Abstract

Two-photon fluorescence microscopy is an influential tool in biology, providing valuable information on the activity of cells deep inside the tissue. However, it is limited by its low speed for imaging volume samples. Here we present the design of a two-photon scanning microscope with an extended and adjustable depth of field, which improves the temporal resolution for sampling thick samples. Moreover, this method implies no loss of optical power and resolution, and can be easily integrated into most commercial laser-scanning microscopy systems. We demonstrate experimentally the gain in performance of the system by comparing volumetric scans of neuronal structures with a standard versus an extended depth of field system.

© 2013 OSA

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  1. F. Helmchen and W. Denk, “Deep tissue two-photon microscopy,” Nat. Methods2, 932–40 (2005).
    [CrossRef] [PubMed]
  2. E. Beaurepaire, M. Oheim, and J. Mertz, “Ultra-deep two-photon fluorescence excitation in turbid media,” Opt. Comm.188, 25–29 (2001).
    [CrossRef]
  3. W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science248, 73–76 (1990).
    [CrossRef] [PubMed]
  4. W. R. Zipfel, R. M. Williams, and W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol.21, 1369–1377 (2003).
    [CrossRef] [PubMed]
  5. G. D. Reddy, K. Kelleher, R. Fink, and P. Saggau, “Three-dimensional random access multiphoton microscopy for functional imaging of neuronal activity,” Nat. Neurosci.11, 713–720 (2008).
    [CrossRef]
  6. W. Göbel, B. M. Kampa, and F. Helmchen, “Imaging cellular network dynamics in three dimensions using fast 3D laser scanning,” Nat. Methods473–79 (2007).
    [CrossRef]
  7. B. F. Grewe, D. Langer, H. Kasper, B. M. Kampa, and F. Helmchen, “High-speed in vivo calcium imaging reveals neuronal network activity with near-millisecond precision,” Nat. Methods7, 399–405 (2010).
    [CrossRef] [PubMed]
  8. E. J. Botcherby, C. W. Smith, M. M. Kohl, D. Débarre, M. J. Booth, R. Juškaitis, O. Paulsen, and T. Wilson, “Aberration-free three-dimensional multiphoton imaging of neuronal activity at kHz rates,” PNAS109, 2919–2924 (2012).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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2012 (1)

E. J. Botcherby, C. W. Smith, M. M. Kohl, D. Débarre, M. J. Booth, R. Juškaitis, O. Paulsen, and T. Wilson, “Aberration-free three-dimensional multiphoton imaging of neuronal activity at kHz rates,” PNAS109, 2919–2924 (2012).
[CrossRef] [PubMed]

2011 (2)

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nature Meth.8, 417–423 (2011).
[CrossRef]

H. Lütcke and F. Helmchen, “Two-photon imaging and analysis of neural network dynamics,” Rep. Prog. Phys.74, 086602 (2011).
[CrossRef]

2010 (2)

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nature Photon.4, 780–785 (2010).
[CrossRef]

B. F. Grewe, D. Langer, H. Kasper, B. M. Kampa, and F. Helmchen, “High-speed in vivo calcium imaging reveals neuronal network activity with near-millisecond precision,” Nat. Methods7, 399–405 (2010).
[CrossRef] [PubMed]

2009 (1)

2008 (2)

O. Brzobohatý, T. Čižmár, and P. Zemánek, “High quality quasi-Bessel beam generated by round-tip axicon,” Opt. Express16, 12688–12700 (2008).
[CrossRef] [PubMed]

G. D. Reddy, K. Kelleher, R. Fink, and P. Saggau, “Three-dimensional random access multiphoton microscopy for functional imaging of neuronal activity,” Nat. Neurosci.11, 713–720 (2008).
[CrossRef]

2007 (1)

W. Göbel, B. M. Kampa, and F. Helmchen, “Imaging cellular network dynamics in three dimensions using fast 3D laser scanning,” Nat. Methods473–79 (2007).
[CrossRef]

2006 (2)

E. J. Botcherby, R. Juškaitis, and T. Wilson, “Scanning two photon fluorescence microscopy with extended depth of field.” Opt. Commun.268, 253–260 (2006).
[CrossRef]

P. Dufour, M. Piché, Y. De Koninck, and N. McCarthy, “Two-photon excitation fluorescence microscopy with a high depth of field using an axicon,” Appl. Opt.45, 9246–9252 (2006).
[CrossRef] [PubMed]

2005 (1)

F. Helmchen and W. Denk, “Deep tissue two-photon microscopy,” Nat. Methods2, 932–40 (2005).
[CrossRef] [PubMed]

2003 (2)

W. R. Zipfel, R. M. Williams, and W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol.21, 1369–1377 (2003).
[CrossRef] [PubMed]

T. A. Pologruto, B. L. Sabatini, and K. Svoboda, “ScanImage: flexible software for operating laser scanning microscopes,” Biomed. Eng. Online2, 13 (2003).
[CrossRef] [PubMed]

2001 (1)

E. Beaurepaire, M. Oheim, and J. Mertz, “Ultra-deep two-photon fluorescence excitation in turbid media,” Opt. Comm.188, 25–29 (2001).
[CrossRef]

1996 (1)

1990 (1)

W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science248, 73–76 (1990).
[CrossRef] [PubMed]

1953 (1)

Arnold, C. B.

Beaurepaire, E.

Betzig, E.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nature Meth.8, 417–423 (2011).
[CrossRef]

Booth, M. J.

E. J. Botcherby, C. W. Smith, M. M. Kohl, D. Débarre, M. J. Booth, R. Juškaitis, O. Paulsen, and T. Wilson, “Aberration-free three-dimensional multiphoton imaging of neuronal activity at kHz rates,” PNAS109, 2919–2924 (2012).
[CrossRef] [PubMed]

Botcherby, E. J.

E. J. Botcherby, C. W. Smith, M. M. Kohl, D. Débarre, M. J. Booth, R. Juškaitis, O. Paulsen, and T. Wilson, “Aberration-free three-dimensional multiphoton imaging of neuronal activity at kHz rates,” PNAS109, 2919–2924 (2012).
[CrossRef] [PubMed]

E. J. Botcherby, R. Juškaitis, and T. Wilson, “Scanning two photon fluorescence microscopy with extended depth of field.” Opt. Commun.268, 253–260 (2006).
[CrossRef]

Brzobohatý, O.

Cižmár, T.

Davidson, M. W.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nature Meth.8, 417–423 (2011).
[CrossRef]

De Koninck, Y.

Débarre, D.

E. J. Botcherby, C. W. Smith, M. M. Kohl, D. Débarre, M. J. Booth, R. Juškaitis, O. Paulsen, and T. Wilson, “Aberration-free three-dimensional multiphoton imaging of neuronal activity at kHz rates,” PNAS109, 2919–2924 (2012).
[CrossRef] [PubMed]

Denk, W.

F. Helmchen and W. Denk, “Deep tissue two-photon microscopy,” Nat. Methods2, 932–40 (2005).
[CrossRef] [PubMed]

W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science248, 73–76 (1990).
[CrossRef] [PubMed]

Dufour, P.

Fahrbach, F. O.

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nature Photon.4, 780–785 (2010).
[CrossRef]

Fink, R.

G. D. Reddy, K. Kelleher, R. Fink, and P. Saggau, “Three-dimensional random access multiphoton microscopy for functional imaging of neuronal activity,” Nat. Neurosci.11, 713–720 (2008).
[CrossRef]

Friberg, A. T.

Galbraith, C. G.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nature Meth.8, 417–423 (2011).
[CrossRef]

Galbraith, J. A.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nature Meth.8, 417–423 (2011).
[CrossRef]

Gao, L.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nature Meth.8, 417–423 (2011).
[CrossRef]

Göbel, W.

W. Göbel, B. M. Kampa, and F. Helmchen, “Imaging cellular network dynamics in three dimensions using fast 3D laser scanning,” Nat. Methods473–79 (2007).
[CrossRef]

Grewe, B. F.

B. F. Grewe, D. Langer, H. Kasper, B. M. Kampa, and F. Helmchen, “High-speed in vivo calcium imaging reveals neuronal network activity with near-millisecond precision,” Nat. Methods7, 399–405 (2010).
[CrossRef] [PubMed]

Helmchen, F.

H. Lütcke and F. Helmchen, “Two-photon imaging and analysis of neural network dynamics,” Rep. Prog. Phys.74, 086602 (2011).
[CrossRef]

B. F. Grewe, D. Langer, H. Kasper, B. M. Kampa, and F. Helmchen, “High-speed in vivo calcium imaging reveals neuronal network activity with near-millisecond precision,” Nat. Methods7, 399–405 (2010).
[CrossRef] [PubMed]

W. Göbel, B. M. Kampa, and F. Helmchen, “Imaging cellular network dynamics in three dimensions using fast 3D laser scanning,” Nat. Methods473–79 (2007).
[CrossRef]

F. Helmchen and W. Denk, “Deep tissue two-photon microscopy,” Nat. Methods2, 932–40 (2005).
[CrossRef] [PubMed]

Juškaitis, R.

E. J. Botcherby, C. W. Smith, M. M. Kohl, D. Débarre, M. J. Booth, R. Juškaitis, O. Paulsen, and T. Wilson, “Aberration-free three-dimensional multiphoton imaging of neuronal activity at kHz rates,” PNAS109, 2919–2924 (2012).
[CrossRef] [PubMed]

E. J. Botcherby, R. Juškaitis, and T. Wilson, “Scanning two photon fluorescence microscopy with extended depth of field.” Opt. Commun.268, 253–260 (2006).
[CrossRef]

Kampa, B. M.

B. F. Grewe, D. Langer, H. Kasper, B. M. Kampa, and F. Helmchen, “High-speed in vivo calcium imaging reveals neuronal network activity with near-millisecond precision,” Nat. Methods7, 399–405 (2010).
[CrossRef] [PubMed]

W. Göbel, B. M. Kampa, and F. Helmchen, “Imaging cellular network dynamics in three dimensions using fast 3D laser scanning,” Nat. Methods473–79 (2007).
[CrossRef]

Kasper, H.

B. F. Grewe, D. Langer, H. Kasper, B. M. Kampa, and F. Helmchen, “High-speed in vivo calcium imaging reveals neuronal network activity with near-millisecond precision,” Nat. Methods7, 399–405 (2010).
[CrossRef] [PubMed]

Kelleher, K.

G. D. Reddy, K. Kelleher, R. Fink, and P. Saggau, “Three-dimensional random access multiphoton microscopy for functional imaging of neuronal activity,” Nat. Neurosci.11, 713–720 (2008).
[CrossRef]

Kohl, M. M.

E. J. Botcherby, C. W. Smith, M. M. Kohl, D. Débarre, M. J. Booth, R. Juškaitis, O. Paulsen, and T. Wilson, “Aberration-free three-dimensional multiphoton imaging of neuronal activity at kHz rates,” PNAS109, 2919–2924 (2012).
[CrossRef] [PubMed]

Langer, D.

B. F. Grewe, D. Langer, H. Kasper, B. M. Kampa, and F. Helmchen, “High-speed in vivo calcium imaging reveals neuronal network activity with near-millisecond precision,” Nat. Methods7, 399–405 (2010).
[CrossRef] [PubMed]

Lütcke, H.

H. Lütcke and F. Helmchen, “Two-photon imaging and analysis of neural network dynamics,” Rep. Prog. Phys.74, 086602 (2011).
[CrossRef]

McCarthy, N.

McLeod, J. H.

Mermillod-Blondin, A.

Mertz, J.

E. Beaurepaire, M. Oheim, and J. Mertz, “Ultra-deep two-photon fluorescence excitation in turbid media,” Opt. Comm.188, 25–29 (2001).
[CrossRef]

Milkie, D. E.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nature Meth.8, 417–423 (2011).
[CrossRef]

Oheim, M.

E. Beaurepaire, M. Oheim, and J. Mertz, “Ultra-deep two-photon fluorescence excitation in turbid media,” Opt. Comm.188, 25–29 (2001).
[CrossRef]

Olivier, N.

Paulsen, O.

E. J. Botcherby, C. W. Smith, M. M. Kohl, D. Débarre, M. J. Booth, R. Juškaitis, O. Paulsen, and T. Wilson, “Aberration-free three-dimensional multiphoton imaging of neuronal activity at kHz rates,” PNAS109, 2919–2924 (2012).
[CrossRef] [PubMed]

Piché, M.

Planchon, T. A.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nature Meth.8, 417–423 (2011).
[CrossRef]

Pologruto, T. A.

T. A. Pologruto, B. L. Sabatini, and K. Svoboda, “ScanImage: flexible software for operating laser scanning microscopes,” Biomed. Eng. Online2, 13 (2003).
[CrossRef] [PubMed]

Reddy, G. D.

G. D. Reddy, K. Kelleher, R. Fink, and P. Saggau, “Three-dimensional random access multiphoton microscopy for functional imaging of neuronal activity,” Nat. Neurosci.11, 713–720 (2008).
[CrossRef]

Rohrbach, A.

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nature Photon.4, 780–785 (2010).
[CrossRef]

Sabatini, B. L.

T. A. Pologruto, B. L. Sabatini, and K. Svoboda, “ScanImage: flexible software for operating laser scanning microscopes,” Biomed. Eng. Online2, 13 (2003).
[CrossRef] [PubMed]

Saggau, P.

G. D. Reddy, K. Kelleher, R. Fink, and P. Saggau, “Three-dimensional random access multiphoton microscopy for functional imaging of neuronal activity,” Nat. Neurosci.11, 713–720 (2008).
[CrossRef]

Simon, P.

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nature Photon.4, 780–785 (2010).
[CrossRef]

Smith, C. W.

E. J. Botcherby, C. W. Smith, M. M. Kohl, D. Débarre, M. J. Booth, R. Juškaitis, O. Paulsen, and T. Wilson, “Aberration-free three-dimensional multiphoton imaging of neuronal activity at kHz rates,” PNAS109, 2919–2924 (2012).
[CrossRef] [PubMed]

Strickler, J. H.

W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science248, 73–76 (1990).
[CrossRef] [PubMed]

Svoboda, K.

T. A. Pologruto, B. L. Sabatini, and K. Svoboda, “ScanImage: flexible software for operating laser scanning microscopes,” Biomed. Eng. Online2, 13 (2003).
[CrossRef] [PubMed]

Webb, W. W.

W. R. Zipfel, R. M. Williams, and W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol.21, 1369–1377 (2003).
[CrossRef] [PubMed]

W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science248, 73–76 (1990).
[CrossRef] [PubMed]

Williams, R. M.

W. R. Zipfel, R. M. Williams, and W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol.21, 1369–1377 (2003).
[CrossRef] [PubMed]

Wilson, T.

E. J. Botcherby, C. W. Smith, M. M. Kohl, D. Débarre, M. J. Booth, R. Juškaitis, O. Paulsen, and T. Wilson, “Aberration-free three-dimensional multiphoton imaging of neuronal activity at kHz rates,” PNAS109, 2919–2924 (2012).
[CrossRef] [PubMed]

E. J. Botcherby, R. Juškaitis, and T. Wilson, “Scanning two photon fluorescence microscopy with extended depth of field.” Opt. Commun.268, 253–260 (2006).
[CrossRef]

Zemánek, P.

Zipfel, W. R.

W. R. Zipfel, R. M. Williams, and W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol.21, 1369–1377 (2003).
[CrossRef] [PubMed]

Appl. Opt. (1)

Biomed. Eng. Online (1)

T. A. Pologruto, B. L. Sabatini, and K. Svoboda, “ScanImage: flexible software for operating laser scanning microscopes,” Biomed. Eng. Online2, 13 (2003).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Nat. Biotechnol. (1)

W. R. Zipfel, R. M. Williams, and W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol.21, 1369–1377 (2003).
[CrossRef] [PubMed]

Nat. Methods (3)

F. Helmchen and W. Denk, “Deep tissue two-photon microscopy,” Nat. Methods2, 932–40 (2005).
[CrossRef] [PubMed]

W. Göbel, B. M. Kampa, and F. Helmchen, “Imaging cellular network dynamics in three dimensions using fast 3D laser scanning,” Nat. Methods473–79 (2007).
[CrossRef]

B. F. Grewe, D. Langer, H. Kasper, B. M. Kampa, and F. Helmchen, “High-speed in vivo calcium imaging reveals neuronal network activity with near-millisecond precision,” Nat. Methods7, 399–405 (2010).
[CrossRef] [PubMed]

Nat. Neurosci. (1)

G. D. Reddy, K. Kelleher, R. Fink, and P. Saggau, “Three-dimensional random access multiphoton microscopy for functional imaging of neuronal activity,” Nat. Neurosci.11, 713–720 (2008).
[CrossRef]

Nature Meth. (1)

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nature Meth.8, 417–423 (2011).
[CrossRef]

Nature Photon. (1)

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nature Photon.4, 780–785 (2010).
[CrossRef]

Opt. Comm. (1)

E. Beaurepaire, M. Oheim, and J. Mertz, “Ultra-deep two-photon fluorescence excitation in turbid media,” Opt. Comm.188, 25–29 (2001).
[CrossRef]

Opt. Commun. (1)

E. J. Botcherby, R. Juškaitis, and T. Wilson, “Scanning two photon fluorescence microscopy with extended depth of field.” Opt. Commun.268, 253–260 (2006).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

PNAS (1)

E. J. Botcherby, C. W. Smith, M. M. Kohl, D. Débarre, M. J. Booth, R. Juškaitis, O. Paulsen, and T. Wilson, “Aberration-free three-dimensional multiphoton imaging of neuronal activity at kHz rates,” PNAS109, 2919–2924 (2012).
[CrossRef] [PubMed]

Rep. Prog. Phys. (1)

H. Lütcke and F. Helmchen, “Two-photon imaging and analysis of neural network dynamics,” Rep. Prog. Phys.74, 086602 (2011).
[CrossRef]

Science (1)

W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science248, 73–76 (1990).
[CrossRef] [PubMed]

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Figures (5)

Fig. 1
Fig. 1

An axicon produces a nondiffractive beam. (a) Schematic representation of an axicon (conical lens) of angle α illuminated by a plane wave. After the axicon, the interference pattern generates a Bessel beam, characterized by the deviation angle, β. The radial profile of a Bessel beam is traced in red. (b) Transverse intensity profile of a Bessel beam.

Fig. 2
Fig. 2

Schematic of the set-up. A Ti:Sapphire laser illuminates an axicon of angle α followed by a lens (focal length fα), which transform the laser beam into an annulus of light of radius R. This annulus is imaged into the back focal plane of the objective lens, which creates a tightly focused Bessel-Gauss beam in the sample. The scanning system (in our case a pair of galvanometric mirrors and relay lenses) enables a beam tilt in the back focal plane of the objective, leading to an x–y scan of the beam in the sample. Fluorescence light is retro-collected with the objective and directed to a photomultiplier tube (PMT) with a dichroic mirror.

Fig. 3
Fig. 3

Longitudinal point-spread function. (a) Two-photon excitation fluorescence (2PEF) signal along the z axis for an extended depth of field set-up (green) compared to a standard set-up (blue) with the same transverse resolution. (b), (c) Calculated and experimental PSF in the x–z plane with standard depth of field set-up (w0 = 2.7 mm). (d) to (g) Calculated and experimental PSF in the x–z plane with extended depth of field set-ups (w0 = 0.27 mm for d–e and w0 = 0.47 mm for f–g). Experimental PSFs were measured with 500 nm fluorescent beads. See text for parameters used. Scale bars, 5 μm.

Fig. 4
Fig. 4

3-micron fluorescent polymer beads in agarose. (a)–(c) Standard two-photon fluorescence images acquired at various depths by translating the sample. (d) Z-summed stack of 13 standard two-photon scans (including those shown in a to c), spanning a depth of 60 μm. (e) An extended depth of field image acquired in a single scan. See text for parameters used. Scale bars, 50 μm.

Fig. 5
Fig. 5

Rat hippocampal neuron in fixed brain slice, stained with Lucifer Yellow. (a)–(c) Standard two-photon fluorescence images acquired at various depths by translating the sample. (d) Z-summed stack of 26 standard two-photon scans (including those shown in a to c), spanning a depth of 60 μm. (e) An extended depth of field image acquired in a single scan. Inset curves: Intensity line scans along the dashed lines. See text for parameters used. Scale bars, 5 μm.

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

I ( r , z ) = I 0 4 π 2 β 2 z λ exp ( 2 β 2 z 2 w 0 2 ) J 0 2 ( 2 π r β λ ) ,
ρ = 2.4048 λ 2 π β .
L = C ( w 0 β ) ,
w f = w 0 F m f α
β f = tan 1 ( m f α tan β F ) ,
I ( r = 0 , z ) = 8 π P z λ ( C L ) 2 exp ( 2 z 2 C 2 L 2 ) .

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