Abstract

We address the general problem of detecting chemical interfaces arbitrarily oriented in space in coherent anti-Stokes Raman scattering (CARS) microscopy. Such a task is accomplished by using a beam reversal scheme, as recently demonstrated experimentally [J. Biomed. Opt. 16, 086006 (2011)]. We develop a full vectorial theoretical analysis of the situation and show that transverse chemical interfaces are readily highlighted without special care in the CARS signal detection. In addition, a finer analysis reveals that adequate angular analysis of the CARS far-field radiation pattern enables the detection of axial interfaces. Background-free CARS microscopy and spectroscopy are thus achievable through the combined application of excitation beam reversal and angular analysis of the CARS far-field radiation pattern. This differential CARS (D-CARS) technique is relevant for fast detection of interfaces between molecularly different media.

© 2011 Optical Society of America

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References

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  1. D. Gachet and H. Rigneault, “Detection of chemical interfaces in coherent anti-Stokes Raman scattering microscopy: Dk-CARS. I. Axial interfaces,” J. Opt. Soc. Am. A 28, 2519–2530 (2011).
    [CrossRef]
  2. C. L. Evans and X. S. Xie, “Coherent anti-Stokes Raman scattering microscopy: chemical imaging for biology and medicine,” Annu. Rev. Anal. Chem. 1, 883–909 (2008).
    [CrossRef]
  3. 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]
  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) .
  5. 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]
  6. V. V. Krishnamachari and E. O. Potma, “Detecting lateral interfaces with focus-engineered coherent anti-Stokes Raman scattering microscopy,” J. Raman Spectrosc. 39, 593–598 (2008).
    [CrossRef]
  7. 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]
  8. 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]
  9. 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]
  10. S. Brustlein, D. Gachet, F. Billard, and H. Rigneault, “Transverse chemical interface detection with coherent anti-Stokes Raman scattering (CARS) microscopy,” J. Biomed. Opt. 16, 086006(2011).
    [CrossRef] [PubMed]
  11. D. Gachet, H. Rigneault, S. Brustlein, and F. Billard, “Méthode pour la détection d’un signal optique non linéaire résonant et dispositif pour la mise en oeuvre de ladite méthode (II),” Brevet CNRS, international patent application (INPI No10/00244)—Extension PCT/EP2011/050619 (CNRS, 22/01/2010).
  12. S. Hell and E. H. K. Stelzer, “Properties of a 4Pi confocal fluorescence microscope,” J. Opt. Soc. Am. A 9, 2159–2166 (1992).
    [CrossRef]
  13. R. W. Hellwarth, “Third-order optical susceptibilities of solids and liquids,” Prog. Quantum Electron. 5, 1–68 (1977).
    [CrossRef]
  14. M. Müller and J. M. Schins, “Imaging the thermodynamic state of lipid membranes with multiplex CARS microscopy,” J. Phys. Chem. B 106, 3715–3723 (2002).
    [CrossRef]

2011 (2)

S. Brustlein, D. Gachet, F. Billard, and H. Rigneault, “Transverse chemical interface detection with coherent anti-Stokes Raman scattering (CARS) microscopy,” J. Biomed. Opt. 16, 086006(2011).
[CrossRef] [PubMed]

D. Gachet and H. Rigneault, “Detection of chemical interfaces in coherent anti-Stokes Raman scattering microscopy: Dk-CARS. I. Axial interfaces,” J. Opt. Soc. Am. A 28, 2519–2530 (2011).
[CrossRef]

2010 (1)

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]

2008 (4)

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]

V. V. Krishnamachari and E. O. Potma, “Detecting lateral interfaces with focus-engineered coherent anti-Stokes Raman scattering microscopy,” J. Raman Spectrosc. 39, 593–598 (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]

C. L. Evans and X. S. Xie, “Coherent anti-Stokes Raman scattering microscopy: chemical imaging for biology and medicine,” Annu. Rev. Anal. Chem. 1, 883–909 (2008).
[CrossRef]

2007 (2)

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]

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]

2002 (1)

M. Müller and J. M. Schins, “Imaging the thermodynamic state of lipid membranes with multiplex CARS microscopy,” J. Phys. Chem. B 106, 3715–3723 (2002).
[CrossRef]

1992 (1)

1977 (1)

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

Billard, F.

D. Gachet, H. Rigneault, S. Brustlein, and F. Billard, “Méthode pour la détection d’un signal optique non linéaire résonant et dispositif pour la mise en oeuvre de ladite méthode (II),” Brevet CNRS, international patent application (INPI No10/00244)—Extension PCT/EP2011/050619 (CNRS, 22/01/2010).

S. Brustlein, D. Gachet, F. Billard, and H. Rigneault, “Transverse chemical interface detection with coherent anti-Stokes Raman scattering (CARS) microscopy,” J. Biomed. Opt. 16, 086006(2011).
[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]

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, H. Rigneault, S. Brustlein, and F. Billard, “Méthode pour la détection d’un signal optique non linéaire résonant et dispositif pour la mise en oeuvre de ladite méthode (II),” Brevet CNRS, international patent application (INPI No10/00244)—Extension PCT/EP2011/050619 (CNRS, 22/01/2010).

S. Brustlein, D. Gachet, F. Billard, and H. Rigneault, “Transverse chemical interface detection with coherent anti-Stokes Raman scattering (CARS) microscopy,” J. Biomed. Opt. 16, 086006(2011).
[CrossRef] [PubMed]

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]

Evans, C. L.

C. L. Evans and X. S. Xie, “Coherent anti-Stokes Raman scattering microscopy: chemical imaging for biology and medicine,” Annu. Rev. Anal. Chem. 1, 883–909 (2008).
[CrossRef]

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, H. Rigneault, S. Brustlein, and F. Billard, “Méthode pour la détection d’un signal optique non linéaire résonant et dispositif pour la mise en oeuvre de ladite méthode (II),” Brevet CNRS, international patent application (INPI No10/00244)—Extension PCT/EP2011/050619 (CNRS, 22/01/2010).

S. Brustlein, D. Gachet, F. Billard, and H. Rigneault, “Transverse chemical interface detection with coherent anti-Stokes Raman scattering (CARS) microscopy,” J. Biomed. Opt. 16, 086006(2011).
[CrossRef] [PubMed]

D. Gachet and H. Rigneault, “Detection of chemical interfaces in coherent anti-Stokes Raman scattering microscopy: Dk-CARS. I. Axial interfaces,” J. Opt. Soc. Am. A 28, 2519–2530 (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, “Focused field symmetries for background-free coherent anti-Stokes Raman spectroscopy,” Phys. Rev. A 77, 061802(R) (2008).
[CrossRef]

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]

Hell, S.

Hellwarth, R. W.

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

Krishnamachari, V. V.

V. V. Krishnamachari and E. O. Potma, “Detecting lateral interfaces with focus-engineered coherent anti-Stokes Raman scattering microscopy,” J. Raman Spectrosc. 39, 593–598 (2008).
[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]

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]

Müller, M.

M. Müller and J. M. Schins, “Imaging the thermodynamic state of lipid membranes with multiplex CARS microscopy,” J. Phys. Chem. B 106, 3715–3723 (2002).
[CrossRef]

Potma, E. O.

V. V. Krishnamachari and E. O. Potma, “Detecting lateral interfaces with focus-engineered coherent anti-Stokes Raman scattering microscopy,” J. Raman Spectrosc. 39, 593–598 (2008).
[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]

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]

Rigneault, H.

D. Gachet, H. Rigneault, S. Brustlein, and F. Billard, “Méthode pour la détection d’un signal optique non linéaire résonant et dispositif pour la mise en oeuvre de ladite méthode (II),” Brevet CNRS, international patent application (INPI No10/00244)—Extension PCT/EP2011/050619 (CNRS, 22/01/2010).

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) .

S. Brustlein, D. Gachet, F. Billard, and H. Rigneault, “Transverse chemical interface detection with coherent anti-Stokes Raman scattering (CARS) microscopy,” J. Biomed. Opt. 16, 086006(2011).
[CrossRef] [PubMed]

D. Gachet and H. Rigneault, “Detection of chemical interfaces in coherent anti-Stokes Raman scattering microscopy: Dk-CARS. I. Axial interfaces,” J. Opt. Soc. Am. A 28, 2519–2530 (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, “Focused field symmetries for background-free coherent anti-Stokes Raman spectroscopy,” Phys. Rev. A 77, 061802(R) (2008).
[CrossRef]

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]

Schins, J. M.

M. Müller and J. M. Schins, “Imaging the thermodynamic state of lipid membranes with multiplex CARS microscopy,” J. Phys. Chem. B 106, 3715–3723 (2002).
[CrossRef]

Stelzer, E. H. K.

Xie, X. S.

C. L. Evans and X. S. Xie, “Coherent anti-Stokes Raman scattering microscopy: chemical imaging for biology and medicine,” Annu. Rev. Anal. Chem. 1, 883–909 (2008).
[CrossRef]

Annu. Rev. Anal. Chem. (1)

C. L. Evans and X. S. Xie, “Coherent anti-Stokes Raman scattering microscopy: chemical imaging for biology and medicine,” Annu. Rev. Anal. Chem. 1, 883–909 (2008).
[CrossRef]

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. Biomed. Opt. (1)

S. Brustlein, D. Gachet, F. Billard, and H. Rigneault, “Transverse chemical interface detection with coherent anti-Stokes Raman scattering (CARS) microscopy,” J. Biomed. Opt. 16, 086006(2011).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (3)

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

J. Phys. Chem. B (1)

M. Müller and J. M. Schins, “Imaging the thermodynamic state of lipid membranes with multiplex CARS microscopy,” J. Phys. Chem. B 106, 3715–3723 (2002).
[CrossRef]

J. Raman Spectrosc. (1)

V. V. Krishnamachari and E. O. Potma, “Detecting lateral interfaces with focus-engineered coherent anti-Stokes Raman scattering microscopy,” J. Raman Spectrosc. 39, 593–598 (2008).
[CrossRef]

Phys. Rev. A (1)

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. (1)

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]

Prog. Quantum Electron. (1)

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

Other (2)

D. Gachet, H. Rigneault, S. Brustlein, and F. Billard, “Méthode pour la détection d’un signal optique non linéaire résonant et dispositif pour la mise en oeuvre de ladite méthode (II),” Brevet CNRS, international patent application (INPI No10/00244)—Extension PCT/EP2011/050619 (CNRS, 22/01/2010).

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) .

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Figures (7)

Fig. 1
Fig. 1

Scheme of two-wave interference with two sources arbitrarily located in space. (a) The observer stands at an initial position defined by the angle θ. The point 0 located in the middle of the two sources defines the origin. (b) The observer stands in the opposite direction as compared to the two sources. (c) Situation equivalent to (b) after a π-angle rotation.

Fig. 2
Fig. 2

Scheme of the studied configurations. (a) α problem: the respective resonant and nonresonant objects 1 and 2 are excited by the pump and Stokes beams and the generated anti-Stokes signal is emitted in the k = ( k x , k y , k z ) direction. (b) β problem: the same objects are excited by the reflected pump and Stokes beams and the anti-Stokes signal is detected in the k = ( k x , k y , k z ) direction. In each case, the center of the excitation volume defines the origin of the Cartesian coordinates.

Fig. 3
Fig. 3

Possible experimental scheme: the pump and Stokes beams are focused a first time into the sample (α problem) through objective lens 1 and are then reflected by a dichroic filter to be focused into the sample at the same point (β problem) through objective lens 2. The generated forward anti-Stokes signals are then collected by objective lenses 2 and 1, respectively, before being detected by detectors 1 and 2, respectively. Objective lenses 1 and 2 define the angular apertures over which the anti-Stokes signals are collected and then detected. The α and β problems consider the anti-Stokes signals generated in the k and k directions and collected by objectives 2 and 1, respectively. Note that, because the referential is defined by the direction of propagation of the excitation beams for each (α or β) problem, the k y and k z axes, for the respective α and β problems, have opposed orientations in the laboratory frame.

Fig. 4
Fig. 4

Detection schemes for highlighting plane interfaces. (a) Detection scheme for highlighting a plane interface perpendicular to the z axis (Z-interface detection mode). The anti-Stokes signals collected by objective lenses 1 and 2 in the α and β problems are fully detected ( I α ( k α ) and I β ( k α ) ) and then subtracted ( | I α ( k α ) I β ( k α ) | ). (b) Detection scheme for highlighting a plane interface perpendicular to the x axis ( X Z -interface detection mode). The anti- Stokes signals collected by objective lenses 1 and 2 in the α and β problems are filtered in the reciprocal space and only the ( k x > 0 ) and ( k x < 0 ) parts are detected [ I α ( k x , α + ) and I β ( k x , β ) , respectively]. Then they are subtracted from each other ( | I α ( k x , α + ) I β ( k x , β ) | ). (c) Equivalent detection scheme for a plane interface perpendicular to the y axis ( Y Z -interface detection mode). (d) Detection scheme for highlighting an interface with an arbitrary orientation ( X Y Z -interface detection mode). The anti-Stokes signals I α ( k x , k y ) and I β ( k x , k y ) collected by objective lenses 1 and 2 in the k = ( k x , k y , k z ) (α problem) and the k = ( k x , k y , k z ) (β problem) directions in the reciprocal space are subtracted individually ( k x , k y | I α ( k x , k y ) I β ( k x , k y ) | ).

Fig. 5
Fig. 5

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), (b), (d), and (e) Contrasts obtained after integration of the anti-Stokes signal over the full NA of the collecting objective lenses for the α [ I α ( k α ) , (a) and (d)] and β [ I β ( k β ) , (b) and (e)] problems. The bead undergoes a vibrational resonance [ ζ = 0 , (a) and (b)] or is out of resonance [ ζ = 10 , (d) and (e)]. (c) and (f) Contrasts obtained after subtraction of the α and β anti-Stokes signals [ | I α ( k α ) I β ( k β ) | ] (c) on and (f) off resonance. The normalized Raman resonance detuning is defined as ζ = ( ω p ω s Ω R ) / Γ . For the bead, 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 surrounding medium, χ x x y y ( 3 ) 2 NR = 2 χ x x y y ( 3 ) 1 NR . The NA of objective lenses 1 and 2 is 1.2. The incident pump and Stokes laser fields are linearly polarized along the x axis.

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) Z- interface detection mode: detection and subtraction of the I α ( k α ) and I β ( k β ) integrated anti-Stokes signals [ | I α ( k α ) I β ( k β ) | ]. (b)  X Z - interface detection mode: detection and subtraction of the I α ( k x , α + ) and I β ( k x , β ) integrated anti-Stokes signals [ | I α ( k x , α + ) I β ( k x , β ) | ]. (c)  Y Z - interface detection mode: detection and subtraction of the I α ( k y , α + ) and I β ( k y , β ) integrated anti-Stokes signals [ | I α ( k y , α + ) I β ( k y , β ) | ]. (d)  X Y Z - interface detection mode: detection and subtraction of the I α ( k x , k y ) and I β ( k x , k y ) anti-Stokes signals [ k x , k y | I α ( k x , k y ) I β ( k x , k y ) | ]. The bead and the surrounding medium have the same spectroscopic properties as in Fig. 5. 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) Z- interface detection mode: detection and subtraction of the I α ( k α ) and I β ( k β ) integrated anti-Stokes signals [ | I α ( k α ) I β ( k β ) | ]. (b)  X Y Z - interface detection mode: detection and subtraction of the I α ( k x , k y ) and I β ( k x , k y ) anti-Stokes signals [ k x , k y | I α ( k x , k y ) I β ( k x , k y ) | ]. The bead and the surrounding medium have the same spectroscopic properties as in Fig. 5. The incident pump and Stokes laser fields are linearly polarized along the x axis.

Equations (38)

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I α ( θ ) = | E 1 | 2 + | E 2 | 2 + 2 | E 1 | | E 2 | cos [ 2 π λ ( d + a tan θ ) + φ ] ,
I β ( θ ) = | E 1 | 2 + | E 2 | 2 + 2 | E 1 | | E 2 | cos [ 2 π λ ( d + a tan θ ) φ ] .
Δ I ( θ ) = 4 | E 1 | | E 2 | sin [ 2 π λ ( d + a tan θ ) ] sin ( φ ) .
χ 1 ( 3 ) = χ 1 R ( 3 ) + χ 1 NR ( 3 ) , χ 2 ( 3 ) = χ 2 NR ( 3 ) ,
E as α ( 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 ] .
M ( k ) S ( r , ρ R ) = A [ k × S ( r , ρ R ) ] × k ,
E as α ( 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 α ( 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 ( 2 ) 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 β ( 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 β ( 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 ( 2 ) 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 ,
I V ( ρ R , k ) = V M ( k ) S ( r , ρ R ) exp ( i k · r ) d r ,
S ( r , ρ R ) = S * ( r , ρ R ) ,
M ( k ) S ( r , ρ R ) = D x M ( k ) S ( r , ρ R ) ,
D x = ( 1 0 0 0 1 0 0 0 1 ) .
M ( k ) S ( r , ρ R ) = D x M ( k ) S * ( r , ρ R ) .
I V ( ρ R , k ) = V D x M ( k ) S * ( r , ρ R ) exp ( i k · r ) d r , I V ( ρ R , k ) = D x [ V M ( k ) S ( r , ρ R ) exp ( i k · r ) d r ] * .
I V ( ρ R , k ) = D x I V * ( ρ R , k ) .
I V i ( ρ R , k ) · I V i * ( ρ R , k ) = I V i * ( ρ R , k ) · I V i ( ρ R , k ) ,
Δ I as ( k ) = I as α ( k ) I as β ( 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 ) ] .
F 1 ( k 1 , V 1 , ρ R ) + F 1 ( k 2 , V 1 , ρ R ) = Im [ I V 1 ( ρ R , k 1 ) · I V 1 * ( 1 / 3 , k 1 ) + I V 1 ( ρ R , k 2 ) · I V 1 * ( 1 / 3 , k 2 ) ] ,
F 2 ( k 1 , V 1 , V 2 , ρ R ) + F 2 ( k 2 , V 1 , V 2 , ρ R ) = Im [ I V 1 ( ρ R , k 1 ) · I V 2 * ( 1 / 3 , k 1 ) + I V 1 ( ρ R , k 2 ) · I V 2 * ( 1 / 3 , k 2 ) ] .
I V ( ρ R , k 2 ) = V M ( k 2 ) S ( r , ρ R ) exp ( i k 2 · r ) d r .
I V ( ρ R , k 2 ) = V M ( k 2 ) S ( r , ρ R ) exp ( i k 1 · r ) d r .
M ( k 2 ) S ( r , ρ R ) = D x M ( k 1 ) S * ( r , ρ R )
I V ( ρ R , k 2 ) = D x I V * ( ρ R , k 1 ) .
I V ( ρ R , k 3 ) = V M ( k 3 ) S ( r , ρ R ) exp ( i k 3 · r ) d r .
I V ( ρ R , k 3 ) = V M ( k 3 ) S ( r , ρ R ) exp ( i k 1 · r ) d r .
S ( r , ρ R ) = D y S * ( r , ρ R ) ,
M ( k 3 ) S ( r , ρ R ) = D y M ( k 1 ) S * ( r , ρ R ) ,
D y = ( 1 0 0 0 1 0 0 0 1 ) .
I V ( ρ R , k 3 ) = D y I V * ( ρ R , k 1 ) .

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