Abstract

We have analyzed how the maximal imaging depth of two-photon microscopy in scattering samples depends on properties of the sample and the imaging system. We find that the imaging depth increases with increasing numerical aperture and staining inhomogeneity and with decreasing excitation-pulse duration and scattering anisotropy factor, but is ultimately limited by near-surface fluorescence with slight improvements possible using special detection strategies.

© 2006 Optical Society of America

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    [CrossRef]
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    [CrossRef]

2003 (3)

W. R. Zipfel, R. M. Williams, and W. W. Webb, "Nonlinear magic: multiphoton microscopy in the biosciences," Nat. Biotechnol. 21, 1368-1376 (2003).
[CrossRef]

O. Funk and K. Pfeilsticker, "Photon path length distributions for cloudy skies--oxygen A-band measurements and model calculations," Ann. Geophys. 21, 615-626 (2003).
[CrossRef]

P. Theer, M. T. Hasan, and W. Denk, "Two-photon imaging to a depth of 1000 μm in living brains by use of a TiAl2O3 regenerative amplifier," Opt. Lett. 28, 1022-1024 (2003).
[CrossRef] [PubMed]

2002 (5)

V. I. Haltrin, "One-parameter two-term Henyey-Greenstein phase function for light scattering in seawater," Appl. Opt. 41, 1022-1028 (2002).
[CrossRef] [PubMed]

E. Beaurepaire and J. Mertz, "Epifluorescence collection in two-photon microscopy," Appl. Opt. 41, 5376-5382 (2002).
[CrossRef] [PubMed]

A. N. Yaroslavsky, P. C. Schulze, I. V. Yaroslavsky, R. Schober, F. Ulrich, and H. J. Schwarzmaier, "Optical properties of selected native and coagulated human brain tissues in vitro in the visible and near infrared spectral range," Phys. Med. Biol. 47, 2059-2073 (2002).
[CrossRef] [PubMed]

J. T. Trachtenberg, B. E. Chen, G. W. Knott, G. P. Feng, J. R. Sanes, E. Welker, and K. Svoboda, "Long-term in vivo imaging of experience-dependent synaptic plasticity in adult cortex," Nature 420, 788-794 (2002).
[CrossRef] [PubMed]

J. Grutzendler, N. Kasthuri, and W. B. Gan, "Long-term dendritic spine stability in the adult cortex," Nature 420, 812-816 (2002).
[CrossRef] [PubMed]

2001 (1)

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

1999 (4)

1998 (3)

V. P. Kalosha, M. Muller, J. Herrmann, and S. Gatz, "Spatiotemporal model of femtosecond pulse generation in Kerr-lens mode-locked solid-state lasers," J. Opt. Soc. Am. B 15, 535-550 (1998).
[CrossRef]

J. W. McLean, J. D. Freeman, and R. E. Walker, "Beam spread function with time dispersion," Appl. Opt. 37, 4701-4711 (1998).
[CrossRef]

D. Kleinfeld, P. P. Mitra, F. Helmchen, and W. Denk, "Fluctuations and stimulus-induced changes in blood flow observed in individual capillaries in layers 2 through 4 of rat neocortex," Proc. Natl. Acad. Sci. U.S.A. 95, 15741-15746 (1998).
[CrossRef] [PubMed]

1997 (1)

K. Svoboda, W. Denk, D. Kleinfeld, and D. W. Tank, "In vivo dendritic calcium dynamics in neocortical pyramidal neurons," Nature 385, 161-165 (1997).
[CrossRef] [PubMed]

1996 (1)

W. Denk, "Two-photon excitation in functional biological imaging," J. Biomed. Opt. 1, 296-304 (1996).
[CrossRef]

1995 (1)

1994 (2)

E. H. K. Stelzer, S. Hell, S. Lindek, R. Stricker, R. Pick, C. Storz, G. Ritter, and N. Salmon, "Nonlinear absorption extends confocal fluorescence microscopy into the ultra-violet regime and confines the illumination volume," Opt. Commun. 104, 223-228 (1994).
[CrossRef]

W. Denk, K. R. Delaney, A. Gelperin, D. Kleinfeld, B. W. Strowbridge, D. W. Tank, and R. Yuste, "Anatomical and functional imaging of neurons using 2-photon laser-scanning microscopy," J. Neurosci. Methods 54, 151-162 (1994).
[CrossRef] [PubMed]

1993 (1)

1990 (2)

W. Denk, J. H. Strickler, and W. W. Webb, "2-photon laser scanning fluorescence microscopy," Science 248, 73-76 (1990).
[CrossRef] [PubMed]

W. F. Cheong, S. A. Prahl, and A. J. Welch, "A review of the optical properties of biological tissues," IEEE J. Quantum Electron. 26, 2166-2185 (1990).
[CrossRef]

1981 (1)

E. Wolf and Y. Li, "Conditions for the validity of the Debye integral-representation of focused fields," Opt. Commun. 39, 205-210 (1981).
[CrossRef]

1977 (1)

1975 (1)

H. A. Haus, C. V. Shank, and E. P. Ippen, "Shape of passively mode-locked laser pulses," Opt. Commun. 15, 29-31 (1975).
[CrossRef]

1973 (1)

1972 (1)

1970 (1)

1965 (1)

1941 (1)

L. C. Henyey and J. L. Greenstein, "Diffuse radiation in the Galaxy," Astrophys. J. 93, 70-83 (1941).
[CrossRef]

1909 (1)

P. Debye, "Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie," Ann. Phys. 30, 755-776 (1909).
[CrossRef]

Alfano, R. R.

Arnush, D.

Beaurepaire, E.

E. Beaurepaire and J. Mertz, "Epifluorescence collection in two-photon microscopy," Appl. Opt. 41, 5376-5382 (2002).
[CrossRef] [PubMed]

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

Blanca, C. M.

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

Bucher, E. A.

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Clarendon, 1950).

Chen, B. E.

J. T. Trachtenberg, B. E. Chen, G. W. Knott, G. P. Feng, J. R. Sanes, E. Welker, and K. Svoboda, "Long-term in vivo imaging of experience-dependent synaptic plasticity in adult cortex," Nature 420, 788-794 (2002).
[CrossRef] [PubMed]

Cheong, W. F.

W. F. Cheong, S. A. Prahl, and A. J. Welch, "A review of the optical properties of biological tissues," IEEE J. Quantum Electron. 26, 2166-2185 (1990).
[CrossRef]

Ciervo, A. P.

Debye, P.

P. Debye, "Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie," Ann. Phys. 30, 755-776 (1909).
[CrossRef]

Delaney, K. R.

W. Denk, K. R. Delaney, A. Gelperin, D. Kleinfeld, B. W. Strowbridge, D. W. Tank, and R. Yuste, "Anatomical and functional imaging of neurons using 2-photon laser-scanning microscopy," J. Neurosci. Methods 54, 151-162 (1994).
[CrossRef] [PubMed]

Denk, W.

P. Theer, M. T. Hasan, and W. Denk, "Two-photon imaging to a depth of 1000 μm in living brains by use of a TiAl2O3 regenerative amplifier," Opt. Lett. 28, 1022-1024 (2003).
[CrossRef] [PubMed]

W. Denk and P. B. Detwiler, "Optical recording of light-evoked calcium signals in the functionally intact retina," Proc. Natl. Acad. Sci. U.S.A. 96, 7035-7040 (1999).
[CrossRef] [PubMed]

D. Kleinfeld, P. P. Mitra, F. Helmchen, and W. Denk, "Fluctuations and stimulus-induced changes in blood flow observed in individual capillaries in layers 2 through 4 of rat neocortex," Proc. Natl. Acad. Sci. U.S.A. 95, 15741-15746 (1998).
[CrossRef] [PubMed]

K. Svoboda, W. Denk, D. Kleinfeld, and D. W. Tank, "In vivo dendritic calcium dynamics in neocortical pyramidal neurons," Nature 385, 161-165 (1997).
[CrossRef] [PubMed]

W. Denk, "Two-photon excitation in functional biological imaging," J. Biomed. Opt. 1, 296-304 (1996).
[CrossRef]

W. Denk, K. R. Delaney, A. Gelperin, D. Kleinfeld, B. W. Strowbridge, D. W. Tank, and R. Yuste, "Anatomical and functional imaging of neurons using 2-photon laser-scanning microscopy," J. Neurosci. Methods 54, 151-162 (1994).
[CrossRef] [PubMed]

W. Denk, J. H. Strickler, and W. W. Webb, "2-photon laser scanning fluorescence microscopy," Science 248, 73-76 (1990).
[CrossRef] [PubMed]

W. Denk, D. W. Piston, and W. W. Webb, "Two-photon molecular excitation in laser-scanning microscopy," in Handbook of Biological Confocal Microscopy, J.B.Pawley, ed. (Plenum, 1995), pp. 445-458.

W. Denk, "Optical beam power controller using a tiltable birefringent plate," U.S. patent 6,249,379 (June 19, 2001).

Detwiler, P. B.

W. Denk and P. B. Detwiler, "Optical recording of light-evoked calcium signals in the functionally intact retina," Proc. Natl. Acad. Sci. U.S.A. 96, 7035-7040 (1999).
[CrossRef] [PubMed]

Dickson, L. D.

Feng, G. P.

J. T. Trachtenberg, B. E. Chen, G. W. Knott, G. P. Feng, J. R. Sanes, E. Welker, and K. Svoboda, "Long-term in vivo imaging of experience-dependent synaptic plasticity in adult cortex," Nature 420, 788-794 (2002).
[CrossRef] [PubMed]

Freeman, J. D.

Funk, O.

O. Funk and K. Pfeilsticker, "Photon path length distributions for cloudy skies--oxygen A-band measurements and model calculations," Ann. Geophys. 21, 615-626 (2003).
[CrossRef]

Gan, W. B.

J. Grutzendler, N. Kasthuri, and W. B. Gan, "Long-term dendritic spine stability in the adult cortex," Nature 420, 812-816 (2002).
[CrossRef] [PubMed]

Gatz, S.

Gelperin, A.

W. Denk, K. R. Delaney, A. Gelperin, D. Kleinfeld, B. W. Strowbridge, D. W. Tank, and R. Yuste, "Anatomical and functional imaging of neurons using 2-photon laser-scanning microscopy," J. Neurosci. Methods 54, 151-162 (1994).
[CrossRef] [PubMed]

Greenstein, J. L.

L. C. Henyey and J. L. Greenstein, "Diffuse radiation in the Galaxy," Astrophys. J. 93, 70-83 (1941).
[CrossRef]

Greiner, W.

W. Greiner, Classical Electrodynamics, Classical Theoretical Physics (Springer-Verlag, 1996), pp. 316-321.

Grutzendler, J.

J. Grutzendler, N. Kasthuri, and W. B. Gan, "Long-term dendritic spine stability in the adult cortex," Nature 420, 812-816 (2002).
[CrossRef] [PubMed]

Hall, G. J.

Haltrin, V. I.

Hasan, M. T.

Haus, H. A.

H. A. Haus, C. V. Shank, and E. P. Ippen, "Shape of passively mode-locked laser pulses," Opt. Commun. 15, 29-31 (1975).
[CrossRef]

Hell, S.

E. H. K. Stelzer, S. Hell, S. Lindek, R. Stricker, R. Pick, C. Storz, G. Ritter, and N. Salmon, "Nonlinear absorption extends confocal fluorescence microscopy into the ultra-violet regime and confines the illumination volume," Opt. Commun. 104, 223-228 (1994).
[CrossRef]

Helmchen, F.

D. Kleinfeld, P. P. Mitra, F. Helmchen, and W. Denk, "Fluctuations and stimulus-induced changes in blood flow observed in individual capillaries in layers 2 through 4 of rat neocortex," Proc. Natl. Acad. Sci. U.S.A. 95, 15741-15746 (1998).
[CrossRef] [PubMed]

Henyey, L. C.

L. C. Henyey and J. L. Greenstein, "Diffuse radiation in the Galaxy," Astrophys. J. 93, 70-83 (1941).
[CrossRef]

Herrmann, J.

Ippen, E. P.

H. A. Haus, C. V. Shank, and E. P. Ippen, "Shape of passively mode-locked laser pulses," Opt. Commun. 15, 29-31 (1975).
[CrossRef]

Kalosha, V. P.

Kasthuri, N.

J. Grutzendler, N. Kasthuri, and W. B. Gan, "Long-term dendritic spine stability in the adult cortex," Nature 420, 812-816 (2002).
[CrossRef] [PubMed]

Kleinfeld, D.

D. Kleinfeld, P. P. Mitra, F. Helmchen, and W. Denk, "Fluctuations and stimulus-induced changes in blood flow observed in individual capillaries in layers 2 through 4 of rat neocortex," Proc. Natl. Acad. Sci. U.S.A. 95, 15741-15746 (1998).
[CrossRef] [PubMed]

K. Svoboda, W. Denk, D. Kleinfeld, and D. W. Tank, "In vivo dendritic calcium dynamics in neocortical pyramidal neurons," Nature 385, 161-165 (1997).
[CrossRef] [PubMed]

W. Denk, K. R. Delaney, A. Gelperin, D. Kleinfeld, B. W. Strowbridge, D. W. Tank, and R. Yuste, "Anatomical and functional imaging of neurons using 2-photon laser-scanning microscopy," J. Neurosci. Methods 54, 151-162 (1994).
[CrossRef] [PubMed]

Knott, G. W.

J. T. Trachtenberg, B. E. Chen, G. W. Knott, G. P. Feng, J. R. Sanes, E. Welker, and K. Svoboda, "Long-term in vivo imaging of experience-dependent synaptic plasticity in adult cortex," Nature 420, 788-794 (2002).
[CrossRef] [PubMed]

Kogelnik, H.

Kumar, S.

Li, Y.

E. Wolf and Y. Li, "Conditions for the validity of the Debye integral-representation of focused fields," Opt. Commun. 39, 205-210 (1981).
[CrossRef]

Lindek, S.

E. H. K. Stelzer, S. Hell, S. Lindek, R. Stricker, R. Pick, C. Storz, G. Ritter, and N. Salmon, "Nonlinear absorption extends confocal fluorescence microscopy into the ultra-violet regime and confines the illumination volume," Opt. Commun. 104, 223-228 (1994).
[CrossRef]

Liu, F.

Lutomirski, R. F.

McLean, J. W.

Mertz, J.

E. Beaurepaire and J. Mertz, "Epifluorescence collection in two-photon microscopy," Appl. Opt. 41, 5376-5382 (2002).
[CrossRef] [PubMed]

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

Mitra, K.

Mitra, P. P.

D. Kleinfeld, P. P. Mitra, F. Helmchen, and W. Denk, "Fluctuations and stimulus-induced changes in blood flow observed in individual capillaries in layers 2 through 4 of rat neocortex," Proc. Natl. Acad. Sci. U.S.A. 95, 15741-15746 (1998).
[CrossRef] [PubMed]

Mobley, J.

J. Mobley and T. Vo-Dinh, "Optical properties of tissue," in Biomedical Photonics Handbook, T.Vo-Dinh, ed. (CRC, 2003).
[CrossRef]

Muller, M.

Oheim, M.

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

Pfeilsticker, K.

O. Funk and K. Pfeilsticker, "Photon path length distributions for cloudy skies--oxygen A-band measurements and model calculations," Ann. Geophys. 21, 615-626 (2003).
[CrossRef]

Pick, R.

E. H. K. Stelzer, S. Hell, S. Lindek, R. Stricker, R. Pick, C. Storz, G. Ritter, and N. Salmon, "Nonlinear absorption extends confocal fluorescence microscopy into the ultra-violet regime and confines the illumination volume," Opt. Commun. 104, 223-228 (1994).
[CrossRef]

Piston, D. W.

W. Denk, D. W. Piston, and W. W. Webb, "Two-photon molecular excitation in laser-scanning microscopy," in Handbook of Biological Confocal Microscopy, J.B.Pawley, ed. (Plenum, 1995), pp. 445-458.

Prahl, S. A.

S. A. Prahl, M. J. C. Vangemert, and A. J. Welch, "Determining the optical-properties of turbid media by using the adding-doubling method," Appl. Opt. 32, 559-568 (1993).
[CrossRef] [PubMed]

W. F. Cheong, S. A. Prahl, and A. J. Welch, "A review of the optical properties of biological tissues," IEEE J. Quantum Electron. 26, 2166-2185 (1990).
[CrossRef]

Ritter, G.

E. H. K. Stelzer, S. Hell, S. Lindek, R. Stricker, R. Pick, C. Storz, G. Ritter, and N. Salmon, "Nonlinear absorption extends confocal fluorescence microscopy into the ultra-violet regime and confines the illumination volume," Opt. Commun. 104, 223-228 (1994).
[CrossRef]

Salmon, N.

E. H. K. Stelzer, S. Hell, S. Lindek, R. Stricker, R. Pick, C. Storz, G. Ritter, and N. Salmon, "Nonlinear absorption extends confocal fluorescence microscopy into the ultra-violet regime and confines the illumination volume," Opt. Commun. 104, 223-228 (1994).
[CrossRef]

Saloma, C.

Sanes, J. R.

J. T. Trachtenberg, B. E. Chen, G. W. Knott, G. P. Feng, J. R. Sanes, E. Welker, and K. Svoboda, "Long-term in vivo imaging of experience-dependent synaptic plasticity in adult cortex," Nature 420, 788-794 (2002).
[CrossRef] [PubMed]

Schober, R.

A. N. Yaroslavsky, P. C. Schulze, I. V. Yaroslavsky, R. Schober, F. Ulrich, and H. J. Schwarzmaier, "Optical properties of selected native and coagulated human brain tissues in vitro in the visible and near infrared spectral range," Phys. Med. Biol. 47, 2059-2073 (2002).
[CrossRef] [PubMed]

Schulze, P. C.

A. N. Yaroslavsky, P. C. Schulze, I. V. Yaroslavsky, R. Schober, F. Ulrich, and H. J. Schwarzmaier, "Optical properties of selected native and coagulated human brain tissues in vitro in the visible and near infrared spectral range," Phys. Med. Biol. 47, 2059-2073 (2002).
[CrossRef] [PubMed]

Schwarzmaier, H. J.

A. N. Yaroslavsky, P. C. Schulze, I. V. Yaroslavsky, R. Schober, F. Ulrich, and H. J. Schwarzmaier, "Optical properties of selected native and coagulated human brain tissues in vitro in the visible and near infrared spectral range," Phys. Med. Biol. 47, 2059-2073 (2002).
[CrossRef] [PubMed]

Shank, C. V.

H. A. Haus, C. V. Shank, and E. P. Ippen, "Shape of passively mode-locked laser pulses," Opt. Commun. 15, 29-31 (1975).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science, 1986), Chap. 16.

Stelzer, E. H. K.

E. H. K. Stelzer, S. Hell, S. Lindek, R. Stricker, R. Pick, C. Storz, G. Ritter, and N. Salmon, "Nonlinear absorption extends confocal fluorescence microscopy into the ultra-violet regime and confines the illumination volume," Opt. Commun. 104, 223-228 (1994).
[CrossRef]

Storz, C.

E. H. K. Stelzer, S. Hell, S. Lindek, R. Stricker, R. Pick, C. Storz, G. Ritter, and N. Salmon, "Nonlinear absorption extends confocal fluorescence microscopy into the ultra-violet regime and confines the illumination volume," Opt. Commun. 104, 223-228 (1994).
[CrossRef]

Stotts, L. B.

Stricker, R.

E. H. K. Stelzer, S. Hell, S. Lindek, R. Stricker, R. Pick, C. Storz, G. Ritter, and N. Salmon, "Nonlinear absorption extends confocal fluorescence microscopy into the ultra-violet regime and confines the illumination volume," Opt. Commun. 104, 223-228 (1994).
[CrossRef]

Strickler, J. H.

W. Denk, J. H. Strickler, and W. W. Webb, "2-photon laser scanning fluorescence microscopy," Science 248, 73-76 (1990).
[CrossRef] [PubMed]

Strowbridge, B. W.

W. Denk, K. R. Delaney, A. Gelperin, D. Kleinfeld, B. W. Strowbridge, D. W. Tank, and R. Yuste, "Anatomical and functional imaging of neurons using 2-photon laser-scanning microscopy," J. Neurosci. Methods 54, 151-162 (1994).
[CrossRef] [PubMed]

Svoboda, K.

J. T. Trachtenberg, B. E. Chen, G. W. Knott, G. P. Feng, J. R. Sanes, E. Welker, and K. Svoboda, "Long-term in vivo imaging of experience-dependent synaptic plasticity in adult cortex," Nature 420, 788-794 (2002).
[CrossRef] [PubMed]

K. Svoboda, W. Denk, D. Kleinfeld, and D. W. Tank, "In vivo dendritic calcium dynamics in neocortical pyramidal neurons," Nature 385, 161-165 (1997).
[CrossRef] [PubMed]

Tank, D. W.

K. Svoboda, W. Denk, D. Kleinfeld, and D. W. Tank, "In vivo dendritic calcium dynamics in neocortical pyramidal neurons," Nature 385, 161-165 (1997).
[CrossRef] [PubMed]

W. Denk, K. R. Delaney, A. Gelperin, D. Kleinfeld, B. W. Strowbridge, D. W. Tank, and R. Yuste, "Anatomical and functional imaging of neurons using 2-photon laser-scanning microscopy," J. Neurosci. Methods 54, 151-162 (1994).
[CrossRef] [PubMed]

Theer, P.

P. Theer, M. T. Hasan, and W. Denk, "Two-photon imaging to a depth of 1000 μm in living brains by use of a TiAl2O3 regenerative amplifier," Opt. Lett. 28, 1022-1024 (2003).
[CrossRef] [PubMed]

P. Theer, "On the fundamental imaging-depth limit in two-photon microscopy," Doctoral dissertation (University of Heidelberg, 2004), http://www.ub.uni-heidelberg.de/archiv/4830.

Trachtenberg, J. T.

J. T. Trachtenberg, B. E. Chen, G. W. Knott, G. P. Feng, J. R. Sanes, E. Welker, and K. Svoboda, "Long-term in vivo imaging of experience-dependent synaptic plasticity in adult cortex," Nature 420, 788-794 (2002).
[CrossRef] [PubMed]

Ulrich, F.

A. N. Yaroslavsky, P. C. Schulze, I. V. Yaroslavsky, R. Schober, F. Ulrich, and H. J. Schwarzmaier, "Optical properties of selected native and coagulated human brain tissues in vitro in the visible and near infrared spectral range," Phys. Med. Biol. 47, 2059-2073 (2002).
[CrossRef] [PubMed]

Vangemert, M. J. C.

Vo-Dinh, T.

J. Mobley and T. Vo-Dinh, "Optical properties of tissue," in Biomedical Photonics Handbook, T.Vo-Dinh, ed. (CRC, 2003).
[CrossRef]

Walker, R. E.

Webb, W. W.

W. R. Zipfel, R. M. Williams, and W. W. Webb, "Nonlinear magic: multiphoton microscopy in the biosciences," Nat. Biotechnol. 21, 1368-1376 (2003).
[CrossRef]

W. Denk, J. H. Strickler, and W. W. Webb, "2-photon laser scanning fluorescence microscopy," Science 248, 73-76 (1990).
[CrossRef] [PubMed]

W. Denk, D. W. Piston, and W. W. Webb, "Two-photon molecular excitation in laser-scanning microscopy," in Handbook of Biological Confocal Microscopy, J.B.Pawley, ed. (Plenum, 1995), pp. 445-458.

Welch, A. J.

S. A. Prahl, M. J. C. Vangemert, and A. J. Welch, "Determining the optical-properties of turbid media by using the adding-doubling method," Appl. Opt. 32, 559-568 (1993).
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W. F. Cheong, S. A. Prahl, and A. J. Welch, "A review of the optical properties of biological tissues," IEEE J. Quantum Electron. 26, 2166-2185 (1990).
[CrossRef]

Welker, E.

J. T. Trachtenberg, B. E. Chen, G. W. Knott, G. P. Feng, J. R. Sanes, E. Welker, and K. Svoboda, "Long-term in vivo imaging of experience-dependent synaptic plasticity in adult cortex," Nature 420, 788-794 (2002).
[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, 1368-1376 (2003).
[CrossRef]

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E. Wolf and Y. Li, "Conditions for the validity of the Debye integral-representation of focused fields," Opt. Commun. 39, 205-210 (1981).
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M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

Yaroslavsky, A. N.

A. N. Yaroslavsky, P. C. Schulze, I. V. Yaroslavsky, R. Schober, F. Ulrich, and H. J. Schwarzmaier, "Optical properties of selected native and coagulated human brain tissues in vitro in the visible and near infrared spectral range," Phys. Med. Biol. 47, 2059-2073 (2002).
[CrossRef] [PubMed]

Yaroslavsky, I. V.

A. N. Yaroslavsky, P. C. Schulze, I. V. Yaroslavsky, R. Schober, F. Ulrich, and H. J. Schwarzmaier, "Optical properties of selected native and coagulated human brain tissues in vitro in the visible and near infrared spectral range," Phys. Med. Biol. 47, 2059-2073 (2002).
[CrossRef] [PubMed]

Ying, J. P.

Yuste, R.

W. Denk, K. R. Delaney, A. Gelperin, D. Kleinfeld, B. W. Strowbridge, D. W. Tank, and R. Yuste, "Anatomical and functional imaging of neurons using 2-photon laser-scanning microscopy," J. Neurosci. Methods 54, 151-162 (1994).
[CrossRef] [PubMed]

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W. R. Zipfel, R. M. Williams, and W. W. Webb, "Nonlinear magic: multiphoton microscopy in the biosciences," Nat. Biotechnol. 21, 1368-1376 (2003).
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[CrossRef]

J. Biomed. Opt. (1)

W. Denk, "Two-photon excitation in functional biological imaging," J. Biomed. Opt. 1, 296-304 (1996).
[CrossRef]

J. Neurosci. Methods (1)

W. Denk, K. R. Delaney, A. Gelperin, D. Kleinfeld, B. W. Strowbridge, D. W. Tank, and R. Yuste, "Anatomical and functional imaging of neurons using 2-photon laser-scanning microscopy," J. Neurosci. Methods 54, 151-162 (1994).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (2)

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

Nat. Biotechnol. (1)

W. R. Zipfel, R. M. Williams, and W. W. Webb, "Nonlinear magic: multiphoton microscopy in the biosciences," Nat. Biotechnol. 21, 1368-1376 (2003).
[CrossRef]

Nature (3)

K. Svoboda, W. Denk, D. Kleinfeld, and D. W. Tank, "In vivo dendritic calcium dynamics in neocortical pyramidal neurons," Nature 385, 161-165 (1997).
[CrossRef] [PubMed]

J. T. Trachtenberg, B. E. Chen, G. W. Knott, G. P. Feng, J. R. Sanes, E. Welker, and K. Svoboda, "Long-term in vivo imaging of experience-dependent synaptic plasticity in adult cortex," Nature 420, 788-794 (2002).
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E. Wolf and Y. Li, "Conditions for the validity of the Debye integral-representation of focused fields," Opt. Commun. 39, 205-210 (1981).
[CrossRef]

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[CrossRef]

Opt. Lett. (1)

Phys. Med. Biol. (1)

A. N. Yaroslavsky, P. C. Schulze, I. V. Yaroslavsky, R. Schober, F. Ulrich, and H. J. Schwarzmaier, "Optical properties of selected native and coagulated human brain tissues in vitro in the visible and near infrared spectral range," Phys. Med. Biol. 47, 2059-2073 (2002).
[CrossRef] [PubMed]

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

D. Kleinfeld, P. P. Mitra, F. Helmchen, and W. Denk, "Fluctuations and stimulus-induced changes in blood flow observed in individual capillaries in layers 2 through 4 of rat neocortex," Proc. Natl. Acad. Sci. U.S.A. 95, 15741-15746 (1998).
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Science (1)

W. Denk, J. H. Strickler, and W. W. Webb, "2-photon laser scanning fluorescence microscopy," Science 248, 73-76 (1990).
[CrossRef] [PubMed]

Other (9)

W. Denk, D. W. Piston, and W. W. Webb, "Two-photon molecular excitation in laser-scanning microscopy," in Handbook of Biological Confocal Microscopy, J.B.Pawley, ed. (Plenum, 1995), pp. 445-458.

MATHEMATICA, Version 5.2 (Wolfram Research, Incorporated, Champaign, Ill., 2005).

P. Theer, "On the fundamental imaging-depth limit in two-photon microscopy," Doctoral dissertation (University of Heidelberg, 2004), http://www.ub.uni-heidelberg.de/archiv/4830.

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W. Greiner, Classical Electrodynamics, Classical Theoretical Physics (Springer-Verlag, 1996), pp. 316-321.

A. E. Siegman, Lasers (University Science, 1986), Chap. 16.

J. Mobley and T. Vo-Dinh, "Optical properties of tissue," in Biomedical Photonics Handbook, T.Vo-Dinh, ed. (CRC, 2003).
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M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

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Figures (8)

Fig. 1
Fig. 1

Excitation beam is focused by an immersion objective and enters the scattering sample through a transparent immersion medium with an almost matched refractive index. The imaging depth is z 0 . Note that for a sufficiently large imaging depth, significant fluorescence is generated near the top of the sample.

Fig. 2
Fig. 2

Effective NA ( A NA ) of a Gaussian beam in an absorbing–scattering sample ( n = 1.33 ) versus focus depth, expressed as multiples of the MFP ( α z 0 ) .

Fig. 3
Fig. 3

(a) Effective NA ( A NA ) as a function of lens fill factor ( β ) for a nominal lens NA of 1.0 and z 0 = 1000 μ m using the paraxial approximation [Eq. (8)] with α = 0 and α = 0.005 μ m , and numerical integration of the Debye integral [Eq. (9)] with α = 0 and α = 0.005 μ m , and (b) the corresponding point-spread functions for β = 1 .

Fig. 4
Fig. 4

Ballistic light-induced 2P absorption as a function of (a) the radius ( r z R ) of a spherical 2P absorber-filled volume, (b) the half-thickness [ d ( 2 z R ) ] of a 2P absorber-filled slab expressed in terms of the Rayleigh range, and (c) as actual distance ( d 2 = z 1 ) for various effective NAs. (d) The same data plotted as a function of the effective NA ( A NA ) for a number of different spherical-volume radii, expressed in terms of the wavelength in the medium ( λ n ) ; curves are normalized to their approximate plateau value.

Fig. 5
Fig. 5

Fluorescence per depth slice versus depth for anisotropy factors (a) g = 0.01 , (b) g = 0.7 , (c) g = 0.9 , and (d) g = 0.98 . The total, [ I scattered ( ρ , z , τ ) + I ballistic ( ρ , z , τ ) ] 2 , and its components due to the ballistic light I ballistic 2 ( ρ , z , τ ) , the scattered light I scattered 2 ( ρ , z , τ ) , and the mixed term 2 I scattered ( ρ , z , τ ) I ballistic ( ρ , z , τ ) are plotted separately. The focus depth is z 0 = 5 l s . Curves are normalized to the maximum fluorescence intensity, which occurs at the focus. The background fluorescence is approximately (a) 2.6, (b) 4.4, (c) 8.8, and (d) 35.1, times that of the perifocal fluorescence. (e), (f) The perifocal (sgn) to out-of-focus (bgr) ratio as a function of the (e) anisotropy factor and (f) pulse length for a focus depth of 1000 μ m , corresponding to five scattering MFPs, and g = 0.9 .

Fig. 6
Fig. 6

Two-photon imaging-depth limit in terms of scattering MFPs ( α z 0 ) versus NA ( A NA ) for g = 0.9 and staining inhomogeneities of 1, 10, 100, and 1000.

Fig. 7
Fig. 7

Radial perifocal (sgn) and out-of-focus fluorescence (bgr) intensity distributions (a) at the surface of a scattering sample and signal-to-background ratio (b) as a function of radius of a circular obstruction centered at the optical axis for a focus depth of 4.5 l s ( 900 μ m ) ; solid curves show the theoretically expected gain when neglecting scattering of near-surface fluorescence, dotted curves are experimental values measured using an NA of 0.95 with a fill factor of 0.63. For details see Ref. [14]. (c), (d) Gamma-corrected (0.7) examples of planar images of a single fluorescent bead (c) without and (d) with an opaque disk placed at the image of the sample surface in the detection path; in this case the signal-to-background ratio increased by a factor of 1.6 .

Fig. 8
Fig. 8

Determination of the depth limit. (a) y z projection of the fluorescent bead distribution in a subvolume of a scattering sample with a staining inhomogeneity of 3660 and individual images at depths of 0.5 l s ( 100 μ m ) above, at, and 0.5 l s below the depth limit, which in this case was at 7 l s ( 1291 μ m ) . (b) The 2P imaging-depth limit (expressed in terms of the scattering MFP) versus staining inhomogeneity. The dots are experimental values, the solid line shows the theoretical expectation. Experiments and theory are based on Gaussian beams, an NA of 0.6 and g = 0.88 .

Equations (23)

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U ( r ) = A 0 q ( z ) exp ( i k ρ 2 2 q ( z ) ) exp ( i k z ) ,
1 q ( z ) = 1 R ( z ) i λ n π w 2 ( z ) = 1 i z R + z .
z R = λ n π tan 2 θ 0 .
I ( z , ρ ) = 2 P π w 2 ( z ) exp ( 2 ρ 2 w 2 ( z ) ) ,
q ( z ) = q ( z = z 0 ) + z n = i z R z 0 + 1 1 + i α λ ( 4 π n ) z = i z R ( z ) z 0 ( z ) + z ,
z 0 ( z 0 ) z 0 = 1 1 + [ 4 n π ( α λ ) ] 2 z 0 ,
z R ( z 0 ) z R = 4 n π α λ 16 n 2 π 2 + α 2 λ 2 z 0 .
A NA ( z 0 ) A NA 0 1 + α z 0 4 ( A NA 0 n ) 2 ,
i λ 0 2 π 0 θ NA P ( θ ) exp ( i k ρ sin θ cos ϕ ) sin θ cos θ d θ d ϕ .
F pf = 1 2 η 2 δ C V pf I ballistic 2 ( ρ , z ) d V ,
V I ballistic 2 ( ρ , z ) d V = z 1 z 2 0 I ballistic 2 ( ρ , z ) 2 π ρ d ρ d z = 2 n 2 π P 2 λ ( 16 n 2 π 2 + α 2 λ 2 ) exp ( 2 α z 0 ) { ( α λ i 4 n π ) exp ( i 2 α z R ) { E i [ 2 α ( z 1 i z R ) ] E i [ 2 α ( z 2 + i z R ) ] } + c . c . 2 α λ exp ( 8 n π z R λ ) [ E i ( 2 α z 1 8 n π z R λ ) E i ( 2 α z 2 8 n π z R λ ) ] } .
z 2 z 2 0 I ballistic 2 ( ρ , z ) 2 π ρ d ρ d z = 2 n 2 π P 2 λ ( 16 n 2 π 2 + α 2 λ 2 ) exp ( 2 α z 0 ) [ ( α λ i 4 n π ) exp ( i 2 α z R ) { E i [ 2 α ( z 2 i z R ) ] E i [ 2 α ( z 2 + i z R ) ] } + c . c . ] .
z 2 z 2 0 I ballistic 2 ( ρ , z ) 2 π ρ d ρ d z z 2 z R P 2 n π λ .
F oof = 1 2 η 2 δ C V oof [ I scattered ( ρ , z , t ) + I ballistic ( ρ , z , t ) ] 2 d t d V .
P ( t ) = P 0 exp ( 2 t 2 τ 0 2 ) ,
s ( ρ , z , τ ) = h ( ρ , z , t ) g ( z , τ ) ,
I scattered ( ρ , z , t ) = I ballistic ( x , y , z , τ ) s ( x x , y y , z , t τ ) d x d y d τ .
h ( z , ρ , τ ) = 3 n 4 π c τ z exp ( 3 n ρ 2 4 c τ z ) .
g ( τ , z ) = 1 τ 2 π S ( z ) exp { [ ln τ M ( z ) ] 2 2 S 2 ( z ) } .
μ ( z ) = n z c 1 exp ( α z v ) α v c n ,
σ 2 ( z ) = 3 2 ( w 2 3 w v ) [ exp ( α z v ) 1 + α z v ] + 2 v 2 [ exp ( α z w ) s 1 + α z w ] w ( w v ) α 2 v 2 c 2 n 2 [ 1 exp ( α z v ) α v c n ] 2 ,
F pf F oof = V pf C ( ρ , z , t ) I ballistic 2 ( ρ , z , t ) d t d V V oof C ( ρ , z , t ) [ I scattered ( ρ , z , t ) + I ballistic ( ρ , z , t ) ] 2 d t d V = 1 .
V oof [ I scattered ( ρ , z , t ) + I ballistic ( ρ , z , t ) ] 2 d t d V = χ n π λ P 2 exp ( 2 α z 0 ) .

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