With its ultrafast time resolution, broadband spectrum, full coherence, and simple generation and selection devices, high-order harmonic generation (HHG) has shown its potential in the fields of ultrafast optics, strong-field physics, and semiconductor imaging. The characterization of the spatial domain properties of the high-order harmonics is important for applications in the attosecond electron dynamics processes. Although spatial measurements such as the Hartmann wavefront sensor (HWFS) and holography have been proposed in many areas, the extreme ultraviolet (EUV) or X-ray wavelength of the high-order harmonics and the difficulty of introducing reference light make spatial measurements of high-order harmonics a challenge. This paper introduces an improved mixed-state ptychography scheme to realize the spatial measurement of high-order harmonics. This approach makes good use of the multiple wavelengths of the high-order harmonics as well as reduces the stability requirements of the light source. We hope that it will be useful for spatial domain measurements of high-order harmonics and for the study of high-order harmonics generation and focusing.

In order to reduce the experimental requirement for the reconstruction of high-order harmonics, we try to make use of known information in our algorithm as more as possible and focus only on the reconstruction of the probe; the reconstruction of the sample is not important. Therefore, we modify the existing mixed-state ptychography scheme and introduce two priori conditions: the spectral distribution at different wavelengths and a specific sample with a certain contrast. These priori conditions are easily obtained experimentally. For the first iteration, the probe size is obtained, and then it is substituted into the initial estimated probe along with the priori spectral distribution. Subsequent iterations are performed with the modulus and frequency domain constraints to achieve complex amplitude reconstruction of the probe. The sample is not updated during the iterative process, thus making it more applicable to high-order harmonics. The effect of sample absorption or contrast on the reconstruction is investigated by varying the sample thickness.

In the reconstruction of high-order harmonics with simple wavefront (Fig. 3), we successfully reconstruct the complex amplitudes of the five EUV wavelengths with very low errors as well as a highly consistent spectral distribution. Also, we perform additional reconstructions of the sample without changing the scan matrix, resulting in poorer reconstruction quality of the probes. This indicates that the priori sample information is important for the probe reconstruction at such a small scan matrix. In the reconstruction of the complex wavefront (Fig. 4), we find that the reconstruction of the phase is great, but the reconstruction of the amplitude is not satisfactory (Fig. 5). In addition, the error curve is larger and the spectral distribution is deviated compared with that of the simple wavefront case (Table 1).

By varying the contrast of the particular sample, we find that different sample absorption has an important effect on the reconstruction and that the best complex amplitude reconstruction is obtained at a sample contrast of about 0.19 (Fig. 6). Despite continuing to increase the sample contrast, the quality of the probe amplitude reconstruction remains good. However, the reconstruction of the phase is affected by introducing a large phase due to the increase in thickness. Furthermore, our investigation using the optimum sample contrast (Fig. 7) reveals that the scan matrix could be reduced while maintaining the reconstruction quality, thus reducing the stability requirements for the light source.

In this work, a scheme that is more favorable for the characterization of spatial domain wavefront of the EUV high-order harmonics is proposed. The scan matrix is reduced by means of known sample information and spectral distribution, thus ensuring the reconstruction quality with reduced stability requirements for the light source. We have succeeded in reconstructing simple wavefronts with a small 3×3 scan matrix. For more generalized complex wavefronts, we have also carried out simulations, illustrating the feasibility and advantages of the scheme for high-order harmonic wavefront recovery. More critically, we find that sample contrast has a significant impact on the reconstruction. The best reconstruction quality is obtained when the sample contrast is around 0.19. And samples with a certain contrast can reduce the scanning matrix while maintaining the reconstruction quality. This scheme makes up for the shortcomings of traditional phase measurement methods and requires low stability of the light source. It is an important tool for recovering high-order harmonic wavefronts.