Abstract

We develop a full vectorial theoretical investigation of the chemical interface detection in conventional coherent anti-Stokes Raman scattering (CARS) microscopy. In Part I, we focus on the detection of axial interfaces (i.e., parallel to the optical axis) following a recent experimental demonstration of the concept [Phys. Rev. Lett. 104, 213905 (2010)]. By revisiting the Young’s double slit experiment, we show that background-free microscopy and spectroscopy is achievable through the angular analysis of the CARS far-field radiation pattern. This differential CARS in k space (Dk-CARS) technique is interesting for fast detection of interfaces between molecularly different media. It may be adapted to other coherent and resonant scattering processes.

© 2011 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. E. Ploetz, S. Laimgruber, S. Berner, W. Zinth, and P. Gilch, “Femtosecond stimulated Raman microscopy,” Appl. Phys. B 87, 389–393 (2007).
    [CrossRef]
  2. C. W. Freudiger, W. Min, B. G. Saar, S. Lu, G. R. Holtom, C. He, J. C. Tsai, J. X. Kang, and X. S. Xie, “Label-free biomedical imaging with high sensitivity by stimulated Raman scattering microscopy,” Science 322, 1857–1861 (2008).
    [CrossRef] [PubMed]
  3. Y. Ozeki, F. Dake, S. Kajiyama, K. Fukui, and K. Itoh, “Analysis and experimental assessment of the sensitivity of stimulated Raman scattering microscopy,” Opt. Express 17, 3651–3658 (2009).
    [CrossRef] [PubMed]
  4. P. Nandakumar, A. Kovalev, and A. Volkmer, “Vibrational imaging based on stimulated Raman scattering microscopy,” New J. Phys. 11, 033026 (2009).
    [CrossRef]
  5. C. W. Freudiger, M. B. J. Roeffaers, X. Zhang, B. G. Saar, W. Min, and X. S. Xie, “Optical heterodyne-detected Raman-induced Kerr effect (OHD-RIKE) microscopy,” J. Phys. Chem. B 115, 5574–5581 (2011).
    [CrossRef] [PubMed]
  6. M. D. Duncan, J. Reintjes, and T. J. Manuccia, “Scanning coherent anti-Stokes Raman scattering microscope,” Opt. Lett. 7, 350–352 (1982).
    [CrossRef] [PubMed]
  7. A. Zumbusch, G. R. Holtom, and X. S. Xie, “Three-dimensional vibrational imaging by coherent anti-Stokes Raman scattering,” Phys. Rev. Lett. 82, 4142–4145 (1999).
    [CrossRef]
  8. H. Lotem, R. T. Lynch, Jr., and N. Bloembergen, “Interference between Raman resonances in four-wave difference mixing,” Phys. Rev. A 14, 1748–1755 (1976).
    [CrossRef]
  9. M. D. Levenson and N. Bloembergen, “Dispersion of the nonlinear optical susceptibility tensor in centrosymmetric media,” Phys. Rev. A 10, 4447–4463 (1974).
  10. D. L. Marks and S. A. Boppart, “Nonlinear interferometric vibrational imaging,” Phys. Rev. Lett. 92, 123905 (2004).
    [CrossRef] [PubMed]
  11. J. P. Ogilvie, E. Beaurepaire, A. Alexandrou, and M. Joffre, “Fourier-transform coherent anti-Stokes Raman scattering microscopy,” Opt. Lett. 31, 480–482 (2006).
    [CrossRef] [PubMed]
  12. C. L. Evans, E. O. Potma, and X. S. Xie, “Coherent anti-Stokes Raman scattering spectral interferometry: determination of the real and imaginary components of nonlinear susceptibility for vibrational microscopy,” Opt. Lett. 29, 2923–2925 (2004).
    [CrossRef]
  13. E. O. Potma, C. L. Evans, and X. S. Xie, “Heterodyne coherent anti-Stokes Raman scattering (CARS) imaging,” Opt. Lett. 31, 241–243 (2006).
    [CrossRef] [PubMed]
  14. E. R. Andresen, S. R. Keiding, and E. O. Potma, “Picosecond anti-Stokes generation in a photonic-crystal fiber for interferometric CARS microscopy,” Opt. Express 14, 7246–7251 (2006).
    [CrossRef] [PubMed]
  15. M. Jurna, J. P. Korterik, C. Otto, J. L. Herek, and H. L. Offerhaus, “Vibrational phase contrast microscopy by use of coherent anti-Stokes Raman scattering,” Phys. Rev. Lett. 103, 043905 (2009).
    [CrossRef] [PubMed]
  16. E. M. Vartiainen, H. A. Rinia, M. Müller, and M. Bonn, “Direct extraction of Raman line-shapes from congested CARS spectra,” Opt. Express 14, 3622–3630 (2006).
    [CrossRef] [PubMed]
  17. Y. Liu, Y. J. Lee, and M. T. Cicerone, “Broadband CARS spectral phase retrieval using a time-domain Kramers-Krönig transform,” Opt. Lett. 34, 1363–1365 (2009).
    [CrossRef] [PubMed]
  18. D. Oron, N. Dudovich, and Y. Silberberg, “Narrow-band coherent anti-Stokes Raman signals from broad-band pulses,” Phys. Rev. Lett. 88, 063004 (2002).
    [CrossRef] [PubMed]
  19. D. Oron, N. Dudovich, and Y. Silberberg, “Single-pulse phase-contrast nonlinear Raman spectroscopy,” Phys. Rev. Lett. 89, 273001 (2002).
    [CrossRef]
  20. D. Oron, N. Dudovich, and Y. Silberberg, “Femtosecond phase-and-polarization control for background-free coherent anti-Stokes Raman spectroscopy,” Phys. Rev. Lett. 90, 213902(2003).
    [CrossRef] [PubMed]
  21. S.-H. Lim, A. G. Caster, and S. R. Leone, “Single-pulse phase-control interferometric coherent anti-Stokes Raman scattering spectroscopy,” Phys. Rev. A 72, 041803(R) (2005).
    [CrossRef]
  22. S.-H. Lim, A. G. Caster, O. Nicolet, and S. R. Leone, “Chemical imaging by single pulse interferometric coherent anti-Stokes Raman scattering microscopy,” J. Phys. Chem. B 110, 5196–5204 (2006).
    [CrossRef] [PubMed]
  23. F. Ganikhanov, C. L. Evans, B. G. Saar, and X. S. Xie, “High-sensitivity vibrational imaging with frequency modulation coherent anti-Stokes Raman scattering (FM CARS) microscopy,” Opt. Lett. 31, 1872–1874 (2006).
    [CrossRef] [PubMed]
  24. V. V. Krishnamachari and E. O. Potma, “Focus-engineered coherent anti-Stokes Raman scattering microscopy: a numerical investigation,” J. Opt. Soc. Am. A 24, 1138–1147 (2007).
    [CrossRef]
  25. V. V. Krishnamachari and E. O. Potma, “Imaging chemical interfaces perpendicular to the optical axis with focus-engineered coherent anti-Stokes Raman scattering microscopy,” Chem. Phys. 341, 81–88 (2007).
    [CrossRef]
  26. V. V. Krishnamachari and E. O. Potma, “Multi-dimensional differential imaging with FE-CARS microscopy,” Vib. Spectrosc. 50, 10–14 (2009).
    [CrossRef]
  27. D. Gachet, S. Brustlein, and H. Rigneault, “Revisiting the Young’s double slit experiment for background-free nonlinear Raman spectroscopy and microscopy,” Phys. Rev. Lett. 104, 213905 (2010).
    [CrossRef] [PubMed]
  28. D. Gachet, H. Rigneault, and S. Brustlein, “Méthode pour la détection d’un signal optique non linéaire résonant et dispositif pour la mise en oeuvre de ladite méthode (I),” Brevet CNRS, international patent application (INPI No10/00245)—Extension PCT/EP2011/050622 (22/01/2010)
  29. R. W. Hellwarth, “Third-order optical susceptibilities of solids and liquids,” Prog. Quantum Electron. 5, 1–68 (1977).
    [CrossRef]
  30. J. Lin, F. Lu, H. Wang, W. Zheng, C. J. R. Sheppard, and Z. Huang, “Improved contrast radially polarized coherent anti-Stokes Raman scattering microscopy using annular aperture detection,” Appl. Phys. Lett. 95, 133703 (2009).
    [CrossRef]
  31. J. Lin, F. Lu, H. Wang, and Z. Huang, “Annular aperture-detected coherent anti-Stokes Raman scattering microscopy for high contrast vibrational imaging,” Appl. Phys. Lett. 97, 083701 (2010).
    [CrossRef]
  32. J.-X. Cheng, A. Volkmer, and X. S. Xie, “Theoretical and experimental characterization of anti-Stokes Raman scattering microscopy,” J. Opt. Soc. Am. B 19, 1363–1375 (2002).
    [CrossRef]
  33. D. Gachet, N. Sandeau, and H. Rigneault, “Influence of the Raman depolarisation ratio on far-field radiation patterns in coherent anti-Stokes Raman scattering (CARS) microscopy,” J. Eur. Opt. Soc. Rapid Pub. 1, 06013 (2006).
    [CrossRef]
  34. D. Gachet, F. Billard, and H. Rigneault, “Background-free coherent anti-Stokes Raman spectroscopy near transverse interfaces: a vectorial study,” J. Opt. Soc. Am. B 25, 1655–1666 (2008).
    [CrossRef]
  35. M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).
  36. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanetic system,” Proc. R. Soc. London Series A 253, 358–379 (1959).
    [CrossRef]
  37. L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).
  38. J. D. Jackson, Classical Electrodynamics (Wiley, 1975).
  39. D. Gachet, F. Billard, N. Sandeau, and H. Rigneault, “Coherent anti-Stokes Raman scattering (CARS) microscopy imaging at interfaces: evidence of interference effects,” Opt. Express 15, 10408–10420 (2007).
    [CrossRef] [PubMed]
  40. G. W. H. Wurpel, J. M. Schins, and M. Müller, “Direct measurement of chain order in single phospholipid mono- and bilayers with multiplex CARS,” J. Phys. Chem. B 108, 3400–3403(2004).
    [CrossRef]
  41. C. L. Evans, E. O. Potma, M. Puoris’haag, D. Côté, C. P. Lin, and X. S. Xie, “Chemical imaging of tissue in vivo with video-rate coherent anti-Stokes Raman scattering microscopy,” Proc. Natl. Acad. Sci. USA 102, 16807–16012 (2005).
    [CrossRef] [PubMed]
  42. N. Djaker, D. Gachet, N. Sandeau, P.-F. Lenne, and H. Rigneault, “Refractive effects in coherent anti-Stokes Raman scattering (CARS) microscopy,” Appl. Opt. 45, 7005 (2006).
    [CrossRef] [PubMed]
  43. D. Gachet, F. Billard, and H. Rigneault, “Focused field symmetries for background-free coherent anti-Stokes Raman spectroscopy,” Phys. Rev. A 77, 061802(R) (2008).
    [CrossRef]
  44. J.-X. Cheng, Y. K. Jia, G. Zheng, and X. S. Xie, “Laser-scanning coherent anti-Stokes Raman scattering microscopy and applications to cell biology,” Biophys. J. 83, 502–509 (2002).
    [CrossRef] [PubMed]
  45. D. Akimov, S. Chatzipapadopoulos, T. Meyer, N. Tarcea, B. Dietzek, M. Schmitt, and J. Popp, “Different contrast information obtained from CARS and nonresonant FWM images,” J. Raman Spectrosc. 40, 941–947 (2009).
    [CrossRef]
  46. C. Liu and D. Y. Kim, “Differential imaging in coherent anti-Stokes Raman scattering microscopy with Laguerre-Gaussian excitation beams,” Opt. Express 15, 10123–10134 (2007).
    [CrossRef]
  47. C. Liu, S. Veetil, and D. Y. Kim, “Differential imaging in coherent anti-Stokes Raman scattering microscopy II: a filter-assisted Laguerre-Gaussian beam detection scheme,” Opt. Express 15, 12050–12059 (2007).
    [CrossRef]
  48. D. Gachet and H. Rigneault, “Detection of chemical interfaces in coherent anti-Stokes Raman scattering microscopy: D-CARS. II. Arbitrary interfaces,” J. Opt. Soc. Am. A 28, 2531–2539 (2011).
    [CrossRef]

2011 (2)

C. W. Freudiger, M. B. J. Roeffaers, X. Zhang, B. G. Saar, W. Min, and X. S. Xie, “Optical heterodyne-detected Raman-induced Kerr effect (OHD-RIKE) microscopy,” J. Phys. Chem. B 115, 5574–5581 (2011).
[CrossRef] [PubMed]

D. Gachet and H. Rigneault, “Detection of chemical interfaces in coherent anti-Stokes Raman scattering microscopy: D-CARS. II. Arbitrary interfaces,” J. Opt. Soc. Am. A 28, 2531–2539 (2011).
[CrossRef]

2010 (2)

D. Gachet, S. Brustlein, and H. Rigneault, “Revisiting the Young’s double slit experiment for background-free nonlinear Raman spectroscopy and microscopy,” Phys. Rev. Lett. 104, 213905 (2010).
[CrossRef] [PubMed]

J. Lin, F. Lu, H. Wang, and Z. Huang, “Annular aperture-detected coherent anti-Stokes Raman scattering microscopy for high contrast vibrational imaging,” Appl. Phys. Lett. 97, 083701 (2010).
[CrossRef]

2009 (7)

V. V. Krishnamachari and E. O. Potma, “Multi-dimensional differential imaging with FE-CARS microscopy,” Vib. Spectrosc. 50, 10–14 (2009).
[CrossRef]

J. Lin, F. Lu, H. Wang, W. Zheng, C. J. R. Sheppard, and Z. Huang, “Improved contrast radially polarized coherent anti-Stokes Raman scattering microscopy using annular aperture detection,” Appl. Phys. Lett. 95, 133703 (2009).
[CrossRef]

D. Akimov, S. Chatzipapadopoulos, T. Meyer, N. Tarcea, B. Dietzek, M. Schmitt, and J. Popp, “Different contrast information obtained from CARS and nonresonant FWM images,” J. Raman Spectrosc. 40, 941–947 (2009).
[CrossRef]

M. Jurna, J. P. Korterik, C. Otto, J. L. Herek, and H. L. Offerhaus, “Vibrational phase contrast microscopy by use of coherent anti-Stokes Raman scattering,” Phys. Rev. Lett. 103, 043905 (2009).
[CrossRef] [PubMed]

P. Nandakumar, A. Kovalev, and A. Volkmer, “Vibrational imaging based on stimulated Raman scattering microscopy,” New J. Phys. 11, 033026 (2009).
[CrossRef]

Y. Ozeki, F. Dake, S. Kajiyama, K. Fukui, and K. Itoh, “Analysis and experimental assessment of the sensitivity of stimulated Raman scattering microscopy,” Opt. Express 17, 3651–3658 (2009).
[CrossRef] [PubMed]

Y. Liu, Y. J. Lee, and M. T. Cicerone, “Broadband CARS spectral phase retrieval using a time-domain Kramers-Krönig transform,” Opt. Lett. 34, 1363–1365 (2009).
[CrossRef] [PubMed]

2008 (3)

D. Gachet, F. Billard, and H. Rigneault, “Background-free coherent anti-Stokes Raman spectroscopy near transverse interfaces: a vectorial study,” J. Opt. Soc. Am. B 25, 1655–1666 (2008).
[CrossRef]

C. W. Freudiger, W. Min, B. G. Saar, S. Lu, G. R. Holtom, C. He, J. C. Tsai, J. X. Kang, and X. S. Xie, “Label-free biomedical imaging with high sensitivity by stimulated Raman scattering microscopy,” Science 322, 1857–1861 (2008).
[CrossRef] [PubMed]

D. Gachet, F. Billard, and H. Rigneault, “Focused field symmetries for background-free coherent anti-Stokes Raman spectroscopy,” Phys. Rev. A 77, 061802(R) (2008).
[CrossRef]

2007 (6)

2006 (9)

S.-H. Lim, A. G. Caster, O. Nicolet, and S. R. Leone, “Chemical imaging by single pulse interferometric coherent anti-Stokes Raman scattering microscopy,” J. Phys. Chem. B 110, 5196–5204 (2006).
[CrossRef] [PubMed]

E. O. Potma, C. L. Evans, and X. S. Xie, “Heterodyne coherent anti-Stokes Raman scattering (CARS) imaging,” Opt. Lett. 31, 241–243 (2006).
[CrossRef] [PubMed]

J. P. Ogilvie, E. Beaurepaire, A. Alexandrou, and M. Joffre, “Fourier-transform coherent anti-Stokes Raman scattering microscopy,” Opt. Lett. 31, 480–482 (2006).
[CrossRef] [PubMed]

E. M. Vartiainen, H. A. Rinia, M. Müller, and M. Bonn, “Direct extraction of Raman line-shapes from congested CARS spectra,” Opt. Express 14, 3622–3630 (2006).
[CrossRef] [PubMed]

F. Ganikhanov, C. L. Evans, B. G. Saar, and X. S. Xie, “High-sensitivity vibrational imaging with frequency modulation coherent anti-Stokes Raman scattering (FM CARS) microscopy,” Opt. Lett. 31, 1872–1874 (2006).
[CrossRef] [PubMed]

E. R. Andresen, S. R. Keiding, and E. O. Potma, “Picosecond anti-Stokes generation in a photonic-crystal fiber for interferometric CARS microscopy,” Opt. Express 14, 7246–7251 (2006).
[CrossRef] [PubMed]

N. Djaker, D. Gachet, N. Sandeau, P.-F. Lenne, and H. Rigneault, “Refractive effects in coherent anti-Stokes Raman scattering (CARS) microscopy,” Appl. Opt. 45, 7005 (2006).
[CrossRef] [PubMed]

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

D. Gachet, N. Sandeau, and H. Rigneault, “Influence of the Raman depolarisation ratio on far-field radiation patterns in coherent anti-Stokes Raman scattering (CARS) microscopy,” J. Eur. Opt. Soc. Rapid Pub. 1, 06013 (2006).
[CrossRef]

2005 (2)

S.-H. Lim, A. G. Caster, and S. R. Leone, “Single-pulse phase-control interferometric coherent anti-Stokes Raman scattering spectroscopy,” Phys. Rev. A 72, 041803(R) (2005).
[CrossRef]

C. L. Evans, E. O. Potma, M. Puoris’haag, D. Côté, C. P. Lin, and X. S. Xie, “Chemical imaging of tissue in vivo with video-rate coherent anti-Stokes Raman scattering microscopy,” Proc. Natl. Acad. Sci. USA 102, 16807–16012 (2005).
[CrossRef] [PubMed]

2004 (3)

C. L. Evans, E. O. Potma, and X. S. Xie, “Coherent anti-Stokes Raman scattering spectral interferometry: determination of the real and imaginary components of nonlinear susceptibility for vibrational microscopy,” Opt. Lett. 29, 2923–2925 (2004).
[CrossRef]

D. L. Marks and S. A. Boppart, “Nonlinear interferometric vibrational imaging,” Phys. Rev. Lett. 92, 123905 (2004).
[CrossRef] [PubMed]

G. W. H. Wurpel, J. M. Schins, and M. Müller, “Direct measurement of chain order in single phospholipid mono- and bilayers with multiplex CARS,” J. Phys. Chem. B 108, 3400–3403(2004).
[CrossRef]

2003 (1)

D. Oron, N. Dudovich, and Y. Silberberg, “Femtosecond phase-and-polarization control for background-free coherent anti-Stokes Raman spectroscopy,” Phys. Rev. Lett. 90, 213902(2003).
[CrossRef] [PubMed]

2002 (4)

D. Oron, N. Dudovich, and Y. Silberberg, “Narrow-band coherent anti-Stokes Raman signals from broad-band pulses,” Phys. Rev. Lett. 88, 063004 (2002).
[CrossRef] [PubMed]

D. Oron, N. Dudovich, and Y. Silberberg, “Single-pulse phase-contrast nonlinear Raman spectroscopy,” Phys. Rev. Lett. 89, 273001 (2002).
[CrossRef]

J.-X. Cheng, Y. K. Jia, G. Zheng, and X. S. Xie, “Laser-scanning coherent anti-Stokes Raman scattering microscopy and applications to cell biology,” Biophys. J. 83, 502–509 (2002).
[CrossRef] [PubMed]

J.-X. Cheng, A. Volkmer, and X. S. Xie, “Theoretical and experimental characterization of anti-Stokes Raman scattering microscopy,” J. Opt. Soc. Am. B 19, 1363–1375 (2002).
[CrossRef]

1999 (1)

A. Zumbusch, G. R. Holtom, and X. S. Xie, “Three-dimensional vibrational imaging by coherent anti-Stokes Raman scattering,” Phys. Rev. Lett. 82, 4142–4145 (1999).
[CrossRef]

1982 (1)

1980 (1)

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

1977 (1)

R. W. Hellwarth, “Third-order optical susceptibilities of solids and liquids,” Prog. Quantum Electron. 5, 1–68 (1977).
[CrossRef]

1976 (1)

H. Lotem, R. T. Lynch, Jr., and N. Bloembergen, “Interference between Raman resonances in four-wave difference mixing,” Phys. Rev. A 14, 1748–1755 (1976).
[CrossRef]

1975 (1)

J. D. Jackson, Classical Electrodynamics (Wiley, 1975).

1974 (1)

M. D. Levenson and N. Bloembergen, “Dispersion of the nonlinear optical susceptibility tensor in centrosymmetric media,” Phys. Rev. A 10, 4447–4463 (1974).

1959 (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanetic system,” Proc. R. Soc. London Series A 253, 358–379 (1959).
[CrossRef]

Akimov, D.

D. Akimov, S. Chatzipapadopoulos, T. Meyer, N. Tarcea, B. Dietzek, M. Schmitt, and J. Popp, “Different contrast information obtained from CARS and nonresonant FWM images,” J. Raman Spectrosc. 40, 941–947 (2009).
[CrossRef]

Alexandrou, A.

Andresen, E. R.

Beaurepaire, E.

Berner, S.

E. Ploetz, S. Laimgruber, S. Berner, W. Zinth, and P. Gilch, “Femtosecond stimulated Raman microscopy,” Appl. Phys. B 87, 389–393 (2007).
[CrossRef]

Billard, F.

Bloembergen, N.

H. Lotem, R. T. Lynch, Jr., and N. Bloembergen, “Interference between Raman resonances in four-wave difference mixing,” Phys. Rev. A 14, 1748–1755 (1976).
[CrossRef]

M. D. Levenson and N. Bloembergen, “Dispersion of the nonlinear optical susceptibility tensor in centrosymmetric media,” Phys. Rev. A 10, 4447–4463 (1974).

Bonn, M.

Boppart, S. A.

D. L. Marks and S. A. Boppart, “Nonlinear interferometric vibrational imaging,” Phys. Rev. Lett. 92, 123905 (2004).
[CrossRef] [PubMed]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

Brustlein, S.

D. Gachet, H. Rigneault, and S. Brustlein, “Méthode pour la détection d’un signal optique non linéaire résonant et dispositif pour la mise en oeuvre de ladite méthode (I),” Brevet CNRS, international patent application (INPI No10/00245)—Extension PCT/EP2011/050622 (22/01/2010)

D. Gachet, S. Brustlein, and H. Rigneault, “Revisiting the Young’s double slit experiment for background-free nonlinear Raman spectroscopy and microscopy,” Phys. Rev. Lett. 104, 213905 (2010).
[CrossRef] [PubMed]

Caster, A. G.

S.-H. Lim, A. G. Caster, O. Nicolet, and S. R. Leone, “Chemical imaging by single pulse interferometric coherent anti-Stokes Raman scattering microscopy,” J. Phys. Chem. B 110, 5196–5204 (2006).
[CrossRef] [PubMed]

S.-H. Lim, A. G. Caster, and S. R. Leone, “Single-pulse phase-control interferometric coherent anti-Stokes Raman scattering spectroscopy,” Phys. Rev. A 72, 041803(R) (2005).
[CrossRef]

Chatzipapadopoulos, S.

D. Akimov, S. Chatzipapadopoulos, T. Meyer, N. Tarcea, B. Dietzek, M. Schmitt, and J. Popp, “Different contrast information obtained from CARS and nonresonant FWM images,” J. Raman Spectrosc. 40, 941–947 (2009).
[CrossRef]

Cheng, J.-X.

J.-X. Cheng, Y. K. Jia, G. Zheng, and X. S. Xie, “Laser-scanning coherent anti-Stokes Raman scattering microscopy and applications to cell biology,” Biophys. J. 83, 502–509 (2002).
[CrossRef] [PubMed]

J.-X. Cheng, A. Volkmer, and X. S. Xie, “Theoretical and experimental characterization of anti-Stokes Raman scattering microscopy,” J. Opt. Soc. Am. B 19, 1363–1375 (2002).
[CrossRef]

Cicerone, M. T.

Côté, D.

C. L. Evans, E. O. Potma, M. Puoris’haag, D. Côté, C. P. Lin, and X. S. Xie, “Chemical imaging of tissue in vivo with video-rate coherent anti-Stokes Raman scattering microscopy,” Proc. Natl. Acad. Sci. USA 102, 16807–16012 (2005).
[CrossRef] [PubMed]

Dake, F.

Dietzek, B.

D. Akimov, S. Chatzipapadopoulos, T. Meyer, N. Tarcea, B. Dietzek, M. Schmitt, and J. Popp, “Different contrast information obtained from CARS and nonresonant FWM images,” J. Raman Spectrosc. 40, 941–947 (2009).
[CrossRef]

Djaker, N.

Dudovich, N.

D. Oron, N. Dudovich, and Y. Silberberg, “Femtosecond phase-and-polarization control for background-free coherent anti-Stokes Raman spectroscopy,” Phys. Rev. Lett. 90, 213902(2003).
[CrossRef] [PubMed]

D. Oron, N. Dudovich, and Y. Silberberg, “Narrow-band coherent anti-Stokes Raman signals from broad-band pulses,” Phys. Rev. Lett. 88, 063004 (2002).
[CrossRef] [PubMed]

D. Oron, N. Dudovich, and Y. Silberberg, “Single-pulse phase-contrast nonlinear Raman spectroscopy,” Phys. Rev. Lett. 89, 273001 (2002).
[CrossRef]

Duncan, M. D.

Evans, C. L.

Freudiger, C. W.

C. W. Freudiger, M. B. J. Roeffaers, X. Zhang, B. G. Saar, W. Min, and X. S. Xie, “Optical heterodyne-detected Raman-induced Kerr effect (OHD-RIKE) microscopy,” J. Phys. Chem. B 115, 5574–5581 (2011).
[CrossRef] [PubMed]

C. W. Freudiger, W. Min, B. G. Saar, S. Lu, G. R. Holtom, C. He, J. C. Tsai, J. X. Kang, and X. S. Xie, “Label-free biomedical imaging with high sensitivity by stimulated Raman scattering microscopy,” Science 322, 1857–1861 (2008).
[CrossRef] [PubMed]

Fukui, K.

Gachet, D.

D. Gachet, H. Rigneault, and S. Brustlein, “Méthode pour la détection d’un signal optique non linéaire résonant et dispositif pour la mise en oeuvre de ladite méthode (I),” Brevet CNRS, international patent application (INPI No10/00245)—Extension PCT/EP2011/050622 (22/01/2010)

D. Gachet and H. Rigneault, “Detection of chemical interfaces in coherent anti-Stokes Raman scattering microscopy: D-CARS. II. Arbitrary interfaces,” J. Opt. Soc. Am. A 28, 2531–2539 (2011).
[CrossRef]

D. Gachet, S. Brustlein, and H. Rigneault, “Revisiting the Young’s double slit experiment for background-free nonlinear Raman spectroscopy and microscopy,” Phys. Rev. Lett. 104, 213905 (2010).
[CrossRef] [PubMed]

D. Gachet, F. Billard, and H. Rigneault, “Background-free coherent anti-Stokes Raman spectroscopy near transverse interfaces: a vectorial study,” J. Opt. Soc. Am. B 25, 1655–1666 (2008).
[CrossRef]

D. Gachet, F. Billard, and H. Rigneault, “Focused field symmetries for background-free coherent anti-Stokes Raman spectroscopy,” Phys. Rev. A 77, 061802(R) (2008).
[CrossRef]

D. Gachet, F. Billard, N. Sandeau, and H. Rigneault, “Coherent anti-Stokes Raman scattering (CARS) microscopy imaging at interfaces: evidence of interference effects,” Opt. Express 15, 10408–10420 (2007).
[CrossRef] [PubMed]

D. Gachet, N. Sandeau, and H. Rigneault, “Influence of the Raman depolarisation ratio on far-field radiation patterns in coherent anti-Stokes Raman scattering (CARS) microscopy,” J. Eur. Opt. Soc. Rapid Pub. 1, 06013 (2006).
[CrossRef]

N. Djaker, D. Gachet, N. Sandeau, P.-F. Lenne, and H. Rigneault, “Refractive effects in coherent anti-Stokes Raman scattering (CARS) microscopy,” Appl. Opt. 45, 7005 (2006).
[CrossRef] [PubMed]

Ganikhanov, F.

Gilch, P.

E. Ploetz, S. Laimgruber, S. Berner, W. Zinth, and P. Gilch, “Femtosecond stimulated Raman microscopy,” Appl. Phys. B 87, 389–393 (2007).
[CrossRef]

He, C.

C. W. Freudiger, W. Min, B. G. Saar, S. Lu, G. R. Holtom, C. He, J. C. Tsai, J. X. Kang, and X. S. Xie, “Label-free biomedical imaging with high sensitivity by stimulated Raman scattering microscopy,” Science 322, 1857–1861 (2008).
[CrossRef] [PubMed]

Hecht, B.

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

Hellwarth, R. W.

R. W. Hellwarth, “Third-order optical susceptibilities of solids and liquids,” Prog. Quantum Electron. 5, 1–68 (1977).
[CrossRef]

Herek, J. L.

M. Jurna, J. P. Korterik, C. Otto, J. L. Herek, and H. L. Offerhaus, “Vibrational phase contrast microscopy by use of coherent anti-Stokes Raman scattering,” Phys. Rev. Lett. 103, 043905 (2009).
[CrossRef] [PubMed]

Holtom, G. R.

C. W. Freudiger, W. Min, B. G. Saar, S. Lu, G. R. Holtom, C. He, J. C. Tsai, J. X. Kang, and X. S. Xie, “Label-free biomedical imaging with high sensitivity by stimulated Raman scattering microscopy,” Science 322, 1857–1861 (2008).
[CrossRef] [PubMed]

A. Zumbusch, G. R. Holtom, and X. S. Xie, “Three-dimensional vibrational imaging by coherent anti-Stokes Raman scattering,” Phys. Rev. Lett. 82, 4142–4145 (1999).
[CrossRef]

Huang, Z.

J. Lin, F. Lu, H. Wang, and Z. Huang, “Annular aperture-detected coherent anti-Stokes Raman scattering microscopy for high contrast vibrational imaging,” Appl. Phys. Lett. 97, 083701 (2010).
[CrossRef]

J. Lin, F. Lu, H. Wang, W. Zheng, C. J. R. Sheppard, and Z. Huang, “Improved contrast radially polarized coherent anti-Stokes Raman scattering microscopy using annular aperture detection,” Appl. Phys. Lett. 95, 133703 (2009).
[CrossRef]

Itoh, K.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, 1975).

Jia, Y. K.

J.-X. Cheng, Y. K. Jia, G. Zheng, and X. S. Xie, “Laser-scanning coherent anti-Stokes Raman scattering microscopy and applications to cell biology,” Biophys. J. 83, 502–509 (2002).
[CrossRef] [PubMed]

Joffre, M.

Jurna, M.

M. Jurna, J. P. Korterik, C. Otto, J. L. Herek, and H. L. Offerhaus, “Vibrational phase contrast microscopy by use of coherent anti-Stokes Raman scattering,” Phys. Rev. Lett. 103, 043905 (2009).
[CrossRef] [PubMed]

Kajiyama, S.

Kang, J. X.

C. W. Freudiger, W. Min, B. G. Saar, S. Lu, G. R. Holtom, C. He, J. C. Tsai, J. X. Kang, and X. S. Xie, “Label-free biomedical imaging with high sensitivity by stimulated Raman scattering microscopy,” Science 322, 1857–1861 (2008).
[CrossRef] [PubMed]

Keiding, S. R.

Kim, D. Y.

Korterik, J. P.

M. Jurna, J. P. Korterik, C. Otto, J. L. Herek, and H. L. Offerhaus, “Vibrational phase contrast microscopy by use of coherent anti-Stokes Raman scattering,” Phys. Rev. Lett. 103, 043905 (2009).
[CrossRef] [PubMed]

Kovalev, A.

P. Nandakumar, A. Kovalev, and A. Volkmer, “Vibrational imaging based on stimulated Raman scattering microscopy,” New J. Phys. 11, 033026 (2009).
[CrossRef]

Krishnamachari, V. V.

V. V. Krishnamachari and E. O. Potma, “Multi-dimensional differential imaging with FE-CARS microscopy,” Vib. Spectrosc. 50, 10–14 (2009).
[CrossRef]

V. V. Krishnamachari and E. O. Potma, “Focus-engineered coherent anti-Stokes Raman scattering microscopy: a numerical investigation,” J. Opt. Soc. Am. A 24, 1138–1147 (2007).
[CrossRef]

V. V. Krishnamachari and E. O. Potma, “Imaging chemical interfaces perpendicular to the optical axis with focus-engineered coherent anti-Stokes Raman scattering microscopy,” Chem. Phys. 341, 81–88 (2007).
[CrossRef]

Laimgruber, S.

E. Ploetz, S. Laimgruber, S. Berner, W. Zinth, and P. Gilch, “Femtosecond stimulated Raman microscopy,” Appl. Phys. B 87, 389–393 (2007).
[CrossRef]

Lee, Y. J.

Lenne, P.-F.

Leone, S. R.

S.-H. Lim, A. G. Caster, O. Nicolet, and S. R. Leone, “Chemical imaging by single pulse interferometric coherent anti-Stokes Raman scattering microscopy,” J. Phys. Chem. B 110, 5196–5204 (2006).
[CrossRef] [PubMed]

S.-H. Lim, A. G. Caster, and S. R. Leone, “Single-pulse phase-control interferometric coherent anti-Stokes Raman scattering spectroscopy,” Phys. Rev. A 72, 041803(R) (2005).
[CrossRef]

Levenson, M. D.

M. D. Levenson and N. Bloembergen, “Dispersion of the nonlinear optical susceptibility tensor in centrosymmetric media,” Phys. Rev. A 10, 4447–4463 (1974).

Lim, S.-H.

S.-H. Lim, A. G. Caster, O. Nicolet, and S. R. Leone, “Chemical imaging by single pulse interferometric coherent anti-Stokes Raman scattering microscopy,” J. Phys. Chem. B 110, 5196–5204 (2006).
[CrossRef] [PubMed]

S.-H. Lim, A. G. Caster, and S. R. Leone, “Single-pulse phase-control interferometric coherent anti-Stokes Raman scattering spectroscopy,” Phys. Rev. A 72, 041803(R) (2005).
[CrossRef]

Lin, C. P.

C. L. Evans, E. O. Potma, M. Puoris’haag, D. Côté, C. P. Lin, and X. S. Xie, “Chemical imaging of tissue in vivo with video-rate coherent anti-Stokes Raman scattering microscopy,” Proc. Natl. Acad. Sci. USA 102, 16807–16012 (2005).
[CrossRef] [PubMed]

Lin, J.

J. Lin, F. Lu, H. Wang, and Z. Huang, “Annular aperture-detected coherent anti-Stokes Raman scattering microscopy for high contrast vibrational imaging,” Appl. Phys. Lett. 97, 083701 (2010).
[CrossRef]

J. Lin, F. Lu, H. Wang, W. Zheng, C. J. R. Sheppard, and Z. Huang, “Improved contrast radially polarized coherent anti-Stokes Raman scattering microscopy using annular aperture detection,” Appl. Phys. Lett. 95, 133703 (2009).
[CrossRef]

Liu, C.

Liu, Y.

Lotem, H.

H. Lotem, R. T. Lynch, Jr., and N. Bloembergen, “Interference between Raman resonances in four-wave difference mixing,” Phys. Rev. A 14, 1748–1755 (1976).
[CrossRef]

Lu, F.

J. Lin, F. Lu, H. Wang, and Z. Huang, “Annular aperture-detected coherent anti-Stokes Raman scattering microscopy for high contrast vibrational imaging,” Appl. Phys. Lett. 97, 083701 (2010).
[CrossRef]

J. Lin, F. Lu, H. Wang, W. Zheng, C. J. R. Sheppard, and Z. Huang, “Improved contrast radially polarized coherent anti-Stokes Raman scattering microscopy using annular aperture detection,” Appl. Phys. Lett. 95, 133703 (2009).
[CrossRef]

Lu, S.

C. W. Freudiger, W. Min, B. G. Saar, S. Lu, G. R. Holtom, C. He, J. C. Tsai, J. X. Kang, and X. S. Xie, “Label-free biomedical imaging with high sensitivity by stimulated Raman scattering microscopy,” Science 322, 1857–1861 (2008).
[CrossRef] [PubMed]

Lynch, R. T.

H. Lotem, R. T. Lynch, Jr., and N. Bloembergen, “Interference between Raman resonances in four-wave difference mixing,” Phys. Rev. A 14, 1748–1755 (1976).
[CrossRef]

Manuccia, T. J.

Marks, D. L.

D. L. Marks and S. A. Boppart, “Nonlinear interferometric vibrational imaging,” Phys. Rev. Lett. 92, 123905 (2004).
[CrossRef] [PubMed]

Meyer, T.

D. Akimov, S. Chatzipapadopoulos, T. Meyer, N. Tarcea, B. Dietzek, M. Schmitt, and J. Popp, “Different contrast information obtained from CARS and nonresonant FWM images,” J. Raman Spectrosc. 40, 941–947 (2009).
[CrossRef]

Min, W.

C. W. Freudiger, M. B. J. Roeffaers, X. Zhang, B. G. Saar, W. Min, and X. S. Xie, “Optical heterodyne-detected Raman-induced Kerr effect (OHD-RIKE) microscopy,” J. Phys. Chem. B 115, 5574–5581 (2011).
[CrossRef] [PubMed]

C. W. Freudiger, W. Min, B. G. Saar, S. Lu, G. R. Holtom, C. He, J. C. Tsai, J. X. Kang, and X. S. Xie, “Label-free biomedical imaging with high sensitivity by stimulated Raman scattering microscopy,” Science 322, 1857–1861 (2008).
[CrossRef] [PubMed]

Müller, M.

E. M. Vartiainen, H. A. Rinia, M. Müller, and M. Bonn, “Direct extraction of Raman line-shapes from congested CARS spectra,” Opt. Express 14, 3622–3630 (2006).
[CrossRef] [PubMed]

G. W. H. Wurpel, J. M. Schins, and M. Müller, “Direct measurement of chain order in single phospholipid mono- and bilayers with multiplex CARS,” J. Phys. Chem. B 108, 3400–3403(2004).
[CrossRef]

Nandakumar, P.

P. Nandakumar, A. Kovalev, and A. Volkmer, “Vibrational imaging based on stimulated Raman scattering microscopy,” New J. Phys. 11, 033026 (2009).
[CrossRef]

Nicolet, O.

S.-H. Lim, A. G. Caster, O. Nicolet, and S. R. Leone, “Chemical imaging by single pulse interferometric coherent anti-Stokes Raman scattering microscopy,” J. Phys. Chem. B 110, 5196–5204 (2006).
[CrossRef] [PubMed]

Novotny, L.

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

Offerhaus, H. L.

M. Jurna, J. P. Korterik, C. Otto, J. L. Herek, and H. L. Offerhaus, “Vibrational phase contrast microscopy by use of coherent anti-Stokes Raman scattering,” Phys. Rev. Lett. 103, 043905 (2009).
[CrossRef] [PubMed]

Ogilvie, J. P.

Oron, D.

D. Oron, N. Dudovich, and Y. Silberberg, “Femtosecond phase-and-polarization control for background-free coherent anti-Stokes Raman spectroscopy,” Phys. Rev. Lett. 90, 213902(2003).
[CrossRef] [PubMed]

D. Oron, N. Dudovich, and Y. Silberberg, “Narrow-band coherent anti-Stokes Raman signals from broad-band pulses,” Phys. Rev. Lett. 88, 063004 (2002).
[CrossRef] [PubMed]

D. Oron, N. Dudovich, and Y. Silberberg, “Single-pulse phase-contrast nonlinear Raman spectroscopy,” Phys. Rev. Lett. 89, 273001 (2002).
[CrossRef]

Otto, C.

M. Jurna, J. P. Korterik, C. Otto, J. L. Herek, and H. L. Offerhaus, “Vibrational phase contrast microscopy by use of coherent anti-Stokes Raman scattering,” Phys. Rev. Lett. 103, 043905 (2009).
[CrossRef] [PubMed]

Ozeki, Y.

Ploetz, E.

E. Ploetz, S. Laimgruber, S. Berner, W. Zinth, and P. Gilch, “Femtosecond stimulated Raman microscopy,” Appl. Phys. B 87, 389–393 (2007).
[CrossRef]

Popp, J.

D. Akimov, S. Chatzipapadopoulos, T. Meyer, N. Tarcea, B. Dietzek, M. Schmitt, and J. Popp, “Different contrast information obtained from CARS and nonresonant FWM images,” J. Raman Spectrosc. 40, 941–947 (2009).
[CrossRef]

Potma, E. O.

V. V. Krishnamachari and E. O. Potma, “Multi-dimensional differential imaging with FE-CARS microscopy,” Vib. Spectrosc. 50, 10–14 (2009).
[CrossRef]

V. V. Krishnamachari and E. O. Potma, “Focus-engineered coherent anti-Stokes Raman scattering microscopy: a numerical investigation,” J. Opt. Soc. Am. A 24, 1138–1147 (2007).
[CrossRef]

V. V. Krishnamachari and E. O. Potma, “Imaging chemical interfaces perpendicular to the optical axis with focus-engineered coherent anti-Stokes Raman scattering microscopy,” Chem. Phys. 341, 81–88 (2007).
[CrossRef]

E. R. Andresen, S. R. Keiding, and E. O. Potma, “Picosecond anti-Stokes generation in a photonic-crystal fiber for interferometric CARS microscopy,” Opt. Express 14, 7246–7251 (2006).
[CrossRef] [PubMed]

E. O. Potma, C. L. Evans, and X. S. Xie, “Heterodyne coherent anti-Stokes Raman scattering (CARS) imaging,” Opt. Lett. 31, 241–243 (2006).
[CrossRef] [PubMed]

C. L. Evans, E. O. Potma, M. Puoris’haag, D. Côté, C. P. Lin, and X. S. Xie, “Chemical imaging of tissue in vivo with video-rate coherent anti-Stokes Raman scattering microscopy,” Proc. Natl. Acad. Sci. USA 102, 16807–16012 (2005).
[CrossRef] [PubMed]

C. L. Evans, E. O. Potma, and X. S. Xie, “Coherent anti-Stokes Raman scattering spectral interferometry: determination of the real and imaginary components of nonlinear susceptibility for vibrational microscopy,” Opt. Lett. 29, 2923–2925 (2004).
[CrossRef]

Puoris’haag, M.

C. L. Evans, E. O. Potma, M. Puoris’haag, D. Côté, C. P. Lin, and X. S. Xie, “Chemical imaging of tissue in vivo with video-rate coherent anti-Stokes Raman scattering microscopy,” Proc. Natl. Acad. Sci. USA 102, 16807–16012 (2005).
[CrossRef] [PubMed]

Reintjes, J.

Richards, B.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanetic system,” Proc. R. Soc. London Series A 253, 358–379 (1959).
[CrossRef]

Rigneault, H.

D. Gachet, H. Rigneault, and S. Brustlein, “Méthode pour la détection d’un signal optique non linéaire résonant et dispositif pour la mise en oeuvre de ladite méthode (I),” Brevet CNRS, international patent application (INPI No10/00245)—Extension PCT/EP2011/050622 (22/01/2010)

D. Gachet and H. Rigneault, “Detection of chemical interfaces in coherent anti-Stokes Raman scattering microscopy: D-CARS. II. Arbitrary interfaces,” J. Opt. Soc. Am. A 28, 2531–2539 (2011).
[CrossRef]

D. Gachet, S. Brustlein, and H. Rigneault, “Revisiting the Young’s double slit experiment for background-free nonlinear Raman spectroscopy and microscopy,” Phys. Rev. Lett. 104, 213905 (2010).
[CrossRef] [PubMed]

D. Gachet, F. Billard, and H. Rigneault, “Background-free coherent anti-Stokes Raman spectroscopy near transverse interfaces: a vectorial study,” J. Opt. Soc. Am. B 25, 1655–1666 (2008).
[CrossRef]

D. Gachet, F. Billard, and H. Rigneault, “Focused field symmetries for background-free coherent anti-Stokes Raman spectroscopy,” Phys. Rev. A 77, 061802(R) (2008).
[CrossRef]

D. Gachet, F. Billard, N. Sandeau, and H. Rigneault, “Coherent anti-Stokes Raman scattering (CARS) microscopy imaging at interfaces: evidence of interference effects,” Opt. Express 15, 10408–10420 (2007).
[CrossRef] [PubMed]

D. Gachet, N. Sandeau, and H. Rigneault, “Influence of the Raman depolarisation ratio on far-field radiation patterns in coherent anti-Stokes Raman scattering (CARS) microscopy,” J. Eur. Opt. Soc. Rapid Pub. 1, 06013 (2006).
[CrossRef]

N. Djaker, D. Gachet, N. Sandeau, P.-F. Lenne, and H. Rigneault, “Refractive effects in coherent anti-Stokes Raman scattering (CARS) microscopy,” Appl. Opt. 45, 7005 (2006).
[CrossRef] [PubMed]

Rinia, H. A.

Roeffaers, M. B. J.

C. W. Freudiger, M. B. J. Roeffaers, X. Zhang, B. G. Saar, W. Min, and X. S. Xie, “Optical heterodyne-detected Raman-induced Kerr effect (OHD-RIKE) microscopy,” J. Phys. Chem. B 115, 5574–5581 (2011).
[CrossRef] [PubMed]

Saar, B. G.

C. W. Freudiger, M. B. J. Roeffaers, X. Zhang, B. G. Saar, W. Min, and X. S. Xie, “Optical heterodyne-detected Raman-induced Kerr effect (OHD-RIKE) microscopy,” J. Phys. Chem. B 115, 5574–5581 (2011).
[CrossRef] [PubMed]

C. W. Freudiger, W. Min, B. G. Saar, S. Lu, G. R. Holtom, C. He, J. C. Tsai, J. X. Kang, and X. S. Xie, “Label-free biomedical imaging with high sensitivity by stimulated Raman scattering microscopy,” Science 322, 1857–1861 (2008).
[CrossRef] [PubMed]

F. Ganikhanov, C. L. Evans, B. G. Saar, and X. S. Xie, “High-sensitivity vibrational imaging with frequency modulation coherent anti-Stokes Raman scattering (FM CARS) microscopy,” Opt. Lett. 31, 1872–1874 (2006).
[CrossRef] [PubMed]

Sandeau, N.

Schins, J. M.

G. W. H. Wurpel, J. M. Schins, and M. Müller, “Direct measurement of chain order in single phospholipid mono- and bilayers with multiplex CARS,” J. Phys. Chem. B 108, 3400–3403(2004).
[CrossRef]

Schmitt, M.

D. Akimov, S. Chatzipapadopoulos, T. Meyer, N. Tarcea, B. Dietzek, M. Schmitt, and J. Popp, “Different contrast information obtained from CARS and nonresonant FWM images,” J. Raman Spectrosc. 40, 941–947 (2009).
[CrossRef]

Sheppard, C. J. R.

J. Lin, F. Lu, H. Wang, W. Zheng, C. J. R. Sheppard, and Z. Huang, “Improved contrast radially polarized coherent anti-Stokes Raman scattering microscopy using annular aperture detection,” Appl. Phys. Lett. 95, 133703 (2009).
[CrossRef]

Silberberg, Y.

D. Oron, N. Dudovich, and Y. Silberberg, “Femtosecond phase-and-polarization control for background-free coherent anti-Stokes Raman spectroscopy,” Phys. Rev. Lett. 90, 213902(2003).
[CrossRef] [PubMed]

D. Oron, N. Dudovich, and Y. Silberberg, “Narrow-band coherent anti-Stokes Raman signals from broad-band pulses,” Phys. Rev. Lett. 88, 063004 (2002).
[CrossRef] [PubMed]

D. Oron, N. Dudovich, and Y. Silberberg, “Single-pulse phase-contrast nonlinear Raman spectroscopy,” Phys. Rev. Lett. 89, 273001 (2002).
[CrossRef]

Tarcea, N.

D. Akimov, S. Chatzipapadopoulos, T. Meyer, N. Tarcea, B. Dietzek, M. Schmitt, and J. Popp, “Different contrast information obtained from CARS and nonresonant FWM images,” J. Raman Spectrosc. 40, 941–947 (2009).
[CrossRef]

Tsai, J. C.

C. W. Freudiger, W. Min, B. G. Saar, S. Lu, G. R. Holtom, C. He, J. C. Tsai, J. X. Kang, and X. S. Xie, “Label-free biomedical imaging with high sensitivity by stimulated Raman scattering microscopy,” Science 322, 1857–1861 (2008).
[CrossRef] [PubMed]

Vartiainen, E. M.

Veetil, S.

Volkmer, A.

P. Nandakumar, A. Kovalev, and A. Volkmer, “Vibrational imaging based on stimulated Raman scattering microscopy,” New J. Phys. 11, 033026 (2009).
[CrossRef]

J.-X. Cheng, A. Volkmer, and X. S. Xie, “Theoretical and experimental characterization of anti-Stokes Raman scattering microscopy,” J. Opt. Soc. Am. B 19, 1363–1375 (2002).
[CrossRef]

Wang, H.

J. Lin, F. Lu, H. Wang, and Z. Huang, “Annular aperture-detected coherent anti-Stokes Raman scattering microscopy for high contrast vibrational imaging,” Appl. Phys. Lett. 97, 083701 (2010).
[CrossRef]

J. Lin, F. Lu, H. Wang, W. Zheng, C. J. R. Sheppard, and Z. Huang, “Improved contrast radially polarized coherent anti-Stokes Raman scattering microscopy using annular aperture detection,” Appl. Phys. Lett. 95, 133703 (2009).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanetic system,” Proc. R. Soc. London Series A 253, 358–379 (1959).
[CrossRef]

Wurpel, G. W. H.

G. W. H. Wurpel, J. M. Schins, and M. Müller, “Direct measurement of chain order in single phospholipid mono- and bilayers with multiplex CARS,” J. Phys. Chem. B 108, 3400–3403(2004).
[CrossRef]

Xie, X. S.

C. W. Freudiger, M. B. J. Roeffaers, X. Zhang, B. G. Saar, W. Min, and X. S. Xie, “Optical heterodyne-detected Raman-induced Kerr effect (OHD-RIKE) microscopy,” J. Phys. Chem. B 115, 5574–5581 (2011).
[CrossRef] [PubMed]

C. W. Freudiger, W. Min, B. G. Saar, S. Lu, G. R. Holtom, C. He, J. C. Tsai, J. X. Kang, and X. S. Xie, “Label-free biomedical imaging with high sensitivity by stimulated Raman scattering microscopy,” Science 322, 1857–1861 (2008).
[CrossRef] [PubMed]

E. O. Potma, C. L. Evans, and X. S. Xie, “Heterodyne coherent anti-Stokes Raman scattering (CARS) imaging,” Opt. Lett. 31, 241–243 (2006).
[CrossRef] [PubMed]

F. Ganikhanov, C. L. Evans, B. G. Saar, and X. S. Xie, “High-sensitivity vibrational imaging with frequency modulation coherent anti-Stokes Raman scattering (FM CARS) microscopy,” Opt. Lett. 31, 1872–1874 (2006).
[CrossRef] [PubMed]

C. L. Evans, E. O. Potma, M. Puoris’haag, D. Côté, C. P. Lin, and X. S. Xie, “Chemical imaging of tissue in vivo with video-rate coherent anti-Stokes Raman scattering microscopy,” Proc. Natl. Acad. Sci. USA 102, 16807–16012 (2005).
[CrossRef] [PubMed]

C. L. Evans, E. O. Potma, and X. S. Xie, “Coherent anti-Stokes Raman scattering spectral interferometry: determination of the real and imaginary components of nonlinear susceptibility for vibrational microscopy,” Opt. Lett. 29, 2923–2925 (2004).
[CrossRef]

J.-X. Cheng, Y. K. Jia, G. Zheng, and X. S. Xie, “Laser-scanning coherent anti-Stokes Raman scattering microscopy and applications to cell biology,” Biophys. J. 83, 502–509 (2002).
[CrossRef] [PubMed]

J.-X. Cheng, A. Volkmer, and X. S. Xie, “Theoretical and experimental characterization of anti-Stokes Raman scattering microscopy,” J. Opt. Soc. Am. B 19, 1363–1375 (2002).
[CrossRef]

A. Zumbusch, G. R. Holtom, and X. S. Xie, “Three-dimensional vibrational imaging by coherent anti-Stokes Raman scattering,” Phys. Rev. Lett. 82, 4142–4145 (1999).
[CrossRef]

Zhang, X.

C. W. Freudiger, M. B. J. Roeffaers, X. Zhang, B. G. Saar, W. Min, and X. S. Xie, “Optical heterodyne-detected Raman-induced Kerr effect (OHD-RIKE) microscopy,” J. Phys. Chem. B 115, 5574–5581 (2011).
[CrossRef] [PubMed]

Zheng, G.

J.-X. Cheng, Y. K. Jia, G. Zheng, and X. S. Xie, “Laser-scanning coherent anti-Stokes Raman scattering microscopy and applications to cell biology,” Biophys. J. 83, 502–509 (2002).
[CrossRef] [PubMed]

Zheng, W.

J. Lin, F. Lu, H. Wang, W. Zheng, C. J. R. Sheppard, and Z. Huang, “Improved contrast radially polarized coherent anti-Stokes Raman scattering microscopy using annular aperture detection,” Appl. Phys. Lett. 95, 133703 (2009).
[CrossRef]

Zinth, W.

E. Ploetz, S. Laimgruber, S. Berner, W. Zinth, and P. Gilch, “Femtosecond stimulated Raman microscopy,” Appl. Phys. B 87, 389–393 (2007).
[CrossRef]

Zumbusch, A.

A. Zumbusch, G. R. Holtom, and X. S. Xie, “Three-dimensional vibrational imaging by coherent anti-Stokes Raman scattering,” Phys. Rev. Lett. 82, 4142–4145 (1999).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. B (1)

E. Ploetz, S. Laimgruber, S. Berner, W. Zinth, and P. Gilch, “Femtosecond stimulated Raman microscopy,” Appl. Phys. B 87, 389–393 (2007).
[CrossRef]

Appl. Phys. Lett. (2)

J. Lin, F. Lu, H. Wang, W. Zheng, C. J. R. Sheppard, and Z. Huang, “Improved contrast radially polarized coherent anti-Stokes Raman scattering microscopy using annular aperture detection,” Appl. Phys. Lett. 95, 133703 (2009).
[CrossRef]

J. Lin, F. Lu, H. Wang, and Z. Huang, “Annular aperture-detected coherent anti-Stokes Raman scattering microscopy for high contrast vibrational imaging,” Appl. Phys. Lett. 97, 083701 (2010).
[CrossRef]

Biophys. J. (1)

J.-X. Cheng, Y. K. Jia, G. Zheng, and X. S. Xie, “Laser-scanning coherent anti-Stokes Raman scattering microscopy and applications to cell biology,” Biophys. J. 83, 502–509 (2002).
[CrossRef] [PubMed]

Chem. Phys. (1)

V. V. Krishnamachari and E. O. Potma, “Imaging chemical interfaces perpendicular to the optical axis with focus-engineered coherent anti-Stokes Raman scattering microscopy,” Chem. Phys. 341, 81–88 (2007).
[CrossRef]

J. Eur. Opt. Soc. Rapid Pub. (1)

D. Gachet, N. Sandeau, and H. Rigneault, “Influence of the Raman depolarisation ratio on far-field radiation patterns in coherent anti-Stokes Raman scattering (CARS) microscopy,” J. Eur. Opt. Soc. Rapid Pub. 1, 06013 (2006).
[CrossRef]

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (2)

J. Phys. Chem. B (3)

S.-H. Lim, A. G. Caster, O. Nicolet, and S. R. Leone, “Chemical imaging by single pulse interferometric coherent anti-Stokes Raman scattering microscopy,” J. Phys. Chem. B 110, 5196–5204 (2006).
[CrossRef] [PubMed]

C. W. Freudiger, M. B. J. Roeffaers, X. Zhang, B. G. Saar, W. Min, and X. S. Xie, “Optical heterodyne-detected Raman-induced Kerr effect (OHD-RIKE) microscopy,” J. Phys. Chem. B 115, 5574–5581 (2011).
[CrossRef] [PubMed]

G. W. H. Wurpel, J. M. Schins, and M. Müller, “Direct measurement of chain order in single phospholipid mono- and bilayers with multiplex CARS,” J. Phys. Chem. B 108, 3400–3403(2004).
[CrossRef]

J. Raman Spectrosc. (1)

D. Akimov, S. Chatzipapadopoulos, T. Meyer, N. Tarcea, B. Dietzek, M. Schmitt, and J. Popp, “Different contrast information obtained from CARS and nonresonant FWM images,” J. Raman Spectrosc. 40, 941–947 (2009).
[CrossRef]

New J. Phys. (1)

P. Nandakumar, A. Kovalev, and A. Volkmer, “Vibrational imaging based on stimulated Raman scattering microscopy,” New J. Phys. 11, 033026 (2009).
[CrossRef]

Opt. Express (6)

Opt. Lett. (6)

Phys. Rev. A (4)

H. Lotem, R. T. Lynch, Jr., and N. Bloembergen, “Interference between Raman resonances in four-wave difference mixing,” Phys. Rev. A 14, 1748–1755 (1976).
[CrossRef]

M. D. Levenson and N. Bloembergen, “Dispersion of the nonlinear optical susceptibility tensor in centrosymmetric media,” Phys. Rev. A 10, 4447–4463 (1974).

S.-H. Lim, A. G. Caster, and S. R. Leone, “Single-pulse phase-control interferometric coherent anti-Stokes Raman scattering spectroscopy,” Phys. Rev. A 72, 041803(R) (2005).
[CrossRef]

D. Gachet, F. Billard, and H. Rigneault, “Focused field symmetries for background-free coherent anti-Stokes Raman spectroscopy,” Phys. Rev. A 77, 061802(R) (2008).
[CrossRef]

Phys. Rev. Lett. (7)

D. Gachet, S. Brustlein, and H. Rigneault, “Revisiting the Young’s double slit experiment for background-free nonlinear Raman spectroscopy and microscopy,” Phys. Rev. Lett. 104, 213905 (2010).
[CrossRef] [PubMed]

D. L. Marks and S. A. Boppart, “Nonlinear interferometric vibrational imaging,” Phys. Rev. Lett. 92, 123905 (2004).
[CrossRef] [PubMed]

A. Zumbusch, G. R. Holtom, and X. S. Xie, “Three-dimensional vibrational imaging by coherent anti-Stokes Raman scattering,” Phys. Rev. Lett. 82, 4142–4145 (1999).
[CrossRef]

M. Jurna, J. P. Korterik, C. Otto, J. L. Herek, and H. L. Offerhaus, “Vibrational phase contrast microscopy by use of coherent anti-Stokes Raman scattering,” Phys. Rev. Lett. 103, 043905 (2009).
[CrossRef] [PubMed]

D. Oron, N. Dudovich, and Y. Silberberg, “Narrow-band coherent anti-Stokes Raman signals from broad-band pulses,” Phys. Rev. Lett. 88, 063004 (2002).
[CrossRef] [PubMed]

D. Oron, N. Dudovich, and Y. Silberberg, “Single-pulse phase-contrast nonlinear Raman spectroscopy,” Phys. Rev. Lett. 89, 273001 (2002).
[CrossRef]

D. Oron, N. Dudovich, and Y. Silberberg, “Femtosecond phase-and-polarization control for background-free coherent anti-Stokes Raman spectroscopy,” Phys. Rev. Lett. 90, 213902(2003).
[CrossRef] [PubMed]

Proc. Natl. Acad. Sci. USA (1)

C. L. Evans, E. O. Potma, M. Puoris’haag, D. Côté, C. P. Lin, and X. S. Xie, “Chemical imaging of tissue in vivo with video-rate coherent anti-Stokes Raman scattering microscopy,” Proc. Natl. Acad. Sci. USA 102, 16807–16012 (2005).
[CrossRef] [PubMed]

Proc. R. Soc. London Series A (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanetic system,” Proc. R. Soc. London Series A 253, 358–379 (1959).
[CrossRef]

Prog. Quantum Electron. (1)

R. W. Hellwarth, “Third-order optical susceptibilities of solids and liquids,” Prog. Quantum Electron. 5, 1–68 (1977).
[CrossRef]

Science (1)

C. W. Freudiger, W. Min, B. G. Saar, S. Lu, G. R. Holtom, C. He, J. C. Tsai, J. X. Kang, and X. S. Xie, “Label-free biomedical imaging with high sensitivity by stimulated Raman scattering microscopy,” Science 322, 1857–1861 (2008).
[CrossRef] [PubMed]

Vib. Spectrosc. (1)

V. V. Krishnamachari and E. O. Potma, “Multi-dimensional differential imaging with FE-CARS microscopy,” Vib. Spectrosc. 50, 10–14 (2009).
[CrossRef]

Other (4)

D. Gachet, H. Rigneault, and S. Brustlein, “Méthode pour la détection d’un signal optique non linéaire résonant et dispositif pour la mise en oeuvre de ladite méthode (I),” Brevet CNRS, international patent application (INPI No10/00245)—Extension PCT/EP2011/050622 (22/01/2010)

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

J. D. Jackson, Classical Electrodynamics (Wiley, 1975).

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1

Scheme of the Young’s double slit experiment. Two sources, separated by a distance a, are located at a distance D from an observer. Each source radiates a field E 1 or E 2 . The field E 1 is phase shifted by φ as compared to E 2 . The observer monitors the interference pattern in a (a) θ or (b)  θ = θ direction as compared to the axis perpendicular to the line defined by the two sources.

Fig. 2
Fig. 2

Scheme of the studied configuration. (a) Excitation volume. The focal volume defines the origin Cartesian coordinate system. (b) Scheme of the axial interface between two vibrationally resonant and nonresonant media 1 and 2, forming two volumes V 1 and V 2 , respectively.

Fig. 3
Fig. 3

Forward anti-Stokes far-field radiation patterns from an axial chemical interface as a function of the normalized Raman resonance detuning ζ = ( ω p ω s Ω R ) / Γ (simulation). For the resonant medium, the probed Raman line is assumed to be Lorentzian following χ x x y y ( 3 ) 1 R = a / [ ω p ω s Ω R + i Γ ] and χ x x y y ( 3 ) 1 NR = a / Γ , and the depolarization ratio of the probed Raman line equals ρ R = 1 / 3 . For the nonresonant medium, χ x x y y ( 3 ) 2 NR = 2 χ x x y y ( 3 ) 1 NR . NAs of the objectives: 1.2 for excitation and 0.5 for collection. The incident pump and Stokes laser fields are linearly polarized along the x axis. (a) Bulk CARS spectrum (solid curve) and χ 1 R ( 3 ) tensor phase (dotted curve) of the resonant medium and specific considered detunings. (b) Off-resonance ( ζ = 10 ). (c) CARS peak ( ζ = 0.6 ). (d) Raman peak ( ζ = 0 ). (e) Phase maximum ( ζ = 0.5 ). (f) CARS dip ( ζ = 1.6 ). (g) Off-resonance ( ζ = 10 ). White disks highlight the direction for which the maximum of emission occurs.

Fig. 4
Fig. 4

Forward anti-Stokes far-field radiation patterns from an axial chemical interface as a function of the position x 0 of the interface in the excitation volume at Raman resonance ( ζ = 0 ) (simulation). (a) Scheme of the interface as located into the excitation volume (the center of the excitation volume defines the origin of the Cartesian coordinate system). (b) Excitation volume entirely in the resonant medium ( x 0 = 500 nm ). (c) Excitation volume mostly in the resonant medium ( x 0 = 100 nm ). (d) Excitation volume centered on the interface ( x 0 = 0 nm ). (e) Excitation volume mostly in the nonresonant medium ( x 0 = 100 nm ). (f) Excitation volume entirely in the nonresonant medium ( x 0 = 500 nm ). White disks highlight the direction for which the maximum of emission occurs. Same numerical parameters and exciting laser polarization as in Fig. 3.

Fig. 5
Fig. 5

Forward anti-Stokes signal generated in the vicinity of an axial chemical interface as a function of the normalized Raman resonance detuning ζ and of the position Δ x of the interface in the excitation volume (simulation). (a) Anti-Stokes signal I ( k x ) integrated over the ( k x < 0 ) half-space (in the Fourier space). (b) Anti-Stokes signal I ( k x + ) integrated over the ( k x > 0 ) half-space. (c) Anti-Stokes signal difference Δ I = I ( k x ) I ( k x + ) . (d) Anti-Stokes signal difference Δ I for Δ x = 0 (black dots) and Im [ χ 1 R ( 3 ) ] (solid curve) as a function of the normalized Raman resonance detuning ζ. Same numerical parameters and exciting laser polarization as in Fig. 3.

Fig. 6
Fig. 6

Forward-CARS imaging of a 3 μm diameter bead embedded in a pure nonresonant medium (simulation). The depicted section is imaged in the equatorial ( z = 0 ) plane of the bead. (a) and (b) Contrasts obtained after integration of the anti-Stokes signal over the full NA of the collecting objective lens ( I ( k x ) + I ( k x + ) ). (c) and (d) Contrasts obtained after detection and subtraction of the I ( k x ) and I ( k x + ) anti-Stokes signals ( | I ( k x ) I ( k x + ) | ). (a) and (c) The bead undergoes a vibrational resonance ( ζ = 0 ). (b) and (d) The bead is out of resonance ( ζ = 10 ). The bead and the surrounding medium have the same spectroscopic properties as media 1 and 2 in Fig. 3, repectively. The incident pump and Stokes laser fields are linearly polarized along the x axis.

Fig. 7
Fig. 7

Forward-CARS imaging of a 3 μm diameter bead embedded in a pure nonresonant medium (simulation). The depicted section is imaged in the ( y = 0 ) plane of the bead. (a) and (b) Contrasts obtained after integration of the anti-Stokes signal over the whole NA of the collecting objective lens ( I ( k x ) + I ( k x + ) ). (c) and (d) Contrasts obtained after detection and subtraction of the I ( k x ) and I ( k x + ) integrated anti-Stokes signals ( | I ( k x ) I ( k x + ) | ). (a) and (c) The bead undergoes a vibrational resonance ( ζ = 0 ). (b) and (d) The bead is out of resonance ( ζ = 10 ). The bead and the surrounding medium have the same spectroscopic properties as media 1 and 2 in Fig. 3, respectively. The incident pump and Stokes laser fields are linearly polarized along the x axis.

Fig. 8
Fig. 8

Axial chemical interface between a 3 μm diameter bead and a pure nonresonant medium detected with various detection modes (simulation). (a) X-interface detection mode: detection and subtraction of the I ( k x ) and I ( k x + ) integrated anti-Stokes signals ( | I ( k x ) I ( k x + ) | ). (b) Y interface detection mode: detection and subtraction of the I ( k y ) [integrated in the ( k y < 0 ) half-space] and I ( k y + ) [integrated in the ( k y > 0 ) half-space] anti-Stokes signals ( | I ( k y ) I ( k y + ) | ). (c)  X Y interface detection mode: detection and subtraction of the anti-Stokes signals I ( k x , k y ) and I ( k x , k y ) individually [ k x , k y | I ( k x , k y ) I ( k x , k y ) | ]. The bead and the surrounding medium have the same spectroscopic properties as media 1 and 2 in Fig. 3, respectively. The incident pump and Stokes laser fields are linearly polarized along the x axis.

Fig. 9
Fig. 9

Illustration of the possibility to retrieve the relative positions of the resonant and nonresonant medium forming an axial chemical interface (simulation). (a) X-interface detection mode: I ( k x + ) I ( k x ) . (b) Y interface detection mode: I ( k y + ) I ( k y ) . The bead and the surrounding medium have the same spectroscopic properties as media 1 and 2 in Fig. 3, respectively. The incident pump and Stokes laser fields are linearly polarized along the x axis.

Tables (1)

Tables Icon

Table 1 Symmetry Relations on the x, y, and z Components of the Focused Field E and the Nonlinear Induced Polarization S for Incident Pump and Stokes Beams Linearly Polarized along the x Axis

Equations (38)

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

I ( θ ) = | E 1 | 2 + | E 2 | 2 + 2 | E 1 | | E 2 | cos ( 2 π a λ tan θ + φ ) .
Δ I ( θ ) = 4 | E 1 | | E 2 | sin ( 2 π a λ tan θ ) sin φ .
χ ( 3 ) = χ R ( 3 ) + χ NR ( 3 ) .
P ( 3 ) ( r , ω as ) = χ ( 3 ) ( r ; ω as ; ω p , ω p , ω s ) E p ( r , ω p ) E p ( r , ω p ) E s * ( r , ω s ) .
P ( 3 ) ( r , ρ R ) = 6 [ χ x x y y ( 3 ) R ( r ) S ( r , ρ R ) + χ x x y y ( 3 ) NR ( r ) S ( r , 1 / 3 ) ] ,
S ( r , ρ R ) = [ E p ( r ) · E s * ( r ) ] E p ( r ) + ρ R 1 ρ R [ E p x 2 ( r ) + E p y 2 ( r ) + E p z 2 ( r ) ] E s * ( r ) ,
E x ( ρ M , φ M , z ) = cos α [ I 0 ( ρ M , z ) + cos ( 2 φ M ) I 2 ( ρ M , z ) ] + sin α sin ( 2 φ M ) I 2 ( ρ M , z ) ,
E y ( ρ M , φ M , z ) = cos α sin ( 2 φ M ) I 2 ( ρ M , z ) + sin α [ I 0 ( ρ M , z ) cos ( 2 φ M ) I 2 ( ρ M , z ) ] ,
E z ( ρ M , φ M , z ) = cos φ M I 1 ( ρ M , z ) .
I 0 ( ρ M , z ) = 0 θ max A ( θ ) sin θ ( 1 + cos θ ) J 0 ( k ρ M sin θ ) exp ( i k z cos θ ) d θ ,
I 1 ( ρ M , z ) = 0 θ max A ( θ ) sin 2 θ J 1 ( k ρ M sin θ ) exp ( i k z cos θ ) d θ ,
I 2 ( ρ M , z ) = 0 θ max A ( θ ) sin θ ( 1 cos θ ) J 2 ( k ρ M sin θ ) exp ( i k z cos θ ) d θ ,
A ( θ ) = i k 0 f 2 π E 0 ( cos θ n ) 1 / 2 exp [ ( n β sin θ NA ) 2 ] ,
χ 1 ( 3 ) = χ 1 R ( 3 ) + χ 1 NR ( 3 ) , χ 2 ( 3 ) = χ 2 NR ( 3 ) .
M ( k ) P ( r , ρ R ) = A [ k × P ( r , ρ R ) ] × k ,
E as ( k ) = M ( k ) P ( 3 ) ( r ) exp ( i k · r ) d r .
I as ( k ) = | E as ( k ) | 2 = E as ( k ) · E as * ( k ) .
E as ( k ) = V 1 M ( k ) P 1 ( 3 ) ( r ) exp ( i k · r ) d r + V 2 M ( k ) P 2 ( 3 ) ( r ) exp ( i k · r ) d r .
E as ( k ) = 6 [ χ x x y y ( 3 ) 1 R V 1 M ( k ) S ( r , ρ R ) exp ( i k · r ) d r + χ x x y y ( 3 ) 1 NR V 1 M ( k ) S ( r , 1 / 3 ) exp ( i k · r ) d r + χ x x y y ( 3 ) 2 NR V 2 M ( k ) S ( r , 1 / 3 ) exp ( i k · r ) d r ] .
E as ( k ) = 6 [ χ x x y y ( 3 ) 1 R I V 1 ( ρ R , k ) + χ x x y y ( 3 ) 1 NR I V 1 ( 1 / 3 , k ) + χ x x y y ( 3 ) 2 NR I V 2 ( 1 / 3 , k ) ] .
I as ( k ) = 36 { | χ x x y y ( 3 ) 1 R | 2 | I V 1 ( ρ R , k ) | 2 + 2 χ x x y y ( 3 ) 1 NR Re [ χ x x y y ( 3 ) 1 R I V 1 ( ρ R , k ) · I V 1 * ( 1 / 3 , k ) ] + ( χ x x y y ( 3 ) 1 NR ) 2 | I V 1 ( 1 / 3 , k ) | 2 + 2 χ x x y y ( 3 ) 2 NR Re [ χ x x y y ( 3 ) 1 R I V 1 ( ρ R , k ) · I V 2 * ( 1 / 3 , k ) ] + ( χ x x y y ( 3 ) 2 NR ) 2 | I V 2 ( 1 / 3 , k ) | 2 + 2 χ x x y y ( 3 ) 1 NR χ x x y y ( 3 ) 2 NR Re [ I V 1 ( 1 / 3 , k ) · I V 2 * ( 1 / 3 , k ) ] } .
E as ( k ) = V M ( k ) P ( 3 ) ( r ) exp ( i k · r ) d r .
E as ( k ) = 6 [ χ x x y y ( 3 ) 1 R V 1 M ( k ) S ( r , ρ R ) exp ( i k · r ) d r + χ x x y y ( 3 ) 1 NR V 1 M ( k ) S ( r , 1 / 3 ) exp ( i k · r ) d r + χ x x y y ( 3 ) 2 NR V 2 M ( k ) S ( r , 1 / 3 ) exp ( i k · r ) d r ] .
I as ( k ) = 36 { | χ x x y y ( 3 ) 1 R | 2 | I V 1 ( ρ R , k ) | 2 + 2 χ x x y y ( 3 ) 1 NR Re [ χ x x y y ( 3 ) 1 R I V 1 ( ρ R , k ) · I V 1 * ( 1 / 3 , k ) ] + ( χ x x y y ( 3 ) 1 NR ) 2 | I V 1 ( 1 / 3 , k ) | 2 + 2 χ x x y y ( 3 ) 2 NR Re [ χ x x y y ( 3 ) 1 R I V 1 ( ρ R , k ) · I V 2 * ( 1 / 3 , k ) ] + ( χ x x y y ( 3 ) 2 NR ) 2 | I V 2 ( 1 / 3 , k ) | 2 + 2 χ x x y y ( 3 ) 1 NR χ x x y y ( 3 ) 2 NR Re [ I V 1 ( 1 / 3 , k ) · I V 2 * ( 1 / 3 , k ) ] } .
I V ( ρ R , k ) = V M ( k ) S ( r , ρ R ) exp ( i k · r ) d r , I V ( ρ R , k ) = V M ( k ) S ( r , ρ R ) exp ( i k · r ) d r ,
S ( r , ρ R ) = D z S * ( r , ρ R ) ,
D z = ( 1 0 0 0 1 0 0 0 1 ) ,
M ( k ) S ( r , ρ R ) = D z M ( k ) S * ( r , ρ R ) .
I V ( ρ R , k ) = D z V M ( k ) S * ( r , ρ R ) exp ( i k · r ) d r .
I V ( ρ R , k ) = D z V M ( k ) S * ( r , ρ R ) exp ( i k · r ) d r .
I V ( ρ R , k ) = D z I V * ( ρ R , k ) .
Δ I as ( k ) = I as ( k ) I as ( k ) , Δ I as ( k ) = 72 { χ x x y y ( 3 ) 1 NR Re { χ x x y y ( 3 ) 1 R [ I V 1 ( ρ R , k ) · I V 1 * ( 1 / 3 , k ) I V 1 * ( ρ R , k ) · I V 1 ( 1 / 3 , k ) ] } + χ x x y y ( 3 ) 2 NR Re { χ x x y y ( 3 ) 1 R [ I V 1 ( ρ R , k ) · I V 2 * ( 1 / 3 , k ) I V 1 * ( ρ R , k ) · I V 2 ( 1 / 3 , k ) ] } + χ x x y y ( 3 ) 1 NR χ x x y y ( 3 ) 2 NR Re { I V 1 ( 1 / 3 , k ) · I V 2 * ( 1 / 3 , k ) I V 1 * ( 1 / 3 , k ) · I V 2 ( 1 / 3 , k ) } } .
T 1 = 2 χ x x y y ( 3 ) 1 NR Im [ I V 1 ( ρ R , k ) · I V 1 * ( 1 / 3 , k ) ] Im ( χ x x y y ( 3 ) 1 R ) ,
T 2 = 2 χ x x y y ( 3 ) 2 NR Im [ I V 1 ( ρ R , k ) · I V 2 * ( 1 / 3 , k ) ] Im ( χ x x y y ( 3 ) 1 R ) ,
T 3 = 0 .
Δ I as ( k ) = 144 [ F 1 ( k , V 1 , ρ R ) χ x x y y ( 3 ) 1 NR + F 2 ( k , V 1 , V 2 , ρ R ) χ x x y y ( 3 ) 2 NR ] Im ( χ x x y y ( 3 ) 1 R ) ,
F 1 ( k , V 1 , ρ R ) = Im [ I V 1 ( ρ R , k ) · I V 1 * ( 1 / 3 , k ) ] ,
F 2 ( k , V 1 , V 2 , ρ R ) = Im [ I V 1 ( ρ R , k ) · I V 2 * ( 1 / 3 , k ) ] .

Metrics