Abstract

Back-focal-plane interferometry is used to measure displacements of optically trapped samples with very high spatial and temporal resolution. However, the technique is closely related to a method that measures the rate of change in light momentum. It has long been known that displacements of the interference pattern at the back focal plane may be used to track the optical force directly, provided that a considerable fraction of the light is effectively monitored. Nonetheless, the practical application of this idea has been limited to counter-propagating, low-aperture beams where the accurate momentum measurements are possible. Here, we experimentally show that the connection can be extended to single-beam optical traps. In particular, we show that, in a gradient trap, the calibration product κ·β (where κ is the trap stiffness and 1/β is the position sensitivity) corresponds to the factor that converts detector signals into momentum changes; this factor is uniquely determined by three construction features of the detection instrument and does not depend, therefore, on the specific conditions of the experiment. Then, we find that force measurements obtained from back-focal-plane displacements are in practice not restricted to a linear relationship with position and hence they can be extended outside that regime. Finally, and more importantly, we show that these properties are still recognizable even when the system is not fully optimized for light collection. These results should enable a more general use of back-focal-plane interferometry whenever the ultimate goal is the measurement of the forces exerted by an optical trap.

© 2012 OSA

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2011 (5)

J. Mas, A. Farré, C. López-Quesada, X. Fernández, E. Martín-Badosa, and M. Montes-Usategui, “Measuring stall forces in vivo with optical tweezers through light momentum changes,” Proc. SPIE 8097, 809726, 809726-10 (2011).
[CrossRef]

I. Verdeny, A.-S. Fontaine, A. Farré, M. Montes-Usategui, and E. Martín-Badosa, “Heating effects on NG108 cells induced by laser trapping,” Proc. SPIE 8097, 809724, 809724-12 (2011).
[CrossRef]

M. Jahnel, M. Behrndt, A. Jannasch, E. Schäffer, and S. W. Grill, “Measuring the complete force field of an optical trap,” Opt. Lett. 36(7), 1260–1262 (2011).
[CrossRef] [PubMed]

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(12), 11759–11768 (2011).
[CrossRef] [PubMed]

T. Godazgar, R. Shokri, and S. N. S. Reihani, “Potential mapping of optical tweezers,” Opt. Lett. 36(16), 3284–3286 (2011).
[CrossRef] [PubMed]

2010 (3)

A. Farré and M. Montes-Usategui, “A force detection technique for single-beam optical traps based on direct measurement of light momentum changes,” Opt. Express 18(11), 11955–11968 (2010).
[CrossRef] [PubMed]

J. Dong, C. E. Castro, M. C. Boyce, M. J. Lang, and S. Lindquist, “Optical trapping with high forces reveals unexpected behaviors of prion fibrils,” Nat. Struct. Mol. Biol. 17(12), 1422–1430 (2010).
[CrossRef] [PubMed]

T. Čižmár, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photonics 4(6), 388–394 (2010).
[CrossRef]

2009 (1)

M. P. Landry, P. M. McCall, Z. Qi, and Y. R. Chemla, “Characterization of photoactivated singlet oxygen damage in single-molecule optical trap experiments,” Biophys. J. 97(8), 2128–2136 (2009).
[CrossRef] [PubMed]

2008 (1)

S. Perrone, G. Volpe, and D. Petrov, “10-fold detection range increase in quadrant-photodiode position sensing for photonic force microscope,” Rev. Sci. Instrum. 79(10), 106101 (2008).
[CrossRef] [PubMed]

2007 (3)

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A, Pure Appl. Opt. 9(8), S196–S203 (2007).
[CrossRef]

W. M. Lee, P. J. Reece, R. F. Marchington, N. K. Metzger, and K. Dholakia, “Construction and calibration of an optical trap on a fluorescence optical microscope,” Nat. Protoc. 2(12), 3226–3238 (2007).
[CrossRef] [PubMed]

E. Schäffer, S. F. Nørrelykke, and J. Howard, “Surface forces and drag coefficients of microspheres near a plane surface measured with optical tweezers,” Langmuir 23(7), 3654–3665 (2007).
[CrossRef] [PubMed]

2006 (2)

J. R. Moffitt, Y. R. Chemla, D. Izhaky, and C. Bustamante, “Differential detection of dual traps improves the spatial resolution of optical tweezers,” Proc. Natl. Acad. Sci. U.S.A. 103(24), 9006–9011 (2006).
[CrossRef] [PubMed]

N. B. Viana, M. S. Rocha, O. N. Mesquita, A. Mazolli, and P. A. Maia Neto, “Characterization of objective transmittance for optical tweezers,” Appl. Opt. 45(18), 4263–4269 (2006).
[CrossRef] [PubMed]

2004 (3)

K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75(3), 594–612 (2004).
[CrossRef]

I.-M. Tolić-Nørrelykke, K. Berg-Sørensen, and H. Flyvbjerg, “MatLab program for precision calibration of optical tweezers,” Comput. Phys. Commun. 159(3), 225–240 (2004).
[CrossRef]

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75(9), 2787–2809 (2004).
[CrossRef] [PubMed]

2003 (4)

S. P. Gross, “Application of optical traps in vivo,” Methods Enzymol. 361, 162–174 (2003).
[CrossRef] [PubMed]

S. B. Smith, Y. Cui, and C. Bustamante, “Optical-trap force transducer that operates by direct measurement of light momentum,” Methods Enzymol. 361, 134–162 (2003).
[CrossRef] [PubMed]

A. Rohrbach, H. Kress, and E. H. K. Stelzer, “Three-dimensional tracking of small spheres in focused laser beams: influence of the detection angular aperture,” Opt. Lett. 28(6), 411–413 (2003).
[CrossRef] [PubMed]

K. Berg-Sørensen, L. Oddershede, E.-L. Florin, and H. Flyvbjerg, “Unintended filtering in a typical photodiode detection system for optical tweezers,” J. Appl. Phys. 93(6), 3167–3176 (2003).
[CrossRef]

2002 (4)

W. Grange, S. Husale, H.-J. Güntherodt, and M. Hegner, “Optical tweezers system measuring the change in light momentum flux,” Rev. Sci. Instrum. 73(6), 2308–2316 (2002).
[CrossRef]

P. Bartlett and S. Henderson, “Three-dimensional force calibration of a single-beam optical gradient trap,” J. Phys. Condens. Matter 14(33), 7757–7768 (2002).
[CrossRef]

M. J. Lang, C. L. Asbury, J. W. Shaevitz, and S. M. Block, “An automated two-dimensional optical force clamp for single molecule studies,” Biophys. J. 83(1), 491–501 (2002).
[CrossRef] [PubMed]

A. Rohrbach and E. H. K. Stelzer, “Three-dimensional position detection of optically trapped dielectric particles,” J. Appl. Phys. 91(8), 5474–5488 (2002).
[CrossRef]

2001 (1)

1999 (2)

K. C. Neuman, E. H. Chadd, G. F. Liou, K. Bergman, and S. M. Block, “Characterization of photodamage to Escherichia coli in optical traps,” Biophys. J. 77(5), 2856–2863 (1999).
[CrossRef] [PubMed]

A. Pralle, M. Prummer, E.-L. Florin, E. H. K. Stelzer, and J. K. H. Hörber, “Three-dimensional high-resolution particle tracking for optical tweezers by forward scattered light,” Microsc. Res. Tech. 44(5), 378–386 (1999).
[CrossRef] [PubMed]

1998 (1)

1996 (2)

K. Visscher, S. P. Gross, and S. M. Block, “Construction of multiple-beam optical traps with nanometer-resolution position sensing,” IEEE J. Sel. Top. Quantum Electron. 2(4), 1066–1076 (1996).
[CrossRef]

S. B. Smith, Y. Cui, and C. Bustamante, “Overstretching B-DNA: the elastic response of individual double-stranded and single-stranded DNA molecules,” Science 271(5250), 795–799 (1996).
[CrossRef] [PubMed]

1994 (4)

L. P. Ghislain, N. A. Switz, and W. W. Webb, “Measurement of small forces using an optical trap,” Rev. Sci. Instrum. 65(9), 2762–2768 (1994).
[CrossRef]

K. Svoboda and S. M. Block, “Biological applications of optical forces,” Annu. Rev. Biophys. Biomol. Struct. 23(1), 247–285 (1994).
[CrossRef] [PubMed]

J. T. Finer, R. M. Simmons, and J. A. Spudich, “Single myosin molecule mechanics: piconewton forces and nanometre steps,” Nature 368(6467), 113–119 (1994).
[CrossRef] [PubMed]

K. Svoboda and S. M. Block, “Force and velocity measured for single kinesin molecules,” Cell 77(5), 773–784 (1994).
[CrossRef] [PubMed]

1993 (6)

S. C. Kuo and M. P. Sheetz, “Force of single kinesin molecules measured with optical tweezers,” Science 260(5105), 232–234 (1993).
[CrossRef] [PubMed]

K. Svoboda, C. F. Schmidt, B. J. Schnapp, and S. M. Block, “Direct observation of kinesin stepping by optical trapping interferometry,” Nature 365(6448), 721–727 (1993).
[CrossRef] [PubMed]

W. H. Wright, G. J. Sonek, and M. W. Berns, “Radiation trapping forces on microspheres with optical tweezers,” Appl. Phys. Lett. 63(6), 715–717 (1993).
[CrossRef]

L. P. Ghislain and W. W. Webb, “Scanning-force microscope based on an optical trap,” Opt. Lett. 18(19), 1678–1680 (1993).
[CrossRef] [PubMed]

S. Hell, G. Reiner, C. Cremer, and E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index,” J. Microsc. 169(3), 391–405 (1993).
[CrossRef]

C. J. R. Sheppard and M. Gu, “Imaging by a high aperture optical system,” J. Mod. Opt. 40(8), 1631–1651 (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(2), 569–582 (1992).
[CrossRef] [PubMed]

S. Kamimura and R. Kamiya, “High-frequency vibration in flagellar axonemes with amplitudes reflecting the size of tubulin,” J. Cell Biol. 116(6), 1443–1454 (1992).
[CrossRef] [PubMed]

1990 (1)

1986 (1)

1977 (1)

1971 (1)

1959 (2)

E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 349–357 (1959).
[CrossRef]

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
[CrossRef]

Asbury, C. L.

M. J. Lang, C. L. Asbury, J. W. Shaevitz, and S. M. Block, “An automated two-dimensional optical force clamp for single molecule studies,” Biophys. J. 83(1), 491–501 (2002).
[CrossRef] [PubMed]

Ashkin, A.

A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61(2), 569–582 (1992).
[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(5), 288–290 (1986).
[CrossRef] [PubMed]

Bartlett, P.

P. Bartlett and S. Henderson, “Three-dimensional force calibration of a single-beam optical gradient trap,” J. Phys. Condens. Matter 14(33), 7757–7768 (2002).
[CrossRef]

Behrndt, M.

Bergman, K.

K. C. Neuman, E. H. Chadd, G. F. Liou, K. Bergman, and S. M. Block, “Characterization of photodamage to Escherichia coli in optical traps,” Biophys. J. 77(5), 2856–2863 (1999).
[CrossRef] [PubMed]

Berg-Sørensen, K.

K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75(3), 594–612 (2004).
[CrossRef]

I.-M. Tolić-Nørrelykke, K. Berg-Sørensen, and H. Flyvbjerg, “MatLab program for precision calibration of optical tweezers,” Comput. Phys. Commun. 159(3), 225–240 (2004).
[CrossRef]

K. Berg-Sørensen, L. Oddershede, E.-L. Florin, and H. Flyvbjerg, “Unintended filtering in a typical photodiode detection system for optical tweezers,” J. Appl. Phys. 93(6), 3167–3176 (2003).
[CrossRef]

Berns, M. W.

W. H. Wright, G. J. Sonek, and M. W. Berns, “Radiation trapping forces on microspheres with optical tweezers,” Appl. Phys. Lett. 63(6), 715–717 (1993).
[CrossRef]

Bjorkholm, J. E.

Block, S. M.

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75(9), 2787–2809 (2004).
[CrossRef] [PubMed]

M. J. Lang, C. L. Asbury, J. W. Shaevitz, and S. M. Block, “An automated two-dimensional optical force clamp for single molecule studies,” Biophys. J. 83(1), 491–501 (2002).
[CrossRef] [PubMed]

K. C. Neuman, E. H. Chadd, G. F. Liou, K. Bergman, and S. M. Block, “Characterization of photodamage to Escherichia coli in optical traps,” Biophys. J. 77(5), 2856–2863 (1999).
[CrossRef] [PubMed]

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J. Dong, C. E. Castro, M. C. Boyce, M. J. Lang, and S. Lindquist, “Optical trapping with high forces reveals unexpected behaviors of prion fibrils,” Nat. Struct. Mol. Biol. 17(12), 1422–1430 (2010).
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M. P. Landry, P. M. McCall, Z. Qi, and Y. R. Chemla, “Characterization of photoactivated singlet oxygen damage in single-molecule optical trap experiments,” Biophys. J. 97(8), 2128–2136 (2009).
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J. Dong, C. E. Castro, M. C. Boyce, M. J. Lang, and S. Lindquist, “Optical trapping with high forces reveals unexpected behaviors of prion fibrils,” Nat. Struct. Mol. Biol. 17(12), 1422–1430 (2010).
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J. Dong, C. E. Castro, M. C. Boyce, M. J. Lang, and S. Lindquist, “Optical trapping with high forces reveals unexpected behaviors of prion fibrils,” Nat. Struct. Mol. Biol. 17(12), 1422–1430 (2010).
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K. C. Neuman, E. H. Chadd, G. F. Liou, K. Bergman, and S. M. Block, “Characterization of photodamage to Escherichia coli in optical traps,” Biophys. J. 77(5), 2856–2863 (1999).
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T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A, Pure Appl. Opt. 9(8), S196–S203 (2007).
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J. Mas, A. Farré, C. López-Quesada, X. Fernández, E. Martín-Badosa, and M. Montes-Usategui, “Measuring stall forces in vivo with optical tweezers through light momentum changes,” Proc. SPIE 8097, 809726, 809726-10 (2011).
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Maia Neto, P. A.

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W. M. Lee, P. J. Reece, R. F. Marchington, N. K. Metzger, and K. Dholakia, “Construction and calibration of an optical trap on a fluorescence optical microscope,” Nat. Protoc. 2(12), 3226–3238 (2007).
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J. Mas, A. Farré, C. López-Quesada, X. Fernández, E. Martín-Badosa, and M. Montes-Usategui, “Measuring stall forces in vivo with optical tweezers through light momentum changes,” Proc. SPIE 8097, 809726, 809726-10 (2011).
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I. Verdeny, A.-S. Fontaine, A. Farré, M. Montes-Usategui, and E. Martín-Badosa, “Heating effects on NG108 cells induced by laser trapping,” Proc. SPIE 8097, 809724, 809724-12 (2011).
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J. Mas, A. Farré, C. López-Quesada, X. Fernández, E. Martín-Badosa, and M. Montes-Usategui, “Measuring stall forces in vivo with optical tweezers through light momentum changes,” Proc. SPIE 8097, 809726, 809726-10 (2011).
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T. Čižmár, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photonics 4(6), 388–394 (2010).
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McCall, P. M.

M. P. Landry, P. M. McCall, Z. Qi, and Y. R. Chemla, “Characterization of photoactivated singlet oxygen damage in single-molecule optical trap experiments,” Biophys. J. 97(8), 2128–2136 (2009).
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Metzger, N. K.

W. M. Lee, P. J. Reece, R. F. Marchington, N. K. Metzger, and K. Dholakia, “Construction and calibration of an optical trap on a fluorescence optical microscope,” Nat. Protoc. 2(12), 3226–3238 (2007).
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J. R. Moffitt, Y. R. Chemla, D. Izhaky, and C. Bustamante, “Differential detection of dual traps improves the spatial resolution of optical tweezers,” Proc. Natl. Acad. Sci. U.S.A. 103(24), 9006–9011 (2006).
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I. Verdeny, A.-S. Fontaine, A. Farré, M. Montes-Usategui, and E. Martín-Badosa, “Heating effects on NG108 cells induced by laser trapping,” Proc. SPIE 8097, 809724, 809724-12 (2011).
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J. Mas, A. Farré, C. López-Quesada, X. Fernández, E. Martín-Badosa, and M. Montes-Usategui, “Measuring stall forces in vivo with optical tweezers through light momentum changes,” Proc. SPIE 8097, 809726, 809726-10 (2011).
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E. Schäffer, S. F. Nørrelykke, and J. Howard, “Surface forces and drag coefficients of microspheres near a plane surface measured with optical tweezers,” Langmuir 23(7), 3654–3665 (2007).
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K. Berg-Sørensen, L. Oddershede, E.-L. Florin, and H. Flyvbjerg, “Unintended filtering in a typical photodiode detection system for optical tweezers,” J. Appl. Phys. 93(6), 3167–3176 (2003).
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W. M. Lee, P. J. Reece, R. F. Marchington, N. K. Metzger, and K. Dholakia, “Construction and calibration of an optical trap on a fluorescence optical microscope,” Nat. Protoc. 2(12), 3226–3238 (2007).
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K. Svoboda, C. F. Schmidt, B. J. Schnapp, and S. M. Block, “Direct observation of kinesin stepping by optical trapping interferometry,” Nature 365(6448), 721–727 (1993).
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M. J. Lang, C. L. Asbury, J. W. Shaevitz, and S. M. Block, “An automated two-dimensional optical force clamp for single molecule studies,” Biophys. J. 83(1), 491–501 (2002).
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K. Svoboda, C. F. Schmidt, B. J. Schnapp, and S. M. Block, “Direct observation of kinesin stepping by optical trapping interferometry,” Nature 365(6448), 721–727 (1993).
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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(9), 2762–2768 (1994).
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I.-M. Tolić-Nørrelykke, K. Berg-Sørensen, and H. Flyvbjerg, “MatLab program for precision calibration of optical tweezers,” Comput. Phys. Commun. 159(3), 225–240 (2004).
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Verdeny, I.

I. Verdeny, A.-S. Fontaine, A. Farré, M. Montes-Usategui, and E. Martín-Badosa, “Heating effects on NG108 cells induced by laser trapping,” Proc. SPIE 8097, 809724, 809724-12 (2011).
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K. Visscher, S. P. Gross, and S. M. Block, “Construction of multiple-beam optical traps with nanometer-resolution position sensing,” IEEE J. Sel. Top. Quantum Electron. 2(4), 1066–1076 (1996).
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S. Perrone, G. Volpe, and D. Petrov, “10-fold detection range increase in quadrant-photodiode position sensing for photonic force microscope,” Rev. Sci. Instrum. 79(10), 106101 (2008).
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B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
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W. H. Wright, G. J. Sonek, and M. W. Berns, “Radiation trapping forces on microspheres with optical tweezers,” Appl. Phys. Lett. 63(6), 715–717 (1993).
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K. Svoboda and S. M. Block, “Biological applications of optical forces,” Annu. Rev. Biophys. Biomol. Struct. 23(1), 247–285 (1994).
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W. H. Wright, G. J. Sonek, and M. W. Berns, “Radiation trapping forces on microspheres with optical tweezers,” Appl. Phys. Lett. 63(6), 715–717 (1993).
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K. C. Neuman, E. H. Chadd, G. F. Liou, K. Bergman, and S. M. Block, “Characterization of photodamage to Escherichia coli in optical traps,” Biophys. J. 77(5), 2856–2863 (1999).
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I.-M. Tolić-Nørrelykke, K. Berg-Sørensen, and H. Flyvbjerg, “MatLab program for precision calibration of optical tweezers,” Comput. Phys. Commun. 159(3), 225–240 (2004).
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A. Pralle, M. Prummer, E.-L. Florin, E. H. K. Stelzer, and J. K. H. Hörber, “Three-dimensional high-resolution particle tracking for optical tweezers by forward scattered light,” Microsc. Res. Tech. 44(5), 378–386 (1999).
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W. M. Lee, P. J. Reece, R. F. Marchington, N. K. Metzger, and K. Dholakia, “Construction and calibration of an optical trap on a fluorescence optical microscope,” Nat. Protoc. 2(12), 3226–3238 (2007).
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J. Dong, C. E. Castro, M. C. Boyce, M. J. Lang, and S. Lindquist, “Optical trapping with high forces reveals unexpected behaviors of prion fibrils,” Nat. Struct. Mol. Biol. 17(12), 1422–1430 (2010).
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Nature (2)

K. Svoboda, C. F. Schmidt, B. J. Schnapp, and S. M. Block, “Direct observation of kinesin stepping by optical trapping interferometry,” Nature 365(6448), 721–727 (1993).
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Opt. Express (2)

Opt. Lett. (6)

Proc. Natl. Acad. Sci. U.S.A. (1)

J. R. Moffitt, Y. R. Chemla, D. Izhaky, and C. Bustamante, “Differential detection of dual traps improves the spatial resolution of optical tweezers,” Proc. Natl. Acad. Sci. U.S.A. 103(24), 9006–9011 (2006).
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Proc. R. Soc. Lond. A Math. Phys. Sci. (2)

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[CrossRef]

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[CrossRef]

Proc. SPIE (2)

J. Mas, A. Farré, C. López-Quesada, X. Fernández, E. Martín-Badosa, and M. Montes-Usategui, “Measuring stall forces in vivo with optical tweezers through light momentum changes,” Proc. SPIE 8097, 809726, 809726-10 (2011).
[CrossRef]

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[CrossRef]

Rev. Sci. Instrum. (5)

S. Perrone, G. Volpe, and D. Petrov, “10-fold detection range increase in quadrant-photodiode position sensing for photonic force microscope,” Rev. Sci. Instrum. 79(10), 106101 (2008).
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[CrossRef]

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[CrossRef]

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Other (3)

S. B. Smith, “Stretch transitions observed in single biopolymer molecules (DNA or protein) using laser tweezers,” Doctoral Thesis, University of Twente, The Netherlands (1998).

See Table 1 in: A. Wozniak, “Characterizing quantitative measurements of force and displacement with optical tweezers on the NanotrackerTM,” JPK Instruments Technical Report, http://www.jpk.com/optical-tweezers.233.en.html ; the product of stiffness and parameter β changes by a factor of more than two, for polystyrene beads between 100 nm and 4260 nm (similarly for silica beads, Table 2). Also, in Table 1 and Fig. 5 in: A. Buosciolo, G. Pesce, and A. Sasso, “New calibration method for position detector for simultaneous measurements of force constants and local viscosity in optical tweezers,” Opt. Commun. 230, 375–368 (2004), a factor of two or more is observed between the values of κ·β obtained for different positions of the trap. Finally, Fig. 9 in: M. Capitanio, G. Romano, R. Ballerini, M. Giuntini, F. S. Pavone, D. Dunlap, and L. Finzi, “Calibration of optical tweezers with differential interference contrast signals,” Rev. Sci. Instrum. 73, 1687–1696 (2002), shows a product factor κ·β that apparently changes more than sevenfold for beads between 500 and 5000 nm in size (position detection is through DIC interferometry).

T. Otaki, “Condenser lens system for use in a microscope,” US Patent no. 5657166 (1997).

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Figures (13)

Fig. 1
Fig. 1

Position and force detection capabilities of the BFPI instrument. (a) Position signal and (b) fraction of total intensity. (c) Comparison between the position reading from the detector and the actual piezo displacement. Two different V-to-µm conversions are shown: one obtained with the correct β (orange dots) and the other by multiplying by the β corresponding to a different bead size, (a 1.16-µm particle; hollow squares). The dark shaded area indicates the region where positions are correctly measured with the first calibration factor. (d) The orange dots in (c) are multiplied by the trap stiffness to indicate the force. The light shaded area shows the region where this matches the theoretical force–displacement curve. The hollow squares are obtained by multiplying the corresponding curve in (c) (that with a mismatched β) by the 1.16-µm-bead stiffness. The product of the two mismatched factors gives a puzzlingly accurate force curve. Finally, the dashed vertical lines indicate the bead limits. (e) Error between theory and experiment in (d). The recorded data match the force curve for values up to 2.8 µm within a 6% error, comparable to the uncertainty of the absolute calibration of the instrument (Table 1). (f) Variation of trap stiffness as a function of bead position. (g) Theoretical and experimental force curves for a 0.61-µm bead corresponding to a measured laser power of 17.5 ± 0.9 mW.

Fig. 2
Fig. 2

Power spectrum calibration. (a) Typical two-sided power spectrum of the Brownian motion of a trapped bead (grey dots) and fitting to a corrected Lorentzian curve (orange line). More details about the data recording and analysis are given in the Methods section. The linear dependence of both stiffness, κ, and sensitivity, 1/β, on the laser power (inset) is evidence of correct measurement of the two parameters. (b) Different effects were taken into account to obtain correct measurements of the two constants κ and β. (c) From the fitting of the power spectrum data to a corrected Lorentzian curve, we obtained a mean value for the 3dB-frequency used to characterize the frequency response of the photodetector as a first-order filter, at λ = 1064 nm (f3dB = 6830 ± 170 Hz; mean + SD; n = 60). We checked this result by fitting a simple Lorentzian function to the power spectrum of the laser alone, obtaining a value of 6.7 kHz. (d) In some cases, the digitization error from the analogue-to-digital converter of our detection system showed up in the spectra. We took this into account in the fitting. The dashed line indicates the noise level in this experiment.

Fig. 3
Fig. 3

Relationship between κ and β. (a) Sensitivity is plotted against stiffness for every pair of parameters; the experiment was repeated for 30 different sets of experimental conditions. The linear fit shows the proportionality of the two constants. The variety of experimental conditions is highlighted in (b-k).

Fig. 4
Fig. 4

Schematic representation of the telecentric system used for position detection (not to scale). After interacting with the sample, the focused laser beam is scattered in all directions, but mostly concentrated in the forward direction. When the sample remains at the centre of the trap, the light pattern at the BFP of the condenser lens is symmetric, as it is for the incident beam, and the detector signal is zero. The distribution of light in this plane changes when the position, x0, of the trapped sample relative to the incident beam varies. The refraction of the beam at the water–glass interface at the exit surface of the microchamber allows a large fraction of all the scattered light to be captured (Appendix and Ref [20].). The initial structure of the beam, observed at the front focal plane of the objective, and the changes due to the sample are shown.

Fig. 5
Fig. 5

Determination of f’. (a) Layout of our BFPI instrument. H and H’ indicate the principal planes of both the condenser and the relay lens. (b) The relay lens (or the PSD) position affects the effective focal length of the system. A change of 1 mm in the relay lens position leads to a variation of 6% in f’, which translates into a similar error in αdetector (δαdetectordetector = δf’/f’). This was observed in the calibration experiments where we found a change from 100 to 109 pN/V. We also found that a further reduction of the distance between the lens and the detector (~10 mm) eventually led to a 100% difference in αdetector. In contrast, such changes in the position of the optical elements did not have any impact on the calibration of the instrument efficiency, ψ. (c) In order to establish the correct position of the relay lens, the photodetector signal was recorded as an empty trap was holographically moved in steps of 10 µm across the field of view between two extreme points separated by 100 µm. Taking advantage of the Fourier transform relation between the sample plane and the BFP of the condenser and its shift property (inset), the proper axial position of the relay lens was identified as the one for which the variation in the voltage was minimum. (d) An alternative Ronchi ruling experiment with the photodetector was used both to measure the focal length of the instrument and to determine the contribution of the asymmetries of the PSD responsivity along its two independent axes. Plane waves with known transverse momentum were generated and sequentially projected onto the PSD; they were selected by means of an iris located at the BFP of the condenser lens. The sequence of points was first along the x-axis, then the y-axis and at 45° between the two. The normalized signal for each plane wave, Sr/Ssum = r/RD, where r is the position of the focused wave on the detector and RD is the detector radius, was plotted against its transverse momentum, that is, n·sinθ. The fitting was used to determine the quotient f’/RD. No significant differences were observed between the results for the three directions (~1%).

Fig. 6
Fig. 6

Determination of the efficiency, ψ, of the detection instrument. The infrared laser (λ = 1064 nm), with circular polarization, was focused by the objective lenses: (a) Nikon CFI Plan Apo VC 60xA WI and (b) Nikon CFI Plan Fluor 100xH. The former is an NA = 1.2 water-immersion lens with an entrance pupil diameter rpupil = 4 mm ( = f’NA, f’ = 3.33 mm); the latter is an NA = 1.3 oil-immersion objective with rpupil = 2.6 mm (f’ = 2 mm). The laser power at the back aperture of the objective (triangles) was measured as the diameter of an iris, r, located in a conjugate plane before the telescope was changed. The beam waist, w = 5.6 ± 0.2 mm, was calculated by fitting the data to P(r) = P0(1 - exp(-2r2/w2)), where P0 is the incident laser power, and it was found to be coincident, to within the error, with the product m·rbeam, where m = 2.22 is the magnification between the laser fiber diameter and the back aperture of the objective in our setup, and rbeam = 2.55 mm is the output laser radius. The power in the sample plane (circles) was then modulated by the transmittance function of the objective (top plots), which was measured using the dual objective method [38]. As pointed out in Ref [39], we found a non-homogeneous radial transmission. The profile, obtained for each value of the pupil radius as the ratio (Pout(r)/Pin(r))1/2, fitted a function Toffset + T0exp(-r2/2 σ2) with Toffset = 52.6 and σ = 3.6 mm for the water-immersion lens, and Toffset = 52.9 and σ = 3.3 mm for the oil-immersion objective. The measured transmissions were 55% and 62%, respectively, in good agreement with Ref [38]. The error between these values, corresponding to 〈T21/2, and the actual transmissions 〈T〉 [39] were 5% and 1.5%. Finally, the detector intensity reading (orange dots) was measured and was used (c) to determine the efficiency, ψ, for both objectives as the ratio Ssum(r)/Psample(r), with values of 56 V/W and 59 V/W, respectively. We found that these values were independent of the laser power (as expected) but they showed a certain (small) dependence on radial distance. (d) The mean value of the efficiency was obtained from the distribution of ψ for all the data analyzed, 58 ± 3 V/W, where the standard deviation represents a 5% error. The value depends on the filters in front of the PSD, but it can be corrected by their attenuation without a recalibration.

Fig. 7
Fig. 7

Comparison between αtrap and αdetector. (a) Values of αtrap in units of αdetector obtained from the power spectra for different experimental conditions. The bead size (1.16 µm, 3.06 µm and 8.06 µm) its refractive index (1.48 and 1.57) and the laser power were varied. The shaded area indicates the 6% error in αdetector, which was determined from the propagation of errors in f’ (3%) and Ψ (5%). The error bars were also obtained from the propagation of errors. (b) The separate distributions for x and y show that the two components follow Gaussian functions and are centered at different values: 98 ± 3 pN/V (mean ± SD; 3% error) and 94 ± 4 pN/V (mean ± SD; 4% error), respectively. The result for the y-component is 5% smaller than that for the x-component.

Fig. 8
Fig. 8

Effect of light losses on force measurements. (a) Sketch of the measurement process for a collecting lens with a small NA. (b) As for the results in Fig. 1, we show experimental curves of force for an 8.1-µm bead for different NAs of the condenser lens. The vertical dashed line indicates the limit where the results with NA~1.1 overlap with the correct force curve to within a 6% error. The shaded area corresponds to the harmonic region of the trap, where the force can be described by the orange (and blue) line (-κx0) to within a 6% error. The deviation from the linear approximation starts at 1.8 µm for NA = 1.1, although exact force measurements can be obtained at up to 2.8 µm. The horizontal dashed lines indicate again a 6% error. The range where measurements are correct for the reduced NA correlates with the amount of light collected. (c) An excessive reduction in the amount of light captured can make the system lose the robustness in the force calibration even for small displacements of the sample, as shown in (b) for NA~1. This may eventually restrict the use of the instrument to position detection only. This plot shows the relationship between stiffness and detector sensitivity for two different bead sizes and several laser powers for small NA values of the condenser (we chose 0.65 for this example, since it is a typical value [53]). The data obtained at different laser powers are still correlated, but show two different slopes for the two beads. There is no single calibration constant, αtrap, that characterizes the instrument, so recalibration for different experimental conditions would be necessary in this case.

Fig. 9
Fig. 9

Fraction of light collected. (a) The value of the force at which the experimental data deviates from the exact force curve in Fig. 8(b) for the low-NA condenser, depends on the sample, and more particularly, on its size. This is observed in a Mie scattering simulation of the fraction of forward-scattered light for different sizes of a polystyrene bead. The beam waist was 0.4 µm. (b) A faint scattering disk is the only evidence of the presence of a trapped sample for Rayleigh scatterers (arrows). (c, d) For large microspheres, the deflected beam (NA~1.1) remains inside a cone of NA = 1.2 (dashed circle) for a large range of displacements.

Fig. 10
Fig. 10

Computer simulation of the setup using ZEMAX. The inset is a magnified view of the sample region showing how the front lens captures plane waves scattered at high angles.

Fig. 11
Fig. 11

Analysis of the aplanatic lens requirement. (a) ZEMAX Simulation of the condenser lens alone. Light rays travelling at angles specified by their numerical aperture at the x-axis hit the BFP at positions given by coordinate r in the y-axis. (b) The residues are small and can be fitted to a power law. (c) Spherical aberrations are probably behind the observed discrepancies as they would shift rays following a five-order power law, close to that found in (b). (d) Experimental results for the compound system (condenser + relay) and simulation.

Fig. 12
Fig. 12

Analysis of the requirement of a collecting lens of long focal length. (a) Experimental light patterns at the BFP of the condenser, for two axial positions of the trap, and plot showing the dependence of the effective numerical aperture of the collecting lens with the axial distance. An increasing, but still small, fraction of the light travelling at large angles is lost. (b) Aplanatism of the condenser lens for increasing cover-glass-to-sample distance, the differences are barely noticeable. (c) A computer simulation of an oil-immersion objective with a focal length f’ = 2 mm, used as a substitute condenser and (d) degree of fulfillment of the Abbe sine condition. This lens is more sensitive to axial changes in the sample position because of its shorter focal length.

Fig. 13
Fig. 13

QPD vs. PSD. (a) A QPD produces outputs that are sensitive to the size of the sample as opposed to (b) a PSD, whose outputs are proportional to the centroid of the light distribution, and thus faithfully represent the net momentum flux when placed at the BFP.

Tables (1)

Tables Icon

Table 1 Values of α.

Equations (9)

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e i (r)= i λ sinθN A obj A i ( θ,φ ) e ik s ^ ·r dΩ = i λ sinθN A obj f'a( θ )· A 0 i ( f'sinθcosφ,f'sinθsinφ )· P ^ ( θ,φ ) e ik s ^ ·r sinθdθdφ ,
F x = p x ( θ,φ ) I Ω ( θ,φ ) E dΩ= n c sin θI Ω ( θ,φ ) dΩ n c P sinθ =κ x 0 .
x' = f'cκ P x 0 ,
f'nP sinθ =f'n sinθ· I Ω ( θ,φ )dΩ = f'nsinθ· nc 8π | f' cosθ A 0 s P ^ | 2 dΩ = 2 x' nc 8π | A 0 s ( x',y' ) | 2 dx'dy' = 2 x'I( x',y' )dx'dy' = x' P.
S x = S sum R D x' = ψf'cκ R D x 0 = 1 β x 0 ,
1 β = ψf'cκ R D .
F x =κ x 0 =κβ S x = R D ψf'c S x α S x .
α trap κ i · β i = R D ψf'c α detector ,
α trap = α detector S sum x' S sum m x ' m .

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