## Temporal control of local plasmon distribution on Au nanocrosses by ultra-broadband femtosecond laser pulses and its application for selective two-photon excitation of multiple fluorophores |

Optics Express, Vol. 19, Issue 14, pp. 13618-13627 (2011)

http://dx.doi.org/10.1364/OE.19.013618

Acrobat PDF (1415 KB)

### Abstract

We theoretically demonstrate spatiotemporal control of local plasmon distribution on Au nanocrosses, which have different aspect ratios, by chirped ultra-broadband femtosecond laser pulses. We also demonstrate selective excitation of fluorescence proteins using this spatiotemporal local plasmon control technique for applications to two-photon excited fluorescence microscopy.

© 2011 OSA

## 1. Introduction

8. Y. N. Xia and N. J. Halas, “Shape-controlled synthesis and surface plasmonic properties of metallic nanostructures,” MRS Bull. **30**(05), 338–348 (2005). [CrossRef]

9. E. Stefan Kooij and B. Poelsema, “Shape and size effects in the optical properties of metallic nanorods,” Phys. Chem. Chem. Phys. **8**(28), 3349–3357 (2006). [CrossRef] [PubMed]

10. T. Brixner, F. J. Garcia de Abajo, J. Schneider, C. Spindler, and W. Pfeiffer, “Ultrafast adaptive optical near-field control,” Phys. Rev. B **73**(12), 125437 (2006). [CrossRef]

12. M. I. Stockman, S. V. Faleev, and D. J. Bergman, “Coherent control of femtosecond energy localization in nanosystems,” Phys. Rev. Lett. **88**(6), 067402–067405 (2002). [CrossRef] [PubMed]

10. T. Brixner, F. J. Garcia de Abajo, J. Schneider, C. Spindler, and W. Pfeiffer, “Ultrafast adaptive optical near-field control,” Phys. Rev. B **73**(12), 125437 (2006). [CrossRef]

12. M. I. Stockman, S. V. Faleev, and D. J. Bergman, “Coherent control of femtosecond energy localization in nanosystems,” Phys. Rev. Lett. **88**(6), 067402–067405 (2002). [CrossRef] [PubMed]

13. R. S. Judson and H. Rabitz, “Teaching lasers to control molecules,” Phys. Rev. Lett. **68**(10), 1500–1503 (1992). [CrossRef] [PubMed]

15. D. Meshulach and Y. Silberberg, “Coherent quantum control of two-photon transitions by a femtosecond laser pulse,” Nature **396**(6708), 239–242 (1998). [CrossRef]

16. T. Brixner and G. Gerber, “Femtosecond polarization pulse shaping,” Opt. Lett. **26**(8), 557–559 (2001). [CrossRef] [PubMed]

18. Y. Esumi, M. D. Kabir, and F. Kannari, “Spatiotemporal vector pulse shaping of femtosecond laser pulses with a multi-pass two-dimensional spatial light modulator,” Opt. Express **17**(21), 19153–19159 (2009). [CrossRef] [PubMed]

19. G. Lévêque and O. J. F. Martin, “Narrow-band multiresonant plasmon nanostructure for the coherent control of light: an optical analog of the xylophone,” Phys. Rev. Lett. **100**(11), 117402 (2008). [CrossRef] [PubMed]

## 2. Spatiotemporal control

20. A. Taflove and K. R. Umashankar, “The finite-difference time-domain (FD-TD) method for electromagnetic scattering and interaction problems,” J. Electromagn. Waves Appl. **1**(3), 243–267 (1987). [CrossRef]

26. G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic field equations,” IEEE Trans. Electromagn. Compat. **EMC-23**(4), 377–382 (1981). [CrossRef]

^{3}with 2-nm grid sizes. Time step

*Δt*of 3.85 attosecond is used to satisfy the Courant-Friedrich-Levy (CFL) stabilization condition defined by Eq. (2) [27

27. A. Taflove and M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problem using time-dependent Maxwell's equations,” IEEE Trans. Microw. Theory Tech. **23**(8), 623–630 (1975). [CrossRef]

26. G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic field equations,” IEEE Trans. Electromagn. Compat. **EMC-23**(4), 377–382 (1981). [CrossRef]

_{2}substrate with a sufficient interval of ~300 nm as shown in Fig. 1(a). The complex refractive index of Au nanorods is defined over a wide spectrum range by the Drude model. We obtained the plasmon resonance wavelength peak for those nanorods with different aspect ratios by separate FDTD simulation runs. The calculated resonance peak locates as 1180, 880, 760 and 650 nm for the nanorod with the aspect ratio of 15, 8, 6, and 4, respectively. The plasmon resonance is not so sharp. The resonance bandwidth is typically 125 nm (FWHM) for these rods. Therefore, in order to clearly distinguish the plasmon resonance in time and space, the plasmon resonance wavelength at each nanostructure must be designed with sufficient wavelength separations from that of the other nanostructures.

^{2}is added to the laser pulse resulting in a pulse width of 150 fs (FWHM). The electric field is calculated at the edge of each Au nanorod.

^{3}. The spectral amplitude of the laser pulse is same as Fig. 1(c). As shown in Figs. 4 (c) and (d) only plasmon resonance of the rod A is excited at 470 fs, whereas plasmon resonances of the others are simultaneously excited at 510 fs. In the femtosecond laser pulse with a third-order dispersion, the group delay is in a quadratic function. At the third-order dispersion of −50 fs

^{3}, the spectral component at ~1180 nm that causes the plasmon resonance peak at the cross A with the x-polarization has a substantial negative group delay (arriving earlier), whereas the other three plasmon resonance wavelengths ranging in 650-880 nm for the nanocrosses B-D arrive at the almost same time at ~510 fs. Therefore, more flexible spatiotemporal control of plasmon resonance can be achieved by employing higher-order dispersion of the excitation pulse. Moreover, since different excitation sequences are obtained just by switching the polarization, polarization shaped femtosecond laser pulses will further increase the flexibility of plasmon excitation sequence.

## 3. Selective excitation of fluorescent proteins with spatiotemporally controlled plasmon

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

37. M. Comstock, V. V. Lozovoy, I. Pastirk, and M. Dantus, “Multiphoton intrapulse interference 6; binary phase shaping,” Opt. Express **12**(6), 1061–1066 (2004). [CrossRef] [PubMed]

35. K. Isobe, A. Suda, M. Tanaka, F. Kannari, H. Kawano, H. Mizuno, A. Miyawaki, and K. Midorikawa, “Multifarious control of two-photon excitation of multiple fluorophores achieved by phase modulation of ultra-broadband laser pulses,” Opt. Express **17**(16), 13737–13746 (2009). [CrossRef] [PubMed]

35. K. Isobe, A. Suda, M. Tanaka, F. Kannari, H. Kawano, H. Mizuno, A. Miyawaki, and K. Midorikawa, “Multifarious control of two-photon excitation of multiple fluorophores achieved by phase modulation of ultra-broadband laser pulses,” Opt. Express **17**(16), 13737–13746 (2009). [CrossRef] [PubMed]

36. H. Hashimoto, K. Isobe, A. Suda, F. Kannari, H. Kawano, H. Mizuno, A. Miyawaki, and K. Midorikawa, “Measurement of two-photon excitation spectra of fluorescent proteins with nonlinear Fourier-transform spectroscopy,” Appl. Opt. **49**(17), 3323–3329 (2010). [CrossRef] [PubMed]

36. H. Hashimoto, K. Isobe, A. Suda, F. Kannari, H. Kawano, H. Mizuno, A. Miyawaki, and K. Midorikawa, “Measurement of two-photon excitation spectra of fluorescent proteins with nonlinear Fourier-transform spectroscopy,” Appl. Opt. **49**(17), 3323–3329 (2010). [CrossRef] [PubMed]

_{2}O solvent environment. The spectrum width of the excitation laser pulse is set to 400 nm (FWHM) with a center wavelength of 800 nm. For these calculations the Fourier transform limited (FTL) x-polarization laser pulse is assumed. From the spectra, Venus is efficiently two-photon excited by the plasmon resonance at the Au nanorod with an aspect ratio of 6. On the other hand, CFP is efficiently two-photon excited by that with an aspect ratio of 4.

_{2}substrate. One is rotated at 90° to the other. Thus, these nanocrosses exhibit the opposite plasmon resonance at the orthogonal polarization excitation. These nanocrosses are assumed to be immersed in a solution containing two fluorescence proteins, Venus and CFP. First, the excitation femtosecond laser pulse is shaped by spectral amplitude modulation so that one of the two plasmon resonances is excited.

35. K. Isobe, A. Suda, M. Tanaka, F. Kannari, H. Kawano, H. Mizuno, A. Miyawaki, and K. Midorikawa, “Multifarious control of two-photon excitation of multiple fluorophores achieved by phase modulation of ultra-broadband laser pulses,” Opt. Express **17**(16), 13737–13746 (2009). [CrossRef] [PubMed]

37. M. Comstock, V. V. Lozovoy, I. Pastirk, and M. Dantus, “Multiphoton intrapulse interference 6; binary phase shaping,” Opt. Express **12**(6), 1061–1066 (2004). [CrossRef] [PubMed]

39. G. Labroille, R. S. Pillai, X. Solinas, C. Boudoux, N. Olivier, E. Beaurepaire, and M. Joffre, “Dispersion-based pulse shaping for multiplexed two-photon fluorescence microscopy,” Opt. Lett. **35**(20), 3444–3446 (2010). [CrossRef] [PubMed]

38. R. S. Pillai, C. Boudoux, G. Labroille, N. Olivier, I. Veilleux, E. Farge, M. Joffre, and E. Beaurepaire, “Multiplexed two-photon microscopy of dynamic biological samples with shaped broadband pulses,” Opt. Express **17**(15), 12741–12752 (2009). [CrossRef] [PubMed]

39. G. Labroille, R. S. Pillai, X. Solinas, C. Boudoux, N. Olivier, E. Beaurepaire, and M. Joffre, “Dispersion-based pulse shaping for multiplexed two-photon fluorescence microscopy,” Opt. Lett. **35**(20), 3444–3446 (2010). [CrossRef] [PubMed]

*ω*

_{1}( = 2.22x10

^{15}rad/s:

*λ*

_{1}= 850 nm) and

*ω*

_{2}( = 2.51x10

^{15}rad/s:

*λ*

_{2}= 750 nm) to excite CFP protein and prevent excitation of Venus (Fig. 7 (d)). We set the third-order dispersion with

*ϕ*

_{2}(

*ω*) = 1/6x1.0x10

^{−43}(

*ω*-

*ω*

_{2})

^{3}and

*ϕ*

_{1}(

*ω*) = 1/6x1.0x10

^{−43}(

*ω*-

*ω*

_{1})

^{3}for

*ω*>ω

_{2}, and

*ω*<

*ω*

_{1}, respectively. The resulting SHG spectrum generated by sum frequency mixing between the spectrum components with the same group delay

*ω*>

*ω*

_{2}and

*ω*<

*ω*

_{1}is centered at the frequency

*ω*

_{1}+

*ω*

_{2}. The calculated distinguish ratios are CFP:Venus = 100: 1.6, and 0.6:100 for the nanorods with the aspect ratio of 4 and 6, respectively (see the summary in Table 1).

## 4. Conclusions

## Acknowledgments

## References and links

1. | U. Kreibig and M. Vollmer, |

2. | L. A. Sweatlock, S. A. Maier, H. A. Atwater, J. J. Penninkhof, and A. Polman, “Highly confined electromagnetic fields in arrays of strongly coupled Ag nanoparticles,” Phys. Rev. B |

3. | A. Bouhelier, R. Bachelot, J. S. Im, G. P. Wiederrecht, G. Lerondel, S. Kostcheev, and P. Royer, “Electromagnetic interactions in plasmonic nanoparticle arrays,” J. Phys. Chem. B |

4. | C. J. Murphy, T. K. Sau, A. M. Gole, C. J. Orendorff, J. X. Gao, L. Gou, S. E. Hunyadi, and T. Li, “Anisotropic metal nanoparticles: Synthesis, assembly, and optical applications,” J. Phys. Chem. B |

5. | M. Maillard, S. Giorgio, and M.-P. Pileni, “Tuning the size of silver nanodisks with similar aspect ratios: Synthesis and optical properties,” J. Phys. Chem. B |

6. | S. Chen, Z. Fan, and D. L. Carroll, “Silver nanodisks: Synthesis, characterization, and self-assembly,” J. Phys. Chem. B |

7. | S. L. Westcott, J. B. Jackson, C. Radloff, and N. J. Halas, “Relative contributions to the plasmon line shape of metal nanoshells,” Phys. Rev. B |

8. | Y. N. Xia and N. J. Halas, “Shape-controlled synthesis and surface plasmonic properties of metallic nanostructures,” MRS Bull. |

9. | E. Stefan Kooij and B. Poelsema, “Shape and size effects in the optical properties of metallic nanorods,” Phys. Chem. Chem. Phys. |

10. | T. Brixner, F. J. Garcia de Abajo, J. Schneider, C. Spindler, and W. Pfeiffer, “Ultrafast adaptive optical near-field control,” Phys. Rev. B |

11. | M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, F. J. García de Abajo, W. Pfeiffer, M. Rohmer, C. Spindler, and F. Steeb, “Adaptive subwavelength control of nano-optical fields,” Nature |

12. | M. I. Stockman, S. V. Faleev, and D. J. Bergman, “Coherent control of femtosecond energy localization in nanosystems,” Phys. Rev. Lett. |

13. | R. S. Judson and H. Rabitz, “Teaching lasers to control molecules,” Phys. Rev. Lett. |

14. | A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle, and G. Gerber, “Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses,” Science |

15. | D. Meshulach and Y. Silberberg, “Coherent quantum control of two-photon transitions by a femtosecond laser pulse,” Nature |

16. | T. Brixner and G. Gerber, “Femtosecond polarization pulse shaping,” Opt. Lett. |

17. | F. Weise and A. Lindinger, “Full control over the electric field using four liquid crystal arrays,” Opt. Lett. |

18. | Y. Esumi, M. D. Kabir, and F. Kannari, “Spatiotemporal vector pulse shaping of femtosecond laser pulses with a multi-pass two-dimensional spatial light modulator,” Opt. Express |

19. | G. Lévêque and O. J. F. Martin, “Narrow-band multiresonant plasmon nanostructure for the coherent control of light: an optical analog of the xylophone,” Phys. Rev. Lett. |

20. | A. Taflove and K. R. Umashankar, “The finite-difference time-domain (FD-TD) method for electromagnetic scattering and interaction problems,” J. Electromagn. Waves Appl. |

21. | J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic-waves,” J. Comput. Phys. |

22. | K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antenn. Propag. |

23. | A. Taflove, |

24. | G. D. Smith, |

25. | D. W. Peaceman and H. H. Rachford Jr., “The numerical solution of parabolic and elliptic differential equations,” J. Soc. Ind. Appl. Math. |

26. | G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic field equations,” IEEE Trans. Electromagn. Compat. |

27. | A. Taflove and M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problem using time-dependent Maxwell's equations,” IEEE Trans. Microw. Theory Tech. |

28. | W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science |

29. | K. König, “Multiphoton microscopy in life sciences,” J. Microsc. |

30. | K. Isobe, W. Watanabe, S. Matsunaga, T. Higashi, K. Fukui, and K. Itoh, “Multi-spectral two-photon excited fluorescence microscopy using supercontinuum light source,” Jpn. J. Appl. Phys. |

31. | C. Xu, W. Zipfel, J. B. Shear, R. M. Williams, and W. W. Webb, “Multiphoton fluorescence excitation: new spectral windows for biological nonlinear microscopy,” Proc. Natl. Acad. Sci. U.S.A. |

32. | G. Y. Fan, H. Fujisaki, A. Miyawaki, R. K. Tsay, R. Y. Tsien, and M. H. Ellisman, “Video-rate scanning two-photon excitation fluorescence microscopy and ratio imaging with cameleons,” Biophys. J. |

33. | P. Allcock and D. L. Andrews, “Two-photon fluorescence: Resonance energy transfer,” J. Chem. Phys. |

34. | K. G. Heinze, A. Koltermann, and P. Schwille, “Simultaneous two-photon excitation of distinct labels for dual-color fluorescence crosscorrelation analysis,” Proc. Natl. Acad. Sci. U.S.A. |

35. | K. Isobe, A. Suda, M. Tanaka, F. Kannari, H. Kawano, H. Mizuno, A. Miyawaki, and K. Midorikawa, “Multifarious control of two-photon excitation of multiple fluorophores achieved by phase modulation of ultra-broadband laser pulses,” Opt. Express |

36. | H. Hashimoto, K. Isobe, A. Suda, F. Kannari, H. Kawano, H. Mizuno, A. Miyawaki, and K. Midorikawa, “Measurement of two-photon excitation spectra of fluorescent proteins with nonlinear Fourier-transform spectroscopy,” Appl. Opt. |

37. | M. Comstock, V. V. Lozovoy, I. Pastirk, and M. Dantus, “Multiphoton intrapulse interference 6; binary phase shaping,” Opt. Express |

38. | R. S. Pillai, C. Boudoux, G. Labroille, N. Olivier, I. Veilleux, E. Farge, M. Joffre, and E. Beaurepaire, “Multiplexed two-photon microscopy of dynamic biological samples with shaped broadband pulses,” Opt. Express |

39. | G. Labroille, R. S. Pillai, X. Solinas, C. Boudoux, N. Olivier, E. Beaurepaire, and M. Joffre, “Dispersion-based pulse shaping for multiplexed two-photon fluorescence microscopy,” Opt. Lett. |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(180.2520) Microscopy : Fluorescence microscopy

(240.6680) Optics at surfaces : Surface plasmons

(320.5540) Ultrafast optics : Pulse shaping

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: March 24, 2011

Revised Manuscript: May 15, 2011

Manuscript Accepted: May 26, 2011

Published: June 29, 2011

**Virtual Issues**

Vol. 6, Iss. 8 *Virtual Journal for Biomedical Optics*

**Citation**

Takuya Harada, Keiichiro Matsuishi, Yu Oishi, Keisuke Isobe, Akira Suda, Hiroyuki Kawan, Hideaki Mizuno, Atsushi Miyawaki, Katsumi Midorikawa, and Fumihiko Kannari, "Temporal control of local plasmon distribution on Au nanocrosses by ultra-broadband femtosecond laser pulses and its application for selective two-photon excitation of multiple fluorophores," Opt. Express **19**, 13618-13627 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-14-13618

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### References

- U. Kreibig and M. Vollmer, Optical Properties of Metal Clusters (Springer-Verlag, Berlin, 1995).
- L. A. Sweatlock, S. A. Maier, H. A. Atwater, J. J. Penninkhof, and A. Polman, “Highly confined electromagnetic fields in arrays of strongly coupled Ag nanoparticles,” Phys. Rev. B 71(23), 235408 (2005). [CrossRef]
- A. Bouhelier, R. Bachelot, J. S. Im, G. P. Wiederrecht, G. Lerondel, S. Kostcheev, and P. Royer, “Electromagnetic interactions in plasmonic nanoparticle arrays,” J. Phys. Chem. B 109(8), 3195–3198 (2005). [CrossRef] [PubMed]
- C. J. Murphy, T. K. Sau, A. M. Gole, C. J. Orendorff, J. X. Gao, L. Gou, S. E. Hunyadi, and T. Li, “Anisotropic metal nanoparticles: Synthesis, assembly, and optical applications,” J. Phys. Chem. B 109(29), 13857–13870 (2005). [CrossRef] [PubMed]
- M. Maillard, S. Giorgio, and M.-P. Pileni, “Tuning the size of silver nanodisks with similar aspect ratios: Synthesis and optical properties,” J. Phys. Chem. B 107(11), 2466–2470 (2003). [CrossRef]
- S. Chen, Z. Fan, and D. L. Carroll, “Silver nanodisks: Synthesis, characterization, and self-assembly,” J. Phys. Chem. B 106(42), 10777–10781 (2002). [CrossRef]
- S. L. Westcott, J. B. Jackson, C. Radloff, and N. J. Halas, “Relative contributions to the plasmon line shape of metal nanoshells,” Phys. Rev. B 66(15), 155431 (2002). [CrossRef]
- Y. N. Xia and N. J. Halas, “Shape-controlled synthesis and surface plasmonic properties of metallic nanostructures,” MRS Bull. 30(05), 338–348 (2005). [CrossRef]
- E. Stefan Kooij and B. Poelsema, “Shape and size effects in the optical properties of metallic nanorods,” Phys. Chem. Chem. Phys. 8(28), 3349–3357 (2006). [CrossRef] [PubMed]
- T. Brixner, F. J. Garcia de Abajo, J. Schneider, C. Spindler, and W. Pfeiffer, “Ultrafast adaptive optical near-field control,” Phys. Rev. B 73(12), 125437 (2006). [CrossRef]
- M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, F. J. García de Abajo, W. Pfeiffer, M. Rohmer, C. Spindler, and F. Steeb, “Adaptive subwavelength control of nano-optical fields,” Nature 446(7133), 301–304 (2007). [CrossRef] [PubMed]
- M. I. Stockman, S. V. Faleev, and D. J. Bergman, “Coherent control of femtosecond energy localization in nanosystems,” Phys. Rev. Lett. 88(6), 067402–067405 (2002). [CrossRef] [PubMed]
- R. S. Judson and H. Rabitz, “Teaching lasers to control molecules,” Phys. Rev. Lett. 68(10), 1500–1503 (1992). [CrossRef] [PubMed]
- A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle, and G. Gerber, “Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses,” Science 282(5390), 919–922 (1998). [CrossRef] [PubMed]
- D. Meshulach and Y. Silberberg, “Coherent quantum control of two-photon transitions by a femtosecond laser pulse,” Nature 396(6708), 239–242 (1998). [CrossRef]
- T. Brixner and G. Gerber, “Femtosecond polarization pulse shaping,” Opt. Lett. 26(8), 557–559 (2001). [CrossRef] [PubMed]
- F. Weise and A. Lindinger, “Full control over the electric field using four liquid crystal arrays,” Opt. Lett. 34(8), 1258–1260 (2009). [CrossRef] [PubMed]
- Y. Esumi, M. D. Kabir, and F. Kannari, “Spatiotemporal vector pulse shaping of femtosecond laser pulses with a multi-pass two-dimensional spatial light modulator,” Opt. Express 17(21), 19153–19159 (2009). [CrossRef] [PubMed]
- G. Lévêque and O. J. F. Martin, “Narrow-band multiresonant plasmon nanostructure for the coherent control of light: an optical analog of the xylophone,” Phys. Rev. Lett. 100(11), 117402 (2008). [CrossRef] [PubMed]
- A. Taflove and K. R. Umashankar, “The finite-difference time-domain (FD-TD) method for electromagnetic scattering and interaction problems,” J. Electromagn. Waves Appl. 1(3), 243–267 (1987). [CrossRef]
- J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic-waves,” J. Comput. Phys. 114(2), 185–200 (1994). [CrossRef]
- K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antenn. Propag. 14(8), 302–307 (1966). [CrossRef]
- A. Taflove,Computational Electrodynamics (Artech House, Norwood, MA, 1995).
- G. D. Smith, Numerical Solution of Partial Differential Equations (Oxford Univ. Press, Oxford, UK. 1965).
- D. W. Peaceman and H. H. Rachford., “The numerical solution of parabolic and elliptic differential equations,” J. Soc. Ind. Appl. Math. 3(1), 28–41 (1955). [CrossRef]
- G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic field equations,” IEEE Trans. Electromagn. Compat. EMC-23(4), 377–382 (1981). [CrossRef]
- A. Taflove and M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problem using time-dependent Maxwell's equations,” IEEE Trans. Microw. Theory Tech. 23(8), 623–630 (1975). [CrossRef]
- W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248(4951), 73–76 (1990). [CrossRef] [PubMed]
- K. König, “Multiphoton microscopy in life sciences,” J. Microsc. 200(2), 83–104 (2000). [CrossRef] [PubMed]
- K. Isobe, W. Watanabe, S. Matsunaga, T. Higashi, K. Fukui, and K. Itoh, “Multi-spectral two-photon excited fluorescence microscopy using supercontinuum light source,” Jpn. J. Appl. Phys. 44(4), L167–L169 (2005). [CrossRef]
- C. Xu, W. Zipfel, J. B. Shear, R. M. Williams, and W. W. Webb, “Multiphoton fluorescence excitation: new spectral windows for biological nonlinear microscopy,” Proc. Natl. Acad. Sci. U.S.A. 93(20), 10763–10768 (1996). [CrossRef] [PubMed]
- G. Y. Fan, H. Fujisaki, A. Miyawaki, R. K. Tsay, R. Y. Tsien, and M. H. Ellisman, “Video-rate scanning two-photon excitation fluorescence microscopy and ratio imaging with cameleons,” Biophys. J. 76(5), 2412–2420 (1999). [CrossRef] [PubMed]
- P. Allcock and D. L. Andrews, “Two-photon fluorescence: Resonance energy transfer,” J. Chem. Phys. 108(8), 3089–3095 (1998). [CrossRef]
- K. G. Heinze, A. Koltermann, and P. Schwille, “Simultaneous two-photon excitation of distinct labels for dual-color fluorescence crosscorrelation analysis,” Proc. Natl. Acad. Sci. U.S.A. 97(19), 10377–10382 (2000). [CrossRef] [PubMed]
- K. Isobe, A. Suda, M. Tanaka, F. Kannari, H. Kawano, H. Mizuno, A. Miyawaki, and K. Midorikawa, “Multifarious control of two-photon excitation of multiple fluorophores achieved by phase modulation of ultra-broadband laser pulses,” Opt. Express 17(16), 13737–13746 (2009). [CrossRef] [PubMed]
- H. Hashimoto, K. Isobe, A. Suda, F. Kannari, H. Kawano, H. Mizuno, A. Miyawaki, and K. Midorikawa, “Measurement of two-photon excitation spectra of fluorescent proteins with nonlinear Fourier-transform spectroscopy,” Appl. Opt. 49(17), 3323–3329 (2010). [CrossRef] [PubMed]
- M. Comstock, V. V. Lozovoy, I. Pastirk, and M. Dantus, “Multiphoton intrapulse interference 6; binary phase shaping,” Opt. Express 12(6), 1061–1066 (2004). [CrossRef] [PubMed]
- R. S. Pillai, C. Boudoux, G. Labroille, N. Olivier, I. Veilleux, E. Farge, M. Joffre, and E. Beaurepaire, “Multiplexed two-photon microscopy of dynamic biological samples with shaped broadband pulses,” Opt. Express 17(15), 12741–12752 (2009). [CrossRef] [PubMed]
- G. Labroille, R. S. Pillai, X. Solinas, C. Boudoux, N. Olivier, E. Beaurepaire, and M. Joffre, “Dispersion-based pulse shaping for multiplexed two-photon fluorescence microscopy,” Opt. Lett. 35(20), 3444–3446 (2010). [CrossRef] [PubMed]

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