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# One-Sided Fourier Transform: Application to Linear Absorption and Emission Spectra

One sometimes needs to numerically evaluate the real part of the one-sided Fourier transform of a function . A simple and efficient way to achieve this is to use a fundamental property of the two-sided Fourier transform of a conjugate-even function (i.e., ) and use the efficient fast Fourier transform (FFT) algorithm. This procedure is applied here to the linear absorption and emission spectra of a single excitation interacting with a thermal environment.

### DETAILS

To calculate the real part of the one-sided Fourier transform of a function , we first sample from to , where is the time step and is half the number of steps (this is to ensure an array of even length). The resulting sample vector of the one-sided (red dots) is then changed to a two-sided sample vector such that (red line). In doing so, we drop the repeating and elements. Passing this vector to the built-in Mathematica function Fourier yields the desired one-sided Fourier transform multiplied by two.
Details of the theory of the linear absorption and emission spectra of a single excitation coupled (with coupling strength given by the reorganization energy) to a thermal environment (with bath vibrational frequency and temperature) be can be found in [1], where the well-known Drude–Lorentz, Gaussian, and Lorentzian lineshapes are considered.
Reference
[1] S. Mukamel, Principles of Nonlinear Optical Spectroscopy, New York: Oxford University Press, 1995.

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