Parametric down-conversion with optimized spectral properties in nonlinear photonic crystals

We study the joint spectral properties of photon pairs generated by spontaneous parametric down-conversion in a one-dimensional nonlinear photonic crystal in a collinear, degenerate, type-II geometry. We show that the photonic crystal properties may be exploited to compensate for material dispersion and obtain photon pairs that are nearly factorable, in principle, for arbitrary materials and spectral regions, limited by the ability to fabricate the nonlinear crystal with the required periodic variation in the refractive indices for the ordinary and extraordinary waves.

Figure 1

(Color online) Schematic of a one-dimensional, nonlinear photonic crystal. (a) Second-order nonlinearity. (b) Refractive index, shown for one of the polarizations. (c) Representation of the periodic material, with $b=\Lambda -a$.

Figure 2

(Color online) Plots as a function of frequency of (a) the Bragg wave number $K\left(\omega \right)$, (b) the group velocity $1∕K^{\prime }\left(\omega \right)$, (c) $K^{\prime }\left(\omega \right)$, and (d) group velocity dispersion coefficient $K^{″}\left(\omega \right)$; for comparison, the dotted horizontal line indicates the magnitude of the GVD coefficient for SF-10, a particularly dispersive glass.

Figure 3

(Color online) Normalized modulus of Fourier coefficients for (a) an extraordinary wave at $425\mathrm{nm}$, (b) an extraordinary wave at $850\mathrm{nm}$, and (c) an ordinary wave at $850\mathrm{nm}$.

Figure 4

(Color online) Graphical representation of (a) the conditions (i) $\Delta K=0$ and (ii) $K_p^{\prime }=K_s^{\prime }$, and (b) the conditions (i) $\Delta K=0$ and (ii) $K_p^{\prime }=K_i^{\prime }$ on $\{\alpha ,{\theta }_{pm}\}$ space, for constant $\Lambda$ [we have used the values $\Lambda =277.8$ (shown in blue), 274.9 (red), and $271.9\mathrm{nm}$ (black)]. Each intersection point indicates a solution for simultaneous phase matching and group velocity matching between the pump and one of the generated photons. (c) Graphical representation of the conditions (i) $\Delta K=0$, (ii) $K_p^{\prime }=K_s^{\prime }$, and (iii) $K_p^{\prime }=K_i^{\prime }$ on $\{\alpha ,{\theta }_{pm}\}$ space, assuming $\Lambda =274.9\mathrm{nm}$. The intersection point indicates a solution for simultaneous phase matching and complete group velocity matching. Note that, in the shaded area, the wave number becomes complex, and therefore light does not propagate. (d) Group velocity vs frequency for the ordinary and extraordinary waves; the dotted (black) lines indicate the resulting complete group velocity matching.

Figure 5

(Color online) Joint spectral intensity, where dispersive effects to all orders are included and where the Gaussian approximation has not been used. (a) Pump spectral envelope with FWHM bandwidth of $10\mathrm{nm}$. (b) Phase-matching function; the two yellow (light gray) bands indicate the band gaps for each of the two polarizations. (c) Joint spectral intensity, given as the product of the pump envelope and phase-matching function.

Figure 6

Schmidt number $K$ plotted vs (a) Bragg period $\Lambda$, (b) permittivity contrast $\alpha$, and (c) crystal propagation angle ${\theta }_{pm}$.