Abstract

We report a theoretical study of third-harmonic generation (THG) microscopy by use of a Green’s function formulation. The third-harmonic signal under a tight-focusing condition is calculated for samples with various shapes and sizes. Our results show that THG signals can be efficiently generated at a sizable interface perpendicular or parallel to the optical axis or from a small object with a size comparable to the width of the axial excitation intensity profile. The signal-generation mechanism of THG microscopy is explained by a modified phase-matching condition, |k3-3(k1+Δkg)|lπ, where Δkg is the wave vector mismatch induced by the Gouy phase shift of the focused excitation field. The relation of the THG power and radiation pattern to the orientation of an interface is investigated. A comparison between signal generation in THG microscopy and that in coherent anti-Stokes Raman scattering microscopy is given.

© 2002 Optical Society of America

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  1. R. Gauderon, P. B. Lukins, and C. J. R. Sheppard, “Three-dimensional second-harmonic generation imaging with femtosecond laser pulses,” Opt. Lett. 23, 1209–1211 (1998).
    [CrossRef]
  2. P. J. Campagnola, M.-D. Wei, A. Lewis, and L. M. Loew, “High-resolution nonlinear optical imaging of live cells by second harmonic generation,” Biophys. J. 77, 3341–3349 (1999).
    [CrossRef] [PubMed]
  3. L. Moreaux, O. Sandre, and J. Mertz, “Membrane imaging by second-harmonic generation microscopy,” J. Opt. Soc. Am. B 17, 1685–1694 (2000).
    [CrossRef]
  4. A. Zumbusch, G. R. Holtom, and X. S. Xie, “Three-dimensional vibrational imaging by coherent anti-Stokes Raman scattering,” Phys. Rev. Lett. 82, 4142–4145 (1999).
    [CrossRef]
  5. A. Volkmer, J.-X. Cheng, and X. S. Xie, “Vibrational imaging with high sensitivity via epidetected coherent anti-Stokes Raman scattering microscopy,” Phys. Rev. Lett. 87, 023901 (2001).
    [CrossRef]
  6. J. X. Cheng, L. D. Book, and X. S. Xie, “Polarization coherent anti-Stokes Raman scattering microscopy,” Opt. Lett. 26, 1341–1343 (2001).
    [CrossRef]
  7. 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]
  8. M. Müller, J. Squier, K. R. Wilson, and G. J. Brakenhoff, “3D microscopy of transparent objects using third-harmonic generation,” J. Microsc. (Oxford) 191, 266–274 (1998).
    [CrossRef]
  9. J. A. Squier and M. Muller, “Third-harmonic generation imaging of laser-induced breakdown in glass,” Appl. Opt. 38, 5789–5794 (1999).
    [CrossRef]
  10. J. A. Squier, M. Muller, G. J. Brakenhoff, and K. R. Wilson, “Third harmonic generation microscopy,” Opt. Express 3, 315–324 (1998), http://www.opticsexpress.org.
    [CrossRef] [PubMed]
  11. D. Yelin and Y. Silberberg, “Laser scanning third-harmonic-generation microscopy in biology,” Opt. Express 5, 169–175 (1999), http://www.opticsexpress.org.
    [CrossRef] [PubMed]
  12. L. Canioni, S. Rivet, L. Sarger, R. Barille, P. Vacher, and P. Voisin, “Imaging Ca2+ intracellular dynamics with a third-harmonic generation microscope,” Opt. Lett. 26, 515–517 (2001).
    [CrossRef]
  13. D. A. Kleinman, A. Ashkin, and G. D. Boyd, “Second-harmonic generation of light by focused laser beams,” Phys. Rev. 145, 338–379 (1966).
    [CrossRef]
  14. 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]
  15. G. C. Bjorklund, “Effects of focusing on third-order nonlinear processes in isotropic media,” IEEE J. Quantum Electron. QE-11, 287–296 (1975).
    [CrossRef]
  16. R. B. Miles and S. E. Harris, “Optical third-harmonic generation in alkali metal vapors,” IEEE J. Quantum Electron. QE-9, 470–484 (1973).
    [CrossRef]
  17. R. W. Boyd, Nonlinear Optics (Academic, Boston, Mass., 1992).
  18. A. E. Siegman, Lasers (University Science, Mill Valley, Calif. 1986).
  19. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, 1999).
  20. The effect of distortion is quite small when the laser beam is focused on small features. The discontinuity of χ(1) (refractive index) at a sizable interface provides an additional mechanism for THG signal generation.
  21. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358–379 (1959).
    [CrossRef]
  22. L. Novotny, Lecture Notes on Nano-Optics (University of Rochester, Rochester, N.Y., 2000).
  23. For a fundamental Gaussian beam, max[Ey2]/max[Ex2]= 0.003 and max[Ez2]/max[Ex2]=0.12 under the tight-focusing (NA=1.4) condition. As THG is a third-order nonlinear process, the contributions from the y and z components are negligible. For the same reason, the azimuth-dependent part of the x-polarized component can be neglected.
  24. W. C. Chew, Waves and Fields in Inhomogeneous Media, 2nd ed. (Institute of Electrical and Electronics Engineers, New York, 1995).
  25. L. Novotny, “Allowed and forbidden light in near-field optics. II. Interacting dipolar particles,” J. Opt. Soc. Am. A 14, 105–113 (1997).
    [CrossRef]
  26. The effect of index dispersion on the phase mismatch can be neglected because of the small excitation volume under the tight-focusing condition. For example, the refractive index of water is 1.339 at 0.4 μm and 1.324 at 1.2 μm. The corresponding coherence length, π/|k3−3k1|, is calculated to be 13.3 μm, which is much larger than the axial length of the focal volume under the tight-focusing condition.
  27. J. X. Cheng, A. Volkmer, L. D. Book, and X. S. Xie, “An epi-detected coherent anti-Stokes Raman scattering (E-CARS) microscope with high spectral resolution and high sensitivity,” J. Phys. Chem. B 105, 1277–1280 (2001).
    [CrossRef]

2001

A. Volkmer, J.-X. Cheng, and X. S. Xie, “Vibrational imaging with high sensitivity via epidetected coherent anti-Stokes Raman scattering microscopy,” Phys. Rev. Lett. 87, 023901 (2001).
[CrossRef]

J. X. Cheng, L. D. Book, and X. S. Xie, “Polarization coherent anti-Stokes Raman scattering microscopy,” Opt. Lett. 26, 1341–1343 (2001).
[CrossRef]

L. Canioni, S. Rivet, L. Sarger, R. Barille, P. Vacher, and P. Voisin, “Imaging Ca2+ intracellular dynamics with a third-harmonic generation microscope,” Opt. Lett. 26, 515–517 (2001).
[CrossRef]

J. X. Cheng, A. Volkmer, L. D. Book, and X. S. Xie, “An epi-detected coherent anti-Stokes Raman scattering (E-CARS) microscope with high spectral resolution and high sensitivity,” J. Phys. Chem. B 105, 1277–1280 (2001).
[CrossRef]

2000

1999

A. Zumbusch, G. R. Holtom, and X. S. Xie, “Three-dimensional vibrational imaging by coherent anti-Stokes Raman scattering,” Phys. Rev. Lett. 82, 4142–4145 (1999).
[CrossRef]

P. J. Campagnola, M.-D. Wei, A. Lewis, and L. M. Loew, “High-resolution nonlinear optical imaging of live cells by second harmonic generation,” Biophys. J. 77, 3341–3349 (1999).
[CrossRef] [PubMed]

J. A. Squier and M. Muller, “Third-harmonic generation imaging of laser-induced breakdown in glass,” Appl. Opt. 38, 5789–5794 (1999).
[CrossRef]

D. Yelin and Y. Silberberg, “Laser scanning third-harmonic-generation microscopy in biology,” Opt. Express 5, 169–175 (1999), http://www.opticsexpress.org.
[CrossRef] [PubMed]

1998

1997

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]

L. Novotny, “Allowed and forbidden light in near-field optics. II. Interacting dipolar particles,” J. Opt. Soc. Am. A 14, 105–113 (1997).
[CrossRef]

1975

G. C. Bjorklund, “Effects of focusing on third-order nonlinear processes in isotropic media,” IEEE J. Quantum Electron. QE-11, 287–296 (1975).
[CrossRef]

1973

R. B. Miles and S. E. Harris, “Optical third-harmonic generation in alkali metal vapors,” IEEE J. Quantum Electron. QE-9, 470–484 (1973).
[CrossRef]

1969

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]

1966

D. A. Kleinman, A. Ashkin, and G. D. Boyd, “Second-harmonic generation of light by focused laser beams,” Phys. Rev. 145, 338–379 (1966).
[CrossRef]

1959

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

Ashkin, A.

D. A. Kleinman, A. Ashkin, and G. D. Boyd, “Second-harmonic generation of light by focused laser beams,” Phys. Rev. 145, 338–379 (1966).
[CrossRef]

Barad, Y.

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]

Barille, R.

Bjorklund, G. C.

G. C. Bjorklund, “Effects of focusing on third-order nonlinear processes in isotropic media,” IEEE J. Quantum Electron. QE-11, 287–296 (1975).
[CrossRef]

Book, L. D.

J. X. Cheng, A. Volkmer, L. D. Book, and X. S. Xie, “An epi-detected coherent anti-Stokes Raman scattering (E-CARS) microscope with high spectral resolution and high sensitivity,” J. Phys. Chem. B 105, 1277–1280 (2001).
[CrossRef]

J. X. Cheng, L. D. Book, and X. S. Xie, “Polarization coherent anti-Stokes Raman scattering microscopy,” Opt. Lett. 26, 1341–1343 (2001).
[CrossRef]

Boyd, G. D.

D. A. Kleinman, A. Ashkin, and G. D. Boyd, “Second-harmonic generation of light by focused laser beams,” Phys. Rev. 145, 338–379 (1966).
[CrossRef]

Brakenhoff, G. J.

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

J. A. Squier, M. Muller, G. J. Brakenhoff, and K. R. Wilson, “Third harmonic generation microscopy,” Opt. Express 3, 315–324 (1998), http://www.opticsexpress.org.
[CrossRef] [PubMed]

Campagnola, P. J.

P. J. Campagnola, M.-D. Wei, A. Lewis, and L. M. Loew, “High-resolution nonlinear optical imaging of live cells by second harmonic generation,” Biophys. J. 77, 3341–3349 (1999).
[CrossRef] [PubMed]

Canioni, L.

Cheng, J. X.

J. X. Cheng, A. Volkmer, L. D. Book, and X. S. Xie, “An epi-detected coherent anti-Stokes Raman scattering (E-CARS) microscope with high spectral resolution and high sensitivity,” J. Phys. Chem. B 105, 1277–1280 (2001).
[CrossRef]

J. X. Cheng, L. D. Book, and X. S. Xie, “Polarization coherent anti-Stokes Raman scattering microscopy,” Opt. Lett. 26, 1341–1343 (2001).
[CrossRef]

Cheng, J.-X.

A. Volkmer, J.-X. Cheng, and X. S. Xie, “Vibrational imaging with high sensitivity via epidetected coherent anti-Stokes Raman scattering microscopy,” Phys. Rev. Lett. 87, 023901 (2001).
[CrossRef]

Eisenberg, H.

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]

Gauderon, R.

Harris, S. E.

R. B. Miles and S. E. Harris, “Optical third-harmonic generation in alkali metal vapors,” IEEE J. Quantum Electron. QE-9, 470–484 (1973).
[CrossRef]

Holtom, G. R.

A. Zumbusch, G. R. Holtom, and X. S. Xie, “Three-dimensional vibrational imaging by coherent anti-Stokes Raman scattering,” Phys. Rev. Lett. 82, 4142–4145 (1999).
[CrossRef]

Horowitz, M.

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]

Kleinman, D. A.

D. A. Kleinman, A. Ashkin, and G. D. Boyd, “Second-harmonic generation of light by focused laser beams,” Phys. Rev. 145, 338–379 (1966).
[CrossRef]

Lewis, A.

P. J. Campagnola, M.-D. Wei, A. Lewis, and L. M. Loew, “High-resolution nonlinear optical imaging of live cells by second harmonic generation,” Biophys. J. 77, 3341–3349 (1999).
[CrossRef] [PubMed]

Loew, L. M.

P. J. Campagnola, M.-D. Wei, A. Lewis, and L. M. Loew, “High-resolution nonlinear optical imaging of live cells by second harmonic generation,” Biophys. J. 77, 3341–3349 (1999).
[CrossRef] [PubMed]

Lukins, P. B.

Mertz, J.

Miles, R. B.

R. B. Miles and S. E. Harris, “Optical third-harmonic generation in alkali metal vapors,” IEEE J. Quantum Electron. QE-9, 470–484 (1973).
[CrossRef]

Moreaux, L.

Muller, M.

Müller, M.

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

New, G. H. C.

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]

Novotny, L.

Richards, B.

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

Rivet, S.

Sandre, O.

Sarger, L.

Sheppard, C. J. R.

Silberberg, Y.

D. Yelin and Y. Silberberg, “Laser scanning third-harmonic-generation microscopy in biology,” Opt. Express 5, 169–175 (1999), http://www.opticsexpress.org.
[CrossRef] [PubMed]

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]

Squier, J.

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

Squier, J. A.

Vacher, P.

Voisin, P.

Volkmer, A.

J. X. Cheng, A. Volkmer, L. D. Book, and X. S. Xie, “An epi-detected coherent anti-Stokes Raman scattering (E-CARS) microscope with high spectral resolution and high sensitivity,” J. Phys. Chem. B 105, 1277–1280 (2001).
[CrossRef]

A. Volkmer, J.-X. Cheng, and X. S. Xie, “Vibrational imaging with high sensitivity via epidetected coherent anti-Stokes Raman scattering microscopy,” Phys. Rev. Lett. 87, 023901 (2001).
[CrossRef]

Ward, J. F.

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]

Wei, M.-D.

P. J. Campagnola, M.-D. Wei, A. Lewis, and L. M. Loew, “High-resolution nonlinear optical imaging of live cells by second harmonic generation,” Biophys. J. 77, 3341–3349 (1999).
[CrossRef] [PubMed]

Wilson, K. R.

J. A. Squier, M. Muller, G. J. Brakenhoff, and K. R. Wilson, “Third harmonic generation microscopy,” Opt. Express 3, 315–324 (1998), http://www.opticsexpress.org.
[CrossRef] [PubMed]

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

Wolf, E.

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

Xie, X. S.

J. X. Cheng, A. Volkmer, L. D. Book, and X. S. Xie, “An epi-detected coherent anti-Stokes Raman scattering (E-CARS) microscope with high spectral resolution and high sensitivity,” J. Phys. Chem. B 105, 1277–1280 (2001).
[CrossRef]

J. X. Cheng, L. D. Book, and X. S. Xie, “Polarization coherent anti-Stokes Raman scattering microscopy,” Opt. Lett. 26, 1341–1343 (2001).
[CrossRef]

A. Volkmer, J.-X. Cheng, and X. S. Xie, “Vibrational imaging with high sensitivity via epidetected coherent anti-Stokes Raman scattering microscopy,” Phys. Rev. Lett. 87, 023901 (2001).
[CrossRef]

A. Zumbusch, G. R. Holtom, and X. S. Xie, “Three-dimensional vibrational imaging by coherent anti-Stokes Raman scattering,” Phys. Rev. Lett. 82, 4142–4145 (1999).
[CrossRef]

Yelin, D.

Zumbusch, A.

A. Zumbusch, G. R. Holtom, and X. S. Xie, “Three-dimensional vibrational imaging by coherent anti-Stokes Raman scattering,” Phys. Rev. Lett. 82, 4142–4145 (1999).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

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]

Biophys. J.

P. J. Campagnola, M.-D. Wei, A. Lewis, and L. M. Loew, “High-resolution nonlinear optical imaging of live cells by second harmonic generation,” Biophys. J. 77, 3341–3349 (1999).
[CrossRef] [PubMed]

IEEE J. Quantum Electron.

G. C. Bjorklund, “Effects of focusing on third-order nonlinear processes in isotropic media,” IEEE J. Quantum Electron. QE-11, 287–296 (1975).
[CrossRef]

R. B. Miles and S. E. Harris, “Optical third-harmonic generation in alkali metal vapors,” IEEE J. Quantum Electron. QE-9, 470–484 (1973).
[CrossRef]

J. Microsc. (Oxford)

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

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

J. Phys. Chem. B

J. X. Cheng, A. Volkmer, L. D. Book, and X. S. Xie, “An epi-detected coherent anti-Stokes Raman scattering (E-CARS) microscope with high spectral resolution and high sensitivity,” J. Phys. Chem. B 105, 1277–1280 (2001).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev.

D. A. Kleinman, A. Ashkin, and G. D. Boyd, “Second-harmonic generation of light by focused laser beams,” Phys. Rev. 145, 338–379 (1966).
[CrossRef]

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]

Phys. Rev. Lett.

A. Zumbusch, G. R. Holtom, and X. S. Xie, “Three-dimensional vibrational imaging by coherent anti-Stokes Raman scattering,” Phys. Rev. Lett. 82, 4142–4145 (1999).
[CrossRef]

A. Volkmer, J.-X. Cheng, and X. S. Xie, “Vibrational imaging with high sensitivity via epidetected coherent anti-Stokes Raman scattering microscopy,” Phys. Rev. Lett. 87, 023901 (2001).
[CrossRef]

Proc. R. Soc. London, Ser. A

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

Other

L. Novotny, Lecture Notes on Nano-Optics (University of Rochester, Rochester, N.Y., 2000).

For a fundamental Gaussian beam, max[Ey2]/max[Ex2]= 0.003 and max[Ez2]/max[Ex2]=0.12 under the tight-focusing (NA=1.4) condition. As THG is a third-order nonlinear process, the contributions from the y and z components are negligible. For the same reason, the azimuth-dependent part of the x-polarized component can be neglected.

W. C. Chew, Waves and Fields in Inhomogeneous Media, 2nd ed. (Institute of Electrical and Electronics Engineers, New York, 1995).

The effect of index dispersion on the phase mismatch can be neglected because of the small excitation volume under the tight-focusing condition. For example, the refractive index of water is 1.339 at 0.4 μm and 1.324 at 1.2 μm. The corresponding coherence length, π/|k3−3k1|, is calculated to be 13.3 μm, which is much larger than the axial length of the focal volume under the tight-focusing condition.

R. W. Boyd, Nonlinear Optics (Academic, Boston, Mass., 1992).

A. E. Siegman, Lasers (University Science, Mill Valley, Calif. 1986).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, 1999).

The effect of distortion is quite small when the laser beam is focused on small features. The discontinuity of χ(1) (refractive index) at a sizable interface provides an additional mechanism for THG signal generation.

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Figures (8)

Fig. 1
Fig. 1

Illustrations of THG by a focused laser beam with definitions of the parameters of the excitation and the THG fields.

Fig. 2
Fig. 2

Integrated THG signal as a function of the diameter (D) of a spherical sample centered at the focus of a tightly focused (NA=1.4) excitation beam.

Fig. 3
Fig. 3

(a) Far-field radiation pattern of THG from a D=1.0 λ1 sphere centered at the focus. (b) Far-field radiation pattern of THG from a D=1.0 λ1 sphere centered at (x=0.5 λ1, y=0, z=0). The x, y, and z axes have the same scale, with arbitrary units.

Fig. 4
Fig. 4

Integrated THG signal from hemispherical samples as a function of diameter (D). The samples are centered at the focus and symmetric with respect to the z (solid curve), x (dotted curve), and y (dashed curve) axes. The polarization of the excitation field is along the x axis.

Fig. 5
Fig. 5

(a) Far-field radiation pattern of THG from a D=6.0 λ1 hemisphere centered at the focus and perpendicular to the optical axis. (b) Far-field radiation pattern of THG from a D=6.0 λ1 hemisphere centered at the focus and perpendicular to the x axis. The x, y, and z axes have the same scale, with arbitrary units. The polarization of the excitation field is along the x axis.

Fig. 6
Fig. 6

Axial phase shift and intensity distribution in the focal region of a Gaussian beam focused by an objective lens with an NA of 1.4.

Fig. 7
Fig. 7

(a), (b) Calculated lateral (x) and axial (z) THG intensity profiles of a D=0.5 λ1 spherical sample embedded in a nonlinear medium. (c), (d) Calculated lateral (x) and axial (z) THG intensity profiles of a D=5.0 λ1 spherical sample embedded in a nonlinear medium.

Fig. 8
Fig. 8

THG signal as a function of diameter (D) for a spherical sample centered at the focus, generated with a tightly focused (NA=1.4) excitation field calculated by Eq. (2) (solid curve) and by Eq. (A1) (dashed curve).

Equations (11)

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Einc(α)=E0 exp(-f2 sin2 α/w2),
Efoc(ρ, z)=ik1f exp(-ik1f)/20αmaxEinc sin αcos α×(1+cos α)J0(k1ρ sin α)×exp(k1z cos α)dα,
××E(3ω)(r)-k32E(3ω)(r)=4πω32c2P(3ω)(r).
P(3ω)(r)=χ(3)(r)Efoc(r)Efoc(r)Efoc(r),
E(3ω)(R)=-4πω32c2V dVI+k32G(R-r)·P(3ω)(r),
E(3ω)(R)=-ω32c2exp(ik3|R|)|R|V dV exp-ik3R·r|R|×000cos Θ cos Φcos Θ sin Φ-sin Θ-sin Φcos Φ0×Px(3ω)(r)Py(3ω)(r)Px(3ω)(r)ıˆRıˆΘıˆΦ.
P(3ω)=n3c8π0αmax dΘ 02π dΦ|E(3ω)(R)|2R2 sin Θ.
ETHG(R)=V ETHG[R,r,χobj(3)]dV+V ETHG[R,r,χsol(3)]dV,
ETHG(R)=V ETHG[R,r,χobj(3)]dV-V ETHG[R,r,χsol(3)]dV=V ETHG[R,r,χobj(3)-χsol(3)]dV.
|(k3-3k1)-3Δkg|lπ,
Efoc(ρ, z)=E0w0w(z) exp-ρ2w(z)2exp{i[k1z-η(z)+k1ρ2/2R(z)]},

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