Resonance lineshapes in two-dimensional Fourier transform spectroscopy

Mark E. Siemens; Galan Moody; Hebin Li; Alan D. Bristow; Steven T. Cundiff

doi:10.1364/OE.18.017699

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 [1,2], 2DFTS has recently been applied at much shorter wavelengths [3,4]. 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 [5,6], while visible 2DFTS has been used to investigate electronic transitions in dye molecules [7], photosynthetic processes [8], and semiconductor nanostructures [9]. 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] or Doppler broadening in atomic vapors [11]. 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]. 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], but in more complicated systems containing both homogeneous and inhomogeneous broadening or multiple resonances, 2DFTS gives clearer insight.

A glance at a 2DFT spectrum can give a qualitative sense of the inhomogeneity in a system: for a given resonance, the linewidth in the cross-diagonal direction is associated with homogeneous broadening, while the diagonal linewidth is related to inhomogeneous broadening, as shown in Fig. 1 . However, acquiring quantitative information about the homogeneous and inhomogeneous broadenings is more difficult because they are coupled along the diagonal and cross-diagonal directions of the spectrum. Consider the 2DFT spectrum of a purely homogeneously broadened resonance, which has the classic star shape familiar from NMR [1]. In this case, the diagonal and cross-diagonal slices of an absolute value spectrum are Lorentzians with identical widths, and although there is no inhomogeneous broadening here, the diagonal width is not zero. If inhomogeneous broadening is added, the diagonal will broaden as expected, but the cross-diagonal will also widen slightly and change shape, as shown in the inset to Fig. 1. Clearly there is coupling between the diagonal and cross-diagonal widths, and additional insight is needed in order to obtain quantitative information about the broadening in a two-dimensional (2D) spectrum.

Fig. 1 2D amplitude lineshapes for rephasing signals. a-c) 2D frequency plots for a fixed value of homogeneous broadening with increasing inhomogeneous broadening. The diagonal (red dashes) and cross-diagonal (blue dots) lines are shown. The vertical scale for ω_τ is negative because of phase-matching requirements, and increases (gets more negative) going down. d-f) Slices of the corresponding 2D frequency plots along the diagonal (red) and cross-diagonal (blue) directions. The inset compares cross-diagonal slices in the limits of strong homogeneous (dashes) and inhomogeneous (dots) broadening.

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Past work on 2D lineshapes has focused on managing the coupling between inhomogeneous and homogeneous broadening rather than understanding and isolating the individual contributions [1,14]. 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]. 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]. Phenomenological fitting to simulations was used to obtain correlation information [17,18], as well as ratios of dephasing parameters in the presence of many-body effects [19,20], 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]. We work within the Markovian approximation, which results in monotonic exponential decays in time [12]. Including Gaussian inhomogeneous broadening, we find the signal in the 2D time domain

s (t, τ) = s_{0, 0} e^{- (γ (t + τ) + i ω_{0} (t - τ) + σ^{2} {(t - τ)}^{2} / 2)} Θ (t) Θ (τ) .

Here, s_0,0 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 , which shows the real part of the signal field from Eq. (1) in the 2D time domain for various values of inhomogeneous broadening.

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.

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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') = s_{0, 0} e^{- (γ t' + i ω_{0} τ' + σ^{2} τ'^{2} / 2)} Θ (t' - τ') Θ (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_τ’ and ω_t’; without it, the homogeneous 2D lineshapes would be simply given by the Fourier transform of the inhomogeneous decay along ω_τ’ and the Fourier transform of the homogeneous decay along ω_t’.

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]. 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 ω_t axis.

A signal oscillating at frequency ω₀ will be shifted along the ω_τ’ axis by ω₀ on a 2D frequency spectrum, as illustrated in Fig. 3b [22]. 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 ω_τ’ 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 ω_τ’ axis. Accounting for the shift and normalizing by s_0,0, the signal in the 2D time domain will be

s_{ω_{0}} (t', τ') = s (t', τ') \frac{e^{i ω_{0} τ'}}{s_{0, 0}} = e^{- (γ t' + σ^{2} τ'^{2} / 2)} Θ (t' - τ') Θ (t' + τ') .

This provides sensitive energy selection: if ω_i≉ 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.

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 ω_t’. d) 2D time projection onto the cross-diagonal corresponding to a slice along ω_τ’. 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).

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Homogeneous and inhomogeneous broadenings are most clearly separated along the diagonal and cross-diagonal in 2D frequency space, which corresponds to slices along the ω_τ’ and ω_t’ 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 ω_τ’ and ω_t’ 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

s_{\Pr o j, ω_{0}} (t') = \int_{- \infty}^{\infty} s_{ω_{0}} (t', τ') d τ' = e^{- γ t'} \int_{- t'}^{t'} e^{- σ^{2} τ'^{2} / 2} d τ' .

The projection onto τ’ is illustrated in Fig. 3d and written as

s_{\Pr o j, ω_{0}} (τ') = \int_{- \infty}^{\infty} s_{ω_{0}} (t', τ') d t' = e^{- σ^{2} τ'^{2} / 2} \int_{| τ' |}^{\infty} e^{- γ t'} d t' .

The projections onto the t’ and τ’ axes are depicted in Fig. 4 for homogeneous, inhomogeneous, and moderate inhomogeneous broadening.

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.

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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:

s_{\Pr o j I n} (t') = Θ (t') e^{- γ t'} \Rightarrow S_{S l i c e I n} (ω_{t'}) = \frac{1}{\sqrt{2 π} (γ - i ω_{t'})},

where we have used the Fourier transform definition

S (ω) = \int_{- \infty}^{\infty} s (t) e^{i ω t} d t / \sqrt{2 π}

. 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:

s_{\Pr o j I n} (τ') = e^{- σ^{2} τ'^{2} / 2} \int_{0}^{\infty} e^{- γ t'} d t' = e^{- σ^{2} τ'^{2} / 2} \Rightarrow S_{S l i c e I n} (ω_{τ'}) = \frac{1}{σ} e^{- ω_{τ'}^{2} / 2 σ^{2}} .

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:

s_{\Pr o j H o} (t') = Θ (t') 2 t' e^{- γ t'} \Rightarrow S_{S l i c e H o} (ω_{t'}) = \frac{1}{\sqrt{2 π} {(γ - i ω_{t'})}^{2}} .

The projection onto the τ’ axis in the homogeneous limit is given by Eq. (5) with the Gaussian term approaching a constant:

s_{\Pr o j H o} (τ') = \frac{1}{γ} e^{- γ | τ' |} \Rightarrow S_{S l i c e I n} (ω_{τ'}) = \sqrt{\frac{2}{π}} \frac{1}{γ^{2} + ω_{τ'}^{2}},

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 summarizes the complex shapes and widths, Fig. 1 shows the amplitude, Fig. 5 the imaginary part, and Fig. 6 the real part of the lineshapes in these limits.

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

View Table

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.

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Fig. 6 As in Fig. 5, but real part of 2D frequency (a, b, and c) and slice (d, e, and f) data.

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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

s_{\Pr o j, ω_{0}} (t') = Θ (t') e^{- γ t^{'}} \int_{- t'}^{t'} e^{- σ^{2} {τ^{'}}^{2} / 2} d τ^{'} = \frac{\sqrt{2 π}}{σ} Θ (t^{'}) e^{- γ t^{'}} Erf (σ t^{'} / \sqrt{2}),

where Erf is the error function. A Fourier transform of this projection yields

S_{\Pr o j, ω_{0}} (ω_{t'}) = \frac{e^{\frac{{(γ - i ω_{t^{'}}^{})}^{2}}{2 σ^{2}}} Erfc (\frac{γ - i ω_{t^{'}}}{\sqrt{2} σ})}{σ (γ - i ω_{t^{'}})},

where Erfc is the complementary error function. A similar treatment will yield the diagonal lineshape. We evaluate Eq. (5) and find

s_{\Pr o j, ω_{0}} (τ') = e^{- σ^{2} {τ^{'}}^{2} / 2} \int_{| τ^{'} |}^{\infty} e^{- γ t^{'}} d t^{'} = \frac{1}{γ} e^{- σ^{2} {τ^{'}}^{2} / 2} e^{- γ | τ^{'} |} .

The Fourier transform of Eq. (12) is given by the convolution of the Fourier transforms of the Gaussian and the exponential decay:

S_{\Pr o j, ω_{0}} (ω_{τ^{'}}) = \sqrt{\frac{2}{π σ^{2}}} e^{- ω_{τ^{'}}^{2} / 2 σ^{2}} * \frac{1}{γ^{2} + ω^{2}_{τ'}} = \frac{\sqrt{2 π}}{γ} Voigt (γ, σ, ω_{τ^{'}}) .

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 [9,23,24]. Many-body effects (MBE) are known to strongly modify the coherent response of semiconductor nanostructures [10,25,26], 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].

A 2D amplitude spectrum obtained from cocircular-polarized excitation is shown in Fig. 7a . The two resonances along the diagonal correspond to the light-hole exciton (X_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.

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).

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Close examination of the cross-diagonal X_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]. 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.

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		Imaginary	Real	Amplitude
Inhomogeneous limit	ω_t' axis	d/dx[Lorentzian], 2γ valley-peak	Lineshape is Lorentzian, 2γ	Sqrt[Lorentzian], Lor width of 2γ
Inhomogeneous limit	ω_τ' axis	0	Gaussian, σ	Gaussian, σ
Homogeneous limit	ω_t' axis	d/dx[Lorentzian], 2γ/√3 valley-peak	d2/dx2[Lorentzian] 2γ above-0 width	Lorentzian, 2γ
Homogeneous limit	ω_τ' axis	0	Lorentzian, 2γ	Lorentzian, 2γ

Resonance lineshapes in two-dimensional Fourier transform spectroscopy

Abstract

1. Introduction

2. 2D time domain

3. Analytical lineshapes in the 2D frequency domain

3.1 Projection-slice theorem

3.2 Inhomogeneous and homogeneous limits

3.3 Arbitrary inhomogeneity

4. Comparison with experimental data

5. Conclusion

Acknowledgements

References and links

Cited By

Figures (7)

Tables (1)

Equations (13)

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