Spectral properties of transverse Laguerre-Gauss modes in parametric down-conversion

The first color photos of the parametric down-conversion (PDC) emission cone illustrate the correlation of longitudinal and transverse momentum in the process, i.e., wavelength-dependent emission angles of PDC photons. However, current experiments and applications are more conveniently described in terms of discrete mode sets, with the most suitable choice depending on the propagation symmetries of the experimental setting. Remarkably, despite the fact that experiments with PDC sources are becoming ever more demanding, e.g., in terms of brightness or state fidelity, a description of spectral-spatial coupling in parametric down-conversion for the case of discrete modal decompositions remains elusive. We present a comprehensive study, in theory and experiment, of the spectral dependence of the transverse Laguerre-Gauss modes in parametric down-conversion. Moreover, we show how the spectral and spatial coupling can be harnessed to tune the purity of the well-known orbital angular momentum entanglement. This paper has implications for efficient collection of entangled photons in a transverse single mode, quantum imaging, and engineering pure states for high-dimensional quantum information processing.