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

CAPTION

One sometimes needs to numerically evaluate the real part of the one-sided Fourier transform of a function f(t). 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., f(-t) = (SuperStar[f])(t)) 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 f(t), we first sample f(t) from t=0 to (Subscript[t, max])=2n dt, where dt is the time step and n is half the number of steps (this is to ensure an array of even length). The resulting sample vector of the one-sided f(t) (red dots) is then changed to a two-sided sample vector such that f(-t)=(SuperStar[f])(t) (red line). In doing so, we drop the repeating t=0 and t=-(Subscript[t, max]) 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.

THIS NOTEBOOK IS THE SOURCE CODE FROM

Liam Cleary

"One-Sided Fourier Transform: Application to Linear Absorption and Emission Spectra"
 http://demonstrations.wolfram.com/OneSidedFourierTransformApplicationToLinearAbsorptionAndEmis/
 Wolfram Demonstrations Project
 Published: July 16, 2012

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