**2.2 Mach-Zehnder interferometer with negative index photonic crystal**

To examine effective index differences between different bands in the band diagram experimentally, we designed and fabricated 100 unit cells of PhC and a geometrically identical homogeneous slab on the two arms of the MZI. Example scanning electron micrographs (SEMs) are shown in Figure 1. Transmission is measured with amplified spontaneous emission source, in-line fiber polarizer with a polarization controller to couple the light in with a tapered lensed fiber, and an optical spectrum analyzer. In the transmission (Figure 3d; black), the MZ interference spectra has two steep variations, first at the end of the first band (negative index band) and second at the start of the second band (positive index band). This is a clear indication of an abrupt refractive index change (Figure 3c) that is only possible when there is an abrupt interband transition between two bands. The non-MZI transmission spectrum of a similar structure is also shown in Figure3d (red) for reference.

To characterize this steep index change further we placed on the two arms of the MZI PhC sections with different radius *r*. We kept *a* unchanged in order to have the same total physical length on both arms, for the same number of unit cells in the PhC sections. With this approach, the MZI sections that do not contain PhC regions are identical and hence one isolates the two PhC sections as the only source for the measured phase difference. For instance, we set *r2* to 5/6 of the original value of the radius *r1* (*r2/a=* 0.283 × 5/6=0.236). Figure 4a illustrates the difference between the band structures of the two PhC designs, namely, a frequency shift of the photonic bands. Due to this shifted band structure, the accumulated phase difference between the two arms is almost independent of wavelength, except for a steep variation that again corresponds to a steep refractive index change (moving from band to band). When we place a section of 62 PhC unit cells in both arms of the MZI, the transmission spectra presents two spectral domains, 1525 nm to 1550 nm and

Negative Index Photonic Crystals Superlattices and Zero Phase Delay Lines 333

Fig. 4. **Mach-Zehnder interferences with negative refraction PhCs. a,** Band diagram shifts to lower frequency when the *r/a* ratio changes. Blue is the original design and gray is the design with *r/a*=0.236. **c.** Red line: MZI transmission with 62 unit cells of PhC on one arm with *r/a*=0.283 and 62 unit cells of PhC with *r/a*=0.236 on the other arm; the lattice period (*a*) is the same in both cases. Blue (grey) line: Transmission spectrum for PhC superlattice with *r/a*=0.283 (0.236) and 80 unit cells of PhC. Different index differences (*Δn*) from 1525 nm to

In addition, we performed high spatial resolution imaging of the radiated input-output ports for the devices that have been used for the experiment presented in Figure 4. Results are illustrated in the Figure 5 as follows: In the case of the reference arm (**i-iii)**, we see light transmission for all three wavelengths, which corroborates the characteristics of the transmission spectrum in Figure 4b. For the device arm (**iv-vi**) there is transmission for 1600 nm and 1530 nm but not for 1570 nm. This agrees with the transmission spectra in Figure 4b. Note that although there is similar transmission for both arms at 1530 and 1600 nm, the

The band diagram in Figure 3a is calculated by using *RSoft's BandSOLVE* [45], a commercially available software that implements a numerical method based on the plane wave expansion of the electromagnetic field. 3D simulations have been performed to calculate 30 bands and for each band the corresponding values of the effective refractive index have also been determined. In all these numerical simulations a convergence tolerance of 10-8 has been used. The photonic bands have been divided into TM-like and TE-like, according to their parity symmetry. The path-averaged index of the superlattice has been calculated by using the negative effective index of the second TM-like band and the effective

In order to investigate numerically the spectral properties of the transmission characterizing a specific photonic superlattice, we have employed three-dimensional (3D) simulations

) and different

1550 nm and from 1580 nm to 1615 nm give different phase difference (

modal index of the homogeneous asymmetric slab waveguide.

interference output.

**2.3 Spatial field distribution** 

interference output has 14dB difference.

**3. Existence of zero-n gap** 

**3.1 Numerical simulations** 

1580 nm to 1615 nm, where the interference transmission is rather constant (red curve in Figure 4b) with ~ 14dB transmission difference between the two domains. In the next section, we show high spatial resolution images for this experiment.

Fig. 3. **Band diagram of the PhC and the calculated effective index. a,** Band diagram of the PhC with the parameters given in Figure1. Insets: first Brillouin zones of the hexagonal PhC (top) and the 1D superlattice (bottom). The TM-like (TE-like) photonic bands are depicted in blue (darker) [red (lighter)]. The light cone is denoted by the green lines. **b,** A zoom-in of the spectral domain corresponding to experimental region of interest. Experiments were performed in the spectral region marked by the two horizontal lines. **c,** Calculated effective index of refraction of the PhC, corresponding to the two TM-like bands shown in Figure2b. Insets: zoom-in of the two bands. **d,** Black (solid) line: MZI transmission with 100 unit cells of PhC on one arm and an homogeneous slab waveguide on the other arm. Red (dashed) line: transmission spectrum for non-MZI PhC superlattice with 60 unit cells.

Fig. 4. **Mach-Zehnder interferences with negative refraction PhCs. a,** Band diagram shifts to lower frequency when the *r/a* ratio changes. Blue is the original design and gray is the design with *r/a*=0.236. **c.** Red line: MZI transmission with 62 unit cells of PhC on one arm with *r/a*=0.283 and 62 unit cells of PhC with *r/a*=0.236 on the other arm; the lattice period (*a*) is the same in both cases. Blue (grey) line: Transmission spectrum for PhC superlattice with *r/a*=0.283 (0.236) and 80 unit cells of PhC. Different index differences (*Δn*) from 1525 nm to 1550 nm and from 1580 nm to 1615 nm give different phase difference () and different interference output.
