Spontaneous parametric down-conversion in asymmetric couplers: Photon purity enhancement and intrinsic spectral filtering

We analyze the process of photon-pair generation via spontaneous parametric down-conversion in a quadratic nonlinear asymmetric waveguide coupler. The two waveguides have different geometry, such that light coupling only occurs within a narrow bandwidth of one of the generated (signal) photon modes, while the other (idler) photon together with the pump stay localized in one (driven) arm of the coupler. We demonstrate that such a setup represents a powerful and flexible tool for engineering spectral properties of generated photon pairs. Mode hybridization and dispersion of coupling can be utilized for shifting the balance between group velocities of interacting pump, signal, and idler fields, subsequently leading to a significant increase of spectral factorizability (purity) of photons. We also show that for interaction lengths shorter than one beat length, generated pairs with signal photon being localized in the auxiliary (not pumped) arm of the coupler appear to be spectrally localized in both signal and idler components. The bandwidth of such intrinsic filtering of generated photons can be controlled by several geometrical parameters.

Figure 1

A scheme of SPDC in an asymmetric coupler: the pump and the idler remain confined in the driven (D) waveguide, and the signal photon can couple to the auxiliary (A) waveguide.

Figure 2

Example asymmetric coupler structures. P, I, S1, and S2 mode profiles (dominant electric field component) in (a) a hybrid microfiber-lithium niobate (LN) waveguide [20, 21, 22] (a microfiber of diameter $D=1.44$ $\mu \mathrm{m}$ on top of a $510\mathrm{nm}\times 300\mathrm{nm}$ LN waveguide, waveguide D) adjacent to a suspended $635\mathrm{nm}\times 300\mathrm{nm}$ LN waveguide (waveguide A), edge-to-edge separation of 620 nm, ${\lambda }_p=0.72$ $\mu \mathrm{m}$, ${\lambda }_i=1.34$ $\mu \mathrm{m}$, and ${\lambda }_s=1.55$ $\mu \mathrm{m}$; (b) $1200\mathrm{nm}\times 460\mathrm{nm}$ (waveguide D) and $2650\mathrm{nm}\times 460\mathrm{nm}$ (waveguide A) LN ridge waveguides [8, 23], edge-to-edge separation of 750 nm, LN film thickness of 100 nm, ${\lambda }_p=775$ nm, ${\lambda }_i=1400$ nm, and ${\lambda }_s=1735$ nm. In both examples S1 and S2 supermodes can be considered as in-phase and antiphase superpositions of modes of individual waveguides D and A.

Figure 3

Mode hybridization in the example structure as in Fig. 2. (a) Propagation constants of isolated waveguide D (dashed) and waveguide A (dotted) anticross in the vicinity of ${\lambda }_s=1735$ nm. The resulting S1 and S2 supermodes are shown with solid curves. For clarity of presentation, shifted propagation constants are plotted as $\tilde{\beta }=\beta -{\beta }_0-{\beta }_1\delta$, where ${\beta }_1=({\beta }_D^{\prime }+{\beta }_A^{\prime })/2$ is the average inverse group velocity of isolated waveguides D and A at the wavelength of anticrossing. (b) The coupling efficiency ${\eta }_c$ and the coupling coefficient $C$ calculated from the splitting of the two modes ($\mathrm{\Lambda }$ and $\mathrm{\Gamma }$ parameters).

Figure 4

Normalized joint spectral intensity, ${\left|f\right|}^2/\left({\alpha }_0^2{\gamma }_0^2\right)$, as a function of dimensionless frequency detunings $\nu =\delta ({\beta }_p^{\prime }L/\pi )$ at a fixed propagation distance $L$ in an isolated waveguide D (a), waveguide coupler with $CL/\pi =2$, $C^{\prime }=0$ (b) and $C^{\prime }/{\beta }_p^{\prime }=-0.02$ (c). The corresponding calculated purities are stated in the top right corner of each picture. Disperson parameters of the driven D and auxiliary A waveguides at the phase-matching point ${\omega }_{s0}$ are ${\beta }_{sD}={\beta }_{sA}$, ${\beta }_{sD}^{\prime }/{\beta }_p^{\prime }=1.05$, ${\beta }_{sA}^{\prime }/{\beta }_p^{\prime }=0.95$, and ${\beta }_i^{\prime }/{\beta }_p^{\prime }=1.1$. Energy and momentum conservation conditions are indicated with solid and dashed lines, respectively.

Figure 5

Phase-matching function ${\mathrm{\Phi }}_A$ [normalized intensity $|{\mathrm{\Phi }}_A{|}^2/{\left(z{\gamma }_0\right)}^2]$ as a function of $\mathrm{\Lambda }z$ and $\mathrm{\Delta }{\beta }_{0}z$ with fixed values of $\mathrm{\Lambda }$ and $\mathrm{\Gamma }$. Dashed lines indicate asymptotes $\mathrm{\Delta }{\beta }_0=0,\pm \mathrm{\Lambda }$.

Figure 6

Phase-matching functions ${\mathrm{\Phi }}_D$ (left column) and ${\mathrm{\Phi }}_A$ (right column) as functions of dimensionless frequency detunings $\nu =\delta ({\beta }_p^{\prime }z/\pi )$ at propagation distances $z=0.5\pi /\mathrm{\Lambda }$ (top row), $z=\pi /\mathrm{\Lambda }$ (middle row). and $z=1.5\pi /\mathrm{\Lambda }$ (bottom row). Other parameters are the same as those in Fig. 4.