**2. Negative refraction photonic crystal superlattices**

#### **2.1 Theory**

328 Photonic Crystals – Innovative Systems, Lasers and Waveguides

frequency, respectively, and n is the averaged refractive index. Because of this property

Fig. 1. **Schematic of a Mach-Zehnder interferometer (MZI) and SEM images of the fabricated device. a,** Schematic representation of the MZI. L1~850 μm and L2~250 μm. **b,**  SEM of a fabricated superlattice with 7 super-periods. Each PhC layer contains 7 unit cells of

μm). **c,** SEM of a sample, showing only the PhC layer with same parameters as in **b** (the scale bar = 10 μm). **d,** Near-field image of a supperlattice with each PhC layer containing 9 unit cells (*d1*=3.297 μm) (the scale bar = 2.5 μm). **e,** SEM of the Y-branch with a zoomed-in

Near-zero index materials have a series of exciting potential applications, such as diffraction-free beam propagation over thousands of wavelengths via beam self-collimation [34], extremely convergent lenses and control of spontaneous emission [35], strong field enhancement in thin-film layered structures [37], and cloaking devices [40]. Moreover, the vanishingly small value of the refractive index of near-zero index materials can be used to engineer the phase front of electromagnetic waves emitted by optical sources or antennas,

=4.51 μm) with *a=*423 nm, *r/a* = 0.276, and *t/a* = 0.756 (the scale bar = 5

of the superlattice; here, *k* and are the wave vector and

irrespective of the period

PhC (*d1*=2.564 μm*,* 

image in the inset (the scale bar = 25 μm).

this photonic bandgap is called zero- n , or zero-order, bandgap [30, 34].

The photonic structures examined consist of dielectric PhC superlattices with alternating layers of negative index PhC and positive index homogeneous slabs, as shown in Figure 1 and Figure 2, that can give rise to the zero- n gaps [29]. The hexagonal PhC region (Figure 1c) is made of air holes etched into a dielectric Si slab (*n*Si=3.48), with a lattice period *a* = 423 nm, a slab thickness *t* = 320 nm, placed on top of a silica substrate (*n*SiO2=1.5). The band diagram of the PhC with a hole-to-lattice constant (*r/a*) ratio of 0.276 (*r* ~ 117 nm) is shown in Figure 3a-b. Particularly the two-dimensional (2D) hexagonal PhC base unit has a negative index within the interested spectral band of 0.271 to 0.278 in normalized frequency of *a/2c*, or 1520 to 1560 nm wavelengths, such as reported earlier for near-field imaging [22]. The zero- n superlattices are then integrated with Mach–Zehnder interferometers (MZI) to facilitate the phase delay measurements. As illustrated in Figure 1a, the unbalanced interferometer is designed such that after splitting from the Y-branch (Figure 1e); a single mode input channel waveguide adiabatically tapers (over ~ 400 μm) to match the width of the superlattice structures. On the reference arm, there is either a slab with exactly the same geometry to match the index variations and hence isolate the additional phase contribution of the PhC structures, or a channel waveguide leading to a large index difference and hence to distinctive Fabry-Perot fringes. For the one-dimensional (1D) binary superlattice of Figure 1b, a near-field scanning optical microscope image is taken (Figure 1d) to confirm transmission near the zero- n gap edge (1560 nm). The period of the superlattice is equal to

Negative Index Photonic Crystals Superlattices and Zero Phase Delay Lines 331

are formed at the corresponding frequencies. However, if the lattice satisfies the special

also note that the 1D binary superlattice and the hexagonal PhC have different symmetry properties and therefore different first Brillouin zones (see Figure 2a-insets). Schematic representation of a superlattice with 3 superperiods is shown in Figure 2. The superlattice

In our fabricated devices, the longitudinal direction of the superlattice (*z*-axis) coincides with the -M axis of the hexagonal PhC. Moreover, within our operating wavelength range (Figure 3b) the PhC has two TM-like bands, one with positive index and the other one with negative index, and an almost complete TE-like bandgap (see Methods). The effective refractive indices corresponding TM-like bands (Figure3b) are determined from the relation *k*= *ω|n*|*/c* and plotted in Figure 3c (note that for the second band the effective index of

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)

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

is an integer multiple of π, which is the Bragg condition and thus photonic bandgaps

unless

, thereby

and thus to a spectral gap [30 – 32]. We

holds. This relation implies that the dispersion relation has no real solution for

condition of a spatially averaged zero refractive index ( n =0), again Tr T 2

consists of alternating layers of hexagonal PhCs and homogeneous slabs.

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

leading to imaginary solutions for the wave vector

refraction is negative since *k* decreases with [22]).

n1 1 ωd c

for reference.

*=d1 + d2* where *d1(2)* is the thickness of the PhC (PIM) layer in the primary unit cell. Since a zero- n bandgap is formed when the spatially averaged index is zero, it is insensitive to the variation of the superlattice period, as long as the condition of zero-average index is satisfied [27 – 31, 34].

Fig. 2. **Schematic representation of the photonic superlattice.** There are two Brillouin zones defined as follows: one for the hexagonal photonic crystal lattice and one for the photonic superlattice. *a* is the lattice period and *r* is the radius of the holes forming the hexagonal lattice. *d1* is the length of the PhC layer and *d2* is the length of the PIM region. *d1* + *d1=Λ* is equal to the superperiod (SP) of the photonic superlattice.

The existence of the zero- n bandgaps can be explained with the Bloch theorem, where for a 1D binary periodic lattice the trace of the transfer matrix, *T*, of a primary unit cell can be expressed as [27,29]

$$\operatorname{Tr}\left[\operatorname{T}\left(\boldsymbol{\omega}\right)\right] = 2\cos\left(\kappa\Lambda\right) = 2\cos\left(\frac{\overline{\mathbf{n}}\boldsymbol{\alpha}\Lambda}{\mathbf{c}}\right) - \left(\frac{\overline{\mathbf{Z}\_1} + \overline{\mathbf{Z}\_2} - 2}{\overline{\mathbf{Z}\_1}}\right)\sin\left(\frac{\mathbf{n}\_1\boldsymbol{\alpha}\mathbf{d}\_1}{\mathbf{c}}\right)\sin\left(\frac{\mathbf{n}\_2\boldsymbol{\alpha}\mathbf{d}\_2}{\mathbf{c}}\right) \tag{1}$$

where *n*1(2) and *Z*1(2) are the refractive index and impedance of the first (second) layer, respectively, is the Bloch wave vector of the electromagnetic mode, n is the average refractive index, <sup>Λ</sup> 0 <sup>1</sup> n x n x dx <sup>Λ</sup> . In the general case, when Ζ Ζ 2 1 , if <sup>0</sup> <sup>n</sup>ωΛ κ Λ <sup>m</sup><sup>π</sup> <sup>c</sup> , with *m* an integer, the relation

$$\left| \operatorname{Tr} \left[ \operatorname{T} (\boldsymbol{\alpha}) \right] \right| = \left| 2 + \left( \frac{\mathcal{Z}\_1}{\mathcal{Z}\_2} + \frac{\mathcal{Z}\_2}{\mathcal{Z}\_1} - 2 \right) \sin^2 \left( \frac{\mathbf{n}\_1 \boldsymbol{\alpha} \mathbf{d}\_1}{\mathbf{c}} \right) \right| \ge 2 \tag{2}$$

*=d1 + d2* where *d1(2)* is the thickness of the PhC (PIM) layer in the primary unit cell. Since a zero- n bandgap is formed when the spatially averaged index is zero, it is insensitive to the variation of the superlattice period, as long as the condition of zero-average index is

Fig. 2. **Schematic representation of the photonic superlattice.** There are two Brillouin zones defined as follows: one for the hexagonal photonic crystal lattice and one for the photonic superlattice. *a* is the lattice period and *r* is the radius of the holes forming the hexagonal lattice. *d1* is the length of the PhC layer and *d2* is the length of the PIM region. *d1* + *d1=Λ* is

The existence of the zero- n bandgaps can be explained with the Bloch theorem, where for a 1D binary periodic lattice the trace of the transfer matrix, *T*, of a primary unit cell can be

1 2 11 22

<sup>Λ</sup> . In the general case, when Ζ Ζ 2 1 , if <sup>0</sup>

ZZ c

where *n*1(2) and *Z*1(2) are the refractive index and impedance of the first (second) layer, respectively, is the Bloch wave vector of the electromagnetic mode, n is the average

> 1 2 <sup>2</sup> 1 1 2 1 Z Z nd Tr T 2 2 sin <sup>2</sup>

<sup>n</sup>ωΛ Ζ Ζ <sup>n</sup> <sup>ω</sup>d n <sup>ω</sup><sup>d</sup> <sup>Τ</sup><sup>r</sup> Τ ω 2 cos <sup>Λ</sup> 2 cos 2 sin sin

2 1

c Ζ Ζ c c

(2)

(1)

<sup>n</sup>ωΛ κ Λ <sup>m</sup><sup>π</sup> <sup>c</sup> ,

equal to the superperiod (SP) of the photonic superlattice.

0 <sup>1</sup> n x n x dx

satisfied [27 – 31, 34].

expressed as [27,29]

refractive index, <sup>Λ</sup>

with *m* an integer, the relation

holds. This relation implies that the dispersion relation has no real solution for unless n1 1 ωd c is an integer multiple of π, which is the Bragg condition and thus photonic bandgaps are formed at the corresponding frequencies. However, if the lattice satisfies the special condition of a spatially averaged zero refractive index ( n =0), again Tr T 2 , thereby leading to imaginary solutions for the wave vector and thus to a spectral gap [30 – 32]. We

also note that the 1D binary superlattice and the hexagonal PhC have different symmetry properties and therefore different first Brillouin zones (see Figure 2a-insets). Schematic representation of a superlattice with 3 superperiods is shown in Figure 2. The superlattice consists of alternating layers of hexagonal PhCs and homogeneous slabs.

In our fabricated devices, the longitudinal direction of the superlattice (*z*-axis) coincides with the -M axis of the hexagonal PhC. Moreover, within our operating wavelength range (Figure 3b) the PhC has two TM-like bands, one with positive index and the other one with negative index, and an almost complete TE-like bandgap (see Methods). The effective refractive indices corresponding TM-like bands (Figure3b) are determined from the relation *k*= *ω|n*|*/c* and plotted in Figure 3c (note that for the second band the effective index of refraction is negative since *k* decreases with [22]).
