Abstract

Two-photon excitation fluorescence (TPEF) microscopy is a powerful technique for sensitive tissue imaging at depths of up to 1000 micrometers. However, due to the shallow penetration, for in vivo imaging of internal organs in patients beam delivery by an endoscope is crucial. Until today, this is hindered by linear and non-linear pulse broadening of the femtosecond pulses in the optical fibers of the endoscopes. Here we present an endoscope-ready, fiber-based TPEF microscope, using nanosecond pulses at low repetition rates instead of femtosecond pulses. These nanosecond pulses lack most of the problems connected with femtosecond pulses but are equally suited for TPEF imaging. We derive and demonstrate that at given cw-power the TPEF signal only depends on the duty cycle of the laser source. Due to the higher pulse energy at the same peak power we can also demonstrate single shot two-photon fluorescence lifetime measurements.

© 2016 Optical Society of America

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References

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2015 (2)

H. Segawa, M. Okuno, P. Leproux, V. Couderc, T. Ozawa, and H. Kano, “Multimodal Imaging of Living Cells with Multiplex Coherent Anti-Stokes Raman Scattering (CARS), Third-order Sum Frequency Generation (TSFG) and Two-Photon Excitation Fluorescence (TPEF) Using A Nanosecond White-light Laser Source,” Anal. Sci. 31(4), 299–305 (2015).
[Crossref] [PubMed]

S. Karpf, M. Eibl, W. Wieser, T. Klein, and R. Huber, “A Time-Encoded Technique for fibre-based hyperspectral broadband stimulated Raman microscopy,” Nat. Commun. 6, 6784 (2015).
[Crossref] [PubMed]

2014 (1)

2012 (1)

F. Knorr, D. R. Yankelevich, J. Liu, S. Wachsmann-Hogiu, and L. Marcu, “Two-photon excited fluorescence lifetime measurements through a double-clad photonic crystal fiber for tissue micro-endoscopy,” J. Biophotonics 5(1), 14–19 (2012).
[Crossref] [PubMed]

2010 (1)

S.-Y. Chen, C. Shee-Uan, W. Hai-Yin, L. Wen-Jeng, Y.-H. Liao, and C.-K. Sun, “In vivo virtual biopsy of human skin by using noninvasive higher harmonic generation microscopy,” IEEE J. Sel. Top. Quantum Electron. 16(3), 478–492 (2010).
[Crossref]

2009 (1)

S. Tang, J. Liu, T. B. Krasieva, Z. Chen, and B. J. Tromberg, “Developing compact multiphoton systems using femtosecond fiber lasers,” J. Biomed. Opt. 14, 030508 (2009).

2008 (1)

2007 (2)

K. Taira, T. Hashimoto, and H. Yokoyama, “Two-photon fluorescence imaging with a pulse source based on a 980-nm gain-switched laser diode,” Opt. Express 15(5), 2454–2458 (2007).
[Crossref] [PubMed]

G. Donnert, C. Eggeling, and S. W. Hell, “Major signal increase in fluorescence microscopy through dark-state relaxation,” Nat. Methods 4(1), 81–86 (2007).
[Crossref] [PubMed]

2003 (1)

1998 (2)

Bewersdorf, Hell, and Bewersdorf and Hell, “Picosecond pulsed two-photon imaging with repetition rates of 200 and 400 MHz,” J. Microsc. 191(1), 28–38 (1998).
[Crossref]

M. J. Booth and S. W. Hell, “Continuous wave excitation two-photon fluorescence microscopy exemplified with the 647-nm ArKr laser line,” J. Microsc. 190(3), 298–304 (1998).
[Crossref] [PubMed]

1990 (1)

W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248(4951), 73–76 (1990).
[Crossref] [PubMed]

1978 (1)

R. H. Stolen and C. Lin, “Self-phase-modulation in silica optical fibers,” Phys. Rev. A 17(4), 1448–1453 (1978).
[Crossref]

1977 (1)

K. A. Selanger, J. Falnes, and T. Sikkeland, “Fluorescence lifetime studies of Rhodamine 6G in methanol,” J. Phys. Chem. 81(20), 1960–1963 (1977).
[Crossref]

1973 (1)

R. H. Stolen and E. P. Ippen, “Raman gain in glass optical waveguides,” Appl. Phys. Lett. 22(6), 276–278 (1973).
[Crossref]

1972 (1)

E. P. Ippen and R. H. Stolen, “Stimulated Brillouin scattering in optical fibers,” Appl. Phys. Lett. 21(11), 539–541 (1972).
[Crossref]

1961 (1)

W. Kaiser and C. G. B. Garrett, “Two-Photon Excitation in CaF2:Eu2+,” Phys. Rev. Lett. 7(6), 229–231 (1961).
[Crossref]

1931 (1)

M. Goeppert-Mayer, “Über Elementarakte mit zwei Quantensprüngen,” Ann. Phys. 9(3), 273–294 (1931).
[Crossref]

Bewersdorf,

Bewersdorf, Hell, and Bewersdorf and Hell, “Picosecond pulsed two-photon imaging with repetition rates of 200 and 400 MHz,” J. Microsc. 191(1), 28–38 (1998).
[Crossref]

Blanquet, V.

C. Lefort, R. P. O’Connor, V. Blanquet, L. Magnol, H. Kano, V. Tombelaine, P. Lévêque, V. Couderc, and P. Leproux, “Multicolor multiphoton microscopy based on a nanosecond supercontinuum laser source,” J. Biophoton. In press (2016).

Booth, M. J.

M. J. Booth and S. W. Hell, “Continuous wave excitation two-photon fluorescence microscopy exemplified with the 647-nm ArKr laser line,” J. Microsc. 190(3), 298–304 (1998).
[Crossref] [PubMed]

Chen, S.-Y.

S.-Y. Chen, C. Shee-Uan, W. Hai-Yin, L. Wen-Jeng, Y.-H. Liao, and C.-K. Sun, “In vivo virtual biopsy of human skin by using noninvasive higher harmonic generation microscopy,” IEEE J. Sel. Top. Quantum Electron. 16(3), 478–492 (2010).
[Crossref]

Chen, Z.

S. Tang, J. Liu, T. B. Krasieva, Z. Chen, and B. J. Tromberg, “Developing compact multiphoton systems using femtosecond fiber lasers,” J. Biomed. Opt. 14, 030508 (2009).

Couderc, V.

H. Segawa, M. Okuno, P. Leproux, V. Couderc, T. Ozawa, and H. Kano, “Multimodal Imaging of Living Cells with Multiplex Coherent Anti-Stokes Raman Scattering (CARS), Third-order Sum Frequency Generation (TSFG) and Two-Photon Excitation Fluorescence (TPEF) Using A Nanosecond White-light Laser Source,” Anal. Sci. 31(4), 299–305 (2015).
[Crossref] [PubMed]

C. Lefort, R. P. O’Connor, V. Blanquet, L. Magnol, H. Kano, V. Tombelaine, P. Lévêque, V. Couderc, and P. Leproux, “Multicolor multiphoton microscopy based on a nanosecond supercontinuum laser source,” J. Biophoton. In press (2016).

Denk, W.

Donnert, G.

G. Donnert, C. Eggeling, and S. W. Hell, “Major signal increase in fluorescence microscopy through dark-state relaxation,” Nat. Methods 4(1), 81–86 (2007).
[Crossref] [PubMed]

Drobizhev, M.

Eggeling, C.

G. Donnert, C. Eggeling, and S. W. Hell, “Major signal increase in fluorescence microscopy through dark-state relaxation,” Nat. Methods 4(1), 81–86 (2007).
[Crossref] [PubMed]

Eibl, M.

S. Karpf, M. Eibl, W. Wieser, T. Klein, and R. Huber, “A Time-Encoded Technique for fibre-based hyperspectral broadband stimulated Raman microscopy,” Nat. Commun. 6, 6784 (2015).
[Crossref] [PubMed]

Falnes, J.

K. A. Selanger, J. Falnes, and T. Sikkeland, “Fluorescence lifetime studies of Rhodamine 6G in methanol,” J. Phys. Chem. 81(20), 1960–1963 (1977).
[Crossref]

Garrett, C. G. B.

W. Kaiser and C. G. B. Garrett, “Two-Photon Excitation in CaF2:Eu2+,” Phys. Rev. Lett. 7(6), 229–231 (1961).
[Crossref]

Goeppert-Mayer, M.

M. Goeppert-Mayer, “Über Elementarakte mit zwei Quantensprüngen,” Ann. Phys. 9(3), 273–294 (1931).
[Crossref]

Hai-Yin, W.

S.-Y. Chen, C. Shee-Uan, W. Hai-Yin, L. Wen-Jeng, Y.-H. Liao, and C.-K. Sun, “In vivo virtual biopsy of human skin by using noninvasive higher harmonic generation microscopy,” IEEE J. Sel. Top. Quantum Electron. 16(3), 478–492 (2010).
[Crossref]

Hasan, M. T.

Hashimoto, T.

Hell,

Bewersdorf, Hell, and Bewersdorf and Hell, “Picosecond pulsed two-photon imaging with repetition rates of 200 and 400 MHz,” J. Microsc. 191(1), 28–38 (1998).
[Crossref]

Hell, S. W.

G. Donnert, C. Eggeling, and S. W. Hell, “Major signal increase in fluorescence microscopy through dark-state relaxation,” Nat. Methods 4(1), 81–86 (2007).
[Crossref] [PubMed]

M. J. Booth and S. W. Hell, “Continuous wave excitation two-photon fluorescence microscopy exemplified with the 647-nm ArKr laser line,” J. Microsc. 190(3), 298–304 (1998).
[Crossref] [PubMed]

Hibi, T.

Huber, R.

S. Karpf, M. Eibl, W. Wieser, T. Klein, and R. Huber, “A Time-Encoded Technique for fibre-based hyperspectral broadband stimulated Raman microscopy,” Nat. Commun. 6, 6784 (2015).
[Crossref] [PubMed]

Ippen, E. P.

R. H. Stolen and E. P. Ippen, “Raman gain in glass optical waveguides,” Appl. Phys. Lett. 22(6), 276–278 (1973).
[Crossref]

E. P. Ippen and R. H. Stolen, “Stimulated Brillouin scattering in optical fibers,” Appl. Phys. Lett. 21(11), 539–541 (1972).
[Crossref]

Kaiser, W.

W. Kaiser and C. G. B. Garrett, “Two-Photon Excitation in CaF2:Eu2+,” Phys. Rev. Lett. 7(6), 229–231 (1961).
[Crossref]

Kano, H.

H. Segawa, M. Okuno, P. Leproux, V. Couderc, T. Ozawa, and H. Kano, “Multimodal Imaging of Living Cells with Multiplex Coherent Anti-Stokes Raman Scattering (CARS), Third-order Sum Frequency Generation (TSFG) and Two-Photon Excitation Fluorescence (TPEF) Using A Nanosecond White-light Laser Source,” Anal. Sci. 31(4), 299–305 (2015).
[Crossref] [PubMed]

C. Lefort, R. P. O’Connor, V. Blanquet, L. Magnol, H. Kano, V. Tombelaine, P. Lévêque, V. Couderc, and P. Leproux, “Multicolor multiphoton microscopy based on a nanosecond supercontinuum laser source,” J. Biophoton. In press (2016).

Karpf, S.

S. Karpf, M. Eibl, W. Wieser, T. Klein, and R. Huber, “A Time-Encoded Technique for fibre-based hyperspectral broadband stimulated Raman microscopy,” Nat. Commun. 6, 6784 (2015).
[Crossref] [PubMed]

Kawakami, R.

Klein, T.

S. Karpf, M. Eibl, W. Wieser, T. Klein, and R. Huber, “A Time-Encoded Technique for fibre-based hyperspectral broadband stimulated Raman microscopy,” Nat. Commun. 6, 6784 (2015).
[Crossref] [PubMed]

Knorr, F.

F. Knorr, D. R. Yankelevich, J. Liu, S. Wachsmann-Hogiu, and L. Marcu, “Two-photon excited fluorescence lifetime measurements through a double-clad photonic crystal fiber for tissue micro-endoscopy,” J. Biophotonics 5(1), 14–19 (2012).
[Crossref] [PubMed]

Kozawa, Y.

Krasieva, T. B.

S. Tang, J. Liu, T. B. Krasieva, Z. Chen, and B. J. Tromberg, “Developing compact multiphoton systems using femtosecond fiber lasers,” J. Biomed. Opt. 14, 030508 (2009).

Kusama, Y.

Lefort, C.

C. Lefort, R. P. O’Connor, V. Blanquet, L. Magnol, H. Kano, V. Tombelaine, P. Lévêque, V. Couderc, and P. Leproux, “Multicolor multiphoton microscopy based on a nanosecond supercontinuum laser source,” J. Biophoton. In press (2016).

Leproux, P.

H. Segawa, M. Okuno, P. Leproux, V. Couderc, T. Ozawa, and H. Kano, “Multimodal Imaging of Living Cells with Multiplex Coherent Anti-Stokes Raman Scattering (CARS), Third-order Sum Frequency Generation (TSFG) and Two-Photon Excitation Fluorescence (TPEF) Using A Nanosecond White-light Laser Source,” Anal. Sci. 31(4), 299–305 (2015).
[Crossref] [PubMed]

C. Lefort, R. P. O’Connor, V. Blanquet, L. Magnol, H. Kano, V. Tombelaine, P. Lévêque, V. Couderc, and P. Leproux, “Multicolor multiphoton microscopy based on a nanosecond supercontinuum laser source,” J. Biophoton. In press (2016).

Lévêque, P.

C. Lefort, R. P. O’Connor, V. Blanquet, L. Magnol, H. Kano, V. Tombelaine, P. Lévêque, V. Couderc, and P. Leproux, “Multicolor multiphoton microscopy based on a nanosecond supercontinuum laser source,” J. Biophoton. In press (2016).

Liao, Y.-H.

S.-Y. Chen, C. Shee-Uan, W. Hai-Yin, L. Wen-Jeng, Y.-H. Liao, and C.-K. Sun, “In vivo virtual biopsy of human skin by using noninvasive higher harmonic generation microscopy,” IEEE J. Sel. Top. Quantum Electron. 16(3), 478–492 (2010).
[Crossref]

Lin, C.

R. H. Stolen and C. Lin, “Self-phase-modulation in silica optical fibers,” Phys. Rev. A 17(4), 1448–1453 (1978).
[Crossref]

Liu, J.

F. Knorr, D. R. Yankelevich, J. Liu, S. Wachsmann-Hogiu, and L. Marcu, “Two-photon excited fluorescence lifetime measurements through a double-clad photonic crystal fiber for tissue micro-endoscopy,” J. Biophotonics 5(1), 14–19 (2012).
[Crossref] [PubMed]

S. Tang, J. Liu, T. B. Krasieva, Z. Chen, and B. J. Tromberg, “Developing compact multiphoton systems using femtosecond fiber lasers,” J. Biomed. Opt. 14, 030508 (2009).

Magnol, L.

C. Lefort, R. P. O’Connor, V. Blanquet, L. Magnol, H. Kano, V. Tombelaine, P. Lévêque, V. Couderc, and P. Leproux, “Multicolor multiphoton microscopy based on a nanosecond supercontinuum laser source,” J. Biophoton. In press (2016).

Makarov, N. S.

Marcu, L.

F. Knorr, D. R. Yankelevich, J. Liu, S. Wachsmann-Hogiu, and L. Marcu, “Two-photon excited fluorescence lifetime measurements through a double-clad photonic crystal fiber for tissue micro-endoscopy,” J. Biophotonics 5(1), 14–19 (2012).
[Crossref] [PubMed]

Nemoto, T.

O’Connor, R. P.

C. Lefort, R. P. O’Connor, V. Blanquet, L. Magnol, H. Kano, V. Tombelaine, P. Lévêque, V. Couderc, and P. Leproux, “Multicolor multiphoton microscopy based on a nanosecond supercontinuum laser source,” J. Biophoton. In press (2016).

Okuno, M.

H. Segawa, M. Okuno, P. Leproux, V. Couderc, T. Ozawa, and H. Kano, “Multimodal Imaging of Living Cells with Multiplex Coherent Anti-Stokes Raman Scattering (CARS), Third-order Sum Frequency Generation (TSFG) and Two-Photon Excitation Fluorescence (TPEF) Using A Nanosecond White-light Laser Source,” Anal. Sci. 31(4), 299–305 (2015).
[Crossref] [PubMed]

Ozawa, T.

H. Segawa, M. Okuno, P. Leproux, V. Couderc, T. Ozawa, and H. Kano, “Multimodal Imaging of Living Cells with Multiplex Coherent Anti-Stokes Raman Scattering (CARS), Third-order Sum Frequency Generation (TSFG) and Two-Photon Excitation Fluorescence (TPEF) Using A Nanosecond White-light Laser Source,” Anal. Sci. 31(4), 299–305 (2015).
[Crossref] [PubMed]

Rebane, A.

Sato, S.

Segawa, H.

H. Segawa, M. Okuno, P. Leproux, V. Couderc, T. Ozawa, and H. Kano, “Multimodal Imaging of Living Cells with Multiplex Coherent Anti-Stokes Raman Scattering (CARS), Third-order Sum Frequency Generation (TSFG) and Two-Photon Excitation Fluorescence (TPEF) Using A Nanosecond White-light Laser Source,” Anal. Sci. 31(4), 299–305 (2015).
[Crossref] [PubMed]

Selanger, K. A.

K. A. Selanger, J. Falnes, and T. Sikkeland, “Fluorescence lifetime studies of Rhodamine 6G in methanol,” J. Phys. Chem. 81(20), 1960–1963 (1977).
[Crossref]

Shee-Uan, C.

S.-Y. Chen, C. Shee-Uan, W. Hai-Yin, L. Wen-Jeng, Y.-H. Liao, and C.-K. Sun, “In vivo virtual biopsy of human skin by using noninvasive higher harmonic generation microscopy,” IEEE J. Sel. Top. Quantum Electron. 16(3), 478–492 (2010).
[Crossref]

Sikkeland, T.

K. A. Selanger, J. Falnes, and T. Sikkeland, “Fluorescence lifetime studies of Rhodamine 6G in methanol,” J. Phys. Chem. 81(20), 1960–1963 (1977).
[Crossref]

Stolen, R. H.

R. H. Stolen and C. Lin, “Self-phase-modulation in silica optical fibers,” Phys. Rev. A 17(4), 1448–1453 (1978).
[Crossref]

R. H. Stolen and E. P. Ippen, “Raman gain in glass optical waveguides,” Appl. Phys. Lett. 22(6), 276–278 (1973).
[Crossref]

E. P. Ippen and R. H. Stolen, “Stimulated Brillouin scattering in optical fibers,” Appl. Phys. Lett. 21(11), 539–541 (1972).
[Crossref]

Strickler, J. H.

W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248(4951), 73–76 (1990).
[Crossref] [PubMed]

Sun, C.-K.

S.-Y. Chen, C. Shee-Uan, W. Hai-Yin, L. Wen-Jeng, Y.-H. Liao, and C.-K. Sun, “In vivo virtual biopsy of human skin by using noninvasive higher harmonic generation microscopy,” IEEE J. Sel. Top. Quantum Electron. 16(3), 478–492 (2010).
[Crossref]

Taira, K.

Tang, S.

S. Tang, J. Liu, T. B. Krasieva, Z. Chen, and B. J. Tromberg, “Developing compact multiphoton systems using femtosecond fiber lasers,” J. Biomed. Opt. 14, 030508 (2009).

Tanushi, Y.

Theer, P.

Tombelaine, V.

C. Lefort, R. P. O’Connor, V. Blanquet, L. Magnol, H. Kano, V. Tombelaine, P. Lévêque, V. Couderc, and P. Leproux, “Multicolor multiphoton microscopy based on a nanosecond supercontinuum laser source,” J. Biophoton. In press (2016).

Tromberg, B. J.

S. Tang, J. Liu, T. B. Krasieva, Z. Chen, and B. J. Tromberg, “Developing compact multiphoton systems using femtosecond fiber lasers,” J. Biomed. Opt. 14, 030508 (2009).

Wachsmann-Hogiu, S.

F. Knorr, D. R. Yankelevich, J. Liu, S. Wachsmann-Hogiu, and L. Marcu, “Two-photon excited fluorescence lifetime measurements through a double-clad photonic crystal fiber for tissue micro-endoscopy,” J. Biophotonics 5(1), 14–19 (2012).
[Crossref] [PubMed]

Webb, W. W.

W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248(4951), 73–76 (1990).
[Crossref] [PubMed]

Wen-Jeng, L.

S.-Y. Chen, C. Shee-Uan, W. Hai-Yin, L. Wen-Jeng, Y.-H. Liao, and C.-K. Sun, “In vivo virtual biopsy of human skin by using noninvasive higher harmonic generation microscopy,” IEEE J. Sel. Top. Quantum Electron. 16(3), 478–492 (2010).
[Crossref]

Wieser, W.

S. Karpf, M. Eibl, W. Wieser, T. Klein, and R. Huber, “A Time-Encoded Technique for fibre-based hyperspectral broadband stimulated Raman microscopy,” Nat. Commun. 6, 6784 (2015).
[Crossref] [PubMed]

Yankelevich, D. R.

F. Knorr, D. R. Yankelevich, J. Liu, S. Wachsmann-Hogiu, and L. Marcu, “Two-photon excited fluorescence lifetime measurements through a double-clad photonic crystal fiber for tissue micro-endoscopy,” J. Biophotonics 5(1), 14–19 (2012).
[Crossref] [PubMed]

Yokoyama, H.

Yokoyama, M.

Anal. Sci. (1)

H. Segawa, M. Okuno, P. Leproux, V. Couderc, T. Ozawa, and H. Kano, “Multimodal Imaging of Living Cells with Multiplex Coherent Anti-Stokes Raman Scattering (CARS), Third-order Sum Frequency Generation (TSFG) and Two-Photon Excitation Fluorescence (TPEF) Using A Nanosecond White-light Laser Source,” Anal. Sci. 31(4), 299–305 (2015).
[Crossref] [PubMed]

Ann. Phys. (1)

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Figures (5)

Fig. 1
Fig. 1 Simple sketch to visualize that at given duty cycle (here 10−4) and average power the peak power (and with it the TPEF rate) is independent of pulse duration. Ns- and ps-pulse train with same duty cycle and same average power: High repetition rate ps-pulses (bottom) versus low rep rate ns-pulses (top). A 1ns pulse can be thought of merging groups of 1000 short 1ps pulses together. Accordingly, the repetition rate is lowered from 100MHz to 100kHz.
Fig. 2
Fig. 2 Setup of the ns-pulse TPEF microscopy system. The fiber-MOPA laser source has adjustable pulse length, repetition rate, peak power and output wavelength. The sample was either mounted on a translational stage [for Fig. 4] or galvanometric mirrors were used for scanning [Fig. 5]. The generated TPEF signal (green) passes the dichroic mirrors (DM) and is detected on a photomultiplier tube (PMT). The signals are processed on a computer (PC) equipped with an analogue-to-digital converter (ADC) card. The ADC sample clock is driven synchronously to the MOPA laser.
Fig. 3
Fig. 3 (A) TPEF spectrum of Rhodamine 6G dissolved in methanol. (B) Log-Log-Plot of the pump peak power dependency of the TPEF signal and a fitted slope of 2.04 (red), showing the quadratic dependency. (C) The SHG signal is used to determine the instrument response function (IRF) to 1.42 ns. (D) The high number of TPEF photons per single pulse and the high time-resolution allow TPEF lifetime measurements already with a single pulse. The fitted lifetime of 3.8ns for the 0.1mM Rhodamine 6G (R6G) dye in methanol agrees very well to literature values [22].
Fig. 4
Fig. 4 TPEF Microscopy image of endogenous autofluorescence of plant leaves. The fiber MOPA was operated at 1064nm. (A1) TPEF image of moss on a logarithmic scale. (A2) zoom-in on a leave showing the cells with high contrast against the background. (B) A standard transmission microscopy image of a moss leave. (C) TPEF image of algae. (D) TPEF image of a snowdrop plant leave.
Fig. 5
Fig. 5 TPEF microscopy image comparison of convalaria majalis (scale bar represents 100µm). The images were obtained using the presented nanosecond MOPA laser (left image) and a commercial fs-OPO system (right image). For image comparison, the powers were adjusted in order to achieve same TPEF photon numbers. The images show very good agreement and almost identical TPEF signal levels.

Equations (3)

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. n a = p 0 2 δ τ p f p 2 ( N A 2 2cλ ) 2 = p 0 2 τ p f p 2 .S,
n a = p peak 2 × τ p ×S.
N a = n group n a = n a × f p × t px =S× p 0 2 DC × t px .

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