Abstract

A theoretical description is presented about a new analysis method to determine three-dimensional (3D) molecular orientation by concurrently analyzing multiple Raman polarization profiles. Conventional approaches to polarization Raman spectroscopy are based on single peaks, and their 2D-projected polarization profiles are limited in providing 3D orientational information. Our new method analyzes multiple Raman profiles acquired by a single polarization scanning measurement of broadband coherent anti-Stokes Raman scattering (BCARS). Because the analysis uses only dimensionless quantities, such as intensity ratios and phase difference between multiple profiles, the results are not affected by sample concentration and the system response function. We describe how to determine the 3D molecular orientation with the dimensionless observables by using two simplified model cases. In addition, we discuss the effect of orientational broadening on the polarization profiles in the two model cases. We find that in the presence of broadening we can still determine the mean 3D orientation angles and, furthermore, the degree of orientational broadening.

© 2015 Optical Society of America

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References

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  1. M. Tanaka and R. J. Young, “Review polarised Raman spectroscopy for the study of molecular orientation distributions in polymers,” J. Mater. Sci. 41(3), 963–991 (2006).
    [Crossref]
  2. S. H. Parekh and K. F. Domke, “Watching orientational ordering at the nanoscale with coherent anti-stokes Raman microscopy,” Chemistry 19(36), 11822–11830 (2013).
    [Crossref] [PubMed]
  3. M. Pigeon, R. E. Prud’homme, and M. Pézolet, “Characterization of molecular orientation in polyethylene by Raman spectroscopy,” Macromolecules 24(20), 5687–5694 (1991).
    [Crossref]
  4. M. Richard-Lacroix and C. Pellerin, “Molecular orientation in electrospun fibers: from mats to single fibers,” Macromolecules 46(24), 9473–9493 (2013).
    [Crossref]
  5. F. Munhoz, H. Rigneault, and S. Brasselet, “High order symmetry structural properties of vibrational resonances using multiple-field polarization coherent anti-Stokes Raman spectroscopy microscopy,” Phys. Rev. Lett. 105(12), 123903 (2010).
    [Crossref] [PubMed]
  6. B. G. Saar, H.-S. Park, X. S. Xie, and O. D. Lavrentovich, “Three-dimensional imaging of chemical bond orientation in liquid crystals by coherent anti- Stokes Raman scattering microscopy,” Opt. Express 15(21), 13585–13596 (2007).
    [Crossref] [PubMed]
  7. J. Duboisset, P. Berto, P. Gasecka, F.-Z. Bioud, P. Ferrand, H. Rigneault, and S. Brasselet, “Molecular orientational order probed by coherent anti-Stokes Raman scattering (CARS) and stimulated Raman scattering (SRS) microscopy: a spectral comparative study,” J. Phys. Chem. B 119(7), 3242–3249 (2015).
    [Crossref] [PubMed]
  8. E. Toprak, J. Enderlein, S. Syed, S. A. McKinney, R. G. Petschek, T. Ha, Y. E. Goldman, and P. R. Selvin, “Defocused orientation and position imaging (DOPI) of myosin V,” Proc. Natl. Acad. Sci. U.S.A. 103(17), 6495–6499 (2006).
    [Crossref] [PubMed]
  9. N. Sandeau, L. Le Xuan, D. Chauvat, C. Zhou, J.-F. Roch, and S. Brasselet, “Defocused imaging of second harmonic generation from a single nanocrystal,” Opt. Express 15(24), 16051–16060 (2007).
    [Crossref] [PubMed]
  10. E. A. Büyüktanir, K. Zhang, A. Gericke, and J. L. West, “Raman imaging of nematic and smectic liquid crystals,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 487(1), 39–51 (2008).
    [Crossref]
  11. A. V. Kachynski, A. N. Kuzmin, P. N. Prasad, and I. I. Smalyukh, “Coherent anti-Stokes Raman scattering polarized microscopy of three-dimensional director structures in liquid crystals,” Appl. Phys. Lett. 91(15), 151905 (2007).
    [Crossref]
  12. Y. J. Lee, C. R. Snyder, A. M. Forster, M. T. Cicerone, and W. Wu, “Imaging the molecular structure of polyethylene blends with broadband coherent Raman microscopy,” ACS Macro Lett. 1(11), 1347–1351 (2012).
    [Crossref]
  13. R. P. Davis, A. J. Moad, G. S. Goeken, R. D. Wampler, and G. J. Simpson, “Selection rules and symmetry relations for four-wave mixing measurements of uniaxial assemblies,” J. Phys. Chem. B 112(18), 5834–5848 (2008).
    [Crossref] [PubMed]
  14. C. Zhang, J. Wang, B. Ding, and J. Jasensky, “Quantitative spectral analysis of coherent anti-Stokes Raman scattering signals: C-H stretching modes of the methyl group,” J. Phys. Chem. B 118(27), 7647–7656 (2014).
    [Crossref] [PubMed]
  15. D. I. Bower, “Investigation of molecular orientation distributions by polarized Raman scattering and polarized fluorescence,” J. Polym. Sci., Polym. Phys. Ed. 10(11), 2135–2153 (1972).
  16. M. T. Cicerone, K. A. Aamer, Y. J. Lee, and E. Vartiainen, “Maximum entropy and time-domain Kramers-Kronig phase retrieval approaches are functionally equivalent for CARS microspectroscopy,” J. Raman Spectrosc. 43(5), 637–643 (2012).
    [Crossref]
  17. C. Khatri and K. Mardia, “The von Mises-Fisher matrix distribution in orientation statistics,” J. R. Stat. Soc. B 39(1), 95–106 (1977).
  18. G. W. H. Wurpel, H. A. Rinia, and M. Müller, “Imaging orientational order and lipid density in multilamellar vesicles with multiplex CARS microscopy,” J. Microsc. 218(1), 37–45 (2005).
    [Crossref] [PubMed]
  19. G. Turrell, “Analysis of polarization measurements in Raman microspectroscopy,” J. Raman Spectrosc. 15(2), 103–108 (1984).
    [Crossref]

2015 (1)

J. Duboisset, P. Berto, P. Gasecka, F.-Z. Bioud, P. Ferrand, H. Rigneault, and S. Brasselet, “Molecular orientational order probed by coherent anti-Stokes Raman scattering (CARS) and stimulated Raman scattering (SRS) microscopy: a spectral comparative study,” J. Phys. Chem. B 119(7), 3242–3249 (2015).
[Crossref] [PubMed]

2014 (1)

C. Zhang, J. Wang, B. Ding, and J. Jasensky, “Quantitative spectral analysis of coherent anti-Stokes Raman scattering signals: C-H stretching modes of the methyl group,” J. Phys. Chem. B 118(27), 7647–7656 (2014).
[Crossref] [PubMed]

2013 (2)

M. Richard-Lacroix and C. Pellerin, “Molecular orientation in electrospun fibers: from mats to single fibers,” Macromolecules 46(24), 9473–9493 (2013).
[Crossref]

S. H. Parekh and K. F. Domke, “Watching orientational ordering at the nanoscale with coherent anti-stokes Raman microscopy,” Chemistry 19(36), 11822–11830 (2013).
[Crossref] [PubMed]

2012 (2)

M. T. Cicerone, K. A. Aamer, Y. J. Lee, and E. Vartiainen, “Maximum entropy and time-domain Kramers-Kronig phase retrieval approaches are functionally equivalent for CARS microspectroscopy,” J. Raman Spectrosc. 43(5), 637–643 (2012).
[Crossref]

Y. J. Lee, C. R. Snyder, A. M. Forster, M. T. Cicerone, and W. Wu, “Imaging the molecular structure of polyethylene blends with broadband coherent Raman microscopy,” ACS Macro Lett. 1(11), 1347–1351 (2012).
[Crossref]

2010 (1)

F. Munhoz, H. Rigneault, and S. Brasselet, “High order symmetry structural properties of vibrational resonances using multiple-field polarization coherent anti-Stokes Raman spectroscopy microscopy,” Phys. Rev. Lett. 105(12), 123903 (2010).
[Crossref] [PubMed]

2008 (2)

R. P. Davis, A. J. Moad, G. S. Goeken, R. D. Wampler, and G. J. Simpson, “Selection rules and symmetry relations for four-wave mixing measurements of uniaxial assemblies,” J. Phys. Chem. B 112(18), 5834–5848 (2008).
[Crossref] [PubMed]

E. A. Büyüktanir, K. Zhang, A. Gericke, and J. L. West, “Raman imaging of nematic and smectic liquid crystals,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 487(1), 39–51 (2008).
[Crossref]

2007 (3)

2006 (2)

M. Tanaka and R. J. Young, “Review polarised Raman spectroscopy for the study of molecular orientation distributions in polymers,” J. Mater. Sci. 41(3), 963–991 (2006).
[Crossref]

E. Toprak, J. Enderlein, S. Syed, S. A. McKinney, R. G. Petschek, T. Ha, Y. E. Goldman, and P. R. Selvin, “Defocused orientation and position imaging (DOPI) of myosin V,” Proc. Natl. Acad. Sci. U.S.A. 103(17), 6495–6499 (2006).
[Crossref] [PubMed]

2005 (1)

G. W. H. Wurpel, H. A. Rinia, and M. Müller, “Imaging orientational order and lipid density in multilamellar vesicles with multiplex CARS microscopy,” J. Microsc. 218(1), 37–45 (2005).
[Crossref] [PubMed]

1991 (1)

M. Pigeon, R. E. Prud’homme, and M. Pézolet, “Characterization of molecular orientation in polyethylene by Raman spectroscopy,” Macromolecules 24(20), 5687–5694 (1991).
[Crossref]

1984 (1)

G. Turrell, “Analysis of polarization measurements in Raman microspectroscopy,” J. Raman Spectrosc. 15(2), 103–108 (1984).
[Crossref]

1977 (1)

C. Khatri and K. Mardia, “The von Mises-Fisher matrix distribution in orientation statistics,” J. R. Stat. Soc. B 39(1), 95–106 (1977).

1972 (1)

D. I. Bower, “Investigation of molecular orientation distributions by polarized Raman scattering and polarized fluorescence,” J. Polym. Sci., Polym. Phys. Ed. 10(11), 2135–2153 (1972).

Aamer, K. A.

M. T. Cicerone, K. A. Aamer, Y. J. Lee, and E. Vartiainen, “Maximum entropy and time-domain Kramers-Kronig phase retrieval approaches are functionally equivalent for CARS microspectroscopy,” J. Raman Spectrosc. 43(5), 637–643 (2012).
[Crossref]

Berto, P.

J. Duboisset, P. Berto, P. Gasecka, F.-Z. Bioud, P. Ferrand, H. Rigneault, and S. Brasselet, “Molecular orientational order probed by coherent anti-Stokes Raman scattering (CARS) and stimulated Raman scattering (SRS) microscopy: a spectral comparative study,” J. Phys. Chem. B 119(7), 3242–3249 (2015).
[Crossref] [PubMed]

Bioud, F.-Z.

J. Duboisset, P. Berto, P. Gasecka, F.-Z. Bioud, P. Ferrand, H. Rigneault, and S. Brasselet, “Molecular orientational order probed by coherent anti-Stokes Raman scattering (CARS) and stimulated Raman scattering (SRS) microscopy: a spectral comparative study,” J. Phys. Chem. B 119(7), 3242–3249 (2015).
[Crossref] [PubMed]

Bower, D. I.

D. I. Bower, “Investigation of molecular orientation distributions by polarized Raman scattering and polarized fluorescence,” J. Polym. Sci., Polym. Phys. Ed. 10(11), 2135–2153 (1972).

Brasselet, S.

J. Duboisset, P. Berto, P. Gasecka, F.-Z. Bioud, P. Ferrand, H. Rigneault, and S. Brasselet, “Molecular orientational order probed by coherent anti-Stokes Raman scattering (CARS) and stimulated Raman scattering (SRS) microscopy: a spectral comparative study,” J. Phys. Chem. B 119(7), 3242–3249 (2015).
[Crossref] [PubMed]

F. Munhoz, H. Rigneault, and S. Brasselet, “High order symmetry structural properties of vibrational resonances using multiple-field polarization coherent anti-Stokes Raman spectroscopy microscopy,” Phys. Rev. Lett. 105(12), 123903 (2010).
[Crossref] [PubMed]

N. Sandeau, L. Le Xuan, D. Chauvat, C. Zhou, J.-F. Roch, and S. Brasselet, “Defocused imaging of second harmonic generation from a single nanocrystal,” Opt. Express 15(24), 16051–16060 (2007).
[Crossref] [PubMed]

Büyüktanir, E. A.

E. A. Büyüktanir, K. Zhang, A. Gericke, and J. L. West, “Raman imaging of nematic and smectic liquid crystals,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 487(1), 39–51 (2008).
[Crossref]

Chauvat, D.

Cicerone, M. T.

M. T. Cicerone, K. A. Aamer, Y. J. Lee, and E. Vartiainen, “Maximum entropy and time-domain Kramers-Kronig phase retrieval approaches are functionally equivalent for CARS microspectroscopy,” J. Raman Spectrosc. 43(5), 637–643 (2012).
[Crossref]

Y. J. Lee, C. R. Snyder, A. M. Forster, M. T. Cicerone, and W. Wu, “Imaging the molecular structure of polyethylene blends with broadband coherent Raman microscopy,” ACS Macro Lett. 1(11), 1347–1351 (2012).
[Crossref]

Davis, R. P.

R. P. Davis, A. J. Moad, G. S. Goeken, R. D. Wampler, and G. J. Simpson, “Selection rules and symmetry relations for four-wave mixing measurements of uniaxial assemblies,” J. Phys. Chem. B 112(18), 5834–5848 (2008).
[Crossref] [PubMed]

Ding, B.

C. Zhang, J. Wang, B. Ding, and J. Jasensky, “Quantitative spectral analysis of coherent anti-Stokes Raman scattering signals: C-H stretching modes of the methyl group,” J. Phys. Chem. B 118(27), 7647–7656 (2014).
[Crossref] [PubMed]

Domke, K. F.

S. H. Parekh and K. F. Domke, “Watching orientational ordering at the nanoscale with coherent anti-stokes Raman microscopy,” Chemistry 19(36), 11822–11830 (2013).
[Crossref] [PubMed]

Duboisset, J.

J. Duboisset, P. Berto, P. Gasecka, F.-Z. Bioud, P. Ferrand, H. Rigneault, and S. Brasselet, “Molecular orientational order probed by coherent anti-Stokes Raman scattering (CARS) and stimulated Raman scattering (SRS) microscopy: a spectral comparative study,” J. Phys. Chem. B 119(7), 3242–3249 (2015).
[Crossref] [PubMed]

Enderlein, J.

E. Toprak, J. Enderlein, S. Syed, S. A. McKinney, R. G. Petschek, T. Ha, Y. E. Goldman, and P. R. Selvin, “Defocused orientation and position imaging (DOPI) of myosin V,” Proc. Natl. Acad. Sci. U.S.A. 103(17), 6495–6499 (2006).
[Crossref] [PubMed]

Ferrand, P.

J. Duboisset, P. Berto, P. Gasecka, F.-Z. Bioud, P. Ferrand, H. Rigneault, and S. Brasselet, “Molecular orientational order probed by coherent anti-Stokes Raman scattering (CARS) and stimulated Raman scattering (SRS) microscopy: a spectral comparative study,” J. Phys. Chem. B 119(7), 3242–3249 (2015).
[Crossref] [PubMed]

Forster, A. M.

Y. J. Lee, C. R. Snyder, A. M. Forster, M. T. Cicerone, and W. Wu, “Imaging the molecular structure of polyethylene blends with broadband coherent Raman microscopy,” ACS Macro Lett. 1(11), 1347–1351 (2012).
[Crossref]

Gasecka, P.

J. Duboisset, P. Berto, P. Gasecka, F.-Z. Bioud, P. Ferrand, H. Rigneault, and S. Brasselet, “Molecular orientational order probed by coherent anti-Stokes Raman scattering (CARS) and stimulated Raman scattering (SRS) microscopy: a spectral comparative study,” J. Phys. Chem. B 119(7), 3242–3249 (2015).
[Crossref] [PubMed]

Gericke, A.

E. A. Büyüktanir, K. Zhang, A. Gericke, and J. L. West, “Raman imaging of nematic and smectic liquid crystals,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 487(1), 39–51 (2008).
[Crossref]

Goeken, G. S.

R. P. Davis, A. J. Moad, G. S. Goeken, R. D. Wampler, and G. J. Simpson, “Selection rules and symmetry relations for four-wave mixing measurements of uniaxial assemblies,” J. Phys. Chem. B 112(18), 5834–5848 (2008).
[Crossref] [PubMed]

Goldman, Y. E.

E. Toprak, J. Enderlein, S. Syed, S. A. McKinney, R. G. Petschek, T. Ha, Y. E. Goldman, and P. R. Selvin, “Defocused orientation and position imaging (DOPI) of myosin V,” Proc. Natl. Acad. Sci. U.S.A. 103(17), 6495–6499 (2006).
[Crossref] [PubMed]

Ha, T.

E. Toprak, J. Enderlein, S. Syed, S. A. McKinney, R. G. Petschek, T. Ha, Y. E. Goldman, and P. R. Selvin, “Defocused orientation and position imaging (DOPI) of myosin V,” Proc. Natl. Acad. Sci. U.S.A. 103(17), 6495–6499 (2006).
[Crossref] [PubMed]

Jasensky, J.

C. Zhang, J. Wang, B. Ding, and J. Jasensky, “Quantitative spectral analysis of coherent anti-Stokes Raman scattering signals: C-H stretching modes of the methyl group,” J. Phys. Chem. B 118(27), 7647–7656 (2014).
[Crossref] [PubMed]

Kachynski, A. V.

A. V. Kachynski, A. N. Kuzmin, P. N. Prasad, and I. I. Smalyukh, “Coherent anti-Stokes Raman scattering polarized microscopy of three-dimensional director structures in liquid crystals,” Appl. Phys. Lett. 91(15), 151905 (2007).
[Crossref]

Khatri, C.

C. Khatri and K. Mardia, “The von Mises-Fisher matrix distribution in orientation statistics,” J. R. Stat. Soc. B 39(1), 95–106 (1977).

Kuzmin, A. N.

A. V. Kachynski, A. N. Kuzmin, P. N. Prasad, and I. I. Smalyukh, “Coherent anti-Stokes Raman scattering polarized microscopy of three-dimensional director structures in liquid crystals,” Appl. Phys. Lett. 91(15), 151905 (2007).
[Crossref]

Lavrentovich, O. D.

Le Xuan, L.

Lee, Y. J.

M. T. Cicerone, K. A. Aamer, Y. J. Lee, and E. Vartiainen, “Maximum entropy and time-domain Kramers-Kronig phase retrieval approaches are functionally equivalent for CARS microspectroscopy,” J. Raman Spectrosc. 43(5), 637–643 (2012).
[Crossref]

Y. J. Lee, C. R. Snyder, A. M. Forster, M. T. Cicerone, and W. Wu, “Imaging the molecular structure of polyethylene blends with broadband coherent Raman microscopy,” ACS Macro Lett. 1(11), 1347–1351 (2012).
[Crossref]

Mardia, K.

C. Khatri and K. Mardia, “The von Mises-Fisher matrix distribution in orientation statistics,” J. R. Stat. Soc. B 39(1), 95–106 (1977).

McKinney, S. A.

E. Toprak, J. Enderlein, S. Syed, S. A. McKinney, R. G. Petschek, T. Ha, Y. E. Goldman, and P. R. Selvin, “Defocused orientation and position imaging (DOPI) of myosin V,” Proc. Natl. Acad. Sci. U.S.A. 103(17), 6495–6499 (2006).
[Crossref] [PubMed]

Moad, A. J.

R. P. Davis, A. J. Moad, G. S. Goeken, R. D. Wampler, and G. J. Simpson, “Selection rules and symmetry relations for four-wave mixing measurements of uniaxial assemblies,” J. Phys. Chem. B 112(18), 5834–5848 (2008).
[Crossref] [PubMed]

Müller, M.

G. W. H. Wurpel, H. A. Rinia, and M. Müller, “Imaging orientational order and lipid density in multilamellar vesicles with multiplex CARS microscopy,” J. Microsc. 218(1), 37–45 (2005).
[Crossref] [PubMed]

Munhoz, F.

F. Munhoz, H. Rigneault, and S. Brasselet, “High order symmetry structural properties of vibrational resonances using multiple-field polarization coherent anti-Stokes Raman spectroscopy microscopy,” Phys. Rev. Lett. 105(12), 123903 (2010).
[Crossref] [PubMed]

Parekh, S. H.

S. H. Parekh and K. F. Domke, “Watching orientational ordering at the nanoscale with coherent anti-stokes Raman microscopy,” Chemistry 19(36), 11822–11830 (2013).
[Crossref] [PubMed]

Park, H.-S.

Pellerin, C.

M. Richard-Lacroix and C. Pellerin, “Molecular orientation in electrospun fibers: from mats to single fibers,” Macromolecules 46(24), 9473–9493 (2013).
[Crossref]

Petschek, R. G.

E. Toprak, J. Enderlein, S. Syed, S. A. McKinney, R. G. Petschek, T. Ha, Y. E. Goldman, and P. R. Selvin, “Defocused orientation and position imaging (DOPI) of myosin V,” Proc. Natl. Acad. Sci. U.S.A. 103(17), 6495–6499 (2006).
[Crossref] [PubMed]

Pézolet, M.

M. Pigeon, R. E. Prud’homme, and M. Pézolet, “Characterization of molecular orientation in polyethylene by Raman spectroscopy,” Macromolecules 24(20), 5687–5694 (1991).
[Crossref]

Pigeon, M.

M. Pigeon, R. E. Prud’homme, and M. Pézolet, “Characterization of molecular orientation in polyethylene by Raman spectroscopy,” Macromolecules 24(20), 5687–5694 (1991).
[Crossref]

Prasad, P. N.

A. V. Kachynski, A. N. Kuzmin, P. N. Prasad, and I. I. Smalyukh, “Coherent anti-Stokes Raman scattering polarized microscopy of three-dimensional director structures in liquid crystals,” Appl. Phys. Lett. 91(15), 151905 (2007).
[Crossref]

Prud’homme, R. E.

M. Pigeon, R. E. Prud’homme, and M. Pézolet, “Characterization of molecular orientation in polyethylene by Raman spectroscopy,” Macromolecules 24(20), 5687–5694 (1991).
[Crossref]

Richard-Lacroix, M.

M. Richard-Lacroix and C. Pellerin, “Molecular orientation in electrospun fibers: from mats to single fibers,” Macromolecules 46(24), 9473–9493 (2013).
[Crossref]

Rigneault, H.

J. Duboisset, P. Berto, P. Gasecka, F.-Z. Bioud, P. Ferrand, H. Rigneault, and S. Brasselet, “Molecular orientational order probed by coherent anti-Stokes Raman scattering (CARS) and stimulated Raman scattering (SRS) microscopy: a spectral comparative study,” J. Phys. Chem. B 119(7), 3242–3249 (2015).
[Crossref] [PubMed]

F. Munhoz, H. Rigneault, and S. Brasselet, “High order symmetry structural properties of vibrational resonances using multiple-field polarization coherent anti-Stokes Raman spectroscopy microscopy,” Phys. Rev. Lett. 105(12), 123903 (2010).
[Crossref] [PubMed]

Rinia, H. A.

G. W. H. Wurpel, H. A. Rinia, and M. Müller, “Imaging orientational order and lipid density in multilamellar vesicles with multiplex CARS microscopy,” J. Microsc. 218(1), 37–45 (2005).
[Crossref] [PubMed]

Roch, J.-F.

Saar, B. G.

Sandeau, N.

Selvin, P. R.

E. Toprak, J. Enderlein, S. Syed, S. A. McKinney, R. G. Petschek, T. Ha, Y. E. Goldman, and P. R. Selvin, “Defocused orientation and position imaging (DOPI) of myosin V,” Proc. Natl. Acad. Sci. U.S.A. 103(17), 6495–6499 (2006).
[Crossref] [PubMed]

Simpson, G. J.

R. P. Davis, A. J. Moad, G. S. Goeken, R. D. Wampler, and G. J. Simpson, “Selection rules and symmetry relations for four-wave mixing measurements of uniaxial assemblies,” J. Phys. Chem. B 112(18), 5834–5848 (2008).
[Crossref] [PubMed]

Smalyukh, I. I.

A. V. Kachynski, A. N. Kuzmin, P. N. Prasad, and I. I. Smalyukh, “Coherent anti-Stokes Raman scattering polarized microscopy of three-dimensional director structures in liquid crystals,” Appl. Phys. Lett. 91(15), 151905 (2007).
[Crossref]

Snyder, C. R.

Y. J. Lee, C. R. Snyder, A. M. Forster, M. T. Cicerone, and W. Wu, “Imaging the molecular structure of polyethylene blends with broadband coherent Raman microscopy,” ACS Macro Lett. 1(11), 1347–1351 (2012).
[Crossref]

Syed, S.

E. Toprak, J. Enderlein, S. Syed, S. A. McKinney, R. G. Petschek, T. Ha, Y. E. Goldman, and P. R. Selvin, “Defocused orientation and position imaging (DOPI) of myosin V,” Proc. Natl. Acad. Sci. U.S.A. 103(17), 6495–6499 (2006).
[Crossref] [PubMed]

Tanaka, M.

M. Tanaka and R. J. Young, “Review polarised Raman spectroscopy for the study of molecular orientation distributions in polymers,” J. Mater. Sci. 41(3), 963–991 (2006).
[Crossref]

Toprak, E.

E. Toprak, J. Enderlein, S. Syed, S. A. McKinney, R. G. Petschek, T. Ha, Y. E. Goldman, and P. R. Selvin, “Defocused orientation and position imaging (DOPI) of myosin V,” Proc. Natl. Acad. Sci. U.S.A. 103(17), 6495–6499 (2006).
[Crossref] [PubMed]

Turrell, G.

G. Turrell, “Analysis of polarization measurements in Raman microspectroscopy,” J. Raman Spectrosc. 15(2), 103–108 (1984).
[Crossref]

Vartiainen, E.

M. T. Cicerone, K. A. Aamer, Y. J. Lee, and E. Vartiainen, “Maximum entropy and time-domain Kramers-Kronig phase retrieval approaches are functionally equivalent for CARS microspectroscopy,” J. Raman Spectrosc. 43(5), 637–643 (2012).
[Crossref]

Wampler, R. D.

R. P. Davis, A. J. Moad, G. S. Goeken, R. D. Wampler, and G. J. Simpson, “Selection rules and symmetry relations for four-wave mixing measurements of uniaxial assemblies,” J. Phys. Chem. B 112(18), 5834–5848 (2008).
[Crossref] [PubMed]

Wang, J.

C. Zhang, J. Wang, B. Ding, and J. Jasensky, “Quantitative spectral analysis of coherent anti-Stokes Raman scattering signals: C-H stretching modes of the methyl group,” J. Phys. Chem. B 118(27), 7647–7656 (2014).
[Crossref] [PubMed]

West, J. L.

E. A. Büyüktanir, K. Zhang, A. Gericke, and J. L. West, “Raman imaging of nematic and smectic liquid crystals,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 487(1), 39–51 (2008).
[Crossref]

Wu, W.

Y. J. Lee, C. R. Snyder, A. M. Forster, M. T. Cicerone, and W. Wu, “Imaging the molecular structure of polyethylene blends with broadband coherent Raman microscopy,” ACS Macro Lett. 1(11), 1347–1351 (2012).
[Crossref]

Wurpel, G. W. H.

G. W. H. Wurpel, H. A. Rinia, and M. Müller, “Imaging orientational order and lipid density in multilamellar vesicles with multiplex CARS microscopy,” J. Microsc. 218(1), 37–45 (2005).
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Xie, X. S.

Young, R. J.

M. Tanaka and R. J. Young, “Review polarised Raman spectroscopy for the study of molecular orientation distributions in polymers,” J. Mater. Sci. 41(3), 963–991 (2006).
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Zhang, C.

C. Zhang, J. Wang, B. Ding, and J. Jasensky, “Quantitative spectral analysis of coherent anti-Stokes Raman scattering signals: C-H stretching modes of the methyl group,” J. Phys. Chem. B 118(27), 7647–7656 (2014).
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Zhang, K.

E. A. Büyüktanir, K. Zhang, A. Gericke, and J. L. West, “Raman imaging of nematic and smectic liquid crystals,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 487(1), 39–51 (2008).
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ACS Macro Lett. (1)

Y. J. Lee, C. R. Snyder, A. M. Forster, M. T. Cicerone, and W. Wu, “Imaging the molecular structure of polyethylene blends with broadband coherent Raman microscopy,” ACS Macro Lett. 1(11), 1347–1351 (2012).
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Appl. Phys. Lett. (1)

A. V. Kachynski, A. N. Kuzmin, P. N. Prasad, and I. I. Smalyukh, “Coherent anti-Stokes Raman scattering polarized microscopy of three-dimensional director structures in liquid crystals,” Appl. Phys. Lett. 91(15), 151905 (2007).
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Chemistry (1)

S. H. Parekh and K. F. Domke, “Watching orientational ordering at the nanoscale with coherent anti-stokes Raman microscopy,” Chemistry 19(36), 11822–11830 (2013).
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J. Mater. Sci. (1)

M. Tanaka and R. J. Young, “Review polarised Raman spectroscopy for the study of molecular orientation distributions in polymers,” J. Mater. Sci. 41(3), 963–991 (2006).
[Crossref]

J. Microsc. (1)

G. W. H. Wurpel, H. A. Rinia, and M. Müller, “Imaging orientational order and lipid density in multilamellar vesicles with multiplex CARS microscopy,” J. Microsc. 218(1), 37–45 (2005).
[Crossref] [PubMed]

J. Phys. Chem. B (3)

J. Duboisset, P. Berto, P. Gasecka, F.-Z. Bioud, P. Ferrand, H. Rigneault, and S. Brasselet, “Molecular orientational order probed by coherent anti-Stokes Raman scattering (CARS) and stimulated Raman scattering (SRS) microscopy: a spectral comparative study,” J. Phys. Chem. B 119(7), 3242–3249 (2015).
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R. P. Davis, A. J. Moad, G. S. Goeken, R. D. Wampler, and G. J. Simpson, “Selection rules and symmetry relations for four-wave mixing measurements of uniaxial assemblies,” J. Phys. Chem. B 112(18), 5834–5848 (2008).
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C. Zhang, J. Wang, B. Ding, and J. Jasensky, “Quantitative spectral analysis of coherent anti-Stokes Raman scattering signals: C-H stretching modes of the methyl group,” J. Phys. Chem. B 118(27), 7647–7656 (2014).
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C. Khatri and K. Mardia, “The von Mises-Fisher matrix distribution in orientation statistics,” J. R. Stat. Soc. B 39(1), 95–106 (1977).

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M. T. Cicerone, K. A. Aamer, Y. J. Lee, and E. Vartiainen, “Maximum entropy and time-domain Kramers-Kronig phase retrieval approaches are functionally equivalent for CARS microspectroscopy,” J. Raman Spectrosc. 43(5), 637–643 (2012).
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Mol. Cryst. Liq. Cryst. (Phila. Pa.) (1)

E. A. Büyüktanir, K. Zhang, A. Gericke, and J. L. West, “Raman imaging of nematic and smectic liquid crystals,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 487(1), 39–51 (2008).
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F. Munhoz, H. Rigneault, and S. Brasselet, “High order symmetry structural properties of vibrational resonances using multiple-field polarization coherent anti-Stokes Raman spectroscopy microscopy,” Phys. Rev. Lett. 105(12), 123903 (2010).
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Proc. Natl. Acad. Sci. U.S.A. (1)

E. Toprak, J. Enderlein, S. Syed, S. A. McKinney, R. G. Petschek, T. Ha, Y. E. Goldman, and P. R. Selvin, “Defocused orientation and position imaging (DOPI) of myosin V,” Proc. Natl. Acad. Sci. U.S.A. 103(17), 6495–6499 (2006).
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Figures (12)

Fig. 1
Fig. 1 Schematic diagram of rotation of a nonlinear molecular polarizability tensor, α(3), from the molecule frame (xyz) into the laboratory frame (XYZ), where it becomes a nonlinear susceptibility χ(3). The rotation is described with R(θ,ψ,ϕ), where θ, ψ, and ϕ are Euler angles. The net polarizability tensor, α(3), is represented as the sum of two linear tensor ellipsoids with resonant frequencies at ω1 and ω2. The Z-axis in the laboratory frame is set to be parallel to the light propagation direction. e ^ E (η) denotes the polarization vector of incident lights with the angle η from the X-axis.
Fig. 2
Fig. 2 (a) Schematic presentation of nonlinear susceptibilities of the primary mode, χ ω 1 (3) , in the spherical coordinate of the laboratory frame. (b) Plots of the polarization profiles of effective nonlinear susceptibilities, χ eff, ω 1 (3) , calculated for the orientations presented in (a), where ψ = 60 °. (c) Schematic presentation of nonlinear susceptibilities of the secondary mode, χ ω 2 (3) , for various ϕ. The secondary mode is perpendicular to the primary mode. (d) Plots of the polarization profiles of effective nonlinear susceptibilities, χ eff, ω 2 (3) , calculated for the orientations presented in (c), where ψ = 60 ° and θ = 30 °.
Fig. 3
Fig. 3 (a) Polarization profiles of χ eff, ω 1 (3) and χ eff, ω 2 (3) , which are simulated from Eqs. (8) and (9) with ψ = 40 °; θ = 50 °; ϕ = 30 °; and χ ω 1 (3) = χ ω 2 (3) =1 . Representation of two dimensionless observables, Δη and rχ, from the polarization profiles of χ eff, ω 1 (3) and χ eff, ω 2 (3) , (b) A contour plot of Δη as a function of θ and ϕ. (c) A contour plot of rχ as a function of θ and ϕ.
Fig. 4
Fig. 4 Determination of the 3D orientation angles (ψ, θ, and ϕ) from of polarization profiles calculated for two examples A (top row) and B (bottom row). (a) Plots of χ eff, ω 1 (3) and χ eff, ω 2 (3) where ψ is determined from η ω 1 max . (b) Contour lines of the Δη and rχ values determined from the plots of (a). (c) θ and ϕ are determined from the crossing point of the two contour lines of Δη and rχ. (d)–(f). Similarly, plots of χ eff, ω 1 (3) and χ eff, ω 2 (3) are used to determine the 3D orientation angles. By assuming uncertainties of 2 ° in relative angle measurement and 5% in intensity ratio measurement, the uncertainties are estimated as ± 2 ° for ψ, ± 1 ° for θ, and ± 2 ° for ϕ for the example A; and ± 2 ° for ψ, ± 0.5 ° for θ, and ± 2 ° for ϕ for the example B.
Fig. 5
Fig. 5 (a) Schematic presentation of Case II, where the secondary mode is uniformly distributed over ϕ about the primary mode and the secondary mode is kept to be perpendicular to the primary mode. Polarization profiles of χ eff, ω 1 (3) and χ eff, ω 2 (3) are calculated from Eqs. (8) and (15) for the examples N and L.
Fig. 6
Fig. 6 (a) Plots of polarization profiles of χ eff, ω 2 (3) calculated for four examples of Case II. Example K: θ = 10 °; example L: θ = 30 °; example M: θ = 40 °; and example N: θ = 70 °. For all examples, ψ = 60 °. (b) mχ is plotted as a function of θ from Eq. (16). The mχ values observed from (a) are used to determine the θ values. Polarization profiles of χ eff, ω 1 (3) and χ eff, ω 2 (3) are calculated (c) for the example N and (d) the example M. (e) rχ is plotted as a function of θ from Eq. (17), demonstrating an alternative method to determine the θ values for the examples M and N.
Fig. 7
Fig. 7 (a) Schematic representation of a uniaxial orientational broadening of α(3) in the molecule frame, followed by rotation into χ(3) in the laboratory frame. (b) and (c) Plots and graphical presentation of the von Mises−Fisher probability distribution function ρ(β) for two different concentration parameters, κ.
Fig. 8
Fig. 8 Broadening effect on the two examples A (Left column) and B (right column), which were presented for Case I in Fig. 4. (a) Polarization profiles of χ eff, ω 1 (3) and χ eff, ω 2 (3) calculated with broadening (Case III, κ = 10, dashed lines) are compared with those with no broadening (Case I, κ = ∞, solid lines). (b) χ eff, ω 1 (3),max and χ eff, ω 2 (3),max are plotted as a function of κ. For the random distribution (κ = 0), both values converge to 0.2. (c) The contour lines corresponding to the rχ (or rχ') and Δη values determined from the plots in (a). The rχ' value, shifted by broadening (κ = 10), and shifts, results in false orientation angles (θ' = 49.8 ° and ϕ' = 30.1 °) while the unshifted rχ value yields the true mean orientation angles (θ = 50 ° and ϕ = 30 °). (d)‑(f) Similar to the example A, broadening effect on the example B on determined orientation values: false angles for κ = 10 are θ' = 36.1 ° and ϕ' = –18.8 °, compared with the true values of θ = 30 ° and ϕ = –20 °.
Fig. 9
Fig. 9 (a) A contour plot of nχ as a function of θ and κ for the ω1 mode of Case III. (b)–(d) Contour plots of mχ for the ω2 mode of Case III for various ϕ.
Fig. 10
Fig. 10 (a) Plots of polarization profiles of χ eff, ω 1 (3) and χ eff, ω 2 (3) calculated for the example B with broadening (Case III, κ = 10). (b) Contour lines of Δη and rχ' determined from the χ eff, ω 1 (3) and χ eff, ω 2 (3) curves in (a) are plotted for θ and κ. Their crossing point yields a preliminary (false) value of ϕ'. (c) The preliminary ϕ' is used to the contour line of mχ corresponding to the mχ value determined from the χ eff, ω 2 (3) curve in (a). Another line corresponding to the nχ value determined from the χ eff, ω 1 (3) curve is plotted together with the contour line of mχ. Their crossing point yields θ and κ. (d) Such determined θ is used to determine the true value of ϕ from the contour line of Δη.
Fig. 11
Fig. 11 Broadening effect on polarization profiles of χ eff, ω 1 (3) and χ eff, ω 2 (3) for the two examples (a) M and (b) N with broadening (Case IV, κ = 10, dashed lines) are compared with those with no broadening (Case II, κ = ∞, solid lines). (c) The θ values are determined from the mχ values from the curves of (a) and (b) by using the mχθ plot, presented in Fig. 6(b). Broadening (κ = 10) makes the θ' values deviate from the true θ values. (d) Similarly, θ values are determined from the rχ (or rχ') values from the curves of (a) and (b) by using the rχθ plot, presented in Fig. 6(e).
Fig. 12
Fig. 12 Contour plots of (a) mχ and (b) rχ' as a function of θ and κ for Case IV. (c) θ and κ are determined from the contour lines of mχ and rχ' acquired from χ eff, ω 1 (3) and χ eff, ω 2 (3) of the example M in Fig. 11(a). The contour line of nχ is plotted together for confirmation. (d) Similarly, θ and κ are determined from the contour lines of mχ, rχ', and nχ acquired from χ eff, ω 1 (3) and χ eff, ω 2 (3) of the example N in Fig. 11(b).

Equations (22)

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χ (3) =NR(θ,ψ,ϕ) α (3)
R(θ,ψ,ϕ)=( cosϕ sinϕ 0 sinϕ cosϕ 0 0 0 1 )( cosθ 0 sinθ 0 1 0 sinθ 0 cosθ )( cosψ sinψ 0 sinψ cosψ 0 0 0 1 )
χ (3) = χ ω 1 (3) + χ ω 2 (3) = χ ω 1 (3) R(θ,ψ,ϕ) e ^ ω 1 + χ ω 2 (3) R(θ,ψ,ϕ) e ^ ω 2
P ω (3) = χ ω (3) E pu E S* E pr ={ N α ω (3) | e ^ E (η)R(θ,ψ,ϕ) e ^ ω | 3 E pu E S* E pr }(R(θ,ψ,ϕ) e ^ ω )
I ω CARS =C | e ^ E (ξ) P ω (3) | 2
I ω CARS =C | ( χ eff,ω (3) + χ NR (3) ) E pu E S* E pr | 2
χ eff,ω (3) = χ ω (3) × ( e ^ E (η)R(θ,ψ,ϕ) e ^ ω ) 4
χ eff, ω 1 (3) = χ ω 1 (3) ( sinθcos(ηψ) ) 4
χ eff, ω 2 (3) = χ ω 2 (3) ( cosϕcosθcos(ηψ)+sinϕsin(ηψ) ) 4
Δη η ω 2 max η ω 1 max
r χ χ eff, ω 1 (3),max / χ eff, ω 2 (3),max
Δη(θ,ϕ)= tan 1 (tanϕsecθ),
r χ (θ,ϕ)=( χ ω 1 (3) χ ω 2 (3) ) ( sinθ cosθcosϕcos(Δη)+sinϕsin(Δη) ) 4
χ eff, ω 2 (3) =( χ ω 2 (3) / 2π ) π π ( e ^ E (η)R(θ,ψ,ϕ) e ^ ω 2 ) 4 dϕ
χ eff, ω 2 (3) =( χ ω 2 (3) / 2π ) π π ( cosϕcosθcos(ηψ)+sinϕsin(ηψ) ) 4 dϕ
m χ χ eff, ω 2 (3),min / χ eff, ω 2 (3),max
r χ (θ)= 8 3 ( χ ω 1 (3) χ ω 2 (3) ) ( sinθ ) 4
χ eff, ω 1 (3) = χ ω 1 (3) ( e ^ E (η)R(θ,ψ,ϕ)R(β,γ,0) e ^ ω 1 ) 4 ρ(β)sinβdβ dγ
ρ(β)= κ 4πsinh(κ) e κcosβ ( 0β<π )
n χ χ eff, ω 1 (3),min / χ eff, ω 1 (3),max
χ eff, ω 2 (3) =( χ ω 2 (3) /2 ) ( e ^ E (η)R(θ,ψ,ϕ)R(β,γ,0) e ^ ω 2 ) 4 ρ(β)sinβdβ dγ dϕ
χ eff,ω (3) = χ ω (3) ×( e ^ E (η+ π 2 )R(θ,ψ,ϕ) e ^ ω ) | e ^ E (η)R(θ,ψ,ϕ) e ^ ω | 3

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