**4. Conclusions**

In summary, we discuss how to select the optimal pump and crystal parameters to obtain the pure state photon triplets by cascaded second-order SPDC using PPLN and MgO-doped PPLN. We chose 520 nm as the pump source with duration of 0.1 ps. After calculating the Schmidt number of photon pairs generated in each SPDC, we determined the crystal lengths of the two PPLN are 0.2 cm and 9.16 cm, respectively. The lengths of the two PPMgLN are 0.2 cm and 10.74 cm, respectively. According to theoretical calculation, the purity of photon pairs from each SPDC can reach 100%, that is, there is no frequency correlation. We have achieved photon triplets with a spectral purity of 100% in the C-band. We firmly believe that in the future development of quantum networks, our scheme can provide reliable pure-state photon triplet sources for various quantum information processes.

**Author Contributions:** Conceptualization and methodology, X.-R.S. and Y.-W.H.; validation, Y.-W.H., T.X. and Y.-H.L.; writing—original draft preparation, X.-R.S.; writing—review and editing, T.X., Y.-H.L., X.-F.C.

**Funding:** This work is supported by the National Key R&D Program of China (2017YFA0303701, 2018YFA0306301,); National Natural Science Foundation of China (NSFC) (11734011, 11804135); The Foundation for Development of Science and Technology of Shanghai (17JC1400400).

**Conflicts of Interest:** The authors declare no conflict of interest.
