Complicated high-order harmonic generation due to the falling edge of a trapezoidal laser pulse

High-order harmonic generation (HHG) is investigated for H$_2^+$ and its isotopomers under seven- and ten-cycle trapezoidal laser pulses at 800 nm wavelength and $I$=4$\times 10^{14}$ W$/$cm$^2$ intensity. We solved numerically full-dimensional electronic time-dependent Schrödinger equation with and without the Born-Oppenheimer approximation. We show that the HHG at the falling edge of a trapezoidal laser pulse can result in redshift and complexity on the total HHG spectrum which can be removed by considering different laser pulse duration and nuclear motion not possible for sin$^2$ and Gaussian laser pulses. We resolve the redshifts and complexities of the HHG spectra into different electronic and vibrational states and their interferences.

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