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Figure 3. The real and imaginary parts of the Schmidt modes n = 1 with the chirp rate f = 56mm �2 and the pump bandwidth is set as( a) r p = 2pGHz;( b) r p = 20pGHz.
the frequency response range Dx with the same chirp rate and pump bandwidth in different modes are almost consistent. A slightly wider of the frequency response range Dx could be slightly wider in modulus index n = 1, but not particularly obvious.
In the bi-rectangular spectral distribution, the absolute value, the real part and the imaginary part also show similar distributions in this mode( see Fig. 3). There are two main differences: First, at the central frequency x p / 2, both the real part and the imaginary part slightly oscillate, but for the absolute value, the oscillation part disappears, forming a bi-rectangular spectral distribution similar to that in Refs. [ 28, 30 ]. Another difference is that at low frequency and high frequency, both the real and the imaginary parts have spectral distribution, but the distribution oscillates near 0. However, for the spectral distribution in absolute value, the value at low frequency and high frequency becomes 0 again.
4 Conclusion
As the investigation of Schmidt mode distributions for biphoton generated by SPDC in the chirped QPM crystals, theoretical results show that the pulses have rich structures in many modes. Most distributions are fulfilled with the frequency response points. These Schmidt modes in frequency domain are significantly reshaped and widened by the crystal at a moderate chirp-rate. These broadband spectral distributions in Schmidt modes help us to obtain temporal ultrashort pulses with few-, single- or even sub- cycles. The results provide some insights into this special optical process, and can be helpful to quantum technologies.
Acknowledgments
This work was supported by the Joint Funds of the National Natural Science Foundation of China under Grant U2341246. The first author would like to sincerely thank S. E. Harris for his Mathematica codes.
Funding
This work was funded by the Joint Funds of the National Natural Science Foundation of China under Grant U2341246.
Conflicts of interest The authors declare no conflict of interest.
Data availability statement
Necessary data are included in this article. Other data will be available from the corresponding authors upon reasonable request.
Author contribution statement
Jinbao Wang simulated and calculated this work, and wrote the article. Zhan Zheng and Helin Wang contributed to design and supervise this work. Qiang Lin participated in the discussions about this work and reviewed the manuscript.
References
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