Abstract

Back-focal plane (BFP) interferometry is a very fast and precise method to track the 3D position of a sphere within a focused laser beam using a simple quadrant photo diode (QPD). Here we present a concept of how to track and recover the 5D state of a cylindrical nanorod (3D position and 2 tilt angles) in a laser focus by analyzing the interference of unscattered light and light scattered at the cylinder. The analytical theoretical approach is based on Rayleigh-Gans scattering together with a local field approximation for an infinitely thin cylinder. The approximated BFP intensities compare well with those from a more rigorous numerical approach. It turns out that a displacement of the cylinder results in a modulation of the BFP intensity pattern, whereas a tilt of the cylinder results in a shift of this pattern. We therefore propose the concept of a local QPD in the BFP of a detection lens, where the QPD center is shifted by the angular coordinates of the cylinder tilt.

© 2014 Optical Society of America

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2013 (1)

2012 (3)

2011 (7)

L. Dixon, F. C. Cheong, D. G. Grier, “Holographic deconvolution microscopy for high-resolution particle tracking,” Opt. Express 19(17), 16410–16417 (2011).
[CrossRef] [PubMed]

D. Ruh, B. Tränkle, A. Rohrbach, “Fast parallel interferometric 3D tracking of numerous optically trapped particles and their hydrodynamic interaction,” Opt. Express 19(22), 21627–21642 (2011).
[CrossRef] [PubMed]

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[CrossRef] [PubMed]

K. Dholakia, T. Cizmar, “Shaping the future of manipulation,” Nat. Photonics 5(6), 335–342 (2011).
[CrossRef]

A. Irrera, P. Artoni, R. Saija, P. G. Gucciardi, M. A. Iatì, F. Borghese, P. Denti, F. Iacona, F. Priolo, O. M. Maragò, “Size-scaling in optical trapping of silicon nanowires,” Nano Lett. 11(11), 4879–4884 (2011).
[CrossRef] [PubMed]

P. J. Reece, W. J. Toe, F. Wang, S. Paiman, Q. Gao, H. H. Tan, C. Jagadish, “Characterization of semiconductor nanowires using optical tweezers,” Nano Lett. 11(6), 2375–2381 (2011).
[CrossRef] [PubMed]

R. Huang, I. Chavez, K. M. Taute, B. Lukic, S. Jeney, M. G. Raizen, E.-L. Florin, “Direct observation of the full transition from ballistic to diffusive Brownian motion in a liquid,” Nat. Phys. 7(7), 576–580 (2011).
[CrossRef]

2010 (4)

S. H. Simpson, S. Hanna, “First-order nonconservative motion of optically trapped nonspherical particles,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 82(3), 031141 (2010).
[CrossRef] [PubMed]

K. I. Mortensen, L. S. Churchman, J. A. Spudich, H. Flyvbjerg, “Optimized localization analysis for single-molecule tracking and super-resolution microscopy,” Nat. Methods 7(5), 377–381 (2010).
[CrossRef] [PubMed]

L. Friedrich, A. Rohrbach, “Improved interferometric tracking of trapped particles using two frequency-detuned beams,” Opt. Lett. 35(11), 1920–1922 (2010).
[CrossRef] [PubMed]

P. B. Bareil, Y. Sheng, “Angular and position stability of a nanorod trapped in an optical tweezers,” Opt. Express 18(25), 26388–26398 (2010).
[CrossRef] [PubMed]

2009 (1)

2008 (3)

C. Selhuber-Unkel, I. Zins, O. Schubert, C. Sönnichsen, L. B. Oddershede, “Quantitative optical trapping of single gold nanorods,” Nano Lett. 8(9), 2998–3003 (2008).
[CrossRef] [PubMed]

F. Borghese, P. Denti, R. Saija, M. A. Iati, O. M. Marago, “Radiation torque and force on optically trapped linear nanostructures,” Phys. Rev. Lett. 100, 163903 (2008).

O. M. Maragò, P. H. Jones, F. Bonaccorso, V. Scardaci, P. G. Gucciardi, A. G. Rozhin, A. C. Ferrari, “Femtonewton force sensing with optically trapped nanotubes,” Nano Lett. 8(10), 3211–3216 (2008).
[CrossRef] [PubMed]

2007 (4)

S. J. Parkin, G. Knöner, T. A. Nieminen, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Picoliter viscometry using optically rotated particles,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76(4), 041507 (2007).
[CrossRef] [PubMed]

Y. Nakayama, P. J. Pauzauskie, A. Radenovic, R. M. Onorato, R. J. Saykally, J. Liphardt, P. Yang, “Tunable nanowire nonlinear optical probe,” Nature 447(7148), 1098–1101 (2007).
[CrossRef] [PubMed]

G. Volpe, G. Kozyreff, D. Petrov, “Backscattering position detection for photonic force microscopy,” J. Appl. Phys. 102(8), 084701 (2007).
[CrossRef]

S. H. Simpson, S. Hanna, “Optical trapping of spheroidal particles in Gaussian beams,” J. Opt. Soc. Am. A 24(2), 430–443 (2007).
[CrossRef] [PubMed]

2006 (3)

2004 (2)

H. Kress, E. H. K. Stelzer, A. Rohrbach, “Tilt angle dependent three-dimensional-position detection of a trapped cylindrical particle in a focused laser beam,” Appl. Phys. Lett. 84(21), 4271–4273 (2004).
[CrossRef]

A. Rohrbach, C. Tischer, D. Neumayer, E. L. Florin, E. H. K. Stelzer, “Trapping and tracking a local probe with a photonic force microscope,” Rev. Sci. Instrum. 75(6), 2197–2210 (2004).
[CrossRef]

2003 (3)

2001 (1)

1999 (2)

A. P. Bartko, R. M. Dickson, “Imaging three-dimensional single molecule orientations,” J. Phys. Chem. B 103(51), 11237–11241 (1999).
[CrossRef]

A. Pralle, M. Prummer, E. L. Florin, E. H. K. Stelzer, 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)

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394(6691), 348–350 (1998).
[CrossRef]

1996 (1)

E. L. Florin, J. K. H. Horber, E. H. K. Stelzer, “High-resolution axial and lateral position sensing using two-photon excitation of fluorophores by a continuous-wave Nd alpha YAG laser,” Appl. Phys. Lett. 69(4), 446–448 (1996).
[CrossRef]

1984 (1)

M. M. Tirado, C. L. Martinez, J. G. Delatorre, “Comparison of theories for the translational and rotational diffusion coefficients of rod-like macromolecules. Applications to short DNA fragments,” J. Chem. Phys. 81(4), 2047–2052 (1984).
[CrossRef]

Artoni, P.

A. Irrera, P. Artoni, R. Saija, P. G. Gucciardi, M. A. Iatì, F. Borghese, P. Denti, F. Iacona, F. Priolo, O. M. Maragò, “Size-scaling in optical trapping of silicon nanowires,” Nano Lett. 11(11), 4879–4884 (2011).
[CrossRef] [PubMed]

Bareil, P. B.

Bartko, A. P.

A. P. Bartko, R. M. Dickson, “Imaging three-dimensional single molecule orientations,” J. Phys. Chem. B 103(51), 11237–11241 (1999).
[CrossRef]

Böhmer, M.

Bonaccorso, F.

O. M. Maragò, P. H. Jones, F. Bonaccorso, V. Scardaci, P. G. Gucciardi, A. G. Rozhin, A. C. Ferrari, “Femtonewton force sensing with optically trapped nanotubes,” Nano Lett. 8(10), 3211–3216 (2008).
[CrossRef] [PubMed]

Borghese, F.

A. Irrera, P. Artoni, R. Saija, P. G. Gucciardi, M. A. Iatì, F. Borghese, P. Denti, F. Iacona, F. Priolo, O. M. Maragò, “Size-scaling in optical trapping of silicon nanowires,” Nano Lett. 11(11), 4879–4884 (2011).
[CrossRef] [PubMed]

F. Borghese, P. Denti, R. Saija, M. A. Iati, O. M. Marago, “Radiation torque and force on optically trapped linear nanostructures,” Phys. Rev. Lett. 100, 163903 (2008).

Bowman, R.

D. B. Phillips, J. A. Grieve, S. N. Olof, S. J. Kocher, R. Bowman, M. J. Padgett, M. J. Miles, D. M. Carberry, “Surface imaging using holographic optical tweezers,” Nanotechnology 22(28), 285503 (2011).
[CrossRef] [PubMed]

Bui, A. A. M.

Cao, Y.

Carberry, D. M.

D. B. Phillips, J. A. Grieve, S. N. Olof, S. J. Kocher, R. Bowman, M. J. Padgett, M. J. Miles, D. M. Carberry, “Surface imaging using holographic optical tweezers,” Nanotechnology 22(28), 285503 (2011).
[CrossRef] [PubMed]

Chavez, I.

R. Huang, I. Chavez, K. M. Taute, B. Lukic, S. Jeney, M. G. Raizen, E.-L. Florin, “Direct observation of the full transition from ballistic to diffusive Brownian motion in a liquid,” Nat. Phys. 7(7), 576–580 (2011).
[CrossRef]

Chen, L.

Cheong, F. C.

Churchman, L. S.

K. I. Mortensen, L. S. Churchman, J. A. Spudich, H. Flyvbjerg, “Optimized localization analysis for single-molecule tracking and super-resolution microscopy,” Nat. Methods 7(5), 377–381 (2010).
[CrossRef] [PubMed]

Cizmar, T.

K. Dholakia, T. Cizmar, “Shaping the future of manipulation,” Nat. Photonics 5(6), 335–342 (2011).
[CrossRef]

Delatorre, J. G.

M. M. Tirado, C. L. Martinez, J. G. Delatorre, “Comparison of theories for the translational and rotational diffusion coefficients of rod-like macromolecules. Applications to short DNA fragments,” J. Chem. Phys. 81(4), 2047–2052 (1984).
[CrossRef]

Denti, P.

A. Irrera, P. Artoni, R. Saija, P. G. Gucciardi, M. A. Iatì, F. Borghese, P. Denti, F. Iacona, F. Priolo, O. M. Maragò, “Size-scaling in optical trapping of silicon nanowires,” Nano Lett. 11(11), 4879–4884 (2011).
[CrossRef] [PubMed]

F. Borghese, P. Denti, R. Saija, M. A. Iati, O. M. Marago, “Radiation torque and force on optically trapped linear nanostructures,” Phys. Rev. Lett. 100, 163903 (2008).

Dholakia, K.

K. Dholakia, T. Cizmar, “Shaping the future of manipulation,” Nat. Photonics 5(6), 335–342 (2011).
[CrossRef]

Dickson, R. M.

A. P. Bartko, R. M. Dickson, “Imaging three-dimensional single molecule orientations,” J. Phys. Chem. B 103(51), 11237–11241 (1999).
[CrossRef]

Dixon, L.

Enderlein, J.

Ferrari, A. C.

O. M. Maragò, P. H. Jones, F. Bonaccorso, V. Scardaci, P. G. Gucciardi, A. G. Rozhin, A. C. Ferrari, “Femtonewton force sensing with optically trapped nanotubes,” Nano Lett. 8(10), 3211–3216 (2008).
[CrossRef] [PubMed]

Florin, E. L.

A. Rohrbach, C. Tischer, D. Neumayer, E. L. Florin, E. H. K. Stelzer, “Trapping and tracking a local probe with a photonic force microscope,” Rev. Sci. Instrum. 75(6), 2197–2210 (2004).
[CrossRef]

A. Pralle, M. Prummer, E. L. Florin, E. H. K. Stelzer, 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]

E. L. Florin, J. K. H. Horber, E. H. K. Stelzer, “High-resolution axial and lateral position sensing using two-photon excitation of fluorophores by a continuous-wave Nd alpha YAG laser,” Appl. Phys. Lett. 69(4), 446–448 (1996).
[CrossRef]

Florin, E.-L.

R. Huang, I. Chavez, K. M. Taute, B. Lukic, S. Jeney, M. G. Raizen, E.-L. Florin, “Direct observation of the full transition from ballistic to diffusive Brownian motion in a liquid,” Nat. Phys. 7(7), 576–580 (2011).
[CrossRef]

Flyvbjerg, H.

K. I. Mortensen, L. S. Churchman, J. A. Spudich, H. Flyvbjerg, “Optimized localization analysis for single-molecule tracking and super-resolution microscopy,” Nat. Methods 7(5), 377–381 (2010).
[CrossRef] [PubMed]

Friedrich, L.

Friese, M. E. J.

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394(6691), 348–350 (1998).
[CrossRef]

Gao, Q.

P. J. Reece, W. J. Toe, F. Wang, S. Paiman, Q. Gao, H. H. Tan, C. Jagadish, “Characterization of semiconductor nanowires using optical tweezers,” Nano Lett. 11(6), 2375–2381 (2011).
[CrossRef] [PubMed]

Grier, D. G.

Grieve, J. A.

D. B. Phillips, J. A. Grieve, S. N. Olof, S. J. Kocher, R. Bowman, M. J. Padgett, M. J. Miles, D. M. Carberry, “Surface imaging using holographic optical tweezers,” Nanotechnology 22(28), 285503 (2011).
[CrossRef] [PubMed]

Gucciardi, P. G.

A. Irrera, P. Artoni, R. Saija, P. G. Gucciardi, M. A. Iatì, F. Borghese, P. Denti, F. Iacona, F. Priolo, O. M. Maragò, “Size-scaling in optical trapping of silicon nanowires,” Nano Lett. 11(11), 4879–4884 (2011).
[CrossRef] [PubMed]

O. M. Maragò, P. H. Jones, F. Bonaccorso, V. Scardaci, P. G. Gucciardi, A. G. Rozhin, A. C. Ferrari, “Femtonewton force sensing with optically trapped nanotubes,” Nano Lett. 8(10), 3211–3216 (2008).
[CrossRef] [PubMed]

Guyot-Sionnest, P.

Hanna, S.

S. H. Simpson, S. Hanna, “First-order nonconservative motion of optically trapped nonspherical particles,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 82(3), 031141 (2010).
[CrossRef] [PubMed]

S. H. Simpson, S. Hanna, “Optical trapping of spheroidal particles in Gaussian beams,” J. Opt. Soc. Am. A 24(2), 430–443 (2007).
[CrossRef] [PubMed]

Heckenberg, N. R.

S. J. Parkin, G. Knöner, T. A. Nieminen, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Picoliter viscometry using optically rotated particles,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76(4), 041507 (2007).
[CrossRef] [PubMed]

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394(6691), 348–350 (1998).
[CrossRef]

Horber, J. K. H.

E. L. Florin, J. K. H. Horber, E. H. K. Stelzer, “High-resolution axial and lateral position sensing using two-photon excitation of fluorophores by a continuous-wave Nd alpha YAG laser,” Appl. Phys. Lett. 69(4), 446–448 (1996).
[CrossRef]

Hörber, J. K. H.

A. Pralle, M. Prummer, E. L. Florin, E. H. K. Stelzer, 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]

Huang, R.

R. Huang, I. Chavez, K. M. Taute, B. Lukic, S. Jeney, M. G. Raizen, E.-L. Florin, “Direct observation of the full transition from ballistic to diffusive Brownian motion in a liquid,” Nat. Phys. 7(7), 576–580 (2011).
[CrossRef]

Iacona, F.

A. Irrera, P. Artoni, R. Saija, P. G. Gucciardi, M. A. Iatì, F. Borghese, P. Denti, F. Iacona, F. Priolo, O. M. Maragò, “Size-scaling in optical trapping of silicon nanowires,” Nano Lett. 11(11), 4879–4884 (2011).
[CrossRef] [PubMed]

Iati, M. A.

F. Borghese, P. Denti, R. Saija, M. A. Iati, O. M. Marago, “Radiation torque and force on optically trapped linear nanostructures,” Phys. Rev. Lett. 100, 163903 (2008).

Iatì, M. A.

A. Irrera, P. Artoni, R. Saija, P. G. Gucciardi, M. A. Iatì, F. Borghese, P. Denti, F. Iacona, F. Priolo, O. M. Maragò, “Size-scaling in optical trapping of silicon nanowires,” Nano Lett. 11(11), 4879–4884 (2011).
[CrossRef] [PubMed]

Irrera, A.

A. Irrera, P. Artoni, R. Saija, P. G. Gucciardi, M. A. Iatì, F. Borghese, P. Denti, F. Iacona, F. Priolo, O. M. Maragò, “Size-scaling in optical trapping of silicon nanowires,” Nano Lett. 11(11), 4879–4884 (2011).
[CrossRef] [PubMed]

Jagadish, C.

P. J. Reece, W. J. Toe, F. Wang, S. Paiman, Q. Gao, H. H. Tan, C. Jagadish, “Characterization of semiconductor nanowires using optical tweezers,” Nano Lett. 11(6), 2375–2381 (2011).
[CrossRef] [PubMed]

Jeney, S.

R. Huang, I. Chavez, K. M. Taute, B. Lukic, S. Jeney, M. G. Raizen, E.-L. Florin, “Direct observation of the full transition from ballistic to diffusive Brownian motion in a liquid,” Nat. Phys. 7(7), 576–580 (2011).
[CrossRef]

Jones, P. H.

O. M. Maragò, P. H. Jones, F. Bonaccorso, V. Scardaci, P. G. Gucciardi, A. G. Rozhin, A. C. Ferrari, “Femtonewton force sensing with optically trapped nanotubes,” Nano Lett. 8(10), 3211–3216 (2008).
[CrossRef] [PubMed]

Kim, H. Y.

Knöner, G.

S. J. Parkin, G. Knöner, T. A. Nieminen, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Picoliter viscometry using optically rotated particles,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76(4), 041507 (2007).
[CrossRef] [PubMed]

Kocher, S. J.

D. B. Phillips, J. A. Grieve, S. N. Olof, S. J. Kocher, R. Bowman, M. J. Padgett, M. J. Miles, D. M. Carberry, “Surface imaging using holographic optical tweezers,” Nanotechnology 22(28), 285503 (2011).
[CrossRef] [PubMed]

Kozyreff, G.

G. Volpe, G. Kozyreff, D. Petrov, “Backscattering position detection for photonic force microscopy,” J. Appl. Phys. 102(8), 084701 (2007).
[CrossRef]

Kress, H.

H. Kress, E. H. K. Stelzer, A. Rohrbach, “Tilt angle dependent three-dimensional-position detection of a trapped cylindrical particle in a focused laser beam,” Appl. Phys. Lett. 84(21), 4271–4273 (2004).
[CrossRef]

A. Rohrbach, H. Kress, 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]

Liphardt, J.

Y. Nakayama, P. J. Pauzauskie, A. Radenovic, R. M. Onorato, R. J. Saykally, J. Liphardt, P. Yang, “Tunable nanowire nonlinear optical probe,” Nature 447(7148), 1098–1101 (2007).
[CrossRef] [PubMed]

P. J. Pauzauskie, A. Radenovic, E. Trepagnier, H. Shroff, P. D. Yang, J. Liphardt, “Optical trapping and integration of semiconductor nanowire assemblies in water,” Nat. Mater. 5(2), 97–101 (2006).
[CrossRef] [PubMed]

Liu, M. Z.

Lukic, B.

R. Huang, I. Chavez, K. M. Taute, B. Lukic, S. Jeney, M. G. Raizen, E.-L. Florin, “Direct observation of the full transition from ballistic to diffusive Brownian motion in a liquid,” Nat. Phys. 7(7), 576–580 (2011).
[CrossRef]

Marago, O. M.

F. Borghese, P. Denti, R. Saija, M. A. Iati, O. M. Marago, “Radiation torque and force on optically trapped linear nanostructures,” Phys. Rev. Lett. 100, 163903 (2008).

Maragò, O. M.

A. Irrera, P. Artoni, R. Saija, P. G. Gucciardi, M. A. Iatì, F. Borghese, P. Denti, F. Iacona, F. Priolo, O. M. Maragò, “Size-scaling in optical trapping of silicon nanowires,” Nano Lett. 11(11), 4879–4884 (2011).
[CrossRef] [PubMed]

O. M. Maragò, P. H. Jones, F. Bonaccorso, V. Scardaci, P. G. Gucciardi, A. G. Rozhin, A. C. Ferrari, “Femtonewton force sensing with optically trapped nanotubes,” Nano Lett. 8(10), 3211–3216 (2008).
[CrossRef] [PubMed]

Martinez, C. L.

M. M. Tirado, C. L. Martinez, J. G. Delatorre, “Comparison of theories for the translational and rotational diffusion coefficients of rod-like macromolecules. Applications to short DNA fragments,” J. Chem. Phys. 81(4), 2047–2052 (1984).
[CrossRef]

Miles, M. J.

D. B. Phillips, J. A. Grieve, S. N. Olof, S. J. Kocher, R. Bowman, M. J. Padgett, M. J. Miles, D. M. Carberry, “Surface imaging using holographic optical tweezers,” Nanotechnology 22(28), 285503 (2011).
[CrossRef] [PubMed]

Mortensen, K. I.

K. I. Mortensen, L. S. Churchman, J. A. Spudich, H. Flyvbjerg, “Optimized localization analysis for single-molecule tracking and super-resolution microscopy,” Nat. Methods 7(5), 377–381 (2010).
[CrossRef] [PubMed]

Nakayama, Y.

Y. Nakayama, P. J. Pauzauskie, A. Radenovic, R. M. Onorato, R. J. Saykally, J. Liphardt, P. Yang, “Tunable nanowire nonlinear optical probe,” Nature 447(7148), 1098–1101 (2007).
[CrossRef] [PubMed]

Neumayer, D.

A. Rohrbach, C. Tischer, D. Neumayer, E. L. Florin, E. H. K. Stelzer, “Trapping and tracking a local probe with a photonic force microscope,” Rev. Sci. Instrum. 75(6), 2197–2210 (2004).
[CrossRef]

Nieminen, T. A.

A. A. M. Bui, A. B. Stilgoe, T. A. Nieminen, H. Rubinsztein-Dunlop, “Calibration of nonspherical particles in optical tweezers using only position measurement,” Opt. Lett. 38(8), 1244–1246 (2013).
[CrossRef] [PubMed]

Y. Cao, A. B. Stilgoe, L. Chen, T. A. Nieminen, H. Rubinsztein-Dunlop, “Equilibrium orientations and positions of non-spherical particles in optical traps,” Opt. Express 20(12), 12987–12996 (2012).
[CrossRef] [PubMed]

S. J. Parkin, G. Knöner, T. A. Nieminen, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Picoliter viscometry using optically rotated particles,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76(4), 041507 (2007).
[CrossRef] [PubMed]

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394(6691), 348–350 (1998).
[CrossRef]

Oddershede, L. B.

C. Selhuber-Unkel, I. Zins, O. Schubert, C. Sönnichsen, L. B. Oddershede, “Quantitative optical trapping of single gold nanorods,” Nano Lett. 8(9), 2998–3003 (2008).
[CrossRef] [PubMed]

Olof, S. N.

D. B. Phillips, J. A. Grieve, S. N. Olof, S. J. Kocher, R. Bowman, M. J. Padgett, M. J. Miles, D. M. Carberry, “Surface imaging using holographic optical tweezers,” Nanotechnology 22(28), 285503 (2011).
[CrossRef] [PubMed]

Onorato, R. M.

Y. Nakayama, P. J. Pauzauskie, A. Radenovic, R. M. Onorato, R. J. Saykally, J. Liphardt, P. Yang, “Tunable nanowire nonlinear optical probe,” Nature 447(7148), 1098–1101 (2007).
[CrossRef] [PubMed]

Padgett, M. J.

D. B. Phillips, J. A. Grieve, S. N. Olof, S. J. Kocher, R. Bowman, M. J. Padgett, M. J. Miles, D. M. Carberry, “Surface imaging using holographic optical tweezers,” Nanotechnology 22(28), 285503 (2011).
[CrossRef] [PubMed]

Paiman, S.

P. J. Reece, W. J. Toe, F. Wang, S. Paiman, Q. Gao, H. H. Tan, C. Jagadish, “Characterization of semiconductor nanowires using optical tweezers,” Nano Lett. 11(6), 2375–2381 (2011).
[CrossRef] [PubMed]

Parkin, S. J.

S. J. Parkin, G. Knöner, T. A. Nieminen, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Picoliter viscometry using optically rotated particles,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76(4), 041507 (2007).
[CrossRef] [PubMed]

Pauzauskie, P. J.

Y. Nakayama, P. J. Pauzauskie, A. Radenovic, R. M. Onorato, R. J. Saykally, J. Liphardt, P. Yang, “Tunable nanowire nonlinear optical probe,” Nature 447(7148), 1098–1101 (2007).
[CrossRef] [PubMed]

P. J. Pauzauskie, A. Radenovic, E. Trepagnier, H. Shroff, P. D. Yang, J. Liphardt, “Optical trapping and integration of semiconductor nanowire assemblies in water,” Nat. Mater. 5(2), 97–101 (2006).
[CrossRef] [PubMed]

Pelton, M.

Petrov, D.

G. Volpe, G. Kozyreff, D. Petrov, “Backscattering position detection for photonic force microscopy,” J. Appl. Phys. 102(8), 084701 (2007).
[CrossRef]

Phillips, D. B.

D. B. Phillips, J. A. Grieve, S. N. Olof, S. J. Kocher, R. Bowman, M. J. Padgett, M. J. Miles, D. M. Carberry, “Surface imaging using holographic optical tweezers,” Nanotechnology 22(28), 285503 (2011).
[CrossRef] [PubMed]

Pralle, A.

A. Pralle, M. Prummer, E. L. Florin, E. H. K. Stelzer, 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]

Priolo, F.

A. Irrera, P. Artoni, R. Saija, P. G. Gucciardi, M. A. Iatì, F. Borghese, P. Denti, F. Iacona, F. Priolo, O. M. Maragò, “Size-scaling in optical trapping of silicon nanowires,” Nano Lett. 11(11), 4879–4884 (2011).
[CrossRef] [PubMed]

Prummer, M.

A. Pralle, M. Prummer, E. L. Florin, E. H. K. Stelzer, 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]

Radenovic, A.

Y. Nakayama, P. J. Pauzauskie, A. Radenovic, R. M. Onorato, R. J. Saykally, J. Liphardt, P. Yang, “Tunable nanowire nonlinear optical probe,” Nature 447(7148), 1098–1101 (2007).
[CrossRef] [PubMed]

P. J. Pauzauskie, A. Radenovic, E. Trepagnier, H. Shroff, P. D. Yang, J. Liphardt, “Optical trapping and integration of semiconductor nanowire assemblies in water,” Nat. Mater. 5(2), 97–101 (2006).
[CrossRef] [PubMed]

Raizen, M. G.

R. Huang, I. Chavez, K. M. Taute, B. Lukic, S. Jeney, M. G. Raizen, E.-L. Florin, “Direct observation of the full transition from ballistic to diffusive Brownian motion in a liquid,” Nat. Phys. 7(7), 576–580 (2011).
[CrossRef]

Reece, P. J.

P. J. Reece, W. J. Toe, F. Wang, S. Paiman, Q. Gao, H. H. Tan, C. Jagadish, “Characterization of semiconductor nanowires using optical tweezers,” Nano Lett. 11(6), 2375–2381 (2011).
[CrossRef] [PubMed]

Rieger, B.

Rohrbach, A.

L. Friedrich, A. Rohrbach, “Tuning the detection sensitivity: a model for axial backfocal plane interferometric tracking,” Opt. Lett. 37(11), 2109–2111 (2012).
[CrossRef] [PubMed]

D. Ruh, B. Tränkle, A. Rohrbach, “Fast parallel interferometric 3D tracking of numerous optically trapped particles and their hydrodynamic interaction,” Opt. Express 19(22), 21627–21642 (2011).
[CrossRef] [PubMed]

L. Friedrich, A. Rohrbach, “Improved interferometric tracking of trapped particles using two frequency-detuned beams,” Opt. Lett. 35(11), 1920–1922 (2010).
[CrossRef] [PubMed]

M. Speidel, L. Friedrich, A. Rohrbach, “Interferometric 3D tracking of several particles in a scanning laser focus,” Opt. Express 17(2), 1003–1015 (2009).
[CrossRef] [PubMed]

P. C. Seitz, E. H. K. Stelzer, A. Rohrbach, “Interferometric tracking of optically trapped probes behind structured surfaces: a phase correction method,” Appl. Opt. 45(28), 7309–7315 (2006).
[CrossRef] [PubMed]

A. Rohrbach, C. Tischer, D. Neumayer, E. L. Florin, E. H. K. Stelzer, “Trapping and tracking a local probe with a photonic force microscope,” Rev. Sci. Instrum. 75(6), 2197–2210 (2004).
[CrossRef]

H. Kress, E. H. K. Stelzer, A. Rohrbach, “Tilt angle dependent three-dimensional-position detection of a trapped cylindrical particle in a focused laser beam,” Appl. Phys. Lett. 84(21), 4271–4273 (2004).
[CrossRef]

A. Rohrbach, H. Kress, 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]

A. Rohrbach, E. H. K. Stelzer, “Optical trapping of dielectric particles in arbitrary fields,” J. Opt. Soc. Am. A 18(4), 839–853 (2001).
[CrossRef] [PubMed]

Rozhin, A. G.

O. M. Maragò, P. H. Jones, F. Bonaccorso, V. Scardaci, P. G. Gucciardi, A. G. Rozhin, A. C. Ferrari, “Femtonewton force sensing with optically trapped nanotubes,” Nano Lett. 8(10), 3211–3216 (2008).
[CrossRef] [PubMed]

Rubinsztein-Dunlop, H.

A. A. M. Bui, A. B. Stilgoe, T. A. Nieminen, H. Rubinsztein-Dunlop, “Calibration of nonspherical particles in optical tweezers using only position measurement,” Opt. Lett. 38(8), 1244–1246 (2013).
[CrossRef] [PubMed]

Y. Cao, A. B. Stilgoe, L. Chen, T. A. Nieminen, H. Rubinsztein-Dunlop, “Equilibrium orientations and positions of non-spherical particles in optical traps,” Opt. Express 20(12), 12987–12996 (2012).
[CrossRef] [PubMed]

S. J. Parkin, G. Knöner, T. A. Nieminen, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Picoliter viscometry using optically rotated particles,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76(4), 041507 (2007).
[CrossRef] [PubMed]

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394(6691), 348–350 (1998).
[CrossRef]

Ruh, D.

Saija, R.

A. Irrera, P. Artoni, R. Saija, P. G. Gucciardi, M. A. Iatì, F. Borghese, P. Denti, F. Iacona, F. Priolo, O. M. Maragò, “Size-scaling in optical trapping of silicon nanowires,” Nano Lett. 11(11), 4879–4884 (2011).
[CrossRef] [PubMed]

F. Borghese, P. Denti, R. Saija, M. A. Iati, O. M. Marago, “Radiation torque and force on optically trapped linear nanostructures,” Phys. Rev. Lett. 100, 163903 (2008).

Saykally, R. J.

Y. Nakayama, P. J. Pauzauskie, A. Radenovic, R. M. Onorato, R. J. Saykally, J. Liphardt, P. Yang, “Tunable nanowire nonlinear optical probe,” Nature 447(7148), 1098–1101 (2007).
[CrossRef] [PubMed]

Scardaci, V.

O. M. Maragò, P. H. Jones, F. Bonaccorso, V. Scardaci, P. G. Gucciardi, A. G. Rozhin, A. C. Ferrari, “Femtonewton force sensing with optically trapped nanotubes,” Nano Lett. 8(10), 3211–3216 (2008).
[CrossRef] [PubMed]

Scherer, N. F.

Schubert, O.

C. Selhuber-Unkel, I. Zins, O. Schubert, C. Sönnichsen, L. B. Oddershede, “Quantitative optical trapping of single gold nanorods,” Nano Lett. 8(9), 2998–3003 (2008).
[CrossRef] [PubMed]

Seitz, P. C.

Selhuber-Unkel, C.

C. Selhuber-Unkel, I. Zins, O. Schubert, C. Sönnichsen, L. B. Oddershede, “Quantitative optical trapping of single gold nanorods,” Nano Lett. 8(9), 2998–3003 (2008).
[CrossRef] [PubMed]

Sheng, Y.

Shroff, H.

P. J. Pauzauskie, A. Radenovic, E. Trepagnier, H. Shroff, P. D. Yang, J. Liphardt, “Optical trapping and integration of semiconductor nanowire assemblies in water,” Nat. Mater. 5(2), 97–101 (2006).
[CrossRef] [PubMed]

Simpson, S. H.

S. H. Simpson, S. Hanna, “First-order nonconservative motion of optically trapped nonspherical particles,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 82(3), 031141 (2010).
[CrossRef] [PubMed]

S. H. Simpson, S. Hanna, “Optical trapping of spheroidal particles in Gaussian beams,” J. Opt. Soc. Am. A 24(2), 430–443 (2007).
[CrossRef] [PubMed]

Smith, G.

Sönnichsen, C.

C. Selhuber-Unkel, I. Zins, O. Schubert, C. Sönnichsen, L. B. Oddershede, “Quantitative optical trapping of single gold nanorods,” Nano Lett. 8(9), 2998–3003 (2008).
[CrossRef] [PubMed]

Speidel, M.

Spudich, J. A.

K. I. Mortensen, L. S. Churchman, J. A. Spudich, H. Flyvbjerg, “Optimized localization analysis for single-molecule tracking and super-resolution microscopy,” Nat. Methods 7(5), 377–381 (2010).
[CrossRef] [PubMed]

Stallinga, S.

Stelzer, E. H. K.

P. C. Seitz, E. H. K. Stelzer, A. Rohrbach, “Interferometric tracking of optically trapped probes behind structured surfaces: a phase correction method,” Appl. Opt. 45(28), 7309–7315 (2006).
[CrossRef] [PubMed]

A. Rohrbach, C. Tischer, D. Neumayer, E. L. Florin, E. H. K. Stelzer, “Trapping and tracking a local probe with a photonic force microscope,” Rev. Sci. Instrum. 75(6), 2197–2210 (2004).
[CrossRef]

H. Kress, E. H. K. Stelzer, A. Rohrbach, “Tilt angle dependent three-dimensional-position detection of a trapped cylindrical particle in a focused laser beam,” Appl. Phys. Lett. 84(21), 4271–4273 (2004).
[CrossRef]

A. Rohrbach, H. Kress, 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]

A. Rohrbach, E. H. K. Stelzer, “Optical trapping of dielectric particles in arbitrary fields,” J. Opt. Soc. Am. A 18(4), 839–853 (2001).
[CrossRef] [PubMed]

A. Pralle, M. Prummer, E. L. Florin, E. H. K. Stelzer, 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]

E. L. Florin, J. K. H. Horber, E. H. K. Stelzer, “High-resolution axial and lateral position sensing using two-photon excitation of fluorophores by a continuous-wave Nd alpha YAG laser,” Appl. Phys. Lett. 69(4), 446–448 (1996).
[CrossRef]

Stilgoe, A. B.

Tan, H. H.

P. J. Reece, W. J. Toe, F. Wang, S. Paiman, Q. Gao, H. H. Tan, C. Jagadish, “Characterization of semiconductor nanowires using optical tweezers,” Nano Lett. 11(6), 2375–2381 (2011).
[CrossRef] [PubMed]

Taute, K. M.

R. Huang, I. Chavez, K. M. Taute, B. Lukic, S. Jeney, M. G. Raizen, E.-L. Florin, “Direct observation of the full transition from ballistic to diffusive Brownian motion in a liquid,” Nat. Phys. 7(7), 576–580 (2011).
[CrossRef]

Tirado, M. M.

M. M. Tirado, C. L. Martinez, J. G. Delatorre, “Comparison of theories for the translational and rotational diffusion coefficients of rod-like macromolecules. Applications to short DNA fragments,” J. Chem. Phys. 81(4), 2047–2052 (1984).
[CrossRef]

Tischer, C.

A. Rohrbach, C. Tischer, D. Neumayer, E. L. Florin, E. H. K. Stelzer, “Trapping and tracking a local probe with a photonic force microscope,” Rev. Sci. Instrum. 75(6), 2197–2210 (2004).
[CrossRef]

Toe, W. J.

P. J. Reece, W. J. Toe, F. Wang, S. Paiman, Q. Gao, H. H. Tan, C. Jagadish, “Characterization of semiconductor nanowires using optical tweezers,” Nano Lett. 11(6), 2375–2381 (2011).
[CrossRef] [PubMed]

Tränkle, B.

Trepagnier, E.

P. J. Pauzauskie, A. Radenovic, E. Trepagnier, H. Shroff, P. D. Yang, J. Liphardt, “Optical trapping and integration of semiconductor nanowire assemblies in water,” Nat. Mater. 5(2), 97–101 (2006).
[CrossRef] [PubMed]

Volpe, G.

G. Volpe, G. Kozyreff, D. Petrov, “Backscattering position detection for photonic force microscopy,” J. Appl. Phys. 102(8), 084701 (2007).
[CrossRef]

Wang, F.

P. J. Reece, W. J. Toe, F. Wang, S. Paiman, Q. Gao, H. H. Tan, C. Jagadish, “Characterization of semiconductor nanowires using optical tweezers,” Nano Lett. 11(6), 2375–2381 (2011).
[CrossRef] [PubMed]

Yang, P.

Y. Nakayama, P. J. Pauzauskie, A. Radenovic, R. M. Onorato, R. J. Saykally, J. Liphardt, P. Yang, “Tunable nanowire nonlinear optical probe,” Nature 447(7148), 1098–1101 (2007).
[CrossRef] [PubMed]

Yang, P. D.

P. J. Pauzauskie, A. Radenovic, E. Trepagnier, H. Shroff, P. D. Yang, J. Liphardt, “Optical trapping and integration of semiconductor nanowire assemblies in water,” Nat. Mater. 5(2), 97–101 (2006).
[CrossRef] [PubMed]

Zins, I.

C. Selhuber-Unkel, I. Zins, O. Schubert, C. Sönnichsen, L. B. Oddershede, “Quantitative optical trapping of single gold nanorods,” Nano Lett. 8(9), 2998–3003 (2008).
[CrossRef] [PubMed]

Appl. Opt. (1)

Appl. Phys. Lett. (2)

E. L. Florin, J. K. H. Horber, E. H. K. Stelzer, “High-resolution axial and lateral position sensing using two-photon excitation of fluorophores by a continuous-wave Nd alpha YAG laser,” Appl. Phys. Lett. 69(4), 446–448 (1996).
[CrossRef]

H. Kress, E. H. K. Stelzer, A. Rohrbach, “Tilt angle dependent three-dimensional-position detection of a trapped cylindrical particle in a focused laser beam,” Appl. Phys. Lett. 84(21), 4271–4273 (2004).
[CrossRef]

J. Appl. Phys. (1)

G. Volpe, G. Kozyreff, D. Petrov, “Backscattering position detection for photonic force microscopy,” J. Appl. Phys. 102(8), 084701 (2007).
[CrossRef]

J. Chem. Phys. (1)

M. M. Tirado, C. L. Martinez, J. G. Delatorre, “Comparison of theories for the translational and rotational diffusion coefficients of rod-like macromolecules. Applications to short DNA fragments,” J. Chem. Phys. 81(4), 2047–2052 (1984).
[CrossRef]

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

J. Opt. Soc. Am. B (1)

J. Phys. Chem. B (1)

A. P. Bartko, R. M. Dickson, “Imaging three-dimensional single molecule orientations,” J. Phys. Chem. B 103(51), 11237–11241 (1999).
[CrossRef]

Microsc. Res. Tech. (1)

A. Pralle, M. Prummer, E. L. Florin, E. H. K. Stelzer, 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]

Nano Lett. (4)

A. Irrera, P. Artoni, R. Saija, P. G. Gucciardi, M. A. Iatì, F. Borghese, P. Denti, F. Iacona, F. Priolo, O. M. Maragò, “Size-scaling in optical trapping of silicon nanowires,” Nano Lett. 11(11), 4879–4884 (2011).
[CrossRef] [PubMed]

P. J. Reece, W. J. Toe, F. Wang, S. Paiman, Q. Gao, H. H. Tan, C. Jagadish, “Characterization of semiconductor nanowires using optical tweezers,” Nano Lett. 11(6), 2375–2381 (2011).
[CrossRef] [PubMed]

O. M. Maragò, P. H. Jones, F. Bonaccorso, V. Scardaci, P. G. Gucciardi, A. G. Rozhin, A. C. Ferrari, “Femtonewton force sensing with optically trapped nanotubes,” Nano Lett. 8(10), 3211–3216 (2008).
[CrossRef] [PubMed]

C. Selhuber-Unkel, I. Zins, O. Schubert, C. Sönnichsen, L. B. Oddershede, “Quantitative optical trapping of single gold nanorods,” Nano Lett. 8(9), 2998–3003 (2008).
[CrossRef] [PubMed]

Nanotechnology (1)

D. B. Phillips, J. A. Grieve, S. N. Olof, S. J. Kocher, R. Bowman, M. J. Padgett, M. J. Miles, D. M. Carberry, “Surface imaging using holographic optical tweezers,” Nanotechnology 22(28), 285503 (2011).
[CrossRef] [PubMed]

Nat. Mater. (1)

P. J. Pauzauskie, A. Radenovic, E. Trepagnier, H. Shroff, P. D. Yang, J. Liphardt, “Optical trapping and integration of semiconductor nanowire assemblies in water,” Nat. Mater. 5(2), 97–101 (2006).
[CrossRef] [PubMed]

Nat. Methods (1)

K. I. Mortensen, L. S. Churchman, J. A. Spudich, H. Flyvbjerg, “Optimized localization analysis for single-molecule tracking and super-resolution microscopy,” Nat. Methods 7(5), 377–381 (2010).
[CrossRef] [PubMed]

Nat. Photonics (1)

K. Dholakia, T. Cizmar, “Shaping the future of manipulation,” Nat. Photonics 5(6), 335–342 (2011).
[CrossRef]

Nat. Phys. (1)

R. Huang, I. Chavez, K. M. Taute, B. Lukic, S. Jeney, M. G. Raizen, E.-L. Florin, “Direct observation of the full transition from ballistic to diffusive Brownian motion in a liquid,” Nat. Phys. 7(7), 576–580 (2011).
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Nature (3)

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[CrossRef]

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Figures (15)

Fig. 1
Fig. 1

Setup scheme for trapping and tracking. A cylinder is optically trapped in a laser focus and changes its center position and orientation due to external forces or thermal fluctuations. The interference intensity pattern of scattered and unscattered light is recorded by a sensor in the back focal plane (BFP) of the detection lens (DL). Zoom: a translated and tilted cylinder, described by a position vector bt and an orientation vector br = (bθ, bϕ).

Fig. 2
Fig. 2

Rayleigh-Gans scattering of infinitely thin cylinder of length L. a) A plane wave with wave-vector ki incident on a tilted cylinder as local field approximation for the center of a focused field. b) The tilted form factor s ˜ 0 ( k x , k z , b θ ) of the cylinder (as background in grey scale) is shifted by ki relative to the Ewald circle with radius ki = ks. The overlap (red circle area) defines the part of the angular spectrum of the forward scattered field s(kx) that is detected by a lens with NAdet = 0.9. c) The scattered field spectrum s(kx) as intersecting line between Ewald circle and form factor.

Fig. 3
Fig. 3

Intensity difference I ˜ d i f f O ( k x , k y ) in the BFP of the detection lens for a tilted cylinder. a-c) The flat-top like intensity maximum shifted sideward if the cylinder is tilted (bθ > 0, bϕ = 0). d) Corresponding intensity line scans. e,f) For bϕ ≠0 the center of mass of I ˜ d i f f O is shifted in direction of the tilt (bθ, bϕ).

Fig. 4
Fig. 4

Influence of a cylinders length on the intensity difference I ˜ d i f f O ( k x , k y ) . With increasing length L the width of the flat-top like intensity maximum (red circle) is decreased. The tilt angle bθ is defined by the length of the circle’s center vector (arrow).

Fig. 5
Fig. 5

Tracking signals for a thin cylinder, which is both shifted and tilted. Left column: Scheme for shifted and tilted cylinders. Center column: Corresponding intensity difference I ˜ d i f f O ( k x , k y ) in the BFP. Right column: intensity linecans I ˜ d i f f O ( k x , 0 ) illustrate the signal shift for a cylinder tilt and the bipolar signal modulation for a cylinder shift.

Fig. 6
Fig. 6

Intensity difference I ˜ d i f f O ( k x , k y , b ) for axially displaced thin cylinders with arbitrary positions and orientations, decribed by the state vector b = (bt, br) = (bx,by,bz,bϕ,bθ). A cylinder shift in axial direction results in a sphercial modulation of the signal.

Fig. 7
Fig. 7

Rigourously computed intensities I ˜ d i f f O ( k x , k y , b ) for states b = (bx,by,bz,bθ,bϕ) of a cylinder with finite thickness. The cylinder length is L = 0.8µm and the diameter is D = 0.1µm. The cylinders displacements are in units of 0.1µm. The round pattern I ˜ d i f f O ( k x , k y , b ) is modulated in three cases and is shifted by the positions SP in e,f) to account for cylinder tilts.

Fig. 8
Fig. 8

Rigourously computed intensity differences I ˜ d i f f O ( k x , k y , b ) for a tilted cylinder with positions b = (bx,by,bz, 20°, 45°) . The cylinder length is L = 0.8µm, its diameter D = 0.1µm and its refractive index ns = 1.57. The shifts are in units of 0.1µm. For a cylinder shift of bz, the blue-red patterns within the area of the local QPD rotate from the top row (bz = 0) to the bottom row (bz = 0.2µm) due to a multiplication with sin(ΔΦt(kx,ky,bz)).

Fig. 9
Fig. 9

Intensity read out with a local QPD. a) Computed intensity difference I ˜ d i f f O ( k x , k y , b ) in the BFPDL for a cylinder with state vector b in a focused laser beam. b) Scheme for a read out using a local QPD: A disc aperture with radius kL, center position (kxc,kyc) and distance |kc|, and the local QPD quadrants A, B, C and D. c) Corresponding shift and tilt of the cylinder.

Fig. 10
Fig. 10

Tracking signals Si(b) for a nanorod with different state vectors b. a) Lateral signals Sx(bx) for two different lateral shifts by and tilts bθ . b) Lateral signals Sx(bx) for two different axial shifts bz and tilts bθ. c) Axial signals Sz(bz) for two different lateral shifts bx and tilts bθ . The linear detection range is marked with a box.

Fig. 11
Fig. 11

Position and orientation signals of a cylinder in a focused laser beam. An assumed probability density of states is underlayed in the background in gray scale. a) Iso-lines Sx(bx,by) and Sz(bx,by) of a vertical cylinder centered in y-direction. b) Iso-lines Sϕ(bϕ, bθ) and Sθ(bϕ, bθ) of a tilted cylinder in the center of the focus. The smallest polar angle is bθ = 10°, since azimuth angles bϕ are not defined for bθ = 0.

Fig. 12
Fig. 12

Linearity and orthogonality of position and orientation signals Sx/z(bθ,bx/z) and Sθ(bθ,bx/z) of a shifted and tilted cylinder in a focused laser beam. a), b) Iso-signal lines (in a.u.) with and without axial cylinder shift. c),d) Iso-signal lines (in a.u.) with and without lateral shift.

Fig. 13
Fig. 13

Linearity and orthogonality of position and orientation signals Sx/z(bϕ,bx/z) and Sϕ(bϕ,bx/z) of a shifted and tilted cylinder in a focused laser beam. a), b) Iso-signal lines (in a.u.) with and without axial shift. c),d) Iso-signal lines (in a.u.) with and without lateral shift.

Fig. 14
Fig. 14

Coupling of linear position and orientation signals Sx(bx,bz) and Sz(bx,bz) of a shifted and tilted cylinder in a focused laser beam. An assumed Gaussian probability density of states as a result of linear restoring forces is underlayed in the background in gray scale.

Fig. 15
Fig. 15

Tracking errors for disaplcements and tilts of a cylinder (L = 0.8, D = 1µm, ns = 1.57) a) Absolute tracking error for lateral displacements bx for different state vectors b. b) Absolute tracking error for tilt angles bθfor different state vectors b. Results were calculated with the simulation software LightWave(R).

Equations (44)

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( S x S y S z S θ S ϕ ) ( S 0 x S 0 y S 0 z S 0 θ S 0 ϕ ) + ( g x x g y y g z z g θ θ g ϕ ϕ ) ( b x b y b z b θ b ϕ )
ΔΦ ( k x , k y , b ) = ΔΦ x ( k x , k y , b x ) + ΔΦ y ( k x , k y , b y ) + ΔΦ z ( k x , k y , b z )
S j ( b ) = I ˜ ( k x , k y , b ) H j ( k x , k y ) d k x d k y
S j ( b j ) = S 0 j + 2 I i I s ( b j )   c o s ( ΔΦ( b j ) ) H j d k x d k y S 0 j + g j b j
S ( b ) S 0 + g ^   b
b = ( b t ,   b r ) ;   b t = ( b x , b y , b z ) ;   b r = ( b θ , b ϕ )
( Δ + k 2 ) E ( r ) = α ^   s ( r ) n s ² k 0 2 E ( r )
E ( r ) = E i ( r ) + E s ( r ) E i ( r ) + α   k 2 s ( r ' )   E i ( r ' )   G ( r r ' )   d 3 r '
E s ( r ) =   α   k 2 ( E i ( r )   s ( r ) ) G ( r )
E ˜ s ( k , b ) = α   k 2 ( 2 π ) 3   G ˜ ( k ) ( E ˜ i ( k ) s ˜ ( k , b ) )
E ˜ s ( k , b ) = α   k 2   G ˜ ( k ) E 0   s ˜ ( k k i , b )
s ˜ 0 ( k ) = 2  J 1 ( k r D / 2 ) k r D / 2 sin c ( k z L / 2 )
s ˜ ( k , b ) = FT { s ( r , b ) } =   s ˜ 0 [ k ' ( b r ) ]   exp ( i   k   b t )
R y' ( θ ) = ( cos (θ) 0 sin (θ) 0 1 0 sin (θ) 0 cos (θ) ) ;   R z ( ϕ ) = ( cos ( ϕ ) sin ( ϕ ) 0 sin ( ϕ ) cos ( ϕ ) 0 0 0 1 )
E ˜ i ( k ) s ˜ ( k , b ) = E ˜ i ( k i ) s ˜ 0 [ k ( b r ) k i ]   exp ( i ( k k i )   b t )   d 3 k i
E ˜ s ( k , b ) = α k 2 E 0   G ˜ ( k )   s ˜ 0 [ k ( k ,   b r ) k i ]   e i   ( k k i )   b t
s ˜ 0 ( k ' ) =   sin c ( k z ' L / 2 )
k z ' ( b θ , b ϕ ) = sin ( b θ ) cos ( b ϕ ) k x sin ( b θ ) sin ( b ϕ ) k y + cos ( b θ ) k z
s ˜ 0 ( k x , k y , b θ ) = sin c ( ( sin ( b θ ) k x + cos ( b θ ) k 2 k x 2 k y 2 ) L / 2 )
G ˜ ( k ) = 1 | k | 2 k 2 = 1 ( k x 2 + k y 2 + k z 2 ) k 2
G ˜ ± ( k ) E ˜ ( k ) = E ˜ ( k x , k y , ± k z = ± k 2 k x 2 k y 2 )
E ˜ s ( k x , k y , b ) = α   k 2 ( 2 π ) 3   E 0   s ˜ 0 [ k ( k x , k y , k 2 k x 2 k y 2 ,   b r ) k i ] e i   ( k k i )   b t
I ˜ d i f f ( b ) =   | E ˜ i + E ˜ s ( b ) | 2 | E ˜ i | 2 = | E ˜ s ( b ) | 2 + 2 | E ˜ i | | E ˜ s ( b ) |   s i n ( Δ Φ t ( b ) )
cos ( Δ Φ ( b ) ) = cos ( Φ i π / 2 Φ s ( b ) ) = sin ( Δ Φ t ( b ) )
E ˜ i ( k x , k y ) = E i 0 s t e p ( k 0 N A k x 2 + k y 2 )
E i ( r ) = 1 ( 2 π ) 3 k < k 0 N A E ˜ i ( k ) exp ( i k r ) d 3 k = 1 ( 2 π ) 2 E ˜ i ( k x , k y ) e i ( k x x + k y y + k z z ) / k z d k x d k y
E i ( x , y , z ) = A 0   W 0 W ( z ) e ( x 2 + y 2 ) / W ( z ) 2 e i Φ G ( x , y , z )
Φ G ( x , y , z ) = k   ( x 2 + y 2 ) 2 R ( z ) + k   z + a tan ( z / z 0 ) z < z 0 + k   z + a tan ( z / z 0 )
I ˜ d i f f ( k , b ) = | E ˜ s ( b ) | 2 + 2 | E i 0 s t e p ( k 0 N A k x 2 + k y 2 ) | | α   k 2 ( 2 π ) 3   E i ( b t )   s ˜ 0 [ k ( k ,   b r ) k i ] |   sin ( Δ Φ t ( k , b t ) )
Δ Φ t ( k x , k y , b t ) = ( k k i )   b t + Φ i ( k , b t ) = [ b x k x + b y k y + b z k z b z k ] + [ a tan ( b z / z 0 ) + b z k ] = b x k x + b y k y + b z k 2 k x 2 k y 2 + a tan ( b z / z 0 )
I ˜ d i f f O ( k , b ) = | E ˜ s ( b ) | 2 + 2 B   | E i ( b t ) |   | sin c ( ( k z ' ( b t ) k ) L / 2 ) |   sin ( Δ Φ t ( k , b t ) )
I ˜ d i f f O ( k x , k x , 0 , b θ ) = B ² | sin c ( ( sin ( b θ ) k x + cos ( b θ ) k 2 k x 2 k y 2 k ) L / 2 ) | 2
I ˜ d i f f O ( k x , k x , 0 , b θ ) = B ²   | sin c ( ( b θ k x k x 2 / 2 k k y 2 / 2 k   ) L / 2 ) | 2
I ˜ d i f f O ( k x , k y , b x , b θ ) = | E ˜ s ( b ) | 2 + B   | E i ( 0 ) | | sin c ( ( b θ k x k x 2 + k y 2 2 k ) L / 2 ) | sin ( b x k x )
( S x S y S z S θ S ϕ ) ( S 0 x S 0 y S 0 z S 0 θ S 0 ϕ ) + ( g x x g y y g z z g θ θ g ϕ ϕ ) ( b x b y b z b θ b ϕ )
I ˜ d i f f   t h r ( k x , k y , b ) = { | I ˜ d i f f ( k x , k y , b ) | , if   | I ˜ d i f f | 1 e   I ˜ d i f f m a x 0 , otherwise
k c = ( k x c , k y c ) = I ˜ d i f f   t h r ( k x k y , b )   ( k x , k y )   d k x d k y I ˜ d i f f   t h r ( k x k y , b )   d k x d k y
H t ( k x , k y , k c ) = s t e p ( k L | k - k c | ) ( 2 s t e p ( k x k x c ) 1 2 s t e p ( k x k x c ) 1 1 )
H r ( k x , k y , k c ) = ( k x c 2 + k y c 2 a tan ( k x c / k y c ) )
S j ( b j ) = I ˜ d i f f   t h r ( k x k y , b ) H j ( k x , k y , k c ) d k x d k y         S 0 j + g j j × b j
S x ( b x , b θ ) B   | E i ( 0 ) | | sin c ( ( b θ k x ( k x 2 + k y 2 ) / 2 k ) L 2 ) sin ( b x k x ) ( 2 s t e p ( k x k x c ) 1 ) | d k x d k y  
p ( b i , b j ) = p 0 i j   e x p [ W ( b i , b j ) / ( k B   T ) ] 1 2 π   σ i σ j   e x p [ 1 2 ( b i / σ i ) 2 1 2 ( b j / σ j ) ]
S i ( b i , b j ) = g i i ( b i ) b i + g i j ( b j ) b j g i i b i + g i j b j
Δ b j S j ( σ j / σ s j ) b j

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