Abstract

Combining several methods for contact free micro-manipulation of small particles such as cells or micro-organisms provides the advantages of each method in a single setup. Optical tweezers, which employ focused laser beams, offer very precise and selective handling of single particles. On the other hand, acoustic trapping with wavelengths of about 1 mm allows the simultaneous trapping of many, comparatively large particles. With conventional approaches it is difficult to fully employ the strengths of each method due to the different experimental requirements. Here we present the combined optical and acoustic trapping of motile micro-organisms in a microfluidic environment, utilizing optical macro-tweezers, which offer a large field of view and working distance of several millimeters and therefore match the typical range of acoustic trapping. We characterize the acoustic trapping forces with the help of optically trapped particles and present several applications of the combined optical and acoustic trapping, such as manipulation of large (75 μm) particles and active particle sorting.

© 2011 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Ashkin, Optical Trapping and Manipulation of Neutral Particles using Lasers (World Scientific, 2006).
    [CrossRef]
  2. A. Jonáš and P. Zemánek, “Light at work: The use of optical forces for particle manipulation, sorting, and analysis,” Electrophoresis29, 4813–4851 (2008).
    [CrossRef]
  3. V. Vandaele, P. Lambert, and A. Delchambre, “Non-contact handling in microassembly: Acoustical levitation,” Precis. Eng.29, 491–505 (2005).
    [CrossRef]
  4. J. Nilsson, M. Evander, B. Hammarström, and T. Laurell, “Review of cell and particle trapping in microfluidic systems,” Anal. Chim. Acta649, 141–157 (2009).
    [CrossRef] [PubMed]
  5. M. Padgett and R. Di Leonardo, “Holographic optical tweezers and their relevance to lab on chip devices,” Lab Chip11, 1196–1205 (2011).
    [CrossRef] [PubMed]
  6. K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nat. Photonics5, 335–342 (2011).
    [CrossRef]
  7. J. Hultström, O. Manneberg, K. Dopf, H. M. Hertz, H. Brismar, and M. Wiklund, “Proliferation and viability of adherent cells manipulated by standing-wave ultrasound in a microfluidic chip.” Ultrasound Med. Biol.33, 145–151 (2007).
    [CrossRef]
  8. D. Bazou, R. Kearney, F. Mansergh, C. Bourdon, J. Farrar, and M. Wride, “Gene expression analysis of mouse embryonic stem cells following levitation in an ultrasound standing wave trap,” Ultrasound Med. Biol.37, 321–330 (2011).
    [CrossRef] [PubMed]
  9. M. Pitzek, R. Steiger, G. Thalhammer, S. Bernet, and M. Ritsch-Marte, “Optical mirror trap with a large field of view,” Opt. Express17, 19414–19423 (2009).
    [CrossRef] [PubMed]
  10. G. Thalhammer, R. Steiger, S. Bernet, and M. Ritsch-Marte, “Optical macro-tweezers: trapping of highly motile micro-organisms,” J. Opt.13, 044024 (2011).
    [CrossRef]
  11. A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett.24, 156–159 (1970).
    [CrossRef]
  12. M. Hill, Y. Shen, and J. J. Hawkes, “Modelling of layered resonators for ultrasonic separation,” Ultrasonics40, 385–392 (2002).
    [CrossRef] [PubMed]
  13. L. P. Gor’kov, “On the forces acting on a small particle in an acoustical field in an ideal fluid,” Sov. Phys. Dokl.6, 773–775 (1962).
  14. R. Barnkob, P. Augustsson, T. Laurell, and H. Bruus, “Measuring the local pressure amplitude in microchannel acoustophoresis,” Lab Chip10, 563–570 (2010).
    [CrossRef] [PubMed]
  15. M. Hill, “The selection of layer thicknesses to control acoustic radiation force profiles in layered resonators,” J. Acoust. Soc. Am.114, 2654–2661 (2003).
    [CrossRef] [PubMed]
  16. R. Bowman, A. Jesacher, G. Thalhammer, G. Gibson, M. Ritsch-Marte, and M. Padgett, “Position clamping in a holographic counterpropagating optical trap,” Opt. Express19, 9908–9914 (2011).
    [CrossRef] [PubMed]

2011 (5)

M. Padgett and R. Di Leonardo, “Holographic optical tweezers and their relevance to lab on chip devices,” Lab Chip11, 1196–1205 (2011).
[CrossRef] [PubMed]

K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nat. Photonics5, 335–342 (2011).
[CrossRef]

D. Bazou, R. Kearney, F. Mansergh, C. Bourdon, J. Farrar, and M. Wride, “Gene expression analysis of mouse embryonic stem cells following levitation in an ultrasound standing wave trap,” Ultrasound Med. Biol.37, 321–330 (2011).
[CrossRef] [PubMed]

G. Thalhammer, R. Steiger, S. Bernet, and M. Ritsch-Marte, “Optical macro-tweezers: trapping of highly motile micro-organisms,” J. Opt.13, 044024 (2011).
[CrossRef]

R. Bowman, A. Jesacher, G. Thalhammer, G. Gibson, M. Ritsch-Marte, and M. Padgett, “Position clamping in a holographic counterpropagating optical trap,” Opt. Express19, 9908–9914 (2011).
[CrossRef] [PubMed]

2010 (1)

R. Barnkob, P. Augustsson, T. Laurell, and H. Bruus, “Measuring the local pressure amplitude in microchannel acoustophoresis,” Lab Chip10, 563–570 (2010).
[CrossRef] [PubMed]

2009 (2)

J. Nilsson, M. Evander, B. Hammarström, and T. Laurell, “Review of cell and particle trapping in microfluidic systems,” Anal. Chim. Acta649, 141–157 (2009).
[CrossRef] [PubMed]

M. Pitzek, R. Steiger, G. Thalhammer, S. Bernet, and M. Ritsch-Marte, “Optical mirror trap with a large field of view,” Opt. Express17, 19414–19423 (2009).
[CrossRef] [PubMed]

2008 (1)

A. Jonáš and P. Zemánek, “Light at work: The use of optical forces for particle manipulation, sorting, and analysis,” Electrophoresis29, 4813–4851 (2008).
[CrossRef]

2007 (1)

J. Hultström, O. Manneberg, K. Dopf, H. M. Hertz, H. Brismar, and M. Wiklund, “Proliferation and viability of adherent cells manipulated by standing-wave ultrasound in a microfluidic chip.” Ultrasound Med. Biol.33, 145–151 (2007).
[CrossRef]

2005 (1)

V. Vandaele, P. Lambert, and A. Delchambre, “Non-contact handling in microassembly: Acoustical levitation,” Precis. Eng.29, 491–505 (2005).
[CrossRef]

2003 (1)

M. Hill, “The selection of layer thicknesses to control acoustic radiation force profiles in layered resonators,” J. Acoust. Soc. Am.114, 2654–2661 (2003).
[CrossRef] [PubMed]

2002 (1)

M. Hill, Y. Shen, and J. J. Hawkes, “Modelling of layered resonators for ultrasonic separation,” Ultrasonics40, 385–392 (2002).
[CrossRef] [PubMed]

1970 (1)

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

1962 (1)

L. P. Gor’kov, “On the forces acting on a small particle in an acoustical field in an ideal fluid,” Sov. Phys. Dokl.6, 773–775 (1962).

Ashkin, A.

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

A. Ashkin, Optical Trapping and Manipulation of Neutral Particles using Lasers (World Scientific, 2006).
[CrossRef]

Augustsson, P.

R. Barnkob, P. Augustsson, T. Laurell, and H. Bruus, “Measuring the local pressure amplitude in microchannel acoustophoresis,” Lab Chip10, 563–570 (2010).
[CrossRef] [PubMed]

Barnkob, R.

R. Barnkob, P. Augustsson, T. Laurell, and H. Bruus, “Measuring the local pressure amplitude in microchannel acoustophoresis,” Lab Chip10, 563–570 (2010).
[CrossRef] [PubMed]

Bazou, D.

D. Bazou, R. Kearney, F. Mansergh, C. Bourdon, J. Farrar, and M. Wride, “Gene expression analysis of mouse embryonic stem cells following levitation in an ultrasound standing wave trap,” Ultrasound Med. Biol.37, 321–330 (2011).
[CrossRef] [PubMed]

Bernet, S.

G. Thalhammer, R. Steiger, S. Bernet, and M. Ritsch-Marte, “Optical macro-tweezers: trapping of highly motile micro-organisms,” J. Opt.13, 044024 (2011).
[CrossRef]

M. Pitzek, R. Steiger, G. Thalhammer, S. Bernet, and M. Ritsch-Marte, “Optical mirror trap with a large field of view,” Opt. Express17, 19414–19423 (2009).
[CrossRef] [PubMed]

Bourdon, C.

D. Bazou, R. Kearney, F. Mansergh, C. Bourdon, J. Farrar, and M. Wride, “Gene expression analysis of mouse embryonic stem cells following levitation in an ultrasound standing wave trap,” Ultrasound Med. Biol.37, 321–330 (2011).
[CrossRef] [PubMed]

Bowman, R.

Brismar, H.

J. Hultström, O. Manneberg, K. Dopf, H. M. Hertz, H. Brismar, and M. Wiklund, “Proliferation and viability of adherent cells manipulated by standing-wave ultrasound in a microfluidic chip.” Ultrasound Med. Biol.33, 145–151 (2007).
[CrossRef]

Bruus, H.

R. Barnkob, P. Augustsson, T. Laurell, and H. Bruus, “Measuring the local pressure amplitude in microchannel acoustophoresis,” Lab Chip10, 563–570 (2010).
[CrossRef] [PubMed]

Cižmár, T.

K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nat. Photonics5, 335–342 (2011).
[CrossRef]

Delchambre, A.

V. Vandaele, P. Lambert, and A. Delchambre, “Non-contact handling in microassembly: Acoustical levitation,” Precis. Eng.29, 491–505 (2005).
[CrossRef]

Dholakia, K.

K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nat. Photonics5, 335–342 (2011).
[CrossRef]

Di Leonardo, R.

M. Padgett and R. Di Leonardo, “Holographic optical tweezers and their relevance to lab on chip devices,” Lab Chip11, 1196–1205 (2011).
[CrossRef] [PubMed]

Dopf, K.

J. Hultström, O. Manneberg, K. Dopf, H. M. Hertz, H. Brismar, and M. Wiklund, “Proliferation and viability of adherent cells manipulated by standing-wave ultrasound in a microfluidic chip.” Ultrasound Med. Biol.33, 145–151 (2007).
[CrossRef]

Evander, M.

J. Nilsson, M. Evander, B. Hammarström, and T. Laurell, “Review of cell and particle trapping in microfluidic systems,” Anal. Chim. Acta649, 141–157 (2009).
[CrossRef] [PubMed]

Farrar, J.

D. Bazou, R. Kearney, F. Mansergh, C. Bourdon, J. Farrar, and M. Wride, “Gene expression analysis of mouse embryonic stem cells following levitation in an ultrasound standing wave trap,” Ultrasound Med. Biol.37, 321–330 (2011).
[CrossRef] [PubMed]

Gibson, G.

Gor’kov, L. P.

L. P. Gor’kov, “On the forces acting on a small particle in an acoustical field in an ideal fluid,” Sov. Phys. Dokl.6, 773–775 (1962).

Hammarström, B.

J. Nilsson, M. Evander, B. Hammarström, and T. Laurell, “Review of cell and particle trapping in microfluidic systems,” Anal. Chim. Acta649, 141–157 (2009).
[CrossRef] [PubMed]

Hawkes, J. J.

M. Hill, Y. Shen, and J. J. Hawkes, “Modelling of layered resonators for ultrasonic separation,” Ultrasonics40, 385–392 (2002).
[CrossRef] [PubMed]

Hertz, H. M.

J. Hultström, O. Manneberg, K. Dopf, H. M. Hertz, H. Brismar, and M. Wiklund, “Proliferation and viability of adherent cells manipulated by standing-wave ultrasound in a microfluidic chip.” Ultrasound Med. Biol.33, 145–151 (2007).
[CrossRef]

Hill, M.

M. Hill, “The selection of layer thicknesses to control acoustic radiation force profiles in layered resonators,” J. Acoust. Soc. Am.114, 2654–2661 (2003).
[CrossRef] [PubMed]

M. Hill, Y. Shen, and J. J. Hawkes, “Modelling of layered resonators for ultrasonic separation,” Ultrasonics40, 385–392 (2002).
[CrossRef] [PubMed]

Hultström, J.

J. Hultström, O. Manneberg, K. Dopf, H. M. Hertz, H. Brismar, and M. Wiklund, “Proliferation and viability of adherent cells manipulated by standing-wave ultrasound in a microfluidic chip.” Ultrasound Med. Biol.33, 145–151 (2007).
[CrossRef]

Jesacher, A.

Jonáš, A.

A. Jonáš and P. Zemánek, “Light at work: The use of optical forces for particle manipulation, sorting, and analysis,” Electrophoresis29, 4813–4851 (2008).
[CrossRef]

Kearney, R.

D. Bazou, R. Kearney, F. Mansergh, C. Bourdon, J. Farrar, and M. Wride, “Gene expression analysis of mouse embryonic stem cells following levitation in an ultrasound standing wave trap,” Ultrasound Med. Biol.37, 321–330 (2011).
[CrossRef] [PubMed]

Lambert, P.

V. Vandaele, P. Lambert, and A. Delchambre, “Non-contact handling in microassembly: Acoustical levitation,” Precis. Eng.29, 491–505 (2005).
[CrossRef]

Laurell, T.

R. Barnkob, P. Augustsson, T. Laurell, and H. Bruus, “Measuring the local pressure amplitude in microchannel acoustophoresis,” Lab Chip10, 563–570 (2010).
[CrossRef] [PubMed]

J. Nilsson, M. Evander, B. Hammarström, and T. Laurell, “Review of cell and particle trapping in microfluidic systems,” Anal. Chim. Acta649, 141–157 (2009).
[CrossRef] [PubMed]

Manneberg, O.

J. Hultström, O. Manneberg, K. Dopf, H. M. Hertz, H. Brismar, and M. Wiklund, “Proliferation and viability of adherent cells manipulated by standing-wave ultrasound in a microfluidic chip.” Ultrasound Med. Biol.33, 145–151 (2007).
[CrossRef]

Mansergh, F.

D. Bazou, R. Kearney, F. Mansergh, C. Bourdon, J. Farrar, and M. Wride, “Gene expression analysis of mouse embryonic stem cells following levitation in an ultrasound standing wave trap,” Ultrasound Med. Biol.37, 321–330 (2011).
[CrossRef] [PubMed]

Nilsson, J.

J. Nilsson, M. Evander, B. Hammarström, and T. Laurell, “Review of cell and particle trapping in microfluidic systems,” Anal. Chim. Acta649, 141–157 (2009).
[CrossRef] [PubMed]

Padgett, M.

Pitzek, M.

Ritsch-Marte, M.

Shen, Y.

M. Hill, Y. Shen, and J. J. Hawkes, “Modelling of layered resonators for ultrasonic separation,” Ultrasonics40, 385–392 (2002).
[CrossRef] [PubMed]

Steiger, R.

G. Thalhammer, R. Steiger, S. Bernet, and M. Ritsch-Marte, “Optical macro-tweezers: trapping of highly motile micro-organisms,” J. Opt.13, 044024 (2011).
[CrossRef]

M. Pitzek, R. Steiger, G. Thalhammer, S. Bernet, and M. Ritsch-Marte, “Optical mirror trap with a large field of view,” Opt. Express17, 19414–19423 (2009).
[CrossRef] [PubMed]

Thalhammer, G.

Vandaele, V.

V. Vandaele, P. Lambert, and A. Delchambre, “Non-contact handling in microassembly: Acoustical levitation,” Precis. Eng.29, 491–505 (2005).
[CrossRef]

Wiklund, M.

J. Hultström, O. Manneberg, K. Dopf, H. M. Hertz, H. Brismar, and M. Wiklund, “Proliferation and viability of adherent cells manipulated by standing-wave ultrasound in a microfluidic chip.” Ultrasound Med. Biol.33, 145–151 (2007).
[CrossRef]

Wride, M.

D. Bazou, R. Kearney, F. Mansergh, C. Bourdon, J. Farrar, and M. Wride, “Gene expression analysis of mouse embryonic stem cells following levitation in an ultrasound standing wave trap,” Ultrasound Med. Biol.37, 321–330 (2011).
[CrossRef] [PubMed]

Zemánek, P.

A. Jonáš and P. Zemánek, “Light at work: The use of optical forces for particle manipulation, sorting, and analysis,” Electrophoresis29, 4813–4851 (2008).
[CrossRef]

Anal. Chim. Acta (1)

J. Nilsson, M. Evander, B. Hammarström, and T. Laurell, “Review of cell and particle trapping in microfluidic systems,” Anal. Chim. Acta649, 141–157 (2009).
[CrossRef] [PubMed]

Electrophoresis (1)

A. Jonáš and P. Zemánek, “Light at work: The use of optical forces for particle manipulation, sorting, and analysis,” Electrophoresis29, 4813–4851 (2008).
[CrossRef]

J. Acoust. Soc. Am. (1)

M. Hill, “The selection of layer thicknesses to control acoustic radiation force profiles in layered resonators,” J. Acoust. Soc. Am.114, 2654–2661 (2003).
[CrossRef] [PubMed]

J. Opt. (1)

G. Thalhammer, R. Steiger, S. Bernet, and M. Ritsch-Marte, “Optical macro-tweezers: trapping of highly motile micro-organisms,” J. Opt.13, 044024 (2011).
[CrossRef]

Lab Chip (2)

R. Barnkob, P. Augustsson, T. Laurell, and H. Bruus, “Measuring the local pressure amplitude in microchannel acoustophoresis,” Lab Chip10, 563–570 (2010).
[CrossRef] [PubMed]

M. Padgett and R. Di Leonardo, “Holographic optical tweezers and their relevance to lab on chip devices,” Lab Chip11, 1196–1205 (2011).
[CrossRef] [PubMed]

Nat. Photonics (1)

K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nat. Photonics5, 335–342 (2011).
[CrossRef]

Opt. Express (2)

Phys. Rev. Lett. (1)

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

Precis. Eng. (1)

V. Vandaele, P. Lambert, and A. Delchambre, “Non-contact handling in microassembly: Acoustical levitation,” Precis. Eng.29, 491–505 (2005).
[CrossRef]

Sov. Phys. Dokl. (1)

L. P. Gor’kov, “On the forces acting on a small particle in an acoustical field in an ideal fluid,” Sov. Phys. Dokl.6, 773–775 (1962).

Ultrasonics (1)

M. Hill, Y. Shen, and J. J. Hawkes, “Modelling of layered resonators for ultrasonic separation,” Ultrasonics40, 385–392 (2002).
[CrossRef] [PubMed]

Ultrasound Med. Biol. (2)

J. Hultström, O. Manneberg, K. Dopf, H. M. Hertz, H. Brismar, and M. Wiklund, “Proliferation and viability of adherent cells manipulated by standing-wave ultrasound in a microfluidic chip.” Ultrasound Med. Biol.33, 145–151 (2007).
[CrossRef]

D. Bazou, R. Kearney, F. Mansergh, C. Bourdon, J. Farrar, and M. Wride, “Gene expression analysis of mouse embryonic stem cells following levitation in an ultrasound standing wave trap,” Ultrasound Med. Biol.37, 321–330 (2011).
[CrossRef] [PubMed]

Other (1)

A. Ashkin, Optical Trapping and Manipulation of Neutral Particles using Lasers (World Scientific, 2006).
[CrossRef]

Supplementary Material (6)

» Media 1: MOV (4936 KB)     
» Media 2: MOV (4223 KB)     
» Media 3: MOV (976 KB)     
» Media 4: MOV (360 KB)     
» Media 5: MOV (708 KB)     
» Media 6: MOV (2167 KB)     

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Probe chamber design. A rectangular capillary, filled with water, is sandwiched between a mirror on top and a microscope slide. The ultrasonic wave is excited by a piezo transducer on top of the stack. Next to the capillary a small prism, which acts as a mirror, provides an additional view from the side. The geometrical path length of the side-view imaging is longer than that of the direct-view. To compensate for this difference so that particles appear simultaneously sharp in the direct and side-view image, we add another mounting slide that covers only the area below the prism but not below the capillary.

Fig. 3
Fig. 3

Side-view images of acoustic trapping of motile micro-organisms (Euglena gracilis), see Media 1. (a) The acoustic trap is off, the micro-organisms are randomly distributed within the probe volume. (b) Acoustic trap switched on (f = 1.95 MHz), specimens are confined in single nodal plane. (c) Trapping with resonance at f = 5.77 MHz with three nodal planes. (d) Aggregation of specimens due to additional horizontal confinement.

Fig. 6
Fig. 6

Measurement of the dependence of the acoustic force on on the excitation frequency for a 5.8 μm diameter polystyrene bead in water. These data are valid for a probe chamber with 0.62 mm fluid layer thickness, the other probe chamber dimensions are given in table 1. The optically trapped bead was placed 400 μm above the bottom, and the driving voltage of the ultrasound transducer was 3Vp–p. Also shown are the results of our model calculations (dashed line). We scaled the calculated force such that the peak value for the most prominent resonance at 2380 kHz matches the observed data.

Fig. 2
Fig. 2

Schematic sketch of the experimental setup. We illuminate the spatial light modulator (SLM) with a collimated laser beam. With two lenses (f1 = 200mm, f2 = 250mm) and a dichroic mirror we couple the first diffraction order into an inverted optical microscope. See text for more details on the individual parts.

Fig. 4
Fig. 4

Combined acoustic and optical trapping of micro-organisms (Euglena gracilis). Shown is a direct-view image, selected from Media 2. The acoustic trap confines all micro-organisms within the focal plane. A single organism (marked by a circle) is trapped in the optical trap. It aligns along the (vertical) laser beam and appears point-like. Within 40 s the optical trap has been moved by a distance of about 1 mm, dragging the organism across the field of view.

Fig. 5
Fig. 5

Trapping of large particles in the combined acoustic and optical trap: (a) polystyrene bead, diameter 75 μm ( Media 3), (b) living Dinoflagellate micro-organism, size approx. 70 μm ( Media 4), (c) potato starch grain, size 95 μm×60 μm×60 μm ( Media 5).

Fig. 7
Fig. 7

(a) Force profiles for polystyrene beads (20 μm diameter) in the acoustic trap for different ultrasonic amplitudes. We performed the measurements in a capillary as depicted in Fig. 1 at an ultrasound frequency of f = 1911 kHz. The solid lines are based on a sinusoidal fit, which has been scaled accordingly to FU2, where U is the driving voltage of the piezo transducer. (b) Force profiles for a 0.33 mm thick probe chamber for two different resonances. At 1990 kHz we observe at the bottom of the probe volume (z = 0) a large lifting force of about 75% of the maximum acoustic force, whereas at 2450 kHz our model predicts a force pushing particles against the bottom. The data was measured with a starch grain of 23 μm diameter and is valid for a driving voltage of 0.73Vp–p, which is sufficient to detach such particles from the bottom.

Fig. 8
Fig. 8

Active particle sorting ( Media 6). In a first trapping stage we prepare 10 μm polystyrene beads in a continuous flow with a speed of about 20 μm/s to occupy a plane centered vertically in the capillary. Then they enter the region shown in this image with three nodal planes, where some particles are selectively pushed with optical forces to the upper plane. Also shown in this image are markers indicating the positions of the nodal planes and the laser beam.

Tables (1)

Tables Icon

Table 1 Material properties and typical layer thicknesses of the individual components of the probe chamber used for the simulation of the acoustic properties of the probe chamber. The thicknesses are measured values, the sound velocities have been adjusted such that the simulated resonance frequencies match the actual ones. The densities are estimates taken from tabulated values.

Equations (3)

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

F ( r ) = V 4 c 2 ρ ( f 1 p 2 ( r ) 3 f 2 2 k 2 ( p ( r ) ) 2 ) ,
F ( z ) = V 4 c 2 ρ p 0 2 k ( f 1 + 3 f 2 2 ) sin ( 2 kz ) .
F drag = 6 π r η ν ,

Metrics