Abstract

Two-photon fluorescence microscopy has become increasingly popular in biomedical research as it allows high-resolution imaging of thick biological specimen with superior contrast and penetration than confocal microscopy. However, two-photon microscopy still faces two fundamental limitations: 1) image-contrast deterioration with imaging depth due to out-of-focus background and 2) diffraction-limited spatial resolution. Herein we propose to create and detect high-order (more than quadratic) nonlinear signals by harnessing the frustrated fluorescence resonance energy transfer (FRET) effect within a specially designed donor-acceptor probe pair. Two distinct techniques are described. In the first method, donor fluorescence generated by a two-photon laser at the focus is preferentially switched on and off by a modulated and focused one-photon laser beam that is able to block FRET via direct acceptor excitation. The resulting image, constructed from the enhanced donor fluorescence signal, turns out to be an overall three-photon process. In the second method, a two-photon laser at a proper wavelength is capable of simultaneously exciting both the donor and the acceptor. By sinusoidally modulating the two-photon excitation laser at a fundamental frequency ω, an overall four-photon signal can be isolated by demodulating the donor fluorescence at the third harmonic frequency 3ω. We show that both the image contrast and the spatial resolution of the standard two-photon fluorescence microscopy can be substantially improved by virtue of the high-order nonlinearity. This frustrated FRET approach represents a strategy that is based on extracting the inherent nonlinear photophysical response of the specially designed imaging probes.

© 2013 OSA

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S. Berning, K. I. Willig, H. Steffens, P. Dibaj, and S. W. Hell, “Nanoscopy in a living mouse brain,” Science335(6068), 551 (2012).
[CrossRef] [PubMed]

P. Bianchini, B. Harke, S. Galiani, G. Vicidomini, and A. Diaspro, “Single-wavelength two-photon excitation-stimulated emission depletion (SW2PE-STED) superresolution imaging,” Proc. Natl. Acad. Sci. U.S.A.109(17), 6390–6393 (2012).
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X. Zhu, Y.-T. Kao, and W. Min, “Molecular-switch-mediated multiphoton fluorescence microscopy with high-order nonlinearity,” J. Phys. Chem. Lett.3(15), 2082–2086 (2012).
[CrossRef]

Y.-T. Kao, X. Zhu, F. Xu, and W. Min, “Focal switching of photochromic fluorescent proteins enables multiphoton microscopy with superior image contrast,” Biomed. Opt. Express3(8), 1955–1963 (2012).
[CrossRef] [PubMed]

Z. Chen, L. Wei, X. Zhu, and W. Min, “Extending the fundamental imaging-depth limit of multi-photon microscopy by imaging with photo-activatable fluorophores,” Opt. Express20(17), 18525–18536 (2012).
[CrossRef] [PubMed]

2011

N. J. Durr, C. T. Weisspfennig, B. A. Holfeld, and A. Ben-Yakar, “Maximum imaging depth of two-photon autofluorescence microscopy in epithelial tissues,” J. Biomed. Opt.16(2), 026008 (2011).
[CrossRef] [PubMed]

D. Kobat, N. G. Horton, and C. Xu, “In vivo two-photon microscopy to 1.6-mm depth in mouse cortex,” J. Biomed. Opt.16(10), 106014 (2011).
[CrossRef] [PubMed]

2010

N. Ji, D. E. Milkie, and E. Betzig, “Adaptive optics via pupil segmentation for high-resolution imaging in biological tissues,” Nat. Methods7(2), 141–147 (2010).
[CrossRef] [PubMed]

C. I. Richards, J. C. Hsiang, A. M. Khalil, N. P. Hull, and R. M. Dickson, “FRET-enabled optical modulation for high sensitivity fluorescence imaging,” J. Am. Chem. Soc.132(18), 6318–6323 (2010).
[CrossRef] [PubMed]

2009

J. B. Ding, K. T. Takasaki, and B. L. Sabatini, “Supraresolution imaging in brain slices using stimulated-emission depletion two-photon laser scanning microscopy,” Neuron63(4), 429–437 (2009).
[CrossRef] [PubMed]

T. Dertinger, R. Colyer, G. Iyer, S. Weiss, and J. Enderlein, “Fast, background-free, 3D super-resolution optical fluctuation imaging (SOFI),” Proc. Natl. Acad. Sci. U.S.A.106(52), 22287–22292 (2009).
[CrossRef] [PubMed]

2008

Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photonics2(2), 110–115 (2008).
[CrossRef] [PubMed]

A. Leray, K. Lillis, and J. Mertz, “Enhanced background rejection in thick tissue with differential-aberration two-photon microscopy,” Biophys. J.94(4), 1449–1458 (2008).
[CrossRef] [PubMed]

N. Chen, C. H. Wong, and C. J. Sheppard, “Focal modulation microscopy,” Opt. Express16(23), 18764–18769 (2008).
[CrossRef] [PubMed]

M. Beutler, K. Makrogianneli, R. J. Vermeij, M. Keppler, T. Ng, T. M. Jovin, and R. Heintzmann, “satFRET: estimation of Forster resonance energy transfer by acceptor saturation,” Eur. Biophys. J.38(1), 69–82 (2008).
[CrossRef] [PubMed]

2007

K. Fujita, M. Kobayashi, S. Kawano, M. Yamanaka, and S. Kawata, “High-resolution confocal microscopy by saturated excitation of fluorescence,” Phys. Rev. Lett.99(22), 228105 (2007).
[CrossRef] [PubMed]

2006

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science313(5793), 1642–1645 (2006).
[CrossRef] [PubMed]

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods3(10), 793–796 (2006).
[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]

P. Theer and W. Denk, “On the fundamental imaging-depth limit in two-photon microscopy,” J. Opt. Soc. Am. A23(12), 3139–3149 (2006).
[CrossRef] [PubMed]

2005

F. Helmchen and W. Denk, “Deep tissue two-photon microscopy,” Nat. Methods2(12), 932–940 (2005).
[CrossRef] [PubMed]

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

J. Lukomska, I. Gryczynski, J. Malicka, S. Makowiec, J. R. Lakowicz, and Z. Gryczynski, “Two-photon induced fluorescence of Cy5-DNA in buffer solution and on silver island films,” Biochem. Biophys. Res. Commun.328(1), 78–84 (2005).
[CrossRef] [PubMed]

2003

W. R. Zipfel, R. M. Williams, and W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol.21(11), 1369–1377 (2003).
[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]

1999

J. Ying, F. Liu, and R. R. Alfano, “Spatial distribution of two-photon-excited fluorescence in scattering media,” Appl. Opt.38(1), 224–229 (1999).
[CrossRef] [PubMed]

A. Schönle, P. E. Hanninen, and S. W. Hell, “Nonlinear fluorescence through intermolecular energy transfer and resolution increase in fluorescence microscopy,” Ann. Phys.8(2), 115–133 (1999).
[CrossRef]

1997

1996

P. E. Hänninen, L. Lehtelä, and S. W. Hell, “Two- and multiphoton excitation of conjugate-dyes using a continuous wave laser,” Opt. Commun.130(1-3), 29–33 (1996).
[CrossRef]

1994

1990

W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science248(4951), 73–76 (1990).
[CrossRef] [PubMed]

Alfano, R. R.

Bates, M.

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods3(10), 793–796 (2006).
[CrossRef] [PubMed]

Ben-Yakar, A.

N. J. Durr, C. T. Weisspfennig, B. A. Holfeld, and A. Ben-Yakar, “Maximum imaging depth of two-photon autofluorescence microscopy in epithelial tissues,” J. Biomed. Opt.16(2), 026008 (2011).
[CrossRef] [PubMed]

Berning, S.

S. Berning, K. I. Willig, H. Steffens, P. Dibaj, and S. W. Hell, “Nanoscopy in a living mouse brain,” Science335(6068), 551 (2012).
[CrossRef] [PubMed]

Betzig, E.

N. Ji, D. E. Milkie, and E. Betzig, “Adaptive optics via pupil segmentation for high-resolution imaging in biological tissues,” Nat. Methods7(2), 141–147 (2010).
[CrossRef] [PubMed]

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science313(5793), 1642–1645 (2006).
[CrossRef] [PubMed]

Beutler, M.

M. Beutler, K. Makrogianneli, R. J. Vermeij, M. Keppler, T. Ng, T. M. Jovin, and R. Heintzmann, “satFRET: estimation of Forster resonance energy transfer by acceptor saturation,” Eur. Biophys. J.38(1), 69–82 (2008).
[CrossRef] [PubMed]

Bianchini, P.

P. Bianchini, B. Harke, S. Galiani, G. Vicidomini, and A. Diaspro, “Single-wavelength two-photon excitation-stimulated emission depletion (SW2PE-STED) superresolution imaging,” Proc. Natl. Acad. Sci. U.S.A.109(17), 6390–6393 (2012).
[CrossRef] [PubMed]

Bonifacino, J. S.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science313(5793), 1642–1645 (2006).
[CrossRef] [PubMed]

Chen, N.

Chen, Z.

Colyer, R.

T. Dertinger, R. Colyer, G. Iyer, S. Weiss, and J. Enderlein, “Fast, background-free, 3D super-resolution optical fluctuation imaging (SOFI),” Proc. Natl. Acad. Sci. U.S.A.106(52), 22287–22292 (2009).
[CrossRef] [PubMed]

Davidson, M. W.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science313(5793), 1642–1645 (2006).
[CrossRef] [PubMed]

Denk, W.

P. Theer and W. Denk, “On the fundamental imaging-depth limit in two-photon microscopy,” J. Opt. Soc. Am. A23(12), 3139–3149 (2006).
[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]

F. Helmchen and W. Denk, “Deep tissue two-photon microscopy,” Nat. Methods2(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]

W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science248(4951), 73–76 (1990).
[CrossRef] [PubMed]

Dertinger, T.

T. Dertinger, R. Colyer, G. Iyer, S. Weiss, and J. Enderlein, “Fast, background-free, 3D super-resolution optical fluctuation imaging (SOFI),” Proc. Natl. Acad. Sci. U.S.A.106(52), 22287–22292 (2009).
[CrossRef] [PubMed]

Diaspro, A.

P. Bianchini, B. Harke, S. Galiani, G. Vicidomini, and A. Diaspro, “Single-wavelength two-photon excitation-stimulated emission depletion (SW2PE-STED) superresolution imaging,” Proc. Natl. Acad. Sci. U.S.A.109(17), 6390–6393 (2012).
[CrossRef] [PubMed]

Dibaj, P.

S. Berning, K. I. Willig, H. Steffens, P. Dibaj, and S. W. Hell, “Nanoscopy in a living mouse brain,” Science335(6068), 551 (2012).
[CrossRef] [PubMed]

Dickson, R. M.

C. I. Richards, J. C. Hsiang, A. M. Khalil, N. P. Hull, and R. M. Dickson, “FRET-enabled optical modulation for high sensitivity fluorescence imaging,” J. Am. Chem. Soc.132(18), 6318–6323 (2010).
[CrossRef] [PubMed]

Ding, J. B.

J. B. Ding, K. T. Takasaki, and B. L. Sabatini, “Supraresolution imaging in brain slices using stimulated-emission depletion two-photon laser scanning microscopy,” Neuron63(4), 429–437 (2009).
[CrossRef] [PubMed]

Durr, N. J.

N. J. Durr, C. T. Weisspfennig, B. A. Holfeld, and A. Ben-Yakar, “Maximum imaging depth of two-photon autofluorescence microscopy in epithelial tissues,” J. Biomed. Opt.16(2), 026008 (2011).
[CrossRef] [PubMed]

Durst, M.

G. Zhu, J. van Howe, M. Durst, W. Zipfel, and C. Xu, Simultaneous Spatial and Temporal Focusing of Femtosecond Pulses (Conference on Lasers and Electro-Optics (CLEO), 2005).
[CrossRef]

Enderlein, J.

T. Dertinger, R. Colyer, G. Iyer, S. Weiss, and J. Enderlein, “Fast, background-free, 3D super-resolution optical fluctuation imaging (SOFI),” Proc. Natl. Acad. Sci. U.S.A.106(52), 22287–22292 (2009).
[CrossRef] [PubMed]

Feld, M. S.

Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photonics2(2), 110–115 (2008).
[CrossRef] [PubMed]

Fujita, K.

K. Fujita, M. Kobayashi, S. Kawano, M. Yamanaka, and S. Kawata, “High-resolution confocal microscopy by saturated excitation of fluorescence,” Phys. Rev. Lett.99(22), 228105 (2007).
[CrossRef] [PubMed]

Galiani, S.

P. Bianchini, B. Harke, S. Galiani, G. Vicidomini, and A. Diaspro, “Single-wavelength two-photon excitation-stimulated emission depletion (SW2PE-STED) superresolution imaging,” Proc. Natl. Acad. Sci. U.S.A.109(17), 6390–6393 (2012).
[CrossRef] [PubMed]

Gryczynski, I.

J. Lukomska, I. Gryczynski, J. Malicka, S. Makowiec, J. R. Lakowicz, and Z. Gryczynski, “Two-photon induced fluorescence of Cy5-DNA in buffer solution and on silver island films,” Biochem. Biophys. Res. Commun.328(1), 78–84 (2005).
[CrossRef] [PubMed]

Gryczynski, Z.

J. Lukomska, I. Gryczynski, J. Malicka, S. Makowiec, J. R. Lakowicz, and Z. Gryczynski, “Two-photon induced fluorescence of Cy5-DNA in buffer solution and on silver island films,” Biochem. Biophys. Res. Commun.328(1), 78–84 (2005).
[CrossRef] [PubMed]

Gustafsson, M. G. L.

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

Hanninen, P. E.

A. Schönle, P. E. Hanninen, and S. W. Hell, “Nonlinear fluorescence through intermolecular energy transfer and resolution increase in fluorescence microscopy,” Ann. Phys.8(2), 115–133 (1999).
[CrossRef]

Hänninen, P. E.

P. E. Hänninen, L. Lehtelä, and S. W. Hell, “Two- and multiphoton excitation of conjugate-dyes using a continuous wave laser,” Opt. Commun.130(1-3), 29–33 (1996).
[CrossRef]

Harke, B.

P. Bianchini, B. Harke, S. Galiani, G. Vicidomini, and A. Diaspro, “Single-wavelength two-photon excitation-stimulated emission depletion (SW2PE-STED) superresolution imaging,” Proc. Natl. Acad. Sci. U.S.A.109(17), 6390–6393 (2012).
[CrossRef] [PubMed]

Hasan, M. T.

Heintzmann, R.

M. Beutler, K. Makrogianneli, R. J. Vermeij, M. Keppler, T. Ng, T. M. Jovin, and R. Heintzmann, “satFRET: estimation of Forster resonance energy transfer by acceptor saturation,” Eur. Biophys. J.38(1), 69–82 (2008).
[CrossRef] [PubMed]

Hell, S. W.

S. Berning, K. I. Willig, H. Steffens, P. Dibaj, and S. W. Hell, “Nanoscopy in a living mouse brain,” Science335(6068), 551 (2012).
[CrossRef] [PubMed]

A. Schönle, P. E. Hanninen, and S. W. Hell, “Nonlinear fluorescence through intermolecular energy transfer and resolution increase in fluorescence microscopy,” Ann. Phys.8(2), 115–133 (1999).
[CrossRef]

P. E. Hänninen, L. Lehtelä, and S. W. Hell, “Two- and multiphoton excitation of conjugate-dyes using a continuous wave laser,” Opt. Commun.130(1-3), 29–33 (1996).
[CrossRef]

S. W. Hell and J. Wichmann, “Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy,” Opt. Lett.19(11), 780–782 (1994).
[CrossRef] [PubMed]

Helmchen, F.

F. Helmchen and W. Denk, “Deep tissue two-photon microscopy,” Nat. Methods2(12), 932–940 (2005).
[CrossRef] [PubMed]

Hess, H. F.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science313(5793), 1642–1645 (2006).
[CrossRef] [PubMed]

Holfeld, B. A.

N. J. Durr, C. T. Weisspfennig, B. A. Holfeld, and A. Ben-Yakar, “Maximum imaging depth of two-photon autofluorescence microscopy in epithelial tissues,” J. Biomed. Opt.16(2), 026008 (2011).
[CrossRef] [PubMed]

Horton, N. G.

D. Kobat, N. G. Horton, and C. Xu, “In vivo two-photon microscopy to 1.6-mm depth in mouse cortex,” J. Biomed. Opt.16(10), 106014 (2011).
[CrossRef] [PubMed]

Hsiang, J. C.

C. I. Richards, J. C. Hsiang, A. M. Khalil, N. P. Hull, and R. M. Dickson, “FRET-enabled optical modulation for high sensitivity fluorescence imaging,” J. Am. Chem. Soc.132(18), 6318–6323 (2010).
[CrossRef] [PubMed]

Hull, N. P.

C. I. Richards, J. C. Hsiang, A. M. Khalil, N. P. Hull, and R. M. Dickson, “FRET-enabled optical modulation for high sensitivity fluorescence imaging,” J. Am. Chem. Soc.132(18), 6318–6323 (2010).
[CrossRef] [PubMed]

Iyer, G.

T. Dertinger, R. Colyer, G. Iyer, S. Weiss, and J. Enderlein, “Fast, background-free, 3D super-resolution optical fluctuation imaging (SOFI),” Proc. Natl. Acad. Sci. U.S.A.106(52), 22287–22292 (2009).
[CrossRef] [PubMed]

Ji, N.

N. Ji, D. E. Milkie, and E. Betzig, “Adaptive optics via pupil segmentation for high-resolution imaging in biological tissues,” Nat. Methods7(2), 141–147 (2010).
[CrossRef] [PubMed]

Jovin, T. M.

M. Beutler, K. Makrogianneli, R. J. Vermeij, M. Keppler, T. Ng, T. M. Jovin, and R. Heintzmann, “satFRET: estimation of Forster resonance energy transfer by acceptor saturation,” Eur. Biophys. J.38(1), 69–82 (2008).
[CrossRef] [PubMed]

Juskaitis, R.

Kao, Y.-T.

X. Zhu, Y.-T. Kao, and W. Min, “Molecular-switch-mediated multiphoton fluorescence microscopy with high-order nonlinearity,” J. Phys. Chem. Lett.3(15), 2082–2086 (2012).
[CrossRef]

Y.-T. Kao, X. Zhu, F. Xu, and W. Min, “Focal switching of photochromic fluorescent proteins enables multiphoton microscopy with superior image contrast,” Biomed. Opt. Express3(8), 1955–1963 (2012).
[CrossRef] [PubMed]

Kawano, S.

K. Fujita, M. Kobayashi, S. Kawano, M. Yamanaka, and S. Kawata, “High-resolution confocal microscopy by saturated excitation of fluorescence,” Phys. Rev. Lett.99(22), 228105 (2007).
[CrossRef] [PubMed]

Kawata, S.

K. Fujita, M. Kobayashi, S. Kawano, M. Yamanaka, and S. Kawata, “High-resolution confocal microscopy by saturated excitation of fluorescence,” Phys. Rev. Lett.99(22), 228105 (2007).
[CrossRef] [PubMed]

Keppler, M.

M. Beutler, K. Makrogianneli, R. J. Vermeij, M. Keppler, T. Ng, T. M. Jovin, and R. Heintzmann, “satFRET: estimation of Forster resonance energy transfer by acceptor saturation,” Eur. Biophys. J.38(1), 69–82 (2008).
[CrossRef] [PubMed]

Khalil, A. M.

C. I. Richards, J. C. Hsiang, A. M. Khalil, N. P. Hull, and R. M. Dickson, “FRET-enabled optical modulation for high sensitivity fluorescence imaging,” J. Am. Chem. Soc.132(18), 6318–6323 (2010).
[CrossRef] [PubMed]

Kobat, D.

D. Kobat, N. G. Horton, and C. Xu, “In vivo two-photon microscopy to 1.6-mm depth in mouse cortex,” J. Biomed. Opt.16(10), 106014 (2011).
[CrossRef] [PubMed]

Kobayashi, M.

K. Fujita, M. Kobayashi, S. Kawano, M. Yamanaka, and S. Kawata, “High-resolution confocal microscopy by saturated excitation of fluorescence,” Phys. Rev. Lett.99(22), 228105 (2007).
[CrossRef] [PubMed]

Lakowicz, J. R.

J. Lukomska, I. Gryczynski, J. Malicka, S. Makowiec, J. R. Lakowicz, and Z. Gryczynski, “Two-photon induced fluorescence of Cy5-DNA in buffer solution and on silver island films,” Biochem. Biophys. Res. Commun.328(1), 78–84 (2005).
[CrossRef] [PubMed]

Lehtelä, L.

P. E. Hänninen, L. Lehtelä, and S. W. Hell, “Two- and multiphoton excitation of conjugate-dyes using a continuous wave laser,” Opt. Commun.130(1-3), 29–33 (1996).
[CrossRef]

Leray, A.

A. Leray, K. Lillis, and J. Mertz, “Enhanced background rejection in thick tissue with differential-aberration two-photon microscopy,” Biophys. J.94(4), 1449–1458 (2008).
[CrossRef] [PubMed]

Lillis, K.

A. Leray, K. Lillis, and J. Mertz, “Enhanced background rejection in thick tissue with differential-aberration two-photon microscopy,” Biophys. J.94(4), 1449–1458 (2008).
[CrossRef] [PubMed]

Lindwasser, O. W.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science313(5793), 1642–1645 (2006).
[CrossRef] [PubMed]

Lippincott-Schwartz, J.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science313(5793), 1642–1645 (2006).
[CrossRef] [PubMed]

Liu, F.

Lukomska, J.

J. Lukomska, I. Gryczynski, J. Malicka, S. Makowiec, J. R. Lakowicz, and Z. Gryczynski, “Two-photon induced fluorescence of Cy5-DNA in buffer solution and on silver island films,” Biochem. Biophys. Res. Commun.328(1), 78–84 (2005).
[CrossRef] [PubMed]

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]

Makowiec, S.

J. Lukomska, I. Gryczynski, J. Malicka, S. Makowiec, J. R. Lakowicz, and Z. Gryczynski, “Two-photon induced fluorescence of Cy5-DNA in buffer solution and on silver island films,” Biochem. Biophys. Res. Commun.328(1), 78–84 (2005).
[CrossRef] [PubMed]

Makrogianneli, K.

M. Beutler, K. Makrogianneli, R. J. Vermeij, M. Keppler, T. Ng, T. M. Jovin, and R. Heintzmann, “satFRET: estimation of Forster resonance energy transfer by acceptor saturation,” Eur. Biophys. J.38(1), 69–82 (2008).
[CrossRef] [PubMed]

Malicka, J.

J. Lukomska, I. Gryczynski, J. Malicka, S. Makowiec, J. R. Lakowicz, and Z. Gryczynski, “Two-photon induced fluorescence of Cy5-DNA in buffer solution and on silver island films,” Biochem. Biophys. Res. Commun.328(1), 78–84 (2005).
[CrossRef] [PubMed]

Mertz, J.

A. Leray, K. Lillis, and J. Mertz, “Enhanced background rejection in thick tissue with differential-aberration two-photon microscopy,” Biophys. J.94(4), 1449–1458 (2008).
[CrossRef] [PubMed]

Milkie, D. E.

N. Ji, D. E. Milkie, and E. Betzig, “Adaptive optics via pupil segmentation for high-resolution imaging in biological tissues,” Nat. Methods7(2), 141–147 (2010).
[CrossRef] [PubMed]

Min, W.

Neil, M. A. A.

Ng, T.

M. Beutler, K. Makrogianneli, R. J. Vermeij, M. Keppler, T. Ng, T. M. Jovin, and R. Heintzmann, “satFRET: estimation of Forster resonance energy transfer by acceptor saturation,” Eur. Biophys. J.38(1), 69–82 (2008).
[CrossRef] [PubMed]

Olenych, S.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science313(5793), 1642–1645 (2006).
[CrossRef] [PubMed]

Patterson, G. H.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science313(5793), 1642–1645 (2006).
[CrossRef] [PubMed]

Psaltis, D.

Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photonics2(2), 110–115 (2008).
[CrossRef] [PubMed]

Richards, C. I.

C. I. Richards, J. C. Hsiang, A. M. Khalil, N. P. Hull, and R. M. Dickson, “FRET-enabled optical modulation for high sensitivity fluorescence imaging,” J. Am. Chem. Soc.132(18), 6318–6323 (2010).
[CrossRef] [PubMed]

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]

Rust, M. J.

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods3(10), 793–796 (2006).
[CrossRef] [PubMed]

Sabatini, B. L.

J. B. Ding, K. T. Takasaki, and B. L. Sabatini, “Supraresolution imaging in brain slices using stimulated-emission depletion two-photon laser scanning microscopy,” Neuron63(4), 429–437 (2009).
[CrossRef] [PubMed]

Schönle, A.

A. Schönle, P. E. Hanninen, and S. W. Hell, “Nonlinear fluorescence through intermolecular energy transfer and resolution increase in fluorescence microscopy,” Ann. Phys.8(2), 115–133 (1999).
[CrossRef]

Sheppard, C. J.

Sougrat, R.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science313(5793), 1642–1645 (2006).
[CrossRef] [PubMed]

Steffens, H.

S. Berning, K. I. Willig, H. Steffens, P. Dibaj, and S. W. Hell, “Nanoscopy in a living mouse brain,” Science335(6068), 551 (2012).
[CrossRef] [PubMed]

Strickler, J. H.

W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science248(4951), 73–76 (1990).
[CrossRef] [PubMed]

Takasaki, K. T.

J. B. Ding, K. T. Takasaki, and B. L. Sabatini, “Supraresolution imaging in brain slices using stimulated-emission depletion two-photon laser scanning microscopy,” Neuron63(4), 429–437 (2009).
[CrossRef] [PubMed]

Theer, P.

van Howe, J.

G. Zhu, J. van Howe, M. Durst, W. Zipfel, and C. Xu, Simultaneous Spatial and Temporal Focusing of Femtosecond Pulses (Conference on Lasers and Electro-Optics (CLEO), 2005).
[CrossRef]

Vermeij, R. J.

M. Beutler, K. Makrogianneli, R. J. Vermeij, M. Keppler, T. Ng, T. M. Jovin, and R. Heintzmann, “satFRET: estimation of Forster resonance energy transfer by acceptor saturation,” Eur. Biophys. J.38(1), 69–82 (2008).
[CrossRef] [PubMed]

Vicidomini, G.

P. Bianchini, B. Harke, S. Galiani, G. Vicidomini, and A. Diaspro, “Single-wavelength two-photon excitation-stimulated emission depletion (SW2PE-STED) superresolution imaging,” Proc. Natl. Acad. Sci. U.S.A.109(17), 6390–6393 (2012).
[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(11), 1369–1377 (2003).
[CrossRef] [PubMed]

W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science248(4951), 73–76 (1990).
[CrossRef] [PubMed]

Wei, L.

Weiss, S.

T. Dertinger, R. Colyer, G. Iyer, S. Weiss, and J. Enderlein, “Fast, background-free, 3D super-resolution optical fluctuation imaging (SOFI),” Proc. Natl. Acad. Sci. U.S.A.106(52), 22287–22292 (2009).
[CrossRef] [PubMed]

Weisspfennig, C. T.

N. J. Durr, C. T. Weisspfennig, B. A. Holfeld, and A. Ben-Yakar, “Maximum imaging depth of two-photon autofluorescence microscopy in epithelial tissues,” J. Biomed. Opt.16(2), 026008 (2011).
[CrossRef] [PubMed]

Wichmann, J.

Williams, R. M.

W. R. Zipfel, R. M. Williams, and W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol.21(11), 1369–1377 (2003).
[CrossRef] [PubMed]

Willig, K. I.

S. Berning, K. I. Willig, H. Steffens, P. Dibaj, and S. W. Hell, “Nanoscopy in a living mouse brain,” Science335(6068), 551 (2012).
[CrossRef] [PubMed]

Wilson, T.

Wong, C. H.

Xu, C.

D. Kobat, N. G. Horton, and C. Xu, “In vivo two-photon microscopy to 1.6-mm depth in mouse cortex,” J. Biomed. Opt.16(10), 106014 (2011).
[CrossRef] [PubMed]

G. Zhu, J. van Howe, M. Durst, W. Zipfel, and C. Xu, Simultaneous Spatial and Temporal Focusing of Femtosecond Pulses (Conference on Lasers and Electro-Optics (CLEO), 2005).
[CrossRef]

Xu, F.

Yamanaka, M.

K. Fujita, M. Kobayashi, S. Kawano, M. Yamanaka, and S. Kawata, “High-resolution confocal microscopy by saturated excitation of fluorescence,” Phys. Rev. Lett.99(22), 228105 (2007).
[CrossRef] [PubMed]

Yang, C.

Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photonics2(2), 110–115 (2008).
[CrossRef] [PubMed]

Yaqoob, Z.

Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photonics2(2), 110–115 (2008).
[CrossRef] [PubMed]

Ying, J.

Zhu, G.

G. Zhu, J. van Howe, M. Durst, W. Zipfel, and C. Xu, Simultaneous Spatial and Temporal Focusing of Femtosecond Pulses (Conference on Lasers and Electro-Optics (CLEO), 2005).
[CrossRef]

Zhu, X.

Zhuang, X.

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods3(10), 793–796 (2006).
[CrossRef] [PubMed]

Zipfel, W.

G. Zhu, J. van Howe, M. Durst, W. Zipfel, and C. Xu, Simultaneous Spatial and Temporal Focusing of Femtosecond Pulses (Conference on Lasers and Electro-Optics (CLEO), 2005).
[CrossRef]

Zipfel, W. R.

W. R. Zipfel, R. M. Williams, and W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol.21(11), 1369–1377 (2003).
[CrossRef] [PubMed]

Ann. Phys.

A. Schönle, P. E. Hanninen, and S. W. Hell, “Nonlinear fluorescence through intermolecular energy transfer and resolution increase in fluorescence microscopy,” Ann. Phys.8(2), 115–133 (1999).
[CrossRef]

Appl. Opt.

Biochem. Biophys. Res. Commun.

J. Lukomska, I. Gryczynski, J. Malicka, S. Makowiec, J. R. Lakowicz, and Z. Gryczynski, “Two-photon induced fluorescence of Cy5-DNA in buffer solution and on silver island films,” Biochem. Biophys. Res. Commun.328(1), 78–84 (2005).
[CrossRef] [PubMed]

Biomed. Opt. Express

Biophys. J.

A. Leray, K. Lillis, and J. Mertz, “Enhanced background rejection in thick tissue with differential-aberration two-photon microscopy,” Biophys. J.94(4), 1449–1458 (2008).
[CrossRef] [PubMed]

Eur. Biophys. J.

M. Beutler, K. Makrogianneli, R. J. Vermeij, M. Keppler, T. Ng, T. M. Jovin, and R. Heintzmann, “satFRET: estimation of Forster resonance energy transfer by acceptor saturation,” Eur. Biophys. J.38(1), 69–82 (2008).
[CrossRef] [PubMed]

J. Am. Chem. Soc.

C. I. Richards, J. C. Hsiang, A. M. Khalil, N. P. Hull, and R. M. Dickson, “FRET-enabled optical modulation for high sensitivity fluorescence imaging,” J. Am. Chem. Soc.132(18), 6318–6323 (2010).
[CrossRef] [PubMed]

J. Biomed. Opt.

N. J. Durr, C. T. Weisspfennig, B. A. Holfeld, and A. Ben-Yakar, “Maximum imaging depth of two-photon autofluorescence microscopy in epithelial tissues,” J. Biomed. Opt.16(2), 026008 (2011).
[CrossRef] [PubMed]

D. Kobat, N. G. Horton, and C. Xu, “In vivo two-photon microscopy to 1.6-mm depth in mouse cortex,” J. Biomed. Opt.16(10), 106014 (2011).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A

J. Phys. Chem. Lett.

X. Zhu, Y.-T. Kao, and W. Min, “Molecular-switch-mediated multiphoton fluorescence microscopy with high-order nonlinearity,” J. Phys. Chem. Lett.3(15), 2082–2086 (2012).
[CrossRef]

Nat. Biotechnol.

W. R. Zipfel, R. M. Williams, and W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol.21(11), 1369–1377 (2003).
[CrossRef] [PubMed]

Nat. Methods

N. Ji, D. E. Milkie, and E. Betzig, “Adaptive optics via pupil segmentation for high-resolution imaging in biological tissues,” Nat. Methods7(2), 141–147 (2010).
[CrossRef] [PubMed]

F. Helmchen and W. Denk, “Deep tissue two-photon microscopy,” Nat. Methods2(12), 932–940 (2005).
[CrossRef] [PubMed]

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods3(10), 793–796 (2006).
[CrossRef] [PubMed]

Nat. Photonics

Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photonics2(2), 110–115 (2008).
[CrossRef] [PubMed]

Neuron

J. B. Ding, K. T. Takasaki, and B. L. Sabatini, “Supraresolution imaging in brain slices using stimulated-emission depletion two-photon laser scanning microscopy,” Neuron63(4), 429–437 (2009).
[CrossRef] [PubMed]

Opt. Commun.

P. E. Hänninen, L. Lehtelä, and S. W. Hell, “Two- and multiphoton excitation of conjugate-dyes using a continuous wave laser,” Opt. Commun.130(1-3), 29–33 (1996).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. Lett.

K. Fujita, M. Kobayashi, S. Kawano, M. Yamanaka, and S. Kawata, “High-resolution confocal microscopy by saturated excitation of fluorescence,” Phys. Rev. Lett.99(22), 228105 (2007).
[CrossRef] [PubMed]

Proc. Natl. Acad. Sci. U.S.A.

P. Bianchini, B. Harke, S. Galiani, G. Vicidomini, and A. Diaspro, “Single-wavelength two-photon excitation-stimulated emission depletion (SW2PE-STED) superresolution imaging,” Proc. Natl. Acad. Sci. U.S.A.109(17), 6390–6393 (2012).
[CrossRef] [PubMed]

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

T. Dertinger, R. Colyer, G. Iyer, S. Weiss, and J. Enderlein, “Fast, background-free, 3D super-resolution optical fluctuation imaging (SOFI),” Proc. Natl. Acad. Sci. U.S.A.106(52), 22287–22292 (2009).
[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]

Science

W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science248(4951), 73–76 (1990).
[CrossRef] [PubMed]

S. Berning, K. I. Willig, H. Steffens, P. Dibaj, and S. W. Hell, “Nanoscopy in a living mouse brain,” Science335(6068), 551 (2012).
[CrossRef] [PubMed]

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science313(5793), 1642–1645 (2006).
[CrossRef] [PubMed]

Other

G. Zhu, J. van Howe, M. Durst, W. Zipfel, and C. Xu, Simultaneous Spatial and Temporal Focusing of Femtosecond Pulses (Conference on Lasers and Electro-Optics (CLEO), 2005).
[CrossRef]

R. Yuste, ed., Imaging: A Laboratory Manual (Cold Spring Harbor Laboratory, 2010).

B. R. Master and P. T. C. So, Handbook of Biomedical Nonlinear Optical Microscopy (Oxford University, 2008).

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Figures (7)

Fig. 1
Fig. 1

Principle of frustrated FRET. (a) When only the donor is excited, the fluorescence resonance energy is transferred from the donor to the acceptor, quenching the donor fluorescence. (b) When the donor and the acceptor are excited at the same time, FRET is inhibited, recovering the donor fluorescence.

Fig. 2
Fig. 2

Frustrated FRET with two excitation beams. (a) The proposed experimental setup in which the 2P laser and the 1P laser are collinearly combined with each other and focused onto the same focal spot. The 1P laser beam is being modulated, and the demodulated donor fluorescence is used as the new signal contrast. (b) A simplified Joblonski diagram illustrating that the energy is blocked by simultaneous excitation of both the donor and the acceptor. (c) Synchronized donor/acceptor excitation pulse trains. The 1P laser pulses are temporally followed by the 2P laser pulses with a time gap shorter than 1ns, which avoids the potential stimulated emission of the donor but still blocks FRET efficiently. The final image is reconstructed from the enhanced fluorescence signal which is demodulated from the lock-in amplifier.

Fig. 3
Fig. 3

Several possible fates of the PD (t) population are considered when the 1P acceptor excitation beam is on. The capital letters denote the excited states of the donor (D) and acceptor (A), and the small letters denote the ground states of the donor (d) and the acceptor (a).

Fig. 4
Fig. 4

Frustrated FRET with one excitation beam (a) The proposed experimental setup in which the 2P laser is sinusoidally modulated at the fundamental frequency ω, and the donor fluorescence detected and then demodulated at 3ω. (b) A simplified Joblonski diagram illustrating that the energy transfer is blocked by simultaneous excitation of both the donor and the acceptor with a two-photon laser.

Fig. 5
Fig. 5

Dependence of demodulated donor fluorescence (after normalization) at ω (Sω), 2ω (S), 3ω (S) and 4ω (S) as a function of β in the one-laser scheme. Note that β itself scales with . Thus under non-saturating condition, 3ω and 4ω harmonic demodulation signals scale with.

Fig. 6
Fig. 6

Numerical estimation of the fundamental imaging-depth limit (where S/B ratio is 1) of the standard 2P microscopy.

Fig. 7
Fig. 7

Both the two-laser and one-laser frustrated FRET techniques can improve the spatial resolution of the regular 2P microscopy in all three dimensions. We assume that the images are taken with an air-objective of N.A. = 0.7, refraction index of 1, and the 2P and 1P laser wavelengths to be 1000 nm and 700 nm, respectively. The color bar linearly depends on the fluorescence intensity: red represents the highest intensity and back represents the lowest intensity.

Tables (1)

Tables Icon

Table 1 Comparison of the standard two-photon microscopy, the two-laser frustrated FRET and the one-laser frustrated FRET at the fundamental imaging depth limit of 2P microscopy (zfocal = 1023µm) . *

Equations (29)

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

k exc,d = σ d,2P ( I 2P,ave λ 2P f rep δ 2P hc ) 2 .
k exc,a = σ a,1P ( I 1P,ave λ 1P f rep δ 1P hc ).
P D = σ d,2P δ 2P ( I 2P,ave λ 2P f rep δ 2P hc ) 2 ,
P A = σ a,1P δ 1P ( I 1P,ave λ 1P f rep δ 1P hc ).
S fl.d =Nε f rep P D k fl.d k fl.d + k FRET =Nε f rep P D 1/τ 1/τ+ k FRET ,
η'=[ 1 P A ] 1/τ 1/τ+ k FRET + P A ( 1 2 + 1 2 1/τ 1/τ+ k FRET ) = 1/τ 1/τ+ k FRET + 1 2 P A k FRET 1/τ+ k FRET ,
S fl.d '=Nε f rep P D η'=Nε f rep P D ( 1/τ 1/τ+ k FRET + 1 2 P E A t ).
S enhanced = 1 2 Nε f rep E t P D P A =α I 2P.ave 2 I 1P,ave ,
I 2P,ave (t)= I 2P,ave [ 1+αcos(ωt) ],
k exc,2P (t)= σ 2P ' ( I 2P,ave (t) λ 2P ' f rep δ pulse hc ) 2 .
P D (t)= β D [ 1+cosωt ] 2 ,
P A (t)= β A [ 1+cosωt ] 2 ,
η(t)= 1/τ 1/τ+ k FRET + 1 2 P A (t) E t .
S fl (t)=Nε f rep P D (t)η(t).
S fl,d (t)=Nε f rep β D 1/τ 1/τ+ k FRET [ 1+cos(ωt) ] 2 + 1 2 Nε f rep E t β D β A [ 1+cos(ωt) ] 4 .
S fl,d (t)=(Nε f rep )[ S 0 + S ω cos(ωt)+ S 2ω cos(2ωt)+ S 3ω cos(3ωt)+ S 4ω cos(4ωt) ],
S 0 = 3 2 1/τ 1/τ+ k FRET β D + 25 16 E t β D β A ,
S ω = 1 2 1/τ 1/τ+ k FRET β D + 3 2 E t β D β A ,
S 2ω =2 1/τ 1/τ+ k FRET β D + 5 2 E t β D β A ,
S 3ω = 1 2 E t β D β A ,
S 4ω = 1 16 E t β D β A .
( S B ) 2P = V in 0 τ C s (r,z) I 2P 2 (r,z,t)dtdV V out 0 τ C B (r,z) I 2P 2 (r,z,t)dtdV =1
I(z)= P(z) A(z) = P 0 e z/ l s ω 0 2 [ 1+ ( (z z focal ) z R ) 2 ]
F 2P = V ( Cε f rep )[ σ 2P,d δ 2P,d ( I 2P,ave λ 2P f rep δ 2P hc ) 2 ] d V .
F enhanced = V ( 1 2 E t Cε f rep )[ σ 2P,d δ 2P ( I 2P,ave λ 2P f rep δ 2P hc ) 2 ] [ σ 1P,a δ 1P ( I 1P,ave λ 1P f rep δ 1P hc ) ] d V ,
F 3ω = V ( 1 2 E t Cε f rep )[ σ 2P,d δ 2P ( I 2P,ave λ 2P f rep δ 2P hc ) 2 ] [ σ 2P,a δ 2P ( I 2P,ave λ 2P f rep δ 2P hc ) 2 ] d V .
PS F 2P (x,y,z)=IPS F 2P (x,y,z) 2
PS F enhanced (x,y,z)=IPS F 2P (x,y,z) 2 IPS F 1P (x,y,z)
PS F 2P (x,y,z)=IPS F 2P (x,y,z) 4

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