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Justifying the thin-crystal approximation in spontaneous parametric down-conversion for collinear phase matching

: Baghdasaryan, B.; Steinlechner, F.; Fritzsche, S.


Physical review. A 103 (2021), Nr.6, Art. 063508
ISSN: 2469-9926 (print)
ISSN: 2469-9934 (online)
Fraunhofer IOF ()
Gaussian beams; Photons; quantum entanglement; thin crystal

Spatially engineered photons from spontaneous parametric down-conversion (SPDC) are a valuable tool for studying and applying photonic entanglement. An advantage of SPDC is that simple expressions for the two-photon state can be obtained using justified approximations. In particular, the thin-crystal approximation has often been invoked in the engineering of high-dimensional entangled states. Knowledge of the conditions under which the thin-crystal approximation remains valid is essential for the realization of experimental setups. We provide a quantitative guideline on the validity of the thin-crystal approximation in calculating the two-photon spatial state. In particular, we show that the applicability of this regime is related to the focusing parameter w¯p=wp/√λpL, where wp and λp are the beam waist and wavelength of the pump beam, respectively, and L is the length of the nonlinear crystal. Additionally, the validity of the thin-crystal regime is investigated concerning the size of a subspace in the Laguerre Gaussian basis, into which the two-photon state can be projected in a given experiment.