Relaxed Phase-Matching Constraints in Zero-Index Waveguides

The utility of all parametric nonlinear optical processes is hampered by phase-matching requirements. Quasi-phase-matching, birefringent phase matching, and higher-order-mode phase matching have all been developed to address this constraint, but the methods demonstrated to date suffer from the inconvenience of only being phase matched for a single, specific arrangement of beams, typically copropagating, resulting in cumbersome experimental configurations and large footprints for integrated devices. Here, we experimentally demonstrate that these phase-matching requirements may be satisfied in a parametric nonlinear optical process for multiple, if not all, configurations of input and output beams when using low-index media. Our measurement constitutes the first experimental observation of direction-independent phase matching for a medium sufficiently long for phase matching to be relevant. We demonstrate four-wave mixing from spectrally distinct co- and counterpropagating pump and probe beams, the backward generation of a nonlinear signal, and excitation by an out-of-plane probe beam. These results explicitly show that the unique properties of low-index media relax traditional phase-matching constraints, which can be exploited to facilitate nonlinear interactions and miniaturize nonlinear devices, thus adding to the established exceptional properties of low-index materials.

Figure 1

Phase matching in a low-index medium. (a) In a conventional medium ($n>1$), the input and output beams must be carefully aligned, typically copropagating, to satisfy the phase-matching condition. In a four-wave mixing interaction, this corresponds to aligning the signal beam with the pump beam to generate a collinear idler beam. (b) In a low-index medium ($n\approx 0$), the constituent beams are free to adopt any orientation and still maintain phase matching. (c) Scanning electron microscope image of a Dirac-cone zero-index waveguide surrounded by photonic band gap materials (triangular lattice of holes). (d) Refractive index profile of one of the waveguides used in the experiment, crossing zero at $\lambda =1600\mathrm{nm}$. This refractive index profile was obtained via the method detailed in Ref. [27], whereby two counterpropagating beams are overlapped within the device and the resulting interference pattern is imaged from out of the plane. The shaded region indicates a refractive index below the measurement threshold of $n<0.02$.

Figure 2

Collinear phase-matching measurements. (a) Example spectra of the pump and signal waves when measured after propagating independently through a $14.8\text{−}\mu \mathrm{m}$-long low-index waveguide. (b) When these same pump and signal beams are simultaneously applied to the waveguide, an idler wave is generated in the forward direction at ${\omega }_i=2{\omega }_p-{\omega }_s$ (1600 nm). The spectrum of the idler wave closely follows that of the pump wave because of the narrowness of the signal beam spectrum. Generated idler wave spectra in the (c) forward and (d) backward directions in a FWM process with copropagating pump and signal beams. The red curves show the spectra of the idler beams for ten different values of the pump and signal wavelengths. For each wavelength pair, the spectral gap between the pump and signal frequencies, as well as the incident power of the pump and signal beams, are held constant. The black curves show the peak power of the pulses predicted by phase-matching constraints, while the vertical dotted black lines in (c) and (d) indicate the $n=0$ wavelength.

Figure 3

Counterpropagating and out-of-plane phase-matching measurements. (a) Spectra showing FWM for counterpropagating pump and signal beams with an idler wave generated in the forward (copropagating with the pump wave, orange) and backward (copropagating with signal wave, red) directions. The signal and pump beams are at 1565 and 1600 nm, respectively, while the idler wave appears at 1635 nm. (b) Generated idler wave spectrum resulting from a signal beam seeding from out of the plane of the waveguide. An idler wave is generated in the backward direction only when the pump and signal beams are simultaneously applied (red curve compared to the black curve). The vertical dotted black lines in both figures indicate the $n=0$ wavelength.