Abstract

An experimental implementation of the nonlinear Stokes-Mueller polarimetric (NSMP) microscopy in third-harmonic generation modality is presented. The technique is able to extract all eight 2D-accessible χ(3) components for any sample from 64 polarization measurements, and can be applied to noninvasive ultrastructural characterization. The polarization signature of an isotropic glass coverslip is presented, and carotenoid crystallites in the root of orange carrot (Daucus carota) are investigated, showing complex χ(3) components with a significant chiral contribution.

© 2017 Optical Society of America

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    [Crossref]

2016 (6)

A. Benoit, G. Latour, M.-C. Schanne-Klein, and J.-M. Allain, “Simultaneous microstructural and mechanical characterization of human corneas at increasing pressure,” J. Mech. Behav. Biomed. Mater. 60, 93–105 (2016).
[Crossref] [PubMed]

B. Weigelin, G.-J. Bakker, and P. Friedl, “Third harmonic generation microscopy of cells and tissue organization,” J. Cell Sci. 129, 245–255 (2016).
[Crossref] [PubMed]

M. Samim, S. Krouglov, and V. Barzda, “Nonlinear Stokes-Mueller polarimetry,” Phys. Rev. A 93, 013847 (2016).
[Crossref]

M. Samim, S. Krouglov, and V. Barzda, “Three-photon Stokes-Mueller polarimetry,” Phys. Rev. A 93, 033839 (2016).
[Crossref]

L. Kontenis, M. Samim, A. Karunendiran, S. Krouglov, B. Stewart, and V. Barzda, “Second harmonic generation double Stokes-Mueller polarimetric microscopy of myofilaments,” Biomed. Opt. Express 7, 559–569 (2016).
[Crossref] [PubMed]

M. Samim, S. Krouglov, D. F. James, and V. Barzda, “Characterization of heterogeneous media using nonlinear Stokes-Mueller polarimetry,” J. Opt. Soc. Am. B 33, 2617–2625 (2016).
[Crossref]

2015 (2)

2014 (3)

D. Tokarz, R. Cisek, S. Krouglov, L. Kontenis, U. Fekl, and V. Barzda, “Molecular organization of crystalline β-carotene in carrots determined with polarization-dependent second and third harmonic generation microscopy,” J. Phys. Chem. B 118, 3814–3822 (2014).
[Crossref] [PubMed]

S. Psilodimitrakopoulos, P. Loza-Alvarez, and D. Artigas, “Fast monitoring of in-vivo conformational changes in myosin using single scan polarization-SHG microscopy,” Biomed. Opt. Express 5, 4362–4673 (2014).
[Crossref]

L. Lemmens, I. Colle, S. Van Buggenhout, P. Palmero, A. Van Loey, and M. Hendrickx, “Carotenoid bioaccessibility in fruit- and vegetable-based food products as affected by product (micro)structural characteristics and the presence of lipids: A review,” Trends Food Sci. Technol. 38, 125–135 (2014).
[Crossref]

2013 (2)

D. Sandkuijl, A. E. Tuer, D. Tokarz, J. E. Sipe, and V. Barzda, “Numerical second- and third-harmonic generation microscopy,” J. Opt. Soc. Am. B 30, 382–395 (2013).
[Crossref]

M. Zimmerley, P. Mahou, D. Débarre, M.-C. Schanne-Klein, and E. Beaurepaire, “Probing ordered lipid assemblies with polarized third-harmonic-generation microscopy,” Phys. Rev. X 3, 011002 (2013).

2012 (1)

2011 (1)

C. Brackmann, A. Bengtsson, M. L. Alminger, U. Svanberg, and A. Enejder, “Visualization of β-carotene and starch granules in plant cells using CARS and SHG microscopy,” J. Raman Spectrosc. 42, 586–5892 (2011).
[Crossref]

2010 (4)

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, 2459–2465 (2010).
[Crossref] [PubMed]

V. Nucciotti, C. Stringari, L. Sacconi, F. Vanzi, L. Fusi, M. Linari, G. Piazzesi, V. Lombardi, and F. S. Pavone, “Probing myosin structural conformation in vivo by second-harmonic generation microscopy,” Proc. Natl. Acad. Sci. 107, 7763–7768 (2010).
[Crossref] [PubMed]

N. Olivier, F. Aptel, K. Plamann, M.-C. Schanne-Klein, and E. Beaurepaire, “Harmonic microscopy of isotropic and anisotropic microstructure of the human cornea,” Opt. Express 18, 5028–5040 (2010).
[Crossref] [PubMed]

D. Sandkuijl, R. Cisek, A. Major, and V. Barzda, “Differential microscopy for fluorescence-detected nonlinear absorption linear anisotropy based on a staggered two-beam femtosecond Yb:KGW oscillator,” Biomed. Opt. Express 1, 895–901 (2010).
[Crossref]

2009 (1)

R. Cicchi, D. Kapsokalyvas, V. De Giorgi, V. Maio, A. Van Wiechen, D. Massi, T. Lotti, and F. S. Pavone, “Scoring of collagen organization in healthy and diseased human dermis by multiphoton microscopy,” J. Biophotonics 3, 34–43 (2009).
[Crossref] [PubMed]

2007 (1)

A.-M. Pena, A. Fabre, D. Débarre, J. Marchal-Somme, B. Crestani, J.-L. Martin, E. Beaurepaire, and M.-C. Schanne-Klein, “Three-dimensional investigation and scoring of extracellular matrix remodeling during lung fibrosis using multiphoton microscopy,” Microsc. Res. Tech. 70, 162–170 (2007).
[Crossref]

2006 (3)

2005 (1)

2004 (3)

C. A. Dailey, B. J. Burke, and G. J. Simpson, “The general failure of Kleinman symmetry in practical nonlinear optical applications,” Chem. Phys. Lett. 390, 8–13 (2004).
[Crossref]

C.-K. Sun, S.-W. Chu, S.-Y. Chen, T.-H. Tsai, T.-M. Liu, C.-Y. Lin, and H.-J. Tsai, “Higher harmonic generation microscopy for developmental biology,” J. Struct. Biol. 147, 19–30 (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]

2003 (1)

P. J. Campagnola and L. M. Loew, “Second-harmonic imaging microscopy for visualizing biomolecular arrays in cells, tissues and organisms,” Nat. Biotechnol. 21, 1356–1360 (2003).
[Crossref] [PubMed]

2000 (1)

E. E. Gussakovsky, Y. Shahak, H. Van Amerongen, and V. Barzda, “Circularly polarized chlorophyll luminescence reflects the macro-organization of grana in pea chloroplasts,” Photosynth. Res. 65, 83–92 (2000).
[Crossref]

1999 (1)

1993 (1)

T. Petralli-Mallow, T. M. Wong, J. D. Byers, H. I. Yee, and J. M. Hicks, “Circular dichroism spectroscopy at interfaces: a surface second harmonic generation study,” J. Phys. Chem. 97, 1383–1388 (1993).
[Crossref]

1980 (1)

1973 (1)

J. P. Hermann, D. Ricard, and J. Ducuing, “Optical nonlinearities in conjugated systems: β-carotene,” Appl. Phys. Lett. 23, 178–180 (1973).
[Crossref]

1965 (1)

A. Frey-Wyssling and F. Schwegler, “Ultrastructure of the chromoplasts in the carrot root,” J. Ultrastruct. Res. 13, 543–559 (1965).
[Crossref] [PubMed]

Aiello, A.

Allain, J.-M.

A. Benoit, G. Latour, M.-C. Schanne-Klein, and J.-M. Allain, “Simultaneous microstructural and mechanical characterization of human corneas at increasing pressure,” J. Mech. Behav. Biomed. Mater. 60, 93–105 (2016).
[Crossref] [PubMed]

Alminger, M. L.

C. Brackmann, A. Bengtsson, M. L. Alminger, U. Svanberg, and A. Enejder, “Visualization of β-carotene and starch granules in plant cells using CARS and SHG microscopy,” J. Raman Spectrosc. 42, 586–5892 (2011).
[Crossref]

Amat-Roldan, I.

Anderson, D. R.

K. P. Burnham and D. R. Anderson, Model Selection and Multimodel Inference (Springer, New York, 2002).

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, 2459–2465 (2010).
[Crossref] [PubMed]

N. Olivier, F. Aptel, K. Plamann, M.-C. Schanne-Klein, and E. Beaurepaire, “Harmonic microscopy of isotropic and anisotropic microstructure of the human cornea,” Opt. Express 18, 5028–5040 (2010).
[Crossref] [PubMed]

Artigas, D.

Aus der Au, J.

Auweter, H.

S. Köhn, H. Kolbe, M. Korger, C. Köpsel, B. Mayer, H. Auweter, E. Lüddecke, H. Bettermann, and H.-D. Martin, “Aggregation and interface behaviour of carotenoids,” in “Carotenoids,”, vol. 4, G. Britton, S. Liaaen-Jensen, and H. Pfander, eds. (BirkhäuserBasel, 2008), chap. 5, pp. 53–98.
[Crossref]

Bakker, G.-J.

B. Weigelin, G.-J. Bakker, and P. Friedl, “Third harmonic generation microscopy of cells and tissue organization,” J. Cell Sci. 129, 245–255 (2016).
[Crossref] [PubMed]

Barzda, V.

M. Samim, S. Krouglov, and V. Barzda, “Nonlinear Stokes-Mueller polarimetry,” Phys. Rev. A 93, 013847 (2016).
[Crossref]

M. Samim, S. Krouglov, and V. Barzda, “Three-photon Stokes-Mueller polarimetry,” Phys. Rev. A 93, 033839 (2016).
[Crossref]

L. Kontenis, M. Samim, A. Karunendiran, S. Krouglov, B. Stewart, and V. Barzda, “Second harmonic generation double Stokes-Mueller polarimetric microscopy of myofilaments,” Biomed. Opt. Express 7, 559–569 (2016).
[Crossref] [PubMed]

M. Samim, S. Krouglov, D. F. James, and V. Barzda, “Characterization of heterogeneous media using nonlinear Stokes-Mueller polarimetry,” J. Opt. Soc. Am. B 33, 2617–2625 (2016).
[Crossref]

R. Cisek, D. Tokarz, M. Steup, I. J. Tetlow, M. J. Emes, K. H. Hebelstrup, A. Blennow, and V. Barzda, “Second harmonic generation microscopy investigation of the crystalline ultrastructure of three barley starch lines affected by hydration,” Biomed. Opt. Express 6, 3694–3700 (2015).
[Crossref] [PubMed]

D. Tokarz, R. Cisek, S. Krouglov, L. Kontenis, U. Fekl, and V. Barzda, “Molecular organization of crystalline β-carotene in carrots determined with polarization-dependent second and third harmonic generation microscopy,” J. Phys. Chem. B 118, 3814–3822 (2014).
[Crossref] [PubMed]

D. Sandkuijl, A. E. Tuer, D. Tokarz, J. E. Sipe, and V. Barzda, “Numerical second- and third-harmonic generation microscopy,” J. Opt. Soc. Am. B 30, 382–395 (2013).
[Crossref]

D. Sandkuijl, R. Cisek, A. Major, and V. Barzda, “Differential microscopy for fluorescence-detected nonlinear absorption linear anisotropy based on a staggered two-beam femtosecond Yb:KGW oscillator,” Biomed. Opt. Express 1, 895–901 (2010).
[Crossref]

V. Barzda, C. Greenhalgh, J. Aus der Au, S. Elmore, J. HGM van Beek, and J. Squier, “Visualization of mitochondria in cardiomyocytes,” Opt. Express 13, 8263–8276 (2005).
[Crossref] [PubMed]

E. E. Gussakovsky, Y. Shahak, H. Van Amerongen, and V. Barzda, “Circularly polarized chlorophyll luminescence reflects the macro-organization of grana in pea chloroplasts,” Photosynth. Res. 65, 83–92 (2000).
[Crossref]

Beaurepaire, E.

M. Zimmerley, P. Mahou, D. Débarre, M.-C. Schanne-Klein, and E. Beaurepaire, “Probing ordered lipid assemblies with polarized third-harmonic-generation microscopy,” Phys. Rev. X 3, 011002 (2013).

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, 2459–2465 (2010).
[Crossref] [PubMed]

N. Olivier, F. Aptel, K. Plamann, M.-C. Schanne-Klein, and E. Beaurepaire, “Harmonic microscopy of isotropic and anisotropic microstructure of the human cornea,” Opt. Express 18, 5028–5040 (2010).
[Crossref] [PubMed]

A.-M. Pena, A. Fabre, D. Débarre, J. Marchal-Somme, B. Crestani, J.-L. Martin, E. Beaurepaire, and M.-C. Schanne-Klein, “Three-dimensional investigation and scoring of extracellular matrix remodeling during lung fibrosis using multiphoton microscopy,” Microsc. Res. Tech. 70, 162–170 (2007).
[Crossref]

Bengtsson, A.

C. Brackmann, A. Bengtsson, M. L. Alminger, U. Svanberg, and A. Enejder, “Visualization of β-carotene and starch granules in plant cells using CARS and SHG microscopy,” J. Raman Spectrosc. 42, 586–5892 (2011).
[Crossref]

Benoit, A.

A. Benoit, G. Latour, M.-C. Schanne-Klein, and J.-M. Allain, “Simultaneous microstructural and mechanical characterization of human corneas at increasing pressure,” J. Mech. Behav. Biomed. Mater. 60, 93–105 (2016).
[Crossref] [PubMed]

Bettermann, H.

S. Köhn, H. Kolbe, M. Korger, C. Köpsel, B. Mayer, H. Auweter, E. Lüddecke, H. Bettermann, and H.-D. Martin, “Aggregation and interface behaviour of carotenoids,” in “Carotenoids,”, vol. 4, G. Britton, S. Liaaen-Jensen, and H. Pfander, eds. (BirkhäuserBasel, 2008), chap. 5, pp. 53–98.
[Crossref]

Blennow, A.

Boyd, R. W.

Brackmann, C.

C. Brackmann, A. Bengtsson, M. L. Alminger, U. Svanberg, and A. Enejder, “Visualization of β-carotene and starch granules in plant cells using CARS and SHG microscopy,” J. Raman Spectrosc. 42, 586–5892 (2011).
[Crossref]

Burke, B. J.

C. A. Dailey, B. J. Burke, and G. J. Simpson, “The general failure of Kleinman symmetry in practical nonlinear optical applications,” Chem. Phys. Lett. 390, 8–13 (2004).
[Crossref]

Burnham, K. P.

K. P. Burnham and D. R. Anderson, Model Selection and Multimodel Inference (Springer, New York, 2002).

Byers, J. D.

T. Petralli-Mallow, T. M. Wong, J. D. Byers, H. I. Yee, and J. M. Hicks, “Circular dichroism spectroscopy at interfaces: a surface second harmonic generation study,” J. Phys. Chem. 97, 1383–1388 (1993).
[Crossref]

Campagnola, P. J.

P. J. Campagnola and L. M. Loew, “Second-harmonic imaging microscopy for visualizing biomolecular arrays in cells, tissues and organisms,” Nat. Biotechnol. 21, 1356–1360 (2003).
[Crossref] [PubMed]

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]

C.-K. Sun, S.-W. Chu, S.-Y. Chen, T.-H. Tsai, T.-M. Liu, C.-Y. Lin, and H.-J. Tsai, “Higher harmonic generation microscopy for developmental biology,” J. Struct. Biol. 147, 19–30 (2004).
[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]

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]

C.-K. Sun, S.-W. Chu, S.-Y. Chen, T.-H. Tsai, T.-M. Liu, C.-Y. Lin, and H.-J. Tsai, “Higher harmonic generation microscopy for developmental biology,” J. Struct. Biol. 147, 19–30 (2004).
[Crossref] [PubMed]

Cicchi, R.

R. Cicchi, D. Kapsokalyvas, V. De Giorgi, V. Maio, A. Van Wiechen, D. Massi, T. Lotti, and F. S. Pavone, “Scoring of collagen organization in healthy and diseased human dermis by multiphoton microscopy,” J. Biophotonics 3, 34–43 (2009).
[Crossref] [PubMed]

Cisek, R.

Clevers, S.

F. Simon, S. Clevers, V. Dupray, and G. Coquerel, “Relevance of the second harmonic generation to characterize crystalline samples,” Chem. Eng. Technol. 38, 971–983 (2015).
[Crossref]

Colle, I.

L. Lemmens, I. Colle, S. Van Buggenhout, P. Palmero, A. Van Loey, and M. Hendrickx, “Carotenoid bioaccessibility in fruit- and vegetable-based food products as affected by product (micro)structural characteristics and the presence of lipids: A review,” Trends Food Sci. Technol. 38, 125–135 (2014).
[Crossref]

Compain, E.

Coquerel, G.

F. Simon, S. Clevers, V. Dupray, and G. Coquerel, “Relevance of the second harmonic generation to characterize crystalline samples,” Chem. Eng. Technol. 38, 971–983 (2015).
[Crossref]

Crestani, B.

A.-M. Pena, A. Fabre, D. Débarre, J. Marchal-Somme, B. Crestani, J.-L. Martin, E. Beaurepaire, and M.-C. Schanne-Klein, “Three-dimensional investigation and scoring of extracellular matrix remodeling during lung fibrosis using multiphoton microscopy,” Microsc. Res. Tech. 70, 162–170 (2007).
[Crossref]

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C. A. Dailey, B. J. Burke, and G. J. Simpson, “The general failure of Kleinman symmetry in practical nonlinear optical applications,” Chem. Phys. Lett. 390, 8–13 (2004).
[Crossref]

De Giorgi, V.

R. Cicchi, D. Kapsokalyvas, V. De Giorgi, V. Maio, A. Van Wiechen, D. Massi, T. Lotti, and F. S. Pavone, “Scoring of collagen organization in healthy and diseased human dermis by multiphoton microscopy,” J. Biophotonics 3, 34–43 (2009).
[Crossref] [PubMed]

Débarre, D.

M. Zimmerley, P. Mahou, D. Débarre, M.-C. Schanne-Klein, and E. Beaurepaire, “Probing ordered lipid assemblies with polarized third-harmonic-generation microscopy,” Phys. Rev. X 3, 011002 (2013).

A.-M. Pena, A. Fabre, D. Débarre, J. Marchal-Somme, B. Crestani, J.-L. Martin, E. Beaurepaire, and M.-C. Schanne-Klein, “Three-dimensional investigation and scoring of extracellular matrix remodeling during lung fibrosis using multiphoton microscopy,” Microsc. Res. Tech. 70, 162–170 (2007).
[Crossref]

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, 2459–2465 (2010).
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Drevillon, B.

Ducuing, J.

J. P. Hermann, D. Ricard, and J. Ducuing, “Optical nonlinearities in conjugated systems: β-carotene,” Appl. Phys. Lett. 23, 178–180 (1973).
[Crossref]

Dupray, V.

F. Simon, S. Clevers, V. Dupray, and G. Coquerel, “Relevance of the second harmonic generation to characterize crystalline samples,” Chem. Eng. Technol. 38, 971–983 (2015).
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Emes, M. J.

Enejder, A.

C. Brackmann, A. Bengtsson, M. L. Alminger, U. Svanberg, and A. Enejder, “Visualization of β-carotene and starch granules in plant cells using CARS and SHG microscopy,” J. Raman Spectrosc. 42, 586–5892 (2011).
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A.-M. Pena, A. Fabre, D. Débarre, J. Marchal-Somme, B. Crestani, J.-L. Martin, E. Beaurepaire, and M.-C. Schanne-Klein, “Three-dimensional investigation and scoring of extracellular matrix remodeling during lung fibrosis using multiphoton microscopy,” Microsc. Res. Tech. 70, 162–170 (2007).
[Crossref]

Fekl, U.

D. Tokarz, R. Cisek, S. Krouglov, L. Kontenis, U. Fekl, and V. Barzda, “Molecular organization of crystalline β-carotene in carrots determined with polarization-dependent second and third harmonic generation microscopy,” J. Phys. Chem. B 118, 3814–3822 (2014).
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Flannery, B.

W. Press, B. Flannery, and S. Teukolsky, Numerical Recipes. The Art of Scientific Computing (Cambride University Press, 2007), 3rd ed.

Frey-Wyssling, A.

A. Frey-Wyssling and F. Schwegler, “Ultrastructure of the chromoplasts in the carrot root,” J. Ultrastruct. Res. 13, 543–559 (1965).
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B. Weigelin, G.-J. Bakker, and P. Friedl, “Third harmonic generation microscopy of cells and tissue organization,” J. Cell Sci. 129, 245–255 (2016).
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V. Nucciotti, C. Stringari, L. Sacconi, F. Vanzi, L. Fusi, M. Linari, G. Piazzesi, V. Lombardi, and F. S. Pavone, “Probing myosin structural conformation in vivo by second-harmonic generation microscopy,” Proc. Natl. Acad. Sci. 107, 7763–7768 (2010).
[Crossref] [PubMed]

Greenhalgh, C.

Gussakovsky, E. E.

E. E. Gussakovsky, Y. Shahak, H. Van Amerongen, and V. Barzda, “Circularly polarized chlorophyll luminescence reflects the macro-organization of grana in pea chloroplasts,” Photosynth. Res. 65, 83–92 (2000).
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Hamaguchi, H.-o.

H. Kano and H.-o. Hamaguchi, “Vibrational imaging of a single pollen grain by ultrabroadband multiplex coherent anti-Stokes Raman scattering microspectroscopy,” Chem. Lett. 35, 1124–1125 (2006).
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Hebelstrup, K. H.

Hendrickx, M.

L. Lemmens, I. Colle, S. Van Buggenhout, P. Palmero, A. Van Loey, and M. Hendrickx, “Carotenoid bioaccessibility in fruit- and vegetable-based food products as affected by product (micro)structural characteristics and the presence of lipids: A review,” Trends Food Sci. Technol. 38, 125–135 (2014).
[Crossref]

Hermann, J. P.

J. P. Hermann, D. Ricard, and J. Ducuing, “Optical nonlinearities in conjugated systems: β-carotene,” Appl. Phys. Lett. 23, 178–180 (1973).
[Crossref]

Hicks, J. M.

T. Petralli-Mallow, T. M. Wong, J. D. Byers, H. I. Yee, and J. M. Hicks, “Circular dichroism spectroscopy at interfaces: a surface second harmonic generation study,” J. Phys. Chem. 97, 1383–1388 (1993).
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Huang, H.-Y.

James, D. F.

Kano, H.

H. Kano and H.-o. Hamaguchi, “Vibrational imaging of a single pollen grain by ultrabroadband multiplex coherent anti-Stokes Raman scattering microspectroscopy,” Chem. Lett. 35, 1124–1125 (2006).
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Kapsokalyvas, D.

R. Cicchi, D. Kapsokalyvas, V. De Giorgi, V. Maio, A. Van Wiechen, D. Massi, T. Lotti, and F. S. Pavone, “Scoring of collagen organization in healthy and diseased human dermis by multiphoton microscopy,” J. Biophotonics 3, 34–43 (2009).
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Karunendiran, A.

Köhn, S.

S. Köhn, H. Kolbe, M. Korger, C. Köpsel, B. Mayer, H. Auweter, E. Lüddecke, H. Bettermann, and H.-D. Martin, “Aggregation and interface behaviour of carotenoids,” in “Carotenoids,”, vol. 4, G. Britton, S. Liaaen-Jensen, and H. Pfander, eds. (BirkhäuserBasel, 2008), chap. 5, pp. 53–98.
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Kolbe, H.

S. Köhn, H. Kolbe, M. Korger, C. Köpsel, B. Mayer, H. Auweter, E. Lüddecke, H. Bettermann, and H.-D. Martin, “Aggregation and interface behaviour of carotenoids,” in “Carotenoids,”, vol. 4, G. Britton, S. Liaaen-Jensen, and H. Pfander, eds. (BirkhäuserBasel, 2008), chap. 5, pp. 53–98.
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Kontenis, L.

L. Kontenis, M. Samim, A. Karunendiran, S. Krouglov, B. Stewart, and V. Barzda, “Second harmonic generation double Stokes-Mueller polarimetric microscopy of myofilaments,” Biomed. Opt. Express 7, 559–569 (2016).
[Crossref] [PubMed]

D. Tokarz, R. Cisek, S. Krouglov, L. Kontenis, U. Fekl, and V. Barzda, “Molecular organization of crystalline β-carotene in carrots determined with polarization-dependent second and third harmonic generation microscopy,” J. Phys. Chem. B 118, 3814–3822 (2014).
[Crossref] [PubMed]

Köpsel, C.

S. Köhn, H. Kolbe, M. Korger, C. Köpsel, B. Mayer, H. Auweter, E. Lüddecke, H. Bettermann, and H.-D. Martin, “Aggregation and interface behaviour of carotenoids,” in “Carotenoids,”, vol. 4, G. Britton, S. Liaaen-Jensen, and H. Pfander, eds. (BirkhäuserBasel, 2008), chap. 5, pp. 53–98.
[Crossref]

Korger, M.

S. Köhn, H. Kolbe, M. Korger, C. Köpsel, B. Mayer, H. Auweter, E. Lüddecke, H. Bettermann, and H.-D. Martin, “Aggregation and interface behaviour of carotenoids,” in “Carotenoids,”, vol. 4, G. Britton, S. Liaaen-Jensen, and H. Pfander, eds. (BirkhäuserBasel, 2008), chap. 5, pp. 53–98.
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Krouglov, S.

M. Samim, S. Krouglov, and V. Barzda, “Three-photon Stokes-Mueller polarimetry,” Phys. Rev. A 93, 033839 (2016).
[Crossref]

M. Samim, S. Krouglov, and V. Barzda, “Nonlinear Stokes-Mueller polarimetry,” Phys. Rev. A 93, 013847 (2016).
[Crossref]

L. Kontenis, M. Samim, A. Karunendiran, S. Krouglov, B. Stewart, and V. Barzda, “Second harmonic generation double Stokes-Mueller polarimetric microscopy of myofilaments,” Biomed. Opt. Express 7, 559–569 (2016).
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M. Samim, S. Krouglov, D. F. James, and V. Barzda, “Characterization of heterogeneous media using nonlinear Stokes-Mueller polarimetry,” J. Opt. Soc. Am. B 33, 2617–2625 (2016).
[Crossref]

D. Tokarz, R. Cisek, S. Krouglov, L. Kontenis, U. Fekl, and V. Barzda, “Molecular organization of crystalline β-carotene in carrots determined with polarization-dependent second and third harmonic generation microscopy,” J. Phys. Chem. B 118, 3814–3822 (2014).
[Crossref] [PubMed]

Latour, G.

A. Benoit, G. Latour, M.-C. Schanne-Klein, and J.-M. Allain, “Simultaneous microstructural and mechanical characterization of human corneas at increasing pressure,” J. Mech. Behav. Biomed. Mater. 60, 93–105 (2016).
[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, 2459–2465 (2010).
[Crossref] [PubMed]

Lemmens, L.

L. Lemmens, I. Colle, S. Van Buggenhout, P. Palmero, A. Van Loey, and M. Hendrickx, “Carotenoid bioaccessibility in fruit- and vegetable-based food products as affected by product (micro)structural characteristics and the presence of lipids: A review,” Trends Food Sci. Technol. 38, 125–135 (2014).
[Crossref]

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.

C.-K. Sun, S.-W. Chu, S.-Y. Chen, T.-H. Tsai, T.-M. Liu, C.-Y. Lin, and H.-J. Tsai, “Higher harmonic generation microscopy for developmental biology,” J. Struct. Biol. 147, 19–30 (2004).
[Crossref] [PubMed]

Linari, M.

V. Nucciotti, C. Stringari, L. Sacconi, F. Vanzi, L. Fusi, M. Linari, G. Piazzesi, V. Lombardi, and F. S. Pavone, “Probing myosin structural conformation in vivo by second-harmonic generation microscopy,” Proc. Natl. Acad. Sci. 107, 7763–7768 (2010).
[Crossref] [PubMed]

Liu, T.-M.

C.-K. Sun, S.-W. Chu, S.-Y. Chen, T.-H. Tsai, T.-M. Liu, C.-Y. Lin, and H.-J. Tsai, “Higher harmonic generation microscopy for developmental biology,” J. Struct. Biol. 147, 19–30 (2004).
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P. J. Campagnola and L. M. Loew, “Second-harmonic imaging microscopy for visualizing biomolecular arrays in cells, tissues and organisms,” Nat. Biotechnol. 21, 1356–1360 (2003).
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Lombardi, V.

V. Nucciotti, C. Stringari, L. Sacconi, F. Vanzi, L. Fusi, M. Linari, G. Piazzesi, V. Lombardi, and F. S. Pavone, “Probing myosin structural conformation in vivo by second-harmonic generation microscopy,” Proc. Natl. Acad. Sci. 107, 7763–7768 (2010).
[Crossref] [PubMed]

Lotti, T.

R. Cicchi, D. Kapsokalyvas, V. De Giorgi, V. Maio, A. Van Wiechen, D. Massi, T. Lotti, and F. S. Pavone, “Scoring of collagen organization in healthy and diseased human dermis by multiphoton microscopy,” J. Biophotonics 3, 34–43 (2009).
[Crossref] [PubMed]

Loza-Alvarez, P.

Lüddecke, E.

S. Köhn, H. Kolbe, M. Korger, C. Köpsel, B. Mayer, H. Auweter, E. Lüddecke, H. Bettermann, and H.-D. Martin, “Aggregation and interface behaviour of carotenoids,” in “Carotenoids,”, vol. 4, G. Britton, S. Liaaen-Jensen, and H. Pfander, eds. (BirkhäuserBasel, 2008), chap. 5, pp. 53–98.
[Crossref]

Mahou, P.

M. Zimmerley, P. Mahou, D. Débarre, M.-C. Schanne-Klein, and E. Beaurepaire, “Probing ordered lipid assemblies with polarized third-harmonic-generation microscopy,” Phys. Rev. X 3, 011002 (2013).

Maio, V.

R. Cicchi, D. Kapsokalyvas, V. De Giorgi, V. Maio, A. Van Wiechen, D. Massi, T. Lotti, and F. S. Pavone, “Scoring of collagen organization in healthy and diseased human dermis by multiphoton microscopy,” J. Biophotonics 3, 34–43 (2009).
[Crossref] [PubMed]

Major, A.

Marchal-Somme, J.

A.-M. Pena, A. Fabre, D. Débarre, J. Marchal-Somme, B. Crestani, J.-L. Martin, E. Beaurepaire, and M.-C. Schanne-Klein, “Three-dimensional investigation and scoring of extracellular matrix remodeling during lung fibrosis using multiphoton microscopy,” Microsc. Res. Tech. 70, 162–170 (2007).
[Crossref]

Martin, H.-D.

S. Köhn, H. Kolbe, M. Korger, C. Köpsel, B. Mayer, H. Auweter, E. Lüddecke, H. Bettermann, and H.-D. Martin, “Aggregation and interface behaviour of carotenoids,” in “Carotenoids,”, vol. 4, G. Britton, S. Liaaen-Jensen, and H. Pfander, eds. (BirkhäuserBasel, 2008), chap. 5, pp. 53–98.
[Crossref]

Martin, J.-L.

A.-M. Pena, A. Fabre, D. Débarre, J. Marchal-Somme, B. Crestani, J.-L. Martin, E. Beaurepaire, and M.-C. Schanne-Klein, “Three-dimensional investigation and scoring of extracellular matrix remodeling during lung fibrosis using multiphoton microscopy,” Microsc. Res. Tech. 70, 162–170 (2007).
[Crossref]

Massi, D.

R. Cicchi, D. Kapsokalyvas, V. De Giorgi, V. Maio, A. Van Wiechen, D. Massi, T. Lotti, and F. S. Pavone, “Scoring of collagen organization in healthy and diseased human dermis by multiphoton microscopy,” J. Biophotonics 3, 34–43 (2009).
[Crossref] [PubMed]

Mayer, B.

S. Köhn, H. Kolbe, M. Korger, C. Köpsel, B. Mayer, H. Auweter, E. Lüddecke, H. Bettermann, and H.-D. Martin, “Aggregation and interface behaviour of carotenoids,” in “Carotenoids,”, vol. 4, G. Britton, S. Liaaen-Jensen, and H. Pfander, eds. (BirkhäuserBasel, 2008), chap. 5, pp. 53–98.
[Crossref]

Nucciotti, V.

V. Nucciotti, C. Stringari, L. Sacconi, F. Vanzi, L. Fusi, M. Linari, G. Piazzesi, V. Lombardi, and F. S. Pavone, “Probing myosin structural conformation in vivo by second-harmonic generation microscopy,” Proc. Natl. Acad. Sci. 107, 7763–7768 (2010).
[Crossref] [PubMed]

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, 2459–2465 (2010).
[Crossref] [PubMed]

N. Olivier, F. Aptel, K. Plamann, M.-C. Schanne-Klein, and E. Beaurepaire, “Harmonic microscopy of isotropic and anisotropic microstructure of the human cornea,” Opt. Express 18, 5028–5040 (2010).
[Crossref] [PubMed]

Palmero, P.

L. Lemmens, I. Colle, S. Van Buggenhout, P. Palmero, A. Van Loey, and M. Hendrickx, “Carotenoid bioaccessibility in fruit- and vegetable-based food products as affected by product (micro)structural characteristics and the presence of lipids: A review,” Trends Food Sci. Technol. 38, 125–135 (2014).
[Crossref]

Pavone, F. S.

V. Nucciotti, C. Stringari, L. Sacconi, F. Vanzi, L. Fusi, M. Linari, G. Piazzesi, V. Lombardi, and F. S. Pavone, “Probing myosin structural conformation in vivo by second-harmonic generation microscopy,” Proc. Natl. Acad. Sci. 107, 7763–7768 (2010).
[Crossref] [PubMed]

R. Cicchi, D. Kapsokalyvas, V. De Giorgi, V. Maio, A. Van Wiechen, D. Massi, T. Lotti, and F. S. Pavone, “Scoring of collagen organization in healthy and diseased human dermis by multiphoton microscopy,” J. Biophotonics 3, 34–43 (2009).
[Crossref] [PubMed]

Pena, A.-M.

A.-M. Pena, A. Fabre, D. Débarre, J. Marchal-Somme, B. Crestani, J.-L. Martin, E. Beaurepaire, and M.-C. Schanne-Klein, “Three-dimensional investigation and scoring of extracellular matrix remodeling during lung fibrosis using multiphoton microscopy,” Microsc. Res. Tech. 70, 162–170 (2007).
[Crossref]

Petralli-Mallow, T.

T. Petralli-Mallow, T. M. Wong, J. D. Byers, H. I. Yee, and J. M. Hicks, “Circular dichroism spectroscopy at interfaces: a surface second harmonic generation study,” J. Phys. Chem. 97, 1383–1388 (1993).
[Crossref]

Piazzesi, G.

V. Nucciotti, C. Stringari, L. Sacconi, F. Vanzi, L. Fusi, M. Linari, G. Piazzesi, V. Lombardi, and F. S. Pavone, “Probing myosin structural conformation in vivo by second-harmonic generation microscopy,” Proc. Natl. Acad. Sci. 107, 7763–7768 (2010).
[Crossref] [PubMed]

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, 2459–2465 (2010).
[Crossref] [PubMed]

N. Olivier, F. Aptel, K. Plamann, M.-C. Schanne-Klein, and E. Beaurepaire, “Harmonic microscopy of isotropic and anisotropic microstructure of the human cornea,” Opt. Express 18, 5028–5040 (2010).
[Crossref] [PubMed]

Poirier, S.

Press, W.

W. Press, B. Flannery, and S. Teukolsky, Numerical Recipes. The Art of Scientific Computing (Cambride University Press, 2007), 3rd ed.

Psilodimitrakopoulos, S.

Puentes, G.

Ricard, D.

J. P. Hermann, D. Ricard, and J. Ducuing, “Optical nonlinearities in conjugated systems: β-carotene,” Appl. Phys. Lett. 23, 178–180 (1973).
[Crossref]

Riley, K. F.

K. F. Riley, Mathematical Methods for the Physical Sciences (Cambridge University Press, Cambridge, 1974).
[Crossref]

Sacconi, L.

V. Nucciotti, C. Stringari, L. Sacconi, F. Vanzi, L. Fusi, M. Linari, G. Piazzesi, V. Lombardi, and F. S. Pavone, “Probing myosin structural conformation in vivo by second-harmonic generation microscopy,” Proc. Natl. Acad. Sci. 107, 7763–7768 (2010).
[Crossref] [PubMed]

Samim, M.

Sandkuijl, D.

Schanne-Klein, M.-C.

A. Benoit, G. Latour, M.-C. Schanne-Klein, and J.-M. Allain, “Simultaneous microstructural and mechanical characterization of human corneas at increasing pressure,” J. Mech. Behav. Biomed. Mater. 60, 93–105 (2016).
[Crossref] [PubMed]

M. Zimmerley, P. Mahou, D. Débarre, M.-C. Schanne-Klein, and E. Beaurepaire, “Probing ordered lipid assemblies with polarized third-harmonic-generation microscopy,” Phys. Rev. X 3, 011002 (2013).

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, 2459–2465 (2010).
[Crossref] [PubMed]

N. Olivier, F. Aptel, K. Plamann, M.-C. Schanne-Klein, and E. Beaurepaire, “Harmonic microscopy of isotropic and anisotropic microstructure of the human cornea,” Opt. Express 18, 5028–5040 (2010).
[Crossref] [PubMed]

A.-M. Pena, A. Fabre, D. Débarre, J. Marchal-Somme, B. Crestani, J.-L. Martin, E. Beaurepaire, and M.-C. Schanne-Klein, “Three-dimensional investigation and scoring of extracellular matrix remodeling during lung fibrosis using multiphoton microscopy,” Microsc. Res. Tech. 70, 162–170 (2007).
[Crossref]

Schwegler, F.

A. Frey-Wyssling and F. Schwegler, “Ultrastructure of the chromoplasts in the carrot root,” J. Ultrastruct. Res. 13, 543–559 (1965).
[Crossref] [PubMed]

Shahak, Y.

E. E. Gussakovsky, Y. Shahak, H. Van Amerongen, and V. Barzda, “Circularly polarized chlorophyll luminescence reflects the macro-organization of grana in pea chloroplasts,” Photosynth. Res. 65, 83–92 (2000).
[Crossref]

Shieh, D.-B.

Simon, F.

F. Simon, S. Clevers, V. Dupray, and G. Coquerel, “Relevance of the second harmonic generation to characterize crystalline samples,” Chem. Eng. Technol. 38, 971–983 (2015).
[Crossref]

Simpson, G. J.

C. A. Dailey, B. J. Burke, and G. J. Simpson, “The general failure of Kleinman symmetry in practical nonlinear optical applications,” Chem. Phys. Lett. 390, 8–13 (2004).
[Crossref]

Sipe, J. E.

Squier, J.

Steup, M.

Stewart, B.

Stringari, C.

V. Nucciotti, C. Stringari, L. Sacconi, F. Vanzi, L. Fusi, M. Linari, G. Piazzesi, V. Lombardi, and F. S. Pavone, “Probing myosin structural conformation in vivo by second-harmonic generation microscopy,” Proc. Natl. Acad. Sci. 107, 7763–7768 (2010).
[Crossref] [PubMed]

Sun, C.-K.

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, 6178–6187 (2006).
[Crossref] [PubMed]

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Proc. Natl. Acad. Sci. (1)

V. Nucciotti, C. Stringari, L. Sacconi, F. Vanzi, L. Fusi, M. Linari, G. Piazzesi, V. Lombardi, and F. S. Pavone, “Probing myosin structural conformation in vivo by second-harmonic generation microscopy,” Proc. Natl. Acad. Sci. 107, 7763–7768 (2010).
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Figures (5)

Fig. 1
Fig. 1

a) THG NSMP microscope setup. PSG, PSA – polarization state generator and analyzer, F – 343 nmbandpass filter, PMT – single-photon-counting photomultiplier tube. b) Relationship between the laboratory (XYZ) and molecular (xyz) coordinate systems. δ – in-plane and α – out-of-plane rotation angles. S and s′ represent incoming and outgoing polarization states.

Fig. 2
Fig. 2

THG NSMP results of an isotropic borosilicate glass coverslip: nonlinear Stokes (s′: a, b) and Mueller (: c, d) matrices, and complex χ(3) tensor values (e, f). Measured values are given in panels a, c, e and theoretical isotropic values are given in panels b, d, f. Image-averaged values are shown for each element, uncertainties are reported as ±2 SD. White and gray denotes values that respectively are or are not significantly different from zero. Channel values are shown in a bipolar color scale – red are positive, black are zero, and blue are negative values.

Fig. 3
Fig. 3

The NSMP dataset of a near-isotropic THG-active chromoplast in orange carrot: measured polarization-resolved Stokes (a) and Mueller (d) images, average Stokes (b) and Mueller (e) channel values and their corresponding best-fit values (c and f, respectively) using a C6 tensor model with χzzzz = 1, χxxxx = 0.78 − 0.19i, χzzxx = 0.27, χxxzz = 0.29 − 0.09i and δ = 73°. The colormaps for values in a, d are shown next to the panels. See Fig. 2 for additional formating details. Images are 26 × 30 px, and the scale bar is 3 μm.

Fig. 4
Fig. 4

The molecular-frame χ(3) tensor components of the near-isotropic object of Fig. 3: measured spatially-resolved component maps (a), their average (b) and best-fit values (c) using a C6 model. See Fig. 2 for additional formating details. Image scale same as in Fig. 3.

Fig. 5
Fig. 5

A THG NSMP dataset of a carotene crystallite in orange carrot root: polarization-resolved Stokes (a) and Mueller (b) images; de-rotated molecular-frame χ(3) tensor images of domains A (c) and B (d), and their respective average values (e and f). The domains are marked in the s′0 Stokes image for VLP input. The colormapping of panels (b), (c) and (d) is the same. See Fig. 2 for additional formatting details. Images are 51 × 53 px, and the scale bar is 3 μm.

Equations (5)

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s ( 3 ω ) = S ( ω )
s I 0 3 p L 2 χ z z z z 2 ( 1 , s 1 , s 2 , s 3 ) T
χ I J K L = R I i R J j R K k R L l χ i j k l
χ Z Z Z Z = χ z z z z cos 4 δ cos 4 α + 3 ( χ z z x x + χ x x z z ) 𝒮 2 cos 2 δ cos 2 α + χ x x x x 𝒮 4 χ X X X X = χ x x x x 𝒞 4 + 3 ( χ z z x x + χ x x z z ) 𝒞 2 sin 2 δ cos 2 α + χ z z z z sin 4 δ cos 4 α χ Z Z Z X = ( χ z z z z 2 χ z z x x χ x x z z ) cos 3 δ sin δ cos 4 α ( χ x x x x χ z z x x 2 χ x x z z ) 𝒮 2 cos δ sin δ cos 2 α + 1 3 χ x y y y 𝒮 2 sin α + χ x y z z cos 2 δ cos 2 α sin α χ X X X Z = ( χ x x x x χ z z x x 2 χ x x z z ) 𝒞 2 cos δ sin δ cos 2 α + ( χ z z z z 2 χ z z x x χ x x z z ) cos δ sin 3 δ cos 4 α 1 3 χ x y y y 𝒞 2 sin α χ x y z z sin 2 δ cos 2 α sin α χ Z Z X X = χ z z x x 𝒞 2 cos 2 δ cos 2 α + χ x x z z 𝒮 2 sin 2 δ cos 2 α + ( χ z z z z + χ x x x x 2 χ z z x x 2 χ x x z z ) cos 2 δ sin 2 δ cos 4 α + 1 3 χ x x x x sin 2 α 2 3 χ x y y y cos δ sin δ cos 2 α sin α + 2 χ x y z z cos δ sin δ cos 2 α sin α χ X X Z Z = χ x x z z 𝒞 2 cos 2 δ cos 2 α + χ z z x x 𝒮 2 sin 2 δ cos 2 α + ( χ z z z z + χ x x x x 2 χ z z x x 2 χ x x z z ) cos 2 α sin 2 δ cos 4 α + 1 3 χ x x x x sin 2 α + 2 3 χ x y y y cos δ sin δ cos 2 α sin α 2 χ x y z z cos δ sin δ cos 2 α sin α χ Z X X X = ( 3 χ z z x x χ x x x x ) 𝒞 2 cos δ sin δ cos 2 α + ( χ z z z z 3 χ x x z z ) cos δ sin 3 δ cos 4 α + χ x y y y 𝒞 2 sin α + 3 χ x y z z sin 2 δ cos 2 α sin α χ X Z Z Z = ( χ z z z z 3 χ x x z z ) cos 3 δ sin δ cos 4 α ( χ x x x x 3 χ z z x x ) 𝒮 2 cos δ sin δ cos 2 α χ x y y y 𝒮 2 sin α 3 χ x y z z cos 2 δ cos 2 α sin α
χ ^ z z z z = χ z z z z cos 4 + 3 ( χ z z x x + χ x x z z ) cos 2 α sin 2 α + χ x x x x sin 4 α χ ^ x x x x = χ x x x x χ ^ z z x x = χ z z x x cos 2 α + 1 3 χ x x x x sin 2 α χ ^ x x z z = χ x x z z cos 2 α + 1 3 χ x x x x sin 2 α χ ^ x y z z = χ x y z z cos 2 α sin α + 1 3 χ x y y y sin 3 α χ ^ x y y y = χ x y y y sin α

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