Abstract

We utilize advanced laser fields to clear a path through a dynamic turbid medium, a concept termed “Optical path clearing (OPC).” Particles are evacuated from a volume of the medium using the gradient and/or scattering forces due to an applied laser field with a suitably tailored spatial profile. Our studies encompass both an analytical model and proof-of-principle experiments where paths are cleared in dense bulk colloidal suspensions. Based on our results we suggest that high-performance and high efficiency OPC will be achieved by multiple-step clearing using dynamic laser fields based on Airy or inverted axicon beams.

© 2010 Optical Society of America

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    [CrossRef]
  2. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11, 288–290 (1986).
    [CrossRef] [PubMed]
  3. A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared-laser beams,” Nature 330, 769–771 (1987).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  5. A. Ashkin, “Optical trapping and manipulation of neutral particles using lasers,” Proc. Natl. Acad. Sci. U.S.A. 94, 4853–4860 (1997).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  7. J. R. Moffit, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Annu. Rev. Biochem. 77, 205–228 (2008).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  10. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
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  13. J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip 9, 1334–1336 (2009).
    [CrossRef] [PubMed]
  14. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
    [CrossRef]
  15. G. A. Siviloglou, and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32, 979–981 (2007).
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  21. J. Baumgartl, G. M. Hannappel, D. J. Stevenson, M. Mazilu, D. Day, M. Gu, and K. Dholakia, “Optical “snowblowing” of microparticles and cells in a microfluidic environment using Airy and parabolic wavepackets,” in “Optical Trapping and Optical Micromanipulation VI,”, vol. 7400 of Proc. SPIE, K. Dholakia and G. C. Spalding, eds. (2009), vol. 7400 of Proc. SPIE, p. 74001R.
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    [CrossRef] [PubMed]

2009 (3)

2008 (4)

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2, 675–678 (2008).
[CrossRef]

K. Dholakia, P. J. Reece, and M. Gu, “Optical micromanipulation,” Chem. Soc. Rev. 37, 42–55 (2008).
[CrossRef] [PubMed]

J. R. Moffit, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Annu. Rev. Biochem. 77, 205–228 (2008).
[CrossRef]

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16, 12880–12891 (2008).
[CrossRef] [PubMed]

2007 (2)

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

G. A. Siviloglou, and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32, 979–981 (2007).
[CrossRef] [PubMed]

2005 (1)

L. Paterson, E. Papagiakoumou, G. Milne, V. Garcés-Chávez, S. A. Tatarkova, W. Sibbett, F. J. Gunn-Moore, P. E. Bryant, A. Riches, and K. Dholakia, “Light-induced cell separation in a tailored optical landscape,” Appl. Phys. Lett. 87, 123901 (2005).
[CrossRef]

2003 (1)

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426, 421–424 (2003).
[CrossRef] [PubMed]

2000 (1)

L. Allen, and M. J. Padgett, “The Poynting vector in Laguerre-Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. 84, 67–71 (2000).
[CrossRef]

1997 (1)

A. Ashkin, “Optical trapping and manipulation of neutral particles using lasers,” Proc. Natl. Acad. Sci. U.S.A. 94, 4853–4860 (1997).
[CrossRef] [PubMed]

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

1990 (1)

1987 (2)

A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared-laser beams,” Nature 330, 769–771 (1987).
[CrossRef] [PubMed]

A. Ashkin, and J. M. Dziedzic, “Optical trapping and manipulation of single cells using infrared-laser beams,” Science 235, 1517–1520 (1987).
[CrossRef] [PubMed]

1986 (1)

1979 (1)

M. V. Berry, and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
[CrossRef]

1970 (1)

A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24, 156–159 (1970).
[CrossRef]

Allen, L.

L. Allen, and M. J. Padgett, “The Poynting vector in Laguerre-Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. 84, 67–71 (2000).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Ashkin, A.

A. Ashkin, “Optical trapping and manipulation of neutral particles using lasers,” Proc. Natl. Acad. Sci. U.S.A. 94, 4853–4860 (1997).
[CrossRef] [PubMed]

A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared-laser beams,” Nature 330, 769–771 (1987).
[CrossRef] [PubMed]

A. Ashkin, and J. M. Dziedzic, “Optical trapping and manipulation of single cells using infrared-laser beams,” Science 235, 1517–1520 (1987).
[CrossRef] [PubMed]

A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11, 288–290 (1986).
[CrossRef] [PubMed]

A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24, 156–159 (1970).
[CrossRef]

Balazs, N. L.

M. V. Berry, and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
[CrossRef]

Bandres, M. A.

Baumgartl, J.

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip 9, 1334–1336 (2009).
[CrossRef] [PubMed]

J. E. Morris, M. Mazilu, J. Baumgartl, T. Čižmár, and K. Dholakia, “Propagation characteristics of Airy beams: dependence upon spatial coherence and wavelength,” Opt. Express 17, 13236–13245 (2009).
[CrossRef] [PubMed]

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2, 675–678 (2008).
[CrossRef]

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Berry, M. V.

M. V. Berry, and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
[CrossRef]

Bjorkholm, J. E.

Broky, J.

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16, 12880–12891 (2008).
[CrossRef] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

Bryant, P. E.

L. Paterson, E. Papagiakoumou, G. Milne, V. Garcés-Chávez, S. A. Tatarkova, W. Sibbett, F. J. Gunn-Moore, P. E. Bryant, A. Riches, and K. Dholakia, “Light-induced cell separation in a tailored optical landscape,” Appl. Phys. Lett. 87, 123901 (2005).
[CrossRef]

Bustamante, C.

J. R. Moffit, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Annu. Rev. Biochem. 77, 205–228 (2008).
[CrossRef]

Chemla, Y. R.

J. R. Moffit, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Annu. Rev. Biochem. 77, 205–228 (2008).
[CrossRef]

Christodoulides, D. N.

Chu, S.

Cižmár, T.

Day, D.

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip 9, 1334–1336 (2009).
[CrossRef] [PubMed]

Dholakia, K.

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip 9, 1334–1336 (2009).
[CrossRef] [PubMed]

J. E. Morris, M. Mazilu, J. Baumgartl, T. Čižmár, and K. Dholakia, “Propagation characteristics of Airy beams: dependence upon spatial coherence and wavelength,” Opt. Express 17, 13236–13245 (2009).
[CrossRef] [PubMed]

K. Dholakia, P. J. Reece, and M. Gu, “Optical micromanipulation,” Chem. Soc. Rev. 37, 42–55 (2008).
[CrossRef] [PubMed]

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2, 675–678 (2008).
[CrossRef]

L. Paterson, E. Papagiakoumou, G. Milne, V. Garcés-Chávez, S. A. Tatarkova, W. Sibbett, F. J. Gunn-Moore, P. E. Bryant, A. Riches, and K. Dholakia, “Light-induced cell separation in a tailored optical landscape,” Appl. Phys. Lett. 87, 123901 (2005).
[CrossRef]

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426, 421–424 (2003).
[CrossRef] [PubMed]

Dogariu, A.

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16, 12880–12891 (2008).
[CrossRef] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

Dziedzic, J. M.

A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared-laser beams,” Nature 330, 769–771 (1987).
[CrossRef] [PubMed]

A. Ashkin, and J. M. Dziedzic, “Optical trapping and manipulation of single cells using infrared-laser beams,” Science 235, 1517–1520 (1987).
[CrossRef] [PubMed]

A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11, 288–290 (1986).
[CrossRef] [PubMed]

Garcés-Chávez, V.

L. Paterson, E. Papagiakoumou, G. Milne, V. Garcés-Chávez, S. A. Tatarkova, W. Sibbett, F. J. Gunn-Moore, P. E. Bryant, A. Riches, and K. Dholakia, “Light-induced cell separation in a tailored optical landscape,” Appl. Phys. Lett. 87, 123901 (2005).
[CrossRef]

Gu, M.

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip 9, 1334–1336 (2009).
[CrossRef] [PubMed]

K. Dholakia, P. J. Reece, and M. Gu, “Optical micromanipulation,” Chem. Soc. Rev. 37, 42–55 (2008).
[CrossRef] [PubMed]

Guizar-Sicairos, M.

Gunn-Moore, F. J.

L. Paterson, E. Papagiakoumou, G. Milne, V. Garcés-Chávez, S. A. Tatarkova, W. Sibbett, F. J. Gunn-Moore, P. E. Bryant, A. Riches, and K. Dholakia, “Light-induced cell separation in a tailored optical landscape,” Appl. Phys. Lett. 87, 123901 (2005).
[CrossRef]

Hannappel, G. M.

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip 9, 1334–1336 (2009).
[CrossRef] [PubMed]

MacDonald, M. P.

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426, 421–424 (2003).
[CrossRef] [PubMed]

Mazilu, M.

Milne, G.

L. Paterson, E. Papagiakoumou, G. Milne, V. Garcés-Chávez, S. A. Tatarkova, W. Sibbett, F. J. Gunn-Moore, P. E. Bryant, A. Riches, and K. Dholakia, “Light-induced cell separation in a tailored optical landscape,” Appl. Phys. Lett. 87, 123901 (2005).
[CrossRef]

Moffit, J. R.

J. R. Moffit, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Annu. Rev. Biochem. 77, 205–228 (2008).
[CrossRef]

Morris, J. E.

Nemoto, S.

Padgett, M. J.

L. Allen, and M. J. Padgett, “The Poynting vector in Laguerre-Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. 84, 67–71 (2000).
[CrossRef]

Papagiakoumou, E.

L. Paterson, E. Papagiakoumou, G. Milne, V. Garcés-Chávez, S. A. Tatarkova, W. Sibbett, F. J. Gunn-Moore, P. E. Bryant, A. Riches, and K. Dholakia, “Light-induced cell separation in a tailored optical landscape,” Appl. Phys. Lett. 87, 123901 (2005).
[CrossRef]

Paterson, L.

L. Paterson, E. Papagiakoumou, G. Milne, V. Garcés-Chávez, S. A. Tatarkova, W. Sibbett, F. J. Gunn-Moore, P. E. Bryant, A. Riches, and K. Dholakia, “Light-induced cell separation in a tailored optical landscape,” Appl. Phys. Lett. 87, 123901 (2005).
[CrossRef]

Reece, P. J.

K. Dholakia, P. J. Reece, and M. Gu, “Optical micromanipulation,” Chem. Soc. Rev. 37, 42–55 (2008).
[CrossRef] [PubMed]

Riches, A.

L. Paterson, E. Papagiakoumou, G. Milne, V. Garcés-Chávez, S. A. Tatarkova, W. Sibbett, F. J. Gunn-Moore, P. E. Bryant, A. Riches, and K. Dholakia, “Light-induced cell separation in a tailored optical landscape,” Appl. Phys. Lett. 87, 123901 (2005).
[CrossRef]

Sibbett, W.

L. Paterson, E. Papagiakoumou, G. Milne, V. Garcés-Chávez, S. A. Tatarkova, W. Sibbett, F. J. Gunn-Moore, P. E. Bryant, A. Riches, and K. Dholakia, “Light-induced cell separation in a tailored optical landscape,” Appl. Phys. Lett. 87, 123901 (2005).
[CrossRef]

Siviloglou, G. A.

Smith, S. B.

J. R. Moffit, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Annu. Rev. Biochem. 77, 205–228 (2008).
[CrossRef]

Spalding, G. C.

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426, 421–424 (2003).
[CrossRef] [PubMed]

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Stevenson, D. J.

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip 9, 1334–1336 (2009).
[CrossRef] [PubMed]

Tatarkova, S. A.

L. Paterson, E. Papagiakoumou, G. Milne, V. Garcés-Chávez, S. A. Tatarkova, W. Sibbett, F. J. Gunn-Moore, P. E. Bryant, A. Riches, and K. Dholakia, “Light-induced cell separation in a tailored optical landscape,” Appl. Phys. Lett. 87, 123901 (2005).
[CrossRef]

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Yamane, T.

A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared-laser beams,” Nature 330, 769–771 (1987).
[CrossRef] [PubMed]

Am. J. Phys. (1)

M. V. Berry, and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
[CrossRef]

Annu. Rev. Biochem. (1)

J. R. Moffit, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Annu. Rev. Biochem. 77, 205–228 (2008).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

L. Paterson, E. Papagiakoumou, G. Milne, V. Garcés-Chávez, S. A. Tatarkova, W. Sibbett, F. J. Gunn-Moore, P. E. Bryant, A. Riches, and K. Dholakia, “Light-induced cell separation in a tailored optical landscape,” Appl. Phys. Lett. 87, 123901 (2005).
[CrossRef]

Chem. Soc. Rev. (1)

K. Dholakia, P. J. Reece, and M. Gu, “Optical micromanipulation,” Chem. Soc. Rev. 37, 42–55 (2008).
[CrossRef] [PubMed]

Lab Chip (1)

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip 9, 1334–1336 (2009).
[CrossRef] [PubMed]

Nat. Photonics (1)

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2, 675–678 (2008).
[CrossRef]

Nature (2)

A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared-laser beams,” Nature 330, 769–771 (1987).
[CrossRef] [PubMed]

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426, 421–424 (2003).
[CrossRef] [PubMed]

Opt. Commun. (1)

L. Allen, and M. J. Padgett, “The Poynting vector in Laguerre-Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. 84, 67–71 (2000).
[CrossRef]

Opt. Express (2)

Opt. Lett. (3)

Phys. Rev. A (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Phys. Rev. Lett. (2)

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24, 156–159 (1970).
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Figures (5)

Fig. 1.
Fig. 1.

OPC using a single-ringed LG beam of azimuthal index l = 1. (a) Transverse LG beam intensity profile in the absence of particles. The dashed black circle indicates the region of maximum intensity and is also included in Figs. (b)–(d). (b) Transverse particle position distribution after 100 time steps. (c) Probe beam intensity profile before LG beam was switched on. (d) Probe beam intensity profile 100 time steps after LG beam has been switched on.

Fig. 2.
Fig. 2.

Multiple Airy beam clearing. (a) Transverse beam intensity profile of an Airy beam. (b)–(d) Micrograph of colloidal sample (b) before, (c) shortly after and (d) ≈ 10 min after the clearing beams were switched on. The two squares indicate the initial position and size of the Airy beams, the lower on being oriented as shown in Fig. (a) and the upper one being flipped by 180°. The two arrows indicate the clockwise relocation of the two beams. Moreover the distance between the beams has been persistently increased throughout the course of the experiment yielding “hollowing” of the sample as shown in Fig. (d).

Fig. 3.
Fig. 3.

Clearing beam created with an inverted axicon. (a) Phase mask corresponding to an inverted axicon. (b) Schematic beam propagation.

Fig. 4.
Fig. 4.

OPC in a dense bulk aqueous colloidal suspension using an inverted axicon beam. The beam propagates perpendicularly to the image plane. Particles are propelled and gathered on the beam’s bright ring as revealed by the dark ring-shaped shade occurring in the right-hand side figure.

Fig. 5.
Fig. 5.

Multiple Airy beams arranged on a triangular pattern. The red circle highlights the inner ring of beams which clear the central region from particles. The red arrows indicate the direction of transverse deflection during propagation of the respective beams.

Tables (1)

Tables Icon

Table 1. Rating of clearing beams. “MA”= Multiple Airy, “IA”= Inverted axicon, “−B”=Beam. “+”= good, “〇”= satisfactory, “−”= not satisfactory. The item “Analytical description” refers to the question whether the relevant beam propagation can be described on the basis of the paraxial approximation.

Equations (16)

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2 u ( x , y , z ) x 2 + 2 u ( x , y , z ) y 2 2 ik u ( x , y , z ) z = 0 ,
u 0 ( x , y , z + z 0 ) = ik 2 π z d x d y
× exp ( ik x 2 + y 2 2 ( x x + y y ) 2 + x 2 + y 2 2 z ) u 0 ( x , y , z 0 ) ,
( A 1 B 1 C 1 D 1 ) .
u 1 ( x , y , z ) = exp ( ik ( L 1 ( z 1 z 1 ) ) ) A 1 + C 1 ( z z 1 ) exp ( ik C 1 ( ( x x 1 ) 2 + ( y y 1 ) 2 ) 2 ( A 1 + C 1 ( z z 1 ) ) )
× u 0 ( x x 1 A 1 + C 1 ( z z 1 ) + x 1 , y y 1 A 1 + C 1 ( z z 1 ) + y 1 , B 1 + D 1 ( z z 1 ) A 1 + C 1 ( z z 1 ) + z 1 )
u ( x , y , z ) = a 0 iz r z + iz r ( x + iy z + iz r ) l exp ( ik ( x 2 + y 2 ) 2 ( z + iz r ) )
u ( x , y , z ) = Π ξ = x , y [ Ai ( ξ s 0 z 2 4 k 2 s 0 4 + i az k s 0 2 )
× exp ( a ξ s 0 az 2 2 k 2 s 0 4 i z 3 12 k 3 s 0 6 + i a 2 z 2 k s 0 2 + i z ξ 2 k s 0 3 ) ]
( 1 0 i ( k ω a 2 ) 1 ) ,
( 2 n 0 n 1 n 1 2 R n 1 2 n 0 ( n 0 n 1 ) Rn 1 2 n 0 n 1 n 1 ) ,
T = ε 0 n 0 2 E E * + μ 0 H H * 1 2 ( ε 0 n 0 2 E E * + μ 0 H H * ) I .
T = ( 0 0 ikn 0 2 ε 0 u u * x 0 0 ikn 0 2 ε 0 u u * y ikn 0 2 ε 0 u * u x ikn 0 2 ε 0 u * u y ε 0 n 0 2 k 2 u 2 ) ,
F = 1 2 ( S Tn d σ ) = kn 0 2 ε 0 2 dxdy ( u * u x ) ( u * u y ) k ( u 2 ) z 1 z 2 ,
u = Σ i = 1 N u i
F = Σ i = 1 N Σ j > i N F ij ,

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