## Resonance lineshapes in two-dimensional Fourier transform spectroscopy |

Optics Express, Vol. 18, Issue 17, pp. 17699-17708 (2010)

http://dx.doi.org/10.1364/OE.18.017699

Acrobat PDF (1866 KB)

### Abstract

We derive an analytical form for resonance lineshapes in two-dimensional (2D) Fourier transform spectroscopy. Our starting point is the solution of the optical Bloch equations for a two-level system in the 2D time domain. Application of the projection-slice theorem of 2D Fourier transforms reveals the form of diagonal and cross-diagonal slices in the 2D frequency data for arbitrary inhomogeneity. The results are applied in quantitative measurements of homogeneous and inhomogeneous broadening of multiple resonances in experimental data.

© 2010 OSA

## 1. Introduction

3. D. M. Jonas, “Two-dimensional femtosecond spectroscopy,” Annu. Rev. Phys. Chem. **54**(1), 425–463 (2003). [CrossRef] [PubMed]

4. M. Cho, “Coherent two-dimensional optical spectroscopy,” Chem. Rev. **108**(4), 1331–1418 (2008). [CrossRef] [PubMed]

5. M. C. Asplund, M. T. Zanni, and R. M. Hochstrasser, “Two-dimensional infrared spectroscopy of peptides by phase-controlled femtosecond vibrational photon echoes,” Proc. Natl. Acad. Sci. U.S.A. **97**(15), 8219–8224 (2000). [CrossRef] [PubMed]

6. O. Golonzka, M. Khalil, N. Demirdöven, and A. Tokmakoff, “Vibrational anharmonicities revealed by coherent two-dimensional infrared spectroscopy,” Phys. Rev. Lett. **86**(10), 2154–2157 (2001). [CrossRef] [PubMed]

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

8. 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]

9. S. T. Cundiff, T. Zhang, A. D. Bristow, D. Karaiskaj, and X. Dai, “Optical two-dimensional fourier transform spectroscopy of semiconductor quantum wells,” Acc. Chem. Res. **42**(9), 1423–1432 (2009). [CrossRef] [PubMed]

10. S. T. Cundiff, “Coherent spectroscopy of semiconductors,” Opt. Express **16**(7), 4639–4664 (2008). [CrossRef] [PubMed]

13. T. Yajima and Y. Taira, “Spatial Optical Parametric Coupling of Picosecond Light Pulses and Transverse Relaxation effect in Resonant Media,” J. Phys. Soc. Jpn. **47**(5), 1620–1626 (1979). [CrossRef]

15. S. M. Gallagher Faeder and D. M. Jonas, “Phase-resolved time-domain nonlinear optical signals,” Phys. Rev. A **62**(3), 033820 (2000). [CrossRef]

16. A. Tokmakoff, “Two-dimensional line shapes derived from coherent third-order nonlinear spectroscopy,” J. Phys. Chem. A **104**(18), 4247–4255 (2000). [CrossRef]

17. K. Kwac and M. Cho, “Molecular dynamics simulation study of N-methylacetamide in water. II. Two-dimensional infrared pump-probe spectra,” J. Chem. Phys. **119**(4), 2256–2263 (2003). [CrossRef]

18. K. Lazonder, M. S. Pshenichnikov, and D. A. Wiersma, “Easy interpretation of optical two-dimensional correlation spectra,” Opt. Lett. **31**(22), 3354–3356 (2006). [CrossRef] [PubMed]

19. I. Kuznetsova, T. Meier, S. T. Cundiff, and P. Thomas, “Determination of homogeneous and inhomogeneous broadening in semiconductor nanostructures by two-dimensional Fourier-transform optical spectroscopy,” Phys. Rev. B **76**(15), 153301 (2007). [CrossRef]

## 2. 2D time domain

*τ*is the time between pulse 1 (incident with wavevector k

_{1}) and pulse 2 (k

_{2}), and

*t*is the time of signal emission after the arrival of the second pulse. This is equivalent to three-pulse excitation with zero delay between the second and third pulses. We begin with the optical Bloch equations (OBEs) for a two-level system, apply perturbation theory and the rotating wave approximation, assume delta-function pulses, and select only the signal emitted in the phase-matched FWM direction 2k

_{2}-k

_{1}[13

13. T. Yajima and Y. Taira, “Spatial Optical Parametric Coupling of Picosecond Light Pulses and Transverse Relaxation effect in Resonant Media,” J. Phys. Soc. Jpn. **47**(5), 1620–1626 (1979). [CrossRef]

*s*is the amplitude at time zero,

_{0,0}*ω*is the center resonance frequency,

_{0}*γ*is the homogeneous linewidth,

*σ*is the inhomogeneous linewidth, and the

*Θ*’s are unit step functions establishing that a signal cannot be emitted before the pulses are arrive. We consider only the rephasing pulse sequence where the conjugate pulse comes first and a photon echo is emitted at a time

*t*=

*τ*after the arrival of the final pulse. This photon echo can be clearly seen as a sharp ridge along the diagonal in Figs. 2b and c , which shows the real part of the signal field from Eq. (1) in the 2D time domain for various values of inhomogeneous broadening.

*t’ = t + τ*and an oscillation multiplied by a Gaussian envelope along the anti-echo

*τ’ = t-τ*:This is an intuitive way to visualize the photon echo signal in the 2D time domain, but unfortunately, the signal is not completely separable along these axes because the

*Θ*functions enforcing causality (or time ordering) involve

*t’*and

*τ’*in an inseparable way. It is this time-ordering that causes the mixing between homogeneous and inhomogeneous broadenings along

*ω*and

_{τ’}*ω*; without it, the homogeneous 2D lineshapes would be simply given by the Fourier transform of the inhomogeneous decay along

_{t’}*ω*and the Fourier transform of the homogeneous decay along

_{τ’}*ω*.

_{t’}## 3. Analytical lineshapes in the 2D frequency domain

### 3.1 Projection-slice theorem

*t*axis yields a slice in the 2D frequency domain, at the same angle θ from the ω

_{t}axis.

*ω*will be shifted along the

_{0}*ω*axis by

_{τ’}*ω*on a 2D frequency spectrum, as illustrated in Fig. 3b [22]. This is a result of the

_{0}*e*term in the 2D time signal in Eq. (2); according to the Fourier shift theorem, this oscillation in time translates to a shift along the

^{-iω0τ’}*ω*axis. In order to obtain a slice in the 2D frequency domain that cuts through the resonance peak, we apply a shift from the origin of

_{τ’}*ω*along the

_{0}*ω*axis. Accounting for the shift and normalizing by

_{τ’}*s*, the signal in the 2D time domain will beThis provides sensitive energy selection: if

_{0,0}*ω*, the resulting oscillations will zero the signal when the projection operation is applied. This demonstrates how 2D spectroscopy is capable of determining properties of multiple resonances or along an inhomogeneous distribution.

_{i}≉ ω_{0}*ω*and

_{τ’}*ω*directions shown in Fig. 3b. We therefore evaluate projections onto the

_{t’}*τ’*(-π/4 from

*t*) and

*t’*(π /4 from

*t*) axes in the time domain shown, in Fig. 3a, in order to determine slices along the

*ω*and

_{τ’}*ω*axes in the 2D frequency domain, shown in Fig. 3b. We project onto a given axis by integrating the signal perpendicular to the axis and adjusting the limits of integration to account for the time-ordering limits. The projection onto the

_{t’}*t’*axis and centered at

*ω*is illustrated in Fig. 3c and written as

_{0}### 3.2 Inhomogeneous and homogeneous limits

*σ*≫γ), the Gaussian term in Eq. (4) is narrow enough to be treated as a delta-function, and the photon echo signal is restored along the diagonal (

*t’ = t + τ*). The projected signal onto

*t’ s*

_{ProjIn}

*(t’)*is purely homogeneous dephasing, and the Fourier transform is straightforward to perform:where we have used the Fourier transform definition

*τ’*axis will yield the lineshape of a diagonal slice in 2D frequency space. In the inhomogeneous limit, the projection is a very narrow Gaussian, so the slice is Gaussian as well:In the opposite limit of purely homogeneous broadening (

*σ*≪γ), enforcing time-ordering (zero signal before

*t*= 0 and

*τ*= 0, as shown in Fig. 3c as the gray triangles) strongly affects the projections. Looking first at the projection onto the

*t’*axis, we return to Eq. (4) and consider the limit of

*σ*➔

*0*. In this case, the Gaussian can be treated as a constant, the projection integral is trivial, and the Fourier transform yields a complex Lorentzian:The projection onto the

*τ’*axis in the homogeneous limit is given by Eq. (5) with the Gaussian term approaching a constant:As expected in the homogeneous limit, the amplitude of a diagonal slice (Eq. (9) is a Lorentzian, equal to the amplitude of a cross-diagonal slice (Eq. (8), as shown in Fig. 1a and d.

### 3.3 Arbitrary inhomogeneity

*σ~γ*), where the homogeneous and inhomogeneous broadenings will each contribute to both diagonal and cross-diagonal lineshapes. The integrals in Eqs. (4) and 5 can be evaluated and Fourier transformed analytically without any restriction on homogeneous or inhomogeneous broadenings.

*t’*axis. Evaluating Eq. (4), we findwhere Erf is the error function. A Fourier transform of this projection yieldswhere Erfc is the complementary error function. A similar treatment will yield the diagonal lineshape. We evaluate Eq. (5) and findThe Fourier transform of Eq. (12) is given by the convolution of the Fourier transforms of the Gaussian and the exponential decay:The expressions for cross-diagonal and diagonal slices of 2D frequency spectra derived in Eq. (11) and Eq. (13) are valid in the homogeneous and inhomogeneous limits discussed earlier, as well as any combination of homogeneous and inhomogeneous broadening. Derived directly from the 2D time signal, these analytical expressions provide a powerful means for visualizing and characterizing 2D frequency signals arising from various sources of broadenings.

## 4. Comparison with experimental data

9. S. T. Cundiff, T. Zhang, A. D. Bristow, D. Karaiskaj, and X. Dai, “Optical two-dimensional fourier transform spectroscopy of semiconductor quantum wells,” Acc. Chem. Res. **42**(9), 1423–1432 (2009). [CrossRef] [PubMed]

23. A. D. Bristow, D. Karaiskaj, X. Dai, R. P. Mirin, and S. T. Cundiff, “Polarization dependence of semiconductor exciton and biexciton contributions to phase-resolved optical two-dimensional Fourier-transform spectra,” Phys. Rev. B **79**(16), 1–4 (2009). [CrossRef]

24. 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]

10. S. T. Cundiff, “Coherent spectroscopy of semiconductors,” Opt. Express **16**(7), 4639–4664 (2008). [CrossRef] [PubMed]

25. D. S. Chemla and J. Shah, “Many-body and correlation effects in semiconductors,” Nature **411**(6837), 549–557 (2001). [CrossRef] [PubMed]

26. X. Li, T. Zhang, C. N. Borca, and S. T. Cundiff, “Many-body interactions in semiconductors probed by optical two-dimensional fourier transform spectroscopy,” Phys. Rev. Lett. **96**(5), 057406 (2006). [CrossRef] [PubMed]

_{LH}~1556 meV) and heavy-hole exciton (X

_{HH}~1547 meV) in a quantum well. The peaks above and below the diagonal indicate coupling from the LH to the HH excitons. Centered at the peak of each resonance, we take diagonal and cross-diagonal slices of the data, shown by the dots in Fig. 7c and d. We then fit Eq. (11) to the cross-diagonal and Eq. (13) to the diagonal slices, using

*γ*and

*σ*as fitting parameters. The fits and extracted homogeneous and inhomogeneous values are shown in Fig. 7c and d. These values are plugged in to Eq. (1) and Fourier transformed to model the expected 2D signal; the results of the model are shown in Fig. 7b. We see excellent agreement with the experimental lineshapes for both LH and HH excitons. All diagonal and cross-diagonal slices of the model match exactly with the slice fits (not shown), confirming that this is an absolute measurement of homogeneous and inhomogeneous linewidths for both resonances.

_{HH}data reveals wings on the experimental data that deviate significantly from the fitted analytical lineshape. These sidebands indicate the presence of non-Markovian behavior in these quantum wells, as observed previously [28

28. S. G. Carter, Z. Chen, and S. T. Cundiff, “Echo peak-shift spectroscopy of non-Markovian exciton dynamics in quantum wells,” Phys. Rev. B **76**(12), 121303 (2007). [CrossRef]

## 5. Conclusion

## Acknowledgements

## References and links

1. | R. R. Ernst, G. Bodenhausen, and A. Wokaun, |

2. | K. Wuthrich, |

3. | D. M. Jonas, “Two-dimensional femtosecond spectroscopy,” Annu. Rev. Phys. Chem. |

4. | M. Cho, “Coherent two-dimensional optical spectroscopy,” Chem. Rev. |

5. | M. C. Asplund, M. T. Zanni, and R. M. Hochstrasser, “Two-dimensional infrared spectroscopy of peptides by phase-controlled femtosecond vibrational photon echoes,” Proc. Natl. Acad. Sci. U.S.A. |

6. | O. Golonzka, M. Khalil, N. Demirdöven, and A. Tokmakoff, “Vibrational anharmonicities revealed by coherent two-dimensional infrared spectroscopy,” Phys. Rev. Lett. |

7. | J. Hybl, A. Ferro, and D. Jonas, “Two-dimensional Fourier transform electronic spectroscopy,” J. Chem. Phys. |

8. | 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 |

9. | S. T. Cundiff, T. Zhang, A. D. Bristow, D. Karaiskaj, and X. Dai, “Optical two-dimensional fourier transform spectroscopy of semiconductor quantum wells,” Acc. Chem. Res. |

10. | S. T. Cundiff, “Coherent spectroscopy of semiconductors,” Opt. Express |

11. | W. Demtroder, |

12. | S. Mukamel, |

13. | T. Yajima and Y. Taira, “Spatial Optical Parametric Coupling of Picosecond Light Pulses and Transverse Relaxation effect in Resonant Media,” J. Phys. Soc. Jpn. |

14. | E. Bartholdi and R. R. Ernst, “Fourier Spectroscopy and the Causality Principle,” J. Magn. Reson. |

15. | S. M. Gallagher Faeder and D. M. Jonas, “Phase-resolved time-domain nonlinear optical signals,” Phys. Rev. A |

16. | A. Tokmakoff, “Two-dimensional line shapes derived from coherent third-order nonlinear spectroscopy,” J. Phys. Chem. A |

17. | K. Kwac and M. Cho, “Molecular dynamics simulation study of N-methylacetamide in water. II. Two-dimensional infrared pump-probe spectra,” J. Chem. Phys. |

18. | K. Lazonder, M. S. Pshenichnikov, and D. A. Wiersma, “Easy interpretation of optical two-dimensional correlation spectra,” Opt. Lett. |

19. | I. Kuznetsova, T. Meier, S. T. Cundiff, and P. Thomas, “Determination of homogeneous and inhomogeneous broadening in semiconductor nanostructures by two-dimensional Fourier-transform optical spectroscopy,” Phys. Rev. B |

20. | I. Kuznetsova, P. Thomas, T. Meier, T. Zhang, and S. T. Cundiff, “Determination of homogeneous and inhomogeneous broadenings of quantum-well excitons by 2DFTS: An experiment-theory comparison,” Phys. Status Solidi., C Curr. Top. Solid State Phys. |

21. | K. Nagayama, P. Bachmann, K. Wuthrich, and R. R. Ernst, “The Use of Cross-Sections and of Projections in Two-dimensional NMR Spectroscopy,” J. Magn. Reson. |

22. | J. W. Goodman, |

23. | A. D. Bristow, D. Karaiskaj, X. Dai, R. P. Mirin, and S. T. Cundiff, “Polarization dependence of semiconductor exciton and biexciton contributions to phase-resolved optical two-dimensional Fourier-transform spectra,” Phys. Rev. B |

24. | 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. |

25. | D. S. Chemla and J. Shah, “Many-body and correlation effects in semiconductors,” Nature |

26. | X. Li, T. Zhang, C. N. Borca, and S. T. Cundiff, “Many-body interactions in semiconductors probed by optical two-dimensional fourier transform spectroscopy,” Phys. Rev. Lett. |

27. | A. D. Bristow, T. Zhang, M. E. Siemens, R. P. Mirin, and S. T. Cundiff, “Dephasing in Weakly Disordered GaAs Quantum Wells,” to be submitted. |

28. | S. G. Carter, Z. Chen, and S. T. Cundiff, “Echo peak-shift spectroscopy of non-Markovian exciton dynamics in quantum wells,” Phys. Rev. B |

**OCIS Codes**

(300.3700) Spectroscopy : Linewidth

(300.6300) Spectroscopy : Spectroscopy, Fourier transforms

**ToC Category:**

Spectroscopy

**History**

Original Manuscript: June 22, 2010

Revised Manuscript: July 14, 2010

Manuscript Accepted: July 21, 2010

Published: August 2, 2010

**Citation**

Mark E. Siemens, Galan Moody, Hebin Li, Alan D. Bristow, and Steven T. Cundiff, "Resonance lineshapes in two-dimensional
Fourier transform spectroscopy," Opt. Express **18**, 17699-17708 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-17-17699

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### References

- R. R. Ernst, G. Bodenhausen, and A. Wokaun, Principles of Nuclear Magnetic Resonance in One and Two Dimensions (Clarendon Press – Oxford, 1988).
- K. Wuthrich, NMR of Proteins and Nucleic Acids (John Wiley and Sons, 1986).
- D. M. Jonas, “Two-dimensional femtosecond spectroscopy,” Annu. Rev. Phys. Chem. 54(1), 425–463 (2003). [CrossRef] [PubMed]
- M. Cho, “Coherent two-dimensional optical spectroscopy,” Chem. Rev. 108(4), 1331–1418 (2008). [CrossRef] [PubMed]
- M. C. Asplund, M. T. Zanni, and R. M. Hochstrasser, “Two-dimensional infrared spectroscopy of peptides by phase-controlled femtosecond vibrational photon echoes,” Proc. Natl. Acad. Sci. U.S.A. 97(15), 8219–8224 (2000). [CrossRef] [PubMed]
- O. Golonzka, M. Khalil, N. Demirdöven, and A. Tokmakoff, “Vibrational anharmonicities revealed by coherent two-dimensional infrared spectroscopy,” Phys. Rev. Lett. 86(10), 2154–2157 (2001). [CrossRef] [PubMed]
- J. Hybl, A. Ferro, and D. Jonas, “Two-dimensional Fourier transform electronic spectroscopy,” J. Chem. Phys. 115(14), 6606–6622 (2001). [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]
- S. T. Cundiff, T. Zhang, A. D. Bristow, D. Karaiskaj, and X. Dai, “Optical two-dimensional fourier transform spectroscopy of semiconductor quantum wells,” Acc. Chem. Res. 42(9), 1423–1432 (2009). [CrossRef] [PubMed]
- S. T. Cundiff, “Coherent spectroscopy of semiconductors,” Opt. Express 16(7), 4639–4664 (2008). [CrossRef] [PubMed]
- W. Demtroder, Laser Spectroscopy (Springer, 2002).
- S. Mukamel, Principles of Nonlinear Optical Spectroscopy (Oxford University Press, 1995).
- T. Yajima and Y. Taira, “Spatial Optical Parametric Coupling of Picosecond Light Pulses and Transverse Relaxation effect in Resonant Media,” J. Phys. Soc. Jpn. 47(5), 1620–1626 (1979). [CrossRef]
- E. Bartholdi and R. R. Ernst, “Fourier Spectroscopy and the Causality Principle,” J. Magn. Reson. 11, 9–19 (1973).
- S. M. Gallagher Faeder and D. M. Jonas, “Phase-resolved time-domain nonlinear optical signals,” Phys. Rev. A 62(3), 033820 (2000). [CrossRef]
- A. Tokmakoff, “Two-dimensional line shapes derived from coherent third-order nonlinear spectroscopy,” J. Phys. Chem. A 104(18), 4247–4255 (2000). [CrossRef]
- K. Kwac and M. Cho, “Molecular dynamics simulation study of N-methylacetamide in water. II. Two-dimensional infrared pump-probe spectra,” J. Chem. Phys. 119(4), 2256–2263 (2003). [CrossRef]
- K. Lazonder, M. S. Pshenichnikov, and D. A. Wiersma, “Easy interpretation of optical two-dimensional correlation spectra,” Opt. Lett. 31(22), 3354–3356 (2006). [CrossRef] [PubMed]
- I. Kuznetsova, T. Meier, S. T. Cundiff, and P. Thomas, “Determination of homogeneous and inhomogeneous broadening in semiconductor nanostructures by two-dimensional Fourier-transform optical spectroscopy,” Phys. Rev. B 76(15), 153301 (2007). [CrossRef]
- I. Kuznetsova, P. Thomas, T. Meier, T. Zhang, and S. T. Cundiff, “Determination of homogeneous and inhomogeneous broadenings of quantum-well excitons by 2DFTS: An experiment-theory comparison,” Phys. Status Solidi., C Curr. Top. Solid State Phys. 6(2), 445–448 (2009).
- K. Nagayama, P. Bachmann, K. Wuthrich, and R. R. Ernst, “The Use of Cross-Sections and of Projections in Two-dimensional NMR Spectroscopy,” J. Magn. Reson. 31, 133–148 (1978).
- J. W. Goodman, Introduction to Fourier Optics,” (McGraw-Hill, 1996).
- A. D. Bristow, D. Karaiskaj, X. Dai, R. P. Mirin, and S. T. Cundiff, “Polarization dependence of semiconductor exciton and biexciton contributions to phase-resolved optical two-dimensional Fourier-transform spectra,” Phys. Rev. B 79(16), 1–4 (2009). [CrossRef]
- 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]
- D. S. Chemla and J. Shah, “Many-body and correlation effects in semiconductors,” Nature 411(6837), 549–557 (2001). [CrossRef] [PubMed]
- X. Li, T. Zhang, C. N. Borca, and S. T. Cundiff, “Many-body interactions in semiconductors probed by optical two-dimensional fourier transform spectroscopy,” Phys. Rev. Lett. 96(5), 057406 (2006). [CrossRef] [PubMed]
- A. D. Bristow, T. Zhang, M. E. Siemens, R. P. Mirin, and S. T. Cundiff, “Dephasing in Weakly Disordered GaAs Quantum Wells,” to be submitted.
- S. G. Carter, Z. Chen, and S. T. Cundiff, “Echo peak-shift spectroscopy of non-Markovian exciton dynamics in quantum wells,” Phys. Rev. B 76(12), 121303 (2007). [CrossRef]

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