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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 17 — Aug. 16, 2010
  • pp: 17699–17708
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Resonance lineshapes in two-dimensional Fourier transform spectroscopy

Mark E. Siemens, Galan Moody, Hebin Li, Alan D. Bristow, and Steven T. Cundiff  »View Author Affiliations


Optics Express, Vol. 18, Issue 17, pp. 17699-17708 (2010)
http://dx.doi.org/10.1364/OE.18.017699


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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

Two-dimensional Fourier transform spectroscopy (2DFTS) has many advantages over one-dimensional techniques, including isolation of quantum interaction pathways and clear distinction of many-body and biexciton effects. Based on the same concept as nuclear magnetic resonance (NMR) experiments that use radio frequency (RF) radiation to reveal the atomic structure of complex proteins [1

1. R. R. Ernst, G. Bodenhausen, and A. Wokaun, Principles of Nuclear Magnetic Resonance in One and Two Dimensions (Clarendon Press – Oxford, 1988).

,2

2. K. Wuthrich, NMR of Proteins and Nucleic Acids (John Wiley and Sons, 1986).

], 2DFTS has recently been applied at much shorter wavelengths [3

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

,4

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

]. Using pulsed lasers as light sources, these optical analogues of NMR enable access to sub-picosecond resonance dynamics in systems insensitive to RF radiation. For example, molecular vibrations have been extensively studied using 2DFTS in the infrared [5

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

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]

], while visible 2DFTS has been used to investigate electronic transitions in dye molecules [7

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

], photosynthetic processes [8

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]

], and semiconductor nanostructures [9

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]

]. 2DFTS provides clear insight into coherent and incoherent coupling processes in each of these systems.

2DFTS can also clearly separate the homogenous broadening of individual oscillators from inhomogeneous broadening due to sample irregularities in semiconductors [10

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

] or Doppler broadening in atomic vapors [11

11. W. Demtroder, Laser Spectroscopy (Springer, 2002).

]. This ability is in contrast to linear spectroscopies such as absorption and photoluminescence, which yield a linewidth that is the combination of the homogeneous and inhomogeneous broadenings [12

12. S. Mukamel, Principles of Nonlinear Optical Spectroscopy (Oxford University Press, 1995).

]. Nonlinear one-dimensional techniques such as four-wave mixing (FWM) can isolate the homogeneous broadening of a single resonance in the homogeneous or inhomogeneous limit [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]

], but in more complicated systems containing both homogeneous and inhomogeneous broadening or multiple resonances, 2DFTS gives clearer insight.

Past work on 2D lineshapes has focused on managing the coupling between inhomogeneous and homogeneous broadening rather than understanding and isolating the individual contributions [1

1. R. R. Ernst, G. Bodenhausen, and A. Wokaun, Principles of Nuclear Magnetic Resonance in One and Two Dimensions (Clarendon Press – Oxford, 1988).

,14

14. E. Bartholdi and R. R. Ernst, “Fourier Spectroscopy and the Causality Principle,” J. Magn. Reson. 11, 9–19 (1973).

]. The coupling degraded frequency resolution in NMR experiments; windowing functions were used to improve the resolution of resonance peaks, but provided no insight into the connection between the lineshapes and resonance dephasings. A different approach for molecular systems considered both rephasing and nonrephasing signals together, which reduced the coupling [15

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

]. In theoretical work, Tokmakoff derived envelope lineshapes in the homogeneous and inhomogeneous limits from the Fourier transform of an absolute-value 2D time-domain solution of the optical Bloch equations [16

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

]. Phenomenological fitting to simulations was used to obtain correlation information [17

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

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]

], as well as ratios of dephasing parameters in the presence of many-body effects [19

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]

,20

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. 6(2), 445–448 (2009).

], but a method for determining absolute (quantitative and physically meaningful) homogeneous and inhomogeneous broadening parameters from 2D lineshapes has not yet been presented, to the best of our knowledge.

In this paper, we derive an analytical form for complex resonance lineshapes in 2DFTS signals for arbitrary homogeneous and inhomogeneous linewidths. We begin in the 2D time domain with the solution of the optical Bloch equations for a two-level system. Instead of Fourier transforming this 2D time signal to get the full 2D frequency signal as is usually done, we apply the projection-slice theorem of 2D Fourier transforms. This approach allows us to determine an analytical form of diagonal and cross-diagonal slices in the 2D frequency data. This result provides a method of extracting the absolute homogeneous and inhomogeneous linewidths from a 2D Fourier-transform spectrum with arbitrary amounts of homogeneous and inhomogeneous broadening. We fit the resulting lineshapes to experimental data from semiconductor quantum wells to obtain quantitative homogeneous and inhomogeneous linewidths of multiple resonances.

2. 2D time domain

In order to calculate 2D frequency domain lineshapes, we first consider the signal in the 2D time domain. We will see that the expected signal is strongly modified by the requirements of causality and pathway selection, which will also significantly affect the 2D frequency lineshape.

We consider a two-pulse excitation scheme in which τ is the time between pulse 1 (incident with wavevector k1) and pulse 2 (k2), 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 2k2-k1 [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]

]. We work within the Markovian approximation, which results in monotonic exponential decays in time [12

12. S. Mukamel, Principles of Nonlinear Optical Spectroscopy (Oxford University Press, 1995).

]. Including Gaussian inhomogeneous broadening, we find the signal in the 2D time domain
s(t,τ)=s0,0e(γ(t+τ)+iω0(tτ)+σ2(tτ)2/2)Θ(t)Θ(τ).
(1)
Here, s0,0 is the amplitude at time zero, ω0 is the center resonance frequency, 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
Fig. 2 Real part of the OBE signal in the 2D time domain in the cases of a) homogeneous broadening, b) moderate inhomogeneity, and c) strong inhomogeneous broadening. The signals exhibit sharp edges along t = 0 and τ = 0 due to causality.
, which shows the real part of the signal field from Eq. (1) in the 2D time domain for various values of inhomogeneous broadening.

The signal can be decomposed into a homogeneous decay along the photon echo direction t’ = t + τ and an oscillation multiplied by a Gaussian envelope along the anti-echo τ’ = t-τ:
s(t',τ')=s0,0e(γt'+iω0τ'+σ2τ'2/2)Θ(t'τ')Θ(t'+τ').
(2)
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 ωτ’ and ωt’; without it, the homogeneous 2D lineshapes would be simply given by the Fourier transform of the inhomogeneous decay along ωτ’ and the Fourier transform of the homogeneous decay along ωt’.

3. Analytical lineshapes in the 2D frequency domain

3.1 Projection-slice theorem

In order to extend these results to the 2D frequency domain, we apply the projection-slice theorem, a fundamental property of 2D Fourier transforms [21

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. 31, 133–148 (1978).

]. A projection onto a line in a particular direction is performed by integrating the signal perpendicular to the line at each point. The projection-slice theorem states that a Fourier transform of this projection is equivalent to a slice in the 2D Fourier pair plane. In our case, Fourier transforming the 2D time domain data projected onto a line at an angle θ with respect to the t axis yields a slice in the 2D frequency domain, at the same angle θ from the ωt axis.

A signal oscillating at frequency ω0 will be shifted along the ωτ’ axis by ω0 on a 2D frequency spectrum, as illustrated in Fig. 3b
Fig. 3 a) 2D time and b) frequency coordinates for photon echo signals. c) 2D time projection onto the diagonal corresponding to a slice along ωt’. d) 2D time projection onto the cross-diagonal corresponding to a slice along ωτ’. A signal with moderate inhomogeneity, simulated from Eq. (1), is shown in the background for reference. The gray triangles indicate areas of zero signal as enforced by the Θ functions in Eq. (1).
[22

22. J. W. Goodman, Introduction to Fourier Optics,” (McGraw-Hill, 1996).

]. This is a result of the e-iω0τ’ 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 ωτ’ 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 ω0 along the ωτ’ axis. Accounting for the shift and normalizing by s0,0, the signal in the 2D time domain will be
sω0(t',τ')=s(t',τ')eiω0τ's0,0=e(γt'+σ2τ'2/2)Θ(t'τ')Θ(t'+τ').
(3)
This provides sensitive energy selection: if ωi≉ ω0, 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.

Homogeneous and inhomogeneous broadenings are most clearly separated along the diagonal and cross-diagonal in 2D frequency space, which corresponds to slices along the ωτ’ and ωt’ directions shown in Fig. 3b. We therefore evaluate projections onto the 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 ωτ’ and ωt’ 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’ axis and centered at ω0 is illustrated in Fig. 3c and written as

sProj,ω0(t')=sω0(t',τ')dτ'=eγt't't'eσ2τ'2/2dτ'.
(4)

The projection onto τ’ is illustrated in Fig. 3d and written as
sProj,ω0(τ')=sω0(t',τ')dt'=eσ2τ'2/2|τ'|eγt'dt'.
(5)
The projections onto the t’ and τ’ axes are depicted in Fig. 4
Fig. 4 Coherent signal amplitude in the 2D time domain for a) pure homogeneous b) moderately inhomogeneous, and c) strongly inhomogeneous broadening. Frames d, e, and f show projections onto the t’ (blue) and τ’ (red) axes in the corresponding cases.
for homogeneous, inhomogeneous, and moderate inhomogeneous broadening.

3.2 Inhomogeneous and homogeneous limits

First we consider the inhomogeneous and homogeneous limits to confirm the suitability of the projection-slice theorem. In the case of dominant inhomogeneity, where the distribution of resonance frequencies is much larger than the dephasing rate of a particular resonance (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:
sProjIn(t')=Θ(t')eγt'SSliceIn(ωt')=12π(γiωt'),
(6)
where we have used the Fourier transform definition S(ω)=s(t)eiωtdt/2π. Fourier transforming a projection onto the 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:
sProjIn(τ')=eσ2τ'2/20eγt'dt'=eσ2τ'2/2SSliceIn(ωτ')=1σeωτ'2/2σ2.
(7)
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 σ0. In this case, the Gaussian can be treated as a constant, the projection integral is trivial, and the Fourier transform yields a complex Lorentzian:
sProjHo(t')=Θ(t')2t'eγt'SSliceHo(ωt')=12π(γiωt')2.
(8)
The projection onto the τ’ axis in the homogeneous limit is given by Eq. (5) with the Gaussian term approaching a constant:
sProjHo(τ')=1γeγ|τ'|SSliceIn(ωτ')=2π1γ2+ωτ'2,
(9)
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.

Equations (6-9) provide the lineshapes in the inhomogeneous and homogeneous limits. Table 1

Table 1. Imaginary, real, and amplitude lineshapes and widths in homogeneous and inhomogeneous limits

table-icon
View This Table
summarizes the complex shapes and widths, Fig. 1 shows the amplitude, Fig. 5
Fig. 5 Imaginary part of the 2D signal. Parts a, b, and c) show a calculated signal in the 2D frequency domain obtained by the 2D Fourier transform of the corresponding panels in Fig. 4. Panels d, e, and f) depict the imaginary part of the lineshapes derived in the text for the homogeneous, moderately inhomogeneous, and strongly inhomogeneous cases.
the imaginary part, and Fig. 6
Fig. 6 As in Fig. 5, but real part of 2D frequency (a, b, and c) and slice (d, e, and f) data.
the real part of the lineshapes in these limits.

3.3 Arbitrary inhomogeneity

Finally, we investigate the case of moderate 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.

First we consider projection onto the t’ axis. Evaluating Eq. (4), we find
sProj,ω0(t')=Θ(t')eγtt't'eσ2τ2/2dτ=2πσΘ(t)eγtErf(σt/2),
(10)
where Erf is the error function. A Fourier transform of this projection yields
SProj,ω0(ωt')=e(γiωt)22σ2Erfc(γiωt2σ)σ(γiωt),
(11)
where Erfc is the complementary error function. A similar treatment will yield the diagonal lineshape. We evaluate Eq. (5) and find
sProj,ω0(τ')=eσ2τ2/2|τ|eγtdt=1γeσ2τ2/2eγ|τ|.
(12)
The Fourier transform of Eq. (12) is given by the convolution of the Fourier transforms of the Gaussian and the exponential decay:
SProj,ω0(ωτ)=2πσ2eωτ2/2σ21γ2+ω2τ'=2πγVoigt(γ,σ,ωτ).
(13)
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

The analytical lineshape expressions in Eq. (11) and Eq. (13) can be fit to experimental data to obtain quantitative measurement of the homogeneous and inhomogeneous linewidths. As a demonstration, we apply this analysis to 2D spectra obtained from exciton resonances in a GaAs quantum well. The sample and technique are described in detail elsewhere [9

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

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

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]

]. Many-body effects (MBE) are known to strongly modify the coherent response of semiconductor nanostructures [10

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

,25

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

,26

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]

], and our lineshape analysis has neglected these interactions. We therefore focus our attention on the lineshape amplitude and acknowledge that the homogeneous linewidths reported here include significant excitation-induced many-body effects. The radiative limit of exciton resonances is narrower than the widths reported here, and a quantitative study of linewidth dependence on excitation is an area of future study [27

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.

].

A 2D amplitude spectrum obtained from cocircular-polarized excitation is shown in Fig. 7a
Fig. 7 Fitting analytical lineshapes to experimental data from excitons in GaAs quantum wells. a) Experimental and b) simulated 2D frequency spectra. c) Diagonal (red) and cross-diagonal (blue) experimental data (dots) and analytical fits (lines) using Eq. (13) and Eq. (11) for the HH excitonic resonance. d) Data and fits for the LH. Homogeneous and inhomogeneous parameters measured from the fits in c) and d) were used in the simulation shown in b).
. The two resonances along the diagonal correspond to the light-hole exciton (XLH~1556 meV) and heavy-hole exciton (XHH~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.

Close examination of the cross-diagonal XHH 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]

]. Clearly, 2DFTS is a powerful tool that may provide insight into the physics governing dynamics in molecular and solid systems, including non-Markovian processes.

5. Conclusion

2D Fourier transform spectroscopy is a powerful tool for separating inhomogeneous and homogeneous broadening in coherent signals, but a means for determining quantitative linewidth information from experimental spectra has been missing. We derived analytical expressions for slices of 2D lineshapes from rephasing signals. These lineshapes were determined in the limits of homogeneous and inhomogeneous broadening, as well as arbitrary inhomogeneity. The results can be applied to extract quantitative values of homogeneous and inhomogeneous broadening from experimental 2D signals.

Acknowledgements

The authors would like to thank David Jonas and Warren Warren for helpful discussions and Richard Mirin for providing samples. This work was supported by National Science Foundation and the Chemical Sciences, Geosciences, and Biosciences Division Office of Basic Energy Sciences, U.S. Department of Energy. MES acknowledges funding from the National Academy of Sciences/National Research Council postdoctoral fellows program.

References and links

1.

R. R. Ernst, G. Bodenhausen, and A. Wokaun, Principles of Nuclear Magnetic Resonance in One and Two Dimensions (Clarendon Press – Oxford, 1988).

2.

K. Wuthrich, NMR of Proteins and Nucleic Acids (John Wiley and Sons, 1986).

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]

11.

W. Demtroder, Laser Spectroscopy (Springer, 2002).

12.

S. Mukamel, Principles of Nonlinear Optical Spectroscopy (Oxford University Press, 1995).

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]

14.

E. Bartholdi and R. R. Ernst, “Fourier Spectroscopy and the Causality Principle,” J. Magn. Reson. 11, 9–19 (1973).

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]

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. 6(2), 445–448 (2009).

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. 31, 133–148 (1978).

22.

J. W. Goodman, Introduction to Fourier Optics,” (McGraw-Hill, 1996).

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]

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]

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 76(12), 121303 (2007). [CrossRef]

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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