Abstract

Polarization-resolved second harmonic generation (P-SHG) microscopy is an efficient imaging modality for in situ observation of biopolymers structure in tissues, providing information about their mean in-plane orientation and their molecular structure and 3D distribution. Nevertheless, P-SHG signal build-up in a strongly focused regime is not throroughly understood yet, preventing reliable and reproducible measurements. In this study, theoretical analysis, vectorial numerical simulations and experiments are performed to understand how geometrical parameters, such as excitation and collection numerical apertures and detection direction, affect P-SHG imaging in homogeneous collagen tissues. A good agreement is obtained in tendon and cornea, showing that detection geometry significantly affects the SHG anisotropy measurements, but not the measurements of collagen in-plane orientation.

© 2015 Optical Society of America

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References

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    [Crossref]
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2014 (1)

2013 (6)

I. Gusachenko and M.-C. Schanne-Klein, “Numerical simulation of polarization-resolved second-harmonic microscopy in birefringent media,” Phys. Rev. A 88, 053811 (2013).
[Crossref]

T. Starborg, N. S. Kalson, Y. Lu, A. Mironov, T. F. Cootes, D. F. Holmes, and K. E. Kadler, “Using transmission electron microscopy and 3View to determine collagen fibril size and three-dimensional organization,” Nature Protocols 8, 1433–1448 (2013).
[Crossref] [PubMed]

S. Psilodimitrakopoulos, V. Petegnief, N. d. Vera, O. Hernandez, D. Artigas, A. M. Planas, and P. Loza-Alvarez, “Quantitative imaging of microtubule alteration as an early marker of axonal degeneration after ischemia in neurons,” Biophys. J. 104, 968975 (2013).

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 (2013).
[Crossref]

D. Rouède, J.-J. Bellanger, E. Schaub, G. Recher, and F. Tiaho, “Theoretical and experimental SHG angular intensity patterns from healthy and proteolysed muscles,” Biophys. J. 104, 1959–1968 (2013).
[Crossref] [PubMed]

M. Rivard, C. A. Couture, A. K. Miri, M. Laliberte, A. Bertrand-Grenier, L. Mongeau, and F. Legare, “Imaging the bipolarity of myosin filaments with interferometric second harmonic generation microscopy,” Biomed. Opt. Express 4, 2078–2086 (2013).
[Crossref] [PubMed]

2012 (4)

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] [PubMed]

I. Gusachenko, V. Tran, Y. Goulam Houssen, J.-M. Allain, and M.-C. Schanne-Klein, “Polarization-resolved second-harmonic generation in tendon upon mechanical stretching,” Biophys. J. 102, 2220–2229 (2012).
[Crossref] [PubMed]

J. Duboisset, A. Dora, and M. Roche, “Generic model of the molecular orientational distribution probed by polarization-resolved Second Harmonic Generation,” Phys. Rev. A 85, 1–25 (2012).
[Crossref]

G. Latour, I. Gusachenko, L. Kowalczuk, I. Lamarre, and M.-C. Schanne-Klein, “In vivo structural imaging of the cornea by polarization-resolved second harmonic microscopy,” Biomed. Opt. Express 3, 1–15 (2012).
[Crossref] [PubMed]

2011 (3)

2010 (4)

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. USA 107, 7763–7768 (2010).
[Crossref] [PubMed]

I. Gusachenko, G. Latour, and M.-C. Schanne-Klein, “Polarization-resolved second harmonic microscopy in anisotropic thick tissues,” Opt. Express 18, 19339–19352 (2010).
[Crossref] [PubMed]

R. LaComb, O. Nadiarnykh, S. Carey, and P. J. Campagnola, “Quantitative second harmonic generation imaging and modeling of the optical clearing mechanism in striated muscle and tendon,” J. Biomed. Opt. 13, 021109 (2010).
[Crossref]

D. Aït-Belkacem, A. Gasecka, F. Munhoz, S. Brustlein, and S. Brasselet, “Influence of birefringence on polarization resolved nonlinear microscopy and collagen SHG structural imaging,” Opt. Express 18, 14859–14870 (2010).
[Crossref] [PubMed]

2009 (3)

2008 (1)

2007 (2)

F. Tiaho, G. Recher, and D. Rouède, “Estimation of helical angles of myosin and collagen by second harmonic generation imaging microscopy,” Opt. Express 15, 12286–12295 (2007).
[Crossref] [PubMed]

A. Erikson, J. Örtegren, T. Hompland, C. Davies de Lange, and M. Lindgren, “Quantification of the second-order nonlinear susceptibility of collagen I using a laser scanning microscope,” J. Biomed. Optics 12, 044002 (2007).
[Crossref]

2006 (1)

S. V. Plotnikov, A. C. Millard, P. J. Campagnola, and W. A. Mohler, “Characterization of the myosin-based source for second-harmonic generation from muscle sarcomeres,” Biophys. J. 90, 693703 (2006).

2005 (1)

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

2003 (2)

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] [PubMed]

W. Zipfel, R. Williams, and W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nature Biotechnol. 21, 1369–1377 (2003).
[Crossref]

2002 (1)

P. Stoller, K. M. Reiser, P. M. Celliers, and A. M. Rubenchik, “Polarization-modulated second harmonic generation in collagen,” Biophys. J. 82, 3330–3342 (2002).
[Crossref] [PubMed]

2000 (1)

A. N. Bashkatov, E. A. Genina, V. I. Kochubey, and V. V. Tuchin, “Estimation of wavelength dependence of refractive index of collagen fibers of scleral tissue,” Proc. SPIE 4162, 265–268 (2000).
[Crossref]

1997 (1)

D. W. Leonard and K. M. Meek, “Refractive indices of the collagen fibrils and extrafibrillar material of the corneal stroma,” Biophys. J. 72, 1382–1387 (1997).
[Crossref] [PubMed]

1957 (1)

D. M. Maurice, “The structure and transparency of the cornea,” J. Physiol. 136, 263–286 (1957).
[Crossref] [PubMed]

Aït-Belkacem, D.

Ajeti, V.

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] [PubMed]

Allain, J.-M.

I. Gusachenko, V. Tran, Y. Goulam Houssen, J.-M. Allain, and M.-C. Schanne-Klein, “Polarization-resolved second-harmonic generation in tendon upon mechanical stretching,” Biophys. J. 102, 2220–2229 (2012).
[Crossref] [PubMed]

Amat-Roldan, I.

Artigas, D.

S. Psilodimitrakopoulos, V. Petegnief, N. d. Vera, O. Hernandez, D. Artigas, A. M. Planas, and P. Loza-Alvarez, “Quantitative imaging of microtubule alteration as an early marker of axonal degeneration after ischemia in neurons,” Biophys. J. 104, 968975 (2013).

S. Psilodimitrakopoulos, D. Artigas, G. Soria, I. Amat-Roldan, A. M. Planas, and P. Loza-Alvarez, “Quantitative discrimination between endogenous SHG sources in mammalian tissue, based on their polarization response,” Opt. Express 17, 10168–10176 (2009).
[Crossref] [PubMed]

Barzda, V.

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 (2013).
[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] [PubMed]

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] [PubMed]

Bashkatov, A. N.

A. N. Bashkatov, E. A. Genina, V. I. Kochubey, and V. V. Tuchin, “Estimation of wavelength dependence of refractive index of collagen fibers of scleral tissue,” Proc. SPIE 4162, 265–268 (2000).
[Crossref]

Bellanger, J.-J.

D. Rouède, J.-J. Bellanger, E. Schaub, G. Recher, and F. Tiaho, “Theoretical and experimental SHG angular intensity patterns from healthy and proteolysed muscles,” Biophys. J. 104, 1959–1968 (2013).
[Crossref] [PubMed]

Bertrand-Grenier, A.

Brabec, T.

Brasselet, S.

Brown, C. P.

Brown, E. B.

Brustlein, S.

Burke, R. M.

Campagnola, P. J.

V. Ajeti, O. Nadiarnykh, S. M. Ponik, P. J. Keely, K. W. Eliceiri, and P. J. Campagnola, “Structural changes in mixed Col I/Col V collagen gels probed by SHG microscopy: implications for probing stromal alterations in human breast cancer,” Biomed. Opt. Express 2, 2307–2316 (2011).
[Crossref] [PubMed]

R. LaComb, O. Nadiarnykh, S. Carey, and P. J. Campagnola, “Quantitative second harmonic generation imaging and modeling of the optical clearing mechanism in striated muscle and tendon,” J. Biomed. Opt. 13, 021109 (2010).
[Crossref]

O. Nadiarnykh and P. J. Campagnola, “Retention of polarization signatures in SHG microscopy of scattering tissues through optical clearing,” Opt. Express 17, 5794–5806 (2009).
[Crossref] [PubMed]

S. V. Plotnikov, A. C. Millard, P. J. Campagnola, and W. A. Mohler, “Characterization of the myosin-based source for second-harmonic generation from muscle sarcomeres,” Biophys. J. 90, 693703 (2006).

Carey, S.

R. LaComb, O. Nadiarnykh, S. Carey, and P. J. Campagnola, “Quantitative second harmonic generation imaging and modeling of the optical clearing mechanism in striated muscle and tendon,” J. Biomed. Opt. 13, 021109 (2010).
[Crossref]

Carr, A. J.

Celliers, P. M.

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] [PubMed]

P. Stoller, K. M. Reiser, P. M. Celliers, and A. M. Rubenchik, “Polarization-modulated second harmonic generation in collagen,” Biophys. J. 82, 3330–3342 (2002).
[Crossref] [PubMed]

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] [PubMed]

Clemmow, P. C.

P. C. Clemmow, “The theory of electromagnetic waves in a simple anisotropic medium,” in “Proceedings of IEEE,”, vol. 110 (1963), pp. 101–106.

Cootes, T. F.

T. Starborg, N. S. Kalson, Y. Lu, A. Mironov, T. F. Cootes, D. F. Holmes, and K. E. Kadler, “Using transmission electron microscopy and 3View to determine collagen fibril size and three-dimensional organization,” Nature Protocols 8, 1433–1448 (2013).
[Crossref] [PubMed]

Couture, C. A.

Craig, A. S.

D. A. D. Parry and A. S. Craig, “Quantitative electron microscope observations of the collagen fibrils in rat-tail tendon,” 16, 1015–1031 (1977).
[PubMed]

Davies de Lange, C.

A. Erikson, J. Örtegren, T. Hompland, C. Davies de Lange, and M. Lindgren, “Quantification of the second-order nonlinear susceptibility of collagen I using a laser scanning microscope,” J. Biomed. Optics 12, 044002 (2007).
[Crossref]

Dora, A.

J. Duboisset, A. Dora, and M. Roche, “Generic model of the molecular orientational distribution probed by polarization-resolved Second Harmonic Generation,” Phys. Rev. A 85, 1–25 (2012).
[Crossref]

Duboisset, J.

J. Duboisset, A. Dora, and M. Roche, “Generic model of the molecular orientational distribution probed by polarization-resolved Second Harmonic Generation,” Phys. Rev. A 85, 1–25 (2012).
[Crossref]

Eliceiri, K. W.

Erikson, A.

A. Erikson, J. Örtegren, T. Hompland, C. Davies de Lange, and M. Lindgren, “Quantification of the second-order nonlinear susceptibility of collagen I using a laser scanning microscope,” J. Biomed. Optics 12, 044002 (2007).
[Crossref]

Fusi, 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. USA 107, 7763–7768 (2010).
[Crossref] [PubMed]

Gasecka, A.

Genina, E. A.

A. N. Bashkatov, E. A. Genina, V. I. Kochubey, and V. V. Tuchin, “Estimation of wavelength dependence of refractive index of collagen fibers of scleral tissue,” Proc. SPIE 4162, 265–268 (2000).
[Crossref]

Gill, H. S.

Goulam Houssen, Y.

I. Gusachenko, V. Tran, Y. Goulam Houssen, J.-M. Allain, and M.-C. Schanne-Klein, “Polarization-resolved second-harmonic generation in tendon upon mechanical stretching,” Biophys. J. 102, 2220–2229 (2012).
[Crossref] [PubMed]

Gusachenko, I.

I. Gusachenko and M.-C. Schanne-Klein, “Numerical simulation of polarization-resolved second-harmonic microscopy in birefringent media,” Phys. Rev. A 88, 053811 (2013).
[Crossref]

I. Gusachenko, V. Tran, Y. Goulam Houssen, J.-M. Allain, and M.-C. Schanne-Klein, “Polarization-resolved second-harmonic generation in tendon upon mechanical stretching,” Biophys. J. 102, 2220–2229 (2012).
[Crossref] [PubMed]

G. Latour, I. Gusachenko, L. Kowalczuk, I. Lamarre, and M.-C. Schanne-Klein, “In vivo structural imaging of the cornea by polarization-resolved second harmonic microscopy,” Biomed. Opt. Express 3, 1–15 (2012).
[Crossref] [PubMed]

I. Gusachenko, G. Latour, and M.-C. Schanne-Klein, “Polarization-resolved second harmonic microscopy in anisotropic thick tissues,” Opt. Express 18, 19339–19352 (2010).
[Crossref] [PubMed]

Hacyan, S.

S. Hacyan and R. Jáuregui, “Evolution of optical phase and polarization vortices in birefringent media,” J. Opt. A 11, 085204 (2009).
[Crossref]

Han, X.

Hecht, B.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2012), 2nd ed.
[Crossref]

Hernandez, O.

S. Psilodimitrakopoulos, V. Petegnief, N. d. Vera, O. Hernandez, D. Artigas, A. M. Planas, and P. Loza-Alvarez, “Quantitative imaging of microtubule alteration as an early marker of axonal degeneration after ischemia in neurons,” Biophys. J. 104, 968975 (2013).

Holmes, D. F.

T. Starborg, N. S. Kalson, Y. Lu, A. Mironov, T. F. Cootes, D. F. Holmes, and K. E. Kadler, “Using transmission electron microscopy and 3View to determine collagen fibril size and three-dimensional organization,” Nature Protocols 8, 1433–1448 (2013).
[Crossref] [PubMed]

Hompland, T.

A. Erikson, J. Örtegren, T. Hompland, C. Davies de Lange, and M. Lindgren, “Quantification of the second-order nonlinear susceptibility of collagen I using a laser scanning microscope,” J. Biomed. Optics 12, 044002 (2007).
[Crossref]

Houle, M. A.

Jáuregui, R.

S. Hacyan and R. Jáuregui, “Evolution of optical phase and polarization vortices in birefringent media,” J. Opt. A 11, 085204 (2009).
[Crossref]

Kadler, K. E.

T. Starborg, N. S. Kalson, Y. Lu, A. Mironov, T. F. Cootes, D. F. Holmes, and K. E. Kadler, “Using transmission electron microscopy and 3View to determine collagen fibril size and three-dimensional organization,” Nature Protocols 8, 1433–1448 (2013).
[Crossref] [PubMed]

Kalson, N. S.

T. Starborg, N. S. Kalson, Y. Lu, A. Mironov, T. F. Cootes, D. F. Holmes, and K. E. Kadler, “Using transmission electron microscopy and 3View to determine collagen fibril size and three-dimensional organization,” Nature Protocols 8, 1433–1448 (2013).
[Crossref] [PubMed]

Keely, P. J.

Kochubey, V. I.

A. N. Bashkatov, E. A. Genina, V. I. Kochubey, and V. V. Tuchin, “Estimation of wavelength dependence of refractive index of collagen fibers of scleral tissue,” Proc. SPIE 4162, 265–268 (2000).
[Crossref]

Kowalczuk, L.

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] [PubMed]

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] [PubMed]

LaComb, R.

R. LaComb, O. Nadiarnykh, S. Carey, and P. J. Campagnola, “Quantitative second harmonic generation imaging and modeling of the optical clearing mechanism in striated muscle and tendon,” J. Biomed. Opt. 13, 021109 (2010).
[Crossref]

Laliberte, M.

Lamarre, I.

Latour, G.

Legare, F.

Leonard, D. W.

D. W. Leonard and K. M. Meek, “Refractive indices of the collagen fibrils and extrafibrillar material of the corneal stroma,” Biophys. J. 72, 1382–1387 (1997).
[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. USA 107, 7763–7768 (2010).
[Crossref] [PubMed]

Lindgren, M.

A. Erikson, J. Örtegren, T. Hompland, C. Davies de Lange, and M. Lindgren, “Quantification of the second-order nonlinear susceptibility of collagen I using a laser scanning microscope,” J. Biomed. Optics 12, 044002 (2007).
[Crossref]

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. USA 107, 7763–7768 (2010).
[Crossref] [PubMed]

Loza-Alvarez, P.

S. Psilodimitrakopoulos, V. Petegnief, N. d. Vera, O. Hernandez, D. Artigas, A. M. Planas, and P. Loza-Alvarez, “Quantitative imaging of microtubule alteration as an early marker of axonal degeneration after ischemia in neurons,” Biophys. J. 104, 968975 (2013).

S. Psilodimitrakopoulos, D. Artigas, G. Soria, I. Amat-Roldan, A. M. Planas, and P. Loza-Alvarez, “Quantitative discrimination between endogenous SHG sources in mammalian tissue, based on their polarization response,” Opt. Express 17, 10168–10176 (2009).
[Crossref] [PubMed]

Lu, Y.

T. Starborg, N. S. Kalson, Y. Lu, A. Mironov, T. F. Cootes, D. F. Holmes, and K. E. Kadler, “Using transmission electron microscopy and 3View to determine collagen fibril size and three-dimensional organization,” Nature Protocols 8, 1433–1448 (2013).
[Crossref] [PubMed]

Maurice, D. M.

D. M. Maurice, “The structure and transparency of the cornea,” J. Physiol. 136, 263–286 (1957).
[Crossref] [PubMed]

Meek, K. M.

D. W. Leonard and K. M. Meek, “Refractive indices of the collagen fibrils and extrafibrillar material of the corneal stroma,” Biophys. J. 72, 1382–1387 (1997).
[Crossref] [PubMed]

Millard, A. C.

S. V. Plotnikov, A. C. Millard, P. J. Campagnola, and W. A. Mohler, “Characterization of the myosin-based source for second-harmonic generation from muscle sarcomeres,” Biophys. J. 90, 693703 (2006).

Miri, A. K.

Mironov, A.

T. Starborg, N. S. Kalson, Y. Lu, A. Mironov, T. F. Cootes, D. F. Holmes, and K. E. Kadler, “Using transmission electron microscopy and 3View to determine collagen fibril size and three-dimensional organization,” Nature Protocols 8, 1433–1448 (2013).
[Crossref] [PubMed]

Mohler, W. A.

S. V. Plotnikov, A. C. Millard, P. J. Campagnola, and W. A. Mohler, “Characterization of the myosin-based source for second-harmonic generation from muscle sarcomeres,” Biophys. J. 90, 693703 (2006).

Mongeau, L.

Munhoz, F.

Nadiarnykh, O.

Nicklaus, M.

Novotny, L.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2012), 2nd ed.
[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. USA 107, 7763–7768 (2010).
[Crossref] [PubMed]

Örtegren, J.

A. Erikson, J. Örtegren, T. Hompland, C. Davies de Lange, and M. Lindgren, “Quantification of the second-order nonlinear susceptibility of collagen I using a laser scanning microscope,” J. Biomed. Optics 12, 044002 (2007).
[Crossref]

Parry, D. A. D.

D. A. D. Parry and A. S. Craig, “Quantitative electron microscope observations of the collagen fibrils in rat-tail tendon,” 16, 1015–1031 (1977).
[PubMed]

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. USA 107, 7763–7768 (2010).
[Crossref] [PubMed]

Petegnief, V.

S. Psilodimitrakopoulos, V. Petegnief, N. d. Vera, O. Hernandez, D. Artigas, A. M. Planas, and P. Loza-Alvarez, “Quantitative imaging of microtubule alteration as an early marker of axonal degeneration after ischemia in neurons,” Biophys. J. 104, 968975 (2013).

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. USA 107, 7763–7768 (2010).
[Crossref] [PubMed]

Planas, A. M.

S. Psilodimitrakopoulos, V. Petegnief, N. d. Vera, O. Hernandez, D. Artigas, A. M. Planas, and P. Loza-Alvarez, “Quantitative imaging of microtubule alteration as an early marker of axonal degeneration after ischemia in neurons,” Biophys. J. 104, 968975 (2013).

S. Psilodimitrakopoulos, D. Artigas, G. Soria, I. Amat-Roldan, A. M. Planas, and P. Loza-Alvarez, “Quantitative discrimination between endogenous SHG sources in mammalian tissue, based on their polarization response,” Opt. Express 17, 10168–10176 (2009).
[Crossref] [PubMed]

Plotnikov, S. V.

S. V. Plotnikov, A. C. Millard, P. J. Campagnola, and W. A. Mohler, “Characterization of the myosin-based source for second-harmonic generation from muscle sarcomeres,” Biophys. J. 90, 693703 (2006).

Ponik, S. M.

Popov, K.

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] [PubMed]

Price, A. J.

Psilodimitrakopoulos, S.

S. Psilodimitrakopoulos, V. Petegnief, N. d. Vera, O. Hernandez, D. Artigas, A. M. Planas, and P. Loza-Alvarez, “Quantitative imaging of microtubule alteration as an early marker of axonal degeneration after ischemia in neurons,” Biophys. J. 104, 968975 (2013).

S. Psilodimitrakopoulos, D. Artigas, G. Soria, I. Amat-Roldan, A. M. Planas, and P. Loza-Alvarez, “Quantitative discrimination between endogenous SHG sources in mammalian tissue, based on their polarization response,” Opt. Express 17, 10168–10176 (2009).
[Crossref] [PubMed]

Ramunno, L.

Recher, G.

D. Rouède, J.-J. Bellanger, E. Schaub, G. Recher, and F. Tiaho, “Theoretical and experimental SHG angular intensity patterns from healthy and proteolysed muscles,” Biophys. J. 104, 1959–1968 (2013).
[Crossref] [PubMed]

F. Tiaho, G. Recher, and D. Rouède, “Estimation of helical angles of myosin and collagen by second harmonic generation imaging microscopy,” Opt. Express 15, 12286–12295 (2007).
[Crossref] [PubMed]

Réfrégier, P.

Reiser, K. M.

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] [PubMed]

P. Stoller, K. M. Reiser, P. M. Celliers, and A. M. Rubenchik, “Polarization-modulated second harmonic generation in collagen,” Biophys. J. 82, 3330–3342 (2002).
[Crossref] [PubMed]

Rivard, M.

Roche, M.

J. Duboisset, A. Dora, and M. Roche, “Generic model of the molecular orientational distribution probed by polarization-resolved Second Harmonic Generation,” Phys. Rev. A 85, 1–25 (2012).
[Crossref]

P. Réfrégier, M. Roche, and S. Brasselet, “Precision analysis in polarization-resolved second harmonic generation microscopy,” Opt. Lett. 36, 2149–2151 (2011).
[Crossref] [PubMed]

Rouède, D.

D. Rouède, J.-J. Bellanger, E. Schaub, G. Recher, and F. Tiaho, “Theoretical and experimental SHG angular intensity patterns from healthy and proteolysed muscles,” Biophys. J. 104, 1959–1968 (2013).
[Crossref] [PubMed]

F. Tiaho, G. Recher, and D. Rouède, “Estimation of helical angles of myosin and collagen by second harmonic generation imaging microscopy,” Opt. Express 15, 12286–12295 (2007).
[Crossref] [PubMed]

Rubenchik, A. M.

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] [PubMed]

P. Stoller, K. M. Reiser, P. M. Celliers, and A. M. Rubenchik, “Polarization-modulated second harmonic generation in collagen,” Biophys. J. 82, 3330–3342 (2002).
[Crossref] [PubMed]

Ruediger, A.

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. USA 107, 7763–7768 (2010).
[Crossref] [PubMed]

Sandkuijl, D.

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 (2013).
[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] [PubMed]

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] [PubMed]

Schanne-Klein, M.-C.

I. Gusachenko and M.-C. Schanne-Klein, “Numerical simulation of polarization-resolved second-harmonic microscopy in birefringent media,” Phys. Rev. A 88, 053811 (2013).
[Crossref]

I. Gusachenko, V. Tran, Y. Goulam Houssen, J.-M. Allain, and M.-C. Schanne-Klein, “Polarization-resolved second-harmonic generation in tendon upon mechanical stretching,” Biophys. J. 102, 2220–2229 (2012).
[Crossref] [PubMed]

G. Latour, I. Gusachenko, L. Kowalczuk, I. Lamarre, and M.-C. Schanne-Klein, “In vivo structural imaging of the cornea by polarization-resolved second harmonic microscopy,” Biomed. Opt. Express 3, 1–15 (2012).
[Crossref] [PubMed]

I. Gusachenko, G. Latour, and M.-C. Schanne-Klein, “Polarization-resolved second harmonic microscopy in anisotropic thick tissues,” Opt. Express 18, 19339–19352 (2010).
[Crossref] [PubMed]

Schaub, E.

D. Rouède, J.-J. Bellanger, E. Schaub, G. Recher, and F. Tiaho, “Theoretical and experimental SHG angular intensity patterns from healthy and proteolysed muscles,” Biophys. J. 104, 1959–1968 (2013).
[Crossref] [PubMed]

Sipe, J. E.

Soria, G.

Starborg, T.

T. Starborg, N. S. Kalson, Y. Lu, A. Mironov, T. F. Cootes, D. F. Holmes, and K. E. Kadler, “Using transmission electron microscopy and 3View to determine collagen fibril size and three-dimensional organization,” Nature Protocols 8, 1433–1448 (2013).
[Crossref] [PubMed]

Stoller, P.

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] [PubMed]

P. Stoller, K. M. Reiser, P. M. Celliers, and A. M. Rubenchik, “Polarization-modulated second harmonic generation in collagen,” Biophys. J. 82, 3330–3342 (2002).
[Crossref] [PubMed]

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. USA 107, 7763–7768 (2010).
[Crossref] [PubMed]

Tang, P.

Tiaho, F.

D. Rouède, J.-J. Bellanger, E. Schaub, G. Recher, and F. Tiaho, “Theoretical and experimental SHG angular intensity patterns from healthy and proteolysed muscles,” Biophys. J. 104, 1959–1968 (2013).
[Crossref] [PubMed]

F. Tiaho, G. Recher, and D. Rouède, “Estimation of helical angles of myosin and collagen by second harmonic generation imaging microscopy,” Opt. Express 15, 12286–12295 (2007).
[Crossref] [PubMed]

Tokarz, D.

Tran, V.

I. Gusachenko, V. Tran, Y. Goulam Houssen, J.-M. Allain, and M.-C. Schanne-Klein, “Polarization-resolved second-harmonic generation in tendon upon mechanical stretching,” Biophys. J. 102, 2220–2229 (2012).
[Crossref] [PubMed]

Tuchin, V. V.

A. N. Bashkatov, E. A. Genina, V. I. Kochubey, and V. V. Tuchin, “Estimation of wavelength dependence of refractive index of collagen fibers of scleral tissue,” Proc. SPIE 4162, 265–268 (2000).
[Crossref]

Tuer, A. E.

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 (2013).
[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] [PubMed]

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] [PubMed]

Vanzi, F.

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. USA 107, 7763–7768 (2010).
[Crossref] [PubMed]

Vera, N. d.

S. Psilodimitrakopoulos, V. Petegnief, N. d. Vera, O. Hernandez, D. Artigas, A. M. Planas, and P. Loza-Alvarez, “Quantitative imaging of microtubule alteration as an early marker of axonal degeneration after ischemia in neurons,” Biophys. J. 104, 968975 (2013).

Webb, W.

W. Zipfel, R. Williams, and W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nature Biotechnol. 21, 1369–1377 (2003).
[Crossref]

Webb, W. W.

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

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] [PubMed]

Williams, R.

W. Zipfel, R. Williams, and W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nature Biotechnol. 21, 1369–1377 (2003).
[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]

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] [PubMed]

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] [PubMed]

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] [PubMed]

Zettel, M. L.

Zipfel, W.

W. Zipfel, R. Williams, and W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nature Biotechnol. 21, 1369–1377 (2003).
[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]

Appl. Opt. (1)

Biomed. Opt. Express (4)

Biophys. J. (8)

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]

D. W. Leonard and K. M. Meek, “Refractive indices of the collagen fibrils and extrafibrillar material of the corneal stroma,” Biophys. J. 72, 1382–1387 (1997).
[Crossref] [PubMed]

D. Rouède, J.-J. Bellanger, E. Schaub, G. Recher, and F. Tiaho, “Theoretical and experimental SHG angular intensity patterns from healthy and proteolysed muscles,” Biophys. J. 104, 1959–1968 (2013).
[Crossref] [PubMed]

S. V. Plotnikov, A. C. Millard, P. J. Campagnola, and W. A. Mohler, “Characterization of the myosin-based source for second-harmonic generation from muscle sarcomeres,” Biophys. J. 90, 693703 (2006).

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] [PubMed]

I. Gusachenko, V. Tran, Y. Goulam Houssen, J.-M. Allain, and M.-C. Schanne-Klein, “Polarization-resolved second-harmonic generation in tendon upon mechanical stretching,” Biophys. J. 102, 2220–2229 (2012).
[Crossref] [PubMed]

P. Stoller, K. M. Reiser, P. M. Celliers, and A. M. Rubenchik, “Polarization-modulated second harmonic generation in collagen,” Biophys. J. 82, 3330–3342 (2002).
[Crossref] [PubMed]

S. Psilodimitrakopoulos, V. Petegnief, N. d. Vera, O. Hernandez, D. Artigas, A. M. Planas, and P. Loza-Alvarez, “Quantitative imaging of microtubule alteration as an early marker of axonal degeneration after ischemia in neurons,” Biophys. J. 104, 968975 (2013).

J. Biomed. Opt. (1)

R. LaComb, O. Nadiarnykh, S. Carey, and P. J. Campagnola, “Quantitative second harmonic generation imaging and modeling of the optical clearing mechanism in striated muscle and tendon,” J. Biomed. Opt. 13, 021109 (2010).
[Crossref]

J. Biomed. Optics (1)

A. Erikson, J. Örtegren, T. Hompland, C. Davies de Lange, and M. Lindgren, “Quantification of the second-order nonlinear susceptibility of collagen I using a laser scanning microscope,” J. Biomed. Optics 12, 044002 (2007).
[Crossref]

J. Opt. A (1)

S. Hacyan and R. Jáuregui, “Evolution of optical phase and polarization vortices in birefringent media,” J. Opt. A 11, 085204 (2009).
[Crossref]

J. Opt. Soc. Am. B (1)

J. Phys. Chem. B (1)

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] [PubMed]

J. Physiol. (1)

D. M. Maurice, “The structure and transparency of the cornea,” J. Physiol. 136, 263–286 (1957).
[Crossref] [PubMed]

Nature Biotechnol. (1)

W. Zipfel, R. Williams, and W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nature Biotechnol. 21, 1369–1377 (2003).
[Crossref]

Nature Protocols (1)

T. Starborg, N. S. Kalson, Y. Lu, A. Mironov, T. F. Cootes, D. F. Holmes, and K. E. Kadler, “Using transmission electron microscopy and 3View to determine collagen fibril size and three-dimensional organization,” Nature Protocols 8, 1433–1448 (2013).
[Crossref] [PubMed]

Opt. Express (6)

Opt. Lett. (1)

Phys. Rev. A (2)

I. Gusachenko and M.-C. Schanne-Klein, “Numerical simulation of polarization-resolved second-harmonic microscopy in birefringent media,” Phys. Rev. A 88, 053811 (2013).
[Crossref]

J. Duboisset, A. Dora, and M. Roche, “Generic model of the molecular orientational distribution probed by polarization-resolved Second Harmonic Generation,” Phys. Rev. A 85, 1–25 (2012).
[Crossref]

Proc. Natl. Acad. Sci. USA (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. USA 107, 7763–7768 (2010).
[Crossref] [PubMed]

Proc. SPIE (1)

A. N. Bashkatov, E. A. Genina, V. I. Kochubey, and V. V. Tuchin, “Estimation of wavelength dependence of refractive index of collagen fibers of scleral tissue,” Proc. SPIE 4162, 265–268 (2000).
[Crossref]

Other (3)

D. A. D. Parry and A. S. Craig, “Quantitative electron microscope observations of the collagen fibrils in rat-tail tendon,” 16, 1015–1031 (1977).
[PubMed]

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2012), 2nd ed.
[Crossref]

P. C. Clemmow, “The theory of electromagnetic waves in a simple anisotropic medium,” in “Proceedings of IEEE,”, vol. 110 (1963), pp. 101–106.

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Figures (6)

Fig. 1
Fig. 1 (a) Setup geometry. Excitation polarization angle α is controlled by two achromatic waveplates. Excitation field E⃗inc is then focused through an objective (NAexc). SH radiation is collected forwards with a condensor (NAdet,F) and backwards through the focusing objective (NAdet,B = NAexc). (b) Zoom in the focal plane. (XYZ) is the laboratory frame, (xyz) is the fibril frame with x the fibril axis.
Fig. 2
Fig. 2 Simulation of vectorial focal field E⃗ for an incident electric field E⃗inc directed along x axis and NA=1.2 focusing. (a) Scheme of wavefront for strong focusing, showing the onset of an axial electric field component. (b) Modulus and phase of each electric field component Ex,‖, Ey,‖ and Ez,‖ in the laboratory frame. The modulus is represented by the color brightness, with the multiplication factor indicated in the upper right corner. The phase is color coded, with respect to Ex phase set to 0 in the whole volume.
Fig. 3
Fig. 3 (a–b) Spatial pattern of additional terms in the nonlinear polarizability due to strong focusing (excitation NA 1.2) compared to usual term E x , 2 (a1): (a) P x 2 ω, (b) P z 2 ω. The modulus is represented by the color brightness, with the multiplication factor indicated in the bottom. The phase is color coded with the same colors as in Fig. 2. (c–d) Backwards and Forward intensity radiation patterns induced by those additional terms, for the same excitation NA and collection NA 1.2 (black circle).
Fig. 4
Fig. 4 (a) Spatial pattern of local intensities ratio I / I backwards (bottom) and forwards (top). Full circle: collection NA=1.2, dashed circle: collection NA=0.8. (b) Anisotropy parameter ρ in respect to excitation and collection NA for B-SHG (dashed line) and F-SHG (full line) signals. Intrinsic value: ρ = 1.36. ρ is overestimated in forward measurements, underestimated in backward measurements. ρ values at collection NAs 0.8 and 1.2 are indicated by the vertical lines.
Fig. 5
Fig. 5 Comparison between experimental results and numerical simulations. ρ intrinsic value is given by the full line in gray. In tendon: (a) numerical simulation, (b) mean experimental values of anisotropy parameter ρ at different depths for each of the three objectives and for F-SHG and B-SHG signals. In cornea: (c) numerical simulation, (d) mean experimental values of ρ at different depths for each of the three objectives and for F-SHG and B-SHG signals. dzlens = 0 μm at the surface of the sample.
Fig. 6
Fig. 6 Precision measurements ( oe-23-7-9313-i001.jpg Fit, F-SHG) and theoretical calculation (—) for a Poisson noise. In light blue: usual total mean number of detected photons, considering binning of approx. 400 pixels (for one sub-area) before extracting φ and ρ in order to increase measurements accuracy.

Tables (1)

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Table 1 Comparison between simulation and experimental measurements in cornea. The ρ value is obtained by averaging over the whole depth of imaging (30 μm to 100 μm depth), with high enough r-squared for experimental values (see section 4.3).

Equations (9)

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P i 2 ω = χ i j k ( 2 ) E j ω E k ω
P x 2 ω = χ x x x ( 2 ) E x 2 + χ x y y ( 2 ) E y 2 + χ x y y ( 2 ) E z 2 , P y 2 ω = 2 χ x y y ( 2 ) E x E y P z 2 ω = 2 χ x y y ( 2 ) E x E z
P x 2 ω ρ cos 2 ( α φ ) + sin 2 ( α φ ) P y 2 ω 2 cos ( α φ ) sin ( α φ )
2 ω ( α ) = β [ A cos ( 4 α 4 φ ) + B cos ( 2 α 2 φ ) + 1 ]
ρ = A + B + 1 A B + 1 =
= β ( A + B + 1 ) = β ( A B + 1 )
P x 2 ω ( ρ E x , 2 + E y , 2 + E z , 2 ) cos 2 α + ( ρ E x , 2 + E y , 2 + E z , 2 ) sin 2 α + 2 ( ρ E x , E x , + E y , E y , + E z , E z , ) cos α sin α P y 2 ω 2 E x , E y , cos 2 α + 2 E x , E y , sin 2 α + 2 ( E x , E y , + E y , E x , ) cos α sin α P z 2 ω 2 E x , E z , cos 2 α + 2 E x , E z , sin 2 α + 2 ( E x , E z , + E z , E x , ) cos α sin α
( Δ φ ) 2 1 2 < N tot > ( B 2 + 4 A 2 )
( Δ ρ ) 2 1 < N tot > ( 1 B + A ) 2 [ 1 ρ 2 + ρ 2 ]

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