**3.2 Electric field distribution**

We have also calculated the electric field distribution in the zero-n superlattice for different wavelengths in our experimental region in continuous wave excitation type. In order to be able to get this distribution, we have run the simulation until it gets to steady state and then save the field. Next, we launched another simulation with an input field by using this saved field and ran for only one cycle by saving electric field in small steps. Then we have averaged the *|E|2* and plotted. Figure 7 shows the results.

To further investigate the nature of these photonic gaps, we next calculate the order, *m,* for the Bragg condition. The average index n and the value of *koΛ/π* for the two deigns are summarized in Table 1. The average indices are -0.007 and 0.001 whereas the corresponding *koΛ/π* values are 0.044 and 0.007 for design 1 and design 2, respectively. It is thus clear that

2) correspond to zero-order gaps. Moreover, for both superlattices, in the frequency range of

0.283, corresponds to an average index of refraction of 0.091 (design 1) and 0.153 (design 2), with the corresponding *koΛ/π* being 0.874 and 0.962, and thus it is a first-order gap. None of these gaps is due solely to the presence of the PhC layers, as the PhC band gap for the TM

negative index of refraction there are other spectral gaps. For example, the gap at

Table 1. **Average refractive index of the corresponding gaps and the gaps' order**

expected, it becomes deeper as the number of stacks increases.

averaged the *|E|2* and plotted. Figure 7 shows the results.

**3.2 Electric field distribution** 

As an additional proof that the invariant gap is not a band gap of the PhC, we present in Figure 6d the calculated transmission spectra of a PhC layer with a number of 15, 30, and 45 unit cells (no layers of homogeneous material is present in this case). Thus, this figure shows that the PhC gap is shifted by almost 40 nm from the location of the zero-order gap in gap Figure 6a. We also examined the dependence of the gap locations on the number of stacks in the superlattice. The results of these calculations are presented in Figure 6c for stack numbers 3, 5, and 8. We observe that the zero- n gap location has not changed and, as

We have also calculated the electric field distribution in the zero-n superlattice for different wavelengths in our experimental region in continuous wave excitation type. In order to be able to get this distribution, we have run the simulation until it gets to steady state and then save the field. Next, we launched another simulation with an input field by using this saved field and ran for only one cycle by saving electric field in small steps. Then we have

*c* = 0.276 for design 1 and

*a/2*

*c* = 0.272 for design

*a/2c* =

*a/2*

*c* = 0.27 (see Figure3b).

these normalized gap frequencies (

*a/2*

polarization is at

Fig. 7. A time-averaged steady-state distribution of the field intensity, *|E|2*, corresponding to a propagating mode with different wavelengths within the experimental region.

Negative Index Photonic Crystals Superlattices and Zero Phase Delay Lines 339

Fig. 8. (a) Measured transmission for a superlattice with *d2/d1*= 0.746, with 7 unit cells in the PhC layers and 5 stacks; for comparison, results of numerical simulations are also shown. (b) The same as in a), but for a superlattice with 0.794. (c) Example of near-infrared top-view image of a device with 3 stacks, from transmission measurement at 1550 nm. Superimposed

superlattice. Scale bar: 2 m. (d) Measured transmission for a superlattice with *d2/d1*= 0.746, with 3, 5 and 8 stacks and 7 unit cells in the PhC layers. Both gaps become deeper as the

It has been pointed out that zero- n gaps can be omnidirectional [28, 41]; however, in our case, due to the anisotropy of the index of refraction of the PhC, the zero- n gap is not omnidirectional. Moreover, varying the lattice period, radius, and the thickness of the superlattice, and thus changing the frequency at which the average effective index of refraction is equal to zero, the frequency of the zero- n gap can be easily tuned as we show in the next section. Importantly, we note the demonstration of these zero- n gap structures can have potential applications as delay lines with zero phase differences which we also

are the locations of the negative refraction PhC and positive index material in the

number of stacks increases.

show later in this chapter.
