Abstract

The dynamics of an optically trapped particle are often determined by measuring intensity shifts of the back-scattered light from the particle using position sensitive detectors. We present a technique which measures the phase of the back-scattered light using balanced detection in an external Mach-Zehnder interferometer scheme where we separate out and beat the scattered light from the particle and that from the top surface of our trapping chamber. The technique has improved axial motion resolution over intensity-based detection, and can also be used to measure lateral motion of the trapped particle. In addition, we are able to track the Brownian motion of trapped 1.1 and 3 μm diameter particles from the phase jitter and show that, similar to intensity-based measurements, phase measurements can also be used to simultaneously determine displacements of the trapped particle as well as the spring constant of the trap. For lateral displacements, we have matched our experimental results with a simulation of the overall phase contour of the back-scattered light by using plane wave decomposition in conjunction with Mie scattering theory. The position resolution is limited by path drifts of the interferometer which we have presently reduced to demonstrate the capability of sub-nm displacement resolution in the axial direction for 1.1 μm diameter particles by locking the interferometer to a frequency stabilized diode laser.

© 2012 OSA

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References

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  1. K. Svoboda, C. F. Schmidt, B. J. Schnapps, and S. M. Block, “Direct observation of kinesin stepping by optical trapping interferometry,” Nature 365, 721–727 (1993).
    [CrossRef] [PubMed]
  2. A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283, 1689–1695 (1999).
    [CrossRef] [PubMed]
  3. D. E. Smith, S. J. Tans, S. B. Smith, S. Grimes, D. L. Anderson, and C. Bustamante, “The bacteriophage ϕ29 portal motor can package dna against a large internal force,” Nature (London) 413, 748–752 (2001).
    [CrossRef]
  4. J.-D. Wen, M. Manosas, P. T. X. Li, S. B. Smith, C. Bustamante, F. Ritort, and I. Tinoco, “Force unfolding kinetics of rna using optical tweezers. i. effects of experimental variables on measured results,” Biophys. J. 92, 2996–3009 (2007).
    [CrossRef] [PubMed]
  5. L. P. Ghislain, N. A. Switz, and W. W. Webb, “Measurement of small forces using an optical trap,” Rev. Sci. Instrum. 65, 2762–2768 (1994).
    [CrossRef]
  6. G. Volpe and D. Petrov, “Torque detection using brownian fluctuations,” Phys. Rev. Lett. 6975, 210603 (2006).
    [CrossRef]
  7. A. Pralle, E.-L. Florin, E. H. K. Stelzer, and J. K. H. Horber, “Photonic force microscopy: a new tool providing new methods to study membranes at the molecular level,” Single Mol. 1, 129–133 (2000).
    [CrossRef]
  8. Y. Deng, J. Bechhoefer, and N. Forde, “Brownian motion in a modulated optical trap,” J. Opt. A. 9, S256–S263 (2007).
    [CrossRef]
  9. G. Volpe, G. Volpe, and D. Petrov, “Brownian motion in a nonhomogeneous force field and photonic force microscope,” Phys. Rev. E 76, 061118 (2007).
    [CrossRef]
  10. L. I. McCann, M. Dykman, and B. Golding, “Thermally activated transitions in a bistable three-dimensional optical trap,” Nature 402, 785–787 (1999).
    [CrossRef]
  11. A. R. Carter, G. M. King, and T. T. Perkins, “Back-scattered detection provides atomic-scale localization precision, stability, and registration in 3d,” Opt. Express 15, 13434–13445 (2007).
    [CrossRef] [PubMed]
  12. W. Singer, S. Bernet, N. Hecker, and M. Ritsch-Marte, “Three-dimensional force calibration of optical tweezers,” J. Mod. Opt. 47, 2921–2931 (2000).
  13. T. Li, S. Kheifets, and M. G. Raizen, “Millikelvin cooling of an optically trapped microsphere in vacuum,” Nature Phys. 7, 527–530 (2011).
    [CrossRef]
  14. K. Berg-Sorensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Inst. 75, 594–612 (2004).
    [CrossRef]
  15. M. Mahamdeh, C. P. Campos, and E. Schäffer, “Under-filling trapping objectives optimizes the use of the available laser power in optical tweezers,” Opt. Express 19, 11759–11768 (2011).
    [CrossRef] [PubMed]
  16. S. B. Pal, A. Haldar, B. Roy, and A. Banerjee, “Measurement of probe displacement to the thermal resolution limit in photonic force microscopy using a miniature quadrant photodetector,” Rev. Sci. Instrum. 83, 023108(2012).
    [CrossRef] [PubMed]
  17. G. Volpe, G. Kozyreff, and D. Petrov, “Backscattering position detection for photonic force microscopy,” J. Appl. Phys. 102, 084701 (2007).
    [CrossRef]
  18. M. Born and E. Wolf, Principles of Optics (Pergamon Press, 1989).
  19. F. Czerwinski, A. C. Richardson, and L. Oddershede, “Quantifying noise in optical tweezers by Allan variance,” Opt. Express 17, 13255–13269 (2009).
    [CrossRef] [PubMed]

2012 (1)

S. B. Pal, A. Haldar, B. Roy, and A. Banerjee, “Measurement of probe displacement to the thermal resolution limit in photonic force microscopy using a miniature quadrant photodetector,” Rev. Sci. Instrum. 83, 023108(2012).
[CrossRef] [PubMed]

2011 (2)

2009 (1)

2007 (5)

A. R. Carter, G. M. King, and T. T. Perkins, “Back-scattered detection provides atomic-scale localization precision, stability, and registration in 3d,” Opt. Express 15, 13434–13445 (2007).
[CrossRef] [PubMed]

G. Volpe, G. Kozyreff, and D. Petrov, “Backscattering position detection for photonic force microscopy,” J. Appl. Phys. 102, 084701 (2007).
[CrossRef]

J.-D. Wen, M. Manosas, P. T. X. Li, S. B. Smith, C. Bustamante, F. Ritort, and I. Tinoco, “Force unfolding kinetics of rna using optical tweezers. i. effects of experimental variables on measured results,” Biophys. J. 92, 2996–3009 (2007).
[CrossRef] [PubMed]

Y. Deng, J. Bechhoefer, and N. Forde, “Brownian motion in a modulated optical trap,” J. Opt. A. 9, S256–S263 (2007).
[CrossRef]

G. Volpe, G. Volpe, and D. Petrov, “Brownian motion in a nonhomogeneous force field and photonic force microscope,” Phys. Rev. E 76, 061118 (2007).
[CrossRef]

2006 (1)

G. Volpe and D. Petrov, “Torque detection using brownian fluctuations,” Phys. Rev. Lett. 6975, 210603 (2006).
[CrossRef]

2004 (1)

K. Berg-Sorensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Inst. 75, 594–612 (2004).
[CrossRef]

2001 (1)

D. E. Smith, S. J. Tans, S. B. Smith, S. Grimes, D. L. Anderson, and C. Bustamante, “The bacteriophage ϕ29 portal motor can package dna against a large internal force,” Nature (London) 413, 748–752 (2001).
[CrossRef]

2000 (2)

W. Singer, S. Bernet, N. Hecker, and M. Ritsch-Marte, “Three-dimensional force calibration of optical tweezers,” J. Mod. Opt. 47, 2921–2931 (2000).

A. Pralle, E.-L. Florin, E. H. K. Stelzer, and J. K. H. Horber, “Photonic force microscopy: a new tool providing new methods to study membranes at the molecular level,” Single Mol. 1, 129–133 (2000).
[CrossRef]

1999 (2)

L. I. McCann, M. Dykman, and B. Golding, “Thermally activated transitions in a bistable three-dimensional optical trap,” Nature 402, 785–787 (1999).
[CrossRef]

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283, 1689–1695 (1999).
[CrossRef] [PubMed]

1994 (1)

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

1993 (1)

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

Anderson, D. L.

D. E. Smith, S. J. Tans, S. B. Smith, S. Grimes, D. L. Anderson, and C. Bustamante, “The bacteriophage ϕ29 portal motor can package dna against a large internal force,” Nature (London) 413, 748–752 (2001).
[CrossRef]

Banerjee, A.

S. B. Pal, A. Haldar, B. Roy, and A. Banerjee, “Measurement of probe displacement to the thermal resolution limit in photonic force microscopy using a miniature quadrant photodetector,” Rev. Sci. Instrum. 83, 023108(2012).
[CrossRef] [PubMed]

Bechhoefer, J.

Y. Deng, J. Bechhoefer, and N. Forde, “Brownian motion in a modulated optical trap,” J. Opt. A. 9, S256–S263 (2007).
[CrossRef]

Berg-Sorensen, K.

K. Berg-Sorensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Inst. 75, 594–612 (2004).
[CrossRef]

Bernet, S.

W. Singer, S. Bernet, N. Hecker, and M. Ritsch-Marte, “Three-dimensional force calibration of optical tweezers,” J. Mod. Opt. 47, 2921–2931 (2000).

Block, S. M.

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

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, 1989).

Bustamante, C.

J.-D. Wen, M. Manosas, P. T. X. Li, S. B. Smith, C. Bustamante, F. Ritort, and I. Tinoco, “Force unfolding kinetics of rna using optical tweezers. i. effects of experimental variables on measured results,” Biophys. J. 92, 2996–3009 (2007).
[CrossRef] [PubMed]

D. E. Smith, S. J. Tans, S. B. Smith, S. Grimes, D. L. Anderson, and C. Bustamante, “The bacteriophage ϕ29 portal motor can package dna against a large internal force,” Nature (London) 413, 748–752 (2001).
[CrossRef]

Campos, C. P.

Carter, A. R.

Czerwinski, F.

Deng, Y.

Y. Deng, J. Bechhoefer, and N. Forde, “Brownian motion in a modulated optical trap,” J. Opt. A. 9, S256–S263 (2007).
[CrossRef]

Dykman, M.

L. I. McCann, M. Dykman, and B. Golding, “Thermally activated transitions in a bistable three-dimensional optical trap,” Nature 402, 785–787 (1999).
[CrossRef]

Florin, E.-L.

A. Pralle, E.-L. Florin, E. H. K. Stelzer, and J. K. H. Horber, “Photonic force microscopy: a new tool providing new methods to study membranes at the molecular level,” Single Mol. 1, 129–133 (2000).
[CrossRef]

Flyvbjerg, H.

K. Berg-Sorensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Inst. 75, 594–612 (2004).
[CrossRef]

Forde, N.

Y. Deng, J. Bechhoefer, and N. Forde, “Brownian motion in a modulated optical trap,” J. Opt. A. 9, S256–S263 (2007).
[CrossRef]

Ghislain, L. P.

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

Golding, B.

L. I. McCann, M. Dykman, and B. Golding, “Thermally activated transitions in a bistable three-dimensional optical trap,” Nature 402, 785–787 (1999).
[CrossRef]

Grimes, S.

D. E. Smith, S. J. Tans, S. B. Smith, S. Grimes, D. L. Anderson, and C. Bustamante, “The bacteriophage ϕ29 portal motor can package dna against a large internal force,” Nature (London) 413, 748–752 (2001).
[CrossRef]

Haldar, A.

S. B. Pal, A. Haldar, B. Roy, and A. Banerjee, “Measurement of probe displacement to the thermal resolution limit in photonic force microscopy using a miniature quadrant photodetector,” Rev. Sci. Instrum. 83, 023108(2012).
[CrossRef] [PubMed]

Hecker, N.

W. Singer, S. Bernet, N. Hecker, and M. Ritsch-Marte, “Three-dimensional force calibration of optical tweezers,” J. Mod. Opt. 47, 2921–2931 (2000).

Horber, J. K. H.

A. Pralle, E.-L. Florin, E. H. K. Stelzer, and J. K. H. Horber, “Photonic force microscopy: a new tool providing new methods to study membranes at the molecular level,” Single Mol. 1, 129–133 (2000).
[CrossRef]

Kheifets, S.

T. Li, S. Kheifets, and M. G. Raizen, “Millikelvin cooling of an optically trapped microsphere in vacuum,” Nature Phys. 7, 527–530 (2011).
[CrossRef]

King, G. M.

Kozyreff, G.

G. Volpe, G. Kozyreff, and D. Petrov, “Backscattering position detection for photonic force microscopy,” J. Appl. Phys. 102, 084701 (2007).
[CrossRef]

Li, P. T. X.

J.-D. Wen, M. Manosas, P. T. X. Li, S. B. Smith, C. Bustamante, F. Ritort, and I. Tinoco, “Force unfolding kinetics of rna using optical tweezers. i. effects of experimental variables on measured results,” Biophys. J. 92, 2996–3009 (2007).
[CrossRef] [PubMed]

Li, T.

T. Li, S. Kheifets, and M. G. Raizen, “Millikelvin cooling of an optically trapped microsphere in vacuum,” Nature Phys. 7, 527–530 (2011).
[CrossRef]

Mahamdeh, M.

Manosas, M.

J.-D. Wen, M. Manosas, P. T. X. Li, S. B. Smith, C. Bustamante, F. Ritort, and I. Tinoco, “Force unfolding kinetics of rna using optical tweezers. i. effects of experimental variables on measured results,” Biophys. J. 92, 2996–3009 (2007).
[CrossRef] [PubMed]

McCann, L. I.

L. I. McCann, M. Dykman, and B. Golding, “Thermally activated transitions in a bistable three-dimensional optical trap,” Nature 402, 785–787 (1999).
[CrossRef]

Mehta, A. D.

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283, 1689–1695 (1999).
[CrossRef] [PubMed]

Oddershede, L.

Pal, S. B.

S. B. Pal, A. Haldar, B. Roy, and A. Banerjee, “Measurement of probe displacement to the thermal resolution limit in photonic force microscopy using a miniature quadrant photodetector,” Rev. Sci. Instrum. 83, 023108(2012).
[CrossRef] [PubMed]

Perkins, T. T.

Petrov, D.

G. Volpe, G. Volpe, and D. Petrov, “Brownian motion in a nonhomogeneous force field and photonic force microscope,” Phys. Rev. E 76, 061118 (2007).
[CrossRef]

G. Volpe, G. Kozyreff, and D. Petrov, “Backscattering position detection for photonic force microscopy,” J. Appl. Phys. 102, 084701 (2007).
[CrossRef]

G. Volpe and D. Petrov, “Torque detection using brownian fluctuations,” Phys. Rev. Lett. 6975, 210603 (2006).
[CrossRef]

Pralle, A.

A. Pralle, E.-L. Florin, E. H. K. Stelzer, and J. K. H. Horber, “Photonic force microscopy: a new tool providing new methods to study membranes at the molecular level,” Single Mol. 1, 129–133 (2000).
[CrossRef]

Raizen, M. G.

T. Li, S. Kheifets, and M. G. Raizen, “Millikelvin cooling of an optically trapped microsphere in vacuum,” Nature Phys. 7, 527–530 (2011).
[CrossRef]

Richardson, A. C.

Rief, M.

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283, 1689–1695 (1999).
[CrossRef] [PubMed]

Ritort, F.

J.-D. Wen, M. Manosas, P. T. X. Li, S. B. Smith, C. Bustamante, F. Ritort, and I. Tinoco, “Force unfolding kinetics of rna using optical tweezers. i. effects of experimental variables on measured results,” Biophys. J. 92, 2996–3009 (2007).
[CrossRef] [PubMed]

Ritsch-Marte, M.

W. Singer, S. Bernet, N. Hecker, and M. Ritsch-Marte, “Three-dimensional force calibration of optical tweezers,” J. Mod. Opt. 47, 2921–2931 (2000).

Roy, B.

S. B. Pal, A. Haldar, B. Roy, and A. Banerjee, “Measurement of probe displacement to the thermal resolution limit in photonic force microscopy using a miniature quadrant photodetector,” Rev. Sci. Instrum. 83, 023108(2012).
[CrossRef] [PubMed]

Schäffer, E.

Schmidt, C. F.

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

Schnapps, B. J.

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

Simmons, R. M.

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283, 1689–1695 (1999).
[CrossRef] [PubMed]

Singer, W.

W. Singer, S. Bernet, N. Hecker, and M. Ritsch-Marte, “Three-dimensional force calibration of optical tweezers,” J. Mod. Opt. 47, 2921–2931 (2000).

Smith, D. A.

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283, 1689–1695 (1999).
[CrossRef] [PubMed]

Smith, D. E.

D. E. Smith, S. J. Tans, S. B. Smith, S. Grimes, D. L. Anderson, and C. Bustamante, “The bacteriophage ϕ29 portal motor can package dna against a large internal force,” Nature (London) 413, 748–752 (2001).
[CrossRef]

Smith, S. B.

J.-D. Wen, M. Manosas, P. T. X. Li, S. B. Smith, C. Bustamante, F. Ritort, and I. Tinoco, “Force unfolding kinetics of rna using optical tweezers. i. effects of experimental variables on measured results,” Biophys. J. 92, 2996–3009 (2007).
[CrossRef] [PubMed]

D. E. Smith, S. J. Tans, S. B. Smith, S. Grimes, D. L. Anderson, and C. Bustamante, “The bacteriophage ϕ29 portal motor can package dna against a large internal force,” Nature (London) 413, 748–752 (2001).
[CrossRef]

Spudich, J. A.

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283, 1689–1695 (1999).
[CrossRef] [PubMed]

Stelzer, E. H. K.

A. Pralle, E.-L. Florin, E. H. K. Stelzer, and J. K. H. Horber, “Photonic force microscopy: a new tool providing new methods to study membranes at the molecular level,” Single Mol. 1, 129–133 (2000).
[CrossRef]

Svoboda, K.

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

Switz, N. A.

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

Tans, S. J.

D. E. Smith, S. J. Tans, S. B. Smith, S. Grimes, D. L. Anderson, and C. Bustamante, “The bacteriophage ϕ29 portal motor can package dna against a large internal force,” Nature (London) 413, 748–752 (2001).
[CrossRef]

Tinoco, I.

J.-D. Wen, M. Manosas, P. T. X. Li, S. B. Smith, C. Bustamante, F. Ritort, and I. Tinoco, “Force unfolding kinetics of rna using optical tweezers. i. effects of experimental variables on measured results,” Biophys. J. 92, 2996–3009 (2007).
[CrossRef] [PubMed]

Volpe, G.

G. Volpe, G. Volpe, and D. Petrov, “Brownian motion in a nonhomogeneous force field and photonic force microscope,” Phys. Rev. E 76, 061118 (2007).
[CrossRef]

G. Volpe, G. Volpe, and D. Petrov, “Brownian motion in a nonhomogeneous force field and photonic force microscope,” Phys. Rev. E 76, 061118 (2007).
[CrossRef]

G. Volpe, G. Kozyreff, and D. Petrov, “Backscattering position detection for photonic force microscopy,” J. Appl. Phys. 102, 084701 (2007).
[CrossRef]

G. Volpe and D. Petrov, “Torque detection using brownian fluctuations,” Phys. Rev. Lett. 6975, 210603 (2006).
[CrossRef]

Webb, W. W.

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

Wen, J.-D.

J.-D. Wen, M. Manosas, P. T. X. Li, S. B. Smith, C. Bustamante, F. Ritort, and I. Tinoco, “Force unfolding kinetics of rna using optical tweezers. i. effects of experimental variables on measured results,” Biophys. J. 92, 2996–3009 (2007).
[CrossRef] [PubMed]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, 1989).

Biophys. J. (1)

J.-D. Wen, M. Manosas, P. T. X. Li, S. B. Smith, C. Bustamante, F. Ritort, and I. Tinoco, “Force unfolding kinetics of rna using optical tweezers. i. effects of experimental variables on measured results,” Biophys. J. 92, 2996–3009 (2007).
[CrossRef] [PubMed]

J. Appl. Phys. (1)

G. Volpe, G. Kozyreff, and D. Petrov, “Backscattering position detection for photonic force microscopy,” J. Appl. Phys. 102, 084701 (2007).
[CrossRef]

J. Mod. Opt. (1)

W. Singer, S. Bernet, N. Hecker, and M. Ritsch-Marte, “Three-dimensional force calibration of optical tweezers,” J. Mod. Opt. 47, 2921–2931 (2000).

J. Opt. A. (1)

Y. Deng, J. Bechhoefer, and N. Forde, “Brownian motion in a modulated optical trap,” J. Opt. A. 9, S256–S263 (2007).
[CrossRef]

Nature (2)

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

L. I. McCann, M. Dykman, and B. Golding, “Thermally activated transitions in a bistable three-dimensional optical trap,” Nature 402, 785–787 (1999).
[CrossRef]

Nature (London) (1)

D. E. Smith, S. J. Tans, S. B. Smith, S. Grimes, D. L. Anderson, and C. Bustamante, “The bacteriophage ϕ29 portal motor can package dna against a large internal force,” Nature (London) 413, 748–752 (2001).
[CrossRef]

Nature Phys. (1)

T. Li, S. Kheifets, and M. G. Raizen, “Millikelvin cooling of an optically trapped microsphere in vacuum,” Nature Phys. 7, 527–530 (2011).
[CrossRef]

Opt. Express (3)

Phys. Rev. E (1)

G. Volpe, G. Volpe, and D. Petrov, “Brownian motion in a nonhomogeneous force field and photonic force microscope,” Phys. Rev. E 76, 061118 (2007).
[CrossRef]

Phys. Rev. Lett. (1)

G. Volpe and D. Petrov, “Torque detection using brownian fluctuations,” Phys. Rev. Lett. 6975, 210603 (2006).
[CrossRef]

Rev. Sci. Inst. (1)

K. Berg-Sorensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Inst. 75, 594–612 (2004).
[CrossRef]

Rev. Sci. Instrum. (2)

S. B. Pal, A. Haldar, B. Roy, and A. Banerjee, “Measurement of probe displacement to the thermal resolution limit in photonic force microscopy using a miniature quadrant photodetector,” Rev. Sci. Instrum. 83, 023108(2012).
[CrossRef] [PubMed]

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

Science (1)

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283, 1689–1695 (1999).
[CrossRef] [PubMed]

Single Mol. (1)

A. Pralle, E.-L. Florin, E. H. K. Stelzer, and J. K. H. Horber, “Photonic force microscopy: a new tool providing new methods to study membranes at the molecular level,” Single Mol. 1, 129–133 (2000).
[CrossRef]

Other (1)

M. Born and E. Wolf, Principles of Optics (Pergamon Press, 1989).

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Figures (7)

Fig. 1
Fig. 1

Schematic of the experiment. Key: PD1: Photodiode 1, PD2: Photodiode 2, BS1 : Beam splitter 1, BS2: Beam splitter 2, Signal Gen.: Signal Generator. The pinhole generates an Airy pattern - the central portion of which contains scattered light from the trapped probe, while the diffuse ring contains unscattered reflection from the top gold slide of the sample chamber. Aperture 1 picks off signal from the ring while Aperture 2 picks off the probe signal at the center of the Airy pattern as is shown in the inset (marked as output from pinhole). Both apertures are circular with diameter about 1 mm. The trapping laser is at 1064 nm and the detection laser is at 780 nm.

Fig. 2
Fig. 2

(a). Balanced detection signal obtained by subtraction of the two out-of-phase signals from PD1 and PD2, the photodiodes kept in the two arms of the Mach-Zehnder interferometer. (b) Typical fringes obtained in the phase measurement as the AOM voltage is changed to displace the trap transversely and cause a phase shift in the backscattered signal that is captured in the balanced detection output.

Fig. 3
Fig. 3

The theoretical model. (a) shows a schematic of the simulation, whereas (b) shows the coordinate system used.

Fig. 4
Fig. 4

Fourier components of the incident Gaussian as the aperture function is offset in the x direction by (a) −1 μm (b) 0 μm (c) 1 μm. The Fourier plane is assumed to be 2 μm from the focus in z direction.

Fig. 5
Fig. 5

Experimental and simulation data for phase change for light scattered off 1.1 and 3 μm diameter microspheres (probes) for known travel in the radial direction. The phase shifts from the simulation were calculated by fitting local peaks with standard Gaussian profiles and calculating relative phase shifts of the peak centers. In the experiment, the probes are moved in the radial direction by the AOM, while in simulation, we translate the aperture across the probe. Each experimental data point is taken over an averaging time of 8.3 ms. The error bars in the experimental data signify 1σ standard deviation. The standard deviation is predominantly due to drifts in the interferometer path length, but at low trapping powers, the Brownian motion of the trapped probe also contributes. The data point for the highest displacement of the 3 μm probe has a large error bar since the backscattered signal to noise itself was low with the probe having been displaced significantly from the detection laser.

Fig. 6
Fig. 6

Phase jitter due to path drifts in the external Mach-Zehnder interferometer. Case (a) shows the drifts due to the free running interferometer with averaging over 8.3 ms, the standard deviation in this case being around 15 deg. In case (b), we have used a side-of-fringe locking technique with 100 ms integration time. The standard deviation of the phase jitter is now around 200 mdeg.

Fig. 7
Fig. 7

(a) Power spectrum obtained by fourier transform of the phase jitter for a trapped probe of diameter 1.1 μm for an optical power of 48 mW. The data is fitted to a Lorentzian using IGOR fitting software. The fit parameters are shown, with the corner frequency given by f c = ( B ) coming out to be about 122 Hz. (b) Noise power spectrum of the interferometer in the absence of a trapped probe. The spectrum is mostly flat except for some structure at low frequencies which could be due to slow path drifts. Note that the amplitude of the noise spectrum is about an order of magnitude lower than that of the signal

Equations (18)

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E ( x , y , z ) = E 0 w 0 w ( z z w ) exp ( r 2 w 2 ( z z w ) ) exp ( ikz ik r 2 2 R ( z z w ) + i ζ ( z z w ) ) i ^
E ( k x , k y ) = n = 0 N m = 0 M E ( x , y , z = 0 ) e 2 π i [ k x ( n N ) + k y ( m M ) ]
E i = e ikz e i
E s = n = 0 E n [ ia n N e 1 n ( 3 ) b n M o 1 n ( 3 ) ]
E n = i n 2 n + 1 n ( n + 1 )
N e 1 n ( 3 ) = cos ϕ n ( n + 1 ) sin θ π n ( cos θ ) h n ( 1 ) ( ρ ) ρ e r + cos ϕ τ n ( cos θ ) d d ρ [ ρ h n ( 1 ) ( ρ ) ] ρ e θ sin ϕ π n ( cos θ ) h n ( 1 ) d d ρ [ ρ h n ( 1 ) ( ρ ) ] ρ e ϕ
M o 1 n ( 3 ) = cos ϕ π n ( cos θ ) h n ( 1 ) ( ρ ) e θ sin ϕ τ n ( cos θ ) h n ( 1 ) ( ρ ) e ϕ
ρ = kr ,
a n = m ψ n ( mx ) ψ n ( x ) ψ n ( x ) ψ n ( mx ) m ψ n ( mx ) ξ n ( x ) ξ n ( x ) ψ n ( mx ) ,
b n = ψ n ( mx ) ψ n ( x ) m ψ n ( x ) ψ n ( mx ) ψ n ( mx ) ξ n ( x ) m ξ n ( x ) ψ n ( mx ) ,
x = ka = 2 π n m a λ
m = k p k m = n p n m
ψ n ( ρ ) = ρ j n ( ρ )
ξ n ( ρ ) = ρ h n ( 1 ) ( ρ )
E s = [ sin θ cos ϕ cos θ cos ϕ sin ϕ sin θ sin ϕ cos θ sin ϕ cos ϕ cos θ sin θ 0 ] [ E r E θ E ϕ ]
E s = k x k y E ( k x , k y )   [ cos θ cos ϕ sin ϕ sin θ cos ϕ cos θ sin ϕ cos ϕ sin θ sin ϕ sin θ 0 cos θ ] E s
δ s = k B T 6 π η r κ 2 t ave ,
f c κ / ( 2 π γ 0 )

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