Abstract

We present the use of optical fibers to form a counter-propagating optical trap as a means of manipulating both solid and liquid aerosols. We explore the use of single and multimode fibers to achieve trapping of various particles in air, present the trapping properties of the different fiber types and compare the observed trends to those predicted by theory. Using fibers, we are able to hold suspended particles for extended periods of time and to precisely manipulate them over distances of several hundred microns. We discuss the difficulties and advantages of each fiber configuration and conclude with a demonstration that fiber based trapping offers a good candidate for studying optical binding in air.

©2008 Optical Society of America

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  1. A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24, 156–159 (1970).
    [Crossref]
  2. K. Dholakia, P Reece, and M Gu, “Optical Micromanipulation” Chem. Soc. Rev. 37, 42–55 (2008).
    [Crossref] [PubMed]
  3. M. J. Lang and S. M Block, “Resource Letter: LBOT-1: Laser-based optical tweezers,” Am. J. Phys. 71, 201–215 (2003).
    [Crossref]
  4. K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787–2809 (2004).
    [Crossref]
  5. D. McGloin, “Optical tweezers: 20 years on,” Phil. Trans. R. Soc. 364, 3521–3537 (2006).
    [Crossref]
  6. R. D. Leonardo, G. Ruocco, L. Leach, M. J. Padgett, A. J. Wright, J. M. Girkin, D. R. Burnham, and D. McGloin, “Parametric Resonance of Optically Trapped Aerosols,” Phys. Rev. Lett. 99, 010601 (2007).
    [Crossref] [PubMed]
  7. M. D. King, K. C. Thompson, and A. D. Ward, “Laser Tweezers Raman Study of Optically Trapped Aerosol Droplets of Seawater and Oleic Acid Reacting with Ozone: Implications for Cloud-Droplet Properties,” J. Am. Chem. Soc. 126, 16710–16711 (2004).
    [Crossref] [PubMed]
  8. J. Buajarern, L. Mitchem, and J. P. Reid, “Characterising the formation of organic layers on the surface of inorganic/aqueous aerosols by Raman spectroscopy,” J. Phys. Chem. A. 11111852–11859 (2007).
    [Crossref] [PubMed]
  9. L. Mitchem and J. P. Reid, “Optical Manipulation and Characterisation of Aerosol Particles,” Chem. Soc. Rev. 37, 756–769 (2008).
    [Crossref] [PubMed]
  10. A. Ashkin, J. M. Dziedzic, J. E. Bjorkhom, and S. Chu, “Observation of a single beam gradient force trap for dielectric particles,” Opt. Lett. 11, 288–290 (1986).
    [Crossref] [PubMed]
  11. G. Roosen and C. Imbert, “Optical levitation by means of 2 horizontal laser beams—Theoretical and experimental study,” Phys. Lett. A. 59, 6–8 (1976).
    [Crossref]
  12. M. D. Summers, D. R. Burnham, and D. McGloin, “Trapping solid aerosols with optical tweezers: A comparison between gas and liquid phase optical traps,” Opt. Express 16, 7739–7747 (2008).
    [Crossref] [PubMed]
  13. N. Magome, M. I. Kohira, E. Hayata, S. Mukai, and K. Yoshikawa, “Optical Trapping of a Growing Water Droplet in Air,” J. Phys. Chem. B. 107, 3988–3990 (2003).
    [Crossref]
  14. R. J. Hopkins, L. Mitchem, A. D. Ward, and J. P. Reid, “Control and characterisation of a single aerosol droplet in a single-beam gradient-force optical trap,” Phys. Chem. Chem. Phys. 6, 4924–4927 (2004).
    [Crossref]
  15. D. R. Burnham and D. McGloin, “Holographic optical trapping of aerosol droplets,” Opt. Express. 14, 4175–4181 (2006).
    [Crossref] [PubMed]
  16. J. Buajarern, L. Mitchem, A. D. Ward, N. H. Nahler, D. McGloin, and J. P. Reid, “Controlling and Characterising the Coagulation of Liquid Aerosol Droplets,” J. Chem. Phys. 125, 114506 (2006).
    [Crossref] [PubMed]
  17. R. Pastel, A. Struthers, R. Ringle, J. Rodgers, C. Rohde, and P. Geiser, “Laser trapping of microscopic particles for undergraduate experiments,” Am. J. Phys. 68, 993–1001 (2000).
    [Crossref]
  18. K. Taji, M. Tachikawa, and K. Nagashima, “Laser trapping of ice crystals,” Appl. Phys. Lett. 88, 141111 (2006).
    [Crossref]
  19. M. Guillon, O. Moine, and B. Stout, “Longitudinal Optical Binding of High Optical Contrast Microdroplets in Air,” Phys. Rev. Lett. 96, 143902 (2006). Erratum, Phys. Rev. Lett. 99, 079901 (2007).
    [Crossref] [PubMed]
  20. A. Constable, J. Kim, J. Mervis, F. Zarinetchi, and M. Prentiss, “Demonstration of a-fiber-optical light-force trap,” Opt. Lett. 18, 1867–1869 (1993).
    [Crossref] [PubMed]
  21. J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The Optical Stretcher: A Novel Laser Tool to Micromanipulate Cells,” Biophys. J. 81, 767–784 (2001).
    [Crossref] [PubMed]
  22. P. R. T. Jess, V. Garcés-Chávez, D. Smith, M. Mazilu, L. Paterson, A. Riches, C. S. Herrington, W. Sibbett, and K. Dholakia, “Dual beam fiber trap for Raman micro-spectroscopy of single cells,” Opt. Express. 14, 5779–5791 (2006).
    [Crossref] [PubMed]
  23. W. Singer, M. Frick, S. Bernet, and M. Ritsch-Marte, “Self-organized array of regularly spaced microbeads in a fiber-optical trap,” J. Opt. Soc. Am. B 20, 1568–1574 (2001).
    [Crossref]
  24. N. K. Metzger, K. Dholakia, and E. M. Wright, “Observation of bistability and hysteresis in optical binding of two dielectric spheres,” Phys. Rev. Lett. 96, 068102 (2006).
    [Crossref] [PubMed]
  25. E. Sidick, S. D. Collins, and A. Knoesen, “Trapping forces in a multiple-beam fiber-optic trap,” Appl. Opt. 36, 6423–6433 (1997).
    [Crossref]
  26. S. Ebert, K. Travis, B. Lincoln, and J. Guck “Fluorescence ratio thermometry in a microfluidic dual-beam laser trap,” Opt. Express 15, 15493–15499 (2007).
    [Crossref] [PubMed]
  27. R. Gussgard and T. Lindmo “Calculation of the trapping force in a strongly focused laser beam,” J. Opt. Soc. Am. B. 9, 1922–1930 (1992).
    [Crossref]
  28. J. H. Dennis, C. A. Pieron, and K. Asai, “Aerosol Output and Size from Omron NE-U22 nebulizer,” in Proceedings of the 14th International Congress International Society for Aerosols in Medicines, Baltimore June 14–18 2003. Journal of Aerosol Medicine 16:2213, (2003)
  29. ButlerJ. R.MitchemL.HanfordK. L. Treuel L.Reid J. P. “In situ comparative measurements of the properties of aerosol droplets of different chemical composition,” Faraday Discuss 137, 351–366 (2008).
    [Crossref] [PubMed]
  30. M. M. Burns, J.-M. Fournier, and J. A. Golovchenko, “Optical Matter: Crystallization and Binding in Intense Optical Fields,” Science 249, 749–754 (1990).
  31. S. A. Tatarkova, A. E. Carruthers, and K. Dholakia, “One-Dimensional Optically Bound Arrays of Microscopic Particles,” Phys. Rev. Lett. 89, 283901 (2002).
    [Crossref]

2008 (4)

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

L. Mitchem and J. P. Reid, “Optical Manipulation and Characterisation of Aerosol Particles,” Chem. Soc. Rev. 37, 756–769 (2008).
[Crossref] [PubMed]

ButlerJ. R.MitchemL.HanfordK. L. Treuel L.Reid J. P. “In situ comparative measurements of the properties of aerosol droplets of different chemical composition,” Faraday Discuss 137, 351–366 (2008).
[Crossref] [PubMed]

M. D. Summers, D. R. Burnham, and D. McGloin, “Trapping solid aerosols with optical tweezers: A comparison between gas and liquid phase optical traps,” Opt. Express 16, 7739–7747 (2008).
[Crossref] [PubMed]

2007 (3)

S. Ebert, K. Travis, B. Lincoln, and J. Guck “Fluorescence ratio thermometry in a microfluidic dual-beam laser trap,” Opt. Express 15, 15493–15499 (2007).
[Crossref] [PubMed]

R. D. Leonardo, G. Ruocco, L. Leach, M. J. Padgett, A. J. Wright, J. M. Girkin, D. R. Burnham, and D. McGloin, “Parametric Resonance of Optically Trapped Aerosols,” Phys. Rev. Lett. 99, 010601 (2007).
[Crossref] [PubMed]

J. Buajarern, L. Mitchem, and J. P. Reid, “Characterising the formation of organic layers on the surface of inorganic/aqueous aerosols by Raman spectroscopy,” J. Phys. Chem. A. 11111852–11859 (2007).
[Crossref] [PubMed]

2006 (7)

D. McGloin, “Optical tweezers: 20 years on,” Phil. Trans. R. Soc. 364, 3521–3537 (2006).
[Crossref]

D. R. Burnham and D. McGloin, “Holographic optical trapping of aerosol droplets,” Opt. Express. 14, 4175–4181 (2006).
[Crossref] [PubMed]

J. Buajarern, L. Mitchem, A. D. Ward, N. H. Nahler, D. McGloin, and J. P. Reid, “Controlling and Characterising the Coagulation of Liquid Aerosol Droplets,” J. Chem. Phys. 125, 114506 (2006).
[Crossref] [PubMed]

K. Taji, M. Tachikawa, and K. Nagashima, “Laser trapping of ice crystals,” Appl. Phys. Lett. 88, 141111 (2006).
[Crossref]

M. Guillon, O. Moine, and B. Stout, “Longitudinal Optical Binding of High Optical Contrast Microdroplets in Air,” Phys. Rev. Lett. 96, 143902 (2006). Erratum, Phys. Rev. Lett. 99, 079901 (2007).
[Crossref] [PubMed]

P. R. T. Jess, V. Garcés-Chávez, D. Smith, M. Mazilu, L. Paterson, A. Riches, C. S. Herrington, W. Sibbett, and K. Dholakia, “Dual beam fiber trap for Raman micro-spectroscopy of single cells,” Opt. Express. 14, 5779–5791 (2006).
[Crossref] [PubMed]

N. K. Metzger, K. Dholakia, and E. M. Wright, “Observation of bistability and hysteresis in optical binding of two dielectric spheres,” Phys. Rev. Lett. 96, 068102 (2006).
[Crossref] [PubMed]

2004 (3)

R. J. Hopkins, L. Mitchem, A. D. Ward, and J. P. Reid, “Control and characterisation of a single aerosol droplet in a single-beam gradient-force optical trap,” Phys. Chem. Chem. Phys. 6, 4924–4927 (2004).
[Crossref]

M. D. King, K. C. Thompson, and A. D. Ward, “Laser Tweezers Raman Study of Optically Trapped Aerosol Droplets of Seawater and Oleic Acid Reacting with Ozone: Implications for Cloud-Droplet Properties,” J. Am. Chem. Soc. 126, 16710–16711 (2004).
[Crossref] [PubMed]

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787–2809 (2004).
[Crossref]

2003 (3)

M. J. Lang and S. M Block, “Resource Letter: LBOT-1: Laser-based optical tweezers,” Am. J. Phys. 71, 201–215 (2003).
[Crossref]

N. Magome, M. I. Kohira, E. Hayata, S. Mukai, and K. Yoshikawa, “Optical Trapping of a Growing Water Droplet in Air,” J. Phys. Chem. B. 107, 3988–3990 (2003).
[Crossref]

J. H. Dennis, C. A. Pieron, and K. Asai, “Aerosol Output and Size from Omron NE-U22 nebulizer,” in Proceedings of the 14th International Congress International Society for Aerosols in Medicines, Baltimore June 14–18 2003. Journal of Aerosol Medicine 16:2213, (2003)

2002 (1)

S. A. Tatarkova, A. E. Carruthers, and K. Dholakia, “One-Dimensional Optically Bound Arrays of Microscopic Particles,” Phys. Rev. Lett. 89, 283901 (2002).
[Crossref]

2001 (2)

W. Singer, M. Frick, S. Bernet, and M. Ritsch-Marte, “Self-organized array of regularly spaced microbeads in a fiber-optical trap,” J. Opt. Soc. Am. B 20, 1568–1574 (2001).
[Crossref]

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The Optical Stretcher: A Novel Laser Tool to Micromanipulate Cells,” Biophys. J. 81, 767–784 (2001).
[Crossref] [PubMed]

2000 (1)

R. Pastel, A. Struthers, R. Ringle, J. Rodgers, C. Rohde, and P. Geiser, “Laser trapping of microscopic particles for undergraduate experiments,” Am. J. Phys. 68, 993–1001 (2000).
[Crossref]

1997 (1)

1993 (1)

1992 (1)

R. Gussgard and T. Lindmo “Calculation of the trapping force in a strongly focused laser beam,” J. Opt. Soc. Am. B. 9, 1922–1930 (1992).
[Crossref]

1986 (1)

1976 (1)

G. Roosen and C. Imbert, “Optical levitation by means of 2 horizontal laser beams—Theoretical and experimental study,” Phys. Lett. A. 59, 6–8 (1976).
[Crossref]

1970 (1)

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

Ananthakrishnan, R.

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The Optical Stretcher: A Novel Laser Tool to Micromanipulate Cells,” Biophys. J. 81, 767–784 (2001).
[Crossref] [PubMed]

Asai, K.

J. H. Dennis, C. A. Pieron, and K. Asai, “Aerosol Output and Size from Omron NE-U22 nebulizer,” in Proceedings of the 14th International Congress International Society for Aerosols in Medicines, Baltimore June 14–18 2003. Journal of Aerosol Medicine 16:2213, (2003)

Ashkin, A.

Bernet, S.

Bjorkhom, J. E.

Block, S. M

M. J. Lang and S. M Block, “Resource Letter: LBOT-1: Laser-based optical tweezers,” Am. J. Phys. 71, 201–215 (2003).
[Crossref]

Block, S. M.

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787–2809 (2004).
[Crossref]

Buajarern, J.

J. Buajarern, L. Mitchem, and J. P. Reid, “Characterising the formation of organic layers on the surface of inorganic/aqueous aerosols by Raman spectroscopy,” J. Phys. Chem. A. 11111852–11859 (2007).
[Crossref] [PubMed]

J. Buajarern, L. Mitchem, A. D. Ward, N. H. Nahler, D. McGloin, and J. P. Reid, “Controlling and Characterising the Coagulation of Liquid Aerosol Droplets,” J. Chem. Phys. 125, 114506 (2006).
[Crossref] [PubMed]

Burnham, D. R.

M. D. Summers, D. R. Burnham, and D. McGloin, “Trapping solid aerosols with optical tweezers: A comparison between gas and liquid phase optical traps,” Opt. Express 16, 7739–7747 (2008).
[Crossref] [PubMed]

R. D. Leonardo, G. Ruocco, L. Leach, M. J. Padgett, A. J. Wright, J. M. Girkin, D. R. Burnham, and D. McGloin, “Parametric Resonance of Optically Trapped Aerosols,” Phys. Rev. Lett. 99, 010601 (2007).
[Crossref] [PubMed]

D. R. Burnham and D. McGloin, “Holographic optical trapping of aerosol droplets,” Opt. Express. 14, 4175–4181 (2006).
[Crossref] [PubMed]

Burns, M. M.

M. M. Burns, J.-M. Fournier, and J. A. Golovchenko, “Optical Matter: Crystallization and Binding in Intense Optical Fields,” Science 249, 749–754 (1990).

Carruthers, A. E.

S. A. Tatarkova, A. E. Carruthers, and K. Dholakia, “One-Dimensional Optically Bound Arrays of Microscopic Particles,” Phys. Rev. Lett. 89, 283901 (2002).
[Crossref]

Chu, S.

Collins, S. D.

Constable, A.

Cunningham, C. C.

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The Optical Stretcher: A Novel Laser Tool to Micromanipulate Cells,” Biophys. J. 81, 767–784 (2001).
[Crossref] [PubMed]

Dennis, J. H.

J. H. Dennis, C. A. Pieron, and K. Asai, “Aerosol Output and Size from Omron NE-U22 nebulizer,” in Proceedings of the 14th International Congress International Society for Aerosols in Medicines, Baltimore June 14–18 2003. Journal of Aerosol Medicine 16:2213, (2003)

Dholakia, K.

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

N. K. Metzger, K. Dholakia, and E. M. Wright, “Observation of bistability and hysteresis in optical binding of two dielectric spheres,” Phys. Rev. Lett. 96, 068102 (2006).
[Crossref] [PubMed]

P. R. T. Jess, V. Garcés-Chávez, D. Smith, M. Mazilu, L. Paterson, A. Riches, C. S. Herrington, W. Sibbett, and K. Dholakia, “Dual beam fiber trap for Raman micro-spectroscopy of single cells,” Opt. Express. 14, 5779–5791 (2006).
[Crossref] [PubMed]

S. A. Tatarkova, A. E. Carruthers, and K. Dholakia, “One-Dimensional Optically Bound Arrays of Microscopic Particles,” Phys. Rev. Lett. 89, 283901 (2002).
[Crossref]

Dziedzic, J. M.

Ebert, S.

Fournier, J.-M.

M. M. Burns, J.-M. Fournier, and J. A. Golovchenko, “Optical Matter: Crystallization and Binding in Intense Optical Fields,” Science 249, 749–754 (1990).

Frick, M.

Garcés-Chávez, V.

P. R. T. Jess, V. Garcés-Chávez, D. Smith, M. Mazilu, L. Paterson, A. Riches, C. S. Herrington, W. Sibbett, and K. Dholakia, “Dual beam fiber trap for Raman micro-spectroscopy of single cells,” Opt. Express. 14, 5779–5791 (2006).
[Crossref] [PubMed]

Geiser, P.

R. Pastel, A. Struthers, R. Ringle, J. Rodgers, C. Rohde, and P. Geiser, “Laser trapping of microscopic particles for undergraduate experiments,” Am. J. Phys. 68, 993–1001 (2000).
[Crossref]

Girkin, J. M.

R. D. Leonardo, G. Ruocco, L. Leach, M. J. Padgett, A. J. Wright, J. M. Girkin, D. R. Burnham, and D. McGloin, “Parametric Resonance of Optically Trapped Aerosols,” Phys. Rev. Lett. 99, 010601 (2007).
[Crossref] [PubMed]

Golovchenko, J. A.

M. M. Burns, J.-M. Fournier, and J. A. Golovchenko, “Optical Matter: Crystallization and Binding in Intense Optical Fields,” Science 249, 749–754 (1990).

Gu, M

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

Guck, J.

S. Ebert, K. Travis, B. Lincoln, and J. Guck “Fluorescence ratio thermometry in a microfluidic dual-beam laser trap,” Opt. Express 15, 15493–15499 (2007).
[Crossref] [PubMed]

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The Optical Stretcher: A Novel Laser Tool to Micromanipulate Cells,” Biophys. J. 81, 767–784 (2001).
[Crossref] [PubMed]

Guillon, M.

M. Guillon, O. Moine, and B. Stout, “Longitudinal Optical Binding of High Optical Contrast Microdroplets in Air,” Phys. Rev. Lett. 96, 143902 (2006). Erratum, Phys. Rev. Lett. 99, 079901 (2007).
[Crossref] [PubMed]

Gussgard, R.

R. Gussgard and T. Lindmo “Calculation of the trapping force in a strongly focused laser beam,” J. Opt. Soc. Am. B. 9, 1922–1930 (1992).
[Crossref]

Hayata, E.

N. Magome, M. I. Kohira, E. Hayata, S. Mukai, and K. Yoshikawa, “Optical Trapping of a Growing Water Droplet in Air,” J. Phys. Chem. B. 107, 3988–3990 (2003).
[Crossref]

Herrington, C. S.

P. R. T. Jess, V. Garcés-Chávez, D. Smith, M. Mazilu, L. Paterson, A. Riches, C. S. Herrington, W. Sibbett, and K. Dholakia, “Dual beam fiber trap for Raman micro-spectroscopy of single cells,” Opt. Express. 14, 5779–5791 (2006).
[Crossref] [PubMed]

Hopkins, R. J.

R. J. Hopkins, L. Mitchem, A. D. Ward, and J. P. Reid, “Control and characterisation of a single aerosol droplet in a single-beam gradient-force optical trap,” Phys. Chem. Chem. Phys. 6, 4924–4927 (2004).
[Crossref]

Imbert, C.

G. Roosen and C. Imbert, “Optical levitation by means of 2 horizontal laser beams—Theoretical and experimental study,” Phys. Lett. A. 59, 6–8 (1976).
[Crossref]

Jess, P. R. T.

P. R. T. Jess, V. Garcés-Chávez, D. Smith, M. Mazilu, L. Paterson, A. Riches, C. S. Herrington, W. Sibbett, and K. Dholakia, “Dual beam fiber trap for Raman micro-spectroscopy of single cells,” Opt. Express. 14, 5779–5791 (2006).
[Crossref] [PubMed]

Käs, J.

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The Optical Stretcher: A Novel Laser Tool to Micromanipulate Cells,” Biophys. J. 81, 767–784 (2001).
[Crossref] [PubMed]

Kim, J.

King, M. D.

M. D. King, K. C. Thompson, and A. D. Ward, “Laser Tweezers Raman Study of Optically Trapped Aerosol Droplets of Seawater and Oleic Acid Reacting with Ozone: Implications for Cloud-Droplet Properties,” J. Am. Chem. Soc. 126, 16710–16711 (2004).
[Crossref] [PubMed]

Knoesen, A.

Kohira, M. I.

N. Magome, M. I. Kohira, E. Hayata, S. Mukai, and K. Yoshikawa, “Optical Trapping of a Growing Water Droplet in Air,” J. Phys. Chem. B. 107, 3988–3990 (2003).
[Crossref]

Lang, M. J.

M. J. Lang and S. M Block, “Resource Letter: LBOT-1: Laser-based optical tweezers,” Am. J. Phys. 71, 201–215 (2003).
[Crossref]

Leach, L.

R. D. Leonardo, G. Ruocco, L. Leach, M. J. Padgett, A. J. Wright, J. M. Girkin, D. R. Burnham, and D. McGloin, “Parametric Resonance of Optically Trapped Aerosols,” Phys. Rev. Lett. 99, 010601 (2007).
[Crossref] [PubMed]

Leonardo, R. D.

R. D. Leonardo, G. Ruocco, L. Leach, M. J. Padgett, A. J. Wright, J. M. Girkin, D. R. Burnham, and D. McGloin, “Parametric Resonance of Optically Trapped Aerosols,” Phys. Rev. Lett. 99, 010601 (2007).
[Crossref] [PubMed]

Lincoln, B.

Lindmo, T.

R. Gussgard and T. Lindmo “Calculation of the trapping force in a strongly focused laser beam,” J. Opt. Soc. Am. B. 9, 1922–1930 (1992).
[Crossref]

Magome, N.

N. Magome, M. I. Kohira, E. Hayata, S. Mukai, and K. Yoshikawa, “Optical Trapping of a Growing Water Droplet in Air,” J. Phys. Chem. B. 107, 3988–3990 (2003).
[Crossref]

Mahmood, H.

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The Optical Stretcher: A Novel Laser Tool to Micromanipulate Cells,” Biophys. J. 81, 767–784 (2001).
[Crossref] [PubMed]

Mazilu, M.

P. R. T. Jess, V. Garcés-Chávez, D. Smith, M. Mazilu, L. Paterson, A. Riches, C. S. Herrington, W. Sibbett, and K. Dholakia, “Dual beam fiber trap for Raman micro-spectroscopy of single cells,” Opt. Express. 14, 5779–5791 (2006).
[Crossref] [PubMed]

McGloin, D.

M. D. Summers, D. R. Burnham, and D. McGloin, “Trapping solid aerosols with optical tweezers: A comparison between gas and liquid phase optical traps,” Opt. Express 16, 7739–7747 (2008).
[Crossref] [PubMed]

R. D. Leonardo, G. Ruocco, L. Leach, M. J. Padgett, A. J. Wright, J. M. Girkin, D. R. Burnham, and D. McGloin, “Parametric Resonance of Optically Trapped Aerosols,” Phys. Rev. Lett. 99, 010601 (2007).
[Crossref] [PubMed]

D. McGloin, “Optical tweezers: 20 years on,” Phil. Trans. R. Soc. 364, 3521–3537 (2006).
[Crossref]

J. Buajarern, L. Mitchem, A. D. Ward, N. H. Nahler, D. McGloin, and J. P. Reid, “Controlling and Characterising the Coagulation of Liquid Aerosol Droplets,” J. Chem. Phys. 125, 114506 (2006).
[Crossref] [PubMed]

D. R. Burnham and D. McGloin, “Holographic optical trapping of aerosol droplets,” Opt. Express. 14, 4175–4181 (2006).
[Crossref] [PubMed]

Mervis, J.

Metzger, N. K.

N. K. Metzger, K. Dholakia, and E. M. Wright, “Observation of bistability and hysteresis in optical binding of two dielectric spheres,” Phys. Rev. Lett. 96, 068102 (2006).
[Crossref] [PubMed]

Mitchem, L.

L. Mitchem and J. P. Reid, “Optical Manipulation and Characterisation of Aerosol Particles,” Chem. Soc. Rev. 37, 756–769 (2008).
[Crossref] [PubMed]

J. Buajarern, L. Mitchem, and J. P. Reid, “Characterising the formation of organic layers on the surface of inorganic/aqueous aerosols by Raman spectroscopy,” J. Phys. Chem. A. 11111852–11859 (2007).
[Crossref] [PubMed]

J. Buajarern, L. Mitchem, A. D. Ward, N. H. Nahler, D. McGloin, and J. P. Reid, “Controlling and Characterising the Coagulation of Liquid Aerosol Droplets,” J. Chem. Phys. 125, 114506 (2006).
[Crossref] [PubMed]

R. J. Hopkins, L. Mitchem, A. D. Ward, and J. P. Reid, “Control and characterisation of a single aerosol droplet in a single-beam gradient-force optical trap,” Phys. Chem. Chem. Phys. 6, 4924–4927 (2004).
[Crossref]

Moine, O.

M. Guillon, O. Moine, and B. Stout, “Longitudinal Optical Binding of High Optical Contrast Microdroplets in Air,” Phys. Rev. Lett. 96, 143902 (2006). Erratum, Phys. Rev. Lett. 99, 079901 (2007).
[Crossref] [PubMed]

Moon, T. J.

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The Optical Stretcher: A Novel Laser Tool to Micromanipulate Cells,” Biophys. J. 81, 767–784 (2001).
[Crossref] [PubMed]

Mukai, S.

N. Magome, M. I. Kohira, E. Hayata, S. Mukai, and K. Yoshikawa, “Optical Trapping of a Growing Water Droplet in Air,” J. Phys. Chem. B. 107, 3988–3990 (2003).
[Crossref]

Nagashima, K.

K. Taji, M. Tachikawa, and K. Nagashima, “Laser trapping of ice crystals,” Appl. Phys. Lett. 88, 141111 (2006).
[Crossref]

Nahler, N. H.

J. Buajarern, L. Mitchem, A. D. Ward, N. H. Nahler, D. McGloin, and J. P. Reid, “Controlling and Characterising the Coagulation of Liquid Aerosol Droplets,” J. Chem. Phys. 125, 114506 (2006).
[Crossref] [PubMed]

Neuman, K. C.

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787–2809 (2004).
[Crossref]

Padgett, M. J.

R. D. Leonardo, G. Ruocco, L. Leach, M. J. Padgett, A. J. Wright, J. M. Girkin, D. R. Burnham, and D. McGloin, “Parametric Resonance of Optically Trapped Aerosols,” Phys. Rev. Lett. 99, 010601 (2007).
[Crossref] [PubMed]

Pastel, R.

R. Pastel, A. Struthers, R. Ringle, J. Rodgers, C. Rohde, and P. Geiser, “Laser trapping of microscopic particles for undergraduate experiments,” Am. J. Phys. 68, 993–1001 (2000).
[Crossref]

Paterson, L.

P. R. T. Jess, V. Garcés-Chávez, D. Smith, M. Mazilu, L. Paterson, A. Riches, C. S. Herrington, W. Sibbett, and K. Dholakia, “Dual beam fiber trap for Raman micro-spectroscopy of single cells,” Opt. Express. 14, 5779–5791 (2006).
[Crossref] [PubMed]

Pieron, C. A.

J. H. Dennis, C. A. Pieron, and K. Asai, “Aerosol Output and Size from Omron NE-U22 nebulizer,” in Proceedings of the 14th International Congress International Society for Aerosols in Medicines, Baltimore June 14–18 2003. Journal of Aerosol Medicine 16:2213, (2003)

Prentiss, M.

Reece, P

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

Reid, J. P.

L. Mitchem and J. P. Reid, “Optical Manipulation and Characterisation of Aerosol Particles,” Chem. Soc. Rev. 37, 756–769 (2008).
[Crossref] [PubMed]

J. Buajarern, L. Mitchem, and J. P. Reid, “Characterising the formation of organic layers on the surface of inorganic/aqueous aerosols by Raman spectroscopy,” J. Phys. Chem. A. 11111852–11859 (2007).
[Crossref] [PubMed]

J. Buajarern, L. Mitchem, A. D. Ward, N. H. Nahler, D. McGloin, and J. P. Reid, “Controlling and Characterising the Coagulation of Liquid Aerosol Droplets,” J. Chem. Phys. 125, 114506 (2006).
[Crossref] [PubMed]

R. J. Hopkins, L. Mitchem, A. D. Ward, and J. P. Reid, “Control and characterisation of a single aerosol droplet in a single-beam gradient-force optical trap,” Phys. Chem. Chem. Phys. 6, 4924–4927 (2004).
[Crossref]

Riches, A.

P. R. T. Jess, V. Garcés-Chávez, D. Smith, M. Mazilu, L. Paterson, A. Riches, C. S. Herrington, W. Sibbett, and K. Dholakia, “Dual beam fiber trap for Raman micro-spectroscopy of single cells,” Opt. Express. 14, 5779–5791 (2006).
[Crossref] [PubMed]

Ringle, R.

R. Pastel, A. Struthers, R. Ringle, J. Rodgers, C. Rohde, and P. Geiser, “Laser trapping of microscopic particles for undergraduate experiments,” Am. J. Phys. 68, 993–1001 (2000).
[Crossref]

Ritsch-Marte, M.

Rodgers, J.

R. Pastel, A. Struthers, R. Ringle, J. Rodgers, C. Rohde, and P. Geiser, “Laser trapping of microscopic particles for undergraduate experiments,” Am. J. Phys. 68, 993–1001 (2000).
[Crossref]

Rohde, C.

R. Pastel, A. Struthers, R. Ringle, J. Rodgers, C. Rohde, and P. Geiser, “Laser trapping of microscopic particles for undergraduate experiments,” Am. J. Phys. 68, 993–1001 (2000).
[Crossref]

Roosen, G.

G. Roosen and C. Imbert, “Optical levitation by means of 2 horizontal laser beams—Theoretical and experimental study,” Phys. Lett. A. 59, 6–8 (1976).
[Crossref]

Ruocco, G.

R. D. Leonardo, G. Ruocco, L. Leach, M. J. Padgett, A. J. Wright, J. M. Girkin, D. R. Burnham, and D. McGloin, “Parametric Resonance of Optically Trapped Aerosols,” Phys. Rev. Lett. 99, 010601 (2007).
[Crossref] [PubMed]

Sibbett, W.

P. R. T. Jess, V. Garcés-Chávez, D. Smith, M. Mazilu, L. Paterson, A. Riches, C. S. Herrington, W. Sibbett, and K. Dholakia, “Dual beam fiber trap for Raman micro-spectroscopy of single cells,” Opt. Express. 14, 5779–5791 (2006).
[Crossref] [PubMed]

Sidick, E.

Singer, W.

Smith, D.

P. R. T. Jess, V. Garcés-Chávez, D. Smith, M. Mazilu, L. Paterson, A. Riches, C. S. Herrington, W. Sibbett, and K. Dholakia, “Dual beam fiber trap for Raman micro-spectroscopy of single cells,” Opt. Express. 14, 5779–5791 (2006).
[Crossref] [PubMed]

Stout, B.

M. Guillon, O. Moine, and B. Stout, “Longitudinal Optical Binding of High Optical Contrast Microdroplets in Air,” Phys. Rev. Lett. 96, 143902 (2006). Erratum, Phys. Rev. Lett. 99, 079901 (2007).
[Crossref] [PubMed]

Struthers, A.

R. Pastel, A. Struthers, R. Ringle, J. Rodgers, C. Rohde, and P. Geiser, “Laser trapping of microscopic particles for undergraduate experiments,” Am. J. Phys. 68, 993–1001 (2000).
[Crossref]

Summers, M. D.

Tachikawa, M.

K. Taji, M. Tachikawa, and K. Nagashima, “Laser trapping of ice crystals,” Appl. Phys. Lett. 88, 141111 (2006).
[Crossref]

Taji, K.

K. Taji, M. Tachikawa, and K. Nagashima, “Laser trapping of ice crystals,” Appl. Phys. Lett. 88, 141111 (2006).
[Crossref]

Tatarkova, S. A.

S. A. Tatarkova, A. E. Carruthers, and K. Dholakia, “One-Dimensional Optically Bound Arrays of Microscopic Particles,” Phys. Rev. Lett. 89, 283901 (2002).
[Crossref]

Thompson, K. C.

M. D. King, K. C. Thompson, and A. D. Ward, “Laser Tweezers Raman Study of Optically Trapped Aerosol Droplets of Seawater and Oleic Acid Reacting with Ozone: Implications for Cloud-Droplet Properties,” J. Am. Chem. Soc. 126, 16710–16711 (2004).
[Crossref] [PubMed]

Travis, K.

Ward, A. D.

J. Buajarern, L. Mitchem, A. D. Ward, N. H. Nahler, D. McGloin, and J. P. Reid, “Controlling and Characterising the Coagulation of Liquid Aerosol Droplets,” J. Chem. Phys. 125, 114506 (2006).
[Crossref] [PubMed]

R. J. Hopkins, L. Mitchem, A. D. Ward, and J. P. Reid, “Control and characterisation of a single aerosol droplet in a single-beam gradient-force optical trap,” Phys. Chem. Chem. Phys. 6, 4924–4927 (2004).
[Crossref]

M. D. King, K. C. Thompson, and A. D. Ward, “Laser Tweezers Raman Study of Optically Trapped Aerosol Droplets of Seawater and Oleic Acid Reacting with Ozone: Implications for Cloud-Droplet Properties,” J. Am. Chem. Soc. 126, 16710–16711 (2004).
[Crossref] [PubMed]

Wright, A. J.

R. D. Leonardo, G. Ruocco, L. Leach, M. J. Padgett, A. J. Wright, J. M. Girkin, D. R. Burnham, and D. McGloin, “Parametric Resonance of Optically Trapped Aerosols,” Phys. Rev. Lett. 99, 010601 (2007).
[Crossref] [PubMed]

Wright, E. M.

N. K. Metzger, K. Dholakia, and E. M. Wright, “Observation of bistability and hysteresis in optical binding of two dielectric spheres,” Phys. Rev. Lett. 96, 068102 (2006).
[Crossref] [PubMed]

Yoshikawa, K.

N. Magome, M. I. Kohira, E. Hayata, S. Mukai, and K. Yoshikawa, “Optical Trapping of a Growing Water Droplet in Air,” J. Phys. Chem. B. 107, 3988–3990 (2003).
[Crossref]

Zarinetchi, F.

Am. J. Phys. (2)

M. J. Lang and S. M Block, “Resource Letter: LBOT-1: Laser-based optical tweezers,” Am. J. Phys. 71, 201–215 (2003).
[Crossref]

R. Pastel, A. Struthers, R. Ringle, J. Rodgers, C. Rohde, and P. Geiser, “Laser trapping of microscopic particles for undergraduate experiments,” Am. J. Phys. 68, 993–1001 (2000).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

K. Taji, M. Tachikawa, and K. Nagashima, “Laser trapping of ice crystals,” Appl. Phys. Lett. 88, 141111 (2006).
[Crossref]

Biophys. J. (1)

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The Optical Stretcher: A Novel Laser Tool to Micromanipulate Cells,” Biophys. J. 81, 767–784 (2001).
[Crossref] [PubMed]

Chem. Soc. Rev. (2)

L. Mitchem and J. P. Reid, “Optical Manipulation and Characterisation of Aerosol Particles,” Chem. Soc. Rev. 37, 756–769 (2008).
[Crossref] [PubMed]

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

Faraday Discuss (1)

ButlerJ. R.MitchemL.HanfordK. L. Treuel L.Reid J. P. “In situ comparative measurements of the properties of aerosol droplets of different chemical composition,” Faraday Discuss 137, 351–366 (2008).
[Crossref] [PubMed]

J. Am. Chem. Soc. (1)

M. D. King, K. C. Thompson, and A. D. Ward, “Laser Tweezers Raman Study of Optically Trapped Aerosol Droplets of Seawater and Oleic Acid Reacting with Ozone: Implications for Cloud-Droplet Properties,” J. Am. Chem. Soc. 126, 16710–16711 (2004).
[Crossref] [PubMed]

J. Chem. Phys. (1)

J. Buajarern, L. Mitchem, A. D. Ward, N. H. Nahler, D. McGloin, and J. P. Reid, “Controlling and Characterising the Coagulation of Liquid Aerosol Droplets,” J. Chem. Phys. 125, 114506 (2006).
[Crossref] [PubMed]

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

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

R. Gussgard and T. Lindmo “Calculation of the trapping force in a strongly focused laser beam,” J. Opt. Soc. Am. B. 9, 1922–1930 (1992).
[Crossref]

J. Phys. Chem. A. (1)

J. Buajarern, L. Mitchem, and J. P. Reid, “Characterising the formation of organic layers on the surface of inorganic/aqueous aerosols by Raman spectroscopy,” J. Phys. Chem. A. 11111852–11859 (2007).
[Crossref] [PubMed]

J. Phys. Chem. B. (1)

N. Magome, M. I. Kohira, E. Hayata, S. Mukai, and K. Yoshikawa, “Optical Trapping of a Growing Water Droplet in Air,” J. Phys. Chem. B. 107, 3988–3990 (2003).
[Crossref]

Journal of Aerosol Medicine (1)

J. H. Dennis, C. A. Pieron, and K. Asai, “Aerosol Output and Size from Omron NE-U22 nebulizer,” in Proceedings of the 14th International Congress International Society for Aerosols in Medicines, Baltimore June 14–18 2003. Journal of Aerosol Medicine 16:2213, (2003)

Opt. Express (2)

Opt. Express. (2)

D. R. Burnham and D. McGloin, “Holographic optical trapping of aerosol droplets,” Opt. Express. 14, 4175–4181 (2006).
[Crossref] [PubMed]

P. R. T. Jess, V. Garcés-Chávez, D. Smith, M. Mazilu, L. Paterson, A. Riches, C. S. Herrington, W. Sibbett, and K. Dholakia, “Dual beam fiber trap for Raman micro-spectroscopy of single cells,” Opt. Express. 14, 5779–5791 (2006).
[Crossref] [PubMed]

Opt. Lett. (2)

Phil. Trans. R. Soc. (1)

D. McGloin, “Optical tweezers: 20 years on,” Phil. Trans. R. Soc. 364, 3521–3537 (2006).
[Crossref]

Phys. Chem. Chem. Phys. (1)

R. J. Hopkins, L. Mitchem, A. D. Ward, and J. P. Reid, “Control and characterisation of a single aerosol droplet in a single-beam gradient-force optical trap,” Phys. Chem. Chem. Phys. 6, 4924–4927 (2004).
[Crossref]

Phys. Lett. A. (1)

G. Roosen and C. Imbert, “Optical levitation by means of 2 horizontal laser beams—Theoretical and experimental study,” Phys. Lett. A. 59, 6–8 (1976).
[Crossref]

Phys. Rev. Lett. (5)

S. A. Tatarkova, A. E. Carruthers, and K. Dholakia, “One-Dimensional Optically Bound Arrays of Microscopic Particles,” Phys. Rev. Lett. 89, 283901 (2002).
[Crossref]

N. K. Metzger, K. Dholakia, and E. M. Wright, “Observation of bistability and hysteresis in optical binding of two dielectric spheres,” Phys. Rev. Lett. 96, 068102 (2006).
[Crossref] [PubMed]

M. Guillon, O. Moine, and B. Stout, “Longitudinal Optical Binding of High Optical Contrast Microdroplets in Air,” Phys. Rev. Lett. 96, 143902 (2006). Erratum, Phys. Rev. Lett. 99, 079901 (2007).
[Crossref] [PubMed]

R. D. Leonardo, G. Ruocco, L. Leach, M. J. Padgett, A. J. Wright, J. M. Girkin, D. R. Burnham, and D. McGloin, “Parametric Resonance of Optically Trapped Aerosols,” Phys. Rev. Lett. 99, 010601 (2007).
[Crossref] [PubMed]

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

Rev. Sci. Instrum. (1)

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787–2809 (2004).
[Crossref]

Other (1)

M. M. Burns, J.-M. Fournier, and J. A. Golovchenko, “Optical Matter: Crystallization and Binding in Intense Optical Fields,” Science 249, 749–754 (1990).

Supplementary Material (2)

» Media 1: MOV (1337 KB)     
» Media 2: MOV (3711 KB)     

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Figures (10)

Fig.1
Fig.1 Theoretical plot of the restoring force perpendicular to the optical axis as a function of the axial offset. The water droplets have a radii 2.5, 5, 7.5 and 10µm and are trapped in air. These forces were calculated using a power of 100mW in each fiber. A fiber separation of 170µm and a refracting index ratio of 1.344. The minimum beam waist is 3.5µm and a wavelength of 532nm. These values remain unchanged unless otherwise stated in the figure. caption.

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Fig. 2.
Fig. 2. (a). Theoretical force curves for the different materials used experimentally at a separation of 170µm. N=1.08 corresponds to a silica sphere in water, N=1.344 a 50g/L saltwater droplet in air, N=1.357 a 20% glycerol-water solution trapped in air and N=1.445 a silica particle in air. (b). Theoretical force curves showing the effect of changing fiber separation a 50g/L salt doped water particle trapped in air (N=1.344). In both A and B the power in each fiber was set to 100mw, the particle radius is 10µm and a minimum beam waist of 3.5µm emerging from the fiber was used. In both case the particle is also considered to be at the midpoint of the fibers.

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Fig. 3.
Fig. 3. Diagram of the experimental setup. From the laser the beam is split into two arms using a half waveplate (1/2 WP) and polarising beam cube. This waveplate can be adjusted to control the relative power down each arm while variable neutral density (ND) filers in each arm provide independent power adjustment. The beams are launched into the fibers using stage mounted 5X microscope objectives. The fibers feed directly into the trapping cell where trapping was observed and recorded using a long working distance objective and CCD camera.

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Fig. 4.
Fig. 4. Beam profile produced from a MMF of the type used. The intensity pattern is that of a super Gaussian with added peaks and troughs.

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Fig. 5.
Fig. 5. Video of a salt-water (20g/L) droplet trapped in air using multimode fibers at a power of 135mW in each arm. The power in the left hand fiber is increased as that in the right is decreased moving the trapped particles to the right. Due to the multimode nature the particles moved different amounts and at different rates. The field of view of the image and video is 218µm by 164 µm. The droplets are all approximately 8µm in diameter. (Media 1)

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Fig. 6.
Fig. 6. Fig. 6. Plot of the maximum droplet sizes trapped using MMF at increasing power in each arm. Each point represents the mean size obtained from a large sample of at least 20 particles trapped at each power. The error bars represent the standard deviation of these droplets. While there is a slight increase in the size and range of the droplets that can be trapped, due to the inconsistency between global and local fields the lines are essentially flat.

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Fig. 7.
Fig. 7. Video of a trapped 8µm salt-water drop (50g/L) being moved 120 micron between the fibers by adjusting the power transmitted by each arm. The droplet is moved at speeds of up to 75µm/s without them falling out of the trap. The fiber separation is also adjusted to demonstrate the effect it has on the equilibrium position. (Media 2)

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Fig. 8.
Fig. 8. Plot of the maximum and minimum droplets sizes trapped at increasing power at 170 and 240µm. NaCl concentration is 50g/L. Solid and dashed lines are approximate guides to the maximum and minimum size of droplet that can be trapped at a power respectively. As can be seen the minimum size of droplet observed remains approximately constant regardless of power, but as power is increased larger droplets can be held. For larger separations the trapping efficiencies are smaller requiring more power to hold a particle of equal size.

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Fig. 9.
Fig. 9. Plot of the maximum and minimum size of airborne glycerol (20% in water) droplets trapped versus power. The maximum line is roughly equivalent to that of the salt doped water show in Fig. 8.

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Fig. 10.
Fig. 10. Example images of optical binding observed with SMF. Images A, B and C show binding with glycerol droplets while D is an example of the binding of salt doped water droplets(20g/L). Image A shows a 16.4 and 5.7µm particle separated by 30 µm. The Field of view is the same in all the images.

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Equations (5)

Equations on this page are rendered with MathJax. Learn more.

Q x = 2 r 0 2 π ∫ 0 π ∂ φ ∫ 0 θ max ∂ θ sin ⁡ 2 θ exp ( − 2 r 2 ω 2 ) ω 2 R c
× { q s ( r 0 sin ⁡ θ cos ⁡ φ − d ) + q g tan ⁡ γ [ r 0 sin ⁡ θ cos ⁡ φ − d ( 1 − R c a cos ⁡ γ ) ] }
Q z = 2 r 0 2 π ∫ 0 π ∂ φ ∫ 0 θ max ∂ θ sin ⁡ 2 θ exp ( − 2 r 2 ω 2 ) ω 2 R c
+ { q s R z + q g tan ⁡ γ [ R z − R c ( R z + r 0 cos ⁡ θ ) a cos ⁡ γ ] }
F x , z = n 1 P c Q x , z

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