**1. Introduction**

326 Photonic Crystals – Innovative Systems, Lasers and Waveguides

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An intense interest in negative index metamaterials (NIMs) [1-2] has been witnessed over the last years. Metal based NIMs [3-11] have been demonstrated at both microwave and infrared frequencies with a motivation mainly coming from the unusual physical properties and potential use in many technological applications [12-21]; however, they usually have large optical losses in their metallic components. As an alternative, dielectric based photonic crystals (PhCs) have been shown to emulate the basic physical properties of NIMs [22-26] and, in addition, have relatively small absorption loss at optical frequencies. Equally important, PhCs can be nanofabricated using currently available silicon chip-scale foundry processing, allowing significant potential in the development of future electronic-photonic integrated circuits.

One particular type of PhC can be obtained by cascading alternating layers of NIMs and positive index materials (PIMs) [27 – 32]. This photonic structure (with an example shown in Figure 1) is postulated to show unusual and unique optical properties including new types of surface states and gap solitons [33], unusual transmission and emission properties [34 – 38], complete photonic bandgaps [39], and phase-invariant field that can be effectively used in cloaking applications [40]. Moreover, a remarkable property of these binary photonic structures is the existence of an omnidirectional bandgap that is insensitive to the wave polarization, angle of incidence, structure periodicity, and structural disorder [41 – 43]. The main reason for the occurrence of a bandgap with such unusual properties is the existence of a frequency band at which the path-averaged refractive index is equal to zero [27 – 32, 34]. Specifically, at this frequency the Bragg condition, *k= (* n *ω/c)= mπ*, is satisfied for *m* = 0,

<sup>\*</sup> S. Kocaman1, M. S. Aras1, P. Hsieh1, J. F. McMillan1, C. G. Biris2, N. C. Panoiu2, M. B. Yu3, D. L. Kwong3 and A. Stein4

<sup>1</sup>*Columbia University, New York, NY,* 

*<sup>2</sup>University College of London, London,* 

*<sup>3</sup>Institute of Microelectronics, Singapore,* 

*<sup>4</sup>Brookhaven National Laboratory, Upton, NY,* 

*<sup>1,4</sup>USA* 

*<sup>2</sup>UK* 

*<sup>3</sup>Singapore* 

Negative Index Photonic Crystals Superlattices and Zero Phase Delay Lines 329

namely, to reshape curved wave fronts into planar ones [36], or to transfer into the far-field the phase information contained in the near-field. In addition, at the frequencies at which the refractive index becomes vanishingly small the electromagnetic field has an unusual dual character, i.e., it is static in the spatial domain (the phase difference between arbitrary spatial locations is equal to zero) while remaining dynamic in the time domain, thus allowing energy transport. This remarkable property, which is also the main topic of our study, has exciting technological applications to delay lines with zero phase difference, information processing devices, and the development of new optical phase control and

In this chapter, we show unequivocally that optical beams propagating in path-averaged zero-index photonic crystal superlattices can simultaneously have zero phase delay. The nanofabricated superlattices consist of alternating stacks of negative index photonic crystals and positive index homogeneous dielectric media, where the phase differences corresponding to consecutive primary unit cells are measured with integrated Mach-Zehnder interferometers. These measurements demonstrate that at path-averaged zeroindex frequencies the phase accumulation remains constant despite increases in the physical path lengths. We further demonstrate experimentally for the first time that these superlattice zero- n bandgaps can either remain invariant to geometrical changes of the photonic structure or have a center frequency which is deterministically tunable. The properties of the zero- n gap frequencies, optical phase, and the effective refractive indices agree well between the series of measurements and the complete theoretical analysis and simulations.

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

*c*, 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

**2. Negative refraction photonic crystal superlattices** 

measurement techniques.

**2.1 Theory** 

*a/2*

irrespective of the period of the superlattice; here, *k* and are the wave vector and frequency, respectively, and n is the averaged refractive index. Because of this property this photonic bandgap is called zero- n , or zero-order, bandgap [30, 34].

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 PhC (*d1*=2.564 μm*,* =4.51 μm) with *a=*423 nm, *r/a* = 0.276, and *t/a* = 0.756 (the scale bar = 5 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 image in the inset (the scale bar = 25 μm).

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, namely, to reshape curved wave fronts into planar ones [36], or to transfer into the far-field the phase information contained in the near-field. In addition, at the frequencies at which the refractive index becomes vanishingly small the electromagnetic field has an unusual dual character, i.e., it is static in the spatial domain (the phase difference between arbitrary spatial locations is equal to zero) while remaining dynamic in the time domain, thus allowing energy transport. This remarkable property, which is also the main topic of our study, has exciting technological applications to delay lines with zero phase difference, information processing devices, and the development of new optical phase control and measurement techniques.

In this chapter, we show unequivocally that optical beams propagating in path-averaged zero-index photonic crystal superlattices can simultaneously have zero phase delay. The nanofabricated superlattices consist of alternating stacks of negative index photonic crystals and positive index homogeneous dielectric media, where the phase differences corresponding to consecutive primary unit cells are measured with integrated Mach-Zehnder interferometers. These measurements demonstrate that at path-averaged zeroindex frequencies the phase accumulation remains constant despite increases in the physical path lengths. We further demonstrate experimentally for the first time that these superlattice zero- n bandgaps can either remain invariant to geometrical changes of the photonic structure or have a center frequency which is deterministically tunable. The properties of the zero- n gap frequencies, optical phase, and the effective refractive indices agree well between the series of measurements and the complete theoretical analysis and simulations.
