Abstract

In a typical two-dimensional electronic spectroscopy (2DES) experiment, the timing errors of the coherence and emission time when determining the absolute time zeros usually introduce extraneous spectral phase slopes and distort the 2D spectrum. In this work, a phase-correction method that merely relies on the data post-processing algorithm is proposed. The method allows reconstructing the spectrum by simply subtracting the artificial linear spectral-phase slopes from the phase component of the 2D spectrum along both coherence and emission frequency axes. The new method has the advantages of ease of implementation and no need for the supplementary experiments and iterative fitting algorithm as commonly-used phasing methods, which may improve the phasing issue in 2DES and serve as a cross-check of now available phasing methods.

© 2017 Optical Society of America

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References

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2016 (2)

Z. Huang, P. Wang, X. Shen, T.-M. Yan, Y. Zhang, and J. Liu, “Compact design for two-dimensional electronic spectroscopy,” Laser Phys. 26(3), 035403 (2016).
[Crossref]

Y. Zhang, T.-M. Yan, and Y. H. Jiang, “Precise phase determination with the built-in spectral interferometry in two-dimensional electronic spectroscopy,” Opt. Lett. 41(17), 4134–4137 (2016).
[Crossref] [PubMed]

2015 (1)

S. Yue, Z. Wang, X. He, G. Zhu, and Y. Weng, “Construction of the Apparatus for Two Dimensional Electronic Spectroscopy and Characterization of the Instrument,” Chin. J. Chem. Phys. 28(4), 509–517 (2015).
[Crossref]

2014 (1)

I. A. Heisler, R. Moca, F. V. Camargo, and S. R. Meech, “Two-dimensional electronic spectroscopy based on conventional optics and fast dual chopper data acquisition,” Rev. Sci. Instrum. 85(6), 063103 (2014).
[Crossref] [PubMed]

2013 (1)

2012 (1)

J. M. Anna, E. E. Ostroumov, K. Maghlaoui, J. Barber, and G. D. Scholes, “Two-dimensional electronic spectroscopy reveals ultrafast downhill energy transfer in photosystem I trimers of the cyanobacterium Thermosynechococcus elongatus,” J. Phys. Chem. Lett. 3(24), 3677–3684 (2012).
[Crossref] [PubMed]

2011 (2)

D. B. Turner, K. E. Wilk, P. M. G. Curmi, and G. D. Scholes, “Comparison of Electronic and Vibrational Coherence Measured by Two-Dimensional Electronic Spectroscopy,” J. Phys. Chem. Lett. 2(15), 1904–1911 (2011).
[Crossref]

R. Augulis and D. Zigmantas, “Two-dimensional electronic spectroscopy with double modulation lock-in detection: enhancement of sensitivity and noise resistance,” Opt. Express 19(14), 13126–13133 (2011).
[Crossref] [PubMed]

2010 (2)

2009 (2)

A. D. Bristow, D. Karaiskaj, X. Dai, T. Zhang, C. Carlsson, K. R. Hagen, R. Jimenez, and S. T. Cundiff, “A versatile ultrastable platform for optical multidimensional Fourier-transform spectroscopy,” Rev. Sci. Instrum. 80(7), 073108 (2009).
[Crossref] [PubMed]

K. W. Stone, K. Gundogdu, D. B. Turner, X. Li, S. T. Cundiff, and K. A. Nelson, “Two-quantum 2D FT electronic spectroscopy of biexcitons in GaAs quantum wells,” Science 324(5931), 1169–1173 (2009).
[Crossref] [PubMed]

2008 (2)

2005 (1)

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]

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]

M. L. Cowan, J. P. Ogilvie, and R. J. D. Miller, “Two-dimensional spectroscopy using diffractive optics based phased-locked photon echoes,” Chem. Phys. Lett. 386(1–3), 184–189 (2004).
[Crossref]

2001 (1)

J. D. Hybl, A. A. Ferro, and D. M. Jonas, “Two-dimensional Fourier transform electronic spectroscopy,” J. Chem. Phys. 115(14), 6606–6622 (2001).
[Crossref]

2000 (1)

Anna, J. M.

J. M. Anna, E. E. Ostroumov, K. Maghlaoui, J. Barber, and G. D. Scholes, “Two-dimensional electronic spectroscopy reveals ultrafast downhill energy transfer in photosystem I trimers of the cyanobacterium Thermosynechococcus elongatus,” J. Phys. Chem. Lett. 3(24), 3677–3684 (2012).
[Crossref] [PubMed]

Augulis, R.

Backus, E. H. G.

Barber, J.

J. M. Anna, E. E. Ostroumov, K. Maghlaoui, J. Barber, and G. D. Scholes, “Two-dimensional electronic spectroscopy reveals ultrafast downhill energy transfer in photosystem I trimers of the cyanobacterium Thermosynechococcus elongatus,” J. Phys. Chem. Lett. 3(24), 3677–3684 (2012).
[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]

Bloem, R.

Bristow, A. D.

A. D. Bristow, D. Karaiskaj, X. Dai, T. Zhang, C. Carlsson, K. R. Hagen, R. Jimenez, and S. T. Cundiff, “A versatile ultrastable platform for optical multidimensional Fourier-transform spectroscopy,” Rev. Sci. Instrum. 80(7), 073108 (2009).
[Crossref] [PubMed]

A. D. Bristow, D. Karaiskaj, X. Dai, and S. T. Cundiff, “All-optical retrieval of the global phase for two-dimensional Fourier-transform spectroscopy,” Opt. Express 16(22), 18017–18027 (2008).
[Crossref] [PubMed]

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]

Camargo, F. V.

I. A. Heisler, R. Moca, F. V. Camargo, and S. R. Meech, “Two-dimensional electronic spectroscopy based on conventional optics and fast dual chopper data acquisition,” Rev. Sci. Instrum. 85(6), 063103 (2014).
[Crossref] [PubMed]

Carlsson, C.

A. D. Bristow, D. Karaiskaj, X. Dai, T. Zhang, C. Carlsson, K. R. Hagen, R. Jimenez, and S. T. Cundiff, “A versatile ultrastable platform for optical multidimensional Fourier-transform spectroscopy,” Rev. Sci. Instrum. 80(7), 073108 (2009).
[Crossref] [PubMed]

Cho, 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]

Courtney, T. L.

Cowan, M. L.

M. L. Cowan, J. P. Ogilvie, and R. J. D. Miller, “Two-dimensional spectroscopy using diffractive optics based phased-locked photon echoes,” Chem. Phys. Lett. 386(1–3), 184–189 (2004).
[Crossref]

Cundiff, S. T.

A. D. Bristow, D. Karaiskaj, X. Dai, T. Zhang, C. Carlsson, K. R. Hagen, R. Jimenez, and S. T. Cundiff, “A versatile ultrastable platform for optical multidimensional Fourier-transform spectroscopy,” Rev. Sci. Instrum. 80(7), 073108 (2009).
[Crossref] [PubMed]

K. W. Stone, K. Gundogdu, D. B. Turner, X. Li, S. T. Cundiff, and K. A. Nelson, “Two-quantum 2D FT electronic spectroscopy of biexcitons in GaAs quantum wells,” Science 324(5931), 1169–1173 (2009).
[Crossref] [PubMed]

A. D. Bristow, D. Karaiskaj, X. Dai, and S. T. Cundiff, “All-optical retrieval of the global phase for two-dimensional Fourier-transform spectroscopy,” Opt. Express 16(22), 18017–18027 (2008).
[Crossref] [PubMed]

Curmi, P. M. G.

D. B. Turner, K. E. Wilk, P. M. G. Curmi, and G. D. Scholes, “Comparison of Electronic and Vibrational Coherence Measured by Two-Dimensional Electronic Spectroscopy,” J. Phys. Chem. Lett. 2(15), 1904–1911 (2011).
[Crossref]

Dai, X.

A. D. Bristow, D. Karaiskaj, X. Dai, T. Zhang, C. Carlsson, K. R. Hagen, R. Jimenez, and S. T. Cundiff, “A versatile ultrastable platform for optical multidimensional Fourier-transform spectroscopy,” Rev. Sci. Instrum. 80(7), 073108 (2009).
[Crossref] [PubMed]

A. D. Bristow, D. Karaiskaj, X. Dai, and S. T. Cundiff, “All-optical retrieval of the global phase for two-dimensional Fourier-transform spectroscopy,” Opt. Express 16(22), 18017–18027 (2008).
[Crossref] [PubMed]

Donaldson, P.

Dorrer, C.

Ferro, A. A.

J. D. Hybl, A. A. Ferro, and D. M. Jonas, “Two-dimensional Fourier transform electronic spectroscopy,” J. Chem. Phys. 115(14), 6606–6622 (2001).
[Crossref]

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]

Garrett-Roe, S.

Gundogdu, K.

K. W. Stone, K. Gundogdu, D. B. Turner, X. Li, S. T. Cundiff, and K. A. Nelson, “Two-quantum 2D FT electronic spectroscopy of biexcitons in GaAs quantum wells,” Science 324(5931), 1169–1173 (2009).
[Crossref] [PubMed]

Hagen, K. R.

A. D. Bristow, D. Karaiskaj, X. Dai, T. Zhang, C. Carlsson, K. R. Hagen, R. Jimenez, and S. T. Cundiff, “A versatile ultrastable platform for optical multidimensional Fourier-transform spectroscopy,” Rev. Sci. Instrum. 80(7), 073108 (2009).
[Crossref] [PubMed]

Hamm, P.

He, X.

S. Yue, Z. Wang, X. He, G. Zhu, and Y. Weng, “Construction of the Apparatus for Two Dimensional Electronic Spectroscopy and Characterization of the Instrument,” Chin. J. Chem. Phys. 28(4), 509–517 (2015).
[Crossref]

Heisler, I. A.

I. A. Heisler, R. Moca, F. V. Camargo, and S. R. Meech, “Two-dimensional electronic spectroscopy based on conventional optics and fast dual chopper data acquisition,” Rev. Sci. Instrum. 85(6), 063103 (2014).
[Crossref] [PubMed]

Huang, Z.

Z. Huang, P. Wang, X. Shen, T.-M. Yan, Y. Zhang, and J. Liu, “Compact design for two-dimensional electronic spectroscopy,” Laser Phys. 26(3), 035403 (2016).
[Crossref]

Hybl, J. D.

J. D. Hybl, A. A. Ferro, and D. M. Jonas, “Two-dimensional Fourier transform electronic spectroscopy,” J. Chem. Phys. 115(14), 6606–6622 (2001).
[Crossref]

Jiang, Y. H.

Jimenez, R.

A. D. Bristow, D. Karaiskaj, X. Dai, T. Zhang, C. Carlsson, K. R. Hagen, R. Jimenez, and S. T. Cundiff, “A versatile ultrastable platform for optical multidimensional Fourier-transform spectroscopy,” Rev. Sci. Instrum. 80(7), 073108 (2009).
[Crossref] [PubMed]

Joffre, M.

Jonas, D. M.

Karaiskaj, D.

A. D. Bristow, D. Karaiskaj, X. Dai, T. Zhang, C. Carlsson, K. R. Hagen, R. Jimenez, and S. T. Cundiff, “A versatile ultrastable platform for optical multidimensional Fourier-transform spectroscopy,” Rev. Sci. Instrum. 80(7), 073108 (2009).
[Crossref] [PubMed]

A. D. Bristow, D. Karaiskaj, X. Dai, and S. T. Cundiff, “All-optical retrieval of the global phase for two-dimensional Fourier-transform spectroscopy,” Opt. Express 16(22), 18017–18027 (2008).
[Crossref] [PubMed]

Kitney, K. A.

Li, X.

K. W. Stone, K. Gundogdu, D. B. Turner, X. Li, S. T. Cundiff, and K. A. Nelson, “Two-quantum 2D FT electronic spectroscopy of biexcitons in GaAs quantum wells,” Science 324(5931), 1169–1173 (2009).
[Crossref] [PubMed]

Likforman, J.-P.

Liu, J.

Z. Huang, P. Wang, X. Shen, T.-M. Yan, Y. Zhang, and J. Liu, “Compact design for two-dimensional electronic spectroscopy,” Laser Phys. 26(3), 035403 (2016).
[Crossref]

Maghlaoui, K.

J. M. Anna, E. E. Ostroumov, K. Maghlaoui, J. Barber, and G. D. Scholes, “Two-dimensional electronic spectroscopy reveals ultrafast downhill energy transfer in photosystem I trimers of the cyanobacterium Thermosynechococcus elongatus,” J. Phys. Chem. Lett. 3(24), 3677–3684 (2012).
[Crossref] [PubMed]

Meech, S. R.

I. A. Heisler, R. Moca, F. V. Camargo, and S. R. Meech, “Two-dimensional electronic spectroscopy based on conventional optics and fast dual chopper data acquisition,” Rev. Sci. Instrum. 85(6), 063103 (2014).
[Crossref] [PubMed]

Meyer, K.

Miller, R. J. D.

M. L. Cowan, J. P. Ogilvie, and R. J. D. Miller, “Two-dimensional spectroscopy using diffractive optics based phased-locked photon echoes,” Chem. Phys. Lett. 386(1–3), 184–189 (2004).
[Crossref]

Moca, R.

I. A. Heisler, R. Moca, F. V. Camargo, and S. R. Meech, “Two-dimensional electronic spectroscopy based on conventional optics and fast dual chopper data acquisition,” Rev. Sci. Instrum. 85(6), 063103 (2014).
[Crossref] [PubMed]

Nelson, K. A.

K. W. Stone, K. Gundogdu, D. B. Turner, X. Li, S. T. Cundiff, and K. A. Nelson, “Two-quantum 2D FT electronic spectroscopy of biexcitons in GaAs quantum wells,” Science 324(5931), 1169–1173 (2009).
[Crossref] [PubMed]

Ogilvie, J. P.

M. L. Cowan, J. P. Ogilvie, and R. J. D. Miller, “Two-dimensional spectroscopy using diffractive optics based phased-locked photon echoes,” Chem. Phys. Lett. 386(1–3), 184–189 (2004).
[Crossref]

Ostroumov, E. E.

J. M. Anna, E. E. Ostroumov, K. Maghlaoui, J. Barber, and G. D. Scholes, “Two-dimensional electronic spectroscopy reveals ultrafast downhill energy transfer in photosystem I trimers of the cyanobacterium Thermosynechococcus elongatus,” J. Phys. Chem. Lett. 3(24), 3677–3684 (2012).
[Crossref] [PubMed]

Ott, C.

Peters, W. K.

Pfeifer, T.

Scholes, G. D.

J. M. Anna, E. E. Ostroumov, K. Maghlaoui, J. Barber, and G. D. Scholes, “Two-dimensional electronic spectroscopy reveals ultrafast downhill energy transfer in photosystem I trimers of the cyanobacterium Thermosynechococcus elongatus,” J. Phys. Chem. Lett. 3(24), 3677–3684 (2012).
[Crossref] [PubMed]

D. B. Turner, K. E. Wilk, P. M. G. Curmi, and G. D. Scholes, “Comparison of Electronic and Vibrational Coherence Measured by Two-Dimensional Electronic Spectroscopy,” J. Phys. Chem. Lett. 2(15), 1904–1911 (2011).
[Crossref]

Shen, X.

Z. Huang, P. Wang, X. Shen, T.-M. Yan, Y. Zhang, and J. Liu, “Compact design for two-dimensional electronic spectroscopy,” Laser Phys. 26(3), 035403 (2016).
[Crossref]

Smith, E. R.

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.

Stone, K. W.

K. W. Stone, K. Gundogdu, D. B. Turner, X. Li, S. T. Cundiff, and K. A. Nelson, “Two-quantum 2D FT electronic spectroscopy of biexcitons in GaAs quantum wells,” Science 324(5931), 1169–1173 (2009).
[Crossref] [PubMed]

Strzalka, H.

Turner, D. B.

D. B. Turner, K. E. Wilk, P. M. G. Curmi, and G. D. Scholes, “Comparison of Electronic and Vibrational Coherence Measured by Two-Dimensional Electronic Spectroscopy,” J. Phys. Chem. Lett. 2(15), 1904–1911 (2011).
[Crossref]

K. W. Stone, K. Gundogdu, D. B. Turner, X. Li, S. T. Cundiff, and K. A. Nelson, “Two-quantum 2D FT electronic spectroscopy of biexcitons in GaAs quantum wells,” Science 324(5931), 1169–1173 (2009).
[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]

Wang, P.

Z. Huang, P. Wang, X. Shen, T.-M. Yan, Y. Zhang, and J. Liu, “Compact design for two-dimensional electronic spectroscopy,” Laser Phys. 26(3), 035403 (2016).
[Crossref]

Wang, Z.

S. Yue, Z. Wang, X. He, G. Zhu, and Y. Weng, “Construction of the Apparatus for Two Dimensional Electronic Spectroscopy and Characterization of the Instrument,” Chin. J. Chem. Phys. 28(4), 509–517 (2015).
[Crossref]

Weng, Y.

S. Yue, Z. Wang, X. He, G. Zhu, and Y. Weng, “Construction of the Apparatus for Two Dimensional Electronic Spectroscopy and Characterization of the Instrument,” Chin. J. Chem. Phys. 28(4), 509–517 (2015).
[Crossref]

Wilk, K. E.

D. B. Turner, K. E. Wilk, P. M. G. Curmi, and G. D. Scholes, “Comparison of Electronic and Vibrational Coherence Measured by Two-Dimensional Electronic Spectroscopy,” J. Phys. Chem. Lett. 2(15), 1904–1911 (2011).
[Crossref]

Yan, T.-M.

Y. Zhang, T.-M. Yan, and Y. H. Jiang, “Precise phase determination with the built-in spectral interferometry in two-dimensional electronic spectroscopy,” Opt. Lett. 41(17), 4134–4137 (2016).
[Crossref] [PubMed]

Z. Huang, P. Wang, X. Shen, T.-M. Yan, Y. Zhang, and J. Liu, “Compact design for two-dimensional electronic spectroscopy,” Laser Phys. 26(3), 035403 (2016).
[Crossref]

Yetzbacher, M. K.

Yue, S.

S. Yue, Z. Wang, X. He, G. Zhu, and Y. Weng, “Construction of the Apparatus for Two Dimensional Electronic Spectroscopy and Characterization of the Instrument,” Chin. J. Chem. Phys. 28(4), 509–517 (2015).
[Crossref]

Zhang, T.

A. D. Bristow, D. Karaiskaj, X. Dai, T. Zhang, C. Carlsson, K. R. Hagen, R. Jimenez, and S. T. Cundiff, “A versatile ultrastable platform for optical multidimensional Fourier-transform spectroscopy,” Rev. Sci. Instrum. 80(7), 073108 (2009).
[Crossref] [PubMed]

Zhang, Y.

Zhu, G.

S. Yue, Z. Wang, X. He, G. Zhu, and Y. Weng, “Construction of the Apparatus for Two Dimensional Electronic Spectroscopy and Characterization of the Instrument,” Chin. J. Chem. Phys. 28(4), 509–517 (2015).
[Crossref]

Zigmantas, D.

Chem. Phys. Lett. (1)

M. L. Cowan, J. P. Ogilvie, and R. J. D. Miller, “Two-dimensional spectroscopy using diffractive optics based phased-locked photon echoes,” Chem. Phys. Lett. 386(1–3), 184–189 (2004).
[Crossref]

Chin. J. Chem. Phys. (1)

S. Yue, Z. Wang, X. He, G. Zhu, and Y. Weng, “Construction of the Apparatus for Two Dimensional Electronic Spectroscopy and Characterization of the Instrument,” Chin. J. Chem. Phys. 28(4), 509–517 (2015).
[Crossref]

J. Chem. Phys. (1)

J. D. Hybl, A. A. Ferro, and D. M. Jonas, “Two-dimensional Fourier transform electronic spectroscopy,” J. Chem. Phys. 115(14), 6606–6622 (2001).
[Crossref]

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

J. Phys. Chem. Lett. (2)

D. B. Turner, K. E. Wilk, P. M. G. Curmi, and G. D. Scholes, “Comparison of Electronic and Vibrational Coherence Measured by Two-Dimensional Electronic Spectroscopy,” J. Phys. Chem. Lett. 2(15), 1904–1911 (2011).
[Crossref]

J. M. Anna, E. E. Ostroumov, K. Maghlaoui, J. Barber, and G. D. Scholes, “Two-dimensional electronic spectroscopy reveals ultrafast downhill energy transfer in photosystem I trimers of the cyanobacterium Thermosynechococcus elongatus,” J. Phys. Chem. Lett. 3(24), 3677–3684 (2012).
[Crossref] [PubMed]

Laser Phys. (1)

Z. Huang, P. Wang, X. Shen, T.-M. Yan, Y. Zhang, and J. Liu, “Compact design for two-dimensional electronic spectroscopy,” Laser Phys. 26(3), 035403 (2016).
[Crossref]

Nature (1)

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]

Opt. Express (3)

Opt. Lett. (4)

Rev. Sci. Instrum. (2)

A. D. Bristow, D. Karaiskaj, X. Dai, T. Zhang, C. Carlsson, K. R. Hagen, R. Jimenez, and S. T. Cundiff, “A versatile ultrastable platform for optical multidimensional Fourier-transform spectroscopy,” Rev. Sci. Instrum. 80(7), 073108 (2009).
[Crossref] [PubMed]

I. A. Heisler, R. Moca, F. V. Camargo, and S. R. Meech, “Two-dimensional electronic spectroscopy based on conventional optics and fast dual chopper data acquisition,” Rev. Sci. Instrum. 85(6), 063103 (2014).
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Science (1)

K. W. Stone, K. Gundogdu, D. B. Turner, X. Li, S. T. Cundiff, and K. A. Nelson, “Two-quantum 2D FT electronic spectroscopy of biexcitons in GaAs quantum wells,” Science 324(5931), 1169–1173 (2009).
[Crossref] [PubMed]

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Figures (4)

Fig. 1
Fig. 1 (a) The pulse sequence in the heterodyne-detected 2DES. The time delay t LO between the signal and LO pulse should be precisely determined in SI. (b) Interferogram by heterodyne detecting the signal (red pulse) with the LO (blue pulse) at t LO = 400 fs. (c) The retrieved amplitude (blue line) and phase when the ac components are shifted 400 fs (black) and 402 fs (red) to the time zero in time domain. (d) The comparison of interferograms when the spectrometer is mis-calibrated ~0.5 nm at 800 nm. (e) The retrieved phase with wavelength mis-calibrated spectrometer (red) and precisely calibrated spectrometer (black).
Fig. 2
Fig. 2 (a) The heterodyne-detected spectral interferogram when the zero delay along τ axis is precisely determined. (b) The interferogram with timing error Δ τ = 1 fs. No matter whether τ = 0 is correctly determined, the interference fringes are quite smooth without phase jumps at intersection τ = 0 (white line), where the rephasing and non-rephasing parts are pieced together. (c) and (d) The corresponding purely absorptive 2D spectra reconstructed from (a) and (b). The 2D spectrum are strongly distorted due to the timing error Δ τ in determining the absolute zero delay.
Fig. 3
Fig. 3 (a) The 2D profile of the phase component of Fig. 2(c) with Δ τ = 0 fs, the white line (remarked with a black arrow) represents the spectral phase at ωt = 2.59 rad/fs for comparison with different experimental situations. (b) The 2D phase profile of Fig. 2(d) with Δ τ = 1 fs. (c) The spectral phase along ωτ at ωt = 2.59 rad/fs. Green line: the benchmark spectral phase with the precise timing. Red and blue line: the spectral phases with sub-cycle timing error Δ τ = 1 fs and Δ τ = 2 fs. Dash lines: the fitting with linear slopes proportional to Δ τ . (d) Red line: the benchmark spectral phase with Δ τ = 0 fs. Black line: the spectral phase by removing the linear phase slope from the spectral phase with Δ τ = 1 fs, which well reproduces the benchmark spectral phase.
Fig. 4
Fig. 4 (a) The 2D phase plot by removing the linear phase slope proportional to Δ τ to remedy the phase artifact owing to the timing error in Fig. 3(b). Compared with the Fig. 3(a), the corrected phase map agrees quite well with the benchmark 2D phase mapping. (b) The corrected purely absorptive 2D spectra, where the phase distortion is effectively eliminated. (c) Black line: The projection of the 2D spectrum in Fig. 2(c) on the ωt axis, with Δ τ = 0. Blue line: the projection of Fig. 2(d) with Δ τ = 1 fs. Red line: the pump-probe projection with corrected 2D absorptive spectrum in Fig. 4(b).

Equations (8)

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P ( 3 ) ( t 3 , t 2 , t 1 ) = 0 d t 3 0 d t 2 0 d t 1 n R n ( t 3 , t 2 , t 1 ) E 3 ( t t 3 ) E 2 ( t t 3 t 2 ) E 1 ( t t 3 t 2 t 1 ) .
I S I ( ω t ) = | ( E L O ( t ) + E s i g ( t ) ) e i ω t d t | 2 .
D ( ω t , τ ) = | E L O ( ω t ) + E s i g ( ω t ) e i ω t τ | 2 = | E L O ( ω t ) | 2 + | E s i g ( ω t ) | 2 + E L O ( ω t ) E s i g ( ω t ) e i ω t τ + E L O ( ω t ) E s i g ( ω t ) e i ω t τ = S 0 ( ω t ) + f ( ω t ) e i ω t τ + f ( ω t ) e i ω t τ ,
S 0 ( ω t ) = | E L O ( ω t ) | 2 + | E s i g ( ω t ) | 2
f ( ω t ) = E L O ( ω t ) E s i g ( ω t ) .
| E L O ( ω t ) | = 1 2 ( S 0 ( ω t ) + 2 | f ( ω t ) | + S 0 ( ω t ) 2 | f ( ω t ) | ) ,
| E s i g ( ω t ) | = 1 2 ( S 0 ( ω t ) + 2 | f ( ω t ) | S 0 ( ω t ) 2 | f ( ω t ) | ) ,
φ ( ω t ) = φ L O ( ω t ) + arg f ( ω t ) .

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