Abstract

The Gaussian function is simple and easy to implement as Point Spread Function (PSF) model for fitting the position of fluorescent emitters in localization microscopy. Despite its attractiveness the appropriateness of the Gaussian is questionable as it is not based on the laws of optics. Here we study the effect of emission dipole orientation in conjunction with optical aberrations on the localization accuracy of position estimators based on a Gaussian model PSF. Simulated image spots, calculated with all effects of high numerical aperture, interfaces between media, polarization, dipole orientation and aberrations taken into account, were fitted with a Gaussian PSF based Maximum Likelihood Estimator. For freely rotating dipole emitters it is found that the Gaussian works fine. The same, theoretically optimum, localization accuracy is found as if the true PSF were a Gaussian, even for aberrations within the usual tolerance limit of high-end optical imaging systems such as microscopes (Marechal’s diffraction limit). For emitters with a fixed dipole orientation this is not the case. Localization errors are found that reach up to 40 nm for typical system parameters and aberration levels at the diffraction limit. These are systematic errors that are independent of the total photon count in the image. The Gaussian function is therefore inappropriate, and more sophisticated PSF models are a practical necessity.

© 2010 Optical Society of America

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2010 (2)

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

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

2009 (3)

D. Evanko, "Primer: fluorescence imaging under the diffraction limit," Nat. Methods 6, 19-20 (2009).
[CrossRef]

S. W. Hell, "Microscopy and its focal switch," Nat. Methods 6, 24-32 (2009).
[CrossRef] [PubMed]

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

2008 (5)

M. R. Foreman, C. M. Romero, and P. Török, "Determination of the three-dimensional orientation of single molecules," Opt. Lett. 33, 1020-1022 (2008).
[CrossRef] [PubMed]

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

M. F. Juette, T. J. Gould, M. D. Lessard, M. J. Mlodzianoski, B. S. Nagpure, B. T. Bennett, S. T. Hess, and J. Bewersdorf, "Three-dimensional sub-100nmresolution fluorescence microscopy of thick samples," Nat. Methods 5, 527-530 (2008).
[CrossRef] [PubMed]

J. Flling, M. Bossi, H. Bock, R. Medda, C. A. Wurm, B. Hein, S. Jakobs, C. Eggeling, and S. W. Hell, "Fluorescence nanoscopy by ground-state depletion and single-molecule return," Nat. Methods 5, 943-945 (2008).
[CrossRef]

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

2007 (7)

A. Egner, C. Geisler, C. von Middendorff, H. Bock, D. Wenzel, R. Medda, M. Andresen, A. C. Stiel, S. Jakobs, C. Eggeling, A. Schnle, and S. W. Hell, "Fluorescence nanoscopy in whole cells by asynchronous localization of photoswitching emitters," Biophys. J. 93, 3285-3290 (2007).
[CrossRef] [PubMed]

S. W. Hell, "Far-Field Optical Nanoscopy," Science 316, 1153-1158 (2007).
[CrossRef] [PubMed]

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

L. Holtzer, T. Meckel, and T. Schmidt, "Nanometric three-dimensional tracking of individual quantum dots in cells," Appl. Phys. Lett. 90, 053902 (2007).
[CrossRef]

B. Zhang, J. Zerubia, and J.-C. Olivo-Marin, "Gaussian approximations of fluorescence microscope point-spread function models," Appl. Opt. 46, 1819-1829 (2007).
[CrossRef] [PubMed]

P. Dedecker, B. Muls, J. Hofkens, J. Enderlein, and J. Hotta, "Orientational effects in the excitation and deexcitation of single molecules interacting with donut-mode laser beams," Opt. Express 15, 3372-3383 (2007).
[CrossRef] [PubMed]

M. R. Foreman, S. S. Sherif, and P. Török, "Photon statistics in single molecule orientational imaging," Opt. Express 15, 13597-13606 (2007).
[CrossRef] [PubMed]

2006 (3)

J. Enderlein, E. Toprak, and P. R. Selvin, "Polarization effect on position accuracy of fluorophore localization," Opt. Express 14, 8111 (2006).
[CrossRef] [PubMed]

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, "Imaging Intracellular Fluorescent Proteins at Nanometer Resolution," Science 313, 1643-1645 (2006).
[CrossRef]

M. J. Rust, M. Bates, and X. Zhuang, "Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM)," Nat. Methods 3, 793-795 (2006).
[CrossRef] [PubMed]

2005 (3)

S. P. Müller, C. K. Abbey, F. J. Rybicki, S. C. Moore, and M. F. Kijewski, "Measures of performance in nonlinear estimation tasks: prediction of estimation performance at low signal-to-noise ratio," Phys. Med. Biol. 50, 3697-3715 (2005).
[CrossRef] [PubMed]

S. Stallinga, "Compact description of substrate-related aberrations in high numerical aperture optical disk readout," Appl. Opt. 44, 949-958 (2005).
[CrossRef]

K. A. Lidke, B. Rieger, T. M. Jovin, and R. Heintzmann, "Superresolution by localization of quantum dots using blinking statistics," Opt. Express 13, 7052-7062 (2005).
[CrossRef] [PubMed]

2004 (2)

S. Stallinga, "Light distribution close to focus in biaxially birefringent media," J. Opt. Soc. Am. A 21, 1785-1798 (2004).
[CrossRef]

R. J. Ober, S. Ram, and E. S. Ward, "Localization Accuracy in Single-Molecule Microscopy," Biophys. J. 86, 1185-1200 (2004).
[CrossRef] [PubMed]

2003 (1)

2002 (2)

J. L. Bakx, "Efficient computation of optical disk readout by use of the chirp z transform," Appl. Opt. 41, 4879-4903 (2002).
[CrossRef]

R. E. Thompson, D. R. Larson, and W. W. Webb, "Precise nanometer localization analysis for individual fluorescent probes," Biophys. J. 82, 2775-2783 (2002).
[CrossRef] [PubMed]

2001 (1)

1999 (1)

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

1998 (1)

P. Török, P. D. Higdon, and T. Wilson, "Theory for confocal and conventional microscopes imaging small dielectric scatterers," J. Mod. Opt. 45, 1681-1698 (1998).
[CrossRef]

1997 (1)

T. Wilson, R. Juskaitis, and P. D. Higdon, "The imaging of dielectric point scatterers in conventional and confocal polarisation microscopes," Opt. Commun. 141, 298-313 (1997).
[CrossRef]

1994 (2)

1992 (1)

M. F. Kijewski, S. P. Müller, and S. C. Moore, "The Barankin bound: a model of detection with location uncertainty," Proc. SPIE 1768, 153-160 (1992).
[CrossRef]

Abbey, C. K.

S. P. Müller, C. K. Abbey, F. J. Rybicki, S. C. Moore, and M. F. Kijewski, "Measures of performance in nonlinear estimation tasks: prediction of estimation performance at low signal-to-noise ratio," Phys. Med. Biol. 50, 3697-3715 (2005).
[CrossRef] [PubMed]

Ammar, M.

Andresen, M.

A. Egner, C. Geisler, C. von Middendorff, H. Bock, D. Wenzel, R. Medda, M. Andresen, A. C. Stiel, S. Jakobs, C. Eggeling, A. Schnle, and S. W. Hell, "Fluorescence nanoscopy in whole cells by asynchronous localization of photoswitching emitters," Biophys. J. 93, 3285-3290 (2007).
[CrossRef] [PubMed]

Bakx, J. L.

J. L. Bakx, "Efficient computation of optical disk readout by use of the chirp z transform," Appl. Opt. 41, 4879-4903 (2002).
[CrossRef]

Balci, H.

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

Bartko, A. P.

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

Bates, M.

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

M. J. Rust, M. Bates, and X. Zhuang, "Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM)," Nat. Methods 3, 793-795 (2006).
[CrossRef] [PubMed]

Beane, G. L.

Bennett, B. T.

M. F. Juette, T. J. Gould, M. D. Lessard, M. J. Mlodzianoski, B. S. Nagpure, B. T. Bennett, S. T. Hess, and J. Bewersdorf, "Three-dimensional sub-100nmresolution fluorescence microscopy of thick samples," Nat. Methods 5, 527-530 (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, and H. F. Hess, "Imaging Intracellular Fluorescent Proteins at Nanometer Resolution," Science 313, 1643-1645 (2006).
[CrossRef]

Bewersdorf, J.

M. J. Mlodzianoski, M. F. Juette, G. L. Beane, and 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, and J. Bewersdorf, "Three-dimensional sub-100nmresolution fluorescence microscopy of thick samples," Nat. Methods 5, 527-530 (2008).
[CrossRef] [PubMed]

Blehm, B. H.

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

Bock, H.

J. Flling, M. Bossi, H. Bock, R. Medda, C. A. Wurm, B. Hein, S. Jakobs, C. Eggeling, and S. W. Hell, "Fluorescence nanoscopy by ground-state depletion and single-molecule return," Nat. Methods 5, 943-945 (2008).
[CrossRef]

A. Egner, C. Geisler, C. von Middendorff, H. Bock, D. Wenzel, R. Medda, M. Andresen, A. C. Stiel, S. Jakobs, C. Eggeling, A. Schnle, and S. W. Hell, "Fluorescence nanoscopy in whole cells by asynchronous localization of photoswitching emitters," Biophys. J. 93, 3285-3290 (2007).
[CrossRef] [PubMed]

Bonifacino, J. S.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, "Imaging Intracellular Fluorescent Proteins at Nanometer Resolution," Science 313, 1643-1645 (2006).
[CrossRef]

Bossi, M.

J. Flling, M. Bossi, H. Bock, R. Medda, C. A. Wurm, B. Hein, S. Jakobs, C. Eggeling, and S. W. Hell, "Fluorescence nanoscopy by ground-state depletion and single-molecule return," Nat. Methods 5, 943-945 (2008).
[CrossRef]

Chipman, R. A.

Churchman, L. S.

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

Davidson, M. W.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, "Imaging Intracellular Fluorescent Proteins at Nanometer Resolution," Science 313, 1643-1645 (2006).
[CrossRef]

Dedecker, P.

Dickson, R. M.

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

Eggeling, C.

J. Flling, M. Bossi, H. Bock, R. Medda, C. A. Wurm, B. Hein, S. Jakobs, C. Eggeling, and S. W. Hell, "Fluorescence nanoscopy by ground-state depletion and single-molecule return," Nat. Methods 5, 943-945 (2008).
[CrossRef]

A. Egner, C. Geisler, C. von Middendorff, H. Bock, D. Wenzel, R. Medda, M. Andresen, A. C. Stiel, S. Jakobs, C. Eggeling, A. Schnle, and S. W. Hell, "Fluorescence nanoscopy in whole cells by asynchronous localization of photoswitching emitters," Biophys. J. 93, 3285-3290 (2007).
[CrossRef] [PubMed]

Egner, A.

A. Egner, C. Geisler, C. von Middendorff, H. Bock, D. Wenzel, R. Medda, M. Andresen, A. C. Stiel, S. Jakobs, C. Eggeling, A. Schnle, and S. W. Hell, "Fluorescence nanoscopy in whole cells by asynchronous localization of photoswitching emitters," Biophys. J. 93, 3285-3290 (2007).
[CrossRef] [PubMed]

Enderlein, J.

Evanko, D.

D. Evanko, "Primer: fluorescence imaging under the diffraction limit," Nat. Methods 6, 19-20 (2009).
[CrossRef]

Flling, J.

J. Flling, M. Bossi, H. Bock, R. Medda, C. A. Wurm, B. Hein, S. Jakobs, C. Eggeling, and S. W. Hell, "Fluorescence nanoscopy by ground-state depletion and single-molecule return," Nat. Methods 5, 943-945 (2008).
[CrossRef]

Flyvbjerg, H.

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

Foreman, M. R.

Furukawa, H.

Geisler, C.

A. Egner, C. Geisler, C. von Middendorff, H. Bock, D. Wenzel, R. Medda, M. Andresen, A. C. Stiel, S. Jakobs, C. Eggeling, A. Schnle, and S. W. Hell, "Fluorescence nanoscopy in whole cells by asynchronous localization of photoswitching emitters," Biophys. J. 93, 3285-3290 (2007).
[CrossRef] [PubMed]

Gould, T. J.

M. F. Juette, T. J. Gould, M. D. Lessard, M. J. Mlodzianoski, B. S. Nagpure, B. T. Bennett, S. T. Hess, and J. Bewersdorf, "Three-dimensional sub-100nmresolution fluorescence microscopy of thick samples," Nat. Methods 5, 527-530 (2008).
[CrossRef] [PubMed]

Haeberlé, O.

Heilemann, M.

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

Hein, B.

J. Flling, M. Bossi, H. Bock, R. Medda, C. A. Wurm, B. Hein, S. Jakobs, C. Eggeling, and S. W. Hell, "Fluorescence nanoscopy by ground-state depletion and single-molecule return," Nat. Methods 5, 943-945 (2008).
[CrossRef]

Heintzmann, R.

Hell, S. W.

S. W. Hell, "Microscopy and its focal switch," Nat. Methods 6, 24-32 (2009).
[CrossRef] [PubMed]

J. Flling, M. Bossi, H. Bock, R. Medda, C. A. Wurm, B. Hein, S. Jakobs, C. Eggeling, and S. W. Hell, "Fluorescence nanoscopy by ground-state depletion and single-molecule return," Nat. Methods 5, 943-945 (2008).
[CrossRef]

A. Egner, C. Geisler, C. von Middendorff, H. Bock, D. Wenzel, R. Medda, M. Andresen, A. C. Stiel, S. Jakobs, C. Eggeling, A. Schnle, and S. W. Hell, "Fluorescence nanoscopy in whole cells by asynchronous localization of photoswitching emitters," Biophys. J. 93, 3285-3290 (2007).
[CrossRef] [PubMed]

S. W. Hell, "Far-Field Optical Nanoscopy," Science 316, 1153-1158 (2007).
[CrossRef] [PubMed]

Hess, H. F.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, "Imaging Intracellular Fluorescent Proteins at Nanometer Resolution," Science 313, 1643-1645 (2006).
[CrossRef]

Hess, S. T.

M. F. Juette, T. J. Gould, M. D. Lessard, M. J. Mlodzianoski, B. S. Nagpure, B. T. Bennett, S. T. Hess, and J. Bewersdorf, "Three-dimensional sub-100nmresolution fluorescence microscopy of thick samples," Nat. Methods 5, 527-530 (2008).
[CrossRef] [PubMed]

Higdon, P. D.

P. Török, P. D. Higdon, and T. Wilson, "Theory for confocal and conventional microscopes imaging small dielectric scatterers," J. Mod. Opt. 45, 1681-1698 (1998).
[CrossRef]

T. Wilson, R. Juskaitis, and P. D. Higdon, "The imaging of dielectric point scatterers in conventional and confocal polarisation microscopes," Opt. Commun. 141, 298-313 (1997).
[CrossRef]

Hofkens, J.

Holtzer, L.

L. Holtzer, T. Meckel, and T. Schmidt, "Nanometric three-dimensional tracking of individual quantum dots in cells," Appl. Phys. Lett. 90, 053902 (2007).
[CrossRef]

Hotta, J.

Huang, B.

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

Jakobs, S.

J. Flling, M. Bossi, H. Bock, R. Medda, C. A. Wurm, B. Hein, S. Jakobs, C. Eggeling, and S. W. Hell, "Fluorescence nanoscopy by ground-state depletion and single-molecule return," Nat. Methods 5, 943-945 (2008).
[CrossRef]

A. Egner, C. Geisler, C. von Middendorff, H. Bock, D. Wenzel, R. Medda, M. Andresen, A. C. Stiel, S. Jakobs, C. Eggeling, A. Schnle, and S. W. Hell, "Fluorescence nanoscopy in whole cells by asynchronous localization of photoswitching emitters," Biophys. J. 93, 3285-3290 (2007).
[CrossRef] [PubMed]

Joseph, N.

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

Jovin, T. M.

Juette, M. F.

M. J. Mlodzianoski, M. F. Juette, G. L. Beane, and 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, and J. Bewersdorf, "Three-dimensional sub-100nmresolution fluorescence microscopy of thick samples," Nat. Methods 5, 527-530 (2008).
[CrossRef] [PubMed]

Juskaitis, R.

T. Wilson, R. Juskaitis, and P. D. Higdon, "The imaging of dielectric point scatterers in conventional and confocal polarisation microscopes," Opt. Commun. 141, 298-313 (1997).
[CrossRef]

Kasper, R.

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

Kijewski, M. F.

S. P. Müller, C. K. Abbey, F. J. Rybicki, S. C. Moore, and M. F. Kijewski, "Measures of performance in nonlinear estimation tasks: prediction of estimation performance at low signal-to-noise ratio," Phys. Med. Biol. 50, 3697-3715 (2005).
[CrossRef] [PubMed]

M. F. Kijewski, S. P. Müller, and S. C. Moore, "The Barankin bound: a model of detection with location uncertainty," Proc. SPIE 1768, 153-160 (1992).
[CrossRef]

Larson, D. R.

R. E. Thompson, D. R. Larson, and W. W. Webb, "Precise nanometer localization analysis for individual fluorescent probes," Biophys. J. 82, 2775-2783 (2002).
[CrossRef] [PubMed]

Lessard, M. D.

M. F. Juette, T. J. Gould, M. D. Lessard, M. J. Mlodzianoski, B. S. Nagpure, B. T. Bennett, S. T. Hess, and J. Bewersdorf, "Three-dimensional sub-100nmresolution fluorescence microscopy of thick samples," Nat. Methods 5, 527-530 (2008).
[CrossRef] [PubMed]

Lidke, K. A.

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

K. A. Lidke, B. Rieger, T. M. Jovin, and R. Heintzmann, "Superresolution by localization of quantum dots using blinking statistics," Opt. Express 13, 7052-7062 (2005).
[CrossRef] [PubMed]

Lindwasser, O. W.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, "Imaging Intracellular Fluorescent Proteins at Nanometer Resolution," Science 313, 1643-1645 (2006).
[CrossRef]

Lippincott-Schwartz, J.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, "Imaging Intracellular Fluorescent Proteins at Nanometer Resolution," Science 313, 1643-1645 (2006).
[CrossRef]

McGuire, J. P.

Meckel, T.

L. Holtzer, T. Meckel, and T. Schmidt, "Nanometric three-dimensional tracking of individual quantum dots in cells," Appl. Phys. Lett. 90, 053902 (2007).
[CrossRef]

Medda, R.

J. Flling, M. Bossi, H. Bock, R. Medda, C. A. Wurm, B. Hein, S. Jakobs, C. Eggeling, and S. W. Hell, "Fluorescence nanoscopy by ground-state depletion and single-molecule return," Nat. Methods 5, 943-945 (2008).
[CrossRef]

A. Egner, C. Geisler, C. von Middendorff, H. Bock, D. Wenzel, R. Medda, M. Andresen, A. C. Stiel, S. Jakobs, C. Eggeling, A. Schnle, and S. W. Hell, "Fluorescence nanoscopy in whole cells by asynchronous localization of photoswitching emitters," Biophys. J. 93, 3285-3290 (2007).
[CrossRef] [PubMed]

Mlodzianoski, M. J.

M. J. Mlodzianoski, M. F. Juette, G. L. Beane, and 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, and J. Bewersdorf, "Three-dimensional sub-100nmresolution fluorescence microscopy of thick samples," Nat. Methods 5, 527-530 (2008).
[CrossRef] [PubMed]

Moore, S. C.

S. P. Müller, C. K. Abbey, F. J. Rybicki, S. C. Moore, and M. F. Kijewski, "Measures of performance in nonlinear estimation tasks: prediction of estimation performance at low signal-to-noise ratio," Phys. Med. Biol. 50, 3697-3715 (2005).
[CrossRef] [PubMed]

M. F. Kijewski, S. P. Müller, and S. C. Moore, "The Barankin bound: a model of detection with location uncertainty," Proc. SPIE 1768, 153-160 (1992).
[CrossRef]

Mortensen, K. I.

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

Mukherjee, A.

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

Müller, S. P.

S. P. Müller, C. K. Abbey, F. J. Rybicki, S. C. Moore, and M. F. Kijewski, "Measures of performance in nonlinear estimation tasks: prediction of estimation performance at low signal-to-noise ratio," Phys. Med. Biol. 50, 3697-3715 (2005).
[CrossRef] [PubMed]

M. F. Kijewski, S. P. Müller, and S. C. Moore, "The Barankin bound: a model of detection with location uncertainty," Proc. SPIE 1768, 153-160 (1992).
[CrossRef]

Muls, B.

Nagpure, B. S.

M. F. Juette, T. J. Gould, M. D. Lessard, M. J. Mlodzianoski, B. S. Nagpure, B. T. Bennett, S. T. Hess, and J. Bewersdorf, "Three-dimensional sub-100nmresolution fluorescence microscopy of thick samples," Nat. Methods 5, 527-530 (2008).
[CrossRef] [PubMed]

Ober, R. J.

R. J. Ober, S. Ram, and E. S. Ward, "Localization Accuracy in Single-Molecule Microscopy," Biophys. J. 86, 1185-1200 (2004).
[CrossRef] [PubMed]

Olenych, S.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, "Imaging Intracellular Fluorescent Proteins at Nanometer Resolution," Science 313, 1643-1645 (2006).
[CrossRef]

Olivo-Marin, J.-C.

Patterson, G. H.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, "Imaging Intracellular Fluorescent Proteins at Nanometer Resolution," Science 313, 1643-1645 (2006).
[CrossRef]

Ram, S.

R. J. Ober, S. Ram, and E. S. Ward, "Localization Accuracy in Single-Molecule Microscopy," Biophys. J. 86, 1185-1200 (2004).
[CrossRef] [PubMed]

Rieger, B.

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

K. A. Lidke, B. Rieger, T. M. Jovin, and R. Heintzmann, "Superresolution by localization of quantum dots using blinking statistics," Opt. Express 13, 7052-7062 (2005).
[CrossRef] [PubMed]

Romero, C. M.

Rust, M. J.

M. J. Rust, M. Bates, and X. Zhuang, "Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM)," Nat. Methods 3, 793-795 (2006).
[CrossRef] [PubMed]

Rybicki, F. J.

S. P. Müller, C. K. Abbey, F. J. Rybicki, S. C. Moore, and M. F. Kijewski, "Measures of performance in nonlinear estimation tasks: prediction of estimation performance at low signal-to-noise ratio," Phys. Med. Biol. 50, 3697-3715 (2005).
[CrossRef] [PubMed]

Sauer, M.

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

Schmidt, T.

L. Holtzer, T. Meckel, and T. Schmidt, "Nanometric three-dimensional tracking of individual quantum dots in cells," Appl. Phys. Lett. 90, 053902 (2007).
[CrossRef]

Schnle, A.

A. Egner, C. Geisler, C. von Middendorff, H. Bock, D. Wenzel, R. Medda, M. Andresen, A. C. Stiel, S. Jakobs, C. Eggeling, A. Schnle, and S. W. Hell, "Fluorescence nanoscopy in whole cells by asynchronous localization of photoswitching emitters," Biophys. J. 93, 3285-3290 (2007).
[CrossRef] [PubMed]

Schttpelz, M.

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

Seefeldt, B.

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

Selvin, P. R.

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

J. Enderlein, E. Toprak, and P. R. Selvin, "Polarization effect on position accuracy of fluorophore localization," Opt. Express 14, 8111 (2006).
[CrossRef] [PubMed]

Sherif, S. S.

Smith, C. S.

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

Sougrat, R.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, "Imaging Intracellular Fluorescent Proteins at Nanometer Resolution," Science 313, 1643-1645 (2006).
[CrossRef]

Spudich, J. A.

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

Stallinga, S.

Stiel, A. C.

A. Egner, C. Geisler, C. von Middendorff, H. Bock, D. Wenzel, R. Medda, M. Andresen, A. C. Stiel, S. Jakobs, C. Eggeling, A. Schnle, and S. W. Hell, "Fluorescence nanoscopy in whole cells by asynchronous localization of photoswitching emitters," Biophys. J. 93, 3285-3290 (2007).
[CrossRef] [PubMed]

Tenjimbayashi, K.

Thompson, R. E.

R. E. Thompson, D. R. Larson, and W. W. Webb, "Precise nanometer localization analysis for individual fluorescent probes," Biophys. J. 82, 2775-2783 (2002).
[CrossRef] [PubMed]

Tinnefeld, P.

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

Toprak, E.

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

J. Enderlein, E. Toprak, and P. R. Selvin, "Polarization effect on position accuracy of fluorophore localization," Opt. Express 14, 8111 (2006).
[CrossRef] [PubMed]

Török, P.

van de Linde, S.

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

von Middendorff, C.

A. Egner, C. Geisler, C. von Middendorff, H. Bock, D. Wenzel, R. Medda, M. Andresen, A. C. Stiel, S. Jakobs, C. Eggeling, A. Schnle, and S. W. Hell, "Fluorescence nanoscopy in whole cells by asynchronous localization of photoswitching emitters," Biophys. J. 93, 3285-3290 (2007).
[CrossRef] [PubMed]

Wang, W.

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

Ward, E. S.

R. J. Ober, S. Ram, and E. S. Ward, "Localization Accuracy in Single-Molecule Microscopy," Biophys. J. 86, 1185-1200 (2004).
[CrossRef] [PubMed]

Webb, W. W.

R. E. Thompson, D. R. Larson, and W. W. Webb, "Precise nanometer localization analysis for individual fluorescent probes," Biophys. J. 82, 2775-2783 (2002).
[CrossRef] [PubMed]

Wenzel, D.

A. Egner, C. Geisler, C. von Middendorff, H. Bock, D. Wenzel, R. Medda, M. Andresen, A. C. Stiel, S. Jakobs, C. Eggeling, A. Schnle, and S. W. Hell, "Fluorescence nanoscopy in whole cells by asynchronous localization of photoswitching emitters," Biophys. J. 93, 3285-3290 (2007).
[CrossRef] [PubMed]

Wilson, T.

P. Török, P. D. Higdon, and T. Wilson, "Theory for confocal and conventional microscopes imaging small dielectric scatterers," J. Mod. Opt. 45, 1681-1698 (1998).
[CrossRef]

T. Wilson, R. Juskaitis, and P. D. Higdon, "The imaging of dielectric point scatterers in conventional and confocal polarisation microscopes," Opt. Commun. 141, 298-313 (1997).
[CrossRef]

Wurm, C. A.

J. Flling, M. Bossi, H. Bock, R. Medda, C. A. Wurm, B. Hein, S. Jakobs, C. Eggeling, and S. W. Hell, "Fluorescence nanoscopy by ground-state depletion and single-molecule return," Nat. Methods 5, 943-945 (2008).
[CrossRef]

Zerubia, J.

Zhang, B.

Zhuang, X.

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

M. J. Rust, M. Bates, and X. Zhuang, "Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM)," Nat. Methods 3, 793-795 (2006).
[CrossRef] [PubMed]

Angew. Chem. Int. Ed. Engl. (1)

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

Appl. Opt. (5)

Appl. Phys. Lett. (1)

L. Holtzer, T. Meckel, and T. Schmidt, "Nanometric three-dimensional tracking of individual quantum dots in cells," Appl. Phys. Lett. 90, 053902 (2007).
[CrossRef]

Biophys. J. (3)

R. E. Thompson, D. R. Larson, and W. W. Webb, "Precise nanometer localization analysis for individual fluorescent probes," Biophys. J. 82, 2775-2783 (2002).
[CrossRef] [PubMed]

R. J. Ober, S. Ram, and E. S. Ward, "Localization Accuracy in Single-Molecule Microscopy," Biophys. J. 86, 1185-1200 (2004).
[CrossRef] [PubMed]

A. Egner, C. Geisler, C. von Middendorff, H. Bock, D. Wenzel, R. Medda, M. Andresen, A. C. Stiel, S. Jakobs, C. Eggeling, A. Schnle, and S. W. Hell, "Fluorescence nanoscopy in whole cells by asynchronous localization of photoswitching emitters," Biophys. J. 93, 3285-3290 (2007).
[CrossRef] [PubMed]

J. Mod. Opt. (1)

P. Török, P. D. Higdon, and T. Wilson, "Theory for confocal and conventional microscopes imaging small dielectric scatterers," J. Mod. Opt. 45, 1681-1698 (1998).
[CrossRef]

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

J. Phys. Chem. B (1)

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

Nano Lett. (1)

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

Nat. Methods (7)

M. F. Juette, T. J. Gould, M. D. Lessard, M. J. Mlodzianoski, B. S. Nagpure, B. T. Bennett, S. T. Hess, and J. Bewersdorf, "Three-dimensional sub-100nmresolution fluorescence microscopy of thick samples," Nat. Methods 5, 527-530 (2008).
[CrossRef] [PubMed]

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

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

M. J. Rust, M. Bates, and X. Zhuang, "Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM)," Nat. Methods 3, 793-795 (2006).
[CrossRef] [PubMed]

J. Flling, M. Bossi, H. Bock, R. Medda, C. A. Wurm, B. Hein, S. Jakobs, C. Eggeling, and S. W. Hell, "Fluorescence nanoscopy by ground-state depletion and single-molecule return," Nat. Methods 5, 943-945 (2008).
[CrossRef]

D. Evanko, "Primer: fluorescence imaging under the diffraction limit," Nat. Methods 6, 19-20 (2009).
[CrossRef]

S. W. Hell, "Microscopy and its focal switch," Nat. Methods 6, 24-32 (2009).
[CrossRef] [PubMed]

Opt. Commun. (1)

T. Wilson, R. Juskaitis, and P. D. Higdon, "The imaging of dielectric point scatterers in conventional and confocal polarisation microscopes," Opt. Commun. 141, 298-313 (1997).
[CrossRef]

Opt. Express (6)

Opt. Lett. (1)

Phys. Med. Biol. (1)

S. P. Müller, C. K. Abbey, F. J. Rybicki, S. C. Moore, and M. F. Kijewski, "Measures of performance in nonlinear estimation tasks: prediction of estimation performance at low signal-to-noise ratio," Phys. Med. Biol. 50, 3697-3715 (2005).
[CrossRef] [PubMed]

Proc. SPIE (1)

M. F. Kijewski, S. P. Müller, and S. C. Moore, "The Barankin bound: a model of detection with location uncertainty," Proc. SPIE 1768, 153-160 (1992).
[CrossRef]

Science (3)

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

S. W. Hell, "Far-Field Optical Nanoscopy," Science 316, 1153-1158 (2007).
[CrossRef] [PubMed]

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, "Imaging Intracellular Fluorescent Proteins at Nanometer Resolution," Science 313, 1643-1645 (2006).
[CrossRef]

Other (1)

A. van den Bos, Parameter Estimation for Scientists and Engineers (Wiley & Sons, New Jersey, 2007).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Dipole orientation averaged PSF for a 100 ×/1.25 water immersion objective, an Airy-distribution for the same NA, and a Gaussian with standard deviation σ = 0.25 λ /NA.

Fig. 2
Fig. 2

Simulated spot shapes for different dipole orientations and aberrations. Left: Image spots for NAob = 1.45 (which is larger than nmed = 1.33) for a fixed dipole with polar angle θd = 0 (a), θd = π/4 (b), θd = π /2 (c). Right: Image spots for a free dipole aberrated by 300 m λ RMS defocus (d), 160 m λ RMS coma (e), and 100 m λ RMS astigmatism (f).

Fig. 3
Fig. 3

Simulated spot shape for NAob = 1.25 (which is less than nmed = 1. 33) for a fixed dipole with polar angle θd = π /4, without aberrations (a) and with 72 m λ RMS defocus (b).

Fig. 4
Fig. 4

Estimated error as a function of detected number of photons for fixed/free dipole orientation and NAob values below and above the refractive index of the medium, and the theoretical Gaussian based CRLB.

Fig. 5
Fig. 5

Estimated error as a function of detected number of photons for different aberration settings. Left (a to d): averaged over dipole orientation, right (e to f): randomly selected fixed dipole orientation. First row: defocus (a and e), second row: astigmatism (b and f), third row: coma (c and g), fourth row: spherical aberration (d and h).

Fig. 6
Fig. 6

Estimated error as a function of detected number of photons for fixed dipole orientation and NAob = 1.25, for different defocus settings in presence of a background of one photon per pixel, and the theoretical Gaussian based CRLB with this background.

Equations (39)

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v x = n med sin θ med N A ob cos ϕ ,
v y = n med sin θ med N A ob sin ϕ .
E pub , j ( v , u d ) = π d 0 ɛ 0 λ 2 F ob ( 1 v 2 NA ob 2 / n med 2 ) 1 / 4 k = x , y , z q j k ( v ) d k e 2 π i u d v ,
q x k = cos ϕ T p p k sin ϕ T s s k ,
q y k = sin ϕ T p p k + cos ϕ T s s k ,
T a = T a , med cov T a , cov imm ,
T a , 1 2 = 2 c a , 1 c a + 1 + c a + 2 ,
C ( v ) = { A ( v ) exp ( i W ( v ) ) , | v | 1 , 0 , | v | > 1 , }
E im , j ( u , u d ) = E 0 k = x , y , z w j k ( u u d ) d k ,
E 0 = π 2 NA ob NA im d 0 i ɛ 0 λ 3 ,
w j k ( u ) = 1 π d 2 v C ( v ) q j k ( v ) ( 1 v 2 NA ob 2 / n med 2 ) 1 / 4 exp ( 2 π i u v ) .
I ( u , u d ) = 1 2 ɛ 0 c j = x , y | E im , j ( u , u d ) | 2 = I 0 k , l = x , y , z [ j = x , y w j k ( u u d ) w j l ( u u d ) * ] d k d l .
I 0 = π 4 c NA ob 2 NA im 2 | d 0 | 2 2 ɛ 0 λ 6 .
P ( u p , u d ) = λ 2 NA im 2 A p d 2 u I ( u , u d ) = P 0 k , l = x , y , z [ j = x , y A p d 2 u w j k ( u u d ) w j l ( u u d ) * ] d k d l ,
P 0 = π 4 c NA ob 2 | d 0 | 2 2 ɛ 0 λ 4 .
d 0 = α E ill d ex ,
E foc , k ( u ) = π F ob i λ j = x , y A j w j k ( u ) ,
w x x ( u , ψ ) = F 0 ( u ) + F 2 ( u ) cos ( 2 ψ ) ,
w x y ( u , ψ ) = F 2 ( u ) cos ( 2 ψ ) ,
w x z ( u , ψ ) = i F 1 ( u ) cos ψ ,
w y x ( u , ψ ) = F 2 ( u ) cos ( 2 ψ ) ,
w y y ( u , ψ ) = F 0 ( u ) F 2 ( u ) cos ( 2 ψ ) ,
w y z ( u , ψ ) = i F 1 ( u ) sin ψ ,
F 0 ( u ) = 2 0 1 d v v ( T s + T p 1 v 2 NA ob 2 / n med 2 ) 2 ( 1 v 2 NA ob 2 / n med 2 ) 1 / 4 J 0 ( 2 π u v ) exp ( i W ( v ) ) ,
F 1 ( u ) = 2 0 1 d v v 2 T p NA ob / n med ( 1 v 2 NA ob 2 / n med 2 ) 1 / 4 J 1 ( 2 π u v ) exp ( i W ( v ) ) ,
F 2 ( u ) = 2 0 1 d v v ( T s T p 1 v 2 NA ob 2 / n med 2 ) 2 ( 1 v 2 NA ob 2 / n med 2 ) 1 / 4 J 2 ( 2 π u v ) exp ( i W ( v ) ) .
E im , x ( u , ψ ) = E 0 [ F 0 ( u ) sin θ d cos ϕ d + F 2 ( u ) sin θ d cos ( 2 ψ ϕ d ) + i F 1 ( u ) cos θ d cos ψ , ]
E im , y ( u , ψ ) = E 0 [ F 0 ( u ) sin θ d sin ϕ d + F 2 ( u ) sin θ d sin ( 2 ψ ϕ d ) + i F 1 ( u ) cos θ d sin ψ ] ,
I ( u , ψ ) = I 0 [ ( | F 0 ( u ) | 2 + | F 2 ( u ) | 2 ) sin 2 θ d + | F 1 ( u ) | 2 cos 2 θ d + 2 Im { ( F 0 ( u ) + F 2 ( u ) ) F 1 ( u ) * } sin ( 2 θ d ) cos ( ψ ϕ d ) 2 Re { F 0 ( u ) F 2 ( u ) * } sin 2 θ d cos ( 2 ψ 2 ϕ d ) ] ,
I ( u ) = 1 3 I 0 ( 2 | F 0 ( u ) | 2 + 2 | F 2 ( u ) | 2 + | F 1 ( u ) | 2 ) ,
I = i j k l C i j A k l d ex , i d ex , j d k d l ,
A k l = I 0 l w j l w k l * ,
C i j = Re ( E ill , i E ill , j * ) E ,
E ill = 1 2 A 0 ( p + i q ) ,
C i j = 1 2 | A 0 | 2 p i p j + q i q j E .
I = 1 9 k l C k k A l l ,
d j d k = 1 3 δ j k ,
d i d j d k d l = 1 15 ( δ i j δ k l + δ i l δ j k + δ i k δ j l ) ,
I = 1 15 k l ( C k k A l l + 2 C k l A k l ) .

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