the gray level. (c) Experimental results of the measured gamma response: uncorrected gamma curve (red hollow dots) and corrected gamma curve (blue solid dots). (d) The corrected image used to generate the Bessel beam. The desired beam is formed behind the mirrors irradiated by an expanded Gaussian beam as long as the corrected image is projected onto the DMD.

Fig. 2.
Fig. 2.

Representative cross-sectional image of generated Bessel beam at a distance of z=30mm from the back focal plane of the convex lens (L3, Fig. 1) behind the DMD. Accompanying graphs are radial intensity profiles (blue line) fitted to zero-order Bessel function squared (red line) at x (bottom) and y (right) axis, respectively. The measured intensity distribution is in accordance with theoretical predictions.

Fig. 3.
Fig. 3.

Simulation and experimental results for nondiffraction behavior of Bessel beam. (a) Simulation results of optical intensity profile at xz plane. A free-space BPM is adopted to calculate the spatial optical intensity distribution. (b) Experimental results of optical intensity profile at xz plane, which demonstrates the intensity evolution of the generated beam along with propagation. (c) The transverse intensity profile (blue) of the generated beam at 25 mm (Line 2) and its fitting curve (red). The FWHM (2.252/kρ) obtained from the fitting result is 60 μm, and the range of nondiffraction can be derived from measurement to be about 50 mm. (d) The normalized on-axis (Line 4) intensity versus propagation distance of the generated Bessel beam (blue curve) and the simulation on-axis (Line 3) intensity (red curve) corresponding to Line 3. (e) 3D plots of the optical distribution of a generated Bessel beam through analysis of the cross-sectional images. The cross-sectional images are captured at an interval of 1 mm by translating the CCD camera from back focal plane of the projection lens along the beam path.

Fig. 4.
Fig. 4.

Self-healing results of the generated Bessel beam. (a) Simulated intensity distribution of the obstructed beam at xz plane. The sizes of the Bessel beam and the circular opaque obstacle are about 700 and 400 μm, respectively. (b) Experimental intensity profile of the Bessel beam at xz plane in presence of an obstacle. An ink dot fixed on a glass slide with diameter of about 400 microns is used as an opaque obstacle placed directly in the beam path to block the central portion of the beam. (c) The simulated (red curve) and measured (blue curve) transverse intensity profile of the obstructed beam just behind the obstacle in the focal plane (1 and 1′). The central portion is obstructed, which can be seen from the plot. (d) The measured (blue) and simulated (red) transverse intensity profiles of the obstructed beam at 14 mm (2 and 2′) behind the obstacle. (e) The transverse intensity profile (blue) of the obstructed beam at 33 mm behind the obstacle, accompanying with the simulating (red) and theoretical (purple) curves (3 and 3′).

Fig. 5.
Fig. 5.

Schematic diagram of DMD modeled as a 2D diffraction grating. Coordinate system (x,y,z) with the origin at the center of the rectangular DMD is established for calculation of diffraction field. (2M+1)×(2N+1) square mirrors with mirror spacing d0 is considered to form the device, and each mirror is encoded by the location of its center. Each mirror addressed individually can tilt either +12 deg (“on”) or 12 deg (“off”) along the main-diagonal (magenta dashed line). With all mirrors on, mirrors along the lines parallel to the main-diagonal are taken as a one-dimensional plane mirror array, while those along the lines parallel to the side-diagonal (blue dashed line) are modeled as a blazed grating with constant of 2d0/2 and blazed angle of φ=12°, depicted at the bottom of the figure. The red lines stand for incident ray and diffraction ray, and the diffraction angle is θ. Top right graph demonstrates a local coordinate system (ξ,η,ς) introduced in the m-n-th mirror, which can be transferred into a spherical coordinate (r,θ,ϕ).

Equations (16)

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(21c22t2)ψ(ρ,ϕ,z,t)=0.
ψ(ρ,ϕ,z,t)=k2πiρdρdϕf(ρ)ei(kRωt)R,
R=z2+ρ2+ρ22ρρcos(ϕϕ)z+ρ2+ρ22ρρcos(ϕϕ)2z.
ψn(ρ,ϕ,z)=Aexp(i(kzzωt))Jn(kρρ)exp(inϕ).
E(ρ,ϕ,z)=Aexp(ikzz)J0(kρρ),
I(ρ,ϕ,z>0)=I(ρ,ϕ,z=0)=12|ψ(ρ,ϕ,z,t)|2|J0(kρρ)|2.
2ψpx2+2ψpy2+2ψpz2+k2ψp=0,
d2Ψpdz2+k2(1kx2k2ky2k2)Ψp=0,
Ψp(kx,ky,z)=Ψp0(kx,ky)exp(ikz1kx2k2ky2k2),
I(ρ)=AJ02(kρ(ρρ0))+B,
θ2.405λ2πr0,
I=A02d04sin2[(2M)u]sin2usin2[(2N)u]sin2u×sinc2(ν)·sinc2(ν),
u=πd0λ(sinθcosϕsinθ0cosϕ0),
u=πd0λ(sinθsinϕsinθ0sinϕ0),
ν=πd0λ[(sinθcosϕsinθ0cosϕ0)+tanφ2(cosθcosθ0)],
ν=πd0λ[(sinθsinϕsinθ0sinϕ0)+tanφ2(cosθcosθ0)].

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