Abstract

A full numerical description of second- and third-harmonic generation (SHG and THG) at the focus of a nonlinear microscope is presented. The numerical implementation takes into account reflections and refraction by an arbitrary number of interfaces perpendicular to the optical axis in the focal region. The calculation of the second- and third-harmonic far-field radiation pattern is based on a Green function approach and is presented for any collection direction. The calculations are sped up by using the chirp-z transform for the focusing fields as well as for the far-field radiation calculation. Numerical evidence is presented for deviations in the measurement of the second-order nonlinear susceptibility ratio ρχyyy(2)/χyxx(2) of collagen fibers in SHG microscopy at high excitation numerical aperture. When interface reflections are taken into account, significant direct backward THG is demonstrated from interfaces and multilayer structures.

© 2013 Optical Society of America

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    [CrossRef]
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2012

H. Kim, G. W. Bryant, and S. J. Stranick, “Superresolution four-wave mixing microscopy,” Opt. Express 20, 6042–6051 (2012).
[CrossRef]

A. E. Tuer, M. K. Akens, S. Krouglov, D. Sandkuijl, B. C. Wilson, C. M. Whyne, and V. Barzda, “Hierarchical model of fibrillar collagen organization for interpreting the second-order susceptibility tensors in biological tissue,” Biophys. J. 103, 2093–2105 (2012).
[CrossRef]

2011

M. J. Farrar, F. W. Wise, J. R. Fetcho, and C. B. Schaffer, “In vivo imaging of myelin in the vertebrate central nervous system using third harmonic generation microscopy,” Biophys. J. 100, 1362–1371 (2011).
[CrossRef]

A. E. Tuer, S. Krouglov, N. Prent, R. Cisek, D. Sandkuijl, K. Yasufuku, B. C. Wilson, and V. Barzda, “Nonlinear optical properties of type I collagen fibers studied by polarization dependent second harmonic generation microscopy,” J. Phys. Chem. B 115, 12759–12769 (2011).
[CrossRef]

2010

2009

S. Y. Chen, H. C. Yu, I. J. Wang, and C. K. Sun, “Infrared-based third and second harmonic generation imaging of cornea,” J. Biomed. Opt. 14, 044012 (2009).
[CrossRef]

2008

J. J. Saarinen and J. E. Sipe, “A Green function approach to surface optics in anisotropic media,” J. Mod. Opt. 55, 13–32 (2008).
[CrossRef]

2007

2006

2005

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]

V. Barzda, “Visualization of mitochondria in cardiomyocytes by simultaneous harmonic generation and fluorescence microscopy,” Opt. Express 13, 8263–8276 (2005).
[CrossRef]

2004

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

2003

P. Stoller, P. M. Celliers, K. M. Reiser, and A. M. Rubenchik, “Quantitative second-harmonic generation microscopy in collagen,” Appl. Opt. 42, 5209–5219 (2003).
[CrossRef]

W. R. Zipfel, R. M. Williams, R. Christie, A. Y. Nikitin, B. T. Hyman, and W. W. Webb, “Live tissue intrinsic emission microscopy using multiphoton-excited native fluorescence and second harmonic generation,” Proc. Natl. Acad. Sci. USA 100, 7075–7080 (2003).
[CrossRef]

2002

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. 82, 493–508 (2002).
[CrossRef]

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]

X. Y. Deng, E. D. Williams, E. W. Thompson, X. Gan, and M. Gu, “Second-harmonic generation from biological tissues: effect of excitation wavelength,” Scanning 24, 175–178 (2002).
[CrossRef]

2000

1998

M. Muller, 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]

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

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]

1989

Y. R. Shen, “Surface-properties probed by second-harmonic and sum-frequency generation,” Nature 337, 519–525 (1989).
[CrossRef]

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]

1985

B. Dick, “Irreducible tensor analysis of sum-frequency and difference-frequency-generation in partially oriented samples,” Chem. Phys. 96, 199–215 (1985).
[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]

L. R. Rabiner, R. W. Schafer, and C. M. Rader, “Chirp z-transform algorithm,” IEEE Trans. Audio Electroacoust. 17, 86–92(1969).
[CrossRef]

1961

G. S. Gotterer, T. E. Thompson, and A. L. Lehninger, “Angular light-scattering studies on isolated mitochondria,” J. Biophys. Biochem. Cytol. 10, 15–21 (1961).
[CrossRef]

Akens, M. K.

A. E. Tuer, M. K. Akens, S. Krouglov, D. Sandkuijl, B. C. Wilson, C. M. Whyne, and V. Barzda, “Hierarchical model of fibrillar collagen organization for interpreting the second-order susceptibility tensors in biological tissue,” Biophys. J. 103, 2093–2105 (2012).
[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]

Barzda, V.

A. E. Tuer, M. K. Akens, S. Krouglov, D. Sandkuijl, B. C. Wilson, C. M. Whyne, and V. Barzda, “Hierarchical model of fibrillar collagen organization for interpreting the second-order susceptibility tensors in biological tissue,” Biophys. J. 103, 2093–2105 (2012).
[CrossRef]

A. E. Tuer, S. Krouglov, N. Prent, R. Cisek, D. Sandkuijl, K. Yasufuku, B. C. Wilson, and V. Barzda, “Nonlinear optical properties of type I collagen fibers studied by polarization dependent second harmonic generation microscopy,” J. Phys. Chem. B 115, 12759–12769 (2011).
[CrossRef]

V. Barzda, “Visualization of mitochondria in cardiomyocytes by simultaneous harmonic generation and fluorescence microscopy,” Opt. Express 13, 8263–8276 (2005).
[CrossRef]

R. Cisek, N. Prent, C. Greenhalgh, D. Sandkuijl, A. Tuer, A. Major, and V. Barzda, “Multicontrast nonlinear imaging microscopy,” in Biochemical Applications of Nonlinear Optical Spectroscopy, V. V. Yakovlev, ed. (CRC, 2009).

Beaurepaire, E.

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1997).

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, 2008).

Brakenhoff, G. J.

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

M. Muller, 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]

Bryant, G. W.

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. 82, 493–508 (2002).
[CrossRef]

Celliers, P. M.

Chang, C. F.

Chang, F. H.

Chen, C. Y.

Chen, H. C.

Chen, M. J.

Chen, S. Y.

S. Y. Chen, H. C. Yu, I. J. Wang, and C. K. Sun, “Infrared-based third and second harmonic generation imaging of cornea,” J. Biomed. Opt. 14, 044012 (2009).
[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 x(2)/x(3) tensors in submicron-scaled bio-tissues by polarization harmonics optical microscopy,” Biophys. J. 86, 3914–3922 (2004).
[CrossRef]

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 x(2)/x(3) tensors in submicron-scaled bio-tissues by polarization harmonics optical microscopy,” Biophys. J. 86, 3914–3922 (2004).
[CrossRef]

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 x(2)/x(3) tensors in submicron-scaled bio-tissues by polarization harmonics optical microscopy,” Biophys. J. 86, 3914–3922 (2004).
[CrossRef]

Christie, R.

W. R. Zipfel, R. M. Williams, R. Christie, A. Y. Nikitin, B. T. Hyman, and W. W. Webb, “Live tissue intrinsic emission microscopy using multiphoton-excited native fluorescence and second harmonic generation,” Proc. Natl. Acad. Sci. USA 100, 7075–7080 (2003).
[CrossRef]

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 x(2)/x(3) tensors in submicron-scaled bio-tissues by polarization harmonics optical microscopy,” Biophys. J. 86, 3914–3922 (2004).
[CrossRef]

Cisek, R.

A. E. Tuer, S. Krouglov, N. Prent, R. Cisek, D. Sandkuijl, K. Yasufuku, B. C. Wilson, and V. Barzda, “Nonlinear optical properties of type I collagen fibers studied by polarization dependent second harmonic generation microscopy,” J. Phys. Chem. B 115, 12759–12769 (2011).
[CrossRef]

R. Cisek, N. Prent, C. Greenhalgh, D. Sandkuijl, A. Tuer, A. Major, and V. Barzda, “Multicontrast nonlinear imaging microscopy,” in Biochemical Applications of Nonlinear Optical Spectroscopy, V. V. Yakovlev, ed. (CRC, 2009).

Debarre, D.

Deng, X. Y.

X. Y. Deng, E. D. Williams, E. W. Thompson, X. Gan, and M. Gu, “Second-harmonic generation from biological tissues: effect of excitation wavelength,” Scanning 24, 175–178 (2002).
[CrossRef]

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]

Dick, B.

B. Dick, “Irreducible tensor analysis of sum-frequency and difference-frequency-generation in partially oriented samples,” Chem. Phys. 96, 199–215 (1985).
[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]

Farrar, M. J.

M. J. Farrar, F. W. Wise, J. R. Fetcho, and C. B. Schaffer, “In vivo imaging of myelin in the vertebrate central nervous system using third harmonic generation microscopy,” Biophys. J. 100, 1362–1371 (2011).
[CrossRef]

Fetcho, J. R.

M. J. Farrar, F. W. Wise, J. R. Fetcho, and C. B. Schaffer, “In vivo imaging of myelin in the vertebrate central nervous system using third harmonic generation microscopy,” Biophys. J. 100, 1362–1371 (2011).
[CrossRef]

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]

Gan, X.

X. Y. Deng, E. D. Williams, E. W. Thompson, X. Gan, and M. Gu, “Second-harmonic generation from biological tissues: effect of excitation wavelength,” Scanning 24, 175–178 (2002).
[CrossRef]

Gotterer, G. S.

G. S. Gotterer, T. E. Thompson, and A. L. Lehninger, “Angular light-scattering studies on isolated mitochondria,” J. Biophys. Biochem. Cytol. 10, 15–21 (1961).
[CrossRef]

Greenhalgh, C.

R. Cisek, N. Prent, C. Greenhalgh, D. Sandkuijl, A. Tuer, A. Major, and V. Barzda, “Multicontrast nonlinear imaging microscopy,” in Biochemical Applications of Nonlinear Optical Spectroscopy, V. V. Yakovlev, ed. (CRC, 2009).

Greenhalgh, C. A.

C. A. Greenhalgh, “Nonlinear multicontrast microscopy for structural and dynamic investigations of myocytes,” in Physics (University of Toronto, 2009).

Gu, M.

X. Y. Deng, E. D. Williams, E. W. Thompson, X. Gan, and M. Gu, “Second-harmonic generation from biological tissues: effect of excitation wavelength,” Scanning 24, 175–178 (2002).
[CrossRef]

Hecht, B.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).

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. 82, 493–508 (2002).
[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]

Hsieh, W. F.

Hsu, C. H.

Hyman, B. T.

W. R. Zipfel, R. M. Williams, R. Christie, A. Y. Nikitin, B. T. Hyman, and W. W. Webb, “Live tissue intrinsic emission microscopy using multiphoton-excited native fluorescence and second harmonic generation,” Proc. Natl. Acad. Sci. USA 100, 7075–7080 (2003).
[CrossRef]

Jackson, J. D.

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

Kim, H.

Krishnamachari, V. V.

Krouglov, S.

A. E. Tuer, M. K. Akens, S. Krouglov, D. Sandkuijl, B. C. Wilson, C. M. Whyne, and V. Barzda, “Hierarchical model of fibrillar collagen organization for interpreting the second-order susceptibility tensors in biological tissue,” Biophys. J. 103, 2093–2105 (2012).
[CrossRef]

A. E. Tuer, S. Krouglov, N. Prent, R. Cisek, D. Sandkuijl, K. Yasufuku, B. C. Wilson, and V. Barzda, “Nonlinear optical properties of type I collagen fibers studied by polarization dependent second harmonic generation microscopy,” J. Phys. Chem. B 115, 12759–12769 (2011).
[CrossRef]

Lasser, T.

Lehninger, A. L.

G. S. Gotterer, T. E. Thompson, and A. L. Lehninger, “Angular light-scattering studies on isolated mitochondria,” J. Biophys. Biochem. Cytol. 10, 15–21 (1961).
[CrossRef]

Leitgeb, R. A.

Leutenegger, M.

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 x(2)/x(3) tensors in submicron-scaled bio-tissues by polarization harmonics optical microscopy,” Biophys. J. 86, 3914–3922 (2004).
[CrossRef]

Liu, W. R.

Major, A.

R. Cisek, N. Prent, C. Greenhalgh, D. Sandkuijl, A. Tuer, A. Major, and V. Barzda, “Multicontrast nonlinear imaging microscopy,” in Biochemical Applications of Nonlinear Optical Spectroscopy, V. V. Yakovlev, ed. (CRC, 2009).

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. 82, 493–508 (2002).
[CrossRef]

Mertz, J.

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. 82, 493–508 (2002).
[CrossRef]

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. 82, 493–508 (2002).
[CrossRef]

Moreaux, L.

Muller, M.

M. Muller, 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]

J. A. Squier, M. Muller, G. J. Brakenhoff, and K. R. Wilson, “Third harmonic generation microscopy,” Opt. Express 3, 315–324 (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]

Nikitin, A. Y.

W. R. Zipfel, R. M. Williams, R. Christie, A. Y. Nikitin, B. T. Hyman, and W. W. Webb, “Live tissue intrinsic emission microscopy using multiphoton-excited native fluorescence and second harmonic generation,” Proc. Natl. Acad. Sci. USA 100, 7075–7080 (2003).
[CrossRef]

Novotny, L.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).

Olivier, N.

Potma, E. O.

Prent, N.

A. E. Tuer, S. Krouglov, N. Prent, R. Cisek, D. Sandkuijl, K. Yasufuku, B. C. Wilson, and V. Barzda, “Nonlinear optical properties of type I collagen fibers studied by polarization dependent second harmonic generation microscopy,” J. Phys. Chem. B 115, 12759–12769 (2011).
[CrossRef]

R. Cisek, N. Prent, C. Greenhalgh, D. Sandkuijl, A. Tuer, A. Major, and V. Barzda, “Multicontrast nonlinear imaging microscopy,” in Biochemical Applications of Nonlinear Optical Spectroscopy, V. V. Yakovlev, ed. (CRC, 2009).

Rabiner, L. R.

L. R. Rabiner, R. W. Schafer, and C. M. Rader, “Chirp z-transform algorithm,” IEEE Trans. Audio Electroacoust. 17, 86–92(1969).
[CrossRef]

Rader, C. M.

L. R. Rabiner, R. W. Schafer, and C. M. Rader, “Chirp z-transform algorithm,” IEEE Trans. Audio Electroacoust. 17, 86–92(1969).
[CrossRef]

Rao, R.

Recher, G.

Reintjes, J. F.

J. F. Reintjes, Nonlinear Optical Parametric Processes in Liquids and Gases (Academic, 1984).

Reiser, K. M.

Rouede, D.

Rubenchik, A. M.

Saarinen, J. J.

J. J. Saarinen and J. E. Sipe, “A Green function approach to surface optics in anisotropic media,” J. Mod. Opt. 55, 13–32 (2008).
[CrossRef]

Sandkuijl, D.

A. E. Tuer, M. K. Akens, S. Krouglov, D. Sandkuijl, B. C. Wilson, C. M. Whyne, and V. Barzda, “Hierarchical model of fibrillar collagen organization for interpreting the second-order susceptibility tensors in biological tissue,” Biophys. J. 103, 2093–2105 (2012).
[CrossRef]

A. E. Tuer, S. Krouglov, N. Prent, R. Cisek, D. Sandkuijl, K. Yasufuku, B. C. Wilson, and V. Barzda, “Nonlinear optical properties of type I collagen fibers studied by polarization dependent second harmonic generation microscopy,” J. Phys. Chem. B 115, 12759–12769 (2011).
[CrossRef]

R. Cisek, N. Prent, C. Greenhalgh, D. Sandkuijl, A. Tuer, A. Major, and V. Barzda, “Multicontrast nonlinear imaging microscopy,” in Biochemical Applications of Nonlinear Optical Spectroscopy, V. V. Yakovlev, ed. (CRC, 2009).

Sandre, O.

Schafer, R. W.

L. R. Rabiner, R. W. Schafer, and C. M. Rader, “Chirp z-transform algorithm,” IEEE Trans. Audio Electroacoust. 17, 86–92(1969).
[CrossRef]

Schaffer, C. B.

M. J. Farrar, F. W. Wise, J. R. Fetcho, and C. B. Schaffer, “In vivo imaging of myelin in the vertebrate central nervous system using third harmonic generation microscopy,” Biophys. J. 100, 1362–1371 (2011).
[CrossRef]

Shen, Y. R.

Y. R. Shen, “Surface-properties probed by second-harmonic and sum-frequency generation,” Nature 337, 519–525 (1989).
[CrossRef]

Silberberg, 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]

Sipe, J. E.

J. J. Saarinen and J. E. Sipe, “A Green function approach to surface optics in anisotropic media,” J. Mod. Opt. 55, 13–32 (2008).
[CrossRef]

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]

Squier, J.

M. Muller, 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]

Squier, J. A.

Stoller, P.

Stranick, S. J.

Sun, C. K.

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[CrossRef]

S. Y. Chen, H. C. Yu, I. J. Wang, and C. K. Sun, “Infrared-based third and second harmonic generation imaging of cornea,” J. Biomed. Opt. 14, 044012 (2009).
[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 x(2)/x(3) tensors in submicron-scaled bio-tissues by polarization harmonics optical microscopy,” Biophys. J. 86, 3914–3922 (2004).
[CrossRef]

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. 82, 493–508 (2002).
[CrossRef]

Thompson, E. W.

X. Y. Deng, E. D. Williams, E. W. Thompson, X. Gan, and M. Gu, “Second-harmonic generation from biological tissues: effect of excitation wavelength,” Scanning 24, 175–178 (2002).
[CrossRef]

Thompson, T. E.

G. S. Gotterer, T. E. Thompson, and A. L. Lehninger, “Angular light-scattering studies on isolated mitochondria,” J. Biophys. Biochem. Cytol. 10, 15–21 (1961).
[CrossRef]

Tiaho, F.

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 x(2)/x(3) tensors in submicron-scaled bio-tissues by polarization harmonics optical microscopy,” Biophys. J. 86, 3914–3922 (2004).
[CrossRef]

Tuer, A.

R. Cisek, N. Prent, C. Greenhalgh, D. Sandkuijl, A. Tuer, A. Major, and V. Barzda, “Multicontrast nonlinear imaging microscopy,” in Biochemical Applications of Nonlinear Optical Spectroscopy, V. V. Yakovlev, ed. (CRC, 2009).

Tuer, A. E.

A. E. Tuer, M. K. Akens, S. Krouglov, D. Sandkuijl, B. C. Wilson, C. M. Whyne, and V. Barzda, “Hierarchical model of fibrillar collagen organization for interpreting the second-order susceptibility tensors in biological tissue,” Biophys. J. 103, 2093–2105 (2012).
[CrossRef]

A. E. Tuer, S. Krouglov, N. Prent, R. Cisek, D. Sandkuijl, K. Yasufuku, B. C. Wilson, and V. Barzda, “Nonlinear optical properties of type I collagen fibers studied by polarization dependent second harmonic generation microscopy,” J. Phys. Chem. B 115, 12759–12769 (2011).
[CrossRef]

Wang, I. J.

S. Y. Chen, H. C. Yu, I. J. Wang, and C. K. Sun, “Infrared-based third and second harmonic generation imaging of cornea,” J. Biomed. Opt. 14, 044012 (2009).
[CrossRef]

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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]

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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]

W. R. Zipfel, R. M. Williams, R. Christie, A. Y. Nikitin, B. T. Hyman, and W. W. Webb, “Live tissue intrinsic emission microscopy using multiphoton-excited native fluorescence and second harmonic generation,” Proc. Natl. Acad. Sci. USA 100, 7075–7080 (2003).
[CrossRef]

Whyne, C. M.

A. E. Tuer, M. K. Akens, S. Krouglov, D. Sandkuijl, B. C. Wilson, C. M. Whyne, and V. Barzda, “Hierarchical model of fibrillar collagen organization for interpreting the second-order susceptibility tensors in biological tissue,” Biophys. J. 103, 2093–2105 (2012).
[CrossRef]

Williams, E. D.

X. Y. Deng, E. D. Williams, E. W. Thompson, X. Gan, and M. Gu, “Second-harmonic generation from biological tissues: effect of excitation wavelength,” Scanning 24, 175–178 (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]

W. R. Zipfel, R. M. Williams, R. Christie, A. Y. Nikitin, B. T. Hyman, and W. W. Webb, “Live tissue intrinsic emission microscopy using multiphoton-excited native fluorescence and second harmonic generation,” Proc. Natl. Acad. Sci. USA 100, 7075–7080 (2003).
[CrossRef]

Wilson, B. C.

A. E. Tuer, M. K. Akens, S. Krouglov, D. Sandkuijl, B. C. Wilson, C. M. Whyne, and V. Barzda, “Hierarchical model of fibrillar collagen organization for interpreting the second-order susceptibility tensors in biological tissue,” Biophys. J. 103, 2093–2105 (2012).
[CrossRef]

A. E. Tuer, S. Krouglov, N. Prent, R. Cisek, D. Sandkuijl, K. Yasufuku, B. C. Wilson, and V. Barzda, “Nonlinear optical properties of type I collagen fibers studied by polarization dependent second harmonic generation microscopy,” J. Phys. Chem. B 115, 12759–12769 (2011).
[CrossRef]

Wilson, K. R.

M. Muller, 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]

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

Wise, F. W.

M. J. Farrar, F. W. Wise, J. R. Fetcho, and C. B. Schaffer, “In vivo imaging of myelin in the vertebrate central nervous system using third harmonic generation microscopy,” Biophys. J. 100, 1362–1371 (2011).
[CrossRef]

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M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1997).

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Yasufuku, K.

A. E. Tuer, S. Krouglov, N. Prent, R. Cisek, D. Sandkuijl, K. Yasufuku, B. C. Wilson, and V. Barzda, “Nonlinear optical properties of type I collagen fibers studied by polarization dependent second harmonic generation microscopy,” J. Phys. Chem. B 115, 12759–12769 (2011).
[CrossRef]

Yu, C. H.

Yu, H. C.

S. Y. Chen, H. C. Yu, I. J. Wang, and C. K. Sun, “Infrared-based third and second harmonic generation imaging of cornea,” J. Biomed. Opt. 14, 044012 (2009).
[CrossRef]

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]

W. R. Zipfel, R. M. Williams, R. Christie, A. Y. Nikitin, B. T. Hyman, and W. W. Webb, “Live tissue intrinsic emission microscopy using multiphoton-excited native fluorescence and second harmonic generation,” Proc. Natl. Acad. Sci. USA 100, 7075–7080 (2003).
[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, 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. 82, 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 x(2)/x(3) tensors in submicron-scaled bio-tissues by polarization harmonics optical microscopy,” Biophys. J. 86, 3914–3922 (2004).
[CrossRef]

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]

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]

A. E. Tuer, M. K. Akens, S. Krouglov, D. Sandkuijl, B. C. Wilson, C. M. Whyne, and V. Barzda, “Hierarchical model of fibrillar collagen organization for interpreting the second-order susceptibility tensors in biological tissue,” Biophys. J. 103, 2093–2105 (2012).
[CrossRef]

M. J. Farrar, F. W. Wise, J. R. Fetcho, and C. B. Schaffer, “In vivo imaging of myelin in the vertebrate central nervous system using third harmonic generation microscopy,” Biophys. J. 100, 1362–1371 (2011).
[CrossRef]

Chem. Phys.

B. Dick, “Irreducible tensor analysis of sum-frequency and difference-frequency-generation in partially oriented samples,” Chem. Phys. 96, 199–215 (1985).
[CrossRef]

IEEE Trans. Audio Electroacoust.

L. R. Rabiner, R. W. Schafer, and C. M. Rader, “Chirp z-transform algorithm,” IEEE Trans. Audio Electroacoust. 17, 86–92(1969).
[CrossRef]

J. Biomed. Opt.

S. Y. Chen, H. C. Yu, I. J. Wang, and C. K. Sun, “Infrared-based third and second harmonic generation imaging of cornea,” J. Biomed. Opt. 14, 044012 (2009).
[CrossRef]

J. Biophys. Biochem. Cytol.

G. S. Gotterer, T. E. Thompson, and A. L. Lehninger, “Angular light-scattering studies on isolated mitochondria,” J. Biophys. Biochem. Cytol. 10, 15–21 (1961).
[CrossRef]

J. Microsc.

M. Muller, 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]

J. Mod. Opt.

J. J. Saarinen and J. E. Sipe, “A Green function approach to surface optics in anisotropic media,” J. Mod. Opt. 55, 13–32 (2008).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

J. Phys. Chem. B

A. E. Tuer, S. Krouglov, N. Prent, R. Cisek, D. Sandkuijl, K. Yasufuku, B. C. Wilson, and V. Barzda, “Nonlinear optical properties of type I collagen fibers studied by polarization dependent second harmonic generation microscopy,” J. Phys. Chem. B 115, 12759–12769 (2011).
[CrossRef]

Nature

Y. R. Shen, “Surface-properties probed by second-harmonic and sum-frequency generation,” Nature 337, 519–525 (1989).
[CrossRef]

Opt. Express

Phys. Rev.

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]

Proc. Natl. Acad. Sci. USA

W. R. Zipfel, R. M. Williams, R. Christie, A. Y. Nikitin, B. T. Hyman, and W. W. Webb, “Live tissue intrinsic emission microscopy using multiphoton-excited native fluorescence and second harmonic generation,” Proc. Natl. Acad. Sci. USA 100, 7075–7080 (2003).
[CrossRef]

Scanning

X. Y. Deng, E. D. Williams, E. W. Thompson, X. Gan, and M. Gu, “Second-harmonic generation from biological tissues: effect of excitation wavelength,” Scanning 24, 175–178 (2002).
[CrossRef]

Other

R. W. Boyd, Nonlinear Optics (Academic, 2008).

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

University of Toronto, “Welcome to T-Space,” https://tspace.library.utoronto.ca .

This ratio is often defined in literature for a cylindrically symmetric structure oriented along the z axis [27], taking the y axis as the optical axis. In this article, however, the z axis is the optical axis, and the cylindrical structures will be oriented in the xy plane (as is common in numerical modeling of the focal volume). Therefore, we opt to define the ratio with respect to the fibril coordinate frame: ρ≡χyyy(2)/χyxx(2), where y refers to the axis along the cylindrical structure.

J. F. Reintjes, Nonlinear Optical Parametric Processes in Liquids and Gases (Academic, 1984).

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1997).

C. A. Greenhalgh, “Nonlinear multicontrast microscopy for structural and dynamic investigations of myocytes,” in Physics (University of Toronto, 2009).

R. Cisek, N. Prent, C. Greenhalgh, D. Sandkuijl, A. Tuer, A. Major, and V. Barzda, “Multicontrast nonlinear imaging microscopy,” in Biochemical Applications of Nonlinear Optical Spectroscopy, V. V. Yakovlev, ed. (CRC, 2009).

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).

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Figures (8)

Fig. 1.
Fig. 1.

Model of the nonlinear microscope layout with excitation and detection pathways. (a) Excitation pathway. The collimated incoming beam is transformed into a focusing beam by the excitation objective, and the focusing beam propagates to the focal point. The sample is located in the focal point. In the sample calculations presented in this work, the sample will be either a single collagen fiber or a layered structure. Example interfaces are shown in the focal region. (b) Emission and collection pathways. The nonlinear polarization distribution is calculated from the electric field distribution at the focus and the nonlinear susceptibility distribution, and the far-field harmonic electric field due to this distribution is calculated for several collection directions (most importantly, forward and backward). The collection objective(s) then collimates the wavefronts. The numbers in parentheses refer to the equations in the text describing the relevant transformations.

Fig. 2.
Fig. 2.

Layout of the multilayer structure located in the focal region. Layer 1 is encountered first by the focusing beam, while layer N is encountered last. The interface between layer m and m+1 is located at dm. Every layer except for layer N contains both a forward- and backward-propagating electric field Em+ and Em, respectively. A single component of the Fourier spectrum of the incoming electric field is shown for illustration purposes.

Fig. 3.
Fig. 3.

Extracted ratio of the yyy and yxx elements of the χ(2) tensor based on a plane-wave approximation, as a function of the collection NA, for three different excitation NAs. Using high NA collection optics leads to an overestimation of the ratio, while high NA excitation combined with low NA detection leads to an underestimation of the ratio. (Inset) Schematic overview of the experimental setup with the collagen fiber at the focus.

Fig. 4.
Fig. 4.

Summary of SHG for determination of the nonlinear susceptibility ratio ρ, using 1.3 NA excitation and 0.8 NA collection objectives. The color map runs from blue (lowest amplitude/intensity) to red (highest amplitude/intensity). (a) Electric field distribution at the focal plane (z=0) for the case of incoming fundamental polarization parallel to the fiber orientation (y), showing y- and z-polarized components separately. (b) Same for the case of incoming fundamental polarization perpendicular to the fiber orientation. (c) Generated nonlinear polarization in the 200 nm diameter collagen fiber for the focal electric field from (a), showing y- and z-polarized components separately. (d) Same for the focal electric field from (b). (e) Far-field SHG intensity distribution in the far field, for excitation polarization along the fiber orientation (y). (g) Same for the nonlinear polarization distribution from (d). (f) Nonlinear susceptibility ratio ρ=I/I along the line kx=0, which is the intensity distribution in the zy plane (this plot remains approximately the same for all values of kx). The nonlinear susceptibility ratio should be equal to 1.4 (dotted line), but the z-polarized electric focal field component is responsible for the deviations. Note that the axis limits for (a)–(d) are identical.

Fig. 5.
Fig. 5.

(a) Forward and (b) backward THG from a single glass/medium interface. The refractive index and third-order nonlinear susceptibility are varied from 1.0 to 1.4632 (nω of glass) and from 0 to χglass(3), respectively. All intensity axes are logarithmic. (c) The ratio of forward to backward THG varies by 5 orders of magnitude from 0.17 to 1.8·104. The asterisk indicates the refractive index and nonlinear susceptibility of water. (d) The forward (top) and backward (bottom) far-field THG intensity distribution for the glass/medium interface with refractive index and nonlinear susceptibility closest to that of water.

Fig. 6.
Fig. 6.

(a) Forward and (b) backward THG intensity from a lipid layer of varying thickness (x axis) and varying position (y axis) with respect to the focal spot center. The approximate interface positions are indicated by the solid lines when the layer center is positioned at the center of the focal spot [dashed lines in (a) and (b)]. The FWHM of the intensity of the fundamental focal field along the optical axis is approximately 3.9 μm. (c) THG intensity dependence as a function of the layer thickness with the layer located exactly at the center of the focal spot, as indicated by the dashed line in (a) and (b). Rapid variation of the backward THG as a function of the layer thickness is evident, and interference with the forward THG can be observed (small rapid amplitude variations).

Fig. 7.
Fig. 7.

Forward and backward THG from 10 water–lipid–water multilayers. Lipid layer thickness is fixed at 10 nm, and the water layer thickness is varied (x axis). Note the logarithmic y axis.

Fig. 8.
Fig. 8.

Schematic overview of sideways radiation calculation. The forward and backward far-field radiation are calculated via the Fourier transform of the nonlinear polarization distribution in the xy plane (blue plane, dashed border), while for sideways radiation the Fourier transform is evaluated in the plane perpendicular to the propagation direction; i.e., for radiation in the ±x direction the Fourier transform is evaluated in the yz plane (red plane, dotted border).

Equations (39)

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

r⃗=xx^+yy^+zz^R⃗+zz^,
k⃗=kxx^+kyy^+kzz^κ⃗+kzz^,wherekz=k2κ2andκκ⃗·κ⃗,
s^(κ⃗)κ^×z^=(kykx0)1κandp^±(κ⃗)k1(κz^kzκ^)=(kzkxkzkyκ2)1kκ.
E⃗inp,HG00(R⃗inp)=E0exp(xinp2+yinp2f02Robj2)x^,
κ⃗=k0NAR⃗inpRobj,
ejs,p(κ⃗;z)=[Ej+s,p(κ⃗)eikz,jzEjs,p(κ⃗)eikz,jz].
E1+s(κ⃗)=2πifeik1fn1kz,1k11κ(kyEinpx(R⃗inp)kxEinpy(R⃗inp)),E1+p(κ⃗)=2πifeik1fn1kz,1k11κ(kxEinpx(R⃗inp)kyEinpy(R⃗inp)),
em(z)=Mm,1(z,z)e1(z).
E1=E1+R1,N(d1,dN1+)e2ikz,1d1,
Efocal,m(r⃗)=14π2kx,ky[Em+eikz,mz+Emeikz,mz]eiκ⃗·R⃗dκ⃗.
PiSHG(r⃗)=j,kχijk(2,SHG)(r⃗)Eω,j(r⃗)Eω,k(r⃗),PiTHG(r⃗)=χ(3,THG)(r⃗)(3Eω,i3(r⃗)+jiEω,i(r⃗)Eω,j2(r⃗)).
P⃗(r⃗)=14π2P⃗(κ⃗;z)eiκ⃗·R⃗dκ⃗.
E⃗FF(r⃗)=14π2E⃗FF(κ⃗;z)eiκ⃗·R⃗dκ⃗.
E1(κ⃗)eikz,1d1=dzTj,1(z,d1)1Rj,1(z,d1)Rj,N(z,dN1+)[v+v+Rj,N(z,dN1+)],EN+(κ⃗)eikz,NdN1=dzTj,N(z,dN1+)1Rj,1(z,d1)Rj,N(z,dN1+)[v++vRj,1(z,d1)].
Tj,N(z,dN1+)[v++vRj,1(z,d1)]1Rj,1(z,d1)Rj,N(z,dN1+)=Tj,N(dj,dN1+)eikz,jdj(v+eikz,jz+vRj,1(dj1+,d1)eikz,j(z2dj1+))1Rj,1(dj1+,d1)Rj,N(dj,dN1+)e2ikz,j(djdj1).
Ecollx(R⃗out)=ieikNf2πfnNkNkz,Nκ(kyEN+s(κ⃗)kxEN+p(κ⃗)),Ecolly(R⃗out)=ieikNf2πfnNkNkz,Nκ(kxEN+s(κ⃗)kyEN+p(κ⃗)),
Xk=n=1NxnAkWkn.
Ax=eiΔkxxinitandWx=eiΔkx(nx1)(xfinalxinit).
ATHG,FW=i3ωAω32c{χgl(3)n3ω,gl0exp(iΔkglz)dz(1+2iz/b)2+χmed(3)n3ω,med0exp(iΔkmedz)dz(1+2iz/b)2},
E⃗j(r⃗)=dκ⃗4π2E⃗j(κ⃗;z)eiκ⃗·R⃗,
E⃗j(κ⃗;z)=E⃗j+(z)+E⃗j(z).
E⃗j±(z)=(s^Ej±s+p^±Ej±p)e±ikz,jz,
ejs,p(z)=[Ej+s,peikz,jzEjs,peikz,jz]=ej(z).
ej(z)=[Ej+eikz,jzEjeikz,jz]=[eikz,j(zz)00eikz,j(zz)][Ej+eikz,jzEjeikz,jz]Mj(zz)ej(z).
ej+1(dj+)=[Ej+1,+eikz,j+1dj+Ej+1,eikz,j+1dj+]=1tj+1,j[1rj+1,jrj+1,j1][Ej+eikz,jdjEjeikz,jdj]Mj+1,jej(dj).
eN(z)=MN(zdN1)MN,N1MN1(dN1dN2)Mj+1,jMj(djz)ej(z)MN,j(z,z)ej(z).
MN,j(z,z)=1TN,j(z,z)[TN,j(z,z)Tj,N(z,z)RN,j(z,z)Rj,N(z,z)RN,j(z,z)Rj,N(z,z)1].
P⃗j(r⃗)=dκ⃗4π2P⃗j(κ⃗;z)eiκ⃗·R⃗.
P⃗j(r⃗)=P⃗δ(zz0)eiκ⃗·R⃗.
E⃗(z)=ik022ε0kz,jθ(zz0)eikz,j(zz0)(s^s^+p^j+p^j+)·P⃗+ik022ε0kz,jθ(z0z)eikz,j(zz0)(s^s^+p^jp^j)·P⃗1ε0εjδ(zz0)z^z^·P⃗.
ej(z0+)=v+ej(z0),
eN(dN1+)=MN,j(dN1+,z0+)ej(z0+)andej(z0)=Mj,1(z0,d1)e1(d1).
eN(dN1+)=MN,j(dN1+,z0+)v+MN,1(dN1+,d1)e1(d1).
eN(dN1+)=[EN+eikz,NdN10]ande1(d1)=[0E1eikz,1d1].
E1eik1,zd1=Tj,1(z0,d1)1Rj,1(z0,d1)Rj,N(z0,dN1+)[v+v+Rj,N(z0,dN1+)],EN+eikN,zdN1=Tj,N(z0,dN1+)1Rj,1(z0,d1)Rj,N(z0,dN1+)[v++vRj,1(z0,d1)].
E1(κ⃗)eikz,1d1=dzTj,1(z,d1)1Rj,1(z,d1)Rj,N(z,dN1+)(v+v+Rj,N(z,dN1+)),EN+(κ⃗)eikz,NdN1=dzTj,N(z,dN1+)1Rj,1(z,d1)Rj,N(z,dN1+)(v++vRj,1(z,d1)).
v±=2πik02kz,ju^j±·P⃗j(κ⃗;z).
Ex(ky,kz)=dxeikxxvx,Ex+(ky,kz)=dxeikxxvx+,Ey(kx,kz)=dyeikyyvy,Ey+(kx,kz)=dyeikyyvy+
vx±=2πik02kxu^x±·P⃗(ky,kz;x),vy±=2πik02kyu^y±·P⃗(kx,kz;y).

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