Abstract

Fluorescence nanoscopy has become an indispensable tool for studying organelle structures, protein dynamics, and interactions in biological sciences. Single-molecule localization microscopy can now routinely achieve 10–50 nm resolution through fluorescently labeled specimens in lateral optical sections. However, visualizing structures organized along the axial direction demands scanning and imaging each of the lateral imaging planes with fine intervals throughout the whole cell. This iterative process suffers from photobleaching of tagged probes, is susceptible to alignment artifacts and also limits the imaging speed. Here, we focused on the axial plane super-resolution imaging which integrated the single-objective light-sheet illumination and axial plane optical imaging with single-molecule localization technique to resolve nanoscale cellular architectures along the axial (or depth) dimension without scanning. We demonstrated that this method is compatible with DNA points accumulation for imaging in nanoscale topography (DNA-PAINT) and exchange-PAINT by virtue of its light-sheet illumination, allowing multiplexed super-resolution imaging throughout the depth of whole cells. We further demonstrated this proposed system by resolving the axial distributions of intracellular organelles such as microtubules, mitochondria, and nuclear pore complexes in both COS-7 cells and glioblastoma patient-derived tumor cells.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2019 (1)

J. Kim, M. Wojcik, Y. Wang, S. Moon, E. A. Zin, N. Marnani, Z. L. Newman, J. G. Flannery, K. Xu, and X. Zhang, “Oblique-plane single-molecule localization microscopy for tissues and small intact animals,” Nat. Methods 16(9), 853–857 (2019).
[Crossref]

2018 (5)

A. K. Gustavsson, P. N. Petrov, M. Y. Lee, Y. Shechtman, and W. E. Moerner, “3D single-molecule super-resolution microscopy with a tilted light sheet,” Nat. Commun. 9(1), 123 (2018).
[Crossref]

M. J. Mlodzianoski, P. J. Cheng-Hathaway, S. M. Bemiller, T. J. McCray, S. Liu, D. A. Miller, B. T. Lamb, G. E. Landreth, and F. Huang, “Active PSF shaping and adaptive optics enable volumetric localization microscopy through brain sections,” Nat. Methods 15(8), 583–586 (2018).
[Crossref]

Y. Wu and H. Shroff, “Faster, sharper, and deeper: structured illumination microscopy for biological imaging,” Nat. Methods 15(12), 1011–1019 (2018).
[Crossref]

P. Zhang, S. Liu, A. Chaurasia, D. Ma, M. J. Mlodzianoski, E. Culurciello, and F. Huang, “Analyzing complex single-molecule emission patterns with deep learning,” Nat. Methods 15(11), 913–916 (2018).
[Crossref]

A. Aristov, B. Lelandais, E. Rensen, and C. Zimmer, “ZOLA-3D allows flexible 3D localization microscopy over an adjustable axial range,” Nat. Commun. 9(1), 2409 (2018).
[Crossref]

2017 (4)

J. Schnitzbauer, M. T. Strauss, T. Schlichthaerle, F. Schueder, and R. Jungmann, “Super-resolution microscopy with DNA-PAINT,” Nat. Protoc. 12(6), 1198–1228 (2017).
[Crossref]

F. Schueder, J. Lara-Gutiérrez, B. J. Beliveau, S. K. Saka, H. M. Sasaki, J. B. Woehrstein, M. T. Strauss, H. Grabmayr, P. Yin, and R. Jungmann, “Multiplexed 3D super-resolution imaging of whole cells using spinning disk confocal microscopy and DNA-PAINT,” Nat. Commun. 8(1), 2090 (2017).
[Crossref]

S. S. Agasti, Y. Wang, F. Schueder, A. Sukumar, R. Jungmann, and P. Yin, “DNA-barcoded labeling probes for highly multiplexed Exchange-PAINT imaging,” Chem. Sci. 8(4), 3080–3091 (2017).
[Crossref]

Y. Wang, J. B. Woehrstein, N. Donoghue, M. Dai, M. S. Avendaño, R. C. Schackmann, J. J. Zoeller, S. S. H. Wang, P. W. Tillberg, D. Park, and S. W. Lapan, “Rapid sequential in situ multiplexing with DNA exchange imaging in neuronal cells and tissues,” Nano Lett. 17(10), 6131–6139 (2017).
[Crossref]

2016 (2)

F. Huang, G. Sirinakis, E. S. Allgeyer, L. K. Schroeder, W. C. Duim, E. B. Kromann, T. Phan, F. E. Rivera-Molina, J. R. Myers, I. Irnov, and M. Lessard, “Ultra-high resolution 3D imaging of whole cells,” Cell 166(4), 1028–1040 (2016).
[Crossref]

M. B. Meddens, S. Liu, P. S. Finnegan, T. L. Edwards, C. D. James, and K. A. Lidke, “Single objective light-sheet microscopy for high-speed whole-cell 3D super-resolution,” Biomed. Opt. Express 7(6), 2219–2236 (2016).
[Crossref]

2015 (2)

T. Li, S. Ota, J. Kim, Z. J. Wong, Y. Wang, X. Yin, and X. Zhang, “Axial plane optical microscopy,” Sci. Rep. 4(1), 7253 (2015).
[Crossref]

R. Galland, G. Grenci, A. Aravind, V. Viasnoff, V. Studer, and J. B. Sibarita, “3D high-and super-resolution imaging using single-objective SPIM,” Nat. Methods 12(7), 641–644 (2015).
[Crossref]

2014 (5)

Z. W. Zhao, R. Roy, J. C. M. Gebhardt, D. M. Suter, A. R. Chapman, and X. S. Xie, “Spatial organization of RNA polymerase II inside a mammalian cell nucleus revealed by reflected light-sheet superresolution microscopy,” Proc. Natl. Acad. Sci. U. S. A. 111(2), 681–686 (2014).
[Crossref]

B. C. Chen, W. R. Legant, K. Wang, L. Shao, D. E. Milkie, M. W. Davidson, C. Janetopoulos, X. S. Wu, J. A. Hammer, Z. Liu, and B. P. English, “Lattice light-sheet microscopy: imaging molecules to embryos at high spatiotemporal resolution,” Science 346(6208), 1257998 (2014).
[Crossref]

R. Jungmann, M. S. Avendaño, J. B. Woehrstein, M. Dai, W. M. Shih, and P. Yin, “Multiplexed 3D cellular super-resolution imaging with DNA-PAINT and Exchange-PAINT,” Nat. Methods 11(3), 313–318 (2014).
[Crossref]

Y. Wang, J. Schnitzbauer, Z. Hu, X. Li, Y. Cheng, Z. L. Huang, and B. Huang, “Localization events-based sample drift correction for localization microscopy with redundant cross-correlation algorithm,” Opt. Express 22(13), 15982–15991 (2014).
[Crossref]

J. Kim, T. Li, Y. Wang, and X. Zhang, “Vectorial point spread function and optical transfer function in oblique plane imaging,” Opt. Express 22(9), 11140–11151 (2014).
[Crossref]

2013 (5)

S. Liu, E. B. Kromann, W. D. Krueger, J. Bewersdorf, and K. A. Lidke, “Three dimensional single molecule localization using a phase retrieved pupil function,” Opt. Express 21(24), 29462–29487 (2013).
[Crossref]

R. McGorty, D. Kamiyama, and B. Huang, “Active microscope stabilization in three dimensions using image correlation,” Opt. Nano. 2(1), 3 (2013).
[Crossref]

X. Li, P. Mooney, S. Zheng, C. R. Booth, M. B. Braunfeld, S. Gubbens, D. A. Agard, and Y. Cheng, “Electron counting and beam-induced motion correction enable near-atomic-resolution single particle cryo-EM,” Nat. Methods 10(6), 584–590 (2013).
[Crossref]

J. C. M. Gebhardt, D. M. Suter, R. Roy, Z. W. Zhao, A. R. Chapman, S. Basu, T. Maniatis, and X. S. Xie, “Single-molecule imaging of transcription factor binding to DNA in live mammalian cells,” Nat. Methods 10(5), 421–426 (2013).
[Crossref]

F. Huang, T. M. Hartwich, F. E. Rivera-Molina, Y. Lin, W. C. Duim, J. J. Long, P. D. Uchil, J. R. Myers, M. A. Baird, W. Mothes, and M. W. Davidson, “Video-rate nanoscopy using sCMOS camera–specific single-molecule localization algorithms,” Nat. Methods 10(7), 653–658 (2013).
[Crossref]

2011 (2)

A. Maizel, D. von Wangenheim, F. Federici, J. Haseloff, and E. H. Stelzer, “High-resolution live imaging of plant growth in near physiological bright conditions using light sheet fluorescence microscopy,” Plant J. 68(2), 377–385 (2011).
[Crossref]

D. Aquino, A. Schönle, C. Geisler, C. v Middendorff, C. A. Wurm, Y. Okamura, T. Lang, S. W. Hell, and A. Egner, “Two-color nanoscopy of three-dimensional volumes by 4Pi detection of stochastically switched fluorophores,” Nat. Methods 8(4), 353–359 (2011).
[Crossref]

2010 (3)

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(5), 377–381 (2010).
[Crossref]

C. S. Smith, N. Joseph, B. Rieger, and K. A. Lidke, “Fast, single-molecule localization that achieves theoretically minimum uncertainty,” Nat. Methods 7(5), 373–375 (2010).
[Crossref]

R. Jungmann, C. Steinhauer, M. Scheible, A. Kuzyk, P. Tinnefeld, and F. C. Simmel, “Single-molecule kinetics and super-resolution microscopy by fluorescence imaging of transient binding on DNA origami,” Nano Lett. 10(11), 4756–4761 (2010).
[Crossref]

2008 (4)

P. J. Keller and E. H. Stelzer, “Quantitative in vivo imaging of entire embryos with digital scanned laser light sheet fluorescence microscopy,” Curr. Opin. Neurobiol. 18(6), 624–632 (2008).
[Crossref]

B. Huang, W. Wang, M. Bates, and X. Zhuang, “Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy,” Science 319(5864), 810–813 (2008).
[Crossref]

S. Manley, J. M. Gillette, G. H. Patterson, H. Shroff, H. F. Hess, E. Betzig, and J. Lippincott-Schwartz, “High-density mapping of single-molecule trajectories with photoactivated localization microscopy,” Nat. Methods 5(2), 155–157 (2008).
[Crossref]

B. Huang, S. A. Jones, B. Brandenburg, and X. Zhuang, “Whole-cell 3D STORM reveals interactions between cellular structures with nanometer-scale resolution,” Nat. Methods 5(12), 1047–1052 (2008).
[Crossref]

2007 (1)

S. W. Hell, “Far-field optical nanoscopy,” Science 316(5828), 1153–1158 (2007).
[Crossref]

2006 (3)

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(5793), 1642–1645 (2006).
[Crossref]

S. T. Hess, T. P. Girirajan, and M. D. Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J. 91(11), 4258–4272 (2006).
[Crossref]

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3(10), 793–796 (2006).
[Crossref]

2005 (2)

M. Hofmann, C. Eggeling, S. Jakobs, and S. W. Hell, “Breaking the diffraction barrier in fluorescence microscopy at low light intensities by using reversibly photoswitchable proteins,” Proc. Natl. Acad. Sci. U. S. A. 102(49), 17565–17569 (2005).
[Crossref]

M. G. Gustafsson, “Nonlinear structured-illumination microscopy: wide-field fluorescence imaging with theoretically unlimited resolution,” Proc. Natl. Acad. Sci. U. S. A. 102(37), 13081–13086 (2005).
[Crossref]

2002 (1)

R. E. Thompson, D. R. Larson, and W. W. Webb, “Precise nanometer localization analysis for individual fluorescent probes,” Biophys. J. 82(5), 2775–2783 (2002).
[Crossref]

2000 (1)

M. G. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198(2), 82–87 (2000).
[Crossref]

1999 (1)

1994 (1)

1991 (1)

Agard, D. A.

X. Li, P. Mooney, S. Zheng, C. R. Booth, M. B. Braunfeld, S. Gubbens, D. A. Agard, and Y. Cheng, “Electron counting and beam-induced motion correction enable near-atomic-resolution single particle cryo-EM,” Nat. Methods 10(6), 584–590 (2013).
[Crossref]

Agasti, S. S.

S. S. Agasti, Y. Wang, F. Schueder, A. Sukumar, R. Jungmann, and P. Yin, “DNA-barcoded labeling probes for highly multiplexed Exchange-PAINT imaging,” Chem. Sci. 8(4), 3080–3091 (2017).
[Crossref]

Allgeyer, E. S.

F. Huang, G. Sirinakis, E. S. Allgeyer, L. K. Schroeder, W. C. Duim, E. B. Kromann, T. Phan, F. E. Rivera-Molina, J. R. Myers, I. Irnov, and M. Lessard, “Ultra-high resolution 3D imaging of whole cells,” Cell 166(4), 1028–1040 (2016).
[Crossref]

Aquino, D.

D. Aquino, A. Schönle, C. Geisler, C. v Middendorff, C. A. Wurm, Y. Okamura, T. Lang, S. W. Hell, and A. Egner, “Two-color nanoscopy of three-dimensional volumes by 4Pi detection of stochastically switched fluorophores,” Nat. Methods 8(4), 353–359 (2011).
[Crossref]

Aravind, A.

R. Galland, G. Grenci, A. Aravind, V. Viasnoff, V. Studer, and J. B. Sibarita, “3D high-and super-resolution imaging using single-objective SPIM,” Nat. Methods 12(7), 641–644 (2015).
[Crossref]

Aristov, A.

A. Aristov, B. Lelandais, E. Rensen, and C. Zimmer, “ZOLA-3D allows flexible 3D localization microscopy over an adjustable axial range,” Nat. Commun. 9(1), 2409 (2018).
[Crossref]

Avendaño, M. S.

Y. Wang, J. B. Woehrstein, N. Donoghue, M. Dai, M. S. Avendaño, R. C. Schackmann, J. J. Zoeller, S. S. H. Wang, P. W. Tillberg, D. Park, and S. W. Lapan, “Rapid sequential in situ multiplexing with DNA exchange imaging in neuronal cells and tissues,” Nano Lett. 17(10), 6131–6139 (2017).
[Crossref]

R. Jungmann, M. S. Avendaño, J. B. Woehrstein, M. Dai, W. M. Shih, and P. Yin, “Multiplexed 3D cellular super-resolution imaging with DNA-PAINT and Exchange-PAINT,” Nat. Methods 11(3), 313–318 (2014).
[Crossref]

Baird, M. A.

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Z. W. Zhao, R. Roy, J. C. M. Gebhardt, D. M. Suter, A. R. Chapman, and X. S. Xie, “Spatial organization of RNA polymerase II inside a mammalian cell nucleus revealed by reflected light-sheet superresolution microscopy,” Proc. Natl. Acad. Sci. U. S. A. 111(2), 681–686 (2014).
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Figures (9)

Fig. 1.
Fig. 1. Working principle schematics of the axial plane SMLM and experimental quantifications. (A) Optical diagram of the axial plane SMLM. The illumination method can be switched between light sheet and epi by adding or removing the light-sheet generating part marked with the blue dashed rectangle (a pair of cylindrical lenses and a slit). (B) Excitation and detection profile at the imaging objective. A vertical light sheet is used for exciting a thin slice of the specimen and the emission fluorescence is collected by the same objective lens. (C) Axial plane image formation diagram using a 45° tilted mirror above the remote objective. The 45° tilted mirror rotates the axial plane image of the specimen to the lateral focal plane of the remote objective. (D) Ray paths schematic showing the axial plane imaging concept (the dashed lines represent the effective incident rays to the tilted mirror, and the solid lines represent the effective reflected rays from the tilted mirror). (E) Effective 3D pupil of the axial plane imaging system. The mirror is placed α/2 degree (45°) to the x-y plane, the effective 3D pupil is shown as the overlapped area (jet) of the original pupil (blue) and the rotated pupil (green). (F) Comparison of a simulated ideal axial plane PSF (left) and an experimental PSF (right) of the SMLM system. (G) Example images from blinking dataset excited by the light-sheet illumination and epi illumination during single-molecule imaging with DNA-PAINT probes. (H) Normalized distribution of background photon estimation of the light-sheet illumination and epi illumination corresponding to G. The light-sheet illumination reduced the background ∼11 fold compared to the epi illumination. L: lens; M: mirror; Di: dichroic mirror; AOTF: acousto-optic tunable filter; AL: aspheric lens; SM Fiber: single-mode optical fiber; CL: cylindrical lens; CF: cleanup filter; Obj1: imaging objective lens; Obj2: remote objective lens; EF: emission filter; PBS: polarizing beam splitter cube; QWP: quarter-wave plate; TM: tilted mirror. Scale bar: 500 nm in F, 5 µm in G.
Fig. 2.
Fig. 2. Super-resolution reconstructions of microtubules and nuclear pore complexes by the axial plane SMLM in fixed COS-7 cells. (A) Diffraction-limited axial plane image of α-tubulin immuno-labeled with Alexa Fluor 647 (AF647). (B) Axial plane super-resolution image of A. (C) Enlarged views of three microtubule segments marked in A and B with the yellow, blue, and magenta squares respectively. (D) Normalized intensity profiles of microtubules marked with the white dashed rectangles in C. (E) Diffraction-limited axial plane image of Nup98 immuno-labeled with AF647. (F) Axial plane super-resolution image of E. (G) Enlarged views of two regions marked in E and F with the orange and green squares respectively. (H) Normalized intensity profiles in the x and z directions of two nuclear pore complexes (NPCs) indicated by the white arrows in G. Scale bar: 2 µm in A, B, E and F, 500 nm in C and G, 100 nm in H (insets).
Fig. 3.
Fig. 3. Multiplexed axial plane super-resolution reconstructions of microtubules and mitochondria in COS-7 cells. (A) Axial plane super-resolution image of α-tubulin labeled with anti-mouse P1 “docking” strands. (B) Axial plane super-resolution image of Tom20 labeled with anti-rabbit P4 “docking” strands (the same axial plane as A). (C) Two-channel combined image of α-tubulin (green) and Tom20 (magenta) located in the same axial cross-section. (D) Enlarged views of two segments marked in A with the orange and red squares, respectively. (E) Normalized intensity profiles of microtubules marked with the white dashed rectangles in D. Two microtubules laying along the x-axis and z-axis showed averaged FWHMs in the z and x directions as 112 nm and 90 nm, respectively. (F) Enlarged views of two regions marked in C with the yellow and blue rectangles, respectively. The axial plane organization of microtubules and mitochondria were resolved. Scale bar: 2 µm in A, B and C, 500 nm in D and F.
Fig. 4.
Fig. 4. Axial plane super-resolution reconstructions of immuno-labeled Nup98 in GBM10 cells. (A, D) Axial plane super-resolution images of Nup98 immuno-labeled with Alexa Fluor 647 (AF647). The insets show the corresponding diffraction-limited images. (B, E) Enlarged views of the regions marked in A with the yellow square and in D with the blue square, and the corresponding diffraction-limited images. (C, F) Normalized intensity profiles in the x and z directions of two NPCs located in different imaging depths (∼2.6 µm and 13.3 µm in C, ∼1.8 µm and 16.0 µm in F) indicated by the white arrows in A and D, respectively. The FWHMs highlight the similar resolution in different imaging depths of the axial plane SMLM. Scale bar: 2 µm in A, D and their insets, 500nm in B and E, 100 nm in C and F (insets).
Fig. 5.
Fig. 5. Simulated and experimental axial plane PSFs of the axial plane SMLM system. (A) Simulated ideal axial plane PSFs, scanning along the y-axis (first row). The simulated axial plane PSFs with a small amount of primary spherical aberration ∼127 mλ (second row). The experimental axial plane PSFs acquired with a 100 nm-diameter crimson bead excited by epi illumination, scanning along the y-axis by sample stage (third row). (B) Experimental in-focus PSF. It is the effective PSF used for single-molecule localization. (C) Intensity profiles along the long (i.e. z) and short (i.e. x) axes of the experimental elliptical PSF in B, respectively. (D) Distributions of σx and σz which is 181.37 ± 2.94 nm for the short axis and 366.08 ± 6.75 nm for the long axis (mean ± std., n = 15), measured from the experimental in-focus PSFs. (E) Image of dense fluorescence beads excited by a light sheet. (F) Intensity profile of the light sheet of E in the y direction, with the FWHM of 1.15 µm. (G) Distribution of σy of the light sheet thickness measured 15 different light sheets, with the σy of 505.39 ± 18.35 nm (mean ± std., n = 15), corresponding to a FWHM of 1.19 ± 0.04 µm. (H) Distribution of the detected PSF intensity (with the localization sub-region as 1.5×1.5 µm2) along the y-axis. The PSF intensity becomes significantly dimmer outside the range of ± 300 nm. (I) Experimental axial plane PSFs acquired with a 100 nm-diameter crimson bead excited by the light sheet, scanning along the y-axis. Scale bar: 5 µm in A and E, 1 µm in B and I.
Fig. 6.
Fig. 6. Localization precision quantifications of the axial plane SMLM. (A) Axial plane super-resolution images of α-tubulin in different COS-7 cells. (B) Distributions of σx (37.39 ± 5.47 nm, blue circles) and σz (43.79 ± 5.11 nm, red triangles), measured from 30 microtubules laying approximately along the z-axis (marked with the blue lines in A) and 30 microtubules laying approximately along the x-axis (marked with the red lines in A), respectively. (C) Axial plane super-resolution images of Nup98 in different COS-7 cells. (D) Distributions of σx (33.63 ± 2.60 nm, blue circles) and σz (43.77 ± 6.88 nm, red triangles), measured from 30 NPCs (marked with the white crosses in C). Scale bar: 2 µm in A and C.
Fig. 7.
Fig. 7. Axial plane super-resolution reconstructions of different targets in fixed COS-7 cells. (A, B, C) Axial plane diffraction-limited (left) and the corresponding super-resolution (right) images of α-tubulin (A), Tom20 (B), and Nup98 (C), all immuno-labeled with Alexa Fluor 647 (AF647). Scale bar: 2 µm.
Fig. 8.
Fig. 8. Single-channel axial plane super-resolution reconstruction by combining the axial plane SMLM with DNA-PAINT. (A) Diffraction-limited axial plane image of α-tubulin labeled with anti-mouse P1 “docking” strands. (B) Axial plane super-resolution image of A. (C) Enlarged views of three microtubule segments marked in A and B with the yellow, blue, and red squares respectively. (D) Normalized intensity profiles of microtubules marked with the white dashed line and rectangles in C. Comparable resolution was achieved as the conventional SMLM probe Alexa Fluor 647 (Fig. 2). Scale bar: 2 µm in A and B, 500 nm in C.
Fig. 9.
Fig. 9. Multiplexed axial plane super-resolution reconstructions of microtubules and mitochondria in fixed COS-7 cells. (A) Axial plane super-resolution image of α-tubulin labeled with anti-mouse P1 “docking” strands. (B) Axial plane super-resolution image of Tom20 labeled with anti-rabbit P4 “docking” strands (the same axial plane as A). (C) Two-channel combined image of α-tubulin (green) and Tom20 (magenta) located in the same axial cross-section. (D) Enlarged views of microtubule segments marked in A with the white dashed lines. (E) Normalized intensity profiles of microtubules laying approximately along the x-axis marked with the white dashed rectangles in D, with FWHMs of 102 nm, 100 nm, 106 nm and 108 nm in the z direction, respectively. (F) Enlarged views of mitochondria segments marked in B with the white dashed lines. (G) Normalized intensity profiles of mitochondria marked with the white dashed lines in F, with different membrane distances of 820 nm, 614 nm, 264 nm and 265 nm, respectively. Scale bar: 2 µm in A, B and C, 500 nm in D and F.

Equations (5)

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u k ( x , z ) = I 0 e x p ( 2 ( x x 0 ) 2 w x 2 2 ( z z 0 ) 2 w z 2 ) + b ,
L ( θ | D ) = k u k ( x , z ) d k e u k ( x , z ) d k ! ,
[ x y z ] = [ 1 0 0 0 cos α sin α 0 sin α cos α ] [ 1 0 0 0 1 0 0 0 1 ] [ x y z ] = [ x y cos α + z sin α y sin α z cos α ] .
I ( x , y , z ) = 1 2 ( | I A ( x , y , z ) | 2 + | I B ( x , y , z ) | 2 ) ,
I j ( x , y , z ) = ( λ n ) 2 E j P ( k x , k y ) e i 2 π [ x k x + ( y cos α + z sin α ) k y + ( y sin α z cos α ) k z ] d k x d k y ,

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