## Third-harmonic generation microscopy with focus-engineered beams: a numerical study

Optics Express, Vol. 16, Issue 19, pp. 14703-14715 (2008)

http://dx.doi.org/10.1364/OE.16.014703

Acrobat PDF (647 KB)

### Abstract

We use a vector field model to analyze third-harmonic generation (THG) from model geometries (interfaces, slabs, periodic structures) illuminated by Hermite-Gaussian (HG) and Laguerre-Gaussian (LG) beams focused by a high NA lens. Calculations show that phase matching conditions are significantly affected by the tailoring of the field distribution near focus. In the case of an interface parallel to the optical axis illuminated by an odd HG mode, the emission patterns and signal level reflect the relative orientation of the interface and the focal field structure. In the case of slabs and periodic structures, the emission patterns reflect the interplay between focal field distribution (amplitude and phase) and sample structure. Forward-to-backward emission ratios using different beam shapes provide sub-wavelength information about sample spatial frequencies.

© 2008 Optical Society of America

## 1. Introduction

1. E. Yew and C. Sheppard, “Second harmonic generation microscopy with tightly focused linearly and radially polarized beams,” Opt. Commun. **275**, 453–457 (2007). [CrossRef]

2. K. Yoshiki, R. Kanamaru, M. Hashimoto, N. Hashimoto, and T. Araki, “Second-harmonic-generation microscope using eight-segment polarization-mode converter to observe three-dimensional molecular orientation,” Opt. Lett. **32**, 1680–1682 (2007). [CrossRef] [PubMed]

3. V. V. Krishnamachari and E. O. Potma, “Focus-engineered coherent anti-Stokes Raman scattering microscopy: a numerical investigation,” J. Opt. Soc. Am. A **24**, 1138–1147 (2007). [CrossRef]

4. V. V. Krishnamachari and E. O. Potma, “Imaging chemical interfaces perpendicular to the optical axis with focus-engineered coherent anti-Stokes Raman scattering microscopy,” Chem. Phys. **341**, 81–88 (2007). [CrossRef]

*χ*

^{(3)}of the sample to provide contrast [5

5. 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]

6. 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]

7. D. Débarre and E. Beaurepaire, “Quantitative characterization of biological liquids for third-harmonic generation microscopy,” Biophys. J. **92**, 603–612 (2007). [CrossRef]

8. D. Yelin and Y. Silberberg, “Laser scanning third-harmonic generation microscopy in biology,” Opt. Express5 (1999). [CrossRef] [PubMed]

9. D. Oron, D. Yelin, E. Tal, S. Raz, R. Fachima, and Y. Silberberg, “Depth-resolved structural imaging by third-harmonic generation microscopy,” J. Struct. Biol. **147**, 3–11 (2004). [CrossRef] [PubMed]

10. 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. **29**, 2881–2883 (2004). [CrossRef]

11. C.-K. Sun, S.-W. Chu, S.-Y. Chen, T.-H. Tsai, T.-M. Liu, C.-Y. Lin, and H.-J. Tsai, “Higher harmonic generation microscopy for developmental biology,” J. Struct. Biol. **147**, 19–30 (2004). [CrossRef] [PubMed]

12. W. Supatto, D. Débarre, B. Moulia, E. Brouzés, J.-L. Martin, E. Farge, and E. Beaurepaire, “In vivo modulation of morphogenetic movements in Drosophila embryos with femtosecond laser pulses,” Proc. Nat. Acad. Sci. USA **102**, 1047–1052 (2005). [CrossRef] [PubMed]

13. D. Débarre, W. Supatto, A.-M. Pena, A. Fabre, T. Tordjmann, L. Combettes, M.-C. Schanne-Klein, and E. Beaurepaire, “Imaging lipid bodies in cells and tissues using third-harmonic generation microscopy,” Nat. Methods **3**, 47–53 (2006). [CrossRef]

5. 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]

*χ*

^{(3)}inhomogeneities, with an efficiency depending on the relative sizes of the inhomogeneity and of the focal volume [14

14. D. Débarre, W. Supatto, and E. Beaurepaire, “Structure sensitivity in third-harmonic generation microscopy,” Opt. Lett. **30**, 2134–2136 (2005). [CrossRef] [PubMed]

*χ*

^{(3)}, THG signal creation for a particular sample geometry is essentially determined by interference effects. Focus engineered THG microscopy is therefore expected to give access to sub-wavelength structural information about the sample. We here present a numerical study of vectorial and phase-matching aspects of THG by tightly focused Gaussian, Hermite-Gaussian (

*HG*), and Laguerre-Gaussian (

*LG*) beams incident on slabs, interfaces, and axially periodic samples. These calculations provide insight on the interplay between field and sample structure in THG microscopy with focused complex beams, and should more generally prove useful for designing coherent nonlinear microscopy (SHG, THG, CARS) experiments with engineered beams. Our strategy for simulations follows the framework described in [15

15. 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]

## 2. Theory and numerical implementation

16. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanetic system.,” Proc. Royal Soc. A **253**, 358–379 (1959). [CrossRef]

15. 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]

## 2.1. Excitation field near focus

16. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanetic system.,” Proc. Royal Soc. A **253**, 358–379 (1959). [CrossRef]

**E**

_{0}(

*θ*,Φ) describes the field phase and intensity distribution at the back aperture of the objective,

*k*=

*k*

*= 2π*

_{ω}*ω*/

*n*

*is the wavenumber,*

_{ω}*f*is the focal length of the objective,

*n*

*is the refractive index at frequency*

_{ω}*ω*, (

*ρ*,

*θ*,

*z*) are cylindrical coordinates near focus, and

*θ*

*=*

_{max}*sin*

^{-1}(

*NA*/

*n*) is the maximum focusing angle of the objective.

**E**

_{0}(

*θ*,Φ) as a polynomial expansion of

*cos*(Φ) and

*sin*(Φ) functions, this 2D integral can be reduced to a 1D integral involving Bessel functions

*J*

*. We can then use the following abbreviations to express the focal fields of the various beam modes considered in this study:*

_{n}*x*- polarized

*HG*

_{00}(Gaussian) mode:

*x*- polarized

*HG*

_{10}mode:

*x*- polarized

*HG*

_{01}mode:

*x*- polarized

*HG*

_{20}mode:

*LG*

^{lin}_{01}(‘donut’) mode:

*LG*

^{az}_{01}mode:

*LG*

^{rad}_{01}mode:

*ik*

_{ω}*z*) in order to highlight the differences between the modes.

## 2.2. Calculation of the induced third-order non-linear polarization

*χ*

^{(3)}

*(*

_{ijkl}**r**), the excitation field induces a polarization density described by:

*E*in cartesian coordinates as:

## 2.3. Propagation of the harmonic field

**r**in the focal region and propagated to a position

**R**in the collection optics aperture can be expressed as [17, 15

15. 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]

*V*spans the excitation volume and

**G**

*is the far field Green’s function:*

_{FF}**R**is the coordinate of a point in the far field (see Fig. 1) and

**I**is the third-order identity tensor.

*E*

*(*

_{FF}**R**)|

^{2}at different positions

**R**. Alternatively, total THG power emitted in the forward (F-THG) or backward (B-THG) directions can be estimated by integrating |

*E*

*(*

_{FF}**R**)|

^{2}over the front aperture of an epicollecting or trans-collecting objective.

## 2.4. Numerical implementation

*λ*/40) grid, and evaluate the excitation field using quadrature algorithms. Unless otherwise stated, we use the following parameters:

*λ*= 1.2

*µm*,

*NA*= 1.4 or 1.2,

*f*

_{0}= 2,

*n*

*= 1.5,*

_{ω}*n*

_{3}

*= 1.52. We note that incorporating positive dispersion in the model is numerically advantageous because smaller focal volumes can be considered, and calculations are generally less noise-sensitive than in the limit case of zero-dispersion. For a given sample/focal field combination, we calculate the projection of the forward- and backward- emission patterns on planes perpendicular to the optical axis located at*

_{ω}*Z*= ±10

*cm*. We choose to present projected far-field patterns rather than angular emission diagrams because they appeared to be more readable in the case of complex emission profiles. For the interface and slab sample geometries, we assume that the focal volume encompasses two homogeneous isotropic media with third-order nonlinear susceptibilities

*χ*

^{(3)}

_{1}= 1 and

*χ*

^{(3)}

_{2}= 0. This choice is motivated by the fact that, for excitation geometries where bulk THG emission is canceled by destructive interference, THG from an interface scales as |

*χ*

^{(3)}

_{1}-

*χ*

^{(3)}

_{2}|

^{2}. For periodic samples, we assume a sine-like variation

*χ*

^{(3)}= 1+

*sin*(2π

*z*/

*δe*)/2 along the optical axis. We then iterate for each beam shape and for various sample positions the calculation of emission patterns, F-THG and B-THG powers. Normalization is done by considering the same total intensity in the focal volume for every mode.

## 3. Results

### 3.1. Vectorial aspect of THG microscopy with tightly focused beams

*a priori*a vector field model because high NA focusing does not preserve linear polarization. Furthermore it is seen from Eq.14 that the induced nonlinear polarization

**P**

^{(3ω)}can

*linearly*depend on a particular field component. For example if

*E*

*is strong at a particular location near focus and spatially overlaps with*

_{z}*E*

*, a cross-term proportional to*

_{x}*E*

_{z}*E*

^{2}

*will significantly contribute to*

_{x}**P**

^{(3ω)}

*. Conversely if*

_{x}*E*

*does not overlap with*

_{z}*E*

*, only the*

_{x}*E*

^{3}

*term will contribute to the TH signal. In particular, in the case of a tightly focused Gaussian beam with initial linear polarization the axial component near focus is important (see Fig. 3):*

_{x}*Max*(

*E*

*) ≈*

_{z}*Max*(

*E*

*)/3 for*

_{x}*NA*= 1.4. However in this case there is little overlap between

*E*

*and*

_{x}*E*

*, so that*

_{z}*E*

*contributes little to THG. Thus, a scalar approximation will usually work well for THG from simple interfaces excited by a focused linearly polarized Gaussian beam. However it will typically not be accurate for higher-order beam shapes or other input polarization patterns. Recalling that the phase distribution (including the Gouy shift) is generally different for the various field components [17, 19*

_{z}19. H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. **5**, 1550 (1966). [CrossRef] [PubMed]

*LG*

^{rad}_{01}mode in Fig. 3) [20

20. K. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express **7**, 77–87 (2000). [CrossRef] [PubMed]

1. E. Yew and C. Sheppard, “Second harmonic generation microscopy with tightly focused linearly and radially polarized beams,” Opt. Commun. **275**, 453–457 (2007). [CrossRef]

2. K. Yoshiki, R. Kanamaru, M. Hashimoto, N. Hashimoto, and T. Araki, “Second-harmonic-generation microscope using eight-segment polarization-mode converter to observe three-dimensional molecular orientation,” Opt. Lett. **32**, 1680–1682 (2007). [CrossRef] [PubMed]

*HG*

_{00}beam with circular polarization is focused on an interface between isotropic media [9

9. D. Oron, D. Yelin, E. Tal, S. Raz, R. Fachima, and Y. Silberberg, “Depth-resolved structural imaging by third-harmonic generation microscopy,” J. Struct. Biol. **147**, 3–11 (2004). [CrossRef] [PubMed]

## 3.2. THG imaging of XY interfaces with HG and LG beams

*HG*

_{00}case is qualitatively well-described by the paraxial approximation (not shown) even at high NA (in contrast with [15

**19**, 1604–1610 (2002). [CrossRef]

*z*) are accurately approximated. Of course, the situation can be quite different with complex field distributions.

## 3.3. THG imaging of XZ/YZ interfaces with focused HG beams

*HG*

_{01}or

*HG*

_{10}beam. When a

*YZ*interface is

*x*-scanned across a focused

*HG*

_{01}beam (Fig. 5(a)), the F-THG response exhibits a double peak reflecting the field distribution in the focal plane (see Fig. 5(c)), contrasting with the case of a focused

*HG*

_{00}or

*HG*

_{10}. Even more striking is the case of a

*XZ*interface being

*y*-scanned across a focused

*HG*

_{10}beam (Fig. 5(b)). In this case the THG response exhibits a triple peak. The central peak results from the presence of a significant axially polarized component in the strongly focused

*x*-polarized

*HG*

_{10}field (

*I*

_{030}term in Eq. 6 which is not present in the

*HG*

_{01}case, see also Fig. 5(c)). This vectorial interpretation is corroborated by the double-peaked shape of the THG

*y*-scan obtained when the axial component is omitted in the simulation (Fig. 5(b)).

*χ*

^{(3)}

_{1}-

*χ*

^{(3)}

_{2}|, maximum emission is obtained when one of the two main excitation peaks is incident on the interface.

## 3.4. Focus-engineered THG from slabs

*HG*

_{00}excitation, the Gouy shift defines a signal coherent construction length of ≈ 0.7

*λ*for forward emission (F-THG) [15

**19**, 1604–1610 (2002). [CrossRef]

*k*defines a construction length of

*π*/Δ

*k*≈

*λ*/12

*n*

*for backward emission (B-THG) [21]. The coherence length for forward emission has a major influence on imaging properties, since it acts as a spatial bandpass filter that highlights objects of a given size in F-THG images [14]. Elaborating on this idea, we point out that when focusing non-Gaussian beams such as higher-order*

_{ω}*HG*and

*LG*modes, focal field components exhibit altered phase distributions [19

19. H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. **5**, 1550 (1966). [CrossRef] [PubMed]

22. E. Y. S. Yew and C. J. R. Sheppard, “Fractional Gouy phase,” Opt. Lett. **33**, 1363–1365 (2008). [CrossRef] [PubMed]

*HG*

_{00}and

*HG*

_{20}excitations. Focused

*HG*

_{20}resembles

*HG*

_{00}because it exhibits a single peak along the optical axis, albeit with a slower phase variation and a broader intensity distribution than focused

*HG*

_{00}. Accordingly, the axial coherence length is increased for F-THG and reduced for B-THG. Reduced B-THG coherence length manifests itself through the reduced oscillation period as a function of slab thickness (Fig. 7(c)). We point out that moving from

*HG*

_{00}to

*HG*

_{20}excitation here produces an effect comparable to changing the excitation NA from 1.4 to ≈ 1.2 (see Fig. 7(d)) and comes at the cost of reduced signal level by a factor ≈ 2. Fig. 7(a) also illustrates the consequence of including/excluding dispersion, for the

*HG*

_{20}case (filled and empty blue triangles). For all the cases studied here, we essentially find that dispersion reduces TH efficiency for large objects without affecting the relative behaviors obtained with different beam shapes.

## 3.5. Focus-engineered THG from axially periodic structures

21. D. Débarre, N. Olivier, and E. Beaurepaire, “Signal epidetection in third-harmonic generation microscopy of turbid media,” Opt. Express **15**, 8913–8924 (2007). [CrossRef] [PubMed]

*k*limits signal creation to a small region (≈ 65

*nm*for

*λ*= 1200

*nm*and

*n*

*= 1.5) around an heterogeneity (see fig 7). However the situation can be quite different in the case of a structured sample: if the sample exhibits appropriate axial periodicity, the density distribution of emitters can provide an additional momentum that puts the emitted waves in phase in a particular direction [21*

_{ω}21. D. Débarre, N. Olivier, and E. Beaurepaire, “Signal epidetection in third-harmonic generation microscopy of turbid media,” Opt. Express **15**, 8913–8924 (2007). [CrossRef] [PubMed]

23. J. Mertz and L. Moreaux, “Second-harmonic generation by focused excitation of inhomogeneously distributed scatterers,” Opt. Commun. **196**, 325–330 (2001). [CrossRef]

*HG*

_{00}excitation focused at 1.4 NA, an axial sine-like

*χ*

^{(3)}modulation with spatial period

*δe*≈ 2

*π*/Δ

*k*≈

*λ*/6

*n*

*= 135*

_{ω}*nm*is expected to produce efficient B-THG emission, and a similar distribution with

*δe*≈ 2

*µm*is expected to produce efficient F-THG emission [21

21. D. Débarre, N. Olivier, and E. Beaurepaire, “Signal epidetection in third-harmonic generation microscopy of turbid media,” Opt. Express **15**, 8913–8924 (2007). [CrossRef] [PubMed]

## 4. Conclusion

24. C. J. R. Sheppard, “High-aperture beams,” J. Opt. Soc. Am. A **18**, 1579–1587 (2001). [CrossRef]

25. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. **179**, 1–7 (2000). [CrossRef]

22. E. Y. S. Yew and C. J. R. Sheppard, “Fractional Gouy phase,” Opt. Lett. **33**, 1363–1365 (2008). [CrossRef] [PubMed]

2. K. Yoshiki, R. Kanamaru, M. Hashimoto, N. Hashimoto, and T. Araki, “Second-harmonic-generation microscope using eight-segment polarization-mode converter to observe three-dimensional molecular orientation,” Opt. Lett. **32**, 1680–1682 (2007). [CrossRef] [PubMed]

26. S. Carrasco, B. E. A. Saleh, M. C. Teich, and J. T. Fourkas, “Second- and third-harmonic generation with vector Gaussian beams,” J. Opt. Soc. Am. B **23**, 2134–2141 (2006). [CrossRef]

14. D. Débarre, W. Supatto, and E. Beaurepaire, “Structure sensitivity in third-harmonic generation microscopy,” Opt. Lett. **30**, 2134–2136 (2005). [CrossRef] [PubMed]

*nm*range are reflected in the emission patterns and can be probed using simple ratiometric measurements. A perspective is to design pupil functions producing a targeted field distribution [27

27. S. S. Sherif, M. R. Foreman, and P. Török, “Eigenfunction expansion of the electric fields in the focal region of a high numerical aperture focusing system,” Opt. Express **16**, 3397–3407 (2008). [CrossRef] [PubMed]

28. M. R. Foreman, S. S. Sherif, P. R. T. Munro, and P. Török, “Inversion of the Debye-Wolf diffraction integral using an eigenfunction representation of the electric fields in the focal region,” Opt. Express **16**, 4901–4917 (2008). [CrossRef] [PubMed]

3. V. V. Krishnamachari and E. O. Potma, “Focus-engineered coherent anti-Stokes Raman scattering microscopy: a numerical investigation,” J. Opt. Soc. Am. A **24**, 1138–1147 (2007). [CrossRef]

## Acknowledgments

## References and links

1. | E. Yew and C. Sheppard, “Second harmonic generation microscopy with tightly focused linearly and radially polarized beams,” Opt. Commun. |

2. | K. Yoshiki, R. Kanamaru, M. Hashimoto, N. Hashimoto, and T. Araki, “Second-harmonic-generation microscope using eight-segment polarization-mode converter to observe three-dimensional molecular orientation,” Opt. Lett. |

3. | V. V. Krishnamachari and E. O. Potma, “Focus-engineered coherent anti-Stokes Raman scattering microscopy: a numerical investigation,” J. Opt. Soc. Am. A |

4. | V. V. Krishnamachari and E. O. Potma, “Imaging chemical interfaces perpendicular to the optical axis with focus-engineered coherent anti-Stokes Raman scattering microscopy,” Chem. Phys. |

5. | Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, “Nonlinear scanning laser microscopy by third harmonic generation,” Appl. Phys. Lett. |

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

7. | D. Débarre and E. Beaurepaire, “Quantitative characterization of biological liquids for third-harmonic generation microscopy,” Biophys. J. |

8. | D. Yelin and Y. Silberberg, “Laser scanning third-harmonic generation microscopy in biology,” Opt. Express5 (1999). [CrossRef] [PubMed] |

9. | D. Oron, D. Yelin, E. Tal, S. Raz, R. Fachima, and Y. Silberberg, “Depth-resolved structural imaging by third-harmonic generation microscopy,” J. Struct. Biol. |

10. | 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. |

11. | C.-K. Sun, S.-W. Chu, S.-Y. Chen, T.-H. Tsai, T.-M. Liu, C.-Y. Lin, and H.-J. Tsai, “Higher harmonic generation microscopy for developmental biology,” J. Struct. Biol. |

12. | W. Supatto, D. Débarre, B. Moulia, E. Brouzés, J.-L. Martin, E. Farge, and E. Beaurepaire, “In vivo modulation of morphogenetic movements in Drosophila embryos with femtosecond laser pulses,” Proc. Nat. Acad. Sci. USA |

13. | D. Débarre, W. Supatto, A.-M. Pena, A. Fabre, T. Tordjmann, L. Combettes, M.-C. Schanne-Klein, and E. Beaurepaire, “Imaging lipid bodies in cells and tissues using third-harmonic generation microscopy,” Nat. Methods |

14. | D. Débarre, W. Supatto, and E. Beaurepaire, “Structure sensitivity in third-harmonic generation microscopy,” Opt. Lett. |

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

16. | B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanetic system.,” Proc. Royal Soc. A |

17. | L. Novotny and B. Hecht, Principles of nano-optics (Cambridge Univ Press, 2006). |

18. | R. W. Boyd |

19. | H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. |

20. | K. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express |

21. | D. Débarre, N. Olivier, and E. Beaurepaire, “Signal epidetection in third-harmonic generation microscopy of turbid media,” Opt. Express |

22. | E. Y. S. Yew and C. J. R. Sheppard, “Fractional Gouy phase,” Opt. Lett. |

23. | J. Mertz and L. Moreaux, “Second-harmonic generation by focused excitation of inhomogeneously distributed scatterers,” Opt. Commun. |

24. | C. J. R. Sheppard, “High-aperture beams,” J. Opt. Soc. Am. A |

25. | S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. |

26. | S. Carrasco, B. E. A. Saleh, M. C. Teich, and J. T. Fourkas, “Second- and third-harmonic generation with vector Gaussian beams,” J. Opt. Soc. Am. B |

27. | S. S. Sherif, M. R. Foreman, and P. Török, “Eigenfunction expansion of the electric fields in the focal region of a high numerical aperture focusing system,” Opt. Express |

28. | M. R. Foreman, S. S. Sherif, P. R. T. Munro, and P. Török, “Inversion of the Debye-Wolf diffraction integral using an eigenfunction representation of the electric fields in the focal region,” Opt. Express |

**OCIS Codes**

(170.3880) Medical optics and biotechnology : Medical and biological imaging

(180.6900) Microscopy : Three-dimensional microscopy

(190.4160) Nonlinear optics : Multiharmonic generation

**ToC Category:**

Microscopy

**History**

Original Manuscript: July 8, 2008

Revised Manuscript: August 21, 2008

Manuscript Accepted: August 21, 2008

Published: September 3, 2008

**Virtual Issues**

Vol. 3, Iss. 11 *Virtual Journal for Biomedical Optics*

**Citation**

Nicolas Olivier and Emmanuel Beaurepaire, "Third-harmonic generation microscopy with focus-engineered beams: a numerical study," Opt. Express **16**, 14703-14715 (2008)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-16-19-14703

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### References

- E. Yew and C. Sheppard, "Second harmonic generation microscopy with tightly focused linearly and radially polarized beams," Opt. Commun. 275, 453-457 (2007). [CrossRef]
- K. Yoshiki, R. Kanamaru, M. Hashimoto, N. Hashimoto, and T. Araki, "Second-harmonic-generation microscope using eight-segment polarization-mode converter to observe three-dimensional molecular orientation," Opt. Lett. 32, 1680-1682 (2007). [CrossRef] [PubMed]
- V. V. Krishnamachari and E. O. Potma, "Focus-engineered coherent anti-Stokes Raman scattering microscopy: a numerical investigation," J. Opt. Soc. Am. A 24, 1138-1147 (2007). [CrossRef]
- V. V. Krishnamachari and E. O. Potma, "Imaging chemical interfaces perpendicular to the optical axis with focus-engineered coherent anti-Stokes Raman scattering microscopy," Chem. Phys. 341, 81-88 (2007). [CrossRef]
- 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]
- M. M¨uller, J. Squier, K. R. Wilson, and G. J. Brakenhoff, "3D-microscopy of transparent objects using thirdharmonic generation," J. Microsc. 191, 266-274 (1998). [CrossRef] [PubMed]
- D. Débarre and E. Beaurepaire, "Quantitative characterization of biological liquids for third-harmonic generation microscopy," Biophys. J. 92, 603-612 (2007). [CrossRef]
- D. Yelin and Y. Silberberg, "Laser scanning third-harmonic generation microscopy in biology," Opt. Express 5 (1999). [CrossRef] [PubMed]
- D. Oron, D. Yelin, E. Tal, S. Raz, R. Fachima, and Y. Silberberg, "Depth-resolved structural imaging by thirdharmonic generation microscopy," J. Struct. Biol. 147, 3-11 (2004). [CrossRef] [PubMed]
- D. Débarre, W. Supatto, E. Farge, B. Moulia, M.-C. Schanne-Klein, and E. Beaurepaire, "Velocimetric thirdharmonic generation microscopy: micrometer-scale quantification of morphogenetic movements in unstained embryos," Opt. Lett. 29, 2881-2883 (2004). [CrossRef]
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