Abstract

We demonstrate that broadband electronic optical activity can be measured with supercontinuum light pulse generated by a femtosecond pump (800 nm). It is the self-heterodyned detection technique that enables us to selectively measure the real (optical rotatory dispersion, ORD) or imaginary (circular dichroism, CD) part of the chiroptical susceptibility by controlling the incident polarization state. The single-shot-based measurement that is capable of correcting power fluctuations of the continuum light is realized by using a fast CCD detector and a polarizing beam splitter. Particularly, non-differential scheme used does not rely on any polarization-switching components. We anticipate that this broadband CD/ORD spectrometry with intrinsically ultrafast time-resolution will be applied to a variety of ultrafast chiroptical dynamics studies.

© 2011 OSA

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    [CrossRef] [PubMed]
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2011 (1)

S. T. Roberts, J. J. Loparo, K. Ramasesha, and A. Tokmakoff, “A Fast-scanning Fourier transform 2D IR interferometer,” Opt. Commun. 284(4), 1062–1066 (2011).
[CrossRef]

2010 (4)

E. Chen, R. A. Goldbeck, and D. S. Kliger, “Nanosecond time-resolved polarization spectroscopies: tools for probing protein reaction mechanisms,” Methods 52(1), 3–11 (2010).
[CrossRef] [PubMed]

A. Trifonov, I. Buchvarov, A. Lohr, F. Würthner, and T. Fiebig, “Broadband femtosecond circular dichroism spectrometer with white-light polarization control,” Rev. Sci. Instrum. 81(4), 043104 (2010).
[CrossRef] [PubMed]

H. Rhee, J.-H. Choi, and M. Cho, “Infrared optical activity: electric field approaches in time domain,” Acc. Chem. Res. 43(12), 1527–1536 (2010).
[CrossRef] [PubMed]

H. Rhee, S.-S. Kim, and M. Cho, “Multichannel array detection of vibrational optical activity free-induction-decay,” J. Anal. Sci. Technol. 1(2), 147–151 (2010).
[CrossRef]

2009 (4)

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,” Nature 458(7236), 310–313 (2009).
[CrossRef] [PubMed]

H. Rhee, Y.-G. June, Z. H. Kim, S.-J. Jeon, and M. Cho, “Phase sensitive detection of vibrational optical activity free-induction-decay: vibrational CD and ORD,” J. Opt. Soc. Am. B 26(5), 1008–1017 (2009).
[CrossRef]

H. Rhee, S.-S. Kim, S.-J. Jeon, and M. Cho, “Femtosecond measurements of vibrational circular dichroism and optical rotatory dispersion spectra,” ChemPhysChem 10(13), 2209–2211 (2009).
[CrossRef] [PubMed]

J. Helbing and M. Bonmarin, “Vibrational circular dichroism signal enhancement using self-heterodyning with elliptically polarized laser pulses,” J. Chem. Phys. 131(17), 174507 (2009).
[CrossRef] [PubMed]

2008 (3)

H. Rhee, J.-H. Ha, S.-J. Jeon, and M. Cho, “Femtosecond spectral interferometry of optical activity: theory,” J. Chem. Phys. 129(9), 094507 (2008).
[CrossRef] [PubMed]

M. Bonmarin and J. Helbing, “A picosecond time-resolved vibrational circular dichroism spectrometer,” Opt. Lett. 33(18), 2086–2088 (2008).
[CrossRef] [PubMed]

C. Niezborala and F. Hache, “Conformational changes in photoexcited (R)-(+)-1,1′-bi-2-naphthol studied by time-resolved circular dichroism,” J. Am. Chem. Soc. 130(38), 12783–12786 (2008).
[CrossRef] [PubMed]

2007 (2)

S. H. Shim, D. B. Strasfeld, Y. L. Ling, and M. T. Zanni, “Automated 2D IR spectroscopy using a mid-IR pulse shaper and application of this technology to the human islet amyloid polypeptide,” Proc. Natl. Acad. Sci. U.S.A. 104(36), 14197–14202 (2007).
[CrossRef] [PubMed]

S.-H. Lim, A. G. Caster, and S. R. Leone, “Fourier transform spectral interferometric coherent anti-Stokes Raman scattering (FTSI-CARS) spectroscopy,” Opt. Lett. 32(10), 1332–1334 (2007).
[CrossRef] [PubMed]

2006 (2)

C. Kolano, J. Helbing, M. Kozinski, W. Sander, and P. Hamm, “Watching hydrogen-bond dynamics in a β-turn by transient two-dimensional infrared spectroscopy,” Nature 444(7118), 469–472 (2006).
[CrossRef] [PubMed]

C. Niezborala and F. Hache, “Measuring the dynamics of circular dichroism in a pump-probe experiment with a Babinet-Soleil Compensator,” J. Opt. Soc. Am. B 23(11), 2418–2424 (2006).
[CrossRef]

2005 (4)

E. Chen, Y. Wen, J. W. Lewis, R. A. Goldbeck, D. S. Kliger, and C. E. M. Strauss, “Nanosecond laser temperature-jump optical rotatory dispersion: application to early events in protein folding/unfolding,” Rev. Sci. Instrum. 76(8), 083120 (2005).
[CrossRef]

T. Dartigalongue and F. Hache, “Observation of sub-100 ps conformational changes in photolyzed carbonmonoxy-myoglobin probed by time-resolved circular dichroism,” Chem. Phys. Lett. 415(4-6), 313–316 (2005).
[CrossRef]

T. Brixner, J. Stenger, H. M. Vaswani, M. Cho, R. E. Blankenship, and G. R. Fleming, “Two-dimensional spectroscopy of electronic couplings in photosynthesis,” Nature 434(7033), 625–628 (2005).
[CrossRef] [PubMed]

T. Zhang, C. N. Borca, X. Li, and S. T. Cundiff, “Optical two-dimensional Fourier transform spectroscopy with active interferometric stabilization,” Opt. Express 13(19), 7432–7441 (2005).
[CrossRef] [PubMed]

2004 (2)

T. Brixner, I. V. Stiopkin, and G. R. Fleming, “Tunable two-dimensional femtosecond spectroscopy,” Opt. Lett. 29(8), 884–886 (2004).
[CrossRef] [PubMed]

K. C. Hannah and B. A. Armitage, “DNA-templated assembly of helical cyanine dye aggregates: a supramolecular chain polymerization,” Acc. Chem. Res. 37(11), 845–853 (2004).
[CrossRef] [PubMed]

2002 (1)

2001 (1)

M. T. Zanni, N. H. Ge, Y. S. Kim, and R. M. Hochstrasser, “Two-dimensional IR spectroscopy can be designed to eliminate the diagonal peaks and expose only the crosspeaks needed for structure determination,” Proc. Natl. Acad. Sci. U.S.A. 98(20), 11265–11270 (2001).
[CrossRef] [PubMed]

1999 (1)

G. D. Goodno and R. J. D. Miller, “Femtosecond heterodyne-detected four-wave-mixing studies of deterministic protein motions. 1. theory and experimental technique of diffractive optics-based spectroscopy,” J. Phys. Chem. A 103(49), 10619–10629 (1999).
[CrossRef]

1998 (1)

1997 (2)

W. J. Walecki, D. N. Fittinghoff, A. L. Smirl, and R. Trebino, “Characterization of the polarization state of weak ultrashort coherent signals by dual-channel spectral interferometry,” Opt. Lett. 22(2), 81–83 (1997).
[CrossRef] [PubMed]

R. M. Esquerra, J. W. Lewis, and D. S. Kliger, “An improved linear retarder for time-resolved circular dichroism studies,” Rev. Sci. Instrum. 68(3), 1372–1376 (1997).
[CrossRef]

1995 (2)

L. Lepetit, G. Cheriaux, and M. Joffre, “Linear techniques of phase measurement by femtosecond spectral interferometry for applications in spectroscopy,” J. Opt. Soc. Am. B 12(12), 2467–2474 (1995).
[CrossRef]

D. B. Shapiro, R. A. Goldbeck, D. Che, R. M. Esquerra, S. J. Paquette, and D. S. Kliger, “Nanosecond optical rotatory dispersion spectroscopy: application to photolyzed hemoglobin-CO kinetics,” Biophys. J. 68(1), 326–334 (1995).
[CrossRef] [PubMed]

1993 (1)

C. F. Zhang, J. W. Lewis, R. Cerpa, I. D. Kuntz, and D. S. Kliger, “Nanosecond circular dichroism spectral measurements - extension to the far-ultraviolet region,” J. Phys. Chem. 97(21), 5499–5505 (1993).
[CrossRef]

1992 (1)

J. W. Lewis, R. A. Goldbeck, D. S. Kliger, X. Xie, R. C. Dunn, and J. D. Simon, “Time-resolved circular dichroism spectroscopy - experiment, theory, and applications to biological systems,” J. Phys. Chem. 96(13), 5243–5254 (1992).
[CrossRef]

1990 (1)

X. Xie and J. D. Simon, “Picosecond time-resolved circular dichroism study of protein relaxation in myoglobin following photodissociation of carbon monoxide,” J. Am. Chem. Soc. 112(21), 7802–7803 (1990).
[CrossRef]

Albrecht, A. W.

Armitage, B. A.

K. C. Hannah and B. A. Armitage, “DNA-templated assembly of helical cyanine dye aggregates: a supramolecular chain polymerization,” Acc. Chem. Res. 37(11), 845–853 (2004).
[CrossRef] [PubMed]

Belabas, N.

Blankenship, R. E.

T. Brixner, J. Stenger, H. M. Vaswani, M. Cho, R. E. Blankenship, and G. R. Fleming, “Two-dimensional spectroscopy of electronic couplings in photosynthesis,” Nature 434(7033), 625–628 (2005).
[CrossRef] [PubMed]

Bonmarin, M.

J. Helbing and M. Bonmarin, “Vibrational circular dichroism signal enhancement using self-heterodyning with elliptically polarized laser pulses,” J. Chem. Phys. 131(17), 174507 (2009).
[CrossRef] [PubMed]

M. Bonmarin and J. Helbing, “A picosecond time-resolved vibrational circular dichroism spectrometer,” Opt. Lett. 33(18), 2086–2088 (2008).
[CrossRef] [PubMed]

Borca, C. N.

Brixner, T.

T. Brixner, J. Stenger, H. M. Vaswani, M. Cho, R. E. Blankenship, and G. R. Fleming, “Two-dimensional spectroscopy of electronic couplings in photosynthesis,” Nature 434(7033), 625–628 (2005).
[CrossRef] [PubMed]

T. Brixner, I. V. Stiopkin, and G. R. Fleming, “Tunable two-dimensional femtosecond spectroscopy,” Opt. Lett. 29(8), 884–886 (2004).
[CrossRef] [PubMed]

Buchvarov, I.

A. Trifonov, I. Buchvarov, A. Lohr, F. Würthner, and T. Fiebig, “Broadband femtosecond circular dichroism spectrometer with white-light polarization control,” Rev. Sci. Instrum. 81(4), 043104 (2010).
[CrossRef] [PubMed]

Caster, A. G.

Cerpa, R.

C. F. Zhang, J. W. Lewis, R. Cerpa, I. D. Kuntz, and D. S. Kliger, “Nanosecond circular dichroism spectral measurements - extension to the far-ultraviolet region,” J. Phys. Chem. 97(21), 5499–5505 (1993).
[CrossRef]

Che, D.

D. B. Shapiro, R. A. Goldbeck, D. Che, R. M. Esquerra, S. J. Paquette, and D. S. Kliger, “Nanosecond optical rotatory dispersion spectroscopy: application to photolyzed hemoglobin-CO kinetics,” Biophys. J. 68(1), 326–334 (1995).
[CrossRef] [PubMed]

Chen, E.

E. Chen, R. A. Goldbeck, and D. S. Kliger, “Nanosecond time-resolved polarization spectroscopies: tools for probing protein reaction mechanisms,” Methods 52(1), 3–11 (2010).
[CrossRef] [PubMed]

E. Chen, Y. Wen, J. W. Lewis, R. A. Goldbeck, D. S. Kliger, and C. E. M. Strauss, “Nanosecond laser temperature-jump optical rotatory dispersion: application to early events in protein folding/unfolding,” Rev. Sci. Instrum. 76(8), 083120 (2005).
[CrossRef]

Cheriaux, G.

Cho, M.

H. Rhee, S.-S. Kim, and M. Cho, “Multichannel array detection of vibrational optical activity free-induction-decay,” J. Anal. Sci. Technol. 1(2), 147–151 (2010).
[CrossRef]

H. Rhee, J.-H. Choi, and M. Cho, “Infrared optical activity: electric field approaches in time domain,” Acc. Chem. Res. 43(12), 1527–1536 (2010).
[CrossRef] [PubMed]

H. Rhee, Y.-G. June, Z. H. Kim, S.-J. Jeon, and M. Cho, “Phase sensitive detection of vibrational optical activity free-induction-decay: vibrational CD and ORD,” J. Opt. Soc. Am. B 26(5), 1008–1017 (2009).
[CrossRef]

H. Rhee, S.-S. Kim, S.-J. Jeon, and M. Cho, “Femtosecond measurements of vibrational circular dichroism and optical rotatory dispersion spectra,” ChemPhysChem 10(13), 2209–2211 (2009).
[CrossRef] [PubMed]

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,” Nature 458(7236), 310–313 (2009).
[CrossRef] [PubMed]

H. Rhee, J.-H. Ha, S.-J. Jeon, and M. Cho, “Femtosecond spectral interferometry of optical activity: theory,” J. Chem. Phys. 129(9), 094507 (2008).
[CrossRef] [PubMed]

T. Brixner, J. Stenger, H. M. Vaswani, M. Cho, R. E. Blankenship, and G. R. Fleming, “Two-dimensional spectroscopy of electronic couplings in photosynthesis,” Nature 434(7033), 625–628 (2005).
[CrossRef] [PubMed]

Choi, J.-H.

H. Rhee, J.-H. Choi, and M. Cho, “Infrared optical activity: electric field approaches in time domain,” Acc. Chem. Res. 43(12), 1527–1536 (2010).
[CrossRef] [PubMed]

Cundiff, S. T.

Dartigalongue, T.

T. Dartigalongue and F. Hache, “Observation of sub-100 ps conformational changes in photolyzed carbonmonoxy-myoglobin probed by time-resolved circular dichroism,” Chem. Phys. Lett. 415(4-6), 313–316 (2005).
[CrossRef]

Dunn, R. C.

J. W. Lewis, R. A. Goldbeck, D. S. Kliger, X. Xie, R. C. Dunn, and J. D. Simon, “Time-resolved circular dichroism spectroscopy - experiment, theory, and applications to biological systems,” J. Phys. Chem. 96(13), 5243–5254 (1992).
[CrossRef]

Esquerra, R. M.

R. M. Esquerra, J. W. Lewis, and D. S. Kliger, “An improved linear retarder for time-resolved circular dichroism studies,” Rev. Sci. Instrum. 68(3), 1372–1376 (1997).
[CrossRef]

D. B. Shapiro, R. A. Goldbeck, D. Che, R. M. Esquerra, S. J. Paquette, and D. S. Kliger, “Nanosecond optical rotatory dispersion spectroscopy: application to photolyzed hemoglobin-CO kinetics,” Biophys. J. 68(1), 326–334 (1995).
[CrossRef] [PubMed]

Fiebig, T.

A. Trifonov, I. Buchvarov, A. Lohr, F. Würthner, and T. Fiebig, “Broadband femtosecond circular dichroism spectrometer with white-light polarization control,” Rev. Sci. Instrum. 81(4), 043104 (2010).
[CrossRef] [PubMed]

Fittinghoff, D. N.

Fleming, G. R.

T. Brixner, J. Stenger, H. M. Vaswani, M. Cho, R. E. Blankenship, and G. R. Fleming, “Two-dimensional spectroscopy of electronic couplings in photosynthesis,” Nature 434(7033), 625–628 (2005).
[CrossRef] [PubMed]

T. Brixner, I. V. Stiopkin, and G. R. Fleming, “Tunable two-dimensional femtosecond spectroscopy,” Opt. Lett. 29(8), 884–886 (2004).
[CrossRef] [PubMed]

Gallagher, S. M.

Ge, N. H.

M. T. Zanni, N. H. Ge, Y. S. Kim, and R. M. Hochstrasser, “Two-dimensional IR spectroscopy can be designed to eliminate the diagonal peaks and expose only the crosspeaks needed for structure determination,” Proc. Natl. Acad. Sci. U.S.A. 98(20), 11265–11270 (2001).
[CrossRef] [PubMed]

Goldbeck, R. A.

E. Chen, R. A. Goldbeck, and D. S. Kliger, “Nanosecond time-resolved polarization spectroscopies: tools for probing protein reaction mechanisms,” Methods 52(1), 3–11 (2010).
[CrossRef] [PubMed]

E. Chen, Y. Wen, J. W. Lewis, R. A. Goldbeck, D. S. Kliger, and C. E. M. Strauss, “Nanosecond laser temperature-jump optical rotatory dispersion: application to early events in protein folding/unfolding,” Rev. Sci. Instrum. 76(8), 083120 (2005).
[CrossRef]

D. B. Shapiro, R. A. Goldbeck, D. Che, R. M. Esquerra, S. J. Paquette, and D. S. Kliger, “Nanosecond optical rotatory dispersion spectroscopy: application to photolyzed hemoglobin-CO kinetics,” Biophys. J. 68(1), 326–334 (1995).
[CrossRef] [PubMed]

J. W. Lewis, R. A. Goldbeck, D. S. Kliger, X. Xie, R. C. Dunn, and J. D. Simon, “Time-resolved circular dichroism spectroscopy - experiment, theory, and applications to biological systems,” J. Phys. Chem. 96(13), 5243–5254 (1992).
[CrossRef]

Goodno, G. D.

G. D. Goodno and R. J. D. Miller, “Femtosecond heterodyne-detected four-wave-mixing studies of deterministic protein motions. 1. theory and experimental technique of diffractive optics-based spectroscopy,” J. Phys. Chem. A 103(49), 10619–10629 (1999).
[CrossRef]

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,” Nature 458(7236), 310–313 (2009).
[CrossRef] [PubMed]

H. Rhee, J.-H. Ha, S.-J. Jeon, and M. Cho, “Femtosecond spectral interferometry of optical activity: theory,” J. Chem. Phys. 129(9), 094507 (2008).
[CrossRef] [PubMed]

Hache, F.

C. Niezborala and F. Hache, “Conformational changes in photoexcited (R)-(+)-1,1′-bi-2-naphthol studied by time-resolved circular dichroism,” J. Am. Chem. Soc. 130(38), 12783–12786 (2008).
[CrossRef] [PubMed]

C. Niezborala and F. Hache, “Measuring the dynamics of circular dichroism in a pump-probe experiment with a Babinet-Soleil Compensator,” J. Opt. Soc. Am. B 23(11), 2418–2424 (2006).
[CrossRef]

T. Dartigalongue and F. Hache, “Observation of sub-100 ps conformational changes in photolyzed carbonmonoxy-myoglobin probed by time-resolved circular dichroism,” Chem. Phys. Lett. 415(4-6), 313–316 (2005).
[CrossRef]

Hamm, P.

C. Kolano, J. Helbing, M. Kozinski, W. Sander, and P. Hamm, “Watching hydrogen-bond dynamics in a β-turn by transient two-dimensional infrared spectroscopy,” Nature 444(7118), 469–472 (2006).
[CrossRef] [PubMed]

Hannah, K. C.

K. C. Hannah and B. A. Armitage, “DNA-templated assembly of helical cyanine dye aggregates: a supramolecular chain polymerization,” Acc. Chem. Res. 37(11), 845–853 (2004).
[CrossRef] [PubMed]

Helbing, J.

J. Helbing and M. Bonmarin, “Vibrational circular dichroism signal enhancement using self-heterodyning with elliptically polarized laser pulses,” J. Chem. Phys. 131(17), 174507 (2009).
[CrossRef] [PubMed]

M. Bonmarin and J. Helbing, “A picosecond time-resolved vibrational circular dichroism spectrometer,” Opt. Lett. 33(18), 2086–2088 (2008).
[CrossRef] [PubMed]

C. Kolano, J. Helbing, M. Kozinski, W. Sander, and P. Hamm, “Watching hydrogen-bond dynamics in a β-turn by transient two-dimensional infrared spectroscopy,” Nature 444(7118), 469–472 (2006).
[CrossRef] [PubMed]

Hochstrasser, R. M.

M. T. Zanni, N. H. Ge, Y. S. Kim, and R. M. Hochstrasser, “Two-dimensional IR spectroscopy can be designed to eliminate the diagonal peaks and expose only the crosspeaks needed for structure determination,” Proc. Natl. Acad. Sci. U.S.A. 98(20), 11265–11270 (2001).
[CrossRef] [PubMed]

Hybl, T. D.

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,” Nature 458(7236), 310–313 (2009).
[CrossRef] [PubMed]

H. Rhee, S.-S. Kim, S.-J. Jeon, and M. Cho, “Femtosecond measurements of vibrational circular dichroism and optical rotatory dispersion spectra,” ChemPhysChem 10(13), 2209–2211 (2009).
[CrossRef] [PubMed]

H. Rhee, Y.-G. June, Z. H. Kim, S.-J. Jeon, and M. Cho, “Phase sensitive detection of vibrational optical activity free-induction-decay: vibrational CD and ORD,” J. Opt. Soc. Am. B 26(5), 1008–1017 (2009).
[CrossRef]

H. Rhee, J.-H. Ha, S.-J. Jeon, and M. Cho, “Femtosecond spectral interferometry of optical activity: theory,” J. Chem. Phys. 129(9), 094507 (2008).
[CrossRef] [PubMed]

Joffre, M.

Jonas, D. 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,” Nature 458(7236), 310–313 (2009).
[CrossRef] [PubMed]

H. Rhee, Y.-G. June, Z. H. Kim, S.-J. Jeon, and M. Cho, “Phase sensitive detection of vibrational optical activity free-induction-decay: vibrational CD and ORD,” J. Opt. Soc. Am. B 26(5), 1008–1017 (2009).
[CrossRef]

Kim, S.-S.

H. Rhee, S.-S. Kim, and M. Cho, “Multichannel array detection of vibrational optical activity free-induction-decay,” J. Anal. Sci. Technol. 1(2), 147–151 (2010).
[CrossRef]

H. Rhee, S.-S. Kim, S.-J. Jeon, and M. Cho, “Femtosecond measurements of vibrational circular dichroism and optical rotatory dispersion spectra,” ChemPhysChem 10(13), 2209–2211 (2009).
[CrossRef] [PubMed]

Kim, Y. S.

M. T. Zanni, N. H. Ge, Y. S. Kim, and R. M. Hochstrasser, “Two-dimensional IR spectroscopy can be designed to eliminate the diagonal peaks and expose only the crosspeaks needed for structure determination,” Proc. Natl. Acad. Sci. U.S.A. 98(20), 11265–11270 (2001).
[CrossRef] [PubMed]

Kim, Z. H.

H. Rhee, Y.-G. June, Z. H. Kim, S.-J. Jeon, and M. Cho, “Phase sensitive detection of vibrational optical activity free-induction-decay: vibrational CD and ORD,” J. Opt. Soc. Am. B 26(5), 1008–1017 (2009).
[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,” Nature 458(7236), 310–313 (2009).
[CrossRef] [PubMed]

Kliger, D. S.

E. Chen, R. A. Goldbeck, and D. S. Kliger, “Nanosecond time-resolved polarization spectroscopies: tools for probing protein reaction mechanisms,” Methods 52(1), 3–11 (2010).
[CrossRef] [PubMed]

E. Chen, Y. Wen, J. W. Lewis, R. A. Goldbeck, D. S. Kliger, and C. E. M. Strauss, “Nanosecond laser temperature-jump optical rotatory dispersion: application to early events in protein folding/unfolding,” Rev. Sci. Instrum. 76(8), 083120 (2005).
[CrossRef]

R. M. Esquerra, J. W. Lewis, and D. S. Kliger, “An improved linear retarder for time-resolved circular dichroism studies,” Rev. Sci. Instrum. 68(3), 1372–1376 (1997).
[CrossRef]

D. B. Shapiro, R. A. Goldbeck, D. Che, R. M. Esquerra, S. J. Paquette, and D. S. Kliger, “Nanosecond optical rotatory dispersion spectroscopy: application to photolyzed hemoglobin-CO kinetics,” Biophys. J. 68(1), 326–334 (1995).
[CrossRef] [PubMed]

C. F. Zhang, J. W. Lewis, R. Cerpa, I. D. Kuntz, and D. S. Kliger, “Nanosecond circular dichroism spectral measurements - extension to the far-ultraviolet region,” J. Phys. Chem. 97(21), 5499–5505 (1993).
[CrossRef]

J. W. Lewis, R. A. Goldbeck, D. S. Kliger, X. Xie, R. C. Dunn, and J. D. Simon, “Time-resolved circular dichroism spectroscopy - experiment, theory, and applications to biological systems,” J. Phys. Chem. 96(13), 5243–5254 (1992).
[CrossRef]

Kolano, C.

C. Kolano, J. Helbing, M. Kozinski, W. Sander, and P. Hamm, “Watching hydrogen-bond dynamics in a β-turn by transient two-dimensional infrared spectroscopy,” Nature 444(7118), 469–472 (2006).
[CrossRef] [PubMed]

Kozinski, M.

C. Kolano, J. Helbing, M. Kozinski, W. Sander, and P. Hamm, “Watching hydrogen-bond dynamics in a β-turn by transient two-dimensional infrared spectroscopy,” Nature 444(7118), 469–472 (2006).
[CrossRef] [PubMed]

Kuntz, I. D.

C. F. Zhang, J. W. Lewis, R. Cerpa, I. D. Kuntz, and D. S. Kliger, “Nanosecond circular dichroism spectral measurements - extension to the far-ultraviolet region,” J. Phys. Chem. 97(21), 5499–5505 (1993).
[CrossRef]

Landin, B. L.

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,” Nature 458(7236), 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,” Nature 458(7236), 310–313 (2009).
[CrossRef] [PubMed]

Leone, S. R.

Lepetit, L.

Lewis, J. W.

E. Chen, Y. Wen, J. W. Lewis, R. A. Goldbeck, D. S. Kliger, and C. E. M. Strauss, “Nanosecond laser temperature-jump optical rotatory dispersion: application to early events in protein folding/unfolding,” Rev. Sci. Instrum. 76(8), 083120 (2005).
[CrossRef]

R. M. Esquerra, J. W. Lewis, and D. S. Kliger, “An improved linear retarder for time-resolved circular dichroism studies,” Rev. Sci. Instrum. 68(3), 1372–1376 (1997).
[CrossRef]

C. F. Zhang, J. W. Lewis, R. Cerpa, I. D. Kuntz, and D. S. Kliger, “Nanosecond circular dichroism spectral measurements - extension to the far-ultraviolet region,” J. Phys. Chem. 97(21), 5499–5505 (1993).
[CrossRef]

J. W. Lewis, R. A. Goldbeck, D. S. Kliger, X. Xie, R. C. Dunn, and J. D. Simon, “Time-resolved circular dichroism spectroscopy - experiment, theory, and applications to biological systems,” J. Phys. Chem. 96(13), 5243–5254 (1992).
[CrossRef]

Li, X.

Lim, S.-H.

Ling, Y. L.

S. H. Shim, D. B. Strasfeld, Y. L. Ling, and M. T. Zanni, “Automated 2D IR spectroscopy using a mid-IR pulse shaper and application of this technology to the human islet amyloid polypeptide,” Proc. Natl. Acad. Sci. U.S.A. 104(36), 14197–14202 (2007).
[CrossRef] [PubMed]

Lohr, A.

A. Trifonov, I. Buchvarov, A. Lohr, F. Würthner, and T. Fiebig, “Broadband femtosecond circular dichroism spectrometer with white-light polarization control,” Rev. Sci. Instrum. 81(4), 043104 (2010).
[CrossRef] [PubMed]

Loparo, J. J.

S. T. Roberts, J. J. Loparo, K. Ramasesha, and A. Tokmakoff, “A Fast-scanning Fourier transform 2D IR interferometer,” Opt. Commun. 284(4), 1062–1066 (2011).
[CrossRef]

Miller, R. J. D.

G. D. Goodno and R. J. D. Miller, “Femtosecond heterodyne-detected four-wave-mixing studies of deterministic protein motions. 1. theory and experimental technique of diffractive optics-based spectroscopy,” J. Phys. Chem. A 103(49), 10619–10629 (1999).
[CrossRef]

Niezborala, C.

C. Niezborala and F. Hache, “Conformational changes in photoexcited (R)-(+)-1,1′-bi-2-naphthol studied by time-resolved circular dichroism,” J. Am. Chem. Soc. 130(38), 12783–12786 (2008).
[CrossRef] [PubMed]

C. Niezborala and F. Hache, “Measuring the dynamics of circular dichroism in a pump-probe experiment with a Babinet-Soleil Compensator,” J. Opt. Soc. Am. B 23(11), 2418–2424 (2006).
[CrossRef]

Paquette, S. J.

D. B. Shapiro, R. A. Goldbeck, D. Che, R. M. Esquerra, S. J. Paquette, and D. S. Kliger, “Nanosecond optical rotatory dispersion spectroscopy: application to photolyzed hemoglobin-CO kinetics,” Biophys. J. 68(1), 326–334 (1995).
[CrossRef] [PubMed]

Rajaram, B.

Ramasesha, K.

S. T. Roberts, J. J. Loparo, K. Ramasesha, and A. Tokmakoff, “A Fast-scanning Fourier transform 2D IR interferometer,” Opt. Commun. 284(4), 1062–1066 (2011).
[CrossRef]

Rhee, H.

H. Rhee, S.-S. Kim, and M. Cho, “Multichannel array detection of vibrational optical activity free-induction-decay,” J. Anal. Sci. Technol. 1(2), 147–151 (2010).
[CrossRef]

H. Rhee, J.-H. Choi, and M. Cho, “Infrared optical activity: electric field approaches in time domain,” Acc. Chem. Res. 43(12), 1527–1536 (2010).
[CrossRef] [PubMed]

H. Rhee, S.-S. Kim, S.-J. Jeon, and M. Cho, “Femtosecond measurements of vibrational circular dichroism and optical rotatory dispersion spectra,” ChemPhysChem 10(13), 2209–2211 (2009).
[CrossRef] [PubMed]

H. Rhee, Y.-G. June, Z. H. Kim, S.-J. Jeon, and M. Cho, “Phase sensitive detection of vibrational optical activity free-induction-decay: vibrational CD and ORD,” J. Opt. Soc. Am. B 26(5), 1008–1017 (2009).
[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,” Nature 458(7236), 310–313 (2009).
[CrossRef] [PubMed]

H. Rhee, J.-H. Ha, S.-J. Jeon, and M. Cho, “Femtosecond spectral interferometry of optical activity: theory,” J. Chem. Phys. 129(9), 094507 (2008).
[CrossRef] [PubMed]

Roberts, S. T.

S. T. Roberts, J. J. Loparo, K. Ramasesha, and A. Tokmakoff, “A Fast-scanning Fourier transform 2D IR interferometer,” Opt. Commun. 284(4), 1062–1066 (2011).
[CrossRef]

Sander, W.

C. Kolano, J. Helbing, M. Kozinski, W. Sander, and P. Hamm, “Watching hydrogen-bond dynamics in a β-turn by transient two-dimensional infrared spectroscopy,” Nature 444(7118), 469–472 (2006).
[CrossRef] [PubMed]

Shapiro, D. B.

D. B. Shapiro, R. A. Goldbeck, D. Che, R. M. Esquerra, S. J. Paquette, and D. S. Kliger, “Nanosecond optical rotatory dispersion spectroscopy: application to photolyzed hemoglobin-CO kinetics,” Biophys. J. 68(1), 326–334 (1995).
[CrossRef] [PubMed]

Shim, S. H.

S. H. Shim, D. B. Strasfeld, Y. L. Ling, and M. T. Zanni, “Automated 2D IR spectroscopy using a mid-IR pulse shaper and application of this technology to the human islet amyloid polypeptide,” Proc. Natl. Acad. Sci. U.S.A. 104(36), 14197–14202 (2007).
[CrossRef] [PubMed]

Simon, J. D.

J. W. Lewis, R. A. Goldbeck, D. S. Kliger, X. Xie, R. C. Dunn, and J. D. Simon, “Time-resolved circular dichroism spectroscopy - experiment, theory, and applications to biological systems,” J. Phys. Chem. 96(13), 5243–5254 (1992).
[CrossRef]

X. Xie and J. D. Simon, “Picosecond time-resolved circular dichroism study of protein relaxation in myoglobin following photodissociation of carbon monoxide,” J. Am. Chem. Soc. 112(21), 7802–7803 (1990).
[CrossRef]

Smirl, A. L.

Stenger, J.

T. Brixner, J. Stenger, H. M. Vaswani, M. Cho, R. E. Blankenship, and G. R. Fleming, “Two-dimensional spectroscopy of electronic couplings in photosynthesis,” Nature 434(7033), 625–628 (2005).
[CrossRef] [PubMed]

Stiopkin, I. V.

Strasfeld, D. B.

S. H. Shim, D. B. Strasfeld, Y. L. Ling, and M. T. Zanni, “Automated 2D IR spectroscopy using a mid-IR pulse shaper and application of this technology to the human islet amyloid polypeptide,” Proc. Natl. Acad. Sci. U.S.A. 104(36), 14197–14202 (2007).
[CrossRef] [PubMed]

Strauss, C. E. M.

E. Chen, Y. Wen, J. W. Lewis, R. A. Goldbeck, D. S. Kliger, and C. E. M. Strauss, “Nanosecond laser temperature-jump optical rotatory dispersion: application to early events in protein folding/unfolding,” Rev. Sci. Instrum. 76(8), 083120 (2005).
[CrossRef]

Tokmakoff, A.

S. T. Roberts, J. J. Loparo, K. Ramasesha, and A. Tokmakoff, “A Fast-scanning Fourier transform 2D IR interferometer,” Opt. Commun. 284(4), 1062–1066 (2011).
[CrossRef]

Trebino, R.

Trifonov, A.

A. Trifonov, I. Buchvarov, A. Lohr, F. Würthner, and T. Fiebig, “Broadband femtosecond circular dichroism spectrometer with white-light polarization control,” Rev. Sci. Instrum. 81(4), 043104 (2010).
[CrossRef] [PubMed]

Vaswani, H. M.

T. Brixner, J. Stenger, H. M. Vaswani, M. Cho, R. E. Blankenship, and G. R. Fleming, “Two-dimensional spectroscopy of electronic couplings in photosynthesis,” Nature 434(7033), 625–628 (2005).
[CrossRef] [PubMed]

Walecki, W. J.

Wen, Y.

E. Chen, Y. Wen, J. W. Lewis, R. A. Goldbeck, D. S. Kliger, and C. E. M. Strauss, “Nanosecond laser temperature-jump optical rotatory dispersion: application to early events in protein folding/unfolding,” Rev. Sci. Instrum. 76(8), 083120 (2005).
[CrossRef]

Würthner, F.

A. Trifonov, I. Buchvarov, A. Lohr, F. Würthner, and T. Fiebig, “Broadband femtosecond circular dichroism spectrometer with white-light polarization control,” Rev. Sci. Instrum. 81(4), 043104 (2010).
[CrossRef] [PubMed]

Xie, X.

J. W. Lewis, R. A. Goldbeck, D. S. Kliger, X. Xie, R. C. Dunn, and J. D. Simon, “Time-resolved circular dichroism spectroscopy - experiment, theory, and applications to biological systems,” J. Phys. Chem. 96(13), 5243–5254 (1992).
[CrossRef]

X. Xie and J. D. Simon, “Picosecond time-resolved circular dichroism study of protein relaxation in myoglobin following photodissociation of carbon monoxide,” J. Am. Chem. Soc. 112(21), 7802–7803 (1990).
[CrossRef]

Zanni, M. T.

S. H. Shim, D. B. Strasfeld, Y. L. Ling, and M. T. Zanni, “Automated 2D IR spectroscopy using a mid-IR pulse shaper and application of this technology to the human islet amyloid polypeptide,” Proc. Natl. Acad. Sci. U.S.A. 104(36), 14197–14202 (2007).
[CrossRef] [PubMed]

M. T. Zanni, N. H. Ge, Y. S. Kim, and R. M. Hochstrasser, “Two-dimensional IR spectroscopy can be designed to eliminate the diagonal peaks and expose only the crosspeaks needed for structure determination,” Proc. Natl. Acad. Sci. U.S.A. 98(20), 11265–11270 (2001).
[CrossRef] [PubMed]

Zhang, C. F.

C. F. Zhang, J. W. Lewis, R. Cerpa, I. D. Kuntz, and D. S. Kliger, “Nanosecond circular dichroism spectral measurements - extension to the far-ultraviolet region,” J. Phys. Chem. 97(21), 5499–5505 (1993).
[CrossRef]

Zhang, T.

Acc. Chem. Res. (2)

H. Rhee, J.-H. Choi, and M. Cho, “Infrared optical activity: electric field approaches in time domain,” Acc. Chem. Res. 43(12), 1527–1536 (2010).
[CrossRef] [PubMed]

K. C. Hannah and B. A. Armitage, “DNA-templated assembly of helical cyanine dye aggregates: a supramolecular chain polymerization,” Acc. Chem. Res. 37(11), 845–853 (2004).
[CrossRef] [PubMed]

Biophys. J. (1)

D. B. Shapiro, R. A. Goldbeck, D. Che, R. M. Esquerra, S. J. Paquette, and D. S. Kliger, “Nanosecond optical rotatory dispersion spectroscopy: application to photolyzed hemoglobin-CO kinetics,” Biophys. J. 68(1), 326–334 (1995).
[CrossRef] [PubMed]

Chem. Phys. Lett. (1)

T. Dartigalongue and F. Hache, “Observation of sub-100 ps conformational changes in photolyzed carbonmonoxy-myoglobin probed by time-resolved circular dichroism,” Chem. Phys. Lett. 415(4-6), 313–316 (2005).
[CrossRef]

ChemPhysChem (1)

H. Rhee, S.-S. Kim, S.-J. Jeon, and M. Cho, “Femtosecond measurements of vibrational circular dichroism and optical rotatory dispersion spectra,” ChemPhysChem 10(13), 2209–2211 (2009).
[CrossRef] [PubMed]

J. Am. Chem. Soc. (2)

X. Xie and J. D. Simon, “Picosecond time-resolved circular dichroism study of protein relaxation in myoglobin following photodissociation of carbon monoxide,” J. Am. Chem. Soc. 112(21), 7802–7803 (1990).
[CrossRef]

C. Niezborala and F. Hache, “Conformational changes in photoexcited (R)-(+)-1,1′-bi-2-naphthol studied by time-resolved circular dichroism,” J. Am. Chem. Soc. 130(38), 12783–12786 (2008).
[CrossRef] [PubMed]

J. Anal. Sci. Technol. (1)

H. Rhee, S.-S. Kim, and M. Cho, “Multichannel array detection of vibrational optical activity free-induction-decay,” J. Anal. Sci. Technol. 1(2), 147–151 (2010).
[CrossRef]

J. Chem. Phys. (2)

J. Helbing and M. Bonmarin, “Vibrational circular dichroism signal enhancement using self-heterodyning with elliptically polarized laser pulses,” J. Chem. Phys. 131(17), 174507 (2009).
[CrossRef] [PubMed]

H. Rhee, J.-H. Ha, S.-J. Jeon, and M. Cho, “Femtosecond spectral interferometry of optical activity: theory,” J. Chem. Phys. 129(9), 094507 (2008).
[CrossRef] [PubMed]

J. Opt. Soc. Am. B (4)

J. Phys. Chem. (2)

J. W. Lewis, R. A. Goldbeck, D. S. Kliger, X. Xie, R. C. Dunn, and J. D. Simon, “Time-resolved circular dichroism spectroscopy - experiment, theory, and applications to biological systems,” J. Phys. Chem. 96(13), 5243–5254 (1992).
[CrossRef]

C. F. Zhang, J. W. Lewis, R. Cerpa, I. D. Kuntz, and D. S. Kliger, “Nanosecond circular dichroism spectral measurements - extension to the far-ultraviolet region,” J. Phys. Chem. 97(21), 5499–5505 (1993).
[CrossRef]

J. Phys. Chem. A (1)

G. D. Goodno and R. J. D. Miller, “Femtosecond heterodyne-detected four-wave-mixing studies of deterministic protein motions. 1. theory and experimental technique of diffractive optics-based spectroscopy,” J. Phys. Chem. A 103(49), 10619–10629 (1999).
[CrossRef]

Methods (1)

E. Chen, R. A. Goldbeck, and D. S. Kliger, “Nanosecond time-resolved polarization spectroscopies: tools for probing protein reaction mechanisms,” Methods 52(1), 3–11 (2010).
[CrossRef] [PubMed]

Nature (3)

T. Brixner, J. Stenger, H. M. Vaswani, M. Cho, R. E. Blankenship, and G. R. Fleming, “Two-dimensional spectroscopy of electronic couplings in photosynthesis,” Nature 434(7033), 625–628 (2005).
[CrossRef] [PubMed]

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,” Nature 458(7236), 310–313 (2009).
[CrossRef] [PubMed]

C. Kolano, J. Helbing, M. Kozinski, W. Sander, and P. Hamm, “Watching hydrogen-bond dynamics in a β-turn by transient two-dimensional infrared spectroscopy,” Nature 444(7118), 469–472 (2006).
[CrossRef] [PubMed]

Opt. Commun. (1)

S. T. Roberts, J. J. Loparo, K. Ramasesha, and A. Tokmakoff, “A Fast-scanning Fourier transform 2D IR interferometer,” Opt. Commun. 284(4), 1062–1066 (2011).
[CrossRef]

Opt. Express (1)

Opt. Lett. (5)

Proc. Natl. Acad. Sci. U.S.A. (2)

S. H. Shim, D. B. Strasfeld, Y. L. Ling, and M. T. Zanni, “Automated 2D IR spectroscopy using a mid-IR pulse shaper and application of this technology to the human islet amyloid polypeptide,” Proc. Natl. Acad. Sci. U.S.A. 104(36), 14197–14202 (2007).
[CrossRef] [PubMed]

M. T. Zanni, N. H. Ge, Y. S. Kim, and R. M. Hochstrasser, “Two-dimensional IR spectroscopy can be designed to eliminate the diagonal peaks and expose only the crosspeaks needed for structure determination,” Proc. Natl. Acad. Sci. U.S.A. 98(20), 11265–11270 (2001).
[CrossRef] [PubMed]

Rev. Sci. Instrum. (3)

R. M. Esquerra, J. W. Lewis, and D. S. Kliger, “An improved linear retarder for time-resolved circular dichroism studies,” Rev. Sci. Instrum. 68(3), 1372–1376 (1997).
[CrossRef]

A. Trifonov, I. Buchvarov, A. Lohr, F. Würthner, and T. Fiebig, “Broadband femtosecond circular dichroism spectrometer with white-light polarization control,” Rev. Sci. Instrum. 81(4), 043104 (2010).
[CrossRef] [PubMed]

E. Chen, Y. Wen, J. W. Lewis, R. A. Goldbeck, D. S. Kliger, and C. E. M. Strauss, “Nanosecond laser temperature-jump optical rotatory dispersion: application to early events in protein folding/unfolding,” Rev. Sci. Instrum. 76(8), 083120 (2005).
[CrossRef]

Other (1)

N. Berova, K. Nakanishi, and R. W. Woody, Circular Dichroism: Principles and Applications, 2nd ed. (Wiley-VCH, 2000).

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Figures (6)

Fig. 1
Fig. 1

Schematic representations of principles of (a) OA FID method and the self-heterodyning (b) CD, (c) ORD measurements. PBS: polarizing beam splitter.

Fig. 2
Fig. 2

Experimental layout of the self-heterodyned electronic CD/ORD spectrometer. (Inset: supercontinuum spectra generated from water) P: polarizer (Glan-Taylor), ND: neutral density filter, SP: strain plate, CS: chiral sample, PBS: polarizing beam splitter (Glan-Thomson). For the ORD measurement, the SP is removed and the P is rotated by δ with respect to the analyzer (PBS).

Fig. 3
Fig. 3

(a) Electronic CD spectra of 22.5 mM Ni-( ± )-(tartrate)2 solutions measured by using the self-heterodyned method at θ ~1/10. (Inset: CD spectra measured by commercial spectrometer) Ni-( + )-(tartrate)2 was prepared from mixture of aqueous solution of NiSO4 and potassium sodium tartrate. Similarly, D-tartaric acid was used for Ni-(―)-(tartrate)2. (b) Electronic ORD spectra measured at δ = + 5 deg for several concentrations of the sample solutions. Frequency-dependent optical rotation directly measured by using analyzer angle scan (45 mM) is overlaid in upper panel for comparison (■).

Fig. 4
Fig. 4

Averaged ORD spectra of 22.5 mM Ni-( + )-(tartrate)2 solution at δ = + 5 deg for several multiple laser shots (1000, 100, 10 and 1).

Fig. 5
Fig. 5

(a) Enhanced ORD signals recorded as a function of δ for Ni-( ± )-(tartrate)2 solutions (□ and ●). It is noted that the second harmonic was used instead of the SC. (b) S/N ratio of the ORD signals monitored at 402 nm. Each point was calculated for 1000 pulsed acquisitions. Dashed line indicates the predicted enhancement (1/θ).

Fig. 6
Fig. 6

Electronic CD (━) and ORD (—) spectra measured at θ ~1/10 and δ = + 5 deg, respectively for an aqueous buffer solution of DNA-cyanine dye aggregate. DNA templated dye aggregates (top panel) were prepared by addition of small volume of concentrated DNA buffer solution (pH = 7.0) to the aqueous solution of cyanine dye. Sample concentration was controlled to absorbance ~0.6 at 600 nm.

Equations (7)

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

E ( ω ) E ( ω ) = [ π ω L c n ( ω ) ] Δ χ ( ω ) ,
Δ A ( ω ) = Δ κ L log 10 = 4 π ω c n ( ω ) Im [ Δ χ ( ω ) ] L log 10 = ( 4 log 10 ) Im [ E ( ω ) E ( ω ) ]
Δ φ ( ω ) = Δ n ω L 2 c = π ω L c n ( ω ) Re [ Δ χ ( ω ) ] = Re [ E ( ω ) E ( ω ) ] ,
S ( ω ) S ( ω ) = | E ( ω ) + E L O ( ω ) | 2 | E ( ω ) | 2 = | E ( ω ) | 2 + | E L O ( ω ) | 2 + 2 Re [ E ( ω ) E L O * ( ω ) ] | E ( ω ) | 2 .
S ( ω ) S ( ω ) θ 2 + 2 θ Re ( [ Δ φ ( ω ) + i { Δ A ( ω ) log 10 4 } ] e i Δ ϕ ( ω ) ) ,
S ( ω ) S a t t ( ω ) = S ( ω ) θ 2 S ( ω ) 1 + 2 θ Re ( [ Δ φ ( ω ) + i { Δ A ( ω ) log 10 4 } ] e i Δ ϕ ( ω ) ) .
log [ S ( ω ) S a t t ( ω ) ] 2 θ Re ( [ Δ φ ( ω ) + i { Δ A ( ω ) log 10 4 } ] e i Δ ϕ ( ω ) ) .

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