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We theoretically study high-order harmonics generation (HHG) and isolated attosecond pulse (IAP) generation in a spatially inhomogeneous chirped two-color (5 fs/800 nm and 12 fs/1600 nm) laser field by solving numerically the time-dependent Schrdinger equation(TDSE) for a one-dimensional (1D) model of He+ ion by the splitting-operator fast-Fourier transform technique. Results show that the inhomogeneity of the laser field plays an important role in the HHG process. The harmonic spectra exhibit a two-plateau structure, and the cutoff of high-order harmonics is extremely extended to 851th order and the smooth supercontinuum harmonic spectrum is obtained in a chirped two-color inhomogeneous laser field. To further understand the physical mechanism of HHG, we give a reasonable explanation for the extension of harmonic plateau by using the semi-classical three-step model, the time-frequency profile of the time-dependent dipole, and the classical electron trajectories. Explicitly, the harmonic order as a function of the ionization time and emission time can be given by the semi-classical three-step model. If we define the path with earlier ionization time and later emission time as a ongelectronic trajectory, and the path with later ionization time and earlier emission time as a short electronic trajectory, then, there exist a few electronic trajectories that contribute to the harmonics in cutoff region. Numerical results show that the short quantum path is enhanced, and the long quantum path is suppressed in spatially inhomogeneous fields, and this is advantageous to generate an IAP. We find that the quantum path can be controlled by increasing inhomogeneity parameter of the laser field. Effects of the time delay on HHG is also discussed. We find that the smooth supercontinuum harmonic spectrum is obtained by adjusting the time delay. When the time delay is t0=1.6up/1, the cutoff of the harmonics can be extended remarkably. By synthesizing the 600th to 680th (80th) order harmonics in the continuum region, an isolated 32 attosecond pulse can be generated by a spatially inhomogeneous chirped two-color laser field with parameters =0.25, =0.00105, t0=1.6/1.

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