Gaussian mode coupling of spectrally broadband photons from bulk spontaneous parametric down-conversion: A spatial-spectral mode analysis of fiber coupling

Abstract

Photon sources based on spontaneous parametric down-conversion (SPDC) are central to experimental quantum optics and quantum technologies. Their performance is commonly quantified by three metrics: pair-collection probability, heralding efficiency, and spectral purity. In bulk-crystal SPDC, these metrics are known to be mutually constrained, yet the physical origin of the resulting trade-offs is often obscured. We show that these trade-offs originate from the frequency-dependent population of discrete spatial modes in the SPDC emission. By performing a Laguerre-Gauss mode decomposition at each frequency component, we show how spectral-spatial non-separability impacts collection probability, heralding efficiency, and purity. We apply this framework to two widely used quasi-phase-matching configurations: collinear degenerate type-0 and type-II SPDC in periodically poled bulk crystals, and quantify how different phase-matching functions shape the spectral-spatial mode structure. In particular, for type-II SPDC we compare standard periodically poled and aperiodically poled Gaussian phase matching. We experimentally validate some of our theoretical results using spatial- and spectral-projection measurements. This spectral-spatial mode analysis provides a quantitative and predictive framework for understanding and engineering bulk-crystal photon sources, enabling systematic multi-parameter optimization beyond qualitative design guidelines.

Figures (7)

• Figure 1: Conceptual sketch of heralding efficiency for SPDC photons coupled into single-mode fiber. A Gaussian pump beam drives SPDC in a nonlinear crystal, generating photon pairs in different spatial mode combinations. Blue spots indicate events in which at least one photon occupies the fundamental Gaussian mode and is therefore coupled into the fiber. Events where both photons populate higher-order spatial modes are not shown, as they do not contribute to the detected single- or coincidence detections.
• Figure 2: (a) Relative brightness $B$ and (b) Heralding efficiency $H$ as a function of the focusing parameters for pump $\xi_\mathrm{p}$ and SPDC photons $\xi_\mathrm{s}$. The colored dotted lines represent different optimization strategies. The orange/cyan lines are obtain when optimizing $B$ by fixing $\xi_\mathrm{s}$/$\xi_\mathrm{p}$ and optimizing over $\xi_\mathrm{p}$/$\xi_\mathrm{s}$. Finding different trade-off with respect to $H$. The red dotted line depict the optimization of $H$ by setting a given value of $B$ and optimizing over $\xi_\mathrm{s}$ and $\xi_\mathrm{p}$. (c) Results of the last optimization, showing the trade-off between $B$ and $H$. (d) Spectral density $P_{0,0}(\Omega)$ for different focusing conditions $\xi_\mathrm{p}=\xi_\mathrm{s}=\xi$.
• Figure 3: The first column show the spectral amplitudes, $C_{0,p}(\Omega)$, for different joint modes depicting how spectrally distinguishable the different mode combination and for different focusing parameters used. The second column quantifies the spectral similarity by calculating the overlap of the normalized spectral amplitudes $C'_{0,x}=C_{0,x}/\sqrt{\int d\Omega P_{0,x}}$. The first row shows the case of loose focusing ($\xi_\mathrm{p}=\xi_\mathrm{s}=0.02$) where the spectra overlap is near to unity. The second row a tighter focusing was selected for the SPDC photons ($\xi_\mathrm{p}=2.8$, leading to greater spectral distinguishability. The last column shows how the case of $\xi_\mathrm{p}=\xi_\mathrm{s}=2.8$, where the effect is stronger almost reaching to spectral orthogonality
• Figure 4: Experimental setup. A 405 nm continuous-wave laser is focused into a type-II ppKTP crystal. Signal and idler photons are separated by the polarizing beamsplitter (PBS). The crystal plane is imaged into the detection system consisting of the SLMs and single-mode fibers to perform joint spatial projective measurements. A monochromator is used in the path of one photon to project the state in the spectral domain. MC: Monochromator, NLC: nonlinear crystal, SLM: Spatial light modulator, BPF: Band-pass filter, SPAD: Single-photon avalanche diode, PBS: Polarizing beamsplitter, HWP: Half-wave plate, TT: Time tagger.
• Figure 5: (a) Effect of focusing in the spectral profile $P_{0,0}(\Omega)$, showing spectral broadening and shift for tighter focusing. (b) The solid and dashed blue lines, show the trade-off between heralding efficiency, $H_{max}$, and relative brightness for aperiodically poled (APNLC), and periodically poled (PPNLC) nonlinear crystals, respectively. The orange line, depicts the maximum spectral purity considering a transformed limited Gaussian beam.