Abstract

The effects of the axial field components of a focused beam under high NA on the second harmonic generation (SHG) in collagen was examined using a vectorial approach. We find that with high NA, the cross-component terms that are most likely to have an effect on SHG will be ExEx, ExEy, ExEz and EzEz as a result of tight focusing. By considering the tensor and the presence of the other electric field components the possibility of different polarization states of the generated second harmonic as a result of the nonlinear susceptibility tensor making it possible to generate radially polarized modes with linearly polarized beams.

© 2006 Optical Society of America

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  1. P. Stoller, B. M. Kim, A. M. Rubenchik, K. M. Reiser, and L. B. Da Silva, “Polarization-dependent optical second-harmonic imaging of rat-tail tendon,” J. Biomed. Opt. 7, 205–214 (2002)
    [CrossRef] [PubMed]
  2. S. W. Chu, S. Y. Chen, G. W. Chern, T. H. Tsai, Y. C. Chen, B. L. Lin, and C. K. Sun, “Studies of χ(2)/χ(3) tensors in submicron-scaled bio-tissues by polarization harmonics optical microscopy,” Biophys. J. 86, 3914–3922 (2004)
    [CrossRef] [PubMed]
  3. 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]
  4. C.K. Sun, S.W. Chu, S.P. Tai, S. Keller, A. Abare, U. K. Mishra, and S. P. DenBaars, “Mapping piezoelectric-field distribution in Gallium Nitride with scanning second-harmonic generation microscopy,” Scanning 23, 182–192 (2001)
    [CrossRef] [PubMed]
  5. P. J. Campagnola, A. C. Millard, M. Terasaki, P.E. Hoppe, C. J. Malone, and W. A. Mohler, “Three-dimensional high-resolution second-harmonic generation imaging of endogenous structural proteins in biological tissues,” Biophys. J. 81, 493–508 (2002)
    [CrossRef]
  6. S. Roth and I. Freund, “Second harmonic generation in collagen,” J. Chem. Phy. 70, 1637–1643 (1979)
    [CrossRef]
  7. E. Brown, T. McKee, E. diTomaso, A. Pluen, B. Seed, Y. Boucher, and R. K. Jain, “Dynamic imaging of collagen and its modulation in tumours in vivo using second-harmonic generation,” Nature Med. 9, 796–800 (2003)
    [CrossRef] [PubMed]
  8. M. Both, M. Vogel, O. Friedrich, F. von Wegner, T. Kunsting, R. H. A. Fink, and D. Uttenweiler, “Second harmonic imaging of intrinsic signals in muscle fibres in situ,” J. Biomed. Opt. 9, 882–892 (2004)
    [CrossRef] [PubMed]
  9. S. W. Chu, S. Y. Chen, T. H. Tsai, T. M. Liu, C. Y. Lin, H. J. Tsai, and C. K. Sun, “In vivo developmental biology study using non-invasive multi-harmonic generation microscopy,” Opt. Express 11, 3093–3099 (2003).
    [CrossRef] [PubMed]
  10. D. A. Kleinmann, “Nonlinear dielectric polarization in optical media,” Phys. Rev. 126, 1977–1979 (1962)
    [CrossRef]
  11. L. Moreaux, O. Sandre, and J. Mertz, “Membrane imaging by second-harmonic generation microscopy,” J. Opt. Soc. Am. B 17, 1685–1694 (2000)
    [CrossRef]
  12. J. Mertz and L. Moreaux, “Second-harmonic generation by focused excitation of inhomogeneously distributed scatterers,” Opt. Commun. 196, 325–330 (2001)
    [CrossRef]
  13. R. W. Boyd, “Nonlinear Optics,” 2nd Ed(Academic Press, Amsterdam, 2003)
  14. 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]
  15. R. M. Williams, W. R. Zipfel, and W. W. Webb, “Interpreting second-harmonic generation images of collagen I fibrils,” Biophys. J. 88, 1377–1386 (2005)
    [CrossRef]
  16. A.A. Asatryan, C. J. R. Sheppard, and C. M. de Sterke, “Vector treatment of second harmonic generation produced by tightly focused vignetted Gaussian beams,” J. Opt. Soc. Am. B 21, 2206–2212 (2004)
    [CrossRef]
  17. T. Yasui, Y. Tohno, and T. Araki, “Characterization of collagen orientation in human dermis by two-dimensional second-harmonic-generation polarimetry,” J. Biomed. Opt. 9, 259–264 (2004)
    [CrossRef] [PubMed]
  18. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. Roy. Soc. Lond. Ser. A,  253, 358–379 (1959)
    [CrossRef]
  19. C. T. Tai, Dyadic Green’s Functions in Electromagnetic Theory, (Intext Educational Publishers, 1971)
  20. S. Kazamias, F. Weihe, D. Douillet, C. Valentin, T. Planchin, S. Sebban, G. Grillon, F. Auge, D. Huilin, and Ph. Baclou, “High order harmonic generation optimization with an apertured laser beam,” Eur. Phys. J. D 21, 353–359 (2002)
    [CrossRef]
  21. R. Gauderon, P. B. Lukins, and C. J. R. Sheppard, “Optimization of second-harmonic generation,” Micron 32, 691–700 (2001)
    [CrossRef] [PubMed]

2005 (1)

R. M. Williams, W. R. Zipfel, and W. W. Webb, “Interpreting second-harmonic generation images of collagen I fibrils,” Biophys. J. 88, 1377–1386 (2005)
[CrossRef]

2004 (4)

T. Yasui, Y. Tohno, and T. Araki, “Characterization of collagen orientation in human dermis by two-dimensional second-harmonic-generation polarimetry,” J. Biomed. Opt. 9, 259–264 (2004)
[CrossRef] [PubMed]

M. Both, M. Vogel, O. Friedrich, F. von Wegner, T. Kunsting, R. H. A. Fink, and D. Uttenweiler, “Second harmonic imaging of intrinsic signals in muscle fibres in situ,” J. Biomed. Opt. 9, 882–892 (2004)
[CrossRef] [PubMed]

S. W. Chu, S. Y. Chen, G. W. Chern, T. H. Tsai, Y. C. Chen, B. L. Lin, and C. K. Sun, “Studies of χ(2)/χ(3) tensors in submicron-scaled bio-tissues by polarization harmonics optical microscopy,” Biophys. J. 86, 3914–3922 (2004)
[CrossRef] [PubMed]

A.A. Asatryan, C. J. R. Sheppard, and C. M. de Sterke, “Vector treatment of second harmonic generation produced by tightly focused vignetted Gaussian beams,” J. Opt. Soc. Am. B 21, 2206–2212 (2004)
[CrossRef]

2003 (2)

E. Brown, T. McKee, E. diTomaso, A. Pluen, B. Seed, Y. Boucher, and R. K. Jain, “Dynamic imaging of collagen and its modulation in tumours in vivo using second-harmonic generation,” Nature Med. 9, 796–800 (2003)
[CrossRef] [PubMed]

S. W. Chu, S. Y. Chen, T. H. Tsai, T. M. Liu, C. Y. Lin, H. J. Tsai, and C. K. Sun, “In vivo developmental biology study using non-invasive multi-harmonic generation microscopy,” Opt. Express 11, 3093–3099 (2003).
[CrossRef] [PubMed]

2002 (4)

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]

S. Kazamias, F. Weihe, D. Douillet, C. Valentin, T. Planchin, S. Sebban, G. Grillon, F. Auge, D. Huilin, and Ph. Baclou, “High order harmonic generation optimization with an apertured laser beam,” Eur. Phys. J. D 21, 353–359 (2002)
[CrossRef]

P. J. Campagnola, A. C. Millard, M. Terasaki, P.E. Hoppe, C. J. Malone, and W. A. Mohler, “Three-dimensional high-resolution second-harmonic generation imaging of endogenous structural proteins in biological tissues,” Biophys. J. 81, 493–508 (2002)
[CrossRef]

P. Stoller, B. M. Kim, A. M. Rubenchik, K. M. Reiser, and L. B. Da Silva, “Polarization-dependent optical second-harmonic imaging of rat-tail tendon,” J. Biomed. Opt. 7, 205–214 (2002)
[CrossRef] [PubMed]

2001 (3)

C.K. Sun, S.W. Chu, S.P. Tai, S. Keller, A. Abare, U. K. Mishra, and S. P. DenBaars, “Mapping piezoelectric-field distribution in Gallium Nitride with scanning second-harmonic generation microscopy,” Scanning 23, 182–192 (2001)
[CrossRef] [PubMed]

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

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

2000 (1)

1998 (1)

1979 (1)

S. Roth and I. Freund, “Second harmonic generation in collagen,” J. Chem. Phy. 70, 1637–1643 (1979)
[CrossRef]

1962 (1)

D. A. Kleinmann, “Nonlinear dielectric polarization in optical media,” Phys. Rev. 126, 1977–1979 (1962)
[CrossRef]

1959 (1)

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

Abare, A.

C.K. Sun, S.W. Chu, S.P. Tai, S. Keller, A. Abare, U. K. Mishra, and S. P. DenBaars, “Mapping piezoelectric-field distribution in Gallium Nitride with scanning second-harmonic generation microscopy,” Scanning 23, 182–192 (2001)
[CrossRef] [PubMed]

Araki, T.

T. Yasui, Y. Tohno, and T. Araki, “Characterization of collagen orientation in human dermis by two-dimensional second-harmonic-generation polarimetry,” J. Biomed. Opt. 9, 259–264 (2004)
[CrossRef] [PubMed]

Asatryan, A.A.

Auge, F.

S. Kazamias, F. Weihe, D. Douillet, C. Valentin, T. Planchin, S. Sebban, G. Grillon, F. Auge, D. Huilin, and Ph. Baclou, “High order harmonic generation optimization with an apertured laser beam,” Eur. Phys. J. D 21, 353–359 (2002)
[CrossRef]

Baclou, Ph.

S. Kazamias, F. Weihe, D. Douillet, C. Valentin, T. Planchin, S. Sebban, G. Grillon, F. Auge, D. Huilin, and Ph. Baclou, “High order harmonic generation optimization with an apertured laser beam,” Eur. Phys. J. D 21, 353–359 (2002)
[CrossRef]

Both, M.

M. Both, M. Vogel, O. Friedrich, F. von Wegner, T. Kunsting, R. H. A. Fink, and D. Uttenweiler, “Second harmonic imaging of intrinsic signals in muscle fibres in situ,” J. Biomed. Opt. 9, 882–892 (2004)
[CrossRef] [PubMed]

Boucher, Y.

E. Brown, T. McKee, E. diTomaso, A. Pluen, B. Seed, Y. Boucher, and R. K. Jain, “Dynamic imaging of collagen and its modulation in tumours in vivo using second-harmonic generation,” Nature Med. 9, 796–800 (2003)
[CrossRef] [PubMed]

Boyd, R. W.

R. W. Boyd, “Nonlinear Optics,” 2nd Ed(Academic Press, Amsterdam, 2003)

Brown, E.

E. Brown, T. McKee, E. diTomaso, A. Pluen, B. Seed, Y. Boucher, and R. K. Jain, “Dynamic imaging of collagen and its modulation in tumours in vivo using second-harmonic generation,” Nature Med. 9, 796–800 (2003)
[CrossRef] [PubMed]

Campagnola, P. J.

P. J. Campagnola, A. C. Millard, M. Terasaki, P.E. Hoppe, C. J. Malone, and W. A. Mohler, “Three-dimensional high-resolution second-harmonic generation imaging of endogenous structural proteins in biological tissues,” Biophys. J. 81, 493–508 (2002)
[CrossRef]

Chen, S. Y.

S. W. Chu, S. Y. Chen, G. W. Chern, T. H. Tsai, Y. C. Chen, B. L. Lin, and C. K. Sun, “Studies of χ(2)/χ(3) tensors in submicron-scaled bio-tissues by polarization harmonics optical microscopy,” Biophys. J. 86, 3914–3922 (2004)
[CrossRef] [PubMed]

S. W. Chu, S. Y. Chen, T. H. Tsai, T. M. Liu, C. Y. Lin, H. J. Tsai, and C. K. Sun, “In vivo developmental biology study using non-invasive multi-harmonic generation microscopy,” Opt. Express 11, 3093–3099 (2003).
[CrossRef] [PubMed]

Chen, Y. C.

S. W. Chu, S. Y. Chen, G. W. Chern, T. H. Tsai, Y. C. Chen, B. L. Lin, and C. K. Sun, “Studies of χ(2)/χ(3) tensors in submicron-scaled bio-tissues by polarization harmonics optical microscopy,” Biophys. J. 86, 3914–3922 (2004)
[CrossRef] [PubMed]

Cheng, J. X.

Chern, G. W.

S. W. Chu, S. Y. Chen, G. W. Chern, T. H. Tsai, Y. C. Chen, B. L. Lin, and C. K. Sun, “Studies of χ(2)/χ(3) tensors in submicron-scaled bio-tissues by polarization harmonics optical microscopy,” Biophys. J. 86, 3914–3922 (2004)
[CrossRef] [PubMed]

Chu, S. W.

S. W. Chu, S. Y. Chen, G. W. Chern, T. H. Tsai, Y. C. Chen, B. L. Lin, and C. K. Sun, “Studies of χ(2)/χ(3) tensors in submicron-scaled bio-tissues by polarization harmonics optical microscopy,” Biophys. J. 86, 3914–3922 (2004)
[CrossRef] [PubMed]

S. W. Chu, S. Y. Chen, T. H. Tsai, T. M. Liu, C. Y. Lin, H. J. Tsai, and C. K. Sun, “In vivo developmental biology study using non-invasive multi-harmonic generation microscopy,” Opt. Express 11, 3093–3099 (2003).
[CrossRef] [PubMed]

Chu, S.W.

C.K. Sun, S.W. Chu, S.P. Tai, S. Keller, A. Abare, U. K. Mishra, and S. P. DenBaars, “Mapping piezoelectric-field distribution in Gallium Nitride with scanning second-harmonic generation microscopy,” Scanning 23, 182–192 (2001)
[CrossRef] [PubMed]

Da Silva, L. B.

P. Stoller, B. M. Kim, A. M. Rubenchik, K. M. Reiser, and L. B. Da Silva, “Polarization-dependent optical second-harmonic imaging of rat-tail tendon,” J. Biomed. Opt. 7, 205–214 (2002)
[CrossRef] [PubMed]

de Sterke, C. M.

DenBaars, S. P.

C.K. Sun, S.W. Chu, S.P. Tai, S. Keller, A. Abare, U. K. Mishra, and S. P. DenBaars, “Mapping piezoelectric-field distribution in Gallium Nitride with scanning second-harmonic generation microscopy,” Scanning 23, 182–192 (2001)
[CrossRef] [PubMed]

diTomaso, E.

E. Brown, T. McKee, E. diTomaso, A. Pluen, B. Seed, Y. Boucher, and R. K. Jain, “Dynamic imaging of collagen and its modulation in tumours in vivo using second-harmonic generation,” Nature Med. 9, 796–800 (2003)
[CrossRef] [PubMed]

Douillet, D.

S. Kazamias, F. Weihe, D. Douillet, C. Valentin, T. Planchin, S. Sebban, G. Grillon, F. Auge, D. Huilin, and Ph. Baclou, “High order harmonic generation optimization with an apertured laser beam,” Eur. Phys. J. D 21, 353–359 (2002)
[CrossRef]

Fink, R. H. A.

M. Both, M. Vogel, O. Friedrich, F. von Wegner, T. Kunsting, R. H. A. Fink, and D. Uttenweiler, “Second harmonic imaging of intrinsic signals in muscle fibres in situ,” J. Biomed. Opt. 9, 882–892 (2004)
[CrossRef] [PubMed]

Freund, I.

S. Roth and I. Freund, “Second harmonic generation in collagen,” J. Chem. Phy. 70, 1637–1643 (1979)
[CrossRef]

Friedrich, O.

M. Both, M. Vogel, O. Friedrich, F. von Wegner, T. Kunsting, R. H. A. Fink, and D. Uttenweiler, “Second harmonic imaging of intrinsic signals in muscle fibres in situ,” J. Biomed. Opt. 9, 882–892 (2004)
[CrossRef] [PubMed]

Gauderon, R.

Grillon, G.

S. Kazamias, F. Weihe, D. Douillet, C. Valentin, T. Planchin, S. Sebban, G. Grillon, F. Auge, D. Huilin, and Ph. Baclou, “High order harmonic generation optimization with an apertured laser beam,” Eur. Phys. J. D 21, 353–359 (2002)
[CrossRef]

Hoppe, P.E.

P. J. Campagnola, A. C. Millard, M. Terasaki, P.E. Hoppe, C. J. Malone, and W. A. Mohler, “Three-dimensional high-resolution second-harmonic generation imaging of endogenous structural proteins in biological tissues,” Biophys. J. 81, 493–508 (2002)
[CrossRef]

Huilin, D.

S. Kazamias, F. Weihe, D. Douillet, C. Valentin, T. Planchin, S. Sebban, G. Grillon, F. Auge, D. Huilin, and Ph. Baclou, “High order harmonic generation optimization with an apertured laser beam,” Eur. Phys. J. D 21, 353–359 (2002)
[CrossRef]

Jain, R. K.

E. Brown, T. McKee, E. diTomaso, A. Pluen, B. Seed, Y. Boucher, and R. K. Jain, “Dynamic imaging of collagen and its modulation in tumours in vivo using second-harmonic generation,” Nature Med. 9, 796–800 (2003)
[CrossRef] [PubMed]

Kazamias, S.

S. Kazamias, F. Weihe, D. Douillet, C. Valentin, T. Planchin, S. Sebban, G. Grillon, F. Auge, D. Huilin, and Ph. Baclou, “High order harmonic generation optimization with an apertured laser beam,” Eur. Phys. J. D 21, 353–359 (2002)
[CrossRef]

Keller, S.

C.K. Sun, S.W. Chu, S.P. Tai, S. Keller, A. Abare, U. K. Mishra, and S. P. DenBaars, “Mapping piezoelectric-field distribution in Gallium Nitride with scanning second-harmonic generation microscopy,” Scanning 23, 182–192 (2001)
[CrossRef] [PubMed]

Kim, B. M.

P. Stoller, B. M. Kim, A. M. Rubenchik, K. M. Reiser, and L. B. Da Silva, “Polarization-dependent optical second-harmonic imaging of rat-tail tendon,” J. Biomed. Opt. 7, 205–214 (2002)
[CrossRef] [PubMed]

Kleinmann, D. A.

D. A. Kleinmann, “Nonlinear dielectric polarization in optical media,” Phys. Rev. 126, 1977–1979 (1962)
[CrossRef]

Kunsting, T.

M. Both, M. Vogel, O. Friedrich, F. von Wegner, T. Kunsting, R. H. A. Fink, and D. Uttenweiler, “Second harmonic imaging of intrinsic signals in muscle fibres in situ,” J. Biomed. Opt. 9, 882–892 (2004)
[CrossRef] [PubMed]

Lin, B. L.

S. W. Chu, S. Y. Chen, G. W. Chern, T. H. Tsai, Y. C. Chen, B. L. Lin, and C. K. Sun, “Studies of χ(2)/χ(3) tensors in submicron-scaled bio-tissues by polarization harmonics optical microscopy,” Biophys. J. 86, 3914–3922 (2004)
[CrossRef] [PubMed]

Lin, C. Y.

Liu, T. M.

Lukins, P. B.

Malone, C. J.

P. J. Campagnola, A. C. Millard, M. Terasaki, P.E. Hoppe, C. J. Malone, and W. A. Mohler, “Three-dimensional high-resolution second-harmonic generation imaging of endogenous structural proteins in biological tissues,” Biophys. J. 81, 493–508 (2002)
[CrossRef]

McKee, T.

E. Brown, T. McKee, E. diTomaso, A. Pluen, B. Seed, Y. Boucher, and R. K. Jain, “Dynamic imaging of collagen and its modulation in tumours in vivo using second-harmonic generation,” Nature Med. 9, 796–800 (2003)
[CrossRef] [PubMed]

Mertz, J.

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

Millard, A. C.

P. J. Campagnola, A. C. Millard, M. Terasaki, P.E. Hoppe, C. J. Malone, and W. A. Mohler, “Three-dimensional high-resolution second-harmonic generation imaging of endogenous structural proteins in biological tissues,” Biophys. J. 81, 493–508 (2002)
[CrossRef]

Mishra, U. K.

C.K. Sun, S.W. Chu, S.P. Tai, S. Keller, A. Abare, U. K. Mishra, and S. P. DenBaars, “Mapping piezoelectric-field distribution in Gallium Nitride with scanning second-harmonic generation microscopy,” Scanning 23, 182–192 (2001)
[CrossRef] [PubMed]

Mohler, W. A.

P. J. Campagnola, A. C. Millard, M. Terasaki, P.E. Hoppe, C. J. Malone, and W. A. Mohler, “Three-dimensional high-resolution second-harmonic generation imaging of endogenous structural proteins in biological tissues,” Biophys. J. 81, 493–508 (2002)
[CrossRef]

Moreaux, L.

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

Planchin, T.

S. Kazamias, F. Weihe, D. Douillet, C. Valentin, T. Planchin, S. Sebban, G. Grillon, F. Auge, D. Huilin, and Ph. Baclou, “High order harmonic generation optimization with an apertured laser beam,” Eur. Phys. J. D 21, 353–359 (2002)
[CrossRef]

Pluen, A.

E. Brown, T. McKee, E. diTomaso, A. Pluen, B. Seed, Y. Boucher, and R. K. Jain, “Dynamic imaging of collagen and its modulation in tumours in vivo using second-harmonic generation,” Nature Med. 9, 796–800 (2003)
[CrossRef] [PubMed]

Reiser, K. M.

P. Stoller, B. M. Kim, A. M. Rubenchik, K. M. Reiser, and L. B. Da Silva, “Polarization-dependent optical second-harmonic imaging of rat-tail tendon,” J. Biomed. Opt. 7, 205–214 (2002)
[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. Roy. Soc. Lond. Ser. A,  253, 358–379 (1959)
[CrossRef]

Roth, S.

S. Roth and I. Freund, “Second harmonic generation in collagen,” J. Chem. Phy. 70, 1637–1643 (1979)
[CrossRef]

Rubenchik, A. M.

P. Stoller, B. M. Kim, A. M. Rubenchik, K. M. Reiser, and L. B. Da Silva, “Polarization-dependent optical second-harmonic imaging of rat-tail tendon,” J. Biomed. Opt. 7, 205–214 (2002)
[CrossRef] [PubMed]

Sandre, O.

Sebban, S.

S. Kazamias, F. Weihe, D. Douillet, C. Valentin, T. Planchin, S. Sebban, G. Grillon, F. Auge, D. Huilin, and Ph. Baclou, “High order harmonic generation optimization with an apertured laser beam,” Eur. Phys. J. D 21, 353–359 (2002)
[CrossRef]

Seed, B.

E. Brown, T. McKee, E. diTomaso, A. Pluen, B. Seed, Y. Boucher, and R. K. Jain, “Dynamic imaging of collagen and its modulation in tumours in vivo using second-harmonic generation,” Nature Med. 9, 796–800 (2003)
[CrossRef] [PubMed]

Sheppard, C. J. R.

Stoller, P.

P. Stoller, B. M. Kim, A. M. Rubenchik, K. M. Reiser, and L. B. Da Silva, “Polarization-dependent optical second-harmonic imaging of rat-tail tendon,” J. Biomed. Opt. 7, 205–214 (2002)
[CrossRef] [PubMed]

Sun, C. K.

S. W. Chu, S. Y. Chen, G. W. Chern, T. H. Tsai, Y. C. Chen, B. L. Lin, and C. K. Sun, “Studies of χ(2)/χ(3) tensors in submicron-scaled bio-tissues by polarization harmonics optical microscopy,” Biophys. J. 86, 3914–3922 (2004)
[CrossRef] [PubMed]

S. W. Chu, S. Y. Chen, T. H. Tsai, T. M. Liu, C. Y. Lin, H. J. Tsai, and C. K. Sun, “In vivo developmental biology study using non-invasive multi-harmonic generation microscopy,” Opt. Express 11, 3093–3099 (2003).
[CrossRef] [PubMed]

Sun, C.K.

C.K. Sun, S.W. Chu, S.P. Tai, S. Keller, A. Abare, U. K. Mishra, and S. P. DenBaars, “Mapping piezoelectric-field distribution in Gallium Nitride with scanning second-harmonic generation microscopy,” Scanning 23, 182–192 (2001)
[CrossRef] [PubMed]

Tai, C. T.

C. T. Tai, Dyadic Green’s Functions in Electromagnetic Theory, (Intext Educational Publishers, 1971)

Tai, S.P.

C.K. Sun, S.W. Chu, S.P. Tai, S. Keller, A. Abare, U. K. Mishra, and S. P. DenBaars, “Mapping piezoelectric-field distribution in Gallium Nitride with scanning second-harmonic generation microscopy,” Scanning 23, 182–192 (2001)
[CrossRef] [PubMed]

Terasaki, M.

P. J. Campagnola, A. C. Millard, M. Terasaki, P.E. Hoppe, C. J. Malone, and W. A. Mohler, “Three-dimensional high-resolution second-harmonic generation imaging of endogenous structural proteins in biological tissues,” Biophys. J. 81, 493–508 (2002)
[CrossRef]

Tohno, Y.

T. Yasui, Y. Tohno, and T. Araki, “Characterization of collagen orientation in human dermis by two-dimensional second-harmonic-generation polarimetry,” J. Biomed. Opt. 9, 259–264 (2004)
[CrossRef] [PubMed]

Tsai, H. J.

Tsai, T. H.

S. W. Chu, S. Y. Chen, G. W. Chern, T. H. Tsai, Y. C. Chen, B. L. Lin, and C. K. Sun, “Studies of χ(2)/χ(3) tensors in submicron-scaled bio-tissues by polarization harmonics optical microscopy,” Biophys. J. 86, 3914–3922 (2004)
[CrossRef] [PubMed]

S. W. Chu, S. Y. Chen, T. H. Tsai, T. M. Liu, C. Y. Lin, H. J. Tsai, and C. K. Sun, “In vivo developmental biology study using non-invasive multi-harmonic generation microscopy,” Opt. Express 11, 3093–3099 (2003).
[CrossRef] [PubMed]

Uttenweiler, D.

M. Both, M. Vogel, O. Friedrich, F. von Wegner, T. Kunsting, R. H. A. Fink, and D. Uttenweiler, “Second harmonic imaging of intrinsic signals in muscle fibres in situ,” J. Biomed. Opt. 9, 882–892 (2004)
[CrossRef] [PubMed]

Valentin, C.

S. Kazamias, F. Weihe, D. Douillet, C. Valentin, T. Planchin, S. Sebban, G. Grillon, F. Auge, D. Huilin, and Ph. Baclou, “High order harmonic generation optimization with an apertured laser beam,” Eur. Phys. J. D 21, 353–359 (2002)
[CrossRef]

Vogel, M.

M. Both, M. Vogel, O. Friedrich, F. von Wegner, T. Kunsting, R. H. A. Fink, and D. Uttenweiler, “Second harmonic imaging of intrinsic signals in muscle fibres in situ,” J. Biomed. Opt. 9, 882–892 (2004)
[CrossRef] [PubMed]

von Wegner, F.

M. Both, M. Vogel, O. Friedrich, F. von Wegner, T. Kunsting, R. H. A. Fink, and D. Uttenweiler, “Second harmonic imaging of intrinsic signals in muscle fibres in situ,” J. Biomed. Opt. 9, 882–892 (2004)
[CrossRef] [PubMed]

Webb, W. W.

R. M. Williams, W. R. Zipfel, and W. W. Webb, “Interpreting second-harmonic generation images of collagen I fibrils,” Biophys. J. 88, 1377–1386 (2005)
[CrossRef]

Weihe, F.

S. Kazamias, F. Weihe, D. Douillet, C. Valentin, T. Planchin, S. Sebban, G. Grillon, F. Auge, D. Huilin, and Ph. Baclou, “High order harmonic generation optimization with an apertured laser beam,” Eur. Phys. J. D 21, 353–359 (2002)
[CrossRef]

Williams, R. M.

R. M. Williams, W. R. Zipfel, and W. W. Webb, “Interpreting second-harmonic generation images of collagen I fibrils,” Biophys. J. 88, 1377–1386 (2005)
[CrossRef]

Wolf, E.

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

Xie, X. S.

Yasui, T.

T. Yasui, Y. Tohno, and T. Araki, “Characterization of collagen orientation in human dermis by two-dimensional second-harmonic-generation polarimetry,” J. Biomed. Opt. 9, 259–264 (2004)
[CrossRef] [PubMed]

Zipfel, W. R.

R. M. Williams, W. R. Zipfel, and W. W. Webb, “Interpreting second-harmonic generation images of collagen I fibrils,” Biophys. J. 88, 1377–1386 (2005)
[CrossRef]

Biophys. J. (3)

P. J. Campagnola, A. C. Millard, M. Terasaki, P.E. Hoppe, C. J. Malone, and W. A. Mohler, “Three-dimensional high-resolution second-harmonic generation imaging of endogenous structural proteins in biological tissues,” Biophys. J. 81, 493–508 (2002)
[CrossRef]

S. W. Chu, S. Y. Chen, G. W. Chern, T. H. Tsai, Y. C. Chen, B. L. Lin, and C. K. Sun, “Studies of χ(2)/χ(3) tensors in submicron-scaled bio-tissues by polarization harmonics optical microscopy,” Biophys. J. 86, 3914–3922 (2004)
[CrossRef] [PubMed]

R. M. Williams, W. R. Zipfel, and W. W. Webb, “Interpreting second-harmonic generation images of collagen I fibrils,” Biophys. J. 88, 1377–1386 (2005)
[CrossRef]

Eur. Phys. J. D (1)

S. Kazamias, F. Weihe, D. Douillet, C. Valentin, T. Planchin, S. Sebban, G. Grillon, F. Auge, D. Huilin, and Ph. Baclou, “High order harmonic generation optimization with an apertured laser beam,” Eur. Phys. J. D 21, 353–359 (2002)
[CrossRef]

J. Biomed. Opt. (3)

T. Yasui, Y. Tohno, and T. Araki, “Characterization of collagen orientation in human dermis by two-dimensional second-harmonic-generation polarimetry,” J. Biomed. Opt. 9, 259–264 (2004)
[CrossRef] [PubMed]

M. Both, M. Vogel, O. Friedrich, F. von Wegner, T. Kunsting, R. H. A. Fink, and D. Uttenweiler, “Second harmonic imaging of intrinsic signals in muscle fibres in situ,” J. Biomed. Opt. 9, 882–892 (2004)
[CrossRef] [PubMed]

P. Stoller, B. M. Kim, A. M. Rubenchik, K. M. Reiser, and L. B. Da Silva, “Polarization-dependent optical second-harmonic imaging of rat-tail tendon,” J. Biomed. Opt. 7, 205–214 (2002)
[CrossRef] [PubMed]

J. Chem. Phy. (1)

S. Roth and I. Freund, “Second harmonic generation in collagen,” J. Chem. Phy. 70, 1637–1643 (1979)
[CrossRef]

J. Opt. Soc. Am. B (3)

Micron (1)

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

Nature Med. (1)

E. Brown, T. McKee, E. diTomaso, A. Pluen, B. Seed, Y. Boucher, and R. K. Jain, “Dynamic imaging of collagen and its modulation in tumours in vivo using second-harmonic generation,” Nature Med. 9, 796–800 (2003)
[CrossRef] [PubMed]

Opt. Commun. (1)

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

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. (1)

D. A. Kleinmann, “Nonlinear dielectric polarization in optical media,” Phys. Rev. 126, 1977–1979 (1962)
[CrossRef]

Proc. Roy. Soc. Lond. Ser. A (1)

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

Scanning (1)

C.K. Sun, S.W. Chu, S.P. Tai, S. Keller, A. Abare, U. K. Mishra, and S. P. DenBaars, “Mapping piezoelectric-field distribution in Gallium Nitride with scanning second-harmonic generation microscopy,” Scanning 23, 182–192 (2001)
[CrossRef] [PubMed]

Other (2)

C. T. Tai, Dyadic Green’s Functions in Electromagnetic Theory, (Intext Educational Publishers, 1971)

R. W. Boyd, “Nonlinear Optics,” 2nd Ed(Academic Press, Amsterdam, 2003)

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Figures (7)

Fig. 1.
Fig. 1.

Schematic of the orientation of the subunits (blue arrows). Each subunit is assumed to possess a C6 symmetry with the axis of symmetry indicated by the direction of the arrows. The subunits can be aligned in the (a) z direction and extending in the x direction or, (b) aligned in the x direction extending in the z axis. Axis of propagation is the z axis. Double headed arrows is the direction of polarization.

Fig. 2.
Fig. 2.

Induced SHG polarization in the xy plane corresponding to equation (1), ie the axis of symmetry lies along the z direction. A) |P x SHG|, B) |P y SHG| and C) |P z SHG|. The z axis is in arbitrary units.

Fig. 3.
Fig. 3.

Induced SHG polarization in the xy plane corresponding to equation (2), ie the axis of symmetry lies along the x direction. A) |P x SHG|, B) |P y SHG| and C) |P z SHG|. The z axis is in arbitrary units.

Fig. 4.
Fig. 4.

Radiation pattern of the SHG in the far-field for a single line geometry extending along the x axis for lengths (a) as a dipole, (b) -2.5 to 2.5 and (c) -5 to 5. The axis of symmetry of the collagen is in the z direction. The x, y and z axes are in arbitrary units. Only the radiation in the range 0≤ Φ ≤π has been shown for clarity. Φ is the azimuthal angle of observation.

Fig. 5.
Fig. 5.

Radiation pattern of the SHG in the far-field for a single line geometry extending along the xaxis for lengths (a) as a dipole, (b) -2.5 to 2.5 and (c) -5 to 5. The axis of symmetry of the collagen is in the x direction. The x, y and z axes are in arbitrary units. Only the radiation in the range 0≤ Φ ≤π has been shown for clarity. Φ is the azimuthal angle of observation.

Fig. 6.
Fig. 6.

Radiation pattern of the SHG in the far-field for a single line geometry extending along the z axis for lengths A) as a dipole, B) -2.5 to 2.5 and C) -5 to 5. The axis of symmetry of the collagen is in the z direction. The x, y and z axes are in arbitrary units.

Fig. 7.
Fig. 7.

Radiation pattern of the SHG in the far-field for a single line geometry extending along the z axis for lengths A) as a dipole, B) -2.5 to 2.5 and C) -5 to 5. The axis of symmetry of the collagen is in the x direction. The x, y and z axes are in arbitrary units.

Tables (1)

Tables Icon

Table 1. Approximate magnitudes of |E i E j |

Equations (11)

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[ P x SHG P y SHG P z SHG ] = [ 0 0 0 0 d xxz 0 0 0 0 d yyz 0 0 d zxx d zyy d zzz 0 0 0 ] [ E x E x E y E y E z E z 2 E y E z 2 E x E z 2 E x E y ] .
P x SHG = d xzz E z E z + d xyy E y E y + d xxx E x E x ,
P y SHG = 2 d yyx E y E x ,
P z SHG = 2 d zzx E z E x .
E x u v = i ( I 0 + I 2 cos 2 ϕ ) ,
E y u v = i I 2 sin 2 ϕ ,
E z u v = 2 I 1 cos ϕ .
I 0 u v = 0 α cos 1 / 2 θ sin θ ( 1 + cos θ ) J 0 ( kr sin θ ) exp ( ikz cos θ ) d θ ,
I 1 u v = 0 α cos 1 / 2 θ sin 2 θ J 1 ( kr sin θ ) exp ( ikz cos θ ) d θ ,
I 2 u v = 0 α cos 1 / 2 θ sin θ ( 1 cos θ ) J 2 ( kr sin θ ) exp ( ikz cos θ ) d θ ,
E 2 ω R Θ Φ = exp ( i 2 k R ) R exp ( i 2 k R ̂ r ) × [ 0 0 0 cos Θ cos Φ cos Θ sin Φ sin Θ sin Θ cos Φ 0 ] P ( r ) d V

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