Abstract

Spatially and temporally dependent optical aberrations induced by the inhomogeneous refractive index of live samples limit the resolution of live dynamic imaging. We introduce an adaptive optical microscope with a direct wavefront sensing method using a Shack-Hartmann wavefront sensor and fluorescent protein guide-stars for live imaging. The results of imaging Drosophila embryos demonstrate its ability to correct aberrations and achieve near diffraction limited images of medial sections of large Drosophila embryos. GFP-polo labeled centrosomes can be observed clearly after correction but cannot be observed before correction. Four dimensional time lapse images are achieved with the correction of dynamic aberrations. These studies also demonstrate that the GFP-tagged centrosome proteins, Polo and Cnn, serve as excellent biological guide-stars for adaptive optics based microscopy.

© 2012 OSA

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References

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  1. M. J. Booth, “Adaptive optics in microscopy,” Phil. Trans. R. Soc. A–Math. Phys. Eng. Sci. 365, 2829–2843 (2007).
  2. R. K. Tyson, Principles of Adaptive Optics (Academic, 1991).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  12. O. Azucena, J. Crest, S. Kotadia, W. Sullivan, X. Tao, M. Reinig, D. Gavel, S. Olivier, and J. Kubby, “Adaptive optics wide-field microscopy using direct wavefront sensing,” Opt. Lett. 36(6), 825–827 (2011).
    [CrossRef] [PubMed]
  13. X. Tao, B. Fernandez, O. Azucena, M. Fu, D. Garcia, Y. Zuo, D. C. Chen, and J. Kubby, “Adaptive optics confocal microscopy using direct wavefront sensing,” Opt. Lett. 36(7), 1062–1064 (2011).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  20. M. Gu, Principles of Three-Dimensional Imaging in Confocal Microscopes (World Scientific, Singapore, 1996).
  21. J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford University Press, 1998).
  22. M. Schwertner, M. J. Booth, M. A. Neil, and T. Wilson, “Measurement of specimen-induced aberrations of biological samples using phase stepping interferometry,” J. Microsc. 213(1), 11–19 (2004).
    [CrossRef] [PubMed]
  23. O. Azucena, J. Crest, J. Cao, W. Sullivan, P. Kner, D. Gavel, D. Dillon, S. Olivier, and J. Kubby, “Wavefront aberration measurements and corrections through thick tissue using fluorescent microsphere reference beacons,” Opt. Express 18(16), 17521–17532 (2010).
    [CrossRef] [PubMed]
  24. L. A. Poyneer and B. A. Macintosh, “Spatially filtered wave-front sensor for high-order adaptive optics,” J. Opt. Soc. Am. A 21(5), 810–819 (2004).
    [CrossRef] [PubMed]
  25. J. Zhang and T. L. Megraw, “Proper recruitment of gamma-tubulin and D-TACC/Msps to embryonic Drosophila centrosomes requires Centrosomin Motif 1,” Mol. Biol. Cell 18(10), 4037–4049 (2007).
    [CrossRef] [PubMed]
  26. T. Moutinho-Santos, P. Sampaio, I. Amorim, M. Costa, and C. E. Sunkel, “In vivo localisation of the mitotic POLO kinase shows a highly dynamic association with the mitotic apparatus during early embryogenesis in Drosophila,” Biol. Cell 91(8), 585–596 (1999).
    [PubMed]
  27. W. F. Rothwell and W. Sullivan, “Fluorescent analysis of drosophila embryos,” in Drosophila Protocols, W. Sullivan, M. Ashburner and R. S. Hawley, eds. (Cold Spring Harbor Laboratory Press, 2000), pp. 141–157.

2012 (1)

2011 (6)

2010 (2)

2009 (2)

2007 (2)

M. J. Booth, “Adaptive optics in microscopy,” Phil. Trans. R. Soc. A–Math. Phys. Eng. Sci. 365, 2829–2843 (2007).

J. Zhang and T. L. Megraw, “Proper recruitment of gamma-tubulin and D-TACC/Msps to embryonic Drosophila centrosomes requires Centrosomin Motif 1,” Mol. Biol. Cell 18(10), 4037–4049 (2007).
[CrossRef] [PubMed]

2006 (2)

M. Rueckel, J. A. Mack-Bucher, and W. Denk, “Adaptive wavefront correction in two-photon microscopy using coherence-gated wavefront sensing,” Proc. Natl. Acad. Sci. U.S.A. 103(46), 17137–17142 (2006).
[CrossRef] [PubMed]

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in a Shack-Hartmann sensor,” Mon. Not. R. Astron. Soc. 371(1), 323–336 (2006).
[CrossRef]

2004 (2)

L. A. Poyneer and B. A. Macintosh, “Spatially filtered wave-front sensor for high-order adaptive optics,” J. Opt. Soc. Am. A 21(5), 810–819 (2004).
[CrossRef] [PubMed]

M. Schwertner, M. J. Booth, M. A. Neil, and T. Wilson, “Measurement of specimen-induced aberrations of biological samples using phase stepping interferometry,” J. Microsc. 213(1), 11–19 (2004).
[CrossRef] [PubMed]

2003 (1)

2002 (2)

M. J. Booth, M. A. Neil, R. Juskaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Natl. Acad. Sci. U.S.A. 99(9), 5788–5792 (2002).
[CrossRef] [PubMed]

L. A. Poyneer, D. T. Gavel, and J. M. Brase, “Fast wave-front reconstruction in large adaptive optics systems with use of the Fourier transform,” J. Opt. Soc. Am. A 19(10), 2100–2111 (2002).
[CrossRef]

1999 (1)

T. Moutinho-Santos, P. Sampaio, I. Amorim, M. Costa, and C. E. Sunkel, “In vivo localisation of the mitotic POLO kinase shows a highly dynamic association with the mitotic apparatus during early embryogenesis in Drosophila,” Biol. Cell 91(8), 585–596 (1999).
[PubMed]

Amorim, I.

T. Moutinho-Santos, P. Sampaio, I. Amorim, M. Costa, and C. E. Sunkel, “In vivo localisation of the mitotic POLO kinase shows a highly dynamic association with the mitotic apparatus during early embryogenesis in Drosophila,” Biol. Cell 91(8), 585–596 (1999).
[PubMed]

Andilla, J.

Artigas, D.

Aviles-Espinosa, R.

Azucena, O.

Beaurepaire, E.

Betzig, E.

D. E. Milkie, E. Betzig, and N. Ji, “Pupil-segmentation-based adaptive optical microscopy with full-pupil illumination,” Opt. Lett. 36(21), 4206–4208 (2011).
[CrossRef] [PubMed]

N. Ji, D. E. Milkie, and E. Betzig, “Adaptive optics via pupil segmentation for high-resolution imaging in biological tissues,” Nat. Methods 7(2), 141–147 (2010).
[CrossRef] [PubMed]

Booth, M. J.

D. Débarre, E. J. Botcherby, T. Watanabe, S. Srinivas, M. J. Booth, and T. Wilson, “Image-based adaptive optics for two-photon microscopy,” Opt. Lett. 34(16), 2495–2497 (2009).
[CrossRef] [PubMed]

M. J. Booth, “Adaptive optics in microscopy,” Phil. Trans. R. Soc. A–Math. Phys. Eng. Sci. 365, 2829–2843 (2007).

M. Schwertner, M. J. Booth, M. A. Neil, and T. Wilson, “Measurement of specimen-induced aberrations of biological samples using phase stepping interferometry,” J. Microsc. 213(1), 11–19 (2004).
[CrossRef] [PubMed]

M. J. Booth, M. A. Neil, R. Juskaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Natl. Acad. Sci. U.S.A. 99(9), 5788–5792 (2002).
[CrossRef] [PubMed]

Botcherby, E. J.

Brase, J. M.

Burns, D.

Cao, J.

Chen, D. C.

Costa, M.

T. Moutinho-Santos, P. Sampaio, I. Amorim, M. Costa, and C. E. Sunkel, “In vivo localisation of the mitotic POLO kinase shows a highly dynamic association with the mitotic apparatus during early embryogenesis in Drosophila,” Biol. Cell 91(8), 585–596 (1999).
[PubMed]

Crest, J.

Débarre, D.

Denk, W.

M. Rueckel, J. A. Mack-Bucher, and W. Denk, “Adaptive wavefront correction in two-photon microscopy using coherence-gated wavefront sensing,” Proc. Natl. Acad. Sci. U.S.A. 103(46), 17137–17142 (2006).
[CrossRef] [PubMed]

Dillon, D.

Facomprez, A.

Fernandez, B.

Fu, M.

Fusco, T.

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in a Shack-Hartmann sensor,” Mon. Not. R. Astron. Soc. 371(1), 323–336 (2006).
[CrossRef]

Garcia, D.

Gavel, D.

Gavel, D. T.

Girkin, J.

Ji, N.

D. E. Milkie, E. Betzig, and N. Ji, “Pupil-segmentation-based adaptive optical microscopy with full-pupil illumination,” Opt. Lett. 36(21), 4206–4208 (2011).
[CrossRef] [PubMed]

N. Ji, D. E. Milkie, and E. Betzig, “Adaptive optics via pupil segmentation for high-resolution imaging in biological tissues,” Nat. Methods 7(2), 141–147 (2010).
[CrossRef] [PubMed]

Juskaitis, R.

M. J. Booth, M. A. Neil, R. Juskaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Natl. Acad. Sci. U.S.A. 99(9), 5788–5792 (2002).
[CrossRef] [PubMed]

Kner, P.

Kotadia, S.

O. Azucena, J. Crest, S. Kotadia, W. Sullivan, X. Tao, M. Reinig, D. Gavel, S. Olivier, and J. Kubby, “Adaptive optics wide-field microscopy using direct wavefront sensing,” Opt. Lett. 36(6), 825–827 (2011).
[CrossRef] [PubMed]

O. Azucena, X. Tao, J. Crest, S. Kotadia, W. Sullivan, D. Gavel, M. Reinig, S. Olivier, and J. Kubby, “Adaptive optics wide-field microscope corrections using a MEMS DM and Shack-Hartmann wavefront sensor,” Proc. SPIE 7931, 79310J (2011).
[CrossRef]

Kubby, J.

Levecq, X.

Loza-Alvarez, P.

Macintosh, B. A.

Mack-Bucher, J. A.

M. Rueckel, J. A. Mack-Bucher, and W. Denk, “Adaptive wavefront correction in two-photon microscopy using coherence-gated wavefront sensing,” Proc. Natl. Acad. Sci. U.S.A. 103(46), 17137–17142 (2006).
[CrossRef] [PubMed]

Marsh, P.

Megraw, T. L.

J. Zhang and T. L. Megraw, “Proper recruitment of gamma-tubulin and D-TACC/Msps to embryonic Drosophila centrosomes requires Centrosomin Motif 1,” Mol. Biol. Cell 18(10), 4037–4049 (2007).
[CrossRef] [PubMed]

Michau, V.

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in a Shack-Hartmann sensor,” Mon. Not. R. Astron. Soc. 371(1), 323–336 (2006).
[CrossRef]

Milkie, D. E.

D. E. Milkie, E. Betzig, and N. Ji, “Pupil-segmentation-based adaptive optical microscopy with full-pupil illumination,” Opt. Lett. 36(21), 4206–4208 (2011).
[CrossRef] [PubMed]

N. Ji, D. E. Milkie, and E. Betzig, “Adaptive optics via pupil segmentation for high-resolution imaging in biological tissues,” Nat. Methods 7(2), 141–147 (2010).
[CrossRef] [PubMed]

Moutinho-Santos, T.

T. Moutinho-Santos, P. Sampaio, I. Amorim, M. Costa, and C. E. Sunkel, “In vivo localisation of the mitotic POLO kinase shows a highly dynamic association with the mitotic apparatus during early embryogenesis in Drosophila,” Biol. Cell 91(8), 585–596 (1999).
[PubMed]

Neil, M. A.

M. Schwertner, M. J. Booth, M. A. Neil, and T. Wilson, “Measurement of specimen-induced aberrations of biological samples using phase stepping interferometry,” J. Microsc. 213(1), 11–19 (2004).
[CrossRef] [PubMed]

M. J. Booth, M. A. Neil, R. Juskaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Natl. Acad. Sci. U.S.A. 99(9), 5788–5792 (2002).
[CrossRef] [PubMed]

Nicolle, M.

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in a Shack-Hartmann sensor,” Mon. Not. R. Astron. Soc. 371(1), 323–336 (2006).
[CrossRef]

Nieto, M.

Olarte, O. E.

Olivier, N.

Olivier, S.

Porcar-Guezenec, R.

Poyneer, L. A.

Reinig, M.

O. Azucena, X. Tao, J. Crest, S. Kotadia, W. Sullivan, D. Gavel, M. Reinig, S. Olivier, and J. Kubby, “Adaptive optics wide-field microscope corrections using a MEMS DM and Shack-Hartmann wavefront sensor,” Proc. SPIE 7931, 79310J (2011).
[CrossRef]

O. Azucena, J. Crest, S. Kotadia, W. Sullivan, X. Tao, M. Reinig, D. Gavel, S. Olivier, and J. Kubby, “Adaptive optics wide-field microscopy using direct wavefront sensing,” Opt. Lett. 36(6), 825–827 (2011).
[CrossRef] [PubMed]

Rousset, G.

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in a Shack-Hartmann sensor,” Mon. Not. R. Astron. Soc. 371(1), 323–336 (2006).
[CrossRef]

Rueckel, M.

M. Rueckel, J. A. Mack-Bucher, and W. Denk, “Adaptive wavefront correction in two-photon microscopy using coherence-gated wavefront sensing,” Proc. Natl. Acad. Sci. U.S.A. 103(46), 17137–17142 (2006).
[CrossRef] [PubMed]

Sampaio, P.

T. Moutinho-Santos, P. Sampaio, I. Amorim, M. Costa, and C. E. Sunkel, “In vivo localisation of the mitotic POLO kinase shows a highly dynamic association with the mitotic apparatus during early embryogenesis in Drosophila,” Biol. Cell 91(8), 585–596 (1999).
[PubMed]

Schwertner, M.

M. Schwertner, M. J. Booth, M. A. Neil, and T. Wilson, “Measurement of specimen-induced aberrations of biological samples using phase stepping interferometry,” J. Microsc. 213(1), 11–19 (2004).
[CrossRef] [PubMed]

Srinivas, S.

Sullivan, W.

Sunkel, C. E.

T. Moutinho-Santos, P. Sampaio, I. Amorim, M. Costa, and C. E. Sunkel, “In vivo localisation of the mitotic POLO kinase shows a highly dynamic association with the mitotic apparatus during early embryogenesis in Drosophila,” Biol. Cell 91(8), 585–596 (1999).
[PubMed]

Tao, X.

Thomas, S.

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in a Shack-Hartmann sensor,” Mon. Not. R. Astron. Soc. 371(1), 323–336 (2006).
[CrossRef]

Tokovinin, A.

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in a Shack-Hartmann sensor,” Mon. Not. R. Astron. Soc. 371(1), 323–336 (2006).
[CrossRef]

Watanabe, T.

Wilson, T.

D. Débarre, E. J. Botcherby, T. Watanabe, S. Srinivas, M. J. Booth, and T. Wilson, “Image-based adaptive optics for two-photon microscopy,” Opt. Lett. 34(16), 2495–2497 (2009).
[CrossRef] [PubMed]

M. Schwertner, M. J. Booth, M. A. Neil, and T. Wilson, “Measurement of specimen-induced aberrations of biological samples using phase stepping interferometry,” J. Microsc. 213(1), 11–19 (2004).
[CrossRef] [PubMed]

M. J. Booth, M. A. Neil, R. Juskaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Natl. Acad. Sci. U.S.A. 99(9), 5788–5792 (2002).
[CrossRef] [PubMed]

Zhang, J.

J. Zhang and T. L. Megraw, “Proper recruitment of gamma-tubulin and D-TACC/Msps to embryonic Drosophila centrosomes requires Centrosomin Motif 1,” Mol. Biol. Cell 18(10), 4037–4049 (2007).
[CrossRef] [PubMed]

Zuo, Y.

Biol. Cell (1)

T. Moutinho-Santos, P. Sampaio, I. Amorim, M. Costa, and C. E. Sunkel, “In vivo localisation of the mitotic POLO kinase shows a highly dynamic association with the mitotic apparatus during early embryogenesis in Drosophila,” Biol. Cell 91(8), 585–596 (1999).
[PubMed]

Biomed. Opt. Express (1)

J. Microsc. (1)

M. Schwertner, M. J. Booth, M. A. Neil, and T. Wilson, “Measurement of specimen-induced aberrations of biological samples using phase stepping interferometry,” J. Microsc. 213(1), 11–19 (2004).
[CrossRef] [PubMed]

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

Mol. Biol. Cell (1)

J. Zhang and T. L. Megraw, “Proper recruitment of gamma-tubulin and D-TACC/Msps to embryonic Drosophila centrosomes requires Centrosomin Motif 1,” Mol. Biol. Cell 18(10), 4037–4049 (2007).
[CrossRef] [PubMed]

Mon. Not. R. Astron. Soc. (1)

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in a Shack-Hartmann sensor,” Mon. Not. R. Astron. Soc. 371(1), 323–336 (2006).
[CrossRef]

Nat. Methods (1)

N. Ji, D. E. Milkie, and E. Betzig, “Adaptive optics via pupil segmentation for high-resolution imaging in biological tissues,” Nat. Methods 7(2), 141–147 (2010).
[CrossRef] [PubMed]

Opt. Express (3)

Opt. Lett. (6)

Phil. Trans. R. Soc. A–Math. Phys. Eng. Sci. (1)

M. J. Booth, “Adaptive optics in microscopy,” Phil. Trans. R. Soc. A–Math. Phys. Eng. Sci. 365, 2829–2843 (2007).

Proc. Natl. Acad. Sci. U.S.A. (2)

M. J. Booth, M. A. Neil, R. Juskaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Natl. Acad. Sci. U.S.A. 99(9), 5788–5792 (2002).
[CrossRef] [PubMed]

M. Rueckel, J. A. Mack-Bucher, and W. Denk, “Adaptive wavefront correction in two-photon microscopy using coherence-gated wavefront sensing,” Proc. Natl. Acad. Sci. U.S.A. 103(46), 17137–17142 (2006).
[CrossRef] [PubMed]

Proc. SPIE (1)

O. Azucena, X. Tao, J. Crest, S. Kotadia, W. Sullivan, D. Gavel, M. Reinig, S. Olivier, and J. Kubby, “Adaptive optics wide-field microscope corrections using a MEMS DM and Shack-Hartmann wavefront sensor,” Proc. SPIE 7931, 79310J (2011).
[CrossRef]

Other (6)

M. Gu, Advanced Optical Imaging Theory (Springer-Verlag, New York, 1999).

M. Gu, Principles of Three-Dimensional Imaging in Confocal Microscopes (World Scientific, Singapore, 1996).

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford University Press, 1998).

R. K. Tyson, Principles of Adaptive Optics (Academic, 1991).

J. Porter, H. Queener, J. Lin, K. Thorn, and A. A. S. Awwal, Adaptive Optics for Vision Science: Principles, Practices, Design and Applications, (Wiley, 2006).

W. F. Rothwell and W. Sullivan, “Fluorescent analysis of drosophila embryos,” in Drosophila Protocols, W. Sullivan, M. Ashburner and R. S. Hawley, eds. (Cold Spring Harbor Laboratory Press, 2000), pp. 141–157.

Supplementary Material (4)

» Media 1: MOV (3417 KB)     
» Media 2: MOV (3898 KB)     
» Media 3: MOV (2041 KB)     
» Media 4: MOV (1643 KB)     

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Figures (7)

Fig. 1
Fig. 1

System setup. (a) Schematic diagram of the AOM: GFPs are illuminated with an excitation laser (488 nm) to produce a guide-star. The reference laser is used for wavefront control. The flip mirrors M1 and M2 control the light path for confocal imaging (red), wavefront measurement (blue) and protected closed-loop control of the DM (green). (b) Experimental set-up of the AOM. L, lens; F, filter; P, polarizer; M, mirror flipper; DM, deformable mirror; DB, dichroic beamplitters.

Fig. 2
Fig. 2

Flowchart for the guide-star searching algorithm. The algorithm is to search and calculate the locations of the possible guide-stars stars in the image.

Fig. 3
Fig. 3

Measurement noise change with the exposure time at different depths. The error bar is the standard deviation for 10 measurements.

Fig. 4
Fig. 4

Wavefront measurement and correction. (a-d) The averaged point spread function (PSF) and wavefront errors over 6 measurements using EGFP-Cnn labeled centrosomes of a cycle 14 Drosophila embryo at four different locations (P1, P2, P3 and P4) at a depth of 60 µm. (e) The averaged coefficient value of the first 15 Zernike polynomial modes at these four locations. The error bar is the standard deviation for 6 measurements. (f-g) The images and PSF without and with correction for a cycle 14 Drosophila embryo with GFP-polo at a depth of 83 μm. Scale bars, 2 µm.

Fig. 5
Fig. 5

Comparison of the three-dimensional imaging without and with correction for imaging of cycle 13 fly embryos with EGFP-Cnn label. (a-b) The maximum intensity projection of the scan series from the top surface to 100 μm without and with AO (Media 1). (c-d) The 3D reconstructions without and with AO. (e-f) The confocal images without and with AO at the depths of 60 μm and 90 μm. The color maps are scaled to show the image data over its full range. Scale bar, 10 µm.

Fig. 6
Fig. 6

Comparison of the wavefront measurements and the PSFs without and with AO for different depths. (a-b) The wavefront measurements and PSF without and with AO at the depth of 90 μm (Media 2). (c) The RMS wavefront errors change with the depth. The red and blue lines indicate the measurement without and with AO respectively. (d) The Zernike coefficient values without AO with the change of depth. (e-f) The Strehl ratio and PSF size change for different depths. The red and blue lines indicate without and with AO respectively. (λ = 509 nm)

Fig. 7
Fig. 7

4D imaging of cycle 13 fly embryos with EGFP-Cnn label at depth of 80 µm. (a) A single frame without and with correction of a video movie (Media 3). (b) The coefficient value changes for Zernike modes z 2 2 (Astigmatism x, dashed line) and z 3 3 (Trefoil y, solid line) with and without AO during 20 minutes. (c) The Strehl ratio change with (blue) and without (red) AO during 20 minutes. (d) PSF size change with (blue) and without (red) AO during 20 minutes (Media 4).

Tables (1)

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Table 1 Zernike Polynomials used in this paper

Equations (6)

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V= A + ( S m + S OL )
d diffraction_limit =2.44 λ 2NA D 0 d l
h( x 2 , y 2 )= i λ Σ P( x 1 , y 1 )exp[ikϕ( x 1 , y 1 ) ] exp[ik(rR)] Rr cos(n,r)dS
PSF= | h ill | 2 | h col | 2
σ m = 2 π 2 K g 4(SNR) [ ( 3 2 ) 2 + ( θd λ ) 2 ]
SNR= n p n p + N D [ n B 2 + (e/G) 2 ]

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