Abstract

The full development of mono- or multi-dimensional time-resolved spectroscopy techniques incorporating optical activity signals has been strongly hampered by the challenge of identifying the small chiral signals over the large achiral background. Here we propose a new methodology to isolate chiral signals removing the achiral background from two commonly used configurations for performing two-dimensional optical spectroscopy, known as BOXCARS and gradient assisted photon echo spectroscopy (GRAPES). It is found that in both cases an achiral signal from an isotropic system can be completely eliminated by small manipulations of the relative angles between the linear polarizations of the four input laser pulses. Starting from the formulation of a perturbative expansion of the signal in the angle between the beams and the propagation axis, we derive analytic expressions that can be used to estimate how to change the polarization angles of the four pulses to minimize achiral contributions in the studied configurations. The generalization to any other possible experimental configurations has also been discussed.

© 2017 Optical Society of America

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References

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2016 (1)

D. I. H. Holdaway, E. Collini, and A. Olaya-Castro, “Coherence specific signal detection via chiral pump-probe spectroscopy,” J. Chem. Phys. 144, 194112 (2016).
[Crossref] [PubMed]

2015 (3)

Z. Zhang, P. H. Lambrev, K. L. Wells, G. Garab, and H.-S. Tan, “Direct observation of multistep energy transfer in LHCII with fifth-order 3D electronic spectroscopy,” Nature Comm. 6, 7914 (2015).
[Crossref]

J. Lim, D. Paleček, F. Caycedo-Soler, C. N. Lincoln, J. Prior, H. von Berlepsch, S. F. Huelga, M. B. Plenio, D. Zigmantas, and J. Hauer, “Vibronic origin of long-lived coherence in an artificial molecular light harvester,” Nature Comm. 6, 7755 (2015).
[Crossref]

F. D. Fuller and J. P. Ogilvie, “Experimental Implementations of Two-Dimensional Fourier Transform Electronic Spectroscopy,” Annu. Rev. Phys. Chem. 66, 667–690 (2015).
[Crossref] [PubMed]

2014 (2)

A. F. Fidler, V. P. Singh, P. D. Long, P. D. Dahlberg, and G. S. Engel, “Dynamic localization of electronic excitation in photosynthetic complexes revealed with chiral two-dimensional spectroscopy,” Nature Comm. 5, 3286 (2014).
[Crossref]

N. Mann, P. Nalbach, S. Mukamel, and M. Thorwart, “Probing chirality fluctuations in molecules by nonlinear optical spectroscopy,” J. Chem. Phys. 141, 234305 (2014).
[Crossref] [PubMed]

2012 (3)

H. Rhee, I. Eom, S.-H. Ahn, and M. Cho, “Coherent electric field characterization of molecular chirality in the time domain,” Chem. Soc. Rev. 41, 4457–4466 (2012).
[Crossref] [PubMed]

M. Ren, E. Plum, J. Xu, and N. I. Zheludev, “Giant nonlinear optical activity in a plasmonic metamaterial,” Nature Comm. 3, 833 (2012).
[Crossref]

S. Furumaki, Y. Yabiku, S. Habuchi, Y. Tsukatani, D. A. Bryant, and M. Vacha, “Circular dichroism measured on single chlorosomal light-harvesting complexes of green photosynthetic bacteria,” J. Phys. Chem. Lett. 3, 3545–3549 (2012).
[Crossref] [PubMed]

2011 (2)

E. Harel, P. D. Long, and G. S. Engel, “Single-shot ultrabroadband two-dimensional electronic spectroscopy of the light-harvesting complex LH2,” Opt. Lett. 36, 1665–1667 (2011).
[Crossref] [PubMed]

G. S. Schlau-Cohen, A. Ishizaki, and G. R. Fleming, “Two-dimensional electronic spectroscopy and photosynthesis: Fundamentals and applications to photosynthetic light-harvesting,” Chem. Phys. 387, 1–22 (2011).
[Crossref]

2010 (1)

E. Harel, A. F. Fidler, and G. S. Engel, “Real-time mapping of electronic structure with single-shot two-dimensional electronic spectroscopy,” Proc. Natl. Acad. Sci. U.S.A. 107, 16444–16447 (2010).
[Crossref] [PubMed]

2009 (2)

F. Hache, M.-T. Khuc, J. Brazard, P. Plaza, M. M. Martin, G. Checcucci, and F. Lenci, “Picosecond transient circular dichroism of the photoreceptor protein of the light-adapted form of Blepharisma japonicum,” Chem. Phys. Lett. 483, 133–137 (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, 310–313 (2009).
[Crossref] [PubMed]

2008 (1)

H. S. Tan, “Theory and phase-cycling scheme selection principles of collinear phase coherent multi-dimensional optical spectroscopy,” J. Chem. Phys. 129, 124501 (2008).
[Crossref] [PubMed]

2007 (1)

D. V. Voronine, D. Abramavicius, and S. Mukamel, “Manipulating multidimensional electronic spectra of excitons by polarization pulse shaping,” J. Chem. Phys. 126, 044508 (2007).
[Crossref] [PubMed]

2006 (4)

D. Abramavicius, W. Zhuang, and S. Mukamel, “Probing molecular chirality via excitonic nonlinear response,” J. Phys. B At Mol. Opt. Phys. 39, 5051–5066 (2006).
[Crossref]

S. Georgakopoulou, R. Van Grondelle, and G. Van Der Zwan, “Explaining the visible and near-infrared circular dichroism spectra of light-harvesting 1 complexes from purple bacteria: A modeling study,” J. Phys. Chem. B 110, 3344–3353 (2006).
[Crossref] [PubMed]

D. Abramavicius and S. Mukamel, “Chirality-induced signals in coherent multidimensional spectroscopy of excitons,” J. Chem. Phys. 124, 034113 (2006).
[Crossref] [PubMed]

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

2004 (1)

J. B. Pendry, “A chiral route to negative refraction,” Science 306, 1353–1355 (2004).
[Crossref] [PubMed]

2003 (1)

M. Cho, “Two-dimensional circularly polarized pump-probe spectroscopy,” J. Chem. Phys. 119, 7003–7016 (2003).
[Crossref]

1997 (1)

G. G. C. Büchel, “Organization of the pigment molecules in the chlorophyll a/c light-harvesting complex of Pleurochloris meiringensis (xanthophyceae). Characterization with circular dichroism and absorbance spectroscopy,” J. Photochem. Photobiol. 37, 118–124 (1997).
[Crossref]

1992 (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, 5243–5254 (1992).
[Crossref]

P. W. Hemelrijk, S. L. Kwa, R. van Grondelle, and J. P. Dekker, “Spectroscopic properties of LHC-II, the main light-harvesting chlorophyll a/b protein complex from chloroplast membranes,” Biochimica et Biophysica Acta 1098, 159–166 (1992).
[Crossref]

1987 (1)

I. Tinoco, W. Mickols, M. F. Maestre, and C. Bustamante, “Absorption, scattering, and imaging of biomolecular structures with polarized light,” Ann. Rev. Biophys. 16, 319–349 (1987).
[Crossref]

1982 (1)

G. Wagnière, “The evaluation of three-dimensional rotational averages,” J. Chem. Phys. 76, 473–480 (1982).
[Crossref]

Abramavicius, D.

D. V. Voronine, D. Abramavicius, and S. Mukamel, “Manipulating multidimensional electronic spectra of excitons by polarization pulse shaping,” J. Chem. Phys. 126, 044508 (2007).
[Crossref] [PubMed]

D. Abramavicius, W. Zhuang, and S. Mukamel, “Probing molecular chirality via excitonic nonlinear response,” J. Phys. B At Mol. Opt. Phys. 39, 5051–5066 (2006).
[Crossref]

D. Abramavicius and S. Mukamel, “Chirality-induced signals in coherent multidimensional spectroscopy of excitons,” J. Chem. Phys. 124, 034113 (2006).
[Crossref] [PubMed]

Ahn, S.-H.

H. Rhee, I. Eom, S.-H. Ahn, and M. Cho, “Coherent electric field characterization of molecular chirality in the time domain,” Chem. Soc. Rev. 41, 4457–4466 (2012).
[Crossref] [PubMed]

Brazard, J.

F. Hache, M.-T. Khuc, J. Brazard, P. Plaza, M. M. Martin, G. Checcucci, and F. Lenci, “Picosecond transient circular dichroism of the photoreceptor protein of the light-adapted form of Blepharisma japonicum,” Chem. Phys. Lett. 483, 133–137 (2009).
[Crossref]

Bryant, D. A.

S. Furumaki, Y. Yabiku, S. Habuchi, Y. Tsukatani, D. A. Bryant, and M. Vacha, “Circular dichroism measured on single chlorosomal light-harvesting complexes of green photosynthetic bacteria,” J. Phys. Chem. Lett. 3, 3545–3549 (2012).
[Crossref] [PubMed]

Büchel, G. G. C.

G. G. C. Büchel, “Organization of the pigment molecules in the chlorophyll a/c light-harvesting complex of Pleurochloris meiringensis (xanthophyceae). Characterization with circular dichroism and absorbance spectroscopy,” J. Photochem. Photobiol. 37, 118–124 (1997).
[Crossref]

Bustamante, C.

I. Tinoco, W. Mickols, M. F. Maestre, and C. Bustamante, “Absorption, scattering, and imaging of biomolecular structures with polarized light,” Ann. Rev. Biophys. 16, 319–349 (1987).
[Crossref]

Caycedo-Soler, F.

J. Lim, D. Paleček, F. Caycedo-Soler, C. N. Lincoln, J. Prior, H. von Berlepsch, S. F. Huelga, M. B. Plenio, D. Zigmantas, and J. Hauer, “Vibronic origin of long-lived coherence in an artificial molecular light harvester,” Nature Comm. 6, 7755 (2015).
[Crossref]

Checcucci, G.

F. Hache, M.-T. Khuc, J. Brazard, P. Plaza, M. M. Martin, G. Checcucci, and F. Lenci, “Picosecond transient circular dichroism of the photoreceptor protein of the light-adapted form of Blepharisma japonicum,” Chem. Phys. Lett. 483, 133–137 (2009).
[Crossref]

Cho, M.

H. Rhee, I. Eom, S.-H. Ahn, and M. Cho, “Coherent electric field characterization of molecular chirality in the time domain,” Chem. Soc. Rev. 41, 4457–4466 (2012).
[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, 310–313 (2009).
[Crossref] [PubMed]

M. Cho, “Two-dimensional circularly polarized pump-probe spectroscopy,” J. Chem. Phys. 119, 7003–7016 (2003).
[Crossref]

Collini, E.

D. I. H. Holdaway, E. Collini, and A. Olaya-Castro, “Coherence specific signal detection via chiral pump-probe spectroscopy,” J. Chem. Phys. 144, 194112 (2016).
[Crossref] [PubMed]

Dahlberg, P. D.

A. F. Fidler, V. P. Singh, P. D. Long, P. D. Dahlberg, and G. S. Engel, “Dynamic localization of electronic excitation in photosynthetic complexes revealed with chiral two-dimensional spectroscopy,” Nature Comm. 5, 3286 (2014).
[Crossref]

Dekker, J. P.

P. W. Hemelrijk, S. L. Kwa, R. van Grondelle, and J. P. Dekker, “Spectroscopic properties of LHC-II, the main light-harvesting chlorophyll a/b protein complex from chloroplast membranes,” Biochimica et Biophysica Acta 1098, 159–166 (1992).
[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, 5243–5254 (1992).
[Crossref]

Engel, G. S.

A. F. Fidler, V. P. Singh, P. D. Long, P. D. Dahlberg, and G. S. Engel, “Dynamic localization of electronic excitation in photosynthetic complexes revealed with chiral two-dimensional spectroscopy,” Nature Comm. 5, 3286 (2014).
[Crossref]

E. Harel, P. D. Long, and G. S. Engel, “Single-shot ultrabroadband two-dimensional electronic spectroscopy of the light-harvesting complex LH2,” Opt. Lett. 36, 1665–1667 (2011).
[Crossref] [PubMed]

E. Harel, A. F. Fidler, and G. S. Engel, “Real-time mapping of electronic structure with single-shot two-dimensional electronic spectroscopy,” Proc. Natl. Acad. Sci. U.S.A. 107, 16444–16447 (2010).
[Crossref] [PubMed]

Eom, I.

H. Rhee, I. Eom, S.-H. Ahn, and M. Cho, “Coherent electric field characterization of molecular chirality in the time domain,” Chem. Soc. Rev. 41, 4457–4466 (2012).
[Crossref] [PubMed]

Fasman, G. D.

G. D. Fasman, Circular Dichroism and the Conformational Analysis of Biomolecules (Plenum Press, 1996).
[Crossref]

Fidler, A. F.

A. F. Fidler, V. P. Singh, P. D. Long, P. D. Dahlberg, and G. S. Engel, “Dynamic localization of electronic excitation in photosynthetic complexes revealed with chiral two-dimensional spectroscopy,” Nature Comm. 5, 3286 (2014).
[Crossref]

E. Harel, A. F. Fidler, and G. S. Engel, “Real-time mapping of electronic structure with single-shot two-dimensional electronic spectroscopy,” Proc. Natl. Acad. Sci. U.S.A. 107, 16444–16447 (2010).
[Crossref] [PubMed]

Fleming, G. R.

G. S. Schlau-Cohen, A. Ishizaki, and G. R. Fleming, “Two-dimensional electronic spectroscopy and photosynthesis: Fundamentals and applications to photosynthetic light-harvesting,” Chem. Phys. 387, 1–22 (2011).
[Crossref]

Fuller, F. D.

F. D. Fuller and J. P. Ogilvie, “Experimental Implementations of Two-Dimensional Fourier Transform Electronic Spectroscopy,” Annu. Rev. Phys. Chem. 66, 667–690 (2015).
[Crossref] [PubMed]

Furumaki, S.

S. Furumaki, Y. Yabiku, S. Habuchi, Y. Tsukatani, D. A. Bryant, and M. Vacha, “Circular dichroism measured on single chlorosomal light-harvesting complexes of green photosynthetic bacteria,” J. Phys. Chem. Lett. 3, 3545–3549 (2012).
[Crossref] [PubMed]

Garab, G.

Z. Zhang, P. H. Lambrev, K. L. Wells, G. Garab, and H.-S. Tan, “Direct observation of multistep energy transfer in LHCII with fifth-order 3D electronic spectroscopy,” Nature Comm. 6, 7914 (2015).
[Crossref]

Georgakopoulou, S.

S. Georgakopoulou, R. Van Grondelle, and G. Van Der Zwan, “Explaining the visible and near-infrared circular dichroism spectra of light-harvesting 1 complexes from purple bacteria: A modeling study,” J. Phys. Chem. B 110, 3344–3353 (2006).
[Crossref] [PubMed]

Goldbeck, R. A.

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, 5243–5254 (1992).
[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, 310–313 (2009).
[Crossref] [PubMed]

Habuchi, S.

S. Furumaki, Y. Yabiku, S. Habuchi, Y. Tsukatani, D. A. Bryant, and M. Vacha, “Circular dichroism measured on single chlorosomal light-harvesting complexes of green photosynthetic bacteria,” J. Phys. Chem. Lett. 3, 3545–3549 (2012).
[Crossref] [PubMed]

Hache, F.

F. Hache, M.-T. Khuc, J. Brazard, P. Plaza, M. M. Martin, G. Checcucci, and F. Lenci, “Picosecond transient circular dichroism of the photoreceptor protein of the light-adapted form of Blepharisma japonicum,” Chem. Phys. Lett. 483, 133–137 (2009).
[Crossref]

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

Harel, E.

E. Harel, P. D. Long, and G. S. Engel, “Single-shot ultrabroadband two-dimensional electronic spectroscopy of the light-harvesting complex LH2,” Opt. Lett. 36, 1665–1667 (2011).
[Crossref] [PubMed]

E. Harel, A. F. Fidler, and G. S. Engel, “Real-time mapping of electronic structure with single-shot two-dimensional electronic spectroscopy,” Proc. Natl. Acad. Sci. U.S.A. 107, 16444–16447 (2010).
[Crossref] [PubMed]

Hauer, J.

J. Lim, D. Paleček, F. Caycedo-Soler, C. N. Lincoln, J. Prior, H. von Berlepsch, S. F. Huelga, M. B. Plenio, D. Zigmantas, and J. Hauer, “Vibronic origin of long-lived coherence in an artificial molecular light harvester,” Nature Comm. 6, 7755 (2015).
[Crossref]

Hemelrijk, P. W.

P. W. Hemelrijk, S. L. Kwa, R. van Grondelle, and J. P. Dekker, “Spectroscopic properties of LHC-II, the main light-harvesting chlorophyll a/b protein complex from chloroplast membranes,” Biochimica et Biophysica Acta 1098, 159–166 (1992).
[Crossref]

Holdaway, D. I. H.

D. I. H. Holdaway, E. Collini, and A. Olaya-Castro, “Coherence specific signal detection via chiral pump-probe spectroscopy,” J. Chem. Phys. 144, 194112 (2016).
[Crossref] [PubMed]

Huelga, S. F.

J. Lim, D. Paleček, F. Caycedo-Soler, C. N. Lincoln, J. Prior, H. von Berlepsch, S. F. Huelga, M. B. Plenio, D. Zigmantas, and J. Hauer, “Vibronic origin of long-lived coherence in an artificial molecular light harvester,” Nature Comm. 6, 7755 (2015).
[Crossref]

Ishizaki, A.

G. S. Schlau-Cohen, A. Ishizaki, and G. R. Fleming, “Two-dimensional electronic spectroscopy and photosynthesis: Fundamentals and applications to photosynthetic light-harvesting,” Chem. Phys. 387, 1–22 (2011).
[Crossref]

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, 310–313 (2009).
[Crossref] [PubMed]

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, 310–313 (2009).
[Crossref] [PubMed]

Khuc, M.-T.

F. Hache, M.-T. Khuc, J. Brazard, P. Plaza, M. M. Martin, G. Checcucci, and F. Lenci, “Picosecond transient circular dichroism of the photoreceptor protein of the light-adapted form of Blepharisma japonicum,” Chem. Phys. Lett. 483, 133–137 (2009).
[Crossref]

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

Kliger, D. S.

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, 5243–5254 (1992).
[Crossref]

Kwa, S. L.

P. W. Hemelrijk, S. L. Kwa, R. van Grondelle, and J. P. Dekker, “Spectroscopic properties of LHC-II, the main light-harvesting chlorophyll a/b protein complex from chloroplast membranes,” Biochimica et Biophysica Acta 1098, 159–166 (1992).
[Crossref]

Lambrev, P. H.

Z. Zhang, P. H. Lambrev, K. L. Wells, G. Garab, and H.-S. Tan, “Direct observation of multistep energy transfer in LHCII with fifth-order 3D electronic spectroscopy,” Nature Comm. 6, 7914 (2015).
[Crossref]

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, 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, 310–313 (2009).
[Crossref] [PubMed]

Lenci, F.

F. Hache, M.-T. Khuc, J. Brazard, P. Plaza, M. M. Martin, G. Checcucci, and F. Lenci, “Picosecond transient circular dichroism of the photoreceptor protein of the light-adapted form of Blepharisma japonicum,” Chem. Phys. Lett. 483, 133–137 (2009).
[Crossref]

Lewis, J. W.

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, 5243–5254 (1992).
[Crossref]

Lim, J.

J. Lim, D. Paleček, F. Caycedo-Soler, C. N. Lincoln, J. Prior, H. von Berlepsch, S. F. Huelga, M. B. Plenio, D. Zigmantas, and J. Hauer, “Vibronic origin of long-lived coherence in an artificial molecular light harvester,” Nature Comm. 6, 7755 (2015).
[Crossref]

Lincoln, C. N.

J. Lim, D. Paleček, F. Caycedo-Soler, C. N. Lincoln, J. Prior, H. von Berlepsch, S. F. Huelga, M. B. Plenio, D. Zigmantas, and J. Hauer, “Vibronic origin of long-lived coherence in an artificial molecular light harvester,” Nature Comm. 6, 7755 (2015).
[Crossref]

Long, P. D.

A. F. Fidler, V. P. Singh, P. D. Long, P. D. Dahlberg, and G. S. Engel, “Dynamic localization of electronic excitation in photosynthetic complexes revealed with chiral two-dimensional spectroscopy,” Nature Comm. 5, 3286 (2014).
[Crossref]

E. Harel, P. D. Long, and G. S. Engel, “Single-shot ultrabroadband two-dimensional electronic spectroscopy of the light-harvesting complex LH2,” Opt. Lett. 36, 1665–1667 (2011).
[Crossref] [PubMed]

Maestre, M. F.

I. Tinoco, W. Mickols, M. F. Maestre, and C. Bustamante, “Absorption, scattering, and imaging of biomolecular structures with polarized light,” Ann. Rev. Biophys. 16, 319–349 (1987).
[Crossref]

Mann, N.

N. Mann, P. Nalbach, S. Mukamel, and M. Thorwart, “Probing chirality fluctuations in molecules by nonlinear optical spectroscopy,” J. Chem. Phys. 141, 234305 (2014).
[Crossref] [PubMed]

Martin, M. M.

F. Hache, M.-T. Khuc, J. Brazard, P. Plaza, M. M. Martin, G. Checcucci, and F. Lenci, “Picosecond transient circular dichroism of the photoreceptor protein of the light-adapted form of Blepharisma japonicum,” Chem. Phys. Lett. 483, 133–137 (2009).
[Crossref]

Mickols, W.

I. Tinoco, W. Mickols, M. F. Maestre, and C. Bustamante, “Absorption, scattering, and imaging of biomolecular structures with polarized light,” Ann. Rev. Biophys. 16, 319–349 (1987).
[Crossref]

Mukamel, S.

N. Mann, P. Nalbach, S. Mukamel, and M. Thorwart, “Probing chirality fluctuations in molecules by nonlinear optical spectroscopy,” J. Chem. Phys. 141, 234305 (2014).
[Crossref] [PubMed]

D. V. Voronine, D. Abramavicius, and S. Mukamel, “Manipulating multidimensional electronic spectra of excitons by polarization pulse shaping,” J. Chem. Phys. 126, 044508 (2007).
[Crossref] [PubMed]

D. Abramavicius and S. Mukamel, “Chirality-induced signals in coherent multidimensional spectroscopy of excitons,” J. Chem. Phys. 124, 034113 (2006).
[Crossref] [PubMed]

D. Abramavicius, W. Zhuang, and S. Mukamel, “Probing molecular chirality via excitonic nonlinear response,” J. Phys. B At Mol. Opt. Phys. 39, 5051–5066 (2006).
[Crossref]

S. Mukamel, Principles of Nonlinear Optical Spectroscopy (Oxford University Press, 1999).

Nalbach, P.

N. Mann, P. Nalbach, S. Mukamel, and M. Thorwart, “Probing chirality fluctuations in molecules by nonlinear optical spectroscopy,” J. Chem. Phys. 141, 234305 (2014).
[Crossref] [PubMed]

Niezborala, C.

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

Ogilvie, J. P.

F. D. Fuller and J. P. Ogilvie, “Experimental Implementations of Two-Dimensional Fourier Transform Electronic Spectroscopy,” Annu. Rev. Phys. Chem. 66, 667–690 (2015).
[Crossref] [PubMed]

Olaya-Castro, A.

D. I. H. Holdaway, E. Collini, and A. Olaya-Castro, “Coherence specific signal detection via chiral pump-probe spectroscopy,” J. Chem. Phys. 144, 194112 (2016).
[Crossref] [PubMed]

Palecek, D.

J. Lim, D. Paleček, F. Caycedo-Soler, C. N. Lincoln, J. Prior, H. von Berlepsch, S. F. Huelga, M. B. Plenio, D. Zigmantas, and J. Hauer, “Vibronic origin of long-lived coherence in an artificial molecular light harvester,” Nature Comm. 6, 7755 (2015).
[Crossref]

Parson, W.

W. Parson, Modern Optical Spectroscopy (Springer, 2006).

Pendry, J. B.

J. B. Pendry, “A chiral route to negative refraction,” Science 306, 1353–1355 (2004).
[Crossref] [PubMed]

Philip Wong, D. A. H.-S.

D. A. H.-S. Philip Wong, Carbon Nanotube and Graphene Device Physics (Cambridge University Press, 2011).

Plaza, P.

F. Hache, M.-T. Khuc, J. Brazard, P. Plaza, M. M. Martin, G. Checcucci, and F. Lenci, “Picosecond transient circular dichroism of the photoreceptor protein of the light-adapted form of Blepharisma japonicum,” Chem. Phys. Lett. 483, 133–137 (2009).
[Crossref]

Plenio, M. B.

J. Lim, D. Paleček, F. Caycedo-Soler, C. N. Lincoln, J. Prior, H. von Berlepsch, S. F. Huelga, M. B. Plenio, D. Zigmantas, and J. Hauer, “Vibronic origin of long-lived coherence in an artificial molecular light harvester,” Nature Comm. 6, 7755 (2015).
[Crossref]

Plum, E.

M. Ren, E. Plum, J. Xu, and N. I. Zheludev, “Giant nonlinear optical activity in a plasmonic metamaterial,” Nature Comm. 3, 833 (2012).
[Crossref]

Prior, J.

J. Lim, D. Paleček, F. Caycedo-Soler, C. N. Lincoln, J. Prior, H. von Berlepsch, S. F. Huelga, M. B. Plenio, D. Zigmantas, and J. Hauer, “Vibronic origin of long-lived coherence in an artificial molecular light harvester,” Nature Comm. 6, 7755 (2015).
[Crossref]

Ren, M.

M. Ren, E. Plum, J. Xu, and N. I. Zheludev, “Giant nonlinear optical activity in a plasmonic metamaterial,” Nature Comm. 3, 833 (2012).
[Crossref]

Rhee, H.

H. Rhee, I. Eom, S.-H. Ahn, and M. Cho, “Coherent electric field characterization of molecular chirality in the time domain,” Chem. Soc. Rev. 41, 4457–4466 (2012).
[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, 310–313 (2009).
[Crossref] [PubMed]

Schlau-Cohen, G. S.

G. S. Schlau-Cohen, A. Ishizaki, and G. R. Fleming, “Two-dimensional electronic spectroscopy and photosynthesis: Fundamentals and applications to photosynthetic light-harvesting,” Chem. Phys. 387, 1–22 (2011).
[Crossref]

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, 5243–5254 (1992).
[Crossref]

Singh, V. P.

A. F. Fidler, V. P. Singh, P. D. Long, P. D. Dahlberg, and G. S. Engel, “Dynamic localization of electronic excitation in photosynthetic complexes revealed with chiral two-dimensional spectroscopy,” Nature Comm. 5, 3286 (2014).
[Crossref]

Tan, H. S.

H. S. Tan, “Theory and phase-cycling scheme selection principles of collinear phase coherent multi-dimensional optical spectroscopy,” J. Chem. Phys. 129, 124501 (2008).
[Crossref] [PubMed]

Tan, H.-S.

Z. Zhang, P. H. Lambrev, K. L. Wells, G. Garab, and H.-S. Tan, “Direct observation of multistep energy transfer in LHCII with fifth-order 3D electronic spectroscopy,” Nature Comm. 6, 7914 (2015).
[Crossref]

Thorwart, M.

N. Mann, P. Nalbach, S. Mukamel, and M. Thorwart, “Probing chirality fluctuations in molecules by nonlinear optical spectroscopy,” J. Chem. Phys. 141, 234305 (2014).
[Crossref] [PubMed]

Tinoco, I.

I. Tinoco, W. Mickols, M. F. Maestre, and C. Bustamante, “Absorption, scattering, and imaging of biomolecular structures with polarized light,” Ann. Rev. Biophys. 16, 319–349 (1987).
[Crossref]

Tsukatani, Y.

S. Furumaki, Y. Yabiku, S. Habuchi, Y. Tsukatani, D. A. Bryant, and M. Vacha, “Circular dichroism measured on single chlorosomal light-harvesting complexes of green photosynthetic bacteria,” J. Phys. Chem. Lett. 3, 3545–3549 (2012).
[Crossref] [PubMed]

Vacha, M.

S. Furumaki, Y. Yabiku, S. Habuchi, Y. Tsukatani, D. A. Bryant, and M. Vacha, “Circular dichroism measured on single chlorosomal light-harvesting complexes of green photosynthetic bacteria,” J. Phys. Chem. Lett. 3, 3545–3549 (2012).
[Crossref] [PubMed]

Van Der Zwan, G.

S. Georgakopoulou, R. Van Grondelle, and G. Van Der Zwan, “Explaining the visible and near-infrared circular dichroism spectra of light-harvesting 1 complexes from purple bacteria: A modeling study,” J. Phys. Chem. B 110, 3344–3353 (2006).
[Crossref] [PubMed]

Van Grondelle, R.

S. Georgakopoulou, R. Van Grondelle, and G. Van Der Zwan, “Explaining the visible and near-infrared circular dichroism spectra of light-harvesting 1 complexes from purple bacteria: A modeling study,” J. Phys. Chem. B 110, 3344–3353 (2006).
[Crossref] [PubMed]

P. W. Hemelrijk, S. L. Kwa, R. van Grondelle, and J. P. Dekker, “Spectroscopic properties of LHC-II, the main light-harvesting chlorophyll a/b protein complex from chloroplast membranes,” Biochimica et Biophysica Acta 1098, 159–166 (1992).
[Crossref]

von Berlepsch, H.

J. Lim, D. Paleček, F. Caycedo-Soler, C. N. Lincoln, J. Prior, H. von Berlepsch, S. F. Huelga, M. B. Plenio, D. Zigmantas, and J. Hauer, “Vibronic origin of long-lived coherence in an artificial molecular light harvester,” Nature Comm. 6, 7755 (2015).
[Crossref]

Voronine, D. V.

D. V. Voronine, D. Abramavicius, and S. Mukamel, “Manipulating multidimensional electronic spectra of excitons by polarization pulse shaping,” J. Chem. Phys. 126, 044508 (2007).
[Crossref] [PubMed]

Wagnière, G.

G. Wagnière, “The evaluation of three-dimensional rotational averages,” J. Chem. Phys. 76, 473–480 (1982).
[Crossref]

Wells, K. L.

Z. Zhang, P. H. Lambrev, K. L. Wells, G. Garab, and H.-S. Tan, “Direct observation of multistep energy transfer in LHCII with fifth-order 3D electronic spectroscopy,” Nature Comm. 6, 7914 (2015).
[Crossref]

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, 5243–5254 (1992).
[Crossref]

Xu, J.

M. Ren, E. Plum, J. Xu, and N. I. Zheludev, “Giant nonlinear optical activity in a plasmonic metamaterial,” Nature Comm. 3, 833 (2012).
[Crossref]

Yabiku, Y.

S. Furumaki, Y. Yabiku, S. Habuchi, Y. Tsukatani, D. A. Bryant, and M. Vacha, “Circular dichroism measured on single chlorosomal light-harvesting complexes of green photosynthetic bacteria,” J. Phys. Chem. Lett. 3, 3545–3549 (2012).
[Crossref] [PubMed]

Zhang, Z.

Z. Zhang, P. H. Lambrev, K. L. Wells, G. Garab, and H.-S. Tan, “Direct observation of multistep energy transfer in LHCII with fifth-order 3D electronic spectroscopy,” Nature Comm. 6, 7914 (2015).
[Crossref]

Zheludev, N. I.

M. Ren, E. Plum, J. Xu, and N. I. Zheludev, “Giant nonlinear optical activity in a plasmonic metamaterial,” Nature Comm. 3, 833 (2012).
[Crossref]

Zhuang, W.

D. Abramavicius, W. Zhuang, and S. Mukamel, “Probing molecular chirality via excitonic nonlinear response,” J. Phys. B At Mol. Opt. Phys. 39, 5051–5066 (2006).
[Crossref]

Zigmantas, D.

J. Lim, D. Paleček, F. Caycedo-Soler, C. N. Lincoln, J. Prior, H. von Berlepsch, S. F. Huelga, M. B. Plenio, D. Zigmantas, and J. Hauer, “Vibronic origin of long-lived coherence in an artificial molecular light harvester,” Nature Comm. 6, 7755 (2015).
[Crossref]

Ann. Rev. Biophys. (1)

I. Tinoco, W. Mickols, M. F. Maestre, and C. Bustamante, “Absorption, scattering, and imaging of biomolecular structures with polarized light,” Ann. Rev. Biophys. 16, 319–349 (1987).
[Crossref]

Annu. Rev. Phys. Chem. (1)

F. D. Fuller and J. P. Ogilvie, “Experimental Implementations of Two-Dimensional Fourier Transform Electronic Spectroscopy,” Annu. Rev. Phys. Chem. 66, 667–690 (2015).
[Crossref] [PubMed]

Biochimica et Biophysica Acta (1)

P. W. Hemelrijk, S. L. Kwa, R. van Grondelle, and J. P. Dekker, “Spectroscopic properties of LHC-II, the main light-harvesting chlorophyll a/b protein complex from chloroplast membranes,” Biochimica et Biophysica Acta 1098, 159–166 (1992).
[Crossref]

Chem. Phys. (1)

G. S. Schlau-Cohen, A. Ishizaki, and G. R. Fleming, “Two-dimensional electronic spectroscopy and photosynthesis: Fundamentals and applications to photosynthetic light-harvesting,” Chem. Phys. 387, 1–22 (2011).
[Crossref]

Chem. Phys. Lett. (1)

F. Hache, M.-T. Khuc, J. Brazard, P. Plaza, M. M. Martin, G. Checcucci, and F. Lenci, “Picosecond transient circular dichroism of the photoreceptor protein of the light-adapted form of Blepharisma japonicum,” Chem. Phys. Lett. 483, 133–137 (2009).
[Crossref]

Chem. Soc. Rev. (1)

H. Rhee, I. Eom, S.-H. Ahn, and M. Cho, “Coherent electric field characterization of molecular chirality in the time domain,” Chem. Soc. Rev. 41, 4457–4466 (2012).
[Crossref] [PubMed]

J. Chem. Phys. (7)

D. V. Voronine, D. Abramavicius, and S. Mukamel, “Manipulating multidimensional electronic spectra of excitons by polarization pulse shaping,” J. Chem. Phys. 126, 044508 (2007).
[Crossref] [PubMed]

N. Mann, P. Nalbach, S. Mukamel, and M. Thorwart, “Probing chirality fluctuations in molecules by nonlinear optical spectroscopy,” J. Chem. Phys. 141, 234305 (2014).
[Crossref] [PubMed]

D. I. H. Holdaway, E. Collini, and A. Olaya-Castro, “Coherence specific signal detection via chiral pump-probe spectroscopy,” J. Chem. Phys. 144, 194112 (2016).
[Crossref] [PubMed]

M. Cho, “Two-dimensional circularly polarized pump-probe spectroscopy,” J. Chem. Phys. 119, 7003–7016 (2003).
[Crossref]

G. Wagnière, “The evaluation of three-dimensional rotational averages,” J. Chem. Phys. 76, 473–480 (1982).
[Crossref]

H. S. Tan, “Theory and phase-cycling scheme selection principles of collinear phase coherent multi-dimensional optical spectroscopy,” J. Chem. Phys. 129, 124501 (2008).
[Crossref] [PubMed]

D. Abramavicius and S. Mukamel, “Chirality-induced signals in coherent multidimensional spectroscopy of excitons,” J. Chem. Phys. 124, 034113 (2006).
[Crossref] [PubMed]

J. Photochem. Photobiol. (1)

G. G. C. Büchel, “Organization of the pigment molecules in the chlorophyll a/c light-harvesting complex of Pleurochloris meiringensis (xanthophyceae). Characterization with circular dichroism and absorbance spectroscopy,” J. Photochem. Photobiol. 37, 118–124 (1997).
[Crossref]

J. Phys. B At Mol. Opt. Phys. (1)

D. Abramavicius, W. Zhuang, and S. Mukamel, “Probing molecular chirality via excitonic nonlinear response,” J. Phys. B At Mol. Opt. Phys. 39, 5051–5066 (2006).
[Crossref]

J. Phys. Chem. (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, 5243–5254 (1992).
[Crossref]

J. Phys. Chem. B (1)

S. Georgakopoulou, R. Van Grondelle, and G. Van Der Zwan, “Explaining the visible and near-infrared circular dichroism spectra of light-harvesting 1 complexes from purple bacteria: A modeling study,” J. Phys. Chem. B 110, 3344–3353 (2006).
[Crossref] [PubMed]

J. Phys. Chem. Lett. (1)

S. Furumaki, Y. Yabiku, S. Habuchi, Y. Tsukatani, D. A. Bryant, and M. Vacha, “Circular dichroism measured on single chlorosomal light-harvesting complexes of green photosynthetic bacteria,” J. Phys. Chem. Lett. 3, 3545–3549 (2012).
[Crossref] [PubMed]

Nature (1)

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, 310–313 (2009).
[Crossref] [PubMed]

Nature Comm. (4)

A. F. Fidler, V. P. Singh, P. D. Long, P. D. Dahlberg, and G. S. Engel, “Dynamic localization of electronic excitation in photosynthetic complexes revealed with chiral two-dimensional spectroscopy,” Nature Comm. 5, 3286 (2014).
[Crossref]

M. Ren, E. Plum, J. Xu, and N. I. Zheludev, “Giant nonlinear optical activity in a plasmonic metamaterial,” Nature Comm. 3, 833 (2012).
[Crossref]

Z. Zhang, P. H. Lambrev, K. L. Wells, G. Garab, and H.-S. Tan, “Direct observation of multistep energy transfer in LHCII with fifth-order 3D electronic spectroscopy,” Nature Comm. 6, 7914 (2015).
[Crossref]

J. Lim, D. Paleček, F. Caycedo-Soler, C. N. Lincoln, J. Prior, H. von Berlepsch, S. F. Huelga, M. B. Plenio, D. Zigmantas, and J. Hauer, “Vibronic origin of long-lived coherence in an artificial molecular light harvester,” Nature Comm. 6, 7755 (2015).
[Crossref]

Opt. Lett. (1)

Opt. Soc. Am. B (1)

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

Proc. Natl. Acad. Sci. U.S.A. (1)

E. Harel, A. F. Fidler, and G. S. Engel, “Real-time mapping of electronic structure with single-shot two-dimensional electronic spectroscopy,” Proc. Natl. Acad. Sci. U.S.A. 107, 16444–16447 (2010).
[Crossref] [PubMed]

Science (1)

J. B. Pendry, “A chiral route to negative refraction,” Science 306, 1353–1355 (2004).
[Crossref] [PubMed]

Other (5)

D. A. H.-S. Philip Wong, Carbon Nanotube and Graphene Device Physics (Cambridge University Press, 2011).

W. Parson, Modern Optical Spectroscopy (Springer, 2006).

G. D. Fasman, Circular Dichroism and the Conformational Analysis of Biomolecules (Plenum Press, 1996).
[Crossref]

R. W. W. E. Nina Berova, ed. NakanishiKoji, ed. Circular Dichroism: Principles and Applications, 2nd Edition (Wiley John and Sons Ltd, 2000).

S. Mukamel, Principles of Nonlinear Optical Spectroscopy (Oxford University Press, 1999).

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Figures (7)

Fig. 1
Fig. 1 Illustration of the physical meaning of the three rotation parameters in a lens based configuration. a) Side view, b) Front view, the red circle shows entry point of the pulse.
Fig. 2
Fig. 2 Scheme of the spatial arrangement of the four pulses labeled from 1 to 4 in 2DES measures performed with BOXCARS (red dots) and rephasing/non rephasing GRAPES (blue/green) configurations when looking along the primary axis of propagation z. dashed lines indicate the square/rectangle in the xy plane identified by the four beams and a midpoint at the focal point having of the four beams within the sample. Solid lines show the components of kj orthogonal to z and are labeled with the values of the absolute distance to the centre point in the xy plane.
Fig. 3
Fig. 3 Amplitude of Ξ̃, the contribution of achiral signal, for a range of Θ when correction different orders of correction are made. Note that 0.1 radians ≈ 5.73 degrees. Noise in the graph below 10−15 are down to elementary precision errors.
Fig. 4
Fig. 4 2D rephasing signal from an excitonic dimer (arbitrary units) in a BOXCARS configuration with Θ = π/18 (10 degrees) (a) Using the polarization shifts suggested in this work and (b) Without the shifts. In (b) the signal has significant achiral contributions with Ξ̃ ≈ 0.06.
Fig. 5
Fig. 5 Changes to angles between the four polarizations in the deformed {yxxx} GRAPES configuration, for ϕ = 23π/100 and a range of Θ, using the 4th order approximation. The small changes to the angles between p1 and the others three polarizations along with α34 are plotted on the left scale, where as α23 and α24 are plotted on the right scale.
Fig. 6
Fig. 6 2DES rephasing signal at zero population time, for an excitonic dimer in a GRAPES configuration with Θ = π/18 (10 degrees) and ϕ = 11π/90 in a warped chiral pump {yxxx} configuration. (a) Using the polarization shifts suggested in this work and (b) Without the shifts. The chiral pump configuration gives a significantly different signal to the chiral probe shown before, providing further information about the system.
Fig. 7
Fig. 7 Histogram of the amplitude of the achiral contribute, Θ̃, to our “chiral pump” GRAPES configuration at Θ = 10 degree, with all δθj having uniform random noise added with an amplitude of 1/120 degrees.

Tables (3)

Tables Icon

Table 1 Chiral pump shifts δθj removing achiral signal contributions to order Θ2 within the BOXCARS geometry, for the polarization configuration {yxxx}.

Tables Icon

Table 2 Factors (bj · pk)(p · pm) which occur in isotropic averages relevant to TDS with magnetic interactions, calculated to order Θ4.

Tables Icon

Table 3 Chiral pump shifts δθj which remove achiral signal contributions up to order Θ2, within the GRAPES geometry for the polarization configuration {yxxx}. For compactness, we have defined V ≡ 5Θ2 sin(2ϕ)/8 and U ≡ 3Θ2 sin(2ϕ)/8, which are used only in this table.

Equations (32)

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μ ^ ( R ) = α C α ( r ^ α R ) ,
m ^ ( R ) = α C α 2 m α c ( r ^ α R ) × q ^ α , Q ^ ( R ) = α C α 2 ( r ^ α R ) × ( r ^ α R ) .
H I = μ ^ E ( r , t ) [ m ^ B ( r , t ) + Q : E ( r , t ) ] ,
S j d t Im [ P j ( n ) ( t ) E n + 1 * ( t ) ] .
μ ^ = k μ k ( | 0 k k | + | k 0 k | ) + k k μ k ( | k k , k | + | k , k k | ) ,
( E 1 μ k 1 ) ( E n + 1 μ k n + 1 ) av = 0 2 π d α 0 π d β 0 2 π d γ n sin ( β ) 8 π 2 ( E 1 T ( α , β , γ ) μ k 1 ) ( E n + 1 T ( α , β , γ ) μ k n + 1 ) .
T ( α , β , γ ) = [ c 1 c 2 c 3 s 1 s 3 c 3 s 1 c 1 c 2 s 3 c 1 s 2 c 1 s 3 + c 2 c 3 s 1 c 1 c 3 c 2 s 1 s 3 s 1 s 2 c 3 s 2 s 2 s 3 c 2 , ]
μ 1 p 1 μ 2 p 2 μ 3 p 3 μ 4 p 4 iso = Ξ ( ( μ 1 μ 2 ) ( μ 3 μ 4 ) ( μ 1 μ 3 ) ( μ 2 μ 4 ) ( μ 3 μ 2 ) ( μ 1 μ 4 ) ) .
Ξ = 1 30 ( 4 1 1 1 4 1 1 1 4 ) ( ( p 1 p 2 ) ( p 3 p 4 ) ( p 1 p 3 ) ( p 2 p 4 ) ( p 1 p 4 ) ( p 2 p 3 ) ) ,
α j , k = cos 1 ( p j p k ) .
Ξ ˜ = | Ξ ( 1 ) | + | Ξ ( 2 ) | + | Ξ ( 3 ) | ,
k ^ 1 = ( k x k y k z ) , k ^ 2 = ( k x k y k z ) , k ^ 3 = ( k x k y k z ) , k ^ 4 = ( k x k y k z ) .
Ξ = Θ 4 192 ( ( 476 sin ( 2 ϕ ) + 76 sin ( 4 ϕ ) + 42 C sin ( 2 ϕ ) + 27 C sin ( 4 ϕ ) ) 484 sin ( 2 ϕ ) + 244 sin ( 4 ϕ ) + 78 C sin ( 2 ϕ ) + 33 C sin ( 4 ϕ ) 2 ( 212 sin ( 2 ϕ ) + 132 sin ( 4 ϕ ) 6 C sin ( 2 ϕ ) + 39 C sin ( 4 ϕ ) ) ) + O ( Θ 6 ) ,
H ^ ( t ) = d r [ J ^ ( r , t ) A ( r , t ) + Q ^ ( r , t ) : A ( r , t ) A ( r , t ) ] .
H ^ ( t ) d k J ^ ( k , t ) A ( k , t ) .
J ^ ( k , t ) = , a ( j ¯ a * ( k ) B ^ a + j ¯ a ( k ) B ^ a ) .
j ¯ a ( k ) = i e i k r j α C α ϕ a | ω [ ( r α r j ) i k ( r α r j ) ( r α r j ) / 2 + ] + k × [ ( r α r j ) × p α / 2 j α + ] | ϕ j g .
H ^ = | 1 1 | E 0 + | 2 2 | ( E 0 + Δ E ) + V | 1 2 | + | 1 , 2 1 , 2 | ( 2 E 0 + Δ E ) + h . c . ,
| ξ = k c k ( ξ ) | k ,
m ˜ ξ = i ( C 1 ( ξ ) μ 1 × R 1 + C 2 ( ξ ) μ 2 × R 2 ) = i ( C 1 ( ξ ) μ 1 × Δ R C 2 ( ξ ) μ 2 × Δ R ) / 2 .
t ρ ( t ) = i [ H ^ , ρ ( t ) ] + γ ( σ 1 ( ρ ) + σ 2 ( ρ ) ) ,
P ( r , t ) = d r 3 d t 3 d r 2 d t 2 d r 1 d t 1 E ( r 3 , t t 3 ) E ( r 2 , t t 3 t 2 ) E ( r 1 , t t 3 t 2 t 1 ) S ( t 3 , t 2 , t 1 ; r , r 3 , r 2 , r 1 ) .
Sig = 2 ω s Re d 3 r d t E LO ( r , t ) P ( r , t ) .
2 z 2 ( E x ( z , t ) E y ( z , t ) ) = 1 c 2 2 t 2 ( E x ( z , t ) + 4 π P x ( z , t ) E y ( z , t ) + 4 π P y ( z , t ) ) .
i k z ( E ˜ n x ( z , ω ω 0 ) E ˜ n y ( z , ω ω 0 ) ) = ( k 2 ω 2 c 2 ) ( E ˜ n x ( z , ω ω 0 ) E ˜ n y ( z , ω ω 0 ) ) 4 π ω 2 c 2 ( P ˜ n x ( z , ω ω 0 ) P ˜ n y ( z , ω ω 0 ) ) .
z E n a ( z , ω ω 0 ) 2 π ω n ( ω ) c [ Im ( P n a ( 1 ) ) ( z , ω ω 0 ) ] .
z ( E n x ( z , ω ω 0 ) E n y ( z , ω ω 0 ) ) 2 π ω n ( ω ) c ( Im [ S x x ( 1 ) ( ω ) ] i S x y ( 1 ) ( ω ) i S y x ( 1 ) ( ω ) Im [ S y y ( 1 ) ( ω ) ] ) ( E n x ( z , ω ω 0 ) E n y ( z , ω ω 0 ) ) .
( E n x ( z , ω ω 0 ) E n y ( z , ω ω 0 ) ) e α z 2 ( cosh ( β z 2 ) i sinh ( β z 2 ) i sinh ( β z 2 ) cosh ( β z 2 ) ) ( E n x ( 0 , ω ω 0 ) E n y ( 0 , ω ω 0 ) ) ,
α ( ω ) = ρ 4 π ω n ( ω ) c Im [ S x x ( 1 ) ( ω ) ]
β ( ω ) = ρ 4 π i ω n ( ω ) c S x y ( 1 ) ( ω ) .
p 1 ( L ) p 2 * ( L ) i [ sinh ( β L / 2 ) cosh ( β * L / 2 ) + cos ( θ ) sinh ( β * L / 2 ) cosh ( β L / 2 ) ] .
p 1 ( L ) p 2 * ( L ) = [ 1 cos ( θ ) ] sin ( δ L ) / 2 .

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