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Optics Express

Resonance lineshapes in two-dimensional Fourier transform spectroscopy

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

Published Aug 2, 2010• 178 citations
Publisher

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.

DOI
10.1364/oe.18.017699

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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.
We present an ultrafast coherent spectroscopy data acquisition scheme that samples slices of the time domain used in multidimensional coherent spectroscopy to achieve faster data collection than full spectra. We derive analytical expressions for resonance lineshapes using this technique that completely separate homogeneous and inhomogeneous broadening contributions into separate projected lineshapes for arbitrary inhomogeneous broadening. These lineshape expressions are also valid for slices taken from full multidimensional spectra and allow direct measurement of the parameters contributing to the lineshapes in those spectra as well as our own.
Two-dimensional (2D) line shapes derived from four types of coherent third-order nonlinear spectroscopies as a function of two independent time or frequency variables are compared. The signals scattered into two wave-vector-matching directions in the limits of high-time or high-frequency resolution are calculated for an inhomogeneously broadened ensemble of two-level systems with phenomenological dephasing and population relaxation times. While one-dimensional (1D) line shape analysis for this model is ambiguous, the contours of these 2D line shapes are shown to be characteristic of the relative time scales of the three relevant line-broadening parameters. These broadening mechanisms are also apparent by comparing 2D spectral profiles along the diagonal and antidiagonal frequency axes. For time domain experiments, in analogy with NMR correlation spectroscopy, the radiated signal is observed as a function of the two coherence periods and is 2D Fourier transformed to obtain a line shape. For experiments based on the photon echo, the ellipticity of the absolute value 2D line shape is related to the ratio of the inhomogeneous width to the dephasing rate, whereas experiments based on the transient grating method are shown to be functionally 1D. For frequency domain experiments, the contours of the 2D line shape can yield information on static and both dynamic broadening mechanisms. While the addition of spectral diffusion to such calculations will modify these results for frequency domain experiments, characterizing these dynamics in the time domain naturally leads to a 3D experiment analogous to the spin diffusion experiment in NMR.
With the development of laser technology in the field of optics, ultra-fast optics has become an important research field. Compared with the traditional technology, ultrafast optics can be realized not only under shorter pulse function, but also on a smaller scale, which can more quickly reflect the dynamic process. We present an analytical calculation of the full three-dimensional (3D) coherent spectrum with a finite duration two-dimensional (2D) Gaussian pulse envelope. Our starting point is the solution of the optical Bloch equations for three-level potassium atomic gas in the 3D time domain by using the projection-slice theorem, error function and Fourier-shift theorem of 3D Fourier transform. These principles are used to calculate and simplify the third-order polarization equation generated by the device, and the analytical calculation of three-dimensional Fourier transform frequency spectrum at T = 0 is obtained. We simulate the analytic solution by using mathematics software. By comparing the simulations with the experimental results, with the homogeneous line-width fixed, we can obtain the relationship among the in-homogeneous broadening, the correlation diagonal coefficients and the three-dimensional spectrum characteristics, which can be identified quantitatively by fitting the slices of three-dimensional Fourier transform spectrum peaks in an appropriate direction. The results show that the three-dimensional Fourier transform spectrum will extend along the diagonal direction with the increasing of the in-homogeneous broadening, and the spectrogram progressively becomes a circle with the increasing of the diagonal correlation coefficient, and the amplitude also gradually turns smaller. According to the analytical solution, we give a complete two-dimensional spectrum of the T = 0 interface. The results can be fit to the experimental 3D coherent spectrum for arbitrary inhomogeneity.
Two-dimensional coherent spectroscopy (2DCS) provides simultaneous measurement of homogeneous and inhomogeneous linewidths through quantitative line shape analysis. However, conventional line shape analysis methods assume Gaussian inhomogeneity, limiting its applicability to systems with non-Gaussian inhomogeneity. We present a quantitative line shape analysis method incorporating arbitrary inhomogeneity using a bivariate spectral distribution function in 2DCS simulations. An algorithm is developed to extract the homogeneous linewidth and arbitrary inhomogeneous distribution from an experimentally measured 2D spectrum. We demonstrate this framework for a quantum-well (QW)-exciton resonance with non-Gaussian inhomogeneity. This work broadens the scope of quantitative line shape analysis for studying materials with non-Gaussian inhomogeneity.
A solution to Maxwell's equations in the three-dimensional frequency domain is used to calculate rephasing two-dimensional Fourier transform (2DFT) spectra of the D2 line of atomic rubidium vapor in argon buffer gas. Experimental distortions from the spatial propagation of pulses through the sample are simulated in 2DFT spectra calculated for the homogeneous Bloch line shape model. Spectral features that appear at optical densities of up to 3 are investigated. As optical density increases, absorptive and dispersive distortions start with peak shape broadening, progress to peak splitting, and ultimately result in a previously unexplored coherent transient twisting of the split peaks. In contrast to the low optical density limit, where the 2D peak shape for the Bloch model depends only on the total dephasing time, these distortions of the 2D peak shape at finite optical density vary with the waiting time and the excited state lifetime through coherent transient effects. Experiment-specific conditions are explored, demonstrating the effects of varying beam overlap within the sample and of pseudo-time domain filtering. For beam overlap starting at the sample entrance, decreasing the length of beam overlap reduces the line width along the ωτ axis but also reduces signal intensity. A pseudo-time domain filter, where signal prior to the center of the last excitation pulse is excluded from the FID-referenced 2D signal, reduces propagation distortions along the ωt axis. It is demonstrated that 2DFT rephasing spectra cannot take advantage of an excitation-detection transformation that can eliminate propagation distortions in 2DFT relaxation spectra. Finally, the high optical density experimental 2DFT spectrum of rubidium vapor in argon buffer gas [J. Phys. Chem. A 2013, 117, 6279-6287] is quantitatively compared, in line width, in depth of peak splitting, and in coherent transient peak twisting, to a simulation with optical density higher than that reported.