We propose a fully analytical method for describing high-order harmonic generation (HHG). This method is based on the strong-field approximation and electron-trajectory theory, but utilizes the perturbation expansion on the Keldysh parameter $\gamma$. This expansion allows us to clearly differentiate the nonadiabatic and adiabatic effects on HHG. We show that the nonadiabatic effect relating to high-order expansion depends on the laser wavelength and remarkably enhances the HHG yields for cases of short wavelengths, providing deeper insights into wavelength-dependent HHG yields which are important in producing attosecond pulses. Especially, our method provides the analytical and accurate descriptions of nonadiabatic exit velocity and position of the tunneling electron at the tunnel exit. These descriptions are meaningful for constructing a fully analytical and quantitative Coulomb-included HHG model, which is crucial in HHG-based attosecond measurement.

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• High-order harmonic generation
• Strong field ionization & excitation
• Ultrafast phenomena
• Perturbative methods
• Strong-field approximation