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 ${\overline{w}}_p=w_p/\sqrt{{\lambda }_{p}L}$, where $w_p$ and ${\lambda }_p$ 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.

中文翻译：

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在共线相位匹配的自发参数下变频中证明薄晶体近似的合理性

来自自发参数下转换 (SPDC) 的空间工程光子是研究和应用光子纠缠的宝贵工具。SPDC 的一个优点是可以使用合理的近似值获得双光子状态的简单表达式。特别是，在高维纠缠态的工程中经常调用薄晶体近似。了解薄晶体近似保持有效的条件对于实现实验设置至关重要。我们提供了关于计算双光子空间状态的薄晶体近似有效性的定量指南。特别是，我们表明该制度的适用性与聚焦参数有关${\overline{瓦}}_{磷}={瓦}_{磷}/\sqrt{{\lambda }_{磷}升}$， 在哪里 ${瓦}_{磷}$ 和 ${\lambda }_{磷}$ 分别是泵浦光束的束腰和波长，和 $升$是非线性晶体的长度。此外，还研究了关于拉盖尔高斯基中子空间的大小的薄晶体体系的有效性，在给定的实验中可以将双光子态投影到该子空间中。