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We study the problem of quantum temporal imaging in the case where the time lens is implemented by a sum frequency generation nonlinear process. We consider the general case where the time lens is characterized by a finite aperture and a not-perfect phase-matching in a regime close to 100% conversion efficiency. In particular we tackle this problem in term of the eigenmodes of the entire transformation of the field in the temporal imaging system. We show that in the case of modeling the phase-matching function by a double Gaussian the eigenmodes are given by chirped Gauss-Hermite functions. The effective number of involved eigenmodes is estimated as the ratio of the temporal aperture of the lens to the walk-off time of the signal and the idler waves in the nonlinear crystal. Our theoretical treatment allows us to identify the criteria for designing imaging schemes with close to unity efficiencies
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G. Patera, D. B. Horoshko, M. I. Kolobov, "Quantum temporal imaging," Proc. SPIE 10771, Quantum Communications and Quantum Imaging XVI, 1077107 (18 September 2018); https://doi.org/10.1117/12.2320516