Abstract

We present a fast and accurate method to calculate the vector-field distribution of a focused Gaussian beam. This method is applied to calculate the second harmonic that is generated by such a beam from a sample in the undepleted pump approximation. These calculations can be used to model second-harmonic imaging in an optical microscope with a wide aperture.

© 2004 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. Hellwarth and P. Christiansen, “Nonlinear optical microscope using second harmonic generation,” Appl. Opt. 14, 247–248 (1975).
    [CrossRef] [PubMed]
  2. J. N. Gannaway and C. J. R. Sheppard, “Second-harmonic imaging in the scanning optical microscope,” Opt. Quantum Electron. 10, 435 (1978).
    [CrossRef]
  3. C. J. R. Sheppard and R. Kompfner, “Resonant scanning optical microscope,” Appl. Opt. 17, 2879–2882 (1978).
    [CrossRef] [PubMed]
  4. I. Freund, M. Deutsch, and A. Sprecher, “Connective tissue polarity, optical second harmonic microscopy, crossed-beam summation and small-angle scattering in rat-tail tendon,” Biophys. J. 50, 693–712 (1986).
    [CrossRef] [PubMed]
  5. Y. Guo, P. P. Ho, A. Tirksliunas, F. Liu, and R. R. Alfano, “Optical harmonic generation from animal tissues by the use of picosecond and femtosecond laser pulses,” Appl. Opt. 35, 6810–6813 (1996).
    [CrossRef] [PubMed]
  6. G. Cox, E. Kable, A. Jones, I. Fraser, F. Manconi, and M. D. Gorrell, “3-dimensional imaging of collagen using second harmonic generation,” J. Struct. Biol. 141, 53–62 (2003).
    [CrossRef] [PubMed]
  7. L. Moreaux, O. Sandre, and J. Mertz, “Membrane imaging by second-harmonic generation microscopy,” J. Opt. Soc. Am. B 17, 1685–1694 (2000).
    [CrossRef]
  8. J. Mertz and L. Moreaux, “Multi-harmonic light microscopy: theory and applications to membrane imaging,” in Multiphoton Microscopy in the Biomedical Sciences, A. M. Periasamy and P. T. C. So, eds., Proc. SPIE 4262, 9–17 (2001).
    [CrossRef]
  9. G. Peleg, A. Lewis, M. Linial, and L. Loew, “Nonlinear optical measurement of membrane potential around single molecules at selected cellular sites,” Proc. Natl. Acad. Sci. U.S.A. 96, 6700–6704 (1999).
    [CrossRef] [PubMed]
  10. P. J. Campagnola, M. 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]
  11. R. Gauderon, P. B. Lukins, and C. J. R. Sheppard, “Optimization of second harmonic generation microscopy,” Micron 32, 691–700 (2001).
    [CrossRef] [PubMed]
  12. J. Vydra and M. Eich, “Mapping of the lateral polar orientation distribution in second-order nonlinear thin films by scanning second-harmonic microscopy,” Appl. Phys. Lett. 72, 275–277 (1998).
    [CrossRef]
  13. G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968).
    [CrossRef]
  14. M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge University, Cambridge, 1997).
  15. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, New York, 1995).
  16. V. S. Ignatovsky, “Diffraction by a parabolic mirror having arbitrary opening,” Trans. Opt. Inst. Petrograd I, paper IV (1920).
  17. 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]
  18. A. Yoshida and T. Asakura, “Electromagnetic field near theFocus of a Gaussian beam,” Optik (Stuttgart) 41, 281–292 (1974).
  19. R. Kant, “An analytical solution of vector diffraction for focusing optical systems,” J. Mol. Spectrosc. 40, 337–347 (1993).
  20. C. J. R. Sheppard and P. Török, “Efficient calculation ofelectromagnetic diffraction in optical systems using a multipole expansion,” J. Mod. Opt. 44, 803–818 (1997).
    [CrossRef]
  21. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1998).
  22. M. Abramowitz, Handbook of Mathematical Functions (Dover, New York, 1972).
  23. R. W. Boyd, Nonlinear Optics (Academic, Boston, 1992).
  24. W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE, New York, 1995).
  25. P. A. Franken and J. F. Ward, “Optical harmonics and nonlinear phenomena,” Rev. Mod. Phys. 35, 23 (1963).
    [CrossRef]
  26. Ji-Xin Cheng and X. Sunney Xie, “Green’s function formulation for third-harmonic generation microscopy,” J. Opt. Soc. Am. B 19, 1604–1610 (2002).
    [CrossRef]

2003

G. Cox, E. Kable, A. Jones, I. Fraser, F. Manconi, and M. D. Gorrell, “3-dimensional imaging of collagen using second harmonic generation,” J. Struct. Biol. 141, 53–62 (2003).
[CrossRef] [PubMed]

2002

2001

R. Gauderon, P. B. Lukins, and C. J. R. Sheppard, “Optimization of second harmonic generation microscopy,” Micron 32, 691–700 (2001).
[CrossRef] [PubMed]

J. Mertz and L. Moreaux, “Multi-harmonic light microscopy: theory and applications to membrane imaging,” in Multiphoton Microscopy in the Biomedical Sciences, A. M. Periasamy and P. T. C. So, eds., Proc. SPIE 4262, 9–17 (2001).
[CrossRef]

2000

1999

G. Peleg, A. Lewis, M. Linial, and L. Loew, “Nonlinear optical measurement of membrane potential around single molecules at selected cellular sites,” Proc. Natl. Acad. Sci. U.S.A. 96, 6700–6704 (1999).
[CrossRef] [PubMed]

P. J. Campagnola, M. 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]

1998

J. Vydra and M. Eich, “Mapping of the lateral polar orientation distribution in second-order nonlinear thin films by scanning second-harmonic microscopy,” Appl. Phys. Lett. 72, 275–277 (1998).
[CrossRef]

1997

C. J. R. Sheppard and P. Török, “Efficient calculation ofelectromagnetic diffraction in optical systems using a multipole expansion,” J. Mod. Opt. 44, 803–818 (1997).
[CrossRef]

1996

1993

R. Kant, “An analytical solution of vector diffraction for focusing optical systems,” J. Mol. Spectrosc. 40, 337–347 (1993).

1986

I. Freund, M. Deutsch, and A. Sprecher, “Connective tissue polarity, optical second harmonic microscopy, crossed-beam summation and small-angle scattering in rat-tail tendon,” Biophys. J. 50, 693–712 (1986).
[CrossRef] [PubMed]

1978

J. N. Gannaway and C. J. R. Sheppard, “Second-harmonic imaging in the scanning optical microscope,” Opt. Quantum Electron. 10, 435 (1978).
[CrossRef]

C. J. R. Sheppard and R. Kompfner, “Resonant scanning optical microscope,” Appl. Opt. 17, 2879–2882 (1978).
[CrossRef] [PubMed]

1975

1974

A. Yoshida and T. Asakura, “Electromagnetic field near theFocus of a Gaussian beam,” Optik (Stuttgart) 41, 281–292 (1974).

1968

G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968).
[CrossRef]

1963

P. A. Franken and J. F. Ward, “Optical harmonics and nonlinear phenomena,” Rev. Mod. Phys. 35, 23 (1963).
[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]

Alfano, R. R.

Asakura, T.

A. Yoshida and T. Asakura, “Electromagnetic field near theFocus of a Gaussian beam,” Optik (Stuttgart) 41, 281–292 (1974).

Boyd, G. D.

G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968).
[CrossRef]

Campagnola, P. J.

P. J. Campagnola, M. 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]

Cheng, Ji-Xin

Christiansen, P.

Cox, G.

G. Cox, E. Kable, A. Jones, I. Fraser, F. Manconi, and M. D. Gorrell, “3-dimensional imaging of collagen using second harmonic generation,” J. Struct. Biol. 141, 53–62 (2003).
[CrossRef] [PubMed]

Deutsch, M.

I. Freund, M. Deutsch, and A. Sprecher, “Connective tissue polarity, optical second harmonic microscopy, crossed-beam summation and small-angle scattering in rat-tail tendon,” Biophys. J. 50, 693–712 (1986).
[CrossRef] [PubMed]

Eich, M.

J. Vydra and M. Eich, “Mapping of the lateral polar orientation distribution in second-order nonlinear thin films by scanning second-harmonic microscopy,” Appl. Phys. Lett. 72, 275–277 (1998).
[CrossRef]

Franken, P. A.

P. A. Franken and J. F. Ward, “Optical harmonics and nonlinear phenomena,” Rev. Mod. Phys. 35, 23 (1963).
[CrossRef]

Fraser, I.

G. Cox, E. Kable, A. Jones, I. Fraser, F. Manconi, and M. D. Gorrell, “3-dimensional imaging of collagen using second harmonic generation,” J. Struct. Biol. 141, 53–62 (2003).
[CrossRef] [PubMed]

Freund, I.

I. Freund, M. Deutsch, and A. Sprecher, “Connective tissue polarity, optical second harmonic microscopy, crossed-beam summation and small-angle scattering in rat-tail tendon,” Biophys. J. 50, 693–712 (1986).
[CrossRef] [PubMed]

Gannaway, J. N.

J. N. Gannaway and C. J. R. Sheppard, “Second-harmonic imaging in the scanning optical microscope,” Opt. Quantum Electron. 10, 435 (1978).
[CrossRef]

Gauderon, R.

R. Gauderon, P. B. Lukins, and C. J. R. Sheppard, “Optimization of second harmonic generation microscopy,” Micron 32, 691–700 (2001).
[CrossRef] [PubMed]

Gorrell, M. D.

G. Cox, E. Kable, A. Jones, I. Fraser, F. Manconi, and M. D. Gorrell, “3-dimensional imaging of collagen using second harmonic generation,” J. Struct. Biol. 141, 53–62 (2003).
[CrossRef] [PubMed]

Guo, Y.

Hellwarth, R.

Ho, P. P.

Jones, A.

G. Cox, E. Kable, A. Jones, I. Fraser, F. Manconi, and M. D. Gorrell, “3-dimensional imaging of collagen using second harmonic generation,” J. Struct. Biol. 141, 53–62 (2003).
[CrossRef] [PubMed]

Kable, E.

G. Cox, E. Kable, A. Jones, I. Fraser, F. Manconi, and M. D. Gorrell, “3-dimensional imaging of collagen using second harmonic generation,” J. Struct. Biol. 141, 53–62 (2003).
[CrossRef] [PubMed]

Kant, R.

R. Kant, “An analytical solution of vector diffraction for focusing optical systems,” J. Mol. Spectrosc. 40, 337–347 (1993).

Kleinman, D. A.

G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968).
[CrossRef]

Kompfner, R.

Lewis, A.

P. J. Campagnola, M. 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]

G. Peleg, A. Lewis, M. Linial, and L. Loew, “Nonlinear optical measurement of membrane potential around single molecules at selected cellular sites,” Proc. Natl. Acad. Sci. U.S.A. 96, 6700–6704 (1999).
[CrossRef] [PubMed]

Linial, M.

G. Peleg, A. Lewis, M. Linial, and L. Loew, “Nonlinear optical measurement of membrane potential around single molecules at selected cellular sites,” Proc. Natl. Acad. Sci. U.S.A. 96, 6700–6704 (1999).
[CrossRef] [PubMed]

Liu, F.

Loew, L.

G. Peleg, A. Lewis, M. Linial, and L. Loew, “Nonlinear optical measurement of membrane potential around single molecules at selected cellular sites,” Proc. Natl. Acad. Sci. U.S.A. 96, 6700–6704 (1999).
[CrossRef] [PubMed]

Loew, L. M.

P. J. Campagnola, M. 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.

R. Gauderon, P. B. Lukins, and C. J. R. Sheppard, “Optimization of second harmonic generation microscopy,” Micron 32, 691–700 (2001).
[CrossRef] [PubMed]

Manconi, F.

G. Cox, E. Kable, A. Jones, I. Fraser, F. Manconi, and M. D. Gorrell, “3-dimensional imaging of collagen using second harmonic generation,” J. Struct. Biol. 141, 53–62 (2003).
[CrossRef] [PubMed]

Mertz, J.

J. Mertz and L. Moreaux, “Multi-harmonic light microscopy: theory and applications to membrane imaging,” in Multiphoton Microscopy in the Biomedical Sciences, A. M. Periasamy and P. T. C. So, eds., Proc. SPIE 4262, 9–17 (2001).
[CrossRef]

L. Moreaux, O. Sandre, and J. Mertz, “Membrane imaging by second-harmonic generation microscopy,” J. Opt. Soc. Am. B 17, 1685–1694 (2000).
[CrossRef]

Moreaux, L.

J. Mertz and L. Moreaux, “Multi-harmonic light microscopy: theory and applications to membrane imaging,” in Multiphoton Microscopy in the Biomedical Sciences, A. M. Periasamy and P. T. C. So, eds., Proc. SPIE 4262, 9–17 (2001).
[CrossRef]

L. Moreaux, O. Sandre, and J. Mertz, “Membrane imaging by second-harmonic generation microscopy,” J. Opt. Soc. Am. B 17, 1685–1694 (2000).
[CrossRef]

Peleg, G.

G. Peleg, A. Lewis, M. Linial, and L. Loew, “Nonlinear optical measurement of membrane potential around single molecules at selected cellular sites,” Proc. Natl. Acad. Sci. U.S.A. 96, 6700–6704 (1999).
[CrossRef] [PubMed]

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]

Sandre, O.

Sheppard, C. J. R.

R. Gauderon, P. B. Lukins, and C. J. R. Sheppard, “Optimization of second harmonic generation microscopy,” Micron 32, 691–700 (2001).
[CrossRef] [PubMed]

C. J. R. Sheppard and P. Török, “Efficient calculation ofelectromagnetic diffraction in optical systems using a multipole expansion,” J. Mod. Opt. 44, 803–818 (1997).
[CrossRef]

J. N. Gannaway and C. J. R. Sheppard, “Second-harmonic imaging in the scanning optical microscope,” Opt. Quantum Electron. 10, 435 (1978).
[CrossRef]

C. J. R. Sheppard and R. Kompfner, “Resonant scanning optical microscope,” Appl. Opt. 17, 2879–2882 (1978).
[CrossRef] [PubMed]

Sprecher, A.

I. Freund, M. Deutsch, and A. Sprecher, “Connective tissue polarity, optical second harmonic microscopy, crossed-beam summation and small-angle scattering in rat-tail tendon,” Biophys. J. 50, 693–712 (1986).
[CrossRef] [PubMed]

Sunney Xie, X.

Tirksliunas, A.

Török, P.

C. J. R. Sheppard and P. Török, “Efficient calculation ofelectromagnetic diffraction in optical systems using a multipole expansion,” J. Mod. Opt. 44, 803–818 (1997).
[CrossRef]

Vydra, J.

J. Vydra and M. Eich, “Mapping of the lateral polar orientation distribution in second-order nonlinear thin films by scanning second-harmonic microscopy,” Appl. Phys. Lett. 72, 275–277 (1998).
[CrossRef]

Ward, J. F.

P. A. Franken and J. F. Ward, “Optical harmonics and nonlinear phenomena,” Rev. Mod. Phys. 35, 23 (1963).
[CrossRef]

Wei, M.

P. J. Campagnola, M. 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]

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]

Yoshida, A.

A. Yoshida and T. Asakura, “Electromagnetic field near theFocus of a Gaussian beam,” Optik (Stuttgart) 41, 281–292 (1974).

Appl. Opt.

Appl. Phys. Lett.

J. Vydra and M. Eich, “Mapping of the lateral polar orientation distribution in second-order nonlinear thin films by scanning second-harmonic microscopy,” Appl. Phys. Lett. 72, 275–277 (1998).
[CrossRef]

Biophys. J.

P. J. Campagnola, M. 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]

I. Freund, M. Deutsch, and A. Sprecher, “Connective tissue polarity, optical second harmonic microscopy, crossed-beam summation and small-angle scattering in rat-tail tendon,” Biophys. J. 50, 693–712 (1986).
[CrossRef] [PubMed]

J. Appl. Phys.

G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968).
[CrossRef]

J. Mod. Opt.

C. J. R. Sheppard and P. Török, “Efficient calculation ofelectromagnetic diffraction in optical systems using a multipole expansion,” J. Mod. Opt. 44, 803–818 (1997).
[CrossRef]

J. Mol. Spectrosc.

R. Kant, “An analytical solution of vector diffraction for focusing optical systems,” J. Mol. Spectrosc. 40, 337–347 (1993).

J. Opt. Soc. Am. B

J. Struct. Biol.

G. Cox, E. Kable, A. Jones, I. Fraser, F. Manconi, and M. D. Gorrell, “3-dimensional imaging of collagen using second harmonic generation,” J. Struct. Biol. 141, 53–62 (2003).
[CrossRef] [PubMed]

Micron

R. Gauderon, P. B. Lukins, and C. J. R. Sheppard, “Optimization of second harmonic generation microscopy,” Micron 32, 691–700 (2001).
[CrossRef] [PubMed]

Opt. Quantum Electron.

J. N. Gannaway and C. J. R. Sheppard, “Second-harmonic imaging in the scanning optical microscope,” Opt. Quantum Electron. 10, 435 (1978).
[CrossRef]

Optik (Stuttgart)

A. Yoshida and T. Asakura, “Electromagnetic field near theFocus of a Gaussian beam,” Optik (Stuttgart) 41, 281–292 (1974).

Proc. Natl. Acad. Sci. U.S.A.

G. Peleg, A. Lewis, M. Linial, and L. Loew, “Nonlinear optical measurement of membrane potential around single molecules at selected cellular sites,” Proc. Natl. Acad. Sci. U.S.A. 96, 6700–6704 (1999).
[CrossRef] [PubMed]

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]

Proc. SPIE

J. Mertz and L. Moreaux, “Multi-harmonic light microscopy: theory and applications to membrane imaging,” in Multiphoton Microscopy in the Biomedical Sciences, A. M. Periasamy and P. T. C. So, eds., Proc. SPIE 4262, 9–17 (2001).
[CrossRef]

Rev. Mod. Phys.

P. A. Franken and J. F. Ward, “Optical harmonics and nonlinear phenomena,” Rev. Mod. Phys. 35, 23 (1963).
[CrossRef]

Other

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1998).

M. Abramowitz, Handbook of Mathematical Functions (Dover, New York, 1972).

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

W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE, New York, 1995).

M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge University, Cambridge, 1997).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, New York, 1995).

V. S. Ignatovsky, “Diffraction by a parabolic mirror having arbitrary opening,” Trans. Opt. Inst. Petrograd I, paper IV (1920).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1
Fig. 1

Intensity distribution in the xz plane near the focus of a convergent Gaussian beam. The initial Gaussian beam is taken to be linearly polarized along the x axis.

Fig. 2
Fig. 2

As in Fig. 1, but in the yz plane.

Fig. 3
Fig. 3

As in Fig. 1, but in the xy plane. The Airy pattern is clearly seen.

Fig. 4
Fig. 4

Intensity distribution I=|E|2 along x (dashed curve) and y axes (solid curve) in the focal plane z=0. We use the optical coordinate scale v=kx2+y2 sin α, and we normalize the intensity with respect to the value at the focus.

Fig. 5
Fig. 5

Intensity distribution I=|E|2 along the optical z axis. We used optical coordinate scale u=kz sin2 α, and we normalize the intensity with respect to the value at the focus.

Fig. 6
Fig. 6

Distribution of log10|Ex2|2 in the xy plane at z/λ=577 from the focus. The coordinates x and y are measured in terms of the wavelength.

Fig. 7
Fig. 7

As in Fig. 6, but for log10|Ey2|2 component of the second harmonic.

Fig. 8
Fig. 8

As in Fig. 6, but for log10|Ez2|2 component of the second harmonic.

Fig. 9
Fig. 9

Intensity distribution log10|E2|2 of the second harmonic in the xy plane.

Fig. 10
Fig. 10

Section of Figs. 68 versus y/λ for |Ex2|2 (solid curve), for |Ey2|2 (dotted curve), and for |Ez2|2 (dashed curve).

Equations (21)

Equations on this page are rendered with MathJax. Learn more.

Ex=-iAπ0α02πB(θ, ϕ)cos θ sin θ×[cos θ+(1-cos θ)sin2 ϕ]exp(iks·rP)dθdϕ,
Ey=iAπ0α02πB(θ, ϕ)cos θ sin θ×(1-cos θ)cos ϕ sin ϕ exp(iks·rP)dθdϕ,
Ez=iAπ0α02πB(θ, ϕ)cos θ  sin2 θ cos ϕ×exp(iks·rP)dθdϕ,
B(θ, ϕ)=exp(-f2  tan2 θ/W2),
exp(iks·rP)=4πl=0iljl(krP)×m=-llYlm*(θ, ϕ)Ylm(θP, ϕP),
Ex=4πl=0Il(0)iljl(krP)Yl,0*(θP, ϕP)+4πl=2Il(2)iljl(krP)×[Yl,2(θP, ϕP)+Yl,2*(θP, ϕP)]2,
Ey=4πl=2Il(2)iljl(krP)[Yl,2(θP, ϕP)-Yl,2*(θP, ϕP)]2i,
Ez=4πl=1Il(1)iljl(krP)[Yl,1(θP, ϕP)+Yl,1*(θP, ϕP)]2,
Il(0)=-iAπ π(2l+1)4  cos α1B(x)x(1+x)Pl(0)(x)dx,
Il(2)=iAπ π(2l+1)4(l-2)!(l+2)!  cos α1B(x)x(1-x)×Pl(2)(x)dx,
Il(1)=iAπ π(2l+1)l(l+1)  cos α1B(x)x(1-x2)Pl(1)(x)dx,
B(x)=exp[-f2(1-x2)/(W2x2)].
lmaxkrP.
××En-ωn2c2(1)(ωn)En=4πωn2c2Pn(NL)(r).
××E2-k22E2=16πk2(1)(ω)P2(NL)(r),
P2i(ω2)=χijk(2)(ω2; ω, ω)Ej(ω)Ek(ω),
E2=16πk2(1)(ω) VG¯(r; r)·P2(NL)(r)dV,
G¯(r; r)=I+1k22 exp(ik2|r-r|)4π|r-r|,
χ131(2)=χ113(2)=χ232(2)=χ223(2)=-82,
χ311(2)=χ322(2)=-86,
χ333(2)=-16,

Metrics