Abstract

We show that the axial spread of the focal volume of a tightly focused beam propagating through a glass–water interface is much reduced for Laguerre–Gaussian (LG) modes as compared to the TEM00 mode. Therefore, use of the LG beam helps in achieving a significant improvement of the axial trapping range in optical tweezers. We demonstrate the use of LG modes to manipulate biological cells from the bottom layer of the medium to the top surface layer. Exposure of the cells to a higher oxygen concentration at the surface layer is used for estimation of the intramembrane oxygen diffusion rate.

© 2011 Optical Society of America

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References

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  1. 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]
  2. T. Ota, T. Sugiura, S. Kawata, M. J. Booth, M. A. A. Neil, R. Juškaitis, and T. Wilson, “Enhancement of laser trapping force by spherical aberration correction using a deformable mirror,” Jpn. J. Appl. Phys. 42, L701–L703 (2003).
    [CrossRef]
  3. S. N. S. Reihani, M. A. Charsooghi, H. R. Khalesifard, and R. Golestanian, “Efficient in-depth trapping with an oil-immersion objective lens,” Opt. Lett. 31, 766–768 (2006).
    [CrossRef] [PubMed]
  4. S. N. S. Reihani and L. B. Oddershede, “Optimizing immersion media refractive index improves optical trapping by compensating spherical aberrations,” Opt. Lett. 32, 1998–2000 (2007).
    [CrossRef] [PubMed]
  5. R. Dasgupta, S. Ahlawat, and P. K. Gupta, “Trapping of micron-sized objects at a liquid–air interface,” J. Opt. A: Pure Appl. Opt. 9, S189–S195 (2007).
    [CrossRef]
  6. A. Jesacher, S. Fürhapter, C. Maurer, S. Bernet, and M. Ritsch-Marte, “Holographic optical tweezers for object manipulations at an air–liquid surface,” Opt. Express 14, 6342–6352 (2006).
    [CrossRef] [PubMed]
  7. A. T. Oneil and M. J. Padgett, “Axial and lateral trapping efficiency of Laguerre–Gaussian modes in inverted optical tweezers,” Opt. Commun. 193, 45–50 (2001).
    [CrossRef]
  8. M. E. J. Friese, H. Rubinsztein-Dunlop, N. R. Heckenberg, and E. W. Dearden, “Determination of the force constant of a single-beam gradient trap by measurement of backscattered light,” Appl. Opt. 35, 7112–7116 (1996).
    [CrossRef] [PubMed]
  9. N. B. Simpson, D. McGloin, K. Dholakia, L. Allen, and M. J. Padgett, “Optical tweezers with increased axial trapping efficiency,” J. Mod. Opt. 45, 1943–1949 (1998).
    [CrossRef]
  10. P. Török and P. R. T. Munro, “The use of Gauss–Laguerre vector beams in STED microscopy,” Opt. Express 12, 3605–3617 (2004).
    [CrossRef] [PubMed]
  11. S. Deng, L. Liul, Y. Cheng, R. Li, and Z. Xu, “Investigation of the influence of the aberration induced by a plane interface on STED microscopy,” Opt. Express 17, 1714–1725 (2009).
    [CrossRef] [PubMed]
  12. M.-T. Wei and A. Chiou, “Three-dimensional tracking of Brownian motion of a particle trapped in optical tweezers with a pair of orthogonal tracking beams and the determination of the associated optical force constants,” Opt. Express 13, 5798–5806 (2005).
    [CrossRef] [PubMed]
  13. “Laurdan generalized polarization: from cuvette to microscope,” http://www.formatex.org/microscopy3/pdf/pp1007-1014.pdf, retrieved on 3 May 2010.
  14. S. Fischokoff and J. M. Vanderkooi, “Oxygen diffusion in biological and artificial membranes determined by the fluorochrome pyrene,” J. Gen. Physiol. 65, 663–676 (1975).
    [CrossRef]
  15. T. Parasassi and E. Gratton, “Packing of phospholipid vesicles studied by oxygen quenching of Laurdan fluorescence,” J. Fluoresc. 2, 167–174 (1992).
    [CrossRef]
  16. A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61, 569–582 (1992).
    [CrossRef] [PubMed]
  17. R. M. Simmons, J. T. Finer, S. Chu, J. A. Spudich, P. Török, and P. R. T. Munro, “Quantitative measurements of force and displacement using an optical trap,” Biophys. J. 70, 1813–1822(1996).
    [CrossRef] [PubMed]
  18. H. Schneckenburger, M. Wagner, M. Kretzschmar, W. S. L. Strauss, and R. Sailer, “Laser-assisted fluorescence microscopy for measuring cell membrane dynamics,” Photochem. Photobiol. Sci. 3, 817–822 (2004).
    [CrossRef] [PubMed]
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    [CrossRef]

2009 (1)

2007 (2)

R. Dasgupta, S. Ahlawat, and P. K. Gupta, “Trapping of micron-sized objects at a liquid–air interface,” J. Opt. A: Pure Appl. Opt. 9, S189–S195 (2007).
[CrossRef]

S. N. S. Reihani and L. B. Oddershede, “Optimizing immersion media refractive index improves optical trapping by compensating spherical aberrations,” Opt. Lett. 32, 1998–2000 (2007).
[CrossRef] [PubMed]

2006 (2)

2005 (1)

2004 (2)

H. Schneckenburger, M. Wagner, M. Kretzschmar, W. S. L. Strauss, and R. Sailer, “Laser-assisted fluorescence microscopy for measuring cell membrane dynamics,” Photochem. Photobiol. Sci. 3, 817–822 (2004).
[CrossRef] [PubMed]

P. Török and P. R. T. Munro, “The use of Gauss–Laguerre vector beams in STED microscopy,” Opt. Express 12, 3605–3617 (2004).
[CrossRef] [PubMed]

2003 (1)

T. Ota, T. Sugiura, S. Kawata, M. J. Booth, M. A. A. Neil, R. Juškaitis, and T. Wilson, “Enhancement of laser trapping force by spherical aberration correction using a deformable mirror,” Jpn. J. Appl. Phys. 42, L701–L703 (2003).
[CrossRef]

2001 (1)

A. T. Oneil and M. J. Padgett, “Axial and lateral trapping efficiency of Laguerre–Gaussian modes in inverted optical tweezers,” Opt. Commun. 193, 45–50 (2001).
[CrossRef]

1998 (1)

N. B. Simpson, D. McGloin, K. Dholakia, L. Allen, and M. J. Padgett, “Optical tweezers with increased axial trapping efficiency,” J. Mod. Opt. 45, 1943–1949 (1998).
[CrossRef]

1996 (2)

R. M. Simmons, J. T. Finer, S. Chu, J. A. Spudich, P. Török, and P. R. T. Munro, “Quantitative measurements of force and displacement using an optical trap,” Biophys. J. 70, 1813–1822(1996).
[CrossRef] [PubMed]

M. E. J. Friese, H. Rubinsztein-Dunlop, N. R. Heckenberg, and E. W. Dearden, “Determination of the force constant of a single-beam gradient trap by measurement of backscattered light,” Appl. Opt. 35, 7112–7116 (1996).
[CrossRef] [PubMed]

1992 (3)

J. R. Dynlacht and M. H. Fox, “Heat-induced changes in membrane fluidity of Chinese hamster ovary cells measured by flow cytometry,” Rad. Res. 130, 48–54 (1992).
[CrossRef]

T. Parasassi and E. Gratton, “Packing of phospholipid vesicles studied by oxygen quenching of Laurdan fluorescence,” J. Fluoresc. 2, 167–174 (1992).
[CrossRef]

A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61, 569–582 (1992).
[CrossRef] [PubMed]

1986 (1)

1975 (1)

S. Fischokoff and J. M. Vanderkooi, “Oxygen diffusion in biological and artificial membranes determined by the fluorochrome pyrene,” J. Gen. Physiol. 65, 663–676 (1975).
[CrossRef]

Ahlawat, S.

R. Dasgupta, S. Ahlawat, and P. K. Gupta, “Trapping of micron-sized objects at a liquid–air interface,” J. Opt. A: Pure Appl. Opt. 9, S189–S195 (2007).
[CrossRef]

Allen, L.

N. B. Simpson, D. McGloin, K. Dholakia, L. Allen, and M. J. Padgett, “Optical tweezers with increased axial trapping efficiency,” J. Mod. Opt. 45, 1943–1949 (1998).
[CrossRef]

Ashkin, A.

Bernet, S.

Bjorkholm, J. E.

Booth, M. J.

T. Ota, T. Sugiura, S. Kawata, M. J. Booth, M. A. A. Neil, R. Juškaitis, and T. Wilson, “Enhancement of laser trapping force by spherical aberration correction using a deformable mirror,” Jpn. J. Appl. Phys. 42, L701–L703 (2003).
[CrossRef]

Charsooghi, M. A.

Cheng, Y.

Chiou, A.

Chu, S.

R. M. Simmons, J. T. Finer, S. Chu, J. A. Spudich, P. Török, and P. R. T. Munro, “Quantitative measurements of force and displacement using an optical trap,” Biophys. J. 70, 1813–1822(1996).
[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]

Dasgupta, R.

R. Dasgupta, S. Ahlawat, and P. K. Gupta, “Trapping of micron-sized objects at a liquid–air interface,” J. Opt. A: Pure Appl. Opt. 9, S189–S195 (2007).
[CrossRef]

Dearden, E. W.

Deng, S.

Dholakia, K.

N. B. Simpson, D. McGloin, K. Dholakia, L. Allen, and M. J. Padgett, “Optical tweezers with increased axial trapping efficiency,” J. Mod. Opt. 45, 1943–1949 (1998).
[CrossRef]

Dynlacht, J. R.

J. R. Dynlacht and M. H. Fox, “Heat-induced changes in membrane fluidity of Chinese hamster ovary cells measured by flow cytometry,” Rad. Res. 130, 48–54 (1992).
[CrossRef]

Dziedzic, J. M.

Finer, J. T.

R. M. Simmons, J. T. Finer, S. Chu, J. A. Spudich, P. Török, and P. R. T. Munro, “Quantitative measurements of force and displacement using an optical trap,” Biophys. J. 70, 1813–1822(1996).
[CrossRef] [PubMed]

Fischokoff, S.

S. Fischokoff and J. M. Vanderkooi, “Oxygen diffusion in biological and artificial membranes determined by the fluorochrome pyrene,” J. Gen. Physiol. 65, 663–676 (1975).
[CrossRef]

Fox, M. H.

J. R. Dynlacht and M. H. Fox, “Heat-induced changes in membrane fluidity of Chinese hamster ovary cells measured by flow cytometry,” Rad. Res. 130, 48–54 (1992).
[CrossRef]

Friese, M. E. J.

Fürhapter, S.

Golestanian, R.

Gratton, E.

T. Parasassi and E. Gratton, “Packing of phospholipid vesicles studied by oxygen quenching of Laurdan fluorescence,” J. Fluoresc. 2, 167–174 (1992).
[CrossRef]

Gupta, P. K.

R. Dasgupta, S. Ahlawat, and P. K. Gupta, “Trapping of micron-sized objects at a liquid–air interface,” J. Opt. A: Pure Appl. Opt. 9, S189–S195 (2007).
[CrossRef]

Heckenberg, N. R.

Jesacher, A.

Juškaitis, R.

T. Ota, T. Sugiura, S. Kawata, M. J. Booth, M. A. A. Neil, R. Juškaitis, and T. Wilson, “Enhancement of laser trapping force by spherical aberration correction using a deformable mirror,” Jpn. J. Appl. Phys. 42, L701–L703 (2003).
[CrossRef]

Kawata, S.

T. Ota, T. Sugiura, S. Kawata, M. J. Booth, M. A. A. Neil, R. Juškaitis, and T. Wilson, “Enhancement of laser trapping force by spherical aberration correction using a deformable mirror,” Jpn. J. Appl. Phys. 42, L701–L703 (2003).
[CrossRef]

Khalesifard, H. R.

Kretzschmar, M.

H. Schneckenburger, M. Wagner, M. Kretzschmar, W. S. L. Strauss, and R. Sailer, “Laser-assisted fluorescence microscopy for measuring cell membrane dynamics,” Photochem. Photobiol. Sci. 3, 817–822 (2004).
[CrossRef] [PubMed]

Li, R.

Liul, L.

Maurer, C.

McGloin, D.

N. B. Simpson, D. McGloin, K. Dholakia, L. Allen, and M. J. Padgett, “Optical tweezers with increased axial trapping efficiency,” J. Mod. Opt. 45, 1943–1949 (1998).
[CrossRef]

Munro, P. R. T.

P. Török and P. R. T. Munro, “The use of Gauss–Laguerre vector beams in STED microscopy,” Opt. Express 12, 3605–3617 (2004).
[CrossRef] [PubMed]

R. M. Simmons, J. T. Finer, S. Chu, J. A. Spudich, P. Török, and P. R. T. Munro, “Quantitative measurements of force and displacement using an optical trap,” Biophys. J. 70, 1813–1822(1996).
[CrossRef] [PubMed]

Neil, M. A. A.

T. Ota, T. Sugiura, S. Kawata, M. J. Booth, M. A. A. Neil, R. Juškaitis, and T. Wilson, “Enhancement of laser trapping force by spherical aberration correction using a deformable mirror,” Jpn. J. Appl. Phys. 42, L701–L703 (2003).
[CrossRef]

Oddershede, L. B.

Oneil, A. T.

A. T. Oneil and M. J. Padgett, “Axial and lateral trapping efficiency of Laguerre–Gaussian modes in inverted optical tweezers,” Opt. Commun. 193, 45–50 (2001).
[CrossRef]

Ota, T.

T. Ota, T. Sugiura, S. Kawata, M. J. Booth, M. A. A. Neil, R. Juškaitis, and T. Wilson, “Enhancement of laser trapping force by spherical aberration correction using a deformable mirror,” Jpn. J. Appl. Phys. 42, L701–L703 (2003).
[CrossRef]

Padgett, M. J.

A. T. Oneil and M. J. Padgett, “Axial and lateral trapping efficiency of Laguerre–Gaussian modes in inverted optical tweezers,” Opt. Commun. 193, 45–50 (2001).
[CrossRef]

N. B. Simpson, D. McGloin, K. Dholakia, L. Allen, and M. J. Padgett, “Optical tweezers with increased axial trapping efficiency,” J. Mod. Opt. 45, 1943–1949 (1998).
[CrossRef]

Parasassi, T.

T. Parasassi and E. Gratton, “Packing of phospholipid vesicles studied by oxygen quenching of Laurdan fluorescence,” J. Fluoresc. 2, 167–174 (1992).
[CrossRef]

Reihani, S. N. S.

Ritsch-Marte, M.

Rubinsztein-Dunlop, H.

Sailer, R.

H. Schneckenburger, M. Wagner, M. Kretzschmar, W. S. L. Strauss, and R. Sailer, “Laser-assisted fluorescence microscopy for measuring cell membrane dynamics,” Photochem. Photobiol. Sci. 3, 817–822 (2004).
[CrossRef] [PubMed]

Schneckenburger, H.

H. Schneckenburger, M. Wagner, M. Kretzschmar, W. S. L. Strauss, and R. Sailer, “Laser-assisted fluorescence microscopy for measuring cell membrane dynamics,” Photochem. Photobiol. Sci. 3, 817–822 (2004).
[CrossRef] [PubMed]

Simmons, R. M.

R. M. Simmons, J. T. Finer, S. Chu, J. A. Spudich, P. Török, and P. R. T. Munro, “Quantitative measurements of force and displacement using an optical trap,” Biophys. J. 70, 1813–1822(1996).
[CrossRef] [PubMed]

Simpson, N. B.

N. B. Simpson, D. McGloin, K. Dholakia, L. Allen, and M. J. Padgett, “Optical tweezers with increased axial trapping efficiency,” J. Mod. Opt. 45, 1943–1949 (1998).
[CrossRef]

Spudich, J. A.

R. M. Simmons, J. T. Finer, S. Chu, J. A. Spudich, P. Török, and P. R. T. Munro, “Quantitative measurements of force and displacement using an optical trap,” Biophys. J. 70, 1813–1822(1996).
[CrossRef] [PubMed]

Strauss, W. S. L.

H. Schneckenburger, M. Wagner, M. Kretzschmar, W. S. L. Strauss, and R. Sailer, “Laser-assisted fluorescence microscopy for measuring cell membrane dynamics,” Photochem. Photobiol. Sci. 3, 817–822 (2004).
[CrossRef] [PubMed]

Sugiura, T.

T. Ota, T. Sugiura, S. Kawata, M. J. Booth, M. A. A. Neil, R. Juškaitis, and T. Wilson, “Enhancement of laser trapping force by spherical aberration correction using a deformable mirror,” Jpn. J. Appl. Phys. 42, L701–L703 (2003).
[CrossRef]

Török, P.

P. Török and P. R. T. Munro, “The use of Gauss–Laguerre vector beams in STED microscopy,” Opt. Express 12, 3605–3617 (2004).
[CrossRef] [PubMed]

R. M. Simmons, J. T. Finer, S. Chu, J. A. Spudich, P. Török, and P. R. T. Munro, “Quantitative measurements of force and displacement using an optical trap,” Biophys. J. 70, 1813–1822(1996).
[CrossRef] [PubMed]

Vanderkooi, J. M.

S. Fischokoff and J. M. Vanderkooi, “Oxygen diffusion in biological and artificial membranes determined by the fluorochrome pyrene,” J. Gen. Physiol. 65, 663–676 (1975).
[CrossRef]

Wagner, M.

H. Schneckenburger, M. Wagner, M. Kretzschmar, W. S. L. Strauss, and R. Sailer, “Laser-assisted fluorescence microscopy for measuring cell membrane dynamics,” Photochem. Photobiol. Sci. 3, 817–822 (2004).
[CrossRef] [PubMed]

Wei, M.-T.

Wilson, T.

T. Ota, T. Sugiura, S. Kawata, M. J. Booth, M. A. A. Neil, R. Juškaitis, and T. Wilson, “Enhancement of laser trapping force by spherical aberration correction using a deformable mirror,” Jpn. J. Appl. Phys. 42, L701–L703 (2003).
[CrossRef]

Xu, Z.

Appl. Opt. (1)

Biophys. J. (2)

A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61, 569–582 (1992).
[CrossRef] [PubMed]

R. M. Simmons, J. T. Finer, S. Chu, J. A. Spudich, P. Török, and P. R. T. Munro, “Quantitative measurements of force and displacement using an optical trap,” Biophys. J. 70, 1813–1822(1996).
[CrossRef] [PubMed]

J. Fluoresc. (1)

T. Parasassi and E. Gratton, “Packing of phospholipid vesicles studied by oxygen quenching of Laurdan fluorescence,” J. Fluoresc. 2, 167–174 (1992).
[CrossRef]

J. Gen. Physiol. (1)

S. Fischokoff and J. M. Vanderkooi, “Oxygen diffusion in biological and artificial membranes determined by the fluorochrome pyrene,” J. Gen. Physiol. 65, 663–676 (1975).
[CrossRef]

J. Mod. Opt. (1)

N. B. Simpson, D. McGloin, K. Dholakia, L. Allen, and M. J. Padgett, “Optical tweezers with increased axial trapping efficiency,” J. Mod. Opt. 45, 1943–1949 (1998).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (1)

R. Dasgupta, S. Ahlawat, and P. K. Gupta, “Trapping of micron-sized objects at a liquid–air interface,” J. Opt. A: Pure Appl. Opt. 9, S189–S195 (2007).
[CrossRef]

Jpn. J. Appl. Phys. (1)

T. Ota, T. Sugiura, S. Kawata, M. J. Booth, M. A. A. Neil, R. Juškaitis, and T. Wilson, “Enhancement of laser trapping force by spherical aberration correction using a deformable mirror,” Jpn. J. Appl. Phys. 42, L701–L703 (2003).
[CrossRef]

Opt. Commun. (1)

A. T. Oneil and M. J. Padgett, “Axial and lateral trapping efficiency of Laguerre–Gaussian modes in inverted optical tweezers,” Opt. Commun. 193, 45–50 (2001).
[CrossRef]

Opt. Express (4)

Opt. Lett. (3)

Photochem. Photobiol. Sci. (1)

H. Schneckenburger, M. Wagner, M. Kretzschmar, W. S. L. Strauss, and R. Sailer, “Laser-assisted fluorescence microscopy for measuring cell membrane dynamics,” Photochem. Photobiol. Sci. 3, 817–822 (2004).
[CrossRef] [PubMed]

Rad. Res. (1)

J. R. Dynlacht and M. H. Fox, “Heat-induced changes in membrane fluidity of Chinese hamster ovary cells measured by flow cytometry,” Rad. Res. 130, 48–54 (1992).
[CrossRef]

Other (1)

“Laurdan generalized polarization: from cuvette to microscope,” http://www.formatex.org/microscopy3/pdf/pp1007-1014.pdf, retrieved on 3 May 2010.

Supplementary Material (1)

» Media 1: MPG (3584 KB)     

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Figures (4)

Fig. 1
Fig. 1

a, Geometric optics representation of the refraction of a light beam focused through a refractive index interface with n 2 < n 1 . b, Grayscale images showing transverse focal profiles as a function of trapping distances for the (i)  TEM 00 and (ii)  LG 01 modes. c, Axial focal spread as a function of trapping distances for the (i)  TEM 00 and (ii)  LG 01 modes. d, Variation of peak intensity of the trap beam with trapping distance for different laser modes. All data shown are for laser wavelength of 532 nm . d, Inset, peak intensity of the trap beam with trapping distance for the TEM 00 and LG 01 modes at the laser wavelength of 1064 nm .

Fig. 2
Fig. 2

Schematic of the experimental setup. Lenses L1–L2 expand the laser beam so that it nearly fills the SLM pixel array. A half-wave plate (HWP) was used to orient the polarization of the input beam for phase-only operation of the SLM. The diffracted first order was selected through the iris. A three-lens zoom was used to resize the beam to suitably fill the entrance pupil of the objective lens. A polarizer (P) and quarter-wave plate (QWP) were used to control the ellipticity of the laser beam. Mirrors (M1–M4) steer the laser light path. Lens L3 was used to compensate for the effect of the tube lens (TL) on the trapping laser beam. The objective lens (MO) and the tube lens together form images of the trapped objects onto the CCD camera. The dichroic mirror (DM1) serves to couple the fluorescence excitation light onto the objective lens, and another dichroic mirror (DM2) was used to selectively reflect the trapped laser beam toward the objective lens and transmit the imaging broadband/fluorescence light to the CCD camera.

Fig. 3
Fig. 3

a, Maximum trapping depth observed for different sizes of silica microspheres using the TEM 00 and LG modes. b, Manipulation of a 3 μm silica sphere from the bottom of the medium up to the top free surface and back using the LG 01 mode (Media 1): (i) The silica sphere was trapped at the bottom layer identifiable by other untrapped spheres. (ii)–(iv) Trapped sphere being lifted from the bottom layer toward the top free surface. The untrapped spheres can be seen to be out of focus. (v) The sphere is trapped at the free top layer. As the trap focus is lifted above the top free surface, the sphere can be seen to be pushed down by surface tension force. (vi)–(viii) The trapped sphere is manipulated back toward the bottom of the medium. The thickness of open sample film is 200 μm . b, The scale bar is 5 μm .

Fig. 4
Fig. 4

a, Axial trapping range observed for COLO cells with TEM 00 and LG 01 modes. The data shown are the mean over five cells. b, Brightfield (i) and fluorescence image (ii) of a trapped COLO cell. c, Fluorescence equilibration profile of the COLO cells when incubated with 30 μg / ml Laurdan. The observed fluorescence is seen to reach a steady state after an incubation period of 25 min . d, Observed mean relative fluorescence intensities of cells when trapped at the bottom ( F 0 ) and at the top ( F eq ) surfaces. Data for cells kept at room temperature ( 25 ° C ) and cells heat treated for 1 h at an elevated temperature of 55 ° C are shown. b, The scale bar is 5 μm .

Equations (6)

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

E ( p ) = 0 α 0 2 π Ψ m , l ( θ ) cos θ P ( θ , ϕ ) exp [ i k 0 ( r p κ + ψ d ) ] exp ( i m ϕ ) sin θ d θ d ϕ ,
Ψ m , l ( θ ) = A 0 exp [ γ 2 sin 2 θ sin 2 α ] ( 2 γ sin θ sin α ) | m | L l | m | [ 2 γ 2 sin 2 θ sin 2 α ] ,
γ = a w 0 ,
r max = w 0 m 2 .
F 0 F = 1 + k τ 0 C eq ,
k = 4 π σ N A D ,

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