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Given the wide range of applications of time-domain spectroscopy, andparticularly THz time-domain spectroscopy, the modelling of a probe pulsepropagating through a multilayered structure is often required. Due to the factthat the multilayers are usually grown on a substrate much thicker than theother layers, the transmission of a probe pulse includes a series of echopulses caused by the multiple reflections at the substrate interfaces.Experiments often measure the time profile and construct the transmittedspectrum only from the first transmitted pulse. Due to the fact that typicalsubstrates lead to times of crossing comparable to the spectral bandwidth, thefirst transmitted pulse's spectrum and the full transmitted spectrum can beimportantly different. It is therefore important to theoretically model thetransmission without the echo, to be able to directly compare with experimentalresults. Here we propose a method to elegantly and easily theoretically removethe echo from the transmission spectrum, without Fourier transforming thesignal twice. The spectrum of the transmitted pulse without echo will beproduced analytically without additional numerical steps, reducingcomputational time and program complexity when dealing with larger scalecomputing.

Removal of spectral distortion due to echo for ultrashort pulses propagating through multilayer structures with thick substrate