Abstract

We propose three-dimensional (3D) subpixel estimation in the position measurement of a nanoparticle held in optical tweezers in water by using an in-line, low-coherence digital holographic microscope. The 3D subpixel estimation was performed with the addition of axial subpixel estimation to the lateral subpixel estimation introduced in our previous work [Appl. Opt. 50, H183 (2011)]. The axial subpixel estimation allowed the step length in the diffraction calculation of a hologram to be increased to 20nm while keeping the axial resolution of 3nm. This drastically decreased the computation time of the diffraction calculation to less than 10% of the two-dimensional subpixel estimation.

© 2012 Optical Society of America

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    [CrossRef]
  45. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, 1969), Chap. 3.
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2011

2010

2009

M. Miyazaki and Y. Hayasaki, “Motion control of low-index microspheres in liquid based on optical repulsive force of a focused beam array,” Opt. Lett. 34, 821–823 (2009).
[CrossRef]

S. I. Eom, Y. Takaya, and T. Hayashi, “Novel contact probing method using single fiber optical trapping probe,” Precis. Eng. 33, 235–242 (2009).
[CrossRef]

2008

O. Otto, C. Gutsche, F. Kremer, and U. F. Keyser, “Optical tweezers with 2.5 kHz bandwidth video detection for single-colloid electrophoresis,” Rev. Sci. Instrum. 79, 023710 (2008).
[CrossRef]

K. C. Neuman and A. Nagy, “Single-molecule force spectroscopy: optical tweezers, magnetic tweezers and atomic force microscopy,” Nat. Meth. 5, 491–505 (2008).
[CrossRef]

Y. Yang, B. S. Kang, and Y. J. Choo, “Application of the correlation coefficient method for determination of the focus plane to digital particle holography,” Appl. Opt. 47, 817–824 (2008).
[CrossRef]

G. M. Gibson, J. Leach, S. Keen, A. J. Wright, and M. J. Padgett, “Measuring the accuracy of particle position and force in optical tweezers using high-speed video microscopy,” Opt. Express 16, 14561–14570 (2008).
[CrossRef]

2007

S. Nakanishi, S. Shoji, S. Kawata, and H.-B. Sun, “Giant elasticity of photopolymer nanowires,” Appl. Phys. Lett. 91, 063112 (2007).
[CrossRef]

S. H. Lee and D. G. Grier, “Holographic microscopy of holographically trapped three-dimensional structures,” Opt. Express 15, 1505–1512 (2007).
[CrossRef]

2006

2005

2004

M. Liebling and M. Unser, “Autofocus for digital Fresnel holograms by use of a Fresnelet-sparsity criterion,” J. Opt. Soc. Am. A 21, 2424–2430 (2004).
[CrossRef]

L. Ma, H. Wang, Y. Li, and H. Jin, “Numerical reconstruction of digital holograms for three-dimensional shape measurement,” J. Opt. A 6, 396–400 (2004).
[CrossRef]

K. Ladavac, K. Kasza, and D. G. Grier, “Sorting mesoscopic objects with periodic potential landscapes: optical fractionation,” Phys. Rev. E 70, 010901(R) (2004).
[CrossRef]

P. Jordan, H. Clare, L. Flendrig, J. Leach, J. Cooper, and M. Padgett, “Permanent 3D microstructures in a polymeric host created using holographic optical tweezers,” J. Mod. Opt. 51, 627–632 (2004).
[CrossRef]

2003

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426, 421–424 (2003).
[CrossRef]

M. Ichikawa, Y. Matsuzawa, Y. Koyama, and K. Yoshikawa, “Molecular fabrication: aligning DNA molecules as building blocks,” Langmuir 19, 5444–5447 (2003).
[CrossRef]

2002

2001

1999

M. Reicherter, T. Haist, E. U. Wagemann, and H. J. Tiziani, “Optical particle trapping with computer-generated holograms written on a liquid-crystal display,” Opt. Lett. 24, 608–610 (1999).
[CrossRef]

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, 378–386 (1999).
[CrossRef]

J. S. Meiners and S. Quake, “Direct measurement of hydrodynamic cross correlations between two particles in an external potential,” Phys. Rev. Lett. 82, 2211–2214 (1999).
[CrossRef]

Y. Hayasaki, M. Itoh, T. Yatagai, and N. Nishida, “Nonmechanical optical manipulation of microparticle using spatial light modulator,” Opt. Rev. 6, 24–27 (1999).
[CrossRef]

1998

E. R. Dufresne, and D. G. Grier, “Optical tweezer arrays and optical substrates created with diffractive optical elements,” Rev. Sci. Instrum. 69, 1974–1977 (1998).
[CrossRef]

F. Gittes, and C. F. Schmidt, “Interference model for back-focal-plane displacement detection in optical tweezers,” Opt. Lett. 23, 7–9 (1998).
[CrossRef]

1996

B. Skarman, K. Wozniac, and J. Becker, “Simultaneous 3D-PIV and temperature measurement using a new CCD based holographic interferometer,” Flow Meas. Instrum. 7, 1–6 (1996).
[CrossRef]

1994

1993

K. Svoboda, C. F. Schmidt, B. J. Schnapp, and S. M. Block, “Direct observation of 18 kinesin stepping by optical trapping interferometry,” Nature 365, 721–727 (1993).
[CrossRef]

1987

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

L. Onural and P. D. Scott, “Digital recording of in-line holograms,” Opt. Eng. 26, 1124–1132 (1987).

1986

Agarwal, R.

Ashkin, A.

Becker, J.

B. Skarman, K. Wozniac, and J. Becker, “Simultaneous 3D-PIV and temperature measurement using a new CCD based holographic interferometer,” Flow Meas. Instrum. 7, 1–6 (1996).
[CrossRef]

Bennink, M. L.

Bjorkholm, J. E.

Block, S. M.

W. J. Greenleaf and S. M. Block, “Single-molecule, motion-based DNA sequencing using RNA polymerase,” Science 313, 801 (2006).
[CrossRef]

K. Svoboda, C. F. Schmidt, B. J. Schnapp, and S. M. Block, “Direct observation of 18 kinesin stepping by optical trapping interferometry,” Nature 365, 721–727 (1993).
[CrossRef]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983), Chap. 4.

Bowman, R.

Cai, L.

Callens, N.

Cheong, F. C.

Choo, Y. J.

Chu, S.

Clare, H.

P. Jordan, H. Clare, L. Flendrig, J. Leach, J. Cooper, and M. Padgett, “Permanent 3D microstructures in a polymeric host created using holographic optical tweezers,” J. Mod. Opt. 51, 627–632 (2004).
[CrossRef]

Collin, D.

I. Tinoco, D. Collin, and P. T. X. Li, “Unfolding single RNA molecules: bridging the gap between equilibrium and non-equilibrium statistical thermodynamics,” Q. Rev. Biophys. 38, 291–301 (2005).
[CrossRef]

Cooper, J.

P. Jordan, H. Clare, L. Flendrig, J. Leach, J. Cooper, and M. Padgett, “Permanent 3D microstructures in a polymeric host created using holographic optical tweezers,” J. Mod. Opt. 51, 627–632 (2004).
[CrossRef]

Dholakia, K.

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426, 421–424 (2003).
[CrossRef]

Dixon, L.

Dubois, F.

Dufresne, E. R.

E. R. Dufresne, and D. G. Grier, “Optical tweezer arrays and optical substrates created with diffractive optical elements,” Rev. Sci. Instrum. 69, 1974–1977 (1998).
[CrossRef]

Dziedzic, J. M.

Eom, S. I.

S. I. Eom, Y. Takaya, and T. Hayashi, “Novel contact probing method using single fiber optical trapping probe,” Precis. Eng. 33, 235–242 (2009).
[CrossRef]

Flendrig, L.

P. Jordan, H. Clare, L. Flendrig, J. Leach, J. Cooper, and M. Padgett, “Permanent 3D microstructures in a polymeric host created using holographic optical tweezers,” J. Mod. Opt. 51, 627–632 (2004).
[CrossRef]

Florin, E. L.

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, 378–386 (1999).
[CrossRef]

Flyvbjerg, H.

S. F. Tolic-Norrelykke, E. Schäffer, J. Howard, F. S. Pavone, F. Jülicher, and H. Flyvbjerg, “Calibration of optical tweezers with positional detection in the back focal plane,” Rev. Sci. Instrum. 77, 103101 (2006).
[CrossRef]

Garcia-Sucerquia, J.

Gibson, G.

Gibson, G. M.

Gittes, F.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), Chap. 3.10.

Greenleaf, W. J.

W. J. Greenleaf and S. M. Block, “Single-molecule, motion-based DNA sequencing using RNA polymerase,” Science 313, 801 (2006).
[CrossRef]

Grier, D. G.

Gutsche, C.

O. Otto, C. Gutsche, F. Kremer, and U. F. Keyser, “Optical tweezers with 2.5 kHz bandwidth video detection for single-colloid electrophoresis,” Rev. Sci. Instrum. 79, 023710 (2008).
[CrossRef]

Haist, T.

Hajizadeh, F.

Hasegawa, S.

Hayasaki, Y.

Hayashi, T.

S. I. Eom, Y. Takaya, and T. Hayashi, “Novel contact probing method using single fiber optical trapping probe,” Precis. Eng. 33, 235–242 (2009).
[CrossRef]

Higuchi, T.

Hörber, J. K. H.

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, 378–386 (1999).
[CrossRef]

Howard, J.

S. F. Tolic-Norrelykke, E. Schäffer, J. Howard, F. S. Pavone, F. Jülicher, and H. Flyvbjerg, “Calibration of optical tweezers with positional detection in the back focal plane,” Rev. Sci. Instrum. 77, 103101 (2006).
[CrossRef]

Hoyos, M.

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983), Chap. 4.

Huisstede, J. H. G.

Ichikawa, M.

M. Ichikawa, Y. Matsuzawa, Y. Koyama, and K. Yoshikawa, “Molecular fabrication: aligning DNA molecules as building blocks,” Langmuir 19, 5444–5447 (2003).
[CrossRef]

Itoh, M.

Y. Hayasaki, M. Itoh, T. Yatagai, and N. Nishida, “Nonmechanical optical manipulation of microparticle using spatial light modulator,” Opt. Rev. 6, 24–27 (1999).
[CrossRef]

Jericho, M. H.

Jericho, S. K.

Jin, H.

L. Ma, H. Wang, Y. Li, and H. Jin, “Numerical reconstruction of digital holograms for three-dimensional shape measurement,” J. Opt. A 6, 396–400 (2004).
[CrossRef]

Jordan, P.

P. Jordan, H. Clare, L. Flendrig, J. Leach, J. Cooper, and M. Padgett, “Permanent 3D microstructures in a polymeric host created using holographic optical tweezers,” J. Mod. Opt. 51, 627–632 (2004).
[CrossRef]

Jülicher, F.

S. F. Tolic-Norrelykke, E. Schäffer, J. Howard, F. S. Pavone, F. Jülicher, and H. Flyvbjerg, “Calibration of optical tweezers with positional detection in the back focal plane,” Rev. Sci. Instrum. 77, 103101 (2006).
[CrossRef]

Jüptner, W.

Kang, B. S.

Kasza, K.

K. Ladavac, K. Kasza, and D. G. Grier, “Sorting mesoscopic objects with periodic potential landscapes: optical fractionation,” Phys. Rev. E 70, 010901(R) (2004).
[CrossRef]

Kawata, S.

S. Nakanishi, S. Shoji, S. Kawata, and H.-B. Sun, “Giant elasticity of photopolymer nanowires,” Appl. Phys. Lett. 91, 063112 (2007).
[CrossRef]

Keen, S.

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, 1969), Chap. 3.

Keyser, U. F.

O. Otto, C. Gutsche, F. Kremer, and U. F. Keyser, “Optical tweezers with 2.5 kHz bandwidth video detection for single-colloid electrophoresis,” Rev. Sci. Instrum. 79, 023710 (2008).
[CrossRef]

Klages, P.

Koyama, Y.

M. Ichikawa, Y. Matsuzawa, Y. Koyama, and K. Yoshikawa, “Molecular fabrication: aligning DNA molecules as building blocks,” Langmuir 19, 5444–5447 (2003).
[CrossRef]

Kremer, F.

O. Otto, C. Gutsche, F. Kremer, and U. F. Keyser, “Optical tweezers with 2.5 kHz bandwidth video detection for single-colloid electrophoresis,” Rev. Sci. Instrum. 79, 023710 (2008).
[CrossRef]

Kreuzer, H. J.

Krishnatreya, B. J.

Kurowski, P.

Ladavac, K.

R. Agarwal, K. Ladavac, Y. Roichman, G. Yu, C. M. Lieber, and D. G. Grier, “Manipulation and assembly of nanowires with holographic optical traps,” Opt. Express 13, 8906–8912 (2005).
[CrossRef]

K. Ladavac, K. Kasza, and D. G. Grier, “Sorting mesoscopic objects with periodic potential landscapes: optical fractionation,” Phys. Rev. E 70, 010901(R) (2004).
[CrossRef]

Leach, J.

G. M. Gibson, J. Leach, S. Keen, A. J. Wright, and M. J. Padgett, “Measuring the accuracy of particle position and force in optical tweezers using high-speed video microscopy,” Opt. Express 16, 14561–14570 (2008).
[CrossRef]

P. Jordan, H. Clare, L. Flendrig, J. Leach, J. Cooper, and M. Padgett, “Permanent 3D microstructures in a polymeric host created using holographic optical tweezers,” J. Mod. Opt. 51, 627–632 (2004).
[CrossRef]

Lee, S. H.

Li, P. T. X.

I. Tinoco, D. Collin, and P. T. X. Li, “Unfolding single RNA molecules: bridging the gap between equilibrium and non-equilibrium statistical thermodynamics,” Q. Rev. Biophys. 38, 291–301 (2005).
[CrossRef]

Li, Y.

L. Ma, H. Wang, Y. Li, and H. Jin, “Numerical reconstruction of digital holograms for three-dimensional shape measurement,” J. Opt. A 6, 396–400 (2004).
[CrossRef]

Lieber, C. M.

Liebling, M.

Ma, L.

L. Ma, H. Wang, Y. Li, and H. Jin, “Numerical reconstruction of digital holograms for three-dimensional shape measurement,” J. Opt. A 6, 396–400 (2004).
[CrossRef]

MacDonald, M. P.

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426, 421–424 (2003).
[CrossRef]

Mallahi, E.

Matsuzawa, Y.

M. Ichikawa, Y. Matsuzawa, Y. Koyama, and K. Yoshikawa, “Molecular fabrication: aligning DNA molecules as building blocks,” Langmuir 19, 5444–5447 (2003).
[CrossRef]

Meiners, J. S.

J. S. Meiners and S. Quake, “Direct measurement of hydrodynamic cross correlations between two particles in an external potential,” Phys. Rev. Lett. 82, 2211–2214 (1999).
[CrossRef]

Miyazaki, M.

Monnom, O.

Nagy, A.

K. C. Neuman and A. Nagy, “Single-molecule force spectroscopy: optical tweezers, magnetic tweezers and atomic force microscopy,” Nat. Meth. 5, 491–505 (2008).
[CrossRef]

Nakanishi, S.

S. Nakanishi, S. Shoji, S. Kawata, and H.-B. Sun, “Giant elasticity of photopolymer nanowires,” Appl. Phys. Lett. 91, 063112 (2007).
[CrossRef]

Neuman, K. C.

K. C. Neuman and A. Nagy, “Single-molecule force spectroscopy: optical tweezers, magnetic tweezers and atomic force microscopy,” Nat. Meth. 5, 491–505 (2008).
[CrossRef]

Nishida, N.

S. Tamano, Y. Hayasaki, and N. Nishida, “Phase-shifting digital holography with a low-coherence light source for reconstruction of a digital relief object hidden behind a light-scattering medium,” Appl. Opt. 45, 953–959 (2006).
[CrossRef]

Y. Hayasaki, M. Itoh, T. Yatagai, and N. Nishida, “Nonmechanical optical manipulation of microparticle using spatial light modulator,” Opt. Rev. 6, 24–27 (1999).
[CrossRef]

Onural, L.

L. Onural and P. D. Scott, “Digital recording of in-line holograms,” Opt. Eng. 26, 1124–1132 (1987).

Otto, O.

O. Otto, C. Gutsche, F. Kremer, and U. F. Keyser, “Optical tweezers with 2.5 kHz bandwidth video detection for single-colloid electrophoresis,” Rev. Sci. Instrum. 79, 023710 (2008).
[CrossRef]

Padgett, M.

R. Bowman, G. Gibson, and M. Padgett, “Particle tracking stereomicroscopy in optical tweezers: control of trap shape,” Opt. Express 18, 11785–11790 (2010).
[CrossRef]

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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, 378–386 (1999).
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S. F. Tolic-Norrelykke, E. Schäffer, J. Howard, F. S. Pavone, F. Jülicher, and H. Flyvbjerg, “Calibration of optical tweezers with positional detection in the back focal plane,” Rev. Sci. Instrum. 77, 103101 (2006).
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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, 378–386 (1999).
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Subramaniam, V.

Sun, H.-B.

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[CrossRef]

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K. Svoboda, C. F. Schmidt, B. J. Schnapp, and S. M. Block, “Direct observation of 18 kinesin stepping by optical trapping interferometry,” Nature 365, 721–727 (1993).
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Xu, W.

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Yourassowsky, C.

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Yu, L.

Appl. Opt.

Appl. Phys. Lett.

S. Nakanishi, S. Shoji, S. Kawata, and H.-B. Sun, “Giant elasticity of photopolymer nanowires,” Appl. Phys. Lett. 91, 063112 (2007).
[CrossRef]

Flow Meas. Instrum.

B. Skarman, K. Wozniac, and J. Becker, “Simultaneous 3D-PIV and temperature measurement using a new CCD based holographic interferometer,” Flow Meas. Instrum. 7, 1–6 (1996).
[CrossRef]

J. Mod. Opt.

P. Jordan, H. Clare, L. Flendrig, J. Leach, J. Cooper, and M. Padgett, “Permanent 3D microstructures in a polymeric host created using holographic optical tweezers,” J. Mod. Opt. 51, 627–632 (2004).
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J. Opt. A

L. Ma, H. Wang, Y. Li, and H. Jin, “Numerical reconstruction of digital holograms for three-dimensional shape measurement,” J. Opt. A 6, 396–400 (2004).
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J. Opt. Soc. Am. A

Langmuir

M. Ichikawa, Y. Matsuzawa, Y. Koyama, and K. Yoshikawa, “Molecular fabrication: aligning DNA molecules as building blocks,” Langmuir 19, 5444–5447 (2003).
[CrossRef]

Microsc. Res. Tech.

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, 378–386 (1999).
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Nat. Meth.

K. C. Neuman and A. Nagy, “Single-molecule force spectroscopy: optical tweezers, magnetic tweezers and atomic force microscopy,” Nat. Meth. 5, 491–505 (2008).
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Nature

K. Svoboda, C. F. Schmidt, B. J. Schnapp, and S. M. Block, “Direct observation of 18 kinesin stepping by optical trapping interferometry,” Nature 365, 721–727 (1993).
[CrossRef]

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426, 421–424 (2003).
[CrossRef]

Opt. Eng.

L. Onural and P. D. Scott, “Digital recording of in-line holograms,” Opt. Eng. 26, 1124–1132 (1987).

Opt. Express

F. Hajizadeh and S. N. S. Reihani, “Optimized optical trapping of gold nanoparticles,” Opt. Express 18, 551–559 (2010).
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R. Bowman, G. Gibson, and M. Padgett, “Particle tracking stereomicroscopy in optical tweezers: control of trap shape,” Opt. Express 18, 11785–11790 (2010).
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G. M. Gibson, J. Leach, S. Keen, A. J. Wright, and M. J. Padgett, “Measuring the accuracy of particle position and force in optical tweezers using high-speed video microscopy,” Opt. Express 16, 14561–14570 (2008).
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Opt. Rev.

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Phys. Rev. Lett.

J. S. Meiners and S. Quake, “Direct measurement of hydrodynamic cross correlations between two particles in an external potential,” Phys. Rev. Lett. 82, 2211–2214 (1999).
[CrossRef]

Precis. Eng.

S. I. Eom, Y. Takaya, and T. Hayashi, “Novel contact probing method using single fiber optical trapping probe,” Precis. Eng. 33, 235–242 (2009).
[CrossRef]

Q. Rev. Biophys.

I. Tinoco, D. Collin, and P. T. X. Li, “Unfolding single RNA molecules: bridging the gap between equilibrium and non-equilibrium statistical thermodynamics,” Q. Rev. Biophys. 38, 291–301 (2005).
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C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983), Chap. 4.

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Figures (7)

Fig. 1.
Fig. 1.

Experimental setup.

Fig. 2.
Fig. 2.

(a) Diffraction image that is clipped around the position of the nanoparticle and (b) the template image.

Fig. 3.
Fig. 3.

(a) Lateral SSD map and the profile through the center line including the minimum SSD, (b) the axial SSD profile. The inset shows a magnified view around the minimum SSD.

Fig. 4.
Fig. 4.

3D position detection of a nanoparticle fixed on a glass substrate: (a) 2D subpixel estimation with Δz=0.13nm, (b) 3D subpixel estimation with Δz=13nm, and (c) the difference.

Fig. 5.
Fig. 5.

SDz and RMSEz of the position measurements of the fixed particle for Δz. The open and the filled circles indicate the 2D and 3D subpixel estimations, respectively. The solid and dashed curves indicate SDz and RMSEz, respectively.

Fig. 6.
Fig. 6.

3D position detection of a 200 nm polystyrene sphere trapped in optical tweezers with intensity I=3.84MW/cm2: (a) 2D subpixel estimation with Δz=0.13nm, (b) 3D subpixel estimation with Δz=13nm, and (c) the difference.

Fig. 7.
Fig. 7.

RMSEz of the position measurements of a 200 nm polystyrene particle trapped in optical tweezers, for Δz. The open and the filled circles indicate the 2D and 3D subpixel estimations, respectively. The solid and dashed curves are linear and squared fittings, respectively.

Equations (6)

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

Ih(x,y,z)=k|uk(x,y,z)|2dk.
u(x,y,z)=Ft1(Ft[uh(x,y,zm)]exp[2πiλ2νx2νy2(zzm)]),
SSD(x,y,z)=ζ=0M1ξ=0M1{I(x+ξΔx,y+ζΔy,z)T(ξΔx,ζΔy,z0)}2.
xsub=xmin+ΔxSSD1SSD12SSD14SSD02SSD1,
SDz=1Nn=1N(zsub(n)zsub¯)2,
RMSEz=1Nn=1N(zsub(n)zmin(n)|Δz=0.13nm)2.

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