Abstract

A double tweezers setup was employed to perform ultra sensitive force measurements and to obtain the full optical force curve as a function of radial position and wavelength. The light polarization was used to select either the transverse electric (TE), or transverse magnetic (TM), or both, modes excitation. Analytical solution for optical trapping force on a spherical dielectric particle for an arbitrary positioned focused beam is presented in a generalized Lorenz-Mie diffraction theory. The theoretical prediction of the theory agrees well with the experimental results. The algorithm presented here can be easily extended to other beam geometries and scattering particles.

© 2006 Optical Society of America

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References

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  1. D. G. Grier, "A revolution in optical manipulation," Nature 424, 810-816 (2003).
    [CrossRef] [PubMed]
  2. G. Gouesbet, B. Maheu, G. Grehan, "Light scattering from a sphere arbitrarily located in a Gaussian beam using Bromwich formulation," J Opt Soc Am A,  5, 1427-1443 (1988).
    [CrossRef]
  3. G. Gouesbet and G. Grehan, "Sur la generalization de la théorie de Lorenz-Mie, " J. Opt. 13, 97-103 (1982).
    [CrossRef]
  4. K. F. Ren, G. Gouesbet, and G. Grehan, "Integral localized approximation in generalized Lorenz-Mie theory," Appl. Opt. 37, 4218-4225 (1998).
    [CrossRef]
  5. P. A. M. Neto, and H. M. Nussenzveig, "Theory of optical tweezers," Europhys. Lett. 50, 702-708 (2000).
    [CrossRef]
  6. A. Mazolli, P. A. M. Neto, and H. M. Nussenzveig, "Theory of trapping forces in optical tweezers," Proc. R. Soc. A-Math. Phys. Eng. Sci. 459, 3021-3041 (2003).
    [CrossRef]
  7. J. A. Lock, "Calculation of the radiation trapping force for laser tweezers by use of generalized Lorenz-Mie theory. I. Localized model description of an on-axis tightly focused laser beam with spherical aberration," Appl. Opt. 43, 2532-2544 (2004).
    [CrossRef] [PubMed]
  8. J. A. Lock, "Calculation of the radiation trapping force for laser tweezers by use of generalized Lorenz-Mie theory. II. On-axis trapping force," Appl. Opt. 43, 2545-2554 (2004).
    [CrossRef] [PubMed]
  9. G. Knöner, S. Parkin, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Measurement of the Index of Refraction of Single Microparticles," Phys. Rev. Lett. 97, 157402 (2006).
    [CrossRef] [PubMed]
  10. A. Fontes, A. A. R. Neves, W. L. Moreira, A. A. de Thomaz, L. C. Barbosa, C. L. Cesar, and A. M. de Paula, "Double optical tweezers for ultrasensitive force spectroscopy in microsphere Mie scattering," App. Phys. Lett. 87, 221109 (2005).
    [CrossRef]
  11. A. Fontes, K. Ajito, A. A. R. Neves, W. L. Moreira, A. A. de Thomaz, L. C. Barbosa, A. M. de Paula, and C. L. Cesar, "Raman, hyper-Raman, hyper-Rayleigh, two-photon luminescence and morphology-dependent resonance modes in a single optical tweezers system," Phys. Rev. E. 72, 012903 (2005).
    [CrossRef]
  12. A. A. R. Neves, A. Fontes, L. A. Padilha, E. Rodriguez, C. H. de B. Cruz, L. C. Barbosa, and C. L. Cesar, "Exact partial wave expansion of optical beams with respect to an arbitrary origin," Opt. Lett. 31, 2477-2479 (2006).
    [CrossRef] [PubMed]
  13. A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, "Analytical results for a Bessel function times Legendre polynomials class integrals," J. Phys. A 39, L293-L296 (2006).
    [CrossRef]
  14. L. W. Davis, "Theory of electromagnetic beams," Phys. Rev. A 19, 1177-1179 (1979).
    [CrossRef]
  15. J. P. Barton and D. R. Alexander, "Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam," J. Appl. Phys. 66, 2800-2802 (1989).
    [CrossRef]
  16. L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge U. Press, 2006).
  17. J. D. Jackson, Classical Electrodynamics (Wiley, 1999).
  18. E. Fallman and O. Axner, "Design for fully steerable dual-trap optical tweezers," Appl. Opt. 36, 2107-2113 (1997).
    [CrossRef] [PubMed]

2006 (3)

G. Knöner, S. Parkin, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Measurement of the Index of Refraction of Single Microparticles," Phys. Rev. Lett. 97, 157402 (2006).
[CrossRef] [PubMed]

A. A. R. Neves, A. Fontes, L. A. Padilha, E. Rodriguez, C. H. de B. Cruz, L. C. Barbosa, and C. L. Cesar, "Exact partial wave expansion of optical beams with respect to an arbitrary origin," Opt. Lett. 31, 2477-2479 (2006).
[CrossRef] [PubMed]

A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, "Analytical results for a Bessel function times Legendre polynomials class integrals," J. Phys. A 39, L293-L296 (2006).
[CrossRef]

2005 (2)

A. Fontes, A. A. R. Neves, W. L. Moreira, A. A. de Thomaz, L. C. Barbosa, C. L. Cesar, and A. M. de Paula, "Double optical tweezers for ultrasensitive force spectroscopy in microsphere Mie scattering," App. Phys. Lett. 87, 221109 (2005).
[CrossRef]

A. Fontes, K. Ajito, A. A. R. Neves, W. L. Moreira, A. A. de Thomaz, L. C. Barbosa, A. M. de Paula, and C. L. Cesar, "Raman, hyper-Raman, hyper-Rayleigh, two-photon luminescence and morphology-dependent resonance modes in a single optical tweezers system," Phys. Rev. E. 72, 012903 (2005).
[CrossRef]

2004 (2)

2003 (2)

D. G. Grier, "A revolution in optical manipulation," Nature 424, 810-816 (2003).
[CrossRef] [PubMed]

A. Mazolli, P. A. M. Neto, and H. M. Nussenzveig, "Theory of trapping forces in optical tweezers," Proc. R. Soc. A-Math. Phys. Eng. Sci. 459, 3021-3041 (2003).
[CrossRef]

2000 (1)

P. A. M. Neto, and H. M. Nussenzveig, "Theory of optical tweezers," Europhys. Lett. 50, 702-708 (2000).
[CrossRef]

1998 (1)

1997 (1)

1989 (1)

J. P. Barton and D. R. Alexander, "Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam," J. Appl. Phys. 66, 2800-2802 (1989).
[CrossRef]

1988 (1)

G. Gouesbet, B. Maheu, G. Grehan, "Light scattering from a sphere arbitrarily located in a Gaussian beam using Bromwich formulation," J Opt Soc Am A,  5, 1427-1443 (1988).
[CrossRef]

1982 (1)

G. Gouesbet and G. Grehan, "Sur la generalization de la théorie de Lorenz-Mie, " J. Opt. 13, 97-103 (1982).
[CrossRef]

1979 (1)

L. W. Davis, "Theory of electromagnetic beams," Phys. Rev. A 19, 1177-1179 (1979).
[CrossRef]

Ajito, K.

A. Fontes, K. Ajito, A. A. R. Neves, W. L. Moreira, A. A. de Thomaz, L. C. Barbosa, A. M. de Paula, and C. L. Cesar, "Raman, hyper-Raman, hyper-Rayleigh, two-photon luminescence and morphology-dependent resonance modes in a single optical tweezers system," Phys. Rev. E. 72, 012903 (2005).
[CrossRef]

Alexander, D. R.

J. P. Barton and D. R. Alexander, "Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam," J. Appl. Phys. 66, 2800-2802 (1989).
[CrossRef]

Axner, O.

Barbosa, L. C.

A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, "Analytical results for a Bessel function times Legendre polynomials class integrals," J. Phys. A 39, L293-L296 (2006).
[CrossRef]

A. Fontes, K. Ajito, A. A. R. Neves, W. L. Moreira, A. A. de Thomaz, L. C. Barbosa, A. M. de Paula, and C. L. Cesar, "Raman, hyper-Raman, hyper-Rayleigh, two-photon luminescence and morphology-dependent resonance modes in a single optical tweezers system," Phys. Rev. E. 72, 012903 (2005).
[CrossRef]

A. Fontes, A. A. R. Neves, W. L. Moreira, A. A. de Thomaz, L. C. Barbosa, C. L. Cesar, and A. M. de Paula, "Double optical tweezers for ultrasensitive force spectroscopy in microsphere Mie scattering," App. Phys. Lett. 87, 221109 (2005).
[CrossRef]

Barton, J. P.

J. P. Barton and D. R. Alexander, "Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam," J. Appl. Phys. 66, 2800-2802 (1989).
[CrossRef]

Cesar, C. L.

A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, "Analytical results for a Bessel function times Legendre polynomials class integrals," J. Phys. A 39, L293-L296 (2006).
[CrossRef]

A. Fontes, K. Ajito, A. A. R. Neves, W. L. Moreira, A. A. de Thomaz, L. C. Barbosa, A. M. de Paula, and C. L. Cesar, "Raman, hyper-Raman, hyper-Rayleigh, two-photon luminescence and morphology-dependent resonance modes in a single optical tweezers system," Phys. Rev. E. 72, 012903 (2005).
[CrossRef]

A. Fontes, A. A. R. Neves, W. L. Moreira, A. A. de Thomaz, L. C. Barbosa, C. L. Cesar, and A. M. de Paula, "Double optical tweezers for ultrasensitive force spectroscopy in microsphere Mie scattering," App. Phys. Lett. 87, 221109 (2005).
[CrossRef]

Cruz, C. H. B.

A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, "Analytical results for a Bessel function times Legendre polynomials class integrals," J. Phys. A 39, L293-L296 (2006).
[CrossRef]

Davis, L. W.

L. W. Davis, "Theory of electromagnetic beams," Phys. Rev. A 19, 1177-1179 (1979).
[CrossRef]

de Paula, A. M.

A. Fontes, A. A. R. Neves, W. L. Moreira, A. A. de Thomaz, L. C. Barbosa, C. L. Cesar, and A. M. de Paula, "Double optical tweezers for ultrasensitive force spectroscopy in microsphere Mie scattering," App. Phys. Lett. 87, 221109 (2005).
[CrossRef]

A. Fontes, K. Ajito, A. A. R. Neves, W. L. Moreira, A. A. de Thomaz, L. C. Barbosa, A. M. de Paula, and C. L. Cesar, "Raman, hyper-Raman, hyper-Rayleigh, two-photon luminescence and morphology-dependent resonance modes in a single optical tweezers system," Phys. Rev. E. 72, 012903 (2005).
[CrossRef]

de Thomaz, A. A.

A. Fontes, A. A. R. Neves, W. L. Moreira, A. A. de Thomaz, L. C. Barbosa, C. L. Cesar, and A. M. de Paula, "Double optical tweezers for ultrasensitive force spectroscopy in microsphere Mie scattering," App. Phys. Lett. 87, 221109 (2005).
[CrossRef]

A. Fontes, K. Ajito, A. A. R. Neves, W. L. Moreira, A. A. de Thomaz, L. C. Barbosa, A. M. de Paula, and C. L. Cesar, "Raman, hyper-Raman, hyper-Rayleigh, two-photon luminescence and morphology-dependent resonance modes in a single optical tweezers system," Phys. Rev. E. 72, 012903 (2005).
[CrossRef]

Fallman, E.

Fontes, A.

A. A. R. Neves, A. Fontes, L. A. Padilha, E. Rodriguez, C. H. de B. Cruz, L. C. Barbosa, and C. L. Cesar, "Exact partial wave expansion of optical beams with respect to an arbitrary origin," Opt. Lett. 31, 2477-2479 (2006).
[CrossRef] [PubMed]

A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, "Analytical results for a Bessel function times Legendre polynomials class integrals," J. Phys. A 39, L293-L296 (2006).
[CrossRef]

A. Fontes, K. Ajito, A. A. R. Neves, W. L. Moreira, A. A. de Thomaz, L. C. Barbosa, A. M. de Paula, and C. L. Cesar, "Raman, hyper-Raman, hyper-Rayleigh, two-photon luminescence and morphology-dependent resonance modes in a single optical tweezers system," Phys. Rev. E. 72, 012903 (2005).
[CrossRef]

A. Fontes, A. A. R. Neves, W. L. Moreira, A. A. de Thomaz, L. C. Barbosa, C. L. Cesar, and A. M. de Paula, "Double optical tweezers for ultrasensitive force spectroscopy in microsphere Mie scattering," App. Phys. Lett. 87, 221109 (2005).
[CrossRef]

Gouesbet, G.

K. F. Ren, G. Gouesbet, and G. Grehan, "Integral localized approximation in generalized Lorenz-Mie theory," Appl. Opt. 37, 4218-4225 (1998).
[CrossRef]

G. Gouesbet, B. Maheu, G. Grehan, "Light scattering from a sphere arbitrarily located in a Gaussian beam using Bromwich formulation," J Opt Soc Am A,  5, 1427-1443 (1988).
[CrossRef]

G. Gouesbet and G. Grehan, "Sur la generalization de la théorie de Lorenz-Mie, " J. Opt. 13, 97-103 (1982).
[CrossRef]

Grehan, G.

K. F. Ren, G. Gouesbet, and G. Grehan, "Integral localized approximation in generalized Lorenz-Mie theory," Appl. Opt. 37, 4218-4225 (1998).
[CrossRef]

G. Gouesbet, B. Maheu, G. Grehan, "Light scattering from a sphere arbitrarily located in a Gaussian beam using Bromwich formulation," J Opt Soc Am A,  5, 1427-1443 (1988).
[CrossRef]

G. Gouesbet and G. Grehan, "Sur la generalization de la théorie de Lorenz-Mie, " J. Opt. 13, 97-103 (1982).
[CrossRef]

Grier, D. G.

D. G. Grier, "A revolution in optical manipulation," Nature 424, 810-816 (2003).
[CrossRef] [PubMed]

Heckenberg, N. R.

G. Knöner, S. Parkin, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Measurement of the Index of Refraction of Single Microparticles," Phys. Rev. Lett. 97, 157402 (2006).
[CrossRef] [PubMed]

Knöner, G.

G. Knöner, S. Parkin, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Measurement of the Index of Refraction of Single Microparticles," Phys. Rev. Lett. 97, 157402 (2006).
[CrossRef] [PubMed]

Lock, J. A.

Maheu, B.

G. Gouesbet, B. Maheu, G. Grehan, "Light scattering from a sphere arbitrarily located in a Gaussian beam using Bromwich formulation," J Opt Soc Am A,  5, 1427-1443 (1988).
[CrossRef]

Mazolli, A.

A. Mazolli, P. A. M. Neto, and H. M. Nussenzveig, "Theory of trapping forces in optical tweezers," Proc. R. Soc. A-Math. Phys. Eng. Sci. 459, 3021-3041 (2003).
[CrossRef]

Moreira, W. L.

A. Fontes, K. Ajito, A. A. R. Neves, W. L. Moreira, A. A. de Thomaz, L. C. Barbosa, A. M. de Paula, and C. L. Cesar, "Raman, hyper-Raman, hyper-Rayleigh, two-photon luminescence and morphology-dependent resonance modes in a single optical tweezers system," Phys. Rev. E. 72, 012903 (2005).
[CrossRef]

A. Fontes, A. A. R. Neves, W. L. Moreira, A. A. de Thomaz, L. C. Barbosa, C. L. Cesar, and A. M. de Paula, "Double optical tweezers for ultrasensitive force spectroscopy in microsphere Mie scattering," App. Phys. Lett. 87, 221109 (2005).
[CrossRef]

Neto, P. A. M.

A. Mazolli, P. A. M. Neto, and H. M. Nussenzveig, "Theory of trapping forces in optical tweezers," Proc. R. Soc. A-Math. Phys. Eng. Sci. 459, 3021-3041 (2003).
[CrossRef]

P. A. M. Neto, and H. M. Nussenzveig, "Theory of optical tweezers," Europhys. Lett. 50, 702-708 (2000).
[CrossRef]

Neves, A. A. R.

A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, "Analytical results for a Bessel function times Legendre polynomials class integrals," J. Phys. A 39, L293-L296 (2006).
[CrossRef]

A. A. R. Neves, A. Fontes, L. A. Padilha, E. Rodriguez, C. H. de B. Cruz, L. C. Barbosa, and C. L. Cesar, "Exact partial wave expansion of optical beams with respect to an arbitrary origin," Opt. Lett. 31, 2477-2479 (2006).
[CrossRef] [PubMed]

A. Fontes, K. Ajito, A. A. R. Neves, W. L. Moreira, A. A. de Thomaz, L. C. Barbosa, A. M. de Paula, and C. L. Cesar, "Raman, hyper-Raman, hyper-Rayleigh, two-photon luminescence and morphology-dependent resonance modes in a single optical tweezers system," Phys. Rev. E. 72, 012903 (2005).
[CrossRef]

A. Fontes, A. A. R. Neves, W. L. Moreira, A. A. de Thomaz, L. C. Barbosa, C. L. Cesar, and A. M. de Paula, "Double optical tweezers for ultrasensitive force spectroscopy in microsphere Mie scattering," App. Phys. Lett. 87, 221109 (2005).
[CrossRef]

Nieminen, T. A.

G. Knöner, S. Parkin, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Measurement of the Index of Refraction of Single Microparticles," Phys. Rev. Lett. 97, 157402 (2006).
[CrossRef] [PubMed]

Nussenzveig, H. M.

A. Mazolli, P. A. M. Neto, and H. M. Nussenzveig, "Theory of trapping forces in optical tweezers," Proc. R. Soc. A-Math. Phys. Eng. Sci. 459, 3021-3041 (2003).
[CrossRef]

P. A. M. Neto, and H. M. Nussenzveig, "Theory of optical tweezers," Europhys. Lett. 50, 702-708 (2000).
[CrossRef]

Padilha, L. A.

A. A. R. Neves, A. Fontes, L. A. Padilha, E. Rodriguez, C. H. de B. Cruz, L. C. Barbosa, and C. L. Cesar, "Exact partial wave expansion of optical beams with respect to an arbitrary origin," Opt. Lett. 31, 2477-2479 (2006).
[CrossRef] [PubMed]

A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, "Analytical results for a Bessel function times Legendre polynomials class integrals," J. Phys. A 39, L293-L296 (2006).
[CrossRef]

Parkin, S.

G. Knöner, S. Parkin, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Measurement of the Index of Refraction of Single Microparticles," Phys. Rev. Lett. 97, 157402 (2006).
[CrossRef] [PubMed]

Ren, K. F.

Rodriguez, E.

A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, "Analytical results for a Bessel function times Legendre polynomials class integrals," J. Phys. A 39, L293-L296 (2006).
[CrossRef]

A. A. R. Neves, A. Fontes, L. A. Padilha, E. Rodriguez, C. H. de B. Cruz, L. C. Barbosa, and C. L. Cesar, "Exact partial wave expansion of optical beams with respect to an arbitrary origin," Opt. Lett. 31, 2477-2479 (2006).
[CrossRef] [PubMed]

Rubinsztein-Dunlop, H.

G. Knöner, S. Parkin, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Measurement of the Index of Refraction of Single Microparticles," Phys. Rev. Lett. 97, 157402 (2006).
[CrossRef] [PubMed]

App. Phys. Lett. (1)

A. Fontes, A. A. R. Neves, W. L. Moreira, A. A. de Thomaz, L. C. Barbosa, C. L. Cesar, and A. M. de Paula, "Double optical tweezers for ultrasensitive force spectroscopy in microsphere Mie scattering," App. Phys. Lett. 87, 221109 (2005).
[CrossRef]

Appl. Opt. (4)

Europhys. Lett. (1)

P. A. M. Neto, and H. M. Nussenzveig, "Theory of optical tweezers," Europhys. Lett. 50, 702-708 (2000).
[CrossRef]

J Opt Soc Am A (1)

G. Gouesbet, B. Maheu, G. Grehan, "Light scattering from a sphere arbitrarily located in a Gaussian beam using Bromwich formulation," J Opt Soc Am A,  5, 1427-1443 (1988).
[CrossRef]

J. Appl. Phys. (1)

J. P. Barton and D. R. Alexander, "Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam," J. Appl. Phys. 66, 2800-2802 (1989).
[CrossRef]

J. Opt. (1)

G. Gouesbet and G. Grehan, "Sur la generalization de la théorie de Lorenz-Mie, " J. Opt. 13, 97-103 (1982).
[CrossRef]

J. Phys. A (1)

A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, "Analytical results for a Bessel function times Legendre polynomials class integrals," J. Phys. A 39, L293-L296 (2006).
[CrossRef]

Nature (1)

D. G. Grier, "A revolution in optical manipulation," Nature 424, 810-816 (2003).
[CrossRef] [PubMed]

Opt. Lett. (1)

Phys. Rev. A (1)

L. W. Davis, "Theory of electromagnetic beams," Phys. Rev. A 19, 1177-1179 (1979).
[CrossRef]

Phys. Rev. E. (1)

A. Fontes, K. Ajito, A. A. R. Neves, W. L. Moreira, A. A. de Thomaz, L. C. Barbosa, A. M. de Paula, and C. L. Cesar, "Raman, hyper-Raman, hyper-Rayleigh, two-photon luminescence and morphology-dependent resonance modes in a single optical tweezers system," Phys. Rev. E. 72, 012903 (2005).
[CrossRef]

Phys. Rev. Lett. (1)

G. Knöner, S. Parkin, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Measurement of the Index of Refraction of Single Microparticles," Phys. Rev. Lett. 97, 157402 (2006).
[CrossRef] [PubMed]

Proc. R. Soc. A-Math. Phys. Eng. Sci. (1)

A. Mazolli, P. A. M. Neto, and H. M. Nussenzveig, "Theory of trapping forces in optical tweezers," Proc. R. Soc. A-Math. Phys. Eng. Sci. 459, 3021-3041 (2003).
[CrossRef]

Other (2)

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge U. Press, 2006).

J. D. Jackson, Classical Electrodynamics (Wiley, 1999).

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Figures (4)

Fig. 1.
Fig. 1.

Complete scheme for the double optical tweezers for ultra-sensitive force spectroscopy.

Fig. 2.
Fig. 2.

Radial optical forces for parallel polarization with respect to radial beam movement for: 3, 6 and 9 µm spheres: (red) experimental; (blue) theory.

Fig. 3.
Fig. 3.

Radial optical forces for perpendicular polarization with respect to radial beam movement for: 3, 6 and 9 µm spheres: (red) experimental; (blue) theory.

Fig. 4.
Fig. 4.

Radial optical forces enhancement due to MDR for a 9 µm sphere and polarizations as a function of: (a) wavelength; (b) radial distance.

Equations (11)

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

E inc = E 0 n , m [ 1 k G n m TM × j n ( k r ) X n m ( θ , ϕ ) + G n m TE j n ( k r ) X n m ( θ , ϕ ) ]
H inc = E 0 Z n , m [ G n m TM j n ( k r ) X n m ( θ , ϕ ) i k G n m TE × j n ( k r ) X n m ( θ , ϕ ) ]
[ G n m TM G n m TE ] = ± 2 π i k f exp ( i k f ) i n m exp ( i m ϕ o ) 2 n + 1 4 π n ( n + 1 ) ( n m ) ! ( n + m ) !
0 α max d α cos α exp ( f 2 sin 2 α ω a 2 ) exp ( i k z o cos α )
{ [ m 2 J m ( k ρ o sin α ) k ρ o sin α P n m ( cos α ) sin 2 α J m ( k ρ o sin α ) P n m ( cos α ) ] cos ϕ o
+ i m [ m J m ( k ρ o sin α ) P n m ( cos α ) sin 2 α J m ( k ρ o sin α ) k ρ o sin α P n m ( cos α ) ] sin ϕ o }
[ C x C y ] = 1 4 k 2 [ Re Im ] n = 1 i ( n + 1 ) { n ( n + 2 ) ( 2 n + 3 ) ( 2 n + 1 ) m = n n ( n + m + 2 ) ( n + m + 1 )
[ ( a n + 1 + a n * 2 a n + 1 a n * ) G n + 1 , ( m + 1 ) TM G n , m TM * + ( a n + a n + 1 * 2 a n a n + 1 * ) G nm TM G n + 1 , m + 1 TM * +
( b n + 1 + b n * 2 b n + 1 b n * ) G n + 1 , ( m + 1 ) TE G n , m TE * + ( b n + b n + 1 * 2 b n b n + 1 * ) G nm TE G n + 1 , m + 1 TE * ]
1 n m = n n ( n + m + 2 ) ( n + m + 1 ) ( n m ) ( n + m + 1 )
[ ( a n + b n * 2 a n b n * ) G nm TM G n , m + 1 TE * ( b n + a n * 2 b n a n * ) G nm TE G n , m + 1 TM * ] }

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