Spatial modes in waveguided parametric down-conversion

The propagation of several spatial modes has a significant impact on the structure of the emission from parametric down-conversion in a nonlinear waveguide. This manifests itself not only in the spatial correlations of the photon pairs but also, due to new phase-matching conditions, in the output spectrum, radically altering the degree of entanglement within each pair. Here we investigate both theoretically and experimentally the results of higher-order spatial-mode propagation in nonlinear waveguides. We derive conditions for the creation of pairs in these modes and present observations of higher-order mode propagation in both the spatial and spectral domains. Furthermore, we observe correlations between the different degrees of freedom and finally we discuss strategies for mitigating any detrimental effects and optimizing pair production in the fundamental mode.

Figure 1

(Color online) Schematic picture of waveguided PDC in a periodically poled rectangular waveguide.

Figure 2

Different spatial modes excited in the down-conversion process lead to a superposition of Gaussian shaped frequency distributions in the generated two-photon state. (a) pump function, (b) phase-matching function, and (c) JSA.

Figure 3

(Color online) Microscope image of the waveguide cross section. The height and the width $\left(H\times W=6\times 4\mu {\text{m}}^2\right)$ given by the numerical aperture measurement are in good agreement with the image. Note the different nomenclature for the biaxial crystal basis and the coordinate system used for the calculations.

Figure 4

(Color online) The first four $z^{\prime }$ polarized spatial field modes at 800 nm propagating in our waveguide.

Figure 5

Frequency distribution of the two-photon state for the 60 most important mode triplets. The PDC spectra are spread over a range of more than 200 nm.

Figure 6

(Color online) Schematic of SHG. The two gray (red) lines represent the two pump fields and the three black curves are three different phase-matching functions for three mode triplets. The black (blue) line at $45°$ is the locus of points picked out by the pump function as the fundamental wavelength is scanned. SH signal can be generated at the intersections of this line and the three phase-matching functions.

Figure 7

(Color online) Observed second-harmonic modes. The modes (1) (0,2), (2) (0,0), (3) (0,1), and (4) (0,0) were recorded at wavelengths of 393.9, 397.9, 400.0, and 404.6 nm, respectively. (Axis labels correspond to the coordinate system in Fig. 3.)

Figure 8

(Color online) Experimental setup for measuring the spectral marginals of the signal and idler. BBO, $\beta$-barium borate crystal for SHG; DM, dichroic mirror; CF, color glass filter; HWP, half-wave plate; WG, waveguide; PBS, polarizing beam splitter; MM, multi-mode fiber; SM, single-mode fiber; and SG, spectrograph with CCD camera.

Figure 9

(Color online) The observed noise-subtracted signal and idler marginals in black (blue) are in very good accordance with the predicted spectra in gray (red).

Figure 10

(Color online) Noise subtracted SM fiber measurement (gray, red) and MM fiber (black, blue) marginal measurements. In these graphs the impact of spatial filtering on the spectral measurements can clearly be seen. The SM fibers select the signal and idler photons in the ground mode and suppress the side peaks from higher-order spatial-mode signal and idler photons.