Abstract

Temporal focusing is a simple approach for achieving tight, optically sectioned excitation in nonlinear microscopy and multiphoton photo-manipulation. Key applications and advantages of temporal focusing involve propagation through scattering media, but the progressive broadening of the temporal focus has not been characterized. By combining a detailed geometrical optics model with Monte-Carlo scattering simulations we introduce and validate a simulation strategy for predicting temporal focusing characteristics in scattering and non-scattering media. The broadening of the temporal focus width with increasing depth in brain tissue is studied using both simulations and experiments for several key optical geometries, and an analytical approximation is found for the dependence of this broadening on the microscope’s parameters in a transparent medium. Our results indicate that a multiphoton temporal focus has radically different broadening characteristics in deep tissue than those of a spatial focus.

© 2011 OSA

Full Article  |  PDF Article

Errata

Hod Dana and Shy Shoham, "Numerical evaluation of temporal focusing characteristics in transparent and scattering media: erratum," Opt. Express 20, 28281-28281 (2012)
https://www.osapublishing.org/oe/abstract.cfm?uri=oe-20-27-28281

References

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2011 (1)

2010 (6)

D. N. Vitek, D. E. Adams, A. Johnson, P. S. Tsai, S. Backus, C. G. Durfee, D. Kleinfeld, and J. A. Squier, “Temporally focused femtosecond laser pulses for low numerical aperture micromachining through optically transparent materials,” Opt. Express 18(17), 18086–18094 (2010).
[CrossRef] [PubMed]

D. Kim and P. T. C. So, “High-throughput three-dimensional lithographic microfabrication,” Opt. Lett. 35(10), 1602–1604 (2010).
[CrossRef] [PubMed]

B. K. Andrasfalvy, B. V. Zemelman, J. Tang, and A. Vaziri, “Two-photon single-cell optogenetic control of neuronal activity by sculpted light,” Proc. Natl. Acad. Sci. U.S.A. 107(26), 11981–11986 (2010).
[CrossRef] [PubMed]

E. Papagiakoumou, F. Anselmi, A. Bègue, V. de Sars, J. Glückstad, E. Y. Isacoff, and V. Emiliani, “Scanless two-photon excitation of channelrhodopsin-2,” Nat. Methods 7(10), 848–854 (2010).
[CrossRef] [PubMed]

S. Shoham, “Optogenetics meets optical wavefront shaping,” Nat. Methods 7(10), 798–799 (2010).
[CrossRef] [PubMed]

A. Vaziri and C. V. Shank, “Ultrafast widefield optical sectioning microscopy by multifocal temporal focusing,” Opt. Express 18(19), 19645–19655 (2010).
[CrossRef] [PubMed]

2009 (1)

C. K. Hayakawa, V. Venugopalan, V. V. Krishnamachari, and E. O. Potma, “Amplitude and phase of tightly focused laser beams in turbid media,” Phys. Rev. Lett. 103(4), 043903 (2009).
[CrossRef] [PubMed]

2008 (4)

D. G. Fischer, S. A. Prahl, and D. D. Duncan, “Monte Carlo modeling of spatial coherence: free-space diffraction,” J. Opt. Soc. Am. A 25(10), 2571–2581 (2008).
[CrossRef]

E. Papagiakoumou, V. de Sars, D. Oron, and V. Emiliani, “Patterned two-photon illumination by spatiotemporal shaping of ultrashort pulses,” Opt. Express 16(26), 22039–22047 (2008).
[CrossRef] [PubMed]

A. Vaziri, J. Tang, H. Shroff, and C. V. Shank, “Multilayer three-dimensional super resolution imaging of thick biological samples,” Proc. Natl. Acad. Sci. U.S.A. 105(51), 20221–20226 (2008).
[CrossRef] [PubMed]

M. E. Durst, G. Zhu, and C. Xu, “Simultaneous spatial and temporal focusing in nonlinear microscopy,” Opt. Commun. 281(7), 1796–1805 (2008).
[CrossRef] [PubMed]

2006 (2)

2005 (6)

Y. Ikegaya, M. Le Bon-Jego, and R. Yuste, “Large-scale imaging of cortical network activity with calcium indicators,” Neurosci. Res. 52(2), 132–138 (2005).
[CrossRef] [PubMed]

D. Oron and Y. Silberberg, “Harmonic generation with temporally focused ultrashort pulses,” J. Opt. Soc. B 22(12), 2660–2663 (2005).
[CrossRef]

F. Helmchen and W. Denk, “Deep tissue two-photon microscopy,” Nat. Methods 2(12), 932–940 (2005).
[CrossRef] [PubMed]

D. Oron, E. Tal, and Y. Silberberg, “Scanningless depth-resolved microscopy,” Opt. Express 13(5), 1468–1476 (2005).
[CrossRef] [PubMed]

E. Tal, D. Oron, and Y. Silberberg, “Improved depth resolution in video-rate line-scanning multiphoton microscopy using temporal focusing,” Opt. Lett. 30(13), 1686–1688 (2005).
[CrossRef] [PubMed]

G. Zhu, J. van Howe, M. Durst, W. Zipfel, and C. Xu, “Simultaneous spatial and temporal focusing of femtosecond pulses,” Opt. Express 13(6), 2153–2159 (2005).
[CrossRef] [PubMed]

2003 (2)

P. Theer, M. T. Hasan, and W. Denk, “Two-photon imaging to a depth of 1000 µm in living brains by use of a Ti:Al2O3 regenerative amplifier,” Opt. Lett. 28(12), 1022–1024 (2003).
[CrossRef] [PubMed]

C. Y. Dong, K. Koenig, and P. So, “Characterizing point spread functions of two-photon fluorescence microscopy in turbid medium,” J. Biomed. Opt. 8(3), 450–459 (2003).
[CrossRef] [PubMed]

2002 (1)

2001 (1)

M. Oheim, E. Beaurepaire, E. Chaigneau, J. Mertz, and S. Charpak, “Two-photon microscopy in brain tissue: parameters influencing the imaging depth,” J. Neurosci. Methods 111(1), 29–37 (2001).
[CrossRef] [PubMed]

1998 (1)

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(26), 15741–15746 (1998).
[CrossRef] [PubMed]

Adams, D. E.

Andrasfalvy, B. K.

B. K. Andrasfalvy, B. V. Zemelman, J. Tang, and A. Vaziri, “Two-photon single-cell optogenetic control of neuronal activity by sculpted light,” Proc. Natl. Acad. Sci. U.S.A. 107(26), 11981–11986 (2010).
[CrossRef] [PubMed]

Anselmi, F.

E. Papagiakoumou, F. Anselmi, A. Bègue, V. de Sars, J. Glückstad, E. Y. Isacoff, and V. Emiliani, “Scanless two-photon excitation of channelrhodopsin-2,” Nat. Methods 7(10), 848–854 (2010).
[CrossRef] [PubMed]

Backus, S.

Beaurepaire, E.

M. Oheim, E. Beaurepaire, E. Chaigneau, J. Mertz, and S. Charpak, “Two-photon microscopy in brain tissue: parameters influencing the imaging depth,” J. Neurosci. Methods 111(1), 29–37 (2001).
[CrossRef] [PubMed]

Bègue, A.

E. Papagiakoumou, F. Anselmi, A. Bègue, V. de Sars, J. Glückstad, E. Y. Isacoff, and V. Emiliani, “Scanless two-photon excitation of channelrhodopsin-2,” Nat. Methods 7(10), 848–854 (2010).
[CrossRef] [PubMed]

Boas, D.

Chaigneau, E.

M. Oheim, E. Beaurepaire, E. Chaigneau, J. Mertz, and S. Charpak, “Two-photon microscopy in brain tissue: parameters influencing the imaging depth,” J. Neurosci. Methods 111(1), 29–37 (2001).
[CrossRef] [PubMed]

Charpak, S.

M. Oheim, E. Beaurepaire, E. Chaigneau, J. Mertz, and S. Charpak, “Two-photon microscopy in brain tissue: parameters influencing the imaging depth,” J. Neurosci. Methods 111(1), 29–37 (2001).
[CrossRef] [PubMed]

Culver, J.

de Sars, V.

E. Papagiakoumou, F. Anselmi, A. Bègue, V. de Sars, J. Glückstad, E. Y. Isacoff, and V. Emiliani, “Scanless two-photon excitation of channelrhodopsin-2,” Nat. Methods 7(10), 848–854 (2010).
[CrossRef] [PubMed]

E. Papagiakoumou, V. de Sars, D. Oron, and V. Emiliani, “Patterned two-photon illumination by spatiotemporal shaping of ultrashort pulses,” Opt. Express 16(26), 22039–22047 (2008).
[CrossRef] [PubMed]

Denk, W.

P. Theer and W. Denk, “On the fundamental imaging-depth limit in two-photon microscopy,” J. Opt. Soc. Am. A 23(12), 3139–3149 (2006).
[CrossRef]

F. Helmchen and W. Denk, “Deep tissue two-photon microscopy,” Nat. Methods 2(12), 932–940 (2005).
[CrossRef] [PubMed]

P. Theer, M. T. Hasan, and W. Denk, “Two-photon imaging to a depth of 1000 µm in living brains by use of a Ti:Al2O3 regenerative amplifier,” Opt. Lett. 28(12), 1022–1024 (2003).
[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(26), 15741–15746 (1998).
[CrossRef] [PubMed]

Dong, C. Y.

C. Y. Dong, K. Koenig, and P. So, “Characterizing point spread functions of two-photon fluorescence microscopy in turbid medium,” J. Biomed. Opt. 8(3), 450–459 (2003).
[CrossRef] [PubMed]

Duncan, D. D.

Dunn, A.

Durfee, C. G.

Durst, M.

Durst, M. E.

M. E. Durst, G. Zhu, and C. Xu, “Simultaneous spatial and temporal focusing in nonlinear microscopy,” Opt. Commun. 281(7), 1796–1805 (2008).
[CrossRef] [PubMed]

M. E. Durst, G. Zhu, and C. Xu, “Simultaneous spatial and temporal focusing for axial scanning,” Opt. Express 14(25), 12243–12254 (2006).
[CrossRef] [PubMed]

Emiliani, V.

E. Papagiakoumou, F. Anselmi, A. Bègue, V. de Sars, J. Glückstad, E. Y. Isacoff, and V. Emiliani, “Scanless two-photon excitation of channelrhodopsin-2,” Nat. Methods 7(10), 848–854 (2010).
[CrossRef] [PubMed]

E. Papagiakoumou, V. de Sars, D. Oron, and V. Emiliani, “Patterned two-photon illumination by spatiotemporal shaping of ultrashort pulses,” Opt. Express 16(26), 22039–22047 (2008).
[CrossRef] [PubMed]

Fischer, D. G.

Glückstad, J.

E. Papagiakoumou, F. Anselmi, A. Bègue, V. de Sars, J. Glückstad, E. Y. Isacoff, and V. Emiliani, “Scanless two-photon excitation of channelrhodopsin-2,” Nat. Methods 7(10), 848–854 (2010).
[CrossRef] [PubMed]

Hasan, M. T.

Hayakawa, C. K.

C. K. Hayakawa, E. O. Potma, and V. Venugopalan, “Electric field Monte Carlo simulations of focal field distributions produced by tightly focused laser beams in tissues,” Biomed. Opt. Express 2(2), 278–299 (2011).
[CrossRef] [PubMed]

C. K. Hayakawa, V. Venugopalan, V. V. Krishnamachari, and E. O. Potma, “Amplitude and phase of tightly focused laser beams in turbid media,” Phys. Rev. Lett. 103(4), 043903 (2009).
[CrossRef] [PubMed]

Helmchen, F.

F. Helmchen and W. Denk, “Deep tissue two-photon microscopy,” Nat. Methods 2(12), 932–940 (2005).
[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(26), 15741–15746 (1998).
[CrossRef] [PubMed]

Ikegaya, Y.

Y. Ikegaya, M. Le Bon-Jego, and R. Yuste, “Large-scale imaging of cortical network activity with calcium indicators,” Neurosci. Res. 52(2), 132–138 (2005).
[CrossRef] [PubMed]

Isacoff, E. Y.

E. Papagiakoumou, F. Anselmi, A. Bègue, V. de Sars, J. Glückstad, E. Y. Isacoff, and V. Emiliani, “Scanless two-photon excitation of channelrhodopsin-2,” Nat. Methods 7(10), 848–854 (2010).
[CrossRef] [PubMed]

Johnson, A.

Kim, D.

Kleinfeld, D.

D. N. Vitek, D. E. Adams, A. Johnson, P. S. Tsai, S. Backus, C. G. Durfee, D. Kleinfeld, and J. A. Squier, “Temporally focused femtosecond laser pulses for low numerical aperture micromachining through optically transparent materials,” Opt. Express 18(17), 18086–18094 (2010).
[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(26), 15741–15746 (1998).
[CrossRef] [PubMed]

Koenig, K.

C. Y. Dong, K. Koenig, and P. So, “Characterizing point spread functions of two-photon fluorescence microscopy in turbid medium,” J. Biomed. Opt. 8(3), 450–459 (2003).
[CrossRef] [PubMed]

Krishnamachari, V. V.

C. K. Hayakawa, V. Venugopalan, V. V. Krishnamachari, and E. O. Potma, “Amplitude and phase of tightly focused laser beams in turbid media,” Phys. Rev. Lett. 103(4), 043903 (2009).
[CrossRef] [PubMed]

Le Bon-Jego, M.

Y. Ikegaya, M. Le Bon-Jego, and R. Yuste, “Large-scale imaging of cortical network activity with calcium indicators,” Neurosci. Res. 52(2), 132–138 (2005).
[CrossRef] [PubMed]

Mertz, J.

M. Oheim, E. Beaurepaire, E. Chaigneau, J. Mertz, and S. Charpak, “Two-photon microscopy in brain tissue: parameters influencing the imaging depth,” J. Neurosci. Methods 111(1), 29–37 (2001).
[CrossRef] [PubMed]

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(26), 15741–15746 (1998).
[CrossRef] [PubMed]

Oheim, M.

M. Oheim, E. Beaurepaire, E. Chaigneau, J. Mertz, and S. Charpak, “Two-photon microscopy in brain tissue: parameters influencing the imaging depth,” J. Neurosci. Methods 111(1), 29–37 (2001).
[CrossRef] [PubMed]

Oron, D.

Papagiakoumou, E.

E. Papagiakoumou, F. Anselmi, A. Bègue, V. de Sars, J. Glückstad, E. Y. Isacoff, and V. Emiliani, “Scanless two-photon excitation of channelrhodopsin-2,” Nat. Methods 7(10), 848–854 (2010).
[CrossRef] [PubMed]

E. Papagiakoumou, V. de Sars, D. Oron, and V. Emiliani, “Patterned two-photon illumination by spatiotemporal shaping of ultrashort pulses,” Opt. Express 16(26), 22039–22047 (2008).
[CrossRef] [PubMed]

Potma, E. O.

C. K. Hayakawa, E. O. Potma, and V. Venugopalan, “Electric field Monte Carlo simulations of focal field distributions produced by tightly focused laser beams in tissues,” Biomed. Opt. Express 2(2), 278–299 (2011).
[CrossRef] [PubMed]

C. K. Hayakawa, V. Venugopalan, V. V. Krishnamachari, and E. O. Potma, “Amplitude and phase of tightly focused laser beams in turbid media,” Phys. Rev. Lett. 103(4), 043903 (2009).
[CrossRef] [PubMed]

Prahl, S. A.

Shank, C. V.

A. Vaziri and C. V. Shank, “Ultrafast widefield optical sectioning microscopy by multifocal temporal focusing,” Opt. Express 18(19), 19645–19655 (2010).
[CrossRef] [PubMed]

A. Vaziri, J. Tang, H. Shroff, and C. V. Shank, “Multilayer three-dimensional super resolution imaging of thick biological samples,” Proc. Natl. Acad. Sci. U.S.A. 105(51), 20221–20226 (2008).
[CrossRef] [PubMed]

Shoham, S.

S. Shoham, “Optogenetics meets optical wavefront shaping,” Nat. Methods 7(10), 798–799 (2010).
[CrossRef] [PubMed]

Shroff, H.

A. Vaziri, J. Tang, H. Shroff, and C. V. Shank, “Multilayer three-dimensional super resolution imaging of thick biological samples,” Proc. Natl. Acad. Sci. U.S.A. 105(51), 20221–20226 (2008).
[CrossRef] [PubMed]

Silberberg, Y.

So, P.

C. Y. Dong, K. Koenig, and P. So, “Characterizing point spread functions of two-photon fluorescence microscopy in turbid medium,” J. Biomed. Opt. 8(3), 450–459 (2003).
[CrossRef] [PubMed]

So, P. T. C.

Squier, J. A.

Stott, J.

Tal, E.

Tang, J.

B. K. Andrasfalvy, B. V. Zemelman, J. Tang, and A. Vaziri, “Two-photon single-cell optogenetic control of neuronal activity by sculpted light,” Proc. Natl. Acad. Sci. U.S.A. 107(26), 11981–11986 (2010).
[CrossRef] [PubMed]

A. Vaziri, J. Tang, H. Shroff, and C. V. Shank, “Multilayer three-dimensional super resolution imaging of thick biological samples,” Proc. Natl. Acad. Sci. U.S.A. 105(51), 20221–20226 (2008).
[CrossRef] [PubMed]

Theer, P.

Tsai, P. S.

van Howe, J.

Vaziri, A.

B. K. Andrasfalvy, B. V. Zemelman, J. Tang, and A. Vaziri, “Two-photon single-cell optogenetic control of neuronal activity by sculpted light,” Proc. Natl. Acad. Sci. U.S.A. 107(26), 11981–11986 (2010).
[CrossRef] [PubMed]

A. Vaziri and C. V. Shank, “Ultrafast widefield optical sectioning microscopy by multifocal temporal focusing,” Opt. Express 18(19), 19645–19655 (2010).
[CrossRef] [PubMed]

A. Vaziri, J. Tang, H. Shroff, and C. V. Shank, “Multilayer three-dimensional super resolution imaging of thick biological samples,” Proc. Natl. Acad. Sci. U.S.A. 105(51), 20221–20226 (2008).
[CrossRef] [PubMed]

Venugopalan, V.

C. K. Hayakawa, E. O. Potma, and V. Venugopalan, “Electric field Monte Carlo simulations of focal field distributions produced by tightly focused laser beams in tissues,” Biomed. Opt. Express 2(2), 278–299 (2011).
[CrossRef] [PubMed]

C. K. Hayakawa, V. Venugopalan, V. V. Krishnamachari, and E. O. Potma, “Amplitude and phase of tightly focused laser beams in turbid media,” Phys. Rev. Lett. 103(4), 043903 (2009).
[CrossRef] [PubMed]

Vitek, D. N.

Xu, C.

Yuste, R.

Y. Ikegaya, M. Le Bon-Jego, and R. Yuste, “Large-scale imaging of cortical network activity with calcium indicators,” Neurosci. Res. 52(2), 132–138 (2005).
[CrossRef] [PubMed]

Zemelman, B. V.

B. K. Andrasfalvy, B. V. Zemelman, J. Tang, and A. Vaziri, “Two-photon single-cell optogenetic control of neuronal activity by sculpted light,” Proc. Natl. Acad. Sci. U.S.A. 107(26), 11981–11986 (2010).
[CrossRef] [PubMed]

Zhu, G.

Zipfel, W.

Biomed. Opt. Express (1)

J. Biomed. Opt. (1)

C. Y. Dong, K. Koenig, and P. So, “Characterizing point spread functions of two-photon fluorescence microscopy in turbid medium,” J. Biomed. Opt. 8(3), 450–459 (2003).
[CrossRef] [PubMed]

J. Neurosci. Methods (1)

M. Oheim, E. Beaurepaire, E. Chaigneau, J. Mertz, and S. Charpak, “Two-photon microscopy in brain tissue: parameters influencing the imaging depth,” J. Neurosci. Methods 111(1), 29–37 (2001).
[CrossRef] [PubMed]

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

J. Opt. Soc. B (1)

D. Oron and Y. Silberberg, “Harmonic generation with temporally focused ultrashort pulses,” J. Opt. Soc. B 22(12), 2660–2663 (2005).
[CrossRef]

Nat. Methods (3)

E. Papagiakoumou, F. Anselmi, A. Bègue, V. de Sars, J. Glückstad, E. Y. Isacoff, and V. Emiliani, “Scanless two-photon excitation of channelrhodopsin-2,” Nat. Methods 7(10), 848–854 (2010).
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Figures (6)

Fig. 1
Fig. 1

Schematic representation of light propagation in a TF setup. Light propagating at an angle α' (with its wavefront tilted by the same angle) hits a diffraction grating, causing each spectral component to diffract at a specific angle and a different tilt angle. After passing through a collimating and a focusing lens each spectral component moves towards the TF plane with a propagation angle β and tilt angle α. Since all optical paths from the grating to the focal plane are equal (Fermat's principle), all the spectral components scan the TF plane simultaneously, and α and β are coupled.

Fig. 2
Fig. 2

Experimental system outline. (a) TF optical setup: 800nm laser, beam expander (BE), diffraction grating, collimating lens (CL), and focusing objective lens (obj 1). Objective 2 (obj 2) and another lens image the sample onto a CCD. (b) Detailed view of the sample region. Scattering samples were set over a thin layer with 10μm fluorescent beads. Measurements were obtained by axially moving objective 2 and the sample. (c) xz section through a stack of experimental images of beads taken at different distances from the TF focal plane and under different scattering phantom depths, using 60x magnification and NA = 1. (d) Non-scattering optical sectioning cross-section used for pulse duration estimation.

Fig. 3
Fig. 3

Evaluation of model accuracy for optical sectioning of 10 µm fluorescent beads in different TF setups and scattering depths. Setup 1 has a magnification of 40, NA = 0.8 and TF plane diameter of 75 µm, while setup 2 has a magnification of 60, NA = 1 and TF plane diameter of 50 µm (τ = 325fs). The experimental error bars indicate the means and standard deviations, while solid lines indicate the model simulations (computed at scattering depth steps of 50µm). Insets: examples of experimental sectioning traces of two individual beads vs. model prediction.

Fig. 4
Fig. 4

The effect of varying magnifications on TF sectioning. Dots show model results for different magnifications and solid lines show a square-root of a Lorentzian fit (optical parameters: NA = 0.8, TF plane diameter of 15μm, pulse duration - 325fs).

Fig. 5
Fig. 5

TF sectioning dependence on optical system parameters and their approximate fit. Dots indicate model calculation results and solid lines indicate the approximation by Eq. (3). (a) Sectioning dependence on varying magnification: when a constant TF plane diameter is illuminated (upper line) and for changing TF plane diameters (lower line). (b) Sectioning dependence on diameter of illuminated TF plane (pulse duration: 150fs). (c) Dependence on the pulse duration. (d) Dependence on NA in non-scattering and scattering samples of various depths; here, the blue solid line shows the analytical approximation while the other lines are linear interpolations of the model’s results (parameters: 40x magnification, TF plane diameter-20μm, pulse duration-325fs).

Fig. 6
Fig. 6

Predicted performance of a single-cell TF optogenetic system (Ref. [10].) in a scattering medium. (a) Optical sectioning under different scattering depths. (b) Comparison of signal attenuation of TF based optical systems inside a scattering medium. Blue: simulation results for the single-cell excitation system (dots) and exponential fit (125µm decay constant, solid line); Green: simulation results for setup 2 in Fig. 3(dots) and exponential fit (190 µm decay constant, solid line). Black: expected exponential decay for TPLSM signal (100µm decay constant). (c) and (d) x-z view of fluorescence distribution inside a non-scattering and a 400 µm deep scattering media, respectively.

Equations (3)

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α = cot 1 ( 1 M sin α cos β tan β ) .
F = 1 1 + ( z z R ) 2 ,
z R = k 1 + τ k 2 τ l + k 3 M N A 2 ,

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