Abstract

A coherent confocal microscope is proposed as a means to fully characterize the elastic scattering properties of a nanoparticle as a function of wavelength. Using a high numerical aperture lens, two-dimensional scanning, and a simple vector-beam shaper, the rank-2 polarizability tensor is estimated from a single confocal image. A method for computationally efficient data processing is described, and numerical simulations show that this algorithm is robust to noise and uncertainty in the focal plane position. The proposed method is a generalization of techniques that provide an estimate of a limited set of scattering parameters, such as a single orientation angle for rodlike particles. The measurement of the polarizability obviates the need for a priori assumptions about the nanoparticle.

© 2008 Optical Society of America

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2008 (4)

J.W.Bulte and M.Modo, eds., Nanoparticles in Biomedical Imaging (Springer, 2008).
[CrossRef]

J. Aaron, E. de la Rosa, K. Travis, N. Harrison, J. Burt, M. José-Yacamán, and K. Sokolov, “Polarization microscopy with stellated gold nanoparticles for robust, in-situ monitoring of biomolecules,” Opt. Express 16, 2153-2167 (2008).
[CrossRef] [PubMed]

D. Haefner, S. Sukhov, and A. Dogariu, “Stochastic scattering polarimetry,” Phys. Rev. Lett. 100, 043901 (2008).
[CrossRef] [PubMed]

S. Sukhov, D. Haefner, and A. Dogariu, “Stochastic sensing of relative aniostropic polarizabilities,” Phys. Rev. A 77, 043820 (2008).
[CrossRef]

2007 (9)

A. V. Failla, S. Jäger, T. Züchner, M. Steiner, and A. J. Meixner, “Topology measurements of metal nanoparticles with 1nm accuracy by confocal interference scattering microscopy,” Opt. Express 15, 8532-8542 (2007).
[CrossRef] [PubMed]

B. J. Davis, A. K. Swan, M. S. Ünlü, W. C. Karl, B. B. Goldberg, J. C. Schotland, and P. S. Carney, “Spectral self-interference microscopy for low-signal nanoscale axial imaging,” J. Opt. Soc. Am. A 24, 3587-3599 (2007).
[CrossRef]

B. J. Davis, S. C. Schlachter, D. L. Marks, T. S. Ralston, S. A. Boppart, and P. S. Carney, “Nonparaxial vector-field modeling of optical coherence tomography and interferometric synthetic aperture microscopy,” J. Opt. Soc. Am. A 24, 2527-2542 (2007).
[CrossRef]

B. J. Davis, T. S. Ralston, D. L. Marks, S. A. Boppart, and P. S. Carney, “Autocorrelation artifacts in optical coherence tomography and interferometric synthetic aperture microscopy,” Opt. Lett. 32, 1441-1443 (2007).
[CrossRef] [PubMed]

J. Nelayah, M. Kociak, O. Stéphan, F. J. García de Abajo, M. Tence, L. Henrard, D. Taverna, I. Pastoriza-Santos, L. Liz-Marzán, and C. Colliex, “Mapping surface plasmons on a single metallic nanoparticle,” Nat. Phys. 3, 348-353 (2007).
[CrossRef]

G.E.Fryxell and G.Cao, eds., Environmental Applications of Nanomaterials: Synthesis, Sorbents and Sensors (Imperial College, 2007).
[CrossRef]

D.Thassu, M.Deleers, and Y.Pathak, eds., Nanoparticle Drug Delivery Systems (Informa Healthcare, 2007).
[CrossRef]

K. Kalantar-Zadeh and B. Fry, Nanotechnology-Enabled Sensors (Springer, 2007).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge U. Press, 2007), Chap. 10.5, pp. 502-507.

2006 (13)

M. Leutenegger, R. Rao, R. A. Leitgeb, and T. Lasser, “Fast focus field calculations,” Opt. Express 14, 11277-11291 (2006).
[CrossRef] [PubMed]

E. Fritsch, E. Gaillou, B. Rondeau, A. Barreau, D. Albertini, and M. Ostroumov, “The nanostructure of fire opal,” J. Non-Cryst. Solids 352, 3957-3960 (2006).
[CrossRef]

R. W. Ziolkowski and N. Engheta, eds., Metamaterials: Physics and Engineering Explorations (Wiley, 2006).

B. K. Canfield, S. Kujala, K. Laiho, K. Jefimovs, J. Turunen, and M. Kauranen, “Chirality arising from small defects in gold nanoparticle arrays,” Opt. Express 14, 950-955 (2006).
[CrossRef] [PubMed]

M. E. Brezinski, Optical Coherence Tomography: Principles and Applications (Academic, 2006).

F. V. Ignatovich and L. Novotny, “Real-time and background-free detection of nanoscale particles,” Phys. Rev. Lett. 96, 013901 (2006).
[CrossRef] [PubMed]

S. Ram, E. S. Ward, and R. J. Ober, “Beyond Rayleigh's criterion: A resolution measure with application to single-molecule microscopy,” Proc. Natl. Acad. Sci. U.S.A. 103, 4457-4462 (2006).
[CrossRef] [PubMed]

E. Toprak, J. Enderlein, S. Syed, S. A. McKinney, R. G. Petschek, T. Ha, Y. E. Goldman, and P. R. Selvin, “Defocused orientation and position imaging (DOPI) of myosin V,” Proc. Natl. Acad. Sci. U.S.A. 103, 6495-6499 (2006).
[CrossRef] [PubMed]

R. Hassey, E. J. Swain, N. I. Hammer, D. Venkataraman, and M. D. Barnes, “Probing the chiroptical response of a single molecule,” Science 314, 1437-1439 (2006).
[CrossRef] [PubMed]

M. A. van Dijk, A. L. Tchebotareva, M. Orrit, M. Lippitz, S. Berciaud, D. Lasne, L. Cognet, and B. Lounis, “Absorption and scattering microscopy of single metal nanoparticles,” Phys. Chem. Chem. Phys. 8, 3486-3495 (2006).
[CrossRef] [PubMed]

A. V. Failla, H. Qian, H. Qian, A. Hartschuh, and A. J. Meixner, “Orientation imaging of subwavelength Au particles with higher order laser modes,” Nano Lett. 6, 1374-1378 (2006).
[CrossRef] [PubMed]

A. F. Abouraddy and K. C. Toussaint, Jr., “Three-dimensional polarization control in microscopy,” Phys. Rev. Lett. 96, 153901 (2006).
[CrossRef] [PubMed]

M. R. Beversluis, L. Novotny, and S. J. Stranick, “Programmable vector point-spread function engineering,” Opt. Express 14, 2650-2656 (2006).
[CrossRef] [PubMed]

2005 (8)

C. Sönnichsen and A. P. Alivisatos, “Gold nanorods as novel nobleaching plasmon-based orientation sensors for polarized single-particle microscopy,” Nano Lett. 5, 301-304 (2005).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, and G. Leuchs, “Generation of a radially polarized doughnut mode of high quality,” Appl. Phys. B 81, 597-600 (2005).
[CrossRef]

K. C. Toussaint Jr., S. Park, J. E. Jureller, and N. F. Scherer, “Generation of optical vector beams with a diffractive optical element interferometer,” Opt. Lett. 30, 2846-2848 (2005).
[CrossRef] [PubMed]

D. Patra, I. Gregor, J. Enderlein, and M. Sauer, “Defocused imaging of quantum-dot angular distribution of radiation,” Appl. Phys. Lett. 87, 101103 (2005).
[CrossRef]

I. J. Cooper, M. Roy, and C. J. R. Sheppard, “Focusing of pseudoradial polarized beams,” Opt. Express 13, 1066-1071 (2005).
[CrossRef] [PubMed]

T. Colomb, F. Dürr, E. Cuche, P. Marquet, H. G. Limberger, R.-P. Salathé, and C. Depeursinge, “Polarization microscopy by use of digital holography: Application to optical-fiber birefringence measurements,” Appl. Opt. 44, 4461-4469 (2005).
[CrossRef] [PubMed]

P. Mühlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science 308, 1607-1609 (2005).
[CrossRef] [PubMed]

I. J. Cooper, C. J. R. Sheppard, and M. Roy, “The numerical integration of fundamental diffraction integrals for converging polarized spherical waves using a two-dimensional form of Simpson's 1/3 rule,” J. Mod. Opt. 52, 1123-1134 (2005).
[CrossRef]

2004 (11)

S. Yazdanfar, L. H. Laiho, and P. T. C. So, “Interferometric second harmonic generation microscopy,” Opt. Express 12, 2739-2745 (2004).
[CrossRef] [PubMed]

B. E. Applegate, C. Yang, A. M. Rollins, and J. A. Izatt, “Polarization-resolved second-harmonic-generation optical coherence tomography in collagen,” Opt. Lett. 29, 2252-2254 (2004).
[CrossRef] [PubMed]

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788-792 (2004).
[CrossRef] [PubMed]

G.Schmid, ed., Nanoparticles: From Theory to Application (Wiley, 2004).

M. Mujat, E. Baleine, and A. Dogariu, “Interferometric imaging polarimeter,” J. Opt. Soc. Am. A 21, 2244-2249 (2004).
[CrossRef]

R. J. Ober, S. Ram, and E. S. Ward, “Localization accuracy in single-molecule microscopy,” Biophys. J. 86, 1185-1200 (2004).
[CrossRef] [PubMed]

R. J. Potton, “Reciprocity in optics,” Rep. Prog. Phys. 67, 717-754 (2004).
[CrossRef]

D. Patra, I. Gregor, and J. Enderlein, “Image analysis of defocused single-molecule images for three-dimensional molecule orientation studies,” J. Phys. Chem. A 108, 6836-6841 (2004).
[CrossRef]

F. Kulzer and M. Orrit, “Single-molecule optics,” Annu. Rev. Phys. Chem. 55, 585-611 (2004).
[CrossRef] [PubMed]

M. Artemyev, D. Kisiel, S. Abmiotko, M. N. Antipina, G. B. Khomutov, V. V. Kislov, and A. A. Rakhnyanskaya, “Self-organized highly luminescent CdSe nanorod-DNA complexes,” J. Am. Chem. Soc. 126, 10594-10597 (2004).
[CrossRef] [PubMed]

M. A. Lieb, J. M. Zavislan, and L. Novotny, “Single-moleculeorientations determined by direct emission pattern imaging,” J. Opt. Soc. Am. B 21, 1210-1215 (2004).
[CrossRef]

2003 (8)

M. Prummer, B. Sick, B. Hecht, and U. P. Wild, “Three-dimensional optical polarization tomography of single molecules,” J. Chem. Phys. 118, 9824-9829 (2003).
[CrossRef]

M. Böhmer and J. Enderlein, “Orientation imaging of single molecules by wide-field epifluorescence microscopy,” J. Opt. Soc. Am. B 20, 554-559 (2003).
[CrossRef]

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A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, “Myosin V walks hand-over-hand: Single fluorophore imaging with 1.5-nm localization,” Science 300, 2061-2065 (2003).
[CrossRef] [PubMed]

W. Rechberger, A. Hohenau, A. Leitner, J. R. Krenn, B. Lamprecht, and F. R. Aussenegg, “Optical properties of two interacting gold nanoparticles,” Opt. Commun. 220, 137-141 (2003).
[CrossRef]

Y. L. Kim, Y. Liu, R. K. Wali, H. K. Roy, M. J. Goldberg, A. K. Kromin, K. Chen, and V. Backman, “Simultaneous measurement of angular and spectral properties of light scattering for characterization of tissue microarchitecture and its alteration in early precancer,” IEEE J. Sel. Top. Quantum Electron. 9, 243-256 (2003).
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H. H. Barrett and K. Myers, Foundations of Image Science (Wiley-Interscience, 2003).

V.Rotello, ed., Nanoparticles: Building Blocks for Nanotechnology (Springer, 2003).

2002 (5)

D. Jembrih-Simbürger, C. Neelmeijer, O. Schalm, P. Fredrickx, M. Schreiner, K. De Vis, M. Mäder, D. Schryvers, and J. Caen, “The colour of silver stained glass--analytical investigations carried out with XRF, SEM/EDX, TEM and IBA,” J. Anal. At. Spectrom. 17, 321-328 (2002).
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J. J. Mock, M. Barbic, D. R. Smith, D. A. Schultz, and S. Schultz, “Shape effects in plasmon resonance of individual colloidal silver nanoparticles,” J. Chem. Phys. 116, 6755-6759 (2002).
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M. A. A. Neil, F. Massoumian, R. Juškaitis, and T. Wilson, “Method for the generation of arbitrary complex vector wave fronts,” Opt. Lett. 27, 1929-1931 (2002).
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2001 (2)

J. T. Fourkas, “Rapid determination of the three-dimensional orientation of single molecules,” Opt. Lett. 26, 211-213 (2001).
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L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251-5254 (2001).
[CrossRef] [PubMed]

2000 (3)

B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85, 4482-4485 (2000).
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Y. Zhao, Z. Chen, C. Saxer, S. Xiang, J. F. de Boer, and J. S. Nelson, “Phase-resolved optical coherence tomography and optical Doppler tomography for imaging blood flow in human skin with fast scanning speed and high velocity sensitivity,” Opt. Lett. 25, 114-116 (2000).
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1999 (2)

A. P. Bartko and R. M. Dickson, “Three-dimensional orientations of polymer-bound single molecules,” J. Phys. Chem. B 103, 3051-3056 (1999).

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1997 (1)

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1996 (1)

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1995 (1)

1994 (1)

1993 (1)

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1991 (1)

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1990 (1)

P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge U. Press, 1990), Chap. 5.2, pp. 131-134.

1987 (1)

E. De Castro and C. Morandi, “Registration of translated and rotated images using finite Fourier transforms,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-9, 700-705 (1987).
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1981 (1)

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

1966 (1)

1965 (1)

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M. H. Nayfeh, O. Akcakir, G. Belomoin, N. Barry, J. Therrien, and E. Gratton, “Second harmonic generation in microcrystallite films of ultrasmall Si nanoparticles,” Appl. Phys. Lett. 77, 4086-4088 (2000).
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W. Rechberger, A. Hohenau, A. Leitner, J. R. Krenn, B. Lamprecht, and F. R. Aussenegg, “Optical properties of two interacting gold nanoparticles,” Opt. Commun. 220, 137-141 (2003).
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Y. L. Kim, Y. Liu, R. K. Wali, H. K. Roy, M. J. Goldberg, A. K. Kromin, K. Chen, and V. Backman, “Simultaneous measurement of angular and spectral properties of light scattering for characterization of tissue microarchitecture and its alteration in early precancer,” IEEE J. Sel. Top. Quantum Electron. 9, 243-256 (2003).
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Barbic, M.

J. J. Mock, M. Barbic, D. R. Smith, D. A. Schultz, and S. Schultz, “Shape effects in plasmon resonance of individual colloidal silver nanoparticles,” J. Chem. Phys. 116, 6755-6759 (2002).
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R. Hassey, E. J. Swain, N. I. Hammer, D. Venkataraman, and M. D. Barnes, “Probing the chiroptical response of a single molecule,” Science 314, 1437-1439 (2006).
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E. Fritsch, E. Gaillou, B. Rondeau, A. Barreau, D. Albertini, and M. Ostroumov, “The nanostructure of fire opal,” J. Non-Cryst. Solids 352, 3957-3960 (2006).
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H. H. Barrett and K. Myers, Foundations of Image Science (Wiley-Interscience, 2003).

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M. H. Nayfeh, O. Akcakir, G. Belomoin, N. Barry, J. Therrien, and E. Gratton, “Second harmonic generation in microcrystallite films of ultrasmall Si nanoparticles,” Appl. Phys. Lett. 77, 4086-4088 (2000).
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Bartko, A. P.

A. P. Bartko and R. M. Dickson, “Three-dimensional orientations of polymer-bound single molecules,” J. Phys. Chem. B 103, 3051-3056 (1999).

A. P. Bartko and R. M. Dickson, “Imaging three-dimensional single molecule orientations,” J. Phys. Chem. B 103, 11237-11241 (1999).
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T. Basché, W. E. Moerner, M. Orrit, and U. P. Wild, Single-Molecule Optical Detection, Imaging and Spectroscopy (VCH, 1997).

Belomoin, G.

M. H. Nayfeh, O. Akcakir, G. Belomoin, N. Barry, J. Therrien, and E. Gratton, “Second harmonic generation in microcrystallite films of ultrasmall Si nanoparticles,” Appl. Phys. Lett. 77, 4086-4088 (2000).
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M. A. van Dijk, A. L. Tchebotareva, M. Orrit, M. Lippitz, S. Berciaud, D. Lasne, L. Cognet, and B. Lounis, “Absorption and scattering microscopy of single metal nanoparticles,” Phys. Chem. Chem. Phys. 8, 3486-3495 (2006).
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M. R. Beversluis, L. Novotny, and S. J. Stranick, “Programmable vector point-spread function engineering,” Opt. Express 14, 2650-2656 (2006).
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L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251-5254 (2001).
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Booker, G. R.

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Brown, T. G.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251-5254 (2001).
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Burt, J.

Butcher, P. N.

P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge U. Press, 1990), Chap. 5.2, pp. 131-134.

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D. Jembrih-Simbürger, C. Neelmeijer, O. Schalm, P. Fredrickx, M. Schreiner, K. De Vis, M. Mäder, D. Schryvers, and J. Caen, “The colour of silver stained glass--analytical investigations carried out with XRF, SEM/EDX, TEM and IBA,” J. Anal. At. Spectrom. 17, 321-328 (2002).
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Carney, P. S.

Chang, W.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178-1181 (1991).
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Chen, K.

Y. L. Kim, Y. Liu, R. K. Wali, H. K. Roy, M. J. Goldberg, A. K. Kromin, K. Chen, and V. Backman, “Simultaneous measurement of angular and spectral properties of light scattering for characterization of tissue microarchitecture and its alteration in early precancer,” IEEE J. Sel. Top. Quantum Electron. 9, 243-256 (2003).
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Chen, Z.

Cognet, L.

M. A. van Dijk, A. L. Tchebotareva, M. Orrit, M. Lippitz, S. Berciaud, D. Lasne, L. Cognet, and B. Lounis, “Absorption and scattering microscopy of single metal nanoparticles,” Phys. Chem. Chem. Phys. 8, 3486-3495 (2006).
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J. Nelayah, M. Kociak, O. Stéphan, F. J. García de Abajo, M. Tence, L. Henrard, D. Taverna, I. Pastoriza-Santos, L. Liz-Marzán, and C. Colliex, “Mapping surface plasmons on a single metallic nanoparticle,” Nat. Phys. 3, 348-353 (2007).
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I. J. Cooper, M. Roy, and C. J. R. Sheppard, “Focusing of pseudoradial polarized beams,” Opt. Express 13, 1066-1071 (2005).
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I. J. Cooper, C. J. R. Sheppard, and M. Roy, “The numerical integration of fundamental diffraction integrals for converging polarized spherical waves using a two-dimensional form of Simpson's 1/3 rule,” J. Mod. Opt. 52, 1123-1134 (2005).
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Cotter, D.

P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge U. Press, 1990), Chap. 5.2, pp. 131-134.

Cuche, E.

Davis, B. J.

de Boer, J. F.

De Castro, E.

E. De Castro and C. Morandi, “Registration of translated and rotated images using finite Fourier transforms,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-9, 700-705 (1987).
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de la Rosa, E.

De Vis, K.

D. Jembrih-Simbürger, C. Neelmeijer, O. Schalm, P. Fredrickx, M. Schreiner, K. De Vis, M. Mäder, D. Schryvers, and J. Caen, “The colour of silver stained glass--analytical investigations carried out with XRF, SEM/EDX, TEM and IBA,” J. Anal. At. Spectrom. 17, 321-328 (2002).
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Depeursinge, C.

Dickson, R. M.

A. P. Bartko and R. M. Dickson, “Imaging three-dimensional single molecule orientations,” J. Phys. Chem. B 103, 11237-11241 (1999).
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A. P. Bartko and R. M. Dickson, “Three-dimensional orientations of polymer-bound single molecules,” J. Phys. Chem. B 103, 3051-3056 (1999).

Dogariu, A.

D. Haefner, S. Sukhov, and A. Dogariu, “Stochastic scattering polarimetry,” Phys. Rev. Lett. 100, 043901 (2008).
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S. Sukhov, D. Haefner, and A. Dogariu, “Stochastic sensing of relative aniostropic polarizabilities,” Phys. Rev. A 77, 043820 (2008).
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M. Mujat, E. Baleine, and A. Dogariu, “Interferometric imaging polarimeter,” J. Opt. Soc. Am. A 21, 2244-2249 (2004).
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S. Quabis, R. Dorn, and G. Leuchs, “Generation of a radially polarized doughnut mode of high quality,” Appl. Phys. B 81, 597-600 (2005).
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Dürr, F.

Eisler, H.-J.

P. Mühlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science 308, 1607-1609 (2005).
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Enderlein, J.

E. Toprak, J. Enderlein, S. Syed, S. A. McKinney, R. G. Petschek, T. Ha, Y. E. Goldman, and P. R. Selvin, “Defocused orientation and position imaging (DOPI) of myosin V,” Proc. Natl. Acad. Sci. U.S.A. 103, 6495-6499 (2006).
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D. Patra, I. Gregor, J. Enderlein, and M. Sauer, “Defocused imaging of quantum-dot angular distribution of radiation,” Appl. Phys. Lett. 87, 101103 (2005).
[CrossRef]

D. Patra, I. Gregor, and J. Enderlein, “Image analysis of defocused single-molecule images for three-dimensional molecule orientation studies,” J. Phys. Chem. A 108, 6836-6841 (2004).
[CrossRef]

M. Böhmer and J. Enderlein, “Orientation imaging of single molecules by wide-field epifluorescence microscopy,” J. Opt. Soc. Am. B 20, 554-559 (2003).
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Engheta, N.

R. W. Ziolkowski and N. Engheta, eds., Metamaterials: Physics and Engineering Explorations (Wiley, 2006).

Failla, A. V.

A. V. Failla, S. Jäger, T. Züchner, M. Steiner, and A. J. Meixner, “Topology measurements of metal nanoparticles with 1nm accuracy by confocal interference scattering microscopy,” Opt. Express 15, 8532-8542 (2007).
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A. V. Failla, H. Qian, H. Qian, A. Hartschuh, and A. J. Meixner, “Orientation imaging of subwavelength Au particles with higher order laser modes,” Nano Lett. 6, 1374-1378 (2006).
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Fercher, A. F.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge U. Press, 2007), Chap. 10.5, pp. 502-507.

Flatau, P. J.

Flotte, T.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Forkey, J. N.

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, “Myosin V walks hand-over-hand: Single fluorophore imaging with 1.5-nm localization,” Science 300, 2061-2065 (2003).
[CrossRef] [PubMed]

Fourkas, J. T.

Fredrickx, P.

D. Jembrih-Simbürger, C. Neelmeijer, O. Schalm, P. Fredrickx, M. Schreiner, K. De Vis, M. Mäder, D. Schryvers, and J. Caen, “The colour of silver stained glass--analytical investigations carried out with XRF, SEM/EDX, TEM and IBA,” J. Anal. At. Spectrom. 17, 321-328 (2002).
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Fritsch, E.

E. Fritsch, E. Gaillou, B. Rondeau, A. Barreau, D. Albertini, and M. Ostroumov, “The nanostructure of fire opal,” J. Non-Cryst. Solids 352, 3957-3960 (2006).
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D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178-1181 (1991).
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Gabor, D.

Gaillou, E.

E. Fritsch, E. Gaillou, B. Rondeau, A. Barreau, D. Albertini, and M. Ostroumov, “The nanostructure of fire opal,” J. Non-Cryst. Solids 352, 3957-3960 (2006).
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J. Nelayah, M. Kociak, O. Stéphan, F. J. García de Abajo, M. Tence, L. Henrard, D. Taverna, I. Pastoriza-Santos, L. Liz-Marzán, and C. Colliex, “Mapping surface plasmons on a single metallic nanoparticle,” Nat. Phys. 3, 348-353 (2007).
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Goldberg, B. B.

Goldberg, M. J.

Y. L. Kim, Y. Liu, R. K. Wali, H. K. Roy, M. J. Goldberg, A. K. Kromin, K. Chen, and V. Backman, “Simultaneous measurement of angular and spectral properties of light scattering for characterization of tissue microarchitecture and its alteration in early precancer,” IEEE J. Sel. Top. Quantum Electron. 9, 243-256 (2003).
[CrossRef]

Goldman, Y. E.

E. Toprak, J. Enderlein, S. Syed, S. A. McKinney, R. G. Petschek, T. Ha, Y. E. Goldman, and P. R. Selvin, “Defocused orientation and position imaging (DOPI) of myosin V,” Proc. Natl. Acad. Sci. U.S.A. 103, 6495-6499 (2006).
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A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, “Myosin V walks hand-over-hand: Single fluorophore imaging with 1.5-nm localization,” Science 300, 2061-2065 (2003).
[CrossRef] [PubMed]

Goss, W. P.

Gratton, E.

M. H. Nayfeh, O. Akcakir, G. Belomoin, N. Barry, J. Therrien, and E. Gratton, “Second harmonic generation in microcrystallite films of ultrasmall Si nanoparticles,” Appl. Phys. Lett. 77, 4086-4088 (2000).
[CrossRef]

Gregor, I.

D. Patra, I. Gregor, J. Enderlein, and M. Sauer, “Defocused imaging of quantum-dot angular distribution of radiation,” Appl. Phys. Lett. 87, 101103 (2005).
[CrossRef]

D. Patra, I. Gregor, and J. Enderlein, “Image analysis of defocused single-molecule images for three-dimensional molecule orientation studies,” J. Phys. Chem. A 108, 6836-6841 (2004).
[CrossRef]

Gregory, K.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Ha, T.

E. Toprak, J. Enderlein, S. Syed, S. A. McKinney, R. G. Petschek, T. Ha, Y. E. Goldman, and P. R. Selvin, “Defocused orientation and position imaging (DOPI) of myosin V,” Proc. Natl. Acad. Sci. U.S.A. 103, 6495-6499 (2006).
[CrossRef] [PubMed]

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, “Myosin V walks hand-over-hand: Single fluorophore imaging with 1.5-nm localization,” Science 300, 2061-2065 (2003).
[CrossRef] [PubMed]

Haefner, D.

D. Haefner, S. Sukhov, and A. Dogariu, “Stochastic scattering polarimetry,” Phys. Rev. Lett. 100, 043901 (2008).
[CrossRef] [PubMed]

S. Sukhov, D. Haefner, and A. Dogariu, “Stochastic sensing of relative aniostropic polarizabilities,” Phys. Rev. A 77, 043820 (2008).
[CrossRef]

Hallen, H. D.

C. L. Jahncke, H. D. Hallen, and M. A. Paesler, “Nano-Raman spectroscopy and imaging with a near-field scanning optical microscope,” J. Raman Spectrosc. 27, 579-586 (1996).
[CrossRef]

Hammer, N. I.

R. Hassey, E. J. Swain, N. I. Hammer, D. Venkataraman, and M. D. Barnes, “Probing the chiroptical response of a single molecule,” Science 314, 1437-1439 (2006).
[CrossRef] [PubMed]

Harrison, N.

Hartschuh, A.

A. V. Failla, H. Qian, H. Qian, A. Hartschuh, and A. J. Meixner, “Orientation imaging of subwavelength Au particles with higher order laser modes,” Nano Lett. 6, 1374-1378 (2006).
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Hassey, R.

R. Hassey, E. J. Swain, N. I. Hammer, D. Venkataraman, and M. D. Barnes, “Probing the chiroptical response of a single molecule,” Science 314, 1437-1439 (2006).
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P. Mühlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science 308, 1607-1609 (2005).
[CrossRef] [PubMed]

M. Prummer, B. Sick, B. Hecht, and U. P. Wild, “Three-dimensional optical polarization tomography of single molecules,” J. Chem. Phys. 118, 9824-9829 (2003).
[CrossRef]

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[CrossRef] [PubMed]

Hee, M. R.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Henrard, L.

J. Nelayah, M. Kociak, O. Stéphan, F. J. García de Abajo, M. Tence, L. Henrard, D. Taverna, I. Pastoriza-Santos, L. Liz-Marzán, and C. Colliex, “Mapping surface plasmons on a single metallic nanoparticle,” Nat. Phys. 3, 348-353 (2007).
[CrossRef]

Hitzenberger, C. K.

Hohenau, A.

W. Rechberger, A. Hohenau, A. Leitner, J. R. Krenn, B. Lamprecht, and F. R. Aussenegg, “Optical properties of two interacting gold nanoparticles,” Opt. Commun. 220, 137-141 (2003).
[CrossRef]

Huang, D.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178-1181 (1991).
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Figures (11)

Fig. 1
Fig. 1

Illustration of the proposed coherent confocal system including fields and tensor operators used in the model derivation. The reference field is denoted by E ( r ) , the field after the beam shaper by E ( b ) , the field after the lens by E ( l ) , the field in the sample by g, and the backscattered field by E ( s ) . The tensor operators V ¯ , A ¯ , and F ¯ give relations between the fields of interest. Note that the dependence on the wavenumber k has been dropped in the notation and that the tensor operators describe the evolution of fields in the direction of the arrows.

Fig. 2
Fig. 2

Fully polarized example fields at the entrance pupil, E ( b ) . The field direction is parallel to the entrance pupil with the (a),(c) x-polarized and (b),(d) y-polarized field components shown separately. (a),(b) The proposed beam is piecewise constant, providing a simplified experimental implementation compared to a more traditional (c),(d) radially polarized beam. The fields are displayed as a function of the cosines of the angles-to-focus over the 0.8 NA lens aperture.

Fig. 3
Fig. 3

Fields at the exit pupil, E ( l ) , (a)–(c) for the proposed beam shape and (d)–(f) radially polarized system. The (a),(d) x-polarized, (b),(e), y-polarized, and (c),(f) z-polarized field components are shown separately. The field is displayed as a function of the cosines of the angles-to-focus over the 0.8 NA lens aperture.

Fig. 4
Fig. 4

Calculated intensities of the focused example field g ( x , y , z ; k ) at (a)–(c) z = 0 , (d)–(f) z = λ , and (g)–(i) z = 2 λ , where the wavelength is given by λ = 2 π k . The intensities of each component are plotted separately, i.e., (a),(d),(g) g x 2 , (b),(e),(h) g y 2 , and (c),(f),(i) g z 2 are shown in different plots.

Fig. 5
Fig. 5

Fourier-domain representations of the focal plane PSFs for the example system. At z ( p ) = 0 and for this system, these functions are real. The function axes are shown in the upper right and the remaining plots show (a) h ̃ x x , (b) h ̃ x y , (c) h ̃ y y , (d) h ̃ x z , (e) h ̃ y z , and (f) h ̃ z z .

Fig. 6
Fig. 6

Fourier-domain magnitudes of the system PSFs at (a),(d),(g),(j) z ( p ) = 0 , (b),(e),(h),(k) z ( p ) = λ 2 , and (c),(f),(i),(l) z ( p ) = λ . Magnitudes for (a)–(c) h ̃ x x , (d)–(f) h ̃ z z , (g)–(i) h ̃ x y , and (j)–(l) h ̃ x z are shown with h ̃ y y being a rotation of h ̃ x x and h ̃ y z being a rotation of h ̃ x z . Note that the first column represents the magnitudes of the plots shown in Fig. 5.

Fig. 7
Fig. 7

Fourier-domain phase profiles of the system PSFs at (a),(d),(g),(j) z ( p ) = λ 4 , (b),(e),(h),(k) z ( p ) = λ 2 , and (c),(f),(i),(l) z ( p ) = λ . These are calculated by dividing h ̃ at z ( p ) into h ̃ at z ( p ) = 0 and taking the complex angle. Phases for (a)–(c) h ̃ x x , (d)–(f) h ̃ z z , (g)–(i) h ̃ x y , and (j)–(l) h ̃ x z are shown with h ̃ y y being a rotation of h ̃ x x and h ̃ y z being a rotation of h ̃ x z . Note that the phase profiles have been unwrapped and that the color scale varies across the columns.

Fig. 8
Fig. 8

Conditioning of the relation between the polarizability elements and the collected data for various values of NA. Higher conditioning gives better estimation of the polarizability. Results for both the proposed quadrant-based beam and a radially polarized beam are plotted. In both cases the center of the beam is blocked as seen in Fig. 3. The diameter of the central blocked region scales with NA.  

Fig. 9
Fig. 9

Simulated data for the example nanoparticle parameters of Eqs. (22, 23). The (a)–(c) real and (d)–(f) imaginary parts of the data are plotted with (a),(d) no noise, (b),(e) SNRs of 14.3 dB , and (c),(f) 5.3 dB .

Fig. 10
Fig. 10

(a), (c), (e) Fourier-domain magnitudes and (b),(d),(f) phases of simulated data. (a),(b) Noise-free and (c),(d) 14.3 dB SNR data from the example nanoparticle parameters of Eqs. (22, 23) are shown along with (e),(f) data corresponding to the parameters of Eqs. (24, 25), which were estimated from the noisy data. Note that noise outside the system band limit ( q = 1.6 ) is not shown and that approximately 0.25% of the pixels in plot (c) saturate the color scale.

Fig. 11
Fig. 11

(a) Average polarizability and (b) position errors for various SNRs and expected nanoparticle distances from focus. Each marked point is calculated from 100 simulations and each simulation has a different random nanoparticle polarizability, nanoparticle location, and noise realization.

Equations (25)

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I ( r ; k ) = μ E ( r ) ( k ) + E ( s ) ( r ; k ) 2 , = μ E ( r ) ( k ) 2 + 2 Re { S ( r ; k ) } + E ( s ) ( r ; k ) 2 ,
S ( r ; k ) = [ μ E ( r ) ( k ) ] E ( s ) ( r ; k ) ,
E ( b ) ( s x , s y ; k ) = V ¯ ( s x , s y ) E ( r ) ( k ) .
E ( l ) ( s x , s y ; k ) = A ¯ ( s x , s y ) E ( r ) ( k ) .
g ( r r ; k ) = k 2 π i Ω d s x d s y A ¯ ( s x , s y ) E ( r ) ( k ) s z ( s x , s y ) e i k s ( r r ) , = F ¯ ( r r ; k ) E ( r ) ( k ) ,
s z ( s x , s y ) = 1 s x 2 s y 2 ,
E ( s ) ( r ; k ) = k 2 F ¯ T ( r ( p ) r ; k ) α ¯ ( k ) g ( r ( p ) r ; k ) .
S ( r ; k ) = μ * k 2 [ E ( r ) ( k ) ] F ¯ T ( r ( p ) r ; k ) α ¯ ( k ) F ¯ ( r ( p ) r ; k ) E ( r ) ( k ) .
S ( ρ ; k ) α ζ β ( k ) h ζ β ( ρ ρ ( p ) ; z ( p ) , k ) ,
h ζ β ( ρ ; z ( p ) , k ) = k 2 W κ γ ( k ) F β γ ( ρ , z ( p ) ; k ) F ζ κ ( ρ , z ( p ) ; k ) ,
W ( k ) = E ( r ) ( k ) [ E ( r ) ( k ) ]
S ̃ ( q ; k ) α ζ β ( k ) h ̃ ζ β ( q ; z ( p ) , k ) e i q ρ ( p ) ,
F ̃ β γ ( q ; z , k ) = 2 π i k A β γ ( q k ) s z ( q k ) e i k s z ( q k ) z
h ̃ ζ β ( q ; z ( p ) , k ) = 4 π 2 W κ γ ( k ) R 2 d 2 q A β γ ( q q k ) A ζ κ ( q k ) s z ( q q k ) s z ( q k ) × exp { i k [ s z ( q q k ) + s z ( q k ) ] z ( p ) } .
h ̃ ζ β ( q ; z ( p ) , k ) 4 π 2 n = 1 N exp { i k s z ( p ( q ; ζ , β , n ) k ) z ( p ) } × exp { i k s z ( q p ( q ; ζ , β , n ) k ) z ( p ) } W κ γ ( k ) × R ( q ; ζ , β , n ) d 2 q A β γ ( q q k ) A ζ κ ( q k ) s z ( q q k ) s z ( q k ) .
h ̃ ζ β ( q ; z ( p ) , k ) 4 π 2 exp { i k s z ( p ( q ; ζ , β , 1 ) k ) z ( p ) } × exp { i k s z ( q p ( q ; ζ , β , 1 ) k ) z ( p ) } × W κ γ ( k ) R 2 d 2 q A β γ ( q q k ) A ζ κ ( q k ) s z ( q q k ) s z ( q k ) ,
h ̃ ζ β ( q ; z ( p ) , k ) H ( q ; ζ , β , k ) e i k Φ ( q ; ζ , β , k ) z ( p ) ,
H ( q ; ζ , β , k ) = 4 π 2 W κ γ ( k ) R 2 d 2 q A β γ ( q q k ) A ζ κ ( q k ) s z ( q q k ) s z ( q k ) ,
Φ ( q ; ζ , β , k ) = s z ( q p ( q ; ζ , β , 1 ) k ) + s z ( p ( q ; ζ , β , 1 ) k ) .
C [ α ¯ ( k ) , r ( p ) ; k ] = S ̃ ( q ; k ) α ζ β ( k ) h ̃ ζ β ( q ; z ( p ) , k ) e i q ρ ( p ) 2
α ¯ = [ v ( j ) ] T c ( j ) v ( j ) .
α ¯ = [ 0.433 + 0.633 i 0.137 0.380 i 0.308 0.424 i 0.137 0.380 i 0.540 + 0.164 i 0.096 0.293 i 0.308 0.424 i 0.096 0.293 i 0.087 + 0.185 i ] ,
r ( p ) = λ [ 1.67 1.24 0.088 ] T .
α ¯ ̂ = [ 0.417 + 0.622 i 0.137 0.401 i 0.339 0.405 i 0.137 0.401 i 0.543 + 0.170 i 0.031 0.279 i 0.339 0.405 i 0.031 0.279 i 0.111 + 0.168 i ]
r ̂ = λ [ 1.66 1.25 0.085 ] T .

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