Abstract

We demonstrate a method to measure Raman optical activity (ROA) by using coherent anti-Stokes Raman scattering (CARS) spectral interferometry. An extremely weak chirality-induced CARS field is amplified through the interference with a strong CARS field generated from an external reference and is extracted by the Fourier transformation. In this interferometric coherent Raman optical activity (iCROA), both the sign and the magnitude of optical active non-resonant background susceptibility can be directly determined. Measurement of a CARS-ROA spectrum with less artifact is obtained because a broad offset artifact due to optical rotatory dispersion is clearly distinguished in iCROA.

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References

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  1. L. D. Barron, Molecular Light Scattering and Optical Activity(Cambridge University Press, Cambridge, 2004).
    [CrossRef]
  2. L. Nafie, Vibrational Optical Activity: Principles and Applications(Wiley, New York, 2011).
    [CrossRef]
  3. L. D. Barron, L. Hecht, E. Blanch, and A. Bell, “Solution structure and dynamics of biomolecules from Raman optical activity,” Prog. Biophys. Mol. Biol.73, 1–49 (2000).
    [CrossRef] [PubMed]
  4. I. H. McColl, E. W. Blanch, L. Hecht, and L. D. Barron, “A Study of α-Helix Hydration in Polypeptides, Proteins, and Viruses Using Vibrational Raman Optical Activity,” J. Am. Chem. Soc.126, 8181–8188 (2004).
    [CrossRef] [PubMed]
  5. I. H. McColl, E. W. Blanch, A. C. Gill, A. G. O. Rhie, M. A. Ritchie, L. Hecht, K. Nielsen, and L. D. Barron, “A New Perspective on β-Sheet Structures Using Vibrational Raman Optical Activity: From Poly(L-lysine) to the Prion Protein.” J. Am. Chem. Soc.125, 10019–10026 (2003).
    [CrossRef] [PubMed]
  6. K. Hiramatsu, M. Okuno, H. Kano, P. Leproux, V. Couderc, and H. Hamaguchi, “Observation of Raman Optical Activity by Heterodyne-Detected Polarization-Resolved Coherent Anti-Stokes Raman Scattering,” Phys. Rev. Lett.109, 083901 (2012).
    [CrossRef] [PubMed]
  7. L. Lepetit, G. Chériaux, and M. Joffre, “Linear techniques of phase measurement by femtosecond spectral interferometry for applications in spectroscopy,” J. Opt. Soc. Am. B12, 2467 (1995).
    [CrossRef]
  8. 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 χ(3)for vibrational microscopy.” Opt. Lett.29, 2923–2925 (2004).
    [CrossRef]
  9. 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]
  10. I. V. Stiopkin, H. D. Jayathilake, A. N. Bordenyuk, and A. V. Benderskii, “Heterodyne-Detected Vibrational Sum Frequency Generation Spectroscopy.” J. Am. Chem. Soc.130, 2271–2275 (2008).
    [CrossRef] [PubMed]
  11. S. Yamaguchi and T. Tahara, “Heterodyne-detected electronic sum frequency generation: “up” versus “down” alignment of interfacial molecules.” J. Chem. Phys.129, 101102 (2008).
    [CrossRef] [PubMed]
  12. H. Rhee, Y.-G. June, J.-S. Lee, K.-K. Lee, J.-H. Ha, Z. H. Kim, S.-J. Jeon, and M. Cho, “Femtosecond characterization of vibrational optical activity of chiral molecules.” Nature458, 310–313 (2009).
    [CrossRef] [PubMed]
  13. I. Eom, S.-H. Ahn, H. Rhee, and M. Cho, “Single-Shot Electronic Optical Activity Interferometry: Power and Phase Fluctuation-Free Measurement,” Phys. Rev. Lett.108, 1–5 (2012).
    [CrossRef]
  14. Although the LO term should be written as E2LO(ω)e−iωτcosθin a precise sense, cosθcan be approximated by 1 because θchanges between −0.125° and 0.125° in this study.

2012

K. Hiramatsu, M. Okuno, H. Kano, P. Leproux, V. Couderc, and H. Hamaguchi, “Observation of Raman Optical Activity by Heterodyne-Detected Polarization-Resolved Coherent Anti-Stokes Raman Scattering,” Phys. Rev. Lett.109, 083901 (2012).
[CrossRef] [PubMed]

I. Eom, S.-H. Ahn, H. Rhee, and M. Cho, “Single-Shot Electronic Optical Activity Interferometry: Power and Phase Fluctuation-Free Measurement,” Phys. Rev. Lett.108, 1–5 (2012).
[CrossRef]

2009

H. Rhee, Y.-G. June, J.-S. Lee, K.-K. Lee, J.-H. Ha, Z. H. Kim, S.-J. Jeon, and M. Cho, “Femtosecond characterization of vibrational optical activity of chiral molecules.” Nature458, 310–313 (2009).
[CrossRef] [PubMed]

2008

I. V. Stiopkin, H. D. Jayathilake, A. N. Bordenyuk, and A. V. Benderskii, “Heterodyne-Detected Vibrational Sum Frequency Generation Spectroscopy.” J. Am. Chem. Soc.130, 2271–2275 (2008).
[CrossRef] [PubMed]

S. Yamaguchi and T. Tahara, “Heterodyne-detected electronic sum frequency generation: “up” versus “down” alignment of interfacial molecules.” J. Chem. Phys.129, 101102 (2008).
[CrossRef] [PubMed]

2006

2004

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 χ(3)for vibrational microscopy.” Opt. Lett.29, 2923–2925 (2004).
[CrossRef]

I. H. McColl, E. W. Blanch, L. Hecht, and L. D. Barron, “A Study of α-Helix Hydration in Polypeptides, Proteins, and Viruses Using Vibrational Raman Optical Activity,” J. Am. Chem. Soc.126, 8181–8188 (2004).
[CrossRef] [PubMed]

2003

I. H. McColl, E. W. Blanch, A. C. Gill, A. G. O. Rhie, M. A. Ritchie, L. Hecht, K. Nielsen, and L. D. Barron, “A New Perspective on β-Sheet Structures Using Vibrational Raman Optical Activity: From Poly(L-lysine) to the Prion Protein.” J. Am. Chem. Soc.125, 10019–10026 (2003).
[CrossRef] [PubMed]

2000

L. D. Barron, L. Hecht, E. Blanch, and A. Bell, “Solution structure and dynamics of biomolecules from Raman optical activity,” Prog. Biophys. Mol. Biol.73, 1–49 (2000).
[CrossRef] [PubMed]

1995

Ahn, S.-H.

I. Eom, S.-H. Ahn, H. Rhee, and M. Cho, “Single-Shot Electronic Optical Activity Interferometry: Power and Phase Fluctuation-Free Measurement,” Phys. Rev. Lett.108, 1–5 (2012).
[CrossRef]

Barron, L. D.

I. H. McColl, E. W. Blanch, L. Hecht, and L. D. Barron, “A Study of α-Helix Hydration in Polypeptides, Proteins, and Viruses Using Vibrational Raman Optical Activity,” J. Am. Chem. Soc.126, 8181–8188 (2004).
[CrossRef] [PubMed]

I. H. McColl, E. W. Blanch, A. C. Gill, A. G. O. Rhie, M. A. Ritchie, L. Hecht, K. Nielsen, and L. D. Barron, “A New Perspective on β-Sheet Structures Using Vibrational Raman Optical Activity: From Poly(L-lysine) to the Prion Protein.” J. Am. Chem. Soc.125, 10019–10026 (2003).
[CrossRef] [PubMed]

L. D. Barron, L. Hecht, E. Blanch, and A. Bell, “Solution structure and dynamics of biomolecules from Raman optical activity,” Prog. Biophys. Mol. Biol.73, 1–49 (2000).
[CrossRef] [PubMed]

L. D. Barron, Molecular Light Scattering and Optical Activity(Cambridge University Press, Cambridge, 2004).
[CrossRef]

Bell, A.

L. D. Barron, L. Hecht, E. Blanch, and A. Bell, “Solution structure and dynamics of biomolecules from Raman optical activity,” Prog. Biophys. Mol. Biol.73, 1–49 (2000).
[CrossRef] [PubMed]

Benderskii, A. V.

I. V. Stiopkin, H. D. Jayathilake, A. N. Bordenyuk, and A. V. Benderskii, “Heterodyne-Detected Vibrational Sum Frequency Generation Spectroscopy.” J. Am. Chem. Soc.130, 2271–2275 (2008).
[CrossRef] [PubMed]

Blanch, E.

L. D. Barron, L. Hecht, E. Blanch, and A. Bell, “Solution structure and dynamics of biomolecules from Raman optical activity,” Prog. Biophys. Mol. Biol.73, 1–49 (2000).
[CrossRef] [PubMed]

Blanch, E. W.

I. H. McColl, E. W. Blanch, L. Hecht, and L. D. Barron, “A Study of α-Helix Hydration in Polypeptides, Proteins, and Viruses Using Vibrational Raman Optical Activity,” J. Am. Chem. Soc.126, 8181–8188 (2004).
[CrossRef] [PubMed]

I. H. McColl, E. W. Blanch, A. C. Gill, A. G. O. Rhie, M. A. Ritchie, L. Hecht, K. Nielsen, and L. D. Barron, “A New Perspective on β-Sheet Structures Using Vibrational Raman Optical Activity: From Poly(L-lysine) to the Prion Protein.” J. Am. Chem. Soc.125, 10019–10026 (2003).
[CrossRef] [PubMed]

Bordenyuk, A. N.

I. V. Stiopkin, H. D. Jayathilake, A. N. Bordenyuk, and A. V. Benderskii, “Heterodyne-Detected Vibrational Sum Frequency Generation Spectroscopy.” J. Am. Chem. Soc.130, 2271–2275 (2008).
[CrossRef] [PubMed]

Chériaux, G.

Cho, M.

I. Eom, S.-H. Ahn, H. Rhee, and M. Cho, “Single-Shot Electronic Optical Activity Interferometry: Power and Phase Fluctuation-Free Measurement,” Phys. Rev. Lett.108, 1–5 (2012).
[CrossRef]

H. Rhee, Y.-G. June, J.-S. Lee, K.-K. Lee, J.-H. Ha, Z. H. Kim, S.-J. Jeon, and M. Cho, “Femtosecond characterization of vibrational optical activity of chiral molecules.” Nature458, 310–313 (2009).
[CrossRef] [PubMed]

Couderc, V.

K. Hiramatsu, M. Okuno, H. Kano, P. Leproux, V. Couderc, and H. Hamaguchi, “Observation of Raman Optical Activity by Heterodyne-Detected Polarization-Resolved Coherent Anti-Stokes Raman Scattering,” Phys. Rev. Lett.109, 083901 (2012).
[CrossRef] [PubMed]

Eom, I.

I. Eom, S.-H. Ahn, H. Rhee, and M. Cho, “Single-Shot Electronic Optical Activity Interferometry: Power and Phase Fluctuation-Free Measurement,” Phys. Rev. Lett.108, 1–5 (2012).
[CrossRef]

Evans, C. L.

Gill, A. C.

I. H. McColl, E. W. Blanch, A. C. Gill, A. G. O. Rhie, M. A. Ritchie, L. Hecht, K. Nielsen, and L. D. Barron, “A New Perspective on β-Sheet Structures Using Vibrational Raman Optical Activity: From Poly(L-lysine) to the Prion Protein.” J. Am. Chem. Soc.125, 10019–10026 (2003).
[CrossRef] [PubMed]

Ha, J.-H.

H. Rhee, Y.-G. June, J.-S. Lee, K.-K. Lee, J.-H. Ha, Z. H. Kim, S.-J. Jeon, and M. Cho, “Femtosecond characterization of vibrational optical activity of chiral molecules.” Nature458, 310–313 (2009).
[CrossRef] [PubMed]

Hamaguchi, H.

K. Hiramatsu, M. Okuno, H. Kano, P. Leproux, V. Couderc, and H. Hamaguchi, “Observation of Raman Optical Activity by Heterodyne-Detected Polarization-Resolved Coherent Anti-Stokes Raman Scattering,” Phys. Rev. Lett.109, 083901 (2012).
[CrossRef] [PubMed]

Hecht, L.

I. H. McColl, E. W. Blanch, L. Hecht, and L. D. Barron, “A Study of α-Helix Hydration in Polypeptides, Proteins, and Viruses Using Vibrational Raman Optical Activity,” J. Am. Chem. Soc.126, 8181–8188 (2004).
[CrossRef] [PubMed]

I. H. McColl, E. W. Blanch, A. C. Gill, A. G. O. Rhie, M. A. Ritchie, L. Hecht, K. Nielsen, and L. D. Barron, “A New Perspective on β-Sheet Structures Using Vibrational Raman Optical Activity: From Poly(L-lysine) to the Prion Protein.” J. Am. Chem. Soc.125, 10019–10026 (2003).
[CrossRef] [PubMed]

L. D. Barron, L. Hecht, E. Blanch, and A. Bell, “Solution structure and dynamics of biomolecules from Raman optical activity,” Prog. Biophys. Mol. Biol.73, 1–49 (2000).
[CrossRef] [PubMed]

Hiramatsu, K.

K. Hiramatsu, M. Okuno, H. Kano, P. Leproux, V. Couderc, and H. Hamaguchi, “Observation of Raman Optical Activity by Heterodyne-Detected Polarization-Resolved Coherent Anti-Stokes Raman Scattering,” Phys. Rev. Lett.109, 083901 (2012).
[CrossRef] [PubMed]

Jayathilake, H. D.

I. V. Stiopkin, H. D. Jayathilake, A. N. Bordenyuk, and A. V. Benderskii, “Heterodyne-Detected Vibrational Sum Frequency Generation Spectroscopy.” J. Am. Chem. Soc.130, 2271–2275 (2008).
[CrossRef] [PubMed]

Jeon, S.-J.

H. Rhee, Y.-G. June, J.-S. Lee, K.-K. Lee, J.-H. Ha, Z. H. Kim, S.-J. Jeon, and M. Cho, “Femtosecond characterization of vibrational optical activity of chiral molecules.” Nature458, 310–313 (2009).
[CrossRef] [PubMed]

Joffre, M.

June, Y.-G.

H. Rhee, Y.-G. June, J.-S. Lee, K.-K. Lee, J.-H. Ha, Z. H. Kim, S.-J. Jeon, and M. Cho, “Femtosecond characterization of vibrational optical activity of chiral molecules.” Nature458, 310–313 (2009).
[CrossRef] [PubMed]

Kano, H.

K. Hiramatsu, M. Okuno, H. Kano, P. Leproux, V. Couderc, and H. Hamaguchi, “Observation of Raman Optical Activity by Heterodyne-Detected Polarization-Resolved Coherent Anti-Stokes Raman Scattering,” Phys. Rev. Lett.109, 083901 (2012).
[CrossRef] [PubMed]

Kim, Z. H.

H. Rhee, Y.-G. June, J.-S. Lee, K.-K. Lee, J.-H. Ha, Z. H. Kim, S.-J. Jeon, and M. Cho, “Femtosecond characterization of vibrational optical activity of chiral molecules.” Nature458, 310–313 (2009).
[CrossRef] [PubMed]

Lee, J.-S.

H. Rhee, Y.-G. June, J.-S. Lee, K.-K. Lee, J.-H. Ha, Z. H. Kim, S.-J. Jeon, and M. Cho, “Femtosecond characterization of vibrational optical activity of chiral molecules.” Nature458, 310–313 (2009).
[CrossRef] [PubMed]

Lee, K.-K.

H. Rhee, Y.-G. June, J.-S. Lee, K.-K. Lee, J.-H. Ha, Z. H. Kim, S.-J. Jeon, and M. Cho, “Femtosecond characterization of vibrational optical activity of chiral molecules.” Nature458, 310–313 (2009).
[CrossRef] [PubMed]

Lepetit, L.

Leproux, P.

K. Hiramatsu, M. Okuno, H. Kano, P. Leproux, V. Couderc, and H. Hamaguchi, “Observation of Raman Optical Activity by Heterodyne-Detected Polarization-Resolved Coherent Anti-Stokes Raman Scattering,” Phys. Rev. Lett.109, 083901 (2012).
[CrossRef] [PubMed]

McColl, I. H.

I. H. McColl, E. W. Blanch, L. Hecht, and L. D. Barron, “A Study of α-Helix Hydration in Polypeptides, Proteins, and Viruses Using Vibrational Raman Optical Activity,” J. Am. Chem. Soc.126, 8181–8188 (2004).
[CrossRef] [PubMed]

I. H. McColl, E. W. Blanch, A. C. Gill, A. G. O. Rhie, M. A. Ritchie, L. Hecht, K. Nielsen, and L. D. Barron, “A New Perspective on β-Sheet Structures Using Vibrational Raman Optical Activity: From Poly(L-lysine) to the Prion Protein.” J. Am. Chem. Soc.125, 10019–10026 (2003).
[CrossRef] [PubMed]

Nafie, L.

L. Nafie, Vibrational Optical Activity: Principles and Applications(Wiley, New York, 2011).
[CrossRef]

Nielsen, K.

I. H. McColl, E. W. Blanch, A. C. Gill, A. G. O. Rhie, M. A. Ritchie, L. Hecht, K. Nielsen, and L. D. Barron, “A New Perspective on β-Sheet Structures Using Vibrational Raman Optical Activity: From Poly(L-lysine) to the Prion Protein.” J. Am. Chem. Soc.125, 10019–10026 (2003).
[CrossRef] [PubMed]

Okuno, M.

K. Hiramatsu, M. Okuno, H. Kano, P. Leproux, V. Couderc, and H. Hamaguchi, “Observation of Raman Optical Activity by Heterodyne-Detected Polarization-Resolved Coherent Anti-Stokes Raman Scattering,” Phys. Rev. Lett.109, 083901 (2012).
[CrossRef] [PubMed]

Potma, E. O.

Rhee, H.

I. Eom, S.-H. Ahn, H. Rhee, and M. Cho, “Single-Shot Electronic Optical Activity Interferometry: Power and Phase Fluctuation-Free Measurement,” Phys. Rev. Lett.108, 1–5 (2012).
[CrossRef]

H. Rhee, Y.-G. June, J.-S. Lee, K.-K. Lee, J.-H. Ha, Z. H. Kim, S.-J. Jeon, and M. Cho, “Femtosecond characterization of vibrational optical activity of chiral molecules.” Nature458, 310–313 (2009).
[CrossRef] [PubMed]

Rhie, A. G. O.

I. H. McColl, E. W. Blanch, A. C. Gill, A. G. O. Rhie, M. A. Ritchie, L. Hecht, K. Nielsen, and L. D. Barron, “A New Perspective on β-Sheet Structures Using Vibrational Raman Optical Activity: From Poly(L-lysine) to the Prion Protein.” J. Am. Chem. Soc.125, 10019–10026 (2003).
[CrossRef] [PubMed]

Ritchie, M. A.

I. H. McColl, E. W. Blanch, A. C. Gill, A. G. O. Rhie, M. A. Ritchie, L. Hecht, K. Nielsen, and L. D. Barron, “A New Perspective on β-Sheet Structures Using Vibrational Raman Optical Activity: From Poly(L-lysine) to the Prion Protein.” J. Am. Chem. Soc.125, 10019–10026 (2003).
[CrossRef] [PubMed]

Stiopkin, I. V.

I. V. Stiopkin, H. D. Jayathilake, A. N. Bordenyuk, and A. V. Benderskii, “Heterodyne-Detected Vibrational Sum Frequency Generation Spectroscopy.” J. Am. Chem. Soc.130, 2271–2275 (2008).
[CrossRef] [PubMed]

Tahara, T.

S. Yamaguchi and T. Tahara, “Heterodyne-detected electronic sum frequency generation: “up” versus “down” alignment of interfacial molecules.” J. Chem. Phys.129, 101102 (2008).
[CrossRef] [PubMed]

Xie, X. S.

Yamaguchi, S.

S. Yamaguchi and T. Tahara, “Heterodyne-detected electronic sum frequency generation: “up” versus “down” alignment of interfacial molecules.” J. Chem. Phys.129, 101102 (2008).
[CrossRef] [PubMed]

J. Am. Chem. Soc.

I. H. McColl, E. W. Blanch, L. Hecht, and L. D. Barron, “A Study of α-Helix Hydration in Polypeptides, Proteins, and Viruses Using Vibrational Raman Optical Activity,” J. Am. Chem. Soc.126, 8181–8188 (2004).
[CrossRef] [PubMed]

I. H. McColl, E. W. Blanch, A. C. Gill, A. G. O. Rhie, M. A. Ritchie, L. Hecht, K. Nielsen, and L. D. Barron, “A New Perspective on β-Sheet Structures Using Vibrational Raman Optical Activity: From Poly(L-lysine) to the Prion Protein.” J. Am. Chem. Soc.125, 10019–10026 (2003).
[CrossRef] [PubMed]

I. V. Stiopkin, H. D. Jayathilake, A. N. Bordenyuk, and A. V. Benderskii, “Heterodyne-Detected Vibrational Sum Frequency Generation Spectroscopy.” J. Am. Chem. Soc.130, 2271–2275 (2008).
[CrossRef] [PubMed]

J. Chem. Phys.

S. Yamaguchi and T. Tahara, “Heterodyne-detected electronic sum frequency generation: “up” versus “down” alignment of interfacial molecules.” J. Chem. Phys.129, 101102 (2008).
[CrossRef] [PubMed]

J. Opt. Soc. Am. B

Nature

H. Rhee, Y.-G. June, J.-S. Lee, K.-K. Lee, J.-H. Ha, Z. H. Kim, S.-J. Jeon, and M. Cho, “Femtosecond characterization of vibrational optical activity of chiral molecules.” Nature458, 310–313 (2009).
[CrossRef] [PubMed]

Opt. Lett.

Phys. Rev. Lett.

I. Eom, S.-H. Ahn, H. Rhee, and M. Cho, “Single-Shot Electronic Optical Activity Interferometry: Power and Phase Fluctuation-Free Measurement,” Phys. Rev. Lett.108, 1–5 (2012).
[CrossRef]

K. Hiramatsu, M. Okuno, H. Kano, P. Leproux, V. Couderc, and H. Hamaguchi, “Observation of Raman Optical Activity by Heterodyne-Detected Polarization-Resolved Coherent Anti-Stokes Raman Scattering,” Phys. Rev. Lett.109, 083901 (2012).
[CrossRef] [PubMed]

Prog. Biophys. Mol. Biol.

L. D. Barron, L. Hecht, E. Blanch, and A. Bell, “Solution structure and dynamics of biomolecules from Raman optical activity,” Prog. Biophys. Mol. Biol.73, 1–49 (2000).
[CrossRef] [PubMed]

Other

L. D. Barron, Molecular Light Scattering and Optical Activity(Cambridge University Press, Cambridge, 2004).
[CrossRef]

L. Nafie, Vibrational Optical Activity: Principles and Applications(Wiley, New York, 2011).
[CrossRef]

Although the LO term should be written as E2LO(ω)e−iωτcosθin a precise sense, cosθcan be approximated by 1 because θchanges between −0.125° and 0.125° in this study.

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Figures (4)

Fig. 1
Fig. 1

Experimental setup. PCF: photonic crystal fiber, PBS: polarized beam splitter, HWP: half wave plate, P: polarizer, LPF1: long pass filter (cutoff @ 1064 nm), LPF2: long pass filter (cutoff @ 1050 nm), REF: reference (water), S: sample ((−)-β-pinene, BS: beam splitter, PZT(τ): piezo stage for optical delay τ = τ0 + τ1n, NF: notch filter (1064 nm), SPF: short pass filter (cutoff @ 1050 nm).

Fig. 2
Fig. 2

Schematic diagram of experimental configuration around the sample and the reference.

Fig. 3
Fig. 3

(a): |S(ω, θ)| spectra of (−)-β-pinene at θ = −0.125°(top, red) and 0.000° (bottom, green).

Fig. 4
Fig. 4

CARS-ROA spectra of (−)-β-pinene before (top) and after (bottom) the compensation to cancel the achiral contribution. The top spectrum is calculated as Re[S(ω, 0.000°)S*(ω, 0.125°)] and the bottom one is calculated as Re[S′(ω, 0.000°)S*(ω, 0.125°)].

Equations (8)

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

E ( ω , θ ) = E 1 sample ( ω ) sin θ + E 2 sample ( ω ) cos θ + E 2 LO ( ω ) e i ω τ = [ A ( χ 1111 sample ( ω ) sin θ + χ 2111 sample ( ω ) cos θ ) + B χ 1111 LO e i ω τ ] E 1 pu ( ω 1 ) E 1 pu ( ω 1 ) E 1 St * ( ω 2 ) ,
I ( ω , θ ) = | E 1 sample ( ω ) sin θ + E 2 sample ( ω ) cos θ + E 2 LO ( ω ) e i ω τ | 2 A 2 | χ 1111 sample ( ω ) | 2 sin 2 θ + A 2 | χ 2111 sample ( ω ) | 2 cos 2 θ + B 2 | χ 1111 LO | 2 + A 2 sin θ cos θ Re [ χ 1111 sample ( ω ) χ 2111 sample * ( ω ) ] + A B sin θ Re [ χ 1111 sample ( ω ) χ 1111 LO * exp ( i ω τ ) ] + A B cos θ Re [ χ 2111 sample ( ω ) χ 1111 LO * exp ( i ω τ ) ] .
S ( ω , θ ) = sin θ χ 1111 sample ( ω ) exp ( i ω τ 0 ) + cos θ χ 2111 sample ( ω ) exp ( i ω τ 0 )
| χ 1111 | = ( | χ 1111 NR + A Ω ω R i Γ | 2 ) 1 2 = ( ( χ 1111 NR ) 2 + 2 χ 1111 NR A ( Ω ω R ) ( Ω ω R ) 2 + Γ 2 + A 2 ( Ω ω R ) 2 + Γ 2 ) 1 2 ,
| χ 2111 | = ( | χ 2111 NR | 2 + Im ( χ 2111 NR ) Δ A ( Ω ω R ) ( Ω ω R ) 2 + Γ 2 + Δ 2 A 2 / 4 ( Ω ω R ) 2 + Γ 2 ) 1 2 ,
Re ( χ 1111 * χ 2111 ) = i = 1 n ( Δ i χ 1111 NR + Im ( χ 2111 NR ) ) A i Γ i ( Ω i ω R ) 2 + Γ i 2 .
Re [ S ( ω , 0.000 ° ) S * ( ω , 0.125 ° ) ] Re [ sin ( 0.125 ° ) χ 1111 e i ω τ ( χ 2111 * + sin α ( ω ) χ 1111 * ) e i ω τ ] Re ( χ 1111 * χ 2111 ) + sin α ( ω ) | χ 1111 | 2 ,
S ( ω , 0.000 ° ) = S ( ω , 0.000 ° ) c ( ω ) S ( ω , 0.125 ° ) χ 2111 exp ( i ω τ ) .

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