Abstract

Imaging molecular structures separated by distances of a few nanometers still represents a complex challenge. Moreover, it is normally restricted to observations on thin (few micrometers) samples. In this work, we rotate the polarization of the excitation beam of two-photon excited fluorescence (TPEF) images to show that fluorescent structures at the molecular scale can be discriminated in a living organism. The polarization rotation generates a modulation of the signal intensity in each pixel of the TPEF images that carry information related to the fluorophore orientation. We analyze the signal modulation in every pixel of the polarization-resolved (PR) TPEF images through a Fourier analysis and generate images for the different Fourier components. Doing that, we show that two fluorophores oriented in different directions can be distinguished. Although by imaging the Fourier components the resolution of the optical system restricts the exact localization of two close molecules, discrimination is still possible even when the molecules are located at sub-diffraction distances. We propose a model that predicts this behavior, and demonstrate it experimentally in the neurons of a living Caenorhabditis elegans nematode, where we distinguish the walls of an axon with a diameter below the objective resolution. Since the technique is based in TPEF, the method can be extended to deep tissue imaging and has potential applications in single molecule detection, biological sensors, or super-resolution imaging techniques.

© 2017 Optical Society of America

Full Article  |  PDF Article
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References

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    [Crossref]
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    [Crossref]

2016 (2)

C. A. V. Cruz, H. A. Shabana, A. Kressa, N. Bertauxa, S. Monnereta, M. Mavrakisa, J. Savatiera, and S. Brasselet, “Quantitative nanoscale imaging of orientational order in biological filaments by polarized superresolution microscopy,” Proc. Natl. Acad. Sci. USA 113, E820–E828 (2016).
[Crossref]

L. Frahm and J. Keller, “Polarization modulation adds little additional information to super-resolution fluorescence microscopy,” Nat. Methods 13, 7–8 (2016).
[Crossref]

2015 (1)

2014 (2)

N. Hafi, M. Grunwald, L. S. van den Heuvel, T. Aspelmeier, J. Chen, M. Zagrebelsky, O. M. Schütte, C. Steinem, M. Korte, A. Munk, and P. J. Walla, “Fluorescence nanoscopy by polarization modulation and polarization angle narrowing,” Nat. Methods 11, 579–584 (2014).
[Crossref]

S. Psilodimitrakopoulos, P. Loza-Alvarez, and D. Artigas, “Fast monitoring of in-vivo conformational changes in myosin using single scan polarization-SHG microscopy,” Biomed. Opt. Express 5, 4362–4373 (2014).
[Crossref]

2013 (3)

Y. Han, V. Raghunathan, R.-R. Feng, H. Maekawa, C.-Y. Chung, Y. Feng, E. O. Potma, and N.-H. Ge, “Mapping molecular orientation with phase sensitive vibrationally resonant sum-frequency generation microscopy,” J. Phys. Chem. B 117, 6149–6156 (2013).
[Crossref]

M. Zimmerley, P. Mahou, D. Débarre, M.-C. Schanne-Klein, and E. Beaurepaire, “Probing ordered lipid assemblies with polarized third-harmonic-generation microscopy,” Phys. Rev. X 3, 011002 (2013).

A. Kress, X. Wang, H. Ranchon, J. Savatier, H. Rigneault, P. Ferrand, and S. Brasselet, “Mapping the local organization of cell membranes using excitation-polarization-resolved confocal fluorescence microscopy,” Biophys. J. 105, 127–136 (2013).
[Crossref]

2012 (4)

A. Gasecka, P. Tauc, A. Bentley, and S. Brasselet, “Investigation of molecular and protein crystals by three photon polarization resolved microscopy,” Phys. Rev. Lett. 108, 263901 (2012).
[Crossref]

D. Aït-Belkacem, M. Guilbert, M. Roche, J. Duboisset, P. Ferrand, G. Sockalingum, P. Jeannesson, and S. Brasselet, “Microscopic structural study of collagen aging in isolated fibrils using polarized second harmonic generation,” J. Biomed. Opt. 17, 080506 (2012).
[Crossref]

S. Psilodimitrakopoulos, I. Amat-Roldan, P. Loza-Alvarez, and D. Artigas, “Effect of molecular organization on the image histograms of polarization SHG microscopy,” Biomed. Opt. Express 3, 2681–2693 (2012).
[Crossref]

T. Ansbacher, H. K. Srivastava, T. Stein, R. Baer, M. Merkxc, and A. Shurkiz, “Calculation of transition dipole moment in fluorescent proteins—towards efficient energy transfer,” Phys. Chem. Chem. Phys. 14, 4109–4117 (2012).
[Crossref]

2011 (2)

J. Lazar, A. Bondar, S. Timr, and S. J. Firestein, “Two-photon polarization microscopy reveals protein structure and function,” Nat. Methods. 8, 684–690 (2011).
[Crossref]

S. Brasselet, “Polarization-resolved nonlinear microscopy: application to structural molecular and biological imaging,” Adv. Opt. Photon. 3, 205–271 (2011).
[Crossref]

2010 (1)

2009 (4)

S. Psilodimitrakopoulos, S. I. C. O. Santos, I. Amat-Roldan, A. K. N. Thayil, D. Artigas, and P. Loza-Alvarez, “In vivo, pixel-resolution mapping of thick filaments’ orientation in nonfibrilar muscle using polarization-sensitive second harmonic generation microscopy,” J. Biomed. Opt. 14, 014001 (2009).
[Crossref]

S. Psilodimitrakopoulos, D. Artigas, G. Soria, I. Amat-Roldan, A. M. Planas, and P. Loza-Alvarez, “Quantitative discrimination between endogenous SHG sources in mammalian tissue, based on their polarization response,” Opt. Express 17, 10168–10176 (2009).
[Crossref]

A. Gasecka, T.-J. Han, C. Favard, B. R. Cho, and S. Brasselet, “Quantitative imaging of molecular order in lipid membranes using two-photon fluorescence polarimetry,” Biophys. J. 97, 2854–2862 (2009).
[Crossref]

J. A. Levitt, D. R. Matthews, S. M. Ameer-Beg, and K. S. Suhling, “Fluorescence lifetime and polarization-resolved imaging in cell biology,” Curr. Opin. Biotechnol. 20, 28–36 (2009).
[Crossref]

2008 (1)

K. A. Serrels, E. Ramsay, R. J. Warburton, and D. T. Reid, “Nanoscale optical microscopy in the vectorial focusing regime,” Nat. Photonics 2, 311–314 (2008).
[Crossref]

2007 (1)

2006 (2)

E. Yew and C. Sheppard, “Effects of axial field components on second harmonic generation microscopy,” Opt. Express 14, 1167–1174 (2006).
[Crossref]

A. M. Vrabioiu and T. J. Mitchison, “Structural insights into yeast septin organization from polarized fluorescence microscopy,” Nature 443, 466–469 (2006).
[Crossref]

2003 (1)

F. I. Rosell and S. G. Boxer, “Polarized absorption spectra of green fluorescent protein single crystals: transition dipole moment directions,” Biochemistry 42, 177–183 (2003).
[Crossref]

2002 (1)

S. Inoue, O. Shimomura, M. Goda, M. Shribak, and P. T. Tran, “Fluorescence polarization of green fluorescence protein,” Proc. Natl. Acad. Sci. USA 99, 4272–4277 (2002).
[Crossref]

2000 (1)

P. Stoller, B. M. Kim, A. M. Rubenchik, K. M. Reiser, and L. B. Da Silva, “Polarization-dependent optical second harmonic imaging of a rat-tail tendon,” J. Biomed. Opt. 7, 205–214 (2000).
[Crossref]

1999 (1)

T. Ha, T. A. Laurence, D. S. Chemla, and S. Weiss, “Polarization spectroscopy of single fluorescent molecules,” J. Phys. Chem. B 103, 6839–6850 (1999).
[Crossref]

1979 (1)

D. Axelrod, “Carbocyanine dye orientation in red-cell membrane studied by microscopic fluorescence polarization,” Biophys. J. 26, 557–573 (1979).
[Crossref]

1978 (1)

L. Stryer, “Fluorescence energy transfer as a spectroscopic ruler,” Annu. Rev. Biochem. 47, 819–846 (1978).
[Crossref]

1974 (1)

S. Brenner, “The genetics of Caenorhabditis elegans,” Genetics 77, 71–94 (1974).

1959 (1)

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

Aït-Belkacem, D.

D. Aït-Belkacem, M. Guilbert, M. Roche, J. Duboisset, P. Ferrand, G. Sockalingum, P. Jeannesson, and S. Brasselet, “Microscopic structural study of collagen aging in isolated fibrils using polarized second harmonic generation,” J. Biomed. Opt. 17, 080506 (2012).
[Crossref]

Amat-Roldan, I.

Ameer-Beg, S. M.

J. A. Levitt, D. R. Matthews, S. M. Ameer-Beg, and K. S. Suhling, “Fluorescence lifetime and polarization-resolved imaging in cell biology,” Curr. Opin. Biotechnol. 20, 28–36 (2009).
[Crossref]

Ansbacher, T.

T. Ansbacher, H. K. Srivastava, T. Stein, R. Baer, M. Merkxc, and A. Shurkiz, “Calculation of transition dipole moment in fluorescent proteins—towards efficient energy transfer,” Phys. Chem. Chem. Phys. 14, 4109–4117 (2012).
[Crossref]

Artigas, D.

Aspelmeier, T.

N. Hafi, M. Grunwald, L. S. van den Heuvel, T. Aspelmeier, J. Chen, M. Zagrebelsky, O. M. Schütte, C. Steinem, M. Korte, A. Munk, and P. J. Walla, “Fluorescence nanoscopy by polarization modulation and polarization angle narrowing,” Nat. Methods 11, 579–584 (2014).
[Crossref]

Axelrod, D.

D. Axelrod, “Carbocyanine dye orientation in red-cell membrane studied by microscopic fluorescence polarization,” Biophys. J. 26, 557–573 (1979).
[Crossref]

Baer, R.

T. Ansbacher, H. K. Srivastava, T. Stein, R. Baer, M. Merkxc, and A. Shurkiz, “Calculation of transition dipole moment in fluorescent proteins—towards efficient energy transfer,” Phys. Chem. Chem. Phys. 14, 4109–4117 (2012).
[Crossref]

Beaurepaire, E.

M. Zimmerley, P. Mahou, D. Débarre, M.-C. Schanne-Klein, and E. Beaurepaire, “Probing ordered lipid assemblies with polarized third-harmonic-generation microscopy,” Phys. Rev. X 3, 011002 (2013).

Bentley, A.

A. Gasecka, P. Tauc, A. Bentley, and S. Brasselet, “Investigation of molecular and protein crystals by three photon polarization resolved microscopy,” Phys. Rev. Lett. 108, 263901 (2012).
[Crossref]

Bertauxa, N.

C. A. V. Cruz, H. A. Shabana, A. Kressa, N. Bertauxa, S. Monnereta, M. Mavrakisa, J. Savatiera, and S. Brasselet, “Quantitative nanoscale imaging of orientational order in biological filaments by polarized superresolution microscopy,” Proc. Natl. Acad. Sci. USA 113, E820–E828 (2016).
[Crossref]

Bondar, A.

J. Lazar, A. Bondar, S. Timr, and S. J. Firestein, “Two-photon polarization microscopy reveals protein structure and function,” Nat. Methods. 8, 684–690 (2011).
[Crossref]

Boxer, S. G.

F. I. Rosell and S. G. Boxer, “Polarized absorption spectra of green fluorescent protein single crystals: transition dipole moment directions,” Biochemistry 42, 177–183 (2003).
[Crossref]

Brasselet, S.

C. A. V. Cruz, H. A. Shabana, A. Kressa, N. Bertauxa, S. Monnereta, M. Mavrakisa, J. Savatiera, and S. Brasselet, “Quantitative nanoscale imaging of orientational order in biological filaments by polarized superresolution microscopy,” Proc. Natl. Acad. Sci. USA 113, E820–E828 (2016).
[Crossref]

H. B. de Aguiar, P. Gasecka, and S. Brasselet, “Quantitative analysis of light scattering in polarization-resolved nonlinear microscopy,” Opt. Express 23, 8960–8973 (2015).
[Crossref]

A. Kress, X. Wang, H. Ranchon, J. Savatier, H. Rigneault, P. Ferrand, and S. Brasselet, “Mapping the local organization of cell membranes using excitation-polarization-resolved confocal fluorescence microscopy,” Biophys. J. 105, 127–136 (2013).
[Crossref]

A. Gasecka, P. Tauc, A. Bentley, and S. Brasselet, “Investigation of molecular and protein crystals by three photon polarization resolved microscopy,” Phys. Rev. Lett. 108, 263901 (2012).
[Crossref]

D. Aït-Belkacem, M. Guilbert, M. Roche, J. Duboisset, P. Ferrand, G. Sockalingum, P. Jeannesson, and S. Brasselet, “Microscopic structural study of collagen aging in isolated fibrils using polarized second harmonic generation,” J. Biomed. Opt. 17, 080506 (2012).
[Crossref]

S. Brasselet, “Polarization-resolved nonlinear microscopy: application to structural molecular and biological imaging,” Adv. Opt. Photon. 3, 205–271 (2011).
[Crossref]

A. Gasecka, T.-J. Han, C. Favard, B. R. Cho, and S. Brasselet, “Quantitative imaging of molecular order in lipid membranes using two-photon fluorescence polarimetry,” Biophys. J. 97, 2854–2862 (2009).
[Crossref]

Brenner, S.

S. Brenner, “The genetics of Caenorhabditis elegans,” Genetics 77, 71–94 (1974).

Chemla, D. S.

T. Ha, T. A. Laurence, D. S. Chemla, and S. Weiss, “Polarization spectroscopy of single fluorescent molecules,” J. Phys. Chem. B 103, 6839–6850 (1999).
[Crossref]

Chen, J.

N. Hafi, M. Grunwald, L. S. van den Heuvel, T. Aspelmeier, J. Chen, M. Zagrebelsky, O. M. Schütte, C. Steinem, M. Korte, A. Munk, and P. J. Walla, “Fluorescence nanoscopy by polarization modulation and polarization angle narrowing,” Nat. Methods 11, 579–584 (2014).
[Crossref]

Cho, B. R.

A. Gasecka, T.-J. Han, C. Favard, B. R. Cho, and S. Brasselet, “Quantitative imaging of molecular order in lipid membranes using two-photon fluorescence polarimetry,” Biophys. J. 97, 2854–2862 (2009).
[Crossref]

Chung, C.-Y.

Y. Han, V. Raghunathan, R.-R. Feng, H. Maekawa, C.-Y. Chung, Y. Feng, E. O. Potma, and N.-H. Ge, “Mapping molecular orientation with phase sensitive vibrationally resonant sum-frequency generation microscopy,” J. Phys. Chem. B 117, 6149–6156 (2013).
[Crossref]

Cruz, C. A. V.

C. A. V. Cruz, H. A. Shabana, A. Kressa, N. Bertauxa, S. Monnereta, M. Mavrakisa, J. Savatiera, and S. Brasselet, “Quantitative nanoscale imaging of orientational order in biological filaments by polarized superresolution microscopy,” Proc. Natl. Acad. Sci. USA 113, E820–E828 (2016).
[Crossref]

Da Silva, L. B.

P. Stoller, B. M. Kim, A. M. Rubenchik, K. M. Reiser, and L. B. Da Silva, “Polarization-dependent optical second harmonic imaging of a rat-tail tendon,” J. Biomed. Opt. 7, 205–214 (2000).
[Crossref]

de Aguiar, H. B.

Débarre, D.

M. Zimmerley, P. Mahou, D. Débarre, M.-C. Schanne-Klein, and E. Beaurepaire, “Probing ordered lipid assemblies with polarized third-harmonic-generation microscopy,” Phys. Rev. X 3, 011002 (2013).

Duboisset, J.

D. Aït-Belkacem, M. Guilbert, M. Roche, J. Duboisset, P. Ferrand, G. Sockalingum, P. Jeannesson, and S. Brasselet, “Microscopic structural study of collagen aging in isolated fibrils using polarized second harmonic generation,” J. Biomed. Opt. 17, 080506 (2012).
[Crossref]

Favard, C.

A. Gasecka, T.-J. Han, C. Favard, B. R. Cho, and S. Brasselet, “Quantitative imaging of molecular order in lipid membranes using two-photon fluorescence polarimetry,” Biophys. J. 97, 2854–2862 (2009).
[Crossref]

Feng, R.-R.

Y. Han, V. Raghunathan, R.-R. Feng, H. Maekawa, C.-Y. Chung, Y. Feng, E. O. Potma, and N.-H. Ge, “Mapping molecular orientation with phase sensitive vibrationally resonant sum-frequency generation microscopy,” J. Phys. Chem. B 117, 6149–6156 (2013).
[Crossref]

Feng, Y.

Y. Han, V. Raghunathan, R.-R. Feng, H. Maekawa, C.-Y. Chung, Y. Feng, E. O. Potma, and N.-H. Ge, “Mapping molecular orientation with phase sensitive vibrationally resonant sum-frequency generation microscopy,” J. Phys. Chem. B 117, 6149–6156 (2013).
[Crossref]

Ferrand, P.

A. Kress, X. Wang, H. Ranchon, J. Savatier, H. Rigneault, P. Ferrand, and S. Brasselet, “Mapping the local organization of cell membranes using excitation-polarization-resolved confocal fluorescence microscopy,” Biophys. J. 105, 127–136 (2013).
[Crossref]

D. Aït-Belkacem, M. Guilbert, M. Roche, J. Duboisset, P. Ferrand, G. Sockalingum, P. Jeannesson, and S. Brasselet, “Microscopic structural study of collagen aging in isolated fibrils using polarized second harmonic generation,” J. Biomed. Opt. 17, 080506 (2012).
[Crossref]

Firestein, S. J.

J. Lazar, A. Bondar, S. Timr, and S. J. Firestein, “Two-photon polarization microscopy reveals protein structure and function,” Nat. Methods. 8, 684–690 (2011).
[Crossref]

Frahm, L.

L. Frahm and J. Keller, “Polarization modulation adds little additional information to super-resolution fluorescence microscopy,” Nat. Methods 13, 7–8 (2016).
[Crossref]

Gasecka, A.

A. Gasecka, P. Tauc, A. Bentley, and S. Brasselet, “Investigation of molecular and protein crystals by three photon polarization resolved microscopy,” Phys. Rev. Lett. 108, 263901 (2012).
[Crossref]

A. Gasecka, T.-J. Han, C. Favard, B. R. Cho, and S. Brasselet, “Quantitative imaging of molecular order in lipid membranes using two-photon fluorescence polarimetry,” Biophys. J. 97, 2854–2862 (2009).
[Crossref]

Gasecka, P.

Ge, N.-H.

Y. Han, V. Raghunathan, R.-R. Feng, H. Maekawa, C.-Y. Chung, Y. Feng, E. O. Potma, and N.-H. Ge, “Mapping molecular orientation with phase sensitive vibrationally resonant sum-frequency generation microscopy,” J. Phys. Chem. B 117, 6149–6156 (2013).
[Crossref]

Goda, M.

S. Inoue, O. Shimomura, M. Goda, M. Shribak, and P. T. Tran, “Fluorescence polarization of green fluorescence protein,” Proc. Natl. Acad. Sci. USA 99, 4272–4277 (2002).
[Crossref]

Grunwald, M.

N. Hafi, M. Grunwald, L. S. van den Heuvel, T. Aspelmeier, J. Chen, M. Zagrebelsky, O. M. Schütte, C. Steinem, M. Korte, A. Munk, and P. J. Walla, “Fluorescence nanoscopy by polarization modulation and polarization angle narrowing,” Nat. Methods 11, 579–584 (2014).
[Crossref]

Guilbert, M.

D. Aït-Belkacem, M. Guilbert, M. Roche, J. Duboisset, P. Ferrand, G. Sockalingum, P. Jeannesson, and S. Brasselet, “Microscopic structural study of collagen aging in isolated fibrils using polarized second harmonic generation,” J. Biomed. Opt. 17, 080506 (2012).
[Crossref]

Ha, T.

T. Ha, T. A. Laurence, D. S. Chemla, and S. Weiss, “Polarization spectroscopy of single fluorescent molecules,” J. Phys. Chem. B 103, 6839–6850 (1999).
[Crossref]

Hafi, N.

N. Hafi, M. Grunwald, L. S. van den Heuvel, T. Aspelmeier, J. Chen, M. Zagrebelsky, O. M. Schütte, C. Steinem, M. Korte, A. Munk, and P. J. Walla, “Fluorescence nanoscopy by polarization modulation and polarization angle narrowing,” Nat. Methods 11, 579–584 (2014).
[Crossref]

Han, T.-J.

A. Gasecka, T.-J. Han, C. Favard, B. R. Cho, and S. Brasselet, “Quantitative imaging of molecular order in lipid membranes using two-photon fluorescence polarimetry,” Biophys. J. 97, 2854–2862 (2009).
[Crossref]

Han, Y.

Y. Han, V. Raghunathan, R.-R. Feng, H. Maekawa, C.-Y. Chung, Y. Feng, E. O. Potma, and N.-H. Ge, “Mapping molecular orientation with phase sensitive vibrationally resonant sum-frequency generation microscopy,” J. Phys. Chem. B 117, 6149–6156 (2013).
[Crossref]

Inoue, S.

S. Inoue, O. Shimomura, M. Goda, M. Shribak, and P. T. Tran, “Fluorescence polarization of green fluorescence protein,” Proc. Natl. Acad. Sci. USA 99, 4272–4277 (2002).
[Crossref]

Jeannesson, P.

D. Aït-Belkacem, M. Guilbert, M. Roche, J. Duboisset, P. Ferrand, G. Sockalingum, P. Jeannesson, and S. Brasselet, “Microscopic structural study of collagen aging in isolated fibrils using polarized second harmonic generation,” J. Biomed. Opt. 17, 080506 (2012).
[Crossref]

Keller, J.

L. Frahm and J. Keller, “Polarization modulation adds little additional information to super-resolution fluorescence microscopy,” Nat. Methods 13, 7–8 (2016).
[Crossref]

Kim, B. M.

P. Stoller, B. M. Kim, A. M. Rubenchik, K. M. Reiser, and L. B. Da Silva, “Polarization-dependent optical second harmonic imaging of a rat-tail tendon,” J. Biomed. Opt. 7, 205–214 (2000).
[Crossref]

Korte, M.

N. Hafi, M. Grunwald, L. S. van den Heuvel, T. Aspelmeier, J. Chen, M. Zagrebelsky, O. M. Schütte, C. Steinem, M. Korte, A. Munk, and P. J. Walla, “Fluorescence nanoscopy by polarization modulation and polarization angle narrowing,” Nat. Methods 11, 579–584 (2014).
[Crossref]

Kress, A.

A. Kress, X. Wang, H. Ranchon, J. Savatier, H. Rigneault, P. Ferrand, and S. Brasselet, “Mapping the local organization of cell membranes using excitation-polarization-resolved confocal fluorescence microscopy,” Biophys. J. 105, 127–136 (2013).
[Crossref]

Kressa, A.

C. A. V. Cruz, H. A. Shabana, A. Kressa, N. Bertauxa, S. Monnereta, M. Mavrakisa, J. Savatiera, and S. Brasselet, “Quantitative nanoscale imaging of orientational order in biological filaments by polarized superresolution microscopy,” Proc. Natl. Acad. Sci. USA 113, E820–E828 (2016).
[Crossref]

Laurence, T. A.

T. Ha, T. A. Laurence, D. S. Chemla, and S. Weiss, “Polarization spectroscopy of single fluorescent molecules,” J. Phys. Chem. B 103, 6839–6850 (1999).
[Crossref]

Lazar, J.

J. Lazar, A. Bondar, S. Timr, and S. J. Firestein, “Two-photon polarization microscopy reveals protein structure and function,” Nat. Methods. 8, 684–690 (2011).
[Crossref]

Levitt, J. A.

J. A. Levitt, D. R. Matthews, S. M. Ameer-Beg, and K. S. Suhling, “Fluorescence lifetime and polarization-resolved imaging in cell biology,” Curr. Opin. Biotechnol. 20, 28–36 (2009).
[Crossref]

Loza-Alvarez, P.

Maekawa, H.

Y. Han, V. Raghunathan, R.-R. Feng, H. Maekawa, C.-Y. Chung, Y. Feng, E. O. Potma, and N.-H. Ge, “Mapping molecular orientation with phase sensitive vibrationally resonant sum-frequency generation microscopy,” J. Phys. Chem. B 117, 6149–6156 (2013).
[Crossref]

Mahou, P.

M. Zimmerley, P. Mahou, D. Débarre, M.-C. Schanne-Klein, and E. Beaurepaire, “Probing ordered lipid assemblies with polarized third-harmonic-generation microscopy,” Phys. Rev. X 3, 011002 (2013).

Matthews, D. R.

J. A. Levitt, D. R. Matthews, S. M. Ameer-Beg, and K. S. Suhling, “Fluorescence lifetime and polarization-resolved imaging in cell biology,” Curr. Opin. Biotechnol. 20, 28–36 (2009).
[Crossref]

Mavrakisa, M.

C. A. V. Cruz, H. A. Shabana, A. Kressa, N. Bertauxa, S. Monnereta, M. Mavrakisa, J. Savatiera, and S. Brasselet, “Quantitative nanoscale imaging of orientational order in biological filaments by polarized superresolution microscopy,” Proc. Natl. Acad. Sci. USA 113, E820–E828 (2016).
[Crossref]

Merkxc, M.

T. Ansbacher, H. K. Srivastava, T. Stein, R. Baer, M. Merkxc, and A. Shurkiz, “Calculation of transition dipole moment in fluorescent proteins—towards efficient energy transfer,” Phys. Chem. Chem. Phys. 14, 4109–4117 (2012).
[Crossref]

Mitchison, T. J.

A. M. Vrabioiu and T. J. Mitchison, “Structural insights into yeast septin organization from polarized fluorescence microscopy,” Nature 443, 466–469 (2006).
[Crossref]

Monnereta, S.

C. A. V. Cruz, H. A. Shabana, A. Kressa, N. Bertauxa, S. Monnereta, M. Mavrakisa, J. Savatiera, and S. Brasselet, “Quantitative nanoscale imaging of orientational order in biological filaments by polarized superresolution microscopy,” Proc. Natl. Acad. Sci. USA 113, E820–E828 (2016).
[Crossref]

Munk, A.

N. Hafi, M. Grunwald, L. S. van den Heuvel, T. Aspelmeier, J. Chen, M. Zagrebelsky, O. M. Schütte, C. Steinem, M. Korte, A. Munk, and P. J. Walla, “Fluorescence nanoscopy by polarization modulation and polarization angle narrowing,” Nat. Methods 11, 579–584 (2014).
[Crossref]

Planas, A. M.

Potma, E. O.

Y. Han, V. Raghunathan, R.-R. Feng, H. Maekawa, C.-Y. Chung, Y. Feng, E. O. Potma, and N.-H. Ge, “Mapping molecular orientation with phase sensitive vibrationally resonant sum-frequency generation microscopy,” J. Phys. Chem. B 117, 6149–6156 (2013).
[Crossref]

Psilodimitrakopoulos, S.

Raghunathan, V.

Y. Han, V. Raghunathan, R.-R. Feng, H. Maekawa, C.-Y. Chung, Y. Feng, E. O. Potma, and N.-H. Ge, “Mapping molecular orientation with phase sensitive vibrationally resonant sum-frequency generation microscopy,” J. Phys. Chem. B 117, 6149–6156 (2013).
[Crossref]

Ramsay, E.

K. A. Serrels, E. Ramsay, R. J. Warburton, and D. T. Reid, “Nanoscale optical microscopy in the vectorial focusing regime,” Nat. Photonics 2, 311–314 (2008).
[Crossref]

Ranchon, H.

A. Kress, X. Wang, H. Ranchon, J. Savatier, H. Rigneault, P. Ferrand, and S. Brasselet, “Mapping the local organization of cell membranes using excitation-polarization-resolved confocal fluorescence microscopy,” Biophys. J. 105, 127–136 (2013).
[Crossref]

Recher, G.

Reid, D. T.

K. A. Serrels, E. Ramsay, R. J. Warburton, and D. T. Reid, “Nanoscale optical microscopy in the vectorial focusing regime,” Nat. Photonics 2, 311–314 (2008).
[Crossref]

Reiser, K. M.

P. Stoller, B. M. Kim, A. M. Rubenchik, K. M. Reiser, and L. B. Da Silva, “Polarization-dependent optical second harmonic imaging of a rat-tail tendon,” J. Biomed. Opt. 7, 205–214 (2000).
[Crossref]

Richards, B.

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

Rigneault, H.

A. Kress, X. Wang, H. Ranchon, J. Savatier, H. Rigneault, P. Ferrand, and S. Brasselet, “Mapping the local organization of cell membranes using excitation-polarization-resolved confocal fluorescence microscopy,” Biophys. J. 105, 127–136 (2013).
[Crossref]

Roche, M.

D. Aït-Belkacem, M. Guilbert, M. Roche, J. Duboisset, P. Ferrand, G. Sockalingum, P. Jeannesson, and S. Brasselet, “Microscopic structural study of collagen aging in isolated fibrils using polarized second harmonic generation,” J. Biomed. Opt. 17, 080506 (2012).
[Crossref]

Rosell, F. I.

F. I. Rosell and S. G. Boxer, “Polarized absorption spectra of green fluorescent protein single crystals: transition dipole moment directions,” Biochemistry 42, 177–183 (2003).
[Crossref]

Rouede, D.

Rubenchik, A. M.

P. Stoller, B. M. Kim, A. M. Rubenchik, K. M. Reiser, and L. B. Da Silva, “Polarization-dependent optical second harmonic imaging of a rat-tail tendon,” J. Biomed. Opt. 7, 205–214 (2000).
[Crossref]

Santos, S. I. C. O.

S. Psilodimitrakopoulos, S. I. C. O. Santos, I. Amat-Roldan, A. K. N. Thayil, D. Artigas, and P. Loza-Alvarez, “In vivo, pixel-resolution mapping of thick filaments’ orientation in nonfibrilar muscle using polarization-sensitive second harmonic generation microscopy,” J. Biomed. Opt. 14, 014001 (2009).
[Crossref]

Savatier, J.

A. Kress, X. Wang, H. Ranchon, J. Savatier, H. Rigneault, P. Ferrand, and S. Brasselet, “Mapping the local organization of cell membranes using excitation-polarization-resolved confocal fluorescence microscopy,” Biophys. J. 105, 127–136 (2013).
[Crossref]

Savatiera, J.

C. A. V. Cruz, H. A. Shabana, A. Kressa, N. Bertauxa, S. Monnereta, M. Mavrakisa, J. Savatiera, and S. Brasselet, “Quantitative nanoscale imaging of orientational order in biological filaments by polarized superresolution microscopy,” Proc. Natl. Acad. Sci. USA 113, E820–E828 (2016).
[Crossref]

Schanne-Klein, M.-C.

M. Zimmerley, P. Mahou, D. Débarre, M.-C. Schanne-Klein, and E. Beaurepaire, “Probing ordered lipid assemblies with polarized third-harmonic-generation microscopy,” Phys. Rev. X 3, 011002 (2013).

Schütte, O. M.

N. Hafi, M. Grunwald, L. S. van den Heuvel, T. Aspelmeier, J. Chen, M. Zagrebelsky, O. M. Schütte, C. Steinem, M. Korte, A. Munk, and P. J. Walla, “Fluorescence nanoscopy by polarization modulation and polarization angle narrowing,” Nat. Methods 11, 579–584 (2014).
[Crossref]

Serrels, K. A.

K. A. Serrels, E. Ramsay, R. J. Warburton, and D. T. Reid, “Nanoscale optical microscopy in the vectorial focusing regime,” Nat. Photonics 2, 311–314 (2008).
[Crossref]

Shabana, H. A.

C. A. V. Cruz, H. A. Shabana, A. Kressa, N. Bertauxa, S. Monnereta, M. Mavrakisa, J. Savatiera, and S. Brasselet, “Quantitative nanoscale imaging of orientational order in biological filaments by polarized superresolution microscopy,” Proc. Natl. Acad. Sci. USA 113, E820–E828 (2016).
[Crossref]

Sheppard, C.

Shimomura, O.

S. Inoue, O. Shimomura, M. Goda, M. Shribak, and P. T. Tran, “Fluorescence polarization of green fluorescence protein,” Proc. Natl. Acad. Sci. USA 99, 4272–4277 (2002).
[Crossref]

Shribak, M.

S. Inoue, O. Shimomura, M. Goda, M. Shribak, and P. T. Tran, “Fluorescence polarization of green fluorescence protein,” Proc. Natl. Acad. Sci. USA 99, 4272–4277 (2002).
[Crossref]

Shurkiz, A.

T. Ansbacher, H. K. Srivastava, T. Stein, R. Baer, M. Merkxc, and A. Shurkiz, “Calculation of transition dipole moment in fluorescent proteins—towards efficient energy transfer,” Phys. Chem. Chem. Phys. 14, 4109–4117 (2012).
[Crossref]

Sockalingum, G.

D. Aït-Belkacem, M. Guilbert, M. Roche, J. Duboisset, P. Ferrand, G. Sockalingum, P. Jeannesson, and S. Brasselet, “Microscopic structural study of collagen aging in isolated fibrils using polarized second harmonic generation,” J. Biomed. Opt. 17, 080506 (2012).
[Crossref]

Soria, G.

Srivastava, H. K.

T. Ansbacher, H. K. Srivastava, T. Stein, R. Baer, M. Merkxc, and A. Shurkiz, “Calculation of transition dipole moment in fluorescent proteins—towards efficient energy transfer,” Phys. Chem. Chem. Phys. 14, 4109–4117 (2012).
[Crossref]

Stein, T.

T. Ansbacher, H. K. Srivastava, T. Stein, R. Baer, M. Merkxc, and A. Shurkiz, “Calculation of transition dipole moment in fluorescent proteins—towards efficient energy transfer,” Phys. Chem. Chem. Phys. 14, 4109–4117 (2012).
[Crossref]

Steinem, C.

N. Hafi, M. Grunwald, L. S. van den Heuvel, T. Aspelmeier, J. Chen, M. Zagrebelsky, O. M. Schütte, C. Steinem, M. Korte, A. Munk, and P. J. Walla, “Fluorescence nanoscopy by polarization modulation and polarization angle narrowing,” Nat. Methods 11, 579–584 (2014).
[Crossref]

Stoller, P.

P. Stoller, B. M. Kim, A. M. Rubenchik, K. M. Reiser, and L. B. Da Silva, “Polarization-dependent optical second harmonic imaging of a rat-tail tendon,” J. Biomed. Opt. 7, 205–214 (2000).
[Crossref]

Stryer, L.

L. Stryer, “Fluorescence energy transfer as a spectroscopic ruler,” Annu. Rev. Biochem. 47, 819–846 (1978).
[Crossref]

Suhling, K. S.

J. A. Levitt, D. R. Matthews, S. M. Ameer-Beg, and K. S. Suhling, “Fluorescence lifetime and polarization-resolved imaging in cell biology,” Curr. Opin. Biotechnol. 20, 28–36 (2009).
[Crossref]

Tauc, P.

A. Gasecka, P. Tauc, A. Bentley, and S. Brasselet, “Investigation of molecular and protein crystals by three photon polarization resolved microscopy,” Phys. Rev. Lett. 108, 263901 (2012).
[Crossref]

Thayil, A. K. N.

S. Psilodimitrakopoulos, S. I. C. O. Santos, I. Amat-Roldan, A. K. N. Thayil, D. Artigas, and P. Loza-Alvarez, “In vivo, pixel-resolution mapping of thick filaments’ orientation in nonfibrilar muscle using polarization-sensitive second harmonic generation microscopy,” J. Biomed. Opt. 14, 014001 (2009).
[Crossref]

Tiaho, F.

Timr, S.

J. Lazar, A. Bondar, S. Timr, and S. J. Firestein, “Two-photon polarization microscopy reveals protein structure and function,” Nat. Methods. 8, 684–690 (2011).
[Crossref]

Tran, P. T.

S. Inoue, O. Shimomura, M. Goda, M. Shribak, and P. T. Tran, “Fluorescence polarization of green fluorescence protein,” Proc. Natl. Acad. Sci. USA 99, 4272–4277 (2002).
[Crossref]

van den Heuvel, L. S.

N. Hafi, M. Grunwald, L. S. van den Heuvel, T. Aspelmeier, J. Chen, M. Zagrebelsky, O. M. Schütte, C. Steinem, M. Korte, A. Munk, and P. J. Walla, “Fluorescence nanoscopy by polarization modulation and polarization angle narrowing,” Nat. Methods 11, 579–584 (2014).
[Crossref]

Vrabioiu, A. M.

A. M. Vrabioiu and T. J. Mitchison, “Structural insights into yeast septin organization from polarized fluorescence microscopy,” Nature 443, 466–469 (2006).
[Crossref]

Walla, P. J.

N. Hafi, M. Grunwald, L. S. van den Heuvel, T. Aspelmeier, J. Chen, M. Zagrebelsky, O. M. Schütte, C. Steinem, M. Korte, A. Munk, and P. J. Walla, “Fluorescence nanoscopy by polarization modulation and polarization angle narrowing,” Nat. Methods 11, 579–584 (2014).
[Crossref]

Wang, X.

A. Kress, X. Wang, H. Ranchon, J. Savatier, H. Rigneault, P. Ferrand, and S. Brasselet, “Mapping the local organization of cell membranes using excitation-polarization-resolved confocal fluorescence microscopy,” Biophys. J. 105, 127–136 (2013).
[Crossref]

Warburton, R. J.

K. A. Serrels, E. Ramsay, R. J. Warburton, and D. T. Reid, “Nanoscale optical microscopy in the vectorial focusing regime,” Nat. Photonics 2, 311–314 (2008).
[Crossref]

Weiss, S.

T. Ha, T. A. Laurence, D. S. Chemla, and S. Weiss, “Polarization spectroscopy of single fluorescent molecules,” J. Phys. Chem. B 103, 6839–6850 (1999).
[Crossref]

Wolf, E.

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

Yew, E.

Zagrebelsky, M.

N. Hafi, M. Grunwald, L. S. van den Heuvel, T. Aspelmeier, J. Chen, M. Zagrebelsky, O. M. Schütte, C. Steinem, M. Korte, A. Munk, and P. J. Walla, “Fluorescence nanoscopy by polarization modulation and polarization angle narrowing,” Nat. Methods 11, 579–584 (2014).
[Crossref]

Zimmerley, M.

M. Zimmerley, P. Mahou, D. Débarre, M.-C. Schanne-Klein, and E. Beaurepaire, “Probing ordered lipid assemblies with polarized third-harmonic-generation microscopy,” Phys. Rev. X 3, 011002 (2013).

Adv. Opt. Photon. (1)

Annu. Rev. Biochem. (1)

L. Stryer, “Fluorescence energy transfer as a spectroscopic ruler,” Annu. Rev. Biochem. 47, 819–846 (1978).
[Crossref]

Biochemistry (1)

F. I. Rosell and S. G. Boxer, “Polarized absorption spectra of green fluorescent protein single crystals: transition dipole moment directions,” Biochemistry 42, 177–183 (2003).
[Crossref]

Biomed. Opt. Express (2)

Biophys. J. (3)

D. Axelrod, “Carbocyanine dye orientation in red-cell membrane studied by microscopic fluorescence polarization,” Biophys. J. 26, 557–573 (1979).
[Crossref]

A. Kress, X. Wang, H. Ranchon, J. Savatier, H. Rigneault, P. Ferrand, and S. Brasselet, “Mapping the local organization of cell membranes using excitation-polarization-resolved confocal fluorescence microscopy,” Biophys. J. 105, 127–136 (2013).
[Crossref]

A. Gasecka, T.-J. Han, C. Favard, B. R. Cho, and S. Brasselet, “Quantitative imaging of molecular order in lipid membranes using two-photon fluorescence polarimetry,” Biophys. J. 97, 2854–2862 (2009).
[Crossref]

Curr. Opin. Biotechnol. (1)

J. A. Levitt, D. R. Matthews, S. M. Ameer-Beg, and K. S. Suhling, “Fluorescence lifetime and polarization-resolved imaging in cell biology,” Curr. Opin. Biotechnol. 20, 28–36 (2009).
[Crossref]

Genetics (1)

S. Brenner, “The genetics of Caenorhabditis elegans,” Genetics 77, 71–94 (1974).

J. Biomed. Opt. (3)

S. Psilodimitrakopoulos, S. I. C. O. Santos, I. Amat-Roldan, A. K. N. Thayil, D. Artigas, and P. Loza-Alvarez, “In vivo, pixel-resolution mapping of thick filaments’ orientation in nonfibrilar muscle using polarization-sensitive second harmonic generation microscopy,” J. Biomed. Opt. 14, 014001 (2009).
[Crossref]

D. Aït-Belkacem, M. Guilbert, M. Roche, J. Duboisset, P. Ferrand, G. Sockalingum, P. Jeannesson, and S. Brasselet, “Microscopic structural study of collagen aging in isolated fibrils using polarized second harmonic generation,” J. Biomed. Opt. 17, 080506 (2012).
[Crossref]

P. Stoller, B. M. Kim, A. M. Rubenchik, K. M. Reiser, and L. B. Da Silva, “Polarization-dependent optical second harmonic imaging of a rat-tail tendon,” J. Biomed. Opt. 7, 205–214 (2000).
[Crossref]

J. Phys. Chem. B (2)

T. Ha, T. A. Laurence, D. S. Chemla, and S. Weiss, “Polarization spectroscopy of single fluorescent molecules,” J. Phys. Chem. B 103, 6839–6850 (1999).
[Crossref]

Y. Han, V. Raghunathan, R.-R. Feng, H. Maekawa, C.-Y. Chung, Y. Feng, E. O. Potma, and N.-H. Ge, “Mapping molecular orientation with phase sensitive vibrationally resonant sum-frequency generation microscopy,” J. Phys. Chem. B 117, 6149–6156 (2013).
[Crossref]

Nat. Methods (2)

N. Hafi, M. Grunwald, L. S. van den Heuvel, T. Aspelmeier, J. Chen, M. Zagrebelsky, O. M. Schütte, C. Steinem, M. Korte, A. Munk, and P. J. Walla, “Fluorescence nanoscopy by polarization modulation and polarization angle narrowing,” Nat. Methods 11, 579–584 (2014).
[Crossref]

L. Frahm and J. Keller, “Polarization modulation adds little additional information to super-resolution fluorescence microscopy,” Nat. Methods 13, 7–8 (2016).
[Crossref]

Nat. Methods. (1)

J. Lazar, A. Bondar, S. Timr, and S. J. Firestein, “Two-photon polarization microscopy reveals protein structure and function,” Nat. Methods. 8, 684–690 (2011).
[Crossref]

Nat. Photonics (1)

K. A. Serrels, E. Ramsay, R. J. Warburton, and D. T. Reid, “Nanoscale optical microscopy in the vectorial focusing regime,” Nat. Photonics 2, 311–314 (2008).
[Crossref]

Nature (1)

A. M. Vrabioiu and T. J. Mitchison, “Structural insights into yeast septin organization from polarized fluorescence microscopy,” Nature 443, 466–469 (2006).
[Crossref]

Opt. Express (5)

Phys. Chem. Chem. Phys. (1)

T. Ansbacher, H. K. Srivastava, T. Stein, R. Baer, M. Merkxc, and A. Shurkiz, “Calculation of transition dipole moment in fluorescent proteins—towards efficient energy transfer,” Phys. Chem. Chem. Phys. 14, 4109–4117 (2012).
[Crossref]

Phys. Rev. Lett. (1)

A. Gasecka, P. Tauc, A. Bentley, and S. Brasselet, “Investigation of molecular and protein crystals by three photon polarization resolved microscopy,” Phys. Rev. Lett. 108, 263901 (2012).
[Crossref]

Phys. Rev. X (1)

M. Zimmerley, P. Mahou, D. Débarre, M.-C. Schanne-Klein, and E. Beaurepaire, “Probing ordered lipid assemblies with polarized third-harmonic-generation microscopy,” Phys. Rev. X 3, 011002 (2013).

Proc. Natl. Acad. Sci. USA (2)

C. A. V. Cruz, H. A. Shabana, A. Kressa, N. Bertauxa, S. Monnereta, M. Mavrakisa, J. Savatiera, and S. Brasselet, “Quantitative nanoscale imaging of orientational order in biological filaments by polarized superresolution microscopy,” Proc. Natl. Acad. Sci. USA 113, E820–E828 (2016).
[Crossref]

S. Inoue, O. Shimomura, M. Goda, M. Shribak, and P. T. Tran, “Fluorescence polarization of green fluorescence protein,” Proc. Natl. Acad. Sci. USA 99, 4272–4277 (2002).
[Crossref]

Proc. R. Soc. London A (1)

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

Other (1)

http://www.wormatlas.org/hermaphrodite/nervous/Neuroframeset.html .

Supplementary Material (1)

NameDescription
» Visualization 1       Change in fluorescence intensity due to the rotation of the polarization in the axon of a living C. Elegans nematode.

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Figures (8)

Fig. 1.
Fig. 1.

(a) Coordinate system and angles defining the orientation of the transition dipole moment. (b) Intensity of the electric field at the focal plane of an objective with NA=1.4. The polarization at the objective back aperture is linear and oriented in the x direction. Local amplitudes for the electric field components are (c) Ex, (d) Ey, and (e) Ez.

Fig. 2.
Fig. 2.

(a) Images of a single fluorophore with μabs oriented in the x vertical direction (θ=0°) obtained from Eq. (3) for 8 input polarizations α. (b) Same as (a) but with μabs out of plane (θ=45°,ϕ=90°). The normalized values of the maximum intensities are shown in the bottom-left corner in each panel. The normalization was performed with respect to the first panel in (a), where Imax=1. Objective with NA=1.4 and λ=810  nm. The scanning step is 32 nm, resulting in ΔV=32  nm×32  nm×32  nm. The scale bar corresponds to 500 nm.

Fig. 3.
Fig. 3.

(a) PR images corresponding to two molecules at the focal plane. The distance between molecules is d=290  nm, and μabs is oriented in the vertical direction (θ=0° and Δθ=0°) in both molecules, and (b) PR analysis with the images for the four first Fourier components fp=0, 1, 2, and 3. The bar corresponds to 500 nm. Wavelength, NA, and scanning step are the same as Fig. 2.

Fig. 4.
Fig. 4.

(a) PR images (left column) and PR analysis with the images for the three Fourier components fp=0, 1 and 2 (right column) corresponding to two molecules at the focal plane. The distance between molecules is d=290  nm. The μabs orientation for the right and left molecule are θ=45°, ϕ=0° and θ=45°, ϕ=180°, respectively, resulting in Δθ=90° in both molecules, as shown by the arrows in the top right corner. (b) PR analysis corresponding to case (a) but decreasing the distance between molecules to d=32  nm (one scanning step). (c) Same as (b) but with μabs oriented parallel to each molecule (Δθ=0°, as shown by the arrows at the top right corner). Wavelength, NA, and scanning step are the same as Fig. 2. The bar corresponds to 500 nm.

Fig. 5.
Fig. 5.

Images for the Fourier components of two molecules with a separation distance of d=32  nm, with μabs laying in the focal plane with a different orientation, denoted by the angle Δθ. Fourier component images for Δθ<35° result in a single spot and are not shown. Wavelength, NA, and scanning step are the same than Fig. 2. Bar corresponds to 500 nm.

Fig. 6.
Fig. 6.

(a) Lab coordinate system with the definition of the angles for two molecules. μabs is the plot in blue and the corresponding projection on the sample plane in grey. In this case Δθ indicates the apparent angle between the projected vectors. (b) Images for the Fourier components fp=1 and 2 for a distance between molecules of d=32  nm and θ=50° (initially Δθ=100°), with a changing angle ϕ (out of the plane). Wavelength, NA, and scanning step are the same as Fig. 2. Bar corresponds to 500 nm.

Fig. 7.
Fig. 7.

(a) Image of a neuron in a living C. elegans nematode using a single polarization, in this case, in the vertical direction (0°). Visualization 1 shows the images for the 9 polarizations α. Images for the Fourier components, (b) fp=0, (c) fp=1, and (d) fp=2, are calculated from the experimental images in the Supplementary Video. The insets show a 3×magnification of the area with the red square. Bar corresponds to 5 μm.

Fig. 8.
Fig. 8.

Numerical images synthesized for two possible probability distributions: (a) conical constant-filled distribution centered at θ=60° and conus aperture θ=20°±5°; (b) conical empty distribution centered at θ=62° with an internal conus aperture (inner wall) of θ=14°±2° and an external conus aperture (outer wall) of θ=20°±2°. (c) Scheme of the lab coordinate system, with the axon wall (light brown cylinder) and two conical distributions: filled and empty blue conus on the right and left side of the cylinder, respectively.

Equations (4)

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PI(r)NNAVΩ|μabs(r,Ω)·Eexc(rr)|4|Erad(r,κ^,Ω)|2f(r,Ω)dΩdVdκ^.
Erad(r,κ^)κ^×[κ^×μem(r)],
PI(r)κ^jri|μabs(ri)·Eexc(rirk)|4|Erad(ri,κ^j)|2ΔVΔκ^.
PF(x,y,fp)=FT{PI(x,y,α}.