## Analysis of the influence of spherical aberration from focusing through a dielectric slab in quantitative nonlinear optical susceptibility measurements using third-harmonic generation

Optics Express, Vol. 14, Issue 1, pp. 260-269 (2006)

http://dx.doi.org/10.1364/OPEX.14.000260

Acrobat PDF (1240 KB)

### Abstract

The third-order nonlinear susceptibility (χ^{(3)}) can be measured quantitatively using third-harmonic generation (THG) from two different interfaces. For the first time it is demonstrated both in experiments and theory that the magnitude of the THG signals from the two interfaces is not only determined by material properties (refractive index and χ^{(3)}), but also by optical aberrations. It is found that this method of χ^{(3)} determination can be applied without additional correction factors only for focusing conditions with a numerical aperture (NA) ≤ 0.35. The implications for general application of THG in three-dimensional microscopy are discussed.

© 2006 Optical Society of America

## 1. Introduction

1. Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, “Nonlinear Scanning Laser Microscopy by Third Harmonic Generation,” Appl. Phys. Lett. **70**, 922–924 (1997). [CrossRef]

2. M. Müller, J. Squier, K. R. Wilson, and G. J. Brakenhoff, “3D Microscopy of transparent objects using third-harmonic generation,” J. Microsc. **191**, 266–274 (1998). [CrossRef] [PubMed]

^{(3)}, it can be shown that for a homogenous, isotropic medium of infinite extension and with normal dispersion, no third-harmonic is generated by a tightly focused beam in case of perfect phase matching [3

3. J. F. Ward and G. H. C. New, “Optical Third Harmonic Generation in Gases by a Focused Laser Beam,” Phys. Rev. **185**, 57–72 (1969). [CrossRef]

^{(3)}, within the focal volume can lead to a measurable third-harmonic signal. This is the contrast generating mechanism in THG microscopy, which has been demonstrated to be a useful imaging tool in biology and the material sciences [7–16

7. J. A. Squier, M. Müller, G. J. Brakenhoff, and K. R. Wilson, “Third Harmonic Generation Microscopy,” Opt. Express **3**, 315–324 (1998). [CrossRef] [PubMed]

17. S. Hell, G. Reiner, C. Cremer, and E.H.K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index,” J. Microsc. **169**, 391–405 (1993). [CrossRef]

17. S. Hell, G. Reiner, C. Cremer, and E.H.K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index,” J. Microsc. **169**, 391–405 (1993). [CrossRef]

^{(3)}of solutions in a simple and accurate manner compared to conventional techniques [19

19. R. Barille, L. Canioni, L. Sarger, and G. Rivoire, “Nonlinearity measurements of thin films by third-harmonic-generation microscopy,” Phys. Rev. E **66** (2002). [CrossRef]

20. V. Shcheslavskiy, G. Petrov, and V. V. Yakovlev, “Nonlinear optical susceptibility measurements of solutions using third-harmonic generation on the interface,” Appl. Phys. Lett. **82**, 3982–3984 (2003). [CrossRef]

^{(3)}using THG. We determine the experimental conditions for which such measurements can be done accurately. In addition we consider the implications of spherical aberration for general three-dimensional THG microscopy.

## 2. Materials and methods

_{B}/I

_{A}depends on the NA of the focusing microscope objective, and the thickness and refractive index of the cover glass (i.e. the medium in region AB).

*Experimental Setup*. The schematic of the experimental setup used for the THG measurements is shown in Fig. 1(c). A laser (

*FemtoTrain*, High-Q Laser GMBH, Austria) produces 1062 nm pulses with a duration of 113 fs with a repetition rate of 72.3 MHz. After collimation, the laser beam is focused onto the sample using an infinity corrected microscope objective. This microscope objective is mounted on a piezo (Physik Instruments, Germany) for axial scanning. The signal is collected in the forward direction using a second microscope objective and imaged onto a spectrometer equipped with a low temperature CCD camera (spectral resolution ~0.15 nm at 351 nm). To ensure full collection of the generated signal, the NA of the collection objective was chosen to be always larger than the required one-third of the numerical aperture of the focusing objective [22

22. J. M. Schins, T. Schrama, J. Squier, G. J. Brakenhoff, and M. Müller, “Determination of material properties by use of third-harmonic generation microscopy,” J. Opt. Soc. Am. B **19**, 1627–1634 (2002). [CrossRef]

^{(3)}measurements using THG. To investigate the influence of aberration, glass types with different dispersion and thickness have been used (see table 1).

22. J. M. Schins, T. Schrama, J. Squier, G. J. Brakenhoff, and M. Müller, “Determination of material properties by use of third-harmonic generation microscopy,” J. Opt. Soc. Am. B **19**, 1627–1634 (2002). [CrossRef]

^{-3}rad). The container is filled with water. Measurements done at different transverse locations at interface C, correspond to measurements at different depths.

*Theory*. The paraxial equations describing third-harmonic generation have an analytical solution when the amplitude of the excitation electric field has a Gaussian profile [3

3. J. F. Ward and G. H. C. New, “Optical Third Harmonic Generation in Gases by a Focused Laser Beam,” Phys. Rev. **185**, 57–72 (1969). [CrossRef]

1. Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, “Nonlinear Scanning Laser Microscopy by Third Harmonic Generation,” Appl. Phys. Lett. **70**, 922–924 (1997). [CrossRef]

20. V. Shcheslavskiy, G. Petrov, and V. V. Yakovlev, “Nonlinear optical susceptibility measurements of solutions using third-harmonic generation on the interface,” Appl. Phys. Lett. **82**, 3982–3984 (2003). [CrossRef]

22. J. M. Schins, T. Schrama, J. Squier, G. J. Brakenhoff, and M. Müller, “Determination of material properties by use of third-harmonic generation microscopy,” J. Opt. Soc. Am. B **19**, 1627–1634 (2002). [CrossRef]

23. J. X. Cheng and X. S. Xie, “Green’s function formulation for third-harmonic generation microscopy,” J. Opt. Soc. Am. B **19**, 1604–1610 (2002). [CrossRef]

23. J. X. Cheng and X. S. Xie, “Green’s function formulation for third-harmonic generation microscopy,” J. Opt. Soc. Am. B **19**, 1604–1610 (2002). [CrossRef]

^{(3)}. This means that in the calculations of the focal field, it is assumed that the focusing is in a uniform medium and that the generated third-harmonic is also propagating through a homogeneous medium towards the detector. Another simplification used in the calculation is to neglect a possible wave vector mismatch (Δk = k

_{3ω}- 3 k

_{ω}), which is usually permitted for tight focusing conditions [23

23. J. X. Cheng and X. S. Xie, “Green’s function formulation for third-harmonic generation microscopy,” J. Opt. Soc. Am. B **19**, 1604–1610 (2002). [CrossRef]

*ϕ*(

*α*) is used. The phase error corresponding to primary spherical aberration introduced by a dielectric slab of thickness

*t*in a converging wavefront is given by [18]

*n*

_{2}is the refractive index of the slab,

*n*

_{1}that of the medium and

*s*= sin(

*α*/2). Hence, for a microscope objective designed for a cover glass of thickness 170 μm, the phase error introduced by focusing through a cover glass with an actual thickness

*t*can be written as:

19. R. Barille, L. Canioni, L. Sarger, and G. Rivoire, “Nonlinearity measurements of thin films by third-harmonic-generation microscopy,” Phys. Rev. E **66** (2002). [CrossRef]

^{(3)}of a material in the BC region (Fig. 1(a)) can be determined from the measured ratio I

_{B}/I

_{A}in an air-glass-material configuration. Using the known χ

^{(3)}of the glass and the refractive indices of both glass and the material at the fundamental and the third-harmonic, the χ

^{(3)}of the material is given by:

*χ*from water to 2-propanol.

^{(3)}## 3. Results

*Spherical aberration from the cover glass*. In this measurement the sample consists of a single air-glass-air configuration. For a given sample configuration the optimal microscope objective illumination conditions are determined by measuring the FWHM of the THG z-response across interface B as function of the position of lens L

_{4}(Fig. 2). The position of L

_{4}that provides a minimum in the FWHM is taken as the configuration with minimal spherical aberration. Indeed, the minimal FWHM L

_{4}lens position coincides with a flip in the asymmetry of the z-response (see insets Fig. 2) which corresponds to a flip in the sign of the induced spherical aberration. The effective z-position is corrected for the focal shift that results from the refraction at the air-glass interface. The FWHM values are obtained by fitting Voigt profiles to both the left-hand and right-hand side of the z-response. The choice for fitting with a Voigt line profile is prompted by the observation from experiments and numerical simulations, that the shape of the z-response varies smoothly between purely Lorentzian for Δk=0, to Gaussian in case of large Δk values or the presence of significant spherical aberration.

_{4}positions vary consistently with the thickness and refractive index of the glass used (data not shown), indicates that this procedure provides comparable minimum aberration conditions at interface B for all experimental conditions.

_{4}- and thus the microscope objective illumination conditions that provides minimal aberrations at interface B - is determined for the different glasses (G1, G2 and G3), the effect of aberrations can be determined from a measurement of I

_{B}/I

_{A}as a function of NA (Fig. 3). The sample configuration is identical to that used for Fig. 2. To permit comparison with earlier work on the ‘THG ratio method’ for the quantitative determination of χ

^{(3)}, peak intensities at the interfaces A and B are used. By definition, spherical aberration is zero at interface B and nonzero at the interface A. The deviation of the ratio I

_{B}/I

_{A}from unity is a direct indication for the magnitude of the aberrations. The ratio I

_{B}/I

_{A}increases sharply with NA and with the thickness of the cover glass (G3 = 50 μm → G1 = 200 μm). The solid line in Fig. 3 represents a numerical calculation for the G2 case.

*The effect of spherical aberration on the determination of χ*. In order to demonstrate the effect of spherical aberration on quantitative χ

^{(3)}^{(3)}measurements using THG, the ratio I

_{B}/I

_{A}is measured for methanol, ethanol and 2-propanol using objectives with NA = 0.35 and NA = 0.65, and G1 glass. The χ

^{(3)}values are determined from the ratio I

_{B}/I

_{A}using the method described above. As a reference point, the χ

^{(3)}value of water (2.8 × 10

^{-14}e.s.u) [24

24. F. Kajzar and J. Messier, “Third-harmonic generation in liquids,” Phys. Rev. A **32**, 2352–2363 (1985). [CrossRef] [PubMed]

^{(3)}of Duran glass (2.87 × 10

^{-14}e.s.u) in a separate measurement. Using this χ

^{(3)}of Duran glass as the reference,, the χ

^{(3)}of the liquids (methanol, ethanol and 2-propanol) was determined. The required refractive indices of the different liquids for the J integral evaluation, were obtained using the Cauchy parameters as determined by Kozma et al. [25

25. I. Z. Kozma, P. Krok, and E. Riedle, “Direct measurement of the group-velocity mismatch and derivatiion of the refractive-index dispersion for a variety of solvents in the ultraviolet,” J. Opt. Soc. Am. B **22**, 1479–1485 (2005). [CrossRef]

^{(3)}values are shown in Fig. 4. From Fig. 3 it follows that for NA = 0.35 the effect of aberrations can be neglected, whereas for NA = 0.65 aberrations significantly affect the measured I

_{B}/I

_{A}ratio. For the high NA case the effect of aberrations can be corrected for as follows. Since aberrations are appearing at interface A alone, only the THG signal from that interface need to be corrected. This correction factor is obviously independent of the material medium in the region BC, provided that the confocal parameter is significantly smaller than the axial extent of region AB, to ensure that the signal from interface A is entirely due to AB. In that case, the correction factor r = (I

_{B}/I

_{A})

_{air-glass-air}as measured using air-glass-air interface and the NA of interest. The resulting χ

^{(3)}value for NA = 0.65 after correction for the effect of aberrations is also shown in Fig. 4. For comparison, χ

^{(3)}values from literature [24

24. F. Kajzar and J. Messier, “Third-harmonic generation in liquids,” Phys. Rev. A **32**, 2352–2363 (1985). [CrossRef] [PubMed]

26. G. R. Meredith, B. Buchalter, and C. Hanzlik, “Third-order susceptibility determination by third harmonic generation. II,” J. Chem. Phys. **78**, 1543–1551 (1983). [CrossRef]

*Spherical aberration from refractive index mismatch*. Equivalent effects resulting from induced aberrations, as observed in quantitative χ

^{(3)}measurements using THG, are encountered in three-dimensional THG microscopy, when a refractive index mismatch is present between the immersion of the microscope objective and the mounting medium of the sample. Again, spherical aberration results from focusing through a dielectric slab. The magnitude of this effect is investigated by measuring the ratio I

_{C}/I

_{B}in a sample configuration as depicted in Fig. 1(b), as a function of the lateral position and hence the depth inside the sample region BC. The medium in region BC is water, while focusing is realised either with an oil immersion (NA = 1.25) or air spaced (NA = 0.65) microscope objective. Thus in both cases there is a refractive index mismatch between either oil (n = 1.5158) or air (n = 1) and water (n = 1.33). Figure 5 shows the ratio I

_{C}/I

_{B}as a function of the thickness of the water layer for two different microscope objectives. The same ratio was calculated theoretically for NA = 0.65 using a uniform profile, and 1.25 using a Gaussian profile [23

**19**, 1604–1610 (2002). [CrossRef]

## 4. Discussion

^{(3)}of a medium is based on the measurement of the ratio (I

_{B}/I

_{A}) of the THG signal at the glass-medium interface and the glass-air interface. This simple and elegant method has been applied to the χ

^{(3)}measurement of various liquids [19

19. R. Barille, L. Canioni, L. Sarger, and G. Rivoire, “Nonlinearity measurements of thin films by third-harmonic-generation microscopy,” Phys. Rev. E **66** (2002). [CrossRef]

20. V. Shcheslavskiy, G. Petrov, and V. V. Yakovlev, “Nonlinear optical susceptibility measurements of solutions using third-harmonic generation on the interface,” Appl. Phys. Lett. **82**, 3982–3984 (2003). [CrossRef]

_{B}/I

_{A}ratio is unity when an identical medium is used at both interfaces. In the measurements presented here, it was found that this assumption is true only for low NA focusing (NA ≤ 0.35) conditions, in contrast to those used in these initial studies (NA = 0.65 and 0.55 respectively). It is well known that focusing through a dielectric slab results primarily in spherical aberration [18], and that this effect becomes more severe with increasing NA and optical path length through the material. Microscope objectives are generally pre-compensated for this effect to yield minimal aberration conditions at the backside (interface B) of the cover glass of a certain specified thickness and refractive index. Slight changes in collimation of the microscope objective’s illumination can be used to correct for different glass types and cover glass thicknesses.

_{4}lens position provided us with the key initial settings required to quantify the effect of aberration as a function of numerical aperture, focusing depth, and the refractive index and thickness of the cover glass used. A clear minimum in the THG z-response across interface B can be identified as a function of the L

_{4}position (Fig. 2), which is taken to coincide with minimum aberration conditions at focus. It should be noted that this method of optimising the microscope objective’s illumination conditions can be used as a quick and effective way to control the effect of aberration and hence the signal level in quantitative as well as qualitative THG microscopy. Especially in quantitative applications, such as in the measurement of material properties as described in this report, it is useful to ensure that the focal field at, at least, one of the interfaces is aberration free. The effect of aberrations at the second interface can then readily be obtained either experimentally or theoretically. Once the effect of induced aberrations is determined, the measured I

_{B}/I

_{A}ratio can be corrected to yield χ

^{(3)}independent of the experimental conditions. Thus, although low NA focusing conditions are generally preferred for quantitative χ

^{(3)}measurement, high NA focusing -with proper correction for the induced aberrations- can be used in specific cases (e.g. for thin samples).

_{B}/I

_{A}ratio converges to unity and becomes independent of NA and glass type only for NA ≤ 0.35. For a moderate NA = 0.65 and G1 glass combination, the I

_{B}/I

_{A}ratio increases by a factor as high as ~8.5 as a result of induced aberrations. Figure 3(b) also shows the close agreement - both in absolute value and in the shape of the THG z-response - that is obtained between the experimental data and those calculated using the theory as described in section 2. In fact, the theoretical calculation of I

_{B}/I

_{A}slightly underestimates the experimental value. This slight difference is most likely due to the assumptions employed in the calculation. The fundamental focal field is calculated under the assumption of a uniform medium in linear optical properties, where as in the actual experimental situation, the focusing is done at an interface of two media of relatively large refractive index difference. In this manner the fundamental focal field is not rigorously calculated. In addition to this assumption, we neglected the dispersion of the material medium and the ensuing wave vector mismatch, even though this is justified only in the case of a tight focusing condition [23

**19**, 1604–1610 (2002). [CrossRef]

^{(3)}measurement of three liquids for two different focusing conditions. At low NA (0.35) the χ

^{(3)}can be determined directly from the I

_{B}/I

_{A}ratio. At high NA (0.65) the effect of aberrations result in a much larger I

_{B}/I

_{A}value and hence in an overestimate of χ

^{(3)}. Using the results of Fig. 3, however, this value can be corrected for the effect of aberration. The measured χ

^{(3)}values correspond well with those reported in the literature, measured using the Maker-fringes technique [24

24. F. Kajzar and J. Messier, “Third-harmonic generation in liquids,” Phys. Rev. A **32**, 2352–2363 (1985). [CrossRef] [PubMed]

_{B}/I

_{A}ratio can be measured with high accuracy (±2%). The error in the measurement of χ

^{(3)}then results from the uncertainty in the refractive index values(±0.5%) of the liquid at the fundamental and third-harmonic wavelength and in the precision with which the initial reference χ

^{(3)}value of water is known (±5%)This translates in an uncertainty in the measured χ

^{(3)}value of ±7%

*in vivo*imaging in extremely thick samples (~1 mm) [15

15. S.-W. Chu, Szu-Yu Chen, Tsung-Han Tsai, Tzu-Ming Liu, Cheng-Yung Lin, Huai-Jen Tsai, and Chi-Kuang Sun, “In vivo developmental biology study using noninvasive multi-harmonic generation microscopy,” Opt. Express **11**, 3093 – 3099 (2003). [CrossRef] [PubMed]

_{C}/I

_{B}slightly overestimates the experimental measurements, for the same reason as it underestimated I

_{B}/I

_{A}(see above). A better agreement between the calculated and measured ratios is found for the 1.25 NA oil immersion objective, where the approximations used in the calculation are in better agreement with the experimental conditions compared to the case of 0.65 NA air objective. We also found that the input beam profile also influences the effect of aberration. A Gaussian beam profile was found to be less susceptible to aberrations compared to a flat profile. We have used a flat profile and a Gaussian profile for the low NA and the high NA cases respectively.

^{(3)}measurements using the ‘THG ratio method’. A theoretical analysis, based on a Green’s function formulation, has been presented that permits quantitative calculation of both the THG peak intensity and the functional shape of the THG z-response in the presence of specimen induced aberrations. The close agreement between experiment and theory indicates that spherical aberration is the prime factor to account for in these measurements. We have shown that measurements at low NA (≤ 0.35) do not require additional correction for aberration. For the case of high NA focusing (NA > 0.35) - required e.g. to measure the χ

^{(3)}of thin samples - a procedure has been developed and demonstrated that can correct for the effects of specimen induced aberrations.

## Acknowledgment

## References

1. | Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, “Nonlinear Scanning Laser Microscopy by Third Harmonic Generation,” Appl. Phys. Lett. |

2. | M. Müller, J. Squier, K. R. Wilson, and G. J. Brakenhoff, “3D Microscopy of transparent objects using third-harmonic generation,” J. Microsc. |

3. | J. F. Ward and G. H. C. New, “Optical Third Harmonic Generation in Gases by a Focused Laser Beam,” Phys. Rev. |

4. | R. W. Boyd, |

5. | M. Born and E. Wolf, |

6. | J. F. Reintjes, |

7. | J. A. Squier, M. Müller, G. J. Brakenhoff, and K. R. Wilson, “Third Harmonic Generation Microscopy,” Opt. Express |

8. | J. A. Squier and M. Müller, “Third - Harmonic Generation Imaging of laser - induced breakdown in glass,” Appl. Opt. |

9. | D. Yelin and Y. Silberberg, “Laser scanning third-harmonic-generation microscopy in biology,” Opt. Express |

10. | C.-K. Sun, S.-W. Chu, S.-P. Tai, S. Keller, U.K. Mishra, and S.P. DenBaars, “Scanning second-harmonic/third-harmonic generation microscopy of gallium nitride,” Appl. Phys. Lett. |

11. | L. Canioni, S. Rivet, L. Sarger, R. Barille, P. Vacher, and P. Voisin, “Imaging of Ca2+ intracellular dynamics with a third-harmonic generation microscope,” Opt. Lett. |

12. | D. Oron, E. Tal, and Y. Silberberg, “Depth-resolved multiphoton polarisation microscopy by third-harmonic generation,” Opt. Lett. |

13. | D. Yelin, Y. Silberberg, Y. Barad, and J. S. Patel, “Depth-resolved imaging of nematic liquid crystals by third-harmonic microscopy,” Appl. Phys. Lett. |

14. | D. Débarre, W. Supatto, E. Farge, B. Moulia, M-C. Schanne-Klein, and E. Beaurepaire, “Velocimetric third-harmonic generation microscopy: micrometer-scale quantification of morphogenetic movements in unstained embryos,” Opt. Lett. |

15. | S.-W. Chu, Szu-Yu Chen, Tsung-Han Tsai, Tzu-Ming Liu, Cheng-Yung Lin, Huai-Jen Tsai, and Chi-Kuang Sun, “In vivo developmental biology study using noninvasive multi-harmonic generation microscopy,” Opt. Express |

16. | W. Supatto, D. Débarre, B. Moulia, E. Brouzés, J-L. Martin, E. Farge, and E. Beaurepaire, “In vivo modulation of morphogenetic movements in Drosophila emryos with femtosecond laser pulses,” PNAS |

17. | S. Hell, G. Reiner, C. Cremer, and E.H.K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index,” J. Microsc. |

18. | C. J. R. Sheppard and C.J. Cogswell., “Effects of aberrating layers and tube length on confocal imaging properties,” Optik |

19. | R. Barille, L. Canioni, L. Sarger, and G. Rivoire, “Nonlinearity measurements of thin films by third-harmonic-generation microscopy,” Phys. Rev. E |

20. | V. Shcheslavskiy, G. Petrov, and V. V. Yakovlev, “Nonlinear optical susceptibility measurements of solutions using third-harmonic generation on the interface,” Appl. Phys. Lett. |

21. | V. Shcheslavskiy, G. I. Petrov, S. Saltiel, and V. V. Yakovlev, “Quantitative characterization of aqueous solutions probed by the third-harmonic generation microscopy,” J. Struct. Biol. |

22. | J. M. Schins, T. Schrama, J. Squier, G. J. Brakenhoff, and M. Müller, “Determination of material properties by use of third-harmonic generation microscopy,” J. Opt. Soc. Am. B |

23. | J. X. Cheng and X. S. Xie, “Green’s function formulation for third-harmonic generation microscopy,” J. Opt. Soc. Am. B |

24. | F. Kajzar and J. Messier, “Third-harmonic generation in liquids,” Phys. Rev. A |

25. | I. Z. Kozma, P. Krok, and E. Riedle, “Direct measurement of the group-velocity mismatch and derivatiion of the refractive-index dispersion for a variety of solvents in the ultraviolet,” J. Opt. Soc. Am. B |

26. | G. R. Meredith, B. Buchalter, and C. Hanzlik, “Third-order susceptibility determination by third harmonic generation. II,” J. Chem. Phys. |

**OCIS Codes**

(080.1010) Geometric optics : Aberrations (global)

(180.6900) Microscopy : Three-dimensional microscopy

(190.4160) Nonlinear optics : Multiharmonic generation

(190.4410) Nonlinear optics : Nonlinear optics, parametric processes

(190.4720) Nonlinear optics : Optical nonlinearities of condensed matter

**ToC Category:**

Nonlinear Optics

**Virtual Issues**

Vol. 1, Iss. 2 *Virtual Journal for Biomedical Optics*

**Citation**

Rajesh S. Pillai, G. J. Brakenhoff, and M. Müller, "Analysis of the influence of spherical aberration from focusing through a dielectric slab in quantitative nonlinear optical susceptibility measurements using third-harmonic generation," Opt. Express **14**, 260-269 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-1-260

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### References

- Y. Barad, H. Eisenberg, M. Horowitz, Y. Silberberg, "Nonlinear Scanning Laser Microscopy by Third Harmonic Generation," Appl. Phys. Lett. 70, 922-924 (1997). [CrossRef]
- M. Müller, J. Squier, K. R. Wilson and G. J. Brakenhoff, "3D Microscopy of transparent objects using third-harmonic generation," J. Microsc. 191, 266-274 (1998). [CrossRef] [PubMed]
- J. F. Ward, G. H. C. New, "Optical Third Harmonic Generation in Gases by a Focused Laser Beam," Phys. Rev. 185, 57-72 (1969). [CrossRef]
- R. W. Boyd, Nonlinear Optics (Academic Press, Inc., New York, 1992).
- M. Born, E. Wolf, Principles of Optics (Pergamon Press, Oxford, 1993).
- J. F. Reintjes, Nonlinear Optical Parametric Processes in Liquids and Gases (Academic Press, Inc., Orlando, 1984).
- J. A. Squier, M. Müller, G. J. Brakenhoff and K. R. Wilson, "Third Harmonic Generation Microscopy," Opt. Express 3, 315-324 (1998). [CrossRef] [PubMed]
- J. A. Squier, M. Müller, "Third - Harmonic Generation Imaging of laser - induced breakdown in glass," Appl. Opt. 38, 5789-5794 (1999). [CrossRef]
- D. Yelin, Y. Silberberg, "Laser scanning third-harmonic-generation microscopy in biology," Opt. Express 5, 169-175 (1999). [CrossRef] [PubMed]
- C.-K. Sun, S.-W. Chu, S.-P. Tai, S. Keller, U.K. Mishra, S.P. DenBaars, "Scanning second-harmonic/third-harmonic generation microscopy of gallium nitride," Appl. Phys. Lett. 77, 2331-2333 (2000). [CrossRef]
- L. Canioni, S. Rivet, L. Sarger, R. Barille, P. Vacher, and P. Voisin, "Imaging of Ca2+ intracellular dynamics with a third-harmonic generation microscope," Opt. Lett. 26, 515-517 (2001). [CrossRef]
- D. Oron, E. Tal and Y. Silberberg, "Depth-resolved multiphoton polarisation microscopy by third-harmonic generation," Opt. Lett. 28, 2315-2317 (2003). [CrossRef] [PubMed]
- D. Yelin, Y. Silberberg, Y. Barad and J. S. Patel, "Depth-resolved imaging of nematic liquid crystals by third-harmonic microscopy," Appl. Phys. Lett. 74, 3107-3109 (1999). [CrossRef]
- D. Débarre, W. Supatto, E. Farge, B. Moulia, M-C. Schanne-Klein, E. Beaurepaire, "Velocimetric third-harmonic generation microscopy: micrometer-scale quantification of morphogenetic movements in unstained embryos," Opt. Lett. 29, 2881-2883 (2004). [CrossRef]
- S.-W. Chu, Szu-Yu Chen, Tsung-Han Tsai, Tzu-Ming Liu, Cheng-Yung Lin, Huai-Jen Tsai, and Chi-Kuang Sun, "In vivo developmental biology study using noninvasive multi-harmonic generation microscopy," Opt. Express 11, 3093-3099 (2003). [CrossRef] [PubMed]
- W. Supatto, D. Débarre, B. Moulia, E. Brouzés, J-L. Martin, E. Farge, E. Beaurepaire, "In vivo modulation of morphogenetic movements in Drosophila embryos with femtosecond laser pulses," PNAS 102, 1047-1052 (2005). [CrossRef] [PubMed]
- S. Hell, G. Reiner, C. Cremer, E.H.K. Stelzer, "Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index," J. Microsc. 169, 391-405 (1993). [CrossRef]
- C. J. R. Sheppard, C.J. Cogswell, "Effects of aberrating layers and tube length on confocal imaging properties," Optik 87, 34-38 (1991).
- R. Barille, L. Canioni, L. Sarger, and G. Rivoire, "Nonlinearity measurements of thin films by third-harmonic-generation microscopy," Phys. Rev. E 66 (2002). [CrossRef]
- V. Shcheslavskiy, G. Petrov, and V. V. Yakovlev, "Nonlinear optical susceptibility measurements of solutions using third-harmonic generation on the interface," Appl. Phys. Lett. 82, 3982-3984 (2003). [CrossRef]
- V. Shcheslavskiy, G. I. Petrov, S. Saltiel, and V. V. Yakovlev, "Quantitative characterization of aqueous solutions probed by the third-harmonic generation microscopy," J. Struct. Biol. 147, 42-49 (2004). [CrossRef] [PubMed]
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