Abstract

In this study we present combined third-harmonic generation (THG) and second-harmonic generation (SHG) microscopy images of intact human corneas, and we analyze experimentally and theoretically the origin of the THG signal. Multiharmonic microscopy provides detailed images of the cornea microstructure over its entire thickness. A component of the THG signal originates from cellular structures and another one originates from anisotropy changes between successive collagen lamellae in the stroma. This anisotropy-related signal can be specifically detected using circular incident polarization, and provide contrasted images of the stacking and tissue-scale heterogeneity of stromal lamellae. Forward-radiated THG and SHG signals are generally anticorrelated, indicating that maximum THG is obtained from lamellar interfaces whereas maximum SHG is obtained from within lamellae. Polarization-resolved THG imaging reflects the alternate anisotropy directions of the lamellae. We present a model for THG imaging of layered anisotropic samples and numerical calculations that account for our observations.

© 2010 Optical Society of America

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2010 (1)

F. Aptel, N. Olivier, A. Deniset-Besseau, J.-M. Legeais, K. Plamann, M.-C. Schanne-Klein, and E. Beaurepaire, "Multimodal nonlinear imaging of the human cornea," Invest. Ophthalmol. Vis. Sci.  51 (2010, in press).
[CrossRef] [PubMed]

2009 (3)

P. Matteini, F. Ratto, F. Rossi, R. Cicchi, C. Stringari, D. Kapsokalyvas, F. S. Pavone, and R. Pini, "Photothermally induced disordered patterns of corneal collagen revealed by SHG imaging," Opt. Express 17(6), 4868-4878 (2009).
[CrossRef] [PubMed]

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(4), 044012 (2009).
[CrossRef] [PubMed]

H. Soong and J. Malta, "Femtosecond lasers in ophthalmology," Am. J. Ophthalmol. 147(2), 189-197 (2009).
[CrossRef]

2008 (6)

R. W. Knighton, X. R. Huang, and L. A. Cavuoto, "Corneal birefringence mapped by scanning laser polarimetry," Opt. Express 16(18), 13,738-13,751 (2008).
[CrossRef]

N. Olivier and E. Beaurepaire, "Third-harmonic generation microscopy with focus-engineered beams: a numerical study," Opt. Express 16(19), 14,703-14,715 (2008).
[CrossRef]

X. Han, R. M. Burke, M. L. Zettel, P. Tang, and E. B. Brown, "Second harmonic properties of tumor collagen: determining the structural relationship between reactive stroma and healthy stroma," Opt. Express 16(3), 1846-1859 (2008).
[CrossRef] [PubMed]

Q. F. Wu and A. T. Yeh, "Rabbit cornea microstructure response to changes intraocular pressure visualized by using nonlinear optical microscopy," Cornea 27(2), 202-208 (2008).
[CrossRef] [PubMed]

L. Jay, A. Brocas, K. Singh, J.-C. Kieffer, I. Brunette, and T. Ozaki, "Determination of porcine corneal layers with high spatial resolution by simultaneous second and third harmonic generation microscopy," Opt. Express 16(21), 16284-16293 (2008).
[CrossRef] [PubMed]

W. Drexler and J. Fujimoto, "State-of-the-art retinal optical coherence tomography," Prog. Retin. Eye Res. 27(1), 45-88 (2008).
[CrossRef]

2007 (3)

N. Morishige, A. Wahlert, M. Kenney, D. Brown, K. Kawamoto, T. Chikama, T. Nishida, and J. Jester, "Secondharmonic imaging microscopy of normal human and keratoconus cornea," Invest. Ophthalmol. Vis. Sci. 48(3), 1087-94 (2007).
[CrossRef] [PubMed]

M. Strupler, A.-M. Pena, M. Hernest, P.-L. Tharaux, J.-L. Martin, E. Beaurepaire, and M.-C. Schanne-Klein, "Second harmonic imaging and scoring of collagen in fibrotic tissues," Opt. Express 15(7), 4054-4065 (2007).
[CrossRef] [PubMed]

D. Debarre and E. Beaurepaire, "Quantitative characterization of biological liquids for third-harmonic generation microscopy," Biophys. J. 92(2), 603-612 (2007).
[CrossRef]

2006 (5)

S.-P. Tai, W.-J. Lee, D.-B. Shieh, P.-C. Wu, H.-Y. Huang, C.-H. Yu, and C.-K. Sun, "In vivo optical biopsy of hamster oral cavity with epi-third-harmonic generation microscopy," Opt. Express 14(13), 6178-6187 (2006).
[CrossRef] [PubMed]

D. Debarre, W. Supatto, A.-M. Pena, A. Fabre, T. Tordjmann, L. Combettes, M.-C. Schanne-Klein, and E. Beaurepaire, "Imaging lipid bodies in cells and tissues using third-harmonic generation microscopy," Nat. Methods 3(1), 47-53 (2006).
[CrossRef]

J. Lyubovitsky, J. Spencer, T. Krasieva, B. Andersen, and B. Tromberg, "Imaging corneal pathology in a transgenic mouse model using nonlinear microscopy," J. Biomed. Opt. 11(1), 014013 (2006).
[CrossRef] [PubMed]

S.-W. Teng, H.-Y. Tan, J.-L. Peng, H.-H. Lin, K. Kim, W. Lo, Y. Sun, W.-C. Lin, S.-J. Lin, S.-H. Jee, P. So, and C.-Y. Dong, "Multiphoton autofluorescence and second-harmonic generation imaging of the ex vivo porcine eye," Invest. Ophtalmol. Vis. Sci. 47(3), 5251-5259 (2006).
[CrossRef]

B.-G. Wang, K. Koenig, I. Riemann, R. Krieg, and K.-J. Halbhuber, "Intraocular multiphoton microscopy with subcellular spatial resolution by infrared femtosecond lasers," Histochem. Cell. Biol. 126(4), 507-515 (2006).
[CrossRef] [PubMed]

2005 (5)

M. Han, G. Giese, and J. F. Bille, "Second harmonic generation imaging of collagen fibrils in cornea and sclera," Opt. Express 13(15), 5791-5797 (2005).
[CrossRef] [PubMed]

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

D. Debarre, W. Supatto, and E. Beaurepaire, "Structure sensitivity in third-harmonic generation microscopy," Opt. Lett. 30(16), 2134-2136 (2005).
[CrossRef] [PubMed]

A. Vogel, J. Noack, G. Huttman, and G. Paltauf, "Mechanisms of femtosecond laser nanosurgery of cells and tissues," Appl. Phys. B 81(8), 1015-1047 (2005).
[CrossRef]

W. Supatto, D. Debarre, B. Moulia, E. Brouzes, J.-L. Martin, E. Farge, and E. Beaurepaire, "In vivo modulation of morphogenetic movements in Drosophila embryos with femtosecond laser pulses," Proc. Nat. Acad. Sci. USA 102(4), 1047-1052 (2005).
[CrossRef] [PubMed]

2004 (1)

D. Oron, D. Yelin, E. Tal, S. Raz, R. Fachima, and Y. Silberberg, "Depth-resolved structural imaging by thirdharmonic generation microscopy," J. Struct. Biol. 147(1), 3-11 (2004).
[CrossRef] [PubMed]

2003 (3)

D. Oron, E. Tal, and Y. Silberberg, "Depth-resolved multiphoton polarization microscopy by third-harmonic generation," Opt. Lett. 28(23), 2315 (2003).
[CrossRef] [PubMed]

W. Zipfel, R. Williams, and W. Webb, "Nonlinear magic:multiphoton microscopy in the biosciences," Nat. Biotechnol. 21(11), 1369-1377 (2003).
[CrossRef] [PubMed]

I. Jalbert, F. Stapleton, E. Papas, D. Sweeney, and M. Coroneo, "In vivo confocal microscopy of the human cornea," Br. J. Ophthalmol. 87, 225-36 (2003).
[CrossRef] [PubMed]

2002 (2)

1998 (1)

M. Muller, J. Squier, K. R. Wilson, and G. Brakenhoff, "3D-microscopy of transparent objects using thirdharmonic generation," J. Microsc. 191, 266-274 (1998).
[CrossRef] [PubMed]

1997 (1)

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]

1995 (1)

D. W. Piston, B. R. Masters, and W. W. Webb, "3-Dimensionally Resolved Nad(P)H Cellular Metabolic Redox Imaging of the in-Situ Cornea with 2-Photon Excitation Laser-Scanning Microscopy," J. Microsc.-Oxford 178, 20-27 (1995).
[CrossRef]

Andersen, B.

J. Lyubovitsky, J. Spencer, T. Krasieva, B. Andersen, and B. Tromberg, "Imaging corneal pathology in a transgenic mouse model using nonlinear microscopy," J. Biomed. Opt. 11(1), 014013 (2006).
[CrossRef] [PubMed]

Aptel, F.

F. Aptel, N. Olivier, A. Deniset-Besseau, J.-M. Legeais, K. Plamann, M.-C. Schanne-Klein, and E. Beaurepaire, "Multimodal nonlinear imaging of the human cornea," Invest. Ophthalmol. Vis. Sci.  51 (2010, in press).
[CrossRef] [PubMed]

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]

Beaurepaire, E.

F. Aptel, N. Olivier, A. Deniset-Besseau, J.-M. Legeais, K. Plamann, M.-C. Schanne-Klein, and E. Beaurepaire, "Multimodal nonlinear imaging of the human cornea," Invest. Ophthalmol. Vis. Sci.  51 (2010, in press).
[CrossRef] [PubMed]

N. Olivier and E. Beaurepaire, "Third-harmonic generation microscopy with focus-engineered beams: a numerical study," Opt. Express 16(19), 14,703-14,715 (2008).
[CrossRef]

M. Strupler, A.-M. Pena, M. Hernest, P.-L. Tharaux, J.-L. Martin, E. Beaurepaire, and M.-C. Schanne-Klein, "Second harmonic imaging and scoring of collagen in fibrotic tissues," Opt. Express 15(7), 4054-4065 (2007).
[CrossRef] [PubMed]

D. Debarre and E. Beaurepaire, "Quantitative characterization of biological liquids for third-harmonic generation microscopy," Biophys. J. 92(2), 603-612 (2007).
[CrossRef]

D. Debarre, W. Supatto, A.-M. Pena, A. Fabre, T. Tordjmann, L. Combettes, M.-C. Schanne-Klein, and E. Beaurepaire, "Imaging lipid bodies in cells and tissues using third-harmonic generation microscopy," Nat. Methods 3(1), 47-53 (2006).
[CrossRef]

D. Debarre, W. Supatto, and E. Beaurepaire, "Structure sensitivity in third-harmonic generation microscopy," Opt. Lett. 30(16), 2134-2136 (2005).
[CrossRef] [PubMed]

W. Supatto, D. Debarre, B. Moulia, E. Brouzes, J.-L. Martin, E. Farge, and E. Beaurepaire, "In vivo modulation of morphogenetic movements in Drosophila embryos with femtosecond laser pulses," Proc. Nat. Acad. Sci. USA 102(4), 1047-1052 (2005).
[CrossRef] [PubMed]

Bille, J. F.

Brakenhoff, G.

M. Muller, J. Squier, K. R. Wilson, and G. Brakenhoff, "3D-microscopy of transparent objects using thirdharmonic generation," J. Microsc. 191, 266-274 (1998).
[CrossRef] [PubMed]

Brocas, A.

Brouzes, E.

W. Supatto, D. Debarre, B. Moulia, E. Brouzes, J.-L. Martin, E. Farge, and E. Beaurepaire, "In vivo modulation of morphogenetic movements in Drosophila embryos with femtosecond laser pulses," Proc. Nat. Acad. Sci. USA 102(4), 1047-1052 (2005).
[CrossRef] [PubMed]

Brown, D.

N. Morishige, A. Wahlert, M. Kenney, D. Brown, K. Kawamoto, T. Chikama, T. Nishida, and J. Jester, "Secondharmonic imaging microscopy of normal human and keratoconus cornea," Invest. Ophthalmol. Vis. Sci. 48(3), 1087-94 (2007).
[CrossRef] [PubMed]

Brown, E. B.

Brunette, I.

Burke, R. M.

Cavuoto, L. A.

R. W. Knighton, X. R. Huang, and L. A. Cavuoto, "Corneal birefringence mapped by scanning laser polarimetry," Opt. Express 16(18), 13,738-13,751 (2008).
[CrossRef]

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(4), 044012 (2009).
[CrossRef] [PubMed]

Cheng, J.-X.

Chikama, T.

N. Morishige, A. Wahlert, M. Kenney, D. Brown, K. Kawamoto, T. Chikama, T. Nishida, and J. Jester, "Secondharmonic imaging microscopy of normal human and keratoconus cornea," Invest. Ophthalmol. Vis. Sci. 48(3), 1087-94 (2007).
[CrossRef] [PubMed]

Cicchi, R.

Combettes, L.

D. Debarre, W. Supatto, A.-M. Pena, A. Fabre, T. Tordjmann, L. Combettes, M.-C. Schanne-Klein, and E. Beaurepaire, "Imaging lipid bodies in cells and tissues using third-harmonic generation microscopy," Nat. Methods 3(1), 47-53 (2006).
[CrossRef]

Coroneo, M.

I. Jalbert, F. Stapleton, E. Papas, D. Sweeney, and M. Coroneo, "In vivo confocal microscopy of the human cornea," Br. J. Ophthalmol. 87, 225-36 (2003).
[CrossRef] [PubMed]

Debarre, D.

D. Debarre and E. Beaurepaire, "Quantitative characterization of biological liquids for third-harmonic generation microscopy," Biophys. J. 92(2), 603-612 (2007).
[CrossRef]

D. Debarre, W. Supatto, A.-M. Pena, A. Fabre, T. Tordjmann, L. Combettes, M.-C. Schanne-Klein, and E. Beaurepaire, "Imaging lipid bodies in cells and tissues using third-harmonic generation microscopy," Nat. Methods 3(1), 47-53 (2006).
[CrossRef]

D. Debarre, W. Supatto, and E. Beaurepaire, "Structure sensitivity in third-harmonic generation microscopy," Opt. Lett. 30(16), 2134-2136 (2005).
[CrossRef] [PubMed]

W. Supatto, D. Debarre, B. Moulia, E. Brouzes, J.-L. Martin, E. Farge, and E. Beaurepaire, "In vivo modulation of morphogenetic movements in Drosophila embryos with femtosecond laser pulses," Proc. Nat. Acad. Sci. USA 102(4), 1047-1052 (2005).
[CrossRef] [PubMed]

Deniset-Besseau, A.

F. Aptel, N. Olivier, A. Deniset-Besseau, J.-M. Legeais, K. Plamann, M.-C. Schanne-Klein, and E. Beaurepaire, "Multimodal nonlinear imaging of the human cornea," Invest. Ophthalmol. Vis. Sci.  51 (2010, in press).
[CrossRef] [PubMed]

Dong, C.-Y.

S.-W. Teng, H.-Y. Tan, J.-L. Peng, H.-H. Lin, K. Kim, W. Lo, Y. Sun, W.-C. Lin, S.-J. Lin, S.-H. Jee, P. So, and C.-Y. Dong, "Multiphoton autofluorescence and second-harmonic generation imaging of the ex vivo porcine eye," Invest. Ophtalmol. Vis. Sci. 47(3), 5251-5259 (2006).
[CrossRef]

Drexler, W.

W. Drexler and J. Fujimoto, "State-of-the-art retinal optical coherence tomography," Prog. Retin. Eye Res. 27(1), 45-88 (2008).
[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]

Fabre, A.

D. Debarre, W. Supatto, A.-M. Pena, A. Fabre, T. Tordjmann, L. Combettes, M.-C. Schanne-Klein, and E. Beaurepaire, "Imaging lipid bodies in cells and tissues using third-harmonic generation microscopy," Nat. Methods 3(1), 47-53 (2006).
[CrossRef]

Fachima, R.

D. Oron, D. Yelin, E. Tal, S. Raz, R. Fachima, and Y. Silberberg, "Depth-resolved structural imaging by thirdharmonic generation microscopy," J. Struct. Biol. 147(1), 3-11 (2004).
[CrossRef] [PubMed]

Farge, E.

W. Supatto, D. Debarre, B. Moulia, E. Brouzes, J.-L. Martin, E. Farge, and E. Beaurepaire, "In vivo modulation of morphogenetic movements in Drosophila embryos with femtosecond laser pulses," Proc. Nat. Acad. Sci. USA 102(4), 1047-1052 (2005).
[CrossRef] [PubMed]

Fujimoto, J.

W. Drexler and J. Fujimoto, "State-of-the-art retinal optical coherence tomography," Prog. Retin. Eye Res. 27(1), 45-88 (2008).
[CrossRef]

Giese, G.

Halbhuber, K.-J.

B.-G. Wang, K. Koenig, I. Riemann, R. Krieg, and K.-J. Halbhuber, "Intraocular multiphoton microscopy with subcellular spatial resolution by infrared femtosecond lasers," Histochem. Cell. Biol. 126(4), 507-515 (2006).
[CrossRef] [PubMed]

Han, M.

Han, X.

Hernest, M.

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]

Huang, H.-Y.

Huang, X. R.

R. W. Knighton, X. R. Huang, and L. A. Cavuoto, "Corneal birefringence mapped by scanning laser polarimetry," Opt. Express 16(18), 13,738-13,751 (2008).
[CrossRef]

Huttman, G.

A. Vogel, J. Noack, G. Huttman, and G. Paltauf, "Mechanisms of femtosecond laser nanosurgery of cells and tissues," Appl. Phys. B 81(8), 1015-1047 (2005).
[CrossRef]

Jalbert, I.

I. Jalbert, F. Stapleton, E. Papas, D. Sweeney, and M. Coroneo, "In vivo confocal microscopy of the human cornea," Br. J. Ophthalmol. 87, 225-36 (2003).
[CrossRef] [PubMed]

Jay, L.

Jee, S.-H.

S.-W. Teng, H.-Y. Tan, J.-L. Peng, H.-H. Lin, K. Kim, W. Lo, Y. Sun, W.-C. Lin, S.-J. Lin, S.-H. Jee, P. So, and C.-Y. Dong, "Multiphoton autofluorescence and second-harmonic generation imaging of the ex vivo porcine eye," Invest. Ophtalmol. Vis. Sci. 47(3), 5251-5259 (2006).
[CrossRef]

Jester, J.

N. Morishige, A. Wahlert, M. Kenney, D. Brown, K. Kawamoto, T. Chikama, T. Nishida, and J. Jester, "Secondharmonic imaging microscopy of normal human and keratoconus cornea," Invest. Ophthalmol. Vis. Sci. 48(3), 1087-94 (2007).
[CrossRef] [PubMed]

Kapsokalyvas, D.

Kawamoto, K.

N. Morishige, A. Wahlert, M. Kenney, D. Brown, K. Kawamoto, T. Chikama, T. Nishida, and J. Jester, "Secondharmonic imaging microscopy of normal human and keratoconus cornea," Invest. Ophthalmol. Vis. Sci. 48(3), 1087-94 (2007).
[CrossRef] [PubMed]

Kenney, M.

N. Morishige, A. Wahlert, M. Kenney, D. Brown, K. Kawamoto, T. Chikama, T. Nishida, and J. Jester, "Secondharmonic imaging microscopy of normal human and keratoconus cornea," Invest. Ophthalmol. Vis. Sci. 48(3), 1087-94 (2007).
[CrossRef] [PubMed]

Kieffer, J.-C.

Kim, K.

S.-W. Teng, H.-Y. Tan, J.-L. Peng, H.-H. Lin, K. Kim, W. Lo, Y. Sun, W.-C. Lin, S.-J. Lin, S.-H. Jee, P. So, and C.-Y. Dong, "Multiphoton autofluorescence and second-harmonic generation imaging of the ex vivo porcine eye," Invest. Ophtalmol. Vis. Sci. 47(3), 5251-5259 (2006).
[CrossRef]

Knighton, R. W.

R. W. Knighton, X. R. Huang, and L. A. Cavuoto, "Corneal birefringence mapped by scanning laser polarimetry," Opt. Express 16(18), 13,738-13,751 (2008).
[CrossRef]

Koenig, K.

B.-G. Wang, K. Koenig, I. Riemann, R. Krieg, and K.-J. Halbhuber, "Intraocular multiphoton microscopy with subcellular spatial resolution by infrared femtosecond lasers," Histochem. Cell. Biol. 126(4), 507-515 (2006).
[CrossRef] [PubMed]

Krasieva, T.

J. Lyubovitsky, J. Spencer, T. Krasieva, B. Andersen, and B. Tromberg, "Imaging corneal pathology in a transgenic mouse model using nonlinear microscopy," J. Biomed. Opt. 11(1), 014013 (2006).
[CrossRef] [PubMed]

Krieg, R.

B.-G. Wang, K. Koenig, I. Riemann, R. Krieg, and K.-J. Halbhuber, "Intraocular multiphoton microscopy with subcellular spatial resolution by infrared femtosecond lasers," Histochem. Cell. Biol. 126(4), 507-515 (2006).
[CrossRef] [PubMed]

Lee, W.-J.

Legeais, J.-M.

F. Aptel, N. Olivier, A. Deniset-Besseau, J.-M. Legeais, K. Plamann, M.-C. Schanne-Klein, and E. Beaurepaire, "Multimodal nonlinear imaging of the human cornea," Invest. Ophthalmol. Vis. Sci.  51 (2010, in press).
[CrossRef] [PubMed]

Lin, H.-H.

S.-W. Teng, H.-Y. Tan, J.-L. Peng, H.-H. Lin, K. Kim, W. Lo, Y. Sun, W.-C. Lin, S.-J. Lin, S.-H. Jee, P. So, and C.-Y. Dong, "Multiphoton autofluorescence and second-harmonic generation imaging of the ex vivo porcine eye," Invest. Ophtalmol. Vis. Sci. 47(3), 5251-5259 (2006).
[CrossRef]

Lin, S.-J.

S.-W. Teng, H.-Y. Tan, J.-L. Peng, H.-H. Lin, K. Kim, W. Lo, Y. Sun, W.-C. Lin, S.-J. Lin, S.-H. Jee, P. So, and C.-Y. Dong, "Multiphoton autofluorescence and second-harmonic generation imaging of the ex vivo porcine eye," Invest. Ophtalmol. Vis. Sci. 47(3), 5251-5259 (2006).
[CrossRef]

Lin, W.-C.

S.-W. Teng, H.-Y. Tan, J.-L. Peng, H.-H. Lin, K. Kim, W. Lo, Y. Sun, W.-C. Lin, S.-J. Lin, S.-H. Jee, P. So, and C.-Y. Dong, "Multiphoton autofluorescence and second-harmonic generation imaging of the ex vivo porcine eye," Invest. Ophtalmol. Vis. Sci. 47(3), 5251-5259 (2006).
[CrossRef]

Lo, W.

S.-W. Teng, H.-Y. Tan, J.-L. Peng, H.-H. Lin, K. Kim, W. Lo, Y. Sun, W.-C. Lin, S.-J. Lin, S.-H. Jee, P. So, and C.-Y. Dong, "Multiphoton autofluorescence and second-harmonic generation imaging of the ex vivo porcine eye," Invest. Ophtalmol. Vis. Sci. 47(3), 5251-5259 (2006).
[CrossRef]

Lyubovitsky, J.

J. Lyubovitsky, J. Spencer, T. Krasieva, B. Andersen, and B. Tromberg, "Imaging corneal pathology in a transgenic mouse model using nonlinear microscopy," J. Biomed. Opt. 11(1), 014013 (2006).
[CrossRef] [PubMed]

Malta, J.

H. Soong and J. Malta, "Femtosecond lasers in ophthalmology," Am. J. Ophthalmol. 147(2), 189-197 (2009).
[CrossRef]

Martin, J.-L.

M. Strupler, A.-M. Pena, M. Hernest, P.-L. Tharaux, J.-L. Martin, E. Beaurepaire, and M.-C. Schanne-Klein, "Second harmonic imaging and scoring of collagen in fibrotic tissues," Opt. Express 15(7), 4054-4065 (2007).
[CrossRef] [PubMed]

W. Supatto, D. Debarre, B. Moulia, E. Brouzes, J.-L. Martin, E. Farge, and E. Beaurepaire, "In vivo modulation of morphogenetic movements in Drosophila embryos with femtosecond laser pulses," Proc. Nat. Acad. Sci. USA 102(4), 1047-1052 (2005).
[CrossRef] [PubMed]

Masters, B. R.

D. W. Piston, B. R. Masters, and W. W. Webb, "3-Dimensionally Resolved Nad(P)H Cellular Metabolic Redox Imaging of the in-Situ Cornea with 2-Photon Excitation Laser-Scanning Microscopy," J. Microsc.-Oxford 178, 20-27 (1995).
[CrossRef]

Matteini, P.

Morishige, N.

N. Morishige, A. Wahlert, M. Kenney, D. Brown, K. Kawamoto, T. Chikama, T. Nishida, and J. Jester, "Secondharmonic imaging microscopy of normal human and keratoconus cornea," Invest. Ophthalmol. Vis. Sci. 48(3), 1087-94 (2007).
[CrossRef] [PubMed]

Moulia, B.

W. Supatto, D. Debarre, B. Moulia, E. Brouzes, J.-L. Martin, E. Farge, and E. Beaurepaire, "In vivo modulation of morphogenetic movements in Drosophila embryos with femtosecond laser pulses," Proc. Nat. Acad. Sci. USA 102(4), 1047-1052 (2005).
[CrossRef] [PubMed]

Muller, M.

M. Muller, J. Squier, K. R. Wilson, and G. Brakenhoff, "3D-microscopy of transparent objects using thirdharmonic generation," J. Microsc. 191, 266-274 (1998).
[CrossRef] [PubMed]

Nassif, N.

Nishida, T.

N. Morishige, A. Wahlert, M. Kenney, D. Brown, K. Kawamoto, T. Chikama, T. Nishida, and J. Jester, "Secondharmonic imaging microscopy of normal human and keratoconus cornea," Invest. Ophthalmol. Vis. Sci. 48(3), 1087-94 (2007).
[CrossRef] [PubMed]

Noack, J.

A. Vogel, J. Noack, G. Huttman, and G. Paltauf, "Mechanisms of femtosecond laser nanosurgery of cells and tissues," Appl. Phys. B 81(8), 1015-1047 (2005).
[CrossRef]

Olivier, N.

F. Aptel, N. Olivier, A. Deniset-Besseau, J.-M. Legeais, K. Plamann, M.-C. Schanne-Klein, and E. Beaurepaire, "Multimodal nonlinear imaging of the human cornea," Invest. Ophthalmol. Vis. Sci.  51 (2010, in press).
[CrossRef] [PubMed]

N. Olivier and E. Beaurepaire, "Third-harmonic generation microscopy with focus-engineered beams: a numerical study," Opt. Express 16(19), 14,703-14,715 (2008).
[CrossRef]

Oron, D.

D. Oron, D. Yelin, E. Tal, S. Raz, R. Fachima, and Y. Silberberg, "Depth-resolved structural imaging by thirdharmonic generation microscopy," J. Struct. Biol. 147(1), 3-11 (2004).
[CrossRef] [PubMed]

D. Oron, E. Tal, and Y. Silberberg, "Depth-resolved multiphoton polarization microscopy by third-harmonic generation," Opt. Lett. 28(23), 2315 (2003).
[CrossRef] [PubMed]

Ozaki, T.

Paltauf, G.

A. Vogel, J. Noack, G. Huttman, and G. Paltauf, "Mechanisms of femtosecond laser nanosurgery of cells and tissues," Appl. Phys. B 81(8), 1015-1047 (2005).
[CrossRef]

Papas, E.

I. Jalbert, F. Stapleton, E. Papas, D. Sweeney, and M. Coroneo, "In vivo confocal microscopy of the human cornea," Br. J. Ophthalmol. 87, 225-36 (2003).
[CrossRef] [PubMed]

Pavone, F. S.

Pena, A.-M.

M. Strupler, A.-M. Pena, M. Hernest, P.-L. Tharaux, J.-L. Martin, E. Beaurepaire, and M.-C. Schanne-Klein, "Second harmonic imaging and scoring of collagen in fibrotic tissues," Opt. Express 15(7), 4054-4065 (2007).
[CrossRef] [PubMed]

D. Debarre, W. Supatto, A.-M. Pena, A. Fabre, T. Tordjmann, L. Combettes, M.-C. Schanne-Klein, and E. Beaurepaire, "Imaging lipid bodies in cells and tissues using third-harmonic generation microscopy," Nat. Methods 3(1), 47-53 (2006).
[CrossRef]

Peng, J.-L.

S.-W. Teng, H.-Y. Tan, J.-L. Peng, H.-H. Lin, K. Kim, W. Lo, Y. Sun, W.-C. Lin, S.-J. Lin, S.-H. Jee, P. So, and C.-Y. Dong, "Multiphoton autofluorescence and second-harmonic generation imaging of the ex vivo porcine eye," Invest. Ophtalmol. Vis. Sci. 47(3), 5251-5259 (2006).
[CrossRef]

Pini, R.

Piston, D. W.

D. W. Piston, B. R. Masters, and W. W. Webb, "3-Dimensionally Resolved Nad(P)H Cellular Metabolic Redox Imaging of the in-Situ Cornea with 2-Photon Excitation Laser-Scanning Microscopy," J. Microsc.-Oxford 178, 20-27 (1995).
[CrossRef]

Plamann, K.

F. Aptel, N. Olivier, A. Deniset-Besseau, J.-M. Legeais, K. Plamann, M.-C. Schanne-Klein, and E. Beaurepaire, "Multimodal nonlinear imaging of the human cornea," Invest. Ophthalmol. Vis. Sci.  51 (2010, in press).
[CrossRef] [PubMed]

Ratto, F.

Raz, S.

D. Oron, D. Yelin, E. Tal, S. Raz, R. Fachima, and Y. Silberberg, "Depth-resolved structural imaging by thirdharmonic generation microscopy," J. Struct. Biol. 147(1), 3-11 (2004).
[CrossRef] [PubMed]

Riemann, I.

B.-G. Wang, K. Koenig, I. Riemann, R. Krieg, and K.-J. Halbhuber, "Intraocular multiphoton microscopy with subcellular spatial resolution by infrared femtosecond lasers," Histochem. Cell. Biol. 126(4), 507-515 (2006).
[CrossRef] [PubMed]

Rossi, F.

Schanne-Klein, M.-C.

F. Aptel, N. Olivier, A. Deniset-Besseau, J.-M. Legeais, K. Plamann, M.-C. Schanne-Klein, and E. Beaurepaire, "Multimodal nonlinear imaging of the human cornea," Invest. Ophthalmol. Vis. Sci.  51 (2010, in press).
[CrossRef] [PubMed]

M. Strupler, A.-M. Pena, M. Hernest, P.-L. Tharaux, J.-L. Martin, E. Beaurepaire, and M.-C. Schanne-Klein, "Second harmonic imaging and scoring of collagen in fibrotic tissues," Opt. Express 15(7), 4054-4065 (2007).
[CrossRef] [PubMed]

D. Debarre, W. Supatto, A.-M. Pena, A. Fabre, T. Tordjmann, L. Combettes, M.-C. Schanne-Klein, and E. Beaurepaire, "Imaging lipid bodies in cells and tissues using third-harmonic generation microscopy," Nat. Methods 3(1), 47-53 (2006).
[CrossRef]

Shieh, D.-B.

Silberberg, Y.

D. Oron, D. Yelin, E. Tal, S. Raz, R. Fachima, and Y. Silberberg, "Depth-resolved structural imaging by thirdharmonic generation microscopy," J. Struct. Biol. 147(1), 3-11 (2004).
[CrossRef] [PubMed]

D. Oron, E. Tal, and Y. Silberberg, "Depth-resolved multiphoton polarization microscopy by third-harmonic generation," Opt. Lett. 28(23), 2315 (2003).
[CrossRef] [PubMed]

Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, "Nonlinear scanning laser microscopy by third harmonic generation," Appl. Phys. Lett. 70, 922-924 (1997).
[CrossRef]

Singh, K.

So, P.

S.-W. Teng, H.-Y. Tan, J.-L. Peng, H.-H. Lin, K. Kim, W. Lo, Y. Sun, W.-C. Lin, S.-J. Lin, S.-H. Jee, P. So, and C.-Y. Dong, "Multiphoton autofluorescence and second-harmonic generation imaging of the ex vivo porcine eye," Invest. Ophtalmol. Vis. Sci. 47(3), 5251-5259 (2006).
[CrossRef]

Soong, H.

H. Soong and J. Malta, "Femtosecond lasers in ophthalmology," Am. J. Ophthalmol. 147(2), 189-197 (2009).
[CrossRef]

Spencer, J.

J. Lyubovitsky, J. Spencer, T. Krasieva, B. Andersen, and B. Tromberg, "Imaging corneal pathology in a transgenic mouse model using nonlinear microscopy," J. Biomed. Opt. 11(1), 014013 (2006).
[CrossRef] [PubMed]

Squier, J.

M. Muller, J. Squier, K. R. Wilson, and G. Brakenhoff, "3D-microscopy of transparent objects using thirdharmonic generation," J. Microsc. 191, 266-274 (1998).
[CrossRef] [PubMed]

Stapleton, F.

I. Jalbert, F. Stapleton, E. Papas, D. Sweeney, and M. Coroneo, "In vivo confocal microscopy of the human cornea," Br. J. Ophthalmol. 87, 225-36 (2003).
[CrossRef] [PubMed]

Stringari, C.

Strupler, M.

Sun, C. K.

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(4), 044012 (2009).
[CrossRef] [PubMed]

Sun, C.-K.

Sun, Y.

S.-W. Teng, H.-Y. Tan, J.-L. Peng, H.-H. Lin, K. Kim, W. Lo, Y. Sun, W.-C. Lin, S.-J. Lin, S.-H. Jee, P. So, and C.-Y. Dong, "Multiphoton autofluorescence and second-harmonic generation imaging of the ex vivo porcine eye," Invest. Ophtalmol. Vis. Sci. 47(3), 5251-5259 (2006).
[CrossRef]

Supatto, W.

D. Debarre, W. Supatto, A.-M. Pena, A. Fabre, T. Tordjmann, L. Combettes, M.-C. Schanne-Klein, and E. Beaurepaire, "Imaging lipid bodies in cells and tissues using third-harmonic generation microscopy," Nat. Methods 3(1), 47-53 (2006).
[CrossRef]

D. Debarre, W. Supatto, and E. Beaurepaire, "Structure sensitivity in third-harmonic generation microscopy," Opt. Lett. 30(16), 2134-2136 (2005).
[CrossRef] [PubMed]

W. Supatto, D. Debarre, B. Moulia, E. Brouzes, J.-L. Martin, E. Farge, and E. Beaurepaire, "In vivo modulation of morphogenetic movements in Drosophila embryos with femtosecond laser pulses," Proc. Nat. Acad. Sci. USA 102(4), 1047-1052 (2005).
[CrossRef] [PubMed]

Sweeney, D.

I. Jalbert, F. Stapleton, E. Papas, D. Sweeney, and M. Coroneo, "In vivo confocal microscopy of the human cornea," Br. J. Ophthalmol. 87, 225-36 (2003).
[CrossRef] [PubMed]

Tai, S.-P.

Tal, E.

D. Oron, D. Yelin, E. Tal, S. Raz, R. Fachima, and Y. Silberberg, "Depth-resolved structural imaging by thirdharmonic generation microscopy," J. Struct. Biol. 147(1), 3-11 (2004).
[CrossRef] [PubMed]

D. Oron, E. Tal, and Y. Silberberg, "Depth-resolved multiphoton polarization microscopy by third-harmonic generation," Opt. Lett. 28(23), 2315 (2003).
[CrossRef] [PubMed]

Tan, H.-Y.

S.-W. Teng, H.-Y. Tan, J.-L. Peng, H.-H. Lin, K. Kim, W. Lo, Y. Sun, W.-C. Lin, S.-J. Lin, S.-H. Jee, P. So, and C.-Y. Dong, "Multiphoton autofluorescence and second-harmonic generation imaging of the ex vivo porcine eye," Invest. Ophtalmol. Vis. Sci. 47(3), 5251-5259 (2006).
[CrossRef]

Tang, P.

Teng, S.-W.

S.-W. Teng, H.-Y. Tan, J.-L. Peng, H.-H. Lin, K. Kim, W. Lo, Y. Sun, W.-C. Lin, S.-J. Lin, S.-H. Jee, P. So, and C.-Y. Dong, "Multiphoton autofluorescence and second-harmonic generation imaging of the ex vivo porcine eye," Invest. Ophtalmol. Vis. Sci. 47(3), 5251-5259 (2006).
[CrossRef]

Tharaux, P.-L.

Tordjmann, T.

D. Debarre, W. Supatto, A.-M. Pena, A. Fabre, T. Tordjmann, L. Combettes, M.-C. Schanne-Klein, and E. Beaurepaire, "Imaging lipid bodies in cells and tissues using third-harmonic generation microscopy," Nat. Methods 3(1), 47-53 (2006).
[CrossRef]

Tromberg, B.

J. Lyubovitsky, J. Spencer, T. Krasieva, B. Andersen, and B. Tromberg, "Imaging corneal pathology in a transgenic mouse model using nonlinear microscopy," J. Biomed. Opt. 11(1), 014013 (2006).
[CrossRef] [PubMed]

A. Yeh, N. Nassif, A. Zoumi, and B. Tromberg, "Selective corneal imaging using combined second-harmonic generation and two-photon excited fluorescence," Opt. Lett. 27(23), 2082-2084 (2002).
[CrossRef]

Vogel, A.

A. Vogel, J. Noack, G. Huttman, and G. Paltauf, "Mechanisms of femtosecond laser nanosurgery of cells and tissues," Appl. Phys. B 81(8), 1015-1047 (2005).
[CrossRef]

Wahlert, A.

N. Morishige, A. Wahlert, M. Kenney, D. Brown, K. Kawamoto, T. Chikama, T. Nishida, and J. Jester, "Secondharmonic imaging microscopy of normal human and keratoconus cornea," Invest. Ophthalmol. Vis. Sci. 48(3), 1087-94 (2007).
[CrossRef] [PubMed]

Wang, B.-G.

B.-G. Wang, K. Koenig, I. Riemann, R. Krieg, and K.-J. Halbhuber, "Intraocular multiphoton microscopy with subcellular spatial resolution by infrared femtosecond lasers," Histochem. Cell. Biol. 126(4), 507-515 (2006).
[CrossRef] [PubMed]

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(4), 044012 (2009).
[CrossRef] [PubMed]

Webb, W.

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

Webb, W. W.

D. W. Piston, B. R. Masters, and W. W. Webb, "3-Dimensionally Resolved Nad(P)H Cellular Metabolic Redox Imaging of the in-Situ Cornea with 2-Photon Excitation Laser-Scanning Microscopy," J. Microsc.-Oxford 178, 20-27 (1995).
[CrossRef]

Williams, R.

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

W. Zipfel, R. Williams, and W. Webb, "Nonlinear magic:multiphoton microscopy in the biosciences," Nat. Biotechnol. 21(11), 1369-1377 (2003).
[CrossRef] [PubMed]

Wilson, K. R.

M. Muller, J. Squier, K. R. Wilson, and G. Brakenhoff, "3D-microscopy of transparent objects using thirdharmonic generation," J. Microsc. 191, 266-274 (1998).
[CrossRef] [PubMed]

Wu, P.-C.

Wu, Q. F.

Q. F. Wu and A. T. Yeh, "Rabbit cornea microstructure response to changes intraocular pressure visualized by using nonlinear optical microscopy," Cornea 27(2), 202-208 (2008).
[CrossRef] [PubMed]

Xie, X.

Yeh, A.

Yeh, A. T.

Q. F. Wu and A. T. Yeh, "Rabbit cornea microstructure response to changes intraocular pressure visualized by using nonlinear optical microscopy," Cornea 27(2), 202-208 (2008).
[CrossRef] [PubMed]

Yelin, D.

D. Oron, D. Yelin, E. Tal, S. Raz, R. Fachima, and Y. Silberberg, "Depth-resolved structural imaging by thirdharmonic generation microscopy," J. Struct. Biol. 147(1), 3-11 (2004).
[CrossRef] [PubMed]

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(4), 044012 (2009).
[CrossRef] [PubMed]

Zettel, M. L.

Zipfel, W.

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

W. Zipfel, R. Williams, and W. Webb, "Nonlinear magic:multiphoton microscopy in the biosciences," Nat. Biotechnol. 21(11), 1369-1377 (2003).
[CrossRef] [PubMed]

Zoumi, A.

Am. J. Ophthalmol. (1)

H. Soong and J. Malta, "Femtosecond lasers in ophthalmology," Am. J. Ophthalmol. 147(2), 189-197 (2009).
[CrossRef]

Appl. Phys. B (1)

A. Vogel, J. Noack, G. Huttman, and G. Paltauf, "Mechanisms of femtosecond laser nanosurgery of cells and tissues," Appl. Phys. B 81(8), 1015-1047 (2005).
[CrossRef]

Appl. Phys. Lett. (1)

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. (2)

D. Debarre and E. Beaurepaire, "Quantitative characterization of biological liquids for third-harmonic generation microscopy," Biophys. J. 92(2), 603-612 (2007).
[CrossRef]

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Supplementary Material (3)

» Media 1: MOV (13164 KB)     
» Media 2: MOV (32341 KB)     
» Media 3: MOV (8335 KB)     

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Figures (7)

Fig. 1.
Fig. 1.

Imaging method. (a) Principles of multiharmonic microscopy, and wavelengths involved. (b) Right: Experimental geometry. In this study, intact excised corneas were mounted between glass coverslips, illuminated from the endothelial side, and imaged in transmission. Left: histological image of a human cornea, shown for comparison with the THG-SHG images. Scale bar 100 μm.

Fig. 2.
Fig. 2.

THG/SHG imaging of human corneas showing organization at different depths. (a) and (Media 1) XZ reprojection of a series of THG-SHG images. NA≈0.75; Z-step size=2 μm; scale bars 20 μm(X)×100 μm(Z). (b) THG image of the epithelium recorded at the depth indicated in (a). Scale bar 100 μm. (c) Higher resolution (NA=1.2) THG image showing signals from nucleus-cytoplasm and cell-cell interfaces. Scale bar 100 μm. (d,e) and (Media 2) THG-SHG images recorded at the depths indicated in (a) showing different stroma organizations. Scale bar 100 μm. (f) and (Media 3) XZ reprojections of high resolution (NA=1.2) THG stacks showing collagen lamellae organization in the anterior (top) and posterior (bottom) stroma. Z-step size=0.5 μm; scale bars 20 μm.

Fig. 3.
Fig. 3.

Anticorrelation of THG and SHG maxima in the stroma. (a) High-resolution (1.2 NA excitation) THG/SHG images recorded at successive depths in the stroma illustrating the difference between the two signals (see text). Scale bar 20 μm. (b) Reprojection and profile along the optical axis of a stack recorded with 0.5 μm axial increment. Scale bar 10 μm.

Fig. 4.
Fig. 4.

Polarization-sensitive THG imaging. (a–b) THG imaging of a cornea with (a) linear and (b) circular incident polarization reveals different structures because circ-THG specifically detects anisotropy. Scale bar 100 μm. (c) Principle of the experiment with polarization-resolved detection. (d,e) THG imaging of the stroma with circular incident polarization and polarization-resolved detection. Scale bars 100 μm.

Fig. 5.
Fig. 5.

Representation of our model stroma and imaging geometry.

Fig. 6.
Fig. 6.

Numerical calculations of THG at the interface between air and an isotropic or anisotropic medium. (a) THG power C 0(z) obtained when the excitation beam is z-scanned across a xy interface between an isotropic medium and air. (b) THG power obtained with circular incident polarization at the interface between an anisotropic medium and air, as a function of the parameters χ and χcr with χ = 1 (see text). Notice that no signal is obtained in the case of an isotropic medium (χ = χcr = χ = 1).NA=1.2, λ=1.2 μm.

Fig. 7.
Fig. 7.

Polarization-resolved THG: experiment and simulation. (a) z-profile of the total and polarization-resolved THG signal across a few stromal lamellae. Black squares represent the total THG signal. Red circles and green triangles correspond to regions where THG emission is elliptical. The total signal maxima (z = 4 and 13) and the ellipticity maxima are generally anticorrelated, even though the contrast is less clear when the lamellae are smaller than the axial resolution of the microscope (e.g. z = 11). (b) Numerical simulation for the case of an interface between two semi-infinite layers with different orientation, assuming that the interface is less structured than the interior of the layers (see text). (c) Distribution of χ /χ within a layer used for the simulation shown in (b) (NA=1.2, λ=12 μm).

Equations (14)

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χ ijkl ( 3 ) = χ 0 ( δ ij δ kl + δ ik δ jl + δ il δ jk )
P ( 3 ω ) = χ 0 [ E x ( E x 2 + E y 2 + E z 2 ) E y ( E x 2 + E y 2 + E z 2 ) E z ( E x 2 + E y 2 + E z 2 ) ]
P ( 3 ω ) = [ E x ( χ . E x 2 + χ cr . E y 2 + χ cr . E z 2 ) E y ( χ cr . E x 2 + χ . E y 2 + χ . E z 2 ) E z ( χ cr . E x 2 + χ . E y 2 + χ . E z 2 ) ]
P ( 3 ω ) = [ E x ( χ E x 2 + χ cr E y 2 ) E y ( χ cr . E x 2 + χ . E y 2 ) 0 ]
P ( 3 ω ) ( r ) = G 0 ( r ) 3 [ χ 0 0 0 ]
P ( 3 ω ) = G 0 3 [ cos θ ( χ . cos 2 θ + χ cr . sin 2 θ ) sin θ ( χ cr . cos 2 θ + χ . sin 2 θ ) 0 ]
I THG C 0 ( z ) [ cos 2 θ χ . cos 2 θ + χ cr . sin 2 θ 2 + sin 2 θ χ cr . cos 2 θ + χ . sin 2 θ 2 ]
P ( 3 ω ) = 1 2 2 G 0 3 [ ( χ χ cr ) i . ( χ cr χ ) 0 ]
P + ( 3 ω ) = G 0 3 2 2 [ ( χ χ cr ) i . ( χ cr χ ) 0 ]
P ( 3 ω ) = G 0 3 2 2 [ ( χ χ cr ) i . ( χ cr χ ) 0 ]
P + ( 3 ω ) = G 0 3 2 2 [ ( χ χ ) 0 0 ]
P ( 3 ω ) = G 0 3 2 2 [ 0 i . ( χ χ ) 0 ]
P + ( 3 ω ) = [ Ex . ( χ ( z ) . Ex . 2 + χ cr . E y 2 + χ cr . E z 2 ) Ey . ( χ cr . Ex . 2 + χ . E y 2 + χ . E z 2 ) Ez . ( χ cr Ex . 2 + χ . Ey 2 + χ . E z 2 ) ]
P ( 3 ω ) = [ Ex . ( χ . Ex . 2 + χ cr . E y 2 + χ . E z 2 ) Ey . ( χ cr . Ex . 2 + χ ( z ) . E y 2 + χ cr . E z 2 ) Ez . ( χ . Ex . 2 + χ cr . Ey 2 + χ . E z 2 ) ]

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