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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References

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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 chi(2)/chi(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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