Single-fluorophore orientation determination with multiview polarized illumination: modeling and microscope design

Talon Chandler; Shalin Mehta; Hari Shroff; Rudolf Oldenbourg; Patrick J. La Rivière

doi:10.1364/OE.25.031309

Single-fluorophore orientation determination with multiview polarized illumination: modeling and microscope design

Talon Chandler, Shalin Mehta, Hari Shroff, Rudolf Oldenbourg, and Patrick J. La Rivière

Optics Express
Vol. 25,
Issue 25,
pp. 31309-31325
(2017)
•https://doi.org/10.1364/OE.25.031309

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Talon Chandler, Shalin Mehta, Hari Shroff, Rudolf Oldenbourg, and Patrick J. La Rivière, "Single-fluorophore orientation determination with multiview polarized illumination: modeling and microscope design," Opt. Express 25, 31309-31325 (2017)

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Related Topics
Table of Contents Category
- Microscopy
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Imaging systems (110.0110)
- Polarization-selective devices (130.5440)
- Microscopy (180.0180)
- Fluorescence microscopy (180.2520)
- Three-dimensional microscopy (180.6900)
- Polarization (260.5430)

About this Article
History
- Original Manuscript: September 29, 2017
- Revised Manuscript: November 13, 2017
- Manuscript Accepted: November 23, 2017
- Published: December 1, 2017
Virtual Issues
Virtual Journal for Biomedical Optics Vol. 13, Iss. 2

Accessible

Open Access

Abstract

We investigate the use of polarized illumination in multiview microscopes for determining the orientation of single-molecule fluorescence transition dipoles. First, we relate the orientation of single dipoles to measurable intensities in multiview microscopes and develop an information-theoretic metric—the solid-angle uncertainty—to compare the ability of multiview microscopes to estimate the orientation of single dipoles. Next, we compare a broad class of microscopes using this metric—single- and dual-view microscopes with varying illumination polarization, illumination numerical aperture (NA), detection NA, obliquity, asymmetry, and exposure. We find that multi-view microscopes can measure all dipole orientations, while the orientations measurable with single-view microscopes is halved because of symmetries in the detection process. We also find that choosing a small illumination NA and a large detection NA are good design choices, that multiview microscopes can benefit from oblique illumination and detection, and that asymmetric NA microscopes can benefit from exposure asymmetry.

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References

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E. J. Peterman, H. Sosa, L. S. Goldstein, and W. E. Moerner, “Polarized fluorescence microscopy of individual and many kinesin motors bound to axonemal microtubules,” Biophys. J. 81, 2851–2863 (2001).
[Crossref] [PubMed]
J. N. Forkey, M. E. Quinlan, M. A. Shaw, J. E. T. Corrie, and Y. E. Goldman, “Three-dimensional structural dynamics of myosin V by single-molecule fluorescence polarization,” Nature 422, 399–404 (2003).
[Crossref] [PubMed]
A. S. Backer, M. Lee, and W. E. Moerner, “Enhanced DNA imaging using super-resolution microscopy and simultaneous single-molecule orientation measurements,” in “Conference on Lasers and Electro-Optics,” (Optical Society of America, 2016), p. JTh4B.4.
S. B. Mehta, M. McQuilken, P. J. La Rivière, P. Occhipinti, A. Verma, R. Oldenbourg, A. S. Gladfelter, and T. Tani, “Dissection of molecular assembly dynamics by tracking orientation and position of single molecules in live cells,” Proc. Natl. Acad. Sci. U.S.A. 113, E6352–E6361 (2016).
[Crossref] [PubMed]
B. DeMay, N. Noda, A. Gladfelter, and R. Oldenbourg, “Rapid and quantitative imaging of excitation polarized fluorescence reveals ordered septin dynamics in live yeast,” Biophys. J. 101, 985–994 (2011).
[Crossref] [PubMed]
M. McQuilken, M. S. Jentzsch, A. Verma, S. B. Mehta, R. Oldenbourg, and A. S. Gladfelter, “Analysis of septin reorganization at cytokinesis using polarized fluorescence microscopy,” Front. Cell Dev. Biol. 5, 42 (2017).
[Crossref] [PubMed]
A. Anantharam, B. Onoa, R. H. Edwards, R. W. Holz, and D. Axelrod, “Localized topological changes of the plasma membrane upon exocytosis visualized by polarized TIRFM,” J. Cell Biol. 188, 415–428 (2010).
[Crossref] [PubMed]
M. P. Backlund, M. D. Lew, A. S. Backer, S. J. Sahl, and W. E. Moerner, “The role of molecular dipole orientation in single-molecule fluorescence microscopy and implications for super-resolution imaging,” ChemPhysChem 15, 587–599 (2014).
[Crossref] [PubMed]
M. A. Lieb, J. M. Zavislan, and L. Novotny, “Single-molecule orientations determined by direct emission pattern imaging,” J. Opt. Soc. Am. B 21, 1210–1215 (2004).
[Crossref]
A. S. Backer and W. E. Moerner, “Extending single-molecule microscopy using optical Fourier processing,” J. Phys. Chem. B 118, 8313–8329 (2014).
[Crossref] [PubMed]
A. Agrawal, S. Quirin, G. Grover, and R. Piestun, “Limits of 3D dipole localization and orientation estimation for single-molecule imaging: towards Green’s tensor engineering,” Opt. Express 20, 26667–26680 (2012).
[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]
A. Débarre, R. Jaffiol, C. Julien, D. Nutarelli, A. Richard, P. Tchénio, F. Chaput, and J.-P. Boilot, “Quantitative determination of the 3D dipole orientation of single molecules,” Eur. Phys. J. D. 28, 67–77 (2004).
[Crossref]
M. Shribak and R. Oldenbourg, “Techniques for fast and sensitive measurements of two-dimensional birefringence distributions,” Appl. Opt. 42, 3009–3017 (2003).
[Crossref] [PubMed]
J. T. Fourkas, “Rapid determination of the three-dimensional orientation of single molecules,” Opt. Lett. 26, 211–213 (2001).
[Crossref]
J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. 45, 5453–5469 (2006).
[Crossref] [PubMed]
C.-Y. Lu and D. A. Vanden Bout, “Analysis of orientational dynamics of single fluorophore trajectories from three-angle polarization experiments,” J. Chem. Phys. 128, 244501 (2008).
[Crossref] [PubMed]
Y. Wu, P. Wawrzusin, J. Senseney, R. S. Fischer, R. Christensen, A. Santella, A. G. York, P. W. Winter, C. M. Waterman, Z. Bao, D. A. Colon-Ramos, M. McAuliffe, and H. Shroff, “Spatially isotropic four-dimensional imaging with dual-view plane illumination microscopy,” Nat. Biotechnol. 31, 1032–1038 (2013).
[Crossref] [PubMed]
R. K. Chhetri, F. Amat, Y. Wan, B. Hockendorf, W. C. Lemon, and P. J. Keller, “Whole-animal functional and developmental imaging with isotropic spatial resolution,” Nat. Methods 12, 1171–1178 (2015).
[Crossref] [PubMed]
Y. Wu, P. Chandris, P. W. Winter, E. Y. Kim, V. Jaumouillé, A. Kumar, M. Guo, J. M. Leung, C. Smith, I. Rey-Suarez, H. Liu, C. M. Waterman, K. S. Ramamurthi, P. J. La Rivière, and H. Shroff, “Simultaneous multiview capture and fusion improves spatial resolution in wide-field and light-sheet microscopy,” Optica 3, 897–910 (2016).
[Crossref] [PubMed]
Y. Wu, A. Kumar, C. Smith, E. Ardiel, P. Chandris, R. Christensen, I. Rey-Suarez, M. Guo, H. Vishwasrao, J. Chen, J. Tang, A. Upadhyaya, P. La Rivière, and H. Shroff, “Reflective imaging improves resolution, speed, and collection efficiency in light sheet microscopy,” bioRxiv (2017).
S. M. Kay, Fundamentals of statistical signal processing (Prentice Hall PTR, 1993).
T. W. Anderson, An introduction to multivariate statistical analysis (Wiley, 1958).
D. Coe, “Fisher matrices and confidence ellipses: a quick-start guide and software,” arXiv preprint arXiv:0906.4123 (2009).
S. van der Walt, S. C. Colbert, and G. Varoquaux, “The NumPy array: a structure for efficient numerical computation,” CoRR abs/1102.1523 (2011).
A. Meurer, C. P. Smith, M. Paprocki, O. Čertík, S. B. Kirpichev, M. Rocklin, A. Kumar, S. Ivanov, J. K. Moore, S. Singh, T. Rathnayake, S. Vig, B. E. Granger, R. P. Muller, F. Bonazzi, H. Gupta, S. Vats, F. Johansson, F. Pedregosa, M. J. Curry, A. R. Terrel, v. Roučka, A. Saboo, I. Fernando, S. Kulal, R. Cimrman, and A. Scopatz, “SymPy: symbolic computing in Python,” PeerJ Computer Science 3, e103 (2017).
[Crossref]
J. D. Hunter, “Matplotlib: A 2D graphics environment,” Comput. Sci. Eng. 9, 90–95 (2007).
[Crossref]
L. Campagnola, A. Klein, C. Larson, N. P. Rossant, and Rougier, “VisPy: Harnessing The GPU For Fast, High-Level Visualization,” in Proceedings of the 14th Python in Science Conference, (2015).
J. C. Bowman and A. Hammerlindl, “Asymptote: A vector graphics language,” TUGBOAT 29, 288–294 (2008).
L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).
[Crossref]
M. Levoy, R. Ng, A. Adams, M. Footer, and M. Horowitz, “Light field microscopy,” in “ACM SIGGRAPH 2006 Papers,” (ACM, New York, NY, USA, 2006), SIGGRAPH ’06, pp. 924–934.

2017 (2)

M. McQuilken, M. S. Jentzsch, A. Verma, S. B. Mehta, R. Oldenbourg, and A. S. Gladfelter, “Analysis of septin reorganization at cytokinesis using polarized fluorescence microscopy,” Front. Cell Dev. Biol. 5, 42 (2017).
[Crossref] [PubMed]

A. Meurer, C. P. Smith, M. Paprocki, O. Čertík, S. B. Kirpichev, M. Rocklin, A. Kumar, S. Ivanov, J. K. Moore, S. Singh, T. Rathnayake, S. Vig, B. E. Granger, R. P. Muller, F. Bonazzi, H. Gupta, S. Vats, F. Johansson, F. Pedregosa, M. J. Curry, A. R. Terrel, v. Roučka, A. Saboo, I. Fernando, S. Kulal, R. Cimrman, and A. Scopatz, “SymPy: symbolic computing in Python,” PeerJ Computer Science 3, e103 (2017).
[Crossref]

2016 (2)

Y. Wu, P. Chandris, P. W. Winter, E. Y. Kim, V. Jaumouillé, A. Kumar, M. Guo, J. M. Leung, C. Smith, I. Rey-Suarez, H. Liu, C. M. Waterman, K. S. Ramamurthi, P. J. La Rivière, and H. Shroff, “Simultaneous multiview capture and fusion improves spatial resolution in wide-field and light-sheet microscopy,” Optica 3, 897–910 (2016).
[Crossref] [PubMed]

S. B. Mehta, M. McQuilken, P. J. La Rivière, P. Occhipinti, A. Verma, R. Oldenbourg, A. S. Gladfelter, and T. Tani, “Dissection of molecular assembly dynamics by tracking orientation and position of single molecules in live cells,” Proc. Natl. Acad. Sci. U.S.A. 113, E6352–E6361 (2016).
[Crossref] [PubMed]

2015 (1)

R. K. Chhetri, F. Amat, Y. Wan, B. Hockendorf, W. C. Lemon, and P. J. Keller, “Whole-animal functional and developmental imaging with isotropic spatial resolution,” Nat. Methods 12, 1171–1178 (2015).
[Crossref] [PubMed]

2014 (2)

A. S. Backer and W. E. Moerner, “Extending single-molecule microscopy using optical Fourier processing,” J. Phys. Chem. B 118, 8313–8329 (2014).
[Crossref] [PubMed]

M. P. Backlund, M. D. Lew, A. S. Backer, S. J. Sahl, and W. E. Moerner, “The role of molecular dipole orientation in single-molecule fluorescence microscopy and implications for super-resolution imaging,” ChemPhysChem 15, 587–599 (2014).
[Crossref] [PubMed]

2013 (1)

Y. Wu, P. Wawrzusin, J. Senseney, R. S. Fischer, R. Christensen, A. Santella, A. G. York, P. W. Winter, C. M. Waterman, Z. Bao, D. A. Colon-Ramos, M. McAuliffe, and H. Shroff, “Spatially isotropic four-dimensional imaging with dual-view plane illumination microscopy,” Nat. Biotechnol. 31, 1032–1038 (2013).
[Crossref] [PubMed]

2012 (1)

A. Agrawal, S. Quirin, G. Grover, and R. Piestun, “Limits of 3D dipole localization and orientation estimation for single-molecule imaging: towards Green’s tensor engineering,” Opt. Express 20, 26667–26680 (2012).
[Crossref] [PubMed]

2011 (1)

B. DeMay, N. Noda, A. Gladfelter, and R. Oldenbourg, “Rapid and quantitative imaging of excitation polarized fluorescence reveals ordered septin dynamics in live yeast,” Biophys. J. 101, 985–994 (2011).
[Crossref] [PubMed]

2010 (1)

A. Anantharam, B. Onoa, R. H. Edwards, R. W. Holz, and D. Axelrod, “Localized topological changes of the plasma membrane upon exocytosis visualized by polarized TIRFM,” J. Cell Biol. 188, 415–428 (2010).
[Crossref] [PubMed]

2008 (2)

C.-Y. Lu and D. A. Vanden Bout, “Analysis of orientational dynamics of single fluorophore trajectories from three-angle polarization experiments,” J. Chem. Phys. 128, 244501 (2008).
[Crossref] [PubMed]

J. C. Bowman and A. Hammerlindl, “Asymptote: A vector graphics language,” TUGBOAT 29, 288–294 (2008).

2007 (1)

J. D. Hunter, “Matplotlib: A 2D graphics environment,” Comput. Sci. Eng. 9, 90–95 (2007).
[Crossref]

2006 (2)

J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. 45, 5453–5469 (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]

2004 (2)

A. Débarre, R. Jaffiol, C. Julien, D. Nutarelli, A. Richard, P. Tchénio, F. Chaput, and J.-P. Boilot, “Quantitative determination of the 3D dipole orientation of single molecules,” Eur. Phys. J. D. 28, 67–77 (2004).
[Crossref]

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

2003 (2)

J. N. Forkey, M. E. Quinlan, M. A. Shaw, J. E. T. Corrie, and Y. E. Goldman, “Three-dimensional structural dynamics of myosin V by single-molecule fluorescence polarization,” Nature 422, 399–404 (2003).
[Crossref] [PubMed]

M. Shribak and R. Oldenbourg, “Techniques for fast and sensitive measurements of two-dimensional birefringence distributions,” Appl. Opt. 42, 3009–3017 (2003).
[Crossref] [PubMed]

2001 (2)

J. T. Fourkas, “Rapid determination of the three-dimensional orientation of single molecules,” Opt. Lett. 26, 211–213 (2001).
[Crossref]

E. J. Peterman, H. Sosa, L. S. Goldstein, and W. E. Moerner, “Polarized fluorescence microscopy of individual and many kinesin motors bound to axonemal microtubules,” Biophys. J. 81, 2851–2863 (2001).
[Crossref] [PubMed]

Adams, A.

M. Levoy, R. Ng, A. Adams, M. Footer, and M. Horowitz, “Light field microscopy,” in “ACM SIGGRAPH 2006 Papers,” (ACM, New York, NY, USA, 2006), SIGGRAPH ’06, pp. 924–934.

Agrawal, A.

Amat, F.

Anantharam, A.

Anderson, T. W.

T. W. Anderson, An introduction to multivariate statistical analysis (Wiley, 1958).

Ardiel, E.

Y. Wu, A. Kumar, C. Smith, E. Ardiel, P. Chandris, R. Christensen, I. Rey-Suarez, M. Guo, H. Vishwasrao, J. Chen, J. Tang, A. Upadhyaya, P. La Rivière, and H. Shroff, “Reflective imaging improves resolution, speed, and collection efficiency in light sheet microscopy,” bioRxiv (2017).

Axelrod, D.

Backer, A. S.

A. S. Backer and W. E. Moerner, “Extending single-molecule microscopy using optical Fourier processing,” J. Phys. Chem. B 118, 8313–8329 (2014).
[Crossref] [PubMed]

A. S. Backer, M. Lee, and W. E. Moerner, “Enhanced DNA imaging using super-resolution microscopy and simultaneous single-molecule orientation measurements,” in “Conference on Lasers and Electro-Optics,” (Optical Society of America, 2016), p. JTh4B.4.

Backlund, M. P.

Bao, Z.

Boilot, J.-P.

Bonazzi, F.

Bowman, J. C.

J. C. Bowman and A. Hammerlindl, “Asymptote: A vector graphics language,” TUGBOAT 29, 288–294 (2008).

Campagnola, L.

L. Campagnola, A. Klein, C. Larson, N. P. Rossant, and Rougier, “VisPy: Harnessing The GPU For Fast, High-Level Visualization,” in Proceedings of the 14th Python in Science Conference, (2015).

Certík, O.

Chandris, P.

Chaput, F.

Chen, J.

Chenault, D. B.

J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. 45, 5453–5469 (2006).
[Crossref] [PubMed]

Chhetri, R. K.

Christensen, R.

Cimrman, R.

Coe, D.

D. Coe, “Fisher matrices and confidence ellipses: a quick-start guide and software,” arXiv preprint arXiv:0906.4123 (2009).

Colbert, S. C.

S. van der Walt, S. C. Colbert, and G. Varoquaux, “The NumPy array: a structure for efficient numerical computation,” CoRR abs/1102.1523 (2011).

Colon-Ramos, D. A.

Corrie, J. E. T.

Curry, M. J.

Débarre, A.

DeMay, B.

Edwards, R. H.

Enderlein, J.

Fernando, I.

Fischer, R. S.

Footer, M.

M. Levoy, R. Ng, A. Adams, M. Footer, and M. Horowitz, “Light field microscopy,” in “ACM SIGGRAPH 2006 Papers,” (ACM, New York, NY, USA, 2006), SIGGRAPH ’06, pp. 924–934.

Forkey, J. N.

Fourkas, J. T.

J. T. Fourkas, “Rapid determination of the three-dimensional orientation of single molecules,” Opt. Lett. 26, 211–213 (2001).
[Crossref]

Gladfelter, A.

Gladfelter, A. S.

Goldman, Y. E.

Goldstein, D. L.

J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. 45, 5453–5469 (2006).
[Crossref] [PubMed]

Goldstein, L. S.

Granger, B. E.

Grover, G.

Guo, M.

Gupta, H.

Ha, T.

Hammerlindl, A.

J. C. Bowman and A. Hammerlindl, “Asymptote: A vector graphics language,” TUGBOAT 29, 288–294 (2008).

Hecht, B.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).
[Crossref]

Hockendorf, B.

Holz, R. W.

Horowitz, M.

M. Levoy, R. Ng, A. Adams, M. Footer, and M. Horowitz, “Light field microscopy,” in “ACM SIGGRAPH 2006 Papers,” (ACM, New York, NY, USA, 2006), SIGGRAPH ’06, pp. 924–934.

Hunter, J. D.

J. D. Hunter, “Matplotlib: A 2D graphics environment,” Comput. Sci. Eng. 9, 90–95 (2007).
[Crossref]

Ivanov, S.

Jaffiol, R.

Jaumouillé, V.

Jentzsch, M. S.

Johansson, F.

Julien, C.

Kay, S. M.

S. M. Kay, Fundamentals of statistical signal processing (Prentice Hall PTR, 1993).

Keller, P. J.

Kim, E. Y.

Kirpichev, S. B.

Klein, A.

L. Campagnola, A. Klein, C. Larson, N. P. Rossant, and Rougier, “VisPy: Harnessing The GPU For Fast, High-Level Visualization,” in Proceedings of the 14th Python in Science Conference, (2015).

Kulal, S.

Kumar, A.

La Rivière, P.

La Rivière, P. J.

Larson, C.

L. Campagnola, A. Klein, C. Larson, N. P. Rossant, and Rougier, “VisPy: Harnessing The GPU For Fast, High-Level Visualization,” in Proceedings of the 14th Python in Science Conference, (2015).

Lee, M.

Lemon, W. C.

Leung, J. M.

Levoy, M.

M. Levoy, R. Ng, A. Adams, M. Footer, and M. Horowitz, “Light field microscopy,” in “ACM SIGGRAPH 2006 Papers,” (ACM, New York, NY, USA, 2006), SIGGRAPH ’06, pp. 924–934.

Lew, M. D.

Lieb, M. A.

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

Liu, H.

Lu, C.-Y.

McAuliffe, M.

McKinney, S. A.

McQuilken, M.

Mehta, S. B.

Meurer, A.

Moerner, W. E.

A. S. Backer and W. E. Moerner, “Extending single-molecule microscopy using optical Fourier processing,” J. Phys. Chem. B 118, 8313–8329 (2014).
[Crossref] [PubMed]

Moore, J. K.

Muller, R. P.

Ng, R.

M. Levoy, R. Ng, A. Adams, M. Footer, and M. Horowitz, “Light field microscopy,” in “ACM SIGGRAPH 2006 Papers,” (ACM, New York, NY, USA, 2006), SIGGRAPH ’06, pp. 924–934.

Noda, N.

Novotny, L.

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

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).
[Crossref]

Nutarelli, D.

Occhipinti, P.

Oldenbourg, R.

M. Shribak and R. Oldenbourg, “Techniques for fast and sensitive measurements of two-dimensional birefringence distributions,” Appl. Opt. 42, 3009–3017 (2003).
[Crossref] [PubMed]

Onoa, B.

Paprocki, M.

Pedregosa, F.

Peterman, E. J.

Petschek, R. G.

Piestun, R.

Quinlan, M. E.

Quirin, S.

Ramamurthi, K. S.

Rathnayake, T.

Rey-Suarez, I.

Richard, A.

Rocklin, M.

Rossant, N. P.

L. Campagnola, A. Klein, C. Larson, N. P. Rossant, and Rougier, “VisPy: Harnessing The GPU For Fast, High-Level Visualization,” in Proceedings of the 14th Python in Science Conference, (2015).

Roucka, v.

Rougier,

L. Campagnola, A. Klein, C. Larson, N. P. Rossant, and Rougier, “VisPy: Harnessing The GPU For Fast, High-Level Visualization,” in Proceedings of the 14th Python in Science Conference, (2015).

Saboo, A.

Sahl, S. J.

Santella, A.

Scopatz, A.

Selvin, P. R.

Senseney, J.

Shaw, J. A.

J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. 45, 5453–5469 (2006).
[Crossref] [PubMed]

Shaw, M. A.

Shribak, M.

M. Shribak and R. Oldenbourg, “Techniques for fast and sensitive measurements of two-dimensional birefringence distributions,” Appl. Opt. 42, 3009–3017 (2003).
[Crossref] [PubMed]

Shroff, H.

Singh, S.

Smith, C.

Smith, C. P.

Sosa, H.

Syed, S.

Tang, J.

Tani, T.

Tchénio, P.

Terrel, A. R.

Toprak, E.

Tyo, J. S.

J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. 45, 5453–5469 (2006).
[Crossref] [PubMed]

Upadhyaya, A.

van der Walt, S.

S. van der Walt, S. C. Colbert, and G. Varoquaux, “The NumPy array: a structure for efficient numerical computation,” CoRR abs/1102.1523 (2011).

Vanden Bout, D. A.

Varoquaux, G.

S. van der Walt, S. C. Colbert, and G. Varoquaux, “The NumPy array: a structure for efficient numerical computation,” CoRR abs/1102.1523 (2011).

Vats, S.

Verma, A.

Vig, S.

Vishwasrao, H.

Wan, Y.

Waterman, C. M.

Wawrzusin, P.

Winter, P. W.

Wu, Y.

York, A. G.

Zavislan, J. M.

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

Appl. Opt. (2)

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Fig. 1 Coordinate systems for a) the absorption dipole moment μ̂_abs, b) the polarizer transmission axis p̂_exc, and c) the dummy integration vector r̂. d) The scene consists of a single molecule at the focal point of an objective lens with a polarizer in the aperture plane.

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Fig. 2 Representative examples of single intensity measurements. Black dots indicate where the Cartesian unit vectors intersect the unit sphere.

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Fig. 3 Single-view microscope with varying illumination and detection NA. a) Schematic of a single-view four-polarization epi-illumination microscope. The red arrows indicate the four illumination polarization orientations—one for each intensity measurement. b) Uncertainty ellipses for the microscope in a). The ellipses indicate the relative size of the uncertainty in different directions, not the absolute size of the uncertainty. c) Solid-angle uncertainty for the microscope in a). The solid-angle uncertainty is a measure of the absolute size of the uncertainty in all directions. d) Median of the solid-angle uncertainty as a function of illumination and detection NA. e) MAD of the solid-angle uncertainty as a function of illumination and detection NA. The microscope configuration in a), b), and c) is indicated by a cross in d) and e).

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Fig. 4 Dual-view symmetric designs with varying NA and angle between objectives. a) Schematic of the microscope. We illuminate with the first objective (red solid) and detect from the second objective (red dashed). Then we illuminate from the second objective (blue solid) and detect from the first objective (blue dashed). b) Uncertainty ellipses for the microscope in a). The ellipses indicate the relative size of the uncertainty in different directions, not the absolute size of the uncertainty. c) Solid-angle uncertainty for the microscope in a). The solid-angle uncertainty is a measure of the absolute size of the uncertainty in all directions. d) Median of the solid-angle uncertainty as a function of NA and the angle between the objectives. e) MAD of the solid-angle uncertainty as a function of NA and the angle between the objectives. The microscope configuration in a), b), and c) is indicated by a cross in d) and e). f) Median of the solid-angle uncertainty as a function of the angle between the objectives when NA=0.6. g) MAD of the solid-angle uncertainty as a function of the angle between the objectives when NA=0.6. The profile in f) and g) is taken along the purple line in d) and e).

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Fig. 5 Dual-view symmetric orthogonal designs with varying illumination and detection NA. a) Schematic of the microscope. b) Uncertainty ellipses for the microscope in a). The ellipses indicate the relative size of the uncertainty in different directions, not the absolute size of the uncertainty. c) Solid-angle uncertainty for the microscope in a). The solid-angle uncertainty is a measure of the absolute size of the uncertainty in all directions. d) Median of the solid-angle uncertainty as a function of illumination and detection NA. e) MAD of the solid-angle uncertainty as a function of illumination and detection NA. The microscope configuration in a), b), and c) is indicated by a cross in d) and e).

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Fig. 6 Dual-view asymmetric light-sheet illumination designs with varying NA and sample exposure asymmetry. a) Schematic of the microscope. b) Uncertainty ellipses for the microscope in a). The ellipses indicate the relative size of the uncertainty in different directions, not the absolute size of the uncertainty. c) Solid-angle uncertainty for the microscope in a). The solid-angle uncertainty is a measure of the absolute size of the uncertainty in all directions. d) Median of the solid-angle uncertainty as a function of NA and sample exposure asymmetry. e) MAD of the solid-angle uncertainty as a function of NA and sample exposure asymmetry. The microscope configuration in a), b), and c) is indicated by a cross in d) and e).

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Tables (1)

Table 1 Comparison of designs in each class of microscope. All solid-angle uncertainty statistics are in steradians.

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Equations (28)

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{\hat{μ}}_{abs} (Θ, Φ) = sin Θ cos Φ \hat{x} + sin Θ sin Φ \hat{y} cos Θ \hat{z} .

{\hat{p}}_{exc} (ϕ_{exc}) = cos ϕ_{exc} \hat{x} + sin ϕ_{exc} \hat{y},

\hat{r} (θ, ϕ) = sin θ cos ϕ \hat{x} + sin θ sin ϕ \hat{y} + cos θ \hat{z},

Ω = {ϕ, θ | 0 < ϕ \leq 2 π, 0 < θ \leq α},

NA = n sin α,

R (\hat{r}) = [\begin{matrix} cos θ {cos}^{2} ϕ + {sin}^{2} ϕ & (cos θ - 1) sin ϕ cos ϕ & sin θ cos ϕ \\ (cos θ - 1) sin ϕ cos ϕ & cos θ {sin}^{2} ϕ + {cos}^{2} ϕ & sin θ sin ϕ \\ - sin θ cos ϕ & - sin θ sin ϕ & cos θ \end{matrix}] .

η_{abs}^{ray} = {| {\hat{μ}}_{abs} \cdot {\hat{p}}_{exc} |}^{2} .

η_{abs} = \frac{\int_{Ω} d \hat{r} {| {\hat{μ}}_{abs} \cdot R (\hat{r}) {\hat{p}}_{exc} |}^{2}}{\int_{Ω} d \hat{r}} .

η_{abs} = D {A + B {sin}^{2} Θ + C {sin}^{2} Θ cos [2 (Φ - ϕ_{exc})]}

A = \frac{1}{4} - \frac{3}{8} cos α + \frac{1}{8} {cos}^{3} α

B = \frac{3}{16} cos α - \frac{3}{16} {cos}^{3} α

C = \frac{7}{32} - \frac{3}{32} cos α - \frac{3}{32} {cos}^{2} α - \frac{1}{32} {cos}^{3} α

D = \frac{4}{3 (1 - cos α)} .

η_{abs}^{ray} = {sin}^{2} Θ {cos}^{2} (Φ - ϕ_{exc}),

η_{det}^{pol} = A + B {sin}^{2} Θ + C {sin}^{2} Θ cos [2 (Φ - ϕ_{det})],

η_{det} = 2 (A + B {sin}^{2} Θ) .

Θ^{'} = arccos (sin ψ cos Φ sin Θ + cos ψ cos Θ)

Φ^{'} = {\begin{array}{l} arccos (\frac{cos ψ cos Θ sin Θ - sin ψ cos Θ}{\sqrt{1 - {(sin ψ cos Φ sin Θ + cos ψ cos Θ)}^{2}}}) & 0 \leq Φ < π \\ - arccos (\frac{cos ψ cos Φ sin Θ - sin ψ cos Θ}{\sqrt{1 - {(sin ψ cos Θ sin Θ + cos ψ cos Θ)}^{2}}}) & - π \leq Φ < 0 \end{array}

I \propto I_{in} η_{abs} η_{det},

F = \sum_{k = 1}^{N} \frac{1}{I_{k}} [\begin{matrix} \frac{\partial I_{k}}{\partial Θ} \frac{\partial I_{k}}{\partial Θ} & \frac{\partial I_{k}}{\partial Θ} \frac{\partial I_{k}}{\partial Φ} \\ \frac{\partial I_{k}}{\partial Θ} \frac{\partial I_{k}}{\partial Φ} & \frac{\partial I_{k}}{\partial Φ} \frac{\partial I_{k}}{\partial Φ} \end{matrix}]

σ_{Ω} \equiv sin Θ \sqrt{det {F^{- 1}}}

\frac{{(x cos γ + y sin γ)}^{2}}{a^{2}} + \frac{{(x sin γ + y cos γ)}^{2}}{b^{2}} = 1 .

σ_{x} = \sqrt{F_{0, 0}^{- 1}}

σ_{y} = sin Θ \sqrt{F_{1, 1}^{- 1}}

σ_{x y} = sin Θ \sqrt{F_{1, 0}^{- 1}}

a^{2} = \frac{σ_{x}^{2} + σ_{y}^{2}}{2} + \sqrt{\frac{{(σ_{x}^{2} + σ_{y}^{2})}^{2}}{4} + σ_{x y}^{2}}

b^{2} = \frac{σ_{x}^{2} + σ_{y}^{2}}{2} - \sqrt{\frac{{(σ_{x}^{2} - σ_{y}^{2})}^{2}}{4} + σ_{x y}^{2}}

γ = \frac{1}{2} arctan (\frac{2 σ_{x y}}{σ_{x}^{2} - σ_{y}^{2}}) .

Microscope Type	Single-viewNA_ill=0NA_det=1.1	Dual-view oblique symmetricNA=0.6β=53	Dual-view orthogonal symmetric light-sheetNA=0.8	Dual-view orthogonal symmetric light-sheetNA=0.94
n-fold Degeneracy	4	2	2	2
Max{σ_Ω}	3.04×10⁰	1.35×10⁻¹	6.23 × 10⁻³	3.13×10⁻³
Median{σ_Ω}	8.20×10⁻⁴	4.49×10⁻³	1.79 × 10⁻³	1.28×10⁻³
MAD{σ_Ω}	1.48×10⁻⁴	9.17×10⁻⁴	1.65 × 10⁻⁴	1.14×10⁻⁴