Abstract

We present and verify a theoretical model that predicts trapping forces (escape forces), force constants (trap stiffnesses), and trapping potential depths for dielectric spheres with diameters smaller than or equal to the wavelength of the trapping light. Optical forces can be calculated for arbitrary incident light distributions with a two-component approach that determines the gradient and the scattering force separately. We investigate the influence of spherical aberrations that are due to refractive-index mismatch on the maximum trapping force, the force constant, and the potential depth of a trap, which are important for optical tweezer applications. The relationships between the three parameters are explained and studied for different degrees of spherical aberration and various spheres (refractive indices n s = 1.39–1.57, radii a = 0.1–0.5 µm, λ0 = 1.064 µm). We find that all three parameters decrease when the distance to the coverslip increases. Effects that could make the interpretation of experimental results ambiguous are simulated and explained. Computational results are compared with the experimental data found in the literature. A good coincidence can be established.

© 2002 Optical Society of America

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2002 (1)

A. Rohrbach, H. K. Stelzer, “Three-dimensional position detection of optically trapped dielectric particles,” J. Appl. Phys. 91, 5474–5488 (2002).
[CrossRef]

2001 (1)

2000 (1)

A. C. Dogariu, R. Rajagopalan, “Optical traps as force transducers: the effects of focusing the trapping beam through a dielectric interface,” Langmuir 16, 2770–2778 (2000).
[CrossRef]

1999 (2)

A. Pralle, M. Prummer, E.-L. Florin, E. H. K. Stelzer, J. K. H. Hörber, “Three-dimensional position tracking for optical tweezers by forward scattered light,” Microsc. Res. Tech. 44, 378–386 (1999).
[CrossRef] [PubMed]

R. Drezek, A. Dunn, R. Richards-Kortum, “Light scattering from cells: finite-difference time-domain simulations and goniometric measurements,” Appl. Opt. 38, 3651–3661 (1999).
[CrossRef]

1998 (3)

E.-L. Florin, A. Pralle, E. H. K. Stelzer, J. K. H. Hörber, “Photonic force microscope calibration by thermal noise analysis,” Appl. Phys. A 66, S75–S78 (1998).
[CrossRef]

P. C. Ke, M. Gu, “Characterization of trapping force in the presence of spherical aberration,” J. Mod. Opt. 45, 2159–2168 (1998).
[CrossRef]

A. Rohrbach, W. Singer, “Scattering of a scalar field at dielectric surfaces by Born series expansion,” J. Opt. Soc. Am. A 15, 2651–2659 (1998).
[CrossRef]

1997 (2)

C. J. R. Sheppard, K. G. Larkin, “Vectorial pupil functions and vectorial transfer functions,” Optik (Stuttgart) 107, 79–87 (1997).

E.-L. Florin, A. Pralle, J. K. H. Hörber, E. H. K. Stelzer, “Photonic force microscope (PFM) based on optical tweezers and two-photon excitation for biological applications,” J. Struct. Biol. 119, 202–211 (1997).
[CrossRef] [PubMed]

1996 (4)

T. Wohland, A. Rosin, E. H. K. Stelzer, “Theoretical determination of the influence of the polarization on forces exerted by optical tweezers,” Optik (Stuttgart) 102, 181–190 (1996).

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

Y. Harada, T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124, 529–541 (1996).
[CrossRef]

M. E. J. Friese, H. Rubinsztein-Dunlop, N. R. Heckenberg, 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]

1995 (4)

1994 (4)

F. Ren, G. Gréhan, G. Gouesbet, “Radiation pressure forces exerted on a particle located arbitrarily in a Gaussian beam by using the generalized Lorentz–Mie theory and associated resonance effects,” Opt. Commun. 108, 343–354 (1994).
[CrossRef]

K. Svoboda, S. M. Block, “Biological applications of optical forces,” Ann. Rev. Biophys. Biomol. Struct. 23, 247–285 (1994).
[CrossRef]

L. Ghislain, N. Switz, W. Webb, “Measurement of small forces using an optical trap,” Rev. Sci. Instrum. 65, 2762–2768 (1994).
[CrossRef]

W. H. Wright, G. J. Sonek, M. W. Berns, “Parametric study of the forces on microspheres held by optical tweezers,” Appl. Opt. 33, 1735–1748 (1994).
[CrossRef] [PubMed]

1993 (5)

L. P. Ghislain, W. W. Webb, “Scanning-force microscope based on an optical trap,” Opt. Lett. 18, 1678–1680 (1993).
[CrossRef] [PubMed]

S. C. Kuo, M. P. Sheetz, “Force of single kinesin molecules measured with optical tweezers,” Science 260, 232–234 (1993).
[CrossRef] [PubMed]

K. Svoboda, C. F. Schmidt, B. J. Schnapp, S. M. Block, “Direct observation of kinesin stepping by optical trapping interferometry,” Nature 365, 721–727 (1993).
[CrossRef] [PubMed]

S. Hell, G. Reiner, C. Cremer, E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index,” J. Microsc. 169, 391–405 (1993).
[CrossRef]

W. H. Wright, G. J. Sonek, M. W. Berns, “Radiation trapping forces on microspheres with optical tweezers,” Appl. Phys. Lett. 63, 715–717 (1993).
[CrossRef]

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

1991 (3)

S. Seeger, S. Monajembashi, K. J. Hutter, G. Futterman, J. Wolfrum, K. O. Greulich, “Application of laser optical tweezers in immunology and molecular genetics,” Cytometry 12, 497–504 (1991).
[CrossRef] [PubMed]

R. W. Steubing, S. Cheng, W. H. Wright, Y. Numajiri, M. W. Berns, “Laser induced cell fusion in combination with optical tweezers: the laser cell fusion trap,” Cytometry 12, 505–510 (1991).
[CrossRef] [PubMed]

T. D. Visser, G. J. Brakenhoff, F. C. A. Groen, “The one-point fluorescence response in confocal microscopy,” Optik (Stuttgart) 87, 39–40 (1991).

1990 (2)

S. M. Block, L. S. Goldstein, B. J. Schnapp, “Bead movement by single kinesin molecules studied with optical tweezers,” Nature 348, 348–352 (1990).
[CrossRef] [PubMed]

A. Ashkin, K. Schutze, J. M. Dziedzic, U. Euteneuer, M. Schliwa, “Force generation of organelle transport measured in vivo by an infrared laser trap,” Nature 348, 346–348 (1990).
[CrossRef] [PubMed]

1989 (5)

S. M. Block, D. F. Blair, H. C. Berg, “Compliance of bacterial flagella measured with optical tweezers,” Nature 338, 514–518 (1989).
[CrossRef] [PubMed]

M. W. Berns, W. H. Wright, B. J. Tromberg, G. A. Profeta, J. J. Andrews, R. J. Walter, “Use of a laser-induced optical force trap to study chromosome movement on the mitotic spindle,” Proc. Natl. Acad. Sci. USA 86, 7914–7918 (1989).
[CrossRef]

M. Mansuripur, “Certain computational aspects of vector diffraction problems,” J. Opt. Soc. Am. A 6, 786–805 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4602 (1989).
[CrossRef]

A. L. Stout, D. Axelrod, “Evanescent field excitation of fluorescence by epiillumination microscopy,” Appl. Opt. 28, 5237–5242 (1989).
[CrossRef] [PubMed]

1987 (1)

A. Ashkin, J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science 235, 1517–1520 (1987).
[CrossRef] [PubMed]

1986 (2)

1985 (1)

1983 (1)

1973 (1)

J. P. Gordon, “Radiation forces and momenta in dielectric media,” Phys. Rev. 8, 14–21 (1973).
[CrossRef]

1964 (1)

C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image,” J. Opt. Soc. Am 54, 240–244 (1964).
[CrossRef]

1908 (1)

G. Mie, “Beitraege zur Optik trueber Medien speziell kolloidaler Metalloesungen,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

Alexander, D. R.

J. P. Barton, D. R. Alexander, S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4602 (1989).
[CrossRef]

Almers, W.

T. D. Parsons, J. R. Coorssen, H. Horstmann, W. Almers, “Docked granules, the exocytotic burst, and the need for ATP hydrolysis in endocrine cells,” Neuron 15, 1085–1096 (1995).
[CrossRef] [PubMed]

Andrews, J. J.

M. W. Berns, W. H. Wright, B. J. Tromberg, G. A. Profeta, J. J. Andrews, R. J. Walter, “Use of a laser-induced optical force trap to study chromosome movement on the mitotic spindle,” Proc. Natl. Acad. Sci. USA 86, 7914–7918 (1989).
[CrossRef]

Asakura, T.

Y. Harada, T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124, 529–541 (1996).
[CrossRef]

Ashkin, A.

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]

A. Ashkin, K. Schutze, J. M. Dziedzic, U. Euteneuer, M. Schliwa, “Force generation of organelle transport measured in vivo by an infrared laser trap,” Nature 348, 346–348 (1990).
[CrossRef] [PubMed]

A. Ashkin, J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science 235, 1517–1520 (1987).
[CrossRef] [PubMed]

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

Axelrod, D.

Barton, J. P.

J. P. Barton, D. R. Alexander, S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4602 (1989).
[CrossRef]

Berg, H. C.

S. M. Block, D. F. Blair, H. C. Berg, “Compliance of bacterial flagella measured with optical tweezers,” Nature 338, 514–518 (1989).
[CrossRef] [PubMed]

Berns, M. W.

W. H. Wright, G. J. Sonek, M. W. Berns, “Parametric study of the forces on microspheres held by optical tweezers,” Appl. Opt. 33, 1735–1748 (1994).
[CrossRef] [PubMed]

W. H. Wright, G. J. Sonek, M. W. Berns, “Radiation trapping forces on microspheres with optical tweezers,” Appl. Phys. Lett. 63, 715–717 (1993).
[CrossRef]

R. W. Steubing, S. Cheng, W. H. Wright, Y. Numajiri, M. W. Berns, “Laser induced cell fusion in combination with optical tweezers: the laser cell fusion trap,” Cytometry 12, 505–510 (1991).
[CrossRef] [PubMed]

M. W. Berns, W. H. Wright, B. J. Tromberg, G. A. Profeta, J. J. Andrews, R. J. Walter, “Use of a laser-induced optical force trap to study chromosome movement on the mitotic spindle,” Proc. Natl. Acad. Sci. USA 86, 7914–7918 (1989).
[CrossRef]

Bjorkholm, J. E.

Blair, D. F.

S. M. Block, D. F. Blair, H. C. Berg, “Compliance of bacterial flagella measured with optical tweezers,” Nature 338, 514–518 (1989).
[CrossRef] [PubMed]

Block, S. M.

K. Svoboda, S. M. Block, “Biological applications of optical forces,” Ann. Rev. Biophys. Biomol. Struct. 23, 247–285 (1994).
[CrossRef]

K. Svoboda, C. F. Schmidt, B. J. Schnapp, S. M. Block, “Direct observation of kinesin stepping by optical trapping interferometry,” Nature 365, 721–727 (1993).
[CrossRef] [PubMed]

S. M. Block, L. S. Goldstein, B. J. Schnapp, “Bead movement by single kinesin molecules studied with optical tweezers,” Nature 348, 348–352 (1990).
[CrossRef] [PubMed]

S. M. Block, D. F. Blair, H. C. Berg, “Compliance of bacterial flagella measured with optical tweezers,” Nature 338, 514–518 (1989).
[CrossRef] [PubMed]

Booker, G. R.

Brakenhoff, G. J.

T. D. Visser, G. J. Brakenhoff, F. C. A. Groen, “The one-point fluorescence response in confocal microscopy,” Optik (Stuttgart) 87, 39–40 (1991).

Brenner, K.-H.

Brevik, I.

Cheng, S.

R. W. Steubing, S. Cheng, W. H. Wright, Y. Numajiri, M. W. Berns, “Laser induced cell fusion in combination with optical tweezers: the laser cell fusion trap,” Cytometry 12, 505–510 (1991).
[CrossRef] [PubMed]

Chu, S.

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

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

Coorssen, J. R.

T. D. Parsons, J. R. Coorssen, H. Horstmann, W. Almers, “Docked granules, the exocytotic burst, and the need for ATP hydrolysis in endocrine cells,” Neuron 15, 1085–1096 (1995).
[CrossRef] [PubMed]

Cremer, C.

S. Hell, G. Reiner, C. Cremer, E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index,” J. Microsc. 169, 391–405 (1993).
[CrossRef]

Dearden, E. W.

Dogariu, A. C.

A. C. Dogariu, R. Rajagopalan, “Optical traps as force transducers: the effects of focusing the trapping beam through a dielectric interface,” Langmuir 16, 2770–2778 (2000).
[CrossRef]

Drezek, R.

Dunn, A.

Dziedzic, J. M.

A. Ashkin, K. Schutze, J. M. Dziedzic, U. Euteneuer, M. Schliwa, “Force generation of organelle transport measured in vivo by an infrared laser trap,” Nature 348, 346–348 (1990).
[CrossRef] [PubMed]

A. Ashkin, J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science 235, 1517–1520 (1987).
[CrossRef] [PubMed]

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

Euteneuer, U.

A. Ashkin, K. Schutze, J. M. Dziedzic, U. Euteneuer, M. Schliwa, “Force generation of organelle transport measured in vivo by an infrared laser trap,” Nature 348, 346–348 (1990).
[CrossRef] [PubMed]

Felgner, H.

Finer, J. T.

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

Florin, E.-L.

A. Pralle, M. Prummer, E.-L. Florin, E. H. K. Stelzer, J. K. H. Hörber, “Three-dimensional position tracking for optical tweezers by forward scattered light,” Microsc. Res. Tech. 44, 378–386 (1999).
[CrossRef] [PubMed]

E.-L. Florin, A. Pralle, E. H. K. Stelzer, J. K. H. Hörber, “Photonic force microscope calibration by thermal noise analysis,” Appl. Phys. A 66, S75–S78 (1998).
[CrossRef]

E.-L. Florin, A. Pralle, J. K. H. Hörber, E. H. K. Stelzer, “Photonic force microscope (PFM) based on optical tweezers and two-photon excitation for biological applications,” J. Struct. Biol. 119, 202–211 (1997).
[CrossRef] [PubMed]

Friese, M. E. J.

Futterman, G.

S. Seeger, S. Monajembashi, K. J. Hutter, G. Futterman, J. Wolfrum, K. O. Greulich, “Application of laser optical tweezers in immunology and molecular genetics,” Cytometry 12, 497–504 (1991).
[CrossRef] [PubMed]

Ghislain, L.

L. Ghislain, N. Switz, W. Webb, “Measurement of small forces using an optical trap,” Rev. Sci. Instrum. 65, 2762–2768 (1994).
[CrossRef]

Ghislain, L. P.

Goldstein, L. S.

S. M. Block, L. S. Goldstein, B. J. Schnapp, “Bead movement by single kinesin molecules studied with optical tweezers,” Nature 348, 348–352 (1990).
[CrossRef] [PubMed]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 1st ed. (McGraw-Hill, San Francisco, 1968), pp. 48–56.

Gordon, J. P.

J. P. Gordon, “Radiation forces and momenta in dielectric media,” Phys. Rev. 8, 14–21 (1973).
[CrossRef]

Gouesbet, G.

F. Ren, G. Gréhan, G. Gouesbet, “Radiation pressure forces exerted on a particle located arbitrarily in a Gaussian beam by using the generalized Lorentz–Mie theory and associated resonance effects,” Opt. Commun. 108, 343–354 (1994).
[CrossRef]

Gréhan, G.

F. Ren, G. Gréhan, G. Gouesbet, “Radiation pressure forces exerted on a particle located arbitrarily in a Gaussian beam by using the generalized Lorentz–Mie theory and associated resonance effects,” Opt. Commun. 108, 343–354 (1994).
[CrossRef]

Greulich, K. O.

S. Seeger, S. Monajembashi, K. J. Hutter, G. Futterman, J. Wolfrum, K. O. Greulich, “Application of laser optical tweezers in immunology and molecular genetics,” Cytometry 12, 497–504 (1991).
[CrossRef] [PubMed]

Groen, F. C. A.

T. D. Visser, G. J. Brakenhoff, F. C. A. Groen, “The one-point fluorescence response in confocal microscopy,” Optik (Stuttgart) 87, 39–40 (1991).

Gu, M.

P. C. Ke, M. Gu, “Characterization of trapping force in the presence of spherical aberration,” J. Mod. Opt. 45, 2159–2168 (1998).
[CrossRef]

Gussgard, R.

Harada, Y.

Y. Harada, T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124, 529–541 (1996).
[CrossRef]

Heckenberg, N. R.

Hell, S.

S. Hell, G. Reiner, C. Cremer, E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index,” J. Microsc. 169, 391–405 (1993).
[CrossRef]

Hörber, J. K. H.

A. Pralle, M. Prummer, E.-L. Florin, E. H. K. Stelzer, J. K. H. Hörber, “Three-dimensional position tracking for optical tweezers by forward scattered light,” Microsc. Res. Tech. 44, 378–386 (1999).
[CrossRef] [PubMed]

E.-L. Florin, A. Pralle, E. H. K. Stelzer, J. K. H. Hörber, “Photonic force microscope calibration by thermal noise analysis,” Appl. Phys. A 66, S75–S78 (1998).
[CrossRef]

E.-L. Florin, A. Pralle, J. K. H. Hörber, E. H. K. Stelzer, “Photonic force microscope (PFM) based on optical tweezers and two-photon excitation for biological applications,” J. Struct. Biol. 119, 202–211 (1997).
[CrossRef] [PubMed]

Horstmann, H.

T. D. Parsons, J. R. Coorssen, H. Horstmann, W. Almers, “Docked granules, the exocytotic burst, and the need for ATP hydrolysis in endocrine cells,” Neuron 15, 1085–1096 (1995).
[CrossRef] [PubMed]

Hutter, K. J.

S. Seeger, S. Monajembashi, K. J. Hutter, G. Futterman, J. Wolfrum, K. O. Greulich, “Application of laser optical tweezers in immunology and molecular genetics,” Cytometry 12, 497–504 (1991).
[CrossRef] [PubMed]

Jackson, J. D.

J. D. Jackson, Classial Electrodynamics, 2nd ed. (Wiley, New York, 1975).

Ke, P. C.

P. C. Ke, M. Gu, “Characterization of trapping force in the presence of spherical aberration,” J. Mod. Opt. 45, 2159–2168 (1998).
[CrossRef]

Kim, J. S.

Kuo, S. C.

S. C. Kuo, M. P. Sheetz, “Force of single kinesin molecules measured with optical tweezers,” Science 260, 232–234 (1993).
[CrossRef] [PubMed]

Laczik, Z.

Larkin, K. G.

C. J. R. Sheppard, K. G. Larkin, “Vectorial pupil functions and vectorial transfer functions,” Optik (Stuttgart) 107, 79–87 (1997).

Lee, S. S.

Lindmo, T.

Mansuripur, M.

McCutchen, C. W.

C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image,” J. Opt. Soc. Am 54, 240–244 (1964).
[CrossRef]

Mie, G.

G. Mie, “Beitraege zur Optik trueber Medien speziell kolloidaler Metalloesungen,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

Monajembashi, S.

S. Seeger, S. Monajembashi, K. J. Hutter, G. Futterman, J. Wolfrum, K. O. Greulich, “Application of laser optical tweezers in immunology and molecular genetics,” Cytometry 12, 497–504 (1991).
[CrossRef] [PubMed]

Müller, O.

Mulser, P.

Numajiri, Y.

R. W. Steubing, S. Cheng, W. H. Wright, Y. Numajiri, M. W. Berns, “Laser induced cell fusion in combination with optical tweezers: the laser cell fusion trap,” Cytometry 12, 505–510 (1991).
[CrossRef] [PubMed]

Parsons, T. D.

T. D. Parsons, J. R. Coorssen, H. Horstmann, W. Almers, “Docked granules, the exocytotic burst, and the need for ATP hydrolysis in endocrine cells,” Neuron 15, 1085–1096 (1995).
[CrossRef] [PubMed]

Pralle, A.

A. Pralle, M. Prummer, E.-L. Florin, E. H. K. Stelzer, J. K. H. Hörber, “Three-dimensional position tracking for optical tweezers by forward scattered light,” Microsc. Res. Tech. 44, 378–386 (1999).
[CrossRef] [PubMed]

E.-L. Florin, A. Pralle, E. H. K. Stelzer, J. K. H. Hörber, “Photonic force microscope calibration by thermal noise analysis,” Appl. Phys. A 66, S75–S78 (1998).
[CrossRef]

E.-L. Florin, A. Pralle, J. K. H. Hörber, E. H. K. Stelzer, “Photonic force microscope (PFM) based on optical tweezers and two-photon excitation for biological applications,” J. Struct. Biol. 119, 202–211 (1997).
[CrossRef] [PubMed]

Profeta, G. A.

M. W. Berns, W. H. Wright, B. J. Tromberg, G. A. Profeta, J. J. Andrews, R. J. Walter, “Use of a laser-induced optical force trap to study chromosome movement on the mitotic spindle,” Proc. Natl. Acad. Sci. USA 86, 7914–7918 (1989).
[CrossRef]

Prummer, M.

A. Pralle, M. Prummer, E.-L. Florin, E. H. K. Stelzer, J. K. H. Hörber, “Three-dimensional position tracking for optical tweezers by forward scattered light,” Microsc. Res. Tech. 44, 378–386 (1999).
[CrossRef] [PubMed]

Rajagopalan, R.

A. C. Dogariu, R. Rajagopalan, “Optical traps as force transducers: the effects of focusing the trapping beam through a dielectric interface,” Langmuir 16, 2770–2778 (2000).
[CrossRef]

Reif, F.

F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, Auckland, New Zealand, 1985), pp.560–565.

Reiner, G.

S. Hell, G. Reiner, C. Cremer, E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index,” J. Microsc. 169, 391–405 (1993).
[CrossRef]

Ren, F.

F. Ren, G. Gréhan, G. Gouesbet, “Radiation pressure forces exerted on a particle located arbitrarily in a Gaussian beam by using the generalized Lorentz–Mie theory and associated resonance effects,” Opt. Commun. 108, 343–354 (1994).
[CrossRef]

Richards-Kortum, R.

Rohrbach, A.

Rosin, A.

T. Wohland, A. Rosin, E. H. K. Stelzer, “Theoretical determination of the influence of the polarization on forces exerted by optical tweezers,” Optik (Stuttgart) 102, 181–190 (1996).

Rubinsztein-Dunlop, H.

Schaub, S. A.

J. P. Barton, D. R. Alexander, S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4602 (1989).
[CrossRef]

Schliwa, M.

H. Felgner, O. Müller, M. Schliwa, “Calibration of light forces in optical tweezers,” Appl. Opt. 34, 977–982 (1995).
[CrossRef] [PubMed]

A. Ashkin, K. Schutze, J. M. Dziedzic, U. Euteneuer, M. Schliwa, “Force generation of organelle transport measured in vivo by an infrared laser trap,” Nature 348, 346–348 (1990).
[CrossRef] [PubMed]

Schmidt, C. F.

K. Svoboda, C. F. Schmidt, B. J. Schnapp, S. M. Block, “Direct observation of kinesin stepping by optical trapping interferometry,” Nature 365, 721–727 (1993).
[CrossRef] [PubMed]

Schnapp, B. J.

K. Svoboda, C. F. Schmidt, B. J. Schnapp, S. M. Block, “Direct observation of kinesin stepping by optical trapping interferometry,” Nature 365, 721–727 (1993).
[CrossRef] [PubMed]

S. M. Block, L. S. Goldstein, B. J. Schnapp, “Bead movement by single kinesin molecules studied with optical tweezers,” Nature 348, 348–352 (1990).
[CrossRef] [PubMed]

Schutze, K.

A. Ashkin, K. Schutze, J. M. Dziedzic, U. Euteneuer, M. Schliwa, “Force generation of organelle transport measured in vivo by an infrared laser trap,” Nature 348, 346–348 (1990).
[CrossRef] [PubMed]

Seeger, S.

S. Seeger, S. Monajembashi, K. J. Hutter, G. Futterman, J. Wolfrum, K. O. Greulich, “Application of laser optical tweezers in immunology and molecular genetics,” Cytometry 12, 497–504 (1991).
[CrossRef] [PubMed]

Sheetz, M. P.

S. C. Kuo, M. P. Sheetz, “Force of single kinesin molecules measured with optical tweezers,” Science 260, 232–234 (1993).
[CrossRef] [PubMed]

Sheppard, C. J. R.

C. J. R. Sheppard, K. G. Larkin, “Vectorial pupil functions and vectorial transfer functions,” Optik (Stuttgart) 107, 79–87 (1997).

Simmons, R. M.

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

Singer, W.

Sonek, G. J.

W. H. Wright, G. J. Sonek, M. W. Berns, “Parametric study of the forces on microspheres held by optical tweezers,” Appl. Opt. 33, 1735–1748 (1994).
[CrossRef] [PubMed]

W. H. Wright, G. J. Sonek, M. W. Berns, “Radiation trapping forces on microspheres with optical tweezers,” Appl. Phys. Lett. 63, 715–717 (1993).
[CrossRef]

Spudich, J. A.

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

Stelzer, E. H. K.

A. Pralle, M. Prummer, E.-L. Florin, E. H. K. Stelzer, J. K. H. Hörber, “Three-dimensional position tracking for optical tweezers by forward scattered light,” Microsc. Res. Tech. 44, 378–386 (1999).
[CrossRef] [PubMed]

E.-L. Florin, A. Pralle, E. H. K. Stelzer, J. K. H. Hörber, “Photonic force microscope calibration by thermal noise analysis,” Appl. Phys. A 66, S75–S78 (1998).
[CrossRef]

E.-L. Florin, A. Pralle, J. K. H. Hörber, E. H. K. Stelzer, “Photonic force microscope (PFM) based on optical tweezers and two-photon excitation for biological applications,” J. Struct. Biol. 119, 202–211 (1997).
[CrossRef] [PubMed]

T. Wohland, A. Rosin, E. H. K. Stelzer, “Theoretical determination of the influence of the polarization on forces exerted by optical tweezers,” Optik (Stuttgart) 102, 181–190 (1996).

S. Hell, G. Reiner, C. Cremer, E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index,” J. Microsc. 169, 391–405 (1993).
[CrossRef]

Stelzer, H. K.

A. Rohrbach, H. K. Stelzer, “Three-dimensional position detection of optically trapped dielectric particles,” J. Appl. Phys. 91, 5474–5488 (2002).
[CrossRef]

A. Rohrbach, H. K. Stelzer, “Optical trapping of dielectric particles in arbitrary fields,” J. Opt. Soc. Am. A 18, 839–853 (2001).
[CrossRef]

Steubing, R. W.

R. W. Steubing, S. Cheng, W. H. Wright, Y. Numajiri, M. W. Berns, “Laser induced cell fusion in combination with optical tweezers: the laser cell fusion trap,” Cytometry 12, 505–510 (1991).
[CrossRef] [PubMed]

Stout, A. L.

Svoboda, K.

K. Svoboda, S. M. Block, “Biological applications of optical forces,” Ann. Rev. Biophys. Biomol. Struct. 23, 247–285 (1994).
[CrossRef]

K. Svoboda, C. F. Schmidt, B. J. Schnapp, S. M. Block, “Direct observation of kinesin stepping by optical trapping interferometry,” Nature 365, 721–727 (1993).
[CrossRef] [PubMed]

Switz, N.

L. Ghislain, N. Switz, W. Webb, “Measurement of small forces using an optical trap,” Rev. Sci. Instrum. 65, 2762–2768 (1994).
[CrossRef]

Tischer, C.

C. Tischer, European Molecular Biology Laboratory, Meyerhofstrasse 1, Heidelberg D-69117, Germany (personal communication, 2000).

Török, P.

Tromberg, B. J.

M. W. Berns, W. H. Wright, B. J. Tromberg, G. A. Profeta, J. J. Andrews, R. J. Walter, “Use of a laser-induced optical force trap to study chromosome movement on the mitotic spindle,” Proc. Natl. Acad. Sci. USA 86, 7914–7918 (1989).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1957).

Varga, P.

Visser, T. D.

T. D. Visser, G. J. Brakenhoff, F. C. A. Groen, “The one-point fluorescence response in confocal microscopy,” Optik (Stuttgart) 87, 39–40 (1991).

Walter, R. J.

M. W. Berns, W. H. Wright, B. J. Tromberg, G. A. Profeta, J. J. Andrews, R. J. Walter, “Use of a laser-induced optical force trap to study chromosome movement on the mitotic spindle,” Proc. Natl. Acad. Sci. USA 86, 7914–7918 (1989).
[CrossRef]

Webb, W.

L. Ghislain, N. Switz, W. Webb, “Measurement of small forces using an optical trap,” Rev. Sci. Instrum. 65, 2762–2768 (1994).
[CrossRef]

Webb, W. W.

Wohland, T.

T. Wohland, A. Rosin, E. H. K. Stelzer, “Theoretical determination of the influence of the polarization on forces exerted by optical tweezers,” Optik (Stuttgart) 102, 181–190 (1996).

Wolfrum, J.

S. Seeger, S. Monajembashi, K. J. Hutter, G. Futterman, J. Wolfrum, K. O. Greulich, “Application of laser optical tweezers in immunology and molecular genetics,” Cytometry 12, 497–504 (1991).
[CrossRef] [PubMed]

Wright, W. H.

W. H. Wright, G. J. Sonek, M. W. Berns, “Parametric study of the forces on microspheres held by optical tweezers,” Appl. Opt. 33, 1735–1748 (1994).
[CrossRef] [PubMed]

W. H. Wright, G. J. Sonek, M. W. Berns, “Radiation trapping forces on microspheres with optical tweezers,” Appl. Phys. Lett. 63, 715–717 (1993).
[CrossRef]

R. W. Steubing, S. Cheng, W. H. Wright, Y. Numajiri, M. W. Berns, “Laser induced cell fusion in combination with optical tweezers: the laser cell fusion trap,” Cytometry 12, 505–510 (1991).
[CrossRef] [PubMed]

M. W. Berns, W. H. Wright, B. J. Tromberg, G. A. Profeta, J. J. Andrews, R. J. Walter, “Use of a laser-induced optical force trap to study chromosome movement on the mitotic spindle,” Proc. Natl. Acad. Sci. USA 86, 7914–7918 (1989).
[CrossRef]

Ann. Phys. (1)

G. Mie, “Beitraege zur Optik trueber Medien speziell kolloidaler Metalloesungen,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

Ann. Rev. Biophys. Biomol. Struct. (1)

K. Svoboda, S. M. Block, “Biological applications of optical forces,” Ann. Rev. Biophys. Biomol. Struct. 23, 247–285 (1994).
[CrossRef]

Appl. Opt. (5)

Appl. Phys. A (1)

E.-L. Florin, A. Pralle, E. H. K. Stelzer, J. K. H. Hörber, “Photonic force microscope calibration by thermal noise analysis,” Appl. Phys. A 66, S75–S78 (1998).
[CrossRef]

Appl. Phys. Lett. (1)

W. H. Wright, G. J. Sonek, M. W. Berns, “Radiation trapping forces on microspheres with optical tweezers,” Appl. Phys. Lett. 63, 715–717 (1993).
[CrossRef]

Biophys. J. (2)

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

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]

Cytometry (2)

S. Seeger, S. Monajembashi, K. J. Hutter, G. Futterman, J. Wolfrum, K. O. Greulich, “Application of laser optical tweezers in immunology and molecular genetics,” Cytometry 12, 497–504 (1991).
[CrossRef] [PubMed]

R. W. Steubing, S. Cheng, W. H. Wright, Y. Numajiri, M. W. Berns, “Laser induced cell fusion in combination with optical tweezers: the laser cell fusion trap,” Cytometry 12, 505–510 (1991).
[CrossRef] [PubMed]

J. Appl. Phys. (2)

J. P. Barton, D. R. Alexander, S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4602 (1989).
[CrossRef]

A. Rohrbach, H. K. Stelzer, “Three-dimensional position detection of optically trapped dielectric particles,” J. Appl. Phys. 91, 5474–5488 (2002).
[CrossRef]

J. Microsc. (1)

S. Hell, G. Reiner, C. Cremer, E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index,” J. Microsc. 169, 391–405 (1993).
[CrossRef]

J. Mod. Opt. (1)

P. C. Ke, M. Gu, “Characterization of trapping force in the presence of spherical aberration,” J. Mod. Opt. 45, 2159–2168 (1998).
[CrossRef]

J. Opt. Soc. Am (1)

C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image,” J. Opt. Soc. Am 54, 240–244 (1964).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (6)

J. Opt. Soc. Am. B (2)

J. Struct. Biol. (1)

E.-L. Florin, A. Pralle, J. K. H. Hörber, E. H. K. Stelzer, “Photonic force microscope (PFM) based on optical tweezers and two-photon excitation for biological applications,” J. Struct. Biol. 119, 202–211 (1997).
[CrossRef] [PubMed]

Langmuir (1)

A. C. Dogariu, R. Rajagopalan, “Optical traps as force transducers: the effects of focusing the trapping beam through a dielectric interface,” Langmuir 16, 2770–2778 (2000).
[CrossRef]

Microsc. Res. Tech. (1)

A. Pralle, M. Prummer, E.-L. Florin, E. H. K. Stelzer, J. K. H. Hörber, “Three-dimensional position tracking for optical tweezers by forward scattered light,” Microsc. Res. Tech. 44, 378–386 (1999).
[CrossRef] [PubMed]

Nature (4)

K. Svoboda, C. F. Schmidt, B. J. Schnapp, S. M. Block, “Direct observation of kinesin stepping by optical trapping interferometry,” Nature 365, 721–727 (1993).
[CrossRef] [PubMed]

S. M. Block, D. F. Blair, H. C. Berg, “Compliance of bacterial flagella measured with optical tweezers,” Nature 338, 514–518 (1989).
[CrossRef] [PubMed]

S. M. Block, L. S. Goldstein, B. J. Schnapp, “Bead movement by single kinesin molecules studied with optical tweezers,” Nature 348, 348–352 (1990).
[CrossRef] [PubMed]

A. Ashkin, K. Schutze, J. M. Dziedzic, U. Euteneuer, M. Schliwa, “Force generation of organelle transport measured in vivo by an infrared laser trap,” Nature 348, 346–348 (1990).
[CrossRef] [PubMed]

Neuron (1)

T. D. Parsons, J. R. Coorssen, H. Horstmann, W. Almers, “Docked granules, the exocytotic burst, and the need for ATP hydrolysis in endocrine cells,” Neuron 15, 1085–1096 (1995).
[CrossRef] [PubMed]

Opt. Commun. (2)

F. Ren, G. Gréhan, G. Gouesbet, “Radiation pressure forces exerted on a particle located arbitrarily in a Gaussian beam by using the generalized Lorentz–Mie theory and associated resonance effects,” Opt. Commun. 108, 343–354 (1994).
[CrossRef]

Y. Harada, T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124, 529–541 (1996).
[CrossRef]

Opt. Lett. (2)

Optik (Stuttgart) (3)

T. Wohland, A. Rosin, E. H. K. Stelzer, “Theoretical determination of the influence of the polarization on forces exerted by optical tweezers,” Optik (Stuttgart) 102, 181–190 (1996).

T. D. Visser, G. J. Brakenhoff, F. C. A. Groen, “The one-point fluorescence response in confocal microscopy,” Optik (Stuttgart) 87, 39–40 (1991).

C. J. R. Sheppard, K. G. Larkin, “Vectorial pupil functions and vectorial transfer functions,” Optik (Stuttgart) 107, 79–87 (1997).

Phys. Rev. (1)

J. P. Gordon, “Radiation forces and momenta in dielectric media,” Phys. Rev. 8, 14–21 (1973).
[CrossRef]

Proc. Natl. Acad. Sci. USA (1)

M. W. Berns, W. H. Wright, B. J. Tromberg, G. A. Profeta, J. J. Andrews, R. J. Walter, “Use of a laser-induced optical force trap to study chromosome movement on the mitotic spindle,” Proc. Natl. Acad. Sci. USA 86, 7914–7918 (1989).
[CrossRef]

Rev. Sci. Instrum. (1)

L. Ghislain, N. Switz, W. Webb, “Measurement of small forces using an optical trap,” Rev. Sci. Instrum. 65, 2762–2768 (1994).
[CrossRef]

Science (2)

A. Ashkin, J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science 235, 1517–1520 (1987).
[CrossRef] [PubMed]

S. C. Kuo, M. P. Sheetz, “Force of single kinesin molecules measured with optical tweezers,” Science 260, 232–234 (1993).
[CrossRef] [PubMed]

Other (5)

J. D. Jackson, Classial Electrodynamics, 2nd ed. (Wiley, New York, 1975).

J. W. Goodman, Introduction to Fourier Optics, 1st ed. (McGraw-Hill, San Francisco, 1968), pp. 48–56.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1957).

C. Tischer, European Molecular Biology Laboratory, Meyerhofstrasse 1, Heidelberg D-69117, Germany (personal communication, 2000).

F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, Auckland, New Zealand, 1985), pp.560–565.

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Figures (9)

Fig. 1
Fig. 1

Spherical wave front with phase ϕ(x, y, z) = const in a medium with refractive index n i impinges on a plane dielectric interface. After transition into the second medium with index n m the wave front and thus the spectrum of k vectors change. The axial component of the k-vector changes according to Δk z = k zi - k mi because of the change in the radius of the Ewald sphere. This results in a phase shift of Δϕ = dΔ k z , where d is the axial distance of the NFP to the interface. The AFP is now at z = -FS.

Fig. 2
Fig. 2

Coordinate system of propagation and scattering. A dielectric sphere with permittivity ε s = n s 2 and radius a is located in a medium with permittivity ε m = n m 2. The sphere is shifted by the vector b = (b x , b y , b z ) from the origin (0, 0, 0). The normalized intensities I ∝ | E m (x, 0, z)|2 of highly focused beams that propagate along the z direction are shown in the background. On the left-hand side an ideal focus is centered at the origin, on the right-hand side a spherical aberrated focus is shifted by the FS distance toward the lens. The line scans through the points of maximal intensities are shown at far right and at the bottom.

Fig. 3
Fig. 3

Change of the peak intensity, the focus shift, and the FWHM as a function of the nominal distance. The total optical power that enters the immersion medium was kept constant. In (a) the peak intensity is plotted for two different illuminations of the back focal plane of the objective lens (aperture radius R and Gaussian beam waist w 0). In (b) we show the small change in lateral focus widths in comparison with the strong increase in axial focus width.

Fig. 4
Fig. 4

Trapping efficiency and trapping potential of a 216-nm polystyrene sphere (n s = 1.57) in water as a function of the axial position. (a) The two-component force profile is the sum of the scattering force and the gradient force. Both are plotted for an ideal focus (n m 0.9 NA) and for an aberrated focus from an oil-immersion lens (n i 0.9 NA, n i = 1.52) at a distance of d = 20 µm to the interface. (b) Corresponding axial trapping potentials at a laser power of P = 10 mW in the focus. The energy differences ΔW and δW are explained in the text.

Fig. 5
Fig. 5

Lateral trapping efficiency of a 216-nm polystyrene sphere (n s = 1.57) in water as a function of lateral position. Trapping efficiencies are stronger in the y direction for the x-polarized electric field. The lateral trapping efficiencies are much smaller in the presence of spherical aberrations (d = 20 µm).

Fig. 6
Fig. 6

Lateral and axial force constants in the presence of spherical aberrations as a function of the nominal distance to the coverslip. The lateral x-force constants are plotted versus the right axis (dashed curves), the axial force constants versus the left axis (solid curves). Please note the different axis scaling in all four graphs.

Fig. 7
Fig. 7

Ratios of the force constant of the trap over the maximum (backward) trapping force as a function of the distance to the coverslip. (a) For various sphere parameters the force profiles in the lateral x direction change slightly when the distance to the interface increases. (b) The decrease of the force constant in the axial direction has a nonlinear dependence on the potential depth decrease.

Fig. 8
Fig. 8

Ratios of the force constant of the trap over the potential depth as a function of the distance to the coverslip. (a) For various sphere parameters the force profiles in the lateral x direction change slightly when the distance to the interface increases. (b) The decrease of the force constant in the axial direction is by no means simply proportional to the decrease in potential depth.

Fig. 9
Fig. 9

Top and side views of projected 3-D paths of a sphere that diffuses inside an optical trap. The left-hand side shows the path (that is due to Brownian motion) in an ideal focus, whereas the right-hand side shows the bead trapped in an aberrated focus (d = 10 µm). The sphere diameter is 216 nm (outlined by a ring in the center of the trap), the refractive index is n = 1.57. The laser power is P = 20 mW in both cases.

Tables (1)

Tables Icon

Table 1 Comparison of Measured and Calculated Values of Force Constants f and Maximum Backward Trapping Efficiencies Q b a

Equations (19)

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|k|2-kn2E˜k=0  |k|2=kx2+ky2+kz2=kn2,
Efx, y, z=12π3 Akx, ky, kn2-kx2-ky21/2exp-ikrdkxdkydkz.
Ak=A+kx, kyδkz-kn2-kx2-ky21/2=E0kTkPkBk.
Pxk=kx2kz/kn+ky2/k2,
Pyk=-kxky1-kz/kn/k2,
Pzk=kx/kn,
E˜kx, ky, ±d=E˜kx, ky, 0×exp±idkn2-kx2-ky21/2.
ki=km=const,
Δϕθi, ni, nm, d=dΔkz=d2π/λni cos θi-nm cos θm.
E˜kx, ky, z=E˜kx, ky, -dexpiΔϕ×exp-id+zkn2-kx2-ky21/2.
|Ex, y, z|2|Exx, y, z|2+|Ezx, y, z|2  Ixx, y, z+Izx, y, z,
fr, t=Pr, t · Emr, t+Pr, tt×Bmr, t,
mr=εm/nmEmr×Bmr
fr=αnm2c Imr+αnmΔmrΔt=fgradr+fscar.
Fr=VfrdV=Fgradr+Fscar=Vαnm2c ImrdV+nmc ImCscag/kn.
Csca=1Imkn4-knkn-knknĨs+kx, ky+Ĩs-kx, kyknkn2-kx2-ky21/2dkx, dky.
Qr=FrcPnm=Qgradr+Qscar.
Wr1-Wr2=--r2r1Fgradrdr--r2r1Fscardr=--r2r1 Fgrad,xrdx--r2r1 Fgrad,yrdy--r2r1Fgrad,zr+Fsca,zrdz.
Wr½fxx-x02+fyy-y02+fzz-z02+const.

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