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

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 modal index of the homogeneous asymmetric slab waveguide.

#### **3.1 Numerical simulations**

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

Negative Index Photonic Crystals Superlattices and Zero Phase Delay Lines 335

that for the second band the effective index is negative since decreases with *k* [31]). At

*a/2*

In Figure 6a, the results of the 3D FDTD simulations are summarized, for all three superlattices (7, 11, and 15 unit cells), each have a ratio *d2/d1*=0.746. In order to observe clear gaps, 5 stacks were used in the case of 7 unit cells in the PhC layer, whereas in the cases with 11 and 15 unit cells in the PhC layer 3 stacks sufficed. The results of similar calculations, for *d2/d1*=0.794 are shown in Figure 6b. The grid size resolution in all our numerical simulations is 0.0833*a* (35 nm). Furthermore, we highlight the region where the PhC has negative index of refraction, which is the region of interest in our study. As illustrated in Figure 6a and Figure 3.6b, the transmission spectra show several gaps; however, except for an invariant

the mid-gap locations of the gaps change with the period of the superlattice*.* The shift in the mid-gap frequency with respect to the period is typical for regular Bragg gaps; the presence

Fig. 6. (a) Transmission for a superlattice with *d2/d1*= 0.746, containing 7, 11 and 15 unit cells in each PhC slab. 3 stacks in the PhC superlattice are used in the case when the PhC layers contain 11 and 15 unit cells, and 5 stacks for the case of PhC layers with 7 unit cells. (b) The same as in a), but *d2/d1* = 0.794. (c) Transmission for a superlattice with *d2/d1*= 0.794, containing 3, 5 and 8 stacks. Each PhC layer contains 7 unit cells. (d) Transmission trough a PhC slab containing 15, 30 and 45 unit cells. All results are obtained through full 3D FDTD numerical

 *a/2*

*c* = 0.276, the effective index of refraction

*c* = 0.272 (Figure 6b; design 2),

*c* = 0.295 and

*c* = 0.276 (Figure 6a; design 1) and

of an invariant gap, however, demonstrates the existence of zero- n gaps.

normalized frequencies of

 *a/2*

gap located at

simulations.

*a/2*

is approximately –1.604 and –1.988, respectively.

based on the finite-difference time-domain (FDTD) algorithm; for this, we have used Rsoft's FullWAVE [45]. In our calculations we have studied three superlattices, each with a different period *Λ*: they have 7, 11 and 15 unit cells along the *z*-axis, in the PhC layer, so that the thickness of this layer is d 3.5 3 a <sup>1</sup> , d 5.5 3 a <sup>1</sup> , and d 7.5 3 a <sup>1</sup> , respectively. The corresponding thickness of the PIM layer has been calculated by requiring that the average of the index of refraction is zero that is \_ 11 22 n nd nd / Λ 0 [28], while keeping the ratio *d*2/*d*1 unchanged. Importantly, *n*1 and *n*2 are the *effective indices* of the modes in the PhC and homogeneous layers, respectively, at the corresponding wavelength. We performed this numerical analysis for superlattices with ratio *d2/d1*=0.746 (design 1) and *d2/d1*=0.794 (design 2). Thus by changing the ratio *d2/d*1 we can investigate the dependence of the wavelength at which the zero- n gap is observed on the period of the superlattice. We expect to see the zero order gap in the design 2 at a different frequency, as compared to the frequency of the zero order gap in the design 1, a frequency at which the effective indices again satisfy the equation \_ 11 22 n nd nd / Λ 0 . The effective index of the PIM layer, *n2*, can be calculated analytically, as it corresponds to the TM modes of an asymmetric slab waveguide; at λ=1550 nm it is *n2*=2.648. Similarly, the effective index corresponding to the second TM-like band shown in Figure 1d is determined from the relation *k*=*ω|n*|*/c* (note

Fig. 5. **Infrared images pertaining to the experiments corresponding to Fig3c. i-iii** is taken from the reference arm containing a PhC structure with *r/a*=0.236 and **iv-vi** is taken from the device arm where the ratio *r/a*=0.283. In all the images the input beam is impinging onto the structure from the right, which means that light scattering at the left facet of the device indicates light transmission.

based on the finite-difference time-domain (FDTD) algorithm; for this, we have used Rsoft's FullWAVE [45]. In our calculations we have studied three superlattices, each with a different period *Λ*: they have 7, 11 and 15 unit cells along the *z*-axis, in the PhC layer, so that the thickness of this layer is d 3.5 3 a <sup>1</sup> , d 5.5 3 a <sup>1</sup> , and d 7.5 3 a <sup>1</sup> , respectively. The corresponding thickness of the PIM layer has been calculated by requiring that the

keeping the ratio *d*2/*d*1 unchanged. Importantly, *n*1 and *n*2 are the *effective indices* of the modes in the PhC and homogeneous layers, respectively, at the corresponding wavelength. We performed this numerical analysis for superlattices with ratio *d2/d1*=0.746 (design 1) and *d2/d1*=0.794 (design 2). Thus by changing the ratio *d2/d*1 we can investigate the dependence of the wavelength at which the zero- n gap is observed on the period of the superlattice. We expect to see the zero order gap in the design 2 at a different frequency, as compared to the frequency of the zero order gap in the design 1, a frequency at which the effective indices

can be calculated analytically, as it corresponds to the TM modes of an asymmetric slab waveguide; at λ=1550 nm it is *n2*=2.648. Similarly, the effective index corresponding to the second TM-like band shown in Figure 1d is determined from the relation *k*=*ω|n*|*/c* (note

Fig. 5. **Infrared images pertaining to the experiments corresponding to Fig3c. i-iii** is taken from the reference arm containing a PhC structure with *r/a*=0.236 and **iv-vi** is taken from the device arm where the ratio *r/a*=0.283. In all the images the input beam is impinging onto the structure from the right, which means that light scattering at the left facet of the device

11 22 n nd nd / Λ 0 . The effective index of the PIM layer, *n2*,

11 22 n nd nd / Λ 0 [28], while

average of the index of refraction is zero that is \_

again satisfy the equation \_

indicates light transmission.

that for the second band the effective index is negative since decreases with *k* [31]). At normalized frequencies of *a/2c* = 0.295 and *a/2c* = 0.276, the effective index of refraction is approximately –1.604 and –1.988, respectively.

In Figure 6a, the results of the 3D FDTD simulations are summarized, for all three superlattices (7, 11, and 15 unit cells), each have a ratio *d2/d1*=0.746. In order to observe clear gaps, 5 stacks were used in the case of 7 unit cells in the PhC layer, whereas in the cases with 11 and 15 unit cells in the PhC layer 3 stacks sufficed. The results of similar calculations, for *d2/d1*=0.794 are shown in Figure 6b. The grid size resolution in all our numerical simulations is 0.0833*a* (35 nm). Furthermore, we highlight the region where the PhC has negative index of refraction, which is the region of interest in our study. As illustrated in Figure 6a and Figure 3.6b, the transmission spectra show several gaps; however, except for an invariant gap located at  *a/2c* = 0.276 (Figure 6a; design 1) and  *a/2c* = 0.272 (Figure 6b; design 2), the mid-gap locations of the gaps change with the period of the superlattice*.* The shift in the mid-gap frequency with respect to the period is typical for regular Bragg gaps; the presence of an invariant gap, however, demonstrates the existence of zero- n gaps.

Fig. 6. (a) Transmission for a superlattice with *d2/d1*= 0.746, containing 7, 11 and 15 unit cells in each PhC slab. 3 stacks in the PhC superlattice are used in the case when the PhC layers contain 11 and 15 unit cells, and 5 stacks for the case of PhC layers with 7 unit cells. (b) The same as in a), but *d2/d1* = 0.794. (c) Transmission for a superlattice with *d2/d1*= 0.794, containing 3, 5 and 8 stacks. Each PhC layer contains 7 unit cells. (d) Transmission trough a PhC slab containing 15, 30 and 45 unit cells. All results are obtained through full 3D FDTD numerical simulations.

Negative Index Photonic Crystals Superlattices and Zero Phase Delay Lines 337

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.

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 these normalized gap frequencies (*a/2c* = 0.276 for design 1 and *a/2c* = 0.272 for design 2) correspond to zero-order gaps. Moreover, for both superlattices, in the frequency range of negative index of refraction there are other spectral gaps. For example, the gap at *a/2c* = 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 polarization is at *a/2c* = 0.27 (see Figure3b).


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

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