Abstract

Localization-based superresolution imaging is dependent on finding the positions of individual fluorophores in a sample by fitting the observed single-molecule intensity pattern to the microscope point spread function (PSF). For three-dimensional imaging, system-specific aberrations of the optical system can lead to inaccurate localizations when the PSF model does not account for these aberrations. Here we describe the use of phase-retrieved pupil functions to generate a more accurate PSF and therefore more accurate 3D localizations. The complex-valued pupil function contains information about the system-specific aberrations and can thus be used to generate the PSF for arbitrary defocus. Further, it can be modified to include depth dependent aberrations. We describe the phase retrieval process, the method for including depth dependent aberrations, and a fast fitting algorithm using graphics processing units. The superior localization accuracy of the pupil function generated PSF is demonstrated with dual focal plane 3D superresolution imaging of biological structures.

© 2013 Optical Society of America

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2013 (3)

M. R.  Foreman, C. L.  Giusca, P.  Török, R. K.  Leach, “Phase–retrieved pupil function and coherent transfer function in confocal microscopy,” J. Microsc. 251, 99–107 (2013).
[CrossRef] [PubMed]

D.  Axelrod, “Evanescent Excitation and Emission in Fluorescence Microscopy,” Biophys. J. 7, 1401–1409 (2013).
[CrossRef]

R. P. J.  Nieuwenhuizen, K. A.  Lidke, M.  Bates, D. L.  Puig, D.  Grünwald, S.  Stallinga, B.  Rieger, “Measuring image resolution in optical nanoscopy,” Nat.Methods 10, 557–562 (2013).
[CrossRef] [PubMed]

2012 (3)

D.  Axelrod, “Fluorescence excitation and imaging of single molecules near dielectric coated and bare surfaces: a theoretical study,” J. Microsc. 247, 147–160 (2012).
[CrossRef] [PubMed]

H.  Kirshner, F.  Aguet, D.  Sage, D.  Unser, “3-D PSF fitting for fluorescence microscopy: implementation and localization application,” J. Microsc. 249, 13–25 (2012).
[CrossRef] [PubMed]

E. B.  Kromann, T. J.  Gould, M. F.  Juette, J. E.  Wilhjelm, J.  Bewersdorf, “Quantitative Pupil Analysis in STED Microscopy Using Phase Retrieval,” Opt. Lett. 37, 1805–1807 (2012).
[CrossRef] [PubMed]

2011 (4)

A. G.  York, A.  Ghitani, A.  Vaziri, M. W.  Davidson, H.  Shroff, “Confined activation and subdiffractive localization enables whole-cell PALM with genetically expressed probes,” Nat. Methods 8, 327–333 (2011).
[CrossRef] [PubMed]

S.  Quirin, S. R. P.  Pavani, R.  Piestun, “Optimal 3D single-molecule localization for superresolution microscopy with aberrations and engineered point spread functions,” Proc. Natl. Acad. Sci. USA. 109, 675–679 (2011).
[CrossRef]

J.  Enderlein, I.  Gregor, T.  Ruckstuhl, “Imaging properties of supercritical angle fluorescence optics,” Opt. Express 19, 8011–8018 (2011).
[CrossRef] [PubMed]

F.  Huang, S. L.  Schwartz, J. M.  Byars, K. A.  Lidke, “Simultaneous multiple-emitter fitting for single molecule super-resolution imaging,” Biomed. Opt. Express 2, 1377–1393 (2011).
[CrossRef] [PubMed]

2010 (4)

C. S.  Smith, N.  Joseph, B.  Rieger, K. A.  Lidke, “Fast, single-molecule localization that achieves theoretically minimum uncertainty,” Nat. Methods 7, 373–375 (2010).
[CrossRef] [PubMed]

M. F.  Juette, J.  Bewersdorf, “Three-dimensional tracking of single fluorescent particles with submillisecond temporal resolution,” Nano Lett. 10, 4657–4663 (2010).
[CrossRef] [PubMed]

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

S.  Stallinga, B.  Rieger, “Accuracy of the gaussian point spread function model in 2D localization microscopy,” Opt. Express 18, 24461–24476 (2010).
[CrossRef] [PubMed]

2009 (2)

M. J.  Mlodzianoski, M. F.  Juette, G. L.  Beane, J.  Bewersdorf, “Experimental characterization of 3D localization techniques for particle-tracking and super-resolution microscopy,” Opt. Express 17, 8264–8277 (2009).
[CrossRef] [PubMed]

S. R. P.  Pavani, M. A.  Thompson, J. S.  Biteen, S. J.  Lord, N.  Liu, R. J.  Twieg, R.  Piestun, W. E.  Moerner, “Three-dimensional, single-molecule fluorescence imaging beyond the diffraction limit by using a double-helix point spread function,” Proc. Natl. Acad. Sci. USA. 106, 2995–2999 (2009).
[CrossRef] [PubMed]

2008 (6)

M.  Heilemann, S.  van de Linde, M.  Schüttpelz, R.  Kasper, B.  Seefeldt, A.  Mukherjee, P.  Tinnefeld, M.  Sauer, “Subdiffraction-resolution fluorescence imaging with conventional fluorescent probes,” Angew. Chem. Int. Ed. 47, 6172–6176 (2008).
[CrossRef]

J.  Fölling, V.  Belov, D.  Riedel, A.  Schönle, A.  Egner, C.  Eggeling, M.  Bossi, S. W.  Hell, “Fluorescence Nanoscopy with Optical Sectioning by Two–Photon Induced Molecular Switching using Continuous–Wave Lasers,” Chem. Phys. Chem 9, 321–326 (2008).
[CrossRef]

B.  Huang, S. A.  Jones, B.  Brandenburg, X.  Zhuang, “Whole-cell 3D STORM reveals interactions between cellular structures with nanometer-scale resolution,” Nat. Methods 5, 1047–1052 (2008).
[CrossRef] [PubMed]

M. F.  Juette, T. J.  Gould, M. D.  Lessard, M. J.  Mlodzianoski, B. S.  Nagpure, B. T.  Bennett, S. T.  Hess, J.  Bewersdorf, “Three-dimensional sub100 nm resolution fluorescence microscopy of thick samples,” Nat. Methods 5, 527–529 (2008).
[CrossRef] [PubMed]

S.  Ram, P.  Prabhat, J.  Chao, E. S.  Ward, R. J.  Ober, “High accuracy 3D quantum dot tracking with multifocal plane microscopy for the study of fast intracellular dynamics in live cells,” Biophys. J. 95, 6025–6043 (2008).
[CrossRef] [PubMed]

B.  Huang, W.  Wang, M.  Bates, X.  Zhuang, “Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy,” Science 319, 810–813 (2008).
[CrossRef] [PubMed]

2007 (2)

E.  Toprak, H.  Balci, B. H.  Blehm, P. R.  Selvin, “Three-dimensional particle tracking via bifocal imaging,” Nano Lett. 7, 2043–2045 (2007)
[CrossRef] [PubMed]

S.  Ram, J.  Chao, P.  Prabhat, E. S.  Ward, R. J.  Ober, “A novel approach to determining the three-dimensional location of microscopic objects with applications to 3D particle tracking,” Proc. SPIE 6443, 6443(2007).
[CrossRef]

2006 (3)

M. J.  Rust, M.  Bates, X.  Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3, 793–796 (2006).
[CrossRef] [PubMed]

E.  Betzig, G. H.  Patterson, R.  Sougrat, O. W.  Lindwasser, S.  Olenych, J. S.  Bonifacino, M. W.  Davidson, J.  Lippincott-Schwartz, H. F.  Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
[CrossRef] [PubMed]

S. T.  Hess, T. P. K.  Girirajan, M. D.  Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J. 91, 4258–4272 (2006).
[CrossRef] [PubMed]

2005 (3)

2004 (1)

B. M.  Hanser, M. G. L.  Gustafsson, D. A.  Agard, J. W.  Sedat, “Phase-retrieved pupil functions in wide-field fluorescence microscopy,” J. Microsc. 216, 32–48 (2004).
[CrossRef] [PubMed]

2003 (1)

2002 (1)

M.  Sambridge, K.  Mosegaard, “Monte Carlo methods in geophysical inverse problems,” Reviews of Geophysics 40, 1–29 (2002).
[CrossRef]

1999 (1)

1991 (1)

S. F.  Gibson, F.  Lanni, “Experimental test of an analytical model of aberration in an oil-immersion objective lens used in three-dimensional light microscopy,” J. Opt. Soc. Am. A 19, 1601–1613 (1991)
[CrossRef]

1972 (1)

R. W.  Gerchberg, W. O.  Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

1959 (1)

B.  Richards, E.  Wolf, “Electromagnetic diffraction in optical systems. II. structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253, 358–379 (1959).
[CrossRef]

Agard, D. A.

B. M.  Hanser, M. G. L.  Gustafsson, D. A.  Agard, J. W.  Sedat, “Phase-retrieved pupil functions in wide-field fluorescence microscopy,” J. Microsc. 216, 32–48 (2004).
[CrossRef] [PubMed]

B. M.  Hanser, M. G. L.  Gustafsson, D. A.  Agard, J. W.  Sedat, “Phase retrieval for high-numerical-aperture optical systems,” Opt. Lett. 28, 801–803 (2003).
[CrossRef] [PubMed]

Aguet, F.

H.  Kirshner, F.  Aguet, D.  Sage, D.  Unser, “3-D PSF fitting for fluorescence microscopy: implementation and localization application,” J. Microsc. 249, 13–25 (2012).
[CrossRef] [PubMed]

F.  Aguet, D.  Van De Ville, M.  Unser, “A maximum-likelihood formalism for sub-resolution axial localization of fluorescent nanoparticles,” Opt. Express 13, 10503–10522 (2005).
[CrossRef] [PubMed]

Alam, M. S.

M. S.  Alam, J. G.  Bognar, R. C.  Hardie, B. J.  Yasuda, “High-resolution infrared image reconstruction using multiple randomly shifted low-resolution aliased frames,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing VIII, Proc. SPIE 3063, 102–112 (SPIE Press1997).
[CrossRef]

Axelrod, D.

D.  Axelrod, “Evanescent Excitation and Emission in Fluorescence Microscopy,” Biophys. J. 7, 1401–1409 (2013).
[CrossRef]

D.  Axelrod, “Fluorescence excitation and imaging of single molecules near dielectric coated and bare surfaces: a theoretical study,” J. Microsc. 247, 147–160 (2012).
[CrossRef] [PubMed]

Balci, H.

E.  Toprak, H.  Balci, B. H.  Blehm, P. R.  Selvin, “Three-dimensional particle tracking via bifocal imaging,” Nano Lett. 7, 2043–2045 (2007)
[CrossRef] [PubMed]

Bates, M.

R. P. J.  Nieuwenhuizen, K. A.  Lidke, M.  Bates, D. L.  Puig, D.  Grünwald, S.  Stallinga, B.  Rieger, “Measuring image resolution in optical nanoscopy,” Nat.Methods 10, 557–562 (2013).
[CrossRef] [PubMed]

B.  Huang, W.  Wang, M.  Bates, X.  Zhuang, “Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy,” Science 319, 810–813 (2008).
[CrossRef] [PubMed]

M. J.  Rust, M.  Bates, X.  Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3, 793–796 (2006).
[CrossRef] [PubMed]

Beane, G. L.

M. J.  Mlodzianoski, M. F.  Juette, G. L.  Beane, J.  Bewersdorf, “Experimental characterization of 3D localization techniques for particle-tracking and super-resolution microscopy,” Opt. Express 17, 8264–8277 (2009).
[CrossRef] [PubMed]

Belov, V.

J.  Fölling, V.  Belov, D.  Riedel, A.  Schönle, A.  Egner, C.  Eggeling, M.  Bossi, S. W.  Hell, “Fluorescence Nanoscopy with Optical Sectioning by Two–Photon Induced Molecular Switching using Continuous–Wave Lasers,” Chem. Phys. Chem 9, 321–326 (2008).
[CrossRef]

Bennett, B. T.

M. F.  Juette, T. J.  Gould, M. D.  Lessard, M. J.  Mlodzianoski, B. S.  Nagpure, B. T.  Bennett, S. T.  Hess, J.  Bewersdorf, “Three-dimensional sub100 nm resolution fluorescence microscopy of thick samples,” Nat. Methods 5, 527–529 (2008).
[CrossRef] [PubMed]

Betzig, E.

E.  Betzig, G. H.  Patterson, R.  Sougrat, O. W.  Lindwasser, S.  Olenych, J. S.  Bonifacino, M. W.  Davidson, J.  Lippincott-Schwartz, H. F.  Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
[CrossRef] [PubMed]

Bewersdorf, J.

E. B.  Kromann, T. J.  Gould, M. F.  Juette, J. E.  Wilhjelm, J.  Bewersdorf, “Quantitative Pupil Analysis in STED Microscopy Using Phase Retrieval,” Opt. Lett. 37, 1805–1807 (2012).
[CrossRef] [PubMed]

M. F.  Juette, J.  Bewersdorf, “Three-dimensional tracking of single fluorescent particles with submillisecond temporal resolution,” Nano Lett. 10, 4657–4663 (2010).
[CrossRef] [PubMed]

M. J.  Mlodzianoski, M. F.  Juette, G. L.  Beane, J.  Bewersdorf, “Experimental characterization of 3D localization techniques for particle-tracking and super-resolution microscopy,” Opt. Express 17, 8264–8277 (2009).
[CrossRef] [PubMed]

M. F.  Juette, T. J.  Gould, M. D.  Lessard, M. J.  Mlodzianoski, B. S.  Nagpure, B. T.  Bennett, S. T.  Hess, J.  Bewersdorf, “Three-dimensional sub100 nm resolution fluorescence microscopy of thick samples,” Nat. Methods 5, 527–529 (2008).
[CrossRef] [PubMed]

Biteen, J. S.

S. R. P.  Pavani, M. A.  Thompson, J. S.  Biteen, S. J.  Lord, N.  Liu, R. J.  Twieg, R.  Piestun, W. E.  Moerner, “Three-dimensional, single-molecule fluorescence imaging beyond the diffraction limit by using a double-helix point spread function,” Proc. Natl. Acad. Sci. USA. 106, 2995–2999 (2009).
[CrossRef] [PubMed]

Blehm, B. H.

E.  Toprak, H.  Balci, B. H.  Blehm, P. R.  Selvin, “Three-dimensional particle tracking via bifocal imaging,” Nano Lett. 7, 2043–2045 (2007)
[CrossRef] [PubMed]

Bognar, J. G.

M. S.  Alam, J. G.  Bognar, R. C.  Hardie, B. J.  Yasuda, “High-resolution infrared image reconstruction using multiple randomly shifted low-resolution aliased frames,” in Infrared Imaging Systems: Design, Analysis, Modeling, and Testing VIII, Proc. SPIE 3063, 102–112 (SPIE Press1997).
[CrossRef]

Bonifacino, J. S.

E.  Betzig, G. H.  Patterson, R.  Sougrat, O. W.  Lindwasser, S.  Olenych, J. S.  Bonifacino, M. W.  Davidson, J.  Lippincott-Schwartz, H. F.  Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
[CrossRef] [PubMed]

Bossi, M.

J.  Fölling, V.  Belov, D.  Riedel, A.  Schönle, A.  Egner, C.  Eggeling, M.  Bossi, S. W.  Hell, “Fluorescence Nanoscopy with Optical Sectioning by Two–Photon Induced Molecular Switching using Continuous–Wave Lasers,” Chem. Phys. Chem 9, 321–326 (2008).
[CrossRef]

Brandenburg, B.

B.  Huang, S. A.  Jones, B.  Brandenburg, X.  Zhuang, “Whole-cell 3D STORM reveals interactions between cellular structures with nanometer-scale resolution,” Nat. Methods 5, 1047–1052 (2008).
[CrossRef] [PubMed]

Byars, J. M.

Chao, J.

S.  Ram, P.  Prabhat, J.  Chao, E. S.  Ward, R. J.  Ober, “High accuracy 3D quantum dot tracking with multifocal plane microscopy for the study of fast intracellular dynamics in live cells,” Biophys. J. 95, 6025–6043 (2008).
[CrossRef] [PubMed]

S.  Ram, J.  Chao, P.  Prabhat, E. S.  Ward, R. J.  Ober, “A novel approach to determining the three-dimensional location of microscopic objects with applications to 3D particle tracking,” Proc. SPIE 6443, 6443(2007).
[CrossRef]

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E.  Betzig, G. H.  Patterson, R.  Sougrat, O. W.  Lindwasser, S.  Olenych, J. S.  Bonifacino, M. W.  Davidson, J.  Lippincott-Schwartz, H. F.  Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
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S. T.  Hess, T. P. K.  Girirajan, M. D.  Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J. 91, 4258–4272 (2006).
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S. R. P.  Pavani, M. A.  Thompson, J. S.  Biteen, S. J.  Lord, N.  Liu, R. J.  Twieg, R.  Piestun, W. E.  Moerner, “Three-dimensional, single-molecule fluorescence imaging beyond the diffraction limit by using a double-helix point spread function,” Proc. Natl. Acad. Sci. USA. 106, 2995–2999 (2009).
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E.  Betzig, G. H.  Patterson, R.  Sougrat, O. W.  Lindwasser, S.  Olenych, J. S.  Bonifacino, M. W.  Davidson, J.  Lippincott-Schwartz, H. F.  Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
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S.  Quirin, S. R. P.  Pavani, R.  Piestun, “Optimal 3D single-molecule localization for superresolution microscopy with aberrations and engineered point spread functions,” Proc. Natl. Acad. Sci. USA. 109, 675–679 (2011).
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S.  Ram, P.  Prabhat, J.  Chao, E. S.  Ward, R. J.  Ober, “High accuracy 3D quantum dot tracking with multifocal plane microscopy for the study of fast intracellular dynamics in live cells,” Biophys. J. 95, 6025–6043 (2008).
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S.  Ram, J.  Chao, P.  Prabhat, E. S.  Ward, R. J.  Ober, “A novel approach to determining the three-dimensional location of microscopic objects with applications to 3D particle tracking,” Proc. SPIE 6443, 6443(2007).
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R. P. J.  Nieuwenhuizen, K. A.  Lidke, M.  Bates, D. L.  Puig, D.  Grünwald, S.  Stallinga, B.  Rieger, “Measuring image resolution in optical nanoscopy,” Nat.Methods 10, 557–562 (2013).
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S.  Quirin, S. R. P.  Pavani, R.  Piestun, “Optimal 3D single-molecule localization for superresolution microscopy with aberrations and engineered point spread functions,” Proc. Natl. Acad. Sci. USA. 109, 675–679 (2011).
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R. P. J.  Nieuwenhuizen, K. A.  Lidke, M.  Bates, D. L.  Puig, D.  Grünwald, S.  Stallinga, B.  Rieger, “Measuring image resolution in optical nanoscopy,” Nat.Methods 10, 557–562 (2013).
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H.  Kirshner, F.  Aguet, D.  Sage, D.  Unser, “3-D PSF fitting for fluorescence microscopy: implementation and localization application,” J. Microsc. 249, 13–25 (2012).
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M.  Sambridge, K.  Mosegaard, “Monte Carlo methods in geophysical inverse problems,” Reviews of Geophysics 40, 1–29 (2002).
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M.  Heilemann, S.  van de Linde, M.  Schüttpelz, R.  Kasper, B.  Seefeldt, A.  Mukherjee, P.  Tinnefeld, M.  Sauer, “Subdiffraction-resolution fluorescence imaging with conventional fluorescent probes,” Angew. Chem. Int. Ed. 47, 6172–6176 (2008).
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R. W.  Gerchberg, W. O.  Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

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J.  Fölling, V.  Belov, D.  Riedel, A.  Schönle, A.  Egner, C.  Eggeling, M.  Bossi, S. W.  Hell, “Fluorescence Nanoscopy with Optical Sectioning by Two–Photon Induced Molecular Switching using Continuous–Wave Lasers,” Chem. Phys. Chem 9, 321–326 (2008).
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M.  Heilemann, S.  van de Linde, M.  Schüttpelz, R.  Kasper, B.  Seefeldt, A.  Mukherjee, P.  Tinnefeld, M.  Sauer, “Subdiffraction-resolution fluorescence imaging with conventional fluorescent probes,” Angew. Chem. Int. Ed. 47, 6172–6176 (2008).
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B. M.  Hanser, M. G. L.  Gustafsson, D. A.  Agard, J. W.  Sedat, “Phase-retrieved pupil functions in wide-field fluorescence microscopy,” J. Microsc. 216, 32–48 (2004).
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E.  Toprak, H.  Balci, B. H.  Blehm, P. R.  Selvin, “Three-dimensional particle tracking via bifocal imaging,” Nano Lett. 7, 2043–2045 (2007)
[CrossRef] [PubMed]

Shroff, H.

A. G.  York, A.  Ghitani, A.  Vaziri, M. W.  Davidson, H.  Shroff, “Confined activation and subdiffractive localization enables whole-cell PALM with genetically expressed probes,” Nat. Methods 8, 327–333 (2011).
[CrossRef] [PubMed]

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C. S.  Smith, N.  Joseph, B.  Rieger, K. A.  Lidke, “Fast, single-molecule localization that achieves theoretically minimum uncertainty,” Nat. Methods 7, 373–375 (2010).
[CrossRef] [PubMed]

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E.  Betzig, G. H.  Patterson, R.  Sougrat, O. W.  Lindwasser, S.  Olenych, J. S.  Bonifacino, M. W.  Davidson, J.  Lippincott-Schwartz, H. F.  Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
[CrossRef] [PubMed]

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

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S. R. P.  Pavani, M. A.  Thompson, J. S.  Biteen, S. J.  Lord, N.  Liu, R. J.  Twieg, R.  Piestun, W. E.  Moerner, “Three-dimensional, single-molecule fluorescence imaging beyond the diffraction limit by using a double-helix point spread function,” Proc. Natl. Acad. Sci. USA. 106, 2995–2999 (2009).
[CrossRef] [PubMed]

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[CrossRef]

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E.  Toprak, H.  Balci, B. H.  Blehm, P. R.  Selvin, “Three-dimensional particle tracking via bifocal imaging,” Nano Lett. 7, 2043–2045 (2007)
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Figures (12)

Fig. 1
Fig. 1

Preprocessing of raw PSF data. All images show the PSF at z = −0.6 μm. On the left side of each image, the pixel intensities have been linearly scaled, while the right side shows the same image with intensity values on a log scale. (a) The raw PSF, cropped to 40 × 40 pixels. (b) Subtraction of background from (a) and application of the circular mask (see section 2.2). (c) PSF from (b) after extension to 128 × 128 pixels. This is the measured PSF used in phase retrieval algorithm at ith iteration where the ii0th iteration (see section 2.3). (d) PSF with additional processing (see section 2.3), which is used as measured PSF in phase retrieval algorithm at the ith iteration, where i > i0.

Fig. 2
Fig. 2

Schematic illustration of aberration caused by refractive index mismatch. (a) Geometric ray tracing from an on-axis point (blue) at the designed position and an emitter (red) located at an arbitrary z position. t g * and t i * are the designed thicknesses of the cover glass and the immersion medium respectively, while tg and ti are the corresponding thicknesses when the objective lens is moved away from the designed position. We assume the refractive index of the cover glass, the immersion medium, and the objective lens are the same and equal to n1, while n2 is refractive index of sample medium. (b) In the paraxial range, the image of a fluorophore located at reference position zo is at the designed focus position P.

Fig. 3
Fig. 3

Schematic illustration of the dual focal plane concept and setup

Fig. 4
Fig. 4

Pupil function and mPR-PSF for the dual focal plane setup. These images were obtained using the methods detailed in section 2 and section 3. mPR-PSFs were generated from Zernike fitted pupil function. Measured PSF images at z positions as indicated were used for phase retrieval. For this particular data, the calculated defocus was 3.9 nm. The plane separation was 334.9 nm, λ = 670 nm, and NA = 1.46, while the magnifications at plane 1 and plane 2 were 147 and 147 × 1.148, respectively. The x–z sections in the rightmost column shows the measured PSF and mPR-PSF in the x–z plane. The bottom images show the corresponding pupil functions in k space. Here, the Zernike fitted pupil functions are constructed from 49 Zernike polynomials. The scale bar is 500 nm. The support of pupil function in k-space is NA/λ

Fig. 5
Fig. 5

Measured PSFs and mPR-PSFs at various depths in the sample medium. The data were acquired at z positions from −2 to 2 μm, with 100 nm step size. The images were generated by summing over the 100 frames taken at each z step. The displayed depths are the z positions found in the sample medium using the method presented in section 4. The mPR-PSF was retrieved from the measured PSF at the cover glass where the depth is 0 μm, and the aberration phase is given by Eq. (9). The S-PSF was generated from an unaberrated pupil function.

Fig. 6
Fig. 6

3D localization of bead data. Data was acquired at 100 frames per z position, from −1 μm to 1 μm. (a) z positions found from the bead data: using the mPR-PSF model on the collected single frame data (blue crosses); using the mPR-PSF model (red solid line), the mS-PSF model (green dashed line), and the S-PSF model (brown dashed line) on the averaged PSF data. (b–d) Deviations in the found x, y, z positions of the averaged PSF data from the true positions. Color representation is the same as in (a), and deviations beyond 50 nm are not shown in the plots. (e–f) Fitting precision in x, y, z, using the mPR-PSF model on the averaged PSF data.

Fig. 7
Fig. 7

Fitting precision and CRLB for simulated data. Simulated data were generated from the mPR-PSF shown in Fig. 4, to which Poisson noise was added. I1 = 4000, bg1 = 4.5, bg2 = 3.5, Iratio = 0.7, z = [−1, −0.9,...,0.7] μm, 100 simulated single emitters per z position.

Fig. 8
Fig. 8

Localization speed and sum of square error (SSE) for various numbers of Zernike coefficients and iterations. The plots were generated from 1000 simulated emitters, which were generated from the mPR-PSF (Fig. 4), with I1 = 800, bg1 = 2, bg2 = 1.5, and with random x, y, z positions. The z range is −0.8 to 0.6 μm. The denoted number of iterations and Zernike coefficients are the ones used in the second fit (see text in section 6.3.3). (a) Fitting time and SSE versus number of Zernike coefficients, with the number of iterations fixed at 15. (b) Fitting time and SSE versus number of iterations, with the number of Zernike coefficients fixed at 9.

Fig. 9
Fig. 9

3D whole cell reconstruction using various PSF models. The sample consists of RBL cells labeled with IgE-Alexa 647 conjugate. Data was taken by scanning through the entire cell along the z axis, with a step size of 1 μm, and recording 2500 frames per step. The basic reconstruction method is described in section 4. All units are in μm. The color map indicates the actual z position and the color bars have the same color scale. (a) Reconstruction using the mPR-PSF and correcting for index-mismatch aberration. (b) Reconstruction using the mPR-PSF without index-mismatch correction. (c) Reconstruction using the S-PSF model. (d) Reconstruction using the S-PSF model and assuming the reference plane is the same as the stage position, without applying Eq. (7). In calculating PSF models, we used NA = 1.46, l = 690 nm, a lateral magnification of M = 148.6 at plane 1 and M = 148.6 × 1.135 at plane 2, and a CCD pixel size of Δd = 16 μm. The refractive index of sample medium and immersion medium are 1.33 and 1.53 respectively.

Fig. 10
Fig. 10

3D reconstruction of microtubules in an RBL cell. Data was acquired at a fixed z position near the cover glass. (a) 2D scatter plots of the 3D localization using the mPR-PSF (left) and S-PSF (right) as PSF models. Color represents the z position. (b) 2D scatter plot of the x–z projection of a single microtubule (the red box in (a)). (c) Probablility plot of the z localization of a small region (the red boxes) of the microtubule in (b). (d) Probability plots of z localization (left) and expected photon counts from plane 1 (right) of the raw fitting results of the region in the white box in (a). The raw fitting results are the total localizations before rejection and drift correction. In calculating both the mPR-PSF and the S-PSF, we used NA = 1.46, l = 682.7 nm, a refractive index of immersion medium n = 1.56, a lateral magnification of M = 149.5 at both plane 1 and plane 2, and a CCD pixel size of Δd = 16 μm.

Fig. 11
Fig. 11

Pupil function magnitude with supercritical effect. (a)–(b) Theoretical calculation from Eq. (34), with depth d = 0 and d = l. (c) Zernike fitted PR pupil function. (d) Pupil function generated by multiplying the decay term in (b) to (c). The PSF data used in phase retrieval process were acquired at z positions from −2 to 2 μm. In generating both theoretical and Zernike fitted pupil functions, we used l = 670 nm, NA = 1.49, a lateral magnification of M = 100, and a CCD pixel size of Δd = 16 μm. The refractive index of sample medium and immersion medium is 1.33 and 1.52 respectively.

Fig. 12
Fig. 12

Zernike expansion of the magnitude of PR pupil function. It uses the set of Zernike coefficients in Fig. 11. n is the order of Zernike coefficients, and each image shows the Zernike expansion up to nth order. The number of Zernike coefficients is (n + 1)2. In generating Zernike fitted pupil functions, we used l = 670 nm, NA = 1.49, a lateral magnification of M = 100, and a CCD pixel size of Δd = 16 μm. The refractive index of immersion medium is 1.52.

Equations (34)

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I PSF ( x , y , z ) = | P ( k x , k y ) e i 2 π ( k x x + k y y ) e i 2 π k z z d k x d k y | 2 ,
x shift = C 2 × M λ 2 π NA Δ d , y shift = C 3 × M λ 2 π NA Δ d ,
f = ( 2 π k z z shift W 4 ϕ 0 ) 2 d k x d k y ,
I PSF = | { P ( k x , k y ) e i 2 π k z z } | 2 { G ( k x , k y ) } ,
O P D = n 2 | O A | + n 1 | A B | n 1 | P C | ,
φ aber = 2 π λ [ z n 2 cos ( θ 2 ) ( t i * t i ) n 1 cos ( θ 1 ) ] .
z o n 2 n 1 z stage ,
z z z o .
φ aber = 2 π λ [ z o n 2 cos ( θ 2 ) z o n 1 2 n 2 cos ( θ 1 ) + z n 2 cos ( θ 2 ) ] .
μ ( x , y , z , I , b g ) = I PSF 0 ( x , y , z ) + b g .
L ( θ ) = q μ 1 q N 1 q e μ 1 q N 1 q ! q μ 2 q N 2 q e μ 2 q N 2 q ! , θ = θ ( x 1 , y 1 , z 1 , I 1 , b g 1 , b g 2 ) ,
f ( θ | D ) = Δ ln ( L ( θ ) ) Δ θ = q N 1 q μ 1 q μ 1 q Δ μ 1 q Δ θ + q N 2 q μ 2 q μ 2 q Δ μ 2 q Δ θ .
θ n + 1 = θ n f ( θ n | D ) f ( θ n | D ) .
Z est = p 1 z + p 2 ,
p 1 = S 2 z o + S 1 , p 2 = S 4 z o + S 3 ,
var ( θ i ) F i i 1 ,
F i j = E [ Δ In ( L ( θ | D ) ) Δ θ i Δ In ( L ( θ | D ) ) Δ θ j ] , θ = θ ( x 1 , y 1 , z 1 , I 1 , b g 1 , b g 2 ) .
F i j = q 1 μ 1 q Δ μ 1 q Δ θ i Δ μ 1 q Δ θ j + q 1 μ 2 q Δ μ 2 q Δ θ i Δ μ 2 q Δ θ j .
S S E fit = q ( N q μ q ) 2 ,
U ( x , y , z ) = P ( k x , k y ) e 2 π i ( k x x + k y y ) e i 2 π k z z d k x d k y ,
U ( r , ϕ , z ) = P ( k r , θ ) e 2 π i k r r cos ( θ ϕ ) e 2 π i z k 2 k r 2 k r d k r d θ .
U ( r , ϕ , z ) = 0 N A λ e 2 π i z k 2 k r 2 k r d k r 0 2 π P ( k r , θ + ϕ ) e 2 π i k r r cos θ d θ .
P ( k r , θ ) = n = 0 N m = 0 n R n m [ C a cos ( m ϕ + m θ ) + C b sin ( m ϕ + m θ ) ] ,
R n m = s = 0 n m ( 1 ) s ( 2 n m s ) ! s ! ( n s ) ! ( n m s ) ! ρ 2 ( n m s ) , ρ = k r N A / λ .
P ( k r , θ ) = n = 0 N m = 0 n R n m [ C a cos ( m ϕ ) cos ( m θ ) C a sin ( m ϕ ) sin ( m θ ) + C b sin ( m ϕ ) cos ( m θ ) + C b cos ( m ϕ ) sin ( m θ ) ] .
0 2 π cos ( m θ ) e 2 π i k r r cos θ d θ = 2 π i m J m ( 2 π k r r ) , 0 2 π sin ( m θ ) e 2 π i k r r cos θ d θ = 0 .
U ( r , ϕ , z ) = 2 π 0 N A / λ d k r k r e 2 π i z k 2 k r 2 n , m R n m i m J m ( 2 π k r r ) [ C a cos ( m ϕ ) + C b sin ( m ϕ ) ] .
U ( r , ϕ , z ) = 2 π 0 N A / λ d k r k r e i φ aber n , m R n m i m J m ( 2 π k r r ) [ C a cos ( m ϕ ) + C b sin ( m ϕ ) ] .
[ E r E i ] = 1 τ 21 [ e i k 2 z d 0 0 e i k 2 z d ] [ 1 ρ 21 ρ 21 1 ] [ 0 E t ] ,
E t = τ 21 e i k 2 z d E i ,
τ = 2 i γ 1 + i γ , τ | | = 2 i γ n 2 n 1 i γ + ( n 2 n 1 ) 2 , γ = ( n 1 sin θ 1 ) 2 n 2 2 n 1 cos θ 1 ,
k 2 z = i 2 π λ ( n 1 sin θ 1 ) 2 n 2 2 = i / δ .
E t = τ + τ | | 2 e d / δ E i ,
P = { ( τ + τ | | ) / 2 n 1 sin θ 1 n 2 e d / δ ( τ + τ | | ) / 2 n 2 < n 1 sin θ 1 N A .

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